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--- abstract: | We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $\pi:\hat{G}\twoheadrightarrow \{1,-1\}$, a $\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B \hat{G}$ and a character $\lambda: \hat{G}\rightarrow U(1)$. The underlying oriented theory is a twisted Dijkgraaf--Witten theory. The construction is based in the $(\hat{G}, \hat{\theta},\lambda)$-twisted Real representation theory of $\ker \pi$. In particular, twisted Real representations are boundary conditions and the generalized Frobenius--Schur element is its crosscap state. address: - | Department of Mathematics and Statistics\ Utah State University\ Logan, Utah 84322\ USA - | Department of Mathematics and Statistics\ Utah State University\ Logan, Utah 84322\ USA author: - Levi Gagnon-Ririe - Matthew B. Young title: Frobenius--Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory --- # Introduction {#introduction .unnumbered} [\[sec:Intro\]]{#sec:Intro label="sec:Intro"} Associated to a finite group $G$ and a $U(1)$-valued $2$-cocycle $\theta$ on its classifying space $BG$ is a two dimensional topological gauge theory known as Dijkgraaf--Witten theory [@dijkgraaf1990]. This is an oriented open/closed topological quantum field theory (TFT) $\mathcal{Z}_{(G,\theta)}$ with boundary conditions the category $\textup{\text{Rep}}^{\theta}(G)$ of finite dimensional $\theta$-twisted complex representations of $G$ [@freed1994; @moore2006]. In particular, $\mathcal{Z}_{(G,\theta)}$ assigns a partition function to each compact oriented $2$-manifold with boundary components labelled by twisted representations. Open/closed TFT was introduced as a framework to axiomatize the structure of topological D-branes in string theory [@lazaroiu2001; @kapustin2004; @moore2006] and has found a variety of applications in pure mathematics [@costello2007; @blumberg2009; @abouzaid2010]. The open/closed structure of Dijkgraaf--Witten theory plays an important role in the descriptions of D-branes in orbifold string theory [@dijkgraaf1990], generalized symmetries in quantum field theory [@sharpe2015; @huang2021] and boundary degrees of freedom in topological phases of matter [@shiozaki2017]. Open/closed TFTs on unoriented---and possibly non-orientable---manifolds play a central role in orientifold string theory [@hori2008] and related mathematics [@mbyoung2020; @freed2021; @georgieva2021; @noohiYoung2022]. In condensed matter physics, unoriented TFTs in general, and Dijkgraaf--Witten theory in particular, model topological phases of matter with time reversal symmetry [@freed2013b; @kapustin2017; @barkeshli2020]. The main result of this paper is an algebraic construction of a class of unoriented lifts of the oriented open/closed Dijkgraaf--Witten theories $\mathcal{Z}_{(G,\theta)}$. **Theorem 1** (Theorem [Theorem 21](#thm:twistRealRepTFT){reference-type="ref" reference="thm:twistRealRepTFT"}). *A triple $(\hat{G}, \hat{\theta}, \lambda)$ consisting of a short exact sequence of finite groups $$1 \rightarrow G\rightarrow \hat{G}\xrightarrow[]{\pi} C_2= \{1,-1\} \rightarrow 1,$$ a $\pi$-twisted $2$-cocycle $\hat{\theta}$ on $B\hat{G}$ which restricts to $\theta$ on $B G$ and a character $\lambda: \hat{G}\rightarrow U(1)$ defines a two dimensional unoriented open/closed topological field theory $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ whose oriented sector is a subtheory of $\mathcal{Z}_{(G,\theta)}$.* A number of authors have studied $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ under the assumption that $\hat{G}= G\times C_2$ is the trivial extension, $\hat{\theta}$ is in the image of the map $H^2(BG;C_2) \rightarrow H^{2}(B\hat{G};U(1)_{\pi})$ and $\lambda$ is trivial [@karimipour1997; @alexeevski2006; @turaev2007; @loktev2011; @snyder2017]. For general $(\hat{G}, \hat{\theta})$ and trivial $\lambda$, a topological construction of the closed sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},1)}$, and its higher dimensional analogues, was given in [@mbyoung2020] while a $G$-equivariant extension of the closed sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},1)}$ was given in [@kapustin2017]. We emphasize that for the applications of unoriented Dijkgraaf--Witten theory mentioned before Theorem [Theorem 1](#thm:twistRealRepTFTIntro){reference-type="ref" reference="thm:twistRealRepTFTIntro"}, general input data $(\hat{G},\hat{\theta},\lambda)$ is required; see Remark [Remark 22](#rem:TypeIITwists){reference-type="ref" reference="rem:TypeIITwists"}. As explained below, general input data is also natural from the representation theoretic and $K$-theoretic perspectives. Theorem [Theorem 1](#thm:twistRealRepTFTIntro){reference-type="ref" reference="thm:twistRealRepTFTIntro"} is proved using an algebraic characterization of unoriented TFTs, Theorem [Theorem 18](#thm:genStruAlg){reference-type="ref" reference="thm:genStruAlg"}, which builds off characterizations of oriented closed and open/closed TFTs [@dijkgraaf1989; @abrams1996; @lazaroiu2001; @moore2006; @alexeevski2006; @lauda2008], unoriented closed TFTs [@turaev2006] and unoriented open/closed TFTs with a single boundary condition [@alexeevski2006]. The algebraic data required to define an unoriented open/closed TFT includes: - A commutative Frobenius algebra $A$; this defines the oriented closed sector. - A Calabi--Yau category $\mathcal{B}$; this defines the oriented open sector. - An isometric involution $p: A \rightarrow A$ and a *crosscap state* $Q \in A$, the latter corresponding to the value of the TFT on the compact Möbius strip; this defines the unoriented closed sector. - A strict contravariant involution of $\mathcal{B}$, that is, a functor $P: \mathcal{B}^{\textup{\text{op}}} \rightarrow \mathcal{B}$ which squares to the identity, which is moreover required to be the identity on objects; this defines the unoriented open sector. The data (and that which we have omitted here) is required to satisfy a number of coherence conditions. The oriented theory $\mathcal{Z}_{(G,\theta)}$ is defined by the commutative Frobenius algebra $HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)) \simeq Z(\mathbb{C}^{\theta^{-1}}[G])$ with the Haar bilinear form $\langle-, -\rangle_{G}$ and Calabi--Yau category $\textup{\text{Rep}}^{\theta}(G)$. Motivated by the search for the data required to define the unoriented lift $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$, in Section [2](#sec:FSInd){reference-type="ref" reference="sec:FSInd"} we construct and study a contravariant involution $(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$ of $\textup{\text{Rep}}^{\theta}(G)$. The functor $P^{(\hat{G},\hat{\theta},\lambda)}$ acts non-trivially on objects and so is not an admissible choice for the defining data of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$. A key representation theoretic observation is that the homotopy fixed points of $(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$ is the category of $(\hat{G},\hat{\theta},\lambda)$-twisted Real representations of $G$. The Real representation theory of $G$ was originally studied by Wigner [@wigner1959] and Dyson [@dyson1962] as a generalization of real and quaternionic representation theory in the context of anti-unitary symmetries in quantum mechanics. More recently, Real representation theory has been developed from the related perspective of twisted equivariant $KR$-theory [@atiyah1969; @karoubi1970; @freed2013b; @noohiYoung2022] and categorical representation theory [@mbyoung2021b; @rumyninYoung2021; @rumynin2021b]. In the $K$-theoretic setting, general pairs $(\hat{\theta}, \lambda)$ are required to realize all $KR$-theory twists. Motivated by the above perspectives, we consider the element $$\nu_{(\hat{G},\hat{\theta},\lambda)} = \sum_{\varsigma \in \hat{G}\setminus G} \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} l_{\varsigma^2} \in HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)).$$ The role of $\nu_{(\hat{G},\hat{\theta},\lambda)}$ in Real representation theory is summarized by the next result. **Theorem 2** (Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"} and Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"}). *Let $V$ be a $\theta$-twisted representation of $G$ with character $\chi_V$. Then $\langle \chi_V, \nu_{(\hat{G},\hat{\theta},\lambda)} \rangle_{G}$ is equal to the trace of the involution $$\textup{\text{Hom}}_{G}(V,P^{(\hat{G},\hat{\theta},\lambda)}(V)) \rightarrow \textup{\text{Hom}}_{G}(V,P^{(\hat{G},\hat{\theta},\lambda)}(V)). \qquad f \mapsto P^{(\hat{G},\hat{\theta},\lambda)}(f) \circ \Theta^{(\hat{G},\hat{\theta},\lambda)}_V.$$ In particular, if $V$ is irreducible, then $$\langle \chi_V, \nu_{(\hat{G},\hat{\theta},\lambda)} \rangle_{G} = \begin{cases} 1 & \mbox{if and only if $V$ lifts to a $(\hat{G},\hat{\theta},\lambda)$-twisted Real representation},\\ -1 & \mbox{if and only if $V$ lifts to a $(\hat{G},\delta\hat{\theta},\lambda)$-twisted Real representation}, \\ 0 & \mbox{otherwise}, \end{cases}$$ where $\delta$ is a representative of the generator of $H^{2}(B C_2;U(1)_{\pi}) \simeq C_2$.* The element $\nu_{(\hat{G},\hat{\theta},\lambda)}$ recovers under various specializations of the data $(\hat{G},\hat{\theta},\lambda)$ other generalized Frobenius--Schur elements [@frobenius1906; @gow1979; @turaev2007; @ichikawa2023]. In particular, the second statement in Theorem [Theorem 2](#thm:LefGroupAlgIntro){reference-type="ref" reference="thm:LefGroupAlgIntro"} shows that $\nu_{(\hat{G},\hat{\theta},\lambda)}$ is a generalization to twisted Real representation theory of the classical Frobenius--Schur element. Theorem [Theorem 2](#thm:LefGroupAlgIntro){reference-type="ref" reference="thm:LefGroupAlgIntro"} and a complete understanding of the $\theta$-twisted representation theory of $G$ suffices to understand the $(\hat{G},\hat{\theta},\lambda)$-twisted Real representation theory of $G$. Returning to the proof of Theorem [Theorem 1](#thm:twistRealRepTFTIntro){reference-type="ref" reference="thm:twistRealRepTFTIntro"}, we take for $\mathcal{B}$ the Calabi--Yau category of $(\hat{G},\hat{\theta},\lambda)$-twisted Real representations of $G$ and their $G$-equivariant linear maps. We view this as an orientifold-type construction, with $\textup{\text{Rep}}^{\theta}(G)$ seen as the category of $D$-branes in an oriented string theory and $\mathcal{B}$ the category of $D$-branes which survive the orientifold projection defined by $(P^{(\hat{G},\hat{\theta},\lambda)},\Theta^{(\hat{G},\hat{\theta},\lambda)})$. The category twisted Real representations is a non-full subcategory of $\mathcal{B}$ and the forgetful functor $\mathcal{B}\rightarrow \textup{\text{Rep}}^{\theta}(G)$ respects Calabi--Yau structures. Moreover, $\mathcal{B}$ inherits a contravariant involution which is the identity on objects and $A= HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))$ inherits an isometric involution $p$. We take for the crosscap state $Q$ the generalized Frobenius--Schur element $\nu_{(\hat{G},\hat{\theta},\lambda)}$. It remains to verify the coherence conditions. A mild generalization of the first equality in Theorem [Theorem 2](#thm:LefGroupAlgIntro){reference-type="ref" reference="thm:LefGroupAlgIntro"} (proved in Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"}) is the unoriented counterpart of the famous Cardy condition, asserting the equality of two ways of evaluating a Möbius strip diagram with boundary condition $V$. The remaining coherence conditions required of the crosscap state, involution $p$ and boundary-bulk and bulk-boundary maps are verified using the calculus of twisted cocycles. In Section [3.3](#sec:partFun){reference-type="ref" reference="sec:partFun"}, we compute partition functions of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$. ## Acknowledgements {#acknowledgements .unnumbered} The work of M. B. Y. was supported by National Science Foundation grant DMS-2302363 and a Simons Foundation Collaboration Grant for Mathematicians (Award ID 853541). # Background material {#sec:background} Throughout the paper the ground field is $\mathbb{C}$ and vector spaces are finite dimensional. Linear duality is $(-)^{\vee} = \textup{\text{Hom}}_{\mathbb{C}}(-,\mathbb{C})$. Denote by $U(1)$ the group of unit norm complex numbers and $C_n$ the cyclic group of order $n$, seen as a multiplicative group. ## Group cohomology Let $K$ be a finite group and $M$ a left $K$-module. We regard the underlying abelian group of $M$ as multiplicative. Let $C^{\bullet}(BK; M)$ be the complex of normalized simplicial cochains on $BK$ with coefficients in $M$. An element $\theta \in C^n(BK; M)$ is a function $$\theta: K^n \rightarrow M, \qquad (k_n, \dots, k_1) \mapsto \theta([k_n \vert \cdots \vert k_1])$$ whose value is the identity if any $k_i$ is the identity. The differential $d \theta$ of an $(n-1)$-cochain $\theta$ is defined so that $d \theta ([k_n \vert \cdots \vert k_1])$ is equal to $$k_n \cdot \theta([k_{n-1} \vert \cdots \vert k_1]) \prod_{j=1}^{n-1} \theta([k_n \vert \cdots \vert k_{j+1} k_j\vert \cdots \vert k_1])^{(-1)^{n-j}} \times \theta([k_n \vert \cdots \vert k_2])^{(-1)^n}.$$ Write $Z^{\bullet}(BK;M)$ and $H^{\bullet}(BK;M)$ for the cocycles and cohomologies of $C^{\bullet}(BK; M)$. When $M = U(1)$ with trivial $K$-action, write $C^{\bullet}(BK)$ for $C^{\bullet}(BK;M)$. When $\pi: \hat{G}\rightarrow C_2$ is a group homomorphism and $M=U(1)$ with $\hat{G}$-action $\omega \cdot z = z^{\pi(\omega)}$, write $C^{\bullet+\pi}(B\hat{G})$ for $C^{\bullet}(B\hat{G};M)$. If $\pi: C_2\rightarrow C_2$ is the identity map, then $H^{2+\pi}(B C_2) \simeq C_2$; a cocycle representative $\delta$ for the generator is given by $$\delta ([\varsigma_2 \vert \varsigma_1]) = \begin{cases} -1 & \mbox{if } \varsigma_1=\varsigma_2=-1,\\ 0 & \mbox{otherwise}. \end{cases}$$ We use the same notation for $\delta$ and its image under $\pi^*: Z^{2+\pi}(B C_2) \rightarrow Z^{2+\pi}(B \hat{G})$. **Lemma 1**. *Let $\pi: \hat{G}\rightarrow C_2$ be a $C_2$-graded finite group and $\hat{\theta} \in Z^{2+\pi}(B \hat{G})$. For all $g_i \in G$, $\omega \in \hat{G}$ and $\varsigma \in \hat{G}\setminus G$, the following equalities hold: $$\label{eq:2cocycleKey} \frac{\hat{\theta}([\omega g_2 \omega^{-1} \vert \omega g_1 \omega^{-1}])}{\hat{\theta}([g_2 \vert g_1])^{\pi(\omega)}} = \frac{\hat{\theta}([\omega g_2 \omega^{-1} \vert \omega])}{\hat{\theta}([\omega \vert g_2])} \frac{\hat{\theta}([\omega g_1 \omega^{-1} \vert \omega])}{\hat{\theta}([\omega \vert g_1])} \left( \frac{\hat{\theta}([\omega g_2 g_1 \omega^{-1} \vert \omega])}{\hat{\theta}([\omega \vert g_2 g_1])} \right)^{-1}$$ $$\label{eq:oddConj} \frac{\hat{\theta}([\omega \varsigma \omega^{-1} \vert \omega \varsigma \omega^{-1}])}{\hat{\theta}([\varsigma \vert \varsigma])^{-\pi(\omega)}} = \frac{\hat{\theta}([\omega \vert \varsigma^2])}{\hat{\theta}([\omega \varsigma^2 \omega^{-1} \vert \omega])}.$$* *Proof.* Both equalities follow from repeated use of the $2$-cocycle condition on $\hat{\theta}$. ◻ ## Twisted representation theory {#sec:twistRepThy} We recall background on twisted representation theory following [@karpilovsky1985]. Let $G$ be a finite group and $\theta \in Z^2(BG)$. **Definition 2**. *A *$\theta$-twisted* (or *$\theta$-projective*) *representation of $G$* is pair $(V,\rho)$ consisting of a vector space $V$ and a map $\rho: G\rightarrow GL(V)$ which satisfies $\rho(e)=\textup{\text{id}}_{V}$ and $$\rho(g_2) \circ \rho(g_1) = \theta([g_2 \vert g_1]) \rho(g_2 g_1), \qquad g_1, g_2 \in G.$$* We often write $V$ or $\rho_V$ for $(V,\rho)$. The category $\textup{\text{Rep}}^{\theta}(G)$ of $\theta$-twisted representations and their $G$-equivariant linear maps is $\mathbb{C}$-linear finite semisimple. The $\theta$-twisted group algebra $\mathbb{C}^{\theta}[G]$ is the $\mathbb{C}$-algebra with basis $\{l_g \mid g \in G\}$ and multiplication $l_{g_2} \cdot l_{g_1} = \theta([g_2 \vert g_1]) l_{g_2 g_1}$. The category of finite dimensional $\mathbb{C}^{\theta}[G]$-modules is equivalent to $\textup{\text{Rep}}^{\theta}(G)$. We sometimes interpret $\mathbb{C}^{\theta}[G]$ as functions on $G$, in which case $l_g$ the $\delta$-function at $g$. The centre $Z(\mathbb{C}^{\theta}[G])$ consists of elements $\sum_{g \in G} a_g l_g$ whose coefficients satisfy $$a_{hgh^{-1}} = \uptau(\theta)([h]g)^{-1} a_g, \qquad g,h \in G.$$ Here $\uptau(\theta)([h]g) = \frac{\theta([h g h^{-1} \vert h])}{\theta([h \vert g])}$ are the components of a $1$-cocycle $\uptau(\theta)$ on the loop groupoid of $B G$ called the *loop transgression* of $\theta$ [@willerton2008 Theorem 3]. Define a non-degenerate symmetric bilinear form on $\mathbb{C}^{\theta}[G]$ by $$\langle \sum_{g \in G} a_g l_g, \sum_{h \in G} b_h l_h \rangle_{G,\theta} = \frac{1}{\vert G \vert} \sum_{g \in G} \theta([g^{-1} \vert g]) a_{g^{-1}} b_g.$$ The character of $(V,\rho) \in \textup{\text{Rep}}^{\theta}(G)$ is the function $\chi_V: G\rightarrow \mathbb{C}$, $g \mapsto \textup{\text{tr}}_V\, \rho(g)$. A short calculation shows that $\chi_V(hgh^{-1}) = \uptau(\theta)([h]g) \chi_V(g)$. Functions $G\rightarrow \mathbb{C}$ with this conjugation equivariance are elements of $Z(\mathbb{C}^{\theta^{-1}}[G])$ and are called $\theta$-twisted class functions. Characters of irreducible $\theta$-twisted representations form an orthonormal basis of $Z(\mathbb{C}^{\theta^{-1}}[G])$ with respect to $\langle-,-\rangle_{G}:=\langle -, -\rangle_{G,\theta^{-1}}$. Given $(V,\rho_V) \in \textup{\text{Rep}}^{\theta}(G)$, define $(V^{\vee}, \rho_{V^{\vee}}) \in \textup{\text{Rep}}^{\theta^{-1}}(G)$ by $\rho_{V^{\vee}}(g) = (\rho_V(g)^{-1})^{\vee}$. For ease of notation, we write $\rho_V(g)^{- \vee}$ for $(\rho_V(g)^{-1})^{\vee}$. ## Categories with duality **Definition 3**. 1. *A *category with duality* is a triple $(\mathcal{C},P,\Theta)$ consisting of a category $\mathcal{C}$, a functor $P: \mathcal{C}^{\textup{\text{op}}} \rightarrow \mathcal{C}$ and a natural isomorphism $\Theta: \textup{\text{id}}_{\mathcal{C}} \Rightarrow P \circ P^{\textup{\text{op}}}$ whose components satisfy $$\label{eq:catWDualCoher} P(\Theta_V) \circ \Theta_{P(V)} = \textup{\text{id}}_{P(V)}, \qquad V \in \mathcal{C}.$$ The duality structure $(P,\Theta)$ is *strict* if $\Theta$ is the identity natural transformation.* 2. *A *homotopy fixed point* of $(\mathcal{C},P,\Theta)$ is a pair $(V,\psi_V)$ consisting of an object $V \in \mathcal{C}$ and an isomorphism $\psi_V: V \rightarrow P(V)$ which satisfies $P(\psi_V) \circ \Theta_V = \psi_V$.* We interpret $(P,\Theta)$ as defining a categorical $C_2$-action on $\mathcal{C}$ in which the generator acts contravariantly. Motivated by this, let $\mathcal{C}^{hC_2}$, $\mathcal{C}^{\tilde{h}C_2}$ be the categories with objects homotopy fixed points and morphisms $$\textup{\text{Hom}}_{\mathcal{C}^{hC_2}}((V,\psi_V),(W,\psi_W)) = \{\phi \in \textup{\text{Hom}}_{\mathcal{C}}(V,W) \mid \psi_V = P(\phi) \circ \psi_W \circ \phi \},$$ $$\textup{\text{Hom}}_{\mathcal{C}^{\tilde{h}C_2}}((V,\psi_V),(W,\psi_W)) = \textup{\text{Hom}}_{\mathcal{C}}(V,W).$$ Let $P^{\tilde{h}C_2}: (\mathcal{C}^{\tilde{h}C_2})^{\textup{\text{op}}} \rightarrow \mathcal{C}^{\tilde{h}C_2}$ be the identity on objects and send a morphism $\phi: (V,\psi_V) \rightarrow (W,\psi_W)$ to $P^{\tilde{h}C_2}(\phi) = \psi_V^{-1} \circ P(\phi) \circ \psi_W$. Let $\Theta^{\tilde{h}C_2}: \textup{\text{id}}_{\mathcal{C}^{\tilde{h}C_2}} \Rightarrow P^{\tilde{h}C_2} \circ (P^{\tilde{h}C_2})^{\textup{\text{op}}}$ be the identity natural transformation. **Lemma 4**. *The triple $(\mathcal{C}^{\tilde{h}C_2},P^{\tilde{h}C_2},\Theta^{\tilde{h}C_2})$ is a category with strict duality. Moreover, $P^{\tilde{h}C_2}$ is the identity on objects.* # A Frobenius--Schur indicator for twisted Real representation theory {#sec:FSInd} ## Twisted Real representation theory {#sec:RealProjRep} The Real representation theory of a finite group has been studied by many authors as a generalization of representation theory over $\mathbb{R}$ or $\mathbb{H}$ [@wigner1959; @dyson1962; @atiyah1969; @karoubi1970; @freed2013b; @mbyoung2021b]. We establish relevant aspects of the twisted form of this theory following [@mbyoung2021b §3.2]. Let $\pi: \hat{G}\rightarrow C_2$ be a $C_2$-graded finite group with $\pi$ surjective. Fix $\hat{\theta} \in Z^{2+\pi}(B \hat{G})$ and a character $\lambda: \hat{G}\rightarrow U(1)$. Note that $\lambda$ can be interpreted as an element of $Z^1(B\hat{G})$. Denote by $G= \ker \pi$ and $\theta \in Z^2(BG)$ the restriction of $\hat{\theta}$ along $BG\rightarrow B \hat{G}$. An element $\varsigma \in \hat{G}\backslash G$ determines a $\mathbb{C}$-linear exact duality structure $(P^{(\hat{\theta},\lambda,\varsigma)}, \Theta^{(\hat{\theta},\lambda,\varsigma)})$ on $\textup{\text{Rep}}^{\theta}(G)$. On objects, we have $P^{(\hat{\theta},\lambda,\varsigma)}(V,\rho) = (V^{\vee},\rho^{(\hat{\theta},\lambda,\varsigma)})$, where $$\rho^{(\hat{\theta},\lambda,\varsigma)}(g) = \frac{\lambda(g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] g)} \rho(\varsigma g^{-1} \varsigma^{-1})^{\vee}, \qquad g \in G.$$ The coefficients $$\label{eq:twistTrans2Cocyc} \uptau^{\textup{\text{refl}}}_{\pi}(\hat{\theta})([\omega]g) = \hat{\theta}([g^{-1} \vert g])^{\frac{\pi(\omega)-1}{2}} \frac{\hat{\theta}([\omega g^{\pi(\omega)} \omega^{-1} \vert \omega])}{\hat{\theta}([\omega \vert g^{\pi(\omega)}])}, \qquad g \in G, \; \omega \in \hat{G}$$ are best understood in terms of orientation-twisted loop transgression [@noohiYoung2022 Theorem 2.8], which is a cochain map $$\uptau^{\textup{\text{refl}}}_{\pi} : C^{\bullet+\pi}(B \hat{G}) \rightarrow C^{\bullet-1}(B (G/\!\!/_R \hat{G})).$$ The codomain is simplicial cochains on the classifying space of the quotient groupoid $G/\!\!/_R \hat{G}$ resulting from the Real conjugation action of $\hat{G}$ on $G$: $\omega \cdot g = \omega g^{\pi(\omega)} \omega^{-1}$, $\omega \in \hat{G}, \; g \in G$. In geometric terms, $G/\!\!/_R \hat{G}$ is the unoriented loop groupoid of $B \hat{G}$, that is, the quotient of the loop groupoid of $BG$ by the $C_2$-action which reverses orientation of loops and acts on $BG$ by deck transformations. Continuing, on morphisms $P^{(\hat{\theta},\lambda,\varsigma)}$ is $\mathbb{C}$-linear duality. The natural isomorphism $\Theta^{(\hat{\theta},\lambda,\varsigma)}$ is defined by its components $$\Theta^{(\hat{\theta},\lambda,\varsigma)}_V = \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} \textup{\text{ev}}_V \circ \rho(\varsigma^{2})^{-1}, \qquad (V,\rho) \in \textup{\text{Rep}}^{\theta}(G)$$ where $\textup{\text{ev}}_V : V \rightarrow V^{\vee \vee}$ is the evaluation isomorphism of underlying vector spaces. The normalization of $\Theta^{(\hat{\theta},\lambda,\varsigma)}_V$ ensures that the coherence condition [\[eq:catWDualCoher\]](#eq:catWDualCoher){reference-type="eqref" reference="eq:catWDualCoher"} holds. **Definition 5**. *The *category $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ of $(\hat{\theta},\lambda)$-twisted Real representations of $G$* is the homotopy fixed point category $\textup{\text{Rep}}^{\theta}(G)^{hC_2}$ of $(P^{(\hat{\theta},\lambda,\varsigma)}, \Theta^{(\hat{\theta},\lambda,\varsigma)})$.* Up to equivalence, $(P^{(\hat{\theta},\lambda,\varsigma)}, \Theta^{(\hat{\theta},\lambda,\varsigma)})$ depends only on $(\hat{G},[\hat{\theta}],\lambda)$. The same is therefore true of $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ and we drop $\varsigma$ from the notation if it is fixed or the particular realization of the duality structure is not important. Concretely, an object $(V,\psi_V) \in \textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is an isomorphism $\psi_V: V \rightarrow P^{(\hat{\theta},\lambda)}(V)$ in $\textup{\text{Rep}}^{\theta}(G)$ which satisfies $P^{(\hat{\theta},\lambda)}(\psi_V) \circ \Theta^{(\hat{\theta},\lambda)}_V = \psi_V$. A morphism $\phi: (V,\psi_V) \rightarrow (W,\psi_W)$ in $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is a morphism in $\textup{\text{Rep}}^{\theta}(G)$ which satisfies $P^{(\hat{\theta},\lambda)}(\phi) \circ \psi_W \circ \phi = \psi_V$. Note that $\phi$ is necessarily injective and $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(\hat{G})$ is neither linear nor abelian. A more standard representation theoretic interpretation of $\textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ is as follows. Given a vector space $V$ and sign $\epsilon \in C_2$, introduce the notation $${^{\epsilon}}V = \begin{cases} V & \mbox{if } \epsilon=1, \\ V^{\vee} & \mbox{if } \epsilon=-1 \end{cases}$$ with similar notation for linear maps. A $(\hat{\theta},\lambda)$-twisted Real representation of $G$ is then a vector space $V$ together with linear maps $\rho(\omega): \prescript{\pi(\omega)}{}{V} \rightarrow V$, $\omega \in \hat{G}$, which satisfy $\rho(e) = \textup{\text{id}}_V$ and $$\label{eq:explicitRealRep} \rho(\omega_2) \circ \prescript{\pi(\omega_2)}{}{\rho(\omega_1)}^{\pi(\omega_2)} \circ \textup{\text{ev}}_V^{\delta_{\pi(\omega_1), \pi(\omega_2), -1}} = \lambda(\omega_1)^{\frac{\pi(\omega_2)-1}{2}}\hat{\theta}([\omega_2 \vert \omega_1]) \rho(\omega_2 \omega_1).$$ The notation $\textup{\text{ev}}_V^{\delta_{\pi(\omega_1), \pi(\omega_2), -1}}$ indicates that $\textup{\text{ev}}_V$ is included exactly when $\pi(\omega_1)= \pi(\omega_2)=-1$. The equivalence of this interpretation with that of homotopy fixed points follows from noting that a homotopy fixed point $((V,\rho_V),\psi_V)$ determines an extension of $\rho_V$ to $\hat{G}\setminus G$ by $$\rho_V(\omega) = \hat{\theta}([\omega \varsigma^{-1} \vert \varsigma])^{-1} \rho_V(\omega \varsigma^{-1}) \circ \psi_V^{-1}, \qquad \omega \in \hat{G}\setminus G.$$ A third interpretation of twisted Real representations will also be useful. **Proposition 6**. *Fix $\varsigma \in \hat{G}\setminus G$. A $(\hat{\theta},\lambda)$-twisted Real representation of $G$ is equivalent to the data of a $\theta$-twisted representation of $G$ on $V$ together with a non-degenerate bilinear form $\langle -, -\rangle : V \times V \rightarrow \mathbb{C}$ which satisfies the twisted $G$-invariance condition $$\langle \rho(g)v_1, \rho(\varsigma g \varsigma^{-1})v_2 \rangle = \lambda(g) \frac{\hat{\theta}([\varsigma \vert g])}{\hat{\theta}([\varsigma g \varsigma^{-1} \vert \varsigma])} \langle v_1, v_2 \rangle, \qquad g \in G$$ and the twisted symmetry condition $$\langle v_1, v_2 \rangle = \lambda(\varsigma) \theta([\varsigma^{-1} \vert \varsigma^{-1}]) \langle \rho(\varsigma^{-2})v_2, v_1 \rangle$$ for all $v_1,v_2 \in V$.* *Proof.* Let $(V,\rho)$ be a $(\hat{\theta},\lambda)$-twisted Real representation of $G$. Fix $\varsigma \in \hat{G}\setminus G$ and define a non-degenerate bilinear form on $V$ by $$\label{eq:bilinFormFromRealRep} \langle v_1, v_2 \rangle = \rho(\varsigma^{-1})^{-1}(v_1)v_2.$$ With this definition, $\langle \rho(g)v_1, \rho(\varsigma g \varsigma^{-1})v_2 \rangle$ is equal to $$\begin{aligned} % \langle \rho(g)v_1, \rho(\varsigma g \varsigma^{-1})v_2 \rangle % &=& && \rho(\varsigma^{-1})^{-1}(\rho(g)(v_1))(\rho(\varsigma g \varsigma^{-1})v_2) \\ &=& \lambda(\varsigma g) \hat{\theta}([\varsigma^{-1} \vert \varsigma g])^{-1} \rho(\varsigma g)^{- \vee}(\textup{\text{ev}}_V(v_1))(\rho(\varsigma g \varsigma^{-1})v_2)\\ &=& \lambda(\varsigma g) \lambda(\varsigma^{-1}) \hat{\theta}([\varsigma^{-1} \vert \varsigma g])^{-1} \hat{\theta}([\varsigma g \vert \varsigma^{-1}])^{-1} \rho(\varsigma g \varsigma^{-1})^{- \vee} \rho(\varsigma^{-1})^{-1}(v_1)(\rho(\varsigma g \varsigma^{-1})v_2)\\ &=& \lambda(g) \frac{\hat{\theta}([\varsigma \vert g])}{\hat{\theta}([\varsigma g \varsigma^{-1} \vert \varsigma])} \langle v_1, v_2 \rangle.\end{aligned}$$ The first two equalities follow from equation [\[eq:explicitRealRep\]](#eq:explicitRealRep){reference-type="eqref" reference="eq:explicitRealRep"} and the third from the $2$-cocycle condition on $\hat{\theta}$. Similarly, we compute $$\begin{aligned} \langle v_1, v_2 \rangle % &=& % \rho(\varsigma^{-1})^{-1}(v_1)v_2 \\ &=& \lambda(\varsigma) \hat{\theta}([\varsigma^{-1} \vert \varsigma])^{-1} \rho(\varsigma)^{-\vee} (\textup{\text{ev}}_V(v_1))v_2 \\ &=& \lambda(\varsigma)\hat{\theta}([\varsigma^{-1} \vert \varsigma])^{-1} \rho(\varsigma^{-2})^{-\vee} \circ \rho(\varsigma)^{-\vee} (\textup{\text{ev}}_V(v_1))(\rho(\varsigma^{-2})v_2) \\ % &=& % \lambda(\varsigma) \hat{\theta}([\varsigma^{-1} \vert \varsigma])^{-1} \hat{\theta}([\varsigma^{-2} \vert \varsigma])^{-1} \rho(\varsigma^{-1})^{-\vee}(\ev_V(v_1))(\rho(\varsigma^{-2})v_2) \\ &=& \lambda(\varsigma) \hat{\theta}([\varsigma^{-1} \vert \varsigma])^{-1} \hat{\theta}([\varsigma^{-2} \vert \varsigma])^{-1} \textup{\text{ev}}_V(v_1)(\rho(\varsigma^{-1})^{-1} \circ \rho(\varsigma^{-2})v_2) \\ % &=& % \lambda(\varsigma) \hat{\theta}([\varsigma^{-1} \vert \varsigma])^{-1} \hat{\theta}([\varsigma^{-2} \vert \varsigma])^{-1} \langle \rho(\varsigma^{-2})(v_2),v_1 \rangle \\ &=& \lambda(\varsigma) \hat{\theta}([\varsigma^{-1} \vert \varsigma^{-1}]) \langle \rho(\varsigma^{-2})v_2,v_1 \rangle.\end{aligned}$$ Conversely, given $(V,\rho) \in \textup{\text{Rep}}^{\theta}(G)$ with non-degenerate bilinear form $\langle -, - \rangle$ satisfying the conditions of the lemma, define $\rho(\varsigma^{-1})$ by equation [\[eq:bilinFormFromRealRep\]](#eq:bilinFormFromRealRep){reference-type="eqref" reference="eq:bilinFormFromRealRep"} and set $$\rho(\omega) = \hat{\theta}([\omega \varsigma \vert \varsigma^{-1}])^{-1} \rho(\omega \varsigma) \circ \rho(\varsigma^{-1}), \qquad \omega \in \hat{G}\setminus G.$$ The verification that $\rho$ is a $(\hat{\theta},\lambda)$-twisted Real representation of $G$ mirrors the calculations from the previous paragraph. ◻ A $(\hat{\theta},\lambda)$-twisted Real representation is called *irreducible* if it has no non-trivial Real subrepresentations. The direct sum $(V,\psi_V) \oplus (W,\psi_W) = (V \oplus W, \psi_V \oplus \psi_W)$ allows for the following formulation of a Real analogue of Maschke's lemma. **Proposition 7**. *Let $V \in \textup{\text{RRep}}^{(\hat{\theta},\lambda)}(G)$ be irreducible. Then the restriction of $V$ to $G$ is irreducible or of the form $U \oplus P^{(\hat{\theta},\lambda)}(U)$ for an irreducible $U \in \textup{\text{Rep}}^{\theta}(G)$.* *Proof.* Interpret $V$ as a $\theta$-twisted representation of $G$ with compatible bilinear form $\langle-,-\rangle$, as in Proposition [Proposition 6](#prop:RealRepBilinearForm){reference-type="ref" reference="prop:RealRepBilinearForm"}, and suppose that the restriction $V_{\vert G}$ has a non-trivial irreducible $\theta$-twisted subrepresentation $U$. The twisted $G$-invariance of $\langle-,-\rangle$ implies that the orthogonal complement $U^{\perp}$ is a $\theta$-twisted subrepresentation of $V_{\vert G}$ and $V_{\vert G} = U \oplus U^{\perp}$ as $\theta$-twisted representations. Since $V$ is irreducible, the map $\rho(\varsigma): V^{\vee} \rightarrow V$ restricts to a map $\rho(\varsigma): U^{\vee} \rightarrow U^{\perp}$ which defines an isomorphism $P^{(\hat{\theta},\lambda)}(U) \xrightarrow[]{\sim} U^{\perp}$ of $\theta$-twisted representations. ◻ ## A Frobenius--Schur indicator {#sec:FSIndDetail} Keep the notation of Section [2.1](#sec:RealProjRep){reference-type="ref" reference="sec:RealProjRep"}. **Definition 8**. *The *$(\hat{\theta},\lambda)$-twisted Frobenius--Schur element* is $$\nu_{(\hat{\theta},\lambda)} = \sum_{\varsigma \in \hat{G}\setminus G} \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} l_{\varsigma^2} \in \mathbb{C}^{\theta^{-1}}[G].$$* When $(\hat{\theta},\lambda)$ is clear from the context, we write $\nu$ for $\nu_{(\hat{\theta},\lambda)}$. Note that $$\nu_{(\hat{\theta},\lambda)}=-\nu_{(\delta\hat{\theta},\lambda)}=-\nu_{(\hat{\theta},\pi\lambda)}.$$ **Lemma 9**. *The element $\nu_{(\hat{\theta},\lambda)}$ is a $\theta$-twisted class function on $G$.* *Proof.* The statement amounts to the identity $$\hat{\theta}([h \varsigma h^{-1} \vert h \varsigma h^{-1}])^{-1} = \uptau(\theta)([h] \varsigma^2) \hat{\theta}([\varsigma \vert \varsigma])^{-1}, \qquad h \in G, \; \varsigma \in \hat{G}\backslash G,$$ which is seen to hold using equation [\[eq:oddConj\]](#eq:oddConj){reference-type="eqref" reference="eq:oddConj"}. ◻ We require the following elementary result from linear algebra. **Lemma 10**. *Let $V$ be a finite dimensional vector space and $\phi \in \textup{\text{Hom}}_{\mathbb{C}}(V,V)$. Then $\textup{\text{tr}}_V \,\phi$ is equal to the trace of the map $$\iota_{\phi}: \textup{\text{Hom}}_{\mathbb{C}}(V, V^{\vee}) \rightarrow \textup{\text{Hom}}_{\mathbb{C}}(V, V^{\vee}), \qquad f \mapsto f^{\vee} \circ \textup{\text{ev}}_V \circ \phi.$$* *Proof.* Let $\dim_{\mathbb{C}} V= v$. Fix a basis of $V$ with induced basis $\{E_{ij}\}_{i,j=1}^v$ of $\textup{\text{Hom}}_{\mathbb{C}}(V,V)$. Writing $\phi = \sum_{i,j=1}^v \phi_{ij} E_{ij}$ in this basis, we compute $\iota_{\phi} (E_{ij}) = \sum_{k=1}^v \phi_{ik} E_{jk}$ so that $$\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathbb{C}}(V,V^{\vee})} \, \iota_{\phi} = \sum_{i,j=1}^v \iota_{\phi} (E_{ij})_{ij} = \sum_{i,j,k=1}^v \phi_{ik}(E_{jk})_{ij} = \sum_{i,j,k=1}^v \phi_{ik} \delta_{ji} \delta_{kj} = \textup{\text{tr}}_V \, \phi. \qedhere$$ ◻ Let $V \in \textup{\text{Rep}}^{\theta}(G)$ and $\phi \in \textup{\text{Hom}}_{G}(V,V)$. Consider the map $$\iota_{\phi}: \textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda,\varsigma)}(V)) \rightarrow \textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda,\varsigma)}(V)), \qquad f \mapsto P^{(\hat{\theta},\lambda,\varsigma)}(f) \circ \Theta^{(\hat{\theta},\lambda,\varsigma)}_V \circ \phi.$$ Independence of the duality structure up to equivalence on $\varsigma \in \hat{G}\setminus G$ implies that $\iota_{\phi}$ is independent of $\varsigma$. The coherence condition [\[eq:catWDualCoher\]](#eq:catWDualCoher){reference-type="eqref" reference="eq:catWDualCoher"} implies that $\iota:= \iota_{\textup{\text{id}}_V}$ is an involution. For each $V \in \textup{\text{Rep}}^{\theta}(G)$, define $$\tau^V : \textup{\text{Hom}}_{G}(V,V) \rightarrow Z(\mathbb{C}^{\theta^{-1}}[G]), \qquad \phi \mapsto \sum_{g \in G} \textup{\text{tr}}_V (\phi \circ \rho_V(g)) l_g.$$ Note that $\tau^V(\textup{\text{id}}_V) = \chi_V$. **Theorem 11**. *For each $V \in \textup{\text{Rep}}^{\theta}(G)$ and $\phi \in \textup{\text{Hom}}_{G}(V,V)$, there is an equality $$\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P^{(\hat{\theta},\lambda)}(V))} \, \iota_{\phi} = \langle \tau^V(\phi), \nu_{(\hat{\theta},\lambda)} \rangle_{G}.$$* *Proof.* Write $(P, \Theta)$ for $(P^{(\hat{\theta},\lambda,\varsigma)},\Theta^{(\hat{\theta},\lambda,\varsigma)})$. We compute $$\begin{aligned} \textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))} \, \iota_{\phi} % &=& % \tr_{\Hom_{\G}(V,P(V))} (f \mapsto f^{\vee} \circ \Theta_V \circ \phi) \\ &=& \textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))} \, (f \mapsto \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} f^{\vee} \circ \textup{\text{ev}}_V \circ \rho(\varsigma^2)^{-1} \circ \phi) \\ &=& \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} \textup{\text{tr}}_V\,(\rho(\varsigma^2)^{-1} \circ \phi),\end{aligned}$$ the second equality following from Lemma [Lemma 10](#lem:traceTensorDual){reference-type="ref" reference="lem:traceTensorDual"}. Since $\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))} \, \iota_{\phi}$ is independent of the choice $\varsigma \in \hat{G}\setminus G$ used in the definition of $\iota_{\phi}$, we average over all such choices to obtain $$\textup{\text{tr}}_{\textup{\text{Hom}}_{G}(V,P(V))} \, \iota_{\phi} = \frac{1}{\vert G\vert} \sum_{\varsigma \in \hat{G}\backslash G}\frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} \textup{\text{tr}}_V\,(\rho(\varsigma^2)^{-1} \circ \phi).$$ On the other hand, we have $$\begin{aligned} \langle \tau^V(\phi), \nu \rangle_{G} % &=& % \frac{1}{\vert G \vert} \sum_{g \in \G} \theta([g \vert g^{-1}])^{-1} \bb^V(\phi)(g^{-1}) \nu(g)\\ % &=& % \frac{1}{\vert G \vert} \sum_{\substack{g \in \G \\ \varsigma \in \Gh \setminus \G \\ \varsigma^2 = g}} \theta([g \vert g^{-1}])^{-1} \tr_V\, \left(\phi \circ \rho_V(g^{-1}) \right) \lambda(\varsigma) \hat{\theta}([\varsigma \vert \varsigma])^{-1}\\ &=& \frac{1}{\vert G\vert} \sum_{\varsigma \in \hat{G}\backslash G} \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} \theta([\varsigma^2 \vert \varsigma^{-2}])^{-1} \textup{\text{tr}}_V\, \left(\phi \circ \rho_V(\varsigma^{-2}) \right) \\ &=& \frac{1}{\vert G\vert} \sum_{\varsigma \in \hat{G}\backslash G} \frac{\lambda(\varsigma)}{\hat{\theta}([\varsigma \vert \varsigma])} \textup{\text{tr}}_V\, \left(\rho_V(\varsigma^{2})^{-1} \circ \phi \right),\end{aligned}$$ thereby proving the desired equality. ◻ Recall that $\delta$ is a cocycle representative of the generator of $H^{2+\pi}(B C_2) \simeq C_2$. **Corollary 12**. *Let $V$ be an irreducible $\theta$-twisted representation of $G$. Then $$\langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_{G} = \begin{cases} 1 & \mbox{if and only if $V$ lifts to a $(\hat{\theta},\lambda)$-twisted Real representation},\\ -1 & \mbox{if and only if $V$ lifts to a $(\delta\hat{\theta},\lambda)$-twisted Real representation}, \\ 0 & \mbox{otherwise}. \end{cases}$$ When $\langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_{G} = \pm 1$, the twisted Real structure on $V$ is unique up to isomorphism.* *Proof.* Schur's Lemma for $\textup{\text{Rep}}^{\theta}(G)$ implies that $\textup{\text{Hom}}_G(V, P^{(\hat{\theta},\lambda)}(V)) \simeq \mathbb{C}$ if $P^{(\hat{\theta},\lambda)}(V) \simeq V$ and $\textup{\text{Hom}}_G(V,P^{(\hat{\theta},\lambda)}(V))=0$ otherwise. Hence, if $\textup{\text{Hom}}_G(V,P^{(\hat{\theta},\lambda)}(V))=0$, then $V$ does not lift to a Real representation and the statement follows by applying Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"} with $\phi = \textup{\text{id}}_V$. If $\textup{\text{Hom}}_G(V, P^{(\hat{\theta},\lambda)}(V)) \simeq \mathbb{C}$, then a non-zero element $\psi_V \in \textup{\text{Hom}}_G(V, P^{(\hat{\theta},\lambda)}(V))$ is an isomorphism which, by Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"}, satisfies $$P^{(\hat{\theta},\lambda)}(\psi_V) \circ \Theta_V = \langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_{G} \cdot \psi_V.$$ The first statement of the corollary now follows from the homotopy fixed point interpretation of twisted Real representations. Uniqueness of the Real structure up to isomorphism follows from one dimensionality of $\textup{\text{Hom}}_G(V, P^{(\hat{\theta},\lambda)}(V))$. ◻ In the setting of Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"}, if $\langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_{G} =0$, then $V$ determines an irreducible $(\hat{\theta},\lambda)$-twisted Real representation by $H^{(\hat{\theta},\lambda)}(V) = V \oplus P^{(\hat{\theta},\lambda)}(V)$ with its hyperbolic homotopy fixed point structure [@mbyoung2021b §7.3]. Note that $H^{(\hat{\theta},\lambda)}(V) \simeq H^{(\hat{\theta},\lambda)}(P^{(\hat{\theta},\lambda)}(V))$. **Corollary 13**. *There are finitely many isomorphism classes of irreducible $(\hat{\theta},\lambda)$-twisted Real representations.* *Proof.* This follows from Proposition [Proposition 7](#prop:RealRestr){reference-type="ref" reference="prop:RealRestr"}, finiteness of isomorphism classes of irreducible $\theta$-twisted representations and the final statement of Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"}. ◻ **Corollary 14**. *There is an equality $$\nu_{(\hat{\theta},\lambda)} = \sum_{\substack{ V \in \textup{\text{Irr}}^{\theta}(G) \\ P^{(\hat{\theta},\lambda)}(V) \simeq V}} \langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_G \chi_V.$$* *Proof.* This follows from the second part of Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"} and the fact that the set $\{\chi_V\}_{V \in \textup{\text{Irr}}^{\theta}(G)}$ of irreducible $\theta$-twisted characters is an orthonormal basis of the space of $\theta$-twisted class function on $G$. ◻ Various instances of the element $\nu_{(\hat{\theta},\lambda)}$ and Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"} are known: 1. The classical setting of Frobenius and Schur [@frobenius1906] corresponds to taking $\hat{G}= G\times C_2$ with $\pi$ the projection to the second factor and the cohomological data $\hat{\theta}$ and $\lambda$ trivial. The conditions on the bilinear form $\langle -, - \rangle$ from Proposition [Proposition 6](#prop:RealRepBilinearForm){reference-type="ref" reference="prop:RealRepBilinearForm"} reduce to $G$-invariance and symmetry; if $\hat{\theta}=\delta$, then $\langle -, - \rangle$ is skew-symmetric. Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"} then gives the standard necessary and sufficient condition for $V$ to admit a $G$-invariant bilinear form and so be defined over $\mathbb{R}$ (in the symmetric case) or $\mathbb{H}$ (in the skew-symmetric case). 2. Taking $\hat{\theta}$ and $\lambda$ to be trivial recovers Gow's generalized Frobenius--Schur element used in the character theoretic study of $2$-regularity of finite groups [@gow1979 §2]. For representation theoretic applications, see [@rumynin2021]. 3. Take $\hat{G}= G\times C_2$. In this case, there is a homomorphism $G\rightarrow \hat{G}$ which splits $\pi$. When $\hat{\theta}$ is in the image of the resulting map $H^2(B G; C_2) \rightarrow H^{2+\pi}(B \hat{G})$ and $\lambda$ is trivial, $\nu_{(\hat{\theta},1)}$ recovers Turaev's generalized Frobenius--Schur element studied in the context of closed unoriented TFT [@turaev2007]. See [@ichikawa2023] for a generalization in the setting of closed $\textup{\text{Pin}}_2^-$ TFT. 4. When $\phi=\textup{\text{id}}_V$ the trace of Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"} is an instance of a Shimizu's Frobenius--Schur indicator in a category with duality [@shimizu2012]. [\[ex:cyclicGroups\]]{#ex:cyclicGroups label="ex:cyclicGroups"} Let $G= C_n$ with generator $r$ and $\zeta= e^{\frac{2\pi \sqrt{-1}}{n}}$. The one dimensional representations $\{\rho_k \mid 0 \leq k \leq n-1\}$, defined by $\rho_k(r)=\zeta^k$, constitute a complete set of irreducible representations of $G$. Take $\hat{\theta}$ and $\lambda$ to be trivial in this example. 1. [\[ite:trivRealStr\]]{#ite:trivRealStr label="ite:trivRealStr"} Let $\hat{G}= C_n \times C_2$ with $\pi$ projection to the second factor. We have $$\langle \chi_k, \nu \rangle_{G} = \begin{cases} 1 & \mbox{if $k=0$ or $k=\frac{n}{2}$}, \\ 0 & \text{otherwise}, \end{cases}$$ whence the trivial and sign representation (which exists when $n$ is even) admit Real structures. These are precisely the irreducible representations which are defined over $\mathbb{R}$. 2. [\[ite:cyclicRealStr\]]{#ite:cyclicRealStr label="ite:cyclicRealStr"} Let $\hat{G}= C_{2n}$ with generator $\varsigma$ satisfying $\varsigma^2=r$ and $C_2$-grading $\pi:\hat{G}\rightarrow C_2$ determined by $\pi(\varsigma)=-1$. Assume that $n$ is even, as otherwise $\hat{G}\simeq C_n \times C_2$ as $C_2$-graded groups. We have $\nu = 2 \sum_{j=0}^{\frac{n}{2}-1} l_{r^{2j}}$ from which we compute $$\langle \chi_k, \nu \rangle_{G} = \frac{2}{n} \sum_{j=0}^{\frac{n}{2}-1} \zeta^{2kj} \\ = \begin{cases} 1 & \mbox{if } k=0, \\ -1 & \mbox{if } k=\frac{n}{2}, \\ 0 & \mbox{otherwise}. \end{cases}$$ The Real structure on $\rho_0$ is given by $\rho_{0}(\varsigma)(1^{\vee}) = 1$. The same formula gives the $\delta$-twisted Real structure on $\rho_{\frac{n}{2}}$. 3. [\[ite:dihedralRealStr\]]{#ite:dihedralRealStr label="ite:dihedralRealStr"} Let $\hat{G}$ be the dihedral group $D_{2n} = \langle r,s \mid r^n=s^2=e, \, srs=r^{-1} \rangle$ with $\pi: \hat{G}\rightarrow C_2$ determined by $\pi(r)=1$ and $\pi(s)=-1$. We have $\nu = nl_{e}$ from which we compute $\langle \chi_k, \nu \rangle_{G} =1$. Each irreducible representation $\rho_k$ can therefore be extended to a Real representation by the formula $\rho_k(s)(1^{\vee})=1$. Let $\hat{G}=Q_8$ be the quaternion group with $C_2$-grading given on the standard generators by $\pi(i)=1$ and $\pi(j)=-1$. Then $G \simeq C_4$ is generated by $i$. We have $\nu = 4 l_{-1}$ so that $\langle \chi_k, \nu \rangle = (-1)^k$. The Real structure on $\rho_k$, which is $\delta$-twisted precisely when $k$ is even, is determined by $\rho_k(j)(1^{\vee}) = 1$. Let $G=A_4$ be the alternating group on $4$ letters. The irreducible representations of $G$ are the trivial representation $U$, two non-trivial one dimensional representations $U^{\prime}$ and $U^{\prime \prime}$ and a three dimensional representation $V$. Writing $\zeta= e^{\frac{2\pi \sqrt{-1}}{3}}$, we take the convention that their characters are $$\begin{aligned} \chi_{U^{\prime}}(123) & = \zeta, & \chi_{U^{\prime}}(132) &= \zeta^2, & \chi_{U^{\prime}}((12)(34))& =1, \\ \chi_{U^{\prime \prime}}(123) & = \zeta^2, & \chi_{U^{\prime \prime}}(132) &= \zeta, & \chi_{U^{\prime\prime}}((12)(34)) & =1, \\ \chi_V(123) & =0, & \chi_V(132) & =0, & \chi_V((12)(34)) & = -1.\end{aligned}$$ 1. Taking $\hat{G}= A_4 \times C_2$ with $\pi$ the projection to the second factor gives $\nu= 4 l_{(1)} + \sum_{3\mbox{\tiny-cycles} \; \sigma} l_{\sigma}$. Using this, we compute $$\langle \chi_U,\nu \rangle = 1, \qquad \langle \chi_{U^{\prime}},\nu \rangle=0, \qquad \langle \chi_{U^{\prime \prime}},\nu \rangle=0, \qquad \langle \chi_V,\nu \rangle= 1.$$ Hence, only $U$ and $V$ admit real structures. 2. Taking $\hat{G}= S_4$ the symmetric group with $\pi$ the sign representation gives $\nu = 6 l_{(1)} +2 (l_{(12)(34)} + l_{(13)(24)} + l_{(14)(23)})$. Using this, we compute $$\langle \chi_U,\nu \rangle = 1, \qquad \langle \chi_{U^{\prime}},\nu \rangle=1, \qquad \langle \chi_{U^{\prime \prime}},\nu \rangle=1, \qquad \langle \chi_V,\nu \rangle= 0.$$ Hence, all one dimensional representations admit Real structures. Taking $\lambda$ to be non-trivial, that is, $\lambda=\pi$, replaces $\nu$ with its negative and leads to $\delta$-twisted Real structures on the one dimensional representations. # Two dimensional unoriented open/closed topological field theory {#sec:TFT} ## Algebraic characterization Following Lazaroiu [@lazaroiu2001] and Moore and Segal [@moore2006], we begin by recalling an algebraic characterization of two dimensional oriented open/closed topological field theories (TFTs). See also [@alexeevski2006; @lauda2008]. In topological terms, such a TFT is a symmetric monoidal functor $\mathcal{Z}: \textup{\text{Bord}}_2^{\textup{\text{or}},D} \rightarrow \textup{\text{Vect}}_{\mathbb{C}}$. Here $\textup{\text{Bord}}_2^{\textup{\text{or}},D}$ two dimensional open/closed bordism category [@lauda2008 §3]. Objects are compact oriented $1$-manifolds with boundary components labelled by elements of a given set $D$. Morphisms are isomorphism classes of oriented bordisms with corners whose free boundaries are $D$-labelled compatibly with the incoming and outgoing boundaries. The monoidal structure of $\textup{\text{Bord}}_2^{\textup{\text{or}},D}$ is disjoint union. **Theorem 15** ([@moore2006 Theorem 1]). *Two dimensional oriented open/closed TFTs are classified by the following data:* 1. *A commutative Frobenius algebra $A$ with identity $1_A$ and trace $\langle- \rangle_{0}: A \rightarrow \mathbb{C}$.* 2. *A Calabi--Yau category $\mathcal{B}$, that is, $\mathbb{C}$-linear additive category with cyclic traces $\langle-\rangle_V: \textup{\text{Hom}}_{\mathcal{B}}(V,V) \rightarrow \mathbb{C}$, $V \in \mathcal{B}$, whose associated pairings $$\langle -, - \rangle_{V,W}: \textup{\text{Hom}}_{\mathcal{B}}(W,V) \otimes \textup{\text{Hom}}_{\mathcal{B}}(V,W) \xrightarrow[]{\circ} \textup{\text{Hom}}_{\mathcal{B}}(V,V) \xrightarrow[]{\langle - \rangle_V} \mathbb{C}$$ are non-degenerate.* 3. *For each $V \in \mathcal{B}$, a linear *boundary-bulk* map $\tau^V : \textup{\text{Hom}}_{\mathcal{B}}(V,V) \rightarrow A$ and linear *bulk-boundary* map $\tau_V: A \rightarrow \textup{\text{Hom}}_{\mathcal{B}}(V,V)$.* *This data is required to satisfy the following conditions:* 1. *$\tau_V$ is a unital algebra homomorphism.* 2. *$\tau_W(a) \circ \phi = \phi \circ \tau_V(a)$ for all $a \in A$ and $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,W)$.* 3. *$\langle \phi, \tau_V (a) \rangle_{V,V} = \langle \tau^V(\phi), a \rangle_0$ for all $a \in A$ and $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$.* 4. *(The *oriented Cardy condition*) Let $\{\psi_i\}_i$ be a basis of $\textup{\text{Hom}}_{\mathcal{B}}(V,W)$ and $\{\psi^i\}_i$ the basis of $\textup{\text{Hom}}_{\mathcal{B}}(W,V)$ which is dual with respect to $\langle-,-\rangle_{V,W}$. Then $\tau_V \circ \tau^W$ is equal to the map $$\textup{\text{Hom}}_{\mathcal{B}}(W,W) \rightarrow \textup{\text{Hom}}_{\mathcal{B}}(V,V), \qquad \phi \mapsto \sum_i \psi^i \circ \phi \circ \psi_i.$$* **Remarks 16**. 1. When $\mathcal{B}$ has a single object, the algebraic data of Theorem [Theorem 15](#thm:ocOriTFT){reference-type="ref" reference="thm:ocOriTFT"} is called a *Cardy--Frobenius* or *knowledgeable* Frobenius algebra [@alexeevski2006; @lauda2008]. 2. Let $\mathcal{Z}$ be an oriented open/closed TFT with object set $D$. The category[^1] $\mathcal{B}$ has objects $D$, morphisms $\textup{\text{Hom}}_{\mathcal{B}}(V,W)$ given by the value of $\mathcal{Z}$ on the closed interval labelled by $V$ and $W$ and oriented from $V$ to $W$ and composition defined by the value of $\mathcal{Z}$ on the flattened pair of pants. The value of $\mathcal{Z}$ on the flattened cap defines the Calabi--Yau traces. 3. By non-degeneracy of the Calabi--Yau pairings, the oriented Cardy condition holds if and only if $$\label{eq:baggyCardy} \textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,W)} \, (f \mapsto \psi \circ f \circ \phi) = \langle \tau^W(\psi), \tau^V(\phi) \rangle_0$$ for all $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$ and $\psi \in \textup{\text{Hom}}_{\mathcal{B}}(W,W)$. Following [@caldararu2010 §7.4], we refer to equation [\[eq:baggyCardy\]](#eq:baggyCardy){reference-type="eqref" reference="eq:baggyCardy"} as the *baggy oriented Cardy condition*. Topologically, the oriented Cardy condition asserts the equality of two ways of evaluating the TFT on the annulus with boundary components labelled by $V$ and $W$. We are interested in the extension of Theorem [Theorem 15](#thm:ocOriTFT){reference-type="ref" reference="thm:ocOriTFT"} to the unoriented bordism category $\textup{\text{Bord}}_2^D$, defined analogously to $\textup{\text{Bord}}_2^{\textup{\text{or}},D}$ except that objects and morphisms are unoriented. Upon restriction to the closed sector, the extension is known. **Theorem 17** ([@turaev2006 Proposition 2.9]). *Two dimensional unoriented TFTs are classified by the data of an *unoriented Frobenius algebra*, that is, a commutative Frobenius algebra $(A,1_A, \langle - \rangle_0)$ with an isometric algebra involution $p: A \rightarrow A$ and an element $Q \in A$, the *crosscap state*, which satisfy the following conditions:* 1. *$p(Q a) = Q a$ for all $a \in A$.* 2. *(*The Klein condition*) Given a basis $\{a_i\}_i$ of $A$ with basis $\{a^i\}_i$ of $A$ dual with respect to $\langle - \rangle_0$, the equality $Q^2 = \sum_i p(a^i ) a_i$ holds.* In terms of bordisms, $Q$ is the image under $\mathcal{Z}$ of the compact Möbius strip $\mathbb{RP}^2 \setminus \mathring{D}^2$, $$\begin{tikzpicture}[very thick,scale=2.5,color=black,baseline=0.75cm] %cap portion \coordinate (q1) at (-0.4,0.175); \coordinate (q2) at (-0.4,0.525); \coordinate (x1) at (-0.63,0.375); \coordinate (x2) at (-0.58,0.325); \coordinate (y1) at (-0.63,0.325); \coordinate (y2) at (-0.58,0.375); % \draw[very thick] (q1) .. controls +(-0.4,0) and +(-0.4,0) .. (q2); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{>}}}, postaction={decorate}] (q1) .. controls +(0.15,0) and +(0.15,0) .. (q2); \draw[very thick, blue!80!black, opacity=0.2] (q1) .. controls +(-0.15,0) and +(-0.15,0) .. (q2); %crosscap "x" \draw[thick] (x1) to (x2); \draw[thick] (y1) to (y2); \draw[thick] (-0.605,0.35) circle (0.05cm); \end{tikzpicture} : \varnothing \rightarrow S^1,$$ and $p$ is the image of the mapping cylinder of circle reflection, $$\begin{tikzpicture}[very thick,scale=2.5,color=black,baseline] % \coordinate (r1) at (0.1,0.15); \coordinate (r2) at (0.1,-0.15); \coordinate (r3) at (0.8,0.15); \coordinate (r4) at (0.8,-0.15); % \draw (r1) to (r3); \draw (r2) to (r4); % \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{<}}}, postaction={decorate}] (r1) .. controls +(0.15,0) and +(0.15,0) .. (r2); \draw[very thick, blue!80!black] (r1) .. controls +(-0.15,0) and +(-0.15,0) .. (r2); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{>}}}, postaction={decorate}] (r3) .. controls +(0.15,0) and +(0.15,0) .. (r4); \draw[very thick, blue!80!black,opacity=0.2] (r3) .. controls +(-0.15,0) and +(-0.15,0) .. (r4); \end{tikzpicture} : S^1 \rightarrow S^1.$$ The Klein condition is illustrated in Figure [\[fig:KleinCond\]](#fig:KleinCond){reference-type="ref" reference="fig:KleinCond"}. = We now come the main classification result. **Theorem 18**. *Two dimensional unoriented open/closed TFTs are classified by the data of an underlying closed theory, as in Theorem [Theorem 17](#thm:cUnoriTFT){reference-type="ref" reference="thm:cUnoriTFT"}, together with the data of a $\mathbb{C}$-linear strict duality $P$ on $\mathcal{B}$. This data is required to satisfy the following conditions:* 1. *[\[ite:identObj\]]{#ite:identObj label="ite:identObj"} The functor $P$ is the identity on objects.* 2. *[\[ite:compatPairing\]]{#ite:compatPairing label="ite:compatPairing"} $\langle P(\phi) \rangle_V = \langle \phi \rangle_V$ for all $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$.* 3. *[\[ite:compatBulkBound\]]{#ite:compatBulkBound label="ite:compatBulkBound"} $P \circ \tau_V = \tau_V \circ p$ for all $V \in \mathcal{B}$.* 4. *[\[ite:compatBoundBulk\]]{#ite:compatBoundBulk label="ite:compatBoundBulk"} $p \circ \tau^V = \tau^V\circ P$ for all $V \in \mathcal{B}$.* 5. *[\[ite:unoriCardy\]]{#ite:unoriCardy label="ite:unoriCardy"} (The *unoriented Cardy condition*) Let $\{\psi_i\}_i$ be a basis of $\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ with dual basis $\{\psi^i\}_{i}$ with respect to $\langle-,-\rangle_{V,V}$. Then there is an equality $$\label{eq:unoriCardy} \tau_V(Q) = \sum_i \psi^i \circ P(\psi_i).$$* *Proof.* The theorem is proved in [@alexeevski2006 §4] under the assumption that $\mathcal{B}$ has a single object, where the above algebraic data is known as a *structure algebra*. This proof generalizes immediately to allow for $\mathcal{B}$ to have many objects, in the same way as the analogous generalization in the oriented case [@lauda2008 §5]. ◻ Topologically, $P$ is the image under $\mathcal{Z}$ of the mapping cylinder of reflection of the closed interval so that $P_{V,W}: \textup{\text{Hom}}_{\mathcal{B}}(V,W) \rightarrow \textup{\text{Hom}}_{\mathcal{B}}(W,V)$ comes from the bordism $$\begin{tikzpicture}[scale=3.0] \draw[very thick, blue!80!black,dashed] (-.25,-1) to (0.25,-1.0); \draw[very thick, blue!80!black,dashed] (-.25,-0.75) to (0.25,-0.75); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{>}}}, postaction={decorate}] (-.25,-1) to (-.25,-0.75); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{<}}}, postaction={decorate}] (.25,-1) to (.25,-0.75); %labels \node at (-.25,-1.05) {\scriptsize $V$}; \node at (-.25,-0.7) {\scriptsize $W$}; \node at (.25,-1.05) {\scriptsize $V$}; \node at (.25,-0.7) {\scriptsize $W$}; \node at (0.375,-0.9) {=}; \end{tikzpicture} \begin{tikzpicture}[scale=3.0] \draw[very thick, blue!80!black,dashed] (-.25,-1) to (0.25,-0.75); \draw[very thick, blue!80!black,dashed] (-.25,-0.75) to (0.25,-1.0); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{>}}}, postaction={decorate}] (-.25,-1) to (-.25,-0.75); \draw[very thick, blue!80!black,decoration={markings, mark=at position 0.5 with {\arrow{>}}}, postaction={decorate}] (.25,-1) to (.25,-0.75); %labels \node at (-.25,-1.05) {\scriptsize $V$}; \node at (-.25,-0.7) {\scriptsize $W$}; \node at (.25,-1.05) {\scriptsize $W$}; \node at (.25,-0.7) {\scriptsize $V$}; \node at (0.35,-0.9) {.}; \end{tikzpicture}$$ As indicated on the right, we will picture this bordism as embedded in $\mathbb{R}^3$ with a half-twist. That $P$ is a strict involution follows from the fact that reflection of the closed interval is an involution. We record two basic consequences of Theorem [Theorem 18](#thm:genStruAlg){reference-type="ref" reference="thm:genStruAlg"}. **Proposition 19**. 1. *The equality $\langle Q^2 \rangle_0 = \textup{\text{tr}}_A \, p$ holds.* 2. *For any $V \in \mathcal{B}$ and $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$, the equality $$\langle \tau^V(\phi), Q \rangle_0 = \textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,V)} \, \iota_{\phi}$$ holds, where $\iota_{\phi}$ is defined analogously to Section [2.2](#sec:FSIndDetail){reference-type="ref" reference="sec:FSIndDetail"}.* *Proof.* Since $p$ is an involution, there exists a basis $\{a_i\}_i$ of $A$ such that $p(a_i) = s_i a_i$ with $s_i \in \{1,-1\}$. Let $\{a^i\}_i$ be a dual basis, so that $\langle a^j, a_i \rangle_0 = \delta^j_i$. Since $p$ is an isometry of $\langle- \rangle_0$, we have $p(a^i) = s_i a^i$. With these preliminaries, we compute $$\langle Q^2 \rangle_0 = \langle \sum_i p(a^i) a_i \rangle_0 = \sum_i s_i \langle a^i a_i \rangle_0 = \sum_i s_i = \textup{\text{tr}}_A \, p.$$ For the second statement, we compute $$\begin{gathered} \langle \tau^V(\phi), Q \rangle_0 = \langle \phi, \tau_V(Q) \rangle_{V,V} = \sum_i \langle \phi \circ \psi^i \circ P(\psi_i) \rangle_V =\\ % &=& % \sum_i \langle \psi^i \circ P(\psi_i) \circ \Theta_V \circ \phi \rangle_V \\ % = \sum_i \langle \psi^i \circ \iota_{\phi}(\psi_i)\rangle_V % &=& % \sum_i \langle \psi^i, \iota_{\phi}(\psi_i) \rangle_{V,P(V)} \\ = \textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,V)} \, \iota_{\phi}.\end{gathered}$$ The first equality is the adjointness of $\tau^V$ and $\tau_V$, the second is the unoriented Cardy condition and the third is cyclicity of traces. ◻ By non-degeneracy of the Calabi--Yau pairings, the unoriented Cardy condition is equivalent to the second equality from Proposition [Proposition 19](#prop:wittenIndices){reference-type="ref" reference="prop:wittenIndices"}, which we term the *baggy unoriented Cardy condition*. The unoriented Cardy condition reflects the equality of two ways of evaluating the TFT on the Möbius strip with boundary component labelled by $V$. See Figure [\[fig:unoriCardy\]](#fig:unoriCardy){reference-type="ref" reference="fig:unoriCardy"}. The next result constructs the algebraic input of Theorem [Theorem 18](#thm:genStruAlg){reference-type="ref" reference="thm:genStruAlg"} from a Calabi--Yau category with a contravariant involution which need not act trivially on objects. = **Proposition 20**. *Let $(\mathcal{B}, \tau^{\bullet},\tau_{\bullet},A)$ define a two dimensional oriented open/closed TFT and $(p,Q)$ an unoriented lift of $A$. Let $(P,\Theta)$ be a duality structure on $\mathcal{B}$ such that $\langle P(\phi) \rangle_{P(V)} = \langle \phi \rangle_V$ for all $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$ and $P \circ \tau_V = \tau_{P(V)} \circ p$ and $p \circ \tau^V = \tau^{P(V)} \circ P$ for all $V \in \mathcal{B}$. If the equality $$\label{eq:LefInd} \textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))} \, \iota_{\phi} = \langle \tau^V(\phi),Q \rangle_0$$ holds for all $\phi \in \textup{\text{Hom}}_{\mathcal{B}}(V,V)$, then $(\mathcal{B}^{\tilde{h}C_2}, \tau^{\bullet},\tau_{\bullet},A,p,Q)$ defines a two dimensional unoriented open/closed TFT.* *Proof.* By Lemma [Lemma 4](#lem:homoFixDual){reference-type="ref" reference="lem:homoFixDual"}, the triple $(\mathcal{B}^{\tilde{h}C_2},P^{\tilde{h}C_2},\Theta^{\tilde{h}C_2})$ is a category with strict duality and $P^{\tilde{h}C_2}$ acts trivially on objects. The category $\mathcal{B}^{\tilde{h}C_2}$ inherits a Calabi--Yau structure from $\mathcal{B}$ with traces $\langle - \rangle_{(V,\psi_V)} := \langle- \rangle_V$. Define boundary-bulk and bulk-boundary maps for $\mathcal{B}^{\tilde{h}C_2}$ by $\tau^{(V,\psi_V)} = \tau^V$ and $\tau_{(V,\psi_V)} = \tau_V$. The assumption that $P$ preserves the Calabi--Yau structure and that $P$ and $p$ are compatible with $\tau_{\bullet}$ and $\tau^{\bullet}$ verifies conditions [\[ite:compatPairing\]](#ite:compatPairing){reference-type="ref" reference="ite:compatPairing"}-[\[ite:compatBoundBulk\]](#ite:compatBoundBulk){reference-type="ref" reference="ite:compatBoundBulk"} of Theorem [Theorem 18](#thm:genStruAlg){reference-type="ref" reference="thm:genStruAlg"} for $P^{\tilde{h} C_2}$. It remains to verify the unoriented Cardy condition. Let $(V,\psi_V) \in \mathcal{B}^{\tilde{h}C_2}$. Let $\{\psi_i\}_i$ be a basis of $\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))$ with dual basis $\{\psi^i\}_{i}$ of $\textup{\text{Hom}}_{\mathcal{B}}(P(V),V)$. Then $\{\psi_V^{-1} \circ \psi_i \}_i$ is a basis of $\textup{\text{Hom}}_{\mathcal{B}}(V,V)$ with dual basis $\{\psi^i \circ \psi_V\}_{i}$. We compute $$\begin{gathered} \tau_{(V,\psi_V)}(Q) % = % \bb_V(Q) = \sum_i \psi^i \circ P(\psi_i) \circ \Theta_V = \\ \sum_i \psi^i \circ \psi_V \circ \psi^{-1}_V \circ P(\psi_i) \circ P(\psi_V^{-1}) \circ \psi_V = \sum_i \psi^i \circ \psi_V \circ P^{\tilde{h} C_2}(\psi_V^{-1} \circ \psi_i).\end{gathered}$$ For the first equality, note that the discussion proceeding Proposition [Proposition 19](#prop:wittenIndices){reference-type="ref" reference="prop:wittenIndices"} shows that equation [\[eq:LefInd\]](#eq:LefInd){reference-type="eqref" reference="eq:LefInd"} implies that $\tau_V(Q) = \sum_i \psi^i \circ P(\psi_i) \circ \Theta_V$. The second equality follows from the coherence condition on homotopy fixed points and the final equality from the definition of $(P^{\tilde{h} C_2},\Theta^{\tilde{h} C_2})$. ◻ We comment on the physical interpretation of Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"}. As mentioned in the introduction, the Calabi--Yau category $\mathcal{B}$ should be seen as a model for the category of D-branes in an oriented string theory. With this interpretation, a duality structure $(P,\Theta)$ which preserves the Calabi--Yau pairings is the categorical data of the orientifold construction; see [@diaconescu2007; @hori2008] in the setting of orientifolds of IIB string theory and Landau--Ginzburg theory. In this context, the quantity $\textup{\text{tr}}_{\textup{\text{Hom}}_{\mathcal{B}}(V,P(V))} \, \iota_{\phi}$ is a *parity-twisted Witten index* [@brunner2004 §2] and it is through its computation via closed sector quantities, namely equation [\[eq:LefInd\]](#eq:LefInd){reference-type="eqref" reference="eq:LefInd"}, that the crosscap state $Q$ naturally appears. The D-branes which survive the orientifold projection are the homotopy fixed points of $(P,\Theta)$, that is, objects of the category $\mathcal{B}^{\tilde{h}C_2}$ above. With these remarks in mind, Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"} is an orientifold-type construction of an unoriented open/closed TFT from an oriented open/closed TFT. ## The Frobenius--Schur element as a crosscap state We give an algebraic construction of a two dimensional unoriented open/closed TFT from twisted Real representation theory. When $\hat{G}= G\times C_2$ and the cohomological data $(\hat{\theta},\lambda)$ is trivial, this generalizes results of [@alexeevski2006; @loktev2011]. When $\lambda$ is trivial, a topological construction of the closed sector of this theory was given in [@mbyoung2020 §4.4]. Fix group theoretic data $(\hat{G}, \hat{\theta},\lambda)$ as in Section [2.1](#sec:RealProjRep){reference-type="ref" reference="sec:RealProjRep"}. Let $A = Z(\mathbb{C}^{\theta^{-1}}[G])$ with Frobenius pairing $\langle - , - \rangle_G$ and $\mathcal{B}= \textup{\text{Rep}}^{\theta}(G)$ the Calabi--Yau category with traces $\langle \phi \rangle_V = \frac{1}{\vert G\vert} \textup{\text{tr}}_V \, \phi$. The boundary-bulk map $\tau^V$ is as in Section [2.2](#sec:FSIndDetail){reference-type="ref" reference="sec:FSIndDetail"} and the bulk-boundary map is defined by $$\tau_V \Big(\sum_{g \in G} a_g l_g \Big) = \sum_{g \in G} a_g \theta([g \vert g^{-1}])^{-1} \rho_V(g^{-1}).$$ This data defines a two dimensional oriented open/closed TFT $\mathcal{Z}_{(G,\theta)}$ via Theorem [Theorem 15](#thm:ocOriTFT){reference-type="ref" reference="thm:ocOriTFT"}. See [@moore2006; @turaev2007; @khoi2011]. The main axiom to be verified is the oriented Cardy condition which, in the present setting, is a mild generalization of the orthogonality of characters of irreducible $\theta$-twisted representations. **Theorem 21**. *The data $(\hat{G},\hat{\theta},\lambda)$ defines a two dimensional unoriented open/closed TFT $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ whose oriented sector is a sub TFT of $\mathcal{Z}_{(G,\theta)}$.* We will prove Theorem [Theorem 21](#thm:twistRealRepTFT){reference-type="ref" reference="thm:twistRealRepTFT"} using the orientifold construction of Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"}. We take $(P^{(\hat{\theta},\lambda)},\Theta^{(\hat{\theta},\lambda)})$ for the duality structure on $\mathcal{B}= \textup{\text{Rep}}^{\theta}(G)$ and $Q=\nu_{(\hat{\theta},\lambda)}$ for the candidate crosscap state. We compute $$\langle P(\phi) \rangle_{P(V)} = \frac{1}{\vert G \vert} \textup{\text{tr}}_{P(V)} \, P(\phi) = \frac{1}{\vert G \vert} \textup{\text{tr}}_{V^{\vee}} \, \phi^{\vee} = \langle \phi \rangle_V,$$ which verifies the open sector assumption of Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"}. The remainder of the proof of Theorem [Theorem 21](#thm:twistRealRepTFT){reference-type="ref" reference="thm:twistRealRepTFT"} is divided into closed sector computations and verification of the open/closed coherence conditions required to apply Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"}. The oriented open sector of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ is the full theory $\mathcal{Z}_{(G,\theta)}$ precisely when the forgetful functor $\textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_2} \rightarrow \textup{\text{Rep}}^{\theta}(G)$ is essentially surjective. By Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"}, this is the case when $\langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_{G}=1$ for each irreducible $V \in \textup{\text{Rep}}^{\theta}(G)$. Otherwise, the oriented open sector is a strict subtheory of $\mathcal{Z}_{(G,\theta)}$. In the context of Example [\[ex:cyclicGroups\]](#ex:cyclicGroups){reference-type="ref" reference="ex:cyclicGroups"}, the forgetful functor is essentially surjective only in subexample [\[ite:dihedralRealStr\]](#ite:dihedralRealStr){reference-type="eqref" reference="ite:dihedralRealStr"}. **Remark 22**. We comment on the relation of $\mathcal{Z}_{(\hat{G}, \hat{\theta},\lambda)}$ to categories of $D$-branes in orientifold string theory on global quotients. Recall that the spacetime of an orientifold string theory is an orbifold double cover $\pi: \mathcal{X} \rightarrow \hat{\mathcal{X}}$. Additional data required to define the theory includes the (gauge equivalence class of a) $B$-field $\check{B} \in \check{H}^{3 + \pi}(\hat{\mathcal{X}})$ [@distler2011b], which is a class in the $\pi$-twisted differential cohomology of $\hat{\mathcal{X}}$, and a complex line bundle with connection $\check{L} \in \check{H}^2(\hat{\mathcal{X}})$ [@gao2011 §8.4.1]. The underlying (oriented) orbifold string theory depends only on $(\mathcal{X}, \pi^* \check{B})$. Consider now the particular case in which the spacetime is a global quotient $\pi: X /\!\!/G\rightarrow X /\!\!/\hat{G}$ associated to a finite $C_2$-graded group $\hat{G}$ acting on a smooth manifold $X$. A special class of $B$-fields arises through the composition $$H^{2+\pi}(B \hat{G}) \rightarrow H^{2+\pi}(X /\!\!/\hat{G}) \hookrightarrow \check{H}^{3+\pi}(X /\!\!/G), \qquad \hat{\theta} \mapsto \check{B}_{\hat{\theta}},$$ where the first map is restriction along the the canonical morphism $X /\!\!/\hat{G}\rightarrow B \hat{G}$ and the second is the inclusion of flat $B$-fields. Similarly, a class $\lambda \in H^1(B \hat{G})$ defines a flat line bundle $\check{L}_{\lambda} \in \check{H}^2(X /\!\!/\hat{G})$. The pair $(\check{B}_{\hat{\theta}},\check{L}_{\lambda})$ can be seen as defining universal twists for global $\hat{G}$-orientifolds. The unoriented TFT $\mathcal{Z}_{(\hat{G}, \hat{\theta},\lambda)}$ is a precise mathematical description of the affects of the twists $(\check{B}_{\hat{\theta}},\check{L}_{\lambda})$ on partition functions. See [@braun2002], [@sharpe2011 §5], [@noohiYoung2022 §4.5] for detailed discussions of these affects in the closed sector. We return to the proof of Theorem [Theorem 21](#thm:twistRealRepTFT){reference-type="ref" reference="thm:twistRealRepTFT"}. ### Closed sector Denote by $\textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$ the group of algebra automorphisms and algebra anti-automorphisms of $\mathbb{C}^{\theta^{-1}}[G]$. The group $\textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$ is $C_2$-graded by sending anti-automorphisms to $-1$. **Lemma 23**. *The function $p: \hat{G}\rightarrow \textup{\text{Aut}}^{\textnormal{gen}}(\mathbb{C}^{\theta^{-1}}[G])$, $\omega \mapsto p^{\omega}$, where $$p^{\omega}(l_g) = \lambda(g)^{\frac{\pi(\omega)-1}{2}}\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega] g) l_{\omega g^{\pi(\omega)} \omega^{-1}}, \qquad g \in G$$ is a $C_2$-graded group homomorphism. Moreover, each $p^{\omega}$ is an isometry of $\langle - \rangle_G$.* *Proof.* We prove that $p^{\omega}$, $\omega \in \hat{G}\backslash G$, is an anti-automorphism and omit the easier calculation that $p^g$, $g \in G$, is an automorphism. For $g,h \in G$, direct calculations give $$p^{\omega}(l_g \cdot l_h) = \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]gh)}{\lambda(gh) \theta([g \vert h])} l_{\omega (gh)^{-1} \omega^{-1}}$$ and $$p^{\omega}(l_h) \cdot p^{\omega}(l_g) = \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]h) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]g)}{\lambda(g) \lambda(h) \theta([\omega h^{-1} \omega^{-1} \vert \omega g^{-1} \omega^{-1}])} l_{\omega h^{-1} g^{-1} \omega^{-1}}.$$ It therefore suffices to prove that $$\frac{\theta([g \vert h])}{\theta([\omega h^{-1} \omega^{-1} \vert \omega g^{-1} \omega^{-1}])} = \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]gh)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]h) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega]g)}.$$ A short calculation using equation [\[eq:2cocycleKey\]](#eq:2cocycleKey){reference-type="eqref" reference="eq:2cocycleKey"} shows that this identity indeed holds. That $p^{\omega}$ is an isometry follows from the equalities $$\langle p^{\omega}(l_g) \rangle_{G} = \frac{1}{\vert G\vert} \delta_{e,g} \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})}{\lambda(e)}([\omega] e) = \frac{\delta_{e,g}}{\vert G\vert} = \langle l_g \rangle_{G}.$$ It remains to prove the homomorphism property, $p^{\omega_2} \circ p^{\omega_1} = p^{\omega_2 \omega_1}$. Recall from Section [2.1](#sec:RealProjRep){reference-type="ref" reference="sec:RealProjRep"} that $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ is a $1$-cocycle on the groupoid $G/\!\!/_R \hat{G}$ whose objects are elements of $G$ and whose morphisms are $\omega: g \rightarrow \omega g^{\pi(\omega)} \omega^{-1}$, $\omega \in \hat{G}$. With this description, closedness of $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ becomes the equalities $$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_2]\omega_1 g^{\pi(\omega_1)} \omega_1^{-1}) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_1]g) = \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\omega_2 \omega_1]g), \qquad g \in G, \; \omega_i \in \hat{G}$$ which are immediately seen to imply the homomorphism property. ◻ **Proposition 24**. *For each $\varsigma \in \hat{G}\backslash G$, the map $p^{\varsigma}$ restricts to an algebra involution of $Z(\mathbb{C}^{\theta^{-1}}[G])$. Moreover, this involution is independent of $\varsigma$.* *Proof.* Using the explicit descriptions of the centre $Z(\mathbb{C}^{\theta^{-1}}[G])$ from Section [1.2](#sec:twistRepThy){reference-type="ref" reference="sec:twistRepThy"} and the $G$-action on $\mathbb{C}^{\theta^{-1}}[G]$ from Lemma [Lemma 23](#lem:genGAct){reference-type="ref" reference="lem:genGAct"} we see that $\mathbb{C}^{\theta^{-1}}[G]^{G}=Z(\mathbb{C}^{\theta^{-1}}[G])$. It follows that the generalized $\hat{G}$-action on $\mathbb{C}^{\theta^{-1}}[G]$ from Lemma [Lemma 23](#lem:genGAct){reference-type="ref" reference="lem:genGAct"} induces an action of $C_2\simeq \hat{G}\slash G$ by algebra automorphisms on $Z(\mathbb{C}^{\theta^{-1}}[G])$. ◻ Denote by $p$ the algebra involution of $Z(\mathbb{C}^{\theta^{-1}}[G])$ induced by any $\varsigma \in \hat{G}\backslash G$. **Remark 25**. Using functoriality of Hochschild homology and invariance under taking opposites, we form the composition $$\label{eq:HHInv} HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)) \xrightarrow[]{\sim} HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)^{\textup{\text{op}}}) \xrightarrow[]{HH_{\bullet}(P^{(\hat{\theta},\lambda)})} HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G)).$$ Since $\textup{\text{Rep}}^{\theta}(G)$ is finite semisimple, $HH_{\bullet}(\textup{\text{Rep}}^{\theta}(G))$ is concentrated in degree zero, where is it isomorphic to $Z(\mathbb{C}^{\theta^{-1}}[G])$. Under this isomorphism, the map [\[eq:HHInv\]](#eq:HHInv){reference-type="eqref" reference="eq:HHInv"} is $p$. The $+1$ (resp. $-1$) eigenspace of $p$ is then the involutive (resp. skew-involutive) Hochschild homology of $(\textup{\text{Rep}}^{\theta}(G), P^{(\hat{\theta},\lambda)},\Theta^{(\hat{\theta},\lambda)})$. See [@braun2014 Theorem 2.14] for an analogous result in the setting of strictly involutive $A_{\infty}$-algebras. The first part of Proposition [Proposition 19](#prop:wittenIndices){reference-type="ref" reference="prop:wittenIndices"} therefore shows that the Klein condition computes the difference in dimensions of involutive and skew-involutive Hochschild homologies. **Proposition 26**. *The element $\nu_{(\hat{\theta},\lambda)} \in Z(\mathbb{C}^{\theta^{-1}}[G])$ is $p$-invariant.* *Proof.* We have seen in Lemma [Lemma 9](#lem:FSClassFun){reference-type="ref" reference="lem:FSClassFun"} that $\nu_{(\hat{\theta},\lambda)} \in Z(\mathbb{C}^{\theta^{-1}}[G])$. For $p$-invariance, we have $$p^{\varsigma}(\nu_{(\hat{\theta},\lambda)}) = \sum_{\mu \in \hat{G}\backslash G} \frac{\lambda(\mu) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] \mu^2)}{\lambda(\mu^2) \hat{\theta}([\mu \vert \mu])} l_{\varsigma \mu^{-2} \varsigma^{-1}}.$$ Equation [\[eq:oddConj\]](#eq:oddConj){reference-type="eqref" reference="eq:oddConj"} gives $\hat{\theta}([\varsigma \mu^{-1} \varsigma^{-1} \vert \varsigma \mu^{-1} \varsigma^{-1}]) = \frac{\hat{\theta}([\varsigma \vert \mu^{-2}])}{\hat{\theta}([\mu^{-1} \vert \mu^{-1}]) \hat{\theta}([\varsigma \mu^{-2} \varsigma^{-1} \vert \varsigma])}$ so that $p^{\varsigma}(\nu_{(\hat{\theta},\lambda)})$ is equal to $$\begin{gathered} \sum_{\mu \in \hat{G}\backslash G} \lambda(\mu)^{-1} \frac{\hat{\theta}([\varsigma \mu^{-2} \varsigma^{-1} \vert \varsigma])}{\hat{\theta}([\mu \vert \mu]) \hat{\theta}([\mu^2 \vert \mu^{-2}]) \hat{\theta}([\varsigma \vert \mu^{-2}])} l_{\varsigma \mu^{-2} \varsigma^{-1}} \\ % &=& % \sum_{\mu \in \Gh \backslash \G} % \lambda(\mu)^{-1} \hat{\theta}([\mu \vert \mu])^{-1} \hat{\theta}([\mu^2 \vert \mu^{-2}])^{-1} \hat{\theta}([\mu^{-1} \vert \mu^{-1}])^{-1} \left( \hat{\theta}([\mu^{-1} \vert \mu^{-1}]) \frac{\hat{\theta}([\varsigma \mu^{-2} \varsigma^{-1} \vert \varsigma])}{\hat{\theta}([\varsigma \vert \mu^{-2}])} \right) l_{\varsigma \mu^{-2} \varsigma^{-1}} \\ = \sum_{\mu \in \hat{G}\backslash G} \lambda(\mu)^{-1} \hat{\theta}([\mu \vert \mu])^{-1} \hat{\theta}([\mu^2 \vert \mu^{-2}])^{-1} \hat{\theta}([\mu^{-1} \vert \mu^{-1}])^{-1} \hat{\theta}([\varsigma \mu^{-1} \varsigma^{-1} \vert \varsigma \mu^{-1} \varsigma^{-1}])^{-1} l_{\varsigma \mu^{-2} \varsigma^{-1}}.\end{gathered}$$ A short calculation shows that $\hat{\theta}([\mu^{-1} \vert \mu^{-1}]) \hat{\theta}([\mu \vert \mu]) \hat{\theta}([\mu^2 \vert \mu^{-2}]) =1$, whence $$p^{\varsigma}(\nu_{(\hat{\theta},\lambda)}) = \sum_{\mu \in \hat{G}\backslash G} \lambda(\mu^{-1}) \hat{\theta}([\varsigma \mu^{-1} \varsigma^{-1} \vert \varsigma \mu^{-1} \varsigma^{-1}])^{-1} l_{\varsigma \mu^{-2} \varsigma^{-1}} =\nu_{(\hat{\theta},\lambda)}. \qedhere$$ ◻ **Lemma 27**. *The following equality holds for all $g \in G$ and $\mu \in \hat{G}\backslash G$: $$\hat{\theta}([\mu \vert \mu])^{-1} l_{\mu^2} \cdot a_g l_g = \lambda(g) p^{\mu}(a_g l_g) \cdot \hat{\theta}([\mu g \vert \mu g])^{-1} l_{(\mu g)^2}.$$* *Proof.* This can be verified directly from the twisted $2$-cocycle condition on $\hat{\theta}$. ◻ **Proposition 28**. *The equality $p(\nu_{(\hat{\theta},\lambda)} f) = \nu_{(\hat{\theta},\lambda)} f$ holds for all $f \in Z(\mathbb{C}^{\theta^{-1}}[G])$.* *Proof.* Write $\sum_{g \in G} a_g l_g$ for $f \in Z(\mathbb{C}^{\theta^{-1}}[G])$. Lemma [Lemma 27](#lem:crosscapHelp){reference-type="ref" reference="lem:crosscapHelp"} gives $$\nu_{(\hat{\theta},\lambda)} \sum_{g \in G} a_g l_g % = % \sum_{\substack{g \in \G \\ \mu \in \Gh \backslash \G}} \lambda(\mu) \hat{\theta}([\mu \vert \mu])^{-1} l_{\mu^2} \cdot a_g l_g = \sum_{\substack{g \in G\\ \mu \in \hat{G}\backslash G}} \lambda(\mu g) p^{\mu}(a_g l_g) \hat{\theta}([\mu g \vert \mu g])^{-1} l_{(\mu g)^2}$$ from which we find that $p^{\varsigma}(\nu_{(\hat{\theta},\lambda)} \sum_g a_g l_g)$ is equal to $$\begin{aligned} && \sum_{\substack{g \in G\\ \mu \in \hat{G}\backslash G}} \lambda(\mu g) p^{\varsigma} (p^{\mu}(a_g l_g) \cdot \hat{\theta}([\mu g \vert \mu g])^{-1} l_{(\mu g)^2}) \\ &=& \sum_{g, \mu} \lambda(\mu g) p^{\varsigma}(\hat{\theta}([\mu g \vert \mu g])^{-1} l_{(\mu g)^2}) p^{\varsigma} p^{\mu}(a_g l_g) \\ &=& \sum_{g, \mu} \lambda(\mu g)^{-1} \hat{\theta}([\varsigma g^{-1} \mu^{-1} \varsigma^{-1} \vert \varsigma g^{-1} \mu^{-1} \varsigma^{-1}])^{-1} l_{\varsigma (g^{-1} \mu^{-1})^2 \varsigma^{-1}} p^{\varsigma} p^{\mu}(a_g l_g) \\ &=& \sum_{g, \mu} \lambda(\mu g) \hat{\theta}([\varsigma g^{-1} \mu^{-1} \varsigma^{-1} \vert \varsigma g^{-1} \mu^{-1} \varsigma^{-1}])^{-1} l_{\varsigma (g^{-1} \mu^{-1})^2 \varsigma^{-1}} \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma \mu]g)^{-1} a_g l_{\varsigma \mu g (\varsigma \mu)^{-1}} \\ &=& \sum_{g, \mu} \lambda(\mu g) \hat{\theta}([\varsigma g^{-1} \mu^{-1} \varsigma^{-1} \vert \varsigma g^{-1} \mu^{-1} \varsigma^{-1}])^{-1} l_{\varsigma (g^{-1} \mu^{-1})^2 \varsigma^{-1}} a_{\varsigma \mu g (\varsigma \mu)^{-1}} l_{\varsigma \mu g (\varsigma \mu)^{-1}} \\ &=& \sum_{\substack{h \in G\\ \eta \in \hat{G}\backslash G}} \frac{\lambda(\eta)}{\hat{\theta}([ \eta \vert \eta])} l_{\eta^2} a_h l_h,\end{aligned}$$ which is $\nu_{(\hat{\theta},\lambda)} \sum_{h \in G} a_h l_h$. The first equality follows from the fact that $p^{\varsigma}$ is an anti-homomorphism (Lemma [Lemma 23](#lem:genGAct){reference-type="ref" reference="lem:genGAct"}), the second from Proposition [Proposition 26](#prop:crosscapInv){reference-type="ref" reference="prop:crosscapInv"}, the third from Lemma [Lemma 23](#lem:genGAct){reference-type="ref" reference="lem:genGAct"} and the definition of $p$, the fourth from the assumed centrality of $\sum_g a_g l_g$ and the fifth from the change of variables $\eta = \varsigma g^{-1} \mu^{-1} \varsigma^{-1}$ and $h = \varsigma \mu g \mu^{-1} \varsigma^{-1}$. ◻ Recall that a conjugacy class $\mathcal{O} \subset G$ is called *$\theta$-regular* if $\frac{\theta([g \vert h])}{\theta([h \vert g])} =1$ for all $g \in \mathcal{O}$ and $h \in C_{G}(g)$. **Proposition 29**. *The Klein condition holds.* *Proof.* The vector space $Z(\mathbb{C}^{\theta^{-1}}[G])$ has a basis $\{ l_{\mathcal{O}} \}_{\mathcal{O}}$ labelled by $\theta$-regular conjugacy classes of $G$. For convenience, set $l_{\mathcal{O}}=0$ if $\mathcal{O}$ is not $\theta$-regular. Writing $l_{\mathcal{O}} = \sum_{g \in \mathcal{O}} a_g l_g$ and $l_{\mathcal{O}^{-1}} = \sum_{h \in \mathcal{O}} b_{h^{-1}} l_{h^{-1}}$, we have $\langle l_{\mathcal{O}}, l_{\mathcal{O}^{-1}} \rangle_{G} = \frac{1}{\vert G\vert} \sum_{g \in \mathcal{O}}\theta([g \vert g^{-1}])^{-1} a_g b_{g^{-1}}.$ Centrality of $l_{\mathcal{O}^{\pm 1}}$ implies that the function $\mathcal{O} \rightarrow \mathbb{C}$, $g \mapsto \theta([g \vert g^{-1}])^{-1} a_g b_{g^{-1}}$, is constant; denote its (necessarily non-zero) value by $c_{\mathcal{O}}$. We also have $\langle l_{\mathcal{O}}, l_{\mathcal{O}^{\prime}} \rangle_{G} = 0$ if $\mathcal{O}^{\prime} \neq \mathcal{O}^{-1}$. It follows that $l_{\mathcal{O}}^{\vee} = \frac{\vert G\vert}{c_{\mathcal{O}} \vert \mathcal{O} \vert} l_{\mathcal{O}^{-1}}$. With this notation, the right hand side of the Klein condition is $R := \sum_{\mathcal{O} \in \pi_0(G/\!\!/G)} l_{\mathcal{O}} p^{\varsigma}(l_{\mathcal{O}}^{\vee})$. We compute $$\begin{aligned} R &=& \vert G\vert \sum_{\mathcal{O} \in \pi_0(G/\!\!/G)} \sum_{g, h \in \mathcal{O}} \frac{a_g l_g p^{\varsigma}(b_{h^{-1}} l_{h^{-1}})}{ c_{\mathcal{O}} \vert \mathcal{O} \vert} \\ % &=& % \vert \G \vert \sum_{\mathcal{O} \in \pi_0(\G \git \G)} \sum_{g, h \in \mathcal{O}} \frac{a_g b_{h^{-1}} \lambda(h^{-1})^{-1} \uptau_{\pi}^{\refl}(\hat{\theta})([\varsigma] h^{-1}) l_g l_{\varsigma h \varsigma^{-1}}}{c_{\mathcal{O}} \vert \mathcal{O} \vert} \\ &=& \vert G\vert \sum_{\mathcal{O} \in \pi_0(G/\!\!/G)} \sum_{g, h \in \mathcal{O}} \frac{a_g b_{h^{-1}}\lambda(h) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] h^{-1}) l_{g\varsigma h \varsigma^{-1}}}{c_{\mathcal{O}} \vert \mathcal{O} \vert \hat{\theta}([g \vert \varsigma h \varsigma^{-1}])} \\ %&=& %\vert \G \vert \sum_{\mathcal{O} \in \pi_0(\G \git \G)} \sum_{g \in \mathcal{O}} \sum_{t \in \G} \frac{\vert \mathcal{O} \vert}{\vert \G \vert} \frac{a_g b_{t g^{-1} t^{-1}} \hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}]) \uptau_{\pi}^{\refl}(\hat{\theta})([\varsigma] t g^{-1} t^{-1})^{-1} l_{g\varsigma t g t^{-1} \varsigma^{-1}}}{c_{\mathcal{O}} \vert \mathcal{O} \vert} \\ &=& \sum_{\mathcal{O} \in \pi_0(G/\!\!/G)} \sum_{g \in \mathcal{O}} \sum_{t \in G} \frac{a_g b_{t g^{-1} t^{-1}}\lambda(g) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] t g^{-1} t^{-1}) l_{g\varsigma t g t^{-1} \varsigma^{-1}}}{c_{\mathcal{O}} \hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}])} \\ &=& \sum_{g, t \in G} \frac{a_g b_{t g^{-1} t^{-1}}}{ c_{\mathcal{O}}} \lambda(g) \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] t g^{-1} t^{-1})}{\hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}])} l_{g\varsigma t g t^{-1} \varsigma^{-1}}.\end{aligned}$$ Above we have set $h = t g t^{-1}$. As $b_{t g^{-1} t^{-1}} = \uptau(\theta)([t]g^{-1}) b_{g^{-1}}$, we can write $$\begin{aligned} R &=& \sum_{g, t \in G} \frac{a_g b_{g^{-1}}}{ c_{\mathcal{O}}} \lambda(g) \frac{\uptau(\theta)([t]g^{-1}) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] t g^{-1} t^{-1})}{\hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}])} l_{g\varsigma t g^{-1} t^{-1} \varsigma^{-1}} \\ &=& \sum_{g, t \in G} \lambda(g) \frac{\theta([g \vert g^{-1}])}{\hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}])} \uptau(\theta)([t]g^{-1}) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] t g^{-1} t^{-1}) l_{g\varsigma t g t^{-1} \varsigma^{-1}} \\ &=& \sum_{g, t \in G} \lambda(g) \frac{\theta([g \vert g^{-1}])}{\hat{\theta}([g \vert \varsigma t g t^{-1} \varsigma^{-1}])} \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma t] g^{-1}) l_{g\varsigma t g t^{-1} \varsigma^{-1}}.\end{aligned}$$ The second equality follows from the definition of $c_{\mathcal{O}}$ and the final from closedness of $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$, as in the proof of Lemma [Lemma 23](#lem:genGAct){reference-type="ref" reference="lem:genGAct"}. Define $\mu, \xi \in \hat{G}\backslash G$ by $\mu = g \varsigma t$ and $\xi = t^{-1} \varsigma^{-1}$, so that $t = \varsigma^{-1} \xi^{-1}$ and $g = \mu \xi$. Making these substitutions, the coefficient of $l_{g\varsigma t g t^{-1} \varsigma^{-1}} = l_{\mu^2 \xi^2}$ in $R$ is $$\lambda(\mu \xi) \hat{\theta}([\mu \xi \vert \xi^{-1} \mu \xi^2])^{-1} \frac{\hat{\theta}([\xi^{-1} \mu \xi^2 \vert \xi^{-1}])}{\hat{\theta}([\xi^{-1} \vert \mu \xi])} = \lambda(\mu \xi) \hat{\theta}([\mu \vert \mu])^{-1} \hat{\theta}([\xi \vert \xi])^{-1} \hat{\theta}([\mu^2 \vert \xi^2])^{-1}.$$ It follows that $$R =\sum_{\mu, \xi \in \hat{G}\backslash G} \lambda(\mu \xi) \hat{\theta}([\mu \vert \mu])^{-1} \hat{\theta}([\xi \vert \xi])^{-1} \hat{\theta}([\mu^2 \vert \xi^2])^{-1} l_{\mu^2 \xi^2},$$ which is exactly $\nu_{(\hat{\theta},\lambda)}^2$. ◻ ### Open/closed coherence Note that Theorem [Theorem 11](#thm:LefGroupAlg){reference-type="ref" reference="thm:LefGroupAlg"} verifies equation [\[eq:LefInd\]](#eq:LefInd){reference-type="eqref" reference="eq:LefInd"}. **Proposition 30**. *The maps $\tau^{\bullet}$, $\tau_{\bullet}$, $P$ and $p$ satisfy the assumptions of Proposition [Proposition 20](#prop:orientifoldTFT){reference-type="ref" reference="prop:orientifoldTFT"}.* *Proof.* We compute $$P \circ \tau_V (p(\sum_{g \in G} a_g l_g)) = \sum_{g \in G} a_g \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)}{\lambda(g)} \rho_V(\varsigma g^{-1} \varsigma^{-1})^{\vee} = \tau_{P(V)}(\sum_{g \in G} a_g l_g),$$ that is, $P \circ \tau_V \circ p = \tau_{P(V)}$. Since $p$ is an involution, this implies $P \circ \tau_V = \tau_{P(V)} \circ p$. We also have $$\begin{aligned} p \circ \tau^V(\phi) % &=& % p( \sum_{g \in \G} \theta([g \vert g^{-1}])^{-1} \tr_V (\phi \circ \rho_V(g^{-1})) l_g) \\ &=& \sum_{g \in G} \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] g)}{\lambda(g)} \textup{\text{tr}}_V (\phi \circ \rho_V(g^{-1})) l_{\varsigma g^{-1} \varsigma^{-1}}\end{aligned}$$ and $$\begin{aligned} \tau^{P(V)} ( P(\phi)) % &=& % \bb^{P(V)} (\phi^{\vee}) \\ &=& \sum_{g \in G} \frac{\lambda(g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma]g)} \textup{\text{tr}}_{V^{\vee}} (\phi^{\vee} \circ \rho_V(\varsigma g^{-1} \varsigma^{-1})^{\vee}) l_g \\ &=& \sum_{g \in G} \frac{\lambda(\varsigma g^{-1} \varsigma^{-1})}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})(\varsigma g^{-1} \varsigma^{-1})} \textup{\text{tr}}_{V^{\vee}} (\phi^{\vee} \circ \rho_V(\varsigma^2 g \varsigma^{-2})^{\vee}) l_{\varsigma g^{-1} \varsigma^{-1}}.\end{aligned}$$ Closedness of $\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})$ implies $$\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] g) \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] \varsigma g^{-1} \varsigma^{-1}) = \uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma^2] g).$$ Since $\phi$ is $G$-equivariant and $$\rho_V(\varsigma^2) \circ \rho_V(g) \circ \rho_V(\varsigma^{-2}) = \theta([\varsigma^2 \vert g]) \theta([\varsigma^2 g \vert \varsigma^{-2}]) \rho_V(\varsigma^2 g \varsigma^{-2}),$$ we have $$\textup{\text{tr}}_{V^{\vee}} (\phi^{\vee} \circ \rho_V(\varsigma^2 g \varsigma^{-2})^{\vee}) = \frac{\theta([\varsigma^{-2} \vert \varsigma^2])}{\theta([\varsigma^2 \vert g]) \theta([\varsigma^2 g \vert \varsigma^{-2}])} \textup{\text{tr}}_V (\phi \circ \rho_V(g)).$$ The coefficient of $\textup{\text{tr}}_V (\phi \circ \rho_V(g)) l_{\varsigma g^{-1} \varsigma^{-1}}$ in $\tau^{P(V)} (P(\phi))$ is therefore $$\lambda(g)^{-1} \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] g)}{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma^2] g)} \frac{\theta([\varsigma^{-2} \vert \varsigma^2])}{\theta([\varsigma^2 \vert g]) \theta([\varsigma^2 g \vert \varsigma^{-2}])} = \frac{\uptau_{\pi}^{\textup{\text{refl}}}(\hat{\theta})([\varsigma] g)}{\lambda(g)}.$$ We conclude that $p \circ \tau^V = \tau^{P(V)} \circ P$. ◻ This completes the proof of Theorem [Theorem 21](#thm:twistRealRepTFT){reference-type="ref" reference="thm:twistRealRepTFT"}. ## Partition functions {#sec:partFun} The algebraic construction of $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}$ allows for the explicit computation of the partition function of an arbitrary surface. ### Closed surfaces For the real projective plane, we have $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^2) = \langle \nu_{(\hat{\theta},\lambda)} \rangle_{G} = %&=& %\sum_{\mu \in \Gh \backslash \G} \hat{\theta}([\mu \vert \mu]) \langle l_{\mu^2} \rangle_{\G} \\ %&=& \frac{1}{\vert G\vert} \sum_{\substack{\mu \in \hat{G}\backslash G\\ \mu^2 = e }} \frac{\lambda(\mu)}{\hat{\theta}([\mu \vert \mu])},$$ the first equality reflecting that $\mathbb{RP}^2$ is a Möbius strip glued to a disk. In particular, $\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^2)$ vanishes unless $\pi: \hat{G}\rightarrow C_2$ splits. Realizing the Klein bottle as two cylinders glued together, with one gluing by circle reflection, and using that $Z(\mathbb{C}^{\theta^{-1}}[G]) = \mathbb{C}^{\theta^{-1}}[G]^{G}$ (see the proof of Proposition [Proposition 24](#prop:algInvolution){reference-type="ref" reference="prop:algInvolution"}), we compute $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{K}) % = % \tr_{Z(\C^{\theta^{-1}}[\G])} p^{\varsigma} = \frac{1}{\vert G\vert} \sum_{h \in G} \textup{\text{tr}}_{\mathbb{C}^{\theta^{-1}}[G]} p^{h\varsigma} = \frac{1}{\vert G\vert} \sum_{\substack{g \in G\\ \omega \in \hat{G}\backslash G\\ \omega g^{-1} \omega^{-1} = g}} \frac{1}{\lambda(g) \hat{\theta}([g^{-1} \vert g])} \frac{\hat{\theta}([g \vert \omega])}{\hat{\theta}([\omega \vert g^{-1}])}.$$ In general, a formula for the partition function of a closed connected non-orientable surface $\Sigma$ can be written in terms of $\hat{\theta}$-weighted counts of $C_2$-graded homomorphisms from the fundamental group of the orientation double cover of $\Sigma^{\textup{\text{or}}} \rightarrow \Sigma$ to $\hat{G}$. See [@mbyoung2020 §4.4]. Alternatively, the primitive orthogonal idempotents of the semisimple algebra $Z(\mathbb{C}^{\theta^{-1}}[G])$ can be used to evaluate the partition functions. Proceeding in this way and writing the crosscap state as in Corollary [Corollary 14](#cor:FSIndDecomp){reference-type="ref" reference="cor:FSIndDecomp"}, we find $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma) = \sum_{\substack{ V \in \textup{\text{Irr}}^{\theta}(G) \\ P^{(\hat{\theta},\lambda)}(V) \simeq V}} \left(\frac{\langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_G \dim_{\mathbb{C}} V}{\vert G\vert} \right)^{\chi(\Sigma)}.$$ For example, $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{RP}^2) = \frac{1}{\vert G\vert} \sum_{\substack{ V \in \textup{\text{Irr}}^{\theta}(G) \\ P^{(\hat{\theta},\lambda)}(V) \simeq V}} \langle \chi_V, \nu_{(\hat{\theta},\lambda)} \rangle_G \dim_{\mathbb{C}} V$$ and $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\mathbb{K}) % = % \sum_{\substack{ V \in \Irr^{\theta}(\G) \\ P^{(\hat{\theta},\lambda)}(V) \simeq V}} 1 = \vert \{ V \in \textup{\text{Irr}}^{\theta}(G) \mid P^{(\hat{\theta},\lambda)}(V) \simeq V\} \vert.$$ Equating these expressions for the partition function of $\Sigma$ relates weighted counts of $C_2$-graded homomorphisms $\pi_1(\Sigma^{\textup{\text{or}}}) \rightarrow \hat{G}$ to Real character theoretic sums. Various specializations of these identities are known [@frobenius1906; @karimipour1997; @mulase2005; @snyder2017; @barkeshli2020; @mbyoung2020] and provide non-orientable counterparts of Mednykh's formulae [@mednykh1978]. ### Surfaces with boundary Let $\Sigma$ be a compact connected non-orientable surface with $b \geq 1$ boundary components. To begin, label each boundary component by the same irreducible twisted Real representation $V \in \textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_2}$. The partition function $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;V) : \textup{\text{Hom}}_{\textup{\text{Rep}}^{\theta}(G)^{\tilde{h}C_2}}(V,V)^{\otimes b} \rightarrow \mathbb{C}$$ can be computed as follows. By Proposition [Proposition 7](#prop:RealRestr){reference-type="ref" reference="prop:RealRestr"}, there are two cases to consider. If $V$ is irreducible as a twisted representation, then the primitive orthogonal idempotent of $\mathbb{C}^{\theta^{-1}}[G]$ corresponding to $V$ is $e_{V^{\vee}} = \frac{\dim_{\mathbb{C}} V}{\vert G\vert} \chi_{V}$, whence $\chi_V^b = \left(\frac{\dim_{\mathbb{C}} V}{\vert G\vert}\right)^{-b} e_{V^{\vee}}$. Using this and the fact that $\langle \chi_V, \nu_{(\hat{\theta},\lambda))} \rangle_{G} =1$ (see Corollary [Corollary 12](#cor:FSInd){reference-type="ref" reference="cor:FSInd"}), we compute $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;V)(\textup{\text{id}}_V^{\otimes b}) = \left(\frac{\dim_{\mathbb{C}} V}{\vert G\vert} \right)^{\chi(\Sigma)}.$$ If instead the underlying twisted representation of $V$ is reducible, then $V \simeq H^{(\hat{\theta},\lambda)}(U)$ with $U \in \textup{\text{Rep}}^{\theta}(G)$ irreducible. It follows that $\chi_{H^{(\hat{\theta},\lambda)}(U)}=\frac{\vert G\vert}{\dim_{\mathbb{C}} U} (e_{U^{\vee}} + e_{P^{(\hat{\theta},\lambda)}(U)^{\vee}})$ and $$\chi_{H^{(\hat{\theta},\lambda)}(U)}^{b} =\left( \frac{\dim_{\mathbb{C}} U}{\vert G\vert} \right)^{-b} (e_{U^{\vee}} + e_{P^{(\hat{\theta},\lambda)}(U)^{\vee}}).$$ There are two further sub-cases: - $P^{(\hat{\theta},\lambda)}(U) \not\simeq U$, in which case $\langle \chi_U, \nu_{(\hat{\theta},\lambda)}\rangle_{G} = \langle \chi_{P^{(\hat{\theta},\lambda)}(U)} \nu_{(\hat{\theta},\lambda)}\rangle_{G} = 0$. Since $\nu_{(\hat{\theta},\lambda)}$ has no $\chi_U$ or $\chi_{P^{(\hat{\theta},\lambda)}(U)}$ components, we find $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;H^{(\hat{\theta},\lambda)}(U))(\textup{\text{id}}_{H^{(\hat{\theta},\lambda)}(U)}^{\otimes b})=0.$$ - $P^{(\hat{\theta},\lambda)}(U) \simeq U$, in which case $\langle \chi_U, \nu_{(\hat{\theta},\lambda)}\rangle_{G}= -1$ and $$\mathcal{Z}_{(\hat{G},\hat{\theta},\lambda)}(\Sigma;H^{(\hat{\theta},\lambda)}(U))(\textup{\text{id}}_{H^{(\hat{\theta},\lambda)}(U)}^{\otimes b}) = 2\left(-\frac{\dim_{\mathbb{C}} U}{\vert G\vert} \right)^{\chi(\Sigma)}.$$ A formula in the general case, with boundary components labelled by arbitrary twisted Real representations $V_i$, $i=1, \dots, b$, can be deduced from the previous formulae by writing $\chi_{V_i}$ as a linear combination of primitive idempotents. DGRKS07 M. Abouzaid, *A geometric criterion for generating the Fukaya category*, Publ. Math. Inst. Hautes Études Sci. 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arxiv_math

--- abstract: | We present a comprehensive theory on the well-posedness of a one-dimensional nonlinear backward stochastic differential equation (1D BSDE for short), where the generator $g$ has a one-sided linear/super-linear growth in the first unknown variable $y$ and an at most quadratic growth in the second unknown variable $z$. We first establish several existence theorems and comparison theorems with the test function method and the a priori estimate technique, and then immediately give several existence and uniqueness results. We also overview relevant known results and introduce some practical applications of our theoretical results. Finally, we list some open problems on the well-posedness of 1D BSDEs. author: - Shengjun Fan - Ying Hu - Shanjian Tang title: A user's guide to 1D nonlinear backward stochastic differential equations with applications and open problems --- Backward stochastic differential equation ,A unified theory ,Feynman-Kac formula\ Existence and uniqueness ,Comparison theorem ,Open problems ,$g$-expectation. 60H10 # Introduction {#sec:1-Introduction} Fix a real $T>0$ and an integer $d\geq 1$. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space equipped with augmented filtration $(\mathcal{F}_t)_{t\in[0,T]}$ generated by a standard $d$-dimensional Brownian motion $(B_t)_{t\in[0,T]}$, and assume that $\mathcal{F}_T=\mathcal{F}$. The equality and inequality between random elements are usually understood in the sense of $\mathbb{P}-a.s.$ We consider the following one-dimensional backward stochastic differential equation (1D BSDE in short): $$\label{eq:1} Y_t=\xi+\int_t^T g(s,Y_s,Z_s){\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s, \ \ t\in[0,T],$$ where $\xi$ is called the terminal condition being an $\mathcal{F}_T$-measurable real random variable, the random field $$g(\omega, t, y, z):\Omega\times[0,T]\times{\mathbb R}\times{\mathbb R}^d \to {\mathbb R}$$ is called the generator of [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"}, which is $(\mathcal{F}_t)$-adapted for each $(y,z)$, and the pair of $(\mathcal{F}_t)$-adapted and ${\mathbb R}\times{\mathbb R}^d$-valued processes $(Y_t,Z_t)_{t\in[0,T]}$ is called a solution of [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} if $\mathbb{P}-a.s.$, $t\mapsto Y_t$ is continuous, $t\mapsto |g(t,Y_t,Z_t)|+|Z_t|^2$ is integrable, and [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} is satisfied. Denote by BSDE$(\xi,g)$ the BSDE with the terminal condition $\xi$ and the generator $g$, which are the parameters of BSDEs. For convenience of exposition, throughout the paper, let us always fix the constants $\alpha\in [1,2]$, $\beta,\bar\beta\geq 0$, $\gamma>0$, $\delta\in [0,1]$, and $\lambda\in {\mathbb R}$, and an $(\mathcal{F}_t)$-progressively measurable ${\mathbb R}_+$-valued stochastic process $(f_t)_{t\in [0,T]}$. We assume that the generator $g$ satisfies ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $$\label{eq:1.2} \forall\ (y,z)\in {\mathbb R}\times{\mathbb R}^d,\ \ \ {\rm sgn}(y)g(\omega,t,y,z)\leq f_t(\omega)+\beta |y|(\ln(e+|y|))^\delta+\gamma |z|^\alpha(\ln(e+|z|))^\lambda.$$ We usually say that $g$ has a one-sided linear growth in the state variable $y$ when $\delta=0$, and a one-sided super-linear growth in $y$ when $\delta\in (0,1]$. Furthermore, for the case of $\lambda=0$, we say that $g$ has a power sub-linear growth in the state variable $z$ when $\alpha\in (0,1)$, a linear growth in $z$ when $\alpha=1$, a sub-quadratic growth in $z$ when $\alpha\in (1,2)$, a quadratic growth in $z$ when $\alpha=2$, a super-quadratic growth in $z$ when $\alpha>2$, and for the case of $\alpha=1$ and $\lambda\neq 0$, we say that $g$ has a logarithmic sub-linear growth in $z$ when $\lambda<0$, and a logarithmic super-linear growth in $z$ when $\lambda>0$. ## Overview of relevant existing results BSDEs were initiated by Bismut [@Bismut1973JMAA; @Bismut1976SICON; @Bismut1978SIAMReview], in particular for the linear case. General nonlinear BSDEs were founded by Pardoux and Peng [@PardouxPeng1990SCL], where an existence and uniqueness result was established on adapted solutions of multidimensional BSDEs with square-integrable parameters and uniformly Lipschitz continuous generators. Subsequently, BSDEs have received an extensive attention due to its various connections to numerous topics such as partial differential equations (PDEs in short), mathematical finance, stochastic control, nonlinear mathematical expectation and so on. The reader is referred to for example [@Peng1990SICON; @Peng1991Stochastics; @Peng1992Stochastics; @Peng1993AMO; @ElKarouiPengQuenez1997MF; @DelbaenTang2010PTRF; @PardouxTang1999PTRF; @Kobylanski2000AP; @HuImkeller2005AAP; @Jiang2005SPA; @Jiang2008AAP; @PardouxRascanu2014Book] for more details. Particularly, much attentions have been paid on the well-posedness of adapted solutions of BSDEs under various growth and/or continuity of the generator $g$ with respect to the two unknown variables $(y,z)$ and various integrability of the parameters $(\xi,f_\cdot)$. Generally speaking, these efforts can be classified into three different directions. The first one focuses on $L^p (p\geq 1)$ solution of BSDEs. Relevant classical results are available in [@LepeltierSanMartin1997SPL; @Pardoux1999Nonlinear; @BriandCarmona2000IJSA; @BriandDelyonHu2003SPA; @Jia2008CRA; @Jia2008PHDThesis; @FanJiangDavison2010CRA; @Jia2010SPA; @FanLiu2010SPL; @FanJiang2012JAMC; @FanJiang2013AMSE; @Fan2015JMAA; @Fan2016SPA; @Fan2016SPL; @Fan2018JOTP; @XiaoFan2020KM] when the generators $g$ have a linear/sub-linear growth in the unknown variable $z$. The reader is also referred to [@Bahlali2001CRAS; @Bahlali2002ECP; @BahlaliEssakyHassaniPardoux2002CRAS; @LepeltierSanMartin2002Bernoulli; @BahlaliElAsri2012BSM; @BahlaliEssakyHassani2015SIAM; @BahlaliKebiri2017Stochastics; @BahlaliEddahbiOuknine2017AoP; @Yanghanlin2017Arxiv] when the generators $g$ have a super-linear growth in the unknown variable $z$. The second one is devoted to the bounded solution of BSDEs when the generators $g$ have a quadratic/super-quadratic growth in the unknown variable $z$, see for example [@LepeltierSanMartin1998Stochsatics; @Kobylanski2000AP; @Fan2016SPA; @BriandLepeltierSanMartin2007Bernoulli; @Tevzadze2008SPA; @DelbaenHuBao2011PTRF; @BriandElie2013SPA; @BarrieuElKaroui2013AoP; @Fan2016SPA; @HuTang2016SPA; @FanLuo2017BKMS] for more details. The last one concerns the weakest possible integrability of $(\xi,f_\cdot)$ for existence and uniqueness of adapted solution of BSDEs when the generators $g$ have some growth and/or continuity in $(y,z)$. Such a study can be dated back to [@BriandHu2006PTRF; @BriandHu2008PTRF; @DelbaenHuRichou2011AIHPPS; @Richou2012SPA; @DelbaenHuRichou2015DCDS] for the quadratic BSDEs, and subsequently continued in [@HuTang2018ECP; @BuckdahnHuTang2018ECP; @FanHu2019ECP; @OKimPak2021CRM] for the linearly growing BSDEs, and recently sprang up in [@FanHu2021SPA; @FanHuTang2023SPA; @FanHuTang2023SCL; @FanHuTang2023SCL2] when the generator $g$ has a sub-quadratic, super-linear or logarithmic sub-linear growth in the unknown variable $z$. The so-called localization procedure, $\theta$-difference technique and the test function method were combined to obtain the following existence and uniqueness of a BSDE when the generator $g$ satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"}. Firstly, suppose that the generator $g$ has a one-sided linear growth in $y$ and a linear growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=0$, $\alpha=1$ and $\lambda=0$. It is well known that if the data $|\xi|+\int_0^T f_s {\rm d}s\in L^p$ for some $p>1$, then BSDE$(\xi,g)$ admits a solution in $\mathcal{S}^p\times\mathcal{M}^p$, and the solution is unique when $g$ further satisfies the uniformly Lipschitz continuity in $(y,z)$. The reader is referred to [@PardouxPeng1990SCL; @ElKarouiPengQuenez1997MF; @LepeltierSanMartin1997SPL; @BriandDelyonHu2003SPA; @FanJiang2012JAMC] for more details. Recently, [@HuTang2018ECP; @BuckdahnHuTang2018ECP; @FanHu2019ECP; @OKimPak2021CRM] obtained existence of an unbounded solution to a linearly growing BSDE$(\xi,g)$ under the more general condition $|\xi|+\int_0^T f_s {\rm d}s\in L\exp(\mu\sqrt{2\ln L})$ for some $\mu\geq \gamma \sqrt{T}$ (which is weaker than $L^p\ (p>1)$-integrability and stronger than $L\ln L$-integrability). They also established uniqueness of the unbounded solution provided that $g$ satisfies a monotonicity in $y$ and the uniformly Lipschitz continuity in $z$. Generally speaking, the generator $g$ allows a general growth in $y$ when $g$ satisfies the monotonicity in $y$. Relevant works are available in [@Pardoux1999Nonlinear; @BriandCarmona2000IJSA; @BriandDelyonHu2003SPA; @LepeltierMatoussiXu2005AdAP; @BriandLepeltierSanMartin2007Bernoulli; @FanJiang2013AMSE; @Fan2015JMAA; @LionnetReisSzpruch2015AAP; @Fan2016SPA; @LionnetReisSzpruch2018AAP]. Secondly, suppose that the generator $g$ has a one-sided linear growth in $y$ and a power sub-linear growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=0$, $\alpha\in (0,1)$ and $\lambda=0$. @BriandDelyonHu2003SPA show that if the data $|\xi|+\int_0^T f_s {\rm d}s\in L^1$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $(Y_t)_{t\in[0,T]}$ is of class (D), and the solution is unique when $g$ further satisfies an extended monotonicity in $y$ and the uniformly Lipschitz continuity in $z$. See for example [@BriandHu2006PTRF; @Fan2016SPA; @Fan2018JOTP] for more details. Very recently, the three authors [@FanHuTang2023SCL] prove the existence and uniqueness result when the generator $g$ has a one-sided linear growth in $y$ and a logarithmic sub-linear growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=0$, $\alpha=1$ and $\lambda\in (-\infty,-1/2)$, see also [@FanHuTang2023SCL2] for deeper discussions. Thirdly, suppose that the generator $g$ has a one-sided linear/super-linear growth in $y$ and a logarithmic super-linear growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta\in [0,1]$, $\alpha=1$ and $\lambda\in [0,+\infty)$. Let $p:=\delta\vee (\lambda+{1\over 2})\vee (2\lambda)\in [{1\over 2},+\infty)$. Very recently, it was shown in [@FanHuTang2023SPA] that if the data $|\xi|+\int_0^T f_s {\rm d}s\in L\exp(\mu (\ln L)^p)$ for some $\mu>\mu_0$ with a certain value $\mu_0$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $(|Y_t|\exp(\mu(t)(\ln(e+|Y_t|))^p))_{t\in[0,T]}$ is of class (D) for some nonnegative and increasing function $\mu(t)$ defined on $[0,T]$ with $\mu(T)=\mu$, and the solution is unique when the generator $g$ further satisfies an extended monotonicity in $y$ and a uniform continuity in $z$, or a convexity/concavity in $(y,z)$, see assumptions (UN1)-(UN3) in [@FanHuTang2023SPA] for more details. Furthermore, [@BahlaliElAsri2012BSM; @BahlaliKebiri2017Stochastics] verified existence of a solution to BSDE$(\xi,g)$ in the space of $\mathcal{S}^p\times\mathcal{M}^2$ for some sufficiently large $p>2$, when the data $|\xi|+\int_0^T f_s {\rm d}s\in L^p$ and the generator $g$ satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=1$, $\lambda=1/2$ and $|g(\omega,t,y,z)|$ instead of the left side of [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"}. They also proved uniqueness of the solution when $g$ further satisfies a local monotonicity in $(y,z)$. Related works on super-linearly growing BSDEs are available in [@Bahlali2001CRAS; @Bahlali2002ECP; @BahlaliEssakyHassaniPardoux2002CRAS; @LepeltierSanMartin1998Stochsatics; @BahlaliEssakyHassani2010CRM; @BahlaliHakassouOuknine2015Stochastics; @BahlaliEssakyHassani2015SIAM; @LionnetReisSzpruch2016ArXiv], where the solution of BSDE$(\xi,g)$ in the space of $\mathcal{S}^p\times\mathcal{M}^p$ is considered under the data $|\xi|+\int_0^T f_s {\rm d}s\in L^p$ for some $p>1$, and several kinds of locally Lipschitz continuity or local monotonicity of $g$ in $(y,z)$ are usually used in order to guarantee uniqueness of the solution of BSDE$(\xi,g)$. Fourthly, suppose that the generator $g$ has a one-sided linear growth in $y$ and a sub-quadratic growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=0$, $\alpha\in (1,2)$ and $\lambda=0$. Let $\alpha^*$ represent the conjugate of $\alpha$. It was proved in [@FanHu2021SPA] that if the data $|\xi|+\int_0^T f_s {\rm d}s\in \exp(\mu L^{2 \over \alpha^*})$ for some $\mu>\mu_0$ with a certain value $\mu_0$, which is weaker than $\exp(\mu L)$-integrability and stronger than $L^p\ (p>1)$-integrability, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $(\exp(\mu(t)|Y_t|^{2\over \alpha^*}))_{t\in[0,T]}$ is of class (D) for some nonnegative and increasing function $\mu(t)$ defined on $[0,T]$ with $\mu(T)=\mu$, and the solution is unique when $|\xi|+\int_0^T f_s {\rm d}s\in \exp(\mu L^{2\over \alpha^*})$ for each $\mu>0$ and the generator $g$ further satisfies an extended convexity/concavity in $(y,z)$, see assumption (H2') in [@FanHu2021SPA] for more details. Finally, suppose that the generator $g$ has a one-sided linear growth in $y$ and a quadratic growth in $z$, i.e., it satisfies [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} with $\delta=0$, $\alpha=2$ and $\lambda=0$. It is well known from [@Kobylanski2000AP] that if the data $|\xi|+\int_0^T |f_s| {\rm d}s\in L^\infty$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $(Y_t)_{t\in[0,T]}\in \mathcal{S}^\infty$, and the solution is unique if $g$ further satisfies the uniformly Lipschitz continuity in $y$ and a locally Lipschitz continuity in $z$. The reader is referred to [@BriandElie2013SPA; @Fan2016SPA; @FanLuo2017BKMS; @LuoFan2018SD] for more details on the bounded solution of quadratic BSDEs. Subsequently, [@BriandHu2006PTRF; @BriandHu2008PTRF; @DelbaenHuRichou2011AIHPPS; @DelbaenHuRichou2015DCDS] proved existence and uniqueness of an unbounded solution to quadratic BSDE$(\xi,g)$ under the data $|\xi|+\int_0^T f_s {\rm d}s\in \exp(\mu L)$ for $\mu:=\gamma e^{\beta T}$, where $g$ is required to be uniformly Lipschitz continuous in $y$ and convex/concave in $z$ for the uniqueness of the solution, see also [@BarrieuElKaroui2013AoP; @FanHuTang2020CRM] for more details. A class of quadratic BSDEs subject to $L^p\ (p>1)$-integrable terminal values are studied in several recent works, see for example [@BahlaliEddahbiOuknine2017AoP; @Yanghanlin2017Arxiv; @Bahlali2019Arxiv; @BahlaliTangpi2019Arxiv]. In addition, [@DelbaenHuBao2011PTRF] show that super-quadratic BSDEs [\[eq:1.2\]](#eq:1.2){reference-type="eqref" reference="eq:1.2"} (here $\delta=0$, $\alpha>2$ and $\lambda=0$), are not solvable in general and the solution is not unique even if the solution exists. Some relevant solvability results under the Markovian setting are available in [@DelbaenHuBao2011PTRF; @MasieroRichou2013EJP; @Richou2012SPA; @CheriditoNam2014JFA]. The preceding brief review is flavored with the authors' own tastes, and is also restricted within the scope of their knowledge. Certainly, it does not exhaust all the developments of BSDEs in the last half a century, which seems to be an impossible task to the authors within such a very limited space, and is also not the objective of the paper. We would apologize to all those authors of possibly neglected papers on BSDEs. ## Organization of the paper The paper will present a comprehensive theory on the well-posedness of 1D nonlinear BSDEs to cover most existing results mentioned above. The rest of this paper is organized as follows. In Section 2, we formulate the test function method and prove with the combined techniques of a priori estimate and localization a general existence result (see [Definition 1](#dfn:2.1){reference-type="ref" reference="dfn:2.1"} and [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"}), which yield via right test functions several existence theorems on the adapted solutions of 1D BSDEs (see [\[thm:2.3,thm:2.6\]](#thm:2.3,thm:2.6){reference-type="ref" reference="thm:2.3,thm:2.6"}) for both cases of logarithmic quasi-linear growth and the sub-quadratic/quadratic growth, respectively. In Section 3, we focus on the comparison theorems of the adapted solutions of 1D BSDEs for both cases of at most linear growth (see [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"}) and super-linear at most quadratic growth (see [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"}), respectively, and establish some existence and uniqueness results (see [\[thm:3.12,thm:3.13\]](#thm:3.12,thm:3.13){reference-type="ref" reference="thm:3.12,thm:3.13"}). We first give a crucial a priori estimate (see [Proposition 10](#pro:3.1){reference-type="ref" reference="pro:3.1"}) associated with the test function, and prove [\[thm:3.3,thm:3.6\]](#thm:3.3,thm:3.6){reference-type="ref" reference="thm:3.3,thm:3.6"} with the proper test function and the $\theta$-difference technique, respectively. This yields naturally the desired uniqueness results. Some examples and remarks are provided in the last two sections to illustrate the preceding results. See [\[rmk:2.4,rmk:2.7\]](#rmk:2.4,rmk:2.7){reference-type="ref" reference="rmk:2.4,rmk:2.7"} and [\[ex:2.1,ex:2.2\]](#ex:2.1,ex:2.2){reference-type="ref" reference="ex:2.1,ex:2.2"} in section 2 as well as [\[rmk:3.2,rmk:3.5,rmk:3.6,rmk:3.8\]](#rmk:3.2,rmk:3.5,rmk:3.6,rmk:3.8){reference-type="ref" reference="rmk:3.2,rmk:3.5,rmk:3.6,rmk:3.8"}, [\[ex:3.1,ex:3.2,ex:3.3\]](#ex:3.1,ex:3.2,ex:3.3){reference-type="ref" reference="ex:3.1,ex:3.2,ex:3.3"} and [Proposition 16](#pro:3.4){reference-type="ref" reference="pro:3.4"} in section 3. In section 4, several practical applications of our results are introduced including the conditional $g$-expectation (see [Definition 23](#def:4.1){reference-type="ref" reference="def:4.1"} and [\[pro:4.2,pro:4.3\]](#pro:4.2,pro:4.3){reference-type="ref" reference="pro:4.2,pro:4.3"}), the dynamic utility process (see [Proposition 27](#pro:4.4){reference-type="ref" reference="pro:4.4"} and [Theorem 28](#thm:4.5){reference-type="ref" reference="thm:4.5"}), risk measure ([Example 30](#ex:4.8){reference-type="ref" reference="ex:4.8"}) and nonlinear Feynman-Kac formula (see [Definition 31](#def:4.7){reference-type="ref" reference="def:4.7"} and [Theorem 32](#thm:4.8){reference-type="ref" reference="thm:4.8"}), and some commentaries on known related works are also made, see [\[rmk:4.3\*,rmk:4.6,rmk:4.11\]](#rmk:4.3*,rmk:4.6,rmk:4.11){reference-type="ref" reference="rmk:4.3*,rmk:4.6,rmk:4.11"}. Finally, in section 5 we list several open problems on 1D BSDEs to be further studied, and in Appendix we prove a key inequality (see [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"}) used in section 2, which is interesting in its own right. ## Notations and spaces In this subsection, we give some necessary notations and spaces used in this paper. Let ${\mathbb R}_+:=[0,+\infty)$. For $a,b\in {\mathbb R}$, we denote $a\wedge b:=\min\{a,b\}$, $a^+:=\max\{a,0\}$ and $a^-:=-\min\{a,0\}$, and ${\rm sgn}(x):={\bf 1}_{x>0}-{\bf 1}_{x\leq 0}$, where ${\bf 1}_A$ is the indicator function of set $A$. Denote by ${\bf S}$ the set of ${\mathbb R}_+$-valued continuously differentiable functions $\phi(s,x)$ defined on $[0,T]\times{\mathbb R}_+$ such that $\phi_s(\cdot,\cdot)\geq 0$, $\phi_x(s,\cdot)>0$ and $\phi_{xx}(s,\cdot)>0$, where $\phi_s(\cdot,\cdot)$ is the first-order partial derivative of $\phi(\cdot,\cdot)$ with respect to the first variable, and by $\phi_x(\cdot,\cdot)$ and $\phi_{xx}(\cdot,\cdot)$ respectively the first- and second-order partial derivative of $\phi(\cdot,\cdot)$ with respect to the second variable. Denote by ${\bf \bar S}$ the set of ${\mathbb R}_+$-valued functions $h(t,x,\bar x)$ defined on $[0,T]\times{\mathbb R}_+\times {\mathbb R}_+$ such that $h(t,\cdot,\bar x)$ is nondecreasing for each $(t,\bar x)\in [0,T]\times {\mathbb R}_+$. Denote by $\mathcal{S}^\infty([0,T];{\mathbb R})$ (or $\mathcal{S}^\infty$) the set of $(\mathcal{F}_t)$-adapted and continuous bounded real processes $(Y_t)_{t\in[0,T]}$. For each $p>0$, let $\mathcal{S}^p([0,T];{\mathbb R})$ (or $\mathcal{S}^p$) be the set of $(\mathcal{F}_t)$-adapted and continuous real processes $(Y_t)_{t\in[0,T]}$ satisfying $$\|Y\|_{{\mathcal{S}}^p}:=\left(\mathbb{E}[\sup_{t\in[0,T]} |Y_t|^p]\right)^{{1\over p}\wedge 1}<+\infty,\vspace{0.2cm}$$ and $\mathcal{M}^p([0,T];{\mathbb R}^d)$ (or $\mathcal{M}^p$) the set of all $(\mathcal{F}_t)$-adapted ${\mathbb R}^d$-valued processes $(Z_t)_{t\in[0,T]}$ satisfying $$\|Z\|_{\mathcal{M}^p}:=\left\{\mathbb{E}\left[\left(\int_0^T |Z_t|^2{\rm d}t\right)^{p/2}\right] \right\}^{{1\over p}\wedge 1}<+\infty.\vspace{0.1cm}$$ Denote by $\Sigma_T$ the set of all $(\mathcal{F}_t)$-stopping times $\tau$ valued in $[0,T]$. For an $(\mathcal{F}_t)$-adapted real process $(X_t)_{t\in[0,T]}$, if the family $\{X_\tau: \tau\in \Sigma_T\}$ is uniformly integrable, then we call that it is of class (D). Now, fix $t\in [0,T]$. For $p,\mu>0$, we denote by $L^p(\mathcal{F}_t)$ and $L^\infty(\mathcal{F}_t)$ the set of $\mathcal{F}_t$-measurable real random variables $\xi$ such that $\mathbb{E}[|\xi|^p]<+\infty$ and $|\xi|\leq M$ for some real $M>0$, respectively, and define the following three spaces of $\mathcal{F}_t$-measurable real random variables: $$L(\ln L)^p (\mathcal{F}_t):=\left\{\xi\in \mathcal{F}_t\left| \mathbb{E}\left[|\xi|(\ln(e+|\xi|))^p\right]<+\infty\right.\right\},$$ $$L\exp[\mu(\ln L)^p](\mathcal{F}_t):=\left\{\xi\in \mathcal{F}_t\left| \mathbb{E}\left[|\xi|\exp{\left(\mu (\ln(e+|\xi|))^p\right)}\right]<+\infty\right.\right\}$$ and $$\exp(\mu L^p)(\mathcal{F}_t):=\left\{\xi\in \mathcal{F}_t\left| \mathbb{E}\left[\exp{\left(\mu |\xi|^p\right)}\right]<+\infty\right.\right\}.$$ It is clear that for each $0<p<q$ and $0<\bar \mu,\tilde \mu<\mu$, we have $$L^\infty(\mathcal{F}_t)\subset L^q(\mathcal{F}_t)\subset L^p(\mathcal{F}_t), \ \ \ L(\ln L)^q(\mathcal{F}_t)\subset L(\ln L)^p(\mathcal{F}_t),$$ $$L\exp[\mu(\ln L)^q](\mathcal{F}_t)\subset L\exp[\bar\mu(\ln L)^q](\mathcal{F}_t)\subset L\exp[\tilde\mu(\ln L)^p](\mathcal{F}_t),$$ and $$\exp(\mu L^q)(\mathcal{F}_t)\subset \exp(\bar\mu L^q)(\mathcal{F}_t)\subset \exp(\tilde\mu L^p)(\mathcal{F}_t).$$ It can also be verified that for each $\mu,\bar\mu,r>0$ and $0<p<1<q$, we have $$L^\infty(\mathcal{F}_t)\subset \exp(\mu L^r)(\mathcal{F}_t)\subset L\exp[\bar\mu(\ln L)^q](\mathcal{F}_t) \subset L\exp[\mu \ln L](\mathcal{F}_t)=L^{1+\mu}(\mathcal{F}_t)$$ and $$L^q(\mathcal{F}_t)\subset L\exp[\mu(\ln L)^p](\mathcal{F}_t)\subset L(\ln L)^r(\mathcal{F}_t)\subset L^1(\mathcal{F}_t).$$ Furthermore, for each $p,\mu>0$ and $0<\bar p\leq 1<\tilde p$, the spaces $$L(\ln L)^p(\mathcal{F}_t),\ \ \ L\exp[\mu(\ln L)^{\bar p}](\mathcal{F}_t),\ \ \ \bigcup\limits_{\bar\mu>\mu} L\exp[\bar\mu(\ln L)^{\tilde p}](\mathcal{F}_t)\ \ \ {\rm and}\ \ \ \bigcap\limits_{\bar\mu>0}\exp(\bar\mu L^p)(\mathcal{F}_t)$$ are all the Orlicz hearts corresponding, respectively, to the following Young functions $$x(\ln(e+x))^p,\ \ \ x\exp[\mu(\ln(e+x))^{\bar p}],\ \ \ x\exp[\mu(\ln(e+x))^{\tilde p}]\ \ \ {\rm and}\ \ \ \exp(x^p)-1.$$ More details on the Orlicz space, the Orlicz class and the Orlicz heart are refereed to [@EdgarSucheston1992book; @CheriditoLi2009MF]. Finally, in all notations of the spaces on random variables, the $\sigma$-algebra $(\mathcal{F}_T)$ is usually omitted when there is no confusion. # Existence results {#sec:2-Existence} ## The test function method and a general existence result Let us first introduce the following assumptions on the generator $g$. 1. ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $g(\omega,t,\cdot,\cdot)$ is continuous. 2. There exist two ${\mathbb R}_+$-valued functions $H(\omega,t,x)$ and $\Gamma(\omega,t,x)$ defined on $\Omega\times[0,T]\times{\mathbb R}_+$, which are $(\mathcal{F}_t)$-progressively measurable for each $x\in{\mathbb R}_+$ and nondecreasing with respect to the variable $x$, such that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z)\in {\mathbb R}\times{\mathbb R}^d$, $$|g(\omega,t,y,z)|\leq H(\omega,t,|y|)+\Gamma (\omega,t,|y|) |z|^2,$$ and $\mathbb{P}-a.s.$, for each $x\in {\mathbb R}_+$, $\Gamma(\omega,\cdot,x)$ is left-continuous on $[0,T]$ and $$\int_0^T H(\omega,t,x){\rm d}t+\sup\limits_{t\in[0,T]}\Gamma(\omega,t,x)<+\infty.$$ 3. There exists a function $h(\cdot,\cdot,\cdot)\in {\bf\bar S}$ such that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z)\in {\mathbb R}\times{\mathbb R}^d$, $${\rm sgn}(y)g(\omega,t,y,z)\leq f_t(\omega)+h(t,|y|,|z|).$$ **Definition 1**. *Assume that the generator $g$ satisfies assumption (EX3). A function $\varphi(\cdot,\cdot)\in {\bf S}$ is called a test function for $g$ or $h$ if it satisfies that for each $(s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+$, $$\label{eq:2.1} -\varphi_x(s,x)h(s,x,\bar x)+{1\over 2}\varphi_{xx}(s,x)|\bar x|^2+\varphi_s(s,x)\geq 0.$$* **Theorem 2**. *Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies the above assumptions (EX1)-(EX3). If there exists a test function $\varphi(\cdot,\cdot)\in {\bf S}$ for $g$ such that $$\label{eq:2.2} \mathbb{E}\left[\varphi\left(T,\ |\xi|+\int_0^T f_s{\rm d}s\right)\right]<+\infty,$$ then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that the process $(\varphi(t, |Y_t|+\int_0^t f_s{\rm d}s))_{t\in [0,T]}$ is of class (D). Furthermore, the process $(\varphi(t, |Y_t|+\int_0^t f_s{\rm d}s))_{t\in [0,T]}$ is a sub-martingale on $[0,T]$. In particular, we have $$\label{eq:2.3} \varphi\left(t, |Y_t|+\int_0^t f_s{\rm d}s\right)\leq \mathbb{E}\left[\left.\varphi\left(T, |\xi|+\int_0^T f_s{\rm d}s\right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].\vspace{0.2cm}$$* *Proof.* The whole proof is divided into the following two steps. **Step 1.** We first prove the inequality [\[eq:2.3\]](#eq:2.3){reference-type="eqref" reference="eq:2.3"} for a solution $(Y_t,Z_t)_{t\in[0,T]}\in \mathcal{S}^\infty([0,T];{\mathbb R})\times \mathcal{M}^2([0,T];{\mathbb R}^d)$ of BSDE$(\xi,g)$ if further $|\xi|+\int_0^T f_s{\rm d}s\in L^\infty$. Define $$\bar Y_t:=|Y_t|+\int_0^t f_s {\rm d}s\ \ \ \ {\rm and}\ \ \ \ \bar Z_t:={\rm sgn}(Y_t)Z_t,\ \ \ \ t\in [0,T]. \vspace{0.1cm}$$ We have from Itô-Tanaka's formula that $$\bar Y_t=\bar Y_T+\int_t^T \left({\rm sgn}(Y_s)g(s,Y_s,Z_s)-f_s\right){\rm d}s-\int_t^T \bar Z_s \cdot {\rm d}B_s-\int_t^T {\rm d}L_s, \ \ \ t\in[0,T],$$ where $L_\cdot$ is the local time of $Y$ at $0$. Applying Itô's formula to the process $\varphi(s, \bar Y_s)$ and using assumption (EX3), we have that for each $s\in[0,T]$, $$\begin{array}{lll} \displaystyle{\rm d}\varphi(s,\bar Y_s) &=&\displaystyle\varphi_x(s,\bar Y_s) \left(-{\rm sgn}(Y_s)g(s,Y_s,Z_s)+f_s\right){\rm d}s+\varphi_x(s,\bar Y_s)\bar Z_s\cdot {\rm d}B_s\vspace{0.1cm}\\ &&\displaystyle+\varphi_x(s,\bar Y_s){\rm d}L_s+{1\over 2}\varphi_{xx}(s,\bar Y_s)|\bar Z_s|^2{\rm d}s+\varphi_s(s,\bar Y_s){\rm d}s\vspace{0.2cm}\\ &\geq &\displaystyle\left[-\varphi_x(s,\bar Y_s)h(s,|Y_s|,|Z_s|)+{1\over 2}\varphi_{xx}(s,\bar Y_s)|Z_s|^2+\varphi_s(s,\bar Y_s)\right]{\rm d}s+\varphi_x(s,\bar Y_s)\bar Z_s\cdot {\rm d}B_s. \end{array}$$ Since $|Y_s|\leq \bar Y_s$ and $h(t,\cdot,\bar x)$ is nondecreasing, we see from [\[eq:2.1\]](#eq:2.1){reference-type="eqref" reference="eq:2.1"} that $${\rm d}\varphi(s,\bar Y_s)\geq \varphi_x(s,\bar Y_s)\bar Z_s\cdot {\rm d}B_s,\ \ s\in [0,T],$$ which yields that $$\varphi(T,\bar Y_T)-\varphi(t,\bar Y_t)\geq \int_t^T \varphi_x(s,\bar Y_s)\bar Z_s\cdot {\rm d}B_s,\ \ t\in [0,T].$$ Since $|\xi|+\int_0^T f_s{\rm d}s\in L^\infty$ and $(Y_t,Z_t)_{t\in[0,T]}\in \mathcal{S}^\infty([0,T];{\mathbb R})\times \mathcal{M}^2([0,T];{\mathbb R}^d)$, taking the expectation conditioned on $\mathcal{F}_t$ on both sides of the last inequality, we have [\[eq:2.3\]](#eq:2.3){reference-type="eqref" reference="eq:2.3"}. **Step 2.** Based on Step 1, we use the localization procedure of @BriandHu2006PTRF to construct the desired solution. For each $n,p\geq 1$ and $(\omega,t,y,z)\in \Omega\times[0,T]\times{\mathbb R}\times{\mathbb R}^d$, denote $$\label{eq:2.4} \xi^{n,p}:=\xi^+\wedge n-\xi^-\wedge p\ \ \ \ {\rm and}\ \ \ \ g^{n,p}(\omega,t,y,z):=g^+(\omega,t,y,z)\wedge n-g^-(\omega,t,y,z)\wedge p.$$ It is clear that $|\xi^{n,p}|\leq |\xi|\wedge (n\vee p)$ and $|g^{n,p}|\leq |g|\wedge (n\vee p)$ for each $(y,z)\in {\mathbb R}\times {\mathbb R}^d$. It can also be verified that the generator $g^{n,p}$ satisfies assumptions (EX1)-(EX3) with $f_\cdot\wedge (n\vee p)$ instead of $f_\cdot$ Then, according to [@Kobylanski2000AP], the following BSDE$(\xi^{n,p},g^{n,p})$ has a maximal bounded solution $(Y^{n,p}_t,Z^{n,p}_t)_{t\in[0,T]}$ in the space of processes $\mathcal{S}^\infty([0,T];{\mathbb R})\times\mathcal{M}^2([0,T];{\mathbb R}^d)$: $$\label{eq:2.5} Y^{n,p}_t=\xi^{n,p}+\int_t^T g^{n,p}(s,Y^{n,p}_s,Z^{n,p}_s){\rm d}s-\int_t^T Z^{n,p}_s \cdot {\rm d}B_s, \ \ t\in[0,T].$$ The comparison theorem shows that $(Y^{n,p}_t)_{t\in [0,T]}$ is nondecreasing in $n$ and non-increasing in $p$. Furthermore, we know from Step 1 that for each $n,p\geq 1$, $$\label{eq:2.6} \begin{array}{lll} \displaystyle\varphi\left(t, |Y_t^{n,p}|+\int_0^t \left[f_s\wedge (n\vee p)\right]{\rm d}s\right) &\leq & \displaystyle\mathbb{E}\left[\left.\varphi\left(T, |\xi^{n,p}|+\int_0^T \left[f_s\wedge (n\vee p)\right]{\rm d}s\right)\right|\mathcal{F}_t\right]\vspace{0.2cm}\\ &\leq & \displaystyle\mathbb{E}\left[\left.\varphi\left(T, |\xi|+\int_0^T f_s{\rm d}s\right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].\vspace{0.1cm} \end{array}$$ Now, for each pair of integers $m,l\geq 1$, we define the following stopping times: $$\tau_m:=\inf\left\{t\in [0,T]: \mathbb{E}\left[\left.\varphi\left(T, |\xi|+\int_0^T f_s{\rm d}s\right)\right|\mathcal{F}_t\right] \geq \varphi(t,m)\right\}\wedge T$$ and $$\sigma_{m,l}:=\inf\bigg\{t\in [0,T]: \int_0^t H(s, m){\rm d}s+\sup\limits_{s\in [0,t]}\Gamma(s,m) \geq l\bigg\}\wedge \tau_m\vspace{0.2cm}$$ with the convention that $\inf\emptyset=+\infty$. Then $(Y^{n,p}_{m,l}(t), Z^{n,p}_{m,l}(t))_{t\in [0,T]}:=(Y^{n,p}_{t\wedge\sigma_{m,l}}, Z^{n,p}_t{\bf 1}_{t\leq\sigma_{m,l}})_{t\in [0,T]}$ is a solution in the space of processes $\mathcal{S}^\infty([0,T];{\mathbb R})\times\mathcal{M}^2([0,T];{\mathbb R}^d)$ to the following BSDE: $$Y^{n,p}_{m,l}(t)=Y^{n,p}_{\sigma_{m,l}}+\int_t^T{{\bf 1}_{s\leq \sigma_{m,l}}g^{n,p}(s, Y^{n,p}_{m,l}(s), Z^{n,p}_{m,l}(s)){\rm d}s}-\int_t^T Z^{n,p}_{m,l}(s) \cdot {\rm d}B_s,\ \ t\in[0,T].$$ Observe that for each fixed $m,l\geq 1$, $(Y^{n,p}_{m,l}(t))_{t\in [0,T]}$ is nondecreasing in $n$ and non-increasing in $p$, and that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $(g^{n,p})_{n,p}$ converges locally uniformly in $(y,z)$ to $g$ as $n,p\rightarrow\infty$. Since $\varphi(t,\cdot)$ is nondecreasing for each $t\in[0,T]$, by [\[eq:2.6\]](#eq:2.6){reference-type="eqref" reference="eq:2.6"} together with the definitions of $\tau_m$ and $\sigma_{m,l}$ we can obtain that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $$\sup\limits_{n,p\geq 1}|Y^{n,p}_{m,l}(t)|\leq m.$$ Furthermore, since $|g^{n,p}|\leq |g|$ and $g$ satisfies assumption (EX2), we know that ${\rm d}\mathbb{P}\times{\rm d} s-a.e.$, $$\forall(y, z)\in [-m, m]\times{\mathbb R}^d,\ \ \ \sup\limits_{n,p\geq 1}\left({\bf 1}_{s\leq \sigma_{m,l}}\left|g^{n,p}(s, y, z)\right|\right)\leq {\bf 1}_{s\leq \sigma_{m,l}}H(s, m)+ l|z|^2$$ with $\int_0^T {\bf 1}_{s\leq \sigma_{m,l}} H(s, m){\rm d}s\leq l$. Thus, for each fixed $m,l\geq 1$, we can apply the stability result for the bounded solutions of BSDEs (see for example Proposition 3.1 in [@LuoFan2018SD]). Setting $Y_{m,l}(t):=\inf_p\sup_n Y^{n,p}_{t\wedge\sigma_{m,l}}$, then $(Y_{m,l}(t))_{t\in[0,T]}$ is continuous and the sequence $(Z^{n,p}_{t}{\bf 1}_{t\le \sigma_{m,l}})_{t\in [0,T]}$ converges to $(Z_{m,l}(t))_{t\in [0,T]}$ strongly in $\mathcal{M}^2([0,T];{\mathbb R}^d)$ as $n,p\rightarrow\infty$ such that $$Y_{m,l}(t)=\inf_{p\geq 1}\sup_{n\geq 1} Y^{n,p}_{\sigma_{m,l}}+\int_t^T{{\bf 1}_{s\le \sigma_{m,l}}g(s,Y_{m,l}(s),Z_{m,l}(s)){\rm d}s}-\int_t^T{Z_{m,l}(s)\cdot {\rm d}B_s},\ \ t\in[0,T].$$ Finally, in view of the last equation and the stability of stopping times $\tau_m$ and $\sigma_{m,l}$, since we have ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $m,l\geq 1$, $$Y_{m+1,l+1}(t\wedge \sigma_{m,l})=Y_{m,l+1}(t\wedge \sigma_{m,l})=Y_{m,l}(t\wedge \sigma_{m,l})=\inf_{p\geq1}\sup_{n\geq 1} Y^{n,p}_{t\wedge\sigma_{m,l}}$$ and $$Z_{m+1,l+1}{\bf 1}_{t\leq \sigma_{m,l}}=Z_{m,l+1}{\bf 1}_{t\leq \sigma_{m,l}}=Z_{m,l}{\bf 1}_{t\leq \sigma_{m,l}}=\lim_{n,p\rightarrow\infty}Z^{n,p}_{t}{\bf 1}_{t\le \sigma_{m,l}},\vspace{0.1cm}$$ we see that $(Y_t,Z_t)_{t\in[0,T]}$ is an adapted solution of BSDE$(\xi,g)$, where $$Y_t:=\inf_p\sup_n Y^{n,p}_{t} \ \ {\rm and}\ \ Z_t:=\sum_{m=1}^{+\infty}\left( \sum_{l=1}^{+\infty} Z_{m,l}(t){\bf 1}_{t\in [\sigma_{m,l-1},\sigma_{m,l})}\right){\bf 1}_{t\in [\tau_{m-1},\tau_m)}, \quad t\in[0,T]$$ with $\tau_0:=0$ and $\sigma_{m,0}:=0$ for each $m\geq 1$. And, [\[eq:2.3\]](#eq:2.3){reference-type="eqref" reference="eq:2.3"} follows from [\[eq:2.6\]](#eq:2.6){reference-type="eqref" reference="eq:2.6"} by sending $n,p\to \infty$. Moreover, according to [\[eq:2.5\]](#eq:2.5){reference-type="eqref" reference="eq:2.5"}, in a way similar to step 1, we also verify that for each $n,p\geq 1$ and $0\leq t\leq r\leq T$, $$\varphi\left(t, |Y_t^{n,p}|+\int_0^t \left[f_s\wedge (n\vee p)\right]{\rm d}s\right) \leq \mathbb{E}\left[\left.\varphi\left(r, |Y_r^{n,p}|+\int_0^r \left[f_s\wedge (n\vee p)\right]{\rm d}s\right)\right|\mathcal{F}_t\right].$$ Thus, in view of [\[eq:2.6\]](#eq:2.6){reference-type="eqref" reference="eq:2.6"}, setting $n, p\to \infty$ and using Lebesgue's dominated convergence theorem in the last inequality, we see that the process $(\varphi(t, |Y_t|+\int_0^t f_s{\rm d}s))_{t\in [0,T]}$ is indeed a sub-martingale. The proof is then complete. ◻ As applications of [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"}, we shall prove Theorems [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} and [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"} below, where the function $h(\cdot,\cdot,\cdot)$ in (EX3) takes the following form: $$\label{eq:2.7} h(t,x,\bar x):=\beta x\left(\ln(e+x)\right)^\delta+\gamma \bar x^\alpha \left(\ln(e+\bar x)\right)^\lambda,\ \ (t,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+.$$ Both theorems can be compared to existing existence results (for example, see [@Kobylanski2000AP; @BriandHu2006PTRF; @HuTang2018ECP; @FanHu2019ECP; @FanHu2021SPA; @FanHuTang2023SCL; @FanHuTang2023SPA]) on adapted solutions of one-dimensional BSDEs. ## The logarithmic quasi-linear growth case Let us first consider the case that the generator $g$ has a logarithmic quasi-linear growth in the unknown variables $(y,z)$, i.e., the case of $\alpha=1$ in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"}. **Theorem 3**. *Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies (EX1)-(EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\alpha=1$. Then, the following assertions hold.* *(i) Let $\delta=0$ and $\lambda\in (-\infty,-{1\over 2})$. If $\xi+\int_0^T f_s{\rm d}s\in L^1$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(Y_t)_{t\in[0,T]}$ is of class (D);* *(ii) Let $\delta=0$ and $\lambda=-{1\over 2}$. If $\xi+\int_0^T f_s{\rm d}s\in L(\ln L)^p$ for some $p>0$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|(\ln (e+|Y_t|))^p)_{t\in [0,T]}$ is of class (D);* *(iii) If $p:=\delta\vee (\lambda+{1\over 2})\vee (2\lambda)\in (0,+\infty)$ and $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0}L\exp[\mu(\ln L)^p]$, then BSDE$(\xi,g)$ has a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(|Y_t|\exp \left(\mu(\ln (e+|Y_t|))^p\right))_{t\in[0,T]}$ is of class (D) for each $\mu>0$;* *(iv) Let $\delta=0$ and $\lambda=0$. If $\xi+\int_0^T f_s{\rm d}s\in L^p$ for some $p>1$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|^p)_{t\in[0,T]}$ is of class (D).* **Remark 4**. *When the generator $g$ grows faster in both unknown variables $(y,z)$, a stronger integrability on $\xi+\int_0^T f_s{\rm d}s$ is required for the existence of a solution of BSDE$(\xi,g)$. In addition, the case of $\lambda\in [-\frac{1}{2},0)$ seems to be first given in (ii) and (iii) of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}.* **Example 5**. *Consider the following simple BSDE: $$\label{eq:2.7*} Y_t=\xi+\int_t^T (f_s+\beta |Y_s|+\gamma |Z_s|){\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T],$$ where $\xi\geq 0$ and $f_\cdot\in L^1(0,T)$. It is the special case of $\delta=0$, $\lambda=0$ and $\alpha=1$ in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"}. Theorem 2.1 of [@HuTang2018ECP] states that BSDE [\[eq:2.7\*\]](#eq:2.7*){reference-type="eqref" reference="eq:2.7*"} admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $Y\geq 0$ if and only if there exists a locally bounded process $\bar Y$ such that $${\rm ess} \sup\limits_{q\in \mathcal{A}\ } \left\{\mathbb{E}_q\left[\left.e^{\beta(T-t)}|\xi|\right|\mathcal{F}_t\right]\right\} +\int_t^T e^{\beta(s-t)} f_s {\rm d}s\leq \bar Y_t,$$ where $\mathcal{A}$ is the set of $(\mathcal{F}_t)$-progressively measurable ${\mathbb R}^d$-valued process $(q_t)_{t\in [0,T]}$ such that $|q|\leq \gamma$, and $$\frac{{\rm d}{\mathbb Q}^q}{{\rm d}{\mathbb P}}:=M^q_T$$ with $$M^q_t:=\exp\left\{\int_0^t q_s\cdot {\rm d}B_s-{1\over 2}\int_0^t |q_s|^2{\rm d}s \right\},\ \ t\in [0,T]\vspace{0.2cm}$$ and $\mathbb{E}_q$ is the expectation operator with respect to ${\mathbb Q}^q$. In particular, if BSDE [\[eq:2.7\*\]](#eq:2.7*){reference-type="eqref" reference="eq:2.7*"} admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $Y\geq 0$, then $\xi e^{\gamma |B_T|}\in L^1$.* *The following example is taken from Example 2.3 of [@HuTang2018ECP]. Set $d=1$, $T=1$, $\gamma=1$ and the terminal variable $\xi:=e^{{1\over 2}(|B_1|-1)^2}-1$. Then, BSDE [\[eq:2.7\*\]](#eq:2.7*){reference-type="eqref" reference="eq:2.7*"} has no solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $Y\geq 0$, as $\xi e^{|B_T|}\notin L^1$: $$\mathbb{E}\left[\xi e^{|B_T|}\right]=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{{1\over 2}(|x|-1)^2} e^{|x|}e^{-{1\over 2}x^2}{\rm d}x=+\infty.$$ Whereas it can be directly checked that this $\xi$ belongs to the space of $\cap_{0<\mu<\sqrt{2}} L\exp[\mu(\ln L)^p]$ and then $\cap_{q>0} L(\ln L)^q$, but does not belong to $L\exp[\sqrt{2}(\ln L)^p]$, where $p={1\over 2}$ is defined in (iii) of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}.* *Furthermore, [@HuTang2018ECP; @FanHu2019ECP] show that for a linearly growing BSDE$(\xi,g)$ to have a solution, an $L\exp(\mu\sqrt{\ln L})$-integrability of $\xi+\int_0^T f_s{\rm d}s$ is sufficient for $\mu=\gamma\sqrt{2T}$, but not for any $\mu\in (0, \gamma\sqrt{2T}\,)$, which can not follow from (iii) of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}.* To prove [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}, we introduce the following [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"}, whose proof is given in Appendix. **Proposition 6**. *For each $p>1$ and $\lambda\in {\mathbb R}$, there exists a sufficiently large positive constant $k_{\lambda,p}\geq e$ depending only on $(\lambda,p)$ such that for each $k\geq k_{\lambda,p}$, $$\label{eq:2.8} 2xy\left(\ln (k+y)\right)^\lambda\leq px^2\left(\ln (k+x)\right)^{2\lambda}+y^2,\ \ \forall x,y>0.$$ In particular, when $p=1$, there is no constant $k$ such that [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"} holds true unless $\lambda=0$.* *Proof of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}.* Since the generator $g$ satisfies (EX1)-(EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\alpha=1$, a function $\varphi(\cdot,\cdot)\in {\bf S}$ is a test function for $g$ if for each $(s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+$, it holds that $$-\varphi_x(s,x)\left(\beta x\left(\ln(e+x)\right)^\delta+\gamma \bar x\left(\ln(e+\bar x)\right)^\lambda\right)+{1\over 2}\varphi_{xx}(s,x)|\bar x|^2+\varphi_s(s,x)\geq 0.$$ It follows from [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"} with $p=2$ that there exists a sufficiently large positive constant $k_{\lambda}\geq e$ depending only on $\lambda$ such that for each $k\geq k_\lambda$ and $(s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+$, $$\begin{array}{l} \displaystyle-\gamma\varphi_x(s,x) \bar x\left(\ln(k+\bar x)\right)^\lambda+{1\over 2}\varphi_{xx}(s,x)|\bar x|^2\vspace{0.2cm}\\ \ \ =\displaystyle\frac{\varphi_{xx}(s,x)}{2}\left(-2\frac{\gamma \varphi_x(s,x)}{\varphi_{xx}(s,x)}\bar x\left(\ln(k+\bar x)\right)^\lambda+|\bar x|^2\right)\vspace{0.2cm}\\ \ \ \geq \displaystyle-\frac{\gamma^2 (\varphi_x(s,x))^2}{\varphi_{xx}(s,x)} \left(\ln\left(k+\frac{\gamma \varphi_x(s,x)}{\varphi_{xx}(s,x)} \right)\right)^{2\lambda}.\vspace{0.1cm} \end{array}$$ Thus, if a function $\varphi(\cdot,\cdot)\in {\bf S}$ satisfies that for some $k\geq k_\lambda\geq e$ and each $(s,x)\in [0,T]\times {\mathbb R}_+$, $$\label{eq:2.9} -\beta \varphi_x(s,x)(k+x)\left(\ln(k+x)\right)^\delta-\frac{\gamma^2 (\varphi_x(s,x))^2}{\varphi_{xx}(s,x)} \left(\ln\left(k+\frac{\gamma \varphi_x(s,x)}{\varphi_{xx}(s,x)} \right)\right)^{2\lambda}+\varphi_s(s,x)\geq 0,$$ then it is a test function for the generator $g$. \(i\) Let $\delta=0$ and $\lambda\in (-\infty,-{1\over 2})$. By a similar computation as in [@FanHuTang2023SCL], one can verify that for sufficiently large $k\geq k_\lambda$ and $c\geq 2\beta-\frac{8\gamma^2}{1+2\lambda}$, the following function $$\varphi(s,x)=(k+x)\left(1-\left(\ln(k+x)\right)^{1+2\lambda}\right)\exp(c s),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the inequality [\[eq:2.9\]](#eq:2.9){reference-type="eqref" reference="eq:2.9"} with $\delta=0$ and $\lambda<-1/2$, and thus is a test function for the generator $g$. Since $$\lim\limits_{x\rightarrow+\infty}\frac{\varphi(s,x)}{x}=\exp(cs)\in [1,\exp(cT)], \quad s\in[0,T],$$ we see from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(Y_t)_{t\in [0,T]}$ is of class (D) if $\xi+\int_0^T f_s{\rm d}s\in L^1$. \(ii\) Let $\delta=0$ and $\lambda=-{1\over 2}$. It is not very hard to verify that for $p>0$, sufficiently large $k\geq k_\lambda$ and $c\geq 2\beta+\frac{4\gamma^2}{p}$, the following function $$\varphi(s,x)=(k+x)\left(\ln(k+x)\right)^p \exp(c s),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the inequality [\[eq:2.9\]](#eq:2.9){reference-type="eqref" reference="eq:2.9"} with $\delta=0$ and $\lambda=-1/2$, and thus is a test function for the generator $g$. Since $$\lim\limits_{x\rightarrow+\infty}\frac{\varphi(s,x)}{x\left(\ln(e+x)\right)^p }=\exp(cs)\in [1,\exp(cT)], \quad s\in[0,T],$$ we see from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that if $\xi+\int_0^T f_s{\rm d}s\in L(\ln L)^p$ for some $p>0$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|(\ln (e+|Y_t|))^p)_{t\in [0,T]}$ is of class (D). \(iii\) Let $p:=\delta\vee (\lambda+{1\over 2})\vee (2\lambda)\in (0,+\infty)$. By a similar computation as in [@FanHuTang2023SPA], one can verify that for sufficiently large $k\geq k_\lambda$, $c_1\geq 1$ and $c_2\geq (p+1)\beta-4^{\lambda^+}\gamma^2$, the following function $$\varphi(s,x)=(k+x)\exp\left(c_1\exp(c_2 s) \left(\ln(k+x)\right)^p\right),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the inequality [\[eq:2.9\]](#eq:2.9){reference-type="eqref" reference="eq:2.9"} with $p>0$, and thus is a test function for the generator $g$. Since $$\lim\limits_{x\rightarrow+\infty}\frac{\varphi(s,x)}{x\exp\left(c_1\exp(c_2 s) \left(\ln(e+x)\right)^p\right)}=1, \quad s\in[0,T],$$ we see from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that if $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0} L\exp[\mu(\ln L)^p]$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that for each $\mu>0$, $(|Y_t|\exp \left(\mu(\ln (e+|Y_t|))^p\right))_{t\in[0,T]}$ is of class (D). \(iv\) Let $\delta=0$ and $\lambda=0$. It is easy to verify that for each $p>1$, $k\geq k_\lambda$ and $c\geq p\beta+\frac{p}{p-1}\gamma^2$, the following function $$\varphi(s,x)=(k+x)^p \exp(c s),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the inequality [\[eq:2.9\]](#eq:2.9){reference-type="eqref" reference="eq:2.9"} with $\delta=0$ and $\lambda=0$, and thus is a test function for the generator $g$. Since $$\lim\limits_{x\rightarrow+\infty}\frac{\varphi(s,x)}{x^p }=\exp(cs)\in [1,\exp(cT)], \quad s\in[0,T],$$ we see from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that if $\xi+\int_0^T f_s{\rm d}s\in L^p$ for some $p>1$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|^p)_{t\in[0,T]}$ is of class (D). ◻ ## The sub-quadratic/quadratic growth case Let us further consider the case that the generator $g$ has a one-sided linear growth in the unknown variable $y$ and has a super-linear at most quadratic growth in the unknown variable $z$. **Theorem 7**. *Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies assumptions (EX1)-(EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\delta=0$, $\lambda=0$ and $\alpha\in (1,2]$. Let $\alpha^*$ be the conjugate of $\alpha$, i.e., $$\frac{1}{\alpha}+\frac{1}{\alpha^*}=1.\vspace{0.1cm}$$ If $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0}\exp(\mu L^{2\over \alpha^*})$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(\exp(\mu |Y_t|^{2\over \alpha^*}))_{t\in [0,T]}$ is of class (D) for each $\mu>0$. In particular, if $\xi+\int_0^T f_s{\rm d}s\in L^{\infty}$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(Y_t)_{t\in[0,T]}\in \mathcal{S}^{\infty}([0,T];{\mathbb R})$.* **Example 8**. *Let us consider the following typical BSDE: $$\label{eq:2.10*} Y_t=\xi+\int_t^T {1\over 2}|Z_s|^2 {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ The change of variables leads to the equation $$e^{Y_t}=e^\xi-\int_t^T e^{Y_s}Z_s\cdot {\rm d}B_s, \ \ t\in [0,T],$$ which has a solution as $e^\xi\in L^1$. On the other hand, since $\{e^{Y_t}\}_{t\in [0,T]}$ is a positive super-martingale, Theorem 3.1 of [@BriandLepeltierSanMartin2007Bernoulli] observes that the inclusion $e^\xi\in L^1$ is also necessary for this BSDE to have a solution.* *Furthermore, we consider the case where the generator $g$ satisfies assumptions (EX1)-(EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\delta=0$, $\lambda=0$ and $\alpha=2$. [@BriandHu2006PTRF] show that for BSDE$(\xi,g)$ to have a solution, an $\exp(\mu L)$-integrability of $\xi+\int_0^T f_s{\rm d}s$ is sufficient for $\mu=2\gamma e^{\beta T}$, but not for any $\mu \in (0, 2\gamma e^{\beta T})$, which can not follow from [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"}.* *Proof of [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"}.* Consider both cases of $\alpha=2$ and $\alpha\in (1,2)$. \(i\) The case of $\alpha=2$. In this case, a function $\varphi(\cdot,\cdot)\in {\bf S}$ is a test function for $g$ if $$-\varphi_x(s,x)\left(\beta x+ \gamma \bar x^2\right)+{1\over 2}\varphi_{xx}(s,x)|\bar x|^2+\varphi_s(s,x)\geq 0, \quad \forall (s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+.$$ It is easy to verify that for each $c_1\geq 2\gamma$ and $c_2\geq \beta$, the following function $$\varphi(s,x)=\exp\left(c_1\exp(c_2 s)x \right),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the last inequality, and thus is a test function for the generator $g$. It follows from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that if $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0}\exp(\mu L)$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(\exp\left(\mu |Y_t|\right))_{t\in [0,T]}$ is of class (D) for each $\mu>0$, which is the desired assertion since $\alpha^*=2$ in this case. \(ii\) The case of $\alpha\in (1,2)$. In this case, a function $\varphi(\cdot,\cdot)\in {\bf S}$ is a test function for $g$ if $$-\varphi_x(s,x)\left(\beta x+\gamma\bar x^\alpha\right)+{1\over 2}\varphi_{xx}(s,x)|\bar x|^2+\varphi_s(s,x)\geq 0, \quad \forall (s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+.$$ Using Young's inequality, we have that for each $(s,x,\bar x)\in [0,T]\times {\mathbb R}_+\times{\mathbb R}_+$, $$\begin{array}{l} \displaystyle-\gamma\varphi_x(s,x) \bar x^\alpha+\frac{1}{2}\varphi_{xx}(s,x)|\bar x|^2 \vspace{0.2cm}\\ \ \ =\displaystyle\varphi_{xx}(s,x)\left(-\frac{\gamma \varphi_x(s,x)}{\varphi_{xx}(s,x)}\bar x^\alpha+\frac{1}{2}|\bar x|^2\right)\vspace{0.2cm}\\ \ \ \geq \displaystyle-\frac{2-\alpha}{2\alpha}\cdot \frac{(\alpha\gamma\varphi_x(s,x))^{2\over 2-\alpha}}{(\varphi_{xx}(s,x))^{\alpha\over 2-\alpha}}\geq -\frac{(2\gamma\varphi_x(s,x))^{2\over 2-\alpha}}{(\varphi_{xx}(s,x))^{\alpha\over 2-\alpha}}.\vspace{0.1cm} \end{array}$$ Thus, if a function $\varphi(\cdot,\cdot)\in {\bf S}$ satisfies $$\label{eq:2.10} -\beta \varphi_x(s,x)x-\frac{(2\gamma \varphi_x(s,x))^{2\over 2-\alpha}}{(\varphi_{xx}(s,x))^{\alpha\over 2-\alpha}}+\varphi_s(s,x)\geq 0, \quad (s,x)\in [0,T]\times {\mathbb R}_+,$$ then it is a test function for the generator $g$. Furthermore, by a similar computation as in [@FanHu2021SPA], it can be verified that for each $c_1\geq 1$, $k\geq k_{\alpha,c_1}$ with $k_{\alpha,c_1}$ being a positive constant depending only on $\alpha$ and $c_1$, and $c_2\geq \beta+(1+c_1)2^{6\over 2-\alpha}(2\alpha-2)^{2-2\alpha\over 2-\alpha}\gamma^{2\over 2-\alpha}$, the following function $$\varphi(s,x)=\exp\left(c_1\exp(c_2 s)(x+k)^{2\over \alpha^*} \right),\ \ (s,x)\in [0,T]\times{\mathbb R}_+$$ satisfies the inequality [\[eq:2.10\]](#eq:2.10){reference-type="eqref" reference="eq:2.10"}, and thus is a test function for the generator $g$. Since $$\exp\left(x^{2\over \alpha^*} \right)\leq \varphi(s,x)\leq \exp\left(c_1\exp(c_2 T)k^{2\over \alpha^*}\right) \exp\left(c_1\exp(c_2 T)x^{2\over \alpha^*} \right),\ \ \forall(s,x)\in [0,T]\times {\mathbb R}_+,$$ we see from [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} that if $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0}\exp(\mu L^{2\over \alpha^*})$, then BSDE$(\xi,g)$ admits a solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(\exp(\mu |Y_t|^{2\over \alpha^*}))_{t\in [0,T]}$ is of class (D) for each $\mu>0$. Finally, by [\[eq:2.3\]](#eq:2.3){reference-type="eqref" reference="eq:2.3"} it can be concluded that if $\xi+\int_0^T f_s{\rm d}s\in L^{\infty}$, then $(\varphi(t,|Y_t|+\int_0^t f_s{\rm d}s))_{t\in [0,T]}$ is a bounded process, and then $(Y_t)_{t\in [0,T]}\in \mathcal{S}^{\infty}([0,T];{\mathbb R})$. The proof is complete. ◻ **Remark 9**. *Some finer integrability of the data $\xi+\int_0^T f_t{\rm d}t$ might be found for the existence of an adapted solution to BSDE$(\xi,g)$ with new test functions different from those in the proof of Theorems [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} and [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"}. The reader is referred to [@BriandHu2006PTRF; @FanHu2019ECP; @FanHu2021SPA; @FanHuTang2023SPA] for more details.* # Comparison theorems and existence and uniqueness results {#sec:3-Uniqueness} In this section, we prove two comparison theorems under different growth of the generator $g$ in the unknown variables $(y,z)$, which immediately yield the desired uniqueness and can be compared to some existing comparison results given in for example [@ElKarouiPengQuenez1997MF; @Kobylanski2000AP; @HuImkeller2005AAP; @BriandHu2008PTRF; @Fan2016SPA; @FanHu2021SPA; @FanHuTang2023SPA; @FanHuTang2023SCL]. First, we have the following a priori estimate. **Proposition 10**. *Assume that there exists a function $h(\cdot,\cdot,\cdot)\in {\bf\bar S}$ such that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $${\bf 1}_{Y_t>0}\ g(t,Y_t,Z_t)\leq f_t+h(t,Y_t^+,|Z_t|),$$ and that $\varphi(\cdot,\cdot)\in {\bf S}$ is a test function for $h$. If $(\varphi(t,Y^+_t+\int_0^t f_s{\rm d}s))_{t\in[0,T]}$ is of class (D), then we have $$\varphi\left(t, \ Y_t^+ +\int_0^t f_s{\rm d}s\right)\leq \mathbb{E}\left[\left.\varphi\left(T, \ \xi^+ +\int_0^T f_s{\rm d}s\right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].\vspace{0.1cm}$$* *Proof.* Note that $(\varphi(t,Y^+_t+\int_0^t f_s{\rm d}s))_{t\in[0,T]}$ belongs to class (D). The desired conclusion can be easily obtained by a similar computation to step 1 in the proof of [Theorem 2](#thm:2.2){reference-type="ref" reference="thm:2.2"} with $Y^+_t$, $Y^+_s$, ${\bf 1}_{Y_t>0}$ and ${\bf 1}_{Y_s>0}$ instead of $|Y_t|$, $|Y_s|$, ${\rm sgn}(Y_t)$ and ${\rm sgn}(Y_s)$, respectively. ◻ ## Comparison results for the case of an at most linear growth Now, let us introduce the following assumptions on the generator $g$, where $g$ has a one-sided linear growth in $y$ and an at most linear growth in $z$. 1. $g$ has an extended monotonicity in $y$, i.e., there exists a continuous, increasing and concave function $\rho(\cdot):{\mathbb R}_+\rightarrow{\mathbb R}_+$ satisfying $\rho(0)=0$, $\rho(u)>0$ for $u>0$ and $$\int_{0^+}\frac{{\rm d}u}{\rho(u)}:=\lim\limits_{\varepsilon\rightarrow 0^+}\int_0^\varepsilon\frac{{\rm d}u}{\rho(u)}=+\infty\vspace{0.1cm}$$ such that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y_1,y_2,z)\in {\mathbb R}\times{\mathbb R}\times{\mathbb R}^d$, $${\rm sgn}(y_1-y_2) \left(g(\omega,t,y_1,z)-g(\omega,t,y_2,z)\right)\leq \rho(|y_1-y_2|).\vspace{0.1cm}$$ 2. $g$ has a logarithmic uniform continuity in $z$, i.e., there exist a non-positive constant $\lambda\in (-\infty,0]$ and a nondecreasing continuous function $\kappa(\cdot):{\mathbb R}_+\rightarrow{\mathbb R}_+$ with linear growth and $\kappa(0)=0$ such that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z_1,z_2)\in {\mathbb R}\times {\mathbb R}^d\times {\mathbb R}^d$, $$|g(\omega,t,y,z_1)-g(\omega,t,y,z_2)|\leq \kappa\left(|z_1-z_2|(\ln (e+|z_1-z_2|))^\lambda\right)\leq \kappa\left(|z_1-z_2|\right).$$ **Remark 11**. *Since the function $\rho(\cdot)$ appearing in (UN1) is nondecreasing and concave with $\rho(0)=0$, we can verify that $\rho(\cdot)$ has a linear growth. We always assume that there exists a $A>0$ such that $$\label{eq:3.1} \forall u\in{\mathbb R}_+, \ \ \ \rho(u)\leq A(u+1)\ \ {\rm and}\ \ \kappa(u)\leq A(u+1).$$ Thus, if the generator $g$ satisfies (UN1) and (UN2), then we have ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z)\in {\mathbb R}\times {\mathbb R}^d$, $$\begin{array}{lll} {\rm sgn}(y)g(\omega,t,y,z)&\leq & \displaystyle|g(t,0,0)|+ \rho(|y|)+\kappa(|z|\left(\ln(e+|z|)\right)^\lambda)\\ &\leq & \displaystyle|g(t,0,0)|+ 2A+A|y|+A|z|\left(\ln(e+|z|)\right)^\lambda, \end{array}$$ which means that the generator $g$ has a one-sided linear growth in $y$ and a logarithmic sub-linear/linear growth in the unknown variable $z$, and then satisfies assumption (EX3) with $f_t:=|g(t,0,0)|+2A$ and $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\beta=\gamma=A$, $\delta=0$, $\alpha=1$ and $\lambda\leq 0$. In addition, in the case of $\lambda=0$, assumption (UN2) is equivalent to saying that the function $g(t,\omega, y,z)$ is uniformly continuous in the variable $z$ uniformly with respect to the variables $(t,\omega,y)$; the assumption (UN2) becomes stronger as $\lambda$ decreases.* **Theorem 12**. *Let $\xi$ and $\xi'$ be two terminal conditions such that $\xi\leq \xi'$, the generator $g$ (resp. $g'$) satisfy assumptions (UN1) and (UN2), and $(Y_t, Z_t)_{t\in[0,T]}$ and $(Y'_t, Z'_t)_{t\in[0,T]}$ be, respectively, adapted solutions to BSDE$(\xi, g)$ and BSDE$(\xi', g')$ such that $$\label{eq:3.2} {\bf 1}_{Y_t>Y'_t}\left(g(t,Y'_t,Z'_t)-g'(t,Y'_t,Z'_t)\right)\leq 0\ \ \ ({\rm resp.}\ \ {\bf 1}_{Y_t>Y'_t}\left(g(t,Y_t,Z_t)-g'(t,Y_t,Z_t)\right)\leq 0\ ).$$ Then, we have $Y_t\leq Y'_t$ for each $t\in[0,T]$, if either of the following four conditions is true:* *(i) $\lambda<-1/2$ and both processes $(Y_t)_{t\in[0,T]}$ and $(Y'_t)_{t\in[0,T]}$ are of class (D).* *(ii) $\lambda=-1/2$ and both processes $(|Y_t|(\ln(e+|Y_t))^p)_{t\in[0,T]}$ and $(|Y'_t|(\ln(e+|Y'_t))^p)_{t\in[0,T]}$ are of class (D) for some constant $p>0$.* *(iii) $\lambda\in (-1/2,0]$ and both processes $(|Y_t|\exp(\mu(\ln(e+|Y_t|))^{\lambda+{1\over 2}}))_{t\in[0,T]}$ and $(|Y'_t|\exp(\mu(\ln(e+|Y'_t|))^{\lambda+{1\over 2}}))_{t\in[0,T]}$ are of class (D) for each $\mu>0$.* *(iv) $\lambda=0$ and both processes $(|Y_t|^p)_{t\in[0,T]}$ and $(|Y'_t|^p)_{t\in[0,T]}$ are of class (D) for some $p>1$.* [@ElKarouiPengQuenez1997MF] show that the strict comparison theorem is true for solutions of two BSDEs when either of both generators is uniformly Lipschitz continuous in $(y,z)$. However, the following two examples indicate that the strict comparison theorem fails to hold in general when the generator $g$ satisfies only assumptions (UN1) and (UN2), which are provided in Section 5.3 of [@PardouxRascanu2014Book] and Example 3.2 of [@Jia2010SPA], respectively. In finance, this means that there are infinitely many opportunities of arbitrage. **Example 13**. *Let $d=1$ and consider the following BSDE: $$\label{eq:3.2*} Y_t=\xi+\int_t^T \left(-2\sqrt{Y_s^+}\right) {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ Clearly, the generator $g(\omega,t,y,z):\equiv -2\sqrt{y^+}$ satisfies assumptions (UN1) and (UN2) with $\rho(x)=x$ and $\kappa(x)\equiv 0$. It is not hard to verify that $(Y_t,Z_t)_{t\in [0,T]}:=(0,0)_{t\in [0,T]}$ and $(Y'_t,Z'_t)_{t\in [0,T]}:=(t^2,0)_{t\in [0,T]}$ are respectively the unique solution to [\[eq:3.2\*\]](#eq:3.2*){reference-type="eqref" reference="eq:3.2*"} with $\xi=0$ and $\xi=T^2$ such that $(|Y_t|^p)_{t\in[0,T]}$ and $(|Y'_t|^p)_{t\in[0,T]}$ belong to class (D) for each $p>0$. Note that $Y'_T=T^2>0=Y_T$, but $Y_0=Y'_0=0$.* **Example 14**. *Let $d=1$ and consider the following BSDE: $$\label{eq:3.3*} Y_t=\xi+\int_t^T (-3|Z_s|^{2\over 3}) {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ Clearly, the generator $g(\omega,t,y,z):\equiv -3|z|^{2\over 3}$ satisfies (UN1) and (UN2) with $\rho(x)=x$ and $\kappa(x)= 3x^{2\over 3}$. It is not hard to verify that $(Y_t,Z_t)_{t\in [0,T]}:=(0,0)_{t\in [0,T]}$ and $(Y'_t,Z'_t)_{t\in [0,T]}:=({1\over 4}B_t^4,B_t^3)_{t\in [0,T]}$ are respectively the unique solution to [\[eq:3.3\*\]](#eq:3.3*){reference-type="eqref" reference="eq:3.3*"} with $\xi=0$ and $\xi={1\over 4}B_T^4$ such that $(|Y_t|^p)_{t\in[0,T]}$ and $(|Y'_t|^p)_{t\in[0,T]}$ belong to class (D) for each $p>0$. Note that ${\mathbb P}(Y'_T>Y_T)=1>0$, but $Y_0=Y'_0=0$.* *Proof of [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"}.* We only prove the case that the generator $g$ satisfies assumptions (UN1) and (UN2), and ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $$\label{eq:3.3} {\bf 1}_{Y_t>Y'_t}\left(g(t,Y'_t,Z'_t)-g'(t,Y'_t,Z'_t)\right)\leq 0.$$ The other case can be proved in the same way. According to Theorem 2.1 in [@Fan2016SPA] and the above assumptions, it suffices to prove that the process $((Y_t-Y'_t)^+)_{t\in[0,T]}$ is bounded whenever either of four conditions (i)-(iv) holds. Define $\hat Y:=Y-Y'$ and $\hat Z:=Z-Z'$. Then, the pair of processes $(\hat Y_t,\hat Z_t)_{t\in[0,T]}$ verifies $$\label{eq:3.4} \hat Y_t=\hat\xi+\int_t^T \hat g(s,\hat Y_s,\hat Z_s) {\rm d}s-\int_t^T \hat Z_s \cdot {\rm d}B_s, \ \ \ \ t\in[0,T],$$ where $\hat\xi:=\xi-\xi'$ and $$\hat g(s,\hat Y_s,\hat Z_s):=g(s,Y_s,Z_s)-g'(s,Y'_s,Z'_s).$$ From assumptions (UN1) and (UN2) together with inequalities [\[eq:3.3\]](#eq:3.3){reference-type="eqref" reference="eq:3.3"} and [\[eq:3.1\]](#eq:3.1){reference-type="eqref" reference="eq:3.1"}, we have $$\label{eq:3.5} \begin{array}{l} {\bf 1}_{\hat Y_t>0}\ \hat g(t,\hat Y_t,\hat Z_t)={\bf 1}_{\hat Y_t>0}\left(g(s,Y_s,Z_s)-g'(s,Y'_s,Z'_s)\right)\vspace{0.2cm}\\ \ \ = \displaystyle{\bf 1}_{\hat Y_t>0}\left( g(s,Y_s,Z_s)-g(s,Y'_s,Z_s)+ g(s,Y'_s,Z_s)-g(s,Y'_s,Z'_s)+g(s,Y'_s,Z'_s)-g'(s,Y'_s,Z'_s)\right) \vspace{0.2cm}\\ \ \ \leq \rho(\hat Y^+_t)+\kappa\left(|\hat Z_t|(\ln(e+|\hat Z_t|))^\lambda\right)\vspace{0.2cm}\\ \ \ \leq 2A+A\hat Y^+_t +A|\hat Z_t|\left(\ln(e+|\hat Z_t|)\right)^\lambda=f_t+h(t,\hat Y^+_t,|\hat Z_t|), \end{array}$$ where $f_t:\equiv 2A$ and for each $(t,x,\bar x)\in [0,T]\times{\mathbb R}_+\times{\mathbb R}_+$, $$\label{eq:3.6} h(t,x,\bar x):=Ax+A\bar x\left(\ln(e+\bar x)\right)^\lambda.$$ Using [Proposition 10](#pro:3.1){reference-type="ref" reference="pro:3.1"} together with [\[eq:3.5\]](#eq:3.5){reference-type="eqref" reference="eq:3.5"} and [\[eq:3.6\]](#eq:3.6){reference-type="eqref" reference="eq:3.6"}, we now verify that $\hat Y_\cdot^+$ is a bounded process whenever either of conditions (i)-(iv) is true. \(i\) Let $\lambda<-1/2$ and both processes $(Y_t)_{t\in[0,T]}$ and $(Y'_t)_{t\in[0,T]}$ be of class (D). For each $k\geq e$ sufficient large and each $c\geq 2A-\frac{8A^2}{1+2\lambda}$, define the following function $$\varphi(s,x)=(k+x)\left(1-\left(\ln(k+x)\right)^{1+2\lambda}\right)\exp(c s),\ \ (s,x)\in [0,T]\times{\mathbb R}_+.$$ Since $0\leq \varphi(s,x)\leq (k+x)\exp(c T)$ for each $(s,x)\in [0,T]\times{\mathbb R}_+$, the process $\{\varphi(t,\hat Y_t^+ +2At)\}_{t\in[0,T]}$ is of class (D). On the other hand, a similar analysis to that in (i) of the proof of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} yields that the last function $\varphi(\cdot,\cdot)$ satisfies [\[eq:2.1\]](#eq:2.1){reference-type="eqref" reference="eq:2.1"}, and thus is a test function for $h$ defined in [\[eq:3.6\]](#eq:3.6){reference-type="eqref" reference="eq:3.6"}. It then follows from [Proposition 10](#pro:3.1){reference-type="ref" reference="pro:3.1"} that $$\varphi\left(t, \ Y_t^+ +2At\right)\leq \mathbb{E}\left[\left.\varphi\left(T, \ \xi^+ +2AT\right)\right|\mathcal{F}_t\right]=\varphi\left(T, 2AT\right),\ \ t\in[0,T],$$ which means that $(\hat Y_t^+)_{t\in[0,T]}$ is a bounded process. \(ii\) Let $\lambda=-1/2$ and both processes $(|Y_t|(\ln(e+|Y_t|))^p)_{t\in[0,T]}$ and $(|Y'_t|(\ln(e+|Y'_t|))^p)_{t\in[0,T]}$ be of class (D) for some constant $p>0$. For each $k\geq e$ sufficient large and each $c\geq 2A+\frac{4A^2}{p}$, define the following function $$\varphi(s,x)=(k+x)\left(\ln(k+x)\right)^p \exp(c s),\ \ (s,x)\in [0,T]\times{\mathbb R}_+.$$ Since $0\leq \varphi(s,x)\leq Kx\left(\ln(e+x)\right)^p$ for each $(s,x)\in [0,T]\times{\mathbb R}_+$ and some positive constant $K>0$ depending only on $(k,T)$, we can deduce that $\{\varphi(t,\hat Y_t^+ +2At)\}_{t\in[0,T]}$ is of class (D). On the other hand, a similar analysis to that in (ii) of the proof of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} yields that the last function $\varphi(\cdot,\cdot)$ satisfies [\[eq:2.1\]](#eq:2.1){reference-type="eqref" reference="eq:2.1"}, and hence is a test function for $h$ defined in [\[eq:3.6\]](#eq:3.6){reference-type="eqref" reference="eq:3.6"}. Thus, the boundedness of the process $(\hat Y_t^+)_{t\in[0,T]}$ follows immediately as in (i). \(iii\) Let $\lambda\in (-{1\over 2},0]$ and both processes $(|Y_t|\exp(\mu(\ln(e+|Y_t|))^{\lambda+{1\over 2}}))_{t\in[0,T]}$ and $(|Y'_t|\exp(\mu(\ln(e+|Y'_t|))^{\lambda+{1\over 2}}))_{t\in[0,T]}$ be of class (D) for any $\mu>0$. For each $k\geq e$ sufficient large, $c_1\geq 1$ and $c_2\geq (\lambda+3/2)\beta-4^{\lambda^+}\gamma^2$, define the following function $$\varphi(s,x)=(k+x)\exp\left(c_1\exp(c_2 s) \left(\ln(k+x)\right)^{\lambda+{1\over 2}}\right),\ \ (s,x)\in [0,T]\times{\mathbb R}_+.$$ Since $0\leq \varphi(s,x)\leq K x\exp\left(K\left(\ln(e+x)\right)^{\lambda+{1\over 2}}\right)$ for each $(s,x)\in [0,T]\times{\mathbb R}_+$ and some positive constant $K>0$ depending only on $(k,T)$, we can deduce that $\{\varphi(t,\hat Y_t^+ +2At)\}_{t\in[0,T]}$ is of class (D). On the other hand, a similar analysis to that in (iii) of the proof of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} yields that the last function $\varphi(\cdot,\cdot)$ satisfies [\[eq:2.1\]](#eq:2.1){reference-type="eqref" reference="eq:2.1"}, and is a test function for $h$ defined in [\[eq:3.6\]](#eq:3.6){reference-type="eqref" reference="eq:3.6"}. Thus, the boundedness of the process $(\hat Y_t^+)_{t\in[0,T]}$ follows immediately as in (i). \(iv\) Let $\lambda=0$ and both processes $(|Y_t|^p)_{t\in[0,T]}$ and $(|Y'_t|^p)_{t\in[0,T]}$ be of class (D) for some $p>1$. Note that for each $\mu>0$, there exists a positive constant $K>0$ depending only on $(\mu,p)$ such that $$0\leq x\exp\left(\mu (\ln(e+x))^{1\over 2}\right)\leq K x^p, \ \ x\geq 1.$$ The desired assertion is a direct consequence of (iii). ◻ The following example is taken from Remark 6 of [@Jia2008CRA], which indicates that the uniform continuity of the generator $g$ in the unknown variable $y$ is not sufficient for the uniqueness of the solution to a BSDE$(\xi,g)$. **Example 15**. *Let us consider the following BSDE: $$\label{eq:3.8*} Y_t=\int_t^T \sqrt{|Y_s|} {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ Clearly, $g(\omega,t,y,z):\equiv \sqrt{|y|}$ is uniformly continuous. It is not hard to check that for each $c\in [0,T]$, $$(Y_t,Z_t)_{t\in [0,T]}:=\left({[(c-t)^+]^2\over 4},0\right)_{t\in[0,T]}\vspace{0.1cm}$$ is a solution to [\[eq:3.8\*\]](#eq:3.8*){reference-type="eqref" reference="eq:3.8*"} such that $(|Y_t|^p)_{t\in[0,T]}$ belongs to class (D) for each $p>0$.* ## Comparison results for the super-linear at most quadratic growth case In the following comparison theorem, we will use the following assumption on the generator $g$, where the generator $g$ admits a super-linear at most quadratic growth in $(y,z)$ in general. 1. It holds that ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $$\label{eq:3.7} \begin{array}{c} \displaystyle{\bf 1}_{\delta_\theta y>0} \frac{g(\omega,t,y_1,z)-\theta g(\omega,t,y_2,z)}{1-\theta}\leq f_t(\omega)+\bar\beta(|y_1|+|y_2|)+h(t,(\delta_\theta y)^+, |\delta_\theta z|),\vspace{0.1cm}\\ \forall(y_1,y_2,z_1,z_2)\in {\mathbb R}\times{\mathbb R}\times{\mathbb R}^d\times{\mathbb R}^d\ \ {\rm and}\ \ \theta\in (0,1), \end{array}$$ or $$\label{eq:3.8} \begin{array}{c} \displaystyle-{\bf 1}_{\delta_\theta y<0} \frac{g(\omega,t,y_1,z)-\theta g(\omega,t,y_2,z)}{1-\theta}\leq f_t(\omega)+\bar\beta(|y_1|+|y_2|)+h(t,(\delta_\theta y)^-, |\delta_\theta z|),\vspace{0.1cm}\\ \forall(y_1,y_2,z_1,z_2)\in {\mathbb R}\times{\mathbb R}\times{\mathbb R}^d\times{\mathbb R}^d\ \ {\rm and}\ \ \theta\in (0,1),\vspace{0.2cm} \end{array}$$ where $h(\cdot,\cdot,\cdot)\in {\bf\bar S}$ is defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\lambda\geq 0$, $$\delta_\theta y:=\frac{y_1-\theta y_2}{1-\theta}\ \ {\rm and}\ \ \delta_\theta z:=\frac{z_1-\theta z_2}{1-\theta}.$$ **Proposition 16**. *Assume that the generator $g$ satisfies (EX3) with $h(\cdot,\cdot,\cdot)\in {\bf\bar S}$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\lambda\geq 0$. Then, assumption (UN3) holds true for $g$ if either of the following three conditions is true:* 1. *${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $g(\omega,t,\cdot,\cdot)$ is convex or concave;* 2. *${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z)\in {\mathbb R}\times{\mathbb R}^d$, $g(\omega,t,\cdot,z)$ is Lipschitz continuous and $g(\omega,t, y,\cdot)$ is convex or concave;* 3. *$g(t,y,z)\equiv l(y)q(z)$, where both $l:{\mathbb R}\rightarrow{\mathbb R}$ and $q:{\mathbb R}^d\rightarrow{\mathbb R}$ are bounded and Lipschitz continuous, and the function $q(\cdot)$ has a bounded support.* The proof is similar to that of Proposition 3.5 of [@FanHu2021SPA], and is omitted here. **Remark 17**. *One typical example of (UN3) is $g(\omega,t,y,z):=g_1(y)+g_2(y)$, where $g_1:{\mathbb R}\rightarrow{\mathbb R}$ is convex or concave with a one-sided logarithmic sup-linear growth, i.e., there exists a nonnegative constant $A\geq 0$ such that for each $y\in{\mathbb R}$, $${\rm sgn}(y)g_1(y)\leq A+\beta |y|(\ln(e+|y|))^\delta,$$ and $g_2:{\mathbb R}\rightarrow{\mathbb R}$ is a Lipschitz continuous function. In other words, $g$ is a Lipschitz continuous perturbation of some convex (concave) function. Another typical example of (UN3) is $\bar g(\omega,t,y,z):=g_3(z)+g_4(z)$, where $g_3:{\mathbb R}^d\rightarrow{\mathbb R}$ is convex or concave with a logarithmic sub-quadratic growth, i.e., there exists a nonnegative constant $A\geq 0$ such that for each $z\in{\mathbb R}^d$, $$|g_3(z)|\leq A+\gamma |z|^\alpha (\ln(e+|z|))^\lambda,$$ and $g_4:{\mathbb R}^d\rightarrow{\mathbb R}$ is a Lipschitz continuous function with a bounded support. In other words, $\bar g$ is a local Lipschitz continuous perturbation of some convex (concave) function. Furthermore, it is easy to verify that if both generators $g_1$ and $g_2$ satisfies [\[eq:3.7\]](#eq:3.7){reference-type="eqref" reference="eq:3.7"} (resp. [\[eq:3.8\]](#eq:3.8){reference-type="eqref" reference="eq:3.8"}), then $g_1+g_2$ also satisfies [\[eq:3.7\]](#eq:3.7){reference-type="eqref" reference="eq:3.7"} (resp. [\[eq:3.8\]](#eq:3.8){reference-type="eqref" reference="eq:3.8"}). Consequently, the generator $g$ satisfying (UN3) may be not necessarily convex (concave) or Lipschitz continuous in $(y,z)$, and it can have a general growth in $y$. Finally, a similar argument to that in Section 4 of [@FanWangYong2022AAP] gives that if the generator $g$ satisfies (UN3), then it has to be local Lipschitz continuous in $(y,z)$, and then satisfies (EX1).* **Theorem 18**. *Suppose that $\xi$ and $\xi'$ are two terminal conditions such that $\xi\leq \xi'$, the generater $g$ (resp. $g'$) satisfies assumption (UN3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\lambda\geq 0$, and $(Y_t, Z_t)_{t\in[0,T]}$ and $(Y'_t, Z'_t)_{t\in[0,T]}$ are, respectively, adapted solutions to BSDE$(\xi, g)$ and BSDE$(\xi', g')$ such that $$g(t,Y'_t,Z'_t)\leq g'(t,Y'_t,Z'_t) \ \ ({\rm resp.}\ \ g(t,Y_t,Z_t)\leq g'(t,Y_t,Z_t)\ ).$$ Then the following two assertions hold true:* *(i) Let $\alpha=1$ and $p:=\delta\vee (\lambda+{1\over 2})\vee (2\lambda)$. If $\int_0^T f_s{\rm d}s\in \cap_{\mu>0} L\exp[\mu(\ln L)^p]$ and both processes $(|Y_t|\exp(\mu(\ln(e+|Y_t|))^p))_{t\in[0,T]}$ and $(|Y'_t|\exp(\mu(\ln(e+|Y'_t|))^p))_{t\in[0,T]}$ are of class (D) for each $\mu>0$, then for each $t\in[0,T]$, we have $Y_t\leq Y'_t$.* *(ii) Let $\delta=0$, $\lambda=0$, $\alpha\in (1,2]$ and $\alpha^*$ be the conjugate of $\alpha$. If $\int_0^T f_s{\rm d}s\in \exp(\mu L^{2\over \alpha^*})$ and both processes $(\exp(\mu(|Y_t|)^{2\over \alpha^*}))_{t\in[0,T]}$ and $(\exp(\mu(|Y'_t|)^{2\over \alpha^*}))_{t\in[0,T]}$ are of class (D) for each $\mu>0$, then for each $t\in[0,T]$, we have $Y_t\leq Y'_t$. In particular, if the random variable $\int_0^T f_s{\rm d}s$ and both processes $Y$ and $Y'$ are all bounded, then for each $t\in[0,T]$, $Y_t\leq Y'_t$.* **Remark 19**. *Let us suppose that the generator $g$ satisfies assumption (EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\delta=0$, $\lambda=0$, $\alpha=2$. For the case of the bounded terminal condition, it has been shown in [@Fan2016SPA] and [@Kobylanski2000AP] that in order to ensure that the (strictly) comparison result in [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"} holds, the assumption (UN3) can be weakened such that the generator $g$ further satisfies assumption (UN1) and the following locally Lipschitz condition in $z$: ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, for each $(y,z_1,z_2)\in {\mathbb R}\times{\mathbb R}^{1\times d}\times{\mathbb R}^{1\times d}$, $$\label{eq:3.12*} |g(\omega,t,y,z_1)-g(\omega,t,y,z_2)|\leq \gamma (1+|z_1|^\delta+|z_2|^\delta)|z_1-z_2|$$ with $\delta\in [0,1]$ and $\gamma>0$. For the case of the unbounded terminal condition, it has been shown in [@FanHuTang2020CRM] that in order to ensure that the comparison result in [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"} holds, the assumption (UN3) can be weakened such that the generator $g$ is uniformly Lipschitz continuous in $y$ and further satisfies [\[eq:3.12\*\]](#eq:3.12*){reference-type="eqref" reference="eq:3.12*"} with $\delta\in [0,1)$ and an additional strictly positive/negative quadratic condition of the generator $g$ in $z$, see assumptions (H3) and (H3') in [@FanHuTang2020CRM] for more details.* *Proof of [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"}.* (i) We first consider the case when the generator $g$ satisfies [\[eq:3.7\]](#eq:3.7){reference-type="eqref" reference="eq:3.7"} in assumption (UN3), and ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $$\label{eq:3.9} g(t,Y'_t,Z'_t)\leq g'(t,Y'_t,Z'_t).$$ The $\theta$-technique put forward initially in @BriandHu2008PTRF will be used in the following argument. For each fixed $\theta\in (0,1)$, define $$\label{eq:3.10} \delta_\theta U:=\frac{Y-\theta Y'}{1-\theta}\ \ {\rm and} \ \ \delta_\theta V:=\frac{Z-\theta Z'}{1-\theta}.$$ Then the pair $(\delta_\theta U_t,\delta_\theta V_t)_{t\in[0,T]}$ verifies the following BSDE: $$\label{eq:3.11} \delta_\theta U_t=\delta_\theta U_T +\int_t^T \delta_\theta g (s,\delta_\theta U_s,\delta_\theta V_s) {\rm d}s-\int_t^T \delta_\theta V_s \cdot {\rm d}B_s, \ \ \ \ t\in[0,T],$$ where ${\rm d}\mathbb{P}\times{\rm d} s-a.e.$, $$\label{eq:3.12} \delta_\theta g (s,\delta_\theta U_s,\delta_\theta V_s):= \frac{(g(s, Y_s, Z_s)-\theta g(s, Y'_s, Z'_s))+\theta (g(s,Y'_s, Z'_s)-g'(s,Y'_s, Z'_s))}{1-\theta}.$$ It follows from the assumptions that ${\rm d}\mathbb{P}\times{\rm d} s-a.e.$, for each $(y,z)\in {\mathbb R}\times {\mathbb R}^d$, $$\label{eq:3.13} {\bf 1}_{\delta_\theta U_s>0} \ \delta_\theta g (s,\delta_\theta U_s,\delta_\theta V_s)\leq \bar f_s+h(s, (\delta_\theta U_s)^+, |\delta_\theta V_s|)$$ with $$\bar f_s:=f_s+\bar\beta (|Y_s|+|Y'_s|).$$ On the other hand, for each $k\geq e$ sufficient large, $c_1\geq 1$ and $c_2\geq (p+1)\beta-4^{\lambda^+}\gamma^2$, define the following function $$\varphi(s,x)=(k+x)\exp\left(c_1\exp(c_2 s) \left(\ln(k+x)\right)^p\right),\ \ (s,x)\in [0,T]\times{\mathbb R}_+.$$ Since $0\leq \varphi(s,x)\leq K x\exp\left(K\left(\ln(e+x)\right)^p\right)$ for each $(s,x)\in [0,T]\times{\mathbb R}_+$ and some positive constant $K>0$ depending only on $(k,T)$, according to the assumptions it is not hard to verify that the process $\{\varphi(t, (\delta_\theta U_t)^+ +\int_0^t \bar f_s {\rm d}s)\}_{t\in[0,T]}$ is of class (D). On the other hand, by a similar analysis to that in (iii) of the proof of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} we can conclude that the last function $\varphi(\cdot,\cdot)$ satisfies [\[eq:2.1\]](#eq:2.1){reference-type="eqref" reference="eq:2.1"}, and hence is a test function for $h$ defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} with $\alpha=1$ and $\lambda\geq 0$. It then follows from [Proposition 10](#pro:3.1){reference-type="ref" reference="pro:3.1"} that $$\label{eq:3.14} (\delta_\theta U_t)^+ \leq \varphi\left(t, (\delta_\theta U_t)^+ +\int_0^t \bar f_s {\rm d}s\right)\leq \mathbb{E}\left[\left.\varphi\left(T, (\delta_\theta U_T)^+ +\int_0^T \bar f_s {\rm d}s \right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].$$ Moreover, since $$\label{eq:3.15} \delta_\theta U_T^+=\frac{(\xi-\theta \xi')^+}{1-\theta}=\frac{\left[\xi-\theta \xi+\theta(\xi-\xi')\right]^+}{1-\theta}\leq \xi^+,$$ it follows that $$(Y_t-\theta Y'_t)^+\leq (1-\theta)\mathbb{E}\left[\left.\varphi\left(T, \xi^+ +\int_0^T \bar f_s {\rm d}s \right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].\vspace{0.2cm}$$ Thus, the desired assertion follows by sending $\theta$ to $1$ in the last inequality. For the case that [\[eq:3.9\]](#eq:3.9){reference-type="eqref" reference="eq:3.9"} holds and the generator $g$ satisfies [\[eq:3.8\]](#eq:3.8){reference-type="eqref" reference="eq:3.8"}, we need to use $\theta Y-Y'$ and $\theta Z-Z'$, respectively, instead of $Y-\theta Y'$ and $Z-\theta Z'$ in [\[eq:3.10\]](#eq:3.10){reference-type="eqref" reference="eq:3.10"}. In this case, the generator $\delta_\theta g$ in [\[eq:3.11\]](#eq:3.11){reference-type="eqref" reference="eq:3.11"} and [\[eq:3.12\]](#eq:3.12){reference-type="eqref" reference="eq:3.12"} should be $$\delta_\theta g (s,\delta_\theta U_s,\delta_\theta V_s):= \frac{(\theta g(s, Y_s, Z_s)-g(s,Y'_s, Z'_s))+(g(s,Y'_s, Z'_s)-g'(s,Y'_s, Z'_s))}{1-\theta}.$$ It follows from [\[eq:3.8\]](#eq:3.8){reference-type="eqref" reference="eq:3.8"} that the generator $\delta_\theta g$ still satisfies [\[eq:3.13\]](#eq:3.13){reference-type="eqref" reference="eq:3.13"}. Consequently, [\[eq:3.14\]](#eq:3.14){reference-type="eqref" reference="eq:3.14"} still holds. Moreover, by using $$\delta_\theta U_T^+=\frac{(\theta \xi-\xi')^+}{1-\theta}=\frac{\left[\theta \xi- \xi+(\xi-\xi')\right]^+}{1-\theta}\leq (-\xi)^+=\xi^-$$ instead of [\[eq:3.15\]](#eq:3.15){reference-type="eqref" reference="eq:3.15"}, by virtue of [\[eq:3.14\]](#eq:3.14){reference-type="eqref" reference="eq:3.14"} we deduce that $$(\theta Y_t-Y'_t)^+\leq (1-\theta)\mathbb{E}\left[\left.\varphi\left(T, \xi^- +\int_0^T \bar f_s {\rm d}s \right)\right|\mathcal{F}_t\right],\ \ t\in[0,T].$$ Thus, the desired assertion follows by sending $\theta$ to $1$ in the last inequality. Finally, in the same way we can prove the desired assertion under the conditions that the generator $g'$ satisfies assumption (UN3) and ${\rm d}\mathbb{P}\times{\rm d} t-a.e.$, $g(t,Y_t,Z_t)\leq g'(t,Y_t,Z_t)$. \(ii\) The desired assertion can be proved in the same way as in (i). The only difference lies in that the test function used in (i) needs to be replaced with those used, respectively, in (i) and (ii) of the proof of [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"} for two different cases of $\alpha=2$ and $\alpha\in (1,2)$. ◻ **Remark 20**. *Both assertions (ii) and (iii) of [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} seem to be new, and the key idea in the proof of Theorems [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} and [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"} can be dated back to [@BriandHu2006PTRF; @Fan2016SPA; @FanHu2021SPA; @FanHuTang2023SCL].* ## Existence and uniqueness Using Theorems [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"}, [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"}, [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} and [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"}, we easily have the following two existence and uniqueness results, whose proofs are omitted here. **Theorem 21**. *Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies (EX1)-(EX3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\alpha=1$. Then, the following assertions hold.* *(i) Let $\delta=0$ and $\lambda\in (-\infty,-{1\over 2})$. If $\xi+\int_0^T f_s{\rm d}s\in L^1$ and the generator $g$ further satisfies (UN1) and (UN2), then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(Y_t)_{t\in[0,T]}$ is of class (D);* *(ii) Let $\delta=0$ and $\lambda=-{1\over 2}$. If $\xi+\int_0^T f_s{\rm d}s\in L(\ln L)^p$ for some $p>0$ and the generator $g$ further satisfies (UN1) and (UN2), then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|(\ln (e+|Y_t|))^p)_{t\in[0,T]}$ is of class (D);* *(iii) Let $p:=\delta\vee (\lambda+{1\over 2})\vee (2\lambda)\in (0,+\infty)$. If $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0} L\exp[\mu(\ln L)^p]$ and the generator $g$ further satisfies (UN1) and (UN2) for the case of $\lambda\in (-{1\over 2}, 0]$ and (UN3) for the case of $\lambda\in [0,+\infty)$, then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(|Y_t|\exp (\mu(\ln (e+|Y_t|))^p))_{t\in[0,T]}$ is of class (D) for each $\mu>0$;* *(iv) Let $\delta=0$ and $\lambda=0$. If $\xi+\int_0^T f_s{\rm d}s\in L^p$ for some $p>1$ and $g$ further satisfies (UN1) and (UN2), then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(|Y_t|^p)_{t\in[0,T]}$ is of class (D).* **Theorem 22**. *Assume that $\xi$ is an $\mathcal{F}_T$-measurable random variable and the generator $g$ satisfies assumptions (EX2), (EX3) and (UN3) with $h(\cdot,\cdot,\cdot)$ being defined in [\[eq:2.7\]](#eq:2.7){reference-type="eqref" reference="eq:2.7"} for $\delta=0$, $\lambda=0$ and $\alpha\in (1,2]$. If $\xi+\int_0^T f_s{\rm d}s\in \cap_{\mu>0}\exp(\mu L^{2\over \alpha^*})$ with $\alpha^*$ being the conjugate of $\alpha$, then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that the process $(\exp(\mu |Y_t|^{2\over \alpha^*}))_{t\in[0,T]}$ is of class (D) for each $\mu>0$. In particular, if $\xi+\int_0^T f_s{\rm d}s\in L^{\infty}(\mathcal{F}_T)$, then BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in [0,T]}$ such that $(Y_t)_{t\in[0,T]}\in \mathcal{S}^{\infty}([0,T];{\mathbb R})$.* # Applications {#sec:4-Application} In this section, we will introduce some applications of our theoretical results obtained in the last two sections, which are enlightened by for example [@Peng1990SICON; @Peng1991Stochastics; @Peng1997Book; @BriandHu2008PTRF; @Jia2008PHDThesis; @FanHu2021SPA; @FanHuTang2023SCL; @FanHuTang2023SPA]. ## The (conditional) $g$-expectation defined on $L^1(\mathcal{F}_T)$ First of all, we extend the notion of (conditional) $g$-expectation of [@Peng1997Book] defined on the space $L^2(\mathcal{F}_T)$ of squarely integrable random variables to the larger one $L^1(\mathcal{F}_T)$ of integrable random variables. **Definition 23**. *Let the generator $g$ satisfy assumptions (EX1)-(EX2) and (UN1)-(UN2) with $\lambda\in (-\infty,-{1\over 2})$ and $\int_0^T f_s {\rm d}s\in L^1$. Assume further that $g$ satisfies the following assumption: $$\label{eq:4.1} {\rm d}\mathbb{P}\times{\rm d} t-a.e.,\ \ \ g(\omega,t,y,0)\equiv 0,\ \ \forall y\in {\mathbb R}.$$ By virtue of (i) in [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"}, for each $\xi\in L^1(\mathcal{F}_T)$ and $t\in[0,T]$, we can denote the conditional $g$-expectation $\mathcal{E}_g[\xi|\mathcal{F}_t]$ of $\xi$ with respect to $\mathcal{F}_t$ by the following formula: $$\label{eq:4.2} \mathcal{E}_g[\xi|\mathcal{F}_t]:=Y_t^\xi,$$ where $(Y_t^\xi,Z_t^\xi)_{t\in[0,T]}$ is the unique solution of BSDE$(\xi,g)$ such that $Y_\cdot^\xi$ belongs to class (D). In particular, we call $\mathcal{E}_g[\xi]:=\mathcal{E}_g[\xi|\mathcal{F}_0]$ the $g$-expectation of $\xi$.* It is clear that the conditional $g$-expectation operator $\mathcal{E}_g[\cdot|\mathcal{F}_t]$ defined by [\[eq:4.2\]](#eq:4.2){reference-type="eqref" reference="eq:4.2"} maps $L^1(\mathcal{F}_T)$ to $L^1(\mathcal{F}_t)$ for each $t\in[0,T]$, which shares the same domain with the classical mathematical expectation operator. Furthermore, proceeding identically as in [@Peng1999PTRF] and [@Jiang2008AAP], from (i) of [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} and (i) of [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"} together with [\[eq:4.1\]](#eq:4.1){reference-type="eqref" reference="eq:4.1"} we easily (thus omitting the proof) have the following two propositions. **Proposition 24**. *$\mathcal{E}_g[\cdot]$ possesses the following properties:* - ***Preserving of constants**: For each constant $c\in{\mathbb R}$, $\mathcal{E}_g[c]=c$;* - ***Monotonicity**: For each $\xi_1,\xi_2\in L^1(\mathcal{F}_T)$, if $\xi_1\geq \xi_2$, then $\mathcal{E}_g[\xi_1]\geq \mathcal{E}_g[\xi_2]$.* **Proposition 25**. *For each $t\in[0,T]$, $\mathcal{E}_g[\cdot|\mathcal{F}_t]$ possesses the following properties:* - *If $\xi\in L^1(\mathcal{F}_t)$, then $\mathcal{E}_g[\xi|\mathcal{F}_t]=\xi$;* - *For each $\xi_1,\xi_2\in L^1(\mathcal{F}_T)$, if $\xi_1\geq \xi_2$, then $\mathcal{E}_g[\xi_1|\mathcal{F}_t]\geq \mathcal{E}_g[\xi_2|\mathcal{F}_t]$;* - *For each $\xi\in L^1(\mathcal{F}_T)$ and $r\in [0,T]$, we have $\mathcal{E}_g[\mathcal{E}_g[\xi|\mathcal{F}_t]|\mathcal{F}_r]=\mathcal{E}_g[\xi|\mathcal{F}_{t\wedge r}]$;* - *For each $A\in \mathcal{F}_t$ and $\xi\in L^1(\mathcal{F}_T)$, $\mathcal{E}_g[{\bf 1}_A\xi|\mathcal{F}_t]={\bf 1}_A\mathcal{E}_g[\xi|\mathcal{F}_t]$ and $\mathcal{E}_g[{\bf 1}_A\xi]=\mathcal{E}_g[{\bf 1}_A\mathcal{E}_g[\xi|\mathcal{F}_t]]$.* It can be indicated from both propositions that the (conditional) $g$-expectation preserves essential properties (but except linearity) of the classical expectations. Some extensive issues on the (conditional) $g$-expectation still remain to be further studied along the lines of [@Peng1997Book; @Chen1998CRAS; @Peng1999PTRF; @CoquetHuMeminPeng2002PTRF; @Peng2004AMAS; @Peng2004CRAS; @Peng2004LectureNotes; @Jiang2008AAP; @Delbaen2009MF]. **Remark 26**. *In the same way as above, by [\[thm:3.3,thm:3.6,thm:3.12,thm:3.13\]](#thm:3.3,thm:3.6,thm:3.12,thm:3.13){reference-type="ref" reference="thm:3.3,thm:3.6,thm:3.12,thm:3.13"} one can define the (conditional) $g$-expectation via the solutions of BSDEs on the spaces $$L(\ln L)^p\ (p>0),\ \ \bigcap\limits_{\mu>0}L\exp[\mu(\ln L)^p]\ (p>0),\ \ L^p\ (p>1), \ \ \bigcap\limits_{\mu>0} \exp(\mu L^{2\over \alpha^*})\ (\alpha^*\geq 2)\ \ {\rm and}\ \ L^\infty$$ respectively. It is clear that the generator $g$ of BSDEs needs to satisfy some stronger conditions as the space becomes larger. In particular, when $g(t,y,z):\equiv \gamma |z|$, the corresponding conditional $g$-expectation $\mathcal{E}_g[\cdot|\mathcal{F}_t]$ for $t\in [0,T]$ can be defined on the space $\bigcap\limits_{\mu>0}L\exp[\mu\sqrt{\ln L}]$, which is bigger than $L^p\ (p>1)$ used for example in [@Peng1997Book; @Chen1998CRAS; @CoquetHuMeminPeng2002PTRF; @Peng2004AMAS; @Peng2004CRAS; @Peng2004LectureNotes; @Tang2006CRA; @Jia2008PHDThesis; @Jia2010SPA]. Furthermore, according to (iii) of [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} and (iii) of [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"}, we can verify that for each $t\in [0,T]$ and $\xi\in \bigcap\limits_{\mu>0}L\exp[\mu\sqrt{\ln L}]$, $$\mathcal{E}_g[\xi|\mathcal{F}_t]:={\rm ess} \sup\limits_{q\in \mathcal{A}} \mathbb{E}_q[\xi|F_t]$$ with $\mathcal{A}$ being defined in [Example 5](#ex:2.1){reference-type="ref" reference="ex:2.1"}. This is just the maximal conditional expectation on $\mathcal{A}$.* ## Dynamic utility process and risk measure In the sequel, we introduce an application of our theoretical results in mathematical finance. For simplicity of notations, we set for each $t\in [0,T]$ and $\alpha\in (1,2]$ $$E^\alpha(\mathcal{F}_t):=\bigcap_{\mu>0} \exp[\mu L^{2\over \alpha^*}](\mathcal{F}_t)$$ with $\alpha^*:={\alpha\over \alpha-1}\geq 2$ being the conjugate of $\alpha$. Clearly, $E^\alpha(\mathcal{F}_t)$ is a linear space containing $L^\infty(\mathcal{F}_t)$ of bounded random variables. The following proposition is a direct consequence of (iii) of [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"}, and the proof is omitted. **Proposition 27**. *Suppose that the generator $g(z):{\mathbb R}^d\rightarrow{\mathbb R}$ is a concave function satisfying $g(0)=0$ and $$\label{eq:4.3} |g(z)|\leq a+\gamma |z|^\alpha$$ with $a\geq0$ and $\alpha\in (1,2]$ being two given constants. Then, for each $\xi\in E^\alpha(\mathcal{F}_T)$, BSDE$(\xi,g)$ admits a unique solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $Y_t\in E^\alpha(\mathcal{F}_t)$ for each $t\in [0,T]$.* Now, by virtue of [Proposition 27](#pro:4.4){reference-type="ref" reference="pro:4.4"}, for each $\xi\in E^\alpha(\mathcal{F}_T)$ we can define $$\label{eq:4.4} U_t^g(\xi):=Y_t^\xi,\ \ t\in [0,T],$$ where $(Y_t^\xi,Z_t^\xi)_{t\in[0,T]}$ is the unique solution of BSDE$(\xi,g)$ such that $Y_t^\xi\in E^\alpha(\mathcal{F}_t)$ for each $t\in [0,T]$. The following theorem indicates that the family of operators $\{U_t^g(\cdot)\}_{t\in [0,T]}$ defined via [\[eq:4.4\]](#eq:4.4){reference-type="eqref" reference="eq:4.4"} constitutes a dynamic utility process defined on $E^\alpha(\mathcal{F}_T)$. **Theorem 28**. *For each $t\in[0,T]$, the mapping $U_t^g(\cdot):E^\alpha(\mathcal{F}_T)\rightarrow E^\alpha(\mathcal{F}_t)$ defined via [\[eq:4.4\]](#eq:4.4){reference-type="eqref" reference="eq:4.4"} satisfies the following properties:* *(i) **Positivity**: $U_t^g(0)=0$ and $U_t^g(\xi)\geq 0$ for each nonnegative random variable $\xi\in E^\alpha(\mathcal{F}_T)$;* *(ii) **Monotonicity**: for each $\xi,\eta\in E^\alpha(\mathcal{F}_T)$, if $\xi\geq \eta$, then $U_t^g(\xi)\geq U_t^g(\eta)$;* *(iii) **Monetary**: $U_t^g(\xi+\eta)=U_t^g(\xi)+\eta$ for each $\xi\in E^\alpha(\mathcal{F}_T)$ and $\eta\in E^\alpha(\mathcal{F}_t)$;* *(iv) **Concavity**: $U_t^g(\theta\xi+(1-\theta)\eta)\geq \theta U_t^g(\xi)+(1-\theta)U_t^g(\eta)$ for all $\xi,\eta\in E^\alpha(\mathcal{F}_T)$ and $\theta\in (0,1)$.* *Proof.* In view of $g(0)=0$ and the fact that $g$ is independent of $y$, (i)-(iii) are the direct consequences of (iii) of [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"} and (i) of [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"}. Furthermore, proceeding identically as Proposition 3.5 in [@ElKarouiPengQuenez1997MF], by virtue of (i) of [Theorem 18](#thm:3.6){reference-type="ref" reference="thm:3.6"} and the concavity of $g$ we can get (iv). ◻ Now, we let the function $f:{\mathbb R}^d\rightarrow{\mathbb R}_+$ be convex, satisfy $f(0)=0$, and $\liminf_{|x|\rightarrow\infty} f(x)/|x|^2>0$. For each $\xi\in L^\infty(\mathcal{F}_T)$, define $$\label{eq:4.5} \bar U_t(\xi):={\rm essinf}\left\{\left.\mathbb{E}_q\left[\left. \xi+\int_t^T f(q_u){\rm d}u\right|\mathcal{F}_t\right]\right| {\mathbb Q}^q\sim{\mathbb P}\right\},\ \ t\in [0,T],$$ where $\mathbb{E}_q[\cdot|\mathcal{F}_t]$ is the conditional expectation operator with respect to $\mathcal{F}_t$ under the probability measure ${\mathbb Q}^q$, which is equivalent to ${\mathbb P}$ and $$\mathbb{E}\left[\left.\frac{{\rm d}{\mathbb Q}^q}{{\rm d}{\mathbb P}}\right|\mathcal{F}_t\right] =\exp\left\{\int_0^t q_u\cdot {\rm d}B_u-\frac{1}{2}\int_0^t |q_u|^2{\rm d}u\right\},\ \ t\in[0,T].$$ It is not difficult to check that $\{\bar U_t(\cdot)\}_{t\in [0,T]}$ defined via [\[eq:4.5\]](#eq:4.5){reference-type="eqref" reference="eq:4.5"} constitutes a dynamic utility process defined on $L^\infty(\mathcal{F}_T)$. And, it follows from Theorems 2.1-2.2 in [@DelbaenHuBao2011PTRF] that there exists a $(Z_t)_{t\in[0,T]}\in \mathcal{M}^2$ such that $(\bar U_t(\xi), Z_t)_{t\in[0,T]}$ is the unique bounded solution of the following BSDE $$\label{eq:2-54} Y_t=\xi+\int_t^T g(Z_s){\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in [0,T],$$ where $$\label{eq:2-55} g(z):=-\sup_{x\in {\mathbb R}^d}(-z\cdot x-f(x))=\inf_{x\in {\mathbb R}^d}(z\cdot x+f(x))\leq 0,\ \ \forall z\in {\mathbb R}^d\vspace{0.1cm}$$ is a concave function with $g(0)=0$, and $\limsup_{|x|\rightarrow\infty} |g(x)|/|x|^2<\infty$. For example, if $c>0$ and $$f(x)=c|x|^{\alpha^*}\geq 0,\ \ x\in {\mathbb R}^d,$$ then $$0\geq g(z)=-\frac{1}{c^{\alpha-1}}\frac{\alpha^*-1}{(\alpha^*)^\alpha}|z|^\alpha,\ \ z\in {\mathbb R}^d,\vspace{0.1cm}$$ which means that $g$ satisfies the conditions in [Proposition 27](#pro:4.4){reference-type="ref" reference="pro:4.4"}. **Remark 29**. *It is well known that [\[eq:4.5\]](#eq:4.5){reference-type="eqref" reference="eq:4.5"} is usually used to define the utility of bounded endowments in mathematical finance, see for example [@DelbaenHuBao2011PTRF] for details. However, it is only defined on the space $L^\infty(\mathcal{F}_T)$. This motivates the definition [\[eq:4.4\]](#eq:4.4){reference-type="eqref" reference="eq:4.4"} via a BSDE, which can be defined on a larger space $E^\alpha(\mathcal{F}_T)$ than $L^\infty(\mathcal{F}_T)$. Some relevant results are available in [@ElKarouiPengQuenez1997MF; @DelbaenPengRosazza2010FS; @FanHuTang2023SPA]. In a symmetric way to the above, we can also define a convex risk measure on $E^\alpha(\mathcal{F}_T)$, see [@FollmerSchied2002FS; @Jiang2008AAP] among others for more details.* **Example 30**. *Let the generator $$g(z):=c({\bf 1}_{\lambda\geq 0}-{\bf 1}_{\lambda<0})|z|(\ln (e+|z|))^\lambda,\ \ \ z\in {\mathbb R}^d,$$ where $c>0$ and $\lambda\in {\mathbb R}$ are two given constants. From [Theorem 21](#thm:3.12){reference-type="ref" reference="thm:3.12"}, we easily have the following three assertions:* - *If $\lambda\in (-\infty,-\frac{1}{2})$, then for each $\xi\in L^1$, BSDE$(\xi,g)$ admits a unique solution $(Y_t^\xi,Z_t^\xi)_{t\in [0,T]}$ such that $(Y_t^\xi)_{t\in[0,T]}$ is of class (D);* - *If $\lambda=-\frac{1}{2}$, then for each $\xi\in L(\ln L)^p$ with $p>0$, BSDE$(\xi,g)$ admits a unique solution $(Y_t^\xi,Z_t^\xi)_{t\in [0,T]}$ such that the process $(|Y_t^\xi|(\ln (e+|Y_t^\xi|))^p)_{t\in[0,T]}$ is of class (D);* - *If $\lambda\in (-\frac{1}{2},+\infty)$ and $p:=(\lambda+{1\over 2})\vee (2\lambda)$, then for each $\xi\in \cap_{\mu>0} L\exp[\mu(\ln L)^p]$, BSDE$(\xi,g)$ admits a unique solution $(Y_t^\xi,Z_t^\xi)_{t\in [0,T]}$ such that the process $(|Y_t^\xi|\exp (\mu(\ln (e+|Y_t^\xi|))^p))_{t\in[0,T]}$ is of class (D) for each $\mu>0$.* *Thus, for the preceding three different ranges of $\lambda$, we can define the following operator $$\varrho(\xi):=Y_0^{-\xi}$$ in three different spaces of contingent claims: $L^1$, $L(\ln L)^p$ and $\cap_{\mu>0} L\exp[\mu(\ln L)^p]$. Moreover, according to the properties of the generator $g$ together with [\[thm:3.3,thm:3.6,thm:3.12\]](#thm:3.3,thm:3.6,thm:3.12){reference-type="ref" reference="thm:3.3,thm:3.6,thm:3.12"}, in the same spirit as in for example [@Artzner1999MF; @FollmerSchied2002FS; @Jiang2008AAP; @CheriditoLi2009MF; @Delbaen2009MF; @Castagnoli2022OR], we verify that whenever $\lambda>0$, $\lambda<0$ and $\lambda=0$, $\varrho(\cdot)$ is respectively a convex risk measure, a star-shaped risk measure (see the precise definition in page 2641 of [@Castagnoli2022OR Definition 1]) and a coherent risk measure on the corresponding space of contingent claims. In addition, in the same way, we can also define the corresponding dynamic risk measure with the solution $Y_t^{-\xi}$ of BSDE$(\xi,g)$ for $t\in [0,T]$.* ## Nonlinear Feynman-Kac formula As another application of our theoretical results, in this subsection we will derive a nonlinear Feynman-Kac formula for PDEs which are at most quadratic with respect to the gradient of the solution. Let us consider the following semi-linear PDE: $$\label{eq:4.8} \partial_t u(t,x)+\mathcal{L} u(t,x)+g(t,x,u(t,x),\sigma^* \nabla_x u(t,x))=0,\ \ \ u(T,\cdot)=h(\cdot),$$ where $\mathcal{L}$ is the infinitesimal generator of the solution $X^{t,x}_\cdot$ to the following SDE: $$\label{eq:4.9} X^{t,x}_s=x+\int_t^s b(r,X^{t,x}_r){\rm d}r+\int_t^s \sigma(r,X^{t,x}_r){\rm d}B_r,\quad t\leq s\leq T.$$ For each $(t_0,x_0)\in [0,T]\times {\mathbb R}^n$, $(Y^{t_0,x_0}_t,Z^{t_0,x_0}_t)_{t\in [t_0,T]}$ is the solution to the BSDE $$\label{eq:4.11} Y_t=h\left(X^{t_0,x_0}_T\right)+\int_t^T g(s,X^{t_0,x_0}_s,Y_s,Z_s){\rm d}s+\int_t^T Z_s\cdot {\rm d}B_s,\quad t\in [t_0,T],$$ The nonlinear Feynman-Kac formula says that the function $$\label{eq:4.10} u(t,x):=Y^{t,x}_t, \quad \forall\ (t,x)\in [0,T]\times {\mathbb R}^n,$$ is a viscosity solution to PDE [\[eq:4.8\]](#eq:4.8){reference-type="eqref" reference="eq:4.8"}. Let us first recall the definition of a continuous viscosity solution in our framework, see e.g. [@CrandallIshiiLions1992BAMS]. **Definition 31**. *A continuous function $u:[0,T]\times {\mathbb R}^n\rightarrow{\mathbb R}$ with $u(T,\cdot)=h(\cdot)$ is said to be a viscosity super-solution (resp. sub-solution) to PDE [\[eq:4.8\]](#eq:4.8){reference-type="eqref" reference="eq:4.8"} if the inequality $$\partial_t u(t_0,x_0)+\mathcal{L} u(t_0,x_0)+g(t_0,x_0,u(t_0,x_0),\sigma^* \nabla_x \varphi(t_0,x_0))\leq 0\ \ \ ({\rm resp.}\ \ \geq 0)$$ holds true for any smooth function $\varphi(\cdot,\cdot)$ such that the function $u-\varphi$ attains a local minimum (resp. maximum) at the point $(t_0,x_0)\in (0,T)\times {\mathbb R}^n$. Moreover, a viscosity super-solution is said to be a viscosity solution if it is also a viscosity sub-solution.* Let us now introduce the following assumptions on the coefficients of SDE [\[eq:4.9\]](#eq:4.9){reference-type="eqref" reference="eq:4.9"}. 1. [\[A1\]]{#A1 label="A1"} Both functions $b:[0,T]\times {\mathbb R}^n\rightarrow{\mathbb R}^n$ and $\sigma:[0,T]\times {\mathbb R}^n\rightarrow{\mathbb R}^{n\times d}$ are jointly continuous and there is a positive constant $K>0$ such that for each $(t,x,x')\in [0,T]\times{\mathbb R}^n\times{\mathbb R}^n$, $$|b(t,0)|+|\sigma(t,x)|\leq K\ \ \ \ {\rm and}\ \ \ \ |b(t,x)-b(t,x')|+|\sigma(t,x)-\sigma(t,x')|\leq K|x-x'|.$$ Classical results on SDEs show that under the assumption [\[A1\]](#A1){reference-type="ref" reference="A1"}, SDE [\[eq:4.9\]](#eq:4.9){reference-type="eqref" reference="eq:4.9"} has a unique solution $X^{t,x}_\cdot\in \cap_{q\geq 1}\mathcal{S}^q$ for each $(t,x)\in [0,T]\times {\mathbb R}^n$. And, since $\sigma$ is bounded, the argument in page 563 of @BriandHu2008PTRF yields that for each $\mu>0$, there is a constant $C>0$, depending only on $(q,\mu,T,K)$, such that for each $q\in [1,2)$ and $(t,x)\in [0,T]\times {\mathbb R}^n$, $$\label{eq:4.12} \mathbb{E}\left[\sup_{s\in [t,T]} \exp\left(\mu |X^{t,x}_s|^q\right)\right]\leq C \exp(\mu C |x|^q).$$ Let us further give our assumptions on the generator $g$ and the terminal condition of BSDE [\[eq:4.11\]](#eq:4.11){reference-type="eqref" reference="eq:4.11"}. 1. [\[A2\]]{#A2 label="A2"} Both functions $g:[0,T]\times {\mathbb R}^n\times{\mathbb R}\times{\mathbb R}^d\rightarrow{\mathbb R}$ and $h:{\mathbb R}^n\rightarrow{\mathbb R}$ are jointly continuous and there are three real constants $k\geq 0$, $\alpha\in (1,2]$ and $p\in [1,\alpha^*)$ with $\alpha^*$ being the conjugate of $\alpha$ such that for each $(t,x,y,z)\in [0,T]\times {\mathbb R}^n\times {\mathbb R}\times{\mathbb R}^d$, $$\label{eq:4.13} {\rm sgn}(y)g(t,x,y,z)\leq k \left(1+|x|^p+|y|+|z|^\alpha\right),$$ $$\label{eq:4.14} |g(t,x,y,z)|+|h(x)|\leq k\left(1+|x|^p+\exp(k |y|^{2\over \alpha^*})+|z|^2\right)$$ and either inequality $$\begin{array}{c} \displaystyle{\bf 1}_{y-\theta y'>0}\left(g(t,x,y,z)-\theta g(t,x, y',z')\right)\leq \displaystyle(1-\theta) k \left(1+|x|^p+|y'|+\left(\frac{y-\theta y'}{1-\theta}\right)^+ +\left|\frac{z-\theta z'}{1-\theta}\right|^\alpha\right) \end{array}$$ or $$\begin{array}{c} \displaystyle-{\bf 1}_{y-\theta y'<0}\left(g(t,x,y,z)-\theta g(t,x, y',z')\right)\leq \displaystyle(1-\theta) k\left(1+|x|^p+|y'|+\left(\frac{y-\theta y'}{1-\theta}\right)^- +\left|\frac{z-\theta z'}{1-\theta}\right|^\alpha\right)\vspace{0.3cm} \end{array}$$ is satisfied for $\forall(t,x,y,y',z,z')\in [0,T]\times {\mathbb R}^n\times {\mathbb R}\times {\mathbb R}\times{\mathbb R}^d\times{\mathbb R}^d$ and $\theta\in (0,1)$. Since $p\in [1,\alpha^*)$, we have from [\[eq:4.12\]](#eq:4.12){reference-type="eqref" reference="eq:4.12"} that for each $(t_0,x_0)\in [0,T]\times{\mathbb R}^n$ and each $\mu>0$, $$\displaystyle\mathbb{E}\left[\exp\left(\mu \left(|X^{t_0,x_0}_T|^p\right)^{2\over \alpha^*}\right)\right] \leq \displaystyle\mathbb{E}\left[\sup_{s\in [t_0,T]} \exp\left(\mu |X^{t_0,x_0}_s|^{2p\over \alpha^*}\right)\right]\leq \displaystyle\bar C \exp(\mu \bar C |x_0|^{2p\over \alpha^*})<+\infty$$ and $$\mathbb{E}\left[\exp\left(\mu \left(\int_{t_0}^T |X^{t_0,x_0}_s|^p{\rm d}s\right)^{2\over \alpha^*}\right)\right] \leq\displaystyle\mathbb{E}\left[\sup_{s\in [t_0,T]} \exp\left(\mu T^{2\over \alpha^*} |X^{t_0,x_0}_s|^{2p\over \alpha^*}\right)\right]<+\infty,\vspace{0.1cm}$$ for a constant $\bar C>0$ depending only on $(p,\alpha,\mu,T,K)$. Then, in view of the assumption [\[A2\]](#A2){reference-type="ref" reference="A2"} and the last two inequalities, we can apply [Theorem 22](#thm:3.13){reference-type="ref" reference="thm:3.13"} to construct a unique solution $(Y^{t_0,x_0}_t,Z^{t_0,x_0}_t)_{t\in [t_0,T]}$ to BSDE [\[eq:4.11\]](#eq:4.11){reference-type="eqref" reference="eq:4.11"} such that $(\exp(\mu |Y^{t_0,x_0}_t|^{2\over \alpha^*}) )_{t\in [t_0,T]}$ is of class (D) for each $\mu>0$. Furthermore, a classical argument yields that the function $u$ defined by [\[eq:4.10\]](#eq:4.10){reference-type="eqref" reference="eq:4.10"} is deterministic. The following theorem constitutes the main result of this subsection. **Theorem 32**. *Let assumptions [\[A1\]](#A1){reference-type="ref" reference="A1"} and [\[A2\]](#A2){reference-type="ref" reference="A2"} hold. Then, the function $u$ defined in [\[eq:4.10\]](#eq:4.10){reference-type="eqref" reference="eq:4.10"} is continuous on $[0,T]\times{\mathbb R}^n$ and there exists a constant $C>0$ such that $$\forall\ (t,x)\in [0,T]\times{\mathbb R}^n,\ \ \ |u(t,x)|\leq C(1+|x|^p).$$ Moreover, $u$ is a viscosity solution to PDE [\[eq:4.8\]](#eq:4.8){reference-type="eqref" reference="eq:4.8"}.* The proof is available in [@FanHu2021SPA; @BriandHu2008PTRF]. Similar results can also be found in [@DalioLey2006SCON; @DalioLey2011AMO]. **Remark 33**. *The nonlinear Feynman-Kac formula for solution of PDEs can be dated back to [@Peng1991Stochastics], in the spirit of which numerical discussions appeared successively in for example [@Peng1992Stochastics; @Peng1993AMO; @Pardoux1999Nonlinear; @PardouxTang1999PTRF; @Kobylanski2000AP; @Jia2008PHDThesis; @BriandHu2008PTRF; @PardouxRascanu2014Book; @FanHu2021SPA]. In fact, according to [\[thm:3.3,thm:3.6,thm:3.12,thm:3.13\]](#thm:3.3,thm:3.6,thm:3.12,thm:3.13){reference-type="ref" reference="thm:3.3,thm:3.6,thm:3.12,thm:3.13"}, we can establish a one-to-one correspondence between solutions of PDEs and BSDEs via the Feynman-Kac formulas, as mentioned in section 4 of [@Jia2008PHDThesis]. In general, the generator $g$ and the terminal condition $h$ of BSDE [\[eq:4.11\]](#eq:4.11){reference-type="eqref" reference="eq:4.11"} can admit a more general growth in the unknown variable $x$ when $g$ has a lower growth in the unknown variable $z$.* # Open problems {#sec:5-Problems} In this section, we describe five open problems. The first two concern the existence of a solution to a BSDE and a PDE, and the last three address the uniqueness. **Problem 5.1.** Consider the following BSDE: $$\label{eq:5.1} Y_t=\xi+\int_t^T \frac{|Z_s|}{\sqrt{\ln(e+|Z_s|)}} {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ For $\xi\in L^1$, does BSDE [\[eq:5.1\]](#eq:5.1){reference-type="eqref" reference="eq:5.1"} admit a solution $(Y_t,Z_t)_{t\in[0,T]}$ such that $Y$ is of class (D)? Note that Assertion (ii) of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} concerns existence for $\xi\in L(\ln L)^p$ only when $p>0$, while Assertion (i) of [Theorem 3](#thm:2.3){reference-type="ref" reference="thm:2.3"} concerns the BSDE of a slower growing generator. **Problem 5.2.** Let assumption [\[A1\]](#A1){reference-type="ref" reference="A1"} be satisfied and that the functions $g$ and $h$ only satisfy [\[eq:4.13\]](#eq:4.13){reference-type="eqref" reference="eq:4.13"} and [\[eq:4.14\]](#eq:4.14){reference-type="eqref" reference="eq:4.14"} of assumption [\[A2\]](#A2){reference-type="ref" reference="A2"} in the last section. Does the semi-linear PDE [\[eq:4.8\]](#eq:4.8){reference-type="eqref" reference="eq:4.8"} admit a viscosity solution? Note that the PDE is associated to a BSDE [\[eq:4.11\]](#eq:4.11){reference-type="eqref" reference="eq:4.11"}, which has (from [Theorem 7](#thm:2.6){reference-type="ref" reference="thm:2.6"}) a solution $(Y^{t_0,x_0}_t,Z^{t_0,x_0}_t)_{t\in [t_0,T]}$ such that $(\exp(\mu |Y^{t_0,x_0}_t|^{2\over \alpha^*}) )_{t\in [t_0,T]}$ is of class (D) for each $\mu>0$. **Problem 5.3.** Let $p>1$ and consider the following BSDE: $$Y_t=\xi+\int_t^T |Z_s|\sin|Z_s| {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T].$$ For each $\xi\in L^p$, is the solution $(Y_t,Z_t)_{t\in[0,T]}$ to the last BSDE with $(|Y_t|^p)_{t\in[0,T]}$ being of class (D) unique? We note that the generator $g$ is not uniformly continuous in $z$, while the uniqueness assertion (iv) of [Theorem 12](#thm:3.3){reference-type="ref" reference="thm:3.3"} requires the uniform continuity of the generator $g$ in $z$. **Problem 5.4.** Let $d=2$ and consider the following BSDE: $$\label{eq:5.4} Y_t=\xi+\int_t^T \frac{1}{2}|Z_s^1|^2\ {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T],$$ where $Z:=(Z^1, Z^2)$. For each $\xi\in \exp(L)$, is the solution $(Y_t,Z_t)_{t\in[0,T]}$ to the last BSDE with $(\exp(|Y_t|))_{t\in[0,T]}$ being of class (D) unique? Note that the generator $g$ is only convex, rather than strongly convex (see assumption B in page 5276 of [@DelbaenHuRichou2015DCDS] for the precise definition) in $z$. Whenever the quadratic generator $g$ is strongly convex in $z$, Theorem 4.1 of [@DelbaenHuRichou2015DCDS] gives the uniqueness of the solution to BSDE$(\xi,g)$ for $\xi\in \exp(L)$. While for a terminal value $\xi\in \exp(pL)$ with $p>1$, Theorem 3.3 of [@DelbaenHuRichou2011AIHPPS] gives the uniqueness of the solution $(Y_t,Z_t)_{t\in[0,T]}$ to BSDE [\[eq:5.4\]](#eq:5.4){reference-type="eqref" reference="eq:5.4"} such that $(\exp(p|Y_t|))_{t\in[0,T]}$ is of class (D). **Problem 5.5.** Let $d=2$ and consider the following BSDE: $$\label{eq:5.5} Y_t=\xi+\int_t^T (|Z_s^1|^2-|Z_s^2|^2)\ {\rm d}s-\int_t^T Z_s\cdot {\rm d}B_s,\ \ t\in[0,T],$$ where $Z:=(Z^1, Z^2)$. For each $\xi\in \cap_{\mu>0}\exp(\mu L)$, is the solution $(Y_t,Z_t)_{t\in[0,T]}$ to BSDE [\[eq:5.5\]](#eq:5.5){reference-type="eqref" reference="eq:5.5"} with $(\exp(\mu |Y_t|))_{t\in[0,T]}$ being of class (D) for each $\mu>0$ unique? We note that the generator $g$ is neither convex nor concave in $z$, but satisfies [\[eq:3.12\*\]](#eq:3.12*){reference-type="eqref" reference="eq:3.12*"} with $\delta=1$ and $\gamma=2$. When $\xi\in L^\infty$, it is well known from  [@Kobylanski2000AP] that the solution $(Y_t,Z_t)_{t\in[0,T]}$ to BSDE [\[eq:5.5\]](#eq:5.5){reference-type="eqref" reference="eq:5.5"} with $Y\in \mathcal{S}^\infty$ is unique. For $\xi\in \cap_{\mu>0}\exp(\mu L)$, if the quadratic generator $g$ has a strictly positive (or negative) quadratic growth and an extended convexity (or concavity) in $z$, Theorem 5 together with Remark 7 of [@FanHuTang2020CRM] already give the uniqueness of the solution $(Y_t,Z_t)_{t\in[0,T]}$ to BSDE$(\xi,g)$ such that $(\exp(\mu |Y_t|))_{t\in[0,T]}$ is of class (D) for each $\mu>0$. # Proof of [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"} {#proof-of-pro2.5} *Proof of [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"}.* The case of $\lambda=0$ is clear. Let us consider the case of $\lambda\neq 0$. Given $k\geq k_{\lambda,p}$ with $\bar p:=p^{1\over 2|\lambda|}>1$, $\bar\lambda:=2|\lambda|(2+|\lambda-1|)$, $k_{\lambda,p}\geq e^{|\lambda-1|+1}$, $$\label{eq:2.8+} \bar\lambda \left(\ln k_{\lambda,p}\right)^{\lambda-1}< k_{\lambda,p}^{1-\frac{1}{\bar p}},\ \ k_{\lambda,p}^{\bar p}-k_{\lambda,p}-2\sqrt{p} k_{\lambda,p}(\ln k_{\lambda,p})^\lambda>0 \ \ {\rm and}\ \ k_{\lambda,p}^{1\over \bar p}<\sqrt{p} k_{\lambda,p}(\ln k_{\lambda,p})^\lambda.$$ For each $(x,y)\in (0,+\infty)\times (0,+\infty)$, define the function $$\label{eq:A-5} \begin{array}{lrl} f(x,y)&:=&\displaystyle y^2-2xy\left(\ln (k+y)\right)^\lambda +px^2\left(\ln (k+x)\right)^{2\lambda}\\ &=&\displaystyle\left(y- x\left(\ln (k+y)\right)^\lambda\right)^2+px^2\left(\ln (k+x)\right)^{2\lambda} -x^2\left(\ln (k+y)\right)^{2\lambda}. \end{array}$$ Clearly, in order to prove [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"}, it is enough to prove that $f(x,y)\geq 0$ for each $x,y>0$. Fix arbitrarily $x\in (0,+\infty)$ and let $\bar f(y):=f(x,y),\ y\in (0,+\infty)$. A simple calculation gives that for each $y\in (0,+\infty)$, $$\label{eq:A-6} \bar f'(y)=2y-2x\left(y\left(\ln (k+y)\right)^\lambda\right)'=2y-2x(\ln (k+y))^\lambda\left[1+\frac{\lambda y}{(k+y)\ln (k+y)}\right]$$ and $$\label{eq:A-7} \begin{array}{lll} \bar f''(y)&=& \displaystyle 2-2x \left(y\left(\ln (k+y)\right)^\lambda\right)''=2-\frac{2\lambda x[(2k+y)\ln(k+y)-(1-\lambda)y]}{(k+y)^2\left(\ln (k+y)\right)^{2-\lambda}}.\vspace{0.1cm} \end{array}$$ Furthermore, let $y_0>0$ be the unique constant depending only on $(p,k,\lambda,x)$ and satisfying $$\label{eq:A-8} p\left(\ln (k+x)\right)^{2\lambda}=\left(\ln (k+y_0)\right)^{2\lambda}\ \ {\rm or \ equivalently},\ x=(k+y_0)^{\frac{1}{p^{1\over 2\lambda}}}-k\ \ {\rm or}\ \ y_0=(k+x)^{p^{1\over 2\lambda}}-k.$$ It then follows from [\[eq:A-5\]](#eq:A-5){reference-type="eqref" reference="eq:A-5"} that $\bar f(y_0)=f(x,y_0)\geq 0$. In the sequel, we will distinguish two different cases to prove the desired inequality [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"}. **Case 1: $\lambda>0$**. In this case, by [\[eq:A-5\]](#eq:A-5){reference-type="eqref" reference="eq:A-5"} and [\[eq:A-8\]](#eq:A-8){reference-type="eqref" reference="eq:A-8"} we know that $$\begin{array}{lll} \bar f(y)&\geq &\displaystyle px^2\left(\ln (k+x)\right)^{2\lambda} -x^2\left(\ln (k+y)\right)^{2\lambda}\\ &\geq &\displaystyle x^2 \left[p\left(\ln (k+x)\right)^{2\lambda} -\left(\ln (k+y_0)\right)^{2\lambda}\right]=0,\ \ y\in (0,y_0]. \end{array}$$ Hence, it suffices to verify that $\bar f(y)\geq 0$ for $y\in [y_0,+\infty)$. In fact, by [\[eq:A-7\]](#eq:A-7){reference-type="eqref" reference="eq:A-7"}, [\[eq:A-8\]](#eq:A-8){reference-type="eqref" reference="eq:A-8"} and [\[eq:2.8+\]](#eq:2.8+){reference-type="eqref" reference="eq:2.8+"} we have $$\begin{array}{lll} \bar f''(y)&\geq & \displaystyle 2-\frac{2|\lambda| x[2(k+y)\ln(k+y)+|\lambda-1|(k+y)\ln(k+y)]}{(k+y)^2\left(\ln (k+y)\right)^{2-\lambda}}\vspace{0.2cm}\\ &= & \displaystyle 2-\frac{\bar\lambda\left(\ln (k+y)\right)^{\lambda-1}}{k+y}\left((k+y_0)^{1\over \bar p}-k\right) \vspace{0.2cm}\\ &\geq & \displaystyle 2-\frac{\bar\lambda\left(\ln (k+y_0)\right)^{\lambda-1}}{(k+y_0)^{1-{1\over \bar p}}}>2-\frac{\bar\lambda (\ln k)^{\lambda-1}}{k^{1-{1\over \bar p}}}>1,\ \ y\in [y_0,+\infty). \end{array}$$ And, by [\[eq:A-6\]](#eq:A-6){reference-type="eqref" reference="eq:A-6"}, [\[eq:A-8\]](#eq:A-8){reference-type="eqref" reference="eq:A-8"} and [\[eq:2.8+\]](#eq:2.8+){reference-type="eqref" reference="eq:2.8+"} we can deduce that, in view of $\bar p>1$, $$\begin{array}{lll} \bar f'(y_0)&\geq & \displaystyle 2y_0-2x(\ln (k+y_0))^\lambda\left[1+\frac{\lambda }{\ln (k+y_0)}\right]\vspace{0.1cm}\\ &\geq & \displaystyle 2y_0-4x(\ln (k+y_0))^\lambda\vspace{0.1cm}\\ &=& \displaystyle 2(k+x)^{\bar p}-2k-4\sqrt{p} x(\ln (k+x))^\lambda\vspace{0.1cm}\\ &\geq & \displaystyle 2(k+x)^{\bar p}-2(k+x)-4\sqrt{p} (k+x)(\ln (k+x))^\lambda\vspace{0.1cm}\\ &\geq & \displaystyle 2k^{\bar p}-2k-4\sqrt{p} k(\ln k)^\lambda>0. \end{array}$$ Consequently, for each $y\in [y_0,+\infty)$, we have $\bar f'(y)\geq \bar f'(y_0)>0$ and then $\bar f(y)\geq \bar f(y_0)\geq 0$. **Case 2: $\lambda<0$**. In this case, by [\[eq:A-5\]](#eq:A-5){reference-type="eqref" reference="eq:A-5"} and [\[eq:A-8\]](#eq:A-8){reference-type="eqref" reference="eq:A-8"} we know that $$\begin{array}{lll} \bar f(y)&\geq &\displaystyle px^2\left(\ln (k+x)\right)^{2\lambda} -x^2\left(\ln (k+y)\right)^{2\lambda}\\ &\geq &\displaystyle x^2 \left[p\left(\ln (k+x)\right)^{2\lambda} -\left(\ln (k+y_0)\right)^{2\lambda}\right]=0,\ \ y\in [y_0,+\infty). \end{array}$$ Hence, it suffices to verify that $\bar f(y)\geq 0$ for $y\in (0,y_0]$. In fact, by [\[eq:A-7\]](#eq:A-7){reference-type="eqref" reference="eq:A-7"} and [\[eq:2.8+\]](#eq:2.8+){reference-type="eqref" reference="eq:2.8+"} we have $$\begin{array}{lll} \bar f''(y)&\geq& \displaystyle 2+\frac{2|\lambda| x\left[(k+y)\ln(k+y)-|1-\lambda|(k+y)\right]}{(k+y)^2\left(\ln (k+y)\right)^{2-\lambda}}\vspace{0.2cm}\\ &=& \displaystyle 2+\frac{2|\lambda| x \left[\ln (k+y)-|1-\lambda|\right]}{(k+y)\left(\ln (k+y)\right)^{2-\lambda}}>2,\ \ y\in (0,y_0]. \end{array}$$ And, by [\[eq:A-6\]](#eq:A-6){reference-type="eqref" reference="eq:A-6"}, [\[eq:A-8\]](#eq:A-8){reference-type="eqref" reference="eq:A-8"} and [\[eq:2.8+\]](#eq:2.8+){reference-type="eqref" reference="eq:2.8+"} we can deduce that, in view of $\bar p>1$, $$\begin{array}{lll} \bar f'(y_0)&\leq & \displaystyle 2y_0-2x(\ln (k+y_0))^\lambda \vspace{0.1cm}\\ &\leq & \displaystyle 2(k+x)^{1\over \bar p}-2k-2\sqrt{p} x(\ln (k+x))^\lambda\vspace{0.1cm}\\ &\leq & \displaystyle 2(k+x)^{1\over \bar p}-2\sqrt{p} (k+x)(\ln (k+x))^\lambda\vspace{0.1cm}\\ &\leq & \displaystyle 2k^{1\over \bar p}-2\sqrt{p} k(\ln k)^\lambda<0. \end{array}$$ Consequently, for each $y\in (0,y_0]$, we have $\bar f'(y)\leq \bar f'(y_0)<0$ and then $\bar f(y)\geq \bar f(y_0)\geq 0$. In conclusion, [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"} holds. Finally, we verify that when $p=1$ and $\lambda\neq 0$, the constant $k$ such that [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"} holds does not exist. In fact, assume that [\[eq:2.8\]](#eq:2.8){reference-type="eqref" reference="eq:2.8"} holds for some $k\geq e$. Let $x,y>0$ satisfy $$y=x\left(\ln (k+y)\right)^\lambda.$$ It is clear that $y>x$ for $\lambda>0$, and $y<x$ for $\lambda<0$. Then, in view of [\[eq:A-5\]](#eq:A-5){reference-type="eqref" reference="eq:A-5"}, $$y^2-2xy\left(\ln (k+y)\right)^\lambda +x^2\left(\ln (k+x)\right)^{2\lambda}=x^2\left[\left(\ln (k+x)\right)^{2\lambda} -\left(\ln (k+y)\right)^{2\lambda}\right]<0,$$ which immediately yields the desired assertion. The proof is then complete. ◻ **Remark 34**. *The case of $\lambda<0$ in [Proposition 6](#pro:2.5){reference-type="ref" reference="pro:2.5"} has been established in Proposition 3.2 of [@FanHuTang2023SCL]. 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--- bibliography: - On_Ruh_In.bib --- # Introduction Harmonic numbers, with a history dating back to antiquity, hold significant importance in various aspects of number theory. On the other hand, the polylogarithm function find practical applications in quantum systems, such as estimating the density of particles and relaxation time in certain magnetic alloys at low temperatures [@bibapp] using the Fermi--Dirac (FD) and Bose--Einstein (BE) integrals [@bib23 §25.12(iii)]. In this paper, we provide a new closed form for the proposed integral by Khalef Al-Ruhemi $$\label{ruh1} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z}\, \mathrm{d}z,$$ where $\mathop{\mathrm{Li}}_2(z)$ denotes the dilogarithm function, defined as [@bib23 (25.12.1)] $$\mathop{\mathrm{Li}}_2(z) = \sum_{k=1}^\infty \frac{z^k}{k^2}, \quad |z| \leq 1.$$ Our approach entails the transformation of integrals into infinite series involving harmonic numbers and the trigamma function, followed by the subsequent evaluation of these resulting series. Through this method, we not only determine a closed form for the aforementioned integral but also discover closed forms for some previously challenging genearlized infinite series that were considered nearly impossible to evaluate. These series include: $$\sum_{k=1}^\infty \frac{H_{2k}}{k+1} z^{k+1}, \quad z \in {\mathbb C}\setminus[1,\infty); \quad \sum_{k=1}^\infty \frac{H_{2k}}{(k+1)^2} z^{k+1}, \quad z \in {\mathbb C}\setminus(-\infty, 0] \cup [1,\infty),$$ $$\sum_{k=1}^\infty \frac{(-1)^{k} H_k z^k}{k^2}, \quad \sum_{k=1}^\infty \frac{(-1)^{k} H_k z^k}{k^3}, \quad z\in {\mathbb C},\, |z| \leq1 \wedge z \neq -1, 0,$$ $$\sum_{n=1}^\infty \frac{\psi_{m-1}\left(\frac{2k-1}{2}\right)}{(2k-1)^m}, \quad \sum_{n=1}^\infty \frac{\psi_{m-1}\left(\frac{4k-1}{4}\right)}{(4k-1)^m}, \quad \sum_{k=1}^\infty \frac{\psi_{m-1}\left(\frac{k+1}{2}\right) - \psi_{m-1}\left(\frac{k+2}{2}\right)}{k^m}, \quad m \geq 2,\, m \in {\mathbb N}.$$ $$\sum_{k=1}^\infty \frac{ \zeta\left(m, \frac{rk+r-s}{r}\right) }{(rk-s)^m}, \quad \Re{m} > 1\wedge\hspace{0.03cm} r, s \in {\mathbb C}\setminus\{0\}, rk \neq s, \forall\hspace{0.1cm} k\, \in {\mathbb N}.$$ $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(m)} + (-1)^{k-1}H_k^{(m)}}{k^m}, \quad \Re{m} > 1.$$ In these expressions, $H_n$ represents the $n$th harmonic number, defined as $$H_n = \sum_{k=1}^n \frac{1}{k}, \quad n \in {\mathbb N},$$ $\psi_{m-1}(z)$ represents the polygamma function, defined as [@bib23 §5.15], $$\psi_{m-1}(z) = (-1)^{m} (m - 1)!\sum_{k=0}^\infty \frac{1}{(k+z)^m}, \quad m \geq 2,\, m \in {\mathbb N}, z \not\in -\mathbb{N}_0,$$ and $\zeta(s, z)$ represents the Hurwitz zeta function, defined as [@bib23 §25.11] $$\zeta(s, z) = \sum_{n=0}^{\infty} \frac{1}{(n+z)^s}, \quad z \not\in -\mathbb{N}_0, \Re\,s > 1.$$ By incorporating Vălean's closed form for $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(z)}{1-az}\, \mathrm{d}z, \quad a \in {\mathbb C}\setminus\{0\},$$ alongside our derived closed forms for $$\label{begeqhh} \int_0^1 \frac{\ln{z}\ln(1+az)\ln(1-z)}{z} \, \mathrm{d}z, \quad \int_0^1 \frac{\ln{z}\ln^2(1 + a z)}{z}\, \mathrm{d}z, \quad a \in {\mathbb C}\setminus(-\infty, -1),$$ we present a generalized version of [\[ruh1\]](#ruh1){reference-type="eqref" reference="ruh1"} in the form $$\label{begeqh1} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+az)}{z}\, \mathrm{d}z, \quad a \in {\mathbb C}\setminus(-\infty, -1).$$ The two integrals in [\[begeqhh\]](#begeqhh){reference-type="eqref" reference="begeqhh"} are equal when $a=-1$. The simplest evaluation of [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"} occurs when $a=-1$. In this case, we have $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1-z)}{z}\, \mathrm{d}z = -\frac{\pi^4}{72}.$$ The exclusion of [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"} for $a \in (-\infty, -1)$ stems from the observation that if $a \in (-\infty, -1)$, the integral diverges at a certain point within the integration domain. To illustrate this, set $a=-b$, where $b \in (1, \infty)$, and notice that for all values of $b$ in this interval, there exists a unique $z = \frac{1}{b} \in (0, 1)$ such that $$\label{beghy1} \frac{\mathop{\mathrm{Li}}_2(z) \ln(1-bz)}{z} \to -\infty.$$ In summary, the results established in this article are outlined as follows. In Section [3.1](#sec3.1){reference-type="ref" reference="sec3.1"}, we unveil two novel closed forms for a generalized infinite series involving the harmonic numbers $H_{2k}$. Additionally, we introduce a crucial lemma pertaining to the interchangeability of summation orders for double finite and infinite series. Furthermore, we derive closed forms for infinite series involving harmonic numbers and the trigamma function. It is worth noting that some of these results were previously documented in Vălean's books [@bib32; @bib9]. However, we offer new proofs for those that appear therein. Moving on to Section [3.2](#sec3.2){reference-type="ref" reference="sec3.2"}, we present an entirely new closed form for [\[ruh1\]](#ruh1){reference-type="eqref" reference="ruh1"}. Notably, this closed form has not been previously documented in the existing literature. Moreover, we establish the relationship $\displaystyle 2\sum_{k=1}^\infty \frac{H_{2k}}{(2k)^3} = \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z}\, \mathrm{d}z$. In Section [3.3](#sec3.3){reference-type="ref" reference="sec3.3"}, we extend our exploration by introducing a novel generalization of [\[ruh1\]](#ruh1){reference-type="eqref" reference="ruh1"}. Additionally, we provide generalizations for related integrals. Furthermore, we offer a representation for the series $\displaystyle\sum_{k=1}^\infty \frac{(-1)^{k} H_k a^k}{k^3}$ that allows us to provide the generalization [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"} for $a \in {\mathbb C}\setminus(-\infty, 0]$ while avoiding logarithm of negative real numbers. Lastly, in Section [3.4](#sec3.4){reference-type="ref" reference="sec3.4"}, we offer a new proof of Jonquière's inversion formula for order 4 with arguments $-1/z$ and $z/(z-1)$. We also introduce a transformative approach for double infinite series, enabling us to express them as sums of the square of an infinite series and another infinite series. We apply this theorem to derive novel generalized identities. The Computer Algebra System (CAS) software employed for result verification throughout this paper is [Mathematica 13]{.sans-serif}. # Notations and Definitions {#sec2} In this manuscript, we consistently use the natural logarithm denoted by $\ln{z}$, referring to the principal value of the logarithm [@bib23 §4.2.2], that is $\Im \ln{z} \in (-\pi, \pi)$. We employ the following abbreviated notations: $B_n$ represents the Bernoulli numbers [@bib23 §24.2(i)], $E_n$ represents the Euler numbers [@bib23 §24.2(ii)], $\gamma \approx 0.5772156649$ represents Euler's constant, $\textbf{\textup{G}}= \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}$ represents Catalan's constant ($\textbf{\textup{G}}\approx 0.9159655941$), while $\mathrm{e} \approx 2.71828182845$ stands for Euler's number. We define $\mathbb{N}_0 := \mathbb{N} \cup \{0\}$ as the set of non-negative integers, where $n\mathbb{N}_0$ denotes all elements in $\mathbb{N}_0$ multiplied by $n$. Additionally, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ represent the sets of integers, rational, real, and complex numbers, respectively.\ \ We denote the gamma [@bib23 (5.2.1)] , digamma [@bib23 (5.2.2)], tetragamma, pentagamma, and hexagamma functions [@bib23 §5.15] of argument $z$ as $\Gamma(z)$, $\psi(z)$, $\psi_2(z)$, $\psi_3(z)$, and $\psi_4(z)$, respectively. Here, $\psi_n(z)$ is the polygamma function, defined as the $n$-th derivative of $\ln{\Gamma(z)}$, and $n\in\mathbb{N}_0$. The digamma function can be expressed as [@bib23 §1.7(6)]: $$\psi(z) = -\gamma +\sum_{k=0}^\infty \left(\frac{1}{k+1} - \frac{1}{k+z}\right), \quad z \in \mathbb{C}\setminus- \mathbb{N}_0.$$ For positive integer values of $z$, the digamma function simplifies to [@bib2 §1.7.1(9)]: $$\label{harmonicn} \psi(k+1) = -\gamma + H_k, \quad k \in \mathbb{N}.$$ The recurrence relation for the digamma function is given by [@bib23 (5.5.2)]: $$\label{recc} \psi(z+1) = \psi(z) + \frac{1}{z}.$$ The duplication formula for the $\psi(z)$ is [@bib23 (5.5.8)]: $$\label{dupl} \psi\left(z + \frac{1}{2}\right) = 2\psi(2z) - \psi(z) - \ln{4}, \quad z \in {\mathbb C}\setminus-\mathbb{N}_0.$$ Euler's beta function is defined as [@bib23 (5.12.1)]: $$\mathrm{B}(a, b) := \int_0^1 t^{a-1} (1-t)^{b-1} \, \mathrm{d}t = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}, \quad a, b \in {\mathbb C}\setminus-\mathbb{N}_0.$$ The Lerch transcendent is defined as [@bib23 (24.14.1)]: $$\Phi(z, s, a) = \sum_{n=0}^{\infty} \frac{z^n}{(n + a)^s}, \quad \lvert z\rvert \leq 1, \Re\,s > 1, a \not\in -\mathbb{N}_0.$$ The polygamma function can be expressed as $\psi_n(z) = (-1)^{n-1} n! \Phi(1, n+1, z)$, where $n \in {\mathbb N}$. The Dirichlet eta function is defined as $\eta(n) := \Phi(-1, n, 1)$, where $\Re n > 0$ [@bib23 §1.12(2)]. The Riemann zeta function [@bib23 §25.2] is defined as: $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s},$$ where $z \not\in -\mathbb{N}_0$, $\Re\,s > 1$. The domain $\Re\,s > 1$ of the Riemann and Hurwitz zeta functions can be extended to $s \in \mathbb{C}\setminus \{1\}$ through analytic continuation, using for instance, the Hermite integral representation for the Hurwitz zeta function [@bib23 (25.11.29)]. The relationship between the Dirichlet eta function and the Riemann zeta function is given by $\eta(n) = \left(1- 2^{1-n}\right)\zeta(n)$ [@bib2 §1.12(2)]. The polylogarithm function [@bib23 §25.12(ii), §25.14.3], $\mathop{\mathrm{Li}}_s(z)$, is defined as: $\mathop{\mathrm{Li}}_s(z) = z\Phi(z, s, 1)$, where $\Re s > 1,\, |z| \leq 1$. The dilogarithm function, $\mathop{\mathrm{Li}}_2(z)$, has the integral representation [@bib23 (25.12.2)] $$\label{intrep} \mathop{\mathrm{Li}}_2(z) = -\int_0^z \frac{\ln(1-t)}{t}\, \mathrm{d}t, \quad z \in {\mathbb C}\setminus(1, \infty).$$ The domain $|z| \leq 1$ can be extended to the entire complex plane through analytic continuation. This can be achieved using, for instance, the integral representation [@bib23 (25.14.6)] $$\mathop{\mathrm{Li}}_s(z) = \frac{1}{2}z + \int_0^\infty \frac{z^{t+1}}{(1+t)^s} \, \mathrm{d}t - 2z\int_0^\infty \frac{\sin\left(t\ln{z} - s\arctan{t}\right)}{\left(1+t^2\right)^{\frac{s}{2}} \left(e^{2\pi t} - 1\right)} \, \mathrm{d}t, \quad \Re(s) > 0\,\,\textup{if}\,\, z\in {\mathbb C}\setminus[1, \infty).$$ Throughout this work, we utilize the property that $\mathop{\mathrm{Li}}_s(z)$ is defined for all complex $z$. # Results In this section, we present the main findings and outcomes of the study. ## Intriguing closed forms for some infinite series {#sec3.1} In this subsection, we unveil and prove a series of identities. While Lemmas [\[lemma1\]](#lemma1){reference-type="ref" reference="lemma1"}--[\[newlemh1\]](#newlemh1){reference-type="ref" reference="newlemh1"} and Corollary [\[newcorh1\]](#newcorh1){reference-type="ref" reference="newcorh1"} unveil entirely new findings, Lemma [\[lemma4\]](#lemma4){reference-type="ref" reference="lemma4"} provides unique rediscoveries approached from a distinct perspective. Let $z \in {\mathbb C}\setminus[1,\infty)$. Then [\[lemma1\]]{#lemma1 label="lemma1"} $$\label{eqh1} \sum_{k=1}^\infty \frac{H_{2k}}{k+1} z^{k+1} = -\frac{1}{2}\mathop{\mathrm{Li}}_2(z) - \ln(1 - z) - \sqrt{z}\ln\left(\frac{1+\sqrt{z}}{1-\sqrt{z}}\right) - \mathop{\mathrm{Li}}_2\left(\frac{\sqrt{z}}{\sqrt{z} - 1}\right) - \mathop{\mathrm{Li}}_2\left(\frac{\sqrt{z}}{\sqrt{z} + 1}\right).$$ *Proof.* Utilizing the integral representation for the harmonic numbers [@bib2 §1.7.2(13)] $$\label{harmonic1} H_k = \int_0^1 \frac{1-t^k}{1-t} \, \mathrm{d}t,$$ we deduce that $$\label{spl1} \begin{split} \sum_{k=1}^\infty \frac{H_{2k}}{k+1} z^{k+1} &= \int_0^1 \frac{\ln(1 - zt^2)}{t^2(1-t)} - \frac{\ln(1 - t)}{1-t}\, \mathrm{d}t \\&= -\frac{1}{2}\mathop{\mathrm{Li}}_2(z) - \ln(1 - z) - \sqrt{z}\ln\left(\frac{1+\sqrt{z}}{1-\sqrt{z}}\right) + \int_0^1 \frac{\ln\left(\frac{1 - zt^2}{1-z}\right)}{1-t} \, \mathrm{d}t. \end{split}$$ Completing the proof involves evaluating the last integral in [\[spl1\]](#spl1){reference-type="eqref" reference="spl1"} in a manner analogous to the approach used in [@bibh §3, (3.12)]. ◻ Let $z \in {\mathbb C}\setminus(-\infty, 0] \cup [1,\infty)$. Then [\[lemma2\]]{#lemma2 label="lemma2"} $$\label{lemh1} \begin{split} \sum_{k=1}^\infty \frac{H_{2k}}{(k+1)^2} z^{k+1} &= -\frac{1}{2}\mathop{\mathrm{Li}}_3(z) + \mathop{\mathrm{Li}}_2(z) - 2\left(1 + \sqrt{z}\right)\ln\left(1 + \sqrt{z}\right) - 2\left(1 - \sqrt{z}\right)\ln\left(1 - \sqrt{z}\right) \\&\quad+ 2\zeta(3) - 2\mathop{\mathrm{Li}}_3\left(1 - \sqrt{z}\right) + 2\mathop{\mathrm{Li}}_3\left(\sqrt{z}\right) - \ln{z} \mathop{\mathrm{Li}}_2\left(\sqrt{z}\right) + 2\ln\left(1 - \sqrt{z}\right) \\&\quad \times \mathop{\mathrm{Li}}_2\left(1 - \sqrt{z}\right) + 2\mathop{\mathrm{Li}}_3\left(-\sqrt{z}\right) - \ln{z} \mathop{\mathrm{Li}}_2\left(-\sqrt{z}\right) -\frac{2}{3}\ln^3\left(1 + \sqrt{z}\right) \\&\quad + 2\zeta(3) - 2\mathop{\mathrm{Li}}_3\left(\frac{1}{1+\sqrt{z}}\right) - 2\ln\left(1 + \sqrt{z}\right) \mathop{\mathrm{Li}}_2\left(\frac{1}{1+\sqrt{z}}\right) \\&\quad- \ln{z}\mathop{\mathrm{Li}}_2\left(\frac{\sqrt{z}}{\sqrt{z} - 1}\right) - \ln{z}\mathop{\mathrm{Li}}_2\left(\frac{\sqrt{z}}{\sqrt{z} + 1}\right). \end{split}$$ *Proof.* By substituting the variable in [\[eqh1\]](#eqh1){reference-type="eqref" reference="eqh1"} with $t$, dividing both sides by $t$, integrating from $t=0$ to $z$, and utilizing the results from [@bibh §3.1, (3.3), (3.4), (3.6)], along with $$\int_0^z \frac{\ln{t} \ln(1 + t)}{t} \, \mathrm{d}t = \mathop{\mathrm{Li}}_3(-z) - \ln{z}\mathop{\mathrm{Li}}_2(-z), \quad z \in {\mathbb C}\setminus (-\infty, 1],$$ we conclude the proof. ◻ [\[newlemh1\]]{#newlemh1 label="newlemh1"} Let $f(k, j)$ be an arbitrary function of two variables. Then $$\label{bigcorlh1} \sum_{k=1}^N \sum_{j=1}^k f(k, j) = \sum_{j=1}^N \sum_{k=j}^N f(k, j), \quad N \in {\mathbb N}.$$ *Proof.* Expanding the series on the left-hand side of [\[bigcorlh1\]](#bigcorlh1){reference-type="eqref" reference="bigcorlh1"}, we obtain $$\begin{aligned} \sum_{k=1}^N \sum_{j=1}^k f(k, j) &= f(1, 1) + f(2, 1) + f(2, 2) + f(3, 1) + f(3, 2) + f(3, 3) + \cdots + f(N, 1) \\&\quad+ f(N, 2) + \cdots + f(N, N) \\&= f(1, 1) + f(2, 1) + f(3, 1) + \cdots + f(N, 1) + f(2, 2) + f(3, 2) + \cdots \\&\quad+ f(N, 2) + f(3, 3) + f(4, 3) + \cdots + f(N, 3) + \cdots + f(N, N) \\&= \sum_{k=1}^N f(k, 1) + \sum_{k=2}^N f(k, 2) + \sum_{k=3}^N f(k, 3) +\cdots+ \sum_{k=N-1}^N f(k, N-1) \\&\quad+ \sum_{k=N}^N f(k, N) = \sum_{j=1}^N \sum_{k=j}^N f(k, j).\end{aligned}$$ ◻ Let $f(k, j)$ be an arbitrary function of two variables such that [\[newcorh1\]]{#newcorh1 label="newcorh1"} $$\lim_{N \to \infty} \sum_{k=1}^N \sum_{j=1}^k f(k, j)\,\,\,\textit{exists.}$$ Then $$\label{anal1ab} \sum_{k=1}^\infty\sum_{j=1}^k f(k, j) = \sum_{j=1}^\infty \sum_{k=j}^\infty f(k, j).$$ *Proof.* By evaluating the limits as $N \to \infty$ on both sides of [\[bigcorlh1\]](#bigcorlh1){reference-type="eqref" reference="bigcorlh1"}, we complete the proof of Corollary [\[newcorh1\]](#newcorh1){reference-type="ref" reference="newcorh1"}. ◻ [\[lemma4\]]{#lemma4 label="lemma4"} We present the following closed forms involving the harmonic numbers and trigamma function $$\begin{aligned} &\sum_{k=1}^\infty \frac{H_k}{k^3} = \frac{\pi^4}{72}, \label{hrmkcu}\\ &\sum_{k=1}^\infty \frac{(-1)^{k-1}H_{k}}{k^3} = -\frac{\ln^4{2}}{12} + \frac{11\pi^4}{360} + \frac{\pi^2 \ln^2{2}}{12} - \frac{7\ln{2}\zeta(3)}{4} - 2\mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right), \label{haf1}\\ &\sum_{k=1}^\infty \frac{H_k}{(2k-1)^3} = -\frac{\pi^2}{4} + \frac{\pi^4}{64} + 2\ln{2} + \frac{7\zeta(3)}{4} - \frac{7\ln{2}\zeta(3)}{4}, \label{ftpv}\\ & \sum_{k=1}^\infty \frac{H_{2k-1}}{(2k-1)^3} = \frac{\pi^4}{45} + \frac{\pi^2}{24}\ln^2{2} - \frac{\ln^4{2}}{24} - \frac{7\ln{2}\zeta (3)}{8} - \mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right), \label{wgt1}\\ & \sum_{k=1}^\infty \frac{H_{2k}}{k^3} = -\frac{\pi ^4}{15}-\frac{\pi ^2 }{3}\ln^2{2} +\frac{\ln ^4{2}}{3} +7 \ln{2} \zeta (3) + 8 \mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right),\label{relf1}\\ &\sum_{k=1}^\infty \frac{\psi_1\left(\frac{2k-1}{2}\right)}{(2k-1)^2}= \frac{5\pi^4}{96},\label{psikhal}\\ &\sum_{k=1}^\infty \frac{\psi_1\left(\frac{k}{2}\right)}{k^2}= \frac{41\pi^4}{720}, \label{deriv1}\\ &\sum_{k=1}^\infty \frac{(-1)^{k-1} \psi_1\left(\frac{k}{2}\right)}{k^2} = \frac{17\pi^4}{360}, \label{jadd}\\ &\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2}= \frac{7\pi^4}{360}, \label{psikksq}\\ &\sum_{k=1}^{\infty} \frac{(-1)^{k-1} \psi_1(k)}{k^2} = \frac{17 \pi ^4}{288} + \frac{\pi ^2}{6} \ln ^2{2} -\frac{\ln^4{2}}{6} -\frac{7\ln{2}\zeta (3)}{2} - 4 \mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right), \label{almd1}\\ &\sum_{k=1}^{\infty} \frac{\psi_1(2k)}{k^2} = -\frac{19 \pi ^4}{240} - \frac{\pi ^2}{3} \ln ^2{2} + \frac{\ln^4{2}}{3} + 7\ln{2} \zeta (3) + 8\mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right),\label{almd2}\\ &\sum_{k=1}^{\infty} \frac{\psi_1(2k-1)}{(2k-1)^2} = \frac{113\pi ^4}{2880} + \frac{\pi ^2}{12} \ln ^2{2} -\frac{\ln^4{2}}{12} -\frac{7\ln{2}\zeta (3)}{4} - 2 \mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right),\label{almd3}\\ &\sum_{k=1}^\infty \frac{\psi_1\left(\frac{k+1}{2}\right) - \psi_1\left(\frac{k+2}{2}\right)}{k^2} = \frac{\pi^4}{120}.\label{psidouble}\end{aligned}$$ *Proof.* Using the series expression for $\psi_1(k)$, we have $$\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2} = \sum_{k=1}^\infty \sum_{j=k}^\infty \frac{1}{k^2 j^2}.$$ By applying Corollary [\[newcorh1\]](#newcorh1){reference-type="ref" reference="newcorh1"} to interchange the order of summation and then swapping the roles of the dummy variables $j$ and $k$, we obtain $$\sum_{j=1}^\infty \sum_{k=1}^j \frac{1}{j^2k^2} = \sum_{k=1}^\infty \sum_{j=k}^\infty \frac{1}{j^2k^2} = \sum_{k=1}^\infty \sum_{j=1}^k \frac{1}{j^2k^2}.$$ Consequently, this leads to $$\begin{aligned} 2\sum_{k=1}^\infty \sum_{j=k}^\infty \frac{1}{j^2k^2} &= \zeta(4) + \sum_{k=1}^\infty \sum_{j=1}^\infty \frac{1}{j^2 k^2} = \zeta(4) + \zeta^2(2).\end{aligned}$$ Hence $$\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2} = \frac{\zeta(4) + \zeta^2(2)}{2} = \frac{7\pi^4}{360},$$ completing the proof of [\[psikksq\]](#psikksq){reference-type="eqref" reference="psikksq"}. By applying analogous operations to the series $$\sum_{j=1}^\infty \sum_{k=1}^j \frac{1}{(2j-1)^2(2k-1)^2}\quad \textup{and}\quad \sum_{j=1}^\infty \sum_{k=1}^j \frac{(-1)^{j+k}}{j^2 k^2},$$ we successfully conclude the proofs of [\[psikhal\]](#psikhal){reference-type="eqref" reference="psikhal"} and [\[psidouble\]](#psidouble){reference-type="eqref" reference="psidouble"}, respectively. Substituting the variable in [@bib9 §4.6, pp. 399, (4.36)] with $t$ and dividing both sides by $t$, we then integrate the resulting expression from $t=0$ to $1$, we derive $$\begin{aligned} \sum_{k=1}^\infty \frac{H_k}{k^3} &= \zeta(4) + \int_0^1 \frac{(\mathop{\mathrm{Li}}_2(1 - t) - \mathop{\mathrm{Li}}_2(t))\ln(1 - t)}{t} \, \mathrm{d}t + \frac{1}{2} \int_0^1 \frac{\ln{t} \ln^2(1 -t)}{t}\, \mathrm{d}t \\&= \zeta(4) + \int_0^1 \frac{\left(\frac{\pi^2}{6} - \ln{t}\ln(1- t) - 2\mathop{\mathrm{Li}}_2(t)\right)\ln(1 - t)}{t} \, \mathrm{d}t + \frac{1}{2} \int_0^1 \frac{\ln{t} \ln^2(1 -t)}{t}\, \mathrm{d}t \\&= \frac{\pi^4}{90} - \frac{1}{2} \int_0^1 \frac{\ln{t} \ln^2(1 -t)}{t}\, \mathrm{d}t = \frac{\pi^4}{90} + \frac{1}{2} \sum_{k=1}^\infty \frac{1}{k} \int_0^1 t^{k-1} \ln{t} \ln(1 - t) \, \mathrm{d}t \\&= \frac{\pi^4}{60} + \frac{1}{2}\sum_{k=1}^\infty \frac{H_k}{k^3} - \frac{1}{2}\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2} = \frac{\pi^4}{72},\end{aligned}$$ completing the proof of [\[hrmkcu\]](#hrmkcu){reference-type="eqref" reference="hrmkcu"}. By substituting the variable in [\[lemh1\]](#lemh1){reference-type="eqref" reference="lemh1"} with $t$, dividing both sides by $t$, integrating from $t=0$ to $1$, and utilizing the following results: $$\int_0^1 \frac{\left(1 + \sqrt{t}\right)\ln\left(1 + \sqrt{t}\right)}{t} = - 2 + 4 \ln{2} + \frac{\pi ^2}{6},\quad \int_0^1 \frac{\left(1 - \sqrt{t}\right)\ln\left(1 - \sqrt{t}\right)}{t} = 2 - \frac{\pi ^2}{3},$$ $$\label{almend1} \int_0^1 \frac{\mathop{\mathrm{Li}}_2\left(\sqrt{t}\right)\ln{t}}{t} \mathrm{d}t = 2\int_0^1 \frac{\ln(1 - t) \ln^2{t}}{t}\, \mathrm{d}t = \frac{2}{3} \int_0^1 \frac{\ln^3{t}}{1-t} \, \mathrm{d}t = -\frac{2\pi^4}{45},$$ $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2\left(-\sqrt{t}\right)\ln{t}}{t} \mathrm{d}t = 2\int_0^1 \frac{\ln(1 + t) \ln^2{t}}{t}\, \mathrm{d}t = -\frac{2}{3} \int_0^1 \frac{\ln^3{t}}{1+ t} \, \mathrm{d}t = \frac{7\pi^4}{180},$$ $$\int_0^1 \frac{\mathop{\mathrm{Li}}_3\left(\sqrt{t}\right)}{t}\, \mathrm{d}t = 2\zeta(4) = \frac{\pi^4}{45}, \quad \int_0^1 \frac{\mathop{\mathrm{Li}}_3\left(-\sqrt{t}\right)}{t}\, \mathrm{d}t = -2\eta(4) = -\frac{7\pi^4}{360},$$ $$\label{foreul1} \int_0^1 \frac{\ln{t} \ln^2(1 -t)}{t}\, \mathrm{d}t = 2\left(\frac{\pi^4}{90} - \sum_{k=1}^\infty \frac{H_k}{k^3}\right) = -\frac{\pi^4}{180},$$ $$\begin{aligned} \int_0^1 \frac{\ln\left(1 - \sqrt{t}\right) \mathop{\mathrm{Li}}_2\left(1 - \sqrt{t}\right)}{t} \, \mathrm{d}t &= 2\int_0^1 \frac{\ln(1 - t)}{t}\left(-\mathop{\mathrm{Li}}_2(t) + \frac{\pi^2}{6} - \ln{t}\ln(1 - t)\right)\, \mathrm{d}t \\&= -\zeta^2(2) - 2\int_0^1 \frac{\ln{t} \ln^2(1 -t)}{t}\, \mathrm{d}t = -\frac{\pi^4}{60},\end{aligned}$$ $$\int_0^1 \frac{\zeta(3) - \mathop{\mathrm{Li}}_3\left(1 - \sqrt{t}\right)}{t}\, \mathrm{d}t = 2\int_0^1 \frac{\zeta(3) - \mathop{\mathrm{Li}}_3(1 - t)}{t}\, \mathrm{d}t = -2\int_0^1 \frac{\ln{t}\mathop{\mathrm{Li}}_2(1 - t)}{1-t}\, \mathrm{d}t = \frac{\pi^4}{36},$$ $$\label{tobeu} \begin{split} &\int_0^1 \frac{\zeta(3) - \mathop{\mathrm{Li}}_3\left(\frac{1}{1+ \sqrt{t}}\right) - \ln\left(1 + \sqrt{t}\right) \mathop{\mathrm{Li}}_2\left(\frac{1}{1 + \sqrt{t}}\right)}{t}\, \mathrm{d}t - \frac{1}{3}\int_0^1 \frac{\ln^3\left(1 + \sqrt{t}\right)}{t} \mathrm{d}t \\&\quad= -2\int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t, \end{split}$$ $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2\left(\frac{\sqrt{t}}{\sqrt{t} + 1}\right) \ln{t}}{t} \, \mathrm{d}t = -\frac{7\pi^4}{180} - 2\int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t, \quad \int_0^1 \frac{\mathop{\mathrm{Li}}_2\left(\frac{\sqrt{t}}{\sqrt{t} - 1}\right) \ln{t}}{t} \, \mathrm{d}t = \frac{\pi^4}{18},$$ we arrive at $$\label{lastm1} \sum_{k=1}^\infty \frac{H_{2k}}{(k+1)^3} = \frac{\pi^2}{3} + \frac{\pi^4}{90} - 8\ln{2} + \zeta(3) -2 \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t.$$ Next, by integrating term-wise, we obtain $$\begin{aligned} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t &= \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \int_0^1 t^{k-1} \ln{t} \ln(1 + t) \, \mathrm{d}t = \frac{7\pi^4}{720} - \sum_{k=1}^{\infty} \frac{(-1)^{k-1} H_k}{k^3} \\&\qquad+ \sum_{k=1}^\infty \frac{(-1)^{k-1} H_{\frac{k}{2}}}{k^3} + \frac{1}{2}\sum_{k=1}^\infty \frac{(-1)^{k-1}\left(2\psi_1(k) - \psi_1\left(\frac{k}{2}\right)\right)}{k^2}.\end{aligned}$$ Utilizing [@bibh §3.1, (3.12)], we arrive at $$\sum_{k=1}^\infty \frac{(-1)^{k-1}H_{k}}{k^3} = \frac{7\pi^4}{720} - \frac{1}{2}\int_0^1 \int_0^z \frac{\ln^2(1 + t)}{t}\, \mathrm{d}t \, \mathrm{d}z.$$ Additionally, with the aid of [@bibh §3.1, (3.6)] and the application of [\[tobeu\]](#tobeu){reference-type="eqref" reference="tobeu"}, we obtain $$\label{haf2} \sum_{k=1}^\infty \frac{(-1)^{k-1}H_{k}}{k^3} = \frac{7\pi^4}{720} + \frac{1}{2} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t.$$ This leads us to the result $$\label{last3} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t = \frac{2}{3}\sum_{k=1}^\infty \frac{(-1)^{k-1} H_{\frac{k}{2}}}{k^3} + \frac{1}{3}\sum_{k=1}^\infty \frac{(-1)^{k-1}\left(2\psi_1(k) - \psi_1\left(\frac{k}{2}\right)\right)}{k^2}.$$ We rewrite the third series as $$\label{afed} \sum_{k=1}^\infty \frac{(-1)^{k-1} \psi_1(k)}{k^2} = \sum_{k=1}^\infty \sum_{j=0}^\infty \frac{(-1)^{k-1}}{k^2 (j + k)^2} = \sum_{j=1}^\infty \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^2 (j + k)^2} + \frac{7\pi^4}{720}.$$ Splitting the summand in [\[afed\]](#afed){reference-type="eqref" reference="afed"} and then summing over $k=1$ to $\infty$, we deduce $$\label{last2} \sum_{k=1}^\infty \frac{(-1)^{k-1} \psi_1(k)}{k^2} = -\frac{\pi^4}{480} + 2\sum_{j=1}^{\infty} \frac{H_{\frac{j}{2}}}{j^3}.$$ Upon splitting the last series in [\[last3\]](#last3){reference-type="eqref" reference="last3"} into its odd and even parts, we get $$\label{last1} \sum_{k=1}^\infty \frac{(-1)^{k-1} \psi_1\left(\frac{k}{2}\right)}{k^2} = -\frac{1}{4}\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2} + \sum_{k=1}^\infty \frac{\psi_1\left(\frac{2k-1}{2}\right)}{(2k-1)^2} = \frac{17\pi^4}{360}.$$ This completes the proof of [\[jadd\]](#jadd){reference-type="eqref" reference="jadd"}. By applying a similar operation as that employed in [\[last1\]](#last1){reference-type="eqref" reference="last1"} to [\[deriv1\]](#deriv1){reference-type="eqref" reference="deriv1"}, we complete the proof of [\[deriv1\]](#deriv1){reference-type="eqref" reference="deriv1"}. Substituting [\[last1\]](#last1){reference-type="eqref" reference="last1"} and [\[last2\]](#last2){reference-type="eqref" reference="last2"} into [\[last3\]](#last3){reference-type="eqref" reference="last3"}, we arrive at $$\label{last0} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t = -\frac{23\pi^4}{1440} - \frac{7\ln{2}\zeta(3)}{2} + 4\sum_{k=1}^\infty \frac{H_{2k}}{(2k-1)^3} - 2\sum_{k=1}^\infty \frac{H_{k}}{(2k-1)^3}.$$ Finally, by substituting [\[last0\]](#last0){reference-type="eqref" reference="last0"} into [\[lastm1\]](#lastm1){reference-type="eqref" reference="lastm1"}, two terms cancel out, allowing us to conclude the proof of [\[ftpv\]](#ftpv){reference-type="eqref" reference="ftpv"}. Evaluating the integral in [\[last0\]](#last0){reference-type="eqref" reference="last0"}, we get $$\label{deriv6} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t = \int_1^2 \frac{\ln(t-1) \ln^2{t}}{t-1} \, \mathrm{d}t = \int_{\frac{1}{2}}^1 \frac{\ln(1 - t)\ln^2{t} - \ln^3{t}}{t(1 - t)} \, \mathrm{d}t.$$ From the first component of the resulting integral, we find $$\label{deriv5} \int_{\frac{1}{2}}^1 \frac{\ln(1 - t)\ln^2{t}}{t(1 - t)} \, \mathrm{d}t = \int_0^1 \frac{\ln^2{t}\ln(1 - t)}{t} \, \mathrm{d}t + \int_0^{\frac{1}{2}} \frac{\ln{t}\ln^2(1 - t)}{t} \, \mathrm{d}t - \int_0^{\frac{1}{2}} \frac{\ln^2{t}\ln(1 - t)}{t} \, \mathrm{d}t.$$ Focusing on the second integral within [\[deriv5\]](#deriv5){reference-type="eqref" reference="deriv5"}, we derive $$\label{deriv7} \begin{split} \int_0^{\frac{1}{2}} \frac{\ln{t}\ln^2(1 - t)}{t} \, \mathrm{d}t &= \int_0^1 \frac{\ln(1 - t)\ln^2{t}}{1 - t} \, \mathrm{d}t - \int_0^{\frac{1}{2}} \frac{\ln(1 - t)\ln^2{t}}{1 - t} \, \mathrm{d}t \\&= \frac{\ln^4{2}}{4} + \frac{1}{2} \int_0^1 \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t. \end{split}$$ Substituting [\[deriv7\]](#deriv7){reference-type="eqref" reference="deriv7"} into [\[deriv5\]](#deriv5){reference-type="eqref" reference="deriv5"}, we arrive at $$\label{deriv4} \int_{\frac{1}{2}}^1 \frac{\ln(1 - t)\ln^2{t}}{t(1 - t)} \, \mathrm{d}t = -\frac{\pi^4}{40} - \frac{\ln^4{2}}{12} - \frac{1}{3}\int_0^{\frac{1}{2}} \frac{\ln^3{t}}{1-t}\, \mathrm{d}t.$$ Next, we evaluate the integral in [\[deriv4\]](#deriv4){reference-type="eqref" reference="deriv4"} term by term, yielding $$\label{deriv3} \int_0^{\frac{1}{2}} \frac{\ln^3{t}}{1-t}\, \mathrm{d}t = -\frac{\ln^4{2}}{2} + \frac{\pi^2 \ln^2{2}}{4} - \frac{21\ln{2}\zeta(3)}{4} - 6\mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right).$$ Utilizing [\[deriv3\]](#deriv3){reference-type="eqref" reference="deriv3"} in [\[deriv4\]](#deriv4){reference-type="eqref" reference="deriv4"} and subsequently substituting the latter into [\[deriv6\]](#deriv6){reference-type="eqref" reference="deriv6"}, we obtain $$\label{deriv2} \int_0^1 \frac{\ln{t}\ln^2(1+t)}{t}\, \mathrm{d}t = -\frac{\ln^4{2}}{6} + \frac{\pi^4}{24} + \frac{\pi^2 \ln^2{2}}{6} - \frac{7\ln{2}\zeta(3)}{2} - 4\mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right).$$ By substituting [\[ftpv\]](#ftpv){reference-type="eqref" reference="ftpv"} and [\[deriv2\]](#deriv2){reference-type="eqref" reference="deriv2"} into [\[last0\]](#last0){reference-type="eqref" reference="last0"}, we successfully establish the proof for [\[wgt1\]](#wgt1){reference-type="eqref" reference="wgt1"}. Next, substituting [\[deriv2\]](#deriv2){reference-type="eqref" reference="deriv2"} into [\[lastm1\]](#lastm1){reference-type="eqref" reference="lastm1"}, we are able to conclude the proof for [\[relf1\]](#relf1){reference-type="eqref" reference="relf1"}. Splitting the series in [\[last2\]](#last2){reference-type="eqref" reference="last2"} into odd and even parts and applying [\[hrmkcu\]](#hrmkcu){reference-type="eqref" reference="hrmkcu"} and [\[wgt1\]](#wgt1){reference-type="eqref" reference="wgt1"}, we conclude the proof of [\[almd1\]](#almd1){reference-type="eqref" reference="almd1"}. Subtracting [\[psikksq\]](#psikksq){reference-type="eqref" reference="psikksq"} and [\[almd1\]](#almd1){reference-type="eqref" reference="almd1"}, we conclude the proof of [\[almd2\]](#almd2){reference-type="eqref" reference="almd2"}. Adding [\[psikksq\]](#psikksq){reference-type="eqref" reference="psikksq"} and [\[almd1\]](#almd1){reference-type="eqref" reference="almd1"}, we conclude the proof of [\[almd3\]](#almd3){reference-type="eqref" reference="almd3"}. Substituting [\[deriv2\]](#deriv2){reference-type="eqref" reference="deriv2"} into [\[haf2\]](#haf2){reference-type="eqref" reference="haf2"}, we conclude the proof of [\[haf1\]](#haf1){reference-type="eqref" reference="haf1"}. ◻ After conducting a thorough review of the existing literature, it came to our attention that the series $\displaystyle\sum_{k=1}^\infty \frac{\psi_1(k)}{k^2}= \frac{7\pi^4}{360}$ , $\displaystyle\sum_{k=1}^\infty \frac{(-1)^{k-1}H_{k}}{k^3}$, and $\displaystyle\sum_{k=1}^{\infty} \frac{(-1)^{k-1} \psi_1(k)}{k^2}$ were first evaluated in [@bib32 §6.12, pp. 358, §6.52, pp. 502] using a distinct methodology and notation. Vălean employed the $n$th generalized harmonic number notation, $H^{(m)}_n$, defined as $\displaystyle H^{(m)}_n = \sum_{k=1}^n \frac{1}{k^m}$. Of particular interest is our observation that $\displaystyle H^{(2)}_n \equiv \frac{\pi^2}{6} - \psi_1(n+1)$, highlighting the relationship between this notation and the trigamma function. ## Closed form for the dilogarithmic integral {#sec3.2} In this subsection, we derive a closed form for [\[ruh1\]](#ruh1){reference-type="eqref" reference="ruh1"}. $$\label{qot1} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z} \, \mathrm{d}z = \frac{\ln^4{2}}{12} - \frac{\pi^2 \ln^2{2}}{12} - \frac{\pi^4}{60} + \frac{7\ln{2}\zeta(3)}{4} + 2\mathop{\mathrm{Li}}_4\left(\frac{1}{2}\right).$$ *Proof.* By utilizing the series representation of $\mathop{\mathrm{Li}}_2(z)$ for $|z| \leq 1$, we can express it as $$\label{abv2} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z} \, \mathrm{d}z = \sum_{k=1}^\infty \frac{1}{k^2} \int_0^1 z^{k-1} \ln(1+z) \, \mathrm{d}z.$$ Upon integrating term by term, we obtain $$\label{abv1} \int_0^1 z^{k-1} \ln(1+z) \, \mathrm{d}z = \frac{\ln{2}}{k} - \frac{1}{k}\int_0^1 \frac{z^k}{1+z} \, \mathrm{d}z = \frac{\ln{2}}{k} - \frac{1}{2k}\left(\psi\left(\frac{k+2}{2}\right) - \psi\left(\frac{k+1}{2}\right)\right).$$ Applying [\[dupl\]](#dupl){reference-type="eqref" reference="dupl"} and subsequently employing [\[harmonicn\]](#harmonicn){reference-type="eqref" reference="harmonicn"} in [\[abv1\]](#abv1){reference-type="eqref" reference="abv1"}, we derive $$\label{abv3} \int_0^1 z^{k-1} \ln(1+z) \, \mathrm{d}z = \frac{H_k - H_{\frac{k}{2}}}{k}.$$ Substituting [\[abv3\]](#abv3){reference-type="eqref" reference="abv3"} into [\[abv2\]](#abv2){reference-type="eqref" reference="abv2"} and applying [\[harmonicn\]](#harmonicn){reference-type="eqref" reference="harmonicn"} and [\[dupl\]](#dupl){reference-type="eqref" reference="dupl"} after splitting into odd and even parts, we obtain $$\label{hafe2} \begin{split} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z} \, \mathrm{d}z &= \sum_{k=1}^\infty \frac{H_k - H_{\frac{k}{2}}}{k^3} \\&= -2 + \frac{7}{8} \sum_{k=1}^\infty \frac{H_k}{k^3} + \frac{7\ln{2}\zeta(3)}{4} - \sum_{k=1}^\infty \frac{2H_{2k+1} - H_k}{(2k+1)^3}. \end{split}$$ Reindexing the series [\[ftpv\]](#ftpv){reference-type="eqref" reference="ftpv"}, we have $$\label{hafe1} \sum_{k=1}^\infty \frac{H_k}{(2k+1)^3} = \frac{\pi^4}{64} - \frac{7\ln{2}}{4} \zeta(3).$$ Substituting [\[hrmkcu\]](#hrmkcu){reference-type="eqref" reference="hrmkcu"}, [\[wgt1\]](#wgt1){reference-type="eqref" reference="wgt1"}, and [\[hafe1\]](#hafe1){reference-type="eqref" reference="hafe1"} into [\[hafe2\]](#hafe2){reference-type="eqref" reference="hafe2"}, we successfully conclude the proof of [\[qot1\]](#qot1){reference-type="eqref" reference="qot1"}. ◻ Alternatively, we can employ the series representation of $\ln(1 + z)$ for $|z| \leq 1$ to establish the proof of [\[qot1\]](#qot1){reference-type="eqref" reference="qot1"}. This approach yields $$\label{haffn} \begin{split} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1+z)}{z} \, \mathrm{d}z &= \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\left(\frac{\pi^2}{6k} - \frac{H_k}{k^2}\right) = \frac{\pi^4}{72} - \sum_{k=1}^\infty \frac{(-1)^{k-1}H_k}{k^3} \\&= \sum_{k=1}^\infty \frac{H_k}{k^3} - \frac{(-1)^{k-1}H_k}{k^3} = \frac{1}{4}\sum_{k=1}^\infty \frac{H_{2k}}{k^3}. \end{split}$$ By substituting [\[relf1\]](#relf1){reference-type="eqref" reference="relf1"} into [\[haffn\]](#haffn){reference-type="eqref" reference="haffn"}, we readily conclude the proof of [\[qot1\]](#qot1){reference-type="eqref" reference="qot1"}. Notably, this approach appears to be more straightforward, as we immediately recognise $\displaystyle\frac{\pi^4}{72}$ as the closed form of $\displaystyle\sum_{k=1}^\infty \frac{H_k}{k^3}$. ## Generalization of the dilogarithmic integral and related integrals {#sec3.3} In the following theorems, we provide the generalization [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"}, and the generalization of integrals related to [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"}. The closed forms for integrals presented in Theorems [\[thmabd1\]](#thmabd1){reference-type="ref" reference="thmabd1"}--[\[thmhaf5\]](#thmhaf5){reference-type="ref" reference="thmhaf5"} are new and have not been presented elsewhere in the literature. In these theorems, we avoid computations of logarithm of negative real numbers. Let $a \in {\mathbb C}\setminus(-\infty, -1)$. Then [\[thmabd1\]]{#thmabd1 label="thmabd1"} $$\label{thmhaf1} \begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z + \frac{1}{2} \int_0^1 \frac{\ln{z}\ln^2(1 + a z)}{z}&= - \frac{\left(\mathop{\mathrm{Li}}_2(-a)\right)^2}{2}+ \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) \\&\quad-2\mathop{\mathrm{Li}}_4(-a). \end{split}$$ $$\label{thmhaf2} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 + az)}{z}\, \mathrm{d}z + \frac{1}{2}\int_0^1 \frac{\ln{z} \ln^2(1 + az)}{z} \, \mathrm{d}z = - \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) + \mathop{\mathrm{Li}}_4(-a) .$$ We have the following special values: $$\int_0^1 \frac{\ln{z} \ln(1 + z) \ln(1 - z)}{z} \, \mathrm{d}z + \frac{1}{2} \int_0^1 \frac{\ln{z}\ln^2(1 + z)}{z} = \frac{\pi^4}{480},$$ $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 + z)}{z}\, \mathrm{d}z + \frac{1}{2}\int_0^1 \frac{\ln{z} \ln^2(1 + z)}{z} \, \mathrm{d}z = \frac{\pi^4}{240},$$ $$\int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 - z)}{z}\, \mathrm{d}z + \frac{1}{2}\int_0^1 \frac{\ln{z} \ln^2(1 - z)}{z} \, \mathrm{d}z = -\frac{\pi^4}{60}.$$ *Proof.* In an effort to circumvent the computation of logarithm of negative real numbers, the initial part of the proof addresses the case where $a \in {\mathbb C}\setminus (-\infty, 0)$, while the subsequent section pertains to the scenario where $a \in {\mathbb C}\setminus (-\infty, -1) \cup (0, \infty)$. In both cases, it follows that $a \in {\mathbb C}\setminus(-\infty, -1)$. Now, we begin by performing integration by parts, resulting in $$\label{major1} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 + az)}{z}\, \mathrm{d}z = \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z - a\int_0^1 \frac{\ln{z} \mathop{\mathrm{Li}}_2(z)}{1 + az} \, \mathrm{d}z.$$ Vălean employed the Cauchy product of two series [@bibch] to derive [@bib9 §3.49, pp. 335, (3.333)] $$\label{cauchy1} \left(\mathop{\mathrm{Li}}_2(a)\right)^2 = 4\sum_{k=1}^\infty \frac{a^k H_k}{k^3} + 2\sum_{k=1}^\infty \frac{a^k H_k^{(2)}}{k^2} - 6 \sum_{k=1}^\infty \frac{a^k}{k^4}, \quad a \in {\mathbb C}.$$ Utilizing [\[cauchy1\]](#cauchy1){reference-type="eqref" reference="cauchy1"}, Vălean [@bib9 §1.49, (1.218)] provided the closed form for the second resulting integral in [\[major1\]](#major1){reference-type="eqref" reference="major1"}, with $a$ in [\[major1\]](#major1){reference-type="eqref" reference="major1"} substituted with $-a$. This yields $$\label{major2} \int_0^1 \frac{\ln{z} \mathop{\mathrm{Li}}_2(z)}{1 + az} \, \mathrm{d}z = -\frac{\left(\mathop{\mathrm{Li}}_2(-a)\right)^2}{2a} + \frac{\pi^2\mathop{\mathrm{Li}}_2(-a)}{3a} - \frac{3\mathop{\mathrm{Li}}_4(-a)}{a}.$$ At this point, our focus narrows down to obtaining an expression for the first resulting integral in [\[major1\]](#major1){reference-type="eqref" reference="major1"}. By carrying out term-by-term integration, we deduce $$\label{major3} \begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z &= \sum_{k=1}^\infty \frac{(-1)^{k-1} a^k }{k} \int_0^1 z^{k-1} \ln{z} \ln(1 - z) \, \mathrm{d}z \\&= -\mathop{\mathrm{Li}}_4(-a) - \sum_{k=1}^\infty \frac{(-1)^k a^k H_k}{k^3} + \sum_{k=1}^\infty \frac{(-1)^k a^k \psi_1(k)}{k^2}. \end{split}$$ By implementing the notation change $H^{(2)}_n = \pi^2/ 6- \psi_1(n+1)$ in [\[cauchy1\]](#cauchy1){reference-type="eqref" reference="cauchy1"} and rearranging, we have $$\label{cauchy2} \sum_{k=1}^\infty \frac{(-1)^k a^k \psi_1(k)}{k^2} = \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) - 2\mathop{\mathrm{Li}}_4(-a) - \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 + 2\sum_{k=1}^\infty \frac{(-1)^k a^k H_k}{k^3}.$$ Upon substituting [\[cauchy2\]](#cauchy2){reference-type="eqref" reference="cauchy2"} into [\[major3\]](#major3){reference-type="eqref" reference="major3"}, we derive $$\label{major4} \begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z &= -3\mathop{\mathrm{Li}}_4(-a) + \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) - \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 \\&\qquad+ \sum_{k=1}^\infty \frac{(-1)^k a^k H_k}{k^3}. \end{split}$$ We employ the relationship [@bib9 §4.6, pp. 504], [@bibh §3, (3.12)] $$\label{major5} \sum_{k=1}^\infty \frac{H_k}{k+1} p^{k} = \frac{\ln^2(1 - p)}{2p}, \quad p \in {\mathbb C}\setminus [1, \infty).$$ By substituting $p$ with $-p$ in [\[major5\]](#major5){reference-type="eqref" reference="major5"}, integrating over the interval from $p = 0$ to $z$, we deduce $$\label{major6} \sum_{k=1}^\infty \frac{(-1)^{k+1}H_k}{(k+1)^2} z^{k+1} = \frac{1}{2} \int_0^z \frac{\ln^2(1 + p)}{p}\, \mathrm{d}p, \quad z \in {\mathbb C}\setminus(-\infty, -1].$$ Subsequently, we perform a change of variable from $z$ to $t$ in [\[major6\]](#major6){reference-type="eqref" reference="major6"}, divide by $t$, and integrate once more from $t = 0$ to $a$, resulting in $$\label{major7} \sum_{k=1}^\infty \frac{(-1)^{k+1}H_k}{(k+1)^3} a^{k+1} = \frac{1}{2} \int_0^a \int_0^t \frac{\ln^2(1 + p)}{p}\, \mathrm{d}p \, \mathrm{d}t, \quad t, a \in {\mathbb C}\setminus(-\infty, -1].$$ Applying [@bibh §3, (3.6)] to [\[major7\]](#major7){reference-type="eqref" reference="major7"} and reindexing the series on the left-hand side, we arrive at $$\label{major8} \begin{split} \sum_{k=1}^\infty \frac{(-1)^{k-1} H_k a^k}{k^3}&= \int_0^a \frac{1}{t}\left(-\mathop{\mathrm{Li}}_3(-t) - \zeta(3) + \frac{\ln^3(1 + t)}{3} + \mathop{\mathrm{Li}}_3\left(\frac{1}{1+t}\right) \right. \\&\qquad \left.+ \ln(1+ t)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+t}\right) - \frac{1}{2}\ln{t}\ln^2(1 + t)\right) \mathrm{d}t, \quad a \in {\mathbb C}\setminus(-\infty, 0]. \end{split}$$ Through integration by parts, we obtain these two generalized results related to [\[tobeu\]](#tobeu){reference-type="eqref" reference="tobeu"}: $$\label{combh1} \begin{split} \int_0^a \frac{\zeta(3) - \mathop{\mathrm{Li}}_3\left(\frac{1}{1+t}\right)}{t} \, \mathrm{d}t &= \ln{a}\left(\zeta(3) - \mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right)\right) - \ln{a}\ln(1 + a) \mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) \\&\quad- \mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) \mathop{\mathrm{Li}}_2(-a) + \ln{a}\ln(1 + a) \mathop{\mathrm{Li}}_2(-a) \\&\quad- \frac{1}{2}\ln^2(1+ a)\mathop{\mathrm{Li}}_2(-a) + \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 + \frac{\ln^2{a}}{2}\ln(1 + a) \\&\quad- \frac{\ln{a}}{3}\ln^3(1 + a) - \frac{1}{6} \int_0^a \frac{\ln^3(1 + t)}{t} \, \mathrm{d}t, \end{split}$$ $$\label{combh2} \begin{split} \int_0^a \frac{\ln(1 + t) \mathop{\mathrm{Li}}_2\left(\frac{1}{1+t}\right)}{t} \, \mathrm{d}t &= -\mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) \mathop{\mathrm{Li}}_2(-a) + \mathop{\mathrm{Li}}_2(-a)\ln{a}\ln(1 + a) \\&\quad + \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 - \frac{\ln^2(1 + a)}{2} \mathop{\mathrm{Li}}_2(-a) \\&\quad- \frac{1}{2} \int_0^a \frac{\ln^3(1 + t)}{t} \, \mathrm{d}t + \int_0^a \frac{\ln{t}\ln^2(1 + t)}{t}\, \mathrm{d}t. \end{split}$$ By employing [\[combh1\]](#combh1){reference-type="eqref" reference="combh1"} and [\[combh2\]](#combh2){reference-type="eqref" reference="combh2"} within [\[major8\]](#major8){reference-type="eqref" reference="major8"}, we obtain $$\label{major9} \begin{split} \sum_{k=1}^\infty \frac{(-1)^{k-1} H_k a^k}{k^3} &= -\mathop{\mathrm{Li}}_4(-a) + \ln{a}\left(\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) - \zeta(3)\right) + \ln{a}\ln(1 + a) \\&\quad \times \mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) + \ln{a}\ln(1 + a)\left(\frac{\ln^2(1 + a)}{3} - \frac{\ln{a}}{2}\right) \\&\quad+ \frac{1}{2}\int_0^a \frac{\ln{t}\ln^2(1 + t)}{t}\, \mathrm{d}t. \end{split}$$ Applying [@bibh §3, (3.6)] to the last integral in [\[major9\]](#major9){reference-type="eqref" reference="major9"}, we arrive at $$\label{major10} \begin{split} \frac{1}{2}\int_0^a \frac{\ln{t}\ln^2(1 + t)}{t}\, \mathrm{d}t &= \ln{a}\left(\zeta(3) - \mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right)\right) + \frac{1}{2}\ln^2{a}\ln(1 + a) \\&\quad - \frac{1}{3}\ln{a}\ln^3(1 + a) - \ln{a}\ln(1 + a)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) \\&\quad+ \frac{1}{2} \int_0^1 \frac{\ln{t}\ln^2(1 + a t)}{t}\, \mathrm{d}t. \end{split}$$ Substituting [\[major10\]](#major10){reference-type="eqref" reference="major10"} into [\[major9\]](#major9){reference-type="eqref" reference="major9"}, we deduce $$\label{major11} \sum_{k=1}^\infty \frac{(-1)^{k} H_k a^k}{k^3} = \mathop{\mathrm{Li}}_4(-a) - \frac{1}{2} \int_0^1 \frac{\ln{z}\ln^2(1 + a z)}{z}\, \mathrm{d}z, \quad a \in {\mathbb C}\setminus(-\infty, 0).$$ Likewise, for $a \in {\mathbb C}\setminus (-\infty, 0] \cup [1, \infty)$, we obtain using [\[major5\]](#major5){reference-type="eqref" reference="major5"} $$\label{simha1} \int_0^a \frac{\ln{z}\ln^2(1 - z)}{z}\, \mathrm{d}z = - 2\ln{a}\mathop{\mathrm{Li}}_3(a) + 2\mathop{\mathrm{Li}}_4(a) -2\sum_{k=1}^\infty \frac{H_k a^k}{k^3} + 2\ln{a}\sum_{k=1}^\infty \frac{H_k a^k}{k^2}.$$ By making use of [@bib9 §4.6, pp. 399, (4.36)] for the last series in [\[simha1\]](#simha1){reference-type="eqref" reference="simha1"}, we deduce $$\label{simha2} \sum_{k=1}^\infty \frac{H_k a^k}{k^3} = \mathop{\mathrm{Li}}_4(a) - \frac{1}{2} \int_0^1 \frac{\ln{z}\ln^2(1 - a z)}{z}\, \mathrm{d}z, \quad a \in {\mathbb C}\setminus (-\infty, 0) \cup (1, \infty).$$ Substituting $a$ with $-a$ in [\[simha2\]](#simha2){reference-type="eqref" reference="simha2"}, we once again arrive at [\[major11\]](#major11){reference-type="eqref" reference="major11"}. However, this time, [\[major11\]](#major11){reference-type="eqref" reference="major11"} is applicable for $a \in {\mathbb C}\setminus (-\infty, -1) \cup (0, \infty)$. Further substituting [\[major11\]](#major11){reference-type="eqref" reference="major11"} into [\[major4\]](#major4){reference-type="eqref" reference="major4"}, we derive $$\label{major12} \begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z &= -2\mathop{\mathrm{Li}}_4(-a) + \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) - \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 \\&\quad-\frac{1}{2} \int_0^1 \frac{\ln{z}\ln^2(1 + a z)}{z}\, \mathrm{d}z. \end{split}$$ Rearranging [\[major12\]](#major12){reference-type="eqref" reference="major12"} concludes the proof of [\[thmhaf1\]](#thmhaf1){reference-type="eqref" reference="thmhaf1"}. Finally, by substituting [\[major2\]](#major2){reference-type="eqref" reference="major2"} and [\[major12\]](#major12){reference-type="eqref" reference="major12"} into [\[major1\]](#major1){reference-type="eqref" reference="major1"}, we complete the proof of [\[thmhaf2\]](#thmhaf2){reference-type="eqref" reference="thmhaf2"}. ◻ Let $a \in {\mathbb C}\setminus(-\infty, 0]$. Then [\[thmhaf22\]]{#thmhaf22 label="thmhaf22"} $$\label{thmhaf2} \begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z &= -\frac{\pi^4}{90} - \frac{\left(\mathop{\mathrm{Li}}_2(-a)\right)^2}{2}+ \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) - \mathop{\mathrm{Li}}_4(-a) \\&\quad+ \frac{\pi^2}{12} \ln^2(1 + a) + \frac{1}{3}\ln{a}\ln^3(1 + a) - \frac{1}{4}\ln^4(1 + a) \\&\quad + \ln(1 + a)\left(\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_3\left(\frac{a}{1+a}\right)\right) \\&\quad+ \mathop{\mathrm{Li}}_4\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_4\left(\frac{a}{1+a}\right). \end{split}$$ *Proof.* We establish the proof by evaluating the second integral in [\[thmhaf1\]](#thmhaf1){reference-type="eqref" reference="thmhaf1"}. Employing analogous substitutions to those used when evaluating the integral for $a=1$ in [\[deriv6\]](#deriv6){reference-type="eqref" reference="deriv6"}, we arrive at $$\label{disin} \int_0^1 \frac{\ln{z}\ln^2(1 + az)}{z}\, \mathrm{d}z = \int_{\frac{1}{1+a}}^1 \frac{\ln(1 - z)\ln^2{z} - \ln^3{z} - \ln{a}\ln^2{z}}{z(1 - z)} \, \mathrm{d}z.$$ For the first resulting integral, we have $$\label{eq365h} \begin{split} \int_{\frac{1}{1+a}}^1 \frac{\ln(1 - z)\ln^2{z}}{z(1- z)} &= \int_0^1 \frac{\ln^2{z}\ln(1 - z)}{z} \, \mathrm{d}z - \int_0^{\frac{1}{1+a}} \frac{\ln^2{z}\ln(1 - z)}{z} \, \mathrm{d}z \\&\quad+ \int_0^{\frac{a}{1+a}} \frac{\ln{z}\ln^2(1 - z)}{z} \, \mathrm{d}z. \end{split}$$ Using the following results [@bib9 §4.6, pp. 399, (4.36)], [@bib9 §4.6, pp. 399, (4.38)] $$\label{harmo1} \sum_{k=1}^\infty \frac{H_k}{k^2} z^{k} = \zeta(3) + \mathop{\mathrm{Li}}_2(1-z)\ln(1-z) + \mathop{\mathrm{Li}}_3(z) - \mathop{\mathrm{Li}}_3(1-z) + \frac{1}{2}\ln{z}\ln^2(1-z),$$ $$\label{harmo2} \begin{split} \sum_{k=1}^\infty \frac{H_k}{k^3} z^{k} &= \frac{\pi^4}{90} + \zeta(3)\ln(1 - z) + \frac{\pi^2}{12}\ln^2(1 - z) + \frac{1}{24}\ln^4(1 - z) - \frac{1}{6}\ln{z}\ln^3(1 - z) \\&\quad- \ln(1 - z)\mathop{\mathrm{Li}}_3(z) + 2\mathop{\mathrm{Li}}_4(z) - \mathop{\mathrm{Li}}_4(1-z) + \mathop{\mathrm{Li}}_4\left(\frac{z}{z-1}\right), \end{split}$$ in [\[simha1\]](#simha1){reference-type="eqref" reference="simha1"}, we deduce $$\label{newinh1} \begin{split} \int_0^z \frac{\ln{t}\ln^2(1 - t)}{t} \, \mathrm{d}t &= -\frac{\pi^4}{45} - 2\zeta(3)\ln(1 - z) - \frac{\pi^2}{6}\ln^2(1 - z) - \frac{1}{12}\ln^4(1 - z) \\&\quad+ \frac{1}{3}\ln{z}\ln^3(1 - z) + 2\ln(1 - z)\mathop{\mathrm{Li}}_3(z) - 2\mathop{\mathrm{Li}}_4(z) + 2\mathop{\mathrm{Li}}_4(1 - z) \\&\quad- 2\mathop{\mathrm{Li}}_4\left(\frac{z}{z-1}\right) + 2\ln{z}\zeta(3) + 2\ln{z}\ln(1 - z)\mathop{\mathrm{Li}}_2(1 - z) \\&\quad- 2\ln{z}\mathop{\mathrm{Li}}_3(1 - z) + \ln^2{z}\ln^2(1 - z). \end{split}$$ The following results are obtained through straightforward term-by-term integration $$\label{frsin1} \begin{split} \int_0^{\frac{1}{1+a}} \frac{\ln^3{z}}{1 - z} \, \mathrm{d}z &= (\ln{a} - \ln(1+ a)) \ln^3(1 + a)- 3\ln^2(1 + a)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) \\&\quad- 6\ln(1 + a)\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) - 6\mathop{\mathrm{Li}}_4\left(\frac{1}{1+a}\right), \end{split}$$ $$\label{frsin2} \begin{split} \int_0^{\frac{a}{1+a}} \frac{\ln^2(1 - z)}{z}\, \mathrm{d}z &= 2\zeta(3) - 2\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) + (\ln{a} - \ln(1 +a))\ln^2(1 + a) \\&\quad- 2\ln(1+ a)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right). \end{split}$$ By replacing $z$ with $a/(a+1)$ in [\[newinh1\]](#newinh1){reference-type="eqref" reference="newinh1"} and employing both [\[almend1\]](#almend1){reference-type="eqref" reference="almend1"} and [\[frsin1\]](#frsin1){reference-type="eqref" reference="frsin1"}, and further substituting the latter in [\[disin\]](#disin){reference-type="eqref" reference="disin"} while taking into account [\[frsin2\]](#frsin2){reference-type="eqref" reference="frsin2"}, we ultimately arrive at $$\label{fimp1} \begin{split} \int_0^1 \frac{\ln{z}\ln^2(1 + az)}{z}\, \mathrm{d}z &= \frac{\pi^4}{45} - \frac{\pi^2}{6} \ln^2(1 + a) - \frac{2}{3}\ln{a}\ln^3(1 + a) + \frac{1}{2}\ln^4(1 + a) \\&\quad- 2\ln(1 + a)\left(\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_3\left(\frac{a}{1+a}\right)\right) - 2\mathop{\mathrm{Li}}_4(-a) \\&\quad - 2\mathop{\mathrm{Li}}_4\left(\frac{1}{1+a}\right) - 2\mathop{\mathrm{Li}}_4\left(\frac{a}{1+a}\right), \quad a \in {\mathbb C}\setminus (-\infty, 0]. \end{split}$$ Substituting [\[fimp1\]](#fimp1){reference-type="eqref" reference="fimp1"} into [\[thmhaf1\]](#thmhaf1){reference-type="eqref" reference="thmhaf1"}, we thereby conclude the proof of Theorem [\[thmhaf22\]](#thmhaf22){reference-type="ref" reference="thmhaf22"}. ◻ Let $a \in {\mathbb C}\setminus(-\infty, 0]$. Then [\[thmhaf3\]]{#thmhaf3 label="thmhaf3"} $$\label{inthmhaf3} \begin{split} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 + az)}{z}\, \mathrm{d}z &= -\frac{\pi^4}{90} - \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) + 2\mathop{\mathrm{Li}}_4(-a) + \frac{\pi^2}{12} \ln^2(1 + a) \\&\quad+ \frac{1}{3}\ln{a}\ln^3(1 + a) - \frac{1}{4}\ln^4(1 + a) + \ln(1 + a)\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) \\&\quad+ \ln(1 + a)\mathop{\mathrm{Li}}_3\left(\frac{a}{1+a}\right)+ \mathop{\mathrm{Li}}_4\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_4\left(\frac{a}{1+a}\right). \end{split}$$ *Proof.* Through the substitution of [\[fimp1\]](#fimp1){reference-type="eqref" reference="fimp1"} into [\[thmhaf2\]](#thmhaf2){reference-type="eqref" reference="thmhaf2"}, we establish the proof of Theorem [\[thmhaf3\]](#thmhaf3){reference-type="ref" reference="thmhaf3"}. ◻ Considering [\[major4\]](#major4){reference-type="eqref" reference="major4"}, we substitute $z=-a$ in [\[harmo2\]](#harmo2){reference-type="eqref" reference="harmo2"}, resulting in [\[greatremark\]]{#greatremark label="greatremark"} $$\label{harmo21} \begin{split} \sum_{k=1}^\infty \frac{(-1)^{k} H_k a^k}{k^3} &= \frac{\pi^4}{90} + \zeta(3)\ln(1 + a) + \frac{\pi^2}{12}\ln^2(1 + a) + \frac{1}{24}\ln^4(1 + a) \\&\quad - \frac{1}{6}\ln{(-a)}\ln^3(1 + a) - \ln(1 + a)\mathop{\mathrm{Li}}_3(-a) + 2\mathop{\mathrm{Li}}_4(-a) - \mathop{\mathrm{Li}}_4(1+a) \\&\quad + \mathop{\mathrm{Li}}_4\left(\frac{a}{a+1}\right). \end{split}$$ While Vălean impressively provided different representations for [\[harmo1\]](#harmo1){reference-type="eqref" reference="harmo1"} and [\[harmo2\]](#harmo2){reference-type="eqref" reference="harmo2"}, it is worth noting that the series [\[harmo1\]](#harmo1){reference-type="eqref" reference="harmo1"} and [\[harmo2\]](#harmo2){reference-type="eqref" reference="harmo2"} are truly convergent for $z \in {\mathbb C}$, where $|z| \leq 1$ and $z \neq 0, 1$, but the polylogarithms on the right-hand side of [\[harmo1\]](#harmo1){reference-type="eqref" reference="harmo1"} and [\[harmo2\]](#harmo2){reference-type="eqref" reference="harmo2"} converge without analytic continuation for only $z \in {\mathbb C}\setminus(-\infty, 0] \cup [1, \infty)$. Consequently, [\[harmo21\]](#harmo21){reference-type="eqref" reference="harmo21"} holds true for $a \in {\mathbb C}\setminus (-\infty, -1] \cup [0, \infty)$ without analytic continuation. In another region where $a \in {\mathbb C}\setminus(-\infty, 0] \cup [1, \infty)$, by employing the relationships [\[major11\]](#major11){reference-type="eqref" reference="major11"} and [\[fimp1\]](#fimp1){reference-type="eqref" reference="fimp1"} that we established, we arrive at $$\label{news1} \begin{split} \sum_{k=1}^\infty \frac{(-1)^{k} H_k a^k}{k^3} &= -\frac{\pi^4}{90} + \frac{\pi^2}{12} \ln^2(1 + a) + \frac{1}{3}\ln{a}\ln^3(1 + a) + \frac{1}{2}\ln^4(1 + a) \\&\quad +\ln(1 + a)\left(\mathop{\mathrm{Li}}_3\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_3\left(\frac{a}{1+a}\right)\right) + 2\mathop{\mathrm{Li}}_4(-a) \\&\quad + \mathop{\mathrm{Li}}_4\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_4\left(\frac{a}{1+a}\right), \end{split}$$ which is valid for all $z \in {\mathbb C}$, where $|z| \leq 1$ and $z \neq -1, 0$. The representation [\[news1\]](#news1){reference-type="eqref" reference="news1"}, applicable to $a \in {\mathbb C}\setminus(-\infty, 0] \cup [1, \infty)$, circumvents the need to compute logarithms of negative numbers. Its advantage over [\[harmo21\]](#harmo21){reference-type="eqref" reference="harmo21"} lies in the avoidance of analytic continuation of polylogarithms for values of $a$ within ${\mathbb C}\setminus(-\infty, 0] \cup [1, \infty)$. This implies that the representation on the right-hand side of [\[news1\]](#news1){reference-type="eqref" reference="news1"} provides analytic continuation for the representation on the right-hand side of [\[harmo21\]](#harmo21){reference-type="eqref" reference="harmo21"} for $a \in {\mathbb C}\setminus(-\infty, 0] \cup [1, \infty)$. Moreover, while taking the analytical continuation of the polylogarithms into consideration, [\[news1\]](#news1){reference-type="eqref" reference="news1"} enables us to extend the generalization [\[begeqh1\]](#begeqh1){reference-type="eqref" reference="begeqh1"} to a broader domain while sidestepping logarithmic operations on negative real numbers. In light of these observations, we reveal two additional closed forms for the integrals [\[thmhaf2\]](#thmhaf2){reference-type="eqref" reference="thmhaf2"} and [\[inthmhaf3\]](#inthmhaf3){reference-type="eqref" reference="inthmhaf3"} in the following theorems. Let $a \in {\mathbb C}\setminus (-\infty, -1] \cup [0, \infty)$. Then [\[thmhaf4\]]{#thmhaf4 label="thmhaf4"} $$\begin{split} \int_0^1 \frac{\ln{z} \ln(1 + az) \ln(1 - z)}{z} \, \mathrm{d}z &= \frac{\pi^4}{90} -\mathop{\mathrm{Li}}_4(-a) + \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) - \frac{1}{2}\left(\mathop{\mathrm{Li}}_2(-a)\right)^2 \\&\quad+ \zeta(3)\ln(1 + a) + \frac{\pi^2}{12}\ln^2(1 + a) + \frac{1}{24}\ln^4(1 + a) \\&\quad- \frac{1}{6}\ln{(-a)}\ln^3(1 + a) - \ln(1+ a)\mathop{\mathrm{Li}}_3(-a) \\&\quad- \mathop{\mathrm{Li}}_4(1+a)+ \mathop{\mathrm{Li}}_4\left(\frac{a}{a+1}\right). \end{split}$$ *Proof.* Upon substituting [\[harmo21\]](#harmo21){reference-type="eqref" reference="harmo21"} into [\[major4\]](#major4){reference-type="eqref" reference="major4"}, we conclude the proof of Theorem [\[thmhaf4\]](#thmhaf4){reference-type="ref" reference="thmhaf4"}. ◻ Let $a \in {\mathbb C}\setminus (-\infty, -1] \cup [0, \infty)$. Then [\[thmhaf5\]]{#thmhaf5 label="thmhaf5"} $$\begin{split} \int_0^1 \frac{\mathop{\mathrm{Li}}_2(z) \ln(1 + az)}{z}\, \mathrm{d}z &= \frac{\pi^4}{90} + 2\mathop{\mathrm{Li}}_4(-a) - \frac{\pi^2}{6}\mathop{\mathrm{Li}}_2(-a) + \zeta(3)\ln(1 + a) + \frac{\pi^2}{12}\ln^2(1 + a) \\&\quad + \frac{1}{24}\ln^4(1 + a) - \frac{1}{6}\ln{(-a)}\ln^3(1 + a) - \ln(1 + a)\mathop{\mathrm{Li}}_3(-a) \\&\quad - \mathop{\mathrm{Li}}_4(1+a) + \mathop{\mathrm{Li}}_4\left(\frac{a}{a+1}\right). \end{split}$$ *Proof.* Considering [\[simha2\]](#simha2){reference-type="eqref" reference="simha2"} and [\[harmo21\]](#harmo21){reference-type="eqref" reference="harmo21"}, we infer the following closed form $$\label{thmhaf5eq} \begin{split} \int_0^1 \frac{\ln{z}\ln^2(1 + a z)}{z}\, \mathrm{d}z &= -\frac{\pi^4}{45} - 2\zeta(3)\ln(1 + a) - \frac{\pi^2}{6}\ln^2(1 + a) - \frac{1}{12}\ln^4(1 + a) \\&\quad + \frac{1}{3}\ln{(-a)}\ln^3(1 + a) + 2\ln(1 + a)\mathop{\mathrm{Li}}_3(-a) - 2\mathop{\mathrm{Li}}_4(-a) \\&\quad + 2\mathop{\mathrm{Li}}_4(1+a) - 2\mathop{\mathrm{Li}}_4\left(\frac{a}{a+1}\right), \quad a \in {\mathbb C}\setminus (-\infty, -1] \cup [0, \infty). \end{split}$$ By substituting [\[thmhaf5eq\]](#thmhaf5eq){reference-type="eqref" reference="thmhaf5eq"} into [\[thmhaf2\]](#thmhaf2){reference-type="eqref" reference="thmhaf2"}, we conclude the proof of Theorem [\[thmhaf5\]](#thmhaf5){reference-type="ref" reference="thmhaf5"}. ◻ ## More intriguing identities {#sec3.4} In this subsection, we unveil identities derived from our earlier established results. The first two theorems lead to a rediscovery of Jonquière's inversion formula for specific cases. Subsequently, we provide two instances of analytic continuation for the series represented by [\[harmo1\]](#harmo1){reference-type="eqref" reference="harmo1"}. Furthermore, we introduce a theorem for double infinite series with symmetric summands, applying it to derive new identities. Let $a \in {\mathbb C}\setminus (-\infty, -1] \cup [0, \infty)$. Then [\[anathm\]]{#anathm label="anathm"} $$\label{myiden1} \begin{split} \mathop{\mathrm{Li}}_4\left(\frac{z-1}{z}\right) + \mathop{\mathrm{Li}}_4\left(\frac{z}{z-1}\right) &= -\frac{7\pi^4}{360} - \frac{1}{4}\ln^2{z}\ln^2(1 - z) - \frac{\pi^2}{12}\ln^2(1 - z) - \frac{1}{24}\ln^4(1 - z) \\&\quad+\frac{\pi^2}{6}\ln{z}\ln(1 - z) + \frac{1}{6}\ln{z}\ln^3(1 - z) - \frac{\pi^2}{12}\ln^2{z} \\&\quad + \frac{1}{6}\ln^3{z}\ln(1 - z) - \frac{1}{24}\ln^4{z}. \end{split}$$ *Proof.* On the transformation $z \longrightarrow 1 - z$, we have $$\label{eulan1} \int_0^z \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t + \int_0^{1-z} \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t = \frac{1}{2}\ln^2{z}\ln^2(1 - z) + \int_0^1 \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t.$$ By substituting [\[foreul1\]](#foreul1){reference-type="eqref" reference="foreul1"} into the right-hand side of [\[eulan2\]](#eulan2){reference-type="eqref" reference="eulan2"}, we obtain $$\label{eulan2} \int_0^z \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t + \int_0^{1-z} \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t = -\frac{\pi^4}{180} + \frac{1}{2}\ln^2{z}\ln^2(1 - z).$$ Applying [\[newinh1\]](#newinh1){reference-type="eqref" reference="newinh1"} to the left-hand side of [\[eulan2\]](#eulan2){reference-type="eqref" reference="eulan2"}, we conclude the proof of Theorem [\[anathm\]](#anathm){reference-type="ref" reference="anathm"}. ◻ Let us define a function $F(z)$ as $$F(z) = \int_0^z \frac{\ln{t}\ln^2(1-t)}{t} \, \mathrm{d}t.$$ With this notation, [\[eulan2\]](#eulan2){reference-type="eqref" reference="eulan2"} can be expressed as $$\label{eulan3} F(z) + F(1 - z) = -\frac{\pi^4}{180} + \frac{1}{2}\ln^2{z}\ln^2(1 - z).$$ We refer to Theorem [\[anathm\]](#anathm){reference-type="ref" reference="anathm"} as an analogue of Euler's reflection formula [@bib23 (25.12.6)], since [\[eulan3\]](#eulan3){reference-type="eqref" reference="eulan3"} contains expressions akin to those found in Euler's formula. It involves $\pi^2$ (squared and multiplied by $-\frac{1}{30}$), $\ln{z}\ln(1-z)$ (squared and multiplied by $-\frac{1}{2}$), and substitutes the dilogarithm $\mathop{\mathrm{Li}}_2(z)$ with our defined function $F(z)$. This resemblance underscores the connection between Theorem [\[anathm\]](#anathm){reference-type="ref" reference="anathm"} and Euler's formula. The identity [\[myiden1\]](#myiden1){reference-type="eqref" reference="myiden1"} is not new; however, we can refer to Theorem [\[anathm\]](#anathm){reference-type="ref" reference="anathm"} as a rediscovery, as the proof we provide is novel. Theorem [\[anathm\]](#anathm){reference-type="ref" reference="anathm"} can be derived by substituting 4 for $m$ and $\frac{z}{z-1}$ for $z$ in Jonquière's inversion formula [@bib2 §1.11.1, pp. 31, (16)]. Jonquière's inversion formula is derived from the Lerch transformation formula [@bib2 §1.11(7), pp. 29], and the Lerch transformation formula is derived using the residue theorem (see [@bib2 §1.11, pp. 28]). This demonstrates the uniqueness of our proof. [\[newthgh\]]{#newthgh label="newthgh"} Let $z \in {\mathbb C}\setminus (-\infty, 0]$. Then $$\mathop{\mathrm{Li}}_4\left(-\frac{1}{a}\right) + \mathop{\mathrm{Li}}_4(-a) = -\frac{7\pi^4}{360} - \frac{\pi^2}{12}\ln^2{a} - \frac{\ln^4{a}}{24}.$$ *Proof.* Define a function $G(z)$ as $$G(a) = \int_0^1 \frac{\ln{z}\ln^2(1 + az)}{z}\, \mathrm{d}z.$$ Now, using [\[disin\]](#disin){reference-type="eqref" reference="disin"} and [\[eq365h\]](#eq365h){reference-type="eqref" reference="eq365h"}, we can establish the following relationship $$\label{geezh} \begin{split} G(a) + G\left(\frac{1}{a}\right) &= -\frac{2\pi^4}{45} - \int_0^{\frac{1}{1+a}} \frac{\ln^2{z}\ln(1 - z)}{z} \, \mathrm{d}z - \int_0^{\frac{a}{1+a}} \frac{\ln^2{z}\ln(1 - z)}{z} \, \mathrm{d}z \\&\quad+ \int_0^{\frac{a}{1+a}} \frac{\ln{z}\ln^2(1 - z)}{z} \, \mathrm{d}z + \int_0^{\frac{1}{1+a}} \frac{\ln{z}\ln^2(1 - z)}{z} \, \mathrm{d}z \\&\quad -\int_{\frac{1}{1+a}}^1 \frac{\ln^3{z} + \ln{a}\ln^2{z}}{z(1 - z)} \, \mathrm{d}z - \int_{\frac{a}{1+a}}^1 \frac{\ln^3{z} - \ln{a}\ln^2{z}}{z(1 - z)} \, \mathrm{d}z. \end{split}$$ By applying [\[eulan3\]](#eulan3){reference-type="eqref" reference="eulan3"} to [\[geezh\]](#geezh){reference-type="eqref" reference="geezh"} and subsequently utilizing [\[newinh1\]](#newinh1){reference-type="eqref" reference="newinh1"}--[\[frsin2\]](#frsin2){reference-type="eqref" reference="frsin2"} in [\[geezh\]](#geezh){reference-type="eqref" reference="geezh"}, we derive $$\label{geezh2} \begin{split} &\mathop{\mathrm{Li}}_4\left(-\frac{1}{a}\right) + \mathop{\mathrm{Li}}_4(-a) + \left(\ln{a}\ln(1 + a) - \ln^2(1+a)\right)\left(\mathop{\mathrm{Li}}_2\left(\frac{1}{1+a}\right) + \mathop{\mathrm{Li}}_2\left(\frac{a}{1+a}\right)\right) \\&= -\frac{7\pi^4}{360} - \frac{\pi^2}{12}\ln^2{a} - \frac{\ln^4{a}}{24} + \frac{\pi^2}{6}\ln{a}\ln(1 + a) - \frac{\pi^2}{6}\ln^2(1 + a) - 2\ln{a}\ln^3(1+ a) \\&\quad+ \ln^2{a}\ln^2(1 + a) + \ln^4(1 + a). \end{split}$$ Now, applying Euler's reflection formula [@bib23 (25.12.6)] to the dilogarithms in [\[geezh2\]](#geezh2){reference-type="eqref" reference="geezh2"}, we conclude the proof of Theorem [\[geezh2\]](#geezh2){reference-type="ref" reference="geezh2"}. Additionally, it's worth noting that the inversion formula can also be derived by substituting 4 for $m$ and $-\frac{1}{z}$ for $z$ in Jonquière's inversion formula [@bib2 §1.11.1, pp. 31, (16)]. However, this newly presented proof offers the advantage of avoiding logarithms of negative real numbers. ◻ Let $z\in {\mathbb C}$, where $|z| \leq1$ and $z \neq -1, 0$. Then [\[nnilh1\]]{#nnilh1 label="nnilh1"} $$\label{lastheq1} \begin{split} \sum_{k=1}^\infty \frac{(-1)^k H_k}{(k+1)^2} z^{k+1} &= -\zeta(3) + \frac{1}{3}\ln^3(1 + z) + \mathop{\mathrm{Li}}_3\left(\frac{1}{1+z}\right) + \ln(1+z)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+z}\right) \\&\quad-\frac{1}{2}\ln{z}\ln^2(1+z). \end{split}$$ *Proof.* The proof follows from [\[major6\]](#major6){reference-type="eqref" reference="major6"} and [@bibh §3, (3.6)]. ◻ Let $z\in {\mathbb C}$, where $|z| \leq1$ and $z \neq -1, 0$. Then [\[onlyc\]]{#onlyc label="onlyc"} $$\begin{split} \sum_{k=1}^\infty \frac{(-1)^k H_{k}}{k^2} z^{k} &= \zeta(3) - \frac{1}{3}\ln^3(1 + z) + \mathop{\mathrm{Li}}_3(-z) - \mathop{\mathrm{Li}}_3\left(\frac{1}{1+z}\right) - \ln(1+z)\mathop{\mathrm{Li}}_2\left(\frac{1}{1+z}\right) \\&\quad+\frac{1}{2}\ln{z}\ln^2(1+z). \end{split}$$ *Proof.* Reindexing the series in the left-hand side of [\[lastheq1\]](#lastheq1){reference-type="eqref" reference="lastheq1"}, we derive $$\label{lastheq2} \sum_{k=1}^\infty \frac{(-1)^k H_k}{(k+1)^2} z^{k+1} = -\sum_{k=1}^\infty \frac{(-1)^k H_{k}}{k^2} z^{k} + \mathop{\mathrm{Li}}_3(-z).$$ Substituting [\[lastheq2\]](#lastheq2){reference-type="eqref" reference="lastheq2"} into [\[lastheq1\]](#lastheq1){reference-type="eqref" reference="lastheq1"} completes the proof of Corollary [\[onlyc\]](#onlyc){reference-type="ref" reference="onlyc"}. ◻ The remarks made in Remark [\[greatremark\]](#greatremark){reference-type="ref" reference="greatremark"} also apply to Theorem [\[nnilh1\]](#nnilh1){reference-type="ref" reference="nnilh1"} and Corollary [\[onlyc\]](#onlyc){reference-type="ref" reference="onlyc"}. Importantly, the findings in Theorem [\[nnilh1\]](#nnilh1){reference-type="ref" reference="nnilh1"} and Corollary [\[onlyc\]](#onlyc){reference-type="ref" reference="onlyc"} are not documented elsewhere in the existing literature. Let $f$ be an arbitrary symmetric function of two variables such that [\[theoremh9\]]{#theoremh9 label="theoremh9"} 1. if $f(k, j) = (-1)^{k+j} g(k, j)$, where $g(k, j)$ is positive, and $\sum_{k=1}^\infty f(k, k)$ converges, or 2. if $f(k, j)$ is positive and $\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j)$ converges. Then $\sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j, j)$ is convergent, and $$\label{bigmastht1} \sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j, j) = \frac{1}{2}\left(\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) + \sum_{k=1}^\infty f(k, k)\right).$$ *Proof.* Applying Corollary [\[newcorh1\]](#newcorh1){reference-type="ref" reference="newcorh1"} to interchange order of summation, we have $$\label{symmh0} \sum_{j=1}^\infty \sum_{k=1}^j f(k, j) = \sum_{k=1}^\infty \sum_{j=k}^\infty f(k, j).$$ Next, we interchange the roles of the dummy variables, resulting in $$\label{symmh1} \sum_{j=1}^\infty \sum_{k=1}^j f(k, j) = \sum_{k=1}^\infty \sum_{j=1}^k f(j,k).$$ Since by hypothesis $f(k, j)$ is a symmetric function, it follows that $f(j, k) = f(k, j)$, and thus, [\[symmh1\]](#symmh1){reference-type="eqref" reference="symmh1"} can be expressed as $$\label{symmh2} \sum_{j=1}^\infty \sum_{k=1}^j f(k, j) = \sum_{k=1}^\infty \sum_{j=1}^k f(k, j).$$ Combining the two resulting expressions from [\[symmh0\]](#symmh0){reference-type="eqref" reference="symmh0"} and [\[symmh2\]](#symmh2){reference-type="eqref" reference="symmh2"}, we deduce $$\label{symmh3} \sum_{j=1}^\infty \sum_{k=1}^j f(k, j) = \frac{1}{2}\left(\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) + \sum_{k=1}^\infty f(k, k)\right),$$ By reindexing the series on the left-hand side of [\[symmh3\]](#symmh3){reference-type="eqref" reference="symmh3"}, we have $$\label{symmh4} \sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) - \sum_{j=1}^\infty \sum_{k=j+1}^\infty f(k, j) = \frac{1}{2}\left(\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) + \sum_{k=1}^\infty f(k, k)\right).$$ Shifting the index in the second series on the left-hand side of [\[symmh3\]](#symmh3){reference-type="eqref" reference="symmh3"}, we obtain $$\label{symmh5} \sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) - \sum_{j=1}^\infty \sum_{k=1}^\infty f(k+j, j) = \frac{1}{2}\left(\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) + \sum_{k=1}^\infty f(k, k)\right).$$ Upon reindexing the second series on the left-hand side of [\[symmh5\]](#symmh5){reference-type="eqref" reference="symmh5"} as $$\sum_{j=1}^\infty \sum_{k=1}^\infty f(k+j, j) = \sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j, j) - \sum_{k=1}^\infty f(k, k),$$ and rearranging, [\[bigmastht1\]](#bigmastht1){reference-type="eqref" reference="bigmastht1"} follows. We must now demonstrate the conditions for convergence. To begin with the first condition, let $f(k, j) = (-1)^{k+j} g(k, j)$. It is evident that the symmetricity of $f$ implies the symmetricity of $g$. Consider the following series $$\sum_{j=1}^\infty \sum_{k=1}^\infty (-1)^{k+j} g(k, j) = \sum_{j=1}^\infty g(k, k) - 2\sum_{1 \leq j < k < \infty} g(k, j)$$ Since, according to our hypothesis, $g(k, j)$ is positive, we can write $$\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) = \sum_{j=1}^\infty g(k, k) - 2\sum_{1 \leq j < k < \infty} g(k, j) \leq \sum_{j=1}^\infty g(k, k) = \sum_{j=1}^\infty f(k, k).$$ By applying the comparison test, we can conclude that $\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j)$ converges if and only if $\sum_{j=1}^\infty f(k, k)$ converges. As the sum of two convergent series is itself convergent, we can now deduce from [\[symmh0\]](#symmh0){reference-type="eqref" reference="symmh0"} that $\sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j, j)$ converges if $\sum_{k=1}^\infty f(k, k)$ converges. Moving on to the second condition, assume $\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j)$ converges. It is clear that since $f(k, j)$ is positive, we can write $$\sum_{k=1}^\infty f(k, k) \leq \sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j) = \sum_{j=1}^\infty f(k, k) + 2\sum_{1 \leq j < k < \infty} f(k, j).$$ By employing a similar argument, we can conclude that $\sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j, j)$ is indeed convergent, provided $\sum_{j=1}^\infty \sum_{k=1}^\infty f(k, j)$ converges. ◻ Let $f$ be an arbitrary function such that 1. if $f(k) = (-1)^{k} g(k)$, where $g(k)$ is positive, and $\sum_{k=1}^\infty f^2(k)$ converges, or 2. if $f(k)$ is positive and $\sum_{k=1}^\infty f(k)$ converges. Then $\sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j)f(j)$ is convergent and [\[coroh12\]]{#coroh12 label="coroh12"} $$\sum_{j=1}^\infty \sum_{k=0}^\infty f(k+j)f(j) = \frac{1}{2}\left(\left(\sum_{k=1}^\infty f(k)\right)^2+ \sum_{k=1}^\infty f^2(k)\right).$$ *Proof.* Taking $f(k, j) = f(k)g(j)$ in Theorem [\[theoremh9\]](#theoremh9){reference-type="ref" reference="theoremh9"}, the symmetricity of $f(k, j)$ implies $f(k) = g(k)$. As such, the proof of Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} is complete. ◻ Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} possesses the remarkable property of transforming a double infinite series into an expression that consists of the sum of the square of an infinite series and another infinite series. It is worth noting that the results established in Theorem [\[theoremh9\]](#theoremh9){reference-type="ref" reference="theoremh9"} and Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} have not been previously presented elsewhere in the existing literature. In the following theorem, we present a new generalization of [\[psikhal\]](#psikhal){reference-type="eqref" reference="psikhal"}. Let $m$ be any positive integer greater than $1$. Then [\[tirtheh1\]]{#tirtheh1 label="tirtheh1"} $$\label{bgthm224} \sum_{j=1}^{\infty} \frac{\psi_{m-1}\left(\frac{2j-1}{2}\right)}{(2j-1)^m} = (-1)^m 2^{m-1} (m-1)! \left(\left(1 - 2^{-m}\right)^2 \zeta^2(m) + \left(1 - 2^{-2m}\right) \zeta(2m)\right).$$ **Example 1**. For $m=3, 4, 5$, we have $$\begin{aligned} &\sum_{j=1}^{\infty} \frac{\psi_{2}\left(\frac{2j-1}{2}\right)}{(2j-1)^3} = -\frac{\pi ^6}{120} - \frac{49 \zeta^2(3)}{8}, \\ &\sum_{j=1}^{\infty} \frac{\psi_{3}\left(\frac{2j-1}{2}\right)}{(2j-1)^4} = \frac{23 \pi^8}{2240},\\ &\sum_{j=1}^{\infty} \frac{\psi_{4}\left(\frac{2j-1}{2}\right)}{(2j-1)^5} = -\frac{31 \pi ^{10}}{7560} - \frac{2883 \zeta^2(5)}{8}.\end{aligned}$$ *Proof.* By setting $f(k) = \frac{1}{(2k-1)^m}$ in Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} and considering the recurrence relation $$\label{genrech1} \psi_{m-1}(z + 1) = \psi_{m-1}(z) - \frac{(-1)^m (m-1)!}{z^m},$$ derived from differentiating [\[recc\]](#recc){reference-type="eqref" reference="recc"} $m-1$ times, we conclude the proof of Theorem [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"}. ◻ Let $m$ be any positive integer greater than $1$. Then [\[coroh13h4\]]{#coroh13h4 label="coroh13h4"} $$\sum_{j=1}^{\infty} \frac{\psi_{2m-1}\left(\frac{2j-1}{2}\right)}{(2j-1)^{2m}} = 2^{2m-2} (2m-1)!\left(\frac{\left(1 - 2^{-2m}\right)^2}{2((2m)!)^2} B^2_{2m}+ \frac{\left(1 - 2^{-4m}\right)}{(4m)!}|B_{4m}|\right) (2\pi)^{4m},$$ where $B_m$ are the Bernoulli numbers. *Proof.* By replacing $m$ with $2m$ in [\[bgthm224\]](#bgthm224){reference-type="eqref" reference="bgthm224"} and utilizing [@bib23 (25.6.2)], we conclude the proof of Corollary [\[coroh13h4\]](#coroh13h4){reference-type="ref" reference="coroh13h4"}. ◻ Let $m$ be any positive integer greater than $1$. Then [\[ayinn1\]]{#ayinn1 label="ayinn1"} $$\begin{split} \sum_{j=1}^{\infty} \frac{\psi_{2m-2}\left(\frac{2j-1}{2}\right)}{(2j-1)^{2m-1}} &= -2^{2m-2} (2m-2)! \left(\left(1 - 2^{-2m+1}\right)^2 \zeta^2(2m-1) \right.\\ &\qquad\qquad\qquad\qquad\qquad \left.+ \frac{\left(1 - 2^{-4m+2}\right)|B_{4m-2}|}{2(4m-2)!} \left(2\pi\right)^{4m-2} \right), \end{split}$$ where $B_m$ are the Bernoulli numbers. *Proof.* By replacing $m$ with $2m-1$ in [\[bgthm224\]](#bgthm224){reference-type="eqref" reference="bgthm224"} and employing [@bib23 (25.6.2)], we conclude the proof of Corollary [\[ayinn1\]](#ayinn1){reference-type="ref" reference="ayinn1"}. ◻ Let $m$ be any positive integer greater than $1$. Then [\[tirtheh2\]]{#tirtheh2 label="tirtheh2"} $$\label{tirtheh2eq} \sum_{j=1}^{\infty} \frac{\psi_{m-1}\left(\frac{4j-1}{4}\right)}{(4j-1)^m} = (-1)^m 2^{-2m-1} \left(\frac{\psi_{m-1}^2\left(\frac{3}{4}\right)}{(m-1)!} +\frac{(m-1)!}{(2m-1)!}\psi_{2m-1}\left(\frac{3}{4}\right)\right).$$ **Example 2**. For $m= 2, 3, 4, 5$, we have $$\begin{aligned} &\sum_{j=1}^{\infty} \frac{\psi_{1}\left(\frac{4j-1}{4}\right)}{(4j-1)^2} = 2\textbf{\textup{G}}^2 - \frac{\textbf{\textup{G}}\pi^2}{2} + \frac{\pi^4}{32} + \frac{\psi_3\left(\frac{3}{4}\right)}{192}, \\ &\sum_{j=1}^{\infty} \frac{\psi_{2}\left(\frac{4j-1}{4}\right)}{(4j-1)^3} = -\frac{\pi^6}{64} + \frac{7\pi^3}{8}\zeta(3) - \frac{49\zeta^2(3)}{4} - \frac{\psi_5\left(\frac{3}{4}\right)}{7680}, \\ &\sum_{j=1}^{\infty} \frac{\psi_{3}\left(\frac{4j-1}{4}\right)}{(4j-1)^4} = \frac{\psi_3^2\left(\frac{3}{4}\right)}{3072} + \frac{\psi_7\left(\frac{3}{4}\right)}{430080},\\ &\sum_{j=1}^{\infty} \frac{\psi_{4}\left(\frac{4j-1}{4}\right)}{(4j-1)^5} = -\frac{25\pi^{10}}{768} + \frac{155 \pi^5}{8} \zeta(5) - 2883\zeta^2(5) - \frac{\psi_9\left(\frac{3}{4}\right)}{30965760}.\end{aligned}$$ *Proof.* By setting $f(k) = \frac{1}{(4k-1)^m}$ in Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} and taking into account the recurrence relation [\[genrech1\]](#genrech1){reference-type="eqref" reference="genrech1"}, we conclude the proof of Theorem [\[tirtheh2\]](#tirtheh2){reference-type="ref" reference="tirtheh2"}. ◻ [\[corsha\]]{#corsha label="corsha"} Let $m$ be any positive integer greater than $1$. Then $$\begin{split} \sum_{j=1}^{\infty} \frac{\psi_{2m-2}\left(\frac{4j-1}{4}\right)}{(4j-1)^{2m-1}} &= - \frac{|E_{2m-2}|}{8}\left(1-2^{2m-1}\right) \pi^{2m-1} \zeta(2m-1) \\&\quad- \frac{\left(1-2^{2m-1}\right)^2(2m-2)!}{8}\zeta^2(2m-1)- \frac{E^2_{2m-2}}{32(2m-2)!}\pi^{4m-2} \\&\quad-\frac{(2m-2)!}{2^{4m-1}(4m-3)!}\psi_{4m-3}\left(\frac{3}{4}\right), \end{split}$$ where $E_m$ are the Euler numbers. *Proof.* Olaikhan [@bibash §1.20.6, §1.20.7, pp. 62--63] expressed $\psi_{2a}\left(\frac{3}{4}\right)$ in terms of $\psi_{2a}\left(\frac{1}{4}\right)$, and $\psi_{2a}\left(\frac{1}{4}\right)$ in terms of the Euler numbers $E_a$, with $a$ as a positive integer. By employing both of these expressions, we derive $$\label{psiabd1} \psi_{2m-2}\left(\frac{3}{4}\right) = 2^{2m-2}\left(\left(1 - 2^{2m-1}\right) (2m-2)! \zeta(2m-1) + \frac{\pi^{2m-1}}{2}|E_{2m-2}|\right).$$ By replacing $m$ with $2m-1$ and subsequently substituting [\[psiabd1\]](#psiabd1){reference-type="eqref" reference="psiabd1"} into [\[tirtheh2eq\]](#tirtheh2eq){reference-type="eqref" reference="tirtheh2eq"}, we conclude the proof of Corollary [\[corsha\]](#corsha){reference-type="ref" reference="corsha"}. ◻ In the following theorem, we present a generalization of [\[psidouble\]](#psidouble){reference-type="eqref" reference="psidouble"}. Let $m$ be any positive integer greater than $1$. Then [\[almdh21\]]{#almdh21 label="almdh21"} $$\label{almdh21eq} \sum_{k=1}^\infty \frac{\psi_{m-1}\left(\frac{k+1}{2}\right) - \psi_{m-1}\left(\frac{k+2}{2}\right)}{k^m} = (-1)^{m} 2^{m-1} (m-1)! \left(\zeta(2m) - \left(1 - 2^{1-m}\right)^2\zeta^2(m)\right).$$ **Example 3**. For $m=3, 4, 5$, we have $$\begin{aligned} &\sum_{k=1}^\infty \frac{\psi_{2}\left(\frac{k+1}{2}\right) - \psi_{2}\left(\frac{k+2}{2}\right)}{k^3} = -\frac{8\pi^6}{945} + \frac{9\zeta^2(3)}{2},\end{aligned}$$ $$\begin{aligned} &\sum_{k=1}^\infty \frac{\psi_{3}\left(\frac{k+1}{2}\right) - \psi_{3}\left(\frac{k+2}{2}\right)}{k^4} = \frac{41\pi^8}{75600},\end{aligned}$$ $$\begin{aligned} &\sum_{k=1}^\infty \frac{\psi_{4}\left(\frac{k+1}{2}\right) - \psi_{4}\left(\frac{k+2}{2}\right)}{k^5} = -\frac{128\pi^{10}}{31185} + \frac{675\zeta^2(5)}{2}.\end{aligned}$$ *Proof.* By setting $f(k) = \frac{(-1)^{k}}{k^m}$ in Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"}, we conclude the proof of Theorem [\[almdh21\]](#almdh21){reference-type="ref" reference="almdh21"}. ◻ Let $m$ be any positive integer greater than $1$. Then [\[almdh22\]]{#almdh22 label="almdh22"} $$\begin{split} \sum_{k=1}^\infty \frac{\psi_{2m-1}\left(\frac{k+1}{2}\right) - \psi_{2m-1}\left(\frac{k+2}{2}\right)}{k^{2m}} &= 2^{2m-2} (2m-1)! \left(\frac{|B_{4m}|}{(4m)!} - \frac{\left(1 - 2^{1-2m}\right)^2}{2 ((2m)!)^2}B^2_{2m}\right) \\&\qquad\times(2\pi)^{4m}, \end{split}$$ where $B_m$ are the Bernoulli numbers. *Proof.* By replacing $m$ with $2m$ in [\[almdh21eq\]](#almdh21eq){reference-type="eqref" reference="almdh21eq"} and utilizing [@bib23 (25.6.2)], we conclude the proof of Corollary [\[almdh22\]](#almdh22){reference-type="ref" reference="almdh22"}. ◻ In the next theorem, we extend the results of Theorems [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"} and [\[tirtheh2\]](#tirtheh2){reference-type="ref" reference="tirtheh2"}. Let $\Re{m} > 1$, $r$ and $s$ be two nonzero complex numbers such that $rk \neq s$, for all $k \in {\mathbb N}$. Then [\[bigthmh1\]]{#bigthmh1 label="bigthmh1"} $$\sum_{k=1}^\infty \frac{ \zeta\left(m, \frac{rk+r-s}{r}\right) }{(rk-s)^m}= \frac{1}{2r^m}\left(\zeta^2\left(m, \frac{r-s}{r}\right) - \zeta\left(2m, \frac{r-s}{r}\right)\right).$$ *Proof.* By setting $f(k) = \frac{1}{(rk-s)^m}$ in Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"}, we conclude the proof of Theorem [\[bigthmh1\]](#bigthmh1){reference-type="ref" reference="bigthmh1"}. ◻ Theorems [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"}--[\[bigthmh1\]](#bigthmh1){reference-type="ref" reference="bigthmh1"} and Corollaries [\[coroh13h4\]](#coroh13h4){reference-type="ref" reference="coroh13h4"} and [\[corsha\]](#corsha){reference-type="ref" reference="corsha"} introduce novel results not found in the existing literature. Upon an exhaustive review of the literature, we observed that Theorem [\[almdh21\]](#almdh21){reference-type="ref" reference="almdh21"} for all positive integers $m \geq 2$ and the case $m=2$ in Theorem [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"} can be derived from Vălean's works [@bib32; @bib9]. However, for values of $m$ beyond 2, no alternative methods are available to establish Theorems [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"}, emphasizing the uniqueness and strength of our approach. To elucidate, we begin with the identity from [@bib9 §4.18, (4.98)]: $$\label{valexp} \begin{split} \psi_m\left(\frac{n}{2}\right) &= (-1)^{m-1} 2^{m+1} \frac{m!}{n^{m+1}} + (-1)^{m+n}\left(2^m - 1\right) m! \zeta(m + 1) \\&\quad + (-1)^{m-1} 2^m m! \left(\zeta(m +1) - H_n^{(m+1)}\right) - (-1)^{m+n} 2^m m! \overline{H}_n^{(m+1)}. \end{split}$$ Here, $H_n^{(m)} := \sum_{k=1}^n \frac{1}{k^m}$ represents the $n$th generalized harmonic number of order $m$, and $\overline{H}_n^{(m)} := \sum_{k=1}^n \frac{(-1)^{k-1}}{k^m}$ denotes the $n$th generalized skew-harmonic number of order $m$, where $m \geq 1$ and $m \in {\mathbb N}$. $$\label{valp} \begin{split} \sum_{k=1}^\infty \frac{\psi_{m-1}\left(\frac{2k-1}{2}\right)}{(2k-1)^m} &= \sum_{k=1}^\infty \frac{1}{(2k-1)^m}\left(\frac{(-1)^m 2^m (m-1)!}{(2k-1)^{m}} - (-1)^m (m-1)! \zeta(m) \right. \\&\qquad\qquad\qquad\qquad+ (-1)^{m} 2^{m-1} (m-1)!\left(2\zeta(m) - H^{(m)}_{2k-1}\right) \\&\qquad\qquad\qquad\qquad\left.- (-1)^{m} 2^{m-1} (m-1)! \overline{H}^{(m)}_{2k-1}\right). \end{split}$$ The challenging series in [\[valp\]](#valp){reference-type="eqref" reference="valp"} can be expressed as $$\begin{aligned} \sum_{k=1}^\infty \frac{H^{(m)}_{2k-1}}{(2k-1)^m} &= \frac{1}{2}\left(\sum_{k=1}^\infty \frac{H^{(m)}_k}{k^m} + \sum_{k=1}^\infty \frac{(-1)^{k-1} H^{(m)}_k}{k^m}\right), \label{valdisc1}\\ \sum_{k=1}^\infty \frac{\overline{H}^{(m)}_{2k-1}}{(2k-1)^m} &= \frac{1}{2}\left(\sum_{k=1}^\infty \frac{\overline{H}^{(m)}_k}{k^m} + \sum_{k=1}^\infty \frac{(-1)^{k-1} \overline{H}^{(m)}_k}{k^m}\right).\label{valdisc2}\end{aligned}$$ By evaluating limits as $n \to \infty$ in [@bib32 §4.12, (4.14), pp. 286] and [@bib9 §1.23, (1.120), pp. 31], we derive $$\begin{aligned} &\sum_{k=1}^\infty \frac{H^{(m)}_k}{k^m} = \frac{1}{2}\left(\zeta^2(m) + \zeta(2m)\right), \label{rajfa1} \\ &\sum_{k=1}^\infty \frac{(-1)^{k-1} \overline{H}^{(m)}_k}{k^m} = \frac{1}{2}\left(\left(1 - 2^{1-m}\right)^2 \zeta^2(m)+ \zeta(2m)\right). \label{rajfa2}\end{aligned}$$ The closed form in [\[rajfa1\]](#rajfa1){reference-type="eqref" reference="rajfa1"} can also be derived by setting $f(k) = \frac{1}{k^m}$ in Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"}. Utilizing [\[rajfa1\]](#rajfa1){reference-type="eqref" reference="rajfa1"} and [\[rajfa2\]](#rajfa2){reference-type="eqref" reference="rajfa2"} and evaluating the remaining terms in [\[valp\]](#valp){reference-type="eqref" reference="valp"}, we simplify [\[valp\]](#valp){reference-type="eqref" reference="valp"} to $$\label{compl1} \begin{split} \sum_{k=1}^\infty \frac{\psi_{m-1}\left(\frac{2k-1}{2}\right)}{(2k-1)^m} &= \frac{(-1)^{m} (m-1)!}{4}\left(\left(3\left(2^m\right)+ 2\left(2^{-m}\right) - 6\right)\zeta^2(m) \right. \\&\quad \left.+ \left(3\left(2^m\right) - 4 \left(2^{-m}\right)\right)\zeta(2m) - 2^m \sum_{k=1}^\infty \frac{\overline{H}_k^{(m)} + (-1)^{k-1}H_k^{(m)}}{k^m} \right). \end{split}$$ Regarding [\[valdisc1\]](#valdisc1){reference-type="eqref" reference="valdisc1"}, Vălean provided closed forms for $\sum_{k=1}^\infty \frac{(-1)^{k-1} H^{(2)}_k}{k^2}$ in [@bib32 §4.52, (4.86), pp. 310]. Similarly, in reference to [\[valdisc2\]](#valdisc2){reference-type="eqref" reference="valdisc2"}, Vălean provided closed forms for $\sum_{k=1}^\infty \frac{\overline{H}^{(2)}_k}{k^2}$ in [@bib9 §6.46, (6.306), pp. 722]. By utilizing these closed forms in [\[compl1\]](#compl1){reference-type="eqref" reference="compl1"} with $m=2$, we can successfully conclude the proof of Theorem [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"}, specifically for the case $m=2$. Furthermore, by incorporating [\[valexp\]](#valexp){reference-type="eqref" reference="valexp"} into the series on the left-hand side of [\[almdh21eq\]](#almdh21eq){reference-type="eqref" reference="almdh21eq"}, along with the closed forms provided in [\[rajfa1\]](#rajfa1){reference-type="eqref" reference="rajfa1"} and [\[rajfa2\]](#rajfa2){reference-type="eqref" reference="rajfa2"}, we can derive the closed form presented in Theorem [\[almdh21\]](#almdh21){reference-type="ref" reference="almdh21"}. [\[finallastt\]]{#finallastt label="finallastt"} Let $\Re{m} > 1$. Then $$\label{hafay1} \sum_{k=1}^\infty \frac{\overline{H}_k^{(m)} + (-1)^{k-1}H_k^{(m)}}{k^m} = \left(1 - 2^{1-m}\right)\zeta^2(m) + \left(1 - 2^{1-2m}\right)\zeta(2m),$$ where $H_k^{(m)} = 1 + \frac{1}{2^m} + \frac{1}{3^m} + \cdots + \frac{1}{k^m}$ represents the $k$th generalized harmonic number of order $m$, and $\overline{H}_k^{(m)} = 1 - \frac{1}{2^m} + \frac{1}{3^m} - \cdots + \frac{(-1)^{k-1}}{k^m}$ denotes the $k$th generalized skew-harmonic number of order $m$. **Example 4**. For $m = 2, 3, 4$, we have $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(2)} + (-1)^{k-1}H_k^{(2)}}{k^2} = \frac{17\pi^4}{720},$$ $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(3)} + (-1)^{k-1}H_k^{(3)}}{k^3} = \frac{31\pi^6}{30240} + \frac{3\zeta^2(3)}{4},$$ $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(4)} + (-1)^{k-1}H_k^{(4)}}{k^4} = \frac{773 \pi^8}{3628800}.$$ *Proof.* The proof of Theorem [\[finallastt\]](#finallastt){reference-type="ref" reference="finallastt"} is established by equating [\[compl1\]](#compl1){reference-type="eqref" reference="compl1"} with our closed form provided in Theorem [\[tirtheh1\]](#tirtheh1){reference-type="ref" reference="tirtheh1"}. ◻ Let $m$ be a positive integer greater than $1$. Then [\[coroth1\]]{#coroth1 label="coroth1"} $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(m)} + (-1)^{k-1}H_k^{(m)}}{k^m} = \left(1 - 2^{1-m}\right)\zeta^2(m) + \frac{\left(1 - 2^{1-2m}\right)}{2(2m)!}|B_{2m}| (2\pi)^{2m},$$ where $B_m$ are the Bernoulli numbers, $H_k^{(m)} = 1 + \frac{1}{2^m} + \frac{1}{3^m} + \cdots + \frac{1}{k^m}$ represents the $k$th generalized harmonic number of order $m$, and $\overline{H}_k^{(m)} = 1 - \frac{1}{2^m} + \frac{1}{3^m} - \cdots + \frac{(-1)^{k-1}}{k^m}$ denotes the $k$th generalized skew-harmonic number of order $m$. *Proof.* Utilizing [@bib23 (25.6.2)] in [\[hafay1\]](#hafay1){reference-type="eqref" reference="hafay1"}, we conclude the proof of Corollary [\[coroth1\]](#coroth1){reference-type="ref" reference="coroth1"}. ◻ Let $m$ be a positive integer greater than $1$. Then [\[coroth2\]]{#coroth2 label="coroth2"} $$\sum_{k=1}^\infty \frac{\overline{H}_k^{(2m)} + (-1)^{k-1}H_k^{(2m)}}{k^{2m}} = \frac{1}{2}\left(\frac{\left(1 - 2^{1-2m}\right) |B_{2m}|^2}{2((2m)!)^2} + \frac{\left(1 - 2^{1-4m}\right)|B_{4m}|}{(4m)!}\right) (2\pi)^{4m},$$ where $B_m$ are the Bernoulli numbers, $H_k^{(m)} = 1 + \frac{1}{2^m} + \frac{1}{3^m} + \cdots + \frac{1}{k^m}$ represents the $k$th generalized harmonic number of order $m$, and $\overline{H}_k^{(m)} = 1 - \frac{1}{2^m} + \frac{1}{3^m} - \cdots + \frac{(-1)^{k-1}}{k^m}$ denotes the $k$th generalized skew-harmonic number of order $m$. *Proof.* By replacing $m$ with $2m$ in [\[hafay1\]](#hafay1){reference-type="eqref" reference="hafay1"} and utilizing [@bib23 (25.6.2)], we conclude the proof of Corollary [\[coroth2\]](#coroth2){reference-type="ref" reference="coroth2"}. ◻ # Conclusion We have introduced twelve new theorems and presented a novel proof for specific cases of Jonquière's inversion formula in two other theorems. Additionally, we demonstrated the application of our double infinite series transformation formula in deriving new identities, which have been expressed using well-known numbers such as the Euler and Bernoulli numbers. Interested readers can further explore the transformation formulas outlined in Theorem [\[theoremh9\]](#theoremh9){reference-type="ref" reference="theoremh9"} and Corollary [\[coroh12\]](#coroh12){reference-type="ref" reference="coroh12"} to potentially uncover additional new results. # Acknowledgment {#acknowledgment .unnumbered} I would like to express my heartfelt gratitude to the Spirit of Ramanujan (SOR) STEM Talent Initiative, directed by Ken Ono, for providing the essential computational tools that facilitated the verification and validation of my results. I extend my sincere appreciation to esteemed researchers specializing in harmonic series, sums, and integrals, Ali S. Olaikhan and Cornel I. Vălean, for their invaluable insights and guidance. I am deeply thankful to my mother, my dedicated lecturers, Mr. G. S. Lawal and Dr. D. A. Dikko, my friends, fellow department members, and my industrial training supervisors, Mr. A. B. Olanrewaju and Mr. O. O. Olubodun, for their unwavering encouragement and support. # Funding {#funding .unnumbered} The author did not receive funding from any organization for the submitted work.

arxiv_math

--- abstract: | Let $\mathbb{F}_q$ be the field with $q$ elements and of characteristic $p$. For $a\in\mathbb{F}_p$ consider the set $$S_a(n)=\{f\in\mathbb{F}_q[x]\mid\deg(f)=n,~f\text{ irreducible, monic and} \operatorname{Tr}(f)=a\}.$$ In a recent paper, Robert Granger proved for $q=2$ and $n\ge 2$ $$|S_1(n)|-|S_0(n)|=\begin{cases} 0,&\text{if }2\nmid n\\ |S_1(n/2)|,&\text{if } 2\mid n \end{cases}$$ We will prove a generalization of this result for all finite fields. This is possible due to an observation about the size of certain subsets of monic irreducible polynomials arising in the context of a group action of subgroups of $\operatorname{PGL}_2(\mathbb{F}_q)$ on monic polynomials. Additionally, it enables us to apply these methods to prove two further results that are very similar in nature. author: - | Max Schulz\ University of Rostock, Germany\ `max.schulz@uni-rostock.de` bibliography: - bab.bib title: On the Iterative Behaviour of the Number of Irreducible Polynomials with Certain Properties over Finite Fields --- # Introduction {#introduction .unnumbered} Let $\mathbb{F}_q$ be the finite field with $q$ elements, $p$ the prime dividing $q$, $\mathcal{I}_q$ the set of monic irreducible polynomials in $\mathbb{F}_q[x]$ and $\mathcal{I}_q^n$ the set of monic irreducible polynomials of degree $n$. Moreover, $\operatorname{Tr}_{q/p}$ is the absolut trace. Since $\operatorname{Tr}_{q^n/p}(\alpha)=\operatorname{Tr}_{q^n/p}(\alpha^q)$ for a root $\alpha$ of $f\in\mathcal{I}_q^n$ we can define $\operatorname{Tr}(f):=\operatorname{Tr}_{q^n/p}(\alpha)$. For an $a\in \mathbb{F}_p$ consider the set $$S_a(n):=\{f\in\mathcal{I}_q^n\mid\operatorname{Tr}(f)=a\}.$$ Let $f\in\mathcal{I}_q^n$ be of the form $f=x^n+\sum_{i=0}^{n-1}a_ix^i$ then the trace is given by $$\operatorname{Tr}(f)=-\operatorname{Tr}_{q/p}(a_{n-1}).$$ In [@granger] it is proved that for $q=2$ and $n\ge 2$ $$|S_1(n)|-|S_0(n)|=\begin{cases}0, &\text{if }n\equiv 1\pmod{2}\\ |S_1(n/2)|, &\text{otherwise}. \end{cases}$$ We are going to prove the following extension of this result: **Theorem 1**. *For all $n\ge 1$ and all finite fields $\mathbb{F}_q$ we have $$\sum\limits_{a\in\mathbb{F}_p^{\ast}} |S_a(n)|-(p-1)|S_0(n)|=\begin{cases} 0,&\text{if }p\nmid n\\ \sum\limits_{a\in\mathbb{F}_p^{\ast}}|S_a(n/p)|,&\text{otherwise} \end{cases}$$* *Remark 2*. Note that the balanced case, that is, $$\sum\limits_{a\in\mathbb{F}_p^{\ast}}|S_a(n)|-(p-1)|S_0(n)|=0,$$ where $p\nmid n$, is not hard to see. Let $f\in S_0(n)$, so $\operatorname{Tr}(f)=0$ which means that $\operatorname{Tr}_{q^n/p}(\alpha)=0$ for $\alpha$ a root of $f$. Let $a\in\mathbb{F}_p^{\ast}$ and consider an element $b_a\in\mathbb{F}_q^{\ast}$ such that $\operatorname{Tr}_{q^n/p}(b_a)=a$. Such an element exists since $\operatorname{Tr}_{q^n/p}=\operatorname{Tr}_{q/p}\circ\operatorname{Tr}_{q^n/q}$ and for all $b\in\mathbb{F}_q^{\ast}$ $$\operatorname{Tr}_{q^n/q}(b)=n\cdot b\neq 0$$ if $p\nmid n$, hence $\operatorname{Tr}_{q^n/p}$ as a map from $\mathbb{F}_q$ to $\mathbb{F}_p$ is surjective as $\operatorname{Tr}_{q/p}:\mathbb{F}_q\to\mathbb{F}_p$ and $\operatorname{Tr}_{q^n/q}\vert_{\mathbb{F}_q}:\mathbb{F}_q\to \mathbb{F}_q$ are surjective. The polynomial $f(x-b_a)$ has trace $a$, so the map $f(x)\mapsto f(x-b_a)$ is a bijection between $S_0(n)$ and $S_a(n)$, thus $|S_0(n)|=|S_a(n)|$ for all $a\in\mathbb{F}_p$ and the balanced case follows. A similar idea does not work for the case that $p\mid n$ since then $\operatorname{Tr}_{q^n/q}\vert_{\mathbb{F}_q}$ is not surjective anymore. Another example that exhibits a similar pattern is the following: Let $q$ be odd and $u,v\in\mathbb{F}_q$ with $u\neq v$. Define the following two sets for $n\ge 2$ $$\begin{aligned} C_{u,v}(n)&:=\left\{f\in\mathcal{I}_q^n\mid \left(\frac{f(u)\cdot f(v)}{q}\right)=-1\right\}\\ D_{u,v}(n)&:=\mathcal{I}_q^n\setminus C_{u,v}(n)=\left\{f\in\mathcal{I}_q^n\mid \left(\frac{f(u)\cdot f(v)}{q}\right)=1\right\}.\end{aligned}$$ Here $\left(\frac{\cdot}{q}\right):\mathbb{F}_q^{\ast}\to \{1,-1\}\le \mathbb{F}_q^{\ast}$ denotes the Legendre-Symbol $$\left(\frac{a}{q}\right)=a^{(q-1)/2}=\begin{cases} 1, &a \text{ is a square in }\mathbb{F}_q^{\ast}\\ -1,&\text{otherwise} \end{cases}$$ We prove the following theorem: **Theorem 3**. *Let $q\equiv 1 \pmod 2$. For all $u,v\in\mathbb{F}_q$ with $u\neq v$ and $n\ge 2$ we have $$|C_{u,v}(n)|-|D_{u,v}(n)|=\begin{cases} 0, &2\nmid n\\ |C_{u,v}(n/2)|,&2\mid n \end{cases}$$* In [@granger] a group action of subgroups of $\operatorname{PGL}_2(\mathbb{F}_q)$ on irreducible polynomials over $\mathbb{F}_q$ played a crucial role in some of the proofs, so we thought that ideas out of our recent paper [@schulz] could be utilized to prove similar results. Our proof of Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"} relies on a general underlying principle which can be used to obtain Theorem [Theorem 3](#quad){reference-type="ref" reference="quad"} as well. Explaining how that principle works is the main goal of this paper. We give a quick overview: For an element $A\in\operatorname{GL}_2(\mathbb{F}_q)$ we write $[A]\in\operatorname{PGL}_2(\mathbb{F}_q)$ as its coset in $\operatorname{PGL}_2(\mathbb{F}_q)$ and if $A$ is of the form $$A=\left(\begin{array}{cc} a& b\\ c& d \end{array}\right)$$ then set $$[A]\circ x=\frac{ax+b}{cx+d}$$ as the corresponding linear rational function. For a subgroup $G\le \operatorname{PGL}_2(\mathbb{F}_q)$ consider the set of $G$-invariant rational functions $$\mathbb{F}_q(x)^G:=\left\{Q(x)\in \mathbb{F}_q(x)\mid Q([A]\circ x)=Q(x)\text{ for all }[A]\in G\right\}.$$ This is a subfield of $\mathbb{F}_q(x)$ with $[\mathbb{F}_q(x):\mathbb{F}_q(x)^G]=|G|$. Moreover, by Lüroth's Theorem, there is a rational function $Q(x)=g(x)/h(x)\in \mathbb{F}_q(x)$ of degree $\deg(Q)=\max\{\deg(g),\deg(h)\}=|G|$ such that $\mathbb{F}_q(x)^G=\mathbb{F}_q(Q(x))$. Note that we always assume that the numerator and denominator of a rational function have no common factors. Every such generator $Q$ of $\mathbb{F}_q(x)^G$ can be normalized so that $Q=g/h$ with $\deg(g)=|G|$ and $0\le \deg(h)<\deg(g)$, we call these rational functions *quotient maps* for $G$ and in what follows we write $Q_G$ for an arbitrary quotient map for $G$. In [@schulz], we studied the factorization of rational transformations with quotient maps. A rational transformation of a polynomial $F$ with a rational function $Q=g/h$ is defined as $$\label{rattrans} F^Q(x):=h(x)^{\deg(F)}\cdot F\left(\frac{g(x)}{h(x)}\right)$$ so it is the numerator polynomial of the rational function $F(Q(x))$. To avoid ambiguity we set the numerator polynomial $g$ of $Q$ to be monic. Define the following two sets $$\begin{aligned} C(Q_G,n)&:=\left\{f\in\mathcal{I}_q^n\mid f^{Q_G}\in\mathcal{I}_q^{|G|n}\right\}\\ D(Q_G,n)&:=\mathcal{I}_q^n\setminus C(Q_G,n). \end{aligned}$$ So $C(Q_G,n)$ is the set of irreducible polynomials $f$ of degree $n$ that yield irreducible polynomials of degree $|G|\cdot n$ after transformation with quotient map $Q_G$. The following theorem will be the backbone of our proofs of Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"} and [Theorem 3](#quad){reference-type="ref" reference="quad"}: **Theorem 4**. *Let $G\le \operatorname{PGL}_2(\mathbb{F}_q)$ be a cyclic subgroup of prime order $s$ and $Q_G\in \mathbb{F}_q(x)$ a quotient map for $G$. For all $n>d(G)$ (the number $d(G)$ will be defined before Example [Example 14](#examp){reference-type="ref" reference="examp"}) we have $$|C(Q_G,n)|-(s-1)|D(Q_G,n)|=\begin{cases} 0,&\text{ if }s\nmid n\\ |C(Q_G,\frac{n}{s})|,&\text{ if } s\mid n \end{cases}$$* Both results are immediate consequences of this theorem by choosing the right cyclic subgroups $G$ and quotient maps $Q_G$. The set $C(Q_G,n)$ can be occasionally described in terms of arithmetic properties that the coefficients of irreducible polynomials in $C(Q_G,n)$ need to have if the quotient map $Q_G$ was chosen carefully. The first part of this paper is mainly a recollection of ideas and results of [@schulz]. Afterwards we are going to prove our main theorem about a new combinatorial relationship between $G$-orbits of irreducible monic polynomials from which Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"} follows. The last part is dedicated to proving Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"} and [Theorem 3](#quad){reference-type="ref" reference="quad"}, as well as looking at one more example. # Invariant Polynomials and Rational Transformations Every $[A]\in\operatorname{PGL}_2(\mathbb{F}_q)$ induces a bijective map on $\overline{\mathbb{F}}_q\cup\{\infty\}$ via $$\circ v=\frac{av+b}{cv+d},$$ i.e. just plugging in $v$ into the linear rational function belonging to $[A]$. This induces a left group action of $\operatorname{PGL}_2(\mathbb{F}_q)$ on $\overline{\mathbb{F}}_q\cup\{\infty\}$. An intimately related group action on polynomials is given by **Definition 5**. Define $\ast: \operatorname{PGL}_2(\mathbb{F}_q)\times \mathbb{F}_q[x]\to \mathbb{F}_q[x]$ with $$\ast f(x)=\lambda_{A,f}(cx+d)^{\deg(f)}f\left(\frac{ax+b}{cx+d}\right).$$ The factor $\lambda_{A,f}\in\mathbb{F}_q^{\ast}$ makes the output-polynomial monic. In other words, $[A]\ast f$ is the normalized $([A]\circ x)$-transformation of $f$. This transformation and its variations are well-studied objects over finite fields, see for example [@GarefalakisGL], [@ReisFQG], [@reisEx], [@sticht] and it has some theoretic applications, see for example [@granger], [@kapetan] and [@reisConstr]. Let $G\le \operatorname{PGL}_2(\mathbb{F}_q)$ be a subgroup and $G\circ \infty:=\{[A]\circ \infty|[A]\in G\}$ be the $G$-orbit of $\infty$. Define $$\mathcal{NR}_q^G:=\{f\in \mathbb{F}_q[x]\mid f \text{ monic and }f(\alpha)\neq 0\text{ for all }\alpha\in G\circ \infty\}$$ where $f(\infty)=\infty$ if $\deg(f)\ge 1$ and $f(\infty)=f$ if $\deg(f)\le 0$. **Lemma 6** ([@schulz Lemma 7]). *Let $G\le\operatorname{PGL}_2(\mathbb{F}_q)$. For all $f,g\in\mathcal{NR}_q^G$ and $[A],[B]\in G$ the following hold:* 1. *$\deg([A]\ast f)=\deg(f)$* 2. *$[AB]\ast f=[B]\ast([A]\ast f)$ and $[I_2]\ast f=f$, so $\ast$ is a right group action of $G$ on $\mathcal{NR}_q^G$* 3. *$[A]\ast (fg)=([A]\ast f)([A]\ast g)$* 4. *$f$ irreducible if and only if $[A]\ast f$ irreducible* The first and forth item show that $G$ also acts on $\mathcal{I}_q^n$ for $n\ge 2$ (since $G\circ\infty\subseteq \mathbb{F}_q\cup\{\infty\}$) and on $$\begin{aligned} \mathcal{I}_q^G&:=\mathcal{NR}_q^G\cap \mathcal{I}_q.\end{aligned}$$ We denote a $G$-orbit in $\mathcal{NR}_q^G$ as $G\ast r:=\{[A]\ast r\mid [A]\in G\}$. **Definition 7**. A polynomial $f\in\mathcal{NR}_q^G$ is called $G$-orbit polynomial if there is an irreducible polynomial $r\in\mathcal{I}_q^G$ such that $$f=\prod\limits_{t\in G\ast r}t=:\prod G\ast r$$ A $G$-invariant polynomial is a polynomial $f\in\mathcal{NR}_q^G$ such that $[A]\ast f=f$ for all $[A]\in G$. Every $G$-invariant polynomial can be written as the product of $G$-orbit polynomials (which are $G$-invariant by Lemma [Lemma 6](#basic){reference-type="ref" reference="basic"} 3.), so $G$-orbit polynomials can be seen as the atoms of $G$-invariant polynomials. Next we want to recollect some facts about rational transformations. For $Q=g/h$ with $\gcd(g,h)=1$ and $F\in \mathbb{F}_q[x]$ we write $F^Q\in \mathbb{F}_q[x]$ as the $Q$-transform of $F$ with $Q$ as in ([\[rattrans\]](#rattrans){reference-type="ref" reference="rattrans"}). It is obvious that if $F^Q$ is irreducible then $F$ has to be irreducible. The following lemma gives a necessary and sufficient condition for the irreducibility of $F^Q$: **Lemma 8** ([@cohensLem Lemma 1]). *Let $Q(x)=g(x)/h(x)\in \mathbb{F}_q(x)$ and $F\in \mathbb{F}_q[x]$. Then $F^{Q}\in\mathbb{F}_q[x]$ is irreducible if and only if $F\in \mathbb{F}_q[x]$ is irreducible and $g(x)-\alpha h(x)$ is irreducible over $\mathbb{F}_q(\alpha)[x]$, where $\alpha$ is a root of $F$.* Now consider a quotient map $Q_G\in\mathbb{F}_q(x)$ of $G\le \operatorname{PGL}_2(\mathbb{F}_q)$. We have **Lemma 9** ([@schulz Lemma 13 and Lemma 14]). *Let $F\in \mathbb{F}_q[x]$ be a monic polynomial, then $F^{Q_G}\in\mathcal{NR}_q^G$ and $F^{Q_G}$ is $G$-invariant. Moreover, $F^{Q_G}$ is of degree $\deg(F^{Q_G})=|G|\cdot \deg(F)$.* **Theorem 10** ([@schulz Main Theorem, Theorem 22 and Corollary 23]). *Let $F\in \mathbb{F}_q[x]$ be monic and irreducible, $G\le\operatorname{PGL}_2(\mathbb{F}_q)$ a subgroup and $Q_G=g/h\in \mathbb{F}_q(x)$ a quotient map for $G$. Then there is an irreducible monic polynomial $r\in \mathbb{F}_q[x]$ with $\deg(F)|\deg(r)$ and an integer $k>0$ such that $$F^{Q_G}(x)=\left(\prod G\ast r \right)^k.$$ Additionally $F^{Q_G}$ is an orbit polynomial, i.e. $k=1$, if $|G\circ v|=|G|$ for a root $v\in\overline{\mathbb{F}}_q$ of $F^{Q_G}$. In the case that $F^{Q_G}$ is an orbit polynomial the degree of every irreducible factor of $F^{Q_G}$ can be calculated via $$\deg(r)=\frac{|G|}{|G\ast r|}\cdot \deg(F).$$* The polynomials $F\in\mathcal{I}_q$ for which $F^{Q_G}=(\prod G\ast r)^k$ with $k>1$ are of degree $\deg(F)\le 2$. To show that we use the fact that $F^{Q_G}$ is an orbit polynomial if every (or equivalently just one) root $v\in\overline{\mathbb{F}}_q$ of $F^{Q_G}$ is contained in a regular $G$-orbit, i.e. $|G\circ v|=|G|$ (Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"}) and irreducible polynomials $F$ not satisfying that condition are of degree less than or equal to $2$, as the following lemma shows: **Lemma 11** ([@bluher1 Lemma 2.1]). *Let $G\le \operatorname{PGL}_2(\mathbb{F}_q)$ and set $$P_G:=\left\{v\in\overline{\mathbb{F}}_q\cup\{\infty\}\mid |G\circ v|<|G|\right\}.$$ We have $P_G\subseteq\mathbb{F}_{q^2}\cup\{\infty\}$ and $|P_G|\le 2(|G|-1)$.* Let $F\in\mathcal{I}_q$. Note that if $F^{Q_G}$ has roots in non-regular $G$-orbits, then it only has irreducible factors of degree less than 3 by the lemma above. Moreover, we know that if $r$ is an irreducible factor of $F^{Q_G}$, then $\deg(F)\mid \deg(r)$, so $\deg(F)\le 2$, which is exactly what we wanted to show. Furthermore, there are only finitely many irreducible monic polynomials $F\in\mathcal{I}_q$ such that $F^{Q_G}$ is not a $G$-orbit polynomial but a proper power thereof as the number of non-regular $G$-orbits in $\overline{\mathbb{F}}_q\cup\{\infty\}$ is finite. The next corollary is one of our main tools we make use of in this paper. Define $\mathcal{I}_q^G/G$ as the set of $G$-orbits in $\mathcal{I}_q^G$, that is, $$\mathcal{I}_q^G/G:=\{G\ast r \mid r\in\mathcal{I}_q^G\}.$$ **Corollary 12** ([@schulz Corollary 25]). *The map $\delta_{Q_G}:\mathcal{I}_q\to\mathcal{I}_q^G/G$ with $F\mapsto G\ast r$ such that $F^{Q_G}=\prod(G\ast r)^k$ is a bijection.* An irreducible monic $G$-invariant polynomial $f$ is a $G$-orbit polynomial, thus $f$ can be written as $f=F^{Q_G}$ for $F$ an irreducible monic polynomial if a root of $f$ is contained in a regular $G$-orbit by Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"} and Corollary [Corollary 12](#main2){reference-type="ref" reference="main2"}. # Combinatorics of Orbits of Irreducible Polynomials Set $d(Q_G)\in \mathbb{N}_0$ as the biggest number such that there exists an irreducible polynomial $F\in\mathcal{I}_q$ of degree $d(Q_G)$ with $F^{Q_G}=\prod(G\ast r)^k$ and $k>1$. If no such polynomial exists then $d(Q_G):=0$. Recall that $d(Q_G)\le 2$. *Remark 13*. If $Q_G,Q_G'\in\mathbb{F}_q(x)$ are quotient maps for $G$, then $d(Q_G)=d(Q_G')$. *Proof.* Let $F\in\mathcal{I}_q$ be of degree $d(Q_G)$ such that $F^{Q_G}=(\prod G\ast r)^k$ and $k>1$. It can be shown that there are $a\in \mathbb{F}_q^{\ast}$ and $b\in \mathbb{F}_q$ such that $Q_G(x)=aQ_G'(x)+b$, for reference see [@bluher1 Proposition 3.4]. Since we want the numerator polynomial of quotient maps to be monic we write for $Q_G'=g'/h'$: $$Q_G(x)=aQ_G'(x)+b=\frac{g'(x)}{a^{-1}h'(x)}+b.$$ Thus we have $$\begin{aligned} F^{Q_G}(x)&=(a^{-1}h'(x))^{\deg(F)}\cdot F(Q_G(x))=h'(x)^{\deg(F)}\cdot \left((a^{-1})^{\deg(F)}F(aQ_G'(x)+b)\right)\\ &=h'(x))^{\deg(F)}\cdot (\underbrace{[A]\ast F}_{=:H(x)})^{Q_G'}, \end{aligned}$$ where $$A=\left(\begin{array}{cc} a& b\\ 0& 1 \end{array}\right).$$ Therefore $H(x)=[A]\ast F(x)$ is an irreducible polynomial of degree $\deg(H)=\deg(F)=d(Q_G)$. Additionally, $H^{Q_G'}=F^{Q_G}=\prod(G\ast r)^k$ with $k>1$, so $d(Q_G')\ge d(Q_G)$. Because of symmetry we get $d(Q_G')=d(Q_G)$. ◻ This is why we can write $d(G)$ instead of $d(Q_G)$. We give an example that shows that all the values {0,1,2} are possible: *Example 14*. As a field we take $\mathbb{F}_2$. For $d(G)=0$ we take $$G=\left\{\left[\left(\begin{array}{cc} 1& 1\\ 0& 1 \end{array}\right)\right],\left[\left(\begin{array}{cc} 1& 0\\ 0& 1 \end{array}\right)\right]\right\}.$$ The two possible quotient maps are $Q_G(x)=x^2+x$ and $Q_G(x)=x^2+x+1$. Set $$A=\left(\begin{array}{cc} 1& 1\\ 0& 1 \end{array}\right),$$ then $$[A]\ast f(x)=f(x+1)$$ and $$[A]\circ v=v+1$$ for all $v\in\overline{\mathbb{F}}_2$ and $[A]\circ \infty=\infty$. Since $v\neq v+1$ for $v\in\overline{\mathbb{F}}_2$ all $G$-orbits in $\overline{\mathbb{F}}_2$ are regular, thus $d(G)=0$ by Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"}. For $d(G)=1$ we choose $$G=\left\{\left[\left(\begin{array}{cc} 0& 1\\ 1& 0 \end{array}\right)\right],\left[\left(\begin{array}{cc} 1& 0\\ 0& 1 \end{array}\right)\right]\right\}$$ with quotient map $Q_G(x)=x+1/x=(x^2+1)/x$. Note that $d(G)<2$ since $(x^2+x+1)^{Q_G}=x^4+x^3+x^2+x+1$ is irreducible. To show that $d(Q_G)\ge 1$ we calculate $$x^{Q_G}=x^2+1=(x+1)^2.$$ For $d(G)=2$ we can look at $$G=\left\{\left[\left(\begin{array}{cc} 1& 1\\ 1& 0 \end{array}\right)\right],\left[\left(\begin{array}{cc} 0& 1\\ 1& 1 \end{array}\right)\right],\left[\left(\begin{array}{cc} 1& 0\\ 0& 1 \end{array}\right)\right]\right\}$$ with quotient map $Q_G(x)=(x^3+x+1)/(x^2+x)$. For this case we conveniently only have to look at $$\begin{aligned} (x^2+x+1)^{Q_G}=(x^2+x+1)^3, \end{aligned}$$ so $d(G)=2$. Later we will often determine $d(G)$ by using the second part of Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"} in a contrapositive way, that means: $$F^{Q_G}=\prod(G\ast r)^k \text{ with } k>1 \Rightarrow |G\circ v|<|G| \text{ for a root }v\in\overline{\mathbb{F}}_q\text{ of }F^{Q_G}.$$ Since there are only finitely many non-regular $G$-orbits we only have to check finitely many irreducible polynomials $F$ of degree $1$ or $2$. In this paper we only have to check at most $2$ polynomials. Let $$\omega_G(n,k):=|\{G\ast r \mid r\in\mathcal{I}_q^n\text{ and }|G\ast r|=k\}|$$ be the number of $G$-orbits in $\mathcal{I}_q^n$ of size $k$ and $N_q(n):=|\mathcal{I}_q^n|$. Our main result is the following: **Theorem 15**. *Let $G\le\operatorname{PGL}_2(\mathbb{F}_q)$. For all $m> d(G)$ we get $$N_q(m)=\sum\limits_{k\mid |G|}\omega_G(m\cdot \frac{|G|}{k},k).$$* *Proof.* Define $$M_m(G):=\{(a,b)\in\mathbb{N}^2:~b\mid |G| \text{ and } a\cdot b=|G|\cdot m\}$$ and for $n\ge 1$ $$\Omega_{G}(n,k):=\{G\ast r \mid r\in\mathcal{I}_q^n \text{ and } |G\ast r|=k\},$$ so $\omega_G(n,k)=|\Omega_{G}(n,k)|$. We want to show that $$\delta_{Q_G}(\mathcal{I}_q^m)=\bigcup_{(n,k)\in M_m(G)}\Omega_{G}(n,k).$$ for $\delta_{Q_G}$ as in Theorem [Corollary 12](#main2){reference-type="ref" reference="main2"}. Let $F\in\mathcal{I}_q^m$, then by Theorem [Corollary 12](#main2){reference-type="ref" reference="main2"} together with $m>d(G)$ we have that $F^{Q_G}=\prod(G\ast r)$ for $r\in\mathcal{I}_q^G$ and by Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"} $$\deg(r)\cdot |G\ast r|=m \cdot |G|.$$ Thus $(\deg(r),|G\ast r|)\in M_m(G)$ and $$\delta(F)\in \Omega_G(\deg(r),|G\ast r|)\subseteq \bigcup\limits_{(n,k)\in M_m(G)}\Omega_{G}(n,k).$$ This shows $\delta(\mathcal{I}_q^m)\subseteq \bigcup_{(n,k)\in M_m(G)}\Omega_{G}(n,k)$. Conversely let $G\ast r\in\Omega_{G}(n,k)$ with $(n,k)\in M_m(G)$, thus $|G\ast r|=k$ and $\deg(r)=n$. Moreover $$\deg(r)\cdot |G\ast r|=n\cdot k=|G|\cdot m$$ By Theorem [Corollary 12](#main2){reference-type="ref" reference="main2"} there exists a polynomial $F\in\mathcal{I}_q$ such that $F^{Q_G}=\prod(G\ast r)^l$ and $l\ge 1$. Hence $$\begin{aligned} \deg(F^{Q_G})&=|G|\cdot \deg(F)\\&=l\cdot \deg(r)\cdot |G\ast r|=l\cdot (|G|\cdot m) \end{aligned}$$ So $\deg(F)=l\cdot m>d(G)$ and as a consequence $l=1$. Therefore the degree of $F$ is $m$ which shows $$\delta_{Q_G}(\mathcal{I}_q^m)\supseteq\bigcup_{(n,k)\in M_m(G)}\Omega_{G}(n,k),$$ so both sets are equal. Since $\delta_{Q_G}$ is a bijection we get $$\begin{aligned} N_q(m)&=|\delta_{Q_G}(\mathcal{I}_q^m)|=|\bigcup_{(n,k)\in M_m(G)}\Omega_{G}(n,k)|=\sum\limits_{(n,k)\in M_m(G)}|\Omega_{G}(n,k)|\\&=\sum\limits_{(n,k)\in M_m(G)} \omega_{G}(n,k)=\sum\limits_{k\mid |G|}\omega_{G}(m\cdot \frac{|G|}{k},k). \end{aligned}$$ ◻ This formula can be used to obtain the number of irreducible monic $G$-invariant polynomials, because $$\omega_G(|G|\cdot m,1)=N_q(m)-\sum\limits_{\substack{k\mid |G|,\\ k\neq 1}}\omega_G(m\cdot \frac{|G|}{k},k)$$ and the $G$-orbits of size 1 are the irreducible monic $G$-invariant polynomials. However, as the numbers $\omega_G(n,k)$ are hard to compute in general this formula is not practical. For the exact counting formulae of $G$-invariant polynomials see [@reisEx]. Some special values of $\omega_G(n,k)$ are: - $\omega_G(n,1)=0$ if $n>2$ and $|G|\nmid n$ (Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"} and Lemma [Lemma 11](#bluherquad){reference-type="ref" reference="bluherquad"}) - $\omega_G(n,1)=0$ if $n>2$ and $G$ is non-cyclic ([@reisEx Theorem 1.3] or [@schulz Corollary 35]) - $\omega_G(n,k)=\frac{1}{k}|N_q(n)|$ if $k\nmid n$ and $|G|=k$ is prime (use first item and standard group action arguments) There is one family of subgroups for which Theorem [Theorem 15](#comb){reference-type="ref" reference="comb"} yields an easy counting formula: **Corollary 16**. *Let $G\le\operatorname{PGL}_2(\mathbb{F}_q)$ be a cyclic subgroup with $|G|=s$ and $s$ is prime. For all $m>d(G)$ we have $$N_q(m)=\omega_G(m,s)+\omega_G(m\cdot s,1)$$* This is enough to prove Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"}. *Proof (Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"})..* If $|G|=s$ is prime then $|C(Q_G,n)|=\omega_G(n\cdot s,1)$ and $|D(Q_G,n)|=N_q(n)-|C(Q_G,n)|=\omega_G(n,s)$ by Corollary [Corollary 16](#easycomb){reference-type="ref" reference="easycomb"} and $n>d(G)$. Thus we have $$\begin{aligned} |C(Q_G,n)|-(s-1)|D(Q_G,n)|&=\omega_G(ns,1)+(s-1)\omega_G(n,s)\\&=\left(\omega_G(ns,1)+\omega_G(n,s)\right)-s\omega_G(n,s)\\&=N_q(n)-s\omega_G(n,s)=\omega_G(n,1) \end{aligned}$$ For all $n>d(G)$ with $s\nmid n$ we have $$|C(Q_G,n)|-(s-1)|D(Q_G,n)|=\omega_G(n,1)=0.$$ If $s\mid n$ and $n/s>d(G)$ then $|C(Q_G,n/s)|=\omega_G(n,1)$ by Corollary [Corollary 16](#easycomb){reference-type="ref" reference="easycomb"}. Moreover, if $n>2$ and $s\mid n$ then also $|C(Q_G,n/s)|=\omega_G(n,1)$ by Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"} since all $G$-invariant irreducible polynomials $f\in \mathcal{I}_q^n$ have roots that lie in regular orbits. That is the case because $[\mathbb{F}_q(\gamma):\mathbb{F}_q]=n>2$ for a root $\gamma$ of $f$ and all non-regular $G$-orbits are contained in $\mathbb{F}_{q^2}\cup\{\infty\}$; see Lemma [Lemma 11](#bluherquad){reference-type="ref" reference="bluherquad"}. The only cases for $n$ that are left to cover are for $d(G)<n\le \min\{2,d(G)\cdot s\}$. If $d(G)\in \{0,2\}$ there are no such $n$. Thus $d(G)=1$ and because $s>1$ also $n=2$. Since we still are in the case $s\mid n$ we have $s=2=|G|$. So the last case we have to cover is $d(G)=1$ and $n=s=2$. Let $F\in\mathcal{I}_q$ be of degree $d(G)=1$ such that $F^{Q_G}=\prod(G\ast r)^k$ with $k\ge 2$. Then $$\deg(F^{Q_G})=|G|\cdot \deg(F)=2=k\cdot |G\ast r|\cdot \deg(r)\ge 2|G\ast r|\cdot \deg(r)$$ so $\deg(r)=1=|G\ast r|$. Hence for all irreducible $G$-invariant polynomials $f$ of degree 2 there is a polynomial $F$ of degree 1 such that $F^{Q_G}=f$ by Corollary [Corollary 12](#main2){reference-type="ref" reference="main2"}, so $|C(Q_G,1)|=\omega_G(2,1)$. In all three subcases of $s\mid n$ we have $|C(Q_G,n/s)|=\omega_G(n,1)$, which concludes the proof. ◻ # Examples ## Proof of Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"} {#proof-of-theorem-ahmad} For proving Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"} we need to consider the cyclic subgroup $$G:=\left\langle\left[\left(\begin{array}{cc} 1& 1\\ 0& 1 \end{array}\right)\right]\right\rangle.$$ It has order $p=\operatorname{char}(\mathbb{F}_q)$ and a quotient map is $Q_G(x)=x^p-x$. Note that the $Q_G$-transformation of polynomials $F\in\mathbb{F}_q[x]$ with $Q_G$ is just the composition of $F$ with $Q_G$, that is, $F^{Q_G}(x)=F(Q_G(x))$. The condition for $F(Q_G(x))=F(x^p-x)$ to be irreducible is well-known and originally due to Varshamov, see for example [@comppoly Lemma 1.1] and the references therein. For $F\in\mathcal{I}_q^n$ and $\alpha\in\overline{\mathbb{F}}_q$ a root of $F$ we have $$F(Q_G(x))\in\mathbb{F}_q[x] \text{ is irreducible} \Leftrightarrow \operatorname{Tr}_{q^n/p}(\alpha)\neq 0.$$ As mentioned in the introduction we can write the condition as follows: $$F(Q_G(x))\in\mathbb{F}_q[x] \text{ is irreducible }\Leftrightarrow \operatorname{Tr}(F)\neq 0.$$ Hence $$\bigcup\limits_{a\in\mathbb{F}_p^{\ast}}S_a(n)=C(Q_G,n)$$ and $D(Q_G,n)=S_0(n)$. The number $d(G)=0$ since the only non-regular $G$-orbit in $\overline{\mathbb{F}}_q\cup\{\infty\}$ is $\{\infty\}$. Applying Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"} gives $$\begin{aligned} \left|\bigcup\limits_{a\in\mathbb{F}_p^{\ast}}S_a(n)\right|-(p-1)|S_0(n)|&=|C(Q_G,n)|-(p-1)|D(Q_G,n)|\\&=\begin{cases} 0, &p\nmid n\\ |C(Q_G,n/p)|, &p\mid n \end{cases}\\&=\begin{cases} 0, &p\nmid n\\ |\bigcup\limits_{a\in\mathbb{F}_p^{\ast}}S_a(n/p)|, &p\mid n \end{cases}\end{aligned}$$ for all $n>d(G)=0$. This proves Theorem [Theorem 1](#ahmad){reference-type="ref" reference="ahmad"}. ## Proof of Theorem [Theorem 3](#quad){reference-type="ref" reference="quad"} {#proof-of-theorem-quad} Assume $q\equiv 1 \pmod 2$ and let $u,v\in\mathbb{F}_q$ such that $u\neq v$. Consider the matrix $$A_{u,v}:=\left(\begin{array}{cc} \frac{1}{2}(u+v)& -uv\\ 1& -\frac{1}{2}(u+v) \end{array}\right).$$ Since $[A_{u,v}]^2=[I_2]$ the cyclic group $G_{u,v}:=\langle [A_{u,v}]\rangle\le \operatorname{PGL}_2(\mathbb{F}_q)$ contains only 2 elements. As a quotient map for $G_{u,v}$ we choose $$Q_{G_{u,v}}(x)=\frac{1}{2}(x+[A_{u,v}]\circ x)=\frac{x^2-uv}{2x-(u+v)}.$$ Note that $Q_{G_{u,v}}(u)=u$ and $Q_{G_{u,v}}(v)=v$ since $[A_{u,v}]\circ u=u$ and $[A_{u,v}]\circ v=v$. Moreover $\{u\}$ and $\{v\}$ are the only non-regular $G_{u,v}$-orbits in $\overline{\mathbb{F}}_q\cup\{\infty\}$, hence $d(G_{u,v})$ is the highest degree of the two polynomials $F_1,F_2\in\mathcal{I}_q$ (if they exist) such that $$\begin{aligned} F_1^{Q_{G_{u,v}}}(x)&=(x-u)^2\\ F_2^{Q_{G_{u,v}}}(x)&=(x-v)^2\end{aligned}$$ by Theorem [Theorem 10](#othermain){reference-type="ref" reference="othermain"}. We chose $Q_{G_{u,v}}$ so that $(x-u)^{Q_{G_{u,v}}}=(x-u)^2$ and $(x-v)^{Q_{G_{u,v}}}=(x-v)^2$, thus $d(G_{u,v})=1$. Let $F\in\mathcal{I}_q$, then $F^{Q_{G_{u,v}}}$ is irreducible if and only if $$\begin{aligned} P(x)&:=(x^2-uv)-\gamma(2x-(u+v))\\ &=x^2-2\gamma x+(\gamma(u+v)-uv)\in\mathbb{F}_q(\gamma)[x]\end{aligned}$$ is irreducible for $\gamma\in\overline{\mathbb{F}}_q$ a root of $F$ by Lemma [Lemma 8](#cohen){reference-type="ref" reference="cohen"}. A quadratic polynomial over $\mathbb{F}_q(\gamma)$ is irreducible if and only if it has no roots in $\mathbb{F}_q(\gamma)$. For $P$ this is equivalent to $4\gamma^2-4(\gamma(u+v)-uv)$ being a non-square in $\mathbb{F}_q(\gamma)=\mathbb{F}_{q^{\deg(F)}}$. Hence $$\begin{aligned} -1&=\left(\frac{4\gamma^2-4(\gamma(u+v)-uv)}{q^{\deg(F)}}\right)=\left(\frac{\gamma^2-(u+v)\gamma+uv}{q^{\deg(F)}}\right)\\&=\left(\frac{(\gamma-u)(\gamma-v)}{q^{\deg(F)}}\right)=\left((\gamma-u)(\gamma-v)\right)^{\frac{q^{\deg(F)}-1}{2}}\\&=\left((u-\gamma)(v-\gamma)\right)^{\frac{q^{\deg(F)}-1}{q-1}\cdot \frac{q-1}{2}}=\left(\prod\limits_{i=0}^{\deg(F)-1}(u-\gamma^{q^i})\right)^{\frac{q-1}{2}}\cdot \left(\prod\limits_{i=0}^{\deg(F)-1}(v-\gamma^{q^i})\right)^{\frac{q-1}{2}}\\&=F(u)^{\frac{q-1}{2}}\cdot F(v)^{\frac{q-1}{2}}=\left(\frac{F(u)\cdot F(v)}{q}\right).\end{aligned}$$ This calculation is very similar to a calculation trick that Meyn used in [@meyn Proof of Theorem 8]. We showed that $C(Q_{G_{u,v}},n)=\{f\in\mathcal{I}_q^n\mid \left(\frac{f(u)f(v)}{q}\right)=-1\}$, the rest of the proof of Theorem [Theorem 3](#quad){reference-type="ref" reference="quad"} follows from Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"}. ## An Example similar to Theorem [Theorem 3](#quad){reference-type="ref" reference="quad"} {#an-example-similar-to-theorem-quad} Let $\mathbb{F}_q$ be an arbitrary finite field, $s$ a prime dividing $q-1$ and $c\in\mathbb{F}_q$. Define $$\begin{aligned} T_c(n)&:=\left\{f\in\mathcal{I}_q^n\mid f(c)^{\frac{q-1}{s}}\neq (-1)^{\frac{(q-1)n}{s}}\right\}\\ U_c(n)&:=\mathcal{I}_q^n\setminus T(n)=\left\{f\in\mathcal{I}_q^n\mid f(c)^{\frac{q-1}{s}}= (-1)^{\frac{(q-1)n}{s}}\right\}.\end{aligned}$$ We are going to prove the following theorem **Theorem 17**. *Let $\mathbb{F}_q$ be an arbitrary finite field and $s$ a prime dividing $q-1$. Moreover let $c\in\mathbb{F}_q$, then we have for all $n\ge 2$ that $$|T_c(n)|-(s-1)|U_c(n)|=\begin{cases} 0,&\text{ if }s\nmid n\\ |T_c(n/s)|,&\text{ if }s\mid n \end{cases}$$* Before we start with the proof we need to formulate a lemma first, which is folklore. **Lemma 18**. *Let $\mathbb{F}_q$ be an arbitrary finite field, $c\in\mathbb{F}_q^{\ast}$ and $s$ a prime dividing $q-1$. The polynomial $x^s-c\in\mathbb{F}_q[x]$ is irreducible if and only if $c^{\frac{q-1}{s}}\neq 1$.* *Proof. (Theorem [Theorem 17](#lastexam){reference-type="ref" reference="lastexam"}).* We consider the following matrix $$A:=\left(\begin{array}{cc} a& b\\ 0& 1 \end{array}\right).$$ where $a\in\mathbb{F}_q^{\ast}$ has order $s>1$ which is prime and $b\in\mathbb{F}_q$ is arbitrary. The group $G:=\langle [A]\rangle$ has order $s$. The fixed points of $[A]$ are $\infty$ and $c:=\frac{-b}{a-1}$ and these are again the only non-regular orbits in $\overline{\mathbb{F}}_q\cup\{\infty\}$. Note that we can obtain every $c\in\mathbb{F}_q$ for fixed $a\in\mathbb{F}_q^{\ast}\setminus\{1\}$ by choosing $b=(1-a)\cdot c$. A quotient map for $G$ is $$Q_G(x)=(x-c)^s+c.$$ and $(x-c)^{Q_G}=(x-c)^s$, so $d(G)=1$. Let $F\in\mathcal{I}_q^n$, then $F^{Q_G}(x)=F(Q_G(x))$ is irreducible if and only if $$P(x)=Q_G(x)-\gamma=(x-c)^s+c-\gamma\in\mathbb{F}_{q^n}[x]$$ is irreducible by Lemma [Lemma 8](#cohen){reference-type="ref" reference="cohen"}, where $\gamma$ is a root of $F$. The polynomial $P(x)\in\mathbb{F}_{q^n}[x]$ is irreducible if and only if $P(x+c)=x^s-(\gamma-c)\in\mathbb{F}_{q^n}[x]$ is irreducible. By Lemma [Lemma 18](#crit){reference-type="ref" reference="crit"} this is the case exactly when $(\gamma-c)^{(q^n-1)/s}\neq 1$. Now we calculate: $$\begin{aligned} 1&\neq (\gamma-c)^{\frac{q^n-1}{s}}=(-1)^{\frac{q^n-1}{q-1}\cdot \frac{q-1}{s}}\cdot (c-\gamma)^{\frac{q^n-1}{q-1}\cdot \frac{q-1}{s}}\\&=\left(\prod\limits_{i=0}^{n-1}(-1)^{q^i}\right)^{\frac{q-1}{s}}\cdot \left(\prod\limits_{i=0}^{n-1}(c-\gamma^{q^i})\right)^{\frac{q-1}{s}}=(-1)^{\frac{(q-1)n}{s}}\cdot f(c)^{\frac{q-1}{s}}\end{aligned}$$ Hence $C(Q_G,n)=T_c(n)$, the rest follows from Theorem [Theorem 4](#backbone){reference-type="ref" reference="backbone"} again. ◻ *Remark 19*. 1. If we take $s=2$ the condition in $T_c(n)$ is $$(-1)^{\frac{(q-1)n}{2}}\neq f(c)^{\frac{q-1}{2}}=\left(\frac{f(c)}{q}\right).$$ This looks quite similar to the defining condition of $C_{u,v}(n)$ in Theorem [Theorem 3](#quad){reference-type="ref" reference="quad"}. 2. In Theorem [Theorem 17](#lastexam){reference-type="ref" reference="lastexam"} we used the criterion of Lemma [Lemma 18](#crit){reference-type="ref" reference="crit"} for the irreducibility of binomials of the form $x^s-c$ where $s$ is a prime dividing $q-1$. If the reader is interested in a recent paper that explains the factorization of polynomials $x^n-c\in\mathbb{F}_q[x]$ for arbitrary $n$ we refer them to [@graner2].

arxiv_math

--- abstract: | We prove a four-term exact sequence of surface orbifold pure braid groups for all genus $\geq 1$, $2$-dimensional orientable orbifolds with cone points. This corrects our earlier result in [@Rou21]. address: | School of Mathematics\ Tata Institute\ Homi Bhabha Road\ Mumbai 400005, India author: - S.K. Roushon title: A four-term exact sequence of surface orbifold pure braid groups --- # Introduction Let $M$ be a connected smooth manifold of dimension $\geq 2$. Let $PB_n(M)$ be the configuration space of ordered $n$-tuples of pairwise distinct points of $M$. Then, the Fadell-Neuwirth fibration theorem ([@FN62]) says that, for $n\geq 2$, the projection map $M^n\to M^{n-1}$ to the first $n-1$ coordinates defines a fibration $f(M):PB_n(M)\to PB_{n-1}(M)$, with fiber homeomorphic to $\widetilde M:=M-\{(n-1)\ \text{points}\}$. Hence $f(M)$ induces the following long exact sequence of homotopy groups. $$\begin{aligned} \label{1.1} \xymatrix@C-1.2pc{\cdots\ar[r]&\pi_2(PB_{n-1}(M))\ar[r]&\pi_1(\widetilde M)\ar[r]&\pi_1(PB_n(M))\ar[r]&\pi_1(PB_{n-1}(M))\ar[r]&1.}\end{aligned}$$ It is an important subject to study the homotopy groups, especially the fundamental groups of the configuration spaces of a manifold. Since in dimension $\geq 3$, the space $PB_n(M)$ and the product manifold $M^n$ have isomorphic fundamental groups, the dimension $2$ case is of much interest. For $M={\mathbb C}$, $\pi_1(PB_n(M))$ is known as the pure braid group on $n$ strings. The braid groups were introduced in [@Ar47], and appear in a wide range of areas in both Mathematics and Physics. In [@Rou20] we studied the possibility of extending the Fadell-Neuwirth fibration theorem for orbifolds, to understand a certain class of Artin groups. Orbifolds are also of fundamental importance in algebraic and differential geometry, topology and string theory. However, to define a fibration between orbifolds, we had to consider the category of Lie groupoids. Since an orbifold can be realized as a Lie groupoid ([@Moe02]), and there are enough tools in this category to define a fibration. There, we defined two notions ($a$ and $b$-types) of a fibration (\[[@Rou20], Definition 2.4\]) and the corresponding ($a$ and $b$-types) configuration Lie groupoids of a Lie groupoid to enable us to state a Fadell-Neuwirth type theorem. For an orbifold $M$, the $b$-type configuration Lie groupoid is the correct model to induce the orbifold structure on $PB_n(M)$. We proved that the Fadell-Neuwirth fibration theorem extends in this generality, under some strong hypothesis ($c$-groupoid). We also showed that this is the best possible extension. For this, we deduced that the map $f(M)$ is not a $a$(or $b$)-type fibration for the $a$(or $b$)-type configuration Lie groupoids of Lie groupoids, corresponding to global quotient compact orbifolds of dimension $\geq 2$, with non-empty singular set (See \[[@Rou23-1], Proposition 3.1\]). Recently, in [@JF23] Flechsig corrected the short exact sequence of fundamental groups of the $b$-type configuration Lie groupoids, we deduced in \[[@Rou20], Theorem 2.14, Remark 2.15\] corresponding to all genus zero $2$-dimensional orbifolds with cone points and at least one puncture. He showed that it is in fact a four-term exact sequence. This also implies that $f(M)$ is not even a 'quasifibration' of orbifolds. See Remark [Remark 3](#les){reference-type="ref" reference="les"}. In this paper we correct the analogous short exact sequence, for all genus $\geq 1$, $2$-dimensional orbifolds with cone points, we gave in [@Rou21], and establish a similar four-term exact sequence. Using [@JF23], we further show that it is not a short exact sequence if the orbifold has a cone point (Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"}). The main ingredients of the proof of Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"} is the presentation of the surface pure braid groups of an orientable surface of genus $\geq 1$, from [@Bel04], and a $\pi_1^{orb}$-injectivity result from [@Rou24] for $2$-dimensional orbifolds. # A Four-term exact sequence {#MR} For a connected $2$-dimensional orbifold $M$, $PB_n(M)$ is again an orbifold. **Definition 1**. * The orbifold fundamental group of $PB_n(M)$ is called the *surface orbifold pure braid* group of $M$ on $n$ strings.* Let ${\mathcal C}_0$ be the class of all connected, genus zero $2$-dimensional orbifolds with cone points and at least one puncture, and let ${\mathcal C}_1$ be the class of all connected, genus $\geq 1$, $2$-dimensional orientable orbifolds with cone points. If $M\in{\mathcal C}_0\cup {\mathcal C}_1$ has nonempty boundary, then we replace each boundary component of $M$ by a puncture. We prove the following theorem. **Theorem 2**. *There is a four-term exact sequence of surface orbifold pure braid groups of $M\in {\mathcal C}_1$, as follows.* *-.6pc1&K(M, n-1)&\_1\^orb(M)&\_1\^orb(PB_n(M))\^f(M)\_\*&\_1\^orb(PB\_n-1(M))&1.* *Here $\widetilde M=M-\{(n-1)\ \text{smooth points}\}$. Furthermore, if $M$ has a cone point, then $K(M, n-1)\neq \langle 1\rangle$.* Flechsig proved that for $M\in {\mathcal C}_0$, if $M$ has a cone point then $K(M, n-1)\neq \langle 1\rangle$. In [@Rou20] we used a stretching technique, and there we did not need to appeal to the Fadell-Neuwirth fibration theorem for punctured complex plane. However, the stretching technique is not applicable for genus $\geq 1$ orbifolds, as in this case the movement of the strings of a braid is more complicated and a pictorial description is not possible. To prove Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"}, first we use the presentation of surface pure braid groups from [@Bel04], to produce an explicit set of generators and relations of the surface orbifold pure braid group of $M\in {\mathcal C}_1$. Then, we apply the Fadell-Neuwirth fibration theorem for surfaces to complete the argument. **Remark 3**. *If $M$ is an aspherical $2$-manifold then using the exact sequence ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) and by an induction on $n$, it follows that $\pi_k(PB_n(M))=1$, for all $k\geq 2$, and hence ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) becomes a short exact sequence of fundamental groups. Now let $M\in {\mathcal C}_0$ with a cone point. Then by [@JF23], $K(M, n-1)\neq \langle 1\rangle$. We have given examples of $M\in {\mathcal C}_0$ with cone points in \[[@Rou23-1], Theorem 1.2\], such that $\pi^{orb}_k(PB_n(M))=1$, for all $k\geq 2$. Therefore, for such $M$ there is no long exact sequence induced by $f(M)$, similar to ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}), of orbifold homotopy groups.* # Proof {#proofs} In this section we give the proof of Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"}. *Proof of Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"}.* We have assumed that the underlying space of the orbifold is orientable. Also, we assume that the orbifold has a cone point, since otherwise the Fadell-Neuwirth fibration theorem will be applicable. First we give the proof in the case when $\pi_1^{orb}(M)$ is finitely generated. **Case 1: $\pi_1^{orb}(M)$ is finitely generated:** Under this hypothesis, we can assume that $M$ is of the type $M_r^s$, where $M_r^s$ is a $2$-dimensional orientable orbifold of genus $g\geq 1$, with $r$ number of punctures and $s\geq 1$ number of cone points of orders $q_1,q_2,\ldots,q_s$. The plan of the proof is as follows. We start with the direct way to define the orbifold fundamental group of an orbifold from [@Thu91]. Then, we give an explicit presentation of the orbifold fundamental group of $PB_n(M)$, using the presentation of the surface pure braid group of a surface, from [@Bel04]. The homomorphism $\pi_1^{orb}(PB_n(M))\to \pi_1^{orb}(PB_{n-1}(M))$ is then described using this presentation. Finally, we apply the Fadell-Neuwirth fibration theorem for surfaces and some group theoretic arguments, to conclude the proof in this case. Let $P=\{1,2,\ldots, n\}$ and $\{x_1,x_2,\ldots, x_n\}\subset M_k^0$ be a fixed subset of $n$ distinct points. Consider the set $S$ of all continuous maps $\gamma:P\times I\to M_k^0$, satisfying the following conditions. - $\gamma(i,t)\neq \gamma(j,t)$ for all $t\in I$ and for $i\neq j$. - $\gamma(i,0)=\gamma(i,1)=x_i$ for all $i\in P$. Note that $\gamma$ is a loop in $PB_n(M_k^0)$ based at $(x_1,x_2,\ldots, x_n)\in PB_n(M_k^0)$. We call it a *braid loop*. Given two braid loops $\gamma_0$ and $\gamma_1$, a homotopy between them, fixing end points, is a map $F:P\times I\times I\to M_k^0$ satisfying the following conditions. - For each $t\in I$, $F|_{P\times I\times \{t\}}\in S$. - $F|_{P\times I\times \{0\}}=\gamma_0$ and $F|_{P\times I\times \{1\}}=\gamma_1$. Composition between the braid loops $\gamma_0$ and $\gamma_1$ is defined as $\gamma=\gamma_0*\gamma_1$, where for $i\in P$, the following are satisfied. - $\gamma(i,t)=\gamma_0(i, 2t)$ for $0\leq t \leq \frac{1}{2}$. - $\gamma(i, t)=\gamma_1(i, 2t-1)$ for $\frac{1}{2}\leq t \leq 1$. Clearly, the homotopy classes under this composition law of braid loops, gives $\pi_1(PB_n(M_k^0))$, with base point $(x_1,x_2,\ldots, x_n)$. We are not including the base point in the notation of the fundamental group, as it will remain fixed during the proof. For $PB_{n-1}(M_k^0)$ the base point will be $(x_1,x_2,\ldots, x_{n-1})$. A presentation of the group $\pi_1(PB_n(M_k^0))$, in terms of generators and relations is given in \[[@Bel04], Theorem 5.1\]. In Figure 1 we show all the generators of this group. Let the list of relations be $\bf R$. We do not reproduce the list of relations here, as we will not need its explicit descriptions. That is, we have the following. $$\pi_1(PB_n(M_k^0))=\langle A^j_i, B^j_i,C_l^m,P_i^p; i=1,2,\ldots, n;$$ $$j=1,2,\ldots, g; m < l, m=1,2,\ldots, n-1; p=1,2,\ldots, k\ |\ {\bf R}\rangle.$$ Now let $k=r+s$ and $s\geq 1$. Next, we replace the punctures from $r+1$ to $r+s$ by cone points of orders $q_1,q_2,\ldots, q_s$, respectively, as shown in Figure 2. We also rename the generators with a 'bar' to avoid confusion. {height="5cm" width="10cm"} Figure 1: Generators of the surface pure braid group of $M_k^0$. Then, a presentation of the surface pure orbifold braid group $\pi_1^{orb}(PB_n(M_r^s))$ is obtained from the presentation of $\pi_1(PB_n(M_{r+s}^0))$ by adding the extra relations $(\overline P_i^{r+j})^{q_j}$, for $i=1,2,\ldots, n; j=1,2,\ldots s$. Since, by definition of orbifold fundamental group (see [@All02] or [@Thu91]), if a loop $\eta$ circles around a cone point of order $q$, then $\eta^q=1$ appears as a relation. The only difference between a puncture and a cone point on a $2$-dimensional orbifold, which reflects on the orbifold fundamental group, is this finite order relation. Hence, we have the following. $$\pi_1^{orb}(PB_n(M_r^s))=$$ $$\langle\overline A^j_i, \overline B^j_i, \overline C_l^m, \overline P_i^p; i=1,2,\ldots, n; j=1,2,\ldots, g;m < l, m=1,2,\ldots, n-1;$$ $$p=1,2,\ldots, r+s\ |\ \overline{{\bf R}}\cup \{(\overline P_i^{r+j})^{q_j},\ \text{for}\ i=1,2,\ldots, n, j=1,2,\ldots s\}\rangle.$$ Here $\overline{{\bf R}}$ is the same set of relations as in $\bf R$ but the generators are replaced with a 'bar'. {height="5cm" width="10cm"} Figure 2: Generators of the surface pure orbifold braid group. Clearly, now there is the following surjective homomorphism, which sends a generator $X$ of $\pi_1(PB_n(M_{r+s}^0))$ to $\overline X$, $$g_n:\pi_1(PB_n(M_{r+s}^0))\to \pi_1^{orb}(PB_n(M_r^s)).$$ This homomorphism is also obtained by applying the $\pi_1^{orb}$ functor on the inclusion $PB_n(M_{r+s}^0)\to PB_n(M_r^s)$. Therefore, we have the following commutative diagram. Here, $f_n=f(M_{r+s}^0)_*$, which is induced by the projection to the first $n-1$ coordinates. Recall that, $f(M_{r+s}^ 0)$ is a fibration by the Fadell-Neuwirth fibration theorem. $f_n^o$ is also induced by a similar projection. However, we have seen in \[[@Rou20], Proposition 2.11\]) that the corresponding homomorphism of this projection, on an associated configuration Lie groupoid is not a fibration. Since $f_n$ and $g_{n-1}$ are both surjective, together with the Fadell-Neuwirth fibration theorem we get the following commutative diagram. Here $\widetilde M_{r+s}^0=M_{r+s+n-1}^0$ is the fiber over the point $(x_1,x_2,\ldots, x_{n-1}$) of the projection map. The points $x_1,x_2,\ldots, x_{n-1}$ are replaced by punctures in $\widetilde M_{r+s}^0$. It is clear that, the kernel of $f_n$ is normally generated by the following generators of $\pi_1(PB_n(M_{r+s}^0))$. See Figure 3. $$\{A^j_n, B^j_n,C_n^m,P_n^p; j=1,2,\ldots, g; m=1,2,\ldots, n-1; p=1,2,\ldots, r+s\}.$$ {height="5cm" width="10cm"} Figure 3: Generators of the kernel of $f_n$. Since $f_n$ is induced by a fibration, $\pi_2(PB_k(M_{r+s}^0))=0$ for all $k$ (Remark [Remark 3](#les){reference-type="ref" reference="les"}). Therefore, the above generators, which generate $\pi_1(\widetilde M_{r+s}^0)$, generate a normal subgroup of $\pi_1(PB_n(M_{r+s}^0))$. On the other hand, the kernel of $f^o_n$ is normally generated by the following generators of $\pi_1^{orb}(PB_n(M_r^s))$. See Figure 4. $$\{\overline A^j_n, \overline B^j_n, \overline C_n^m, \overline P_n^p; j=1,2,\ldots, g; m=1,2,\ldots, n-1; p=1,2,\ldots, r+s\}.$$ Next, note that the image of $\pi_1^{orb}(\widetilde M_r^s)$ in $\pi_1^{orb}(PB_n(M_r^s))$ is obtained from the above presentation of $\pi_1(\widetilde M_{r+s}^0)$, by adding the finite order relations $(\overline P_n^{r+j})^{q_j}$, for $j=1,2,\ldots s$. But this is the presentation of $\pi_1^{orb}(\widetilde M_r^s)$, where $\widetilde M_r^s=M_{r+n-1}^s$. Here also, $x_1,x_2,\ldots, x_{n-1}$ are replaced by punctures in $\widetilde M_r^s$. {height="5cm" width="10cm"} Figure 4: Generators of the kernel of $f_n^o$. Since $g_n$ is a surjective homomorphism, and it sends the image of $\pi_1(\widetilde M_{r+s}^0)$ onto the image of $\pi_1^{orb}(\widetilde M_r^s)$, we see that the above generators generate a normal subgroup of $\pi_1^{orb}(PB_n(M_r^s))$. Hence, the kernel of $f^o_n$ is exactly the image of $\pi_1^{orb}(\widetilde M_r^s)$. Therefore, from the above argument we get the following commutative diagram. Here, $K(M_r^s, n-1)$ is denoting the kernel of $\pi_1^{orb}(\widetilde M_r^s)\to \pi_1^{orb}(PB_n(M_r^s))$. This completes the proof of Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"} in the finitely generated case. Next, we give the proof in the case when $\pi_1^{orb}(M)$ is infinitely generated. We will use the finitely generated case in a direct limit argument. **Case 2: $\pi_1^{orb}(M)$ is infinitely generated:** We write $M$ as an increasing union $\bigcup_{i\in {\mathbb N}}M_{r_i}^{s_i}$ of suborbifolds $M_{r_i}^{s_i}$. Here $r_1\leq r_2\leq\cdots$, $s_1\leq s_2\leq\cdots$ and $M_{r_i}^{s_i}$ has finite ($\geq 1$) genus. Clearly, $\pi_1^{orb}(M_{r_i}^{s_i})$ is infinite and finitely generated. Now, we have inclusions $PB_n(M_{r_i}^{s_i})\subset PB_n(M_{r_j}^{s_j})$ for $i \leq j$, and $PB_n(M)=\bigcup_{i\in {\mathbb N}} PB_n(M_{r_i}^{s_i})$. Then, there is the following commutative diagram. +1pcPB_n(M\_r_i\^s_i)\^ f(M\_r_i\^s_i)&PB\_n-1(M\_r_i\^s_i)\ PB_n(M\_r_j\^s_j)\^f(M\_r_j\^s_j)&PB\_n-1(M\_r_j\^s_j). Using Case $1$, the above diagram induces the following commutative diagram. Next, we take the direct limit of this directed system of four-term exact sequences, and complete the proof of the four-term exact sequence of Theorem [Theorem 2](#ig){reference-type="ref" reference="ig"}. Next, we proceed to prove that $K(M, n-1)\neq \langle 1\rangle$, if $M\in {\mathcal C}_1$ has a cone point. Let, $D$ be an open disc in $M$ containing a cone point. Then, we have the following commutative diagram, induced by the inclusion $D\subset M$. -.7pc1&K(D, n-1)&\_1\^orb(D)&\_1\^orb(PB_n(D))&\_1\^orb(PB\_n-1(D))&1\ 1&K(M, n-1)&\_1\^orb(M)&\_1\^orb(PB_n(M))&\_1\^orb(PB\_n-1(M))&1. Note that, $\pi_1^{orb}(\widetilde D)\to \pi_1^{orb} (\widetilde M)$ is injective (see \[[@Rou24], Proposition 3.1\]). Since Flechsig proved in [@JF23] that $K(D, n-1)$ is non-trivial, we conclude that $K(M, n-1)$ is also non-trivial. This completes the proof of the theorem. ◻ See [@All02] or \[[@Rou20], Section 4\] for a different way of looking at elements of the surface pure orbifold braid groups. But that way of presentation is useful for the surface pure orbifold braid groups for members of ${\mathcal C}_0$. **Remark 4**. *Recall that for a manifold $M$ of dimension $\geq 2$, the Fadell-Neuwirth fibration theorem also says that the projection to the first $k$-coordinates, for $k < n$, gives a fibration $PB_n(M)\to PB_k(M)$, with fibers homeomorphic to $PB_{n-k}(M-\{k\ \text{points}\})$. Our proof can be suitably modified to give a proof of the following exact sequence, where $M\in {\mathcal C}_1$ and $\widetilde M=M-\{k\ \text{smooth points}\}$.* *-.7pc1&K(M,k)&\_1\^orb(PB\_n-k(M))&\_1\^orb(PB_n(M))&\_1\^orb(PB_k(M))&1.* 60 D. Allcock, Braid pictures of Artin groups, 354 (2002), no. 9, 3455-3474. E. Artin, Theory of braids, (2) 48 (1947), 101-126. P. Bellingeri, On presentations of surface braid groups, 274 (2004), no. 2, 543-563. E. Fadell and L. Neuwirth, Configuration spaces, 10 (1962), 111-118. J. Flechsig, Braid groups and mapping class groups for $2$-orbifolds, https://doi.org/10.48550/arXiv.2305.04273 I. Moerdijk, Orbifolds as groupoids: an introduction, (Madison, WI, 2001), 205-222, Contemp. Math., 310, *Amer. Math. Soc., Providence, RI*, 2002. S.K. Roushon, Configuration Lie groupoids and orbifold braid groups, 171 (2021), 103028, 35pp. doi:https://doi.org/10.1016/j.bulsci.2021.103028 S.K. Roushon, Quasifibrations in configuration Lie groupoids and orbifold braid groups, https://doi.org/10.48550/arXiv.2106.08110 S.K. Roushon, Orbifold braid groups, https://doi.org/10.48550/arXiv.2301.02043 S.K. Roushon, On aspherical configuration Lie groupoids, https://doi.org/10.48550/arXiv.2309.12198 W. P. Thurston, Three-dimensional Geometry & Topology, December 1991 version, Mathematical Sciences Research Institute Notes, Berkeley, California.

arxiv_math

--- abstract: | We show that a Killing field on a compact pseudo.7pt\--.4ptKähler ddbar manifold is necessarily (real) holomorphic. Our argument works without the ddbar assumption in real dimension four. The claim about holomorphicity of Killing fields on compact pseudo.7pt\--.4ptKähler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada's argument is incomplete. address: - Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA - Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA author: - Andrzej Derdzinski - Ivo Terek title: Killing fields on compact pseudo.7pt\--.7ptKähler manifolds --- =cmr8 =cmti8 =cmtt8 =cmss10 # Introduction {#introduction .unnumbered} By a pseudo.7pt\--.4ptKähler manifold we mean a pseudo.7pt-Riemannian manifold $\,(M\hskip-.7pt,g)$ endowed with a $\,\nabla\hskip-.7pt$-parallel almost-complex structure $\,J$, for the Levi-Civita connection $\,\nabla\hskip.7pt$ of $\,g$, such that the operator $\,J\hskip-1.5pt_x^{\phantom i}:{T\hskip-2.9pt_x^{\phantom i}\hskip-.4ptM}\to{T\hskip-2.9pt_x^{\phantom i}\hskip-.4ptM}\,$ is a linear $\,g_x^{\phantom i}\hskip-.7pt$-isometry (or is, equivalently, $\,g_x^{\phantom i}\hskip-.7pt$-skew-adjoint) at every point $\,x\in M\hskip-.7pt$. We then call $\,(M\hskip-.7pt,g)$ a *pseudo.7pt\--.4ptKähler $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ manifold* if the underlying complex manifold $\,M\,$ has the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ property, or "satisfies the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ lemma" (Section [2](#pa){reference-type="ref" reference="pa"}). The $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ property follows if $\,M$ is compact and admits a Riemannian Kähler metric. **Theorem 1**. *Every Killing vector field on a compact pseudo.7pt\--.4ptKähler $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ manifold is real holomorphic.* We provide two proofs of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"}, in Sections [2](#pa){reference-type="ref" reference="pa"} and [3](#aa){reference-type="ref" reference="aa"}. The former is derived directly from the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ condition; the latter, shorter, relies on the Hodge decomposition, which is equivalent to the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ property [@deligne-griffiths-morgan-sullivan p. 296, subsect. (5.21)]. The Riemannian\--.4ptKähler case of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"} is well known, and straightforward [@ballmann the lines following Remark 4.83 on pp. 60--61]. See also Remark [Remark 4](#riemc){reference-type="ref" reference="riemc"}. For pseudo.7pt\--.4ptKähler surfaces, our argument yields a stronger conclusion. **Theorem 2**. *In real dimension four the assertion of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"} holds without the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ hypothesis.* The authors wish to express their gratitude to Kirollos Masood for bringing Yamada's paper [@yamada] to the first author's attention and discussing with him issues involving Theorem [Theorem 2](#thrmb){reference-type="ref" reference="thrmb"}, formula ([\[tfr\]](#tfr){reference-type="ref" reference="tfr"}), and the Appendix. We also thank Fangyang Zheng for very useful suggestions about Lemma [Lemma 5](#ddbar){reference-type="ref" reference="ddbar"}, and Takumi Yamada for a brief but helpful communication. # Proof of Theorem [Theorem 2](#thrmb){reference-type="ref" reference="thrmb"} {#pb} All manifolds, mappings, tensor fields and connections are assumed smooth. **Lemma 3**. *Given a connection $\,\nabla$ on a manifold $\,M\hskip-.7pt$, let a vector field $\,v\,$ on $\,M\,$ be affine in the sense that its local flow preserves $\,\nabla\hskip-1pt$. Then, for any $\,\nabla\hskip-.7pt$-parallel tensor field $\,\varTheta\hskip.4pt$ on $\,M\hskip-.7pt$, of any type, the Lie derivative $\,\pounds\hskip-1pt_v^{\phantom i}\varTheta\,$ is $\,\nabla\hskip-.7pt$-parallel as well. If $\,\varTheta\,$ happens to be a closed differential form, $\,\pounds\hskip-1pt_v^{\phantom i}\varTheta =\hskip.4ptd\hskip.7pt[\varTheta(v,\,\cdot\,,\ldots,\,\cdot\,)]$.* *Proof.* Clearly, $\,-\hskip-1pt\pounds\hskip-1pt_v^{\phantom i}\varTheta\,$ is the derivative with respect to the real variable $\,t$, at $\,t=0$, of the push-forwards $\,[d\phi\hskip-.4pt_t^{\phantom i}]\varTheta\,$ under the local flow $\,t\mapsto\phi\hskip-.4pt_t^{\phantom i}$ of $\,v$. All $\,[d\phi\hskip-.4pt_t^{\phantom i}]\varTheta$ being $\,\nabla\hskip-.7pt$-parallel, so is $\,\pounds\hskip-1pt_v^{\phantom i}\varTheta$. For the final clause, use Cartan's homotopy formula $\,\pounds\hskip-1pt_v^{\phantom i}=\imath_v^{\phantom i}\hskip-.4ptd+d\hskip.7pt\imath_v^{\phantom i}$ for $\,\pounds\hskip-1pt_v^{\phantom i}$ acting on differential forms [@lee Thm.  14.35, p. 372]. ◻ Lemma [Lemma 3](#ldpar){reference-type="ref" reference="ldpar"} also follows from the Leibniz rule: $\,\pounds\hskip-1pt_v^{\phantom i}(\nabla\hskip-.7pt\varTheta) =(\pounds\hskip-1pt_v^{\phantom i}\nabla)\hskip.7pt\varTheta +\nabla\hskip-.7pt(\pounds\hskip-1pt_v^{\phantom i}\varTheta)$. Let $\,(M\hskip-.7pt,g)\,$ now be a fixed pseudo.7pt\--.4ptKähler manifold. If $\,v\,$ is any vector field on $\,M\,$ then, with $\,J\,$ and $\,\nabla\hskip-.7ptv\,$ treated as bundle morphisms $\,T\hskip-.7ptM\to T\hskip-.7ptM\hskip-.7pt$, $$\label{com} \mathrm{for\ }\,B=\nabla\hskip-.7ptv\,\mathrm{\ and\ }\,A\hskip-.7pt =\hskip.7pt\pounds\hskip-1pt_v^{\phantom i}J\,\mathrm{\ one\ has\ }\,A=[J,B\hskip.4pt]\,\mathrm{\ and\ }\,J\hskip-.7ptA=-\hskip-.7ptAJ\hskip.4pt,$$ which is immediate from the Leibniz rule. For the Kähler form $\,\omega=g(J\hskip.7pt\cdot\,,\,\cdot\,)\,$ of $\,(M\hskip-.7pt,g)\,$ and any $\,g$\--.4ptKilling vector field $\,v$, it follows from ([\[com\]](#com){reference-type="ref" reference="com"}) and Lemma [Lemma 3](#ldpar){reference-type="ref" reference="ldpar"} that $$\label{pre} \begin{array}{rl} \mathrm{i)}&A\,=\hskip.7pt\pounds\hskip-1pt_v^{\phantom i}J\,\mathrm{\ and\ }\,\,\alpha\, =\hskip.7pt\pounds\hskip-1pt_v^{\phantom i}\hskip.7pt\omega\hskip.7pt\,\mathrm{\ are\ related\ by\ }\,\alpha =g(A\hskip.7pt\cdot\,,\,\cdot\,)\hskip.4pt,\mathrm{\ while}\\ \mathrm{ii)}&A^*\hskip-.7pt=-\hskip-.7ptA\hskip.4pt,\hskip7.6ptJ\hskip-.7ptA=-\hskip-.7ptAJ\hskip.4pt,\hskip7.6pt\nabla\hskip-1ptA =0\hskip.4pt,\hskip7.6pt\nabla\hskip-.7pt\alpha=0\hskip.4pt,\mathrm{\ \ and\ }\,\alpha\,\mathrm{\ is\ exact.} \end{array}$$ Given an exact $\,p\hskip.7pt$-form $\,\alpha\,$ on a compact pseudo.7pt-Riemannian manifold $\hskip.7pt(M\hskip-.7pt,g)$, $$\label{ltw} \alpha\,\mathrm{ \ is\ }\,L\hskip-.7pt^2\hskip-.7pt\hskip.5pt\vbox {\hbox{\vrule width3ptheight0.5ptdepth0pt}\vskip 2.2pt}\hskip.5pt\mathrm{or\-thog\-o\-nal\ to\ all\ parallel\ }\,p\,\mathrm{\ times\ co\-var\-i\-ant\ tensor\ fields\ }\,\theta\,\mathrm{\ on\ }\,M.$$ Namely, $\,(\theta,\alpha)=(\mu,\alpha)=(\mu,d\hskip.4pt\beta) =(d\hskip.4pt^*\hskip-1.5pt\mu,\beta)\,$ for $\,\beta\,$ with $\,\alpha=d\hskip.4pt\beta$, the skew-symmetric part $\,\mu\,$ of $\,\theta$, and the $\,L\hskip-.7pt^2$ inner product $\,(\,,\hskip.7pt)$, while $\,d\hskip.4pt^*\hskip-1.5pt\mu=0$, as $\,\nabla\hskip-.7pt\mu=0$. *Remark 4*. By ([\[pre\]](#pre){reference-type="ref" reference="pre"}.7pt-ii) and ([\[ltw\]](#ltw){reference-type="ref" reference="ltw"}), for a Killing field $\,v\,$ on a compact *Riemannian* Kähler manifold, $\,\pounds\hskip-1pt_v^{\phantom i}\hskip.7pt\omega\,$ is $\,L\hskip-.7pt^2\hskip-.7pt$-orthogonal to itself, and so, as a consequence of ([\[pre\]](#pre){reference-type="ref" reference="pre"}.7pt-i), $\,v\,$ must be real holomorphic. Let $\,(M\hskip-.7pt,g)\,$ be, again, a pseudo.7pt\--.4ptKähler manifold. The vector-bundle morphisms $\,C:T\hskip-.7ptM\to T\hskip-.7ptM\,$ having $\,C^*=-C\,$ (that is, $\,g_x^{\phantom i}\hskip-.7pt$-skew-adjoint at every point $\,x\in M$) constitute the sections of $$\label{vsb} \mathrm{\ the\ vector\ sub\-bundle\ }\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)\,\mathrm{\ of\ }\,\mathrm{End}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}(T\hskip-.7ptM) =\mathrm{Hom}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}(T\hskip-.7ptM\hskip-.7pt,T\hskip-.7ptM)\hskip.4pt.$$ We denote by $\,\mathcal{E}\hskip.7pt$ the vector subbundle of $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)$, the sections $\,C\,$ of which are also complex-antilinear (so that $\,JC=-C\hskip-.4ptJ$, in addition to $\,C^*=-C$). Then $$\label{cvb} \begin{array}{l} \mathcal{E}\hskip.7pt\mathrm{\ is\ a\ complex\ vector\ bundle\ of\ rank\ }\,m(m-1)/2\mathrm{,\ where\ }\,m=\dim_{\hskip.4pt{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}}\hskip-1.2pt}^{\phantom i}M\hskip-1pt,\\ \mathrm{with\ a\ pseu\-do\hskip.7pt}\hskip.5pt\vbox {\hbox{\vrule width3ptheight0.5ptdepth0pt}\vskip 2.2pt}\hskip.5pt\mathrm{Her\-mit\-i\-an\ fibre\ metric\ having\ the\ real\ part\ induced\ by\ }g. \end{array}$$ In fact, $\,C\mapsto JC\,$ provides the complex structure for $\,\mathcal{E}\hskip-.7pt$. Nondegeneracy of $\,g$ restricted to $\,\mathcal{E}\hskip.7pt$ follows from $\,g$-orthogonality of the decomposition $\,\mathrm{End}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}(T\hskip-.7ptM) =\,\mathrm{End}_{\hskip.4pt{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}}\hskip0pt}^{\phantom i}(T\hskip-.7ptM)\oplus\mathcal{E}\hskip-.7pt\oplus\mathcal{D}\hskip-.7pt$, the sections $\,C\,$ of the subbundle $\,\mathcal{D}\,$ being characterized by $\,JC=-C\hskip-.4ptJ\,$ and $\,C^*\hskip-.7pt=C$, with $\,\mathrm{End}_{\hskip.4pt{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}}\hskip0pt}^{\phantom i}(T\hskip-.7ptM)\,$ orthogonal to $\,\mathcal{E}\hskip.7pt\oplus\mathcal{D}\hskip.7pt$ since any antilinear morphism $\,C:T\hskip-.7ptM\to T\hskip-.7ptM\,$ is conjugate, via $\,J$, to $\,-C$, and so $\,\mathrm{tr}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}C=0$. The pseudo.7pt-Hermitian fibre metric in $\,\mathcal{E}\hskip.7pt$ arises by restricting $\,\langle\hskip.4pt\cdot\hskip.7pt,\hskip-.4pt\cdot\hskip.4pt\rangle-i\langle J\hskip.7pt\cdot\hskip.7pt,\hskip-.4pt\cdot\hskip.4pt\rangle\,$ to $\,\mathcal{E}\hskip-.7pt$, for the pseudo.7pt-Riemannian fibre metric $\,\langle\hskip.4pt\cdot\hskip.7pt,\hskip-.4pt\cdot\hskip.4pt\rangle\,$ in $\,\mathrm{End}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}(T\hskip-.7ptM)\,$ induced by $\,g$. The rank $\,m(m-1)/2\,$ follows since $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)=\mathfrak{u}\hskip.4pt(T\hskip-.7ptM)\oplus\mathcal{E}\hskip-.7pt$, with $\,\mathfrak{u}\hskip.4pt(T\hskip-.7ptM)\subseteq\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)$ characterized by having sections $\,C:T\hskip-.7ptM\to T\hskip-.7ptM\,$ that commute with $\,J\,$ (which, due to their $\,g$-skew-adjointness, makes them also $\,g\hskip-.4pt^{\mathbf{c}}\hskip-.7pt$-skew-adjoint, for $\,g\hskip-.4pt^{\mathbf{c}}\hskip-.7pt =g-i\hskip.7pt\omega$): $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)$ and $\,\mathfrak{u}\hskip.4pt(T\hskip-.7ptM)\,$ have the real ranks $\,m(2m-1)\,$ and $\,m^2\hskip-.7pt$. *Proof of Theorem [Theorem 2](#thrmb){reference-type="ref" reference="thrmb"}.* By ([\[cvb\]](#cvb){reference-type="ref" reference="cvb"}), with $\,m=2$, the pseudo.7pt-Hermitian fibre metric in the *line* bundle $\,\mathcal{E}\hskip.7pt$ must be positive or negative definite. Hence so is its $\,g$-induced real part. For any Killing field $\,v$, ([\[pre\]](#pre){reference-type="ref" reference="pre"}.7pt-ii) implies that $\,A=\pounds\hskip-1pt_v^{\phantom i}J\,$ is a section of $\,\mathcal{E}\hskip.7pt$ which, due to ([\[pre\]](#pre){reference-type="ref" reference="pre"}) -- ([\[ltw\]](#ltw){reference-type="ref" reference="ltw"}), is $\,L\hskip-.7pt^2\hskip-.7pt$-orthogonal to itself, and so $\,\pounds\hskip-1pt_v^{\phantom i}J=0$. ◻ The above proof does not extend to compact pseudo.7pt\--.4ptKähler manifolds $\hskip.7pt(M\hskip-.7pt,g)\,$ of complex dimensions $\,m>2\,$ with indefinite metrics. Namely, if the pair $\,(j,k)$ represents the metric signature of $\,g$, with $\,j\,$ minuses and $\,k\,$ pluses (both $\,j,k$ even, $\,j+k=2m$), then the analogous signature of the real part (induced by $\,g$) of the pseudo.7pt-Hermitian fibre metric in $\,\mathcal{E}\hskip.7pt$ is $\,(jk/2,\,[\hskip.7ptj^2\hskip-.7pt+k^2\hskip-.7pt-2(j+k)]/4)$, with both components (indices) positive unless $\,jk=0\,$ or $\,j=k=2$. One easily verifies this last claim, about the signature, by using a $\,J\hskip-1.5pt_x^{\phantom i}$-invariant timelike-spacelike orthogonal decomposition of $\,{T\hskip-2.9pt_x^{\phantom i}\hskip-.4ptM}\hskip-.7pt$, at any $\,x\in M\hskip-.7pt$, to obtain obvious three-summand orthogonal decompositions of both $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)\,$ and $\,\mathfrak{u}\hskip.4pt(T\hskip-.7ptM)$ at $\,x$, two summands being spacelike, and one timelike. # Proof of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"} {#pa} We denote by $\,\varOmega\hskip.7pt^{p,q}\hskip-1ptM\,$ the space of complex-valued differential $\,(p,q)\,$ forms on a complex manifold $\,M\hskip-.7pt$. On such $\,M\hskip-.7pt$, as $\,\overline{\hskip-2pt\partial}\hskip.7pt\zeta=0\,$ whenever $\,d\hskip.7pt\zeta=0$, $$\label{clh} \mathrm{closedness\ of\ a\ }\,(p,0)\,\mathrm{\ form\ }\,\zeta\,\mathrm{\ implies\ its\ hol\-o\-mor\-phic\-i\-ty.}$$ Conversely, according to [@deligne-griffiths-morgan-sullivan p. 296, subsect. (5.21)] and [@ueno p. 101, Corollary 9.5], on a compact complex $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ manifold, $$\label{hcl} \mathrm{all\ hol\-o\-mor\-phic\ differential\ forms\ are\ closed.}$$ Every compact complex manifold admitting a Riemannian Kähler metric has the following $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ *property*, also referred to as *the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\hskip.7pt$ lemma* [@voisin Prop. 6.17 on p. 144]: any closed $\,\partial\hskip.7pt$-exact or $\,\overline{\hskip-2pt\partial}\hskip.7pt$-exact $\,(p,q)\,$ form equals $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\hskip.4pt\lambda\,$ for some $\,(p-1,q-1)\,$ form $\,\lambda$. Since many expositions do not state what happens when $\,p\,$ or $\,q\,$ is $\,0$, we note that -- as Fangyang Zheng pointed out to us -- the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property for $\,(p,0)\,$ forms easily follows from the case where $\,p$ and $\,q\,$ are positive. **Lemma 5**. *On a compact complex manifold $\hskip.7ptM$ with the ".4ptpositive $p,q$ version" of the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property, if $\,\xi\in\varOmega\hskip.7pt^{p,0}\hskip-1ptM\hskip-.7pt$, for $\,p\ge1$, and $\,\partial\hskip.7pt\xi\,$ is closed, then $\,\partial\hskip.7pt\xi=0$.* *Proof.* As $\,0=d\hskip.4pt\partial\hskip.7pt\xi=\overline{\hskip-2pt\partial}\hskip-.3pt\partial\hskip.7pt\xi=-\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\hskip.7pt\xi$, the "positive" $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\,$ lemma applied to the closed $\,\overline{\hskip-2pt\partial}\hskip.7pt$-exact $\,(p,1)\,$ form $\,\overline{\hskip-2pt\partial}\hskip.7pt\xi\,$ gives $\,\overline{\hskip-2pt\partial}\hskip.7pt\xi=\overline{\hskip-2pt\partial}\hskip-.3pt\partial\hskip.7pt\eta\,$ for some $\,\eta\in\varOmega\hskip.7pt^{p-1,0}\hskip-1ptM\hskip-.7pt$. Being thus holomorphic, $\,\xi-\partial\hskip.7pt\eta\in\varOmega\hskip.7pt^{p,0}\hskip-1ptM\,$ is closed by ([\[hcl\]](#hcl){reference-type="ref" reference="hcl"}), and $\,0=\partial\hskip.7pt(\xi-\partial\hskip.7pt\eta)=\partial\hskip.7pt\xi$. ◻ Lemma [Lemma 5](#ddbar){reference-type="ref" reference="ddbar"} implies, via complex conjugation, its analog for $\,(0,q)\,$ forms. Also by Lemma [Lemma 5](#ddbar){reference-type="ref" reference="ddbar"}, on a compact complex manifold $\,M\,$ with the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property, $$\label{exz} \mathrm{the\ only\ exact\ }\,(p,0)\,\mathrm{\ form\ }\,\zeta\,\mathrm{\ on\ }\,M\,\mathrm{\ is\ }\,\zeta=0\hskip.4pt,$$ since exactness of $\,\zeta\in\varOmega\hskip.7pt^{p,0}\hskip-1ptM\,$ amounts to its $\,\partial\hskip.7pt$-exactness and implies its closedness. For a pseudo.7pt\--.4ptKähler manifold $\,(M\hskip-.7pt,g)$, a bundle morphism $\,A:T\hskip-.7ptM\to T\hskip-.7ptM\hskip-.7pt$, and the corresponding twice-covariant tensor field $\,\alpha=g(A\hskip.7pt\cdot\,,\,\cdot\,)$, one clearly has $$\label{iff} \alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=\pm\hskip.7pt\alpha\,\mathrm{\ if\ and\ only\ if\ }\,J\hskip-.7ptA=\pm\hskip.4ptAJ\hskip.4pt,\quad\mathrm{with\ either\ sign\ }\,\pm\hskip.4pt.$$ Given a pseudo.7pt\--.4ptKähler manifold $\,(M\hskip-.7pt,g)$, vector fields $\,u,v\,$ on $\,M\,$ and sections $\,A,C\,$ of $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)$, cf. ([\[vsb\]](#vsb){reference-type="ref" reference="vsb"}), may be used to represent a complex-valued $\,1$-form $\,\xi$ and $\,2$-form $\,\zeta\,$ on $\,M\hskip-.7pt$, as follows, $$\label{xie} \xi=u+i\hskip.4ptv\hskip.4pt,\quad\zeta=A+i\hskip.4ptC\hskip.4pt,$$ meaning that $\,\xi=g(u,\,\cdot\,)+ig(v,\,\cdot\,)\,$ and $\,\zeta=g(B\hskip.7pt\cdot\,,\,\cdot\,)+ig(C\hskip.7pt\cdot\,,\,\cdot\,)$. We prefer not to think of ([\[xie\]](#xie){reference-type="ref" reference="xie"}) as sections of the complexifications of $\,T\hskip-.7ptM\,$ or $\,\mathfrak{so}\hskip.4pt(T\hskip-.7ptM)$. For a vector field $\,v\,$ treated via ([\[xie\]](#xie){reference-type="ref" reference="xie"}) as a real $\,1$-form, and $\,B=\nabla\hskip-.7ptv$, our factor convention for the exterior derivative gives $$\label{fcv} dv\,=\,B\,-\,B^*,\quad\mathrm{\ and\ so\ }\,d\hskip.4pt(J\hskip-.4ptv)\,=\,\nabla(J\hskip-.4ptv)\,-[\nabla(J\hskip-.4ptv)]^*\hskip.7pt =\,JB+B^*\hskip-3ptJ\hskip.4pt.$$ *Remark 6*. On a complex manifold, a real-valued $\,2$-form $\,\alpha\,$ is the real part of a complex-bilinear complex-valued $\,2$-form $\,\zeta\,$ if and only if $\,\alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\alpha$, and then necessarily $\,\zeta=\alpha-i\hskip.4pt\alpha(J\hskip.7pt\cdot\,,\,\cdot\,)$. (This clearly remains valid for arbitrary twice-covariant tensor fields, without skew-symmetry.) *Remark 7*. For a complex-valued $\,2$-form $\,\zeta\,$ on a complex manifold $\,M\hskip-.7pt$, having bidegree $\,(2,0)$, or $\,(0,2)$, or $\,(1,1)\,$ clearly amounts to its being complex-bilinear, or bi-antilinear or, respectively, $\,J$-invariant: $\,\zeta(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=\zeta$. Sums $\,\zeta\,$ of $\,(2,0)$ and $\,(0,2)\,$ forms are similarly characterized by $\,J$-anti-invariance: $\,\zeta(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\hskip.7pt\zeta$. Thus, by ([\[iff\]](#iff){reference-type="ref" reference="iff"}), in the pseudo.7pt\--.4ptKähler case, $\,\zeta=A+i\hskip.4ptC\,$ in ([\[xie\]](#xie){reference-type="ref" reference="xie"}) is a $\,(1,1)\,$ form if and only if $\,A\,$ and $\,C\,$ commute with $\,J$. **Lemma 8**. *For a Killing vector field $\,v\,$ on a pseudo.7pt\--.4ptKähler manifold $\,(M\hskip-.7pt,g)$, using the notation of ([\[xie\]](#xie){reference-type="ref" reference="xie"}), we have $$\label{dxe} \begin{array}{l} \xi\in\varOmega\hskip.7pt^{1,0}\hskip-1ptM\hskip.4pt,\quad\zeta\in\varOmega\hskip.7pt^{2,0}\hskip-1ptM\hskip.4pt,\quad\partial\hskip.7pt\xi\, =\,\zeta\hskip.4pt,\quad\overline{\hskip-2pt\partial}\hskip.7pt\xi\,=\,i\hskip.4pt(J\hskip-.4ptB\hskip-.4ptJ-B)\hskip.4pt,\mathrm{\ \ where}\\ \xi\,=\,J\hskip-.4ptv\,-\,iv\hskip.4pt,\quad\zeta\,=\,A\,-\,iAJ\hskip.4pt,\quad\mathrm{with\ \ }\,A =[J,B\hskip.4pt]\,\mathrm{\ \ for\ \ }\hskip.7ptB=\nabla\hskip-.7ptv\hskip.4pt. \end{array}$$* *Proof.* First, $\,J\hskip-.4ptB\hskip-.4ptJ-B$, as well as $\,A=[J,B\hskip.4pt]\,$ and $\,AJ$, are $\,g_x^{\phantom i}\hskip-.7pt$-skew-adjoint at every point $\,x\in M\hskip-.7pt$, since so is $\,B=\nabla\hskip-.7ptv$, and $\,A\,$ anticommutes with $\,J$, cf. ([\[com\]](#com){reference-type="ref" reference="com"}). Thus, $\,\xi,\zeta\,$ and $\,\gamma=i\hskip.4pt(J\hskip-.4ptB\hskip-.4ptJ-B)\,$ are indeed differential forms of degrees $\,1,\,2,\,2$. Furthermore, $\,\xi\,$ is complex-linear, and $\,\zeta\,$ complex-bilinear. This is immediate for $\,\xi$. For $\,\zeta$, note that $\,\zeta=\alpha-i\hskip.4pt\alpha(J\hskip.7pt\cdot\,,\,\cdot\,)$, where $\,\alpha=g(A\hskip.7pt\cdot\,,\,\cdot\,)$, while ([\[com\]](#com){reference-type="ref" reference="com"}) and ([\[iff\]](#iff){reference-type="ref" reference="iff"}) give $\,\alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\alpha$. Now we can use Remark [Remark 6](#realp){reference-type="ref" reference="realp"}. Thus, $\,\xi\in\varOmega\hskip.7pt^{1,0}\hskip-1ptM\hskip-.7pt$. Also, according to Remark [Remark 7](#twozr){reference-type="ref" reference="twozr"}, $\,\zeta\in\varOmega\hskip.7pt^{2,0}\hskip-1ptM\,$ and $\,\gamma\in\varOmega\hskip.7pt^{1,1}\hskip-1ptM\hskip-.7pt$, since $\,J\hskip-.4ptB\hskip-.4ptJ-B\,$ obviously commutes with $\,J$. Finally, for $\,A=[J,B\hskip.4pt]$, ([\[fcv\]](#fcv){reference-type="ref" reference="fcv"}) with $\,B^*\hskip-.7pt=-B\,$ gives $\,d\hskip.4pt\xi=A-2i\hskip.4ptB=[A-i(J\hskip-.4ptB\hskip-.4ptJ+B)]+i\hskip.4pt(J\hskip-.4ptB\hskip-.4ptJ-B)$, while the summands $\,A-i(J\hskip-.4ptB\hskip-.4ptJ+B)=A-iAJ=\hskip.7pt\zeta\,$ and $\,i\hskip.4pt(J\hskip-.4ptB\hskip-.4ptJ-B)=\gamma\,$ lie in $\,\varOmega\hskip.7pt^{2,0}\hskip-1ptM$ and $\,\varOmega\hskip.7pt^{1,1}\hskip-1ptM\hskip-.7pt$, which completes the proof. ◻ *Proof of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"}.* By ([\[pre\]](#pre){reference-type="ref" reference="pre"}) and ([\[iff\]](#iff){reference-type="ref" reference="iff"}), the $\,\partial\hskip.7pt$-exact $\,(2,0)\,$ form $\,\zeta=\partial\hskip.7pt\xi$ in ([\[dxe\]](#dxe){reference-type="ref" reference="dxe"}) is parallel, and hence closed. Lemma [Lemma 5](#ddbar){reference-type="ref" reference="ddbar"} now gives $\,\zeta=0$, so that $\,\,\pounds\hskip-1pt_v^{\phantom i}J=A=0\,$ due to ([\[com\]](#com){reference-type="ref" reference="com"}) and ([\[dxe\]](#dxe){reference-type="ref" reference="dxe"}). ◻ # Another proof of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"} {#aa} On a compact complex manifold $\,M\,$ with the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property, every cohomology space $\,H^k\hskip-1pt(M,{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}})\,$ has the Hodge decomposition [@deligne-griffiths-morgan-sullivan p. 296, subsect. (5.21)]: $$\label{hdc} H^k\hskip-1pt(M,{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}})=H^{k,0}\hskip-1ptM\oplus H^{k-1,1}\hskip-1ptM\oplus\ldots \oplus H^{1,k-1}\hskip-1ptM\oplus H^{0,k}\hskip-1ptM\hskip.4pt,$$ with each $\,H^{p,q}\hskip-1ptM\,$ consisting of cohomology classes of closed $\,(p,q)\,$ forms. The complex conjugation of differential forms descends to a real-linear involution of $\,H^k\hskip-1pt(M,{\mathchoice {\setbox 0=\hbox{$\displaystyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}} {\setbox 0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern 0.4\wd 0\vrule height0.9\ht 0\hss}\box 0}}})$, the fixed points of which obviously are the real cohomology classes (those containing real closed differential forms). In terms of the decomposition ([\[hdc\]](#hdc){reference-type="ref" reference="hdc"}), a complex cohomology class $$\label{crl} \begin{array}{l} \mathrm{is\ real\ if\ and\ only\ if,\ for\ all\ }\,p\hskip.4pt\mathrm{\ and\ }\,q\mathrm{,\hskip-1pt\ its\ }\hskip.7ptH^{q,p}\hskip-.7pt\mathrm{\ com}\hskip.5pt\vbox {\hbox{\vrule width3ptheight0.5ptdepth0pt}\vskip 2.2pt}\hskip.5pt\\ \mathrm{ponent\ equals\ the\ conjugate\ of\ its\ }\hskip.7pt\,H^{p,q}\hskip-.4pt\mathrm{\ component.} \end{array}$$ The standard formula $\,N(u,v)=[u,v]+J[Ju,v]+J[u,Jv]-[Ju,Jv]$, for the Nijenhuis tensor $\,N\,$ of an almost-complex structure $\,J\,$ on a manifold $\,M\,$ and any vector fields $\,u,v$, clearly becomes $$\label{tfr} N(u,v)\,=\,[\nabla_{\!\!J\hskip-.4ptv}^{\phantom i}J]u\,-\,[\nabla_{\!\!J\hskip-.4ptu}^{\phantom i}J]v\, +\,J[\nabla_{\hskip-1pt\!u}^{\phantom i}\hskip-1ptJ]v\,-\,J[\nabla_{\hskip-.7pt\!v}^{\phantom i}J]u$$ when one uses any fixed torsionfree connection $\,\nabla\hskip.7pt$ on $\,M\hskip-.7pt$. We call $\,\nabla\hskip.7pt$ a *Kähler connection* for the given almost-complex structure $\,J\,$ if it is torsionfree and $\,\nabla\hskip-1ptJ=0$. By ([\[tfr\]](#tfr){reference-type="ref" reference="tfr"}), $\,J\,$ then must be integrable. **Lemma 9**. *For any $\,\nabla\hskip-.7pt$-parallel real $\,2$-form $\,\alpha\,$ on a complex manifold $\,M$ with a Kähler connection $\,\nabla\hskip-1pt$, such that $\,\alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\alpha$, the complex-valued $\,2$-form $\,\zeta=\alpha-i\alpha(J\hskip.7pt\cdot\,,\,\cdot\,)$ is holomorphic. If, in addition, $\,M\,$ is also compact and has the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property, while $\,\alpha\,$ is exact, then $\,\alpha=0$.* *Proof.* The relation $\,\alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\alpha\,$ amounts to complex-bilinearity of $\,\zeta$, and so $\,\zeta\in\varOmega\hskip.7pt^{2,0}\hskip-1ptM\,$ (Remarks [Remark 6](#realp){reference-type="ref" reference="realp"} -- [Remark 7](#twozr){reference-type="ref" reference="twozr"}). Being $\,\nabla\hskip-.7pt$-parallel, $\,\zeta\,$ is closed, and hence holomorphic due to ([\[clh\]](#clh){reference-type="ref" reference="clh"}). The final clause: exactness of $\,\alpha\,$ makes $\,[i\zeta]\in\varOmega\hskip.7pt^{2,0}\hskip-1ptM\,$ a real cohomology class, so that, by ([\[crl\]](#crl){reference-type="ref" reference="crl"}), $\,\zeta\,$ is exact, and ([\[exz\]](#exz){reference-type="ref" reference="exz"}) gives $\,\zeta=0$. ◻ *Another proof of Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"}.* Given a Killing field $\,v$, the differential $\,2$-form $\,\alpha=\pounds\hskip-1pt_v^{\phantom i}\hskip.7pt\omega\,$ is parallel and exact by ([\[pre\]](#pre){reference-type="ref" reference="pre"}), while ([\[pre\]](#pre){reference-type="ref" reference="pre"}) gives $\,J\hskip-.7ptA=-\hskip-.7ptAJ\,$ for $\,A=\pounds\hskip-1pt_v^{\phantom i}J$, related to $\,\alpha\,$ via $\,\alpha=g(A\hskip.7pt\cdot\,,\,\cdot\,)$, and so $\,\alpha(J\hskip.7pt\cdot\,,J\hskip.7pt\cdot\hskip.7pt)=-\alpha\,$ due to ([\[iff\]](#iff){reference-type="ref" reference="iff"}). Lemma [Lemma 9](#expar){reference-type="ref" reference="expar"} and ([\[pre\]](#pre){reference-type="ref" reference="pre"}.7pt-i) now yield $\,\pounds\hskip-1pt_v^{\phantom i}\hskip.7pt\omega=\alpha=0\,$ and $\,\pounds\hskip-1pt_v^{\phantom i}J=0$. ◻ We do not know whether -- aside from Theorem [Theorem 2](#thrmb){reference-type="ref" reference="thrmb"} and the Riemannian case -- Theorem [Theorem 1](#thrma){reference-type="ref" reference="thrma"} remains valid without the $\hskip.7pt\partial\hskip 1.7pt\overline{\hskip-2pt\partial}\hskip.7pt$ hypothesis. For possible future reference, let us note that, as shown above, one has the following conclusions about a Killing field $\,v\,$ on a compact pseudo.7pt\--.7ptKähler manifold, whether or not the $\,\partial\hskip 1.7pt\overline{\hskip-2pt\partial}$ property is assumed. First, for $\,\alpha=\pounds\hskip-1pt_v^{\phantom i}\hskip.7pt\omega$, the complex-valued $\,2$-form $\,\zeta=\alpha-i\alpha(J\hskip.7pt\cdot\,,\,\cdot\,)\,$ is parallel and holomorphic (see the preceding proof and Lemma [Lemma 9](#expar){reference-type="ref" reference="expar"}). Also, by ([\[pre\]](#pre){reference-type="ref" reference="pre"}), $\,\alpha\,$ is exact, while $\,A=\pounds\hskip-1pt_v^{\phantom i}J:T\hskip-.7ptM\to T\hskip-.7ptM\,$ is parallel and complex-antilinear, as well as nilpotent at every point. This last conclusion follows since the constant function $\,\mathrm{tr}_{\hskip.4pt\mathrm{I\!R}\hskip0pt}^{\phantom i}A\hskip-.4pt^k\hskip-.7pt$, with any integer $\,k\ge1$, has zero integral as a consequence of ([\[ltw\]](#ltw){reference-type="ref" reference="ltw"}) applied to $\,\alpha=g(A\hskip.7pt\cdot\,,\,\cdot\,)\,$ and $\,\theta=g(A^{k-1}\cdot\,,\,\cdot\,)$. # Appendix: Yamada's argument {#ya .unnumbered} Yamada's claim [@yamada Proposition 3.1] that on a compact pseudo.7pt\--.4ptKähler manifold, Killing fields are real holomorphic, has a proof which reads, *verbatim*, $$\label{vrb} \begin{array}{l} \mathrm{Let\ }\,X\,\mathrm{\ be\ a\ Kil\-ling\ vector\ field.\ From\ Propositions\ 1.2\ and}\\ \mathrm{2.12,\ }\,\,Z\hskip.7pt=\hskip.7ptX\hskip.7pt-\,\sqrt{-\hskip-1pt1\,}JX\,\,\mathrm{\ is\ hol\-o\-mor\-phic.\ Because\ the\ real}\\ \mathrm{part\hskip.7pt\ of\hskip.7pt\ a\hskip.7pt\ hol\-o\-mor\-phic\hskip.7pt\ vector\hskip.7pt\ field\hskip.7pt\ is\hskip.7pt\ an\ infinitesimal\ auto}\hskip.5pt\vbox {\hbox{\vrule width3ptheight0.5ptdepth0pt}\vskip 2.2pt}\hskip.5pt\\ \mathrm{mor\-phism\hskip-.7pt\ of\hskip-.7pt\ the\hskip-.7pt\ complex\hskip-.7pt\ structure,\hskip-.2ptwe\hskip-.7pt\ have\hskip-.7pt\ our\hskip-.7pt\ proposition.} \end{array}$$ Proposition 1.2 of [@yamada], cited from Kobayashi's book [@kobayashi], amounts to the well-known *harmonic-flow condition* satisfied by Killing fields $\,v\,$ on pseudo.7pt-Riemannian manifolds. Thus, 2.12 in ([\[vrb\]](#vrb){reference-type="ref" reference="vrb"}) should read 2.14, since Propositions 1.2 and 2.14 refer to the Ricci tensor quite prominently, while 2.12 does not mention it at all; also, Proposition 2.14 contains, in its second part, a holomorphicity conclusion. In the ninth line of the proof of the second part of Proposition 2.14, it is established -- correctly -- that, for every $\,(1,0)\,$ vector field $\,Y\hskip-.7pt$, and $\,Z\,$ in ([\[vrb\]](#vrb){reference-type="ref" reference="vrb"}), $\,\nabla''\hskip-1ptZ$ is $\,L\hskip-.7pt^2\hskip-.7pt$-orthogonal to $\,\nabla''Y\hskip-1pt$. Then an attempt is made to conclude that $\,\nabla''\hskip-1ptZ=0$, arguing by contradiction: if $\,\nabla''\hskip-1ptZ\ne0\,$ at some point $\,z_0^{\phantom i}$, one can -- again correctly -- find $\,Y\hskip.4pt$ having $\,g(\nabla''\hskip-1ptZ,\nabla''Y)\ne0\,$ everywhere in some neighborhood of $\,z_0^{\phantom i}$. As a next step, it is claimed that a contradiction arises: cited *verbatim*, $$\label{ctr} \begin{array}{l} \mathrm{By\ considering\ a\ cut}\hskip.5pt\vbox {\hbox{\vrule width3ptheight0.5ptdepth0pt}\vskip 2.2pt}\hskip.5pt\mathrm{off\ function,\ we\ see\ that\ there\ exists}\\ \mathrm{a\ complex\ vector\ field\ }\hskip.7ptY\hskip-1pt\mathrm{\ such\ that\,}\int\hskip-4.2pt_{_M}^{\phantom i}g(\nabla''\hskip-1ptZ,\hskip-1pt\nabla''Y)\,dv\ne0. \end{array}$$ It is here that the argument seems incomplete: such a cut-off function $\,\varphi\,$ equals $\,1\,$ on some small "open ball" $\,B\,$ centered at $\,z_0^{\phantom i}$, and vanishes outside a larger "concentric ball" $\,B'\hskip-.7pt$, and after the original choice of $\,Y\hskip.4pt$ has been replaced by $\,\varphi Y\hskip-1pt$, there is no way to control the integral of $\,g(\nabla''\hskip-1ptZ,\nabla''\hskip-.7pt(\varphi Y))\,$ over $\,B'\hskip-.7pt\smallsetminus B\,$ (while the integrals over $\,B\,$ and $\,M\smallsetminus B'$ have fixed values). More precisely, the sum of the three integrals must be zero, $\,\nabla''\hskip-1ptZ\,$ being $\,L\hskip-.7pt^2\hskip-.7pt$-orthogonal to all $\,\nabla''Y\hskip-1pt$. 99 W.2.3ptBallmann, *Lectures on Kähler Manifolds*, ESI Lectures in Mathematics and Physics, European Mathematical Society, Zürich, 2006. P.2.3ptDeligne, P.2.3ptGriffiths, J.2.3ptMorgan and D.2.3ptSullivan, *Real homotopy theory of Kähler manifolds*, Invent. Math. **29** (1975), 245--274. S.2.3ptKobayashi, *Transformation Groups in Differential Geometry*, Springer-Verlag, Berlin-Heidelberg-New York, 1972. J.2.3ptM.2.3ptLee, *Introduction to Smooth Manifolds*, 2nd ed., Grad. Texts in Math. **218**, Springer-Verlag, New York, 2013. C.2.3ptVoisin, *Hodge Theory and Complex Algebraic Geometry, I*, Cambridge Stud. Adv. Math. **76**, Cambridge University Press, Cambridge, 2002. K.2.3ptUeno, *Classification Theory of Algebraic Varieties and Compact Complex Spaces*. Lecture Notes in Math. **439**, Springer-Verlag, 1975. T.2.3ptYamada, *Holomorphic vector fields of compact pseudo\--.4ptKähler manifolds*, J. Geom. Phys. **62** (2012), 740--750.

arxiv_math

--- abstract: | Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$. Let $\Gamma\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\Gamma$ satisfying standard assumptions. Under an algebraic condition on $\Gamma$, which is satisfied for example by the Gårding cones $\Gamma_k^+$ when $k<\frac{n}{2}$, we prove the existence of a Lipschitz viscosity solution $g_u = e^{2u}g_0$ to the fully nonlinear Loewner-Nirenberg problem associated to $(f,\Gamma)$, $$\begin{aligned} \begin{cases} f(\lambda(-g_u^{-1}A_{g_u})) = 1, \quad \lambda(-g_u^{-1}A_{g_u}) \in \Gamma & \mathrm{on~}M\backslash\partial M \\ u(x)\rightarrow+\infty & \mathrm{as~}\operatorname{dist}_{g_0}(x,\partial M)\rightarrow 0, \end{cases} \end{aligned}$$ where $A_{g_u}$ is the Schouten tensor of $g_u$. Previous results on Euclidean domains show that, in general, $u$ is not differentiable. The solution $u$ is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing $(1,0,\dots,0)$, for which we also have uniqueness. In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest. An important feature of our paper is that the existence of a conformal metric $g$ satisfying $\lambda(-g^{-1}A_g)\in\Gamma$ on $M$ is a *consequence* of our results, rather than an assumption. author: - "Jonah A. J. Duncan[^1]  and Luc Nguyen[^2]" bibliography: - references.bib title: The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$ --- # Introduction A pertinent theme in conformal geometry is to establish the existence of conformal metrics satisfying some notion of constant curvature. For example, given a compact Riemannian manifold $(M^n,g_0)$ of dimension $n\geq3$ with non-empty boundary $\partial M$, a natural question is whether there exists a conformal metric which is complete on $M\backslash \partial M$ and has constant negative scalar curvature on $M\backslash \partial M$. In the seminal work of Loewner & Nirenberg [@LN74], the authors proved among other results the existence and uniqueness of such a metric when $M\backslash \partial M$ is a bounded Euclidean domain with smooth boundary[^3] and $g_0$ is the flat metric. Aviles & McOwen [@AM88] later extended this result to the Riemannian setting, and for some further related results we refer e.g. to the work of Allen et. al. [@AILA18], Andersson et. al. [@ACF92], Aviles [@Av82], Finn [@Finn98], Gover & Waldron [@GW17], Graham [@Gr17], Han et. al. [@HJS20], Han & Shen [@HS20], Jiang [@Jia21], Li [@Li22], Mazzeo [@Maz91] and Véron [@Ver81]. We note that the related problem of finding conformal metrics with constant scalar curvature on closed manifolds, known as the Yamabe problem, was solved by the combined works of Yamabe [@Yam60], Trudinger [@Tru68], Aubin [@Aub70] and Schoen [@Sch84]. Since the work of Viaclovsky [@Via00a] and Chang, Gursky & Yang [@CGY02a], there has been significant interest in fully nonlinear generalisations of Yamabe-type problems, including on manifolds with boundary. Suppose that $$\begin{aligned} & \Gamma\subset\mathbb{R}^n\text{ is an open, convex, connected symmetric cone with vertex at 0}, \label{21'} \\ & \Gamma_n^+ = \{\lambda\in\mathbb{R}^n: \lambda_i > 0 ~\forall ~1\leq i \leq n\} \subseteq \Gamma \subseteq \Gamma_1^+ = \{\lambda\in\mathbb{R}^n : \lambda_1+\dots+\lambda_n > 0\}, \label{22'} \\ & f\in C^\infty(\Gamma)\cap C^0(\overline{\Gamma}) \text{ is concave, 1-homogeneous and symmetric in the }\lambda_i, \label{23'} \\ & f>0 \text{ in }\Gamma, \quad f = 0 \text{ on }\partial\Gamma, \quad f_{\lambda_i} >0 \text{ in } \Gamma \text{ for }1 \leq i \leq n. \label{24'} \end{aligned}$$ In this paper, we study the natural generalisation of the Loewner-Nirenberg problem to the fully nonlinear setting on Riemannian manifolds. That is, for $(f,\Gamma)$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and a compact Riemannian manifold $(M,g_0)$ with non-empty boundary $\partial M$, we study the existence and uniqueness of a conformal metric $g_u = e^{2u}g_0$ satisfying $$\label{113} \begin{cases} f(\lambda(-g_u^{-1}A_{g_u})) = 1, \quad \lambda(-g_u^{-1}A_{g_u}) \in \Gamma & \mathrm{on~}M\backslash \partial M \\ u(x)\rightarrow+\infty & \mathrm{as~}\operatorname{d}(x,\partial M)\rightarrow 0. \end{cases}$$ Here, $$\begin{aligned} A_g = \frac{1}{n-2}\bigg(\operatorname{Ric}_g - \frac{R_g}{2(n-1)}g\bigg)\end{aligned}$$ denotes the $(0,2)$-Schouten tensor of a Riemannian metric $g$, $\operatorname{Ric}_g$ and $R_g$ respectively denote the Ricci curvature tensor and scalar curvature of $g$, $\lambda(T)$ denotes the vector of eigenvalues of a $(1,1)$-tensor $T$, and $\operatorname{d}(x,\partial M)$ is the distance from $x\in M$ to $\partial M$ with respect to $g_0$. Typical examples of $(f,\Gamma)$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} are given by $(\sigma_k^{1/k},\Gamma_k^+)$ for $1\leq k \leq n$, where $\sigma_k$ is the $k$'th elementary symmetric polynomial and $\Gamma_k^+ = \{\lambda\in\mathbb{R}^n:\sigma_j(\lambda)>0~\forall~1\leq j \leq k\}$. When $f=\sigma_1$, [\[113\]](#113){reference-type="eqref" reference="113"} reduces to the original Loewner-Nirenberg problem on Riemannian manifolds discussed above. Much of the motivation to study [\[113\]](#113){reference-type="eqref" reference="113"} stems from the fact that, as a consequence of the Ricci decomposition, the Schouten tensor fully determines the conformal transformation properties of the full Riemann curvature tensor. We note that for $g_u = e^{2u}g_0$, one has the conformal transformation law $$\label{118} A_{g_u} = -\nabla_{g_0}^2 u - \frac{1}{2}|\nabla_{g_0} u|_{g_0}^2 g_0 + du\otimes du + A_{g_0},$$ which demonstrates the fully nonlinear nature of [\[113\]](#113){reference-type="eqref" reference="113"} when $f\not=c\sigma_1$. Moreover, [\[113\]](#113){reference-type="eqref" reference="113"} is non-uniformly elliptic when $f\not=c\sigma_1$. By the 1-homogeneity of $f$, without loss of generality we may assume $$\begin{aligned} \label{25'} f\bigg(\frac{1}{2},\dots, \frac{1}{2}\bigg) = 1. \end{aligned}$$ As in [@LN14b], we define $\mu_\Gamma^+$ to be the number satisfying $$\begin{aligned} (-\mu_\Gamma^+,1,\dots,1)\in\partial\Gamma. \end{aligned}$$ We note that $\mu_\Gamma^+$ is uniquely determined by $\Gamma$ and is easily seen to satisfy $\mu_\Gamma^+\in[0,n-1]$. When $\Gamma = \Gamma_k^+$, one has $\mu_{\Gamma_k^+}^+ = \frac{n-k}{k}$. Our first main result concerns the solution to the Loewner-Nirenberg problem [\[113\]](#113){reference-type="eqref" reference="113"} under the assumption $$\begin{aligned} \label{418} \mu_\Gamma^+>1. \end{aligned}$$ Observe that for $\Gamma = \Gamma_k^+$, [\[418\]](#418){reference-type="eqref" reference="418"} holds if and only if $k<\frac{n}{2}$. The role of condition [\[418\]](#418){reference-type="eqref" reference="418"} will be discussed later in the introduction. **Theorem 1**. *Let $(M,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$, and suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Then there exists a locally Lipschitz viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"} satisfying $$\begin{aligned} \label{114} \lim_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) = 0, \end{aligned}$$ which is maximal in the sense that if $\widetilde{u}$ is any continuous viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"}, then $\widetilde{u} \leq u$ on $M\backslash \partial M$. Moreover, when $(1,0,\dots,0)\in\Gamma$, the solution $u$ is smooth and is the unique continuous viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"}.* We recall that a continuous function $u$ on $M\backslash \partial M$ is a viscosity subsolution (resp. viscosity supersolution) to the equation in [\[113\]](#113){reference-type="eqref" reference="113"} if for any $x_0\in M\backslash \partial M$ and $\varphi\in C^2(M\backslash\partial M)$ satisfying $u(x_0) = \varphi(x_0)$ and $u(x)\leq \varphi(x)$ near $x_0$ (resp. $u(x)\geq\varphi(x)$ near $x_0$), it holds that $\lambda(-g_\varphi^{-1} A_{g_{\varphi}})(x_0)\in\{\lambda\in\Gamma:f(\lambda)\geq 1\}$ (resp. $\lambda(-g_\varphi^{-1} A_{g_{\varphi}})(x_0)\in\mathbb{R}^n\backslash\{\lambda\in\Gamma:f(\lambda)> 1\}$). We say that $u$ is a viscosity solution to the equation in [\[113\]](#113){reference-type="eqref" reference="113"} if it is both a viscosity subsolution and a viscosity supersolution. **Remark 2**. In previous work studying equations of the form $f(\lambda(-g_u^{-1}A_{g_u}))=1$, it has been typical to assume that the background metric $g_0$ satisfies $\lambda(-g_0^{-1}A_{g_0})\in\Gamma$ on $M$ (a notable exception is a result of Gursky, Streets & Warren [@GSW11], which will be discussed later in the introduction). In contrast, one of the key points of this paper is that we do not assume the existence of such a metric in Theorem [Theorem 1](#A){reference-type="ref" reference="A"}. Rather, the existence of such a metric is established at an intermediate stage of the proof of Theorem [Theorem 1](#A){reference-type="ref" reference="A"} (see Theorem [Theorem 6](#55){reference-type="ref" reference="55"}). **Remark 3**. In the case that $M\backslash \partial M$ is a Euclidean domain, the existence of a Lipschitz viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"} was established by Gonzáles, Li & Nguyen in [@GLN18]. It was also shown in [@GLN18] that this solution is unique among continuous viscosity solutions. We note that the uniqueness of the viscosity solution obtained in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} remains an open problem when $M\backslash \partial M$ is not a Euclidean domain and $(1,0,\dots,0)\in\partial\Gamma$. **Remark 4**. In [@LN20b; @LNX22] it was shown that if $M\backslash \partial M$ is a Euclidean domain with disconnected boundary and $\Gamma\subset\Gamma_2^+$ (in particular, this implies $(1,0,\dots,0)\in\partial\Gamma$), then the Lipschitz viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"} is not differentiable. Thus, in general, the Lipschitz regularity of the solution in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} cannot be improved to $C^1$ regularity when $(1,0,\dots,0)\in\partial \Gamma$. On the other hand, the smoothness of the solution in Theorem [Theorem 1](#A){reference-type="ref" reference="A"} when $(1,0,\dots,0)\in\Gamma$ is new even when $M\backslash \partial M$ is a Euclidean domain. This smoothness result can be viewed as an analogue of the result of Gursky & Viaclovsky [@GV03b] on the existence of a smooth solution to the $\sigma_k$-Yamabe problem for the trace-modified Schouten tensor on closed manifolds. To describe the proof of Theorem [Theorem 1](#A){reference-type="ref" reference="A"}, we first introduce some notation and an equivalent formulation of the result. For $\tau\in[0,1]$, $\lambda\in\mathbb{R}^n$ and $e=(1,\dots,1)\in\mathbb{R}^n$, we define $$\lambda^\tau \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =\tau\lambda + (1-\tau)\sigma_1(\lambda)e, \quad f^\tau(\lambda) \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =\frac{1}{\tau+n(1-\tau)} f(\lambda^\tau) \quad \text{and} \quad \Gamma^\tau \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =\{\lambda:\lambda^\tau\in\Gamma\}.$$ As shown in [@DN22 Appendix A], $\Gamma$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}, [\[22\'\]](#22'){reference-type="eqref" reference="22'"} and $(1,0,\dots,0)\in\Gamma$ if and only if there exists $\widetilde{\Gamma}$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"}, [\[22\'\]](#22'){reference-type="eqref" reference="22'"} and a number $\tau<1$ for which $\Gamma = (\widetilde{\Gamma})^\tau$. Note that [\[25\'\]](#25'){reference-type="eqref" reference="25'"} implies $f^\tau(\frac{1}{2},\dots,\frac{1}{2})=1$. An equivalent formulation of Theorem [Theorem 1](#A){reference-type="ref" reference="A"} is then as follows: [\[A\'\]]{#A' label="A'"} *Let $(M,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$, and suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Then for each $\tau<1$, there exists a smooth solution $u$ to $$\label{-113} \begin{cases} f^\tau(\lambda(-g_u^{-1}A_{g_u})) = 1, \quad \lambda(-g_u^{-1}A_{g_u}) \in \Gamma^\tau & \mathrm{on~}M\backslash\partial M \\ u(x)\rightarrow+\infty & \mathrm{as~}\operatorname{d}(x,\partial M)\rightarrow 0, \end{cases}$$ and moreover $u$ satisfies [\[114\]](#114){reference-type="eqref" reference="114"} and is the unique continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}. When $\tau=1$, there exists a Lipschitz viscosity solution $u$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"} satisfying [\[114\]](#114){reference-type="eqref" reference="114"}, which is maximal in the sense that if $\widetilde{u}$ is any continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}, then $\widetilde{u} \leq u$ on $M\backslash \partial M$.* **Remark 5**. If we label the solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} in Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"} as $u^\tau$ for each $\tau\leq 1$, then we will show that for each compact set $K\subset M\backslash \partial M$, there exists a constant $C$ which is independent of $\tau$ but dependent on $M,g_0, f,\Gamma$ and $K$ such that $$\begin{aligned} \|u^\tau\|_{C^{0,1}(K)} \leq C \quad \text{for all } \tau\in[0,1]. \end{aligned}$$ In the proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}, we will first prove the existence of a unique smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$. The Lipschitz viscosity solution in the case $\tau=1$ is then obtained in the limit as $\tau\rightarrow 1$. In turn, for each $\tau<1$, the existence of a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} is obtained as the limit of smooth solutions to Dirichlet boundary value problems with finite boundary data. Although we only need to consider constant boundary data in the proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}, we will prove the following more general result: **Theorem 6**. *Let $(M,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with non-empty boundary $\partial M$, and suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Let $\psi\in C^\infty(M)$ be positive and $\xi\in C^\infty(\partial M)$. Then for each $\tau<1$, there exists a smooth solution $u$ to $$\label{12} \begin{cases} f^\tau(\lambda(-g_u^{-1}A_{g_u})) = \psi, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma^\tau & \mathrm{on~}M\backslash\partial M \\ u = \xi & \mathrm{on~}\partial M, \end{cases}$$ and moreover $u$ is the unique continuous viscosity solution to [\[12\]](#12){reference-type="eqref" reference="12"}. When $\tau=1$, there exists a Lipschitz viscosity solution to [\[12\]](#12){reference-type="eqref" reference="12"}.* **Remark 7**. If we label the solution to [\[12\]](#12){reference-type="eqref" reference="12"} in Theorem [Theorem 6](#55){reference-type="ref" reference="55"} as $u^\tau$ for each $\tau\leq1$, then we will show that there exists a constant $C$ which is independent of $\tau$ but dependent on $M,g_0, f,\Gamma, \psi$ and $\xi$ such that $$\begin{aligned} \|u^\tau\|_{C^{0,1}(M)}\leq C \quad \text{for all }\tau\in[0,1]. \end{aligned}$$ The existence of a smooth solution to [\[12\]](#12){reference-type="eqref" reference="12"} when $\tau<1$ is achieved using the continuity method, which relies on obtaining *a priori* estimates. To keep the introduction concise, we only discuss the $C^0$ estimates here and postpone the discussion of the other estimates to the main body of the paper. Now, if one assumes $\lambda(-g_0^{-1}A_{g_0})\in\Gamma$ on $M$, then it is straightforward to obtain both the *a priori* upper and lower bounds on solutions to [\[12\]](#12){reference-type="eqref" reference="12"}. Since we do not make such an assumption on $g_0$, a large portion of our work involves proving the lower bound. The *a priori* lower bound is obtained in two independent stages, which can be summarised as follows: 1. First, in Section [2](#14){reference-type="ref" reference="14"}, we prove a local interior gradient estimate on solutions to [\[12\]](#12){reference-type="eqref" reference="12"} of the form $$\label{69'} |\nabla_{g_0} u|_{g_0}(x) \leq C(r^{-1} + e^{\sup_{B_r}u}) \quad\text{for }x\in B_{r/2},$$ where $B_r$ is a geodesic ball contained in the interior of $M$. An important feature is that the estimate [\[69\'\]](#69'){reference-type="eqref" reference="69'"} does not depend on a lower bound for $u$. 2. Second, in Section [3.2](#147){reference-type="ref" reference="147"}, we construct suitable barrier functions to prove a lower bound for $u$ in a *uniform neighbourhood of $\partial M$* -- this is one of the key new ideas in this paper. We note that the assumption $\mu_\Gamma^+>1$ is used in both stages above. Once the lower bound in a uniform neighbourhood of $\partial M$ is established in the second step, the local interior gradient estimate from the first step and a trivial global upper bound in Proposition [Proposition 20](#75){reference-type="ref" reference="75"} then allows one to propagate the lower bound to all of $M$ -- see the proof of Proposition [Proposition 21](#151){reference-type="ref" reference="151"} for the details. We now discuss the two steps above in more detail. Our local interior gradient estimate, which is also of independent interest, is as follows: **Theorem 8**. *Let $(M,g_0)$ be a smooth Riemannian manifold of dimension $n\geq 3$, possibly with non-empty boundary, and suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Fix $\tau\in(0,1]$, a positive function $\psi\in C^\infty(M)$ and suppose that $u\in C^3(B_r)$ satisfies $$\label{10} f^\tau(\lambda(-g_u^{-1}A_{g_u})) = \psi, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma^\tau$$ in a geodesic ball $B_r$ contained in the interior of $M$. Then $$\label{69} |\nabla_{g_0} u|_{g_0}(x) \leq C(r^{-1} + e^{\sup_{B_r}u}) \quad\text{for }x\in B_{r/2}$$ where $C$ is a constant depending on $n, f, \Gamma$, $\|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$ but independent of $\tau$ and $\inf_{B_r}\psi$.* We note that Theorem [Theorem 8](#40){reference-type="ref" reference="40"} was previously obtained for $(f,\Gamma)=(\sigma_k^{1/k}, \Gamma_k^+)$ when $k<\frac{n}{2}$ and $\tau=1$ in the thesis of Khomrutai [@Kho09][^4]. Roughly speaking, one important observation in [@Kho09] is as follows: if $\rho|\nabla_{g_0} u|_{g_0}^2$ attains its maximum at $x_0$ (here $\rho$ is a cutoff function satisfying standard assumptions), then in a 'worst case scenario' (i.e. in a situation where the gradient estimate cannot be obtained somewhat directly), the ordered eigenvalues $\lambda_1(x_0) \geq \dots \geq \lambda_n(x_0)$ of $(-g_0^{-1}A_{g_u})(x_0)$ are greater than or equal to a perturbation of $(1,\dots,1,-1)\frac{|\nabla u|^2}{2}(x_0)$. But when $k<\frac{n}{2}$, the vector $(1,\dots,1,-1)$ belongs to $\Gamma_k^+$, and so by the equation [\[10\]](#10){reference-type="eqref" reference="10"} and homogeneity of $\sigma_k^{1/k}$, the gradient estimate follows. In our proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, we show that this phenomenon persists for general cones satisfying $\mu_\Gamma^+>1$. In order to circumvent certain arguments of Khomrutai that rely on algebraic properties of the $\sigma_k$ operators, we appeal to some general cone properties recently observed by Yuan in [@Yuan22]. **Remark 9**. For gradient estimates on solutions to equations of the form [\[10\]](#10){reference-type="eqref" reference="10"} which depend on two-sided $C^0$ bounds, see for instance [@GV03b; @Guan08]. For gradient estimates for the related positive cone equation, see e.g. [@Che05; @GW03b; @JLL07; @LL03; @Li09; @Wan06; @Via02]. **Remark 10**. We have been informed that in an upcoming work of Baozhi Chu, YanYan Li and Zongyuan Li [@CLL23], a Liouville-type theorem for a fully nonlinear, degenerate elliptic Yamabe-type equation on negative cones is proved for all $\mu_\Gamma^+ \neq 1$. As an application of this Liouville-type theorem and the method in [@Li09] (which dealt with local gradient estimates for equations on positive cones), the authors obtain local interior gradient estimates for solutions to [\[10\]](#10){reference-type="eqref" reference="10"} depending only on one-sided $C^0$ bounds for all $\mu_\Gamma^+ \neq 1$, without assuming concavity of $f$. Counterexamples to both results are also given when $\mu_\Gamma^+ = 1$. This proof is entirely different from our proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. We now turn to the second step mentioned above, namely the lower bound in a neighbourhood of $\partial M$. This is achieved through constructing suitable comparison functions on small annuli; the main step here is to prove the following proposition (see Proposition [Proposition 23](#54){reference-type="ref" reference="54"} for a more precise version): **Proposition 11**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}, let $g_0$ be a Riemannian metric defined on a neighbourhood $\Omega$ of the origin in $\mathbb{R}^n$, and let $m\in\mathbb{R}$. Then there exist constants $S>1$ and $0<R<1$ depending on $g_0, f, \Gamma$ and $m$ such that whenever $1<\frac{r_+}{r_-}<S$ and $r_+<R$, there exists a solution to $$\begin{aligned} \begin{cases} f(\lambda(-g_w^{-1}A_{g_w})) \geq 1, \quad \lambda(-g_w^{-1}A_{g_w})\in\Gamma & \text{on }A_{r_-, r_+}\mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =\{x : r_-<\operatorname{d}_{g_0}(x,0)<r_+\} \\ w(x) = m & \text{for }x\in\mathbb{S}_{r_-} \\ w(x)\rightarrow -\infty & \text{as }\operatorname{d}_{g_0}(x,\mathbb{S}_{r_+})\rightarrow 0. \end{cases} \end{aligned}$$* Our construction of $w$ in Proposition [Proposition 11](#54'){reference-type="ref" reference="54'"} is modelled on the radial solutions of Chang, Han & Yang [@CHY05] to the $\sigma_k$-Yamabe equation on annular domains in $\mathbb{R}^n$ when $k<\frac{n}{2}$. To apply Proposition [Proposition 11](#54'){reference-type="ref" reference="54'"} to complete the second step, we attach a collar neighbourhood $N$ to $\partial M$, and cover a neighbourhood of $\partial M$ in $M$ by sufficiently small annuli whose centres lie in $N$ and whose inner boundaries touch $\partial M$. On each of these annuli, the solutions constructed in Proposition [Proposition 11](#54'){reference-type="ref" reference="54'"} then serve as the desired lower bound by the comparison principle. We refer the reader to the proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"} for the details. **Remark 12**. The assumption $\mu_\Gamma^+>1$ plays an important role in our proof of Proposition [Proposition 11](#54'){reference-type="ref" reference="54'"}, and in fact a similar construction is not possible when $\mu_\Gamma^+\leq 1$. More precisely, given a smooth metric $g_0$ defined on an annulus $A_{r,R}$, and given a cone $\Gamma$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"}, [\[22\'\]](#22'){reference-type="eqref" reference="22'"} and $\mu_\Gamma^+\leq 1$, there is no smooth metric $g_w = e^{2w}g_0$ satisfying $\lambda(-g_w^{-1}A_{g_w})\in\Gamma$ on $A_{r,R}$ and for which $w\rightarrow-\infty$ at either boundary component of $A_{r,R}$. The proof of this non-existence result uses arguments different in nature to those considered in this paper, and will appear elsewhere. For the remainder of the introduction, we discuss in more detail how our results and methods compare to previous work on fully nonlinear problems of Loewner-Nirenberg type. As mentioned before, when $M\backslash \partial M$ is a Euclidean domain, the existence of a Lipschitz viscosity solution to [\[113\]](#113){reference-type="eqref" reference="113"}, as well as uniqueness of this solution among continuous viscosity solutions, was established in [@GLN18]. Moreover, counterexamples to $C^1$ regularity were given in [@LN20b; @LNX22]. The proof in [@GLN18] uses Perron's method, which in turn uses canonical solutions on interior/exterior balls and a comparison principle on Euclidean domains established in [@LNW18]. Since one cannot use exterior balls in the Riemannian setting, and since it is not currently known whether the comparison principle in [@LNW18] extends to the Riemannian setting, a different approach to that in [@GLN18] is required to prove Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}. On the other hand, for $(f,\Gamma) = (\sigma_k^{1/k},\Gamma_k^+)$ ($2\leq k \leq n$), Gursky, Streets & Warren proved in [@GSW11] the existence of a unique smooth solution to [\[113\]](#113){reference-type="eqref" reference="113"} with the Ricci tensor in place of the Schouten tensor (see Remark [Remark 13](#168){reference-type="ref" reference="168"} below for the relation between this result and Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}, and see also the work of Wang [@W21] and Li [@L22] for some further related results). As in the present paper, the solution of Gursky, Streets & Warren is constructed as a limit of solutions with finite boundary data, and these solutions are in turn obtained using the continuity method. Their method for obtaining an *a priori* lower bound on solutions is different to ours, and is instead based on the explicit construction of a global subsolution. Roughly speaking, the subsolution construction in [@GSW11] uses the fact that, in the analogous formula to [\[118\]](#118){reference-type="eqref" reference="118"} for the Ricci tensor, the gradient terms are collectively nonnegative definite and so can be neglected in certain computations. In our case, the gradient terms do not have an overall sign, thus leading to our new approach for the lower bound discussed above. **Remark 13**. Since $\mu_{\Gamma_k^+}^+ = \frac{n-k}{k}$, it is easy to see that $\mu_{(\Gamma_k^+)^\tau}^+ = \frac{n-k}{k} + (n-1)(1-\tau)$. Thus $\mu_{(\Gamma_k^+)^\tau}^+ >1$ if and only if $\tau < a_{n,k} \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =\frac{n-k+k(n-2)}{k(n-1)}$. On the other hand, for $\tau=\frac{n-2}{n-1}$ we have $(\sigma_k^{1/k})^\tau(\lambda(-g_u^{-1}A_{g_u})) = \frac{1}{n-1}\sigma_k^{1/k}(\lambda(-g_u^{-1}\operatorname{Ric}_{g_u}))$. Since $\frac{n-2}{n-1}<a_{n,k}$ if and only if $k<n$, we therefore see that Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"} recovers the result of [@GSW11] for $k<n$. The plan of the paper is as follows. In Section [2](#14){reference-type="ref" reference="14"} we prove the local interior gradient estimate stated in Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. In Section [3](#15){reference-type="ref" reference="15"} we consider the Dirichlet boundary value problem [\[12\]](#12){reference-type="eqref" reference="12"}, proving Theorem [Theorem 6](#55){reference-type="ref" reference="55"}. Finally, in Section [4](#135){reference-type="ref" reference="135"} we turn to the fully nonlinear Loewner-Nirenberg problem [\[-113\]](#-113){reference-type="eqref" reference="-113"}, proving Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"} (and hence Theorem [Theorem 1](#A){reference-type="ref" reference="A"}). **Notation:** Throughout the rest of the paper, if $X$ is a $(1,1)$-tensor satisfying $\lambda(X)\in\Gamma$ then we frequently denote $f(X)\mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =f(\lambda(X))$. **Acknowledgements:** The authors would like to thank Prof. YanYan Li for stimulating discussions and his constant support. # Proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}: the local interior gradient estimate {#14} In this section we prove the local interior gradient estimate stated in Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. Throughout the section, unless otherwise stated all derivatives and norms are taken with respect to $g_0$. Moreover, $C$ will denote a constant that may change from line to line and depends only on $n, f, \Gamma, \|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$. ## Set-up and main ideas of the proof {#302} Our set-up for the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"} is similar to that in the related works [@Che05; @GW03b; @JLL07; @LL03; @Kho09; @Li09; @Wan06] on local gradient estimates. Throughout this section we denote $S=A_{g_0}$ and $$W = \nabla^2 u + \frac{1}{2}|\nabla u|^2 g_0 - du\otimes du - S.$$ By a standard argument it suffices to consider the case $r=1$ in the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. Suppose $\rho\in C_c^\infty(B_1)$ is a cutoff function in $B_1$ with $\rho = 1$ on $B_{1/2}$, $|\nabla \rho| \leq C \rho^{1/2}$ and $|\nabla^2 \rho|\leq C$. Set $H=\rho|\nabla u|^2$ and suppose $H$ attains a maximum at $x_0$. We may assume that $|\nabla u|\geq 1$ at $x_0$, otherwise we are done. Choosing suitable normal coordinates centred at $x_0$, we may also assume $W = (w_{ij})$ is diagonal at $x_0$ with $w_{11} \geq \dots \geq w_{nn}$, and hence at $x_0$ we have $$\begin{aligned} \label{21} \begin{cases}w_{ii} = u_{ii} - u_i^2 + \frac{1}{2}|\nabla u|^2 - S_{ii} & \text{for all }1\leq i \leq n, \\ u_{ij} = u_i u_j + S_{ij} &\text{for }i\not=j. \end{cases} \end{aligned}$$ Using the fact that $H_i(x_0)=0$ for each $i$, we obtain at $x_0$ $$\label{120} \sum_{l=1}^n u_{il}u_l = -\frac{\rho_i}{2\rho}|\nabla u|^2$$ and hence $$\label{4} \bigg|\sum_{l=1}^n u_{il}u_l \bigg| \leq C\rho^{-1/2}|\nabla u|^2.$$ For $A_0$ a large number to be fixed later, we may assume at $x_0$ that $$\label{5} \rho^{-1/2} \leq C\frac{|\nabla u|}{A_0}\quad\text{and}\quad|S|\leq \frac{|\nabla u|^2}{A_0},$$ otherwise we are done. Note that by combining [\[4\]](#4){reference-type="eqref" reference="4"} with the first estimate in [\[5\]](#5){reference-type="eqref" reference="5"}, we have $$\begin{aligned} \label{26} \bigg|\sum_{l=1}^n u_{il}u_l \bigg| \leq C\frac{|\nabla u|^3}{A_0}. \end{aligned}$$ Denote by $F_\tau^{ij}$ the coefficients of the linearised operator at $(g_0^{-1}W)(x_0)$, that is $$\begin{aligned} F_\tau^{ij}= \frac{\partial f^\tau}{\partial A_{ij}}\bigg|_{A=(g_0^{-1}W)(x_0)}. \end{aligned}$$ Then $(F^{ij}_\tau)$ is a positive definite, diagonal matrix. Also denote $$\begin{aligned} \mathcal{F}_\tau = \sum_{i=1}^n F^{ii}_\tau \quad \text{and} \quad \widetilde{u}_{ij} \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =u_{ij} - S_{ij}. \end{aligned}$$ By homogeneity and concavity of $f$, it is easy to see that $\mathcal{F}_\tau \geq \frac{1}{C}>0$: indeed, denoting $\lambda = \lambda(g_0^{-1}W)(x_0)$, we have $$\begin{aligned} \label{183} \mathcal{F}_\tau = \sum_{i=1}^n \frac{\partial f^\tau}{\partial \lambda_i}(\lambda) = f^\tau(\lambda) + \sum_{i=1}^n \frac{\partial f^\tau}{\partial \lambda_i}(\lambda)(1-\lambda_i) \geq f^\tau(1,\dots,1). \end{aligned}$$ With our set-up and notation established, we now briefly discuss the main ideas in the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. The first step is to obtain the following lemma: **Lemma 14**. *Under the same hypotheses as Theorem [Theorem 8](#40){reference-type="ref" reference="40"} but without the restriction $\mu_\Gamma^+>1$, there exists a constant $C$ such that $$\begin{aligned} \label{1''} 0 \geq- C\mathcal{F}_\tau(1+e^{2u})|\nabla u|^2 - C\rho\mathcal{F}_\tau\frac{|\nabla u|^4}{A_0} + \rho\sum_{i,l}F_\tau^{ii}\widetilde{u}_{il}^2 \quad \text{at }x_0. \end{aligned}$$* The proof of Lemma [Lemma 14](#34){reference-type="ref" reference="34"} is by now standard and will be given in Section [2.2](#132){reference-type="ref" reference="132"}. Now, in the case that the positive term on the RHS of [\[1\'\'\]](#1''){reference-type="eqref" reference="1''"} dominates $|\nabla u|^4\mathcal{F}_\tau$, in the sense that $$\begin{aligned} \label{300} \sum_{i,l}F^{ii}_\tau \widetilde{u}_{il}^2 \geq \varepsilon|\nabla u|^4\mathcal{F}_\tau \quad \text{at }x_0 \end{aligned}$$ for a suitably chosen small constant $\varepsilon>0$, then the desired gradient estimate is routine (the details will be given later). On the other hand, if [\[300\]](#300){reference-type="eqref" reference="300"} fails for our suitably chosen small constant $\varepsilon>0$, we will see that the ordered eigenvalues $w_{11} \geq \dots \geq w_{nn}$ of $W$ at $x_0$ are greater than or equal to a perturbation of $(1,\dots,1,-1)\frac{|\nabla u|^2}{2}$. As mentioned in the introduction, this phenomenon was previously observed in the case $(f,\Gamma) = (\sigma_k^{1/k},\Gamma_k^+)$ when $k<\frac{n}{2}$ in the thesis of Khomrutai [@Kho09]. Using the fact that $(1,\dots,1,-1)\in\Gamma$ (this is the only place in the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"} where the assumption $\mu_\Gamma^+>1$ is used), the gradient estimate again follows. The details will be given in Section [2.3](#301){reference-type="ref" reference="301"}. ## Proof of Lemma [Lemma 14](#34){reference-type="ref" reference="34"} {#132} In this section we give the proof of Lemma [Lemma 14](#34){reference-type="ref" reference="34"}: *Proof.* We follow closely the proof of Guan & Wang [@GW03b]. In what follows, all computations are implicitly carried out at $x_0$. First observe that by [\[120\]](#120){reference-type="eqref" reference="120"}, $$\begin{aligned} H_{ij} = \bigg(\rho_{ij} - \frac{2\rho_i\rho_j}{\rho}\bigg)|\nabla u|^2 + 2\rho\sum_{l=1}^n u_{lij}u_l + 2\rho\sum_{l=1}^n u_{il}u_{jl}, \end{aligned}$$ and hence by positivity of $(F^{ij}_\tau)$ and non-positivity of $(H_{ij})$, $$\begin{aligned} \label{9} 0 \geq \sum_{i=1}^nF_\tau^{ii}H_{ii} & = \sum_{i=1}^nF_\tau^{ii}\bigg[\bigg(\rho_{ii} - \frac{2\rho_i^2}{\rho}\bigg)|\nabla u|^2 + 2\rho\sum_{l=1}^n u_{lii}u_l + 2\rho\sum_{l=1}^n u_{il}^2\bigg] \nonumber \\ & = -C |\nabla u|^2 \mathcal{F}_\tau + 2\rho\sum_{i,l} F_\tau^{ii}u_{lii}u_l +2\rho\sum_{i,l}F_\tau^{ii} u_{il}^2. \end{aligned}$$ Now, commuting derivatives yields $$\begin{aligned} \label{8} \sum_{i,l} F_\tau^{ii}u_{lii}u_l & \geq \sum_{i,l} F_\tau^{ii}u_{iil} u_l - C|\nabla u|^2 \mathcal{F}_\tau \nonumber \\ & = \sum_{i,l} F_\tau^{ii}\bigg[(w_{ii})_l - \bigg(\frac{1}{2}|\nabla u|^2 - u_i^2\bigg)_l + (S_{ii})_l \bigg]u_l - C|\nabla u|^2\mathcal{F}_\tau \nonumber \\ & = \sum_{l=1}^n (\psi e^{2u})_l u_l - \mathcal{F}_\tau\sum_{k,l} u_{kl}u_k u_l + 2\sum_{i,l} F_\tau^{ii} u_{il}u_i u_l + \sum_{i,l} F_\tau^{ii}(S_{ii})_l u_l - C|\nabla u|^2 \mathcal{F}_\tau, \end{aligned}$$ where to reach the last line we have used the fact that $f^\tau$ is homogeneous of degree one to assert that $\sum_i F^{ii}_\tau (w_{ii})_l = (f^\tau(g_0^{-1}W))_l= (\psi e^{2u})_l$. Also, since $|\nabla u|\geq 1$, we can bound the penultimate term in [\[8\]](#8){reference-type="eqref" reference="8"} from below by $-C|\nabla u|^2\mathcal{F}_\tau$, and also observe that $$\begin{aligned} \label{121} \sum_{l=1}^n (\psi e^{2u})_l u_l & = \sum_{l=1}^n e^{2u}\psi_l u_l + 2 e^{2u}\psi|\nabla u|^2 \geq -Ce^{2u}|\nabla u|^2. \end{aligned}$$ Also, by [\[26\]](#26){reference-type="eqref" reference="26"} we have $$\label{6} - \mathcal{F}_\tau \sum_{k,l} u_{kl}u_k u_l \geq -C\frac{|\nabla u|^4}{A_0}\mathcal{F}_\tau,$$ and likewise $$\label{7} 2\sum_{i,l} F_\tau^{ii} u_{il}u_i u_l = 2\sum_i \bigg(F^{ii}_\tau u_i\sum_l u_{il}u_l\bigg) \geq - 2\sum_i \bigg(\big|F^{ii}_\tau u_i\big|\Big|\sum_l u_{il}u_l\Big|\bigg)\geq -C \frac{|\nabla u|^4}{A_0}\mathcal{F}_\tau.$$ Substituting [\[121\]](#121){reference-type="eqref" reference="121"}--[\[7\]](#7){reference-type="eqref" reference="7"} back into [\[8\]](#8){reference-type="eqref" reference="8"} and recalling $\mathcal{F}_\tau \geq \frac{1}{C}$, we get $$\begin{aligned} \sum_{i,l} F_\tau^{ii}u_{lii}u_l & \geq -C(1+e^{2u})|\nabla u|^2 \mathcal{F}_\tau - C\mathcal{F}_\tau\frac{|\nabla u|^4}{A_0}, \end{aligned}$$ and substituting this back into [\[9\]](#9){reference-type="eqref" reference="9"} we see $$\begin{aligned} \label{1} 0 & \geq - C\mathcal{F}_\tau(1+e^{2u})|\nabla u|^2 - C\rho\mathcal{F}_\tau\frac{|\nabla u|^4}{A_0} + 2\rho \sum_{i,l}F_\tau^{ii}u_{il}^2. \end{aligned}$$ The desired estimate [\[1\'\'\]](#1''){reference-type="eqref" reference="1''"} then follows from [\[1\]](#1){reference-type="eqref" reference="1"} and the following inequality, which is a consequence of the Cauchy-Schwarz inequality and the second inequality in [\[5\]](#5){reference-type="eqref" reference="5"}: $$\begin{aligned} \sum_{i,l} F_\tau^{ii}u_{il}^2 \geq \frac{1}{2}\sum_{i,l} F_\tau^{ii} \widetilde{u}_{il}^2 - \frac{1}{A_0}\mathcal{F}_\tau|\nabla u|^4. \end{aligned}$$ ◻ ## Proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"} {#301} We begin this section by stating a central result in our argument, namely Proposition [Proposition 15](#303){reference-type="ref" reference="303"}. The proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"} is then given assuming the validity of Proposition [Proposition 15](#303){reference-type="ref" reference="303"} -- this should serve to elucidate the ideas outlined at the end of Section [2.1](#302){reference-type="ref" reference="302"}. The proof of Proposition [Proposition 15](#303){reference-type="ref" reference="303"} will be given later in the section, and consists of a series of technical lemmas. To this end, for $1>\delta_0 \geq A_0^{-1/10}$ a small number to be fixed later, define the set $$\begin{aligned} \mathcal{I} = \bigg\{ i\in \{1,\dots,n\}: \bigg|w_{jj} + \frac{|\nabla u|^2}{2}\bigg| < 2\delta_0^2 |\nabla u|^2\bigg\}. \end{aligned}$$ We remind the reader that all computations are implicitly carried out at $x_0$, and that we have the ordering $w_{11} \geq \dots \geq w_{nn}$. We will prove: **Proposition 15**. *There exists a constant $\widetilde{C}>1$ depending only on $n, f, \Gamma, \|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$ such that if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$ and $$\begin{aligned} \label{125} \sum_{i,l} F_\tau^{ii} \widetilde{u}_{il}^2 < \widetilde{C}^{-1}\delta_0^4|\nabla u|^4 \mathcal{F}_\tau, \end{aligned}$$ then:* 1. *$\mathcal{I} = \{n\}$, and* 2. *$\big| w_{n-1,n-1} - \frac{|\nabla u|^2}{2}\big|<2\delta_0|\nabla u|^2$.* Assuming the validity of Proposition [Proposition 15](#303){reference-type="ref" reference="303"} for now, let us complete the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}: *Proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}.* We start by fixing $\widetilde{C}$ sufficiently large so that Proposition [Proposition 15](#303){reference-type="ref" reference="303"} applies. Then for $A_0> \widetilde{C}^{10}$ to be fixed later, if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$ and [\[125\]](#125){reference-type="eqref" reference="125"} is satisfied, then $$w_{n-1, n-1} = (1+a_{n-1})\frac{|\nabla u|^2}{2} \quad \text{and} \quad w_{nn} =-(1+a_n)\frac{|\nabla u|^2}{2}$$ for some $|a_{n-1}|, |a_n| \leq 4\delta_0$. On the other hand, since $w_{11}\geq \dots \geq w_{nn}$, for each $\alpha = 1,\dots,n-2$ we can write $w_{\alpha\alpha} = w_{n-1,n-1} + X_\alpha$ for some $X_\alpha\geq 0$. Therefore $$\begin{aligned} \label{39} & \begin{pmatrix} w_{11} \\ \vdots \\ w_{n-2, n-2} \\ w_{n-1, n-1} \\ w_{nn} \end{pmatrix} = \begin{pmatrix} X_1 \\ \vdots \\ X_{n-2} \\ 0 \\ 0 \end{pmatrix} + \frac{|\nabla u|^2}{2}\underbrace{\begin{pmatrix} 1+a_{n-1} \\ \vdots \\ 1+a_{n-1} \\ 1+a_{n-1} \\ -(1+a_n) \end{pmatrix}}_{\mathcal{B}}, \end{aligned}$$ with the first vector on the RHS of [\[39\]](#39){reference-type="eqref" reference="39"} clearly belonging to $\overline{\Gamma^\tau}$ for each $\tau \leq 1$, since each entry is nonnegative. We also observe that $\mathcal{B}$ is a perturbation of $\mathcal{B}_0 \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =(1,\dots,1,-1)$, and that $\mathcal{B}_0\in\Gamma^\tau$ for any $\tau\leq 1$ since we assume $\mu_\Gamma^+>1$. Therefore, since $|a_{n-1}|, |a_n| \leq 4\delta_0$, for $\widetilde{C}$ sufficiently large it will hold that $\mathcal{B}\in\Gamma$ with $f^\tau(\mathcal{B}) \geq \frac{1}{2}f^\tau(\mathcal{B}_0)$. Monotonicity of $f$ then implies $$\begin{aligned} \psi e^{2u} = f^\tau(w_{11},\dots,w_{nn}) \geq \frac{|\nabla u|^2}{2}f^\tau(\mathcal{B}) \geq \frac{|\nabla u|^2}{4}f^\tau(\mathcal{B}_0), \end{aligned}$$ which implies the desired gradient estimate. It remains to address the case that, for the value of $\widetilde{C}$ fixed in the foregoing argument, [\[125\]](#125){reference-type="eqref" reference="125"} is not satisfied. Then $$\label{2} \sum_{i,l} F_\tau^{ii} \widetilde{u}_{il}^2 \geq \widetilde{C}^{-1}A_0^{-2/5}|\nabla u|^4 \mathcal{F}_\tau,$$ and substituting [\[2\]](#2){reference-type="eqref" reference="2"} into [\[1\'\'\]](#1''){reference-type="eqref" reference="1''"} we therefore have $$\begin{aligned} 0 \geq- C\mathcal{F}_\tau(1+e^{2u})|\nabla u|^2 - C\rho\mathcal{F}_\tau\frac{|\nabla u|^4}{A_0} + \widetilde{C}^{-1}A_0^{-2/5}\rho|\nabla u|^4 \mathcal{F}_\tau. \end{aligned}$$ Multiplying through by $\widetilde{C}A_0^{2/5}\rho$ then yields the estimate $$\begin{aligned} \label{419} 0 & \geq - \widetilde{C}CA_0^{2/5}\rho (1+e^{2u})|\nabla u|^2 - \frac{\widetilde{C}C}{A_0^{3/5}} \rho^2|\nabla u|^4 + \rho^2 |\nabla u|^4. \end{aligned}$$ It follows that if we choose $A_0 \geq \max\{(2\widetilde{C}C)^{5/3}, \widetilde{C}^{10}\}$ (where $C$ and $\widetilde{C}$ are the constants in [\[419\]](#419){reference-type="eqref" reference="419"}), then we have (for a possibly different constant $C$) $$\begin{aligned} 0 & \geq -C\rho(1+e^{2u})|\nabla u|^2 + \frac{1}{2}\rho^2 |\nabla u|^4, \end{aligned}$$ and therefore $$H^2 = \rho^2 |\nabla u|^4 \leq C(1+e^{2u}) H.$$ After dividing through by $H$ we again arrive at the desired gradient estimate. ◻ The rest of the section is devoted to the proof of Proposition [Proposition 15](#303){reference-type="ref" reference="303"}, which we obtain through a series of three lemmas. In the first of these lemmas we show that if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$ for $\widetilde{C}$ sufficiently large, then $\mathcal{I}\not=\emptyset$: **Lemma 16**. *There exists a constant $\widetilde{C} >1$ depending only on $n, f, \Gamma, \|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$ such that if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$, then $\mathcal{I}\not=\emptyset$.* *Proof.* It is clear that for $\delta_0\leq \sqrt{1/n}$, there is at least one index $j\in \{1,\dots,n\}$ such that $u_j^2 \geq \delta_0^2|\nabla u|^2$. We claim that for such an index $j$, it holds that $j\in\mathcal{I}$. We follow the method of Guan & Wang [@GW03b]. We know that for $l\not=j$, $u_{jl} = u_j u_l + S_{jl}$ and therefore $$\sum_{l\not=j} u_{jl}u_l = \sum_{l\not=j}u_ju_l^2 + \sum_{l\not=j} S_{jl}u_l.$$ It follows that $$\begin{aligned} \sum_{l=1}^n u_{jl}u_l & = \sum_{l\not=j} u_ju_l^2 + \sum_{l\not=j} S_{jl}u_l + u_{jj}u_j \nonumber \\ & = u_j|\nabla u|^2 + \sum_{l\not=j} S_{jl}u_l + u_{jj}u_j - u_j^3 \nonumber \\ & = \sum_{l\not=j} S_{jl}u_l - u_j\bigg(\big(u_j^2 - |\nabla u|^2\big)- u_{jj}\bigg). \end{aligned}$$ Hence $$\begin{aligned} \bigg|u_j\bigg(\big(u_j^2 - |\nabla u|^2\big)- u_{jj}\bigg) - \sum_{l\not=j}S_{jl}u_l\bigg| = \bigg|\sum_{l=1}^n u_{jl}u_l\bigg| \stackrel{\eqref{26}}{\leq} C\frac{|\nabla u|^3}{A_0}. \end{aligned}$$ It follows that $$\begin{aligned} \label{30} \big|u_j\big|\big|\big(u_j^2 - |\nabla u|^2\big)- u_{jj}\big| \leq C\frac{|\nabla u|^3}{A_0} + \bigg|\sum_{l\not=j}S_{jl}u_l\bigg| \stackrel{\eqref{5}}{\leq} C\frac{|\nabla u|^3}{A_0} \leq C\delta_0^{10}|\nabla u|^3, \end{aligned}$$ where to reach the last inequality we have used $A_0^{-1/10} \leq \delta_0$. Substituting $|u_j| \geq \delta_0|\nabla u|$ back into [\[30\]](#30){reference-type="eqref" reference="30"} yields $$\begin{aligned} \label{31} \big|\big(u_j^2 - |\nabla u|^2\big)- u_{jj}\big| \leq C\delta_0^9|\nabla u|^2. \end{aligned}$$ Next, substituting $u_{jj} = w_{jj} + u_j^2 - \frac{1}{2}|\nabla u|^2 + S_{jj}$ into [\[31\]](#31){reference-type="eqref" reference="31"} and again applying [\[5\]](#5){reference-type="eqref" reference="5"} we obtain $$\begin{aligned} \label{128} \bigg|w_{jj} + \frac{1}{2}|\nabla u|^2\bigg| \leq C\delta_0^9|\nabla u|^2 + \frac{|\nabla u|^2}{A_0} = C\delta_0^9|\nabla u|^2 + \delta_0^{10}|\nabla u|^2. \end{aligned}$$ It is clear that one can then choose $\widetilde{C}$ sufficiently large so that the right hand side of [\[128\]](#128){reference-type="eqref" reference="128"} is less than $2\delta_0^2|\nabla u|^2$ for $\delta_0 \leq \widetilde{C}^{-1}$. Once such a choice is made, we see that [\[128\]](#128){reference-type="eqref" reference="128"} implies $j\in\mathcal{I}$, which proves the claim and therefore the lemma. ◻ In our subsequent arguments we will use the following proposition, which is essentially a consequence of [@Yuan22 Theorem 1.4] -- see Appendix [5](#AA){reference-type="ref" reference="AA"} for a summary of the proof. **Proposition 17**. *Suppose $\Gamma$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"} and [\[22\'\]](#22'){reference-type="eqref" reference="22'"} with $\Gamma\not=\Gamma_n^+$ (equivalently, $\mu_\Gamma^+>0$). Then there exists a constant $\theta=\theta(n,\Gamma)>0$ such that for any $\lambda\in\Gamma$ with $\lambda_1\geq \dots \geq \lambda_n$, $$\label{37'} \frac{\partial f}{\partial \lambda_i}(\lambda) \geq \theta \sum_{j=1}^n \frac{\partial f}{\partial\lambda_j}(\lambda) \quad \text{if }i\in\{n-1,n\} \text{ or }\lambda_i\leq 0.$$* We are now in a position to show that if one additionally assumes [\[125\]](#125){reference-type="eqref" reference="125"} holds for $\widetilde{C}$ sufficiently large, then $|\mathcal{I}|=\{n\}$ (recall once again the ordering $w_{11} \geq \dots \geq w_{nn}$): **Lemma 18**. *There exists a constant $\widetilde{C}>1$ depending only on $n, f, \Gamma, \|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$ such that if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$ and [\[125\]](#125){reference-type="eqref" reference="125"} is satisfied, then $|\mathcal{I}|=\{n\}$.* *Proof.* We first claim that if $\widetilde{C}$ is sufficiently large and [\[125\]](#125){reference-type="eqref" reference="125"} holds, then $u_{jj}> -2\delta_0^2 |\nabla u|^2$ for $j\in \mathcal{I}$. Indeed, suppose for a contradiction that this is not the case. Then we would have $$\begin{aligned} \label{56} \sum_{i,l} F_\tau^{ii} \widetilde{u}_{il}^2 \geq F_\tau^{jj}\widetilde{u}_{jj}^2 \stackrel{\eqref{5}}{\geq} \frac{1}{2}F^{jj}_\tau u_{jj}^2 - F^{jj}_\tau\frac{|\nabla u|^4}{A_0^2} & \geq 2F_\tau^{jj}\delta_0^4|\nabla u|^4 - F^{jj}_\tau\delta_0^{20}|\nabla u|^4 \nonumber \\ & \geq F^{jj}_\tau \delta_0^4|\nabla u|^4 \nonumber \\ & \geq \theta \delta_0^4|\nabla u|^4 \mathcal{F}_\tau, \end{aligned}$$ with the last inequality following from Proposition [Proposition 17](#46){reference-type="ref" reference="46"} -- note that Proposition [Proposition 17](#46){reference-type="ref" reference="46"} applies in this case, since $w_{jj}<0$ by virtue of $j\in\mathcal{I}$ if $\widetilde{C}$ is sufficiently large. But this contradicts [\[125\]](#125){reference-type="eqref" reference="125"} if $\widetilde{C}$ is sufficiently large, proving the claim. By the claim, we may therefore suppose that $\widetilde{C}$ is large enough so that $u_{jj}> -2\delta_0^2 |\nabla u|^2$ whenever $j\in \mathcal{I}$. Then for $j\in\mathcal{I}$, we therefore have $$\begin{aligned} -2\delta_0^2|\nabla u|^2 - u_j^2 + \frac{1}{2}|\nabla u|^2 - S_{jj} < u_{jj} - u_j^2 + \frac{1}{2}|\nabla u|^2 - S_{jj} = w_{jj} < -\frac{1}{2}|\nabla u|^2 + 2\delta_0^2|\nabla u|^2, \end{aligned}$$ with the last inequality following from the definition of $\mathcal{I}$. That is, $$\begin{aligned} \label{57} -u_j^2 < (-1+4\delta_0^2)|\nabla u|^2 + S_{jj} \stackrel{\eqref{5}}{<} (-1+4\delta_0^2)|\nabla u|^2 + \delta_0^{10}|\nabla u|^2 < (-1+5\delta_0^2)|\nabla u|^2. \end{aligned}$$ Clearly [\[57\]](#57){reference-type="eqref" reference="57"} cannot hold for more than one index if $10\delta_0^2<1$. Hence $|\mathcal{I}| \leq 1$ for $\widetilde{C}$ sufficiently large, and after increasing $\widetilde{C}$ further if necessary so that $\mathcal{I}\not=\emptyset$ (recall that this is possible by Lemma [Lemma 16](#68){reference-type="ref" reference="68"}), then it must be the case that $|\mathcal{I}|=1$, i.e. $\mathcal{I} = \{n\}$. ◻ To finish the proof of Proposition [Proposition 15](#303){reference-type="ref" reference="303"} it remains to show (after taking $\widetilde{C}$ larger if necessary) that $| w_{n-1,n-1} - \frac{|\nabla u|^2}{2}|<2\delta_0|\nabla u|^2$. This is the focus of the next lemma: **Lemma 19**. *There exists a constant $\widetilde{C}>1$ depending only on $n, f, \Gamma, \|g_0\|_{C^3(B_r)}$ and $\|\psi\|_{C^1(B_r)}$ such that if $A_0^{-1/10} \leq \delta_0 \leq \widetilde{C}^{-1}$ and [\[125\]](#125){reference-type="eqref" reference="125"} is satisfied, then $$\begin{aligned} \bigg| w_{n-1,n-1} - \frac{|\nabla u|^2}{2}\bigg|<2\delta_0|\nabla u|^2 \end{aligned}$$* *Proof.* **Step 1:** In this first step we show $$\begin{aligned} \label{58} w_{n-1,n-1} > \bigg(\frac{1}{2}-2\delta_0\bigg)|\nabla u|^2. \end{aligned}$$ Suppose for a contradiction that $w_{n-1,n-1} \leq (\frac{1}{2}-2\delta_0)|\nabla u|^2$, i.e. $$\begin{aligned} \label{59} u_{n-1,n-1} - u_{n-1}^2 - S_{n-1,n-1} \leq - 2\delta_0|\nabla u|^2. \end{aligned}$$ Either $u_{n-1}^2 < \delta_0|\nabla u|^2$ or $u_{n-1}^2 \geq \delta_0|\nabla u|^2$. In the former case, [\[59\]](#59){reference-type="eqref" reference="59"} then implies $$\begin{aligned} u_{n-1,n-1} < -\delta_0|\nabla u|^2 + S_{n-1,n-1} \stackrel{\eqref{5}}{<} -\delta_0|\nabla u|^2 + \delta_0^{10}|\nabla u|^2 < -\frac{1}{2}\delta_0|\nabla u|^2 \quad \text{if }\delta_0<\frac{1}{2}, \end{aligned}$$ and one obtains a contradiction as in [\[56\]](#56){reference-type="eqref" reference="56"} if $\widetilde{C}$ is sufficiently large -- note that Proposition [Proposition 17](#46){reference-type="ref" reference="46"} is again justified, since $w_{n-1,n-1}$ is the second lowest eigenvalue. If instead $u_{n-1}^2 \geq \delta_0|\nabla u|^2$, the proof of Lemma [Lemma 16](#68){reference-type="ref" reference="68"} shows that $n-1\in\mathcal{I}$. This contradicts the conclusion $|\mathcal{I}|=\{n\}$ of Lemma [Lemma 18](#63){reference-type="ref" reference="63"} if $\widetilde{C}$ is sufficiently large. Thus [\[58\]](#58){reference-type="eqref" reference="58"} is established, which completes the proof of Step 1. **Step 2:** In this second step we show $$\begin{aligned} \label{64'} w_{n-1,n-1} < \bigg(\frac{1}{2}+2\delta_0\bigg)|\nabla u|^2. \end{aligned}$$ Indeed, we have $$\begin{aligned} w_{n-1,n-1} = u_{n-1,n-1} - u_{n-1}^2 + \frac{1}{2}|\nabla u|^2 - S_{n-1,n-1} \stackrel{\eqref{5}}{\leq} |u_{n-1,n-1}| + \frac{1}{2}|\nabla u|^2 + \delta_0^{10}|\nabla u|^2. \end{aligned}$$ But $|u_{n-1,n-1}| \leq \delta_0 |\nabla u|^2$, else one would obtain a contradiction as in [\[56\]](#56){reference-type="eqref" reference="56"} if $\widetilde{C}$ is sufficiently large (again we are using the fact $w_{n-1,n-1}$ is the second lowest eigenvalue, so Proposition [Proposition 17](#46){reference-type="ref" reference="46"} applies). The estimate [\[64\'\]](#64'){reference-type="eqref" reference="64'"} thus follows, which completes the proof of Step 2. With [\[58\]](#58){reference-type="eqref" reference="58"} and [\[64\'\]](#64'){reference-type="eqref" reference="64'"} established, the proof of Lemma [Lemma 19](#65){reference-type="ref" reference="65"} is complete. ◻ *Proof of Proposition [Proposition 15](#303){reference-type="ref" reference="303"}.* This is an immediate consequence of Lemmas [Lemma 16](#68){reference-type="ref" reference="68"}, [Lemma 18](#63){reference-type="ref" reference="63"} and [Lemma 19](#65){reference-type="ref" reference="65"}. ◻ # Proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"}: the Dirichlet boundary value problem {#15} As discussed in the introduction, in the proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"} we will first address the corresponding Dirichlet boundary value problem with finite boundary data. To this end, in this section we prove Theorem [Theorem 6](#55){reference-type="ref" reference="55"}. Our proof uses the continuity method, and we proceed according to the following steps: 1. In Section [3.1](#146){reference-type="ref" reference="146"} we give a routine proof of the global upper bound on solutions for $\tau\leq 1$, independently of whether or not $\mu_\Gamma^+>1$. 2. In Section [3.2](#147){reference-type="ref" reference="147"} we prove the global lower bound on solutions for $\tau\leq 1$ when $\mu_\Gamma^+>1$. As outlined in the introduction, we use two main ingredients: our local interior gradient estimate obtained in Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, and a lower bound in a uniform neighbourhood of $\partial M$, which is obtained by constructing suitable comparison functions on small annuli (see Propositions [Proposition 22](#41){reference-type="ref" reference="41"} and [Proposition 23](#54){reference-type="ref" reference="54"}). 3. In Section [3.3](#148){reference-type="ref" reference="148"} we prove the global gradient estimate for $\tau\leq 1$ when $\mu_\Gamma^+>1$. To obtain the lower bound for the normal derivative on $\partial M$ we use our comparison functions on small annuli constructed in Section [3.2](#147){reference-type="ref" reference="147"}, and to obtain the upper bound for the normal derivative on $\partial M$ we use comparison functions similar to that of Guan [@Guan08] (this latter argument does not use $\mu_\Gamma^+>1$). For the interior estimates we use Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, and for estimates near $\partial M$ we appeal to the proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"}. 4. In Section [3.4](#149){reference-type="ref" reference="149"} we prove the global Hessian estimate for $\tau< 1$, following arguments of Guan [@Guan08]. These estimates apply independently of whether or not $\mu_\Gamma^+>1$. 5. In Section [3.5](#150){reference-type="ref" reference="150"}, we complete the proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"}: we first prove the existence of a unique smooth solution when $\tau<1$ using the continuity method, and we then obtain a Lipschitz viscosity solution in the case $\tau=1$ in the limit as $\tau\rightarrow 1$. We point out that, in order to obtain a Lipschitz viscosity solution in the limit $\tau\rightarrow 1$ in Section [3.5](#150){reference-type="ref" reference="150"}, it is important that our *a priori* $C^1$ estimates obtained in Sections [3.1](#146){reference-type="ref" reference="146"}--[3.3](#148){reference-type="ref" reference="148"} are uniform in $\tau\in[0,1]$. On the other hand, the global Hessian estimate in Section [3.4](#149){reference-type="ref" reference="149"} deteriorates as $\tau\rightarrow 1$; this is to be expected in view of the work in [@LN20b; @LNX22], where the non-existence of $C^2$ solutions is established for all Euclidean domains with disconnected smooth boundary when $\tau=1$. ## Upper bound {#146} The global upper bound on solutions to [\[12\]](#12){reference-type="eqref" reference="12"} is routine and does not require the assumption $\mu_\Gamma^+>1$: **Proposition 20**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and let $\tau\leq 1$. Let $\psi\in C^\infty(M)$ be positive and $\xi\in C^\infty(\partial M)$. Then there exists a constant $C$ which is independent of $\tau$ but dependent on $g_0, f, \Gamma$, a lower bound for $\inf_M \psi$ and an upper bound for $\sup_{\partial M} \xi$ such that any $C^2$ solution to [\[12\]](#12){reference-type="eqref" reference="12"} satisfies $u\leq C$ on $M$.* *Proof.* Suppose the maximum of $u$ occurs at $x_0\in M$. If $x_0\in\partial M$, then $u(x_0)\leq \xi(x_0)$. If $x_0\in M\backslash \partial M$, then $\nabla_{g_0}^2 u(x_0) \leq 0$ and $du(x_0)=0$, and hence $$\psi(x_0)e^{2u(x_0)} \leq f^\tau(-g_0^{-1}A_{g_0})(x_0),$$ which yields $u(x_0) \leq \frac{1}{2}\ln\big(\frac{f^\tau(-g_0^{-1}A_{g_0})}{\psi}\big)(x_0)$. ◻ ## Lower bound {#147} In this section we obtain the global lower bound on solutions to [\[12\]](#12){reference-type="eqref" reference="12"}: **Proposition 21**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}, and let $\tau\leq 1$. Let $\psi\in C^\infty(M)$ be positive and $\xi\in C^\infty(\partial M)$. Then there exists a constant $C$ which is independent of $\tau$ but dependent on $g_0, f, \Gamma$, an upper bound for $\|\psi\|_{C^1(M)}$ and a lower bound for $\inf_{\partial M}\xi$ such that any $C^3$ solution to [\[12\]](#12){reference-type="eqref" reference="12"} satisfies $u\geq C$ on $M$.* There are two main ingredients in our proof of Proposition [Proposition 21](#151){reference-type="ref" reference="151"}: our local interior gradient estimate from Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, and a lower bound in a uniform neighbourhood of $\partial M$; the assumption $\mu_\Gamma^+>1$ plays a role at both stages. As pointed out before, a delicate point is that we do not assume that the background metric satisfies $\lambda(-g_0^{-1}A_{g_0})\in\Gamma$ on $M$ -- if such an assumption is made, then the proof of the lower bound is as straightforward as the proof of Proposition [Proposition 20](#75){reference-type="ref" reference="75"}. In our case, the global lower bound requires more work and is one of the key steps in this paper. To state our result concerning the lower bound near $\partial M$, for $\delta>0$ we denote $$\begin{aligned} M_\delta = \{x\in M:\operatorname{d}(x,\partial M)<\delta\},\end{aligned}$$ where $\operatorname{d}(x,\partial M)$ is the distance from $x$ to $\partial M$ with respect to $g_0$. It is well-known that for $\delta>0$ sufficiently small, $M_\delta$ is a tubular neighbourhood of $\partial M$. We show: **Proposition 22**. *Under the same hypotheses as Proposition [Proposition 21](#151){reference-type="ref" reference="151"}, there exists a constant $\delta>0$ which is independent of $\tau$ but dependent on $g_0, f, \Gamma$, an upper bound for $\sup_{M}\psi$ and a lower bound for $\inf_{\partial M}\xi$ such that any $C^3$ solution $u$ to [\[12\]](#12){reference-type="eqref" reference="12"} satisfies $u\geq \inf_{\partial M}\xi - 1$ in $M_{\delta}$.* Assuming the validity of Proposition [Proposition 22](#41){reference-type="ref" reference="41"} for now, we give the proof of Proposition [Proposition 21](#151){reference-type="ref" reference="151"}: *Proof of Proposition [Proposition 21](#151){reference-type="ref" reference="151"}.* Let $\delta>0$ be as in the statement of Proposition [Proposition 22](#41){reference-type="ref" reference="41"}, so that $u$ satisfies the lower bound $u\geq \inf_{\partial M}\xi - 1$ in $M_{\delta}$. It follows that $$\begin{aligned} \label{163} u \geq \inf_{\partial M}\xi - 1 - \operatorname{diam}(M,g_0)\sup_{M\backslash M_\delta}|\nabla_{g_0} u|_{g_0} \quad \text{in }M. \end{aligned}$$ On the other hand, by Theorem [Theorem 8](#40){reference-type="ref" reference="40"} and the uniform upper bound for $u$ obtained in Proposition [Proposition 20](#75){reference-type="ref" reference="75"}, we have $$\begin{aligned} \label{162} |\nabla_{g_0} u|_{g_0} \leq C(\delta^{-1}+1) \quad \text{in }M\backslash M_\delta. \end{aligned}$$ Substituting [\[162\]](#162){reference-type="eqref" reference="162"} into [\[163\]](#163){reference-type="eqref" reference="163"}, the proof of Proposition [Proposition 21](#151){reference-type="ref" reference="151"} is complete. ◻ Roughly speaking, to prove Proposition [Proposition 22](#41){reference-type="ref" reference="41"} we cover a neighbourhood of $\partial M$ by small annuli on which we construct suitable comparison functions. The construction of such comparison functions is given in the following proposition (which is a more precise version of Proposition [Proposition 11](#54'){reference-type="ref" reference="54'"} stated in the introduction). For a Riemannian metric $g_0$ defined on a neighbourhood of the origin in $\mathbb{R}^n$, let $r(x) = \operatorname{d}_{g_0}(0,x)$, let $\mathbb{S}_r = \partial \mathbb{B}_r$ denote the geodesic sphere of radius $r$ centred at the origin, and denote by $A_{r_1, r_2}$ the annulus $\mathbb{B}_{r_2}\backslash\overline{\mathbb{B}}_{r_1}$. We also denote $$\begin{aligned} \beta = \frac{2}{\mu_\Gamma^+ - 1},\end{aligned}$$ and recall the convention $g_w = e^{2w}g_0$. **Proposition 23**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}, let $g_0$ be a Riemannian metric defined on a neighbourhood $\Omega$ of the origin in $\mathbb{R}^n$, and fix a constant $\varepsilon>0$. Then there exists a constant $C>1$ depending only on $g_0, f$ and $\Gamma$, and a constant $0<R<1$ depending additionally on $\varepsilon$, such that for each $m\in\mathbb{R}$, $$\begin{aligned} \label{129} w(r) \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =(\beta+\varepsilon)\ln\bigg(\frac{r_+ - r}{r_+ - r_-}\bigg) +m \end{aligned}$$ satisfies $$\begin{aligned} \label{76} \begin{cases} f(\lambda(-g_w^{-1}A_{g_w})) \geq \frac{f(-\mu_\Gamma^+ + C^{-1}\varepsilon,\,1,\, \dots\,,\,1)}{Ce^{2m}(r_+-r_-)^2}>0, \quad \lambda(-g_w^{-1}A_{g_w})\in\Gamma & \text{on }A_{r_-, r_+} \\ w(x) = m & \text{for }x\in\mathbb{S}_{r_-} \\ w(x)\rightarrow -\infty & \text{as }d(x,\mathbb{S}_{r_+})\rightarrow 0 \end{cases} \end{aligned}$$ whenever $1<\frac{r_+}{r_-}<1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<R$.* **Remark 24**. Our choice of $w$ in [\[129\]](#129){reference-type="eqref" reference="129"} is motivated by the work of Chang, Han & Yang [@CHY05] on radial solutions to the $\sigma_k$-Yamabe equation on annular domains in $\mathbb{R}^n$. Indeed, when $\varepsilon=0$ and $\mu_\Gamma^+ = \frac{n-k}{k}$, [\[51\]](#51){reference-type="eqref" reference="51"} corresponds to the leading order term in the solution to the $\sigma_k$-Yamabe equation in $\Gamma_k^-$ on annular domains in $\mathbb{R}^n$ for $k<\frac{n}{2}$. **Remark 25**. We reiterate that Proposition [Proposition 23](#54){reference-type="ref" reference="54"} relies crucially on the assumption $\mu_\Gamma^+>1$, and that a similar construction is not possible when $\mu_\Gamma ^+ \leq 1$ -- see Remark [Remark 12](#182){reference-type="ref" reference="182"} in the introduction. Assuming the validity of Proposition [Proposition 23](#54){reference-type="ref" reference="54"} for now, we first give the proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"} -- the reader may wish to refer to Figure [\[145\]](#145){reference-type="ref" reference="145"} in the following argument: *Proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"}.* We attach a collar neighbourhood $N$ to $\partial M$ such that $g_0$ extends smoothly to $M\cup N$; we denote this extension also by $g_0$. Let $$D = \inf_{x\in \partial M} \operatorname{d}_{g_0}(x,\partial(M\cup N))$$ denote the thickness of $N$. Fix $\varepsilon>0$ and let $m = \inf_{\partial M}\xi$, and cover a neighbourhood of $\partial M$ in $M$ by a finite collection of annuli $\{A_{r_-^i, r_+^i}(x_i)\}_{1\leq i \leq K}$ centred at $x_i$ such that the collection $\{A_{r_-^i, \frac{1}{2}(r_-^i + r_+^i)}(x_i)\}$ still covers a neighbourhood of $\partial M$ in $M$, and such that for each $i$: 1. $x_i\in N$, 2. $r_-^i + r_+^i< D$, 3. $r_-^i = \operatorname{d}_{g_0}(x_i, \partial M)$, 4. The closed ball $\overline{B_{r_+^i}(x_i)}$ is contained in a single normal coordinate chart $(U_i, \zeta_i)$ mapping $x_i$ to the origin, 5. $$\begin{aligned} \frac{r_+^i}{r_-^i} \leq 1+\frac{\varepsilon}{2(\beta+2)}, \end{aligned}$$ 6. $r_i^+ < R$ is sufficiently small so that $\frac{f(-\mu_\Gamma^+ + C^{-1}\varepsilon,\,1,\, \dots\,,\,1)}{Ce^{2m}(r_+^i - r_-^i)^2} \geq \sup_M \psi$ (here $C$ and $R$ are as in the statement of Proposition [Proposition 23](#54){reference-type="ref" reference="54"}, where we are implicitly identifying the annulus $A_{r_-^i, r_+^i}^i(x_i)$ with its image under $\zeta_i$, which is possible by Property 4). In what follows, we continue to implicitly make the identification between $A_{r_-^i, r_+^i}^i(x_i)$ and its image under $\zeta_i$. Let $w_i$ denote the solution obtained in Proposition [Proposition 23](#54){reference-type="ref" reference="54"} on $A_{r_-^i, r_+^i}(x_i)$ with $\varepsilon>0$ and $m=\inf_{\partial M}\xi$ as fixed above. Since $w_i$ is radially decreasing and $w_i(x) =\inf_{\partial M}\xi$ for $x\in \mathbb{S}_{r_-^i}(x_i)$, we have $w_i \leq \inf_{\partial M}\xi$ on $A_{r_-^i, r_+^i}^i(x_i)\cap\partial M$. On the other hand, $w_i = -\infty < u$ on $\mathbb{S}_{r_+^i}(x_i)$. Therefore, the comparison principle (see Proposition [Proposition 26](#174){reference-type="ref" reference="174"} below) yields $u\geq w_i$ on $A_{r_-^i, r_+^i}(x_i)\cap M$ for each $i$. This yields a finite lower bound for $u$ on $A_{r_-^i, \frac{1}{2}(r_-^i + r_+^i)}(x_i)$. Since we assume the collection $\{A_{r_-^i, \frac{1}{2}(r_-^i + r_+^i)}(x_i)\}$ still covers a neighbourhood of $\partial M$ in $M$, we may piece together the estimates for $u$ on each annulus $A_{r_-^i, \frac{1}{2}(r_-^i + r_+^i)}(x_i)$ to obtain the desired estimate for $u$ on a uniform neighbourhood of $\partial M$ in $M$. ◻ In the above proof we made use of the following comparison principle: **Proposition 26** (Comparison principle). *Let $\alpha>0$ be a positive constant and $(M,g)$ a compact Riemannian manifold with non-empty boundary $\partial M$. Suppose $u,v \in C^0(M)$ with at least one of $u$ or $v$ belonging to $C^2(M\backslash \partial M)$. If $f(-g_u^{-1}A_{g_u}) \geq f(-g_v^{-1}A_{g_v})\geq \alpha > 0$ in the viscosity sense on $M\backslash \partial M$ and $u\leq v$ on $\partial M$, then $u\leq v$ in $M$.* In the proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"}, we only needed Proposition [Proposition 26](#174){reference-type="ref" reference="174"} in the case that both $u,v \in C^2(M\backslash \partial M)$. In this case, the proof of Proposition [Proposition 26](#174){reference-type="ref" reference="174"} is standard in light of the fact that if $f(-g_v^{-1}A_{g_v})>0$, $c$ is a positive constant and $w = v + c$, then $f(-g_w^{-1}A_{g_w}) < f(-g_v^{-1}A_{g_v})$. The case when $u\in C^0(M)$ in Proposition [Proposition 26](#174){reference-type="ref" reference="174"} will be needed later in the paper. When $u\in C^2(M\backslash \partial M)$, Proposition [Proposition 26](#174){reference-type="ref" reference="174"} follows from [@CLN13 Theorem 2.1], since the proof on page 130 therein applies also on Riemannian manifolds with boundary. When $v\in C^2(M\backslash \partial M)$, Proposition [Proposition 26](#174){reference-type="ref" reference="174"} again follows from [@CLN13 Theorem 2.1], therein considering $\widetilde{F}(x,s,p,M) \mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =-F(x,-s,-p,-M)$ in place of $F$. We now give the proof of Proposition [Proposition 23](#54){reference-type="ref" reference="54"}: *Proof of Proposition [Proposition 23](#54){reference-type="ref" reference="54"}.* It will be more convenient to write our conformal metrics in the form $g^v = v^{-2}g_0$, so that $g_w = g^v$ for $e^{2w}=v^{-2}$. Then the $(0,2)$-Schouten tensor of $g^v$ is given by $$\begin{aligned} (A_{g^v})_{ij}= v^{-1}(\nabla_{g_0}^2 v)_{ij} - \frac{1}{2}v^{-2}|\nabla_{g_0} v|_{g_0}^2 (g_0)_{ij} + (A_{g_0})_{ij}. \end{aligned}$$ In a fixed normal coordinate system based at the origin, it follows that if $v=v(r)$ then $$\begin{aligned} \label{50} ((g^v)^{-1}A_{g^v})^p_j & =v^2\bigg(\lambda\delta_j^p + \chi\frac{x^px_j}{r^2}\bigg) + O(r^2)v|v_{rr}| + O(r)\big(v^2 + v|v_r|\big) \quad \text{as }r\rightarrow 0,\end{aligned}$$ where $$\label{16} \lambda = \frac{v_r}{rv}\bigg(1-\frac{rv_r}{2v}\bigg) \quad\text{and}\quad \chi = \frac{v_{rr}}{v} - \frac{v_r}{vr};$$ we refer the reader to Appendix [6](#appb){reference-type="ref" reference="appb"} for the derivation of [\[50\]](#50){reference-type="eqref" reference="50"}. Therefore $$\begin{aligned} \label{130} (-(g^v)^{-1}A_{g^v})^p_j \geq -v^2\bigg(\lambda\delta_j^p + \chi\frac{x^px_j}{r^2}\bigg) - |\Psi|\delta_j^p\end{aligned}$$ in the sense of matrices, where $|\Psi| = O(r^2)v|v_{rr}| + O(r)\big(v^2 + v|v_r|\big)$ as $r\rightarrow 0$. **Step 1:** In this first step we compute and estimate the quantities on the RHS of [\[130\]](#130){reference-type="eqref" reference="130"} for our particular choice of $w$ in [\[129\]](#129){reference-type="eqref" reference="129"}, i.e. for $$\begin{aligned} \label{51} v(r) = e^{-\Lambda} (r_+ - r)^{-\beta-\varepsilon},\end{aligned}$$ where we have denoted $\Lambda = m - (\beta+\varepsilon)\ln(r_+ - r_-)$. For shorthand we denote $\varphi(r) = r_+ - r$. Then $$\begin{aligned} \label{414} v_r = e^{-\Lambda}(\beta+\varepsilon) \varphi^{-\beta - \varepsilon-1} \quad \text{and} \quad v_{rr} = e^{-\Lambda}(\beta +\varepsilon)(\beta+\varepsilon+1)\varphi^{-\beta - \varepsilon-2},\end{aligned}$$ from which it follows that $$\begin{aligned} \frac{v_r}{rv} = (\beta+\varepsilon) r^{-1}\varphi^{-1}, \quad \frac{rv_r}{2v} = \frac{\beta+\varepsilon}{2}r\varphi^{-1} \quad \text{and} \quad \frac{v_{rr}}{v} = (\beta+\varepsilon)(\beta+\varepsilon+1)\varphi^{-2}.\end{aligned}$$ Therefore $$\begin{aligned} \label{133} \lambda = \frac{v_r}{rv}\bigg(1-\frac{rv_r}{2v}\bigg) = (\beta+\varepsilon) r^{-1}\varphi^{-1}\bigg(1 - \frac{\beta+\varepsilon}{2}r\varphi^{-1}\bigg)\end{aligned}$$ and $$\begin{aligned} \label{134} \chi = \frac{v_{rr}}{v} - \frac{v_r}{vr} = -(\beta+\varepsilon)r^{-1}\varphi^{-1} \bigg(1 - (\beta+\varepsilon+1)r\varphi^{-1}\bigg).\end{aligned}$$ For $\Psi$ we estimate using [\[414\]](#414){reference-type="eqref" reference="414"} to get $$\begin{aligned} |\Psi| & \leq Cr^2v|v_{rr}| + Cr(v^2 + v|v_r|) \nonumber \\ & \leq Cre^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2}\Big(r + \varphi+ \varphi^2\Big) \nonumber \\ & \leq C_1 r e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2} =\mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% \eta. \end{aligned}$$ **Step 2:** We now use the computations from Step 1 to analyse the eigenvalues of the matrix on the RHS of [\[130\]](#130){reference-type="eqref" reference="130"}, or more precisely the eigenvalues of $-v^2(\lambda\delta_j^p + \chi\frac{x^px_j}{r^2}) - \eta\delta^p_j$, which are given by $$\begin{aligned} -(\chi v^2+\lambda v^2 +\eta, \lambda v^2 + \eta, \dots, \lambda v^2 +\eta ). \end{aligned}$$ We write this vector of eigenvalues more conveniently as $$\begin{aligned} (-\lambda v^2 - \eta)\bigg(\frac{\chi v^2}{\lambda v^2 + \eta}+1, 1, \dots, 1\bigg).\end{aligned}$$ We make the following two claims: *[Claim 1:]{.ul}* There exist constants $c_1>0$ and $0<R_1<1$ depending only on $g_0, f$ and $\Gamma$ such that $$\begin{aligned} \label{164} -\lambda v^2 - \eta >c_1e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2} \quad \text{in }\{r_-<r<r_+\}\end{aligned}$$ whenever $1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<R_1$. *[Claim 2:]{.ul}* There exists a constant $c_2>0$ depending only on $g_0, f$ and $\Gamma$, and a constant $0< R_2<1$ depending additionally on $\varepsilon$, such that $$\begin{aligned} \label{202} \frac{\chi v^2}{\lambda v^2 + \eta}+1 > - \mu_\Gamma^+ + c_2\varepsilon\quad \text{in }\{r_-<r<r_+\}\end{aligned}$$ whenever $1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<R_2$. Once the claims are proved, Proposition [Proposition 23](#54){reference-type="ref" reference="54"} is obtained as follows. First fix $r_+$ and $r_-$ such that $1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<\min\{R_1,R_2\}$. By Claim 2 and the definition of $\mu_{\Gamma}^+$, $$\begin{aligned} f\bigg(\frac{\chi v^2}{\lambda v^2 + \eta}+1, 1, \dots, 1\bigg)> f(-\mu_\Gamma^+ + c_2\varepsilon, 1,\dots,1)>0 \quad \text{in }\{r_-<r<r_+\}.\end{aligned}$$ Then by Claim 1, it follows that $$\begin{aligned} f\bigg((-\lambda v^2 - \eta)\bigg(&\frac{\chi v^2}{\lambda v^2 + \eta}+1, 1, \dots, 1\bigg)\bigg)\nonumber \\ & >c_1e^{-2\Lambda}\varphi^{-2\beta-2\varepsilon- 2}f(-\mu_\Gamma^+ + c_2\varepsilon, 1,\dots,1) \quad \text{in }\{r_-<r<r_+\},\end{aligned}$$ from which [\[76\]](#76){reference-type="eqref" reference="76"} follows. To complete the proof of Proposition [Proposition 23](#54){reference-type="ref" reference="54"}, it therefore remains to prove Claims 1 and 2. **Note:** We will use at various stages the fact that $$\begin{aligned} \label{156} 1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)} \quad \iff \quad 0<\varphi r^{-1} < \frac{\varepsilon}{2(\beta+2)} \quad \text{in }\{r_-<r<r_+\}.\end{aligned}$$ *Proof of Claim 1.* Suppose $1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<1$. We start by computing $$\begin{aligned} \label{200} -\lambda v^2 & = e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon-2}(\beta+\varepsilon)\bigg( \frac{\beta+\varepsilon}{2} - \varphi r^{-1}\bigg). \end{aligned}$$ By [\[156\]](#156){reference-type="eqref" reference="156"} and [\[200\]](#200){reference-type="eqref" reference="200"}, it follows that $$\begin{aligned} \label{157} -\lambda v^2 \geq \frac{1}{C}e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2} \quad \text{in }\{r_-<r<r_+\}. \end{aligned}$$ Recalling also that $$\begin{aligned} \label{201} \eta = C_1re^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon-2}, \end{aligned}$$ we see that [\[157\]](#157){reference-type="eqref" reference="157"} and [\[201\]](#201){reference-type="eqref" reference="201"} imply $$\begin{aligned} \label{77} -\lambda v^2 - \eta \geq (C^{-1}-C_1r)e^{-2\Lambda}\varphi^{-2\beta- 2\varepsilon-2} \quad \text{in }\{r_- < r < r_+\}. \end{aligned}$$ The inequality [\[164\]](#164){reference-type="eqref" reference="164"} then follows from [\[77\]](#77){reference-type="eqref" reference="77"} after taking $r_+$ sufficiently small. This completes the proof of Claim 1. ◻ *Proof of Claim 2.* Suppose $1<\frac{r_+}{r_-}< 1+\frac{\varepsilon}{2(\beta+2)}$ and $r_+<1$. By [\[200\]](#200){reference-type="eqref" reference="200"} and the fact that $\mu_\Gamma^+ = \frac{2+\beta}{\beta}$ we have $$\begin{aligned} \label{423} -\lambda v^2 - \mu_\Gamma^+\lambda v^2 = -\frac{2+2\beta}{\beta}\lambda v^2 = \frac{2+2\beta}{\beta}e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon-2}(\beta+\varepsilon)\bigg( \frac{\beta+\varepsilon}{2} - \varphi r^{-1}\bigg), \end{aligned}$$ and by the formula for $\chi$ in [\[134\]](#134){reference-type="eqref" reference="134"} we have $$\begin{aligned} \label{424} -\chi v^2 = e^{-2\Lambda}(\beta+\varepsilon)\varphi^{-2\beta-2\varepsilon-2}\big(\varphi r^{-1} - (\beta+\varepsilon+1)\big). \end{aligned}$$ It follows from [\[423\]](#423){reference-type="eqref" reference="423"} and [\[424\]](#424){reference-type="eqref" reference="424"} that $$\begin{aligned} \label{159} -\chi v^2 - \lambda v^2 - \mu_\Gamma^+\lambda v^2 = e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2}\frac{\beta+\varepsilon}{\beta}\big(\varepsilon-(\beta+2)r^{-1}\varphi\big). \end{aligned}$$ On the other hand, by [\[156\]](#156){reference-type="eqref" reference="156"} we have $$\begin{aligned} \frac{\beta+\varepsilon}{\beta}\Big(\varepsilon-(\beta+2)r^{-1}\varphi\Big) > \frac{\varepsilon}{2} \quad \text{in }\{r_- < r< r_+\}, \end{aligned}$$ which when substituted into [\[159\]](#159){reference-type="eqref" reference="159"} yields $$\begin{aligned} \label{160} -\chi v^2 - \lambda v^2 - \mu_\Gamma^+ \lambda v^2 > \frac{\varepsilon}{2}e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2} \quad \text{in }\{r_- < r< r_+\}. \end{aligned}$$ Recalling [\[201\]](#201){reference-type="eqref" reference="201"}, the estimate [\[160\]](#160){reference-type="eqref" reference="160"} therefore implies $$\begin{aligned} \label{44} -\chi v^2 -\lambda v^2 -\mu_\Gamma^+&\lambda v^2 - \eta - \mu_\Gamma^+ \eta \nonumber \\ & \geq \bigg(\frac{\varepsilon}{2}- Cr\bigg)e^{-2\Lambda}\varphi^{-2\beta-2\varepsilon- 2} \quad \text{in }\{r_- < r< r_+\}. \end{aligned}$$ After taking $r_+$ smaller if necessary (but in a way that only depends on $\varepsilon$ and the constant $C$ in [\[44\]](#44){reference-type="eqref" reference="44"}), we therefore have $$\begin{aligned} -\chi v^2 -\lambda v^2 -\mu_\Gamma^+\lambda v^2 - \eta - \mu_\Gamma^+ \eta \geq \frac{\varepsilon}{4}e^{-2\Lambda}\varphi^{-2\beta - 2\varepsilon- 2}\quad \text{in }\{r_- < r< r_+\}, \end{aligned}$$ or equivalently $$\begin{aligned} \label{203} \frac{\chi v^2}{\lambda v^2 + \eta}+1 \geq -\mu_\Gamma^++ \frac{\frac{\varepsilon}{4}e^{-2\Lambda}\varphi^{-2\beta-2\varepsilon-2}}{-\lambda v^2 - \eta}. \end{aligned}$$ On the other hand, by [\[200\]](#200){reference-type="eqref" reference="200"} we have $$\begin{aligned} 0<-\lambda v^2 - \eta \leq - \lambda v^2 \leq Ce^{-2\Lambda}\varphi^{-2\beta-2\varepsilon-2}\quad\text{in }\{r_- < r< r_+\}.\end{aligned}$$ Thus, if $r_+$ is chosen sufficiently small (but depending only on $g_0, f, \Gamma$ and $\varepsilon$), we see $$\begin{aligned} \frac{\chi v^2}{\lambda v^2 + \eta}+1 \geq -\mu_\Gamma^+ + c\varepsilon\quad \text{in }\{r_- < r< r_+\},\end{aligned}$$ as required. This completes the proof of Claim 2. ◻ As explained above, with Claims 1 and 2 established, the proof of Proposition [Proposition 23](#54){reference-type="ref" reference="54"} is complete. ◻ ## Gradient estimate {#148} In this section we prove the global gradient estimate: **Proposition 27**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[418\]](#418){reference-type="eqref" reference="418"}, and let $\tau\leq 1$. Let $\psi\in C^\infty(M)$ be positive and $\xi\in C^\infty(\partial M)$. Then there exists a constant $C$ which is independent of $\tau$ but dependent on $g_0, f, \Gamma$ and upper bounds for $\|\psi\|_{C^1(M)}, \|\xi\|_{C^2(\partial M)}$ and $\|u\|_{C^0(M)}$ such that any $C^3$ solution to [\[12\]](#12){reference-type="eqref" reference="12"} satisfies $|\nabla_{g_0} u|_{g_0}\leq C$ on $M$.* *Proof.* By our interior local gradient estimate in Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, we only need to prove the gradient estimate near the boundary, say in $B_{1/2}(y_0)\cap M$ where $y_0\in \partial M$ is arbitrary. Consider $H=\rho|\nabla_{g_0} u|_{g_0}^2$, where $\rho$ is a smooth cutoff function satisfying $\rho=1$ on $B_{1/2}(y_0)$, $\rho=0$ outside $B_1(y_0)$, $|\nabla_{g_0} \rho|_{g_0}\leq C\rho^{1/2}$ and $|\nabla_{g_0}^2\rho|_{g_0}\leq C$. Suppose that $H$ attains its maximum at $x_0\in M$. If $x_0\not\in B_1(y_0)\cap M$, then $\nabla_{g_0} u=0$ in $B_{1/2}(y_0)\cap M$ and we are done. If $x_0\in B_1(y_0)\cap (M\backslash \partial M)$, then our proof of Theorem [Theorem 8](#40){reference-type="ref" reference="40"} applies and we again obtain the desired estimate. It remains to consider the case that $x_0\in B_1(y_0)\cap \partial M$. We first observe that, since the tangential derivatives of $u$ on $\partial M$ are bounded by $\|\xi\|_{C^1(\partial M)}$, we only need to bound the normal derivative $\nabla_{\nu}u(x_0)$, where $\nu$ denotes the inward pointing unit normal to $\partial M$ at $x_0$. We first consider the lower bound for $\nabla_{\nu}u(x_0)$. With the same setup and notation as in the proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"}, except now with $m=u(x_0)$ and $\varepsilon>0$ to be fixed later, consider the set $\mathcal{S}$ of annuli $A_{r_-, r_+}(y)$ satisfying $\mathbb{S}_{r_-}(y)\cap\partial M = \{x_0\}$ and conditions 1)--6) in the proof of Proposition [Proposition 22](#41){reference-type="ref" reference="41"}. By taking $\varepsilon$ sufficiently large and $r_+$ sufficiently small, we may choose an annulus $A_{r_-, r_+}(y)$ in $\mathcal{S}$ such that the function $w$ on $A_{r_-, r_+}(y)$, as defined in [\[129\]](#129){reference-type="eqref" reference="129"}, satisfies $w \leq u$ on $A_{r_-,r_+}(y)\cap \partial M$. By the comparison principle stated in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}, it follows that $w\leq u$ on $A_{r_-,r_+}(y) \cap M$. Thus for $x\in A_{r_-,r_+}(y) \cap M$ we have $$\begin{aligned} \frac{u(x) - u(x_0)}{\operatorname{d}(x,x_0)} = \frac{u(x) - w(x_0)}{\operatorname{d}(x,x_0)} \geq \frac{w(x) - w(x_0)}{\operatorname{d}(x,x_0)}, \end{aligned}$$ which implies $\nabla_\nu u(x_0) \geq \nabla_\nu w(x_0)$. For the upper bound for $\nabla_\nu u(x_0)$, we construct a barrier function similar to that of Guan [@Guan08]. First observe that since $\Gamma\subset\Gamma_1^+$, we have $$\begin{aligned} 0 < \sigma_1(-g_0^{-1}A_{g_u}) & = \Delta_{g_0} u + \frac{n-2}{2}|\nabla_{g_0} u|_{g_0}^2 -\sigma_1(g_0^{-1}A_{g_0}). \end{aligned}$$ Now let $d(x) = \operatorname{d}(x,\partial M)$ and recall $M_{\delta} = \{x\in M: d(x)<\delta\}$. It is well-known that for sufficiently small $\delta>0$, $d$ is smooth in $M_\delta$ with $|\nabla_{g_0} d|_{g_0} = 1$. To obtain an upper bound for $\nabla_\nu u(x_0)$, it suffices to find a function $\bar{u}\in C^3(M_\delta)$ satisfying $$\label{10'} \begin{cases} \sigma_1(-g_0^{-1}A_{g_{\bar{u}}}) \leq 0 & \text{in }M_\delta\\ \bar{u} = u & \text{on }\partial M\\ \bar{u}\geq u & \text{on }\partial M_\delta \backslash \partial M. \end{cases}$$ Indeed, once such a function $\bar{u}$ is obtained, the maximum principle implies $\bar{u}\geq u$ on $M_\delta$, and it follows that for any $x\in M_\delta$, we have $$\frac{u(x) - u(x_0)}{\operatorname{d}(x,x_0)} = \frac{u(x) - \bar{u}(x_0)}{\operatorname{d}(x,x_0)} \leq \frac{\bar{u}(x) - \bar{u}(x_0)}{\operatorname{d}(x,x_0)},$$ which implies $\nabla_\nu u(x_0) \leq \nabla_\nu \bar{u}(x_0)$. To construct $\bar{u}\in C^3(M_\delta)$ satisfying [\[10\'\]](#10'){reference-type="eqref" reference="10'"}, we first extend $\xi$ to a smooth function $\overline{\xi}$ on $M_\delta$ by defining $\overline{\xi}$ to be constant along geodesics normal to $\partial M$; such a construction is always possible for sufficiently small $\delta>0$. We then define $$\bar{u}(x) = \overline{\xi} + \frac{1}{n-2} \ln \frac{d(x)+\delta^2}{\delta^2}.$$ We first observe that $\bar{u}|_{\partial M} = \overline{\xi}|_{\partial M} = \xi = u|_{\partial M}$. Next we calculate $\sigma_1(-g_0^{-1}A_{g_{\bar{u}}})$. In what follows, we denote by $\nabla d$ the differential of $d$ (whereas $\nabla_{g_0}d$ will continue to denote the gradient of $d$ with respect to $g_0$). Routine computations yield $$\begin{aligned} \nabla_{g_0}\bar{u}(x) = \nabla_{g_0}\overline{\xi}(x) + \frac{1}{n-2}\frac{\nabla_{g_0}d(x)}{d(x) +\delta^2} \end{aligned}$$ and $$\begin{aligned} \nabla_{g_0}^2 \bar{u}(x) = \nabla_{g_0}^2 \overline{\xi}(x) + \frac{1}{n-2}\bigg(\frac{ \nabla_{g_0}^2 d(x) }{d(x) + \delta^2} - \frac{\nabla d(x)\otimes \nabla d(x)}{(d(x)+\delta^2)^2}\bigg), \end{aligned}$$ from which it follows that $$\begin{aligned} \label{74} \sigma_1(-g_0^{-1}A_{g_{\bar{u}}}) & = \Delta_{g_0} \bar{u} + \frac{n-2}{2}|\nabla_{g_0}\bar{u}|_{g_0}^2 -\sigma_1(g_0^{-1}A_{g_0}) \nonumber \\ & \leq C - \frac{1}{2(n-2)}\frac{1}{(d(x) + \delta^2)^2} + \frac{C}{d(x) + \delta^2}, \end{aligned}$$ where we have used the fact that $|\nabla_{g_0}d|_{g_0} = 1$ and $|\Delta_{g_0}d|\leq C$ in $M_{\delta}$ for $\delta$ sufficiently small, and the fact that $\|\overline{\xi}\|_{C^2(M_\delta)}$ is bounded by a constant depending only on $g_0$ and $\|\xi\|_{C^2(\partial M)}$. We then see that the negative term on the last line of [\[74\]](#74){reference-type="eqref" reference="74"} dominates the remaining terms for $\delta>0$ sufficiently small. Therefore, for $\delta>0$ sufficiently small, we have $\sigma_1(-g_0^{-1}A_{g_{\bar{u}}})\leq 0$ in $M_\delta$. Finally, we observe that on $\partial M_\delta\backslash \partial M$ we have $$\begin{aligned} \bar{u} = \overline{\xi} + \frac{1}{n-2}\ln \bigg(\frac{\delta+\delta^2}{\delta^2}\bigg) \geq \overline{\xi} + \frac{1}{n-2}\ln (1/\delta). \end{aligned}$$ Choosing $\delta$ smaller if necessary so that $\overline{\xi} + \frac{1}{n-2}\ln (1/\delta) \geq \max_{M} u$ on $\partial M_\delta\backslash\partial M$, the construction of $\bar{u}$ is complete. This completes the proof of Proposition [Proposition 27](#154){reference-type="ref" reference="154"}. ◻ ## Hessian estimate {#149} In this section we give the global Hessian estimate assuming $\tau<1$: **Proposition 28**. *Suppose $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and let $\tau<1$. Let $\psi\in C^\infty(M)$ be positive and $\xi\in C^\infty(\partial M)$. Then there exists a constant $C$ depending on $g_0, f, \Gamma,(1-\tau)^{-1}$ and upper bounds for $\|\psi\|_{C^2(M)}, \|\xi\|_{C^2(M)}$ and $\|u\|_{C^1(M)}$ such that any solution to [\[12\]](#12){reference-type="eqref" reference="12"} satisfies $|\nabla_{g_0}^2 u|_{g_0} \leq C$ on $M$.* We point out that we do not require $\mu_\Gamma^+>1$ in Proposition [Proposition 28](#83){reference-type="ref" reference="83"}. *Proof.* If the maximum of $|\nabla^2_{g_0} u|_{g_0}$ occurs in $M\backslash \partial M$, then one can appeal to the proof of the global estimate of Gursky & Viaclovsky [@GV03b] if $f=\sigma_k^{1/k}$, or the proof of the global estimate of Guan [@Guan08] for general $(f,\Gamma)$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}. So we suppose that the maximum occurs at a point $x_0\in\partial M$. Let $e_n$ denote the interior unit normal vector field on $\partial M$, and fix an orthonormal frame $\{e_1,\dots,e_{n-1}\}$ for the tangent bundle of $\partial M$ near $x_0$. By parallel transporting along geodesics normal to $\partial M$, we may extend this to an orthonormal frame $\{e_1,\dots,e_n\}$ for the tangent bundle of $M$ near $x_0$. Since $(\nabla_{g_0}^2 u)_{ij}(x_0) = (\nabla_{g_0}^2\xi)_{ij}(x_0)$ for $i,j\not=n$, we only need to estimate $(\nabla_{g_0}^2 u)_{ij}(x_0)$ when at least one of $i$ or $j$ are equal to $n$. The proof is almost identical to that of Guan in [@Guan08], but for the convenience of the reader we summarise the argument here. In what follows, all computations are carried out in a neighbourhood of $x_0$ on which the frame $\{e_1,\dots,e_n\}$ is defined. Still with the convention $g_u = e^{2u}g_0$, it will be convenient to write the equation in [\[12\]](#12){reference-type="eqref" reference="12"} in the equivalent form $$\begin{aligned} \label{417} f(\lambda(-g_0^{-1}A_{g_u}^\tau)) = \psi e^{2u}, \quad \lambda(-g_u^{-1}A_{g_u}^\tau)\in\Gamma \quad \text{on }M\backslash \partial M, \end{aligned}$$ where $$\begin{aligned} A_{g_u}^\tau & = \tau A_{g_u} + (1-\tau)\sigma_1(-g_u^{-1}A_{g_u})g_u \nonumber \\ & = -\tau\nabla_{g_0}^2 u - (1-\tau)\Delta_{g_0} u\,g_0 - b_{n,\tau}|\nabla_{g_0} u|_{g_0}^2 g_0 + \tau du\otimes du + A_{g_0}^\tau \end{aligned}$$ and $b_{n,\tau} = \frac{1}{2}\big(n-2-(n-3)\tau\big)$. Denoting $F[u] = f(\lambda(-g_0^{-1}A_{g_u}^\tau))$ and $$\begin{aligned} F^{ij} = \frac{\partial f}{\partial A_{ij}}\bigg|_{A=-g_0^{-1}A_{g_u}^\tau}, \end{aligned}$$ the linearisation of $F$ at $u$ in the direction $\eta$ (excluding zeroth order terms) is given by $$\begin{aligned} \label{165} \mathcal{L}\eta & = F^{ij}\bigg(\tau(\nabla_{g_0}^2 \eta)_{ij} +(1-\tau)\Delta_{g_0} \eta \,(g_0)_{ij} + 2b_{n,\tau}\langle \nabla_{g_0} u,\nabla_{g_0} \eta\rangle_{g_0}(g_0)_{ij} - 2\tau \partial_i u\,\partial_j\eta\bigg) \nonumber \\ & = F^{ij}\bigg(\tau(\nabla_{g_0}^2 \eta)_{ij} - 2\tau\partial_i u\,\partial_j\eta \bigg) + \bigg((1-\tau)\Delta_{g_0} \eta + 2b_{n,\tau}\langle \nabla_{g_0} u,\nabla_{g_0} \eta\rangle_{g_0}\bigg)\sum_iF^{ii}. \end{aligned}$$ Now suppose $\delta>0$ is sufficiently small so that $d(x) = \operatorname{d}(x,\partial M)$ is smooth in $M_\delta= \{x\in M: d(x)<\delta\}$. For a positive constant $N$ to be determined later, define $$\begin{aligned} \label{82} v = \frac{N}{2}d^2 - d. \end{aligned}$$ A routine compute shows that for $\delta>0$ sufficiently small, $$\begin{aligned} \label{80} |\mathcal{L}d| \leq C_0\sum_i F^{ii} \quad\text{in }M_\delta, \end{aligned}$$ where $C_0$ is a constant independent of $\tau$ but depending on $g_0$ and an upper bound for $\|u\|_{C^1(M)}$. It follows that $$\begin{aligned} \label{81} \mathcal{L}d^2 & = 2d \mathcal{L}d + 2(1-\tau)|\nabla_{g_0} d|_{g_0}^2\sum_i F^{ii} + 2F^{ij}\partial_i d\,\partial_j d \nonumber \\ & \geq 2d \mathcal{L}d + 2(1-\tau)\sum_i F^{ii} \nonumber \\ & \geq 2\big((1-\tau)-C_0d\big)\sum_i F^{ii} \quad \text{in }M_\delta. \end{aligned}$$ Choosing $N\geq \frac{4(1+C_0)}{1-\tau}$ and subsequently $\delta \leq \min\{N^{-1}, C_0^{-1}\}$, one sees from [\[80\]](#80){reference-type="eqref" reference="80"} and [\[81\]](#81){reference-type="eqref" reference="81"} that the function $v$ defined in [\[82\]](#82){reference-type="eqref" reference="82"} satisfies $$\begin{aligned} \label{152} \mathcal{L}v \geq \sum_i F^{ii} \quad \text{and} \quad v \leq -\frac{d}{2}\quad\text{in }M_\delta. \end{aligned}$$ With [\[152\]](#152){reference-type="eqref" reference="152"} in hand, one can then show: **Lemma 29**. *Fix $\delta>0$ sufficiently small as in the foregoing argument. If $h\in C^2(\overline{M_\delta})$ satisfies $h\leq 0$ on $\partial M$, $h(z_0) = 0$ for some $z_0\in \partial M$ and $$\begin{aligned} \label{85} -\mathcal{L}h \leq C_1\sum_i F^{ii} \quad \text{in }M_\delta \end{aligned}$$ for some constant $C_1$, then $$\begin{aligned} \label{86} (\nabla_{g_0} h)_n(z_0) \leq C, \end{aligned}$$ where $C$ is a constant depending on $g_0$, $C_1$, $(1-\tau)^{-1}$ and upper bounds for $\|h\|_{C^0(\overline{M_\delta})}$ and $\|u\|_{C^1(M)}$.* *Proof.* It is clear from the definition of $v$ that we can choose $A>0$ large (depending on $\|h\|_{C^0(\overline{M_\delta})})$ such that $-Av - h \geq 0$ on $\partial M_\delta$. On the other hand, using [\[152\]](#152){reference-type="eqref" reference="152"} and [\[85\]](#85){reference-type="eqref" reference="85"}, we have $$\begin{aligned} \mathcal{L}(-Av - h)\leq (-A+C_1)\sum_i F^{ii} \quad \text{in }M_\delta, \end{aligned}$$ and hence $\mathcal{L}(-Av - h) \leq 0$ in $M_\delta$ for $A$ sufficiently large. Thus, for $A$ sufficiently large the maximum principle yields $-Av - h \geq 0$ in $M_\delta$, and since $(-Av - h)(z_0) = 0$, it follows that $(\nabla_{g_0}(-Av - h))_n(z_0) \geq 0$, i.e. $(\nabla_{g_0} h)_n(z_0) \leq -A(\nabla_{g_0} v)_n(z_0)$. The estimate [\[86\]](#86){reference-type="eqref" reference="86"} then follows. ◻ We now continue the proof of Proposition [Proposition 28](#83){reference-type="ref" reference="83"}. Suppose $i\in \{1,\dots,n-1\}$ and define $h=\pm(\nabla_{g_0}(u-\overline{\xi}))_i$, where (as in the proof of Proposition [Proposition 27](#154){reference-type="ref" reference="154"}) $\overline{\xi}$ denotes the extension of $\xi$ to $M_\delta$ such that $\overline{\xi}$ is constant along geodesics normal to $\partial M$. By differentiating the equation [\[417\]](#417){reference-type="eqref" reference="417"}, one can show directly that $|\mathcal{L}(\nabla_{g_0} u)_i| \leq C\sum_i F^{ii}$, and by [\[183\]](#183){reference-type="eqref" reference="183"} we also have $|\mathcal{L}\overline{\xi}| \leq C \leq C\sum_i F^{ii}$. Therefore $h$ satisfies the assumptions of Lemma [Lemma 29](#84){reference-type="ref" reference="84"}, and it follows from Lemma [Lemma 29](#84){reference-type="ref" reference="84"} that $$\begin{aligned} |(\nabla_{g_0}^2 u)_{in}(x_0)| \leq C.\end{aligned}$$ It remains to estimate the double normal derivative $(\nabla_{g_0}^2 u)_{nn}(x_0)$. Note that since $\{e_1,\dots,e_{n}\}$ is an orthonormal frame and $(\nabla_{g_0}^2 u)_{ii}(x_0) = (\nabla_{g_0}^2\xi)_{ii}(x_0)$ for $i\in\{1,\dots,n-1\}$, to obtain an upper (resp. lower) bound for $(\nabla_{g_0}^2 u)_{nn}(x_0)$, it is equivalent to obtain an upper (resp. lower) bound for $\Delta_{g_0} u(x_0)$. Now, since $\Gamma\subseteq \Gamma_1^+$, the lower bound $\Delta_{g_0} u \geq -C$ in $M$ is immediate. To obtain the upper bound for $(\nabla_{g_0}^2 u)_{nn}(x_0)$, we may assume $(\nabla_{g_0}^2 u)_{nn}(x_0) \geq 1$, otherwise we are done. We may also assume that with respect to the frame $\{e_1,\dots,e_n\}$, the Hessian of $u$ at $x_0$ is given by $\nabla_{g_0}^2 u(x_0) = \operatorname{diag}((\nabla_{g_0}^2 u)_{11}(x_0),\dots, (\nabla_{g_0}^2 u)_{nn}(x_0))$. Then by the equation [\[417\]](#417){reference-type="eqref" reference="417"}, monotonicity of $f$ and our estimates for $(\nabla_{g_0}^2 u)_{ij}(x_0)$ when $i$ and $j$ are not both equal to $n$, we have $$\begin{aligned} \label{425} \psi(x_0)e^{2u(x_0)} = f(-g_0^{-1}A_{g_u}^\tau(x_0)) \geq f\big((1-\tau)(\nabla_{g_0}^2 u)_{nn}(x_0)g_0 + B\big),\end{aligned}$$ where $B$ is a symmetric matrix bounded in terms of $\|u\|_{C^1(M)}$. The upper bound for $(\nabla_{g_0}^2 u)_{nn}(x_0)$ then follows from [\[425\]](#425){reference-type="eqref" reference="425"} and homogeneity of $f$. ◻ ## Proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"} {#150} We now complete the proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"}: *Proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"}.* We first prove the existence of a smooth solution to [\[12\]](#12){reference-type="eqref" reference="12"} when $\tau<1$. Fix $\varepsilon>0$ and let $S_\varepsilon= \{\tau\in[0,1-\varepsilon]\,:\, \eqref{12} \text{ admits a solution in }C^{2,\alpha}(M)\}$. Since [\[12\]](#12){reference-type="eqref" reference="12"} admits a unique smooth solution when $\tau=0$, $S_\varepsilon$ is non-empty. A computation as in [\[165\]](#165){reference-type="eqref" reference="165"} (but now including zeroth order terms) shows that the linearised operator is invertible as a mapping from $C^{2,\alpha}(M)$ to $C^\alpha(M)$, from which openness of $S_\varepsilon$ follows. By Propositions [Proposition 20](#75){reference-type="ref" reference="75"} and [Proposition 21](#151){reference-type="ref" reference="151"}, solutions to [\[12\]](#12){reference-type="eqref" reference="12"} admit a global $C^0$ estimate. By Proposition [Proposition 27](#154){reference-type="ref" reference="154"}, solutions to [\[12\]](#12){reference-type="eqref" reference="12"} therefore admit a global $C^1$ estimate. Note that, at this point, the estimates are independent of $\varepsilon$. By Proposition [Proposition 28](#83){reference-type="ref" reference="83"}, one then obtains the global $C^2$ estimate on solutions to [\[12\]](#12){reference-type="eqref" reference="12"}, which do now depend on $\varepsilon$. With the $C^2$ estimate established, [\[12\]](#12){reference-type="eqref" reference="12"} becomes uniformly elliptic, and the regularity theory of Evans-Kyrlov [@Ev82; @Kry82; @Kry83] then implies a $C^{2,\alpha}$ estimate. Thus $S_\varepsilon$ is also closed, and so $S_\varepsilon= [0,1-\varepsilon]$. Since $\varepsilon>0$ was arbitrary, existence of a $C^{2,\alpha}$ solution to [\[12\]](#12){reference-type="eqref" reference="12"} for any $\tau<1$ then follows. Higher regularity then follows from classical Schauder theory, and uniqueness is a consequence of the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}. Now, since the solutions obtained to [\[12\]](#12){reference-type="eqref" reference="12"} are uniformly bounded in $C^1(M)$ as $\tau\rightarrow 1$, along a sequence $\tau_i\rightarrow 1$ these solutions converge uniformly to some $u\in C^{0,1}(M)$. The proof that $u$ is a viscosity solution to [\[12\]](#12){reference-type="eqref" reference="12"} when $\tau=1$ is exactly the same as in the proof of Theorem 1.3 in [@LN20b], and is omitted here. ◻ # Proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}: the fully nonlinear Loewner-Nirenberg problem {#135} In this section we prove Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}. Our proof proceeds according to the following steps: 1. In Section [4.1](#137){reference-type="ref" reference="137"} we construct a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$. The solution is obtained as the limit of solutions with constant finite boundary data $m\in\mathbb{R}$ (which we know to exist by Theorem [Theorem 6](#55){reference-type="ref" reference="55"}) as $m\rightarrow\infty$. 2. In Section [4.2](#138){reference-type="ref" reference="138"} we prove that there exists a smooth solution $u$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$ satisfying the asymptotics stated in [\[114\]](#114){reference-type="eqref" reference="114"}. 3. In Section [4.3](#139){reference-type="ref" reference="139"} we prove that *any* smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} must satisfy [\[114\]](#114){reference-type="eqref" reference="114"} when $\tau<1$. When combined with the maximum principle, this will imply that the solution $u$ obtained to [\[-113\]](#-113){reference-type="eqref" reference="-113"} is unique when $\tau<1$. 4. In Section [4.4](#166){reference-type="ref" reference="166"} we complete the proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}. ## Existence of a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$ {#137} Fix $\tau<1$ and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. By Theorem [Theorem 6](#55){reference-type="ref" reference="55"}, we know that for each $m\in\mathbb{R}$, there exists a unique smooth solution $u_m$ to $$\label{13''} \begin{cases} f^\tau(\lambda(-g_{u_m}^{-1}A_{g_{u_m}})) = 1, \quad \lambda(-g_{u_m}^{-1}A_{g_{u_m}})\in\Gamma^\tau & \text{on }M\backslash\partial M \\ u_m = m & \text{on }\partial M. \end{cases}$$ In this section we show that in the limit $m\rightarrow\infty$, one obtains a smooth solution $u$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"}. **Proposition 30**. *Fix $\tau<1$ and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Let $u_m$ denote the unique smooth solution to [\[13\'\'\]](#13''){reference-type="eqref" reference="13''"}. Then a subsequence of $\{u_m\}_m$ converges uniformly as $m\rightarrow \infty$ to a solution $u\in C^\infty(M\backslash\partial M)$ of [\[-113\]](#-113){reference-type="eqref" reference="-113"}. Moreover, given any constant $\alpha>0$, there exists a constant $\delta>0$ independent of $\tau$ but dependent on $g_0, \alpha, f$ and $\Gamma$ such that $u \geq \alpha$ in $M_\delta\backslash\partial M$.* *Proof.* Since the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"} implies $u_{m+1} \geq u_m$, to prove the existence of a limit $u\in C^\infty(M\backslash \partial M)$ solving [\[-113\]](#-113){reference-type="eqref" reference="-113"}, it suffices to show that for each compact set $K\subset M\backslash \partial M$ there exists a constant $C$ independent of $m$ such that $\|u_m\|_{C^2(K)} \leq C$; higher order estimates then follow from the work of Evans-Krylov [@Ev82; @Kry82] and classical Schauder theory. The lower bound is trivial (and in fact global), since $u_m \geq u_1$ for all $m$. Next we address the local upper bound -- note that whilst we obtained a global upper bound in Proposition [Proposition 20](#75){reference-type="ref" reference="75"}, the bound therein depends on $m$, which is insufficient for our current purposes. Recalling the normalisation $f(\frac{1}{2}, \dots, \frac{1}{2}) =1$, we have by concavity and homogeneity of $f$ $$\begin{aligned} \label{98} f(\lambda) \leq f\bigg(\frac{\sigma_1(\lambda)}{n}e\bigg) + \nabla f\bigg(\frac{\sigma_1(\lambda)}{n}e\bigg)\cdot\bigg(\lambda - \frac{\sigma_1(\lambda)}{n}e\bigg) = \frac{f(e)}{n}\sigma_1(\lambda) = \frac{2}{n}\sigma_1(\lambda) \quad \text{for }\lambda\in\Gamma, \end{aligned}$$ and thus any solution to the equation in [\[13\'\'\]](#13''){reference-type="eqref" reference="13''"} satisfies $R_{g_{u_m}} \leq -n(n-1)$. On the other hand, by the work of Aviles & McOwen [@AM88], there exists a smooth metric $g_w = e^{2w}g_0$ satisfying $$\begin{aligned} \label{185} \begin{cases} R_{g_{w}} = -n(n-1) & \text{on }M\backslash\partial M \\ w(y)\rightarrow+\infty & \text{as } \operatorname{d}(y,\partial M)\rightarrow 0. \end{cases} \end{aligned}$$ By the comparison principle for the semilinear equation [\[185\]](#185){reference-type="eqref" reference="185"}, $u_m \leq w$ in $M\backslash\partial M$ for each $m$, which yields a finite upper bound for $u_m$ on any compact subset of $M\backslash\partial M$ which is independent of $m$. The local gradient estimate then follows from Theorem [Theorem 8](#40){reference-type="ref" reference="40"}, or alternatively one can appeal to [@Guan08 Theorem 2.1] since we have the two-sided $C^0$ bound at this point. For the local Hessian estimate, we appeal to [@Guan08 Theorem 3.1]. We therefore obtain the full $C^2$ estimate $\|u_m\|_{C^2(K)} \leq C(K)$ on any compact set $K\subset M\backslash \partial M$, as required. It remains to prove the second assertion in the statement of Proposition [Proposition 30](#90){reference-type="ref" reference="90"}. Fix $\alpha>0$ and consider the solution $u_{\alpha+1}$ to [\[13\'\'\]](#13''){reference-type="eqref" reference="13''"} with $m=\alpha+1$. Since $u_{\alpha+1}$ admits a global $C^0$ estimate depending only $g_0, \alpha, f$ and $\Gamma$, there exists a constant $\delta>0$ depending only on $g_0, \alpha, f$ and $\Gamma$ such that $u_{\alpha+1} \geq \alpha$ in $M_\delta$. By the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}, $u\geq u_{\alpha+1}$ in $M\backslash\partial M$, and in particular $u \geq \alpha$ in $M_\delta\backslash\partial M$, as required. ◻ ## Asymptotics {#138} Fix $\tau<1$ and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. In this section we show that there exists a smooth solution $u$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"} satisfying [\[114\]](#114){reference-type="eqref" reference="114"}, that is $$\begin{aligned} \label{114'} \lim_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) = 0. \end{aligned}$$ **Remark 31**. At this point of the argument, we do not know that this constructed solution coincides with the one obtained in Section [4.1](#137){reference-type="ref" reference="137"}, although we will later see in Section [4.3](#139){reference-type="ref" reference="139"} that this is the case. We start by proving an upper bound on the growth of any smooth solution to the equation in [\[-113\]](#-113){reference-type="eqref" reference="-113"}, irrespective of the boundary data or whether $\tau<1$ or $\mu_\Gamma^+>1$: **Proposition 32**. *Let $(M,g_0)$ be a smooth Riemannian manifold with non-empty boundary and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"} and [\[25\'\]](#25'){reference-type="eqref" reference="25'"}. Then there exist constants $\delta>0$ and $C>0$ depending only on $g_0$ such that any continuous metric $g_u=e^{2u}g_0$ satisfying $$\begin{aligned} \label{102} f(\lambda(-g_u^{-1}A_{g_u})) \geq 1, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma \quad \text{in the viscosity sense on }M\backslash \partial M \end{aligned}$$ satisfies $$\begin{aligned} \label{186} u(x) + \ln \operatorname{d}(x,\partial M) \leq C\operatorname{d}(x,\partial M)^{1/2} \quad \text{in }M_\delta\backslash \partial M. \end{aligned}$$ In particular, any continuous metric $g_u=e^{2u}g_0$ satisfying [\[102\]](#102){reference-type="eqref" reference="102"} satisfies $$\begin{aligned} \label{187} \limsup_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) \leq 0. \end{aligned}$$* *Proof.* By [\[98\]](#98){reference-type="eqref" reference="98"}, the comparison principle for viscosity sub-/supersolutions to uniformly elliptic equations implies that if $g_w = e^{2w}g_0$ satisfies $$\begin{aligned} \label{99} \begin{cases} \sigma_1(-g_w^{-1}A_{g_w}) \leq \frac{n}{2} & \text{in }\Omega\Subset M\backslash \partial M \\ w(x) \rightarrow +\infty & \text{as } \operatorname{d}(x,\partial \Omega)\rightarrow 0, \end{cases} \end{aligned}$$ then $u\leq w$ in $\Omega$. Since $\sigma_1(-g_w^{-1}A_{g_w}) = -\frac{1}{2(n-1)}R_{g_w}$, the transformation law for scalar curvature implies that the equation in [\[99\]](#99){reference-type="eqref" reference="99"} is equivalent to $$\begin{aligned} \label{100} -\frac{S_{g_0}}{n-1} + 2\Delta_{g_0} w + (n-2)|\nabla_{g_0} w|_{g_0}^2 \leq n e^{2w}. \end{aligned}$$ We follow an argument of Gursky, Streets & Warren [@GSW11], in turn based on the original argument of Loewner & Nirenberg [@LN74], to construct such local supersolutions near $\partial M$. For a point $x_0$ distance $d$ from $\partial M$, consider a point $z_0$ a distance $R>d$ from $\partial M$, which lies along the shortest path geodesic from $x_0$ to $\partial M$. We may assume $R$ is small enough so that $\Delta_{g_0} d^2(z_0, \cdot) \geq 1$ on $B_R(z_0)$, and so that there exists a function $h$ defined on $[0,R^2]$ satisfying $$\begin{aligned} \label{167} (n-2)(h')^2 + 2h'' \leq 0, \quad h' > \operatorname{max}_M |S_{g_0}| + \widetilde{C}(g_0), \quad h(0) = 0, \end{aligned}$$ where $\widetilde{C}(g_0)$ is a sufficiently large constant to be fixed in the proof. Indeed, once $\widetilde{C}(g_0)$ is fixed, the function $h(t) = \sqrt{t+\varepsilon^2} -\varepsilon$ satisfies [\[167\]](#167){reference-type="eqref" reference="167"} for $\varepsilon$ sufficiently small and $t$ in a sufficiently small interval $[0,R^2]$. Let $r$ denote the distance from $z_0$, and define on $B_R(z_0)$ the radial function $$\begin{aligned} w(r) = -\ln (R^2 - r^2) + h(R^2 - r^2) + \ln \alpha, \end{aligned}$$ where $\alpha>0$ is to be determined. Exactly as in the proof of Lemma 5.2 in [@GSW11], a direct computation shows that, for $R$ sufficiently small and $\widetilde{C}(g_0)$ sufficiently large, the LHS of [\[100\]](#100){reference-type="eqref" reference="100"} satisfies $$\begin{aligned} -\frac{S_{g_0}}{n-1} + 2\Delta_{g_0} w + (n-2)|\nabla_{g_0} w|_{g_0}^2 & \leq \frac{4nR^2}{(R^2 - r^2)^2}e^{2h} = \frac{4nR^2}{\alpha^2}e^{2w}. \end{aligned}$$ Therefore, if we take $\alpha = 2R$, we see $w$ indeed satisfies [\[100\]](#100){reference-type="eqref" reference="100"}. We then obtain $$\begin{aligned} u(x_0) & \leq w(x_0) \nonumber \\ & = -\ln (R^2 - (R-d)^2) + h(R^2 - (R-d)^2) + \ln (2R) \nonumber \\ & = -\ln (d(2R-d)) + h(d(2R-d)) + \ln (2R) \nonumber \\ & = -\ln d - \ln \bigg(1-\frac{d}{2R}\bigg) + h(d(2R-d)). \end{aligned}$$ But $h(d(2R-d)) = \sqrt{d(2R-d) + \varepsilon^2} - \varepsilon\leq \sqrt{d(2R-d)} \leq C\sqrt{d}$ and $\ln(1-\frac{d}{2R}) \geq -\frac{d}{2R} \geq -C\sqrt{d}$ for sufficiently small $d$, and thus [\[186\]](#186){reference-type="eqref" reference="186"} follows. The inequality [\[187\]](#187){reference-type="eqref" reference="187"} is a clear consequence of [\[186\]](#186){reference-type="eqref" reference="186"}. ◻ We are now in a position to prove the existence of a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$ with the desired asymptotic behaviour in [\[114\'\]](#114'){reference-type="eqref" reference="114'"}. **Proposition 33**. *Fix $\tau<1$ and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Then there exists a smooth solution $g_v =e^{2v}g_0$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"} and a constant $C$ independent of $\tau$ but dependent on $g_0, f$ and $\Gamma$ such that the following holds: for each $\varepsilon>0$ sufficiently small, there exists a constant $a\gg0$ independent of $\tau$ but dependent on $g_0, \varepsilon, C, f$ and $\Gamma$ such that $$\begin{aligned} \label{184} v(x) + \ln \operatorname{d}(x,\partial M) \geq \ln\sqrt{1-2\varepsilon} - \ln\big(1+a\operatorname{d}(x,\partial M)\big) \quad \text{in } A^a_\varepsilon\subset M, \end{aligned}$$ where $$\begin{aligned} A^a_\varepsilon= \bigg\{x\in M\backslash\partial M: d(x)+ad(x)^2 \leq \frac{\varepsilon}{C}\bigg\}. \end{aligned}$$ In particular, $$\begin{aligned} \label{103} \lim_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(v(x) + \ln \operatorname{d}(x,\partial M)\big) = 0. \end{aligned}$$* *Proof.* Consider an exhaustion of $M$ by smooth compact manifolds with boundary defined by $M_{(j)} = \{x\in M: \operatorname{d}(x,\partial M) \geq j^{-1}\}$. By Proposition [Proposition 30](#90){reference-type="ref" reference="90"}, for each $j$ there exists a smooth solution $g_{v_{(j)}} = e^{2v_{(j)}}g_0$ to $$\begin{aligned} \begin{cases} f^\tau(-g_{v_{(j)}}^{-1}A_{g_{v_{(j)}}}) = 1, \quad \lambda(-g_{v_{(j)}}^{-1}A_{g_{v_{(j)}}})\in\Gamma^\tau & \text{on }M_{(j)}\backslash\partial M_{(j)} \\ v_{(j)}(x)\rightarrow+\infty & \text{as }\operatorname{d}(x,\partial M_{(j)})\rightarrow 0. \end{cases} \end{aligned}$$ (Note that we put parentheses around the index $j$ to avoid confusion with the solutions $u_m$ to [\[13\'\'\]](#13''){reference-type="eqref" reference="13''"}). Since $v_{(j)}(x)\rightarrow +\infty$ as $\operatorname{d}(x,\partial M_{(j)})\rightarrow 0$, the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"} implies that if $j<m$, then $$\begin{aligned} \label{140} v_{(m)}\big|_{M_{(j)}} < v_{(j)}. \end{aligned}$$ Now, as justified in the proof of Proposition [Proposition 30](#90){reference-type="ref" reference="90"}, a subsequence of $\{v_{(j)}\}_j$ converges locally uniformly to some $v \in C^\infty(M\backslash\partial M)$. We claim that $v$ is our desired function. It is clear that $v$ solves the equation in [\[-113\]](#-113){reference-type="eqref" reference="-113"}. We now establish [\[184\]](#184){reference-type="eqref" reference="184"}, which we split into two steps: in the first step we show $v(x)\rightarrow+\infty$ as $\operatorname{d}(x,\partial M)\rightarrow 0$, and in the second step we prove [\[184\]](#184){reference-type="eqref" reference="184"}. **Step 1:** In this first step we show that $v(x)\rightarrow+\infty$ as $\operatorname{d}(x,\partial M)\rightarrow 0$. To this end, let $d(x) = \operatorname{d}(x,\partial M)$ and define $\varphi= - \ln (B(d+ad^2))$, $g_{\varphi} = e^{2\varphi}g_0$, where $a$ and $B$ are positive constants to be determined. Writing $e^{2\varphi} = \psi^{-2}$, so that $\psi = B(d+ad^2)$, we compute near $\partial M$ $$\begin{aligned} |\nabla_{g_0} \psi|_{g_0}^2 = B^2(1+2ad)^2 \quad \text{and} \quad \nabla_{g_0}^2 \psi = B(1+2ad)\nabla_{g_0}^2 d + 2aB\,\nabla d \otimes \nabla d, \end{aligned}$$ where $\nabla d$ denotes the differential of $d$. It follows that near $\partial M$, $$\begin{aligned} \label{104} -g_\varphi^{-1}A_{g_\varphi} & = g_0^{-1}\bigg(-\psi\nabla_{g_0}^2 \psi + \frac{1}{2}|\nabla_{g_0} \psi|_{g_0}^2 g_0 - \psi^2 A_{g_0}\bigg) \nonumber \\ & = B^2 g_0^{-1}\bigg(\frac{1}{2}g_0 + 2a^2d^2\Big[g_0 - \nabla d\otimes \nabla d - d\nabla_{g_0}^2d\Big] - d(1+3ad)\nabla_{g_0}^2 d \nonumber \\ & \qquad \qquad + 2ad\Big[g_0 - \nabla d\otimes \nabla d\Big] - d^2(1+ad)^2A_{g_0}\bigg). \end{aligned}$$ Taking for instance $a=1$, we then see that for $\delta$ fixed sufficiently small and $B$ fixed sufficiently large, it holds that $$\begin{aligned} \label{169} f^\tau(-g_\varphi^{-1}A_{g_\varphi}) \geq 1 \quad\text{in }M_\delta\backslash\partial M. \end{aligned}$$ To use [\[169\]](#169){reference-type="eqref" reference="169"} to show $v(x)\rightarrow+\infty$ as $\operatorname{d}(x,\partial M)\rightarrow 0$, we follow the proof of [@LN74 Theorem 5]. For $m\gg 1$, denote by $S_m$ the set where $\varphi(x)= -\ln (B(d+d^2)) \geq m$. We may assume (by taking $m$ sufficiently large) that $S_m$ is a tubular neighbourhood of $\partial M$ contained in $M_\delta$. Let $\Sigma_m = \partial S_m \backslash\partial M$ and $D_m = \min_{\Sigma_m} v$, and suppose $J$ is sufficiently large so that $\Sigma_m \subset M_{(j)}$ for all $j\geq J$. Then $\varphi=m$ and $v \geq D_m$ on $\Sigma_m$, and by the monotonicity in [\[140\]](#140){reference-type="eqref" reference="140"} we also have $v_{(j)} \geq D_m$ on $\Sigma_m$ for each $j\geq J$. Therefore $$\begin{aligned} \label{170} v_{(j)} + \operatorname{max}\{0, m-D_m\}\geq m = \varphi\text{ on }\Sigma_m \end{aligned}$$ and $$\begin{aligned} \label{171} v_{(j)} + \operatorname{max}\{0, m-D_m\} = \infty > \varphi\text{ on }\partial M_{(j)}. \end{aligned}$$ In light of [\[169\]](#169){reference-type="eqref" reference="169"}--[\[171\]](#171){reference-type="eqref" reference="171"}, the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"} implies that $v_{(j)} + \operatorname{max}\{0, m-D_m\} \geq \varphi$ on $M_{(j)} \cap S_m$. Sending $j\rightarrow\infty$, it follows that $v+ \operatorname{max}\{0, m-D_m\}\geq \varphi$ in $S_m$, and in particular $v(x)\rightarrow+\infty$ as $\operatorname{d}(x,\partial M)\rightarrow 0$. **Step 2:** In this second step we show that $v$ satisfies [\[184\]](#184){reference-type="eqref" reference="184"}. The method is essentially a quantitative version of Step 1, requiring a more careful choice of parameters $a$ and $B$ in the definition of $\varphi$. We first claim that the two quantities in the square parentheses in [\[104\]](#104){reference-type="eqref" reference="104"} are nonnegative definite for sufficiently small $d$. Indeed, observe that $g_0(x) - \nabla d(x)\otimes \nabla d(x)$ is the induced metric on $\partial M_{d(x)}\backslash\partial M$, and is therefore nonnegative definite. Moreover, $\nabla_{g_0}^2 d$ is a bounded tensor near $\partial M$ whose kernel contains $\nabla d$. Hence $\nabla_{g_0}^2 d$ is bounded from above by $C(g_0 - \nabla d \otimes \nabla d)$ for some constant $C$ depending only on $(M,g_0)$. Therefore $g_0 - \nabla d\otimes \nabla d - d\nabla_{g_0}^2d$ is nonnegative definite for $d$ sufficiently small, as claimed. In light of [\[104\]](#104){reference-type="eqref" reference="104"} and the above claim, we see that for $\delta$ chosen sufficiently small independently of $a$ (but depending on $(M,g_0)$), and $\widehat{C}\geq 1$ a constant such that $|A_{g_0}|_{g_0}, |\nabla_{g_0}^2 d|_{g_0} \leq \widehat{C}$ on $M_\delta$, we have $$\begin{aligned} \label{108} -g_\varphi^{-1}A_{g_\varphi} & \geq B^2 g_0^{-1}\bigg(\frac{1}{2}g_0 - d(1+3ad)\nabla_{g_0}^2 d - d^2(1+ad)^2A_{g_0} \bigg) \nonumber \\ & \geq B^2 g_0^{-1}\bigg(\frac{1}{2} - \widehat{C}d - \widehat{C}(1+3a)d^2 - 2\widehat{C}ad^3 - \widehat{C}a^2 d^4\bigg)g_0 \quad \text{in }M_\delta\backslash\partial M. \end{aligned}$$ Since we will eventually take $a$ large, we may assume $a\geq1$, in which case [\[108\]](#108){reference-type="eqref" reference="108"} implies $$\begin{aligned} \label{111} -g_\varphi^{-1}A_{g_\varphi} & \geq B^2 \bigg(\frac{1}{2} - \widehat{C}\Big[d + 4ad^2 +2ad^3 +a^2 d^4\Big]\bigg)\operatorname{Id} \quad \text{in } M_\delta\backslash\partial M. \end{aligned}$$ Now fix $\varepsilon>0$ small, define $B = \frac{1}{\sqrt{1-2\varepsilon}}$ and denote by ${\widehat{A}}_\varepsilon^{a}$ the set $$\begin{aligned} \widehat{A}_\varepsilon^{a} &= \bigg\{x\in M\backslash\partial M: \varphi(x)= -\ln (B(d+ad^2)) \geq -\ln \bigg(\frac{\varepsilon}{100\widehat{C}}\bigg) \bigg\} \nonumber \\ & = \bigg\{x\in M\backslash\partial M: d+ad^2 \leq \frac{\varepsilon\sqrt{1-2\varepsilon}}{100\widehat{C}}\bigg\}, \end{aligned}$$ where $\widehat{C}$ is the constant in [\[111\]](#111){reference-type="eqref" reference="111"}. It is easily verified that in $\widehat{A}^a_\varepsilon$, we have $\widehat{C}(d + 4ad^2 +2ad^3 +a^2 d^4)\leq \varepsilon$. Moreover, if we define $$\begin{aligned} \Sigma_\varepsilon^{a} = \partial \widehat{A}_\varepsilon^{a} \backslash \partial M, \end{aligned}$$ then $\Sigma_\varepsilon^a$ converges to $\partial M$ as $a$ increases. It follows from these two facts and [\[111\]](#111){reference-type="eqref" reference="111"} that for $a$ sufficiently large (depending only on $(M,g_0)$), $$\begin{aligned} -g_\varphi^{-1}A_{g_\varphi} & \geq B^2\operatorname{diag}\bigg(\frac{1}{2}-\varepsilon, \dots, \frac{1}{2}-\varepsilon\bigg) = \operatorname{diag}\bigg(\frac{1}{2},\dots,\frac{1}{2}\bigg) \quad \text{in }\widehat{A}^a_\varepsilon. \end{aligned}$$ It then follows from our normalisation $f(\frac{1}{2},\dots,\frac{1}{2})=1$ that $$\begin{aligned} \label{110} f^\tau(-g_{\varphi}^{-1}A_{g_\varphi}) \geq 1\quad \text{in }\widehat{A}^a_\varepsilon. \end{aligned}$$ We now let $$\begin{aligned} \quad C_\varepsilon^{a} = \min_{\Sigma_\varepsilon^{a}} v. \end{aligned}$$ Since $v(x)\rightarrow +\infty$ as $\operatorname{d}(x,\partial M)\rightarrow 0$ (by Step 1), and since $\Sigma_\varepsilon^a$ converges to $\partial M$ as $a$ increases, we can choose $a$ large enough so that $C_\varepsilon^{a} \geq -\ln (\frac{\varepsilon}{100\widehat{C}})$. Moreover, this choice of $a$ depends only on $g_0, \varepsilon, \widehat{C}, f$ and $\Gamma$: since each $v_{(j)}$ was constructed according to the procedure in the proof of Proposition [Proposition 30](#90){reference-type="ref" reference="90"}, we know from the second statement in Proposition [Proposition 30](#90){reference-type="ref" reference="90"} that there exists $\delta=\delta(g_0, \varepsilon, \widehat{C}, f, \Gamma)>0$ such that $v_{(j)} \geq -\ln(\frac{\varepsilon}{100\widehat{C}})$ in $(M_{(j)})_\delta\backslash\partial M_{(j)}$ for each $j$. Taking $j\rightarrow\infty$, we see $v \geq -\ln(\frac{\varepsilon}{100\widehat{C}})$ in $M_\delta\backslash\partial M$. Therefore, to ensure $C^a_\varepsilon\geq -\ln(\frac{\varepsilon}{100\widehat{C}})$, one only needs to pick $a$ large depending on $\delta = \delta(g_0, \varepsilon, \widehat{C}, f, \Gamma)$. We now fix such a value of $a$ and suppose $J$ is sufficiently large so that $\Sigma^a_\varepsilon\subset M_{(j)}$ for all $j\geq J$. Then $\varphi=-\ln (\frac{\varepsilon}{100\widehat{C}})$ and $v \geq C_\varepsilon^{a}$ on $\Sigma_\varepsilon^{a}$, and by the monotonicity in [\[140\]](#140){reference-type="eqref" reference="140"} we also have $v_{(j)} \geq C_\varepsilon^{a}$ on $\Sigma_\varepsilon^{a}$ for each $j\geq J$. Therefore, $$\begin{aligned} \label{172} v_{(j)} \geq -\ln \bigg(\frac{\varepsilon}{100\widehat{C}}\bigg) = \varphi\text{ on }\Sigma_\varepsilon^a \end{aligned}$$ and $$\begin{aligned} \label{173} v_{(j)} = \infty > \varphi\text{ on }\partial M_{(j)}. \end{aligned}$$ In light of [\[110\]](#110){reference-type="eqref" reference="110"}--[\[173\]](#173){reference-type="eqref" reference="173"}, the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"} then yields $$\begin{aligned} v_{(j)} \geq \varphi\quad \text{in }\widehat{A}_\varepsilon^{a} \cap M_{(j)}. \end{aligned}$$ Sending $j\rightarrow\infty$, it follows that $v\geq \varphi$ in $\widehat{A}_\varepsilon^a$, i.e. $$\begin{aligned} v \geq \varphi& = - \ln (B(d+ad^2)) \nonumber \\ & = \ln \sqrt{1-2\varepsilon} - \ln d - \ln (1+ad) \quad \text{in }\widehat{A}_\varepsilon^{a}. \end{aligned}$$ This is precisely [\[184\]](#184){reference-type="eqref" reference="184"} after relabelling constants, and thus the second step is complete. To complete the proof of the proposition, we observe that [\[184\]](#184){reference-type="eqref" reference="184"} implies $$\begin{aligned} \liminf_{\operatorname{d}(x, \partial M)\rightarrow 0}\big(v(x) + \ln \operatorname{d}(x,\partial M)\big) & \geq \ln \sqrt{1-2\varepsilon}, \end{aligned}$$ and since $\varepsilon>0$ is arbitrary, it follows that $$\begin{aligned} \label{115} \liminf_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(v(x) + \ln \operatorname{d}(x,\partial M)\big) \geq 0. \end{aligned}$$ By [\[115\]](#115){reference-type="eqref" reference="115"} and Proposition [Proposition 32](#101){reference-type="ref" reference="101"}, we therefore see that $v$ satisfies [\[103\]](#103){reference-type="eqref" reference="103"}. ◻ ## Uniqueness {#139} Having established the existence of a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} satisfying [\[114\'\]](#114'){reference-type="eqref" reference="114'"} when $\tau<1$ and $\mu_{\Gamma}^+>1$ in the previous section, we now turn to uniqueness of solutions. We first show: **Proposition 34**. *Fix $\tau<1$ and suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}. Then any continuous viscosity solution $g_u = e^{2u}g_0$ to [\[-113\]](#-113){reference-type="eqref" reference="-113"} satisfies [\[114\'\]](#114'){reference-type="eqref" reference="114'"}.* *Proof.* Let $u$ be a continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}. By Proposition [Proposition 32](#101){reference-type="ref" reference="101"}, we know that $u$ satisfies $\limsup_{\operatorname{d}(x,\partial M)\rightarrow 0}(u(x) + \ln \operatorname{d}(x,\partial M)) \leq 0$, so it remains to show $$\begin{aligned} \label{142} \liminf_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) \geq 0. \end{aligned}$$ To prove [\[142\]](#142){reference-type="eqref" reference="142"}, we attach a collar neighbourhood $N$ to $\partial M$, extend $g_0$ smoothly to $M \cup N$ and consider the sequence $\{M^{(j)}\}_j$ of smooth compact manifolds with boundary given by $M^{(j)} = \{x\in M\cup N:\operatorname{d}(x,M)\leq j^{-1}\}$. Note that for $x\in M$ and $j$ sufficiently large, $\operatorname{d}(x,\partial M^{(j)}) = d(x,\partial M) + j^{-1}$. Fix $\varepsilon>0$. By Proposition [Proposition 33](#105){reference-type="ref" reference="105"}, there exist constants $\delta>0$ and $a>0$ depending on $g_0, \varepsilon, f,\Gamma$ but independent of $j$, and a smooth metric $g_{u^{(j)}} = e^{2u^{(j)}}g_0$ for each $j$, such that $$\begin{aligned} f^\tau(-g_{u^{(j)}}^{-1}A_{g_{u^{(j)}}}) = 1, \quad \lambda(-g_{u^{(j)}}A_{g_{u^{(j)}}})\in\Gamma^\tau \quad \text{on }M^{(j)}\backslash \partial M^{(j)} \end{aligned}$$ and $$\begin{aligned} u^{(j)}(x) + \ln \operatorname{d}(x,\partial M^{(j)}) \geq \ln\sqrt{1-2\varepsilon} -\ln \big(1+a\operatorname{d}(x,\partial M^{(j)})\big) \quad \text{in }(M^{(j)})_\delta \backslash\partial M^{(j)}. \end{aligned}$$ In particular, for $j$ sufficiently large so that $(M^{(j)})_\delta\cap M\not=\emptyset$, we have $$\begin{aligned} \label{420} u^{(j)}(x) + \ln \bigg(\operatorname{d}(x,\partial M) + \frac{1}{j}\bigg) \geq \ln\sqrt{1-2\varepsilon} -\ln \bigg(1+a\operatorname{d}(x,\partial M) + \frac{a}{j}\bigg) \quad \text{in }M_{\delta-\frac{1}{j}}. \end{aligned}$$ Now, by the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}, $u^{(j)}|_M \leq u$ for each $j$, and thus [\[420\]](#420){reference-type="eqref" reference="420"} implies $$\begin{aligned} \label{421} u(x) + \ln \bigg(\operatorname{d}(x,\partial M) + \frac{1}{j}\bigg) \geq \ln\sqrt{1-2\varepsilon} -\ln \bigg(1+a\operatorname{d}(x,\partial M) + \frac{a}{j}\bigg) \quad \text{in }M_{\delta-\frac{1}{j}}\backslash\partial M. \end{aligned}$$ After taking $j\rightarrow \infty$ in [\[421\]](#421){reference-type="eqref" reference="421"}, it follows that $$\begin{aligned} \liminf_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) \geq \ln\sqrt{1-2\varepsilon}, \end{aligned}$$ and since $\varepsilon>0$ is arbitrary, we obtain [\[142\]](#142){reference-type="eqref" reference="142"}. ◻ Finally we prove uniqueness of solutions to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau<1$: **Proposition 35**. *Fix $\tau<1$, suppose that $(f,\Gamma)$ satisfies [\[21\'\]](#21'){reference-type="eqref" reference="21'"}--[\[24\'\]](#24'){reference-type="eqref" reference="24'"}, [\[25\'\]](#25'){reference-type="eqref" reference="25'"} and [\[418\]](#418){reference-type="eqref" reference="418"}, and let $v$ denote the smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} obtained in Proposition [Proposition 33](#105){reference-type="ref" reference="105"}. Then $v$ is the unique continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}.* *Proof.* Suppose that $w$ is a continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}. By Proposition [Proposition 34](#144){reference-type="ref" reference="144"}, both $v$ and $w$ satisfy [\[114\'\]](#114'){reference-type="eqref" reference="114'"}. For $\delta \geq 0$, define $\Sigma_\delta = \{d=\delta\}$. Then for each $\varepsilon>0$, there exists a minimal $\delta_\varepsilon> 0$ such that $w \leq v + \varepsilon$ on $\Sigma_{\delta_\varepsilon}$. Denoting $v_\varepsilon= v+\varepsilon$, we have $$\begin{aligned} f^\tau(-g_0^{-1}A_{g_{v_\varepsilon}}) = f^\tau(-g_0^{-1}A_{g_v}) = e^{2v}< e^{2v_\varepsilon} \end{aligned}$$ and thus $v_\varepsilon$ is a supersolution the equation in [\[-113\]](#-113){reference-type="eqref" reference="-113"}. By the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}, it follows that $w \leq v+\varepsilon$ on $M\backslash M_{\delta_\varepsilon}$. By minimality of $\delta_\varepsilon$, we have $\delta_\varepsilon\rightarrow 0$ as $\varepsilon\rightarrow 0$, thus $w \leq v$ on $M\backslash\partial M$. Reversing the roles of $w$ and $v$, we see also that $w \geq v$ on $M\backslash\partial M$, and therefore $w=v$. ◻ ## Proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"} {#166} In this final section we complete the proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}: *Proof of Theorem [\[A\'\]](#A'){reference-type="ref" reference="A'"}.* The existence of a smooth solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} for each $\tau<1$, the asymptotic behaviour stated in [\[114\]](#114){reference-type="eqref" reference="114"} and uniqueness in the class of continuous viscosity solutions follow from Propositions [Proposition 33](#105){reference-type="ref" reference="105"} and [Proposition 35](#422){reference-type="ref" reference="422"}. Let us denote these solutions by $u^\tau$. As observed previously, these solutions $u^\tau$ satisfy a locally uniform $C^1$ estimate which is independent of $\tau$, i.e. for each compact set $K\subset M\backslash \partial M$, there exists a constant $C$ independent of $\tau$ but dependent on $g_0, f, \Gamma$ and $K$ such that $$\begin{aligned} \|u^\tau\|_{C^1(K)} \leq C.\end{aligned}$$ It follows that a subsequence of $\{u^\tau\}$ converges locally uniformly in $C^{0,\alpha}$ to some $u\in C_{\operatorname{loc}}^{0,1}(M,g_0)$ for each $\alpha\in(0,1)$. As noted in the proof of Theorem [Theorem 6](#55){reference-type="ref" reference="55"} in Section [3.5](#150){reference-type="ref" reference="150"}, the fact that $u$ is a viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"} when $\tau=1$ follows from exactly the same argument as in the proof of [@LN20b Theorem 1.4]. So it remains to show that $u$ satisfies the asymptotics in [\[114\]](#114){reference-type="eqref" reference="114"} and is maximal. To this end, first note that since we only require $u$ to be a viscosity subsolution in Proposition [Proposition 32](#101){reference-type="ref" reference="101"}, we have $$\begin{aligned} \label{136} \limsup_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) \leq 0.\end{aligned}$$ To show that $$\begin{aligned} \label{155} \liminf_{\operatorname{d}(x,\partial M)\rightarrow 0}\big(u(x) + \ln \operatorname{d}(x,\partial M)\big) \geq 0,\end{aligned}$$ we first recall that $u$ is the $C^{0,\alpha}$ limit of the solutions $u^\tau$ as $\tau\rightarrow 1$. By Proposition [Proposition 33](#105){reference-type="ref" reference="105"}, for each $\varepsilon>0$ sufficiently small, there exist constants $\delta>0$ and $a>0$ independent of $\tau$ (but dependent on $g_0, \varepsilon, f$ and $\Gamma$) such that $$\begin{aligned} \label{409} u^\tau(x) + \ln \operatorname{d}(x,\partial M) \geq \ln \sqrt{1-2\varepsilon} - \ln \big(1+a\operatorname{d}(x,\partial M)\big) \quad \text{in }M_\delta\backslash\partial M. \end{aligned}$$ Taking $\tau\rightarrow 1$ in [\[409\]](#409){reference-type="eqref" reference="409"}, we obtain $$\begin{aligned} u(x) + \ln \operatorname{d}(x,\partial M) \geq \ln \sqrt{1-2\varepsilon} - \ln \big(1+a\operatorname{d}(x,\partial M)\big) \quad \text{in }M_\delta\backslash\partial M,\end{aligned}$$ and [\[155\]](#155){reference-type="eqref" reference="155"} then follows exactly as in the proof of Proposition [Proposition 33](#105){reference-type="ref" reference="105"}. Finally, to see that $u$ is maximal, suppose that $\widetilde{u}$ is another continuous viscosity solution to [\[-113\]](#-113){reference-type="eqref" reference="-113"}. By Proposition [Proposition 32](#101){reference-type="ref" reference="101"}, [\[136\]](#136){reference-type="eqref" reference="136"} holds with $\widetilde{u}$ in place of $u$, and we also know that [\[114\]](#114){reference-type="eqref" reference="114"} is satisfied with $u^\tau$ in place of $u$ for each $\tau\leq 1$. Combining these facts, it follows that for each $\tau\leq 1$ and $\varepsilon>0$, there exists $\delta>0$ such that $\widetilde{u} \leq u^\tau_\varepsilon\mathrel{\rlap{% \raisebox{0.3ex}{$\m@th\cdot$}}% \raisebox{-0.3ex}{$\m@th\cdot$}}% =u^\tau + \varepsilon$ in $M_\delta\backslash\partial M$. On the other hand, $f^\tau(-g_{u^\tau_\varepsilon}^{-1}A_{g_{u^\tau_\varepsilon}}) = e^{-2\varepsilon} f^\tau(-g_{u^\tau}^{-1}A_{g_{u^\tau}})<1$ on $M\backslash\partial M$ and $f^{\tau}(-g_{\tilde{u}}^{-1}A_{g_{\tilde{u}}}) \geq 1$ in the viscosity sense on $M\backslash\partial M$; to see this latter inequality, observe $$\begin{aligned} f^\tau(\lambda) & = \frac{1}{\tau+n(1-\tau)}f\big(\tau\lambda + (1-\tau)\sigma_1(\lambda)e\big) \nonumber \\ & \geq \frac{1}{\tau+n(1-\tau)} \big(\tau f(\lambda) + (1-\tau)\sigma_1(\lambda)f(e)\big) \nonumber \\ & \stackrel{\eqref{98}}{\geq} \frac{1}{\tau+n(1-\tau)} \bigg(\tau f(\lambda) + (1-\tau)\frac{nf(\lambda)}{f(e)}f(e)\bigg) = f(\lambda). \end{aligned}$$ By the comparison principle in Proposition [Proposition 26](#174){reference-type="ref" reference="174"}, it follows that $\widetilde{u} \leq u^\tau_\varepsilon$ in $M\backslash M_\delta$, and therefore $\widetilde{u} \leq u_\varepsilon^\tau$ in $M\backslash \partial M$. Taking $\varepsilon\rightarrow 0$ and then $\tau\rightarrow 1$, it follows that $\widetilde{u} \leq u$ in $M\backslash \partial M$, as claimed. ◻ # Proof of Proposition [Proposition 17](#46){reference-type="ref" reference="46"}: a cone property {#AA} *Proof of Proposition [Proposition 17](#46){reference-type="ref" reference="46"}.* The result is essentially a consequence of [@Yuan22 Theorem 1.4]. We summarise the details here for the convenience of the reader. Let $\Gamma$ be any cone satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"} and [\[22\'\]](#22'){reference-type="eqref" reference="22'"}, and denote $$\kappa_\Gamma = \max\{k:(\underbrace{0,\dots,0}_{k},\underbrace{1,\dots,1}_{n-k})\in\Gamma\}.$$ Assume for now that there exists a constant $\theta=\theta(n,\Gamma)>0$ such that, whenever $\lambda\in\Gamma$ with $\lambda_1\geq \dots \geq \lambda_n$, it holds that $$\label{37} \frac{\partial f}{\partial \lambda_i}(\lambda) \geq \theta \sum_{j=1}^n \frac{\partial f}{\partial\lambda_j}(\lambda) \quad\text{for }i \geq n-\kappa_\Gamma.$$ Since $\kappa_\Gamma=0$ if and only $\Gamma = \Gamma_n^+$, we see that $\kappa_\Gamma \geq 1$ whenever $\Gamma\not=\Gamma_n^+$, and thus [\[37\'\]](#37'){reference-type="eqref" reference="37'"} holds for $i\in\{n-1,n\}$. Also, it is easy to see that $\kappa_\Gamma$ is equal to the maximum number of negative entries a vector in $\Gamma$ can have, i.e. $$\begin{aligned} \kappa_\Gamma = \max\{k: (-\alpha_1,\dots,-\alpha_k, \alpha_{k+1}, \dots, \alpha_n)\in\Gamma,~\alpha_j>0 \text{ for all } 1\leq j \leq n\}. \end{aligned}$$ Thus [\[37\'\]](#37'){reference-type="eqref" reference="37'"} also holds if $\lambda_i\leq 0$. It remains to justify [\[37\]](#37){reference-type="eqref" reference="37"}, for which we follow [@Yuan22]. By concavity, $f_i(\lambda) \geq f_j(\lambda)$ whenever $\lambda_i \leq \lambda_j$. In particular, our ordering implies $$\begin{aligned} \frac{\partial f}{\partial \lambda_n}(\lambda) \geq \frac{1}{n}\sum_{j=1}^n\frac{\partial f}{\partial\lambda_j}(\lambda), \end{aligned}$$ which establishes [\[37\]](#37){reference-type="eqref" reference="37"} for $\Gamma = \Gamma_n^+$. On the other hand, for a general cone $\Gamma$ satisfying [\[21\'\]](#21'){reference-type="eqref" reference="21'"} and [\[22\'\]](#22'){reference-type="eqref" reference="22'"}, we have $$\label{42} \sum_{i=1}^n f_i(\lambda)\mu_i>0 \quad \text{whenever }\lambda, \mu\in\Gamma.$$ Suppose $\Gamma\not=\Gamma_n^+$, in which case it is clear that $\kappa_\Gamma>0$, and fix any $\alpha_1,\dots,\alpha_n>0$ such that $$(-\alpha_1, \dots, -\alpha_{\kappa_\Gamma}, \alpha_{\kappa_\Gamma + 1}, \dots, \alpha_n)\in\Gamma.$$ Then [\[42\]](#42){reference-type="eqref" reference="42"} implies $$\begin{aligned} \label{43} \sum_{i=\kappa_\Gamma + 1}^n \alpha_i f_{n-i+1}(\lambda) - \sum_{i=1}^{\kappa_\Gamma} \alpha_i f_{n-i+1}(\lambda)>0. \end{aligned}$$ We may assume $\alpha_1 \geq \dots \geq \alpha_{\kappa_\Gamma}$, in which case [\[43\]](#43){reference-type="eqref" reference="43"} implies $$f_{n-\kappa_\Gamma }(\lambda) > \frac{\alpha_1}{\sum_{i=\kappa_\Gamma+1}^n \alpha_i} f_n(\lambda).$$ The desired estimate then follows for all $i\geq n-\kappa_\Gamma$, again by our ordering. ◻ # The Schouten tensor for a radial conformal factor {#appb} In this appendix we prove the formula [\[50\]](#50){reference-type="eqref" reference="50"}. In normal coordinates, $r = \sqrt{x_1^2 + \dots + x_n^2}$, and therefore $\partial_i v(r) = \frac{x_i}{r}v_r$. It follows that $$|\nabla_{g_0} v|_{g_0}^2 = g_0^{ij}\partial_i v\partial_j v = \frac{g_0^{ij}x_ix_j}{r^2}v_r^2 = v_r^2,$$ where we have used the fact that $\frac{\partial}{\partial r} = \frac{x_i}{\sqrt{x_1^2 + \dots + x_n^2}}\frac{\partial }{\partial x_i}$ has unit magnitude. Moreover, $$\begin{aligned} (\nabla_{g_0}^2 v)_{ij} = \partial_i\partial_j v - \Gamma_{ij}^k \partial_k v = \frac{\delta_{ij}}{r}v_r + \frac{x_ix_j}{r}\bigg(\frac{v_{rr}}{r}- \frac{v_r}{r^2}\bigg) - \Gamma_{ij}^k \partial_k v. \end{aligned}$$ Combining the above, we therefore see that $$\begin{aligned} (g_v^{-1}A_{g_v})^p_j & = v^2(g_0^{-1}A_{g_v})^p_j = v^2 g_0^{pi}(A_{g_v})_{ij} \nonumber \\ & = v^2\bigg[\frac{g_0^{pi}\delta_{ij}}{vr}v_r + g_0^{pi}\frac{x_ix_j}{vr}\bigg(\frac{v_{rr}}{r}- \frac{v_r}{r^2}\bigg) - g_0^{pi}\frac{\Gamma_{ij}^kx_kv_r}{vr} - \frac{v_r^2}{2v^2}\delta_j^p + (g_0^{-1}A_{g_0})_j^p \bigg].\end{aligned}$$ Now write $g_0^{pi} = \delta^{pi} + \chi^{pi}$ where $\chi = O(r^2)$ as $r\rightarrow 0$. Then $$\begin{aligned} (g_v^{-1}A_{g_v})^p_j & = v^2\bigg[\frac{\delta_j^p}{vr}v_r + \frac{x^px_j}{vr}\bigg(\frac{v_{rr}}{r}- \frac{v_r}{r^2}\bigg) - \frac{v_r^2}{2v^2}\delta_j^p \bigg] \nonumber \\ & \qquad + \underbrace{v^2\bigg[\frac{\chi^{pi}\delta_{ij}}{vr}v_r + \chi^{pi}\frac{x_ix_j}{vr}\bigg(\frac{v_{rr}}{r}- \frac{v_r}{r^2}\bigg) - g_0^{pi}\frac{\Gamma_{ij}^kx_kv_r}{vr} + (g_0^{-1}A_{g_0})_j^p\bigg]}_{= \Psi_j^p} \nonumber \\ & = v^2\bigg(\lambda\delta_j^p + \chi\frac{x^px_j}{r^2}\bigg) + \Psi_j^p,\end{aligned}$$ where $\lambda$ and $\chi$ are as in [\[16\]](#16){reference-type="eqref" reference="16"}. Now, since $\chi = O(r^2)$ we have $$v^2\frac{\chi^{pi}\delta_{ij}}{vr}v_r = O(r)v|v_r| \quad \text{and}\quad v^2\chi^{pi}\frac{x_ix_j}{vr}\bigg(\frac{v_{rr}}{r}-\frac{v_r}{r^2}\bigg) = O(r^2)v|v_{rr}| + O(r)v|v_r|,$$ and since $\Gamma_{ij}^k = O(r)$ and $(g_0^{-1}A_{g_0})^p_j = O(r)$, we also have $$v^2g_0^{pi}\frac{\Gamma_{ij}^kx_kv_r}{vr} = O(r)v|v_r| \quad\text{and}\quad v^2(g_0^{-1}A_{g_0})_j^p = O(r)v^2.$$ The claim [\[50\]](#50){reference-type="eqref" reference="50"} then follows. [^1]: Johns Hopkins University, 404 Krieger Hall, Department of Mathematics, 3400 N. Charles Street, Baltimore, MD 21218, US. Email: jdunca33\@jhu.edu. [^2]: Mathematical Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG, UK. Email: luc.nguyen\@maths.ox.ac.uk. [^3]: Loewner & Nirenberg also considered in [@LN74] the problem on a class of non-smooth Euclidean domains, but we will not be concerned with such generalisations in this paper. [^4]: We would like to thank Baozhi Chu, YanYan Li and Zongyuan Li for bringing [@Kho09] to our attention.

arxiv_math

--- abstract: | It is provided a local characterization of quasi-crystal graphs, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems. It is also shown that quasi-crystal graphs satisfying these axioms are closed under the tensor product recently introduced by Cain, Guilherme and Malheiro. It is deduced that each connected component of such a graph has a unique highest weight element, whose weight is a composition, and it is isomorphic to a quasi-crystal graph of semistandard quasi-ribbon tableaux. address: - | Center for Mathematics and Applications (NOVA Math)\ NOVA School of Science and Technology\ NOVA University of Lisbon\ 2829--516 Caparica\ Portugal - | Center for Mathematics and Applications (NOVA Math) / Department of Mathematics\ NOVA School of Science and Technology\ NOVA University of Lisbon\ 2829--516 Caparica\ Portugal - | Center for Mathematics and Applications (NOVA Math) / Department of Mathematics\ NOVA School of Science and Technology\ NOVA University of Lisbon\ 2829--516 Caparica\ Portugal - | Center for Mathematics and Applications (NOVA Math)\ NOVA School of Science and Technology\ NOVA University of Lisbon\ 2829--516 Caparica\ Portugal author: - Alan J. Cain - António Malheiro - Fátima Rodrigues - Inês Rodrigues bibliography: - local_characterization_quasicrystal_graphs.bib title: A local characterization of quasi-crystal graphs --- # Introduction In the 1980's, Drinfeld [@Dri85] and Jimbo [@Jim85] independently introduced quantum groups, which are quantized deformations of universal enveloping algebras of semisimple Lie algebras, and which played a significant role in theoretical physics. Kashiwara, building upon their work, developed the theory of crystal bases and crystal graphs as a framework for studying representations of quantum groups [@Kas94; @Kas90; @Kas91], at $q=0$ limit. Crystal bases are (informally) combinatorial objects associated with representations, and crystal graphs are directed graphs that encode the combinatorial data of crystal bases. Kashiwara has showed that, the crystal graph structure has very interesting properties such as being stable under tensor products [@Kas90]. The theory of crystal bases has proven to be a valuable tool in representation theory and has found applications in various fields such as algebraic geometry, mathematical physics, and combinatorics. Indeed, crystal bases are related to Young tableaux through a categorification process that involves replacing the crystal operators of a crystal base by certain combinatorial operations, known as the Kashiwara operators. These operators correspond to edges in the crystal graph allowing for transitions between different tableaux. Through this connection, the combinatorial properties of Young tableaux, such as their shapes, content, and row and column insertion operations, can be related to the combinatorial structures of crystal bases [@KN94]. The monoid whose elements are identified with Young tableaux is called the plactic monoid. It has origin in the works of Schensted [@Sch61] and Knuth [@Knu70], and it was later studied in depth by Lascoux and Schützenberger [@LS81]. Kashiwara showed that the plactic monoid arises from the crystal bases associated with the vector representation of the quantized universal enveloping general linear Lie algebra. It emerges as the quotient of a free monoid on a given alphabet $A_n=\{1,\ldots, n\}$ by a congruence that identifies words that have the same position in isomorphic components of the crystal graph [@KN94]. The plactic monoid also plays a significant role in the theory of symmetric polynomials, particularly in connection with Schur polynomials. These polynomials, which serve as the irreducible polynomial characters of the general linear group $\mathrm{GL}_n(\mathbb{C})$, are indexed by shapes of Young tableaux with entries in the alphabet $A_n$. They form a basis for the ring of symmetric polynomials in $n$ indeterminates. The application of the plactic monoid has yielded the first rigorous proof of the Littlewood--Richardson rule, a combinatorial rule that expresses a product of two Schur polynomials as a linear combination of Schur polynomials [@Lit34]. In addition to the classical plactic monoid, there exists another monoid known as the hypoplactic monoid that emerges in the realm of quasi-symmetric functions and non-commutative symmetric functions. It was first introduced by Krob and Thibon [@KT97] and studied in depth by Novelli [@Nov00]. The hypoplactic monoid provides an analogue to the classical plactic monoid, but with quasi-ribbon tableaux as elements. The quasi-ribbon polynomials serve as a basis for the ring of quasi-symmetric polynomials, analogous to how Schur polynomials form a basis for the ring of symmetric polynomials. Cain and Malheiro [@CM17] introduced a purely combinatorial quasi-crystal structure for the hypoplactic monoid, similar to the crystal structure for the plactic monoid. They show that many of the intriguing connections observed between the crystal graph, Kashiwara operators, Young tableaux, and the plactic monoid are mirrored in the interaction of the analogous quasi-crystal graph, quasi-Kashiwara operators, quasi-ribbon tableaux, and the hypoplactic monoid. In particular, the hypoplactic monoid is defined as the quotient of the free monoid on the alphabet $A_n$ by the congruence that identifies words in the same position in isomorphic connected components of the quasi-crystal graph. Recently, Maas-Gariépy [@MG23] independently introduced an equivalent quasi-crystal structure by considering the decomposition of Schur functions into fundamental quasi-symmetric functions. The first two authors, together with Guilherme [@CGM23], introduced the concept of a hypoplactic congruence that can be defined for any seminormal quasi-crystal, leading to a broader notion of a hypoplactic monoid. They demonstrate that the hypoplactic monoid construction proposed by Cain and Malheiro can be viewed within the context of the hypoplactic monoid associated with the general linear Lie algebra. In the same paper, it is also defined a quasi-tensor product of quasi-crystals, in a similar way as it was done for crystals. In this paper, the authors aim to further advance the theory of quasi-crystals in parallel with the existing theory of crystals. This endeavor is motivated by the desire to extend the rich interplay between the crystal graph, Kashiwara operators, Young tableaux, and the plactic monoid to the realm of quasi-crystals. In [@Kas95], Kashiwara presented an abstract notion of crystal associated to a root system as an edge-coloured graph satisfying certain specific axioms, levering this notion to a more general setting that goes beyond crystals of representations -- see [@BumpSchi17]. While explicit constructions of crystals exist for certain quantum algebras, such as those described in [@KN94] and [@Lit95], the characterization of those arising from representations was obtained by Stembridge in [@Stem03]. More specifically, Stembridge gave a set of local structural properties on crystal graphs of simply-laced root systems that permit to identify which crystal graphs correspond to the crystal of a representation. The simply-laced cases include all quantum Kac--Moody algebras having a Cartan matrix with off-diagonal entries of $0$ or $-1$. It is worth noting that these simply-laced crystals hold significant significance as they encompass all highest weight crystals of finite or affine type, which are of immense interest in the field. Other crystal-like structures also exhibit local characterizations. For instance, Gillespie and Levinson [@GL19] provided local axioms for a crystal of shifted tableaux, Gillespie, Hawkes, Poh and Schilling [@GHPS20] gave a characterization of crystals for the quantum queer superalgebra, building on local axioms introduced by Assaf and Oğuz [@AO20], and Tsuchioka [@Tsu21] introduced a local characterization of $B_2$ regular crystals. Paralleling the previous work, the authors present in this paper a set of local axioms, similar to those presented by Stembridge for crystals, that characterize quasi-crystal graphs of simply-laced root systems that arise from the quasi-crystal of type $A_n$. This characterization answers a question (Question 1) posed by the referee of [@CM17]. This paper is organized as follows. In Section [2](#sec:background){reference-type="ref" reference="sec:background"} we recall the notion of crystals, focusing on Stembridge crystals, and quasi-crystals. In Section [3](#sec:local_axs){reference-type="ref" reference="sec:local_axs"}, we introduce local axioms for quasi-crystal graphs (Definition [Definition 6](#def:local_axioms){reference-type="ref" reference="def:local_axioms"}), and prove that connected quasi-crystals satisfying these axioms are completely characterized by their unique highest weight elements (Theorems [Theorem 11](#thm:uniq_hw){reference-type="ref" reference="thm:uniq_hw"} and [Theorem 12](#thm:wt_iso){reference-type="ref" reference="thm:wt_iso"}). We also prove that the quasi-crystal graphs satisfying the said axioms are closed under the quasi-tensor product introduced in [@CGM23]. In Section [4](#sec:stem_crystals_quasi){reference-type="ref" reference="sec:stem_crystals_quasi"}, we introduce an algorithm to obtain quasi-crystal graphs satisfying the local axioms from a connected Stembridge crystal (Theorem [Theorem 30](#thm:crystal_quasi_ax){reference-type="ref" reference="thm:crystal_quasi_ax"}). # Background {#sec:background} We begin by recalling the notion of (abstract) crystals and Stembridge crystals, following mainly [@BumpSchi17]. Then, we recall the notion of quasi-crystals, first introduced in [@CM17] and further developed in [@CGM23]. ## Crystals **Definition 1**. Let $\Phi$ be a root system, with weight lattice $\Lambda$, index set $I$ and simple roots $\alpha_i$, for $i \in I$. A *crystal* of type $\Phi$ is a non-empty set $\mathcal{C}$ together with maps $\widetilde{e}_i, \widetilde{f}_i: \mathcal{C}\longrightarrow \mathcal{C} \sqcup \{ \bot \}$, $\widetilde{\varepsilon}_i, \widetilde{\varphi}_i : \mathcal{C}\longrightarrow \mathbb{Z} \sqcup \{-\infty\}$, and $\mathrm{wt}: \mathcal{C}\longrightarrow \Lambda$ for $i \in I := \{1, \ldots, n-1\}$, satisfying the following: C1. : For any $x,y \in \mathcal{C}$, $\widetilde{e}_i (x) = y$ if and only $x = \widetilde{f}_i (y)$, and in that case, $$\begin{aligned} \mathrm{wt}(y) &= \mathrm{wt}(x) + \alpha_i,\\ \widetilde{\varepsilon}_i(y) &= \widetilde{\varepsilon}_i (x) - 1,\\ \widetilde{\varphi}_i (y) &= \widetilde{\varphi}_i (x)+1,\end{aligned}$$ C2. : $\widetilde{\varphi}_i (x) = \widetilde{\varepsilon}_i (x) + \langle \mathrm{wt}(x), \alpha_i^{\vee} \rangle$. C3. : If $\widetilde{\varepsilon}_i (x) = -\infty$, then $\widetilde{e}_i (x) = \widetilde{f}_i (x) = \bot$. The maps $\widetilde{e}_i$ and $\widetilde{f}_i$ are called the *Kashiwara operators* (respectively, the *raising* and *lowering* operators), $\widetilde{\varepsilon}_i$ and $\widetilde{\varphi}_i$ are the *length functions* and $\mathrm{wt}$ is the *weight function*. A crystal is said to be *seminormal* if $$\begin{aligned} \widetilde{\varepsilon}_i (x) &= \max \{k: \widetilde{e}_i^{\,k} (x) \neq \bot\},\\ \widetilde{\varphi}_i (x) &= \max \{k: \widetilde{f}_i^{\,k} (x) \neq \bot\},\end{aligned}$$ for all $i \in I$, $x \in \mathcal{C}$. In particular, in a seminormal crystal, we have $\widetilde{\varepsilon}_i(x), \widetilde{\varphi}_i (x) \geq 0$, for all $x \in \mathcal{C}$, $i \in I$. We say that $x \in \mathcal{C}$ is a *highest weight element* if $\widetilde{e}_i(x) = \bot$, for all $i \in I$. If $\mathcal{C}$ is seminormal, this is equivalent to have $\widetilde{\varepsilon}_i (x)= 0$, for all $i \in I$. We associate a crystal to its *crystal graph*, a directed graph, whose edges are labelled in $I$ and whose vertices are weighted in $\Lambda$, and such that there exists an $i$-labelled edge $y \overset{i}{\longrightarrow} x$ if and only if $\widetilde{f}_i (y) = x$. **Definition 2**. Let $\Phi$ be a simply-laced root system. A crystal of type $\Phi$ is a *weak Stembridge crystal* if the following axioms are satisfied, for all $i, j \in I$ such that $i \neq j$: 1. If $\widetilde{e}_i (x)=y$, then $\widetilde{\varepsilon}_j (y)$ is equal to either $\widetilde{\varepsilon}_j(x)$ or $\widetilde{\varepsilon}_j (x)+1$, and the latter happens only if $\alpha_i$ and $\alpha_j$ are orthogonal roots. 2. If $\widetilde{e}_i (x) = y$ and $\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x) > 0$, then $$\widetilde{e}_i \widetilde{e}_j (x) = \widetilde{e}_j \widetilde{e}_i (x) \neq \bot,$$ and $\widetilde{\varphi}_i (\widetilde{e}_j (x)) = \widetilde{\varphi}_i (x)$. 3. If $\widetilde{f}_i (x) = y$ and $\widetilde{\varphi}_j (y) = \widetilde{\varphi}_j (x) > 0$, then $$\widetilde{f}_i \widetilde{f}_j (x) = \widetilde{f}_j \widetilde{f}_i (x) \neq \bot,$$ and $\widetilde{\varepsilon}_i (\widetilde{f}_j (x)) = \widetilde{\varepsilon}_i (x)$. 4. If $\widetilde{e}_i (x) = y$ and $\widetilde{e}_j (x)=z$, and $\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x)+1$ and $\widetilde{\varepsilon}_i (z)= \widetilde{\varepsilon}_i (x)+1$, then $$\widetilde{e}_i \widetilde{e}_j^{\,2} \widetilde{e}_i (x) = \widetilde{e}_j \widetilde{e}_i^{\,2} \widetilde{e}_j (x) \neq \bot,$$ and $\widetilde{\varphi}_i (\widetilde{e}_j (x)) = \widetilde{\varphi}_i (\widetilde{e}_j^{\,2} \widetilde{e}_i (x))$ and $\widetilde{\varphi}_j (\widetilde{e}_i (x)) = \widetilde{\varphi}_j (\widetilde{e}_i^{\,2} \widetilde{e}_j (x))$. 5. If $\widetilde{f}_i (x) = y$ and $\widetilde{f}_j (x)=z$, and $\widetilde{\varphi}_j (y) = \widetilde{\varphi}_j (x)+1$ and $\widetilde{\varphi}_i (z)= \widetilde{\varphi}_i (x)+1$, then $$\widetilde{f}_i \widetilde{f}_j^{\,2} \widetilde{f}_i (x) = \widetilde{f}_j \widetilde{f}_i^{\,2} \widetilde{f}_j (x) \neq \bot,$$ and $\widetilde{\varepsilon}_i (\widetilde{f}_j (x)) = \widetilde{\varepsilon}_i (\widetilde{f}_j^{\,2} \widetilde{f}_i (x))$ and $\widetilde{\varepsilon}_j (\widetilde{f}_i (x)) = \widetilde{\varepsilon}_j (\widetilde{f}_i^{\,2} \widetilde{f}_j (x))$. A *Stembridge crystal* is a weak Stembridge crystal that is also seminormal. **Proposition 3** ([@BumpSchi17 Proposition 4.5]). *Let $\mathcal{C}$ be a crystal graph satisfying axiom **S1** and let $\widetilde{e}_i (x) = y$. Then, exactly one of the following possibilities is true:* 1. *$\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x)$, $\widetilde{\varphi}_j (y) = \widetilde{\varphi}_j (x)-1$, for $\langle \alpha_i, \alpha_j \rangle = -1$.* 2. *$\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x)+1$, $\widetilde{\varphi}_j (y) = \widetilde{\varphi}_j (x)$, for $\langle \alpha_i, \alpha_j \rangle = -1$.* 3. *$\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x)$, $\widetilde{\varphi}_j (y) = \widetilde{\varphi}_j (x)$, for $\langle \alpha_i, \alpha_j \rangle = 0$.* ## Quasi-crystals Consider $\mathbb{Z}\sqcup\{-\infty, +\infty\}$ the set of integers with two additional symbols, a minimal element $-\infty$ and a maximal element $+\infty$. In this set consider also the usual addition between integers, and set $m+(-\infty)=(-\infty)+m= - \infty$ and $m+(+\infty)=(+\infty)+m= + \infty$, for all $m \in \mathbb{Z}$. **Definition 4** ([@CGM23 Definition 3.1]). A *quasi-crystal* of type $\Phi$ is a non-empty set $\mathcal{Q}$ together with maps $\ddot{e}_i, \ddot{f}_i: \mathcal{Q}\longrightarrow \mathcal{Q}\sqcup \{ \bot \}$, $\ddot{\varepsilon}_i, \ddot{\varphi}_i : \mathcal{Q}\longrightarrow \mathbb{Z} \sqcup \{-\infty, + \infty\}$, and $\mathrm{wt}: \mathcal{Q}\longrightarrow \Lambda$ for $i \in I := \{1, \ldots, n-1\}$, satisfying the following: Q1. : For any $x,y \in \mathcal{Q}$, $\ddot{e}_i (x) = y$ if and only $x = \ddot{f}_i (y)$, and in that case, $$\begin{aligned} \mathrm{wt}(y) &= \mathrm{wt}(x) + \alpha_i,\\ \ddot{\varepsilon}_i(y) &= \ddot{\varepsilon}_i (x) -1,\\ \ddot{\varphi}_i (y) &= \ddot{\varphi}_i (x)+1.\end{aligned}$$ Q2. : $\ddot{\varphi}_i (x) = \ddot{\varepsilon}_i (x) + \langle \mathrm{wt}(x), \alpha_i^{\vee} \rangle$. Q3. : If $\ddot{\varepsilon}_i (x) = - \infty$, then $\ddot{e}_i (x) = \ddot{f}_i (x) = \bot$. Q4. : If $\ddot{\varepsilon}_i (x) = + \infty$, then $\ddot{e}_i (x) = \ddot{f}_i (x) = \bot$. We use the same terminology as of crystals, except the maps $\ddot{e}_i$ and $\ddot{f}_i$ are called the *quasi-Kashiwara operators*. A quasi-crystal is said to be *seminormal* if $$\begin{aligned} \ddot{\varepsilon}_i (x) &= \max \{k: \ddot{e}_i^k (x) \neq \bot\},\\ \ddot{\varphi}_i (x) &= \max \{k: \ddot{f}_i^k (x) \neq \bot\},\end{aligned}$$ for all $i \in I$, $x \in \mathcal{C}$, whenever $\ddot{\varepsilon}_i (x) \neq + \infty$. In particular, in a seminormal quasi-crystal $\ddot{\varepsilon}_i(x),\ddot{\varphi}_i(x)\in\mathbb{Z}_{\geq 0} \sqcup \{+ \infty\}$. It follows from Definition [Definition 4](#def:quasicrystal){reference-type="ref" reference="def:quasicrystal"} that a crystal is a quasi-crystal such that $\ddot{\varepsilon}_i (x), \ddot{\varphi}_i (x) \neq + \infty$, for all $x \in \mathcal{C}, i \in I$ [@CGM23 Remark 3.2]. We say that $x \in \mathcal{Q}$ is a *highest weight element* if $\ddot{e}_i (x) = \bot$ for all $i \in I$. Note that, in a seminormal quasi-crystal, this is not equivalent to have $\ddot{\varepsilon}_i (x) = 0$ for all $i \in I$, as one might have $\ddot{e}_j (x) = \bot$ and $\ddot{\varepsilon}_j (x) = + \infty$. Similarly to crystals, we associate a quasi-crystal with its *quasi-crystal graph*, defined in the same way as the crystal graph, but such that $x \in \mathcal{Q}$ has an $i$-labelled loop if and only if $\ddot{\varepsilon}_i (x) = \ddot{\varphi}_i (x)=+ \infty$. # Local axioms for quasi-crystals {#sec:local_axs} In what follows, we will consider quasi-crystals of type $A_{n-1}$, where the weight lattice is $\Lambda = \mathbb{Z}^n$, the index set $I = \{1 < \cdots < n-1\}$, and the simple roots are $\alpha_i = (0, \ldots, 0, 1, -1, 0, \ldots, 0)$, and satisfy $\alpha_i^{\vee} = \alpha_i$, for $i\in I$. We will also use the symbols $\mathcal{Q}$, $\ddot{e}_i$, $\ddot{f}_i$, $\ddot{\varepsilon}_i$, $\ddot{\varphi}_i$ and $\mathrm{wt}$ as in Definition [Definition 4](#def:quasicrystal){reference-type="ref" reference="def:quasicrystal"}. *Remark 5*. We focus our study on type $A_{n-1}$ quasi-crystals, because, unlike the crystal case, the other simply-laced types do not exhibit the same nice properties. For instance, connected quasi-crystals of type $D_n$ might have more that one highest weight element [@CGM23]. We remark that, for seminormal crystals (or, more generally, for upper seminormal ones, where it is only required that $\ddot{\varepsilon}_i (x) = \max \{k: \ddot{e}_i^k (x) \neq \bot\}$), the condition $\widetilde{\varepsilon}_i (x) > 0$ is equivalent to $\widetilde{e}_i (x) \neq \bot$. For seminormal quasi-crystals, one may have $\ddot{\varepsilon}_i (x) = + \infty > 0$ and, by **Q4**, $\ddot{e}_i (x) = \bot$. Therefore, in the following statements, differing from [@BumpSchi17], we will require specifically that $\ddot{e}_i (x) \neq \bot$, instead of $\ddot{\varepsilon}_i (x) > 0$. For this reason, for certain results we do not require that the quasi-crystal graphs are seminormal, but rather that $\ddot{\varepsilon}_i (x), \ddot{\varphi}_i (x) \in \mathbb{Z}_{\geq 0} \sqcup \{+ \infty\}$ (which is satisfied by seminormal quasi-crystals). **Definition 6** (Local quasi-crystal axioms). Let $\mathcal{Q}$ be a quasi-crystal graph. Let $i,j\in I$ and $x,y\in \mathcal{Q}$. LQ1. : For $i+1\in I$, $\ddot{\varepsilon}_i(x) = 0 \Leftrightarrow \ddot{\varphi}_{i+1}(x)=0$. LQ2. : If $\ddot{e}_i(x)=y$, then: 1. $\ddot{\varepsilon}_j(x) = \ddot{\varepsilon}_j(y)$, for $|i-j|>1$. 2. For $i+1\in I$, $\ddot{\varepsilon}_{i+1}(x) \neq \ddot{\varepsilon}_{i+1}(y) \Leftrightarrow \big( \ddot{\varepsilon}_{i+1}(x) = + \infty \wedge \ddot{\varepsilon}_i(y) =0 \big) \Rightarrow \ddot{\varepsilon}_{i+1}(y)\neq 0$. 3. For $i-1\in I$, $\ddot{\varphi}_{i-1}(x) \neq \ddot{\varphi}_{i-1}(y) \Leftrightarrow \big( \ddot{\varphi}_{i-1}(y) = + \infty \wedge \ddot{\varphi}_i(x) = 0 \big) \Rightarrow \ddot{\varphi}_{i-1}(x)\neq 0$. LQ3. : For $i\neq j$, if both $\ddot{e}_i(x)$ and $\ddot{e}_j(x)$ are defined, then $\ddot{e}_i \ddot{e}_j(x)=\ddot{e}_j \ddot{e}_i(x)\neq \bot$. LQ3$'$. : For $i\neq j$, if both $\ddot{f}_i(x)$ and $\ddot{f}_j(x)$ are defined, then $\ddot{f}_i \ddot{f}_j(x)=\ddot{f}_j \ddot{f}_i(x)\neq \bot$. The three cases of axiom **LQ2** are depicted in Figures [1](#fig:LQ2_1){reference-type="ref" reference="fig:LQ2_1"}, [2](#fig:LQ2_2){reference-type="ref" reference="fig:LQ2_2"} and [3](#fig:LQ2_3){reference-type="ref" reference="fig:LQ2_3"}. {#fig:LQ2_1} {#fig:LQ2_2} {#fig:LQ2_3} We remark that, in the previous definition, the condition in **LQ2** (1) can be replaced with $\ddot{\varphi}_j$. We prove this equivalence in the next result. **Lemma 7**. *Let $\mathcal{Q}$ be a quasi-crystal graph satisfying ***LQ1*** and ***LQ2***. Let $i,j\in I$ and $x,y\in \mathcal{Q}$, and suppose that $\ddot{e}_i(x) = y$.* 1. *If $|i-j|>1$, then $\big( \ddot{\varepsilon}_j(y)=\ddot{\varepsilon}_j(x) \Leftrightarrow \ddot{\varphi}_j(y)=\ddot{\varphi}_j(x)\big)$.* 2. *If $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$, then $\big(\ddot{\varepsilon}_{i+1} (y) = \ddot{\varepsilon}_{i+1} (x) \Leftrightarrow \ddot{\varphi}_{i+1} (y) = \ddot{\varphi}_{i+1} (x)-1\big)$.* 3. *If $\ddot{\varepsilon}_{i-1} (x) \neq + \infty$, then $\big(\ddot{\varepsilon}_{i-1} (y) = \ddot{\varepsilon}_{i-1} (x)+1 \Leftrightarrow \ddot{\varphi}_{i-1} (y) = \ddot{\varphi}_{i-1} (x)\big)$.* *Proof.* Suppose that $|i-j|>1$. Then, since $\mathcal{Q}$ is of type $A_{n-1}$, we have $\langle \alpha_i, \alpha_j \rangle = 0$. Suppose that $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x)$. Then, $$\begin{aligned} \ddot{\varphi}_j(y) &= \ddot{\varepsilon}_j(y) + \langle \mathrm{wt}(y), \alpha_j \rangle & (\textrm{by \textbf{Q2} and since $\alpha_i^{\vee}=\alpha_i$})\\ &= \ddot{\varepsilon}_j (x) + \langle \mathrm{wt}(x) + \alpha_i, \alpha_j \rangle & (\textrm{by \textbf{Q1}})\\ &= \ddot{\varepsilon}_j(x) + \langle \mathrm{wt}(x), \alpha_j \rangle + \langle \alpha_i, \alpha_j\rangle \\ &= \ddot{\varepsilon}_j(x) + \langle \mathrm{wt}(x), \alpha_j \rangle \\ &= \ddot{\varphi}_j(x) & (\textrm{by \textbf{Q2} and since $\alpha_i^{\vee}=\alpha_i$}).\end{aligned}$$ Similarly, $\ddot{\varphi}_j (y) = \ddot{\varphi}_j (x)$, implies that $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x)$. Now suppose that $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$, and suppose that $\ddot{\varepsilon}_{i+1} (y) = \ddot{\varepsilon}_{i+1} (x)$. Then, $$\begin{aligned} \ddot{\varphi}_{i+1} (y) &= \ddot{\varepsilon}_{i+1}(y) + \langle \mathrm{wt}(y), \alpha_{i+1} \rangle & (\textrm{by \textbf{Q2} and since $\alpha_i^{\vee}=\alpha_i$})\\ &= \ddot{\varepsilon}_{i+1} (x) + \langle \mathrm{wt}(x) + \alpha_i, \alpha_{i+1} \rangle & (\textrm{by \textbf{Q1}})\\ &= \ddot{\varepsilon}_{i+1} (x) + \langle \mathrm{wt}(x), \alpha_{i+1} \rangle + \langle \alpha_i, \alpha_{i+1} \rangle\\ &= \ddot{\varepsilon}_{i+1} (x) + \langle \mathrm{wt}(x), \alpha_{i+1} \rangle - 1\\ &= \ddot{\varphi}_{i+1} (x)-1 & (\textrm{by \textbf{Q2} and since $\alpha_i^{\vee}=\alpha_i$})\end{aligned}$$ and similarly, one shows that $\ddot{\varphi}_{i+1} (y) = \ddot{\varphi}_{i+1} (x)-1$ implies that $\ddot{\varepsilon}_{i+1} (y) = \ddot{\varepsilon}_{i+1} (x)$. The proof for the third case is analogous. ◻ **Proposition 8**. *Let $\mathcal{Q}$ be a quasi-crystal graph satisfying ***LQ1*** and ***LQ2*** and such that $\ddot{\varepsilon}_i (x)$, $\ddot{\varphi}_i (x) \in \mathbb{Z}_{\geq 0} \sqcup \{+ \infty\}$, for all $x \in Q, i \in I$. Let $i,j\in I$ and $x,y\in \mathcal{Q}$, and suppose that $\ddot{e}_i(x) = y$. Then:* 1. *If $|i-j|>1$, $\ddot{\varepsilon}_j(y)=\ddot{\varepsilon}_j(x)$ and $\ddot{\varphi}_j(y)=\ddot{\varphi}_j(x)$.* 2. *If $i-j=-1$, then:* 1. *If $\ddot{\varepsilon}_{i+1}(x) \neq + \infty$, then $\ddot{\varepsilon}_{i+1}(y)=\ddot{\varepsilon}_{i+1}(x)$ and $\ddot{\varphi}_{i+1}(y)=\ddot{\varphi}_{i+1}(x)-1$.* 2. *If $\ddot{\varepsilon}_{i+1}(x)=+\infty$ and $\ddot{\varepsilon}_i(y)>0$, then $\ddot{\varepsilon}_{i+1}(y)=\ddot{\varepsilon}_{i+1}(x)=+\infty$ and $\ddot{\varphi}_{i+1}(y)=\ddot{\varphi}_{i+1}(x)=+\infty$.* 3. *If $\ddot{\varepsilon}_{i+1}(x)=+\infty$ and $\ddot{\varepsilon}_i(y)=0$, then $\ddot{\varepsilon}_{i+1}(y)= {- \langle \mathrm{wt}(y), \alpha_{i+1} \rangle} > 0$ and $\ddot{\varphi}_{i+1}(y)=0$.* 3. *If $i-j = 1$, then:* 1. *If $\ddot{\varphi}_{i-1}(y) \neq + \infty$, then $\ddot{\varepsilon}_{i-1}(x)=\ddot{\varepsilon}_{i-1}(y)-1$ and $\ddot{\varphi}_{i-1}(x)=\ddot{\varphi}_{i-1} (y)$.* 2. *If $\ddot{\varphi}_{i-1}(y)=+\infty$ and $\ddot{\varphi}_i(x)>0$, then $\ddot{\varepsilon}_{i-1}(x)=\ddot{\varepsilon}_{i-1}(y)=+\infty$ and $\ddot{\varphi}_{i-1}(x)=\ddot{\varphi}_{i-1}(y)=+\infty$.* 3. *If $\ddot{\varphi}_{i-1}(y)=+\infty$ and $\ddot{\varphi}_i(x)=0$, then $\ddot{\varepsilon}_{i-1}(x)=0$ and $\ddot{\varphi}_{i-1}(x)= {\langle \mathrm{wt}(x), \alpha_{i-1} \rangle} >0$.* We remark that, in the previous result, exactly one of the cases holds. *Proof of Proposition [Proposition 8](#prop:local_ax){reference-type="ref" reference="prop:local_ax"}.* Since $\ddot{e}_i(x) = y$, we have $\mathrm{wt}(y) = \mathrm{wt}(x) + \alpha_i$, by **Q1**. The first condition is a direct consequence of Lemma [Lemma 7](#lem:lem_ij){reference-type="ref" reference="lem:lem_ij"}. Now suppose that $j = i+1$ (the proof for $j=i-1$ is analogous). Then, $\langle \alpha_i, \alpha_{i+1} \rangle = -1$. If $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$, the result follows from Lemma [Lemma 7](#lem:lem_ij){reference-type="ref" reference="lem:lem_ij"}. If $\ddot{\varepsilon}_{i+1} (x) = + \infty$, and since $\ddot{\varepsilon}_{i}(y) \geq 0$, we have two cases to consider: 1. If $\ddot{\varepsilon}_i(y)>0$, axiom **LQ2** implies that $\ddot{\varepsilon}_{i+1} (y) = \ddot{\varepsilon}_{i+1} (x) = + \infty$. Consequently, by **Q2**, we have $\ddot{\varphi}_{i+1} (y) = +\infty$. 2. If $\ddot{\varepsilon}_i(y)=0$, then **LQ2** implies that $\ddot{\varepsilon}_{i+1} (x) \neq \ddot{\varepsilon}_{i+1}(y)$ and $\ddot{\varepsilon}_{i+1} (y) > 0$. By axiom **LQ1**, $\ddot{\varepsilon}_i(y) = 0$ implies that $\ddot{\varphi}_{i+1} (y) = 0$. In particular, by **Q2**, we have that $\ddot{\varepsilon}_{i+1}(y)$ and $\ddot{\varphi}_{i+1} (y)$ are finite, and thus, since $\alpha_i^{\vee}=\alpha_i$, $$\ddot{\varepsilon}_{i+1} (y) = \ddot{\varphi}_{i+1} (y) - \langle \mathrm{wt}(y), \alpha_{i+1} \rangle = - \langle \mathrm{wt}(y), \alpha_{i+1} \rangle. \qedhere$$  ◻ The next result is illustrated in Figure [5](#fig:local_infs){reference-type="ref" reference="fig:local_infs"}. **Corollary 9**. *Let $\mathcal{Q}$ be a seminormal quasi-crystal graph satisfying ***LQ1*** and ***LQ2***, and suppose that $\ddot{e}_i (x) = y$, for some $x,y\in \mathcal{Q}$ and $i\in I$.* 1. *Let $i+1\in I$. If $\ddot{\varepsilon}_{i+1}(y) = + \infty$, then $\ddot{\varepsilon}_{i+1} (x) = + \infty$ and, there exists $k>0$ such that $\ddot{\varepsilon}_{i+1} (\ddot{e}_i^k (y)) \notin \{0, + \infty\}$, and for $0\leq l<k$, we have $\ddot{\varepsilon}_i(\ddot{e}_i^l(y))\notin \{0, + \infty\}$.* 2. *Let $i-1\in I$. If $\ddot{\varphi}_{i-1} (x) = + \infty$, then $\ddot{\varphi}_{i-1} (y) = + \infty$ and there exists $k>0$ such that $\ddot{\varphi}_{i-1} (\ddot{f}_i^k (x)) \notin \{0,+\infty\}$, and for $0\leq l<k$, we have $\ddot{\varphi}_{i-1}(\ddot{f}_i^l(y))\notin \{0, + \infty\}$.* *Proof.* We prove the first statement; the second one is proved similarly. Suppose that $\ddot{\varepsilon}_{i+1} (y) = + \infty$. If $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$, then axiom **LQ2** (2) implies that $\ddot{\varepsilon}_{i+1} (y)= \ddot{\varepsilon}_{i+1} (x) \neq + \infty$, which is a contradiction. Therefore, $\ddot{\varepsilon}_{i+1} (x)= +\infty$. Now suppose that $\ddot{\varepsilon}_i (y)=0$. By **LQ1**, we have $\ddot{\varphi}_{i+1} (y)=0$, which implies that $\ddot{\varepsilon}_{i+1} (y)$ is finite, contradicting the hypothesis. Thus, $\ddot{\varepsilon}_i (y) \neq 0$. By **Q1**, $\ddot{e}_i (x) = y$ implies that $\ddot{f}_i (y)=x$, hence $\ddot{f}_i (y) \neq \bot$, and by **Q4**, $\ddot{\varepsilon}_i (y) \neq + \infty$. Therefore, $\ddot{\varepsilon}_i (x) \not\in \{0, + \infty\}$, and thus, there exists $k > 0$ such that $\ddot{\varepsilon}_i (y)=k$. Since $\mathcal{Q}$ is seminormal, we have $\ddot{e}_i^l (y) \neq \bot$, for all $l \leq k$ and $\ddot{e}_i^s (y)= \bot$, for $s > k$. Therefore, $\ddot{\varepsilon}_i (\ddot{e}_i^l (y)) \not\in \{0, +\infty\}$, for $l < k$. Now, iteratively, applying **LQ2** (2), we obtain $\ddot{\varepsilon}_{i+1} (\ddot{e}_i^l (y))=\ddot{\varepsilon}_{i+1} (y) = + \infty$, for $1\leq l < k$. Finally, since $\ddot{\varepsilon}_i(\ddot{e}_i^k(y))=0$ and $\ddot{\varepsilon}_{i+1} (\ddot{e}_i^{k-1} (y))=+\infty$ we conclude by **LQ2** (2), that $\ddot{\varepsilon}_{i+1} (\ddot{e}_i^{k}(y)) \notin \{0,+ \infty\}$. ◻ {reference-type="ref" reference="cor:infs"}.](local_infs_1.pdf "fig:"){#fig:local_infs} {reference-type="ref" reference="cor:infs"}.](local_infs_2.pdf "fig:"){#fig:local_infs} We have recalled that a crystal $\mathcal{C}$ is a quasi-crystal such that $\ddot{\varepsilon}_i (x) \neq + \infty$ for all $x \in \mathcal{C}, i \in I$ [@CGM23]. Therefore, we have the following result. **Proposition 10**. *Let $\mathcal{Q}$ be a crystal graph satisfying the local quasi-crystal axioms of Definition [Definition 6](#def:local_axioms){reference-type="ref" reference="def:local_axioms"}. Then $\mathcal{Q}$ is a weak Stembridge crystal.* *Proof.* Suppose that $\ddot{e}_i (x) = y$, for $x, y \in \mathcal{Q}$, $i \in I$ and let $j \in I$ be such that $j \neq i$. Since $\mathcal{Q}$ is a crystal, we have $\ddot{\varepsilon}_j (x) \neq + \infty$. Thus, if $|i-j|>1$ or $j=i+1$, by axiom **LQ2** we have $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x)$. Otherwise, if $j=i-1$ (and, in this case, $\alpha_j$ and $\alpha_i$ are orthogonal roots), we have $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x)+1$. Therefore, $\mathcal{Q}$ satisfies axiom **S1**. Now suppose that $\ddot{e}_i (x)=y$ and $\ddot{e}_j (x)=z$ are both defined. Then, axiom **LQ3** ensures that $\ddot{e}_i \ddot{e}_j (x) = \ddot{e}_j \ddot{e}_i (x) \neq \bot$. We now consider three cases: 1. If $|i-j|>1$, we have $\ddot{\varepsilon}_i (z) = \ddot{\varepsilon}_i (x)>0$ and $\ddot{\varepsilon}_j(y)=\ddot{\varepsilon}_j (x)>0$, and thus, by Lemma [Lemma 7](#lem:lem_ij){reference-type="ref" reference="lem:lem_ij"}, we have $\ddot{\varphi}_i (z) = \ddot{\varphi}_i (x)$ and $\ddot{\varphi}_j (y) = \ddot{\varphi}_j (x)$. 2. If $j=i+1$, and since $\ddot{e}_i (x)=y$ and $\ddot{\varepsilon}_j (x) \neq + \infty$ (because $\mathcal{Q}$ is a crystal and thus $\ddot{\varepsilon}_i (x) \neq + \infty$, for any $x \in Q$, $i \in I$), it follows from axiom **LQ2** (2) that $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x) > 0$. By the same reasoning, since $\ddot{e}_j (x)=z$ and $\ddot{\varphi}_i (z) \neq + \infty$, axiom **LQ2** (3) ensures that $\ddot{\varepsilon}_i (z) = \ddot{\varepsilon}_i (x)+1$, and thus, by Lemma [Lemma 7](#lem:lem_ij){reference-type="ref" reference="lem:lem_ij"}, we have $\ddot{\varphi}_i (z)=\ddot{\varphi}_i (x)$. 3. If $j=i-1$, since $\ddot{e}_i (x)=y$ and $\ddot{\varphi}_j (y) \neq + \infty$, axiom **LQ2** (3) implies that $\ddot{\varphi}_j (y)=\ddot{\varphi}_j (x)$, and thus, by Lemma [Lemma 7](#lem:lem_ij){reference-type="ref" reference="lem:lem_ij"}, we have $\ddot{\varepsilon}_j (y)=\ddot{\varepsilon}_j (x)+1$. Similarly, since $\ddot{e}_j (x)=z$ and $\ddot{\varepsilon}_i (x) \neq + \infty$, axiom **LQ2** (2) implies that $\ddot{\varepsilon}_i (z) = \ddot{\varepsilon}_i (x) > 0$. In all cases, $\mathcal{Q}$ satisfies axiom **S2**. Moreover, the conditions $\ddot{\varepsilon}_i (z) = \ddot{\varepsilon}_i (x)+1$ and $\ddot{\varepsilon}_j (y) = \ddot{\varepsilon}_j (x)+1$ do not occur simultaneously, hence the hypotheses of axiom **S3** never occur. With similar reasoning, we obtain the same conclusions for the dual axioms. ◻ Recall that a highest weight element in a quasi-crystal is an element $x$ such that $\ddot{e}_i(x) = \bot$, for all $i \in I$. We define a partial order in $\mathcal{Q}$ where $x \prec y$ if there exists a sequence $i_1, \dots, i_N$ in $I$ such that $\ddot{e}_{i_1} \dots \ddot{e}_{i_N} (x)=y$. Similarly to the crystal case, we say that a quasi-crystal graph $\mathcal{Q}$ is *bounded above* if, for every $x \in \mathcal{Q}$, there exists and highest weight element $x_h \in \mathcal{Q}$ such that $x \preceq x_h$. **Theorem 11**. *Let $\mathcal{Q}$ be a non-empty and bounded above connected quasi-crystal graph satisfying the local quasi-crystal axioms. Then, $\mathcal{Q}$ has a unique highest weight element.* *Proof.* Let $S$ be the subset of vertices $w \in \mathcal{Q}$ for which there exist distinct highest weight elements $x$ and $y$, such that $w \preceq x$ and $w \preceq y$. Suppose that $S \neq \emptyset$ and let $w_0$ be a maximal element of $S$. Then, by maximality, there are vertices $x, y \in \mathcal{Q}$, such that $w_0 \prec x$ and $w_0 \prec y$, where $x$ and $y$ are each comparable to only one highest weight element, say $x_h$ and $y_h$, respectively. Without loss of generality, assume that $x = \ddot{e}_i (w_0)$ and $y = \ddot{e}_j (w_0)$, for $i, j \in I$. Then, by axiom **LQ3**, there exists $z \in C$ such that $\ddot{e}_i \ddot{e}_j (w_0) = \ddot{e}_j \ddot{e}_i (w_0) = z$. Then, we have $x \prec z$, and by hypothesis, $x$ is comparable to a unique highest weight element $x_h$. Thus, we have $z \preceq x_h$. By the same reasoning, we have $z \preceq y_h$, and thus $z \in S$. But since $w_0 \prec x$, we have $w_0 \prec x \prec z$, which contradicts the maximality of $w_0$. Therefore, $S = \emptyset$. ◻ **Theorem 12**. *Let $\mathcal{Q}$ and $\mathcal{Q}'$ be connected components of seminormal quasi-crystal graphs satisfying the local quasi-crystal axioms, with highest weight elements $u$ and $u'$, respectively. Then, if $\mathrm{wt}(u)=\mathrm{wt}(u')$, there exists an isomorphism between $\mathcal{Q}$ and $\mathcal{Q}'$.* To prove Theorem [Theorem 12](#thm:wt_iso){reference-type="ref" reference="thm:wt_iso"}, we recall the notion of *rank* of $x \in \mathcal{Q}$ [@BumpSchi17 Section 4.4], where $\mathcal{Q}$ is a connected component of a quasi-crystal graph with unique highest weight element $u$, which is defined as $\mathrm{rank} (x) := \langle \mathrm{wt}(u) - \mathrm{wt}(x), \rho \rangle$, where $\rho$ is any vector such that $\langle \alpha_i, \rho \rangle = 1$, for all $i \in I$. This is well defined, and, in particular, if $\ddot{e}_{i_N} \cdots \ddot{e}_{i_1} (x) = u$, and all $\ddot{e}_{i_1} (x), \ddot{e}_{i_2} \ddot{e}_{i_1} (x), \ldots, \ddot{e}_{i_N} \cdots \ddot{e}_{i_1} (x)$ are defined, then $\mathrm{rank}(x) = N$. *Proof of Theorem [Theorem 12](#thm:wt_iso){reference-type="ref" reference="thm:wt_iso"}.* Let $\Omega$ be the set of subsets $S \subseteq \mathcal{Q}$ such that: - $u \in S$. - If $x \in S$ and $\ddot{e}_i (x) \in Q$, then $\ddot{e}_i (x) \in S$. - There exists a subset $S' \subseteq \mathcal{Q}'$ and a bijection $\theta : S \longrightarrow S'$ such that $\theta(u) = u'$ and, given $x \in S$, then, for every $i \in I$, $$\big(\ddot{e}_i (x) \neq \bot \Leftrightarrow \ddot{e}_i (\theta(x)) \neq \bot \big) \wedge \big( \ddot{e}_i(x) \neq \bot \Rightarrow \theta (\ddot{e}_i (x)) = \ddot{e}_i (\theta (x)) \big).$$ We have $\Omega \neq \emptyset$, since $\{u\} \in \Omega$. Let $S$ be a maximal element of $\Omega$, with respect to set inclusion. We will show that $S = \mathcal{Q}$. We claim that $\theta$ preserves the length and weight functions, that is, for all $x \in S$ and $i \in I$, $\ddot{\varepsilon}_i (\theta(x)) = \ddot{\varepsilon}_i (x)$, $\ddot{\varphi}_i (\theta(x)) = \ddot{\varphi}_i (x)$ and $\mathrm{wt}(\theta(x)) = \mathrm{wt}(x)$. Given $x \in S$ and $i \in I$, there exist $i_1, \ldots, i_N \in I$, for some $N$, such that $$\ddot{e}_{i_N} \cdots \ddot{e}_{i_1} (x) = u.$$ By the definition of $S$, we have $$u' = \theta (u) = \theta (\ddot{e}_{i_N} \cdots \ddot{e}_{i_1} (x)) = \ddot{e}_{i_N} \cdots \ddot{e}_{i_1} ( \theta (x) )$$ and thus, $$\mathrm{wt}(\theta (x)) = \mathrm{wt}(u') - \sum\limits_{j=1}^N \alpha_{i_j} = \mathrm{wt}(u) - \sum\limits_{j=1}^N \alpha_{i_j} = \mathrm{wt}(x).$$ Given $x \in S$, since $\mathcal{Q}$ is seminormal, if $\ddot{\varepsilon}_i (x) \neq + \infty$, then $\ddot{\varepsilon}_i (x) = \max \{k: \ddot{e}_i^k(x) \neq \bot\}$. By the definition of $S$, we have $\ddot{e}_i^k (x) \in S$, for $k = 1, \ldots, \ddot{\varepsilon}_i (x)$. In particular, $\ddot{e}_i^k (\theta(x)) = \theta (\ddot{e}_i^k (x))$. Thus, we have $$\begin{aligned} \ddot{\varepsilon}_i (\theta(x)) &= \max \{k: \ddot{e}_i^k (\theta(x)) \neq \bot\}\\ &= \max \{k: \theta(\ddot{e}_i^k (x)) \neq \bot\}\\ &= \max \{k: \ddot{e}_i^k (x) \neq \bot\} = \ddot{\varepsilon}_i (x).\end{aligned}$$ Consequently, $\ddot{\varepsilon}_i (x) = + \infty$ if and only if $\ddot{\varepsilon}_i (\theta(x))=+ \infty$, and therefore, $\ddot{\varepsilon}_i (x) = \ddot{\varepsilon}_i (\theta(x))$. From **Q2**, we get $\ddot{\varphi}_i (x) = \ddot{\varphi}_i (\theta(x))$. Now suppose that $\mathcal{Q}\neq S$, and let $z \in \mathcal{Q}\setminus S$ be an element of minimal rank. Since $z \not\in S$, then $z \neq u$, and thus, there exists $i \in I$ such that $\ddot{e}_i (z) \neq \bot$. Moreover, $\mathrm{rank} (\ddot{e}_i (z)) < \mathrm{rank} (z)$, hence $\ddot{e}_i (z) \in S$, as $z$ has minimal rank. Therefore, $\ddot{\varphi}_i ( \theta(\ddot{e}_i(z))) = \ddot{\varphi}_i (\ddot{e}_i (z)) > 0$, and thus, there exists $z' \in \mathcal{Q}'$ such that $\ddot{e}_i (z') = \theta (\ddot{e}_i (z))$, as depicted in the following diagram:  We will show that $z'$ does not depend on the choice of $i$, that is, if there exists $z'' \in \mathcal{Q}'$ such that $\ddot{e}_j (z'') = \theta (\ddot{e}_j (z))$, with $j \neq i$, then $z' = z''$. Since $\ddot{e}_i (z)$ and $\ddot{e}_j (z)$ are both defined, axiom **LQ3** implies that there exists $w \in S$ such that $$w = \ddot{e}_i \ddot{e}_j (z) = \ddot{e}_j \ddot{e}_i (z).$$ Therefore, we have $\theta (w) = \theta \ddot{e}_i \ddot{e}_j (z) = \theta \ddot{e}_j \ddot{e}_i (z)$, and by definition of $S$, $\theta(w) = \ddot{e}_i \theta \ddot{e}_j (z) = \ddot{e}_j \theta \ddot{e}_i (z)$. Thus, we have $\ddot{f}_i (\theta(w)) = \theta (\ddot{e}_j (w))$ and $\ddot{f}_j (\theta(w)) = \theta (\ddot{e}_i (w))$. It then follows from axiom **LQ3$'$** that $$z' = \ddot{f}_i \ddot{f}_j (\theta (w)) = \ddot{f}_j \ddot{f}_i (\theta (w)) = z''.$$ Then, if $Q \neq S$, the map $\theta$ may be extend by defining $\theta (z) := z'$. This contradicts the maximality of $S$. Therefore, $Q = S$ and the map $\theta$ is a weight-preserving isomorphism. ◻ **Proposition 13**. *Let $\mathcal{Q}$ be a connected component of a seminormal quasi-crystal graph satisfying the local axioms, with highest weight element $u$. Then, $u$ has degree at most one.* *Proof.* If $\mathcal{Q}$ has only one vertex, then the result is trivial. Suppose that $\mathcal{Q}$ has at least two vertices. Therefore, there exists $k \in I$ such that $\ddot{f}_k (u) \neq \bot$, and consequently, $\ddot{\varphi}_k (u), \ddot{\varepsilon}_k (u) \neq + \infty$, by **Q4**. Since $u$ is the highest weight element, we have $\ddot{e}_k (u) = \bot$, and since $\mathcal{Q}$ is seminormal, this implies that $\ddot{\varepsilon}_k (u)=0$. We claim that $\ddot{\varphi}_j (u) = 0$, for $j \geq k$. If $k=n-1$, the result is trivial. Otherwise, axiom **LQ1** implies that $\ddot{\varphi}_{k+1} (u) = 0$, and, in particular, $\ddot{\varphi}_{k+1} (u) \neq + \infty$. Thus, by **Q2**, $\ddot{\varepsilon}_{k+1} (u) \neq + \infty$, and since $u$ is the highest weight element and $\mathcal{Q}$ is seminormal, we get $\ddot{\varepsilon}_{k+1} (u) = 0$. Then, by axiom **LQ1** we obtain $\ddot{\varphi}_{k+2} (u) = 0$, and applying the same reasoning, we have $\ddot{\varphi}_j (u) =0$, for $j \geq k$. Therefore, $\ddot{f}_j (u) = \bot$, for $j \geq k$. If $k = 1$, the proof is done. Thus, suppose that $k > 1$, and we will prove that $\ddot{f}_j (u) = \bot$, for $j < k$. Since $\ddot{\varphi}_k (u) \neq 0$ (because $\ddot{f}_k (u) \neq \bot$), by axiom **LQ1** we have $\ddot{\varepsilon}_{k-1} (u) \neq 0$. Then, as $\ddot{e}_{k-1} (u) = \bot$ and $\mathcal{Q}$ is seminormal, we must have $\ddot{\varepsilon}_{k-1} (u) = + \infty$. Consequently, by **Q4**, $\ddot{f}_{k-1} (u) = \bot$. Now suppose that $\ddot{f}_{k-2} (u)$ is defined. Then, since $\ddot{\varepsilon}_{k-1} (u) = + \infty$, by Corollary [Corollary 9](#cor:infs){reference-type="ref" reference="cor:infs"}, there exists $l >0$ such that $\ddot{\varepsilon}_{k-1} (\ddot{e}_{k-2}^l (u)) \neq +\infty$. Since $l > 0$, this also contradicts the fact that $u$ is the highest weight element. Therefore, $\ddot{f}_{k-2} (u)$ is undefined. Applying the same reasoning, we conclude that $\ddot{f}_{j} (u)$ is undefined, for $j < k$. Thus, there is exactly one edge coming from $u$, and thus $u$ has degree one. ◻ ## Quasi-tensor products We recall the notion of quasi-tensor product introduced by Cain, Guilherme and Malheiro [@CGM23]. Throughout this section, we will only consider seminormal quasi-crystals. **Definition 14** ([@CGM23 Theorem 5.1]). Let $\mathcal{Q}$ and $\mathcal{Q}'$ be seminormal quasi-crystals of type $A_{n-1}$. The quasi-tensor product $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ is the Cartesian product $\mathcal{Q}\times \mathcal{Q}'$ together with the maps defined by: 1. $\mathrm{wt}(x \mathbin{\ddot{\otimes}}x') = \mathrm{wt}(x) + \mathrm{wt}(x')$, for $x \in \mathcal{Q}$, $x' \in \mathcal{Q}'$. 2. If $\ddot{\varphi}_i(x) >0$ and $\ddot{\varepsilon}_i(x')>0$, for $i \in I$, $$\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = \ddot{f}_i (x \mathbin{\ddot{\otimes}}x') = \bot \qquad \text{and} \qquad \ddot{\varepsilon}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{\varphi}_i (x \mathbin{\ddot{\otimes}}x') = + \infty.$$ 3. Otherwise, $$\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = \begin{cases} \ddot{e}_i(x) \mathbin{\ddot{\otimes}}x' & \text{if}\; \ddot{\varphi}_i(x) \geq \ddot{\varepsilon}_i(x')\\ x \mathbin{\ddot{\otimes}}\ddot{e}_i(x') & \text{if}\; \ddot{\varphi}_i(x) < \ddot{\varepsilon}_i(x') \end{cases}$$ $$\ddot{f}_i(x \mathbin{\ddot{\otimes}}x') = \begin{cases} \ddot{f}_i(x) \mathbin{\ddot{\otimes}}x' & \text{if}\; \ddot{\varphi}_i(x) > \ddot{\varepsilon}_i(x')\\ x \mathbin{\ddot{\otimes}}\ddot{f}_i(x') & \text{if}\; \ddot{\varphi}_i(x) \leq \ddot{\varepsilon}_i(x') \end{cases}$$ and $$\begin{aligned} \ddot{\varepsilon}_i(x) &= \max\{ \ddot{\varepsilon}_i(x), \ddot{\varepsilon}_i(x') - \langle \mathrm{wt}(x), \alpha_i \rangle \}\\ \ddot{\varphi}_i (x) &= \max \{ \ddot{\varphi}_i(x) + \langle \mathrm{wt}(x'), \alpha_i \rangle, \ddot{\varphi}_i(x') \}\end{aligned}$$ where $x \mathbin{\ddot{\otimes}}\bot = \bot \mathbin{\ddot{\otimes}}x' = \bot$. The quasi-tensor product $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ is a seminormal quasi-crystal of type $A_{n-1}$ [@CGM23 Theorem 5.1]. We remark that, with this convention, $x \mathbin{\ddot{\otimes}}y$ is identified with the word $yx$. The following Lemma will be often used and is a direct consequence of [@CGM23 Theorem 5.1]. **Lemma 15**. *Let $x \in \mathcal{Q}$ and $x' \in \mathcal{Q}'$. If $\ddot{\varepsilon}_i (x \mathbin{\ddot{\otimes}}x') = + \infty$, then $\ddot{\varphi}_{i}(x) > 0$ or $\ddot{\varepsilon}_{i}(x') > 0$.* **Proposition 16** ([@CGM23 Proposition 5.3]). *Let $\mathcal{Q}$ and $\mathcal{Q}'$ be seminormal quasi-crystals of the same type, and let $x \in \mathcal{Q}$ and $x' \in \mathcal{Q}'$ be such that $\ddot{\varphi}_i(x) = 0$ or $\ddot{\varepsilon}_i(x')=0$. Then: $$\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = \begin{cases} \ddot{e}_i(x) \mathbin{\ddot{\otimes}}x' & \text{if } \ddot{\varepsilon}_i(x') =0\\ x \mathbin{\ddot{\otimes}}\ddot{e}_i(x') & \text{if } \ddot{\varepsilon}_i(x') >0 \end{cases} \qquad \ddot{f}_i(x \mathbin{\ddot{\otimes}}x') = \begin{cases} \ddot{f}_i(x) \mathbin{\ddot{\otimes}}x' & \text{if } \ddot{\varphi}_i(x) > 0\\ x \mathbin{\ddot{\otimes}}\ddot{f}_i(x') & \text{if } \ddot{\varphi}_i(x) = 0 \end{cases}$$ and $\ddot{\varepsilon}_i(x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_i (x)+ \ddot{\varepsilon}_i (x')$, $\ddot{\varphi}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{\varphi}_i (x)+ \ddot{\varphi}_i (x')$.* We may now state the following result, which is proved in the next section. **Theorem 17**. *Let $\mathcal{Q}$ and $\mathcal{Q}'$ be seminormal quasi-crystal graphs satisfying the local quasi-crystal axioms of Definition [Definition 6](#def:local_axioms){reference-type="ref" reference="def:local_axioms"}. Then, $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies the same axioms.* Recall that the *standard crystal* $\mathcal{B}_n$ of type $A_{n-1}$, depicted in Figure [\[fig:std_crystal\]](#fig:std_crystal){reference-type="ref" reference="fig:std_crystal"}, is the crystal structure on $\{1 < \cdots < n\}$ in which $\widetilde{f}_i (j)=j+1$ and $\widetilde{\varphi}_i (j) = \delta_{i,j}$, for $j \in I$. {#fig:quasi_tens} {#fig:crystal_tens} As a consequence of Theorem [Theorem 17](#thm:qtens){reference-type="ref" reference="thm:qtens"}, we have the following result. Figures [6](#fig:quasi_tens){reference-type="ref" reference="fig:quasi_tens"} and [7](#fig:crystal_tens){reference-type="ref" reference="fig:crystal_tens"} illustrate the differences between the usual tensor product of crystals and the quasi-tensor product, on the standard crystal $\mathcal{B}_3$ of type $A_2$. **Corollary 18**. *Let $\mathcal{B}_n$ be the standard crystal of type $A_{n-1}$. Then, every connected component of $\mathcal{B}_n^{\mathbin{\ddot{\otimes}}k}$ satisfies the local quasi-crystal axioms of Definition [Definition 6](#def:local_axioms){reference-type="ref" reference="def:local_axioms"}.* *Proof.* The standard crystal $\mathcal{B}_n$ clearly satisfies the local quasi-crystal axioms. If follows from Theorem [Theorem 17](#thm:qtens){reference-type="ref" reference="thm:qtens"} that $\mathcal{B}_n^{\mathbin{\ddot{\otimes}}k}$ satisfies the same axioms as well. ◻ ## Proof of Theorem [Theorem 17](#thm:qtens){reference-type="ref" reference="thm:qtens"} {#proof-of-theorem-thmqtens} **Proposition 19**. *$\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies axiom ***LQ1***.* *Proof.* Suppose that $\ddot{\varepsilon}_i (x \mathbin{\ddot{\otimes}}x') = 0$. Then, by Definition [Definition 14](#def:qtens){reference-type="ref" reference="def:qtens"}, we must have $\ddot{\varphi}_i (x) = 0$ or $\ddot{\varepsilon}_i (x')=0$. Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"} then implies that $\ddot{\varepsilon}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_i (x) + \ddot{\varepsilon}_i (x') =0$. Since $\mathcal{Q}$ and $\mathcal{Q}'$ are seminormal, we have $\ddot{\varepsilon}_i (x), \ddot{\varepsilon}_i (x') \geq 0$. Therefore, we must have $\ddot{\varepsilon}_i (x) = \ddot{\varepsilon}_i (x') = 0$, and then, axiom **LQ1** implies that $\ddot{\varphi}_{i+1} (x) = \ddot{\varphi}_{i+1} (x') = 0$. In particular, since $\ddot{\varphi}_{i+1}(x) =0$, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $\ddot{\varphi}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varphi}_{i+1} (x) + \ddot{\varphi}_{i+1} (x') =0$. The reverse implication is proved similarly. ◻ **Proposition 20**. *$\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies ***LQ2***.* *Proof.* Suppose that $\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = y \mathbin{\ddot{\otimes}}y'$. Then, by Definition [Definition 14](#def:qtens){reference-type="ref" reference="def:qtens"}, we have $\ddot{\varphi}_i (x) =0$ or $\ddot{\varepsilon}_i (x')=0$. **We show that $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies LQ2** (1). Let $i, j \in I$ be such that $|i-j| > 1$. Then, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have two cases. We prove the case where $\ddot{\varepsilon}_i (x') = 0$. The case where $\ddot{\varepsilon}_i(x') > 0$ is proved similarly. Suppose that $\ddot{\varepsilon}_i(x') = 0$. Then, $\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}x' = y \mathbin{\ddot{\otimes}}y'$, and we have $$\ddot{e}_i(x) = y, \quad x' = y'.$$ If $\ddot{\varphi}_j (y), \ddot{\varepsilon}_j(y') > 0$, then, as $\ddot{e}_i(x) = y$, we have $\ddot{\varphi}_j (x) = \ddot{\varphi}_j (y)$ by Proposition [Proposition 8](#prop:local_ax){reference-type="ref" reference="prop:local_ax"}, and since $x' = y'$, we have $\ddot{\varepsilon}_j (x') = \ddot{\varepsilon}_j (y')$. Then, by Definition [Definition 14](#def:qtens){reference-type="ref" reference="def:qtens"}, we have $\ddot{\varepsilon}_j (y \mathbin{\ddot{\otimes}}y') = \ddot{\varepsilon}_j(x \mathbin{\ddot{\otimes}}x') = + \infty$. Otherwise, suppose that $\ddot{\varphi}_j (y) = 0$ or $\ddot{\varepsilon}_j (y') = 0$. Then, by axiom **LQ1**, we have $\ddot{\varepsilon}_j (x) = \ddot{\varepsilon}_j (y)$, and since $x'=y'$, we have $\ddot{\varepsilon}_j (x') = \ddot{\varepsilon}_j (y')$. Thus, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\ddot{\varepsilon}_j (y \mathbin{\ddot{\otimes}}y') = \ddot{\varepsilon}_j (y) + \ddot{\varepsilon}_j (y') = \ddot{\varepsilon}_j (x) + \ddot{\varepsilon}_j (x') = \ddot{\varepsilon}_j(x \mathbin{\ddot{\otimes}}x').$$ **We show that $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies LQ2** (2). The proof for **LQ2** (3) is done similarly. 1. We first suppose that $\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') \neq + \infty$, and we prove that $\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1} (y \mathbin{\ddot{\otimes}}y')$. By Definition [Definition 14](#def:qtens){reference-type="ref" reference="def:qtens"}, we have $\ddot{\varphi}_{i+1} (x) =0$ or $\ddot{\varepsilon}_{i+1} (x') = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1}(x) + \ddot{\varepsilon}_{i+1}(x')$. In particular, $\ddot{\varepsilon}_{i+1}(x)$ and $\ddot{\varepsilon}_{i+1}(x')$ must be finite. Moreover, since $\ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = y \mathbin{\ddot{\otimes}}y'$, we have $\ddot{\varphi}_i(x) = 0$ or $\ddot{\varepsilon}_i (x')=0$. Following Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have two cases to consider: **Case 1.1.** Suppose that $\ddot{\varepsilon}_i (x') = 0$. Then, $$\ddot{e}_i(x) = y, \quad x' = y'.$$ Since $\ddot{\varepsilon}_{i+1}(x)$ is finite, Proposition [Proposition 8](#prop:local_ax){reference-type="ref" reference="prop:local_ax"} implies that $$\label{eq:2_2_eps_phs} \ddot{\varepsilon}_{i+1}(y) = \ddot{\varepsilon}_{i+1}(x), \quad \ddot{\varphi}_{i+1}(y) = \ddot{\varphi}_{i+1}(x)-1.$$ Therefore, since $\mathcal{Q}$ is seminormal, we have $\ddot{\varphi}_{i+1}(x)>0$, and consequently, $\ddot{\varepsilon}_{i+1} (x') = 0$. Since $x' = y'$, we have $$\label{eq:2_2_zero} \ddot{\varepsilon}_{i+1} (x') = \ddot{\varepsilon}_{i+1}(y') = 0.$$ Then, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, and equations [\[eq:2_2\_eps_phs\]](#eq:2_2_eps_phs){reference-type="eqref" reference="eq:2_2_eps_phs"} and [\[eq:2_2\_zero\]](#eq:2_2_zero){reference-type="eqref" reference="eq:2_2_zero"}, we have $$\begin{aligned} \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') &= \ddot{\varepsilon}_{i+1} (y) + \ddot{\varepsilon}_{i+1}(y') \\ &=\ddot{\varepsilon}_{i+1} (x) + \ddot{\varepsilon}_{i+1}(x') \\ &=\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x')\end{aligned}$$ **Case 1.2.** Now suppose that $\ddot{\varepsilon}_i(x') > 0$. Then, we have $\ddot{\varphi}_i(x) = 0$ and $$x =y, \quad \ddot{e}_i(x')=y'.$$ Since $\ddot{\varepsilon}_{i+1}(x')$ is finite, by axiom **LQ2** (2) we have $$\label{eq:2_2_eps_p} \ddot{\varepsilon}_{i+1}(x') = \ddot{\varepsilon}_{i+1}(y').$$ If $\ddot{\varphi}_{i+1}(x)=0$, since $x=y$, we get $\ddot{\varphi}_{i+1}(y)=0$. Otherwise, we must have $\ddot{\varepsilon}_{i+1}(x')=0$, and by [\[eq:2_2\_eps_p\]](#eq:2_2_eps_p){reference-type="eqref" reference="eq:2_2_eps_p"}, we have $\ddot{\varepsilon}_{i+1}(y') = 0$. Thus, we have $\ddot{\varphi}_{i+1}(y)=0$ or $\ddot{\varepsilon}_{i+1}(y') = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\begin{aligned} \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') &= \ddot{\varepsilon}_{i+1} (y) + \ddot{\varepsilon}_{i+1}(y')\\ &=\ddot{\varepsilon}_{i+1} (x) + \ddot{\varepsilon}_{i+1}(x') \\ &=\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x')\end{aligned}$$ 2. Now suppose that $\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = +\infty$. By Lemma [Lemma 15](#lem:tens_inf){reference-type="ref" reference="lem:tens_inf"}, we have $\ddot{\varphi}_{i+1}(x)>0$ or $\ddot{\varepsilon}_{i+1}(x')>0$. We will show that $\ddot{\varepsilon}_{i+1}(x \mathbin{\ddot{\otimes}}x') \neq \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y')$ if, and only if, $\ddot{\varepsilon}_{i} (y \mathbin{\ddot{\otimes}}y') = 0$. Suppose that $\ddot{\varepsilon}_i(y \mathbin{\ddot{\otimes}}y') = 0$. By Proposition [Proposition 19](#prop:LQ1_qtens){reference-type="ref" reference="prop:LQ1_qtens"}, we have $\ddot{\varphi}_{i+1} (y \mathbin{\ddot{\otimes}}y') = 0 \neq + \infty$. Therefore, by **Q2**, $\ddot{\varepsilon}_{i+1} (y \mathbin{\ddot{\otimes}}y') \neq +\infty$ and hence $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') \neq \ddot{\varepsilon}_{i+1}(x \mathbin{\ddot{\otimes}}x') = + \infty$. Now suppose that $\ddot{\varepsilon}_{i+1}(x \mathbin{\ddot{\otimes}}x') \neq \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y')$. Then, $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') \neq + \infty$, which implies, in particular, that $\ddot{\varepsilon}_{i+1}(y)$ and $\ddot{\varepsilon}_{i+1}(y')$ are finite, and thus, $\ddot{\varphi}_{i+1}(y) = 0$ or $\ddot{\varepsilon}_{i+1}(y') = 0$. By Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have two cases to consider. **Case 2.1.** Suppose that $\ddot{\varepsilon}_{i}(x') = 0$. Then, $$\ddot{e}_i(x)=y, \quad x'=y'.$$ This implies that $\ddot{\varepsilon}_i(y') = \ddot{\varepsilon}_i(x') =0$, and thus, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\label{eq:2_2_y1} \ddot{\varepsilon}_i(y \mathbin{\ddot{\otimes}}y') = \ddot{\varepsilon}_i (y)+\ddot{\varepsilon}_i(y') = \ddot{\varepsilon}_i (y).$$ If $\ddot{\varphi}_{i+1}(y)=0$, then axiom **LQ1** implies that $\ddot{\varepsilon}_i (y) = 0$. Otherwise, if $\ddot{\varphi}_{i+1}(y) >0$, we must have $\ddot{\varepsilon}_{i+1} (y') = 0$. Since $x'=y'$, we have $\ddot{\varepsilon}_{i+1}(x') = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1}(x) + \ddot{\varepsilon}_{i+1} (x') = \ddot{\varepsilon}_{i+1} (x) = +\infty.$$ Since $\ddot{\varepsilon}_{i+1}(y) \neq + \infty$, we have $\ddot{\varepsilon}_{i+1}(x) \neq \ddot{\varepsilon}_{i+1}(y)$, and then, axiom **LQ2** (2) implies that $\ddot{\varepsilon}_i(y)=0$. In either case we get $\ddot{\varepsilon}_i(y) = 0$, and thus, by [\[eq:2_2\_y1\]](#eq:2_2_y1){reference-type="eqref" reference="eq:2_2_y1"}, $\ddot{\varepsilon}_{i}(y \mathbin{\ddot{\otimes}}y') = 0$. **Case 2.2.** Now suppose that $\ddot{\varepsilon}_i(x') > 0$. Then, we have $\ddot{\varphi}_i(x) = 0$ and $$x=y, \quad \ddot{e}_i(x')=y'.$$ Since $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') \neq +\infty$, we have $\ddot{\varphi}_{i+1}(y) = 0$ or $\ddot{\varepsilon}_{i+1}(y')=0$. We claim that $\ddot{\varepsilon}_{i+1}(y') > 0$. If $\ddot{\varepsilon}_{i+1}(y') = 0$, then, as $\ddot{e}_i (x') = y'$, axiom **LQ2** (2) implies that $\ddot{\varepsilon}_{i+1}(x') = \ddot{\varepsilon}_{i+1}(y') = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\ddot{\varepsilon}_{i+1} (x) = \ddot{\varepsilon}_{i+1}(x) + \ddot{\varepsilon}_{i+1}(x') = \ddot{\varepsilon}_{i+1}(x \mathbin{\ddot{\otimes}}x') = + \infty.$$ But since $x=y$, we have $\ddot{\varepsilon}_{i+1}(y) = + \infty$, and thus $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') = + \infty$, which contradicts the hypothesis that $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') \neq + \infty$. Thus, we have $\ddot{\varepsilon}_{i+1}(x') > 0$ and consequently, $\ddot{\varphi}_{i+1}(y) = 0$. Axiom **LQ1** then implies that $\ddot{\varepsilon}_i(y) = 0$. Since $\ddot{\varphi}_i(x) = 0$ and $x = y$, we get $\ddot{\varphi}_i (y) = 0$. Thus, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\label{eq:2_2_y2} \ddot{\varepsilon}_{i}(y \mathbin{\ddot{\otimes}}y') = \ddot{\varepsilon}_{i}(y) + \ddot{\varepsilon}_{i}(y') = \ddot{\varepsilon}_{i} (y').$$ Moreover, since $\ddot{\varphi}_{i+1}(y) = 0$ and $x = y$, we have $\ddot{\varphi}_{i+1}(x) = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\label{eq:2_2_x_inf} \ddot{\varepsilon}_{i+1} (x) + \ddot{\varepsilon}_{i+1}(x') = \ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = +\infty,$$ which implies that $\ddot{\varepsilon}_{i+1}(x) = + \infty$ or $\ddot{\varepsilon}_{i+1} (x') = + \infty$. Since $\ddot{\varphi}_{i+1}(x) = 0$ (in particular, it is finite), we get that $\ddot{\varepsilon}_{i+1}(x)$ is finite. Thus, [\[eq:2_2\_x_inf\]](#eq:2_2_x_inf){reference-type="eqref" reference="eq:2_2_x_inf"} implies that $\ddot{\varepsilon}_{i+1}(x') = + \infty$. Since $\ddot{\varepsilon}_{i+1}(y') \neq + \infty$, we have $\ddot{\varepsilon}_{i+1} (x') \neq \ddot{\varepsilon}_{i+1}(y')$, and thus, by axiom **LQ2** (2), we have $\ddot{\varepsilon}_i(y') = 0$. Therefore, by [\[eq:2_2\_y2\]](#eq:2_2_y2){reference-type="eqref" reference="eq:2_2_y2"}, we have $\ddot{\varepsilon}_{i}(y \mathbin{\ddot{\otimes}}y') = 0$. 3. Finally, we show that $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') = 0$ implies that $\ddot{\varepsilon}_{i+1}(x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y')$. If $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y') = 0$, then, in particular, $\ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y')$ is finite, and thus $\ddot{\varphi}_{i+1}(y)=0$ or $\ddot{\varepsilon}_{i+1}(y')=0$. Thus, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"} we have $$\ddot{\varepsilon}_{i+1}(y) + \ddot{\varepsilon}_{i+1}(y') = \ddot{\varepsilon}_{i+1} (y \mathbin{\ddot{\otimes}}y') = 0.$$ This implies, since $\mathcal{Q}$ and $\mathcal{Q}'$ are seminormal, that $\ddot{\varepsilon}_{i+1} (y) = \ddot{\varepsilon}_{i+1} (y') = 0$. If $\ddot{\varepsilon}_i(x')=0$, we get $\ddot{e}_i(x)=y$ and $\quad x'=y'$. Since $\ddot{\varepsilon}_{i+1}(y) = 0$, axiom **LQ2** (2) implies that $\ddot{\varepsilon}_{i+1}(x) = \ddot{\varepsilon}_{i+1}(y) = 0$. And since $x' = y'$, we have $\ddot{\varepsilon}_{i+1}(x') = \ddot{\varepsilon}_{i+1}(y') = 0$. Therefore, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1} (x) + \ddot{\varepsilon}_{i+1} (x') = \ddot{\varepsilon}_{i+1} (y) + \ddot{\varepsilon}_{i+1} (y') = \ddot{\varepsilon}_{i+1} (y \mathbin{\ddot{\otimes}}y').$$ If $\ddot{\varepsilon}_i(x')>0$, then $x=y$ and $\ddot{e}_i(x') = y'$. Since $\ddot{\varepsilon}_{i+1}(y')=0$, as before, axiom **LQ2** (2) implies that $\ddot{\varepsilon}_{i+1} (x') = \ddot{\varepsilon}_{i+1} (y') = 0$, and as $x=y$, we get $\ddot{\varepsilon}_{i+1} (x)=\ddot{\varepsilon}_{i+1}(y)=0$. Reasoning as before, we conclude that $\ddot{\varepsilon}_{i+1} (x \mathbin{\ddot{\otimes}}x') = \ddot{\varepsilon}_{i+1}(y \mathbin{\ddot{\otimes}}y')$.  ◻ **Proposition 21**. *$\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies ***LQ3*** and ***LQ3$'$***.* *Proof.* **We show that $\mathcal{Q}\mathbin{\ddot{\otimes}}\mathcal{Q}'$ satisfies LQ3.** Let $i, j \in I$, with $i \neq j$, and suppose that $\ddot{e}_i(x \mathbin{\ddot{\otimes}}x')$ and $\ddot{e}_j(x \mathbin{\ddot{\otimes}}x')$ are both defined. By Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have four cases to consider. **Case 1.** Suppose that $\ddot{\varepsilon}_i(x') = 0$ and $\ddot{\varepsilon}_j(x') = 0$. Then, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\label{eq:3_1_qt} \ddot{e}_i(x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}x', \quad \ddot{e}_j(x \mathbin{\ddot{\otimes}}x') = \ddot{e}_j (x) \mathbin{\ddot{\otimes}}x'.$$ Since $\ddot{e}_i(x) = y$ and $\ddot{e}_j(x)= z$ are both defined, by axiom **LQ3** we have $$\label{eq:3_1_LQ} \ddot{e}_i \ddot{e}_j (x) = \ddot{e}_j \ddot{e}_i (x) \neq \bot.$$ Then, since $\ddot{\varepsilon}_i (x') = \ddot{\varepsilon}_j(x') = 0$, we have $$\begin{aligned} \ddot{e}_i \ddot{e}_j (x \mathbin{\ddot{\otimes}}x') &= \ddot{e}_i (\ddot{e}_j(x) \mathbin{\ddot{\otimes}}x') &\text{(by \eqref{eq:3_1_qt})}\\ &= \ddot{e}_i \ddot{e}_j (x) \mathbin{\ddot{\otimes}}x' &\text{(by Proposition \ref{prop:ricardo})}\\ &= \ddot{e}_j \ddot{e}_i (x) \mathbin{\ddot{\otimes}}x' &\text{(by \eqref{eq:3_1_LQ})}\\ &= \ddot{e}_j (\ddot{e}_i(x) \mathbin{\ddot{\otimes}}x') &\text{(by Proposition \ref{prop:ricardo})}\\ &= \ddot{e}_j \ddot{e}_i (x \mathbin{\ddot{\otimes}}x'). &\text{(by \eqref{eq:3_1_qt})}\end{aligned}$$ **Case 2.** Suppose that $\ddot{\varepsilon}_i (x') =0$ and $\ddot{\varepsilon}_j(x') > 0$. Then, we have $\ddot{\varphi}_j(x)=0$ and $$\label{eq:3_2_qt} \ddot{e}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}x', \quad \ddot{e}_j(x \mathbin{\ddot{\otimes}}x') = x \mathbin{\ddot{\otimes}}\ddot{e}_j(x').$$ We claim that $j \neq i+1$. If $j = i + 1$, then, since $\ddot{e}_i(x)$ is defined, we have $\ddot{\varepsilon}_i(x)>0$, and axiom **LQ1** implies that $\ddot{\varphi}_{i+1}(x)>0$. And since $\ddot{e}_j (x') = \ddot{e}_{i+1} (x')$ is defined, we have $\ddot{\varepsilon}_{i+1} (x')>0$. Thus, by Definition [Definition 14](#def:qtens){reference-type="ref" reference="def:qtens"}, we have $\ddot{e}_j(x \mathbin{\ddot{\otimes}}x') = \ddot{e}_{i+1} (x \mathbin{\ddot{\otimes}}x')=\bot$, which contradicts the hypothesis. Thus, $j \neq i+1$, and we have two cases to consider. Case 2.1 : Suppose that $|i-j|>1$. By axiom **LQ2** (1), we have $\ddot{\varepsilon}_i(\ddot{e}_j (x')) = \ddot{\varepsilon}_i (x') = 0$. Thus, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\label{eq:3_2_ij} \ddot{e}_i \ddot{e}_j (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x \mathbin{\ddot{\otimes}}\ddot{e}_j(x')) = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}\ddot{e}_j (x').$$ Axiom **LQ2** (1) also implies that $\ddot{\varepsilon}_j (\ddot{e}_i (x)) = \ddot{\varepsilon}_j (x) > 0$, and therefore, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\label{eq:3_2_ji} \ddot{e}_j \ddot{e}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_j (\ddot{e}_i (x) \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}\ddot{e}_j (x').$$ From [\[eq:3_2\_ij\]](#eq:3_2_ij){reference-type="eqref" reference="eq:3_2_ij"} and [\[eq:3_2\_ji\]](#eq:3_2_ji){reference-type="eqref" reference="eq:3_2_ji"}, we get $\ddot{e}_i \ddot{e}_j (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_j \ddot{e}_i (x \mathbin{\ddot{\otimes}}x')$. Case 2.2 : Now suppose that $j=i-1$. We claim that $\ddot{\varphi}_{i-1}(\ddot{e}_i (x))$ is finite. If $\ddot{\varphi}_{i-1} (\ddot{e}_i(x)) = + \infty$, then, as $\ddot{\varphi}_{i-1} (x) = \ddot{\varphi}_j (x) = 0$, we would have $\ddot{\varphi}_{i-1} (\ddot{e}_i (x)) \neq \ddot{\varphi}_{i-1} (x)$. Thus, by axiom **LQ2** (3), we would have $\ddot{\varphi}_{i-1}(x) > 0$, which is a contradiction. Therefore, we have $\ddot{\varphi}_{i-1}(\ddot{e}_i(x)) \neq + \infty$, and then, axiom **LQ2** (3) implies that $\ddot{\varphi}_{i-1} (\ddot{e}_i (x)) = \ddot{\varphi}_{i-1} (x) = 0$. Consequently, $\ddot{e}_{i-1} (\ddot{e}_i (x) \mathbin{\ddot{\otimes}}x')$ is defined. Moreover, $\ddot{\varepsilon}_j (x') = \ddot{\varepsilon}_{i-1}(x') > 0$, and thus Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"} implies that $$\label{eq:3_2_i1_i} \ddot{e}_{i-1} \ddot{e}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_{i-1} (\ddot{e}_i (x) \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}\ddot{e}_{i-1} (x').$$ Since $\ddot{\varepsilon}_i(x') = 0$, in particular, $\ddot{\varepsilon}_i (x') \neq + \infty$. Thus, by axiom **LQ2** (2), we have $\ddot{\varepsilon}_i (\ddot{e}_{i-1}(x')) = \ddot{\varepsilon}_i (x') = 0$, and by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, $$\label{eq:3_2_i_i1} \ddot{e}_i \ddot{e}_{i-1} (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i (x \mathbin{\ddot{\otimes}}\ddot{e}_{i-1} (x')) = \ddot{e}_i (x) \mathbin{\ddot{\otimes}}\ddot{e}_{i-1} (x').$$ From [\[eq:3_2\_i1_i\]](#eq:3_2_i1_i){reference-type="eqref" reference="eq:3_2_i1_i"} and [\[eq:3_2\_i_i1\]](#eq:3_2_i_i1){reference-type="eqref" reference="eq:3_2_i_i1"}, we get $\ddot{e}_{i-1} \ddot{e}_i (x \mathbin{\ddot{\otimes}}x') = \ddot{e}_i \ddot{e}_{i-1} (x \mathbin{\ddot{\otimes}}x').$ **Case 3.** Suppose that $\ddot{\varepsilon}_i (x') > 0$ and $\ddot{\varepsilon}_j (x') = 0$. This is similar to the previous case, except now we have $j \neq i-1$. **Case 4.** Suppose that $\ddot{\varepsilon}_i (x') > 0$ and $\ddot{\varepsilon}_j(x')>0$. Then, we have $\ddot{\varphi}_i (x) = \ddot{\varphi}_j (x) = 0$, and $$\label{eq:3_4_qt} \ddot{e}_i (x \mathbin{\ddot{\otimes}}x') = x \mathbin{\ddot{\otimes}}\ddot{e}_i(x'), \quad \ddot{e}_j (x \mathbin{\ddot{\otimes}}x') = x \mathbin{\ddot{\otimes}}\ddot{e}_j(x').$$ From [\[eq:3_4\_qt\]](#eq:3_4_qt){reference-type="eqref" reference="eq:3_4_qt"}, we get that $\ddot{\varepsilon}_i (\ddot{e}_j (x')) >0$ and $\ddot{\varepsilon}_j (\ddot{e}_i (x'))>0$. Since $\ddot{e}_i (x')$ and $\ddot{e}_j (x')$ are both defined, we have $\ddot{e}_i \ddot{e}_j (x') = \ddot{e}_j \ddot{e}_i (x') \neq \bot$, by axiom **LQ3**. And since $\ddot{\varphi}_i (x) = \ddot{\varphi}_j (x) = 0$, by Proposition [Proposition 16](#prop:ricardo){reference-type="ref" reference="prop:ricardo"}, we have $$\begin{aligned} \ddot{e}_i \ddot{e}_j (x \mathbin{\ddot{\otimes}}x') &= \ddot{e}_i (x \mathbin{\ddot{\otimes}}\ddot{e}_j (x'))\\ &= x \mathbin{\ddot{\otimes}}\ddot{e}_i \ddot{e}_j (x')\\ &= x \mathbin{\ddot{\otimes}}\ddot{e}_j \ddot{e}_i (x')\\ &= \ddot{e}_j (x \mathbin{\ddot{\otimes}}\ddot{e}_i (x'))\\ &= \ddot{e}_j \ddot{e}_i (x \mathbin{\ddot{\otimes}}x'). \qedhere\end{aligned}$$ ◻ # Stembridge crystals and quasi-crystals {#sec:stem_crystals_quasi} In what follows we consider $\mathcal{C}$ to be a crystal graph of type $A_{n-1}$. *Remark 22*. We consider the weight lattice of $\mathcal{C}$ to be $\mathbb{Z}^n$ modulo $\mathbf{e_1} + \cdots + \mathbf{e_n}$. Therefore, we may fix a representative such that highest weights are always partitions, followed by a possible sequence of zeros. This implies that the entries of $\mathrm{wt}(x)$ are non-negative, for any $x \in \mathcal{C}$. We now state some auxiliary results. **Lemma 23**. *Let $\mathcal{C}$ be a Stembridge crystal, and suppose that $\widetilde{e}_i(x)=y \in \mathcal{C}$. Let $j\neq i$. Then,* 1. *$\widetilde{\varepsilon}_j (y) \geq \widetilde{\varepsilon}_j (x)$,* 2. *$\widetilde{\varphi}_j (y) \leq \widetilde{\varphi}_j (x)$.* *Proof.* This is a direct consequence of Stembridge axioms. ◻ **Lemma 24**. *Let $\mathcal{C}$ be a seminormal crystal. Then,* 1. *$\widetilde{\varepsilon}_i (x) \leq \mathrm{wt}_{i+1} (x)$,* 2. *$\widetilde{\varphi}_i (x) \leq \mathrm{wt}_i (x)$.* *Proof.* We prove the first statement; the second is similar. Since $\mathcal{C}$ is seminormal, let $k_0 := \widetilde{\varepsilon}_i (x) = \max \{k: \widetilde{e}_i^{\,k} (x) \neq \bot\}$. If $k_0 > \mathrm{wt}_{i+1} (x)$, then, as $\mathrm{wt}(\widetilde{e}_i^{\,k_0} (x)) = \mathrm{wt}(x) + k_0 \alpha_i$, we would have $\mathrm{wt}_{i+1} (\widetilde{e}_i^{\,k_0} (x)) = \mathrm{wt}_{i+1} (x)-k_0 < 0$, which is a contradiction by Remark [Remark 22](#rmk:wt_lattice){reference-type="ref" reference="rmk:wt_lattice"}. ◻ **Lemma 25**. *$\widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x)$ if and only if $\widetilde{\varphi}_i (x) = \mathrm{wt}_i (x)$.* *Proof.* Suppose that $\widetilde{\varepsilon}_i(x) = \mathrm{wt}_{i+1} (x)$. Then, by **C2**, $\widetilde{\varphi}_i (x) = \widetilde{\varepsilon}_i (x) + \langle \mathrm{wt}(x), \alpha_i \rangle = \mathrm{wt}_{i+1} (x) + \mathrm{wt}_i (x) - \mathrm{wt}_{i+1} (x) = \mathrm{wt}_i (x)$. The other implication is proved analogously. ◻ Next, we provide a construction to obtain quasi-crystal graphs from a connected crystal graph. Recently, Maas-Gariépy introduced an equivalent construction, by considering the induced subgraphs corresponding to fundamental quasi-symmetric functions [@MG23]. **Definition 26**. Let $(\mathcal{C}, \widetilde{e}_i, \widetilde{f}_i, \widetilde{\varepsilon}_i, \widetilde{\varphi}_i)$ be a connected Stembridge crystal. We construct a quasi-crystal $(\mathcal{Q}_{\mathcal{C}}, \ddot{e}_i, \ddot{f}_i, \ddot{\varepsilon}_i, \ddot{\varphi}_i)$ as follows: $\mathcal{Q}_{\mathcal{C}}$ has the same underlying set, index set $I$ and weight function $\mathrm{wt}$ as $\mathcal{C}$, and we define $$\ddot{\varepsilon}_i (x) := \begin{cases} \widetilde{\varepsilon}_i (x) & \text{if}\; \widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x)\\ + \infty & \text{otherwise} \end{cases},$$ $$\ddot{e}_i (x) := \begin{cases} \widetilde{e}_i (x) & \text{if}\; \widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x)\\ \bot & \text{otherwise} \end{cases}$$ and set $\ddot{\varphi}_i (x) := \ddot{\varepsilon}_i (x) + \langle \mathrm{wt}(x), \alpha_i \rangle$, and $\ddot{f}_i (x) := y$ if and only if $\ddot{e}_i (y)=x$. If $\widetilde{f}_i (x) \neq \bot$ and $\ddot{f}_i (x) = \bot$, we say that $\widetilde{f}_i$ is a *strict Kashiwara operator* on $x \in \mathcal{C}$. Otherwise, it is simply called a quasi-Kashiwara operator. In what follows we consider $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$ as in Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"}. **Lemma 27**. *Let $x, y \in \mathcal{C}$ be such that $\widetilde{e}_i (x) = y$. Then, using the notation of Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"}, $\ddot{\varepsilon}_i (x) = + \infty$ if and only if $\ddot{\varepsilon}_i (y) = + \infty$.* *Proof.* By Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"} and Lemma [Lemma 24](#lem:leq_wt){reference-type="ref" reference="lem:leq_wt"}, $\ddot{\varepsilon}_i (x) = + \infty$ implies that $\widetilde{\varepsilon}_i (x) < \mathrm{wt}_{i+1} (x)$, and consequently $$\widetilde{\varepsilon}_i (x) -1 < \mathrm{wt}_{i+1} (x) -1.$$ Since $\widetilde{e}_i (x) = y$, by **C1** we have $\widetilde{\varepsilon}_i (y) = \widetilde{\varepsilon}_i (x)-1$ and $\mathrm{wt}_{i+1} (y) = \mathrm{wt}_{i+1}(x)-1$, and thus $\widetilde{\varepsilon}_i (y) < \mathrm{wt}_{i+1} (y)$ and $\ddot{\varepsilon}_i(y) = + \infty$. The proof of the reverse implication is analogous. ◻ **Lemma 28**. *Let $i, j \in I$ be such that $|i-j|>1$ and suppose that $\ddot{e}_i(x) = y$. Then, $\ddot{\varepsilon}_j(x) = + \infty$ if and only if $\ddot{\varepsilon}_j (y) = + \infty$.* *Proof.* Since $\ddot{e}_i(x) = y$, we have $\widetilde{e}_i(x) = y$ (and $\widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1}(x)$), and by axiom **S1**, $\widetilde{\varepsilon}_j (x) = \widetilde{\varepsilon}_j (y)$. Suppose that $\ddot{\varepsilon}_j (x) = + \infty$. Then, $\widetilde{\varepsilon}_j (x) < \mathrm{wt}_{j+1} (x)$ and thus, $\widetilde{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (x) < \mathrm{wt}_{i+1} (x)$. Since $|i-j|>1$, we have $\mathrm{wt}_{j+1} (x) = \mathrm{wt}_{j+1} (y)$. Therefore, $\widetilde{\varepsilon}_j(y) < \mathrm{wt}_{j+1}(y)$ and $\ddot{\varepsilon}_j (y) = + \infty$. The other implication is proved similarly. ◻ **Lemma 29**. *Suppose that $\ddot{\varepsilon}_i(x) = + \infty$. Then, $\mathrm{wt}_i (x), \mathrm{wt}_{i+1}(x) >0$.* *Proof.* Since $\ddot{\varepsilon}_i (x) = + \infty$, we have $\widetilde{\varepsilon}_i (x) < \mathrm{wt}_{i+1} (x)$ and $\widetilde{\varphi}_i(x) < \mathrm{wt}_i (x)$, by Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"} and Lemma [Lemma 25](#lem:eps_phis_eq){reference-type="ref" reference="lem:eps_phis_eq"}. Since $\mathcal{C}$ is seminormal, we have $\widetilde{\varepsilon}_i (x),\, \widetilde{\varphi}_i (x) \geq 0$, which implies that $\mathrm{wt}_i (x),\, \mathrm{wt}_{i+1}(x) >0$. ◻ We prove the following result in subsection [4.1](#subsec:proof_thm){reference-type="ref" reference="subsec:proof_thm"} below. **Theorem 30**. *The quasi-crystal graph $\mathcal{Q}_{\mathcal{C}}$ is seminormal and satisfies the axioms of Definition [Definition 6](#def:local_axioms){reference-type="ref" reference="def:local_axioms"}.* An example of the construction of Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"} is depicted in Figure [8](#fig:crystal_quasi_3121){reference-type="ref" reference="fig:crystal_quasi_3121"}. {reference-type="ref" reference="def:crystal_quasi"}.](crystal_quasi_3121.pdf){#fig:crystal_quasi_3121} **Corollary 31**. *Let $\mathcal{C}$ be a connected Stembridge crystal having highest weight $\lambda$. Then the number of connected components in $\mathcal{Q}_{\mathcal{C}}$ is given by $f_{\lambda}$, the number of standard Young tableaux of shape $\lambda$.* *Proof.* The character of $\mathcal{C}$ is the Schur function $s_{\lambda}$, where $\lambda$ is the highest weight (see, for instance, [@BumpSchi17]). Since the character of a quasi-crystal connected component is a fundamental quasi-symmetric function $F_{\alpha}$, taking the characters of $\mathcal{Q}_{\mathcal{C}}$, one obtains a decomposition of $s_{\lambda}$ as a sum of fundamental quasi-symmetric functions. The result then follows from the following decomposition [@Gess19]: $$s_{\lambda} = \sum\limits_{T \in \mathsf{SYT}(\lambda)} F_{\mathsf{DesComp}(T)},$$ where $\mathsf{DesComp}(T)$ denotes the descent composition of a standard tableau $T$, together with the fact that the fundamental quasi-symmetric functions $F_{\alpha}$ form a basis for the ring of quasi-symmetric functions. ◻ ## Proof of Theorem [Theorem 30](#thm:crystal_quasi_ax){reference-type="ref" reference="thm:crystal_quasi_ax"} {#subsec:proof_thm} In this section we prove Theorem [Theorem 30](#thm:crystal_quasi_ax){reference-type="ref" reference="thm:crystal_quasi_ax"}. For simplicity, we let $\mathcal{Q}:= \mathcal{Q}_{\mathcal{C}}$. By construction, $\mathcal{Q}$ is seminormal. **Proposition 32**. *$\mathcal{Q}$ satisfies axiom ***LQ1***.* *Proof.* Let $i, i+1 \in I$. Suppose that $\ddot{\varepsilon}_i (x) = 0$. Since $\ddot{\varepsilon}_i (x) \neq + \infty$, by Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"}, we have $\ddot{\varepsilon}_i (x) = \widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x) = 0$. By Lemma [Lemma 24](#lem:leq_wt){reference-type="ref" reference="lem:leq_wt"}, we have $\widetilde{\varphi}_{i+1} (x) \leq \mathrm{wt}_{i+1} (x) = 0$. Therefore, $\widetilde{\varphi}_{i+1}(x) = 0 = \mathrm{wt}_{i+1} (x)$ and thus $\ddot{\varphi}_{i+1} (x) = 0$. The other implication is proved analogously. ◻ **Proposition 33**. *$\mathcal{Q}$ satisfies axiom ***LQ2***.* *Proof.* Suppose that $\ddot{e}_i (x) = y$. By Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"}, we have $\widetilde{e}_i (x) = y$. **We show that $\mathcal{Q}$ satisfies LQ2** (1). Suppose that $|i-j|>1$. We claim that $\ddot{\varepsilon}_j (x) = + \infty$ if and only if $\ddot{\varepsilon}_j (y) = + \infty$. Indeed, since $\widetilde{e}_i(x) = y$, by **C1** we have $\mathrm{wt}_{j+1} (x) = \mathrm{wt}_{j+1} (y)$, and by axiom **S1**, we have $\widetilde{\varepsilon}_j(x) = \widetilde{\varepsilon}_j (y)$. Therefore, $\widetilde{\varepsilon}_j (x) < \mathrm{wt}_{j+1} (x)$ if and only if $\widetilde{\varepsilon}_j (y) < \mathrm{wt}_{j+1} (y)$. Thus, if $\ddot{\varepsilon}_j (x) = + \infty$, then $\ddot{\varepsilon}_j (y) = + \infty = \ddot{\varepsilon}_j (x)$. If $\ddot{\varepsilon}_j (x) \neq + \infty$, then $\ddot{\varepsilon}_j (y) \neq + \infty$, therefore $\ddot{\varepsilon}_j (x) = \widetilde{\varepsilon}_j (x)$ and $\ddot{\varepsilon}_j (y) = \widetilde{\varepsilon}_j (y)$. By axiom **S1**, we have $\widetilde{\varepsilon}_j (x) = \widetilde{\varepsilon}_j (y)$, which concludes the proof of **LQ2** (1). **We show that $\mathcal{Q}$ satisfies LQ2** (2). The proof of **LQ2** (3) is analogous. Let $i+1 \in I$ 1. Suppose that $\ddot{\varepsilon}_{i+1} (x) = + \infty$ and that $\ddot{\varepsilon}_i (y) = 0$. By Proposition [Proposition 32](#prop:q_LQ1){reference-type="ref" reference="prop:q_LQ1"}, $\ddot{\varepsilon}_i (y)=0$ implies that $\ddot{\varphi}_{i+1} (y) = 0$. In particular, $\ddot{\varphi}_{i+1} (y) \neq + \infty$, and by Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"}, we have $\ddot{\varepsilon}_{i+1} (y) \neq + \infty$. Therefore, $\ddot{\varepsilon}_{i+1} (x) = + \infty \neq \ddot{\varepsilon}_{i+1} (y)$. 2. Now suppose that $\ddot{\varepsilon}_{i+1} (y) = 0$. We will show that $\ddot{\varepsilon}_{i+1} (x) = \ddot{\varepsilon}_{i+1} (y)$. By Definition [Definition 26](#def:crystal_quasi){reference-type="ref" reference="def:crystal_quasi"} and Lemma [Lemma 25](#lem:eps_phis_eq){reference-type="ref" reference="lem:eps_phis_eq"}, as $\ddot{\varepsilon}_{i+1} (y) = 0 \neq + \infty$, we have $\widetilde{\varepsilon}_{i+1} (y) = \mathrm{wt}_{i+2} (y) = 0$ and $\widetilde{\varphi}_{i+1}(y) = \mathrm{wt}_{i+1} (y)$. Since $\widetilde{e}_i(x)=y$, Lemma [Lemma 23](#lem:ax_s1){reference-type="ref" reference="lem:ax_s1"} implies that $\widetilde{\varepsilon}_{i+1}(y) \geq \widetilde{\varepsilon}_{i+1}(x)$. Thus, since $\widetilde{\varepsilon}_{i+1} (y) = 0$, we have $\widetilde{\varepsilon}_{i+1} (x) \leq 0$, which implies, as $\mathcal{C}$ is seminormal, that $\widetilde{\varepsilon}_{i+1} (x)=0$. Moreover, $\widetilde{e}_i(x) = y$ implies that $\mathrm{wt}_{i+2} (x) = \mathrm{wt}_{i+2} (y)$. Thus, we have $\widetilde{\varepsilon}_{i+1} (x) = \mathrm{wt}_{i+2} (y) = \mathrm{wt}_{i+2} (x) = 0$ and, consequently, $\ddot{\varepsilon}_{i+1} (x) = 0 = \ddot{\varepsilon}_{i+1}(y)$. 3. Suppose that $\ddot{\varepsilon}_{i+1} (x) \neq \ddot{\varepsilon}_{i+1} (y)$. We will show that $\ddot{\varepsilon}_{i+1} (x) = + \infty$ and $\ddot{\varepsilon}_i (y)=0$. Since $\ddot{\varepsilon}_{i+1} (x) \neq \ddot{\varepsilon}_{i+1} (y)$, we claim that $$\label{eq:sq_1} \ddot{\varepsilon}_{i+1} (x) = + \infty, \quad \ddot{\varepsilon}_{i+1} (y) \neq + \infty.$$ If $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$ and $\ddot{\varepsilon}_{i+1}(y) = + \infty$, we would have $\widetilde{\varepsilon}_{i+1} (x) = \mathrm{wt}_{i+2} (x)$ and $\widetilde{\varepsilon}_{i+1} (y) < \mathrm{wt}_{i+2} (y)$. Since $\widetilde{e}_i(x)=y$, we have $\mathrm{wt}(y) = \mathrm{wt}(x) + \alpha_{i}$, hence $\mathrm{wt}_{i+2} (x) = \mathrm{wt}_{i+2} (y)$. This would imply that $$\widetilde{\varepsilon}_{i+1} (y) < \mathrm{wt}_{i+2} (y) = \mathrm{wt}_{i+2} (x) = \widetilde{\varepsilon}_{i+1}(x),$$ which contradicts Lemma [Lemma 23](#lem:ax_s1){reference-type="ref" reference="lem:ax_s1"}. If $\ddot{\varepsilon}_{i+1} (x), \,\ddot{\varepsilon}_{i+1} (y) \neq + \infty$, then we would have $$\widetilde{\varepsilon}_{i+1} (y) = \mathrm{wt}_{i+2} (y) = \mathrm{wt}_{i+2} (x) = \widetilde{\varepsilon}_{i+1} (x),$$ and consequently, $\widetilde{\varepsilon}_{i+1} (x) = \widetilde{\varepsilon}_{i+1} (y)$, which implies $\ddot{\varepsilon}_{i+1} (x) = \ddot{\varepsilon}_{i+1} (y)$, since both are finite, contradicting the hypothesis that $\ddot{\varepsilon}_{i+1} (x) \neq \ddot{\varepsilon}_{i+1} (y)$. Therefore, equation [\[eq:sq_1\]](#eq:sq_1){reference-type="eqref" reference="eq:sq_1"} holds. In particular, as $\widetilde{\varepsilon}_{i+1} (x) < \mathrm{wt}_{i+2} (x) = \mathrm{wt}_{i+2} (y) = \widetilde{\varepsilon}_{i+1} (y)$, by axiom **S1** we have $$\label{eq:1_eps_1} \widetilde{\varepsilon}_{i+1} (y) = \widetilde{\varepsilon}_{i+1} (x)+1.$$ Since $\ddot{\varepsilon}_{i+1} (x) = + \infty$, it remains to show that $\ddot{\varepsilon}_i (y) = 0$. So, suppose that $\ddot{\varepsilon}_i (y) = k > 0$. In particular, as $\ddot{\varepsilon}_i (y) \neq +\infty$ (because $\ddot{e}_i (x) = y$), we have $\widetilde{\varepsilon}_{i} (y) = \mathrm{wt}_{i+1} (y) = k$. Consider the connected component consisting of only $i$-labelled edges, containing $y$. Clearly, that component has a unique highest weight element, which is not $y$, since $\widetilde{\varepsilon}_i (y) > 0$. Let $z$ be the highest weight element of that component. Then, there exists $z_1, \ldots, z_k = z \in \mathcal{Q}$ such that $$\label{eq:1_z} \begin{cases} \ddot{e}_i (y) = z_1 \;\text{(and thus,}\; \widetilde{e}_i (y) = z_1),\\ \ddot{e}_i (z_l) = z_{l+1} \;\text{(and thus,}\; \widetilde{e}_i (z_l) = z_{l+1}), \;\text{for}\; l = 1, \ldots, k-1,\\ \ddot{\varepsilon}_i (z_k) = 0 \;\text{(and thus,}\; \widetilde{\varepsilon}_i (z_k)=0), \end{cases}$$ as shown in the following diagram:  By **LQ1**, $\ddot{\varepsilon}_i (z_k) = 0$ implies that $\ddot{\varphi}_{i+1} (z_k)=0$, and thus, $$\label{eq:phi_z} \widetilde{\varphi}_{i+1}(z_k) = 0.$$ By [\[eq:sq_1\]](#eq:sq_1){reference-type="eqref" reference="eq:sq_1"}, we have $\ddot{\varepsilon}_{i+1} (y) \neq + \infty$ and hence, $$\label{eq:phi_i1_k} \widetilde{\varphi}_{i+1} (y) = \mathrm{wt}_{i+1} (y) = k.$$ Thus, there exists $y_1, \ldots, y_k \in \mathcal{Q}$ such that $$\label{eq:1_y} \begin{cases} \widetilde{f}_{i+1} (y) = y_1\\ \widetilde{f}_{i+1} (y_l) = y_{l+1}, \;\text{for}\; l = 1, \ldots, k-1\\ \widetilde{\varphi}_{i+1} (y_k) = 0, \end{cases}$$ as shown in the following diagram:  From equations [\[eq:1_eps_1\]](#eq:1_eps_1){reference-type="eqref" reference="eq:1_eps_1"} and [\[eq:phi_i1_k\]](#eq:phi_i1_k){reference-type="eqref" reference="eq:phi_i1_k"}, and Proposition [Proposition 3](#prop:stem_prop){reference-type="ref" reference="prop:stem_prop"}, we have $$\label{eq:phi_xy} \widetilde{\varphi}_{i+1} (x) = \widetilde{\varphi}_{i+1} (y) = k.$$ Then, there exists $x_1, \ldots, x_k \in \mathcal{Q}$ such that $$\label{eq:1_x} \begin{cases} \widetilde{f}_{i+1} (x) = x_1,\\ \widetilde{f}_{i+1} (x_l) = x_{l+1}, \;\text{for}\; l = 1, \ldots, k-1,\\ \widetilde{\varphi}_{i+1} (x_k) = 0. \end{cases}$$ We have $\widetilde{f}_i(y) = x$ and $\widetilde{f}_{i+1} (y)= y_1$. From [\[eq:1_eps_1\]](#eq:1_eps_1){reference-type="eqref" reference="eq:1_eps_1"} and [\[eq:phi_xy\]](#eq:phi_xy){reference-type="eqref" reference="eq:phi_xy"}, axiom **S2$'$** implies that $\widetilde{f}_i \widetilde{f}_{i+1} (y) = \widetilde{f}_{i+1} \widetilde{f}_i (y)$ and thus $$\begin{aligned} \widetilde{f}_i (y_1) &= \widetilde{f}_i \widetilde{f}_{i+1} (y) &\text{by \eqref{eq:1_y}}\\ &= \widetilde{f}_{i+1} \widetilde{f}_i(y)\\ &= \widetilde{f}_{i+1} (x) = x_1 &\text{by \eqref{eq:1_x}}\end{aligned}$$ Then, we have $\widetilde{f}_{i} (y_1)=x_1$, or equivalently, $y_1 = \widetilde{e}_i (x_1)$, as illustrated in the diagram below:  Since $\widetilde{f}_{i+1} (x) = x_1$, we have $x = \widetilde{e}_{i+1} (x_1)$. By axiom **S1**, we either have $\widetilde{\varepsilon}_{i} (x) = \widetilde{\varepsilon}_{i} (x_1)$ or $\widetilde{\varepsilon}_{i} (x) = \widetilde{\varepsilon}_{i} (x_1)+1$. We will show that both cases lead to contradictions. Suppose that $\widetilde{\varepsilon}_{i} (x) = \widetilde{\varepsilon}_{i} (x_1)$. Then, since $\widetilde{e}_i (x) = y$ and $\widetilde{e}_i (x_1) = y_1$, **C1** implies that $$k = \widetilde{\varepsilon}_{i} (y) = \widetilde{\varepsilon}_i (x) - 1 = \widetilde{\varepsilon}_i(x_1) -1 = \widetilde{\varepsilon}_{i} (y_1).$$ Therefore, there exists $w_1, \ldots, w_k$ such that $$\label{eq:1_w} \begin{cases} \widetilde{e}_i (y_1) = w_1,\\ \widetilde{e}_i (w_l) = w_{l+1}, \;\text{for}\; l=1, \ldots, k-1,\\ \widetilde{\varepsilon}_i (w_k) = 0. \end{cases}$$ Thus, we have $\widetilde{e}_{i+1} (y_1) = y$ and $\widetilde{e}_i (y_1) = w_1$. Since $\widetilde{\varepsilon}_i (y) = \widetilde{\varepsilon}_i (y_1)$, axiom **S2** implies that $\widetilde{e}_i \widetilde{e}_{i+1} (y_1) = \widetilde{e}_{i+1} \widetilde{e}_i (y_1)$ and thus $$\begin{aligned} \widetilde{e}_{i+1} (w_1) &= \widetilde{e}_{i+1} \widetilde{e}_i (y_1) &\text{by \eqref{eq:1_w}}\\ &= \widetilde{e}_i \widetilde{e}_{i+1} (y_1)\\ &= \widetilde{e}_i (y) &\text{by \eqref{eq:1_y}}\\ &= z_1 &\text{by \eqref{eq:1_z}}\end{aligned}$$ Applying this reasoning iteratively, we get $\widetilde{e}_{i+1} (w_l) = z_l$, for $l = 1, \ldots, k$. In particular, we have $\widetilde{e}_{i+1} (w_k) = z_k$ and thus, $\widetilde{f}_{i+1} (z_k) = w_k$, as depicted in the following diagram:  This implies that $\widetilde{\varphi}_{i+1} (z_k) > 0$, which contradicts [\[eq:phi_z\]](#eq:phi_z){reference-type="eqref" reference="eq:phi_z"}. Now suppose that $\widetilde{\varepsilon}_i (x) = \widetilde{\varepsilon}_i (x_1) +1$. Since we have $\widetilde{e}_i (x_1) = y_1$ and $\widetilde{e}_{i+1} (x_1) = x$, this implies that $$\begin{aligned} \widetilde{\varepsilon}_{i+1} (y_1) &= \widetilde{\varepsilon}_{i+1} (y) + 1 &\text{by \textbf{C1}}\\ &= (\widetilde{\varepsilon}_{i+1} (x) + 1 ) +1 &\text{by \eqref{eq:1_eps_1}}\\ &= \widetilde{\varepsilon}_{i+1} (x_1) +1 &\text{by \eqref{eq:1_x}}\end{aligned}$$ Therefore, axiom **S3** implies that $$\widetilde{e}_i \widetilde{e}_{i+1}^{\,2} \widetilde{e}_i (x_1) = \widetilde{e}_{i+1} \widetilde{e}_i^{\,2} \widetilde{e}_{i+1} (x_1),$$ and furthermore, $$y = \widetilde{e}_i \widetilde{e}_{i+1} (x_1) \neq \widetilde{e}_{i+1} \widetilde{e}_i (x_1) = y.$$ which is again a contradiction. Thus, the original assumption that $\ddot{\varepsilon}_i(y) > 0$ is false, and we have $\ddot{\varepsilon}_i (y) = 0$.  ◻ **Proposition 34**. *$\mathcal{Q}$ satisfies axioms ***LQ3*** and ***LQ3$'$***.* *Proof.* We will prove that $\mathcal{Q}$ satisfies axiom **LQ3**, the proof for **LQ3$'$** is similar. Let $x \in \mathcal{Q}$ and $i,j \in I$, such that $i \neq j$, and suppose that $\ddot{e}_i (x)$ and $\ddot{e}_j (x)$ are both defined. This implies that $\widetilde{e}_i (x)$ and $\widetilde{e}_j (x)$ are both defined as well and that $$\label{eq:q3_wts} \widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x), \quad \widetilde{\varepsilon}_j(x) = \mathrm{wt}_{j+1} (x).$$ **Case 1.** Suppose that $|i-j|>1$. Then, axiom **S1** implies that $\widetilde{\varepsilon}_i (\widetilde{e}_j (x)) = \widetilde{\varepsilon}_i (x)$, and therefore, by axiom **S2**, $$\label{eq:q3_ij} \widetilde{e}_i \widetilde{e}_j (x) = \widetilde{e}_j \widetilde{e}_i (x).$$ Since $|i-j|>1$, we have $\mathrm{wt}_{i+1} (x) = \mathrm{wt}_{i+1} (\widetilde{e}_j (x))$. Thus, [\[eq:q3_wts\]](#eq:q3_wts){reference-type="eqref" reference="eq:q3_wts"} implies that $$\widetilde{\varepsilon}_i (\widetilde{e}_j (x)) = \widetilde{\varepsilon}_i (x) = \mathrm{wt}_{i+1} (x) = \mathrm{wt}_{i+1} (\widetilde{e}_j (x)),$$ and consequently, $\ddot{\varepsilon}_i (\widetilde{e}_j (x)) = \ddot{\varepsilon}_i (\ddot{e}_j (x)) \neq + \infty$. Thus, $\ddot{e}_i \ddot{e}_j (x) = \widetilde{e}_i \widetilde{e}_j (x)$. Applying the same reasoning, we get $\ddot{e}_j \ddot{e}_i (x) = \widetilde{e}_j \widetilde{e}_i (x)$. Therefore, by [\[eq:q3_ij\]](#eq:q3_ij){reference-type="eqref" reference="eq:q3_ij"}, we have $\ddot{e}_i \ddot{e}_j (x) = \ddot{e}_j \ddot{e}_i (x)$. **Case 2.** Suppose that $|i-j|=1$, and without loss of generality, suppose that $$\ddot{e}_i (x) = \widetilde{e}_i (x) = y, \quad \ddot{e}_{i+1} (x) = \widetilde{e}_{i+1} (x) = z.$$ Since $\ddot{e}_{i+1} (x)$ is defined, we have $\ddot{\varepsilon}_{i+1} (x) \neq + \infty$. Therefore, as $\ddot{e}_i (x) =y$, Proposition [Proposition 33](#prop:q_LQ2){reference-type="ref" reference="prop:q_LQ2"} implies that $$\label{eq:q3_i1_xy} \ddot{\varepsilon}_{i+1} (x) = \ddot{\varepsilon}_{i+1} (y) \neq +\infty,$$ and, consequently, $\widetilde{\varepsilon}_{i+1} (x) = \widetilde{\varepsilon}_{i+1} (y)$. Thus, it follows from axiom **S2** that $$\label{eq:q3_i1_com} \widetilde{e}_i \widetilde{e}_{i+1} (x) = \widetilde{e}_{i+1} \widetilde{e}_i (x).$$ From [\[eq:q3_i1_xy\]](#eq:q3_i1_xy){reference-type="eqref" reference="eq:q3_i1_xy"} and [\[eq:q3_i1_com\]](#eq:q3_i1_com){reference-type="eqref" reference="eq:q3_i1_com"}, we have $\ddot{e}_{i+1} \ddot{e}_i (x) = \widetilde{e}_{i+1} \widetilde{e}_i (x) = \widetilde{e}_i \widetilde{e}_{i+1} (x)$. Thus, it remains to show that $\ddot{e}_i \ddot{e}_{i+1} (x) = \widetilde{e}_i \widetilde{e}_{i+1} (x)$. We claim that $\ddot{\varphi}_i (z) \neq + \infty$. If $\ddot{\varphi}_i (z) = + \infty$ and $\ddot{\varphi}_{i+1} (x) > 0$, Propositions [Proposition 8](#prop:local_ax){reference-type="ref" reference="prop:local_ax"} (3.2) and [Proposition 33](#prop:q_LQ2){reference-type="ref" reference="prop:q_LQ2"} would imply that $\ddot{\varphi}_{i} (x) = + \infty$, which contradicts $\ddot{e}_i (x)$ being defined. If $\ddot{\varphi}_i (z) = + \infty$ and $\ddot{\varphi}_{i+1} (x) = 0$, then Proposition [Proposition 32](#prop:q_LQ1){reference-type="ref" reference="prop:q_LQ1"} would imply that $\ddot{\varepsilon}_i (x) = 0$, which also contradicts $\ddot{e}_i (x)$ being defined. Therefore, we have $\ddot{\varphi}_i (z) \neq + \infty$, and thus $\ddot{\varepsilon}_i (z) \neq +\infty$. Therefore, $\ddot{e}_i \ddot{e}_{i+1} (x) = \widetilde{e}_i \widetilde{e}_{i+1} (x)$. ◻

arxiv_math

--- abstract: | In this paper, we propose an alternative technique to dynamic programming for solving stochastic control problems. We consider a weak formulation that is written as an optimization (minimization) problem on the space of probabilities. We then propose a regularized version of this problem obtained by splitting the minimization variables and penalizing the entropy between the two probabilities to be optimized. We show that the regularized problem provides a good approximation of the original problem when the weight of the entropy regularization term is large enough. Moreover, the regularized problem has the advantage of giving rise to optimization problems that are easy to solve in each of the two optimization variables when the other is fixed. We take advantage of this property to propose an alternating optimization algorithm whose convergence to the infimum of the regularized problem is shown. The relevance of this approach is illustrated by solving a high-dimensional stochastic control problem aimed at controlling consumption in electrical systems. author: - "Thibaut BOURDAIS [^1] , Nadia OUDJANE [^2]  and Francesco RUSSO [^3]" bibliography: - ../../../../BIBLIO_FILE/ThesisBourdais.bib date: September 2023 title: An entropy penalized approach for stochastic control problems. Complete version. --- Stochastic control; optimization; Donsker-Varadhan representation; exponential twist; relative entropy. : 49M99; 49J99; 60H10; 60J60; 65C05. # Introduction Stochastic control problems appear in many fields of application such as robotics [@RobotPathIntegral], economics and finance [@touzibook]. Their numerical solution is most often based on the dynamic programming principle allowing the representation of the value function via nonlinear Hamilton-Jacobi-Bellman PDEs or Backward Stochastic Differential Equations (BSDEs). This permits to estimate recursively the value (Bellman) functions backwardly from the terminal instant to the initial instant. However, when the state space is large, estimating the Bellman functions becomes challenging due to the curse of dimensionality. In the last twenty years, mainly motivated by applications in finance, important progress has been made in this field, especially around the numerical resolution of BSDEs or PDEs. We can mention in particular variance reduction techniques [@bender10; @gobet10bis; @gobet17], neural network based approaches [@Hure20; @Germain22], time reversal techniques [@FullyBackward] or Lagrangian decomposition techniques [@Carpentier20; @Seguret]. The idea of this paper is to propose a radically different approach based on a weak reformulation of the stochastic control problem as an optimization problem on the space of probabilities. Interest in optimization problems on the space of probabilities has increased strongly during the recent years with the Monge-Kantorovitch optimal transport problem, which, for two fixed Borel probabilities on $\mathbb R^d$, $\nu_1$ and $\nu_2$ consists in determining a joint law whose marginals are precisely $\nu_1$ and $\nu_2$, minimizing an expected given cost. Benamou and Brenier in [@benamou] propose a dynamical formulation of this problem: it consists in an optimal control problem where the aim is to minimize the integrated kinetic energy of a deterministic dynamical system over a given time horizon, in order to go from the initial law $\nu_1$ to $\nu_2$ as terminal law. Mikami and Thieullen in [@ThieullenMikami] replace the deterministic dynamical system with a diffusion introducing the so called stochastic mass transportation problem. This consists in controlling the drift of the diffusion to minimize over a given finite horizon a mean integrated cost depending on the drift and the state of the process, while imposing the initial and final distribution of the diffusion. Those authors formulate their problem as an optimization on a space of probabilities, for which they make use of convex duality techniques. Tan and Touzi generalize these techniques in [@TanTouzi], controlling the volatility as well. Those authors also propose a numerical scheme in order to approximate the dual formulation of their stochastic mass transport problem. In the same spirit as in [@ThieullenMikami], in this paper, we formulate a stochastic optimal control problem as a minimization on the space of probability measures. We propose an entropic regularization of this optimization problem which suitably approximates the original control problem. Under mild convexity conditions, we prove the convergence of an alternating optimization algorithm to the infimum of the regularized problem and the performance of this algorithm is shown to be competitive in simulation with existing regression-based Monte Carlo approaches relying on dynamic programming. The proof of the convergence of our algorithm relies on geometric arguments rather than classical convex optimization techniques. More precisely, on some filtered probability space $(\Omega, {\cal F},\mathbb P)$, we are interested in a problem of the type $$\label{eq:strongControlIntro} \inf_{\nu } \mathbb E\left[\int_0^T f(r, X_r^{\nu}, \nu_r) dr + g(X_T^\nu)\right],$$ where $\nu$ is a progressively measurable processes taking values in some fixed convex compact domain $\mathbb U\subset \mathbb R^d$. $X = X^\nu$ will be a controlled diffusion process taking values in $\mathbb R^d$ of the form $$\label{eq:nu} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \nu_r dr + \int_0^t \sigma(r,X_r) dW_r.$$ The above problem corresponds to the strong formulation of a stochastic control problem in the sense of Problem (II.4.1.SS) in [@YongZhou2000], and can be generally associated with a weak formulation in the sense of Problem (II.4.2.WS) in [@YongZhou2000], resulting in an optimization problem on a space of probability measures of the form $$\label{eq:controlProblemIntro} J^*:= \inf_{\mathbb P\in {\mathcal P}_{\mathbb U}} J(\mathbb P),\quad\textrm{with}\quad J(\mathbb P):=\mathbb E^\mathbb P\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr + g(X_T)\right],$$ with ${\mathcal P}_{\mathbb U}$ a set of probability measures defined in Definition [Definition 5](#def:PU){reference-type="ref" reference="def:PU"}, such that under $\mathbb P\in {\mathcal P}_\mathbb U$ the canonical process $X$ is decomposed as $$\label{eq:nu_u} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u^\mathbb P_r dr + \int_0^t \sigma(r,X_r) dW_r,$$ where $u^\mathbb P$ is a progressively measurable process with respect to the canonical filtration ${\mathcal F}^X$ of $X$ taking values in $\mathbb U$ and $W$ is some standard Brownian motion. We refer to [@krylov; @touzibook] for a detailed account of stochastic optimal control in strong form and to [@YongZhou2000; @ElKarouiCompactification] for more details on the weak formulation of stochastic optimal control. The link between those two formulations is discussed in Appendix [7.4](#app:equiControl){reference-type="ref" reference="app:equiControl"}. Our essential hypothesis is here that the running cost $f$ is convex in the control variable $u$. Besides the question of the existence of a probability $\mathbb P^*$ for which $J(\mathbb P^*) = J^*$ in [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}, the problem of approximation is crucial. One major difficulty is the lack of convexity of the functional $J$ in [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} with respect to $\mathbb P$, even though the literature includes some techniques to transform the original problem into a minimization of a convex functional, see e.g. [@benamou]. For that reason, we cannot rely on classical convex analysis techniques, see e.g. [@ekeland], in order to perform related algorithms, see e.g. [@Bonnans_Gilbert]. As announced above, our method consists in replacing Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} with the regularized version $$\label{eq:regularizedProblemIntro} {\cal J}_\epsilon^*:= \inf_{(\mathbb P, \mathbb Q) \in \mathcal{A}} {\cal J}_\epsilon(\mathbb Q,\mathbb P),\quad \textrm{with}\quad {\cal J}_\epsilon (\mathbb Q,\mathbb P):= \mathbb E^\mathbb Q\left[\int_0^T f(r, X_r, u^\mathbb P_r)dr + g(X_T)\right] + \frac{1}{\epsilon}H(\mathbb Q| \mathbb P),$$ where $\mathcal{A}$ is a subset of elements $(\mathbb P, \mathbb Q) \in {\mathcal P}(\Omega)^2$ defined in Definition [Definition 10](#def:A){reference-type="ref" reference="def:A"}, $H$ is the relative entropy, see Definition [\[def:klDiv\]](#def:klDiv){reference-type="ref" reference="def:klDiv"}, and the regularization parameter $\epsilon >0$ is intended to vanish to zero in order to impose $\mathbb Q= \mathbb P$. In Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"} one shows that previous infimum is indeed a minimum ${\cal J}_\epsilon^*={\cal J}_\epsilon (\mathbb Q^*_\epsilon,\mathbb P^*_\epsilon)$ (attained on some admissible couple of probability measures $(\mathbb P^*_\epsilon, \mathbb Q^*_\epsilon)\in \mathcal{A}$). Given one solution $(\mathbb P^*_\epsilon, \mathbb Q^*_\epsilon)$ of Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}, Proposition [Proposition 13](#prop:approximateControl){reference-type="ref" reference="prop:approximateControl"} shows that $\mathbb P_\epsilon^*$ is an approximate solution of Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} in the sense that $\mathbb P^*_\epsilon \in {\mathcal P}_\mathbb U$ and the infimum $J^*$ can be indeed approached by $J(\mathbb P^*_\epsilon)$ when $\epsilon \rightarrow 0$ and more precisely $J(\mathbb P^*_\epsilon) - J^* = O(\epsilon)$. The interest of the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} with respect to the original Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} is that the minimization of the functional ${\cal J}_\epsilon$ with respect to one variable $\mathbb Q$ or $\mathbb P$ (the other variable being fixed) can be provided explicitly, see Section [5.1](#sec:pointwiseMinimization){reference-type="ref" reference="sec:pointwiseMinimization"} (for the minimization with respect to $\mathbb P$) and Section [5.2](#sec:markovDrift){reference-type="ref" reference="sec:markovDrift"} (for the minimization with respect to $\mathbb Q$). Indeed, on the one hand, the resolution with respect to $\mathbb Q$ is a well-known problem in the area of large deviations, see [@DupuisEllisLargeDeviations]. It gives rise to a variational representation formulas relating log-Laplace transform of the costs and relative entropy which is linked to a specific case of stochastic optimal control for which it is possible to linearize the HJB equation by an exponential transform, see [@FlemingExpoTrans; @FlemingPathIntegral]. This type of problem is known as path integral control and has been extensively studied with many applications, see [@PathIntegralControl; @RobotPathIntegral; @ControlFlexibility]. On the other hand, the minimization with respect to $\mathbb P$ can be reduced to pointwise minimization. Indeed $u^\mathbb P$ can be expressed as a function $(t, x) \mapsto u^\mathbb P(t, x)$ such that for all $(t, x) \in [0, T] \times \mathbb R^d$, $u^\mathbb P(t, x)$ is independently obtained as the minimum of a strictly convex function. Concerning the convergence of the algorithm we insist again on the fact that ${\cal J}_\epsilon$ is not jointly convex with respect to $(\mathbb Q, \mathbb P)$, so, in Section [4](#sec:algo){reference-type="ref" reference="sec:algo"} we rely on geometric arguments developed in [@CsiszarAlternating] to prove that the iterated values of the algorithm converge to the minimum value ${\cal J}^*$. In Section [6](#sec:example){reference-type="ref" reference="sec:example"}, we show the relevance of this algorithm compared with classical Monte Carlo based regression techniques by considering an application dedicated to the control of thermostatic loads in power systems. # Notations and definitions {#S2} In this section we introduce the basic notions and notations used throughout this document. In what follows, $T \in \mathbb R^+$ will be a fixed time horizon. - All vectors $x \in \mathbb R^d$ are column vectors. Given $x \in \mathbb R^d$, $|x|$ will denote its Euclidean norm. - Given a matrix $A \in \mathbb R^{d \times d}$, $\|A\| := \sqrt {Tr[AA^\top]}$ will denote its Frobenius norm. - Given $\phi \in C^{1, 2}([0, T] \times \mathbb R^d)$, $\partial_t\phi$, $\nabla_x \phi$ and $\nabla_x^2\phi$ will denote respectively the partial derivative of $\phi$ with respect to (w.r.t.) $t \in [0, T]$, its gradient and its Hessian matrix w.r.t. $x \in \mathbb R^d$. - Given any bounded function $\Phi: U \rightarrow V$, $U, V$ being Banach spaces, we denote by $\vert \Phi \vert_\infty$ its supremum. - $\mathbb U$ will denote the closure of a bounded and convex open subset of $\mathbb R^d$ (in particular $\mathbb U$ is a **convex compact** subset of $\mathbb R^d$). $diam(\mathbb U)$ will denote its diameter. - For any topological spaces $E$ and $F,$ $\mathcal{B}(E)$ will denote the Borel $\sigma$-field of $E;$ $C(E, F)$ (${\mathcal B}(E, F)$) will denote the linear space of functions from $E$ to $F$ that are continuous (resp. Borel). ${\mathcal P}(E)$ will denote the Borel probability measures on $E$. Given $\mathbb P\in {\mathcal P}(E),$ $\mathbb E^\mathbb P$ will denote the expectation with respect to (w.r.t.) $\mathbb P.$ - Except if differently specified, $\Omega$ will denote the space of continuous functions from $[0, T]$ to $\mathbb R^d.$ For any $t \in [0, T]$ we denote by $X_t : \omega \in \Omega \mapsto \omega_t$ the coordinate mapping on $\Omega.$ We introduce the $\sigma$-field ${\mathcal F}:= \sigma(X_r, 0 \le r \le T)$. On the measurable space $(\Omega, {\mathcal F}),$ we introduce the **canonical process** $X : \omega \in ([0, T] \times \Omega, {\mathcal B}([0, T])\otimes {\mathcal F}) \mapsto X_t(\omega) = \omega_t \in (\mathbb R^d, {\mathcal B}(\mathbb R^d))$.\ We endow $(\Omega, {\mathcal F})$ with the right-continuous filtration ${\mathcal F}_t := \underset{t \le r \le T}{\bigcap} \sigma(X_r), \ t \in [0,T].$ The filtered space $(\Omega, {\mathcal F}, ({\mathcal F}_t))$ will be called the **canonical space** (for the sake of brevity, we denote $({\mathcal F}_t)_{t \in [0, T]}$ by $({\mathcal F}_t)$). - Given a continuous (locally) square integrable martingale $M$, $\langle M \rangle$ will denote its **quadratic variation**. - Equality between stochastic processes are in the sense of **indistinguishability**. **Definition 1**. *(Relative entropy). [\[def:klDiv\]]{#def:klDiv label="def:klDiv"} Let $E$ be a topological space. Let $\mathbb P, \mathbb Q\in {\mathcal P}(E).$ The **relative entropy** $H(\mathbb Q| \mathbb P)$ between the measures $\mathbb P$ and $\mathbb Q$ is defined by $$\label{eq:relativeEntropy} H(\mathbb Q| \mathbb P) := \left\{ \begin{aligned} & \mathbb E^{\mathbb Q}\left[\log \frac{d\mathbb Q}{d\mathbb P}\right] &\text{if $\mathbb Q\ll \mathbb P$}\\ & + \infty &\text{otherwise.} \end{aligned} \right.$$ with the convention $\log(0/0) = 0$.* **Remark 2**. *The relative entropy $H$ is **non negative** and **jointly convex**, that is for all $\mathbb P_1, \mathbb P_2, \mathbb Q_1, \mathbb Q_2 \in {\mathcal P}(E)$, for all $\lambda \in [0, 1]$, $H(\lambda \mathbb Q_1 + (1 - \lambda) \mathbb Q_2 | \lambda \mathbb P_1 + (1 - \lambda)\mathbb P_2) \le \lambda H(\mathbb Q_1 | \mathbb P_1) + (1 - \lambda)H(\mathbb Q_2| \mathbb P_2)$. Moreover, $(\mathbb P, \mathbb Q) \mapsto H(\mathbb Q| \mathbb P)$ is lower semicontinous with respect to the weak convergence on Polish spaces. We refer to [@DupuisEllisLargeDeviations] Lemma 1.4.3 for a proof of these properties.* **Definition 3**. *(Minimizing sequence, solution and $\epsilon$-solution). Let $E$ be a generic set. Let $J : E \mapsto \mathbb R$ be a function. Let $J^* := \underset{x \in E}{\inf} J(x)$ (which can be finite or not). A **minimizing sequence** for $J$ is a sequence $(x_n)_{n \ge 0}$ of elements of $E$ such that $J(x_n) \underset{n \rightarrow + \infty}{\longrightarrow} J^*$. We will say that $x^* \in E$ is a **solution** to the optimization Problem $$\label{eq:MinSeq} \underset{x \in E}{\inf} J(x), %\ {\rm if} \ J(x^*) = J^*.$$ if $J(x^*) = J^*.$ In this case, $J^* = \underset{x \in E}{\min} J(x)$. For $\epsilon \ge 0$, we will say that $x^\epsilon \in E$ is an **$\epsilon$-solution** to the optimization Problem [\[eq:MinSeq\]](#eq:MinSeq){reference-type="eqref" reference="eq:MinSeq"} if $0 \le J(x^\epsilon) - J^* \le \epsilon$. We also say that $x^\epsilon$ is $\epsilon$-optimal for the (optimization) Problem [\[eq:MinSeq\]](#eq:MinSeq){reference-type="eqref" reference="eq:MinSeq"}.* We remark that a $0$-solution is a solution of the optimization Problem [\[eq:MinSeq\]](#eq:MinSeq){reference-type="eqref" reference="eq:MinSeq"}. # From the stochastic optimal control problem to a regularized optimization problem {#S3} In this section we consider a stochastic control problem that we reformulate in terms of an optimization problem on a space of probabilities. Later we propose a regularized version of that problem whose solutions are $\varepsilon$-optimal for the original problem. ## The stochastic optimal control problem We specify the assumptions and the formulation of the stochastic optimal control Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} stated in the introduction. Let us first consider a drift $b \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb R^d)$ and a diffusion matrix $\sigma \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb R^{d \times d})$ following the assumptions below. **Hypothesis 4**. *(SDE Diffusion coefficients). [\[hyp:coefDiffusion\]]{#hyp:coefDiffusion label="hyp:coefDiffusion"}* 1. *There exists a constant $C_{b, \sigma} > 0$ such that for all $(t, x) \in [0, T] \times \mathbb R^d$ $$|b(t, x)| + \|\sigma(t, x)\| \le C_{b, \sigma}(1 + |x|).$$* 2. *There exists $c > 0$ such that for all $(t, x) \in [0, T] \times \mathbb R^d, \xi \in \mathbb R^d.$ $$\xi^\top \sigma\sigma^\top(t, x)\xi \ge c |\xi|^2.$$ $\sigma$ is referred in the rest of the paper as **elliptic**.* 3. *For all $x \in \mathbb R^d$, $$\lim_{y \rightarrow x} \sup_{0 \le r \le T} \|\sigma(r, x) - \sigma(r, y)\| = 0.$$* Let us define the admissible set of probabilities ${\mathcal P}_\mathbb U$ for Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}. **Definition 5**. *Let ${\mathcal P}_\mathbb U$ be the set of probability measures on $(\Omega, {\mathcal F})$ such that for all $\mathbb P\in {\mathcal P}_\mathbb U$, under $\mathbb P$ the canonical process decomposes as $$\label{eq:decompP} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u_r^\mathbb Pdr + M_t^\mathbb P,$$ with $x \in \mathbb R^d$, $M^\mathbb P$ is a local martingale such that $\langle M^\mathbb P\rangle_t = \int_0^t \sigma\sigma^\top(r, X_s)dr$, $u^\mathbb P$ is a progressively measurable process with values in $\mathbb U$. If in addition there exists $u \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$ such that $u_t^\mathbb P= u(t, X_t)$ $d\mathbb P\otimes dt$-a.e, we will denote $\mathbb P\in {\mathcal P}_\mathbb U^{Markov}$.* **Remark 6**. *If $\mathbb P\in {\mathcal P}_\mathbb U^{Markov}$ in the sense of Definition [Definition 5](#def:PU){reference-type="ref" reference="def:PU"}, then the following equivalent properties hold.* 1. *One has $$\label{eq:decompPBis} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u^\mathbb P(r,X_r) dr + M_t^\mathbb P,$$ with $x \in \mathbb R^d$, $\langle M^\mathbb P\rangle_t = \int_0^t \sigma\sigma^\top(r, X_s)dr$.* 2. *$\mathbb P$ is solution of the **martingale problem** (in the sense of Stroock and Varadhan in [@stroock]) associated with the initial condition $(0, x)$ and the operator $L_u$ defined for all $\phi \in C_b^{1, 2}([0, T] \times \mathbb R^d)$, $(t, y) \in [0, T] \times \mathbb R^d$ by $$\label{eq:generatorU} L_u\phi(t, y) = \partial_t \phi(t, y) + \langle \nabla_x\phi(t, y), b(t, y) + u(t, y)\rangle + \frac{1}{2}Tr[\sigma\sigma^\top(t, y)\nabla_x^2\phi(t, y)],$$ with $u = u^\mathbb P$.* 3. *$\mathbb P$ is a solution (in law) of $$\label{eq:decompPTer} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u^\mathbb P(r,X_r) dr + \int_0^t \sigma(s,X_s) dW_s,$$ for some suitable Brownian motion $W$.* We will often make use of the following proposition. **Proposition 7**. *Assume Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} holds. Let $u \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$. There exists a unique probability measure $\mathbb P^u \in {\mathcal P}_\mathbb U^{Markov}$ solution to the martingale problem with initial condition $(0, x)$, and the operator $L_{u}$ defined in [\[eq:generatorU\]](#eq:generatorU){reference-type="eqref" reference="eq:generatorU"} in the sense of Remark [Remark 6](#rmk:stroockVaradhan){reference-type="ref" reference="rmk:stroockVaradhan"}.* *Proof.* This result follows from Theorem 10.1.3 in [@stroock]. ◻ Let then $f \in {\mathcal B}([0, T] \times \mathbb R^d \times \mathbb U, \mathbb R^d), ~g \in {\mathcal B}(\mathbb R^d, \mathbb R)$, referred to as the **running cost** and the **terminal cost** respectively, and assume that the following holds. **Hypothesis 8**. *(Cost functions). [\[hyp:costFunctionsControl\]]{#hyp:costFunctionsControl label="hyp:costFunctionsControl"}* 1. *The functions $f, g$ are positive. There exists $C_{f, g} > 0$, $p \ge 1$ such that for all $(t, x, u) \in [0, T] \times \mathbb R^d \times \mathbb U$ $$|f(t, x, u)| + |g(x)| \le C_{f, g}(1 + |x|^p).$$* 2. *$f$ is continuous in $(t, x, u)$, $f(t,x, \cdot)$ is convex for all $(t, x) \in [0, T] \times \mathbb R^d$ and $g$ is continuous.* We conclude this section by a moment estimate, see e.g. Corollary 12 in Section 5.2 in [@krylov], which will be often used in the rest of the paper. **Lemma 9**. *Let $(\Omega, {\mathcal F}, ({\mathcal F}_t), \mathbb P)$ be a filtered probability space. Let $u : [0, T] \times \Omega \rightarrow \mathbb U$ be an $({\mathcal F}_t)$-progressively measurable process. Let $X$ be an Itô process on $(\Omega, {\mathcal F}, \mathbb P)$ which decomposes as $$X_t = x + \int_0^t b(r, X_r)dr + \int_0^tu_rdr + M_t^\mathbb P,$$ where $M^\mathbb P$ is a martingale such that $\langle M^\mathbb P\rangle_t = \int_0^t \sigma\sigma^\top(r, X_r)dr$. Let $q \ge 1$. Under Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} there exists a constant $C_{\mathbb U}(q) > 0,$ which depends only on $T, C_{b, \sigma}$, $diam(\mathbb U)$ (and $q$), such that for all $\mathbb P\in {\mathcal P}_{\mathbb U}$, $$\mathbb E^{\mathbb P}\left[\sup_{0 \le t \le T} |X_t|^q\right] \le C_{\mathbb U}(q).$$* Under Hypotheses [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} and [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"}, by the moment estimate given by Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} one has $$\mathbb E^{\mathbb P}\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr + g(X_T)\right] < + \infty,$$ for all $\mathbb P\in {\mathcal P}_{\mathbb U}$. Then under Hypotheses [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} and [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} the function $J$ introduced in [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} is well-defined on ${\mathcal P}_\mathbb U$. ## The regularized optimization problem As mentioned earlier, finding a numerical approximation of the solution of a stochastic optimal control problem often relies on solving the associated Hamilton-Jacobi-Bellman (HJB) equation. This is typically done via finite difference schemes when $d \le 3$ and by Monte Carlo methods for estimating Forward BSDE (i.e. a BSDE whose underlying is a Markov diffusion) when $d > 3.$ We aim at finding another way to compute an optimal strategy that does not require the approximation of the solution of the HJB equation. To this aim we regularize Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} by doubling the decision variables and adding a relative entropy term in the objective function. We get the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} where $\mathcal{A}$ is the subset of elements $(\mathbb P, \mathbb Q) \in {\mathcal P}(\Omega)^2$ defined below. **Definition 10**. *Let $\mathcal{A}$ be the set of probability measures $(\mathbb P, \mathbb Q) \in {\mathcal P}(\Omega)^2$ such that* 1. *$\mathbb P\in \mathcal{P}_{\mathbb U}$,* 2. *$H(\mathbb Q| \mathbb P) < + \infty.$* In the perspective of solving the regularized optimization Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} we will introduce in Sections [5.1](#sec:pointwiseMinimization){reference-type="ref" reference="sec:pointwiseMinimization"} and [5.2](#sec:markovDrift){reference-type="ref" reference="sec:markovDrift"} two subproblems. The regularization is justified by the fact that each of these subproblems $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf} {\cal J}_{\epsilon}(\mathbb Q, \mathbb P)$ and $\underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\inf} {\cal J}_{\epsilon}(\mathbb Q, \mathbb P)$ can be treated by classical techniques of the literature and will build the two steps of our alternating minimization algorithm. The one in Section [5.2](#sec:markovDrift){reference-type="ref" reference="sec:markovDrift"} is a minimization on $\mathbb Q$, the probability $\mathbb P$ being fixed and it is related to a variational representation formula whose solution is expressed as a so called exponential twist, see e.g. [@DupuisEllisLargeDeviations]. In particular we will make use of the following result. **Proposition 11**. *Let ${\varphi}: \Omega \rightarrow \mathbb R$ be a Borel function and $\mathbb P\in {\mathcal P}(\Omega)$. Assume that ${\varphi}$ is bounded below. Then $$\label{eq:klOpti} \inf_{\mathbb Q\in {\mathcal P}(\Omega)} \mathbb E^{\mathbb Q}[{\varphi}(X)] + \frac{1}{\epsilon} H(\mathbb Q| \mathbb P) = - \log \mathbb E^{\mathbb P}\left[\exp(-{\varphi}(X))\right].$$ Moreover there exists a unique minimizer $\mathbb Q^* \in {\mathcal P}(\Omega)$ given by $$d\mathbb Q^* = \frac{\exp(-\epsilon{\varphi}(X))}{\mathbb E^{\mathbb P}[\exp(-\epsilon{\varphi}(X))]}d\mathbb P.$$* *Proof.* The random variable ${\varphi}(X)$ is bounded below, hence satisfies condition $(FE)$ of [@EntropyWeighted]. The statement then follows from Proposition 2.5 in [@EntropyWeighted]. ◻ Applying Proposition [Proposition 11](#prop:markovExistenceMinimizer){reference-type="ref" reference="prop:markovExistenceMinimizer"} to our framework for $\mathbb P\in {\mathcal P}_\mathbb U$ we get that under Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} the subproblem $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf} {\cal J}_\epsilon(\mathbb Q, \mathbb P)$ admits a unique solution $\mathbb Q^*$ given by $$\label{eq:minimizerGeneralOptimizationProblem} d\mathbb Q^* = \frac{\exp\left(-\epsilon\int_0^T f(r, X_r, u_r^{\mathbb P})dr - \epsilon g(X_T)\right)}{\mathbb E^{\mathbb P}\left[\exp\left(-\epsilon\int_0^T f(r, X_r, u_r^{\mathbb P})dr - \epsilon g(X_T)\right)\right]}d\mathbb P,$$ and that the optimal value is $$\label{eq:optimalValueMinQ} {\cal J}_\epsilon(\mathbb Q^*, \mathbb P) = - \frac{1}{\epsilon}\log \mathbb E^{\mathbb P}\left[\exp\left(-\epsilon\int_0^T f(r, X_r, u_r^{\mathbb P})dr - \epsilon g(X_T)\right)\right].$$ This subproblem is further analyzed in Section [5.2](#sec:markovDrift){reference-type="ref" reference="sec:markovDrift"}. In particular Proposition [Proposition 25](#prop:markovianDrift){reference-type="ref" reference="prop:markovianDrift"} allows to identify $\mathbb Q^*$ as the law of a semimartingale with Markovian drift. On the other hand, the subproblem $\underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\inf} {\cal J}_\epsilon(\mathbb Q, \mathbb P)$ in Section [5.1](#sec:pointwiseMinimization){reference-type="ref" reference="sec:pointwiseMinimization"} is a minimization on $\mathbb P$, the probability $\mathbb Q$ remaining unchanged. The solution arises via a pointwise real minimization providing the function $u^\mathbb P\in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$ associated with the optimal probability $\mathbb P$ by Proposition [Proposition 24](#prop:pointwiseMinimization){reference-type="ref" reference="prop:pointwiseMinimization"}. The next theorem proves that the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} has as Markovian solution. **Theorem 12**. *Assume Hypotheses [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} and [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} hold. Then the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} has a *solution* $(\mathbb P^*_\epsilon, \mathbb Q^*_\epsilon) \in {\cal A}$, in the sense that ${\cal J}^*_\epsilon = {\cal J}(\mathbb Q^*_\epsilon, \mathbb P^*_\epsilon).$ Moreover, under $\mathbb P^*_\epsilon$, the canonical process is a Markov process and $u^{\mathbb P_\epsilon^*}$ related to $\mathbb P_\epsilon^*$ by Definition [Definition 5](#def:PU){reference-type="ref" reference="def:PU"} is such that $u^{\mathbb P^*_\epsilon}(r, X) = u^{\mathbb P^*_\epsilon}(r, X_r)$.* In fact by a slight abuse of notation $u^{\mathbb P^*_\epsilon}$ denotes a function on $[0,T] \times C([0,T])$ and $[0,T] \times \mathbb R^d$ at the same time. The proof of this result relies on technical lemmas. For the convenience of the reader it is postponed to Appendix [7.3](#app:proofThEx){reference-type="ref" reference="app:proofThEx"}. The following proposition justifies the use of the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} to approximatively solve the initial stochastic optimal control Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}. **Proposition 13**. *We suppose Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} and item $1.$ of Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"}. Let $\epsilon >0, \epsilon' \ge 0$ and let $\mathbb P_\epsilon^{\epsilon'}$ be the first component of an $\epsilon'$-solution of Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} in the sense of Definition [Definition 3](#def:MinSeq){reference-type="ref" reference="def:MinSeq"} with $E = {\cal A}.$ We set $Y^{\epsilon'}_\epsilon := \int_0^T f(r, X_r, u^{\epsilon'}_\epsilon(r, X))dr + g(X_T),$ where $u^{\epsilon'}_\epsilon$ corresponds to the $u^{\mathbb P_\epsilon^{\epsilon'}}$ appearing in decomposition [\[eq:decompP\]](#eq:decompP){reference-type="eqref" reference="eq:decompP"}. Then the following holds.* 1. *There is a constant $C^*$ depending only on $C_{b, \sigma}, C_{f, g}, p, d, T$ of Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} $1.$ and the diameter of $\mathbb U$ such that $Var^{\mathbb P^{\epsilon'}_\epsilon}(Y_\epsilon^{\epsilon'}) \le C^*,$ where $Var^{\mathbb P^{\epsilon'}_\epsilon}(Y^{\epsilon'}_\epsilon)$ denotes the variance of $Y^{\epsilon'}_\epsilon$ under ${\mathbb P^{\epsilon'}_\epsilon}$.* 2. *We have $$0 \le J(\mathbb P_\epsilon^{\epsilon'}) - J^* \le \frac{\epsilon}{2} Var^{\mathbb P^{\epsilon'}_\epsilon}(Y^{\epsilon'}_\epsilon) + \epsilon',$$* *where we recall that $J$ was defined in [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}.* **Remark 14**. 1. *Let $(\mathbb P_\epsilon^*, \mathbb Q_\epsilon^*)$ be a solution of Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"} given by Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"}. Applying Proposition [Proposition 13](#prop:approximateControl){reference-type="ref" reference="prop:approximateControl"} with $\epsilon' = 0$ implies that $\mathbb P_\epsilon^*$ is an $\frac{\epsilon}{2} Var^{\mathbb P^*_\epsilon}(Y^{0}_\epsilon)$-solution of the original Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}.* 2. *By definition of infimum, for $\epsilon' > 0$, the existence of an $\epsilon'$-solution is always guaranteed without any convex assumption on the running cost $f$ w.r.t. the control variable.* 3. *In the sequel, assuming that $f$ is convex w.r.t. the control variable, we will propose an algorithm providing a sequence of $\epsilon'_n$-solutions of the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}, where $\epsilon'_n \rightarrow 0$ as $n \rightarrow + \infty$. This will also provide a sequence of $(\frac{\epsilon}{2} Var^{\mathbb P^{\epsilon'_n}_\epsilon}(Y^{\epsilon'_n}_\epsilon) + \epsilon'_n)$-solutions to the original Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} (with a fixed $\epsilon > 0$).* *Proof*(of Proposition [Proposition 13](#prop:approximateControl){reference-type="ref" reference="prop:approximateControl"}). We first prove item $1.$ Let $(\mathbb P_{\epsilon}^{\epsilon'},\mathbb Q_\epsilon^{\epsilon'})$ be an $\epsilon'$-solution of Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}. By Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"}, for all $\epsilon > 0$, one has $$Var^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon] \le \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}\left[(Y^{\epsilon'}_\epsilon )^2\right] \le 8C_{f, g}^2(T \vee 1)\left(1 + \mathbb E^{\mathbb P_\epsilon^*}\left[\sup_{0 \le t \le T}|X_t|^{2p}\right]\right).$$ Combining this inequality with Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} implies the existence of a constant $C^*$ depending only on $C_{b, \sigma}, C_{f, g}, p, d, T$ and the diameter of $\mathbb U$ such that $Var^{\mathbb P_\epsilon^{\epsilon'}}[Y_\epsilon^{\epsilon'}] \le C^*$. We go on with the proof of item $2.$ First a direct application of Lemma [Lemma 42](#lemma:squareIntVar){reference-type="ref" reference="lemma:squareIntVar"} with $\eta = Y_{\epsilon}^{\epsilon'}$ yields $$0 \le \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ] - \left(-\frac{1}{\epsilon}\log \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}[\exp(-\epsilon Y^{\epsilon'}_\epsilon )]\right) \le \frac{\epsilon}{2}Var^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ].$$ Let then $\tilde \mathbb Q$ be the solution of $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf} {\cal J}_\epsilon(\mathbb Q, \mathbb P_\epsilon^{\epsilon'})$ given by [\[eq:minimizerGeneralOptimizationProblem\]](#eq:minimizerGeneralOptimizationProblem){reference-type="eqref" reference="eq:minimizerGeneralOptimizationProblem"}. Then by [\[eq:optimalValueMinQ\]](#eq:optimalValueMinQ){reference-type="eqref" reference="eq:optimalValueMinQ"} ${\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) = -\frac{1}{\epsilon}\log \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}[\exp(-\epsilon Y^{\epsilon'}_\epsilon )],$ which implies $$\label{eq:interEpsilonOpti1} 0 \le \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ] - {\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) \le \frac{\epsilon}{2}Var^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ].$$ Observe that ${\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) \le {\cal J}_\epsilon(\mathbb Q_\epsilon^{\epsilon'}, \mathbb P_\epsilon^{\epsilon'}) \le {\cal J}^*_\epsilon + \epsilon'$. Besides, as Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} rewrites $\underset{(\mathbb P, \mathbb Q) \in {\cal A}}{\inf} {\cal J}_\epsilon(\mathbb Q, \mathbb P)~\text{s.t.}~\mathbb Q= \mathbb P$, it holds that ${\cal J}_\epsilon^* \le J^*$. Then $$\label{eq:interEpsilonOpti2} {\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) - J^* \le {\cal J}_\epsilon^* + \epsilon' - J^* \le \epsilon'.$$ Using [\[eq:interEpsilonOpti1\]](#eq:interEpsilonOpti1){reference-type="eqref" reference="eq:interEpsilonOpti1"} and [\[eq:interEpsilonOpti2\]](#eq:interEpsilonOpti2){reference-type="eqref" reference="eq:interEpsilonOpti2"} finally yields $$\label{eq:ineqVariance} 0 \le J(\mathbb P_\epsilon^{\epsilon'}) - J^* = \mathbb E^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ] - {\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) + {\cal J}_\epsilon(\tilde \mathbb Q, \mathbb P_\epsilon^{\epsilon'}) - J^* \le \frac{\epsilon}{2}Var^{\mathbb P_\epsilon^{\epsilon'}}[Y^{\epsilon'}_\epsilon ] + \epsilon'.$$ This concludes the proof of item $2.$ 0◻ From now on, $\epsilon$ will be implicit in the cost function ${\cal J}_\epsilon$ to alleviate notations. # Alternating minimization algorithm {#sec:algo} In this section we present an alternating algorithm for solving the regularized Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}. Let $(\mathbb P_0, \mathbb Q_0) \in \mathcal{A}$. We will define a sequence $(\mathbb P_k, \mathbb Q_k)_{k \ge 0}$ verifying by the alternating minimization procedure $$\label{eq:alternateMinimizationProcedure} \mathbb Q_{k + 1} = \underset{\mathbb Q\in {\mathcal P}(\Omega)}{\mathop{\mathrm{arg\,min}}}~{\cal J}(\mathbb Q, \mathbb P_{k}), ~~ \mathbb P_{k + 1} \in \underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\mathop{\mathrm{arg\,min}}}~{\cal J}(\mathbb Q_{k + 1}, \mathbb P).$$ ## Convergence result The convergence of alternating minimization algorithms has been extensively studied in particular in Euclidean spaces. In general the proof of convergence results requires joint convexity and smoothness properties of the objective function, see [@auslender]. The major difficulty in our case is that the convexity only holds w.r.t $\mathbb Q$ (in fact the set ${\mathcal P}_\mathbb U$ is not even convex). To prove the convergence we need to rely on other techniques which exploit the properties of the entropic regularization. Let us first assume that the initial probability measure $\mathbb P_0 \in {\mathcal P}_\mathbb U$ is Markovian in the following sense. **Hypothesis 15**. *$\mathbb P_0 \in {\mathcal P}_\mathbb U^{Markov}$. In particular, there exists $u^0 \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$ such that $\mathbb P_0 = \mathbb P^{u^0}$, i.e. $\mathbb P_0$ is solution of a martingale problem with operator $L_{u^0}$ given by [\[eq:generatorU\]](#eq:generatorU){reference-type="eqref" reference="eq:generatorU"}, see Remark [Remark 6](#rmk:stroockVaradhan){reference-type="ref" reference="rmk:stroockVaradhan"}.* For a fixed Borel function $\beta: [0,T]\times \mathbb R^d \rightarrow \mathbb R^d$ we set $$\label{eq:fBeta} F_{\beta} : (t, x, u) \in [0, T] \times \mathbb R^d \times \mathbb U\mapsto f(t, x, u) + \frac{1}{2\epsilon}|\sigma^{-1}(t, x)(\beta(t, x) - u)|^2.$$ Let $\mathbb P_0 \in {\mathcal P}_\mathbb U$ satisfying Hypothesis [Hypothesis 15](#hyp:initialPoint){reference-type="ref" reference="hyp:initialPoint"}. We set $\mathbb Q_0 = \mathbb P_0$. We build a sequence $(\mathbb P_k, \mathbb Q_k)_{k \ge 0}$ of elements of ${\cal A}$ according to the following procedure. Let $k \ge 1$. - Let $$\label{eq:defQk} d\mathbb Q_k := \frac{\exp\left(-\epsilon\int_0^T f(r, X_r, u^{k - 1}(r, X_r))dr - \epsilon g(X_T)\right)}{\mathbb E^{\mathbb P_{k - 1}}\left[\exp\left(-\epsilon\int_0^T f(r, X_r, u^{k - 1}(r, X_r))dr - \epsilon g(X_T)\right)\right]}d\mathbb P_{k - 1}.$$ By Proposition [Proposition 25](#prop:markovianDrift){reference-type="ref" reference="prop:markovianDrift"} there exists a measurable function $\beta^k : [0, T] \times \mathbb R^d \rightarrow \mathbb R^d$ such that under $\mathbb Q_k$ the canonical process decomposes as $$\label{eq:decompQn} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta^k(r, X_r)dr + M_t^{\mathbb Q_k},$$ where $M^{\mathbb Q_k}$ is a martingale such that $\langle M^{\mathbb Q_k}\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$. - Let $$\label{eq:defPk} \mathbb P_k := \mathbb P^{u^k},~\text{where}~(t, x) \mapsto u^k(t, x) := \mathop{\mathrm{arg\,min}}_{\nu \in \mathbb U} F_{\beta^k}(t, x, \nu),$$ and $F_{\beta^k}$ is given by [\[eq:fBeta\]](#eq:fBeta){reference-type="eqref" reference="eq:fBeta"}. By Proposition [Proposition 24](#prop:pointwiseMinimization){reference-type="ref" reference="prop:pointwiseMinimization"} $u_k$ is measurable, $\mathbb P_k$ is well-defined and under $\mathbb P_k$ the canonical process decomposes as $$\label{eq:decompPn} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u^k(r, X_r)dr + M_t^{\mathbb P_k},$$ where $M^{\mathbb P_k}$ is a martingale such that $\langle M^{\mathbb P_k}\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$. Lemma [Lemma 16](#lemma:sequenceAlternateDirection){reference-type="ref" reference="lemma:sequenceAlternateDirection"} below states that the sequence $(\mathbb P_k, \mathbb Q_k)_{k \ge 0}$ defined above verifies the alternating minimization procedure [\[eq:alternateMinimizationProcedure\]](#eq:alternateMinimizationProcedure){reference-type="eqref" reference="eq:alternateMinimizationProcedure"}. **Lemma 16**. *Let $\mathbb P_0 = \mathbb Q_0\in {\mathcal P}_\mathbb U$ satisfying Hypothesis [Hypothesis 15](#hyp:initialPoint){reference-type="ref" reference="hyp:initialPoint"}. Let $(\mathbb P_k, \mathbb Q_k)_{k\ge 0}$ be given by the recursion [\[eq:defQk\]](#eq:defQk){reference-type="eqref" reference="eq:defQk"} and [\[eq:defPk\]](#eq:defPk){reference-type="eqref" reference="eq:defPk"}. The following holds for $k \ge 1$.* 1. *$\mathbb Q_k = \underset{\mathbb Q\in {\mathcal P}(\Omega)}{\mathop{\mathrm{arg\,min}}}~{\cal J}(\mathbb Q, \mathbb P_{k-1})$, and ${\cal J}(\mathbb Q_{k}, \mathbb P_{k - 1}) = -\frac{1}{\epsilon}\log \mathbb E^{\mathbb P_{k - 1}}\left[\exp\left(-\epsilon\int_0^T f(r, X_r, u^{k - 1}(r, X_r))dr - \epsilon g(X_T)\right)\right]$. Moreover, under $\mathbb Q_k$ the canonical process is a Markov process and $\beta^k \in L^q(dt \otimes \mathbb Q^k)$ for all $1 < q < 2$.* 2. *$\mathbb P_{k} \in \underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\mathop{\mathrm{arg\,min}}}~{\cal J}(\mathbb Q_{k}, \mathbb P)$.* The proof is a direct application of Proposition [Proposition 25](#prop:markovianDrift){reference-type="ref" reference="prop:markovianDrift"} for item $(i)$ and Proposition [Proposition 24](#prop:pointwiseMinimization){reference-type="ref" reference="prop:pointwiseMinimization"} for item $(ii)$. The main result of this section is given below. **Theorem 17**. *Let $\mathbb P_0 = \mathbb Q_0\in {\mathcal P}_\mathbb U$ satisfying Hypothesis [Hypothesis 15](#hyp:initialPoint){reference-type="ref" reference="hyp:initialPoint"}. Assume also that Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"}, [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} hold. Let $(\mathbb P_k, \mathbb Q_k)_{k\ge 0}$ be given by the recursion [\[eq:defQk\]](#eq:defQk){reference-type="eqref" reference="eq:defQk"} and [\[eq:defPk\]](#eq:defPk){reference-type="eqref" reference="eq:defPk"}. Then ${\cal J}(\mathbb Q_k, \mathbb P_k) \underset{k \rightarrow + \infty}{\searrow} {\cal J}^*$, where ${\cal J}^* = \underset{(\mathbb P, \mathbb Q) \in \mathcal{A}}{\inf} {\cal J}(\mathbb Q, \mathbb P)$.* The proof of the theorem uses the so-called three and four-points properties introduced in [@CsiszarAlternating]. The whole convergence proof makes use of the specific features of the sub-problems $\underset{\mathbb P\in {\mathcal P}_\mathbb U}{\inf}~{\cal J}(\mathbb Q, \mathbb P)$, whose study is the object of Section [5.1](#sec:pointwiseMinimization){reference-type="ref" reference="sec:pointwiseMinimization"}, and $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf}~{\cal J}(\mathbb Q, \mathbb P)$ which is the object of Section [5.2](#sec:markovDrift){reference-type="ref" reference="sec:markovDrift"}. **Lemma 18**. *(Three points property). [\[lemma:3Points\]]{#lemma:3Points label="lemma:3Points"} For all $\mathbb Q\in {\mathcal P}(\Omega)$, $$\label{eq:3points} \frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k+1}) + {\cal J}(\mathbb Q_{k+1}, \mathbb P_k) \le {\cal J}(\mathbb Q, \mathbb P_k).$$* *Proof.* We can suppose that $H(\mathbb Q| \mathbb P_{k}) < + \infty$, otherwise ${\cal J}(\mathbb Q, \mathbb P_k) = + \infty$ and the inequality holds trivially. Let $$\varphi : X \mapsto \int_0^T f(s, X_s, u^k(s, X_s))ds + g(X_T),$$ where $u^k$ (and $\mathbb P_k$) have been defined in [\[eq:defPk\]](#eq:defPk){reference-type="eqref" reference="eq:defPk"}. By the definition [\[eq:defQk\]](#eq:defQk){reference-type="eqref" reference="eq:defQk"} we have $$d \mathbb P_k = \frac{\exp(-\epsilon \varphi(X))} {\mathbb E^{\mathbb P_k}[\exp(-\epsilon {\varphi}(X))]}d\mathbb Q_{k + 1} .$$ and we get $$\begin{aligned} \frac{1}{\epsilon} H(\mathbb Q\vert \mathbb Q_{k + 1}) & = \frac{1}{\epsilon}\mathbb E^\mathbb Q\left[\log \frac{d\mathbb Q}{d\mathbb P_k} + \log \frac{d\mathbb P_k}{d\mathbb Q_{k+1}}\right]\\ & = \frac{1}{\epsilon}H(\mathbb Q| \mathbb P_k) + \frac{1}{\epsilon}\log \mathbb E^{\mathbb P_k}\left[\exp(-\epsilon {\varphi}(X))\right] + \mathbb E^{\mathbb Q}[{\varphi}(X)]\\ & = {\cal J}(\mathbb Q, \mathbb P_k) + \frac{1}{\epsilon}\log \mathbb E^{\mathbb P_k}\left[\exp(-\epsilon {\varphi}(X))\right]. \end{aligned}$$ By Lemma [Lemma 16](#lemma:sequenceAlternateDirection){reference-type="ref" reference="lemma:sequenceAlternateDirection"} item $(i)$ ${\cal J}(\mathbb Q_{k + 1}, \mathbb P_k) = -\frac{1}{\epsilon}\log \mathbb E^{\mathbb P_k}\left[\exp(-\epsilon {\varphi}(X))\right].$ Thus $$\frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k+1}) + {\cal J}(\mathbb Q_{k + 1}, \mathbb P_k) = {\cal J}(\mathbb Q, \mathbb P_k).$$ ◻ **Remark 19**. *Whenever $H(\mathbb Q| \mathbb P_{k}) < + \infty$, previous proof shows that [\[eq:3points\]](#eq:3points){reference-type="eqref" reference="eq:3points"} is indeed an equality.* **Lemma 20**. *(Four points property). [\[lemma:4Points\]]{#lemma:4Points label="lemma:4Points"} For all $(\mathbb P, \mathbb Q) \in \mathcal{A}$ $$\label{eq:4PP} {\cal J}(\mathbb Q, \mathbb P_{k+1}) \le \frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k+1}) + {\cal J}(\mathbb Q, \mathbb P).$$* *Proof.* Let $(\mathbb P, \mathbb Q) \in \mathcal{A}.$ If $H(\mathbb Q| \mathbb Q_{k + 1}) = + \infty$ or ${\cal J}(\mathbb Q, \mathbb P) = + \infty$, the inequality is trivial. We then assume until the end of the proof that $H(\mathbb Q| \mathbb Q_{k+1}) < + \infty$ and ${\cal J}(\mathbb Q, \mathbb P) < + \infty$. By construction (see [\[eq:decompQn\]](#eq:decompQn){reference-type="eqref" reference="eq:decompQn"}), there exists a measurable function $\beta^{k +1} : [0, T] \times \mathbb R^d \rightarrow \mathbb R^d$ such that under $\mathbb Q_{k + 1}$ the canonical process has decomposition $$X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta^{k+1}(r, X_r)dr + M_t^{\mathbb Q_{k + 1}},$$ where $M^{\mathbb Q_{k + 1}}$ is a martingale under $\mathbb Q_{k + 1}$ and $\langle M^{\mathbb Q_{k + 1}}\rangle_t = \int_0^t \sigma \sigma^\top(r, X_r)dr$. We now characterize the probability measure $\mathbb Q$. By Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} $1.$ in the Appendix with $\mathbb P= \mathbb Q_{k + 1}$ and the fact that $H(\mathbb Q| \mathbb Q_{k + 1}) < + \infty$, there exists a progressively measurable process with respect to the canonical filtration $\alpha = \alpha(\cdot,X)$ such that under $\mathbb Q$ the canonical process has the decomposition $$\label{eq:Q} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta^{k+1}(r, X_r)dr + \int_0^t \sigma\sigma^\top(r, X_r)\alpha(r, X) dr + M_t^\mathbb Q,$$ where $M^\mathbb Q$ is a martingale such that $\langle M^\mathbb Q\rangle_t = \int_0^t \sigma \sigma^\top(r, X_r)dr$, and $$\label{eq:Qk1} H(\mathbb Q| \mathbb Q_{k+1}) \ge \frac{1}{2}\mathbb E^\mathbb Q\left[\int_0^T |\sigma^\top(r, X_r)\alpha(r, X)|^2dr\right].$$ We set $$\label{eq:Q1} \beta(t, X) := \beta^{k+1}(t, X_t) + \sigma\sigma^\top(t, X_t)\alpha(t, X),$$ so that [\[eq:Qk1\]](#eq:Qk1){reference-type="eqref" reference="eq:Qk1"} can be rewritten $$\label{eq:lowerBoundEntropy} H(\mathbb Q| \mathbb Q_{k+1}) \ge \frac{1}{2}\mathbb E^\mathbb Q\left[\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X) - \beta^{k+1}(r, X_r))|^2dr\right].$$ We now prove the four-points property [\[eq:4PP\]](#eq:4PP){reference-type="eqref" reference="eq:4PP"}. Let then $u^{k+1}$ be the function introduced in [\[eq:defPk\]](#eq:defPk){reference-type="eqref" reference="eq:defPk"} replacing $k$ with $k+1$. Let $F_\beta$ be given by [\[eq:fBeta\]](#eq:fBeta){reference-type="eqref" reference="eq:fBeta"}. Since $f$ is convex in the $u$ variable, for all $(r, X) \in [0, T] \times \Omega$ one has $$\label{eq:4PointsEq1} \begin{aligned} F_\beta(r, X, u^\mathbb P(r, X)) - F_\beta(r, X, u^{k + 1}(r, X_r)) & \ge \langle \partial_u f(r, X_r, u^{k + 1}(r, X_r)), u^{\mathbb P}(r, X_r) - u^{k + 1}(r, X_r)\rangle\\& + \frac{1}{2\epsilon}|\sigma^{-1}(r, X_r)(\beta(r, X) - u^\mathbb P(r, X))|^2 \\ & - \frac{1}{2\epsilon}|\sigma^{-1}(r, X_r)(\beta(r, X) - u^{k + 1}(r, X_r))|^2, \end{aligned}$$ where $\partial_u f(r, X_r, \nu)$ denotes a subgradient of $f$ in $\nu \in \mathbb U$. We focus on the last two terms in the previous inequality. Applying the algebraic equality $|a|^2 - |b|^2 = |a - b|^2 + 2\langle a - b, b\rangle,$ with $$a = \sigma^{-1}(\beta - u^{\mathbb P}), b = \sigma^{-1}(\beta - u^{k+1}),$$ where we have omitted the dependencies in $(r, X)$ of all the quantities at hand for conciseness, we have $$\begin{aligned} \frac{1}{2\epsilon}|\sigma^{-1}(\beta - u^\mathbb P)|^2 - \frac{1}{2\epsilon}|\sigma^{-1}(\beta - u^{k + 1})|^2= \frac{1}{2\epsilon}|\sigma^{-1}(u^\mathbb P- u^{k + 1})|^2 + \frac{1}{\epsilon}\langle \sigma^{-1}(u^\mathbb P- u^{k + 1}), \sigma^{-1}(u^{k + 1} - \beta) \rangle. \end{aligned}$$ On the other hand $$\begin{aligned} \frac{1}{\epsilon}\langle \sigma^{-1}(u^\mathbb P- u^{k + 1}), \sigma^{-1}(u^{k + 1} - \beta) \rangle %& = \frac{1}{\epsilon}\langle \sigma^{-1}(u^\P - u^{k + 1}), \sigma^{-1}(u^{k + 1} - \beta^{k + 1} + \beta^{k + 1} - \beta) \rangle\\ & = \frac{1}{\epsilon}\langle \sigma^{-1}(u^\mathbb P- u^{k + 1}), \sigma^{-1}(u^{k + 1} - \beta^{k + 1}) \rangle\\ & + \frac{1}{\epsilon}\langle \sigma^{-1}(u^\mathbb P- u^{k + 1}), \sigma^{-1}(\beta^{k + 1} - \beta) \rangle. \end{aligned}$$ Combining what precedes yields $$\begin{aligned} \frac{1}{2\epsilon}|\sigma^{-1}(\beta - u^\mathbb P)|^2 - \frac{1}{2\epsilon}|\sigma^{-1}(\beta - u^{k + 1})|^2 & = \frac{1}{2\epsilon}|\sigma^{-1}(u^\mathbb P- u^{k + 1})|^2 + \frac{1}{\epsilon}\langle u^\mathbb P- u^{k + 1}, (\sigma^{-1})^\top\sigma^{-1}(u^{k + 1} - \beta^{k + 1}) \rangle\\ & + \frac{1}{\epsilon}\langle \sigma^{-1}(u^\mathbb P- u^{k + 1}), \sigma^{-1}(\beta^{k + 1} - \beta) \rangle. \end{aligned}$$ From the inequality [\[eq:4PointsEq1\]](#eq:4PointsEq1){reference-type="eqref" reference="eq:4PointsEq1"} we then get $$\label{eq:ineqsub} \begin{aligned} & F_\beta(r, X, u^\mathbb P(r, X)) - F_\beta(r, X, u^{k + 1}(r, X_r)) \ge \frac{1}{2\epsilon}|\sigma^{-1}(r, X_r)(u^\mathbb P(r, X) - u^{k+1}(r, X_r))|^2\\ & + \frac{1}{\epsilon}\langle \sigma^{-1}(r, X_r)(\beta^{k+1}(r, X) - \beta(r, X)), \sigma^{-1}(r, X_r)(u^{\mathbb P}(r, X_r) - u^{k+1}(r, X_r))\rangle\\ & + \langle \partial_u f(r, X_r, u^{k+1}(r, X_r)) + \frac{1}{\epsilon}(\sigma^{-1})^\top\sigma^{-1}(r, X_r)(u^{k+1}(r, X_r) - \beta^{k+1}(r, X_r)), u^{\mathbb P}(r, X) - u^{k+1}(r, X_r)\rangle. \end{aligned}$$ By definition [\[eq:defPk\]](#eq:defPk){reference-type="eqref" reference="eq:defPk"} $u^{k + 1}(t, x)$ is the minimum of $F_{\beta^{k + 1}}(t, x, .)$ for all $(t, x) \in [0, T] \times \mathbb R^d$, where the application $F_{\beta^{k + 1}}$ is the one defined in [\[eq:fBeta\]](#eq:fBeta){reference-type="eqref" reference="eq:fBeta"}. We recall that $F_{\beta^{k + 1}}$ is (strictly) convex in $u$ with subgradient $\partial_u f + \frac{1}{\epsilon}(\sigma^{-1})^\top\sigma^{-1}(u - \beta^{k + 1})$. Consequently, for the generic probability $\mathbb P$ we get that the term on third line of inequality [\[eq:ineqsub\]](#eq:ineqsub){reference-type="eqref" reference="eq:ineqsub"} is non-negative by the first order optimality condition for subdifferentiable functions at $u^{k + 1}$. Next by the classical inequality $|ab| \le a^2/2 + b^2/2$ for all $(a, b) \in \mathbb R^2$, term on the second line of [\[eq:ineqsub\]](#eq:ineqsub){reference-type="eqref" reference="eq:ineqsub"} gives $$\begin{aligned} & \frac{1}{\epsilon}\langle \sigma^{-1}(r, X_r)(\beta^{k+1}(r, X_r) - \beta(r, X)), \sigma^{-1}(r, X_r)(u^{\mathbb P}(r, X) - u^{k+1}(r, X_r))\rangle \\ \ge & -\frac{1}{2\epsilon}|\sigma^{-1}(r, X_r)(u^\mathbb P(r, X) - u^{k+1}(r, X_r))|^2\\ & - \frac{1}{2\epsilon} |\sigma^{-1}(r, X_r)(\beta(r, X) - \beta^{k+1}(r, X_r))|^2. \end{aligned}$$ From inequality [\[eq:ineqsub\]](#eq:ineqsub){reference-type="eqref" reference="eq:ineqsub"} we get $$F_\beta(r, X, u^\mathbb P(r, X)) + \frac{1}{2\epsilon}|\sigma^{-1}(r, X_r)(\beta(r, X) - \beta^{k+1}(r, X_r))|^2 \ge F_\beta(r, X, u^{k + 1}(r, X_r)),$$ and integrating the previous inequality with respect to $r \in [0, T]$ yields $$\label{eq:4pointLastStep} \int_0^T F_\beta(r, X, u^\mathbb P(r, X))dr + \frac{1}{2\epsilon} \int_0^T|\sigma^{-1}(r, X_r)(\beta(r, X) - \beta^{k+1}(r, X_r))|^2dr \ge \int_0^T F_\beta(r, X, u^{k + 1}(r, X_r))dr.$$ By [\[eq:Q\]](#eq:Q){reference-type="eqref" reference="eq:Q"} and [\[eq:Q1\]](#eq:Q1){reference-type="eqref" reference="eq:Q1"}, under $\mathbb Q$, the canonical process decomposes as $$\label{eq:lastDecompQ} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta(r, X)dr + M_t^\mathbb Q,$$ where $M^\mathbb Q$ is a martingale verifying $\langle M^\mathbb Q\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$. We recall the decomposition [\[eq:decompP\]](#eq:decompP){reference-type="eqref" reference="eq:decompP"}. As $H(\mathbb Q| \mathbb P) < + \infty$ by assumption, Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} item $1.$ applied to $\mathbb P$ with $\delta = b + u^\mathbb P$ states the existence of a predictable process $\tilde \alpha$ such that $$\label{eq:841bis} X_t = x + \int_0^t (b + u^\mathbb P)(r, X)dr + \int_0^t\sigma\sigma^\top(r, X_r) \tilde \alpha(r, X)dr + \tilde M_t^\mathbb Q,$$ where $\tilde M^\mathbb Q$ is a local martingale (with respect to the canonical filtration). Identifying the bounded variation component between [\[eq:841bis\]](#eq:841bis){reference-type="eqref" reference="eq:841bis"} and decomposition [\[eq:lastDecompQ\]](#eq:lastDecompQ){reference-type="eqref" reference="eq:lastDecompQ"} under $\mathbb Q$, yields $u^{\mathbb P}(r, X_r) - \beta(r, X) = \sigma\sigma^\top(r, X_r) \tilde \alpha(r, X)$ and [\[eq:inequalityEntropy\]](#eq:inequalityEntropy){reference-type="eqref" reference="eq:inequalityEntropy"} in Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} item $1.$ implies that $$\label{eq:entropQP} H(\mathbb Q| \mathbb P) \ge \frac{1}{2}\mathbb E^{\mathbb P}\left[\int_0^T |\sigma^{- 1}(r, X_r)(u^{\mathbb P}(r, X_r) - \beta(r, X))|^2dr\right].$$ Then recalling the definition of ${\cal J}$ in [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}, by [\[eq:entropQP\]](#eq:entropQP){reference-type="eqref" reference="eq:entropQP"} $$\label{eq:costQP} {\cal J}(\mathbb Q, \mathbb P) \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X, u^\mathbb P(r, X))dr \right].$$ From [\[eq:costQP\]](#eq:costQP){reference-type="eqref" reference="eq:costQP"} and [\[eq:lowerBoundEntropy\]](#eq:lowerBoundEntropy){reference-type="eqref" reference="eq:lowerBoundEntropy"} it holds $${\cal J}(\mathbb Q, \mathbb P) + \frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k + 1}) \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X, u^\mathbb P(r, X))dr \right] + \frac{1}{2\epsilon}\mathbb E^{\mathbb Q}\left[\int_0^T|\sigma^{-1}(r, X_r)(\beta(r, X) - \beta^{k+1}(r, X_r))|^2dr\right],$$ and by [\[eq:4pointLastStep\]](#eq:4pointLastStep){reference-type="eqref" reference="eq:4pointLastStep"} $${\cal J}(\mathbb Q, \mathbb P) + \frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k + 1}) \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X, u^{k + 1}(r, X))dr \right].$$ In particular, $\mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X, u^{k + 1}(r, X))dr \right] < + \infty$, hence $\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{- 1}(r, X_r)(u^{k + 1}(r, X_r) - \beta(r, X))|^2dr\right] < + \infty$. Then by Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} item $2.$ applied to $\mathbb P= \mathbb P_{k + 1}$ with $\delta = b + u^{k + 1}$ and $\gamma = b + \beta$, we have $$\label{eq:entropQPk} H(\mathbb Q| \mathbb P_{k + 1}) = \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{- 1}(r, X_r)(u^{k + 1}(r, X_r) - \beta(r, X))|^2dr\right],$$ and by [\[eq:entropQPk\]](#eq:entropQPk){reference-type="eqref" reference="eq:entropQPk"} $$\label{eq:costQPk} {\cal J}(\mathbb Q, \mathbb P_{k + 1}) = \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X, u^{k + 1}(r, X))dr \right].$$ Finally applying [\[eq:costQPk\]](#eq:costQPk){reference-type="eqref" reference="eq:costQPk"} in the previous inequality we get $${\cal J}(\mathbb Q, \mathbb P) + \frac{1}{\epsilon}H(\mathbb Q| \mathbb Q_{k + 1}) \ge {\cal J}(\mathbb Q, \mathbb P_{k + 1}).$$ This concludes the proof. ◻ **Remark 21**. *The three points property holds even if the set $\mathbb U$ is not compact, but our proof of the four points property crucially relies on the uniqueness in law of $\mathbb P_{k + 1}$ which is in particular guaranteed when $\mathbb U$ is compact. Otherwise it is unclear how to adapt the proof.* *Proof*(of Theorem [Theorem 17](#th:convAlgo){reference-type="ref" reference="th:convAlgo"}). Lemmas [\[lemma:3Points\]](#lemma:3Points){reference-type="ref" reference="lemma:3Points"} and [\[lemma:4Points\]](#lemma:4Points){reference-type="ref" reference="lemma:4Points"} state that the objective function has the three and four points property. It follows from Theorem 2 in [@CsiszarAlternating] that $({\cal J}(\mathbb Q_k, \mathbb P_k))_{k \ge 0}$ converges and $$\lim_{k \rightarrow + \infty} {\cal J}(\mathbb Q_k, \mathbb P_k) = {\cal J}^*.$$ 0◻ We conclude the section by stating a lemma which is a reformulation in our setting of Proposition 3.9 in [@EntropyWeighted]. This allows us to estimate the drift $\beta_k$ in the algorithm via a conditional derivative. **Lemma 22**. *Assume Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"}. For almost all $0 \le t < T$, it holds that $$\label{eq:approxDrift} \lim_{h \downarrow 0} \mathbb E^{\mathbb Q_k}\left[\frac{X_{t + h} - X_t}{h}~\Big| ~X_t\right] = b(t, X_t) + \beta^k(t, X_t)~\text{in}~L^1(\mathbb Q_k).$$* *Proof.* We fix some $1 < p < 2.$ By decomposition [\[eq:decompQn\]](#eq:decompQn){reference-type="eqref" reference="eq:decompQn"}, in order to apply Lemma [Lemma 43](#lemma:nelsonDerivative){reference-type="ref" reference="lemma:nelsonDerivative"}, we are going to prove that $\|b \|_{L^p(dt \otimes \mathbb Q_k)} + \| \beta^k\|_{L^p(dt \otimes \mathbb Q_k)} < + \infty$. On the one hand, as $f, g \ge 0$, by [\[eq:defQk\]](#eq:defQk){reference-type="eqref" reference="eq:defQk"} one has $C_\infty := \|d\mathbb Q_k/d\mathbb P_{k - 1}\|_\infty < + \infty$. By Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} and having $b$ linear growth we get $$\| b \|_{L^p(dt \otimes \mathbb Q_k)} = \mathbb E^{\mathbb Q_k}\left[\int_0^T |b(r, X_r)|^pdr\right] \le C_\infty\mathbb E^{\mathbb P_{k - 1}}\left[\int_0^T |b(r, X_r)|^pdr\right] < + \infty.$$ On the other hand, $\|\beta^k\|_{L^p(dt \otimes d \mathbb Q^k)} < + \infty$ by Lemma [Lemma 16](#lemma:sequenceAlternateDirection){reference-type="ref" reference="lemma:sequenceAlternateDirection"} item $(i)$. Consequently Lemma [Lemma 43](#lemma:nelsonDerivative){reference-type="ref" reference="lemma:nelsonDerivative"} and Remark [Remark 44](#remark:nelsonDerivative){reference-type="ref" reference="remark:nelsonDerivative"} yield the result. ◻ ## Entropy penalized Monte Carlo algorithm The previous alternating minimization procedure suggests a Monte Carlo algorithm to approximate a solution to Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}. In the following, $0 = t_0 \le t_1 < ... < t_M = T$ is a regular subdivision of the time interval $[0, T]$ with step $\Delta t,$ $N \ge 0$ the number of particles and $K$ the number of descent steps of the algorithm. $P_r$ will denote the set of $\mathbb R^d$ valued polynomials defined on $\mathbb R^d$ of degree $\le r$. Recall that for all $\hat u \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$, $\mathbb P^{\hat u}$ is the probability measure given by Proposition [Proposition 7](#prop:existencePu){reference-type="ref" reference="prop:existencePu"}. The estimation of the drift ${\hat \beta}^k$ in Step 2 of the algorithm below is performed via regression. It is inspired by [\[eq:approxDrift\]](#eq:approxDrift){reference-type="eqref" reference="eq:approxDrift"} in Lemma [Lemma 22](#lemma:regDrift){reference-type="ref" reference="lemma:regDrift"}, which is a reformulation in our setting of Proposition 3.9 in [@EntropyWeighted]. The term in the argmin is a weighted Monte Carlo approximation of the expectation of $\frac{X_{m + 1}^n - X_m^n}{\Delta t}$ under the exponential twist probability of $\mathbb P^{\hat u^{k-1}}$. $M, N, K \in \mathbb N^*,~r \in \mathbb N, ~\Delta t := \frac{T}{M},~x \in \mathbb R^d$, $\hat u^0 \in {\mathcal B}([0,T] \times \mathbb R^d, \mathbb U)$. $(X^n)_{1 \le n \le N}$, $N$ iid Monte Carlo path simulations under ${\hat \mathbb P}_0 = \mathbb P^{{\hat u}^0}$ on the time-grid $(t_m)_{0 \le m \le M}$ with $X^n = (X^n_m)_{0 \le m \le M}$ and $X^n_0 = x$ for all $1 \le n \le N$. **Step 1.  ** Compute the weights $(D_n)_{1 \le n \le N}$ by $$D_n = \exp\left(-\epsilon \sum_{m = 0}^{M - 1} f(t_m, X_m^n, {\hat u}^{k - 1}(t_m,X_m^n))\Delta t - \epsilon g(X_M^n)\right).$$ **Step 2.  ** Compute ${\hat \beta}^k = ({\hat \beta}^k_m)_{0 \le m \le M-1}$ in [\[eq:decompQn\]](#eq:decompQn){reference-type="eqref" reference="eq:decompQn"} by the weighted Monte Carlo approximation of [\[eq:approxDrift\]](#eq:approxDrift){reference-type="eqref" reference="eq:approxDrift"} $${\hat \beta}_m^k = \mathop{\mathrm{arg\,min}}_{{\varphi}\in P_r} \frac{1}{\sum_{\ell= 1}^N D_\ell} \sum_{n = 1}^ND_n\left|{\varphi}(X_m^n) - \left(\frac{X_{m + 1}^n - X_m^n}{\Delta t} - b(t_m, X_m^n)\right) \right|^2.$$ **Step 3.  ** Simulate new iid Monte Carlo paths $(X^n)_{1 \le n \le N}$ under $\mathbb P^{\hat u^k}$ where for $0 \le m \le M - 1$ $${\hat u}^{k}(t, x) = \underset{\nu \in \mathbb U}{\mathop{\mathrm{arg\,min}}}~ f(t_m, x, \nu) + \frac{1}{2\epsilon}|\sigma^{-1}(t_m, x) ({\hat \beta}_m^k( x) - \nu)|^2, \ t \in [t_m,t_{m+1}[.$$\ ${\hat u}^K$ [\[algo:mcEntropy\]]{#algo:mcEntropy label="algo:mcEntropy"} An interest of the entropy penalized Monte Carlo algorithm is that in Lemma [Lemma 22](#lemma:regDrift){reference-type="ref" reference="lemma:regDrift"}, [\[eq:approxDrift\]](#eq:approxDrift){reference-type="eqref" reference="eq:approxDrift"} can be independently estimated by regression techniques at each time step $t_m$, $1 \le m \le M$, while in dynamic programming approaches, conditional expectations are recursively computed in time, implying an error accumulation from time $t_M = T$ to $t_m$. Moreover one can expect that the trajectories simulated under $\mathbb P^{\hat u^k}$ localize around the optimally controlled trajectories when the number of iterations $k$ of the algorithm increases to $+ \infty$. Hence the computation effort to estimate the optimal control focuses on this specific region of the state space, whereas standard regression based Monte Carlo approaches are blindly exploring the state space with forward Monte Carlo simulations of the process. # Solving the subproblems In this short section we aim at describing the two subproblems $\underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\inf} {\cal J}(\mathbb Q, \mathbb P)$ and $\underset{\mathbb Q\in {\mathcal P}_{\Omega}}{\inf} {\cal J}(\mathbb Q, \mathbb P)$ appearing in the alternating minimization algorithm proposed in Section [4](#sec:algo){reference-type="ref" reference="sec:algo"}. ## Pointwise minimization subproblem {#sec:pointwiseMinimization} Let us first describe the minimization $\underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\inf} {\cal J}(\mathbb Q, \mathbb P)$ where the probability $\mathbb Q\in {\mathcal P}(\Omega)$ is fixed and is such that, under $\mathbb Q$, the canonical process is a fixed Itô process. In this section we assume that Hypotheses [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} and [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} are fulfilled. We introduce the following assumption for a given probability $\mathbb Q$ on the canonical space. **Hypothesis 23**. *There is a Borel function $\beta:[0,T] \times \mathbb R^d \rightarrow \mathbb R$ for which the canonical process $X$ decomposes as $$\label{eq:decompQ} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta(r, X_r)dr +M^\mathbb Q_t,$$ where $\langle M^\mathbb Q\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^T(r, X_r)dr$.* For the proposition below we recall that if $u:[0,T] \times \mathbb R^d \rightarrow \mathbb R$ is a bounded measurable function then $\mathbb P^u \in {\mathcal P}_\mathbb U^{Markov}$ denotes the associated probability measure given by Proposition [Proposition 7](#prop:existencePu){reference-type="ref" reference="prop:existencePu"}. **Proposition 24**. *Assume that Hypotheses [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"}, [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} are fulfilled. Suppose also that $\mathbb Q$ fulfills Hypothesis [Hypothesis 23](#hyp:QMarkov){reference-type="ref" reference="hyp:QMarkov"}. Then, the function $(t, x) \mapsto u(t, x)$ where $$\label{eq:pointwiseMinimization} u(t, x) := \mathop{\mathrm{arg\,min}}_{\nu \in \mathbb U} F_\beta(t, x, \nu),$$ and $F_\beta$ is given by [\[eq:fBeta\]](#eq:fBeta){reference-type="eqref" reference="eq:fBeta"}, is well-defined and measurable. Moreover ${\cal J}(\mathbb Q, \mathbb P^{u}) = \underset{\mathbb P\in {\mathcal P}_{\mathbb U}}{\inf} {\cal J}(\mathbb Q, \mathbb P)$.* *Proof.* As $F_\beta(t, x, .)$ is continuous and strongly convex on the convex compact set $\mathbb U$, it admits a unique minimum on $\mathbb U,$ denoted $u(t, x).$ The measurability of the application $(t, x) \mapsto u(t, x)$ follows e.g. from Theorem 18.19 in [@chara]. Let then $\mathbb P\in {\mathcal P}_\mathbb U$. We want to show that $$\label{eq:Ineq} {\cal J}(\mathbb Q, \mathbb P) \ge {\cal J}(\mathbb Q, \mathbb P^u).$$ Recall that, by Definition [Definition 5](#def:PU){reference-type="ref" reference="def:PU"}, there exists a progressively measurable process $u^\mathbb P$ taking values in $\mathbb U$ such that under $\mathbb P$ the canonical process has decomposition $$\label{eq:decompQuinque} X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u_r^\mathbb Pdr + M_t^\mathbb P,$$ where $M^\mathbb P$ is a martingale and $\langle M^\mathbb P\rangle_t = \int_0^t \sigma\sigma^\top(r, X_r)dr$. If ${\cal J}(\mathbb Q, \mathbb P) = \infty$ then inequality [\[eq:Ineq\]](#eq:Ineq){reference-type="eqref" reference="eq:Ineq"} is trivially fulfilled. We consider now $\mathbb P\in {\mathcal P}_\mathbb U$ such that ${\cal J}(\mathbb Q, \mathbb P) < + \infty$. Then $H(\mathbb Q| \mathbb P) < + \infty$ and by Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} item $1.$ there exists a process $\alpha$ of the form $\alpha = \alpha(\cdot, X)$ such that, under $\mathbb Q$, $X$ decomposes as $$\label{eq:entropyDecompQ} X_t = x + \int_0^t (b(r, X_r) + u_r^\mathbb P) dr + \int_0^t\sigma\sigma^\top(r, X_r) \alpha(r, X)dr + \tilde M_t^\mathbb Q,$$ and $$H(\mathbb Q| \mathbb P) \ge \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^\top(r, X_r)\alpha(r, X)|^2dr\right].$$ Identifying the bounded variation and the martingale components in decompositions [\[eq:decompQ\]](#eq:decompQ){reference-type="eqref" reference="eq:decompQ"} and [\[eq:entropyDecompQ\]](#eq:entropyDecompQ){reference-type="eqref" reference="eq:entropyDecompQ"} under $\mathbb Q$ we get $\sigma^\top(t, X_t)\alpha(t, X) = \sigma^{-1}(t, X_r)(\beta(t, X_t) - u_t^\mathbb P), \ dt \otimes d\mathbb Q$-a.e. and $\tilde M^\mathbb Q= M^\mathbb Q$. Hence $$H(\mathbb Q| \mathbb P) \ge \frac{1}{2\epsilon}\mathbb E^\mathbb Q\left[\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X_r) - u_r^\mathbb P)|^2dr\right].$$ Previous inequality yields $$\label{eq:interPointwiseMin} \begin{aligned} {\cal J}(\mathbb Q, \mathbb P) & = \mathbb E^{\mathbb Q}\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr\right] + \frac{1}{\epsilon}H(\mathbb Q| \mathbb P)\\ & \ge \mathbb E^{\mathbb Q}\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr + \frac{1}{2\epsilon}\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X_r) - u_r^\mathbb P)|^2dr\right]. \end{aligned}$$ Proposition 5.1 in [@MimickingItoGeneral] and the tower property of the conditional expectation gives the existence of $v \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$ such that $v(t, X_t) = \mathbb E^\mathbb Q[u_t^\mathbb P| X_t]$ $dt \otimes d\mathbb Q$-a.e. Fubini's theorem and Jensen's inequality for the conditional expectation applied to [\[eq:interPointwiseMin\]](#eq:interPointwiseMin){reference-type="eqref" reference="eq:interPointwiseMin"} then yields $$\label{eq:globalMinU} \begin{aligned} {\cal J}(\mathbb Q, \mathbb P) & \ge \mathbb E^{\mathbb Q}\left[\int_0^T f(r, X_r, v(r, X_r))dr + \frac{1}{2\epsilon}\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X_r) - v(r, X_r))|^2dr\right]\\ & = \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X_r, v(r, X_r))dr\right]. \end{aligned}$$ By the definition [\[eq:pointwiseMinimization\]](#eq:pointwiseMinimization){reference-type="eqref" reference="eq:pointwiseMinimization"} of $u$, it holds that $$\label{eq:ineqPointwise} \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X_r, v(r, X_r))dr\right] \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X_r, u(r, X_r))dr\right].$$ In particular $$\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X_r) - u(r, X_r))|^2dr\right] < + \infty.$$ By item $2.$ of Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} with $\delta(t,X) = u(t,X_t) + b(t,X_t)$ and $\gamma(t,X) = \beta(t,X_t) + b(t,X_t),$ we have that $H(\mathbb Q| \mathbb P^u) = \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\beta(r, X_r) - u(r, X_r))|^2dr\right]$. Hence $${\cal J}(\mathbb Q, \mathbb P) \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X_r, v(r, X_r))dr\right] \ge \mathbb E^{\mathbb Q}\left[\int_0^T F_\beta(r, X_r, u(r, X_r))dr\right] = {\cal J}(\mathbb Q, \mathbb P^u).$$ This concludes the proof of inequality [\[eq:Ineq\]](#eq:Ineq){reference-type="eqref" reference="eq:Ineq"}. ◻ ## Exponential twist subproblem {#sec:markovDrift} In this section we focus on the minimization $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf} {\cal J}(\mathbb Q, \mathbb P)$, $\mathbb P\in {\mathcal P}_\mathbb U^{Markov}$ being the reference probability. We recall that $\mathbb Q^*$ is the solution of $\underset{\mathbb Q\in {\mathcal P}(\Omega)}{\inf} {\cal J}(\mathbb Q, \mathbb P)$ given by Proposition [Proposition 11](#prop:markovExistenceMinimizer){reference-type="ref" reference="prop:markovExistenceMinimizer"}. **Proposition 25**. *Assume that, under $\mathbb P$, the canonical process decomposes as $$X_t = x + \int_0^t b(r, X_r)dr + \int_0^t u(r, X_r)dr + M_t^\mathbb P,$$ where $M^\mathbb P$ is a martingale such that $\langle M^\mathbb P\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$ and $u \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb U)$. Then there exists $\beta \in {\mathcal B}([0, T] \times \mathbb R^d, \mathbb R^d)$ such that under $\mathbb Q^*$ the canonical process decomposes as $$X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \beta(r, X_r)dr + M_t^{\mathbb Q^*},$$ where $M^{\mathbb Q^*}$ is a martingale such that $\langle M^{\mathbb Q^*}\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$. Moreover, $X$ is a Markov process under $\mathbb Q^*$ and $\beta \in L^q(dt\otimes d\mathbb Q^*)$ for all $1 < q < 2$.* *Proof.* Recall that by Remark [Remark 6](#rmk:stroockVaradhan){reference-type="ref" reference="rmk:stroockVaradhan"}, $\mathbb P$ is a solution in law of the SDE $$dX_t = b(t, X_t)dt + u(t, X_t)dt + \sigma(t, X_t)dW_t, ~X_0 = x.$$ The result is stated in Section 5 of [@BORMarkov2023]. ◻ # Application to the control of thermostatic loads in power systems {#sec:example} We consider in this section the problem of controlling a large, heterogeneous population of $N$ air-conditioners in order that their overall consumption tracks a given target profile $r = (r_t)_{0\leq t\leq T}$ on a given time horizon $[0,T]$. This problem was introduced in [@FullyBackward]. Air-conditioners are aggregated in $d$ clusters indexed by $1 \le i \le d$ depending on their characteristics. We denote by $N_i$ the number of air-conditioners in the cluster $i$. Individually, the temperature $X^{i, j}$ in the room with air-conditioner $j$ in cluster $i$ is assumed to evolve according to the following dynamics $$\label{eq:individualTempSDE} dX_t^{i, j} = -\theta^i(X_t^{i, j} - x_{out}^i)dt - \kappa^iP_{max}^iu^{i, j}_t dt + \sigma^{i, j}dW_t^{i, j}, ~X_0^{i, j} = x_0^{i, j}, 1 \le i \le d, 1 \le j \le N_i,$$ where : $x_{out}^i$ is the outdoor temperature; $\theta^i$ is a positive thermal constant; $\kappa^i$ is the heat exchange constant; $P^i_{max}$ is the maximal power consumption of an air-conditioner in cluster $i$. $W^{i, j}$ are independent Brownian motion that represent random temperature fluctuations inside the rooms, such as a window or a door opening. For each cluster, a **local controller** decides at each time step to turn $ON$ or $OFF$ some conditioners in the cluster $i$ by setting $u^{i, j} = 1$ or $0$ in order to satisfy a **prescribed proportion** of active air-conditioners. We are interested in the global planner problem which consists in computing the prescribed proportion $u^i = \frac{1}{N_i}\sum_{j = 1}^{N_i} u^{i, j}$ of air conditioners ON in each cluster in order to track the given target consumption profile $r = (r_t)_{0\leq t\leq T}$. For each $1 \le i \le d$ the average temperature $X^i = \frac{1}{N}\sum_{j = 1}^{N_i} X^{i, j}$ in the cluster $i$ follows the aggregated dynamics $$\label{eq:aggregatedTempSDE} dX_t^{i} = -\theta^i(X_t^{i} - x_{out}^i)dt - \kappa^iP_{max}^iu^{i}_t dt + \sigma^{i}dW_t^{i}, ~X_0^{i} = x_0^{i},$$ with $$W_t^i = \frac{1}{N_i}\sum_{j = 1}^{N_i}W_t^{i, j}, ~\sigma^i = \frac{1}{N_i}\sum_{j = 1}^{N_i} \sigma^{i, j} ~\text{and}~x_0^{i} = \frac{1}{N_i}\sum_{j = 1}^{N_i}x_0^{i, j}.$$ We consider the stochastic control Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"} on the time horizon $[0,T]$ with $\mathbb U= [0, 1]^d$ and $T = 2h$. The running cost $f$ is defined for any $(t, x, u) \in [0, T] \times \mathbb R^d \times \mathbb U$ such that $$\label{eq:runningCostExample} f(t, x, u) := \mu\left(\sum_{i=1}^d \rho_i u_i - r_t\right)^2 + \frac{1}{d}\sum_{i=1}^d\left(\gamma_i(\rho_i u_i)^2 + \eta_i (x_i - x_{max}^i)^2_+ + \eta_i (x_{min}^i - x_i)^2_+\right) ,$$ where $\rho_i = N_iP_{max}^i/(\sum_{j = 1}^d N_jP_{max}^j)$, the first term in the above cost function penalizes the deviation of the the overall consumption $\sum_i \rho_i u^i_t$ with respect to the target consumption $r_t$, $\gamma_i$ quantifies the penalization for irregular controls in cluster $i$ while $\eta_i$ penalizes the exits of the mean temperatures in the cluster $i$ from a comfort band $[x^i_{min}, x_{max}^i]$. Finally the terminal cost is given by $g(x) = \frac{1}{d}\sum_{i = 1}^d |x^i - x_{target}^i|^2$ where $x_{target}^i$ is a target temperature for cluster $i$. Clearly the cost functions $f$ and $g$ satisfy Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"}. To estimate an optimal policy $u^*$ for this problem we use Algorithm [\[algo:mcEntropy\]](#algo:mcEntropy){reference-type="ref" reference="algo:mcEntropy"} with a time step $t_{m+1}-t_m=60s$ for $m=0,\cdots M$. The parameters of the problem are the same as in [@FullyBackward]. We perform $N_{grid} = 100$ independent runs of the algorithm, providing $(\hat u^i)_{1 \le i \le N_{grid}}$ estimations of an optimal control on the whole period $t_0,t_1,\cdots t_M$. For each estimation $\hat u^i$, we simulate $N_{simu} = 1000$ iid trajectories of the process controlled by $\hat u^i$ and compute the associated costs $(\mathcal{J}_\ell (\hat u^i))_{1 \le \ell \le N_{simu}}$. The average cost is finally estimated by $\mathcal{J} = \frac{1}{N_{grid}N_{simu}}\sum_{i = 1}^{N_{grid}}\sum_{\ell = 1}^{N_{simu}}\mathcal{J}_\ell (\hat u^i)$. To evaluate the performances of our approach, we compare it with the classical regression-based Monte Carlo technique relying on a BSDE representation of the problem implemented in [@FullyBackward]. We underline that we only aim to obtain lower costs compared to the BSDE technique in [@FullyBackward], there are no benchmark costs. The results are reported in Table [\[tab:resultsSimu\]](#tab:resultsSimu){reference-type="ref" reference="tab:resultsSimu"} for dimensions $d = 1, ~2, ~5, ~10, ~15, ~20$. For both methods, $N = 10^3, ~10^4, ~5 \times 10^4, ~10^5$ particles are used to estimate an optimal policy for each dimension $d$. For the entropy penalized Monte Carlo algorithm, we use a regularization parameter $\epsilon = 70$ and $K = 20$ iterations for dimensions $d = 1, ~2, ~5, ~10$ and $\epsilon = 20$ and $K = 60$ iterations for dimensions $d = 15, ~20$; concerning the approximation in Step 1 of the Algorithm [\[algo:mcEntropy\]](#algo:mcEntropy){reference-type="ref" reference="algo:mcEntropy"} we limit ourselves to the set ${\mathcal P}_0$ of polynomials of degree $0$ as the problem is very localized in space. On Table [\[tab:resultsSimu\]](#tab:resultsSimu){reference-type="ref" reference="tab:resultsSimu"} we can observe very good performances that seem to be weakly sensitive to the dimensions of the problem. On Figure [1](#fig:illustrationAlternate){reference-type="ref" reference="fig:illustrationAlternate"}, we have reported the cost ${\cal J}(\mathbb Q_k,\mathbb P_k)$ and ${\cal J}(\mathbb P_k,\mathbb P_k) =\mathbb E^{\mathbb P_k}\left[\int_0^T f(r, X_r, u^k(r, X_r))dr + g(X_T)\right]$ as a function of the iteration number $k$ obtained on one run of the algorithm with $d = 20$ and $N =50000$. Theses costs are compared to a reference cost obtained with a run of our algorithm with $N = 100000$ particles. As expected ${\cal J}(\mathbb Q_k,\mathbb P_k)$ is decreasing and converging to a limiting value. It is interesting to notice that ${\cal J}(\mathbb P_k,\mathbb P_k)$ is also decreasing and very close to ${\cal J}(\mathbb Q_k,\mathbb P_k)$. Hence, it seems that the parameter $\epsilon$ does not need to be so small to obtain a good approximation of the original control Problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}. {#fig:illustrationAlternate width="12cm"} # Appendices {#appendix} ## Decomposition of a semimartingale in its own filtration We give here a proposition discussing the decomposition of a semimartingale in its own filtration. Even though it is a natural result, we have decided to carefully write its proof, as it raises several measurability issues. Recall that given a process $X$ defined on a probability space $(\Omega, {\mathcal F}, \mathbb P)$, $({\mathcal F}_t^X)_{t \in [0, T]}$ denotes the natural filtration of $X$. **Proposition 26**. *Let $(\Omega, {\mathcal F}, ({\mathcal F}_t), \mathbb P)$ be a filtered probability space. Let $u$ be a progressively measurable process such that $\mathbb E^\mathbb P\left[\int_0^T |u_r|dr\right] < + \infty$. Then there exists a measurable function $\phi : [0, T] \times C([0, T], \mathbb R^d) \rightarrow \mathbb R^d$ such that the following properties hold.* 1. *The map $(t, X) \mapsto \phi(t, X)$ is progressively measurable with respect to $({\mathcal F}^X_t)$ $$\label{eq:phi_u} %% La mesure de Lebesgue n'a pas lieu ici il me semble % \E^{\P}[u_t~|~\F_t^X] = \phi(t, X), \ {\rm } \ d\P \otimes dt {\rm -a.e.} \mathbb E^{\mathbb P}[u_t~|~{\mathcal F}_t^X] = \phi(t, X), \ {\rm } \ d\mathbb Pa.s. \forall t \in [0,T],$$ so that $\phi(t,X)$ is a version of the conditional expectation.* 2. *Let $X$ be a continuous $({\mathcal F}_t)$-semimartingale. with decomposition $$X_t = x + \int_0^t u_rdr + M_t^\mathbb P,$$ where $M^\mathbb P$ is an $({\mathcal F}_t)$-martingale. Then $X$ has the decomposition $$X_t = x + \int_0^t \phi(r, X)dr + M^X_t,$$ where $M^X$ is an $(({\mathcal F}_t^X), \mathbb P)$-martingale with $\langle M^X\rangle = \langle M \rangle$.* *Proof.* We follow closely the proof of Theorem 7.17 in [@LipsterShiryaev]. As $t \mapsto \mathbb E^{\mathbb P}[u_t~|~{\mathcal F}_t^X]$ has a ${\mathcal B}([0, T])\otimes {\mathcal F}_T^X$-measurable (adapted) version $(t,\omega) \mapsto \xi_t(\omega)$ (see e.g. [@IrleMeasurability]), it follows from Chapter IV Theorem T46 in [@meyer], see also Theorem 0.1 in [@OndrejatSeidler], that $\xi$ has a a progressively measurable modification w.r.t $({\mathcal F}_t^X)$ that we denote precisely $\phi(t,X)$, which proves $(i)$. Let then $M^X := X - x - \int_0^{\cdot} \phi(s, X)ds.$ Let $0 \le s \le t \le T.$ Then $$\begin{aligned} \mathbb E^{\mathbb P}[M^X_t - M^X_s~|~{\mathcal F}_s^X] & = \mathbb E^{\mathbb P}\left[\int_s^t (u_r - \phi(r, X))dr + M^\mathbb P_t - M^\mathbb P_s~\Big |~{\mathcal F}_s^X\right]\\ & = \int_s^t \mathbb E^\mathbb P[u_r - \phi(r, X)~|~{\mathcal F}_s^X]dr + \mathbb E^{\mathbb P}\left[\mathbb E^\mathbb P[M_t^\mathbb P- M_s^\mathbb P~|~{\mathcal F}_s]~|~{\mathcal F}_s^X\right]\\ & = \int_s^t \mathbb E^\mathbb P\left[\mathbb E^\mathbb P[u_r - \phi(r, X)~|~{\mathcal F}_r^X]~|~{\mathcal F}_s^X\right]dr\\ & = 0, \end{aligned}$$ where we used Fubini's theorem for conditional expectation as well as the tower property. The process $M^X$ is an $(({\mathcal F}^X_t), \mathbb P)$-martingale. Furthermore, $\langle M^X\rangle = \langle M \rangle$ since they are both equal to $[X]$. So $(ii)$ is also proved. ◻ ## Relative entropy related results The theorem below is a Girsanov's type theorem based on finite relative entropy assumptions. It is an adaptation of Theorem 2.1 in [@GirsanovEntropy] on a general probability space instead of the canonical space. **Theorem 27**. *Let $(\Omega, {\mathcal F}, ({\mathcal F}_t)_{t \in [0, T]}, \mathbb P)$ be a filtered probability space. Let $\delta = (\delta_t)_{t \in [0,T]}$ (resp. $a = (a_t)_{t \in [0,T]}$) be a progressively measurable process with values in $\mathbb R^d$ (resp. in the set of square $d \times d$ non-negative defined symmetric matrices $S_d^+$). Let $X$ be a continuous process which decomposes as $$\label{eq:decXP} X_t = x + \int_0^t \delta_r dr + M^\mathbb P_t, ~0 \le t \le T,$$ where $M^\mathbb P$ is a continuous $(({\mathcal F}_t), \mathbb P)$-local martingale such that $\langle M^\mathbb P\rangle_{\cdot} = \int_0^{\cdot} a_rdr$. Let $\mathbb Q$ be a probability measure on $(\Omega, {\mathcal F})$. Assume that $H(\mathbb Q| \mathbb P) < + \infty$.* *Then there exists an $\mathbb R^d$-valued progressively measurable process $\alpha$ such that $$\label{eq:alphaInt} \mathbb E^{\mathbb Q}\left[\int_0^T \alpha_r^\top a_r \alpha_rdr\right] < + \infty,$$ and such that, under $\mathbb Q$, the process $X$ is still a continuous semimartingale with decomposition $$\label{eq:alpha} X_t = x + \int_0^t \delta_rdr + \int_0^t a_r\alpha_rdr + M^\mathbb Q_t, ~0 \le t \le T,$$ where $M^\mathbb Q$ is a continuous $\mathbb Q$-local martingale and $\langle M^\mathbb Q\rangle_{\cdot} = \int_0^{\cdot} a_rdr$. Furthermore, $$\label{eq:subBoundEntropy} \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T \alpha_r^\top a_r \alpha_rdr\right] \le H(\mathbb Q| \mathbb P).$$* *Proof.* The existence of the process $\alpha$ is given by Theorem 2.1 in [@GirsanovEntropy], noticing that the proof of this result relies on a variational formulation of the relative entropy, which does not depend on the probability space, see Proposition 3.1 in [@GirsanovEntropy]. For the proof of [\[eq:subBoundEntropy\]](#eq:subBoundEntropy){reference-type="eqref" reference="eq:subBoundEntropy"} we closely follows the proof of Theorem 2.3 in [@GirsanovEntropy]. 1. Assume first that $\mathbb Q\sim \mathbb P$. Let $\tau_k := \inf\{t \ge 0~:~\int_0^t\alpha_r^\top a_r \alpha_rdr \ge k\}$ (with the convention that the $\inf$ is $+\infty$ if $\{ \}$ is empty). Setting $M^k := \int_0^{.\wedge \tau_k}\alpha_r^\top dM^\mathbb P_r$ and $Z^k$ the Doléans exponential $\mathcal{E}(M^k)$, we define $d\mathbb Q_k := Z^k_Td\mathbb P$. By Novikov's criterion, $Z^k$ is a martingale, therefore $\mathbb Q_k$ is a probability measure on $(\Omega, {\mathcal F})$ equivalent to $\mathbb Q$ since $Z^k_T$ is strictly positive and $\mathbb Q\sim \mathbb P$. It follows that $$\begin{aligned} H(\mathbb Q| \mathbb P) & = \mathbb E^\mathbb Q\left[\log \frac{d\mathbb Q}{d\mathbb P}\right] = \mathbb E^\mathbb Q\left[\log \frac{d\mathbb Q}{d\mathbb Q_k}\right] + \mathbb E^\mathbb Q\left[\log\frac{d\mathbb Q_k}{d\mathbb P}\right]\\ & = H(\mathbb Q|\mathbb Q_k) + \mathbb E^{\mathbb Q}[\log Z^k_T]\\ & \ge \mathbb E^{\mathbb Q}\left[\int_0^{T \wedge \tau_k}\alpha^\top_rdM_r^\mathbb P- \frac{1}{2}\int_0^{T \wedge \tau_k}\alpha_r^\top a_r \alpha_r dr\right]\\ & = \mathbb E^\mathbb Q\left[\int_0^{T \wedge \tau_k}\alpha_r^\top dM_r^\mathbb Q+ \frac{1}{2}\int_0^{T \wedge \tau_k}\alpha_r^\top a_r \alpha_r dr\right]. \end{aligned}$$ By definition of $\tau_k$ the process $\int_0^{t \wedge \tau_k}\alpha_r^\top dM_r^\mathbb Q$ is a genuine martingale under $\mathbb Q$. Hence $$H(\mathbb Q|\mathbb P) \ge \frac{1}{2}\mathbb E^\mathbb Q\left[\int_0^{T \wedge \tau_k}\alpha^\top a_r \alpha_rdr\right].$$ Letting $\tau_k \rightarrow T$ increasingly as $k \rightarrow + \infty$, a direct application of the monotone convergence theorem then yields $$\label{eq:HQP} H(\mathbb Q|\mathbb P) \ge \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T \alpha_r^\top a_r \alpha_rdr\right].$$ 2. We consider now the general case. Since $H(\mathbb Q| \mathbb P) < \infty$ we know that $\mathbb Q\ll \mathbb P$ and set $\mathbb Q_n := \left(1 - \frac{1}{n}\right)\mathbb Q+ \frac{1}{n}\mathbb P$. Then $\mathbb Q_n \sim \mathbb P$ and by convexity of the relative entropy, see Remark [Remark 2](#rmk:relativeEntropy){reference-type="ref" reference="rmk:relativeEntropy"}, $H(\mathbb Q_n | \mathbb P) \le \left(1 - \frac{1}{n}\right)H(\mathbb Q| \mathbb P) < + \infty$. By item $1.$, and the first part of the statement, using [\[eq:HQP\]](#eq:HQP){reference-type="eqref" reference="eq:HQP"} with $\mathbb Q_n$ instead of $\mathbb Q$, there exists a progressively measurable process $\alpha^n$ such that $$\label{eq:interSubBound} \begin{aligned} \left(1 - \frac{1}{n}\right)H(\mathbb Q| \mathbb P) \ge H(\mathbb Q_n | \mathbb P) & \ge \frac{1}{2}\mathbb E^{\mathbb Q_n}\left[\int_0^T (\alpha^n_r)^\top a_r \alpha_r^n\right]\\ & = \frac{1}{2}\left(1 - \frac{1}{n}\right)\mathbb E^{\mathbb Q}\left[\int_0^T (\alpha_r^n)^\top a_r \alpha^n_rdr\right] + \frac{1}{2n}\mathbb E^{\mathbb P}\left[\int_0^T (\alpha_r^n)^\top a_r \alpha^n_rdr\right]\\ & \ge \frac{1}{2}\left(1 - \frac{1}{n}\right)\mathbb E^{\mathbb Q}\left[\int_0^T (\alpha_r^n)^\top a_r \alpha^n_rdr\right]. \end{aligned}$$ Using the crucial estimate (33) in [@GirsanovEntropy], whose proof once again can be carried out on any probability space, one has $$\lim_{n \rightarrow + \infty} \mathbb E^{\mathbb Q}\left[\int_0^T (\alpha_r^n - \alpha_r)^\top a_r (\alpha_r^n - \alpha_r)dr\right] = 0,$$ which implies that $$\lim_{n \rightarrow + \infty} \mathbb E^{\mathbb Q}\left[\int_0^T (\alpha_r^n)^\top a_r \alpha^n_rdr\right] = \mathbb E^{\mathbb Q}\left[\int_0^T (\alpha_r)^\top a_r \alpha_rdr\right].$$ Letting $n \rightarrow + \infty$ in [\[eq:interSubBound\]](#eq:interSubBound){reference-type="eqref" reference="eq:interSubBound"} yields the desired result.  ◻ For the following lemma we keep the notations and assumptions of Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"}. Let in particular $X$ be a process fulfilling [\[eq:decXP\]](#eq:decXP){reference-type="eqref" reference="eq:decXP"} with $a_t = \sigma\sigma^\top(t, X_t)$. Then by Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"} there is a progressively measurable process $\alpha$ such that [\[eq:alpha\]](#eq:alpha){reference-type="eqref" reference="eq:alpha"} holds. For that we have the following estimates. **Lemma 28**. *We suppose the existence of $1 < p < 2$ such that $$C_p := \mathbb E^{\mathbb P}\left[\int_0^T \|\sigma(r, X_r)\|^{2p/(2 - p)}dr\right] < + \infty.$$* 1. *If $C_\infty := \|d\mathbb Q/d\mathbb P\|_\infty < + \infty$, there exists a constant $L > 0$ which depends only on $C_p$ and $C_\infty$ such that $$\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] \le L(1 + H(\mathbb Q| \mathbb P)).$$* 2. *Suppose moreover $H(\mathbb P| \mathbb Q) < + \infty$. Then it holds that $$\frac{1}{2}\mathbb E^{\mathbb P}\left[\int_0^T |\sigma^\top(r, X_r)\alpha_r|^2 dr\right] \le H(\mathbb P|\mathbb Q),$$ and $L$ can be chosen such that $$\label{eq:L2} \mathbb E^{\mathbb P}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] \le L(1 + H(\mathbb P| \mathbb Q)).$$* *Proof.* 1. We recall that $H(\mathbb Q,\mathbb P) < \infty.$ By Hölder's inequality applied on the measure space $([0, T] \times \Omega, {\mathcal B}([0, T])\otimes {\mathcal F}, dt\otimes d\mathbb Q)$, it holds that $$\label{eq:ineqLambdaIntermediate} \begin{aligned} \mathbb E^{\mathbb Q}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] & \le \mathbb E^{\mathbb Q}\left[\int_0^T \|\sigma(r, X_r)\|^p|\sigma^\top(r, X_r)\alpha_r|^pdr\right]\\ & \le \left(\mathbb E^{\mathbb Q}\left[\int_0^T \|\sigma(r, X_r)\|^{2p/(2 - p)}\right]\right)^{1 - p/2}\left(\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^\top(r, X_r)\alpha_r|^2dr\right]\right)^{p/2}. \end{aligned}$$ On the one hand, $$\label{eq:ineqLambda} \mathbb E^{\mathbb Q}\left[\int_0^T \|\sigma(r, X_r)\|^{2p/(p - 2)}dr\right] = \mathbb E^{\mathbb P}\left[\frac{d\mathbb Q}{d\mathbb P}\int_0^T \|\sigma(r, X_r)\|^{2p/(p - 2)}dr\right] \le C_\infty C_p.$$ On the other hand, by [\[eq:subBoundEntropy\]](#eq:subBoundEntropy){reference-type="eqref" reference="eq:subBoundEntropy"} $$\label{eq:ineqEntrop} \mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^\top(r, X_r)\alpha_r|^2dr\right] \le 2H(\mathbb Q| \mathbb P).$$ Combining [\[eq:ineqLambda\]](#eq:ineqLambda){reference-type="eqref" reference="eq:ineqLambda"} and [\[eq:ineqEntrop\]](#eq:ineqEntrop){reference-type="eqref" reference="eq:ineqEntrop"} with [\[eq:ineqLambdaIntermediate\]](#eq:ineqLambdaIntermediate){reference-type="eqref" reference="eq:ineqLambdaIntermediate"}, we get $$\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] \le 2^{p/2}(C_\infty C_p)^{1 - p/2}H(\mathbb Q| \mathbb P)^{p/2},$$ and as $p < 2$, using the inequality $$\label{eq:aq} \vert a\vert^q \le (1 + \vert a \vert), \ {\rm if} \ q \in ]0,1],$$ we have $$\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] \le 2 C_\infty C_p (1 + H(\mathbb Q| \mathbb P)).$$ Setting $L := 2 C_p \max (C_\infty, 1)$ one concludes the proof. 2. Just before the statement of the present lemma we have mentioned the decomposition [\[eq:alpha\]](#eq:alpha){reference-type="eqref" reference="eq:alpha"} holds, where the martingale $M^\mathbb Q$ verifies $\langle M^\mathbb Q\rangle_{\cdot} = \int_0^\cdot \sigma\sigma^\top(r, X_r)dr$. As $H(\mathbb P| \mathbb Q) < + \infty$, again Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"}, interchanging $\mathbb P$ and $\mathbb Q$, yields the existence of a progressively measurable process $\tilde \alpha$ such that under $\mathbb P$ the process $X$ decomposes as $$X_t = x + \int_0^t \delta_rdr + \int_0^t \sigma\sigma^\top(r, X_r)\alpha_rdr + \int_0^t \sigma\sigma^\top(r, X_r)\tilde \alpha_rdr + \tilde M_t,$$ where $\tilde M$ is a martingale and $$\frac{1}{2}\mathbb E^{\mathbb P}\left[\int_0^T |\sigma^\top(r, X_r)\tilde \alpha_r|^2dr\right] \le H(\mathbb P| \mathbb Q).$$ Identifying the bounded variation and the martingale components of $X$ under $\mathbb P$, we get that $\tilde M = M^\mathbb P$ and $\sigma\sigma^\top(r, X_r)\tilde \alpha_r = - \sigma\sigma^\top(r, X_r) \alpha_r$ $d\mathbb P\otimes dr$-a.e. In particular, $$\label{eq:reverseEntropy} \frac{1}{2}\mathbb E^{\mathbb P}\left[\int_0^T |\sigma^\top(r, X_r)\alpha_r|^2dr\right] \le H(\mathbb P| \mathbb Q).$$ Then, as in the proof of item $1.$, Hölder's inequality, [\[eq:ineqLambdaIntermediate\]](#eq:ineqLambdaIntermediate){reference-type="eqref" reference="eq:ineqLambdaIntermediate"} with $\mathbb Q$ replaced by $\mathbb P$, and [\[eq:reverseEntropy\]](#eq:reverseEntropy){reference-type="eqref" reference="eq:reverseEntropy"} yield $$\begin{aligned} \mathbb E^{\mathbb P}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\alpha_r|^pdr\right] & \le \left(\mathbb E^{\mathbb P}\left[\int_0^T \|\sigma(r, X_r)\|^{2p/(2 - p)}\right]\right)^{1 - p/2}\left(\mathbb E^{\mathbb P}\left[\int_0^T |\sigma^\top(r, X_r)\alpha_r|^2dr\right]\right)^{p/2}\\ & \le 2^{p/2}C_p^{1 - p/2}H(\mathbb P| \mathbb Q)^{p/2} \le 2 C_p (1+H(\mathbb P| \mathbb Q)), \end{aligned}$$ where, for the last inequality we have used [\[eq:aq\]](#eq:aq){reference-type="eqref" reference="eq:aq"} $p < 2$. This finally also implies the result [\[eq:L2\]](#eq:L2){reference-type="eqref" reference="eq:L2"}.  ◻ The results of Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"} can be specified if one considers probability measures on the canonical space $\Omega = C([0, T], \mathbb R^d)$. In the following, $\delta, \gamma : [0, T] \times C([0, T], \mathbb R^d) \mapsto \mathbb R^d$ are progressively measurable functions w.r.t. their corresponding Borel $\sigma$-fields. Let us reformulate Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"} in our setting. **Lemma 29**. *Let $\mathbb P\in {\mathcal P}(\Omega)$ such that, under $\mathbb P$ the canonical process can be decomposed as $$\label{eq:lemmaSDE} X_t = x + \int_0^t\delta(r, X)dr + M_t^\mathbb P,$$ where $M^\mathbb P$ is a martingale with $\langle M^\mathbb P\rangle = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$, where $\sigma$ verifies item $(ii)$ of Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"}. Let $\mathbb Q\in {\mathcal P}(\Omega)$.* 1. *Assume that $H(\mathbb Q|\mathbb P) < + \infty.$ Then we have the following.* 1. *There exists a progressively measurable process $\alpha$, with respect with natural filtration of $X$ (in particular of the form $\alpha = \alpha(\cdot, X)$) such that, under $\mathbb Q$, $X$ decomposes as $$\label{eq:841} X_t = x + \int_0^t \delta(r, X)dr + \int_0^t\sigma\sigma^\top(r, X_r) \alpha(r, X)dr + M_t^\mathbb Q,$$ where $M^\mathbb Q$ is a martingale with $\langle M^\mathbb Q\rangle_t = \int_0^t \sigma\sigma^\top(r, X_r)dr$ and $$\label{eq:inequalityEntropy} H(\mathbb Q| \mathbb P) \ge \frac{1}{2}\mathbb E^\mathbb Q\left[\int_0^T |\sigma^\top(r, X_r)\alpha(r, X)|^2dr\right].$$* 2. *If moreover uniqueness in law holds for [\[eq:lemmaSDE\]](#eq:lemmaSDE){reference-type="eqref" reference="eq:lemmaSDE"}, equality holds in [\[eq:inequalityEntropy\]](#eq:inequalityEntropy){reference-type="eqref" reference="eq:inequalityEntropy"}.* 2. *Assume that under $\mathbb Q$ the canonical process writes $$\label{eq:842} X_t = x + \int_0^t\gamma(r, X)dr + M_t^\mathbb Q,$$ where $M^\mathbb Q$ is a martingale with $\langle M^\mathbb Q\rangle = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$ and that uniqueness in law holds for [\[eq:lemmaSDE\]](#eq:lemmaSDE){reference-type="eqref" reference="eq:lemmaSDE"}.* *If $$\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{- 1}(r, X_r)(\delta(r, X) - \gamma(r, X))|^2dr\right] < + \infty,$$ then $H(\mathbb Q| \mathbb P) < + \infty$ and $$\label{eq:equalityEntropy} H(\mathbb Q|\mathbb P) = \frac{1}{2}\mathbb E^{\mathbb Q}\left[\int_0^T |\sigma^{- 1}(r, X_r)(\delta(r, X) - \gamma(r, X))|^2dr\right].$$* *Proof.* Part $(a)$ of item $1.$ of Lemma [\[eq:lemmaSDE\]](#eq:lemmaSDE){reference-type="ref" reference="eq:lemmaSDE"} is constituted by Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"} applied to the canonical space equipped with the natural filtration of the canonical process. Item 2. is the object of Lemma 4.4 $(iii)$ in [@LackerHierarchies]. As far as item 1.$(b)$ is concerned, we apply item 2. with $\gamma(r,X) = \delta(r, X) + \sigma\sigma^\top(r, X_r) \alpha(r, X)$ in [\[eq:842\]](#eq:842){reference-type="eqref" reference="eq:842"} so that $(\gamma-\delta)(r,X) = \sigma\sigma^\top(r, X_r) \alpha(r, X)$. So $\sigma^{-1}(r,X_r) (\delta-\gamma)(r,X)$ and the equality in [\[eq:inequalityEntropy\]](#eq:inequalityEntropy){reference-type="eqref" reference="eq:inequalityEntropy"} holds because of [\[eq:equalityEntropy\]](#eq:equalityEntropy){reference-type="eqref" reference="eq:equalityEntropy"}. ◻ **Remark 30**. *By Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} on the diffusion coefficient $\sigma$, uniqueness in law for the SDE [\[eq:lemmaSDE\]](#eq:lemmaSDE){reference-type="eqref" reference="eq:lemmaSDE"} holds e.g. if $\delta$ is bounded, or if $\delta(r, X) = b(r, X_r) + u(r, X)$ where $b$ has linear growth and $u$ is bounded. If $u = 0$, this follows from Theorem 10.1.3 of [@stroock] and the general case holds by Girsanov theorem.* ## Proof of Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"} {#app:proofThEx} To simplify the formalism of the proof we will assume that $b = 0$, as well as $\epsilon = 1$. Recall that in what follows, the filtration $({\mathcal F}_t)_{t \in [0, T]}$ is the canonical filtration on the canonical space $\Omega = C([0, T], \mathbb R^d)$, see the notations in Section [2](#S2){reference-type="ref" reference="S2"}. Following Section 3.2 in [@TanTouzi] we will make use of an enlarged probability space $\bar \Omega$ as well as an analogous form of the set ${\cal A}$ on this enlarged space, denoted $\bar {\cal A}$. This is stated in the definitions below. **Definition 31**. *Let $\bar \Omega := C([0, T], \mathbb R^d \times \mathbb R^d)$ and we denote $(X, U)$ its canonical process and $(\bar {\mathcal F}_t)_{0 \le t \le T}$ the associated canonical filtration. We also denotes $(\bar {\mathcal F}_t^X)_{t \in [0, T]}$ the natural filtration generated by the first component $X$ of the canonical process.* **Definition 32**. *Let $\bar {\cal A}$ be the subset of ${\mathcal P}(\bar \Omega)^2$ such that $(\bar \mathbb P, \bar \mathbb Q) \in \bar {\cal A}$ if the following holds.* 1. *$H(\bar \mathbb Q| \bar \mathbb P) < + \infty.$* 2. *Under $\bar \mathbb P$, $X$ decomposes as $$\label{eq:decompAugmentedSpaceP} X_t = x + U_t + M_t^{\bar \mathbb P},$$ such that $M^{\bar \mathbb P}$ is an $(\bar {\mathcal F}_t)$-local martingale, with $\langle M^{\bar \mathbb P} \rangle_t = \int_0^t \sigma\sigma^\top(r, X_r)dr$.* 3. *The processes $U$ is absolutely continuous w.r.t. the Lebesgue measure $\bar \mathbb P$-a.s. and $$\label{eq:ULambda1} U_t = \int_0^t u_r dr, \ d\bar \mathbb P\otimes dt {\rm-a.e.}$$* 4. *$u_r \in \mathbb U,$ $d\bar \mathbb P\otimes dr$-a.e.* **Remark 33**. 1. *The property $H(\bar \mathbb Q| \bar \mathbb P) < + \infty$ in Definition [Definition 32](#def:enlargedMinSet){reference-type="ref" reference="def:enlargedMinSet"} implies in particular that $\bar \mathbb Q\ll \bar \mathbb P$. Hence any property which is verified almost surely w.r.t. $\bar \mathbb P$ in the above definition also holds $\bar \mathbb Q$-a.s.* 2. *Clearly we have $$u_t := \underset{n \rightarrow + \infty}{\limsup}~n(U_t - U_{t - 1/n}), \ d\bar \mathbb P\otimes dt \ {\rm a.e.},$$ where we recall that previous $\limsup$ is defined for all $t$ and all $\omega$.* For $(\bar \mathbb P, \bar \mathbb Q) \in \bar {\cal A}$ we introduce the functional $\bar {\cal J}$ defined by $$\label{eq:barFunctional} \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P) := \mathbb E^{\bar \mathbb Q}\left[\int_0^T f(r, X_r, u_r)dr + g(X_T)\right] + H(\bar \mathbb Q| \bar \mathbb P).$$ The proof of Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"} requires several lemmas. We first need to establish a correspondence between the functional ${\cal J}$ defined on $\mathcal{A}$ and the functional $\bar {\cal J}$ defined on $\bar {\cal A}$. Let $(\mathbb P, \mathbb Q) \in {\cal A}$. By Definitions [Definition 10](#def:A){reference-type="ref" reference="def:A"} and [Definition 5](#def:PU){reference-type="ref" reference="def:PU"}, under $\mathbb P,$ the canonical process decomposes as $$X_t = x + \int_0^t u_r^\mathbb Pdr + M_t^\mathbb P,$$ where $M^\mathbb P$ is an $({\mathcal F}_t)$-martingale such that $\langle M^\mathbb P\rangle = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$ and $u^\mathbb P$ is a progressively measurable process with respect to the canonical filtration $({\mathcal F}_t)$ with values in $\mathbb U$. We rely on this decomposition in the following lemma for the association of the functional $\bar {\cal J}$ to ${\cal J}$. **Lemma 34**. *Let $(\mathbb P, \mathbb Q) \in \mathcal{A}$ introduced above. Let $\bar \mathbb P$ (resp. $\bar \mathbb Q$) be the law of $\left(X, \int_0^{\cdot} u_r^\mathbb Pdr\right)$ under $\mathbb P$ (resp. $\mathbb Q$). Then $(\bar \mathbb P, \bar \mathbb Q) \in \bar {\cal A}$, $\bar \mathbb Q$ is absolutely continuous with respect to $\bar \mathbb P$ with $d\bar \mathbb Q/d\bar \mathbb P= d\mathbb Q/d\mathbb P\circ \pi_X$ where $\pi_X$ is the projection on the first component of the space $\bar \Omega$, $H(\mathbb Q| \mathbb P) = H(\bar \mathbb Q| \bar \mathbb P)$ and ${\cal J}(\mathbb Q, \mathbb P) = \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P)$.* *Proof.* Letting $\bar \mathbb P$ (resp. $\bar \mathbb Q$) be the law of $\left(X, \int_0^{\cdot}u_r^{\mathbb P}dr\right)$ induced by $\mathbb P$ (resp. $\mathbb Q$) on $\bar \Omega$ we get $$\mathbb E^{\bar \mathbb Q}\left[\int_0^T f(r, X_r, u_r)dr\right] = \mathbb E^{\mathbb Q}\left[\int_0^T f(r, X_r, u_r^{\mathbb P})dr\right].$$ Furthermore, recalling that $\pi_X$ is the first coordinate projection on $\bar \Omega$, one has $d\bar \mathbb Q/d\bar \mathbb P= d\mathbb Q/d\mathbb P\circ \pi_X$ and this yields $$H(\bar \mathbb Q| \bar \mathbb P) = \mathbb E^{\bar \mathbb Q}\left[\log \frac{d\bar \mathbb Q}{d\bar \mathbb P}\right] = \mathbb E^{\bar \mathbb Q}\left[\log \frac{d\mathbb Q}{d\mathbb P} \circ \pi_X \right] = \mathbb E^{\mathbb Q}\left[\log \frac{d\mathbb Q}{d\mathbb P}(X)\right] = H(\mathbb Q| \mathbb P).$$ It follows that ${\cal J}(\mathbb Q, \mathbb P) = \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P).$ ◻ We establish a partial converse of the connection between ${\cal J}$ and $\bar {\cal J}$ in Lemma [Lemma 35](#lemma:fromOmegabarToOmega){reference-type="ref" reference="lemma:fromOmegabarToOmega"} below, whose proof crucially relies on the convexity of the functions $f(t, x, \cdot)$ and Jensen's inequality. **Lemma 35**. *Let $(\bar \mathbb P, \bar \mathbb Q) \in \bar {\cal A}$. Assume that $\|d\bar \mathbb Q/d\bar \mathbb P\|_\infty < + \infty$. There exists $(\mathbb P, \mathbb Q) \in \mathcal{A}$ such that $\bar {\cal J}(\bar \mathbb Q, \bar \mathbb P) \ge {\cal J}(\mathbb Q, \mathbb P)$ and $\mathbb P\in {\mathcal P}_\mathbb U^{Markov}$.* *Proof.* The proof of this result consists in two steps. In the first step we provide a lower bound of the cost $\bar{\cal J}(\bar \mathbb Q, \bar \mathbb P)$ in term of an expectation under the marginal law $\tilde \mathbb Q$ of the first component $X$ of the vector $(X, U)$. In the second step, we introduce a probability $\mathbb P$ such that $\bar{\cal J}(\bar \mathbb Q, \bar \mathbb P) \ge {\cal J}(\mathbb Q, \mathbb P)$ where $\mathbb Q$ is the law of a diffusion process mimicking the marginals of $\tilde \mathbb Q$ at each fixed time $t \in [0, T]$. We keep in mind the characterization of $\bar \mathbb P$ given by [\[eq:decompAugmentedSpaceP\]](#eq:decompAugmentedSpaceP){reference-type="eqref" reference="eq:decompAugmentedSpaceP"} and [\[eq:ULambda1\]](#eq:ULambda1){reference-type="eqref" reference="eq:ULambda1"}. In particular, under $\bar \mathbb P$ $X$ decomposes as $$X_t = x + \int_0^t u_r + M_t^{\bar \mathbb P}.$$ 1. Consequently, since $H(\bar \mathbb Q| \bar \mathbb P) < + \infty,$ see item $(i)$ of Definition [Definition 32](#def:enlargedMinSet){reference-type="ref" reference="def:enlargedMinSet"}, by Theorem [Theorem 27](#th:girsanovEntropy){reference-type="ref" reference="th:girsanovEntropy"}, on the space $\bar \Omega$ equipped with the probabilities $\bar \mathbb P$ and $\bar \mathbb Q$ with $\delta_r = u_r, a_r = \sigma \sigma^\top(r,X_r)$, there exists a progressively measurable process $\bar \alpha$ w.r.t. $(\bar {\mathcal F}_t)_{t \in [0, T]}$ on $\bar \Omega$ such that under $\bar \mathbb Q$ the process $X$ writes as $$\label{eq:decompAugmentedSpaceQTer} X_t = x + \int_0^t u_r dr + \int_0^t \sigma\sigma^\top(r, X_r)\bar\alpha_rdr + M_t^{\bar \mathbb Q},$$ where $M^{\bar \mathbb Q}$ is (again under $\bar \mathbb Q$) an $(\bar{{\mathcal F}}_t)$-local martingale with $\langle M^{\bar \mathbb Q} \rangle_{\cdot} = \int_0^{\cdot}\sigma\sigma^\top(r, X_r)dr$ and $$\label{eq:PrevEntropy} H(\bar \mathbb Q| \bar \mathbb P) \ge \frac{1}{2}\mathbb E^{\bar \mathbb Q}\left[\int_0^T |\sigma^\top(r, X_r)\bar \alpha_r|^2dr\right].$$ Moreover, under $\bar \mathbb Q,$ the process $X$ has some integrability properties. Indeed, by Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} we have that for all $q \ge 1$, $\mathbb E^{\bar \mathbb P}\left[\underset{0 \le r \le T}{\sup} |X_r|^q\right] < + \infty.$ In particular, by linear growth of $\sigma$ it holds that $$\label{eq:momentSigma} \mathbb E^{\bar \mathbb P}\left[\int_0^T \|\sigma(r, X_r)\|^qdr\right] < + \infty$$ for all $q \ge 1$. Then we can apply Lemma [Lemma 28](#lemma:estimateBetaEntrop){reference-type="ref" reference="lemma:estimateBetaEntrop"} item $1.$ which implies that for any $1 < p < 2$ $$\label{eq:boundEntropyPart} \mathbb E^{\bar \mathbb Q}\left[\int_0^T |\sigma\sigma^\top(r, X_r)\bar \alpha_r|^pdr\right] < + \infty.$$ Let us now decompose the semimartingale $X$ under $\bar \mathbb Q$ in its own filtration. To this aim, we denote by $(\beta_t)$ the process $\beta_t = u_t + \sigma\sigma^\top(t, X_t)\bar \alpha_t$. In particular, as $u$ is $\bar \mathbb Q$-essentially bounded, we get from [\[eq:boundEntropyPart\]](#eq:boundEntropyPart){reference-type="eqref" reference="eq:boundEntropyPart"} that $\mathbb E^{\bar \mathbb Q}\left[\int_0^T |\beta_r|dr\right] < + \infty$. Then by Proposition [Proposition 26](#prop:decompItoProcess){reference-type="ref" reference="prop:decompItoProcess"} item $(i)$, there exist progressively measurable functions $\tilde \beta, \tilde u : [0, T] \times \Omega \rightarrow \mathbb R^d$ with respect to $(\bar {\mathcal F}_t^X)_{t \in [0, T]}$, such that $$\label{eq:tildebeta-u} \tilde \beta(t, X) = \mathbb E^{\bar \mathbb Q}[\beta_t~|~\bar {\mathcal F}_t^X], \ \tilde u(t, X) = \mathbb E^{\bar \mathbb Q}[u_t | \bar {\mathcal F}_t^X]), \ \bar \mathbb Q\ {a.s.} \ \forall t \in [0,T].$$ Consequently, under $\bar \mathbb Q$, by [\[eq:decompAugmentedSpaceQTer\]](#eq:decompAugmentedSpaceQTer){reference-type="eqref" reference="eq:decompAugmentedSpaceQTer"}, the process $X$ is also an $(\bar {\mathcal F}_t^X)_{t \in [0, T]}$-semimartingale with decomposition $$\label{eq:interDecompQ} X_t = x + \int_0^t\tilde \beta(r, X)dr + \tilde M_t,$$ where $\tilde M$ is an $(\bar {\mathcal F}_t^X)_{t \in [0, T]}$-martingale and $\langle \tilde M \rangle_{\cdot} = \int_0^{\cdot}\sigma\sigma^\top(r, X_r)dr.$ By Fubini's theorem and Jensen's inequality for the conditional expectation $$\label{eq:1} \begin{aligned} \mathbb E^{\bar \mathbb Q}\left[\int_0^T f(r, X_r, u_r)dr + g(X_T)\right] & \ge \mathbb E^{\bar \mathbb Q}\left[\int_0^T f(r, X_r, \mathbb E^{\bar \mathbb Q}[u_r|\bar{\mathcal F}_r^X])dr + g(X_T)\right]\\ & = \mathbb E^{\bar \mathbb Q}\left[\int_0^T f(r, X_r, \tilde u(r, X))dr + g(X_T)\right] \\ & = \mathbb E^{\tilde\mathbb Q}\left[\int_0^T f(r, X_r, \tilde u(r, X))dr + g(X_T)\right], \end{aligned}$$ where $\tilde \mathbb Q$ is the first marginal of $\bar \mathbb Q$, i.e. the law of the first component $X$ of the vector $(X, U)$ under $\bar \mathbb Q$. Moreover, the entropy inequality [\[eq:PrevEntropy\]](#eq:PrevEntropy){reference-type="eqref" reference="eq:PrevEntropy"} rewrites $$\label{eq:PEntropyIneq} H(\bar \mathbb Q| \bar \mathbb P) \ge \frac{1}{2}\mathbb E^{\bar \mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\beta_r - u_r)|^2dr\right],$$ and again Fubini's theorem and Jensen's inequality for the conditional expectation gives $$\label{eq:2} H(\bar \mathbb Q| \bar \mathbb P) \ge \frac{1}{2}\mathbb E^{\tilde\mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\tilde \beta(r, X) - \tilde u(r, X))|^2dr\right].$$ 2. Next by Fubini's theorem and Jensen's inequality for the conditional expectation and taking into account [\[eq:tildebeta-u\]](#eq:tildebeta-u){reference-type="eqref" reference="eq:tildebeta-u"}, it holds that $$\label{eq:estimateL1TildeBeta} \mathbb E^{\tilde \mathbb Q}\left[\int_0^T |\tilde \beta(r, X)|dr\right] = \mathbb E^{\bar \mathbb Q}\left[\int_0^T |\tilde \beta(r, X)|dr\right] \le \mathbb E^{\bar \mathbb Q}\left[\int_0^T |\beta_r|dr\right] < + \infty.$$ First, observe that as $\|d\bar\mathbb Q/d\bar \mathbb P\|_\infty < + \infty$, [\[eq:momentSigma\]](#eq:momentSigma){reference-type="eqref" reference="eq:momentSigma"} is also true replacing $\bar \mathbb P$ by $\bar \mathbb Q$. Hence from [\[eq:estimateL1TildeBeta\]](#eq:estimateL1TildeBeta){reference-type="eqref" reference="eq:estimateL1TildeBeta"} and [\[eq:momentSigma\]](#eq:momentSigma){reference-type="eqref" reference="eq:momentSigma"} with $q = 2$ we deduce that $\mathbb E^{\tilde \mathbb Q}\left[\int_0^T (|\tilde \beta(r, X)| + \|\sigma\sigma^\top(r, X_r)\|)dr\right] < + \infty$, and by Corollary 3.7 in [@MimickingItoGeneral] there exist a measurable function $\hat \beta : [0, T] \times \mathbb R^d \mapsto \mathbb R^d$ and a probability measure $\mathbb Q$ on $(\Omega, {\mathcal F})$ such that the following holds. - For all $0 \le t \le T$, $\hat \beta(t, X_t) = \mathbb E^{\tilde \mathbb Q}[\tilde \beta(t, X)~|~X_t]$ $d\tilde \mathbb Q\otimes dt$-a.e. - Under $\mathbb Q$ the canonical process can be expressed as $X_t = x + \int_0^t \hat \beta(r, X_r)dr + M^\mathbb Q_t,$ where $M^\mathbb Q$ is a $({\mathcal F}_t)$-local martingale with $\langle M^\mathbb Q\rangle_{\cdot} = \int_0^{\cdot}\sigma\sigma^\top(r, X_r)dr$. - $\mathcal{L}^\mathbb Q(X_t) = \mathcal{L}^{\tilde \mathbb Q}(X_t), \ \forall t \in [0,T].$ Finally Proposition 5.1 in [@MimickingItoGeneral] provides a measurable function $\hat u$ such that $$\label{eq:hatu} \hat u(t, X_t) = \mathbb E^{\tilde \mathbb Q}[\tilde u(t, X)~|~X_t], \ d\tilde \mathbb Q\otimes dt \ {\rm a.e.}$$ We modify $\hat u$ on the Borel set $N = \{(t, x) \in [0, T] \times \mathbb R^d~:~\hat u(t, x) \notin \mathbb U\}$ so that $\hat u(t, x) \in \bar \mathbb U$ for all $(t, x) \in [0, T] \times \mathbb R^d$. Then once again by Fubini's theorem and Jensen's inequality for the conditional expectation, we get that $$\label{eq:3} \begin{aligned} \mathbb E^{\tilde\mathbb Q}\left[\int_0^T f(r, X_r, \tilde u(r, X))dr + g(X_T)\right] & \ge \mathbb E^{\tilde\mathbb Q}\left[\int_0^T f(r, X_r, \mathbb E^{\tilde \mathbb Q}[\tilde u(r, X)~|~X_r])dr + g(X_T)\right]\\ & = \mathbb E^{\tilde\mathbb Q}\left[\int_0^T f(r, X_r, \hat u(r, X_r))dr + g(X_T)\right]\\ & = \mathbb E^{\mathbb Q}\left[\int_0^T f(r, X_r, \hat u(r, X_r))dr + g(X_T)\right] \end{aligned}$$ and also $$\label{eq:Jensen_again} \begin{aligned} \frac{1}{2}\mathbb E^{\tilde\mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\tilde \beta(r, X) - \tilde u(r, X))|^2dr\right] & \ge \frac{1}{2}\mathbb E^{\tilde\mathbb Q}\left[\int_0^T |\sigma^{-1}(r, X_r)(\hat \beta(r, X_r) - \hat u(r, X_r))|^2dr \right] \\ & = \frac{1}{2}\mathbb E^{\mathbb Q} \left[\int_0^T |\sigma^{-1}(r, X_r)(\hat \beta(r, X_r) - \hat u(r, X_r))|^2dr \right]. \end{aligned}$$ As $\hat u$ is bounded and $b$, chosen here for simplicity equal to zero, has also linear growth, Theorem 10.1.3 in [@stroock] proves existence and uniqueness of a solution $\mathbb P\in {\mathcal P}_\mathbb U^{Markov}$ to the martingale problem with initial condition $(0, x)$, and the operator $L_{\hat u}$ defined in [\[eq:generatorU\]](#eq:generatorU){reference-type="eqref" reference="eq:generatorU"} so, by Remark [Remark 6](#rmk:stroockVaradhan){reference-type="ref" reference="rmk:stroockVaradhan"} $$X_t = x + \int_0^t \hat u(r,X_r) dr + M^\mathbb P_t, \ t\in [0,T],$$ where $M^\mathbb P$ is a local martingale vanishing at zero such that $\langle M^\mathbb P\rangle_t = \int_0^t \sigma\sigma^\top(r,X_r) dr.$ By Lemma [Lemma 29](#lemma:girsanovEntropy){reference-type="ref" reference="lemma:girsanovEntropy"} item 2., the right-hand side of [\[eq:Jensen_again\]](#eq:Jensen_again){reference-type="eqref" reference="eq:Jensen_again"} is equal to $H(\mathbb Q| \mathbb P)$. Combining alltogether the expressions [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"}, [\[eq:2\]](#eq:2){reference-type="eqref" reference="eq:2"}, [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} and [\[eq:Jensen_again\]](#eq:Jensen_again){reference-type="eqref" reference="eq:Jensen_again"} we get $\bar {\cal J}(\bar \mathbb Q, \bar \mathbb P) \ge {\cal J}(\mathbb Q, \mathbb P)$. This concludes the proof.  ◻ We emphasize that, even though the condition $\|d\bar \mathbb Q/d\bar \mathbb P\|_\infty < + \infty$ in Lemma [Lemma 35](#lemma:fromOmegabarToOmega){reference-type="ref" reference="lemma:fromOmegabarToOmega"} is very restrictive, we will see at the end of this section that it will be enough to prove Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"}. The connection between ${\cal J}$ and $\bar {\cal J}$ is thus established. To prove the theorem we also need tightness results on our enlarged space. This is stated in the following lemma and proposition. **Lemma 36**. *Let $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ be a sequence of elements of $\mathcal{A}$. Then $(\mathbb P_n)_{n \ge 1}$ is tight.* *Proof.* In this proof, $C$ denotes a generic non-negative constant. By definition, under $\mathbb P_n$, the canonical process has decomposition $$X_t = x + \int_0^t u_r^{\mathbb P_n}dr + M_t^{\mathbb P_n},$$ where $u^{\mathbb P^n}$ takes values in $\mathbb U$, $M^{\mathbb P_n}$ is a martingale and $\langle M^{\mathbb P_n}\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma^\top(r, X_r)dr$. Let $p > 1$. For $0 \le s \le t$ we have $$\begin{aligned} \mathbb E^{\mathbb P_n}[|X_t - X_s|^{2p}] & \le C\left(\mathbb E^{\mathbb P_n}\left[\left|\int_s^t u_r^{\mathbb P_n}dr\right|^{2p}\right] + \mathbb E^{\mathbb P_n}\left[|M^{\mathbb P_n}_t - M^{\mathbb P_n}_s|^{2p}\right]\right)\\ & \le C\left((t - s)^{2p} + \mathbb E^{\mathbb P_n}\left[\left(\int_s^t\|\sigma(r, X_r)\|^2dr\right)^{p}\right]\right) & \text{(BDG inequality)}\\ & \le C\left((t - s)^{2p} + (t - s)^p + \mathbb E^{\mathbb P_n}\left[\left(\int_s^t |X_r|^2dr\right)^p\right]\right) & \text{(Hypothesis \ref{hyp:coefDiffusion})}\\ & \le C\left((t - s)^{2p} + (t - s)^p +(t - s)^p\mathbb E^{\mathbb P_n}\left[\int_s^t |X_r|^{2p/(p - 1)}dr\right]^{p - 1}\right) & \text{(H\"older inequality)}\\ & \le C(t - s)^p. & \text{(Lemma \ref{lemma:classicalEstimates})}, \end{aligned}$$ so $(\mathbb P^n)_{n \ge 1}$ is a tight sequence by Kolmogorov criteria, see e.g. Problem 4.11 of [@karatshreve]. ◻ We will need in the following a simple technical observation. **Lemma 37**. *Let $(\mathbb P_n)_{n \ge 1}$ be a sequence of Borel probability measures on a Polish space $Y$ that weakly converges towards a probability measure $\mathbb P_{\infty}.$ Let $\phi : Y \rightarrow \mathbb R$ be a continuous function. Assume that there exists $\alpha, C > 0$ such that $$\label{eq:uniformInt} \sup_{n\ge 1} \int_{Y} |\phi(y)|^{1 + \alpha}(y)\mathbb P_n(dy) \le C.$$ Then $$\int_{Y}\phi(y)\mathbb P_n(dy) \underset{n \rightarrow + \infty}{\longrightarrow} \int_{Y}\phi(y)\mathbb P_{\infty}(dy).$$* *Proof.* By Skorokhod's representation theorem, there exists a probability space $( \Omega, {\mathcal F}, \mathbb{Q})$, a sequence of random variable $(X_n)_{n \ge 1}$ on $\Omega$ and a random variable $X$ such that $\mathcal{L}^{\mathbb{Q}}(X_n) = \mathbb P_n$ and $X_n \rightarrow X$ $\mathbb{Q}$-a.s. Condition [\[eq:uniformInt\]](#eq:uniformInt){reference-type="eqref" reference="eq:uniformInt"} implies that the sequence $(\phi(X_n))_{n \ge 1}$ is uniformly integrable. Furthermore, by continuity of $\phi$, $\phi(X_n) \underset{n \rightarrow + \infty}{\longrightarrow} \phi(X)$ $\mathbb{Q}$-a.s. Thus $$\mathbb E^{\mathbb{Q}}[\phi(X_n)] \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\mathbb{Q}}[\phi(X)]$$ or equivalently $$\int_{Y}\phi(y)\mathbb P_n(dy) \underset{n \rightarrow + \infty}{\longrightarrow} \int_{Y}\phi(y)\mathbb P_{\infty}(dy).$$ ◻ **Remark 38**. *Let $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ be a sequence of couples of probability measures on a measurable space, where both the sequences $(\mathbb P_n)_{n \ge 1}$ and $(\mathbb Q_n)_{n \ge 1}$ are tight. Then there is a couple of probability measures $(\mathbb P,\mathbb Q)$ and a subsequence $(\mathbb P_{n_k}, \mathbb Q_{n_k})$ such that $(\mathbb P_{n_k})$ (resp. $(\mathbb Q_{n_k})$) converges weakly to $\mathbb P$ (resp. $(\mathbb Q)$). Such a couple $(\mathbb P,\mathbb Q)$ will be called **limit point** of the sequence $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$.* **Proposition 39**. *Let $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ be a sequence of elements of $\mathcal{A}$ such that $\sup_{n \ge 1} \Vert d\mathbb Q_n/d\mathbb P_n \Vert_{\infty} < + \infty$ and $\sup_{n \ge 1} H(\mathbb Q_n | \mathbb P_n) < + \infty$. Let $(\bar \mathbb P_n, \bar \mathbb Q_n)_{n \ge 1}$ be the corresponding sequence of probability measures on $(\bar \Omega, \bar {\mathcal F})$ given by Lemma [Lemma 34](#lemma:fromOmegaToOmegabar){reference-type="ref" reference="lemma:fromOmegaToOmegabar"}. Then the following properties hold.* 1. *The sequences $(\bar \mathbb P_n)_{n \ge 1}$ and $(\bar \mathbb Q_n)_{n \ge 1}$ are tight and under any corresponding limit point $\bar \mathbb P$ of $(\bar \mathbb P_n)_{n \ge 1}$, the process $U$ has absolutely continuous paths.* 2. *Any limit point $(\bar \mathbb P, \bar \mathbb Q)$ of $(\bar \mathbb P_n, \bar \mathbb Q_n)_{n \ge 1}$ belongs to $\bar {\cal A}$.* *Proof.* 1. Let $1 < p$. By Definitions [Definition 10](#def:A){reference-type="ref" reference="def:A"} and [Definition 5](#def:PU){reference-type="ref" reference="def:PU"} there is a progressively measurable process $u^{\mathbb P_n}$ with values in $\mathbb U$ such that $$\label{eq:PUn} X_t = x + \int_0^t u^{\mathbb P_n}_rdr + M^{\mathbb P_n}_t, ~0 \le t \le T,$$ for some local martingale $M^\mathbb P_n$. As $\mathbb U$ is bounded, we have $$\label{eq:tightnessUP} \sup_{n \ge 1}\mathbb E^{\mathbb P_n}\left[\int_0^T |u_r^{\mathbb P_n}|^{p}dr\right] < + \infty.$$ We set $U^n_{\cdot} := \int_0^{\cdot} u_r^{\mathbb P_n}dr$. As [\[eq:tightnessUP\]](#eq:tightnessUP){reference-type="eqref" reference="eq:tightnessUP"} holds, Lemma 2 in [@ZhengTightness] yields tightness of the laws $(\mu_n := \mathcal{L}^{\mathbb P_n}(U^n))_{n \ge 1}$ and under any limit point $\bar \mu$, the second component $U$ of the canonical process $(X,U)$ on $\bar \Omega$, has absolutely continuous path w.r.t. the Lebesgue measure. Moreover by Lemma [Lemma 36](#lemma:tightness){reference-type="ref" reference="lemma:tightness"} the sequence $\mathbb P_n)_{n \ge 1}$ is tight. It follows from what precedes that each marginal of $\left(X, \int_0^{\cdot}u_r^{\mathbb P_n}dr\right)$ under $(\mathbb P_n)_{n \ge 1}$ is tight, hence tightness of the laws $(\bar \mathbb P_n)_{n \ge 1}$ of previous vector on the product space $\bar \Omega$, and under any limit point $\bar \mathbb P$ the paths of $U$ are absolutely continuous. Finally Lemma [Lemma 34](#lemma:fromOmegaToOmegabar){reference-type="ref" reference="lemma:fromOmegaToOmegabar"} states that $d\bar \mathbb Q_n/d\bar \mathbb P_n = d\mathbb Q_n/d\mathbb P_n \circ \pi_X$, where $\pi_X$ denotes the projection on the first component of the space $\bar \Omega$. This implies that $\sup_{n \ge 1} \|d\bar \mathbb Q_n/d\bar \mathbb P_n\|_\infty < + \infty$, and tightness of the sequence $(\bar \mathbb Q_n)_{n \ge 1}$ then follows from the tightness of $(\bar \mathbb P_n)_{n \ge 1}$. 2. Let $(\bar \mathbb P, \bar \mathbb Q)$ be any limit point of the sequence $(\bar \mathbb P_n, \bar \mathbb Q_n)_{n \ge 1}$, see Remark [Remark 38](#rmk:tightness){reference-type="ref" reference="rmk:tightness"}. One can assume that both sequences $(\bar \mathbb P_n)_{n \ge 1}$ and $(\bar \mathbb Q_n)_{n \ge 1}$ converges weakly towards $\bar \mathbb P$ and $\bar \mathbb Q$ respectively. We are going to prove that $(\bar \mathbb P, \bar \mathbb Q) \in \bar {\cal A}$. By item $1.$ item $(iii)$ of Definition [Definition 32](#def:enlargedMinSet){reference-type="ref" reference="def:enlargedMinSet"} of $\bar {\cal A}$ holds. Let us verify item $(i)$ of the same definition. Indeed by Lemma [Lemma 34](#lemma:fromOmegaToOmegabar){reference-type="ref" reference="lemma:fromOmegaToOmegabar"} we have $H(\bar \mathbb Q_n | \bar \mathbb P_n) = H(\mathbb Q_n | \mathbb P_n)$. We recall that $(\mathbb Q, \mathbb P) \mapsto H(\mathbb Q| \mathbb P)$ is lower semicontinous with respect to the weak convergence on Polish spaces, see Remark [Remark 2](#rmk:relativeEntropy){reference-type="ref" reference="rmk:relativeEntropy"}, $\bar\Omega$ being the Polish space. Consequently $$H(\bar \mathbb Q| \bar \mathbb P) \le \liminf_{n \rightarrow + \infty} H(\bar \mathbb Q_n | \bar \mathbb P_n) = \liminf_{n \rightarrow + \infty} H(\mathbb Q_n | \mathbb P_n) \le \sup_{n \ge 1} H(\mathbb Q_n | \mathbb P_n) < + \infty.$$ Let us now check item $(ii)$. Let $0 \le u < t \le T$. Let $h$ belonging to the space $C_c^{\infty}(\mathbb R^d)$ of smooth functions with compact support on $\mathbb R^d$. By [\[eq:decompP\]](#eq:decompP){reference-type="eqref" reference="eq:decompP"}, under $\bar \mathbb P_n$ we have $X = x + M + U$ where $U$ is an absolutely continuous process. By Itô's formula applied to [\[eq:decompP\]](#eq:decompP){reference-type="eqref" reference="eq:decompP"} under $\bar \mathbb P_n$, the process $$N [h]_{\cdot} := h(X_{\cdot} - U_{\cdot}) - h(x) - \frac{1}{2}\int_0^{\cdot} Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]dr$$ is a martingale under $\mathbb P_n$. We want to prove that $N(h)$ is a martingale under $\bar \mathbb P$. Let $\psi : C([0, u]; \mathbb R^d \times \mathbb R^d) \rightarrow \mathbb R$ be a bounded continuous function. Then $$\label{eq:martingaleEquality} %\begin{aligned} \mathbb E^{\bar\mathbb P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) N[h]_t\right] = %& = %\E^{\P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) % N[h]_t\right] \\ % & = %\E^{\P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) % N[h]_u\right] \\ & = \mathbb E^{\bar\mathbb P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) N[h]_u\right]. % \end{aligned}$$ On the one hand, the function $$(X, U) \mapsto \psi\left((X_r,U_r)_{r \in [0,u]}\right) \left(h(X_t - U_t) - h(x)\right)$$ is bounded and continuous. Hence $$\label{eq:weakLimit1} \mathbb E^{\bar \mathbb P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) \left(h(X_t - U_t) - h(x)\right)\right] \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\bar \mathbb P}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) \left(h(X_t - U_t) - h(x)\right)\right].$$ On the other hand, there exists a constant $C$ which only depends on $d$ such that for all $r \in [0, T]$, $$\left|Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]\right| \le C\|\nabla_x^2 h\|_\infty\|\sigma(r, X_r)\| \le CC_{b, \sigma}\|\nabla_x^2 h\|_\infty(1 + |X_r|). %% J'ai rajouté Nabla à h$$ Combining the previous inequality with Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} we get that for some $\alpha > 0$, $$\label{eq:estimateWeakConv} \sup_{n \in \mathbb N}\sup_{r \in [0, T]} \mathbb E^{\bar \mathbb P_n}\left[\left|Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]\right|^{1 + \alpha}\right] < + \infty.$$ Hence it holds $$\sup_{n \in \mathbb N} \mathbb E^{\bar \mathbb P_n}\left[\left|\psi\left((X_r,U_r)_{r \in [0,u]}\right)\int_0^t Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]dr\right|^{1 + \alpha}\right] < + \infty,$$ and by Lemma [Lemma 37](#lemma:weakConvUnbounded){reference-type="ref" reference="lemma:weakConvUnbounded"} with $Y = C([0,t])$, taking into account Hypothesis [\[hyp:coefDiffusion\]](#hyp:coefDiffusion){reference-type="ref" reference="hyp:coefDiffusion"} $(ii)$, we get $$\label{eq:weakLimit2} \begin{aligned} &\mathbb E^{\bar \mathbb P_n}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right)\int_0^t Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]dr\right]\\ & \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\bar \mathbb P}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right)\int_0^t Tr[\sigma\sigma^\top(r, X_r)\nabla_x^2h(X_r - U_r)]dr\right]. \end{aligned}$$ Combining [\[eq:weakLimit1\]](#eq:weakLimit1){reference-type="eqref" reference="eq:weakLimit1"} and [\[eq:weakLimit2\]](#eq:weakLimit2){reference-type="eqref" reference="eq:weakLimit2"} and letting $n \mapsto + \infty$ in [\[eq:martingaleEquality\]](#eq:martingaleEquality){reference-type="eqref" reference="eq:martingaleEquality"} yields $$\mathbb E^{\bar\mathbb P}\left[\psi\left((X_r,U_r)_{r \in [0,u]}\right) N[h]_t\right].$$ Hence the process $N[h]$ is an $((\bar {\mathcal F}_t), \bar \mathbb P)$-martingale for all $h \in C_c^{\infty}(\mathbb R^d)$. By standard usual stochastic calculus arguments, this implies that under $\bar \mathbb P$ the process writes $$X_t = x + U_t + M_t^{\bar \mathbb P},$$ where $M^{\bar \mathbb P}$ is a $(\bar {\mathcal F}_t)$-local martingale with $$\langle M^{\bar \mathbb P} \rangle_t = \int_0^t \sigma\sigma^\top(r, X_r)dr ~d\bar\mathbb P\otimes dt\text{-a.e.},$$ hence item $(ii)$. Finally, it remains to prove $(iv)$. Let $t \in ]0, T[$, $k \in \mathbb N^*$ large enough. As $\mathbb U$ is convex and closed, one has $\bar \mathbb P_n(k (U_{t + 1/k} - U_t) \in \mathbb U) % = \P_n\left(k\int_t^{t + 1/k}u_r^{\P_n}dr \in \U \right) = 1$ for all $n \ge 1$. Moreover, as $\mathbb U$ is closed, the set $\left\{U \in C([0, T], \mathbb R^d)~:~k (U_{t + 1/k} - U_t) \in \mathbb U\right\}$ is closed under the uniform convergence. By Portmanteau Theorem, see Theorem 2.1 in [@BillingsleyConv], $1 = \underset{n \ge 1}{\sup}~ \bar \mathbb P_n(k (U_{t + 1/k} - U_t) \in \mathbb U) \le \bar \mathbb P(k (U_{t + 1/k} - U_t) \in \mathbb U)$. Hence $\bar \mathbb P$-a.s., $k(U_{t + 1/k} - U_t) \in \mathbb U$. As $U$ is continuous, $\bar \mathbb P$-a.s., for all $t \in [0, T]$ and $k \in \mathbb N^*$, $k(U_{t + 1/k} - U_t) \in \mathbb U$. $\mathbb U$ being closed, letting $k \rightarrow + \infty$ yields $$\limsup_{k \rightarrow + \infty}k(U_{t + 1/k} - U_t) \in \mathbb U, ~\bar \mathbb P\text{-a.s.},$$ and we conclude that $d\bar \mathbb P\otimes dt$-a.e. $$u_t = \lim_{k \rightarrow + \infty} k(U_{t + 1/k} - U_t) = \limsup_{k \rightarrow + \infty}k(U_{t + 1/k} - U_t) \in \mathbb U.$$  ◻ To apply Proposition [Proposition 39](#prop:tightness){reference-type="ref" reference="prop:tightness"} we will need the lemma below. **Lemma 40**. *There exists a minimizing sequence $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ for ${\cal J}$ such that the following holds.* 1. *$\underset{n \ge 1}{\sup}\Vert d\mathbb Q_n/d\mathbb P_n \Vert_{\infty} < + \infty$ and $\underset{n \ge 1}{\sup} H(\mathbb Q_n | \mathbb P_n) < + \infty$.* 2. *$\underset{n \in \mathbb N}{\sup}~\mathbb E^{\mathbb Q_n}\left[\underset{0 \le t \le T}{\sup} |X_t|^q\right] < + \infty$ for all $q \ge 1$.* *Proof.* Let $(\mathbb P_n, \tilde \mathbb Q_n)_{n \ge 1}$ be a minimizing sequence for ${\cal J}$. Let $$\label{eq:Interm} d\mathbb Q_n := \frac{\exp\left(-\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr - g(X_T)\right)}{\mathbb E^{\mathbb P_n}\left[\exp\left(-\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr - g(X_T)\right)\right]}d\mathbb P_n.$$ Then from Proposition [Proposition 11](#prop:markovExistenceMinimizer){reference-type="ref" reference="prop:markovExistenceMinimizer"} applied to $\mathbb P= \mathbb P_n$, we get that ${\cal J}(\mathbb Q_n, \mathbb P_n) \le {\cal J}(\tilde \mathbb Q_n, \mathbb P_n),$ thus $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ is also a minimizing sequence for ${\cal J}.$ It follows from Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} that $$\label{eq:Interm1} \exp\left(-\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr - g(X_T)\right) \le 1.$$ Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} and Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} imply the existence of $C > 0$ independent of $n$ such that $$\label{eq:Interm2} 0 \le \mathbb E^{\mathbb P_n}\left[\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr + g(X_T)\right] \le C.$$ By Jensen's inequality, we get for all $n \ge 1$ $$\mathbb E^{\mathbb P_n}\left(\exp\left(-\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr - g(X_T)\right)\right) \ge \exp\left( \mathbb E^{\mathbb P_n} \left(-\int_0^T f(r, X_r, u_r^{\mathbb P_n})dr - g(X_T)\right)\right) \ge \exp(- C).$$ Consequently $$\frac{d\mathbb Q_n}{d\mathbb P_n} \le e^{C},$$ hence $$\label{eq:Hexp} \sup_{n \ge 1} \left\Vert \frac{d\mathbb Q_n}{d\mathbb P_n} \right\Vert_{\infty} \le e^{C} < + \infty$$ and $$\sup_{n \ge 1} H(\mathbb Q_n | \mathbb P_n) \le C < + \infty.$$ This establishes item $(i)$. Furthermore for all $q \ge 1$, by [\[eq:Hexp\]](#eq:Hexp){reference-type="eqref" reference="eq:Hexp"} we have $$\mathbb E^{\mathbb Q_n}\left[\underset{0 \le t \le T}{\sup} |X_t|^q\right] = \mathbb E^{\mathbb P_n}\left[\left(\underset{0 \le t \le T}{\sup} |X_t|^q\right)\frac{d\mathbb Q_n}{d\mathbb P_n}\right] \le e^{ C}\mathbb E^{\mathbb P_n}\left[\underset{0 \le t \le T}{\sup} |X_t|^q\right]$$ and the estimate $(ii)$ follows from Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"}. ◻ We are finally ready to prove Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"}. *Proof*(of Theorem [Theorem 12](#th:existenceSolutionRegProb){reference-type="ref" reference="th:existenceSolutionRegProb"}). Let $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ be a minimizing sequence as provided by Lemma [Lemma 40](#lemma:exisGoodMinSeq){reference-type="ref" reference="lemma:exisGoodMinSeq"}. Let $(\bar \mathbb P_n, \bar \mathbb Q_n)_{n \ge 1}$ be the corresponding sequence of probability measures induced by $(\mathbb P_n, \mathbb Q_n)_{n \ge 1}$ on $\bar \Omega$ given by Lemma [Lemma 34](#lemma:fromOmegaToOmegabar){reference-type="ref" reference="lemma:fromOmegaToOmegabar"}. By Proposition [Proposition 39](#prop:tightness){reference-type="ref" reference="prop:tightness"} $(i)$ the sequences $(\bar \mathbb P_n)_{n \ge 1}$ and $(\bar \mathbb Q_n)_{n \ge 1}$ are tight. Let then $(\bar \mathbb P, \bar \mathbb Q)$ be a limit point of $(\bar \mathbb P_n, \bar \mathbb Q_n)_{n \ge 1}$), see Remark [Remark 38](#rmk:tightness){reference-type="ref" reference="rmk:tightness"}. By Proposition [Proposition 39](#prop:tightness){reference-type="ref" reference="prop:tightness"} $(ii)$, $(\mathbb P,\mathbb Q) \in \bar {\cal A}$. By Hypothesis [\[hyp:costFunctionsControl\]](#hyp:costFunctionsControl){reference-type="ref" reference="hyp:costFunctionsControl"} and Lemma [Lemma 40](#lemma:exisGoodMinSeq){reference-type="ref" reference="lemma:exisGoodMinSeq"} *(ii)* for any given $\alpha > 0$ we have $$\label{eq:intermediateWeakConv} \sup_{n \ge 1}\sup_{r \in [0, T]} \mathbb E^{\bar \mathbb Q_n}\left[|f(r, X_r, u_r)|^{1 + \alpha}\right] < + \infty.$$ Since $\bar \mathbb Q_n$ converges weakly to $\bar \mathbb Q$, by Lemma [Lemma 37](#lemma:weakConvUnbounded){reference-type="ref" reference="lemma:weakConvUnbounded"}, for all $r \in [0, T]$ we have $$\mathbb E^{\bar \mathbb Q_n}[f(r, X_r, u_r)] \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\bar \mathbb Q}[f(r, X_r, u_r)], \quad \mathbb E^{\bar \mathbb Q_n}[g(X_T))] \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\bar \mathbb Q}[g(X_T))]. %% J'ai rajouté$$ By [\[eq:intermediateWeakConv\]](#eq:intermediateWeakConv){reference-type="eqref" reference="eq:intermediateWeakConv"}, Fubini's and dominated convergence theorems $$\mathbb E^{\bar \mathbb Q_n} \left[\int_0^T f(r, X_r, u_r)dr + g(X_T)\right] = \int_0^T \mathbb E^{\bar \mathbb Q_n}[f(r, X_r, u_r)]dr + \mathbb E^{\bar \mathbb Q_n}[g(X_T)] \underset{n \rightarrow + \infty}{\longrightarrow} \mathbb E^{\bar \mathbb Q} \left[\int_0^T f(r, X_r, u_r)dr + g(X_T)\right].$$ We recall now that the relative entropy $H$ is lower semicontinuous with respect to the two variables for the topology of weak convergence on Polish spaces, see Remark [Remark 2](#rmk:relativeEntropy){reference-type="ref" reference="rmk:relativeEntropy"}. Hence, keeping in mind [\[eq:barFunctional\]](#eq:barFunctional){reference-type="eqref" reference="eq:barFunctional"} $$\label{eq:lscBarJ} \liminf_{n \rightarrow + \infty} \bar {\cal J}(\bar \mathbb Q_n, \bar \mathbb P_n) \ge \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P).$$ Since $(\mathbb P_n,\mathbb Q_n)$ is a minimizing sequence, $$\label{eq:ineqExistenceMin} {\cal J}^* = \lim_{n \rightarrow + \infty} {\cal J}(\mathbb Q_n, \mathbb P_n) = \liminf_{n \rightarrow + \infty} {\cal J}(\mathbb Q_n, \mathbb P_n) = \liminf_{n \rightarrow + \infty} \bar {\cal J}(\bar \mathbb Q_n, \bar \mathbb P_n) \ge \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P),$$ where for the third equality we have used Lemma [Lemma 34](#lemma:fromOmegaToOmegabar){reference-type="ref" reference="lemma:fromOmegaToOmegabar"}. We set ${\varphi}: (X, U) \mapsto \int_0^T f(r, X_r, u_r)dr + g(X_T)$, where $u$ corresponds to the one in item $(iii)$ in Definition [Definition 32](#def:enlargedMinSet){reference-type="ref" reference="def:enlargedMinSet"} and we define the probability measure $\bar{\bar \mathbb Q}$ such that $$\label{eq:Quotient} d\bar{\bar \mathbb Q} = \frac{\exp\left(-{\varphi}(X, U)\right)}{\mathbb E^{\bar \mathbb P}\left[\exp\left(-{\varphi}(X, U)\right)\right]}d\bar \mathbb P.$$ By Proposition [Proposition 11](#prop:markovExistenceMinimizer){reference-type="ref" reference="prop:markovExistenceMinimizer"} with $\mathbb P= \bar \mathbb P$ we obtain $\bar {\cal J}(\bar{\bar \mathbb Q}, \bar \mathbb P) \le \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P)$. As ${\varphi}\ge 0$, the nominator of the Radon-Nykodim density of [\[eq:Quotient\]](#eq:Quotient){reference-type="eqref" reference="eq:Quotient"} is smaller or equal to $1$ and the denominator is strictly positive number so that $\|d\bar{\bar \mathbb Q}/d\bar\mathbb P\|_\infty < + \infty$. Hence by Lemma [Lemma 35](#lemma:fromOmegabarToOmega){reference-type="ref" reference="lemma:fromOmegabarToOmega"} there exists $(\mathbb P^*, \mathbb Q^*) \in {\cal A}$ such that $\bar {\cal J}(\bar{\bar \mathbb Q}, \bar \mathbb P) \ge {\cal J}(\mathbb Q^*, \mathbb P^*)$ and $\mathbb P^* \in {\mathcal P}_\mathbb U^{Markov}$. Finally combining what precedes with [\[eq:ineqExistenceMin\]](#eq:ineqExistenceMin){reference-type="eqref" reference="eq:ineqExistenceMin"} yields ${\cal J}^* \ge \bar {\cal J}(\bar \mathbb Q, \bar \mathbb P) \ge \bar {\cal J}(\bar{\bar \mathbb Q}, \bar \mathbb P) \ge {\cal J}(\mathbb Q^*, \mathbb P^*)$ and this implies that $(\mathbb P^*, \mathbb Q^*)$ is a solution to Problem [\[eq:regularizedProblemIntro\]](#eq:regularizedProblemIntro){reference-type="eqref" reference="eq:regularizedProblemIntro"}. 0◻ **Remark 41**. *Taking $b \neq 0$ amounts to add another dimension to our enlarged space, to notice in the proof of Proposition [Proposition 39](#prop:tightness){reference-type="ref" reference="prop:tightness"} that by linear growth of $b$ and classical estimates given by Lemma [Lemma 9](#lemma:classicalEstimates){reference-type="ref" reference="lemma:classicalEstimates"} one has $$\sup_{n \ge 1} \mathbb E^{\mathbb P_n}\left[\int_0^T |b(r, X_r)|^pdr\right] < + \infty$$ for all $p \ge 1$ and to apply Lemma 4 in [@ZhengTightness] to get that any limit point $(\bar \mathbb P, \bar \mathbb Q)$ is in $\bar {\cal A}$.* ## Strong and weak controls {#app:equiControl} We give here some details on the equivalence between a strong formulation of our stochastic optimal control [\[eq:strongControlIntro\]](#eq:strongControlIntro){reference-type="eqref" reference="eq:strongControlIntro"} and our optimization problem [\[eq:controlProblemIntro\]](#eq:controlProblemIntro){reference-type="eqref" reference="eq:controlProblemIntro"}. We assume in this section that the coefficients of the diffusion $b$ and $\sigma$ are Lipschitz continuous which ensures in particular well-posedness for the SDEs for each strong control. Given some filtered probability space $(\tilde \Omega, \tilde {\mathcal F}, (\tilde {\mathcal F}_t)_{t \in [0, T]}, \tilde \mathbb P)$ endowed with a Brownian motion $W$, let $\mathcal{V}$ be the set of $(\Tilde {\mathcal F}_t)$-progressively measurable processes $\nu$ on $(\tilde \Omega, \tilde {\mathcal F}, \tilde \mathbb P)$ taking values in $\mathbb U$. By Theorem 3.1 in [@touzibook], there exists a unique process $X^\nu$ on $(\tilde \Omega, \tilde {\mathcal F}, \tilde \mathbb P)$ such that $$X^\nu_t = x + \int_0^t b(r, X_r^\nu)dr + \int_0^t \nu_r dr + \int_0^t\sigma(r, X_r^\nu)dW_r.$$ An application of Proposition [Proposition 26](#prop:decompItoProcess){reference-type="ref" reference="prop:decompItoProcess"} $(ii)$ yields the existence of an $(\tilde {\mathcal F}_t^{X^\nu})$-progressively measurable function $u : [0, T] \times C([0, T], \mathbb R^d) \rightarrow \mathbb R^d$ such that $\mathbb E^{\tilde \mathbb P}[\nu_t | \tilde {\mathcal F}_t^{X^\nu}] = u(t, X^\nu)$ $d\tilde \mathbb P\otimes dt$-a.e. and, under $\mathbb P$ $X^\nu,$ it has decomposition $$X^\nu_t = x + \int_0^t b(r, X_r^\nu)dr + \int_0^t u(r, X^\nu) dr + M_t^\nu,$$ where $\langle M^\nu\rangle = \int_0^{\cdot} \sigma\sigma^\top(r, X_r^\nu)dr$. Setting $\mathbb P^\nu := \mathcal{L}^{\tilde \mathbb P}(X^\nu)$ it is then clear that $\mathbb P^\nu \in {\mathcal P}_\mathbb U$. Hence $$J^*_{strong} := \underset{\nu \in \mathcal{V}}{\inf}\mathbb E^{\mathbb P}\left[\int_0^T f(r, X_r^\nu, \nu_r)dr + g(X_T)\right] \ge \inf_{\mathbb P\in {\mathcal P}_\mathbb U}\mathbb E^{\mathbb P}\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr + g(X_T)\right].$$ Assume now that the previous inequality is strict. Then there exists a probability measure $\mathbb P\in {\mathcal P}_\mathbb U$ such that $J(\mathbb P) < J^*_{strong}$. Then again, by Corollary 3.7 in [@MimickingItoGeneral] there exist a measurable function $\hat u : [0, T] \times \mathbb R^d \rightarrow \mathbb R^d$ and a probability measure $\hat \mathbb P$ on $(\Omega, {\mathcal F})$ such that the following holds. - For all $0 \le t \le T$, $\hat u(t, X_t) = \mathbb E^{\mathbb P}[u_t^\mathbb P| X_t]$. - Under $\hat \mathbb P$ the canonical process decomposes as $$X_t = x + \int_0^t b(r, X_r)dr + \int_0^t \hat u(r, X_r)dr + M_t^{\hat \mathbb P},$$ where $M^{\hat \mathbb P}$ is an $(F_t)$-local martingale such that $\langle M^{\hat \mathbb P}\rangle_{\cdot} = \int_0^{\cdot} \sigma\sigma(r, X_r)dr$. - $\mathcal{L}^\mathbb P(X_t) = \mathcal{L}^{\hat \mathbb P}(X_t)$. We modify $\hat u$ on the Borel set $N = \{(t, x) \in [0, T] \times \mathbb R^d~:~u(t, x) \notin \mathbb U\}$ so that $\hat u(t, x) \in \mathbb U$ for all $(t, x) \in [0, T] \times \mathbb R^d$. On the one hand, Fubini's theorem and Jensen's inequality for conditional expectation yields $$\mathbb E^{\mathbb P}\left[\int_0^T f(r, X_r, u_r^\mathbb P)dr + g(X_T)\right] \ge \mathbb E^{\hat \mathbb P}\left[\int_0^T f(r, X_r, \hat u(r, X_r))dr + g(X_T)\right].$$ On the other hand, Theorem 1.1 in [@ZhangStrong] ensures the existence of a unique (strong) solution $X^{\hat u}$ (on the space $(\tilde \Omega, \tilde {\mathcal F}, (\tilde {\mathcal F}_t)_{t \in [0, T]}, \tilde \mathbb P)$ to the SDE $dX_t = b(t, X_t)dt + \hat u(t, X_t)dt + \sigma(t, X_t)dW_t, ~X_0 = x$. In particular the process $\hat \nu := \hat u(., X^{\hat u}_.) \in \mathcal{V}$, and we have $$J^*_{strong} > J(\mathbb P) \ge J(\hat \mathbb P) = \mathbb E^{\tilde \mathbb P}\left[\int_0^T f(r, X_r^{\hat u}, \hat u(r, X_r^{\hat u}))dr + g(X_T^{\hat u})\right],$$ hence a contradiction and we conclude that $J^{*}_{strong} = J^*$. ## Miscellaneous We gather in this section two useful technical results. In the following, all the random variables are defined on a filtered probability space $(\Omega, {\mathcal F}, ({\mathcal F}_t)_{t \in [0, T]}, \mathbb P)$. **Lemma 42**. *Let $\eta$ be a squared-integrable, non negative random variable. Then for all $\epsilon > 0$, $$0 \le \mathbb E[\eta] - \left(-\frac{1}{\epsilon}\log\mathbb E\left[\exp(-\epsilon \eta)\right]\right) \le \frac{\epsilon}{2}Var[\eta].$$* *Proof.* For all $a, b \in \mathbb R$, it holds by Taylor's formula with integral remainder that $$e^{- b} = e^{- a} - (b - a)e^{- a} + \frac{(b - a)^2}{2}e^{- a} - \frac{1}{2}\int_{\mathbb R}{\mathds 1}_{\{a \le t \le b\}}(b - t)^2e^{-t}dt \le e^{- a} - (b - a)e^{- a} + \frac{(b - a)^2}{2}e^{- a}.$$ Let $\omega \in \Omega$. A direct application of this formula with $a = 0$, $b = \epsilon(\eta(\omega) - \mathbb E[\eta])$ yields $$e^{-\epsilon(\eta(\omega) - \mathbb E[\eta])} \le 1 - \epsilon(\eta(\omega) - \mathbb E[\eta]) + \frac{\epsilon^2}{2}(\eta(\omega) - \mathbb E[\eta])^2. % - \frac{1}{2}\int_\R\1_{0 \le t \le \epsilon(\eta(\omega) - \E[\eta])}(\epsilon(\eta(\omega) - \E[\eta]) - t)^2dt.$$ Taking the expectation in the previous inequality we get $$% \begin{aligned} % \E\left[e^{-\epsilon(\eta - \E[\eta])}\right] & \le 1 + \frac{\epsilon}{2}Var[\eta] - \frac{1}{2}\int_\R\E[\1_{0 \le t \le \epsilon(\eta(\omega) - \E[\eta])}(\epsilon(\eta(\omega) - \E[\eta]) - t)^2]dt\\ % & \le 1 + \frac{\epsilon}{2}Var[\eta], % \end{aligned} \mathbb E\left[e^{-\epsilon(\eta - \mathbb E[\eta])}\right] \le 1 + \frac{\epsilon^2}{2}Var[\eta], % \end{aligned}$$ and as $\log(1 + x) \le x$ for all $x > -1,$ we have $$\frac{1}{\epsilon}\log \mathbb E\left[e^{-\epsilon(\eta - \mathbb E[\eta])}\right] \le \frac{\epsilon}{2}Var[\eta].$$ Notice that $\mathbb E[\eta]$ is a constant, hence $\frac{1}{\epsilon}\log \mathbb E\left[e^{-\epsilon(\eta - \mathbb E[\eta])}\right] = \mathbb E[\eta] - \left(-\frac{1}{\epsilon}\log \mathbb E\left[e^{-\epsilon\eta}\right]\right)$. We then have $$0 \le \mathbb E[\eta] - \left(-\frac{1}{\epsilon}\log \mathbb E\left[e^{-\epsilon\eta}\right]\right) \le \frac{\epsilon}{2}Var[\eta],$$ where the first inequality follows from Jensen's inequality. ◻ **Lemma 43**. *Let $(X_t)_{t \in [0, T]}$ be an $({\mathcal F}_t)$-adapted process of the form $$X_t = x + \int_0^t b_rdr + M_t,$$ where $\mathbb E\left[\int_0^T |b_r|^pdr\right] < + \infty$ for some $p > 1$ and where $M$ is a martingale. For Lebesgue almost all $0 \le t < T$ $$\lim_{h \downarrow 0}\mathbb E\left[\frac{X_{t + h} - X_t}{h}~\Big|~{\mathcal F}_t\right] = b_t~\text{in}~L^1(\mathbb P).$$* *Proof.* In this proof we extend the process $X$ by continuity after $T$ and $b_t$ by zero for $t > T$. Let $0 < h \le 1$. Notice first that $$\mathbb E\left[\int_0^T \left|\mathbb E\left[\frac{X_{t + h} - X_t}{h}~\Big|~{\mathcal F}_t\right] - b_t\right|dt\right] \le \mathbb E\left[\int_0^T\left|\frac{1}{h}\int_t^{t + h}b_rdr -b_t\right|dt\right],$$ and that for all $\omega \in \Omega$, for almost all $0 \le t < T$, by Lebesgue differentiation theorem, $$\label{eq:lebesgueDiff} \frac{1}{h}\int_t^{t + h}b_r(\omega)dr \underset{n \rightarrow + \infty}{\longrightarrow} b_t.$$ To conclude by a uniform integrability argument w.r.t. $d\mathbb P\otimes dt$ we need to prove that $$\sup_{0 < h \le 1} \mathbb E\left[\int_0^T \left|\frac{1}{h}\int_t^{t + h}b_rdr\right|^pdt\right] < + \infty.$$ Previous expectation, by Hölder inequality, is bounded above by $$\mathbb E\left[\int_0^T \frac{1}{h}\int_t^{t + h}|b_r|^pdrdt\right] = \mathbb E\left[\int_0^T |b_r|^p \frac{1}{h}\int_{(r - h)_+}^{r}dtdr\right] \le \mathbb E\left[\int_0^T |b_r|^pdr\right] < + \infty,$$ where interchanging the integral inside the expectation is justified by Fubini's theorem. The family $\left(\frac{1}{h}\int_t^{t + h}b_rdr\right)_{0 < h \le 1}$ is uniformly integrable with respect to $d\mathbb P\otimes dt$ and we conclude using the Lebesgue's dominated convergence theorem. ◻ **Remark 44**. *If $b_t$ is a.e. $\sigma(X_t)$-measurable then the statement of Lemma [Lemma 43](#lemma:nelsonDerivative){reference-type="ref" reference="lemma:nelsonDerivative"} still holds replacing the $\sigma$-field ${\cal F}_t$ with $\sigma(X_t)$. This is an obvious property of the tower property of the conditional expectation.* # Acknowledgments {#acknowledgments .unnumbered} The research of the first named author is supported by a doctoral fellowship PRPhD 2021 of the Région Île-de-France. The research of the second and third named authors was partially supported by the ANR-22-CE40-0015-01 project (SDAIM). [^1]: ENSTA Paris, Institut Polytechnique de Paris. Unité de Mathématiques Appliquées (UMA). E-mail: `thibaut.bourdais@ensta-paris.fr` [^2]: EDF R&D, and FiME (Laboratoire de Finance des Marchés de l'Energie (Dauphine, CREST, EDF R&D) www.fime-lab.org). E-mail:` nadia.oudjane@edf.fr` [^3]: ENSTA Paris, Institut Polytechnique de Paris. Unité de Mathématiques Appliquées (UMA). E-mail:`francesco.russo@ensta-paris.fr`.

arxiv_math

--- author: - "Geir Bogfjellmo[^1], Charles Curry[^2], Sylvie Vega--Molino[^3]" bibliography: - References.bib title: The Prytz connections --- # Introduction and historical background A *planimeter* is a mechanical or electronic device for measuring the area of a region $\Omega$, typically by tracing its outline. The first planimeter was invented by Jakob Amsler-Laffon in 1854. For more background on planimeters, see also Prof. Foote's webpage <http://persweb.wabash.edu/facstaff/footer/Planimeter/PLANIMETER.HTM>. The key operating principle for planimeters is the Moving Segment Theorem [@foote2006volume]. **Theorem 1**. *Let $p,q\colon[0, T]\to \mathbb{R}^2$ be two parametrized closed curves in the plane, and let $\ell(t)$ denote the moving line segment from $p(t)$ to $q(t)$. Let $A_\ell$ be the signed area swept out by $\ell(t)$, $t\in [0,T]$, and $A_p, A_q$ the signed areas of the regions enclosed by $p$ and $q$, respectively. Then $$A_\ell=A_p-A_q.$$* {#fig: mst width="60%"} In a planimeter, the line segment $\ell$ is realized as a rigid rod of a fixed length, and one end of the rod is made to follow the perimeter of a region, $p\colon[0,T]\to \partial \Omega_1\subset \mathbb{R}^2.$ The planimeter is then constructed such that such that $A_\ell$ and $A_q$ are either restricted to be zero or possible to measure. In the Amsler planimeters, $q$ is restricted to a curve in $\mathbb{R}^2$ (a circle for the polar planimeters, a line for the linear planimeters). The two restrictions imposed on $q$ work together to ensure that $q(T)=q(0)$, and that $A_q=0$. Thus the area of $A_p=\mathop{\mathrm{area}}(\Omega_1)$ can be read directly from the signed area swept by the line segment. This signed area is then measured via a gauge driven by a perpendicular wheel on the rod. (See <https://mathweb.ucsd.edu/~jeggers/Planimeter/Amsler_Radial/Amsler_Radial_gallery.html> or [@bryant2008round Section 8.4] for pictures.) The Amsler planimeters were complicated pieces of equipment, and were quite costly. A competitor to Amsler's planimeters appeared in 1875. In contrast to Amsler's planimeters, Holger Prytz' planimeter can be constructed by "a country blacksmith" (Prytz' own suggestion [@prytz1896prytz]) or by yourself from a metal coathanger (see [@bryant2008round Chapter 8] for a recipe.) The Prytz planimeter (also known as a "stang" or "hatchet" planimeter) is mathematically not as precise as the Amsler planimeter. Strictly speaking, it does not measure the area of $\Omega_1$, but a more complicated geometric quantity approximating the area. Despite its apparent simplicity, the Prytz planimeter is a geometrically interesting object. In the present article, we will show how the motion of the planimeter induces a sub-Riemannian structure on its configuration space, how the motion of the planimeter can be described as the horizontal lift of various connections in differential geometry, and how this can be used to understand what the Prytz planimeter measures and its relation to the area of $\Omega_1$. We conclude by noting that the planimeter is a type of kinematic linkeage of more general interest, being for example closely related to the motion of articulated vehicles. # The Prytz planimeter  The Prytz planimeter is a rigid metal rod with a perpendicular prong at either end. One prong, the *tracer end* is sharpened into a point. The other prong, which we will call the *chisel end*, is sharpened into an edge parallel to the rod. In operation, the tracer end $p$ is made to follow the boundary of a (simply connected) region $\Omega_1$ drawn on a paper. The motion induces the chisel end $q$ to follow a curve in the plane limited by nonholonomic constraints. When tracer end returns to its starting point, the chisel end will not have returned to its starting point, but will have rotated by an angle $\Delta \theta$. See Figure [\[fig: planimeter\]](#fig: planimeter){reference-type="ref" reference="fig: planimeter"} for examples of paths traced out by the planimeter. By the moving segment theorem, it is possible to show that $$\mathop{\mathrm{area}}(\Omega_1) = l^2\Delta \theta + A_q$$ where $A_q$ is the signed area between the path of the chisel and $\Delta\theta$ and the circular arc between $q(0)$ and $q(T)$ with radius $l$. The proof can be found in [@bryant2008round Section 8.7] or [@foote1998geometry]. To increase the accuracy of the measurement, i.e, reduce $A_q$, one possibility suggested by Prytz is to start with the tracer at the centroid[^4] of $\Omega_1$, move the tracer out to the boundary along a straight line, then trace the boundary of $\Omega_1$, before returning to the start point by retracing the straight line in reverse. **Remark 1**. $\mathop{\mathrm{area}}(\Omega_1)$ can approximated as either $$\mathop{\mathrm{area}}(\Omega_1) \approx l^2\Delta \theta \label{eq: areaformulaangle}$$ or $$\mathop{\mathrm{area}}(\Omega_1)\approx l \cdot d \label{eq: areaformulacord}$$ where $d=2l\sin\left(\frac{\Delta \theta}{2}\right)= \|q(0)-q(T)\|$ is the distance between the start and end point of the chisel end. The simplicity of measuring $d$ makes [\[eq: areaformulacord\]](#eq: areaformulacord){reference-type="eqref" reference="eq: areaformulacord"} preferable from a practical point of view, and a simple series expansion shows that $ld-l^2\Delta \theta = \mathcal{O}\left(\frac{1}{l^4}\right)$. In light of the other errors inherently present in the Prytz planimeter, this difference does not matter in practice. This did not prevent engineers Goodman and Scott from patenting and marketing two separate "improved" hatch planimeters which could accurately measure [\[eq: areaformulaangle\]](#eq: areaformulaangle){reference-type="eqref" reference="eq: areaformulaangle"}. This led to a sometimes heated debate in the form of letters to the magazine *Engineering*. Excerpts of this debate can be found in [@bryant2008round Chapter 8.6]. ## The configuration space Let $p\in \mathbb{R}^2$ denote the tracer end and $q\in \mathbb{R}^2$ the chisel end of the planimeter. When the planimeter is used, $p$ and $q$ will move around in the plane restricted by a holonomic constraint $\|p-q\|=l$ and a nonholonomic constraint $\dot{q}\parallel q-p$. These two constraints are sufficient to ensure that for a given path of the tracer $p\colon[0,T]\to \mathbb{R}^2$, and an allowed initial position $q_0$, there is a unique path of the chisel $q\colon[0,T]\to \mathbb{R}^2$ satisfying the constraints and $q(0)=q_0$. Let $C$ denote all possible configurations of the planimeter. We have the following descriptions of $C$: 1. As a submanifold of $\mathbb{R}^2$: $C=\{(p, q)\in \mathbb{R}^2\times \mathbb{R}^2 \mid \|p-q\|=l\}$. 2. As a (trivial) fibre bundle $C=\mathbb{R}^2\times \mathbb{S}_1$ with base $\mathbb{R}^2$ and fibres isomorphic to $\mathbb{S}_1$. We use coordinates $(x,y,\theta)$, where $(x,y)$ are the coordinates of $p$ and $\theta$ is the angle between the positive $x$-axis and the vector $q-p$. The canonical projection is $$\pi \colon C\to \mathbb{R}^2, \qquad \pi(x,y,\theta)=(x,y).$$ Let $e\colon[0,T]\to C, e(t)=(x(t), y(t), \theta(t))$ be a possible path for the planimeter. The nonholonomic constraint $\dot{q}(t)\parallel q(t)-p(t)$ can be written as $-\sin \theta \dot{x}+\cos \theta \dot{y}+l\dot{\theta}=0$ or $\eta_e(\dot{e})=0$, where $\eta$ is the one-form $$\eta=-\sin \theta dx+\cos \theta dy +ld\theta. \label{eq: so2}$$ ## The motion of the planimeter Let $p\colon[0,T]\to \mathbb{R}^2$ form the boundary of some area we want to measure the area of. When the tracer end is moved according along the curve $p(t)=(x(t),y(t))$, the chisel end is dragged or pushed according to the nonholonomic constraint. The resulting angle $\theta\colon[0,T]\to \mathbb{S}_1$ solves the ordinary differential equation $$\dot{\theta}= \frac{1}{l}\left(\sin \theta \dot{x}-\cos \theta \dot{y}\right), \label{eq: thetaODE}$$ with initial value $\theta(0)=\theta_0$ given by the initial orientation of the planimeter. At the end, the area is approximated as $A\approx l^2(\theta(T)-\theta(0))$. This approximation is dependent on the initial angle $\theta_0$. # Prytz connections and sub-Riemannian geometry The above equation describes how a curve in the plane induces a motion of a planimeter. In the fibre bundle interpretation of the configuration space, this amounts to a prescription of how to lift any curve on the base of the bundle to the total space. Identifying this with a "horizontal lift" we have arrived at the heart of differential geometry. The standard procedure is to define a connection, from which we obtain horizontal lifts. An alternative perspective arising from Cartan geometry leads to the closely related notion of development. It is often desirable to define connections on a principal bundle satisfying an equivariance -- a principal connection. We will return to a principal connection for the planimeter motions in section [4](#sec: principalPrytz){reference-type="ref" reference="sec: principalPrytz"}. In this section, however, we consider connections which lack this equivariance. ## Infinitesimal connection In this section, we describe how the Prytz planimeter describe a connection on the fibre bundle $C$. Recall that for a fibre bundle $C$ over the base $\mathbb{R}^2$, the *vertical bundle* is the vector subbundle $\mathcal{V}=\ker T\pi \subset TC$. In our case, the vertical bundle is spanned by the vector field $\frac{\partial}{\partial \theta}$. For every $e\in C$, we can form the vector space $\mathcal{H}_e\subset T_eC$ consisting of all tangent vectors $v$ satisfying the nonholonomic constraint $\eta_e(v)=0$. Then $\mathcal{H}_e$ is a complement to $\mathcal{V}_e$, and $\mathcal{H}=\bigcup_{e\in C} \mathcal{H}_e$ is a vector subbundle everywhere transversal to $\mathcal{V}$ that we call the *horizontal bundle*. Such a splitting of $TC$ into a vertical and horizontal bundle is what Ehresmann called an infinitesimal connection on a fibre bundle. **Definition 1** ([@marle2014works; @ehresmann1950connexions]). Let $E(M,F)$ denote a fibre bundle with base manifold $M$ and fibres diffeomorphic to $F$, and let $\pi\colon E \to M$ be the canonical projection. An *infinitesimal connection* on $E$ is a vector sub-bundle $\mathcal{H}\subset TE$, that is transversal to the vertical bundle $\mathcal{V}=\ker T\pi \subset TE$, i.e. such that for each $e\in E$, $T_e E=\mathcal{V}_e \oplus \mathcal{H}_e$. Equivalently, we can consider an infinitesimal connection to be defined by a smooth section of linear maps $\Phi_e\colon T_eE\to T_eE$ with constant rank satisfying $\Phi_e\circ \Phi_e=\Phi_e$. The horizontal and vertical bundles are given by $\mathcal{H}_e=\ker \Phi_e$ and $\mathcal{V}_e=\mathop{\mathrm{ran}}\Phi_e$. Equipped with an infinitesimal connection, we can define horizontal lifts of paths in the base manifold $\mathbb{R}^2$. **Definition 2**. Let $E(M,F)$ be a fibre bundle equipped with horizontal bundle $H$, let $\gamma \colon[0,T]\to M$ be a smooth curve in the base $M$ with $\gamma(0)=p$. and let $e$ be point in the fibre $\pi^{-1})(p).$ The horizontal lift of $\gamma$ through $e$ is the unique smooth curve $\tilde{\gamma}\colon[0,T]\to E$ satisfying $\pi(\tilde{\gamma(t)})=\gamma(t)$, $\dot{\tilde{\gamma}}(t)\in H_{\tilde{\gamma}(t)}$ for all $t$ and $\tilde{\gamma}(0)=e$. We also define the horizontal lift of a vector field $U$ over $M$ to be the vector field $\tilde{U}$ over $E$ that is everywhere horizontal and $\pi$-related to $U$. For the Prytz planimeter, the horizontal lift of a path $p\colon[0,T]\to \mathbb{R}$ describes the motion of the planimeter when the tracer end follows the path $p(t)$. **Remark 2**. To get a first understanding of why the Prytz planimeter measures area, consider what happens when the tracer follows the boundary of an infinitesimal square: In the picture, $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ denote the coordinate vector fields on $\mathbb{R}^2$. The horizontal lift will follow the horizontally lifted vector fields $$\begin{aligned} \widetilde{\frac{\partial}{\partial x}}=X&=\frac{\partial}{\partial x} + \frac{1}{l}\sin \theta \frac{\partial}{\partial \theta}\\ \widetilde{\frac{\partial}{\partial y}}=Y&=\frac{\partial}{\partial y} - \frac{1}{l}\cos \theta \frac{\partial}{\partial \theta} \end{aligned} \label{eq: horizontalvectorfields}$$ In contrast to the original path, the lifted path is not closed. The failure of such an infinitesimal square to be closed is measured by the *Jacobi--Lie* bracket of vector fields. For the vector fields [\[eq: horizontalvectorfields\]](#eq: horizontalvectorfields){reference-type="eqref" reference="eq: horizontalvectorfields"}, we have $\llbracket X, Y\rrbracket= \frac{1}{l^2}\frac{\partial}{\partial \theta}$. Loosely speaking: *When the tracer traces out the boundary of an infinitesimal region with area $\epsilon^2$, the planimeter rotates by the angle $\epsilon^2/l^2$.* More precisely, the curvature of an infinitesimal connection, as defined in [@kolar1993natural p. 73], is a two-form $R$ on $C$ taking values in $TC$. On arbitrary vector fields $U$ and $V$ over $C$, $R$ is given by the function $$R(U,V) = \Phi \left(\llbracket U-\Phi(U), V-\Phi(V)\rrbracket\right)$$ where $\Phi\colon TC\to TC$ is the section of linear maps with $\ker \Phi=\mathcal{H}$, $\mathop{\mathrm{ran}}\Phi=\mathcal{V}$, and $\llbracket\cdot, \cdot \rrbracket$ is the Jacobi--Lie bracket of vector fields. In our case, this tracks the rotation of the planimeter $$R(U,W) = \pi^{*}\mathop{\mathrm{vol}}(U,W)\cdot \frac{1}{l^2} \frac{\partial}{\partial \theta},$$ where $\pi^{*}\mathop{\mathrm{vol}}$ is the pull-back of the area form on $\mathbb{R}^2$ to a two-form on $C$. In other words, for two vectors $U_e$ and $W_e$ in $T_eC$, the curvature $R_e(U_e,W_e)$ is a vertical vector with length proportional to the area of the parallelogram spanned by the projection of the two vectors onto $T\mathbb{R}^2$. ## Sub-Riemannian perspective In the setting of sub-Riemannian geometry, one works with a smooth manifold $C$ equipped with a pair $(\mathcal{E},g_\mathcal{E})$ where $\mathcal{E}$ is a subbundle of $TC$ and $g_\mathcal{E}$ is a symmetric, positive-definite (2,0)-tensor on $\mathcal{E}$. One insists that the bracket-generating condition holds, which is said to be satisfied if at every point $p \in C$ one can generate all of $T_pC$ by taking sufficiently many Lie brackets of vector fields in $\Gamma(\mathcal{E})$ at $p$. Notably, this is equivalent to Hörmander's condition in PDEs. One can define the sub-Riemannian (Carnot-Carathéodory) distance $d_{cc}(p,q)$ for two points $p,q \in C$ by the usual infimum formula taken over the space of smooth paths $C_{\mathcal{E},p,q}$ connecting $p$ to $q$ such that $\dot\gamma(t) \in \mathcal{E}_{\gamma(t)}$ at almost every point along $\gamma$. The now famous theorem of Chow and Rashevsky [@C39; @R38] tells us that $(C,d_{cc})$ is a complete metric space precisely when $\mathcal{E}$ is bracket-generating. Such a triple $(C,\mathcal{E},g_\mathcal{E})$ is called a sub-Riemannian manifold. We introduce a sub-Riemannian structure on the description of the Prytz planimeter by defining $\mathcal{E}$ to be the span of the vector fields, just as in [\[eq: horizontalvectorfields\]](#eq: horizontalvectorfields){reference-type="eqref" reference="eq: horizontalvectorfields"} $$\begin{aligned} X &= \frac{\partial}{\partial x} + \frac{1}{l}\sin \theta \frac{\partial}{\partial \theta}, \\ Y& = \frac{\partial}{\partial y} - \frac{1}{l}\cos \theta \frac{\partial}{\partial \theta},\end{aligned}$$ and letting the sub-Riemannian metric $g_\mathcal{E}$ be such that $X,Y$ are orthonormal. The first bracket is $[X,Y] = \frac{1}{l^2} \frac{\partial}{\partial \theta}$ from which we see that $\mathcal{E}$ is indeed bracket-generating. **Corollary 1**. There is a path between any two configurations of a planimeter via planimeter motions. *Proof.* This follows from Chow-Rashevskii. ◻ **Remark 3**. Observe that the sub-Riemannian horizontal distribution $\mathcal{E}$ and the horizontal sub-bundle $\mathcal{H}$ determining the Ehresmann connection in the previous section coincide. This is to say that the connection is adapted to the sub-Riemannian structure, see [@Vega-Molino_2020] for an overview. We also observe that the form $\eta$ in [\[eq: so2\]](#eq: so2){reference-type="eqref" reference="eq: so2"} is contact, and the sub-Riemannian structure we have defined is the associated contact sub-Riemannian structure. It follows that there can be no abnormal geodesics. ### Hamiltonian Mechanics {#hamiltonian-mechanics .unnumbered} Following the Hamiltonian perspective (c.f. [@M02; @ABB20]) it is of interest to consider the Hamiltonian $$\begin{aligned} H &= \frac{1}{2} \left( P_X^2 + P_Y^2 \right) \\ &= \frac{1}{2} \left( p_x^2 + p_y^2 + \frac{1}{l^2}p_\theta^2 \right) + \frac{1}{l}\left( \sin\theta p_x - \cos\theta p_y \right) p_\theta\end{aligned}$$ which then induces the Hamiltonian system $$\begin{aligned} \dot x &= p_x + \frac{1}{l} \sin\theta p_\theta & \dot p_x &= 0 \\ \dot y &= p_y - \frac{1}{l} \cos\theta p_\theta & \dot p_y &= 0 \\ \dot\theta &= \frac{1}{l^2} p_\theta + \frac{1}{l} \left( \sin\theta p_x - \cos\theta p_y \right) & \dot p_\theta &= -\frac{1}{l}\left( \cos\theta p_x + \sin\theta p_y \right) p_\theta \end{aligned}$$ the solutions of which are the normal sub-Riemannian geodesics. The problem reduces to solving only the last line of the system, and moreover it can be shown that $\theta$ is the solution of the autonomous differential equation $$\ddot\theta = \frac{1}{2l^2}\left(2p_xp_y \cos(2\theta) + (p_x^2 - p_y^2)\sin(2\theta) \right).$$ To give some intution, recall that a sub-Riemannian geodesic is a curve which is locally length-minimizing (for the sub-Riemannian metric). For example, when tracing a planimeter around a closed loop the initial and final points in $C$ differ only in the $\theta$ coordinate; a sub-Riemannian geodesic projecting onto a closed loop therefore minimizes $\Delta\theta$ and so by [\[eq: areaformulaangle\]](#eq: areaformulaangle){reference-type="eqref" reference="eq: areaformulaangle"} it also approximates a minimization of area. Therefore the projection must approximate a circle. The precise shape is determined by the higher-order error terms. **Remark 4**. We note that the vertical vector field $Z = \frac{1}{l^2}\frac{\partial}{\partial\theta}$ determines a foliation of $C$, however the Lie derivatives $(\mathcal{L}_U \tilde g_\mathcal{E})(Z,Z)$ do not vanish for $U \in \mathcal{E}$ and so the foliation is not totally-geodesic (here $\tilde g_\mathcal{E}$ is the Riemannian extention of $g_\mathcal{E}$ making $X,Y,Z$ an orthonormal frame). As a consequence the Eulerian approach to sub-Riemannian geometry via penalty metrics does not define an H-type foliation (see [@BGRV18]). Equivalently, we can understand that the contact structure is not $K$-contact. ## Linear and affine pseudoconnections We move a step closer to mainstream differential geometry by noting that the fibres of the bundle $C$ can be identified with $SO(2)$. It is a principal bundle, where moreover the fibres are a subgroup of the general linear group, with its natural representation on the base $\mathbb{R}^2$. The connection form ([\[eq: so2\]](#eq: so2){reference-type="ref" reference="eq: so2"}) is $\mathfrak{so}(2)$-valued, and can therefore be considered a linear pseudoconnection, i.e. the standard representation of $SO(2)$ on $\mathbb{R}^2$ induces an associated connection on the tangent bundle of $\mathbb{R}^2$. The $\mathrm{Ad}\,SO(2)$-equivariance is lacking however, hence the name pseudoconnection. More interesting perhaps is to take the perspective of Cartan geometry and enrich the connection form with a solder form to obtain a $\mathfrak{se}(2)$-valued pseudoconnection. This can be done in such a way that the curve traced out by the chisel end of the planimeter is the development of the curve traced out by the tracer. For this purpose, note that the chisel has coordinates $q=(\tilde{x}, \tilde{y}) = (x + l \cos (\theta), y + l\sin (\theta))$. Differentiating and rearranging, we obtain $$\begin{aligned} \dot{\tilde{x}} & = & \dot{x} \cos^2 \theta + \dot{y} \sin \theta \cos \theta,\\ \dot{\tilde{y}} & = & \dot{x} \sin \theta \cos \theta + \dot{y} \sin^2 \theta.\end{aligned}$$ In other words, if we define a basis of $\mathfrak{se} (2)$ by $$e_1=\begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}, \quad e_2=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 1 & 0 & 0 \end{pmatrix}, \quad e_3=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{pmatrix},$$ so that $e_3$ is the rotational basis vector and $e_1$ and $e_2$ the translational basis vectors, we can define an affine pseudoconnection $\tilde{\omega}\colon T\mathbb{R}^2\to \mathfrak{se}(2)$ by $$\begin{aligned} \tilde{\omega} =&e_1\left(l \cos \theta (\cos \theta dx + \sin \theta dy) - l\sin \theta d \theta\right)\\ &+e_2\left(l \sin \theta (\cos \theta dx + \sin \theta dy) + l\cos \theta d \theta\right)\\ &+e_3\left( l d \theta + \sin \theta dx - \cos \theta dy\right). \end{aligned} %=\begin{pmatrix} % l \cos \theta (\cos \theta dx + \sin \theta dy) - l\sin \theta d \theta\\ % l \sin \theta (\cos \theta dx + \sin \theta dy) + l\cos \theta d \theta\\ % l d \theta + \sin \theta dx - \cos \theta dy$$ # The principal Prytz connection {#sec: principalPrytz} To obtain an equivariant connection on a principal bundle we must deal with the dependency of the subsequent motion on the initial angle. ## Principal connections One approach, following Foote is to look at the subgroup of diffeomorphisms on $\mathbb{S}_1$ generated by the planimeter. We refer to [@foote1998geometry] for further details. First, let us recall *principal bundles* and *principal connections* (See [@kobayashi1963foundations].) **Definition 3**. A *principal bundle* $P(M,G)$ is a fibre bundle with fibres diffeomorphic to a Lie group $G$, equipped with a right group action $P\times G\to P$. We will write the right action $(p,g)\mapsto R_g(p)$. Differentiating the right action with respect to $g$ at the identity $I\in G$, we get a Lie algebra morphism $\mathfrak{g}\to \cal V$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\cal V\subset TP$ is the vertical subbundle. We write this Lie algebra morphism $\xi \mapsto X_\xi$. **Definition 4**. A *principal connection* on $P$ is a $\mathfrak{g}$-valued one-form $\omega$ on $P$ satisfying $$\begin{aligned} \omega(X_\xi)&=\xi, &\text{for all }\xi &\in \mathfrak{g},\\ R_{g}^*\omega &= Ad_{g^{-1}} \circ \omega, &\text{for all } g&\in G. \end{aligned}$$ A principal connection $\omega$ defines an infinitesimal connection in the sense of Definition [Definition 1](#def: infinitesmalconnection){reference-type="ref" reference="def: infinitesmalconnection"} via the projection $\Phi\colon TP\to TP$, $\Phi(\cdot)=X_{\omega(\cdot)}.$ In the case of a trivial principal bundle, $P=M\times G$, a principal connection $\omega$ induces a $\mathfrak{g}$-valued form $\varpi$ on $M$ via the pullback of $\omega$ to the trivial section $M\simeq M\times \{I\} \subset P$. The principal connection $\omega\colon TP\to \mathfrak{g}$ is in turn uniquely defined from $\varpi\colon TM\to \mathfrak{g}$ (See [@kobayashi1963foundations Proposition II.1.4]). Given a connection $\omega$ on a principal bundle $P$ and a curve $\gamma(t)$ on the base $M$, one can constructs a horizontal lift as follows [@kobayashi1963foundations II.3]: 1. Take an arbitrary $C^1$ lift $v(t)\in P$ of $\gamma(t)$ to the principal bundle. 2. Solve the equation of Lie type $$\dot{\Gamma}(t) = -\omega\big(\dot{v}(t)\big) \Gamma(t),\quad \Gamma(0)=I.$$ 3. The horizontal lift is $u(t)=v(t)\Gamma(t)$. In the case of a trivial principal bundle, we can take the curve $v(t)=(\gamma(t),I)$, so the equation to be solved becomes $$\dot{\Gamma}(t) = -\xi(t) \Gamma(t), \quad \xi(t)= \varpi(\dot{\gamma}(t)). \label{eq: evolution}$$ ## The principal bundle We consider $C=\mathbb{R}^2\times \mathbb{S}_1$ as a trivial fibre bundle with coordinates $(x,y,\theta)$. Recall the ODE [\[eq: thetaODE\]](#eq: thetaODE){reference-type="eqref" reference="eq: thetaODE"} $$\dot{\theta}= \frac{1}{l}\sin \theta \dot{x}-\frac{1}{l}\cos \theta \dot{y}.$$ We can view this as a linear function from the tangent vector $(\dot{x}, \dot{y})\in T_{p}\mathbb{R}^2$ to the Lie algebra of smooth vector fields on $\mathbb{S}_1$. By letting the initial value $\theta(0)=\theta_0$ vary, the ODE induces a family of flow maps $\Gamma(t)\in \mathop{\mathrm{Diff}}(\mathbb{S}_1)$ via $\Gamma(t)(\theta_0)=\theta(t).$ We want to view the flow maps as the horizontal lifts of the curve $\gamma(t)=(x(t), y(t))$ in $\mathbb{R}^2$. We can avoid dealing with the infinite-dimensional Lie group of diffeomorphisms and instead work with a finite-dimensional subgroup of $\mathop{\mathrm{Diff}}(\mathbb{S}_1)$. Let $G\subset \mathop{\mathrm{Diff}}(\mathbb{S}_1)$ be the subgroup of diffeomorphisms that can be written as flow maps of the ODE [\[eq: thetaODE\]](#eq: thetaODE){reference-type="eqref" reference="eq: thetaODE"}. Then $G$ is the group generated by the two vector fields on $\mathbb{S}_1$: $$\begin{cases} X=\frac{1}{l}\sin \theta \frac{\partial}{\partial \theta}, \\ Y=-\frac{1}{l}\cos \theta \frac{\partial}{\partial \theta}. \end{cases}$$ The Lie algebra of $G$ is generated by the same vector fields. We first compute the Lie bracket of $X$ and $Y$, which we define as[^5] $[X,Y]=-\llbracket X, Y\rrbracket.$ $$[X,Y]=-\frac{1}{l^2}\frac{\partial}{\partial \theta}.$$ The vector fields $X,Y, [X,Y]$ are in involution and form a three-dimensional simple Lie algebra, $\mathfrak{g} = \mathop{\mathrm{Span}}\{X, Y, [X,Y]\}$. It can be shown that $$\mathfrak{g}\simeq \mathfrak{su}(1,1)= \left\{\begin{pmatrix} i\gamma & \beta \\ \beta^{*} & -i\gamma \end{pmatrix} \middle| \gamma\in \mathbb{R}, \beta \in \mathbb{C}\right\},$$ with isomorphism given by $$\begin{aligned} e_1= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}& \mapsto -2\sin \theta \frac{\partial}{\partial \theta},\\ e_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}& \mapsto 2\cos \theta \frac{\partial}{\partial \theta}, \\ e_3 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}& \mapsto 2 \frac{\partial}{\partial \theta}. \end{aligned} \label{eq: isomorphisms}$$ It can also be shown (See [@foote1998geometry]) that the corresponding Lie group $G$ is isomorphic to the projective special unitary group of signature $(1,1)$. $$G\simeq PSU(1,1)= \left\{ \begin{pmatrix} a & b\\ b^* & a^* \end{pmatrix} \middle| |a|^2-|b|^2=1 \right\}\Large{/} \left\{\pm I\right\},$$ and that the corresponding diffeomorphisms on $\mathbb{S}_1$ are given by: $$\begin{pmatrix} a & b\\ b^* & a^* \end{pmatrix}(e^{i\theta})= \frac{ae^{i\theta}+b}{b^{*}e^{i\theta}+a^*}, \label{eq: groupaction}$$ where we have identified $\mathbb{S}_1$ with the unit circle in $\mathbb{C}$. ## The connection We now define a principal connection on the trivial principal bundle $P=\mathbb{R}^2\times G$. The relation between the principal connection and the planimeter is as follows: Let $\gamma\colon[0,T]\to \mathbb{R}^2$ be a curve in $\mathbb{R}^2$ that the tracer end follows. One horizontal lift of $\gamma$ is the curve $\gamma, \Gamma \colon[0,T] \to \mathbb{R}^2\times G$ with $\Gamma(0)=I$ and $$\Gamma(t)= \begin{pmatrix} a(t) & b(t)\\ b(t)^* & a(t)^*\end{pmatrix} \in G.$$ If $\theta(0)=\theta_0$ denotes the initial angle of the planimeter, then $\theta(t)$ is given by $$e^{i\theta(t)}= \Gamma(t)(e^{i\theta_0})=\frac{a(t)e^{i\theta_0}+b(t)}{b^*(t)e^{i\theta_0}+a^*(t)}.$$ As a $\mathfrak{g}$-valued connection on $\mathbb{R}^2$, we can write the connection as $\varpi = \frac{1}{2l}(e_1 dx+ e_2 dy)$ or $$\varpi(v)= \frac{1}{2l}\begin{pmatrix} 0 & \kappa(v) \\ \kappa(v)^* & 0 \end{pmatrix}\in \mathfrak{g}$$ where $\kappa\colon T_p \mathbb{R}^2 \simeq \mathbb{R}^2 \to \mathbb{C}$ is the standard identification. The corresponding principal connection on $P=\mathbb{R}^2\times G$ satisfies $\omega(X_\xi)=\xi$ for left-invariant vector fields $X_{\xi}$ and $R_g^{*}\omega = Ad_{g^{-1}}\circ \omega$. Explicitly, if $(p, g)\in \mathbb{R}^2\times G =P$ is a point on the principal bundle, we can write a tangent vector as $(v, X_\xi)\in T_p\mathbb{R}^2\times T_g G \simeq T_{(p,g)}P$. Then $$\omega_{(p,g)}(v, X_\xi)= \xi+ Ad_{g^{-1}} \varpi(v).$$ On the trivial bundle, the curvature 2-form of $\varpi$ can be computed as $$\bar{\Omega}_p(u,v)= d \varpi(u,v)+ [\varpi(u), \varpi(v)] =- \frac{\mathop{\mathrm{vol}}(u,v)}{2l^2}\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix},$$ where $u, v \in T_p \mathbb{R}^2$ and $\mathop{\mathrm{vol}}$ is the area form on $\mathbb{R}^2$. This also determines the curvature of the principal connection $\omega$ (see [@kobayashi1963foundations II.5], noting that the curvature is a tensorial 2-form of type $(\mathrm{Ad},\mathfrak{g})$) $$\Omega_{(p,g)}\left((u, X_\xi), (v, X_\eta)\right) = Ad_{g^{-1}}\circ \bar{\Omega}_p(u,v)$$ where $(p,g)\in P=\mathbb{R}^2\times G$ and $(u, X_\xi), (v, X_\eta)$ are two vectors in $T_{(p,g)}P=T_p \mathbb{R}^2\times T_g G$. # Horizontal lifts and the Magnus expansion We have now defined a connection on a principal bundle such that the relationship between the tracer curve and the motion of the planimeter may be understood as a horizontal lift. The key observation is that the related notions of horizontal lifts and developments typically amount to solving differential equations in a Lie group. For this purpose it is profitable to employ techniques of Lie group integration. It should be noted that when the tracer end follows a closed curve, the horizontal lift to $P$ describes an element of the *holonomy group* of the connection. There exists a non-abelian version of the Stokes theorem [@schreiber2011smooth Corollary 3.6] that links the holonomy and curvature of a principal connection. We will here take a more pedestrian approach using the Magnus expansion. ## Magnus Expansion Let $\gamma \colon[0,T]\to \mathbb{R}^2$ be the curve traced out by the tracer end, i.e., a parametrization of a boundary of the region $\Omega_1$. For simplicity, assume that the curve is closed and beginning at the origin, so $\gamma(0)=\gamma(T)=(0,0)$. The horizontal lift to $P=\mathbb{R}^2\times G$ is given by the solution to [\[eq: evolution\]](#eq: evolution){reference-type="eqref" reference="eq: evolution"}. The Magnus expansion (see [@magnus1954exponential; @iserles1999solution]) is an expansion of the evolution as the exponential of a series in $\mathfrak{g}$, $$\Gamma(T) = \exp(U_1 + U_2 + U_3 +U_4\dotsc)$$ where $U_1,U_2,\dotsc$ are given as integrals of Lie polynomials. In our case: $$\begin{aligned} U_1 &= -\int_{0}^T \xi(t) dt \\ &= -\frac{1}{2l} \oint_\gamma \left(dx e_1 + dy e_2\right)=0\\ U_2 &= \frac{1}{2}\int_{0}^T\int_0^{t_1} [\xi(t_1),\xi(t_2)] dt_2 dt_1 \\ &= \frac{1}{8l^2} \int_0^T \left[ \dot{x}(t_1) e_1+\dot{y}(t_1) e_2, \int_0^{t_1} \dot{x}(t_2) e_1+\dot{y}(t_2) e_2 dt_2\right] d t_1\\ &= \frac{1}{8l^2} \int_0^T \left[ \dot{x}(t_1) e_1+\dot{y}(t_1) e_2, x(t_1)e_1+y(t_1) e_2\right] d t_1\\ &= \frac{1}{4l^2} \oint_\gamma \left( xdy-ydx\right) e_3\\ &= \frac{A}{2l^2} \cdot e_3, \end{aligned}$$ where $A= \mathop{\mathrm{area}}(\Omega_1)$. This again shows how the Prytz planimeter measures area: If we truncate the Magnus series after the leading term, we have $$\begin{aligned} \Gamma_T &=\exp(U_1+U_2)+\mathcal{O}\left(\frac{1}{l^3}\right) \\ &= \exp\left(\frac{A}{2l^2} e_3\right)+\mathcal{O}\left(\frac{1}{l^3}\right) \\ &= \begin{bmatrix} e^{i\frac{A}{2l^2}}& 0 \\ 0 & e^{-i\frac{A}{2l^2}}\end{bmatrix}+\mathcal{O}\left(\frac{1}{l^3}\right). \end{aligned}$$ The corresponding group action from [\[eq: groupaction\]](#eq: groupaction){reference-type="eqref" reference="eq: groupaction"} is $$\Gamma_T\cdot e^{i\theta}= e^{i(\theta+\Delta \theta)}, \quad \text{where}\quad \Delta \theta= \frac{A}{l^2}+\mathcal{O}\left(\frac{1}{l^3}\right).$$ By rearrangement, we get $$A=l^2\Delta \theta+ \mathcal{O}\left(\frac{1}{l}\right),$$ which shows that the "Prytz area" and actual area agree to an error term of order $\mathcal{O}\left(\frac{1}{l}\right)$ Further computations show that the leading error terms in the "Prytz area" are controlled by the Magnus expansion terms: $$\begin{aligned} U_3&=\frac{1}{2l^3}\left(M_y e_1-M_x e_2\right),\\ U_4&=\frac{1}{4l^4}M_2 e_3, \end{aligned}$$ where $$\begin{aligned} M_x&=\int_{\Omega_1} x dx\wedge dy\\ M_y &=\int_{\Omega_1} y dx \wedge dy\\ M_2&=\int_{\Omega_1} (x^2+y^2)dx \wedge dy \end{aligned}$$ are first and second moments of area of the region $\Omega_1$. The term $U_3=\frac{1}{2l^3}\left(M_y e_1-M_x e_2\right)$ explains why starting in the centroid is an advantage: If $M_x=M_y=0$, the first error term is eliminated. # Postscript: development and trailers The relationship between front and back end of the planimeter is of broader interest than simply measuring areas. We begin by noting that the curve induced in the back end by a straight line was already considered Huygens and Leibniz in the 17th century [@hairer2000analysis The tractrix, p. 135] Moreover, note that a bicycle is essentially a planimeter of length $l$ equal to the distance between the centres of the front and back wheel. A car can be approximated by the same construction. In practice, the radius of curvature of the tracer curve $\gamma(t)$ will be large compared to the length $l$, so that front and back follow each other closely. What is of more interest is vehicles with trailers. A vehicle with a system of $n$ trailers can be approximated by $n+1$ planimeters, chained together (see [@ljungqvist2019motion], note that the planimeter is here a "kinematic bicycle"). In this case the Cartan picture of development becomes natural: the configuration space of each planimeter is isomorphic to $SE(2)$, and the motion of the ($m+1$)th planimeter is the development the the $m$th. It is hoped that this perspective can be usefully applied to control problems within trailer system, an issue of increasing practical importance due to the increased automation of vehicles and requirement to improve efficiency of road haulage [@ljungqvist2019motion]. [^1]: Corresponding author, Norwegian University of Life Sciences [^2]: Norwegian University of Science and Technology [^3]: University of Bergen [^4]: In practice: at an estimation of the centroid. [^5]: The sign is due to equation [\[eq: evolution\]](#eq: evolution){reference-type="eqref" reference="eq: evolution"}: it is convenient to identify $X$ and $Y$ with right-invariant vector fields on $G$ instead of the usual left-invariant vector fields.

arxiv_math

--- abstract: | In this paper we study embeddings between de Branges-Rovnyak spaces $H(b)$ and harmonically weighted Dirichlet spaces $\mathcal{D}(\mu)$ in terms of the boundary spectrum of $b$ and the support of the measure $\mu$, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces $K_u$ and the local Dirichlet spaces $\mathcal{D}_\zeta$, and we discuss some applications. address: - Department of Mathematics, Aristotle University of Thessaloniki, 54124, Greece - Department of Mathematics, Università degli studi di Milano, 20133, Italy author: - Carlo Bellavita - Eugenio Dellepiane bibliography: - mybibliography.bib title: Embedding Model and de Branges-Rovnyak spaces in Dirichlet Spaces ---     # Introduction In this article we deal with spaces of analytic functions on the unit disk $\mathbb{D}:=\{z\in\mathbb{C}\colon |z|<1\}.$ In this theory, a prominent role is played by the Hardy spaces $H^p(\mathbb{D})$, see for example [@garnett]. We briefly introduce the different spaces of interest for this work. Given a bounded analytic function $b$ on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})}\leq1$, we define the *de Branges-Rovnyak space* $H(b)$ as the reproducing kernel Hilbert space having for reproducing kernel the function $$k^b(z,\omega):= \frac{1-\overline{b(\omega)}b(z)}{1-\overline{\omega}z}, \quad \omega,z\in\mathbb{D}.$$ These spaces were originally introduced by Louis de Branges and James Rovnyak in 1966 as a generalization of the orthogonal complement of the range of multiplication by $b$ on $H^2(\mathbb{D})$, see [@Debrangesrovnyak]. For a complete introduction of such spaces see [@sarHb] and [@hb2]. Another space of interest in this paper is the *local Dirichlet space* $\mathcal{D}_\zeta$. For a fixed point $\zeta$ on the torus $\mathbb{T}:=\partial\mathbb{D}$, we define the *local Dirichlet integral* at $\zeta$ of a function $f$ in $\text{Hol}(\mathbb{D})$ as $$D_\zeta(f) := \frac{1}{\pi}\int_\mathbb{D}|f'(z)|^2 \frac{1-|z|^2}{|z-\zeta|^2} \text{ d}A(z),$$ where $\text{d}A$ is the bidimensional Lebesgue measure. We call $\mathcal{D}_\zeta$ the space of functions $f$ in $\text{Hol}(\mathbb{D})$ such that $D_\zeta(f)<\infty$. These spaces are studied in [@RS] and they belong to a more general class, the so-called *harmonically weighted Dirichlet spaces*. We will discuss this later. In Section $3$ we provide a sufficient condition and a necessary one in order to have an embedding between de Branges-Rovnyak spaces and local Dirichlet spaces, i.e. a bounded inclusion $H(b)\hookrightarrow\mathcal{D}_\zeta$. Both these conditions involve the notion of *boundary spectrum*: given a bounded analytic function $b$ with $\|b\|_{H^\infty(\mathbb{D})}=1,$ we define its boundary spectrum as the set $$\sigma(b):= \{\lambda\in\mathbb{T}\colon \liminf_{z\to\lambda} |b(z)|<1\}.$$ As we will explain later, this set carries information about regularity of the function $b$ and of all elements of $H(b)$. In particular, we proved the following results. **Theorem 1**. *Let $b$ be a bounded analytic function with $\|b\|_{H^\infty(\mathbb{D})}=1$, and let $\zeta\in\mathbb{T}$ be such that $\zeta\notin\overline{\sigma(b)}$. Then, the embedding $H(b)\hookrightarrow \mathcal{D}_\zeta$ holds.* **Theorem 2**. *Let $b$ be a bounded analytic function with $\|b\|_{H^\infty(\mathbb{D})}=1$, and let $\zeta\in\mathbb{T}$ be such that $\zeta\in\sigma(b)$. Then, the de Branges-Rovnyak space $H(b)$ does not embed into the local Dirichlet space $\mathcal{D}_\zeta$.* Later in the article, we restrict our attention to the *model spaces*. Given an inner function $u$, i.e. a bounded analytic function on $\mathbb{D}$ with $|u|=1$ a.e. on $\mathbb{T}$, we define the model space $K_u$ as the complementary space $K_u := H^2(\mathbb{D})\ominus uH^2(\mathbb{D})$. These spaces naturally arise as the closed invariant subspaces of the backward shift operator $S^*$ on $H^2(\mathbb{D})$. For a complete introduction on this subject, we refer to [@nikolski] and [@1]. The model spaces are a particular class of de Branges-Rovnyak spaces: when one considers an inner function $u$, one has that $H(u)=K_u$ with equality of norms. As we mentioned before, we also consider the class of harmonically weighted Dirichlet spaces $\mathcal{D}(\mu)$. Given a finite positive Borel measure $\mu$ on the torus $\mathbb{T}$, the associated $\mathcal{D}(\mu)$ space is the space of holomorphic functions on $\mathbb{D}$ having finite *harmonically weighted* Dirichlet integral $$\label{Dm} D_\mu (f) :=\frac{1}{\pi} \int_\mathbb{D}|f'(z)|^2 \, P\mu(z) \text{ d}A(z),$$ where $P\mu$ is the Poisson integral of $\mu$, $$P\mu(z) := \int_\mathbb{T}\frac{1-|z|^2}{|\lambda -z|^2} \text{ d}\mu(\lambda), \qquad z \in \mathbb{D}.$$ These spaces were introduced by Stefan Richter in 1991 for the representation of cyclic analytic two-isometries, see [@R]. Also, they play a key role in the description of the closed shift-invariant subspaces of the classical Dirichlet space $\mathcal{D}:= \mathcal{D}(m)$, where $m$ is the Lebesgue measure on $\mathbb{T}$, see [@RS]. In Section $5$ we deal with the embedding $K_u \hookrightarrow \mathcal{D}(\mu)$. Again, we provide a sufficient condition and a necessary one for the embedding to hold, involving the support of the measure $\mu$ and the boundary spectrum $\sigma(u)$ of the inner function $u$. **Theorem 3**. *Let $\mu$ be a finite positive Borel measure on $\mathbb{T}$ and let $u$ be an inner function. If $\text{supp}(\mu)\cap\sigma(u) = \emptyset$, then the embedding $K_u \hookrightarrow \mathcal{D}(\mu)$ holds.* **Theorem 4**. *Let $\mu$ be a finite positive Borel measure on $\mathbb{T}$ and let $u$ be an inner function. If the embedding $K_u \hookrightarrow \mathcal{D}(\mu)$ holds, then $\mu\big(\sigma(u)\big)=0$.* The paper is organized as follows. Section $2$ is devoted to some well-known preliminaries. In Section $3$ we describe the embedding $H(b)\hookrightarrow \mathcal{D}_\zeta$ for general $b$'s. In Section $4$ we discuss some applications of the embedding $K_u\hookrightarrow\mathcal{D}_\zeta$. In the fifth section, we prove Theorems [Theorem 3](#KuinDm){reference-type="ref" reference="KuinDm"} and [Theorem 4](#KuinDmnec){reference-type="ref" reference="KuinDmnec"}. We conclude with an open problem. # Preliminaries We introduce the main spaces involved in this article. Let us start with the harmonically weighted Dirichlet spaces. Given a finite positive Borel measure $\mu$ on the torus $\mathbb{T}$, its Poisson integral is the harmonic function $$P\mu(z) := \int_\mathbb{T}\frac{1-|z|^2}{|\lambda -z|^2} \text{ d}\mu(\lambda), \qquad z \in \mathbb{D}.$$ The associated harmonically weighted Dirichlet space $\mathcal{D}(\mu)$ is $$\mathcal{D}(\mu):= \{ f\in\text{Hol}(\mathbb{D}) \colon D_\mu(f) <\infty\},$$ where $$\label{Dm} D_\mu (f) :=\frac{1}{\pi} \int_\mathbb{D}|f'(z)|^2 \, P\mu(z) \text{ d}A(z)$$ is the harmonically weighted Dirichlet integral. Notice that $D_\mu$ is a seminorm that annihilates the constants. We recall a few basic properties; for a treatise of Dirichlet spaces we refer to [@primer]. If $\mu$ is a finite measure on $\mathbb{T}$ such that $\mu(\mathbb{T}) >0$, then $\mathcal{D}(\mu)$ is a subset of $H^2(\mathbb{D})$ which contains all polynomials. Moreover, $\mathcal{D}(\mu)$ is a Hilbert space with respect to the inner product induced by the norm $$\|f\|_\mu^2 := \|f\|_{H^2}^2 + D_\mu(f).$$ For $\zeta\in\mathbb{T}$, considering the Dirac delta $\delta_\zeta$ we obtain the so-called local Dirichlet space, which we simply denote by $\mathcal{D}_\zeta$. Also, we write $D_\zeta(f)$ instead of $D_{\delta_\zeta}(f)$. For $f \in H^2(\mathbb{D})$, by Fubini's theorem, $D_\mu(f)$ given in [\[Dm\]](#Dm){reference-type="eqref" reference="Dm"} can be expressed as $$\label{eqfubini} D_\mu(f) = \int_{\mathbb{T}} D_\zeta(f) \, d\mu(\zeta).$$ In [@RS] Richter and Sundberg proved the following useful formula for $D_\zeta(f)$, which includes the boundary value $f(\zeta)$ defined as the radial limit $\lim_{r\to1^-}f(r\zeta)$, whenever it exists. **Theorem 5** (Local Douglas formula). *Let $f\in H^2(\mathbb{D})$ and $\zeta\in\mathbb{T}$. If the boundary value $f(\zeta)$ exists, then $$\label{localdouglas} D_\zeta(f)=\int_\mathbb{T}\,\,\bigg| \frac{f(\lambda)-f(\zeta)}{\lambda-\zeta}\bigg|^2 \text{ d}m(\lambda).$$ On the other hand, if $f(\zeta)$ does not exist, then $D_\zeta(f)=\infty$. In particular, all functions in $\mathcal{D}_\zeta$ admit boundary value at $\zeta$.* This formula shows that the quotient ratio at $\zeta$ plays an important role for membership in the local Dirichlet space. Richter and Sundberg also proved the following characterization of $\mathcal{D}_\zeta$. One has $$\label{second defn} \mathcal{D}_\zeta =\left\lbrace f \in \text{Hol}(\mathbb{D})\ : f(z)=c+(z-\zeta)g(z), \text{where } c \in \mathbb{C} \text{ and } g\in H^2(\mathbb{D})\right\rbrace ,$$ with the equality $D_\zeta(f)=\| g\|_{H^2}^2$. Some aspects of the structure of local Dirichlet spaces have been recently studied by Fricain and Mashreghi in [@7]. Finally, we point out that local Dirichlet spaces of order $m\in\mathbb{N}$ have been recently introduced by Luo, Gu and Richter in [@LGR] and further developed in [@9] and [@8]. In the rest of this section we provide some preliminary information about de Branges-Rovnyak spaces. There are many equivalent ways to define these spaces: we will follow the reproducing kernel approach. As shown in the classic work of Aronszajn in [@aronszajn], given a positive definite function $k$ on $\mathbb{D}\times\mathbb{D}$ one can construct a Hilbert space $H_k$ of functions on $\mathbb{D}$ such that for all $\omega\in\mathbb{D}$ the function $k(\cdot,\omega)$ belongs to $H_k$ and it holds the so-called *reproducing kernel property*, i.e. $$f(\omega) = \langle f,k(\cdot,\omega) \rangle_{H_k}, \quad f\in H_k.$$ Given a bounded analytic function $b$ on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})}\leq1$, the de Branges-Rovnyak space $H(b)$ is the reproducing kernel Hilbert space having for reproducing kernel the function $$k^b(z,\omega):= \frac{1-\overline{b(\omega)}b(z)}{1-\overline{\omega}z}, \quad \omega,z\in\mathbb{D}.$$ We denote by $\langle \cdot,\cdot\rangle_b$ the inner product of $H(b)$ and by $\|\cdot\|_b$ its induced norm. $H(b)$ is a space of analytic functions contained in $H^2(\mathbb{D})$ and it holds the norm inequality $$\label{HbinH2} \|f\|_{H^2} \leq \|f\|_{b}, \qquad f\in H(b).$$ However, in general $H(b)$ is not complete with respect to the $H^2$ norm. If $b=u$ is an inner function, then $H(u)$ coincides with the *model space* $K_u$, defined as the orthogonal complement $K_u:=H^2(\mathbb{D}) \ominus uH^2(\mathbb{D})$. Therefore, $H(u)=K_u$ is closed in $H^2(\mathbb{D})$, and it holds the norm identity $\|\cdot\|_b=\|\cdot\|_{H^2}$. As a corollary of a classic result of Beurling (see Theorem $4.3$ in [@1]), the closed $S^\ast$-invariant subspaces of $H^2(\mathbb{D})$ are exactly the model spaces. More in general, all de Branges-Rovnyak spaces are $S^\ast$-invariant. The operator $$X_b : H(b) \ni f \mapsto S^\ast f \in H(b)$$ is well-defined and bounded. In order to introduce the notion of boundary spectrum, first we recall a key factorization result (see for example Theorem $3.20$ in [@1]): every analytic function $b$ with $\|b\|_{H^\infty(\mathbb{D})} = 1$ can be factorized as $b=Ou$, where $O$ is the outer function $$\label{hinf function} O(z):= \exp{\bigg\{\int_\mathbb{T}\frac{\zeta+z}{\zeta-z} \log|b(\zeta)| \text{ d}m(\zeta)\bigg\}}$$ and $u$ is an inner function. In particular, according to the Nevanlinna factorization, we can write $$\label{inner function} u(z) = \prod_{n=1}^\infty \frac{\overline{a_n}}{|a_n|}\frac{a_n-z}{1-\overline{a_n}z} \,\exp{\bigg\{-\int_\mathbb{T}\frac{\zeta+z}{\zeta-z} \text{ d}\tau(\zeta)\bigg\}},$$ where $\{a_n\}_{n\geq 1}$ are the zeros of $u$ and $\tau$ a positive singular measure. **Definition 6**. For a bounded analytic function $b$ on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})} = 1$, we define its boundary spectrum as the set $$\sigma(b) := \{\lambda\in\mathbb{T}\colon \liminf_{z\to\lambda} |b(z)|<1\}.$$ As stated in [@10], the closure $\overline{\sigma(b)}$ is the smallest closed subset of $\mathbb{T}$ containing the closure of the zero set $\{a_n\}_n$ and the supports of the (positive finite) measures $\tau$ and $-\log|b(\zeta)|dm(\zeta)$. It is known that $b$ has an analytic extension through any arc of the open set $\mathbb{T}\setminus \overline{\sigma(b)}$ with unimodular values on such arcs, see again [@10]. If $b=u$ is an inner function, thenit holds $$\sigma(u)=\{\lambda\in\mathbb{T}\colon \liminf_{z\to\lambda} |u(z)|=0\}.$$ In particular, the spectrum of inner functions is a closed set. We also note that there exist bounded functions with closed spectrum that are not necessarily inner, for example one-component bounded functions, defined and studied in [@12]. The name *spectrum* comes from the following fact. An easy proof follows from Theorem $20.13$ in [@hb2]. **Theorem 7**. *Let $b$ be a bounded analytic function on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})}=1$. Then, the intersection of the spectrum of the operator $X_b^*$ and the torus $\mathbb{T}$ coincides with the closure of the boundary spectrum of $b$. In symbols, $$\sigma(X_b^*) \cap \mathbb{T}= \overline{\sigma(b)}.$$* The boundary regularity of the function $b$ results in properties of functions in $H(b)$. The notion we need is the *angular derivative in the sense of Caratheodory* (ADC). We say that an analytic function $b$ on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})}\leq1$ admits ADC at $\zeta\in\mathbb{T}$ if the derivative $b'$ admits non-tangential limit at $\zeta$ and $|b(\zeta)|=1$. The result that follows is Theorem $21.1$ in [@hb2]. **Theorem 8**. *Let $b$ be an analytic function on $\mathbb{D}$ with $\|b\|_{H^\infty(\mathbb{D})}\leq1$ and let $\zeta\in \mathbb{T}$. The following are equivalent:* 1. *There exists $\lambda\in\mathbb{T}$ such that the function $$\mathbb{D}\ni z \mapsto \frac{b(z)-\lambda}{z-\zeta}$$ belongs to $H(b)$.* 2. *Every function $f$ in $H(b)$ admits non-tangential limit at $\zeta$.* 3. *$b$ has ADC at $\zeta$.* *Furthermore, under these conditions, $\lambda=b(\zeta)$ and for every $f\in H(b)$ one has $f(\zeta) = \langle f,k_{\zeta}^b\rangle_b$, where $$k_\zeta^b(z) = \frac{1-\overline{b(\zeta)}b(z)}{1-\overline{\zeta}z}\in H(b).$$* Also, by Theorem $18.21$ in [@hb2], the operator $X^*_b$ intertwines the reproducing kernels, in the sense that $$k_z^b = (I-\overline{z} X^*_b)^{-1}k_0^b, \qquad z\in\mathbb{D}.$$ One can easily prove that the same formula still holds replacing $z\in\mathbb{D}$ with $\zeta\in\mathbb{T}\setminus\overline{\sigma(b)}$, that is $$\label{rel kernel} k_\zeta^b = (I-\overline{\zeta} X^*_b)^{-1}k_0^b, \qquad \zeta\in\mathbb{T}\setminus\overline{\sigma(b)}.$$ # The embedding $H(b) \hookrightarrow \mathcal{D}_\zeta$ We come now to our two main results. By assumption, $b$ extends analytically in a neighbourhood of $\zeta$. Then, by Theorem [Theorem 8](#thm211debranges){reference-type="ref" reference="thm211debranges"}, every function in $H(b)$ admits non-tangential boundary value at $\zeta$. Also, since $\zeta\notin \overline{\sigma(b)}$, by Theorem [Theorem 7](#spectrumshift){reference-type="ref" reference="spectrumshift"} the operator $I-\overline{\zeta} X^*_b$ is boundedly invertible in $H(b)$ and, by [\[rel kernel\]](#rel kernel){reference-type="eqref" reference="rel kernel"}, $$k_\zeta^b = (I-\overline{\zeta} X^*_b)^{-1}k_0^b.$$ Thus, the operator $$\label{Q_u} Q_\zeta^b := ( I-\zeta X_b)^{-1} X_b$$ is bounded on $H(b)$. By an algebraic computation, for $z\in\mathbb{D}$ it holds the operator identity $$(I-zX_b)^{-1}(I-\zeta X_b)^{-1} X_b=\frac{(I-zX_b)^{-1}-(I-\zeta X_b)^{-1}}{z-\zeta}.$$ For every $f\in H(b)$ it holds the formula $$Q_\zeta^b f(z) = \frac{f(z)-f(\zeta)}{z-\zeta}, \qquad z\in\mathbb{D},$$ since $$\begin{aligned} Q_\zeta^b f (z) &= \langle Q_\zeta^b f,k_z^b \rangle_b = \langle Q_\zeta^b f, (I-\overline{z} X^*_b)^{-1}k_0^b \rangle_b = \langle (I-zX_b)^{-1} Q^b_\zeta f, k_0^b\rangle_b \\ &= \langle (I-zX_b)^{-1}(I-\zeta X_b)^{-1} X_b f,k_0^b\rangle_b = \frac{1}{z-\zeta} \langle (I-zX_b)^{-1}f - (I-\zeta X_b)^{-1} f,k_0^b\rangle_b \\ &= \frac{1}{z-\zeta} \langle f, (I-\overline{z}X^*_b)^{-1} k_0^b\rangle_b - \frac{1}{z-\zeta} \langle f,(I-\overline{\zeta}X^*_b)^{-1}k_0^b\rangle_b \\ &= \frac{1}{z-\zeta} \langle f, k_z^b \rangle_b - \frac{1}{z-\zeta} \langle f, k_\zeta^b \rangle_b = \frac{f(z)-f(\zeta)}{z-\zeta}.\end{aligned}$$ This proves the boundedness of the embedding $H(b) \hookrightarrow \mathcal{D}_\zeta$, since by Theorem [Theorem 5](#localdouglasthm){reference-type="ref" reference="localdouglasthm"} and [\[HbinH2\]](#HbinH2){reference-type="eqref" reference="HbinH2"} $$\|f\|_{H^2}^2 + D_\zeta(f) = \|f\|_{H^2}^2 + \|Q_\zeta^b f\|_{H^2}^2 \leq \|f\|_b^2 + \|Q_\zeta^b f\|_b^2 \leq \big( 1+ \|Q_\zeta^b\|^2\big) \|f\|_b^2. \qed$$ *Proof of Theorem [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"}.* By contradiction, let us suppose that the embedding $H(b) \hookrightarrow \mathcal{D}_\zeta$ holds. Let $C>0$ be such that $$\label{boundassurdob} D_\zeta(f) \leq C\|f\|_b^2, \qquad f\in H(b).$$ By assumption, $\zeta\in\sigma(b)$, hence there exists a sequence $(\omega_n)_n$ in $\mathbb{D}$ converging to $\zeta$ such that $$\beta:= \lim_n |b(\omega_n)|<1.$$ Let us consider the family of kernels $$k_n(z) := k_{\omega_n}^b(z) = \frac{1-\overline{b(\omega_n)}b(z)}{1-\overline{\omega_n}z}.$$ Since $H(b) \subseteq \mathcal{D}_\zeta$, by Theorem [Theorem 5](#localdouglasthm){reference-type="ref" reference="localdouglasthm"} every function of $H(b)$ admits boundary value at $\zeta$. By Theorem [Theorem 8](#thm211debranges){reference-type="ref" reference="thm211debranges"}, $b(\zeta)$ is well defined and unimodular. Therefore, one can compute $$\begin{aligned} k_n(z) - k_n(\zeta) &= \frac{1-\overline{b(\omega_n)}b(z)}{1-\overline{\omega_n}z} - \frac{1-\overline{b(\omega_n)}b(\zeta)}{1-\overline{\omega_n}\zeta} \\ %&=\frac{1-\overline{\om_n}\z -\overline{b(\om_n)}b(z) +\overline{b(\om_n)}b(z)\overline{\om_n}\z -1 +\overline{\om_n}z+\overline{b(\om_n)}b(\z)-\overline{b(\om_n)}b(\z)\overline{\om_n}z }{(1-\overline{\om_n}z)(1-\overline{\om_n}\z)}\\ &= \frac{\overline{\omega_n}(z-\zeta) - \overline{b(\omega_n)}\big(b(z)-b(\zeta)\big) + \overline{\omega_n b(\omega_n)}\big(\zeta b(z)-zb(\zeta)\big)}{(1-\overline{\omega_n}z)(1-\overline{\omega_n}\zeta)}\\ %&=\frac{\overline{\om_n}(z-\z) - \overline{b(\om_n)}\big(b(z)-b(\z)\big) + \overline{\om_n b(\om_n)}\big[\z\big(b(z)-b(\z)\big)+b(\z)(\z-z)\big]}{(1-\overline{\om_n}z)(1-\overline{\om_n}\z)}\\ &=\frac{\overline{\omega_n}(z-\zeta)(1-\overline{b(\omega_n)}b(\zeta)) - \overline{b(\omega_n)}\big(b(z)-b(\zeta)\big)(1-\overline{\omega_n}\zeta) }{(1-\overline{\omega_n}z)(1-\overline{\omega_n}\zeta)} .\end{aligned}$$ Consequently, $$\begin{aligned} \nonumber \frac{k_n(z) - k_n(\zeta)}{z-\zeta} &= \frac{\overline{\omega_n}}{1-\overline{\omega_n}z}\frac{1-\overline{b(\omega_n)}b(\zeta)}{1-\overline{\omega_n}\zeta} - \frac{\overline{b(\omega_n)}}{1-\overline{\omega_n}z}\frac{b(z)-b(\zeta)}{z-\zeta}\\ \label{quotientku2} &= \overline{\omega_n} c_{\omega_n}(z) k_n(\zeta) - \overline{b(\omega_n)} c_{\omega_n}(z) b(\zeta) \overline{\zeta} k_\zeta^b(z),\end{aligned}$$ where $$c_{\omega_n}(z) = \frac{1}{1-\overline{\omega_n}z}$$ is the usual Szegö kernel, the reproducing kernel of the Hardy space $H^2(\mathbb{D})$. The local Dirichlet integral can be computed as in [\[localdouglas\]](#localdouglas){reference-type="eqref" reference="localdouglas"}, yielding $$\begin{aligned} D_\zeta(k_n) &= \bigg\| \frac{k_n-k_n(\zeta)}{\cdot-\zeta}\bigg\|_{H^2}^2 \\ &= \big\langle \overline{\omega_n}\, k_n(\zeta) c_{\omega_n} - \overline{b(\omega_n)} b(\zeta) \overline{\zeta} c_{\omega_n} k_\zeta^b, \overline{\omega_n}\, k_n(\zeta) c_{\omega_n} - \overline{b(\omega_n)} b(\zeta) \overline{\zeta} c_{\omega_n} k_\zeta^b \big\rangle_{H^2} \\ &= |\omega_n|^2 |k_n(\zeta)|^2 \|c_{\omega_n}\|_{H^2}^2 -2\Re\Big(\overline{\omega_n}k_n(\zeta) b(\omega_n) \overline{b(\zeta)}\zeta\langle c_{\omega_n},c_{\omega_n}k_\zeta^b\rangle_{H^2} \Big) +|b(\omega_n)|^2 \|c_{\omega_n}k_\zeta^b\|_{H^2}^2. %&\geq |\om_n|^2 |k_n(\z)|^2 \|c_{\om_n}\|_{H^2}^2 -2\Re\Big(\overline{\om_n}k_n(\z) b(\om_n) \overline{b(\z)}\z \langle c_{\om_n},c_{\om_n}k_\z^b\rangle_{H^2} \Big).\end{aligned}$$ We have written the local Dirichlet integral $D_\zeta(k_n)$ as a sum of three terms. We leave the first one as it is and work on the other two. We use the reproducing property of the Szegö kernel, the fact that $c_{\omega_n}k_\zeta^b$ is an $H^2$ function and we estimate the real part with the modulus, obtaining $$\begin{aligned} \Re\Big(\overline{\omega_n}k_n(\zeta) b(\omega_n) \overline{b(\zeta)}\zeta\langle c_{\omega_n},c_{\omega_n}k_\zeta^b\rangle_{H^2} \Big) &= \Re\Big(\overline{\omega_n}k_n(\zeta) b(\omega_n) \overline{b(\zeta)}\zeta\overline{c_{\omega_n}(\omega_n)k_\zeta^b(\omega_n) }\Big) \\ &=\|c_{\omega_n}\|_{H^2}^2 \Re\big(\overline{\omega_n} b(\omega_n) \overline{b(\zeta)}\zeta k_n(\zeta)^2 \big) \\ &\leq \|c_{\omega_n}\|_{H^2}^2 |k_n(\zeta)|^2 \, |\omega_n b(\omega_n)|.\end{aligned}$$ For the third summand, using the triangular inequality, we have $$\begin{aligned} \|c_{\omega_n}k_\zeta^b\|_{H^2}^2 &= \int_\mathbb{T}\bigg| \frac{1}{1-\overline{\omega_n}\lambda}\frac{1-\overline{b(\zeta)}b(\lambda)}{1-\overline{\zeta}\lambda}\bigg|^2 \text{ d}m(\lambda) \\ &= \int_\mathbb{T}\bigg|\frac{1}{1-\omega_n\overline{\lambda}} \frac{1}{1-\overline{\omega_n}\lambda}\bigg(\frac{1-\overline{b(\zeta)}b(\lambda)}{1-\overline{\zeta}\lambda}\bigg)^2\bigg| \text{ d}m(\lambda) \\ &\geq \bigg| \int_\mathbb{T}\frac{1}{1-\omega_n\overline{\lambda}} \frac{1}{1-\overline{\omega_n}\lambda}\bigg(\frac{1-\overline{b(\zeta)}b(\lambda)}{1-\overline{\zeta}\lambda}\bigg)^2 \text{ d}m(\lambda) \bigg|.\end{aligned}$$ The function $c_{\omega_n} \big(k_\zeta^b\big)^2$ belongs to $H^1(\mathbb{D})$ and in particular the Cauchy integral formula holds $$\begin{aligned} \int_\mathbb{T}\frac{1}{1-\omega_n\overline{\lambda}} \frac{1}{1-\overline{\omega_n}\lambda}\bigg(\frac{1-\overline{b(\zeta)}b(\lambda)}{1-\overline{\zeta}\lambda}\bigg)^2 \text{ d}m(\lambda) &= \int_\mathbb{T}\frac{c_{\omega_n}(\lambda) \big(k_\zeta^b\big)^2(\lambda)}{1-\omega_n\overline{\lambda}} \text{ d}m(\lambda) \\ &= c_{\omega_n}(\omega_n) \big(k_\zeta^b\big)^2(\omega_n).\end{aligned}$$ Using this, we obtain $$|b(\omega_n)|^2 \|c_{\omega_n}k_\zeta^b\|_{H^2}^2 \geq |b(\omega_n)|^2 \|c_{\omega_n}\|_{H^2}^2 |k_n(\zeta)|^2.$$ Now, computing the norms of the kernels $$\|c_{\omega_n}\|_{H^2}^2 = \frac{1}{1-|\omega_n|^2}, \qquad \|k_n\|_b^2 = \frac{1-|b(\omega_n)|^2}{1-|\omega_n|^2},$$ we obtain the lower bound $$\begin{aligned} \frac{D_\zeta(k_n)}{\|k_n\|_b^2} &\geq \frac{|\omega_n|^2 |k_n(\zeta)|^2}{1-|b(\omega_n)|^2} -\frac{2 |k_n(\zeta)|^2|\omega_n b(\omega_n)|}{1-|b(\omega_n)|^2} + \frac{|b(\omega_n)|^2 |k_n(\zeta)|^2}{1-|b(\omega_n)|^2}\\ &= |k_n(\zeta)|^2\frac{|\omega_n|^2 -2 |\omega_n b(\omega_n)| + |b(\omega_n)|^2}{1-|b(\omega_n)|^2} \\ &=\bigg| \frac{1-\overline{b(\omega_n)}b(\zeta)}{1-\overline{\omega_n}\zeta}\bigg|^2\frac{\big( |\omega_n| - |b(\omega_n)| \big)^2}{1-|b(\omega_n)|^2} \\ &\geq \frac{\big(1-|b(\omega_n)|\big)^2}{|1-\overline{\omega_n}\zeta|^2} \frac{\big( |\omega_n| - |b(\omega_n)| \big)^2}{1-|b(\omega_n)|^2}.\end{aligned}$$ Since $\lim_n \omega_n = \zeta$ and $\lim_n |b(\omega_n)|=\beta \in [0,1),$ we conclude that $$\liminf_n \frac{D_\zeta(k_n)}{\|k_n\|_b^2} \geq \liminf_n \frac{(1-\beta)^2}{|1-\overline{\omega_n}\zeta|^2} \frac{(1-\beta)^2}{1-\beta^2}= +\infty ,$$ contradicting the uniform bound in [\[boundassurdob\]](#boundassurdob){reference-type="eqref" reference="boundassurdob"}. ◻ *Remark 1*. In Theorem [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"} it is shown that there cannot be a (bounded) embedding $H(b)\hookrightarrow \mathcal{D}_\zeta$, if $\zeta\in\sigma(b)$. By the closed graph theorem, even a set inclusion $H(b) \subseteq \mathcal{D}_\zeta$ cannot hold. Also, we point out that during the proof of Theorem [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"}, we have verified that if $b$ admits ADC at a point $\zeta\in\mathbb{T}$, then for all $\omega\in\mathbb{D}$ the reproducing kernel $k^b_\omega$ belongs to $\mathcal{D}_\zeta$, for $$\mathcal{D}_\zeta(k^b_\omega)= \bigg\| \frac{k^b_\omega-k^b_\omega(\zeta)}{\cdot-\zeta}\bigg\|_{H^2}^2 \leq |k^b_\omega(\zeta)|^2 \|c_{\omega}\|_{H^2}^2 + 2| k^b_\omega(\zeta)|^2 \|c_\omega\|_{H^2} + \|c_{\omega}k_\zeta^b\|_{H^2}^2<\infty.$$ We have proved a positive result, that is, that $H(b) \hookrightarrow \mathcal{D}_\zeta$ when $\zeta\notin\overline{\sigma(b)}$, and a negative one, that is, that if $\zeta\in\sigma(b)$, $H(b) \nsubseteq \mathcal{D}_\zeta$. We now present some examples to show that for the remaining case $\zeta\in\overline{\sigma(b)}\setminus\sigma(b)$, anything can happen. *Example 1*. Set $$\label{constantw0} w_0 := \frac{3-\sqrt{5}}{2},$$ let $\zeta\in\mathbb{T}$ and define the function $$b_\zeta(z) = \frac{(1-w_0)\overline{\zeta} z}{1-w_0\overline{\zeta} z}.$$ By Proposition 2 in [@5], $H(b_\zeta) = \mathcal{D}_\zeta$ with equality of norms, guaranteeing the embedding. Since $b_\zeta$ is continuous up to the boundary $\mathbb{T}$, it holds $$\sigma(b_\zeta) =\{ \lambda\in\mathbb{T}\colon |b_\zeta(\lambda)|<1\}.$$ Writing $\zeta=e^{i\eta}$ and $\lambda=e^{i\theta}$, one can easily see that $$|1-w_0 \overline{\zeta}\lambda|^2 = 1-2w_0\cos(\theta-\eta)+w_0^2 > 1-2w_0+w_0^2 = |(1-w_0) \overline{\zeta}\lambda|^2, \qquad \text{if} \quad e^{ i \theta}\neq e^{i\eta},$$ whereas $$|1-w_0 \overline{\zeta}\lambda|^2 = |(1-w_0) \overline{\zeta}\lambda|^2, \qquad \text{if} \quad e^{ i \theta}= e^{i\eta}.$$ This means that $\sigma(b_\zeta) = \mathbb{T}\setminus \{\zeta\}$, providing a function in $H^\infty(\mathbb{D})$ such that $\zeta\in\overline{\sigma(b_\zeta)}\setminus\sigma(b_\zeta)$ while the embedding $H(b_\zeta)\hookrightarrow\mathcal{D}_\zeta$ holds. Now we provide an example of a case with $1\in\overline{\sigma(b)}\setminus\sigma(b)$ such that $H(b)\hookrightarrow \mathcal{D}_1$ doesn't hold. We use the following proposition as a criterion for the inclusion, see Corollary $27.18$ in [@hb2]: **Proposition 9**. *Suppose $b_1$ is a non-extreme point of the closed unit ball of $H^\infty(\mathbb{D})$, and assume $b_1$ is continuous on the closed unit disk. Let $b_2$ be a function in $H^\infty(\mathbb{D})$ and $\theta_2$ its inner factor. Then the following are equivalent:* 1. *It holds the inclusion of de Branges-Rovnyak spaces $H(b_2) \subset H(b_1)$.* 2. *The following conditions hold:* - *$\{\lambda\in\mathbb{T}\colon |b_1(\lambda)| =1\}\cap \sigma(\theta_2) = \emptyset$.* - *There exists $\gamma >0$ such that $1-|b_2|^2 \leq \gamma (1-|b_1|^2)$ a.e. on $\mathbb{T}$.* *Example 2*. Let $$b_1(z):= \frac{(1-w_0) z}{1-w_0 z},$$ where $w_0$ is the constant in [\[constantw0\]](#constantw0){reference-type="eqref" reference="constantw0"}, so that $H(b_1)=\mathcal{D}_1$, and we construct an outer function $b_2$ as follows. We start by considering the function $\varphi$ defined on $\mathbb{T}$ as $$\varphi(\lambda)= \begin{cases} \log\Big(\sqrt{1-|1-\lambda|^{\frac{3}{2}}}\Big), & \text{if} \quad |\arg(\lambda)|\leq\frac{\pi}{6} , \\ 0, & \text{elsewhere}. \end{cases}$$ The function $\varphi$ is in $L^{\infty}(\mathbb{T})$ and real-valued, and this allows us to define the outer function $$b_2(z) := \exp\bigg\{ \int_\mathbb{T}\frac{\lambda+z}{\lambda-z} \, \varphi(\lambda) \!\text{ d}m(\lambda) \bigg\}.$$ The first condition of $(ii)$ in Proposition [Proposition 9](#cormashreghi){reference-type="ref" reference="cormashreghi"} is trivially true, since $b_2$ is outer and therefore $\sigma(\theta_2)=\emptyset$. For the second condition of $(ii)$, one can show that being $\varphi$ continuous on the set $\{|\arg(\lambda)|<\pi/6\}$ and bounded, $b_2$ extends continuously to $\{|\arg(\lambda)|<\pi/6\}$ and it holds $$\label{example2} |b_2(\lambda)| = e^{\varphi(\lambda)} = \sqrt{1-|1-\lambda|^{\frac{3}{2}}}, \qquad \text{for} \,\,\, |\arg(\lambda)|<\frac{\pi}{6},$$ so that $$1-|b_2(\lambda)|^2= |1-\lambda|^{\frac{3}{2}},\qquad \text{for} \,\,\, |\arg(\lambda)|<\frac{\pi}{6}.$$ Since for all $\lambda\in\mathbb{T}$ it holds $$1-|b_1(\lambda)|^2 = \frac{(1-w_0)^2|1-\lambda|^2}{|1-w_0\lambda|^2},$$ it follows that in proximity of the point $1$ the condition $(ii)$ of Proposition [Proposition 9](#cormashreghi){reference-type="ref" reference="cormashreghi"} fails, meaning that the inclusion $H(b_2) \subset H(b_1) = \mathcal{D}_1$ cannot hold. Finally, from [\[example2\]](#example2){reference-type="eqref" reference="example2"} and the fact that $b_2$ is continuous on $\{|\arg(\lambda)|<\pi/6\}$, one can see that $$\sigma(b)\cap\{\lambda\in\mathbb{T}\colon|\arg(\lambda)|<\pi/6\}= \{\lambda\in\mathbb{T}\colon|\arg(\lambda)|<\pi/6\}\setminus\{1\},$$ so that $1\in\overline{\sigma(b_2)}\setminus\sigma(b_2)$ while $H(b_2) \not\subset \mathcal{D}_1$. # Applications of $K_u \hookrightarrow \mathcal{D}_\zeta$ In the last two sections, we focus on the model space $K_u$. **Corollary 10**. *Let $b$ be an analytic function with $\|b\|_{H^\infty}=1$ with closed boundary spectrum, and let $\zeta\in\mathbb{T}$. Then, the embedding $H(b)\hookrightarrow\mathcal{D}_\zeta$ holds if and only if $\zeta\notin\sigma(b)$. In particular, if $u$ is an inner function, then the embedding $K_u\hookrightarrow\mathcal{D}_\zeta$ holds if and only if $\zeta\notin\sigma(u)$.* *Proof.* The result follows using Theorems [Theorem 1](#HbinDz){reference-type="ref" reference="HbinDz"} and [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"} and the fact that the spectrum of $b$ is closed. ◻ We can rewrite the embedding $K_u \hookrightarrow \mathcal{D}_\zeta$ in terms of the boundedness of the derivative operator, providing a corollary which is somehow related to the results of Baranov about the boundedness of the differentiation operator acting on model spaces, see [@BARANOV2013541]. **Corollary 11**. *Let $u$ be an inner function and $\zeta\in\mathbb{T}$. Let $D$ be the derivative operator $$D \colon K_u \to L^2\big(P\delta_\zeta\!\text{ d}A \big), \quad f \mapsto f',$$ acting from the model space to the Lebesgue space $L^2(\mathbb{D}, P\delta_\zeta\!\text{ d}A)$. Then, $D$ is bounded if and only if $\zeta\notin\sigma(u)$.* It follows at once from Corollary [Corollary 10](#KuinDz){reference-type="ref" reference="KuinDz"}, for $$\|f'\|_{L^2\big(P\delta_\zeta\!\text{ d}A \big)} =\int_\mathbb{D}|f'(z)|^2 \frac{1-|z|^2}{|z-\zeta|^2} \text{ d}A(z) = \pi D_\zeta(f). \qed$$ As already said in the introduction, the embedding $K_u \hookrightarrow \mathcal{D}_\zeta$ allows one to find some Carleson measures for $K_u$. First, let us recall the definition. **Definition 12**. Let $H$ be a Hilbert space of holomorphic functions on $\mathbb{D}$. We say that a positive Borel measure $\nu$ on $\mathbb{D}$ is a *Carleson measure* for $H$ if there exists a constant $C>0$ such that $$\label{Carlesondef} \int_\mathbb{D}|f|^2 \text{ d}\nu\leq C\|f\|_H^2, \qquad f\in H.$$ Carleson measures for $H^2(\mathbb{D})$ appeared in a very natural and powerful way in the proof of the Corona Theorem for $H^\infty(\mathbb{D})$, see [@garnett]. Such measures have been well studied, and they admit a nice geometric characterization in terms of *Carleson boxes*. **Proposition 13**. *Let $\nu$ be a finite positive Borel measure on $\mathbb{D}$. Given an arc $I\subseteq\mathbb{T}$, the Carleson box associated to $I$ is $$S(I) := \{re^{i\theta} \colon e^{i\theta}\in I, \, 1-|I| < r < 1 \},$$ where $|I|$ denotes the arc length of $I$. Then, $\nu$ is Carleson for $H^2(\mathbb{D})$ if and only if there exists a constant $C>0$ such that $$\label{carmeasH2eq} \nu(S(I))\leq C|I|, \qquad I\subset\mathbb{T}.$$* Carleson measures of $D_\zeta$ have been characterized in [@20] in terms of Carleson measures of $H^2(\mathbb{D})$, as follows: **Proposition 14**. *Let $\nu$ be a finite positive Borel measure on $\mathbb{D}$. Then, $\nu$ is a Carleson measure for $\mathcal{D}_\zeta$ if and only if the measure $|z-\zeta|^2 \text{ d}\nu(z)$ is Carleson for $H^2(\mathbb{D})$.* Note that every Carleson measure of $\mathcal{D}_\zeta$ has to be finite, since $1\in\mathcal{D}_\zeta$. Having mentioned these preliminary facts, we can state our result. **Corollary 15**. *Let $u$ be an inner function with $\sigma(u) \neq \mathbb{T}$, and $\nu$ a finite positive Borel measure on $\mathbb{D}$. If there exists $\zeta\in\mathbb{T}\setminus\sigma(u)$ such that $|z-\zeta|^2\text{ d}\nu(z)$ is a Carleson measure for $H^2(\mathbb{D})$, then $\nu$ is a Carleson measure for the model space $K_u$.* *Proof.* Since $\zeta\notin\sigma(u)$, by Theorem [Corollary 10](#KuinDz){reference-type="ref" reference="KuinDz"} the embedding $K_u\hookrightarrow\mathcal{D}_\zeta$ holds. Also, by Proposition [Proposition 14](#carmeasDz){reference-type="ref" reference="carmeasDz"}, the measure $\nu$ is a Carleson measure for $\mathcal{D}_\zeta$. Then, for every $f\in K_u$ it holds $$\int_\mathbb{D}|f|^2 \text{ d}\nu \leq C \|f\|_{\mathcal{D}_\zeta}^2 \leq C' \|f\|_{K_u}^2,$$ for some positive constants $C,C'$, meaning that $\nu$ is a Carleson measure for $K_u$. ◻ We conclude this part with an example of a Carleson measure for $\mathcal{D}_1$ (and thus for every model space $K_u$ with $1\notin\sigma(u)$) which is not Carleson for $H^2(\mathbb{D})$. *Example 3*. Let $\nu$ be the measure defined on Borel sets of $\mathbb{D}$ as $$\nu(A) := \int_{A\cap[0,1]} \frac{1}{\sqrt{1-s}}\text{ d}s.$$ We use the characterization in Proposition [Proposition 13](#carmeasH2){reference-type="ref" reference="carmeasH2"} to prove that $\nu$ is not a Carleson measure for $H^2(\mathbb{D})$. For $\delta>0$, consider the arc $I_\delta$ centered at $1$ with arc length $\delta$. One can compute the measure of the Carleson boxes $S(I_\delta)$ and obtain $$\nu\big(S(I_\delta)\big) = \int_{1-\delta}^1 \frac{1}{\sqrt{1-s}}\text{ d}s = 2\sqrt{\delta},$$ showing that the bound in [\[carmeasH2eq\]](#carmeasH2eq){reference-type="eqref" reference="carmeasH2eq"} cannot hold as $\delta \to 0$. However, the measure $\nu$ is a Carleson measure for the local Dirichlet space $\mathcal{D}_1$. We use Proposition [Proposition 14](#carmeasDz){reference-type="ref" reference="carmeasDz"}, and because of the definition of $\nu$ it suffices to consider only the arcs that contain $1$, and one can show that the measure $|z-1|^2\text{ d}\nu(z)$ satisfies [\[carmeasH2eq\]](#carmeasH2eq){reference-type="eqref" reference="carmeasH2eq"}. We move now to the description of multipliers. **Definition 16**. Let $H_1,H_2$ be Hilbert spaces of holomorphic functions on $\mathbb{D}$. The multipliers from $H_1$ to $H_2$ are defined as $$M(H_1,H_2) := \{\phi\in\text{Hol}(\mathbb{D}) \colon \phi H_1 \subseteq H_2\}.$$ When $H_1=H_2$ we simply write $M(H_1)$. The multiplier algebra $M(\mathcal{D}_\zeta)$ of the local Dirichlet space is characterized as follows. **Lemma 17**. *For $\zeta\in\mathbb{T}$, the multiplier algebra of $\mathcal{D}_\zeta$ is $\mathcal{D}_\zeta\cap H^\infty(\mathbb{D})$.* The fact that the multipliers of $\mathcal{D}_\zeta$ are in $\mathcal{D}_\zeta\cap H^\infty(\mathbb{D})$ follows from the standard argument which holds for many other reproducing kernel Hilbert spaces of analytic functions, see for example Proposition $3.1$ in [@modelmultipliers]. Let us move to the other inclusion: let $\phi \in \mathcal{D}_\zeta\cap H^\infty(\mathbb{D})$, and let $f\in\mathcal{D}_\zeta$. In light of the characterization in [\[second defn\]](#second defn){reference-type="eqref" reference="second defn"}, there exist functions $\eta, g\in H^2(\mathbb{D})$ such that $$\label{multDzeq} \phi(z) = \phi(\zeta)+(z-\zeta)\eta(z), \qquad f(z)=f(\zeta) + (z-\zeta)g(z), \qquad z\in\mathbb{D}.$$ Then, for $z\in\mathbb{D}$ it holds $$\begin{aligned} \phi(z)f(z) &= \big( \phi(\zeta)+(z-\zeta)\eta(z) \big) \big( f(\zeta) + (z-\zeta)g(z) \big) \\ &= \phi(\zeta)f(\zeta) + (z-\zeta)[ \phi(\zeta)g(z)+\eta(z)f(\zeta) + (z-\zeta) \eta(z)g(z)]. \end{aligned}$$ Again by [\[second defn\]](#second defn){reference-type="eqref" reference="second defn"}, membership of the product $\phi f$ in $\mathcal{D}_\zeta$ is equivalent to the membership in $H^2(\mathbb{D})$ of the function $$\phi(\zeta)g(z)+\eta(z)f(\zeta) + (z-\zeta) \eta(z)g(z).$$ Since $\eta,g\in H^2(\mathbb{D})$, it suffices to show that $(z-\zeta) \eta(z)g(z)$ belongs to $H^2(\mathbb{D})$, and this follows from [\[multDzeq\]](#multDzeq){reference-type="eqref" reference="multDzeq"} and the assumption that $\phi\in H^\infty(\mathbb{D})$, for $$(z-\zeta) \eta(z)g(z) = \big(\phi(z)-\phi(\zeta)\big)g(z). \qed$$ In [@modelmultipliers], multipliers between model spaces are studied. It is shown that $M(K_u)=\mathbb{C}$, meaning that every function multiplying any model space into itself must be constant. Furthermore, multipliers from model spaces to the Hardy space $H^2(\mathbb{D})$ are characterized in terms of a Carleson condition on the torus. More precisely, $\phi\in M\big(K_u,H^2(\mathbb{D})\big)$ if and only if the measure $|\phi|^2\!\text{ d}m$ is a Carleson measure for $K_u$, i.e. there exists a constant $C>0$ such that $$\int_\mathbb{T}|f\phi|^2 \text{ d}m \leq C \|f\|_{K_u}^2, \qquad f\in K_u .$$ Assuming the inclusion $K_u\subseteq \mathcal{D}_\zeta$, the local Dirichlet space $\mathcal{D}_\zeta$ is an intermediate space between $K_u$ and $H^2(\mathbb{D})$. This is reflected in our following multiplier theorem. **Theorem 18**. *Let $u$ be an inner function, $\zeta\in\mathbb{T}$ such that $\zeta\notin\sigma(u)$, and $\phi\in\emph{Hol}(\mathbb{D})$. Then $\phi$ is a multiplier from $K_u$ to $\mathcal{D}_\zeta$ if and only if the measure $|\phi|^2\!\text{ d}m$ is Carleson for $K_u$ and $\phi$ belongs to $\mathcal{D}_\zeta$.* *Proof of Theorem [Theorem 18](#thmmoltKuDz){reference-type="ref" reference="thmmoltKuDz"} .* Let us assume that $\phi\in M(K_u,\mathcal{D}_\zeta)$. Then, in particular, the measure $|\phi|^2 \text{ d}m$ is a Carleson measure for $K_u$, so it suffices to show that every multiplier from $K_u$ to $\mathcal{D}_\zeta$ belongs to $\mathcal{D}_\zeta$. If $u(0)=0$, then $1\in K_u$, implying that the multiplier $\phi$ belongs to $\mathcal{D}_\zeta$. If $u(0)\neq 0$, we consider the kernel $$k_0^u= 1-\overline{u(0)}u.$$ Using Theorem [Theorem 5](#localdouglasthm){reference-type="ref" reference="localdouglasthm"}, one can check that $1/k_0^u \in H^\infty(\mathbb{D})\cap\mathcal{D}_\zeta$, so that by Lemma [Lemma 17](#multDz){reference-type="ref" reference="multDz"} the function $1/k_0^u$ is a multiplier of $\mathcal{D}_\zeta$. Thus, $$\phi = \frac{1}{k_0^u} \, \phi k_0^u \in \mathcal{D}_\zeta$$ which implies the statement. Let us now prove the other implication. We assume that $|\phi|^2 \text{ d}m$ is a Carleson measure for $K_u$ and that $\phi$ belongs to $\mathcal{D}_\zeta$. Since $\phi\in M\big(K_u,H^2(\mathbb{D})\big)$, for every $f\in K_u$ the product $\phi f$ belongs to $H^2(\mathbb{D})$. We compute the local Dirichlet integral. $$\begin{aligned} D_\zeta(f\phi) &= \int_\mathbb{T}\bigg| \frac{f(\lambda)\phi(\lambda)-f(\zeta)\phi(\zeta)}{\lambda-\zeta}\bigg|^2 \text{ d}m(\lambda) \\ &= \int_\mathbb{T}\bigg| \frac{f(\lambda)\phi(\lambda)-\phi(\lambda)f(\zeta)+\phi(\lambda)f(\zeta) - f(\zeta)\phi(\zeta)}{\lambda-\zeta}\bigg|^2 \text{ d}m(\lambda) \\ &\leq \int_\mathbb{T}|\phi(\lambda)|^2 \bigg| \frac{f(\lambda)- f(\zeta)}{\lambda-\zeta} \bigg|^2 \text{ d}m(\lambda) + |f(\zeta)|^2 \int_\mathbb{T}\bigg| \frac{\phi(\lambda)- \phi(\zeta)}{\lambda-\zeta} \bigg|^2 \text{ d}m(\lambda) \\ &= \bigg\|\frac{f- f(\zeta)}{\cdot-\zeta} \bigg\|_{L^2(|\phi|^2\!\text{ d}m)}^2 + |f(\zeta)|^2\bigg\| \frac{\phi- \phi(\zeta)}{\cdot-\zeta} \bigg\|_{H^2}^2 \\ &\leq C \bigg\|\frac{f- f(\zeta)}{\cdot-\zeta} \bigg\|_{K_u}^2 + |f(\zeta)|^2 D_\zeta(\phi) \\ &\leq \big( C+D_\zeta(\phi)\big) \|f\|_{\mathcal{D}_\zeta}^2, \end{aligned}$$ concluding the proof. ◻ It is natural to ask whether the condition in Theorem [Theorem 18](#thmmoltKuDz){reference-type="ref" reference="thmmoltKuDz"} guarantees the boundedness of the multipliers, in other words, whether $M(K_u,\mathcal{D}_\zeta)$ is contained or not in $H^\infty(\mathbb{D})$. The next example shows that the answer to this question is negative. *Example 4*. For simplicity, we take $\zeta=1$. We consider the function $$\phi(z) := \frac{z-1}{(z+1)^{\frac{1}{3}}},$$ and a finite Blaschke product $$B(z) = \prod_{j=1}^N \frac{\overline{a_j}}{|a_j|}\frac{a_j-z}{1-\overline{a_j}z},$$ so that it holds the embedding $K_B \hookrightarrow \mathcal{D}_1$. We also assume that $a_j\neq a_l$ when $j\neq l$. We recall that $K_B$ is the finite-dimensional vector space spanned by the Szegö kernels $$K_B = \text{span}\{c_{a_j} \colon j=1,\ldots,N\}.$$ In particular, $K_B\subseteq H^\infty(\mathbb{D}).$ We show that $\phi$ is an unbounded multiplier from $K_B$ to $\mathcal{D}_1$. First, we check that the measure $\text{d}\mu_\phi := |\phi|^2 \text{ d}m$ is a Carleson measure for $K_B$. For $f\in K_B$, $$\|f\|_{L^2(\text{d}\mu_\phi)}^2 = \int_\mathbb{T}|f(\lambda)|^2\frac{|\lambda-1|^2}{|\lambda+1|^{\frac{2}{3}}} \text{ d}m(\lambda) \leq \|f\|_{H^\infty(\mathbb{D})}^2 \int_\mathbb{T}\frac{|\lambda-1|^2}{|\lambda+1|^{\frac{2}{3}}} \text{ d}m(\lambda) <\infty,$$ so that, by the closed graph theorem, the inclusion $K_B \hookrightarrow L^2(\mu_\phi)$ is bounded. Membership of $\phi$ in $\mathcal{D}_1$ follows from Theorem [Theorem 5](#localdouglasthm){reference-type="ref" reference="localdouglasthm"}. # The embedding $K_u \hookrightarrow \mathcal{D}(\mu)$ In this section we study the embedding of $K_u$ into $\mathcal{D}(\mu)$, for an arbitrary measure $\mu$. In this case, the sufficient condition we obtain is different from the necessary one. We start with the proof of Theorem [Theorem 3](#KuinDm){reference-type="ref" reference="KuinDm"}. *Proof of Theorem [Theorem 3](#KuinDm){reference-type="ref" reference="KuinDm"}..* By assumption, $supp(\mu)$ and $\sigma(u)$ are disjoint compact sets, therefore $$\delta := \text{dist}\big(\text{supp}(\mu),\sigma(u)\big) >0.$$ We consider the open set $$U:=\bigcup_{x\in\sigma(u)} \bigg\{z\in\mathbb{C}\colon |z-x|<\frac{\delta}{2} \bigg\}.$$ We split the harmonically weighted Dirichlet integral into $$D_\mu(f) = \frac{1}{\pi} \int_{\mathbb{D}\cap U} |f'|^2 P\mu\text{ d}A+ \frac{1}{\pi} \int_{\mathbb{D}\setminus U} |f'|^2 P\mu\text{ d}A.$$ For the first summand, we use a classical Littlewood-Paley estimate, see Proposition $3.2$ in [@garnett]: $$\begin{aligned} \frac{1}{\pi}\int_{\mathbb{D}\cap U} |f'|^2 P\mu\text{ d}A &=\frac{1}{\pi}\int_{\mathbb{D}\cap U} |f'(z)|^2 (1-|z|^2) \bigg( \int_{\text{supp}(\mu)} \frac{\text{ d}\mu(\zeta)}{|z-\zeta|^2}\bigg) \text{ d}A(z) \\ &\leq \frac{4}{\pi \delta^2}\mu(\mathbb{T}) \int_{\mathbb{D}} |f'(z)|^2 (1-|z|^2) \text{ d}A(z) \\ &\leq \frac{2}{\delta^2}\mu(\mathbb{T})\int_\mathbb{T}|f(\lambda)-f(0)|^2\text{ d}m(\lambda) \\ &\leq \frac{8}{\delta^2}\mu(\mathbb{T}) \|f\|_{H^2}^2.\end{aligned}$$ For the second summand, we recall that every function in the model space $K_u$ admits an analytic extension across $\mathbb{T}\setminus\sigma(u)$. Hence, we have $$\begin{aligned} \frac{1}{\pi} \int_{\mathbb{D}\setminus U} |f'(z)|^2 P\mu(z)\text{ d}A(z) &\leq \max_{\overline{\mathbb{D}} \setminus U} |f'|\, \frac{1}{\pi}\int_\mathbb{D}\int_\mathbb{T}\frac{1-|z|^2}{|z-\zeta|^2} \text{ d}\mu(\zeta) \text{ d}A(z) \\ &= \max_{\overline{\mathbb{D}} \setminus U} |f'| \, \int_\mathbb{T}\text{ d}\mu(\zeta) \\ &= \max_{\overline{\mathbb{D}} \setminus U} |f'|\, \mu(\mathbb{T}).\end{aligned}$$ We have proved that for every $f\in K_u$ $$D_\mu(f) \leq \frac{8}{\delta^2}\mu(\mathbb{T}) \|f\|_{H^2}^2 + \max_{\overline{\mathbb{D}} \setminus U} |f'| \mu(\mathbb{T}) <\infty.$$ The boundedness of the embedding $K_u \hookrightarrow\mathcal{D}(\mu)$ follows from the closed graph theorem. ◻ Now we prove Theorem [Theorem 4](#KuinDmnec){reference-type="ref" reference="KuinDmnec"}, giving a necessary condition for the considered embedding. *Proof.* For the proof, we introduce the function $V_\mu\colon\mathbb{C}\to[0,+\infty]$ defined as $$V_\mu(\omega):= \int_{\mathbb{T}}\frac{1}{|\zeta-\omega|^2}\text{ d}\mu(\zeta), \qquad \omega\in\mathbb{C}.$$ First, we prove that $V_\mu$ is bounded on the boundary spectrum $\sigma(u)$, which we can assume to be non-empty without loss of generality. Let $C>0$ be a constant such that $$D_\mu(f) \leq C\|f\|_{H^2}^2, \qquad f\in K_u.$$ Let $\lambda\in\sigma(u)$ and, as we did in the proof of Theorem [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"}, let us consider a sequence $(\omega_n)_n$ in $\mathbb{D}$ such that $u(\omega_n) \to 0$ as $\omega_n \to \lambda$. By the disintegration formula in [\[eqfubini\]](#eqfubini){reference-type="eqref" reference="eqfubini"} and the lower estimate for $\mathcal{D}_\zeta(k_n)$ obtained in the proof of Theorem [Theorem 2](#KuinDzb){reference-type="ref" reference="KuinDzb"}, we have that $$\begin{aligned} C\|k_n\|_{H^2}^2&\geq D_\mu(k_n)= \int_{\mathbb{T}} D_\zeta(k_n) \text{ d}\mu(\zeta) \\ &\geq \int_{\mathbb{T}} \|k_n\|_{H^2}^2 \frac{\big(1-|u(\omega_n)|\big)^2 \big(|\omega_n| - |u(\omega_n)|\big)^2}{|\zeta-\omega_n|^2 \big(1-|u(\omega_n)|^2\big)} \text{ d}\mu(\zeta) \\ &= \|k_n\|^2_{H^2} \frac{\big(1-|u(\omega_n)|\big)\big(|\omega_n| - |u(\omega_n)|\big)^2}{1+|u(\omega_n)|} \int_{\mathbb{T}} \frac{1}{|\zeta -\omega_n|^2} \text{ d}\mu(\zeta) .\end{aligned}$$ Hence, by Fatou's Lemma, it holds that $$C \geq \liminf_{n} \int_{\mathbb{T}} \frac{1}{|\zeta -\omega_n|^2} \text{ d}\mu(\zeta) \geq \int_{\mathbb{T}} \frac{1}{|\zeta -\lambda|^2} \text{ d}\mu(\zeta)=V_\mu(\lambda),$$ which proves that $\sup_{\lambda\in \sigma(u)} V_\mu(\lambda) <\infty$. Now the theorem follows from the fact that $V_\mu = \infty$ $\mu$-a.e. on $\mathbb{T}$ and therefore, necessarily, we have that $\mu\big(\sigma(u)\big)=0$. ◻ *Remark 2*. We note that a similar necessary condition holds also for the embedding $H(b)\hookrightarrow \mathcal{D}(\mu)$. Let $b_i$ be the inner factor associated to the bounded function $b$, and we consider a point $\zeta\in\sigma(b_i)$. If $\lim_n \omega_n = \zeta$ and $\lim_n |b(\omega_n)|=0$ we note that $$C\|k_n\|_{b}^2\geq \int_\mathbb{T}D_\zeta(k_n)d\mu(\zeta)\geq \|k_n\|^2_b \frac{\big(1-|b(\omega_n)|\big)\big(|\omega_n| - |b(\omega_n)|\big)^2}{1+|b(\omega_n)|} \int_{\mathbb{T}} \frac{1}{|\zeta -\omega_n|^2} d\mu(\zeta),$$ and once again by Fatou's Lemma we conclude that $V_\mu$ is bounded on $\sigma(b_i)$ and therefore $\mu\big(\sigma(b_i)\big)=0$. We conclude this section discussing the compactness of the embeddings. Due to the trivial norm inequality $\|\cdot\|_{K_u}\leq\|\cdot\|_\mu$, the compactness of the embedding $K_u\hookrightarrow\mathcal{D}(\mu)$ implies the compactness of the identity map $I_{K_u}$. Therefore, it is easy to see that the embedding $K_u\hookrightarrow\mathcal{D}(\mu)$ is compact if and only if $K_u$ is finite dimensional, that is, if and only if $u$ is a finite Blaschke product. # Final remarks and open questions Given $\mu$ a finite positive Borel measure on $\mathbb{T}$ and an inner function $u$, we have provided a sufficient condition and a necessary condition for the embedding $K_u\hookrightarrow\mathcal{D}(\mu)$, respectively $\text{supp}(\mu)\cap\sigma(u)=\emptyset$ and $\mu\big(\sigma(u)\big)=0$. If $\mu=\delta_\zeta$, both these conditions are equivalent to $\zeta\notin\sigma(u)$. For the Lebesgue measure, the two conditions do not coincide, but the sufficient one is also necessary: if the inclusion $K_u \hookrightarrow \mathcal{D}=\mathcal{D}(m)$ holds, then, using the reproducing kernels of $K_u$ one can see that necessarily $u$ belongs to $\mathcal{D}$. However, it is shown in [@primer] that the only inner functions in the classical Dirichlet space are finite Blaschke products, resulting in the boundary spectrum $\sigma(u)$ being empty. In future works we will investigate whether the sufficient condition $\text{supp}(\mu)\cap\sigma(u)=\emptyset$ is in general necessary as well for the embedding $K_u\hookrightarrow\mathcal{D}(\mu)$. For the time being, we leave this as an open problem. # Acknowledgements {#acknowledgements .unnumbered} The authors want to thank Professor Maria Nowak for the interest in discussing some problems addressed in this article. They also want to thank Professor Marco M. Peloso for his constant help and guidance.

arxiv_math

--- abstract: | We prove that a substitution is aperiodic if and only if some of its associated generating functions are transcendental. These generating functions have a recursive structure arising from the substitution which we use to study their roots in the case of the Fibonacci substitution. address: - Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada - Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada - Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada, and Institute of Mathematics "Simon Stoilow", Bucharest, Romania author: - Aisling Pouti - Christopher Ramsey - Nicolae Strungaru title: Generating functions of substitutions --- # Introduction One of the challenges of mathematics is to develop objects that both have enough and not too much structure, that are far-reaching in their connections, and raise many tractable questions. Substitutions readily fit these requirements, with a far ranging and well-studied theory [@AA; @TAO; @DOP; @Queff]. The aim of this paper is to add another connection, between transcendental generating functions and aperiodic substitutions. Power series under reasonable conditions provide a very nice environment to study aperiodic structures and vice versa to study transcendentality. This is because Fatou [@PF] showed that a rational power series with radius of convergence 1 is either rational or transcendental over ${\mathbb Q}(X)$, completely removing the possibility of being irrational and algebraic. This has been used to study number-theoretic sequences [@BBC; @BC] and walks on the slit plane [@BM; @Rubey]. Section 2 collects the necessary background for algebraic and transcendental power series for our context, including the aforementioned theorem of Fatou, Theorem [Theorem 3](#thm:fatou){reference-type="ref" reference="thm:fatou"}. Section 3 studies generating functions for one-sided infinite words on a finite alphabet in two ways. The first is to consider the characteristic generating function of a letter in the finite alphabet, that is, taking the generating function of the integer sequence obtained by replacing that letter with a 1 and all others with a 0 in the infinite word. The second way is to consider the position-generating function of a letter, which uses the sequence of integers that give the positions of the letter in the infinite word. These generating functions encode the aperiodicity of a letter in the infinite word as transcendentality, Propositions [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"} and [Proposition 13](#prop:position){reference-type="ref" reference="prop:position"}. While much can be said about infinite words in this context, Section 4 studies words that arise as the fixed points of substitutions. These are self-maps of the words in the alphabet that respect concatenation, and so a substitution is given uniquely by what it does to the letters in the alphabet. For every primitive substitution, there exist infinite fixed words of a power of the substitution, that arises naturally. Moreover, the fixed word can be chosen uniquely if we fix the starting letter in the alphabet. The aperiodicity of a substitution is determined from this infinite word, but can also be more simply determined by its linear algebraic structure. Theorem [Theorem 17](#thm:aperiodictrans){reference-type="ref" reference="thm:aperiodictrans"} relates the aperiodicity of a substitution with the transcendentality of its affiliated characteristic and position-generating functions. Substitutions can also be viewed as a tiling of the infinite half-line using a finite set of tiles and a so-called inflation rule. This geometric version is in one-to-one correspondence with the symbolic substitutions. Using the coordinates of the left endpoints of such a tiling as coefficients, we get another generating function but now with possibly irrational coefficients. In the case of two letters, and algebraic tile lengths, Theorem [Theorem 24](#thm:geometric){reference-type="ref" reference="thm:geometric"} proves that the substitution is aperiodic if and only if this geometric generating function is transcendental over ${\mathbb Q}(X)$. The last section studies the Fibonacci substitution and what can be accomplished through the recursive structure of polynomials induced by the substitution. We show that the position generating functions can be given in terms of the characteristic generating functions, Corollary [Corollary 28](#cor:fibpostochar){reference-type="ref" reference="cor:fibpostochar"}. The rest of the section shows that this recursive structure can give us specific knowledge about the zeroes of the characteristic generating function. In particular, the characteristic function for the first letter of the Fibonacci substitution is shown to have no zeroes on $(-0.99729758,1)$, almost its whole interval of convergence, and we conjecture that it has no zeroes. # Preliminaries: power series ## Algebraic and transcendental power series As usual, for a field $F$ we denote by $F[X]$ the ring of polynomials with coefficients in $F$ and by $F[[X]]$ the ring of power series with coefficients in $F$. Note here that both $F[X]$ and $F[[X]]$ are integral domains. $F(X)$ denotes the field of fractions of $F[X]$, that is $$F(X)=\left\{ \frac{P(X)}{Q(X)}: P, Q \in F[X], Q \neq 0 \right\} \,.$$ Recall here that given a field extension $F \subseteq K$, an element $a \in K$ is called *algebraic over $F$* if there exists a nonzero polynomial $P \in F[X]$ such that $P(a)=0$. If $a \in K$ is not algebraic, then we say that $a$ is *transcendental*. Given a field $F$, the ring $F[[X]]$ is an integral domain, and hence a subring of its field of fractions $$F((X)):= \left\{ \frac{A}{B} : A,B \in F[[X]], B \neq 0\right\} \,.$$ It is well known that a power series $$A(X)=a_0+a_1X+a_2X^2+ \ldots +a_nX^n + \ldots \in F[[X]]$$ is a unit if and only if $a_0 \neq 0$. This immediately implies that each non-zero $B(X) \in F[[X]]$ can be written uniquely as $$B(X)=X^n C(X)$$ where $C(X)$ is a unit in $F[[X]]$. It follows that $$F((X))= \left\{ \sum_{n=-N}^\infty a_nX^n : N \in \mathbb N, a_n \in F \right\} \,,$$ that is, $F((X))$ is exactly the ring of formal Laurent series with coefficients in $F$. Now, with the field extension $$F(X) \subseteq F((X)) \,,$$ we can talk about elements $S \in F((X))$ being algebraic or transcendental over $F(X)$. A standard common denominator procedure shows that an element $S \in F((X))$ is algebraic over $F(X)$ if and only if there exists some polynomials $P_0, P_1, \ldots, P_n \in F[X]$, not all zero, such that $$P_0(X)+ P_1(X)S(X)+ P_2(X) \left(S(X)\right)^2+\ldots + P_n(X) \left(S(X)\right)^n=0 \,.$$ Below we will often be interested in whether some $A \in {\mathbb R}[[X]]\subseteq {\mathbb R}((X))$ is algebraic or transcendental over ${\mathbb Q}(X)$. Note here that we have $${\mathbb Q}(X) \subseteq {\mathbb R}(X) \subseteq {\mathbb R}[[X]] \subseteq {\mathbb R}((X)) \,.$$ ## Radius of convergence of power series Consider now a power series $$S(X) =a_0+a_1X+ \ldots +a_nX^n+ \ldots \in {\mathbb C}[[X]]$$ and let $$R:= \frac{1}{\limsup_n \sqrt[n]{|a_n|}} \,.$$ Then the series $$a_0+a_1z+ \ldots +a_nz^n+ \ldots$$ is absolutely convergent for all $|z| < R$ and divergent for all $|z| >R$. $R$ is called the *radius of convergence* of $S(X)$. On the boundary of this disc, the series can be convergent at none, some or all the points. Therefore the domain $D$ of convergence satisfies $$\{ z \in {\mathbb C}: |z| < R \} \subseteq D \subseteq \{ z \in {\mathbb C}: |z| \leq R \} \,.$$ Throughout the paper, we will be interested in series for which $R \geq 1$, specifically this is a requirement for a theorem of Fatou introduced in the next subsection. The following is a trivial result. **Lemma 1**. *Let $a_n \in {\mathbb C}$ be a sequence which only takes finitely many values. Then exactly one of the following holds:* - *There exists some $n,N \in \mathbb N$ such that $a_n =0$ for all $n >N$. In this case $$A(X)=\sum_{n=0}^\infty a_nX^n$$ is a polynomial and has $R=\infty$.* - *$a_n \neq 0$ for infinitely many $n \in \mathbb N$. In this case $$A(X)=\sum_{n=0}^\infty a_nX^n$$ has a radius of convergence $R= 1$.* *0◻* ## A theorem of Fatou and applications In this section we restrict to series $$A(X) = \sum_{n=0}^\infty a_n X^n \in {\mathbb Z}[[X]] \,.$$ Note here that for each $n \in \mathbb N$ we have $$\sqrt[n]{|a_n|} = 0 \mbox{ or } \sqrt[n]{|a_n|} \geq 1 \,.$$ It follows immediately that either $A(X)$ is a polynomial or its radius of convergence satisfies $R \leq 1$. For this reason, the conditions that the series converges inside the unit disk below force $A(X)$ to either be a polynomial or have the largest possible radius of convergence, that is $R=1$. Let us note here in passing the following standard result, the proof of which boils down to the comparison test against a $p$-series where $p = \deg(P)$. **Lemma 2**. *Let $$A(X) = \sum_{n=0}^\infty a_n X^n \in {\mathbb Z}[[X]] \,.$$ If there exists some polynomial $P(X)$ such that $|a_n| \leq |P(n)|$ for all $n \in \mathbb N$ then $R \geq 1$.* *In particular, the conclusion holds when $a_n$ is a bounded sequence.* *0◻* We can now review the Fatou Theorem. **Theorem 3** (Fatou [@PF]). *Let $a_n \in {\mathbb Z}$ be a sequence and let $F(X)= \sum_{n=0}^\infty a_n X^n$ be its generating function. If $F$ is convergent inside the unit disk, then exactly one of the following holds:* - *There exists some $P, Q \in {\mathbb Q}(X)$ such that $$F(X)=\frac{P(X)}{Q(X)} \,,$$ and all roots of $Q$ are roots of unity; or,* - *$F$ is transcendental over ${\mathbb Q}(X)$.* *0◻* Now, the following is a generalization of a well-known fact about generating functions of $0-1$ sequences. Fatou proved a version of this [@PF] which has had many variations over the years (cf. [@BC]). For an excellent history of the subject look to the introduction of [@BellChen]. Since we could not find a good reference for our specific formulation, we include the proof for completeness. **Proposition 4**. *Let $a_n \in {\mathbb Q}$ and let $$A(X)= \sum_{n=0}^\infty a_n X^n$$ be the generating function of $a_n$. If $a_n$ only takes finitely many values, then:* - *$a_n$ is eventually periodic if and only if $A(X) \in {\mathbb Q}(X)$. Moreover, in this case, there exists some $P(X) \in {\mathbb Q}[X]$ and $d \in \mathbb N$ such that $$A(X)=\frac{P(X)}{1-X^d} \,.$$* - *$a_n$ is not eventually periodic if and only if $A(X)$ is transcendental over ${\mathbb Q}(X)$.* *Proof.* **(a)**$\Longrightarrow$ Let $a_n \in {\mathbb Q}$ be eventually periodic so that $a_n=a_{n+d}$ for all $n\geqslant N$ where $n,N,d \in \mathbb N$. Let $A(X)=\sum_{n=0}^{\infty}a_nX^n$ be the generating function of $a_n$. Then we have that $$\begin{aligned} A(X)=& \underbrace{a_0+a_1X+a_2X^2+\ldots++a_{N-1}X^{N-1}}_{P_1(X)}+a_NX^N+a_{N+1}X^{N+1}+a_{N+2}X^{N+2}+\ldots+a_{N+d}X^{N+d}\\ &\quad\quad + a_{N+d+1}X^{N+d+1}+a_{N+d+2}X^{N+d+2}+\ldots+a_{N+2d}X^{N+2d}+\ldots \\ =& P_1(X)+X^N(a_N+a_{N+1}X+a_{N+2}X^2+\ldots+a_{N+d}X^d+ a_{N+d+1}X^{d+1}+a_{N+d+2}X^{d+2}\\ &\quad\quad +\ldots+a_{N+2d}X^{2d}+\ldots)\\ =&P_1(X)+X^N(a_N+a_{N+1}X+a_{N+2}X^2+\ldots+a_{N}X^d+ a_{N+1}X^{d+1}+a_{N+2}X^{d+2}\\ &\quad\quad +\ldots+a_{N}X^{2d}+\ldots)\\ =&P_1(X)+X^NP_2(X)(1+X^d+X^{2d}+\ldots+)=P_1(X)+\frac{X^NP_2(X)}{1-X^d}=\frac{P(X)}{1-X^d} \,,\end{aligned}$$ where $$P_2(X)=a_N+a_{N+1}X+\ldots+a_{N+d-1}X^{d-1} \,,$$ and $$P(X)=P_1(X)(1-X^d)+X^NP_2(X)\,.$$.\ $\Longleftarrow$ We first prove the claim when $a_n \in \mathbb N$ and then the general case. *Case 1:* $a_n \in \mathbb N$ for all $n$. Fix some $b \in \mathbb N$ such that, for all $n$ we have $$0 \leq a_n \leq b-1 \,.$$ Now, since $A(X) \in {\mathbb Q}(X)$, there exists $P,Q \in {\mathbb Q}[X]$ such that $$A(X)= \frac{P(X)}{Q(X)} \,.$$ Since $\frac{1}{b} <1$ and the series $A$ converges on the unit disk, we get $$\sum_{n=0}^\infty a_n \left(\frac{1}{b}\right)^n = \frac{P(\frac{1}{b})}{Q(\frac{1}{b})} \in {\mathbb Q}\,.$$ Now, the left hand side is just the base $b$ representation of the number $$\alpha = a_0.a_1a_2a_3 \ldots a_n \ldots \qquad \,.$$ Since $a_0,\ldots, a_n, \ldots$ are digits in base $b$, and since $\alpha \in {\mathbb Q}$ we get that $a_n$ is eventually periodic. This proves the claim. *Case 2:* The general case. Let $a_n \in {\mathbb Q}$ taking only finitely many values and let $m$ be the common denominator of these values. Then, the sequence $$b_n :=m a_n$$ is a sequence of integers taking only finitely many values. Set $$k = \min \{ b_n : n \in \mathbb N\} \in {\mathbb Z}\,.$$ and $$c_n:= b_n - k \,.$$ Then, $c_n$ is a sequence of non-negative integers, taking only finitely many values. Moreover, $$\begin{aligned} C(X) &= \sum_{n=0}^\infty c_n X^n = \sum_{n=0}^\infty (ma_n-k) X^n =m A(X) - \frac{k}{1-X } \in {\mathbb Q}(X) \,.\end{aligned}$$ Therefore, by Case 1, $c_n$ is eventually periodic, and so is $a_n$. **(b)** Follows immediately from (a), the theorem of Fatou, and Lemma [Lemma 1](#lem:1){reference-type="ref" reference="lem:1"}. ◻ **Remark 5**. *Under the conditions of Prop. [Proposition 4](#fact2){reference-type="ref" reference="fact2"}, if $a_n$ is a periodic sequence of integers taking only finitely many values, it follows from the proof of Prop. [Proposition 4](#fact2){reference-type="ref" reference="fact2"} that there exists some $P(X) \in {\mathbb Z}[X]$ and $d \in \mathbb N$ such that $$A(X)=\frac{P(X)}{1-X^d} \,.$$* As an immediate consequence we get: **Corollary 6**. *Let $b_n \in \{ 0,1 \}$ and let $$G(X)= \sum_{n=0}^\infty b_n X^n$$ be the generating function of $b_n$. Then,* - *$b_n$ is eventually periodic if and only if $G(X) \in {\mathbb Q}(X)$. Moreover, in this case, there exists some $P(X) \in {\mathbb Q}[X]$ and $d \in \mathbb N$ such that $$G(X)=\frac{P(X)}{1-X^d} \,.$$* - *$b_n$ is not eventually periodic if and only if $G(X)$ is transcendental over ${\mathbb Q}(X)$.* *0◻* # Generating functions for one-sided words {#worddef} Consider an alphabet $\mathcal A= \{ a_1, \ldots, a_k \}$, that is, a finite set of symbols. A one-sided infinite word $w$ is simply $$w=w_0w_1w_2\ldots w_n \ldots$$ where $w_0, \ldots ,w_n, \ldots \in \mathcal A$. We will primarily be interested in words coming from substitutions but we first develop the general theory for arbitrary words. ## The characteristic function of a letter {#charfunk} Consider an arbitrary infinite word $$w=w_0w_1 \ldots$$ on a finite alphabet $\mathcal A$. Now, to each function $g: \mathcal A\to {\mathbb C}$ we can assign a formal power series $$C_{g}(X)= \sum_{n=0}^\infty g(w_n) X^n \,.$$ In the particular case when $g$ is the characteristic function of $a_j$, we will simply denote the power series by $C_{a_j}$, that is $$C_{a_j}(X)= \sum_{n=0}^\infty 1_{a_j}(w_n) X^n \,.$$ For all $g : \mathcal A\to {\mathbb C}$ we have $$\label{eq11} C_g(X) = \sum_{j=1}^k g(a_j) C_{a_j}(X) \,,$$ that is $C_{a_1}, \ldots, C_{a_k}$ is a ${\mathbb C}$-basis for the vector space of such series. For this reason, properties of $C_{a_1}, \ldots, C_{a_k}$ will be of particular interest. Note also in passing that if $w$ is the fixed word of a primitive substitution tiling, then each $a_j \in \mathcal A$ appears infinitely many times, with bounded gaps. This implies that no $C_{a_j}$ is a polynomial in this case. We also have $$\label{eq12} \sum_{j=1}^k C_{a_j}(X) = \frac{1}{1-X} \,.$$ Note that each $C_{a_j}$ has coefficients in $\{ 0,1 \}$, and hence, Corollary [Corollary 6](#cor1){reference-type="ref" reference="cor1"} yields: **Proposition 7**. *For each $1 \leq j \leq k$, the positions of $a_j$ are not eventually periodic if and only if $C_{a_j}$ is transcendental over ${\mathbb Q}(X)$. 0◻* Similarly, by Proposition [Proposition 4](#fact2){reference-type="ref" reference="fact2"}, we have the following two results: **Proposition 8**. *Let $g: \mathcal A\to {\mathbb Q}$ be arbitrary. The sequence $g(w_n)$ is not eventually periodic if and only if $C_g(X)$ is transcendental over ${\mathbb Q}(X)$.0◻* In the case the alphabet consists of only two letters, we can also allow $g$ to take algebraic values: **Proposition 9**. *Let $\mathcal A= \{a_1, a_2 \}$ and $g:\mathcal A\rightarrow {\mathbb C}$ such that $g(a_1)$ and $g(a_2)$ are algebraic over ${\mathbb Q}$. The sequence $g(w_n)$ is not eventually periodic if and only if $C_g(X)$ is transcendental over ${\mathbb Q}(X)$.* *Proof.* Equations [\[eq11\]](#eq11){reference-type="eqref" reference="eq11"} and [\[eq12\]](#eq12){reference-type="eqref" reference="eq12"} yield $$\begin{aligned} C_g(X) &=g(a_1)C_{a_1}(X)+ g(a_2)C_{a_2}(X) \\ & = g(a_1)C_{a_1}(X) +g(a_2)\left(\frac{1}{1-X}-C_{a_1}(X)\right) \\ & = (g(a_1)-g(a_2))C_{a_1}(X) + g(a_1)\frac{1}{1-X} \,.\end{aligned}$$ The result now follows from Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}. ◻ ## The position function of a letter There are an enormous amount of ways to encode the letters of a one-sided infinite word into ${\mathbb C}[[X]]$. The previous subsection outlined a very sensible encoding, and we turn to a second choice here. Again let $\mathcal A= \{a_1,\dots, a_k\}$ be our finite alphabet and $w$ be a one-sided infinite word in $\mathcal A$. For $1 \leq j \leq k$, define $$p_j(n)= \mbox{ position of } n^{\mbox{th}}\ a_j \mbox{ in } w,$$ sometimes referred to as $p_{a_j}(n)$, and $$P_{a_j}(X)= \sum_{n=1}^{N_j} p_j(n)X^n \,.$$ Here $N_j \in \mathbb N\cup \{ \infty \}$ is the number of times $a_j$ appears in $w$. This sequence of positions is quite natural as $$C_{a_j}(X) = \sum_{n=1}^{N_j} X^{p_j(n)}\,.$$ With these position-generating functions, one can certainly define $P_g$ for any $g:\mathcal A\rightarrow {\mathbb C}$ by linear combinations of the functions above. However, in many cases $\{P_{a_1},\dots, P_{a_k}\}$ will not form a basis, as they may be linearly dependent. **Example 10**. *Let $\mathcal A= \{a,b,c\}$ and $w$ be the $4$-periodic word $$w=abcbabcbabcb\ldots \ldots$$ Then, $$\begin{aligned} p_1(n) & = 4n-4\\ p_2(n) & = 2n-1 \\ p_3(n) & = 4n-2\end{aligned}$$ are linearly dependent.* **Remark 11**. *Note here that there is a strong connection between the summatory function of the characteristic function $1_{a_j}(w_n)$ and the position function $p_j$:* *For an arbitrary word $w$ and some $1 \leq j \leq k$, denote by $$S_j(n):=1_{a_j}(w_0)+1_{a_j}(w_1)+\ldots + 1_{a_j}(w_n)$$ the summatory function of $1_{a_j}(n)$. Then $$S_j \circ p_j = \mbox{Id}$$ that is $p_j$ is a left inverse of $S_j$.* *It is also easy to see that $$p_j \circ S_j (n) = \mbox{ position of the last } a_j \mbox{ up to position } n \,.$$* We wish to examine the rationality/transcendentality of these generating functions similarly as to the last subsection. Obviously, some new techniques will be required. The point of all of this is that a letter in a word can be not eventually periodic but still have an eventually periodic structure if viewed by its position function. If $a_j$ appears only a finite number of times then $P_{a_j}(X)$ is a polynomial. Otherwise, $\{p_j(n)\}$ is a strictly increasing non-negative sequence of integers. This implies that the radius of convergence is $$\frac{1}{R} = \limsup_{n\rightarrow \infty} \sqrt[n]{p_j(n)} \geq 1.$$ Thus, for the radius of convergence to be 1, the criterion for the theorem of Fatou, the growth of $\{p_j(n)\}$ must be subexponential. This is still a very large class of sequences, including $\{\lfloor 2^{\sqrt n}\rfloor\}$ for example. The most tractable position functions are those which are within a finite bound of polynomial growth. Let us prove some preliminary results about such sequences. **Lemma 12**. *Let $q(X)$ be a non-constant polynomial and $C\geq0$ be a constant. If $\{a_n\}$ is a sequence of complex numbers such that $|q(n) - a_n| \leq C$, then $$|(q(n) - q(n-1)) - (a_n - a_{n-1})| \leq 2C$$ and $\deg(q(X) - q(X-1)) = \deg(q(X)) - 1$.0◻* **Proposition 13**. *Let $w$ be a one-sided infinite word in the alphabet $\mathcal A= \{a_1,\dots, a_k\}$. For a fixed $1\leq j\leq k$, suppose there is a polynomial $q(X)$ of degree $m$ and a constant $C\geq0$ such that $$|q(n) - p_j(n)| \leq C\,.$$ If $$\sum_{n=1}^\infty b_n X^n := (1-X)^m\sum_{n=1}^\infty p_j(n)X^n$$ then $$b_n = \sum_{l=0}^{\min\{n,m\}-1} (-1)^l\: {\binom{n}{l}}\: p_j(n-l)$$ and $b_n$ can only take a finite number of integer values. Moreover, $P_{a_j}(X)$ is rational or transcendental over ${\mathbb Q}(X)$ if and only if $b_n$ are eventually periodic or not, respectively.* *Proof.* First, let $$\begin{aligned} \sum_{n=1}^\infty b_n X^n & := (1-X)\sum_{n=1}^\infty p_j(n)X^n \\ &= p_j(1)X + \sum_{n=2}^\infty (p_j(n) - p_j(n-1))X^n\,.\end{aligned}$$ Let $r(X) = q(X) - q(X - 1)$ which has degree $m-1$. As well, let $D = \max\{2C, |r(0) - p_j(0)|\}$. Then by the previous lemma we have that for $n\geq 0$ $$|r(n) - b_n| \leq D.$$ Simply repeat this argument $m$ times to reach the first conclusion since the $\{b_n\}$ sequence of integers will be a constant away from a degree 0 polynomial, that is a constant. The last conclusion then follows from Proposition [Proposition 4](#fact2){reference-type="ref" reference="fact2"}. ◻ The opposite action of taking differences is of course taking summatory series. Namely, if $$\sum_{n=1}^\infty b_n X^n = (1-X)\sum_{n=1}^\infty a_n X^n \ \ \ \textrm{then} \ \ \ \frac{1}{1-X}\sum_{n=1}^\infty b_n X^n = \sum_{n=1}^\infty a_n X^n$$ That is, $a_n = \sum_{l=1}^n b_l$. It is of note that the summatory sequence of a sequence of integers $\{a_n\}$ that takes only a finite number of values need not be within a bound of a linear polynomial. For example, consider $$a_{n} =\left\{ \begin{array}{cc} 1& \mbox{ if } n=2^m \\ 0& \mbox{otherwise } \end{array} \right. \,.$$ However, we can expand on the previous proposition. **Proposition 14**. *Let $w$ be a one-sided infinite word in the alphabet $\mathcal A= \{a_1,\dots, a_k\}$. For a fixed $1\leq j\leq k$, suppose that there is an $m\in \mathbb N$ such that the sequence $$b_n = \sum_{l=0}^{\min\{n,m\}-1} (-1)^l\: {\binom{n}{l}}\: p_j(n-l)$$ takes only a finite number of values. Then $P_{a_j}(X)$ is rational or transcendental if and only if the $b_n$ are eventually periodic or not, respectively.* *Proof.* From the previous proposition we saw that $$\sum_{n=1}^\infty b_n X^n = (1-X)^m\sum_{n=1}^\infty p_j(n)X^n$$ and so flipping this around we get that $$\sum_{n=1}^\infty p_j(n)X^n = \frac{1}{(1-X)^m}\sum_{n=1}^\infty b_n X^n\,.$$ Hence, $P_{a_j}(X)$ has radius of convergence 1 and so the $p_j(n)$ are subexponential. The conclusion follows immediately. ◻ As a consequence, we get: **Corollary 15**. *Let $w$ be a one-sided infinite word in the alphabet $\mathcal A= \{a_1,\dots, a_k\}$. If for a fixed $1\leq j\leq k$, if the difference sequence $\{p_j(n) - p_j(n-1)\}$ takes on only a finite number of values, then $P_{a_j}(X)$ is rational or transcendental over ${\mathbb Q}(X)$ if and only if $a_j$ is eventually periodic in $w$ or not, respectively.* *Proof.* The difference sequence of the positions is the same as the gaps between successive appearances of the letter $a_j$. ◻ Let us complete this section by remarking that the difference sequence $\{p_j(n) - p_j(n-1)\}$ takes on only a finite number of values if and only if the support of the letter $a_j$ is relatively dense in $\mathbb N$. # Generating functions for substitutions Recall that a substitution $\sigma$ on a finite alphabet $\mathcal A= \{a_1,\dots, a_k\}$ is a mapping $\sigma$ of $\mathcal A$ into the set $\mathcal A^+$ of non-empty finite words in $\mathcal A$. Via concatenation, $\sigma$ extends naturally to a map from $\mathcal A^+$ into $\mathcal A^+$. We typically add simple extra assumptions which force repeated application of $\sigma$ on every letter or word to grow. To every such $\sigma$, we can associate an $k\times k$ matrix $A_\sigma$ where the $(i,j)$ entry equals the number of $a_j$ letters in the $\sigma(a_i)$ word. The best-behaved substitutions are those which are primitive, that is, there is an $m\in\mathbb N$ such that $A_\sigma^m$ has only strictly positive entries. This is equivalent to requiring the existence of some power $m$ such that each letter $a_j$ appears in $\sigma^m(a_i)$ for every letter $a_i$. Given such a primitive substitution, the Perron-Frobenius Theorem states that there exists a unique real eigenvalue of $A_\sigma$, called the PF-eigenvalue $\lambda_{PF}$, which is positive, has a positive eigenvector and is larger than the modulus of any other eigenvalue. We are interested in the one-sided infinite words that are fixed points of the substitution, meaning $\sigma(w) = w$. For primitive substitutions, fixed points always exist for some power of $\sigma$ [@TAO Lemma 4.3]. A substitution $\sigma$ on $\mathcal A$ is called aperiodic if it has a one-sided infinite fixed word that is not eventually periodic. Note that if one is not eventually periodic then they all are. In particular, [@TAO Theorem 4.6] gives that if the PF-eigenvalue of a primitive substitution $\sigma$ is irrational then $\sigma$ is aperiodic. Do note that the converse is false, cf. Thue-Morse substitution [@TAO Section 4.6]. The reader looking for a comprehensive discussion of substitutions is directed to Chapter 4 of [@TAO]. We start by showing that in any fixed word of a primitive substitution, all letters appear within bounded gaps. **Proposition 16**. *Let $\sigma$ be a primitive substitution of a finite alphabet $\mathcal A= \{a_1,\dots, a_k\}$ of at least two letters and let $w$ be a one-sided fixed point of the substitution. There exists an $N\in \mathbb N$ such that for all $1\leq i\leq k$ the gap between successive copies of $a_i$ in $w$ is never more than $N$.* *Proof.* By primitivity, there exists an $m\in \mathbb N$ such that every $a_i$ is in every $\sigma^m(a_j)$. Since $w = w_0w_1w_2\dots$, $w_n\in \mathcal A$, is fixed under $\sigma$ then we see that $$w = \sigma^m(w) = \sigma^m(w_0)\sigma^m(w_1)\sigma^m(w_2)\dots\,.$$ Thus, for each $1\leq i\leq k$ the letter $a_i$ appears in every $\sigma^m(w_n)$. Therefore, by taking $N$ to be twice the length of the longest word $\sigma^m(a_j), 1\leq j\leq k$, the result follows. ◻ Now, we can prove the following: **Theorem 17**. *Let $\sigma$ be a primitive substitution of a finite alphabet $\mathcal A= \{a_1,\dots, a_k\}$ of at least two letters and let $w$ be a one-sided fixed point of the substitution. The following are equivalent:* - *$\sigma$ is aperiodic.* - *There exists $1\leq i\leq k$ such that $C_{a_i}(X)$ is transcendental over ${\mathbb Q}(X)$.* - *There exist $1\leq i<j\leq k$ such that $C_{a_i}(X)$ and $C_{a_j}(X)$ are transcendental over ${\mathbb Q}(X)$.* - *There exists $1\leq i\leq k$ such that $P_{a_i}(X)$ is transcendental over ${\mathbb Q}(X)$.* - *There exist $1\leq i<j\leq k$ such that $P_{a_i}(X)$ and $P_{a_j}(X)$ are transcendental over ${\mathbb Q}(X)$.* *Proof.* **(i) $\Longleftrightarrow$ (ii)** The only way $\sigma$ is aperiodic is if at least one of the letters in $\mathcal A$, say $a_i$ is not eventually periodic in $w$. By Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"} this is equivalent to $C_{a_i}$ being transcendental over ${\mathbb Q}(X).$ **(iii) $\Longrightarrow$ (ii)** is obvious, while **(ii) $\Longrightarrow$ (iii)** follows trivially from [\[eq12\]](#eq12){reference-type="eqref" reference="eq12"}: $$\sum_{j=1}^k C_{a_j}(X) = \frac{1}{1-X} \,,$$ which implies that the sum of the characteristic generating functions is a rational function over ${\mathbb Q}(X)$. Hence, $C_{a_i}(X)$ is transcendental over ${\mathbb Q}(X)$ if and only if at least two of these generating functions are transcendental over ${\mathbb Q}(X)$. **(iii) $\Longleftrightarrow$ (v)** and **(ii) $\Longleftrightarrow$ (iv)**: By Proposition [Proposition 16](#prop:bd){reference-type="ref" reference="prop:bd"}, for every $1\leq i\leq k$ the difference sequence of the positions $\{p_i(n) - p_i(n-1)\}$ takes on a finite number of integers. Hence, by Corollary [Corollary 15](#cor:finitegaps){reference-type="ref" reference="cor:finitegaps"}, $a_i$ is not eventually periodic in $w$ if and only if $P_{a_i}$ is transcendental over ${\mathbb Q}(X)$. By Corollary [Corollary 6](#cor1){reference-type="ref" reference="cor1"} we also have $a_i$ is not eventually periodic in $w$ if and only if $C_{a_i}$ is transcendental over ${\mathbb Q}(X)$. The equivalence follows. ◻ Let us now look at some examples: **Example 18**. *The Fibonacci substitution is given by $\mathcal A= \{a,b\}$ with $$\sigma \ : \ \begin{array}{l} a \mapsto ab \\ b\mapsto a \end{array}$$ and let $$w \ = \ abaababaabaababaababaabaababaabaab\dots \ = \ \lim_{m\rightarrow \infty} \sigma^m(a)\,.$$ Then $$A_\sigma\ = \ \left[\begin{matrix} 1&1\\1&0 \end{matrix}\right],$$ and $\lambda_{PF} = \frac{1 + \sqrt 5}{2}$, the golden ratio, which is irrational. Therefore, $\sigma$ is primitive and aperiodic and so, by Theorem [Theorem 17](#thm:aperiodictrans){reference-type="ref" reference="thm:aperiodictrans"}, all of $C_a(X), C_b(X), P_a(X)$, and $P_b(X)$ are all transcendental over ${\mathbb Q}(X)$.* It is possible for $w$ to be aperiodic but for the positions of one letter to be periodic. This can happen even for primitive substitutions, as the next example shows. **Example 19**. *Consider the substitution $$\sigma \ : \ \begin{array}{l}x \mapsto xyzy \\ y \mapsto xy \\ z \mapsto zy \end{array}$$ and let $$w \ = \ xyzyxyzyxyxyzy.... \ = \ \lim_{m\rightarrow \infty} \sigma^m(x)$$ be its one-sided fixed point starting from $x$. Then, it is easy to see that $$\begin{aligned} w_{2k} & \in \{x,z \} \\ w_{2k+1} & = y \,.\end{aligned}$$ This shows that the positions of $y$'s are periodic. On another hand, the substitution matrix is $$A_\sigma = \begin{bmatrix} 1& 2 & 1\\ 1& 1 & 0\\ 0 & 1 & 1 \end{bmatrix}$$ with $\lambda_{PF} = \frac{3 + \sqrt{5}}{2}$. Hence, $\sigma$ is primitive and aperiodic.* *A fast computation shows $$\begin{aligned} C_y(X)&=\sum_{n=0}^\infty X^{2n+1} =\frac{X}{1-X^2} \in {\mathbb Q}(X)\\ P_y(X)&=\sum_{n=1}^\infty (2n-1)X^{n} =\frac{X^2+X}{(1-X)^2} \in {\mathbb Q}(X) \,. \end{aligned}$$ Note that by Theorem [Theorem 17](#thm:aperiodictrans){reference-type="ref" reference="thm:aperiodictrans"} and Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}, $C_x(X), C_z(X), P_x(X)$, and $P_z(X)$ are transcendental over ${\mathbb Q}(X)$.* ## Recursive structure It will come as no surprise that a substitution carries over into the structure of the $C_{a_j}(X)$ and $P_{a_j}(X)$. First we need more notation. For each finite word $w = w_0\dots w_l$ in the alphabet $\mathcal A= \{a_1,\dots, a_k\}$ define, for $1\leq i\leq k$, the polynomials $$C_{a_i, w}(X) = \sum_{n=0}^l 1_{a_i}(w_n)X^n$$ and $$P_{a_i, w}(X) = \sum_{n=1}^{\# \: a_i \: \textrm{in} \: w} p_i(n) X^n\,.$$ Then, we have the following recursive formulas: **Proposition 20**. *Let $\sigma$ be a substitution on $\mathcal A= \{a_1,\dots, a_k\}$. For $m\in\mathbb N$ and $1\leq i,j\leq k$, if $\sigma(a_j) = b_1\dots b_l$, then $$C_{a_i, \sigma^m(a_j)}(X) \ = \ \sum_{r=1}^{|\sigma(a_j)|} X^{|\sigma^{m-1}(b_1\cdots b_{r-1})|}C_{a_i, \sigma^{m-1}(b_r)}(X)$$ and $$P_{a_i, \sigma^m(a_j)}(X) \ = \ \sum_{r=1}^{|\sigma(a_j)|} X^{|\sigma^{m-1}(b_1\cdots b_{r-1})|}\left( |\sigma^{m-1}(b_1\cdots b_{r-1})|\frac{1-X^{1+\deg(P_{a_i,\sigma^{m-1}(b_r)})}}{1-X} + P_{a_i, \sigma^{m-1}(b_r)}(X)\right)$$0◻* Let us note here in passing that all numbers $|\sigma(a_j)|, |\sigma^{m-1}(b_1\cdots b_{r-1})|$ and $\deg(P_{a_i,\sigma^{m-1}(b_r)})$ can be written explicitly in terms of entries of $A_\sigma$ and its powers. Both of the above equalities are straightforward conversions of $\sigma$ into polynomials. A careful proof of the Fibonacci case can be found in [@PP], noting that they start their polynomials from the $X$ term and not the constant term. As we have seen, any one-sided infinite fixed point $w$ of $\sigma$ will be of the form $$w = \lim_{m\rightarrow \infty} \sigma^m(a_1),$$ after possibly replacing $\sigma$ with a power and shuffling the alphabet, where convergence is in the product topology. This implies that for $1\leq i\leq k$ $$C_{a_i}(X) = \lim_{m\rightarrow \infty} C_{a_i, \sigma^m(a_1)}(X)$$ in the product topology of ${\mathbb Q}[[X]]$ and pointwise in $(-1,1)$. In fact, the convergence is uniform on each interval $[-r,r]$ for all $0 < r <1$. ## Geometric realisation Recall that substitutions give rise to one-dimensional tilings of the real line (or the half line in our one-sided case). Chapter 5 and 6 of [@TAO] give a very thorough development of this subject (compare [@AA]). Here we provide enough detail to support our results. Let $\sigma$ be a substitution on the alphabet $\mathcal A=\{a_1, a_2, \ldots, a_k \}$. Replace each letter $a_j$ in the alphabet by an interval $I_j$. The substitution leads to an inflation rule, or substitution tiling, if there is a common inflation factor $\lambda$ such that for each $1\leq j\leq k$, $\lambda |I_j|$ is equal to the lengths of the intervals arising from $\sigma(a_j)$. If $\sigma$ is primitive then the PF-eigenvalue is the inflation factor and the lengths of the $I_j$ are given by the left eigenvector of $A_\sigma$ for $\lambda_{PF}$. **Remark 21**. *It is of note that every inflation factor of a primitive substitution is always algebraic, since it arises as the PF-eigenvalue of an integer matrix. If one moves to a substitution tiling using an infinite number of tiles, then transcendental inflation factors are possible [@FGM].* Now suppose that $w$ is a fixed one-sided infinite word for $\sigma$. Replace each letter $a_j$ in the alphabet by a translated copy of the interval $I_j$. Position these intervals to start at the origin, proceed to the right and have no gaps, and let $$\Lambda =\{ 0=t_0<t_1 < \ldots < t_n < \ldots \}$$ be the left-end points of these intervals. We will call this point set the geometric realisation of $w$. As before we have a number of choices for generating functions. One reasonable option is $$G(X) \ = \ \sum_{n=0}^\infty t_n X^n \,.$$ We have the following simple lemma: **Lemma 22**. *If $g: \mathcal A\to {\mathbb C}, \: g(a_j)=|I_j|$, then $$t_n = \sum_{i=0}^{n-1}g(w_i)\, , \quad \textrm{and}$$ $$G(X) \ = \ \frac{X}{1-X}C_g(X) \ = \ \sum_{j=1}^k \frac{X}{1-X} |I_j| C_{a_j}(X) \,.$$* *Proof.* By construction, $t_n$ is the left-end point of the $n$th interval. Since $t_0=0$ this means that $t_n$ is equal to the sum of lengths of the first $n-1$ intervals and the formula follows. This implies that the summatory series of $C_g(X)$ is $$\frac{1}{1-X}C_g(X) = \sum_{n=0}^\infty t_{n+1}X^n.$$ Therefore, multiplying by $X$, which shifts the sequence of coefficients, yields the desired result. ◻ As a consequence, the next proposition immediately follows from the last lemma and from Proposition [Proposition 8](#prop:1){reference-type="ref" reference="prop:1"}. **Proposition 23**. *Let $w$ be a fixed word of a substitution and let $g: \mathcal A\to {\mathbb C}, \: g(a_j)=|I_j|$. Suppose all interval lengths are rational. Then the interval length sequence $\{g(w_n)\}$ is not eventually periodic if and only if $G(X)$ is transcendental over ${\mathbb Q}(X)$. 0◻* The next result is important for substitution tilings on two letters, as in this case the lengths of the intervals can be chosen to be algebraic numbers. **Theorem 24**. *If the alphabet $\mathcal A$ has two letters and the tile lengths are algebraic numbers, then exactly one of the following holds.* - *$|I_1|=|I_2|$. In this case $$G(X)= \frac{|I_1|X}{(1-X)^2} \,.$$* - *$|I_1| \neq |I_2|$ but $w$ is eventually periodic. Then, there exists some $P \in {\mathbb Q}[X]$ and $d \in \mathbb N$ such that $$G(X)=(|I_1|-|I_2|)\frac{XP(X)}{(1-X)(1-X^d)}+ |I_1|\frac{X}{(1-X)^2} \,.$$* - *$|I_1| \neq |I_2|$ but $w$ is not eventually periodic. Then $G(X)$ is transcendental over ${\mathbb Q}(X)$.* *Proof.* This follows directly from Proposition [Proposition 9](#prop:complex){reference-type="ref" reference="prop:complex"}. For the extra detail in (b) we use the formula from Lemma [Lemma 22](#lem:geometric){reference-type="ref" reference="lem:geometric"}, the proof of Proposition [Proposition 9](#prop:complex){reference-type="ref" reference="prop:complex"} and Proposition [Proposition 4](#fact2){reference-type="ref" reference="fact2"}. ◻ One could also split the geometric realization generating function $G(X)$ into its constituent letters. This would combine aspects of the characteristic and position generating functions. # The Fibonacci substitution As we have seen, the structure of a substitution passes to its generating functions. This will be used to give us precise information about how the characteristic and position generating functions are related and information about the roots of these functions. While this can be done for any substitution we will focus our efforts on the Fibonacci substitution, Example [Example 18](#ex:fib){reference-type="ref" reference="ex:fib"}. First, we need some extra language to deal with the structure of the Fibonacci substitution. **Definition 25**. *Let $A_n$ and $B_n$ denote the $n$-level supertiles of the Fibonacci substitution where $$\begin{aligned} A_n&=\sigma^n(a)\\ B_n&=\sigma^n(b)=A_{n-1}.\end{aligned}$$* We can now give the following formula relating the position function of each letter to the summatory function of $1_a$: **Proposition 26**. *For the Fibonacci substitution $\sigma$ on $\mathcal A= \{a,b\}$ and the one-sided infinite fixed word $w = \lim_{m\rightarrow \infty} \sigma^m(a)$, we have $$p_a(n)=n-2+\sum_{j=0}^{n-1} 1_{a}(w_j) \,.$$ and $$p_b(n)=2n-2+\sum_{j=0}^{n-1} 1_a(w_j) \,.$$* *Proof.* Consider the level one supertiles $A_1 = ab$ and $B_1=a$. The substitution allows us to write $$abaababaabaab \ldots \ =\ w \ =\ \sigma(w) \ =\ A_1B_1A_1A_1B_1A_1B_1A_1A_1B_1A_1A_1B_1 \ldots$$ Now, both $A_1, B_1$ start with an $a$ and contain exactly one $a$. Thus, the $n^{th}$ $a$ appears at the beginning of the $n^{th}$ level one supertile. There are $n-1$ supertiles before the $n^{th}$ $a$. Each supertile contains a letter, and we have to add an extra spot for each $A$ supertile. Therefore, there are $n-1+\sum_{j=1}^{n-1} 1_a(w_j)$ positions before the $n^{th}$ $a$. However, the positions start at 0, so it follows that $$p_a(n)= n-1+\sum_{j=0}^{n-1} 1_a(w_j) - 1 \,.$$ Next, consider the level two supertiles $$A_2=aba, \ B_2=ab$$ Each of these has exactly one $b$ which appears in the second position. Hence, the $n^{th}$ $b$ appears on the second position of the $n^{th}$ level two supertile. There are $n-1$ supertiles before the $n^{th}$ $b$. Each supertile contains two letters, and we have to add an extra spot for each $A$ supertile. Therefore, there are $2(n-1)+\sum_{j=0}^{n-1} 1_{a}(w_j)$ positions before the $n^{th}$ $b$. Again, since we start at position 0 the b is in position 1, it follows that $$p_b(n)= 2(n-1)+\sum_{j=0}^{n-1} 1_{a}(w_j) \,.\qedhere$$ ◻ **Remark 27**. *Let $S_a(n)=\sum_{j=0}^{n-1} 1_{a}(w_j)$. Then, for all $n$ we have: $$\begin{aligned} p_a(n)&=n-2+S_a(n-1) \\ S_a(p_a(n))&=n \,.\end{aligned}$$ In particular, we have $$S_a(n-2+S_a(n-1))=n \qquad \forall n \in \mathbb N\,.$$* As an immediate consequence, we get the following relations between the polynomials $P_a, P_b$ and $C_a$. **Corollary 28**. *$$\begin{aligned} P_a(X)& \ = \ \frac{X}{(1-X)^2} - \frac{2X}{1-X} + \frac{X}{1-X}C_a(X) \\ P_b(X)& \ =\ \frac{2X}{(1-X)^2} - \frac{2X}{1-X} + \frac{X}{1-X}C_a(X)\end{aligned}$$ In particular, $P_b(X)-P_a(X)= X/(1-X)^2$ is rational.* Before moving to the study of roots of $C_a(X)$, let us briefly discuss the generating function of the geometric realisation. Starting from the one-sided fixed word $abaababa\ldots$, replacing each $a$ letter by an interval of lenght $\tau=\frac{1+\sqrt{5}}{2}$ and each $b$ letter by an interval of length $1$, and picking the left end point of this interval we end up with a sequence $$\Lambda = \{ 0 < \tau < \tau+1 <2 \tau+1 < 3 \tau +1 < \ldots \}$$ representing the geometric realisation of the one-sided Fibonacci word. This can also be described as the Fibonacci model set intersected with the positive real axis. The generating function of the geometric realisation is then $$G(X)= \tau X + (\tau+1)X^2+ (2\tau+1)X^3+ (3 \tau+1)X^4+ \ldots \,.$$ Now, [\[eq12\]](#eq12){reference-type="eqref" reference="eq12"} and Lemma [Lemma 22](#lem:geometric){reference-type="ref" reference="lem:geometric"} give $$\begin{aligned} G(X) &= \frac{X}{1-X} \left( \tau C_{a}(X) + C_{b}(X)\right) \\ &= \frac{X}{1-X} \left( \tau C_{a}(X) + \frac{1}{1-X}-C_{a}(X)\right) \\ &= \frac{X}{(1-X)^2} + \frac{(\tau-1)X}{1-X} C_{a}(X) \,.\end{aligned}$$ ## The roots of the characteristic generating function. Now we turn to the roots of $C_a(X)$. Consider $$\begin{aligned} A_{3}&=\sigma^3(a)=\sigma^2(ab)=\sigma(aba)=abaab\\ B_{3}&=\sigma^3(b)=\sigma^2(a)=\sigma(ab)=aba\, ,\end{aligned}$$ the level 3 supertiles of the Fibonacci substitution. Note that both of these have odd lengths, with $|A_3|=5$ and $|B_3|=3$. The level 3 supertiles are the first level where both $A_n$ and $B_n$ have the same parity. As the odd parity of both $A_n$ and $B_n$ repeats every three levels, this fact is extended to the $3n$-level supertiles $A_{3n}$ and $B_{3n}$, which will be of importance for the remainder of the section. Due to the recursive nature of substitutions, $\sigma$ can be applied to $3n$-level supertiles giving $$\begin{aligned} A_{3n+3}&=A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}\\ B_{3n+3}&=A_{3n}B_{3n}A_{3n} \,.\end{aligned}$$ Now using the information stated above, consider the one-sided infinite Fibonacci word made up of $3n$-level supertiles: $$\begin{aligned} w = \lim_{m\rightarrow \infty} \sigma^m(a) = A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}A_{3n}B_{3n}A_{3n}\ldots \,.\end{aligned}$$ Split the infinite word into pairs of two consecutive $3n$-level supertiles, noting that $B_{3n}B_{3n}$ can not appear as $bb$ never occurs in $w$. The following definition states possible pairings. **Definition 29**. *Let the following represent the pairings of the $3n$-level supertiles found in the Fibonacci substitution: $$\begin{aligned} R_n=A_{3n}B_{3n}\\ S_n=A_{3n}A_{3n}\\ T_n=B_{3n}A_{3n} \,.\end{aligned}$$ Then their lengths are $$\begin{aligned} |R_n|&=f_{3n+2}+f_{3n+1}=f_{3n+3}\\ |S_n|&=f_{3n+2}+f_{3n+2}=2f_{3n+2}\\ |T_n|&=f_{3n+1}+f_{3n+2}=f_{3n+3} \,.\end{aligned}$$* Note that every third number in the Fibonacci sequence is even. Thus $|R_n|$, $|S_n|$, and $|T_n|$ are even. Similar to Proposition [Proposition 20](#prop:recursive){reference-type="ref" reference="prop:recursive"} for finite words $w_1,w_2$ in $\mathcal A= \{a,b\}$ and $f \in \mathcal A$, we have $$\begin{aligned} C_{f, w_1w_2}(X)=C_{f, w_1}(X)+X^{|w_1|}C_{f, w_2}(X)\,.\end{aligned}$$ The recursive structure of these polynomials will be heavily used. **Proposition 30**. *There exist sequences of even integers $j_k,l_k,m_k$ such that $$\begin{aligned} C_{a}(X)=C_{a,{R_n}}(X)\sum_{k=0}^{\infty}X^{j_k}+C_{a,{S_n}}(X)\sum_{k=0}^{\infty}X^{l_k}+C_{a,{T_n}}(X)\sum_{k=0}^{\infty}X^{m_k}\end{aligned}$$* *Proof.* As stated above, the one-sided infinite fixed point of the Fibonacci substitution can be described in terms of its $3n$-supertiles: $$\begin{aligned} w & = \lim_{m\rightarrow \infty} \sigma^m(a) \\ & = \lim_{m\rightarrow \infty} \sigma^m(A_{3n}) \\ & = A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}A_{3n}B_{3n}A_{3n}\ldots \\ & = R_nS_nT_nT_n\ldots \qquad \,. \end{aligned}$$ Thus, $$C_a(X) \ = \ C_{a,R_n}(X) + X^{|R_n|}C_{a,S_n}(X) + X^{|R_n|+|S_n|}C_{a,T_n}(X) + X^{|R_n|+|S_n|+|T_n|}C_{a,T_n}(X)+\cdots$$ Therefore, since $|R_n|, |S_n|$ and $|T_n|$ are all even, the result follows. ◻ The great advantage of this description of $C_a(X)$ is that we can get some information about its roots since we can analyze polynomials in greater detail than power series in general. It is immediate without any finer theory that $C_a(X) \geq 1$ on $[0,1)$. The following are the results obtained by taking increasingly larger supertiles of $R_n$, $S_n$ and $T_n$ and using their generating functions to obtain a better understanding of the roots of $C_a(X)$ on the interval $(-1,0)$. **Example 31**. *Let $n=1$, then $$\begin{aligned} R_1&\ =\ A_3B_3\ =\ abaababa\\ S_1&\ =\ A_3A_3\ =\ abaababaab\\ T_1&\ =\ B_3A_3\ =\ abaabaab\end{aligned}$$ which gives $$\begin{aligned} C_{a, {R_1}}(X)&\ =\ 1+X^2+X^3+X^5+X^7\\ C_{a, {S_1}}(X)&\ =\ 1+X^2+X^3+X^5+X^7+X^8=(1+X^2+X^3)(1+X^5)\\ C_{a, {T_1}}(X) &\ =\ 1+X^2+X^3+X^5+X^6=(1+X^2)(1+X^3)+X^6 \,.\end{aligned}$$ In this case, $C_{a, {S_1}}(X)>0$, $C_{a, {T_1}}(X)>0$ on $(-1,0)$ and $C_{a, {R_1}}(X)>0$ on $(-0.901593,0)$. Note that all results are verifiable via Wolfram Alpha/Maple/Mathematica. Thus, by Proposition [Proposition 30](#prop:newtiling){reference-type="ref" reference="prop:newtiling"} $$C_a(X)>0, \ \ X\in(-0.901593,1)\,.$$* **Example 32**. *Let $n=2$, then $$\begin{aligned} R_2&\ =\ A_6B_6\ =\ abaababaabaababaababaabaababaabaab\\ S_2&\ =\ A_6A_6\ =\ abaababaabaababaababaabaababaabaababaababa\\ T_2&\ =\ B_6A_6\ =\ abaababaabaababaababaabaababaababa\end{aligned}$$ which gives $$\begin{aligned} C_{a,{R_2}}(X) &=1+X^2+X^3+X^5+X^7+X^8+X^{10}+X^{11}+X^{13}+X^{15}+X^{16}+X^{18}+X^{20}+X^{21}+X^{23}\\ &\ \ +X^{24}+X^{26}+X^{28}+X^{29}+X^{31}+X^{32}\\ C_{a,{S_2}}(X) &=1+X^2+X^3+X^5+X^7+X^8+X^{10}+X^{11}+X^{13}+X^{15}+X^{16}+X^{18}+X^{20}+X^{21}+X^{23}\\ &\ \ +X^{24}+X^{26}+X^{28}+X^{29}+X^{31}+X^{32}+X^{34}+X^{36}+X^{37}+X^{39}+X^{41}\\ C_{a,{T_2}}(X) &=1+X^2+X^3+X^5+X^7+X^8+X^{10}+X^{11}+X^{13}+X^{15}+X^{16}+X^{18}+X^{20}+X^{21}+X^{23}\\ &\ \ +X^{24}+X^{26}+X^{28}+X^{29}+X^{31}+X^{33} \,.\end{aligned}$$ Therefore, $C_{a,{R_2}}(X)>0$ and $C_{a,{S_2}}(X)>0$ on $(-1,0)$ while $C_{a,{T_2}}(X)>0$ on $(-0.951699,0)$; giving that $$C_a(X)>0, \ \ X\in(-0.951699,1)\,.$$* These polynomials become hard to manage quite quickly. Hence, we should look at their recursive structure. **Proposition 33**. *For $n\in\mathbb N$, $$\begin{aligned} R_{n+1}&\ = \ R_nS_nT_nT_n\\ S_{n+1}&\ =\ R_nS_nT_nT_nR_n\\ T_{n+1}&\ =\ R_nS_nT_nR_n\end{aligned}$$ which gives $$\begin{aligned} C_{a,{R_{n+1}}}(X)&=C_{a,{R_n}}(X)+X^{f_{3n+3}}C_{a,{S_n}}(X)+X^{f_{3n+3}+2f_{3n+2}}C_{a,{T_n}}(X)(1+X^{f_{3n+3}})\\ C_{a,{S_{n+1}}}(X)&=C_{a,{R_n}}(X)+X^{f_{3n+3}}C_{a,{S_n}}(X)+X^{f_{3n+3}+2f_{3n+2}}C_{a,{T_n}}(X)(1+X^{f_{3n+3}})+X^{3f_{3n+3}+2{f_{3n+2}}}C_{a,{R_n}}(X)\\ &=C_{a,{R_{n+1}}}(X)+X^{3f_{3n+3}+2f_{3n+2}}C_{a,{R_n}}(X)\\ C_{a,{T_{n+1}}}(X)&=C_{a,{R_n}}(X)+X^{f_{3n+3}}C_{a,{S_n}}(X)+X^{f_{3n+3}+2f_{3n+2}}C_{a,{T_n}}(X)+X^{2f_{3n+4}}C_{a,{R_n}}(X) \,. \end{aligned}$$* *Proof.* Everything follows by the following supertile calculations: $$\begin{aligned} R_{n+1}&=A_{3n+3}B_{3n+3}=A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}A_{3n}B_{3n}A_{3n}\\&=R_nS_nT_nT_n\\ S_{n+1}&=A_{3n+3}A_{3n+3}=A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}\\&=R_nS_nT_nT_nR_n\\ T_{n+1}&=B_{3n+3}A_{3n+3}=A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}A_{3n}A_{3n}B_{3n}\\&=R_nS_nT_nR_n \,.\end{aligned}$$ ◻ Note that the first part of this proposition is really giving a new substitution in three letters $\{r,s,t\}$: $$\begin{aligned} r & \mapsto rstt \\ s & \mapsto rsttr \\ t & \mapsto rstr \,.\end{aligned}$$ This has the substitution matrix $$A \ = \ \left[\begin{matrix} 1 & 1 & 2 \\ 2 & 1 & 2 \\ 2& 1 & 1 \end{matrix}\right]$$ which is certainly primitive and has $\lambda_{PF} = 2 + \sqrt 5$. The fact that this substitution is obtained from $\sigma^3$, where $\sigma$ is still the Fibonacci substitution, is recorded in the fact that $2+\sqrt 5 = \left(\frac{1+\sqrt 5}{2}\right)^3$. **Example 34**. *With this recursion one can confidently calculate the behaviour of $n=3$ and $n=4$ but the polynomials become very large. We verified with Maple that $C_{a,{S_3}}(X)>0$ and $C_{a,{T_3}}(X)>0$ on $(-1,0)$ and $C_{a,{R_3}}(X)>0$ on $(-0.99436269,0)$, giving that $$C_a(X)>0, \ \ X\in(-0.951699,1)\,.$$* *Similarly, $C_{a,{R_4}}(X)>0$ and $C_{a,{S_4}}(X)>0$ on $(-1,0)$ while $C_{a,{T_4}}(X)>0$ on $(-0.99729758,0)$ giving that $$C_a(X)>0, \ \ X\in(-0.99729758,1)\,.$$ Note that the degrees of these polynomials are $609, \ 752$, and $608$ respectively, easily found by the formulas in Definition [Definition 29](#def:triplefib){reference-type="ref" reference="def:triplefib"}. By the previous proposition, the number of terms and the degrees of these polynomials are growing approximately by $2+\sqrt 5$. Perhaps one could computationally squeeze out another round or two but certainly not too many more.* Let us now state what will happen when one looks at larger and larger values of $n$. **Proposition 35**. *For $n\in \mathbb N$, let $$\alpha_n=inf\big\{ s\in[-1,0)\ :\ C_{a,{R_n}}(X)>0, C_{a,{S_n}}(X)>0, C_{a,{T_n}}(X)>0, \hspace{0.3cm}\forall X\in(s,0)\big\}\,.$$ Then $\alpha_{n+1} \leq \alpha_n$. Moreover, $\alpha_n = \alpha_{n+1}$ if and only if one of the following two situations happens:* - *$\alpha_m=-1$ for all $m\geq n$. In this case $C_a(X)>0$ on $(-1,1)$.* - *$\alpha_m = \alpha_n > -1$ for all $m\geq n$ and $$C_{a,{R_m}}(\alpha_m)=C_{a,{S_m}}(\alpha_m)=C_{a,{T_m}}(\alpha_m)=0\,.$$ In this case, $$\begin{aligned} C_a(\alpha_n) = 0 \ \ \textrm{and} \ \ C_a(X)>0, \ \forall X \in (-\alpha_n,0). \end{aligned}$$* *Proof.* By the recursive formulas for $C_{a,R_n}(X), C_{a,S_n}(X)$ and $C_{a,T_n}(X)$ given in Proposition [Proposition 33](#prop:recursiveRST){reference-type="ref" reference="prop:recursiveRST"} we see that $$C_{a,R_n}(X), C_{a,S_n}(X), C_{a,T_n}(X) \geq 0 \ \ \Rightarrow \ \ C_{a,R_{n+1}}(X), C_{a,S_{n+1}}(X), C_{a,T_{n+1}}(X) \geq 0\,.$$ If all three of the polynomials on the left are strictly positive at $X$ then all three polynomials on the right are strictly positive at $X$. This implies that $\alpha_{n+1} \leq \alpha_n$. In the same way, if at least one of the polynomials on the left is strictly positive at $X$ and the others are non-negative at $X$, then all three polynomials on the right are strictly positive at $X$. Hence, the only way $\alpha_{n+1} = \alpha_n > -1$ is if all three polynomials become zero at the same time, $$C_{a,{R_n}}(\alpha_n)=C_{a,{S_n}}(\alpha_n)=C_{a,{T_n}}(\alpha_n)=0\,.$$ Thus, by Proposition [Proposition 30](#prop:newtiling){reference-type="ref" reference="prop:newtiling"} this implies that $C_a(\alpha_n) = 0$ and $C_a(X) > 0$ on $(\alpha_n,1)$. Lastly, by the recursive formulas we have for all $m>n$ $$C_{a,{R_m}}(\alpha_m)=C_{a,{S_m}}(\alpha_m)=C_{a,{T_m}}(\alpha_m)=0\,.\qedhere$$ ◻ With all of this evidence we make the following conjecture: **Conjecture 36**. *$C_a(X) > 0$ on $(-1,1)$.* # Acknowledgements {#acknowledgements .unnumbered} A.P. was supported by the NSERC USRA grant 2023-581566, A.P. and C.R. were supported by the NSERC Discovery grant 2019-05430, and N.S. was supported by the NSERC Discovery grant 2020-00038. 99 S. Akiyama and P. Arnoux (eds.), *Substitution and Tiling Dynamics: Introduction to Self-inducing Structures*, Lecture Notes in Mathematics **2273**, Springer,Cham, 2020. M. Baake and U. Grimm, *Aperiodic Order. Vol. 1: A Mathematical Invitation*, Cambridge University Press, Cambridge, 2013. J. Bell, N. Bruin, and M. Coons, *Transcendence of generating functions whose coefficients are multiplicative*, Trans. Amer. Math. Soc. **364** (2012), 933-959. J. Bell and S. Chen, *Power series with coefficients from a finite set*, J. Combin. Theory Ser. A **151** (2017), 241-253. P. Borwein and M. Coons, *Transcendence of power series for some number theoretic functions*, Proc. Amer. Math. Soc. **137** (2009), 1303-1305. M. Bousquet-Melou, *Walks on the slit plane: other approaches*, Advances in Applied Mathematics **27** (2001), 243-288. F. Durand, N. Ormes, and S. Petite, *Self-induced systems* J. Anal. Math. **135** (2018), 725--756. P. Fatou, *Séries trigonométriques et séries de Taylor*, Acta Math. **30** (1906), no.1, 355-400. D. Frettlöh, A. Garber, and N. Mañibo, *Substitution tilings with transcendental inflation factor*, preprint, (2022), `arXiv:2208.01327`. N. Phan and A. Pouti, *Generating functions related to the Fibonacci substitution*, MacEwan University Student eJournal **7** (2023). M. Queffélec, *Substitution Dynamical Systems -- Spectral Analysis (2nd ed.)*, Springer, Berlin, 2010. M. Rubey, *Transcendence of generating functions of walks on the slit plane*. In: Mathematics and Computer Science III. Trends in Mathematics. Birkhäuser, Basel, 2004.

arxiv_math

--- abstract: | We develop a theory of generalized solutions of the nonlinear evolution equations for complex-valued functions of a real positive time variable and $p$-adic spatial variable, which can be seen as non-Archimedean counterparts of the fractional porous medium equation. In this case, we face the problem that a $p$-adic ball is simultaneously open and closed, thus having an empty boundary. To address this issue, we use the algebraic structure of the field of $p$-adic numbers and apply the Pontryagin duality theory to construct the appropriate fractional Sobolev type spaces. We prove the existence and uniqueness results for the corresponding nonlinear equation and define an associated nonlinear semigroup. author: - | **Alexandra V. Antoniouk**\ Institute of Mathematics of the National Academy of Sciences of Ukraine,\ Tereshchenkivska 3, Kyiv, 01024 Ukraine,\ American University Kyiv\ Poshtova Sq 3, Kyiv, 04070 Ukraine E-mail: antoniouk.a\@gmail.com - | **Anatoly N. Kochubei**\ Institute of Mathematics of the National Academy of Sciences of Ukraine,\ Tereshchenkivska 3, Kyiv, 01024 Ukraine,\ E-mail: kochubei\@imath.kiev.ua - | **Oleksii L. Nikitchenko**\ The Ohio State University,\ 231 W 18th Ave, Columbus, OH 43210,\ Kyiv Academic University of the National Academy of Sciences of Ukraine\ Vernadsky blvd., 36, 03142, Kyiv, Ukraine\ E-mail: oleksiinikitchenko94\@gmail.com title: Existence and uniqueness for $p$-adic counterpart of the porous medium equation --- **Keywords:** fractional differential operator, $p$-adic analysis, porous medium equation, maximal monotone operators **MSC 2010**. Primary: 35S10; 47J35. Secondary: 11S80; 60J25; 76S05. # Introduction By Ostrowski's theorem [@Koblitz:1984 Thm. 1, Ch. I], the field of rational numbers $\mathbb{Q}$ permits only two essentially different topological completion: with respect to the metric topology induced by the usual absolute value of $x\in\mathbb{Q}$ or ultrametric topology induced by non-Archimedean $p$-adic absolute value. As a result, there exist two parallel mathematical worlds that evolve entirely independently. While classical differential equation theory, which is founded on the analysis of functions on the real (complex) number field, has already yielded a broad array of mathematical methods and concepts, the theory of such equations in $p$-adic function spaces is not as deeply developed. Simultaneously, as proposed by Stephen Hawking and other authors, it is suggested that the intricate nature of spacetime at smaller scales could exhibit fractal characteristics, including the possibility of being $p$-adic [@GibHaw:1993; @Not:1993; @WheelerFord:1998; @Volovich:2010]. Therefore, the development of mathematical foundations for such objects is considered exceptionally important. While the linear theory of such equations has already seen some development (see e.g. [@VVZ; @Ko:2001; @ZG] and the references therein), then nonlinear equations in $p$-adic functional spaces require further development of corresponding techniques. It appears that the first papers in this direction have only appeared relatively recently [@Ko:2018; @AKK]. At the same time, there is a lot of unsolved questions. A non-Archimedean counterpart of the fractional porous medium equation, that is the equation $$\label{1-1}\ D_tu+D^\alpha\big(\varphi(u)\big)=0,\ \ u=u(t,x),\ \ t>0,\ x\in\mathbb Q_p,$$ was first studied in the paper [@Ko:2018]. Here $\mathbb Q_p$ is the field of $p$-adic numbers, $D^\alpha$, $\alpha>0$ is Vladimirov's fractional differentiation operator $$\label{DA}\ \big(D^\alpha u\big) (x)=\frac{1-p^\alpha}{1-p^{-\alpha-1}}\int\limits_{\mathbb Q_p}\dfrac{u(x-y)-u(x)}{\vert y\vert_p^{\alpha+1}}\,dy$$ acting on complex-valued functions on $\mathbb Q_p$. Above in [\[1-1\]](#1-1){reference-type="eqref" reference="1-1"} $\varphi$ is a strictly monotone increasing smooth continuous function, such that $\varphi^\prime>0$, $\varphi(\pm\infty)=\pm\infty$ and $\varphi(0)=0$. We keep in mind the example $\varphi(u)=\vert u\vert^{m-1}u$, $m>0$. Equation [\[1-1\]](#1-1){reference-type="eqref" reference="1-1"} is a nonlinear evolution equation for complex-valued functions of a real positive time variable and a $p$-adic spatial variable. The additional complexity for such problem lies in the non-local nature of the corresponding pseudo-differential operator. Even ordinary non-local problems for real-variable functions are the subject of modern challenging research (see e.g. [@Vazquez:2007; @Bonforte:2015] and references therein). Developing an $L^1$-theory of Vladimirov's $p$-adic fractional differentiation operator, in [@Ko:2018] it was proved the $m$-accretivity of the corresponding nonlinear operator and obtained the existence and uniqueness of a mild solution. That was motivated both by the existence of a $p$-adic model of a porous medium [@KOJ] and by the fact that the equation ([\[1-1\]](#1-1){reference-type="ref" reference="1-1"}) was the first natural example of a strongly nonlinear $p$-adic pseudo-differential equation. In this paper, we study a stronger notion of solvability for the Cauchy problem on the $p$-adic ball, which serves as a model for a bounded open set in the $p$-adic context: $$\label{1-2} \left\{ \begin{array}{lc} D_tu + D^\alpha_N\big(\varphi(u)\big)=0,&t\in[0,T];\\ u(0)=u_0,& \end{array} \right.$$ with the Vladimirov operator $D^\alpha_N$ on the $p$-adic ball $$B_N=\{x\in \mathbb Q_p\colon\Vert x\Vert_p\leq p^N\}$$ defined, as in [@VVZ] by restricting the Vladimirov operator $D^\alpha$ to functions $u_N$ supported in the ball $B_N$ and considering the resulting function $D^\alpha u_N$ only on the ball $B_N$. We prove the existence and uniqueness of generalized solutions of the problem ([\[1-2\]](#1-2){reference-type="ref" reference="1-2"}), developing with the appropriate modifications, the method by Brezis [@Brezis:1971]; see also [@Barbu:2010]. A necessary prerequisite having independent interest is the investigation of a "proper" $p$-adic analog for Sobolev spaces, which is suitable for the study of boundary value problems (BVP) in the non-Archimedean case. Sobolev spaces play an extraordinary role in the theory of partial differential equations since they give a natural and important tool for the investigation of different properties of BVP. For the $p$-adic analysis, different authors [@Taib; @RVZG:2010; @GorkaKostrRey:2014; @GorkaKostr:2015; @GorkaKost:2020] proposed suitable candidates for the role of Sobolev spaces based on the analogy or in the framework of pseudodifferential equations approach. At the same time, there is a fundamental difference between the study of boundary value problems in the Archimedean and non-Archimedean situations: the finite ball in $p$-adic field has no boundary in the usual sense. That's why the study of $p$-adic BVP requires elaboration of a special and natural approach to the description of the spaces, which should play the role of the analog to the Sobolev spaces in usual PDE analysis. For this purpose, in Section [3](#sec3){reference-type="ref" reference="sec3"} we prove the equivalence of different analogs of $p$-adic Sobolev spaces and their connection with the domain of operator restricted to a $p$-adic ball. The investigation of this case is based on a purely non-Archimedean effect -- a $p$-adic ball centered at the origin is an additive group, so that we can use the harmonic analysis provided by the Pontryagin duality theory. Building upon this developed technique, we can establish our main result Theorem [Theorem 8](#main){reference-type="ref" reference="main"} on the existence and uniqueness of the weak solution to the nonlinear boundary value problem in $p$-adic ball and state the properties of the corresponding nonlinear semigroup. # Preliminaries *2.1. $p$-Adic numbers* [@Ko:2001; @VVZ]. Let $p$ be a prime number. The field of $p$-adic numbers is the completion $\mathbb Q_p$ of the field $\mathbb Q$ of rational numbers, with respect to the absolute value $|x|_p$ defined by setting $|0|_p=0$, $$|x|_p=p^{-\nu }\ \mbox{if }x=p^\nu \frac{m}n,$$ where $\nu ,m,n\in \mathbb Z$, and $m,n$ are prime to $p$. $\mathbb Q_p$ is a locally compact topological field. Note that by Ostrowski's theorem there are no absolute values on $\mathbb Q$, which are not equivalent to the "Euclidean" one, or one of $|\cdot |_p$. The absolute value $|x|_p$, $x\in \mathbb Q_p$, has the following properties: $$\begin{gathered} |x|_p=0\ \mbox{if and only if }x=0;\\ |xy|_p=|x|_p\cdot |y|_p;\\ |x+y|_p\le \max (|x|_p,|y|_p).\end{gathered}$$ The latter property called the ultra-metric inequality (or the non-Archimedean property) implies the total disconnectedness of $\mathbb Q_p$ in the topology determined by the metric $|x-y|_p$, as well as many unusual geometric properties. Note also the following consequence of the ultra-metric inequality: $$|x+y|_p=\max (|x|_p,|y|_p)\quad \mbox{if }|x|_p\ne |y|_p.$$ The absolute value $|x|_p$ takes the discrete set of non-zero values $p^N$, $N\in \mathbb Z$. If $|x|_p=p^N$, then $x$ admits a (unique) canonical representation $$\label{2.1}\ x=p^{-N}\left( x_0+x_1p+x_2p^2+\cdots \right) ,$$ where $x_0,x_1,x_2,\ldots \in \{ 0,1,\ldots ,p-1\}$, $x_0\ne 0$. The series converges in the topology of $\mathbb Q_p$. For example, $$-1=(p-1)+(p-1)p+(p-1)p^2+\cdots ,\quad |-1|_p=1.$$ The *fractional part* of element $x\in\mathbb Q_p$ in canonical representation [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"} is given by: $$\{x\}_p= \left\{\begin{array}{ll} 0,&\text{if}\quad N\leq 0 \ \ \ \text{or}\ \ \ x=0;\\ p^{-N}\big(x_0+x_1p+\ldots+x_{N-1}p^{N-1}\big),&\text{if} \quad N>0. \end{array} \right.$$ The function $\chi(x)=\exp(2\pi i\{x\}_p)$ is an additive character of the field $\mathbb Q_p$, i.e. the character of its additive group. Let us remark that $\chi(x) = 1$ if and only if $\vert x\vert_p\leq 1$. We denote by $dx$ the Haar measure on the additive group $\mathbb Q_p$ normalized on $\mathbb{Z}_p$ with the requirement: $\int_{\mathbb{Z}_p} dx = 1$, where $\mathbb Z_p= \{x\in\mathbb Q_p\colon\vert x\vert_p\leq 1\}$ is the unit ball in $\mathbb Q_p$. *2.2. Fourier transformation and distributions on $\mathbb Q_p$*. Let us denote by $\mathcal{D}(\mathbb Q_p)$ the vector space of *test functions*, locally constant functions with compact supports. Recall that a function $\psi: \mathbb Q_p\to \mathbb{C}$ is *locally constant* if there exist such an integer $\ell \geq 0$ that for any $x\in\mathbb Q_p$ $$\psi (x+y)=\psi (x),\quad \text{if}\quad \Vert y\Vert_p\leq p^{-\ell},\quad \text{($\ell$ is independent on $x$).}$$ The smallest number $\ell$ with this property is called *the exponent of constancy of the function $\psi$.* Typical examples of locally constant functions are additive characters, and also cutoff functions like $\Omega(\Vert x\Vert_p)$, where $$\Omega(t)=\left\{\begin{array}{ll} 1,& \ \text{if}\ \ 0 \leq t \leq 1;\\ 0,& \ \text{if}\ \ t >1. \end{array}\right.$$ It is worth remarking that $\Omega$ is continuous, which is an expression of the non-Archimedean properties of $\mathbb Q_p$. Note also that $\mathcal{D}(\mathbb Q_p)$ is dense in $L_q(\mathbb Q_p)$ for each $q\in [1,\infty)$. Let us also introduce the subspace $D_N^\ell\subset \mathcal{D}(\mathbb Q_p)$ consisting of functions with supports in a ball $B_N$, $N\in\mathbb{Z}$ and with the exponents of local constancy less than $\ell\in\mathbb{Z}$. Then the topology in $\mathcal{D}(\mathbb Q_p)$ is defined as the double inductive limit topology, so that $$\mathcal{D}(\mathbb Q_p)=\lim\limits_{\longrightarrow\atop{N\to\infty}} \lim\limits_{\longrightarrow\atop{\ell\to\infty}}D_N^\ell.$$ If $V\subset \mathbb Q_p$ is an open set, the space $\mathcal{D}(V)$ of test functions on $V$ is defined as a subspace of $\mathcal{D}(\mathbb Q_p)$ consisting of functions with supports in $V$. For a ball $V=B_N$, we can identify $\mathcal{D}(B_N)$ with the set of all locally constant functions on $B_N$. The *Fourier transform* of a test function $\psi\in\mathcal{D}(\mathbb Q_p)$ is defined by the formula $$\big(\mathcal{F}_{x\to \xi}\,\psi\big)(\xi)=\int\limits_{\mathbb Q_p}\chi(\xi\, x)\psi(x)\,dx,\quad \xi \in \mathbb Q_p.$$ Remark that the additive group of $\mathbb Q_p$ is self-dual, so that the Fourier transform of a complex-valued function $\psi\in \mathbb Q_p$ is again a function on $\mathbb Q_p$ and if $\mathcal{F}_{x\to \xi}\psi\in L_1(\mathbb Q_p)$ then we have the inversion formula $$\psi(x)=\int\limits_{\mathbb Q_p}\chi(-x\, \xi)\mathcal{F}_{x\to \xi}\psi(\xi)\,d\xi.$$ Let us also remark that, in contrast to the Archimedean situation, the Fourier transform $\psi \to \mathcal{F}_{x\to \xi}\psi$ is a linear and continuous automorphism of the space $\mathcal{D}(\mathbb Q_p)$ (cf. [@AKK:book Lemma 4.8.2], see also [@Gel Ch. II,§2.4.], [@Taib III,(3.2)], [@VVZ VII.2.], i.e. $\psi(x)=\mathcal{F}^{\,-1}_{\xi\to x}\Big(\mathcal{F}_{x\to\xi} \psi \Big).$ The space $\mathcal{D}^\prime(\mathbb Q_p)$ of Bruhat-Schwartz distributions on $\mathbb Q_p$ is defined as a strong conjugate space to $\mathcal{D}(\mathbb Q_p)$. By duality, the Fourier transform is extended to a linear (and therefore continuous) automorphism of $\mathcal{D}^\prime(\mathbb Q_p)$. For a detailed theory of convolutions and direct product of distributions on $\mathbb Q_p$ closely connected with the theory of their Fourier transforms see [@AKK:book; @Ko:2001; @VVZ] For the needs of this article, we also require some facts from the harmonic analysis on the $p$-adic ball $B_N$. At first, remark that $B_N$ is a compact subgroup of $\mathbb Q_p$ and its annihilator $\{\xi\in\mathbb Q_p\colon\chi(\xi\,x)=1\ \text{for all}\ x\in B_N\}$ coincides with the ball $B_{-N}$. By the duality theorem (see, e.g. [@Morris:1977 Theorem 27], the dual group $\widehat{B}_N$ to $B_N$ is isomorphic to the discrete group $\mathbb Q_p/B_{-N}$ consisting of cosets $$p^m (r_0+r_1p+\cdots + r_{N-m-1}p^{N-m-1})+B_{-N}, \ \ r_j\in \{0,1, \ldots, p-1\}, \ m\in \mathbb{Z}, \ m<N.$$ This isomorphism means that any nontrivial continuous character of $B_N$, which has the form $\chi (\xi\cdot x)$, $x\in B_N$, where $\vert \xi \vert_p > p^{-N}$ and $\xi \in \mathbb Q_p$, is considered as a representative of the class $\xi + B_{-N}$. Moreover the value $\vert \xi\vert_p$ is the same for any representative of the class. Let us recall that the normalized Haar measure on $B_N$ is $d\mu=p^{-N}\, dx$. The normalization of the Haar measure on $\mathbb Q_p/B_{-N}$ can be made in such a way that the equality $$\int\limits_{\mathbb Q_p} f(x)\, dx = \int\limits_{\mathbb Q_p/ B_{-N}}\Big (p^N \int\limits_{B_{-N}} f(x+h)\, dh\Big)\, d\hat{\mu} (x+B_{-N})$$ is valid for any $f\in \mathcal{D}(\mathbb Q_p)$. (See, for example, [@Bourbaki:2004 Chapter VII, Proposition 10] or [@Hewitt-Ross:1979:II Th. (28.54), p. 51(91)]). Above $\hat{\mu}(x+B_{-N})$ denotes the normalized Haar measure on $\mathbb Q_p/ B_{-N}$. With this normalization it is also true the Plancherel identity for the corresponding Fourier transform on $B_N$ given by the formula: $$\label{FN}\ \widehat{f}\equiv(\mathcal{F}_N f)(\xi) = p^{-N}\int\limits_{B_N} \chi(x\, \xi)f(x)\, dx, \quad \xi\in (\mathbb Q_p/ B_{-N})\cup \{0\}.$$ It follows from [\[FN\]](#FN){reference-type="eqref" reference="FN"} that $\mathcal{F}_N f$ can be understood as a function on $\mathbb Q_p/ B_{-N}$. Since $\mathcal{F}\colon\mathcal{D}(\mathbb Q_p)\to \mathcal{D}(\mathbb Q_p)$, the Fourier transform $\mathcal{F}$ maps $\mathcal{D}(B_N)$ onto the set of functions on the discrete set $\widehat{B_N}$ with only a finite number of nonzero values. Thus the set $\mathcal{D}(B_N)$ with natural locally convex topology may be considered as the set of test functions on $\widehat{B_N}=\mathbb Q_p/ B_{-N}$. The conjugate space $\mathcal{D}^\prime(\widehat{B_N})$ consists of all functions on $\widehat{B_N}$, see e.g. [@Helem:2006]. Therefore the Fourier transform is extended, via duality, to the mapping from $\mathcal{D}^\prime(B_N)$ to $\mathcal{D}^\prime(\widehat{B_N})$ and the theory of distributions on locally compact group, in particular $B_N$, developed by Bruhat [@Bruhat:1961] as well applicable. The corresponding Plancherel identity has the form (see [@Hewitt-Ross:1979:II (31.46)]): $$\label{31-46}\ \dfrac{1}{p^N}\int\limits_{B_N}\vert h\vert^2\,dx=\Vert \mathcal{F}_N h\Vert^2_{L_2(\widehat B_N)},$$ for $h\in L_2(B_N)$. *2.3 Spectrum of the Vladimirov operator on the $p$-adic ball*. In [@Kochubei:2018:Ball] it was proven that Vladimirov operator $D_N^\alpha$ has the representation: $$\label{1-3} \left( D^\alpha_Nu\right) (x)=\lambda_0 u (x)+\frac{1-p^{\alpha}}{1-p^{-\alpha -1}}\int\limits_{B_N}|y|_p^{-\alpha -1}[u (x-y)-u (x)]\,dy$$ on functions $u$ from the space $\mathcal D(B_N)$ of locally constant functions with compact support in $B_N$. Operator $D_N^\alpha$ is a positive definite self-adjoint operator on $L_2(B_N)$ and $$\label{lla}\ \lambda_0 =\frac{p-1}{p^{\alpha +1} -1}p^{\alpha (1-N)}$$ is its smallest eigenvalue. It is also well known that operator $D^\alpha_N$ on $B_N$ has a complete orthonormal system of eigenfunctions, consisting of so called Vladimirov functions (see e.g. [@Ko:2001 Ch.3, §3.3.2], [@VVZ Section 10.4]): $$\varPsi_0 (x) = \begin{cases} p^{-N/2}, &\text{if $x \in B_N$}\\ 0, &\text{if $x \notin B_N$}, \end{cases}$$ corresponding to the eigenvalue $\lambda_0$ of multiplicity 1, and $${\varPsi_{1-N,j,0}^1(x) = p^{\frac{-N}{2}}\Omega(p^{-N} \Vert x \Vert_p) \chi(jp^{N-1}x)} \, ,$$ where ${j \in \{1,\ldots,p-1\}} \,, j \notin \mathbb Z_p^0,$ with eigenvalue $\lambda_1=p^{\alpha(1-N)}\ \ (\text{multiplicity} \ p-1),$ $${\varPsi_{\mu-N,j,\varepsilon}^{l}(x) = p^{\frac{\mu-N-l}{2}}\sqrt{\frac{p}{p-1}}\delta(\Vert x \Vert_p - p^{l+N-\mu}) \chi(\varepsilon p^{l-2(\mu-N)}x^2+jp^{l+N-\mu-1}x) \, ,}$$ where $2 \leq l \leq \mu, j \in \{1,\ldots,p-1\}, j \notin \mathbb Z_p^0, \varepsilon = \varepsilon_0 + \varepsilon_1 \beta + ... + \varepsilon_{l-2} \beta^{l-2}$ and $$\varPsi_{\mu-N,j,0}^{1}(x)=p^{\frac{\mu-N-1}{2}}\Omega(p^{\mu-N-1}\Vert x \Vert_p)\chi(jp^{N-\mu}x),$$ where $j \in \{1,\ldots,p-1\},$ $j \notin \mathbb Z_p^0,$ with the eigenvalue $\lambda_{\mu}=p^{\alpha(\mu-N)}$ of the total multiplicity $p^{\mu-1}(p-1), \mu = 2,3,...$. Here $\mathbb Z_p^0 = \{ x \in B_N \colon \Vert x \Vert_p < 1\}$, and $$\delta(t) = \begin{cases} 1, &\text{if $t=0$}\\ 0, &\text{if $t\neq 0$}. \end{cases}$$ For other systems of the orthogonal eigenvectors see e.g. [@Koz:2007; @BGPW:2014; @BZ:2019]. # Sobolev spaces over $p$-adic ball {#sec3} Consider the additive group of $B_N$. The dual group $\widehat{B}_N$ is isomorphic to the discrete group $\mathbb Q_p/B_{-N}$ consisting of the cosets $$\label{K1}\ \xi = p^m (r_0+r_1p+\ldots+r_{N-m-1}p^{N-m-1}) + B_{-N},$$ where $r_j \in \{0,1,\ldots,p-1\}$, $m\in\mathbb{Z}$, $m<N.$ For $\xi\in \mathbb Q_p/B_{-N}$ we set $\Vert \xi\Vert = p^{-m}$. **Definition 1**. * The Sobolev space $H^\alpha(B_N)$ consists of such functions $f\in L_2(B_N)$ that $$\Vert f\Vert^2_{H^\alpha(B_N)}=\int\limits_{\widehat B_N}\vert \widehat f(\xi)\vert^2\big(1+\vert \xi\vert^2_p\big)^\alpha\,d\xi=\sum\limits_{\xi\in\widehat{B}_N}\vert \widehat{f}(\xi)\vert^2 (1+\Vert \xi\Vert^2)^\alpha< \infty$$ where $\widehat{f}=\mathcal{F}_N f$ is the Fourier transform in the ball $B_N$ [\[FN\]](#FN){reference-type="eqref" reference="FN"}.* **Definition 2**. * Let $u\in L_2(B_N)$ and $s\in (0,1)$. We say that function $u$ belongs to the Aronszain-Gagliardo-Slobodecki space $H_{AGS}^s(B_N)$ if the following norm is finite: $$\Vert u\Vert_{H_{AGS}^s}=\Vert u\Vert_{L_2(B_N)}+[u]_s,$$ where $$\label{Us}\ [u]_s^2=\int\limits_{B_N}\int\limits_{B_N}\frac{\vert u(x)-u(y)\vert^2}{\vert x-y\vert_p^{2s+1}}\,dx\,dy.$$* For the case $\mathbb{R}^n$ the corresponding space was introduced by Aronszain, Gagliardo and Slobodecki independently in [@Aron:1955; @Gagl:1958; @Slob:1958]. The properties of these spaces for the locally compact abelian groups were recently investigated in [@GorkaKost:2020]. Below we prove some results about these spaces, which we apply further. **Theorem 3**. * If $0<s<1$ then the spaces $H^s_{AGS}(B_N)$ and $H^s(B_N)$ are isomorphic, i.e. there are some constants $C_1$ and $C_2$ such that $$\label{CC1}\ C_2\Vert u\Vert_{H^s(B_N)}\leq \Vert u\Vert_{H^s_{AGS}(B_N)} \leq C_1\Vert u\Vert_{H^s(B_N)}.$$* *Proof.* In fact, $$[u]^2_s=\sum\limits_{\xi\in \widehat{B}_N}\vert \widehat{u}(\xi)\vert^2 A_s(\xi),$$ where $$\label{2K}\ A_s(\xi)=\int\limits_{B_N}\dfrac{\vert \chi(z\,\xi)-1\vert^2}{\vert z\vert_p^{2s+1}}\, dz.$$ Indeed, by substituting $z= x-y$ into [\[Us\]](#Us){reference-type="eqref" reference="Us"} and using Fubini theorem and then Plancherel equality we have: $$\begin{aligned} ^2_s&=\int\limits_{B_N}\int\limits_{B_N}\frac{\vert u(z+y)-u(y)\vert^2}{\vert z\vert_p^{2s+1}}\,dy\,dz=\int\limits_{B_N}\Big\Vert \dfrac{u(z+\cdot)-u(\cdot)}{\vert z\vert^{s+1/2}}\Big\Vert^2_{L_{2}(B_N)} dz=\\ &=\int\limits_{B_N}\Big\Vert \mathcal{F}_N\Big(\dfrac{u(z+\cdot)-u(\cdot)}{\vert z\vert^{s+1/2}}\Big)\Big\Vert^2_{L_{2}(\widehat{B}_N)} dz=\int\limits_{B_N}\Big\Vert \dfrac{\chi(z\,\xi) \widehat{u}(\xi)-\widehat{u}(\xi)}{\vert z\vert^{s+1/2}}\Big\Vert^2_{L_{2}(\widehat{B}_N)} dz=\\ &=\int\limits_{B_N}\sum\limits_{\xi\in\widehat{B}_N} \dfrac{\vert\chi(z\, \xi) -1\vert^2}{\vert z\vert^{2s+1}}\vert \widehat{u}(\xi)\vert^2 dz=\sum\limits_{\xi\in \widehat{B}_N}\vert \widehat{u}(\xi)\vert^2 A_s(\xi).\end{aligned}$$ Similar to the inequality (13) from [@GorkaKost:2020], we may prove that there exists such a constant $C_1$ that $$\label{3-4}\ A_s(\xi)\leq C_1\Vert \xi\Vert^{2s}, \quad \forall \xi \in \widehat{B}_N.$$ This implies that $$[u]_s\leq C_1\sum\limits_{\xi\in\widehat{B}_N}\vert \widehat{u}(\xi)\vert^2\Vert \xi\Vert^{2s}\leq C_1 \Vert u\Vert^2_{H^s(B_N)}.$$ To prove left part of the inequality [\[CC1\]](#CC1){reference-type="eqref" reference="CC1"} let us make the change of variables in [\[2K\]](#2K){reference-type="eqref" reference="2K"}, where $\xi \in \widehat{B}_N$ is identified with its principal part in [\[K1\]](#K1){reference-type="eqref" reference="K1"}, $\vert \xi\vert_p\geq p^{-N+1}$. We write $z=\eta\,\xi^{-1}$, so that $$\begin{aligned} A_s(\xi)&=\int\limits_{\vert \eta\vert\leq p^N\Vert\xi\Vert}\dfrac{\vert\chi(\eta)-1\vert^2}{\vert \eta\vert_p^{2s+1}\Vert\xi\Vert^{-2s-1}}\Vert \xi\Vert^{-1}\,d\eta=\\ &=\Vert \xi\Vert^{2s}\int\limits_{\vert \eta \vert_p\leq p^N\Vert \xi\Vert}\dfrac{\vert \chi(\eta)-1\vert^2}{\vert \eta\vert_p^{2s+1}}\, d\eta.\end{aligned}$$ Since $\Vert \xi\Vert \geq p^{-N+1}$, we have $$\big\{\vert \eta\vert_p \leq p^N\Vert \xi\Vert \big\}\supset\big\{\vert \eta \vert_p\leq p\big\},$$ therefore $$A_s(\xi)\geq C_2 \Vert \xi\Vert^{2s},$$ where $$C_2 = \int\limits_{\vert \eta\vert_p\leq p}\dfrac{\vert\chi(\eta)-1\vert^2}{\vert \eta\vert_p^{2s+1}}\,d\eta.$$ ◻ # The existence of a weak solution As we mentioned in Section 2.3 operator $D_N^\alpha$ [\[1-3\]](#1-3){reference-type="eqref" reference="1-3"} is a self-adjoint operator on $L_2(B_N)$ with discrete spectrum. Let us denote by $\{\mathfrak{a}_k\}, k = 1, 2, \ldots$, $\mathfrak{a}_k >0$ its eigenvalues written in increasing order and repeating according to their multiplicity, and by $\{\psi_k\}_{k\geq 1}$ corresponding set of eigenfunctions, normalized in $L_2(B_N)$. They form an orthogonal basis in $L_2(B_N)$ (see e.g. [@Koz:2007; @BGPW:2014; @BZ:2019]). **Definition 4**. * Let us define a Hilbert space $H_1$ as the space of all functions $u\in L_2(B_N)$ such that $$\label{H1}\ u=\sum\limits_{k=1}^\infty c_k \psi_k \in L_2(B_N), \quad \Vert u \Vert^2_{H_1} = \sum\limits_{k=1}^\infty \mathfrak{a}_k \vert c_k\vert^2 <\infty.$$ Thus the dual space $H_{-1}$ to $H_1$ is defined as the completion of the finite sums of the form $$f=\sum\limits_{k=1}^Nc'_k\psi_k$$ with respect to the dual norm $$||f||^2_{H_{-1}}=\sum\limits_{k=1}^{\infty} \mathfrak{a}_k^{-1}|c'_k|^2.$$* The operator $D_N^{\alpha}$ in $L_2(B_N)$, defined on the dense domain $\mathcal D(B_N)$ of locally constant functions with compact support in the $p$-adic ball $B_N$, is semi-bounded from below $$( D_N^{\alpha} u,u)_{L_2(B_N)} \geq \lambda_0\Vert u \Vert_{L_2(B_N)}$$ for any $u \in \mathcal D(B_N)$. Therefore the Hilbert space $H_1$ is isomorphic to the closure of $\mathcal D(B_N)$ with respect to the scalar product $$\label{H1-1}\ (u,v)_{1} = (u,v)_{L_2(B_N)}+(D_N^{\alpha}u, v)_{L_2(B_N)}.$$ Let operator $\mathcal{P}_{N,\alpha}$ acting on $\mathcal{D}(B_N)$ given by $$\label{PNal}\ \mathcal{P}_{N,\alpha}u(x)=D_N^\alpha u(x)-\lambda_0u(x)=\dfrac{1-p^\alpha}{1-p^{-\alpha-1}}\int\limits_{B_N}\dfrac{u(x-y)-u(x)}{\vert y\vert^{\alpha+1}_p}\,dy,$$ with $\lambda_0$ from [\[lla\]](#lla){reference-type="eqref" reference="lla"}. **Theorem 5**. * $$\label{PN}\ \mathcal{F}_N \big(\mathcal{P}_{N,\alpha}u\big)(\xi)=\frac{1}{p^N}\vert\xi\vert^\alpha_p\mathcal{F}_Nu(\xi),\quad \xi\in\widehat B_N, \alpha\in(0,1).$$* *Proof.* Let us find $\mathcal{F}_N\big(\mathcal{P}_{N,\alpha}u\big)$. $$\begin{aligned} \nonumber \mathcal{F}_N\big(\mathcal{P}_{N,\alpha}u\big)&=\dfrac{1}{p^N}\int\limits_{B_N}\chi(x\xi)\mathcal{P}_{N,\alpha}u(x)\,dx=\\ \nonumber &=\dfrac{a_p}{p^N}\int\limits_{B_N}\chi(x\xi)\int\limits_{B_N}\dfrac{u(x-y)-u(x)}{\vert y\vert_p^{\alpha+1} }\,dy\,dx=\\ \nonumber &=\dfrac{a_p}{p^N}\int\limits_{B_N}\dfrac{1}{\vert y\vert_p^{\alpha+1} } \int\limits_{B_N}\chi(x\xi)[u(x-y)-u(x)]\,dx\,dy=\\ \nonumber &=a_p\int\limits_{B_N}\dfrac{\chi(y\,\xi)-1}{\vert y\vert_p^{\alpha+1} } \,dy \,\mathcal{F}_N u(\xi),\end{aligned}$$ where $\xi\in\mathbb Q_p/B_{-N}$ and $a_p=\dfrac{1-p^\alpha}{1-p^{-\alpha-1}}$. Let us prove that $$\label{norm-alpha}\ \frac{1-p^\alpha}{1-p^{-\alpha-1}}\int\limits_{B_N}\dfrac{\chi(y\,\xi)-1}{\vert y\vert_p^{\alpha+1} } \,dy =\vert \xi\vert^\alpha_p, \quad \xi\in \widehat{B}_N, \alpha\in(0,1).$$ To prove [\[norm-alpha\]](#norm-alpha){reference-type="eqref" reference="norm-alpha"} let us consider the Riesz kernel $$f_\gamma^N(x)=\dfrac{\vert x\vert_p^{\gamma-1}}{\Gamma_p(\gamma)}, \quad x\in B_N, \gamma\in (0,1),$$ where $\Gamma_p(\gamma)=\dfrac{1-p^{\gamma-1}}{1-p^{-\gamma}}$ denotes $p$-adic Gamma function (see e.g. [@Gel Ch. II, §2.6 (14)]). Its Fourier transform equals to: $$\begin{aligned} \nonumber\ \mathcal{F}_Nf_\gamma^N(\xi)&=\dfrac{1}{p^N \Gamma_p(\gamma)}\int\limits_{B_N}\chi(x\xi)\vert x\vert_p^{\gamma-1}\, dx=\\ \nonumber &=\dfrac{1}{p^N \Gamma_p(\gamma)}\int\limits_{\vert y\vert_p\leq p^N\vert \xi\vert_p}\chi(y)\vert y\vert_p^{\gamma-1}\vert \xi\vert_p^{-\gamma}\, dy=\\ \nonumber\ &=\dfrac{\vert \xi\vert_p^{-\gamma}}{p^N \Gamma_p(\gamma)}\int\limits_{\vert y\vert_p\leq p^N\vert \xi\vert_p}\chi(y)\vert y\vert_p^{\gamma-1}\, dy,\quad \gamma\in(0,1),\end{aligned}$$ $\xi\in \mathbb Q_p/ B_{-N}\simeq\widehat{B}_N$. Above we used the change of variables $x=y\xi^{-1}$, therefore $dx=\vert\xi\vert_p^{-1}dy.$ Remark that due to (12.40) in [@Vladimirov:tables] for any $r\geq 1$: $$\int_{B_r}\vert x\vert_p^{\gamma-1}\chi(x)\,dx=\Gamma_p(\gamma).$$ Since $\vert \xi\vert_p=p^m$ with $m<N$, see [\[K1\]](#K1){reference-type="eqref" reference="K1"}, we have that $p^N\vert \xi\vert_p>1$. Therefore considering $f_\gamma^N(x)$ as a distribution from $\mathcal{D}^\prime(B_N)$ we may identify the Fourier transform $\mathcal{F}_N$ of $f_\gamma^N$ with $$\widetilde{f}_\gamma^N(\xi)=p^{-N}\vert \xi\vert_p^{-\gamma}, \quad\xi \in \widehat{B}_N.$$ Now for any $\varphi\in \mathcal{D}(\widehat B_N)$ we denote $\psi=\mathcal{F}^{-1}_N\varphi$ and applying the regularization theory for homogeneous distributions [@Gel-Shilov] (see also [@VVZ; @Ko:2001]) we find that for $\gamma\in (0,1)$: $$\begin{aligned} \langle\vert \xi\vert^{-\gamma}_p,\varphi(\xi)\rangle_{L_2(\widehat{B}_N)}&= p^N\,\langle \widetilde{f}_{\gamma}^N,\varphi(\xi)\rangle_{L_2(\widehat{B}_N)}= p^N\,\langle \mathcal{F}_N f_{\gamma}^N,\mathcal{F}_N\psi\rangle_{L_2(\widehat{B}_N)}=\\ &=\frac{ p^{N}}{\Gamma_p(\gamma)}\, \langle \vert x\vert_p^{\gamma-1},\psi (x)-\psi(0)\rangle_{L_2(B_N)}=\\ &=\frac{ p^{N}}{\Gamma_p(\gamma)}\, \int\limits_{B_N}\vert x\vert_p^{\gamma-1}\int\limits_{\widehat B_N}\big(\chi(x\xi)-1\big)\varphi(\xi)\,d\xi\,dx=\\ &=\int\limits_{\widehat B_N}\varphi(\xi)\int\limits_{B_N}\frac{ p^{N}}{\Gamma_p(\gamma)}\vert x\vert_p^{\gamma-1}\big(\chi(x\xi)-1\big)\, dx\,d\xi.\end{aligned}$$ Therefore in terms of distributions we have for $\gamma\in (0,1)$, $\xi \in \widehat B_N$: $$\label{4-7}\ \dfrac{1}{\Gamma_p(\gamma)}\int\limits_{B_N}\vert x\vert_p^{\gamma-1}\big(\chi(-x\xi)-1\big)\, dx =\vert \xi\vert_p^{-\gamma}.$$ The distribution $f_\gamma^N(x)=\dfrac{\vert x\vert_p^{\gamma-1}}{\Gamma_p(\gamma)}$ is holomorphic on $\gamma$ everywhere except poles $1+\alpha_k$, $\alpha_k=\dfrac{2k\pi}{\ln p}i$ (e.g. [@VVZ Ch.I, §8]). Thus both sides of the equality [\[4-7\]](#4-7){reference-type="eqref" reference="4-7"} are well-defined for $\gamma\in (-1,0)$ and also for $\gamma= -\alpha$. Finally applying [\[norm-alpha\]](#norm-alpha){reference-type="eqref" reference="norm-alpha"} we have the statement. ◻ **Theorem 6**. * Let $\alpha\in(0,1)$. Then $$\label{eq-prop-3-4}\ \Vert u\Vert^2_{H^\alpha(B_N)}\asymp \Vert u\Vert^2_{H^\alpha_{AGS}}\asymp \Vert u\Vert ^2_{H_1}.$$* *Proof.* The first part of the isomorphism relation [\[eq-prop-3-4\]](#eq-prop-3-4){reference-type="eqref" reference="eq-prop-3-4"} follows from Theorem [Theorem 3](#prop-3-3){reference-type="ref" reference="prop-3-3"}. To prove second part first remark that $$\Vert u \Vert_{H_1}\asymp\Vert u\Vert^2_{L_2(B_N)}+\Vert [\mathcal{P}_{N,\alpha}]^{1/2}u\Vert^2_{L_2(B_N)}.$$ From Definition [Definition 1](#defSob){reference-type="ref" reference="defSob"} it follows that $u\in H^\alpha(B_N)$ is equivalent to $\vert \xi\vert_p^\alpha\, \widehat u(\xi)\in L_2(\widehat B_N)$. Finally from [\[PN\]](#PN){reference-type="eqref" reference="PN"} we have the result. ◻ The dual space $H_{-1}$ to $H_1$ is obtained as the closure of $L_2(B_N)$ w.r.t. the scalar product $$\label{sc} \ (f,g)_{-1}=(f,g)_{L_2(B_N)}+([D_N^{\alpha}]^{-1}f, g)_{L_2(B_N)}.$$ The dual space $H_{-1}$ is also called the distribution space. For the construction of the dual space in terms of unbounded self-adjoint operator with discrete spectrum acting in some Hilbert space we refer to [@BER Ch. XIV]. If $u,v \in H_1$ and $f=(D^\alpha_Nu)(x)$ then since $D^\alpha_N$ acts as an isomorphism between $H_1$ and $H_{-1}$ we have $$(f,v)_{H_{-1}\times H_1}=([D^\alpha_N]^{-1}f,v)_{H_1\times H_1}=(u,v)_{H_1\times H_1}=\sum\limits_{k=1}^{\infty} \mathfrak{a}_k u_kv_k.$$ If $f \in L_2(B_N)$, then $(f,v)_{H_{-1}\times H_1}= \int_{B_N}fv\,dy$ for $v\in H_1$. **Definition 7**. *A function $u \in C([0,T], H_{-1})$ is called an $H_{-1}$ - solution to equation [\[1-2\]](#1-2){reference-type="eqref" reference="1-2"} if $\varphi(u) \in L_1([0,T], H_1)$ and* *$$\label{1-8}\ \int\limits_0^T\int\limits_{B_N}u\,D_{t}\psi\,dy\,dt = \int\limits_{0}^T\int\limits_{B_N}\varphi(u)\,D^\alpha_N\psi \,dy\,dt$$ for any $\psi \in \mathcal{D}(B_N)$, where $\mathcal{D}(B_N)$ is the space of locally constant functions with compact support in $B_N$.* Our main result is the following theorem about the existence of a weak solution to nonlinear problem [\[1-2\]](#1-2){reference-type="eqref" reference="1-2"} in sense of Definition [Definition 7](#def-4-4){reference-type="ref" reference="def-4-4"} for any $T>0$. **Theorem 8** (**Main result**). *For any $u_0 \in H_{-1}$ there exists a unique solution $u \in C([0,T], H_{-1})$ of problem [\[1-2\]](#1-2){reference-type="eqref" reference="1-2"} for every $T>0$. Moreover we have $$t \varphi(u) \in L_{\infty}([0,T],H_{-1})$$ $$t\,\partial_tu \in L_{\infty}([0,T],H_{-1})$$ We also have that $u\varphi(u) \in L_1([0,T]\times B_N )$ and the solution map $S_t: u_0 \rightarrow u(t)$ defines a semigroup of non-strict contractions in $H_{-1}$ $$\label{eq4-5}\ \Vert u(t)-v(t)\Vert_{H^{-1}}\leq\Vert u(0) - v(0)\Vert_{H^{-1}},$$ which turns out to be also compact in $H_{-1}$.* Let us introduce some notations. Let $j:~\mathbb{R}\rightarrow \mathbb{R}^+$ be a convex, lower semi-continuous function such that its subdifferential $\partial j=\varphi$, $j(0) = 0$ and $j(r)/|r| \rightarrow \infty$ as $|r| \rightarrow \infty$. For $u\in H_{-1}$ we define $$\Psi(u)=\int_{B_N}j(u)\,dx,$$ if $u \in L_1(B_N)$ and $j(u) \in L_1(B_N)$, and $\Psi(u):= +\infty$ otherwise. Let us recall that the *subdifferential of function* $j:~\mathbb{R}\rightarrow \mathbb{R}^+$ is given by $$\label{gradj}\ \partial j(u) =\big \{\rho \in \mathbb{R}\colon\, j(v)-j(u)\geq \rho\cdot(v-u), \ \forall v \in Dom(j) \},$$ where $Dom(j) = \{ x \in \mathbb{R}\colon\, j(x)< +\infty\}$. The proof of the main result Theorem [Theorem 8](#main){reference-type="ref" reference="main"} follows the principal approach of Brezis [@Brezis:1971] with necessary modifications dictated by specific features of the $p$-adic situation, where classical properties of Sobolev spaces require a separate investigation. The proof is given in a set of lemmas. **Lemma 9**. *The function $\Psi(u)$ is convex lower semi-continuous on $H_{-1}$.* *Proof.* Let $u_n$ be a sequence of function such that $u_n \in H_{-1}\cap L_1(B_N)$, $u_n \to u$ in $H_{-1}$ as $n \to \infty$ and $\int_{B_N}j(u_n(x))\,dx \leq C$. Remark that for each $n \in \mathbb{N}$ $\int_{B_N} u_n(x)\,dx$ is uniformly absolutely continuous, i.e. $\forall \varepsilon>0\ \exists\,\delta>0$ such that $\forall E\colon\ \mu(E)<\delta$ implies $\int_{E}u_n(x)\,dx < \varepsilon$. Indeed, let $A>\frac{2C}{\varepsilon}$ and $R$ be such that $j(r)/|r| \geq A$ for $|r|>R$. Let $\delta< \frac{\varepsilon}{2R}$, then we have $$\begin{aligned} \nonumber \int_{E}|u_n|\,dx &\leq \int\limits_{\{ x\ \in E \colon|u_n(x)|\geq R \}}|u_n(x)|\,dx + \int\limits_{\{ x\ \in E \colon|u_n(x)|< R \}}|u_n(x)|\,dx \leq\\ \nonumber &\leq \int_{B_N}\frac{j(u_n(x))}{A}\,dx + R\delta \leq \frac{C}{A}+ R\delta < \varepsilon.\end{aligned}$$ By Dunford-Pettis theorem [@Dun Theorem 9, Ch. IV, §8], which is valid for any measurable space, there is a subsequence such that $u_{n_k} \to \tilde{u}$ weakly in $L_1(B_N)$. Since we know that $u_n \to u$ in $H_{-1}$ as $n\to \infty$, we conclude that $u_n \to u$ weakly in $L_1(B_N)$. Finally the function $\Psi(u)$ is convex and by Fatou's lemma, $\int\limits_{B_N}j(u)\,dx \leq \liminf\limits_{n \to \infty}\int\limits_{B_N}j(u_n)\,dx$, it is lower semi-continuous function on $L_1(B_N)$ and thus is weakly lower semi-continuous on $H_{-1}$. ◻ **Corollary 10**. *The sub-differential $\partial\Psi$ is a maximal monotone operator in $H_{-1}$.* *Proof.* The function $\Psi$ is convex and lower semi-continious in $H_{-1}$, so that its sub-differential $\partial\Psi$ is a maximal monotone operator in $H_{-1}$ [@Min]. ◻ Let us define operator $A$ on $H_{-1}$ by the following: $$\label{Anon}\ Au = \{D^\alpha_Nw\colon\ w \in H_1,\, w(x) \in \varphi(u(x))\ \text{for a.e.}\ x\in B_N \}$$ with $u\in Dom (A)$ iff there is some $w\in H$ such that $w(x)\in \varphi(u(x))$ for a.e. $x\in B_N$. Our main goal is to prove that $A$ is maximal monotone, then this gives the main statement of Theorem [Theorem 8](#main){reference-type="ref" reference="main"} on the existence of the solution. Inequality [\[eq4-5\]](#eq4-5){reference-type="eqref" reference="eq4-5"} is the consequence of maximal monotonicity by standard arguments similar to [@Kato:1970](see also [@Komura:1967; @CrandallPasy:1969; @Brezis:1971]). The maximal monotonicity of $A$ will follow from the fact that $A \subset \partial\varPsi$. To prove the letter we need the following result. **Theorem 11**. *$$f \in \partial\Psi(u) \Leftrightarrow [D^\alpha_N]^{-1}(f)(x) \in \varphi(u(x)) \text{\ for a.e \ } x\in B_N.$$* **Lemma 12**. *Let $F\in H_{-1} \cap L_1(B_N)$ and let $w \in H_1$. Let $g \in L_1(B_N)$ and $h$ be measurable function such that $$\ \label{28}\ g \leq h \leq F \cdot w, \ \text{for a.e.}\ x\in B_N.$$ Then $h \in L_1(B_N)$ and $$\label{43}\ \int_{B_N}h\,dx \leq \int_{B_N}F(x)\cdot w(x)\,dx = (F,w)_{H_{-1}\times H_1 }$$ holds. Here $(F,w)_{H_{-1}\times H_1}$ means the duality pairing between $H_{-1}$ and $H_1$.* *Proof.* Let $$\label{1-14-0}\ w_n = \begin{cases} n, &\text{if $w \geq n$}\\ w, &\text{if $|w| \leq n$}\\ -n, &\text{if $w \leq -n$} \end{cases}$$ and $h_n := h\frac{w_n}{w}, g_n :=g\frac{w_n}{w}$. Let us remark that $w_n\in H_1$ for $w\in H_1$. To check this first of all note that $w_n\in L_2(B_N)$ for $w\in H_1$ due to the inequality: $\vert w_n\vert \leq \vert w\vert$ a.e. on $B_N$. Moreover, by Definition [Definition 4](#def1-2){reference-type="ref" reference="def1-2"} of $H_1$ $w_n\in H_1$ iff $\vert w_n\vert \in H_1$. For the orthogonal basis $\{\psi_k\}_{k\geq 1}$ in $L_2(B_N)$ the Fourier coefficients $c_k = (\vert w\vert ,\psi_k)_{L_2(B_N)}$ and $c_k^n=(\vert w_n\vert ,\psi_k)_{L_2(B_N)}$ in the decompositions $$\vert w\vert = \sum\limits_{k=0}^{\infty} c_k \psi_k, \quad \vert w_n\vert = \sum\limits_{k=0}^{\infty} c^n_{k} \psi_{k}$$ are connected by the following inequality $\vert c^n_{k} \vert \leq \vert c_k \vert.$ Thus $w_n \in H_1$, if $w\in H_1$. Moreover the sequence $\{\vert w_n\vert \}_{n\geq 1}$ and therefore $\{w_n\}_{n\geq 1}$ are bounded in $H_1$. Multiplying inequality [\[28\]](#28){reference-type="eqref" reference="28"} by $\frac{w_n}{w}$, we get $$g_n \leq h_n\leq F\cdot w_n \ \text{for a.e.}\ x\in B_N.$$ Therefore $$0\leq h_n-g_n\leq F\cdot w_n - g_n\ \text{ for a.e.}\ x \in B_N.$$ Since $w_n \to w$ a.e. on $B_N$, we have $h_n - g_n \rightarrow h-g$ a.e. on $B_N$ as $n\to \infty$. We also have that $$\label{wn}\ \int_{B_N}(h_n-g_n)\,dx\leq \int_{B_N}F\cdot w_n\, dx -\int_{B_N} g_n \,dx= (F,w_n)_{H^{-1}\times H} - \int_{B_N} g_n \,dx.$$ Further, we will apply Theorem 11 in [@GorkaKostrRey:2014], which in our notations states the following. Since $\dfrac{1}{1+\vert \xi\vert_p^2}\in L_q(\widehat B_N)$ for any $q>1$, we have for $\alpha\in (0,1)$ (thus $\alpha< q$) the compact embedding of the space $H^\alpha(B_N)$ into $L_r(B_N)$ for all $2<r<q^*$, where $q^*=\frac{2q}{q-\alpha} > 2$. From the definition of $w_n$ [\[1-14-0\]](#1-14-0){reference-type="eqref" reference="1-14-0"} we have that $$\label{eq14-1}\ \vert w_n(x)-w_n(y)\leq \vert w(x)-w(y)\vert, \quad \text{\ for all}\ x,y \in B_N.$$ Therefore, due to the Theorem [Theorem 6](#isom){reference-type="ref" reference="isom"} the sequence $\{w_n\}_{n\geq 1}$ is uniformly bounded in $H^\alpha_{AGS}$, and we may assume its weak convergence. Moreover, it follows from definition that $\frac{w_n}{w}\to 1$ as $n\to\infty$ and $\vert\frac{w_n}{w}\vert\leq 1$. Therefore $w_n\to w$, a.e. on $B_N$ as $n\to\infty$. Using the isomorphism properties (Theorem [Theorem 6](#isom){reference-type="ref" reference="isom"}) and the above-mentioned compact embedding, we have that $w_n\to w$, $n\to\infty$ in $L_r(B_N)$ strongly, and therefore in $L_2(B_N)$ $(r>2)$. Using [\[eq14-1\]](#eq14-1){reference-type="eqref" reference="eq14-1"} together with the dominated convergence theorem in $H^\alpha_{AGS}$ we can pass to the limit. Together with the strong $L_2$ convergence this gives the strong convergence in $H^\alpha_{AGS}$ and therefore in $H_1$. We have also that $g_n\to g$, $n\to \infty$ in $L_1(B_N)$ and from [\[wn\]](#wn){reference-type="eqref" reference="wn"} by Fatou's Lemma we conclude that $h-g \in L_1(B_N)$ and thus $h\in L_1(B_N)$ together with $$\int_{B_N}(h-g)\,dx\leq (F,w)_{H_{-1} \times H_1} - \int_{B_N}g\,dx<\infty,$$ which implies the statement. ◻ **Lemma 13**. *$A \subset \partial\Psi.$* *Proof.* To prove this statement we need to show that for any $f$ such that $f = Au$ for some $u\in Dom(A)$, it is also true $f\in \partial\Psi$. Recalling the definition of $\partial\Psi$: $$\,\partial\Psi (u) =\big \{g\in H_{-1}\colon\, \Psi (v)-\Psi (u)\geq (g,v-u)_{H^{-1}}, \ \forall v \in \mathcal{D}(\Psi)\big\},$$ where $Dom(\Psi)=\{u\in H_{-1}\colon\Psi(u)< +\infty\}$, we conclude that it suffices to prove that for each $f= Au$ (i.e. such that there exists $u \in H_{-1}\cap L_1(B_N)$ with $f =D^\alpha_Nw$, where $w\in H_1$ and $w(x) \in \varphi(u(x))$ for a.e $x \in B_N$) we have $$\int_{B_N}j(v)\,dx -\int_{B_N}j(u)\,dx \geq (f,v-u)_{H_{-1}}=(w,v-u)_{H_1\times H_{-1} }$$ for any $v\in Dom(\Psi)$. Above we have used that $D^\alpha_N$ acts as an isomorphism between $H_1$ and $H_{-1}$. To prove Lemma [Lemma 13](#L1-11){reference-type="ref" reference="L1-11"} for $v\in Dom(\Psi)$ let us take $v \in H^{-1} \cap L_1(B_N)$ and $j(v) \in L_1(B_N)$. Since $w \in \varphi(u)$ and $\varphi = \partial j$, we have $w \in \partial j(u)$. Therefore according to the definition of subdifferential ([\[gradj\]](#gradj){reference-type="ref" reference="gradj"}) we have $j(v)-j(u) \geq w\cdot(v-u)$ a.e on $B_N$. Since $j(r)/|r| \rightarrow \infty$ as $|r| \rightarrow \infty$, we can find constants $C_1\ C_2$, such that $j(r)\geq C_1|r|+C_2$. Let $F=u-v$, $h=j(u)-j(v)$, $g=j(v)-C_1| u|-C_2$. Then $F \cdot w \geq h \geq g$ a.e on $B_N$, and we can apply Lemma [Lemma 12](#L1-10){reference-type="ref" reference="L1-10"}. We get $h \in L_1(B_N)$. Also, $(F,w)_{H_{-1} \times H_1} \geq \int\limits_{B_N}h\,dx$. From that, we get inequality $$\int\limits_{B_N}j(u)\,dx -\int\limits_{B_N}j(v)\,dx \geq (v-u,w)_{H_{-1}\times H_1}=(w,v-u)_{H_1 \times H_{-1}},$$ it means that $\Psi(v)-\Psi(u)\geq(w,v-u)_{H_1 \times H_{-1}}= (f,v-u)_{H_{-1}}$, so $f \in \partial\Psi(u)$. ◻ This finished the proof of Theorem [Theorem 11](#L1-6){reference-type="ref" reference="L1-6"}. **Lemma 14**. *$A$ is a maximal monotone operator.* *Proof.* Due to [\[Anon\]](#Anon){reference-type="eqref" reference="Anon"}, we need to proof that for a given $f \in H_{-1}$ there exists $u \in H_{-1}\cap L_1(B_N)$ and $w\in H_1$ such that $$\label{eq1}\ u+D^\alpha_Nw =f,$$ and $w(x)=\varphi(u(x))$ a.e. on $B_N$. As $\varphi$ is a strictly monotone increasing smooth continuous function we introduce $\eta := \varphi^{-1}$ and rewrite equation [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} as $$\label{eq2}\ \eta(w)+ D^\alpha_Nw=f.$$ Let $\eta_{\mu}=\dfrac{1}{\mu}(1-J_\mu)$ be the Yosida approximation of the nonlinear map $\eta$, where its resolvent $J_\mu$ is given by $J_\mu = (1+\mu \eta)^{-1}$. Remark that for any $\mu \in \mathbb{R}^+$ there is a solution $w_{\mu}\in H_1$ of equation $$\label{eq3}\ \eta_{\mu}(w_\mu)+ D^\alpha_Nw_\mu=f.$$ Indeed, let us denote $\widetilde{D^\alpha_N}= D^\alpha_N- \frac{\lambda_0}{2}$, where $\lambda_0$ is the minimal non-zero eigenvalue of operator $D_N^\alpha$ (see e.g. [@VVZ Chapter II 10 4]). Since $\widetilde{D^\alpha_N}$ is a maximal monotone operator, due to the Browder Theorem [@Bro Th. 2] we have that operator $\eta_{\mu}+\widetilde{D^\alpha_N}$ is maximal monotone. Then from Minty Theorem [@Min] it follows that the map $\eta_{\mu}+\widetilde{D^\alpha_N}+\frac{\lambda I}{2}$ is surjective. Therefore equation $$\eta_{\mu}(w_\mu)+ \widetilde{D^\alpha_N} w_\mu+\frac{\lambda}{2}w_\mu=f$$ has a unique solution for any right-hand side $f \in H_{-1}$ and this implies solvability of equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"}. The equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} may be rewritten in form $$\label{eq4}\ D^\alpha_N(w_\mu)=\mathfrak{f}_\mu,$$ where $\mathfrak{f}_\mu = f- \eta_{\mu}(w_\mu)$. Let us show that $w_\mu$ is uniformly bounded in $H_1$ with respect to $\mu \to 0$. Firstly, operator $(D^\alpha_N)^{-1} : H_{-1} \to H_1$ is continuous. This follows from closed graph theorem. For this we need to check that operator $(D^\alpha_N)^{-1} : H_{-1} \to H_1$ has closed graph, i.e from convergence $\varphi_k \to \varphi$ in $H_{-1}$ and $(D^\alpha_N)^{-1}\varphi_k \to f$ in $H_1$ it follows that $(D^\alpha_N)^{-1}\varphi = f$. Indeed, from these conditions, due to continuity of embedding $H_{1} \subset H_{-1}$ (see Appendix) we have that $(D^\alpha_N)^{-1}\varphi_k \to f$ in $H_{-1}$. Since operator $(D^\alpha_N)^{-1} : H_{-1} \to H_{-1}$ is continuous (see Appendix), we have that $(D^\alpha_N)^{-1}\varphi_k \to (D^\alpha_N)^{-1}\varphi$ in $H_{-1}$, therefore $(D^\alpha_N)^{-1}\varphi =f$ as required to prove. Secondly, $\mathfrak{f}_\mu$ is uniformly bounded in $H_{-1}$ as $\mu \to 0$. Indeed, due to [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"} $$\label{eq5}\ \Vert \mathfrak{f}_\mu \Vert_{H_{-1}} \leq \Vert f \Vert_{H_{-1}} + \Vert \eta_\mu(w_\mu) \Vert_{H_{-1}} \leq C + \Vert \eta_\mu(w_\mu) \Vert_{H_{-1}},$$ and uniform boundness of $\eta_\mu(w_\mu)$ in $H_{-1}$ as $\mu \to 0$ follows from Theorem 2.1 [@BCP]. The continuity of operator $(D^\alpha_N)^{-1} : H_{-1} \to H_1$ and the uniform boundedness of $\mathfrak{f}_\mu$ in $H_{-1}$ as $\mu \to 0$ due to $$w_\mu = (D^\alpha_N)^{-1}(\mathfrak{f}_\mu)$$(see [\[eq4\]](#eq4){reference-type="eqref" reference="eq4"}), gives the uniform boundedness of $w_\mu$ in $H_1$ with respect to $\mu \to 0$. Thus we can find a subsequence $\mu_n \to 0$ such that $w_{\mu_n} \to w$ in $H_1$ and in $L_2(B_N)$, i.e. $$\begin{aligned} \label{eq8}\ w_{\mu_n} \to w &\text{ a.e. on } B_N,\\ \label{eq8-1}\ (I+\mu_n\eta)^{-1}w_{\mu_n} \to w &\text{ a.e. on } B_N.\end{aligned}$$ Indeed, by [@BCP Lemma 1.3.d] $$J_{\mu_n} (w):=(I+\mu_n\eta)^{-1}w \to w$$ as $\mu_n \to 0$ in $H_1$. Due to the inequality $\Vert J_\mu(x)-J_\mu(y)\Vert_{H_1}\leq \Vert x - y\Vert_{H_1}$ (see e.g. [@Barbu:2010 Prop. 2.3]) we have that $J_{\mu_n}(w_{\mu_n})\to w$ in $H_1$ and therefore [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"}. Multiplying equation [\[eq3\]](#eq3){reference-type="eqref" reference="eq3"} by $w_{\mu_n}$, we see that for all $\mu_n > 0$ $$\begin{aligned} \label{eq11}\, \int\limits_{B_N}\eta_{\mu_n}(w_{\mu_n}) \cdot w_{\mu_n} \, dx &\leq \int\limits_{B_N}\eta_{\mu_n}(w_{\mu_n}) \cdot w_{\mu_n} \, dx + \int\limits_{B_N} D_N^\alpha w_{\mu_n}\cdot w_{\mu_n}\, dx \leq\\ & \leq\Vert f\Vert_{H_{-1}}\cdot\Vert w_{\mu_n}\Vert_{H_1} \leq C,\end{aligned}$$ due to uniform boundedness of $w_{\mu_n}$ in $H_1$ with respect to $\mu_n \to 0$. Let us define $g_{\mu_n}:= \eta_{\mu_n}(w_{\mu_n})\subset \eta (w_{\mu_n})$ (see e.g. [@BrPa:1970 Lemma 2.1]). To finish the proof of Lemma [Lemma 14](#L1-13){reference-type="ref" reference="L1-13"} we need the following lemma. **Lemma 15**. *  Let $\eta$ be a maximal monotone graph in $\mathbb{R}\times \mathbb{R}$ such that $Dom\,(\eta)=\mathbb{R}$ and $0 \in \eta(0)$, such that $\varphi= \eta^{-1}$ may be represented as $\varphi= \partial j$, $j(0)=0$, $Dom\, (j)=\mathbb{R}.$* *Let $g_{n}$ and $w_{n}$ be measurable functions on $B_N$ such that $w_{n} \to w$ a.e. on $B_N$ as $n\to \infty$, $g_{n}\in \eta(w_{n})$ a.e. on $B_N$ and $g_{n}\cdot w_{n} \in L_1(B_N)$ with $$\label{eqTh18}\, \int\limits_{B_N}g_{n}\cdot w_{n}\, dx \leq C.$$ Then, there is a subsequence $n_k \to \infty$ such that $g_{n_k} \to u$ weakly in $L_1(B_N)$.* *Proof.* Since $\varphi = \partial j$, $j(0)=0$ we may write $$\label{1-20}\ \partial j(u)=\varphi(u)=\{a\in \mathbb{R}\colon j(v)-j(u)\geq a\cdot (v-u), \forall v\in Dom\,(j)\}.$$ By the definition of the inverse map $$T^{-1}(x^*)=\{x\in X\colon x^*\in T(x)\}$$ for nonlinear map $T\colon X\to X^*$ acting from some Banach space $X$ into its dual $X^*$, from inclusion $g_n\in \eta(w_n)$ (see e.g. [@BrPa:1970 Lemma 2.1]) we have that $w_n \in \eta^{-1}(g_n)=\varphi(g_n)$. Taking in [\[1-20\]](#1-20){reference-type="eqref" reference="1-20"} $v=0$, $u=g_n(x)$, then $a= w_n\in \varphi(g_n)$ therefore we have $$-j(g_n(x))\geq w_n\cdot\big(-g_n(x)\big).$$ Thus from [\[eqTh18\]](#eqTh18){reference-type="eqref" reference="eqTh18"} we obtain that for all $n\geq 1$ $$\int_{B_N} j(g_{n}(x))\, dx \leq \int_{B_N}g_{n}\cdot w_{n}\, dx \leq C.$$ Applying the Vallée Poussin reformulation of Dunford-Pettis Theorem ([@EkelTemam Th.1.3(d), Ch. VIII]) to the sequence $\{g_{\mu_n}\}$ we have its weak compactness in $L_1(B_N)$. Therefore, there is a subsequence $n_k \to \infty$ such that $$\label{eq6}\ g_{\mu_{n_k}} \to \, u \ \text{weakly in} \ L_1(B_N)$$ to some $u\in L_1(B_N)$, which finishes the proof of Lemma [Lemma 15](#Th18){reference-type="ref" reference="Th18"}. ◻ To finish the proof of Lemma [Lemma 14](#L1-13){reference-type="ref" reference="L1-13"} and thus the statement of the existence of the solution in the main Theorem, it remains to prove that $$\label{eq6-1}\ u(x) \in \eta(w(x)) \ \text{a.e. on} \ B_N,$$ which implies that $w(x) \in \varphi(u(x))$ a.e. on $B_N$. It suffices to prove that for every $M>0$, $u(x)\in \eta (w(x))$ a.e. for $x\in B_N^M \colon= \{x\in B_N\colon\vert w(x)\vert \leq M\}.$ From [\[eq8\]](#eq8){reference-type="eqref" reference="eq8"} and Egorov theorem, it follows that for every $\varepsilon >0$ there exists a measurable subset $E_\varepsilon \subset B^M_N$ such that $mes\, (B^M_N \setminus E_\varepsilon) \leq \varepsilon$ and $$\label{eq7}\ w_{\mu_n} \to w \text{ uniformly in } E_\varepsilon \text{ as } \mu_n \to 0,$$ and $w\in L_\infty(B_N)$. Note that operator $$\tilde{\eta} = \{ [u,w] \in L_1(E_\varepsilon)\times L_{\infty}(E_\varepsilon) \colon u(x) \in \eta(w(x)) \text{ a.e. on } E_\varepsilon \}$$ is maximal monotone in $L_1(E_\varepsilon)\times L_{\infty}(E_\varepsilon)$. Indeed, let $\tilde{w} \in L_{\infty}(E_\varepsilon)$ and $\tilde{u} \in L_{1}(E_\varepsilon)$ be such that $$\label{eq9}\ \tilde{u}(x) \in \eta(\tilde{w}(x))\ \text{a.e. on}\ E_\varepsilon.$$ Without loss of generality we may assume that $Dom(\eta)$ is bounded. If $Dom(\eta)$ is not bounded, consider $\tilde{\eta}=\eta+\partial I_B$, where $I_B$ is the indicator function of a ball centered at $0$ of large radius: $$I_B(x)=\left\{ \begin{array}{lc} 0&x\in B;\\ +\infty& x\notin B. \end{array} \right.$$ By the monotonicity of $\eta$, since $g_{n_k}\in \eta(w_{n_k})$, we have $$(\tilde{u}-g_{\mu_{n_k}},\tilde{w}-w_{\mu_{n_k}})\geq 0$$ a.e. on $E_\varepsilon$, thus $$\int_{E_\varepsilon}(\tilde{u}-g_{\mu_{n_k}},\tilde{w}-w_{\mu_{n_k}})\, dx \geq 0.$$ Consequently $\int\limits_{E_\varepsilon}(\tilde{u}-u,\tilde{w}-w)\, dx \geq 0$. Let $\tilde{w} = (I+\eta)^{-1}(w+u)$. Remark that $\tilde{w} \in L_{\infty}(E_\varepsilon)$ since $Dom(\eta)$ is bounded. We have $\tilde{w}+\eta\tilde{w} \in w+u$ a.e. on $E_\varepsilon$. Choosing $$\label{eq10}\ \tilde{u}=w+u-\tilde{w}$$ we get $\int\limits_{E_\varepsilon}|w-\tilde{w}|^2\,dx \leq 0$ so that $\tilde{w}=w$ a.e. Therefore from [\[eq10\]](#eq10){reference-type="eqref" reference="eq10"} we have $\tilde{u} = u$. Substituting both of this equalities in [\[eq9\]](#eq9){reference-type="eqref" reference="eq9"} we have $u(x) \in \eta(w(x))$ a.e. on $E_\varepsilon$. Therefore $w(x) \in \varphi(u(x))$ a.e. on $E_\varepsilon$. Since $\varepsilon$ is arbitrary, we conclude that $w(x) \in \varphi(u(x))$ a.e. on $B_N$. Therefore, proof of Lemma [Lemma 14](#L1-13){reference-type="ref" reference="L1-13"} and thus Theorem [Theorem 11](#L1-6){reference-type="ref" reference="L1-6"} is finished. ◻ From maximal monotonicity of operator $\partial \Psi (u)$ in $H_{-1}$ and it's characterization in terms of $D^\alpha_N$, we obtain that statement of Theorem [Theorem 8](#main){reference-type="ref" reference="main"} holds. **Remark 16**. *The above results remain valid, if we consider, instead of $\mathbb Q_p$, an arbitrary non-Archimedean local field. In turn, this makes it easy to develop multi-dimensional generalizations of our results. 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arxiv_math

--- abstract: | We use high-low frequency methods developed in the context of decoupling to prove sharp (up to $C_\varepsilon R^\varepsilon$) square function estimates for the moment curve $(t,t^2,\ldots,t^n)$ in $\mathbb R^n$. Our inductive scheme incorporates sharp square function estimates for auxiliary conical sets, which allows us to fully exploit lower dimensional information. address: - | Department of Mathematics\ Massachusetts Institute of Technology\ Cambridge, MA 02142-4307, USA - | Department of Mathematics\ Massachusetts Institute of Technology\ Cambridge, MA 02142-4307, USA author: - Larry Guth - Dominique Maldague bibliography: - arxivdraft.bib title: A sharp square function estimate for the moment curve in $\mathbb R^n$ --- # Introduction Let $\gamma_n:[0,1]\to\mathbb R^n$ be the $n$-dimensional moment curve defined by $(t,t^2,\ldots,t^n)$ and consider functions whose Fourier transforms are supported in a neighborhood of the moment curve. Writing the neighborhood as a finitely-overlapping union $\cup\theta$, we study square function estimates of the form $$\label{sqfnform} \int_{\mathbb R^n}|\sum_\theta f_\theta|^p\le S(\cup\theta)\int_{\mathbb R^n}(\sum_\theta|f_\theta|^2)^{\frac{p}{2}}$$ for any Schwartz $f_\theta:\mathbb R^n\to\mathbb C$ with $\mathrm{supp\,}\widehat{f}_\theta\subset \theta$. We prove bounds of the form $S(\cup\theta)\le C_\varepsilon(\#\theta)^\varepsilon$ for any $\varepsilon>0$, for the optimal range of exponents $2\le p\le p_{n}$. When $p=2$, [\[sqfnform\]](#sqfnform){reference-type="eqref" reference="sqfnform"} follows directly from Plancherel's theorem. When $n=2$ and $p=4$, [\[sqfnform\]](#sqfnform){reference-type="eqref" reference="sqfnform"} follows from a counting argument [@feffL4; @cordoba]. This approach was extended to higher dimensions in [@gpsqfn], but does not apply for non-even integers and only gives the sharp range of exponents when $n=2$. This manuscript builds on the work in [@locsmooth] for the cone in $\mathbb R^3$ and [@maldagueM3] for the moment curve in $\mathbb R^3$ by further developing tools from decoupling theory to prove sharp square function estimates. Although the critical exponent for the cone in $\mathbb R^3$ is $p=4$, counting arguments are not sufficient to prove the sharp square function estimate. Guth, Wang, and Zhang introduced tools like wave envelope estimates which gave very strong information about the cone in $\mathbb R^3$. However, there was and is the question whether these ideas apply to any other manifolds besides the $2$-dimensional cone in $\mathbb R^3$. This paper generalizes these ideas to all moment curves, cones over moment curves, and some other shapes. The proof is by induction on dimension. In order to make the induction work, we need to work with not only moment curves and cones over moment curves but also more general shapes which we call $m$th order Taylor cones. Figuring out the right shapes to include to make the induction work is one of the main new ingredients in the paper. For $R\in 2^{n\mathbb N}$, define the anisotropic neighborhood $\mathcal{M}^n(R)\subset\mathbb R^n$ of the $n$-dimensional moment curve by $$\label{mmomcurve} \mathcal{M}^n(R):=\big\{\gamma_n(t)+\sum_{j=1}^n\lambda_j\gamma_n^{(j)}(t):t\in[0,1],\quad|\lambda_j|\le R^{-\frac{j}{n}} \big\}.$$ Use $I_\theta$ to denote the intervals of length $R^{-\frac{1}{n}}$ with endpoints in $R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$ and define $\Theta^n(R)$ to be the collection of $\theta:=\mathcal{M}^n(R)\cap\{t\in I_\theta\}$, so that $\mathcal{M}^n(R)=\bigcup_{\theta\in\Theta^n(R)}\theta$. Each $\theta$ is approximately the convex hull of the subarc of $\gamma_n$ corresponding to the interval $I_\theta$. Let $p_{n}=\frac{n(n+1)}{2}+1$. Our main theorem is the following. **Theorem 1**. *For any $\varepsilon>0$, there exists $C_\varepsilon\in(0,\infty)$ so that $$\int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)} f_\theta|^p\le C_\varepsilon R^\varepsilon\int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}|f_\theta|^2|^{\frac{p}{2}}$$ for any $2\le p\le p_n$ and any Schwartz functions $f_\theta:\mathbb R^n\to\mathbb C$ with $\mathrm{supp\,}\widehat{f}_\theta\subset\theta$.* Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} is the $n$-dimensional generalization of the $3$-dimensional result in [@maldagueM3]. Adapting the sharp example discussed in [@maldagueM3] to $n$-dimensions is straightforward, and shows that the range of $p$ in Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"} is sharp. The argument for $\mathcal{M}^3(R)$ in [@maldagueM3] used sharp wave envelope estimates (more refined versions of square function estimates) for the parabola in $\mathbb R^2$ and for the cone in $\mathbb R^3$. Our inductive scheme for $\mathcal{M}^n(R)$ involves analyzing more complicated shapes which do not admit wave envelope estimates, but do nevertheless satisfy sharp square function estimates. We provide some descriptions for the geometries and intuition for the inductive scheme in [\[geointro\]](#geointro){reference-type="ref" reference="geointro"}-[\[intro2\]](#intro2){reference-type="ref" reference="intro2"}. Define $\phi_n:[0,1]\to\mathbb R^{n+1}$ by $\phi_n(t)=(1,\frac{t}{1!},\frac{t^2}{2!},\ldots,\frac{t^n}{n!})$. For $0\le m\le n-1$, define the $m$th order Taylor cone $\Gamma^{n+1}_m(R)\subset\mathbb R^{n+1}$ to be $$\label{mcone} \Gamma_m^{n+1}(R):=\Big\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(t):t\in[0,1],\quad|\lambda_i|\le R^{-\frac{i}{n}}\quad\forall\,i,\quad \max_{0\le j\le m}|\lambda_j|2^{m+1-j}R^{\frac{j}{n}}\ge 1 \Big\}.$$ Write $\Gamma_m^{n+1}(R)=\bigcup_{\theta\in\Xi_m^{n+1}(R)}\theta$ where $\Xi_m^{n+1}(R)$ is the collection of subsets $\theta=\Gamma_m^{n+1}(R)\cap\{t\in J_\theta\}$, with $J_\theta$ varying over intervals of length $2^{-100n}R^{-\frac{1}{n}}$ with endpoints $2^{-100n}R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$. We define multi-scale constants which are part of the inductive scheme. For the moment curve, let $\mathbb{M}^n(R)$ denote that smallest constant such that $$\int_{\mathbb R^{n}}|\sum_{\theta\in\Theta^n(R)}f_\theta|^{p}\le \mathbb{M}^n(R) \int_{\mathbb R^{n}}|\sum_{\theta\in\Theta^n(R)}|f_\theta|^2|^{\frac{p}{2}}$$ for any $2\le p\le p_n$ and any Schwartz functions $f_\theta:\mathbb R^{n}\to\mathbb C$ satisfying $\mathrm{supp\,}\widehat{f}_\theta\subset\theta\in\Theta^n(R)$. Similarly, for $0\le m<n-1$, define $\mathbb C_m^{n+1}(R)$ be the smallest constant such that $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R)}g_\theta|^{p}\le \mathbb{C}_m^{n+1}(R) \int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R)}|g_\theta|^2|^{\frac{p}{2}}$$ for any $2\le p\le p_{n-m}$ and any Schwartz functions $g_\theta:\mathbb R^{n+1}\to\mathbb C$ satisfying $\mathrm{supp\,}\widehat{g}_\theta\subset\theta\in\Xi^{n+1}_m(R)$. If $m=n-1$, then define $\mathbb C_{n-1}^{n+1}(R)$ to be the smallest constant such that $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}g_\theta|^{p}\le \mathbb{C}_m^{n+1}(R) \int_{\mathbb R^{n+1}}\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}|g_\theta|^p$$ for any $1\le p\le 2$ and any Schwartz functions $g_\theta:\mathbb R^{n+1}\to\mathbb C$ satisfying $\mathrm{supp\,}\widehat{g}_\theta\subset\theta\in\Xi^{n+1}_{n-1}(R)$. Write $a\lesssim b$ to denote $a\le C b$ for some absolute constant $C$, meaning $C$ only depends on the fixed parameters of the set-up. The notation $a\sim b$ means $a\lesssim b$ and $b\lesssim a$. We write $a\lesssim_\varepsilon b$ to emphasize that the implicit constant depends on some parameter $\varepsilon$. The following two propositions together with the $L^4$ square function estimate for the parabola [@cordoba; @feffL4] as a base case, imply Theorem [Theorem 1](#maintheorem){reference-type="ref" reference="maintheorem"}. **Proposition 1**. *Let $n\ge 2$. Suppose that for all $\varepsilon>0$ and all $R>1$, $$\begin{aligned} \mathbb{M}^k(R)&\lesssim_\varepsilon R^\varepsilon\quad\text{for all}\quad 1\le k\le n \label{conecon1}\\ \mathbb C^{k+1}_m(R)&\lesssim_\varepsilon R^\varepsilon\quad\text{for all}\quad 1\le k\le n-1\quad\text{and}\quad 0\le m\le k-1. \label{conecon2}\end{aligned}$$ Then for all $\varepsilon>0$ and $R>1$, $\mathbb C_m^{n+1}(R)\lesssim_\varepsilon R^\varepsilon$ for any $0\le m\le n-1$.* **Proposition 2**. *Let $n\ge 2$. Suppose that for all $\varepsilon>0$ and all $R>1$, $$\begin{aligned} \mathbb{M}^k(R)&\lesssim_\varepsilon R^\varepsilon\quad\text{for all}\quad 1\le k\le n-1 \\ \mathbb C^{k}_m(R)&\lesssim_\varepsilon R^\varepsilon\quad\text{for all}\quad 1\le k\le n\quad\text{and}\quad 0\le m\le k-1. \end{aligned}$$ Then for all $\varepsilon>0$ and all $R>1$, $\mathbb{M}^n(R)\lesssim_\varepsilon R^\varepsilon$.* The first half of the paper ([\[prelimsec\]](#prelimsec){reference-type="ref" reference="prelimsec"},[\[conesec\]](#conesec){reference-type="ref" reference="conesec"}) concerns Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"}. We define local, multi-scale constants and introduce a key iteration to control how different scales are related to one another. In the second half of the paper ([\[tools\]](#tools){reference-type="ref" reference="tools"}-[\[mainsec\]](#mainsec){reference-type="ref" reference="mainsec"}), we prove Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} by following a high-low inductive scheme which also incorporates the key iteration. We describe relevant geometries in [\[geointro\]](#geointro){reference-type="ref" reference="geointro"} and features of a general multi-scale approach to proving square function estimates in [\[intro1\]](#intro1){reference-type="ref" reference="intro1"}. In [\[intro2\]](#intro2){reference-type="ref" reference="intro2"}, we describe the key iteration in a special case. **Acknowledgements.** LG is supported by a Simons Investigator grant. DM is supported by the National Science Foundation under Award No. 2103249. ## Description of geometries [\[geointro\]]{#geointro label="geointro"} Let $0\le m\le n-1$. The following are properties of the $m$th order Taylor cone $\Gamma_m^{n+1}(R)$: 1. $\Gamma_m^{n+1}(R)$ is contained in a ball of radius $\sim_n1$ centered at the origin. 2. Each $\theta\in\Xi_m^{n+1}(R)$ is a subset of the convex set $$\overline{\theta}:=\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{i}{n}}\quad\forall i\},$$ where $\theta$ has associated $t$-interval $[a,a+2^{-100n}R^{-\frac{1}{n}}]$. The displayed set is comparable to (meaning contains and is contained in a dilated copy of) a box of dimensions $1\times R^{-\frac{1}{n}}\times\cdots\times R^{-1}$. To see an element of $\Xi_0^{n+1}(R)$ from $\overline{\theta}$, imagine removing a box of dimensions $1/2\times CR^{-\frac{1}{n}}\times\cdots CR^{-1}$ from $\overline{\theta}$. If we remove a box of dimension $1/2\times R^{-\frac{1}{n}}/2\times\cdots\times R^{-\frac{m}{n}}/2\times CR^{\frac{m+1}{n}}\times\cdots\times CR^{-1}$ from $\overline{\theta}$, then we obtain an element of $\Xi_m^{n+1}(R)$. 3. $\Gamma_m^{n+1}(R)$ contains the following copies of $\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})$: - $\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})\times\{0\}^m$ - $\{0\}\times R^{-\frac{1}{n}}\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})\times \{0\}^{m-1}$ - $\{0\}^2\times R^{-\frac{2}{n}}\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})\times\{0\}^{m-2}$ $\vdots$ - $\{0\}^m\times R^{-\frac{m}{n}}\Gamma_{0}^{n-m+1}(R^{\frac{n-m}{n}})$ . 4. If $r<R$, then $\Gamma_0^{n+1}(R)\subset\Gamma_0^{n+1}(r)$. 5. If $r<R$ and $m>0$, then $\Gamma_m^{n+1}(R)$ and $\Gamma_m^{n+1}(r)$ have a nontrivial symmetric difference. See the beginning of [\[conesec\]](#conesec){reference-type="ref" reference="conesec"} for the technical significance of this observation. ## A basic inductive scheme for square function estimates [\[intro1\]]{#intro1 label="intro1"} To demonstrate the usefulness of an inductive structure for square function estimates, we first review an earlier counting argument that applies for certain even exponents. Consider $\mathcal{M}^2(R)$ (the parabola), for which the critical exponent is $p=4$. If $f_\theta:\mathbb R^2\to\mathbb C$ has Fourier support in $\theta\in \Theta^2(R)$, we have by Plancherel's theorem $$\label{L4exp} \int_{\mathbb R^2}|\sum_{\theta\in\Theta^2(R)}f_\theta|^4=\sum_{i=1}^4\sum_{\theta_i\in\Theta^2(R)}\int_{\mathbb R^2}\widehat{f}_{\theta_1}*\widehat{f}_{\theta_2}\overline{\widehat{f}_{\theta_1}*\widehat{f}_{\theta_2}}.$$ Since $\widehat{f}_{\theta_i}*\widehat{f}_{\theta_j}$ is supported in the sum-set $\theta_i+\theta_j$, bounding the number of $\theta_l,\theta_k$ for which $(\theta_i+\theta_j)\cap(\theta_l+\theta_k)\not=\emptyset$ leads to a square function estimate. These observations were first made by Fefferman and later recorded by Cordoba [@feffL4; @cordoba]. In [@gpsqfn], the authors extended the counting approach to moment curves in all dimensions, for even exponents in a certain range. However, for exponents $p$ which are not even integers or which are too large, there is no counting approach available. It may also happen that the sets $\theta_i+\theta_j$ have large overlap, but there is other cancellation in [\[L4exp\]](#L4exp){reference-type="eqref" reference="L4exp"} which still leads to a square function estimate. This is the case for the cone in $\mathbb R^3$, which is partitioned into angular sectors $\theta$. The sharp square function estimate for the cone in $\mathbb R^3$ has critical exponent $p=4$ [@locsmooth], but it does not follow from a counting argument. Now we describe the structure of an inductive set-up which applies to square function estimates for arbitrary $p$. For $R>1$, consider an abstract subset $X(R)\subset\mathbb R^n$ which is partitioned into $\sqcup_{\theta\in \mathcal{S}(R)}\theta$. Suppose that if $r\le R$, $\theta\in\mathcal{S}(R)$, and $\tau\in\mathcal{S}(r)$, then either $\theta\subset \tau$ or $\theta\cap\tau=\emptyset$. If $g:\mathbb R^n\to\mathbb C$ is a function with Fourier support in $X(R)$, then for each $r\le R$, we have $g=\sum_{\tau\in\mathcal{S}(r)}g_\tau$, where $g_\tau=(\widehat{g}\chi_{\tau})^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,\,}{\scalebox{-1}{\tmpbox}}% }$. We are interested in bounding the smallest constant $D_{n,p}(R)$ satisfying $$\int_{\mathbb R^n}|g|^p\le D_{n,p}(R) \int_{\mathbb R^n}(\sum_{\theta\in\mathcal{S}(R)}|g_\theta|^2)^{\frac{p}{2}}$$ for any $g$ with Fourier transform supported in $X(R)$. In particular, we would like to show that for any $\varepsilon>0$, there exists $C_\varepsilon>0$ such that $D_{n,p}(R)\le C_\varepsilon R^\varepsilon$ for all $R$. We will describe some ideas introduced by Guth, Wang, and Zhang to incorporate multi-scale analysis, though the following is not precisely how their argument is presented in [@locsmooth]. The idea is to study a two-parameter quantity $D_{n,p}(r,R)$ (for $1<r<R$), which is the smallest constant satisfying $$\int_{\mathbb R^n}(\sum_{\tau\in\mathcal{S}(r)}|g_\tau|^2)^{\frac{p}{2}}\le D_{n,p}(r,R) \int_{\mathbb R^n}(\sum_{\theta\in\mathcal{S}(R)}|g_\theta|^2)^{\frac{p}{2}}$$ for all $g$ with $\mathrm{supp\,}\widehat{g}\subset X(R)$. Note that $D_{n,p}(1,R)=D_{n,p}(R)$. Induction on scales says that to prove $D_{n,p}(R)\le C_\varepsilon R^\varepsilon$, we may assume that $D_{n,p}(r)\le C_\varepsilon r^\varepsilon$ for all $r<R/2$. Working instead with $D_{n,p}(r,R)$, it suffices to prove that $D_{n,p}(r,R)\le C_\varepsilon(R/r)^\varepsilon$ with the assumption that $D_{n,p}(s,S)\le C_\varepsilon(S/s)^\varepsilon$ for all $(S/s)<(R/r)/2$, which is a stronger inductive hypothesis. One may also think of bounding $D_{n,p}(r,R)$ via an iterative argument. Let $K>0$ be a large parameter we will choose later. Let $\mathrm{supp\,}\widehat{g}\subset X(R)$. There are three key ingredients: 1. (Base case) For each $1\le r\le K^2$, $\int_{\mathbb R^n}|\sum_{\tau\in\mathcal{S}(r)}|g_\tau|^2|^{\frac{p}{2}}\lesssim A_\delta K^\delta\int_{\mathbb R^n}\big(\sum_{\tau'\in\mathcal{S}(K^2)}|g_{\tau'}|^2\big)^{\frac{p}{2}}$. 2. (Progress) If $K^3\le Kr\le R$, then one of the following holds: 1. $\int_{\mathbb R^n}(\sum_{\tau\in\mathcal{S}(r)}|g_\tau|^2)^{\frac{p}{2}}\le B_\delta K^\delta\int_{\mathbb R^n}(\sum_{\tau'\in\mathcal{S}(Kr)}|g_{\tau'}|^2)^{\frac{p}{2}}$, or 2. $\int_{\mathbb R^n}(\sum_{\tau\in\mathcal{S}(r)}|g_\tau|^2)^{\frac{p}{2}}\le B_\delta K^\delta\sum_{\tau'\in \mathcal{S}(r/K)}\int_{\mathbb R^n}(\sum_{\substack{\tau\in\mathcal{S}(r)\\ \tau\subset\tau'}}|g_{\tau}|^2)^{\frac{p}{2}}$. 3. (Rescaling) For any $\rho\le r\le R$, $$\sum_{\tau'\in\mathcal{S}(\rho)}\int_{\mathbb R^n}(\sum_{\substack{\tau\in\mathcal{S}(r)\\\tau\subset\tau'}}|g_\tau|^2)^{\frac{p}{2}} \le D_{n,p}(r/\rho,R/\rho)\int_{\mathbb R^n}(\sum_{\theta\in\mathcal{S}(R)}|g_\theta|^2)^{\frac{p}{2}}.$$ We can then show that $D_{n,p}(r,R)\le C_\varepsilon R^\varepsilon$. Indeed, by using the above ingredients in succession, we have for $K^3\le Kr\le R$ that $$D_{n,p}(r,R)\le A_{\delta}B_{\delta} K^{2\delta} \left[ D_{n,p}(rK,R)+D_{n,p}(K^2,RK/r)\right].$$ Iterating and choosing $\delta>0$ and $K>0$ depending on $\varepsilon>0$ leads to the upper bound $D_{n,p}(r,R)\le C_\varepsilon R^\varepsilon$. Using lower dimensional results, a base case (ingredient (1)) is available for the cone-like regions $\Gamma_m^{n+1}(R)$, which enables analysis of multi-scale constants as described above. There is no similar base case for $\mathcal{M}^n(R)$, which is why the proof techniques bounding $\mathbb C_m^{n+1}(R)$ and $\mathbb{M}^n(R)$ are organized differently. See the beginning of [\[tools\]](#tools){reference-type="ref" reference="tools"} for more about the strategy to bound $\mathbb{M}^n(R)$. The main work of establishing Propositions [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} and [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} lies in justifying the second *progress* ingredient above. We use algorithms adapted to the specific geometries being considered, in Lemmas [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}, [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"}, and [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"} for $\Gamma_m^n(R)$ and in Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"} for $\mathcal{M}^n(R)$. In the following subsection, we describe a simplified example of the algorithm which also demonstrates how $m$th order Taylor cones naturally arise. ## A simplified explanation of the multi-scale algorithm and $m$th order Taylor cones [\[intro2\]]{#intro2 label="intro2"} In addition to induction on the scale $R$, there is another important type of induction we employ: induction on the integers $m$ and $k$ in $\mathbb C_m^{k+1}(R)$ and $\mathbb{M}^k(R)$. One type of base case is when $m=k-1$, where we bound $\mathbb C_{k-1}^{k+1}(R)$ directly using $L^2$-based analysis. The other base case is the $L^4$ square function estimate for the parabola. Let $\mathbb{M}^k$ be the statement that $\mathbb{M}^k(R)\lesssim_\varepsilon R^\varepsilon$ for all $\varepsilon>0$ and $R>1$ (and similarly for $\mathbb C_m^{k+1}$). Then the sequence of implications is as follows, with the non-blue terms depending on all previous inputs: $$\textcolor{blue}{\mathbb{M}^2}\overset{\textcolor{blue}{\mathbb C_1^{2+1}}}{\implies} \mathbb C_0^{2+1}\implies {\mathbb{M}^3} \overset{\textcolor{blue}{\mathbb C_2^{3+1}}}{\implies} \mathbb C_1^{3+1}\implies \mathbb C_0^{3+1}\implies \mathbb{M}^4 \overset{\textcolor{blue}{\mathbb C_3^{4+1}}}{\implies}\mathbb C_2^{4+1}\implies\mathbb C_1^{4+1}\implies \mathbb C_0^{4+1}\implies\mathbb{M}^5\overset{\textcolor{blue}{\mathbb C_4^{5+1}}}{\implies} \cdots$$ For concreteness, let us consider how to prove $\mathbb{M}^4$. We focus just on the critical exponent $p_4=11$. Given an intermediate expression $$\label{inteqn} \int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)}|f_\tau|^2)^{\frac{11}{2}}$$ where $r<R$ and $\mathrm{supp\,}\widehat{f}\subset \mathcal{M}^4(R)$, we would like to make progress towards an upper bound of the form $$\label{goalintro} C_\varepsilon R^\varepsilon\int_{\mathbb R^4}(\sum_{\theta\in\Theta^4(R)}|f_\theta|^2)^{\frac{11}{2}} ,$$ where *progress* means an inequality of type 2(a) or 2(b) from the key ingredients listed in [\[intro1\]](#intro1){reference-type="ref" reference="intro1"}. Consider the Fourier support of $\sum_{\tau\in\Theta^4(r)}|f_\tau|^2$ in [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"}. Each summand $|f_{\tau}|^2$ has Fourier transform $\widehat{f}_\tau*\widehat{\overline{f}}_{\tau}$, which is supported in $\tau-\tau$. If $\tau$ is associated to the $t$-interval $[a,a+r^{-\frac{1}{4}}]$, then $\tau-\tau$ is contained in $$\begin{aligned} \label{tau-tauintro} \{\sum_{i=1}^4\lambda_i\gamma_4^{(i)}(a):|\lambda_i|\lesssim r^{-\frac{i}{4}} \}. \end{aligned}$$ Note that the exponent $\frac{11}{2}$ lies in the range $2\le \frac{11}{2}\le 7$, which is the range of $p$ for which $\Gamma_0^{3+1}(r)$ satisfies a square function estimate. The subset of $\tau-\tau$ corresponding to $|\lambda_1|\sim r^{-\frac{1}{4}}$ is essentially an element of $\Xi_0^{3+1}(r^{\frac{3}{4}})$ dilated by a factor of $r^{-\frac{1}{4}}$. A high-low frequency decomposition and the triangle inequality bounds [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} by $$\begin{aligned} \label{hilo} C\int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}(\widehat{|f_{\tau}|^2}\eta_r)^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,}{\scalebox{-1}{\tmpbox}}% }|^{\frac{11}{2}} +C\int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}(\widehat{|f_{\tau}|^2}(1-\eta_r))^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,}{\scalebox{-1}{\tmpbox}}% }|^{\frac{11}{2}},\end{aligned}$$ where $\eta_r$ is identically $1$ in the ball of radius $C_0^{-1}r^{-\frac{1}{4}}$ centered at the origin (we will be vague about the selection of a constant $C_0$ here). Label the case where [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} is dominated by the first term in [\[hilo\]](#hilo){reference-type="eqref" reference="hilo"} as Case $(l_{11/2})$ to reflect that the integrand is dominated by low frequencies and the exponent is $11/2$. Case $(h_{11/2})$ is when [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} is dominated by the second term in [\[hilo\]](#hilo){reference-type="eqref" reference="hilo"}. [Case $(l_{11/2})$]{.ul} The low-frequency integrand satisfies a favorable pointwise inequality by local $L^2$ orthogonality: $$||f_\tau|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _r|\lesssim \sum_{\substack{\tau'\in\Theta^4(C_0r)\\\tau'\subset\tau}} |f_{\tau'}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _r|\qquad\text{for each $\tau\in\Theta^4(r)$}.$$ Then by Young's convolution inequality, since $\|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _r\|_1\sim 1$, conclude that [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} is bounded by $$C\int_{\mathbb R^4}|\sum_{\tau'\in\Theta^4(C_0r)}|f_{\tau'}|^2|^{\frac{11}{2}}.$$ The refinement of $\tau$ into $\tau'$ with controlled implicit constants represents progress of type (2a) towards [\[goalintro\]](#goalintro){reference-type="eqref" reference="goalintro"}, and Case $(l_{11/2})$ concludes. [Case $(h_{11/2})$]{.ul} Assume that [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} is bounded by the second, high-frequency term in [\[hilo\]](#hilo){reference-type="eqref" reference="hilo"}. Since the exponent $\frac{11}{2}$ is in the range $2\le \frac{11}{2}\le 7$ and the support of each $\widehat{|f_\tau|^2}(1-\eta_r)$ is identified with an element of $\Xi_0^{3+1}(r^{\frac{3}{4}})$, we have $$\int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}(\widehat{|f_{\tau}|^2}(1-\eta_r))^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,}{\scalebox{-1}{\tmpbox}}% }|^{\frac{11}{2}}\lesssim C_0^{3+1}(r^{\frac{3}{4}}) \int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}|(\widehat{|f_{\tau}|^2}(1-\eta_r))^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,}{\scalebox{-1}{\tmpbox}}% }|^2|^{\frac{11}{4}}.$$ We are done using the high-frequency cutoff, and the integral on the right hand side is bounded using Cauchy-Schwarz and Young's inequality by $$\int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)}|f_{\tau}|^4)^{\frac{11}{4}}.$$ This is similar to the initial expression since the summands $|f_\tau|^4$ have approximately the same Fourier support [\[tau-tauintro\]](#tau-tauintro){reference-type="eqref" reference="tau-tauintro"} as $|f_\tau|^2$ (since $\widehat{|f_\tau|^4}$ may be written as a 4-fold convolution of two copies of $\widehat{f}_\tau$ and two copies of the reflection of $\widehat{f}_\tau$). However, we have made some progress by reducing the exponent from $\frac{11}{2}$ to $\frac{11}{4}$, which lies in the range $2\le \frac{11}{4}\le 4$. The next step is to iterate the previous argument by again performing a high-low decomposition of frequency space. Since we are working in $\mathbb R^4$ and in the range $[2,4]$, both the geometries $\Gamma_0^{3+1}$ and $\Gamma_1^{3+1}$ admit square function estimates. We will define the high frequency part of $\mathrm{supp\,}\widehat{|f_{\tau}|^4}$ so that it may be identified with an element of $\Xi_1^{3+1}(r^{\frac{3}{4}})$. See the comments about key decisions at the end of this subsection for an explanation of why we chose $\Xi_1^{3+1}(r^{\frac{3}{4}})$ instead of $\Xi_0^{3+1}(r^{\frac{3}{4}})$. If the $t$-interval associated to $\tau\in\Theta^4(r)$ has initial point $a$, we divide the Fourier support of $|f_\tau|^4$ into the high part, $$\label{newhi} \tau_h:=\{\sum_{i=1}^4\lambda_i\gamma_4^{(i)}(a):|\lambda_i|\lesssim r^{-\frac{i}{n}}\,\forall i,\quad \max(|\lambda_1|C_1^{2}r^{\frac{1}{4}},|\lambda_2|C_1r^{\frac{2}{4}})\ge 1\},$$ which is identified with an element of $\Xi_1^{3+1}(r^{\frac{3}{4}})$ dilated by $r^{-\frac{1}{4}}$, and the low part $$\label{newlo} \tau_l:= \{\sum_{i=1}^4\lambda_i\gamma_4^{(i)}(a):|\lambda_i|\lesssim r^{-\frac{i}{n}}\,\forall i,\quad \max(|\lambda_1|C_1^2r^{\frac{1}{4}},|\lambda_2|C_1r^{\frac{2}{4}})\le 1\} ,$$ where $C_1$ is a large constant that we do not specify. Let $\eta_{\tau_h}+\eta_{\tau_l}$ be smooth bump functions which are identically one on $\mathrm{supp\,}\widehat{|f_\tau|^4}$ and supported in $\tau_h$ and $\tau_l$ respectively. Consider the cases where the $\tau_h$ and $\tau_l$ frequencies dominate separately. [Case $(h_{11/2},l_{11/4})$]{.ul} In this case, we have the inequality $$\int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)}|f_\tau|^2)^{\frac{11}{2}} \le C \int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}|f_\tau|^4*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|^{\frac{11}{4}} .$$ Consider each summand one at a time. By Fubini's theorem, we have $$|f_{\tau}|^4*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|(x,y)= \int_{\mathbb R^2}\int_{\mathbb R^2}|f_{\tau}(x',y')|^4|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|(x-x',y-y')dx' dy'$$ where $x,y\in\mathbb R^2$. We will consider upper bounds for the inner integral above. Because the Fourier transform on $\mathbb R^4$ factors into the composition of 2-dimensional Fourier transforms, we may regard $f_{\tau}$ as a function in the first two variables with Fourier support in $\mathcal{M}^2(R^{\frac{1}{2}})$. The weight function $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|$ is localized in the first two variables to a rectangle of dimensions $C_1^2r^{\frac{1}{4}}\times C_1r^{\frac{2}{4}}$. Invoking a local $L^4$ square function estimate for $\mathcal{M}^2(R^{\frac{1}{2}})$ then yields the pointwise inequality $$|f_{\tau}|^4*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|\lesssim \mathbb{M}^2(C_1) |\sum_{\substack{\tau'\in\Theta^4(C_1r)\\\tau'\subset\tau}}|f_{\tau'}|^2|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|.$$ In summary, we have an upper bound for [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} of the form $$C_0 \mathbb C_0^{3+1}(r^{\frac{3}{4}})\mathbb{M}^2(C_1) \int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)}|\sum_{\substack{\tau'\in\Theta^4(C_1r)\\\tau'\subset\tau}}|f_{\tau'}|^2|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}|)^{\frac{11}{4}}.$$ We remark that $C_1$ should be chosen to be significantly larger than $C_0$ in order to make progress of type 2(a). Also, since $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_l}$ depends on each $\tau$, we cannot simply invoke Young's convolution inequality to eliminate the auxiliary convolutions. This is a technical point, but by carefully defining the functions $\eta_{\tau_l}$, we show that it is safe to substitute the integral displayed above by $$\int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)}|\sum_{\substack{\tau'\in\Theta^4(C_1r)\\\tau'\subset\tau}}|f_{\tau'}|^2|^2)^{\frac{11}{4}}\le \int_{\mathbb R^4}( \sum_{\substack{\tau'\in\Theta^4(C_1r)}}|f_{\tau'}|^2)^{\frac{11}{2}},$$ which completes the *progress* step for this line of reasoning. [Case $(h_{11/2},h_{11/4})$]{.ul} The high-frequency part that we have not yet addressed is the case that $$\eqref{inteqn} \lesssim \mathbb C_0^{3+1}(r^{\frac{3}{4}})\int_{\mathbb R^4}|\sum_{\tau\in\Theta^4(r)}|f_\tau|^4*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_h}|^{\frac{11}{4}}.$$ Recall that each $\tau_h$ is identified with an (almost unique, depending on $C_1$) element of $\Xi_1^{3+1}(r^{\frac{3}{4}})$ dilated by a fixed factor, and so the right hand side above is $$\lesssim \mathbb C_0^{3+1}(r^{\frac{3}{4}})\mathbb C_1^{3+1}(r^{\frac{3}{4}})\int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)} ||f_\tau|^4*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau_h}|^2)^{\frac{11}{8}} .$$ Although we can no longer cite standard inequalities like Cauchy-Schwarz or Young's inequality, we show that the integral above can be bounded by $$\label{l8} \int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)} |f_\tau|^8)^{\frac{11}{8}} .$$ At this stage, we see something different from the previous cases. One difference is that the exponent lies in the range $1\le \frac{11}{8}\le 2$, so we should consider using $\mathbb C_2^{3+1}(r^{\frac{3}{4}})$. Since $8$ is an even integer, the Fourier support of $|f_\tau|^8$ is roughly the same as the Fourier support of $|f_\tau|^2$, so we could perform one more high-low frequency decomposition $$\sum_{\tau\in\Theta^4(r)}|f_\tau|^8*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tilde{\tau}_l}+\sum_{\tau\in\Theta^4(r)}|f_\tau|^8*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tilde{\tau}_h}.$$ The difficulty now is dealing with the low part. By regarding the convolution with $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tilde{\tau}_l}$ as an $L^8$ integral against a weight, we would like to invoke previously proven square function estimates for lower dimensional moment curves. The problem is that even for one lower dimension, $\mathcal{M}^3$ only admits a sharp square function estimate in the range $2\le p\le 7$, not $2\le p\le 8$. To address this issue, we use the fact that $\|\cdot\|_{\ell^p}\le \|\cdot\|_{\ell^q}$ whenever $p\ge q$ to bound [\[l8\]](#l8){reference-type="eqref" reference="l8"} by $$\int_{\mathbb R^4}(\sum_{\tau\in\Theta^4(r)} |f_\tau|^6)^{\frac{11}{6}}.$$ If we had chosen to replace the $\ell^8$ expression by an $\ell^7$ expression, then the exponent for the low analysis is no longer an issue, but there is no simple formula for the Fourier transform of $|f_{\tau}|^7$ as there is for $|f_{\tau}|^6$. Furthermore, we remark that the exponent $\frac{11}{6}$ remains in the same range $1\le \frac{11}{8}\le \frac{11}{6}\le 2$ as the previous exponent $\frac{11}{8}$, so we should still identify high-frequency pieces which may be bounded using $\mathbb C_2^{3+1}(r^{\frac{3}{4}})$. We verify that similar arguments involving the exponents $p$ may be made in the general setting in Lemma [Lemma 12](#pprops){reference-type="ref" reference="pprops"}. Define the high and low subsets of $\mathrm{supp\,}\widehat{|f_\tau|^6}$ as $$\tilde{\tau}_h:=\{\sum_{i=1}^4\lambda_i\gamma_4^{(i)}(a):|\lambda_i|\lesssim r^{-\frac{i}{n}} \,\,\forall i,\quad \max(|\lambda_1|C_2^3r^{\frac{1}{4}},|\lambda_2|C_2^2r^{\frac{2}{4}},|\lambda_3|C_2r^{\frac{3}{4}}) \ge 1 \}$$ and $$\tilde{\tau}_l:= \{\sum_{i=1}^4\lambda_i\gamma_4^{(i)}(a):|\lambda_i|\lesssim r^{-\frac{i}{n}} \,\,\forall i,\quad \max(|\lambda_1|C_2^3r^{\frac{1}{4}},|\lambda_2|C_2^2r^{\frac{2}{4}},|\lambda_3|C_2r^{\frac{3}{4}}) \le 1 \},$$ and let $\eta_{\tilde{\tau}_h}$ and $\eta_{\tilde{\tau}_l}$ be adapted to their respective sets. We consider this high-low frequency decomposition in the following final two cases. [Case $(h_{11/2},h_{11/4},l_{11/8})$]{.ul} In the case that the $\tilde{\tau}_l$ frequencies dominate, we have the pointwise inequality $$|\sum_{\tau\in\Theta^4(r)}|f_\tau|^6*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tilde{\tau}_l}|\lesssim \mathbb{M}^3(C_2)|\sum_{\tau\in\Theta^4(r)}|\sum_{\substack{\tau'\subset\tau \\\tau'\in\Theta^4(C_2r)}}|f_{\tau'}|^2|^3*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tilde{\tau}_l}| .$$ A few more technical steps leads to the final inequality bounding [\[inteqn\]](#inteqn){reference-type="eqref" reference="inteqn"} by $$\lesssim \mathbb C_0^{3+1}(r^{\frac{3}{4}})\mathbb C_1^{3+1}(r^{\frac{3}{4}})\mathbb{M}_3(C_2) \int_{\mathbb R^4}(\sum_{\tau'\in \Theta^4(C_2r)}|f_{\tau'}|^2)^{\frac{11}{2}},$$ which is progress of type 2(a) towards [\[goalintro\]](#goalintro){reference-type="eqref" reference="goalintro"}. [Case $(h_{11/2},h_{11/4},h_{11/8})$]{.ul} Now we consider when the $\tilde{\tau}_h$ frequencies dominate. We have the inequality $$\eqref{inteqn}\lesssim \mathbb C_0^{3+1}(r^{\frac{3}{4}})\mathbb C_1^{3+1}(r^{\frac{3}{4}})\mathbb C_2^{3+1}(r^{\frac{3}{4}})\sum_{\tau\in\Theta^4(r)}\int_{\mathbb R^n}|f_\tau|^{11}.$$ This is the one case where we see progress of type 2(b), and the algorithm is concluded. [**Comments about key decisions**]{.ul} In each of the cases in our algorithm, we identify high and low subsets of the Fourier support of $\sum_\tau|f_\tau|^{2k}$ for some $k$, which is approximately contained in $\cup_\tau (\tau-\tau)$. The initial division is the simplest: $$\tag{$\star$} \begin{cases} (\cup_\tau(\tau-\tau))_{\text{low}}:=(\cup_\tau(\tau-\tau))\cap\{|\xi|<C_0^{-1}r^{-\frac{1}{4}}\} \\ (\cup_\tau(\tau-\tau))_{\text{high}}:=(\cup_\tau(\tau-\tau))\cap\{|\xi|\ge C_0^{-1}r^{-\frac{1}{4}}\} \end{cases}.$$ While it would be convenient to use this definition in all of the cases, our criterion for a good high/low decomposition is changing at each stage, depending on the integer $k$ in the expression $(\sum_\tau|f_\tau|^{2k})^{\frac{11}{2k}}$. We have two requirements: 1. there is an $L^{\frac{11}{2k}}$ square function estimate for $(\sum_{\tau}|f_\tau|^{2k})_{\text{high}}$, and 2. there is a pointwise $L^{2k}$ estimate for each $(|f_\tau|^{2k})_{\text{low}}$. It is easier to satisfy (1) when the high set is smaller (for example, the high set from ($\star$)) and it is easier to satisfy (2) when the low set is smaller. Identifying copies of $\Gamma_1^{3+1}(r)$ and $\Gamma_2^{3+1}(r)$ within $\cup_\tau(\tau-\tau)$ as the high sets in the cases $k=2$ and $k=3$ respectively allows for both (1) and (2) to be satisfied. The set $\Gamma_1^{2+1}(r)$ was also studied in [@locsmooth], but only an $L^2(R^3)$ estimate was required in that set-up. It is fairly easy to get a similar $L^2(\mathbb R^4)$ estimate for the set $\Gamma_1^{3+1}(r)$ by finite overlapping on the Fourier side, but this is no longer enough. At some point, we realized that $\Gamma_1^{3+1}(r)$ is closely related to $\Gamma_0^{2+1}(r)$ (the cone in $\mathbb R^3$) and there is an $L^p(\mathbb R^4)$ square function estimate for $2\le p\le 4$ that is similar to the one proven in [@locsmooth]. This allows us to use an $L^{\frac{11}{4}}(\mathbb R^4)$ square function estimate to bound the high part of $\sum_\tau|f_\tau|^4$, despite the rather large high-frequency set identified with $\Gamma_1^{3+1}(r)$. # Preliminaries for the proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} [\[prelimsec\]]{#prelimsec label="prelimsec"} {#preliminaries-for-the-proof-of-proposition-coneinduct-prelimsec} In this section, we prove preliminary technical results which are inputs to the proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"}. In [\[smallgen\]](#smallgen){reference-type="ref" reference="smallgen"}, we describe some flexibility in the definition of the sets in $\Xi_m^{n+1}(R)$ which justifies later appeals to $\mathbb C_m^{n+1}(R)$ to bound functions which are only approximately supported in elements of $\Xi_m^{n+1}(R)$. In [\[L2sec\]](#L2sec){reference-type="ref" reference="L2sec"}, we directly prove key $L^2$ based inequalities which directly bound the constants $\mathbb C_{n-1}^{n+1}(R)$. We also introduce weight functions which will be used extensively to write local versions of inequalities. ## Small generalizations of $\mathbb C_m^{n+1}(R)$ [\[smallgen\]]{#smallgen label="smallgen"} We show that the definition we used for the elements of $\Xi_m^{n+1}(R)$ is not sensitive to changes by constant factors. **Proposition 3** (A general $\mathbb C_m^{n+1}(R)$). *Let $n\ge 2$, $0\le m\le n-1$, and $L\ge 2$. Suppose that $\mathbb C_m^{n+1}(R)\lesssim_{n,\varepsilon}R^\varepsilon$. There is a constant $c_n>0$ so that the following holds. If for $l=0,\ldots,R^{1/n}-1$, $g_{R,L,l}:\mathbb R^n\to\mathbb C$ are Schwartz functions with $$\begin{aligned} \nonumber \mathrm{supp\,}\widehat{g_{R,L,l}}\subset \{\sum_{i=0}^n\lambda_i&\phi_n^{(i)}(t):t\in[\frac{l}{100L2^nR^{\frac{1}{n}}},\frac{l+1}{100L2^nR^{\frac{1}{n}}}],\quad |\lambda_i|\le R^{-\frac{i}{n}}\,\forall \,i,\quad\max_{0\le j\le m}|\lambda_j|2^{m+1-j}R^{\frac{j}{n}}\ge\frac{1}{L}\}, \end{aligned}$$ then $$\int_{\mathbb R^{n+1}}|\sum_{l=0}^{R^{1/n}-1} g_{R,L,l}|^{p}\lesssim_{n,\varepsilon} L^{c_n} R^\varepsilon\int_{\mathbb R^{n+1}}|\sum_{l=0}^{R^{1/n}-1}|g_{R,L,l}|^2|^{\frac{p} {2}}$$ for any $2\le p\le p_{n-m}$.* *Proof.* It suffices to assume that $R\ge L^C$ for a constant $C$ that we may choose. By Taylor expansion, the Fourier support of $g_{R,L,l}$ is contained in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}((2^{100n}L)^{-1}lR^{-\frac{1}{n}}):|\lambda_i|\le (1+\frac{1}{100L})R^{-\frac{i}{n}},\quad\max_{0\le i\le m}|\lambda_i|2^{m+1-i}R^{\frac{i}{n}}\ge \frac{99}{100L}\}.$$ Let $s:\mathbb R\to[0,1]$ be a smooth bump function satisfying $s(t)=1$ when $|t|\le \frac{3}{2}$ and $\mathrm{supp\,}s\subset(-2,2)$. The function $\psi(x_0,\ldots,x_n):=s(x_0)\cdots s(R^{\frac{m}{n}}x_m) s(x_{m+1})\cdots s(x_n)$ is identically equal to $1$ on the box $[-1,1]\times[-R^{-\frac{1}{n}},R^{-\frac{1}{n}}]\times\cdots\times[-R^{-\frac{m}{n}},R^{-\frac{m}{n}}]\times[-1,1]^{n-m}$. Let $A_l:\mathbb R^{n+1}\to\mathbb R^{n+1}$ be a linear map which, in $x_0,\ldots,x_m$, takes $\phi_{m}^{(i)}(l(2^{100n}L)^{-1}R^{-\frac{1}{n}})$ to the standard basis vector ${\bf{e}}_{i+1}$ (with a $1$ in the $(i+1)$st coordinate) for $i=0,\ldots,m$, and which is the identity in the remaining variables $x_{m+1},\ldots,x_n$. The function $\psi\circ A_l$ is supported in $$%\label{ksupp2} \{\sum_{i=0}^m\mu_i\phi_m^{(i)}((2^{100n}L)^{-1}lR^{-\frac{1}{n}}): |\mu_i|\le 2R^{-\frac{i}{n}}\}\times[-2,2]^{n-m}$$ and identically one in $$\{\sum_{i=0}^m\mu_i\phi_m^{(i)}((2^{100n}L)^{-1}lR^{-\frac{1}{n}}): |\mu_i|\le (3/2)R^{-\frac{i}{n}}\}\times[-2,2]^{n-m}.$$ For $j\in\mathbb N$, define $\eta_{l,j}(x_0,\ldots,x_n)$ to be the difference $$\begin{aligned} \label{auxbump} \eta_{l,j}(x_0,\ldots,x_n):=&\psi\circ A_l\Big(\Big(\frac{3}{4}\Big)^j2^{j(m+1)}x_0,\ldots,\Big(\frac{3}{4}\Big)^j2^jx_m,x_{m+1},\ldots,x_n\Big)\\ &\qquad -\psi\circ A_l\Big(\Big(\frac{3}{4}\Big)^{j+1}2^{(j+1)(m+1)}x_0,\ldots,\Big(\frac{3}{4}\Big)^{j+1}2^{j+1}x_m,x_{m+1},\ldots,x_n\Big). \nonumber\end{aligned}$$ Note that $\eta_{l,j}$ is supported in $$\label{ksupp2} \{\sum_{i=0}^m\mu_i\phi_m^{(i)}((2^{100n}L)^{-1}lR^{-\frac{1}{n}}): \max_{0\le i\le m}|\mu_i|[2(4/3)^j2^{-j(m+1)}]^{-1}2^{m-j}(R/2^j)^{\frac{i}{n}}\in[\frac{1}{2},1] \}\times[-2,2]^{n-m}.$$ For the support of $\eta_{l,j}$ to have nonempty intersection with the support of $\widehat{g_{R,L,l}}$, we have $3^j2^{jm}\lesssim L$. Since there are fewer than $C\log L$ many $j$, by dyadic pigeonholing, there is a $j\lesssim\log L$ which satisfies $$\label{RHS} \int_{\mathbb R^{n+1}}|\sum_l g_{R,L,l}|^{p }\le (\log L)^{O(1)} \int_{\mathbb R^{n+1}}|\sum_{l} g_{R,L,l}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta_{l,j}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta_{l,j}}{\scalebox{-1}{\tmpbox}}% |^{p }.$$ Each $\widehat{g_{R,L,l}}\eta_{l,j}$ is supported in the $2\cdot\big(\frac{4}{3}\big)^j2^{-j(m+1)}$-dilation of an element of $\Xi_m^{n+1}(2^{-j}R)$. Using the hypothesis that $\mathbb C_m^{n+1}(R)\lesssim_\varepsilon R^\varepsilon$ and Cauchy-Schwarz, it follows that $$\int_{\mathbb R^{n+1}}|\sum_{l} g_{R,L,l}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta_{l,j}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta_{l,j}}{\scalebox{-1}{\tmpbox}}% |^{p }\lesssim_\varepsilon R^\varepsilon L^{O(1)} \int_{\mathbb R^{n+1}}|\sum_{l} |g_{R,L,l}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta_{j,l}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta_{j,l}}{\scalebox{-1}{\tmpbox}}% |^2|^{\frac{p }{2}}.$$ It remains to prove **Lemma 4**. *$$\int_{\mathbb R^{n+1}}|\sum_{l} |g_{R,L,l}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{l,j}|^2|^{\frac{p}{2}}\lesssim_\varepsilon R^\varepsilon\int_{\mathbb R^{n+1}}|\sum_{l} |g_{R,L,l}|^2|^{\frac{p}{2}}.$$* We delay this proof until [\[truncsec\]](#truncsec){reference-type="ref" reference="truncsec"}. ◻ **Lemma 5** (Localized $\mathbb C_m^{n+1}(R)$). *Let $\psi:\mathbb R^{n+1}\to\mathbb C$ be a smooth function with Fourier transform supported in a ball of radius $R^{-1}$ centered at the origin. If $\mathbb C_m^{n+1}(R)\lesssim_\varepsilon R^\varepsilon$, then $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R)}f_\theta \psi|^{p}\lesssim_\varepsilon R^\varepsilon\int_{\mathbb R^{n+1}} |\sum_{\theta\in\Xi_m^{n+1}(R)}|f_\theta \psi|^2|^{\frac{p}{2}}$$ for any $2\le p\le p_{n-m}$ and any Schwartz functions $f_\theta$ which have Fourier support in $\theta$.* *Proof.* Each summand $f_\theta \psi$ has Fourier transform supported in $$\{\sum_{i=0}^n\lambda_i \phi_n^{(i)}(l2^{-100n}R^{-\frac{1}{n}}):|\lambda_i|\le 2 R^{-\frac{i}{n}},\quad\max_{0\le j\le m}|\lambda_j|2^{m-j}R^{\frac{j}{n}}\ge \frac{1}{4}\}.$$ Therefore, Proposition [Proposition 3](#gencone){reference-type="ref" reference="gencone"} applies. ◻ ## Weight functions and the locally constant property [\[wtsec\]]{#wtsec label="wtsec"} We introduce weight functions which will be used to localize the integrals. Let $\psi_n:\mathbb R^{n}\to[0,\infty)$ be a radial, smooth bump function supported in $|\xi|\le 1$ and satisfying $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _n(x)|>0$ for all $|x|\le 1$. For any $d\in\mathbb N^+$, define $W^{n,d}:\mathbb R^{n}\to[0,\infty)$ by $$W^{n,d}(x)=\sum_{j=0}^\infty \frac{1}{2^{100n^2jd}}|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _n|^2(2^{-j}x).$$ Let $B_0$ be the unit ball centered at the origin in $\mathbb R^n$. For any set $U=T(B_0)$ where $T$ is an affine transformation $T:\mathbb R^n\to\mathbb R^n$, define $$W_{U}^{n,d}(x)=W^{n,d}(T^{-1}(x)).$$ Note that specifying either the spatial localization $U$ or its Fourier dual $U^*:=\{\xi\in\mathbb R^n:|\langle\xi,x\rangle|\le \frac{1}{2}\quad\forall x\in U-U\}$ (assuming $W_U^{n,d}$ is centered at the origin) suffices to define the weight function $W_U^{n,d}$. We write $\tilde{W}^{n,d}_U$ to mean the $L^1$-normalized version of $W_U^{n,d}$, so $\tilde{W}_U^{n,d}(x):=\|W_U^{n,d}\|_1^{-1}W_U^{n,d}$. If $U\subset\mathbb R^n$ is a convex set, then we write $W_U^{n,d}$ to mean $W_{\tilde{U}}^{n,d}$, where $\tilde{U}=T(B_0)$ for some affine transformation $T$ and $\tilde{U}$ is comparable to $U$. This defines $W_U^{n,d}$ up to a bounded constant, which is sufficient for our arguments. The following are some important properties of $W^{n,d}$ that we will repeatedly use. 1. The function $W^{n,d}(x)$ is $\sim 1$ if $|x|\le 1$ and $\sim |x|^{-100n^2d}$ if $|x|>1$. 2. The Fourier transform $\widehat{W^{n,d}}$ is supported in $|\xi|\le 2$. 3. If $A_1,A_2$ are affine transformations of the unit ball and $A_1\subset A_2$, then the following hold: 1. $\tilde{W}_{A_1}^{n,d}*\tilde{W}_{A_1}^{n,d}\lesssim_{n,d} \tilde{W}_{A_1}^{n,d}$, 2. ${W}_{A_1}^{n,d}\le {W}_{A_2}^{n,d}$, 3. $\tilde{W}_{A_1}^{n,d_1}*\tilde{W}_{A_2}^{n,d_2}\lesssim_{n,d_1,d_2} \tilde{W}_{A_2}^{n,\min(d_1,d_2-(100n)^{-1})}$. Let $f:\mathbb R^n\to\mathbb C$ be Schwartz and suppose that $\widehat{f}$ is supported in $T(B_0)$ for some affine transformation $T$. By *locally constant property*, we mean the pointwise inequality $|f(x)|\lesssim_{n,d}|f|*\tilde{W}_{(T(B_0))^*}^{n,d}(x)$ for each $d\ge 1$. This follows from writing $\widehat{f}=\widehat{f}(\psi\circ T^{-1})$ where $\psi$ is a fixed, appropriately chosen smooth bump function, and then using $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi\circ T^{-1}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi\circ T^{-1}}{\scalebox{-1}{\tmpbox}}% |\lesssim_{n,d}\tilde{W}_{(T(B_0))^*}^{n,d}$. ### A key local $L^2$ estimate [\[L2sec\]]{#L2sec label="L2sec"} Let $1\le r\le R$ with $r,R\in2^{n\mathbb N}$. For each $\theta\in\Xi_{n-1}^{n+1}(R)$, let $I(\theta)$ be the $t$-interval of length $2^{-100n}R^{-\frac{1}{n}}$ associated to $\theta$. We consider estimates of the form $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_n^{n+1}(R)}f_\theta|^2W^{n+1,d}_{B_{\tilde{r}}} \lesssim_\varepsilon r^\varepsilon\int_{\mathbb R^{n+1}} \sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_{n-1}^{n+1}(R)}}f_\theta|^2 W_{B_{\tilde{r}}}^{n+1,d},$$ for an appropriate radius $\tilde{r}$. We may use different radii $\tilde{r}$ depending on which part of the Fourier support within $\Gamma_{n-1}^{n+1}(R)$ is dominant. Consider the following subsets of $\Gamma_{n-1}^{n+1}(R)$: for a dyadic parameter $\lambda$, $R^{-\frac{n-1}{n}}\le\lambda\le 1$, let $\Xi_{n-1}^{n+1}(R,\lambda)$ be the collection of $$\theta=\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{i}{n}}\quad\forall i,\quad\max_{0\le i\le n-1}|\lambda_i|\ge\lambda/2,\quad\max_{0\le i\le n-1}|\lambda_i|2^{n-i}R^{\frac{i}{n}}\ge 1 \},$$ where $a\in 2^{-100n}R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$ is the initial point of $I(\theta)$. **Theorem 6** (Local $L^2$ orthogonality). *For any $d\ge 1$ and ball $B\subset\mathbb R^{n+1}$of radius $\lambda^{-1}r$, we have $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_{n-1}^{n+1}(R,\lambda)}f_\theta |^2W_B^{n+1,d}\lesssim \int_{\mathbb R^{n+1}}\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{n-1}^{n+1}(R,\lambda)}}f_\theta |^2W_B^{n+1,d}.$$* To prove Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}, it suffices to consider functions with Fourier support contained in further specialized subsets of the elements of $\Xi_{n-1}^{n+1}(R,\lambda)$. By the triangle inequality, it suffices to work with $\theta\in\Xi_{n-1}^{n+1}(R,\lambda)$ with $I(\theta)$ all contained in a single $L^{-1}$-intervals (where $L$ is a large constant we are free to choose). For $0\not=(k_0,\ldots,k_{n-1})\in\{-1,0,1\}^n$, let $\Xi_{n-1}^{n+1}(R,\lambda,{\bf{k}})$ be the collection of $$\theta=\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le \min(\lambda,R^{-\frac{i}{n}}),\quad k_i\lambda_i\ge |k_i|\lambda/2,\quad\max_{0\le i\le n-1}|\lambda_i|2^{n-i}R^{\frac{i}{n}}\ge 1 \},$$ where $a\in 2^{-100n}R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$ is the initial point of $I(\theta)$. There are bump functions $\psi_{{\bf{k}}}$ with $\mathrm{supp\,}\psi_{\bf{k}}\cap\theta$ equal to the displayed set for each $\theta$, and we may assume that for some $0\not={\bf{k}}$, $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_{n-1}^{n+1}(R,\lambda)}f_\theta |^2W_B^{n+1,d} \lesssim \int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_{n-1}^{n+1}(R,\lambda)}f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{{\bf{k}}} |^2W_B^{n+1,d} .$$ Suppose that the right hand side is bounded by $$C\int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_{n-1}^{n+1}(R,\lambda)}}f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{{\bf{k}}} |^2W_B^{n+1,d}.$$ Since the $\psi_{{\bf{k}}}$ satisfies $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{{\bf{k}}}|\lesssim \tilde{W}_{\tilde{B}}^{n+1,d}$, where $\tilde{B}\subset\mathbb R^{n+1}$ is the ball centered at the origin of radius $\lambda^{-1}$. Then we have $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_{n-1}^{n+1}(R,\lambda)}}f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{{\bf{k}}} |^2W_B^{n+1,d}&\lesssim \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_{n-1}^{n+1}(R,\lambda)}}f_\theta |^2(\tilde{W}_{\tilde{B}}^{n+1,d} *W_B^{n+1,d})\\ &\sim \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_{n-1}^{n+1}(R,\lambda)}}f_\theta |^2 W_B^{n+1,d}. \end{aligned}$$ We use the following technical lemma in the proof of Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}. **Lemma 7**. *Let $C,c>0$. There is a constant $L=L(n,C,c)>0$ so that the following holds. Let $\lambda$ be a dyadic value in the range $R^{-\frac{n-1}{n}}\le \lambda\le 1$. Let $k_0\in\{0,\ldots,n-1\}$ and let $0\not={\bf{k}}\in\{-1,0,1\}^n$. Suppose that $|T|\le \frac{1}{L_n}$ and $$\label{assump2} |H_{k,k}^1-\sum_{i=0}^{k}H_{k,i}^2\frac{1}{(k-i)!}T^{k-i}|\le C \lambda r^{-1} \quad \text{for each $k=1,\ldots,n$}$$ where $|H_{k,i}^j|\le C\min(R^{-\frac{i}{n}},\lambda)$ for all $k,i,j$, $|H_{k,k_0}^j|\ge cR^{-\frac{k_0}{n}}$ for all $k,j$, and $H_{k,i}^jk_i\ge c|k_i|\lambda$ for each $i=0,\ldots,n-1$. Then $$|T|\lesssim_{n,C,c} r^{-\frac{1}{n}}.$$* *Proof.* We induct on $n$. If $n=1$, then $k_0=0$, $\lambda\sim 1$, and [\[assump2\]](#assump2){reference-type="eqref" reference="assump2"} implies that $|H_{1,0}^2 T|\le C\lambda r^{-1}+|H_{1,1}^1|+|H_{1,1}^2|$. By assumption, we have $|H_{1,0}^2|\gtrsim 1$ and $|H_{1,i}^j|\lesssim R^{-1}$. It follows that $|T|\lesssim r^{-1}$. Now assume that $n>1$ and that [\[assump2\]](#assump2){reference-type="eqref" reference="assump2"} holds. If $k_0<n-1$, then the result follows from the $(n-1)$-dimensional case. Therefore, we assume that $k_0=n-1$. Let $l_0\in\{0,\ldots,n-1\}$ be the the smallest index with $k_{l_0}\not=0$. If $l_0=0$, then the lemma is proved using the $n=1$ case. Suppose that $l_0>0$. Rewrite the hypotheses [\[assump2\]](#assump2){reference-type="eqref" reference="assump2"} for $k=l_0+1,\ldots,n$ as $$\label{revhyp} |H_{l_0+l,l_0+l}^1-\frac{1}{l!}\tilde{H}_{l_0+l,l_0}^2 T^l+\sum_{i=1}^{l}H_{l_0+l,l_0+i}^2\frac{1}{(l-i)!}T^{l-i}|\le C\lambda r^{-1}\qquad\text{for all $l=1,\ldots,n-l_0$},$$ where $\tilde{H}_{l_0+l,l_0}^2=\sum_{i=0}^{l_0}H_{l_0+l,i}^2\frac{l!}{(l_0+l-i)!}T^{l_0-i}$. Note that since $|T|\le \frac{1}{L}$, $|H_{l_0+l,j}^2|\le \lambda$ for all $j$, and $|H_{l_0+l,l_0}^2|\ge c\lambda$, we have $|\tilde{H}_{l_0+l,l_0}^2|\sim \lambda$ if $L$ is sufficiently large. Since $l_0>0$, we may invoke the $n-l_0$ case to conclude that for $L$ sufficiently large, [\[revhyp\]](#revhyp){reference-type="eqref" reference="revhyp"} implies that $$|T|\lesssim r^{-\frac{1}{n-l_0}}\lesssim r^{-\frac{1}{n}},$$ as desired. ◻ *Proof of Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}.* As discussed after the statement of Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}, it suffices to consider functions $\sum_{\theta\in\Xi^{n+1}_{n-1}(R,\lambda,{\bf{k}})}f_\theta$. For each $J\in\mathcal{J}(r^{-\frac{1}{n}})$, the Fourier support of $\sum_{I(\theta)\subset J}f_\theta$ is contained in $$\begin{aligned} \tau_J=\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a+t)&:|\lambda_j|\le \min(\lambda,R^{-\frac{j}{n}})\quad\forall 0\le j\le n, \\ &\qquad\qquad\qquad\qquad|t|\le 2^{-100n}r^{-\frac{1}{n}},\quad k_i\lambda_i\ge |k_i|\lambda/2\quad\forall 0\le i\le n-1\}. \end{aligned}$$ where $a$ is the initial point of $J$. By Plancherel's theorem, $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_n^{n+1}(R,\lambda,{\bf{k}})}f_\theta|^2W^{n+1,d}_B=\int_{\mathbb R^{n+1}}\big(\sum_{\theta,\theta'\in\Xi_n^{n+1}(R,\lambda,{\bf{k}})}\widehat{f_\theta}*\widehat{\overline{f}}_\theta\big)% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{W}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{W}{\scalebox{-1}{\tmpbox}}% ^{n+1,d}_B.$$ By the triangle inequality (absorbing some factors of $L$), it suffices to count the overlap of the $\lambda r^{-1}$ neighborhood of the $\theta$ and $\theta'$, where $I(\theta)$ and $I(\theta')$ are within the same $L^{-1}$-interval. Suppose that $|t_1-t_2|\le L^{-1}$, and for each $j=1,2$, $|\mu_i^j|\le \min(\lambda,R^{-\frac{i}{n}})$, $k_i\mu_{i}^j|\gtrsim |k_i|\lambda$, $\max_{0\le i\le n-1}|\mu_i^j|R^{\frac{i}{n}}\gtrsim 1$, and $$|\sum_{i=0}^n\mu_i^1\phi_n^{(i)}(t_1)-\sum_{i=0}^n\mu_i^2\phi_n^{(i)}(t_2)|\le \lambda r^{-1}.$$ By Taylor expansion, this is equivalent to $$|\sum_{i=0}^n\big(\mu_i^1-\sum_{j=0}^{i}\frac{1}{(i-j)!}\mu_j^1(t_2-t_1)^{i-j}\big)\phi_n^{(i)}(t_1)|\le \lambda r^{-1}.$$ Since $\det[\phi_n^{(0)}(t_1)\cdots\phi_n^{(n)}(t_1)]\sim_n1$ uniformly in $t_1\in[0,1]$, by Cramer's rule, the previous displayed inequality implies that $$|\mu_i^1-\sum_{j=0}^{i}\frac{1}{(i-j)!}\mu_j^1(t_2-t_1)^{i-j}|\lesssim \lambda r^{-1}\qquad\text{for each }i=1,\ldots,n.$$ It follows from Lemma [Lemma 7](#L2techlem){reference-type="ref" reference="L2techlem"} that $|t_1-t_2|\lesssim r^{-\frac{1}{n}}$. Then $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_n^{n+1}(R,\lambda,{\bf{k}})}f_\theta |^2W^{n+1,d}_B\lesssim_{L} \int_{\mathbb R^{n+1}}\sum_{J\in \mathcal{J}(r^{-\frac{1}{n}})} |\sum_{I(\theta)\subset J}f_\theta|^2 W^{n+1,d}_B,$$ which finishes the proof. ◻ ### Global estimates for $L^p$, $1\le p\le 2$ [\[globL2\]]{#globL2 label="globL2"} **Theorem 2**. *For each $1\le p\le 2$, $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}f_\theta|^p\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)} \int_{\mathbb R^{n+1}}|f_\theta|^p.$$* *Proof.* For each $\theta\in\Xi_{n-1}^{n+1}(R)$ and $j=0,\ldots,n-1$, let $\theta^j=\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a)\in\theta:|\lambda_j|2^{m-j}R^{\frac{j}{n}}\sim 1\}$ and note that $\theta=\cup_{j=0}^{n-1}\theta^j$. There are smooth bump functions $\psi_{\theta^j}$ which satisfy $\sum_j\psi_{\theta_j}\equiv 1$ on $\theta$ and $\mathrm{supp\,}\psi_{\theta_j}\subset\theta_j$. Suppose that $$\int_{\mathbb R^{n+1}}|\sum_{\theta}f_\theta|^{p}\lesssim_n \int_{\mathbb R^{n+1}}|\sum_{\theta}f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{\theta^j} |^p.$$ If $j<n-1$, then since $\theta^j\subset\tau \times\mathbb R^{n-j-1}$ for an element $\tau\in\Xi_{j+1}^{j+2}(R^{\frac{j+1}{n}})\times\mathbb R^{n-j-1}$, the theorem follows from a cylindrical version of the $n=j+1$ dimensional theorem (see Theorem 4.1 from [@maldagueM3] for a precise cylindrical version of square function estimate; the proof also applies to general geometries). Therefore, it suffices to consider the case where $\theta^{n-1}$ dominates for each $\theta$. It also suffices to show that $$\int_{\mathbb R^{n+1}}|\sum_{\theta}f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{\theta^{n-1}} |^p\lesssim_\varepsilon R^\varepsilon\int_{\mathbb R^{n+1}}\sum_{\theta}|f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{\theta^{n-1}} |^p$$ since for each $\theta$, by Young's inequality, $$\int_{\mathbb R^{n+1}}|f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{\theta^{n-1}}|^p\lesssim \int_{\mathbb R^{n+1}}|f_\theta|^p.$$ We will slightly abuse notation by using $f_\theta$ to denote $f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}% _{\theta^{n-1}}$ for the remainder of the proof. Each function $f_\theta$ then has a wave packet decomposition adapted to translates $T$ of a dual set $\theta^*$ which tile $\mathbb R^{n+1}$. By a standard pigeonholing procedures outlined, for example, in [@maldagueM3], it suffices to prove that for $\alpha>0$, $A>0$, and each ball $B_R\subset\mathbb R^{n+1}$ of radius $R$, $$\label{p=2case} \alpha^p|\{x\in B_R:|f(x)|\sim\alpha\}|\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}\int_{\mathbb R^{n+1}}|f_\theta|^p,$$ where $f_\theta=\sum_{T\in\mathbb T_\theta}c_T\psi_T$ is a wave packet decomposition with the properties that $\mathbb T_\theta$ is a finite set, $|c_T|\sim A$ for all $T$, and $\psi_T$ are $L^\infty$-normalized wave packets subordinate to $T$ from a tiling of $\mathbb R^{n+1}$ by translates of $\theta^*$. The $p=2$ case of [\[p=2case\]](#p=2case){reference-type="eqref" reference="p=2case"} follows directly from Plancherel's theorem. It follows that $$\begin{aligned} \alpha^p|\{x\in\mathbb R^{n+1}:|f(x)|\sim\alpha\}|&\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}\alpha^{p-2}\int_{\mathbb R^{n+1}}|f_\theta|^2 \\ &\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}\frac{A^{2-p}}{\alpha^{2-p}}A^p\#\mathbb T_\theta|\theta^*|\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}\frac{A^{2-p}}{\alpha^{2-p}}\int_{\mathbb R^{n+1}}|f_\theta|^p. \end{aligned}$$ If $\frac{A}{\alpha}\le 1$, then we are done. Suppose now that $\alpha\le A$. Then $$\begin{aligned} \alpha^p|\{x\in\mathbb R^{n+1}:|f(x)|\sim\alpha\}|&\le A^{p-1} \sum_{\theta\in\Xi_{n-1}^{n+1}(R)} \int_{\mathbb R^{n+1}}|f_\theta| \\ &\lesssim_\varepsilon R^\varepsilon A^{p-1}\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}A\#\mathbb T_\theta|\theta^*|\lesssim_\varepsilon R^\varepsilon\sum_{\theta\in\Xi_{n-1}^{n+1}(R)}\int_{\mathbb R^{n+1}}|f_\theta|^p. \end{aligned}$$ ◻ # Proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} [\[conesec\]]{#conesec label="conesec"} {#proof-of-proposition-coneinduct-conesec} We induct on the difference $n-m$. The base case of $m=n-1$ is treated in [\[globL2\]](#globL2){reference-type="ref" reference="globL2"}. The main work of the proof is to analyze a multi-scale constant for $0\le m\le n-2$. A significant component of the analysis uses local versions of the square function estimates defining $\mathbb C_m^{n+1}(R)$. Local estimates follow from global estimates for $\Gamma_0^{n+1}(R)$, but not for $\Gamma_m^{n+1}(R)$ for $m>0$. This is because the sets $\Gamma_m^{n+1}(R)$, indexed by $R$, are not nested. For comparison, $\Gamma_0^{n+1}(R)\subset \Gamma_0^{n+1}(r)$ and also the $r^{-1}$-neighborhood of each $\theta\in\Xi_0^{n+1}(R)$ is approximately contained in one $\tau\in\Xi_0^{n+1}(r)$. If we localize the spatial side to a ball $B_r$ of radius $r$ using a weight function $W_{B_r}^{n+1,d}$ (see [\[wtsec\]](#wtsec){reference-type="ref" reference="wtsec"} for a precise definition) with $\mathrm{supp\,}\widehat{W_{B_r}^{n+1,d}}$ contained in a ball of radius $r^{-1}$, then for $2\le p\le p_n$, $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_0^{n+1}(R)}f_\theta W_{B_r}^{n+1,d}|^p\lesssim \mathbb C_0^{n+1}(r)\int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_0^{n+1}(R)}}f_\theta W_{B_r}^{n+1,d}|^2)^{\frac{p}{2}},$$ so $\mathbb C_0^{n+1}(\cdot)$ also bounds local square function estimates. This is no longer true for $\Gamma_m^{n+1}(R)$ when $m>0$ since $\Gamma_m^{n+1}(R)$ and $\Gamma_m^{n+1}(r)$ have nontrivial symmetric difference. Indeed, we have $\Gamma_m^{n+1}(R)\cap\Big(\{0\}^m\times\mathbb R^{n-m+1}\Big)=\{0\}^m\times R^{-\frac{m}{n}}\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})$ and it follows that - $\Gamma_m^{n+1}(R)\setminus\Gamma_m^{n+1}(r)\supset \{0\}^m\times R^{-\frac{m}{n}}\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})$, - $\Gamma_m^{n+1}(r)\setminus\Gamma_m^{n+1}(R)\supset \{0\}^m\times r^{-\frac{m}{n}}\Gamma_0^{n-m+1}(R^{\frac{n-m}{n}})$. Since global estimates do not imply local estimates when $m>0$, we explicitly analyze multi-scale constants which are localized to spatial balls of some scale. In order to show that these constants are bounded, we first treat a special case. The special case deals with functions with refined, truncated Fourier support that is described in the following section. ## [\[multidef\]]{#multidef label="multidef"}Truncated Fourier support and a local multi-scale quantity To prove that $\mathbb C_m^{n+1}(R)\lesssim_\varepsilon R^\varepsilon$, we work with localized multi-scale quantities and various truncated versions of $\Gamma_m^{n+1}(R)$. We use the weight functions $W^{n+1,d}_B$ and $\tilde{W}_B^{n+1,d}$ introduced in [\[wtsec\]](#wtsec){reference-type="ref" reference="wtsec"} to localize the integrals. We introduce the first truncated version of $\Gamma_m^{n+1}(R)$. Define $\mathcal{H}_m^n(R,K)$ to be the tuples ${\bf{h}}=(h_0,\ldots,h_m)\in (2^{-\mathbb N})^{m+1}$ satisfying the following properties: 1. $2^{-m-1}R^{-\frac{m}{n}}\le |h_m|\le R^{-\frac{m}{n}}$. 2. $K^{-1}R^{-\frac{m}{n}}\le |h_i|\le R^{-\frac{m}{n}}$ for all $i$. Define the truncated $m$-cone $\Gamma_m^{n+1}(R,K,{\bf{h}})\subset\mathbb R^{n+1}$ to be the union of $$\begin{aligned} \label{Gammah} \theta_a=\big\{\sum_{i=0}^{n}\lambda_i\phi_n^{(i)}(a):&\,\,|\lambda_i-h_i|\le K^{-\beta}|h_i|\quad\text{for all }0\le i\le m,\,\, |\lambda_j|\le R^{-\frac{j}{n}}\,\, \text{if}\,\, m+1\le j\le n \}\end{aligned}$$ where $a\in 2^{-200n}R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$ and we choose $\beta$ in Proposition [Proposition 9](#initscale){reference-type="ref" reference="initscale"}. Let $\Xi_m^{n+1}(R,K,{\bf{h}})$ denote the collection of these $\theta_a$. For a scale $s\in2^{-\mathbb N}$, let $\mathcal{J}(s)=\{[l2^{-100n}s,(l+1)2^{-100n}s):l=0,\ldots,2^{100n}s^{-1}-1\}$. Define $S_{K}^{n,m}(r,\rho,R)$ to be the smallest constant such that for any vector ${\bf{h}}\in\mathcal{H}_m^n(R,K)$, ball $B\subset\mathbb R^{n+1}$ of radius $R^{\frac{m}{n}}\rho$, $d\ge 1$, and any $2\le p\le p_{n-m}$, $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}}& f_\theta|^2|^{\frac{p}{2}} W_B^{n+1,d} \\ \le S_{K}^{n,m}(r,\rho,R)&\times\int_{\mathbb R^{n+1}}|\sum_{I\in\mathcal{J}(\rho^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset I\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}} f_\theta|^2|^{\frac{p}{2}} W_B^{n+1,d} \end{aligned}$$ for all Schwartz functions $f_\theta:\mathbb R^{n}\to\mathbb C$ with $\mathrm{supp\,}\widehat{f}_\theta\subset \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})$. **Proposition 8**. *Let $\varepsilon>0$. Suppose that $R\ge C_K$ and $K$ is sufficiently large depending on $\varepsilon$. Then $S_{K}^{n,m}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$.* We prove Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"} in [\[SnmmKbdsec\]](#SnmmKbdsec){reference-type="ref" reference="SnmmKbdsec"}. After bounding $S_K^{n,m}(r,\rho,R)$, we will work with a more general local multi-scale constant. We do not prove a local estimate in the full generality of functions with Fourier support in $\Gamma_m^{n+1}(R)$ because we require some truncation to justify a base case for the induction on scales. Since our truncations are somewhat complex (compared to truncating to balls or annuli), it is not easy to prove that local square function estimates for truncated functions imply local square function estimates for non-truncated functions with Fourier support in all of $\Gamma_m^{n+1}(R)$. However, we show that local estimates for truncated functions imply global estimates for non-truncated functions, which is sufficient to prove Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"}. We define a more general truncated version of $\Gamma_m^{n+1}(R)$, for $1\le m\le n-2$. Let $\sigma=(\sigma_0,\ldots,\sigma_m)\in2^{\mathbb Z}$ with $\sigma_m=1$ and $\sigma_k\in[\sigma_{k+1}^{-1}\cdots\sigma_m^{-1}R^{-\frac{1}{n}},1]$ for all $k<m$. It is also notationally convenient to define $\sigma_{m+1}:=1$. Define $\Xi_m^{n+1}(R,\sigma)$ to be the union of sets $$\begin{aligned} \theta_a(\sigma)&=\Big\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(t):a\le t\le a+2^{-100n} R^{-\frac{1}{n}},\quad|\lambda_i|\le \sigma_0^{\max(0,1-i)}\cdots\sigma_m^{\max(0,m+1-i)}R^{-\frac{i}{n}}\text{ for all $i$},%\quad \max_{0\le i\le m}2^i|\lambda_i|R^{\frac{i}{n}}\in[\frac{1}{2},1], \\ &\quad\text{for each $0\le k\le m$,} \begin{cases} \underset{0\le i\le k}{\max}2^{-i}|\lambda_i|\sigma_k^{-k-1+i}\cdots\sigma_m^{-m-1+i}R^{\frac{i}{n}}\ge 2^{-k-1} &\,\, \text{if}\,\,\sigma_k>\sigma_{k+1}^{-1}\cdots\sigma_{m+1}^{-1}R^{-\frac{1}{n}} \\ \underset{0\le i\le k}{\max}|\lambda_i|\le \sigma_{k+1}\cdots\sigma_{m}^{m-k}R^{-\frac{k+1}{n}} &\,\, \text{if}\,\,\sigma_k=\sigma_{k+1}^{-1}\cdots\sigma_{m+1}^{-1}R^{-\frac{1}{n}} \end{cases} \Big\}, \end{aligned}$$ where $a\in 2^{-100n} R^{-\frac{1}{n}}\mathbb Z\cap[0,1]$. Although the above definition looks complicated, we describe how to obtain a truncation from functions with Fourier support in $\Gamma_m^{n+1}(R)$ in the proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} in [\[truncsec\]](#truncsec){reference-type="ref" reference="truncsec"}. For each $2\le r\le \rho\le R$ and $\sigma$, let $S_{\sigma}^{n,m}(r,\rho,R)$ be the corresponding local multi-scale quantity defined to be the smallest constant such that for any ball $B$ of radius $R^{\frac{m}{n}}\rho$, $d\ge 1$, and any $2\le p\le p_{n-m}$, $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_m^{n+1}(R,\sigma)}} f_\theta|^2|^{\frac{p}{2}} W_B^{n+1,d} \le S_{\sigma}^{n,m}(r,\rho,R)&\int_{\mathbb R^{n+1}}|\sum_{I\in\mathcal{J}(\rho^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset I\\ \theta\in\Xi_m^{n+1}(R,\sigma)}} f_\theta|^2|^{\frac{p}{2}} W_B^{n+1,d} \end{aligned}$$ for all Schwartz functions $f_\theta:\mathbb R^{n}\to\mathbb C$ with $\mathrm{supp\,}\widehat{f}_\theta\subset \theta\in\Xi_m^{n+1}(R,\sigma)$. ## Estimate for an initial scale The following proposition is based on the Pramanik-Seeger approximation of cones by neighborhoods of cylinders over curves, which was also used by Bourgain-Demeter in the proof of decoupling for the cone. **Proposition 9** (Initial scale for $0\le m\le n-2$). *Suppose that the hypotheses of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} hold. Suppose that $\beta$ is a large enough dimensional constant. Let $\psi:\mathbb R^{n+1}\to\mathbb R^{n+1}$ be a smooth function with Fourier transform supported in a ball of radius $R^{\frac{m}{n}} K^{-\beta}$. For any $\delta>0$, $R\ge C_K$, and $2\le p\le p_{n-m}$, $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}g_\theta\psi|^{p}\lesssim_\delta K^\delta\int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(K^{-1})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}}g_\theta\psi|^2|^{\frac{p}{2}}.$$* *Proof.* Each element of $\Xi_m^{n+1}(R,K,{\bf{h}})$ is within a distance $\lesssim K^{-B}R^{-\frac{m}{n}}$ of a point $\sum_{i=0}^m\tilde{h}_i\phi_n^{(i)}(t)$, for some $t\in[0,1]$ and $\tilde{h}_i=h_i$ if $|h_i|> K^{-1}R^{-\frac{m}{n}}$ and $\tilde{h}_i=0$ if $|h_i|\le K^{-1}R^{-\frac{m}{n}}$. We initiate an algorithm that allows us to use lower dimensional square function estimates to certain neighborhoods of one of curves $\gamma_n^{(0)}(t),\ldots,\gamma_n^{(m)}(t)$. The function $\psi$ is harmless since it has the effect of taking a $K^{-\beta}R^{-\frac{m}{n}}$-neighborhood on the Fourier side. We will choose constants $B_0=\beta>(n+1)B_1>(n+1)^2B_2>\cdots>(n+1)^{k} B_{k}$ at the end of the proof. Suppose that $|\tilde{h}_0|\ge K^{-B_1}|h_m|$. Let $A_0$ be inverse of a lower triangular $(n+1)\times (n+1)$ matrix with $1$'s along the diagonal satisfying $$\sum_{i=0}^m\tilde{h}_iA_0\phi_n^{(i)}(t)=\tilde{h}_0\phi_n(t)\quad\text{for all}\quad t\in[0,1].$$ Note that by Cramer's rule, the entries of $A_0$ have norm bounded by $\lesssim K^{nB_1}$. Therefore, $$A_0\big(\Gamma_m^{n+1}(R,K,{\bf{h}})\big)\subset\mathcal{N}_{C|h_m|K^{nB_1-B_0}}\{\tilde{h}_0\phi_n(t):t\in[0,1]\},$$ where $|\tilde{h}_0|\ge K^{-B_1}|h_m|$. After rescaling the set on the right hand side by $\frac{1}{\tilde{h}_0}$, it is contained in $$\mathbb R\times \mathcal{N}_{CK^{(n+1)B_1-B_0}}\{\gamma_{n-m}(t):t\in[0,1]\}\times \mathbb R^{m}.$$ We will choose $B_1$ and $B_0$ so that $CK^{(n+1)B_1-B_0)}\le K^{-(n-m)}$, so that the proposition follows from $\mathbb{M}^{n-m}(K^{n-m})\lesssim_\delta K^\delta$. Now suppose that $|\tilde{h}_0|\le K^{-B_1}|h_m|$ and $|\tilde{h}_1|\ge K^{-B_2}|h_m|$. Then $$\Gamma_m^{n+1}(R,K,{\bf{h}})\subset\mathcal{N}_{CK^{-B_1}|h_m|}\{\sum_{i=1}^m\tilde{h}_iR^{-\frac{i}{n}}\phi_n^{(i)}(t):t\in[0,1]\}.$$ Similar to the previous case, we can define an $(n+1)\times (n+1)$ matrix $A_1$ satisfying $$A_1\Big(\sum_{i=1}^m\tilde{h}_i\phi_n^{(i)}(t)\Big)= \tilde{h}_1\phi_n^{(1)}(t)\quad\text{for all}\quad t\in[0,1]$$ and $A_1\big(\Gamma_m^{n+1}(R,K,{\bf{h}})\big)\subset\mathbb R\times \mathcal{N}_{CK^{nB_2-B_1}|h_m|}\{\tilde{h}_1\gamma_{n-m+1}^{(1)}(t):t\in[0,1]\}\times \mathbb R^{m-1}$. After dilating by $\frac{1}{\tilde{h}_1}$, we have $$\frac{1}{\tilde{h}_1}A_1\big(\Gamma_m^{n+1}(R,K,{\bf{h}})\big)\subset \mathcal{N}_{CK^{(n+1)B_2-B_1}}\{\phi_{n}^{(1)}(t):t\in[0,1]\}.$$ The curve $\phi_{n}^{(1)}(t)$ is contained in the product of $\mathbb R^{m+1}$ with an anisotropic rescaling of $\gamma_{n-m}(t)$, so if $(n+1)B_1-B_2\le -(n-m)$, then the proposition again follows from $\mathbb{M}^{n-m}(K^{n-m})\lesssim_\delta K^\delta$. We iterate this procedure until either the proposition is proved, or we are in the final case that $|\tilde{h}_i|\ge K^{-B_i}|h_m|$ for each $i=0,\ldots,m-1$. Then $$\Gamma^{n+1}(R,K,{\bf{h}})\subset \mathcal{N}_{K^{-B_{k-1}}|h_m|}\{{h}_m\phi_n^{(m)}(t):t\in[0,1]\}.$$ Again, use an affine transformation $A_m$ which transforms $$A_m\big(\Gamma^{n+1}(R,K,{\bf{h}})\big)\subset\mathcal{N}_{CK^{-B_{k-1}}|h_m|}\{h_m\phi_n^{(m)}(t):t\in[0,1]\}.$$ Since $\phi_n^{(k)}$ contains an anisotropically rescaled version of $\varphi_{n-m}$, we are done as long as $B_k\ge n-m$. The choice $B_0=(n-m)(n+2)^n$ and $B_i=(n+2)^{n-i}(n-m)$ satisfies the necessary properties. ◻ ### The multi-scale inequality We prove a key multi-scale inequality in Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}. Note that when $m=n-1$, we have a bound for $S_{K}^{n,n-1}(r,\rho,R)$ from the $L^2$-based analysis in [\[L2sec\]](#L2sec){reference-type="ref" reference="L2sec"}. **Proposition 10**. *Let $1\le n$. Then for $K$ larger than a dimensional constant, $S_{K}^{n,n-1}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$.* *Proof.* This follows immediately from Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}. ◻ **Proposition 11**. *Suppose that the hypotheses of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} hold and that $0\le m\le n-2$. Suppose also that $S_K^{n,k}(r,\rho,R)\lesssim_\delta\rho^\delta$ for each $m+1\le k\le n-1$. Then is an absolute constant $C>0$ such that for any $\delta_0\in(0,2^{-100n})$, $$S_{K}^{n,m}(r,\rho,R)\lesssim_{\delta_0} r^{\delta_0} S_{K}^{n,m}(1,\rho/r,R/r)+\sum_{k=1}^{n-m} r^{C\delta_0^{n-k+1}}\Big[S_{K}^{n,m}(\min(\rho,r^{1+\delta_0^{n-k}}),\rho,R) \Big].$$* The implicit constant is uniform in ${\bf{h}}\in\mathcal{H}_m^{n}(R,K)$. We work with a sequence of exponents $\tilde{p}_n$ which is an even version of the sequence $p_n$. Let $\tilde{p}_1=2$ and for $n\ge2$, let $$\tilde{p}_n:=\begin{cases}\sum_{i=1}^ni+1\quad&\text{if} \quad n\equiv1\text{ or }2\pmod{4}\\ \sum_{i=1}^ni\quad&\text{if} \quad n\equiv0\text{ or }3\pmod{4} \end{cases} .$$ Before proving Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}, we need to verify some properties about the sequence of exponents $\tilde{p}_n$, which are recorded in the following lemma. **Lemma 12**. 1. *$2\le\tilde{p}_n\le p_n$,* 2. *$\tilde{p}_n\le \tilde{p}_{n+1}$,* 3. *$\frac{\tilde{p}_{n}}{\tilde{p}_{n-1}}\le 2$,* 4. *for $n\ge k\ge 1$, if $\frac{p_n}{\tilde{p}_k}>2$, then $\frac{p_n}{\tilde{p}_k}\le p_{n-k}$.* *Proof.* The first two properties are clear. To verify property (3), first observe that that $\frac{2}{1}$, $\frac{4}{2}$, and $\frac{6}{4}$ are all $\le 2$. Therefore, it suffices to check that $\frac{k^2+k+2}{k^2-k}\le 2$ for $k\ge 4$. This is equivalent to $0\le k^2-3k-2$, which is true when $k\ge 4$. Now we verify the last property. The cases $k=1,2$ are easy to check, so we assume that $k\ge 3$. We need to show that if $p_n-2\tilde{p}_k>0$, then $p_{n-k}\tilde{p}_k-p_n\ge 0$. The inequality $p_n-2\tilde{p}_k>0$ is a quadratic inequality in $n$ with a positive and a negative root. Since $n\ge 0$, the inequality is equivalent to the condition that $$\begin{aligned} \label{equiv1} n> -\frac{1}{2}+\frac{1}{2}\sqrt{8 k^2 + 8 k - 7}.\end{aligned}$$ The inequality we need to verify is also quadratic in $n$: $0\le \frac{1}{2}(k^2+k-2)n^2+[\frac{1}{2}(k^2+k)(-2k+1)-1]n+\frac{1}{2}k^4+\frac{1}{2}k^2+k-2$. The roots of this inequality are $$\frac{2 k^3 + k^2 - k + 2\pm \sqrt{k^4 - 6 k^3 + 21 k^2 + 28 k - 28}}{2 (k^2 + k - 2)}.$$ It suffices to verify that the larger root is less than or equal to the right hand side of [\[equiv1\]](#equiv1){reference-type="eqref" reference="equiv1"}. Indeed, we have for $k\ge 3$, $$\begin{aligned} -(k^2+k-2)+(k^2+k-2)&\sqrt{8k^2+8k-8}-\big[2k^3+k^2-k+2+\sqrt{k^4-6k^3+21k^2+28k-28}\big] \\ &= (k^2+k-2)\sqrt{8k^2+8k-8}-2k^3-2k^2-\sqrt{k^4-6k^3+21k^2+28k-28} \\ &\ge(2\sqrt{2}-2)k^3+(k-2)\sqrt{8k^2+8k-8}-2k^2-\sqrt{k^4+3k^2+28k} \\ &\ge(2\sqrt{2}-2)k^3+(k-2)2\sqrt{2}k-(2+\frac{8}{3\sqrt{3}})k^2\\ &\ge k\left[(2\sqrt{2}-1)k^2-(2+\frac{8}{3\sqrt{3}}-2\sqrt{2})k-2\sqrt{2} \right]\\ &\ge k\left[\frac{8}{5}k-2\sqrt{2} \right]\\ &\ge 0. \end{aligned}$$ ◻ Next, we prove some key lemmas that we will use in the proof of Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}. Let $2\le p\le p_{n-m}$. We describe the set-up needed to state the lemmas. Consider the left hand side of the defining inequality for $S_{K}^{n,m}(r,\rho,R)$: $$\label{now2} \int_{\mathbb R^{n+1}} |\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}} f_\theta|^2|^{\frac{p}{2}}W,$$ where $W=W_{B}^{n+1,d}$ for a ball $B\subset\mathbb R^{n+1}$ a ball of radius $R^{\frac{m}{n}}\rho$. Let $n_{p}=\min\{l\ge 2:1\le \frac{p}{\tilde{p}_{l-1}}\le 2\}$. Our iteration will be indexed by a parameter $k$ in the range $2\le k\le n_p$. Consider $J\in \mathcal{J}(r^{-\frac{1}{n}})$ with initial point $a\in[0,1]$. The Fourier support of $|\sum_{I(\theta)\subset J}f_\theta|^2$ is contained in $$\label{claim} \{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}} \}.$$ We define auxiliary functions which we will use to decompose the above Fourier support. Let $s:\mathbb R\to[0,1]$ be the bump function used in [\[auxbump\]](#auxbump){reference-type="eqref" reference="auxbump"} and note that the function $$\psi^k(x_0,\ldots,x_n):=s(R^{\frac{m}{n}}r^{-\frac{m}{n}}x_0)s(R^{\frac{m}{n}}r^{\frac{1-m}{n}} x_1)\cdots s(R^{\frac{m}{n}}r^{\frac{k-1}{n}}x_{m+k-1})s(x_{m+k})\cdots s(x_n)$$ is identically $1$ on the rectangle $[-R^{-\frac{m}{n}}r^{\frac{m}{n}},R^{-\frac{m}{n}}r^{\frac{m}{n}}]\times\cdots\times[-R^{-\frac{m}{n}}r^{-\frac{k-1}{n}},R^{-\frac{m}{n}}r^{-\frac{k-1}{n}}]\times[-1,1]^{n+1-m-k}$. Let $A^k_{J}:\mathbb R^{n+1}\to \mathbb R^{n+1}$ be a linear map satisfying $$\begin{aligned} A_{J}^k\big([-R^{-\frac{m}{n}}r^{\frac{m}{n}},R^{-\frac{m}{n}}r^{\frac{m}{n}}]&\times\cdots\times[-R^{-\frac{m}{n}}r^{-\frac{k-1}{n}},R^{-\frac{m}{n}}r^{\frac{m}{n}}r^{-\frac{k-1}{n}}]\times[-1,1]^{n+1-m-k}\big) \nonumber\\ \label{etaFsupp} &=\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\}\times[-1,1]^{n+1-m-k} . \end{aligned}$$ Define $\psi_{J_{m+1}}^k(x_0,\ldots,x_n)$ by $\psi^k\circ (A_{J}^k)^{-1}$. For each dyadic $\sigma\in[r^{-1/n},1]$, define $$\psi_{J,\sigma}^k(x_0,\ldots,x_n)=\psi_{J_{m+1}}^k(\sigma^{-m-k}x_0,\sigma^{-m-k+1}x_1,\ldots,\sigma^{-1}x_{m+k-1},x_{m+k},\ldots,x_n)$$ and note that $$\psi_{J}^k=\psi_{J,r^{-\delta_k}}^k+\sum_{r^{-\delta_k}<\sigma\le 1}(\psi_{J,\sigma}^k-\psi_{J,\sigma/2}^k)=:\eta_{J,r^{-\delta_k}}^k+\sum_{r^{-\delta_k}<\sigma\le 1}\eta_{J,\sigma}^k.$$ In the following lemmas, $\theta$ always indexes the sets in $\Xi_m^{n+1}(R,K,{\bf{h}})$. Our iteration involves convolving weight functions, which loses a factor in the decay rate. There are at most $n$ many steps in the iteration, so we let $d_k=4(n-k)d$. For each $J=[a,a+2^{-100n}r^{-\frac{1}{n}})$, let $\tilde{W}_{J,m+k-1}^{n+1,d_k}$ be an $L^1$-normalized weight function (as was introduced in [\[multidef\]](#multidef){reference-type="ref" reference="multidef"}) which is Fourier supported in $$\{\sum_{i=0}^{m+k-2}\lambda_i\phi_{m+k-2}^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}} r^{\frac{m-i}{n}}\}\times [-2,2]^{n+2-m-k}.$$ Let $\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}$ be a weight function that is centered at the origin and that has the same Fourier support as $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k$. Let $0<\delta_0$ and define $\delta_k=\delta_0^{n-k}$. **Lemma 13**. *The integral [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by $$\begin{aligned} \sum_{2\le k\le n_p} &C_{\delta_{k-1}}r^{C\delta_{k-1}}\int_{\mathbb R^{n+1}}|\sum_{J_{m+1}}|\sum_{\substack{I(\theta)\subset J%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,m+k-1}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|^{\frac{p}{\tilde{p}_{k-1}}}W\\ &+C_{\delta_{n_p}}r^{\delta_{n_p}}\int_{\mathbb R^{n+1}}\sum_{J_{m+1}}|\sum_{\substack{I(\theta)\subset J_{m+1}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^p W. \end{aligned}$$* **Lemma 14**. *For each $1\le k\le n_p$ and $J\in\mathcal{J}(r^{-\frac{1}{n}})$, we have $$||\sum_{\substack{I(\theta)\subset J_{m+1}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^{\tilde{p}_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|\lesssim_\delta r^\delta|\sum_{I_k \subset J_{m+1}}|\sum_{I(\theta)\subset I_k}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}*\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k},$$ where $I_k\in\mathcal{J}\big(\max(\rho^{-\frac{1}{n}},r^{-\frac{(1+\delta_k)}{n}})\big)$.* **Lemma 15**. *Let $I_k$ vary over intervals in $\mathcal{J}\big(\max(\rho^{-\frac{1}{n}},r^{-\frac{1+\delta_k}{n}})\big)$. Then $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}|^{\frac{p}{\tilde{p}_{k-1}}}W\lesssim_\delta r^\delta\int_{\mathbb R^{n+1}}|\sum_{I_{k}}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{p}{2}}W. \end{aligned}$$* We remark that we are free to choose any $\delta>0$ in the notation $\lesssim_\delta r^\delta$ from Lemmas [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"} and [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"}, and this parameter is independent of $\delta_0>0$. Now we show how Lemmas [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}, [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"}, and [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"} imply Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}. *Proof of Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}.* Our objective is to bound [\[now2\]](#now2){reference-type="eqref" reference="now2"}. Begin by using Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}. There are two possible outcomes. Suppose first that [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by $$C_{\delta_{n_p}}r^{\delta_{n_p}}\int_{\mathbb R^{n+1}}\sum_{J_{m+1}}|\sum_{\substack{I(\theta)\subset J%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^p W^p.$$ Let $J$ be the interval with initial point $a$ and of length $2^{-100n}r^{-\frac{1}{n}}$. Let $A$ be the affine map mapping $\phi_{n}^{(i)}(a)\mapsto r^{\frac{i}{n}}\phi_n^{(i)}(0)$ for all $i$. Then the function $f_\theta\circ (A^{-1})^t$ has Fourier transform supported in $A(\theta)\in\Xi_m^{n+1}(R/r,K,{\bf{\tilde{h}}})$, where $\tilde{h}=(\tilde{h}_0,\ldots,\tilde{h}_m)$ is defined by $\tilde{h}_i=r^{\frac{i}{n}}h_i$ if $r^{\frac{i}{n}}|h_i|\ge K^{-1}r^{\frac{m}{n}}R^{-\frac{m}{n}}$ and $\tilde{h}_i=K^{-1}r^{\frac{m}{n}},R^{-\frac{m}{n}}$ otherwise. Note that ${\bf{\tilde{h}}}$ is an element of $\mathcal{H}_m^n(R/r,K)$. The weight function is localized to an ellipsoid which may be approximated with finitely overlapping balls of radius $R^{\frac{m}{n}}r^{-\frac{m}{n}}(\rho/r)$, so we have $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{\substack{I(\theta)\subset J }}&f_\theta\circ(A^{-1})^t|^p W\circ(A^{-1})^t\\ &\lesssim S^{n,m}_{K}(1,\rho/r,R/r)\int_{\mathbb R^{n+1}}|\sum_{\substack{I\in\mathcal{J}(\rho^{-\frac{1}{n}})\\ I\subset J}} |\sum_{\substack{I(\theta)\subset I }}f_\theta\circ(A^{-1})^t|^2|^{\frac{p}{2}} W\circ(A^{-1})^t. \end{aligned}$$ Summing over $J$ and using $\|\cdot\|_{\ell^{p/2}}\le\|\cdot\|_{\ell^1}$, conclude in this case that [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by $$C_{\delta_{n_p}}r^{\delta_{n_p}} S^{n,m}_{K}(1,\rho/r,R/r)\int_{\mathbb R^{n+1}}|\sum_{\substack{I\in\mathcal{J}(\rho^{-\frac{1}{n}})}} |\sum_{\substack{I(\theta)\subset I }}f_\theta|^2|^{\frac{p}{2}} W.$$ The remaining case is if for some $2\le k\le n_p$, [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by $$C_{\delta_{k-1}}r^{C\delta_{k-1}}\int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{\substack{I(\theta)\subset J}}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,m+k-1}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|^{\frac{p}{\tilde{p}_{k-1}}}W .$$ Then apply Lemma [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"} pointwise to bound the above expression by $$C_{\delta_{k-1}}r^{C\delta_{k-1}}\int_{\mathbb R^{n+1}}|\sum_{J_{m+1}}|\sum_{I_k\subset J_{m+1}}|\sum_{\substack{I(\theta)\subset I_k}}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}*\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}|^{\frac{p}{\tilde{p}_{k-1}}}W$$ where $I_k\in\mathcal{J}\big(\max(\rho^{-\frac{1}{n}},r^{-\frac{(1+\delta_k)}{n}})\big)$ and $\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$ has the same Fourier support as $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_{m+1},r^{-\delta_k}}^k$. Finally, by Lemma [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"}, we have bounded [\[now2\]](#now2){reference-type="eqref" reference="now2"} by $$C_{\delta_{k-1}}r^{C\delta_{k-1}}\int_{\mathbb R^{n+1}}|\sum_{I_k}|\sum_{\substack{I(\theta)\subset I_k}}f_\theta|^2|^{\frac{p}{2}}W.$$ Conclude in this case that $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_{k-1}}r^{C\delta_{k-1}} S_K^{n,m}(\min(\rho,r^{1+\delta_k}),\rho,R).$$ ◻ It remains to prove the lemmas. *Proof of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}.* We initiate an iteration to bound [\[now2\]](#now2){reference-type="eqref" reference="now2"}. For each $J$ with $J=[a,a+2^{-100n}r^{-\frac{1}{n}})$, the Fourier support of $|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}} f_\theta|^2$ is contained in $$\begin{aligned} & \label{disabo} \big\{\sum_{i=0}^{n}\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}} \big\}\subseteq \{\sum_{i=0}^m\lambda_i\phi_{m}^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}} r^{\frac{m-i}{n}}\}\times [-2,2]^{n-m} . \end{aligned}$$ It follows from the locally constant property, [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by $$\int_{\mathbb R^{n+1}} |\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}} f_\theta|^2*\tilde{W}_{J,{m+1}}^{n+1,d_1}|^{\frac{p}{2}}W,$$ where $\tilde{W}_{J,{m+1}}^{n+1,d}$ is an $L^1$-normalized weight function that is centered at the origin and Fourier supported in the right hand side of [\[disabo\]](#disabo){reference-type="eqref" reference="disabo"}. Let $n_{p}=\min\{l\ge 2:1\le \frac{p}{\tilde{p}_{l-1}}\le 2\}$. Let $2\le k\le n_p$ and define $\delta_k=\delta_0^{n-k}$. The input for step $k$ is a bound for [\[now2\]](#now2){reference-type="eqref" reference="now2"} of the form $$\label{stepkeqn} C_{\delta_{k-1}} r^{C\delta_{k-1}} \int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}|^{\frac{p}{\tilde{p}_{k-1}}} W,$$ where $\tilde{W}_{J,m+k-1}^{n+1,d_{k-1}}$ is Fourier supported in $$\{\sum_{i=0}^{m+k-2}\lambda_i\phi_{m+k-2}^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}} r^{\frac{m-i}{n}}\}\times [-2,2]^{n+2-m-k}.$$ Suppose that $2\le k< n_p$, so $2\le \frac{p}{\tilde{p}_{k-1}}$, and that [\[now2\]](#now2){reference-type="eqref" reference="now2"} is bounded by [\[stepkeqn\]](#stepkeqn){reference-type="eqref" reference="stepkeqn"}. We will use the auxiliary functions described before Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"} to decompose the Fourier support of each summand $|\sum_{I(\theta)\subset J} f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}$. Since $\tilde{p}_{k-1}$ is even, the Fourier support of $|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k-1}}$ is contained in the left hand side of [\[disabo\]](#disabo){reference-type="eqref" reference="disabo"} (dilated by a harmless constant factor). If [\[stepkeqn\]](#stepkeqn){reference-type="eqref" reference="stepkeqn"} is bounded by $$CC_{\delta_{k-1}}r^{C\delta_{k-1}}\int_{\mathbb R^{n+1}}|\sum_{J \in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|^{\frac{p}{\tilde{p}_{k-1}}} W,$$ then the lemma is proved and the iteration halts. Otherwise, assume that [\[stepkeqn\]](#stepkeqn){reference-type="eqref" reference="stepkeqn"} is bounded by $$\label{displayedabovek} (\log r)^CC_{k-1}r^{C\delta_{k-1}} \int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})}|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k|^{\frac{p}{\tilde{p}_{k-1}}} W$$ for some dyadic $\sigma$, $r^{-\delta_k}\le \sigma\le 1$. The function $\eta_{J,\sigma}^k$ is supported in $$\begin{aligned} \label{J-Jsigma} \big\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(a):\max_{0\le i\le m+k-1}&|\lambda_i|2^{-i}\sigma^{i-m-k}R^{\frac{m}{n}}r^{\frac{i-m}{n}}\in[2^{-m-k},1]\big\}\times[-2,2]^{n+1-m-k}. \end{aligned}$$ Therefore, the Fourier support of each $|\sum_{I(\theta)\subset J }f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k$ is contained in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le \sigma^{m+k-i}R^{-\frac{m}{n}}r^{\frac{m-i}{n}},\quad\max_{0\le i\le m+k}|\lambda_i|2^{-i}\sigma^{i-m-k}R^{\frac{m}{n}}r^{\frac{i-m}{n}}\in[2^{-m-k},1]\},$$ which is contained in a unique element $\tau\in\Xi_{m+k-1}^{n+1}(\sigma^{n}r)$ dilated by a factor of $\sigma^{m+k}R^{-\frac{m}{n}}r^{\frac{m}{n}}$. There are $\sim\sigma^{-1}$ many adjacent intervals $J_{m+1}$ whose corresponding sets [\[J-Jsigma\]](#J-Jsigma){reference-type="eqref" reference="J-Jsigma"} are identified with a single $\tau$. Note that after rescaling the spatial side by a factor of $\sigma^{m+k}R^{-\frac{m}{n}}r^{\frac{m}{n}}$, $W$ becomes a weight function localized to a ball of radius $\sigma^{m+k}r^{\frac{m}{n}}\rho\ge \sigma^n r$. We assumed in this case that $2\le \frac{p}{\tilde{p}_{k-1}}$ and by (4) of Lemma [Lemma 12](#pprops){reference-type="ref" reference="pprops"}, we have $\frac{p}{\tilde{p}_{k-1}}\le p_{n-m-k+1}$. Therefore, we may apply the hypothesis that $\mathbb C_{m+k-1}^{n+1}(r)\lesssim_\delta r^\delta$ and Lemma [Lemma 5](#loclem){reference-type="ref" reference="loclem"} to bound [\[displayedabovek\]](#displayedabovek){reference-type="eqref" reference="displayedabovek"} by $$C_{\delta_k} r^{C\delta_k} \int_{\mathbb R^{n+1}} |\sum_{J } ||\sum_{I(\theta)\subset J }f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k|^2|^{\frac{p}{2\tilde{p}_{k-1}}} W.$$ Since $2\ge \frac{\tilde{p}_k}{\tilde{p}_{k-1}}$, use $\|\cdot\|_{\ell^2}\le \|\cdot\|_{\ell^{\tilde{p}_k/\tilde{p}_{k-1}}}$ to bound the above displayed expression by $$C_{\delta_k} r^{C\delta_k} \int_{\mathbb R^{n+1}} |\sum_{J} ||\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k|^{\frac{\tilde{p}_k}{\tilde{p}_{k-1}}} |^{\frac{p}{\tilde{p}_k}} W.$$ Since $\sigma\ge r^{-\delta_k}$, for each $J$, we have $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k|\lesssim r^{C\delta_k}\tilde{W}_{J,{m+k}}^{n+1,d_k}$. The weight functions $\tilde{W}_{J,{m+k}}^{n+1,d_k}$ are $L^1$ normalized and satisfy $\tilde{W}_{J,{m+k-1}}^{n+1,d_{k-1}}*\tilde{W}_{J,{m+k}}^{n+1,d_k}\lesssim \tilde{W}_{J,{m+k}}^{n+1,d_k}$, so by Cauchy-Schwarz (applied pointwise to the integral from the convolution), the previous displayed expression is bounded by $$C_{\delta_k} r^{C\delta_k} \int_{\mathbb R^{n+1}} |\sum_{J} |\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{k}}*\tilde{W}_{J,{m+k}}^{n+1,d_k}|^{\frac{p}{\tilde{p}_k}} W.$$ This concludes Step $k$. The iteration reaches step $n_p$ if it has produced a bound of [\[now2\]](#now2){reference-type="eqref" reference="now2"} by [\[stepkeqn\]](#stepkeqn){reference-type="eqref" reference="stepkeqn"} with $k=n_p$. Note that by the definition of $n_p$, we have $1\le \frac{p}{\tilde{p}_{n_p-1}}\le 2$. As in Step $k$, we decompose the Fourier supports of each summand $|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{n_p-1}}*\tilde{W}_{J,{m+n_p-1}}^{n+1,d_{n_p-1}}$ into $\lesssim (\log r)$ many pieces. However, instead of performing this decomposition in the first $(m+n_p)$ many coordinates, we decompose the first $n$ many coordinates according to the dyadic parameter $\sigma$, $r^{-\delta_{n_p}}\le \sigma\le 1$, into sets $$\label{sets} \{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):\max_{0\le i\le n-1}|\lambda_i|\sigma^{i-n}R^{\frac{m}{n}}r^{\frac{i-m}{n}}\in[2^{-n},1],\quad|\lambda_n|\le R^{-\frac{m}{n}}r^{\frac{m-n}{n}}\}$$ (and $\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-n}{n}}\}$). Suppose that [\[stepkeqn\]](#stepkeqn){reference-type="eqref" reference="stepkeqn"} is bounded by $$\label{fromhere} (\log r)^CC_{n_p-1}r^{C\delta_{n_p-1}}\int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{n_p-1}}*\tilde{W}_{J,{m+n_p-1}}^{n+1,d_{n_p-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^{n-m}|^{\frac{p}{\tilde{p}_{n_p-1}}} W.$$ If $\sigma=r^{-\delta_{n_p}}$, then the lemma is proved and the iteration halts. The alternative is that $\sigma>r^{-\delta_{n_p}}$. The sets [\[sets\]](#sets){reference-type="eqref" reference="sets"} (for a fixed $\sigma$ and $a$ varying over $r^{-\frac{1}{n}}\mathbb Z\cap[0,1]$) may be organized into elements $\tau\in\Xi_{n-1}^{n+1}(\sigma^nr)$ dilated by $\sigma^nR^{-\frac{m}{n}}r^{\frac{m}{n}}$. Then invoke Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"} to bound [\[fromhere\]](#fromhere){reference-type="eqref" reference="fromhere"} by $$\label{almostdone} (\log r)^CC_{\delta_{n_p-1}}r^{C\delta_{n_p-1}}\sum_{I\in\mathcal{J}(\sigma^{-1}r^{-\frac{1}{n}})}\int_{\mathbb R^{n+1}}|\sum_{J\subset I}|\sum_{I(\theta)\subset J}f_\theta|^{\tilde{p}_{n_p-1}}*\tilde{W}_{J,{m+n_p-1}}^{n+1,d_{n_p-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^{n-m}|^{\frac{p}{\tilde{p}_{n_p-1}}}W .$$ For each $J$, we have $\tilde{W}_{J,m+n_p-1}^{n+1,d_{n_p-1}}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^{n-m}|*W\lesssim r^{C\delta_{n_p}}W$. Therefore, by Hölder's inequality (applied to the integral from the convolution) and using the fact that $\sigma>r^{-\delta_{n_p}}$, the previous displayed expression is bounded by $$\begin{aligned} (\log R)^C&C_{\delta_{n_p}} r^{C\delta_{n_p}}\sum_{J} \int_{\mathbb R^{n+1}} |\sum_{I(\theta)\subset J}f_\theta|^{p}W. \end{aligned}$$ ◻ *Proof of Lemma [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"}.* Begin by using $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|\lesssim \tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$, where $\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$ is Fourier supported in $$\label{a&e} \big\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}}(r^{-\delta_k})^{m+k-i}\big\}\times[-2,2]^{n+1-m-k}.$$ We have $$||\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}}f_\theta|^{\tilde{p}_{k-1}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,r^{-\delta_k}}^k|\lesssim |\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}})}}f_\theta|^{\tilde{p}_{k-1}}*\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}.$$ Write the integral on the right hand side evaluated at a point $x\in\mathbb R^{n+1}$: $$\label{conv} \int_{\mathbb R^{n+1}}|\sum_{I(\theta)\subset J}f_\theta(y)|^{\tilde{p}_{k-1}}\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}(x-y)dy.$$ This integral looks like a local $L^{\tilde{p}_{k-1}}$ expression of a function with Fourier support in $\cup_{I(\theta)\subset J}\theta$. The next step is to rescale this integral. Let $J$ be the interval with initial point $a$ and of length $2^{-100n}r^{-\frac{1}{n}}$. Let $A:\mathbb R^{n+1}\to\mathbb R^{n+1}$ be the linear map acting as $\phi_{m+k-1}^{(i)}(a)\mapsto r^{\frac{i}{n}}\phi_{m+k-1}^{(i)}(0)$ in the first $m+k$ coordinates and as the identity in the remaining coordinates. Applying $A$ to the Fourier support of $\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}$ [\[a&e\]](#a&e){reference-type="eqref" reference="a&e"} yields $$\label{AJ-J} \big\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(0): |\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m}{n}}(r^{-\delta_k})^{m+k-i} \quad\forall i\big\}\times[-2,2]^{n+1-m-k}.$$ This set is contained in $B\times \mathbb R^{n+1-m-k}$, where $B\subset\mathbb R^{m+k}$ is a ball of radius $R^{-\frac{m}{n}}r^{\frac{m}{n}}r^{-\delta_k}$ centered at the origin. It follows that in the first $m+k$ coordinates, $\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$ is localized to a finitely overlapping dual balls $B^*$ of radius $R^{\frac{m}{n}}r^{-\frac{m}{n}}r^{\delta_k}$. For each $\theta$ with $I(\theta)\subset J$, $A(\theta)\subset \tau_\theta\times[-2,2]^{n+1-m-k}$ where $\tau_\theta$ is a unique element of $\Xi_m^{m+k}((R/r)^{\frac{m+k-1}{n}},K,{\bf{\tilde{h}}})$, where $\tilde{h}_i=h_ir^{\frac{i}{n}}$ if $|h_i|r^{\frac{i}{n}}\ge K^{-1}R^{-\frac{m}{n}}r^{\frac{m}{n}}$ and $\tilde{h}_i=R^{-\frac{m}{n}}r^{\frac{m}{n}}$ if $|h_i|r^{\frac{i}{n}}<K^{-1}R^{-\frac{m}{n}}r^{\frac{m}{n}}$. Thus $A$ maps $\cup_{I(\theta)\subset J}\theta$ into a cylindrical neighborhood of $\Gamma_m^{m+k}((R/r)^{\frac{m+k-1}{n}},K,{\bf{\tilde{h}}})$. The rescaled integral we wish to bound is $$\int_{\mathbb R^{n+1}}|\sum_{I(\theta)\subset J_{m+1}}f_\theta(A^ty)|^{\tilde{p}_{k-1}}\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}(x-A^ty)dy.$$ For each element $y\in\mathbb R^{n+1}$, write $y=(y_{m+k},y')$ where $y_{m+k}\in\mathbb R^{m+k}$ and $y'\in\mathbb R^{n+1-m-k}$. Rewrite the rescaled integral above as $$\label{rescaled} \int_{\mathbb R^{n+1-m-k}}\int_{\mathbb R^{m+k}}|\sum_{I(\theta)\subset J}f_\theta(A^t(y_{m+k},y'))|^{\tilde{p}_{k-1}}W_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}(x-A^t(y_{m+k},y'))dy_{m+k}dy'.$$ For each $\theta$, we have by Fourier inversion and Fubini's theorem that $$\begin{aligned} f_\theta(A^ty)&=\int_{\mathbb R^{n+1}}\widehat{f}_\theta(\xi) e(y\cdot A\xi)d\xi = |\det A|^{-1}\int_{\mathbb R^{n+1}}\widehat{f}_\theta(A^{-1}\xi) e(y\cdot \xi)d\xi \\ &= \int_{\tau_\theta} \left[ |\det A|^{-1}\int_{\mathbb R^{n+1-k}}\widehat{f}_\theta(A^{-1}(\xi_{m+k},\xi')) e(y'\cdot \xi')d\xi' \right] e(y_{m+k}\cdot\xi_{m+k})d\xi_{m+k} \end{aligned}$$ where $\tau_\theta$ is the associated element of $\Xi_m^{m+k}((R/r)^{\frac{m+k-1}{n}},K,{\bf{\tilde{h}}})$. Using the displayed equation, for each $y'$, we may view $f_\theta(A^t(y_{m+k},y'))$ as a function with Fourier transform supported in $\tau_\theta$. Recall that for each $y'\in \mathbb R^{n+1-m-k}$ and $x\in\mathbb R^{n+1}$, the weight function $\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k}(x-A^t(y_{m+k},y'))$ is localized to a union of translates of $B^*$, which has radius $R^{\frac{m}{n}}r^{-\frac{m}{n}}r^{\delta_k}$. Since $2\le \tilde{p}_{k-1}\le p_{k-1}$, we may use $$S_K^{m+k-1,m}(1,\min(r^{\delta_k(m+k-1)/n},(\rho/r)^{(m+k-1)/n}),(R/r)^{(m+k-1)/n})\lesssim_\delta r^\delta$$ to bound [\[rescaled\]](#rescaled){reference-type="eqref" reference="rescaled"} by $$C_\delta r^\delta\int_{\mathbb R^{n+1-m-k}}\int_{\mathbb R^{m+k}}|\sum_{\substack{I_k\subset J}}|\sum_{I(\theta)\subset I_k} f_\theta(A^t(y_{m+k},y'))|^2|^{\frac{\tilde{p}_{k-1}}{2}}\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}(x-A^t(y_{m+k},y'))dy_{m+k}dy',$$ where $I_k$ varies over intervals in $\mathcal{J}(\max(\rho^{-1/n},r^{-(1+\delta_k)/n}))$ contained in $J$. Undoing the change of variables finishes the proof. ◻ *Proof of Lemma [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"}.* Recall that $I_k$ denotes intervals in $\mathcal{J}\big(\max(\rho^{-\frac{1}{n}},r^{-\frac{(1+\delta_k)}{n}})\big)$. Our goal is to show that $$\begin{aligned} \label{goal3} \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}|^{\frac{p}{\tilde{p}_{k-1}}}W\lesssim_\delta r^\delta\int_{\mathbb R^{n+1}}|\sum_{I_{k}}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{p}{2}}W. \end{aligned}$$ Our strategy is to use an iteration that is similar to the proof of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}. Begin by understanding the Fourier support of each $|\sum_{I_k\subset J} |\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$. For each $J$ with initial point $a$, $\sum_{I_k\subset J}|\sum_{I(\theta)\subset I_k } f_\theta|^2$ is Fourier supported in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}} \}.$$ Since $\tilde{p}_{k-1}$ is even, $|\sum_{I_k\subset J_{m+1}}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}$ is Fourier supported in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le (\tilde{p}_{k-1}/2)R^{-\frac{m}{n}}r^{\frac{m-i}{n}} \}.$$ The intersection of this set with the Fourier support of $\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$ is contained in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{\frac{m-i}{n}}(r^{\delta_k})^{m+k-i} \}.$$ Use the auxiliary functions $\sum_{r^{-\frac{1}{n}}\le \sigma_k\le r^{-\delta_k}}\eta^k_{J,\sigma_k}$ to decompose the above Fourier support and suppose that the left hand side of [\[goal3\]](#goal3){reference-type="eqref" reference="goal3"} is bounded by $$\begin{aligned} \label{case1it} C(\log r)^C \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma_k}^k|^{\frac{p}{\tilde{p}_{k-1}}}W. \end{aligned}$$ If $\sigma_k=r^{-\frac{1}{n}}$, then $\eta_{J,r^{-\frac{1}{n}}}^k$ is a smooth bump function supported in $\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{-\frac{k}{n}} \}\times[-1,1]^{n+1-m-k}$, which is contained in the Fourier support of each $\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}$. If $$B_k=\{(R^{\frac{m}{n}}r^{\frac{k}{n}}x_0)^2+\cdots+(R^{\frac{m}{n}}r^{\frac{k}{n}}x_{m+k-1})^2+x_{m+k}^2+\cdots+x_n^2\le 1\},$$ then the weight function $\tilde{W}_{B_k}^{n+1,d_{k+1}}$ satisfies $|\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma}^k|\lesssim \tilde{W}_{B_k}^{n+1,d_{k+1}}$ for each $J$. Finally, by Cauchy-Schwarz, [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is bounded by $$C(\log r)^C \int_{\mathbb R^{n+1}}|\sum_{J}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}|^{\frac{p}{\tilde{p}_{k-1}}}\tilde{W}_{B_k}^{n+1,d_{k+1}}*W.$$ Since $R^{\frac{m}{n}}\rho\ge R^{\frac{m}{n}}r^{\frac{k}{n}}$, we have $\tilde{W}_{B_k}^{n+1,d_{k+1}}*W\lesssim W$. It remains to note that since $\|\cdot\|_{\ell^{\tilde{p}_{k-1}/2}}\le\|\cdot\|_{\ell^1}$, the previous displayed expression is bounded by $$C(\log r)^C \int_{\mathbb R^{n+1}}|\sum_{I_k} |\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{p}{2}}W,$$ which proves the lemma and halts the iteration. Next, consider the case that [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} holds with $\sigma_k>r^{-\frac{1}{n}}$. As in the proof of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}, there are $\sim \sigma_k^{-1}$ many neighboring $J$ for which the Fourier support of $$|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J_{m+1},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_{m+1},\sigma_k}^k$$ is contained in an element $\tau\in\Xi_{m+k}^{n+1}(\sigma_k^nr)$ dilated by a factor of $R^{-\frac{m}{n}}r^{\frac{m}{n}}\sigma_k^{m+k}$. If we dilate frequency side by $R^{\frac{m}{n}}r^{-\frac{m}{n}}\sigma_k^{-(m+k)}$, then the weight function $W$ gets dilated by a factor of $R^{-\frac{m}{n}}r^{\frac{m}{n}}\sigma_k^{m+k}$, and so is localized to a ball of radius $r^{\frac{m}{n}}\sigma_k^{m+k}\rho\ge \sigma_k^nr$. Using the hypothesis that $\mathbb C_{m+k-1}^{n+1}(\sigma_k^n r)\lesssim_\delta(\sigma_k^n r)^\delta$ and Lemma [Lemma 5](#loclem){reference-type="ref" reference="loclem"}, [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is bounded above by $$C_{\delta}r^{\delta} \int_{\mathbb R^{n+1}}|\sum_{J_k}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma_k}^k|^2|^{\frac{p}{2\tilde{p}_{k-1}}}W ,$$ where $J_k$ are intervals in $\mathcal{J}(\sigma_k^{-1}r^{-\frac{1}{n}})$. There are $L^1$-normalized weight functions $\tilde{W}_{J_k,\sigma_k}^{n+1,d_k-1}$ with Fourier transform supported in $$\{\sum_{i=0}^{m+k-1}\lambda_i\phi_{m+k-1}^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\sigma_k^{m+k-i} \}\times[-1,1]^{n+1-m-k}$$ (where $a$ is the initial point of $J_k$) which satisfy $|\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_{m+1},\sigma_k}^k|\lesssim \tilde{W}_{J_k,\sigma_k}^{n+1,d_k-1}$. Thus [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is bounded by $$C_{\delta}r^{\delta} \int_{\mathbb R^{n+1}}|\sum_{J_k}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}*\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1} |^2|^{\frac{p}{2\tilde{p}_{k-1}}}W .$$ The Fourier support of each summand $|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1}|^2$ is contained in $$\{\sum_{i=0}^{n}\lambda_i\phi_{n}^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\sigma_k^{m+k-i} \},$$ where $a$ is the initial point of $J_k$. If $1\le \frac{p}{2\tilde{p}_{k-1}}\le 2$, then use $\eta_{J_k}^{n-m}=\sum_{\sigma_k^{-1}r^{-\frac{1}{n}}\le \sigma\le 1}\eta_{J_k,\sigma}^{n-m}$ to decompose the Fourier support into subsets $$\{\sum_{i=0}^{n}\lambda_i\phi_{n}^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\sigma_k^{m+k-i}\sigma^{n-i} \}$$ and $\{\sum_{i=0}^{n}\lambda_i\phi_{n}^{(i)}(a):|\lambda_i|\lesssim R^{-\frac{m}{n}}r^{\frac{m-n}{n}}\sigma_k^{m+k-n}\}$. Then since $\mathbb C_{n-1}^{n+1}(\sigma_k^n\sigma^nr)\lesssim_\delta(\sigma_k^n\sigma^nr)^\delta$, [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is bounded by $$C_{\delta}r^{\delta}\sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}*\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1} |^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_k,\sigma}^{n-m}|^{\frac{p}{2\tilde{p}_{k-1}}}W$$ where the initial sum is over $J_{n-m}\in\mathcal{J}(\sigma^{-1}\sigma_k^{-1}r^{-\frac{1}{n}})$. It remains to observe that there are weight functions $\tilde{W}_{J_{n-m}}^{n+1,d_{k+1}}$ satisfying $\tilde{W}_{J_k,\sigma_k}^{n+1,d_k-1}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_k,\sigma}^{n-m}|\lesssim \tilde{W}_{J_{n-m}}^{n+1,d_{k+1}}$ for all $J_k\subset J_{n-m}$ and $$\begin{aligned} \sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|&\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}}*\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1} |^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_k,\sigma}^{n-m}|^{\frac{p}{2\tilde{p}_{k-1}}}W \\ &\le \sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} |^2*\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_k,\sigma}^{n-m}||^{\frac{p}{2\tilde{p}_{k-1}}}W \\ &\lesssim \sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} |^2*\tilde{W}_{J_{n-m}}^{n+1,d_{k+1}}|^{\frac{p}{2\tilde{p}_{k-1}}}W\\ &\lesssim \sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} |^2|^{\frac{p}{2\tilde{p}_{k-1}}}(\tilde{W}_{J_{n-m}}^{n+1,d_{k+1}}*W)\\ &\lesssim \sum_{J_{n-m}}\int_{\mathbb R^{n+1}}|\sum_{J_k\subset J_{n-m}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} |^2|^{\frac{p}{2\tilde{p}_{k-1}}}W\\ &\lesssim \int_{\mathbb R^{n+1}}|\sum_{I_k}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{p}{2}}W,\end{aligned}$$ which proves the lemma. Now suppose that $\frac{p}{2\tilde{p}_{k-1}}> 2$. Let $p_{n-m-k_2}$ be the minimal exponent such that $2\le\frac{p}{2\tilde{p}_{k-1}}\le p_{n-m-k_2}$ and $n-m\ge k_2> k$. Use $\eta^{k_2}_{J_k,\sigma_{k_2}}$ to decompose the Fourier supports of $|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J_k,\sigma_k}^{n+1,d_k-1}|^2$ into $$\begin{aligned} \Big\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):&|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\sigma_k^{m+k-i}\sigma^{m+k_2-i}\quad\forall i,\\ &\qquad |\lambda_j|\ge 2^{-m-k_2+j}R^{-\frac{m}{n}}r^{\frac{m-j}{n}}\sigma_k^{m+k-j}\sigma^{m+k_2-j}\quad\text{for some }j \text{ with } 0\le j\le m+k_2-1 \Big\} \end{aligned}$$ and $\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{-\frac{k_2}{n}} \sigma_k^{k-k_2}\quad\forall i\}$. Using that $\mathbb C_{m+l}^{n+1}(\sigma_{k_2}^n\sigma_k^nr)\lesssim_\delta(\sigma_{k_2}^n\sigma_k^nr)^\delta$, [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is then bounded by $$\begin{aligned} C_{\delta}r^{\delta} \int_{\mathbb R^{n+1}}&|\sum_{J_{k_2}}|\sum_{J_k\subset J_{k_2}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1}|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J_k,\sigma_{k_2}}^{k_2}|^2|^{\frac{p}{2^2\tilde{p}_{k-1}}}W\\ &\lesssim C_{\delta}r^{\delta} \int_{\mathbb R^{n+1}}|\sum_{J_{k_2}}|\sum_{J_k\subset J_{k_2}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J_{k},\sigma_k}^{n+1,d_k-1}|^2*\tilde{W}_{J_{k_2},\sigma_{k_2}}^{n+1,d_{k_2}+1}|^2|^{\frac{p}{2^2\tilde{p}_{k-1}}}W\end{aligned}$$ where the initial sum is over $J_{k_2}\in\mathcal{J}(\sigma_k^{-1}\sigma_{k_2}^{-1}r^{-\frac{1}{n}})$. Iterate this process $j-2$ more times until either the lemma has been proved (corresponding to a low frequency case) or the outermost exponent $\frac{p}{2^j\tilde{p}_{k-1}}$ satisfies $1\le \frac{p}{2^j\tilde{p}_{k-1}}\le 2$ and, after repeating the case $1\le \frac{p}{2\tilde{p}_{k-1}}\le 2$ argument, [\[case1it\]](#case1it){reference-type="eqref" reference="case1it"} is bounded by $$\label{ben} C_{\delta_k}r^{C\delta_k} \sum_{J_{k_{j+1}}}\int_{\mathbb R^{n+1}}|\sum_{J_{k_j}\subset J_{k_{j+1}}}|F_{J_{k_j}}|^2*\tilde{W}_{J_{k_{j+1}},\sigma_{j+1}}^{n+1,d_{k_{j+1}+1}}|^{\frac{p}{2^j\tilde{p}_{k-1}}}W$$ where $k=k_1$, $\sigma_{k_i}\in[(\sigma_{k_1}\cdots\sigma_{k_{i-1}})^{-1}r^{-\frac{1}{n}},1]$, $J_{k_i}\in\mathcal{J}((\sigma_{k_1}\cdots\sigma_{k_{i-1}})^{-1}r^{-\frac{1}{n}})$, and $$F_{J_i}=\sum_{J_{i-1}\subset J_i}|F_{J_{i-1}}|^2*W_{J_i,\sigma_i}^{n+1,d_{k_i}+1}\qquad\text{with}\qquad F_{J_{k_1}}=\sum_{J\subset J_{k_1}} |\sum_{I_{k_1}\subset J}|\sum_{\substack{I(\theta)\subset I_{k_1}}} f_\theta|^2|^{\frac{\tilde{p}_{k_1-1}}{2}} *\tilde{W}_{J_{k_1},\sigma_{k_1}}^{n+1,d_{k_1}-1}.$$ The weight functions satisfy the property that if $J_{k_i}\subset J_{k_{i+l}}$, then $\tilde{W}_{J_{k_i},\sigma_i}^{n+1,d_{k_i}-1}*\tilde{W}_{J_{k_{i+l}},\sigma_{i+l}}^{n+1,d_{k_{i+l}}+1}\sim \tilde{W}_{J_{k_{i+l}},\sigma_{i+l}}^{n+1,d_{k_{i+l}}}$. Therefore, by applying Cauchy-Schwarz pointwise in the convolutions, we have that $$\begin{aligned} F_{J_{k_2}}=\sum_{J_{k_1}\subset J_{k_2}}|F_{J_{k_1}}|^2*\tilde{W}_{J_{k_2},\sigma_{k_2}}^{n+1,d_{k_2}+1} &\lesssim \sum_{J_{k_1}\subset J_{k_2}}|\sum_{J\subset J_{k_1}} |\sum_{I_{k_1}\subset J}|\sum_{\substack{I(\theta)\subset I_{k_1}}} f_\theta|^2|^{\frac{\tilde{p}_{k_1-1}}{2}} |^2*\tilde{W}_{J_{k_2},\sigma_{k_2}}^{n+1,d_{k_2}}\\ &\lesssim |\sum_{I_k\subset J_{k_2}}|\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{\tilde{p}_{k-1}} *\tilde{W}_{J_{k_2},\sigma_{k_2}}^{n+1,d_{k_2}}\end{aligned}$$ and by induction, $$\begin{aligned} \sum_{ J_{k_{i-1}}\subset J_{k_i} }|F_{J_{k_{i-1}}}|^2*\tilde{W}_{J_{k_i},\sigma_{k_i}}^{n+1,d_{k_i}+1} &\lesssim \sum_{ J_{k_{i-1}}\subset J_{k_i}}||\sum_{I_k\subset J_{k_{i-1}}} |\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{2^{i-3}\tilde{p}_{k-1}} *\tilde{W}_{J_{k_{i-1}},\sigma_{i-1}}^{n+1,d_{k_{i-1}}}|^2*\tilde{W}_{J_{k_i},\sigma_{k_i}}^{n+1,d_{k_i}+1}\\ &\lesssim \sum_{ J_{k_{i-1}}\subset J_{k_i}}|\sum_{I_k\subset J_{k_{i-1}}} |\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{2^{i-2}\tilde{p}_{k-1}}*\tilde{W}_{J_{k_i},\sigma_{k_i}}^{n+1,d_{k_i}}\\ &\lesssim |\sum_{I_k\subset J_{k_{i}}} |\sum_{\substack{I(\theta)\subset I_{k}}} f_\theta|^2|^{2^{i-2}\tilde{p}_{k-1}}*\tilde{W}_{J_{k_i},\sigma_{k_i}}^{n+1,d_{k_i}}. \end{aligned}$$ Conclude that [\[ben\]](#ben){reference-type="eqref" reference="ben"} is therefore bounded by $$\begin{aligned} C_{\delta_k}r^{C\delta_k} \sum_{J_{k_{j+1}}}\int_{\mathbb R^{n+1}}&||\sum_{I_k\subset J_{k_{j+1}}}|\sum_{I(\theta)\subset I_k}|f_\theta|^2|^{2^{j-1}\tilde{p}_{k-1}}*\tilde{W}_{J_{k_{j+1}},\sigma_{j+1}}^{n+1,d_{j+1}}|^{\frac{p}{2^j\tilde{p}_{k-1}}}W\\ &\lesssim C_{\delta_k}r^{C\delta_k} \sum_{J_{k_{j+1}}}\int_{\mathbb R^{n+1}}|\sum_{I_k\subset J_{k_{j+1}}}|\sum_{I(\theta)\subset I_k}|f_\theta|^2|^{\frac{p}{2}}W\\ &\lesssim C_{\delta_k}r^{C\delta_k} \int_{\mathbb R^{n+1}}|\sum_{I_k}|\sum_{I(\theta)\subset I_k}|f_\theta|^2|^{\frac{p}{2}}W,\end{aligned}$$ as desired. This final case is analogous to the beginning of the $2< \frac{p}{2\tilde{p}_{k-1}}$ case. We use auxiliary functions $\sum_{r^{-\frac{1}{n}}\le\sigma\le r^{-\delta_k}}\eta_{J,\sigma_k}^{k}$ to decompose the Fourier supports. By using a cylindrical version of the bound $\mathbb C_{m+k-2}^{m+k}(\sigma_k^{m+k-1}r^{\frac{m+k-1}{n}})\lesssim_\delta(\sigma_k^{m+k-1}r^{\frac{m+k-1}{n}})^\delta$, the left hand side of [\[goal3\]](#goal3){reference-type="eqref" reference="goal3"} is bounded by $$\begin{aligned} C_\delta r^\delta\sum_{J_k\in\mathcal{J}(\sigma_k^{-1}r^{-\frac{1}{n}})} \int_{\mathbb R^{n+1}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} *\tilde{W}_{J,{m+k},r^{-\delta_k}}^{n+1,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma_k}^k|^{\frac{p}{\tilde{p}_{k-1}}}W.\end{aligned}$$ Since $\tilde{W}_{J,m+k,r^{-\delta_k}}^{n+1,d_k-1}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{J,\sigma_k}^k|\lesssim \tilde{W}_{J_k,\sigma_k}^{n+1,d_{k+1}}$, where $\tilde{W}_{J_k,\sigma_k}^{n+1,d_{k+1}}$ is $L^1$-normalized, and $\tilde{W}_{J_k,\sigma_k}^{n+1,d_{k+1}}*W\lesssim W$, the previous displayed math is bounded by $$C_\delta r^\delta\sum_{J_k\in\mathcal{J}(\sigma_k^{-1}r^{-\frac{1}{n}})} \int_{\mathbb R^{n+1}}|\sum_{J\subset J_k}|\sum_{I_k\subset J}|\sum_{\substack{I(\theta)\subset I_{k}%\\ \theta\in\Xi_m^{n+1}(R,K,{\bf{h}}) }}f_\theta|^2|^{\frac{\tilde{p}_{k-1}}{2}} |^{\frac{p}{\tilde{p}_{k-1}}}W.$$ Finally, observing using that $\|\cdot\|_{\ell^{\tilde{p}_{k-1}/2}}\le \|\cdot\|_{\ell^1}$ and that $\|\cdot\|_{\ell^{p/2}}\le \|\cdot\|_{\ell^1}$ finishes the proof. ◻ The argument used to prove Lemma [Lemma 15](#multilem3){reference-type="ref" reference="multilem3"} actually justifies a more general proposition, which we state here for future use. Let $\tau\in\Xi_m^{n+1}(r)$ and consider $r^{-\frac{1}{n}}\le \sigma\le 1$. For each $k\ge 1$ and $d\ge 1$, let $\tilde{W}^k_{\tau,\sigma}$ be the $L^\infty$-normalized weight function with decay factor $d$ (this dependence is suppressed because it is not important) which is centered at the origin and Fourier supported in $$\{\sum_{i=0}^{k-1}\lambda_i\phi_{k-1}^{(i)}(a):|\lambda_i|\le \sigma^{k-i} r^{\frac{-i}{n}} \}\times[-2,2]^{n+1-k}$$ where $a$ is the initial point of $I(\tau)$. **Lemma 16**. *Suppose that $\mathbb C^{n'+1}_{m'}(r)\lesssim_\varepsilon r^\varepsilon$ for all $0\le m'\le n'-1<n-1$ and $\mathbb C^{n+1}_{m'}(r)\lesssim_\varepsilon r^\varepsilon$ for all $m<m'\le n-1$. Let $p_{n,m}(k)=p_{n-k}$ for $k=1,\ldots,m+1$ and $p_{n,m}(k)=p_{n+1-k}$ for $k=m+2,\ldots,n$. For any $2\le p\le p_{n,m}(k)$ and any $r^{-\frac{1}{n}}\le \sigma\le 1$, $$\int_{\mathbb R^{n+1}}|\sum_{\tau\in\Xi_m^{n+1}(r)} G_\tau*\tilde{W}^{k}_{\tau,\sigma}|^{p}\lesssim_\delta r^\delta\int_{\mathbb R^{n+1}}|\sum_{\tau\in\Xi_m^{n+1}(r)} G_\tau |^{p}$$ whenever $G_{\tau}:\mathbb R^n\to[0,\infty)$ has Fourier transform supported in $\tau-\tau$.* ## Inductive proof of Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"} [\[SnmmKbdsec\]]{#SnmmKbdsec label="SnmmKbdsec"} {#inductive-proof-of-proposition-snmmkbd-snmmkbdsec} *Inductive proof of Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"} .* We will show that for any $\varepsilon>0$, there exists $\tilde{C}_\varepsilon$ such that $S_{K}^{n,m}(r,\rho,R)\le \tilde{C}_\varepsilon\rho^\varepsilon$ for all $1\le r\le\rho\le R$. We are permitted to choose $K$ to be a constant depending on $\varepsilon$. By Cauchy-Schwarz, if $R\le C_K$, then $S_{K}^{n,m}(r,\rho,R)\lesssim_K 1$. Suppose that for any $1\le r\le \rho\le R$ satisfying $R<R_0/K$, we have $S_{K}^{n,m}(r,\rho,R)\le \tilde{C}_\varepsilon\rho^\varepsilon$. We will show that for any $1\le r\le \rho\le R$ satisfying $R<R_0$, $S_{K}^{n,m}(r,\rho,R)\le \tilde{C}_\varepsilon\rho^\varepsilon$. It suffices to consider $\rho\ge C_K$. If $r\le K$, then by Proposition [Proposition 9](#initscale){reference-type="ref" reference="initscale"}, $S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1} K^{\delta_1} S_{K}^{n,m}(K,\rho,R)$, where we may choose any $\delta_1>0$. Assume from now on that $r\ge K$. By Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}, $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}C_{\delta_2}\left[r^{\delta_2}S_{K}^{n,m}(1,\rho/r,R/r)+\sum_{k=1}^{n-m} r^{\delta_2^{n-k+1}}\Big[S_{K}^{n,m}(\min(\rho,r^{1+\delta_2^{n-k}}),\rho,R) \Big]\right],$$ where we may choose any $\delta_2>0$. We will perform an iteration that depends on which term dominates the right hand side. Suppose that $$S_{K}^{n,m}(r,R)\le C_{\delta_1}K^{\delta_1}nC_{\delta_2}r^{\delta_2}S_{K}^{n,m}(1,\rho/r,R/r).$$ Note that by assumption, $\frac{R}{r}< \frac{R_0}{K}$, so the inductive hypothesis applies and we have $$S_{K}^{n,m}(r,R)\le \big(nC_{\delta_1}K^{\delta_1}C_{\delta_2}r^{\delta_2}r^{-{\varepsilon}}\big)\tilde{C}_\varepsilon\rho^\varepsilon,$$ which halts our iteration. The other case is that for some $1\le k\le n-m$, $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}nC_{\delta_2}r^{\delta_2^{n-k+1}}S_{K}^{n,m}(\min(\rho,r^{1+\delta_2^{n-k}}),\rho,R).$$ If $r^{1+\delta_2^{n-k}}\ge \rho$, then $S_{K}^{n,m}(\rho,\rho,R)=1$ and we have $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}nC_{\delta_2}r^{\delta_2^{n-k+1}}\le \big(C_{\delta_1}K^{\delta_1}C_{\delta_2}r^{\delta_2^{n-k+1}}\rho^{-\varepsilon}\big)\tilde{C}_\varepsilon\rho^\varepsilon$$ and again, our iteration halts. Finally, suppose that $r^{1+\delta_0^{n-k}}<\rho$. Write $\eta_1=\delta_2^{n-k}$ so that we have $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}nC_{\delta_2}r^{\delta_2\eta_1}S_{K}^{n,m}(r^{(1+\eta_1)},\rho,R).$$ Now proceed to the second step of the iteration. The input for step $k$ of the iteration (supposing that it has not halted at a previous step) is the inequality $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k-1}r^{\delta_2[\eta_1+\cdots+\eta_{k-1}\prod_{i=1}^{k-2}(1+\eta_i)]} S_{K}^{n,m}(r^{\prod_{i=1}^{k-1}(1+\eta_i)},\rho,R).$$ Apply Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} again to bound $S_{K}^{n,m}(r^{\prod_{i=1}^{k-1}(1+\eta_i)},\rho,R)$ on the right hand side above by $$\begin{aligned} C_{\delta_2}&\left[r^{\delta_2\prod_{i=1}^{k-1}(1+\eta_i)}S_{K}^{n,m}(1,\rho/r^{\prod_{i=1}^{k-1}(1+\eta_i)},R/r^{\prod_{i=1}^{k-1}(1+\eta_i)})\right.\\ &\qquad\qquad\qquad\left.+\sum_{l=1}^{n-m} r^{\delta_2^{n-l+1}\prod_{i=1}^{k-1}(1+\eta_i)}S_{K}^{n,m}(\min(\rho,r^{(1+\delta_2^{n-l})\prod_{i=1}^{k-1}(1+\eta_i)}),\rho,R)\right]. \end{aligned}$$ If the first term dominates, then by the inductive hypothesis, $$S_{K}^{n,m}(r,\rho,R)\le (C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k-1}r^{\delta_2[\eta_1+\cdots+\eta_{k-1}\prod_{i=1}^{k-2}(1+\eta_i)]} C_{\delta_2}r^{\delta_2\prod_{i=1}^{k-1}(1+\eta_i)}r^{-\varepsilon\prod_{i=1}^{k-1}(1+\eta_i)})\tilde{C}_\varepsilon\rho^\varepsilon,$$ and the iteration halts. If one of the other terms dominates, then for some $1\le l\le n-m$, $\eta_k:=\delta_2^{n-l}$ satisfies $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k}r^{\delta_2[\eta_1+\cdots+\eta_{k}\prod_{i=1}^{k-1}(1+\eta_i)]}S_{K}^{n,m}(\min(\rho,r^{\prod_{i=1}^k(1+\eta_i)}),\rho,R).$$ If $\rho\le r^{\prod_{i=1}^k(1+\eta_i)}$, then the iteration halts with the inequality $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k}r^{\delta_2[\eta_1+\cdots+\eta_{k}\prod_{i=1}^{k-1}(1+\eta_i)]}.$$ If $\rho>r^{\prod_{i=1}^k(1+\eta_i)}$, then $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k}r^{\delta_2[\eta_1+\cdots+\eta_{k}\prod_{i=1}^{k-1}(1+\eta_i)]}S_{K}^{n,m}(r^{\prod_{i=1}^k(1+\eta_i)},\rho,R)$$ and we proceed to step $k+1$. There are two possible outcomes of the iteration. The first is that for some $k\ge 1$ with $r^{\prod_{i=1}^{k-1}(1+\eta_i)}\le \rho$, we have $$S_{K}^{n,m}(r,\rho,R)\le C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k}r^{\delta_2[\eta_1+\cdots+\eta_{k-1} \prod_{i=1}^{k-2}(1+\eta_i)]}r^{\delta_2\prod_{i=1}^{k-1}(1+\eta_i)-\varepsilon\prod_{i=1}^{k}(1+\eta_i)}\tilde{C}_\varepsilon\rho^\varepsilon.$$ It suffices to verify that $$C_{\delta_1}K^{\delta_1}(nC_{\delta_2})^{k}r^{(2\delta_2-\varepsilon)\prod_{i=1}^{k-1}(1+\eta_i)}\le 1.$$ Choose $\delta_1=\frac{\varepsilon}{8}$ and $\delta_2=\frac{\varepsilon}{4}$ so that it suffices to verify that $$C_{\varepsilon/8}K^{\varepsilon/8}(nC_{\varepsilon/4})^k r^{-\varepsilon/2(1+(\varepsilon/4)^n)^k}\le 1 .$$ This follows from $K\le r$ and choosing $K=K(\varepsilon)$ large enough so that the above inequality holds for all $k\ge 1$. The second possible outcome of the iteration is that for $r^{\prod_{i=1}^{k-1}(1+\eta_i)}\le \rho\le r^{\prod_{i=1}^k(1+\eta_i)}$, we have the inequality $$S_{K}^{n,m}(r,\rho,R)\le C_{\varepsilon/8}K^{\varepsilon/8}(nC_{\varepsilon/4})^{k}r^{(\varepsilon/4)[\eta_1+\cdots+\eta_k\prod_{i=1}^{k-1}(1+\eta_i)]},$$ where we inputted the definitions of $\delta_1$ and $\delta_2$. The exponent of $r$ satisfies $\eta_1+\cdots+\eta_k\prod_{i=1}^{k-1}(1+\eta_i)\le (1+\max_{1\le j\le k}\eta_j)\prod_{i=1}^{k-1}(1+\eta_i)$, so it suffices to verify that $$C_{\varepsilon/8}K^{\varepsilon/8}(nC_{\varepsilon/4})^{k}\rho^{(\varepsilon/4)(1+\varepsilon/4)} \le \tilde{C}_\varepsilon\rho^\varepsilon.$$ Using that $r^{(1+(\varepsilon/4)^n)^k}\le \rho$ and $K\le \rho$, it further suffices to verify that $$C_{\varepsilon/8}(\log \rho)^{4^n\varepsilon^{-n}\log (nC_{\varepsilon/4})}\rho^{(5\varepsilon/8)(1+\varepsilon/4)} \le \tilde{C}_\varepsilon\rho^\varepsilon.$$ It is no loss of generality to assume that $\tilde{C}_\varepsilon$ and $K$ are large enough so that the above inequality is true, which concludes the proof. ◻ ## Bounding $S_{\sigma}^{n,m}(r,\rho,R)$ for $m>0$ We use the boundedness of $S_K^{n,m}(r,\rho,R)$ proved in Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"} to bound $S_\sigma^{n,m}(r,\rho,R)$. Call the $(m+1)$-tuples $\sigma$ considered in the definition of $\Gamma_m^{n+1}(R,\sigma)$ *admissible* and write $\mathcal{S}_R^{n,m}$ for the collection of admissible $\sigma$. **Proposition 17**. *Let $1\le n$. Then $S_{\sigma}^{n,n-1}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$ uniformly in $\sigma\in\mathcal{S}^{n,m}_R$.* *Proof.* This follows immediately from Theorem [Theorem 6](#locL2thm){reference-type="ref" reference="locL2thm"}. ◻ **Proposition 18**. *Assume that the hypotheses of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} hold. Also assume that $S^{n',m'}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$ for each $0\le m'\le n'-1$ when $n'<n$, and $S^{n,m'}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$ for each $m< m'\le n-1$. Then $S^{n,m}_{\sigma}(r,\rho,R)\lesssim_\varepsilon\rho^\varepsilon$ uniformly in $\sigma\in\mathcal{S}^{n,m}_R$.* If $m=0$, then it is easy to show that $S^{n,m}_\sigma(r,\rho,R)\le K^{O(1)}S_K^{n,m}(r,\rho,R)$, so Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"} follows from Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"}. For the rest of this section, assume that $0<m$. We describe how to modify the iteration used to bound $S_{K}^{n,m}(r,\rho,R)$ in Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"} to treat the $S_{\sigma}^{n,m}(r,\rho,R)$. To prove Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"}, consider a Schwartz function $f=\sum_{\theta\in\Xi_{m}^{n+1}(R,\sigma)} f_\theta$ with $\mathrm{supp\,}\widehat{f}_\theta\subset \theta$, $W=W_B^{n+1,d}$ for a ball of radius $R^{\frac{m}{n}}\rho$, and $2\le p\le p_{n-m}$. The following lemma uses a cylindrical version of a lower-dimensional inequality to obtain an initial square function estimate into intervals of length $|J|=2^{-100n} \sigma_{m-1}^{-1}R^{-\frac{1}{n}}$. **Lemma 19**. *Assume that $0<m$. For each $1\le r\le \min(\rho,\sigma^nR)$ We have $$\begin{aligned} \int_{\mathbb R^{n+1}}(\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})} |\sum_{\substack{\theta\in\Xi_{m}^{n+1}(R,\sigma)\\I(\theta)\subset J}} f_\theta&|^2)^{\frac{p}{2}}W\lesssim S_{\sigma'}^{n-1,m-1}(1,\min(\rho,\sigma^nR)^{\frac{n-1}{n}},\min(\rho,\sigma^n R)^{\frac{n-1}{n}}) \\ &\times \int_{\mathbb R^{n+1}}|\sum_{J\in\mathcal{J}(\max(\rho^{-\frac{1}{n}},\sigma^{-1}R^{-\frac{1}{n}}))} |\sum_{\substack{I(\theta)\subset J}\theta\in\Xi_{m}^{n+1}(R,\sigma)}f_\theta|^2|^{\frac{p}{2}}W. \end{aligned}$$ where $\sigma'$ is an $m$-tuple with $\sigma_i'=\sigma_i$ for $i<m-1$ and $\sigma_{m-1}'=1$* **Lemma 20**. *Assume the hypotheses of Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"} and that $0<m$. For each $r^{-\frac{1}{n}}\le \sigma_{m-1}^{-1}R^{-\frac{1}{n}}$, let $J$ denote intervals in $\mathcal{J}( r^{-\frac{1}{n}})$ and let $J'$ denote intervals in $\mathcal{J}( \rho^{-\frac{1}{n}})$. Then for any $\delta_0\in(0,1)$, $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}&f_\theta|^2|^{\frac{p}{2}}W\lesssim_{\delta_0} r^{\delta_0}\sum_{J\in\mathcal{J}(r^{-\frac{1}{n}})} \int_{\mathbb R^{n+1}} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^pW \\ & + \sum_{k=1}^{n-m} r^{\delta_0^{n-k+1}}\Big[S_{\sigma}^{n,m}(\min(\rho,r^{1+\delta_0^{n-k}}),\rho,R) \Big]\int_{\mathbb R^{n+1}}|\sum_{J'} |\sum_{\substack{I(\theta)\subset J'\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W. \end{aligned}$$* **Lemma 21**. *Assume that $r^{-\frac{1}{n}}\le \sigma_{m-1}^{-1}R^{-\frac{1}{n}}$. For each $J\in\mathcal{J}(r^{-\frac{1}{n}})$, $$\begin{aligned} \int_{\mathbb R^{n+1}} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^pW&\lesssim S_{\sigma(r)}^{n,m}(1,\rho/r,R/r) \int_{\mathbb R^{n+1}} |\sum_{J'\subset J}|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W\end{aligned}$$ where $\sigma(r)$ is an $(m+1)$-tuple with $\sigma_m(r)=1$ and $\sigma_i(r)=\max(\sigma_i,\sigma_{i+1}^{-1}\cdots\sigma_m^{-1}(R/r)^{-1/n})$ for $i=0,\ldots,m-1$.* First we show how the previous lemmas imply Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"}. *Proof of Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"}.* Let $\sigma\in\mathcal{S}^{n,m}_R$. Assume that $\sigma_{m-1}^{-1}R^{-\frac{1}{n}}>K^{-1}$. In this case, $S_\sigma^{n,m}(r,\rho,R)\lesssim K^CS_K^{n,m}(r,\rho,R)$. This is because after using the triangle inequality and rescaling (for the cost of $K^C$), it suffices to suppose that each $f_\theta$ is supported in $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a_\theta):R^{-\frac{m}{n}}\ge |\lambda_m|\ge R^{-\frac{m}{n}}/2,\quad |\lambda_i|\le \min(K^{-1}R^{-\frac{m}{n}},R^{-\frac{i}{n}})\quad\forall i\not=m\}.$$ By dyadic pigeonholing, there is a bump function $\eta_{h_m}$ supported in an annulus $||\xi|-h_m|<K^{-\beta}R^{-\frac{m}{n}}$, for some $R^{-\frac{m}{n}}/2\le |h_m|\le R^{-\frac{m}{n}}$ which satisfies $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R,\sigma)}f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{h_m}|^pW.$$ The Fourier support of each $f_\theta*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{h_m}$ is contained in an element of $\Xi_m^{n+1}(R,K,{\bf{h}})$ for ${\bf{h}}=(K^{-1}R^{-\frac{m}{n}},\ldots,K^{-1}R^{-\frac{m}{n}},h_m)$. Then by Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"}, $$\int_{\mathbb R^{n+1}}|\sum_{\substack{\theta\in\Xi_m^{n+1}(R,\sigma)}} f_{\theta}|^{p}W\lesssim_\varepsilon K^{O(1)}\rho^\varepsilon\int_{\mathbb R^{n+1}}(\sum_{J\in\mathcal{J}(\rho^{-\frac{1}{n}})}|\sum_{\substack{\theta\in\Xi_m^{n+1}(R,\sigma)\\I(\theta)\subset J}} f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{h_m}|^2)^{\frac{p}{2}}W.$$ By Cauchy-Schwarz and properties of weight functions, the integrand is $\lesssim (\sum_{\theta\in\Xi_m^{n+1}(R,\sigma)}|f_\theta|^2)^{\frac{p}{2}}W$. Since $K$ is a constant depending on $\varepsilon$, the proposition is proved in this case. It remains to consider the $\sigma\in\mathcal{S}^{n,m}_R$ satisfying $\sigma_{m-1}^{-1}R^{-\frac{1}{n}}<K^{-1}$. By Lemma [Lemma 19](#lem1k){reference-type="ref" reference="lem1k"}, it suffices to consider $r^{-\frac{1}{n}}\le \sigma_{m-1}^{-1}R^{-\frac{1}{n}}$. Then the combination of Lemmas [Lemma 20](#lem2k){reference-type="ref" reference="lem2k"} and [Lemma 21](#lem3k){reference-type="ref" reference="lem3k"} implies that for each $0<\delta_0<1$, $$\begin{aligned} S_{\sigma}^{n,m}(r,\rho,R)&\lesssim_{\delta_0} r^{\delta_0} S_{\sigma(r)}^{n,m}(1,\rho/r,R/r) +\sum_{k=1}^{n-m} r^{\delta_0^{n-k+1}}\Big[S_{\sigma}^{n,m}(\min(\rho,r^{1+\delta_0^{n-k}}),\rho,R) \Big]. \end{aligned}$$ This is similar to the multi-scale inequality satisfied by $S_{K}^{n,m}(r,\rho,R)$. In this set-up, we may bound the first term immediately since $(\sigma(r))_{m-1}\le K(R/r)^{-\frac{1}{n}}$. It thus suffices to suppose that $$S_{\sigma}^{n,m}(r,\rho,R)\lesssim_{\delta_0} \sum_{k=1}^{n-m} r^{\delta_0^{n-k+1}}\Big[S_{\sigma}^{n,m}(\min(\rho,r^{1+\delta_0^{n-k}}),\rho,R) \Big].$$ This implies the desired bound for $S_\sigma^{n,m}(r,\rho,R)$ by the same argument as in the proof of Proposition [Proposition 8](#SnmmKbd){reference-type="ref" reference="SnmmKbd"}. ◻ It remains to verify Lemmas [Lemma 19](#lem1k){reference-type="ref" reference="lem1k"}, [Lemma 20](#lem2k){reference-type="ref" reference="lem2k"}, and [Lemma 21](#lem3k){reference-type="ref" reference="lem3k"}. *Proof of Lemma [Lemma 19](#lem1k){reference-type="ref" reference="lem1k"}.* By Khintchine's inequality, it suffices to bound $$\int_{\mathbb R^{n+1}}|\sum_{\theta\in\Xi_m^{n+1}(R,\sigma)}f_\theta|^pW.$$ Organize the $f_\theta$ into subcollections based on $I(\theta)$ being in the same $2^{-100n}\sigma_{m-1}^{-1}R^{-\frac{1}{n}}$-interval. If $J\in\mathcal{J}(\sigma^{-1}R^{-\frac{1}{n}})$, then $\sum_{I(\theta)\subset J}f_\theta$ is Fourier supported in a set $\sigma^m\tau\times \mathbb R$ for some $\tau\in\Xi_{m-1}^{n}(\sigma^{n-1}R^{\frac{n-1}{n}},\sigma')$. Here, $\sigma'$ is an $m$-tuple with $\sigma_i'=\sigma_i$ for $i<m-1$ and $\sigma_{m-1}'=1$. After dilating the spatial side by a factor of $\sigma^m$, the weight function $W$ becomes localized to a ball of radius $\sigma^mR^{\frac{m}{n}}\rho\ge \sigma^{m-1}R^{\frac{m-1}{n}}\min(\rho,\sigma^nR)^{\frac{n-1}{n}}$. Therefore, the lemma follows from a cylindrical version of the defining inequality for $S_{\sigma'}^{n-1,m-1}(1,\min(\rho,\sigma^{n}R)^{\frac{n-1}{n}},\min(\rho,\sigma^{n}R))^{\frac{n-1}{n}}$. ◻ *Proof of Lemma [Lemma 20](#lem2k){reference-type="ref" reference="lem2k"}.* Lemma [Lemma 20](#lem2k){reference-type="ref" reference="lem2k"} is a small variation of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}, so we only describe how to adapt the proof of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}. We consider $\sigma^nR\le r\le \rho\le R$. For each $J$, the summand $|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2$ is Fourier supported in a set of the form $$\label{typeof} \{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\}$$ for some $a\in 2^{-100n}r^{-\frac{1}{n}}\mathbb Z\cap[0,1]$. As before, let $\delta_k=\delta_0^{n-k}$ and $d_k=4(n-k)d$. As in the proof of Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}, we dyadically decompose the Fourier supports into subsets $$\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le\mu^{m+2-i}R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\}\setminus \{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le(\mu/2)^{m+2-i}R^{-\frac{m}{n}}r^{\frac{m-i}{n}} \} ,$$ where $r^{\delta_2}\le \mu\le 1$ and we take the smallest set to be $\{\sum_{i=0}^n\lambda_i\phi_n^{(i)}(a):|\lambda_i|\le (r^{-\delta_2})^{m+2-i}R^{-\frac{m}{n}}r^{\frac{m-i}{n}}\}$. Suppose that $\eta^{2}_{J,\mu}$ are bump functions associated to the above sets and which satisfy $$\int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W\lesssim (\log r)^c \int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% ^{2}_{J,\mu}|^{\frac{p}{2}}W .$$ If $\mu>r^{-\delta_2}$, then since $r^{\frac{m}{n}}\rho\ge \mu^nr$, we have by $\mathbb C^{n+1}_{m+1}(r)\lesssim_\delta r^\delta$ and Lemma [Lemma 5](#loclem){reference-type="ref" reference="loclem"} that $$\begin{aligned} \int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% ^{2}_{J,\mu}|^{\frac{p}{2}}W &\lesssim_{\delta_0}r^{C\delta_2} \int_{\mathbb R^{n+1}}|\sum_{J} ||\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% ^{2}_{J,\mu}|^2|^{\frac{p}{4}}W \\ &\lesssim_{\delta_0} r^{C\delta_2} \int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^4*\tilde{W}_{J,{m+2}}^{n+1,d_2}|^{\frac{p}{4}}W . \end{aligned}$$ Since the Fourier support of each summand $|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^4*\tilde{W}_{J,{m+2}}^{n+1,d_2}$ is contained in the same type of set as [\[typeof\]](#typeof){reference-type="eqref" reference="typeof"}, we may iterate this procedure. If $1\le \frac{p}{4}\le 2$, then decompose the first $n$ many coordinates. If $2\le \frac{p}{4}\le \tilde{p}_{n-m-2}$, then decompose the first $m+3$ many coordinates. If each step of the iteration produces high-frequency dominating cases, we will obtain the inequality of the form $$\int_{\mathbb R^{n+1}}|\sum_J |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W\lesssim_{\delta_0}r^{\delta_0}\sum_J\int_{\mathbb R^{n+1}}|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^p W ,$$ which proves the lemma. If one of the steps of the iteration is low-frequency dominating, then we use a pointwise bound. We now describe one of those cases. In the case that $\mu=r^{-\delta_2}$, which we call a low-frequency case, we use an argument analogous to Lemma [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"}. We bound each summand $$||\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% ^{2}_{J,\mu}(x)|\lesssim |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*\tilde{W}^{n+1,d_2}_{J,m+2,\mu}(x)$$ pointwise. After rescaling, the expression on the right hand side becomes $$|\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}\underline{f_\theta}|^2*\underline{\tilde{W}}^{n+1,d_2}_{J,m+2,\mu}(x) .$$ If we write $x=(x_2,x')\in\mathbb R^{m+2}\times\mathbb R^{n-m-1}$, then for fixed $x'$, $\underline{\tilde{W}}^{n+1,d_2}_{J,\mu}(\cdot,x')$ is a weight function localized to a union of balls of radius $r^{\delta_2}R^{\frac{m}{n}}r^{-\frac{m}{n}}$ and each $\underline{f}_\theta(\cdot,x')$ is Fourier supported in an element of $\Xi_m^{m+2}((R/r)^{\frac{m+1}{n}},\sigma(r))$, where $\sigma(r)$ is an $(m+1)$-tuple with $\sigma_m(r)=1$ and $\sigma_i(r)=\max(\sigma_i,\sigma_{i+1}^{-1}\cdots\sigma_m^{-1}(R/r)^{-1/n})$ for $i=0,\ldots,m-1$. Then by Proposition [Proposition 17](#L2conekaklocgen){reference-type="ref" reference="L2conekaklocgen"}, the previous displayed expression is bounded by $$C_\delta r^\delta\sum_{J''\subset J}|\sum_{\substack{I(\theta)\subset J'\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}\underline{f_\theta}|^2*\underline{\tilde{W}}^{n+1,d_2}_{J,m+2,\mu}(x),$$ where $J''\in\mathcal{J}(\max(\rho^{-\frac{1}{n}},r^{-\frac{1+\delta_2}{n}}))$. (Note that in other low-frequency dominating cases that appear elsewhere in this algorithm, we would invoke the hypothesized boundedness of $S_{\sigma'}^{m+k,m}(\cdot,\cdot,\cdot)$ instead of Proposition [Proposition 17](#L2conekaklocgen){reference-type="ref" reference="L2conekaklocgen"} to obtain pointwise bounds.) The summary so far of this low-frequency case is that $$\int_{\mathbb R^{n+1}}|\sum_{J} |\sum_{\substack{I(\theta)\subset J\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W\lesssim C_{\delta_1}r^{O(\delta_1)} \int_{\mathbb R^{n+1}}|\sum_J\sum_{J''\subset J} |\sum_{\substack{I(\theta)\subset J''\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2*\tilde{W}_{J,m+2,r^{-\delta_2}}^{n+1,d_2}|^{\frac{p}{2}}W.$$ By Lemma [Lemma 16](#multilem3pf){reference-type="ref" reference="multilem3pf"}, the integral on the right hand side is bounded above by $$C_{\delta_1}r^{\delta_1} \int_{\mathbb R^{n+1}}|\sum_{J''} |\sum_{\substack{I(\theta)\subset J''\\\theta\in\Xi_{m}^{n+1}(R,\sigma)}}f_\theta|^2|^{\frac{p}{2}}W ,$$ which we may then bound using $S_\sigma^{n,m}(\min(\rho,r^{1+\delta_0^{n-2}}),\rho,R)$ to prove the lemma. ◻ *Proof of Lemma [Lemma 21](#lem3k){reference-type="ref" reference="lem3k"}.* We perform a rescaling analogous to the proof of Proposition [Proposition 11](#conemulti){reference-type="ref" reference="conemulti"}. Fix $J\in\mathcal{J}(r^{-\frac{1}{n}})$. After rescaling, the integral $$\int_{\mathbb R^{n+1}}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,\sigma)}}f_\theta|^pW$$ becomes $$\int_{\mathbb R^{n+1}}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,\sigma)}}\underline{f}_\theta|^p\underline{W}$$ where $\underline{f}_\theta$ has Fourier support in an element of $\Xi_m^{n+1}(R/r,\sigma(r))$. The weight function $W$ which is initially adapted to a ball of radius $R^{\frac{m}{n}}\rho$ is dilated by factors of $1,r^{-\frac{1}{n}},\ldots,r^{-1}$ in $n$ many directions determined by $J$. The rescaled weight function $\underline{W}$ may be approximated by a sum of weight functions which are localized to finitely overlapping $(R/r)^{\frac{m}{n}}(\rho/r)$-balls. It follows that $$\int_{\mathbb R^{n+1}}|\sum_{\substack{I(\theta)\subset J\\ \theta\in\Xi_m^{n+1}(R,\sigma)}}\underline{f}_\theta|^p\underline{W}\lesssim S_{\sigma(r)}^{n,m}(1,\rho/r,R/r) \int_{\mathbb R^{n+1}}|\sum_{\substack{J'\subset J\\ J'\in\mathcal{J}(\rho^{-\frac{1}{n}}}} |\sum_{\substack{I(\theta)\subset J'\\ \theta\in\Xi_m^{n+1}(R,\sigma)}}\underline{f}_\theta|^p\underline{W}.$$ Undoing the change of variables proves the lemma. ◻ ## Proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} [\[truncsec\]]{#truncsec label="truncsec"} {#proof-of-proposition-coneinduct-truncsec} We prove Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} by reducing to the set-up from $S_\sigma^{n,m}(1,R,R)$ for some $\sigma$. *Proof of Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"}.* We will show that $\mathbb C_m^{n+1}(R)$ satisfies $$\mathbb C_m^{n+1}(R)\le (\log R)^C\sup_{\sigma\in\mathcal{S}^{n,m}_R}S_\sigma^{n,m}(1,R,R) .$$ Then Proposition [Proposition 1](#coneinduct){reference-type="ref" reference="coneinduct"} follows from Proposition [Proposition 18](#multiconek){reference-type="ref" reference="multiconek"}. By definition, for each $\theta\in\Xi_m^{n+1}(R)$, $f_\theta$ has Fourier support contained in $$\label{thetaapprox} \{\sum_{i=0}^m\lambda_i\phi_n^{(i)}(a_\theta): |\lambda_i|\le 2R^{-\frac{i}{n}}\,\,\forall i,\quad \max_{0\le i\le m}|\lambda_i|2^{m-i}R^{\frac{i}{n}}\ge \frac{1}{4}\}$$ for some $a_\theta=2^{-100n}R^{-\frac{1}{n}}\mathbb Z$. We use an analogous construction as in the proof of Proposition [Proposition 3](#gencone){reference-type="ref" reference="gencone"}. For $\sigma\in 2^\mathbb Z\cap [2^{-1}R^{-\frac{1}{n}},1]$, let $\eta_{\theta,\sigma}^{m-1}$ be bump functions supported on $$\{\sum_{i=0}^m\lambda_i\phi_n^{(i)}(a_\theta): |\lambda_i|\le \sigma^{m-i} (R/2)^{-\frac{i}{n}}\,\,\forall i\}\setminus \{\sum_{i=0}^m\lambda_i\phi_n^{(i)}(a_\theta): |\lambda_i|\le (\sigma/2)^{m-i}(R/2)^{-\frac{i}{n}}\,\,\forall i\},$$ and $\eta_{\theta,2^{-1}R^{-1/n}}$ supported in $\{\sum_{i=0}^m\lambda_i\phi_n^{(i)}(a_\theta): |\lambda_i|\le R^{-\frac{m}{n}}\}$. By the triangle inequality, there is some tuple $\sigma$ which satisfies $$\int_{\mathbb R^{n+1}}|\sum_\theta f_\theta|^{p}\le (\log R)^C \int_{\mathbb R^{n+1}}|\sum_{\substack{\theta}} f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_0}^{0}\cdots*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_{m-1}}^{m-1}|^{p}.$$ The summands $f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_0}^{0}\cdots*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_{m-1}}^{m-1}$ are supported on elements of $\Xi_m^{n+1}(R,\sigma)$, where $\sigma=(\sigma_0,\ldots,\sigma_{m-1},1)$. In this case, we have $$\int_{\mathbb R^{n+1}}|\sum_\theta f_\theta|^{p}\le (\log R)^CS_\sigma^{n,m}(1,R,R) \int_{\mathbb R^{n+1}}(\sum_{\substack{\theta}}| f_{\theta}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_0}^{0}\cdots*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_{m-1}}^{m-1}|^2)^{\frac{p}{2}} ,$$ so it suffices to prove that the integral on the right hand side is bounded above by $C_\delta r^\delta\int_{\mathbb R^{n+1}}|\sum_{\theta}|f_\theta|^2|^{\frac{p}{2}}$. The auxiliary functions satisfy $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\theta,\sigma_i}^i|\lesssim \tilde{W}_{I(\theta),i+1,\sigma_i}^{n,1}$ for each $i$. By Cauchy-Schwarz, it suffices to show that $$\int_{\mathbb R^{n+1}}|\sum_{\substack{\theta}} |f_{\theta}|^2*\tilde{W}_{I(\theta),1,\sigma_0}^{n,1}*\cdots*\tilde{W}_{I(\theta),m,\sigma_{m-1}}^{n,1}*\tilde{W}_{I(\theta),m+1}^{n,1}|^{\frac{p}{2}} \lesssim_\delta R^\delta\int_{\mathbb R^{n+1}}|\sum_{\substack{\theta}} |f_{\theta}|^2|^{\frac{p}{2}}.$$ This follows directly from $m$ applications of Lemma [Lemma 16](#multilem3pf){reference-type="ref" reference="multilem3pf"}, so we are done. ◻ # High-low analysis set-up for Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} [\[tools\]]{#tools label="tools"} {#high-low-analysis-set-up-for-proposition-momcurveinduct-tools} Recall from [\[intro1\]](#intro1){reference-type="ref" reference="intro1"} that there are three key ingredients to a general multi-scale analysis proving a square function estimate. To see the role of the first ingredient (a *base case*), consider the following special case of the iteration. Our goal is to prove $$\int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}f_\theta|^{p_n}\le C_\varepsilon R^\varepsilon\int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}|f_\theta|^2|^{\frac{p_n}{2}}$$ for any Schwartz functions with $\mathrm{supp\,}\widehat{f_\theta}\subset\theta$. Temporarily use $\lessapprox$ to hide implicit constants that we are not worrying about here. One iteration of the key ingredients from [\[intro1\]](#intro1){reference-type="ref" reference="intro1"} uses $\mathbb{M}^n(K)$, a special version of type 2(b) progress, and rescaling, yielding $$\begin{aligned} (\text{def. of $\mathbb{M}^n(K)$}) \qquad \qquad \int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}f_\theta|^{p_n}&\le \mathbb{M}^n(K)\int_{\mathbb R^n}(\sum_{\tau\in\Theta^n(K)}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^2)^{\frac{p_n}{2}}\\ (\text{type 2(b) progress})\qquad\qquad\qquad &\lessapprox \mathbb{M}^n(K)\sum_{\tau\in\Theta^n(K)}\int_{\mathbb R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^{p_n} \\ (\text{rescaling $+$ $\mathbb{M}^n(K)$})\qquad\qquad\qquad &\lessapprox [\mathbb{M}^n(K)]^2\sum_{\tau\in\Theta^n(K)}\int_{\mathbb R^n}(\sum_{\substack{\tau'\in\Theta^n(K^2)\\\tau'\subset\tau}}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau'}}f_\theta|^2)^{\frac{p_n}{2}} .\end{aligned}$$ If these steps are iterated, then we obtain $$\begin{aligned} \int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}f_\theta|^{p_n} \lessapprox [\mathbb{M}^n(K)]^s \sum_{\tau\in\Theta^n(K^s)} \int_{\mathbb R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^{p_n}. \end{aligned}$$ If $K^s=R$ (and $p\ge 2$), then $$\sum_{\tau\in\Theta^n(K^s)} \int_{\mathbb R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^{p_n}=\sum_{\theta\in\Theta^n(R)} \int_{\mathbb R^n}|f_\theta|^{p_n} \le \int_{\mathbb R^n}(\sum_{\theta\in\Theta^n(R)}|f_\theta|^2)^{\frac{p_n}{2}}.$$ We have accumulated $\log R/\log K$ many factors of the factor $\mathbb{M}^n(K)$. There is no *a priori* bound for $\mathbb{M}^n(K)$ since $p_n>p_{n-1}$. We could try induction on scales. Our goal is to show that $\mathbb{M}^n(R)\le \tilde{C}_\varepsilon R^\varepsilon$. By induction, suppose that $\mathbb{M}^n(K)\le \tilde{C}_\varepsilon K^\varepsilon$. Then the above algorithm accumulates a bound of the form $\mathbb{M}^n(R)\lessapprox (\tilde{C}_\varepsilon K^\varepsilon)^{\frac{\log R}{\log K}}=\tilde{C}_\varepsilon^{\frac{\log R}{\log K}} R^\varepsilon$ and the induction does not close since $\tilde{C}_\varepsilon^{\frac{\log R}{\log K}}\gg \tilde{C}_\varepsilon$. The high-low frequency argument will give us a way to bound $\mathbb{M}^n(R)$ by an expression that does not have too many factors of $\mathbb{M}^n(K)$, which will allow us to close the induction. A highly abbreviated version of the high-low frequency analysis is as follows. First we approximate the $L^p$ integral with a level set $$\int_{\mathbb R^n}|\sum_{\theta\in\Theta^n(R)}f_\theta|^{p_n}\approx \alpha^{p_n}|U_\alpha|,$$ where $U_\alpha=\{x\in\mathbb R^n:|\sum_{\theta\in\Theta^N(R)}f_\theta|\sim \alpha\}$. Then there is a stopping time algorithm which produces a scale $1<r<R$ satisfying 1. $\sum_{\tau\in\Theta^n(r)}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}} f_\theta|^2\sim \sum_{\theta\in\Theta^n(R)}|f_\theta|^2$ on most of $U_\alpha$, and 2. the inequality $$\alpha^{p_n}|U_\alpha|\lessapprox \int_{\mathbb R^n}|\sum_{\tau\in\Theta^n(r)}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^2|^{\frac{p_n}{2}}.$$ If $r$ is small, say $r\le R^\delta$ for some $\delta>0$, then we have a good bound just using (1) and Cauchy-Schwarz: $$\alpha^{p_n}|U_\alpha|\lesssim R^{\delta\frac{p_n}{n}}\int_{U_\alpha}(\sum_{\tau\in\Theta^n(r)}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^2)^{\frac{p_n}{2}} \lesssim R^{\delta\frac{p_n}{n}}\int_{U_\alpha}(\sum_{\theta\in\Theta^n(R)}|f_\theta|^2)^{\frac{p_n}{2}}.$$ If $r$ is large, so $r>R^\delta$, we use the inequality from (2). If we consider the special iteration discussed above, then the right hand side of the inequality in (2) is bounded by $$\begin{aligned} (\text{type 2(b) progress}) \qquad \qquad \int_{\mathbb R^n}|\sum_{\tau\in\Theta^n(r)}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^2|^{\frac{p_n}{2}}&\lessapprox \sum_{\tau\in\Theta^n(r)}\int_{\mathbb R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^{p_n}\\ (\text{rescaling $+$ $\mathbb{M}^n(K)$})\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad &\lessapprox \mathbb{M}^n(K)\sum_{\tau\in\Theta^n(r)}\int_{\mathbb R^n}(\sum_{\substack{\tau'\in\Theta^n(rK)\\\tau'\subset\tau}}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau'}}f_\theta|^2)^{\frac{p_n}{2}} \\ %(\text{type 2(b) progress})\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad % &\lessapprox \M^n(K)\sum_{\tau'\in\Theta^n(rK)}\int_{\R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau'}}f_\theta|^{p_n} \\ &\lessapprox \cdots \lessapprox [\mathbb{M}^n(K)]^s \sum_{\tau\in\Theta^n(rK^s)} \int_{\mathbb R^n}|\sum_{\substack{\theta\in\Theta^n(R)\\\theta\subset\tau}}f_\theta|^{p_n}. \end{aligned}$$ Assuming by induction that $\mathbb{M}^n(K)\le \tilde{C}_\varepsilon K^\varepsilon$, the above argument gives the inequality $\mathbb{M}^n(R)\lessapprox (\tilde{C}_\varepsilon K^\varepsilon)^{\frac{\log(R/r)}{\log K}} =(\tilde{C}_\varepsilon)^{\frac{\log (R/r)}{\log K}}r^{-\varepsilon}R^\varepsilon$. If $r$ is large enough, then $(\tilde{C}_\varepsilon)^{\frac{\log (R/r)}{\log K}}r^{-\varepsilon}\le \tilde{C}_\varepsilon$. Finally, we choose $\delta$ and $K$ so that both cases of the high-low analysis allow us to close the induction. This is carried out carefully in the proof of Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"}. Now we begin laying out the set-up for the high-low analysis, beginning with a detailed definition of the anisotropic neighborhoods. The set-up is analogous to the one used in [@gmw]. The following alternative definition of canonical moment curve blocks has the advantage that the blocks at various scales are nested. **Definition 1** (Canonical moment curve blocks). *For $R\in 2^{n\mathbb N}$, define ${\bf{S}}_n(R^{-\frac{1}{n}})$ to be the following collection of canonical moment curve blocks at scale $R$ which partition $\mathcal{M}^n(R)$: $$\bigsqcup\limits_{l=0}^{R^{\frac{1}{n}}-1}\mathcal{M}^n(R)\cap\{\xi\in\mathbb R^N:lR^{-\frac{1}{n}}\le \xi_1<(l+1)R^{-\frac{1}{n}}\}.$$* If $\tau\in{\bf{S}}_n(R^{-\frac{1}{n}})$ is the $\ell$th moment curve block, then $\tau$ is comparable to the set $$\{\gamma_n(lR^{-\frac{1}{n}})+\sum_{i=1}^n\lambda_i\gamma_n^{(i)}(lR^{-\frac{1}{n}}): |\lambda_i|\le R^{-\frac{i}{n}}\} .$$ By comparable, we mean that there is an absolute constant $C>0$ for which $C^{-1}\tau$ is contained in the displayed set and $C\tau$ contains the displayed set, where the dilations are taken with respect to the centroid of $\tau$. Since the moment curve blocks in $\Theta^n(R)$ are comparable to those in ${\bf{S}}_n(R^{-\frac{1}{n}})$, it suffices to prove a square function estimate with either definition of moment curve blocks. For $i\ge 1$, let $\tilde{\gamma}_n^{(i)}(t)$ denote the projection of $\gamma_n^{(i)}(t)$ onto the orthogonal complement of $\text{Span}(\gamma_n(t),\ldots,\gamma_n^{(i-1)}(t))$. Define the dual set $\tau^*$ by $$\label{dualdef} \tau^* = \{\sum_{i=1}^n\lambda_i\tilde{\gamma}_n^{(i)}(t):|\lambda_i|\le R^{\frac{i}{n}}\}.$$ We sometimes refer to the set $\tau^*$ as well as its translates as wave packets. Next, we fix some notation for the scales. Let $\varepsilon>0$. To prove Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"}, it suffices to assume that $R$ is larger than a constant which depends on $\varepsilon$. Consider scales $R_k\in 8^\mathbb N$ closest to $R^{k\varepsilon}$, for $k=1,\ldots,N$ and $R_N\le R\le R^\varepsilon R_N$. Since $R$ differs from $R_N$ at most by a factor of $R^\varepsilon$, we will assume that $R=R_N$. The relationship between the parameters is $$1=R_0\le R_k\le R_{k+1}\le R_N=R.$$ Fix notation for moment curve blocks of various sizes. 1. Let $\theta$ denote elements of ${\bf{S}}_n(R^{-1/n})$. 2. Let $\tau_k$ denote elements of ${\bf{S}}_n(R_k^{-1/n})$. The definitions of $\theta,\tau_k$ provide the additional property that if $\tau_k\cap\tau_{k+m}\not=\emptyset$, then $\tau_{k+m}\subset\tau_k$. We will use two square function constants: $\text{T}_n(r,R)$ and (the weighted version) $\text{T}^w_{n,d}(R)$ [\[const\]]{#const label="const"}, where the weights are defined in Definition [Definition 5](#M3ballweight){reference-type="ref" reference="M3ballweight"}. **Definition 2**. *Let $n,d\in\mathbb N_{>0}$. Let $1\le r\le R$. Let $\emph{T}^w_{n,d}(R)$ be the infimum of $A>0$ such that $$\int_{\mathbb R^n}|f|^{p} \le A\int_{\mathbb R^n}|\sum_{\theta\in{\bf{S}}_n(R)}|f_\theta|^2*\omega_{\theta,d}|^{\frac{p}{2}}$$ for any $2\le p\le p_n$ and any Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$.* **Definition 3**. *Let $R\ge 1$. Let $\emph{T}_n(R)$ be the infimum of $B>0$ such that $$\int_{\mathbb R^n}|f|^{p}\le B\int_{\mathbb R^n}|\sum_{\theta\in {\bf{S}}_n(R)} |f_\theta|^2|^{\frac{p}{2}}$$ for any $2\le p\le p_n$ and any Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$.* Fix a ball $B_R\subset\mathbb R^n$ of radius $R$ as well as a Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$. Also fix $D\in\mathbb N_{>0}$. The parameters $\alpha,\beta>0$ describe the set $$U_{\alpha,\beta}=\{x\in B_{R}:|f(x)|\ge \alpha,\quad\frac{\beta}{2}\le \sum_{\theta\in{\bf{S}}_n(R^{-1/n})}|f_\theta|^2*\omega_{\theta,D}(x)\le \beta\}.$$ The weight function $\omega_{\theta,D}$ is defined in Definition [Definition 5](#M3ballweight){reference-type="ref" reference="M3ballweight"} below. We assume throughout this section (and until [\[M3pigeon\]](#M3pigeon){reference-type="ref" reference="M3pigeon"}) that the $f_\theta$ satisfy the extra condition that $$\label{unihyp} \frac{1}{2}\le \|f_\theta\|_{L^\infty(\mathbb R^n)}\le 2\qquad\text{or}\qquad \|f_\theta\|_{L^\infty(\mathbb R^n)}=0.$$ ## A pruning step [\[prusec\]]{#prusec label="prusec"} We define wave packets associated to $f_{\tau_k}$ and sort them according to an amplitude condition which depends on the parameters $\alpha$ and $\beta$. For each $\tau_k$, let $\mathbb T_{\tau_k}$ be the collection of $\tau_k^*$ its translates $T_{\tau_k}$ which form a tiling of $\mathbb R^n$. Fix an auxiliary function $\varphi(\xi)$ which is a bump function supported in $[-\frac{1}{4},\frac{1}{4}]^n$. For each $m\in\mathbb Z^n$, let $$\psi_m(x)=c\int_{[-\frac{1}{2},\frac{1}{2}]^n}|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\varphi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\varphi}{\scalebox{-1}{\tmpbox}}% |^2(x-y-m)dy,$$ where $c$ is chosen so that $\sum_{m\in\mathbb Z^n}\psi_m(x)=c\int_{\mathbb R^n}|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\varphi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\varphi}{\scalebox{-1}{\tmpbox}}% |^2=1$. Since $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\varphi}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\varphi}{\scalebox{-1}{\tmpbox}}% |$ is a rapidly decaying function, for any $k\in\mathbb N$, there exists $C_k>0$ such that $$\psi_m(x)\le c\int_{[-\frac{1}{2},\frac{1}{2}]^3}\frac{C_k}{(1+|x-y-m|^2)^n}dy \le \frac{\tilde{C}_k}{(1+|x-m|^2)^k}.$$ Define the partition of unity $\psi_{T_{\tau_k}}$ associated to ${\tau_k}$ to be $\psi_{T_{\tau_k}}(x)=\psi_m\circ A_{\tau_k}$, where $A_{\tau_k}$ is a linear transformations taking $\tau_k^*$ to $[-\frac{1}{2},\frac{1}{2}]^n$ and $A_{\tau_k}(T_{\tau_k})=m+[-\frac{1}{2},\frac{1}{2}]^n$. The important properties of $\psi_{T_{\tau_k}}$ are (1) rapid decay off of $T_{\tau_k}$ and (2) Fourier support contained in $\tau_k$ translated to the origin. We sort the wave packets $\mathbb T_{\tau_k}=\mathbb T_{\tau_k}^g\sqcup\mathbb T_{\tau_k}^b$ into "good\" and "bad\" sets, and define corresponding versions of $f$, as follows. **Remark 1**. *In the following definitions, let $K\ge 1$ be a large parameter which will be used to define the broad set in Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"}. Also, $A=A(\varepsilon)\gg 1$ is a large enough constant (determined by Lemma [Lemma 27](#ftofk){reference-type="ref" reference="ftofk"}) which also satisfies $A\ge \tilde{D}$, where $\tilde{D}$ is from Lemma [Lemma 25](#low){reference-type="ref" reference="low"}.* **Definition 4** (Pruning with respect to $\tau_k$). *Let $f^{N}=f$, $f^{N}_{\tau_N}=f_{\theta}$. For each $1\le k\le N-1$, let $$\begin{aligned} \mathbb T_{\tau_k}^{g}&=\{T_{\tau_k}\in\mathbb T_{\tau_{k}}:\|\psi_{T_{\tau_{k}}}^{1/2}f_{\tau_{k}}^{k+1}\|_{L^\infty(R^3)}\le K^3A^{N-k+1}\frac{\beta}{\alpha}\}, \\ f_{\tau_{k}}^{k}=\sum_{T_{\tau_k}\in\mathbb T_{\tau_k}^{g}}&\psi_{T_{\tau_k}}f^{k+1}_{\tau_k}\qquad\text{and}\qquad f_{\tau_{k-1}}^{k}=\sum_{\tau_k\subset\tau_{k-1}}f_{\tau_k}^k .\end{aligned}$$* For each $k$, the $k$th version of $f$ is $f^k=\underset{\tau_k}{\sum} f_{\tau_k}^k$. **Remark 2**. *We may assume that $\alpha\lesssim R^{C_0 \varepsilon}\beta$. This will be discussed in Proposition [Proposition 37](#wpd){reference-type="ref" reference="wpd"} and Corollary [Corollary 38](#wpdcor){reference-type="ref" reference="wpdcor"}, which involve pigeonholing the wave packets of $f$.* **Lemma 22** (Properties of $f^k$). 1. *[\[item1\]]{#item1 label="item1"} $| f_{\tau_{k}}^k (x) | \le |f_{ \tau_{k}}^{k+1}(x)|\lesssim \#\theta\subset\tau_k.$* 2. *[\[item2\]]{#item2 label="item2"} $\| f_{\tau_k}^k \|_{L^\infty(\mathbb R^n)} \le K^3A^{N-k+1}\frac{\beta}{\alpha}$.* 3. *[\[item3\]]{#item3 label="item3"} For $R$ sufficiently large depending on $\varepsilon$, $\text{supp} \widehat{f_{\tau_k}^{k}}\subset3\tau_k.$* *Proof.* For the first property, recall that $\sum_{T_{\tau_k} \in \mathbb T_{\tau_k}}\psi_{T_{\tau_k}}$ is a partition of unity so we may iterate the inequalities $$\begin{aligned} |f_{\tau_k}^k|\le |f_{\tau_k}^{k+1}|&\le \sum_{\tau_{k+1}\subset\tau_k}|f_{\tau_{k+1}}^{k+1}|\le\cdots\le \sum_{\tau_N\subset\tau_k}|f_{\tau_N}^N|= \sum_{\theta \subset\tau_k}|f_{\theta}|. \end{aligned}$$ The first property follows from our assumption [\[unihyp\]](#unihyp){reference-type="eqref" reference="unihyp"} that each $\|f_\theta\|_{L^\infty(\mathbb R^n)}\lesssim 1$. For the $L^\infty$ bound in the second property, write $$|f_{ \tau_k}^k(x)| = |\sum_{\substack{T_{\tau_k} \in \mathbb T_{\tau_k^h}}} \psi_{T_{\tau_k}}(x) f_{ \tau_k}^{k+1}(x)|\le \sum_{\substack{T_{\tau_k} \in \mathbb T_{\tau_k^h}}} \psi_{T_{\tau_k}}^{1/2}(x) \|\psi_{T_{\tau_k}}^{1/2}f_{ \tau_k}^{k+1}\|_\infty\lesssim \|\psi_{T_{\tau_k}}^{1/2}f_{ \tau_k}^{k+1}\|_\infty.$$ By the definition of $\mathbb T_{\tau_k}^h$, $\|\psi_{T_{\tau_k}}^{1/2}f_{\tau_k}^{k+1}\|_\infty\le K^3 A^{N-k+1}\frac{\beta}{\alpha}$. The third property depends on the Fourier support of $\psi_{T_{\tau_k}}$, which is contained in $\tau_k$ shifted to the origin. Note if each $f_{\tau_k}^{k+1}$ has Fourier support in $\cup_{\tau_{k+1}\subset\tau_k}3\tau_{k+1}$, then $\mathrm{supp\,}\widehat{f_{\gamma_k}^k}$ is contained in $3\tau_k$. ◻ **Definition 5**. *Let $d\in\mathbb N_{>0}$ and let $W^{n,d}$ be the weight function defined in [\[wtsec\]](#wtsec){reference-type="ref" reference="wtsec"}. Since we consider one dimension $n$ at a time, we suppress the $n$ notation in the weights. Let $B_0\subset\mathbb R^n$ denote the unit ball centered at the origin. For any set $U=T(B_0)$ where $T$ is an affine transformation $T:\mathbb R^n\to\mathbb R^n$, define $$w_{U,d}(x)=|U|^{-1}W^{n,d}(T^{-1}(x)).$$ For each $\tau_k$, let $A_{\tau_k}$ be a linear transformation mapping $\tau_k^*$ to the unit cube and define $\omega_{\tau_k,d}$ by $$\omega_{\tau_k,d}(x)=|\tau_k^*|^{-1}W^{n,d}(A_{\tau_k}(x)).$$ Assume that $d=1$ whenever the $d$ is not specified in a weight function.* Let the capital-W version of weight functions denote the $L^\infty$-normalized (as opposed to $L^1$-normalized) versions, so for example, for any ball $B_s$, $W_{B_s,d}(x)=|B_s|w_{B_s,d}(x)$. If a weight function replaces the convex set with only a scale, say $s$, then the functions $w_{s,d},W_{s,d}$ are weight functions localized to the $s$-ball centered at the origin. Next, we record the locally constant property. By locally constant property, we mean that if a function $f$ has Fourier transform supported in a convex set $A$, then $|f|$ is bounded above by an averaged version of $|f|$ over a dual set $A^*$. **Lemma 23** (Locally constant property). *For each $\tau_k$ and $T_{\tau_k}\in\mathbb T_{\tau_k}$, $$\begin{aligned} \|f_{\tau_k}\|_{L^\infty(T_{\tau_k})}^2\lesssim_d |f_{\tau_k}|^2*\omega_{\tau_k,d}(x)\qquad\text{for any}\quad x\in T_{\tau_k} .\end{aligned}$$ Also, for any $R_k^{1/n}$-ball $B_{R_k^{1/n}}$, $$\begin{aligned} \|\sum_{\tau_k}|f_{\tau_k}|^2\|_{L^\infty(B_{R_k^{1/n}})}\lesssim_d |f_{\tau_k}|^2*w_{B_{R_k^{1/n}},d}(x)\qquad\text{for any}\quad x\in B_{R_k^{1/n}} .\end{aligned}$$* Because the pruned versions of $f$ and $f_{\tau_k}$ have essentially the same Fourier supports as the unpruned versions, the locally constant lemma applies to the pruned versions as well. *Proof of Lemma [Lemma 23](#locconst){reference-type="ref" reference="locconst"}.* For the first claim, we write the argument for $f_{\tau_k}$ in detail. Let $\rho_{\tau_k}$ be a bump function equal to $1$ on $\tau_k$ and supported in $2\tau_k$. Then using Fourier inversion and Hölder's inequality, $$|f_{\tau_k}(y)|^2=|f_{\tau_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% (y)|^2\le\|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% \|_1 |f_{\tau_k}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% |(y).$$ Since $\rho_{\tau_k}$ may be taken to be an affine transformation of a standard bump function adapted to the unit ball, $\|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% \|_1$ is a constant. The function $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}%$ decays rapidly off of $\tau_k^*$, so $|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% |\lesssim_d \omega_{{\tau_k},d}$. Since for any $T_{\tau_k}\in\mathbb T_{\tau_k}$, $\omega_{\tau_k,d}(y)\sim\omega_{\tau_k}(y')$ for all $y,y'\in T_{\tau_k}$, we have $$\begin{aligned} \sup_{x\in T_{\tau_k}}|f_{\tau_k}|^2*\omega_{\tau_k,d}(x)&\le \int|f_{\tau_k}|^2(y)\sup_{x\in T_{\tau_k}}\omega_{\tau_k,d}(x-y)dy\\ &\sim \int|f_{\tau_k}|^2(y)\omega_{\tau_k,d}(x-y)dy\qquad \text{for all}\quad x\in T_{\tau_k}. \end{aligned}$$ For the second part of the lemma, repeat analogous steps as above, except begin with $\rho_{\tau_k}$ which is identically $1$ on a ball of radius $2R_k^{-1/n}$ containing $\tau_k$. Then $$\sum_{\tau_k}|f_{\tau_k}(y)|^2=\sum_{\tau_k}|f_{\tau_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{\tau_k}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{\tau_k}}{\scalebox{-1}{\tmpbox}}% (y)|^2\lesssim \sum_{\tau_k}|f_{\tau_k}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho_{R_k^{-1/n}}}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho_{R_k^{-1/n}}}{\scalebox{-1}{\tmpbox}}% |(y),$$ where we used that each $\rho_{\tau_k}$ is a translate of a single function $\rho_{R^{-1/n}}$. The rest of the argument is analogous to the first part. ◻ The following local $L^2$-orthogonality lemma is Lemma 3 in [@M3smallcap]. **Lemma 24** (Local $L^2$ orthogonality). *Let $U=T(B)$ where $B$ is the unit ball centered at the origin and $T:\mathbb R^n\to\mathbb R^n$ is an affine transformation. Let $h:\mathbb R^n\to\mathbb C$ be a Schwartz function with Fourier transform supported in a disjoint union $X=\sqcup_k X_k$, where $X_k\subset B$ are Lebesgue measurable. If the maximum overlap of the sets $X_k+U^*$ is $L$, then $$\int |h_X|^2w_{U,d}\lesssim L\sum_{X_k}\int|h_{X_k}|^2w_{U,d},$$ where $h_{X_k}=\int_{X_k}\widehat{h}(\xi)e^{2\pi i x\cdot\xi}d\xi$.* Here, we may take $\{x:|x\cdot\xi|\le 1\quad\forall \xi\in U-U\}$ as the definition of $U^*$. We will include a sketch of the proof for future reference. *Proof.* By Plancherel's theorem, we have $$\begin{aligned} \int|h_X|^2w_U&=\int h_X \overline{h_Xw_U}=\int \widehat{h_X}\overline{\widehat{h_X}*\widehat{w_U}}.\end{aligned}$$ Since $\widehat{h_X}=\sum_k\widehat{h_{X_k}}$, $\int \widehat{h_X}\overline{\widehat{h_X}*\widehat{w_U}}=\sum_{X_k}\sum_{X_k'}\int\widehat{h_{X_k}}\overline{\widehat{h_{X_k'}}*\widehat{w_U}}$. For each $X_k$, the integral on the right hand side vanishes except for $\lesssim L$ many choices of $X_k'$. ◻ ## High-low frequency decomposition of square functions **Definition 6** (Auxiliary functions). *Let $\eta:\mathbb R^n\to[0,\infty)$ be a radial, smooth bump function satisfying $\eta(x)=1$ on $B_{1/2}$ and $\mathrm{supp\,}\eta\subset B_1$. Then for each $s>0$, let $$\eta_{\le s}(\xi) =\eta(s^{-1}\xi) .$$ We will sometimes abuse notation by denoting $h*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{>s}=h-h*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le s}$, where $h$ is some Schwartz function. Also define $\eta_{s}(x)=\eta_{\le s}-\eta_{\le s/2}$.* Fix $N_0<N-1$ which will be specified in [\[ind\]](#ind){reference-type="ref" reference="ind"}. Also fix $d_0=d_0(n,\varepsilon,D)\in\mathbb N_{>0}$ which will be specified in Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"}. Let $d_k=(4\varepsilon^{-1}-k)d_0$ **Definition 7**. *For $N_0\le k\le N-1$, let $$g_k(x)=\sum_{\tau_k}|f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}, \qquad g_k^{\ell}(x)=g_k*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R_{k+1}^{-1/n}}, \qquad\text{and}\qquad g_k^h=g_k-g_k^{\ell}.$$* In the following definition, $A\gg 1$ is the same constant that goes into the pruning definition of $f^k$. **Definition 8**. *Define the high set by $$H=\{x\in U_{\alpha,\beta}: A \beta\le g_{N-1}(x)\}.$$ For each $k=N_0,\ldots,N-2$, let $H=\Omega_{N-1}$ and let $$\Omega_k=\{x\in U_{\alpha,\beta}\setminus \cup_{l=k+1}^{N-1}\Omega_{l}: A^{N-k}\beta\le g_k(x) \}.$$ Define the low set to be $$L=U_{\alpha,\beta}\setminus[\cup_{k=N_0}^{N-1}\Omega_k].$$* **Lemma 25** (Low lemma). *There is an absolute constant $\tilde{D}>0$ so that for each $x$, $|g_k^\ell(x)|\le \tilde{D} g_{k+1}(x)$.* *Proof.* We perform a pointwise version of the argument in the proof of local/global $L^2$-orthogonality (Lemma [Lemma 24](#L2orth){reference-type="ref" reference="L2orth"}). For each $\tau_k^{k+1}$, by Plancherel's theorem, $$\begin{aligned} |f_{\tau_k}^{k+1}|^2*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}(x)&= \int_{\mathbb R^n}|f_{\tau_k}^{k+1}|^2(x-y)% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}(y)dy \nonumber \\ &= \int_{\mathbb R^n}\widehat{f_{\tau_k}^{k+1}}*\widehat{\overline{f_{\tau_k}^{k+1}}}(\xi)e^{-2\pi i x\cdot\xi}\eta_{<R_{k+1}^{-1/n}}(\xi)d\xi \nonumber \\ &= \sum_{\tau_{k+1},\tau_{k+1}'\subset\tau_k}\int_{\mathbb R^n}e^{-2\pi i x\cdot\xi}\widehat{f_{\tau_{k+1}}^{k+1}}*\widehat{\overline{f_{\tau_{k+1}'}^{k+1}}}(\xi)\eta_{<R_{k+1}^{-1/n}}(\xi)d\xi .\label{dis2}\nonumber\end{aligned}$$ The integrand is supported in $(2\tau_{k+1}-2\tau_{k+1}')\cap B_{R_{k+1}^{-1/n}}$. This means that the integral vanishes unless $\tau_{k+1}$ is within $\sim R_{k+1}^{-1/n}$ of $\tau_{k+1}'$, in which case we write $\tau_{k+1}\sim\tau_{k+1}'$. Then $$\sum_{\tau_{k+1},\tau_{k+1}'\subset\tau_k}\int_{\mathbb R^2}e^{-2\pi i x\cdot\xi}\widehat{f}_{\tau_{k+1}}^{k+1}*\widehat{\overline{f}_{\tau_{k+1}'}^{k+1}}(\xi)\eta_{<R_{k+1}^{-1/n}}(\xi)d\xi=\sum_{\substack{\tau_{k+1},\tau_{k+1}'\subset\tau_k\\ \tau_{k+1}\sim\tau_{k+1}'}}\int_{\mathbb R^2}e^{-2\pi i x\cdot\xi}\widehat{f}_{\tau_{k+1}}^{k+1}*\widehat{\overline{f}_{\tau_{k+1}'}^{k+1}}(\xi)\eta_{<R_{k+1}^{-1/n}}(\xi)d\xi.$$ Use Plancherel's theorem again to get back to a convolution in $x$ and conclude that $|g_k*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}(x)|$ equals $$\begin{aligned} &\Big|\sum_{\substack{\tau_{k+1},\tau_{k+1}'\subset\tau_k\\ \tau_{k+1}\sim\tau_{k+1}'}}(f_{\tau_{k+1}}^{k+1}\overline{f_{\tau_{k+1}'}^{k+1}})*\omega_{\tau_k,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}(x) \Big|\lesssim \sum_{\tau_k} \sum_{\tau_{k+1}\subset\tau_k}|f_{\tau_{k+1}}^{k+1}|^2*\omega_{\tau_k,d_k}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}|(x) . \end{aligned}$$ By the locally constant property (Lemma [Lemma 23](#locconst){reference-type="ref" reference="locconst"}) and [\[item1\]](#item1){reference-type="eqref" reference="item1"} of Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, the right hand side above is $$\lesssim \sum_{\tau_k} \sum_{\tau_{k+1}\subset\tau_k}|f_{\tau_{k+1}}^{k+2}|^2*\omega_{\tau_{k+1},d_{k}}*\omega_{\tau_k,d_k}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}|(x)\lesssim g_{k+1}(x).$$ It remains to note that $$w_{\tau_{k+1},d_{k+1}}*\omega_{\tau_k,d_k}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}|(x)\lesssim w_{\tau_{k+1},d_{k+1}}(x)$$ since $\tau_k^*\subset\tau_{k+1}^*$ and $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{<R_{k+1}^{-1/n}}$ is an $L^1$-normalized function that is rapidly decaying away from $B_{R_{k+1}^{1/n}}(0)$. ◻ **Corollary 26** (High-dominance on $\Omega_k$). *For $R$ large enough depending on $\varepsilon$, $g_k(x)\le 2|g_k^h(x)|$ for all $x\in\Omega_k$.* *Proof.* This follows directly from Lemma [Lemma 25](#low){reference-type="ref" reference="low"}. Indeed, since $g_k(x)=g_k^{\ell}(x)+g_k^h(x)$, the inequality $g_k(x)>2|g_k^h(x)|$ implies that $g_k(x)<2|g_k^{\ell}(x)|$. Then by Lemma [Lemma 25](#low){reference-type="ref" reference="low"}, $|g_k(x)|<2\tilde{D} g_{k+1}(x)$. Since $x\in\Omega_k$, $g_{k+1}(x)\le A^{N-k-1}\beta$, which altogether gives the upper bound $$g_k(x)\le 2\tilde{D} A^{N-k-1}\beta.$$ The contradicts the property that on $\Omega_k$, $A^{N-k}\beta\le g_k(x)$, for $A$ sufficiently larger than $\tilde{D}$, which finishes the proof. ◻ **Lemma 27** (Pruning lemma). *For any $s\ge R^{-\varepsilon/n}$ and $\tau\in{\bf{S}}_n(s)$, $$\begin{aligned} |\sum_{\tau_k\subset\tau}f_{\tau_k}-\sum_{\tau_k\subset\tau}f_{\tau_k}^{k+1}(x)|&\le \frac{\alpha}{A^{1/2}K^3} \qquad\text{for all $x\in \Omega_k$}, \qquad N_0\le k\le N-1,\\ \text{and}\qquad |\sum_{\tau_B\subset\tau}f_{\tau_B}-\sum_{\tau_B\subset\tau}f_{\tau_B}^{B}(x)|&\le \frac{\alpha}{A^{1/2}K^3}\qquad \text{ for all $x\in L$}. \end{aligned}$$* *Proof.* Begin by proving the first claim about $\Omega_k$. By the definition of the pruning process, we have $$\label{diffs} f_{\tau}=f^{N-1}_{\tau}+(f_{\tau}^N-f^{N-1}_{\tau})=\cdots=f^{k+1}_{\tau}(x)+\sum_{m=k+1}^{N-1}(f^{m+1}_{\tau}-f^{m}_{\tau})$$ where formally, the subscript $\tau$ means $f_\tau=\sum_{\theta\subset\tau}f_\theta$ and $f_{\tau}^m=\sum_{\tau_m\subset\tau}f_{\tau_m}^m$. We will show that each difference in the sum is much smaller than $\alpha$. For each $N-1\ge m\ge k+1$ and $\tau_m$, $$\begin{aligned} |f_{\tau_m}^m(x)-f_{\tau_m}^{m+1}(x)|&=|\sum_{T_{\tau_m}\in\mathbb T_{\tau_m}^{b}}\psi_{T_{\tau_m}}(x)f_{\tau_m}^{m+1}(x)| = \sum_{T_{\tau_m}\in T_{\tau_m}^b} |\psi_{T_{\tau_m}}^{1/2}(x)f_{\tau_m}^{m+1}(x)|\psi_{T_{\tau_m}}^{1/2}(x) \\ & \le\sum_{T_{\tau_m}\in \mathbb T_{\tau_m}^b} K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta} \| \psi_{T_{\tau_m}}^{1/2}f_{{\tau_m}}^{m+1} \|_{L^\infty(\mathbb R^n)}^2 \psi_{T_{\tau_m}}^{1/2}(x) \\ %& \lesssim K^{-3}A^{-(N-m+1)}\frac{\a}{\b} \sum_{T_{\tau_m}\in \T_{\tau_m}^b} \| \s_{T_{\tau_m}}^{1/2}f_{{\tau_m}}^{m+1} \|_{L^\infty(\R^n)}^2 \s_{T_{\tau_m}}^{1/2}(x) \\ & \lesssim K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta}\sum_{T_{\tau_m}\in \mathbb T_{\tau_m}^b} \sum_{\tilde{T}_{{\tau_m}}\in\mathbb T_{\tau_m}} \| \psi_{T_{\tau_m}}|f_{{\tau_m}}^{m+1}|^2 \|_{L^\infty(\tilde{T}_{{\tau_m}})} \psi_{T_{\tau_m}}^{1/2}(x) \\ & \lesssim K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta} \sum_{T_{\tau_m},\tilde{T}_{\tau_m}\in \mathbb T_{\tau_m}} \| \psi_{T_{\tau_m}}\|_{L^\infty(\tilde{T}_{\tau_m})}\||f_{{\tau_m}}^{m+1} |^2\|_{{L}^\infty(\tilde{T}_{{\tau_m}})} \psi_{T_{\tau_m}}^{1/2}(x) .\end{aligned}$$ Let $c_{\tilde{T}_{\tau_m}}$ denote the center of $\tilde{T}_{\tau_m}$ and note the pointwise inequality $$\sum_{{T}_{\tau_m}}\|\psi_{T_{\tau_m}}\|_{L^\infty(\tilde{T}_{\tau_m})}\psi_{T_{\tau_m}}^{1/2}(x)\lesssim_{d_m} |\tau_m^*|\omega_{\tau_m,d_m}(x-c_{\tilde{T}_{\tau_m}}) ,$$ which means that $$\begin{aligned} |f_{\tau_m}^m(x)-f_{\tau_m}^{m+1}(x)| & \lesssim_{d_m} K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta} |\tau_m^*|\sum_{\tilde{T}_{\tau_m}\in \mathbb T_{\tau_m}} \omega_{\tau_m,{d_m}}(x-c_{\tilde{T}_{\tau_m}})\||f_{{\tau_m}}^{m+1} |^2\|_{{L}^\infty(\tilde{T}_{{\tau_m}})} \\ &\lesssim_{d_m} K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta}|\tau_m^*| \sum_{\tilde{T}_{\tau_m}\in \mathbb T_{\tau_m}} \omega_{\tau_m,{d_m}}(x-c_{\tilde{T}_{\tau_m}})|f_{{\tau_m}}^{m+1} |^2*\omega_{\tau_m,{d_m}}(c_{\tilde{T}_{\tau_m}})\\ &\lesssim_{d_m} K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta} |f_{{\tau_m}}^{m+1} |^2*\omega_{\tau_m,{d_m}}(x)\end{aligned}$$ where we used the locally constant property in the second to last inequality. The last inequality is justified by the fact that $\omega_{\tau_m,{d_m}}(x-c_{\tilde{T}_{\tau_m}})\sim \omega_{\tau_m,{d_m}}(x-y)$ for any $y\in\tilde{T}_{\tau_m}$, and we have the pointwise relation $\omega_{\tau_m,{d_m}}*\omega_{\tau_m,{d_m}}\lesssim \omega_{\tau_m,{d_m}}$. Then $$|\sum_{\tau_m\subset\tau}(f_{\tau_m}^m(x)-f_{\tau_m}^{m+1}(x))|\lesssim_{d_m} K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta}\sum_{\tau_m\subset\tau}|f_{\tau_m}^{m+1}|^2*\omega_{\tau_m,{d_m}}(x)\sim K^{-3}A^{-(N-m+1)}\frac{\alpha}{\beta}g_m(x).$$ We choose $A$ sufficiently large, determined by the proof of Corollary [Corollary 26](#highdom){reference-type="ref" reference="highdom"} and the proof of Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"} (where we choose $d_0=d_0(\varepsilon)$), so that if $g_m(x)\le A^{N-m}\beta$, then the above inequality implies that $$|\sum_{\tau_m\subset\tau}(f_{\tau_m}^m(x)-f_{\tau_m}^{m+1}(x))|\le \varepsilon K^{-3}A^{-1/2}\alpha.$$ This finishes the proof since the number of terms in [\[diffs\]](#diffs){reference-type="eqref" reference="diffs"} is bounded by $N\le \varepsilon^{-1}$. The argument for the pruning on $L$ is analogous. ◻ ## High-frequency analysis Now that we have identified moment curve blocks with cone planks and moment curve wave envelopes with cone wave envelopes, we are prepared to use a square function estimate for a cone. **Lemma 28** (High lemma). *Let $4\le p\le p_{n-1}$. For each $\delta>0$, there is $B_\delta\in(0,\infty)$ so that the following holds. We have $$\int_{\mathbb R^n}|g_k^h|^{\frac{p}{2}}\lesssim_\varepsilon R^{2\varepsilon} \int_{\mathbb R^{n}}(\sum_{\tau_k}||f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{>R_{k+1}^{-1/n}}|^2)^{\frac{p}{4}}.$$* *Proof.* First describe the Fourier support of $g_k^h$. By [\[item3\]](#item3){reference-type="eqref" reference="item3"} of Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, the support of $\widehat{|f_{\tau_k}^{k+1}|^2}$ is $2(\tau_k-\tau_k)$. The high-frequency cutoff removes a ball of radius $R_{k+1}^{-1/n}$, so $g_k^h$ is Fourier supported within the annulus $R_{k+1}^{-1/n}\le |\xi|\le 10 R_k^{-1/n}$. By dyadic pigeonholing, there is some dyadic $s\in[R_{k+1}^{-1/n},2R_k^{-1/n}]$ for which $$\int|g_k^h|^4\lesssim(\log R) \int|g_k^h*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _s|^4,$$ where $\eta_s:\mathbb R^n\to[0,\infty)$ is a smooth function supported in the annulus $s/4\le |\xi|\le s$. In the proof of Lemma [Lemma 25](#low){reference-type="ref" reference="low"}, we showed the pointwise equality $$\label{ptwiselo} g_k^h*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _s(x)=\sum_{\tau_s}\sum_{\tau_s'\sim\tau_s}(f_{\tau_s}^{k+1}\overline{f_{\tau_s'}^{k+1}})*\omega_{\tau_k,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{>R_{k+1}^{-1/n}}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _s(x)$$ where $\tau_s\in\Theta^n(s^{-n})$ and $\tau_s'\sim\tau_s$ means that $\tau_s'\in\Theta^n(s^{-n})$ and $\text{dist}(2\tau_s,2\tau_s')\le 2s$. For each $\tau_s$, the sub-sum on the right hand side has Fourier transform supported in $2(\tau_{2s}-\tau_{2s})\setminus B_s(0)$ where $\tau_{2s}\in\Theta^n((2s)^{-n})$ contains $\tau_s$. Now write $$\label{formg} g_k^h*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _s(x)=|\det T|^{-1}(\widehat{g_k^h}\eta_s\circ T^{-1})^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,\,}{\scalebox{-1}{\tmpbox}}% }((T^{-1})^t x) ,$$ where $T$ is an affine transformation mapping $\mathrm{supp\,}\widehat{g_k^h}{\eta}_s$ to $\Gamma_0^n(s^{-n})$. It suffices to bound $$\int|(\widehat{g_k^h}\eta_s\circ T^{-1})^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,\,}{\scalebox{-1}{\tmpbox}}% }(x)|^{\frac{p}{2}}dx.$$ We may view the sub-sums corresponding to each $\tau_k$ on the right hand side of [\[ptwiselo\]](#ptwiselo){reference-type="eqref" reference="ptwiselo"} as having Fourier support contained in an element of $\Xi_0^n(s^{-n})$, dilated by a factor of $s$. Therefore, since $\mathbb C_0^n(s^{-1})\lesssim_\varepsilon R^\varepsilon$, we have $$\begin{aligned} \int|(\widehat{g_k^h}\eta_s*\circ T^{-1})^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,\,}{\scalebox{-1}{\tmpbox}}% }&(x)|^{\frac{p}{2}}dx\lesssim_\varepsilon R^{C\varepsilon}\int(\sum_{\tau_k}||\sum_{\tau_s\subset\tau_{2s}}\sum_{\tau_s'\sim\tau_s}(\widehat{|f_{\tau_k}^{k+1}|^2}\widehat{\omega}_{\tau_k,d_k}{\eta}_{>R_{k+1}^{-1/n}}{\eta}_s\circ T^{-1})^{% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\,\,\,}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\,\,\,}{\scalebox{-1}{\tmpbox}}% }(x)|^2)^{\frac{p}{4}}. \end{aligned}$$ After undoing the change of variables, the lemma is proved. ◻ # Key iterations that unwind the pruning process[\[keyalgo\]]{#keyalgo label="keyalgo"} In the process of unwinding the pruning process, we will encounter expressions of the form $$\int_{\mathbb R^n}(\sum_{\tau\in\Theta^n(s^{-n})}|f_\tau|^{\tilde{p}_l}*\omega_{\tau,d})^{\frac{p}{\tilde{p}_l}}$$ for various values of $d$ and $R^{-\frac{1}{n}}<s<1$, and where $\frac{p}{\tilde{p}_l}\ge 1$. To analyze this expression, we further decompose the Fourier support of each summand, which is contained in (a constant dilate of) $\tau-\tau$, or the set $$\{\sum_{i=1}^{n}\lambda_i\gamma_n^{(i)}(a):|\lambda_i|\le s^{i}\,\,\forall i\},$$ where $a$ is the initial point of $I(\tau')$. Fix $s<\sigma_0<1$. Decompose the above set according to the dyadic parameter $\sigma_0<\sigma<1$ into subsets $$\label{summands} \{\sum_{i=1}^{n}\lambda_i \gamma_n^{(i)}(a): |\lambda_i|\le \min(1,\sigma^{l+1-i})s^{i}\quad\forall i\}\setminus\{\sum_{i=1}^{n}\lambda_i \gamma_n^{(i)}(a): |\lambda_i|\le \min(1,(\sigma/2)^{l+1-i})s^{i}\quad\forall i\}$$ and $\{\sum_{i=1}^{n}\lambda_i \gamma_n^{(i)}(a): |\lambda_i|\le \min(1,\sigma_0^{l+1-i})s^{i}\quad\forall i\}$. We carry out this decomposition using smooth bump functions $\eta_{\tau,\sigma}^l$ which are part of a partition of unity and constructed analogously to those introduced before Lemma [Lemma 13](#multilem1){reference-type="ref" reference="multilem1"}. The function $\eta_{\tau,\sigma}^l$ is supported in $$\Big(\{\sum_{i=1}^{l}\lambda_i \gamma_l^{(i)}(a): |\lambda_i|\le \sigma^{l+1-i}s^{i}\,\,\forall i\}\setminus\{\sum_{i=1}^{l}\lambda_i \gamma_l^{(i)}(a): |\lambda_i|\le (\sigma/2)^{l+1-i}s^{i}\,\,\forall i\}\Big)\times[-1,1]^{n-l}$$ if $\sigma_0< \sigma\le 1$ and supported in $$\{\sum_{i=1}^{l}\lambda_i \gamma_l^{(i)}(a): |\lambda_i|\le \sigma_0^{l+1-i}s^{i}\quad\forall i\}\times[-1,1]^{n-l}$$ if $\sigma=\sigma_0$. Write $\tilde{W}_{\tau,\sigma}^l$ for the $L^1$-normalized weight function centered at the origin that is Fourier supported in $\{\sum_{i=1}^{l}\lambda_i \gamma_l^{(i)}(a): |\lambda_i|\le \sigma^{l+1-i}s^{i}\,\,\forall i\}\times[-1,1]^{n-l}$. ## Incorporating square function estimates for lower dimensional moment curves. We use the following cylindrical, pointwise version of square function estimates for lower dimensional moment curves. **Lemma 29**. *Suppose that $\mathbb{M}^{m_1}(R)\lesssim_\delta R^\delta$ for all $R\ge 2$. Let $R_k^{-\frac{1}{n}}<s\le R_{k-1}^{-\frac{1}{n}}$. For each $0<\sigma<1$, and $\tau'\in{\bf{S}}_n(s)$, let $\tilde{W}_{\tau',\sigma}^{m_1}$ be a weight function that is Fourier supported in $\{\sum_{i=1}^{m_1}\lambda_i\gamma_{m_1}^{(i)}(a):|\lambda_i|\le \sigma^{m_1+1-i}\quad\forall i\}\times[-1,1]^{n-m_1}$, where $a$ is the initial point of $I(\tau')$. For any $s\le r\le 1$, $\tau\in{\bf{S}}(r)$ with $\tau'\subset\tau$, and $2\le p\le p_{m_1}\le p_n$, we have $$\begin{aligned} |f_{\tau'}^{k}|^{p}*\omega_{\tau,d}*\tilde{W}_{\tau',\sigma}^{m_1} \lesssim_{\delta,\varepsilon} R^\delta(\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*\omega_{\tau,2p_n^{-1}d})^{\frac{p}{2}}* \tilde{W}_{\tau,\sigma}^{m_1} \end{aligned}$$ where $k_m\ge k$ satisfies $R_{k_m}^{-\frac{1}{n}}< \sigma^{\frac{1}{m_1}}s\le R_{k_m-1}^{-\frac{1}{n}}$.* *Proof of Lemma [Lemma 29](#algo1){reference-type="ref" reference="algo1"}.* The $k=k_m$ case follows easily from the argument when $k<k_m$, so assume that $k<k_m$. By the same argument as was used in the proof of Lemma [Lemma 14](#ptwise){reference-type="ref" reference="ptwise"} (using moment curve rescaling in place of cone rescaling and invoking the definition of $\mathbb{M}^{m_1}(\cdot)$ in place of $\mathbb C_m^{n+k}(\cdot)$), we have $$|f_{\tau'}^{k}|^{p}*\omega_{\tau,d}*\tilde{W}_{\tau',\sigma}^{m_1} \lesssim_{\delta,\varepsilon}R^{\delta\varepsilon} (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(R_k^{-\frac{1}{n}})}}|f_{\tau''}^{k}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1} .$$ By Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, the right hand side is bounded above by $C_{\delta,\varepsilon}R^{\delta\varepsilon} (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(R_k^{-\frac{1}{n}})}}|f_{\tau''}^{k+1}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1}$. By Khintchine's inequality, there is a choice of signs $e_{\tau''}\in\{\pm1\}$ (permitted to depend on the point we are evaluating the two-fold convolution at) which satisfy $$(\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(R_k^{-\frac{1}{n}})}}|f_{\tau''}^{k+1}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1} \sim |\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(R_k^{-\frac{1}{n}})}}e_{\tau''}f_{\tau''}^{k+1}|^{p}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1} .$$ Then use a cylindrical version of $\mathbb{M}^{m_1}(\cdot)$ again to bound the right hand side above by $$C_{\delta,\varepsilon}R^{\delta\varepsilon}(\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(R_{k+1}^{-\frac{1}{n}})}}|f_{\tau''}^{k+1}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1} .$$ Iterate this process (which is at most $\varepsilon^{-1}$ many steps) until we obtain the inequality $$|f_{\tau'}^{k}|^{p}*\omega_{\tau,d}*\tilde{W}_{\tau',\sigma}^{m_1} \lesssim_{\delta,\varepsilon}R^{\delta} (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1} .$$ Then, by the locally constant property, we have $$(\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau',\sigma}^{m_1}\lesssim (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho}{\scalebox{-1}{\tmpbox}}% _\tau|)^{\frac{p}{2}}*\omega_{\tau,d} *\tilde{W}_{\tau,\sigma}^{m_1}$$ where $\rho_\tau$ is a bump function localized to $\tau$, which contains all of the $\tau''\subset\tau'$. It suffices to show that $$(\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho}{\scalebox{-1}{\tmpbox}}% _\tau|)^{\frac{p}{2}}*\omega_{\tau,d} \lesssim (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*\omega_{\tau,p_n^{-1}d})^{\frac{p}{2}} .$$ Let $T\|\tau^*$ denote a tiling of $\mathbb R^n$ by translates of $\tau^*$. Then, using $\frac{p}{2}\ge 1$ and $\|\cdot\|_{\ell^{p/2}}\le\|\cdot\|_1$, we have $$\begin{aligned} (\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2&*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho}{\scalebox{-1}{\tmpbox}}% _\tau|)^{\frac{p}{2}}*\omega_{\tau,d}(x)\le \sum_{T\|\tau^*}\|\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho}{\scalebox{-1}{\tmpbox}}% _\tau|(x-y)\|_{L^\infty_y(T)}^{\frac{p}{2}} \|\omega_{\tau,d}\|_{L^\infty(T)}|T| \\ &\le \Big(\int_{\mathbb R^n}\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2(z)\sum_{T\|\tau^*}\||% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\rho}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\rho}{\scalebox{-1}{\tmpbox}}% _\tau|(x-y-z)\|_{L^\infty_y(T)} \|\omega_{\tau,d}\|_{L^\infty(T)}^{\frac{2}{p}}|T|^{\frac{2}{p}} dz\Big)^{\frac{p}{2}} \\ &\lesssim_d \Big(\int_{\mathbb R^n}\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2(z)\omega_{\tau,2p^{-1}d}*\omega_{\tau,2p^{-1}d}(x-z) dz\Big)^{\frac{p}{2}}\\ &\lesssim_d \Big(\int_{\mathbb R^n}\sum_{\substack{\tau''\subset\tau'\\\tau''\in{\bf{S}}_n(\sigma^{\frac{1}{m_1}}s)}}|f_{\tau''}^{k_m}|^2*\omega_{\tau,2p_n^{-1}d}(x)\Big)^{\frac{p}{2}}, \end{aligned}$$ as desired. ◻ ## Auxiliary estimates related to Taylor cones. **Lemma 30**. *Assume that $\mathbb{M}^{n'}(R)\lesssim_\delta R^\delta$ and $\mathbb C^{n'+1}_{m'}(R)\lesssim_\delta R^\delta$ for all $n'<n$, $0\le m'\le n'-1$, $R\ge 1$, and $\delta>0$. Suppose that $2\le \tilde{p}_l\le p\le p_n$ and $2\le \frac{p}{\tilde{p}_l}$. For $R_{k}^{-\frac{1}{n}}< s\le R_{k-1}^{-\frac{1}{n}}$, there exists some dyadic $s'$, $R^{-\frac{1}{n}}\le s'\le s$ such that $$\begin{aligned} \int_{\mathbb R^n}(&\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_l}*\omega_{\tau',d})^{\frac{p}{\tilde{p}_l}} \le (C\log R)^{C}B_{\delta}R^{\delta}%\left[\int_{\R^n}(\sum_{\theta\in\Theta^n(R)}|f_\theta|^2*\w_{\theta,D})^{\frac{p}{2}} \right. \label{LHS}\\ &\times \left[ \int_{\mathbb R^n}(\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}\sum_{\substack{\tau\subset\tau'\\\tau\in{\bf{S}}_n(\max(R^{-\frac{1}{n}},R^{-\frac{\varepsilon^3}{n}}s))}} |f_{\tau}^{k_m}|^2*\omega_{\tau',p_n^{-1}d})^{\frac{p}{2}} +R^{C\varepsilon^3}\int_{\mathbb R^n}(\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l+1}}*\omega_{\tau',d})^{\frac{p}{\tilde{p}_{l+1}}} \right] \nonumber\end{aligned}$$ where $k_m\ge k$ satisfies $R_{k_m}^{-\frac{1}{n}}< \max(R^{-\frac{1}{n}},R^{-\frac{\varepsilon^3}{n}}s)\le R_{k_m-1}^{-\frac{1}{n}}$.* *Proof of Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"}.* We use the bump functions $\eta_{\tau',\sigma}^{n-1}$ introduced at the beginning of [\[keyalgo\]](#keyalgo){reference-type="ref" reference="keyalgo"} to decompose the Fourier support of the integrand. Using dyadic pigeonholing, let $\sigma$ be a dyadic parameter, $R^{-\varepsilon^3}\le \sigma\le 1$, which satisfies $$\int_{\mathbb R^n}(\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l}}*\omega_{\tau',d})^{\frac{p}{\tilde{p}_{l}}}\lesssim (\log R)^C\int_{\mathbb R^n}|\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^k|^{\tilde{p}_{l}}*\omega_{\tau,d}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{l}|^{\frac{p}{\tilde{p}_{l}}}.$$ If $\sigma= R^{-\varepsilon^3}$, then by Lemma [Lemma 29](#algo1){reference-type="ref" reference="algo1"}, $$\int_{\mathbb R^n}|\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l}}*\omega_{\tau',d}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{l}|^{\frac{p}{\tilde{p}_{l}}}\lesssim_\delta R^{\delta} \int_{\mathbb R^n}|\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|\sum_{\substack{\tau\subset\tau'\\ \tau\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s)}} |f_{\tau}^{k_m}|^2*\omega_{\tau',p_n^{-1}d}|^{\frac{\tilde{p}_{l}}{2}}*\tilde{W}_{\tau',\sigma}^{l}|^{\frac{p}{\tilde{p}_{l}}}.$$ Apply Proposition [Lemma 16](#multilem3pf){reference-type="ref" reference="multilem3pf"} to eliminate the weights $\tilde{W}_{\tau,\sigma}^{n-1}$, producing the upper bound $$D_\delta R^\delta\int_{\mathbb R^n}|\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|\sum_{\substack{\tau\subset\tau'\\ \tau\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s)}} |f_{\tau}^{k_m}|^2*\omega_{\tau',p_n^{-1}d}|^{\frac{\tilde{p}_{l}}{2}}|^{\frac{p}{\tilde{p}_{l}}},$$ which, since $\|\cdot\|_{\ell^{\tilde{p}_l/2}}\lesssim 1$, is bounded by $$D_\delta R^\delta\int_{\mathbb R^n}\big(\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}\sum_{\substack{\tau\subset \tau'\\ \tau\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s)}} |f_{\tau}^{k_m}|^2*\omega_{\tau',p_n^{-1}d} \big)^{\frac{p}{2}}.$$ It remains to consider the case that $\sigma>R^{-\varepsilon^3}$. The Fourier supports of the summands are contained in sets [\[summands\]](#summands){reference-type="eqref" reference="summands"}, which are a fixed dilate of elements of $\Xi_{l-1}^{n}(\sigma^{-\frac{n-1}{n}}s^{-\frac{n-1}{n}})$. Since we assumed that $2\le \frac{p}{\tilde{p}_l}$, by Lemma [Lemma 12](#pprops){reference-type="ref" reference="pprops"} we have $2\le \frac{p}{\tilde{p}_l}\le p_{n-l}$. Therefore, by the hypothesized boundedness of $\mathbb C_{l-1}^n(\cdot)$, we have $$\int_{\mathbb R^n}|\sum_{\substack{\tau'\in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l}}*\omega_{\tau',d}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{l}|^{\frac{p}{\tilde{p}_{l}}}\lesssim_\delta R^\delta\int_{\mathbb R^n}|\sum \sum_{\tau\in{\bf{S}}_n(\sigma^{-1}s)}|\sum_{\substack{\tau'\subset\tau \\ \tau' \in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l}}*\omega_{\tau',d}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{l}|^2|^{\frac{p}{2\tilde{p}_{l}}}.$$ Since $\sigma>R^{-\varepsilon^3}$, we have the pointwise inequality $\omega_{\tau',d}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^l|\lesssim_\varepsilon R^{C\varepsilon^3}\omega_{\tau',d}$. Then by Cauchy-Schwarz, the right hand side above is bounded by $$D_\delta R^\delta R^{C\varepsilon^3}\int_{\mathbb R^n}\big(\sum_{\substack{ \tau' \in{\bf{S}}_n(s)}}||f_{\tau'}^{k}|^{\tilde{p}_{l}}*\omega_{\tau',d}|^2\big)^{\frac{p}{2\tilde{p}_{l}}}.$$ Use that $1\le \frac{\tilde{p}_{l+1}}{\tilde{p}_l}\le 2$ and the pointwise inequality $||f_{\tau'}^k|^{\tilde{p}_l}*\omega_{\tau',d}|^{\frac{\tilde{p}_{l+1}}{\tilde{p}_l}}\lesssim |f_{\tau'}^k|^{\tilde{p}_l}*\omega_{\tau',d}$ to bound the right hand side above by $$D_\delta R^\delta R^{C\varepsilon^3}\int_{\mathbb R^n}\big(\sum_{\substack{ \tau' \in{\bf{S}}_n(s)}}|f_{\tau'}^{k}|^{\tilde{p}_{l+1}}*\omega_{\tau',d}\big)^{\frac{p}{\tilde{p}_{l+1}}},$$ as desired. ◻ ## Algorithm to fully unwind the pruning process [\[hisec\]]{#hisec label="hisec"} **Proposition 31**. *Let $D\in\mathbb N_{>0}$. There exists $d_0=d_0(n,\varepsilon,D)$ such that the following holds. Suppose that $\mathbb{M}^{m_1}(r)\lesssim_\delta r^\delta$ for all $1\le m_1<n$. For each $k$ and each $2\le p\le p_n$, we have $$\begin{aligned} \label{algline1} &\int_{\mathbb R^n}|\sum_{\tau_k}|f_{\tau_k}^{k+1}|^{2}*\omega_{\tau_k,d_k}|^{\frac{p}{2}}\le C_\varepsilon R^{3\varepsilon^2}\emph{T}_n(R^\varepsilon)^{N-k+1}\int_{\mathbb R^n} |\sum_\theta|f_\theta|^2*\omega_{\theta,D}|^{\frac{p}{2}}.\end{aligned}$$* Propositon [Proposition 31](#algo){reference-type="ref" reference="algo"} will follow from an algorithm which uses Lemmas [Lemma 29](#algo1){reference-type="ref" reference="algo1"} and [Lemma 30](#algo2){reference-type="ref" reference="algo2"} as building blocks. *Proof of Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"}.* We will define an algorithm which at intermediate step $m$, produces an inequality $$\begin{aligned} \label{stepm} (\text{L.H.S. of \eqref{algline1}})\le (C_\varepsilon\log R)^{4\varepsilon^{-1}m}&(R^{\varepsilon^7})^{m}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_m-k} \int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_m)}}|f_{\tau}^{k_m}|^{2}*\omega_{\tau,p_n^{-m}d_k})^{\frac{p}{2}} \end{aligned}$$ in which $0\le a\le m$ and $R^{-\frac{1}{n}}\le s_m\le \max(R_k^{-\frac{1}{n}}R^{-\frac{\varepsilon^3}{n}a},R_{k_{m-1}}^{-\frac{1}{n}})$ and $N\ge k_m\ge k+1$ satisfies $R_{k_m}^{-\frac{1}{n}}< s_m\le R_{k_m-1}^{-\frac{1}{n}}$. Notice that [\[stepm\]](#stepm){reference-type="eqref" reference="stepm"} clearly holds with $m=0$, taking $k_m=k+1$ and $s_m=R_k^{-\frac{1}{n}}$. Assuming [\[stepm\]](#stepm){reference-type="eqref" reference="stepm"} holds for $m-1$, we will show that either the algorithm terminates and the proposition is proved or [\[stepm\]](#stepm){reference-type="eqref" reference="stepm"} holds for $m\ge 1$. Suppose that [\[stepm\]](#stepm){reference-type="eqref" reference="stepm"} holds with $m-1$, so the left hand side of [\[algline1\]](#algline1){reference-type="eqref" reference="algline1"} is bounded by $$\begin{aligned} (C_\varepsilon\log &R)^{4\varepsilon^{-1}(m-1)}(R^{\varepsilon^7})^{m-1}(R^{\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_{m-1}-k} \int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}|f_{\tau}^{k_{m-1}}|^{2}*\omega_{\tau,p_n^{-(m-1)}d_k})^{\frac{p}{2}} \end{aligned}$$ in which $0\le a\le m-1$ and $R^{-\frac{1}{n}}\le s_{m-1}\le \max(R_k^{-\frac{1}{n}}R^{-\frac{\varepsilon^3}{n}a},R_{k_{m-1}}^{-\frac{1}{n}})$ and $N\ge k_{m-1}\ge k+1$ satisfies $R_{k_{m-1}}^{-\frac{1}{n}}< s_{m-1}\le R_{k_{m-1}-1}^{-\frac{1}{n}}$. Apply Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"} to the integral, yielding the inequality $$\begin{aligned} \label{stephyp} &(\text{L.H.S. of \eqref{algline1}})\le (C_\varepsilon\log R)^{4\varepsilon^{-1}m}(R^{\varepsilon^7})^{m-1+\frac{1}{n}}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_{m-1}-k} \\ &\times \left[ \int_{\mathbb R^n}(\sum_{\substack{\tau'\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s_{m-1})}}|f_{\tau'}^{k_m}|^2*\omega_{\tau',p_n^{-m}d_k})^{\frac{p}{2}} +R^{C\varepsilon^3}\int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}} |f_{\tau}^{k_{m-1}}|^{\tilde{p}_{2}}*\omega_{\tau,p_n^{-(m-1)}d_k})^{\frac{p}{\tilde{p}_{2}}}\right] \nonumber\end{aligned}$$ where $R_{k_m}^{-\frac{1}{n}}<R^{-\frac{\varepsilon^3}{n}}s_{m-1}\le R_{k_m-1}^{-\frac{1}{n}}$, $N\ge k_m\ge k+1$. If the first term on the right hand side dominates, then the inner loop terminates, producing the step $m$ inequality $$(\text{L.H.S. of \eqref{algline1}})\le (C_\varepsilon\log R)^{4\varepsilon^{-1}m}(R^{\varepsilon^7})^{m}(R^{\varepsilon^3}\text{\text{T}}_n(R^\varepsilon))^{k_{m-1}-k} \int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s_{m-1})}}|f_{\tau}^{k_m}|^2*\omega_{\tau,p_n^{-m}d_k})^{\frac{p}{2}} .$$ If the second term dominates, we have the following upper bound for the left hand side of [\[algline1\]](#algline1){reference-type="eqref" reference="algline1"}: $$(C_\varepsilon\log R)^{4\varepsilon^{-1}m}(R^{\varepsilon^7})^{m-1+\frac{1}{n}}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_{m-1}-k} R^{n^{-1}C\varepsilon^3}\int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{2}}*\omega_{\tau,p_n^{-(m-1)}d_k})^{\frac{p}{\tilde{p}_{2}}} .$$ Note that we may again apply Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"} to the integral above. We iterate this process until either the inner loop terminates with proving step $m$ or we have shown that the left hand side of [\[algline1\]](#algline1){reference-type="eqref" reference="algline1"} is bounded by $$(C_\varepsilon\log R)^{4\varepsilon^{-1}m}(R^{\varepsilon^7})^{m-1+\frac{l}{n}}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_{m-1}-k} R^{ln^{-1}C\varepsilon^3}\int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k})^{\frac{p}{\tilde{p}_{l}}}$$ with $1\le \frac{p}{\tilde{p}_l}\le 2$ and $2\le \frac{p}{\tilde{p}_{l-1}}$, which means that $l\le n$. As in the proof of Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"}, use bump functions $\eta_{\tau,\sigma}^{n-1}$ and suppose that $$\int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k})^{\frac{p}{\tilde{p}_{l}}}\lesssim (\log R)^C\int_{\mathbb R^n}|\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}.$$ If $\sigma= R^{-\varepsilon^3}s_{m-1}$, then by Lemma [Lemma 29](#algo1){reference-type="ref" reference="algo1"}, $$\begin{aligned} \int_{\mathbb R^n}|\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}}&|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}\\ &\lesssim C_\varepsilon R^{n^{-1}\varepsilon^7} \int_{\mathbb R^n}|\sum_{\substack{\tau\in{\bf{S}}_n(s_{m-1})}} |\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(R^{-\frac{\varepsilon^3}{n}}s_{m-1})}} |f_{\tau'}^{k_m}|^2*\omega_{\tau,p_n^{-m}d_k}|^{\frac{\tilde{p}_{l}}{2}}*\tilde{W}_{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}. \end{aligned}$$ By the locally constant property and $\omega_{\tau',p_n^{-m}d_k}*\omega_{\tau,2p_n^{-m}d_k}\lesssim\omega_{\tau',p_n^{-m}d_k}$ whenever $\tau'\subset \tau$, so we have $|f_{\tau'}^{k_m}|^2*\omega_{\tau,2p_n^{-1}d_k}\lesssim |f_{\tau'}^{k_m}|^2*\omega_{\tau',p_n^{-m}d_k}$ for each $\tau'$. Finally, apply Proposition [Lemma 16](#multilem3pf){reference-type="ref" reference="multilem3pf"} to eliminate $\tilde{W}_{\tau,\sigma}^{n-1}$, concluding step $m$. The other case is that $\sigma>R^{-\varepsilon^3}s_{m-1}$. Using the boundedness of $\mathbb C^n_{n-2}(\cdot)$ and Cauchy-Schwarz, we have $$\int_{\mathbb R^n}|\sum_{\substack{\tau\in\Theta^n(s_{m-1}^{-n})}}|f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}\le C_\varepsilon R^{n^{-1}\varepsilon^7}\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_n(s_{m-1})} ||f_{\tau}^{k_{m-1}}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}} .$$ By Young's convolution inequality, the right hand side is bounded by $$C_\varepsilon R^{n^{-1}\varepsilon^7}\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_n(s_{m-1})} |f_{\tau}^{k_{m-1}}|^{p} ,$$ which, by the definition of $\text{T}_n(\cdot)$ and rescaling, is itself bounded by $$C_\varepsilon R^{n^{-1}\varepsilon^7}\text{T}_n(R^\varepsilon)\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_b(s_{m-1})} (\sum_{\tau_{k_{m-1}}\subset\tau}|f_{\tau_{k_{m-1}}}^{k_{m-1}}|^2 )^{\frac{p}{2}} .$$ By Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, $\|\cdot\|_{\ell^{p/2}}\le \|\cdot\|_{\ell^1}$, and the locally constant property, this expression is bounded by $$\begin{aligned} C_\varepsilon R^{n^{-1}\varepsilon^7}\text{T}_n(R^\varepsilon)\int_{\mathbb R^n}\sum_{\tau\in\Theta^n(s_{m-1}^{-n})} &(\sum_{\tau_{k_{m-1}}\subset\tau}|f_{\tau_{k_{m-1}}}^{k_{m-1}+1}|^2)^{\frac{p}{2}}\\ &\le C_\varepsilon R^{n^{-1}\varepsilon^7}\text{T}_n(R^\varepsilon)\int_{\mathbb R^n}(\sum_{\tau\in\Theta^n(s_{m-1}^{-n})} |f_{\tau_{k_{m-1}}}^{k_{m-1}+1}|^2*\omega_{\tau_{k_{m-1}},p_n^{-m}d_k})^{\frac{p}{2}}. \end{aligned}$$ Taking $k_m=k_{m-1}+1$, this concludes the justification of step $m$. By taking $m$ large enough so that $s_m=R^{-\frac{1}{n}}$, the algorithm terminates with the inequality $$\label{term1} (\text{L.H.S. of \eqref{algline1}})\le (C_\varepsilon\log R)^{4\varepsilon^{-1}m}(R^{\varepsilon^7})^{m+1}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_m-k} \int_{\mathbb R^n}(\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}|f_\theta|^2*\omega_{\theta,p_n^{-m}d_k})^{\frac{p}{2}}$$ where $0\le a<m$ and $R^{-\frac{1}{n}}= s_m\le \max(R_k^{-\frac{1}{n}}R^{-\frac{\varepsilon^3}{n}a},R_{k_{m-1}}^{-\frac{1}{n}})$. Then $R^{-\frac{1}{n}}\le R_k^{-\frac{1}{n}}R^{-\frac{\varepsilon^3}{n}a}$ implies that $a\le\varepsilon^{-3}(N-k)$ and $N\ge k_m\ge k+1$ implies that $(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{k_m-k}\le(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{N-k}$. The total number of steps $m$ is bounded by $2\varepsilon^{-3}$ since each step either refines $s_m$ by a factor of $R^{-\frac{\varepsilon^3}{n}}$ or replaces $s_m$ by $R_{k_m}^{-\frac{1}{n}}$, with $k_m>k_{m-1}$. Thus the constants in the upper bound from [\[term1\]](#term1){reference-type="eqref" reference="term1"} are bounded by $$\begin{aligned} (C_e\log R)^{4\varepsilon^{-1}m+2\varepsilon^{-1}+1}&(R^{\varepsilon^7})^{m-1}(R^{C\varepsilon^3}\text{T}_n(R^\varepsilon))^{N-k} \\ &\le (C_\varepsilon R^{\varepsilon^7})^{\varepsilon^{-5}}R^{\varepsilon^2}\text{T}_n(R^\varepsilon)^{N-k}\le C_\varepsilon R^{3\varepsilon^2}\text{T}_n(R^\varepsilon)^{N-k+1} .\end{aligned}$$ Finally, since $m\le 2\varepsilon^{-3}$, it suffices to choose $d_0$ (from the definition of $d_k$) satisfying $p_n^{-2\varepsilon^{-3}}d_0\ge D$, so the proposition is proved. ◻ # Proof of Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} [\[mainsec\]]{#mainsec label="mainsec"} {#proof-of-proposition-momcurveinduct-mainsec} We use the set-up for the high-low method to prove a broad estimate, Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"}. Then, we prove Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"}, which says that $\text{T}_{n,D}^w(R)\lesssim_\varepsilon R^\varepsilon$. This follows using various reductions from pigeonholing, a broad-narrow argument, and Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"}. Finally, in [\[S2\]](#S2){reference-type="ref" reference="S2"}, we use induction to show that Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"} implies that $\text{T}_n(R)\lesssim_\varepsilon R^\varepsilon$, which is equivalent to Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"}. ## Bounding the broad part of $U_{\alpha,\beta}$ [\[broad\]]{#broad label="broad"} For $n$ canonical blocks $\tau^1,\ldots,\tau^n$ (with dimensions $\sim R^{-\varepsilon/n}\times R^{-2\varepsilon/n}\times \cdots\times R^{-\varepsilon}$) which are pairwise $\ge R^{-\varepsilon/n}$-separated, define the broad part of $U_{\alpha,\beta}$ to be $$\text{Br}_{\alpha,\beta}^K=\{x\in U_{\alpha,\beta}: \alpha\le K|\prod_{i=1}^nf_{\tau^i}(x)|^{\frac{1}{n}},\quad\max_{\tau^i}|f_{\tau^i}(x)|\le \alpha\}.$$ We bound the broad part of $U_{\alpha,\beta}$ in the following proposition. Recall that the parameter $N_0$ was used in the definition of the sets $\Omega_k$ and $L$. **Proposition 32**. *Let $R,K\ge1$ and $2\le p\le p_n$. Suppose that $\|f_\theta\|_{L^\infty(\mathbb R^n)}\le 2$ for all $\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})$. Then $$\alpha^{p}|\text{\emph{Br}}_{\alpha,\beta}^{K}|\le \big[CR^{10 \varepsilon N_0}A^{\varepsilon^{-1}} + K^{50}R^{4\varepsilon^2+10\varepsilon}A^{\varepsilon^{-1}} \emph{T}_n(R^\varepsilon)^{\varepsilon^{-1}-N_0}\big]\int\Big|\sum_\theta|f_\theta|^2*\omega_{\theta,D}\Big|^{\frac{p}{2}} .$$* We will use the following version of a (well-known) local multilinear restriction inequality for the moment curve. The weight function $W_{B_r,d}$ is defined in Definition [Definition 5](#M3ballweight){reference-type="ref" reference="M3ballweight"}. **Theorem 33**. *Let $s\ge 10r\ge10$ and let $f:\mathbb R^n\to\mathbb C$ be a Schwartz function with Fourier transform supported in $\mathcal{N}_{r^{-1}}(\mathcal{M}^n)$, the $r^{-1}$-neighborhood of the moment curve in $\mathbb R^n$. Suppose that $\tau^1,\tau^2,\ldots,\tau^n\in{\bf{S}}_n(R^{-\varepsilon/n})$ satisfy $\text{dist}(\tau^i,\tau^j)\ge s^{-1}$ for $i\not=j$. Then $$\alpha^p |\emph{Br}_{\alpha,\beta}^K\cap B_r| \lesssim_d s^n|B_r| \prod_{i=1}^n\big(|B_r|^{-1}\int|f_{\tau^i}|^2W_{B_r,d}\big)^{\frac{p}{2n}}$$ for any $2\le p\le 2n$ and any Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(r)$.* *Proof.* In the case that $p=2n$, $$\alpha^{2n}|\emph{Br}_{\alpha,\beta}^K\cap B_r|\lesssim_d s^n|B_r|^{-(n-1)}\prod_{i=1}^n\big(\int|f_{\tau^i}|^2W_{B_r,d}\big)$$ follows from a straightforward adaptation of the proof of Proposition 6 from [@M3smallcap] to $n$-dimensions. The $2\le p\le 2n$ case follows from rearranging the above inequality: $$\label{intp} \alpha^{p}|\emph{Br}_{\alpha,\beta}^K\cap B_r|\lesssim_d s^n|B_r|\left(\prod_{i=1}^n\big(|B_r|^{-1}\int|f_{\tau^i}|^2W_{B_r,d}\right)^{\frac{p}{2n}}\left[\frac{1}{\alpha}\prod_{i=1}^n\big(|B_r|^{-1}\int|f_{\tau^i}|^2W_{B_r,d}\big)^{\frac{1}{2n}}\right]^{2n-p}.$$ If $$\prod_{i=1}^n\big(|B_r|^{-1}\int|f_{\tau^i}|^2W_{B_r,d}\big)^{\frac{1}{2n}}\le \alpha,$$ then the theorem follows from [\[intp\]](#intp){reference-type="eqref" reference="intp"}. If the above inequality does not hold, then we have $$\begin{aligned} \alpha^p|\text{Br}_{\alpha,\beta}^K\cap B_r|\le |B_r| \prod_{i=1}^n\big(|B_r|^{-1}\int|f_{\tau^i}|^2W_{B_r,d}\big)^{\frac{p}{2n}},\end{aligned}$$ which proves the theorem directly. ◻ *Proof of Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"}.* Fix $2\le p\le p_n$ and the decay rate $D\in\mathbb N_{>0}$ for the weights. Note that $$\text{Br}_{\alpha,\beta}^K=(L\cap \text{Br}_{\alpha,\beta}^K)\cup( \sqcup_{k=B}^{N-1}\Omega_k\cap \text{Br}_{\alpha,\beta}^K)$$ We bound each of the sets $\text{Br}_{\alpha,\beta}^K\cap\Omega_k$ and $\text{Br}_{\alpha,\beta}^K\cap L$ in separate cases. It suffices to consider the case that $R$ is at least some constant depending on $\varepsilon$ since if $R\le C_\varepsilon$, we may prove the proposition using trivial inequalities. [Case 1: bounding $| \text{Br}_{\alpha,\beta}^{K}\cap\Omega_k|$. ]{.ul} By Lemma [Lemma 27](#ftofk){reference-type="ref" reference="ftofk"}, $$|\text{Br}_{\alpha,\beta}^K\cap\Omega_k|\le|\{x\in U_{\alpha,\beta}\cap\Omega_k:\alpha\lesssim K|\prod_{i=1}^nf_{\tau^i}^{k+1}(x)|^{\frac{1}{n}},\quad\max_{\tau^i}|f_{\tau^i}(x)|\le \alpha\}|.$$ By Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, the Fourier supports of the $f_{\tau^i}^{k+1}$ are contained in $2\tau^i$, which are pairwise $\ge5 R^{-\varepsilon}$-separated blocks of the moment curve. Let $\{B_{R_k^{\frac{1}{n}}}\}$ be a finitely overlapping cover of $\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k$ by $R_k^{\frac{1}{n}}$-balls. For $R$ large enough depending on $\varepsilon$, apply Theorem [Theorem 33](#trirestprop){reference-type="ref" reference="trirestprop"} with $r:= \min(2n,p)$ to get $$\begin{aligned} \alpha^{r}|\text{Br}_{\alpha,\beta}^K\cap B_{R_k^{\frac{1}{n}}}|&\lesssim_\varepsilon R^{\varepsilon} |B_{R_k^{\frac{1}{n}}}|\prod_{i=1}^n\Big(|B_{R_k^{\frac{1}{n}}}|^{-1}\int|f_{\tau^i}^{k+1}|^2W_{B_{R_k^{\frac{1} {n}}},d_k+1}\Big)^{\frac{r}{2n}}.\end{aligned}$$ Using local $L^2$-orthogonality (Lemma [Lemma 24](#L2orth){reference-type="ref" reference="L2orth"}), each integral on the right hand side above is bounded by $$\lesssim \int\sum_{\tau_k}|f_{\tau_k}^{k+1}|^2W_{B_{R_k^{\frac{1}{n}},d_k+1}}.$$ If $x\in \text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}$, then the above integral is bounded by $$\lesssim \int \sum_{\tau_k}|f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k+1}W_{B_{R_k^{\frac{1}{n}}},d_k+1}\lesssim C |B_{R_k^{\frac{1}{n}}}| \sum_{\tau_k}|f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}(x)$$ by the locally constant property (Lemma [Lemma 23](#locconst){reference-type="ref" reference="locconst"}) and properties of the weight functions. The summary of the inequalities so far is that $$\alpha^{r}|\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}|\lesssim_\varepsilon R^\varepsilon K^{2n} |B_{R_k^{\frac{1}{n}}}|g_k(x)^{\frac{r}{2}}$$ where $x\in \text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}$. Recall that since $x\in\Omega_k$, we have the lower bound $A^{M-k}\beta\le g_k(x)$ (where $A$ is from Definition [Definition 8](#impsets){reference-type="ref" reference="impsets"}), which leads to the inequality $$\alpha^{r}|\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}|\lesssim_\varepsilon K^{2n} R^{\varepsilon} \frac{1}{[A^{M-k}\beta]^{q-\frac{r}{2}}}|B_{R_k^{\frac{1}{n}}}|g_k(x)^{q} ,$$ where $q$ is defined by $\frac{r}{2}+q=p$. There are two possibilities for $q$, depending on whether $r=p$ or $r=2n$. If $q=\frac{p}{2}$, then $r=p$ and since $g_k\lesssim R^{C\varepsilon}g_{k+1}\lesssim_\varepsilon R^{C\varepsilon}\beta$ on $\Omega_k$, we have $$\sum_{B_{R_k^{\frac{1}{n}}}} \alpha^p|\text{Br}_{\alpha,\beta}^K\cap B_{R_k^{\frac{1}{n}}}|\lesssim_\varepsilon R^{C\varepsilon}K^{2n} |\mathcal{N}_{R_k^{\frac{1}{n}}}(\text{Br}_{\alpha,\beta}^K)|\beta^{\frac{p}{2}}\lesssim_\varepsilon R^{C\varepsilon}K^{2n} \int_{\mathbb R^n}|\sum_\theta|f_\theta|^2*\omega_{\theta,D}|^{\frac{p}{2}} ,$$ where we used the locally constant property in the final inequality. It remains to treat the case that $r=2n$, meaning that $n+q=p\ge 2n$ and $q$ therefore satisfies $2\le q\le p_{n-1}$. By Corollary [Corollary 26](#highdom){reference-type="ref" reference="highdom"}, we also have the upper bound $|g_k(x)|\le 2|g_k^h(x)|$, so that $$\alpha^{2n}|\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}|\lesssim_\varepsilon K^{2n} R^{\varepsilon} \frac{1}{[A^{M-k}\beta]^{q-n}} |B_{R_k^{\frac{1}{n}}}||g_k^h(x)|^{q} .$$ By the locally constant property applied to $g_k^h$, we have $|g_k^h|^{q}\lesssim_\varepsilon|g_k^h*w_{ B_{R_k^{\frac{1}{n}}}}|^{q}$. By Cauchy-Schwarz, we also have $|g_k^h*w_{ B_{R_k^{\frac{1}{n}}}}|^{q}\lesssim |g_k^h|^{q}*w_{B_{R_k^{\frac{1}{n}}}}$. Combine this with the previous displayed inequality to get $$\alpha^{2n}|\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k\cap B_{R_k^{\frac{1}{n}}}|\lesssim_\varepsilon K^{2n} R^{\varepsilon} \frac{1}{[A^{M-k}\beta]^{q-n}}\int|g_k^h|^{q}W_{ B_{R_k^{\frac{1}{n}}}} .$$ Summing over the balls $B_{R_k^{\frac{1}{n}}}$ in our finitely-overlapping cover of $\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k$, we conclude that $$\label{peqn} \alpha^{2n}|\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k|\lesssim_\varepsilon K^6 R^{\varepsilon} \frac{1}{[A^{M-k}\beta]^{q-n}}\int_{\mathbb R^n}|g_k^h|^{q} .$$ We are done using the properties of the set $\text{Br}_{\alpha,\beta}^{K}\cap\Omega_k$, which is why we now integrate over all of $\mathbb R^n$ on the right hand side. We will now use Lemma [Lemma 28](#high1){reference-type="ref" reference="high1"} to analyze the high part $g_k^h$. In particular, Lemma [Lemma 28](#high1){reference-type="ref" reference="high1"} gives $$\label{highapp} \int|g_k^h|^{q} \lesssim_\varepsilon R^{\varepsilon} \int_{\mathbb R^n}\big|\sum_{\tau_k}||f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{R_{k+1}^{-\frac{1}{n}}}|^2\big|^{\frac{q}{2}} .$$ By Young's convolution inequality, since $% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{>R_{k+1}^{-\frac{1}{n}}}$ is $L^1$-normalized, the right hand side above is bounded by $$C_\varepsilon R^{\varepsilon} \int_{\mathbb R^n}\big|\sum_{\tau_k}||f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}|^2\big|^{\frac{q}{2}}$$ Next use [\[item2\]](#item2){reference-type="eqref" reference="item2"} from Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"} to note that $\|f_{\tau_k}^{k+1}\|_\infty\le \sum_{\tau_{k+1}\subset\tau_k}\|f_{\tau_{k+1}}^{k+1}\|_\infty\lesssim R^\varepsilon A^{\varepsilon^{-1}}K^n\frac{\beta}{\alpha}$: $$\begin{aligned} &\int_{\mathbb R^n}\big|\sum_{\tau_k}||f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}|^2\big|^{\frac{q}{2}} \lesssim_\varepsilon[(A^{\varepsilon^{-1}}R^{\varepsilon}K^n\frac{\beta}{\alpha})]^{q-n}\int_{\mathbb R^n}\big|\sum_{\tau_k}||f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}|^{2-\frac{q-n}{q}}\big|^{\frac{q}{2}}. \end{aligned}$$ Since the exponent $2-\frac{q-n}{q}=\frac{p}{q}\ge 1$, the integral on the right hand side above is bounded by $$\int_{\mathbb R^n}|\sum_{\tau_k}|f_{\tau_k}^{k+1}|^2*\omega_{\tau_k,d_k}|^{\frac{p}{2}}.$$ Combining this with [\[peqn\]](#peqn){reference-type="eqref" reference="peqn"}, [\[highapp\]](#highapp){reference-type="eqref" reference="highapp"}, and Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"}, the summary of the argument from this case is $$\alpha^{p}|U_{\alpha,\beta}|\lesssim_\varepsilon K^{2n} R^{2\varepsilon}(A^{\varepsilon^{-1}}R^{\varepsilon}K^n)\big(\text{T}_n(R^\varepsilon)\big)^{N-k+1}\int_{\mathbb R^n}\big|\sum_{\theta\in{\bf{S}}_n(R^{-1/n})}|f_\theta|^2*\omega_{\theta,D}\big|^{\frac{p}{2}}.$$ Since $k>N_0$, this upper bound has the desired form. [Case 2: bounding $|U_{\alpha,\beta}\cap L|$.]{.ul} Begin by using Lemma [Lemma 27](#ftofk){reference-type="ref" reference="ftofk"} to bound $$\alpha^{p}|\text{Br}_{\alpha,\beta}^K\cap L|\lesssim K^{p} \int_{U_{\alpha,\beta}\cap L}|f^{N_0+1}|^{p}.$$ Then use Cauchy-Schwarz and the locally constant property for $g_{N_0}$ : $$\int_{U_{\alpha,\beta}\cap L}|f^{N_0+1}|^{p}\lesssim (R^{ N_0\varepsilon/n})^{p/2} \int_{U_{\alpha,\beta}\cap L}(\sum_{\tau_{N_0}}|f_{\tau_{N_0}}^{N_0+1}|^2)^{p/2}\lesssim (R^{ N_0\varepsilon/n})^{p/2} \int_{U_{\alpha,\beta}\cap L}(g_{N_0})^{p/2}.$$ Using the definition the definition of $L$, we bound the factors of $g_{N_0}$ by $$\int_{U_{\alpha,\beta}\cap L} (A^{\varepsilon^{-1}}\beta)^{p/2}.$$ Finally, by the definition of $U_{\alpha,\beta}$, conclude that $$\alpha^{p}|\text{Br}_{\alpha,\beta}^K\cap L|\lesssim_\varepsilon K^{p}R^{2N_0\varepsilon}A^{\varepsilon^{-1}}\int_{\mathbb R^n}|\sum_\theta|f_\theta|^2*\omega_{\theta,D}|^{p/2}.$$ ◻ ## Proof of Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} from Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"} {#proof-of-proposition-momcurveinduct-from-proposition-mainprop} **Proposition 34**. *For any $\varepsilon>0$, $R\ge 1$, and $D\in\mathbb N_{>0}$, $$\emph{T}_{n,D}^w(R)\lesssim_{\varepsilon,D} R^\varepsilon.$$* In order to make use of Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"}, we need to reduce to the case that our function $f$ is localized to a ball, its wave packets have been pigeonholed so that $\|f_\theta\|_\infty\lesssim 1$ for all $\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})$, and we have approximated $\|f\|_p$ by an expression involving a superlevel set. This is the content of the following subsection. ## Wave packet decomposition and pigeonholing [\[M3pigeon\]]{#M3pigeon label="M3pigeon"} We describe standard reductions from dyadic pigeonholing. The following lemmas have direct analogues in [@maldagueM3], so we omit their proofs. The following spatial localization lemma is analogous to Lemma 6.2 in [@maldagueM3]. **Lemma 35**. *For any $R$-ball $B_R\subset\mathbb R^n$, suppose that $$\|f\|_{L^p(B_R)}^p\lesssim_{\varepsilon,D} R^\varepsilon\int\big|\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}|f_\theta|^2*\omega_{\theta,D}\big|^{\frac{p}{2}}$$ for any $2\le p\le p_n$ and any Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$. Then Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"} is true.* The following weak, level-set version of Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"} is analogous to Lemma 6.3 in [@maldagueM3]. **Lemma 36**. *For each $B_R$ and Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$, and each $2\le p\le p_n$, there exists $\alpha>0$ such that $$\|f\|_{L^p(B_R)}^p\lesssim (\log R)\alpha^p|\{x\in B_R:\alpha\le |f(x)|\}|+R^{-500n}\int|\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})} |f_\theta|^2*\omega_{\theta,D}|^{p/2}.$$* Continue to use the notation $$U_{\alpha}=\{x\in B_R:\alpha\le |f(x)|\}.$$ We will show that to estimate the size of $U_{\alpha}$, it suffices to replace $f$ with a version whose wave packets at scale $\theta$ have been pigeonholed. Write $$\begin{aligned} \label{sum} f=\sum_\theta\sum_{T\in\mathbb T_\theta}\psi_Tf_\theta \end{aligned}$$ where for each $\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})$, $\{\psi_T\}_{T\in\mathbb T_\theta}$ is the partition of unity from [\[prusec\]](#prusec){reference-type="ref" reference="prusec"}. If $\alpha\le C_\varepsilon R^{-100n}\max_\theta\|f_\theta\|_{\infty}$, then using a similar argument that bounds the second expression in the proof of Lemma [Lemma 36](#alph){reference-type="ref" reference="alph"}, the inequality $$\alpha^p|U_{\alpha}|\lesssim_\varepsilon R^\varepsilon\int|\sum_\theta|f_\theta|^2*\omega_{\theta,D}|^{p/2}$$ is trivial. The following proposition about wave packet decomposition is analogous to Lemma 6.4 in [@maldagueM3]. **Proposition 37** (Wave packet decomposition). *Let ${\alpha}>C_\varepsilon R^{-100n}\max_\theta\|f_\theta\|_{L^\infty(\mathbb R^n)}$. There exist subsets $\tilde{\mathbb T}_\theta\subset\mathbb T_\theta$, as well as a constant $A>0$ with the following properties: $$\begin{aligned} |U_{\alpha}|\lesssim (\log R)|\{x\in U_{\alpha}:\,\,{{\alpha}}&\lesssim |\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}\sum_{T\in\tilde{\mathbb T}_\theta}\psi_T(x)f_\theta (x)|\,\,\}|, \\ R^\varepsilon T\cap U_{\alpha}\not=\emptyset\qquad&\text{for all}\quad\theta\in{\bf{S}}_n(R^{-\frac{1}{n}}),\quad T\in\tilde{\mathbb T}_\theta\\ A\lesssim \|\sum_{T\in\tilde{\mathbb T}_\theta}\psi_Tf_\theta\|_{L^\infty(\mathbb R^n)}&\lesssim R^{3\varepsilon} A\qquad\text{for all}\quad \theta\in{\bf{S}}_n(R^{-\frac{1}{n}})\label{propM}\\ \|\psi_Tf_\theta\|_{L^\infty(\mathbb R^n)}&\sim A\qquad\text{for all}\quad \theta\in{\bf{S}}_n(R^{-\frac{1}{n}}),\quad T\in\tilde{\mathbb T}_\theta. \label{prop'M}\end{aligned}$$* The following is analogous to Corollary 6.5 in [@maldagueM3]. **Corollary 38**. *Let $f$, $\alpha$, $\tilde{T}_\theta$ and $A>0$ be as in Proposition [Proposition 37](#wpd){reference-type="ref" reference="wpd"}. Then for each $x\in U_\alpha$, $$\alpha\le R^{103n\varepsilon}\frac{1}{A} \sum_{\theta\in\mathcal{S}}|\sum_{T\in\tilde{\mathbb T}_\theta}\psi_T f_\theta|^2*\omega_{\theta,D}(x).$$* Using the locally constant property for $\sum_\theta|f_\theta|^2$ and dyadic pigeonholing, we have the following, which is analogous to Lemma 6.6 in [@maldagueM3]. **Lemma 39**. *For each $2\le p\le p_n$, $\alpha>0$, $B_R$, and Schwartz function $f:\mathbb R^n\to\mathbb C$ with Fourier transform supported in $\mathcal{M}^n(R)$, there exists $\beta>0$ such that $\alpha^p|\{x\in B_R:\alpha\le|f(x)|\}|$ is bounded by $$C(\log R)\alpha^{p}|\{x\in B_R:\alpha\le|f(x)|,\quad\beta/2\le\sum_{\theta}|f_\theta|^2*\omega_{\theta,D}(x)\le \beta\}|+R^{-500n}\int|\sum_{\theta}|f_\theta|^2*\omega_{\theta,D}|^{p/2} .$$* ## A multi-scale inequality for $\text{T}_{n,D}^w(R)$ implying Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"} [\[ind\]]{#ind label="ind"} {#a-multi-scale-inequality-for-textt_ndwr-implying-proposition-s1bd-ind} First we use a broad/narrow analysis to prove a multi-scale inequality for $\text{T}_n^w(R)$. **Lemma 40**. *For any $D\in\mathbb N_{>0}$, $1\le K^n\le R$, and $1\le N_0\le \varepsilon^{-1}$, $$\emph{T}_{n,D}^w(R)\lesssim(\log R)^2\left( K^{53}\big[R^{10 \varepsilon N_0}A^{\varepsilon^{-1}} + R^{4\varepsilon^2+200\varepsilon}A^{\varepsilon^{-1}} \emph{T}_{n,D}^w(R^\varepsilon)^{\varepsilon^{-1}-N_0}\big]+\emph{T}_{n,D}^w(R/K^3)\right) .$$* *Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"} implies Lemma [Lemma 40](#multi-scale'){reference-type="ref" reference="multi-scale'"}.* Let $f:\mathbb R^n\to\mathbb C$ be a Schwartz function with Fourier transform supported in $\mathcal{M}^n(R)$. Let $2\le p\le p_n$. By Lemma [Lemma 35](#loc){reference-type="ref" reference="loc"}, it suffices to bound $\|f\|_{L^p(B_R)}^p$ instead of $\|f\|_{L^p(\mathbb R^n)}^p$. By Lemma [Lemma 36](#alph){reference-type="ref" reference="alph"}, we may fix $\alpha>0$ so that $\|f\|_{L^p(B_R)}^p\lesssim(\log R)^2\alpha^p|U_{\alpha}|$. By Proposition [Proposition 37](#wpd){reference-type="ref" reference="wpd"}, we may replace $\alpha$ by $\alpha/A$ and replace $f$ by $\tilde{f}=\frac{1}{A}\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}\sum_{T\in\tilde{\mathbb T}_\theta}\psi_Tf_\theta$ where $\tilde{\mathbb T}_\theta$ satisfies the properties in that proposition. From here, we will take $f$ to mean $\tilde{f}$. By Lemma [Lemma 39](#bet){reference-type="ref" reference="bet"}, we may fix $\beta>0$ so that $\alpha^p|U_\alpha|\lesssim (\log R)\alpha^p|U_{\alpha,\beta}|$. Finally, by Corollary [Corollary 38](#wpdcor){reference-type="ref" reference="wpdcor"}, we have $\alpha\lesssim R^{103n\varepsilon}$. Write $f=\sum_{\tau\in{\bf{S}}_n(K^{-1})}f_\tau$. The broad-narrow inequality is $$\begin{aligned} \label{brnar} |f(x)|&\le 2n\max_{\tau\in{\bf{S}}_n(K^{-1})}|f_{\tau}(x)|+K^3\max_{\substack{d(\tau^i,\tau^j)\ge K^{-1}\\\tau^i\in{\bf{S}}_n(K^{-1})}}|\prod_{i=1}^nf_{\tau^i}(x)|^{\frac{1}{n}} .\end{aligned}$$ Indeed, suppose that the set $\{\tau\in{\bf{S}}_n(K^{-1}):|f_{\tau}(x)|\ge K^{-1}\max_{\tau'\in{\bf{S}}_n(K^{-1})}|f_{\tau'}(x)|\}$ has at least $2n-1$ elements. Then we can find $n$ many $\tau^i$ which are pairwise $\ge K^{-1}$-separated and satisfy $|f(x)|\le K^3|\prod_{i=1}^nf_{\tau^i}(x)|^{\frac{1}{n}}$. If there are fewer than $2n-1$ elements, then $|f(x)|\le 2n \max_{\tau\in{\bf{S}}_n(K^{-1})}|f_{\tau}(x)|$. The broad-narrow inequality leads to two possibilities. In one case, we have $$\label{case1brn} |U_{\alpha,\beta}|\lesssim |\{x\in U_{\alpha,\beta}:|f(x)|\le 2n\max_{\tau\in{\bf{S}}_n(K^{-1})}|f_{\tau}(x)|\}| .$$ Then the summary of inequalities from this case is $$\begin{aligned} \int_{B_R}|f|^p\lesssim(\log R)^2\alpha^p|U_{\alpha,\beta}|\lesssim(\log R)^2\sum_{\tau\in{\bf{S}}_n(K^{-1})}\int_{\mathbb R^n}|f_{\tau\in{\bf{S}}_n(K^{-1})}|^p .\end{aligned}$$ By rescaling for the moment curve and the definition of $\text{T}_{n,D}^w(\cdot)$, we may bound each integral in the final upper bound by $$\int_{\mathbb R^n}|f_{\tau}|^p\le \text{T}_{n,D}^w(R/K^n)\int_{\mathbb R^n}|\sum_{\theta\subset\tau}|f_\theta|^2*\omega_{\theta,D}|^{p/2}.$$ Noting that $\sum_\tau\int_{\mathbb R^n}|\sum_{\theta\subset\tau}|f_\theta|^2*\omega_{\theta,D}|^{7/2}\le\int_{\mathbb R^n}|\sum_{\theta}|f_\theta|^2*\omega_{\theta,D}|^{7/2}$ finishes this case. The remaining case from the broad-narrow inequality is that $$|U_{\alpha,\beta}|\lesssim |\{x\in U_{\alpha,\beta}:|f(x)|\le K^3 \max_{\substack{%(\tau^1,\tau^2,\tau^3)\\ d(\tau^i,\tau^j)\ge K^{-1}\\\tau^i\in{\bf{S}}_n(K^{-1})}}|\prod_{i=1}^nf_{\tau^i}(x)|^{\frac{1}{n}}\}| .$$ We may further assume that $$|U_{\alpha,\beta}|\lesssim |\{x\in U_{\alpha,\beta}:|f(x)|\le K^3 \max_{\substack{%(\tau^1,\tau^2,\tau^3)\\ d(\tau^i,\tau^j)\ge K^{-1}\\\tau^i\in{\bf{S}}_n(K^{-1})}}|\prod_{i=1}^nf_{\tau^i}(x)|^{\frac{1}{n}},\quad \max_{\tau\in{\bf{S}}_n(K^{-1})}|f_\tau(x)|\le \alpha\}|$$ since otherwise, we would be in the first case [\[case1brn\]](#case1brn){reference-type="eqref" reference="case1brn"}. The size of the set above is now bounded by a sum over pairwise $K^{-1}$-separated $n$-tuples $(\tau^1,\ldots,\tau^n)$ which form $|\text{Br}_{\alpha,\beta}^{K^3}|$ from [\[broad\]](#broad){reference-type="ref" reference="broad"}. Using Proposition [Proposition 32](#mainprop){reference-type="ref" reference="mainprop"} to bound $|\text{Br}_{\alpha,\beta}^{K^3}|$, the summary of the inequalities from this case is $$\int_{B_R}|f|^p\lesssim(\log R)^2 K^{53}\big[R^{10 \varepsilon N_0}A^{\varepsilon^{-1}} + R^{4\varepsilon^2}A^{\varepsilon^{-1}} \text{T}_{n,D}^w(R^\varepsilon)^{\varepsilon^{-1}-N_0}\big]\int|\sum_\theta|f_\theta|^2*\omega_{\theta,D}|^{p/2},$$ where we used that $\text{T}_n(r)\lesssim_D\text{T}_{n,D}^w(r)$, which follows from the locally constant property. ◻ With Lemma [Lemma 40](#multi-scale'){reference-type="ref" reference="multi-scale'"} in hand, we may now prove Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"}. *Proof of Proposition [Proposition 34](#S1bd){reference-type="ref" reference="S1bd"}.* Let $\eta$ be the infimum of the set $$\mathcal{S}=\{\delta\ge 0:\sup_{R\ge 1}\frac{\text{T}_{n,D}^w(R)}{R^\delta}<\infty\}.$$ Suppose that $\eta>0$. Let $\varepsilon_1$, $\eta>\varepsilon_1>0$, be a parameter we will specify later. By Lemma [Lemma 40](#multi-scale'){reference-type="ref" reference="multi-scale'"}, we have $$\begin{aligned} \sup_{R\ge 1}\frac{\text{T}_{n,D}^w(R)}{R^{\eta-\varepsilon_1}}&\lesssim_\varepsilon\sup_{R\ge 1}\frac{1}{R^{\eta-\varepsilon_1}}\Big[(\log R)^2\left( K^{53}\big[R^{10 \varepsilon N_0} + R^{4\varepsilon^2+10\varepsilon} \text{T}_{n,D}^w(R^\varepsilon)^{\varepsilon^{-1}-N_0}\big]+\text{T}_{n,D}^w(R/K^n)\right) \Big] \end{aligned}$$ where we are free to choose $\varepsilon>0$, $1\le N_0\le\varepsilon^{-1}$, and $1\le K^3\le R$. Continue to bound the expression on the right hand side by $$\begin{aligned} \sup_{R\ge 1}(\log R)^2\Big( K^{53}\frac{R^{10 \varepsilon N_0}}{R^{\eta-\varepsilon_1}} +& K^{53}\frac{R^{4\varepsilon^2+10\varepsilon}}{R^{N_0\varepsilon(\eta+\varepsilon_1)-2\varepsilon_1}} \big[\frac{\text{T}_{n,D}^w(R^\varepsilon)}{R^{\varepsilon(\eta+\varepsilon_1)}}\big]^{\varepsilon^{-1}-N_0}+\frac{1}{K^{3(\eta+\varepsilon_1)}R^{-2\varepsilon_1}}\frac{{\text{T}_{n,D}^w(R/K^n)}}{(R/K^n)^{\eta+\varepsilon_1}} \Big).\end{aligned}$$ By definition of $\eta$, $$\sup_{R\ge 1}\frac{\text{T}_{n,D}^w(R^\varepsilon)}{R^{\varepsilon(\eta+\varepsilon_1)}}+\sup_{R\ge 1}\frac{{\text{T}_{n,D}^w(R/K^n)}}{(R/K^n)^{\eta+\varepsilon_1}} <\infty,$$ so it suffices to check that $$\begin{aligned} \sup_{R\ge 1}(\log R)^2\Big( K^{53}\frac{R^{10 \varepsilon N_0}}{R^{\eta-\varepsilon_1}} +& K^{53}\frac{R^{4\varepsilon^2+10\varepsilon}}{R^{N_0\varepsilon(\eta+\varepsilon_1)-2\varepsilon_1}} +\frac{1}{K^{n(\eta+\varepsilon_1)}R^{-2\varepsilon_1}} \Big)<\infty\end{aligned}$$ to obtain a contradiction. From here, choose $N_0=\varepsilon^{-1/2}$ and $K=R^{\varepsilon_1}$ so that it suffices to check $$\begin{aligned} \sup_{R\ge 1}(\log R)^2\Big( \frac{1}{R^{\eta-10\varepsilon^{1/2}-54\varepsilon_1}} +& \frac{1}{R^{\varepsilon^{1/2}\eta-4\varepsilon^2-10\varepsilon-55\varepsilon_1}} +\frac{1}{R^{(n-2)\varepsilon_1}} \Big)<\infty .\end{aligned}$$ This is clearly true if we choose $\varepsilon>0$ to satisfy $\min(\eta-10\varepsilon^{1/2},\eta-4\varepsilon^{3/2}-10\varepsilon^{1/2})>\eta/2$ and then choose $\varepsilon_1$ to be smaller than $\frac{1}{55}\varepsilon^{1/2}\eta/4$. Our reasoning has shown that $\eta-\varepsilon_1\in\mathcal{S}$, which is a contradiction. Conclude that $\eta=0$, as desired. ◻ ## Proof of Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} [\[S2\]]{#S2 label="S2"} {#proof-of-proposition-momcurveinduct-s2} We will show that $\text{T}_{n,D}^w(R)\lesssim_\varepsilon R^\varepsilon$ implies $\text{T}_n(R)\lesssim_\varepsilon R^\varepsilon$, which proves Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"}. See Definitions [Definition 2](#TnRw){reference-type="ref" reference="TnRw"} and [Definition 3](#TnR){reference-type="ref" reference="TnR"} at the beginning of [\[tools\]](#tools){reference-type="ref" reference="tools"} for the definitions of $\text{T}_n(R)$ and $\text{T}_n^w(R)$. The following is a multi-scale inequality bounding $\text{T}_n(R)$ by $\text{T}_{n,D}^w(\cdot)$, and $\text{T}_n(\cdot)$ evaluated at parameters smaller than $R$. **Proposition 41**. *For $R\ge 10$ and $D\ge p_n^{n\varepsilon^{-n}}$, $$\emph{T}_n(R)\lesssim_\varepsilon R^\varepsilon\emph{T}_{n,D}^w(R^{\frac{1}{n}})\max_{1\le \lambda\le R^{1-\frac{1}{n}}}\emph{T}_n(\lambda).$$* Proposition [Proposition 2](#momcurveinduct){reference-type="ref" reference="momcurveinduct"} follows directly from Proposition [Proposition 41](#multi-scaleS2){reference-type="ref" reference="multi-scaleS2"} by the proof of Theorem 1 in [@maldagueM3]. The proof of Proposition [Proposition 41](#multi-scaleS2){reference-type="ref" reference="multi-scaleS2"} has the same overall structure as the proof of Proposition 6.8 in [@maldagueM3]. *Proof of Proposition [Proposition 41](#multi-scaleS2){reference-type="ref" reference="multi-scaleS2"}.* Let $2\le p\le p_n$. Let $f:\mathbb R^n\to\mathbb C$ be a Schwartz function with Fourier transform supported in $\mathcal{M}^n(R)$. By the defining inequality for $\text{T}_{n,D}^w(R^{\frac{1}{n}})$, $$\label{S1app} \int_{\mathbb R^n}|f|^p\le \text{T}_n^w(R^{\frac{1}{n}})\int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}|f_\tau|^2*\omega_{\tau,D}|^{p/2}.$$ We choose the scale $R^{\frac{1}{n}}$ because each $\omega_{\tau}$ is localized to an $R^{\frac{1}{n}^2}\times R^{2/n^2}\times\cdots\times R^{n/n^2}$ plank, which is contained in an $R^{\frac{1}{n}}$ ball. The square function we are aiming for, $\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}|f_\theta|^2$, is locally constant on $R^{\frac{1}{n}}$ balls, so we will be able to eliminate the weights and therefore obtain a bound for $\text{T}_n(R)$. The idea for going from the right hand side of [\[S1app\]](#S1app){reference-type="eqref" reference="S1app"} to our desired right hand side is to perform a simplified version of the proof of Proposition [Proposition 31](#algo){reference-type="ref" reference="algo"}. Begin with the assumption that $$\label{assS2} \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}|f_\tau|^2*\omega_{\tau,D}|^{p/2}\lesssim \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}|f_\tau|^2*\omega_{\tau,D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{>R^{-\frac{1}{n}}}|^{p/2}.$$ Indeed, if this does not hold, then $$\int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}|f_\tau|^2*\omega_{\tau,D}|^{p/2}\lesssim \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}|f_\tau|^2*\omega_{\tau,D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R^{-\frac{1}{n}}}|^{p/2}.$$ This is a termination criterion case since by local $L^2$-orthogonality, for each $\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})$, $$||f_\tau|^2*\omega_{\tau,D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R^{-\frac{1}{n}}}(x)|\lesssim \sum_{\theta\subset\tau}|f_\theta|^2*\omega_{\tau,D}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{R^{-\frac{1}{n}}}|(x).$$ Finally, simply note that $\omega_{\tau,D}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{R^{-\frac{1}{n}}}|\lesssim w_{R^{\frac{1}{n}}}$ and by Young's convolution inequality, $$\int_{\mathbb R^n}|\sum_\theta|f_\theta|^2*w_{R^{\frac{1}{n}}}|^{p/2}\lesssim \int_{\mathbb R^n}|\sum_\theta|f_\theta|^2|^{p/2}.$$ From here on, assume that [\[assS2\]](#assS2){reference-type="eqref" reference="assS2"} holds. Now we describe the simplified algorithm. Let $\delta>0$ be a constant that we specify later in the proof. At intermediate step $m$, we have the inequality $$\begin{aligned} \label{stepmS2} \text{(R.H.S. of \eqref{assS2})} \le (C_\delta R^{C\delta})^m s_m^{-C\varepsilon}\int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\tau'\subset\tau \\\tau'\in{\bf{S}}_n(s_m)}}|f_{\tau'}|^2*\omega_{\tau,p_n^{-m}D} \Big)^{\frac{p}{2}}\end{aligned}$$ in which $R^{-\frac{1}{n}}\le s_m\le R^{-\frac{1}{n^2}}$ and $s_m\le R^{-\varepsilon^n/n}s_{m-1}$. Note that [\[stepmS2\]](#stepmS2){reference-type="eqref" reference="stepmS2"} clearly holds with $m=0$ and $s_m=R^{-\frac{1}{n^2}}$. We will show that assuming [\[stepmS2\]](#stepmS2){reference-type="eqref" reference="stepmS2"} for $m-1$, either [\[stepmS2\]](#stepmS2){reference-type="eqref" reference="stepmS2"} holds for $m$ or the iteration terminates. Suppose that [\[stepmS2\]](#stepmS2){reference-type="eqref" reference="stepmS2"} holds with $m-1$, so $$\begin{aligned} (\text{L.H.S. of \eqref{assS2}})\le (C_\delta R^{C\delta})^{m-1}s_{m-1}^{-C\varepsilon} & \int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}}\sum_{\substack{\tau'\subset\tau\\\tau'\in{\bf{S}}_n(s_{m-1})}} |f_{\tau'}|^{2}*\omega_{\tau,p_n^{-(m-1)}D})^{\frac{p}{2}} .\end{aligned}$$ Apply Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"} to the integral, yielding $$\begin{aligned} \label{stephypS2} (\text{L.H.S. of \eqref{assS2}})&\le (C_\delta R^{2\delta})^{m-1}R^{\delta/n} s_{m-1}^{-C\varepsilon} \left[ \int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(R^{-\frac{\varepsilon^n}{n}}s_{m-1})}}|f_{\tau'}|^2*\omega_{\tau,p_n^{-m}D})^{\frac{p}{2}}\right.\\ & \left.+R^{C\varepsilon^n/n}\int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_2}*\omega_{\tau,p_n^{-(m-1)}D})^{\frac{p}{\tilde{p}_{2}}}\right] .\nonumber\end{aligned}$$ If the first term on the right hand side dominates, then we have produced the step $m$ inequality $$(\text{L.H.S. of \eqref{assS2}})\le (C_\delta R^{2\delta})^{m} s_m^{-\varepsilon}\int_{\mathbb R^n}(\sum_{\substack{\tau\in{\bf{S}}_n(R^{-\frac{1}{n}})}}\sum_{\substack{\tau'\subset\tau\\\tau'\in{\bf{S}}_n(R^{-\frac{\varepsilon^n}{n}}s_{m-1})}}|f_{\tau'}|^2*\omega_{\tau,p_n^{-m}D})^{\frac{p}{2}} .$$ If the second term dominates, we have $$(\text{L.H.S. of \eqref{assS2}})\le (C_\delta R^{2\delta})^{m-1}R^{\delta/n}s_{m-1}^{-C\varepsilon}R^{C\varepsilon^n/n}\int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau \\\tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_{2}}*\omega_{\tau,p_n^{-(m-1)}D})^{\frac{p}{\tilde{p}_{2}}} .$$ Note that we may again apply Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"}, with $\varepsilon^n$ replaced by $\varepsilon^{n-1}$, to the integral on the right hand side. We iterate this process until we have one of two outcomes. The first outcome is that for some $j<n$, $$(\text{L.H.S. of \eqref{assS2}})\le (C_\delta R^{2\delta})^{m-1}R^{j\delta/n}s_{m-1}^{-C\varepsilon}R^{C\varepsilon^{n-j}/n} \int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m})}}|f_{\tau'}|^2*\omega_{\tau,p_n^{-m}D})^{\frac{p}{2}}$$ where $R^{C\varepsilon^{n-j}/n}=(s_{m-1}/s_m)^{C\varepsilon}$. Step $m$ is proved in this case. The second outcome is that $$(\text{L.H.S. of \eqref{assS2}})\le (C_\delta R^{2\delta})^{m-1}R^{j\delta/n}s_{m-1}^{-C\varepsilon}R^{C\varepsilon^{n+2-l}} \int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,D})^{\frac{p}{\tilde{p}_{l}}}$$ with $1\le \frac{p}{\tilde{p}_l}\le 2$ and $2\le \frac{p}{\tilde{p}_{l-1}}$, which means that $l< n$. As in the proof of Lemma [Lemma 30](#algo2){reference-type="ref" reference="algo2"}, use bump functions $\eta_{\tau',\sigma}^{n-1}$ and suppose that $$\begin{aligned} \int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}&\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}D})^{\frac{p}{\tilde{p}_{l}}}\lesssim (\log R)^C\\ &\times \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}. \end{aligned}$$ If $\sigma\le R^{-\varepsilon^{n+1-l}}s_{m-1}$, then by Lemma [Lemma 29](#algo1){reference-type="ref" reference="algo1"}, $$\begin{aligned} \int_{\mathbb R^n}|&\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\ \tau'\in{\bf{S}}_n(s_{m-1})}}|f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,p_n^{-(m-1)}D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}} \\ &\le C_\delta R^{\delta/n} \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau \\\tau'\in{\bf{S}}_n(s_{m-1})}}|\sum_{\substack{\tau''\subset\tau'\\ \tau''\in{\bf{S}}_n(R^{-\frac{\varepsilon^{n+1-l}}{n}}s_{m-1})}} |f_{\tau''}|^2*\omega_{\tau,p_n^{-m}D}|^{\frac{\tilde{p}_{l}}{2}}*\tilde{W}_{\tau',\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}}. \end{aligned}$$ Apply Proposition [Lemma 16](#multilem3pf){reference-type="ref" reference="multilem3pf"} to get rid of $\tilde{W}_{\tau,\sigma}^{n-1}$, concluding step $m$. The other case is that $\sigma>R^{-\varepsilon^{n+1-l}}s_{m-1}$. Using the boundedness of $\mathbb C^n_{n-2}(\cdot)$ and Cauchy-Schwarz, we have $$\begin{aligned} \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\\tau'\in{\bf{S}}_n(s_{m-1})}}&|f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau',\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}} \\ &\lesssim_\varepsilon R^{C\varepsilon^2}\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\tau'\subset\tau \\ \tau'\in{\bf{S}}_n(s_{m-1})}} ||f_{\tau'}|^{\tilde{p}_{l}}*\omega_{\tau,D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\tau,\sigma}^{n-1}|^{\frac{p}{\tilde{p}_{l}}} .\end{aligned}$$ By Young's convolution inequality, the right hand side is bounded by $$C_\varepsilon R^{C\varepsilon^2}\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})}\sum_{\substack{\tau'\subset\tau\\\tau'\in{\bf{S}}_n(s_{m-1})}} |f_{\tau'}|^{p} ,$$ which, by the definition of $\text{T}_n(\cdot)$ and rescaling, is itself bounded by $$C_\varepsilon R^{C\varepsilon^2}\text{T}_n(s_{m-1}^nR)\int_{\mathbb R^n}\sum_{\tau\in{\bf{S}}_n(s_{m-1})} (\sum_{\substack{\theta\subset\tau\\\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2 )^{\frac{p}{2}} .$$ By Lemma [Lemma 22](#pruneprop){reference-type="ref" reference="pruneprop"}, $\|\cdot\|_{\ell^{p/2}}\le \|\cdot\|_{\ell^1}$, this expression is bounded by $$\begin{aligned} C_\varepsilon R^{C\varepsilon^2}\text{T}_n(s_{m-1}^n R)\int_{\mathbb R^n}(\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})} |f_{\theta}|^2)^{\frac{p}{2}}, \end{aligned}$$ which terminates the algorithm. This concludes the justification of step $m$. There are two outcomes of the algorithm. The first, as we have just seen, is that $$\text{(R.H.S. of \eqref{assS2})} \le (C_\delta R^{2\delta})^{m}s_{m-1}^{-\varepsilon} C_\varepsilon R^{C\varepsilon^2}\text{T}_n(s_{m-1}^nR) \int_{\mathbb R^n}(\sum_{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})} |f_{\theta}|^2)^{\frac{p}{2}}$$ in which $mn\varepsilon^{-n}$ and $s_{m-1}^{-\varepsilon}\le R^\varepsilon$. Choose $\delta<\varepsilon^n$ guarantees an upper bound of the form $C_\varepsilon R^\varepsilon\text{T}_n(s_{m-1}^nR)$. The second outcomes is that in fewer than $n\varepsilon^{-n}$ many steps, the algorithm terminates with the inequality $$\text{(R.H.S. of \eqref{assS2})} \le (C_\delta R^{C\delta})^m R^\varepsilon\int_{\mathbb R^n}\Big(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\theta\subset\tau \\\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2*\omega_{\tau,p_n^{-n\varepsilon^{-n}}D} \Big)^{\frac{p}{2}}.$$ By the locally constant property, we have $$\int_{\mathbb R^n}(\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\theta\subset\tau \\\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2*\omega_{\tau,p_n^{-n\varepsilon^{-n}}D} \Big)^{\frac{p}{2}}\lesssim \int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\theta\subset\tau \\\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2*\omega_{\tau,p_n^{-n\varepsilon^{-n}}D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R^{-\frac{1}{n}}} |^{\frac{p}{2}}.$$ As before, $\omega_{\tau,p_n^{-n\varepsilon^{-n}}D}*|% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R^{-\frac{1}{n}}}|\lesssim w_{R^{-\frac{1}{n}}}$ for each $\tau$. Therefore, by Young's convolution inequality, $$\int_{\mathbb R^n}|\sum_{\tau\in{\bf{S}}_n(R^{-\frac{1}{n^2}})} \sum_{\substack{\theta\subset\tau \\\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2*\omega_{\tau,p_n^{-n\varepsilon^{-n}}D}*% \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{\ensuremath{\eta}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{\eta}{\scalebox{-1}{\tmpbox}}% _{\le R^{-\frac{1}{n}}} |^{\frac{p}{2}}\lesssim \int_{\mathbb R^n}| \sum_{\substack{\theta\in{\bf{S}}_n(R^{-\frac{1}{n}})}}|f_{\theta}|^2|^{\frac{p}{2}},$$ which finishes the proof. ◻

arxiv_math

--- abstract: | This article is intended towards the study of the bidual of generalized group algebra $L^1(G,\mathcal{A})$ equipped with two Arens product, where $G$ is any locally compact group and $\mathcal{A}$ is a Banach algebra. We show that the left topological center of $(L^1(G)\hat\otimes\mathcal{A})^{**}$ is a Banach $L^1(G)$-module if $G$ is abelian. Further it also holds permanance property with respect to the unitization of $\mathcal{A}$. We then use this fact to extend the remarkable result of A.M Lau and V. Losert[@Lau-losert], about the topological center of $L^1(G)^{**}$ being just $L^1(G)$, to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of $L^1(G,\mathcal{A})$ for non-reflexive Banach algebras $\mathcal{A}$ and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when $\mathcal{A}$ holds the Radon-Nikodym property and weak sequential completeness. author: - Lav Kumar Singh title: Geometry of Banach algebra $\mathcal{A}$ and the Bidual of $L^1(G)\hat{\otimes}\mathcal{A}$ --- [^1] # Introduction Arens regularity is an important tool for classifying Banach algebras. R. Arens in [@Arens] defined two products $% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}%$ and $\diamond$ on the bidual of a Banach algebra $\mathcal{A}$. When these two products are same, the Banach algebra $\mathcal{A}$ is said to be Arens regular. He went on to prove that the algebra $\ell^1(\mathbb Z)$ equipped with point-wise multiplication is Arens regular. Further it can easily be seen that every reflexive Banach algebra is Arens regular. Remarkably, every $C^*$-algebra is Arens regular ([@Palacios]). On the other end, for any locally compact infinite group $G$, the Banach algebra $L^1(G)$ is known to be Arens irregular. Ülger in [@Ulger] showed that the projective tensor product of Arens regular Banach algebras may not be regular. but if $\mathcal{A}\hat\otimes\mathcal B$ is Arens regular then both $\mathcal{A}$ and $\mathcal B$ must be Arens regular. He also explored the connections between geometric properties (like RNP, WCG, WSC etc) of the underlying Banach space of an algebra $\mathcal{A}$ and its Arens regularity in his subsequent articles. A.M Lau and V. Losert showed in [@Lau-losert] that the left topological center of $L^1(G)^{**}$ is exactly $L^1(G)$ for any locally compact group $G$.\ We consider the vector valued Banach algebra $L^1(G,\mathcal{A})=L^1(G)\hat\otimes \mathcal{A}$, where $\mathcal{A}$ is a Banach algebra and $G$ is a locally compact group. The natural question- whether the topological center of $L^1(G,\mathcal{A})^{**}$ is $L^1(G,\mathcal{A})$ remains un-answered even for the case when $\mathcal{A}$ is finite dimensional Banach algebra, due to the lack of structure theorem for finite dimensional Banach algberas. We explore this aspect in this article and settle the proof that for a compact abelian group $G$ and reflexive Banach algebra $\mathcal{A}$, the Banach algebra $L^1(G,\mathcal{A})$ is Strongly Arens irregular. To do so, we first show that the topological center $Z(L^1(G,\mathcal{A})^{**})$ is an $L^1(G)$-module and holds permanance property with respect to unitization. This will enable us to assume that $\mathcal{A}$ is a unital algebra and avoid approximate idenitity arguments at each step. Many of the techniques in the proof are motivated from Lau and Losert's methods in [@Lau-losert], with significant generalization to the vector valued setting. In the last section we give a nice application of Cohen's factorization theorem to give a partial characterization of elements in pseudo-center(defined in section 2). Finally we end the article with a weak sequential completeness(WSC) like property for Banach space $\mathcal{A}$ which have RNP and are WSC. For studying the vector valued functions spaces, it is inevitable to encounter Bochner integrals and vector measures and hence we revisit the basics in the next section. # Preliminaries to Vector valued functions and integration Let $(S,\mathscr{A},\mu)$ be a measure space and $X$ be a Banach space. A function $h:S\to X$ is said to be *$\mu$-simple* if $h=\sum_{i=1}^n\chi_{A_i}x_i$, where $A_i\in \mathscr{A}$ for each $i$ such that $\mu(A_i)<\infty$ and $x_1,x_2,..,x_n\in X$. Now if a function $f:S\to X$ is said to be *$\mu$-strongly measurable* if there exists a sequence of $\mu$-simple functions converging pointwise to $f$ almost everywhere. The Pettis measurability theorem states that $f$ is $\mu$-strongly measurable if and only if it is $\mu$-Borel measurable and $\mu$-essentially separably valued (see [@Ryan] and [@Tuomas] for more details). For each $\mu$-simple function $h=\sum_{i=1}^r\chi_{A_i}x_i$, we define $\int hd\mu=\sum_{i=1}^r\mu(A_i)x_i$. A $\mu$-strongly measurable function $f:S\to X$ is said to be *Bochner integrable* with respect to $\mu$ if there exists a sequence of $\mu$-simple functions $f_n:S\to X$ such that $\lim_n \int||f(s)-f_n(s)||d\mu \to 0$. And the Bochner integral is given by $$\int fd\mu=\lim_n \int f_nd\mu$$ For $1\leq p<\infty$ we denote by $L^p(S,X)$ the collection of equivalance classes of $\mu$-strongly measurable functions $f$ which are $\mu$-almost everywhere equal and $\int||f(s)||^pd\mu<\infty$. $L^p(S,X)$ becomes a Banach space with respect to the norm $||f||_{L^p(S,X)}=\left(\int ||f(s)||^pd\mu\right)^{1/p}$. The space $L^\infty(S,X)$ is the collection of equivalence classes of functions which are essentially bounded in the natural sense. $L^\infty(S,X)$ is also a Banach space with respect to the norm $$||f||_{L^\infty(S,X)}=\inf\{M~:~\mu\left(\{s:~||f(s)||\leq M~\text{almost everywhere}\}\right)=0\}$$ **Definition 1**. A function $F:\mathscr{A}\to X$ is called an *$X$-valued measure* on $S,\mathscr{A}$ if it is countably additive in the sense that for any disjoint sequence $\{A_i\}$ of sets in $\mathscr A$, we have $F(\cup A_i)=\sum F(A_i)$, where convergence on the right hand side is given in norm. The *variation* of an $X$-valued measure is the map $||F||:S\to [0,\infty]$ given by $$||F||(A)=sup\Big\{\sum_{i=1}^n||F(A_i)||~:~A_i\in \mathscr A, \cup_{i=1}^n A_i=A~\text{and~} A_i\cap A_j=\phi \text{~for~}i\neq j, n\in \mathbb N\Big\}$$ We say that $F$ has bounded variation if $||F||(S)<\infty$. Further, $F$ is said to be absolutely continuous with respect to $\mu$ if for any $A\in \mathscr{A}$ with $\mu(A)=0$, we have $F(A)=0$. We denote the Banach space of all $X$-valued measures on $(S,\mathscr A )$ of bounded variation by $M(S,X)$ with respect to the variation norm $||F||=||F||(S)$. Further, $B(S,X)$ denote the closed subspace of $M(S,X)$ consisting of $X$-valued measures of bounded variation which are aboslutely coninuous with respect to $\mu$. A $X$-valued measure $F$ is said to be *regular* if $\varphi\mu$ is a regular measure for each $\phi\in X^*$. Equivalently for every set $E\in \mathscr{A}$ and $\epsilon>0$, there exists closed set $A\in\mathscr A$ contained in $E$ and an open set $B\in\mathscr A$ containing $E$ such that $||F(E\setminus A)||<\epsilon$ and $||F(B\setminus E)||<\epsilon$. We denote the Banach subspace of all regular measures in $M(S,X)$ by $M_r(S,X)$. Clearly $B(S,X)$ is a subspace of $M_r(S,X)$ if $X$ has RNP. **Theorem 1**. [@Ryan Section 5.3][\[RRT1\]]{#RRT1 label="RRT1"} If $(S,\mathscr B(S),\mu)$ is a compact Hausdorff measure space and $X$ is a Banach space then, $C(S,X)^*$ can be isometrically identified with $M_r(S,X^*)$ of all regular measures of bounded variation on Borel susbsets of $S$. *Proof.* An element $\phi$ of $C(S,X)^*$ gives rise to a family of $X^*$-valued measures on $S$ in the following way. Fixing $\xi \in X$, one can define a linear functional $L_{\phi,\xi}$ on $C(S)$ by sending the function $f$ on $S$, to the value of $\phi$ on the function $S \to X$ given by $s \mapsto f(x) \xi$. $$L_{\phi,\xi}(f) = \phi\left(f\otimes \xi\right).$$ From the usual Riesz theorem, there is then a measure $\mu_{\phi, \xi}$ defined on the Borel subsets of $S$ satisfying $$L_{\xi,\phi}(f) = \int_S f \, d\mu_{\phi, \xi}.$$ Corresponding to each pair $\phi\in C(S,X)$ and $\xi\in X$, we have obtained a measure on $S$.. Now define a map $\mu_{\phi}$ from the Borel subsets of $S$ to $X^*$ as follows: for any Borel subset $E$ of $S$, define $m_{\phi}(E)$ to be the linear functional on $X$ given by $$\mu_{\phi}(E)(\xi) = \int_E \, d\mu_{\phi, \xi}.$$ The map $\mu_{\phi}$ is regular $X^*$-valued measure on $S$. Since the functions of the form $x \mapsto f(x) \xi$, with $f \in C(S)$ and $\xi \in X$, are dense in $C(S,X)$, it is easy to show that $\phi$ is uniquely determined by $m_{\phi}$. Conversely, starting with a $X^*$-valued regular measure, one can show that it must be $m_{\phi}$ for some $\phi$ in $C(X,Y)^*$. Thus, $\phi\mapsto m_\phi$ is an isometric isomorphism with respect to the variation norm. ◻ Notice that for any $\phi\in L^1(S,X)$, the function $F:\mathscr{A}\to X$ defined as $$F(A)=\int_A\phi(s)d\mu(s)$$ for each $A\in \mathscr{A}$ is a well defined vector measure of bounded variation which is absolutely continuous with respect to $\mu$ and $||F||(A)=\int_A||\phi(s)||d\mu(s)$. It turns out that if $X$ has a nice geometric property called Radon-Nikodym Property(RNP), then all vector measures of bounded variation that is absolutely continuous with respect to $\mu$ arises in the same fashion. **Definition 2**. A Banach space $X$ is said to have the Radon-Nikodym Property(RNP) with respect to a measure space $(S,\mathscr A,\mu)$, if for every $X$-valued measure of $F$ of bounded variation on $(S,\mathscr A)$ that is absolutely continuous with respect to $\mu$, there exists a function $\phi\in L^1(S,X)$ such that $$F(A)=\int_A\phi d\mu,~~~~~ A\in \mathscr A$$ Thus, if $X$ has RNP then $B(S,X)$ can be isometrically identified with $L^1(G,X)$. **Definition 3**. A bounded operator $T:L^1(S)\to X$ is said to representable if there exists a function $\phi\in L^\infty(X,S)$ such that $$Tf=\int_S f\phi d\mu,~~~~f\in L^1(S,X)$$ **Theorem 2**. [@Tuomas Th. 1.3.10][\[RNP1\]]{#RNP1 label="RNP1"} Let $(S,\mathscr A,\mu)$ be a $\sigma$-finite measure space and $X$ be a Banach space, and let $1\leq p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. The following assertions are equivalenet. 1. $X^*$ has RNP with respect to $(S,\mathscr A,\mu)$. 2. The mapping $g\mapsto\phi_g$ establishes an isometric isomorphism of Banach space $$L_q(S,X^*)\simeq(L^p(S,X))^*$$ where $\left<\phi_g,f\right>=\int\left<g(s),f(s)\right>d\mu(s)$ for each $f\in L^p(S,X)$. **Remark 1**. *If $X^*$ does not have the RNP w.r.t $(S,\mathscr{A},\mu)$, then $g\to \phi_g$ is still an isometry onto a norming subspace of $L^p(S,X)^*$, but it may not be surjective in general.* **Theorem 4**. [@Tuomas Th. 1.3.15][\[representable\]]{#representable label="representable"} Let $(S,\mathscr A,\mu)$ be a $\sigma$-finite measure space. For a Banach space $X$, the following assertions are equivalent. 1. $X$ has RNP with respect to $(S,\mathscr A,\mu)$. 2. Every bounded linear operator $T:L^1(S)\to X$ is representable. It is a standard fact that if $X$ is reflexive or is a separable dual space, then $X$ posess RNP with respect to any $\sigma$-finite measure space. The fourier algebra $A(G)$ has RNP iff $G$ is compact. The algebra of Trace class operators on a Hilbert space Space has the RNP. Dual of a $C^*$-algebra $\mathcal{A}$ have RNP iff and only if $\mathcal{A}$ is Scattered. Pre-dual of a Von-neumann algebra has RNP if and only if it is a direct sum of type-$I$ factors (see [@Chu]). Space $c_0,\ell^\infty, C[0,1]$ and $L^1([0,1])$ does not have RNP. Proof of these facts can be found in any standard text on geometry of Banach spaces. The RNP defined above is dependent is relative to a measure space. Interestingly RNP is more well behaved with respect to $\sigma$-finite measure spaces. **Theorem 3**. [@Tuomas Th. 1.3.26] For a Banach space $X$ the following are equivalent. 1. $X$ has RNP with respect to $[0,1]$. 2. $X$ has RNP with respect to any $\sigma$-finite measure space. In view of the above theorem, we will say that $X$ has RNP without refererring to a measure space whenever the measure space involved is $\sigma$-finite. **Definition 4**. A Banach space $X$ is said to be *Weakly Sequentially Complete*(WSC), if every weakly Cauchy sequence $\{\phi_n\}_{n=1}^\infty$ of functions in $X$ converges weakly to some element in $X$. It is a well known and easy to prove fact that $L^1(S)$ is WSC for any measure space $S$. All reflexive spaces are WSC by default. If $X$ is WSC and $\mu$ is a finite measure, then $L^1(S,X)$ is also WSC [@Talagrand Th. 11]. The fourier algebra $A(G)$ is WSC for any locally compact Hausdorff group. A $C^*$ algebra is WSC if and only if is finite dimensional. Now, if $\mathcal{A}$ is a Banach algebra and $G$ is any locally compact Hausdorff group, then $L^1(G,\mathcal{A})$ can be given an algebra structure through convolution $f\ast g(t)=\int_G f(s)g(s^{-1}t)d\mu(s)$, with respect to which $L^1(G,\mathcal{A})$ becomes a Banach algebra. This generalized group algebra can be identified with the projective tensor product $L^1(G)\hat{\otimes} \mathcal{A}$ (see [@Kaniuth] for the proof). We shall make use of the Cohen's factorization theorem as stated below, in coming sections. **Theorem 5** (Cohen's Factorization). Let $\mathscr{A}$ be a Banach algebra and $\mathscr{K}$ be a left Banach $\mathscr{A}$-module and suppose that $\mathscr{A}$ has a left approximate identity $\{e_\beta\}_{\beta\in B}$ that is also a left approximate identity for $\mathscr{K}$ bounded by some constant $\delta\geq 1$. Then for any $x_o\in \mathscr{K}$ and any $\epsilon>0$, there exists an $a\in \mathscr A$ and $x\in \mathscr{K}$ such that $x_o=ax$, $||a||\leq \delta$ and $||x-x_o||\leq \epsilon$. # Topological Center of $L^1(G,\mathcal{A})^{**}$ as $L^1(G)$-module. Given any Banach algebra $A$, due to Hahn-Banach, we have an isometric embedding $J:A\to A^{**}$ as Banach spaces. For each $a\in A$ and $f\in A^*$, we define two functionals $f_a,{_a}f\in A^*$ as $$f_a(b)=f(ab)~,~{_a}f(b)=f(ba)~~~\text{for all } b\in A$$ Further, for each $f\in A^*$ and $m\in A^{**}$ we define $f_m, {_m}f\in A^*$ as $$f_m(a)=m({_a}f)~,~{_m}f(a)=m(f_a)~\forall a\in A$$ Now, the two Arens product on $A^**$ are defined as $$m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(f)=m({_n}f)~,~m\diamond n(f)=n(f_m)$$ for each $m,n\in A^{**}$ and $f\in A^*$. It is a trivial fact that $A^{**}$ becomes a Banach algebra with respect both these Arens product $% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}%$ and $\diamond$ and the two Arens prodcuct agree on $J(A)$, and the embedding $J$ becomes an algebra Homomorphism with respect to both these products. We denote by $Z(A^{**})$ the (left) topological center- $$\begin{aligned} Z(A^{**})&=\{m\in A^{**}~:~m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n=m\diamond n~\forall n\in A^{**}\}\\&=\{m\in L^1(G,\mathcal{A})^{**}~:~n\mapsto m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n~\text{is~} w^\ast\text~\text{continuous}\}\nonumber\end{aligned}$$ Clearly, $Z(A^{**})$ is a Banach algebra itself naturally. For Banach algebras $\mathcal{A}$ and $\mathcal B$, their direct sum $\mathcal{A}\oplus \mathcal B$ is a Banach algebra when equipped with co-ordinate wise multiplication $(a,b)\cdot (c,d)=(ac,bd)$ and norm $||(a,b)||=||a||+||b||$. Further we have identification for dual Banach space $(A\oplus B)^*=A^*\oplus_\infty B^*$ and $$((\mathcal{A}\oplus \mathcal B)^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% )=(\mathcal{A}^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% )\oplus (\mathcal B^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% )$$ $$((\mathcal{A}\oplus \mathcal B)^{**},\diamond)=(\mathcal{A}^{**},\diamond)\oplus (\mathcal B^{**},\diamond)$$ The following permanence property is elementary in nature but will help us significantly in calculating the topological center. **Theorem 6**. Let $\mathcal{A}$ and $\mathcal B$ be Banach algebras. Then $A\oplus B$ is SAI if and only if both $\mathcal{A}$ and $\mathcal B$ are SAI. Recall that the minimal unitization of a non-unital Banach algebra $\mathcal{A}$ is $\tilde{\mathcal{A}}=\mathcal{A}\oplus \mathbb C$ equipped with norm $||(a,\alpha)||=||a||+|\alpha|$ and multiplication $(a,\alpha)(b,\beta)=(ab+\beta a+\alpha b,\alpha\beta)$. If Banach algebra $\mathcal{A}$ and $\mathcal B$ are such that $\mathcal{A}$ is a right Banach $\mathcal B$-module, i.e $\exists$ continuous bilinear map $\mathfrak m:\mathcal{A}\times \mathcal B\to \mathcal{A}$ of norm one giving an algebraic module structure, then we form a direct sum Banach algebra $\mathcal{A}\tilde{\oplus}\mathcal B$ with norm $||(a,b)||=||a||+||b||$ and multiplication $(a,b)(c,d)=(ac+bc+ad,bd)$. We can consider $L^1(G,\mathcal{A})$ as right $L^1(G)$-module naturally by $\mathfrak{m}:L^1(G,\mathcal{A})\times L^1(G)\to L^1(G,\mathcal{A})$ defined as $\mathfrak{m}(\phi,F)=\phi\ast F$. **Theorem 7**. The map $\theta:L^1(G,\tilde{\mathcal{A}})\to L^1(G,\mathcal{A})\tilde\oplus L^1(G)$ is an isometric isomorphism of algebras, where $\theta$ is defined as $\theta(\phi)=(\pi_1\phi,\pi_2\phi)$ for each $\phi\in L^1(G,\mathcal{A})$. *Proof.* Notice that $$\begin{aligned} ||\theta(\phi)||&=||\pi_1\phi||+||\pi_2\phi||\\&=\int||\pi_2\phi(t)||dt+\int ||\pi_2\phi(t)||dt\\&=\int||\phi(t)||dt\\&=||\phi||_{L^1(G,\tilde{\mathcal{A}})} \end{aligned}$$ i.e $\theta$ is isometry. $\theta$ is surjective can be seen easily. Further, for any two $\phi_1,\phi_2\in L^1(G,\tilde{\mathcal{A}})$, notice that $$\begin{aligned} \phi_1\ast\phi_2(t)&=\int\phi_1(s)\phi_2(s^{-1}t)dt\\&=\int\big(\pi_1\phi_1(s),\pi_2\phi_1(s)\big)\big(\pi_1\phi_2(s^{-1}t),\pi_2\phi_2(s^{-1}t)\big)dt\\&=\int \big(\pi_1\phi_1(s)\pi_1\phi_2(s^{-1}t)+\pi_1\phi_1(s)\pi_2\phi_2(s^{-1}t)+\pi_2\phi_1(s)\pi_1\phi_2(s^{-1}t),\pi_2\phi_1(s)\pi_2\phi_2(s^{-1}t)\big)dt\\&=\big(\pi_1\phi_1\ast\pi_1\phi_2(t)+\pi_1\phi_1\ast\pi_2\phi_2(t)+\pi_2\phi_1\ast\pi_1\phi_2(t),\pi_2\phi_1\ast\pi_2\phi_2(t)\big)\end{aligned}$$ Thus, we can see that $\theta(\phi_1\ast\phi_2)=\theta(\phi_1)\theta(\phi_2)$. Hence, $\theta$ is an algebra isomorphism. ◻ **Theorem 8**. If $\mathcal{A}$ is a Banach algebra and $G$ is a locally compact abelian group then $Z(L^1(G,\mathcal{A})^{**})$ is an $L^1(G)$-module. *Proof.* Since $G$ is abelian, the algebra $L^1(G)$ is commutative. Consider the following adjoints of biliniar map $\mathfrak{m}$, where $\mathfrak{m}^*(f,\phi)(F)=\left<f,\mathfrak{m}(\phi,F)\right>$ $$\begin{aligned} \mathfrak{m}&:L^1(G,\mathcal{A})\times L^1(G)\to L^1(G,\mathcal{A})\\ \mathfrak{m}^*&:L^1(G,\mathcal{A})^*\times L^1(G,\mathcal{A})\to L^1(G)^*\\ \mathfrak{m}^{**}&:L^1(G)^{**}\times L^1(G,\mathcal{A})^*\to L^1(G,\mathcal{A})^*\\ \mathfrak{m}^{***}&:L^1(G,\mathcal{A})^{**}\times L^1(G)^{**}\to L^1(G,\mathcal{A})^{**} \end{aligned}$$ Let $m\in Z( L^1(G,\mathcal{A})^{**})$, $n\in L^1(G,\mathcal{A})^{**}$, $F\in L^1(G)$ and $f\in L^1(G,\mathcal{A})^*$. Further let $\{m_\alpha\}_{\alpha\in \wedge_1}$ and $\{n_\beta\}_{\beta\in \wedge_2}$ be nets in $L^1(G,\mathcal{A})$ converging to $m$ and $n$ respectively in the $w^\ast$-topology of $L^1(G,\mathcal{A})^{**}$.Then $$\begin{aligned} \mathfrak{m}^{***}(m,F)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(f)&=\left<m,(\mathfrak{m}^{**}(F,{_n}f))\right>\nonumber\\&=\lim_\alpha \left<\mathfrak{m}^{**}(F,{_n}f),m_\alpha\right>\nonumber\\&=\lim_\alpha \left<\mathfrak{m}^{*}({_n}f,m_\alpha),F\right>\nonumber\\&=\lim_{\alpha}\left<{_n}f,m_\alpha\ast F\right>\nonumber\\&=\left<{_F}({_n}f),m\right> \end{aligned}$$ Now, $$\begin{aligned} \mathfrak{m}^{***}(m,F)\diamond n(f)&=\lim_\beta \mathfrak{m}^{***}(m,F)({_{n_\beta}}f)\nonumber\\&=\lim_\beta m(\mathfrak{m}^{**}(F,{_{n_\beta}}f))\nonumber\\&=\lim_\beta\lim_\alpha\left<\mathfrak{m}^{**}(F,{_{n_\beta}}f),m_\alpha\right>\nonumber\\&=\lim_\beta\lim_\alpha \left<F,\mathfrak{m}^*({_{n_\beta}f},m_\alpha)\right>\nonumber\\&=\lim_\beta\lim_\alpha\left<{_{n_\beta}}f,m_\alpha\ast F\right>\nonumber\\&=\lim_\beta\left<{_F}({_{n_\beta}}f),m\right>\nonumber\\&=\lim_\beta \left<{_{n_\beta}}({_F}f),m\right>&&(\because n_\beta\ast F=F\ast n_\beta)\nonumber\\&=\lim_\beta m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n_\beta({_F}f)&&(\text{here we use ~}m\in Z(L^1(G,\mathcal{A})^{**}))\nonumber\\&=\left<{_n}({_F}f),m\right>\nonumber\\&=\left<{_F}({_n}f),m\right>&&(\because {_F}({_n}f)={_n}({_F}f) ) \end{aligned}$$ Thus we see that $\mathfrak{m}^{***}(m,F)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(f)=\mathfrak{m}^{***}(m,F)\diamond n(f)$ for all $f\in L^1(G,\mathcal{A})^*$. Hence, $\mathfrak{m}^{***}(m,F)\in Z(L^1(G,\mathcal{A})^{**})$. Further it is easy to verify that $\mathfrak{m}^{***}(m,F_1\ast F_2)=\mathfrak{m}^{***}(\mathfrak{m}^{***}(m,F_1),F_2)$ holds for each $m\in Z(L^1(G,\mathcal{A})^{**})$ and $F_1,F_2\in L^1(G)$. Thus, the restriction of $m^{***}$ to $Z(L^1(G,\mathcal{A})^{**})\times L^1(G)$ gives the required $L^1(G)$-module structure on $Z(L^1(G,\mathcal{A})^{**})$. ◻ **Theorem 9**. If $G$ is a locally compact abelian group and $\mathcal{A}$ is a Banach algebra, then $$Z(L^1(G,\tilde{\mathcal{A}})^{**})\cong Z(L^1(G,\mathcal{A})^{**})\tilde{\oplus} L^1(G)$$ is an isometric isomorphism of algebras. *Proof.* Consider the double adjoint map of $\theta$ $$\theta^{**}:L^1(G,\tilde\mathcal{A})^{**}\to L^1(G,\mathcal{A})^{**}\oplus L^1(G)^{**}$$ Clearly $\theta^{**}$ is an isometric isomorphism of Banach spaces, because $\theta$ is. We claim that the restriction of $\theta^{**}$ to $Z(L^1(G,\tilde\mathcal{A})^{**})$ gives us the isometric isomorphism of algebras $Z(L^1(G,\tilde\mathcal{A})^{**})$ and $Z(L^1(G,\mathcal{A})^{**})\tilde{\oplus} L^1(G)$. To see this, let $\tilde m\in Z(L^1(G,\mathcal{A})^{**})$, $n\in L^1(G,\mathcal{A})^{**}$ and $f\in L^1(G,\mathcal{A})^*$. It is easy to verify that $$\begin{aligned} \label{lav} {_{(n,0)}}(f,0)(\phi,\psi)&={_n}f(\phi)+{_n}f(\psi)&&\forall (\phi,\psi)\in L^1(G,\mathcal{A})\tilde\oplus L^1(G)\end{aligned}$$ Thus, $$\begin{aligned} \label{lav1} \tilde m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (n,0)(f,0)&=\tilde m \left({_{(n,0)}}(f,0)\right)\nonumber\\&=\lim_\alpha {_{(n,0)}}(f,0)(\pi_1\theta (\tilde m_\alpha),\pi_2\theta(\tilde m_\alpha))\nonumber\\&=\pi_1\theta^{**}(\tilde m)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(f)+{_n}f(\pi_2\theta^{**}(\tilde m))&&(\text{using~}\cref{lav})\end{aligned}$$ Similarly, one can prove that $$\begin{aligned} \label{lav2} \tilde m\diamond(n,0)(f,0)=\pi_1\theta^{**}(\tilde m)\diamond n(f)+{_n}f(\pi_2\theta^{**}(\tilde m))\end{aligned}$$ Since, $\tilde{m}\in Z(L^1(G,\tilde\mathcal{A})^{**})$, from [\[lav1\]](#lav1){reference-type="ref" reference="lav1"} and [\[lav2\]](#lav2){reference-type="ref" reference="lav2"} we deduce that $\pi_1\theta^{**}% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n=\pi_1\theta^{**}\diamond n$ for each $n\in L^1(G,\mathcal{A})^{**}$ i.e $\pi_1\theta^{**}(\tilde m)\in Z(L^1(G,\mathcal{A})^{**})$. Thus, $\pi_1\theta^{**}(\tilde m)\in Z(L^1(G,\mathcal{A})^{**})$. And similarly, one can show that $\pi_2\theta^{**}(\tilde m)\in Z(L^1(G)^{**})=L^1(G)$. Hence, $\theta^{**}_{\restriction Z(L^1(G,\mathcal{A})^{**})}$ is a well defined isometric linear map into $Z(L^1(G,\mathcal{A})^{**})\oplus L^1(G)$. Further, it can be easily verified that this map is surjective and is a Homomorphism with respect to $\tilde\oplus$ structure on the right side. ◻ **Corollary 2**. *For a locally compact abelian group $G$ and a Banach algebra $\mathcal{A}$, the group algebra $L^1(G,\mathcal{A})$ is SAI if and only if $L^1(G,\tilde{\mathcal{A}})$ is SAI.* **Definition 3**. *For $L^1(G,\mathcal{A})$, the (left) *topological pseudo-center* is defined as $$\begin{aligned} Z_s(L^1(G,\mathcal{A})^{**})&=&\{m\in L^1(G,\mathcal{A})^{**}~:~m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(f)=m\diamond n(f)~\forall f\in L^\infty(G,\mathcal{A}^*), n\in L^1(G,\mathcal{A})^{**}\}\\&=&\{m\in L^1(G,\mathcal{A})^{**}~:~n\mapsto m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n~\text{is~}\sigma(L^1(G,\mathcal{A})^{**},L^\infty(G,\mathcal{A}^*)) \text{-continuous}\}\end{aligned}$$* Clearly, $Z_s(L^1(G,\mathcal{A})^{**})=Z(L^1(G,\mathcal{A})^{**})$ if $\mathcal{A}^*$ has RNP. In general the pseudo center is a bigger class.\ We can consider $L^1(G,\mathcal{A})$ as a subalgebra of both $L^1(G,(\mathcal{A}^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% ))$ and $L^1(G,(\mathcal{A}^{**},\diamond))$ naturally since $\mathcal{A}$ is a subalgebra of both $(\mathcal{A}^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% )$ and $(\mathcal{A}^{**},\diamond)$. We notice the following consequence of $\mathcal{A}^*$ having RNP.. **Theorem 10**. Let $\mathcal{A}$ be a Banach algebra such that $\mathcal{A}^*$ has RNP, then $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% : L^1(G,(\mathcal{A}^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% ))\to \left(L^1(G,\mathcal{A})^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \right)$ and $\theta_\diamond: L^1(G,(\mathcal{A}^{**},\diamond))\to \left(L^1(G,\mathcal{A})^{**},\diamond\right)$ are isometric homomorphisms, where $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi)(f)=\theta_\diamond(\phi)(f)=\int_G \left<f(t),\phi(t)\right>d\mu(t)$ for all $\phi\in L^1(G,\mathcal{A}^{**})$ and $f\in L^\infty(G,\mathcal{A}^*)$. *Proof.* Since, $\mathcal{A}^*$ has RNP, we have the dual identification $L^1(G,\mathcal{A})^*=L^\infty(G,\mathcal{A}^*)$. Clearly, $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}%$ and $\theta_\diamond$ are well defined linear maps which are same at the Banach space level. Further, $||\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi)||\leq ||\phi||_{L^1(G,\mathcal{A}^{**})}$. Since, $\phi$ is Bochner integrable, there exists a sequence of $\mu$-simple functions $\{\psi_n\}$ in $L^1(G,\mathcal{A}^{**})$ which converges pointwise to $\phi$ almost everywhere such that $\lim_n\int_G||\psi_n(t)-\phi(t)||d\mu(t)\to 0$. For $\epsilon>0$, fix a natural number $N$ such that $\int_G||\psi_Nt)-\phi(t)||d\mu(t)<\epsilon/3$. Let $\psi_N=\sum_{i=1}^r\chi_{A_i}a^{**}_i$ for some disjoint $A_1,A_2,..,A_r\in \mathscr{B}(G)$ with $\mu(A_i)<\infty$ for each $i$ and $a^{**}_1,...,a^{**}_r\in \mathcal{A}^{**}$. For each $i$, choose a $b^*_i\in \mathcal{A}^*$ such that $\left| a^{**}_i(b^*_i)-||a^{**}_i||\right|<\epsilon/3$. Now define $f:G\to \mathcal{A}^*$ as $f=\sum_{i=1}^r\chi_{A_i}b^*_i$. Clearly $f$ is a $\mu$-simple function and $f\in L^\infty(G,\mathcal{A}^*)$ such that $||f||_{L^\infty(G,\mathcal{A}^*)}\leq 1$. Now notice that $$\begin{aligned} \left|\int_G\left<f(t),\phi(t)\right>d\mu(t)-\int_G||\phi(t)||d\mu(t)\right|&=&\left|\int_G\left<\sum_{i=1}^r\chi_{A_i}b^*_i,\phi(t)\right>-||\phi(t)||d\mu(t)\right|\\ &\leq&\sum_{i=1}^r\int_{A_i}\bigl\lvert\left<b^*_i,\phi(t)\right>-||\phi(t)||\bigr\rvert d\mu(t)\\ &\leq&\sum_{i=1}^r\int_{A_i}\bigl\lvert \left<b^*_i,\phi(t)\right>-||a^{**}_i||\bigr\rvert d\mu(t)+\sum_{i=1}^r\int_{A_i}\bigl\lvert||a^{**}_i||-||\phi(t)||\bigr\rvert d\mu(t)\\&\leq&\sum_{i=1}^r\int_{A_i}\left|\left<b_i^*,\phi(t)\right>-a^{**}_i(b^*_i)\right|d\mu(t)+\sum_{i=1}^r\int_{A_i}\bigl\lvert a^{**}_i(b^*_i)-||a^{**}_i||\bigr\rvert d\mu(t)+\frac{\epsilon}{3}\\&< &\sum_{i=1}^r\int_{A_i}||\phi(t)-a^{**}_i||d\mu(t)+\frac{\epsilon}{3}+\frac{\epsilon}{3}\\&=&\int_G||\phi(t)-\psi_n(t)||d\mu(t)+\frac{2\epsilon}{3}\\& <&\epsilon \end{aligned}$$ Thus, we see that $||\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi)||=||\phi||_{L^1(G,\mathcal{A})}$ and $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% ,\theta_\diamond$ are isometry. Now we prove that $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}%$ and $\theta_\diamond$ are algebra homomorphisms. For $\phi_1,\phi_2\in L^1(G,(\mathcal{A}^{**},% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% ))$ and $f\in L^\infty(G,\mathcal{A}^*)$, we have $$\begin{aligned} \theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_1\ast\phi_2)(f)&=&\int\left<\phi_1\ast\phi_2(t),f(t)\right>d\mu(t)\\ &=&\int\int\left<\phi_1(s)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \phi_(s^{-1}t),f(t)\right>d\mu(s)d\mu(t)\\&=&\int\left<\phi_1(s),\int {_{\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_2)(s^{-1}t)}}(f(t))d\mu(t)\right>d\mu(s)\\&=&\int\left<\phi_1(s),{_{\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_2)}}f\right>d\mu(s)\\&=&\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_1)({_{\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_2)}}f)\\&=&\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_1)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_2)(f) \end{aligned}$$ Thus, $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_1\ast\phi_2)=\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_1)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% (\phi_2)$. Hence, $\theta_% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}%$ is an homomorphism of algebras. Similarly, $\theta_\diamond$ is a homomorphism. ◻ # The bidual of $L^1(G,\mathcal{A})$ for compact group $G$. {#4} In this section, hereafter we will assume $G$ to a be a compact Hausdorff group with normalized Haar measure $\mu$ and $\mathcal{A}$ will denote a Banach algebra, unless stated otherwise.\ **Lemma 4**. *If $f\in L^\infty(G,\mathcal{A}^*)$ and $\phi\in L^1(G,\mathcal{A})$, then $f_\phi(t)=\int L_{\phi(s)}(f(st))ds$ for all $t\in G$.* *Proof.* Suppose $\psi \in L^1(G,\mathcal{A})$. Then $$\begin{aligned} f_\phi(\psi)&=&f(\phi\ast\psi)\\ &=&\int \left<f(t),\phi\ast\psi(t)\right>dt\\&=&\int\left<f(t),\int\phi(s)\psi(s^{-1}t)ds\right>dt\\&=&\int\int\left<f(st),\phi(s)\psi(t)\right>dtds\\&=&\int\left<\int (f(st))_{\phi(s)}ds,\psi(t)\right>dt \end{aligned}$$ Since, $\psi$ was arbitrary, we conclude that $f_\phi(t)=\int (f(st))_{\phi(s)}ds$ ◻ **Lemma 5**. *If $f\in L^\infty(G,\mathcal{A}^*)$ and $\phi\in L^1(G,\mathcal{A})$, we have ${_\phi}f(t)=\int {_{\phi(t^{-1}s)}}(f(s))ds$* *Proof.* Proved similarly as previous lemma. ◻ We say that $f\in L^\infty(G,X)$ is left uniformly continuous if $||L_a(f)-f||\to 0$ as $a\to e$, where $L_a(f)(x)=f(ax)$. Similarly, $f$ is said to be right uniformly continuous if $||R_a(f)-f||\to 0$ as $a\to e$. Let $LUC(G,X),RUC(G,X)$ denote the collection of all left/right uniformly continuous functions in $L^\infty(G,X)$. Clearly $LUC(G,X), RUC(G,X)$ are closed subspaces. **Lemma 6**. *Continuous functions are left/right uniformly continuous, i.e $C(G,X)\subseteq LUC(G,X)\cap RUC(G,X)$ for any Banach space $X$ and a compact group $G$.* *Proof.* Proof runs ditto as in the case of scalar valued functions. See for instance [@Folland Prop. 2.6]. ◻ **Lemma 7**. *Let $G$ be a compact group and $X$ be a Banach space. If $f\in C(G,X)$ then $||L_z(f)-f||_{L^1(G,X)}$ and $||R_z(f)-f||_{L^1(G,X)}$ tends to $0$ as $z\to e$.* *Proof.* Fix a compact neighborhood $V$ of $e$. Let $K=(supp(f))V^{-1}\cap V(supp(f))$. Then $K$ is compact and $L_z(f)$ is supported in $K$ when $z\in V$. Hence, $||L_z(f)-f||_{L^1(G,X)}\leq \mu(G)||L_zf-f||_{L^{\infty}(G,X)}\to 0$ as $z\to e$ ◻ **Lemma 8**. *Let $\mathcal{A}$ be any Banach algebra. If $f\in L^\infty(G,\mathcal{A}^*)$ and $\phi\in C(G,A)$, then $f_\phi\in LUC(G,\mathcal{A}^*)$.* *Proof.* For any $z\in G$, it is straighforward to verify that $L_z(f_\phi)=f_{\widetilde{L_z\tilde{\phi}}}$. For any $t\in G$, $$\begin{aligned} \vert\vert L_z(f_\phi)(t)-f_\phi(t)\vert\vert&=&\left|\left|\int(f(st))_{\widetilde{L_z\tilde{\phi}}-\phi(s)}d\mu(s)\right|\right|\\ &\leq&||f||_{L^\infty(G,\mathcal{A}^*)}\int||(\widetilde{L_z\tilde{\phi}})(s)-\phi(s)||d\mu(s)\\&=&||f||_{L^\infty(G,\mathcal{A}^*)}\int||\phi(sz^{-1})-\phi(s)||d\mu(s)\end{aligned}$$ Thus, $||L_z(f_\phi)-f_\phi||_{L^\infty(G,\mathcal{A}^*)}\leq ||f||_{L^\infty(G,\mathcal{A}^*)}\left|\left|R_{z^{-1}}\phi-\phi\right|\right|_{L^1(G,\mathcal{A})}$ and by previous lemma, $f_\phi$ is left uniformly continuous. ◻ For $f\in L^\infty(G,X)$ and $t\in G$, write $(tf)(x)=f(xt)$. **Lemma 9**. *Let $m\in Z_s$ and $f\in L^\infty(G,\mathcal{A}^*)$, then $f_m\in LUC(G,\mathcal{A}^*)$ and $f_m(t)(a)=\left<m,f^{(t,a)}\right>$ for each $t\in G$ and $a\in A$, where $f^{(t,a)}(s)={_a}(f(st))$* *Proof.* Using the Goldstein's theorem and the denseness of compactly supported functions in $L^1(G,\mathcal{A})$, we choose a net $\{\phi_{\alpha}\}_{\alpha\in \wedge}$ of compactly supported functions converging to $m$ in the $weak^\ast$-topology of $L^1(G,A)^{**}$. Then, $$\begin{aligned} \left<n,f_m\right>&=&\left<m\diamond n,f\right>=\left< m\Box n,f\right>=\left<m,{_n}f\right>\\&=&\lim_\alpha\left<\phi_\alpha,{_n}f\right>=\lim_\alpha\left<n,f_{\phi_\alpha}\right>\end{aligned}$$ for each $n\in L^1(G,\mathcal{A})^{**}$. Hence, $\{f_{\phi_\alpha}\}$ converges weakly to $f_m$. Using Mazur's lemma, we can choose a net of suitable convex combinations of $\{\phi_\alpha\}_{\alpha\in \wedge}$ such that $f_{\phi_\alpha}$ converges to $f_m$ in norm. Thus we assume that $f_{\phi_\alpha}$ converges to $f_m$ in norm. But $f_{\phi_\alpha}\in LUC(G,\mathcal{A}^*)$ by previous lemma. Thus, it follows that $f_m\in LUC(G,\mathcal{A}^*)$. Further, for any $t\in G$ and $a\in \mathcal{A}$ $$\begin{aligned} f_m(t)(a)&=&\lim_\alpha\int L_{\phi_\alpha(s)}(f(st))(a)d\mu(s)\\&=&\lim_{\alpha}\int f(st)(\phi_\alpha(s)a)d\mu(s)\\&=&\lim_{\alpha}\int\left<f(st),\phi_\alpha(s)a\right>d\mu(s)\\&=&\lim_{\alpha}\left<\phi_\alpha,f^{(t,a)}\right>\\&=&\left<m,f^{(t,a)}\right>\end{aligned}$$ ◻ **Lemma 10**. *Let $\mathcal{A}$ be a unital Banach algebra. If $m\in Z_s$ is such that $m(f)=0$ for all $f\in C(G,\mathcal{A}^*)$ then $m(f)=0$ for all $f\in L^\infty(G,\mathcal{A}^*)$. Further if $\mathcal{A}^*$ has RNP then $m=0$.* *Proof.* To see this, let $f\in L^\infty(G,\mathcal{A}^*)$. Using the [Lemma 9](#imp1){reference-type="ref" reference="imp1"}, for $\epsilon>0$, we choose $V\subset \{x:||f_m(x)-f_m(e)||<\epsilon\}$ such that $V$ is open and relatively compact. Consider $v=\frac{1_V}{\mu(V)}\otimes \mathds{1}_A$. One can easily see that ${_v}f\in C(G,\mathcal{A}^*)$. Hence, ${_v}(f_m)(\phi)=({_v}f)_m(\phi)=m({_{\phi\ast v}}f)=0$, because ${_{\phi\ast v}}f\in C(G,\mathcal{A}^*)$ for each $\phi\in L^1(G,\mathcal{A})$. Thus, ${_v}(f_m)=0$. $$\begin{aligned} |m(f)|&=&\left|f_m(e)(\mathds{1}_\mathcal{A})-{_v}(f_m)(e)(\mathds{1}_\mathcal{A})\right|\\&=&\frac{1}{\mu(V)}\left|\int_V\left(f_m(e)(\mathds{1}_\mathcal{A})-f_m(x)(\mathds{1}_\mathcal{A})\right)dx\right|\\&\leq& \frac{1}{\mu(V)}||f_m(e)-f_m(x)||\mu(V)\\&=&\epsilon \end{aligned}$$ Hence, $m(f)=0$ for all $f\in L^\infty(G,\mathcal{A}^*)$, proving the first part of assertion. Now if $\mathcal{A}^*$ has RNP then $L^\infty(G,\mathcal{A}^*)$ is the full dual space of $L^1(G,\mathcal{A})$ and hence $m=0$. ◻ Let $S^\infty(G,\mathcal{A}^*)$ denotes the closure of the space of all $\mu$-simple functions in $L^\infty(G,\mathcal{A}^*)$. Clearly $C(G,\mathcal{A}^*)$ is contained in $S^\infty(G,\mathcal{A}^*)$ . For each $\nu\in M_r(G,A)$ we define $\varphi_{\nu}:S(G,\mathcal{A}^*)\to \mathbb C$ such that for $\sum_{i=1}^r\chi_{E_i}\otimes a^*_i\in S(G,\mathcal{A}^*)$. $$\left<\varphi_{\nu},\sum_{i=1}^r\chi_{E_i}\otimes a^*_i\right>=\sum_{i=1}^r\left<\nu(E_i),a^*_i\right>$$ It is an easy exercise to verify that this action is well defined(independent of representation of simple functions) and is linear. Further, it can be verified that $||\varphi_{\nu}||=||\nu||$. Hence, $M_r(G,A)$ sits inside $S^\infty(G,\mathcal{A}^*)^*$ isometrically. **Lemma 11**. *Let $\mathcal{A}$ be a reflexive Banach algebra. Suppose that $m\in Z(L^1(G,\mathcal{A})^{**})$ and $\nu\in M_r(G,\mathcal{A})$ be such that there exists a sequence $\{\nu_n\}\in L^1(G,\mathcal{A})$ converging to $\nu$ in the $\sigma(M_r(G,\mathcal{A}),C(G,\mathcal{A}^*))$ topology. Then $m\diamond n_\nu\in L^1(G,\mathcal{A})$ for any continuous extension $n_\nu$ of $\nu$ to $L^1(G,\mathcal{A})^*$ .* *Proof.* Let us first assume that $u\in L^1(G,\mathcal{A})$. Then by [\[RRT1\]](#RRT1){reference-type="ref" reference="RRT1"}, restriction of $m$ to $C(G,\mathcal{A}^*)$ is given by a measure $\eta\in M_r(G,\mathcal{A})$. For $f\in C_0(G,\mathcal{A}^*)$ $$\begin{aligned} \left<m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% u,f\right> &=\left<m,{_u}f\right>\\&=\left<\eta,{_u}f\right>\end{aligned}$$ Since, regular $\mathcal{A}$-valued measures are weakly compact in the sense that the associated operator $C(K)\to X$ is weakly compact (see [@Ryan Th. 5.2]), and space of weakly compact $\mathcal{A}$-valued measures form a Banach algebra with respect to the convolution (see [@White Th. 3.2]), we conclude that $\left<m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% u,f\right> =\left<\eta\ast u,f\right>$ for all $f\in C(G,\mathcal{A}^*)$. Notice that $m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% u\in Z(L^1(G,\mathcal{A})^{**})$. Thus, by [Lemma 10](#Imp2){reference-type="ref" reference="Imp2"}, $m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% u=\eta\ast u\in L^1(G,\mathcal{A})$ (because $\eta\ast u$ is a regular measure of bounded variation and $\nu\ast u\ll \mu$).\ Now if $f\in L^\infty(G,\mathcal{A}^*)$, then $f_m\in LUC(G)$ by [Lemma 9](#imp1){reference-type="ref" reference="imp1"} and hence $$\begin{aligned} \left<m\diamond n_\nu,f\right>&=n(f_m)\\&=\lim_n \nu_n(f_m)\\&=\lim_n m\diamond \nu_n(f)\\&=\lim_n m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \nu_n(f)\end{aligned}$$ But $m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% \nu_n\in L^1(G,\mathcal{A})$ as proved above and $L^1(G,\mathcal{A})$ is WSC, hence $m\diamond n_\nu\in L^1(G,\mathcal{A})$. ◻ **Corollary 12**. *Let $\mathcal{A}$ be a unital refelexive Banach algebra $K$ be a closed subgroup of $G$ such that $G/K$ is metrizable, then $m\diamond \mu_k\in L^1(G,\mathcal{A})$ where $\mu_k$ is the $\mathcal{A}$-valued vector measure on Borel subsets of $G$ such that $\mu_k(E)=\frac{\mu(E\cap K)}{\mu(K)}\mathds{1}_\mathcal{A}$ for each Borel subset $E$ of $G$.* *Proof.* Since, $G/K$ is metrizable, we can choose a decreasing sequence $\{U_n\}$ of neighborhoods of $K$ such that $K=\cap_n U_n$. Let $u_n=\frac{1}{\mu(U_n)}\chi_{U_n}\mathds 1_\mathcal{A}$. Clearly $u_n\in L^1(G,\mathcal{A})$ and $u_n\to \mu_K$ in the $\sigma(M_r(G,\mathcal{A}),C(G,\mathcal{A}^*))$ topology. By previous lemma, $m\diamond \mu_k\in L^1(G,\mathcal{A})$. ◻ For a subgroup $K$ of $G$, we say that $f\in L^\infty(G,\mathcal{A}^*)$ is right *$K$-periodic* if $kf=f$ for all $k\in K$. **Lemma 13**. *Let $K$ be a compact subgroup of $G$ and $m\in Z$. If $f\in L^\infty(G,\mathcal{A}^*)$ is $K$-periodic then $\left<m,f\right>=\left<m\diamond\mu_K,f\right>$ for all $f\in L^\infty(G,\mathcal{A}^*)$.* *Proof.* Since, $f$ is $K$-periodic, by [Lemma 9](#imp1){reference-type="ref" reference="imp1"}, $f_m$ is also $K$-periodic. Hence, $$\begin{aligned} \left<m,f\right>&=f_m(e)(\mathds 1_\mathcal{A})\\&=\left<\mu_k,f_m\right>\\&=\left<m\diamond \mu_K,f\right> \end{aligned}$$ ◻ Now, we have all the required tools to prove that for an abelian compact group $G$ and a reflexive Banach algebra $\mathcal{A}$, the generalized group algebra $L^1(G,\mathcal{A})$ is left Strongly Arens irregular. **Theorem 11**. Let $G$ be a compact group and $\mathcal{A}$ be a unital reflexive Banach algebra. Then, $$Z(L^1(G,\mathcal{A})^{**})=L^1(G,\mathcal{A}).$$ *Proof.* The inclusion $L^1(G,\mathcal{A})\subset Z(L^1(G,\mathcal{A})^{**})$ holds trivially true. To prove the reverse inclusion, let $m\in Z(L^1(G,\mathcal{A})^{**})$ and $\nu_m\in M_r(G,\mathcal{A})$ denote its restriction to $C(G,\mathcal{A}^*)$. Due to [Lemma 9](#imp1){reference-type="ref" reference="imp1"}, it will be sufficient to show that $\nu_m\in L^1(G,\mathcal{A})$. Let $B$ be a compact subset of $G$ such that $\mu(B)=0$. Then we choose a decreasing sequence of open sets $U_n\supset B$ such that $(\mu+|\nu_m|)(U_n\setminus B)\to 0$. By induction, we construct a sequence $\{\phi_n\}$ in $C(G)$ such that $0\leq \phi_n\leq 1$, $\phi_n(x)=1$ for $x\in B$ and $\phi_n(x)=0$ for $x\notin U_n\cap V_{n-1}$ (where $V_0=G$,$V_n=\{y:\phi_n(y)\neq 0\}$, $n=1,2,...$). For each $n$, $$d_n(x,y)=||x\phi_n-y\phi_n||_\infty$$ defines a continuous pseudo metric on $G$, and $K=\cap_{n=1}^\infty K_n$, then $G/K$ is metrizable and hence $m\diamond \mu_K\in L^1(G,\mathcal{A})$ by [Corollary 12](#metrizable){reference-type="ref" reference="metrizable"}. Consequently, by [Lemma 13](#periodic){reference-type="ref" reference="periodic"} $\left<\nu_m,f\right>=\left<m\diamond \mu_K,f\right>$ for each right $K$-periodic function function $f\in L^\infty(G,\mathcal{A}^*)$. Since, $\{V_n\}$ is decreasing, $\mu(V_n)\to \mu(B)=0$. Further, for each $a^*\in \mathcal{A}^*$, the function $\chi_{V_N}\otimes a^*$ is right $K$-periodic. Thus, $$\nu_{m,a^*}(V)=\left<\nu,\chi_{V_n}\otimes a^*\right>=\left<m\diamond\mu_K,\chi_{V_n}\otimes a^*\right>\to 0\hspace{0.3in}(\because m\diamond\mu_K\in L^1(G,\mathcal{A}))$$ Since, $B\subset V_n\subset U_n$, and we have $\nu_{m,a^*}(V_n)\to \nu_{m,a^*}(B)$. Thus, $\nu_{m,a^*}(B)=0$ for each $a^*\in \mathcal{A}^*$. Hence, $\nu_m(B)=0$ and due to regularity of $\nu_m$, we conclude that $\nu_m\ll \mu$, i.e $\nu_m\in L^1(G,\mathcal{A})$ and required. ◻ **Corollary 14**. *Let $G$ be a compact abelian group and $\mathcal{A}$ be any reflexive Banach algebra(not necessarily unital), then $L^1(G,\mathcal{A})$ is SAI.* *Proof.* The preceeding theorem combined with the [Corollary 2](#unitization1){reference-type="ref" reference="unitization1"} proves the assertion. ◻ # Bidual of $L^1(G,\mathcal{A})$ for non-reflexive Banach algebra $\mathcal{A}$. As we have noticed in the previous sections that when $\mathcal{A}^*$ does not have RNP, then the dual $L^1(G,\mathcal{A})^*$ strictly contains a copy of $L^\infty(G,\mathcal{A}^*)$. This makes it difficult to access the topological center. It would be too demanding to expect $L^1(G,\mathcal{A})$ to be SAI in such cases, even when $\mathcal{A}$ itself is SAI. But pseudo-center seems to be accessible in certain cenarios. We shall see that in certain situations, the elements of pseudo center can be identified with $Z(\mathcal{A}^{**})$-valued measures.\ Let $\mathcal{A}$ be a Banach algebra and $a^*\in \mathcal{A}^*$ be a fixed element. For each $m\in Z_s(L^1(G,\mathcal{A})^{**})$, there is an associated map $\Delta_{m,a^*}: C(G)\to \mathcal{A}^*$ defined as $\Delta_{m,a^*}(f)(a)=m(f\otimes {_a}a^*)$. **Lemma 15**. *Let $G$ be compact Hausdorff group, $\mathcal{A}$ be a Banach algebra,and $m\in Z_s(L^1(G,\mathcal{A})^{**})$be such that $\Delta_{m,a^*}$ is compact for every $a^*\in \mathcal{A}^*$, then the restriction of $m$ to $C(G,\mathcal{A}^*)$ is a $Z(\mathcal{A}^{**})$-valued measure.* *Proof.* Let $m\in Z_s(L^1(G,\mathcal{A})^{**})$ be any any arbitrary element. We denote the restriction of $m$ to $C(G,\mathcal{A}^*)$ by the measure $\mu_m\in M_r(G,\mathcal{A}^{**})$. Due to regularity of $\mu_m$, it would be sufficient to show that $\mu_m(E)\in Z(\mathcal{A}^{**})$ for any open subset $E$ of $G$. Let $E$ by an open susbset of $G$. For any $a^*\in\mathcal{A}^*$ and $a^{**}\in \mathcal{A}^{**}$, we have $$\begin{aligned} \mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a^{**}(a^*)&=\mu_m(E)({_{a^{**}}}a^*) \end{aligned}$$ Let $b^*={_{a^{**}}}a^*$. Invoking, the Riesz Representation theorem [\[RRT1\]](#RRT1){reference-type="ref" reference="RRT1"}, we have $$\begin{aligned} \mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a^{**}(a^*)=\mu_{m,b^*}\left(E\right) \end{aligned}$$ where $\mu_{m,b^*}$ is the regular borel measure corresponding to the linear functional $L_{m,b^*}:C(G)\to \mathbb C$ defined as $L_{m,b^*}(f)=m(f\otimes b^*)$. Then there exists a compactly supported continuous function $f$ such that $0\leq f\preceq \chi_E$ and $\left|\mu_{m,b^*}(E)-\int fd\mu_{m,b^*}\right|<\epsilon$. But $\int fd\mu_{m,b^*}=m(f\otimes b^*)=m(f\otimes {_{a^{**}}}a^*)$. Now notice that for any $\phi\in L^1(G,\mathcal{A})$, we have $$\begin{aligned} (f\otimes {_{a^{**}}}a^*)(\phi)&=\int\left<f(s){_{a^{**}}}a^*,\phi(s)\right>dt\\&=\left<{_{a^{**}}}a^*,\int f(s)\phi(s)dt\right> \end{aligned}$$ Using the module version of Cohen's factorization theorem [Theorem 5](#Cohen){reference-type="ref" reference="Cohen"}, we know that $L^1(G)\ast L^\infty(G)=C_{lu}(G)$ and since compactly supported continuous functions are left/right uniformly continuous, we can choose $g\in L^1(G)$ and $h\in L^\infty(G)$ such that $g\ast \check{h}= \check{f}$. Let $n$ denotes any Hahn-Banach extension of $g\otimes a^{**}$ to $L^1(G,\mathcal{A})^{**}$. Now for any any net $\{a_\gamma\}_{\gamma\in \wedge}$ in $\mathcal{A}$ converging to $a^{**}$ in the $w^\ast$ topology of $\mathcal{A}^{**}$, we see that $g\otimes a_\gamma$ converges to $g\otimes a^{**}$ in the $\sigma(L^1(G,\mathcal{A})^{**},L^\infty(G,\mathcal{A}^*))$ topology and hence, $$\begin{aligned} {_n}(h\otimes a^*)(\phi)&=n((h\otimes a^*)_\phi)\\&=\left<g\otimes a^{**},(h\otimes a^{*})_\phi\right>\\&=\lim_\gamma \left<g\otimes a_\gamma,(h\otimes a^*)_\phi\right>\\&=\lim_\gamma\int g(t)(h\otimes a^*)_\phi(t)(a_\gamma)dt\\&=\lim_\gamma\int g(t)\int\left<h(st)a^*,\phi(s)a_\gamma\right>dsdt\\&=\lim_\gamma\left<a^*,\left(\int g\ast \check{h}(s^{-1})\phi(s)ds\right) a_\gamma\right>\\&=\lim_\gamma\left<a^*,\left(\int f(s)\phi(s)ds\right) a_\gamma\right>\\&=\left<{_{a^{**}}}a^*,\int f(s)\phi(s)ds\right> \end{aligned}$$ Thus, we see that ${_n}(h\otimes a^*)=f\otimes {_{a^{**}}}a^*$. And hence $$\begin{aligned} \left|\mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a^{**}(a^*)-m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(h\otimes a^*)\right|<\epsilon \end{aligned}$$ Now we turn to the second Arens product and show that it is also in the arbitrarily small neighbourhood of $m% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% n(h\otimes a^*)$. Notice that $$\begin{aligned} \mu_m(E)\diamond a^{**}(a^*)&=\lim_\gamma \mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a_\gamma(a^*)\\&=\lim_\gamma \mu_{m,b_\gamma^*}(E)\\&=\lim_\gamma \int \chi_Ed\mu_{m,b_\gamma^*} \end{aligned}$$ where $b_\gamma^*={_{a_\gamma}}a^*$. Since, $\Delta_{m,a^*}$ is compact, its adjoint $\Delta_{m,a^*}^{*}:A^{**}\to M_r(G)$ given by $\Delta_{m,a^{*}}^*(a^{**})=\mu_{m,{_{a^{**}}a^*}}$ satisfies the property that for any bounded net $\{a_\lambda^{**}\}$ in $\mathcal{A}^{**}$, the net $\{\Delta_{m,a^*}^*(a_\lambda^{**})\}$ converges in norm. Hence, $\lim_{\gamma}\int\chi_Ed\mu_{m,b_\lambda^*}=\mu_{m,b_\gamma^*}(E)$ $$\begin{aligned} \left|\mu_m(E)\diamond a^{**}(a^*)-\int fd\mu_{m,b^*}\right|\leq |\mu_{m,b^*}|(U\setminus E)<\epsilon\\& \end{aligned}$$ Thus, we see that $|\mu_m(E)\diamond a^{**}(a^*)-\mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a^{**}(a^*)|<\epsilon$. Since $\epsilon$ was arbitrarily chosen, we conclude that $\mu_m(E)\diamond a^{**}(a^*)=\mu_m(E)% \text{\fboxsep=-.1pt\fbox{\rule{0pt}{1ex}\rule{1ex}{0pt}}}% a^{**}(a^*)$ for all $a^{**}\in \mathcal{A}^{**}$ and all $a^*\in \mathcal{A}^*$. Thus, $\mu_m(E)\in Z(\mathcal{A}^{**})$. ◻ **Remark 16**. *It is expected that if $\mathcal{A}$ is SAI and holds RNP and WSC, then for any compact group $G$ and $m\in Z_s(L^1(G,\mathcal{A})^{**})$, the map $\Delta_{m,a^{*}}$ is compact for each $a^*\in \mathcal{A}^*$. And it is possible to do computations similar to those in [4](#4){reference-type="ref" reference="4"} to conclude that $L^1(G,\mathcal{A})$ is pseudo-strongly irregular. For doing this, one might need something stronger than WSC of $L^1(G,\mathcal{A})$.* Although it is known that $L^1(S,X)$ is WSC for any WSC Banach space $X$ and finite measure space $S$, the addition constraint of RNP on $X$ results into something remarkably stronger. **Lemma 17**. *If $X$ is WSC Banach space such that $X$ has RNP w.r.t a finite measure space $(S,\mathscr{A},\nu)$, then $L^1(S,X)$ is $\sigma(L^1(S,X),S^\infty(S,X^*))$-sequentially complete. In other words, if $\{\phi_n\}\}_{n\in \mathbb N}$ is sequence in $L^1(S,X)$ such that for each $f\in S^\infty(S,X^*)$ the sequence $\{\phi_n(f)\}_{n=1}^\infty$ is Cauchy, then there exists $\phi\in L^1(S,X)$ such that $\phi_n\to\phi$ in the $\sigma(L^1(S,X),S^\infty(S,X^*))$-topology.* *Proof.* Suppose $\{\phi_n\}_{n\in \mathbb N}$ is a sequence in $L^1(S,X)$ such that for each $f\in S^\infty(S,X^*)$ the sequence $\{\phi_n(f)\}_{n=1}^\infty$ is Cauchy. Using the uniform boundedness principle and the fact that $S^\infty(S,X^*)$ is norming for $L^1(S,X)$, we can conclude that the sequence $\{\phi_n\}_{n=1}^\infty$ is norm bounded. For any subset $E$ of $S$ with $\nu(E)\neq 0$ and any $x^*\in X^*$, consider the function $\chi_E\otimes x^*\in S^\infty(S,X^*)$. Then $\{\left<\phi_n,\chi_E\otimes x^*\right>\}$ is a cauchy sequence and $$\begin{aligned} \left<\phi_n,\chi_E\otimes x^*\right>&=&\int_E\left<\phi_n(t),x^*\right>dt\\ &=&x^*\left(\int_E\phi_n(t)dt\right) \end{aligned}$$ Since, above equality holds for each $x^*\in X^*$, we conclude that $\{\int_E\phi_n(t)dt\}$ is a weakly cauchy sequence in $X$. Since $X$ is WSC, there must exists $a_E\in X$ such that $\int_E\phi_n(t)dt\to a_E$ weakly. Now we define a $X$ valued measure $F:\mathscr A\to X$ as $F(E)=a_E$. Clearly $F$ is a measure of bounded variation. Further, for each $x^*\in X^*$ we see that $x^*F$ is a finite measure wich is a limit of a sequence of measures absolutely continuous w.r.t to $\mu$ and hence, $x^*F$ is countably additive due to Vitali-Hahn-Saks theorem for each $x^*\in X^*$. Thus, $F$ is weakly countably additive and hence countably additive due to Orlicz-Pettis theorem([@Diestel Sec. I.4,Corr.4]). Further, $F\ll\mu$ can be seen from the fact that $F$ vanishes on $\mu$-null sets. Thus, by RNP there exists $\phi\in L^1(S,X)$ such that $F(E)=\int_E \phi(t)d\mu(t)$. Thus, we have that $$\lim_n\left<\phi_n,\chi_E\otimes x^*\right>=\left<\phi,\chi_E\otimes x^*\right>$$ for each borel subset $E$ of $G$ and each $x^*\in X^*$. Since, functions of the type $\chi_E\otimes x^*$ spans $S^\infty(S,X^*)$, we conclude that $\phi_n\to \phi$ in $\sigma(L^1(S,X),S^\infty(S,X^*))$-topology. ◻ 45 R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. **26** (1951), 839-848. CH. Chu . A Note on Scattered C\*-Algebras and the Radon-Nikodym Property, Journal of The London Mathematical Society-second Series, 533-536, 1981. T. Hytönen, J.Neerven, M. Veraar, L. Weis, Analysis in Banach space, Volume-1, Springer. C. C. Graham, A B.A.I. proof of the non-Arens regularity of $L^{1}(G)$, Proc. Amer. Math. Soc. **133** (2005), no. 1, 163--165. E. Kaniuth, A Course in Commutative Banach Algebras, Springer New York, 2009. A. T.-M. Lau and V. Losert, On the second conjugate algebra of $L^1(G)$ of a locally compact group, J. London Math. Soc. **37** (1988), no. 2, 464-470. V. Montesinos, M. Fabian, P. Hájek, P. Habala and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, Springer New York, 2011. R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer London, 2013. A. Ülger, Arens regularity of the algebra $A \otimes^{\gamma}B$, Trans. Amer. Math. Soc. **305** (1988), no. 2, 623-639. A. Ülger, Erratum to: "Arens regularity of the algebra $A \otimes^{\gamma}B$" \[Trans. Amer. Math. Soc. 305 (1988), no. 2, 623--639\], Trans. Amer. Math. Soc. **355** (2003), no. 9, 3839. N.J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) , 24 (1973) pp. 59--62 G.B Folland, A course in abstract harmonic analysis,second edition, Taylor and Francis 2016. N. Dunford, J.T. Schwartz, Linear Operators, Part 1:General Theory, John-Wiley and sons inc.1958. A.J White, Convolution of vector measures,Proceedings of the Royal Society of Edinburgh, 73A. 7, 1974/75. Michel Talagrand, Weak Cauchy sequences in $L^1(E)$, American Journal of mathematics, Vol. 106 No.3,1984. J. Diestel, J. Uhl, Vector measures, American Mathematical Soc., 1977. A. Rodríguez-Palacios, A note on Arens regularity, Quart. J. Math. Oxford Ser. (2) 38:149 (1987), 91--93 [^1]: This research was supported by the Intitute post-doctoral fellowship provided to the author by the Indian Institute of Science Education and Research-Bhopal(India)

arxiv_math

--- abstract: | In this paper, combining Nash-Moser iteration and Sallof-Coste type Sobolev ineualities, we establish fundamental and concise $C^0$ and $C^1$ estimates for solutions to a class of nonlinear elliptic equations of the form $$\Delta u(x)+a(x)u(x)\ln u(x)+b(x)u(x)=0,$$ which possesses abundant geometric backgrounds. Utilizing these estimates which retrieve more geometric information, we obtain some further properties of such solutions. Especially, we prove a local Liouville type theorem of corresponding constant coefficient equation. author: - Jie Wang and Youde Wang - "Jie Wang and Youde Wang[^1]" title: Boundedness and gradient estimates for solutions to $\Delta u + a(x)u\log u + b(x)u = 0$ on Riemannian manifolds --- [^2]\ [^3] # Introduction This is a further study based on our previous works [@Wj; @WW]. In this paper, we continue to study the following equation on a smooth complete Riemannian n-manifold $(M^n,g)(n\geq 3)$: $$\label{1.1} \Delta u(x)+a(x)u(x)\ln u(x)+b(x)u(x)=0,$$ where $\Delta$ is the Laplace-Beltrami operator, $a(x)$ and $b(x)$ are coefficient functions on $(M^n,g)$. The existence of such solution $u$ on compact domains is a classical result proved by Rothaus [@Ro]. The type of equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) is closely related to that of Euler-Lagrange equations of the $\mathcal{W}$ entropy and Log-Sobolev functional on $(M^n, g)$. $\mathcal{W}$ entropy [@P] is defined for $f\in W^{1,2}(M^n, e^{-f} dV)$ as $$\mathcal{W}(g, f, \tau) = \int_M[\tau(| \nabla f|^2 + R) + f - n](4\pi\tau)^{-\frac{n}{2}}e^{-f} dV,$$ where $\tau > 0$ is a scale parameter and $R \in L^1(M^n, e^{-f} dV)$ is supposed. Setting $u^2 = (4\pi\tau)^{-\frac{n}{2}}e^{-f}$, we may rewrite the above $\mathcal{W}$-entropy for $u \in W^{1,2}(M^n,g)$ as $$\mathcal{W}(g, u, \tau)=\int_M\left(\tau(4|\nabla u|^2 + Ru^2) - u^2\log u^2 - nu^2 -\frac{n}{2}\log(4\pi\tau)u^2\right)dV.$$ By the scale invariance property of $\mathcal{W}$ entropy, without loss of generality, we may assume $\tau=1$. Then, it follows $$\mathcal{W}(g, u, 1) = \mathcal{L}(u, g) - \frac{n}{2}(\log(4\pi)+2),$$ if $\|u\|_{L^2(M^n,g)}=1$. Here $\mathcal{L}(u, g)=:\mathcal{L}(u, M^n, g)$ is the Log-Sobolev functional on $(M^n, g)$ perturbed by $R$, the scalar curvature of the manifold $(M^n, g)$, which is defined for $u\in W^{1,2}(M^n,g)$ as $$\mathcal{L}(u, M^n, g)=\int_M (4|\nabla u|^2 + Ru^2 - u^2\log u^2)dV.$$ We define the best Log-Sobolev constant of a domain $\Omega\subset M^n$ as $$\lambda(\Omega) = \inf\{\mathcal{L}(u, g): u\in C_c^\infty(\Omega),\,\,\|u\|_{L^2(\Omega)}=1\}.$$ When $\Omega = M^n$, we denote by $\lambda(M)$ the best Log-Sobolev constant of $(M^n, g)$. The Euler-Lagrange equation for the Log-Sobolev functional $\mathcal{L}$ is given by $$\label{1.2*} \Delta u + \frac{1}{2}u \log u + \frac{1}{4}(\lambda- R)u =0.$$ Under certain geometric conditions, [@RV; @Z*] proved some existence results of the extremal functions of Log-Sobolev functional defined on a complete non-compact Riemannian manifold. Especially, corresponding $C^0$ and $C^1$ estimates of $u$ are indispensable in their studies. Besides ([\[1.2\*\]](#1.2*){reference-type="ref" reference="1.2*"}), after normalization, the potential function $f$ of a Ricci soliton also satisfies $$\label{1.3*} \Delta\left( e^{-f}\right) -\varepsilon e^{-f}\log\left(e^{-f}\right) = 0,$$ where $\varepsilon\in\left\lbrace-1,0,1\right\rbrace$ and the soliton is called shrinking, steady, expanding respectively (cf. [@CL]). As we know, Ricci solitons are crucial for the study of singularity analysis of Ricci flows. In short, ([\[1.2\*\]](#1.2*){reference-type="ref" reference="1.2*"}) and ([\[1.3\*\]](#1.3*){reference-type="ref" reference="1.3*"}) imply that equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) possesses abundant geometric backgrounds. For constant $a(x)\equiv a$ and $b(x)\equiv0$, one has discussed the bounds of the solutions $u$ to equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) via proving the Li-Yau type gradient estimates of corresponding parabolic equation and has obtained many interested results, for details we refer to [@C-C*; @CL; @Q1; @Yang] and references therein. For equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) with variable coefficients, J. Wang [@Wj] showed the bounds via elliptic gradient estimates and all of these estimates were based on the maximum principle. As for gradient estimates for the solution $u$ to equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}), most of the previous results relied on the maximum principle, such as [@CL; @Q1; @Q2; @Yang]. Moreover, in most of the relevant work one chose $\ln u$ as a test function, but by taking a further observation, we find that $\ln u$ is not a good function used to derive the gradient estimates (cf. Proposition [Proposition 8](#p3.3){reference-type="ref" reference="p3.3"}). In the previous work [@WW] due to the authors of this paper, for a bounded positive solution $u$, under integral Ricci curvature conditions we employed the Nash-Moser iteration method to show some new local gradient estimates by analyzing the function $u^{\frac{1}{q}}(q>1)$ instead of $\ln u$. Especially, this type of gradient estimate was applied to study the properties of Ricci flow under integral Ricci curvature conditions by Ma-Wang [@MW] successfully. Recently, Y. Wang (one of the authors of this paper) and his coauthors considered $$\Delta_p u+ a u^q=0$$ in [@HWW; @WWei] and combined Nash-Moser iteration method and Saloff-Coste's inequalities to derive some unified Cheng-Yau type inequalities (also see [@WZ]). Shortly after, inspired by [@WW], Han, He and Wang in [@HHW] adopted a similar method with that in [@HWW; @WWei] to appraoch the gradient estimates of solutions to some quasilinear elliptic equations, for instance $$\Delta_p u-|\nabla u|^q+b(x)|u|^{r-1}u=0$$ which is defined on a complete Riemannian manifold $(M,g)$. In particular, in the case $b(x)\equiv0$, a unified Cheng-Yau type estimate of the solutions to this equation is derived. Naturally, one wants to know whether or not combining Saloff-Coste type Sobolev inequalities and Nash-Moser iteration leads to some more refined gradient estimates for solutions to [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"}? The goal of the present paper is to answer the above question, i.e., we can obtain some general estimates of bounds on solutions to equation [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} by the method mentioned in the above. To compare with the maximum principle, we find that Nash-Moser iteration owns some unique advantages, for instance, one does not need to look for a test function painstakingly. Besides, this method still works for more general curvature conditions such as integral Ricci curvature conditions, for details we refer to [@WW] for related discussions. In order to state our results, we need to introduce some notations. In the following, we will use $B(x,r)$ or $B_r$ and $\left| B(x, r)\right|$ or $\left| B_r\right|$ to denote the geodesic ball with radius $r$ in $M$ centered at $x$ and its volume, respectively. $(f)^+$ denotes the positive part of $f$ and $(f)^-$ denotes the negative part of $f$. For convenience, for $p\geq1$, we define the average $L^p$ norm as $$\left| \left| f\right| \right|^*_{p,B(x,r)}= \left( \fint_{B(x,r)}\left| f\right|^p \right)^{\frac{1}{p}}.$$ Furthermore, when $p\geq1$, it's well-known that the norm $\left| \left| f\right| \right|^*_{p,B(x,r)}$ is non-decreasing in $p$ for fixed $f$ and $B(x,r)$. We also assume that $\partial B_r$ does not intersect with the boundary $\partial M^n$. Now, we are ready to state our main results: **Theorem 1**. *Let $B_{2r}\subset M^n$ be some geodesic ball with $0<r\leq1$ and its Ricci curvature $Ric\geq-(n-1)Kg$ for some $K\geq0$. Let $u>0$ be a positive (weak) solution to equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B_{2r}$ and $p>\frac{n}{2}$.* *(1). Suppose $a(x)>0$, then there exists a constant $$C=C\left( n,K,p,\left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b^-\right| \right|^*_{p, B_r}, \left| \left| \frac{(\Delta a)^- }{a}\right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that on $B_{\frac{r}{2}}$, $$\label{1.2} a\ln u\leq \frac{C}{r^2}.$$ Especially, if $a\geq A_1>0$, then on $B_{\frac{r}{2}}$, $$\label{1.3} u\leq e^{\frac{C}{A_1r^2}}.$$* *(2). (i) Suppose $a(x)<0$, then there exists a constant $$C=C\left( n,K,p,\left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b^-\right| \right|^*_{p, B_r}, \left| \left| \frac{(\Delta a)^+}{a}\right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that on $B_{\frac{r}{2}}$, $$\label{1.4} a\ln u\leq \frac{C}{r^2}.$$ Especially, if $a\leq A_2<0$, then on $B_{\frac{r}{2}}$, $$\label{1.5} u\geq e^{\frac{C}{A_2r^2}}.$$* *(ii) Suppose $a<0$ is constant, then there exists a constant $$C=C\left( n,K,a,p,\left| \left|(\ln u)^+\right| \right|^*_{p, B_r}, \left| \left| b\right| \right|^*_{p, B_r}, \left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that on $B_{\frac{r}{2}}$, $$\ln u\leq \frac{C}{r^2}.$$* By the $C^0$ estimates obtained in the above Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"} and the techniques used in our previous paper [@WW], we are able to conclude the following general $C^1$ estimates which are helpful for further studies of the properties of solutions $u$. **Theorem 2**. *Let $B_{4r}\subset M^n$ be some geodesic ball with $0<r\leq1$ and its Ricci curvature $Ric\geq-(n-1)Kg$ for some $K\geq0$. Let $u>0$ be a positive solution to equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B_{2r}$, $p>\frac{n}{2}$ and $q>1$.* *(1). Suppose $a\geq A_1>0$, there exists a constant $$C=C\left( n,K,p,q,A_1, \left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b\right| \right|^*_{p, B_r}, \left| \left|(\Delta a)^- \right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that on $B_{\frac{r}{2}}$, $$\label{1.6} \frac{\left|\nabla u\right|}{u^{1-\frac{1}{2q}}} \leq \frac{C}{r}.$$* *(2). Suppose $a\leq A_2<0$ and $u\leq D$, there exists a constant $$C=C\left( n,K,p,q,D,A_2, \left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b\right| \right|^*_{p, B_r}, \left| \left|(\Delta a)^+\right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that on $B_{\frac{r}{2}}$, $$\label{1.7} \frac{\left|\nabla u\right|}{u^{1-\frac{1}{2q}}} \leq \frac{C}{r}.$$* If the coefficients $a(x)$ and $b(x)$ of equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) are constants, one would expect better behaviors of $u$, such as Liouville type theorems. Moreover, in this situation, $u\equiv e^{-\frac{b}{a}}$ is the trivial positive solution. **Theorem 3**. *For constant coefficients $a>0$ and $b$, there exists a constant $\varepsilon=\varepsilon\left(n,a,b\right)>0$ such that if $$\label{Seo} 0<e^{-\frac{b}{a}}-\varepsilon\leq u\leq e^{-\frac{b}{a}}\quad\mbox{or}\quad e^{-\frac{b}{a}}\leq u\leq e^{-\frac{b}{a}}+\varepsilon$$ is a solution to ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B\left(x,\varepsilon^{-1}\right)$ with $Ric\geq 0$, then $u\equiv e^{-\frac{b}{a}}$. Especially, in these situations, except for $u\equiv e^{-\frac{b}{a}}$, there is no nonconstant solution $u$, which satisfies the above pinching condition [\[Seo\]](#Seo){reference-type="eqref" reference="Seo"}, such that $u\rightarrow e^{-\frac{b}{a}}$ at infinity on a complete non-compact Riemannian manifold with $Ric\geq 0$.* This paper is organized as follows. We give the $C^0$ estimates and gradient estimates of solutions to ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) respectively in Section [2](#sec2){reference-type="ref" reference="sec2"} and Section [3](#sec3){reference-type="ref" reference="sec3"}. In section [4](#sec4){reference-type="ref" reference="sec4"}, we are devoted to study the global properties of the solutions, and prove mainly the above local Liouville type Theorem [Theorem 3](#t1.3){reference-type="ref" reference="t1.3"}. # $C^0$ estimates {#sec2} First we recall the Saloff-Coste's Sobolev inequality which is crucial for the Nash-Moser iteration. **Theorem 4** ([@SC], Theorem 3.1). *Let $(M^n,g)$ be a complete Riemannian manifold with $Ric\geq-(n-1)K g$ on some $B_{2r}$. Then for $n\geq3$, there exists a dimensional constant $C_n$ such that for all $\varphi\in C^\infty_0(B_r)$, $$\left( \int_{B_r}\varphi^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq \frac{C_Sr^2}{\left|B_r\right|^{\frac{2}{n}}}\int_{B_r}\left(\left|\nabla \varphi \right|^2+r^{-2}\varphi^2\right),$$ where the Sobolev constant $$C_S=e^{C_n\left( 1+r\sqrt{K}\right)}.$$ Moreover, for $n=2$, the above inequality still holds with $n$ replaced by any $n'>2$. For convenience, we also write the Sobolev inequality as $$\label{2.1} \left( \fint_{B_r}\varphi^{\frac{2n}{n-2}}\right)^{\frac{n-2}{n}}\leq C_Sr^2\fint_{B_r}\left(\left|\nabla \varphi \right|^2+r^{-2}\varphi^2\right).$$* Here we remark that this type of Sobolev inequalities can be also derived by heat kernel estimates (cf. [@Li section 14]). Now we are going to prove Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"}. ***Proof of Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"}**.* **(1)** Let $w=\ln u$ and $G=\left|\nabla w\right|^2+aw+1$. By ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}), direct computations imply $$\label{2.2} \Delta w+\left|\nabla w\right|^2+aw+b=0,$$ and hence $$\label{2.3} \Delta w+G+b-1=0.$$ Without loss of generality, we may assume $w>0\, (i.e. \hspace*{0.5em}a(x)w(x)>0,\hspace*{0.5em}u(x)>1 \hspace*{0.5em}\mbox{and} \hspace*{0.5em}G(x)>1)$, since otherwise we can replace $w$ by $w^+$ in $G$. In the following, we only show the case of $n\geq 3$, and the case of $n=2$ can be proved by the same arguments. To apply Nash-Moser iteration, we need to estimate $\Delta G$. By ([\[2.2\]](#2.2){reference-type="ref" reference="2.2"}) and ([\[2.3\]](#2.3){reference-type="ref" reference="2.3"}), $$\begin{aligned} \Delta G&=\Delta\left|\nabla w\right|^2+\Delta(aw)\nonumber\\ &=2\left|D^2w\right|^2+2\left\langle\nabla w,\nabla \Delta w\right\rangle+2Ric(\nabla w,\nabla w)+a\Delta w+w\Delta a+2\left\langle \nabla a,\nabla w\right\rangle\nonumber\\ &\geq\frac{2(\Delta w)^2}{n}-2\left\langle\nabla w,\nabla (G+b)\right\rangle-2(n-1)K\left|\nabla w\right|^2-a(G+b-1)+w\Delta a-\left|\nabla a\right|^2-\left|\nabla w\right|^2\nonumber\\ &\geq \frac{2(G+b-1)^2}{n}-2\left\langle\nabla w,\nabla G\right\rangle-\left|\nabla b\right|^2-(G-aw-1)-2(n-1)K(G-aw-1)\nonumber\\ &\quad-a(G+b-1)+w\Delta a-\left|\nabla a\right|^2-(G-aw-1).\nonumber \end{aligned}$$ After recombining, $$\begin{aligned} \Delta G&\geq\frac{2G^2+2(b-1)^2}{n}-\left\lbrace 2(n-1)K+a+2-\frac{4(b-1)}{n}\right\rbrace G- 2\left\langle\nabla w,\nabla G\right\rangle\nonumber\\ &\quad +\left\lbrace 2(n-1)Ka+2a+\Delta a\right\rbrace w-\left|\nabla a\right|^2-\left|\nabla b\right|^2-ab+a\nonumber\\ &\geq\frac{2G^2}{n}-\left\lbrace 2(n-1)K+a+2-\frac{4(b-1)}{n}\right\rbrace G- 2\left\langle\nabla w,\nabla G\right\rangle\nonumber\\ &\quad +\left\lbrace 2(n-1)Ka+2a+\Delta a\right\rbrace w-\left|\nabla a\right|^2-\left|\nabla b\right|^2+\left(\sqrt{\frac{2}{n}}(b-1)-\frac{1}{2}\sqrt{\frac{n}{2}}a\right)^2-\frac{n}{8}a^2.\label{2.4} \end{aligned}$$ Since $a>0,0<aw<G, G>1$ and $K\geq0$, then by ([\[2.4\]](#2.4){reference-type="ref" reference="2.4"}), $$\begin{aligned} \Delta G&\geq\frac{2G^2}{n}-\left\lbrace 2(n-1)K+a-\frac{4b-4}{n}+2+\frac{(\Delta a)^- }{a}\right\rbrace G\nonumber\\ &- 2\left\langle\nabla w,\nabla G\right\rangle-\left|\nabla a\right|^2-\left|\nabla b\right|^2-\frac{n}{8}a^2.\label{2.5} \end{aligned}$$ Next, for any $l\geq0$ and $\eta\in C^\infty_0(B_{r})$, we multiply by $\eta^2G^l$ on both sides of ([\[2.5\]](#2.5){reference-type="ref" reference="2.5"}) and integrate on $B_{r}$, then $$\begin{aligned} &\int\eta^2G^l\Delta G\nonumber\\ &\geq\int\eta^2G^l\left( \frac{2G^2}{n}-\left\lbrace 2(n-1)K+a-\frac{4b-4}{n}+2+\frac{(\Delta a)^- }{a}\right\rbrace G- 2\left\langle\nabla w,\nabla G\right\rangle\right) \nonumber\\ &\quad-\int\eta^2G^{l+1}\left( \left|\nabla a\right|^2+\left|\nabla b\right|^2+\frac{n}{8}a^2\right).\label{2.6} \end{aligned}$$ By Green formula, we have $$\int\eta^2G^{l+1}\Delta w=-\int \left\langle \nabla\left( \eta^2G^{l+1}\right), \nabla w \right\rangle =-\int (l+1)\eta^2G^l\left\langle \nabla G, \nabla w\right\rangle-\int 2\eta G^{l+1}\left\langle \nabla\eta, \nabla w\right\rangle,$$ therefore $$\begin{aligned} &\int\eta^2G^l\left\langle \nabla G, \nabla w\right\rangle\nonumber\\&=-\frac{1}{l+1}\int\eta^2G^{l+1}\Delta w-\frac{2}{l+1}\int G^{l+1}\left\langle \nabla\eta, \eta\nabla w\right\rangle\nonumber\\ &\leq -\frac{1}{l+1}\int\eta^2G^{l+1}\left(-G-b+1\right)+\frac{1}{l+1}\int G^{l+1}\left(\left|\nabla\eta \right|^2 +\eta^2\left|\nabla w \right|^2\right)\nonumber\\ &\leq \frac{1}{l+1}\int\eta^2G^{l+2}+\frac{1}{l+1}\int \left( b\eta^2+\left|\nabla\eta \right|^2\right) G^{l+1}+\frac{1}{l+1}\int\eta^2G^{l+1}(G-aw-1)\nonumber\\ &\leq \frac{2}{l+1}\int\eta^2G^{l+2}+\frac{1}{l+1}\int b\eta^2G^{l+1}+\frac{1}{l+1}\int G^{l+1}\left|\nabla\eta \right|^2.\label{2.7} \end{aligned}$$ Combining ([\[2.6\]](#2.6){reference-type="ref" reference="2.6"}) and ([\[2.7\]](#2.7){reference-type="ref" reference="2.7"}) yields $$\begin{aligned} \label{2.8} &\int\eta^2G^l\Delta G\nonumber\\ &\geq \int \left(\frac{2}{n}-\frac{4}{l+1}\right)\eta^2G^{l+2}-\left(2(n-1)K+2+a+\frac{(\Delta a)^- }{a}+\frac{2b}{l+1} -\frac{4b-4}{n}\right)\eta^2G^{l+1}\nonumber\\ &\quad-\int\left( \left|\nabla a\right|^2+\left|\nabla b\right|^2+\frac{n}{8}a^2\right)\eta^2G^{l+1}-\frac{2}{l+1}\int\left|\nabla\eta \right|^2G^{l+1}. \end{aligned}$$ Next we let $l\geq 2n-1$, then by ([\[2.8\]](#2.8){reference-type="ref" reference="2.8"}), $$\begin{aligned} \label{2.9} &\int\eta^2G^l\Delta G\nonumber\\ &\geq-\left(2(n-1)K+2+a+\frac{4b^-+4}{n}+\frac{(\Delta a)^- }{a}+\left|\nabla a\right|^2+\left|\nabla b\right|^2+\frac{n}{8}a^2\right)\eta^2G^{l+1}\nonumber\\ &\quad-\frac{2}{l+1}\int\left|\nabla\eta \right|^2G^{l+1}. \end{aligned}$$ By (5.11) of [@DWZ], utilizing integration by parts, we have the following general integral inequality, $$\begin{aligned} \label{2.10} &\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2\nonumber\\ &\leq-\frac{\left( l+1\right)^2 }{2l}\int\eta^2G^l\Delta G+\frac{\left( l+1\right)^2+l }{l^2} \int G^{l+1}\left| \nabla\eta\right|^2-\frac{l+1}{l}\int \eta G^{l+1}\Delta \eta. \end{aligned}$$ Notice that $$\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2=\int\left(\eta^2\left| \nabla\left(G^{\frac{l+1}{2}}\right) \right|^2+\left|\nabla\eta\right|^2G^{l+1}+2\eta G^{\frac{l+1}{2}}\left\langle\nabla\eta,\nabla\left(G^{\frac{l+1}{2}}\right)\right\rangle\right),$$ $$\begin{aligned} -\int \eta G^{l+1}\Delta \eta=\int\left\langle \nabla\eta,\nabla\left(\eta G^{l+1} \right)\right\rangle&=\int\left( \left| \nabla\eta\right|^2G^{l+1}+\eta\left\langle\nabla\eta,\nabla G^{l+1} \right\rangle \right)\\ &=\int\left( \left| \nabla\eta\right|^2G^{l+1}+2\eta G^{\frac{l+1}{2}}\left\langle\nabla\eta,\nabla\left(G^{\frac{l+1}{2}}\right)\right\rangle\right). \end{aligned}$$ Also, $$\begin{aligned} &\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2+\frac{l+1}{l}\int2\eta G^{\frac{l+1}{2}}\left\langle\nabla\eta,\nabla\left(G^{\frac{l+1}{2}}\right)\right\rangle\\ &=\int\left(\eta^2\left| \nabla\left(G^{\frac{l+1}{2}}\right) \right|^2+\left|\nabla\eta\right|^2G^{l+1}-\frac{2}{l}\eta G^{\frac{l+1}{2}}\left\langle\nabla\eta,\nabla\left(G^{\frac{l+1}{2}}\right)\right\rangle\right)\\ &\geq\frac{l-1}{l}\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2. \end{aligned}$$ Combining these relational expressions and ([\[2.10\]](#2.10){reference-type="ref" reference="2.10"}), we have $$\begin{aligned} \label{2.11} &\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2\nonumber\\ &\leq-\frac{\left( l+1\right)^2 }{2(l-1)}\int\eta^2G^l\Delta G+\frac{\left( l+1\right)^2+l(l+2)}{l(l-1)} \int G^{l+1}\left| \nabla\eta\right|^2. \end{aligned}$$ Then by ([\[2.11\]](#2.11){reference-type="ref" reference="2.11"}) and the assumption $G>1$, we infer from ([\[2.9\]](#2.9){reference-type="ref" reference="2.9"}) that $$\begin{aligned} &\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2\nonumber\\ &\leq\frac{\left( l+1\right)^2 }{2(l-1)}\int \left(2(n-1)K+2+a+\frac{4b^-+4}{n}+\frac{(\Delta a)^- }{a}+\left|\nabla a\right|^2+\left|\nabla b\right|^2+\frac{n}{8}a^2\right)\eta^2G^{l+1}\nonumber\\ &\quad+\frac{l\left(2l+3\right)+(l+1)^2}{l(l-1)}\int \left| \nabla\eta\right|^2G^{l+1}.\label{2.12} \end{aligned}$$ By the Sobolev inequality ([\[2.1\]](#2.1){reference-type="ref" reference="2.1"}) with $\varphi=\eta G^{\frac{l+1}{2}}$ and $0<r\leq1$, ([\[2.11\]](#2.11){reference-type="ref" reference="2.11"}) gives $$\begin{aligned} \label{2.13} &\left( \fint_{B_r}\left(\eta^2 G^{l+1}\right)^{\frac{n}{n-2}}\right)^{\frac{n-2}{n}}\nonumber\\ &\leq C_Sl\fint_{B_r}\left(2(n-1)K+3+a+\frac{4b^-+4}{n}+\frac{(\Delta a)^- }{a}+\left|\nabla a\right|^2+\left|\nabla b\right|^2+\frac{n}{8}a^2\right) \eta^2G^{l+1}\nonumber\\ &\quad+C_Sr^2l\fint_{B_r}\left| \nabla\eta\right|^2G^{l+1}. \end{aligned}$$ Furthermore, by interpolation inequality(cf. [@WW section 3]) with $p>\frac{n}{2}$ and $\mu=\frac{n}{n-2}$, there exists a constant $C_2=C_2\left(n,p\right)$ such that $$\label{2.13*} C_Sl\fint a\eta^2G^{l+1}\leq\frac{1}{10}\left(\fint\left( \eta^2G^{l+1}\right)^\mu\right)^{\frac{1}{\mu}}+C_2\left(C_Sl \left| \left| a\right| \right|^*_{p, B_r}\right)^{\frac{2p}{2p-n}}\fint\eta^2G^{l+1}.$$ Applying similar interpolation inequalities to the remainder terms and plugging these inequalities into ([\[2.13\]](#2.13){reference-type="ref" reference="2.13"}) implies that there exists a constant $$C_3=C_3\left( C_2,n,p,K,\left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b^-\right| \right|^*_{p, B_r}, \left| \left| \frac{(\Delta a)^- }{a}\right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)$$ such that $$\begin{aligned} \left( \fint_{B_r}\left(\eta^2 G^{l+1}\right)^{\frac{n}{n-2}}\right)^{\frac{n-2}{n}}\leq C_3\left(C_Sl\right)^{\frac{2p}{2p-n}}\fint\eta^2v^{l+1}+10C_Sr^2l\fint_{B_r}\left| \nabla\eta\right|^2G^{l+1}.\label{2.14} \end{aligned}$$ Here we used the monotonic inequality $$\left| \left|f\right| \right|^*_{p, B_r}\leq\left| \left| f\right| \right|^*_{2p, B_r}.$$ Let $\phi(s)$ be a non-negative $C^2$-smooth function on $\left[0,+\infty \right)$ such that $\phi(s)=1$ for $s\leq \frac{1}{2}$, $\phi(s)=0$ for $s\geq1$, and $-4\leq\phi'\leq0$. Next, let $\eta(y)=\phi\left(\frac{d(y,x)}{r}\right)$ where $d(y,x)$ denotes the distance from $y$ to $x$ and it is obvious that $\eta(y)$ is supported in $B_r$: $$\begin{aligned} &\eta|_{B_\frac{r}{2}}=1,\\ &\eta|_{M\backslash B_r}=0. \end{aligned}$$ Clearly, $$\begin{aligned} \left| \nabla \eta\right|^2\leq\frac{16}{r^2}. \end{aligned}$$ Recall $l\geq 2n-1$, then by standard iteration(e.g. see [@WW section 3]), for some constant $C_4=C_4(n,C_3,C_S,p)$, $$\label{2.15} \sup\limits_{B_\frac{r}{2}}G\leq C_5\left\|G\right\|^*_{2n-1, B_r}.$$ Since $Ric\geq-(n-1)Kg$, by Bishop-Gromov volume comparison theorem, the geodesic ball $B_r$ satisfies the volume doubling property, hence by the standard trick of lowering power(cf. [@WW Lemma 2.5]), ([\[2.15\]](#2.15){reference-type="ref" reference="2.15"}) can be improved to $$\label{2.16} \sup\limits_{B_\frac{r}{2}}G\leq C_5\left\|G\right\|^*_{1, B_\frac{4r}{5}},$$ where $C_5=C_5\left(C_3,C_4,C_S,n,K\right).$ Next, we choose some $\eta\in C^\infty_0(B_r)$ such that $$\eta\equiv1\quad\mbox{on}\hspace*{0.3em}B_\frac{4r}{5}\quad\mbox{and}\quad \left| \nabla\eta\right|\leq\frac{10}{r}.$$ For $\eta^2G$, since $aw>0$ and by ([\[2.3\]](#2.3){reference-type="ref" reference="2.3"}), there holds $$\begin{aligned} \int\eta^2G=-\int\eta^2\left( \Delta w+b-1\right)&=2\int\left\langle \nabla\eta, \eta\nabla w\right\rangle-\int\eta^2(b-1)\nonumber\\ &\leq2\int \left| \nabla\eta\right|^2+\frac{1}{2}\int\eta^2\left| \nabla w\right|^2-\int\eta^2(b-1)\nonumber\\ &=2\int\left|\nabla\eta\right|^2+\frac{1}{2}\int\eta^2G-\frac{1}{2}\int\eta^2(aw+1)-\int\eta^2(b-1)\\ &\leq 2\int \left| \nabla\eta\right|^2+\frac{1}{2}\int\eta^2G+\int\eta^2b^-+\frac{1}{2}\int\eta^2. \end{aligned}$$ Therefore, $$\begin{aligned} \label{2.17} \int\eta^2G\leq4\int \left| \nabla\eta\right|^2+2\int\eta^2b^-+\int\eta^2. \end{aligned}$$ As a consequence of ([\[2.17\]](#2.17){reference-type="ref" reference="2.17"}), $$\begin{aligned} \label{2.18} 0<\left\|G\right\|^*_{1, B_\frac{4r}{5}}\leq\frac{\int\eta^2G}{\left| B_\frac{4r}{5}\right|}\leq\frac{\left( 400+2\left| \left| b^-\right| \right|^*_{p, B_r}+1\right)\left| B_r\right|}{\left| B_\frac{4r}{5}\right|r^2}. \end{aligned}$$ Finally, by the definition of $G$, combining the volume comparison theorem and ([\[2.18\]](#2.18){reference-type="ref" reference="2.18"}) completes the proof. **(2)** (i) The proof is almost the same as above with a few minor modifications. Let $w=\ln u$ and $G=\left|\nabla w\right|^2+aw+1$. Without loss of generality, we may assume $w<0(i.e. \hspace*{0.5em}a(x)w(x)>0,\hspace*{0.5em}u(x)<1 \hspace*{0.5em}\mbox{and} \hspace*{0.5em}G(x)>1)$. Then the estimate ([\[1.4\]](#1.4){reference-type="ref" reference="1.4"}) follows verbatim. As for (ii), we need a few minor modifications based on the proof as above. Here for simplicity, we only consider the case of $a(x)<0$ is constant. Let $w=\ln u$ and $G=\left|\nabla w\right|^2-aw+1$. By ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}), direct computations imply $$\Delta w+\left|\nabla w\right|^2+aw+b=0,$$ and hence $$\Delta w+G+2aw+b-1=0.$$ Without loss of generality, we may assume $w>0(i.e. \hspace*{0.5em}a(x)w(x)<0,\hspace*{0.5em}u(x)>1 \hspace*{0.5em}\mbox{and} \hspace*{0.5em}G(x)>1)$. Then the same arguments as above give the desired estimate of the upper bound of $\ln u$. The only difference is that during the iterative process, we need $\left\|w^+\right\|^*_{p, B_r}$ to control the term $-\fint_{B_r}w\eta^2G^{l+1}$ as we did in ([\[2.13\*\]](#2.13*){reference-type="ref" reference="2.13*"}). ◻ **Remark 5**. *Since all the elements we need, such as the Sobolev inequality and comparison geometry, have counterparts under integral Ricci curvature conditions, hence the above local $C^0$ and the following $C^1$ estimates of the solution $u$ are still valid under integral Ricci curvature conditions. See [@WW] for related discussions.* # $C^1$ estimates {#sec3} ***Proof of Theorem [Theorem 2](#t1.2){reference-type="ref" reference="t1.2"}**.* **(1)** By Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"}, there exists a constant $D>0$ such that $u\leq D$ on $B_r$. Set $w=u^{\frac{1}{q}}$ for some $q>1$, then by ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}), $w$ satisfies $$\label{3.1} \Delta w+(q-1)\frac{\left| \nabla w\right|^2}{w}+aw\log w+\frac{bw}{q}=0.$$ Let $v=(q-1)\frac{\left| \nabla w\right|^2}{w}$ and $b'=\frac{b}{q}$. Without loss of generality, we may assume $v\geq1$, since otherwise we may consider $v+1$ instead of $v$. By Bochner formula, $$\begin{aligned} \Delta v=&(q-1)\Delta\left( \frac{\left| \nabla w\right|^2}{w}\right)\nonumber\\ =&(q-1)\left[ \frac{\Delta\left| \nabla w\right|^2}{w}+\left| \nabla w\right|^2\Delta\left( \frac{1}{w}\right)+2\left\langle \nabla(\left| \nabla w\right|^2), \nabla\left(\frac{1}{w} \right) \right\rangle \right] \nonumber\\ \geq&(q-1)\left[ \frac{2\left| D^2 w\right|^2+2\left\langle\nabla w, \nabla\Delta w \right\rangle - 2(n-1)K\left| \nabla w\right|^2 }{w}+\left| \nabla w\right|^2\left( -\frac{\Delta w}{w^2}+\frac{2\left| \nabla w\right|^2}{w^3}\right)\right]\nonumber\\ & +2(q-1)\left\langle \nabla\left| \nabla w\right|^2, -\frac{\nabla w}{w^2} \right\rangle.\label{3.2} \end{aligned}$$ Substituting the equations $\Delta w=-v-aw\log w-b'w$ and $\left| \nabla w\right|^2=\frac{vw}{q-1}$ into ([\[3.2\]](#3.2){reference-type="ref" reference="3.2"}) gives $$\begin{aligned} \label{3.3} \Delta v\geq&(q-1)\left[ \frac{2\left| D^2 w\right|^2}{w}+\frac{v^2}{(q-1)w}-\left( \frac{2}{w}+\frac{2}{(q-1)w}\right)\left\langle\nabla w,\nabla v \right\rangle\right]\nonumber\\ &-2(q-1)\left( \ln w\left\langle\nabla a,\nabla w\right\rangle+\left\langle \nabla w, \nabla b'\right\rangle\right) -\left(2(n-1)K+a\log w+b'+2a\right) v. \end{aligned}$$ Moreover, $$-2\left\langle \nabla w, \nabla b'\right\rangle\geq-\frac{\left| \nabla w\right|^2}{w}-w\left| \nabla b'\right|^2,$$ $$\begin{aligned} -2\ln w\left\langle\nabla a,\nabla w\right\rangle\geq-2 \left|\ln w\right|\left|\nabla a\right|\left|\nabla w\right|&=-2\left|\nabla a\right|\left|\ln w\right|\sqrt{\frac{wv}{q-1}}\\ &=-2\left|\nabla a\right|\left(\sqrt{w}\left|\ln w\right|\right)\sqrt{\frac{v}{q-1}}\\ &\geq-\frac{\max\left\lbrace 2e^{-1},D^{\frac{1}{2}}\left| \ln D\right| \right\rbrace }{(q-1)^{\frac{1}{2}}}\left(\left|\nabla a\right|^2+v\right)\\ &=:-C_1\left(\left|\nabla a\right|^2+v\right), \end{aligned}$$ hence ([\[3.3\]](#3.3){reference-type="ref" reference="3.3"}) with $v\geq1$ leads to $$\begin{aligned} \label{3.4} \Delta v\geq&(q-1)\left[ \frac{v^2}{(q-1)w}-\left( \frac{2}{w}+\frac{2}{(q-1)w}\right)\left\langle\nabla w,\nabla v \right\rangle\right]-\left\lbrace C_2+a\left(2+\ln D\right)+b'\right\rbrace v\nonumber\\ &-2\left(q-1\right)\left(D^{\frac{1}{q}}\left| \nabla b'\right|^2+C_1\left| \nabla a\right|^2\right)v, \end{aligned}$$ where $C_2=C_2\left(n,q,K,D,C_1\right)$. Next, for any $l\geq0$ and $\eta\in C^\infty_0\left( B_r\right)$, we multiply by $\eta^2v^l$ on the both sides of ([\[3.4\]](#3.4){reference-type="ref" reference="3.4"}), then $$\begin{aligned} \label{3.5} \int\eta^2v^l\Delta v\geq&\int\left( \frac{\eta^2v^{l+2}}{w}-2q\eta^2v^l\left\langle\nabla \log w, \nabla v \right\rangle-\left\lbrace C_2+a\left(2+(\ln D)^+\right)+(b')^+\right\rbrace \eta^2v^{l+1}\right)\nonumber\\ &-2\int\left(q-1\right)\left(D\left| \nabla b'\right|^2+C_1\left| \nabla a\right|^2\right)\eta^2v^{l+1}. \end{aligned}$$ By Green's formula and Cauchy-Schwartz inequality, $$\begin{aligned} &\int\eta^2v^l\left\langle \nabla v, \nabla \log w\right\rangle\nonumber\\ &=-\frac{1}{l+1}\int\eta^2v^{l+1}\Delta \log w-\frac{2}{l+1}\int v^{l+1}\left\langle \nabla\eta, \eta\nabla \log w\right\rangle\nonumber\\ &\leq -\frac{1}{l+1}\int\eta^2v^{l+1}\left( \frac{\Delta w}{w}-\frac{\left| \nabla w\right|^2}{w^2}\right)+\frac{1}{l+1}\int v^{l+1}\left(\left|\nabla\eta \right|^2 +\eta^2\frac{\left| \nabla w\right|^2}{w^2}\right)\label{3.6}. \end{aligned}$$ Substituting the equations $\Delta w=-v-aw\log w-b'w$ and $\left| \nabla w\right|^2=\frac{vw}{q-1}$ into ([\[3.6\]](#3.6){reference-type="ref" reference="3.6"}) gives $$\label{3.7} \begin{split} &\int\eta^2v^l\left\langle \nabla v, \nabla \log w\right\rangle\\ &\leq\left( \frac{1}{l+1}+\frac{2}{(l+1)(q-1)}\right)\int\frac{\eta^2v^{l+2}}{w}+\frac{1}{l+1}\left( \int C_3\eta^2v^{l+1}+\int\left| \nabla\eta \right|^2v^{l+1}\right), \end{split}$$ where $$C_3=\left( a\log D^{\frac{1}{q}}\right)^++(b')^+.$$ Combining ([\[3.5\]](#3.5){reference-type="ref" reference="3.5"}) and ([\[3.7\]](#3.7){reference-type="ref" reference="3.7"}), for some constant $C_4=C_4(q,D,C_1,C_2)$, $$\begin{aligned} &\int\eta^2v^l\Delta v\nonumber\\ &\geq\int\left[1-\left(\frac{2q}{l+1}+\frac{4q}{(l+1)(q-1)} \right)\right]\frac{\eta^2v^{l+2}}{w}\\ &\quad-C_4\left( \int \left(a+(b)^++\left| \nabla a\right|^2+\left| \nabla b\right|^2\right)\eta^2v^{l+1}+\int\left| \nabla\eta \right|^2v^{l+1}\right). \end{aligned}$$ Let $l+1\geq2q+\frac{4q}{q-1}$, then $1-\left(\frac{2q}{l+1}+\frac{4q}{(l+1)(q-1)}\right)\geq0$, and hence $$\label{3.8} \int\eta^2v^l\Delta v\geq-C_4\left( \int \left(a+(b)^++\left| \nabla a\right|^2+\left| \nabla b\right|^2\right)\eta^2v^{l+1}+\int\left| \nabla\eta \right|^2v^{l+1}\right).$$ Plugging ([\[3.8\]](#3.8){reference-type="ref" reference="3.8"}) into the integral inequality ([\[2.11\]](#2.11){reference-type="ref" reference="2.11"}) implies $$\begin{aligned} &\int\left| \nabla\left( \eta v^{\frac{l+1}{2}}\right) \right|^2\nonumber\\ &\leq\frac{\left(l+1\right)^2C_4}{2(l-1)}\int \left(a+(b)^++\left| \nabla a\right|^2+\left| \nabla b\right|^2\right) \eta^2v^{l+1}+\frac{(lC_4+2)(l+1)^2+2l(l+2)}{2l(l-1)}\int\left| \nabla\eta\right|^2v^{l+1}. \end{aligned}$$ Then following the same iteration steps as to get ([\[2.16\]](#2.16){reference-type="ref" reference="2.16"}), there exists a constant $$C_5=C_5\left(n,p,q,K,C_S,C_4,\left| \left| a\right| \right|^*_{p, B_r},\left| \left| b^+\right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r}, \left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)>0$$ such that $$\label{3.10} \left| \left| v\right| \right|_{\infty,B_{\frac{r}{2}}}\leq C_5\left| \left| v\right| \right|^*_{1,B_\frac{4r}{5}}.$$ Next, we choose some $\eta\in C^\infty_0(B_r)$ such that $$\eta\equiv1\quad\mbox{on}\hspace*{0.3em}B_\frac{4r}{5}\quad\mbox{and}\quad \left| \nabla\eta\right|\leq\frac{10}{r}.$$ For $\eta^2v$, there holds $$\label{3.11} \begin{split} \int\eta^2v=-\int\eta^2\left( \Delta w+aw\log w+b'w\right). \end{split}$$ By Green's formula, $$-\int\eta^2\Delta w=2\int\left\langle \nabla\eta, \eta\nabla w\right\rangle\leq\frac{1}{2}\int\eta^2v+2\int\frac{w\left| \nabla\eta\right|^2}{q-1}\leq\frac{1}{2}\int\eta^2v+\frac{200D^{\frac{1}{q}}}{(q-1)r^2}\left| B_r\right|.$$ On the other hand, we also have $$-\int\eta^2(aw\log w+b'w)\leq\left( \frac{\left| \left| a\right| \right|^*_{p, B_r}}{e}+\frac{\left| \left| b^-\right| \right|^*_{p, B_r}D^{\frac{1}{q}}}{q}\right)\left|B_r \right|.$$ Here, we have used the following facts that for $w\in(0, +\infty)$ there holds $$-w\log w\leq\frac{1}{e}.$$ Now, by ([\[3.11\]](#3.11){reference-type="ref" reference="3.11"}) and $0<r\leq1$, we estimate $$\begin{aligned} \label{3.12} \left| \left| v\right| \right|^*_{1,B_\frac{4r}{5}}\leq\frac{\int\eta^2v}{\left|B_{\frac{4r}{5}}\right|}&\leq\frac{\left(\frac{400D^{\frac{1}{q}}}{(q-1)r^2}+\frac{2\left| \left| a\right| \right|^*_{p, B_r}}{e}+\frac{2\left| \left|b^-\right| \right|^*_{p, B_r}D^{\frac{1}{q}}}{q}\right)\left|B_r \right|}{\left|B_{\frac{4r}{5}} \right|}\nonumber\\ &\leq\frac{\left(\frac{400D^{\frac{1}{q}}}{(q-1)}+\frac{2\left| \left| a\right| \right|^*_{p, B_r}}{e}+\frac{2\left| \left| b^-\right| \right|^*_{p, B_r}D^{\frac{1}{q}}}{q}\right)\left|B_r \right|}{r^2\left|B_{\frac{4r}{5}} \right|}. \end{aligned}$$ Substituting ([\[3.12\]](#3.12){reference-type="ref" reference="3.12"}) into ([\[3.10\]](#3.10){reference-type="ref" reference="3.10"}) and noticing $$\frac{\left|\nabla w \right|^2}{w}=\frac{\left|\nabla u \right|^2u^{\frac{1}{q}-2}}{q^2},$$ then by volume comparison theorem, we get the desired estimate $$\sup\limits_{B\left( x,\frac{r}{2}\right) }\frac{\left|\nabla u\right|}{u^{1-\frac{1}{2q}}} \leq \frac{C}{r},$$ where $C=C\left( n,K,p,q,A_1, \left| \left| a\right| \right|^*_{2p, B_r}, \left| \left| b\right| \right|^*_{p, B_r}, \left| \left|(\Delta a)^- \right| \right|^*_{p, B_r}, \left| \left|\nabla a\right| \right|^*_{2p, B_r},\left| \left|\nabla b\right| \right|^*_{2p, B_r}\right)$. **(2)** The proof is the same as above with a few minor modifications and the extra condition $u\leq D$ is used to control $\left\langle\nabla b,\nabla w\right\rangle$ and $\ln w\left\langle\nabla a,\nabla w\right\rangle$. ◻ Due to the previous $C^0$, $C^1$ estimates and standard bootstrapping arguments for elliptic PDEs, we immediately have following regularity of $u$. **Corollary 6**. *Let $u>0$ be a positive (weak) solution to the equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B_r\subset(M^n,g)$ with $a(x), b(x)\in C^\infty\left(B_r\right)$. Suppose $a(x)>0$, or $a(x)<0$ and $u$ is bounded from above, then $u\in C^\infty\left(B_{r'}\right)$ for any $B_{r'}\subset\subset B_r$.* *Proof.* By our estimates of Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"} and [Theorem 2](#t1.2){reference-type="ref" reference="t1.2"}, we know $u, \left|\nabla u\right|,u\ln u,\left| \nabla(u\ln u)\right|$ are all locally bounded, where for $q>1$, we estimate $$\left| \nabla(u\ln u)\right|\leq\left|\ln u\right| \left| \nabla u\right|+\left| \nabla u\right|\leq\frac{C\left( \left|\ln u\right|+1\right)u^{1-\frac{1}{2q}}}{r}\leq\frac{C}{r}.$$ Then standard bootstrapping arguments give the desired regularity (cf. [@GT section 8]). ◻ From the view of specific geometric problems, integrable solutions to ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) with $a(x)>0$ are the most important ones such as the minimizers of $\mathcal{W}$ entropy. Based on the previous estimates, we find some further properties of these solutions on non-compact manifolds. **Corollary 7**. *Let $u>0$ be a positive solution to the equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on a complete non-compact and non-collapsing Riemannian manifold $(M^n,g)$ with $Ric\geq-(n-1)Kg$ for some $K\geq0$. Suppose on $(M^n,g)$, $0<A_1\leq a\leq A_2$, $\left|\nabla a\right|\leq A_3$, $(\Delta a)^-\leq A_4$, $\left|b\right|\leq B_1$, $\left|\nabla b\right|\leq B_2$ and $\int_{M^n}u^k<+\infty$ for some $k>0$, then $u(x)\longrightarrow0$ as $x\longrightarrow\infty$ uniformly. Especially, in this situation, $u$ has a maximum point $x_0$ such that $$u(x_0)\geq e^{-\frac{B_1}{A_1}}.$$* *Proof.* Since $\int_{M^n}u^k<+\infty$, then for $x\longrightarrow\infty$, $$\label{3.13} \int_{B(x,1)}u^k\longrightarrow0.$$ However, by Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"} and [Theorem 2](#t1.2){reference-type="ref" reference="t1.2"}, $\left|\nabla u\right|$ is uniformly bounded on $(M^n,g)$. Hence if there is a sequence $x_k\longrightarrow\infty$ such that $u\geq c>0$, then by the gradient estimate and the non-collapsing condition, $\int_{B(x,1)}u^k$ has a strict positive lower bound which contradicts ([\[3.13\]](#3.13){reference-type="ref" reference="3.13"}). Hence $u\longrightarrow0$ at infinity. Moreover, since $u\longrightarrow0$ at infinity, $u$ must have a maximum point $x_0$ and then by maximum principle, $\Delta u(x_0)\leq 0$, hence $$a(x_0)\ln u(x_0)+b(x_0)\geq0.$$ Finally, $$u(x_0)\geq e^{-\frac{b(x_0)}{a(x_0)}}\geq e^{-\frac{B_1}{A_1}}.$$ ◻ In form, the solution $u$ to equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) is a generalization of positive harmonic functions, so one may expect that its gradient behaves like that of positive harmonic functions. Especially, one may ask whether there exists a constant $C>0$ such that on $(M^n,g)$ with $Ric\geq-(n-1)Kg$, $$\label{3.14} \frac{\left|\nabla u\right|}{u}\leq C.$$ As a supplement of our Theorem [Theorem 2](#t1.2){reference-type="ref" reference="t1.2"}, in the following, we claim that ([\[3.14\]](#3.14){reference-type="ref" reference="3.14"}) is impossible if $u\longrightarrow0$. **Proposition 8**. *Let $u>0$ be a positive solution to the equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B(x,r)\subset(M^n,g)$ with $0<r\leq 1$ and $Ric\geq-(n-1)Kg$ for some $K\geq0$. Suppose on $B(x,r)$, $0<A_1\leq a\leq A_2$, $\left|b\right|\leq B_1$ and $$\label{3.15} \frac{\left|\nabla u\right|}{u}\leq C_1,$$ then there exists a constant $C=C\left( n,K,A_1,B_1,C_1\right)>0$ such that $$\label{3.15*} u(x)\geq e^{-C}.$$ Consequently, there is no $u$ such that ([\[3.15\]](#3.15){reference-type="ref" reference="3.15"}) is valid but $u\longrightarrow0$. Especially, if ([\[3.15\]](#3.15){reference-type="ref" reference="3.15"}) holds on $(M^n,g)$, there is no $u$ satisfying the conditions of Corollary [Corollary 7](#c3.3){reference-type="ref" reference="c3.3"}.* *Proof.* Let $w=\ln u$. Then by Green's formula, $$\begin{aligned} \label{3.16} \int_{B(x,r)}\Delta w=\int_{\partial B(x,r)}\left\langle \textbf{n}, \nabla w\right\rangle, \end{aligned}$$ where $\textbf{n}$ is the outward normal vector of $\partial B(x,r)$. By volume comparison theorem and co-area formula, as well as mean value theorem for integral, for some $C_2=C_2(n,K)>0$, we have $\left|\partial B(x,r)\right|\leq C_2\left| B(x,r)\right|$. Since $w$ satisfies equation ([\[2.2\]](#2.2){reference-type="ref" reference="2.2"}) and $\left|\nabla w\right|\leq C_1$, therefore by Jensen's inequality, $$\label{3.17*} \ln\left(\fint_{B(x,r)}u\right)\geq\fint_{B(x,r)}w\geq-C.$$ Utilizing $\left|\nabla w\right|\leq C_1$ again, it's easy to know the following Harnack inequality for $y\in B(x,r)$: $$\label{3.18*} \frac{u(y)}{u(x)}\leq e^{C_1}.$$ Then combining ([\[3.17\*\]](#3.17*){reference-type="ref" reference="3.17*"}) and ([\[3.18\*\]](#3.18*){reference-type="ref" reference="3.18*"}) gives ([\[3.15\*\]](#3.15*){reference-type="ref" reference="3.15*"}). ◻ # A Liouville type theorem {#sec4} In this section, we focus on equation ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) with constant coefficients $a(x)>0$ and $b(x)$. In this situation, ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) becomes $$\label{4.1} \Delta u(x)+au(x)\ln u(x)+bu(x)=0,$$ and has a constant solution $u(x)\equiv e^{-\frac{b}{a}}$. First we observe that if the upper bound of $u$ is small enough, then $u\equiv0$. **Proposition 9**. *Let $u>0$ be a positive solution to the equation ([\[4.1\]](#4.1){reference-type="ref" reference="4.1"}) on non-compact $(M^n,g)$ with $Ric\geq0$. Then there is no $u$ such that $u\leq e^{-\frac{b}{a}-1}$.* *Proof.* If $u\leq e^{-\frac{b}{a}-1}$, then by ([\[4.1\]](#4.1){reference-type="ref" reference="4.1"}), $\Delta u\geq0$. On the other hand, we compute $$\begin{aligned} \left|\nabla u\right|\Delta\left|\nabla u\right|+\left\langle \nabla\left|\nabla u\right|, \nabla\left|\nabla u\right| \right\rangle=\frac{1}{2}\Delta\left|\nabla u\right|^2&=\left|D^2u\right|^2+\left\langle \nabla u, \nabla\Delta u\right\rangle+Ric\left(\nabla u,\nabla u\right)\\ &\geq \left|D^2u\right|^2-\left\langle \nabla u, \nabla\left(au\ln u+bu\right)\right\rangle\\ &\geq \left|D^2u\right|^2-\left(a+a\ln u+b\right)\left|\nabla u\right|^2\\ &\geq \left|D^2u\right|^2, \end{aligned}$$ hence $\Delta\left|\nabla u\right|\geq0$. Then by the mean value inequality for subharmonic functions due to Li-Schoen [@LS](see also [@SY chapter 2,Theorem 6.1]), there exists a uniform constant $C_1>0$ such that for any $B(x,r)$, $$\left|\nabla u\right|^2(x)\leq C_1\fint_{B(x,r)}\left|\nabla u\right|^2.$$ On the other hand, by the integral estimate of non-negative sub-harmonic functions [@SY chapter 2, Lemma 6.3], there exists a uniform constant $C_2>0$ such that $$\left|\nabla u\right|^2(x)\leq\frac{C_2}{r^2}\fint_{B(x,r)}u^2.$$ In light of Theorem [Theorem 1](#t1.1){reference-type="ref" reference="t1.1"}, we know $u\leq D$ where $D$ is a uniform upper bound of $u$, hence by the above arguments, there exists a uniform constant $C>0$ such that $$\left|\nabla u\right|^2(x)\leq\frac{C}{r^2}.$$ Then setting $r\longrightarrow +\infty$ implies $u$ must be constant and contradicts $0<u\leq e^{-\frac{b}{a}-1}<e^{-\frac{b}{a}}$. ◻ The above result implies that around $u\equiv0$, there is no positive solution to equation ([\[4.1\]](#4.1){reference-type="ref" reference="4.1"}). In fact, this type of gap phenomenon also occurs to the constant solution $u\equiv e^{-\frac{b}{a}}$. **Theorem 10** (=Theorem [Theorem 3](#t1.3){reference-type="ref" reference="t1.3"}). *For constant coefficients $a>0$ and $b$, there exists a constant $\varepsilon=\varepsilon\left(n,a,b\right)>0$ such that if $$\label{4.2*} 0<e^{-\frac{b}{a}}-\varepsilon\leq u\leq e^{-\frac{b}{a}}\quad\mbox{or}\quad e^{-\frac{b}{a}}\leq u\leq e^{-\frac{b}{a}}+\varepsilon$$ is a solution to ([\[1.1\]](#1.1){reference-type="ref" reference="1.1"}) on $B\left(x,\varepsilon^{-1}\right)$ with $Ric\geq 0$, then $u\equiv e^{-\frac{b}{a}}$. Especially, in these situations, except for $u\equiv e^{-\frac{b}{a}}$, there is no nonconstant solution $u$, which satisfies the above pinching condition ([\[4.2\*\]](#4.2*){reference-type="ref" reference="4.2*"}), such that $u\rightarrow e^{-\frac{b}{a}}$ at infinity on a complete non-compact Riemannian manifold with $Ric\geq 0$.* *Proof.* Since the proofs of two cases are almost the same, here we only need to give the proof of the later case, i.e. $e^{-\frac{b}{a}}\leq u\leq e^{-\frac{b}{a}}+\varepsilon$. For $u> e^{-\frac{b}{a}}$, we rewrite equation ([\[4.1\]](#4.1){reference-type="ref" reference="4.1"}) as $$\label{4.2} \Delta\left(u-e^{-\frac{b}{a}}\right)+V\left(u-e^{-\frac{b}{a}}\right)=0,$$ where $$V=\frac{u(a\ ln u+b)}{\left(u-e^{-\frac{b}{a}}\right)}.$$ Moreover, it's easy to check that $\lim\limits_{u\longrightarrow e^{-\frac{b}{a}}}V(u)=a>0$ and $\lim\limits_{u\longrightarrow e^{-\frac{b}{a}}}V'(u)=\frac{a}{2e^{-\frac{b}{a}}}>0$, hence if we set $w=u-e^{-\frac{b}{a}}\geq 0$, then by ([\[4.1\]](#4.1){reference-type="ref" reference="4.1"}) and ([\[4.2\]](#4.2){reference-type="ref" reference="4.2"}), the following Schrödinger equation is well defined for all $w\geq0$: $$\label{4.3} \Delta w+Vw=0.$$ Also, we may choose a small $\varepsilon=\varepsilon(a,b)>0$ such that $V\geq \frac{2a}{3}$. As a consequence of the strong maximum principle, either $w\equiv0$ or $w>0$ in $B\left(x,\varepsilon^{-1}\right)$, where $\varepsilon$ is determined later, therefore we only need to consider the case of $w>0$. Similarly, if $u\leq e^{-\frac{b}{a}}$, then we set $w=e^{-\frac{b}{a}}-u\geq 0$ and consider $$\Delta w+Uw=0,$$ where $$U=\frac{-u(a\ ln u+b)}{\left(e^{-\frac{b}{a}}-u\right)}.$$ As before, let $v=\ln w=\ln\left(u-e^{-\frac{b}{a}}\right)$ and $G=\left| \nabla v\right|^2+V$, then by ([\[4.3\]](#4.3){reference-type="ref" reference="4.3"}), $$\begin{aligned} \label{4.4} \Delta v+G=0. \end{aligned}$$ First we compute $$\begin{aligned} \Delta V&=\Delta\left(\frac{u(a\ ln u+b)}{\left(u-e^{-\frac{b}{a}}\right)}\right)\\ &=\frac{\Delta \left( u(a\ ln u+b)\right)}{u-e^{-\frac{b}{a}}}+ u(a\ ln u+b)\Delta \left(u-e^{-\frac{b}{a}}\right)^{-1}+2\left\langle\nabla \left( u(a\ ln u+b)\right), \nabla \left(u-e^{-\frac{b}{a}}\right)^{-1} \right\rangle\\ &=\left\lbrace\frac{a(1+\ln u)+b}{u-e^{-\frac{b}{a}}}-\frac{u(a\ln u+b)}{\left(u-e^{-\frac{b}{a}} \right)^2}\right\rbrace\Delta u\\ &\quad+\left\lbrace\frac{a}{u\left( u-e^{-\frac{b}{a}}\right)}+\frac{2u(a\ln u+b)}{\left(u-e^{-\frac{b}{a}}\right)^3}-\frac{2a(1+\ln u)+2b}{\left(u-e^{-\frac{b}{a}} \right)^2}\right\rbrace\left| \nabla u\right|^2\\ &=: \uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}. \end{aligned}$$ Recall $\Delta u=-au\ln u-bu$, then we can rewrite $\uppercase\expandafter{\romannumeral1}=V^2-(a+a\ln u+b)V$. Notice that $$\lim\limits_{u\longrightarrow e^{-\frac{b}{a}}}\frac{\uppercase\expandafter{\romannumeral1}}{V}=\lim\limits_{u\longrightarrow e^{-\frac{b}{a}}}\left(V-(a+a\ln u+b)\right)=0,$$ hence it's easy to know, for all $\varepsilon=\varepsilon(a,b)>0$ small enough, if $u\leq e^{-\frac{b}{a}}+\varepsilon$, then $$\label{4.5} \frac{\uppercase\expandafter{\romannumeral1}}{V}\geq -C(\varepsilon,a,b),$$ and moreover, $C\longrightarrow0$ as $\varepsilon\longrightarrow0$. As for $\uppercase\expandafter{\romannumeral2}$, we consider $$\overline{\uppercase\expandafter{\romannumeral2}}=\frac{a}{u}+\frac{2u(a\ln u+b)}{\left(u-e^{-\frac{b}{a}}\right)^2}-\frac{2a(1+\ln u)+2b}{\left(u-e^{-\frac{b}{a}} \right)}.$$ Also, it's easy to check $\lim\limits_{u\longrightarrow e^{-\frac{b}{a}}}\overline{\uppercase\expandafter{\romannumeral2}}=0$. Hence for $\left| \nabla u\right|>0$ and all $\varepsilon=\varepsilon(a,b)>0$ small enough, if $u\leq e^{-\frac{b}{a}}+\varepsilon$, then $$\begin{aligned} \label{4.6} \frac{\uppercase\expandafter{\romannumeral2}}{\left| \nabla v\right|^2}=\overline{\uppercase\expandafter{\romannumeral2}}\left(u-e^{-\frac{b}{a}} \right)\geq-C(\varepsilon,a,b), \end{aligned}$$ and also, $C\longrightarrow0$ as $\varepsilon\longrightarrow0$. Now by ([\[4.5\]](#4.5){reference-type="ref" reference="4.5"}), ([\[4.6\]](#4.6){reference-type="ref" reference="4.6"}) and recall $G=\left| \nabla v\right|^2+V$, we have $$\begin{aligned} \label{4.7} \Delta V\geq-C(\varepsilon,a,b)G. \end{aligned}$$ Next, since $Ric\geq0$, then by ([\[4.4\]](#4.4){reference-type="ref" reference="4.4"}), ([\[4.7\]](#4.7){reference-type="ref" reference="4.7"}), we estimate $$\begin{aligned} \label{4.8} \Delta G&=\Delta\left(\left| \nabla v\right|^2+V\right)\nonumber\\ &=\Delta \left| \nabla v\right|^2+\Delta V\nonumber\\ &\geq 2\left|D^2 v\right|^2+2\left\langle\nabla v,\nabla\Delta v\right\rangle-C(\varepsilon,a,b)G\nonumber\\ &\geq \frac{2G^2}{n}-2\left\langle\nabla v,\nabla G\right\rangle-C(\varepsilon,a,b)G\nonumber\\ &\geq \frac{G^2}{n}-2\left\langle\nabla v,\nabla G\right\rangle, \end{aligned}$$ here we choose $\varepsilon=\varepsilon(a,b,n)$ sufficiently small such that $G\geq \frac{2a}{3} \geq C(\varepsilon,a,b)n$. Next, for $r>0$, $l\geq0$ and $\eta\in C^\infty_0(B_{r})$, we multiply by $\eta^2G^l$ on both sides of ([\[4.8\]](#4.8){reference-type="ref" reference="4.8"}) and integrate on $B_{r}$. Since $V>0$, then by the arguments as to obtain ([\[2.12\]](#2.12){reference-type="ref" reference="2.12"}), $$\begin{aligned} &\int\left| \nabla\left( \eta G^{\frac{l+1}{2}}\right) \right|^2\nonumber\\ &\leq\frac{\left( l+1\right)^2 }{2(l-1)}\int \left(\frac{4}{l+1}-\frac{1}{n}\right)\eta^2G^{l+2}+\frac{l\left(2l+3\right)+(l+1)^2}{l(l-1)}\int \left| \nabla\eta\right|^2G^{l+1}. \end{aligned}$$ Therefore by the Sobolev inequality ([\[2.1\]](#2.1){reference-type="ref" reference="2.1"}) with $\varphi=\eta G^{\frac{l+1}{2}}$, $$\begin{aligned} \label{4.9} &\left( \fint_{B_r}\left(\eta^2 G^{l+1}\right)^{\frac{n}{n-2}}\right)^{\frac{n-2}{n}}\nonumber\\ &\leq C_Sr^2l\fint_{B_r}\left(\frac{4}{l+1}-\frac{1}{n}\right)\eta^2G^{l+2}+C_S\fint_{B_r}\eta^2 G^{l+1}+C_Sr^2l\fint_{B_r}\left| \nabla\eta\right|^2G^{l+1}. \end{aligned}$$ Let $r\geq 1$ and $l\geq 8n+\frac{2n}{a}$, then ([\[4.9\]](#4.9){reference-type="ref" reference="4.9"}) gives $$\begin{aligned} \label{4.10} &\left( \fint_{B_r}\left(\eta^2 G^{l+1}\right)^{\frac{n}{n-2}}\right)^{\frac{n-2}{n}}\nonumber\\ &\leq C_Sr^2l\fint_{B_r}\left| \nabla\eta\right|^2G^{l+1}. \end{aligned}$$ Consequently, by standard iteration as to get ([\[2.17\]](#2.17){reference-type="ref" reference="2.17"}), there exists a constant $C_1=C_1(n,a)>0$ independent of $r$ such that $$\label{4.11} \sup\limits_{B_\frac{r}{2}}G\leq C_1\left\|G\right\|^*_{1, B_\frac{4r}{5}}.$$ Next, we choose some $\eta\in C^\infty_0(B_r)$ such that $$\eta\equiv1\quad\mbox{on}\hspace*{0.3em}B_\frac{4r}{5}\quad\mbox{and}\quad \left| \nabla\eta\right|\leq\frac{10}{r}.$$ For $\eta^2G$, since $V>0$, there holds $$\begin{aligned} \int\eta^2G=-\int\eta^2\Delta v&=2\int\left\langle \nabla\eta, \eta\nabla v\right\rangle\nonumber\\ &\leq2\int \left| \nabla\eta\right|^2+\frac{1}{2}\int\eta^2\left| \nabla v\right|^2\nonumber\\ &=2\int\left|\nabla\eta\right|^2+\frac{1}{2}\int\eta^2G-\frac{1}{2}\int\eta^2V\\ &\leq 2\int \left| \nabla\eta\right|^2+\frac{1}{2}\int\eta^2G. \end{aligned}$$ Therefore, $$\begin{aligned} \label{4.12} \int\eta^2G\leq4\int \left| \nabla\eta\right|^2. \end{aligned}$$ As a consequence of ([\[4.12\]](#4.12){reference-type="ref" reference="4.12"}), $$\begin{aligned} \label{4.13} 0<\left\|G\right\|^*_{1, B_\frac{4r}{5}}\leq\frac{\int\eta^2G}{\left| B_\frac{4r}{5}\right|}\leq\frac{400\left| B_r\right|}{\left| B_\frac{4r}{5}\right|r^2}. \end{aligned}$$ Finally, combining the volume comparison theorem and ([\[4.13\]](#4.13){reference-type="ref" reference="4.13"}), if we set $r= r(n,a)$ large enough, then $G\leq \frac{a}{2}$, but this contradicts the fact that $G\geq\frac{2a}{3}>0$ at the beginning. Hence we complete the proof. ◻ : The author Y. Wang is supported partially by NSFC (Grant No.11971400) and National key Research and Development projects of China (Grant No. 2020YFA0712500). 2 L. Chen and W.-Y. Chen, *Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds*, Ann. Global Anal. Geom. **35** (2009), no. 4, 397-404. X. Cao, B. Fayyazuddin Ljungberg and B. Liu, *Differential Harnack estimates for a nonlinear heat equation*, J. Funct. Anal. **265**(2013), no.10, 2312-2330. X. Dai, G. Wei and Z. Zhang, *Local Sobolev constant estimate for integral Ricci curvature bounds*, Adv. in Math. **325**(2018), 1-33. D. Gilbarg and N. S. Trudinger, *Elliptic partial differential equations of second order*, Springer, Berlin, 1998. A. Grigor'yan, S. Ishiwata and L. Saloff-Coste, *Poincaré constant on manifolds with ends*, Proc. Lond. Math. Soc. **126**(2023), no. 6, 1961-2012. D. Han, J. He and Y.-D. Wang, *Gradient estimates for $\Delta_pu-|\nabla u|^q+b(x)|u|^{r-1}u=0$ on a complete Riemannian manifold and Liouville type theorems*, arXiv:2309.03510. J. He, Y.-D. Wang and G.-D. Wei, *Gradient estimates for solutions of the equation $\Delta_pu + a u^q=0$ on a complete Riemannian manifold*, arXiv:2304.08238. P. Li, *Geometric analysis*, Cambridge Univ. Press, Cambridge, 2012. P. Li and R. Schoen, *$L^p$ and mean value properties of subharmonic functions on Riemannian manifolds*, Acta Math. **153**(1984), no.3-4, 279-301. Y. Ma and B. Wang, *Ricci curvature integrals, local functionals, and the Ricci flow*, arXiv:2109.02449, to appear in Trans. Amer. Math. Soc. Grisha Perelman, *The entropy formula for the Ricci flow and its geometric applications*, arXiv:math.DG/0211159. B. Qian, *Hamilton-type Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds*, Acta Mathematica Sinica, English Series, **27**(2011), 1071--1078. B. Qian, *Yau's gradient estimates for a nonlinear elliptic equation*, Arch. Math. **108**(2017), 427--435. O. S. Rothaus, *Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators*, J. Funct. Anal. **42(1)**(1981), 110-120. M. Rimoldi and G. Veronelli, *Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures*, Calc. Var. Partial Differential Equations **58** (2019), no. 2, Paper No. 66, 26 pp. L. Saloff-Coste, *Uniformly elliptic operators on Riemannian manifolds*, J. Differ. Geom. **36**(1992), no.2, 417-450. R. Schoen and S.T. Yau, *Lectures on differential geometry*, International Press, Boston, 2012. J. Wang, *Gradient estimates for a class of nonlinear elliptic and parabolic equations on Riemannian manifolds*, arXiv:2010.08059, to appear in Frontiers of Mathematics. J. Wang and Y. Wang, *Gradient Estimates For $\Delta u + a(x)u\log u + b(x)u = 0$ and its Parabolic Counterpart Under Integral Ricci Curvature Bounds*, arXiv:2109.05235, to appear in Comm. Anal. Geom. X.-D. Wang and L. Zhang, *Local gradient estimate for $p$-harmonic functions on Riemannian manifolds*, Comm. Anal. Geom., **19(4)** (2011), 759--771. Y.-D. Wang and G.-D. Wei, *On the nonexistence of positive solution to $\Delta u + au^{p+1} = 0$ on Riemannian manifolds*, J. Differential Equations, **362**(2023), 74--87. Y.-Y. Yang, *Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds*, Proc. Amer. Math. Soc. **136** (2008), 4095-4102. Q. S. Zhang, *Extremal of Log Sobolev inequality and $W$ entropy on noncompact manifolds*, J. Funct. Anal. **263**(2012), 2051-2101. Jie Wang, Institute of Geometry and Physics, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, Anhui Province, 230026, China. Email: wangjie9math\@163.com Youde Wang, 1. School of Mathematics and Information Sciences, Guangzhou University; 2. Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; 3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. Email: wyd\@math.ac.cn [^1]: corresponding author [^2]: Key words: gradient estimate; Nash-Moser iteration; Liouville type theorem [^3]: MSC 2020: 58J05; 35B45

arxiv_math

--- abstract: | In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation $$\begin{aligned} \Delta^2_{X}u=Vu,\end{aligned}$$ where $\Delta_{X}=\Delta_{x}+|x|^{2\alpha}\Delta_{y}$ ($0<\alpha\leq1$), with $x\in\mathbb{R}^{m}, y\in\mathbb{R}^{n}$, denotes the Baouendi-Grushin type subelliptic operators, and the potential $V$ satisfies the strongly singular growth assumption $|V|\leq \frac{c_0}{\rho^4}$, where $$\begin{aligned} \rho=\left(|x|^{2(\alpha+1)}+(\alpha+1)^2|y|^2\right)^{\frac{1}{2(\alpha+1)}}\end{aligned}$$ is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms. :Strong unique continuation property; Fourth order Baouendi-Grushin type subelliptic operator; Local Hardy-Rellich type inequality.\ **AMS Subject Classifications.** 35H20, 35J70. author: - Hairong Liu$^{1}$ - Xiaoping Yang$^{2}$ title: Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential --- [^1] [^2] # Introduction and main results This paper is devoted to studying the strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential. A differential operator $L$ is said to have the strong unique continuation property in $\Omega$ if the only solution of $Lu=0$ which vanishes of infinite order at a point $x_0\in\Omega$ is $u\equiv0$. In the past decades the strong unique continuation property for solutions to various kinds of partial differential equations has attracted a large number of researchers and induced many interesting and intensive results. The results for the second order operator $$\begin{aligned} \label{sch} -\Delta u=Vu\end{aligned}$$ go back to the work of Carleman [@c1939], who solved the uniqueness problem in $\mathbb{R}^{2}$ with bounded potentials. Cordes [@c1956] and Aronszajn [@a1957] extended the strong unique continuation property to second-order equations in $\mathbb{R}^{n}$. Subsequent developments in this direction is Jerison and Kenig's result [@jk1985] on the strong unique continuation property for ([\[sch\]](#sch){reference-type="ref" reference="sch"}) with $V\in L_{loc}^{n/2}(\mathbb{R}^{n})$. There is a large amount of work on strong unique continuation and quantitative unique continuation for second order elliptic operators, achieved by different Carleman type estimates (cf. [@DZ2019; @DLW2021; @h2001] and references therein). On the other hand, Garofalo and Lin [@gl1; @gl2] presented a geometric-variational approach to the strong unique continuation by using the frequency function. Their method is based on establishing the doubling estimate which in turn depends on the monotonicity property of the frequency function. It is worth pointing out that the frequency function was first introduced by Almgren [@alm] for harmonic functions. On the contrary, it turns out that unique continuation property is generically not true for subelliptic operators. Bahouri [@B1986] showed that unique continuation property is not true for even smooth and compactly supported perturbations of the sub-Laplacian on the Heisenberg group. Garofalo and Lanconelli [@GL] showed some positive results of the strong unique continuation property to the sub-Laplace $\Delta_{H}u=Vu$ on the Heisenberg group provided $u$ has some symmetries. Closely related to sub-Laplacian on the Heisenberg group is the following Baouendi-Grushin operator $$\begin{aligned} \label{Grushin0} \Delta_{X}=\Delta_{x}+|x|^{2\alpha}\Delta_{y}, \quad \alpha>0,\end{aligned}$$ which is the operator of a sum of squares of the following vector fields $X=\{X_1,\cdots,X_{m+n}\}$ $$\label{vector1} X_i=\partial_ {x_i}, \hspace{1mm}i=1,\cdots,m,\hspace{2mm} X_{m+j}=|x|^{\alpha} \partial_{y_j},\hspace{1mm}j=1,\cdots,n, \quad m,n\geq1.$$ Here $\alpha>0$ is a fixed parameter, $x=(x_1,\cdots,x_{m})\in \mathbb{R}^{m}$, $y=(y_1,\cdots,y_{n})\in \mathbb{R}^{n}$ and $\Delta_{x}, \Delta_{y}$ denote the standard Laplacian. When $\alpha=0$, $\Delta_{X}$ is just the standard Laplacian. For $\alpha>0$, $\Delta_{X}$ is elliptic for $x\neq 0$ and degenerate on the submanifold $\{0\}\times \mathbb{R}^{n}$, and it is not translation invariant in $\mathbb{R}^{N}$. We mention that when $\alpha=1$, operator $\Delta_{X}$ is connected to the sub-Laplacians on the Heisenberg-type groups (see e.g., [@G2015]). We recall that a more general class of operators modelled on $\Delta_{X}$ was first introduced by Baouendi, who studied the Dirichlet problem in weighted Sobolev spaces in [@B1967]. When $\alpha\in\mathbb{N}$ the operator in ([\[Grushin0\]](#Grushin0){reference-type="ref" reference="Grushin0"}) was studied by Grushin [@Grushin1970; @Grushin1971], who established its hypoellipticity. If $\alpha=2n$, with $n\in \mathbb{N}$, then $\Delta_{X}$ is a sum of squares of $C^{\infty}$ vector field satisfying Hörmander finite rank. Franchi and Lanconelli [@FL] studied embedding theorems for Sobolev spaces related to general vector fields ([\[vector1\]](#vector1){reference-type="ref" reference="vector1"}). For some other interesting properties related to the Baouendi-Grushin operators see [@grushin2020; @chen2017; @liu]. The strong unique continuation of the following Baouendi-Grushin operators $$\begin{aligned} \label{second-order} -\Delta_{X}u=Vu\end{aligned}$$ was established by Garofalo [@G1993] by introducing an Almgren-type frequency function, under the assumptions $$\label{V2} |V|\leq \frac{f(\rho)}{\rho^2}\psi,$$ for some non-decreasing $f:(0,R_0)\rightarrow \mathbb{R}^{+}$ such that $$\label{f} \int_0^{R_0}\frac{f(r)}{r}dr< \infty, \quad \mbox{for some }\quad R_0>0.$$ Here $\rho$ means a gauge norm associated to the vector field ([\[vector1\]](#vector1){reference-type="ref" reference="vector1"}), and $0\leq\psi\leq1$ is a weight function (see ([\[gauge\]](#gauge){reference-type="ref" reference="gauge"}) and ([\[psi\]](#psi){reference-type="ref" reference="psi"}) below for details). In [@G1993], for the case $V=V^{+}-V^{-}$ satisfies $$\begin{aligned} \label{V3} 0<V^{+}\leq C\frac{\psi}{\rho^2},\quad 0\leq V^{-}\leq\delta\frac{\psi}{\rho^2},\end{aligned}$$ for some $\delta>0$ small enough, the author established the unique continuation property in the sense of solution decays exponentially at one point (see ([\[exp\]](#exp){reference-type="ref" reference="exp"}) below). Note that the weight $\psi$ degenerates on the submanifold $\{x=0\}$ and so the result in [@G1993] does not allow to take $V\in L^{\infty}$. Recently, Banerjee, Mallick [@BM2020] obtained the strong unique continuation property of ([\[second-order\]](#second-order){reference-type="ref" reference="second-order"}) in the special case $\alpha=1, n=1$ under the weaker assumptions $$\label{V4} |V|\leq \frac{f(\rho)}{\rho^2}, \quad \mbox{or}\quad |V|\leq C\frac{\psi^{\varepsilon}}{\rho^2}, \quad \mbox{for some}\quad \varepsilon>0,$$ where $f$ is as in ([\[f\]](#f){reference-type="ref" reference="f"}). In this paper, we consider the following fourth order Baouendi-Grushin type subelliptic equation $$\label{L} Lu\equiv -\Delta^2_{X}u+Vu=0,$$ with $0<\alpha\leq 1$, where the potential $V$ satisfies the following strongly singular, $$\label{V} |V|\leq \frac{c_0}{\rho^4},$$ for some $c_0$ small depending only on $m, n$ and $\alpha$. Even for higher order uniform elliptic operators, the strong unique continuation property is quite different with and more complex than second order operators. Alinhac [@a] constructed a strong uniqueness counterexample for differential operators $P$ of any order in $\mathbb{R}^2$, under the condition that $P$ has two simple, nonconjugate complex characteristics. Colombini and Grammatico [@cg1999] gave strong uniqueness for a differential inequality related to bi-Laplacian $$\begin{aligned} \label{his1} |\Delta^{2}u(x)|\leq C_1\frac{|u|}{|x|^4}+C_2\frac{|\nabla u|}{|x|^3}+C_3\sum_{|\alpha|=2}\frac{|D^{\alpha}u|}{|x|^2},\quad \mbox{where}\quad C_3<\frac{3}{2}.\end{aligned}$$ Borgne [@b2001] studied this problem for solutions to ([\[his1\]](#his1){reference-type="ref" reference="his1"}) additionally with the third order derivatives with the singular potential $|x|^{-1+\varepsilon}$ $(\varepsilon>0)$. Colombini, Koch [@ck2010] studied the strong unique continuation for solutions to the differential inequality ([\[his1\]](#his1){reference-type="ref" reference="his1"}) additionally with the third order derivatives with the singular potential $\big|\ln|x|\big|^{-1-\varepsilon}|x|^{-1}$ for $\varepsilon>0$. Recently, the present authors [@liu2022] proved the strong unique continuation to the following bi-Laplacian $$-\Delta^2 u+b_1(x)\cdot\nabla (\Delta u)+V_1(x)\Delta u+b_2(x)\cdot\nabla u+V_2(x)u=0,$$ with the strongly singular terms $b_1, b_2, V_1$ and $V_2$. We refer to [@i2012; @linc2007; @zhu2018] for the quantitative unique continuation properties of uniformly elliptic operators with higher order. Before describing our main results, we will explain the definition of vanishing of infinite order in the subelliptic setting. Usually, in subelliptic cases, a function $u\in L_{loc}^{2}$ is said to vanish to infinite order at the origin if $$\begin{aligned} \label{order} \int_{B_r}u^2\psi dxdy=O(r^{k}), \quad \mbox{for every} \quad k\in\mathbb{N},\end{aligned}$$ as $r\rightarrow 0$. Moreover, we say that $u$ vanishes at the origin more rapidly than any power of $k$, namely that $$\begin{aligned} \label{exp} \int_{B_r}u^2\psi dxdy=O\left(\exp(-Br^{-\gamma})\right),\end{aligned}$$ as $r\rightarrow0$ for some constants $B,\gamma>0$. The main results of this paper are the following theorems. **Theorem 1**. *Assume the potential $V$ satisfies ([\[V\]](#V){reference-type="ref" reference="V"}), where $c_0$ is small in the sense of $$\begin{aligned} \label{small-1} \frac{4c_0}{(m-2)^2(Q-6)}+\frac{4}{(m-2)^4}<1,\end{aligned}$$ or $$\begin{aligned} \label{small-2} \frac{4(c_0+1)}{(m-2)^2(Q-6)}<1,\end{aligned}$$ where $Q=m+(\alpha+1)n$ denotes the homogenous dimension. Let $u$ be a weak solution of ([\[L\]](#L){reference-type="ref" reference="L"}). Then there exist $C, A, \gamma>0$, with $C$ and $\gamma$ only depending on $m$, $Q$, and $A$ depending on $u$, $m$, $Q$ such that for every ball $B_r$, for which $B_{2r}\subset\Omega$, it holds $$\begin{aligned} \label{doubling} \int_{B_{2r}}\left(u^2+(\Delta_{X}u)^2\right)\psi dxdy\leq C\exp\left(A r^{-\gamma}\right)\int_{B_{r}}\left(u^2+(\Delta_{X}u)^2\right)\psi dxdy.\end{aligned}$$* Using the doubling estimate in Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} above, we obtain the following strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with singular potential. **Theorem 2**. *Let $\Omega$ be a connected open subset of $\mathbb{R}^{m+n}$, with $m>4, Q>6$, containing the origin. Assume that $V$ satisfies the same assumptions of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}. Let $u$ be a weak solution of ([\[L\]](#L){reference-type="ref" reference="L"}) in $\Omega$. If $u$ vanishes to infinite order at the origin in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}), then $u\equiv 0$ in $\Omega$.* **Remark 3**. *With the help of the refined Hardy-Rellich inequalities (Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"}), similar to the proof of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}, for equation ([\[second-order\]](#second-order){reference-type="ref" reference="second-order"}), we can prove that\ (i) if $|V|\leq\frac{f(\rho)}{\rho^2}$ ($f$ is as in ([\[f\]](#f){reference-type="ref" reference="f"})) and $u$ vanishes to infinite order in the sense of ([\[order\]](#order){reference-type="ref" reference="order"}), then $u\equiv 0$;\ (ii) if $V=V^{+}-V^{-}$, and $0<V^{+}\leq \frac{C}{\rho^2}$, $0\leq V^{-}\leq\frac{\delta}{\rho^2}$ for some $\delta$ small, assume that $u$ vanishes to infinite order in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}), then $u\equiv 0$.\ That is, we can prove the strong unique continuation property of ([\[second-order\]](#second-order){reference-type="ref" reference="second-order"}) under the case of removing the weight $\psi$ in the assumptions ([\[V2\]](#V2){reference-type="ref" reference="V2"}) and $(\ref{V3})$ in [@G1993].* The argument we will adopt in this paper is to introduce an Almgren's type frequency function for the solutions $u$, and show its monotonicity to obtain a doubling estimate. In order to prove the monotonicity of the frequency function and deal with the strongly singular potential such as $\int_{B_r}\frac{Vuw}{|\nabla\rho|}$ with $w=\Delta_{X}u$, we should set up some refined Hardy-Rellich type inequalities for $\int_{B_r}\frac{u^2\psi}{\rho^6}$ and $\int_{B_r}\frac{w^2}{\rho^2\psi}$ on the gauge balls. On the other hand, the boundary term $\int_{\partial B_r}\frac{Vuw}{|\nabla\rho|}$ can not be directly controlled by $\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}$ and $\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}$ under the assumption ([\[V\]](#V){reference-type="ref" reference="V"}), we solve it by combining the divergence theorem and the higher order Hardy-Rellich inequalities which we established in Section 3 (see ([\[R6\]](#R6){reference-type="ref" reference="R6"})). The rest of the paper is organized as follows. In Section 2 we recall some notations and direct results about Baouendi-Grushin vector fields. In order to control the strongly singular potentials, we prove some versions of Hardy-Rellich type inequalities on the gauge balls in Section 3. In Section 4, we firstly introduce an Almgren's type frequency function related to solutions of $Lu=0$, and prove the monotonicity property of it. Using the monotonicity property, we obtain a doubling estimate including the solutions $u$ as well as $\Delta_{X}u$. In Section 5, we estimate a Caccioppoli type estimate to show that $\Delta_{X}u$ vanishes to infinite order provided $u$ does, and finally prove the strong unique continuous property. # Notations and preliminary results We begin this section by giving some notations and some basic results about the Baouendi-Grushin type vector fields. Let $(x,y)\in \mathbb{R}^{m}\times\mathbb{R}^{n}\equiv \mathbb{R}^{N}$. Consider the vector fields $$\label{vector} X_i=\partial_ {x_i}, \hspace{1mm}i=1,\cdots,m,\hspace{2mm} X_{m+j}=|x|^{\alpha} \partial_{y_j},\hspace{1mm}j=1,\cdots,n.$$ We denote the Baouendi-Grushin gradient and the corresponding divergence operator as $$\begin{aligned} &\nabla_{X}u=(X_1u,\cdots,X_{m+n}u), \quad \mbox{for a function}\ u, \\ &\mbox{div}_{X}F=\sum_{i=1}^{m+n}X_i F_i,\quad \mbox{for a vector field}\ F=(F_1,\cdots, F_{m+n}),\end{aligned}$$ and the Baouendi-Grushin type Laplacian as $$\begin{aligned} \label{Grushin} \Delta_{X}u=\mbox{div}_{X}(\nabla_{X}u)=\Delta_{x}u+|x|^{2\alpha}\Delta_{y}u.\end{aligned}$$ We note that the vector fields ([\[vector\]](#vector){reference-type="ref" reference="vector"}) are homogeneous of degree 1 with respect to the anisotropic dilations $$\begin{aligned} \label{dilation} \delta_{\lambda}(x,y)=(\lambda x,\lambda^{\alpha+1} y),\quad \lambda>0,\end{aligned}$$ which leads to a homogeneous dimension $Q=m+(\alpha+1)n$. The infinitesimal generator of the family of dilations ([\[dilation\]](#dilation){reference-type="ref" reference="dilation"}) is given by the vector field $$\label{Z0} Z=\sum^{m}_{i=1}x_i\partial_{x_i}+(\alpha+1)\sum^{n}_{j=1}y_j\partial_{y_j}.$$ Associated to the vector fields ([\[vector\]](#vector){reference-type="ref" reference="vector"}), for $(x,y)\in \mathbb{R}^{N}$, there is a gauge norm given by $$\begin{aligned} \label{gauge} \rho(x,y)=\left(|x|^{2(\alpha+1)}+(\alpha+1)^2|y|^2\right)^{\frac{1}{2(\alpha+1)}}.\end{aligned}$$ A gauge-ball and its gauge-sphere with respect to $\rho$ centered at the origin with radius $r$ are $$\begin{aligned} B_r=\{(x,y)\mid \rho(x,y)< r\}, \quad \partial {B_r}=\{(x,y)\mid \rho(x,y)=r\},\end{aligned}$$ respectively. Since $\rho\in C^{\infty}(\mathbb{R}^{N}\backslash\{(0,0)\})$, the outer unit normal on $\partial {B_r}$ is given by $\vec{n}=\frac{\nabla\rho}{|\nabla\rho|}$, where $\nabla\rho$ means the ordinary Euclidean gradient of $\rho$. The gauge-ball with respect to $\rho$ centered at $(0,y_0)$ with radius $r$ is $$\begin{aligned} B_{r}(0,y_0)=\left\{(x,y)\Large\mid \left(|x|^{2(\alpha+1)}+(\alpha+1)^2|y-y_0|^2\right)^{\frac{1}{2(\alpha+1)}}<r\right\}.\end{aligned}$$ Introducing the angle function $$\begin{aligned} \label{psi} \psi\equiv |\nabla_{X}\rho|^2=\frac{|x|^{2\alpha}}{\rho^{2\alpha}}.\end{aligned}$$ The function $\psi$ vanishes on the submanifold $\{0\}\times\mathbb{R}^{n}$ and clearly $0\leq\psi\leq1$. In the following, we collect some important identities (see [@G1993]), which will be used frequently in later sections. $$\begin{aligned} \label{psi2} \Delta_{X}\rho=\frac{Q-1}{\rho} \psi, \quad \left|X_iX_j\rho\right|\leq \frac{C}{\rho}.\end{aligned}$$ $$\begin{aligned} \label{Z} Z\rho=\rho.\end{aligned}$$ $$\begin{aligned} \label{Z2} \nabla_{X}u\cdot \nabla_{X}\rho= \frac{Zu}{\rho}\psi.\end{aligned}$$ $$\begin{aligned} \label{com} [X_i,Z]=Z,\quad i=1,\cdots, m+n.\end{aligned}$$ In order to study the weak solution to equation ([\[L\]](#L){reference-type="ref" reference="L"}), we introduce a function space associated with vector fields ([\[vector1\]](#vector1){reference-type="ref" reference="vector1"}), which is analogue to the corresponding Sobolev space [@Xu]. For any integer $k\geq 1$, $p\geq 1$ and $\Omega\subset\mathbb{R^{N}}$, we define $$\begin{aligned} M^{k,p}(\Omega)=\left\{ f\in L^{p}(\Omega) \mid X^{J}f\in L^{p}(\Omega), \forall J=(j_1,\cdots,j_s), |J|\leq k\right\}\end{aligned}$$ where $X^{J}f=X_{j_1}\cdots X_{j_s}f$ and define the norm in $M^{k,p}$ to be $$\begin{aligned} \|f\|_{M^{k,p}(\Omega)}=\left(\sum_{|J|\leq k}\|X^{J}f\|^{p}_{L^{p}(\Omega)}\right)^{\frac{1}{p}}.\end{aligned}$$ We denote by $M_0^{k,p}(\Omega)$ the closure of $C_0^{\infty}(\Omega)$ in $M^{k,p}(\Omega)$. A function $u\in M_{loc}^{2,2}(\Omega)$ is called a weak solution of ([\[L\]](#L){reference-type="ref" reference="L"}), it means $$\begin{aligned} \int_{\Omega}\Delta_{X}u\Delta_{X}\phi=\int_{\Omega}Vu\phi,\end{aligned}$$ for any test function $\phi\in M_0^{2,2}(\Omega)$. In the above integral, and henceforth, we omit indicating the Lebesgue measure $dxdy$ in the relevant integrals. In the rest of this section, we recall the Sobolev embedding inequality for the vector fields, and apply which to show a equivalent definition of vanishing to infinite order in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}). The Sobolev embedding inequality for the vector fields (see [@M2006]) states $$\begin{aligned} \label{sobolev} \|u\|_{L^{2^{*}}(\Omega)}\leq C\|u\|_{M^{1,2}(\Omega)},\end{aligned}$$ where $2^{*}=\frac{2Q}{Q-2}$ and constant $C=C(Q,\Omega)>0$. **Proposition 4**. *For any function $u\in M^{1,2}(B_r)$, then $u$ vanishes to infinite order at the origin in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}) if and only if it vanishes to infinite order in the following sense, $$\begin{aligned} \label{exp2} \int_{B_r}u^2 = O\left(\exp(-B_1r^{-\gamma})\right),\end{aligned}$$ as $r\rightarrow0$ for some constants $B_1, \gamma>0$.* *Proof.* The proof is similar to Lemma 2.4 in [@BM2020]. We give it here for the sake of completeness. Indeed, for some $q\in(2, 2^{*})$, Hölder inequality infers that $$\begin{aligned} \label{q} \int_{B_r}|u|\leq \left(\int_{B_r}u^q\psi\right)^{\frac{1}{q}} \left(\int_{B_r}\psi^{-\frac{1}{q-1}}\right)^{\frac{q-1}{q}}.\end{aligned}$$ Noting that $\frac{1}{q-1}<1$, so $\frac{2\alpha}{q-1}<2$, which together with $m>2$ yields that for $r<1$, $$\begin{aligned} \int_{B_r}\psi^{-\frac{1}{q-1}}dxdy\leq C(n)\int_{|x|\leq r}|x|^{-\frac{2\alpha}{q-1}}dx<\infty.\end{aligned}$$ By using an interpolation inequality and ([\[sobolev\]](#sobolev){reference-type="ref" reference="sobolev"}), for $\frac{\theta}{2}+\frac{1-\theta}{2^{*}}=\frac{1}{q}$, it holds $$\begin{aligned} \left(\int_{B_r}u^q\psi\right)^{\frac{1}{q}}&\leq \left(\int_{B_r}u^2\psi\right)^{\frac{\theta}{2}}\left(\int_{B_r}u^{2^{*}}\psi\right)^{\frac{1-\theta}{2^{*}}}\\[2mm] &\leq \left(\int_{B_r}u^2\psi\right)^{\frac{\theta}{2}} \|u\|^{1-\theta}_{M^{1,2}(B_r)}\\[2mm] &\leq C e^{-\frac{\theta}{2}Br^{-\gamma}},\end{aligned}$$ by ([\[exp\]](#exp){reference-type="ref" reference="exp"}). Therefore, ([\[q\]](#q){reference-type="ref" reference="q"}) implies $$\begin{aligned} \int_{B_r}|u|\leq C e^{-\frac{\theta}{2}Br^{-\gamma}}.\end{aligned}$$ Using an interpolation inequality again, for $\tau+\frac{1-\tau}{2^*}=\frac{1}{2}$, it holds $$\begin{aligned} \|u\|_{L^2}\leq \|u\|^{\tau}_{L^1}\|u\|^{1-\tau}_{L^{2^{*}}} \leq e^{-\tau\frac{\theta}{2}Br^{-\gamma}},\end{aligned}$$ that is ([\[exp2\]](#exp2){reference-type="ref" reference="exp2"}) with $B_1=\tau\theta B$. ◻ # Some Hardy-Rellich inequalities in gauge balls In this section, we prove some local versions of Hardy-Rellich type inequalities related to vector fields ([\[vector\]](#vector){reference-type="ref" reference="vector"}) on gauge balls. They will play an important role in showing doubling estimates and are also independently interesting themselves. There are many versions of Hardy type inequalities related the vector fields (see e.g., [@D2004; @hardy2004; @R2017; @2019PAMS]). However, those inequalities are established for functions with compact support. What we need is a local version. In [@GL] the authors obtained the following inequality for the Heisenberg group, which can be proved for Baouendi-Grushin vector fields ([\[vector\]](#vector){reference-type="ref" reference="vector"}), $$\begin{aligned} \int_{B_r}\frac{ u^2\psi}{\rho^2}\leq \left(\frac{2}{Q-2}\right)^2\int_{B_r}|\nabla_{X}u|^2+\frac{2}{Q-2}r^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$ Our task is to work under the assumption $|V|\leq \frac{C}{\rho^4}$ for $(\ref{L})$, we need to estimate some terms or quantities such as $\int_{B_r}\frac{ u^2}{\rho^2\psi}$ and $\int_{B_r}\frac{u^2\psi}{\rho^6}$. Let us start from the following refined Hardy inequality for the vector fields ([\[vector\]](#vector){reference-type="ref" reference="vector"}) in a gauge ball. **Lemma 5**. *Assume $m>2$, for every $u\in M^{1,2}(B_r)$, it holds $$\begin{aligned} \label{hhardy-1} \int_{B_r}\frac{ u^2}{|x|^2}\leq \left(\frac{2}{m-2}\right)^2\int_{B_r}|\nabla_{X}u|^2+\frac{2}{m-2}r^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$ Moreover, if $0<\alpha\leq1$, then the following inequality is valid, $$\begin{aligned} \label{hhardy-2} \int_{B_r}\frac{ u^2}{\rho^2\psi}\leq \left(\frac{2}{m-2}\right)^2\int_{B_r}|\nabla_{X}u|^2+\frac{2}{m-2}r^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$* *Proof.* Let $$h=\frac{1}{|x|^2} \left( \begin{array}{lll} x \\ 0 \end{array}\right) \in \mathbb{R}^{m+n}$$ A direct calculation shows that $$\label{h-1} \mbox{div}_{X}h=\frac{m-2}{|x|^2}, \hspace{3mm} \quad h\cdot \nabla_{X}\rho=\rho^{-(2\alpha+1)}|x|^{2\alpha}=\rho^{-1}\psi.$$ Hence, applying integration by parts, using ([\[h-1\]](#h-1){reference-type="ref" reference="h-1"}), and the Cauchy inequality, one has $$\begin{aligned} \int_{B_r}\frac{ u^2}{|x|^2}&=\frac{1}{m-2}\int_{B_r}\mbox{div}_{X}h u^2 \\[2mm] &=-\frac{1}{m-2}\int_{B_r}\nabla_{X}u^2\cdot h +\frac{1}{m-2}\int_{\partial B_r}u^2 h\cdot\frac{\nabla_{X}\rho}{|\nabla\rho|}\\[2mm] &\leq\frac{2}{m-2}\int_{B_r}\frac{|u||\nabla_{X}u|}{|x|}+\frac{1}{m-2}r^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|} \\[2mm] &\leq\frac{1}{2}\int_{B_r}\frac{ u^2}{|x|^2}+\frac{2}{(m-2)^2}\int_{B_r}|\nabla_Xu|^2+\frac{1}{m-2}r^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|},\end{aligned}$$ which yields the desired result ([\[hhardy-1\]](#hhardy-1){reference-type="ref" reference="hhardy-1"}). Furthermore, if $\alpha\leq 1$, then $\rho^2\psi\geq |x|^2$, so ([\[hhardy-1\]](#hhardy-1){reference-type="ref" reference="hhardy-1"}) yields ([\[hhardy-2\]](#hhardy-2){reference-type="ref" reference="hhardy-2"}) directly. ◻ **Lemma 6**. *Assume $Q>6$. Then, for any $u\in M^{3,2}(B_r)$, it holds $$\begin{aligned} \label{ineq6-1} \int_{B_r} \frac{u^2\psi}{\rho^6}\leq \frac{1}{(Q-6)^2}\int_{B_r}\frac{|\Delta_{X}u|^2}{\rho^2\psi}+\frac{2}{Q-6}r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$ Moreover, if $m>2$ and $0<\alpha\leq1$, it holds $$\begin{aligned} \label{ineq6-2} \int_{B_r} \frac{u^2\psi}{\rho^6}&\leq \frac{4}{(m-2)^2(Q-6)^2}\int_{B_r}|\nabla_X(\Delta_{X}u)|^2\nonumber\\[2mm] &+\frac{2}{(m-2)(Q-6)^2}r^{-1}\int_{\partial B_r} \frac{(\Delta _{X}u)^2\psi}{|\nabla\rho|}+\frac{2}{Q-6}r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$* *Proof.* Noting $$\Delta_{X}\rho^{-4}=-4(Q-6)\rho^{-6}\psi.$$ Integrating by parts, one obtain $$\begin{aligned} \label{6-1} \int_{B_r}\frac{u^2\psi}{\rho^6}&=-\frac{1}{4(Q-6)}\int_{B_r}\Delta_{X}\left(\frac{1}{\rho^4}\right)u^2\nonumber\\[2mm] &=-\frac{1}{4(Q-6)}\int_{B_r}\frac{1}{\rho^4} \Delta_{X}u^2+\frac{1}{4(Q-6)}\int_{\partial B_r}\frac{1}{\rho^4}\nabla_{X}u^2\cdot\frac{\nabla_{X}\rho}{|\nabla\rho|}\nonumber\\[2mm] &-\frac{1}{4(Q-6)}\int_{\partial B_r}u^2\nabla_{X}\left(\frac{1}{\rho^4}\right)\cdot\frac{\nabla_{X}\rho}{|\nabla\rho|}\nonumber\\[2mm] &\equiv J_1+J_2+J_3.\end{aligned}$$ Since $Q>6$, the term $J_1$ can be estimated as $$\begin{aligned} \label{J1} J_1 &=-\frac{1}{2(Q-6)}\int_{B_r}\frac{u\Delta_{X}u}{\rho^4}-\frac{1}{2(Q-6)}\int_{B_r}\frac{|\nabla_{X}u|^2}{\rho^4}\nonumber\\[2mm] &\leq \frac{1}{2(Q-6)}\int_{B_r}\frac{|u||\Delta_{X}u|}{\rho^4}-\frac{1}{2(Q-6)}\int_{B_r}\frac{|\nabla_{X}u|^2}{\rho^4},\end{aligned}$$ By using the divergence theorem, $$\begin{aligned} \label{J2} J_2 &=\frac{1}{2(Q-6)}r^{-4}\int_{\partial B_r} u \frac{\nabla_Xu\cdot\nabla_{X}\rho}{|\nabla\rho|}\nonumber\\[2mm] &=\frac{1}{2(Q-6)}r^{-4}\int_{B_r}\mbox{div}_{X}\left(u\nabla_Xu\right)\nonumber\\[2mm] &=\frac{1}{2(Q-6)}r^{-4}\int_{B_r}u\Delta_{X}u + \frac{1}{2(Q-6)}r^{-4}\int_{B_r}|\nabla_{X}u|^2\nonumber\\[2mm] &\leq \frac{1}{2(Q-6)}\int_{B_r}\frac{|u||\Delta_{X}u|}{\rho^4}+\frac{1}{2(Q-6)}\int_{B_r}\frac{|\nabla_{X}u|^2}{\rho^4},\end{aligned}$$ where we have used $Q>6$ again in the last inequality, then the last term on the right-hand side of ([\[J2\]](#J2){reference-type="ref" reference="J2"}) can cancel out that of ([\[J1\]](#J1){reference-type="ref" reference="J1"}). The term $J_3$ becomes $$\begin{aligned} \label{J3} J_3 &=\frac{1}{(Q-6)}\int_{\partial B_r} u^2 \rho^{-5} \frac{|\nabla_X\rho|^2}{|\nabla\rho|}\nonumber\\[2mm] &=\frac{1}{(Q-6)}r^{-5}\int_{\partial B_r} u^2 \frac{\psi}{|\nabla\rho|}.\end{aligned}$$ Substituting ([\[J1\]](#J1){reference-type="ref" reference="J1"})-([\[J3\]](#J3){reference-type="ref" reference="J3"}) into ([\[6-1\]](#6-1){reference-type="ref" reference="6-1"}), $$\begin{aligned} \int_{B_r}\frac{u^2\psi}{\rho^6} &\leq \frac{1}{Q-6}\int_{B_r}\frac{|u||\Delta_{X}u|}{\rho^4}+ \frac{1}{Q-6}r^{-5}\int_{\partial B_r} u^2\frac{\psi}{|\nabla\rho|}\nonumber\\[2mm] &\leq \frac{1}{2} \int_{B_r}\frac{u^2\psi}{\rho^6}+ \frac{1}{2(Q-6)^2}\int_{B_r}\frac{|\Delta_{X}u|^2}{\rho^2\psi}+\frac{1}{Q-6}r^{-5}\int_{\partial B_r} u^2\frac{\psi}{|\nabla\rho|},\end{aligned}$$ which deduces ([\[ineq6-1\]](#ineq6-1){reference-type="ref" reference="ineq6-1"}). Furthermore, noting that $\rho^2\psi\geq |x|^2$ thanks to $\alpha\leq 1$, then we can apply Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} for the first term on the right-hand side of ([\[ineq6-1\]](#ineq6-1){reference-type="ref" reference="ineq6-1"}) to deduce ([\[ineq6-2\]](#ineq6-2){reference-type="ref" reference="ineq6-2"}). ◻ **Lemma 7**. *Assume $m>2$, $Q>6$ and $\alpha\leq1$, and let $u\in M^{3,2}(\Omega)$. Then $$\begin{aligned} \label{hardy-4} \int_{B_r} \frac{|\nabla_{X}u|^2}{\rho^4}\leq C\left(r^{-4}\int_{B_r}|\nabla_{X} u|^2+\int_{B_r}|\nabla_{X}(\Delta_{X}u)|^2+r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+r^{-1}\int_{\partial B_r}\frac{(\Delta _{X}u)^2\psi}{|\nabla\rho|}\right),\end{aligned}$$ and $$\begin{aligned} \label{hhardy2} &\int_{B_r}\frac{ u^2}{\rho^6}\leq\int_{B_r}\frac{ u^2}{\rho^4|x|^2}\nonumber\\[2mm] &\leq C\left( \int_{B_r}|\nabla_X(\Delta_{X}u)|^2+r^{-4}\int_{B_r}|\nabla_{X} u|^2+r^{-1}\int_{\partial B_r} \frac{(\Delta _{X}u)^2\psi}{|\nabla\rho|}+r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}\right). \end{aligned}$$* *Proof.* Using integration by parts and the Cauchy inequality, we have $$\begin{aligned} \int_{B_r}\frac{|\nabla_{X} u|^2}{\rho^4}&=-\int_{B_r}\mbox{div}_{X}\left(\frac{\nabla_{X} u}{\rho^4}\right)u+r^{-4}\int_{\partial B_r} u \frac{\nabla_{X}u\cdot\nabla_{X}\rho}{|\nabla\rho|}\nonumber\\[2mm] &=-\int_{B_r}\frac{u \Delta_{X} u}{\rho^4}+5\int_{B_r}\rho^{-5}u\nabla_{X} u\cdot\nabla_{X}\rho+r^{-4}\int_{B_r}\mbox{div}_{X}\left(u\nabla_{X} u\right)\nonumber\\[2mm] &=-\int_{B_r}\frac{u \Delta_{X} u}{\rho^4}+5\int_{B_r}\rho^{-5}u\nabla_{X} u\cdot\nabla_{X}\rho+r^{-4}\int_{B_r}|\nabla_{X} u|^2 +r^{-4}\int_{B_r} u\Delta_{X} u \nonumber\\[2mm] &\leq 2 \int_{B_r}\frac{|u| |\Delta_{X} u|}{\rho^4} +\varepsilon \int_{B_r}\frac{|\nabla_{X} u|^2}{\rho^4}+C(\varepsilon)\int_{B_r}\frac{u^2|\nabla_{X}\rho|^2}{\rho^6}+ r^{-4}\int_{B_r}|\nabla_{X} u|^2\\[2mm] &\leq \int_{B_r}\frac{u^2\psi}{\rho^6} +\int_{B_r}\frac{|\Delta_{X}u|^2}{\rho^2\psi}+\varepsilon \int_{B_r}\frac{|\nabla_{X} u|^2}{\rho^4}+C(\varepsilon)\int_{B_r}\frac{u^2\psi}{\rho^6}+ r^{-4}\int_{B_r}|\nabla_{X} u|^2,\end{aligned}$$ which yields $$\begin{aligned} \int_{B_r}\frac{|\nabla_{X} u|^2}{\rho^4} \leq C\left(r^{-4}\int_{B_r}|\nabla_{X} u|^2 +\int_{B_r}\frac{|\Delta_{X}u|^2}{\rho^2\psi}+\int_{B_r}\frac{u^2\psi}{\rho^6}\right).\end{aligned}$$ Applying Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} and Lemma [Lemma 6](#lem4){reference-type="ref" reference="lem4"} implies the desired result ([\[hardy-4\]](#hardy-4){reference-type="ref" reference="hardy-4"}). Using the same function $h$ in Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} and using ([\[h-1\]](#h-1){reference-type="ref" reference="h-1"}) again, integrating by parts, one has $$\begin{aligned} \int_{B_r}\frac{ u^2}{\rho^4|x|^2}&=\frac{1}{m-2}\int_{B_r}\mbox{div}_{X}h \frac{u^2}{\rho^4} \\[2mm] &=-\frac{1}{m-2}\int_{B_r}\nabla_{X}(\rho^{-4}u^2)\cdot h +\frac{1}{m-2}\int_{\partial B_r}\frac{u^2}{\rho^4} h\cdot\frac{\nabla_{X}\rho}{|\nabla\rho|}\\[2mm] &=-\frac{2}{m-2}\int_{B_r}\frac{u\nabla_{X}u\cdot h}{\rho^4}+ \frac{4}{m-2}\int_{B_r}\frac{u^2\nabla_{X}\rho\cdot h}{\rho^5}+\frac{1}{m-2}r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|} \\[2mm] &\leq\frac{1}{2} \int_{B_r}\frac{ u^2}{\rho^4|x|^2}+\frac{2}{(m-2)^2}\int_{B_r}\frac{|\nabla_{X}u|^2}{\rho^4}+ \frac{4}{m-2}\int_{B_r}\frac{u^2\psi}{\rho^6}+\frac{1}{m-2}r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}.\end{aligned}$$ Therefore, $$\begin{aligned} \int_{B_r}\frac{ u^2}{\rho^4|x|^2} \leq \frac{4}{(m-2)^2}\int_{B_r}\frac{|\nabla_{X}u|^2}{\rho^4}+ \frac{8}{m-2}\int_{B_r}\frac{u^2\psi}{\rho^6}+\frac{2}{m-2}r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|},\end{aligned}$$ which together with Lemma [Lemma 6](#lem4){reference-type="ref" reference="lem4"} and Lemma [Lemma 7](#lem5){reference-type="ref" reference="lem5"} yields the estimate ([\[hhardy2\]](#hhardy2){reference-type="ref" reference="hhardy2"}). ◻ **Remark 8**. *(i) The integrands on the the boundary integrals of these inequalities have the weight function $\psi$. This is an important point and will be shown in the next section when we prove the monotonicity of the frequency function.\ (ii) All the local Hardy-Rellich type inequalities in this Section also hold on the Heisenberg-type groups.* # The frequency function and doubling estimate In this section, we first introduce an Almgren's type frequency function and study the behavior of it. After obtaining a monotonicity of the frequency function, we establish a doubling estimate for a combination of $u$ and $\Delta_{X}u$. First, we decompose the fourth order equation ([\[L\]](#L){reference-type="ref" reference="L"}) into a system of two second order equations, that is, $$\label{w} \left\{\begin{array}{lll} \Delta_{X} u=w,\\[2mm] \Delta_{X} w=V u. \end{array}\right.$$ Let $u\in M_{loc}^{2,2}(\Omega)$ be a weak solution of ([\[L\]](#L){reference-type="ref" reference="L"}). Define $$H_1(r)=\int_{\partial B_r}u^2\frac{\psi}{|\nabla\rho|},\quad \quad H_2(r)=\int_{\partial B_r}w^2\frac{\psi}{|\nabla\rho|},$$ and $$\label{h} H(r)=H_1(r)+r^{4}H_2(r).$$ **Lemma 9**. *Let $u$ be a nonzero solution of ([\[L\]](#L){reference-type="ref" reference="L"}), where $V$ satisfies the assumption ([\[V\]](#V){reference-type="ref" reference="V"}), with $c_0$ is small in the sense of $$\begin{aligned} \label{small-0} \frac{4c_0}{(m-2)^2(Q-6)}<1.\end{aligned}$$ Then there exists $r_0$, such that $$H(r)\neq 0\quad \mbox{for every}\quad r\in(0,r_0).$$* **Remark 10**. *We observe that, if $u\in M_{loc}^{2,2}(\Omega)$ is a weak solution of ([\[L\]](#L){reference-type="ref" reference="L"}), then we can check that $u\in M_{loc}^{3,2}(\Omega)$ provided $c_0$ is small in the sense ([\[small-0\]](#small-0){reference-type="ref" reference="small-0"}) by using the difference quotient method and the Hardy-Rellich inequality.* *Proof.* We prove it by contradiction. Suppose that $H(r_{*})=0$ for some $r_{*}\in(0,r_0)$. Then the definition of $H(r)$ implies that $u|_{\partial B_{r_{*}}}=0$ and $w|_{\partial B_{r_*}}=0$. Multiplying the second equation of ([\[w\]](#w){reference-type="ref" reference="w"}) by $w$ and then integrating over $B_{r_*}$, integrating by parts with $w|_{\partial B_{r_*}}=0$, we get $$\begin{aligned} \int_{B_{r_*}}Vuw=\int_{B_{r_*}}\Delta_X w w=-\int_{B_{r_*}}|\nabla_X w|^2.\end{aligned}$$ Then, by using the Hardy-Rellich inequalities ([\[hhardy-2\]](#hhardy-2){reference-type="ref" reference="hhardy-2"}) and ([\[ineq6-2\]](#ineq6-2){reference-type="ref" reference="ineq6-2"}) with $u|_{\partial {B_{r_*}}}=w|_{\partial {B_{r_*}}}=0$, we have $$\begin{aligned} \int_{B_{r_*}}|\nabla_X w|^2&\leq \int_{B_{r_*}}|V||u||w|\leq c_0\int_{B_{r_*}}\frac{|u||w|}{\rho^4}\\[2mm] &\leq\varepsilon \int_{B_{r_*}}\frac{u^2\psi}{\rho^6}+ \frac{1}{4}c_0^2\varepsilon^{-1}\int_{B_{r_*}}\frac{w^2}{\rho^2\psi}\\[2mm] &\leq \frac{4}{(m-2)^2(Q-6)^2}\varepsilon r_*^{4}\int_{B_{r_*}}|\nabla_{X}w|^2 +\frac{4}{(m-2)^2}\frac{1}{4}c_0^2\varepsilon ^{-1}r_*^{4}\int_{B_{r_*}}|\nabla_{X}w|^2\\[2mm] &=\left(\frac{4}{(m-2)^2(Q-6)^2}\varepsilon+\frac{4}{(m-2)^2}\frac{1}{4}c_0^2\varepsilon ^{-1}\right)r_*^{4}\int_{B_{r_*}}|\nabla_{X}w|^2.\end{aligned}$$ Let $$\begin{aligned} g(\varepsilon)\equiv\frac{4}{(m-2)^2(Q-6)^2}\varepsilon+ \frac{4}{(m-2)^2}\frac{1}{4}c_0^2\varepsilon ^{-1},\end{aligned}$$ it is easy to see that $$\begin{aligned} g_{min}=\frac{4c_0}{(m-2)^2(Q-6)},\end{aligned}$$ which deduces $$\begin{aligned} \int_{B_{r_*}}|\nabla_X w|^2\leq \frac{4c_0}{(m-2)^2(Q-6)} \int_{B_{r_*}}|\nabla_X w|^2.\end{aligned}$$ This is a contradiction thanks to the assumption ([\[small-0\]](#small-0){reference-type="ref" reference="small-0"}) under the case $\int_{B_{r_*}}|\nabla_X w|^2\neq0$. If $\int_{B_{r_*}}|\nabla_X w|^2=0$, then applying Sobolev-Poincaré inequalities related to the vector fields ([\[vector1\]](#vector1){reference-type="ref" reference="vector1"}) (see, e.g.,[@FG1994]), one has $w=0$ in $B_{r_*}$. That is $$\begin{aligned} \left\{ \begin{array}{lll} \Delta_{X}u=0, \quad \mbox{in}\ B_{r_*},\\[2mm] u\mid_{\partial{B_{r_*}}}=0. \end{array} \right.\end{aligned}$$ Applying the maximum principle for Baouendi-Grushin operator [@max], there holds $u=0$ in $B_{r_*}$. This contradicts the hypothesis of a non-zero solution. The proof of Lemma [Lemma 9](#lemma-H){reference-type="ref" reference="lemma-H"} is concluded. ◻ Define $$\label{I3} I_1(r)=\int_{ B_r} |\nabla_X u|^2+\int_{B_r}uw, \quad \quad I_2(r)=\int_{ B_r} |\nabla_X w|^2+\int_{B_r} V wu,$$ and $$\label{I} I(r)=I_1(r)+ r^4 I_2(r).$$ Moreover, by using the divergence theorem, equations ([\[w\]](#w){reference-type="ref" reference="w"}) and the fact ([\[Z2\]](#Z2){reference-type="ref" reference="Z2"}), $I_1(r)$ and $I_2(r)$ can be rewritten as $$\begin{aligned} \label{I1} &I_1(r)=\int_{\partial B_r}u \frac{\nabla_{X}u\cdot\nabla_{X}\rho}{|\nabla\rho|}=\frac{1}{r}\int_{\partial B_r}\frac{u Zu \psi}{|\nabla\rho|},\nonumber\\[2mm] &I_2(r)=\int_{\partial B_r}w \frac{\nabla_{X}w\cdot\nabla_{X}\rho}{|\nabla\rho|}=\frac{1}{r}\int_{\partial B_r}\frac{w Zw \psi}{|\nabla\rho|}.\end{aligned}$$ **Lemma 11**. *Let $u$ be a solution of ([\[L\]](#L){reference-type="ref" reference="L"}). Then $$\begin{aligned} \label{H'(r)} H'(r)=\frac{Q-1}{r}H(r)+2I(r)+4r^3H_{2}(r).\end{aligned}$$* *Proof.* By using the divergence theorem and fact ([\[psi\]](#psi){reference-type="ref" reference="psi"}), one has $$\begin{aligned} H_1(r)&=\int_{\partial B_r}u^2 \frac{\nabla_{X}\rho\cdot\nabla_{X}\rho}{|\nabla\rho|} =\int_{B_r}\mbox{div}_{X}\left(u^2\nabla_{X}\rho\right)\\[2mm] &=\int_{B_r} u^2\Delta_{X}\rho+2\int_{B_r} u\nabla_{X}u\cdot\nabla_{X}\rho.\end{aligned}$$ Using the co-area formula, we get $$\label{h} H_1(r)=\int_{0}^{r}ds\int_{\partial B_{s}}u^2\frac{\Delta_{X}\rho}{\left|\nabla\rho\right|}+2\int_{0}^{r}ds\int_{\partial B_s} u \frac{ \nabla_{X}u\cdot\nabla_{X}\rho}{\left|\nabla\rho\right|}$$ Differentiating ([\[h\]](#h){reference-type="ref" reference="h"}) with respect to $r$ and according to ([\[psi2\]](#psi2){reference-type="ref" reference="psi2"}), ([\[I1\]](#I1){reference-type="ref" reference="I1"}), we get $$\begin{aligned} \label{H1} H_1'(r)&=\int_{\partial B_r}u^2\frac{\Delta_{X}\rho}{|\nabla\rho|}+2\int_{\partial B_r}u\frac{\nabla_{X}u\cdot\nabla_{X}\rho}{|\nabla \rho|}\nonumber\\[2mm] &=\frac{Q-1}{r}H_1(r)+2I_1(r).\end{aligned}$$ In a similar way, one has $$\begin{aligned} \label{H2} H_2'(r)=\frac{Q-1}{r}H_2(r)+2 I_2(r).\end{aligned}$$ Combining ([\[H1\]](#H1){reference-type="ref" reference="H1"}) and ([\[H2\]](#H2){reference-type="ref" reference="H2"}) implies the Lemma. ◻ Due to Lemma [Lemma 9](#lemma-H){reference-type="ref" reference="lemma-H"}, we can define the frequency function as $$\label{n} N(r)=\frac{rI(r)}{H(r)}, \quad \mbox{for every}\quad r\in(0,r_0).$$ Lemma [Lemma 9](#lemma-H){reference-type="ref" reference="lemma-H"} also implies that the function $r\rightarrow N(r)$ is absolutely continuous on $(0,r_0)$. Let $$\begin{aligned} \label{ome} \Omega_{r_0}=\left\{r\in(0,r_0):N(r)>\max(1, N(r_0))\right\}.\end{aligned}$$ Then $\Omega_{r_0}$ is an open subset of $\mathbb{R}$. Therefore, $$\begin{aligned} \label{omega} \Omega_{r_0}=\bigcup_{j=1}^{\infty}(a_j,b_j),\quad a_j,b_j\notin \Omega_{r_0},\end{aligned}$$ and $$\begin{aligned} \label{h<i} \frac{H(r)}{r}<I(r),\quad \mbox{for all} \quad r\in \Omega_{r_0}.\end{aligned}$$ The main purpose of this section is to prove the doubling estimate of solutions to ([\[L\]](#L){reference-type="ref" reference="L"}) (Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}). To achieve this goal, we should build the monotonicity of the frequency function. Precisely, **Theorem 12**. *Let $u$ be a solution of ([\[L\]](#L){reference-type="ref" reference="L"}), where $V$ satisfies the assumption ([\[V\]](#V){reference-type="ref" reference="V"}), with $c_0$ is small in the sense of ([\[small-1\]](#small-1){reference-type="ref" reference="small-1"}) or ([\[small-2\]](#small-2){reference-type="ref" reference="small-2"}). There exists a constant $\beta>0$, depending only on $m, Q$ and $c_0$, such that for every $r\in(0,r_0)$ for which ([\[h\<i\]](#h<i){reference-type="ref" reference="h<i"}) holds, we have $$\begin{aligned} \label{key6} \frac{N'(r)}{N(r)}\geq -\beta\frac{1}{r}.\end{aligned}$$* We postpone the proof of Theorem [Theorem 12](#thm4){reference-type="ref" reference="thm4"} for the moment and are first devoted to set up a doubling estimate for solutions $u$ to ([\[L\]](#L){reference-type="ref" reference="L"}). This proof is standard (see, e.g., [@GL]), we include it for the sake of completeness. **Proof of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}**. Integrating ([\[key6\]](#key6){reference-type="ref" reference="key6"}) on $(r,b_j)$ where $r\in(a_j,b_j)$, we get $$\ln N(b_j)-\ln N(r)\geq -\beta(\ln b_j-\ln r),$$ which yields $$N(r)\leq N(b_j)b_j^{\beta}r^{-\beta}.$$ Recalling that $N(b_{j})\leq \max(1,N(r_0))$, we infer that $$\label{9} N(r)\leq C\max(1,N(r_0))r^{-\beta}, \quad \mbox{for}\quad r\in\Omega_{r_0}.$$ By the definitions of $H(r)$ and $I(r)$, one deduces from ([\[H\'(r)\]](#H'(r)){reference-type="ref" reference="H'(r)"}) that $$\label{ineq} \left(\ln\frac{H(r)}{r^{Q+3}}\right)'\leq \frac{2N(r)}{r}.$$ Integrating ([\[ineq\]](#ineq){reference-type="ref" reference="ineq"}) from $r$ to $2r$, with $r\leq \frac{r_0}{2}$, we deduce $$\begin{aligned} \label{10} \ln\frac{H(2r)}{(2r)^{Q+3}}-\ln\frac{H(r)}{r^{Q+3}}\leq 2\int_{r}^{2r}\frac{N(t)}{t}dt.\end{aligned}$$ Let $J_{r}=\{t\in(r,2r): t\notin\Omega_{r_0}, N(t)\geq0\}$, where $\Omega_{r_0}$ is as in ([\[ome\]](#ome){reference-type="ref" reference="ome"}). Then $$\begin{aligned} \label{est} \int_{r}^{2r}\frac{N(t)}{t}dt&\leq \int_{(r,2r)\cap\Omega_{r_0}}\frac{N(t)}{t}dt+\int_{J_r}\frac{N(t)}{t}dt\nonumber\\[2mm] &\leq C\max(1,N(r_0))\int_{r}^{2r}t^{-\beta+1}dt+\max(1,N(r_0))\int_{r}^{2r}\frac{1}{t}dt\nonumber\\[2mm] %&\leq C\max(1,N(r_0))r^{-\beta+2}+\max(1,N(r_0))\ln2\nonumber\\[2mm] &\leq C\max(1,N(r_0))r^{-\beta+2}.\end{aligned}$$ Plugging ([\[est\]](#est){reference-type="ref" reference="est"}) into ([\[10\]](#10){reference-type="ref" reference="10"}), it follows that $$\begin{aligned} \label{dou2} H(2r)\leq 2^{Q+3}\exp\left(C\max(1,N(r_0))r^{-\beta+2}\right)H(r).\end{aligned}$$ Integrating ([\[dou2\]](#dou2){reference-type="ref" reference="dou2"}) from $\frac{r}{2}$ to $r$, we derive the desired estimate ([\[doubling\]](#doubling){reference-type="ref" reference="doubling"}) with $\gamma=\beta-2$ and $A=C\max(1,N(r_0))$.0◻ Now, we turn to prove the monotonicity of $N(r)$. We first estimate $I'(r)$ in the following lemma, which is the key step to show $N(r)$'s monotonicity. The refined Hardy-Rellich type inequalities plays a very important role in this calculation. **Lemma 13**. *Let $u$ be a solution of ([\[L\]](#L){reference-type="ref" reference="L"}), where $V$ satisfies the assumption ([\[V\]](#V){reference-type="ref" reference="V"}), with $c_0$ is small in the sense of ([\[small-1\]](#small-1){reference-type="ref" reference="small-1"}) or ([\[small-2\]](#small-2){reference-type="ref" reference="small-2"}). Then for every $r\in(0,r_0)$ for which ([\[h\<i\]](#h<i){reference-type="ref" reference="h<i"}) holds, the following estimate is valid, $$\begin{aligned} \label{I'(r)} I'(r)=\frac{Q-2}{r}I(r)+\frac{2}{r^2}\int_{\partial{B_r}}\frac{(Zu)^2\psi}{|\nabla\rho|} +2r^2\int_{\partial{B_r}}\frac{(Zw)^2\psi}{|\nabla\rho|}+O(r^{-1})I(r).\end{aligned}$$* *Proof.* At first, we prove the following assertion: if $c_0$ is small in the sense of ([\[small-1\]](#small-1){reference-type="ref" reference="small-1"}) or ([\[small-2\]](#small-2){reference-type="ref" reference="small-2"}), then it holds $$\begin{aligned} \label{claim} \int_{B_r}|\nabla_{X}u|^2+r^{4}\int_{B_r}|\nabla_{X}w|^2\leq CI(r).\end{aligned}$$ Indeed, by the definition of $I(r)$, one has $$\begin{aligned} \label{claim-0} \int_{B_r}|\nabla_{X}u|^2+r^{4}\int_{B_r}|\nabla_{X}w|^2=I(r)-\int_{B_r}uw-r^4\int_{B_r}Vuw.\end{aligned}$$ Using the Cauchy's inequality with $\varepsilon$ as in Lemma [Lemma 9](#lemma-H){reference-type="ref" reference="lemma-H"}, and applying Lemmas [Lemma 5](#lem3){reference-type="ref" reference="lem3"}, [Lemma 6](#lem4){reference-type="ref" reference="lem4"}, we obtain $$\begin{aligned} \label{claim-1} r^4\int_{B_r}Vuw&\leq c_0 r^4\int_{B_r}\frac{|u||w|}{\rho^4} \leq \varepsilon r^4\int_{B_r}\frac{u^2\psi}{\rho^6}+ \frac{1}{4}c_0^2\varepsilon ^{-1}r^4\int_{B_r}\frac{w^2}{\rho^2\psi}\nonumber\\[2mm] &\leq \frac{4}{(m-2)^2(Q-6)^2}\varepsilon r^{4}\int_{B_r}|\nabla_{X}w|^2 +\frac{4}{(m-2)^2}\frac{1}{4}c_0^2\varepsilon ^{-1}r^{4}\int_{B_r}|\nabla_{X}w|^2\nonumber\\[2mm] &+Cr^{-1}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+Cr^{3}\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}\nonumber\\[2mm] &\leq \frac{4c_0}{(m-2)^2(Q-6)}r^4\int_{B_r}|\nabla_{X}w|^2+Cr^{-1}\left(\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+r^{4}\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ For the term $\int_{B_r}uw$, we can estimate as above with $c_0=1$, that is, $$\begin{aligned} \label{claim-2} \int_{B_r}uw&\leq r^4\int_{B_r}\frac{|u||w|}{\rho^4}\nonumber\\[2mm] &\leq \frac{4}{(m-2)^2(Q-6)}r^4\int_{B_r}|\nabla_{X}w|^2+Cr^{-1}\left(\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+r^{4}\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ Putting ([\[claim-1\]](#claim-1){reference-type="ref" reference="claim-1"}), ([\[claim-2\]](#claim-2){reference-type="ref" reference="claim-2"}) into ([\[claim-0\]](#claim-0){reference-type="ref" reference="claim-0"}), and using ([\[h\<i\]](#h<i){reference-type="ref" reference="h<i"}) and the assumption ([\[small-2\]](#small-2){reference-type="ref" reference="small-2"}), it holds $$\begin{aligned} &\int_{B_r}|\nabla_{X} u|^2+r^{4}\int_{B_r}|\nabla_{X} w|^2\\[2mm] &\leq \frac{4c_0+4}{(m-2)^2(Q-6)}r^4\int_{B_r}|\nabla_{X}w|^2+Cr^{-1}\left(\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+r^{4}\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}\right)\\[2mm] &<r^4\int_{B_r}|\nabla_{X}w|^2+CI(r),\end{aligned}$$ so, the claim ([\[claim\]](#claim){reference-type="ref" reference="claim"}) is proved provided ([\[small-2\]](#small-2){reference-type="ref" reference="small-2"}). On the other hand, applying Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} to $u$ and $w$, the term $\int_{B_r}uw$ can also be controlled as $$\begin{aligned} \int_{B_r}uw&\leq r^2\int_{B_r}\frac{|u||w|}{\rho^2}\leq \frac{\epsilon}{2}\int_{B_r}\frac{u^2}{\rho^2}+\frac{1}{2\epsilon}r^4\int_{B_r}\frac{w^2}{\rho^2}\nonumber\\[2mm] &\leq \frac{2\epsilon}{(m-2)^2}\int_{B_r}|\nabla_{X}u|^2 +\frac{2}{(m-2)^2\epsilon}r^4\int_{B_r}|\nabla_{X}w|^2+Cr^{-1}\left(\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}+r^4\int_{\partial B_r}\frac{w^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ Choosing $\epsilon$ such that $\frac{2\epsilon}{(m-2)^2}<1$, i.e. $\epsilon=\frac{(m-2)^2}{2+\delta}$ for some $\delta>0$ to be determined later. Then the above inequality becomes $$\begin{aligned} \label{claim-3} \int_{B_r}uw \leq \frac{2}{2+\delta}\int_{B_r}|\nabla_{X}u|^2 +\frac{4+2\delta}{(m-2)^4}r^4\int_{B_r}|\nabla_{X}w|^2+Cr^{-1}H(r).\end{aligned}$$ Substituting ([\[claim-1\]](#claim-1){reference-type="ref" reference="claim-1"}) and ([\[claim-3\]](#claim-3){reference-type="ref" reference="claim-3"}) into ([\[claim-0\]](#claim-0){reference-type="ref" reference="claim-0"}), and using ([\[h\<i\]](#h<i){reference-type="ref" reference="h<i"}), $$\begin{aligned} \label{claim-4} &\int_{B_r}|\nabla_{X} u|^2+r^{4}\int_{B_r}|\nabla_{X} w|^2\nonumber\\[2mm] &\leq \frac{2}{2+\delta}\int_{B_r}|\nabla_{X}u|^2 +\left(\frac{4+2\delta}{(m-2)^4}+\frac{4c_0}{(m-2)^2(Q-6)}\right)r^4\int_{B_r}|\nabla_{X}w|^2+ CI(r).\end{aligned}$$ Since ([\[small-1\]](#small-1){reference-type="ref" reference="small-1"}), let $\delta=\frac{1}{4}(m-2)^4\left(1-\frac{4c_0}{(m-2)^2(Q-6)}-\frac{4}{(m-2)^4}\right)>0$, then $$\begin{aligned} \frac{4+2\delta}{(m-2)^4}+\frac{4c_0}{(m-2)^2(Q-6)}<1.\end{aligned}$$ Therefore, we prove ([\[claim\]](#claim){reference-type="ref" reference="claim"}) in the case ([\[small-1\]](#small-1){reference-type="ref" reference="small-1"}). Now, we calculate $I'(r)$. Differentiating ([\[I\]](#I){reference-type="ref" reference="I"}) with respect to $r$, one gets $$\begin{aligned} \label{I-1} I'(r)&=I'_{1}(r)+r^4I'_{2}(r)+4r^3I_{2}(r)\nonumber\\[2mm] &=\int_{ \partial{B_r}} \frac{|\nabla_X u|^2}{|\nabla\rho|}+\int_{\partial{B_r}}\frac{uw}{|\nabla\rho|} +r^{4}\int_{ \partial{B_r}} \frac{|\nabla_X w|^2}{|\nabla\rho|} \nonumber\\[2mm] &+r^{4}\int_{\partial{B_r}}\frac{V uw}{|\nabla\rho|} +4r^{3}\int_{ B_r} |\nabla_X w|^2+4r^{3}\int_{B_r} V uw \nonumber\\[2mm] & \equiv K_1+K_2+K_3+r^{4}\int_{\partial{B_r}}\frac{V uw}{|\nabla\rho|} +4r^{3}\int_{ B_r} |\nabla_X w|^2+4r^{3}\int_{B_r} V uw.\end{aligned}$$ By using the fact ([\[Z\]](#Z){reference-type="ref" reference="Z"}) and the divergence theorem , one has $$\begin{aligned} K_1&\equiv\int_{ \partial{B_r}} \frac{|\nabla_X u|^2}{|\nabla\rho|}=\frac{1}{r}\int_{ \partial{B_r}} \frac{|\nabla_X u|^2}{|\nabla\rho|}Z\rho\nonumber\\[2mm] &=\frac{1}{r}\int_{ \partial{B_r}} |\nabla_X u|^2 Z\cdot\frac{\nabla\rho}{|\nabla\rho|}= \frac{1}{r}\int_{B_r}\mbox{div}(|\nabla_{X}u|^2Z) \nonumber\\[2mm] &=\frac{Q}{r}\int_{B_r}|\nabla_{X}u|^2+\frac{2}{r}\int_{B_r}\sum_{i}X_iu Z(X_iu).\nonumber\end{aligned}$$ Recalling $[X_i,Z]=X_i$, using the divergence theorem and equation ([\[w\]](#w){reference-type="ref" reference="w"}), the last term on $K_1$ becomes $$\begin{aligned} \frac{2}{r}\int_{B_r}\sum_{i}X_iu Z(X_iu)&=\frac{2}{r}\int_{B_r}\sum_{i}X_iu X_iZu-\frac{2}{r}\int_{B_r}\sum_{i}(X_iu)^2\nonumber\\[2mm] &=-\frac{2}{r}\int_{B_r}w Zu+\frac{2}{r}\int_{\partial{B_r}}Zu\frac{\nabla_{X}u\cdot\nabla_{X}\rho}{|\nabla\rho|} -\frac{2}{r}\int_{B_r}|\nabla_{X}u|^2.\nonumber\end{aligned}$$ Hence, by using the identity ([\[Z2\]](#Z2){reference-type="ref" reference="Z2"}), we obtain $$\begin{aligned} K_1&=\frac{Q-2}{r}\int_{B_r}|\nabla_{X}u|^2-\frac{2}{r}\int_{B_r}w Zu+\frac{2}{r}\int_{\partial{B_r}} Zu \frac{\nabla_{X}u\cdot\nabla_{X}\rho}{|\nabla\rho|}\nonumber\\[2mm] &=\frac{Q-2}{r}\int_{B_r}|\nabla_{X}u|^2-\frac{2}{r}\int_{B_r}w Zu +\frac{2}{r^2}\int_{\partial{B_r}} (Zu)^2 \frac{\psi}{|\nabla\rho|}.\end{aligned}$$ The term $K_3$ can be calculated in the same way as $K_1$, $$\begin{aligned} K_3&\equiv r^{4}\int_{ \partial{B_r}} \frac{ |\nabla_X w|^2}{|\nabla\rho|}\nonumber\\[2mm] &=\frac{Q-2}{r}r^{4}\int_{B_r}|\nabla_Xw|^2-2r^{3}\int_{B_r}V u Zw +2r^{2} \int_{\partial{B_r}}(Zw)^2\frac{\psi}{|\nabla\rho|}.\end{aligned}$$ By using the fact ([\[Z\]](#Z){reference-type="ref" reference="Z"}) and the divergence theorem again, one has $$\begin{aligned} K_2&\equiv \int_{\partial{B_r}}\frac{uw}{|\nabla\rho|}=\frac{1}{r}\int_{\partial{B_r}}uw\frac{Z\rho}{|\nabla\rho|} =\frac{1}{r}\int_{B_r}\mbox{div}(uw Z)\nonumber\\[2mm] &=\frac{Q}{r}\int_{B_r}uw+\frac{1}{r}\int_{B_r}w Zu+ \frac{1}{r}\int_{B_r}u Zw.\end{aligned}$$ Putting $K_i$ (i=1,2,3) into ([\[I-1\]](#I-1){reference-type="ref" reference="I-1"}) and recalling ([\[I1\]](#I1){reference-type="ref" reference="I1"}), it holds $$\begin{aligned} \label{sum1} I'(r)&=\frac{Q-2}{r} I(r)+\frac{2}{r^2}\int_{\partial{B_r}}\frac{(Zu)^2\psi}{|\nabla\rho|} +2r^2\int_{\partial{B_r}}\frac{(Zw)^2\psi}{|\nabla\rho|}+4r^3\int_{B_r}|\nabla_{X}w|^2\nonumber\\[2mm] &+\frac{2}{r}\int_{B_r}uw-(Q-6)r^3\int_{B_r}Vuw -\frac{1}{r}\int_{B_r} Zu w\nonumber\\[2mm] &+\frac{1}{r}\int_{B_r}Zw u-2r^{3}\int_{B_r}VuZw+r^{4}\int_{\partial B_r}\frac{Vuw}{|\nabla\rho|} \nonumber\\[2mm] &\equiv\frac{Q-2}{r} I(r)+\frac{2}{r^2}\int_{\partial{B_r}}\frac{(Zu)^2\psi}{|\nabla\rho|} +2r^2\int_{\partial{B_r}}\frac{(Zw)^2\psi}{|\nabla\rho|}+4r^3\int_{B_r}|\nabla_{X}w|^2\nonumber\\[2mm] &+\sum_{i=1}^6R_{i}.\end{aligned}$$ In the following, we estimate $R_i$. Applying Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} to $w$ and Lemma [Lemma 6](#lem4){reference-type="ref" reference="lem4"} to $u$, term $R_1$ can be estimated as follows: $$\begin{aligned} \label{mon-1} R_1&\equiv2r^{-1}\int_{B_r}uw\leq2r^{3}\int_{B_r}\frac{|u||w|}{\rho^4}\nonumber\\[2mm] &\leq r^{3} \int_{B_r} \frac{u^2\psi}{\rho^6}+ r^{3} \int_{B_r} \frac{w^2}{\rho^2\psi}\nonumber\\[2mm] &\leq C\left(r^{3} \int_{B_r}|\nabla_{X}w|^2 +r^{2}\int_{\partial{B_r}}\frac{w^2\psi}{|\nabla\rho|} + r^{-2}\int_{\partial{B_r}}\frac{u^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ Applying the assumption on $V$ ([\[V\]](#V){reference-type="ref" reference="V"}), using Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"} and Lemma [Lemma 6](#lem4){reference-type="ref" reference="lem4"} again, one has $$\begin{aligned} R_2&\equiv -(Q-6)r^{3}\int_{B_r}Vuw\leq C r^{3}\int_{B_r} \frac{|u||w|}{\rho^4}\nonumber\\[2mm] &\leq Cr^{3} \left(\int_{B_r} \frac{u^2\psi}{\rho^6}+ \int_{B_r} \frac{w^2}{\rho^2\psi}\right)\nonumber\\[2mm] &\leq C r^{3} \left(\int_{B_r}|\nabla_{X}w|^2 +r^{-1}\int_{\partial{B_r}}\frac{w^2\psi}{|\nabla\rho|} + r^{-5}\int_{\partial{B_r}}\frac{u^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ In order to estimate the terms with $Zu$ or $Zw$, we show the following fact: $$\label{fact} |Zu|\leq \frac{\rho^{\alpha+1}}{|x|^{\alpha}}|\nabla_{X}u|\leq \rho \psi^{-1/2}|\nabla_{X}u|.$$ Indeed, $$Zu=(x,(\alpha+1)y)\cdot(\partial_{x}u,\partial_{y}u)=\left(x,\frac{(\alpha+1)y}{|x|^{\alpha}}\right)\cdot (\partial_{x}u,|x|^{\alpha}\partial_{y}u).$$ According to ([\[fact\]](#fact){reference-type="ref" reference="fact"}), and applying Lemma [Lemma 5](#lem3){reference-type="ref" reference="lem3"}, it holds $$\begin{aligned} \label{R3} R_3&\equiv-r^{-1}\int_{B_r} Zu w \leq r^{-1}\int_{B_r} \rho \psi^{-1/2}|\nabla _{X}u||w|\leq r\int_{B_r} \frac{|\nabla _{X}u||w|}{\rho \psi^{1/2}}\nonumber\\[2mm] &\leq C \left(r^{-1}\int_{B_r} |\nabla _{X}u|^2+ r^3\int_{B_r}\frac{w^2}{\rho^2\psi}\right)\nonumber\\[2mm] &\leq C \left(r^{-1}\int_{B_r} |\nabla _{X}u|^2+ r^3\int_{B_r} |\nabla_{X}w|^2+ r^{2} \int_{\partial{B_r}}\frac{w^2\psi}{|\nabla\rho|}\right),\end{aligned}$$ and $$\begin{aligned} \label{R4} R_4&\equiv r^{-1}\int_{B_r}Zw u \leq r\int_{B_r} \frac{|u| |\nabla _{X}w|}{\rho\psi^{1/2}}\nonumber\\[2mm] &\leq C\left( r^3\int_{B_r} |\nabla _{X}w|^2+r^{-1} \int_{B_r} \frac{|u|^2}{\rho^2\psi}\right)\nonumber\\[2mm] &\leq C \left(r^3\int_{B_r} |\nabla _{X}w|^2+r^{-1} \int_{B_r} |\nabla_{X}u|^2+ r^{-2} \int_{\partial{B_r}}\frac{u^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ Using ([\[fact\]](#fact){reference-type="ref" reference="fact"}) again, and applying Lemma [Lemma 7](#lem5){reference-type="ref" reference="lem5"}, one has $$\begin{aligned} \label{R5} R_5&\equiv-2r^{3}\int_{B_r}VuZw \leq Cr^{3}\int_{B_r}\frac{|u||Zw|}{\rho^4} \leq Cr^{3}\int_{B_r}\frac{|u||\nabla_{X}w|}{\rho^3\psi^{1/2}}\nonumber\\[2mm] &\leq Cr^{3}\left(\int_{B_r}\frac{u^2}{\rho^6\psi} + \int_{B_r}|\nabla_{X}w|^2\right) \nonumber\\[2mm] &\leq Cr^{3}\left(\int_{B_r}|\nabla_Xw|^2+r^{-4}\int_{B_r}|\nabla_{X} u|^2+r^{-1}\int_{\partial B_r} \frac{w^2\psi}{|\nabla\rho|}+r^{-5}\int_{\partial B_r}\frac{u^2\psi}{|\nabla\rho|}\right).\end{aligned}$$ Under assumption ([\[V\]](#V){reference-type="ref" reference="V"}), we estimate the boundary term $R_6$ as follows, $$\begin{aligned} \label{R6} R_6&\equiv r^{4}\int_{\partial{B_r}}\frac{V wu}{|\nabla\rho|}\leq c_0\int_{\partial{B_r}}\frac{|w||u|}{|\nabla\rho|}\nonumber\\[2mm] &=c_0 r^{-1}\int_{\partial{B_r}}\frac{|w||u|Z\cdot\nabla\rho}{|\nabla\rho|} =c_0r^{-1}\int_{{B_r}}\mbox{div}\left(Z|w||u|\right)\nonumber\\[2mm] &= c_0r^{-1}\left(\int_{{B_r}}\mbox{div}Z|w||u| +\int_{{B_r}}|u|Z|w|+\int_{{B_r}}|w|Z|u| \right)\nonumber\\[2mm] &\equiv R_{61}+ R_{62}+ R_{63}.\end{aligned}$$ Recalling the fact ([\[Z0\]](#Z0){reference-type="ref" reference="Z0"}), we have $$\begin{aligned} R_{61}&=c_0Qr^{-1}\int_{{B_r}}|w||u| \leq C r^{3} \int_{{B_r}}\frac{|w||u|}{\rho^4} \nonumber\\[2mm] &\leq C r^{3}\left(\int_{B_r} \frac{u^2\psi}{\rho^6}+ \int_{B_r} \frac{w^2}{\rho^2\psi}\right)\nonumber\\[2mm] &\leq Cr^{3}\left(\int_{B_r}|\nabla_{X}w|^2 + r^{-1}\int_{\partial{B_r}}\frac{w^2\psi}{|\nabla\rho|} + r^{-5}\int_{\partial{B_r}}\frac{u^2\psi}{|\nabla\rho|}\right). \end{aligned}$$ By ([\[fact\]](#fact){reference-type="ref" reference="fact"}) again, similarly as for ([\[R3\]](#R3){reference-type="ref" reference="R3"}), one has $$\begin{aligned} R_{62}&\equiv C r^{-1}\int_{{B_r}}|u|Z|w| \leq C r^{-1}\int_{B_r}|Zw||u|\nonumber\\[2mm] &\leq C \left( r^{3}\int_{B_r} |\nabla _{X}w|^2+ r^{-1} \int_{B_r} |\nabla_{X}u|^2+ r^{-2} \int_{\partial{B_r}}\frac{u^2\psi}{|\nabla\rho|}\right), \end{aligned}$$ and $$\begin{aligned} \label{mon-2} R_{63}&\equiv C r^{-1}\int_{{B_r}}|w|Z|u|\leq C r^{-1}\int_{B_r}|Zu||w|\nonumber\\[2mm] &\leq C \left( r^{-1}\int_{B_r} |\nabla _{X}u|^2+ r^{3} \int_{B_r} |\nabla_{X}w|^2+ r^{2} \int_{\partial{B_r}}\frac{w^2\psi}{|\nabla\rho|}\right). \end{aligned}$$ Putting ([\[mon-1\]](#mon-1){reference-type="ref" reference="mon-1"})-([\[mon-1\]](#mon-1){reference-type="ref" reference="mon-1"}) into ([\[sum1\]](#sum1){reference-type="ref" reference="sum1"}), and according to the claim ([\[claim\]](#claim){reference-type="ref" reference="claim"}) and ([\[h\<i\]](#h<i){reference-type="ref" reference="h<i"}), we arrive at $$\begin{aligned} \left|I'(r)-\left( \frac{Q-2}{r}I(r)+ \frac{2}{r^2}\int_{\partial{B_r}}\frac{(Zu)^2\psi}{|\nabla\rho|} +2r^2\int_{\partial{B_r}}\frac{(Zw)^2\psi}{|\nabla\rho|}\right)\right|\leq C\frac{I(r)}{r}.\end{aligned}$$ This finishes the proof of Lemma [Lemma 13](#lemma2){reference-type="ref" reference="lemma2"}. ◻ Based on Lemmas [Lemma 11](#lemma1){reference-type="ref" reference="lemma1"}, [Lemma 13](#lemma2){reference-type="ref" reference="lemma2"}, the monotonicity of the frequency function can be be easily established. **Proof of Theorem [Theorem 12](#thm4){reference-type="ref" reference="thm4"}**. Using ([\[H\'(r)\]](#H'(r)){reference-type="ref" reference="H'(r)"}) and ([\[I\'(r)\]](#I'(r)){reference-type="ref" reference="I'(r)"}), we finally obtain $$\begin{aligned} \frac{N'(r)}{N(r)}&=\frac{1}{r}+\frac{I'(r)}{I(r)}-\frac{H'(r)}{H(r)}\\ &\geq \frac{2\left(\int_{\partial B_r}u_n^2+r^4\int_{\partial B_r}w_n^2\right)}{\int_{\partial B_r}u u_n+r^4\int_{\partial B_r}w w_n} -\frac{2\left(\int_{\partial B_r}u u_n+r^4\int_{\partial B_r}w w_n\right)}{\int_{\partial B_r}u^2+r^{4}\int_{\partial B_r}w^2}-\frac{C}{r}-\frac{4}{r}\\ &\geq-\frac{\beta}{r},\end{aligned}$$ where we have used the Schwarz's inequality. This finishes the proof.0◻ # Vanishing order and strong unique continuation property To prove Theorem [Theorem 2](#thm0){reference-type="ref" reference="thm0"}, we need to check that if the vanishing order of a solution $u$ to $Lu=0$ is infinite, then $\Delta_{X}u$ also vanishes to infinite order. Precisely, **Theorem 14**. *Let $u\in M^{2,2}_{loc}(\Omega)$ be a solution of ([\[L\]](#L){reference-type="ref" reference="L"}), where $V$ satisfies the growth assumption ([\[V\]](#V){reference-type="ref" reference="V"}). If $u$ vanishes to infinite order at the origin in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}), then $\Delta_{X}u$ also vanishes to infinite order at the origin, that is $$\begin{aligned} \label{exp-2} \int_{B_r}(\Delta_{X}u)^2\psi=O\left(\exp(-\widetilde{B}r^{-\gamma})\right),\end{aligned}$$ as $r\rightarrow0$ for some constants $\widetilde{B},\gamma>0$.* *Proof.* Let $\eta(t)\in C_0^{\infty}(\mathbb{R})$ be a cut-off function such that $$\eta(t)=\left\{ \begin{array}{lll}1\quad 0\leq t\leq r,\\ 0\quad t\geq 2r. \end{array}\right.$$ Moreover, $0\leq\eta\leq1$, and $$\begin{aligned} | \eta'(t)|\leq \frac{C}{r},\quad \mbox{and}\hspace{2mm}|\eta''(t)|\leq \frac{C}{r^2}, \quad \mbox{for} \hspace{2mm} r\leq t\leq 2r.\end{aligned}$$ Choosing cut-off function $\eta(\rho(z))$, from the properties of the gauge norm ([\[psi\]](#psi){reference-type="ref" reference="psi"}), ([\[psi2\]](#psi2){reference-type="ref" reference="psi2"}), it holds $$\begin{aligned} \label{test} %|\nabla_{X}\rho|=\psi^{1/2},\quad \mbox{and}\quad | X_iX_j\rho|\leq \frac{C}{\rho}, \quad z\in B_{2r}\backslash B_{r} &\left|X \eta(\rho(z))\right|=\left|\eta' X \rho\right|\leq \frac{C}{r}\psi^{1/2}\nonumber\\[2mm] &\left|X_iX_j\eta(\rho(z))\right|=\left|\eta'' X_i\rho X_j\rho+\eta' X_iX_j\rho\right|\leq \frac{C}{r^2},\quad z\in B_{2r}\backslash B_{r}.\end{aligned}$$ Since $u$ is a solution of ([\[L\]](#L){reference-type="ref" reference="L"}), it holds $$\begin{aligned} \label{L1} \int_{\Omega}\Delta_{X}^2 u\phi=\int_{\Omega}Vu\phi,\end{aligned}$$ for any $\phi\in M^{2,2}_0(\Omega)$. Taking the test function $\phi=u\eta^4$ in ([\[L1\]](#L1){reference-type="ref" reference="L1"}) and integrating by parts twice, we deduce that $$\begin{aligned} &\int_{B_{2r}}Vu^2\eta^4=\int_{B_{2r}}\Delta_{X}^2u (u\eta^4) =\int_{B_{2r}}\Delta_{X} u \Delta_{X}(u\eta^4),\end{aligned}$$ which yields $$\begin{aligned} \label{inest} \int_{B_{2r}}(\Delta_{X} u)^2\eta^4&=\int_{B_{2r}}Vu^2\eta^4-4\int_{B_{2r}}\eta^3u\Delta_{X} u \Delta_{X}\eta\nonumber\\[2mm] &-12\int_{B_{2r}}\eta^2u\Delta_{X} u|\nabla_{X} \eta|^2-8\int_{B_{2r}}\eta^3\Delta_{X} u\nabla_{X} u\cdot\nabla_{X} \eta\nonumber\\[2mm] &\equiv \int_{B_{2r}}Vu^2\eta^4+\sum_{i=1}^{3}J_i.\end{aligned}$$ Using the Cauchy inequality and the properties of the test function ([\[test\]](#test){reference-type="ref" reference="test"}), it holds $$\begin{aligned} \label{inestJ1} J_1&\leq C\int_{B_{2r}}\eta^3\left|u\Delta_{X} u \Delta_{X}\eta\right| \nonumber\\[2mm] &\leq \frac{1}{8} \int_{B_{2r}}(\Delta_{X} u)^2\eta^4+C \int_{B_{2r}}u^2 \eta^2(\Delta_{X}\eta)^2, %&\leq \varepsilon \int_{B_{2r}}(\Delta_{X} u)^2\eta^4+C(\varepsilon)r^{-4} \int_{B_{2r}}u^2 \eta^2 \psi^{2},\end{aligned}$$ $$\begin{aligned} \label{inestJ2} J_2&\leq C\int_{B_{2r}}\eta^2\left|u\Delta_{X} u\right| |\nabla_{X} \eta|^2\nonumber\\[2mm] &\leq \frac{1}{8} \int_{B_{2r}}(\Delta_{X} u)^2\eta^4+C \int_{B_{2r}}u^2 |\nabla_{X} \eta|^4, %&\leq \varepsilon \int_{B_{2r}}(\Delta_{X} u)^2\eta^4+C(\varepsilon)r^{-4} \int_{B_{2r}}u^2 \psi^{2}.\end{aligned}$$ and $$\begin{aligned} \label{inestJ3} J_3&\leq C\int_{B_{2r}}\eta^3\Delta_{X} u\nabla_{X} u\cdot\nabla_{X} \eta \nonumber\\[2mm] &\leq \frac{1}{8} \int_{B_{2r}}(\Delta_{X} u)^2\eta^4+C \int_{B_{2r}} \eta^2|\nabla_{X}\eta|^2|\nabla_{X}u|^2.\end{aligned}$$ Next, we estimate the last term on the right-hand side of ([\[inestJ3\]](#inestJ3){reference-type="ref" reference="inestJ3"}). Integrating by parts deduces that $$\begin{aligned} &\int_{B_{2r}}\eta^2|\nabla_{X} u|^2|\nabla_{X}\eta|^2=-\int_{B_{2r}}\mbox{div}_{X}\left(\nabla_{X}u|\nabla_{X}\eta|^2\eta^2\right)u\\[2mm] &=-\int_{B_{2r}}\Delta_{X}u|\nabla_{X}\eta|^2\eta^2u-2\int_{B_{2r}}X_kuX_{k}(X_j\eta)X_{j}\eta \eta^2u-2\int_{B_{2r}}\eta u|\nabla_{X}\eta|^2\nabla_{X}u\cdot\nabla_{X}\eta\\[2mm] &\leq\frac{1}{16}\int_{B_{2r}}|\Delta_{X}u|^2\eta^4+\frac{1}{2}\int_{B_{2r}}|\nabla_{X}u|^2|\nabla_{X}\eta|^2\eta^2+ C\left(\int_{B_{2r}}u^2|\nabla_{X}\eta|^4+\int_{B_{2r}}|\nabla^2_{X}\eta|^2\eta^2u^2\right),\end{aligned}$$ which gives $$\begin{aligned} \label{key} \int_{B_{2r}}\eta^2|\nabla_{X} u|^2|\nabla_{X}\eta|^2 \leq \frac{1}{8}\int_{B_{2r}}|\Delta_{X}u|^2\eta^4+C\left(\int_{B_{2r}}u^2|\nabla_{X}\eta|^4+\int_{B_{2r}}u^2|\nabla^2_{X}\eta|^2\eta^2\right).\end{aligned}$$ Putting ([\[inestJ1\]](#inestJ1){reference-type="ref" reference="inestJ1"})-([\[key\]](#key){reference-type="ref" reference="key"}) into ([\[inest\]](#inest){reference-type="ref" reference="inest"}), we see that $$\begin{aligned} \label{vanish} \int_{B_{r}}(\Delta_{X} u)^2\leq 2\int_{B_{2r}}Vu^2\eta^4+Cr^{-4}\int_{B_{2r}}u^2.\end{aligned}$$ Since $u$ vanishes to infinite order at the origin in the sense of ([\[exp\]](#exp){reference-type="ref" reference="exp"}), the assumption on $V$ ([\[V\]](#V){reference-type="ref" reference="V"}) and Proposition [Proposition 4](#prop){reference-type="ref" reference="prop"} infer that $$\begin{aligned} \int_{B_{2r}}Vu^2\eta^4&\leq c_0\int_{B_{2r}}\frac{u^2}{\rho^4}=c_0\sum_{j=0}^{\infty}\int_{2^{-j}r\leq\rho(z)\leq 2^{-(j-1)}r}\frac{u^2}{\rho^4} \\[2mm] &\leq c_0\sum_{j=0}^{\infty}2^{4j}r^{-4} \int_{\rho(z)\leq 2^{-(j-1)}r}u^2 \\[2mm] &\leq C \sum_{j=0}^{\infty}2^{4j}r^{-4} e^{-B2^{(j-1)\gamma}r^{-\gamma}} \\[2mm] %&\leq Cr^{-4}\sum_{j=0}^{\infty}e^{(ln 16j-B2^{j\gamma})(2r)^{-\gamma}} &\leq Ce^{-\widetilde{B}(2r)^{-\gamma}}.\end{aligned}$$ Then, both terms on the right side of ([\[vanish\]](#vanish){reference-type="ref" reference="vanish"}) vanish to infinite order and the theorem follows. ◻ With Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}, Theorem [Theorem 14](#thm3){reference-type="ref" reference="thm3"} at hand, the strong unique continuation property can be obtained directly. The argument is standard (see e.g. [@gl1]), we include it for the sake of completeness. **Proof of Theorem [Theorem 2](#thm0){reference-type="ref" reference="thm0"}**. Making use of Theorem [Theorem 14](#thm3){reference-type="ref" reference="thm3"} and Proposition [Proposition 4](#prop){reference-type="ref" reference="prop"}, the assumption ([\[exp\]](#exp){reference-type="ref" reference="exp"}) imply that $$\begin{aligned} \int_{B_r}\left(u^2+(\Delta_{X}u)^2\right)\psi\leq C \exp(-Br^{-\gamma}), \quad \mbox{for some constants}\quad B,\gamma>0.\end{aligned}$$ Now, for fixed $R$, after $k$ times iterations of ([\[doubling\]](#doubling){reference-type="ref" reference="doubling"}), we infer $$\begin{aligned} \int_{B_{R}}\left(u^2 + (\Delta_{X}u)^2\right)\psi&\leq \Big(C\exp\left(A R^{-\gamma}\right)\Big)^{k}\int_{B_{2^{-k}R}}\left(u^2 + (\Delta_{X}u)^2\right)\psi\\[2mm] &\leq\Big(C\exp\left(A R^{-\gamma}\right)\Big)^{k}\exp\left(-B(2^{-k}R)^{-\gamma}\right)\\[2mm] &\leq \exp\left((k\ln C+kA-B2^{k\gamma})R^{-\gamma}\right)\rightarrow 0\quad \mbox{as} \hspace{1mm} k\rightarrow \infty.\end{aligned}$$ Then $u\equiv 0$ in $B_{R}$. Moreover, for some point $(0,y_0)\in B_{R}$, we have $u\equiv0$ in the neighbourhood of $(0,y_0)$, so $u$ vanishes to infinite order at $(0,y_0)$. Since the operator $\Delta^2_{X}$ is translation invariant in $y$, we can repeat the previous process in the gauge ball $B_{r}(0,y_0)$ to show that $u\equiv 0$ in $B_{R}(0,y_0)$. On the other hand, outside of $B_{R} (0,y_0)$, the equation ([\[L\]](#L){reference-type="ref" reference="L"}) can be seen as bi-Laplace with bounded potential $V$, therefore we can apply the results in [@liu2022] to conclude that $u\equiv0$ in $\Omega$. 0◻ # Funding {#funding .unnumbered} The research of the first author was supported by the National Natural Science Foundation of China (No.12071219). The research of the second author was supported by the National Natural Science Foundation of China (No.11971229). 99 Almgren, F J Jr. Dirichlet's problem for muptiple valued functions and the regularity of mass minimizing integral currents, In: Obata M, ed. Minimal Submanifolds and Geodesics. Amsterdam: North Holland, 1979, 1-6. Alinhac, S. Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. École Norm. Sup. 1980, 13(3): 385-393. Aronszajn, N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures et Appl., 1957, 36: 235-249. Bahouri, H. Non prolongement unique des solutions d'opérateurs "somme de carrés", Ann. Inst. Fourier, 1986, 36(4): 137-155. Bahrouni, A., Rădulescu, V.D., Winkert, P. Double phase problems with variable growth and convection for the Baouendi-Grushin operator, Zeitschrift für angewandte Mathematik und Physik, 2020, 71(6): 183. Banerjee, A., Mallick, A. On the strong unique continuation property of a degenerate elliptic operator with Hardy-type potential, Annali di Matematica Pura ed Applicata (1923-), 2020, 199(1): 1-21. Baouendi, M.S. Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. 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Ha, 1956, 239-258. Davey, B., Zhu, J.Y. Quantitative uniqueness of solutions to second-order elliptic equations with singular lower order terms, Communications in Partial DifferentialEquations, 2019, 44:11, 1217-1251. Davey, B., Lin, C.L., Wang, J.N. Strong unique continuation for the Lamé system with less regular coefficients, Math. Ann., 2021, 381: 1005-1029. D'Ambrosio, L. Hardy inequalities related to Grushin type operators, Proceedings of the American Mathematical Society, 2004, 132(3): 725-734. Franchi, B., Lanconelli, E., An embedding theorem for sobolev spaces related to non-smooth vector fieldsand harnack inequality, Communications in Partial Differential Equations, 1984, 9(13): 1237-1264. Franchi, B., Gutiérrez, C.E., Wheeden, R.L. Weighted Sobolev-Poincaré inequalities for Grushin type operators, Communications in Partial Differential Equations, 1994, 19(3-4): 523-604. Garofalo, N., Lin, F.H. Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 1986, 35: 245-268. Garofalo, N., Lin, F.H. Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 1987, 40: 347-366. Garofalo, N., Rotz, K. Properties of a frequency of Almgren type for harmonic functions in Carnot groups, Calculus of Variations and Partial Differential Equations, 2015, 54(2): 2197-2238. Garofalo, N., Lanconelli, E. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier, 1990, 44(2): 313-356. Garofalo, N. Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Differential Equations, 1993, 104(1): 117-146. Gazzola, F., Grunau, H.C., Mitidieri, E. Hardy inequalities with optimal constants and remainder terms, Transactions of the American Mathematical Society, 2004, 356(6): 2149-2168. Grushin, V.V. On a class of hypoelliptic operators, Math. USSR Sbornik., 1970, 12(3):458-476. Grushin, V.V. On a class of hypoelliptic pseudodifferential operators degenerate on submanifold, Math. USSR Sbornik., 1971, 13(2):155-186. Ignatova, M., Kukavica, I. Strong unique continuation for higher order elliptic equations with Gevrey coefficients, J. Differential Equations, 2012, 252: 2983-3000. Jerison, D., Kenig, C. Unique continuation and absence of positive eigenvalues for Schröing-operators, Ann. of Math., 1985, 12(2): 463-494. Koch, H., Tataru, D., Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 2001, 54(3): 339-360. Lin, C.L. Strong unique continuation for $m$-th powers of a Laplacian operator with singular coefficients, Proc. Am. Math. Soc., 2007, 135(2): 569-578. Liu, H.R., Yang, X.P. Critical points and level sets of Grushin-Harmonic functions in the plane, Journal d'Analyse Mathématique, 2021, 143(2): 435-460. Liu, H.R., Yang, X.P. Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials, Science China Mathematics, 2022, 65: 707-730. Monti, R., Morbidelli, D. Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J., 2006, 131 (1): 167-202. Monticelli, D.D. Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, Journal of the European Mathematical Society, 2010, 12(3): 611-654. Ruzhansky, M., Yessirkegenov, N. Rellich inequalities for sub-Laplacians with drift, Proceedings of the American Mathematical Society, 2019, 147(3): 1335-1349. Ruzhansky, M., Suragan, D. On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups, J. Differential Equations, 2017, 262: 1799-1821. Xu, C.J. Subelliptic variational problems, Bulletin de la Société Mathématique de France, 1990, 118(2): 147-169. Zhu, J.Y. Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients, Calculus of Variations and Partial Differential Equations, 2018, 57(2): 58. [^1]: $^{1}$School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094 P.R.China. E-mail : hrliu\@njust.edu.cn [^2]: $^{2}$School of Mathematics, Nanjing University, Nanjing 210093, P.R. China. E-mail: xpyang\@nju.edu.cn

arxiv_math

--- abstract: | By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize those pairs of fuzzy sets which form fuzzy rough sets w.r.t. a t-similarity relation $\theta$ on $U$, for certain t-norms and implicators. Then we establish conditions under which fuzzy rough sets form lattices. We show that for the $\min$ t-norm and any S-implicator defined by the $\max$ co-norm with an involutive negator, the fuzzy rough sets form a complete lattice, whenever $U$ is finite or the range of $\theta$ and of the fuzzy sets is a fixed finite chain. address: Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary author: - Dávid Gégény - Sándor Radeleczki bibliography: - rough.bib title: On the lattice of fuzzy rough sets --- Fuzzy rough sets; Fuzzy relations; Lower and upper approximation; Self-dual poset # Introduction {#sec:intro} Rough sets were introduced by Zdzisław Pawlak [@Pawlak1], by defining the lower and upper approximations of a (crisp) set based on a so-called indiscernibility relation of the elements. Originally, Pawlak assumed that this relation is an equivalence, but later several other types of relations were also examined (see e.g. [@JR1], [@JRV09], or [@JPR], [@YY]). For a relation $\varrho \subseteq U\times U$ and any element $u\in U$, denote $\varrho(u):=\{x\in U\mid(u,x)\in R\}$. Now, for any subset $A\subseteq U$, the *lower approximation* of $A$ is defined as $$A_{\varrho}:=\{x\in U\mid \varrho(x)\subseteq A\},$$ and the *upper approximation* of $A$ is given by $$A^{\varrho}:=\{x\in U\mid \varrho(x)\cap A\neq\emptyset\}.$$ If $\varrho$ is reflexive and transitive, i.e. it is a *quasiorder*, then the properties $A_{\varrho}\subseteq A\subseteq A^{\varrho}$ and $(A_{\varrho})_{\varrho}=A_{\varrho}$, $\left( A^{\varrho}\right) ^{\varrho}=A^{\varrho}$ hold for all $A\subseteq U$. The rough sets induced by $\varrho$ can be ordered w.r.t. the component-wise inclusion, and for an equivalence, or more generally, for a quasiorder $\varrho$, they form a complete distributive lattice with several particular properties, see e.g. [@JRV09] or [@Po88]. The notion of a fuzzy set was introduced by Lotfi Zadeh [@Zh]. A fuzzy set is defined by a mapping $f: U \rightarrow [0, 1]$. We say that $f$ has a *finite range*, whenever the (crisp) set $\{f(x)\mid x\in U\}$ is finite. The collection of all fuzzy sets on $U$ is denoted by $\mathcal{F}(U)$. Ordering any elements $f,g\in\mathcal{F}(U)$ as follows $$f\leq g\Leftrightarrow f(x)\leq g(x)\text{, for all }x\in U\text{,}%$$ we obtain a completely distributive (complete) lattice $\mathcal{F}(U)$. For any system $f_{i}\in\mathcal{F}(U)$, $i\in I$, its infimum and supremum are given by the formulas $$\left( \bigwedge_{i\in I}f_{i}\right) (x)=\bigwedge_{i\in I}f_{i}(x)\text{; }\left( \bigvee_{i\in I}f_{i}\right) (x)=\bigvee_{i\in I}f_{i}(x)\text{,} \tag{1}%$$ where $\bigwedge$ and $\bigvee$ denote the infimum and the supremum, respectively, in the complete lattice $([0,1],\leq)$. The first step to integrate the two main theories relates to the works of del Cerro and Prade [@CP], Nakamura [@NK] and Dubois and Prade [@DP]. In [@DP] the fuzzy rough sets are defined as pairs $(\underline{f},\overline{f}) \in\mathcal{F}(U)\times \mathcal{F}(U)$ of lower and upper approximations of the fuzzy sets $f \in \mathcal{F}(U)$. These fuzzy approximations were defined by using a similarity relation, the t-norm min and conorm max. Their approach was generalized in several papers, like [@CCK; @GMS; @HYPC; @In; @Pei; @RK; @WLM; @WMZ] and [@DVCG], where fuzzy rough sets are defined on the basis of different t-norms (or conjunctors) and related implicators. A detailed study of these approximation operators was developed in [@DVCG], [@RK] and in [@MH], [@RKL], where the structure of the lower and upper approximations of $L$-fuzzy sets is also investigated. An axiomatic approach of these properties was elaborated e.g. in [@BJR], [@Liu], [@MZ], and [@MY]. In [@GKR] it was shown that for crisp reference sets (i.e. for $f(x)\in\{0,1\}$, $\forall x\in U$) the fuzzy rough sets defined by a t-similarity relation $\theta$ with a well-ordered spectrum form a completely distributive lattice. The goal of the present paper is to find conditions under which fuzzy rough sets form lattices. With this purpose, by the means of lower and upper fuzzy approximations we define (crisp) quasiorders on $U$. The properties of these quasiorders and of the equivalences determined by them are discussed in Sections [3](#sec:quasiorders){reference-type="ref" reference="sec:quasiorders"}, [4](#sec:eqbyqo){reference-type="ref" reference="sec:eqbyqo"} and [6](#sec:propofeqc){reference-type="ref" reference="sec:propofeqc"}. These properties will be used to prove our main results, Theorems [Theorem 15](#thm:frs){reference-type="ref" reference="thm:frs"} and [Theorem 23](#thm:lattice){reference-type="ref" reference="thm:lattice"}. Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"} contains the essential prerequisites of our study. In Section [5](#sec:frs){reference-type="ref" reference="sec:frs"} by using singleton equivalence classes, we characterize those pairs of fuzzy sets which form a fuzzy rough set with respect to a t-similarity relation $\theta$ for certain t-norms and related implicators. In Section [7](#sec:latticefrs){reference-type="ref" reference="sec:latticefrs"}, we establish conditions under which fuzzy rough sets with a finite range form lattices. For instance, we show that for the min t-norm and any S-implicator defined by the max co-norm with an involutive negator, the fuzzy rough sets form a complete lattice, whenever $U$ is finite, or whenever the range of $\theta$ and of the fuzzy reference sets is a fixed finite chain $L\subseteq \lbrack0,1]$. # Preliminaries {#sec:prelim} ## T-norms, implicators and fuzzy relations *A triangular norm* $\odot$ (*t-norm* for short) is a commutative, associative and monotone increasing binary operation $\odot$ defined on $[0,1]$ satisfying $1\odot x=x\odot1=x$, for all $x\in\lbrack0,1]$. The t-norm $\odot$ is called *(left) continuous,* if it is (left) continuous as a function $\odot\colon\lbrack0,1]^{2}\rightarrow\lbrack0,1]$ in the usual interval topology on $[0,1]^{2}$. Every t-norm $\odot$ satisfies $x\odot0=0\odot x=0$, for all $x\in L$. The most known t-norms are: \- the *standard min operator*: $x\odot y:= \min(x,y)$;\ - the *arithmetical product*: $x\odot y:= x\cdot y$;\ - the *Łukasiewicz t-norm* $x\odot y:= \max(0,x+y-1)$.\ A *negator* is a decreasing map $n\colon\lbrack0,1]\rightarrow \lbrack0,1]$ with $n(0)=1$ and $n(1)=0$. $n$ is called *involutive* if $n(n(x))=x$, for all $x\in\lbrack0,1]$ (see e.g. [@Fodor]). The so-called *standard negator* $n(x):=1-x$, $x\in\lbrack0,1]$ is an involutive negator. *A triangular conorm* $\oplus$ (shortly *t-conorm*) is a commutative, associative and monotone increasing binary operation $\oplus$ defined on $[0,1]$, that satisfies $0\oplus x=x\oplus0=x$, for all $x\in\lbrack0,1]$. The t-conorm $\oplus$ is *(left) continuous,* if it is (left) continuous as a function $\oplus\colon\lbrack0,1]^{2}\rightarrow \lbrack0,1]$ in the usual topology. Given an involutive negator $n$, a t-norm $\odot$ and a t-conorm $\oplus$, we say that $\odot$ and $\oplus$ form an $n$*-dual pair* if for all $x,y \in [0,1]$ $$n(x\oplus y)=n(x)\odot n(y)\text{.}%$$ Clearly, this identity also implies the identity $$n(x\odot y)=n(x)\oplus n(y)\text{.}%$$ For instance, min$(x,y)$, max$(x,y)$ form a well-known $n$-dual pair w.r.t. any involutive negator on $[0,1]$. An *implicator* is a binary operation (mapping) $\vartriangleright \colon\lbrack0,1]^{2}\rightarrow\lbrack0,1]$ that is decreasing in the first and increasing in the second argument and that satisfies the boundary conditions $$0\vartriangleright0=0\vartriangleright1=1\vartriangleright1=1\text{ and }1\vartriangleright0=0\text{.}%$$ $\vartriangleright$ is called a *border implicator* if $1\vartriangleright x=x$ holds for all $x\in\lbrack0,1]$. There are two important classes of border implicators. The *R-implicato*r based on a t-norm $\odot$ is defined by $$x\vartriangleright y:=\bigvee\{z\in\lbrack0,1]\mid x\odot z\leq y\}\text{, for all }x,y\in\lbrack0,1].$$ If $\odot$ is a continuous t-norm, then the algebra $([0,1],\vee,\wedge,\odot,\vartriangleright ,0,1)$ is a so-called *commutative (integral) residuated lattice* (see [@GJ]). Let $\oplus$ be a t-conorm and $n$ a negator on $[0,1]$, then the *S-implicato*r based on them is defined by $$x\vartriangleright y:=n(x)\oplus y$$ The *Łukasiewicz implicator* $\vartriangleright _{L}$ is both an R-implicator and S-implicator defined by $x$ $\vartriangleright_{L}y:=$ min$(1,1-x+y)$, $\forall x\in\lbrack0,1]$. The *Kleene-Dienes (KD) implicator* $\vartriangleright_{KD}$ is an S-implicator given by $x\vartriangleright_{KD}y$ $:=\ $max$(1-x,y)$, $\forall x\in\lbrack0,1]$. If $\vartriangleright$ is an implicator, then a corresponding negator is defined by $n(x)=x\vartriangleright0$. If $n$ is an involutive negator and $\vartriangleright$ is an R-implicator defined by a left-continuous t-norm, then $\vartriangleright$ is called an *ITML-implicator.* A *fuzzy binary relation* on $U$ is a fuzzy set $\theta\colon U\times U\rightarrow\lbrack0,1]$. The pair $(U,\theta)$ is usually called a *fuzzy approximation space*. $\theta$ is called *reflexive* if $\theta(x,x)=1$ for all $x\in U$, and it is called *symmetric* if for all $x,y\in U$, $\theta(x,y)=\theta(y,x)$. Given a t-norm $\odot$, the relation $\theta$ is called $\odot$*-transitive* if $$\theta(x,y)\odot\theta(y,z)\leq\theta(x,z)$$ holds for every $x,y,z\in U$. If a relation $\theta$ is reflexive and $\odot$-transitive, then it is called a (fuzzy) $\odot$*-quasiorder.* A symmetric $\odot$-quasiorder $\theta$ is called a (fuzzy) $\odot$*-similarity relation*. When $x\odot y=$ min$(x,y)$, then $\theta$ is simply named a *similarity relation*. Since the minimum t-norm is the largest t-norm, a similarity relation is always $\odot$-transitive for any t-norm $\odot$. We say that $\theta$ is of a *finite range*, if the (crisp) set $\{\theta(x,y)\mid x,y\in U\}$ of its values is finite. ## Fuzzy rough sets Let $(U,\theta)$ be a fuzzy approximation space with a relation $\theta\colon U\times U\rightarrow\lbrack0,1]$. The precise notion of a fuzzy rough set was introduced by D. Dubois and H. Prade in [@DP]. They defined for any fuzzy set $f\in\mathcal{F}(U)$ its *lower* *approximation* $\underline{\theta}(f)$ and its *upper approximations* $\overline{\theta}(f)$ *relative to* $\theta$ by the formulas $\underline{\theta}(f)(x):=\bigwedge\{\text{max}(1-\theta(x,y),f(y))\mid y\in U\}\text{, for all }x\in U\text{;}$ $\overline{\theta}(f)(x):=\bigvee\{\text{min}(\theta(x,y),f(y))\mid y\in U\}\text{, for all }x\in U\text{.}$ The *fuzzy rough set of* $f$ is identified by the pair $(\underline{\theta}(f),\overline{\theta}(f))\in\mathcal{F}(U)\times \mathcal{F}(U)$ (see [@DP]). This definition was generalized in several papers. Here we will use the approach based on implicators and t-norms from [@DVCG] and [@RK]. Hence, in what follows, let $\odot$ be a t-norm and $\vartriangleright$ a border implicator on $[0,1]$. **Definition 1**. *If $(U,\theta)$ is a fuzzy approximation space, then for any fuzzy set $f\in\mathcal{F}(U)$ its *fuzzy lower approximation* $\underline{\theta}(f)$ and its *fuzzy upper approximation* $\overline{\theta}(f)$ are defined as follows:* *$$\underline{\theta}(f)(x):=\bigwedge\{\theta(x,y)\vartriangleright f(y)\mid y\in U\}\text{, for all }x\in U\text{.}\tag{2}%$$* *$$\overline{\theta}(f)(x):=\bigvee\{\theta(x,y)\odot f(y)\mid y\in U\}\text{, for all }x\in U\text{.}\tag{2'}%$$* *The pair $(\underline{\theta}(f),\overline{\theta}(f))\in \mathcal{F}(U)\times\mathcal{F}(U)$ is called a *fuzzy rough set in* $(U,\theta)$.* This definition also includes the one of Dubois and Prade, where $\odot$ is the min t-norm and $\vartriangleright$ is the Kleene-Dienes implicator $x\vartriangleright_{KD}y$ $=\ $max$(1-x,y)$. Notice that $\underline{\theta}$ and $\overline{\theta}$ are *order-preserving* operators, i.e. $f\leq g$ implies $\underline{\theta}(f)\leq\underline{\theta}(g)$ and $\overline {\theta}(f)\leq\overline{\theta}(g)$. In addition, if $\theta$ is a reflexive fuzzy relation, then $\underline{\theta}(f)\leq f\leq\overline{\theta}(f)$ holds for all $f\in\mathcal{F}(U)$ (see e.g. [@DVCG] or [@RK]) The following properties will have a special importance in our proofs: \(D\) Let $\odot$ be a left-continuous t-norm such that its induced R-implicator $\vartriangleright$ is an ITML implicator, i.e. $n(x):=x\vartriangleright0$, $x\in U$ is an involutive negator, or let $n$ be an involutive negator, $\oplus$ a t-conorm $n$-dual to $\odot$ and $\vartriangleright$ the S-implicator defined by them (i.e. $x\vartriangleright y=n(x)\oplus y$). Then $n(\overline{\theta}(f))=\underline{\theta}(n(f))$ and $n(\underline{\theta }(f))=\overline{\theta}(n(f))$ (see e.g. [@DVCG], [@MH] or [@RK]). (ID) Let $\odot$ be a left-continuous t-norm and $\vartriangleright$ the R-implicator induced by it, or $n$ an involutive negator, $\oplus$ a t-conorm $n$-dual to $\odot$ and $\vartriangleright$ the S-implicator corresponding to them. If $\theta$ is $\odot$-transitive, then for any $f,g\in\mathcal{F}(U)$ we have $\overline{\theta}(\overline{\theta }(f))=\overline{\theta}(f)$ and $\underline{\theta}(\underline{\theta }(g))=\underline{\theta}(g)$ (see [@DVCG], [@MH], [@RK]). In other words, for $F=\overline{\theta}(f)$ and $G=\underline{\theta}(g)$ we have $F=\overline {\theta}(F)$ and $G=\underline{\theta}(G)$. **Lemma 2**. *Let $(U,\theta)$ be a fuzzy approximation space such that the relation $\theta$ is of a finite range. If $f\in\mathcal{F}(U)$ has a finite range, then the fuzzy sets $\underline{\theta}(f)$ and $\overline{\theta}(f)$ are also of a finite range.* *Proof.* Since $\{\theta(x,y)\mid x,y\in U\}$ and $\{f(y)\mid y\in U\}$ are finite sets, their Cartesian product $\{(\theta (x,y),f(y))\mid x,y\in U\}$ is finite, hence the sets $\mathcal{C}% =\{\theta(x,y)\odot f(y)\mid x,y\in U\}$ and $\mathcal{I}=\{\theta (x,y)\vartriangleright f(y)\mid x,y\in U\}$ are also finite. In particular, this means that the sets $\mathcal{C}$ and $\mathcal{I}$ have finitely many (different) subsets of the form $\{\theta(x,y)\vartriangleright f(y)\mid y\in U\}$ and $\{\theta(x,y)\odot f(y)\mid y\in U\}$, and this immediately implies that both $\underline{\theta}(f)$ and $\overline{\theta}(f)$ have finitely many different values, i.e. they have finite ranges. ◻ # Quasiorders induced by lower and upper approximations {#sec:quasiorders} In what follows, suppose that conditions in (ID) hold and $n(x):=x\vartriangleright0$. For any $f,g\in\mathcal{F}(U)$, denote $F=\overline{\theta}(f)$ and $G=\underline {\theta}(g)$. Using $F$ and $G$ we define two binary relations $R(F)$ and $\varrho(G)$ on $U$ as follows: **Definition 3**. *Let $(U,\theta)$ be a fuzzy approximation space, $a,b\in U$ and $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$. Then* *(i) $(a,b)\in R(F)\Leftrightarrow F(a)=$ $\theta(a,b)\odot F(b)$;* *(ii) $(a,b)\in\varrho(G)\Leftrightarrow G(a)=$ $\theta (a,b)\vartriangleright G(b)$.* **Proposition 4**. *(i) $(a,b)\in R(F)$ implies $F(a)\leq \theta(a,b)$, and $(a,b)\in\varrho(G)$ implies $G(a)\geq n(\theta(a,b))$.* *(ii) If $\theta$ is reflexive, then for any $f,g\in\mathcal{F}(U)$, $R(F)$ and $\varrho(G)$ are reflexive.* *(iii) If $\theta$ is a $\odot$-quasiorder, then $R(F)$, $\varrho(G)$ are crisp quasiorders, and\ $F(a)\geq\theta(a,y)\odot F(y)$, $G(a)\leq \theta(a,y)\vartriangleright G(y)$, for any $a,y\in U$.* *(iv) If $n(x)$ is involutive, then $R(F)=\varrho (n(F))$, and $\varrho(G)=R(n(G))$.* *(v) Let $\odot$ be the minimum t-norm, $n$ an involutive negator, and $x\vartriangleright y:=$ max$(n(x),y)$. If $\theta$ is a similarity relation, then $(a,b)\in R(F)\Leftrightarrow F(a)\leq\theta(a,b)$ and $(a,b)\in \varrho(G)\Leftrightarrow G(a)\geq n(\theta(a,b))$.* *Proof.* (i) By definition $(a,b)\in R(F)$ implies $F(a)=\theta(a,b)\odot F(b)\leq\theta(a,b)$ and $(a,b)\in\varrho(G)$ yields $G(a)=\theta(a,b)\vartriangleright G(b)\geq\theta(a,b)\vartriangleright 0=n(\theta(a,b))$. \(ii\) If $\theta$ is reflexive, then $\theta(a,a)=1$ implies $F(a)=$ $\theta(a,a)\odot F(a)$ and $G(a)=$ $\theta(a,a)\vartriangleright G(a)$, i.e. $(a,a)\in R(F)$ and $(a,a)\in\varrho(G)$ hold for all $a\in U$. Thus $R(F)$ and $\varrho(G)$ are reflexive. \(iii\) Let $\theta$ be a $\odot$-quasiorder. Then $R(F)$, $\varrho(G)$ are reflexive, property (ID) holds, and hence $F(a)=\overline{\theta}(F)(a)$, $G(a)=\underline{\theta }(G)(a)$ imply $F(a)=\bigvee\{\theta(x,y)\odot F(y)\mid y\in U\}\geq \theta(a,y)\odot F(y)$ and $G(a)=\bigwedge\{\theta(x,y)\vartriangleright G(y)\mid y\in U\}\leq\theta(a,y)\vartriangleright G(y)$, $\forall y\in U$. Take $a,b,c\in U$ with $(a,b),(b,c)\in R(F)$. Then $F(a)=\theta (a,b)\odot F(b)$ and $F(b)=\theta(b,c)\odot F(c)$ imply $F(a)=(\theta (a,b)\odot\theta(b,c))\odot F(c)\leq\theta(a,c)\odot F(c)$, because $\theta$ is $\odot$-transitive. Now $F(a)\geq\ \theta(a,c)\odot F(c)$ yields $F(a)=\theta(a,c)\odot F(c)$, i.e. $(a,c)\in R(F)$. Thus $R(F)$ is also transitive, hence it is a quasiorder. Let $(a,b),(b,c)\in\varrho(G)$. Then $G(a)=\theta(a,b)\vartriangleright G(b)$ and $G(b)=\theta(b,c)\vartriangleright G(c)$ imply $G(a)=\theta (a,b)\vartriangleright(\theta(b,c)\vartriangleright G(c))$. If $\vartriangleright$ is an R-implicator, then $\theta(a,b)\vartriangleright (\theta(b,c)\vartriangleright G(c))=(\theta(a,b)\odot\theta (b,c))\vartriangleright G(c)$. If $\vartriangleright$ is an S-implicator $x\vartriangleright y=n(x)\oplus y$, then $\theta(a,b)\vartriangleright (\theta(b,c)\vartriangleright G(c))=n(\theta(a,b))\oplus(n(\theta(b,c))\oplus G(c))=(n(\theta(a,b))\oplus n(\theta(b,c))\oplus G(c)=n(\theta(a,b)\odot\theta(b,c))\oplus G(c)=(\theta(a,b)\odot \theta(b,c))\vartriangleright G(c)$. Hence in both cases $G(a)=(\theta (a,b)\odot\theta(b,c))\vartriangleright G(c)$. Because $\theta$ is $\odot$-transitive (and $\vartriangleright$ is decreasing in the first variable) we get $G(a)\geq\theta(a,c)\vartriangleright G(c)$. Then $G(a)\leq\theta (a,c)\vartriangleright G(c)$ yields $G(a)=\theta(a,c)\vartriangleright G(c)$, i.e. $(a,c)\in\varrho(G)$. Thus $\varrho(G)$ is a $\odot$-quasiorder. \(iv\) Observe, that in this case property (D) holds, i.e., $n(\overline{\theta}(f))=\underline{\theta}(n(f))$. This yields $n(F)=\underline{\theta}(n(f))$, and $(a,b)\in R(F)$ means $F(a)=\theta(a,b)\odot F(b)$. As $n$ is involutive, this is equivalent to $n(F(a))=n(\theta(a,b)\odot F(b))$. If $\vartriangleright$ is an ITML implicator, then $n(\theta(a,b)\odot F(b))=(\theta(a,b)\odot F(b))\vartriangleright0=\theta(a,b)\vartriangleright(F(b)\vartriangleright0)$ $=\theta(a,b)\vartriangleright n(F(b))$. If $\vartriangleright$ is an S-implicator, then $n(\theta(a,b)\odot F(b))=n(\theta(a,b))\oplus n(F(b))=\theta(a,b)\vartriangleright n(F(b))$. Hence in both cases $n(F(a))=n(\theta(a,b)\odot F(b))\Leftrightarrow n(F)(a)=\theta (a,b)\vartriangleright n(F)(b)$. The right side means $(a,b)\in\varrho(n(F))$. Thus we get $(a,b)\in R(F)\Leftrightarrow(a,b)\in\varrho(n(F))$, proving $R(F)=\varrho(n(F))$. Let $g=n(h)$ for some $h\in\mathcal{F}(U)$. Then $n(G)=n(\underline{\theta }(g))=\overline{\theta}(n(g))=\overline{\theta}(h)$, and $G=n(\overline {\theta}(h))$. Hence $(a,b)\in\varrho(G)\Leftrightarrow$ $(a,b)\in\varrho (n(\overline{\theta}(h)))\Leftrightarrow(a,b)\in R(\overline{\theta }(h))=R(n(G))$, and this proves $\varrho(G)=R(n(G))$. \(v\) In view of (i) $(a,b)\in R(F)$ yields $F(a)\leq\theta(a,b)$ and $(a,b)\in\varrho(G)$ implies $G(a)\geq n(\theta(a,b))$. We need only to prove the converse implications. Let $F(a)\leq\theta(a,b)$. Since $\theta$ is also a $\odot$-quasiorder, in view of (iii) $F(b)\geq\theta(b,a)\odot F(a)=\ $ min$(\theta(b,a),F(a))=\ $min$(\theta(a,b),F(a))=F(a)$. Hence $F(a)\leq \ $min$(\theta(a,b),F(b))$. As $F(a)\geq\theta(a,b)\odot F(b)=\ $ min$(\theta(a,b),F(b))$ also holds, we get $F(a)=\ $min$(\theta(a,b),F(b))$, i.e. $(a,b)\in R(F)$. Now let $G(a)\geq n(\theta(a,b))$. As $\theta(b,a)=\theta(a,b)$, and by (iii), $G(b)\leq\theta(b,a)\vartriangleright G(a)=\ $ max$(n(\theta(a,b)),G(a))=G(a)$, we get $G(a)\geq\ $max$(n(\theta(a,b),G(b))$. Since $G(a)\leq\theta(a,b)\vartriangleright G(b)=\ $max$(n(\theta(a,b)),G(b))$ also holds, we obtain $G(a)=\ $max$(n(\theta(a,b)),G(b))=\theta (a,b)\vartriangleright G(b)$, i.e. $(a,b)\in\varrho(G)$. ◻ **Corollary 5**. *If the conditions in Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(v) are satisfied, then $(a,b)\notin R(F)\Leftrightarrow F(a)>\theta(a,b)$ and $(a,b)\notin\varrho(G)\Leftrightarrow G(a)<n(\theta(a,b))$.* **Proposition 6**. *Let $\theta$ be a $\odot$-quasiorder and $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$, for some $f,g\in \mathcal{F}(U)$, and $a,b\in U$. The following hold true:* *(i) If $(a,b)\in R(F)$, then for any $h\in\mathcal{F}(U)$ with $h\leq F$, $h(b)=F(b)$ implies $\overline{\theta}(h)(a)=F(a)$.* *(ii) If $(a,b)\in\varrho(G)$, then for any $h\in\mathcal{F}(U)$ with $h\geq G$, $h(b)=G(b)$ implies $\underline{\theta}(h)(a)=G(a)$.* *Proof.* (i) By definition, we have $F(a)=$ $\theta(a,b)\odot F(b)$. Hence we get: $\overline{\theta}(h)(a)=\bigvee\{\theta(a,y)\odot h(y)\mid y\in U\}\geq\ $ $\theta(a,b)\odot h(b)=\ \theta(a,b)\odot F(b)=F(a)$ On the other hand, $\overline{\theta}(h)\leq\overline{\theta}(F)=F$ implies $\overline{\theta}(h)(a)\leq F(a)$. Thus we obtain $\overline{\theta }(h)(a)=F(a)$. \(ii\) Now, analogously we have $G(a)=$ $\theta(a,b)\vartriangleright G(b)$. Therefore, we get: $\underline{\theta}(h)(a)=\bigwedge\{\theta(a,y)\vartriangleright h(y)\mid y\in U\}\leq\ $ $\theta(a,b)\vartriangleright h(b)=\theta(a,b)\vartriangleright G(b)=G(a)$. Now $\underline{\theta}(h)\geq\underline{\theta}(G)=G$ yields $\underline{\theta}(h)(a)\geq G(a)$, whence $\underline{\theta }(h)(a)=G(a)$. ◻ # The equivalences induced by the quasiorders $R(F)$ **and** $\varrho(G)$ {#sec:eqbyqo} In this section we assume that conditions in (ID) are satisfied, i.e. that $\odot$ is a left-continuous t-norm, $\vartriangleright$ is the R-implicator induced by it and $n(x)=x\vartriangleright0$, or $n$ is an involutive negator, $\oplus$ is the t-conorm $n$-dual to $\odot$ and $\vartriangleright$ is the S-implicator defined by them. We also suppose that $(U,\theta)$ is an approximation space with a $\odot$-similarity relation $\theta$ and $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$, for some $f,g\in \mathcal{F}(U)$. Now, by Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iii), $R(F)$, $\varrho (G)\subseteq U\times U$ are (crisp) quasiorders. It is known that for any quasiorder $q\subseteq U\times U$ the relation $\varepsilon _{q}:=q\cap q^{-1}$ is an equivalence and $q$ induces a *natural partial order* $\leq_{q}$ on the factor-set $U/\varepsilon_{q}$ as follows: for any equivalence classes $A,B\in U/\varepsilon_{q}$ we say that $A\leq_{q}B$, whenever there exists $a\in A$ and $b\in B$ with $(a,b)\in q$. This is equivalent to the fact that $(x,y)\in q$ holds for all $x\in A$ and $y\in B$. Thus we can introduce two equivalence relations $E(F)$ and $\varepsilon (G)$ as follows: $$E(F):=R(F)\cap R(F)^{-1}\text{and }\varepsilon% (G):=\varrho(G)\cap\varrho(G)^{-1}\text{.}%$$ The corresponding (natural) partial orders on the factor-sets $U/E(F)$ and $U/\varepsilon (G)$ can be defined as follows: For any $E_{1},E_{2}\in U/E(F)$, we have $E_{1}\leq_{R(F)}% E_{2}\Leftrightarrow(a_{1},a_{2})\in R(F)$ for some $a_{1}\in E_{1}$ and $a_{2}\in E_{2}$, and for any $\mathcal{E}_{1},\mathcal{E}_{2}\in U/\varepsilon (G)$ we have $\mathcal{E}_{1}\leq_{\varrho(G)}% \mathcal{E}_{2}\Leftrightarrow(b_{1},b_{2})\in\varrho(G)$ for some $b_{1}% \in\mathcal{E}_{1}$ and $b_{2}\in\mathcal{E}_{2}$. $E$ is called a *maximal* $E(F)$* class*, if it is a maximal element of the poset $(U/E(F),\leq_{R(F)})$ and $\mathcal{E}$ is a *maximal* $\varepsilon (G)$* class* if it is maximal in $(U/\varepsilon (G),\leq_{\varrho(G)})$. The $E(F)$ and $\varepsilon (G)$ class of an $a\in U$ is denoted by $[a]_{E(F)}$ and $[a]_{\varepsilon (G)}$, respectively. In this section we prove several properties of these classes used to characterize pairs of fuzzy sets which together form fuzzy rough sets. **Lemma 7**. *The following assertions hold true:* *(i) If $E\subseteq U$ is an $E(F)$ class, then $F(a)=F(b)\leq\theta(a,b)$, for all $a,b\in E$;* *(ii) If $\mathcal{E}\subseteq U$ is an $\varepsilon (G)$ class, then $G(a)=G(b)\geq n(\theta(a,b))$, for all $a,b\in\mathcal{E}$;* *(iii) If $E\subseteq U$ is a maximal $E(F)$ class, then $\theta(a,z)\odot F(z)<F(a)=F(b)\leq\theta(a,b)$ and $\theta(a,z)<\theta (a,b)$, for all $a,b\in E$ and $z\notin E$;* *(iv) If $\mathcal{E}\subseteq U$ is a maximal $\varepsilon (G)$ class, then $n(\theta(a,b))\leq G(a)=G(b)<\theta(a,z)\vartriangleright G(z)$ and $\theta(a,z)<\theta(a,b)$, for all $a,b\in\mathcal{E}$ and $z\notin\mathcal{E}$;* *(v) Assume that $n(x)=x\vartriangleright0$ is involutive. Then the $E(F)$ classes and the $\varepsilon (n(F))$ classes are the same, and $E\subseteq U$ is a maximal $E(F)$ class if and only if it is also a maximal $\varepsilon (n(F))$ class.* *Proof.* (i) If $E\subseteq U$ is an $E(F)$ class, then $(a,b),(b,a)\in R(F)$ holds for any $a,b\in E$. Therefore, $F(a)=$ $\theta(a,b)\odot F(b)\leq\theta(a,b)$, $F(b)$ and $F(b)=\theta(b,a)\odot F(a)\leq F(a)$. Thus we obtain $F(b)=F(a)\leq\theta(a,b)$. \(ii\) If $\mathcal{E}\subseteq U$ is an $\varepsilon (G)$ class, then $(a,b),(b,a)\in\varrho(G)$ imply $G(a)=\theta(a,b)\vartriangleright G(b)\geq1\vartriangleright G(b)=G(b)$ and $G(b)\geq\theta (b,a)\vartriangleright G(a)\geq1\vartriangleright G(a)=G(a)$, for any $a,b\in\mathcal{E}$. Hence $G(a)=G(b)$. By Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(i) we obtain $G(a)=G(b)\geq n(\theta(a,b))$, for all $a,b\in\mathcal{E}$. \(iii\) Let $E$ be a maximal $E(F)$ class. Then $F(a)=\theta (a,b)\odot F(b)$, $F(b)=\theta(a,b)\odot F(a)$, and in view of (i), $F(a)=F(b)\leq\theta(a,b)$. We also have $F(a)\geq\ \theta(a,z)\odot F(z)$, according to Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iii). As $E\nleq\lbrack z]_{E(F)}$ implies $(a,z)\notin R(F)$ for each $z\notin E$, we obtain $F(a)>\theta (a,z)\odot F(z)$, for all $z\notin E$. Now, suppose that $\theta (a,c)\geq\theta(a,b)$, for some $c\notin E$. Then $F(c)\geq\ \theta(c,a)\odot F(a)=\theta(a,c)\odot F(b)\geq\theta(a,b)\odot F(b)=F(a)$. This further yields $F(a)>\theta(a,c)\odot F(c)\geq\theta(a,b)\odot F(a)=F(b)$, a contradiction. Thus $\theta(a,z)<\theta(a,b)$, for each $z\notin E$. \(iv\) If $\mathcal{E}\subseteq U$ is a maximal $\varepsilon (G)$ class, then $G(a)=\theta(a,b)\vartriangleright G(b)$, $G(b)=\theta (a,b)\vartriangleright G(a)$, and in view of (i), $n(\theta(a,b))\leq G(a)=G(b)$, for all $a,b\in\mathcal{E}$ and $\mathcal{E}\nleq\lbrack z]_{\varepsilon (G)}$, for any $z\notin\mathcal{E}$. Now, by Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iii), $G(a)\leq\theta (a,z)\vartriangleright G(z)$, hence $(a,z)\notin\varrho(G)$ yields $G(a)<\theta (a,z)\vartriangleright G(z)$. By way of contradiction, assume $\theta(a,c)\geq\theta(a,b)$, for some $c\notin E$. Then $G(c)\leq \theta(a,c)\vartriangleright G(a)\leq\theta(a,b)\vartriangleright G(b)=G(a)$. This further yields $G(a)<\theta(a,c)\vartriangleright G(c)\leq\theta (a,b)\vartriangleright G(a)=G(b)$, a contradiction again. \(v\) If $n(x)=x\vartriangleright0$ is involutive, then property (D) means that $n(F)=n(\overline{\theta}(f))=\underline{\theta}(n(f))$. Hence, relation $\varrho(n(F))=R(F)$ is well defined, and $E(F)=R(F)\cap R(F)^{-1}=\varrho(n(F))\cap\varrho(n(F))^{-1}=\varepsilon (n(F))$. Thus the equivalence classes of $E(F)$ and $\varepsilon (n(F))$ coincide. $R(F)=\varrho(n(F))$ also yields $\leq_{R(F)}=\leq_{\varrho(n(F))}$, i.e. the posets $(U/E(F),\leq_{R(F)})$ and $(U/\varepsilon (n(F)),\leq_{\varrho(n(F))})$ are the same. Therefore, the maximal $E% (F)$ and $\varepsilon (n(F))$ classes coincide. ◻ **Corollary 8**. *Let $E$ be an $E(F)$ class and $\mathcal{E}$ be an $\varepsilon (G)$ class such that $E\cap \mathcal{E}\neq\emptyset$. Then the following assertions hold:* *(i) If $E\subseteq U$ is a maximal $E(F)$ class and $\mathcal{E}\subseteq U$ is a maximal $\varepsilon (G)$ class, then $E\subseteq\mathcal{E}$ or $\mathcal{E}\subseteq E$ holds.* *(ii) If $\theta$ is a similarity relation, then $(x,y)\in R(F)$ or $(y,x)\in\varrho(G)$ holds for all $x\in E$ and $y\in\mathcal{E}$.* *Proof.* Let $a\in E\cap\mathcal{E}$. (i) Assume that neither $E\subseteq\mathcal{E}$ nor $\mathcal{E}\subseteq E$ hold. Then there exist elements $b\in E\setminus\mathcal{E}$, $c\in\mathcal{E}\setminus E$. As $a,b\in E$ but $c\notin E$, in view of Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}(iii) we have $\theta (a,c)<\theta(a,b)$. Similarly, $a,c\in\mathcal{E}$ and $b\notin\mathcal{E}$ imply $\theta(a,b)<\theta(a,c)$, a contradiction to the previous result. \(ii\) If $\mathcal{E}\subseteq E$ or $E\subseteq\mathcal{E}$ then (ii) is clearly satisfied. Hence we may assume $\mathcal{E}\setminus E\neq\emptyset$ and $E\setminus\mathcal{E}\neq\emptyset$. Suppose that there exist $x\in E$ and $y\in\mathcal{E}$ with $(x,y)\notin R(F)$. We claim that $(y,x)\in\varrho(G)$. Assume by contradiction $(y,x)\notin\varrho(G)$. Since $x,a\in E$, $y\notin E$ and $y,a\in\mathcal{E}$, $x\notin\mathcal{E}$, in view of Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}(iii) and (iv) we get $\theta(x,y)<\theta(x,a)$ and $\theta (x,y)=\theta(y,x)<\theta(y,a)=\theta(a,y)$. Thus we obtain $\theta (x,y)<\ $min$(\theta(x,a),\theta(a,y))\leq\theta(x,y)$, a contradiction. This proves $(y,x)\in\varrho(G)$. ◻ **Proposition 9**. *(i) If $E_{1},E_{2}$ are different $E(F)$ classes with $E_{1}\leq_{R(F)}E_{2}$, then for any $a_{1}\in E_{1}$ and $a_{2}\in E_{2}$ we have $F(a_{1})<F(a_{2})$.* *(ii) If $\mathcal{E}_{1},\mathcal{E}_{2}$ are different $\varepsilon (G)$ classes with $\mathcal{E}_{1}\leq_{\varrho (G)}\mathcal{E}_{2}$ then for any $b_{1}\in\mathcal{E}_{1}$ and $b_{2}% \in\mathcal{E}_{2}$ we have $G(b_{1})>G(b_{2})$.* *Proof.* (i) Assume $E_{1}\leq_{R(F)}E_{2}$. Then for any $a_{1}\in E_{1}$ and $a_{2}\in E_{2}$ we have $(a_{1},a_{2})\in R(F)$, i.e. $F(a_{1})=\theta(a_{1},a_{2})\odot F(a_{2})\leq F(a_{2})$. Observe that $F(a_{2})\neq F(a_{1})$. Indeed, $F(a_{2})=F(a_{1})$ would imply $F(a_{2})=\theta(a_{1},a_{2})\odot F(a_{1})=\theta(a_{2},a_{1})\odot F(a_{1}% )$, i.e. $(a_{2},a_{1})\in R(F)$, which means $E_{2}\leq_{R(F)}E_{1}$. As $\leq_{R(F)}$ is a partial order, this would yield $E_{1}=E_{2}$, a contradiction. Thus we deduce $F(a_{1})<F(a_{2})$. \(ii\) Let $\mathcal{E}_{1}\leq_{\varrho(G)}\mathcal{E}_{2}$. Then for any $b_{1}\in\mathcal{E}_{1}$, $b_{2}\in\mathcal{E}_{2}$ we have $(b_{1},b_{2})\in\varrho(G)$, which gives $G(b_{1})=\theta(b_{1},b_{2}% )\vartriangleright G(b_{2})\geq G(b_{2}).$ We claim $G(b_{1})>G(b_{2})$. Indeed, $G(b_{2})=G(b_{1})$ would imply $G(b_{2}% )=\theta(b_{1},b_{2})\vartriangleright G(b_{1})=\theta(b_{2},b_{1}% )\vartriangleright G(b_{1})$, i.e. $(b_{2},b_{1})\in\varrho(G)$, which would yield $\mathcal{E}_{1}=\mathcal{E}_{2}$, a contradiction. ◻ Clearly, if each chain in the posets $(U/E(F),\leq_{R(F)})$ and $(U/\varepsilon (G),\leq_{\varrho(G)})$ is finite, then any element of them is less than or equal to a maximal element in the corresponding poset. By using this observation we deduce **Corollary 10**. *Assume that the relation $\theta$ and the fuzzy sets $f,g\in\mathcal{F}(U)$ have a finite range, and let $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$. Then for any $E(F)$ class $E$, there exists a maximal $E(F)$ class $E_{M}$ such that $E\leq_{R(F)}E_{M}$, and for any $\varepsilon (G)$ class $\mathcal{E}$, there is a maximal $\varepsilon (G)$ class $\mathcal{E}_{M}$ with $\mathcal{E}\leq_{\varrho(G)}\mathcal{E}_{M}$.* *Proof.* If the above conditions hold, then the fuzzy sets $F$ and $G$ also have a finite range. Now let $\{E_{i}\mid i\in I\}$ be an arbitrary (nonempty) chain of $E(F)$ classes. In view of Proposition [Proposition 9](#prop:twodiffeqclass){reference-type="ref" reference="prop:twodiffeqclass"}, for any $a_{i}\in E_{i}$, $i\in I$, the values $\{F(a_{i})\mid i\in I\}$ also form a chain, and for $E_{i}\leq_{R(F)}E_{j}$, $E_{i}\neq E_{j}$ we have $F(a_{i})<F(a_{j})$, and vice versa. This means that the chains $\{E_{i}\mid i\in I\}$ and $\{F(a_{i})\mid i\in I\}$ are order-isomorphic. Since $F$ has a finite range, the chain $\{F(a_{i})\mid i\in I\}$ has a finite length. Hence the chain $\{E_{i}\mid i\in I\}$ is also finite. As every chain in the poset $(U/E(F),\leq_{R(F)})$ is finite, any element $E$ of it is less than or equal to a maximal element $E_{M}$ of it, i.e. $E\leq_{R(F)}E_{M}$. The second statement is proved analogously. ◻ The importance of maximal classes in this case is shown by the following: **Proposition 11**. *Suppose that $\theta$ and the fuzzy sets $f,g\in\mathcal{F}(U)$ have a finite range, and let $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$. Then the following assertions hold:* *(i) If $h\leq F$ for some $h\in\mathcal{F}(U)$ and for any maximal $E(F)$ class $E_{M}$ there exists an element $u\in E_{M}$ with $h(u)=F(u)$, then $\overline{\theta}(h)=F$.* *(ii) If $h\geq G$ for some $h\in\mathcal{F}(U)$ and for any maximal $\varepsilon (G)$ class $\mathcal{E}_{M}$ there exists an element $v\in\mathcal{E}_{M}$ with $h(v)=G(v)$, then $\underline{\theta}(h)=G$.* *Proof.* (i) Let $x\in U$ be arbitrary. As $\theta$ and $f$ have finite ranges, in view of Corollary [Corollary 10](#cor:maxeqclexists){reference-type="ref" reference="cor:maxeqclexists"}, there exists a maximal $E(F)$ class $E_{M}$ such that $[x]_{E(F)}\leq_{R(F)}E_{M}$. Then $(x,y)\in R(F)$ for all $y\in E_{M}$. By assumption, there exists an element $u\in E_{M}$ with $h(u)=F(u)$. Since $h\leq F$ and $(x,u)\in R(F)$, in view of Proposition [Proposition 6](#prop:prop2){reference-type="ref" reference="prop:prop2"}(i) we obtain $\overline{\theta}(h)(x)=F(x)$. This proves $\overline{\theta}(h)=F$. \(ii\) is proved dually, by using Corollary [Corollary 10](#cor:maxeqclexists){reference-type="ref" reference="cor:maxeqclexists"} and Proposition [Proposition 6](#prop:prop2){reference-type="ref" reference="prop:prop2"}(ii). ◻ **Proposition 12**. *Suppose that the relation $\theta$ and the fuzzy sets $f,g\in\mathcal{F}(U)$ have a finite range, and let $F=\overline {\theta}(f)$, $G=\underline{\theta}(g)$.* *(i) If $E$ is a maximal $E(F)$ class, then for any $a\in E$ we have* *$F(a)=\ $max$\{f(y)\mid y\in E\}$.* *(ii) If $\mathcal{E}$ is a maximal $\varepsilon (G)$ class, then for any $a\in\mathcal{E}$ we have* *$G(a)=\ $min$\{g(y)\mid y\in\mathcal{E}\}$.* *Proof.* (i) By definition, $F(a)=\overline{\theta }(f)(a)=\bigvee\{\theta(a,y)\odot f(y)\mid y\in U\}$. If $y\notin E$, then $(a,y)\notin R(F)$, because $E$ is a maximal $E(F)$ class. This means that $F(a)=\theta(a,y)\odot F(y)$ is not possible, and hence $F(a)>\theta (a,y)\odot F(y)$, according to Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iii). Since $f\leq \overline{\theta}(f)=F$, we obtain $F(a)>\theta(a,y)\odot f(y)$, for all $y\in U\setminus E$. As $\theta$ and $f$ are of a finite range, the set $\{\theta(a,y)\odot f(y)\mid y\in U\setminus E\}$ has finitely many different elements, and hence $\bigvee\{\theta(a,y)\odot f(y)\mid y\in U\setminus E\}<F(a)$. This implies $F(a)=(\bigvee\{\theta(a,y)\odot f(y)\mid y\in E\})\vee\left( \bigvee \{\theta(a,y)\odot f(y)\mid y\in U\setminus E\}\right) =$ $\bigvee\{\theta(a,y)\odot f(y)\mid y\in E\}$. If $y\in E$, then $F(y)=F(a)$. As $\theta(a,y)\odot f(y)\leq f(y)\leq F(y)=F(a)$, we obtain: $F(a)=$ $\bigvee\{\theta(a,y)\odot f(y)\mid y\in E\}\leq\bigvee\{f(y)\mid y\in E\}\leq F(a)$. This implies $F(a)=$ $\bigvee\{f(y)\mid y\in E\}$. Because $f$ has a finite range, the set $\{f(y)\mid y\in E\}$ is finite, and hence we can write $F(a)=\ $max$\{f(y)\mid y\in E\}$. \(ii\) By definition $G(a)=\underline{\theta}(g)(a)=\bigwedge \{\theta(a,y)\vartriangleright g(y)\mid y\in U\}$. If $y\notin\mathcal{E}$, then $(a,y)\notin\varrho(G)$, because $\mathcal{E}$ is a maximal $\varepsilon (G)$ class, and hence $G(a)\neq\theta (a,y)\vartriangleright G(y)$. Thus we have $G(a)<\theta(a,y)\vartriangleright G(y)$, according to Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iii). Since $G=\underline{\theta}(g)\leq g$, we obtain $G(a)<\theta(a,y)\vartriangleright g(y)$, for all $y\in U\setminus\mathcal{E}$. As $\theta$ and $g$ are of a finite range, the set $\{\theta(a,y)\vartriangleright g(y)\mid y\in U\setminus E\}$ is finite, whence we get $G(a)<\bigwedge\{\theta(a,y)\odot g(y)\mid y\in U\setminus E\}$. This yields $G(a)=(\bigwedge\{\theta(a,y)\vartriangleright g(y)\mid y\in\mathcal{E}% \})\wedge(\bigwedge\{\theta(a,y)\vartriangleright g(y)\mid y\in U\setminus \mathcal{E}\})=$ $\bigwedge\{\theta(a,y)\vartriangleright g(y)\mid y\in\mathcal{E}\}$. If $y\in\mathcal{E}$, then $G(y)=G(a)$. Since $\theta (a,y)\vartriangleright g(y)\geq1\vartriangleright g(y)=g(y)\geq G(y)=G(a)$, we obtain $G(a)=\bigwedge\{\theta(a,y)\vartriangleright g(y)\mid y\in\mathcal{E}% \}\geq\bigwedge\{g(y)\mid y\in\mathcal{E}\}\geq G(a)$, $\ $ and this implies $G(a)=\bigwedge\{g(y)\mid y\in\mathcal{E}\}$. Since $\{g(y)\mid y\in\mathcal{E}\}$ is a finite set, we can write: $G(a)=\ $ min$\{g(y)\mid y\in\mathcal{E}\}$. ◻ The following corollary is immediate: **Corollary 13**. *Assume that $\theta$ and $f,g\in \mathcal{F}(U)$ are of a finite range, and let $a\in U$ and $F=\overline {\theta}(f)$, $G=\underline{\theta}(g)$.* *(i) If $\{a\}$ is a maximal $E(F)$ class, then $F(a)=f(a)$.* *(ii) If $\{a\}$ is a maximal $\varepsilon (G)$ class, then $G(a)=g(a)$.* **Corollary 14**. *Assume that $\theta$ and $f\in\mathcal{F}(U)$ are of a finite range, $F=\overline{\theta}(f)$, $G=\underline{\theta}(f)$, and let $E$ be a maximal $E(F)$ class and $\mathcal{E}$ a maximal $\varepsilon (G)$ class. (i) If every $\{x\}\subseteq E$ is a maximal $\varepsilon (G)$ class, then there exists a $u\in E$ with $F(u)=G(u)$.* *(ii) If every $\{x\}\subseteq\mathcal{E}$ is a maximal $E(F)$ class, then there exists a $v\in\mathcal{E}$ with $F(v)=G(v)$.* *Proof.* (i) In view of Proposition [Proposition 12](#prop:famaxfy){reference-type="ref" reference="prop:famaxfy"}(i), for each $x\in E$ we have $F(x)=\ $max$\{f(y)\mid y\in E\}$, i.e. $F(x)=f(u)$, for some $u\in E$. As $\{u\}$ is a maximal $\varepsilon (G)$ class, by applying Corollary [Corollary 13](#cor:fafagaga){reference-type="ref" reference="cor:fafagaga"}(ii) with $g:=f$ we get $G(u)=f(u)$. Hence $F(u)=G(u)$. (ii) is proved dually. ◻ [\[ex:factor\]]{#ex:factor label="ex:factor"} Let us consider the similarity relation $\theta$, a fuzzy set $h$ and its approximations $F=\overline{\theta}(h)$, $G=\underline{\theta}(h)$ given on Figure [\[fig:theta2\]](#fig:theta2){reference-type="ref" reference="fig:theta2"} and Table [\[tab:fuzzy2\]](#tab:fuzzy2){reference-type="ref" reference="tab:fuzzy2"}. \|P0.75cm\|P0.75cm\|P0.75cm\|P0.75cm\|P0.75cm\|P0.75cm\|P0.75cm\| $u$ & $a$ & $b$ & $c$ & $d$ & $e$ & $f$\ $h(u)$ & 0 & 1 & 0.25 & 0.5 & 0.5 & 0.75\ $F(u)$ & 1 & 1 & 0.75 & 0.5 & 0.5 & 0.75\ $G(u)$ & 0 & 0 & 0.25 & 0.5 & 0.5 & 0.5\ The quasiorders $R(F)$ and $\varrho(G)$, their $E(F)$ and $\varepsilon (G)$ equivalence classes and the partial orders induced on the factor-sets are given in Figure [\[fig:factor2\]](#fig:factor2){reference-type="ref" reference="fig:factor2"}. Loops are not drawn for any relation. As $\theta$ is symmetric, its edges are undirected, and those with $\theta(x,y)=0$ are not shown either. The maximal $\varepsilon (G)$ classes are $\mathcal{E}_1$, $\mathcal{E}_4$, and the maximal $E(F)$ classes are $E_1$ and $E_5$. In all our examples the approximations are defined by a min t-norm and the KD-implicator max$(1-x,y)$. Clearly, all statements in [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}, [Corollary 8](#cor:assertions){reference-type="ref" reference="cor:assertions"}, [Corollary 14](#cor:fugufvgv){reference-type="ref" reference="cor:fugufvgv"} hold. # A characterization of fuzzy rough sets {#sec:frs} In case of an equivalence $\varrho\subseteq U\times U$, the sets $\{X\subseteq U\mid X_{\varrho}=X\}$ and $\{X\subseteq U\mid X^{\varrho}=X\}$ coincide and their members are called $\varrho$*-definable* subsets of $U$. They can be described as those subsets of $U$ which are the union of some $\varrho$-equivalence classes, and their set is denoted by Def$(U,\varrho )$. The rough sets induced by an equivalence relation $\varrho\subseteq U\times U$ can be characterized by using the *set of* its *singletons* $S=\{s\in U\mid\varrho(s)=\{s\}\}$, as follows: $(A,B)$ is rough set of $\varrho$, if and only if $(A,B)\in\ $Def$(U,\varrho)\times\ $ Def$(U,\varrho)$, $A\subseteq B$ and $A\cap S=B\cap S$ (see e.g. [@JPR]). In this section we will derive an analogous characterisation for the fuzzy rough sets with finite ranges and satisfying conditions in (ID). For a fuzzy approximation space $(U,\theta)$ we will introduce the notations: $$\text{Fix}(\underline{\theta})=\{f\in\mathcal{F}(U)\mid\underline{\theta }(f)=f\}\text{, Fix}(\overline{\theta})=\{f\in\mathcal{F}(U)\mid \overline{\theta}(f)=f\}\text{.}%$$ Unfortunately, in case of a $\odot$-similarity relation Fix$(\underline{\theta})$ and Fix$(\overline{\theta})$ coincide only for a left-continuous t-norm $\odot$ and the R-implicator $\vartriangleright$ induced by it. **Theorem 15**. *Assume that conditions in (ID) are satisfied and let $(U,\theta)$ be a fuzzy approximation space with a $\odot$-similarity relation $\theta$ of a finite range, and $F,G\in\mathcal{F}(U)$. Then $(F,G)$ is a fuzzy rough set induced by a fuzzy set with a finite range, if and only if the following conditions hold:* *(1) $G\in\ $Fix$(\underline{\theta})$, $F\in\ $Fix$(\overline{\theta})$, $G\leq F$, and $F$ and $G$ have finite ranges;* *(2) If $\mathcal{E}$ is a maximal $\varepsilon (G)$ class such that each $\{a\}\subseteq\mathcal{E}$ is a maximal $E(F)$* *class, then there exists an element $u\in\mathcal{E}$ such that $G(u)=F(u)$;* *(3) If $E$ is a maximal $E(F)$ class such that each $\{a\}\subseteq E$ is a maximal $\varepsilon (G)$* *class, then there exists an element $v\in E$ such that $G(v)=F(v)$.* *Proof.* By definition, $(F,G)$ is a fuzzy rough set if there exists a map $f\in\mathcal{F}(U)$ such that $F=\overline{\theta}(f)$, $G=\underline{\theta}(f)$. Suppose that $f$ has a finite range. We prove that the conditions of Theorem [Theorem 15](#thm:frs){reference-type="ref" reference="thm:frs"} are satisfied. Indeed, \(1\) Property (ID) implies $\overline{\theta}(F)=F$, $\underline{\theta}(G)=G$, hence $F\in \ $Fix$(\overline{\theta})$ and $G\in\ $Fix$(\underline{\theta})$. Clearly, $G=\underline{\theta}(f)\leq\overline{\theta}(f)=F$. Because $f$ has a finite range, in view of Lemma [Lemma 2](#lem:frange){reference-type="ref" reference="lem:frange"}, $F$ and $G$ also have finite ranges. In view of Corollary [Corollary 14](#cor:fugufvgv){reference-type="ref" reference="cor:fugufvgv"}, conditions (2) and (3) are also satisfied. Conversely, suppose that conditions (1), (2) and (3) are satisfied by $F$ and $G$. In order to prove that $(F,G)$ is a fuzzy rough set, we will construct a fuzzy set $f\in\mathcal{F}(U)$ with $F=\overline{\theta}(f)$, $G=\underline {\theta}(f)$. Since $F$ and $G$ are of a finite range, in view of Corollary [Corollary 10](#cor:maxeqclexists){reference-type="ref" reference="cor:maxeqclexists"}, for each $E(F)$ class $E$ there exists a maximal $E(F)$ class $E_{M}$ such that $E\leq_{R(F)}E_{M}$, and for any $\varepsilon (G)$ class $\mathcal{E}$ there is a maximal $\varepsilon (G)$ class $\mathcal{E}_{M}$ with $\mathcal{E}\leq_{\varrho(G)}\mathcal{E}_{M}$. Denote the family of maximal $\varepsilon (G)$ classes by $\{\mathcal{E}_{t}\mid t\in T\}$. As a first step, from each class $\mathcal{E}_{t}$, $t\in T$ we select exactly one element $a_{t}\in\mathcal{E}_{t}$ as follows: ) If $\mathcal{E}_{t}$ contains an element $p_{t}$ which does not belong to any maximal $E(F)$ class, then we select it and set $a_{t}:=p_{t}$. ) If in $\mathcal{E}_{t}$ there is no element of type 1), however there exists an element $q_{t}\in\mathcal{E}_{t}$ with $G(q_{t})=F(q_{t})$, then we select it and set $a_{t}:=q_{t}$. ) If there are no elements of type 1) or 2) in $\mathcal{E}_{t}$, then we select an element $r_{t}\in\mathcal{E}_{t}$ such that $\{r_{t}\}$ is not a maximal $E(F)$ class, and we set $a_{t}:=r_{t}$. First, we show that we can always effectuate such a selection: assume by contradiction that in some class $\mathcal{E}_{t}$ there are no elements of type 1), 2) or 3). This means that for each $a_{t}\in\mathcal{E}_{t}$ the set $\{a_{t}\}$ is a maximal $E(F)$ class. Then by Corollary [Corollary 14](#cor:fugufvgv){reference-type="ref" reference="cor:fugufvgv"}(ii) there exists an element $v\in\mathcal{E}_{t}$ with $F(v)=G(v)$. Since this means that $v\in\mathcal{E}_{t}$ is of type 2), this is a contradiction. As next step, we construct a fuzzy set $f\in\mathcal{F}(U)$ as follows: $$f(x)=\left\{ \begin{array} [c]{l}% G(x)\text{, if }x\in\{a_{t}\mid t\in T\}\text{;}\\ F(x)\text{, if }x\in U\setminus\{a_{t}\mid t\in T\} \end{array} \right. \tag{3}$$ By its construction, $f$ also has a finite range. Now, we prove that in any maximal $E(F)$ class $E$ there exists an element $u\in E$ with $f(u)=F(u)$. By our construction, this would mean that any such class $E$ contains an element $x_{0}\in U\setminus\{a_{t}\mid t\in T\}$ or an element $a_{t}=q_{t}\in E$ of type 2) with $f(q_{t})=G(q_{t})=F(q_{t})$. By way of contradiction, assume that there is a maximal $E(F)$ class $E_{M}$ with $E_{M}\subseteq\{a_{t}\mid t\in T\}$ and $F(x)\neq f(x)=G(x)$, for all $x\in E_{M}$. Then $E_{M}=\{a_{s}\mid s\in S\}$, for some nonempty $S\subseteq T$. Observe that in this case $E_{M}$ cannot contain elements of type 1) and 2). Hence, by our construction, for any element $a_{s}$, $s\in S$ the set $\{a_{s}\}$ is not a maximal $E(F)$ class. Thus $E_{M}$ is not a one-element set, i.e. $|S|\geq2$. Observe also, that we can exclude the case when each element $a_{s},s\in S$ belongs to an $\mathcal{E}_{t}$ class with a single element. Indeed, as in such case each $\{a_{s}\}\subseteq E_{M}$ would be a maximal $\varepsilon (G)$ class, and by Corollary [Corollary 14](#cor:fugufvgv){reference-type="ref" reference="cor:fugufvgv"}(i) we would obtain $G(a_{s_{0}})=F(a_{s_{0}})$, for some $a_{s_{0}}\in E_{M}$, contrary to our assumption. Hence there exists an element $a_{s^{\ast}}\in E_{M}$ which was chosen from a maximal $\varepsilon (G)$ class $\mathcal{E}_{s^{\ast}}$ with $| \mathcal{E}_{s^{\ast}}|\geq2$. Since $a_{s^{\ast}}\in E_{M}\cap \mathcal{E}_{s^{\ast}}$, in view of Corollary [Corollary 8](#cor:assertions){reference-type="ref" reference="cor:assertions"}(i), we have $E_{M}% \subseteq\mathcal{E}_{s^{\ast}}$ or $\mathcal{E}_{s^{\ast}}\subseteq E_{M}$. Since both $E_{M}$ and $\mathcal{E}_{s^{\ast}}$ have at least two elements, both cases would imply that from the class $\mathcal{E}_{s^{\ast}}$ at least two elements had been inserted into the set $\{a_{t}\mid t\in T\}$, in contradiction to our construction for $\{a_{t}\mid t\in T\}$. Thus we proved that in any maximal $E(F)$ class $E$ there is an element $u\in E$ with $f(u)=F(u)$. It is also clear, that by our construction from each maximal $\varepsilon (G)$ class $\mathcal{E}_{t}$, $t\in T$ an element $v=a_{t}\in\mathcal{E}_{t}$ had been selected with $f(v)=G(v)$. Since by definition $G\leq f\leq F$, applying Proposition [Proposition 11](#prop:hufuhvgv){reference-type="ref" reference="prop:hufuhvgv"} we obtain $F=\overline{\theta}(f)$, $G=\underline{\theta}(f)$, and our proof is completed. ◻ # Further properties of $E(F)$ and $\varepsilon (G)$ classes {#sec:propofeqc} In this section we deduce some additional properties of $E(F)$ and $\varepsilon (G)$ classes which will be used to prove our main Theorem [Theorem 23](#thm:lattice){reference-type="ref" reference="thm:lattice"}. In the whole section we assume that for all $x,y\in\lbrack0,1]$ condition $$x\odot y=\text{min}(x,y)\text{, }x\vartriangleright y:=\ \text{max}% (n(x),y)\text{, }n\text{ is an involutive negator} \tag{C}%$$ holds, and that $\theta$ is a similarity relation. Then $\vartriangleright$ is an S-implicator, and for $n(x)=1-x$, we re-obtain the Kleene-Dienes implicator, therefore our $\vartriangleright$ is an extension of it. Clearly, if (C) holds then (D) and (ID) are also satisfied. **Proposition 16**. *(i) $E\subseteq U$ is a maximal $E(F)$ class if and only if $\theta(a,z)<F(a)=F(b)\leq\theta(a,b)$ , for all $a,b\in E$ and $z\notin E$;* *(ii) $\mathcal{E}\subseteq U$ is a maximal $\varepsilon (G)$ class, if and only if $n(\theta(a,b))\leq G(a)=G(b)<n(\theta(a,z))$, for all $a,b\in\mathcal{E}$ and $z\notin\mathcal{E}$.* *Proof.* (i) If $E\subseteq U$ is a maximal $E(F)$ class, then by Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}(i) and (iii), we have $F(a)=F(b)\leq \theta(a,b)$ and $\theta(a,z)<F(a)$, for all $a,b\in E$ and $z\notin E$. Conversely, let $E\subseteq U$ and assume that all the relations from (i) are satisfied. Then, $F(a)=$ min$(\theta(a,b),F(b))$, for all $a,b\in E$, i.e. we get $(a,b)\in R(F)$ for all $a,b\in E$, and in view of Corollary [Corollary 5](#cor:cor1){reference-type="ref" reference="cor:cor1"}, we have $(a,z)\notin R(F)$ for all $z\notin E$. Hence $(a,b)\in R(F)\cap R(F)^{-1}=E(F)$ holds for all $a,b\in E$, and $(a,z)\notin E(F)$ for each $z\notin E$. This means that $E$ is an $E(F)$ class. We also get $E\nleq\lbrack z]_{E(F)}$ for all $z\notin E$, because $(a,z)\notin R(F)$. Thus $E$ is a maximal $E(F)$ class. \(ii\) If $\mathcal{E}\subseteq U$ is a maximal $\varepsilon (G)$ class, then in view of Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}(iv) $n(\theta(a,b))\leq G(a)=G(b)$, for all $a,b\in\mathcal{E}$ and we have $\mathcal{E}\nleq\lbrack z]_{\varepsilon (G)}$, for any $z\notin\mathcal{E}$. Then $(a,z)\notin\varrho(G)$ implies $G(a)<n(\theta(a,z))$, according to Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"}(iv). The converse implication is proved analogously as in (i). ◻ **Lemma 17**. *Let $F=\overline{\theta}(f)$, $G=\underline {\theta}(g)$, for some $f,g\in\mathcal{F}(U)$, and let $E$ be an $E(F)$ class and $\mathcal{E}$ be an $\varepsilon (G)$ class such that $E\cap\mathcal{E}\neq\emptyset$. Then the following assertions hold:* *(i) If $(x,y)\notin R(F)$, for some $x\in E$ and $y\in\mathcal{E}$, then $(x,z)\in R(F)$ implies $(y,z)\in\varrho(G)$, for all $z\notin E\cup\mathcal{E}$.* *(ii) If $(x,y)\notin\varrho(G)$, for some $x\in\mathcal{E}$ and $y\in E$, then $(x,z)\in\varrho(G)$ implies $(y,z)\in R(F)$, for all $z\notin E\cup\mathcal{E}$.* *Proof.* Let $a\in E\cap\mathcal{E}$. (i) Observe that the relations $(x,y)\notin R(F)$ and $(x,a)\in R(F)$ exclude $(a,y)\in R(F)$. Thus $(a,y)\notin R(F)$ yields $F(a)>\theta(a,y)$, by Corollary [Corollary 5](#cor:cor1){reference-type="ref" reference="cor:cor1"}. Now, let $(x,z)\in R(F)$ and assume by contradiction $(y,z)\notin\varrho(G)$. Then $a,y\in\mathcal{E}$ and $z\notin\mathcal{E}$ imply $\theta(a,z)<\theta(a,y)$. On the other hand, $(a,x)\in R(F)$ and $(x,z)\in R(F)$ imply $(a,z)\in R(F)$. Hence, Proposition [Proposition 16](#prop:maxiff){reference-type="ref" reference="prop:maxiff"}(i) yields $F(a)\leq\theta(a,z)<\theta(a,y)$, a contradiction to $F(a)>\theta(a,y)$. This proves $(y,z)\in\varrho(G)$. \(ii\) By Proposition [Proposition 4](#prop:prop1){reference-type="ref" reference="prop:prop1"}(iv) we have $\varrho(G)=R(n(G))$, $R(F)=\varrho(n(F))$, and by Lemma [Lemma 7](#lem:assertions){reference-type="ref" reference="lem:assertions"} (v), $\mathcal{E}$ is an $E% (n(G))$ class, and $E$ is $\ $an $\varepsilon (n(F))$ class. Hence $(x,y)\notin\varrho(G)$ for some $x\in\mathcal{E}$ and $y\in E$ and $(x,z)\in \varrho(G)$ is equivalent to $(x,y)\notin R(n(G))$ and $(x,z)\in R(n(G))$, therefore, $n(G)=n(\underline{\theta}(g))=\overline{\theta}(n(g))$ and $n(F)=n(\overline{\theta}(f))=\underline{\theta}(n(f))$ form a pair that replaces in the context of (ii) the pair $(F,G)$ from (i). Thus $(y,z)\in\varrho(n(F))=R(F)$, in view of (i). ◻ **Corollary 18**. *Let $F=\overline{\theta}(f)$, $G=\underline {\theta}(g)$, for some $f,g\in\mathcal{F}(U)$ and let $E$ be an $E% (F)$ class and $\mathcal{E}$ an $\varepsilon (G)$ class such that $E\cap\mathcal{E}\neq\emptyset$.* *(i) If $\mathcal{E}$ is a maximal $\varepsilon (G)$ class, then $(x,y)\in R(F)$ for all $x\in E$ and $y\in\mathcal{E}$ or $E\subseteq \mathcal{E}$ and there is no $t\in E$ and $z\notin\mathcal{E}$ with $(t,z)\in R(F)$.* *(ii) If $\mathcal{E}$ is a maximal $\varepsilon (G)$ class with $E\varsubsetneqq\mathcal{E}$ and there is no element $x\in E$ and $y\in\mathcal{E}\setminus E$ with $(x,y)\in R(F)$, then $E$ is a maximal $E(F)$ class.* *(iii) If $E$ is a maximal $E(F)$ class, then $(x,y)\in\varrho(G)$ for all $x\in\mathcal{E}$ and $y\in E$ or $\mathcal{E}\subseteq E$ and there is no $t\in\mathcal{E}$ and $z\notin E$ with $(t,z)\in\varrho(G)$.* *(iv) If $E$ is a maximal $E(F)$ class such that $\mathcal{E}\varsubsetneqq E$ and there is no element $x\in\mathcal{E}$ and $y\in E\setminus\mathcal{E}$ with $(x,y)\in\varrho(G)$, then $\mathcal{E}$ is a maximal $\varepsilon (G)$ class.* *Proof.* (i) Let $\mathcal{E}$ be a maximal $\varepsilon (G)$ class and assume $E\nsubseteqq\mathcal{E}$. Then there exists $a\in E\setminus\mathcal{E}$, and for all $y\in\mathcal{E}$, $(y,a)\notin\varrho(G)$ by maximality of $\mathcal{E}$. Hence, in view of Corollary [Corollary 8](#cor:assertions){reference-type="ref" reference="cor:assertions"}(ii) we have $(a,y)\in R(F)$, and because $(x,a)\in R(F)$ for each $x\in E$, we get $(x,y)\in R(F)$ for all $x\in E$ and $y\in\mathcal{E}$. Consider now the case when $E\subseteq\mathcal{E}$ and there are $x\in E$, $y\in\mathcal{E}$ with $(x,y)\notin R(F).$ Then $E\cup\mathcal{E=E}$. Assume that there exist some elements $t\in E$ and $z\notin\mathcal{E}$ with $(t,z)\in R(F)$. Then $(x,t)\in R(F)$ also yields $(x,z)\in R(F)$. Now, applying Lemma [Lemma 17](#lem:outer){reference-type="ref" reference="lem:outer"}(i) we obtain $(y,z)\in\varrho(G)$. Since $\mathcal{E}$ is a maximal $\varepsilon (G)$ class and $z\notin\mathcal{E}$, this is not possible, and this means that the second part of (i) holds. \(ii\) Suppose that for all $x\in E$ and $y\in\mathcal{E\setminus}E$ we have $(x,y)\notin R(F)$, and let $z\notin\mathcal{E}$. In view of Lemma [Lemma 17](#lem:outer){reference-type="ref" reference="lem:outer"}(i), $(x,z)\in R(F)$ for some $x\in E$ would imply $(y,z)\in\varrho(G)$, for all $y\in\mathcal{E}$ - in contradiction to the fact that $\mathcal{E}$ is a maximal $\varepsilon (G)$ class. Thus we deduce $(x,z)\notin R(F)$, for all $x\in E$ and $z\notin E$. This means that $E$ is a maximal $E(F)$ class. The proofs of (iii) and (iv) are duals of the proofs of (i) and (ii). ◻ **Proposition 19**. *Let $\theta$ be a similarity relation with a finite range, and $F=\overline{\theta}(f)$, $G=\underline{\theta}(g)$, for some $f,g\in\mathcal{F}(U).$* *(i) If $\{a\}\subseteq U$ is a maximal $E(F)$ class, then for any $h\in\mathcal{F}(U)$ with $\overline{\theta}(h)(a)\geq F(a)$, we have $\overline{\theta}(h)(a)=h(a)$.* *(ii) If $\{b\}\subseteq U$ is a maximal $\varepsilon (G)$ class, then for any $h\in\mathcal{F}(U)$ with $\underline{\theta}((h)(b)\leq G(b)$, we have $\underline{\theta}(h)(b)=h(b)$.* *Proof.* (i) If $\{a\}\subseteq U$ is a maximal $E% (F)$ class, then $F(a)>\theta(a,y)$, for all $y\in U$, $y\neq a$, according to Corollary [Corollary 5](#cor:cor1){reference-type="ref" reference="cor:cor1"}. Now, we can write: $\overline{\theta}(h)(a)=\bigvee\{$min$(\theta(a,y),h(y))\mid y\in U\}=$ $h(a)\vee\left( \bigvee\{\text{min}(\theta(a,y),h(y))\mid y\in U\setminus \{a\}\}\right)$, and $\bigvee\{$min$(\theta(a,y),h(y))\mid y\in U\setminus\{a\}\}\leq \bigvee\{\theta(a,y)\mid y\in U\setminus\{a\}\}<F(a)\leq\overline{\theta }(h)(a)$, because $\theta$ is of a finite range. This implies $\overline {\theta}(h)(a)=h(a)$. \(ii\) If $\{b\}\subseteq U$ is a maximal $\varepsilon (G)$ class, then $G(b)<n(\theta(b,y))$, for all $y\in U$, $y\neq b$, according to Corollary [Corollary 5](#cor:cor1){reference-type="ref" reference="cor:cor1"}. We can write: $\underline{\theta}((h)(b)=\bigwedge\{$max$(n(\theta(b,y)),h(y))\mid y\in U\}=$ $h(b)\wedge\left( \bigwedge\{\text{max}(n(\theta(b,y)),h(y))\mid y\in U\setminus\{b\}\}\right)$, and $\bigwedge\{$max$(n(\theta(b,y)),h(y))\mid y\in U\setminus\{b\}\}\geq \bigwedge\{n(\theta(b,y))\mid y\in U\setminus\{b\}\}>G(b)\geq$ $\underline {\theta}(h)(b)$, since $\theta$ is of a finite range. This yields $\underline{\theta}(h)(b)=h(b)$. ◻ # The lattice of fuzzy rough sets {#sec:latticefrs} Clearly, fuzzy rough sets corresponding to an approximation space $(U,\theta )$, can be ordered as follows: $$\left( \underline{\theta}(f),\overline{\theta}(f)\right) \leq\left( \underline{\theta}(g),\overline{\theta}(g)\right) \Leftrightarrow \underline{\theta}(f)\leq\underline{\theta}(g)\text{ and }\overline{\theta }(f)\leq\overline{\theta}(g)\text{,}% \tag{4}$$ obtaining a poset $(\mathcal{FR}% (U,\theta),\leq)$. If $\theta$ is reflexive, then $(\mathbf{0},\mathbf{0})$ and $(\mathbf{1},\mathbf{1})$ are its least and greatest elements. If conditions in (D) hold, $n(\overline {\theta}(f))=\underline{\theta}(n(f))$ and $n(\underline{\theta}% (f))=\overline{\theta}(n(f))$ imply $(n(\overline{\theta}(f),n(\underline {\theta}(f))\in\mathcal{FR}(U,\theta)$, for all $f\in\mathcal{F}(U)$. As $n$ is an involutive negator, $\Phi\colon\mathcal{FR}(U,\theta )\rightarrow\mathcal{FR}(U,\theta)$, $\Phi(\left( \underline{\theta }(f),\overline{\theta}(f)\right) )=(n(\overline{\theta}(f),n(\underline {\theta}(f))$ is an involution, i.e. $\Phi(\Phi\left( \underline{\theta }(f),\overline{\theta}(f)\right) )=\left( \underline{\theta}(f),\overline {\theta}(f)\right)$. Since $\ \left( \underline{\theta}(f),\overline {\theta}(f)\right) \leq\left( \underline{\theta}(g),\overline{\theta }(g)\right) \Leftrightarrow(n(\overline{\theta}(g),n(\underline{\theta }(g))\leq(n(\overline{\theta}(f),n(\underline{\theta}(f))$, we have $$\left( \underline{\theta}(f),\overline{\theta}(f)\right) \leq\left( \underline{\theta}(g),\overline{\theta}(g)\right) \Leftrightarrow\Phi\left( \underline{\theta}(g),\overline{\theta}(g)\right) \leq\Phi\left( \underline{\theta}(f),\overline{\theta}(f)\right) \text{,}%$$ meaning that $\Phi$ is a dual order-isomorphism. Thus $(\mathcal{FR}(U,\theta),\leq)$ is a self-dual poset, whenever conditions in (D) hold. In this section we will deduce some conditions under which $(\mathcal{FR}(U,\theta),\leq)$ is a lattice. Now let $L$ be a complete sublattice of $[0,1]$, and let $\mathcal{F}(U,L)$ stand for the family of all fuzzy sets $f\colon U\rightarrow L$. The system of all $f\in\mathcal{F}(U,L)$ with a finite range is denoted by $\mathcal{F}_{fr}(U,L)$. If $L=[0,1]$, then we write simply $\mathcal{F}% _{fr}(U)$. As $0,1\in L$, we have $\mathbf{0,1}\in\mathcal{F}_{fr}(U,L)$. It is obvious that for any $f_{1},f_{2}\in\mathcal{F}_{fr}(U,L)$, $f_{1}\vee f_{2}=$ max$\left( f_{1},f_{2}\right)$ and $f_{1}\wedge f_{2}=$ min$\left( f_{1},f_{2}\right)$ are of a finite range and their values are in $L$, hence $(\mathcal{F}_{fr}(U,L),\leq)$ is a bounded distributive lattice. Clearly, for any $f\in\mathcal{F}(U,L)$ with a finite range and any negator $n$, the fuzzy set $n(f)$ also has a finite range, i.e. $n(f) \in \mathcal{F}_{fr}(U,L)$. Further, if relation $\theta$ has a finite range, then in view of Lemma [Lemma 2](#lem:frange){reference-type="ref" reference="lem:frange"}, for any $f\in\mathcal{F}_{fr}(U)$: $\underline{\theta}(f),\overline{\theta}(f)\in\mathcal{F}_{fr}(U)$.  In all what follows, suppose that condition (C) holds with $n(L) \subseteq L$, and $\theta\colon U\times U\rightarrow L$ is a similarity relation. Then $\overline{\theta}(f)(x)=\bigvee\{$min$(\theta(x,y),f(y))\mid y\in U\}$ and $\underline{\theta}(f)(x)=\bigwedge\{$max$(n(\theta(x,y)),f(y))\mid y\in U\}$,  for all $x\in U$. As $L$ is closed w.r.t. arbitrary joins and meets, and $n(L) \subseteq L$, we get that $\underline{\theta}(f),\overline{\theta }(f) \in\mathcal{F}_{fr}(U,L)$. Now consider the poset defined on $$\mathcal{H}:=\{\left( \underline{\theta}(f),\overline{\theta}(f)\right) \mid f\in\mathcal{F}_{fr}(U,L)\}\text{.}%$$ We will prove that $(\mathcal{H},\leq)$ is a lattice, moreover if $U$ or $L$ is finite, then it is a complete lattice. This approach is motivated by the following examples: 1\) If $U$ is a finite set, then $\theta$ and all $f\in\mathcal{F}(U)$ have finite ranges. Hence for $L=[0,1]$ we have $\mathcal{F}_{fr}% (U,L)=\mathcal{F}(U)$, and $(\mathcal{H},\leq)$ equals to $(\mathcal{FR}(U,\theta),\leq)$. 2\) If $L$ is a finite chain with $0,1\in L$, then any $f\in\mathcal{F}(U,L)$ has a finite range, hence $\mathcal{F}_{fr}(U,L)=\mathcal{F}(U,L)$, and $(\mathcal{H},\leq)$ is the same as $(\mathcal{F}\mathcal{R}(U,L),\leq)$. **Remark 20**. *(a) The relations $\underline{\theta}(f_{1}% )\wedge\underline{\theta}(f_{1})=$ $\underline{\theta}\left( f_{1}\wedge f_{2}\right)$ and $\overline{\theta}(f_{1})\vee\overline{\theta}% (f_{2})=\overline{\theta}(f_{1}\vee f_{2})$ always hold (see e.g. [@DVCG]) for any $f_{1},f_{2}\in\mathcal{F}(U)$. Assume now that condition (C) holds, or $\odot$ is a left continuous t-norm and $\vartriangleright$ is its $R$-implicator. It is known (see e.g. [@MH]) that in this case the equalities* *$\bigwedge\{$ $\underline{\theta}(f_{i})\mid i\in I\}=\ \underline{\theta }\left( \bigwedge\{f_{i}\mid i\in I\}\right)$, $\bigvee\{\overline{\theta }(f_{i})\mid i\in I\}=\overline{\theta}\left( \bigvee\{f_{i}\mid i\in I\}\right)$* *also hold for any (nonempty) system $f_{i}\in\mathcal{F}(U)$, $i\in I$.* *(b) If now $L\subseteq\lbrack0,1]$ is a complete lattice and $\theta\colon U\times U\rightarrow L$, then clearly, for any $f_{i}% \in\mathcal{F}(U,L)$, $i\in I$ we get $\bigwedge\{f_{i}\mid i\in I\},\bigvee\{f_{i}\mid i\in I\}\in\mathcal{F}(U,L)$ and $\bigwedge\{$ $\underline{\theta}(f_{i})\mid i\in I\}=\underline {\theta}\left( \bigwedge\{f_{i}\mid i\in I\}\right) \in\mathcal{F}(U,L)$, $\overline{\theta }\left( \bigvee\{f_{i}\mid i\in I\}\right) \in\mathcal{F}(U,L)$.\ (c) As in this case conditions from (ID) also hold, in view of \[4\], for a $\odot$-similarity relation $\theta$, $f\mapsto\underline{\theta}(f)$, $f\in\mathcal{F}(U,L)$ is an interior operator, and the map $f\mapsto\overline{\theta}(f)$, $f\in\mathcal{F}(U,L)$ is a closure operator. Hence $\left( \text{Fix}_{L}\left( \underline{\theta}\right) ,\leq\right)$ and $\left( \text{Fix}_{L}\left( \overline{\theta}\right) ,\leq\right)$ are complete lattices, where Fix$_{L}\left( \underline{\theta }\right) :=\{f\in\mathcal{F}(U,L)\mid\underline{\theta}(f)=f\}$ and Fix$_{L}\left( \overline{\theta}\right) :=\{f\in\mathcal{F}(U,L)\mid \overline{\theta}(f)=f\}$.* **Proposition 21**. *Assume that conditions in (ID) are satisfied, and let $L\subseteq\lbrack0,1]$ be a complete lattice and $\theta\colon U\times U\rightarrow L$ be a $\odot$-similarity relation. Then $\left( \text{Fix}_{L}\left( \overline{\theta}\right) ,\leq\right)$ and $\left( \text{Fix}_{L}\left( \underline{\theta}\right) ,\leq\right)$ are complete sublattices of $\mathcal{F}(U,L)$.* *Proof.* Let $f_{i}\in\ $Fix$_{L}\left( \overline{\theta }\right)$, $i\in I$ arbitrary. Then, in view of Remark [Remark 20](#rem:complete){reference-type="ref" reference="rem:complete"}, $\bigvee \{f_{i}\mid i\in I\}\in\mathcal{F}(U,L)$, and $\overline{\theta}\left( \bigvee\{f_{i}\mid i\in I\}\right) =\bigvee\{\overline{\theta}(f_{i})\mid i\in I\}=\bigvee\{f_{i}\mid i\in I\}$. Hence $\bigvee\{f_{i}\mid i\in I\}\in$ Fix$_{L}\left( \overline{\theta}\right)$. As Fix$_{L}\left( \overline{\theta}\right)$ is the system of closed sets of the operator $f\mapsto\overline{\theta }(f)$ and $f_{i} \in\ $Fix$_{L}\left( \overline{\theta }\right)$, $i\in I$, we also have $\bigwedge\limits_{i\in I}f_{i}\in\ $ Fix$_{L}\left( \overline{\theta}\right)$. Hence $\left( \text{Fix}% _{L}\left( \overline{\theta}\right) ,\leq\right)$ is a complete sublattice of $\left( \mathcal{F}(U,L),\leq\right)$. The claim that $\left( \text{Fix}_{L}\left( \underline{\theta}\right) ,\leq\right)$ is complete sublattice of $\left( \mathcal{F}(U,L),\leq\right)$ is proved dually. ◻ **Corollary 22**. *Let $\theta\colon U\times U\rightarrow L$ be a similarity relation with a finite range on $U$, $f_{i}\in\mathcal{F}(U,L)$, $i\in I$, $F=% %TCIMACRO{\tbigwedge }% %BeginExpansion {\textstyle\bigwedge} %EndExpansion \{\overline{\theta}(f_{i})\mid i\in I\}$ and let $\{a\}\subseteq U$ be a maximal $E(F)$ class. Then $F(a)=\bigwedge\{f_{i}(a)\mid i\in I\}$.* *Proof.* In view of Proposition [Proposition 21](#prop:completesubl){reference-type="ref" reference="prop:completesubl"} we have $F=% %TCIMACRO{\tbigwedge }% %BeginExpansion {\textstyle\bigwedge} %EndExpansion \{\overline{\theta}(f_{i})\mid i\in I\}\in$ Fix$_{L}(\overline{\theta})$, i.e. $F=\overline{\theta}(F)$. Since $\overline{\theta}(f_{i})(a)\geq F(a)$, $i\in I$, by using Proposition [Proposition 19](#prop:lowerupper){reference-type="ref" reference="prop:lowerupper"}(i) we obtain $\overline{\theta}% (f_{i})(a)=f_{i}(a)$, for all $i\in I$. This yields $F(a)=\bigwedge \{f_{i}(a)\mid i\in I\}$. ◻ **Theorem 23**. *Let $\theta\colon U\times U\rightarrow L$ be a similarity relation of a finite range, and assume that condition (C) holds with a negator satisfying $n(L)\subseteq L$.* *(i) If the fuzzy sets $\bigwedge\limits_{i\in I}f_{i}$, $\underset{i\in I}{\bigwedge}\overline{\theta}(f_{i})$, $f_{i}\in\mathcal{F}(U,L)$, $i\in I$ have finite ranges, then the infimum of fuzzy rough sets $\left( \underline{\theta}(f_{i}),\overline{\theta}(f_{i})\right)$, $i\in I$ exists in $(\mathcal{FR}(U,L),\leq)$ and its components have finite ranges.* *(ii) $(\mathcal{H},\leq)=(\{\left( \underline{\theta}(f),\overline{\theta}(f)\right) \mid f\in\mathcal{F}_{fr}(U,L)\},\leq)$ is a lattice.* *(iii) If $U$ or $L$ is finite, then $(\mathcal{FR}(U,L),\leq)$ is a complete lattice.* *Proof.* (i) Denote $G=\underline{\theta}(\bigwedge \limits_{i\in I}f_{i})$ and $F=\underset{i\in I}{\bigwedge}\overline{\theta }(f_{i})$. Then $G,F \in \mathcal{F}(U,L)$, by Remark [Remark 20](#rem:complete){reference-type="ref" reference="rem:complete"}(b), and we have $\underline{\theta}(G)=G$ and $\overline{\theta}(\overline{\theta }(f_{i}))=\overline{\theta}(f_{i})$, $i\in I$, according to Remark [Remark 20](#rem:complete){reference-type="ref" reference="rem:complete"}(c). Thus $G\in$ Fix$_{L}(\underline{\theta })$. Since $\theta$ and $\bigwedge\limits_{i\in I}f_{i}$ have finite ranges, $G$ also has a finite range. As $\overline{\theta}(f_{i}) \in$ Fix$_{L}(\overline{\theta })$, Proposition [Proposition 21](#prop:completesubl){reference-type="ref" reference="prop:completesubl"} gives $F\in$ Fix$_{L}(\overline{\theta })$, and by assumption $F$ has a finite range. Clearly, $G=\underline{\theta }(\bigwedge\limits_{i\in I}f_{i})\leq\overline{\theta}(f_{i})$, for all $i\in I$, whence $G\leq F$. Using $G$ and $F$ we will construct a fuzzy set $f\in\mathcal{F}(U,L)$ such that $\left( \underline{\theta}(f),\overline{\theta}(f)\right)$ equals to inf$\{(\underline{\theta}(f_{i}),\overline{\theta}(f_{i}))\mid i\in I\}$. First, from each maximal $\varepsilon (G)$ class $\mathcal{E}_{t}$, $t\in T$ we select exactly one element $b_{t}\in\mathcal{E}_{t}$ as follows: ) If $\mathcal{E}_{t}$ contains an element $q_{t}\in\mathcal{E}_{t}$ with $G(q_{t})=F(q_{t})$, then we set $b_{t}:=q_{t}$. ) If there are no such elements in $\mathcal{E}_{t}$, however there exists an $s_{t}\in\mathcal{E}_{t}$ such that $\{s_{t}\}$ is not an $E(F)$ class, then we choose it and set $b_{t}:=s_{t}$. ) If there are no elements of type 1) or 2) in in $\mathcal{E}_{t}$, then we select an element $r_{t}\in\mathcal{E}_{t}$ such that $\{r_{t}\}$ is not a maximal $E(F)$ class, and we set $b_{t}:=r_{t}$. Now we show that we can always manage such a selection. Indeed, assume by contradiction that in some class $\mathcal{E}_{z}$ there are no elements of type 1), 2) and 3). This means that for each $x\in\mathcal{E}_{z}$ the set $\{x\}$ is a maximal $E(F)$ class. Then in view of Corollary [Corollary 22](#cor:fafia){reference-type="ref" reference="cor:fafia"}, we have $F(x)=\bigwedge\limits_{i\in I}f_{i}(x)$, for each $x\in\mathcal{E}_{z}$. As $\bigwedge\limits_{i\in I}f_{i}(x)$ has a finite range, by Proposition [Proposition 12](#prop:famaxfy){reference-type="ref" reference="prop:famaxfy"}(ii) we get $G(y)=\ $min$\{\bigwedge\limits_{i\in I}% f_{i}(x)\mid x\in\mathcal{E}_{z}\}$, for all $y\in\mathcal{E}_{z}$, because $G=\underline{\theta}(\bigwedge\limits_{i\in I}f_{i})$. Hence, there exists an element $v\in\mathcal{E}_{z}$ such that $G(v)=\bigwedge\limits_{i\in I}f_{i}(v)=F(v)$. Since this result means that $v$ is an element of type 1) in $\mathcal{E}_{z}$, this is a contradiction. As next step, we construct a fuzzy set $f\in\mathcal{F}(U,L)$ as follows: $$f(x)=\left\{ \begin{array} [c]{l}% G(x)\text{, if }x\in\{b_{t}\mid t\in T\}\text{;}\\ F(x)\text{, if }x\in U\setminus\{b_{t}\mid t\in T\} \end{array} \right. \tag{5}$$ As $G,F\in\mathcal{F}(U,L)$, we have $f\in\mathcal{F}(U,L)$. Since $F$ and $G$ have finite ranges, $f$ also has a finite range. As from each maximal $\varepsilon (G)$ class $\mathcal{E}_{t}$, $t\in T$ an element $b_{t}\in\mathcal{E}_{t}$ was selected and $f(b_{t})=G(b_{t})$, $f\geq G$ hold, by Proposition [Proposition 11](#prop:hufuhvgv){reference-type="ref" reference="prop:hufuhvgv"}(ii) we have $\underline{\theta}(f)=G=\underline {\theta}(\bigwedge\limits_{i\in I}f_{i})$. We prove that $\left( \underline{\theta}(f),\overline{\theta}(f)\right)$ is the infimum of the system $(\underline{\theta}(f_{i}),\overline{\theta}(f_{i})),i\in I$. Thus we are going to show that $\left( \underline{\theta}(f),\overline {\theta}(f)\right)$ is a lower bound of $(\underline{\theta}(f_{i}% ),\overline{\theta}(f_{i})),i\in I$ and for any $h\in\mathcal{F}(U,L)$ with $(\underline{\theta}(h),\overline{\theta}(h))\leq(\underline{\theta}% (f_{i}),\overline{\theta}(f_{i}))$, $i\in I$ we have $(\underline{\theta }(h),\overline{\theta}(h))\leq\left( \underline{\theta}(f),\overline{\theta }(f)\right)$. As by definition $f\leq F$, we also have $\overline{\theta }(f)\leq\overline{\theta}(F)=F \leq\overline{\theta}(f_{i})$, $i\in I$. Since $\underline{\theta}(f)=\underline{\theta}(\bigwedge\limits_{i\in I}f_{i})\leq\underline{\theta}(f_{i})$, $i\in I$, now $\left( \underline {\theta}(f),\overline{\theta}(f)\right)$ is a lower bound of $(\underline {\theta}(f_{i}),\overline{\theta}(f_{i})),i\in I$ and condition $\overline {\theta}(h)\leq\overline{\theta}(f_{i})$, $i\in I$ is equivalent to $\overline{\theta}(h)\leq\ \underset{i\in I}{% %TCIMACRO{\tbigwedge }% %BeginExpansion {\textstyle\bigwedge} %EndExpansion }\overline{\theta}(f_{i})=F$. Since $\underline{\theta}(f)=\underline{\theta }(\bigwedge\limits_{i\in I}f_{i})=\bigwedge\limits_{i\in I}\underline{\theta }(f_{i})$, we also have $\underline{\theta}(h)\leq\underline{\theta}(f_{i})$, $i\in I \Longleftrightarrow \underline{\theta}(h)\leq\bigwedge \limits_{i\in I}\underline{\theta}(f_{i})=\underline{\theta}(f)=G$. Hence to prove $(\underline{\theta}(h),\overline{\theta}(h))\leq\left( \underline {\theta}(f),\overline{\theta}(f)\right)$, for all sets $h\in\mathcal{F}(U,L)$ with\ $(\underline{\theta}(h),\overline{\theta}(h))\leq(\underline{\theta}% (f_{i}),\overline{\theta}(f_{i}))$, $i\in I$, it is enough to show that $\overline{\theta}(h)\leq$ $\overline{\theta}(f)$ holds for any $h\in \mathcal{F}(U,L)$ with $\underline{\theta}(h)\leq G$ and $\overline{\theta }(h)\leq F$. Take any $h$ with this property and any $x\in U$. If $x\in U\setminus \{b_{t}\mid t\in T\}$ or $x=b_{t_{0}}$ for some $t_{0}\in$ $T$ with $G(b_{t_{0}})=F(b_{t_{0}})$, then $f(x)=F(x)$, hence $h(x)\leq\overline {\theta}(h)(x)\leq F(x)=f(x)\leq$ $\overline{\theta}(f)(x)$. Let $x=b_{t_{0}}$, for some $t_{0}\in T$ such that $G(b_{t_{0}})\neq F(b_{t_{0}})$. Then $f(b_{t_{0}})=G(b_{t_{0}})$, by our construction. If $\{b_{t_{0}}\}$ is a maximal $\varepsilon (G)$ class, then in view of Proposition [Proposition 19](#prop:lowerupper){reference-type="ref" reference="prop:lowerupper"}(ii), $\underline{\theta}(h)(b_{t_{0}})\leq G(b_{t_{0}})$ implies $h(b_{t_{0}})=\underline{\theta}(h)(b_{t_{0}})\leq G(b_{t_{0}})$, i.e. we obtain $h(x)\leq G(x)=f(x)\leq\overline{\theta}(f)(x)$. Assume now that $\mathcal{E}_{t_{0}}$, the maximal $\varepsilon (G)$ class containing $b_{t_{0}}$, has at least two elements. Denote the $E(F)$ class containing $b_{t_{0}}$ by $E_{0}$. If $E_{0}\nsubseteqq\mathcal{E}_{t_{0}}$, then there exists a $z_0\in E_{0}\setminus\mathcal{E}_{t_{0}}$, and we have $(y,z_0)\notin\varrho(G)$ for each $y\in\mathcal{E}_{t_{0}}$, because $\mathcal{E}_{t_{0}}$ is a maximal $\varepsilon (G)$ class. Hence, by Corollary [Corollary 8](#cor:assertions){reference-type="ref" reference="cor:assertions"}(ii), we get $(z,y)\in R(F)$ for all $z\in E_{0}$ and $y\in\mathcal{E}_{t_{0}}$. Thus $(b_{t_{0}},c)\in R(F)$ for any $c\in\mathcal{E}_{t_{0}}$, $c\neq b_{t_{0}}$. Clearly, $c\notin\{b_{t}\mid t\in T\}$, because only a single element $b_{t_{0}}$ was selected from $\mathcal{E}_{t_{0}}$, and hence $f(c)=F(c)$. Since $\left( b_{t_{0}},c\right) \in R(F)$ and $f\leq F$, by applying Proposition [Proposition 6](#prop:prop2){reference-type="ref" reference="prop:prop2"}(i) we get $\overline{\theta}(f)(b_{t_{0}})=F(b_{t_{0}})$. Thus we obtain $h(b_{t_{0}})\leq\overline{\theta}(h)(b_{t_{0}})\leq F(b_{t_{0}})=\overline{\theta}(f)(b_{t_{0}})$, i.e. $h(x)\leq\overline{\theta }(f)(x)$. If $E_{0}\subseteq\mathcal{E}_{t_{0}}$, then we claim that $(b_{t_{0}},e)\in R(F)$ for some element $e\in\mathcal{E}_{t_{0}}\setminus\{b_{t_{0}}\}$ (such an element exists, because $|\mathcal{E}_{t_{0}}|\geq2$). Clearly, if $E_{0}$ has at least two elements, then $e$ can be chosen as any element from $E_{0}\setminus\{b_{t_{0}}\}$. If $E_{0}=\{b_{t_{0}}\}$, then in view of our construction, the element $b_{t_{0}}$ is of type 3), i.e. $\{b_{t_{0}}\}$ is an $E(F)$ class which is not maximal. However, if $(b_{t_{0}},e)\notin R(F)$ would hold for each $e\in\mathcal{E}_{t_{0}% }\setminus\{b_{t_{0}}\}$, then in view of Corollary [Corollary 18](#cor:outer){reference-type="ref" reference="cor:outer"}(ii), $\{b_{t_{0}}\}$ would be a maximal $E(F)$ class, contrary to our hypothesis. As no element different from $b_{t_{0}}$ was selected from $\mathcal{E}_{t_{0}}$, we have $e\notin\{b_{t}\mid t\in T\}$, and hence $f(e)=F(e)$. Since $(b_{t_{0}},e)\in R(F)$, repeating now the previous argument, we obtain again $h(b_{t_{0}})\leq$ $\overline{\theta }(f)(b_{t_{0}})$, i.e. $h(x)\leq\overline{\theta}(f)(x)$. Hence for each $x\in U$ we obtained $h(x)\leq\overline{\theta}(f)(x)$. Thus $h\leq\overline{\theta}(f)$. In view of [@DVCG], this implies $\overline{\theta}(h)\leq\overline{\theta}\left( \overline{\theta}(f)\right) =\overline{\theta}(f)$. Thus $\left( \underline{\theta}(f),\overline{\theta }(f)\right)$ is the infimum of $(\underline{\theta}(f_{i}),\overline{\theta }(f_{i}))$, $i\in I$. Since $f\in\mathcal{F}(U,L)$ has a finite range, $\underline{\theta}(f),\overline{\theta}(f)\in\mathcal{F}(U,L)$ also have finite ranges. \(ii\) For any $f_{1},f_{2}\in\mathcal{F}_{fr}(U,L)$, $f_{1}\wedge f_{2}$ has a finite range. By Lemma [Lemma 2](#lem:frange){reference-type="ref" reference="lem:frange"}, as $\overline{\theta}(f_{1}% ),\overline{\theta}(f_{2})\in\mathcal{F}_{fr}(U,L)$, $\overline{\theta}% (f_{1})\wedge\overline{\theta}(f_{2})$ also has a finite range. Applying now (i) with $I=\{1,2\}$, we get that $(\mathcal{H},\leq)$ is a $\wedge$-semilattice. Since condition (C) implies property (D), $(\mathcal{H},\leq)$ is self-dual, and hence it is a lattice. \(iii\) If $U$ or $L$ is finite, then $\theta$ and each $f\in\mathcal{F}(U,L)$ have finite ranges, i.e. $\mathcal{F}(U,L)=$ $\mathcal{F}_{fr}(U,L)$. As for any $f_{i}$ $\in\mathcal{F}(U,L)$, $i\in I$ we have $\bigwedge\limits_{i\in I}f_{i}$, $\underset{i\in I}{% %TCIMACRO{\tbigwedge }% %BeginExpansion {\textstyle\bigwedge} %EndExpansion }\overline{\theta}(f_{i})\in\mathcal{F}(U,L)$, the fuzzy sets $\bigwedge \limits_{i\in I}f_{i}$ and $\underset{i\in I}{% %TCIMACRO{\tbigwedge }% %BeginExpansion {\textstyle\bigwedge} %EndExpansion }\overline{\theta}(f_{i})$ also have finite ranges. Hence, in view of (i), inf$\{(\underline{\theta}(f_{i}),\overline{\theta}(f_{i}))\mid i\in I\}$ always exists, i.e. $(\mathcal{H}$,$\leq)$ is a complete $\wedge$-semilattice. Since $(\mathcal{H},\leq)$ is self-dual, it is a complete lattice. ◻ **Remark 24**. *If for a system $f_{i}\in\mathcal{F}(U,L)$, $i\in I$ we have $\left( \underline{\theta}(f),\overline{\theta}(f)\right)$ = $\left( \bigwedge\{\underline{\theta}\left( f_{i}\right) \mid i\in I\},\bigwedge\{\overline{\theta}\left( f_{i}\right) \mid i\in I\}\right)$, for an $f\in\mathcal{F}(U,L)$, then $\left( \underline{\theta}(f),\overline {\theta}(f)\right)$ equals to the infimum of $\left( \underline{\theta}(f_{i}),\overline{\theta}(f_{i})\right), i\in I$. Indeed, for any $h\in\mathcal{F}(U,L)$ with $(\underline {\theta}(h),\overline{\theta}(h))\leq(\underline{\theta}(f_{i}),\overline {\theta}(f_{i}))$, $i\in I$ we get $(\underline{\theta}(h),\overline{\theta}(h))\leq\left( \underline{\theta}(f),\overline{\theta}(f)\right)$, meaning that $\left( \underline{\theta}(f),\overline{\theta}(f)\right)$ is the infimum of $\left( \underline{\theta}(f_{i}),\overline{\theta}(f_{i})\right), i\in I$. Analogously, $\left( %TCIMACRO{\tbigvee }% %BeginExpansion {\textstyle\bigvee} %EndExpansion \{\underline{\theta}\left( f_{i}\right) \mid i\in I\},% %TCIMACRO{\tbigvee }% %BeginExpansion {\textstyle\bigvee} %EndExpansion \{\overline{\theta}\left( f_{i}\right) \mid i\in I\}\right)$ is the supremum of $\left( \underline{\theta}(f_{i}),\overline{\theta }(f_{i})\right), i\in I$ whenever $\left( %TCIMACRO{\tbigvee }% %BeginExpansion {\textstyle\bigvee} %EndExpansion \{\underline{\theta}\left( f_{i}\right) \mid i\in I\},% %TCIMACRO{\tbigvee }% %BeginExpansion {\textstyle\bigvee} %EndExpansion \{\overline{\theta}\left( f_{i}\right) \mid i\in I\}\right) \in \mathcal{FR}(U,L)$.* [\[ex:notfrs\]]{#ex:notfrs label="ex:notfrs"} Here we show how a meet $\left( \underline{\theta}(f_{1}),\overline{\theta}(f_{1})\right) \wedge\left( \underline{\theta}(f_{2}),\overline{\theta}(f_{2})\right)$ can be calculated by using construction (5) in the proof of Theorem [Theorem 23](#thm:lattice){reference-type="ref" reference="thm:lattice"}. The similarity relation $\theta$ is given on Figure [\[fig:theta4\]](#fig:theta4){reference-type="ref" reference="fig:theta4"}, and $L=\{0, 0.1, 0.25, 0.5, 0.75, 1 \}$. The fuzzy sets $f_{1},f_{2}$ and their approximations are given in Table [\[tab:fuzzy3\]](#tab:fuzzy3){reference-type="ref" reference="tab:fuzzy3"}. \|P1.5cm\|P0.75cm\|P0.75cm\|P0.75cm\| $u$ & $a$ & $b$ & $c$\ $f_1(u)$ & 1 & 0.1 & 0.5\ $\overline{\theta}(f_1)(u)$ & 1 & 0.75 & 0.5\ $\underline{\theta}(f_1)(u)$ & 0.25 & 0.1 & 0.5\ \|P1.5cm\|P0.75cm\|P0.75cm\|P0.75cm\| $u$ & $a$ & $b$ & $c$\ $f_2(u)$ & 0.1 & 1 & 0.5\ $\overline{\theta}(f_2)(u)$ & 0.75 & 1 & 0.5\ $\underline{\theta}(f_2)(u)$ & 0.1 & 0.25 & 0.5\ The corresponding fuzzy rough sets are represented in the form $\alpha_1= \begin{pmatrix} 1 & 0.75 & 0.5\\ 0.25 & 0.1 & 0.5 \end{pmatrix}$ and $\alpha_2= \begin{pmatrix} 0.75 & 1 & 0.5\\ 0.1 & 0.25 & 0.5 \end{pmatrix}$, where the first row stands for the upper approximations and the second row shows their lower approximations. Computing the meets $F = \overline{\theta}(f_1) \wedge \overline{\theta}(f_2)$ and $G = \underline{\theta}(f_1) \wedge \underline{\theta}(f_2)$, we obtain the pair $\begin{pmatrix} F \\ G \end{pmatrix}= \begin{pmatrix} 0.75 & 0.75 & 0.5\\ 0.1 & 0.1 & 0.5 \end{pmatrix}$, which is not a fuzzy rough set. The quasiorders induced by $F$ and $G$ are given in Figure [\[fig:rhor\]](#fig:rhor){reference-type="ref" reference="fig:rhor"}. Observe that each element is a maximal $\varepsilon (G)$ class. Hence, applying formula (5) from the proof of Theorem [Theorem 23](#thm:lattice){reference-type="ref" reference="thm:lattice"}, we obtain the reference set $f:=G$, and as a corresponding fuzzy rough set $\begin{pmatrix} 0.25 & 0.25 & 0.5\\ 0.1 & 0.1 & 0.5 \end{pmatrix}$. [\[ex:distlattice\]]{#ex:distlattice label="ex:distlattice"} Let us consider the similarity relation $\theta$ on Figure [\[fig:theta3\]](#fig:theta3){reference-type="ref" reference="fig:theta3"}, and set $L= \{0, 0.5, 1 \}$. The lattice $(\mathcal{H},\leq)$ of fuzzy rough sets is shown on Figure [\[fig:lattice\]](#fig:lattice){reference-type="ref" reference="fig:lattice"}. # Conclusions {#sec:conc .unnumbered} The properties of a poset formed by fuzzy rough sets depend strongly both on the framework in which the approximations are defined (t-norm $\odot$ - implicator $\vartriangleright$), and on the properties of the approximation space $(U,\theta)$. The majority of our arguments work only under some finiteness conditions imposed on the domain or range of the fuzzy reference sets and of the relation $\theta$. We hope that these conditions can be replaced with weaker ones (see e.g. [@St]) or with conditions related to some topology defined on $U$. In case of a finite universe or range set $L$, we were able to show that $(\mathcal{FR}(U,L),\leq)$ is a lattice only for a similarity relation $\theta$ in a particular context (min t-norm and S-implicator), by using property (D). It would be interesting to check if the proof can be extended for fuzzy quasiorders or other types of relations. Theorem [Theorem 15](#thm:frs){reference-type="ref" reference="thm:frs"} seems to suggest that such a result can be obtained even in a general context (of a t-norm and a related implicator) for a t-similarity relation $\theta$ with some (strong) particular properties, even in the absence of the property (D). This can serve as a further research goal. Even in conditions of Theorem [Theorem 23](#thm:lattice){reference-type="ref" reference="thm:lattice"}, the lattices formed by fuzzy rough sets are not distributive in general - this is shown in Example [\[ex:notdistr\]](#ex:notdistr){reference-type="ref" reference="ex:notdistr"} below. Hence an interesting question could be if these lattices have any characteristic common properties. We can see that for some particular approximation spaces as in Example [\[ex:notfrs\]](#ex:notfrs){reference-type="ref" reference="ex:notfrs"}, we even obtain a particular distributive lattice (a so-called double Stone lattice). Therefore, it makes sense to ask under what conditions imposed on $(U,\theta)$ will we obtain a distributive lattice $\mathcal{FR}(U,L)$. [\[ex:notdistr\]]{#ex:notdistr label="ex:notdistr"} Let $U$, $L$, the similarity relation $\theta$ be as in Example [\[ex:distlattice\]](#ex:distlattice){reference-type="ref" reference="ex:distlattice"}, and let us consider the fuzzy rough sets $\alpha_{1},\alpha_{2}$ from [\[ex:distlattice\]](#ex:distlattice){reference-type="ref" reference="ex:distlattice"} and $c=% \begin{pmatrix} 0.5 & 0.5 & 0.5\\ 0.5 & 0.5 & 0.5 \end{pmatrix}$. We prove that $(\alpha_{1}\wedge\alpha_{2})\vee c\neq(\alpha_{1}\vee c)\wedge(\alpha_{2}\vee c)$: Indeed, by Example [\[ex:distlattice\]](#ex:distlattice){reference-type="ref" reference="ex:distlattice"}, $\alpha_{1}\wedge\alpha_{2}=% \begin{pmatrix} 0.25 & 0.25 & 0.5\\ 0.1 & 0.1 & 0.5 \end{pmatrix} <c$, and hence $(\alpha_{1}\wedge\alpha_{2})\vee c=c$. In view of Remark [Remark 24](#rem:infsup){reference-type="ref" reference="rem:infsup"} we have $\alpha_{1}\vee c=% \begin{pmatrix} 1 & 0.75 & 0.5\\ 0.5 & 0.5 & 0.5 \end{pmatrix}$ and $\alpha_{2}\vee c=% \begin{pmatrix} 0.75 & 1 & 0.5\\ 0.5 & 0.5 & 0.5 \end{pmatrix}$, because $\alpha_{1}\vee c=% \begin{pmatrix} \overline{\theta}(h_{1})\\ \underline{\theta}(h_{1}) \end{pmatrix}$ and $\alpha_{2}\vee c=% \begin{pmatrix} \overline{\theta}(h_{2})\\ \underline{\theta}(h_{2}) \end{pmatrix}$, where $h_{1} = 1/a + 0.5/b + 0.5/c$ and $h_{2} = 0.5/a + 1/b + 0.5/c$. Now, observe that $% \begin{pmatrix} \overline{\theta}(h_{1})\wedge\overline{\theta}(h_{2})\\ \underline{\theta}(h_{1})\wedge\underline{\theta}(h_{2}) \end{pmatrix} =% \begin{pmatrix} 0.75 & 0.75 & 0.5\\ 0.5 & 0.5 & 0.5 \end{pmatrix}$ is a fuzzy rough set induced by the fuzzy set $m = 0.75/a + 0.5/b + 0.5/c$. In view of Remark [Remark 24](#rem:infsup){reference-type="ref" reference="rem:infsup"} this means that $(\alpha_{1}\vee c)\wedge(\alpha _{2}\vee c)=% \begin{pmatrix} 0.75 & 0.75 & 0.5\\ 0.5 & 0.5 & 0.5 \end{pmatrix} \neq c$.

arxiv_math

--- address: - ", , " - ", , " author: - Anwar Ahmad - Muhammad Ali - Salman A. Malik title: "Unraveling Forward and Backward Source Problems for a Nonlocal Integrodifferential Equation: A Journey through Operational Calculus for Dzherbashian-Nersesian Operator" --- # Introduction and Problems Statement {#intro.mokin} Merit for developing operational calculus is credited to the famous Polish mathematician Jan Mikusińki. He presented the new approach for constructing an operational calculus for the differential operator during the 1950s [@Mikusinski]. This algebraic scheme revolves around the idea that the Laplace convolution can be interpreted as a multiplication within the continuous functions ring in the upper half-plane. This approach, known as the Mikusiński operational calculus, has been effectively applied to various mathematical fields, such as the study of ordinary and partial differential equations. Many mathematicians (see, for example [@Ditkin-1957; @Ditkin-Prudnikov; @Dimovski; @Meller]) extended the Mikusińki scheme to create operational calculi for differential equations with variable coefficients, including the Bessel differential operator defined as follows $$(\mathcal{B}g)(x):=x^{-\alpha}\prod^{n}_{i=1}\Big(\beta_{i}+\frac{1}{\alpha}x\frac{d}{dx}\Big)g(x).$$ In year 1994, Luchko and Srivastva developed an operational calculus for the Riemann-Liouville derivative (RLD), which allowed them to solve a Cauchy boundary value problem associated with a linear equation that involved RLDs [@Luchko-Srivastava-CAMWA]. The following year, Al-Bassam and Luchko constructed an operational calculus pertaining to the multiple Erdelyi-Kober operators [@Al-Bassam]. This operational calculus enabled them to solve the nonlocal integrodifferential equations with multiple Erdelyi-Kober operators. In 1999, Luchko and Gorenflo published an article in Acta Mathematica in which they introduced Mikusińki's operational calculus for Caputo derivative [@Luchko-Gorenflo-1999]. In 2009, Hilfer and coauthors developed an operational calculus for the generalized RLD [@Hilfer-Luchko-Tomovski]. This scheme of operational calculus entails solving linear differential equations featuring constant coefficients that incorporate generalized RLD. Despite being introduced in 1968 [@Dzhrbashyan-Nersesyan-1968], the Dzherbashian-Nersesian operator has not received significant attention from the mathematical community. However, the appearance of an English translation of Dzhrbashyan et al. [@Dzhrbashyan-Nersesyan-1968] in an article published in FCAA [@Dzhrbashyan-Nersesyan-2020] has sparked interest and discussion related to this operator. In addition to presenting several interesting results related to this operator, such as the construction of Laplace transform in Ahmad et al. [@Anwar-Ali-Malik-2021], the authors have also explored the specific parameter choices that allow Dzherbashian-Nersesian operator to recover the renowned fractional operators like Riemann-Liouville, Caputo, and Hilfer derivatives. Forward and backward problems for FDEs arise in various areas of science. These kind of problems have also been studied for fractional diffusion equations in [@Kerbal-Kadirkulov-Kirane; @Restrepo-Suragan; @Ali-Malik-2017]. Recently, backward problems for FDEs involving Caputo and generalized RLDs have been extensively studied [@Hamidi-Kirane-Tfayli; @Dib-Kirane; @Al-Salti-Karimov]. However, there is a limited body of literature pertaining to the forward and backward problems associated with FDEs that feature the Dzherbashian-Nersesian operator. To our knowledge, this marks the inaugural attempt to address a forward problem involving this operator, with only a handful of articles in the existing literature addressing backward problems for FDEs incorporating this operator. A few such instances include Berdyshev et al.[@Berdyshev-Kadirkulov-2016] and Karimov et al. [@Karimov-Bakhodirjon-2022]. Additionally, in [@Anwar-Ali-Malik-2021], authors used the Fourier method to investigate backward problems involving source terms that are both space and time dependent. Motivated by the persistent research activity in operational calculus, the renewed interest in Dzherbashian-Nersesian operator, and applications of backward problems, our study in this paper is of dual purpose. First of all, this chapter addresses the construction of operational calculus for Dzherbashian-Nersesian operator. Second, this calculus is used to address both forward and backward problems for FDE with Dzherbashian-Nersesian operator. The subsequent sections of the article are structured in the following manner. Further in this section, we formulate the problems. Section [2](#Preliminaries.Operational){reference-type="ref" reference="Preliminaries.Operational"} is dedicated to some basics definitions and results related to nonlocal integrodifferential operators and Mittag-Leffler function. Section [3](#sec: Operational){reference-type="ref" reference="sec: Operational"} focuses the development of operational calculus for Dzherbashian-Nersesian operator. Both forward and backward problems for a FDE are addressed in Section [4](#sec: FSP and BSP){reference-type="ref" reference="sec: FSP and BSP"}. Finally, the concluding remarks are given in Section [5](#sec: Conclusion){reference-type="ref" reference="sec: Conclusion"}. ## Formulation of Forward and Backward Problems {#formulation-of-forward-and-backward-problems .unnumbered} This paper focuses on investing the following FDE: $$\label{probeq.oper} \mathcal{D}^{\varrho_{m}}_{t,0+}u(t,x)=u_{xx}(t,x)+F(t,x),\quad (t,x)\in (0,T)\times (0,1),$$ contingent upon the subsequent boundary conditions $$\label{probbcs.oper} u(t,1)=0,\quad u_{x}(t,0)=u_{x}(t,1),\quad t\in (0,T),$$ and initial condition given by $$\label{probic.oper} J^{1-\alpha_{1}}_{t,0+}u(t,x)\Big|_{t=0}=\phi(x),\quad \alpha_{0}, x\in (0,1).$$ The expression $\mathcal{D}^{\varrho_{m}}_{t,0+}$, such that $\varrho_{m} \in (0,1)$, denotes Dzherbashian-Nersesian operator (See Definition [\[DherNer-def\]](#DherNer-def){reference-type="ref" reference="DherNer-def"}). The pursuit of finding $u(t,x)$, which adheres to the initial boundary value problem (IBVP) described by ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}) while satisfying the conditions $t^{\alpha_{1}}u(t,.)\in C^{2}(0,1)(.,x)$ and $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+}u(.,x) \in C(0,T)$, given the source term $F(t,x)$ and the initial data $\phi(x)$ are provided, is recognized as the establishment of a strong or classical solution to ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}). This is commonly referred to as the forward problem. We also explore the BSP within the context of IBVP ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}). It is important to note that the source term $F(t,x)$ ,is considered to be solely dependent on spatial coordinate, denoted as $F(t,x):=f(x)$. Let us formally introduce the concept of solving BSP: We aim to identify the set $\big\{u(t,x), f(x)\big\}$ that satisfy the conditions, specifically, $t^{\alpha_{1}}u(t,.)$ $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+} u(.,x)$, and $f$ belonging to the spaces $C^{2}[0,1]$, $C(0,T)$, and $\in C[0,1]$ respectively, for the problem ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}). This solution of BSP presents challenges as it falls under the category of ill-posed problems according to Hadamard's definition. To uniquely determine the source term, an additional condition is required, commonly referred to as the overdetermined condition. We propose the following overdetermination condition, thereby enhancing the uniqueness of the source term determination: $$\label{probover.oper} u(T,x)=\psi(x).$$ # Preliminaries and Notations {#Preliminaries.Operational} This section is dedicated to presenting the fundamental definitions and properties of some well-known nonlocal integrodifferential operators and Mittag-Leffler function. [@Kilbas-Srivastava], [@Samko-book] The RLI denoted as $J^{\alpha}_{t,0+}$, where $\alpha \in \mathbb{C}$ and $\mathcal{R}(\alpha)>0$, is given by $$\begin{aligned} \label{defRLI.oper} J^{\alpha}_{t,0+}g(t):=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\frac{g(\tau)}{(t-\tau)^{1-\alpha}}d\tau, \quad t>0. \end{aligned}$$ The function denoted by $\Gamma(\cdot)$ refers to the second kind of Euler integral. [@Kilbas-Srivastava], [@Samko-book] [\[RL-def\]]{#RL-def label="RL-def"} The RLD denoted as $D^{\alpha}_{t,0+}$, with order $\alpha \in (m-1,m)$ such that $\mathcal{R}(\alpha)\geqq 0$, is given by $$\begin{aligned} D^{\alpha}_{t,0+}g(t):= \frac{d^{m}}{dt^{m}}J^{m-\alpha}_{t,0+}g(t), \quad m=\mathcal{R}(\alpha)+1, \end{aligned}$$ where $[\mathcal{R}(\alpha)]$ represents the integer part of $\mathcal{R}(\alpha)$. [@Dzhrbashyan-Nersesyan-1968][\[DherNer-def\]]{#DherNer-def label="DherNer-def"} The Dzherbashian-Nersesian operator $\mathcal{D}^{\varrho_{m}}_{t,0+}$ in relation to time with order $\varrho_{m} \in (0,m)$ defined as follows $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+} g(t):=J^{1-\alpha_{m}}_{t,0+}D^{\alpha_{m-1}}_{t,0+}D^{\alpha_{m-2}}_{t,0+}...D^{\alpha_{1}}_{t,0+}D^{\alpha_{0}}_{t,0+}g(t),\quad m\in \mathbb{Z^{+}},\quad t>0,\label{DzhrNer-mathematical} \end{aligned}$$ where $$\begin{aligned} \varrho_{m}=\sum_{j=0}^{m}\alpha_{j}-1>0,\quad \alpha_{j}\in (0,1]. \end{aligned}$$ [\[mittag\]]{#mittag label="mittag"}[@Mittag-Leffler-Sur] Defined by Magnus Gösta Mittag-Leffler, the classical Mittag-Leffler function can be expressed in the following manner: $$\begin{aligned} E_{\alpha}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+ 1)},\quad \mathcal{R}(\alpha)>0,\; z\in \mathbb{C}. \end{aligned}$$ Wiman provided the generalization of ([\[mittag\]](#mittag){reference-type="ref" reference="mittag"}) in [@Wiman] as follows: $$\begin{aligned} E_{\alpha, \beta}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+ \beta)},\quad \mathcal{R}(\alpha),\:\mathcal{R}(\beta)>0, \:z \in\mathbb{C}. \end{aligned}$$ Moreover, the MLTF is defined in the following manner $$\begin{aligned} e_{\alpha,\beta}(t,\lambda):=t^{\beta-1}E_{\alpha,\beta}(-\lambda t^{\alpha}),\quad \mathcal{R}(\alpha),\:\mathcal{R}(\beta),\;t,\;\lambda>0.\end{aligned}$$ [@Podlubny][\[podlem\]]{#podlem label="podlem"} The relation given below is valid subject to certain conditions that $\alpha<2$, $\beta$ is any real number, $\mu$ is a value satisfying $\pi\alpha/2<\mu<min\{\pi,\pi\alpha\}$, $z$ is a complex number in a manner that it satisfies $|z|\geq0$, $\mu\leq |arg(z)|\leq\pi$ and $C_{1}$ is a real constant. $$\begin{aligned} \big|E_{\alpha,\beta}(z)\big|\leq \frac{C_{1}}{1+|z|}. \end{aligned}$$ [@Ali-Aziz-Salman-FCAA1][\[AliMittagtypeLemma1\]]{#AliMittagtypeLemma1 label="AliMittagtypeLemma1"} The MLTF $e_{\alpha,\alpha+1}(t,\lambda)$ exhibits the subsequent property: $$\begin{aligned} e_{\alpha,\alpha+1}(t,\lambda)=\frac{1}{\lambda}\big(1-e_{\alpha,1}(t,\lambda)\big),\quad t, \lambda>0. \end{aligned}$$ [@Ali-Aziz-Salman-FCAA1][\[AliMittagtypeLemma1\'\]]{#AliMittagtypeLemma1' label="AliMittagtypeLemma1'"} The subsequent property holds for the MLTF $e_{\alpha,1}(T,\lambda)$, for any $\alpha$ within the interval $(0,1)$: $$\begin{aligned} \frac{1}{1-e_{\alpha,1}(T,\lambda)}\leq C_{2},\quad T,\: \lambda, \: C_{2}>0. \end{aligned}$$ # Operational calculus for Dzherbashian-Nersesian operator {#sec: Operational} The section showcases the construction of operational calculus for the Dzherbashian-Nersesian operator. This calculus is afterwards in the paper plays a pivotal role in investigating ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}) and ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}). Furthermore, under certain fixation of parameters, the formula of operational calculus for Dzherbashian-Nersesian operator $\mathcal{D}^{\varrho_{m}}_{t,0+}$ interpolate those of fractional derivatives namely Riemann-Liouville, Caputo and Hilfer. However, we start with providing some important operational relations from the existing literature. [@Luchko-Srivastava-CAMWA] A function $g$ with real or complex values is considered to be a member of the space $C_{\gamma}$, where $\gamma$ is a real number, only if a real number $p$ greater than $\gamma$ exists in a manner that $$\begin{aligned} g(z)=z^{p}g_{1}(z) ,\quad t>0, \end{aligned}$$ where $g_{1}\in C[0,\infty)$. The fact that $C_{\gamma}$ forms a vector space is evident. Moreover, the collection of $C_{\gamma}$ is arranged in an order based on inclusion given by the subsequent relation $$\begin{aligned} C_{\gamma}\subset C_{\delta},\quad \iff \gamma \ge \delta.\end{aligned}$$ [@Luchko-Srivastava-CAMWA] The RLI $J^{\alpha}_{t,0+}$, such that $\alpha>0$, is linear mapping of the space $C_{\gamma}$, $\gamma \geq -1$, into itself, i.e., $$\begin{aligned} J^{\alpha}_{t,0+}:C_{\gamma}\rightarrow C_{\gamma+\alpha}\subset C_{\gamma}. \end{aligned}$$ It is a well established fact that $J^{\alpha}_{t,0+}$, $\alpha>0$, possesses the subsequent convolution representation $C_{\gamma}$, $\gamma \geq -1$: $$\begin{aligned} (J^{\alpha}_{t,0+}y)(z)=(h_{\alpha}* y)(z), \quad h_{\alpha}(z):=\frac{z^{\alpha}-1}{\Gamma(\alpha)}, \quad y\in C_{\gamma}.\end{aligned}$$ where $*$ denotes the Laplace convolution defined as under: $$\begin{aligned} (g* f)(z)=\int_{0}^{z}g(z-t)f(t)dt,\quad z>0, \end{aligned}$$ [@Hilfer-Luchko-Tomovski] If $D^{\alpha}_{t,0+}y\in C_{-1}$ for all $\alpha$ between $n-1$ and $n$, then $y$ belongs to space $\Omega^{\alpha}_{-1}$ and has $\alpha\geq 0$. [@Luchko-Srivastava-CAMWA] The commutative ring $(C_{-1}, *, +)$ formed by the Laplace convolution $*$ and ordinary addition in the space $C_{-1}$ is free from zero divisors. By applying traditional Mikusiński operational calculus, one can extend this ring to the field of convolution quotients denoted as $M_{-1}$: $$\begin{aligned} M_{-1}:=C_{-1}\times (C_{-1}\cup \{0\})/\sim,\end{aligned}$$ with the relation of equivalence $(\sim)$ described as $$\begin{aligned} (f,g)\sim (f_{1},g_{1}) \equiv (f* g_{1})(t)=(g* f_{1})(t).\end{aligned}$$ The elements within the field $M_{1}$ can be viewed as convolution quotients $f/g$, for the sake of simplicity. However, the standard operations of addition and multiplication continue to hold their definitions in $M_{1}$ in the following manner: $$\begin{aligned} \frac{f}{g}+\frac{f_{1}}{g_{1}}:=\frac{f* g_{1}+g* f_{1}}{g* g_{1}},\label{add}\\ \frac{f}{g}\circ\frac{f_{1}}{g_{1}}:=\frac{f* f_{1}}{g* g_{1}}.\label{multi}\end{aligned}$$ [@Luchko-Srivastava-CAMWA] The commutative field $(M_{-1},.,+)$ can be constructed by defining the addition ([\[add\]](#add){reference-type="ref" reference="add"}) and multiplication ([\[multi\]](#multi){reference-type="ref" reference="multi"}) operations within $M_{1}$. Using a mapping that assigns $(\alpha > 0)$, we can embed the ring $C_{-1}$ in $M_{-1}$ as follows: $$\begin{aligned} f \longmapsto \frac{h_{\alpha}* f}{h_{\alpha}}. \end{aligned}$$ It is possible to define the scalar multiplication operation in $M_{-1}$ using the relation given below where the scalar $\lambda$ is taken from $\mathbb{R}$ or $\mathbb{C}$ $$\begin{aligned} \lambda\frac{f}{g}:=\frac{\lambda f}{g},\quad \frac{f}{g}\in M_{-1}.\end{aligned}$$ In $M_{-1}$, the definition of the operation involving a constant function and multiplication is as follows: $$\begin{aligned} \{\lambda\} \circ \frac{f}{g}=\frac{\lambda h_{\alpha+1}}{h_{\alpha}}* \frac{f}{g}=\{1\}* \frac{f}{g}.\end{aligned}$$ Specifically, we assume an element in $M_{-1}$ denoted by $I$. This element can be defined as the ratio of $h_{\alpha}$ to itself and serves as the element of identity in $M_{-1}$ when using the multiplication operation given as follows: $$\begin{aligned} I \circ \frac{f}{g}=\frac{h_{\alpha}* f}{h_{\alpha}* g}=\frac{f}{g}.\end{aligned}$$ In the conventional theory of generalized functions, the Dirac $\delta$-function is analogous to the element of identity $I$ in $M_{-1}$, as illustrated by the last formula. [@Luchko-Srivastava-CAMWA] In $M_{-1}$, the reciprocal to the element $h_{\alpha}$ is denoted as $S_{\alpha}$. It is recognized as the algebraic inverse of the RLI operator $J^{\alpha}_{t,0+}$, i.e., $$\begin{aligned} S_{\alpha}=\frac{I}{h_{\alpha}}\equiv \frac{h_{\alpha}}{h_{\alpha}* h_{\alpha}}\equiv \frac{h_{\alpha}}{2h_{\alpha}}, \end{aligned}$$ where $I$ represents the element of identity in $M_{-1}$ under the operation of multiplication, given by the ratio of $h_{\alpha}$ to itself. The representation of RLI $J^{\alpha}_{t,0+}$ involves multiplication (convolution) with the function $h_{\alpha}$, as mentioned in [@Dimovski]. This multiplication (convolution) takes place in the ring $C_{-1}$. By the virtue of embedding of $C_{-1}$ in $M_{-1}$ of convolution quotients, it can be expressed in alternative form as: $$\begin{aligned} (J^{\alpha}_{t,0+})y(x)=\frac{1}{S_{\alpha}}\circ y.\end{aligned}$$ The operational calculus for the Dzherbashian-Nersesian operator $\mathcal{D}^{\varrho_{m}}_{t,0+}$ is significantly aided by the following Theorem in solving the nonlocal integrodifferential equations: [@Luchko-2020-FCAA; @Luchko-2020-Mathematics] Let $y\in \Omega^{\varrho_{m}}_{-1}$, $0<\varrho_{m}\leq m$, $m \in \mathbb{Z^{+}}$. Then the RLI ([\[defRLI.oper\]](#defRLI.oper){reference-type="ref" reference="defRLI.oper"}) and the Dzherbashian-Nersesian operator ([\[DzhrNer-mathematical\]](#DzhrNer-mathematical){reference-type="ref" reference="DzhrNer-mathematical"}) are connected with each other by the following relation $$\begin{aligned} \label{RLIDzh.oper} (J^{\varrho_{m}}_{t,0+}\mathcal{D}^{\varrho_{m}}_{t,0+}y)(x)=y(x)-\sum_{k=0}^{m-1}\frac{x^{\varrho_{k}}}{\Gamma(\varrho_{k}+1)}(\mathcal{D}^{\varrho_{k}}_{t,0+}y)(x)\Big|_{x=0}, \quad x>0.\end{aligned}$$ [\[inverse.oper\]]{#inverse.oper label="inverse.oper"} Let function $y \in \Omega^{\varrho_{m}}_{-1}$, where $0<\varrho_{m}\leq m$, $m\in \mathbb{Z^{+}}$. In this case, the Dzherbashian-Nersesian operator $\mathcal{D}^{\varrho_{m}}_{t,0+}$ can be expressed as multiplication in $M_{-1}$ of convolution quotients as follows: $$\begin{aligned} \label{dzhrmain.oper} (\mathcal{D}^{\varrho_{m}}_{t,0+}y)(x)=S_{\varrho_{m}}\circ y(x)-\sum_{k=0}^{m-1}S_{\varrho_{m}-\varrho_{m-k}-1}\circ (\mathcal{D}^{\varrho_{k}}_{t,0+}y)(x)\Big|_{x=0}, \quad x>0. \end{aligned}$$ *Proof.* To obtain the formula ([\[dzhrmain.oper\]](#dzhrmain.oper){reference-type="ref" reference="dzhrmain.oper"}), initially, the embedding of the ring $C_{-1}$ in $M_{-1}$ is carried out. Subsequently, the relationship ([\[RLIDzh.oper\]](#RLIDzh.oper){reference-type="ref" reference="RLIDzh.oper"}) undergoes multiplication with the algebraic inverse of the RLI, denoted by $S_{\alpha}$. This gives the desired relation without any differences from formula ([\[dzhrmain.oper\]](#dzhrmain.oper){reference-type="ref" reference="dzhrmain.oper"}). ◻ On setting the parameters $\alpha_{0}=1+\alpha-m$ and $\alpha_{1}=\alpha_{2}=...=\alpha_{k}=1$ in Equation ([\[dzhrmain.oper\]](#dzhrmain.oper){reference-type="ref" reference="dzhrmain.oper"}), we obtain the relation of operational calculus for Riemann-Liouville derivative [@Luchko-Srivastava-CAMWA], i.e., $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+} y=S_{\alpha}\circ y-S_{\alpha}\circ y_{\alpha}, \end{aligned}$$ where $$y_{\alpha}(x):=\sum_{k=0}^{m-1}\frac{x^{k+\alpha-m}}{\Gamma(k+\alpha-m+1)}\lim_{x\rightarrow 0+}D^{k+\alpha-m}_{t,0+}y(x).$$ The Equation ([\[dzhrmain.oper\]](#dzhrmain.oper){reference-type="ref" reference="dzhrmain.oper"}) admits the relation of operational calculus for Caputo derivative [@Luchko-Gorenflo-1999]if we take parameters $\alpha_{1}=\alpha_{2}=...=\alpha_{k-1}=1$ and $\alpha_{k}=1+\alpha-k$, i.e., $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+}y=S_{\alpha}\circ y-S_{\alpha}\circ y_{\alpha}, \end{aligned}$$ where $$y_{\alpha}(x):=\sum_{k=0}^{m-1}\frac{x^{k}}{\Gamma(k+1)}\lim_{x\rightarrow 0+}y^{(k)}(x).$$ The relation for operational calculus for Hilfer derivative [@Hilfer-Luchko-Tomovski] is interpolated if we fix the parameters $\alpha_{0}=1-(m-\alpha)(1-\beta)$ and $\alpha_{1}=\alpha_{2}=...=\alpha_{k}=1$ in Equation ([\[dzhrmain.oper\]](#dzhrmain.oper){reference-type="ref" reference="dzhrmain.oper"}), i.e., $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+}y=S_{\alpha}\circ y-S_{\alpha}\circ y_{\alpha,\beta}, \end{aligned}$$ where $$y_{\alpha,\beta}(x):=\sum_{k=0}^{m-1}\frac{x^{k-m+\alpha-\beta\alpha+\beta m}}{\Gamma(k-m+\alpha-\beta\alpha+\beta m+1)}\lim_{x\rightarrow 0+}\frac{d^{k}}{dx^{k}}(J^{(1-\beta)(m-\alpha)}_{t,0+}y)(x).$$ [@Luchko-Gorenflo-1999][\[Luchko-Caputo-Lemma\]]{#Luchko-Caputo-Lemma label="Luchko-Caputo-Lemma"} For $\alpha_{i}$, $m_{i}$, $\sigma$, $\eta$ $>0$, we have $$\begin{aligned} \frac{S_{-\eta\sigma}}{1-\sum_{k=1}^{\infty}}=z^{\eta\sigma-1}E_{\alpha_{1}\sigma,...,\alpha_{n}\sigma},\eta\sigma(m_{1}z^{\alpha_{1}\eta},...,m_{n}z^{\alpha_{n}\eta}).\end{aligned}$$ # Forward and Backward Source Problems {#sec: FSP and BSP} In this section, we will utilize the operational calculus derived in the previous section to examine both the forward and backward problems ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}) and ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}), respectively. However, we begin our discussion by exploring the spectral problem associated with ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"}) and ([\[probbcs.oper\]](#probbcs.oper){reference-type="ref" reference="probbcs.oper"}), as it plays a crucial role in addressing the problems delineated by ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}) and ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}). ## Spectral Problem {#spectral.oper} Our approach to solving the BSP ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}) involves the Fourier method, also referred to as separation of variables. The nonlocal boundary conditions in this problem render the spectral non-self-adjoint. Below is the spectral problem for ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"}) contingent upon boundary conditions ([\[probbcs.oper\]](#probbcs.oper){reference-type="ref" reference="probbcs.oper"}) $$\label{probspec.oper} \begin{cases} X^{''}(x)=-\lambda X(x), \quad x\in (0,1),\\ X(1)=0,\quad X'(0)=X'(1). \end{cases}$$ The set of eigenfunctions denoted as $\{X_{0}, X_{1k}\}$ for ([\[probspec.oper\]](#probspec.oper){reference-type="ref" reference="probspec.oper"}) with respective eigenvalues $\lambda_{0}=0$, $\lambda_{k}=(2\pi k)^{2}$, along with the associated eigenfunction $X_{2k}$ (see [@Moiseev-1999]), can be expressed as follows $$\begin{aligned} \label{eigenfpec.oper} X_{0}(x)=2(1-x),\quad X_{1k}(x) =4(1-x)\cos (2\pi kx),\quad X_{2k}(x)=4\sin (2\pi kx).\end{aligned}$$ Under the spectral problem ([\[probspec.oper\]](#probspec.oper){reference-type="ref" reference="probspec.oper"}), the adjoint problem is presented as follows $$\label{probadj.oper} \begin{cases} Y^{''}(x)=-\lambda Y(x), \quad x\in (0,1),\\ Y(0)=0,\quad Y'(0)=Y'(1). \end{cases}$$ The set of eigenfunctions denoted as $\{Y_{0}, Y_{1k}\}$ for ([\[probadj.oper\]](#probadj.oper){reference-type="ref" reference="probadj.oper"}) with respective eigenvalues $\lambda_{0}=0$, $\lambda_{k}=(2\pi k)^{2}$, together with the associated eigenfunction $Y_{2k}$ (see [@Moiseev-1999]), can be expressed as follows $$\begin{aligned} \label{eigenfadj.oper} Y_{0}(x)=1,\quad Y_{1k}(x)=\cos(2\pi kx), \quad Y_{2k}(x)=x\sin(2\pi kx).\end{aligned}$$ [\[Riesz.oper\]]{#Riesz.oper label="Riesz.oper"}[@Ionkin-Moiseev] The systems of functions illustrated in ([\[eigenfpec.oper\]](#eigenfpec.oper){reference-type="ref" reference="eigenfpec.oper"}) and ([\[eigenfadj.oper\]](#eigenfadj.oper){reference-type="ref" reference="eigenfadj.oper"}) establish Riesz basis in $L^{2}(0,1)$. ## Forward Problem This subsection is dedicated to the study of forward problem described by ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}). We demonstrate, subject to specific assumptions given in Theorem [\[fp.oper\]](#fp.oper){reference-type="ref" reference="fp.oper"}, the existence of a unique classical solution to the forward problem. We will outline the outcomes related to the existence and uniqueness of solutions for the forward problem. The forward problem ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probic.oper\]](#probic.oper){reference-type="ref" reference="probic.oper"}) is linear. This allows us to express the solution using the following formulation: $$\begin{aligned} u(t,x)=v(t,x)+w(t,x),\end{aligned}$$ where $v(t,x)$ is the solution of following associated homogeneous problem, i.e., $$\label{probeq.homo.oper} \mathcal{D}^{\varrho_{m}}_{t,0+}v(t,x)=v_{xx}(t,x),\quad (t,x)\in (0,T)\times (0,1),$$ $$\label{probbcs.homo.oper} v(t,1)=0,\quad v_{x}(t,0)=v_{x}(t,1),\quad t\in (0,T),$$ $$\label{probic.homo.oper} J^{1-\alpha_{1}}_{t,0+}v(t,x)\Big|_{t=0}=\phi(x),\quad \alpha_{1}, x\in (0,1),$$ and $w(t,x)$ is the solution of following associated non-homogeneous problem, i.e., $$\label{probeq.nonhomo.oper} \mathcal{D}^{\varrho_{m}}_{t,0+}w(t,x)=w_{xx}(t,x)+f(x),\quad (t,x)\in (0,T)\times (0,1),$$ $$\label{probbcs.nonhomo.oper} w(t,1)=0,\quad w_{x}(t,0)=u_{x}(t,1),\quad t\in (0,T),$$ $$\label{probic.nonhomo.oper} J^{1-\alpha_{1}}_{t,0+}w(t,x)\Big|_{t=0}=0,\quad \alpha_{1}, x\in (0,1).$$ [\[fp.oper\]]{#fp.oper label="fp.oper"} For $\varrho_{m}\in (0,1)$ and $\phi(x) \in C^{2}(0,1)$ in a manner that it satisfies $\phi(1)=0$, $\phi'(0)=\phi'(1)$, the forward problem ([\[probeq.homo.oper\]](#probeq.homo.oper){reference-type="ref" reference="probeq.homo.oper"})-([\[probic.homo.oper\]](#probic.homo.oper){reference-type="ref" reference="probic.homo.oper"}) possesses a unique classical solution given by: $$\begin{aligned} v(t,x)=&\varphi_{0}\frac{t^{\alpha_{0}-1}}{\Gamma(\alpha_{0})} 2(1-x)+\sum_{k=1}^{\infty} \Big[\Big\{\varphi_{1k} e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\Big\}4(1-x)\cos (2\pi kx)\notag\\&+\Big\{e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{2k}+2\sqrt{\lambda_{k}}e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}-1}(t,\lambda_{k})*e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{1k}\Big\}4\sin (2\pi kx)\Big], \end{aligned}$$ where $\varphi_{i}:=\langle \varphi(x),Y_{i}(x)\rangle, \quad i \in \mathbb{Z}^{+}\cup \{0\}.$ *Proof.* The solution to the ([\[probeq.homo.oper\]](#probeq.homo.oper){reference-type="ref" reference="probeq.homo.oper"})-([\[probic.homo.oper\]](#probic.homo.oper){reference-type="ref" reference="probic.homo.oper"}) is succeeded by the establishment of the existence and uniqueness of the obtained solution. ### Construction of the Solution: {#construction-of-the-solution .unnumbered} Due to the Lemma [\[Riesz.oper\]](#Riesz.oper){reference-type="ref" reference="Riesz.oper"}, the set $\big\{X_{0}(x), X_{1k}(x), X_{2k}(x)\big\}$ forms a Riesz basis within the space $L^{2}(0,1)$, we are enabled to present the expression for $v(t,x)$ as follows: $$\begin{aligned} v(t,x)=v_{0}(t)X_{0}(x)+\sum_{k=1}^{\infty} \big(v_{1k}(t)X_{1k}(x)+v_{2k}(t)X_{2k}(x)\big),\label{u.homo.oper} \end{aligned}$$ Utilizing the Equation ([\[u.homo.oper\]](#u.homo.oper){reference-type="ref" reference="u.homo.oper"}) within the framework of the Equation ([\[probeq.homo.oper\]](#probeq.homo.oper){reference-type="ref" reference="probeq.homo.oper"}), and considering the fact the bi-orthogonality of the function of sets $\big\{X_{0}(x), X_{1k}(x), X_{2k}(x)\big\}$ and $\big\{Y_{0}(x), Y_{1k}(x), Y_{2k}(x)\big\}$ in $L^{2}(0,1)$, we proceed to derive the subsequent system of FDEs: $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+}v_{0}(t)&=0,\label{fdeu0.homo.oper}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}v_{1k}(t)+\lambda_{k}v_{1k}(t)&=0,\label{fdeu1k.homo.oper}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}v_{2k}(t)-2\sqrt{\lambda_{k}}v_{1k}(t)+\lambda_{k}v_{2k}(t)&=0.\label{fdeu2k.homo.oper} \end{aligned}$$ Using Theorem [\[inverse.oper\]](#inverse.oper){reference-type="ref" reference="inverse.oper"} in ([\[fdeu0.homo.oper\]](#fdeu0.homo.oper){reference-type="ref" reference="fdeu0.homo.oper"})-([\[fdeu2k.homo.oper\]](#fdeu2k.homo.oper){reference-type="ref" reference="fdeu2k.homo.oper"}), we have $$\begin{aligned} S_{\alpha_{0}+\alpha_{1}-1}\circ v_{0}(t)&=S_{\alpha_{1}}\circ \phi_{0},\label{invuo.homo.oper}\\ (S_{\alpha_{0}+\alpha_{1}-1}+\lambda_{k})\circ v_{1k}(t)&=S_{\alpha_{1}}\circ \phi_{1k},\label{invu1k.homo.oper}\\ (S_{\alpha_{0}+\alpha_{1}-1}+\lambda_{k})\circ v_{2k}(t)-2\sqrt{\lambda_{k}}\circ v_{1k}(t)&=S_{\alpha_{1}}\circ \phi_{2k}.\label{invu2k.homo.oper} \end{aligned}$$ Using Lemma [\[Luchko-Caputo-Lemma\]](#Luchko-Caputo-Lemma){reference-type="ref" reference="Luchko-Caputo-Lemma"} in ([\[invuo.homo.oper\]](#invuo.homo.oper){reference-type="ref" reference="invuo.homo.oper"})-([\[invu2k.homo.oper\]](#invu2k.homo.oper){reference-type="ref" reference="invu2k.homo.oper"}), we have $$\begin{aligned} v_{0}(t)=&\frac{t^{\alpha_{0}-1}}{\Gamma(\alpha_{0})}\varphi_{0},\label{spaceu0.homo.oper}\\ v_{1k}(t)=&e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{1k},\label{spaceu1k.homo.oper}\\ v_{2k}(t)=&e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{2k}+2\sqrt{\lambda_{k}}e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}-1}(t,\lambda_{k})*v_{1k}(t).\label{spaceu2k.homo.oper} \end{aligned}$$ Prior to discussing the existence of the solution for the IBVP described by Equations ([\[probeq.homo.oper\]](#probeq.homo.oper){reference-type="ref" reference="probeq.homo.oper"})-([\[probic.homo.oper\]](#probic.homo.oper){reference-type="ref" reference="probic.homo.oper"}), it is pertinent to state the following Lemma: [\[estlemma.oper\]]{#estlemma.oper label="estlemma.oper"} For $g\in C^{n}([0,1])$ in a manner that $g(1)=0$, $g^{(1)}(0)=g^{(1)}(1)$, $g(1)=0$ $g^{(3)}(0)=g^{(3)}(1)$ and we have the following relations: $$\begin{aligned} \big|g_{1k}\big|\leq \frac{1}{k^{n}}\|g^{(n)}\|, \quad \big|g_{2k}\big|\leq \frac{n+1}{k^{n}}\|g^{(n)}\|,\quad n=1,2,3,4. \end{aligned}$$ Representation $\|.\|$ denotes the norm in $L^{2}(0,1)$ which is defined by $$\begin{aligned} \|.\|:=\sqrt{\langle .,. \rangle} \end{aligned}$$ where $\langle .,. \rangle$ denotes the inner product which is given by $\langle f,g \rangle:=\int_{0}^{1}f(x)g(x)dx$. ### Existence of the Solution: {#existence-of-the-solution .unnumbered} To prove the existence of the classical solution, we will prove that the series corresponding to $t^{\alpha_{1}}u(t,x)$, $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}u(t,x)$, and $t^{\alpha_{1}}u_{xx}(t,x)$ represent continuous functions. Taking into account the CBSI, ([\[spaceu0.homo.oper\]](#spaceu0.homo.oper){reference-type="ref" reference="spaceu0.homo.oper"}) and the fact that $Y_{0}(x) = 1$, we obtain $$\begin{aligned} t^{\alpha_{1}}\big|v_{0}(t)\big|\leq \frac{t^{\alpha_{0}+\alpha_{1}-1}\|\phi\|}{\Gamma(\alpha_{0})}. \end{aligned}$$ On using Lemma [\[podlem\]](#podlem){reference-type="ref" reference="podlem"}, Equation ([\[spaceu1k.homo.oper\]](#spaceu1k.homo.oper){reference-type="ref" reference="spaceu1k.homo.oper"}), we have $$\begin{aligned} \big|v_{1k}(t)\big|\leq \frac{C_{1}\big|\phi_{1k}\big|}{\lambda_{k}t^{\alpha_{1}}}. \end{aligned}$$ Using the Lemma [\[estlemma.oper\]](#estlemma.oper){reference-type="ref" reference="estlemma.oper"}, CBSI and the property $\big|\frac{t}{T}\big|\leq 1$, we have $$\begin{aligned} \label{spaceu1k'.homo.oper} t^{\alpha_{1}}\big|v_{1k}(t)\big|\leq \frac{C_{1}\|\phi\|}{k^{2}}. \end{aligned}$$ Likewise due to the Lemma [\[podlem\]](#podlem){reference-type="ref" reference="podlem"} along with Equations ([\[spaceu2k.homo.oper\]](#spaceu2k.homo.oper){reference-type="ref" reference="spaceu2k.homo.oper"}) and ([\[spaceu1k\'.homo.oper\]](#spaceu1k'.homo.oper){reference-type="ref" reference="spaceu1k'.homo.oper"}), we have $$\begin{aligned} t^{\alpha_{1}}\big|v_{2k}(t)\big|\leq \frac{C_{1}\|\phi\|}{k^{2}}+\frac{2C^{2}_{1}\|\phi\|}{k^{3}}.\label{spaceu2k'.homo.oper} \end{aligned}$$ We still need to demonstrate the uniform convergence of the series associated with $\mathcal{D}^{\varrho_{m}}_{t,0+}v(t,x)$ within the interval $[\varepsilon, T]$. To establish this we employ the following lemma from [@Ali-Aziz-Salman-FCAA1]: [\[DzhrSamkolemma\]]{#DzhrSamkolemma label="DzhrSamkolemma"} Suppose that for each $i\in\mathbb{Z^{+}}$, we have a set $g_{i}$ of functions defined on $(0,b]$, that adhere to the subsequent conditions: 1. derivatives $D^{\alpha_{0}}_{t,0+}g_{i}(t)$, $D^{\alpha_{1}}_{t,0+}D^{\alpha_{0}}_{t,0+}g_{i}(t)$,\..., $D^{\alpha_{m-1}}_{t,0+}...D^{\alpha_{0}}_{t,0+}g_{i}(t)$ for $i\in \mathbb{Z^{+}}, t\in (0,b]$ exist, 2. the series $\sum_{i=1}^{\infty}g_{i}(t)$ and $\sum_{i=1}^{\infty}D^{\alpha_{0}}_{t,0+}g_{i}(t)$, $\sum_{i=1}^{\infty}D^{\alpha_{1}}_{t,0+}D^{\alpha_{0}}_{t,0+}g_{i}(t)$,\..., $\sum_{i=1}^{\infty}D^{\alpha_{m-1}}_{t,0+}... D^{\alpha_{1}}_{t,0+}D^{\alpha_{0}}_{t,0+}g_{i}(t)$ are uniformly convergent on the interval $[a+\varepsilon,b]$ for any $\varepsilon>0$. Then $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+} \displaystyle\sum_{i=1}^{\infty}g_{i}(t)=\displaystyle\sum_{i=1}^{\infty}\mathcal{D}^{\varrho_{m}}_{t,0+}g_{i}(t). \end{aligned}$$ Therefore, to ensure the continuity of $\mathcal{D}^{\varrho_{m}}_{t,0+}v(t,x)$, it is necessary for the series representations of $v(t,x)$ and $D^{\alpha_{0}}v(t,x)$ to exhibit uniform convergence. In view of ([\[spaceu1k\'.homo.oper\]](#spaceu1k'.homo.oper){reference-type="ref" reference="spaceu1k'.homo.oper"}) and ([\[spaceu2k\'.homo.oper\]](#spaceu2k'.homo.oper){reference-type="ref" reference="spaceu2k'.homo.oper"}), $v(t,x)$ is already continuous. Therefore, it suffices to prove the continuity of $D^{\alpha_{0}}_{t,0+}v(t,x)$. One can observe that $D^{\alpha_{0}}_{t,0+}v(t,x)$ is bounded from above by a series that converges. Therefore, using the WMT, it can be concluded that $D^{\alpha}_{t,0+}v(t,x)$ denotes a function which is continuous. Moreover, from Equations ([\[fdeu0.homo.oper\]](#fdeu0.homo.oper){reference-type="ref" reference="fdeu0.homo.oper"})-([\[fdeu2k.homo.oper\]](#fdeu2k.homo.oper){reference-type="ref" reference="fdeu2k.homo.oper"}), we have the following estimates $$\begin{aligned} t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}v_{0}(t)\big|\leq& \frac{\Gamma(\alpha_{0}+\alpha_{1})\|\phi\|}{\Gamma(\alpha_{0})},\label{estsumu0.homo.oper}\\ %\end{align} %\begin{align} t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}v_{1k}(t)\big|\leq& \sum_{k=1}^{\infty}\frac{C_{1}\|\phi^{(2)}\|}{k^{2}},\label{estsumu1k.homo.oper}\\ %\end{align} %\begin{align} t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}v_{2k}(t)\big|\leq& \sum_{k=1}^{\infty}\Big(\frac{C_{1}\|\phi^{(2)}\|}{k^{2}}+\frac{2C_{1}\|\phi^{(2)}\|}{k^{3}}\Big).\label{estsumu2k.homo.oper} \end{aligned}$$ By virtue of the Lemma [\[DzhrSamkolemma\]](#DzhrSamkolemma){reference-type="ref" reference="DzhrSamkolemma"} and ([\[estsumu0.homo.oper\]](#estsumu0.homo.oper){reference-type="ref" reference="estsumu0.homo.oper"})-([\[estsumu2k.homo.oper\]](#estsumu2k.homo.oper){reference-type="ref" reference="estsumu2k.homo.oper"}), we notice that $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+}v(t,x)$ has a finite upper bound. Therefore, by WMT $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+}v(t,x)$ converges. Similarly for $v_{xx}(t,x)$, we have $$\begin{aligned} t^{\alpha_{1}}\big|v_{xx}(t,x)\big|\leq 48\sum_{k=1}^{\infty}\frac{C_{1}\|\phi\|}{k^{2}}, \end{aligned}$$ which on employing WMT denotes a continuous function. ### Uniqueness of the Solution: {#uniqueness-of-the-solution .unnumbered} Suppose $\bar{v}(t,x)=v_{1}(t,x)-v_{2}(t,x)$, where $v_{1}(t,x)$ and $v_{2}(t,x)$ represent two solutions of IBVP defined by ([\[probeq.homo.oper\]](#probeq.homo.oper){reference-type="ref" reference="probeq.homo.oper"})-([\[probic.homo.oper\]](#probic.homo.oper){reference-type="ref" reference="probic.homo.oper"}). The function $v(t,x)$ adheres to the following system: $$\label{probeq.homo.oper.uniq} \mathcal{D}^{\varrho_{m}}_{t,0+}\bar{v}(t,x)=\bar{v}_{xx}(t,x),\quad (t,x)\in (0,T)\times (0,1),$$ $$\label{probbcs.homo.oper.uniq} \bar{v}(t,1)=0,\quad \bar{v}_{x}(t,0)=\bar{v}_{x}(t,1),\quad t\in (0,T),$$ $$\label{probic.homo.oper.uniq} J^{1-\alpha_{1}}_{t,0+}\bar{v}(t,x)\Big|_{t=0}=0,\quad \alpha_{1}, x\in (0,1),$$ Taking into account the bi-orthogonality property within sets of functions $\{X_{0}(x), X_{1k}(x), X_{2k}(x)\}$ and $\{Y_{0}(x), Y_{1k}(x), Y_{2k}(x)\}$ in $L^{2}(0,1)$, we proceed to formulate the following system of FDEs: $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+}\bar{v}_{0}(t)&=0,\label{fdeu0.homo.oper.uniq}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}\bar{v}_{1k}(t)+\lambda_{k}\bar{v}_{1k}(t)&=0,\label{fdeu1k.homo.oper.uniq}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}\bar{v}_{2k}(t)-2\sqrt{\lambda_{k}}\bar{v}_{1k}(t)+\lambda_{k}\bar{v}_{2k}(t)&=0.\label{fdeu2k.homo.oper.uniq} \end{aligned}$$ In the view of ([\[probeq.homo.oper.uniq\]](#probeq.homo.oper.uniq){reference-type="ref" reference="probeq.homo.oper.uniq"}), we obtain $\bar{v}_{0}(t)=0,\quad \bar{v}_{1k}(t)=0, \quad \bar{v}_{2k}(t)=0$. As a result, we have $$\bar{v}_{0}(t)=0$$. The solution of the problem outlined in ([\[probeq.nonhomo.oper\]](#probeq.nonhomo.oper){reference-type="ref" reference="probeq.nonhomo.oper"})--([\[probic.nonhomo.oper\]](#probic.nonhomo.oper){reference-type="ref" reference="probic.nonhomo.oper"}) can be obtained through the application of Duhamel's principle as referenced in [@Umarov-2012; @Umarov-Saidamatov-2007] and is given by: $$\begin{aligned} w(t,x)=\int_{0}^{t} u^{w}(t,\tau,x)d\tau, \end{aligned}$$ where $u^{w}(t,\tau,x)$ is the solution of the subsequent system $$\label{probeq.nonhomo.duhamel.oper} \mathcal{D}^{\varrho_{m}}_{t,0+}u^{w}(t,\tau,x)=u^{w}_{xx}(t,\tau,x),\quad (t,x)\in (0,T)\times (0,1),$$ $$\label{probbcs.nonhomo.duhamel.oper} u^{w}(t,\tau,1)=0,\quad u^{w}_{x}(t,\tau,0)=u^{w}_{x}(t,\tau,1),\quad t\in (0,T),$$ $$\label{probic.nonhomo.duhamel.oper} J^{1-\alpha_{1}}_{t,0+}u^{w}(t,\tau,x)\Big|_{t=\tau}=D^{2-\alpha_{0}-\alpha_{1}}_{t,0+}f(\tau,x),\quad \alpha_{0}, \alpha_{1}, x\in (0,1),$$ Applying the similar approach, we derive the solution for initial boundary value problem described in ([\[probeq.nonhomo.duhamel.oper\]](#probeq.nonhomo.duhamel.oper){reference-type="ref" reference="probeq.nonhomo.duhamel.oper"})-([\[probic.nonhomo.duhamel.oper\]](#probic.nonhomo.duhamel.oper){reference-type="ref" reference="probic.nonhomo.duhamel.oper"}), resulting in: $$\begin{aligned} w(t,x)=&\frac{t^{\alpha_{0}-1}}{\Gamma(\alpha_{0})}\varphi_{0}X_{0}(x)D^{2-\alpha_{0}-\alpha_{1}}_{t,0+}f_{0}(\tau)+\sum_{k=1}^{\infty} \Big(e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{1k}X_{1k}(x)D^{2-\alpha_{0}-\alpha_{1}}_{t,0+}f_{1k}(\tau)\\ &+\big(e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{2k}D^{2-\alpha_{0}-\alpha_{1}}_{t,0+}f_{2k}(\tau)+2\sqrt{\lambda_{k}}e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}-1}(t,\lambda_{k})*v_{1k}(t)D^{2-\alpha_{0}-\alpha_{1}}_{t,0+}f_{2k}(\tau)\big)X_{2k}(x)\Big). \end{aligned}$$ ◻ ## Backward Problem Operational calculus constructed in the previous subsection will be used to solve the space dependent BSP ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}). We begin by establishing the theorem addressing the BSP represented by ([\[probeq.nonhomo.oper\]](#probeq.nonhomo.oper){reference-type="ref" reference="probeq.nonhomo.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}). For $\varrho_{m}\in (0,1)$ and\ $1$. $\phi(x) \in C^{2}(0,1)$ in a manner that it satisfies $\phi(1)=0$, $\phi'(0)=\phi'(1)$,\ $2$. $\psi(x) \in C^{4}(0,1)$ in a manner that it satisfies $\psi(1)=0$, $\psi'(0)=\psi'(1)$, $\psi''(1)=0$, $\psi'''(0)=0=\psi'''(1)$,\ the backward problem ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}) possesses a unique regular solution given by: *Proof.* The solution to the BSP ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}) is constructed followed by the demonstration of the existence and uniqueness of the obtained solution. ### Construction of the Solution: {#construction-of-the-solution-1 .unnumbered} Due to the establishment in Lemma [\[Riesz.oper\]](#Riesz.oper){reference-type="ref" reference="Riesz.oper"} that the collection $\{X_{0}(x), X_{1k}(x), X_{2k}(x)\}$ makes up the Riesz basis for the space $L^{2}(0,1)$, it becomes possible for us to express $u(t,x)$ and $f(x)$ in the subsequent manner $$\begin{aligned} u(t,x)&=u_{0}(t)X_{0}(x)+\sum_{k=1}^{\infty} \big(u_{1k}(t)X_{1k}(x)+u_{2k}(t)X_{2k}(x)\big),\label{u.oper}\\ f(x)&=f_{0}X_{0}(x)+\sum_{k=1}^{\infty} \big(f_{1k}X_{1k}(x)+f_{2k}X_{2k}(x)\big).\label{f.oper} \end{aligned}$$ Making use of Equations ([\[u.oper\]](#u.oper){reference-type="ref" reference="u.oper"}) and ([\[f.oper\]](#f.oper){reference-type="ref" reference="f.oper"}) in Equation ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"}) and owing to the fact that the sets $\big\{X_{0}(x), X_{1k}(x), X_{2k}(x)\big\}$ and $\big\{Y_{0}(x), Y_{1k}(x), Y_{2k}(x)\big\}$ form a bi-orthogonal system of functions for the space $L^{2}(0,1)$, the subsequent set of fractional differential equations is derived $$\begin{aligned} \mathcal{D}^{\varrho_{m}}_{t,0+}u_{0}(t)&=f_{0},\label{fdeu0.oper}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}u_{1k}(t)+\lambda_{k}u_{1k}(t)&=f_{1k},\label{fdeu1k.oper}\\ \mathcal{D}^{\varrho_{m}}_{t,0+}u_{2k}(t)-2\sqrt{\lambda_{k}}u_{1k}(t)+\lambda_{k}u_{2k}(t)&=f_{2k}.\label{fdeu2k.oper} \end{aligned}$$ Using Theorem [\[inverse.oper\]](#inverse.oper){reference-type="ref" reference="inverse.oper"} in ([\[fdeu0.oper\]](#fdeu0.oper){reference-type="ref" reference="fdeu0.oper"})-([\[fdeu2k.oper\]](#fdeu2k.oper){reference-type="ref" reference="fdeu2k.oper"}), we have $$\begin{aligned} S_{\alpha_{0}+\alpha_{1}-1}\circ u_{0}(t)-S_{\alpha_{1}}\circ \phi_{0}&=f_{0},\label{invuo.oper}\\ (S_{\alpha_{0}+\alpha_{1}-1}+\lambda_{k})\circ u_{1k}(t)-S_{\alpha_{1}}\circ \phi_{1k}&=f_{1k},\label{invu1k.oper}\\ (S_{\alpha_{0}+\alpha_{1}-1}+\lambda_{k})\circ u_{2k}(t)-2\sqrt{\lambda_{k}}\circ u_{1k}(t)-S_{\alpha_{1}}\circ \phi_{2k}&=f_{2k}.\label{invu2k.oper} \end{aligned}$$ Using Lemma [\[Luchko-Caputo-Lemma\]](#Luchko-Caputo-Lemma){reference-type="ref" reference="Luchko-Caputo-Lemma"} in ([\[invuo.oper\]](#invuo.oper){reference-type="ref" reference="invuo.oper"})-([\[invu2k.oper\]](#invu2k.oper){reference-type="ref" reference="invu2k.oper"}), we can write $$\begin{aligned} u_{0}(t)=&\frac{t^{\alpha_{0}-1}}{\Gamma(\alpha_{0})}\varphi_{0}+\frac{t^{\alpha_{0}+\alpha_{1}-1}}{\Gamma(\alpha_{0}+\alpha_{1})}f_{0},\label{spaceu0.oper}\\ u_{1k}(t)=&e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{1k}+e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}}(t,\lambda_{k})f_{1k}\;,\label{spaceu1k.oper}\\ u_{2k}(t)=&e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(t,\lambda_{k})\varphi_{2k}+2\sqrt{\lambda_{k}}e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}-1}(t,\lambda_{k})*u_{1k}(t)x+e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}}(t,\lambda_{k})f_{2k}.\label{spaceu2k.oper} \end{aligned}$$ Taking into consideration the overdetermined condition $u(T,x)=\psi(x)$, $$\begin{aligned} f_{0}&=\frac{\Gamma(\alpha_{0}+\alpha_{1})}{T^{\alpha_{0}+\alpha_{1}-1}}\Big(\psi_{0}-\varphi_{0}\frac{T^{\alpha_{0}-1}}{\Gamma(\alpha_{0})}\Big),\label{spacef0.oper}\\ f_{2k-1}&= \frac{\psi_{2k-1}-\varphi_{2k-1}\;e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(T,\lambda_{k})}{e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}}(T,\lambda_{k})},\label{spacef1k.oper}\\ f_{2k}&= \frac{\psi_{2k}-2\sqrt{\lambda_{k}}e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}-1}(T,\lambda_{k})*u_{1k}(T)-\varphi_{2k}\;e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}}(T,\lambda_{k})}{e_{\alpha_{0}+\alpha_{1}-1,\alpha_{0}+\alpha_{1}}(T,\lambda_{k})}.\label{spacef2k.oper} \end{aligned}$$ ### Existence of the Solution: {#existence-of-the-solution-1 .unnumbered} To establish the existence of the regular solution, we will demonstrate the continuity of the series representing $t^{\alpha_{1}}u(t,x)$, $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}u(t,x)$, $t^{\alpha_{1}}u_{xx}(t,x)$ and $f(x)$. On using Equation ([\[spacef0.oper\]](#spacef0.oper){reference-type="ref" reference="spacef0.oper"}), the Cauchy-Bunyakovsky-Schwarz inequality (CBSI) and $Y_{0}(x)=1$, we have $$\begin{aligned} \big|f_{0}\big|\leq \frac{\Gamma (\alpha_{0}+\alpha_{1})}{T^{\alpha_{0}+\alpha_{1}-1}}\Big(\|\psi\|+\frac{T^{\alpha_{0}-1}}{\Gamma(\alpha_{0})}\|\phi\|\Big),\label{estf0.oper} \end{aligned}$$ Due to Lemmas [\[podlem\]](#podlem){reference-type="ref" reference="podlem"}, [\[AliMittagtypeLemma1\]](#AliMittagtypeLemma1){reference-type="ref" reference="AliMittagtypeLemma1"}, [\[AliMittagtypeLemma1\'\]](#AliMittagtypeLemma1'){reference-type="ref" reference="AliMittagtypeLemma1'"} and Equation ([\[spacef1k.oper\]](#spacef1k.oper){reference-type="ref" reference="spacef1k.oper"}), we obtain $$\begin{aligned} \big|f_{1k}\big|\leq \frac{C_{1}C_{2}|\phi_{1k}|}{T^{\alpha_{1}}}+C_{2}\lambda_{k}\big|\psi_{1k}\big|. \end{aligned}$$ On using the Lemma [\[estlemma.oper\]](#estlemma.oper){reference-type="ref" reference="estlemma.oper"}, we get $$\begin{aligned} \big|f_{1k}\big|\leq \frac{C_{1}C_{2}\|\phi^{(2)}\|}{k^{2}T^{\alpha_{1}}}+\frac{C_{2}\|\psi^{(4)}\|}{k^{2}}.\label{estf1k.oper} \end{aligned}$$ Likewise, from ([\[spacef2k.oper\]](#spacef2k.oper){reference-type="ref" reference="spacef2k.oper"}), we have $$\begin{aligned} \big|f_{2k}\big|\leq \frac{3C_{1}C_{2}\|\phi^{(2)}\|}{k^{2}T^{\alpha_{1}}}+\frac{5C_{2}\|\psi^{(4)}\|}{k^{2}}+\frac{2C_{1}\|\psi^{(3)}\|}{k^{2}}.\label{estf2k.oper} \end{aligned}$$ By means of Equations ([\[f.oper\]](#f.oper){reference-type="ref" reference="f.oper"}) and ([\[estf0.oper\]](#estf0.oper){reference-type="ref" reference="estf0.oper"}) -- ([\[estf2k.oper\]](#estf2k.oper){reference-type="ref" reference="estf2k.oper"}), sum of the series dominating $f(x)$ has a finite upper bound. Hence, according to the WMT, $f(x)$ denotes a function that is continuous. Considering the CBSI along with ([\[estf0.oper\]](#estf0.oper){reference-type="ref" reference="estf0.oper"}) and taking into account $Y_{0}(x) = 1$, we obtain $$\begin{aligned} \label{spaceu0'.oper} t^{\alpha_{1}}\big|u_{0}(t)\big|\leq \frac{t^{\alpha_{0}+\alpha_{1}-1}\|\phi\|}{\Gamma(\alpha_{0})}+t^{\alpha_{1}}\|\psi\|+\frac{t^{\alpha_{1}}T^{\alpha_{0}-1}\|\phi\|}{\Gamma(\alpha_{0})}. \end{aligned}$$ On using Lemma [\[podlem\]](#podlem){reference-type="ref" reference="podlem"}, Equation ([\[spaceu1k.oper\]](#spaceu1k.oper){reference-type="ref" reference="spaceu1k.oper"}) and ([\[estf1k.oper\]](#estf1k.oper){reference-type="ref" reference="estf1k.oper"}), we have $$\begin{aligned} \big|u_{1k}(t)\big|\leq \frac{C_{1}\big|\phi_{1k}\big|}{\lambda_{k}t^{\alpha_{1}}}+\frac{C_{1}\big|f_{1k}\big|}{\lambda_{k}}. \end{aligned}$$ Using the Lemma [\[estlemma.oper\]](#estlemma.oper){reference-type="ref" reference="estlemma.oper"}, CBSI and the property $\big|\frac{t}{T}\big|\leq 1$, we have $$\begin{aligned} \label{spaceu1k'.oper} t^{\alpha_{1}}\big|u_{1k}(t)\big|\leq \frac{C_{1}\|\phi\|}{k^{2}}+\frac{C_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{4}}+\frac{C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{4}}. \end{aligned}$$ Likewise due to the Lemma [\[podlem\]](#podlem){reference-type="ref" reference="podlem"}, Equation ([\[spaceu2k.oper\]](#spaceu2k.oper){reference-type="ref" reference="spaceu2k.oper"}), ([\[estf2k.oper\]](#estf2k.oper){reference-type="ref" reference="estf2k.oper"}) and ([\[spaceu1k\'.oper\]](#spaceu1k'.oper){reference-type="ref" reference="spaceu1k'.oper"}), we have $$\begin{aligned} t^{\alpha_{1}}\big|u_{2k}(t)\big|\leq& \frac{C_{1}\|\phi\|}{k^{2}}+\frac{2C^{2}_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{5}}+\frac{2C^{2}_{1}C_{2}\|\phi\|}{k^{5}}+\frac{5C_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{4}}+\frac{3C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{4}}+\frac{2C^{2}_{1}C_{2}t^{\alpha_{1}}\|\psi^{(3)}\|}{k^{4}}.\label{spaceu2k'.oper} \end{aligned}$$ By means of ([\[spaceu0\'.oper\]](#spaceu0'.oper){reference-type="ref" reference="spaceu0'.oper"}), ([\[spaceu1k\'.oper\]](#spaceu1k'.oper){reference-type="ref" reference="spaceu1k'.oper"}), and [\[spaceu2k\'.oper\]](#spaceu2k'.oper){reference-type="ref" reference="spaceu2k'.oper"}) alongside the application of WMT, it is evident that $t^{\alpha_{1}}u(t,x)$ is a continuous function. One can observe that $D^{\alpha_{0}}_{t,0+}u(t,x)$ is bounded from above by a series that converges. Therefore, using the WMT, it can be concluded that $D^{\alpha}_{t,0+}u(t,x)$ denotes a function which is continuous. Moreover, from Equations ([\[fdeu0.oper\]](#fdeu0.oper){reference-type="ref" reference="fdeu0.oper"})-([\[fdeu2k.oper\]](#fdeu2k.oper){reference-type="ref" reference="fdeu2k.oper"}), we have the following estimates $$\begin{aligned} t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}u_{0}(t)\big|\leq& \frac{\Gamma(\alpha_{0}+\alpha_{1})\|\phi\|}{\Gamma(\alpha_{0})}+\frac{\Gamma(\alpha_{0}+\alpha_{1})\|\psi\|}{T^{\alpha_{0}-1}},\label{estsumu0.oper}\\ %\end{align} %\begin{align} t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}u_{1k}(t)\big|\leq& \sum_{k=1}^{\infty}\Big(\frac{C_{1}\|\phi^{(2)}\|}{k^{2}}+\frac{C_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{2}}+\frac{C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{2}}+\frac{C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{2}}+\frac{C_{1}C_{2}\|\phi^{(2)}\|}{k^{2}}\Big),\label{estsumu1k.oper}\\ t^{\alpha_{1}}\big|\mathcal{D}^{\varrho_{m}}_{t,0+}u_{2k}(t)\big|\leq& \sum_{k=1}^{\infty}\Big(\frac{3C_{1}\|\phi^{(2)}\|}{k^{2}}+\frac{4C^{2}_{1}\|\phi^{(1)}\|}{k^{2}}+\frac{2C^{2}_{1}C_{2}\|\psi^{(4)}\|}{k^{3}}+\frac{2C^{3}_{1}C_{2}\|\phi^{(2)}\|}{k^{3}}+\frac{5C_{1}C_{2}\|\psi^{(4)}\|}{k^{2}}+\frac{3C^{2}_{1}C_{2}\|\phi\|}{k^{2}}\notag\\ &+\frac{2C^{2}_{1}C_{2}\|\psi^{(3)}\|}{k^{3}}+\frac{2C_{1}\|\phi^{(1)}\|}{k^{2}}+\frac{2C^{2}_{1}C_{2}\|\psi^{(4)}\|}{k^{2}}+\frac{2C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{2}}+\frac{5C_{2}\|\psi^{(4)}\|}{k^{2}}+\frac{3C_{1}C_{2}\|\phi^{(2)}\|}{k^{2}}\notag\\ &+\frac{2C_{1}C_{2}\|\psi^{(2)}\|}{k^{2}}\Big).\label{estsumu2k.oper} \end{aligned}$$ By virtue of the Lemma [\[DzhrSamkolemma\]](#DzhrSamkolemma){reference-type="ref" reference="DzhrSamkolemma"} i.e. $\mathcal{D}^{\varrho_{m}}_{t,0+}\sum_{k=1}^{\infty}g_{i}(t)=\sum_{k=1}^{\infty}\mathcal{D}^{\varrho_{m}}_{t,0+}g_{i}(t)$ and ([\[estsumu0.oper\]](#estsumu0.oper){reference-type="ref" reference="estsumu0.oper"})-([\[estsumu2k.oper\]](#estsumu2k.oper){reference-type="ref" reference="estsumu2k.oper"}), we notice that $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+}(t,x)$ is bounded above by convergent series. Therefore, by WMT $t^{\alpha_{1}}\mathcal{D}^{\varrho_{m}}_{t,0+}(t,x)$ represents a continuous function. Similarly for $u_{xx}(t,x)$, we have $$\begin{aligned} \big|u_{xx}(t,x)\big|\leq& \sum_{k=1}^{\infty}\Bigg[\Bigg(\frac{C_{1}\|\phi\|}{k^{2}}+\frac{C_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{4}}+\frac{C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{4}}\Big)+16\Big(\frac{C_{1}\|\phi\|}{k^{2}}+\frac{2C^{2}_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{5}}+\frac{2C^{2}_{1}C_{2}\|\phi\|}{k^{5}}\notag\\ &+\frac{5C_{1}C_{2}t^{\alpha_{1}}\|\psi^{(4)}\|}{k^{4}}+\frac{3C^{2}_{1}C_{2}\|\phi^{(2)}\|}{k^{4}}+\frac{2C^{2}_{1}C_{2}t^{\alpha_{1}}\|\psi^{(3)}\|}{k^{4}}\Bigg)\Bigg], \end{aligned}$$ which on employing WMT denotes a continuous function. ### Uniqueness of the Solution: {#uniqueness-of-the-solution-1 .unnumbered} In order to demonstrate the uniqueness of $\{u(t,x), f(x)\}$, we consider two separate solutions of the system ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}), assuming they are distinct. Following the similar approach as outlined in Theorem [\[fp.oper\]](#fp.oper){reference-type="ref" reference="fp.oper"}, we reach the required result. ◻ The recovery of the results of Kirane et al. [@Kirane-Salman-APC] and Furati et al. [@Furati-Tyiola-Kirane] can be achieved from BSP ([\[probeq.oper\]](#probeq.oper){reference-type="ref" reference="probeq.oper"})-([\[probover.oper\]](#probover.oper){reference-type="ref" reference="probover.oper"}) by utilizing $\alpha_{0}=1$, $\alpha_{1}=\alpha$ and $\alpha_{0}=1-(1-\alpha)(1-\beta)$, $\alpha_{1}=1-\beta(1-\alpha)$, where $\alpha_{0},\: \alpha_{1} \in (0,1)$, respectively. # Concluding Remarks {#sec: Conclusion} In conclusion, this article has provided a comprehensive exploration of Mikusiński's type operational calculus for the Dzherbashian-Nersesian operator and its application in solving forward and backward problems. Through the utilization of this operational calculus, we derive solutions for FDEs which is expressed as infinite series of Mittag-Leffler type functions. The significance of this article lies in its potential to address complex problems in various fields, including mathematical physics and engineering, where the FDEs are encountered. Furthermore, by imposing certain conditions of the consistency and regularity of the provided data, we employed the WMT to establish the existence of a regular solution. The rigorous analysis adds a layer of confidence to the applicability of the derived solution in the practical scenarios. In summary, the operational calculus for the Dzherbashian-Nersesian operator, coupled with the use of Mittag-Leffler function and the WMT, offers a powerful toolset for solving challenging problems of fractional calculus and it's potential applications, paving the way for future advancements. 00 Mikusiński, J.: Operational Calculus. Pergamon Press, London (1959). 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--- abstract: | We consider invertible linear maps with additive spherical bounded noise. We show that minimal attractors of such random dynamical systems are unique, strictly convex and have a continuously differentiable boundary. Moreover, we present an auxiliary finite-dimensional deterministic *boundary map* for which the unit normal bundle of this boundary is globally attracting. address: - Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom - International Research Center for Neurointelligence (IRCN), The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-0033 Japan - Centre for Applied Mathematics and Bioinformatics, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, Halwally 32093, Kuwait - Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom - International Research Center for Neurointelligence (IRCN), The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-0033 Japan author: - Jeroen S.W. Lamb - Martin Rasmussen - Wei Hao Tey bibliography: - reference.bib title: Attractors of linear maps with bounded noise --- [^1] [^2] [^3] # Introduction and main results The development of dynamical systems theory is one of the scientific revolutions of the 20th century, and many important insights from this field are now embedded within computational methodologies in many branches of science and engineering as well as at the heart of deep and abstract mathematics. During the last few decades, the importance of noise and uncertainty has become evident in real-world applications of dynamical systems, but the corresponding random dynamical systems theory is still only in its early stages of development [@arnold98]. In random dynamical systems with bounded noise, trajectories are typically attracted by bounded invariant sets, from which escape is not possible. These attractors are important for the understanding of the long-term dynamical behaviour. In particular, they support ergodic stationary measures which describe statistical features of the random dynamics [@zmarrou2007bifurcations]. It has been observed [@Lamb2023numerical] that nonlinear random dynamical systems with bounded noise may have multiple coexisting attractors with complicated shapes whose boundaries may have points of non-differentiability, yielding the general study of such attractors challenging [@Kourliouros2023; @lamb2020boundaries; @lamb2021boundaries]. In this paper, we study geometric properties of minimal invariant sets for the elementary class of random dynamical systems consisting of invertible linear maps with spherical bounded noise. We choose this setting since linear random dynamical systems provide insight into attractors when the underlying dynamics is approximately linear, like when perturbing a nonlinear system near a periodic sink with small bounded noise. We study iterations in $\mathbb{R}^m$ of the form $$\label{eq:rds} x_{i+1} = L(x_i) + \xi_i\,,$$ where $i\in\mathbb{N}_0$, $x_i\in\mathbb{R}^m$ and $L\in GL(m,\mathbb{R})$ is an invertible linear map. $\{\xi_i\}_{i\in\mathbb N_0}$ is a sequence of i.i.d. random variables that are supported on a ball of radius $\varepsilon$ centered at $0$. We say that a compact subset $A\subset\mathbb{R}^m$ is an attractor of the random dynamical system [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}, if it is attracting and minimally forward invariant.[^4] $A$ is minimally forward invariant if $x_i\in A$ implies that $x_{i+1}\in A$ and there does not exist a proper subset of $A$ that is also forward invariant. $A$ is attracting if there exists a neighbourhood $N$ of $A$ such that all forward orbits starting in $N$ tend to $A$, in the sense that $\lim_{i\to\infty}\inf_{a\in A} d(x_{i},a)=0$ for all $x_0\in N$, with $d$ denoting the Euclidean metric in $\mathbb{R}^m$. The long term behaviour of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"} ($i\to\infty$) is characterized by its dynamics on attractors. In this paper, we establish that [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"} has an attractor if and only if $L$ is eventually contracting, i.e. if all its eigenvalues lie inside the unit circle, and that in this case the attractor is unique and globally attracting (in the sense that forward obits of all initial conditions tend to the attractor). Moreover, concerning the shape of minimal attractors of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}, we show that these are strictly convex sets with a continuously differentiable boundary. It remains an open problem whether there exists a more concise geometric, analytical or algebraic characterisation of these attractors. For instance, we note that the attractor is a solid sphere of constant radius if and only if $L$ is a scalar multiple of the identity. Interestingly, this is also the only instance in which the attractor is a solid ellipsoid [@homburg2010bifurcations]. This paper is organized as follows. We establish existence and uniqueness in Section [2](#sec:eau){reference-type="ref" reference="sec:eau"}, by taking an infinite-dimensional set-valued approach, following [@lamb2015topological]. In Section [4](#sec:global){reference-type="ref" reference="sec:global"} we show that a finite-dimensional boundary map, first introduced in [@Kourliouros2023; @tey2022minimal], has a globally attracting object, representing the unit normal bundle of the boundary of the unique attractor of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}. The latter boundary map helps us to establish the strict convexity of the attractor but serves also as an efficient computational tool to approximate the boundary of attractors. # Existence and uniqueness {#sec:eau} To understand the structure of attractors of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}, it suffices to consider the compound evolution of trajectories of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}, represented by the set-valued map $L_\varepsilon(x)=\overline{B_\varepsilon(L(x))}$, where $B_\varepsilon(x):=\{y\in\mathbb{R}^m~|~d(x,y)<\varepsilon\}$ denotes the open ball of radius $\varepsilon$. This map naturally extends to the space $\mathcal{K}(\mathbb{R}^m)$ of non-empty compact subsets of $\mathbb{R}^m$ as $$L_\varepsilon(C) := \overline{B_{\varepsilon}(L(C))} = \{x+y \in \mathbb{R}^m \mid x \in L(C), \|y\|\le \varepsilon\},$$ where $L(C):=\{L(x)~|~ x\in C\}$. There exists a useful correspondence between attractors of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"} and attracting fixed points of $L_\varepsilon$. To give context to the latter notion, we consider $\mathcal{K}(\mathbb{R}^m)$ equipped with the Hausdorff metric $$d_H(A,B):=\max\{\sup_{x\in A} \inf_{y\in B} d(x,y), \sup_{y\in B} \inf_{x\in A} d(x,y)\}.$$ We say that $A$ is an attracting fixed point of $L_\varepsilon$ if $L_\varepsilon(A)=A$ and $\lim_{i\to\infty}L_\varepsilon^i(C)= A$ for all $C\in \mathcal{K}(\mathbb{R}^m)$ in some open neighbourhood of $A$, and it is called globally attracting if the convergence holds for all $C\in\mathcal{K}(\mathbb{R}^m)$. **Lemma 1**. *If $A\in \mathcal{K}(\mathbb{R}^m)$ is an attractor of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"} then $A$ is a fixed point of $L_\varepsilon$. If $A$ is a globally attracting fixed point of $L_\varepsilon$, then $A$ is the unique global attractor of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}.* *Proof.* Let $A$ be an attractor of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}. We show that $L_\varepsilon(A)=A$. Let $A^c:=\mathbb{R}^m\setminus A$. Then $L_\varepsilon(A)\cap A^c = \emptyset$, otherwise there exists $x_i\in A$ such that $x_{i+1}\not\in A$. Moreover, $A\setminus L_\varepsilon(A)=\emptyset$, otherwise $A$ would not be a minimal trapping region, since $L_\varepsilon(A)\subseteq A$ implies that $L_\varepsilon^i(A)\subseteq L_\varepsilon(A)$ for all $i\in\mathbb{N}$. The second claim holds, as $\lim_{i\to\infty}L_\varepsilon^i(C)=A$ for all $C\in\mathcal{K}(\mathbb{R}^m)$ implies that $\lim_{i\to\infty}\inf_{a\in A} d(x_{i},a)=0$ for all $x_0\in A^c$. ◻ The existence and uniqueness of an attractor for [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"}and its global attractivity, follow from Lemma [Lemma 1](#lem:rdsvssetv){reference-type="ref" reference="lem:rdsvssetv"} and the following proposition. **Proposition 2** (Existence and uniqueness). *$L_\varepsilon$ has a fixed point $A\in \mathcal{K}(\mathbb{R}^m)$ if and only if all eigenvalues of $L$ have magnitude less than 1. The fixed point $A$ is unique and globally attracting for $L_\varepsilon$.* *Proof.* The condition on the eigenvalues of $L$ implies that there exists a metric $d_w$ on $\mathbb{R}^m$, equivalent to the Euclidean metric, such that $L$ is a contraction (i.e. there exists $0<c<1$ such that $d_w(L(x),L(y))\leq c d(x,y)$ for all $x,y\in\mathbb{R}^m$) [@belitskii2013matrix]\*Corollary 2.1.1. If $L$ is a contraction on $\mathbb{R}^m$ with metric $d_w$ then so is its set-valued analogue $L_\varepsilon$ on $\mathcal{K}(\mathbb{R}^m)$, equipped with the associated Hausdorff metric [@lamb2015topological]\*Lemma 3.3. Finally, as $\mathcal{K}(\mathbb{R}^m)$ is complete , if $L_\varepsilon$ is a contraction, then by Banach's Fixed Point Theorem $L_\varepsilon$ has a unique fixed point $A$ and $\lim_{i\to\infty}L_\varepsilon^i(C)=A$, for any $C\in\mathcal{K}(\mathbb{R}^m)$. It remains to be shown that the condition on the eigenvalues of $L$ is necessary. Suppose $L$ has an eigenvalue $\lambda$ with $|\lambda|>1$, then for every $x$ in the corresponding (real) eigenspace $E_\lambda$, $\|L^i(x)\|= |\lambda|^i\|x\|$. Hence, $\{\|L^i(x)\|~|~i\in\mathbb{N}\}$ is unbounded. We show that this is incompatible with the existence of a bounded attractor, exploiting the linearity of $L$. Namely, every attractor $A$ contains a point $y\in A$ such that $\overline{B_\varepsilon(y)}\subset A$. This ball contains a point $z$, whose direct sum into elements of (generalized) eigenspaces of $L$, contains a nonzero component in $E_\lambda$. This implies, by $L$-invariance of $E_\lambda$, that $\{\|L^i(z)\|~|~i\in\mathbb{N}\}$ is unbounded, which contradicts the boundedness of the attractor $A$. If $L$ has an eigenvalue $\lambda$ with $|\lambda|=1$, by the preceding argument, there exists a point $z$ in the attractor with nonzero component, say $x$, in the (real) eigenspace $E_\lambda$. In this case, $\{\|L^i(x)\|~|~i\in\mathbb{N}\}$ is not unbounded, as $\|L(x)\|=\|x\|$ if $x\in E_\lambda$. However, we observe that $(1+\frac{\varepsilon}{\|L(x)\|})L(x)\in L_\varepsilon(x)$. This implies, by induction, that $L_\varepsilon^i(x)$ contains a point of distance $i\varepsilon$ from the origin, implying that $\{\|L_\varepsilon ^i(x)\|~|~i\in\mathbb{N}\}$ is unbounded, contradicting the assumption that $z$ lies inside a bounded attractor of $L_\varepsilon$. ◻ In the remainder of this paper, we assume that all eigenvalues of $L$ have norm less than 1, and let $A$ denote the unique fixed point of $L_{\varepsilon}$. # Convexity and differentiability {#sec:convex} In this section, we show that the minimal attractor $A$ of [\[eq:rds\]](#eq:rds){reference-type="eqref" reference="eq:rds"} (which is equal to the unique fixed point of $L_\varepsilon$) is convex, its boundary is always continuously differentiable. **Proposition 3** (Convexity). *$A$ is convex.* *Proof.* $A$ contains the origin $0$, since this point is a global attractor of $L$. By induction, it follows that $L^i_{\varepsilon}(0) \subset L^{i+1}_{\varepsilon}(0) \subset A$ for all $i \in \mathbb N_0$. By Proposition [Proposition 2](#prop:unique){reference-type="ref" reference="prop:unique"}, the attractor $A$ is the limit of this nested sequence: $A=\lim_{i\to\infty}L^i_{\varepsilon}(0).$ All elements of this sequence are convex. Namely, $L_\varepsilon(0)=\overline{B_\varepsilon(0)}$ is convex, and for any convex $C\in\mathcal{K}(\mathbb{R}^m)$ containing the origin $0$, both $L(C)$ and $\overline{B_{\varepsilon}(C)}$ are convex. The proof concludes by noting that the limit of a set of nested convex sets is convex. ◻ **Proposition 4** (Differentiability). *The boundary of $A$ is continuously differentiable.* *Proof.* We make use of so-called supporting hyperplanes. Recall that a hyperplane is an $(n-1)$-dimensional affine subspace of $\mathbb{R}^m$, dividing $\mathbb{R}^m$ into two closed halfspaces, $H^-$ and $H^+$, bounded by $H$. A hyperplane $H(x)$ is a supporting hyperplane of a set $C\in\mathcal{K}(\mathbb{R}^m)$ at a boundary point $x \in \partial C$ if either $C \subset H^-$ or $C \subset H^+$. Importantly, if $C$ is convex, then there exists a supporting hyperplane at every boundary point [@schneider2014convex]\*Theorem 1.3.2. Moreover, $\partial C$ is a $C^1$ submanifold of $\mathbb{R}^m$ if and only if the supporting hyperplanes are unique [@schneider2014convex]\*Theorem 2.2.4. It thus suffices to demonstrate the uniqueness of the supporting hyperplane at every point $x\in \partial A$. Since $\overline{B_{\varepsilon}(L(A))} = A$ and $L$ is invertible, the boundary $\partial A$ of $A$ satisfies $\partial A = \partial B_{\varepsilon}(L(A))$ $\subset \partial B_{\varepsilon}(L(\partial A))$. Hence, for all $x \in \partial A$, there exists $y \in L(\partial A)$ such that $x \in \partial B_{\varepsilon}(y)$. As $B_\varepsilon(y)\subset A$, the supporting hyperplane at $x$ must also be a supporting hyperplane for $B_\varepsilon(y)$. As the latter is bounded by a smooth manifold, this hyperplane is unique, equal to the plane tangent to $B_\varepsilon(y)$ at $x$. ◻ It turns out that $A$ is even strictly convex. This refinement of Proposition [Proposition 3](#prop:conv){reference-type="ref" reference="prop:conv"} is presented in Proposition [Proposition 6](#prop:strict_convex){reference-type="ref" reference="prop:strict_convex"}. Its proof uses the so-called boundary map, which is introduced and discussed in the next section. # The boundary map and its dynamics {#sec:global} In this section we introduce the *boundary map*, as an alternative tool to study the boundary of the attractor $A$. This map was first introduced in [@Kourliouros2023]. We provide a self-contained introduction, tailored to the specific context of this paper. The observation that leads to the consideration of a boundary map, is that if the boundary of $A$ is smooth, the set-valued map $L_\varepsilon$ induces an invertible map of $\partial A$ to itself. **Lemma 5**. *The self-mapping of $\partial A$ induced by $L_\varepsilon$ is invertible.* *Proof.* Recall the proof of Proposition [Proposition 4](#prop:diff){reference-type="ref" reference="prop:diff"}, where a relationship was established between a point $x\in \partial A$ and a point $y\in L(A)$ such that $x\in\partial B_\varepsilon(y)$. As $L$ is invertible, the restriction of $L$ from $\partial A$ to $\partial L(A)$ is also invertible. If the relationship between $y\in L(A)$ and $x\in \partial A$ is one-to-one, then the invertibility of the restriction of $L_\varepsilon$ to $\partial A$ is established. To establish the one-to-one relationship, suppose for contradiction that there are two diferent points $y,y'\in L(A)$ whose $\varepsilon$-balls contribute to the same point $x\in\partial A$. Since there is only one supporting hyperplane at $x$ for $A$, and $y\neq y'$, $y$ and $y'$ can only lie on opposite sides of $x$, i.e. $y'=2x-y$. But the combination of two $\varepsilon$-balls touching each other from opposite sides, implies that $\partial A$ would be non-convex. ◻ {#fig:boundary_map width=".5\\textwidth"} The induced action of $L_\varepsilon$ on the boundary of the attractor does not provide any information about the shape of the boundary. Information about the tangent space (supporting hyperplane) to the boundary, contains additional geometric information about the boundary. Tangent planes are naturally represented by their direction, a unit normal vector. This leads us to define a boundary map from the unit normal bundle $\mathbb{R}^m\times\mathbb{S}^{m-1}$ to itself, where $\mathbb{R}^m$ represents the ambient space of the attractor boundary $\partial A$ and $\mathbb{S}^{m-1}:=\{x\in\mathbb{R}^m~|~\|x\|=1\}$ the sphere of unit vectors in $\mathbb{R}^m$, representing normal directions. The boundary of the attractor $\partial A$ has a natural representation in $\mathbb{R}^m\times\mathbb{S}^{m-1}$ as the section $$N_1^+\partial A:=\{(x,n(x))~|x\in\partial A\},$$ where $n(x)\in \mathbb{S}^{m-1}$ is the (unique) unit normal to $\partial A$ at $x\in\partial A$. The boundary map $b:\mathbb{R}^m\times\mathbb{S}^{m-1}\to \mathbb{R}^m\times\mathbb{S}^{m-1}$ is defined so as to leave the (outward) unit normal bundle $N^+_1\partial A$ of $\partial A$ invariant. Like $L_\varepsilon$, it is the composition of two maps $$b(x,n):=t_\varepsilon\circ {L}(x,n),$$ Here, with slight abuse of notation, $L:\mathbb{R}^m\times\mathbb{S}^{m-1}\to \mathbb{R}^m\times\mathbb{S}^{m-1}$ denotes the natural extension of $L\in GL(m,\mathbb{R})$ to the unit normal bundle, in the sense that for any compact convex set $C$ with smooth boundary $\partial C$, ${L}(N_1^+\partial C)=N_1^+\partial L(C)$. $t_\varepsilon$ denotes $\varepsilon$-translation induced by $\varepsilon$-balls, mapping $N_1^+\partial L(C)$ to $N_1^+\partial L_\varepsilon(C)$. We proceed to discuss the action of $L$ on the unit normal bundle. We may write $$\label{eq:b1} {L}(x,n):=(L(x), L_\perp(n))$$ reflecting in the first component the basic action of $L$ on $\mathbb{R}^m$. The induced action $L_\perp$ on the space $\mathbb{S}^{m-1}$ of unit normal vectors is given by $$\label{eq:b2} L_\perp(n):=P\circ(L^T)^{-1}n,$$ where $P:\mathbb{R}^m\setminus \{0\}\to \mathbb{S}^{m-1}$ denotes the canonical projection to the unit sphere, $P(x)=\frac{x}{||x||}$, and $L^T$ the transpose of $L$. To verify that the action on the unit normal induced by $L$ is indeed $L_\perp$, we note that the normal direction is defined to be orthogonal to a tangent space $T$, i.e. $\langle v,n\rangle=0$ for all $v\in T$. By linearity, $L$ acts on the tangent space $T$ in the same way as it acts on $\mathbb{R}^m$. Thus the image of the normal, $L_\perp(n)$, satisfies $\langle L(v),L_\perp(n)\rangle=0$ for all $v\in T$. This implies that $L^T L_\perp(n)=n$, from which the expression in [\[eq:b2\]](#eq:b2){reference-type="eqref" reference="eq:b2"} follows. The $\varepsilon$-translation $t_\varepsilon$ from $\partial L(A)$ to $\partial A$ is precisely in the direction of the (outward) normal $L_\perp(x)$ of $\partial L(A)$ at $L(x)$: $$t_\varepsilon(x,n)=(x+\varepsilon n,n).$$ In summary, the boundary map is thus derived to have the form $$\label{eq:bmap} b(x,n)=(L(x)+\varepsilon L_\perp(n), L_\perp(n)).$$ Importantly, by construction, the unit normal bundle of the attractor boundary $N_1^+\partial A$ is $b$-invariant. See also Figure [1](#fig:boundary_map){reference-type="ref" reference="fig:boundary_map"}. We use the boundary map to strengthen the result in Proposition [Proposition 3](#prop:conv){reference-type="ref" reference="prop:conv"} and show that $A$ is strictly convex. **Proposition 6**. *$A$ is strictly convex.* *Proof.* Assume for contradiction that $A$ is not strictly convex. Then there exist two distinct boundary points $x,y \in \partial A$ with the same outward unit normal $n$. Consider the inverse boundary mapping $b^{-1}(x,n)=(L^{-1}(x-\varepsilon n),L_\perp(n))$. Then $d(b^{-1}(x,n),b^{-1}(y,n))=d(L^{-1}(x),L^{-1}(y))$. As $L$ is eventually contracting, $L^{-1}$ is eventually expanding, so that $\lim_{k\to\infty}d(L^{-k}(x),L^{-k}(y))=\infty$, which contradicts the boundedness of $A$. ◻ It turns out that $N_1^+\partial A$ exhibits global attractivity under the boundary map $b$ and admits an explicit series expansion. **Theorem 7** (Global attraction of the normal bundle). *$$\label{eq:att} N_1^+\partial A=\{(x(n),n)~|~n\in\mathbb{S}^{m-1}\},\mbox{~with~} x(n):=\varepsilon\sum_{k=0}^\infty L^k\circ L_\perp^{-k}(n),$$ is globally attracting under the boundary map $b$.* *Proof.* Because of the strict convexity of $A$ (Proposition [Proposition 3](#prop:conv){reference-type="ref" reference="prop:conv"}) and smoothness of $\partial A$ (Proposition [Proposition 4](#prop:diff){reference-type="ref" reference="prop:diff"}), for each outward normal $n\in\mathbb{S}^{m-1}$ there exists a unique point $x(n)\in\partial A$ such that $(x(n),n)$ lies on the unit normal bundle of $\partial A$. Consider any initial condition $(x,n)$ for the boundary map $b$, then $$\lim_{k\to\infty} d(b^k(x,n),b^k(x(n),n))=\lim_{k\to\infty} d(L^k(x),L^k(x(n)))=0.$$ because $L$ is eventually contracting. Hence, the distance between the $b(x,n)$-orbit of $(x,n)$ to the unit normal bundle of $\partial A$ converges to $0$. From the proof of Proposition [Proposition 3](#prop:conv){reference-type="ref" reference="prop:conv"}, we recall that $A=\lim_{k\to \infty} L^k_\varepsilon(0)$. As $L^k_\varepsilon(0)$ is convex for all $k\in\mathbb{N}$, it follows from the construction of the boundary map that $$b(N_1^+(\partial L_\varepsilon^n(0))=N_1^+\partial(L^{n+1}_\varepsilon(0)).$$ Hence, $$N_1^+\partial A=\lim_{k\to\infty}b^k(N_1^+(\partial L_\varepsilon(0)).$$ As $\partial L_\varepsilon(0)=\{x=\varepsilon n~|~n\in\mathbb{S}^{m-1}\}$ we finally observe that $$\begin{aligned} b(\{x=\varepsilon n~|~n\in\mathbb{S}^{m-1}\}) &=& \{x=\varepsilon L(n)+\varepsilon L_\perp(n)~|~L_\perp(n)\in\mathbb{S}^{m-1}\}\\&=& \{x=\varepsilon L\circ L_\perp^{-1}(n)+\varepsilon n~|~n\in\mathbb{S}^{m-1}\}, \end{aligned}$$ from which the expression of $N_1^+\partial A=\lim_{k\to\infty}b^k(\{x=\varepsilon n~|~n\in\mathbb{S}^{m-1}\})$ follows by induction. ◻ The explicit equation for $x(n)$ in Theorem [\[eq:att\]](#eq:att){reference-type="ref" reference="eq:att"} offers a explicit series expansion for the attractor's boundary, presenting an efficient means of boundary approximation. By systematically sampling points from the unit sphere $\mathbb{S}^{m-1}$, it becomes possible to approximate boundary points by utilising their corresponding unit normals through the series expansion. Notably, the approximation procedure converges exponentially to the actual boundary. This convergence rate aligns with that of the linear map $L$. We now consider the dynamics on the boundary mapping $N_1^+\partial A$, which is by construction invariant. We recall that on this normal bundle, due to the strict convexity of $A$ and smoothness of its boundary, there is a one-to-one continuous relationship between points $x\in\partial A$ and its normal $n(x)$ to $\partial A$. Hence, on the one hand, this restriction of the boundary mapping is topologically conjugate to the invertible mapping induced by $L_\varepsilon$ on $\partial A$, cf. Lemma [Lemma 5](#lem:bminv){reference-type="ref" reference="lem:bminv"}. On the other hand, it is also topologically conjugate to $L_\perp$, the action induced by $L$ on its outward unit normal vectors. **Corollary 8**. *$b|_{N_1^+\partial A}$ is topologically conjugate to $L_\perp$.* *Proof.* Let $\pi (x,n):=n$ be the natural projection, then $$\pi\circ b(x,n)=L_\perp(n)=L_\perp\circ\pi(x,n).$$ This implies the topological conjugacy, since $\pi|_{N_1^+\partial A}$ is a homeomorphism. ◻ # Outlook The results of this paper highlight some important geometric properties of attractors of random dynamical systems with bounded noise in a very specific setting. Some interesting problems remain open, for instance the precise smoothness of the boundary of attractors. For instance, we need $C^2$ smoothness to apply the results of [@Kourliouros2023] to establish smooth persistence of attractor boundaries (under nonlinear perturbations), but additional smoothness requires a deeper understanding of the dynamics of the boundary map restricted to the boundary. Finally, it would be interesting to extend our results to more general noise with non-spherical reach. [^1]: [^2]: [^3]: The first two authors were supported by the EPSRC grant EP/W009455/1. The third author was supported by the Project of Intelligent Mobility Society Design, Social Cooperation Program, UTokyo, JST Moonshot R & D Grant Number JPMJMS2021, EPSRC grant EP/S515085/1 [^4]: Attracting random subsets for specific noise realisations are referred to as random attractors cf. [@arnold98]. The attractors defined here are deterministic and the union of random attractors over all noise realisations.

arxiv_math

--- abstract: | In this paper, the Laplacian characteristic polynomial of uniform hypergraphs with cut vertices or pendant edges and the Laplacian matching polynomial of uniform hypergraphs are characterized. The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees is given, which proves the conjecture in [@zheng2023zero] (The zero eigenvalue of the Laplacian tensor of a uniform hypergraph, Linear and Multilinear Algebra, (2023) Doi:10.1080/03081087.2023.2172541). address: College of Mathematical Sciences, Harbin Engineering University, Harbin, PR China author: - Ge Lin - Changjiang Bu bibliography: - atbib.bib title: "**The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees**" --- 1.15 hypertree, Laplacian tensor, multiplicity, characteristic polynomial, matching polynomial\ *AMS classification(2020):*05C65, 05C50. # Introduction A hypergraph is called $k$-uniform if its each edge contains exactly $k$ vertices. For a $k$-uniform hypergraph $H=(V(H),E(H))$, its adjacency tensor $\mathcal{A}_{H}=(a_{i_{1}i_{2}\cdots i_{k}})$ is a $k$-order $|V(H)|$-dimensional tensor [@cooper2012spectra], where $$a_{i_{1}i_{2}\cdots i_{k}}=\begin{cases} \frac{1}{(k-1)!}&\text{if $\{i_{1},i_{2},\ldots,i_{k}\}\in E(H)$},\\ 0&\text{otherwise}. \end{cases}$$ The Laplacian tensor of $H$ is $\mathcal{L}_{H}=\mathcal{D}_{H}-\mathcal{A}_{H}$ [@qi2014h], where $\mathcal{D}_{H}$ is the diagonal tensor of vertex degrees of $H$. The eigenvalues of $\mathcal{A}_{H}$ and $\mathcal{L}_{H}$ are called the eigenvalues and Laplacian eigenvalues of $H$, respectively. The characteristic polynomial of $\mathcal{A}_{H}$ and $\mathcal{L}_{H}$ are called the characteristic polynomial and the Laplacian characteristic polynomial of $H$, respectively. The characteristic polynomials of uniform hypergraphs are a research area that has attached much attention in spectral hypergraph theory. In 2012, Cooper and Dutle [@cooper2012spectra] characterized some properties on the characteristic polynomials of uniform hypergraphs and gave the characteristic polynomial of the one-edge hypergraph. In 2015, Cooper and Dutle [@cooper2015computing] gave the characteristic polynomial of the $3$-uniform hyperstar. In 2020, Bao et al. [@bao2020combinatorial] provided a combinatorial method for computing the characteristic polynomial of uniform hypergraphs with cut vertices, and gave the characteristic polynomial of the $k$-uniform hyperstar. In 2021, Chen and Bu [@chen2021reduction] gave a reduction formula for the characteristic polynomial of uniform hypergraphs with pendant edges. Besides, they used the reduction formula to derive the characteristic polynomial of the uniform hyperpath. However, there are few results on the Laplacian characteristic polynomials of uniform hypergraphs. In 2023, Zheng [@zheng2023zero] gave the Laplacian characteristic polynomial of uniform hyperstar, and obtained the multiplicity of the zero Laplacian eigenvalue of uniform hyperstar and hyperpath. Moreover, the following conjecture was proposed in [@zheng2023zero]. **Conjecture 1**. *[@zheng2023zero][\[caixiang1.1\]]{#caixiang1.1 label="caixiang1.1"} Let $T=(V(T),E(T))$ be a $k$-uniform hypertree for $k\geq3$. Then the multiplicity of the zero Laplacian eigenvalue of $T$ is $k^{|E(T)|(k-2)}$.* The eigenvalues of uniform hypertrees can be studied by the matching polynomial. In 2017, Zhang et al. [@zhang2017spectra] showed that the roots of the matching polynomial of a uniform hypertree are its eigenvalues. For a $k$-uniform hypertree $T$ with $k\geq3$, Clark and Cooper [@clark2018hypertrees] determined all eigenvalues of $T$ by roots of the matching polynomials of all sub-hypertrees of $T$. In 2022, Wan et al. [@wan2022spectra] defined the Laplacian matching polynomial of uniform hypergraphs, and used the roots of the Laplacian matching polynomials of all sub-hypertrees of $T$ to obtain all Laplacian eigenvalues of $T$ (without multiplicity). In this paper, we give a expression for the Laplacian characteristic polynomial of uniform hypergraphs with cut vertices or pendant edges (Section [2](#section2){reference-type="ref" reference="section2"}). And we characterize some properties on the Laplacian matching polynomial of uniform hypergraphs (Section [3](#section3){reference-type="ref" reference="section3"}). Further, we use these results to give the multiplicity of the zero Laplacian eigenvalue of uniform hypertrees, which shows that Conjecture [\[caixiang1.1\]](#caixiang1.1){reference-type="ref" reference="caixiang1.1"} is true (Section [4](#section4){reference-type="ref" reference="section4"}). # The Laplacian characteristic polynomial of uniform hypergraphs {#section2} ## Preliminaries In this subsection, we present some notation and lemmas about the eigenvalue of tensors and the formula of resultants. A $k$-order $n$-dimensional tensor $\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{k}})$ refers to a multi-dimensional array with entries $a_{i_{1}i_{2}\cdots i_{k}}$ for all $i_{j}\in[n]:=\{1,\ldots,n\}$ and $j\in[k]$. If there exists $\lambda\in\mathbb{C}$ and a non-zero vector $\mathbf{x}=(x_{1},\ldots,x_{n})^{\mathrm{T}}\in\mathbb{C}^{n}$ such that $$\begin{aligned} \mathcal{A}\mathbf{x}^{k-1}=\lambda\mathbf{x}^{[k-1]},\end{aligned}$$ where $\mathcal{A}\mathbf{x}^{k-1}$ is an $n$-dimensional vector with $\sum_{i_{2},\ldots,i_{k}=1}^{n}a_{ii_{2}\ldots i_{k}}x_{i_{2}}\cdots x_{i_{k}}$ as its $i$-th component and $\mathbf{x}^{[k-1]}=(x_{1}^{k-1},\ldots,x_{n}^{k-1})^{\mathrm{T}}$, then $\lambda$ is called an eigenvalue of $\mathcal{A}$ and $\mathbf{x}$ is an eigenvector of $\mathcal{A}$ corresponding to $\lambda$ (see [@lim2005singular; @qi2005eigenvalues]). The resultant of the polynomials system $(\lambda\mathbf{x}^{[k-1]}-\mathcal{A}\mathbf{x}^{k-1})$ is called the characteristic polynomial of $\mathcal{A}$, denoted by $\phi(\mathcal{A})$. In the following, we introduce some formulas of resultants required for proofs in this section. **Lemma 2**. *[@gelfand1994discriminants Poisson Formula for resultants][\[yinli2.1\]]{#yinli2.1 label="yinli2.1"} Let $F_{1},F_{2},\ldots,F_{n}\in\mathbb{C}[x_{1},\ldots,x_{n}]$ be homogeneous polynomials of respective degrees $d_{1},d_{2},\ldots,d_{n}$. For each $i\in[n]$, let $\overline{F}_{i}=F_{i}|_{x_{1}=0}$ and $f_{i}=F_{i}|_{x_{1}=1}$. Let $\mathcal{V}$ be the affine variety defined by the polynomials $f_{2},\ldots,f_{n}$. If $\mathrm{Res}(\overline{F}_{2},\ldots,\overline{F}_{n})\neq0$, then $$\begin{aligned} \mathrm{Res}(F_{1},F_{2},\ldots,F_{n}) =\mathrm{Res}(\overline{F}_{2},\ldots,\overline{F}_{n})^{d_{1}}\prod_{\mathbf{p}\in\mathcal{V}}f_{1}(\mathbf{p})^{m(\mathbf{p})},\end{aligned}$$ where $m(\mathbf{p})$ is the multiplicity of a point $\mathbf{p}$ in $\mathcal{V}$.* **Lemma 3**. *[@cooper2012spectra lemma 3.2][\[yinli2.2\]]{#yinli2.2 label="yinli2.2"} Let $F_{1},\ldots,F_{n}\in\mathbb{C}[x_{1},\ldots,x_{n}]$ be homogeneous polynomials of respective degrees $d_{1},\ldots,d_{n}$, and let $G_{1},\ldots,G_{m}\in\mathbb{C}[y_{1},\ldots,y_{m}]$ be homogeneous polynomials of respective degrees $\delta_{1},\ldots,\delta_{m}$. Then $$\begin{aligned} \mathrm{Res}(F_{1},\ldots,F_{n},G_{1},\ldots,G_{m})=\mathrm{Res}(F_{1},\ldots,F_{n})^{\prod_{j=1}^{m}\delta_{j}}\mathrm{Res}(G_{1},\ldots,G_{m})^{\prod_{i=1}^{n}d_{i}}.\end{aligned}$$* Let $H=(V(H),E(H))$ be a $k$-uniform hypergraph with $V(H)=[n]$. For a vertex $v\in V(H)$, let $E_{H}(v)$ denote the set of edges of $H$ containing $v$ and $d_{H}(v)$ denote the degree of $v$ in $H$. Given an edge $e\in E(H)$ and a vector $\mathbf{x}=(x_{1},\ldots,x_{n})^{\mathrm{T}}\in \mathbb{C}^{n}$, let $\mathbf{x}_{e}=\prod_{v\in e}x_{v}$. Then the eigenvalue equation $\mathcal{L}_{H}\mathbf{x}^{k-1}=\lambda\mathbf{x}^{[k-1]}$ corresponding to the Laplacian tensor of $H$ can be written as $$\begin{aligned} d_{H}(v)x_{v}^{k-1}-\sum_{e\in E_{H}(v)}\mathbf{x}_{e\setminus\{v\}}=\lambda x_{v}^{k-1}, v=1,\ldots,n.\end{aligned}$$ For each $v\in V(H)$, define $$\begin{aligned} F_{v}=(\lambda-d_{H}(v))x_{v}^{k-1}+\sum_{e\in E_{H}(v)}\mathbf{x}_{e\setminus\{v\}}.\end{aligned}$$ For a fixed vertex $w\in V(H)$, let $$\begin{aligned} \overline{F}_{v}=F_{v}|_{x_{w}=0}, f_{v}=F_{v}|_{x_{w}=1}.\end{aligned}$$ Let $\mathcal{V}^{H}$ be the affine variety defined by the polynomials $f_{v}$ for all $v\in V(H)\setminus\{w\}$. We use $\mathcal{L}_{H}(w)=(l_{i_{1}\cdots i_{k}})$ to denote a $k$-order $n-1$-dimensional principal sub-tensor of $\mathcal{L}_{H}$, where $i_{1},\ldots,i_{k}\in V(H)\setminus\{w\}$. By the Poisson Formula for resultants, we obtain the following lemma about the Laplacian characteristic polynomial of $H$. **Lemma 4**. *Let $H$ be a $k$-uniform hypergraph and $w$ be a vertex on $H$. Then the Laplacian characteristic polynomial $$\begin{aligned} \label{shi2.1} \phi(\mathcal{L}_{H}) =\phi(\mathcal{L}_{H}(w))^{k-1}\prod_{\mathbf{p}\in\mathcal{V}^{H}}(\lambda-d_{H}(w)+\sum_{e\in E_{H}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})},\end{aligned}$$ where $m(\mathbf{p})$ is the multiplicity of $\mathbf{p}$ in $\mathcal{V}^{H}$.* *Proof.* By the definition of the Laplacian characteristic polynomial, we know that $\phi(\mathcal{L}_{H})=\mathrm{Res}(F_{v}:v\in V(H))$, where $F_{v}=(\lambda-d_{H}(v))x_{v}^{k-1}+\sum_{e\in E_{H}(v)}\mathbf{x}_{e\setminus\{v\}}$. For the vertex $w\in V(H)$, by Lemma [\[yinli2.1\]](#yinli2.1){reference-type="ref" reference="yinli2.1"}, we have $$\begin{aligned} \phi(\mathcal{L}_{H})=\mathrm{Res}(\overline{F}_{v}:v\in V(H)\setminus\{w\})^{k-1} \prod_{\mathbf{p}\in\mathcal{V}^{H}}f_{w}(\mathbf{p})^{m(\mathbf{p})}.\end{aligned}$$ For all $v\in V(H)\setminus\{w\}$, $\overline{F}_{v}=F_{v}|_{x_{w}=0}=(\lambda-d_{H}(v))x_{v}^{k-1}+\sum_{e\in E_{H-w}(v)}\mathbf{x}_{e\setminus\{v\}}=0$ are the eigenvalue equations of $\mathcal{L}_{H}(w)$, where $H-w$ denote the hypergraph obtained from $H$ by removing the vertex $w$ and all edges incident to it, so we have $$\begin{aligned} \label{shi2.2} \mathrm{Res}(\overline{F}_{v}: v\in V(H)\setminus\{w\})=\phi(\mathcal{L}_{H}(w)).\end{aligned}$$ Note that $f_{w}=F_{w}|_{x_{w}=1}=\lambda-d_{H}(w)+\sum_{e\in E_{H}(w)}\mathbf{x}_{e\setminus\{w\}}$. Then we obtain $$\begin{aligned} \phi(\mathcal{L}_{H}) =\phi(\mathcal{L}_{H}(w))^{k-1}\prod_{\mathbf{p}\in\mathcal{V}^{H}}(\lambda-d_{H}(w)+\sum_{e\in E_{H}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})}.\end{aligned}$$ ◻ When $H$ is a uniform hypergraph with cut vertices, we can give a description of the affine variety $\mathcal{V}^{H}$ for this case and obtain a more explicit expression for the Laplacian characteristic polynomial of $H$ than ([\[shi2.1\]](#shi2.1){reference-type="ref" reference="shi2.1"}). ## Main results Let $H=(V(H),E(H))$ be a $k$-uniform connected hypergraph and $w\in V(H)$. Denote $\widehat{E}_{H}(w)=\{e\setminus\{w\}:e\in E_{H}(w)\}$. Deleting the vertex $w$, it can get a non-uniform hypergraph $\widehat{H}$ with vertex set $V(\widehat{H})=V(H)\setminus\{w\}$ and edge set $E(\widehat{H})=(E(H)\setminus E_{H}(w))\cup \widehat{E}_{H}(w)$. The vertex $w$ is called a cut vertex if $\widehat{H}$ is not connected [@bao2020combinatorial]. Suppose that $w$ is a cut vertex on $H$ and $\widehat{H}_{1},\ldots,\widehat{H}_{s}$ are connected components of $\widehat{H}$. For each $i\in[s]$, denote the induced sub-hypergraph of $H$ on $V(\widehat{H}_{i})\cup\{w\}$ by $\widetilde{H}_{i}$, and we call $\widetilde{H}_{i}$ a branch of $H$ associated with $w$. Clearly, $H$ can be obtained by coalescing the branches $\widetilde{H}_{1},\ldots,\widetilde{H}_{s}$ to the vertex $w$. Recall that the affine variety $\mathcal{V}^{H}$ is defined by the polynomials $f_{v}=(\lambda-d_{H}(v))x_{v}^{k-1}+\sum_{e\in E_{H}(v)}\mathbf{x}_{e\setminus\{v\}}|_{x_{w}=1}$ for all $v\in V(H)\setminus\{w\}$. Then, for each $v_{i}\in V(\widetilde{H}_{i})\setminus\{w\}$ and $i\in[s]$, we have $$\begin{aligned} f_{v_{i}}&=(\lambda-d_{H}(v_{i}))x_{v_{i}}^{k-1}+\sum_{e\in E_{H}(v_{i})}\mathbf{x}_{e\setminus\{v_{i},w\}}\\ &=(\lambda-d_{\widetilde{H}_{i}}(v_{i}))x_{v_{i}}^{k-1}+\sum_{e\in E_{\widetilde{H}_{i}}(v_{i})}\mathbf{x}_{e\setminus\{v_{i},w\}}.\end{aligned}$$ It is known that $\mathcal{V}^{\widetilde{H}_{i}}$ is the affine variety defined by the polynomials $f_{v_{i}}$ for all $v_{i}\in V(\widetilde{H}_{i})\setminus\{w\}$ and each $i\in[s]$. So $$\begin{aligned} \label{shi2.3} \mathcal{V}^{H}=\mathcal{V}^{\widetilde{H}_{1}}\times\cdots\times\mathcal{V}^{\widetilde{H}_{s}}.\end{aligned}$$ Combining Lemma [\[yinli2.1\]](#yinli2.1){reference-type="ref" reference="yinli2.1"} with ([\[shi2.3\]](#shi2.3){reference-type="ref" reference="shi2.3"}), an expression for the Laplacian characteristic polynomial of uniform hypergraphs with cut vertices is derived as follows. **Theorem 5**. *Let $H$ be a $k$-uniform hypergraph and $w$ be a cut vertex on $H$. Let $\widetilde{H}_{1},\ldots,\widetilde{H}_{s}$ are the branches of $H$ associated with $w$. Denote $\mathcal{V}^{(i)}=\mathcal{V}^{\widetilde{H}_{i}}$ and $E_{i}(w)=E_{\widetilde{H} _{i}}(w)$. Then $$\begin{aligned} \phi(\mathcal{L}_{H})=\prod_{i=1}^{s}\phi\left(\mathcal{L}_{\widetilde{H}_{i}}(w)\right)^{(k-1)^{2-s+\sum_{j\neq i}|V(\widetilde{H}_{j})|}} \prod_{\substack{\mathbf{p}^{(i)}\in\mathcal{V}^{(i)}\\i\in[s]}} (\lambda-\sum_{i=1}^{s}d_{\widetilde{H}_{i}}(w)+\sum_{\substack{e\in E_{i}(w)\\i\in[s]}}\mathbf{p}^{(i)}_{e\setminus\{w\}})^{\prod_{i=1}^{s}m(\mathbf{p}^{(i)})},\end{aligned}$$ where $m(\mathbf{p}^{(i)})$ is the multiplicity of $\mathbf{p}^{(i)}$ in $\mathcal{V}^{(i)}$ for each $i\in[s]$.* *Proof.* By Lemma [Lemma 4](#yinli2.3){reference-type="ref" reference="yinli2.3"}, the Laplacian characteristic polynomial $$\begin{aligned} \label{shi2.4} \phi(\mathcal{L}_{H})=\phi(\mathcal{L}_{H}(w))^{k-1}\prod_{\mathbf{p}\in\mathcal{V}^{H}}(\lambda-d_{H}(w)+\sum_{e\in E_{H}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})}.\end{aligned}$$ From ([\[shi2.2\]](#shi2.2){reference-type="ref" reference="shi2.2"}), we know that $\phi(\mathcal{L}_{H}(w))=\mathrm{Res}(\overline{F}_{v}: v\in V(H)\setminus\{w\})$. Recall that $\overline{F}_{v}=(\lambda-d_{H}(v))x_{v}^{k-1}+\sum_{e\in E_{H}(v)}\mathbf{x}_{e\setminus\{v\}}|_{x_{w}=0}$ for each $v\in V(H)\setminus\{w\}$, and note that $H$ can be obtained by coalescing the branches $\widetilde{H}_{1},\ldots,\widetilde{H}_{s}$ to the vertex $w$. For all $v_{i}\in V(\widetilde{H}_{i})\setminus\{w\}$ and each $i\in[s]$, we have $$\begin{aligned} \overline{F}_{v_{i}}&=(\lambda-d_{H}(v_{i}))x_{v_{i}}^{k-1}+\sum_{e\in E_{H}(v_{i})}\mathbf{x}_{e\setminus\{v_{i}\}}|_{x_{w}=0}\\ &=(\lambda-d_{\widetilde{H}_{i}}(v_{i}))x_{v_{i}}^{k-1}+\sum_{e\in E_{\widetilde{H}_{i}}(v_{i})}\mathbf{x}_{e\setminus\{v_{i}\}}|_{x_{w}=0}\\ &=(\lambda-d_{\widetilde{H}_{i}}(v_{i}))x_{v_{i}}^{k-1}+\sum_{e\in E_{\widetilde{H}_{i}-w}(v_{i})}\mathbf{x}_{e\setminus\{v_{i}\}},\end{aligned}$$ where $\widetilde{H}_{i}-w$ denote the hypergraph obtained from $\widetilde{H}_{i}$ by removing the vertex $w$ and all edges incident to it. So $\phi(\mathcal{L}_{H}(w))=\mathrm{Res}(\overline{F}_{v}: v\in V(H)\setminus\{w\})=\mathrm{Res}(\overline{F}_{v_{i}}: v_{i}\in V(\widetilde{H}_{i})\setminus\{w\}, i\in[s])$. By Lemma [\[yinli2.2\]](#yinli2.2){reference-type="ref" reference="yinli2.2"}, we get $$\begin{aligned} \phi(\mathcal{L}_{H}(w))=\prod_{i=1}^{s}\mathrm{Res}(\overline{F}_{v_{i}}:v_{i}\in V(\widetilde{H}_{i})\setminus\{w\})^{(k-1)^{1-s+\sum_{j\neq i}|V(\widetilde{H}_{j})|}}.\end{aligned}$$ For all $v_{i}\in V(\widetilde{H}_{i})\setminus\{w\}$ and each $i\in[s]$, $\overline{F}_{v_{i}}=0$ are the eigenvalue equations of $\mathcal{L}_{\widetilde{H}_{i}}(w)$. Then we have $\mathrm{Res}(\overline{F}_{v_{i}}: v_{i}\in V(\widetilde{H}_{i})\setminus\{w\})=\phi(\mathcal{L}_{\widetilde{H}_{i}}(w))$, which implies that $$\begin{aligned} \label{shi2.5} \phi(\mathcal{L}_{H}(w)) =\prod_{i=1}^{s}\phi(\mathcal{L}_{\widetilde{H}_{i}}(w))^{(k-1)^{1-s+\sum_{j\neq i}|V(\widetilde{H}_{j})|}}.\end{aligned}$$ For any $\mathbf{p}\in\mathcal{V}^{H}$, by ([\[shi2.3\]](#shi2.3){reference-type="ref" reference="shi2.3"}), we have $\mathbf{p}=\left(\begin{matrix} \mathbf{p}^{(1)}\\ \vdots \\ \mathbf{p}^{(s)} \end{matrix}\right)$, where $\mathbf{p}^{(i)}\in\mathcal{V}^{(i)}$ for all $i\in[s]$. Then we obtain $$\begin{aligned} \label{shi2.6}\notag \prod_{\mathbf{p}\in\mathcal{V}^{H}}(\lambda-d_{H}(w)+\sum_{e\in E_{H}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})} &=\prod_{\mathbf{p}\in\mathcal{V}^{H}}(\lambda-d_{H}(w)+\sum_{\substack{e\in E_{i}(w)\\i\in[s]}}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})}\\ &=\prod_{\substack{\mathbf{p}^{(i)}\in\mathcal{V}^{(i)}\\i\in[s]}}(\lambda-\sum_{i=1}^{s}d_{\widetilde{H}_{i}}(w)+\sum_{\substack{e\in E_{i}(w)\\i\in[s]}}\mathbf{p}^{(i)}_{e\setminus\{w\}})^{\prod_{i=1}^{s}m(\mathbf{p}^{(i)})}.\end{aligned}$$ Substituting ([\[shi2.5\]](#shi2.5){reference-type="ref" reference="shi2.5"}) and ([\[shi2.6\]](#shi2.6){reference-type="ref" reference="shi2.6"}) into ([\[shi2.4\]](#shi2.4){reference-type="ref" reference="shi2.4"}), the proof is completed. ◻ An edge on $k$-uniform hypergraph is called a pendant edge if it contains exactly $k-1$ vertices with degree one. When $k$-uniform hypergraph $H$ has a pendant edge incident to $w$, it implies that $w$ is a cut vertex on $H$ and one of the branches is the one-edge hypergraph. We use Theorem [Theorem 5](#dingli2.4){reference-type="ref" reference="dingli2.4"} to give a more explicit expression for the Laplacian characteristic polynomial of uniform hypergraphs with pendant edges. **Corollary 6**. *Let $H$ be a $k$-uniform hypergraph with a pendant edge incident to the non-pendent vertex $w$, and we define $\widetilde{H}$ as the $k$-uniform hypergraph obtained by removing the pendant edge and pendent vertices on it from $H$. Then $$\begin{aligned} \phi(\mathcal{L}_{H})=&(\lambda-1)^{(k-1)^{|V(\widetilde{H})|+k-1}}\phi(\mathcal{L}_{\widetilde{H}}(w))^{(k-1)^{k}} \prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}}(\lambda-d_{\widetilde{H}}(w)-1+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})K_{1}}\\ &\times\prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}}(\lambda-d_{\widetilde{H}}(w)-1+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})K_{2}},\end{aligned}$$ where $K_{1}=(k-1)^{k-1}-k^{k-2}$ and $K_{2}=k^{k-2}$.* *Proof.* Clearly, $w$ is a cut vertex on $H$. Suppose that the branches of $H$ associated with $w$ are $\widetilde{H}$ and the one-edge hypergraph with $k$ vertices, denoted by $H'$. By Theorem [Theorem 5](#dingli2.4){reference-type="ref" reference="dingli2.4"}, we have $$\begin{aligned} \label{shi2.7}\notag \phi(\mathcal{L}_{H})=&\phi\left(\mathcal{L}_{\widetilde{H}}(w)\right)^{(k-1)^{k}}\phi\left(\mathcal{L}_{H'}(w)\right)^{(k-1)^{|V(\widetilde{H})|}}\\ &\times\prod_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}\\\mathbf{q}\in\mathcal{V}^{H'}}} (\lambda-d_{\widetilde{H}}(w)-1+\mathbf{q}_{e'\setminus\{w\}}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})m(\mathbf{q})},\end{aligned}$$ where $e'$ is the edge of $H'$. Since $\mathcal{L}_{H'}(w)$ is a $k$-order $k-1$-dimensional identity tensor for the one-edge hypergraph $H'$, we get $$\begin{aligned} \label{shi2.8} \phi(\mathcal{L}_{H'}(w))=(\lambda-1)^{(k-1)^{k-1}}.\end{aligned}$$ It is shown that the Laplacian characteristic polynomial of $H'$ is $\phi(\mathcal{L} _{H'})=(\lambda-1)^{k(k-1)^{k-1}-k^{k-1}}((\lambda-1)^{k}+(-1)^{k-1})^{k^{k-2}}$ in the [@zheng2023zero Theorem 4.2]. It follows from ([\[shi2.1\]](#shi2.1){reference-type="ref" reference="shi2.1"}) that $$\begin{aligned} \prod_{\mathbf{q}\in\mathcal{V}^{H'}}(\lambda-1+\mathbf{q}_{e'\setminus\{w\}})^{m(\mathbf{q})}&=\frac{\phi(\mathcal{L}_{H'})}{\phi\left(\mathcal{L}_{H'}(w)\right)^{k-1}}\\ &=(\lambda-1)^{(k-1)^{k-1}-k^{k-2}}(\lambda-1+(\frac{-1}{\lambda-1})^{k-1})^{k^{k-2}}.\end{aligned}$$ Then we have $$\begin{aligned} \label{shi2.9} \mathbf{q}_{e'\setminus\{w\}}=\begin{cases} 0,& \text{if $\mathbf{q}=\mathbf{0}$,}\\ (\frac{-1}{\lambda-1})^{k-1},& \text{if $\mathbf{q}\neq\mathbf{0}$,} \end{cases}\end{aligned}$$ for $\mathbf{q}\in\mathcal{V}^{H'}$, and we have $m(\mathbf{0})=(k-1)^{k-1}-k^{k-2}$, $\sum_{\mathbf{0}\neq\mathbf{q}\in\mathcal{V}^{H'}}m(\mathbf{q})=k^{k-2}$ for $\mathbf{0}\in\mathcal{V}^{H'}$. By ([\[shi2.9\]](#shi2.9){reference-type="ref" reference="shi2.9"}), the equation in ([\[shi2.7\]](#shi2.7){reference-type="ref" reference="shi2.7"}) is derived as follows: $$\begin{aligned} \label{shi2.10}\notag &\prod_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}\\\mathbf{q}\in\mathcal{V}^{H'}}}(\lambda-d_{\widetilde{H}}(w)-1+\mathbf{q}_{e'\setminus\{w\}}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})m(\mathbf{q})}\\ \notag =&\prod_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}\\ \mathbf{0}=\mathbf{q}\in\mathcal{V}^{H'}}}(\lambda-d_{\widetilde{H}}(w)-1+\mathbf{q}_{e'\setminus\{w\}}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})m(\mathbf{q})}\\ \notag &\times\prod_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}\\ \mathbf{0}\neq\mathbf{q}\in\mathcal{V}^{H'}}}(\lambda-d_{\widetilde{H}}(w)-1+\mathbf{q}_{e'\setminus\{w\}}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})m(\mathbf{q})}\\ \notag =&\prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}}(\lambda-d_{\widetilde{H}}(w)-1+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})((k-1)^{k-1}-k^{k-2})}\\ &\times\prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{H}}}(\lambda-d_{\widetilde{H}}(w)-1+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{H}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})k^{k-2}}.\end{aligned}$$ Substituting ([\[shi2.8\]](#shi2.8){reference-type="ref" reference="shi2.8"}) and ([\[shi2.10\]](#shi2.10){reference-type="ref" reference="shi2.10"}) into ([\[shi2.7\]](#shi2.7){reference-type="ref" reference="shi2.7"}), the proof is completed. ◻ # The Laplacian matching polynomial of uniform hypergraphs {#section3} Let $H=(V(H),E(H))$ be a $k$-uniform hypergraph. Let $M$ be a sub-set of $E(H)$. Denote by $V(M)$ the set of vertices of $H$ each of which is an endpoint of one of the edges in $M$. If no two distinct edges in $M$ share a common vertex, then $M$ is called a matching of $H$. The set of matchings (including the empty set) of $H$ is denoted by $\mathcal{M}(H)$. Let $\mathbf{w}:V(H)\cup E(H)\rightarrow \mathbb{C}$ be a weighting function on $H$. In 2022, Wan et al. [@wan2022spectra] defined the weighted matching polynomial of $H$ as $$\begin{aligned} \sum_{M\in\mathcal{M}(H)}(-1)^{|M|}\prod_{e\in M}\mathbf{w}(e)^{k} \prod_{v\in V(H)\setminus V(M)}(\lambda-\mathbf{w}(v)).\end{aligned}$$ For any sub-hypergraph $\widetilde{H}$ of $H$, if we choose the weighting function on $\widetilde{H}$ such that $\mathbf{w}(v) = d_{H}(v)$ for all $v\in V(\widetilde{H})$ and $\mathbf{w}(e)=-1$ for all $e\in E(\widetilde{H})$, then the weighted matching polynomial of $\widetilde{H}$ can be derived as $$\begin{aligned} \label{shi3.1} \sum_{M\in\mathcal{M}(\widetilde{H})}(-1)^{(k-1)|M|}\prod_{v\in V(\widetilde{H})\setminus V(M)}(\lambda-d_{H}(v))=:\varphi_{H}(\widetilde{H}).\end{aligned}$$ In [@wan2022spectra], the polynomial ([\[shi3.1\]](#shi3.1){reference-type="ref" reference="shi3.1"}) is called the Laplacian matching polynomial of $\widetilde{H}$ with respect to $H$. The goal of this section is to characterize some properties on the Laplacian matching polynomial of uniform hypergraphs, which will be used to prove the main results in Section [4](#section4){reference-type="ref" reference="section4"}. Firstly, we introduce some related notation. For a sub-set $S\subseteq V(H)$, we use $H-S$ to denote the hypergraph obtained from $H$ by deleting the vertices in $S$ and the edges incident to them. For a sub-set $I\subseteq E(H)$, let $H\setminus I$ denote the hypergraph obtained from $H$ by deleting the edges in $I$ (no deletion of resultant isolated vertices). When $S=\{v\}$ and $I=\{e\}$, $H-S$ and $H\setminus I$ are simply written as $H-v$ and $H\setminus e$, respectively. **Theorem 7**. *Let $H$ be a $k$-uniform hypergraph, and $\widetilde{H}$ be a sub-hypergraph of $H$. Then the following statements hold.\ (1) If $\widetilde{H}$ is not connected and its connected components is $\widetilde{H}_{1}$ and $\widetilde{H}_{2}$, then $\varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}_{1})\varphi_{H}(\widetilde{H}_{2})$;\ (2) For $e\in E(\widetilde{H})$, we have $\varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}\setminus e)+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e))$;\ (3) For $v\in V(\widetilde{H})$ and $I\subseteq E_{\widetilde{H}}(v)$, we have $$\begin{aligned} \varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}\setminus I)+(-1)^{k-1}\sum_{e\in I}\varphi_{H}(\widetilde{H}-V(e)),\end{aligned}$$ and $$\begin{aligned} \varphi_{H}(\widetilde{H})=(\lambda-d_{H}(v))\varphi_{H}(\widetilde{H}-v)+(-1)^{k-1}\sum_{e\in E_{\widetilde{H}}(v)}\varphi_{H}(\widetilde{H}-V(e));\end{aligned}$$ (4) $\frac{\mathrm{d}}{\mathrm{d}\lambda}\varphi_{H}(\widetilde{H})=\sum_{v\in V(\widetilde{H})}\varphi_{H}(\widetilde{H}-v)$.* *Proof.* (1) For any $M\in\mathcal{M}(\widetilde{H})$, there exists $M_{1}\in\mathcal{M}(\widetilde{H}_{1})$ and $M_{2}\in\mathcal{M}(\widetilde{H}_{2})$ such that $M=M_{1}\cup M_{2}$. It is easy to check that $\varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}_{1})\varphi_{H}(\widetilde{H}_{2})$. \(2\) For any $M\in\mathcal{M}(\widetilde{H})$, if $M$ does not contain edge $e$, then $M$ is a matching of $\widetilde{H}\setminus e$; if $M$ contain edge $e$, then $M\setminus\{e\}$ is a matching of $\widetilde{H}-V(e)$. Thus, we have $$\begin{aligned} \varphi_{H}(\widetilde{H})=&\sum_{e\notin M\in\mathcal{M}(\widetilde{H})}(-1)^{(k-1)|M|}\prod_{v\in V(\widetilde{H})\setminus V(M)}(\lambda-d_{H}(v))\\ &+\sum_{e\in M\in\mathcal{M}(\widetilde{H})}(-1)^{(k-1)|M|}\prod_{v\in V(\widetilde{H})\setminus V(M)}(\lambda-d_{H}(v))\\ =&\sum_{M\in\mathcal{M}(\widetilde{H}\setminus e)}(-1)^{(k-1)|M|}\prod_{v\in V(\widetilde{H}\setminus e)\setminus V(M)}(\lambda-d_{H}(v))\\ &+\sum_{M\setminus\{e\}\in\mathcal{M}(\widetilde{H}-V(e))}(-1)^{(k-1)(|M\setminus\{e\}|+1)}\prod_{v\in V(\widetilde{H}-V(e))\setminus V\left(M\setminus\{e\}\right)}(\lambda-d_{H}(v))\\ =&\varphi_{H}(\widetilde{H}\setminus e)+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e)).\end{aligned}$$ \(3\) Suppose that $I=\{e_{1},\ldots,e_{s}\}$. It follows from Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (2) that $$\begin{aligned} \varphi_{H}(\widetilde{H})&=\varphi_{H}(\widetilde{H}\setminus e_{1})+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e_{1}))\\ &=\varphi_{H}(\widetilde{H}\setminus\{e_{1},e_{2}\})+(-1)^{k-1}\varphi_{H}(\widetilde{H}\setminus e_{1}-V(e_{2}))+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e_{1})).\end{aligned}$$ Since $\widetilde{H}\setminus e_{1}-V(e_{2})=\widetilde{H}-V(e_{2})$, we have $$\begin{aligned} \varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}\setminus\{e_{1},e_{2}\})+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e_{2}))+(-1)^{k-1}\varphi_{H}(\widetilde{H}-V(e_{1})).\end{aligned}$$ Repeatedly using Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (2), we get $$\begin{aligned} \label{shi3.2} \varphi_{H}(\widetilde{H})=\varphi_{H}(\widetilde{H}\setminus I)+(-1)^{k-1}\sum_{e\in I}\varphi_{H}(\widetilde{H}-V(e)).\end{aligned}$$ When $I=E_{\widetilde{H}}(v)$, the vertex $v$ is an isolated vertex on $H\setminus I$. By ([\[shi3.2\]](#shi3.2){reference-type="ref" reference="shi3.2"}) and Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1), we thus have that $$\begin{aligned} \varphi_{H}(\widetilde{H})=(\lambda-d_{H}(v))\varphi_{H}(\widetilde{H}-v)+(-1)^{k-1}\sum_{e\in E_{\widetilde{H}}(v)}\varphi_{H}(\widetilde{H}-V(e)).\end{aligned}$$ \(4\) By ([\[shi3.1\]](#shi3.1){reference-type="ref" reference="shi3.1"}), we have $$\begin{aligned} \label{shi3.3}\notag \frac{\mathrm{d}}{\mathrm{d}\lambda}\varphi_{H}(\widetilde{H})&= \sum_{M\in\mathcal{M(\widetilde{H})}}\sum_{v\in V(\widetilde{H})\setminus V(M)}(-1)^{(k-1)|M|}\prod_{v\neq u\in V(\widetilde{H})\setminus V(M)}(\lambda-d_{H}(u))\\ &=\sum_{M\in\mathcal{M(\widetilde{H})}}\sum_{v\in V(\widetilde{H})\setminus V(M)}(-1)^{(k-1)|M|}\prod_{u\in V(\widetilde{H}-v)\setminus V(M)}(\lambda-d_{H}(u)).\end{aligned}$$ For any $v\in V(\widetilde{H})$, a matching of $\widetilde{H}$ without $v$ is a matching of $\widetilde{H}-v$. So $\mathcal{M}(\widetilde{H}-v)$ can be seen as the set of all matchings without $v$ in $\widetilde{H}$. From ([\[shi3.3\]](#shi3.3){reference-type="ref" reference="shi3.3"}), we obtain $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\lambda}\varphi_{H}(\widetilde{H})&= \sum_{v\in V(\widetilde{H})}\sum_{M\in \mathcal{M}(\widetilde{H}-v)}(-1)^{(k-1)|M|}\prod_{u\in V(\widetilde{H}-v)\setminus V(M)}(\lambda-d_{H}(u))\\ &=\sum_{v\in V(\widetilde{H})}\varphi_{H}(\widetilde{H}-v).\end{aligned}$$ ◻ Next, we will give a result about the zero roots of the Laplacian matching polynomial of uniform hypertrees. For this we need a result about the eigenvalues of principal sub-tensor of Laplacian tensor and the relationship between the eigenvalue of weighted adjacency tensor and the weighted matching polynomial. For a non-empty $S\subseteq V(H)$, let $\mathcal{L}_{H}[S]=(l_{i_{1}\cdots i_{k}})$ denote the $k$-order $|S|$-dimensional principal sub-tensor of $\mathcal{L}_{H}$, where $i_{1},\ldots,i_{k}\in S$. When $S=V(H)\setminus\{v\}$, $\mathcal{L}_{H}[S]$ is simply written as $\mathcal{L}_{H}(v)$. A tensor is called a $\mathcal{Z}$-tensor if all of its off-diagonal entries are non-positive. Clearly, $\mathcal{L}_{H}[S]$ is a $\mathcal{Z}$-tensor for any non-empty $S\subseteq V(H)$. Applying some properties of $\mathcal{Z}$-tensor, we obtain the following result. **Lemma 8**. *Let $H$ be a uniform connected hypergraph. For any non-empty proper sub-set $S\subset V(H)$, the real eigenvalues of $\mathcal{L}_{H}[S]$ are all greater than zero.* *Proof.* For any non-empty proper sub-set $S\subset V(H)$, let $\tau(\mathcal{L}_{H}[S])$ denote the minimum real part of all eigenvalues of $\mathcal{L}_{H}[S]$. For a non-empty proper sub-set $U\subset V(H)$ satisfying $U\supseteq S$, it is known that $\tau(\mathcal{L}_{H}[U])\leq\tau(\mathcal{L}_{H}[S])$ [@shen2019some Theorem 3.1]. Thus, we have $$\begin{aligned} \min_{v\in V(H)}\tau(\mathcal{L}_{H}(v))\leq\tau(\mathcal{L}_{H}[S]).\end{aligned}$$ By [@gowda2015z Proposition 2.4], $\tau(\mathcal{L}_{H}(v))$ is the minimum $\mathrm{H}$-eigenvalue of $\mathcal{L}_{H}(v)$ for any $v\in V(H)$. It is shown that the minimum $\mathrm{H}$-eigenvalue of $\mathcal{L}_{H}(v)$ is greater than zero for uniform connected hypergraph $H$ in [@bu2018Inverse Lemma 2.1 and Theorem 3.1]. Then we have $\tau(\mathcal{L}_{H}(v))>0$. Thus $0<\min_{v\in V(H)}\tau(\mathcal{L}_{H}(v))\leq\tau(\mathcal{L}_{H}[S])$, which implies that the real eigenvalues of $\mathcal{L}_{H}[S]$ are all greater than zero. ◻ For a $k$-uniform hypergraph $H$ and the weighting function $\mathbf{w}:V(H)\cup E(H)\rightarrow \mathbb{C}$, Wan et al. [@wan2022spectra] defined the weighted adjacency tensor $\mathcal{A}_{H,\mathbf{w}}=(a_{i_{1}\ldots i_{k}})$, where $$a_{i_{1}\cdots i_{k}}=\begin{cases} \mathbf{w}(v) &\text{if $i_{1}=\cdots=i_{k}=v\in V(H)$}, \\ \frac{\mathbf{w}(e)}{(k-1)!} &\text{if $\{i_{1},\ldots,i_{k}\}=e\in E(H)$}, \\ 0 &\text{otherwise}. \end{cases}$$ They determined all eigenvalues of the weighted adjacency tensor of uniform hypertrees by means of the weighted matching polynomial. **Lemma 9**. *[@wan2022spectra Theorem2][\[yinli3.3\]]{#yinli3.3 label="yinli3.3"} Let $T=(V(T),E(T))$ be a $k$-uniform hypertree for $k\geq3$. Let $\mathbf{w}:V(T)\cup E(T)\rightarrow\mathbb{C}$ be a weighting function on $T$. Then $\lambda$ is an eigenvalue of $\mathcal{A}_{T,\mathbf{w}}$ if and only if there exists a sub-hypertree $\widetilde{T}$ of $T$ (including isolated vertices) such that $\lambda$ is a root of the weighted matching polynomial $$\begin{aligned} \sum_{M\in\mathcal{M}(\widetilde{T})}(-1)^{|M|}\prod_{e\in M}\mathbf{w}(e)^{k}\prod_{v\in V(\widetilde{T})\setminus V(M)}(\lambda-\mathbf{w}(v)).\end{aligned}$$* We are now ready to derive the result as follows. **Theorem 10**. *Let $T$ be a $k$-uniform hypertree. Then zero is a simple root of the polynomial $\varphi_{T}(T)$. Moreover, zero is not a root of the polynomial $\varphi_{T}(\widetilde{T})$ for any non-trivial sub-hypertree $\widetilde{T}$ of $T$.* *Proof.* When $k=2$, $\varphi_{T}(T)$ is the Laplacian matching polynomial of tree $T$. It is shown that $\varphi_{T}(T)$ is equal to the Laplacian characteristic polynomial of $T$ in the [@mohammadian2020laplacian Theorem3.3]. Since zero is a simple root of the Laplacian characteristic polynomial of $T$, zero is a simple root of $\varphi_{T}(T)$. By [@wan2022Onthelocation Theorem 2.7], for any non-trivial sub-tree $\widetilde{T}$ of $T$, it is easy to check that $\varphi_{T}(\widetilde{T})$ is equal to the characteristic polynomial of the Laplacian principal sub-matrix $L_{T}(w)$ of $T$. Since zero is not a root of the characteristic polynomial of $L_{T}(w)$, zero is not a root of $\varphi_{T}(\widetilde{T})$. In the following, we consider the case $k\geq3$. Clearly, for any sub-hypertree $\widetilde{T}$ of $T$, if we choose the weighting function $\mathbf{w}$ on $\widetilde{T}$ such that $\mathbf{w}(v) = d_{T}(v)$ for all $v\in V(\widetilde{T})$ and $\mathbf{w}(e)=-1$ for all $e\in E(\widetilde{T})$, then $\mathcal{A}_{\widetilde{T},\mathbf{w}}$ is exactly the principal sub-tensor $\mathcal{L}_{T}[V(\widetilde{T})]$ of $\mathcal{L}_{T}$, and the weighted matching polynomial of $\widetilde{T}$ is exactly $\varphi_{T}(\widetilde{T})$. It follows from Lemma [\[yinli3.3\]](#yinli3.3){reference-type="ref" reference="yinli3.3"} that the roots of $\varphi_{T}(\widetilde{T})$ is the eigenvalues of $\mathcal{L}_{T}[V(\widetilde{T})]$. When $\widetilde{T}$ is a non-trivial sub-hypertree of $T$, by Lemma [Lemma 8](#yinli3.2){reference-type="ref" reference="yinli3.2"}, we know that zero is not the eigenvalue of $\mathcal{L}_{T}[V(\widetilde{T})]$, which implies that zero is not a root of the polynomial $\varphi_{T}(\widetilde{T})$. Since zero is a Laplacian eigenvalue of $T$, by [@wan2022spectra Corollary4], there exists a sub-hypertree of $T$ such that zero is the root of the Laplacian matching polynomial of it with respect to $T$. It is known that zero is not a root of $\varphi_{T}(\widetilde{T})$ for any non-trivial sub-hypertree $\widetilde{T}$ of $T$, which implies that zero is a root of $\varphi_{T}(T)$. Next, we prove that zero is a simple root of $\varphi_{T}(T)$. By Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (4), we have $$\begin{aligned} \label{shi3.4} \frac{\mathrm{d}}{\mathrm{d}\lambda}\varphi_{T}(T)=\sum_{v\in V(T)}\varphi_{T}(T-v).\end{aligned}$$ Given a vertex $v\in V(T)$, we know that $T-v$ is not connected and each connected component is sub-hypertree of $T$. By Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1), the roots of $\varphi_{T}(T-v)$ are the eigenvalues of $\mathcal{L}_{T}[V(T-v)]$. By Lemma [Lemma 8](#yinli3.2){reference-type="ref" reference="yinli3.2"}, the real eigenvalues of $\mathcal{L}_{T}[V(T-v)]$ are all greater than zero, which implies that all real roots of $\varphi_{T}(T-v)$ are greater than zero. Note that $\varphi_{T}(T-v)$ is a real coefficient polynomial, whose all of imaginary part non-zero complex roots occur in pairs. So the product of all roots of $\varphi_{T}(T-v)$ is greater than zero. Let $\lambda_{1}^{(v)},\ldots,\lambda_{|V(T)|-1}^{(v)}$ denote the roots of $\varphi_{T}(T-v)$ for each $v\in V(T)$ and we have $\lambda_{1}^{(v)}\cdots\lambda_{|V(T)|-1}^{(v)}>0$. Then the constant term of the polynomial $\sum_{v\in V(T)}\varphi_{T}(T-v)$ is $(-1)^{|V(T)|-1}\sum_{v\in V(T)}\lambda_{1}^{(v)}\cdots\lambda_{|V(T)|-1}^{(v)}\neq0$, which implies that zero is not a root of $\sum_{v\in V(T)}\varphi_{T}(T-v)$. By ([\[shi3.4\]](#shi3.4){reference-type="ref" reference="shi3.4"}), zero is not a root of $\frac{\mathrm{d}}{\mathrm{d}\lambda}\varphi_{T}(T)$. Thus, zero is a simple root of $\varphi_{T}(T)$. ◻ # The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees {#section4} In this section, we apply the Laplacian characteristic polynomial and the Laplacian matching polynomial to give the multiplicity of the zero Laplacian eigenvalue of uniform hypertrees, which shows that Conjecture [\[caixiang1.1\]](#caixiang1.1){reference-type="ref" reference="caixiang1.1"} is true. For a $k$-uniform hypertree $T=(V(T),E(T))$ and a vertex $w\in V(T)$, recall that $F_{v}=F_{v}(x_{i}:i\in V(T))=(\lambda-d_{T}(v))x_{v}^{k-1}+\sum_{e\in E_{T}(v)}\mathbf{x}_{e\setminus\{v\}}$ and $f_{v}=F_{v}|_{x_{w}=1}$ for all $v\in V(T)$. Let $\mathcal{V}^{T}$ be the affine variety defined by the polynomials $f_{v}$ for all $v\in V(T)\setminus\{w\}$. By Lemma [\[yinli2.1\]](#yinli2.1){reference-type="ref" reference="yinli2.1"}, the Laplacian characteristic polynomial of $T$ is $$\begin{aligned} \label{shi4.1}\notag \phi(\mathcal{L}_{T})&=\phi(\mathcal{L}_{T}(w))^{k-1}\prod_{\mathbf{p}\in\mathcal{V}^{T}}(\lambda-d_{T}(w)+\sum_{e\in E_{T}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})}\\ &=\phi(\mathcal{L}_{T}(w))^{k-1}\prod_{\mathbf{p}\in\mathcal{V}^{T}}f_{w}(\mathbf{p})^{m(\mathbf{p})}.\end{aligned}$$ From Lemma [Lemma 8](#yinli3.2){reference-type="ref" reference="yinli3.2"}, we know that zero is not the eigenvalue of $\mathcal{L}_{T}(w)$. Hence, in order to determine the multiplicity of the zero Laplacian eigenvalue of $T$, we only need to consider $\prod_{\mathbf{p}\in\mathcal{V}^{T}}f_{w}(\mathbf{p})^{m(\mathbf{p})}$ in ([\[shi4.1\]](#shi4.1){reference-type="ref" reference="shi4.1"}). Let $\mathbf{p}=(p_{i})$ be a point in affine variety $\mathcal{V}^{T}$, and let $\mathbf{q}=(q_{i})$ be a $|V(T)|$-dimensional vector with components $q_{w}=1$ and $q_{i}=p_{i}$ for all $i\in V(T)\setminus\{w\}$. Then we have $$\begin{aligned} f_{w}(\mathbf{p})=F_{w}(q_{i}:i\in V(T))=F_{w}(\mathbf{q}),\end{aligned}$$ and $f_{v}(\mathbf{p})=F_{v}(q_{i}:i\in V(T))=F_{v}(\mathbf{q})=0$ for all $v\in V(T)\setminus\{w\}$. When $\lambda=0$. If $F_{w}(\mathbf{q})=0$, then $\mathbf{q}$ is an eigenvector corresponding to the zero Laplacian eigenvalue of $T$. It is shown that all components of the eigenvector corresponding to the zero Laplacian eigenvalue of a connected uniform hypergraph are non-zero in the [@hu2014eigenvectors Theorem 4.1 (i)]. Therefore, the all components of $\mathbf{p}\in\mathcal{V}^{T}$ satisfying $f_{w}(\mathbf{p})=0$ are non-zero when $\lambda=0$. It implies that the multiplicity of the zero Laplacian eigenvalue of $T$ is only related to the points having all components non-zero in $\mathcal{V}^{T}$. **Lemma 11**. *Let $T$ be a $k$-uniform hypertree and $w$ be a vertex on $T$. If $\mathbf{p}\in\mathcal{V}^{T}$ have all components non-zero, then $$\begin{aligned} \mathbf{p}_{e\setminus\{w\}}=\frac{(-1)^{k-1}\varphi_{T}(T-V(e))}{\varphi_{T}(T-w)}\end{aligned}$$ for each $e\in E_{T}(w)$.* *Proof.* We prove the result by the induction on the number of edges of $T$. When $|E(T)|=1$, we have $\varphi_{T}(T-w)=(\lambda-1)^{k-1}$ and $\varphi_{T}(T-V(e))=1$ for the edge $e\in E_{T}(w)$. From ([\[shi2.9\]](#shi2.9){reference-type="ref" reference="shi2.9"}), we know that $\mathbf{p}_{e\setminus\{w\}}=(\frac{-1}{\lambda-1})^{k-1}$, which implies that $$\begin{aligned} \mathbf{p}_{e\setminus\{w\}}=\frac{(-1)^{k-1}\varphi_{T}(T-V(e))}{\varphi_{T}(T-w)}.\end{aligned}$$ So the assertion holds. Assuming that the result holds for any $|E(T)|\leq r$, we consider the case $|E(T)|=r+1$. When $w$ is a cut vertex of $T$, $T$ has $d_{T}(w)$$(>1)$ branches associated with $w$ and each $e\in E_{T}(w)$ belongs to a distinct branch. Let $\widetilde{T}_{i}$ be the branch of $T$ with edge $e_{i}\in E_{T}(w)$ for each $i\in[d_{T}(w)]$ and we know that $|E(\widetilde{T}_{i})|\leq r$. By the induction hypothesis, for $\mathbf{p}^{(i)}\in\mathcal{V}^{\widetilde{T}_{i}}$ having all components non-zero, we have $$\begin{aligned} \mathbf{p}^{(i)}_{e_{i}\setminus\{w\}}=\frac{(-1)^{k-1}\varphi_{\widetilde{T}_{i}}(\widetilde{T}_{i}-V(e_{i}))}{\varphi_{\widetilde{T}_{i}}(\widetilde{T}_{i}-w)}.\end{aligned}$$ By the definition of the Laplacian matching polynomial, we have $\varphi_{\widetilde{T}_{i}}(\widetilde{T}_{i}-V(e_{i}))=\varphi_{T}(\widetilde{T}_{i}-V(e_{i}))$ and $\varphi_{\widetilde{T}_{i}}(\widetilde{T}_{i}-w)=\varphi_{T}(\widetilde{T}_{i}-w)$. Then $$\begin{aligned} \label{shi4.2}\notag \mathbf{p}^{(i)}_{e_{i}\setminus\{w\}}&=\frac{(-1)^{k-1}\varphi_{T}(\widetilde{T}_{i}-V(e_{i}))}{\varphi_{T}(\widetilde{T}_{i}-w)}\\ &=\frac{(-1)^{k-1}\varphi_{T}(\widetilde{T}_{i}-V(e_{i}))\prod_{\substack{j\in[d_{T}(w)]\\j\neq i}}\varphi_{T}(\widetilde{T}_{j}-w)}{\prod_{j\in[d_{T}(w)]}\varphi_{T}(\widetilde{T}_{j}-w)}.\end{aligned}$$ Note that $T-w$ is the disjoint union of $\widetilde{T}_{i}-w$ for all $i\in[d_{T}(w)]$, and $T-V(e_{j})$ is the disjoint union of $\widetilde{T}_{j}-V(e_{j})$ and $\widetilde{T}_{i}-w$ for all $i\neq j\in[d_{T}(w)]$. It follows from Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1) that $$\begin{aligned} \prod_{j\in[d_{T}(w)]}\varphi_{T}(\widetilde{T}_{j}-w)=\varphi_{T}(T-w),\end{aligned}$$ and $$\begin{aligned} \varphi_{T}(\widetilde{T}_{i}-V(e_{i}))\prod_{\substack{j\in[d_{T}(w)]\\j\neq i}}\varphi_{T}(\widetilde{T}_{j}-w)=\varphi_{T}(T-V(e_{i})).\end{aligned}$$ By Theorem [Theorem 5](#dingli2.4){reference-type="ref" reference="dingli2.4"} and ([\[shi4.2\]](#shi4.2){reference-type="ref" reference="shi4.2"}), for $\mathbf{p}\in\mathcal{V}^{T}$ having all components non-zero, we get $$\begin{aligned} \mathbf{p}_{e_{i}\setminus\{w\}}=\mathbf{p}^{(i)}_{e_{i}\setminus\{w\}}=\frac{(-1)^{k-1}\varphi_{T}(T-V(e_{i}))}{\varphi_{T}(T-w)}.\end{aligned}$$ When $w$ is not a cut vertex of $T$, the degree of $w$ is clearly one. Let the edge $\widehat{e}=\{v_{1},\ldots,v_{k-1},w\}$. Then $T\setminus\widehat{e}$ has $k$ connected components and we use $\widehat{T}_{t}$ to denote the connected component containing $v_{t}$ for each $t\in[k]$. For all $v\in V(T)$, recall that $F_{v}=F_{v}(x_{i}:i\in V(T))=(\lambda-d_{T}(v))x_{v}^{k-1}+\sum_{e\in E_{T}(v)}\mathbf{x}_{e\setminus\{v\}}$ and $f_{v}=F_{v}|_{x_{w}=1}$. For all $t\in[k-1]$ and any $v\in V(\widehat{T}_{t})\setminus\{v_{t}\}$, note that $f_{v}=f_{v}(x_{i}:i\in V(\widehat{T}_{t}))$ is a homogeneous polynomial. Since $\mathbf{p}=(p_{i})\in\mathcal{V}^{T}$ have all components non-zero, we get $$\begin{aligned} \label{shi4.3} f_{v}(\mathbf{p})=f_{v}(p_{i}:i\in V(\widehat{T}_{t}))=f_{v}\left(\frac{p_{i}}{p_{v_{t}}}:i\in V(\widehat{T}_{t})\right)=0.\end{aligned}$$ Fix $t\in[k-1]$, we consider the sub-hypertree $\widehat{T}_{t}$. For all $v\in V(\widehat{T}_{t})\setminus\{v_{t}\}$, let $\widehat{F}_{v}=\widehat{F}_{v}(x_{i}:i\in V(\widehat{T}_{t}))=(\lambda-d_{\widehat{T}_{t}}(v))x_{v}^{k-1}+\sum_{e\in E_{\widehat{T}_{t}}(v)}\mathbf{x}_{e\setminus\{v\}}$ and $\widehat{f}_{v}=\widehat{F}_{v}|_{x_{v_{t}}=1}$. It is easy to check that $\widehat{F}_{v}=f_{v}$. Let $q_{i}=\frac{p_{i}}{p_{v_{t}}}$ for all $i\in V(\widehat{T}_{t})$ and note that $q_{v_{t}}=1$. By ([\[shi4.3\]](#shi4.3){reference-type="ref" reference="shi4.3"}), we have $$\begin{aligned} \label{shi4.4} \widehat{f}_{v}(q_{i}:i\in V(\widehat{T}_{t})\setminus\{v_{t}\})=\widehat{F}_{v}(q_{i}:i\in V(\widehat{T}_{t}))=f_{v}(q_{i}:i\in V(\widehat{T}_{t}))=0\end{aligned}$$ for all $v\in V(\widehat{T}_{t})\setminus\{v_{t}\}$. Let the vector $\mathbf{q}=(q_{i})$ for $i\in V(\widehat{T}_{t})\setminus\{v_{t}\}$. Then $\mathbf{q}$ is a point in the affine variety $\mathcal{V}^{\widehat{T}_{t}}$ defined by the polynomials $\widehat{f}_{v}$ for all $v\in V(\widehat{T}_{t})\setminus\{v_{t}\}$, and the all components of $\mathbf{q}$ are non-zero. By the induction hypothesis, for each $e\in E_{\widehat{T}_{t}}(v_{t})$, we have $$\begin{aligned} \mathbf{q}_{e\setminus\{v_{t}\}}=\frac{(-1)^{k-1}\varphi_{\widehat{T}_{t}}(\widehat{T}_{t}-V(e))}{\varphi_{\widehat{T}_{t}}(\widehat{T}_{t}-v_{t})}.\end{aligned}$$ By the definition of the Laplacian matching polynomial, we have $\varphi_{\widehat{T}_{t}}(\widehat{T}_{t}-V(e))=\varphi_{T}(\widehat{T}_{t}-V(e))$ and $\varphi_{\widehat{T}_{t}}(\widehat{T}_{t}-v_{t})=\varphi_{T}(\widehat{T}_{t}-v_{t})$. Then $$\begin{aligned} \mathbf{q}_{e\setminus\{v_{t}\}}=\frac{(-1)^{k-1}\varphi_{T}(\widehat{T}_{t}-V(e))}{\varphi_{T}(\widehat{T}_{t}-v_{t})}=\frac{\mathbf{p}_{e\setminus\{v_{t}\}}}{p_{v_{t}}^{k-1}}.\end{aligned}$$ Thus, for $\mathbf{p}\in\mathcal{V}^{T}$ having all components non-zero and each $e\in E_{\widehat{T}_{t}}(v_{t})$, we get $$\begin{aligned} \label{shi4.5} \mathbf{p}_{e\setminus\{v_{t}\}}=\frac{(-1)^{k-1}\varphi_{T}(\widehat{T}_{t}-V(e))}{\varphi_{T}(\widehat{T}_{t}-v_{t})}p_{v_{t}}^{k-1}.\end{aligned}$$ For each $t\in[k-1]$, recall that $$\begin{aligned} f_{v_{t}}(\mathbf{p})=(\lambda-d_{T}(v_{t}))p_{v_{t}}^{k-1}+\mathbf{p}_{\widehat{e}\setminus\{v_{t},w\}}+\sum_{e\in E_{\widehat{T}_{t}}(v_{t})}\mathbf{p}_{e\setminus\{v_{t}\}}=0.\end{aligned}$$ By ([\[shi4.5\]](#shi4.5){reference-type="ref" reference="shi4.5"}) and Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (3), we have $$\begin{aligned} \mathbf{p}_{\widehat{e}\setminus\{v_{t},w\}}&=-\left(\lambda-d_{T}(v_{t})+\sum_{e\in E_{\widehat{T}_{t}}(v_{t})}\frac{(-1)^{k-1}\varphi_{T}(\widehat{T}_{t}-V(e))}{\varphi_{T}(\widehat{T}_{t}-v_{t})}\right)p_{v_{t}}^{k-1}\\ &=-\frac{\varphi_{T}(\widehat{T}_{t})}{\varphi_{T}(\widehat{T}_{t}-v_{t})}p_{v_{t}}^{k-1}.\end{aligned}$$ Combining these equations for all $t\in[k-1]$, we get $$\begin{aligned} \prod_{t=1}^{k-1}\mathbf{p}_{\widehat{e}\setminus\{v_{t},w\}}&=(-1)^{k-1}\prod_{t=1}^{k-1}\frac{\varphi_{T}(\widehat{T}_{t})}{\varphi_{T}(\widehat{T}_{t}-v_{t})}p_{v_{t}}^{k-1}.\end{aligned}$$ Since $\prod_{t=1}^{k-1}\mathbf{p}_{\widehat{e}\setminus\{v_{t},w\}}=\prod_{t=1}^{k-1}p_{v_{t}}^{k-2}$, we have $$\begin{aligned} \mathbf{p}_{\widehat{e}\setminus\{w\}}=\frac{\prod_{t=1}^{k-1}p_{v_{t}}^{k-1}}{\prod_{t=1}^{k-1}\mathbf{p}_{\widehat{e}\setminus\{v_{t},w\}}}=(-1)^{k-1}\prod_{t=1}^{k-1}\frac{\varphi_{T}(\widehat{T}_{t}-v_{t})}{\varphi_{T}(\widehat{T}_{t})}.\end{aligned}$$ Note that for all $t\in[k-1]$, the disjoint union of $\widehat{T}_{t}-v_{t}$ is $T-V(\widehat{e})$ and the disjoint union of $\widehat{T}_{t}$ is $T-w$. It follows from Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1) that $$\begin{aligned} \mathbf{p}_{\widehat{e}\setminus\{w\}}=\frac{(-1)^{k-1}\varphi_{T}(T-V(\widehat{e}))}{\varphi_{T}(T-w)}.\end{aligned}$$ ◻ For the point $\mathbf{p}\in\mathcal{V}^{T}$, we have $f_{w}(\mathbf{p})=\lambda-d_{T}(w)+\sum_{e\in E_{T}(w)}\mathbf{p}_{e\setminus\{w\}}$. If $\mathbf{p}$ have all components non-zero, by Lemma [Lemma 11](#yinli4.1){reference-type="ref" reference="yinli4.1"} and Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (3), we get $$\begin{aligned} \label{shi4.6}\notag f_{w}(\mathbf{p})&=\lambda-d_{T}(w)+\sum_{e\in E_{T}(w)}\frac{(-1)^{k-1}\varphi_{T}(T-V(e))}{\varphi_{T}(T-w)}\\ &=\frac{\varphi_{T}(T)}{\varphi_{T}(T-w)}.\end{aligned}$$ Note that $T-w$ is not connected and each connected component is a non-trivial sub-hypertree of $T$. From Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1) and Theorem [Theorem 10](#dingli3.4){reference-type="ref" reference="dingli3.4"}, we know that zero is not the root of $\varphi_{T}(T-w)$ and is a simple root of $\varphi_{T}(T)$. Let $n_{0}(T)$ denote the multiplicity of the zero Laplacian eigenvalue of $T$. Since $n_{0}(T)$ is only related to $\mathbf{p}$ having all components non-zero in $\mathcal{V}^{T}$, combining ([\[shi4.1\]](#shi4.1){reference-type="ref" reference="shi4.1"}) with ([\[shi4.6\]](#shi4.6){reference-type="ref" reference="shi4.6"}), we have $$\begin{aligned} \label{shi4.7} n_{0}(T)=\sum_{\substack{\mathbf{p}\in\mathcal{V}^{T}\\\forall p_{i}\neq0}}m(\mathbf{p}),\end{aligned}$$ where $m(\mathbf{p})$ is the multiplicity of $\mathbf{p}=(p_{i})$ in $\mathcal{V}^{T}$. We are now ready to determine the multiplicity of the zero Laplacian eigenvalue of $T$. **Theorem 12**. *Let $T=(V(T),E(T))$ be a $k$-uniform hypertree. Then the multiplicity of the zero Laplacian eigenvalue of $T$ is $k^{|E(T)|(k-2)}$.* *Proof.* We prove the result by the induction on the number of edges of $T$. When $|E(T)|=1$. It is shown that the multiplicity of the zero Laplacian eigenvalue of $T$ is $k^{k-2}$ in the [@zheng2023zero Theorem 4.9]. So the assertion holds. Assuming that the result holds when $|E(T)|=r$, we consider the case $|E(T)|=r+1$. Let $w$ be a non-pendent vertex on a pendant edge of $T$, and $\widetilde{T}$ denote the $k$-uniform hypertree obtained by removing this pendant edge and pendent vertices on it from $T$. By Corollary [Corollary 6](#tuilun2.5){reference-type="ref" reference="tuilun2.5"}, the Laplacian characteristic polynomial of $T$ is $$\begin{aligned} \label{shi4.8}\notag \phi(\mathcal{L}_{T})=&(\lambda-1)^{(k-1)^{(r+1)(k-1)+1}}\phi(\mathcal{L}_{\widetilde{T}}(w))^{(k-1)^{k}} \prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{T}}}(\lambda-d_{T}(w)+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})K_{1}}\\ &\times\prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{T}}}(\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})K_{2}},\end{aligned}$$ where $K_{1}=(k-1)^{k-1}-k^{k-2}$ and $K_{2}=k^{k-2}$. Clearly, $w$ is a cut vertex on $T$. Suppose that the branches of $T$ associated with $w$ are $\widetilde{T}$ and a one-edge hypergraph, denoted by $T'$. By ([\[shi2.3\]](#shi2.3){reference-type="ref" reference="shi2.3"}), we know that $\mathcal{V}^{T}=\mathcal{V}^{\widetilde{T}}\times\mathcal{V}^{T'}$. Then we have $\mathbf{r}=\left(\begin{matrix} \mathbf{p} \\ \mathbf{q} \end{matrix}\right)$ for any $\mathbf{r}\in\mathcal{V}^{T}$, where $\mathbf{p}\in\mathcal{V}^{\widetilde{T}}$, $\mathbf{q}\in\mathcal{V}^{T'}$. It is known from ([\[shi4.7\]](#shi4.7){reference-type="ref" reference="shi4.7"}) that the multiplicity of the zero Laplacian eigenvalue of $T$ is only related to $\mathbf{r}\in\mathcal{V}^{T}$ having all components non-zero. By ([\[shi2.10\]](#shi2.10){reference-type="ref" reference="shi2.10"}), it implies that we only need to consider $$\begin{aligned} \label{shi4.9} \prod_{\mathbf{p}\in\mathcal{V}^{\widetilde{T}}}(\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}})^{m(\mathbf{p})K_{2}}\end{aligned}$$ in ([\[shi4.8\]](#shi4.8){reference-type="ref" reference="shi4.8"}) and $\mathbf{p}$ have all components non-zero in $\mathcal{V}^{\widetilde{T}}$. By Lemma [Lemma 11](#yinli4.1){reference-type="ref" reference="yinli4.1"}, for $\mathbf{p}\in\mathcal{V}^{\widetilde{T}}$ having all components non-zero, we have $$\begin{aligned} &\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}}\\ =&\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\frac{(-1)^{k-1}\varphi_{\widetilde{T}}(\widetilde{T}-V(e))}{\varphi_{\widetilde{T}}(\widetilde{T}-w)}.\end{aligned}$$ By the definition of the Laplacian matching polynomial, we know that$\varphi_{\widetilde{T}}(\widetilde{T}-w)=\varphi_{T}(\widetilde{T}-w)$ and $\varphi_{\widetilde{T}}(\widetilde{T}-V(e))=\varphi_{T}(\widetilde{T}-V(e))$ for each $e\in E_{\widetilde{T}}(w)$. It follows from Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (3) that $$\begin{aligned} \label{shi4.10}\notag &\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}}\\ =&\frac{(\lambda-1)^{k-1}\varphi_{T}(\widetilde{T})+(-1)^{k-1}\varphi_{T}(\widetilde{T}-w)}{(\lambda-1)^{k-1}\varphi_{T}(\widetilde{T}-w)}.\end{aligned}$$ Let pendant edge $\widetilde{e}=\{v_{1},\ldots,v_{k-1},w\}$, where $v_{1},\ldots,v_{k-1}$ are the pendent vertices. Note that the Laplacian matching polynomial of $v_{i}$ with respect to $T$ is $\lambda-1$ for each $i\in[k-1]$. Since the disjoint union of $\widetilde{T}-w$ and $v_{i}$ for all $i\in[k-1]$ is $T-w$, by Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1), we have $$\begin{aligned} (\lambda-1)^{k-1}\varphi_{T}(\widetilde{T}-w)=\varphi_{T}(T-w).\end{aligned}$$ Since $(\lambda-1)^{k-1}\varphi_{T}(\widetilde{T})+(-1)^{k-1}\varphi_{T}(\widetilde{T}-w)=(\lambda-d_{T}(v_{i}))\varphi_{T}(T-v_{i})+(-1)^{k-1}\varphi_{T}(T-V(\widetilde{e}))$ for any $i\in[k-1]$, by Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (3), we have $$\begin{aligned} (\lambda-1)^{k-1}\varphi_{T}(\widetilde{T})+(-1)^{k-1}\varphi_{T}(\widetilde{T}-w)=\varphi_{T}(T).\end{aligned}$$ From ([\[shi4.10\]](#shi4.10){reference-type="ref" reference="shi4.10"}), for $\mathbf{p}\in\mathcal{V}^{\widetilde{T}}$ having all components non-zero, we obtain $$\begin{aligned} &\lambda-d_{T}(w)+(\frac{-1}{\lambda-1})^{k-1}+\sum_{e\in E_{\widetilde{T}}(w)}\mathbf{p}_{e\setminus\{w\}}\\ &=\frac{\varphi_{T}(T)}{\varphi_{T}(T-w)}.\end{aligned}$$ Note that $T-w$ is not connected and each connected component is a non-trivial sub-hypertree of $T$. It is known from Theorem [Theorem 7](#dingli3.1){reference-type="ref" reference="dingli3.1"} (1) and Theorem [Theorem 10](#dingli3.4){reference-type="ref" reference="dingli3.4"} that zero is not the root of $\varphi_{T}(T-w)$ and is a simple root of $\varphi_{T}(T)$. By ([\[shi4.9\]](#shi4.9){reference-type="ref" reference="shi4.9"}), we get $$\begin{aligned} n_{0}(T)=k^{k-2}\sum_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{T}}\\\forall p_{i}\neq0}}m(\mathbf{p}).\end{aligned}$$ It follows from ([\[shi4.7\]](#shi4.7){reference-type="ref" reference="shi4.7"}) that $\sum_{\substack{\mathbf{p}\in\mathcal{V}^{\widetilde{T}}\\\forall p_{i}\neq0}}m(\mathbf{p})=n_{0}(\widetilde{T})$. By the induction hypothesis, we have $n_{0}(\widetilde{T})=k^{r(k-2)}$. Thus, $n_{0}(T)=k^{k-2}n_{0}(\widetilde{T})=k^{(r+1)(k-2)}$. ◻ # References {#references .unnumbered}

arxiv_math

# Realizing the symplectic group as a Galois group over the function field $\mathop{\mathrm{\mathbb{F}}}_q(t)$ {#realizing-the-symplectic-group-as-a-galois-group-over-the-function-field-mathopmathrmmathbbf_qt .unnumbered} Rod Gow and Gary McGuire\  \ School of Mathematics and Statistics\ University College Dublin\ Ireland ## Abstract {#abstract .unnumbered} We realize the symplectic group $Sp(2m,q)$ as a Galois group over the function field $\mathop{\mathrm{\mathbb{F}}}_q(t)$, where $q$ is a power of the prime $p$. Our proof is based on the study of certain so-called $q$-palindromic polynomials, whose roots form a vector space over $\mathop{\mathrm{\mathbb{F}}}_q$. The main thrust of our approach is to find symplectic transvections in the Galois group and to exploit the fact that the Galois group acts transitively on the nonzero roots in the irreducible case. MSC 11R58 Keywords Galois group, linearized polynomial, symplectic group # Introduction Let $p$ be a prime number, and let $q$ be a positive power of $p$. Let $\mathop{\mathrm{\mathbb{F}}}_q$ denote the finite field with $q$ elements and let $\mathop{\mathrm{\mathbb{F}}}_q(t)$ denote the function field over $\mathop{\mathrm{\mathbb{F}}}_q$ in the single variable $t$. A $q$-linearized polynomial over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ is a polynomial of the form $$\label{lin1} L(x)=a_n x^{q^n}+a_{n-1}x^{q^{n-1}}+\cdots +a_1x^q+a_0x \in \mathop{\mathrm{\mathbb{F}}}_q(t)[x].$$ If $a_n\not=0$ we say that $n$ is the $q$-degree of $f$. In this case, we usually assume that $a_n=1$, so that $L$ is monic. We will usually talk of a $q$-polynomial or of a linearized polynomial, the $q$-term being understood. The set of roots in a splitting field of the $q$-polynomial $L$ forms an $\mathop{\mathrm{\mathbb{F}}}_q$-vector space. We make this statement more precise in the following lemma, which is straightforward to prove (see [@SL] for example). **Lemma 1**. *Let $L$ be a $q$-polynomial of $q$-degree $n$ in $\mathop{\mathrm{\mathbb{F}}}_q(t)[x]$, with $$L=x^{q^n}+a_{n-1}x^{q^{n-1}}+\cdots +a_1x^q+a_0x.$$ Let $E$ be a splitting field for $L$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ and let $V$ be the set of roots of $L$ in $E$. Let $G$ be the Galois group of $E$ over $F$. Suppose that $a_0\neq 0$. Then $V$ is an $\mathop{\mathrm{\mathbb{F}}}_q$-vector space of dimension $n$ and $G$ is naturally a subgroup of $GL(n,q)$.* In this statement we are denoting the general linear group of invertible $n\times n$ matrices over $\mathop{\mathrm{\mathbb{F}}}_q$ by $GL(n,q)$. We will assume that our $q$-polynomial $L$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ has coefficients in the ring $\mathop{\mathrm{\mathbb{F}}}_q[t]$ (that is, they are polynomials in $t$) and usually that the $x$ term in $L(x)$ is nonzero. This guarantees that $L$ has no repeated roots. Our aim is to investigate the Galois group $G$, say, of $L(x)$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$. We will concentrate on polynomials $L$ expressible in the form $$L(x)=\Phi(x)+tx^{q^m}$$ where $\Phi$ is a special type of $q$-polynomial of even $q$-degree $2m$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$. Write $$\Phi=\sum_{i=0}^{2m} c_i x^{q^i},$$ where the coefficients $c_i$ are in $\mathop{\mathrm{\mathbb{F}}}_q$, $c_0=c_{2m}=1$, and $$c_{2m-i}=c_i$$ for all $i$. We say that $\Phi$ is $q$-palindromic in this case. It means that the associated polynomial $$f(x)=\sum_{i=0}^{2m} c_i x^i$$ is self-reciprocal, in other words, $f(x)=x^{2m}f(x^{-1})$. While we know in general that the Galois group is a subgroup of $GL(2m,q)$, a result due to Elkies [@E] shows that in the $q$-palindromic case, the Galois group is a subgroup of the symplectic group $Sp(2m,q)$. The main result of this paper, Theorem [Theorem 14](#symplectic_group_as_Galois_group){reference-type="ref" reference="symplectic_group_as_Galois_group"} in Section 5, is that there is a large number (of the order of $q^{m-1})$ of polynomials $\Phi$ for which the Galois group of $L$ is the full symplectic group. The key to the proof of Theorem [Theorem 14](#symplectic_group_as_Galois_group){reference-type="ref" reference="symplectic_group_as_Galois_group"} is Corollary [Corollary 12](#is_the_whole_symplectic_group){reference-type="ref" reference="is_the_whole_symplectic_group"} whose proof is developed in Sections 3 and 4. The main structural result used is the fact that the symplectic group modulo its centre is a simple group, with three small exceptions, and the proof is elementary in that sense. The Inverse Galois Problem for a field $K$ is the problem of determining which finite groups can occur as the Galois group of a Galois extension of $K$. In this article we present an elementary and explicit construction of polynomials over $K=\mathop{\mathrm{\mathbb{F}}}_q(t)$ with Galois group $Sp(2m,q)$. Elkies shows in [@E], Theorem 7, that if the coefficients of certain $q$-palindromic polynomials are algebraically independent transcendentals, the resulting Galois group is the symplectic group. Using the Hilbert Irreducibility Theorem, specialization of the coefficients in these polynomials to elements of $\mathop{\mathrm{\mathbb{F}}}_q$ in some suitable way shows the existence of polynomials over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ with Galois group $Sp(2m,q)$, but we hope our self-contained approach is reasonably accessible and is also constructive. # Galois theory relating to ring extensions Let $L$ be a $q$-polynomial of $q$-degree $n$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$. As is often the case when working over the field $\mathop{\mathrm{\mathbb{F}}}_q(t)$, we specialize the indeterminate $t$ to some value in $\mathop{\mathrm{\mathbb{F}}}_q$ and then examine the properties of the resulting polynomial. This section is devoted to establishing the framework required for this type of study, and it relies on the exposition given by Lang, [@L], pp.340-346. (We remark that the proofs given by Lang are apparently flawed with respect to dealing with problems of inseparability but these do not arise for us, as the residue fields we use are finite.) Let $I=I_\mu$ be the prime ideal in $\mathop{\mathrm{\mathbb{F}}}_q[t]$ generated by the linear polynomial $t-\mu$, where $\mu\in \mathop{\mathrm{\mathbb{F}}}_q$. Let $R$ be the integral closure of $\mathop{\mathrm{\mathbb{F}}}_q[t]$ in the splitting field $E$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$. There are only finitely many prime ideals of $R$ lying above $I$, and the Galois group $G$ permutes these ideals transitively in its action on $R$. Thus, let $J$ be a prime ideal of $R$ lying over $I$ and let $D=D_J$ be the subgroup of $G$ that fixes $J$. We call $D$ the decomposition group connected to this configuration. If we choose a different prime ideal lying above $I$, we obtain another decomposition group but it is conjugate to $D$ on account of the transitive $G$-action on prime ideals. Thus we have a uniquely determined $G$-conjugacy class of decomposition groups, and we have chosen one member of the class. We consider the quotient ring $\overline{R}=R/J$. This is a finite field extension of $\mathop{\mathrm{\mathbb{F}}}_q[t]/I=\mathop{\mathrm{\mathbb{F}}}_q$. Given $z$ in $R$, we let $\overline{z}=z+J\in \overline{R}$. The map sending $z$ to $\overline{z}$ is a ring homomorphism from $R$ onto $\overline{R}$, and it is also $\mathop{\mathrm{\mathbb{F}}}_q$-linear. We call this the reduction homomorphism. We extend the reduction homomorphism to a ring homomorphism from $R[x] \longrightarrow \overline{R}[x]$ by acting on the coefficients. We apply this to $L$ (which has coefficients in $\mathop{\mathrm{\mathbb{F}}}_q[t]$ by assumption) to obtain a polynomial $\overline{L}$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$. Given the assumption that $L$ is monic of $q$-degree $n$, $\overline{L}$ is also a monic $q$-polynomial of $q$-degree $n$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$. There is an induced action of $D$ on $\overline{R}$, defined as follows. Given $\sigma\in D$, define $\overline{\sigma}$ acting on $\overline{R}$ by $$\overline{\sigma}(\overline{z})=\overline{\sigma(z)}$$ for $z$ in $R$. This is well-defined, since $D$ fixes $J$ as a set. The map sending $\sigma$ to $\overline{\sigma}$ is a homomorphism from $D$ onto a subgroup $\overline{D}$ of $\mathop{\mathrm{\mathbb{F}}}_q$-automorphisms of the finite field $\overline{R}$. The kernel of this homomorphism is called the inertia group. Provided that $\overline{L}$ has no repeated roots (equivalently, provided that the $x$ term of $\overline{L}$ is nonzero), the inertia group is trivial and $D$ is isomorphic to $\overline{D}$, and $\overline{D}$ is the full group of $\mathop{\mathrm{\mathbb{F}}}_q$-automorphisms of $\overline{R}$. Thus, in this unramified case, since $\overline{D}$ is cyclic, $D$ is also cyclic, and we may choose a generator $\sigma$ of $D$ so that $$\overline{\sigma}(\overline{z})=(\overline{z})^q$$ for all $z$ in $R$. We call the element $\sigma$ the Frobenius element of $G$ for this configuration, as its induced action is the Frobenius $q$-power map. Note that $\mu\in \mathop{\mathrm{\mathbb{F}}}_q$ determines the element. Different powers of $\sigma$ determine other powers of $q$ in the induced action. Technically, we really only have a Frobenius conjugacy class in $G$, but this is sufficient for our purposes. ## Galois Theory and Linear Algebra In the context of $q$-polynomials, we have an enhanced structure which we intend to exploit to obtain more information about the conjugacy class of $\sigma$ and about its action on the space of roots of $L$. Recall that we have a vector space of roots of $L$ in $E$. More specifically, $V$ is an $n$-dimensional $\mathop{\mathrm{\mathbb{F}}}_q$-vector space contained in $R$, and $D$ acts as $\mathop{\mathrm{\mathbb{F}}}_q$-linear automorphisms of $V$. Under the assumption that the $x$ term of $\overline{L}$ is nonzero, we also have a vector space $\overline{V}\subseteq \overline{R}$ of roots of $\overline{L}$, which is $n$-dimensional over $\mathop{\mathrm{\mathbb{F}}}_q$. Note that given $\alpha$ in $V$, $\overline{\alpha}$ is in $\overline{V}$. **Lemma 2**. *The map $\phi:V\to \overline{V}$ given by $$\phi(\alpha)=\overline{\alpha}$$ is an $\mathop{\mathrm{\mathbb{F}}}_q$-linear isomorphism.* This is clear, since the reduction map is $\mathop{\mathrm{\mathbb{F}}}_q$-linear and the two spaces have the same dimension, given the assumption on the $x$ term in $\overline{L}$. Additionally, we have an action of $D$ on $V$ and of $\overline{D}$ on $\overline{V}$. These actions are $\mathop{\mathrm{\mathbb{F}}}_q$-isomorphic, as we now show. **Lemma 3**. *The map $\phi:V\to \overline{V}$ is an $\mathop{\mathrm{\mathbb{F}}}_q$-linear isomorphism between the actions of $D$ and $\overline{D}$ on the respective vector spaces.* *Proof.* We have by definition $$\overline{\sigma}(\overline{\alpha})=\overline{\sigma(\alpha)}$$ for $\sigma$ in $D$ and $\alpha$ in $V$. Thus, $$\phi(\sigma\alpha)=\overline{\sigma(\alpha)}=\overline{\sigma}(\overline{\alpha})= \overline{\sigma}\phi(\alpha)$$ and this is what we require for an $\mathop{\mathrm{\mathbb{F}}}_q$-isomorphism of actions. ◻ We are going to study the $\mathop{\mathrm{\mathbb{F}}}_q$-linear actions of $D$ and $\overline{D}$ on the respective vector spaces, and in particular we are going to study the minimal polynomials of the group elements considered as linear transformations. **Corollary 4**. *With the notation as before, suppose that the $x$ term of the reduced polynomial $\overline{L}$ is nonzero. Then the minimal polynomial of the Frobenius element $\sigma$ in $G$ (defined with respect to the degree $1$ ideal $I$) acting on the space of roots $V$ of $L$ is equal to the minimal polynomial of the Frobenius $q$-power map acting on the space of roots $\overline{V}$ of $\overline{L}$.* Thus although $\sigma$ need not act as a $q$-power mapping on $V$, we can determine its minimal polynomial by looking at the action of $\overline{\sigma}$ on $\overline{V}$, which is a $q$-power mapping. What we think is the main significance of this explanation is that, in principle, it is straightforward to find the minimal polynomial of the $q$-power mapping on the space of roots of a $q$-polynomial with coefficients in $\mathop{\mathrm{\mathbb{F}}}_q$, as the next result reveals. **Theorem 5**. *Let $P$ be a monic $q$-polynomial of $q$-degree $n$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ with $$P(x)= x^{q^n}+a_{n-1}x^{q^{n-1}}+\cdots +a_0 x,$$ where $a_0\neq 0$. Let $E$ be a splitting field for $P$ over $\mathop{\mathrm{\mathbb{F}}}_q$ and let $W$ be the space of roots of $P$ in $E$. Let $F$ be the Frobenius $q$-power mapping of $W$ into itself. Then the minimal polynomial $m(x)$ of $F$ acting on $W$ is $$x^n+a_{n-1}x^{n-1}+\cdots +a_0.$$ Moreover, $W$ is cyclic with respect to the action of $F$, so that there is a root $\alpha$, say, of $P$ such that $\alpha$, $F(\alpha)$, ..., $F^{n-1}(\alpha)$ are an $\mathop{\mathrm{\mathbb{F}}}_q$-basis of $W$.* *Proof.* Let $\alpha$ be an element of $W$. Then we have $F^i(\alpha)=\alpha^{q^i}$ for all positive integers $i$. Since $P(\alpha)=0$, it is clear that $$m(F)(\alpha)=0,$$ so that $m(F)$ is the zero transformation of $W$. Suppose if possible that there is a monic polynomial $g$, say, in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ of degree $d$ less than $n$ such that $g(F)=0$ on $W$. Then if we set $$g(x)=x^d+b_{d-1}x^{d-1}+\cdots +b_0,$$ we find that the elements of $W$ are roots of the $q$-polynomial $$Q(x)=x^{q^d}+b_{d-1}x^{q^{d-1}}+\cdots +b_0 x,$$ of $q$-degree $d$. But $Q$ has at most $q^d$ roots in its splitting field, whereas we have $q^n$ roots in $W$ which are also roots of $Q$. This is a contradiction and hence $m(x)$ is the minimal polynomial of $F$. Finally, concerning cyclicity, the minimal polynomial of $F$ acting on $W$ has degree $n=\dim W$. It is a basic theorem of linear algebra, [@J], Exercise 7, p.202, for example, that $W$ is cyclic with respect to $F$. (The result is a simple consequence of the theory of the rational canonical form.) This implies in addition that $m(x)$ is also the characteristic polynomial of $F$. ◻ The appearance of the polynomial $m(x)$ in this description is well known in the theory of $q$-polynomials over $\mathop{\mathrm{\mathbb{F}}}_q$, as described for example in [@LN], Chapter 3, Section 3, where $m(x)$ is called the conventional associate of the $q$-polynomial. Much of this theory was developed by Ore in the 1930's. In our opinion, the interpretation of the conventional associate as the minimal polynomial of the Frobenius mapping makes more sense and enables various theorems relating to $q$-polynomials to be proved by simple principles of theory of linear transformations. # Some results from linear algebra To prove our main theorem, we require some results from elementary linear algebra, most of which we will prove. We start with what is called the primary decomposition of a vector space with respect to a linear transformation. See, for example, [@J], Theorem 3.11, p.191. **Lemma 6**. *Let $W$ be a finite dimensional vector space over $\mathop{\mathrm{\mathbb{F}}}_q$ and let $\sigma$ be an $\mathop{\mathrm{\mathbb{F}}}_q$-linear transformation of $W$ into itself. Let $m(x)$ be the minimal polynomial of $\sigma$ acting on $W$. Let $$m(x)=p_1(x)^{r_1}\cdots p_t(x)^{r_t}$$ be the factorization of $m(x)$ into powers of different irreducible monic polynomials $p_i(x)$, with multiplicity $r_i$ for $1\leq i\leq t$. Then there is a unique decomposition $$W=W_1\oplus \cdots \oplus W_t,$$ of $W$ into $\sigma$-invariant subspaces $W_i$, where the minimal polynomial of $\sigma$ acting on $W_i$ is $p_i(x)^{r_i}$, $1\leq i\leq t$.* Note that, along similar lines, if we have a factorization $m(x)=m_1(x)m_2(x)$ of the minimal polynomial of $\sigma$, where $m_1$ and $m_2$ are relatively prime, there is a corresponding decomposition $$W=W_1\oplus W_2$$ of $W$ into $\sigma$-invariant subspaces $W_i$, where the minimal polynomial of $\sigma$ acting on $W_i$ is $m_i$. The following result is well known, but we include a proof for the sake of completeness. **Lemma 7**. *Let $W$ be a finite dimensional vector space over $\mathop{\mathrm{\mathbb{F}}}_q$ and let $\sigma$ be an $\mathop{\mathrm{\mathbb{F}}}_q$-linear transformation of $W$ into itself. Suppose that $\sigma$ acts on $W$ as a cyclic transformation, with minimal polynomial $m(x)$. Suppose that $$m(x)=p_1(x)\cdots p_t(x)$$ where the $p_i(x)$ is a monic irreducible polynomial of degree $d_i$, for $1\leq i\leq t$, with no $p_i$ equal to $x$. Then the order of $\sigma$ in the group of automorphisms of $W$ is a divisor of $q^d-1$, where $d$ is the least common multiple of the $d_i$. In particular the order of $\sigma$ is relatively prime to the characteristic $p$ of $\mathop{\mathrm{\mathbb{F}}}_q$.* *Proof.* We first prove the lemma when $t=1$, so that $m(x)$ is irreducible in this case. Then if $\deg m=r$, $m$ divides $x^{q^r-1}-1$ (note that we are of course assuming that $m$ is different from $x$). Thus since $m(\sigma)=0$, it follows that $$\sigma^{q^r-1}=I$$ and the result is proved in this case. Now we consider the general case. By the primary decomposition, $$W=W_1\oplus \cdots \oplus W_t,$$ where $\sigma$ acts on $W_i$ with minimal polynomial $p_i$. Let $\sigma_i$ denote the restriction of $\sigma$ to $W_i$. Then by the previous argument, $$\sigma_i^{q^{d_i}-1}=I_{W_i}$$ for each $i$. It follows that $\sigma^{q^d-1}$ is the identity on $W$, where $d$ is the lcm of the $d_i$. ◻ We consider further refinements of these ideas, which again are well known in the theory of finite groups and of algebraic groups. Let $\sigma$ be an invertible linear transformation of the finite dimensional $\mathop{\mathrm{\mathbb{F}}}_q$-vector space $W$. Let $m(x)$ be the minimal polynomial of $\sigma$ and write $$m(x)=(x-1)^rm_2(x),$$ where $x-1$ is relatively prime to $m_2$. Suppose that $m_2$ is a product of different monic irreducible polynomials (none of which are equal to $x$). Let $$W=W_1\oplus W_2$$ be the corresponding decomposition of $W$ into $\sigma$-invariant subspaces $W_1$ and $W_2$. Let $\sigma_i$ be restriction of $\sigma$ to $W_i$ for each $i$. Extend $\sigma_1$ to act on the whole of $W$ by defining the extension to act trivially on $W_2$, and extend $\sigma_2$ by defining the extension to act trivially on $W_1$. Retaining this notation, we have the following result. **Lemma 8**. *We have the commuting factorization $$\sigma=\sigma_1\sigma_2=\sigma_2\sigma_1.$$ The order of $\sigma_1$ is a power of $p$ and the order of $\sigma_2$ is relatively prime to $p$. Each of $\sigma_1$ and $\sigma_2$ is a power of $\sigma$.* *Proof.* It is clear from the definition that the extensions commute and $\sigma$ is their product. It is similarly clear from its definition that $\sigma_1$ has minimal polynomial $(x-1)^r$. Thus $(\sigma_1-I)^r=0$. Let $p^a$ be a power of $p$ that satisfies $p^a\geq r$. Then we have $$(\sigma_1-I)^{p^a}=(\sigma_1-I)^{r}(\sigma_1-I)^{p^a-r}=0.$$ But as we are working in characteristic $p$, $$0=(\sigma_1-I)^{p^a}=\sigma_1^{p^a}-I$$ and thus we see that $\sigma_1$ has order dividing $p^a$. Lemma [Lemma 7](#order_prime_to_p){reference-type="ref" reference="order_prime_to_p"} shows that $\sigma_2$ has order dividing $q^d-1$ for some positive integer $d$. Now since $p^a$ and $q^d-1$ are relatively prime, we can find integers $\alpha$ and $\beta$ such that $$\alpha p^a+\beta(q^d-1)=1.$$ We then deduce that $$\sigma_1=\sigma_1^{\beta(q^d-1)},\quad \sigma_2=\sigma_2^{\alpha p^a}.$$ Since $\sigma_1$ and $\sigma_2$ commute, we obtain $$\sigma^{\alpha p^a}=\sigma_1^{\alpha p^a}\sigma_2^{\alpha p^a}=\sigma_2$$ and likewise $$\sigma^{\beta (q^d-1)}=\sigma_1^{\beta (q^d-1)}\sigma_2^{\beta (q^d-1)}=\sigma_1.$$ Thus we have shown that $\sigma_1$ and $\sigma_2$ are powers of $\sigma$, as required. ◻ The element $\sigma_1$ is called the unipotent or $p$-part of $\sigma$ and $\sigma_2$ is called the semisimple or $p$-regular part of $\sigma$. # Transvections and symplectic transvections We will make use of a special type of linear transformation, known as a transvection, to force our Galois groups to be large known groups. Transvections have been used as a key tool in the study of linear groups since the foundations of the subject and so it is not surprising that they prove useful to us. We begin therefore with a discussion of transvections, all of which is well known, but we hope this will assist nonspecialists to follow the arguments. Let $W$ be a vector space of finite dimension $n$ over $\mathop{\mathrm{\mathbb{F}}}_q$. A linear transformation $\sigma$ of $W$ into itself is called a *transvection* if its minimal polynomial is $(x-1)^2$ and $\sigma-I$ has rank one. Note that in characteristic $p$, a transvection $\sigma$ has order $p$. For if $(\sigma-I)^2=0$, then $(\sigma-I)^p=\sigma^p-I=0$, and thus $\sigma^p=I$. Let $W_1$ be the kernel of $\sigma-I$. It is a subspace of dimension $n-1$ in $W$ and hence can be realized as the kernel of a nonzero linear functional $\phi$, say, defined on $W$. (Recall that a linear functional is an $\mathop{\mathrm{\mathbb{F}}}_q$-linear map from $W$ into $\mathop{\mathrm{\mathbb{F}}}_q$.) Let $u$ be a nonzero vector in the image of $\sigma-I$. Then we must have $$(\sigma-I)(w)=\phi(w)u$$ for all $w$ in $W$. As we want $(\sigma-I)^2=0$, $u$ must be in the kernel of $\phi$. We can reverse this procedure. We start with a nonzero linear functional $\phi$ defined on $W$ and a nonzero element $u$ in $\ker \phi$. Then we define $\sigma=\sigma_{\phi,u}$ by $$\sigma(w)=w+\phi(w)u$$ for all $w$ in $W$. The resulting $\sigma$ is a transvection. There is some ambiguity here, for if we replace $\phi$ by a nonzero scalar multiple $\mu \phi$ and $u$ by $\mu^{-1}u$ we obtain the same transvection but otherwise we have a unique procedure. We can calculate the number of transvections defined on $W$ from this observation. We have $q^n-1$ choices for $\phi$ and $q^{n-1}-1$ choices for nonzero $u$ in $\ker \phi$. This gives $(q^n-1)(q^{n-1}-1)$ choices for ordered pairs of $\phi$ and $u$, but we divide this number by $q-1$ to obtain the number of transvections because of the lack of uniqueness just explained. The total number of transvections is $$\frac{(q^n-1)(q^{n-1}-1)}{q-1}.$$ ## Symplectic Transvections Suppose that there is a nondegenerate alternating bilinear form $B$ defined on $W\times W$. Recall that $n=\dim W$ must be even for a nondegenerate $B$ to exist. We say that a transvection $\sigma$ defined on $W$ is a symplectic transvection (with respect to $B$) if it is also an isometry of $B$, meaning that we have $$B(\sigma u,\sigma w)=B(u,w)$$ for all $u$ and $w$ in $W$. Suppose then that $\sigma$ is a symplectic transvection. We know that we can represent $\sigma$ in the form $$\sigma(w)=w+\phi(w)u$$ for some $u\in W$, where $u\in \ker \phi$. Since $B$ is nondegenerate, we can find some $z$ in $W$ such that $$\phi(w)=B(w,z)$$ for all $w$ in $W$. A small manipulation shows that $z$ is a scalar multiple of $u$, say $z=cu$ for some nonzero $c$ in $\mathop{\mathrm{\mathbb{F}}}_q$. Then we have $$\sigma(w)=w+cB(w,u)u$$ and this is the usual expression for a symplectic transvection. Conversely, any $\sigma$ defined in this way is an isometry of $B$ and thus is a symplectic transvection. Concerning the scalar $c$, we make the following observation. Suppose that $c$ is a square in $\mathop{\mathrm{\mathbb{F}}}_q$, say $c=d^2$. Then we set $v=du$ and we obtain $$\sigma(w)=w+B(w,v).$$ In this case, we see that there is no need for the appearance of the scalar $c$ (since $v$ serves as well as $u$ for our purpose). We deduce that if $q$ is a power of 2, all symplectic transvections are expressible by the formula $$\sigma(w)=w+B(w,u)u$$ as $u$ runs over the nonzero elements of $W$. Consequently, there are $q^n-1$ of these transvections, where $n=\dim W$. Suppose now that $q$ is odd. It is easy to see that $u$ and $-u$ determine the same symplectic transvection and this is the only way in which duplication occurs. Thus there are $(q^n-1)/2$ symplectic transvections of this form when $c$ is a square. Let $c$ be a nonsquare in $\mathop{\mathrm{\mathbb{F}}}_q$. Then symplectic transvections of the form $$\sigma(w)=w+cB(w,u)u$$ cannot be expressed as $w\to w+B(w,v)v$ for any choice of $v$. Thus, for this fixed nonsquare $c$, we obtain a further $(q^n-1)/2$ symplectic transvections and we have thus accounted for all $q^n-1$ symplectic transvections. As we now explain, for odd $q$ they fall into two conjugacy classes under the action of the symplectic group (which is the isometry group of $B$). **Lemma 9**. *With the notation as developed, let $\sigma$ and $\tau$ be symplectic transvections acting on $W$, defined by $$\sigma(w)=w+cB(w,u)u, \quad \tau(w)=w+cB(w,v)v,$$ where $u$ and $v$ are nonzero elements of $W$ and $c$ is a nonzero scalar. Then there is an isometry $g$ of $B$ (in other words, an element of the symplectic group defined by $B$) such that $$\tau=g \sigma g^{-1}.$$* *Proof.* It is well known, and easy to prove, that the symplectic group acts transitively on the nonzero elements of $W$. Thus there is an isometry, $g$, say, of $B$ such that $gu=v$. We now calculate as follows. We have $$\tau g(w)=g(w)+cB(gw,v)v$$ and likewise $$g\sigma(w)=g(w)+cB(\sigma w,u)g(u).$$ Thus since $gu=v$, all we have to do to complete the proof is to show that $$B(gw, v)=B(\sigma w, u)$$ holds for all $w$. Now as $g$ is an isometry of $B$, $$B(gw,v)=B(w, g^{-1}v)=B(w,u).$$ But $\sigma$ is also an isometry and it fixes $u$. Therefore, $$B(\sigma w, u)=B(\sigma w, \sigma u)=B(w,u),$$ and this completes the proof. ◻ **Corollary 10**. *Suppose that $q$ is a power of $2$. Then the symplectic group $Sp(2m,q)$ contains a single conjugacy class of symplectic transvections, which has size $q^{2m}-1$.* *If $q$ is odd, $Sp(2m,q)$ contains exactly two conjugacy classes of symplectic transvections, and each such class has size $(q^{2m}-1)/2$.* An interesting point emerges in the proof of this lemma. Let $G$ be a subgroup of $Sp(2m,q)$ that acts transitively on the nonzero elements of $W$. Then $G$ has transitive actions on the conjugacy classes of symplectic transvections in $Sp(2m,q)$. Thus, if $G$ contains a symplectic transvection, it contains the entire conjugacy class of symplectic transvections corresponding to the given scalar $c$. In view of the fact that $Sp(2m,q)$ modulo its central subgroup of order 2 (or order 1 in characteristic 2) is a finite simple group (with three small exceptions), this is enough to prove that $G$ is equal to $Sp(2m,q)$. This is an idea which we will use to identify some Galois groups in the next section. We now provide the details. **Lemma 11**. *Let $G$ be a finite group and let $H$ be a subgroup of $G$. Suppose that $H$ contains a conjugacy class $K$ of $G$. Then $H$ contains a normal subgroup of $G$ that contains $K$.* *Proof.* Given $g\in G$, the subgroup $g^{-1}Hg$ of $G$ is called a conjugate of $H$. The intersection of all the conjugates of $H$ is a normal subgroup of $G$ contained in $H$. It is called the core of $H$. Since $K$ is contained in all conjugates of $H$, as it is invariant under $G$-conjugation, $K$ is contained in the core of $H$, as required. ◻ **Corollary 12**. *Let $G$ be a subgroup of the symplectic group $Sp(2m,q)$. Suppose that $G$ acts transitively on the nonzero elements of the underlying vector space of dimension $2m$. Suppose also that $G$ contains a transvection. Then $G=Sp(2m,q)$.* *Proof.* Let $\sigma$ be a symplectic transvection contained in $G$. The proof of Lemma [Lemma 9](#conjugate_elements){reference-type="ref" reference="conjugate_elements"} implies that $G$ contains the $Sp(2m,q)$ conjugacy class $K$, say, that contains $\sigma$. Then Lemma [Lemma 11](#contains_conjugacy_class){reference-type="ref" reference="contains_conjugacy_class"} implies that $G$ contains a normal subgroup of $Sp(2m,q)$ containing $K$. We consider various cases. Suppose that $q$ is a power of 2. Then $Sp(2m,q)$ is simple unless $q=2$ and $m=1$ or $m=2$. See, for example,[@J], Theorem 6.16, p.396. Thus the corollary is clear when $q$ is a power of 2, except possibly when $m=1$ or $m=2$ and $q=2$. The only proper normal subgroup of $Sp(2,2)$ has order 3 and hence does not contain a transvection, which has order 2. The group $Sp(4,2)$ is isomorphic to the symmetric group $S_6$, whose only normal subgroup is the alternating group $A_6$, of index 2. Thus, we wish to know whether $A_6$ contains a transvection. This can be solved quickly by appealing to the fact that $Sp(4,2)$ is generated by symplectic transvections and there is a single conjugacy class of these. See, for example, [@J], Lemma 1, p.392. Thus, no symplectic transvection can be contained in a proper normal subgroup. Alternatively, we can argue from the identification made above of the groups involved. The group $S_6$ contains three conjugacy classes of involutions (elements of order two). Two conjugacy classes have size 15, the third has size 45. Since this third class consists of elements expressible as products of two disjoint transpositions, it lies in $A_6$, but the other two conjugacy classes do not lie in $A_6$. The conjugacy class of two disjoint transpositions forms the single class of involutions in $A_6$, and thus this class has size 45. Since the size of a conjugacy class of transvections is 15, it follows that $A_6$ contains no transvections. Finally, we consider odd values of $q$. Let $Z$ be the central subgroup of $Sp(2m,q)$ generated by $-I$. Then, apart from the case when $m=1$ and $q=3$, $Z$ is the only proper normal subgroup of $Sp(2m,q)$. See, again, [@J], Theorem 6.16, p.396. Since $-I$ is not a transvection, we are finished apart from examining what happens in $Sp(2,3)$. In addition to $Z$, $Sp(2,3)$ has one other proper normal subgroup, which has order 8. Since the order of a transvection is 3 in characteristic 3, no proper normal subgroup of $Sp(2,3)$ contains a transvection. This completes the proof. ◻ # The symplectic group as a Galois group Let $\Phi$ be a monic $q$-polynomial of even $q$-degree $2m$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$. Write $$\Phi=\sum_{i=0}^{2m} c_i x^{q^i},$$ where the coefficients $c_i$ are in $\mathop{\mathrm{\mathbb{F}}}_q$, $c_0=c_{2m}=1$, and $$c_{2m-i}=c_i$$ for all $i$. Thus $\Phi$ is $q$-palindromic. We form the $q$-polynomial $$L=L(x)=\Phi+tx^{q^m}$$ in $\mathop{\mathrm{\mathbb{F}}}_q(t)[x]$. It is straightforward to see that $L(x)/x$ is irreducible over $\mathop{\mathrm{\mathbb{F}}}_q(t)$. Let $G$ be its Galois group over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ and let $V$ be its space of roots in its splitting field. By Galois theory, as $L(x)/x$ is irreducible, $G$ acts transitively on the nonzero elements of $V$. Since $\Phi$ is $q$-palindromic, so also is $L$. Thus, by Elkies's theorem, there is a nondegenerate alternating bilinear form defined on $V\times V$ which is invariant under $G$-action. Thus $G$ is a subgroup of the symplectic group $Sp(2m,q)$. We intend to specialize $t$ to the value 0. Then the corresponding reduced polynomial $\overline{L}$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ is $\Phi$. Since $\Phi$ has no repeated roots, there is a corresponding Frobenius element $\sigma$ in $G$. Using Theorem [Theorem 5](#minimal_polynomial){reference-type="ref" reference="minimal_polynomial"}, this element acts as cyclic element on $V$ and its minimal polynomial is $$x^{2m}+c_{2m-1}x^{2m-1}+\cdots + c_1x+1.$$ Call this polynomial $M(x)$. Because the coefficients $c_i$ satisfy $c_i=c_{2m-i}$ for all $i$, $M(x)$ is self-reciprocal, as we already observed. Our intention is to show that there are many choices of $\Phi$, and corresponding $M(x)$, that guarantee that some power of the corresponding Frobenius element $\sigma$ is a symplectic transvection. In these cases, Corollary [Corollary 12](#is_the_whole_symplectic_group){reference-type="ref" reference="is_the_whole_symplectic_group"} will enable us to conclude that $G$ is $Sp(2m,q)$. To prepare the ground for this task, we want to show how the self-reciprocal nature of $M(x)$ imposes restrictions on its irreducible factors. ## Some analysis of self-reciprocal polynomials Given a monic polynomial $f$ of degree $d$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ with $f(0)\neq 0$, we set $$f^*(x)=x^df(0)^{-1}f(x^{-1}).$$ We note that $f^*$ is also monic of degree $d$ and it is irreducible if $f$ is itself irreducible. We say that $f$ is self-reciprocal if $f=f^*$. Consider now a monic self-reciprocal polynomial $F$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ with $F(0)\neq 0$. It is an automatic consequence of the definition that $F(0)=\pm 1$. We shall also assume that $F(0)=1$, which is thus assured when $q$ is even. Let $f$ be a monic irreducible factor of $F$. It follows that $f^*$ is also an irreducible factor of $F$. Therefore, either $f^*$ is relatively prime to $f$, and each of $f$ and $f^*$ is a factor of $F$ with the same multiplicity, or $f=f^*$ and thus $f$ is self-reciprocal. Suppose now that $f$ is a monic self-reciprocal irreducible factor of $F$ of degree $d$. Then by definition $$f(x)=f(0)^{-1}x^df(x^{-1}).$$ We substitute $x=1$ to obtain $$f(1)=f(0)^{-1}f(1).$$ This yields two possibilities: either $f(1)=0$ or $f(0)=1$. Since $f$ is irreducible, $f(1)=0$ implies that $f(x)=x-1$. Similarly, we substitute $x=-1$ to obtain $$f(-1)=f(0)^{-1}(-1)^df(-1).$$ Thus we either have $f(-1)=0$ or $f(0)=(-1)^d$. The first possibility implies that $f(x)=x+1$, given that $f$ is irreducible. Putting together these possibilities we deduce that $f(x)=x\pm 1$ or $f$ is different from these two polynomials and $f(0)=1=(-1)^d$. Thus if $f$ is not equal to $x\pm 1$ and $q$ is odd, $d$ is even. This tells us that an irreducible self-reciprocal factor of $F$ different from the two linear polynomials described has even degree in odd characteristic. In fact the even degree conclusion holds in characteristic 2 as well, as we now indicate. Recalling that $f(0)=1$, write $$f(x)=x^d+b_{d-1}x^{d-1}+\cdots +b_1x+1.$$ Then we have $b_i=b_{d-i}$ for all $i$, as $f$ is self-reciprocal. Suppose that $q$ is even and, by way of contradiction, $d$ is odd. Then pairing each coefficient $b_i$ with its equal coefficient $b_{d-i}$, we find that $f(1)=0$. This implies that $f=x+1$. Therefore, $f$ has even degree if it is irreducible, self-reciprocal and different from $x+1$ when we are in even characteristic. So far, we have not used the assumption that $F(0)=1$, which as we have noted above is guaranteed when $q$ is even. Write $$F(x)=(x-1)^r(x+1)^sF_1(x),$$ where $x\pm 1$ do not divide $F_1$. It is easy to verify, using the arguments that we have employed above, that $F_1$ is self-reciprocal of even degree and $F_1(0)=1$. Thus, assuming that $q$ is odd, the assumption that $F(0)=1$ implies that $r$ is even. Then $s$ is also even, as the degree of $F$ is even. On the other hand, if $q$ is even, we omit the $x-1$ factor, as it is the same as the $x+1$ factor, and argue again that $s$ is even. We summarize this discussion in the form of a theorem. The result is well known to specialists in the theory of classical linear groups, but perhaps less familiar to nonspecialists. **Theorem 13**. *Let $F(x)$ be a monic self-reciprocal polynomial of even degree in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ with $F(0)=1$. Then the polynomials $x\pm 1$ are factors of $F$ each with even multiplicity. Any monic self-reciprocal irreducible factor of $F$ different from $x\pm 1$ has even degree and its constant term is $1$. Any monic irreducible factor $f$ of $F$ that is different from $f^*$ occurs with the same multiplicity as $f^*$.* This theorem applies to the minimal polynomial $M(x)$ of a Frobenius element $\sigma$ in the Galois group $G$. We note that the process is reversible, in the sense that we can construct monic self-reciprocal polynomials of even degree in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ with $F(0)=1$ in this way. We do this in the next section. ## Main Theorem Let us summarize the procedure so far. We take any monic self-reciprocal polynomial $M(x)$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$ of even degree $2m$ and satisfying $M(0)=1$. Say $$M(x)=x^{2m}+c_{2m-1}x^{2m-1}+\cdots +c_1 x+1,$$ where the coefficients $c_i$ satisfy $c_i=c_{2m-i}$. Define the $q$-polynomial $\Phi$ by $$\Phi(x)=x^{q^{2m}}+c_{2m-1}x^{q^{2m-1}}+\cdots +c_1 x^q+x$$ which is the linearized associate of $M(x)$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$. Define $L$ by $$L(x)=\Phi(x)+tx^{q^m}$$ in $\mathop{\mathrm{\mathbb{F}}}_q(t)[x]$. Then $L(x)/x$ is irreducible. If $G$ denotes the Galois group of $L$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$, we know from Sections 2 and 3 that the Frobenius element of $G$ with respect to the specialization of $t$ to 0 has minimal polynomial $M(x)$ in its action on the space of roots of $L$. In order to prove that $G$ is the full symplectic group, we now adopt the following specific construction. We choose $M(x)$ so that $x-1$ is a factor with multiplicity two and $M(x)/(x-1)^2$ is a self-reciprocal polynomial with no factor equal to $x\pm 1$ and all irreducible factors having multiplicity one. This presents no obstacles. Here is our main theorem. **Theorem 14**. *Let $$M(x)=x^{2m}+c_{2m-1}x^{2m-1}+\cdots +c_1 x+1,$$ be a monic self-reciprocal polynomial of degree $2m$ in $\mathop{\mathrm{\mathbb{F}}}_q[x]$, where the coefficients $c_i$ satisfy $c_i=c_{2m-i}$. Suppose that $x-1$ is a factor of $M(x)$ with multiplicity two and suppose that $M(x)/(x-1)^2$ has no factor equal to $x\pm 1$ and all its irreducible factors have multiplicity one. Define the $q$-polynomial $\Phi \in \mathop{\mathrm{\mathbb{F}}}_q[x]$ by $$\Phi(x)=x^{q^{2m}}+c_{2m-1}x^{q^{2m-1}}+\cdots +c_1 x^q+x$$ and define the $q$-polynomial $L\in \mathop{\mathrm{\mathbb{F}}}_q(t)[x]$ by $$L(x)=\Phi(x)+tx^{q^m}.$$ Then the Galois group $G$ of $L$ over $\mathop{\mathrm{\mathbb{F}}}_q(t)$ is $Sp(2m,q)$.* *Proof.* Let $V$ be the space of roots of $L$ in its splitting field over $\mathop{\mathrm{\mathbb{F}}}_q(t)$. As $\Phi$ is $q$-palindromic, there is a nondegenerate alternating bilinear form defined on $V\times V$ which is $G$-invariant. Thus $G$ is a subgroup of $Sp(2m,q)$ that acts transitively on the nonzero elements of the underlying vector space, since as we have observed, $L(x)/x$ is irreducible in $\mathop{\mathrm{\mathbb{F}}}_q(t)[x]$ . Let $\sigma$ be the Frobenius element of $G$ defined by the specialization of $t$ to 0. We know then that $\sigma$ has minimal polynomial $M(x)$ when it acts on $V$, by Theorem [Theorem 5](#minimal_polynomial){reference-type="ref" reference="minimal_polynomial"}. Lemma [Lemma 8](#extension_lemma){reference-type="ref" reference="extension_lemma"} and its proof imply that we can write $$\sigma=\sigma_1\sigma_2=\sigma_2\sigma_1,$$ where $\sigma_1$ has minimal polynomial $(x-1)^2$ and fixes pointwise a hyperplane of $V$, and $\sigma_2$ has order prime to $p$. Thus $\sigma_1$ is a transvection. In addition, Lemma [Lemma 8](#extension_lemma){reference-type="ref" reference="extension_lemma"} shows that $\sigma_1$ is a power of $\sigma$ and hence $\sigma_1$ is in $G$. Thus $\sigma_1$ is a symplectic transvection. Corollary [Corollary 12](#is_the_whole_symplectic_group){reference-type="ref" reference="is_the_whole_symplectic_group"} implies that $G$ is $Sp(2m,q)$. ◻ Note that when $m=1$, there is a unique polynomial $L=x^{q^2}+tx^q+x$, and its Galois group is $SL(2,q)=Sp(2,q)$. If we examine this construction, $\sigma_2$ defines an element of $Sp(2m-2,q)$, as it fixes elementwise a two-dimensional subspace $V_1$, say, and acts as a symplectic isometry on a complementary subspace of dimension $2m-2$, $V_2$, say. $V$ is the orthogonal direct sum of $V_1$ and $V_2$. The minimal polynomial of $\sigma_2$ acting on $V_2$ is $M_1(x)$. The properties of $M_1(x)$ in respect of its irreducible factors guarantee that $\sigma_2$ is a regular semisimple element of $Sp(2m-2,q)$. This means that its centralizer in $Sp(2m-2,q)$ has order relatively prime to $p$. It is a fact that two regular semisimple elements in the symplectic group are conjugate in the group if and only they have the same minimal polynomial. Thus we can say that our construction is parametrized by regular semisimple conjugacy classes in $Sp(2m-2,q)$. The number of such conjugacy classes is $O(q^{m-1})$, but we do not know if there is an explicit known value for their number. In the case of $Sp(2,q)$ the number of such classes is $q-1$ if $q$ is even and $q-2$ if $q$ is odd. We have also calculated that when $q$ is even, the number of these classes is $(q-1)^2$ in $Sp(4,q)$ and $(q-1)(q^2-q+1)$ for $Sp(6,q)$. We could choose other polynomials for $M(x)$ but it is not clear how easy it would be to characterize the resulting Galois group. One attractive possibility occurs when $M(x)$ equals $(x-1)^{2m}$, in which case the Frobenius element is regular unipotent (its order is a power of $p$) but proper subgroups of $Sp(2m,q)$ that contain such elements and act transitively on the nonzero vectors exist in some special circumstances. 99 Elkies, N. D. Linearized algebra and finite groups of Lie type. I. Linear and symplectic groups. Applications of curves over finite fields (Seattle, WA, 1997), 77-107, Contemp. Math., 245, Amer. Math. Soc., Providence, RI, 1999. Gow, R; McGuire, G. On Galois groups of linearized polynomials related to the special linear group of prime degree, to appear, Linear Algebra Appl. (2023). https://doi.org/10.1016/j.laa.2023.03.004. Nathan Jacobson, *Basic Algebra I* Second ed. W. H. Freeman and Co. New York, 1985. Serge Lang, *Algebra*. Third ed. Addison-Wesley. Reading, Mass.,1993. Rudolf Lidl and Harald Niederreiter, *Finite Fields*. Addison-Wesley. Reading, Mass., 1983.

arxiv_math

--- abstract: | This article answers the question of V.M. Buchstaber about the growth function of a particular $n$-valued group. This question is closely related to discrete integrable systems. In this paper, we will find a formula for the growth function in the case when $n$ is prime. In addition, we will prove a polynomial asymptotic estimate for the growth function in the general case. Finally, we will pose new problems and hypotheses about growth functions. author: - "M. Chirkov [^1]" bibliography: - main.bib title: Growth function for an $n$-valued dynamics --- # Introduction The theory of $n$-valued groups was introduced by S. P. Novikov and V. M. Buchstaber in 1971 [@Buhtaber1971] in connection with the study of the theory of characteristic classes of vector bundles, in which the product of two elements is an $n$-multiset with multiplicities taken into account. Later, the theory of 2-valued formal groups was developed, which found applications to addition theorems related to elliptic functions [@Bukhshtaber1990]. Further, V. M. Buchstaber developed the topological and algebraic theory of $n$-valued groups, thanks to which it was possible to find a solution [@Buchstaber2004] of the problem of rings of functions on a space of symmetric power. We introduce an $n$-valued multiplication [@Buchstaber2006] over the field $\mathbb{C}$ by the following rule: $$x \ast y = [(\sqrt[n]{x} + \varepsilon^r \sqrt[n]{y})^n, r=1,\ldots,n], \text{ where } \varepsilon = e^{\frac{2 \pi i}{n}}.$$ Consider the function $T: \mathbb{C} \to (\mathbb{C})^n$, which defines the $n$-valued dynamics [@Veselov1991; @Buchstaber1996] as follows: $$\label{dynamics} T(y) = x \ast y \text{ for some fixed } x \in \mathbb{C}.$$ Then, in a natural way, we can define $T^k(y)$ as the result of applying the function $T$ to all elements of $T^{k-1}(y)$. For example, $T^2(y) = x \ast (x \ast y)$. The question that was first posed by V. M. Buchstaber in his work [@Buchstaber2006] is how many different elements will be in the set $\{ 0 \} \, \cup \, T(0) \cup \ldots \cup \, T ^k(0)$. Let us denote this number $P_n(k)$. This function is related to counting the number of elements of the cyclic $n$-valued group $\langle x \rangle$ generated by an arbitrary $x \in \mathbb{C}$ after $k$ iterations. The dynamics of a cyclic $n$-valued group was first studied in [@Veselov1992]. V.M. Buchstaber and A.P. Veselov found [@Buchstaber2019] an application of the theory of 2-valued groups and their dynamics to the classical problem of adding points on elliptic curves. In the case of a free group without bonds, growth with such dynamics turns out to be exponential. The fact of polynomial growth is not accidental and is related to the integrability of the corresponding systems with discrete time. An integrable system closely related to the 2-valued group is an integrable billiard [@Buchstaber2018]. In the next section, we will introduce all the necessary definitions for describing the dynamics of [\[dynamics\]](#dynamics){reference-type="eqref" reference="dynamics"}. In Section 3, we prove an explicit formula for the growth function in the case where $n$ is a prime number. Then, in section 4, some results of numerical calculations of growth polynomials will be described. Finally, in section 5 we will make some new hypotheses and pose some questions. # Description of the dynamics Note that $T(0) = [x, \ldots, x]$, i.e. on the first iteration the new element is only one, namely $x$. We assume that $P_n(0) = 1$, since the element $0$ is put there by definition, and $P_n(1) = 2$, since we always get only $x$ on the first iteration. Let us now look at the second iteration: $$(\sqrt[n]{x} + \varepsilon^r \sqrt[n]{x})^n = x (1 + \varepsilon^r)^n.$$ That is, at the second iteration, only elements of the form $\alpha x$ appear, where $\alpha \in \mathbb{C}$. At the next iteration, we have $$(\sqrt[n]{x} + \varepsilon^r \sqrt[n]{\alpha x})^n = x (1 + \varepsilon^r \sqrt[n]{k})^n.$$ Thus, we again obtain elements of the form $\overline{\alpha} x$. So $x$ can always be taken out of the bracket and only the following dynamics can be considered $$\alpha_{r_1,\ldots,r_k} = (1 + \varepsilon^{r_k} \sqrt[n]{\alpha_{r_1, \ldots, r_{k-1}}})^n,$$ since the new number in dynamics is given by the number from the previous iteration and the choice of the parameter $r$. If we write down all the previous members of the sequence, we obtain: $$\alpha_{r_1,\ldots,r_k} = (1 + \varepsilon^{r_k} + \varepsilon^{r_k + r_{k-1}} + \ldots + \varepsilon^{r_k + r_{k-1} + \ldots + r_1})^n.$$ Further, it will be convenient for us to change the parametrization as follows. **Lemma 1**. *For any $\{a_i\} \in \mathbb{Z}^{k+1}_n$ there exists $\{r_i\} \in \mathbb{Z}^k_n$, such that $$\alpha_{r_1,\ldots,r_k} = (\varepsilon^{a_1} + \varepsilon^{a_2} + \ldots + \varepsilon^{a_{k+1}})^n$$* *Proof.* We use the fact that $\varepsilon^n = 1$, and multiply $(\varepsilon^{a_1} + \varepsilon^{a_2} + \ldots + \varepsilon^{a_{k+1}})^n$ by $\varepsilon^{-a_1 n}$: $$(\varepsilon^{a_1} + \varepsilon^{a_2} + \ldots + \varepsilon^{a_{k+1}})^n = (1 + \varepsilon^{a_2 - a_1} + \varepsilon^{a_3 - a_1} + \ldots + \varepsilon^{a_{k+1} - a_1})^n.$$ Thus, we obtain the following system of equalities modulo $n$: $$\begin{cases} r_k \equiv a_2 - a_1, \\ r_k + r_{k-1} \equiv a_3 - a_1, \\ \ldots \\ r_k + \ldots + r_1 \equiv a_{k+1} - a_1 \end{cases}.$$ It possesses the folowing solution: $$\begin{cases} r_k = a_2 - a_1 \mod n, \\ r_{k-1} = a_3 - a_2 \mod n, \\ r_{k-2} = a_4 - a_3 \mod n, \\ \ldots \\ r_1 = a_{k+1} - a_k \mod n \end{cases}$$ ◻ Now define $A_j \stackrel{def}{=} \{ (\varepsilon^{a_1} + \ldots + \varepsilon^{a_j})^n \mid a_i \in \mathbb{Z}_n \}$, $A_0 \stackrel{def}{=} \{ 0 \}$. Then, by Lemma $\ref{lemma1}$, we have $P_n(k) = | A_0 \cup \ldots \cup A_k |$. Let us split these sets into disjoint ones: $\widetilde{A_j} \stackrel{def}{=} A_j \setminus {A_0 \cup \ldots \cup A_{j-1}}$, thus $P_n(k) = |\widetilde{A_0}| + \ldots + |\widetilde{A_k}|$. It will be convenient to introduce function $Q_n(k) \stackrel{def}{=} P_n(k) - P_n(k-1) = |\widetilde{A_k}|$. It shows how many elements are added at the new step of dynamics. Next, we focus on the description of the elements of the set $\widetilde{A_j}$ in the case where $n$ is a prime number, which will be denoted by $p$ in what follows. # Main result for prime $p$ The elements of the set $A_k$ can be specified a bit differently using grouping by powers of $\varepsilon$: $$A_k = \{ (b_0 + b_1 \varepsilon + \ldots + b_{p-1} \varepsilon^{p-1})^p \mid b_0 + \ldots + b_{p-1} = k, \ b_i \in \mathbb{Z}_{\geq 0}\}$$ The following classical definitions follow naturally from this notation. **Definition 1**. *An ordered set of numbers $\lambda = (b_0, \ldots, b_{n-1})$ is called a composition of $m$ into $n$ parts, if $b_i \in \mathbb{Z}_{>0}$ and $b_0 + \ldots + b_{n-1} = m$.* **Definition 2**. *$C(m, n)$ is the set of compositions of $m$ on $n$ positive terms.* Number compositions are a classical constructions. It is known that $|C(m, n)| = {{m-1} \choose {n-1}}$. It can be seen from our construction that we need to allow $b_i = 0$, so the following definitions are required. **Definition 3**. *An ordered set of numbers $\lambda = (b_0, \ldots, b_{n-1})$ is called a weak composition of $m$ into $n$ parts, if $b_i \in \mathbb{Z}_{\geq 0}$ and $b_0 + \ldots + b_{n-1} = m$.* **Definition 4**. *$\overline{C(m, n)}$ is the set of weak compositions of $m$ on $n$ non-negative terms.* **Lemma 2**. *$C(m+n, n) \cong \overline{C(m, n)}$* *Proof.* Any composition on the left side can be associated with a weak composition on the right, if 1 is subtracted from each term. Similarly, to the right side, any weak composition on the right side can be compared to a composition on the left if one is added to each term. ◻ Hence, we obtain that $| \overline{C(m, n)} | = {{m + n - 1} \choose {n-1}}$. Next, we introduce the following equivalence relation. We say that $z_1 \sim z_2$, if $z_1 = \varepsilon z_2$. We need this equivalence relation to get rid of exponentiation to the power $p$ of elements of the set $A_k$, because $z_1 \sim z_2 \iff z_1^p = z_2^p$. So we can define the following set $$B_k \stackrel{def}{=} \{ b_0 + b_1 \varepsilon + \ldots + b_{p-1} \varepsilon^{p-1} \mid b_0 + \ldots + b_{p-1} = k, \ b_i \in \mathbb{Z}_{\geq 0}\}.$$ In connection with the introduced equivalence relation, we have that $A_k = B_k/ z \sim \varepsilon z$. Now we want to describe the elements of the set $\widetilde{B_k}$, which is defined analogously to $\widetilde{A_k}$ by splitting into a disjunctive union. In addition, we need the concept of cyclic composition, because the multiplication of an element of a set $B_k$ by $\varepsilon$ corresponds to a cyclic shift $(b_{p-1}, b_0, \ldots, b_{p-2})$. **Definition 5**. *$\left\langle \begin{matrix} m \\ n \\ \end{matrix} \right\rangle$ is the number of equivalence classes $C(m, n)$ relative to cyclic shift.* To calculate the number of equivalence classes, the formula is known [@KnopfmacherRobbins]: $$\left\langle \begin{matrix} m \\ n \\ \end{matrix} \right\rangle = \dfrac{1}{m} \sum\limits_{j | \gcd( m, n) } \varphi(j) {m/j \choose n/j}.$$ **Lemma 3**. *Each element of the set $\widetilde{A_k}$ is uniquely represented by the weak composition equivalence class $(b_0, \ldots, b_{p-1})$ relative to cyclic shift, and in this representation there is an index $j$, that $b_j = 0$.* *Proof.* To begin with, suppose that, on the contrary, there is a number $z_0 = \tilde{b}_0 + \ldots + \tilde{b}_{p-1} \varepsilon^{p-1} \in \widetilde{B_k}$ (the same as $z^n_0 \in \widetilde{A_k}$), that $\min b_i = d > 0$. For any $p$ (not only prime number) there is always a relation on the roots of unity $1 + \varepsilon + \ldots + \varepsilon^{p-1} = 0$, then $z_0 = (\tilde{b}_0 - d) + \ldots + (\tilde{b}_{p-1} - d) \varepsilon^{p-1}$. Hence we get that $z_0 \in B_{k - p d}$, which is contrary to $z_0 \in \widetilde{B_k}$, because by construction we have $\widetilde{B_k} \cap B_{k - p d} = \varnothing$. Let us now show that the representation is unique up to a cyclic shift of $(b_0, \ldots, b_{p-1})$. Assume, on the contrary, that $z_0 = \hat{b}_0 + \ldots + \hat{b}_{p-1} \varepsilon^{p-1} = \hat{d}_0 + \ldots + \hat{d}_{p-1} \varepsilon^{p-1}$, where $(\hat{b}_0, \ldots, \hat{b}_{p-1})$ and $(\hat{d}_0, \ldots, \hat{d}_{p-1})$ are different non-equivalent weak compositions of $k$ on $p$ parts, and without loss of generality we can assume that $\hat{b}_{0} = 0$ and $\hat{d}_{0} = 0$. Subtracting one representation from the other, we obtain that $$0 = (\hat{b}_1 - \hat{d}_1) \varepsilon + \ldots + (\hat{b}_{p-1} - \hat{d}_{p-1}) \varepsilon^{p-1}.$$ From the theory of cyclotomic fields [@Ireland1982] we have, that the degree of field extension $\mathbb{Q}(\varepsilon)$, where $\varepsilon = e^{2 \pi i / n}$ is equal to $\varphi(n)$, where $\varphi$ is the Euler totient function. Since we are working with a prime number $p$, we have $\varphi(p) = p-1$. It means that $p-1$ successive roots of unity $\varepsilon, \varepsilon^2, \ldots, \varepsilon^{p-1}$ are linearly independent over the field $\mathbb{Q}$. Hence we get a contradiction, since $\hat{b}_i = \hat{d}_i, i = 0, \ldots, p-1$. ◻ In connection with the lemma just proved, it is useful to introduce the following definition. **Definition 6**. *We denote $\widetilde{C(k, p)}$ as the set of equivalence classes with respect to cyclic shifts of weak compositions $\overline{C(k, p)}$, that contain at least one zero.* From lemma [Lemma 3](#lemma3){reference-type="ref" reference="lemma3"} we obtain that $\widetilde{A_k} \cong \widetilde{C(k, p)}$. **Lemma 4**. *$\widetilde{C(k, p)} \ \sqcup \ C(k, p)/\!\sim \ \cong C(k+p, p)/\!\sim$, where $\sim$ is an equivalence relation with respect to cyclic shift.* *Proof.* The disjoint union symbol does indeed take place here, because ordinary compositions contain only positive integers by definition, but representatives of equivalence classes from $\widetilde{C(k, p)}$ have at least one zero. In total, the left side contains the equivalence classes of arbitrary weak compositions of $k$ on $p$ parts. Further on the lemma $\ref{lemma2}$ we establish the final isomorphism. ◻ Hence, we obtain that $$Q_k(p) = | \widetilde{A_k} | = | \widetilde{C(k, p)} | = | C(k+p, p)/\!\sim| \ - \ | C(k, p)/\!\sim | = \left\langle \begin{matrix} k+p \\ p \\ \end{matrix} \right\rangle - \left\langle \begin{matrix} k \\ p \\ \end{matrix} \right\rangle.$$ When $k < p$, it is natural to set $\left\langle \begin{matrix} k \\ p \\ \end{matrix} \right\rangle = {k \choose p} = 0$. Thus, we have proved the following theorem. **Theorem 1**. *If $p$ is a prime number, then $Q_p(k) = \left\langle \begin{matrix} k+p \\ p \\ \end{matrix} \right\rangle - \left\langle \begin{matrix} k \\ p \\ \end{matrix} \right\rangle$.* **Corollary 1**. *If $p$ is a prime number, then $$Q_p(k) = \dfrac{1}{p} \left[ {{k + p - 1} \choose {p-1}} - {{k-1} \choose {p-1}} \right].$$* *Proof.* Since $p$ is prime $\gcd(p, k)$ is equal to 1 or $p$: $$\left\langle \begin{matrix} k \\ p \\ \end{matrix} \right\rangle = \begin{cases} \dfrac{1}{k} {k \choose p}, \ p \nmid k, \\ \\ \dfrac{1}{k} \left( {k \choose p} + \varphi(p) \dfrac{k}{p} \right), \ p \mid k \end{cases} \left\langle \begin{matrix} k+p \\ p \\ \end{matrix} \right\rangle = \begin{cases} \dfrac{1}{k+p} {{k+p} \choose p}, \ p \nmid k, \\ \\ \dfrac{1}{k+p} \left( {{k+p} \choose p} + \varphi(p) \dfrac{k + p}{p} \right), \ p \mid k \end{cases}$$ In both cases the same result is obtained: $$Q_p(k) = \dfrac{1}{k+p} {{k+p} \choose p} - \dfrac{1}{k} {k \choose p}.$$ We can expand the binomial coefficients by the definition and get the formula from the statement: $$Q_p(k) = \dfrac{1}{k+p} \dfrac{(k+p)!}{k! \ p!} - \dfrac{1}{k} \dfrac{k!}{(k-p)! \ p!} = \dfrac{(k+p-1)!}{k! \ p!} - \dfrac{(k-1)!}{(k-p)! \ p!} = \dfrac{1}{p} \left[ {{k + p - 1} \choose {p-1}} - {{k-1} \choose {p-1}} \right].$$ ◻ **Corollary 2**. *If $p$ is a prime number, then $$P_p(k) = 1 + \sum\limits_{j = 1}^{k} \left\langle \begin{matrix} j+p \\ p \\ \end{matrix} \right\rangle - \left\langle \begin{matrix} j \\ p \\ \end{matrix} \right\rangle = 1 + \dfrac{1}{p} \sum\limits_{j = 1}^{k} {{j+p-1} \choose {p-1}} - {{j-1} \choose {p-1}}.$$* *Proof.* It is because $P_p(k) = P_p(0) + Q_p(1) + \ldots + Q_p(k)$. ◻ **Corollary 3**. *If $p$ is a prime number, then $Q_p(k) = O(k^{p-2})$ and $P_p(k) = O(k^{p-1})$ for $k \to +\infty$.* *Proof.* This result follows from the formula $$Q_p(k) = \dfrac{1}{p} \left[ {{k + p - 1} \choose {p-1}} - {{k-1} \choose {p-1}} \right].$$ To do this, we expand the binomial coefficients $$\dfrac{1}{p}{{k + p - 1} \choose {p-1}} - {{k-1} \choose {p-1}} = \dfrac{1}{p!} \left[ (k + p - 1) \ldots (k + 1) - (k-1) \ldots (k - p + 1) \right].$$ From this we obtain that the coefficient at $k^{p-1}$ is equal to 0, hence $Q_p(k) = O(k^{p-2})$. The asymptotic for $P_p(k)$ can be obtained from a rough estimate $P_p(k) \leqslant 1 + k Q_p(k)$. ◻ # Numerical calculation of growth polynomials **Lemma 5**. *$Q_n(k) = O(k^{n-2})$ and $P_n(k) = O(k^{n-1})$ for $k \to +\infty$.* *Proof.* In the case of an arbitrary $n$, Lemma $\ref{lemma3}$ is only partially true; that is, the part about uniqueness is not true. So there is an injection from $\widetilde{A_k}$ into $\widetilde{C(k, n)}$, hence $$Q_n(k) \leqslant \left\langle \begin{matrix} k+n \\ n \\ \end{matrix} \right\rangle - \left\langle \begin{matrix} k \\ n \\ \end{matrix} \right\rangle$$ If $\gcd(k, n) = 1$ then, as we have already found out in the proof, in the case of a prime $n$ we obtain $Q_n(k) = O(k^{n-2})$ for $k \to + \infty$. In the general case, additional terms of the form ${{(k+n)/j} \choose {n/j}}$ and ${{k/j} \choose {n/j}}$ are added, but they have smaller asymptotic of $O(k^{n/j})$. ◻ Now, we calculate the interpolation polynomial $\overline{P_n(k)}$ for $P_n(k)$ over the nodes $0, \ldots, n-1$. Since from the estimate we have $P_n(k) = O(k^{n-1})$, for prime $n$ we can guarantee that $\overline{P_n(k)} \equiv P_n(k)$: $$\begin{aligned} \overline{P_2(k)} &= k+1, \\ \overline{P_3(k)} &= 1 + \frac{1}{2}(k + k^2) = \overline{P_4(k)} = \overline{P_6(k)}, \\ \overline{P_5(k)} &= 1 + \frac{5}{12}(k + k^2) + \frac{1}{24}(k + k^2)^2, \\ \overline{P_7(k)} &= 1 + \frac{7}{20}(k + k^2) + \frac{13}{180}(k + k^2)^2 + \frac{1}{720}(k + k^2)^3, \\ \overline{P_8(k)} &= 1 + \frac{1}{3}(k + k^2) + \frac{1}{12}(k + k^2)^2, \\ \overline{P_9(k)} &= 1 + \frac{3}{10}(k + k^2) + \frac{11}{120}(k + k^2)^2 + \frac{1}{240}(k + k^2)^3, \\ \overline{P_{10}(k)} &= 1 + \frac{1}{4}(k + k^2) + \frac{1}{8}(k + k^2)^2 = \overline{P_{12}(k)}.\end{aligned}$$ # Conclusions and further study We found an interesting relation of cyclic compositions to the task of counting images of n-valued map $T^k(0)$ and using this relation found a specific formula for the case, when $n$ is prime. From our numerical results, the following hypotheses and questions arise: 1. The degree of $\overline{P_n(k)}$ is $\varphi(n)$ (we see it from numerical results), we suggest that it is related to the fact that the degree of cyclotomic field $\mathbb{Q}(\varepsilon)$ as an extension of $\mathbb{Q}$ is equal to $\varphi(n)$; 2. We have the same polynomials for some $n$, we want to know for which exactly; 3. If $n$ is prime, the coefficient at the highest degree is $\frac{1}{(n-1)!}$; 4. Polynomials depend only on $(k + k^2)$ (or equivalently ${{k+1} \choose 2}$) (exception -- $n=2$). 5. Polynomials $\overline{P_n(k)}$ are integer-valued (it can be easily proven using Theorem 3.2.1 from [@Prasolov2004]). So we can write any integer-valued polynomial $p_d$ of degree $d$ as $$p_d(x) = c_0 {x \choose d} + c_1 {x \choose {d-1}} + \ldots + c_d, \ c_j \in \mathbb{Z}$$ We are interested in $c_j$ for our polynomials $\overline{P_n(k)}$. In further works we are going to prove hypothesis 1 (that $P_n(k)$ is polynomial and $\deg P_n(k) = \varphi(n)$), and answer some questions about the form of $P_p(k)$, where $p$ is prime. # Acknowledgements {#acknowledgements .unnumbered} Special thanks to V. M. Buchstaber for formulation of the problem, our fruitful discussions and his helpful advises. This work was carried out within the framework of a development programme for the Regional Scientific and Educational Mathematical Center of the Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement on provision of subsidy from the federal budget No. 075-02-2023-948). [^1]: mikhlchirkov\@gmail.com

arxiv_math

--- abstract: | We consider the $L^{2}$-boundedness of the solution itself of the Cauchy problem for wave equations with time-dependent wave speeds. We treat it in the one-dimensional Euclidean space ${\bf R}$. To study these, we adopt a simple multiplier method by using a special property equiped with the one dimensional space. author: - | Ryo Ikehata[^1]\ Department of Mathematics, Division of Educational Sciences\ Graduate School of Humanities and Social Sciences\ Hiroshima University\ Higashi-Hiroshima 739-8524, Japan\ title: | **Some attempts on $L^2$ boundedness for $1$-D\ wave equations with time variable coefficients** --- # Introduction [^2] [^3] We consider the Cauchy problem for wave equations with time dependent wavespeeds in one-dimensional Euclidean space ${\bf R}$ $$u_{tt}(t,x) - a(t)^{2}u_{xx}(t,x) = 0,\ \ \ (t,x)\in (0,\infty)\times {\bf R},\label{eqn}$$ $$u(0,x)= u_{0}(x),\ \ u_{t}(0,x)= u_{1}(x),\ \ \ x\in {\bf R},\label{initial}$$ where the initial data $[u_{0},u_{1}]$ are taken from the test function space (for simplicity) $$u_{0} \in C_{0}^{\infty}({\bf R}),\quad u_{1} \in C_{0}^{\infty}({\bf R}),$$ $a \in {\rm C}^{1}([0,\infty))$ and we denote $$u_{t}=\frac{\partial u}{\partial t},\quad u_{tt}=\frac{\partial^2 u}{\partial t^2},\quad u_{xx}=\frac{\partial^2 u}{\partial x^2}.$$ Throughout this paper, $\| \cdot\|$ stands for the usual $L^2({\bf R})$-norm. The total energy $E_{u}(t)$ of the solution $u(t,x)$ to problem (1.1) is defined by $$E_{u}(t)=\frac{1}{2}(\| u_t(t,\cdot)\|^2+a(t)^{2}\| u_x(t,\cdot)\|^2).$$ We shall impose the following two assumptions on $a(t)$:\ (**A.1**) $a \in {\rm C}^{1}([0,\infty))$ and $a(t) > 0$ for all $t \geq 0$, and $a_{m} := \sup\{a(t)\,:\,t \geq 0\} < +\infty$,\ (**A.2**) $a'(t) \geq 0$ for all $t \geq 0$.\ Under these conditions, it is known that the problem (1.1)-(1.2) admits a unique strong solution $u \in {\rm C}([0,\infty);H^{2}({\bf R})) \cap {\rm C}^{1}([0,\infty);H^{1}({\bf R}))\cap {\rm C}^{2}([0,\infty);L^{2}({\bf R}))$, which has finite speed propagation of waves with its propagation speed $a_{m} > 0$ (cf. [@ikawa]). This is a sufficient class to deploy the multiplier method. In fact, the solution is much smoother than is convenient.\ The purpose of the present paper is to consider whether the $L^{2}$-boundedness of the solution itself for problem (1.1)-(1.2) can be observed or not in the one-dimensional case. The problem itself is never trivial in the sense that one has no Hardy or Poincaré inequality. Furthermore, one can imagine that time-dependent coefficients $a(t)$ can be an obstacle in many ways in partial integration. There have been many interesting papers published on the estimate of the total energy and asymptotic behavior of solutions of the wave equation, not only in the case of constant coefficients, but also in the case of time and space variable dependence (see [@AR; @C; @E; @H; @HW; @Ma; @RS; @Y] and the references therein). Furthermore, a series of sharp research results have been published on the $L^p$-$L^q$-estimate of the solution itself, starting with Strichartz [@S] (see e.g. [@BS; @B; @M; @P; @RY; @W]). However, when we look at the $L^2$ estimate of the solution itself, there seems to be a little gap from the viewpoint of bestness. In such a situation, the author [@JHDE-ike] published the following results for the case $a(t) = 1$ (constant coefficient) in problems (1.1)-(1.2): $$\label{1.4} \int_{-\infty}^{\infty}u_{1}(x)dx \ne 0 \Rightarrow \Vert u(t,\cdot)\Vert \sim \sqrt{t},\quad (t \to \infty).$$ This shows a growth estimate of the $L^{2}$-norm of the solution itself to problem (1.1)-(1.2) with $a(t) = 1$. Thus, when one wishes to observe the $L^2$ boundedness of the solution itself, it will be necessary to consider the general time-variable coefficient problem (1.1) by factoring in the information that the zero-order moment of the initial velocity may or may not vanish. Herein lies the difficulty of deriving the solution's own $L^2$ estimate in the case of general variable coefficients. Furthermore, with variable coefficients, it is not possible to describe and evaluate the solution \"explicitly\" using the Fourier transform as in the method of [@JHDE-ike], and this makes us imagine a very hopeless prediction for deriving the $L^2$ estimate formula from \"under\" the solution itself. Therefore, in order to construct a \"general theory\" involving constant coefficients to watch the $L^2$ boundedness of the solution itself, the condition that the zero-order moment of the initial velocity vanishes, can never be removed.\ It's time to introduce the results. To state our results, we define $$v_{1}(x) := \int_{-\infty}^{x}u_{1}(y)dy.$$ Our result then can be stated as follows. **Theorem 1**. *Suppose (**A.1**), (**A.2**) and assume $v_{1} \in L^{2}({\bf R})$. Then, the corresponding solution $u(t,x)$ to problem (1.1)-(1.2) with initial data $[u_{0},u_{1}] \in C_{0}^{\infty}({\bf R})\times C_{0}^{\infty}({\bf R})$ satisfies $$\Vert u(t,\cdot)\Vert^{2} \leq I_{0}^{2}a(0)^{-2},\quad (t \geq 0)$$ with a constant $I_0 \geq 0$ defined by $$I_{0} := \left( \Vert v_{1}\Vert^{2} + a(0)^{2}\Vert u_{0}\Vert^{2} \right)^{1/2}.$$* **Remark 1**. *In this Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}, it is essentially imposing a new condition on $u_1$ through $v_{1} \in L^{2}({\bf R})$. For example, if $u_{1} \in C_{0}^{\infty}({\bf R})$ is odd about the origin, then we see that $\int_{{\bf R}}u_{1}(x)dx = 0$, and in this case $v_{1} \in C_{0}^{\infty}({\bf R})$. Thus one has $v_{1} \in L^{2}({\bf R})$. The condition $v_{1} \in L^{2}({\bf R})$ on the initial velocity $u_{1}$ of the theorem makes sense.* **Example 1.**  We can find an appropriate function $a(t)$ satisfying (**A.1**) and (**A.2**).\ $$a(t) = \left\{ \begin{array}{ll} \displaystyle {1+e^{-1/t}},& \qquad t > 0,\\[0.2cm] \\ \displaystyle {1},& \qquad t = 0, \end{array} \right.$$ From the proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} we see that the condition **(A.2)** can be replaced by the following one:\ (**A.3**) $a'(t) \leq 0$ for all $t \geq 0$, and $A_{0} := \inf\{a(t)\,:\,t \geq 0\} > 0$.\ Then, one can also derive the following corollary. A monotone \"decreasing\" function $a(t)$ can be also covered in our theory. **Corollary 1**. *Suppose (**A.1**) and (**A.3**) and assume $v_{1} \in L^{2}({\bf R})$. Then, the corresponding solution $u(t,x)$ to problem (1.1)-(1.2) with initial data $[u_{0},u_{1}] \in C_{0}^{\infty}({\bf R})\times C_{0}^{\infty}({\bf R}))$ satisfies $$\Vert u(t,\cdot)\Vert^{2} \leq I_{0}^{2},\quad (t \geq 0).$$* **Example 2.**  We can also choose $a(t) := 1+e^{-t}$, and/or $a(t) := \frac{2+t}{1+t}$. Then, the statement of Corollary [Corollary 1](#th3){reference-type="ref" reference="th3"} implies $$\Vert u(t,\cdot)\Vert^{2} \leq I_{0}^{2},\quad (t \geq 0).$$ **Remark 2**. *The conditions $\sup\{a(t)\,:\,t \geq 0\} < +\infty$ and $\inf\{a(t)\,:\,t \geq 0\} > 0$ assumed in **(A.2)** and **(A.3)** express a propagation speed of the wave, and the ellipticity of the operator $u \mapsto a(t)^{2}\partial_{xx}u$, respectively.* By modifying the proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} one can also present another version of the result. For this purpose we set one more assumption.\ (**A.4**) $A_{0} > 0$, and $a' \in L^{1}(0,\infty)$,\ where $A_{0}$ is the constant already defined in **(A.3)**. Then, one can derive one more corollary. **Corollary 2**. *Suppose (**A.1**) and (**A.4**) and assume $v_{1} \in L^{2}({\bf R})$. Then, the corresponding solution $u(t,x)$ to problem (1.1)-(1.2) with initial data $[u_{0},u_{1}] \in C_{0}^{\infty}({\bf R})\times C_{0}^{\infty}({\bf R}))$ satisfies $$\Vert u(t,\cdot)\Vert^{2} \leq \frac{1}{A_{0}^{2}}I_{0}^{2}e^{\frac{2}{A_{0}}\int_{0}^{\infty}\vert a'(s)\vert ds},\quad (t \geq 0).$$* **Example 3.**  We can present an additional oscillating example: $$a(t) := 2 + \frac{\sin t}{(1+t)^{2}}.$$ The above theorem and examples include the case of constant coefficients $a(t) = 1$. Therefore, what is still a concern is the fear that some moment conditions for the initial velocity $u_{1}$ may contradict (1.4). Let us discuss this situation below. Suppose $u_{1} \in C_{0}^{\infty}({\bf R})$. Then, there is a large number $L > 0$ such that one can assume that ${\rm supp}\,u_{1} \subset [-L,L]$. Since $$v_{1}(x) = \int_{-\infty}^{x}u_{1}(y)dy,$$ if $x > 2L$, then one sees that $$v_{1}(x) = \int_{-L}^{L}u_{1}(y)dy = \int_{-\infty}^{\infty}u_{1}(y)dy =: c_{0}\quad (\forall x > 2L).$$ Assume for the moment that $c_{0} \ne 0$. Then, it follws that $$\Vert v_{1}\Vert^{2} = \int_{-\infty}^{2L}\vert v_{1}(x)\vert^{2}dx + \int_{2L}^{\infty}\vert v_{1}(x)\vert^{2}dx \geq \int_{2L}^{\infty}\vert v_{1}(x)\vert^{2}dx = \int_{2L}^{\infty}c_{0}^{2}dx = \infty,$$ which shows a contradiction to the assumption that $v_{1} \in L^{2}({\bf R})$ in Theorem 1.1 and Corollary 1.1. Thus it must be $c_{0} = 0$. Note that $u_{1} \in C_{0}^{\infty}({\bf R})$ and $v_{1} \in L^{2}({\bf R})$ impliy $v_{1} \in C_{0}^{\infty}({\bf R})$ and $v_{1}(\infty) = 0$. Thus, under the conditions of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}, there is no contradiction with [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} because of the additional assumption that the zero-order moment of the initial velocity vanishes. Conversely, if the zero-order moment of the initial velocity does not vanish, then the augmentation property as in [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} may still be derived, but this remains unresolved at this time. Note that the theorem took the initial value class to be the test function, but this is not the essence of the theorem. This is off the topic of this issue, though. **Remark 3**. *With our method, it is difficult to handle high dimensional cases now and in our time. Also, by analogy with the results in [@JHDE-ike] for the constant coefficients, the moment vanishing condition for initial velocity $u_{1}$ would be required even in two dimensions, but that condition would not be necessary for three or more dimensions. The $L^2$-bounddedness is more likely to be used for three or more dimensions.* **Remark 4**. *From the above discussion, we expect the following in the $1$-D case (and probably in the $2$-D case as well): under the condition such that $$\int_{{\bf R}}u_{1}(x)dx \ne 0,$$ then $$\lim_{t \to \infty}\Vert u(t,\cdot)\Vert = +\infty$$ with some growth rate. This is still open in the time variable coefficient case (cf. [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"}).* The remainder of this paper is organized as follows. In Section 2, we shall prove Theorem 1.1 and Corollaries 1.1 and 1.2. # Proof of results. In this section, let us prove Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} and then Corollary [Corollary 1](#th3){reference-type="ref" reference="th3"} by using multiplier method inspired from the idea in [@Z]. Incidentally, it seems that the method used in [@Z] itself is partially inspired from an idea developed in [@IM] as can be observed from their proofs.\ *Proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}.*\ First of all, define a function $v(t,x)$ by $$v(t,x) := \int_{-\infty}^{x}u(t,y)dy.$$ Note that the function $v(t,x)$ is well-defined because of finite speed propagation of waves. Furthermore, for each $j = 0,1$ we set $$v_{j}(x) := \int_{-\infty}^{x}u_{j}(y)dy.$$ Incidentally, we see that $v_{j} \in C^{\infty}({\bf R})$ ($j = 0,1$). Then, the function $v(t,x)$ satisfies $$v_{tt}(t,x) - a(t)^{2}v_{xx}(t,x) = 0,\ \ \ (t,x)\in (0,\infty)\times {\bf R},\label{eqn-v}$$ $$v(0,x) = v_{0}(x),\ \ v_{t}(0,x) = v_{1}(x),\ \ \ x\in {\bf R},\label{initial-v}$$ where one has just used the fact that $$\lim_{y \to -\infty}u_{y}(t,y) = 0\quad (t \geq 0).$$ Multiplying both sides of [\[eqn-v\]](#eqn-v){reference-type="eqref" reference="eqn-v"} by $v_{t}$, and integration by parts yield the equality: $$\frac{d}{dt}E_{v}(t) = a(t)a'(t)\Vert v_{x}(t,\cdot)\Vert^{2},$$ where one has used the fact that $v_{x}(t,x) = u(t,x) = 0$ for large $\vert x\vert \gg 1$ and each $t \geq 0$. Integrating the above equality on $[0,t]$ it follows that $$\label{ike-1} E_{v}(t) = E_{v}(0) + \int_{0}^{t}a(s)a'(s)\Vert v_{x}(s,\cdot)\Vert^{2}ds$$ $$\label{ike-2} = E_{v}(0) + 2\int_{0}^{t}\frac{a'(s)}{a(s)}\left(\frac{1}{2}a(s)^{2}\Vert v_{x}(s,\cdot)\Vert^{2}\right)ds$$ $$\leq E_{v}(0) + 2\int_{0}^{t}\frac{a'(s)}{a(s)}E_{v}(s)ds,$$ where we have just used the assumption **(A.2)**. Thus, by using the Gronwall inequality one has $$\label{i1} E_{v}(t) \leq E_{v}(0)e^{2\int_{0}^{t}\frac{a'(s)}{a(s)}ds}.$$ Therefore, from the definition of the total energy and [\[i1\]](#i1){reference-type="eqref" reference="i1"} one can get the inequality: $$\Vert v_{x}(t,\cdot)\Vert^{2} \leq 2E_{v}(0)a(t)^{-2}e^{2W(t)},$$ where $$W(t) := \int_{0}^{t}\frac{a'(s)}{a(s)}ds = \int_{0}^{t}\frac{d}{ds}\log a(s) ds = \log \frac{a(t)}{a(0)}.$$ Thus one has $$\Vert v_{x}(t,\cdot)\Vert^{2} \leq 2E_{v}(0)a(t)^{-2}\left( \frac{a(t)}{a(0)}\right)^{2} = 2E_{v}(0)a(0)^{-2}.$$ Now, since $v_{x}(t,x) = u(t,x)$ (This is a crucial idea in [@IM] and [@Z]), we can arrive at the desired estimate: $$\Vert u(t,\cdot)\Vert^{2} \leq 2E_{v}(0)a(0)^{-2} \quad (t \geq 0).$$ Set $$\label{i2} I_{0} := \left( 2E_{v}(0) \right)^{1/2} = \left(\Vert v_{1}\Vert^{2} + a(0)^{2}\Vert u_{0}\Vert^{2} \right)^{1/2}.$$ These imply the desired statement of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"}. $\Box$\ **Remark 5**. * The function $v(t,x) := \int_{-\infty}^{x}u(t,y)dy$ used in the proof above can be considered as $$v(t,x) \sim v(t,\infty) \quad (x \to \infty),$$ and $$v(t,\infty) = \int_{{\bf R}}u(t,y)dy = \left. \int_{{\bf R}}e^{-iy\xi}u(t,y)dy\right\vert_{\xi = 0} = {\cal F}(u(t,\cdot))(0) = \hat{u}(t,0),$$ where ${\cal F}$ denotes the Fourier transform. Thus, the function $v(t,x)$ is equal to $\hat{u}(t,0)$ as $x \to \infty$ approximately. The estimate for $v(t,x)$ may correspond to the low frequency estimate near $\xi = 0$ of $\hat{u}(t,\xi)$. * *Proof of Corollary [Corollary 1](#th3){reference-type="ref" reference="th3"}.*\ It follows from [\[ike-1\]](#ike-1){reference-type="eqref" reference="ike-1"} of the proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} and the assumption **(A.3)** we see that $$E_{v}(t) \leq E_{v}(0).$$ The other part of proof is similar. $\Box$\ *Proof of Corollary [Corollary 2](#th4){reference-type="ref" reference="th4"}.*\ It follows from [\[ike-2\]](#ike-2){reference-type="eqref" reference="ike-2"} of the proof of Theorem [Theorem 1](#th1){reference-type="ref" reference="th1"} and the assumption **(A.4)** we see that $$E_{v}(t) \leq E_{v}(0) + 2\int_{0}^{t}\frac{\vert a'(s) \vert}{a(s)}E_{v}(s)ds \leq E_{v}(0) + \frac{2}{A_{0}}\int_{0}^{t}\vert a'(s) \vert E_{v}(s)ds.$$ The use of the Gronwall inequality yields the desired estimate similarly. $\Box$ The work of the author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C) 22540193 of JSPS. 99 H. S. Aslan and M. Reissig, $L^{p}$-$L^{q}$ estimates for wave equations with strong time-dependent oscillations, Hokkaido Math. J. 51 (2022), 57--106. M. Beals and W. Strauss, $L^{p}$ estimates for the wave equation with a potential, Commun. Partial Differ. Equ. **18**, no. 7 and 8, (1993), 1365--1397. P. Brenner, On $L_{p}-L_{p'}$ estimates for the wave equation, Math. Z. **145** (1975), 251--254. F. Colombini, Energy estimates at infinity for hyperbolic equations with oscillating coefficients, J. Diff. Eqns **231** (2006), 598--610. T. Crin-Barat, L-Y. Shou and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped P-system, arXiv: 2308.08280v1 \[math. AP\] 16 Aug 2023. M-R. Ebert, L. Fitriana and F. Hirosawa, On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions, J. Math. Anal. Appl. 432 (2015), 654--677. M. Ikawa, Hyperbolic Partial Differential Equations and Wave Phenomena; 2000. Translations of Mathematical Monographs, American Mathematical Society. F. Hirosawa, On the asymptotic behavior of the energy for the wave equation with time depending coefficients, Math. Ann. 339 (2007), 819--838. F. Hirosawa and J. Wirth, Generalized energy conservation law for wave equations with variable propagation speed, J. Math. Anal. Appl. 358 (2009), 56-74. R. Ikehata, $L^{2}$-blowup estimates of the wave equation and its application to local energy decay, J. Hyperbolic Differ. Equ. **20**, no.1, (2023), 259--275. R. Ikehata and T. Matsuyama, $L^{2}$-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002), 33--42. T. Matsuyama, Asymptotic behaviour for wave equation with time-dependent coefficients. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no. 2, 383--393. A. Miyachi, On some estimates for the wave equation in $L_p$ and $H_p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. **27**(1980), no.2, 331--354. H. Pecher, $L^{p}$-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. **150** (1976), 159--183. M. Reissig and J. Smith, $L^{p}$-$L^{q}$ estimate for the wave equation with bounded time dependent coefficient, Hokkaido Math. J. 34 (2005), 541--586. M. Reissig and K. Yagdjian, $L_{p}$-$L_q$ decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, Math. Nachr. 214 (2000), 71--104. R. S. Strichartz, Convolutions with kernels having singularities on a sphere. Trans. Amer. Math. Soc. 148 (1970), 461--471. W. Von Wahl, $L^{p}$-decay rates for homogeneous wave equations, Math. Z. **120**(1971), 93--106. T. Yamazaki, Asymptotically free property of the solutions of an abstract linear hyperbolic equation with time-dependent coefficients. Asymptot. Anal. 97 (2016), no. 1-2, 165--187. [^1]: ikehatar\@hiroshima-u.ac.jp [^2]: Keywords and Phrases: wave equation; $1$-D space; time-variable coefficient; $L^{2}$-boundedness; multiplier method. [^3]: 2020 Mathematics Subject Classification. Primary 35L05; Secondary 35L10, 35B45, 35B40.

arxiv_math

--- abstract: | We show that the local Burkholder functional ${\mathcal B}_K$ is quasiconvex. In the limit of $p$ going to $2$ we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers. address: - Department of Mathematics and Statistics, University of Helsinki, Finland - Departamento de Matematicas, Universidad Autonoma de Madrid, Spain and ICMAT CSIC, Madrid, Spain - Institute for Theoretical Studies ETH-ITS, Zürich, Switzerland - Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland - Mathematical Institute, University of Oxford, United Kingdom author: - K. Astala - D.  Faraco - A. Guerra - A. Koski - J. Kristensen title: | The Local Burkholder functional, quasiconvexity and\ Geometric Function Theory --- # Introduction In 1952, C. Morrey coined the notion of quasiconvexity [@Morrey] to investigate the scope of the direct method of Calculus of Variations for vectorial variational problems of the type $$\label{eq:variational} {\mathcal E }[f] \,\equiv \int_\Omega { \bf E}(\textup{D}f(x)) \, \mathrm{d}x$$ where $f \colon \Omega \to \mathbb{R}^m$ is a suitably regular map and the functional ${ \bf E}\colon \mathbb{R}^{m \times n} \to \mathbb{R}$ is assumed to be at least Borel measurable. Here, and throughout the paper, $\Omega\subset \mathbb{R}^n$ is a bounded domain and $m,n\geq 2$. Under natural growth assumptions, quasiconvexity is equivalent [@AcerbiFusco; @Morrey] to the sequential weak lower semicontinuity of [\[eq:variational\]](#eq:variational){reference-type="eqref" reference="eq:variational"}, and (strong) quasiconvexity is equivalent to the coercivity of [\[eq:variational\]](#eq:variational){reference-type="eqref" reference="eq:variational"}, see [@ChJK]. The interest in quasiconvexity was largely increased when J.M. Ball discovered its relevance to the theory of Nonlinear Elasticity [@Ball1; @Ball1981], opening a new era in the field. In hyperelasticity one considers variational problems as in [\[eq:variational\]](#eq:variational){reference-type="eqref" reference="eq:variational"} with $m=n$, but to exclude interpenetration of matter one additionally assumes $J_f(x) \geq 0$ a.e. for the admissible maps, or even the stronger condition $$\label{eq:blowup} %J_f(x) \geq 0 \qquad {\rm almost \; everywhere.} { \bf E}(A) \to +\infty \quad \textup{as } \det A \to 0^+,$$ which penalizes the compressibility severely, see for example [@Ciarlet §4.6] for a detailed discussion in the elasticity context. Nowadays, condition [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"} has become classical in the mathematical treatment of variational problems arising in hyperelasticity see e.g [@Cartesian]. In any case, both conditions ask that ${ \bf E}$ be defined in $\mathbb{R}^{n\times n}_+\equiv \mathbb{R}^{n\times n}\cap \{\det >0\}$. Somewhat independently, the concept of quasiconvexity has attracted the attention of researchers in Geometric Function Theory, for instance because of its relations to the sharp $\mathrm{L}^p$ theory of singular integrals and quasiconformal mappings. Prime among the relevant objects here is the celebrated Burkholder functional, defined for $A\in \mathbb{R}^{2 \times 2}$ by $$\label{Burk} {\bf B}_p(A) \equiv \Bigl( (\tfrac{p}{2} - 1)|A|^2 - \tfrac{p}{2}\det A \Bigr) |A|^{p-2}, \quad 2 \leq p < \infty,$$ where $|\cdot|$ denotes the operator norm. This functional was introduced by D. Burkholder [@Burk1; @Burk2] in the context of martingale inequalities; see also [@BarnsteinMontgomerySmith; @Sverak91] and [@Tade] for its relation with quasiconvexity, geometric function theory and the Morrey problem discussed below. The Burkholder functional is both $p$-homogeneous and isotropic, and moreover ${\bf B}_2(A)=-\det(A)$. Thus one can interpret ${\bf B}_p$ as an $\mathrm{L}^p$-version of the determinant. This analogy goes in fact much deeper: the determinant is a null-Lagrangian, while the Burkholder functional has a large class of extremals, i.e. maps for which the integral bound [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} below holds as an equality, see e.g. [@AIPS12; @AIPS15a; @BarnsteinMontgomerySmith; @Tade]. Furthermore, the Burkholder functional and its (potential) quasiconvexity is intimately related to the problem of determining the $\mathrm{L}^p$-norms of the Beurling--Ahlfors transform, cf. [\[Beur1\]](#Beur1){reference-type="eqref" reference="Beur1"}. We refer the reader to [@BJ; @BJV; @NV] and to the survey [@Banuelos] for further results and information in this direction. The main objects of study in hyperelasticity are *deformations*, i.e. regular maps with regular inverses [@Ball1981; @Ciarlet; @Cartesian], so that no distinction is made between the properties of the map and its inverse. Precisely the same view holds also for the quasiconformal maps and their applications [@IO]. Thus, as in [@Ball1; @Cartesian], and motivated by our interest in Nonlinear Elasticity and Geometric Function Theory, for functionals defined on $\mathbb{R}^{n\times n}_+$ it is natural to set: **Definition 1**. Let ${\bf E}\colon \mathbb{R}^{n\times n}_+\to \mathbb{R}$ be locally bounded and Borel measurable. Then ${\bf E}$ is said to be quasiconvex at $A\in \mathbb{R}^{n\times n}_+$ if $$\label{eq:quasiconvexity} {\bf E}(A)\leq \fint_{\Omega} {\bf E}(\textup{D}f) \, \mathrm{d}x$$ whenever $\Omega \subset \mathbb{R}^n$ is a bounded domain and $f$ a $\mathrm{C}^1$-diffeomorphism with $f=A$ in $\partial\Omega$; we clarify that $f \in \mathrm{C}^1(\overline \Omega)$ and $f^{-1}\in \mathrm{C}^1\Bigl(\overline{A(\Omega)} \Bigr)$ are required. *Remark 2*. From the general point of view, it is important to note that in this paper we study functionals ${\bf E}$ defined on $\mathbb{R}^{2 \times 2}_+$ and their quasiconvexity properties, so that the test functions are typically homeomorphisms. Some of these functionals satisfy [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"}, some not, and thus applications to non-linear elasticity are one particular (even if very interesting) special case. Another important remark is that for a given functional, inequality [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} typically holds for a much larger class of Sobolev functions. For instance, for all explicit functionals studied in this work we will prove [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} for homeomorphisms such that both $f \in \mathrm{W}^{1,2}(\Omega, \Omega')$ and $f^{-1} \in \mathrm{W}^{1,2}(\Omega', \Omega)$, where $\Omega' = A(\Omega)$. It is still a question of interest to study [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} when testing the inequality for general Lipschitz maps with a.e. $\det(\textup{D}f) > 0$. Then, however, the question becomes a problem of approximation of Sobolev homeomorphisms, an active area of its own [@HP; @IKO]. See Section [9](#sec:shield){reference-type="ref" reference="sec:shield"} for a discussion on this and for some positive partial results. Testing the quasiconvexity inequality with smooth approximations of a planar wave, i.e. a map which only takes two values, shows that that quasiconvexity implies convexity along rank-one directions, abbreviated as *rank-one convexity*. Whether conversely rank-one convexity implies quasiconvexity is a famous problem, going back to Morrey's work [@Morrey; @Morrey2]. The celebrated work of V. Šverák [@Sverak92] gives a counterexample in dimensions $m\geqslant 3$. His example consists of a map which is a superposition of three planar waves (see also [@Grabovsky] for a different example when $m\geq 8$). When $m=2$ Šverák's example does not work [@PedregalSverak98] and indeed three waves cannot provide a counterexample [@SebestyenSzekelyhidi17], see also [@GTdC]. In fact, in two dimensions there are partial positive results [@FaracoSzekelyhidi08; @HKL18; @KirchheimSzekelyhidi; @MullerDiag] which suggest that rank-one convexity might imply quasiconvexity. The purpose of this paper is to investigate whether for $n=m=2$ rank-one convexity might imply quasiconvexity, at least for functionals with symmetries and additional structure, such as the Burkholder functionals. In the context of Definition [Definition 1](#quasiconvexity){reference-type="ref" reference="quasiconvexity"} it is natural to set ${\bf E}(A) = +\infty$ when $\det(A) \leq 0$. In this case quasiconvexity as defined above is no longer a sufficient condition for weak lower semicontinuity, cf. Section [2](#sec:prelims){reference-type="ref" reference="sec:prelims"}. Thus, in addition, one also needs an exploration towards the properties of the functionals on existence of minimizers. Here we realised that a stronger quasiconvexity inequality, one which for the Burkholder functional ${\bf B}_p(A)$ can be viewed as a version of the classical area formula, is then needed, see Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"}. Moreover, versions of such inequalities in the limit $p \to 2$ lead to new lower semicontinuity and existence theorems of interest in their own. ## The Burkholder functional The Burkholder functional is known to be rank-one convex, since the original work [@Burk1; @Burk2]. For other approaches on this see [@BarnsteinMontgomerySmith; @Sverak91] or [@Tade]. Our first theorem asserts that the Burkholder functional is quasiconvex when restricted to the set where it takes *non-positive* values. To interpret this setting, note that $$\label{Burkh46} {\bf B}_p(A) \leq 0 \iff |A|^2 \leq \frac{p}{p-2} \det(A),$$ thus such a map $A \in \mathbb{R}^{2 \times 2}$ is $K$-quasiconformal with $K = \frac{p}{p-2}$, equivalently $$\label{K46} p = p_K \equiv \frac{2K}{K-1}.$$ In particular, if [\[Burkh46\]](#Burkh46){reference-type="eqref" reference="Burkh46"} holds then $A \in \mathbb{R}^{2 \times 2}_+$, unless $A = 0$. **Theorem 3**. *Let $p\geq 2$. For any $A \in \mathbb{R}^{2 \times 2}$ and any $f \in A+ \mathrm{W}^{1,2}_0(\Omega , \mathbb{R}^2)$ such that ${\bf B}_p(\textup{D}f) \leq 0$ a.e. in $\Omega$, we have $$\label{Burkh} {\bf B}_p(A) \; \leq \; \fint_{\Omega} \! {\bf B}_p\bigl(\textup{D}f(z)\bigr) \, \mathrm{d}m(z).$$* On the other hand, it was shown in [@GK] that the positive part $\mathbf{B}_{p}^{+} \equiv \max \{ \mathbf{B}_{p},0 \}$ is polyconvex. In particular, it follows that [\[Burkh\]](#Burkh){reference-type="eqref" reference="Burkh"} persists for maps $f \in A+ \mathrm{W}^{1,2}_0(\Omega , \mathbb{R}^2)$ satisfying $\mathbf{B}_{p}(\textup{D}f) \geq 0$ almost everywhere in $\Omega$. Thus combined, these two results provide strong evidence towards the full quasiconvexity of ${\bf B}_p$. Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} was established in the special case $A=\mathrm{Id}$ in [@AIPS12]. Also, as we will explain later, the result entails a sharp integrability statement for quasiconformal maps. At the moment let us emphasize that it comprises delicate cancellation properties: for any $K$-quasiregular map $f$ and for $p = p_K$, we have ${\bf B}_{p_K}(\textup{D}f(z)) \in \mathrm{L}^1_{\mathop{\mathrm{loc}}}$, even if in general the map $f \in \mathrm{W}^{1,s}_{\mathop{\mathrm{loc}}}$ only for $s < p_{K}$ [@Astala1994]. In order to establish the existence of minimizers for the induced Burkholder energy we actually need a stronger form of quasiconvexity. To this end, it is convenient to introduce the *local Burkholder functional*, defined by $$\label{eq:locBurk} {\mathcal B}_K (A) \equiv \begin{cases} \mathbf{B}_{p_K}(A), & \textup{if } |A|^2 \leq K \det A,\\ +\infty & \textup{otherwise}. \end{cases}$$ Thus ${\mathcal B}_K$ equals the Burkholder functional $\mathbf{B}_p$ with the largest $p$ for which $\mathbf{B}_p(\textup{D}f) \leq 0$ for every $K$-quasiconformal map $f$, and it becomes defined in all of $\mathbb{R}^{2 \times 2}$ at the cost of admitting the value $+\infty$ outside the $K$-quasiconformal cone. However, since ${\mathcal B}_K$ assumes the value $+\infty$, the notion of quasiconvexity needs to be strengthened to *closed quasiconvexity* [@JK; @Pe1]: briefly, one requires the Jensen inequality, that is [\[Burkh\]](#Burkh){reference-type="eqref" reference="Burkh"}, to hold not just for maps but also for gradient Young measures, cf. Section [2](#sec:prelims){reference-type="ref" reference="sec:prelims"} for the precise definition. With this terminology, we obtain the following stronger version of Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}. **Theorem 4**. *Let $K\geq 1$ and $p>\tfrac{2K}{K+1}$. Then the local Burkholder functional ${{\mathcal B}_{K}\colon \mathbb{R}^{2 \times 2}\to \mathbb{R}\cup \{+\infty\}}$ is closed $\mathrm{W}^{1,p}$-quasiconvex.* By combining Theorem [Theorem 4](#main){reference-type="ref" reference="main"} with standard results from the theory of Young measures and the Direct Method of the Calculus of Variations we obtain the existence of minimizers: **Corollary 5**. *Let $K\geq 1$ and $2 \leq p < \frac{2K}{K-1}$. Then for any $K$-quasiregular map $g\colon \mathbb{C}\to\mathbb{C}$, the problem $$\inf\left\{\int_\Omega {\mathbf{B}}_p(\textup{D}f)\, \mathrm{d}m(z): f\in g+\mathrm{W}^{1,p}_0(\Omega) \text{ is } K\text{-quasiregular}\right\}$$ admits a minimizer $f\in g+\mathrm{W}^{1,p}_0(\Omega)$.* Note that here we do not need to require the maps to be homeomorphisms. ## The Burkholder area inequality We next turn to a further refinement of Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}, of independent interest, but also one of the key points in the study of the associated weak lower semicontinuity and minimization problems, see Section [12](#sec:swlsc){reference-type="ref" reference="sec:swlsc"}. Namely, given a $\mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$-homeomorphism $f$, analytic outside $\mathbb D$, we say that $f$ is a *principal map* if it has the Laurent expansion $$\label{eq:principal} f(z)=z + \frac{b_1}{z} + \sum_{j=2}^\infty \frac{b_j}{z^j},\qquad |z|>1.$$ It follows from the classical Grönwall--Bieberbach area formula that in this expansion $|b_1| < 1$, see Section [3](#sec:principal){reference-type="ref" reference="sec:principal"}. This allows us to interpret the first two terms of the above series, i.e. the main asymptotics of $f$, in terms of the invertible linear map $$\label{eq:asympt} A_f(z) = z+b_1 \bar z, \qquad {\rm equivalently} \quad A_f\equiv \fint_{\mathbb D} \textup{D}f(w) \, \mathrm{d}m(w).$$ As we will see, in many respects $A_f$ plays the role which the linear boundary values have in the standard definition of quasiconvexity. For instance, with this notation the classical area formula, cf. [\[areafmla\]](#areafmla){reference-type="eqref" reference="areafmla"}, asserts that $$\label{eq:area} \fint_{\mathbb D} \left[ - \det \textup{D}f + \det A_f\right]\mathrm{d}m(z)= \sum_{j=2}^\infty j |b_j|^2,$$ and one can think of this identity as a sharpening of the well-known null-Lagrangian property of the Jacobian determinant. In the same spirit one can consider also the Burkholder functional, recalling that ${\mathbf{B}}_2(A) = - \det(A)$. In fact, with Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} we find an $\mathrm{L}^p$ version of [\[eq:area\]](#eq:area){reference-type="eqref" reference="eq:area"}: **Theorem 6** (Burkholder Area Inequality). *Let $f$ be a $K$-quasiconformal principal map as in [\[eq:principal\]](#eq:principal){reference-type="eqref" reference="eq:principal"}. Then, for any $2\leq p\leq p_K$, we have $$\fint_\mathbb D \left[ {\mathbf{B}}_p(\textup{D}f)- {\mathbf{B}}_p(A_f)\right] \mathrm{d}m(z) \geqslant\gamma_p(A_f) \sum_{j=2}^\infty j |b_j|^2$$ where $\gamma_p(A_f) \equiv \frac{p}{2} \frac{{\mathbf{B}}_p(A_f)}{{\mathbf{B}}_2(A_f)}>0$. Note that $\gamma_2=1$.* Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"} sharpens the main result in [@AIPS12], since ${\mathbf{B}}_p(A_f) \geqslant{\mathbf{B}}_p(\mathrm{Id})$ for any principal map $f$ as above (note, however, the different sign-convention for the Burkholder functional in [@AIPS12]). Since $A_f=\fint_{\mathbb D} \textup{D}f(z) \, \mathrm{d}m(z)$, a surprising feature of Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"} is that it establishes a Jensen inequality even without requiring that the map takes affine boundary values, as in the definition of quasiconvexity! In fact, the result shows that, if $$\int_{\mathbb D} {\mathbf{B}}_p(\textup{D}f) \, \mathrm{d}m(z) = \int_{\mathbb D} {\mathbf{B}}_p(A_f) \, \mathrm{d}m(z),$$ then $f(z)=z+\frac{b_1}{z}$ for $|z|>1$; in particular, $f|_{\mathbb S^1}$ is linear. ## Functionals for Nonlinear Elasticity, as $p \to 2$. As observed in [@Tade] the theory of Burkholder functionals has very interesting consequences at the limit when the index $p$ goes to $2$. Namely, we have $\mathbf{B}_2(A)=-\det (A)$ and for the next order of approximation $$\lim_{p \to 2} \, \frac{p}{p-2} \left[ {\bf B}_p(A) - {\bf B}_2(A)\right] = {\mathscr F}(A),$$ where the functional $$\label{F2} { \mathscr F}(A) \equiv \, |A|^2\, - \,\left(1\, + \,\log |A|^2 \,\right )\, \det(A),$$ is rank-one convex in $\mathbb{R}^{2\times 2}$, but not polyconvex. On the other hand, the local quasiconvexity of the Burkholder functional, as formulated in Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}, allows us to show that $\mathscr F$ satisfies the quasiconvexity inequality [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} for all $\mathrm{W}^{1,2}$-homeomorphisms with linear boundary values $A \in \mathbb{R}^{2\times 2}_+$, see Corollary [Corollary 55](#cor:LlogL){reference-type="ref" reference="cor:LlogL"} for the precise statement. Notice that this for instance implies a sharp and quantitative version of the celebrated $\mathrm{L}\log \mathrm{L}$-higher integrability properties for the derivatives of $\mathrm{W}^{1,2}$-homeomorphisms. Another consequence of these relations concerns the quasiconvexity of the functional $$\label{eq:Wfcn} \mathscr W(A)\equiv \frac{|A|^2}{\det A} - \log \left( \frac{|A|^2}{\det A}\right) + \log \det A, \quad A \in \mathbb{R}^{2 \times 2}_+.$$ This is an example of a rank-one convex but non-polyconvex functional, which diverges as the $\det(A) \to 0.$ It was introduced in [@AIPS12] and further studied in the recent works [@Voss1; @Voss2], where its quasiconvexity remained undecided. In fact, $\mathscr W$ arises from ${\mathscr F}$ by applying the Shield transform [@Shield]. Therefore Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} leads us to the following: **Corollary 7**. *The functional $\mathscr{W} \colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ is quasiconvex.* In Section [12](#sec:swlsc){reference-type="ref" reference="sec:swlsc"} we investigate weak lower semicontinuity properties of $\mathscr W$ and establish that some features similar to closed quasiconvexity hold also for $\mathscr{W}$; for precise formulations see Proposition [Proposition 76](#thm:closedqc.new){reference-type="ref" reference="thm:closedqc.new"}. Originating from ${\bf B}_p$, the initial functional [\[eq:Wfcn\]](#eq:Wfcn){reference-type="eqref" reference="eq:Wfcn"} assumes all real values, and even tends to $-\infty$ along suitable directions when the determinant goes to zero. However, it allows easy modifications creating quasiconvex and non-polyconvex functionals that satisfy [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"}. The following is perhaps the easiest example: $$\label{modiW} \widetilde{\, \mathscr W \,} (A) \equiv \frac{|A|^2}{\det A} - \log \left( \frac{|A|^2}{\det A}\right) + |\log \det A|,$$ see Remark [Remark 80](#weehat){reference-type="ref" reference="weehat"}. The interest in [@Voss1; @Voss2] on $\mathscr W$ originates from the fact that it spans the only non-polyconvex extreme ray in a class of functionals satisfying the *additive volumetric-isochoric split*, see also [@Guerra19] for further information on extremal functionals. Thus, as a consequence of Corollary [Corollary 7](#cor:Wintro){reference-type="ref" reference="cor:Wintro"}, we obtain a solution to Morrey's problem in a class of elastic functionals: **Theorem 8**. *Let $\, {\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be a functional of the form $${\bf E}(A)=g(\det A)+h(K_A), \qquad K_A\equiv \frac{|A|^2}{\det A},$$ where $h\colon[1,+\infty)\to \mathbb{R}$ is convex and $g\colon (0,+\infty)\to \mathbb{R}$. Then $${\bf E} \text{ is rank-one convex} \iff {\bf E} \text{ is quasiconvex.}$$* The additive volumetric-isochoric split goes back at least to the work of Flory [@Flory] and since then it has been used extensively to model slightly compressible materials, see for instance [@HartmannNeff; @Ogden] and the references therein. ## Lower semicontinuity and existence of minimizers As discussed above the notion of quasiconvexity was introduced by Morrey to characterize sequential weak lower semicontinuity for integral functionals in the vectorial calculus of variations. On the other hand, for functionals ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ the condition [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"} expresses the intuitive and natural requirement of hyperelasticity that *an infinite amount of energy is required to compress a finite volume of material into zero volume* [@Ball1981; @Ciarlet]. However, for such functionals with [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"} it is not clear if quasiconvexity suffices for lower semicontinuity results. The stronger notion of polyconvexity does suffice, allowing a wealth of interesting models for hyperelastic materials [@Ball1]. In fact, Theorem [Theorem 8](#thm:morreysplit){reference-type="ref" reference="thm:morreysplit"} already provides a natural class of functionals that have been considered before in the engineering literature [@Ciarlet], but their weak lower semicontinuity and minimization properties had not been established. In addition to Theorem [Theorem 8](#thm:morreysplit){reference-type="ref" reference="thm:morreysplit"} or Corollary [Corollary 7](#cor:Wintro){reference-type="ref" reference="cor:Wintro"} we shall also address this point here. In this connection, the extended Stoilow factorization due to Iwaniec and Šverák [@IwaSve] suggested to us that Jensen inequality with respect to principal maps might be sufficient for lower semicontinuity. Indeed, this was our original indication that a theorem like the Burkholder area inequality might be true. Applying this line of thought leads us to the following Jensen inequality for principal maps: **Theorem 9**. *Let $f \in \mathrm{W}^{1,1}_{\mathop{\mathrm{loc}}}(\mathbb{C})$ be a homeomorphism, and a principal map with integrable distortion $K_f \in \mathrm{L}^1(\mathbb D)$. Then $$\fint_{\mathbb D} \bigl[ \mathscr W\bigl( \textup{D}f(z) \bigr) - \mathscr W\bigl( A_f \bigr) \bigr]\mathrm{d}m(z) \geq 0.$$* Here the assumption $K_f \in \mathrm{L}^1$ is optimal mathematically. Moreover, notice that in the study of incompressible neo-Hookean materials, the first invariant of the isochoric part of the Cauchy-Green tensor of the deformation $f$ is $\widehat{I_1} = K_f + 1/K_f$, see [@Ogden]. Since it is unclear how to measure experimentally the response of materials as the determinant tends to zero [@Ciarlet], the condition $K_f \in \mathrm{L}^1$ might in fact be the right postulate within our current knowledge. In addition, recall that the norm $\| K_f\|_{\mathrm{L}^1}$ equals the $\mathrm{W}^{1,2}$-Sobolev norm of $f^{-1}$; thus it is plausible that the condition $K_f \in \mathrm{L}^1$ is the right regularity requirement in order to have a flexible lower semicontinuity theory. It turns out that, for general functionals with the blow-up [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"}, proving lower semicontinuity requires two properties: the Jensen inequality for principal maps, allowing analysis via gradient Young measures, and secondly, control of concentration, typically via suitable equiintegrability. With these properties available we easily obtain the following lower semicontinuity result. **Theorem 10**. *Let $g\in \mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$ be a homeomorphism. Consider a sequence $(f_j )$ in $g + \mathrm{W}^{1,2}_0(\Omega)$ such that $f_j\rightharpoonup f$ in $\mathrm{W}^{1,2}(\Omega)$ and for some $q>1$ we have $\|K_{f_j}\|_{\mathrm{L}^{q}(\Omega)} \leq C < \infty$. Then $$\liminf_{j\to \infty} \int_\Omega {\mathscr W}(\textup{D}f_j(z)) \, \mathrm{d}m(z) \geq \int_\Omega {\mathscr W}(\textup{D}f(z))\, \mathrm{d}m(z).$$* Similar weak lower semicontinuity holds for the rank-one convex functionals from Theorem [Theorem 8](#thm:morreysplit){reference-type="ref" reference="thm:morreysplit"} with the appropriate volumetric-isochoric split, see Corollary [Corollary 77](#cor:lscsplit){reference-type="ref" reference="cor:lscsplit"}. Notice that as in the Burkholder setting the endpoint result, which in this context is $K_f \in \mathrm{L}^1$, is missing. This amounts to study possible concentration effects and we will investigate it in a future work. Building on ${\mathscr W}$ or on $\widetilde {\, \mathscr W\,}$ we obtain in Subsection [12.3](#sec:elasticity){reference-type="ref" reference="sec:elasticity"} a number of quasiconvex non-polyconvex functionals for which the direct method of the Calculus of Variation gives existence of minimizers, see Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"} and Example [Example 79](#nonpoly){reference-type="ref" reference="nonpoly"}. Morever since, by the work of Koskela and Onninen [@KO], the norm $\|K_f\|_{\mathrm{L}^q}$ controls $\|\log J_f\|_{\mathrm{L}^q}$, many of these functionals allow the blow-up condition [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"}. This, in particular, sheds light on the problem of existence of minimizers in hyperelasticity, see [@Ball2002] and in particular Problem 1 there. On the other hand, this class also contains functionals which degenerate in various ways when the determinant vanishes, like the Burkholder functional itself. Therefore its interest is not restricted to the elasticity ecosystem but is relevant also to the geometric function theory interpretation of our work. ## Outline {#outline .unnumbered} Finally, we conclude the introduction with a description of the organization of the paper. Section [2](#sec:prelims){reference-type="ref" reference="sec:prelims"} revisits the standard notions of the vectorial calculus of variations, with an emphasis on the special care needed to treat extended-real valued functionals. Section [3](#sec:principal){reference-type="ref" reference="sec:principal"} reviews relevant parts of the basic quasiconformal theory, with special focus on principal maps. Section [4](#Young){reference-type="ref" reference="Young"} adapts the theory of quasiregular Young measures, initiated in [@AstalaFaraco02] and applied for instance in [@FaracoSzekelyhidi08], to the case of principal maps. It also extends this theory to maps of integrable distortion. Section [5](#sec:Bf){reference-type="ref" reference="sec:Bf"} provides the preliminary results needed in the proof of the local quasiconvexity of the Burkholder functional, in particular an extremality argument in the spirit of [@AIPS12]. Section [6](#sec:Maintheorem){reference-type="ref" reference="sec:Maintheorem"} presents the proof of Theorems [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} and [Theorem 4](#main){reference-type="ref" reference="main"}. Section [7](#sec:areaBp){reference-type="ref" reference="sec:areaBp"} contains the proof of Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"}. Section [8](#sec:LlogL){reference-type="ref" reference="sec:LlogL"} studies the functional $\mathscr F$, which is the derivative of $B_p$ at $p=2$, and is closely related to the higher integrability of the Jacobian. Section [9](#sec:shield){reference-type="ref" reference="sec:shield"} revisits the classical Shield transformation, which uses inverses to define new integral functionals and, in particular, presents $\mathscr W$ as a transformation of $\mathscr F$, which leads to the proof of Corollary [Corollary 7](#cor:Wintro){reference-type="ref" reference="cor:Wintro"}. For the sake of completeness we also investigate for which class of test functions the quasiconvexity inequality for $\mathscr W$ can be verified. Section [10](#sec:inequalities){reference-type="ref" reference="sec:inequalities"} proves Theorem [Theorem 9](#thm:Wareaintro){reference-type="ref" reference="thm:Wareaintro"}, which gives quasiconvexity of $\mathscr W$ in the class of principal maps. Similar inequalities are established for $\mathscr F$. Section [11](#sec:advolum){reference-type="ref" reference="sec:advolum"} proves the quasiconvexity of functionals with volumetric isochoric split after that of $\mathscr W$, streamlining the arguments in [@Voss1] and proving Theorem [Theorem 8](#thm:morreysplit){reference-type="ref" reference="thm:morreysplit"}. In Section [12](#sec:swlsc){reference-type="ref" reference="sec:swlsc"} we are then in position to apply a version of the direct method of calculus of variations to prove lower semicontinuity theorems and existence of minimizers for a quite large family of functionals, see in particular Theorem [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"}. In this section we also prove Corollary [Corollary 5](#cor:Bpminims){reference-type="ref" reference="cor:Bpminims"} and Theorem [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"}. ## Notation {#notation .unnumbered} We denote by $\mathbb D\subset \mathbb{C}$ the unit disk and by $\mathbb A(r,R)$ the annulus $\{z:r<|z|<R\}$. Unless explicitly stated otherwise, $\Omega\subset \mathbb{R}^n$ is a bounded domain such that $\mathscr{L}^n(\partial\Omega)=0$; in most of the paper we will take ${n=2}$. Given a map $g\colon \mathbb{R}^n\to \mathbb{R}^n$, we sometimes use the notation $\mathrm{W}^{1,p}_g(\Omega)\equiv g+\mathrm{W}^{1,p}_0(\Omega,\mathbb{R}^n)$; in particular, this space is well-defined even if $\partial\Omega$ is irregular. We also use the standard notation $$\fint_\Omega \varphi(x) \,\mathrm{d}x \equiv \frac{1}{|\Omega|} \int_\Omega \varphi(x) \,\mathrm{d}x.$$ A matrix $A\in \mathbb{R}^{2\times 2}$ is naturally identified with a linear map $A\colon \mathbb{C}\to \mathbb{C}$. It will also be useful to use conformal coordinates, whereby we identify $A\in \mathbb{R}^{2\times 2}$ with a pair $(a_+,a_-)\in \mathbb{C}^2$ according to the rule $$\label{eq:confcoords} A(z) = a_+ z + a_- \bar z.$$ Here, on the left-hand side we see $z$ as an element of $\mathbb{R}^2$, while on the right-hand side $z\in \mathbb{C}$. In these coordinates we have $$\label{eq:detconformal} \det A=|a_+|^2-|a_-|^2, \qquad |A|=|a_+|+|a_-|,$$ where $|A|\equiv \max_{z\in \mathbb S^1} |A(z)|$ denotes the operator norm. Given a map $f\in \mathrm{W}^{1,1}_\textup{loc}(\Omega)$, we write $J_f\equiv \det \textup{D}f$ for its Jacobian. If $J_f>0$ a.e.  in $\Omega$ then there is a measurable function $K\colon \Omega\to [1,+\infty]$ such that $K<\infty$ a.e.  and $$\label{distortion2} |\textup{D}f(z)|^2\leqslant K(z)J_f(z),\qquad {\rm a.e.\ in} \;\; \Omega.$$ The *distortion function* of $f$, which we denote by $K_f$, is the smallest such function $K$. ## Acknowledgments {#acknowledgments .unnumbered} D.F, K.A, A.K acknowledge the financial support of QUAMAP, the ERC Advanced Grant 834728, and of the Severo Ochoa Programme CEX2019-000904-S. A.G. was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. D.F and A.K were partially supported by CM and UAM, and A.K by Academy of Finland CoE Randomness and Structures, and Academy Fellowship Grant 355840. D.F acknowledges financial support by PI2021-124-195NB-C32. K.A, D.F, A.G, A.K acknowledge the hospitality and financial support of the Institute of Advanced studies during various periods in 2021-2022 and the discussions there with L. Székelyhidi, V. Šverák and C. De Lellis on topics related to the paper. K.A, A.G, J.K ackowledge the hospitality of Universidad Autónoma de Madrid and ICMAT during the autum of 2022. D.F also wants to acknowledge the hospitality of the Mathematical Institute of Oxford during the summer of 2023. # Extended real-valued functionals and Young measures {#sec:prelims} ## Quasiconvexity and rank-one convexity Put $\overline{\mathbb{R}}\equiv \mathbb{R}\cup \{ \pm \infty \}$ and let $F \colon \mathbb{R}^d \to \overline{\mathbb{R}}$ be an extended real-valued function. The epigraph of $F$ is the subset $$\mathrm{epi}(F) \equiv \bigl\{ (x,y) \in \mathbb{R}^{d+1} : \, y \geq F(x) \bigr\}$$ of $\mathbb{R}^{d+1}$. We say that $F$ is *convex* if its epigraph is a convex subset of $\mathbb{R}^{d+1}$. The following result is an easy consequence of the definitions and we leave the proof to the interested reader. **Lemma 11**. *Assume $F \colon \mathbb{R}\to \overline{\mathbb{R}}$ is convex and that for some $x_0 \in \mathbb{R}$ we have $F(x_{0})=-\infty$. Then there exist $\alpha$, $\beta \in \overline{\mathbb{R}}$ with $\alpha \leq x_{0} \leq \beta$ such that $$F = \left\{ \begin{array}{ll} -\infty & \mbox{ in } ( \alpha , \beta ),\\ +\infty & \mbox{ on } \mathbb{R}\setminus [ \alpha , \beta ] . \end{array} \right.$$ Furthermore, when $\alpha \in \mathbb{R}$ or $\beta \in \mathbb{R}$, the values of $F$ there are unrestricted.* Let ${\bf E} \colon \mathbb{R}^{m\times n} \to \overline{\mathbb{R}}$ be an extended real-valued function (henceforth we refer to functions defined on $\mathbb{R}^{m\times n}$ as *functionals*). The *effective domain* of ${\bf E}$ is the subset $$\label{effective} \mathrm{dom}({\bf E}) \equiv \bigl\{ A \in \mathbb{R}^{m\times n}: \, {\bf E}(A) < +\infty \bigr\}$$ of $\mathbb{R}^{m\times n}$, where we emphasize that the value $-\infty$ is allowed for ${\bf E}$ on $\mathrm{dom}({\bf E})$. **Definition 12**. A functional ${\bf E} \colon \mathbb{R}^{m\times n} \to \overline{\mathbb{R}}$ is *rank-one convex* if, for all $A$, $X \in \mathbb{R}^{m\times n}$ with $\mathrm{rank}(X)=1$, the function $t \mapsto {\bf E}(A+tX)$ is convex. We emphasize that in the considered generality this notion of rank-one convexity is rather weak, see already Example [Example 32](#ex:4pts){reference-type="ref" reference="ex:4pts"} below. For the next result we need the notion of algebraic interior or *core* of a subset $S \subseteq \mathbb{R}^{m\times n}$, namely $A \in \mathrm{core}(S)$ provided for each $X \in \mathbb{R}^{m\times n}$ we can find $\delta > 0$ such that $A+tX \in S$ for all $t \in [0,\delta )$. **Lemma 13**. *Assume ${\bf E} \colon \mathbb{R}^{m\times n} \to \overline{\mathbb{R}}$ is a rank-one convex functional and that for some $A_{0} \in \mathrm{core}\bigl( \mathrm{dom}({\bf E}) \bigr)$ we have ${\bf E}(A_{0}) \in \mathbb{R}$. Then ${\bf E} > -\infty$ on all lines through $A_0$ that are parallel to a rank one matrix.* *Proof.* If there exists $X \in \mathbb{R}^{m\times n}$ of rank one such that ${\bf E}(A_{0}-X) = -\infty$, then by convexity in the direction of $X$ we infer that ${\bf E}(A_{0}+tX) = +\infty$ for all $t>0$. But this is impossible when $A_{0} \in \mathrm{core}\bigl( \mathrm{dom}({\bf E}) \bigr)$, so ${\bf E}> -\infty$ must hold on all lines through $A_0$ that are parallel to a rank one matrix. ◻ **Corollary 14**. *Assume ${\bf E} \colon \mathbb{R}^{m\times n} \to \overline{\mathbb{R}}$ is a rank-one convex functional and that for some $A_0$ in the topological interior of $\mathrm{dom}({\bf E})$ we have ${\bf E}(A_{0}) \in \mathbb{R}$. Then ${\bf E} > -\infty$ on the connected component $\mathcal{U}$ of the interior of $\mathrm{dom}({\bf E})$ that contains $A_0$. In fact, ${\bf E} > -\infty$ on any line through a point of $\mathcal{U}$ that is parallel to a rank one matrix.* *Proof.* It is well-known that any two points of an open connected set in $\mathbb{R}^{m\times n}$ can be connected by a piecewise linear curve with each line segment parallel to a rank one matrix. It therefore follows using Lemma [Lemma 13](#lem2){reference-type="ref" reference="lem2"} that ${\bf E} > -\infty$ on $\mathcal{U}$ and hence that ${\bf E}>-\infty$ on any line which is parallel to a rank one matrix and which intersects $\mathcal{U}$. ◻ In relation to rank-one convexity it is natural to consider quasiconvexity, which is an integral condition. In our key results we are considering extended real-valued functionals ${\bf E}$, and thus given a Borel measure $\nu$ in $\mathbb{R}^{m\times n}$ we write for such functionals $$\label{upperint} \int_{\mathbb{R}^{m\times n}}^* \! {\bf E} \, \mathrm{d}\nu \equiv \begin{cases} \int_{\mathbb{R}^{2 \times 2}} \! {\bf E} \, \mathrm{d}\nu \quad &\text{if} \;\; {\bf E} \in \mathrm{L}^{1}(\nu), \\ + \infty \quad &\text{if} \;\; {\bf E}^+ \notin \mathrm{L}^{1}(\nu),\\ -\infty & \text{if} \;\; {\bf E}^+ \in \mathrm{L}^{1}(\nu) \; {\rm but } \; {\bf E}^- \notin \mathrm{L}^{1}(\nu). \end{cases}$$ One can interpret [\[upperint\]](#upperint){reference-type="eqref" reference="upperint"} as an *upper $\nu$-integral* of ${\bf E}$. In the setting of extended real-valued functionals, a natural counterpart to the notion [\[quasiconvexity\]](#quasiconvexity){reference-type="eqref" reference="quasiconvexity"} of quasiconvexity is to require integral bounds such as $$\label{eq:qc23} {\bf E}(A)\leq \fint^*_\Omega {\bf E}(A+\textup{D}\varphi) \,\mathrm{d}x$$ to hold for all bounded domains $\Omega\subset \mathbb{R}^n$ and, say, for all $\varphi \in \mathrm{W}^{1,p}_0(\Omega,\mathbb{R}^m)$. However, the above version of quasiconvexity is very weak. In fact, it does not even imply rank-one convexity [@BM1 Example 3.5]: **Example 15** (Two points). Let $X\in \mathbb{R}^{m\times n}$ have $\textup{rank}(X)=1$. The functional $${\bf E}(0)={\bf E}(X)=0,\qquad {\bf E}=+\infty \textup{ otherwise},$$ is not rank-one convex, yet it satisfies [\[eq:qc23\]](#eq:qc23){reference-type="eqref" reference="eq:qc23"} for all $\varphi \in \mathrm{W}^{1,1}_0$ and $A \in \mathbb{R}^{m\times n}$. Thus, arguably, the correct notions to consider are those of closed rank-one convexity and closed quasiconvexity [@Pe1], see also [@JK]. In order to define these concepts we need to introduce the gradient Young measures, and this is the purpose of the next subsection. ## Gradient Young measures Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $\mathscr L^n(\partial\Omega)=0$. Informally, a *gradient Young measure* on $\Omega$ is a parametrized family of probability measures $\nu=(\nu_x)_{x\in \Omega}$ on $\mathbb{R}^{m \times n}$, where at each point $x \in \Omega$, the measure $\nu_x$ describes the oscillations of the derivatives $\textup{D}\varphi_j(y)$ for points $y$ near $x$, in the limit $j \to \infty$ for a sequence $( \varphi_j )$ that converges weakly in $\mathrm{W}^{1,p}(\Omega, \mathbb{R}^m)$. More precisely, if $\mathscr{M}(\mathbb{R}^{m \times n})$ denotes the space of finite signed Borel measures on $\mathbb{R}^{m \times n}$, then to any $\varphi \in \mathrm{W}^{1,p}(\Omega, \mathbb{R}^m)$ we can associate the function $$T_\varphi \colon \Omega \mapsto \mathscr{M}(\mathbb{R}^{m \times n}), \quad T_\varphi (x) = \delta_{\textup{D}\varphi(x)}.$$ Here $T_\varphi$ is bounded and weakly$\mbox{}^\ast$ measurable, that is, $T_\varphi \in \mathrm{L}^{\infty}_{\omega}\bigl(\Omega, \mathscr{M}(\mathbb{R}^{m \times n})\bigr)$. Furthermore, with the natural duality pairing, the Lebesgue--Pettis space $\mathrm{L}^{\infty}_{\omega}\bigl(\Omega, \mathscr{M}(\mathbb{R}^{m \times n})\bigr)$ is the dual of the Lebesgue--Bochner space $\mathrm{L}^1\bigl(\Omega, \mathrm{C}_0(\mathbb{R}^{m \times n})\bigr)$ and under this duality the $\mathrm{W}^{1,p}$-gradient Young measures $\nu=(\nu_x)_{x\in \Omega}$ are precisely the weak$^\ast$-limits of the sequences $( T_{\varphi_j} )$, where $( \varphi_j )$ is a sequence converging weakly in $\mathrm{W}^{1,p}(\Omega, \mathbb{R}^m)$. In explicit terms, this means that for all ${\bf E}\in \mathrm{C}_0( \mathbb{R}^{m\times n})$, $$\label{Young1} \; {\bf E}( \textup{D}\varphi_j (x) ) \; \stackrel{w^*}{\longrightarrow} \; \; \langle \nu_x , {\bf E} \rangle \, \equiv \int_{\mathbb{R}^{m \times n}} {\bf E}(A) \, \mathrm{d}\nu_x(A) \qquad {\rm in } \; \; \mathrm{L}^\infty(\Omega).$$ When [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} holds, we say that the sequence $(D\varphi_j)$ of gradients *generates* the Young measure $\nu$. Formalising this concept we have: **Definition 16**. Let $p\in [1,\infty]$. A parametrized family of probability measures $\nu=(\nu_x)_{x\in \Omega}$ on $\mathbb{R}^{m \times n}$ is a *$\mathrm{W}^{1,p}$-gradient Young measure* if there exists a weakly converging sequence $(\varphi_j)$ in $\mathrm{W}^{1,p}(\Omega,\mathbb{R}^m)$ whose gradients generate $\nu$, that is, [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} holds for all ${\bf E}\in \mathrm{C}_0( \mathbb{R}^{m\times n})$. *Remark 17*. We collect a few immediate consequences of the definition of $\mathrm{W}^{1,p}$ gradient Young measure. First, as a limit of weak$^\ast$-measurable functions also the family $( \nu_x )$ is weak$^*$-measurable. This means that for every ${\bf E}\in \mathrm{C}_0( \mathbb{R}^{m\times n})$ the function $$\Omega \ni x \mapsto \int_{\mathbb{R}^{m\times n}} {\bf E}(A) \, \mathrm{d}\nu_x(A)$$ is Lebesgue measurable. This property ensures that standard constructions involving $( \nu_x )$ result in measurable functions. Next, the fact that $( \nu_x )$ is generated by an $\mathrm{L}^p$-bounded sequence implies that the measure satisfies the *$p$-moment condition*:   For almost all $x \in \Omega$, $$\label{moments} \int_{\Omega} \! \langle \nu_{x} , | \cdot |^{p} \rangle \, \mathrm{d}x < \infty \quad \mbox{ when } p \in [1,\infty ),$$ whereas when $p=\infty$ there exists $R>0$ such that the supports $\mathrm{supp}( \nu_{x}) \subset B_{R}(0)$ for $\mathscr{L}^n$ almost all $x \in \Omega$. The moment condition in particular means that for $\mathscr{L}^n$ almost all $x \in \Omega$ the probability measure $\nu_x$ has a centre of mass, and it is not difficult to see from [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} and Remark [Remark 20](#Young31){reference-type="ref" reference="Young31"} that $$\label{limit} \langle \nu_x, \mathrm{Id}\rangle \equiv \int_{\mathbb{R}^{m \times n}} \! A \, \mathrm{d}\nu_x(A) = \textup{D}\phi (x) \qquad {\rm for} \; a. e. \; x \in \Omega ,$$ where $\phi$ is the weak limit of the generating sequence $( \phi_{j})$. Finally we note that since for any bounded open set $\Omega$ with $\mathscr{L}^{n}( \partial \Omega )=0$ the inclusion $\mathrm{W}^{1,s}(\Omega,\mathbb{R}^m) \subset \mathrm{W}^{1,p}(\Omega,\mathbb{R}^m)$ holds if $p \leq s$, every $\mathrm{W}^{1,s}$-gradient Young measure is a $\mathrm{W}^{1,p}$-gradient Young measure whenever $p \leq s$. Any norm-bounded sequence of gradients admits a subsequence that generates a gradient Young measure [@Muller Theorem 3.1]: **Theorem 18**. *Let $p\in [1,\infty]$ and let $( \varphi_j )$ be a bounded sequence in $\mathrm{W}^{1,p}(\Omega,\mathbb{R}^m)$. Then there is a subsequence, which we do not relabel, such that $(\textup{D}\varphi_j)$ generates a $\mathrm{W}^{1,p}$-gradient Young measure $\nu=(\nu_x)_{x\in \Omega}$.* However, even if the original sequence $(\varphi_j)$ converges weakly in $\mathrm{W}^{1,p}(\Omega,\mathbb{R}^m)$, the gradient Young measures its subsequences generate need not be unique. On the other hand, since the center of mass or barycenter is the weak limit of [\[limit\]](#limit){reference-type="eqref" reference="limit"}, this is of course the same for all Young measures generated by the subsequences of a given weakly converging sequence. In view of [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} the gradient Young measures provide flexible methods to compute weak limits of nonlinear quantities. Hence the notion is very useful also in our setting. For later purposes we list here some of their basic and well-known properties. For instance, given a Young measure one can often modify or improve the generating sequence. **Theorem 19**. *[@Pe Theorem 8.15 and Lemma 6.3] Suppose $\nu=(\nu_x)_{x\in \Omega}$ is a $\mathrm{W}^{1,p}$-gradient Young measure in $\Omega$ and suppose $\langle \nu_{x},\mathrm{Id} \rangle = \textup{D}g(x)$ a.e. in $\Omega$, where $g \in \mathrm{W}^{1,p}( \mathbb{R}^n , \mathbb{R}^m )$.* *(1) Then there is a bounded sequence $(\varphi_j)$ in $g+\mathrm{W}^{1,p}_{0}(\Omega,\mathbb{R}^m)$ whose gradients generate $\nu$ and for which $$\label{equi} ( | \textup{D}\varphi_j |^p ) \quad {\rm is \; equiintegrable \; on} \; \Omega.$$* *(2) If $(\varphi_j)$ is a bounded sequence in $\mathrm{W}^{1,p}(\Omega,\mathbb{R}^m)$ that generates $\nu$, and $(\psi_j)$ is another sequence for which $\textup{D}\psi_j-\textup{D}\varphi_j\to 0$ in $\mathrm{L}^p(\Omega)$, then also $(\textup{D}\psi_j)$ generates $\nu$.* The fact that the generating sequence can be chosen so that $(|\textup{D}\varphi_j|^p)$ is equiintegrable will be important for us and is related to the so-called Decomposition Lemma [@FMP; @JK-1; @JK0]. *Remark 20*. On the other hand, the limit [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} exists even for a general continuous ${\bf E}\in \mathrm{C}(\mathbb{R}^{m\times n})$ which does not vanish at $\infty$, but where, for the generating sequence, $( {\bf E}(\textup{D}\varphi_j) )$ is equiintegrable, see [@Pe Theorem 6.2]. The literature on Gradient Young measures is by now quite extensive and for further properties we refer to the monographs [@KrRa; @Muller; @Pe; @Rindler]. *Remark 21*. It is not difficult to extend the convergence [\[Young1\]](#Young1){reference-type="eqref" reference="Young1"} also to lower semicontinuous functionals ${\bf E}\colon \mathbb{R}^{m\times n}\to\mathbb{R}\cup\{+\infty\}$ which are continuous on $\textup{dom}({\bf E})$, provided that $\left({\bf E}(\textup{D}\varphi_j)\right)$ is equiintegrable. This can be proved by approximating ${\bf E}$ with the sequence of truncations ${\bf E}_k\equiv \min\{{\bf E},k\}$ and applying the standard lower semicontinuity result $$\label{eq:lscYM} \liminf_{j\to \infty} \int_\Omega \eta(x) {\bf E}_k(\textup{D}\varphi_j)\, \mathrm{d}x\geq \int_\Omega \eta(x) \int_{\mathbb{R}^{m\times n}} {\bf E}_k(A) \,\mathrm{d}\nu_x(A) \, \mathrm{d}x,$$ where $\eta\in \mathrm{L}^\infty(\Omega)$ is arbitrary, cf. [@Muller Corollary 3.3]; the opposite direction simply follows from the equi-integrability assumption. We note that [\[eq:lscYM\]](#eq:lscYM){reference-type="eqref" reference="eq:lscYM"} can also be proved by approximating lower semicontinuous functionals with continuous ones, using the Scorza--Dragoni theorem. In this paper, only in Section [12](#sec:swlsc){reference-type="ref" reference="sec:swlsc"} we really use gradient Young measures in the full generality of Definition [Definition 16](#def:gYM){reference-type="ref" reference="def:gYM"}. Instead, most of the time we will be content to work with homogeneous gradient Young measures: **Definition 22**. A $\mathrm{W}^{1,p}$-gradient Young measure $(\nu_x)_{x\in \Omega}$ is *homogeneous* if there is a probability measure $\nu$ on $\mathbb{R}^{m\times n}$ such that $\nu_x=\nu$ for a.e. $x\in \Omega$. In this case we naturally identify $(\nu_x)_{x\in \Omega}$ with $\nu$. We often denote by $\mathscr{M}^p_\textup{qc}$ the class of homogeneous $\mathrm{W}^{1,p}$-gradient Young measures; the notation is motivated by Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"} below. For a subset $\mathcal{U}\subset\mathbb{R}^{m\times n}$ we write $\mathscr{M}^{p}_{\mathrm{qc}}( \mathcal{U})$ for the set of measures in $\mathscr{M}^{p}_{\mathrm{qc}}$ whose support is contained in $\mathcal{U}$. *Remark 23*. There are many natural ways to construct homogeneous $\mathrm{W}^{1,p}$-gradient Young measure. For instance, given $A \in \mathbb{R}^{m \times n}$ and $\phi \in \mathrm{W}^{1,p}_{0}( \Omega , \mathbb{R}^m )$ we associate to them the measure $\nu_{A+\textup{D}\phi}$ defined by the rule $$\label{exhgym} \nu_{A+\textup{D}\phi}(\mathcal{S}) \equiv \frac{\mathscr{L}^{n}\bigl( \{ x \in \Omega : \, A+\textup{D}\phi (x) \in \mathcal{S} \} \bigr)}{\mathscr{L}^{n}( \Omega )},$$ where $\mathcal{S} \subset \mathbb{R}^{m \times n}$ is a Borel set. By inspection, $\nu_{A+\textup{D}\phi}$ is a Borel probability measure with a finite $p$-th moment $\langle \nu_{A+\textup{D}\phi} , | \cdot |^{p} \rangle < +\infty$ and centre of mass $\langle \nu_{A+\textup{D}\phi},\mathrm{Id} \rangle =A$. In particular, the measure $\nu_{A+\textup{D}\phi}$ describes the distribution of values of $A+\textup{D}\phi (x)$ in $\mathbb{R}^{m \times n}$ when $x$ varies over $\Omega$ and we use the normalized volume $\mathscr{L}^n$ as weight. To represent $\nu_{A+\textup{D}\phi}$ as a homogeneous Young measure, since $\Omega$ is a bounded domain with $\mathscr{L}^{n}( \partial \Omega )=0$, we can realize this distribution on any other open bounded subset of $\mathbb{R}^n$, and for later purposes we choose a realization on the open unit cube, $\mathbf{Q} \equiv \bigl( -\tfrac{1}{2},\tfrac{1}{2} \bigr)^n$. Indeed, a standard exhaustion argument allows us to write $\mathbf{Q}$ as a disjoint union of scaled and translated copies of $\Omega$: $$\mathbf{Q} = N \cup \bigcup_{s \in \mathbb{N}} \bigl( x_{s}+r_{s}\Omega \bigr) \quad \bigl(\mbox{disjoint union!} \bigr)$$ where $\mathscr{L}^{n}(N)=0$. Next we import $\phi$ on $\mathbf{Q}$ by the definition $$\varphi (x) \equiv \left\{ \begin{array}{ll} r_{s}\phi \bigl( \frac{x-x_{s}}{r_{s}} \bigr) & \mbox{ if } x \in x_{s}+r_{s}\Omega , \, s \in \mathbb{N}\\ 0 & \mbox{ if } x \in N. \end{array} \right.$$ It is routine to check that hereby $\varphi \in \mathrm{W}^{1,p}_{0}( \mathbf{Q}, \mathbb{R}^m )$ and that $\nu_{A+\textup{D}\varphi} = \nu_{A+\textup{D}\phi}$, if $\nu_{A+\textup{D}\varphi}$ is defined as in [\[exhgym\]](#exhgym){reference-type="eqref" reference="exhgym"} with the obvious modifications. It is now easy to check that $\nu_{A+\textup{D}\phi}$ is a homogeneous $\mathrm{W}^{1,p}$-gradient Young measure with centre of mass at $A$: Namely, extend $\varphi$ to $\mathbb{R}^n$ by $\mathbf{Q}$-periodicity and define $u_{j}(x) \equiv Ax+\varphi (jx)/j$, $x \in \Omega$. Using the Riemann-Lebesgue lemma it follows that $u_{j} \rightharpoonup A$ in $\mathrm{W}^{1,p}( \Omega , \mathbb{R}^m )$ and that $( \textup{D}u_j )$ generates the Young measure $( \nu_x )_{x \in \Omega}$, where $\nu_x = \nu_{A+\textup{D}\phi}$ for all $x \in \Omega$. The examples provided in the next Subsection [2.3](#closedqc){reference-type="ref" reference="closedqc"} show that $\mathscr{M}^{p}_{\mathrm{qc}}$, the set of homogeneous measures, contains probability measures that cannot be represented as $\nu_{A+\textup{D}\phi}$ for any $A$, $\phi$. On the other hand, using a variant of the above construction and a diagonalization argument it is not too difficult to see that $\mathscr{M}^{p}_{\mathrm{qc}}$ can be defined as a suitable closure of the set $\bigl\{ \nu_{A+\textup{D}\phi} : \, \phi \in \mathrm{W}^{1,p}_{0}( \Omega , \mathbb{R}^m ), \, A \in \mathbb{R}^{m \times n} \bigr\}$. However, we will not need this in the sequel. The usefulness of homogeneous gradient Young measures stems from the fact that a *general* gradient Young measure is essentially a collection of homogeneous gradient measures: **Proposition 24**. *[(The Localization Principle [@KP1])]{.roman}[\[prop:homogenization\]]{#prop:homogenization label="prop:homogenization"} Given a $\mathrm{W}^{1,p}$-gradient Young measure $\nu=(\nu_x)_{x\in \Omega}$, we have $\nu_x\in \mathscr M^p_\textup{qc}$ for a.e. $x\in \Omega$.* *Remark 25*. In fact, if $\nu$ is generated by a sequence $(\varphi_j)$ then, for a.e. $x_0$, the measure $\nu_{x_0}$ is generated by a diagonal subsequence of the sequence $\psi_{j,\lambda}(x) \equiv \lambda^{-1} (\varphi_j(x_0+\lambda x) - \varphi(x_0))$, as $j\to \infty$ and $\lambda\to 0$. Note that $$\textup{D}\psi_{j, \lambda}(x) = \textup{D}\varphi_j(x_0+\lambda x).$$ Since $\lambda\to 0$, we can assume that the maps $\psi_{j,\lambda}$ are defined on any bounded open set $\Omega$ for which $\mathscr L^n(\partial\Omega)=0$. Also, the basic invariance properties of $\mathscr M^p_\textup{qc}$ follow quickly: For $t>0$, let $$\langle \nu_t,{\bf E}\rangle \equiv \langle \nu, {\bf E}(t\cdot)\rangle = \int_{\mathbb{R}^{n\times n}} \! {\bf E}(t A) \, \mathrm{d}\nu(A),$$ and similarly, if $m=n$ and $Q,R\in \textup{SO}(n)$, we define $$\langle \nu_{Q,R},{\bf E}\rangle \equiv \int_{\mathbb{R}^{n\times n}} \! {\bf E}(Q A R) \, \mathrm{d}\nu(A).$$ **Lemma 26**. *Fix $p\in [1,\infty]$.* 1. *[\[it:YMhomogeneous\]]{#it:YMhomogeneous label="it:YMhomogeneous"} For any $t>0$, the map $\nu\mapsto \nu_t$, maps $\mathscr M^p_\textup{qc}$ bijectively onto itself.* 2. *[\[it:YMisotropic\]]{#it:YMisotropic label="it:YMisotropic"} Similarly, if $m=n$ and $Q,R\in \textup{SO}(n)$, the map $\nu\mapsto \nu_{Q,R},$ maps $\mathscr M^p_\textup{qc}$ bijectively onto itself.* *Proof.* Let $(\varphi_j)$ be a bounded sequence in $\mathrm{W}^{1,p}( \Omega ,\mathbb{R}^m )$ such that $( \textup{D}\varphi_j )$ generates $\nu$. For part ([\[it:YMhomogeneous\]](#it:YMhomogeneous){reference-type="ref" reference="it:YMhomogeneous"}) we consider a new sequence $\psi_j\equiv \varphi_j(t \cdot)$ and for part ([\[it:YMisotropic\]](#it:YMisotropic){reference-type="ref" reference="it:YMisotropic"}) we take $\eta_j \equiv Q \varphi_j(R\cdot)$, so $$\textup{D}\psi_j = t \textup{D}\varphi_j(t \cdot), \qquad \textup{D}\eta_j = Q \textup{D}\varphi_j(R\cdot) R.$$ It is easy to verify that the sequences $(\textup{D}\psi_j)$ and $(\textup{D}\eta_j)$ generate the measures $\nu_t$ and $\nu_{Q,R}$, respectively. Since $(\nu_t)_{t^{-1}}= \nu$ and $(\nu_{Q,R})_{Q^{-1},R^{-1}}= \nu$, the claim follows. ◻ Homogeneous gradient Young measures admit another characterization which is often more convenient to work with [@KP; @KP1]. **Theorem 27**. *Fix $p\in [1,\infty)$ and let $\nu$ be a Borel probability measure on $\mathbb{R}^{m\times n}$. Here and throughout we denote its center of mass by $$\left\langle \nu, \mathrm{Id} \right\rangle \equiv \int_{\mathbb{R}^{m\times n}} A \,\mathrm{d}\nu(A),$$ so that in particular $\langle \nu, \mathrm{Id} \rangle \in \, \mathbb{R}^{m\times n}$.* *Then we have $\nu\in \mathscr{M}^p_\textup{qc}$ if and only if $\nu$ satisfies the following two conditions:* 1. *[\[it:jensen\]]{#it:jensen label="it:jensen"} For all quasiconvex ${\bf E} \colon \mathbb{R}^{m\times n} \to \mathbb{R}$ with $\sup_{A\in \mathbb{R}^{m\times n}} \frac{|{\bf E}(A)|}{1+|A|^p} < \infty$, (no condition if $p=\infty$), the Jensen inequality $${\bf E}( \langle \nu, \mathrm{Id} \rangle ) \, \leq \; \int_{\mathbb{R}^{m\times n}} \! {\bf E}(A) \, \mathrm{d}\nu(A)$$ holds;* 2. *[\[it:pmoment\]]{#it:pmoment label="it:pmoment"} $\nu$ has finite $p$-th moment, in the sense [\[moments\]](#moments){reference-type="eqref" reference="moments"}.* In this connection we also recall that $$\label{momentcondition} \mathscr{M}_{\mathrm{qc}}^{p} = \mathscr{M}^{1}_{\mathrm{qc}} \cap \bigl\{ \nu : \, \nu \mbox{ has a finite $p$-th moment } \bigr\}$$ for $p \in (1,\infty ]$, see [@JK0 Corollary 1.8] for $p<\infty$ and [@KZ] for $p=\infty$. *Remark 28*. Note also that if $p\in (1,\infty)$, $\nu \in \mathscr{M}^p_\textup{qc}(\mathbb{R}^{m\times n})$ and $\varphi$ is the weak limit of a sequence $(\varphi_j)$ generating $\nu$, then by [\[limit\]](#limit){reference-type="eqref" reference="limit"} $$\textup{D}\varphi(x) \equiv \langle \nu, \mathrm{Id} \rangle$$ and therefore $\varphi$ is affine. As a last remark, Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"} also motivates the definition of the class of measures which satisfy Jensen's inequality with respect to rank-one convex instead of quasiconvex functionals. **Definition 29**. For $p\in [1,\infty]$, $\mathscr{M}^p_\textup{rc}$ is the set of those Borel probability measures $\nu$ in $\mathbb{R}^{m\times n}$ such that conditions ([\[it:jensen\]](#it:jensen){reference-type="ref" reference="it:jensen"}) and ([\[it:pmoment\]](#it:pmoment){reference-type="ref" reference="it:pmoment"}) of Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"} hold, with the word *quasiconvex* replaced by *rank-one convex*. The set $\mathscr{M}^p_\textup{rc}$ agrees with the set of *$p$-laminates*, see e.g. [@Faraco2004 Definition 5.3], although we will not use laminates in the sequel. ## Closed quasiconvexity and closed rank-one convexity {#closedqc} We are now ready to introduce what we believe are the correct notions of rank-one convexity and quasiconvexity for extended-real valued functionals. **Definition 30**. A functional ${{\bf E}}\colon \mathbb{R}^{m \times n} \to \overline \mathbb{R}$ is *closed $\mathrm{W}^{1,p}$-quasiconvex* (respectively *closed $p$-rank-one convex*) if for all $A\in \mathbb{R}^{m\times n}$ and all $\nu \in \mathscr{M}^p_\textup{qc}$ (respectively all $\nu \in \mathscr{M}^p_\textup{rc}$) with $\langle \nu, \mathrm{Id} \rangle =A$ we have $$\label{closed} {\bf E}(A) = {\bf E}\left( \int_{\mathbb{R}^{m\times n}} \lambda \,\mathrm{d}\nu(\lambda) \right) \leq \int_{\mathbb{R}^{m\times n}}^* {\bf E}(\lambda) \,\mathrm{d}\nu(\lambda).$$ In particular we note that if ${{\bf E}}\colon \mathbb{R}^{m \times n} \to \overline \mathbb{R}$ is closed $\mathrm{W}^{1,p}$-quasiconvex, then it is also closed $\mathrm{W}^{1,s}$-quasiconvex for every $s > p$. Furthermore, closed $\mathrm{W}^{1,p}$-quasiconvexity implies the standard $\mathrm{W}^{1,p}$-quasiconvexity. In order to verify this, let $\phi \in \mathrm{W}^{1,p}_{0}( \Omega , \mathbb{R}^m )$ and $A \in \mathbb{R}^{m \times n}$ and recall Remark [Remark 23](#graddist){reference-type="ref" reference="graddist"}, where the probability measure $\nu_{A+\textup{D}\phi}$ was defined and shown to be a homogeneous $\mathrm{W}^{1,p}$-gradient Young measure with centre of mass $A$. Hence if $\mathbf{E}$ is a closed $\mathrm{W}^{1,p}$-quasiconvex functional, then $$\fint_{\Omega}^* \! \mathbf{E}(A + \textup{D}\phi (x)) \, \mathrm{d}x = \int_{\mathbb{R}^{m \times n}}^* \! \mathbf{E} \, \mathrm{d}\nu_{A+\textup{D}\phi} \geq \mathbf{E}(A).$$ For other approaches see for instance [@Muller; @Pe; @Rindler]. However, the converse is not true, and it is instructive to exemplify this point here. Similarly to Example [Example 15](#ex:2pts){reference-type="ref" reference="ex:2pts"}, the standard counterexamples are based on the following functional: Given a subset $\mathcal{U}\subset\mathbb{R}^{m\times n}$ we define its characteristic function, in the sense of convex analysis, by $$\label{eq:indicator} % \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\,\mathcal{U}}(A) \equiv \left\{ \begin{array}{ll} 0 & \mbox{ if } A \in \mathcal{U},\\ +\infty & \mbox{ if } A \in \mathbb{R}^{2 \times 2}\setminus \mathcal{U}. \end{array} \right.$$ We then choose the set $\mathcal U$ to be a special finite set of matrices called a *$T_N$-configuration*, see [@Forster2018 Definition 2.1]. In particular such sets contain no rank-one connections, meaning that if $\mathcal T_N=\{A_1,\dots, A_N\} \subset \mathbb{R}^{2\times 2}$ then $$\label{eq:noconnection} \textup{rank}(A_i-A_j)>1\quad \textup{ for all } i\neq j.$$ Kirchheim and Preiss [@K1; @K2] provide the following optimal example concerning $T_N$-configurations with $N \geq 5$, see also [@Forster2018]. **Example 31** (five points). It is possible to construct a set $\mathcal T_5=\{A_1,\dots, A_5\}$ of five matrices with the following property: $$\label{eq:trivsols} \textup{There is } A\not \in \mathcal{T}_5 \textup{ and }\varphi \in W^{1,\infty}_0(\Omega,\mathbb{R}^2) \textup{ with } A + \textup{D}\varphi \in \mathcal{T}_5 \textup{ a.e.}$$ Thus the functional $% \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\,\mathcal T_5}$ is rank-one convex but not quasiconvex, it does not satisfy [\[eq:qc23\]](#eq:qc23){reference-type="eqref" reference="eq:qc23"}. Note that this in particular provides us with an example of a rank-one convex functional on symmetric $2$-by-$2$ matrices which is not quasiconvex. It is an extended real-valued and lower semicontinuous functional and it is not clear if one could construct such an example which is continuous, let alone real-valued. The property [\[eq:trivsols\]](#eq:trivsols){reference-type="eqref" reference="eq:trivsols"} fails for all configurations of four matrices as proved by Chlebik and Kirchheim in [@Chlebik]. This gives us the following example: **Example 32** (four points). Let $\mathcal T_4=\{A_1,\dots, A_4\}$ be any set in $\mathbb{R}^{2 \times 2}$ satisfying [\[eq:noconnection\]](#eq:noconnection){reference-type="eqref" reference="eq:noconnection"}. Then [\[eq:trivsols\]](#eq:trivsols){reference-type="eqref" reference="eq:trivsols"} fails for the set. Thus the functional $% \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\,\mathcal T_4}$ is both quasiconvex and rank-one convex. Moreover, as observed in [@JK Example 1.3] the corresponding variational integral $\int_{\Omega} \! % \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\, \mathcal{T}_{4}} \! (Du) \, \mathrm{d}x$ is sequentially weakly lower semicontinuous on $\mathrm{W}^{1,1}( \Omega , \mathbb{R}^{2})$. Nonetheless, if we assume the set $\mathcal{T}_4$ is a $T_4$ configuration, then there exists nontrivial $\nu \in \mathscr{M}^\infty_\textup{rc}(\mathcal T_4)\subseteq \mathscr{M}^\infty_\textup{qc}(\mathcal T_4)$ (see for instance [@Muller]), so $% \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\,\mathcal T_4}$ is neither closed $\mathrm{W}^{1,p}$-quasiconvex nor closed $p$-rank-one convex for any $p \in [1,\infty ]$. To summarize, the above examples with Example [Example 15](#ex:2pts){reference-type="ref" reference="ex:2pts"} show for the extended real-valued functionals in $\mathbb{R}^{m\times n}$, with $m,n\geq 2$, the following relations $$\begin{tikzcd}[row sep=tiny,arrows=Rightarrow] & \textup{quasiconvex} \arrow[dr, shift left=0.9ex,degil]\arrow[dd,degil]\arrow[dl,shift left=0.9ex,degil]& \\ \textup{closed quasiconvex}\arrow[ur,shift left=0.9ex] \arrow[dr] & & \textup{rank-1 convex} \arrow[shift left=0.9ex, ul,degil] \arrow[shift left=0.9ex,dl,degil]\\ & \textup{closed rank-1 convex} \arrow[ur,shift left=0.9ex] & \end{tikzcd}$$ and we emphasize that whether or not closed rank-one convexity implies either quasiconvexity or closed quasiconvexity is unknown if $m=2$, $n \geq 2$. If $m>2$ then closed rank-one convexity is a strictly weaker notion [@Sverak92]. As observed already by Morrey [@Morrey], quasiconvexity of a given functional ${\bf E}\colon \mathbb{R}^{m\times n}\to \mathbb{R}\cup\{+\infty\}$ is intimately tied to the sequential weak lower semicontinuity of the associated integral (compare also Remark [Remark 21](#rmk:moreintegrands){reference-type="ref" reference="rmk:moreintegrands"}, Proposition [\[prop:homogenization\]](#prop:homogenization){reference-type="ref" reference="prop:homogenization"} and Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"}). In particular, see e.g. [@Dacorogna2007], for functionals with standard growth $|{\bf E}| \leq C(1+|\cdot|^p)$, the $\mathrm{W}^{1,p}$-quasiconvexity is equivalent to sequential weak lower semicontinuity in $\mathrm{W}^{1,p}$. Here to be precise, for signed functionals such as the determinant which take also negative values, one must restrict to sequences with fixed boundary values, as otherwise concentration effects at the boundary can destroy the lower semicontinuity (see [@Dacorogna2007] for an example and [@ChJK] for some positive results). It is not known what are the weakest possible growth properties for functionals still guaranteing this equivalence. In any case, all growth conditions imply that $\mathbf{E}$ is real valued. Without going into too much detail, we remark that for *non-negative* functionals ${\bf E} \geq 0$ taking the value $+\infty$ and for $1\leq p<\infty$, we have, $$\begin{tikzcd}[row sep=tiny,arrows=Rightarrow] \textup{closed $\mathrm{W}^{1,p}$-quasiconvex}\arrow[r,shift left=0.9ex] & \arrow[l,shift left=0.9ex,degil]\textup{$\mathrm{W}^{1,p}$-seq. wlsc}\arrow[r,shift left=0.9ex] & \arrow[l,shift left=0.9ex,degil]\textup{$\mathrm{W}^{1,p}$-quasiconvex} \\ \end{tikzcd}$$ To see that the implications above cannot be reversed, see Example [Example 32](#ex:4pts){reference-type="ref" reference="ex:4pts"} for the first and Example [Example 15](#ex:2pts){reference-type="ref" reference="ex:2pts"} for the second. Indeed, by [@BM1 Example 3.5] the functional in Example [Example 15](#ex:2pts){reference-type="ref" reference="ex:2pts"} is quasiconvex but with planar waves, described in the introduction, one sees that the functional is not sequentially weak lower semicontinuous. The case of signed functionals that are allowed to assume also the value $+\infty$, such as the local Burkholder functionals arising from our work, has not been considered before. Consequently, this requires a separate study of their lower semicontinuity properties, covered in the last Section 12. Finally, note that our approach here with *pointwise* definitions of the associated variational integrals, such as [\[eq:variational\]](#eq:variational){reference-type="eqref" reference="eq:variational"}, is not the only possibility. Following an old tradition going back to H. Lebesgue, J. Serrin and introduced in the current context by P. Marcellini [@Marcellini], a definition by *relaxation from smooth maps* is often more natural and desirable as it leads to variational integrals with better properties. We intend to return to this elsewhere. The characteristic function in [\[eq:indicator\]](#eq:indicator){reference-type="eqref" reference="eq:indicator"} also leads us to a natural notion of quasiconvexity for sets: **Definition 33**. A subset $\mathcal{U}\subset\mathbb{R}^{m\times n}$ is said to be *$\mathrm{W}^{1,p}$-quasiconvex* if $$\nu \in \mathscr{M}^p_{\mathrm{qc}}(\mathcal U) \quad \implies \quad \langle \nu, \mathrm{Id} \rangle \in \mathcal U.$$ Thus a closed set $\mathcal{U}$ is $\mathrm{W}^{1,p}$-quasiconvex if and only if $% \mathrel{\raisebox{1.1pt}{$\chi$}}^\infty% _{\,\mathcal U}$ is a closed $\mathrm{W}^{1,p}$-quasiconvex functional. We have been discussing $\mathrm{W}^{1,p}$-quasiconvex functionals, but we are yet to mention the role played by $p$. For later reference we record the following example, which displays the dependence of quasiconvexity on $p$. **Example 34**. Let $K\geq 1$. The $K$-quasiconformal cone $$Q_{2}(K)\equiv \bigl\{ A\in \mathbb{R}^{2\times2}:|A|^2\leq K \det A \bigr\}$$ is a $\mathrm{W}^{1,p}$-quasiconvex set if and only if $p \geq \tfrac{2K}{K+1}$. The statement for $p>\frac{2K}{K+1}$ was shown in [@AstalaFaraco02] and follows easily from Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} below, while the reader can find a proof of the case $p<\frac{2K}{K+1}$ in [@Faraco2004]. We also refer the reader to [@YZ] where higher dimensional versions of Example [Example 34](#ex:qccone){reference-type="ref" reference="ex:qccone"} are discussed in detail. The borderline case $p=\frac{2K}{K+1}$ is more subtle and follows from a variant of Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}, as will be shown elsewhere. ## Rank-one convexity and radial maps To conclude this section we relate rank-one convexity to radial maps. For simplicity and because it is our main focus we will only consider planar maps. **Definition 35**. A *radial map* is a map $\phi \in \mathrm{W}^{1,\infty}(\mathbb D)$ of the form $$\phi(z)=\rho(r)\frac{z}{r}, \qquad r\equiv |z|,$$ where $\rho\colon[0,1]\to \mathbb{R}$ is a Lipschitz function such that $\rho(0)=0$. If $\rho\colon [0,1]\to [0,+\infty)$ we say that $\phi$ is a *radial stretching*. It is easy to verify that for a radial map $\phi$ we have a.e. in $\mathbb D$ the identities $$\begin{gathered} \label{eq:derivativephi} \partial_z \phi = \frac 1 2 \Big(\dot \rho(r) +\frac{\rho(r)}{r}\Big),\qquad \partial_{\bar z} \phi = \frac 1 2 \Big(\dot \rho(r) -\frac{\rho(r)}{r}\Big) \frac{\, z \, }{\bar z}.\end{gathered}$$ Rank-one convex functionals are quasiconvex along radial maps, see e.g. [@Sivaloganathan1988] or [@Ball1990]. Here we state a slightly more general version of this result in order to account for extended real-valued functionals. **Lemma 36**. *Let ${\bf E}\colon \mathbb{R}^{2\times 2}\to \overline \mathbb{R}$ be a rank-one convex functional such that $\{t \,\mathrm{Id}: t>0\}$ is in the interior of $\textup{dom}({\bf E})$. Then, for a radial map $\phi\colon \mathbb D\to \mathbb{C}$ such that $\textup{D}\phi \in \textup{int}(\textup{dom}({\bf E}))$ a.e., we have $${\bf E}(\mathrm{Id})\leq \fint_{\mathbb D}^* {\bf E}(\textup{D}\phi(z)) \,\mathrm{d}m(z).$$* *Proof.* The essential range of $\textup{D}\phi$ is the smallest closed set $\mathcal U$ such that $\textup{D}\phi \in \mathcal U$ a.e. in $\mathbb D$; since $\phi$ is Lipschitz, clearly this set is compact. Thus, since $\{t\, \mathrm{Id}: t>0\}$ is contained in the interior of $\textup{dom}({\bf E})$, after mollifying ${\bf E}$ we can without loss of generality assume that ${\bf E}$ is smooth in the interior of its domain. Identifying $\textup{D}\phi=(\partial_z \phi, \partial_{\bar z} \phi)$, by [\[eq:derivativephi\]](#eq:derivativephi){reference-type="eqref" reference="eq:derivativephi"} we may write $$\textup{D}\phi = \left(\frac{\rho(r)}{r},0\right)+\frac 1 2\left(\dot \rho(r)-\frac{\rho(r)}{r}\right) \left(1,\frac{z}{\bar z}\right)$$ and notice that the second term corresponds to a matrix with rank one, since it is singular, cf. [\[eq:detconformal\]](#eq:detconformal){reference-type="eqref" reference="eq:detconformal"}. Thus, by rank-one convexity, we obtain $$\label{eq:rcinequality} {\bf E}(\textup{D}\phi)\geq {\bf E}\left(\frac{\rho(r)}{r} ,0\right) + \frac 1 2 \Big\langle {\bf E}'\left(\frac{\rho(r)}{r},0\right), \left(\dot \rho(r) -\frac{\rho(r)}{r}\right)\left(1,\frac{z}{\bar z}\right)\Big\rangle.$$ Integrating over $\mathbb D$, and using $\int_{\mathbb S^{1}} (1,\frac{z}{\bar z}) \,\mathrm{d}\theta=\int_{\mathbb S^1}(1,0) \,\mathrm{d}\theta$, we find $$\begin{aligned} & \int_{\mathbb D} {\bf E}(\textup{D}\phi) \,\mathrm{d}m(z) \\ & \geq \int_{\mathbb S^{1}} \int_0^1 \left[{\bf E}\left(\frac{\rho(r)}{r} ,0\right) + \frac 1 2 \Big\langle {\bf E}'\left(\frac{\rho(r)}{r},0\right), \left(\dot \rho(r)-\frac{\rho(r)}{r}\right)\left(1,0\right)\Big\rangle\right]r\, \mathrm{d}r \,\mathrm{d}\theta\\ & = \frac 1 2 \int_{\mathbb S^{1}} \int_0^1 \frac{\mathrm{d}}{\mathrm{d}r}\left(r^2 {\bf E}\bigg(\frac{\rho(r)}{r} \mathrm{Id}\bigg)\right) \mathrm{d}r\, \mathrm{d}\theta \\ & = \pi {\bf E}(\mathrm{Id}).\end{aligned}$$ Note that, since $\rho$ is Lipschitz continuous and ${\bf E}$ is locally Lipschitz continuous in a neighborhood of $\{t\, \mathrm{Id}:t>0\}$, see e.g. [@Dacorogna2007], the application of the fundamental theorem of calculus in the last step is justified. ◻ # Principal mappings {#sec:principal} Throughout this paper we will use principal quasiconformal maps to test quasiconvexity and to generate the required gradient Young measures. One can of course always use the flexibility allowed by Theorem [Theorem 19](#generates){reference-type="ref" reference="generates"} in choosing the generating sequences. However, in proving quasiconvexity for the local Burkholder functional this is a delicate issue, since here the natural deformations of maps typically deform also the boundary values, c.f. Subsection [5.1](#deformations){reference-type="ref" reference="deformations"} for a discussion. We solve this issue by using the principal maps, which allow deformations yet carry enough information to control the center of mass of the limiting gradient Young measures; for typical cases see e.g. the next section. In the end, it turns out that quasiconvexity with respect to the principal maps determines the lower semicontinuity properties for a large class of functionals, similarly as the classical quasiconvexity does for functionals with standard growth. In this section we hence briefly recall the basic properties of such maps. First, for $K\geq 1$, a planar ${\mathrm{W}}^{1,2}_{\mathop{\mathrm{loc}}}$ map $f\colon \Omega \to \mathbb{C}$ is *$K$-quasiregular* if $$|\textup{D}f(z)|^2 \leq K J_f(z) \qquad \textup{for a.e.\ } z \textup{ in } \Omega;$$ if in addition $f$ is a homeomorphism, we say that $f$ is *$K$-quasiconformal*. Any such mapping satisfies the Beltrami equation $$\label{Belt1} \, f_{\bar{z}} = \mu(z) f_z$$ where $\mu$ is a measurable function with $\| \mu\|_\infty \leq k \equiv \frac{K-1}{K+1} <1$. We are especially interested in homeomorphic solutions of [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"} with a certain normalization: **Definition 37**. A map $f\colon \mathbb{C}\to \mathbb{C}$ is said to be a *principal map* if: 1. $f$ is a $\mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$-homeomorphism and 2. $f$ is conformal outside $\mathbb{D}$, with Laurent series $$\label{princip1} f(z) = z + \frac{b_1}{z} + \sum_{j=2}^\infty \frac{b_j}{z^j}, \qquad |z| > 1.$$ The classical area formula, a quick consequence of Green's theorem [@AIM Theorem 2.10.1] gives for any $\mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$-principal map the identity $$\label{areafmla} \int_{\mathbb D} \! J_f(z) \, \mathrm{d}m(z) = \pi \left( 1 - \sum_{j=1}^\infty j |b_j|^2 \right),$$ which controls the size of the coefficients in [\[princip1\]](#princip1){reference-type="eqref" reference="princip1"}. For instance, the Jacobian of a Sobolev homeomorphism does not change sign, thus $J_f(z) \geq 0$ for any principal map and hence $|b_1| \leq1$ in [\[princip1\]](#princip1){reference-type="eqref" reference="princip1"}. Indeed, even if the principal map has only $\mathrm{W}^{1,1}_\textup{loc}$-regularity we still have the bound $$\sum_{j=1}^\infty j |b_j|^2 \leq 1,$$ and in particular the condition $|b_1| \leq 1$ holds. But if in either case $|b_1| = 1$, then the area formula forces all other coefficients to vanish, and that would force $f$ affine and non-injective on the unit circle. Thus $|b_1| < 1$ for every principal homeomorphism as in Definition [Definition 37](#prmap){reference-type="ref" reference="prmap"}. We hence find that for any principal map as in [\[princip1\]](#princip1){reference-type="eqref" reference="princip1"}, the associated linear operator $$A_f(z) \equiv z+b_1 \bar z$$ is a homeomorphism with $$\label{asympto} \det( A_f ) > 0.$$ Principal maps are therefore, in a sense, close to having affine boundary values on the unit circle, but are yet flexible enough to allow deformations of maps. As another aspect of this view, the area formula [\[areafmla\]](#areafmla){reference-type="eqref" reference="areafmla"} implies [@AIM Corollary 2.10.3] that $$\label{Kareafmla} f\; \mbox{ is $K$-quasiconformal } \quad \Longrightarrow \quad A_f \; \mbox{ is $K$-quasiconformal}.$$ This is not immediate since the set of $K$-quasiconformal linear maps is not convex. The notion of a principal map is very natural also since for each coefficient $\mu$ supported in the unit disk with $\| \mu\|_\infty < 1$, there is a unique $\mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$ principal solution $f = f_\mu$ to [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"}, cf. [@AIM Theorem 5.3.2]. This extends even to suitable degenerate Beltrami equations, see [@IwaSve]. Indeed, a simple way to find principal solutions for the given coefficient $\mu$ is via the Cauchy transform $$\mathbf C \varphi (z) = \frac{1}{\pi} \int_\mathbb C \frac{\varphi(\xi)}{ z - \xi }\, \textnormal d \xi.$$ One now looks for a solution in the form $$\label{Cauchy} f(z) = z + (\mathbf C \omega) (z), \;\;\;\textnormal{with} \;\; \omega %\equiv f_{\bar{z}} \, \in \, \mathscr \mathrm{L}^2(\mathbb D),$$ and the derivative $f_{\bar{z}} \equiv \omega$ is then found by a Neumann-series argument. Namely, if $\mathbf S$ is the *Beurling--Ahlfors transform*, i.e. a Calderón--Zygmund singular integral operator bounded in $\mathrm{L}^s(\mathbb{C})$ for all $1 < s < \infty$ and defined by $$\label{Beur1} \mathbf S \varphi (z) \; \equiv \;- \frac{1}{\pi} \int_\mathbb C \frac{\varphi(\xi)}{ (z - \xi)^2 }\mathrm{d}\xi,$$ then [@AIM (5.8)] shows that $$\label{Beur2} f_{\bar{z}} = (I - \mu \mathbf S)^{-1} \mu, \qquad f_z = 1 + (I - \mu \mathbf S)^{-1} \mathbf S \mu.$$ Here $\| \mathbf S\|_{L^2(\mathbb{C})} = 1$ while by [@AIM Theorem 14.0.4], the operator $I - \mu \mathbf S$ is invertible on $\mathrm{L}^{s}(\mathbb{C})$ whenever $$\label{eq:invertBelt} 1 + \| \mu\|_\infty < s < 1+1/ \| \mu\|_\infty,$$ with the operator-norm of the inverse $\| (I - \mu \mathbf S)^{-1} \|_{\mathrm{L}^s(\mathbb{C})}$ bounded by a constant that depends only on $s$ and $\| \mu\|_\infty$. Finally, to show that the mapping defined by [\[Beur2\]](#Beur2){reference-type="eqref" reference="Beur2"} is a homeomorphism requires more work; for details see [@AIM]. From [\[Beur2\]](#Beur2){reference-type="eqref" reference="Beur2"} we obtain global higher integrability bounds for the derivatives of principal solutions to the Beltrami equation [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"}, in particular $$\label{Belt2} \| f_{\bar{z}} \|_{\mathrm{L}^{s}(\mathbb{C})} + \| f_z -1 \|_{\mathrm{L}^{s}(\mathbb{C})} \leq C_s(K) < \infty,$$ whenever $$\label{Holder} \frac{2K}{K+1} < s < \frac{2K}{K-1},$$ where this last condition comes from [\[eq:invertBelt\]](#eq:invertBelt){reference-type="eqref" reference="eq:invertBelt"}, recalling that $\|\mu\|_\infty\leq \frac{K-1}{K+1}$. Note also that for exponents $s > 2$ and $\alpha < 1 -2/s$, the Cauchy operator $\mathbf{C} \colon \mathrm{L}^{s}(\mathbb D) \to \mathrm{C}^\alpha(\mathbb{C})$ is compact. Therefore from [\[Cauchy\]](#Cauchy){reference-type="eqref" reference="Cauchy"} and [\[Beur2\]](#Beur2){reference-type="eqref" reference="Beur2"} we see that for each $k <1$ the family $$\label{Belt3} {\mathscr F}_k \equiv \biggl\{ f \mbox{ is a principal solution to \,} \eqref{Belt1} \, \mbox{ with } \| \mu \|_{\infty} \leq k \biggr\}$$ is normal, i.e. every sequence of ${\mathscr F}_k$ contains a subsequence converging uniformly on $\mathbb{C}$. The limit, too, belongs to ${\mathscr F}_k$ since any (non-constant) limit of a uniformly converging sequence of $K$-quasiconformal maps is $K$-quasiconformal. # Quasiregular gradient Young measures {#Young} In this section we consider homogeneous gradient Young measures which are generated by sequences with suitable bounds on their distortion; our goal is to show that such measures can be generated also by principal maps with the same distortion bounds. The case where the sequence has uniformly bounded distortion was studied by the first two authors in [@AstalaFaraco02], but here we are also interested in the more general case where the distortion of the sequence is only bounded in $\mathrm{L}^1$. To start with, recall that the effective domain [\[effective\]](#effective){reference-type="eqref" reference="effective"} of the local Burkholder functional ${\mathcal B}_K (A)$ in [\[eq:locBurk\]](#eq:locBurk){reference-type="eqref" reference="eq:locBurk"} is the set of all $K$-quasiconformal matrices $$Q_2(K) \equiv \{ A \in \mathbb{R}^{2 \times 2} : |A|^2 \leq K \det A \}.$$ This takes us to study the gradient Young measures supported on $Q_2(K)$. We begin with a modification of results from [@AstalaFaraco02]: **Theorem 38**. *Let $\frac{2K}{K+1}< s < \frac{2K}{K-1}$ and let $\nu\in \mathscr{M}^s_\textup{qc}(Q_2(K))$ be such that $$\label{CM2} \langle \nu, \mathrm{Id} \rangle = A, \qquad Az = z + a \bar z.$$ Then there is a sequence of $K$-quasiconformal principal maps which generates in $\mathbb{D}$ the Young measure $\nu$.* Here recall that any matrix in $Q_2(K)$ is a scalar multiple of $Az = z + a \bar z$ where $|a| \leq \frac{K-1}{K+1}$. Thus the choice [\[CM2\]](#CM2){reference-type="eqref" reference="CM2"} is merely a normalisation. Moreover, recall that if $f \in \mathrm{W}^{1,2}_{\mathop{\mathrm{loc}}}(\mathbb{C})$ is the weak limit of the generating sequence given by Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"}, then $f$ is $K$-quasiconformal and principal; this follows from the compactness of the family ${\mathscr F}_k$ in [\[Belt3\]](#Belt3){reference-type="eqref" reference="Belt3"}. In addition, [\[limit\]](#limit){reference-type="eqref" reference="limit"} shows that $$\textup{D}f(x) = \langle \nu, \mathrm{Id}\rangle \qquad {\rm a.e.} \; x \in \mathbb D.$$ Thus also the center of mass $\langle \nu, \mathrm{Id}\rangle$ is a $K$-quasiconformal matrix and for some $|a| \leq \frac{K-1}{K+1}$, $$f(z) = \left\{ \begin{array}{ll} z+a\bar{z} & \mbox{ if } |z| \leq 1,\\ z+\tfrac{a}{z} & \mbox{ if } |z| > 1. \end{array} \right.$$ For the proof of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} we borrow an auxiliary result from [@AstalaFaraco02 Lemma 4.1]: **Lemma 39**. *Suppose $\frac{2K}{K+1}< s < \frac{2K}{K-1}$ and $\nu\in \mathscr{M}^s_\textup{qc}(Q_2(K))$ is generated by a sequence $(\phi_j)$ such that $( |\textup{D}\phi_j|^s)$ is equiintegrable in $\mathbb D$.* *Then there are measurable functions $\mu_j\colon\mathbb D\to \mathbb{C}$, with $\| \mu_j\|_\infty \leq \frac{K-1}{K+1}$, such that $$\lim_{j\to \infty} \| {\partial_{\bar z}} \phi_j - \mu_j \partial_z \phi_j \|_{\mathrm{L}^s(\mathbb D)} = 0.$$* *Proof of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"}.* Let $(\phi_j) \subset \mathrm{W}^{1,s}(\mathbb D, \mathbb{C})$ be a sequence generating $\nu$, given by Theorem [Theorem 19](#generates){reference-type="ref" reference="generates"}, for which $(|\textup{D}\phi_j|^s)$ is equiintegrable. We then set $$\eta_j \equiv \bigl( \partial_{\bar z}\phi_j - \mu_j \partial_z \phi_{j}\bigr) \chi_{\mathbb D} \in \mathrm{L}^s(\mathbb{C}),$$ where the $\mu_j$ are given by Lemma [Lemma 39](#gen.seq){reference-type="ref" reference="gen.seq"}. As in [\[Cauchy\]](#Cauchy){reference-type="eqref" reference="Cauchy"}, via the Cauchy transform we find global solutions $\omega_j$ to $$\partial_{\bar z} \omega_j - \mu_j \partial_z \omega_j = \eta_j$$ with $|\textup{D}\omega_j| \in \mathrm{L}^s(\mathbb{C})$, simply by letting $$\partial_{\bar z} \omega_j = (I - \mu_j \mathbf S)^{-1} \eta_j, \quad \omega_j = \mathbf C( \partial_{\bar z} \omega_j).$$ In particular, $\| \textup{D}\omega_j\|_{\mathrm{L}^s(\mathbb{C})} \leq C_s(K) \| \eta_j \|_{\mathrm{L}^s(\mathbb{C})} \to 0$ as $j \to \infty$. By Theorem [Theorem 19](#generates){reference-type="ref" reference="generates"}, also the sequence of maps $$\psi_j \equiv \phi_j - \omega_j \in \mathrm{W}^{1,s}(\mathbb D)$$ generates the given Young measure $\nu$, since $\| \textup{D}\psi_j - \textup{D}\phi_j \|_{\mathrm{L}^s (\mathbb D)} \to 0$. Further, the maps $\psi_j$ are all $K$-quasiregular, $\partial_{\bar z} \psi_j - \mu_j \partial_z \psi_j = 0$ in $\mathbb D$, but they need not be homeomorphisms. On the other hand, in Section [3](#sec:principal){reference-type="ref" reference="sec:principal"} we saw that there is a principal solution $f_j \in \mathrm{W}^{1,p}_\textup{loc}(\mathbb{C}, \mathbb{C})$ to the Beltrami equation $$\partial_{\bar z} f_j - \chi_{\mathbb D} \mu_j \partial_{z} f_j = 0.$$ Then by Stoilow's factorization [@AIM Theorem 5.5.1] we have $$\psi_j = h_j \circ f_j,$$ where the maps $h_j$ are holomorphic in $f_j(\mathbb D)$. We can next estimate as in [@AstalaFaraco02 (4.11)]. Namely, as $J_{f_j}(z) \leq |\textup{D}f_j(z)|^2$, by the change of variables formula and [\[Belt2\]](#Belt2){reference-type="eqref" reference="Belt2"} we have $$\begin{aligned} \label{holom.factor} \int_{f_j(\mathbb D)} |h_j'(w)| & \, \mathrm{d}m(w) = \int_\mathbb D |h_j'(f_j (z))| J_{f_j}(z) \, \mathrm{d}m(z)\\ & \leq \left( \int_\mathbb D |h_j'(f_j (z))|^s |\textup{D}f_j(z)|^s \right)^{1/s} \left( \int_\mathbb D |\textup{D}f_j(z)|^t\right)^{1/t} \nonumber \\ & \leq C_t(K) \left( \int_\mathbb D |\textup{D}\psi_j (z)|^s \right)^{1/s}, \nonumber\end{aligned}$$ where $t$, the Hölder conjugate of $s$, also satisfies [\[Holder\]](#Holder){reference-type="eqref" reference="Holder"}. Thus the norms $\| h_j' \|_{\mathrm{L}^{1}(f_j(\mathbb D))}$ of the derivatives of the holomorphic factors are uniformly bounded. Choosing a point $x_0 \in \mathbb D$ and adding a constant to elements of the generating sequence $\phi_j$, we can assume that $\psi_j(x_0) = x_0$, i.e. that $h_j$ takes $f_j(x_0)$ to $x_0$. Choosing then a subsequence such that $f_j(x_0)$ converge, we see that the holomorphic functions $h_j \colon f_j(\mathbb D) \to \mathbb{C}$ form a normal family. All in all, taking subsequences we can assume that $f_j \to f \in {\mathscr F}_k$ uniformly on $\mathbb{C}$ and $h_j \to h$ locally uniformly on $f(\mathbb D)$, where $h$ is analytic on $f(\mathbb D)$. Further, we have the weak convergence $$\label{CM3} \textup{D}\psi_j = h_j'(f_j(z)) \textup{D}f_j \rightharpoonup \textup{D}(h \circ f) \quad \mbox{ in } \; \mathrm{L}^{s}(\mathbb D).$$ Since $(\psi_j)$ generate the given Young measure $\nu$, we see from [\[CM2\]](#CM2){reference-type="eqref" reference="CM2"} and Remark [Remark 28](#affine){reference-type="ref" reference="affine"} that $\textup{D}(h \circ f)(z)=A$ for a.e. $z \in \mathbb D$. But this means that $\mu_f = \mu_A = a$ from [\[CM2\]](#CM2){reference-type="eqref" reference="CM2"}, and so as a principal map, $f$ must be equal to $$%\begin{equation} \label{hups} f(z)=\begin{cases} A(z)= z + a \bar z, & \textup{if } |z|\leq 1, \\ z+ \frac a z, & \textup{if } |z|\geq 1. \end{cases}$$ It follows that $\textup{D}h = \mathrm{Id}$ in $f(\mathbb D)$ and so we see that $h_j'(f_j(z)) \to 1$ locally uniformly on $\mathbb D$ as $j \to \infty$. Thus with [\[Belt2\]](#Belt2){reference-type="eqref" reference="Belt2"}, for any $R<1$, $$\|\textup{D}\psi_j - \textup{D}f_j\|_{\mathrm{L}^{s}(\mathbb D(0,R))} = \| (h_{j}' \circ f_j - 1) \textup{D}f_j\|_{\mathrm{L}^{s}(\mathbb D(0,R))} \to 0.$$ Therefore, again by Theorem [Theorem 19](#generates){reference-type="ref" reference="generates"}, $( f_j)$ and $(\psi_j)$ generate the same (homogeneous) Young measure $\nu$ in every disc compactly contained in $\mathbb D$. Since the sequence $(|\textup{D}f_j|^s)$ is equiintegrable over $\mathbb D$, and as $\mathscr L^2(\mathbb S^1)=0$, this finally shows that the sequence of principal maps $( f_j )$ generates $\nu$ in $\mathbb D$. ◻ Combining Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} with [\[Belt2\]](#Belt2){reference-type="eqref" reference="Belt2"} we obtain the following consequence, cf. [@AstalaFaraco02 Corollary 1.6]: **Corollary 40**. *If $\nu\in\mathscr{M}^s_\textup{qc}(Q_2(K))$ for some $\frac{2K}{K+1}< s$, then $\nu \in \mathscr{M}^p_\textup{qc}(Q_2(K))$ for all $p<\frac{2K}{K-1}$.* We conclude this section with a version of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} for gradient Young measures which are generated by sequences with integrable distortion. **Theorem 41**. *Let $\nu \in \mathscr M^{2}_\textup{qc}(\mathbb{R}^{2\times 2}_+)$ be such that, for some $|a|<1$, $$\langle \nu, \mathrm{Id}\rangle = A, \qquad A(z)=z + a \bar z.$$ Assume that $\nu$ is generated by a bounded sequence $\{\psi_j\}\subset \mathrm{W}^{1,{2}}(\mathbb D)$ of homeomorphisms such that $$\|K_{\psi_j}\|_{\mathrm{L}^{1}(\mathbb D)}\leq C. %\qquad {\color{red} {\rm Prop.\; 13.4. \; needs } \; q \geq 1 \; ??}$$ Then there is a sequence of maps $f_j\colon \mathbb{C}\to \mathbb{C}$ such that:* 1. *$f_j$ are principal maps;* 2. *$(f_j)\subset \mathrm{W}^{1,2}(\mathbb D)$ is bounded and $(f_j |_{\mathbb D(0,R)})$ generates $\nu$ for all $R<1$;* 3. *$\psi_j= h_j \circ f_j$ for some conformal maps $h_j\colon f_j(\mathbb D)\to \psi_j(\mathbb D)$.* The proof strategy is similar to that of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"}. However, since we are in a setting where the Beltrami equations are degenerate elliptic the argument is more subtle. Also, for the applications we have in mind it suffices to consider generating sequences which consist of homeomorphisms. The argument relies crucially on the Stoilow factorization for maps with integrable distortion, due to Iwaniec and Šverák [@IwaSve]. *Proof.* The maps $\psi_j$ give us homeomorphic solutions to the Beltrami equations $$\label{integrable.dist} \partial_{\bar z} \psi_j -\mu_j \partial_z \psi_j=0 \quad \text{in } \mathbb D,$$ where $K_{\psi_j} =\frac{1+|\mu_j|}{1-|\mu_j|}\in \mathrm{L}^1(\mathbb D)$. By the general Iwaniec-Šverák theorem, given any non-constant solution $F \in \mathrm{W}^{1,2}(\mathbb D)$ to [\[integrable.dist\]](#integrable.dist){reference-type="eqref" reference="integrable.dist"} where $K_{\psi_j} \in \mathrm{L}^1(\mathbb D)$, there is a principal solution $f_j\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ to the Beltrami equation $$\partial_{\bar z} f_j - \chi_{\mathbb D} \mu_j\partial_z f_j=0.$$ Moreover, see [@AIM Theorem 20.2.1], these solutions admit the Stoilow's factorization $$F = h_j\circ f_j$$ where the $h_j$ are holomorphic in $f_j(\mathbb D)$, and in addition, if $E_j$ stands for the set of critical points of $h_j$, we have the higher regularity $f_j \in \mathrm{W}^{1,2}_{\mathop{\mathrm{loc}}}(\mathbb{C}\setminus E_j)$. In our case [\[integrable.dist\]](#integrable.dist){reference-type="eqref" reference="integrable.dist"} and $\psi_j$, the maps are homeomorphism and hence so are the $h_j$. In particular, they have no critical points. We thus see that the principal maps $f_j \in \mathrm{W}^{1,2}(\mathbb D)$. It only remains to show that the $(f_j|_{\mathbb D(0,R)})$ generate $\nu$ for every $R<1$. For this, arguing as in [\[holom.factor\]](#holom.factor){reference-type="eqref" reference="holom.factor"} with $s = t = 2$, we see that for any $0 < R < 1$, $$\int_{f_j(\mathbb{D}(0,R))} |h_j'| \, \mathrm{d}m(w)\leq C(R) \, \left( \int_{\mathbb D(0,R)} |\textup{D}\psi_j(z)|^2\right)^{1/2}, % \left( \int_{\mb D(0,R)} |\D f_j(z)|^2\right)^{1/2}$$ and similarly as in the proof of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} this implies that $h_j \colon f_j(\mathbb{D}(0,R)) \to \mathbb{C}$ form a normal family of holomorphic functions. At this point we can apply exactly the same reasoning as in the proof of Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} to conclude that $h'_j\circ f_j \to 1$ locally uniformly, and so ${\|\textup{D}\psi_j-\textup{D}f_j\|_{\mathrm{L}^2(\mathbb D(0,R))}\to 0}$, which shows that $(\psi_j)|_{\mathbb D(0,R)}$ and $(f_j)|_{\mathbb D(0,R)}$ generate the same gradient Young measure. ◻ # The Burkholder functionals {#sec:Bf} In the literature there are a few slightly varying versions of the Burkholder functional. For instance, Iwaniec studies in [@Tade] the functionals $${\mathscr B}^{\pm}_p(A) = \left| 1 - \frac{n}{p}\right| \, |A|^p \, \pm |A|^{p-n} \det A, \quad A\in \mathbb{R}^{n \times n}.$$ Here, see [@Tade Theorem 5], both functionals ${\mathscr B}^{+}_p$ and ${\mathscr B}^{-}_p$ are *rank-one convex*, and in two dimensions ${\mathscr B}^{+}_p$ is convex in the directions of matrices with nonnegative determinant, while ${\mathscr B}^{-}_p$ is convex in the directions of matrices with nonpositive determinant, see [@Tade Proposition 12.1]. In fact, precomposing with a reflection one can interchange the functionals ${\mathscr B}^{\pm}_p$. In any case, our choice [\[Burk\]](#Burk){reference-type="eqref" reference="Burk"} amounts to $${\bf B}_p(A) = (p/2){\mathscr B}^{-}_p(A),$$ with $n=2 \leq p < \infty$, which is a rank-one convex functional. In conformal coordinates, cf. [\[eq:confcoords\]](#eq:confcoords){reference-type="eqref" reference="eq:confcoords"}, the Burkholder functional can be written as ${\bf B}_p(A) = |A|^{p-1} \bigl( (p-1)|a_-| - |a_+| \bigr)$ and, in particular, $$\label{Bconf} {\bf B}_p(\textup{D}f) = |\textup{D}f|^{p-1} \left( (p-1)|f_{\overline{z}}| - |f_z| \right).$$ One easily checks that ${\bf B}_p( \mathrm{Id}) = -1$ and that, for $p= \frac{2K}{K-1}$, $$\label{eq:levelsetBp} \bigl\{ {\bf B}_p \leq 0 \bigr\} = \bigl\{ A \in \mathbb{R}^{2 \times 2} : \, |A|^2 \leq K \det(A) \bigr\} = Q_2(K),$$ the cone of $K$-quasiconformal matrices introduced in Example [Example 34](#ex:qccone){reference-type="ref" reference="ex:qccone"}. ## Weighted integral estimates and the Burkholder functionals {#deformations} The proof of closed quasiconvexity of the local Burkholder functional [\[eq:locBurk\]](#eq:locBurk){reference-type="eqref" reference="eq:locBurk"}, with domain the quasiconformal cone $Q_2(K)$, is based on two fundamentally different methods, both essential for the argument. The first, developed in [@AIPS12], establishes optimal weighted integral bounds for $K$-quasiconformal principal mappings, via holomorphic motions and special complex interpolation. On the other hand, the second method, developed in this work, analyses the interaction of the Burkholder functional with the gradient Young measures generated by principal mappings. For the optimal integral estimates, recall first that principal solutions $f = f_\mu$ to the Beltrami equation [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"} allow holomorphic deformations [@AIM Section 12]. That is, given the coefficient $\mu(z)$ with $\| \mu \|_\infty < 1$, one can construct families of coefficients $\mu_\lambda(z)$ depending holomorphically on the parameter $\lambda \in \mathbb D$, such that $\| \mu_\lambda \|_\infty < 1$ for all $\lambda \in \mathbb D$, and $\mu_{\lambda_0} = \mu$ for a suitable $\lambda_0 \in \mathbb D$. Typically one also requires that $\mu_0 \equiv 0$. In this setting, for each $\lambda \in \mathbb D$ there is a unique principal solution to [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"} with coefficient $\mu_\lambda(z)$. In fact, this family of solutions defines a *holomorphic motion* $$\label{holomotion} \Phi(\lambda,z) \equiv f_{\mu_{\lambda}}(z),$$ that is $\lambda \mapsto \Phi(\lambda,z)$ is holomorphic, $z \mapsto \Phi(\lambda,z)$ is injective and $\Phi(0,z) = z$, c.f. [@AIM Section 12]. Furthermore, the gradients of the above holomorphic deformations [\[holomotion\]](#holomotion){reference-type="eqref" reference="holomotion"} and their $\mathrm{L}^p$-norms allow optimal interpolation bounds, similar to those in the classical Riesz-Thorin complex interpolation. For details see [@AIPS12 Lemmas 1.4 and 1.6]. To combine all this with the Burkholder functional, note that for any given solution $f$ to the Beltrami equation [\[Belt1\]](#Belt1){reference-type="eqref" reference="Belt1"} the Burkholder functional may be written equivalently as $$\label{Beltqc} {\bf B}_p\bigl(Df(z)\bigr) = \Bigl(\frac{p |\mu(z)}{1+|\mu(z)|} - 1\Bigr) \bigl(|f_z(z)| + |f_{\overline{z}}(z)| \bigr)^{p}.$$ This suggests that for quasiconformal principal mappings one should approach the Burkholder integrals via $\mathrm{L}^p$-estimates in the appropriate weighted spaces. Indeed, using the deformations [\[holomotion\]](#holomotion){reference-type="eqref" reference="holomotion"} and combining [\[Beltqc\]](#Beltqc){reference-type="eqref" reference="Beltqc"} with the optimal weighted $\mathrm{L}^p$-bounds from the above complex interpolation leads to the following result. **Theorem 42**. *[@AIPS12 Theorem 3.5] [\[closedQC2\]]{#closedQC2 label="closedQC2"} Suppose $f \colon \mathbb{C}\to \mathbb{C}$ is a principal solution to the Beltrami equation $$\label{Belt4} f_{\overline{z}} = \mu f_z, \quad | \mu(z)| \leq k \chi_{\mathbb{D}}(z), \quad 0 \leq k < 1,$$ Then for all exponents $2 \leq p \leq 1 + 1/k$, we have $$\label{Belt55} \fint_{\mathbb{D}} \Bigl( 1-\frac{p |\mu(z)|}{1+|\mu(z)|} \Bigr) \left(|f_z(z)| + |f_{\overline{z}}(z)| \right)^{p} \mathrm{d}m(z) \leq 1. %|\DD|.$$* Finally, comparing now [\[Beltqc\]](#Beltqc){reference-type="eqref" reference="Beltqc"} and [\[Belt55\]](#Belt55){reference-type="eqref" reference="Belt55"}, we may rewrite Theorem [\[closedQC2\]](#closedQC2){reference-type="ref" reference="closedQC2"} as follows **Theorem 43**. *[@AIPS12 Theorem 1.3] [\[closedQC\]]{#closedQC label="closedQC"} Suppose $f \colon \mathbb{C}\to \mathbb{C}$ is a principal solution to the Beltrami equation [\[Belt4\]](#Belt4){reference-type="eqref" reference="Belt4"}. Then for all $p \in [2,1+1/k]$ we have $$\label{Burk123} {\bf B}_p(\mathrm{Id})\leq \fint_\mathbb D {\bf B}_p(\textup{D}f(z)) \,\mathrm{d}m(z).$$* In particular, if a quasiconformal map $f$ on $\mathbb D$ has identity boundary values, it extends trivially to a principal map of $\mathbb{C}$. Thus the above result shows that the Burkholder functional ${\bf B}_p$ is quasiconvex at identity when tested with $K$-quasiconformal maps, under the condition that $2\leq p \leq \frac{2K}{K-1}$. Furthermore, the equality in [\[Burk123\]](#Burk123){reference-type="eqref" reference="Burk123"} occurs for a large class of radial mappings, see for instance Subsection [5.2](#Bradial){reference-type="ref" reference="Bradial"} below. *Remark 44*. Note that holomorphic deformations of solutions $f$ to [\[Belt4\]](#Belt4){reference-type="eqref" reference="Belt4"}, such as $\lambda \mapsto f_{\mu_{\lambda}}(z)$ above, in general change the boundary values of the mapping, even if the original map $f$ has identity boundary values on $\partial \mathbb D$. For this reason, in particular, the principal mappings and their integral bounds are indispensable for the quasiconvexity estimates [\[Burk123\]](#Burk123){reference-type="eqref" reference="Burk123"}. On the other hand, to prove quasiconvexity bounds such as [\[Burk123\]](#Burk123){reference-type="eqref" reference="Burk123"} for quasiconformal maps on $\mathbb D$ with linear boundary values $A \neq \mathrm{Id}$, different methods appear necessary. Here we will make extensive use of the gradient Young measures discussed in Sections [2](#sec:prelims){reference-type="ref" reference="sec:prelims"} and [4](#Young){reference-type="ref" reference="Young"}. However, even for these Theorem [\[closedQC\]](#closedQC){reference-type="ref" reference="closedQC"} is required as their basis and the starting point for the estimates they provide, see e.g. Proposition [Proposition 48](#Id){reference-type="ref" reference="Id"}. *Remark 45*. Note that, in general, at the endpoint exponent $p=1 + 1/k$ we have $\textup{D}f \notin \mathrm{L}^{p}_{\mathop{\mathrm{loc}}}$ for the solutions [\[Belt4\]](#Belt4){reference-type="eqref" reference="Belt4"}: simple examples are obtained, for instance, by considering radial stretchings. Nonetheless, we always have $\textup{D}f \in \textup{weak-}\mathrm{L}^p_\textup{loc}$ [@AIM Theorem 13.2.1]. A surprising feature of Theorem [\[closedQC\]](#closedQC){reference-type="ref" reference="closedQC"} is that, even if ${\bf B}_p$ has $p$-growth, we are able to test the quasiconvexity inequality [\[Burk123\]](#Burk123){reference-type="eqref" reference="Burk123"} with maps which are *not* in $\mathrm{W}^{1,p}_{\mathop{\mathrm{loc}}}$, and in particular ${\bf B}_p$ is *integrable* along such maps. One can regarded this as an extension to the planar quasiconformal setting of the results of [@IS], where it is shown that $\det \textup{D}f\in L^1_\textup{loc}$ if $f\in \textup{weak-}W^{1,n}_\textup{loc}$ is orientation-preserving. ## Burkholder functional and radial mappings {#Bradial} In Lemma [Lemma 36](#lemma:rcradialstretchings){reference-type="ref" reference="lemma:rcradialstretchings"} we saw that rank-one convex functionals are quasiconvex along radial maps. The Burkholder functional is special, as it is *quasiaffine* on a large class of radial stretchings: **Lemma 46**. *Let $\phi(z)=\rho(r) \frac{z}{r}$ be a radial stretching satisfying the condition $$\label{eq:nonexpanding} |\dot \rho(r)|\leq \frac{\rho(r)}{r}.$$ The functional ${\bf B}_p$ is quasiaffine along such radial stretchings: $$-1={\bf B}_p(\mathrm{Id})= \fint_{\mathbb D} {\bf B}_p(\textup{D}\phi) \,\mathrm{d}m(z).$$* *Proof.* We refer the reader to [@AIPS15a Theorem 8.1] or [@Guerra19] for a proof. ◻ We also have the following refinement of [@AIPS15a Theorem 8.1] to the $\overline \mathbb{R}$-valued setting, showing that ${\bf B}_p$ is an extreme point in a natural class of functionals. **Theorem 47**. *Let ${\bf E}\colon \mathbb{R}^{2\times 2} \to \overline \mathbb{R}$ be an functional such that:* 1. *[\[it:domain\]]{#it:domain label="it:domain"} for some $K\geq 1$, $Q_2(K)\subset\textup{int}\bigl(\textup{dom}({\bf E})\bigr)$;* 2. *${\bf E}$ is rank-one convex and ${\bf E}(\mathrm{Id})=-1$;* 3. *${\bf E}$ is positively $p$-homogeneous, for some $p\geq 2$;* 4. *${\bf E}$ is isotropic, that is ${\bf E}(QAR)={\bf E}(A)$ for all $A\in \mathbb{R}^{2\times 2}$ and all $Q,R\in \textup{SO}(2)$.* *Then ${\bf E}(A)\geq {\bf B}_p(A)$ for all $A\in Q_2(K)$. [\[thm:extremality\]]{#thm:extremality label="thm:extremality"}* *Proof.* Let $\phi(z)=\rho(r) \frac{z}{r}$ be a $K$-quasiconformal radial stretching satisfying [\[eq:nonexpanding\]](#eq:nonexpanding){reference-type="eqref" reference="eq:nonexpanding"}. Combining Lemmas [Lemma 36](#lemma:rcradialstretchings){reference-type="ref" reference="lemma:rcradialstretchings"} and [Lemma 46](#lemma:Bpradialstretchings){reference-type="ref" reference="lemma:Bpradialstretchings"}, we estimate $$\fint_{\mathbb D}^* {\bf E}(\textup{D}\phi) \,\mathrm{d}m(z) \geq {\bf E}(\mathrm{Id}) = -1 ={\bf B}_p(\mathrm{Id})=\fint_{\mathbb D} {\bf B}_p(\textup{D}\phi) \,\mathrm{d}m(z);$$ here note that condition [\[it:domain\]](#it:domain){reference-type="eqref" reference="it:domain"} ensures that Lemma [Lemma 36](#lemma:rcradialstretchings){reference-type="ref" reference="lemma:rcradialstretchings"} is applicable. We now take, for some $\alpha\in [-1,1]$, $$\rho(r)\equiv \begin{cases} \frac{r}{2^{\alpha-1}} & \text{if } r\leq \frac 1 2,\\ r^\alpha &\text{if } r\geq \frac 1 2. \end{cases}$$ In particular, $\rho$ satisfies [\[eq:nonexpanding\]](#eq:nonexpanding){reference-type="eqref" reference="eq:nonexpanding"} whenever $|\alpha|\leq 1$. With this choice the map $\phi$ is $\frac{1}{|\alpha|}$-quasiconformal, since we have a.e. the identities $$|\textup{D}\phi(x)|^2= r^{2(\alpha-1)}, \qquad \det \textup{D}\phi(x)=\alpha r^{2(\alpha-1)},$$ cf. [\[eq:derivativephi\]](#eq:derivativephi){reference-type="eqref" reference="eq:derivativephi"}. Since ${\bf E}(\mathrm{Id})={\bf B}_p(\mathrm{Id})=-1$ and both ${\bf E}$ and ${\bf B}_p$ are positively $p$-homogeneous, $$\int_{\frac 1 2\mathbb D} {\bf E}(\textup{D}\phi)\,\mathrm{d}m(z) = \int_{\frac 1 2 \mathbb D} {\bf B}_p(\textup{D}\phi)\, \mathrm{d}m(z).$$ For $\frac{1}{2} \leq |z| \leq 1$ we have $\rho(r)=r^\alpha$. Thus $$\label{annulus} \int_{\mathbb A(\frac 1 2, 1)} {\bf E}(\textup{D}\phi) \,\mathrm{d}m(z) \geq \int_{\mathbb A(\frac 1 2, 1)} {\bf B}_p(\textup{D}\phi)\, \mathrm{d}m(z).$$ Moreover, it follows from [\[eq:derivativephi\]](#eq:derivativephi){reference-type="eqref" reference="eq:derivativephi"} that for ${\bf E}$ as in the statement of the lemma (hence for ${\bf B}_p$ as well), $${\bf E}(\textup{D}\varphi(x))=r^{p(\alpha-1)}{\bf E}(\alpha,1),$$ where we have identified $(x,y)\equiv \textup{diag}(x,y)$. Thus [\[annulus\]](#annulus){reference-type="eqref" reference="annulus"} gives $${\bf E}(\alpha,1)\int_{\frac 1 2}^1 r^{p(\alpha-1)+1}\mathrm{d}r \geq {\bf B}_p(\alpha,1)\int_{\frac 1 2}^1 r^{p(\alpha-1)+1}\mathrm{d}r \implies {\bf E}(\alpha,1)\geq {\bf B}_p(\alpha,1).$$ Finally, varying $\alpha\in [1/K,1]$, we have ${\bf E}\geq {\bf B}_p$ on the rank-one segment $[\mathrm{Id}, \textup{diag}(1/K,1)]\subset Q_2(K)$. Since ${\bf E}$ and ${\bf B}_p$ are positively $p$-homogeneous and isotropic, it follows that ${\bf E}\geq {\bf B}_p$ in $Q_2(K)$. ◻ # Proof of Theorem [Theorem 4](#main){reference-type="ref" reference="main"} {#sec:Maintheorem} The purpose of this section is to prove Theorem [Theorem 4](#main){reference-type="ref" reference="main"}, the closed quasiconvexity of the local Burkholder functional. In fact, it will be convenient to consider slightly more general versions of the local Burkholder functional $\mathcal B_{K}$ introduced in [\[eq:locBurk\]](#eq:locBurk){reference-type="eqref" reference="eq:locBurk"}, so let us define $$\label{locB} {\bf B}_{K,p}(A) \equiv \begin{cases} {\bf B}_p(A),\quad &\text{if} \;\; |A|^2 \leq K \det(A), \\ +\infty, &\text{otherwise}. \end{cases}$$ Thus these functionals all have the same domain $\,\mathrm{dom} \left({\bf B}_{K,p} \right)$ = $Q_2(K)$, while $\mathcal B_{K} = {\bf B}_{K,p_{K}}$ when $\,p\,$ equals the limiting exponent $p_K = 2K/(K-1)$. Also, for this exponent $$% \mathrm{dom} \left({\mathcal B}_K \right) = Q_2(K) = \{ A\in \mathbb{R}^{2 \times 2}: {\bf B}_{p_K}(A) \leq 0 \}.$$ We first consider the closed $\mathrm{W}^{1,p}$-quasiconvexity at the identity, for these modified functionals and for $2 < p < p_K$. **Proposition 48**. *Let $K > 1$ and fix $p \in (2,\tfrac{2K}{K-1})$. Then the functional ${\bf B}_{K,p}\colon \mathbb{R}^{2 \times 2}\to \mathbb{R}\cup \{+\infty\}$ is closed $\mathrm{W}^{1,p}$-quasiconvex at $\mathrm{Id}$.* *Proof.* Let $\nu \in \mathscr{M}_{\mathrm{qc}}^{p}$ be a homogeneous gradient Young measure with center of mass $\langle \nu, \mathrm{Id}\rangle = \mathrm{Id}$; our goal is to establish the Jensen inequality $${\bf B}_{K,p}(\mathrm{Id})\leq \langle \nu, {\bf B}_{K,p}\rangle = \int_{\mathbb{R}^{2 \times 2}} {\bf B}_{K,p}(A) d\nu(A).$$ There is nothing to prove when $\langle \nu , {\bf B}_{K,p} \rangle = +\infty$ so without loss of generality we assume that $\langle \nu , {\bf B}_{K,p} \rangle < +\infty$. Under this assumption, from the definition [\[locB\]](#locB){reference-type="eqref" reference="locB"} we see that $\nu$ is supported in the $K$-quasiconformal cone $Q_{2}(K)$. Therefore Theorem [Theorem 38](#thm:YMqc){reference-type="ref" reference="thm:YMqc"} gives us a sequence $( f_j )$ of $K$-quasiconformal principal maps generating the measure $\nu$. In particular, each map $f_j$ is conformal outside $\mathbb{D}$, and by [\[Belt2\]](#Belt2){reference-type="eqref" reference="Belt2"} we have $(\textup{D}f_j )$ bounded in $\mathrm{L}^s(\mathbb{D})$, for each $s < p_K$. Furthermore, for this sequence ${\bf B}_{K,p}\bigl(\textup{D}f_j(z)\bigr) = {\bf B}_{p}\bigl(\textup{D}f_j(z)\bigr)$, so that with Theorem [\[closedQC\]](#closedQC){reference-type="ref" reference="closedQC"} and [\[Burk123\]](#Burk123){reference-type="eqref" reference="Burk123"} we have $${\bf B}_{K,p}\bigl( \langle \nu, \mathrm{Id}\rangle \bigr) = {\bf B}_{p}( \mathrm{Id}) \leq \, \liminf_{j \to \infty} \fint_{\mathbb{D}} \! {\bf B}_{K,p}\bigl(\textup{D}f_j(z)\bigr) \, \mathrm{d}m(z).$$ On the other hand, as pointed out above, $(\textup{D}f_j )$ bounded in $\mathrm{L}^s(\mathbb{D})$ for $p < s < p_K$ and since ${\bf B}_{p}$ is $p$-homogeneous, it follows that the sequence $\bigl( {\bf B}_{K,p}(\textup{D}f_j )\bigr) = \bigl( {\bf B}_{p}(\textup{D}f_j )\bigr)$ is equiintegrable in $\mathbb D$. We can thus apply Remark [Remark 21](#rmk:moreintegrands){reference-type="ref" reference="rmk:moreintegrands"}, which implies $$\lim_{j\to \infty} \fint_{\mathbb{D}} \! {\bf B}_{K,p}\bigl(\textup{D}f_j(z)\bigr) \, \mathrm{d}m(z) = \int_{\mathbb{R}^{2 \times 2}} \! {\bf B}_{K,p}(A) \, \mathrm{d}\nu (A).$$ This completes the proof. ◻ To complete the proof of Theorem [Theorem 4](#main){reference-type="ref" reference="main"}, our task is then to extend the closed quasiconvexity from $A = \mathrm{Id}$ to a general matrix $A \in \mathbb{R}^{2 \times 2}$. We start with: **Proposition 49**. *Let $K>1$. Then for each $\, 2 < p < \tfrac{2K}{K-1}$, the local Burkholder functional ${\bf B}_{K,p} \colon \mathbb{R}^{2 \times 2}\to \mathbb{R}\cup \{ +\infty \}$ is closed $\mathrm{W}^{1,p}$-quasiconvex.* *Proof.* Define the relaxation $\mathscr{R}\colon \mathbb{R}^{2 \times 2}\to \overline{\mathbb{R}}$ of the Burkholder functional ${\bf B}_{K,p}$ by $$\mathscr{R}(A) \equiv \inf \left\{ \int_{\mathbb{R}^{2 \times 2}}^{\ast} \! {\bf B}_{K,p} \; \mathrm{d}\nu : \, \nu \in \mathscr{M}^{p}_{\mathrm{qc}} \, \mbox{ and } \, \langle \nu, \mathrm{Id}\rangle = A \right\} \, , \quad A \in \mathbb{R}^{2 \times 2}.$$ Since $p > \tfrac{2K}{K+1}$, the $K$-quasiconformal cone $Q_{2}(K)$ is $\mathrm{W}^{1,p}$-quasiconvex, cf. Example [Example 34](#ex:qccone){reference-type="ref" reference="ex:qccone"} and the discussion before Lemma [Lemma 39](#gen.seq){reference-type="ref" reference="gen.seq"}. Thus we have for the effective domain $\mathrm{dom} (\mathscr{R}) = Q_{2}(K)$. Obviously, $$\mathscr{R} \leq {\bf B}_{K,p} \quad {\rm on} \quad \mathbb{R}^{2 \times 2},$$ and according to Proposition [Proposition 48](#Id){reference-type="ref" reference="Id"}, we have the equality at the identity matrix: $$\label{finiteid} \mathscr{R}( \mathrm{Id}) = {\bf B}_{K,p}( \mathrm{Id}) = -1.$$ We then claim that $\mathscr{R}$ is rank-one convex. To this end, let us fix $A_0$, $A_1 \in \mathbb{R}^{2 \times 2}$ with $\mathrm{rank}(A_1-A_0) = 1$, $\lambda \in (0,1)$ and let $A_{\lambda} \equiv (1-\lambda )A_{0}+\lambda A_1$: we will show that $$\label{rkconvex} \mathscr R(A_\lambda)\leq \lambda \mathscr R(A_0)+(1-\lambda) \mathscr R(A_1).$$ There is nothing to prove if one of the matrices $A_i \notin Q_{2}(K)$, so we can assume that $A_0$, $A_1 \in Q_{2}(K)$. Under this assumption, take $t_i \in \mathbb{R}$ such that $t_i > \mathscr{R}(A_i )$. By the definition of $\mathscr R$, we may then find $\nu_i \in \mathscr{M}^{p}_{\mathrm{qc}}$ with $\langle \nu_i, \mathrm{Id}\rangle = A_i$ such that $t_i > \langle \nu_{i},{\bf B}_{K,p} \rangle$. In particular, it follows that both measure $\nu_0$ and $\nu_1$ are supported on $Q_{2}(K)$. Next, define $\nu_{\lambda} \equiv (1-\lambda )\nu_{0}+\lambda \nu_1$. Hereby $\nu_\lambda$ is a probability measure on $\mathbb{R}^{2 \times 2}$, supported on $Q_{2}(K)$, with $\langle \nu_\lambda, \mathrm{Id}\rangle=A_\lambda$. It clearly also has a finite $p$-th moment and, if ${\bf E}$ is a quasiconvex functional of at most $p$-th growth in the sense of Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"}.(1), then $$\begin{aligned} \bigl\langle \nu_{\lambda},{\bf E} \bigr\rangle &=& (1-\lambda ) \bigl\langle \nu_{0},{\bf E} \bigr\rangle + \lambda \bigl\langle \nu_{1},{\bf E} \bigr\rangle\\ &\geq& (1-\lambda ){\bf E}(A_{0})+\lambda {\bf E}(A_{1}) \geq {\bf E}(A_{\lambda}),\end{aligned}$$ where the first inequality follows from Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"}.(1), while second holds since as a real-valued quasiconvex functional, ${\bf E}$ is rank-one convex. Thus all conditions required by Theorem [Theorem 27](#thm:KP){reference-type="ref" reference="thm:KP"} are fulfilled. Consequently, $\nu_\lambda \in \mathscr{M}^{p}_{\mathrm{qc}}$ and we get $$(1-\lambda )t_0 + \lambda t_1 > \bigl\langle \nu_{\lambda},{\bf B}_{K,p} \bigr\rangle \geq \mathscr{R}(A_{\lambda}).$$ Since $t_i>\mathscr R(A_i)$ was arbitrary, the desired inequality [\[rkconvex\]](#rkconvex){reference-type="eqref" reference="rkconvex"} follows and proves that $\mathscr{R}$ is rank-one convex. By inspection we also deduce that $\mathscr{R}$ is positively $p$-homogeneous and isotropic. Indeed, this is a simple consequence of Lemma [Lemma 26](#lemma:invYM){reference-type="ref" reference="lemma:invYM"} and the fact that ${\bf B}_{K,p}$ has the same properties. Let us just check positive homogeneity, as isotropy is similar. Fix $t>0$ and, for a measure $\nu$, let $\langle \nu_t,{\bf E}\rangle \equiv \langle \nu, {\bf E}(t \cdot)\rangle$ for any continuous ${\bf E}$ with $p$-growth. Since the map $\nu\mapsto \nu_t$ is a bijection of $\mathscr M^p_\textup{qc}$ onto itself, we have $$\begin{aligned} t^p \mathscr R(A) & = \inf\left\{\int^\ast {\bf B}_{K,p}(t\cdot) \, \mathrm{d}\nu : \nu \in \mathscr M^p_\textup{qc}, \, \langle \nu, \mathrm{Id}\rangle = A\right\}\\ & = \inf\left\{\int^\ast {\bf B}_{K,p} \, \mathrm{d}\mu : \mu \in \mathscr M^p_\textup{qc}, \, \langle \mu, \mathrm{Id}\rangle = tA\right\} = \mathscr R( t A),\end{aligned}$$ where we used positive homogeneity of ${\bf B}_{K,p}$ in the first equality. We have so far shown that $\mathscr{R} \colon \mathbb{R}^{2\times 2} \to \overline\mathbb{R}$ is positively $p$-homogeneous, isotropic, rank-one convex and $-\infty<\mathscr{R}\leq {\bf B}_{p}$ in $Q_2(K)$, with equality at the identity matrix. It follows from Theorem [\[thm:extremality\]](#thm:extremality){reference-type="ref" reference="thm:extremality"} that in fact $\mathscr{R} = {\bf B}_{p}$ on $Q_{2}(K)$ and, consequently, that $\mathscr{R} = {\bf B}_{K,p}$ on $\mathbb{R}^{2 \times 2}$. But this is exactly what we set out to prove: $${\bf B}_{K,p}(A) = \mathscr{R}(A) \leq \int_{\mathbb{R}^{2 \times 2}} \! {\bf B}_{K,p} \, \mathrm{d}\nu,$$ where the inequality holds, by definition of $\mathscr R$, for all $A \in \mathbb{R}^{2 \times 2}$ and all $\nu \in \mathscr{M}_{\mathrm{qc}}^{p}$ with $\langle \nu, \mathrm{Id}\rangle =A$. In brief, we have shown that for any $2 < p < \tfrac{2K}{K-1}$ the functional ${\bf B}_{K,p}$ is closed $\mathrm{W}^{1,p}$-quasiconvex. ◻ *Remark 50*. In fact, combined with Corollary [\[cor:higherintegrability\]](#cor:higherintegrability){reference-type="eqref" reference="cor:higherintegrability"}, the above argument shows that ${\bf B}_{K,p}$ is closed $\mathrm{W}^{1,s}$-quasiconvex for every $s > \frac{2K}{K+1}$. We have done most of the hard work, so that the proof of Theorem [Theorem 4](#main){reference-type="ref" reference="main"} now follows easily. *Proof of Theorem [Theorem 4](#main){reference-type="ref" reference="main"}.* Let $p > \tfrac{2K}{K+1}$ and $\nu \in \mathscr{M}^{p}_{\mathrm{qc}}$. If $\int_{\mathbb{R}^{2 \times 2}}^{\ast} \! \mathcal B_K \, \mathrm{d}\nu = +\infty$, then there is nothing to prove, so we can assume that $\int_{\mathbb{R}^{2 \times 2}}^{\ast} \! \mathcal B_{K} \, \mathrm{d}\nu < +\infty$. This assumption implies that $\nu$ is supported in $Q_{2}(K)$ and therefore by Corollary [Corollary 40](#cor:higherintegrability){reference-type="ref" reference="cor:higherintegrability"} we have that $\nu \in \mathscr{M}_{\mathrm{qc}}^{s}\bigl( Q_{2}(K) \bigr)$ for every $s < p_{K} = \tfrac{2K}{K-1}$. Next, we can use Theorem [Proposition 49](#kqc){reference-type="ref" reference="kqc"} to get $${\bf B}_{s}( \langle \nu, \mathrm{Id}\rangle ) \leq \int_{\mathbb{R}^{2 \times 2}} \! {\bf B}_{s} \, \mathrm{d}\nu$$ for each $2< s < p_{K}$. Also, observe that ${\bf B}_{s} \leq 0$ on $Q_{2}(K)$ and that ${\bf B}_{s} \to {\bf B}_{p_{K}}$ pointwise as $s \nearrow p_K$. Consequently, Fatou's lemma yields $${\bf B}_{p_K}\bigl( \langle \nu, \mathrm{Id}\rangle \bigr) = \lim_{s\nearrow p_K} {\bf B}_{s}\bigl( \langle \nu, \mathrm{Id}\rangle \bigr) \leq \limsup_{s \nearrow p_K} \int_{\mathbb{R}^{2 \times 2}} \! {\bf B}_{s} \, \mathrm{d}\nu \leq \int_{\mathbb{R}^{2 \times 2}} \! {\bf B}_{p_K} \, \mathrm{d}\nu.$$ This completes the proof, since $\mathcal B_K = {\bf B}_{p_K}$ in the support of $\nu$. ◻ In particular, as closed $\mathrm{W}^{1,p}$-quasiconvexity implies $\mathrm{W}^{1,p}$-quasiconvexity, see for instance the discussion after Definition [Definition 30](#def:closedqc){reference-type="ref" reference="def:closedqc"}, Theorem [Theorem 4](#main){reference-type="ref" reference="main"} implies Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}. # The Burkholder area inequality {#sec:areaBp} The purpose of this section is to prove Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"}. We restate result here in a slightly different (but equivalent) form: **Theorem 51**. *Let $f$ be a $K$-quasiconformal principal map, conformal outside $\mathbb{D}$ with expansion $$\label{eq:expansionarea} f(z) = z + \frac {b_1}{z} + \sum_{j=2}^\infty \frac{b_j}{z^j} \equiv z+ \frac{b_1}{z} + \phi(z), \quad |z| > 1.$$ Then with the linear asymptotics $A_f(z) \equiv z+b_1 \bar z$, we have $$\label{eq:Bparea} \int_\mathbb D \Big({\bf B}_p(\textup{D}f)-{\bf B}_p(A_f)\Big)\, \mathrm{d}m(z) \geqslant- \frac{p}{2} \frac{{\bf B}_p(A_f)}{ \det(A_f)}\int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \, \mathrm{d}m(z),$$ provided that $2\leq p \leq \frac{2K}{K-1}$.* We begin with a rather general lemma which improves the asymptotics of the map in a controlled manner, while keeping the map unchanged in the disk. **Lemma 52**. *Suppose $f \in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ is a principal map with expansion [\[eq:expansionarea\]](#eq:expansionarea){reference-type="eqref" reference="eq:expansionarea"}. Then the map $\tilde f \colon \mathbb{C}\to \mathbb{C}$ defined by $$\tilde f \equiv \begin{cases} f & \textup{in } \mathbb D,\\ h \circ A_f & \textup{in } \mathbb{C}\backslash \mathbb D,\end{cases}$$ is a $\mathrm{W}^{1,1}_\textup{loc}$-homeomorphism, where $h\colon A_f(\mathbb{C}\setminus \mathbb{D}) \to f(\mathbb{C}\setminus \mathbb{D})$ is a conformal map defined by $$h \equiv f \circ R^{-1}, \qquad R(z)\equiv z+ b_1/z.$$ Moreover, $$h(z) = z + {\mathcal O}\left(z^{-2}\right) \text{ as } |z| \to \infty.$$ Finally, if $f$ is $K$-quasiconformal for some $K \geq 1$, then so is $\tilde f$.* *Proof.* We saw in Section [3](#sec:principal){reference-type="ref" reference="sec:principal"} that $|b_1| < 1$. Hence the rational map $R$ is injective in $\mathbb{C}\setminus \mathbb{D}$ and equals the linear map $A_f$ on the unit circle. In particular, $R^{-1} \circ A_f$ is a homeomorphism of the exterior disc $\mathbb{C}\setminus \mathbb{D}$ and equals the identity on $\mathbb S^1$. Hence the map $\tilde f$ is a $\mathrm{W}^{1,1}_\textup{loc}$-homeomorphism of $\mathbb{C}$. It is clear that $h\equiv f \circ R^{-1}$ is conformal in $\mathbb{C}\setminus A_f(\mathbb{D})$ and by [\[Kareafmla\]](#Kareafmla){reference-type="eqref" reference="Kareafmla"} $\tilde f$ is $K$-quasiconformal when $f$ is, so we only need to prove the decay of $h$. But this is easy, since $$R^{-1}(z) = \frac{1}{2} \left( z + \sqrt{z^2 - 4b_1} \right) = z - \frac{b_1}{z} + {\mathcal O}\left(\frac{1}{z^3} \right)\qquad \textup{as } |z| \to \infty,$$ which shows that $$\begin{aligned} h(z) = f \circ R^{-1}(z) & = R^{-1}(z) + \frac{b_1}{R^{-1}(z) } + {\mathcal O}\left(\frac{1}{z^2} \right)\\ & = z - \frac{b_1}{z} + \frac{b_1}{ z - \frac{b_1}{z}} + {\mathcal O}\left(\frac{1}{z^2} \right) = z + {\mathcal O}\left(\frac{1}{z^2} \right).\end{aligned}$$ completing the proof. ◻ The next lemma establishes a global quasiconvexity inequality for the modified map $\tilde f$; here the fast decay of $h$, provided by Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"}, is crucial. **Lemma 53**. *Let $f$ be as in Theorem [Theorem 51](#thm:Bparea){reference-type="ref" reference="thm:Bparea"} and define $\tilde f$ as in Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"}. Then $$\int_\mathbb{C}\Big({\bf B}_p(\textup{D}\tilde f) - {\bf B}_p(A_f)\Big)\, \mathrm{d}m(z) \geq 0.$$* *Proof.* Choose first a sequence of exponents ${p_n} \nearrow p \leq \frac{2K}{K-1}$, and write $K_{p_n} \equiv \frac{{p_n}}{{p_n}-2}$ so that $K < K_{p_n}$. Then define an auxiliary function $\psi(z)$, $|z|>1$, via $$\tilde f(z)=h(A_f(z))=A_f(z) + \mathcal O(A_f(z)^{-2})\equiv A_f(z) + \psi(z).$$ Finally choose, for each large integer $j\geq 2$, a smooth radially symmetric cutoff $0\leq \eta_j\leq 1$ such that $\eta_j(z)= 1$ if $|z|\leq j$, $\eta_j(z)=0$ if $|z|\geq j+1$ and $|\nabla \eta_j|\leq 2$. We claim that for any given index $n$, for all $j$ large enough the map $A_f+ \eta_j \psi$ is $K_{p_n}-$quasiconformal in $\mathbb{C}\backslash \mathbb D$. Indeed, note first that via [\[Kareafmla\]](#Kareafmla){reference-type="eqref" reference="Kareafmla"} the linear map $A_f$ is $K$-quasiconformal. Thus by construction, also $\tilde f(z)$ is $K$-quasiconformal in all of $\mathbb{C}$. We then set $$\label{tildej} \tilde f_j \equiv \begin{cases} f & \textup{in } \mathbb D,\\ A_f+\eta_j \psi & \textup{in } \mathbb{C}\backslash \mathbb D,\end{cases}$$ so that the function equals $\tilde f$ whenever $|z| < j$. In addition, since we chose $K_{p_n} > K$, for any $n$ we can take small $\varepsilon_n > 0$, converging to $0$ as $n \to \infty$, so that $X\in Q_2(K_{p_n})$ for every matrix with $|A_f-X|<\varepsilon_n$. On the other hand, for all $j$ sufficiently large we can estimate $$\label{decay13} |\textup{D}(\eta_j \psi)(z)| \leq |\eta_j \textup{D}\psi|(z)+ |\psi \otimes \nabla \eta_j|(z) \leq C |z|^{-2}$$ where the constant $C$ is independent of $j$. Thus, taking $j = j_n$ so large that $C j_n^{-2}\leq \frac{\varepsilon_n}{ 2}$, one obtains $|\textup{D}(\eta_{j_n} \psi)(z)| \leq \frac{\varepsilon_n }{2}$ for $|z|\geq j_n$. This means that by [\[Burkh46\]](#Burkh46){reference-type="eqref" reference="Burkh46"} and by what we have just shown, the map in [\[tildej\]](#tildej){reference-type="eqref" reference="tildej"} satisfies ${\bf B}_{p_n}(\textup{D}\tilde f_{j_n}) \leq 0$. We are now in a position to apply Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"}: Since $\tilde f_{j_n}(z)=A_f$ for $|z|\geq {j_n}+1$, we conclude that $$\label{preliBp} \int_{\mathbb{C}} \Big({\bf B}_{p_n}(\textup{D}\tilde f_{j_n}) - {\bf B}_{p_n}(A_f)\Big)\, \mathrm{d}m(z)\geqslant 0, \qquad n \geq 1.$$ To complete the argument we decompose the integral in [\[preliBp\]](#preliBp){reference-type="eqref" reference="preliBp"} as follows. Since $\tilde f_{j_n} = \tilde f$ in $\{ |z|\leq {j_n}\}$, we choose some fixed $\rho > 1$ and from [\[preliBp\]](#preliBp){reference-type="eqref" reference="preliBp"} obtain for all ${j_n} > \rho$ the lower bound $$\begin{aligned} \label{preliBp7} %VIRHE ?? & \int_{|z|\leq \rho} \left({\bf B}_{p_n}(\textup{D}\tilde f) - {\bf B}_{p_n}(A_f)\right) \mathrm{d}m + \int_{\rho \leq |z|\leq j_n} \left({\bf B}_{p_n}(\textup{D}\tilde f) - {\bf B}_{p_n}(A_f)\right) \mathrm{d}m \\ &\; \qquad + \int_{j_n\leq |z|\leq j_n+1} \left({\bf B}_{p_n}(\textup{D}\tilde f_{j_n}) - {\bf B}_{p_n}(A_f)\right) \mathrm{d}m\geqslant 0. \nonumber \end{aligned}$$ For the first term above, Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"} tells that $\tilde f$ is $K$-quasiconformal and hence ${\bf B}_{p_n}(\textup{D}\tilde f) \leq {\bf B}_p(\textup{D}\tilde f) \leq 0$. We can thus use Fatou's lemma to see that $$\int_{|z|\leq \rho} \Big({\bf B}_p(\textup{D}\tilde f) - {\bf B}_p(A_f)\Big)\mathrm{d}m\geq \limsup_{p_n \nearrow p} \int_{|z|\leq \rho} \Big({\bf B}_{p_n}(\textup{D}\tilde f) - {\bf B}_{p_n}(A_f)\Big)\mathrm{d}m.$$ For the second term in [\[preliBp7\]](#preliBp7){reference-type="eqref" reference="preliBp7"} recall that in the annuli $1 \leq |z| \leq j_n$ we have $\tilde f(z) = h \circ A_f(z)$, where for the conformal factor $h'$ is bounded on the set $A_f(\{ |z| \geq \rho \})$. Therefore in these annuli the functions $${\bf B}_{p_n}(\textup{D}\tilde f) - {\bf B}_{p_n}(A_f) = {\bf B}_{p_n}(A_f)\left( |h'\circ A_f|^{p_n}-1 \right)= {\bf B}_{p_n}(A_f) \mathcal O(A_f(z)^{-3})$$ are uniformly bounded and decay like $|z|^{-3}$ at infinity. Thus the dominated convergence theorem gives $$\lim_{n \to \infty} \int_{\rho \leq |z|\leq j_n} \left({\bf B}_{p_n}(\textup{D}\tilde f) - {\bf B}_{p_n}(A_f)\right) \mathrm{d}m = \int_{|z| \geq \rho} \Big({\bf B}_p(\textup{D}\tilde f) - {\bf B}_p(A_f)\Big)\mathrm{d}m.$$ Finally, we claim that when $n\to \infty$, the third term in [\[preliBp7\]](#preliBp7){reference-type="eqref" reference="preliBp7"} vanishes. Indeed, from the explicit expression [\[Burk\]](#Burk){reference-type="eqref" reference="Burk"}, or from the general properties of positively $p$-homogeneous and rank-one convex functionals [@BKK], we have the Lipschitz estimate $$\left| {\bf B}_{p_n}(A_f+ \textup{D}(\eta_{j_n} \psi) ) - {\bf B}_{p_n}(A)\right|\leq C(K) | \textup{D}(\eta_{j_n} \psi)| \left(|A_f|+| \textup{D}(\eta_{j_n} \psi)|\right)^{p-1}.$$ Thus we can use the estimate [\[decay13\]](#decay13){reference-type="eqref" reference="decay13"} which gives $$\begin{aligned} & \left|\int_{{j_n}\leq |z|\leq {j_n}+1} \left({\bf B}_{p_n}(\textup{D}\tilde f_{j_n}) - {\bf B}_{p_n}(A_f)\right) \mathrm{d}m(z) \right| \\ & \qquad \leq C(K)\int_{{j_n}\leq |z|\leqslant{j_n}+1} | \textup{D}(\eta_{j_n} \psi)| \left(|A_f|+| \textup{D}(\eta_{j_n} \psi)|\right)^{p-1}\mathrm{d}m(z) \\ & \qquad \leq C(K, A_f) {j_n}^{-1} \to 0,\end{aligned}$$ since $|\textup{D}(\eta_{j_n}\psi)|\leq C {j_n}^{-2}$ in $\{{j_n}\leq |z|\leq {j_n}+1\}$ and the area of this annulus is $(2{j_n}+1)\pi$. Combining now the above estimates and letting $n\to\infty$, we conclude that $$\begin{aligned} \int_{\mathbb C} \left({\bf B}_p(\textup{D}\tilde f) - {\bf B}_p(A_f) \right) \mathrm{d}m(z) \geq 0,\end{aligned}$$ as claimed. ◻ *Proof of Theorem [Theorem 51](#thm:Bparea){reference-type="ref" reference="thm:Bparea"}.* Using the definition of $\tilde f$ from Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"} and applying Lemma [Lemma 53](#lemma:globalqcBp){reference-type="ref" reference="lemma:globalqcBp"}, we obtain: $$\int_{\mathbb{D}} \Big( {\bf B}_p(\textup{D}f ) - {\bf B}_p(A_f) \Big)\, \mathrm{d}m(z) \geq - \int_{\mathbb{C}\setminus \mathbb{D}} \Big( {\bf B}_p(\textup{D}(h \circ A_f)) - {\bf B}_p(A_f) \Big)\, \mathrm{d}m(z).$$ Our goal is to give a lower bound on the right-hand side of this estimate. We saw already in the previous lemma that ${\bf B}_p(\textup{D}(h \circ A_f)) - {\bf B}_p(A_f) = {\bf B}_p(A_f) \Bigl(\bigl| h' \circ A_f \bigr|^p \, - 1 \Bigr)$. Hence changing the variables twice, we have: $$\begin{aligned} & \int_{\mathbb{C}\setminus \mathbb{D}} \Bigl( {\bf B}_p(\textup{D}(h \circ A_f)) - {\bf B}_p(A_f)\Bigr) \, \mathrm{d}m(z) \\ %& \qquad = \int_{\C\setminus \DD} \Bigl( \bigl| h' \circ A_f \bigr|^p \, - 1 \Bigr) {\bf B}_p(A_f)\, \dd m(z) \\ & \qquad = \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{A_f(\mathbb{C}\setminus \mathbb{D})} \Bigl( \bigl| h'(z) \bigr|^p \, - 1 \Bigr) \, \mathrm{d}m(z)\\ & \qquad = \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{R(\mathbb{C}\setminus \mathbb{D})} \Bigl( \bigl| (f \circ R^{-1})'(z) \bigr|^p \, - 1 \Bigr) \, \mathrm{d}m(z) \\ & \qquad = \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{R(\mathbb{C}\setminus \mathbb{D})} \Bigl( \bigl| f'( R^{-1}(z))\bigr|^{p} \, \bigl| (R^{-1})' (z)\bigr|^p \, - 1 \Bigr) \, \mathrm{d}m(z) \\ & \qquad = \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{\mathbb{C}\setminus \mathbb{D}} \Bigl( \bigl| f'(z)\bigr|^{p} \, \bigl| R' (z)\bigr|^{-p} \, - 1 \Bigr) \, |R'(z)|^2 \, \mathrm{d}m(z). \end{aligned}$$ Finally, in the notation of [\[eq:expansionarea\]](#eq:expansionarea){reference-type="eqref" reference="eq:expansionarea"}, $f = R + \phi$ in $\mathbb{C}\setminus \mathbb{D}$. In particular, $$\begin{aligned} & \int_{\mathbb{D}} \Big({\bf B}_p(\textup{D}f(z) ) - {\bf B}_p(A_f)\Big) \, \mathrm{d}m(z) \\ & \qquad \geq - \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{\mathbb{C}\setminus \mathbb{D}} \Big(| R'(z) + \phi'(z)|^p |R'(z)|^{2-p} - |R'(z)|^2\Big) \, \mathrm{d}m(z).\end{aligned}$$ This suggests us to define $$H_p(z,w) \equiv | R'(z) + w|^p |R'(z)|^{2-p} - |R'(z)|^2$$ to estimate the integrand on the right-hand side. We claim that $$\label{eq:areaclaim} \int_{\mathbb{C}\setminus \mathbb{D}} H_p\bigl(z,\phi'(z) \bigr) \, \mathrm{d}m(z)\geq \frac{p}{2} \int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \,\mathrm{d}m(z),$$ To this end, we need a further lemma. **Lemma 54**. *Let $G_p(w) = |1+ w|^p - 1$. Then for any $2 < p < \infty$, $$G_p(w) \geq p \operatorname{Re}w + \frac{p}{2} |w|^2, \qquad w \in \mathbb{C},$$ where the coefficient in front of $|w|^2$ is the largest possible.* *Proof.* Letting $\zeta = w + 1$, the task is to show that $$|\zeta|^p \geq 1 + p \operatorname{Re}(\zeta-1) + \frac{p}{2} |\zeta -1|^2 = 1-\frac{p}{2} + \frac{p}{2}|\zeta|^2,$$ which is easy to verify. The derivatives of the left-hand side and the right-hand side both evaluate to $p$ at $\zeta=1$, hence the constant $\frac p 2$ is optimal. ◻ Note that $$H_p(z,w) = |R'(z)|^2 G_p\bigl(w/R'(z)\bigr),$$ and thus Lemma [Lemma 54](#lemma:convexity){reference-type="ref" reference="lemma:convexity"} yields $$\begin{aligned} \begin{split} \label{eq:auxarea1} H_p\bigl(z,\phi'(z)\bigr) & \geq p |R'(z)|^2\operatorname{Re}\bigl(\phi'(z) / R'(z) \bigr) + \frac{p}{2} |\phi'(z) |^2 \\ & = p \operatorname{Re}\left(\phi'(z) \, \overline{R'(z)} \right) + \frac{p}{2} |\phi'(z) |^2. \end{split}\end{aligned}$$ To use this inequality note that as $\phi'(z) =- \sum_{j=2}^\infty \frac{j \,b_j}{z^{j+1}}$ and ${R'(z) = 1 -b_1/z^2}$, by integration in polar coordinates it follows that $$\int_{\mathbb{C}\setminus \mathbb D} \phi'(z)\overline{R'(z)} \,\mathrm{d}m(z) = 0.$$ Thus we obtain the required claim [\[eq:areaclaim\]](#eq:areaclaim){reference-type="eqref" reference="eq:areaclaim"} upon integrating the estimate [\[eq:auxarea1\]](#eq:auxarea1){reference-type="eqref" reference="eq:auxarea1"}. We have thus shown that $$\int_{\mathbb D}\Big( {\bf B}_p(\textup{D}f ) - {\bf B}_p(A_f)\Big) \, \mathrm{d}m(z) \geq - \frac{p}{2} \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \, \mathrm{d}m(z),$$ which completes the proof. ◻ *Proof of Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"}.* In the notation of [\[eq:expansionarea\]](#eq:expansionarea){reference-type="eqref" reference="eq:expansionarea"}, $\phi'(z) =- \sum_{j=2}^\infty \frac{j \,b_j}{z^{j+1}}$ where the powers $z^{-(j+1)}$ are orthogonal in $L^2(\mathbb{C}\setminus \mathbb{D})$. Thus $$\int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \, \mathrm{d}m(z) = \sum_{j=2} j^2 |b_j|^2 \Big\| z^{-j-1} \Big\|^2_{L^2(\mathbb{C}\setminus \mathbb{D})} = \pi \sum_{j=2} j |b_j|^2.$$ Since ${\bf B}_2(A_f) = - \det(A_f)$, this combined with [\[eq:Bparea\]](#eq:Bparea){reference-type="eqref" reference="eq:Bparea"} proves Theorem [Theorem 6](#thm:Bpareaintro){reference-type="ref" reference="thm:Bpareaintro"}. ◻ # The sharp higher integrability of the Jacobian {#sec:LlogL} As usual, in this section $\Omega\subset \mathbb{R}^2$ denotes an arbitrary bounded domain such that $\mathscr L^2(\partial\Omega)=0$. By letting $\,p\to 2\,$ in Theorems [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} and [Theorem 4](#main){reference-type="ref" reference="main"}, we will obtain new quasiconvex functionals defined on $$\mathbb{R}^{2\times 2}_+\equiv \mathbb{R}^{2\times 2}\cap\{\det>0\}.$$ To be precise, any functional ${\bf E}\colon \mathbb{R}^{2\times2}_+\to \mathbb{R}$ can be naturally extended to $\mathbb{R}^{2\times 2}$ by setting $$\label{eq:extenddetfunc} {\bf E}(A)=+\infty, \quad \text{ whenever } \det A\leq 0.$$ We will always identify ${\bf E}$ with this extension. It is important to note that the set $\mathbb{R}^{2\times 2}\backslash \mathbb{R}^{2\times 2}_+=\{\det \leq 0\}$ is $\mathrm{W}^{1,2}$-quasiconvex, recall Definition [Definition 33](#qcset){reference-type="ref" reference="qcset"}, and hence the extension of ${\bf E}$ is trivially $\mathrm{W}^{1,2}$-quasiconvex at matrices with non-positive determinant. It is for this reason that there is no ambiguity in talking about quasiconvexity of functionals defined on $\mathbb{R}^{2\times 2}_+$. Let us first consider the functional $$\label{F2} { \mathscr F}(A) \equiv \, |A|^2\, - \,\left(1\, + \,\log |A|^2 \,\right )\, \det(A),$$ which is rank-one convex in $\mathbb{R}^{2\times 2}$, cf. [@Tade Corollary 5.1] or [\[eq:derivativeBp\]](#eq:derivativeBp){reference-type="eqref" reference="eq:derivativeBp"} below, but not polyconvex: indeed, for $t>0$ we have $\mathscr F(t\, \mathrm{Id})=-2t^2 \log t$, which decreases faster than any quadratic function [@Dacorogna2007 Corollary 5.9]. Here we succeed in proving that $\mathscr F$ is quasiconvex in $\mathbb{R}^{2\times 2}_+$: **Corollary 55**. *Let $A \in \mathbb{R}^{2 \times 2}_+$ and $f \in A + \mathrm{W}^{1,2}_0(\Omega)$ be a homeomorphism. Then $$\label{LlogL} { \mathscr F}(A) \leq \fint _\Omega \, { \mathscr F}(\textup{D}f(z)) \, \mathrm{d}m(z). % {eq:qc23}$$* *Proof.* Suppose first that $f$ is a smooth diffeomorphism on $\overline{\Omega}$, with boundary values $f|_{ \partial \Omega} = A$. Since ${\bf B}_2(A) = - \det(A)$ is a null Lagrangian and $$\label{eq:derivativeBp} \frac{p}{p-2} \left[ {\bf B}_p(A) - {\bf B}_2(A)\right] \, = \, |A|^2 - \det(A)(1+ \log(|A|^2) + {\mathcal O}(p-2),$$ taking the limit $p \to 2$ we see from Theorem [Theorem 3](#thm:soft){reference-type="ref" reference="thm:soft"} that [\[LlogL\]](#LlogL){reference-type="eqref" reference="LlogL"} holds for such a map $f$. In the case of a general ${\mathrm{W}}^{1,2}$-homeomorphism $f$ with boundary values $A$, we apply [@IKO Theorem 1.1] to obtain a sequence of diffeomorphisms $f_j$ such that $f_j \to f$ in $\mathrm{W}^{1,2}(\Omega)$ and $f_j=A$ on $\partial\Omega$. We now argue similarly as in the proof of [@AIPS12 Corollary 1.7], see also [@IwaniecVerde]. Applying the quasiconvexity inequality for each $f_j$, we obtain $$\begin{aligned} \mathscr F(A) & \leq \limsup_{j\to \infty} \fint_\Omega \mathscr F(\textup{D}f_j(z)) \,\mathrm{d}m(z)\\ & = \lim_{j\to \infty} \fint_\Omega \left(|\textup{D}f_j|^2 - J_{f_j}\right)\mathrm{d}m(z) - \liminf_{j\to \infty} \fint_\Omega J_{f_j} \log|\textup{D}f_j|^2 \,\mathrm{d}m(z).\end{aligned}$$ There is no difficulty in passing to the limit in the first term. For the second term, we estimate $$\begin{aligned} & \liminf_{j\to \infty} \fint_\Omega J_{f_j} \log|\textup{D}f_j|^2\,\mathrm{d}m\\ &\quad =\liminf_{j\to \infty} \fint_\Omega J_{f_j} \log(1+|\textup{D}f_j|^2) \,\mathrm{d}m - \lim_{j\to \infty} \fint_\Omega J_{f_j} \log(1+|\textup{D}f_j|^{-2})\, \mathrm{d}m\\ &\quad \geq \fint_\Omega J_{f} \log(1+|\textup{D}f|^2) \, \mathrm{d}m-\fint_\Omega J_f \log(1+|\textup{D}f_j|^{-2})\,\mathrm{d}m\\ & \quad = \fint_\Omega J_{f} \log|\textup{D}f|^2 \, \mathrm{d}m,\end{aligned}$$ where in the third line we applied Fatou's lemma for the first term, since the integrand is non-negative, and the Dominated Convergence Theorem for the second term, since the integrand is dominated pointwise by $J_{f_j}|\textup{D}f_j|^{-2}\leq 1$. The desired inequality follows. ◻ *Remark 56*. Exactly the same argument shows that, when $\Omega=\mathbb D$, inequality [\[LlogL\]](#LlogL){reference-type="eqref" reference="LlogL"} holds more generally if $f$ is a *monotone map* in the topological sense of Morrey, i.e. if $f^{-1}(w)$ is connected for all $w$. In fact, by a Sobolev version of a classical theorem of Youngs, a map in $A + \mathrm{W}^{1,2}_0(\mathbb D)$ is monotone if and only if it is the strong $\mathrm{W}^{1,2}$-limit of a sequence of smooth diffeomorphisms with the same boundary condition [@IO2016]. For maps with identity boundary values the estimate [\[LlogL\]](#LlogL){reference-type="eqref" reference="LlogL"} was obtained in [@AIPS12 Corollary 1.7]. The fact that the Jacobian determinant possesses higher integrability was first noticed by Müller [@Muller1990] and then generalized by Coifman, Lions, Meyer and Semmes [@CLMS]. From these results one deduces that if $f\in W^{1,n}_\textup{loc}(\mathbb{R}^n)$ is an orientation-preserving mapping then $J_f \log |\textup{D}f|^n\in L^1_\textup{loc}(\mathbb{R}^n)$, see [@IwaniecGreco]. Corollary [Corollary 55](#cor:LlogL){reference-type="ref" reference="cor:LlogL"} expresses a sharp, global version of this result when $n=2$: indeed, we may rewrite [\[LlogL\]](#LlogL){reference-type="eqref" reference="LlogL"} as the sharp inequality $$\label{eq:sharpLlogL} \int_\Omega J_f(z)(1+\log|\textup{D}f(z)|^2) \,\mathrm{d}m(z) \leq \int_\Omega |\textup{D}f(z)|^2 \,\mathrm{d}m(z) -\mathscr F(A)\mathscr L^2(\Omega),$$ to hold whenever $f\in A + \mathrm{W}^{1,2}_0(\Omega)$ is a homeomorphism and $A \in \mathbb{R}^{2 \times 2}_+$. As mentioned above, the functional $\mathscr F\colon \mathbb{R}^{2\times 2}\to \mathbb{R}$ is rank-one convex and it is conjectured to be quasiconvex as well [@Tade (6.5)], in which case [\[eq:sharpLlogL\]](#eq:sharpLlogL){reference-type="eqref" reference="eq:sharpLlogL"} would hold for general maps $f\in A +\mathrm{W}^{1,2}_0(\Omega)$ which are not necessarily orientation-preserving. For such maps one cannot give meaning to $\int_\Omega J_f \log|\textup{D}f|^2\,\mathrm{d}m(z)$ as a Lebesgue integral; instead, this integral needs to be interpreted distributionally [@IwaniecVerde]. # New sharp bounds via the Shield transformation {#sec:shield} The optimal integral bounds of the previous Section lead to new ones via the *Shield transformation*. This is a general procedure for generating new functionals, by applying a given functional to the inverse map: **Definition 57**. Given ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ we define $\widehat {\bf E}\colon \mathbb{R}^{2\times2}_+\to \mathbb{R}$ by $$\widehat {\bf E}(A)\equiv {\bf E}(A^{-1}) \det A.$$ In the Elasticity literature, the involution $\,\widehat \cdot \,$ is sometimes referred to as the *Shield transformation*, after Schield's work [@Shield]. Since $\widehat{1}=\det$, $\widehat{\cdot}$ does not preserve convexity; yet it preserves the usual semi-convexity notions from the vectorial Calculus of Variations. Indeed, it is easy to verify that polyconvexity and rank-one convexity are preserved by $\,\widehat \cdot \,$, see [@Ball1977 Theorem 2.6] for further details. The case of quasiconvexity, which is the one concerning us here, is more subtle. Indeed, as in Definition [Definition 1](#quasiconvexity){reference-type="ref" reference="quasiconvexity"}, in the case of deformations a crucial point is that one needs to address also the regularity of the inverse map when discussing quasiconvexity of $\widehat {\bf E}$, here see also Example [Example 62](#ex:radialmapW){reference-type="ref" reference="ex:radialmapW"}. In fact, the inverse of a planar $\mathrm{W}^{1,1}_\textup{loc}$-homeomorphism $f$ is in $\mathrm{W}^{1,2}_\textup{loc}$ if and only if $K_f\in \mathrm{L}^1_\textup{loc}$ [@HK Theorem 1.7], and hence this is the assumption that we shall make. **Proposition 58**. *Let ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be a $\mathrm{W}^{1,2}$-quasiconvex functional, in the sense that ${\bf E}$ satisfies [\[eq:quasiconvexity\]](#eq:quasiconvexity){reference-type="eqref" reference="eq:quasiconvexity"} for any $A\in \mathbb{R}^{2\times 2}_+$ and any homeomorphism $f\in A + \mathrm{W}_0^{1,2}(\Omega)$. Then the Shield transformation $\widehat{{\bf E}}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ satisfies $$\widehat{{\bf E}}(A)\leq \fint_\Omega \widehat{{\bf E}}(\textup{D}f(z)) \,\mathrm{d}m(z)$$ for all $A\in \mathbb{R}^{2\times 2}_+$ and all homeomorphisms $f\in A + \mathrm{W}_0^{1,1}(\Omega)$ with $K_f\in \mathrm{L}^1(\Omega)$.* *Proof.* Let $f\in A + \mathrm{W}^{1,1}_0(\Omega)$ be a homeomorphism with integrable distortion. Clearly we may assume that $J_f>0$ a.e. in $\Omega$, as otherwise there is nothing to prove. By [@HKO Theorem 2.1], the inverse homeomorphism $g \equiv f^{-1}$ satisfies[^1] $$\label{eq:aimo} \int_{A(\Omega )} |\textup{D}g(w)|^2 \, \mathrm{d}m(w)=\int_\Omega K_f(z) \, \mathrm{d}m(z)$$ and thus $g\in A^{-1} + W^{1,2}_{0}\bigl(A(\Omega)\bigr)$. Since $\mathrm{W}^{1,2}$-homeomorphisms satisfy Lusin's condition (N), we can apply the change of variables formula to obtain $$\begin{aligned} \int_\Omega \widehat {\bf E}(\textup{D}f(z))\,\mathrm{d}m(z) & =\int_{A(\Omega)}{\bf E}(\textup{D}g(w))\,\mathrm{d}m(w)\\& \geq \mathscr L^2(A(\Omega)){\bf E}(A^{-1}) = \mathscr L^2(\Omega) \widehat {\bf E}(A), \end{aligned}$$ where in the last equality we used $\mathscr L^2(A(\Omega))=\det(A) \,\mathscr L^2(\Omega)$. ◻ *Remark 59*. The same proof as in Proposition [Proposition 58](#prop:shield){reference-type="ref" reference="prop:shield"} applies more generally to $\mathrm{W}^{1,n}$-quasiconvex functionals ${\bf E}\colon \mathbb{R}^{n\times n}_+\to \mathbb{R}$. The natural assumptions are that $f\in A + \mathrm{W}^{1,1}_0(\Omega)$ is a homeomorphism such that $K_f\in \mathrm{L}^{n-1}(\Omega)$, where $\Omega\subset \mathbb{R}^n$ is a domain. *Remark 60*. We note that any map $f\in A + \mathrm{W}^{1,2}_0(\Omega)$ such that $K_f\in \mathrm{L}^1(\Omega)$ is necessarily a homeomorphism: such maps are automatically continuous and, by the results of Iwaniec and Šverák [@IwaSve], they are also open, hence the claim follows from degree theory. We then apply the Shield transform to generate a new quasiconvex functional, simply by setting $$\mathscr W\equiv \widehat {\mathscr F}+1.$$ Using $|A^{-1}| = |A|/\det(A)$, one easily computes $$\label{W} \mathscr W(A) = \begin{cases} \frac{|A|^2}{{\rm det}\, A} - \log \left( \frac{|A|^2}{{\rm det}\, A} \right) + \log \det(A) \quad {\rm if} \; \det(A) > 0,\\ + \infty \hspace{4,9cm} {\rm otherwise}. \end{cases}$$ This functional had already been considered in [@AIPS12] and was also studied in the recent works [@Voss1; @Voss2]. In [@Voss1] it was suggested that $\mathscr W$, denoted by $W^+_{\rm magic}$ in that paper, could be an example of a functional which is rank-one convex but not quasiconvex. Later in [@Voss2] numerical evidence was found which instead supports the quasiconvexity of $\mathscr W$. As a direct consequence of Corollary [Corollary 55](#cor:LlogL){reference-type="ref" reference="cor:LlogL"} and Proposition [Proposition 58](#prop:shield){reference-type="ref" reference="prop:shield"}, we now obtain that indeed $\mathscr W$ *is quasiconvex*: **Corollary 61**. *Let $A \in \mathbb{R}^{2 \times 2}_+$ and let $f \in A+\mathrm{W}^{1,1}_0 (\Omega)$ be a homeomorphism. If $K_f \in \mathrm{L}^1(\Omega)$ then $$\label{Wqconv} \mathscr W(A) \leq \fint_{\Omega} \mathscr W\bigl(\textup{D}f(z)\bigr) \, \mathrm{d}m(z).$$* Koskela and Onninen showed in [@KO] that for a map $f\in \mathrm{W}^{1,2}_\textup{loc}(\Omega)$ with $K_f\in \mathrm{L}^1_\textup{loc}(\Omega)$, we have $\log J_f \in \mathrm{L}^1_\textup{loc}(\Omega)$, see also Proposition [Proposition 73](#prop:KOR){reference-type="ref" reference="prop:KOR"} below and [@KOR] for an analogue in space. We can interpret [\[Wqconv\]](#Wqconv){reference-type="eqref" reference="Wqconv"} as a sharp, global version of their result: indeed, we may rewrite this estimate as $$\int_\Omega -\log J_f(z) \,\mathrm{d}m(z) \leq \int_\Omega \left(K_f(z) -\log K_f(z)\right) \mathrm{d}m(z) -\mathscr W(A)\mathscr L^2(\Omega)$$ for homeomorphisms $f\in A+ \mathrm{W}_0^{1,1}(\Omega)$ with $K_f\in \mathrm{L}^1(\Omega)$. Moreover, $\mathscr W$ has also some closed quasiconvexity features, for details see Proposition [Proposition 76](#thm:closedqc.new){reference-type="ref" reference="thm:closedqc.new"} below. ## Quasiconvexity beyond integrable distortion So far we have been discussing quasiconvexity of $\mathscr W$ for maps with integrable distortion. This assumption is natural since, as we saw in [\[eq:aimo\]](#eq:aimo){reference-type="eqref" reference="eq:aimo"}, it is *equivalent* to the $\mathrm{W}^{1,2}$-regularity of the inverse map. Nonetheless, one can ask whether $\mathscr W$ satisfies the quasiconvexity inequality when tested with smooth homeomorphisms with less regular inverses. Indeed there are smooth homeomorphisms with finite $\mathscr W$-energy for which the inverse is not in $\mathrm{W}^{1,2}$; this is possible since the distortion and the Jacobian terms in $\mathscr W$ may cancel each other: **Example 62**. Consider the radial stretching $$f(z)=\exp(1-1/r^2) \frac{z}{r}, \qquad r=|z|.$$ It is easy to see that $f$ is a smooth homeomorphism which equals the identity on $\mathbb S^1$. We compute $$\begin{aligned} K_f(z) = \frac{2}{r^2};\end{aligned}$$ in particular, $K_f \in \textup{weak-}\mathrm{L}^1(\mathbb D)$ but $K_f \not\in \mathrm{L}^1(\mathbb D)$. Nonetheless $f$ has finite $\mathscr W$-energy: indeed, $$J_f(z)= \frac{2}{r^4} \exp(2-2/r^2) \quad \implies \quad \log J_f (z) = 2-\frac{2}{r^2} + \log \frac{2}{r^4}$$ and hence the divergent terms $\frac{2}{r^2}$ cancel each other: $$\mathscr W(\textup{D}f(z)) = 2(1-\log r) \quad \implies \quad \int_{\mathbb D} \mathscr W(\textup{D}f(z)) \, \mathrm{d}m(z) = 3 \pi.$$ In view of the above example, if no further restrictions are made for the homeomorphisms $f$ in Corollary [Corollary 61](#cor:W){reference-type="ref" reference="cor:W"} beyond the assumptions that $f \in A + \mathrm{W}^{1,1}_0(\Omega)$ and that $\mathscr W(\textup{D}f) \in \mathrm{L}^1(\Omega)$, the inequality [\[Wqconv\]](#Wqconv){reference-type="eqref" reference="Wqconv"} reduces to a question of approximation, i.e. whether for any such $f$ there exists a sequence of diffeomorphisms $f_j \in C^\infty(\Omega)$ equal to $A$ on the boundary, with $$\lim_{j\to\infty} \int_{\Omega} |\mathscr W(\textup{D}f(z)) - \mathscr W(\textup{D}f_j(z))| \mathrm{d}m(z) = 0.$$ In the context of Sobolev homeomorphisms, the main tools for constructing a diffeomorphic approximation employ either the harmonic (or $p$-harmonic) extension [@IKO] or a shortest curve extension method [@HP]. Unfortunately, neither of these tools work in the present setting due to the existence of boundary maps for which the standard extension methods produce the wrong integrability of the distortion, see [@KosO Example 1.3]. Hence it is clear that to prove the appropriate result for the functional $\mathscr W$, completely new methods of approximation need to be developed first. On the other hand, for radial maps as in Example [Example 62](#ex:radialmapW){reference-type="ref" reference="ex:radialmapW"} there is no difficulty in proving the quasiconvexity inequality [\[Wqconv\]](#Wqconv){reference-type="eqref" reference="Wqconv"} directly, using e.g. the argument in Lemma [Lemma 36](#lemma:rcradialstretchings){reference-type="ref" reference="lemma:rcradialstretchings"}. In addition, as we shall next see, Corollary [Corollary 61](#cor:W){reference-type="ref" reference="cor:W"} extends to all monotone maps. Recall that a map is *monotone* in $\Omega$ if $$\langle f(z) - f(w), z - w \rangle \geq 0, \qquad \forall \; z, w \in \Omega.$$ If furthermore, say, $f\in \mathrm{W}^{1,1}_\textup{loc}(\Omega)$, then monotonocity is equivalent to $$\label{eq:monotone} |\partial_{\bar z} f |\leq \operatorname{Re}\partial_z f\quad \textup{ a.e.\ in } \Omega,$$ see for instance [@AIM §3.11]. Note that any orientation-preserving radial stretching is monotone, since for such maps $\partial_z f\in \mathbb{R}$. In the monotone case, the main point of proof is to find an approximation of a given monotone map by maps with smaller $\mathscr W$-energy, at least in the region where the map has large distortion. In order to construct such an approximating sequence we rely on a clever trick due to Chlebík and Kirchheim [@Chlebik]. **Proposition 63**. *Let $A\in \mathbb{R}^{2\times 2}_+$ and let $f\in A+\mathrm{W}^{1,2}_0(\Omega)$ be monotone with $\mathscr W(\textup{D}f)\in \mathrm{L}^1(\mathbb D)$. Then $$\mathscr W(A) \leq \fint_{\mathbb D} \mathscr W(\textup{D}f(z))\,\mathrm{d}m(z).$$* *Proof.* By the definition of $\mathscr W$ as an extended real-valued functional in [\[eq:extenddetfunc\]](#eq:extenddetfunc){reference-type="eqref" reference="eq:extenddetfunc"}, we may assume that $J_f>0$ a.e. in $\mathbb D$. For $\delta\in (0,1)$ let us consider the map $$f_\delta (z)\equiv f(z)+ \delta z,$$ where we compute $$|\partial_z f_\delta| = \sqrt{|\partial_z f|^2+2 \delta \operatorname{Re}\partial_z f + \delta^2}, \qquad |\partial_{\bar z} f_\delta|= |\partial_{\bar z} f|.$$ Notice that since $\operatorname{Re}(\partial_z f) \geq 0$, the expression $|\partial_z f_\delta|$ is an increasing function of $\delta$. Furthermore, we obtain $$\label{eq:derivativefdelta} |\partial_z f|^2 + \delta^2 \leq |\partial_z f_\delta|^2 \leq 2(|\partial_z f|^2 + \delta^2).$$ Let us set $E_\delta \equiv \{z\in \mathbb D: |\partial_z f_\delta| \leq 2|\partial_{\bar z} f|\}$. We claim that if $0 < \delta < 1$, then $$\label{eq:comparison} \begin{cases} \mathscr W(\textup{D}f_\delta)\leq \mathscr W(\textup{D}f) & \textup{ in } E_\delta,\\ \mathscr W(\textup{D}f_\delta)\leq \mathscr W(\textup{D}f_1) & \textup{ in } \mathbb D \backslash E_\delta. \end{cases}$$ To see this, note that $$\mathscr W(\textup{D}f)=\frac{|\partial_z f|+|\partial_{\bar z} f|}{|\partial_z f|-|\partial_{\bar z} f|}+ 2 \log\left(|\partial_z f|-|\partial_{\bar z} f|\right)=\omega(|\partial_z f|),$$ where $$\omega(t)\equiv \frac{t+|\partial_{\bar z} f|}{t-|\partial_{\bar z} f|}+ 2 \log(t-|\partial_{\bar z} f|), \qquad t > |\partial_{\bar z} f|.$$ Since $\omega'(t)=2(t-2|\partial_{\bar z}f|)/(t-|\partial_{\bar z} f|)^2$, we see that $\omega$ is decreasing on $(|\partial_{\bar z} f|,2|\partial_{\bar z} f|)$ and increasing on $(2|\partial_{\bar z} f|,+\infty)$, thus $$\begin{cases} \omega(|\partial_z f_\delta|) \leq \omega(|\partial_z f|) &\textup{ in } E_\delta,\\ \omega(|\partial_z f_\delta|) \leq \omega(|\partial_z f_1|) &\textup{ in } \mathbb D \backslash E_\delta, \qquad \end{cases}$$ which yields [\[eq:comparison\]](#eq:comparison){reference-type="eqref" reference="eq:comparison"}. Since $J_f>0$ a.e. in $\mathbb D$, using [\[eq:derivativefdelta\]](#eq:derivativefdelta){reference-type="eqref" reference="eq:derivativefdelta"} we can estimate $$K_{f_\delta}= \frac{|\textup{D}f_\delta|^2}{J_{f_\delta}} \leq 2 \frac{|\partial_z f_\delta|^2 + |\partial_{\bar z} f|^2}{|\partial_z f_\delta|^2 - |\partial_{\bar z} f|^2} \leq 4 \frac{|\partial_z f|^2 + |\partial_{\bar z} f|^2 + \delta^2}{J_f + \delta^2} \leq 4\frac{|\textup{D}f|^2+\delta^2}{\delta^2},$$ thus $K_{f_\delta}\in \mathrm{L}^1(\mathbb D)$. With $A_\delta\equiv A + \delta \,\mathrm{Id}$ we have that $f_\delta\in A_\delta + \mathrm{W}^{1,2}_{0}(\mathbb D)$ is a homeomorphism, cf. Remark [Remark 60](#rmk:automatichomeo){reference-type="ref" reference="rmk:automatichomeo"}, and hence, by Corollary [Corollary 61](#cor:W){reference-type="ref" reference="cor:W"} and [\[eq:comparison\]](#eq:comparison){reference-type="eqref" reference="eq:comparison"}, $$\begin{aligned} \label{eq:auxmonotone} \begin{split} \pi \mathscr W(A_\delta)& \leq \int_\mathbb D \mathscr W(\textup{D}f_\delta(z))\, \mathrm{d}m(z)\\ & = \int_{E_\delta} \mathscr W(\textup{D}f_\delta(z))\, \mathrm{d}m(z)+\int_{\mathbb D \setminus E_\delta} \mathscr W(\textup{D}f_\delta(z))\, \mathrm{d}m(z)\\ & \leq \int_{E_\delta} \mathscr W(\textup{D}f(z))\, \mathrm{d}m(z) + \int_{\mathbb D \setminus E_\delta} \mathscr W(\textup{D}f_\delta(z))\, \mathrm{d}m(z) . \end{split}\end{aligned}$$ We can now apply the reverse Fatou lemma to the last term in [\[eq:auxmonotone\]](#eq:auxmonotone){reference-type="eqref" reference="eq:auxmonotone"} since we have the estimate $\mathscr W(\textup{D}f_\delta(z)) \leq \mathscr W(\textup{D}f_1(z))$ for $z \in \mathbb D \setminus E_\delta$, where $\mathscr W(\textup{D}f_1) \in \mathrm{L}^1(\mathbb D)$ since $K_{f_1} \in \mathrm{L}^1(\mathbb D)$. Therefore $$\begin{aligned} \limsup_{\delta \to 0} \int_{\mathbb D \setminus E_\delta} \mathscr W(\textup{D}f_\delta(z))\, \mathrm{d}m(z) &\leq \int_{\mathbb D} \limsup_{\delta \to 0} \chi_{\mathbb D \setminus E_\delta}(z) \mathscr W(\textup{D}f_\delta(z)) \, dz \\ &= \int_{\mathbb D \setminus E_0} \mathscr W(\textup{D}f(z)) \, dz .\end{aligned}$$ The conclusion follows by sending $\delta \to 0$ in [\[eq:auxmonotone\]](#eq:auxmonotone){reference-type="eqref" reference="eq:auxmonotone"}, since $\mathscr W(\textup{D}f)\in \mathrm{L}^1(\mathbb D)$. ◻ # Improved integral inequalities {#sec:inequalities} The purpose of this section is to prove further integral inequalities for the functionals $\mathscr F$ and $\mathscr W$, similar to the Burkholder area inequality of Section [7](#sec:areaBp){reference-type="ref" reference="sec:areaBp"}. As a first observation, the precise value of the constant on the right-hand side of [\[eq:Bparea\]](#eq:Bparea){reference-type="eqref" reference="eq:Bparea"} allows us to improve the bound in Corollary [Corollary 55](#cor:LlogL){reference-type="ref" reference="cor:LlogL"} for principal homeomorphisms of Definition [Definition 37](#prmap){reference-type="ref" reference="prmap"}. **Corollary 64**. *Let $f$ be a $\mathrm{W}^{1,2}_\textup{loc}$-principal mapping, with the linear asymptotics $A_f(z) = z + b_1 \overline{z}$. Then $$\label{Ffunc} \int_{\mathbb{D}} \left( {\mathscr F}(\textup{D}f(z)) - {\mathscr F}(A_f) \right) \mathrm{d}m(z) \geq \left(1- \frac{{\mathscr F}(A_f)}{\det(A_f)}\right)\int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \mathrm{d}m(z).$$* *Proof.* Let us begin by assuming that $f$ is a smooth diffeomorphism. The classical area formula, cf. [\[eq:area\]](#eq:area){reference-type="eqref" reference="eq:area"} and [\[areafmla\]](#areafmla){reference-type="eqref" reference="areafmla"}, can be written in the form $$\label{eq:areaB2} \int_{\mathbb{D}} \left({\bf B}_2(\textup{D}f) - {\bf B}_2(A_f) \right) \mathrm{d}m(z) = \int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \,\mathrm{d}m(z),$$ since ${\bf B}_2=-\det$. By [\[eq:Bparea\]](#eq:Bparea){reference-type="eqref" reference="eq:Bparea"} and [\[eq:areaB2\]](#eq:areaB2){reference-type="eqref" reference="eq:areaB2"}, for $p>2$ sufficiently close to 2 we have that $f$ is a principal $\frac{p}{p-2}$-quasiconformal map and so $$\begin{aligned} \begin{split} \label{eq:auxareaF} 0 & \leq \int_{\mathbb{D}} \left({\bf B}_p(\textup{D}f) - {\bf B}_p(A_f) \right) \mathrm{d}m(z) + \frac{p}{2} \frac{{\bf B}_p(A_f)}{\det(A_f)} \int_{\mathbb{C}\setminus \mathbb{D}} |\phi'|^2 \, \mathrm{d}m(z)\\ & = \int_{\mathbb{D}} \left({\bf B}_p(\textup{D}f) - {\bf B}_2(\textup{D}f)\right) \,\mathrm{d}m(z) - \int_{\mathbb{D}} ({\bf B}_p(A_f) - {\bf B}_2(A_f))\, \mathrm{d}m(z)\\ & \qquad + \left(1+ \frac{p}{2} \frac{{\bf B}_p(A_f)}{\det(A_f)}\right) \int_{\mathbb{C}\setminus \mathbb{D}} |\phi'(z)|^2 \,\mathrm{d}m(z). \end{split}\end{aligned}$$ Recalling that $$\frac{p}{p-2} \left[ {\bf B}_p(A) - {\bf B}_2(A)\right] \, = \, { \mathscr F}(A) + {\mathcal O}(p-2),$$ multiplying the above estimate by $\frac{p}{p-2}$ and taking the limit $p \searrow 2$, since $$\frac{p}{p-2}\left(1+ \frac{p}{2} \frac{{\bf B}_p(A_f)}{\det(A_f)}\right)= \frac{p}{2 \det(A_f)} \left(\frac{p}{p-2} [{\bf B}_p(A_f)-{\bf B}_2 (A_f)]\right)- \frac p 2,$$ we obtain the desired inequality. The case of general orientation-preserving $\mathrm{W}^{1,2}$ maps follows as in the proof of Corollary [Corollary 55](#cor:LlogL){reference-type="ref" reference="cor:LlogL"}. ◻ The reader can easily check that [\[Ffunc\]](#Ffunc){reference-type="eqref" reference="Ffunc"} remains true under the scalings $f \mapsto t f$. Also, writing $A(z)=a_+z +a_-\bar z$ we compute $$1-\frac{ { \mathscr F}(A) }{\det(A)} = 1 - \frac{2 |a_-|}{|a_+|-|a_-|} + 2 \log(|a_+| + |a_-|) \equiv c_p(|a_+|, |a_-|).$$ The sign of $c_p$ is not constant in the set $\{|a_+|>|a_-|\}$: for instance, the reader can verify that $c_p(1,|a_-|) > 0$ for $|a_-|$ small while $c_p(1,|a_-|)<0$ when $|a_-|$ is close to 1. We next prove Theorem [Theorem 9](#thm:Wareaintro){reference-type="ref" reference="thm:Wareaintro"}, which is a version of the area inequality for $\mathscr W$. This result will be very useful in Section [12](#sec:swlsc){reference-type="ref" reference="sec:swlsc"} since, in combination with Theorem [Theorem 41](#thm:intdistort){reference-type="ref" reference="thm:intdistort"}, it shows that one can test quasiconvexity of $\mathscr W$ with an appropriate class of gradient Young measures. **Theorem 65**. *Let $f\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ be a homeomorphism with $K_f\in \mathrm{L}^1(\mathbb D)$ and suppose that $f$ is conformal outside $\mathbb D$ with expansion $$\label{princip3} f(z) = z + \frac{b_1}{z} + %\phi(z), \qquad \phi(z)= \sum_{j=2}^\infty \frac {b_j}{z^j}, \qquad |z| > 1.$$ Letting $A_f\in \mathbb{R}^{2\times 2}$ be given by $A_f(z) = z + b_1 \bar z$, we have $$\mathscr W(A_f)\leq \fint_{\mathbb D} \mathscr W(\textup{D}f(z)) \, \mathrm{d}m(z).$$* Here recall from [\[asympto\]](#asympto){reference-type="eqref" reference="asympto"} that for any principal homeomorphism $\det(A_f) > 0$. In fact, under the normalisation [\[princip3\]](#princip3){reference-type="eqref" reference="princip3"} we have $1 \leq \mathscr W(A_f) < +\infty$. The proof of the above Theorem follows the same broad strategy as for the Burkholder area inequality in Theorem [Theorem 51](#thm:Bparea){reference-type="ref" reference="thm:Bparea"}, but some of the details are quite different. As before, the first step is to establish a quasiconvexity inequality over the full space, as in Lemma [Lemma 53](#lemma:globalqcBp){reference-type="ref" reference="lemma:globalqcBp"}. **Lemma 66**. *Given $f$ as in Theorem [Theorem 65](#thm:Warea){reference-type="ref" reference="thm:Warea"}, define the auxiliary function $\tilde f$ as in Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"}. Then we have $$0 \leq \int_{\mathbb{C}} \left(\mathscr W(\textup{D}\tilde f(z)) - \mathscr W(A_f) \right) \mathrm{d}m(z).$$* *Proof.* Like in the proof of Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"}, define for $|z|>1$ a new function $\psi(z)$ by the identity $$\label{tilde2} \tilde f(z)=h(A_f(z))=A_f(z) + \mathcal O(A_f(z)^{-2})\equiv A_f(z) + \psi(z).$$ Let also $0\leq \eta_j\leq 1$ be a smooth, radially symmetric cutoff such that $\eta_j(z)= 1$ if $|z|\leq j$, $\eta_j(z)=0$ if $|z|\geq j+1$ and $|\nabla \eta_j|\leq 2$. As in the proof of Lemma [Lemma 53](#lemma:globalqcBp){reference-type="ref" reference="lemma:globalqcBp"} we have the basic estimate, that for any given $\varepsilon>0$ and for all $j$ sufficiently large, depending on $\varepsilon$, $$\label{eq:decaycutoff} |\textup{D}(\eta_j \psi)(z)| \leq |\eta_j \textup{D}\psi|(z)+ |\psi \otimes \nabla \eta_j|(z) \leq C |z|^{-2} \leq \varepsilon$$ whenever $|z|\geq j$. With these we set $$\tilde f_j =\begin{cases} f & \textup{in } \mathbb D,\\ A_f+\eta_j \psi & \textup{in } \mathbb{C}\backslash \mathbb D.\end{cases}$$ Our first task is to show that for $j$ large enough the distortion functions $K(\tilde f_j) \in \mathrm{L}^1_\textup{loc}(\mathbb{C})$. Indeed, in the unit disc $K(\tilde f_j) = K(f) \in \mathrm{L}^1(\mathbb{D})$ by assumption, while by [\[tilde2\]](#tilde2){reference-type="eqref" reference="tilde2"} we have $K(\tilde f_j) = K(A_f)$ when $1 \leq |z| \leq j$ or $|z| \geq j+1$. Hence we only need to cover the annulus $\{ z : j < |z| < j+1 \}$, where by [\[eq:decaycutoff\]](#eq:decaycutoff){reference-type="eqref" reference="eq:decaycutoff"} $$K(\tilde f_j) = \frac{\bigl| A_f + \textup{D}(\eta_j \psi) \bigr|^2}{\, \det\bigl(A_f + \textup{D}(\eta_j \psi)\bigr)\,} \leq 2K(A_f ),$$ when $j = j_\varepsilon$ is large enough. Thus $K(\tilde f_j(z)) \in \mathrm{L}^1_\textup{loc}(\mathbb{C})$, and since $\tilde f_j(z)=A_f$ for $|z|\geq j+1$, we may now apply Corollary [Corollary 61](#cor:W){reference-type="ref" reference="cor:W"} to conclude that $$\begin{aligned} \int_{\mathbb{C}} \left(\mathscr W(\textup{D}\tilde f_j(z)) - \mathscr W(A_f)\right) \mathrm{d}m(z)\geqslant 0 \end{aligned}$$ or, rearranging, $$\int_{|z|\leq j} \left(\mathscr W(\textup{D}\tilde f) - \mathscr W(A_f)\right) \mathrm{d}m + \int_{j\leq |z|\leq j+1}\left( \mathscr W(\textup{D}\tilde f_j) - \mathscr W(A_f)\right) \mathrm{d}m\geqslant 0.$$ Our claim is that, when $j\to \infty$, the second term above vanishes. For this we need quantitative estimates. Recalling that $$\label{Wdecom} \mathscr W(\textup{D}f) = K_f - \log K_f + \log J_f ,$$ we will estimate each term separately. First, since $$|\det A_f- \det(A_f+\textup{D}(\eta_j \psi))| \leq C |\textup{D}(\eta_j \psi)| (|A_f|+|\textup{D}(\eta_j \psi)|),$$by taking $\varepsilon$ small enough we have $$\label{eq:lowerboundjac} \frac 1 2\det A_f\leq \det(A_f+\textup{D}(\eta_j \psi)) \quad \textup{for all } |z|\geq j$$ Second, given matrices $A, B \in \mathbb{R}^{2\times 2}_+$, we have the Lipschitz estimate $$\begin{aligned} |K_A- K_B| & = \left|\frac{\det B (|A|^2-|B|^2) + |B|^2 (\det B - \det A)}{\det A \det B}\right|\\ & \leq \frac{3|B|^2(|A|+|B|)}{\det A \det B} \left||A|-|B|\right|.\end{aligned}$$ Thus, applying this estimate with $A=A_f$ and $B=\textup{D}\tilde f_j=A_f+\textup{D}(\eta_j \psi)$, by [\[eq:decaycutoff\]](#eq:decaycutoff){reference-type="eqref" reference="eq:decaycutoff"} and [\[eq:lowerboundjac\]](#eq:lowerboundjac){reference-type="eqref" reference="eq:lowerboundjac"} we find $$\label{eq:estimateK} |K_{A_f} - K_{f_j}| \leq C(A_f) j^{-2}.$$ For the second term in [\[Wdecom\]](#Wdecom){reference-type="eqref" reference="Wdecom"} we use the estimate $|\log x - \log y|\leq c^{-1} |x-y|$ which holds provided that $0<c\leq x,y$. Thus, using [\[eq:estimateK\]](#eq:estimateK){reference-type="eqref" reference="eq:estimateK"} twice, we get $$\label{eq:estimatelogK} |\log K_{A_f} - \log K_{f_j} | \leq C(K_{A_f}) |K_{A_f} - K_{f_j} | \leq C(A_f) j^{-2}.$$ Finally, for the last term in [\[Wdecom\]](#Wdecom){reference-type="eqref" reference="Wdecom"} we have $$\begin{aligned} \begin{split} \label{eq:estimatelogJ} |\log J_{A_f} - \log J_{f_j} | & \leq C(A_f) |J_{A_f} - J_{f_j}| \\ & \leq C(A_f) |A_f-\textup{D}f_j| \leq C(A_f) j^{-2}, \end{split}\end{aligned}$$ where the constant $C(A_f)$ changes in each inequality. Combining [\[eq:estimateK\]](#eq:estimateK){reference-type="eqref" reference="eq:estimateK"}--[\[eq:estimatelogJ\]](#eq:estimatelogJ){reference-type="eqref" reference="eq:estimatelogJ"}, since $\mathscr L^2(\{j\leq |z|\leq j+1\}) \leq C j$, we finally obtain $$\begin{aligned} \left|\int_{j\leq |z|\leq j+1} \left(\mathscr W(\textup{D}\tilde f_j(z)) - \mathscr W(A_f)\right)\, \mathrm{d}m(z) \right| \leq \frac{C(p,A_f)}{j} \to 0,\end{aligned}$$ as claimed. Therefore, letting $j\to\infty$, we have $$\begin{aligned} \int_{\mathbb C} \left( \mathscr W(\textup{D}\tilde f(z)) - \mathscr W(A_f) \right) \mathrm{d}m(z) \geq 0.\end{aligned}$$ As in the proof of Lemma [Lemma 53](#lemma:globalqcBp){reference-type="ref" reference="lemma:globalqcBp"}, note that this is indeed a Lebesgue integral: as $|z|\to \infty$, with $h'(z) = 1+ \mathcal O(z^{-3})$ we see from [\[tilde2\]](#tilde2){reference-type="eqref" reference="tilde2"} that $$\mathscr W(\textup{D}\tilde f) - \mathscr W(A_f) = \log( |h'\circ A_f|^2 )= \mathcal O(z^{-3}),$$ which is integrable. ◻ Continuing with the proof of Theorem [Theorem 65](#thm:Warea){reference-type="ref" reference="thm:Warea"}, instead of Lemma [Lemma 54](#lemma:convexity){reference-type="ref" reference="lemma:convexity"} we now rely on the following well-known fact, whose proof we recall for the convenience of the reader: **Lemma 67**. *Let ${\mathscr E} \subset \mathbb{C}$ be an ellipse and suppose $u \in L^{1} (\mathbb{C}\setminus {\mathscr E})$ is holomorphic in $\mathbb{C}\setminus {\mathscr E}$. Then $$\int_{\mathbb{C}\setminus {\mathscr E}} u(z) \,\mathrm{d}m(z) = 0.$$* *Proof.* We may assume that $\Omega \equiv \mathbb{C}\setminus {\mathscr E} = R(\mathbb{C}\setminus \mathbb{D})$, where $R(z)= z + c/z$ and $|c| < 1$. Since $u$ is Lebesgue-integrable and holomorphic in $\Omega$, we have $u(z) = {\mathcal O}\left(\frac{1}{z^3} \right)$ as $|z| \to \infty$. Now $\psi(w)\equiv R(1/w)$ is a conformal map from $\mathbb{D}$ to $\Omega$, and thus by Stokes' theorem and the Residue Theorem we have $$\begin{aligned} \int_{\Omega} u(z) \, \mathrm{d}m(z) & = \frac{1}{2i} \int_{\partial \Omega} u(z) \, \overline{z} \, \mathrm{d}z \\ & = \frac{1}{2i} \int_{\mathbb S^1} (u \circ \psi)(w) \, \left( w + \frac{\; \overline{ c }\, }{w} \, \right) \, \left( c - \frac{1}{w^2} \right) \mathrm{d}w = 0,\end{aligned}$$ since at origin $u \circ \psi$ has a zero of order at least $3$. ◻ Lemma [Lemma 67](#lemma:nullquaddomain){reference-type="ref" reference="lemma:nullquaddomain"} asserts that the complement of an ellipse is a *null quadrature domain* [@GS]; such domains have been completely classified in the plane by Sakai [@Sakai] and very recently in [@Eberle] in higher dimensions. We can finally proceed to the proof of the main result. *Proof of Theorems [Theorem 65](#thm:Warea){reference-type="ref" reference="thm:Warea"} and [Theorem 9](#thm:Wareaintro){reference-type="ref" reference="thm:Wareaintro"}.* By Lemma [Lemma 66](#lemma:globalqcW){reference-type="ref" reference="lemma:globalqcW"}, recalling the definition of $\tilde f$ from Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"}, we have $$\int_{\mathbb D} \Big(\mathscr W(\textup{D}f(z)) - \mathscr W(A_f) \Big) \mathrm{d}m(z) \geq -\int_{\mathbb{C}\backslash \mathbb{D}}\left( \mathscr W(\textup{D}\tilde f(z)) - \mathscr W(A_f) \right) \mathrm{d}m(z).$$ We claim that the right-hand side of this inequality vanishes, and so the conclusion will follow. Indeed, in $\mathbb{C}\setminus \mathbb{D}$ we have $\tilde f = h \circ A_f$, where $h$ is holomorphic, and thus $$\mathscr W(\textup{D}\tilde f) - \mathscr W(A_f) = (\log J_h) \circ A_f \quad {\rm in} \; \mathbb{C}\setminus \mathbb{D}.$$ A change of variables then gives $$\int_{\mathbb{C}\setminus \mathbb{D}} (\log J_h) \circ A_f(z) \, \mathrm{d}m(z) = \frac{1}{\det(A_f)} \int_{\mathbb{C}\setminus A_f (\mathbb{D})} \log J_h(w)\, \mathrm{d}m(w)$$ Here $A_f(\mathbb{D})$ is an ellipse and $h$ is a conformal map in $\mathbb{C}\setminus A_f (\mathbb{D})$, while Lemma [Lemma 52](#lemma:extension){reference-type="ref" reference="lemma:extension"} gives the decay $h'(z) = 1 + {\mathcal O}\left(\frac{1}{z^3} \right)$ as $|z| \to \infty$. It follows that $\log h'(z)$ is analytic in $\mathbb{C}\setminus A_f (\mathbb{D})$ with the decay $$\log h'(z) = {\mathcal O}\left(\frac{1}{z^3} \right) \quad \textup{ as } |z| \to \infty.$$ By Lemma [Lemma 67](#lemma:nullquaddomain){reference-type="ref" reference="lemma:nullquaddomain"}, the integrals of $\log h'(z)$ and of $\log J_h = 2 \operatorname{Re}\log h'(z)$ over $\mathbb{C}\setminus A_f(\mathbb{D})$ vanish, which completes the proof for Theorem [Theorem 9](#thm:Wareaintro){reference-type="ref" reference="thm:Wareaintro"}, as well as for Theorem [Theorem 65](#thm:Warea){reference-type="ref" reference="thm:Warea"}, which is a reformulation of it. ◻ # The additive volumetric-isochoric split {#sec:advolum} In this section we consider general functionals satisfying the so-called *additive volumetric-isochoric split*, that is, we consider functionals defined on $\mathbb{R}^{2\times 2}_+\equiv \{A\in \mathbb{R}^{2\times 2}: \det A>0\}$ which have the form $$\label{eq:split} {\bf E}(A)={\bf G}(\det A)+{\bf H}(K_A), \qquad K_A\equiv \frac{|A|^2}{\det A},$$ where ${\bf G}\colon (0,+\infty)\to \mathbb{R}$ and ${\bf H}\colon [1,+\infty)\to \mathbb{R}$ are given functions. The term ${\bf G}$ corresponds to the *volumetric* part of ${\bf E}$, while ${\bf H}$ represents the *isochoric* part of ${\bf E}$; note also that ${\bf H}$ is invariant under the left- and right-actions of the conformal group $Q_2(1)$. Of course, any functional as in [\[eq:split\]](#eq:split){reference-type="eqref" reference="eq:split"} extends naturally to an functional ${\bf E}\colon \mathbb{R}^{2\times 2}\to \mathbb{R}\cup \{+\infty\}$ by setting ${\bf E}(A)=+\infty$ if $\det A\leq 0$, cf.  the discussion in Section [8](#sec:LlogL){reference-type="ref" reference="sec:LlogL"}. We also have that $\mathscr W$, as defined in [\[W\]](#W){reference-type="eqref" reference="W"}, can be written in the form [\[eq:split\]](#eq:split){reference-type="eqref" reference="eq:split"} by taking $${\bf G}(t)=\log(t),\quad {\bf H}(t)=t-\log t.$$ In addition to rank-one convexity of ${\bf E}$, we will assume that the isochoric part ${\bf H}$ is *convex*: **Theorem 68**. *Let ${\bf E}\colon\mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be a rank-one convex functional of the form [\[eq:split\]](#eq:split){reference-type="eqref" reference="eq:split"}, where ${\bf G}\colon (0,\infty)\to \mathbb{R}$ and ${\bf H}\colon [1,+\infty)\to \mathbb{R}$ is convex. Then there is a polyconvex functional ${\bf F}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ and a constant $c\geq 0$ such that $${\bf E}={\bf F}+c \mathscr W.$$* We recall that a functional ${\bf F}\colon \mathbb{R}^{2\times 2}\to \overline{\mathbb{R}}$ is said to be *polyconvex* if there is a convex function $\tilde{\bf F}\colon \mathbb{R}^5\to \overline{\mathbb{R}}$ such that ${\bf F}(A)=\tilde {\bf F}(A, \det(A))$, see also [@Dacorogna2007] for further details. Since the determinant is a null Lagrangian, Jensen's inequality easily implies that polyconvex functionals are quasiconvex. Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"} was proved implicitly in [@Voss1] and in this section we give a short, direct proof. Combining Corollary [Corollary 61](#cor:W){reference-type="ref" reference="cor:W"} and Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"}, we obtain: **Corollary 69**. *Any functional ${\bf E}\colon \mathbb{R}^{2\times 2}\to \mathbb{R}$ as in Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"} is quasiconvex: if $A\in \mathbb{R}^{2\times 2}_+$ and if $f\in A+\mathrm{W}^{1,1}_0(\Omega,\mathbb{R}^2)$ is a homeomorphism such that $K_f\in \mathrm{L}^1(\Omega)$, then $${\bf E}(A)\leq \fint_\Omega^* {\bf E}(\textup{D}f(z)) \,\mathrm{d}m(z).$$* The proof of Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"} relies on the classical Baker--Ericksen inequality. Given $A\in \mathbb{R}^{2\times 2}$, we write $\lambda(A)\equiv (\lambda_1(A),\lambda_2(A))$ for the vector of *singular values* of $A$, which is are the eigenvalues of the positive-definite matrix $\sqrt{A^\textup{T} A}$. The Baker--Ericksen inequality read as follows: **Lemma 70**. *Let ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be an isotropic rank-one convex functional: thus there is a symmetric function $\Phi\colon (0,\infty)^2 \to \mathbb{R}$ such that $${\bf E}(A)=\Phi(\lambda_1(A), \lambda_2(A)).$$ If $\Phi$ is $C^1$ and $\lambda_1\neq \lambda_2$ then $$\frac{\lambda_1 \partial_1 \Phi(\lambda) - \lambda_2 \partial_2 \Phi(\lambda)}{\lambda_1-\lambda_2}\geq 0$$ for all $\lambda=(\lambda_1,\lambda_2)\in \mathbb{R}^2$ such that $\lambda_1,\lambda_2>0$.* Lemma [Lemma 70](#lemma:BE){reference-type="ref" reference="lemma:BE"} is well-known and the interested reader can find a short proof for instance in [@GK Proposition 3.2]. We will also require the following result: **Lemma 71**. *Let ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be a rank-one convex functional with the representation [\[eq:split\]](#eq:split){reference-type="eqref" reference="eq:split"}.* 1. *[\[it:h=0\]]{#it:h=0 label="it:h=0"} If ${\bf H}=0$ then ${\bf G}\colon (0,\infty)\to \mathbb{R}$ is convex and ${\bf E}$ is polyconvex.* 2. *[\[it:g=0\]]{#it:g=0 label="it:g=0"} If ${\bf G}=0$ then ${\bf H}\colon [1,\infty)\to \mathbb{R}$ is convex and non-decreasing, and ${\bf E}$ is polyconvex.* The first claim in Lemma [Lemma 71](#trivlemma){reference-type="ref" reference="trivlemma"} is classical, see e.g. [@Dacorogna2007 Theorem 5.46]. The result in Lemma [Lemma 71](#trivlemma){reference-type="ref" reference="trivlemma"}[\[it:g=0\]](#it:g=0){reference-type="eqref" reference="it:g=0"} is not difficult to obtain, see e.g. [@Neff2017]. Here we present a short proof for the sake of completeness. The crucial point is the easily-checked fact that $A\mapsto K_A$ is a polyconvex functional. We refer the reader to [@IM] for a systematic study of polyconvexity properties of distortion functions in higher dimensions. *Proof of Lemma [Lemma 71](#trivlemma){reference-type="ref" reference="trivlemma"}([\[it:g=0\]](#it:g=0){reference-type="ref" reference="it:g=0"}).* Since ${\bf E}$ is rank-one convex, for $\lambda_1\geq 1$ we have that $\lambda_1\mapsto {\bf E}(\textup{diag}(\lambda_1,1))={\bf H}(\lambda_1)$ is convex. To prove the monotonicity, fix $1\leq s <t$ and let $\theta\in (0,1)$ be such that $\theta t + (1-\theta) t^{-1} =s$. Thus, by rank-one convexity, $$\begin{aligned} {\bf H}(s) & ={\bf E}(\textup{diag}(s,1))\\ &\leq \theta {\bf E}(\textup{diag}(t,1))+(1-\theta){\bf E}(\textup{diag}(t^{-1},1))\\ & =\theta {\bf H}(t) + (1-\theta){\bf H}(t)= {\bf H}(t)\end{aligned}$$ and hence ${\bf H}$ has the claimed properties. Since ${\bf H}$ is non-decreasing and convex, and $K_A$ is polyconvex, it follows that ${\bf E}(A)={\bf H}(K_A)$ is polyconvex as well. ◻ *Proof of Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"}.* Since polyconvexity and rank-one convexity are preserved under pointwise limits, there is no loss of generality in assuming that both ${\bf G}\colon (0,\infty)\to \mathbb{R}$ and ${\bf H}\colon (1,\infty)\to \mathbb{R}$ are smooth. Note, however, that we do not assume that ${\bf H}$ is smooth up to $t=1$. We consider arbitrary $x>y>0$. Since ${\bf E}$ is rank-one convex, a simple calculation yields $$0\leq x^2 \partial_{xx} {\bf E}(\textup{diag}(x,y))=(x y )^2 {\bf G}''(x y ) + \Big(\frac x y \Big)^2 {\bf H}''(x/y).$$ By changing variables $t=x y , s = x/y$, we deduce the inequality $$\inf_{t>0} t^2 {\bf G}''(t)+\inf_{s>1} s^2 {\bf H}''(s) \equiv G_0+H_0\geq 0.$$ Similarly, with $\Phi(x,y)={\bf E}(\textup{diag}(x,y))$ as in Lemma [Lemma 70](#lemma:BE){reference-type="ref" reference="lemma:BE"}, we calculate $$\frac{ x \partial_x \Phi(x,y)-y\partial_y \Phi(x,y)}{x-y} = \frac{2 x}{y} \frac{{\bf H}'(x/y)}{x-y}$$ and thus the Baker--Ericksen inequality implies the condition $${\bf H}'(t)\geq 0 \quad \textup{ for } t>1.$$ By assumption $H_0\geq 0$. Suppose that $G_0\geq 0$ as well; in this case, both ${\bf H}$ and ${\bf G}$ are convex and Lemma [Lemma 71](#trivlemma){reference-type="ref" reference="trivlemma"} shows that ${\bf E}$, being the sum of two polyconvex functionals, is itself polyconvex, so we may take $c=0$. Hence we now assume that $G_0\leq 0$ and we take $c\equiv -G_0$. We claim that $F\equiv {\bf E}-c \mathscr W$ is polyconvex. In fact, $F$ can be written as $$\begin{aligned} {\bf F}(A)& =[{\bf G}(\det A)- c \log(\det A)]+ [{\bf H}(K_A)- c (K_A-\log K_A)]\\ & \equiv \widetilde {\bf G}(\det A)+\widetilde {\bf H}(K_A)\end{aligned}$$ and we claim that both terms are polyconvex functionals. This will follow from Lemma [Lemma 71](#trivlemma){reference-type="ref" reference="trivlemma"}. That $\widetilde {\bf G}$ is convex follows from the definition of $c$: $$\widetilde {\bf G}''(t)={\bf G}''(t)+c/t^2 \geq 0.$$ Again from the definition of $c$, we have $$\widetilde {\bf H}''(t)={\bf H}''(t)-c/t^2\geq (H_0-c)/t^2\geq 0,$$ so $\widetilde {\bf H}$ is convex. Suppose now that $\widetilde {\bf H}$ is not non-decreasing, so in particular there is $t_0>1$ such that $\widetilde {\bf H}'(t_0)<0$. For $t>1$, since ${\bf H}'(t)\geq 0$, $$\widetilde {\bf H}'(t)={\bf H}'(t)+c(1-1/t)\geq c(1-1/t).$$ The right-hand side vanishes in the limit $t\to 1$; so, by choosing $t$ sufficiently close to $1$, we may suppose that $t<t_0$ and that $\widetilde {\bf H}'(t)\geq \widetilde {\bf H}'(t)/2>\widetilde {\bf H}'(t_0)$. This contradicts the fact that $\widetilde {\bf H}'$ is non-decreasing in $(1,\infty)$, since $\widetilde {\bf H}$ is convex in the same interval. ◻ # Sequential weak lower semicontinuity and minimizers {#sec:swlsc} In this last section we apply the previous results and methods to establish existence of minimisers for the Burkholder functionals as well as for a quite large class of functionals directly related to Nonlinear Elasticity. All functionals considered here are non-polyconvex. ## Existence of minimizers for the Burkholder energy As usual, throughout this section $\Omega\subset\mathbb{C}$ denotes a bounded domain. It is well known that for functionals with standard growth properties, quasiconvexity is equivalent to sequential weak lower semicontinuity [@AcerbiFusco; @Morrey]. However, the equivalence need not hold for $\overline{\mathbb{R}}$-valued functionals, cf. the discussion in Section [2](#sec:prelims){reference-type="ref" reference="sec:prelims"}. Therefore, in finding minimizers for the Burkholder functional, we start by showing that under natural assumptions the functional is sequentially weakly lower semicontinuous. **Proposition 72**. *Let $K\geq 1$ and fix $2\leq p \leq \frac{2K}{K-1}$. Given a sequence $(f_j)\subset \mathrm{W}^{1,p}(\Omega)$ of $K$-quasiregular maps such that $f_j\rightharpoonup f$ in $\mathrm{W}^{1,p}(\Omega)$ and $({\bf B}_p(\textup{D}f_j))$ is equiintegrable, we have $$\liminf_{j\to \infty} \int_\Omega {\bf B}_p(\textup{D}f_j(z))\, \mathrm{d}m(z) \geq \int_\Omega {\bf B}_p(\textup{D}f(z))\,\mathrm{d}m(z).$$* *Proof.* Theorem [Theorem 18](#thm:fundYM){reference-type="ref" reference="thm:fundYM"} shows that up to a subsequence, which we do not relabel, $(\textup{D}f_j)$ generates a $\mathrm{W}^{1,p}$-gradient Young measure $(\nu_z)_{z\in \Omega}$. Since by hypothesis $({\bf B}_p(\textup{D}f_j))$ is equiintegrable, we have $$\begin{aligned} \lim_{j\to \infty} \int_\Omega {\bf B}_p(\textup{D}f_j) \, \mathrm{d}m(z) & = \int_\Omega \int_{\mathbb{R}^{2\times 2}} {\bf B}_p(A) \,\mathrm{d}\nu_z(A) \, \mathrm{d}m(z). %& = \int_\Omega \int_{\R^{2\times 2}} {\bf B}_{K,p}(A) \,\dd \nu_z(A) \, \dd m(z)\end{aligned}$$ Moreover, here $\textup{supp}\,\nu_z \subset Q_2(K)$ a.e., as seen from $$\label{eq:FK} 0 = \int_\Omega {\bf F}_K(\textup{D}f_j) \, \mathrm{d}m(z) \to \int_\Omega \int_{\mathbb{R}^{2\times 2}} {\bf F}_K(A) \, \mathrm{d}\nu_z(A) \, \mathrm{d}m(z),$$ with ${\bf F}_K\equiv \min\{0,K \det(\cdot) - |\cdot|^2\}$. In addition, ${\bf B}_{K,p}={\bf B}_p$ on the $K$-quasiconformal cone $Q_2(K)$, and hence $$\lim_{j\to \infty} \int_\Omega {\bf B}_p(\textup{D}f_j) \, \mathrm{d}m(z) = \int_\Omega \int_{\mathbb{R}^{2\times 2}} {\bf B}_{K,p}(A) \,\mathrm{d}\nu_z(A) \, \mathrm{d}m(z).$$ Continuing the calculation, by Proposition [\[prop:homogenization\]](#prop:homogenization){reference-type="ref" reference="prop:homogenization"} we have $\nu_z\in \mathscr M^p_\textup{qc}$ for a.e. $z$ and thus, by Theorem [Theorem 4](#main){reference-type="ref" reference="main"}, we find $$\int_{\mathbb{R}^{2\times 2}} {\bf B}_{K,p}(A) \, \mathrm{d}\nu_z(A) \geq {\bf B}_{K,p}(\langle \nu_z, \mathrm{Id}\rangle) \quad \textup{ for a.e.\ } z\in \Omega.$$ Finally via [\[limit\]](#limit){reference-type="eqref" reference="limit"} this yields $$\lim_{j\to \infty} \int_\Omega {\bf B}_p(\textup{D}f_j) \, \mathrm{d}m(z) \geq \int_\Omega {\bf B}_p(\langle \nu_z, \mathrm{Id}\rangle) \, \mathrm{d}m(z) = \int_\Omega {\bf B}_p(\textup{D}f) \, \mathrm{d}m(z),$$ as wished. ◻ Proposition [Proposition 72](#prop:lscBp){reference-type="ref" reference="prop:lscBp"} gives us the tools to prove existence of minimizers for the ${\bf B}_p$-energy in suitable Dirichlet classes, as stated in Corollary [Corollary 5](#cor:Bpminims){reference-type="ref" reference="cor:Bpminims"}: *Proof of Corollary [Corollary 5](#cor:Bpminims){reference-type="ref" reference="cor:Bpminims"}.* The main point is that each $K$-quasiregular map $f \in g + \mathrm{W}^{1,p}_0(\Omega,\mathbb{C})$ admits the uniform bound $$\label{eq:sobouniformf}||f||_{\mathrm{W}^{1,q}(\Omega)} \leq C(\Omega,q,K,g)$$ for all $p \leq q < \frac{2K}{K-1}$. Indeed, we extend $f$ as a $K$-quasiregular mapping of the whole plane by setting $f \equiv g$ in $\mathbb{C}\setminus \Omega$, and then we use the higher integrability of quasiregular mappings [@AIM Corollary 13.2.5]. The corollary now follows by the Direct Method of the Calculus of Variations. Indeed, take a sequence $(f_j)\subset g + \mathrm{W}^{1,p}_0(\Omega)$ of $K$-quasiregular maps such that $$\int_\Omega {\bf B}_p(\textup{D}f_j) \, \mathrm{d}m \to \inf\left\{\int_\Omega {\bf B}_p(\textup{D}h) \, \mathrm{d}m: h\in g + \mathrm{W}^{1,p}_0(\Omega) \textup{ is } K\textup{-quasiregular}\right\}.$$ By [\[eq:sobouniformf\]](#eq:sobouniformf){reference-type="eqref" reference="eq:sobouniformf"} we may assume that $f_j \rightharpoonup f$ in $\mathrm{W}^{1,p}(\Omega)$ for some map $f$ which is then automatically $K$-quasiregular. Moreover, [\[eq:sobouniformf\]](#eq:sobouniformf){reference-type="eqref" reference="eq:sobouniformf"} also shows that $|\textup{D}f_j|^p$ is equiintegrable, hence Proposition [Proposition 72](#prop:lscBp){reference-type="ref" reference="prop:lscBp"} applies and gives $$\liminf_{j\to \infty} \int_\Omega {\bf B}_p(\textup{D}f_j(z)) \, \mathrm{d}m(z) \geq \int_\Omega {\bf B}_p(\textup{D}f(z)) \, \mathrm{d}m(z).$$ It follows that $f$ is a minimizer, as claimed. ◻ ## Sequential weak lower-semicontinuity of $\mathscr W$ Our next goal is to prove Theorem [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"}, which is an analogue of Proposition [Proposition 72](#prop:lscBp){reference-type="ref" reference="prop:lscBp"} for the functional $\mathscr W$ introduced in [\[W\]](#W){reference-type="eqref" reference="W"}. Before proceeding with the main part of the proof, we need the following result, which is essentially proved in [@KOR]: **Proposition 73**. *Let $g\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ be a homeomorphism with $K_g\in \mathrm{L}^q_\textup{loc}$ for $q\geq 1$. If $f\in g + \mathrm{W}^{1,1}_0(\Omega)$ is a homeomorphism such that $K_f\in \mathrm{L}^q(\Omega),$ then $$\int_\Omega \log^q\left(e+\frac{1}{J_f(z)}\right) \, \mathrm{d}m(z) \leq C(q,g,\Omega) \left(1+\int_\Omega K_f(z)^q \, \mathrm{d}m(z)\right).$$* *Proof.* In [@KOR] the authors only prove a local estimate; however setting $f(x) =g(x)$ on $\mathbb{C}\setminus \Omega$ defines a global mapping and reduces the required bound to the estimate in [@KOR]. ◻ As a quick consequence of Proposition [Proposition 73](#prop:KOR){reference-type="ref" reference="prop:KOR"} we have **Lemma 74**. *Let $(f_j)\subset \mathrm{W}^{1,1}(\Omega)$ be a sequence of homeomorphisms that $\sup_j \|K_{f_j}\|_{\mathrm{L}^q(\Omega)}<\infty$ for some $q>1$. Suppose that either* 1. *[\[it:trace\]]{#it:trace label="it:trace"} $f_j=g$ on $\partial\Omega$ for a homeomorphism $g\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ with $K_g \in \mathrm{L}^q_\textup{loc}$, or* 2. *[\[it:principal\]]{#it:principal label="it:principal"} $\Omega=\mathbb D$ and $f_j$ are principal maps.* *Then $(\mathscr W(\textup{D}f_j))$ is equiintegrable.* *Proof.* Since $K_{f_j}\geq 1$ we have $K_{f_j} - \log K_{f_j} \leq 2 K_{f_j}$ which is equiintegrable by assumption. To deal with the Jacobian term in $\mathscr W$, we use the pointwise estimate $$\label{eq:elementaryestimatelog} |\log(J_{f_j})| \leq \log\left(e+\frac{1}{J_{f_j}}\right) + 2(J_{f_j})^{\frac 1 2}.$$ The first term is clearly equiintegrable by Proposition [Proposition 73](#prop:KOR){reference-type="ref" reference="prop:KOR"}. Given a measurable set $U\subset \Omega$, we have $$\int_U (J_{f_j})^{\frac 1 2}\, \mathrm{d}m(z) \leq \mathscr L^2(U)^{\frac 1 2} \left(\int_{\Omega} J_{f_j} \, \mathrm{d}m(z)\right)^{\frac 1 2} \leq C \mathscr L^2(U)^{\frac 1 2},$$ where $C=\mathscr L^2(g(\Omega))$ in case ([\[it:trace\]](#it:trace){reference-type="ref" reference="it:trace"}) and $C=\mathscr L^2(\mathbb D(0,2))$ in case ([\[it:principal\]](#it:principal){reference-type="ref" reference="it:principal"}), by the area formula. Thus the second term is also equiintegrable. ◻ *Remark 75*. A small variant of the above proof, combined with Proposition [Proposition 73](#prop:KOR){reference-type="ref" reference="prop:KOR"}, gives the estimate $$\int_\Omega |\log(J_f(z))|^q \, \mathrm{d}m(z) \leq C(g,\Omega)\left(1+\int_\Omega K_f(z)^q\, \mathrm{d}m(z)\right)$$ for homeomorphisms $f\in g + \mathrm{W}^{1,1}_0(\Omega)$ such that $K_f\in \mathrm{L}^q(\Omega)$, where $q\geq 1$. For the proof of Theorem [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"} we also need some closed-quasiconvexity features for the $\mathscr W$-functional. **Proposition 76**. *Let $\nu \in \mathscr M^{2}_\textup{qc}(\mathbb{R}^{2\times 2}_+)$ be a gradient Young measure generated by a bounded sequence $(\psi_j)\subset \mathrm{W}^{1,{2}}(\mathbb D)$ of homeomorphisms such that $$\|K_{\psi_j}\|_{\mathrm{L}^{q}(\mathbb D)}\leq C,$$ for some $q > 1$. Then $$\mathscr W(\langle \nu, \mathrm{Id}\rangle) \leq \int_{\mathbb{R}^{2\times 2}} \mathscr W(A) \,\mathrm{d}\nu(A).$$* *Proof.* To start with, momentarily assume that in conformal coordinates $$\label{eq:normalizationnuz0} \langle \nu, \mathrm{Id}\rangle = A; \quad A(z) = z + a \overline{z}, \quad {\rm with} \; \; |a|<1.$$ Applying Theorem [Theorem 41](#thm:intdistort){reference-type="ref" reference="thm:intdistort"}, we find a sequence of maps $f_j\colon \mathbb{C}\to \mathbb{C}$ such that: 1. $f_j$ are principal maps; 2. $(f_j)\subset \mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$ is bounded and $(f_j|_{\mathbb D(0,R)})$ generates $\nu$ for all $R<1$; 3. $\psi_j = h_j\circ f_j$ for some conformal maps $h_j\colon f_j(\mathbb D) \to \psi_j(\mathbb{D})$. In particular, we have $K_{f_j}=K_{\psi_j}$ a.e. in $\mathbb D$ and so by Lemma [Lemma 74](#lemma:equiintegrability){reference-type="ref" reference="lemma:equiintegrability"} the sequence $(\mathscr W(\textup{D}f_j))$ is again equiintegrable over $\mathbb D$. Thus $$\begin{aligned} \int_{\mathbb{R}^{2 \times 2}} \mathscr W(A) \,\mathrm{d}\nu(A) &=\lim_{j\to \infty} \fint_{\mathbb D(0,R)} \mathscr W(\textup{D}f_j(z))\,\mathrm{d}m(z)\\ & \to\lim_{j\to \infty} \fint_{\mathbb D} \mathscr W(\textup{D}f_j(z))\,\mathrm{d}m(z) \geqslant\lim_{j\to \infty} \mathscr W(A_{f_j}) = \mathscr W(\langle \nu, \mathrm{Id}\rangle),\end{aligned}$$ as $R\nearrow 1$, where the inequality follows from Theorem [Theorem 65](#thm:Warea){reference-type="ref" reference="thm:Warea"} and the last identity from [\[eq:asympt\]](#eq:asympt){reference-type="eqref" reference="eq:asympt"} and Remark [Remark 28](#affine){reference-type="ref" reference="affine"}. To deal with the general case where we do not have [\[eq:normalizationnuz0\]](#eq:normalizationnuz0){reference-type="eqref" reference="eq:normalizationnuz0"}, note that since $\nu \in \mathscr M^{2}_\textup{qc}(\mathbb{R}^{2\times 2}_+)$ one still has $\det(\langle \nu, \mathrm{Id}\rangle) > 0$. Thus we simply replace $\nu$ by a suitable normalized measure. Namely if $\langle \nu, \mathrm{Id}\rangle = A \in \mathbb{R}^{2\times 2}_+$, choose $t>0$ and $Q\in \textup{SO}(2)$ such that $$t Q A (z) = z + a \bar z, \qquad |a| < 1.$$ Here notice that $\mathscr W$ satisfies $$\mathscr W(QA)=\mathscr W(A) \qquad \textup{and} \qquad \mathscr W(tA)=\mathscr W(A)+\log t^2$$ for all $Q\in \textup{SO}(2)$ and $t>0$. Hence, if in the notation of Lemma [Lemma 26](#lemma:invYM){reference-type="ref" reference="lemma:invYM"} one defines $\mu \equiv \left(\nu_{(Q,\mathrm{Id})}\right)_t$, then $\mu$ satisfies [\[eq:normalizationnuz0\]](#eq:normalizationnuz0){reference-type="eqref" reference="eq:normalizationnuz0"} so that $$\begin{aligned} \langle \nu, \mathscr W\rangle +\log t^2 = \langle \nu, \mathscr W(t\cdot)\rangle = \langle \mu, \mathscr W \rangle \geqslant\mathscr W(\langle \mu,\mathrm{Id}\rangle) = \mathscr W(\langle \nu, \mathrm{Id}\rangle )+ \log t^2.\end{aligned}$$ This completes the proof. ◻ We are now ready for the main result of this Subsection. *Proof of Theorem [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"}.* Given a homeomorphism $g\in \mathrm{W}^{1,2}_\textup{loc}(\mathbb{C})$ and a sequence $(f_j)\subset g + \mathrm{W}^{1,2}_0(\Omega)$ such that $f_j\rightharpoonup f$ in $\mathrm{W}^{1,2}(\Omega)$ and $\|K_{f_j}\|_{\mathrm{L}^{q}(\Omega)} \leq C < \infty$ for some $q>1$, we are to show that $$\label{eq:Wwlsc} \liminf_{j\to \infty} \int_\Omega {\mathscr W}(\textup{D}f_j(z)) \, \mathrm{d}m(z) \geq \int_\Omega {\mathscr W}(\textup{D}f(z))\, \mathrm{d}m(z).$$ For this we follow a similar strategy as in Proposition [Proposition 72](#prop:lscBp){reference-type="ref" reference="prop:lscBp"}. First note that by Remark [Remark 60](#rmk:automatichomeo){reference-type="ref" reference="rmk:automatichomeo"} the sequence $f_j$ consists of homeomorphisms. Also, by passing to subsequences we may assume that $f_j \rightharpoonup f$ in $\mathrm{W}^{1,2}(\Omega)$ and, by Theorem [Theorem 18](#thm:fundYM){reference-type="ref" reference="thm:fundYM"}, that $f_j$ generates the $\mathrm{W}^{1,2}$-gradient Young measure $(\nu_z)_{z\in \Omega}$. Since $(\mathscr W(\textup{D}f_j))$ is equiintegrable by Lemma [Lemma 74](#lemma:equiintegrability){reference-type="ref" reference="lemma:equiintegrability"}, we have $$\label{eq:repW} \lim_{j\to \infty}\int_\Omega \mathscr W(\textup{D}f_j)\,\mathrm{d}m(z)= \int_\Omega \int_{\mathbb{R}^{2\times 2}} \mathscr W(A) \,\mathrm{d}\nu_z(A) \, \mathrm{d}m(z).$$ Moreover, for a.e. $z\in \Omega$ $$\label{eq:conditionsbarycenter} \det \langle \nu_z, \mathrm{Id}\rangle >0\quad \textup{and} \quad \nu_{z}\in \mathscr{M}^2_\textup{qc},$$ where the former claim follows by arguing similarly to [\[eq:FK\]](#eq:FK){reference-type="eqref" reference="eq:FK"}, but with $F_K$ replaced with $\min\{0,\det\}$, and the latter claim by Proposition [\[prop:homogenization\]](#prop:homogenization){reference-type="ref" reference="prop:homogenization"}. Let us then fix a point $z_0\in \Omega$ for which [\[eq:conditionsbarycenter\]](#eq:conditionsbarycenter){reference-type="eqref" reference="eq:conditionsbarycenter"} holds. The measure $\nu_{z_0}$ is generated by taking a diagonal subsequence $(\psi_j)$ of $\psi_{j,\lambda}(z)=\lambda^{-1} f_j(z_0+\lambda z)$, where $j\to \infty$ and $\lambda\to 0$, cf.  Remark [Remark 25](#rmk:homogenization){reference-type="ref" reference="rmk:homogenization"}. In particular, $\psi_j\colon \mathbb D \to \mathbb{C}$ is a sequence of homeomorphisms such that $\sup_j\|K_{\psi_j}\|_{\mathrm{L}^q(\mathbb D)} <\infty$. We can now apply Proposition [Proposition 76](#thm:closedqc.new){reference-type="ref" reference="thm:closedqc.new"} which says that $$\label{Wqconv2} \int_{\mathbb{R}^{2 \times 2}} \mathscr W(A) \,\mathrm{d}\nu_{z_0}(A) \geqslant\mathscr W(\langle \nu_{z_0}, \mathrm{Id}\rangle ).$$ On the other hand, by [\[limit\]](#limit){reference-type="eqref" reference="limit"} we have $\langle \nu_{z}, \mathrm{Id}\rangle = \textup{D}f(z)$ for a.e. $z \in \Omega$. Combining this with [\[eq:repW\]](#eq:repW){reference-type="eqref" reference="eq:repW"} and [\[Wqconv2\]](#Wqconv2){reference-type="eqref" reference="Wqconv2"} completes the proof. ◻ It remains an interesting open question whether [\[eq:Wwlsc\]](#eq:Wwlsc){reference-type="eqref" reference="eq:Wwlsc"} still holds at $q =1$, i.e. under the natural assumption $\|K_{f_j}\|_{\mathrm{L}^{1}(\Omega)} \leq C < \infty$. ## Existence of minimizers in Nonlinear Elasticity {#sec:elasticity} As a last theme, let us collect the previous results to prove existence of minimisers for a quite large family of functionals in Nonlinear Elasticity. We again emphasize that these are non-polyconvex, see Example [Example 79](#nonpoly){reference-type="ref" reference="nonpoly"}. First, as is well-known, polyconvex functionals are lower semicontinuous, see e.g. [@DacorognaMarcellini]. Thus combining Theorems [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"} and [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"} gives: **Corollary 77**. *Let $g\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ be a homeomorphism with $K_g\in \mathrm{L}^q_\textup{loc}(\mathbb{C})$ for some $q>1$ and let ${\bf E}\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$ be a functional as in Theorem [Theorem 68](#thm:decomposition){reference-type="ref" reference="thm:decomposition"}, $${\bf E}(A) = {\bf G}(\det A)+ {\bf H}(K_A),$$ which we assume to be rank-one convex.* *If $f_j \rightharpoonup f$ in $g + \mathrm{W}^{1,2}_0(\Omega)$ and $\sup_j \|K_{f_j}\|_{\mathrm{L}^q(\Omega)}<\infty$ then $$\liminf_{j\to \infty} \int_\Omega {\bf E}(\textup{D}f_j(z)) \, \mathrm{d}m(z) \geq \int_\Omega {\bf E}(\textup{D}f(z)) \, \mathrm{d}m(z).$$* In conclusion, to promote the lower semicontinuity to the existence of minimizers requires now some form of coercivity, and this takes us to the following examples. **Corollary 78**. *Suppose $q > q_0 \geq 1$ with $p\geq 2$, and let $${\bf E}(A)\equiv {\bf G}(\det A)+ {\bf H}(K_A) + |A|^p,$$ where we assume that for some $C>0$,* 1. *$A\mapsto {\bf G}(\det A)+{\bf H}(K_A)$ is rank-one convex;* 2. *$|{\bf G}(t)| \leq C (1+ |\log(t)|^{q_0} )$;* 3. *${\bf H}$ is convex and ${\bf H}(t)\geq t^q/C - C$.* *Then for any homeomorphism $g\in \mathrm{W}^{1,1}_\textup{loc}(\mathbb{C})$ with $K_g\in \mathrm{L}^q_\textup{loc}(\mathbb{C})$ there is a minimizer $f\in \mathrm{W}^{1,p}(\Omega)$ of the problem $$\inf\left\{\int_\Omega {\bf E}(\textup{D}h(z))\, \mathrm{d}m(z) : h \in g + \mathrm{W}^{1,p}_0(\Omega)\right\}.$$ In addition, $f$ is a homeomorphism such that $f^{-1}\in \mathrm{W}^{1,2}(g(\Omega))$.* *Proof.* By our hypothesis we have the lower bound $$\begin{aligned} \int_\Omega {\bf E}(\textup{D}f(z)) \, \mathrm{d}m(z) & \geq \int_\Omega |\textup{D}f(z)|^p \, \mathrm{d}m(z) + \frac{1}{C} \int_\Omega K_f(z)^q \, \mathrm{d}m(z)\\ & \qquad - C \int_\Omega |\log(J_f(z))|^{ q_0} \, \mathrm{d}m(z) - 2C|\Omega| \end{aligned}$$ for any orientation-preserving map $f\in \mathrm{W}^{1,p}_g(\Omega)$. Next, for any $\varepsilon > 0$ and $q > q_0 \geq 1$, we have $C x^{q_0} \leq \varepsilon x^q + M$ where the constant $M = M(\varepsilon,C,q_0/q) < \infty$. With Remark [Remark 75](#rmk:explicitKO){reference-type="ref" reference="rmk:explicitKO"} this gives us $$C \int_\Omega |\log J_f(z)|^{ q_0} \, \mathrm{d}m(z) \leq \frac{1}{2C} \int_\Omega K_f(z)^q \,\mathrm{d}m(z)+ C_1(M, g,\Omega).$$ That is, we have the coercivity $$\label{eq:coercivity} \int_\Omega {\bf E}(\textup{D}f) \, \mathrm{d}m(z) \geq \int_\Omega |\textup{D}f|^p \, \mathrm{d}m(z) + \frac{1}{2C} \int_\Omega K_f(z)^q \,\mathrm{d}m(z) - C_2(M, g,\Omega).$$ The result is now a consequence of the Direct Method: we take a minimizing sequence $(f_j)\subset g + \mathrm{W}^{1,2}_0(\Omega)$ and by the last estimate we have $$\sup_j\|\textup{D}f_j\|_{\mathrm{L}^p(\Omega)} <\infty, \qquad \sup_j\|K_{f_j} \|_{\mathrm{L}^q(\Omega)} <\infty.$$ Since $p\geq 2$, up to a subsequence we have $f_j\rightharpoonup f$ in $\mathrm{W}^{1,2}(\Omega)$ and hence Corollary [Corollary 77](#cor:lscsplit){reference-type="ref" reference="cor:lscsplit"} yields $$\liminf_{j\to \infty} \int_\Omega {\bf E} ( \textup{D}f_j(z)) \, \mathrm{d}m(z) \geq \int_\Omega {\bf E}(\textup{D}f(z)) \, \mathrm{d}m(z),$$ thus $f$ is a minimizer. Since $f\in g + \mathrm{W}^{1,p}_0(\Omega)$ is a map of integrable distortion, we have $f^{-1}\in \mathrm{W}^{1,2}(g(\Omega))$. ◻ The main point in the proof of Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"} is that, once the condition $\int_\Omega {\bf E}(\textup{D}f) \,\mathrm{d}m(z)<\infty$ imposes bounds on the $\mathrm{L}^p$ norms of both $\textup{D}f$ and $K_f$, the existence of minimizers follows from the sequential lower semicontinuity result of Corollary [Corollary 77](#cor:lscsplit){reference-type="ref" reference="cor:lscsplit"}, which in turn is essentially a consequence of Theorem [Theorem 10](#thm:lscW){reference-type="ref" reference="thm:lscW"}. In particular, it is not difficult to write more general versions of Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"}: for instance, to ${\bf E}$ as in Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"} one can add a polyconvex term ${\bf P}\geq 0$, as well as a quasiconvex term ${\bf Q}$ satisfying $0\leq {\bf Q}\leq c(1+|\cdot|^p)$ for some $c \geq 0$, and still obtain existence of minimizers. Instead of pursuing the maximal degree of generality, we give here a simple, concrete example, where we add a neo-Hookean term to $\mathscr W$, and leave more complicated examples to the interested reader: **Example 79**. For $c\geq 0$, consider the functional ${\bf E}_c\colon \mathbb{R}^{2\times 2}_+\to \mathbb{R}$, $${\bf E}_c(A) = \mathscr W(A) + c \left(\frac{1}{\det A} + |A|^2\right)^2.$$ Then the ${\bf E}$-energy admits minimizers, as in Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"}. Moreover, ${\bf E}_c$ is bounded from below and it satisfies [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"}. If $c$ is small enough then ${\bf E}_c$ is non-polyconvex. *Proof of the above claims.* The functional ${\bf E}_c$ satisfies the assumptions of Corollary [Corollary 78](#thm:miniselasticity){reference-type="ref" reference="thm:miniselasticity"}, hence one obtains a minimizer for ${\bf E}_c$ in $\mathrm{W}^{1,2}$. It is not difficult to check that ${\bf E}_c$ is bounded from below; for instance, $$\mathscr W(A)+\frac{c}{(\det A)^2}\geq \frac 3 2 + \log(\sqrt{2 c}).$$ The blow-up condition [\[eq:blowup\]](#eq:blowup){reference-type="eqref" reference="eq:blowup"} is easy to verify. We then show that ${\bf E}_c$ is non-polyconvex for small enough $c$. First, consider points $$A_1 = \textup{diag}(3/10,3/10),\quad A_2=\textup{diag}(2/3, 8), \quad A_3 = \textup{diag}(8,2/3).$$ These points satisfy the so-called minors relations, that is, $$\begin{gathered} \frac{100}{121} A_1 + \frac{21}{242} A_2 + \frac{21}{242} A_3 = \mathrm{Id}, \\ \frac{100}{121} \det A_1 + \frac{21}{242} \det A_2 + \frac{21}{242} \det A_3 = 1,\end{gathered}$$ yet we have $$\frac{100}{121} \mathscr W(A_1) + \frac{21}{242} \mathscr W(A_2 )+ \frac{21}{242} \mathscr W(A_3) \approx 0.78 < 1=\mathscr W(\mathrm{Id}),$$ which in particular shows that $\mathscr W$ is non-polyconvex. Clearly we still have $$\frac{100}{121} \mathscr {\bf E}_c(A_1) + \frac{21}{242} {\bf E}_c(A_2 )+ \frac{21}{242} {\bf E}_c(A_3) < {\bf E}_c(\mathrm{Id}),$$ provided that $c$ is chosen sufficiently small. ◻ *Remark 80*. In the same token, the non-negative functional $$\widetilde{\, \mathscr W \, } (A) = \frac{|A|^2}{\det A} - \log \left( \frac{|A|^2}{\det A}\right) + |\log \det A|$$ from [\[modiW\]](#modiW){reference-type="eqref" reference="modiW"} is quasiconvex but not polyconvex. Indeed, the functional is the sum of $\mathscr W$ and a polyconvex functional, thus quasiconvex by Theorem [Theorem 8](#thm:morreysplit){reference-type="ref" reference="thm:morreysplit"}. Moreover, $\widetilde{\, \mathscr W \, } (A) = \mathscr W(A)$ when $\det(A) > 1$. Hence we can use the above minors relations and multiply the $A_j$'s by a number $t >1$ so that each has determinant $> 1$. As $\mathscr W(t A) = \mathscr W(A) + \log(t^2)$ with $\mathscr W(t \mathrm{Id}) = 1 + \log(t^2)$, the new minors relations with the multiplied matrices show that $\widetilde{\, \mathscr W \,}$ is non-polyconvex. Also, similarly as in Example [Example 79](#nonpoly){reference-type="ref" reference="nonpoly"} considering, say, $\widetilde{\, \mathscr W \, }(A) + c(|A|^2 + K_A^2)$ gives a non-negative and non-polyconvex functional which admits minimizers. Having established the existence of minimizers for the above class of functionals it is natural to inquire about their regularity properties, but this appears to be a very difficult problem. Away from perturbative regimes, almost nothing is known concerning regularity of minimizers in nonlinear elasticity, even for polyconvex functionals, but see [@Bauman; @IKO2] for some interesting results. 9 E. Acerbi, and N. Fusco: *Semicontinuity problems in the calculus of variations.* Arch. Ration. Mech. Anal. **86** (1984), 125--145. 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arxiv_math

--- abstract: | Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to constructing a filtration for cluster analysis via persistent homology. The key advantages of the new filtration is that (a) it provides richer signatures for connected components by introducing non-trivial birth times, and (b) it is robust to outliers. The key idea is that nodes are ignored until they belong to sufficiently large clusters. We demonstrate the computational efficiency of our filtration, its practical effectiveness, and explore into its properties when applied to random graphs. author: - Omer Bobrowski - Primoz Skraba bibliography: - zotero.bib title: Cluster-Persistence for Weighted Graphs --- # Introduction **Clustering** data is a fundamental task in unsupervised machine learning and exploratory data analysis. It has been the subject of countless studies over the last 50 years with many definitions and algorithms proposed, e.g.,  [@jain_data_1999; @mcinnes_umap_2018]. **Persistent homology** [@edelsbrunner_computational_2010; @zomorodian_topology_2005] is a powerful topological tool that provides multi-scale structural information about data. Given an increasing sequence of spaces (filtration), persistent homology tracks the formation of connected components ($0$-dimensional cycles), holes ($1$-dimensional cycles), cavities ($2$-dimensional cycles), and their higher-dimensional extensions. The information encoded in persistent homology is often represented by a *persistence diagram* -- a collection of points in $\mathbb{R}^2$, representing the birth and death of homology classes, and providing an intuitive numerical representation for topological information (see Figure [1](#fig:example_diagram){reference-type="ref" reference="fig:example_diagram"}). The connection between clustering and $0$-dimensional persistent homology has been well-established under a various different scenarios including the relationship with functoriality  [@carlsson_classifying_2013; @carlsson_characterization_2010], and density-based methods [@bobrowski_topological_2017; @chazal_persistence-based_2013]. An important motivating factor for connecting these methods is *stability*. Namely, given small perturbations of the input data, persistent homology and can provide guarantees on the number of the output clusters. One important drawback of this topological approach is that statistical tests for persistent homology and clustering based on persistence, have been lacking. Recently, for persistent homology in dimensions $1$ and above (i.e., excluding connected components), persistent homology based on distance filtrations was experimentally shown to exhibit a strong sense of universal behavior [@bobrowski_universal_2023]. Suppose we are given as input a point-cloud generated by some unknown distribution. If we compute the distance-based persistent homology, under an appropriate transformation, the distribution of persistence values was shown to be independent of the original point-cloud distribution. This phenomenon was then used to develop a statistical test to detect statistically significant homology classes. A key point in [@bobrowski_universal_2023] is that in order to obtain such universal behavior, the measure of persistence is given by the value of $\mathop{\mathrm{death}}/\mathop{\mathrm{birth}}$, which makes the measure of persistence scale-invariant. However, in distance-based filtrations, the $0$-dimensional persistent homology (tracking clusters) does not fit into this universality framework, as the birth time of all the $0$-dimensional homology classes is set to $0$. To address this issue, and to enable the study of universality in the context of clustering, we introduce a new filtration, which we call the $k$-cluster filtration. This is a novel, non-local, construction, where vertices become "alive" only once they belong to a sufficiently large cluster. In other words, while traditional persistent homology considers every vertex as an individual clusters and tracks its evolution, in the $k$-cluster filtration we only consider components of $k$ or more vertices as 'meaningful' clusters. We note that while the motivation for this new filtration is distance-based filtration, the $k$-cluster filtration can be constructed over any weighted graph. It generally provides two key advantages to the traditional filtration. Firstly, it results in a 'richer' persistence diagram, in the sense that components have non-trivial birth times. This improves our ability to compare between the different features within the same diagram, or across different diagrams. Secondly, the $k$-cluster filtration provides a more 'focused' view on connected components, by discarding those that are considered small (determined by application). In particular it easily allows to remove outliers from the persistence diagram. The paper is organized as follows. Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"} provides essential background about persistent homology. In Section [3](#sec:filt){reference-type="ref" reference="sec:filt"} we introduce the $k$-cluster filtration and give some preliminary properties. In Section [4](#sec:alg){reference-type="ref" reference="sec:alg"} we provide an algorithm for computing the filtration and corresponding persistence diagram in a single pass. In Section [5](#sec:exp){reference-type="ref" reference="sec:exp"} we show some experimental results comparing the clustering method to some other approaches. Finally, in Section [6](#sec:prob){reference-type="ref" reference="sec:prob"} we discuss some probabilistic aspects of this filtration, in comparison with known properties of random graphs and simplicial complexes. #### Remark. As we deal exclusively with $0$-dimensional homology, we phrase all the statements in this paper terms of weighted graphs rather than simplicial complexes. # Graph Filtrations and Persistent Homology {#sec:prelim} In this section, we introduce the required topological notions. As we focus on the special case of graphs and connected components, i.e. 0-dimensional homology, we restrict our definitions to this case. For a general description of $k$-dimensional homology we refer the reader to [@hatcher_algebraic_2002; @munkres_elements_1984]. Let $G = (V,E)$ be an undirected graph. Our main object of study is a *graph filtration* or an increasing sequence of graphs. This can be constructed by defining a function $\tau:(V\cup E)\to [0,\infty)$, under the restriction that if $e=(u,v)\in E$, then $\tau(e) \ge \max(\tau(u),\tau(v))$. This restriction ensures that the sublevel sets of $\tau$ define a subgraph. The filtration $\left\{\mathcal{G}_t\right\}_{t\ge 0}$ is then defined via $$\mathcal{G}_t = \left\{\sigma \in V\cup E : \tau(\sigma) \le t\right\}.$$ As we increase $t$ from $0$ to $\infty$, we can track connected components of $\mathcal{G}_t$ as they appear and merge, which are referred to as *birth* and *deaths*, respectively. When two components merge, we use the 'elder rule' to determine that the later born component is the one that dies. Note that at least one component has an infinite death time in any graph filtration. We refer the reader to [@edelsbrunner_computational_2010] for further details on this. These birth-death events can be tracked by an algebraic object called a $0$-dimensional persistent homology. Its most common visualization is via a *persistence diagram* -- a collection of points in $\mathbb{R}^{2+}$, where each corresponds to a single connected component. The coordinates of a points encode the information with the $x$-coordinate representing the birth time, and the $y$-coordinate representing the death time. An example for a function on a line-graph is shown in Figure [1](#fig:example_diagram){reference-type="ref" reference="fig:example_diagram"}. Note that one component is infinite which we denote with a dashed line at the top of the diagram. In a more general context, given a filtration of higher-dimensional objects (e.g., simplicial complexes), we can study the $k$-dimensional persistent homology. This object tracks the formation of $k$-dimensional cycles (various types of holes), and its definition is a natural extension of the $0$-dimensional persistent homology we study here. We refer the reader to [@edelsbrunner_computational_2010] for more information. ![[\[fig:example_diagram\]]{#fig:example_diagram label="fig:example_diagram"} An example of a graph filtration on a line-graph. The filtration value of the vertices are given by $\tau$ (the $y$-axis). The filtration value of each edge is taken as the highest value between its plotted endpoints. The bars in the middle represent the tracking of the components. The vertices which are local minima, i.e. $a$, $c$, $f$, and $i$, generate new components and so $\tau(a)$, $\tau(c)$, $\tau(f)$, and $\tau(i)$ correspond to birth times. The first merge occurs at $\tau(b) = \tau((a,b)) = \tau((b,c))$ merging $\left\{a\right\}$ with $\left\{c,d\right\}$. In this case we declare the latter as dead since $\tau(a)<\tau(c)$. Next, at $\tau((d,e))$, the components $\left\{a,b,c,d\right\}$ and $\left\{e,f\right\}$ are merged, and the latter dies. Finally, at $\tau((g,h))$, the components $\left\{a,b,c,d,e,f,g\right\}$ and $\left\{h,i\right\}$ are merged, killing the former. The component containing $i$ has the earliest birth time, and thus is declared infinite. ](Figures/tracking.pdf){#fig:example_diagram width="95%"} # The $k$-Cluster Filtration {#sec:filt} Let $G = (V,E,W)$ be an undirected weighted graph. In computing $0$-dimensional persistent homology, the filtration values are commonly taken to be $\tau(v) = 0$ for all $v\in V$, and $\tau(e) = W(e)$ for all $e\in E$. We will denote this filtration by $\mathcal{G}^*_t$. In other words, we assume all vertices are present at time zero, and edges are gradually added according to the weight function $W$. This has been the practice in the TDA literature in almost all studies, and in particular in the geometric settings where $W$ represents the distance between points (i.e., the geometric graph, which is the skeleton of both the Čech and Vietoris-Rips complexes). While in many models, this choice of $\tau$ seems reasonable, it has two significant drawbacks: - The produced persistence diagrams are *degenerate*, as the birth times of all 0-cycles is $t=0$. This significantly reduces the amount of information we can extract from persistence diagrams. - The generated persistence diagrams are *superfluous*, in the sense that they contains a point for each vertex $V$, while obviously not all vertices contribute significant structural information. In this paper we propose a modification to the standard graph filtration, that will resolve both of these issues, and will lead to a more concise and informative persistence diagrams. We will first define the filtration value for the vertices. For every vertex, and a value $t>0$ we define $N_t(v)$ to be the number of vertices in the connected component of $\mathcal{G}^*_t$ that contains $v$. Fix $k\ge 1$, and define \_k(v) := {t : N_t(v) k}. The edges values are then \_k((u,v)) = (\_k(u),\_k(v), W((u,v))). Denoting the corresponding filtration by $\mathcal{G}^{(k)}_t$, note that $\mathcal{G}^{(1)}_t \equiv \mathcal{G}^*_t$. In other words, compared to $\mathcal{G}^*_t$, in $\mathcal{G}_t^{(k)}$ we delay the vertices appearance, until the first time each vertex is contained in a component with at least $k$ vertices (and adjust the edge appearance to be compatible). Effectively, the assignment of the new filtration values to the vertices introduces two changes to the persistence diagrams: 1. All the points that are linked to components of size smaller than $k$ are removed. 2. Each birth time corresponds to an edge merging two components $C_1,C_2$ in $\mathcal{G}_t^*$, such that $|C_1|,|C_2| < k$, and $|C_1|+|C_2|\ge k$. 3. Each death time corresponds to an edge merging two components larger than $k$. We call this filtration the '$k$-cluster filtration', to represent the fact that it tracks the formation and merging of clusters of size at least $k$. The parameter $k$ determines what we consider as a sufficiently meaningful cluster. In $\mathcal{G}_t^*$, every vertex is considered a cluster, but statistically speaking, this is an overkill. The chosen value of $k$ should depend on the application as well as the sample size. We conclude this section showing that the $k$-cluster filtrations are decreasing (in a set sense) as we increase $k$. This can be useful, for example, in the context of multi-parameter persistence, which we briefly mention but leave for future work. **Lemma 1**. *The filtrations $\mathcal{G}_t^{(k)}$ are decreasing in $k$, i.e., $$\tau_{k-1}(x) \leq \tau_k(x),\quad \forall x\in V\cup E.$$* *Proof.* For any vertex $v\in V$, if $|N_t(v)|\geq k$, then $|N_t(v)|\geq k-1$. From [\[eqn:def_tau_v\]](#eqn:def_tau_v){reference-type="eqref" reference="eqn:def_tau_v"} we therefore have that $\tau_{k-1}(v) \le \tau_k(v)$. Using [\[eqn:def_tau_e\]](#eqn:def_tau_e){reference-type="eqref" reference="eqn:def_tau_e"}, we have $\tau_{k-1}(e) \le \tau_k(e)$ for all $e\in E$. ◻ # Algorithm {#sec:alg} In this section, we describe an efficient one-pass algorithm for computing the filtration and persistence diagram at the same time. The time complexity of the algorithm is $O(|E|\times \alpha(|V|))$, where $\alpha(\cdot)$ is the inverse Ackermann function [@cormen_introduction_2022]. This is the same complexity as computing the 0-dimensional persistence diagram if we were given the filtration as input. We begin with the (standard) terminology and data structures. For simplicity of the description, we assume that the weights on the edges are unique and the vertices have a lexicographical order. We first define a total order on the vertices as follows: the filtration function determines the ordering. Undefined filtration functions are assumed to be $\infty$. If the function is the same or undefined for both vertices, the order is then determined by lexicographical ordering. It is straightforward to check this is a total ordering. **Remark 2**. *In the case of a total ordering, one can choose a representative of 0-dimensional persistent homology classes -- notably, in the total ordering a unique vertex is the earliest generator for the homology class (i.e., the cluster) which we denote as the canonical representative of the persistent component.* To track components as we proceed incrementally through the filtration, we use the union-find data structure, which supports two operations: - $\text{ROOT}(v)$: returns the canonical representative for the connected component containing $v$. - $\text{MERGE}(u,v)$: merges the connected components containing $u$ and $v$ into one component -- including updating the root. We augment the data structure by keeping track of two additional records: - $\text{SIZE}(v)$: returns the size of the connected component containing $v$. - $\text{COMPONENT}(v)$: returns the list of vertices in the same component as $v$. To track the size of the component, we store the size at the root (i.e., the canonical representative) of each component, updating each time a merge occurs. To access a connected component, recall that the union-find data structure is implemented as a rooted tree. For each vertex, we store a list of children in the tree. To recover the list of vertices in the component, we perform a depth-first search of the tree starting from the root (although any other traversal method could be used). All update operations have $O(1)$ cost (cf., [@cormen_introduction_2022]). Note that when $k=1$, the filtration value for all vertices is $0$ and so the problem reduces to finding the minimum spanning tree of a weighted graph. Hence, we will assume that $k>1$. Initially, we set the filtration function $\tau(v)=0$ for all vertices, and $\tau(e) = W(e)$ for all edges, and assume the edges are sorted by increasing weight. Note that if this is not the case, this step will be the bottleneck, with a cost of $O(|E|\log |E|)$. Thus, we begin with a forest where each component is a single vertex, i.e. all components are initially born at 0. We proceed as in the case of standard 0-dimensional persistence, adding edges incrementally. As no components are added, we are only concerned with merges, the problem is reduced to updating the birth times as we proceed by keeping track of "active\" components (i.e., larger than $k$). We omit points in the persistence diagram which are on the diagonal (birth=death), but these can be included with some additional book-keeping. Assume we are adding the edge $e = (u,v)$. If $e$ is internal to a connected component (i.e., $\text{ROOT}(u) = \text{ROOT}(v)$), then it does not affect the $0$-persistence. Otherwise, it connects two components denoted $C_u, C_v$. There are a few cases to consider: 1. $|C_u\cup C_v| < k$: The merged component is too small to affect the persistence diagram. We only perform a merge of the components. 2. $|C_u\cup C_v| \geq k$ and $|C_u| < k$: In this case, $C_u$ becomes active. Thus, we merge the components, and update the value of $\tau$ for all vertices in $C_u$. $$\tau(x) \leftarrow W(e) \quad\quad \forall x\in C_u$$ is performed. We take similar action if $|C_v|<k$ (or both are less than $k$). 3. $|C_u|,|C_v| \geq k$: Both components are already active and so a new point $(\mathop{\mathrm{birth}},\mathop{\mathrm{death}})$ is added to the persistence diagram, with $$\splite&= \max\left\{\tau(\text{ROOT}(u)), \tau(\text{ROOT}(v))\right\},\\ \mathop{\mathrm{death}}&= W(e). \splite$$ The components are again merged. We note that for any $v$, $$\tau(\text{ROOT}(v)) = \min_{x\in C_v} \tau(x).$$ The full procedure is given in Algorithm [\[alg:main\]](#alg:main){reference-type="ref" reference="alg:main"}. Note that we only compute the filtration for the vertices, as the correct edge values can then be computed by Equation [\[eqn:def_tau_e\]](#eqn:def_tau_e){reference-type="ref" reference="eqn:def_tau_e"}. $G=(V,E,W)$ $\tau: V \rightarrow [0,\infty)$ Initialize union-find data structure: $\text{ROOT}(v)=v$ for all $v\in V$ $\mathrm{Dgm},\text{MST}=\emptyset$ $\text{MST}\leftarrow \text{MST}\cup e$ $\tau(x)\leftarrow W(e)$ $\tau(x)\leftarrow W(e)$ $\mathop{\mathrm{birth}}= \max\{\tau(\text{ROOT}(u)),\tau(\text{ROOT}(v))\}$ $\mathop{\mathrm{death}}= W(e)$ $\mathrm{Dgm}\leftarrow \mathrm{Dgm}\cup (\text{birth},\mathop{\mathrm{death}})$ $\text{MERGE}(u,v)$\ $\mathrm{Dgm}, \text{MST}, \tau$ #### Proof of Correctness. We first argue that the function $\tau$ is correctly computed. This follows directly from the fact that the algorithm explicitly tests when the component contains at least $k$ vertices. The fact that the persistence diagram is correctly computed is a consequence of the following result. **Lemma 3**. *The minimum spanning tree for $k=1$ is a minimum spanning tree for any $k$.* *Proof.* The key observation is that until a component contains $k$ vertices, any spanning tree is a minimum spanning tree, as all the edges will be assigned the value when the component becomes active. The remaining edges do not have their values changed and so remain in the MST. ◻ The equivalence of the MST and the persistence diagram [@skraba_randomly_2017] then implies correctness of the algorithm. #### Proof of Running Time. The analysis of the merging is covered verbatim from the standard analysis of the union-find data structure. As described above, the update to the size of the component and updating the list of children in the merge are $O(1)$ operations. All that remains is to prove is the cost of updating the function $\tau$. We observe that each vertex is only updated once. This therefore has a total cost of $O(|V|)$, and the edges can be updated at a cost of $O(1)$ per edge (however, there is no practical need for that). This implies the overall running time is $O(|E|\times \alpha(|V))$. #### Extracting the Clusters. To obtain clusters, we can use the algorithm in [@chazal_persistence-based_2013]. This algorithm extracts the $\ell$-most persistent clusters by performing merges only when the resulting persistence is less than a threshold. This threshold can be chosen such that there are only $\ell$ points above the threshold in the diagram. Finally, we note that the cluster extraction can be done on the MST rather than the full graph. # Experiments and Applications {#sec:exp} ## Simulated point-clouds We start by generating point-clouds from a mixture of Gaussians, resulting in several blobs of points (Figure [2](#fig:example_points){reference-type="ref" reference="fig:example_points"}). We first show the effect of the parameter $k$ on the filtration function and the corresponding persistence diagrams. For the two point-clouds in Figure [2](#fig:example_points){reference-type="ref" reference="fig:example_points"}, we show the resulting persistence diagrams for the $k$-cluster filtrations in Figure [3](#fig:k-comp-rel){reference-type="ref" reference="fig:k-comp-rel"}. Notice that the correct number of persistent clusters is evident, especially for $k=10,20,$ and $50$. An important phenomenon that is evident in the figures is that higher values of $k$ filter out more of the 'noise'. ![[\[fig:example_points\]]{#fig:example_points label="fig:example_points"} Two examples point-clouds consisting of an i.i.d. sampling from a mixture of three and four Gaussians and consisting of 1000 and 2000 points respectively.](Figures/example_points.pdf){#fig:example_points width="75%"} ![[\[fig:k-comp-rel\]]{#fig:k-comp-rel label="fig:k-comp-rel"} The persistence diagrams with death/birth on the $y$-axis with different choices of $k$ for the points sampled from the two mixtures of Gaussians (top row) 3 blobs (bottom row) 4 blobs. Note that the number of outstanding features in the diagrams correspond to the number of clusters in the data.](Figures/k-comp-rel.pdf){#fig:k-comp-rel width="99%"} ![[\[fig:twomoons\]]{#fig:twomoons label="fig:twomoons"} A comparison of the $k$-cluster and $k$-degree for the two moons data set. On the right we have the death/birth ratios for different values of $k$. ](Figures/twomoons.pdf){#fig:twomoons width="90%"} To place the behaviour of the persistence diagrams into further context, we compare the $k$-cluster filtration with a related construction from the applied topology literature, which has been suggested for dealing with outliers in clustering (and in higher homological dimensions) -- the $k$-degree Vietoris-Rips filtration [@lesnick_interactive_2015]. Given a weighted graph $G = (V,E,W)$, we define the $k$-degree filtration, denoted $\delta_k:(V\cup E)\to [0,\infty)$ as follows. For every vertex $v\in V$ we take $\delta_k(v)$ to be its $k$-nearest neighbor distance. The values of the edges, is then determined the same as in [\[eqn:def_tau_e\]](#eqn:def_tau_e){reference-type="eqref" reference="eqn:def_tau_e"}. The $k$-degree filtration has been used in the context of multi-parameter persistence, with the bifiltration induced by decreasing $k$ and increasing the edge weight (commonly, Euclidean distance). In this paper, we do not explore the multi-parameter setting. Rather, we focus the properties of the persistence diagrams for a fixed $k$. We make two observations before investigating the differences: 1. The $k$-degree filtration function is determined completely by the local neighborhood of a vertex (i.e., its immediate neighbors in the graph). The same is not true for the $k$-cluster filtration. 2. For a fixed value of $k$ we have $\tau_k(v)\le \delta_{k-1}(v)$ for all $v\in V$. In other words, the value of $k$-cluster function is less than or equal to than the value of the $(k-1)$-degree function. This follows from the fact that if a vertex has $k-1$ neighbors, then it is part of a cluster of at least $k$ vertices. In Figure [4](#fig:twomoons){reference-type="ref" reference="fig:twomoons"}, we show the relative persistence diagrams for two non-convex clusters for both the $k$-degree and $k$-cluster filtrations, for different values of $k$. In this example, especially for larger $k$, the persistent clusters are much more prominent in the $k$-cluster filtration compared to the $k$-degree filtration. This may be explained by the fact that a much larger radius is needed to obtain the required number of neighbors. In Figure [5](#fig:deg_vs_cluster){reference-type="ref" reference="fig:deg_vs_cluster"}, we show the same comparison for relative persistence diagrams for 3 and 4 blobs, where the difference between the two methods is less clear. However, Figure [6](#fig:kstability){reference-type="ref" reference="fig:kstability"} highlights an additional difference in the behaviors of the two filtrations. In this figure, we compare the persistence (death/birth) for the second most persistent cluster, for a wide range of $k$ values. In the left and center plots, the second most persistent cluster corresponds to a true cluster in the data. We observe that the persistence value decays much more slowly for the $k$-cluster filtration, i.e. the true cluster remains more persistent for increasing values of $k$. The plot on the right presents the same comparison, but for uniformly distributed random points. In this case, the second most persistent cluster is by construction noise (i.e., not a real cluster in the data). Here although the $k$-cluster filtration decays more slowly, it is comparable to the $k$-filtration. Hence we can conclude that persistent clusters show a more stable behavior over ranges of $k$ for the $k$-cluster filtration compared to the $k$-degree filtration. ![[\[fig:deg_vs_cluster\]]{#fig:deg_vs_cluster label="fig:deg_vs_cluster"} (top row) 3 blobs, (bottom row) 4 blobs. The relative persistence diagrams for each point cloud, with the $k$-degree filtration in yellow and the $k$-cluster filtration in blue for $k=5,10,20,$ and $50$. ](Figures/rel-comp.pdf){#fig:deg_vs_cluster width="99%"} ![[\[fig:kstability\]]{#fig:kstability label="fig:kstability"} The effect on the second most persistent cluster for different values of $k$. On the left and center, this corresponds to a true cluster (left -- two moons and center -- mixture of 3 Gaussians). On the right --uniform random points. Here the noise cluster drops nearly as quickly in both cases. ](Figures/k-stability.pdf){#fig:kstability width="90%"} ## Universality {#sec:univ} In [@bobrowski_universal_2023], we published a comprehensive experimental work, showing that the distribution of persistence values is universal. We consider a persistence diagram as a finite collection of points in $\mathbb{R}^2$, $\mathrm{dgm}= \left\{(b_1,d_1),\ldots, (b_M,d_M)\right\}$. For each point $p_i = (b_i,d_i)$ we consider the multiplicative persistence value $\pi(p_i) = d_i/b_i$. Our goal is to study the distribution of the $\pi$-values across an entire diagram. Our results in [@bobrowski_universal_2023] are divided into two main parts. Given a point cloud of size $n$, we compute the persistence diagram for either the Čech or the Vietoris-Rips filtrations. In *weak universality* we consider the empirical measure $$\Pi_{n} := \frac{1}{|\mathrm{dgm}_k|}\sum_{p\in\mathrm{dgm}_k}\delta_{\pi(p)},$$ and we conjecture that for iid samples, we have $$\lim_{n\to\infty}\Pi_n = \Pi^*_{d,\mathcal{T},k},$$ where $d$ is the dimension of the point-cloud, $k$ is the degree of homology, and $\mathcal{T}$ is the filtration type (i.e., Čech or Vietoris-Rips). In other words, the limiting distribution for the $\pi$-values depends on $d,k,\mathcal{T}$ but is independent of probability distribution generating the point-cloud. In *strong universality* we present a much more powerful and surprising conjecture. Here, we define $\ell(p) := A\mathop{\mathrm{\log\log}}(\pi(p)) + B$ (the values of $A$ and $B$ are speficied in [@bobrowski_universal_2023]), and the empirical measure $$\mathcal{L}_n := \frac{1}{|\mathrm{dgm_k}|}\sum_{p\in\mathrm{dgm}_k}\delta_{\ell(p)}.$$ Our conjecture is that for wide class of random point-clouds (including non-iid and real-data), we have $$\lim_{n\to\infty}\mathcal{L}= \mathcal{L}^*,$$ where $\mathcal{L}^*$ is a unique universal limit. Furthermore, we conjecture that $\mathcal{L}^*$ might be the left-skewed Gumbel distribution. Originally, the results in [@bobrowski_universal_2023] are irrelevant for the $0$-th persistence diagram of random point-clouds, as the birth times are all zeros. However, once we replace the standard filtration with the $k$-cluster filtration, we have a new persistence diagrams with non-trivial birth time that we can study. In Figure [\[fig:universal\]](#fig:universal){reference-type="ref" reference="fig:universal"} we demonstrate both weak and strong universality properties for the $k$-cluster persistent homology. We generated iid point-clouds across different dimensions, with different distributions (uniform in a box, exponential, normal). The results show that both weak and strong universality hold in these cases as well. We note that for weak universality, the limiting distribution depends on both $d$ (dimension of point-cloud) and $k$ (minimum cluster size). ![[\[fig:universal\]]{#fig:universal label="fig:universal"}Universal distribution for $k$-cluster persistence. The labels in the legend are structured as distribution/$d$/$k$, where $d$ is the point-cloud dimension, and $k$ is the cluster size. The distributions taken are uniform in a unit box, exponential, and normal. The first two plots show that weak universality holds, and that the limit depends on $d,k$, but not on the distribution. The rightmost plot, demonstrates that strong universality holds under a proper normalization. We also included the left-skewed Gumbel distribution (dashed line) for comparison. ](Figures/univ_all.png){#fig:enter-label width="99%"} ## Clustering As mentioned in the introduction, a key motivation for this work was to apply the $k$-cluster filtration to clustering. To obtain a clustering from a 0-dimensional persistence diagram, we use the algorithm proposed in [@chazal_persistence-based_2013]. Roughly speaking, given a threshold $\alpha$, it extracts all clusters which are more than $\alpha$-persistent. We note that the original measure for persistence in [@chazal_persistence-based_2013] was given by $d-b$, however the change to use $d/{b}$ in the algorithm is trivial. #### Statistical Testing. An important consequence of the universality results in Section [5.2](#sec:univ){reference-type="ref" reference="sec:univ"} is that the limiting distribution (after normalization) appears to be a known distribution, i.e. left-skewed Gumbel. We can thus perform statistical testing on the number of clusters as in [@bobrowski_universal_2023]. The null-hypothesis denoted by $\mathcal{H}_0^{(i)}$, is that the $i$-th most persistent cluster is due to noise. Assuming the universality conjectures hold, the null hypothesis is given in terms of the $\ell$-values as $$\mathcal{H}_0^{(i)}\ :\ \ell(p_i) \sim \mathrm{LGumbel}.$$ where $p_i$ represents the $i$-th most persistent cluster in terms of death/birth. The corresponding p-value is given by $$\text{p-value}_i = \mathbb{P}\left(\ell(p_i) \ge x\;|\;\mathcal{H}_0^{(i)}\right) = e^{-e^x}.$$ Note that since we are testing sorted values, we must use a multiple hypothesis testing correction. In the experiments we describe below, we use the Bonferroni correction. ![[\[fig:clustering\]]{#fig:clustering label="fig:clustering"} A comparison of standard clustering examples for different clustering approaches. In the case of the $k$-cluster filtration (PD) and $k$-degree filtration (Deg), the number of clusters was chosen using statistical significance testing. ](Figures/clustering2-crop.pdf){#fig:clustering height="0.95\\textheight"} In Figure [8](#fig:clustering){reference-type="ref" reference="fig:clustering"}, we compared the $k$-cluster filtration and the $k$-degree filtration using persistence based clustering from [@chazal_persistence-based_2013] with other common algorithms for clustering. For the other approaches, we used the standard implementations found in [@pedregosa_scikit-learn_2011], which have associated techniques for choosing the number of clusters. In the cases of the $k$-cluster filtration and the $k$-degree filtration, the number of clusters was chosen using the statistical testing described above. Note that since the number of points in the standard examples was quite small, we limited $k$ to $5$ and $10$. The best result is for the $k$-cluster filtration with $k=10$ ($k=5$ fails to identify one of the clusters in the third example). The $k$-degree filtration performs well but the additional "noise\" points in the diagram, mean that some clusters are not identified as significant. #### Clustering on Trees. As a second example, we describe clustering on weighted trees. We generated a uniform random tree on $n$ vertices, and assigned uniformly distributed random weights on the edges (between $0$ and $1$). We show an example in Figure [9](#fig:tree){reference-type="ref" reference="fig:tree"}. The methods seems to capture certain structure about the tree, although we leave further investigation of this structure as future work. Note that in the tree case, it is often impossible to use $k$-degree filtrations, as the tree will have vertices with degree smaller than $k$ that will never be included in the filtration, whereas for the $k$-clustering filtration, all nodes are included as long as the underlying graph is connected (or all components have at least $k$ vertices). We note that it is possible to use an alternative definition for the $k$-degree filtrations, by embedding the tree into a metric space (i.e., using the graph metric induced by the weights). However, this is similar to studying a complete graph induced by the metric which is somewhat different than studying the graph directly. We use this method in the rightmost plot of Figure [9](#fig:tree){reference-type="ref" reference="fig:tree"}. ![[\[fig:tree\]]{#fig:tree label="fig:tree"} A clustering on a uniform random tree. The threshold with $k$-clustering gives 4 clusters while only 3 with the (metric) $k$-degree. ](Figures/tree.pdf){#fig:tree width="99%"} # Probabilistic Analysis {#sec:prob} In this section we wish to revisit some of the fundamental results known for the (persistent) homology of random graphs and simplicial complexes, and show that analogous statements hold for our new $k$-cluster filtration. We provide here the main statements. Proofs are available in the appendix. ## Connectivity We will consider two models here. In the $G(n,p)$ random graph we have $n$ vertices, and each edge is placed independently with probability $p$. In the $G(n,r)$ random geometric graph, we take a homogeneous Poisson process $\mathcal{P}_n$ on the $d$-dimensional flat torus, with rate $n$. Edges are then placed between vertices that are less than distance $r$ apart. In both models, connectivity results are tied to the expected degree. For the $G(n,p)$ model we define $\Lambda =np$, and for the $G(n,r)$ we take $\Lambda = n\omega_dr^d$. Then in [@erdos_random_1959] and [@penrose_longest_1997] the following was proved. **Theorem 4**. *Let $G_n$ be either $G(n,p)$ or $G(n,r)$. Then $$\lim_{n\to\infty}\mathbb{P}\left(G_n\text{ is connected}\right) = \begin{cases} 1 & \Lambda = \log n + w(n),\\ 0 & \Lambda = \log n - w(n).\end{cases}$$* A key element in proving connectivity (for either models) is to show that around $\Lambda = \log n$, the random graph consists of a single giant component, a few isolated vertices, and nothing else. Thus, connectivity is achieved when the last isolated vertices gets connected. Our goal in this section is to analyze connectivity in the $G(n,p)$ and $G(n,r)$ model, via our new $k$-cluster filtration. Note that for a fixed $n$, we can view both models as filtrations over the complete graph. For the $G(n,p)$ model the weights of the edges, are independent random variables, uniformly distributed in $[0,1]$. For the $G(n,r)$ the weight of an edge is given by the distance between the corresponding points in the torus. We define $G^{(k)}(n,p)$ and $G^{(k)}(n,r)$ to be the random filtrations generated by changing the filtration function to be $\tau_k$. Our goal here is to explore the phase transition for the $k$-cluster connectivity. As opposed to connectivity in the original random graphs, the results here differ between the models. **Theorem 5**. *For the $G^{(k)}(n,p)$ filtered graph we have, $$\lim_{n\to\infty}\mathbb{P}\left(G^{(k)}(n,p)\text{ is connected}\right) = \begin{cases}1 & \Lambda = \frac1k(\log n + (k-1)\mathop{\mathrm{\log\log}}n) +w(n),\\ 0 & \Lambda = \frac1k(\log n + (k-1)\mathop{\mathrm{\log\log}}n) - w(n),\end{cases}$$ for any $w(n) = o(\mathop{\mathrm{\log\log}}n)$ such that $w(n)\to\infty$.* For the $G^{(k)}(n,r)$ model, proving the connectivity is a much more challenging task and beyond the scope of this paper. The following statement, however, is relatively straightforward to prove. **Proposition 6**. *Let $N_k = N_k(n,r)$ be the number of connected components of size $k$ in $G(n,r)$. Then, $$\lim_{n\to\infty}\mathbb{P}\left(N_k =0\right) = \begin{cases}1 & \Lambda = \log n-(d-1)(k-1)\mathop{\mathrm{\log\log}}n+w(n),\\ 0 & \Lambda = \log n - (d-1)(k-1)\mathop{\mathrm{\log\log}}n - w(n).\end{cases}$$ for any $w(n) = o(\mathop{\mathrm{\log\log}}n)$ such that $w(n)\to\infty$.* From this lemma we conclude that when $\Lambda = \log n -(d-1)(k-1)\mathop{\mathrm{\log\log}}n -w(n)$ the graph $G(n,r)$ has components of size $k$, which implies that $G^{(k)}(n,r)$ is not connected. On the other hand, when $\Lambda = \log n -(d-1)(k-1)\mathop{\mathrm{\log\log}}n +w(n)$, we have $N_j=0$ for all fixed $j\ge k$. Which indicates that $G^{(k)}(n,r)$ should be connected. This leads to the following conjecture. **Conjecture 7**. *For the $G^{(k)}(n,r)$ filtered graph we have, $$\lim_{n\to\infty}\mathbb{P}\left(G^{(k)}(n,r)\text{ is connected}\right) = \begin{cases}1 & \Lambda = \log n-(d-1)(k-1)\mathop{\mathrm{\log\log}}n+w(n),\\ 0 & \Lambda = \log n - (d-1)(k-1)\mathop{\mathrm{\log\log}}n - w(n).\end{cases}$$* Note that both phase transitions occur before the ones for the original graph models. This is due to the fact that for $k>1$ the $k$-cluster filtration does not allow having any isolated vertices. Also note that taking $k=1$ both results coincide with Theorem [Theorem 4](#thm:rg_conn){reference-type="ref" reference="thm:rg_conn"}. ## Limiting Persistence Diagrams In [@hiraoka_limit_2018], it was shown that for stationary point processes, persistence diagrams have a non-random limit (in the vague convergences of measures). A similar statement will hold for the $k$-cluster persistence diagrams. Let $\mathrm{Dgm}^{(k)}(\mathcal{P})$ be the $k$-cluster persistence diagram for a point-cloud $\mathcal{P}$. We define the discrete measure on $\mathbb{R}^2$, $$\xi^{(k)}(\mathcal{P}) := \sum_{(b,d)\in\mathrm{Dgm}^{(k)}(\mathcal{P})} \delta_{(b,d)}.$$ Let $Q_L = [-L/2,L/2]^d$. The following is an analogue of Theorem 1.5 in [@hiraoka_limit_2018]. **Theorem 8**. *Assume that $\mathcal{P}$ is a stationary point process in $\mathbb{R}^d$ with all finite moments. For any $k$, there exists a deterministic measure $\mu_k$, such that $$\lim_{L\to\infty}\frac{1}{L^d}\mathbb{E}\left\{{\xi^{(k)}(\mathcal{P}\cap Q_L)}\right\} = \mu_k,$$ where the limit is in the sense of vague convergence. Furthermore, if $\mathcal{P}$ is ergodic, then almost surely $$\lim_{L\to\infty}\frac{1}{L^d}{\xi^{(k)}(\mathcal{P}\cap Q_L)} = \mu_k.$$* ## Maximal Cycles In [@bobrowski_maximally_2017] the largest cycles in persistence diagrams were studied. Specifically, for every point $p= (b,d)$ in a diagram, we compute the so-called $\pi$-value - $\pi(p) = d/b$, as a scale-invariant measure of size. Considering the homogeneous Poisson process $\mathcal{P}_n$, we define $\Pi_{k,\max}$ as the largest $\pi$-value in the $k$-th persistent homology. The main result in [@bobrowski_maximally_2017] then states that with high probability $$A_k \Delta_k(n) \le \Pi_{k,\max} \le B_k\Delta_k(n),$$ where $A_k,B_k>0$ are constants, and $$\Delta_k(n) = \left(\frac{\log n}{\mathop{\mathrm{\log\log}}n}\right)^{1/k}.$$ For the $k$-cluster persistence, we will show that the largest $\pi$-value has a completely different scaling. **Theorem 9**. *Let $\mathcal{P}_n$ be a homogeneous Poisson process in the flat torus, with rate $n$. Let $\Pi^{(k)}_{\max}$ denote the maximum $\pi$-value in the $k$-cluster persistence diagram (excluding the infinite cluster). Then, for every $\epsilon>0$ we have $$\lim_{n\to\infty}\mathbb{P}\left(n^{\frac 1{d(k-1)}-\epsilon} \le \Pi^{(k)}_{\max} \le n^{\frac 1{d(k-1)}+\epsilon}\right) = 1.$$* **Remark 10**. *We observe that the largest $\pi$-value in the $k$-cluster persistence, is significantly larger than that of the $k$-dimensional homology. The main reason for that is the following. In [@bobrowski_maximally_2017], our upper bound for $\Pi_{k,\max}$ used an iso-perimetric inequality, which implies that large $\pi$-values require large connected components. However, the $\pi$-values in the $k$-cluster persistence, only require a cluster of size $k$ to be formed, and thus can be generated by much smaller connected components.* # Appendix {#appendix .unnumbered} In this appendix we provide the proofs for the statements made in Section [6](#sec:prob){reference-type="ref" reference="sec:prob"}. # Connectivity *Proof of Theorem [Theorem 5](#thm:pt_er){reference-type="ref" reference="thm:pt_er"}.* Note that for the $k$-cluster filtration, connectivity is equivalent to the original $G(n,p)$ graph having no components of size $j$ for any $k\le j \le n/2$. Let $N_j = N_j(n,p)$ be the number of components of size $j$ in $G(n,p)$. Taking similar steps to the proof of connectivity for random graphs (e.g., [@frieze_introduction_2016]), we have {N_j} j\^j-2p\^j-1(1-p)\^j(n-j). For $k+1\le j< 4k$, we have $$\mathbb{E}\left\{{N_j}\right\}\le C n^j \left(\frac{\Lambda}{n}\right)^{j-1} e^{-j(n-4k)(\Lambda/n)},$$ for some $C>0$. Taking $\Lambda = \frac1k (\log n + (k-1)\mathop{\mathrm{\log\log}}n)+c$, we have $$\mathbb{E}\left\{{N_j}\right\} \le C n^{-1/k} (\log n)^{j-1} e^{4j\log n /n }.$$ For $4k \le j \le n/2$, we have $$\mathbb{E}\left\{{N_j}\right\} \le \left(\frac{ne}{j}\right)^j j^{j-2} \left(\frac{\Lambda}{n}\right)^{j-1} e^{-j\Lambda/2}\le \frac{n}{j^2} \left(\frac{e^{1-c/2}\log n}{n^{1/2k}}\right)^j \le \frac{n}{j^2} n^{-j/3k}.$$ Therefore, $$\sum_{j=4k}^{n/2} \mathbb{E}\left\{{N_j}\right\} \le \frac{1}{8k^2} n^{-1/3}.$$ To conclude, we showed that $$\lim_{n\to\infty}\sum_{j=k+1}^{n/2} \mathbb{E}\left\{{N_j}\right\} = 0.$$ This implies that for $\Lambda = \frac1k(\log n + (k-1)\mathop{\mathrm{\log\log}}n)+c$, we have $$\mathbb{P}\left(G^{(k)}(n,p) \text{ is connected}\right) \approx \mathbb{P}\left(N_k > 0\right).$$ Similar estimates to the ones above, show that $$\mathbb{E}\left\{{N_k}\right\} \approx e^{-kc}.$$ Therefore, when $c=w(n)\to\infty$, we have $\mathbb{P}(N_k>0) \to 0$. Together with a second-order argument, we can similarly show that when $c=-w(n)$, we have $\mathbb{P}\left(N_k>0\right) \to 1$. This concludes the proof. ◻ *Proof of Proposition [Proposition 6](#prop:N_k){reference-type="ref" reference="prop:N_k"}.* Recall that $N_k$ is the number of components of size $k$ in $G(n,r)$. In [@penrose_k_2022]\[Theorem 3.3\], it was shown that $$\mathbb{E}\left\{{N_k}\right\}\approx \mathrm{Var}\left({N_k}\right) \approx C_kn \Lambda^{-(d-1)(k-1)}e^{-\Lambda},$$ for some constant $C_k>0$. When $\Lambda = \log n -(d-1)(k-1)\mathop{\mathrm{\log\log}}n + w(n)$, we have $\mathbb{E}\left\{{N_k}\right\} \to 0$, implying that $\mathbb{P}\left(N_k>0\right) \to 0$. When $\Lambda = \log n -(d-1)(k-1)\mathop{\mathrm{\log\log}}n -w(n)$, we can use Chebyshev's inequality, $$\mathbb{P}\left(N_k=0\right) \le \mathbb{P}\left(|N_k-\mathbb{E}\left\{{N_k}\right\}| \ge \mathbb{E}\left\{{N_k}\right\}\right) \le \frac{\mathrm{Var}\left({N_k}\right)}{(\mathbb{E}\left\{{N_k}\right\})^2} \approx \frac1{\mathbb{E}\left\{{N_k}\right\}} \to 0.$$ This completes the proof. ◻ # Maximal $\pi$-value *Proof.* Let $r,R$ denote the birth and death radii of a cluster in the $k$-cluster persistence diagram, and recall that $\pi = R/r$. For an upper bound, we denote by $N_k(r)$ the number of connected subsets of size $k$, at radius $r$. Using Mecke's formula (cf. [@penrose_random_2003]), $$\splite{N_k(r)} &= \frac{n^k}{k!}\int_{(\mathbb{T}^d)^k} \boldsymbol{\mathbbm{1}}\left\{G(\mathbf{x},r)\text{ is connected}\right\}d\mathbf{x}\\ &=\frac{n\lambda^{k-1}}{k!} \int_{(\mathbb{R}^d)^{k-1}}\boldsymbol{\mathbbm{1}}\left\{G((0,\mathbf{y}),1)\text{ is connected}\right\}d\mathbf{y},\\ &=C_k n\lambda^{k-1}, \splite$$ where $C_k$ is a positive constant, $\lambda = nr^d$, and we used the change of variables $x_i \to x_1+ry_i$ ($i=2,\ldots,k$). For any $\epsilon>0$, if $\lambda = n^{-1/(k-1)-\epsilon}$ then $\mathbb{E}\left\{{N_k(r)}\right\}\to 0$. Thus, we can assume that with high probability the birth time of all $k$-clusters has $\lambda \ge n^{-1/(k-1)-\epsilon}$. In addition, from Theorem [Theorem 4](#thm:rg_conn){reference-type="ref" reference="thm:rg_conn"}, if we denote $\Lambda = nR^d$, then when $\Lambda = C\log n$ the graph $G(n,r)$ is connected. This implies that with high probability all death times of $k$-clusters have $\Lambda \le C\log n$. These bounds together imply that with high probability, for all the points in the $k$-cluster persistence diagram, for any $\epsilon>0$, we have $$\pi = \left(\frac{\Lambda}{\lambda}\right)^{1/d} \le \left(\frac{C\log n}{n^{-1/(k-1)-\epsilon}}\right)^{1/d}.$$ Therefore, for any $\epsilon>0$, we have $$\lim_{n\to\infty}\mathbb{P}\left(\Pi_{\max}^{(k)} \le n^{\frac{1}{d(k-1)}+\epsilon}\right) = 1.$$ For the lower bound, we denote by $\hat N_k(r,R)$ the number of components of size $k$, born before $r$, that are isolated at radius $R$ (and hence die after $R$). Then $$\splite{\hat N_k(r,R)} &= \frac{n^k}{k!}\int_{(\mathbb{T}^d)^k}\boldsymbol{\mathbbm{1}}\left\{G(\mathbf{x},r)\text{ is connected}\right\} e^{-n\mathop{\mathrm{Vol}}(B_R(\mathbf{x}))}d\mathbf{x},\\ &=\frac{n\lambda^{k-1}}{k!}\int_{(\mathbb{R}^d)^{k-1}}\boldsymbol{\mathbbm{1}}\left\{G((0,\mathbf{y}),1)\text{ is connected}\right\} e^{-n\mathop{\mathrm{Vol}}(B_R(0,r\mathbf{y}))}d\mathbf{y},\\ \splite$$ where $B_R(\mathbf{x})$ is the union of balls of radius $R$ around $\mathbf{x}$. We will apply the dominated convergence theorem, using the fact that when $r/R\to 0$, we have $$\lim_{n\to\infty}\frac{\mathop{\mathrm{Vol}}(B_R(0,0+r\mathbf{y}))}{\omega_d R^d} = 1.$$ This leads to $${\mathbb{E}\{{\hat N_k(r,R)}\}}\approx C_k n\lambda^{k-1}e^{-\omega_d\Lambda}.$$ Taking $\Lambda = C>0$, and $\lambda = n^{-1/(k-1)+\epsilon}$, we have $$\mathbb{E}\{{\hat N_k(r,R)}\} \to \infty.$$ Using a second moment argument will show that for all $\epsilon>0$ $$\mathbb{P}\left(\Pi_{\max}^{(k)} \ge n^{\frac 1{d(k-1)} -\epsilon}\right) \to 1,$$ completing the proof. ◻ # Limiting Persistence Diagram The key part of the proof in [@hiraoka_limit_2018], is bounding the add-one cost of the *persistent* Betti numbers. Let $G=(V,E,W)$ be a weighted graph, and $\{G^{(k)}_t\}$ be the corresponding $k$-cluster filtration. Define $\beta_0^{r,s}(G^{(k)})$ as the $0$-th persistent Betti number, i.e., the number of components born in $t\in [0,r]$ and die at $(s,\infty]$ (for a formal definition, see [@hiraoka_limit_2018]). Fix an edge $e_0\not\in E$, with a given weight $W(e_0) = w_0$, and let $\tilde G = (V,\tilde E, \tilde W)$ be a weighted graph with $\tilde E = E\cup\{e_0\}$, and $$\tilde W(e) := \begin{cases} W(e) & e\ne e_0,\\ w_0 & e = e_0. \end{cases}$$ Let $\{\tilde G^{(k)}_t\}$ denote the corresponding $k$-cluster filtration. The entire proof Theorem [Theorem 8](#thm:limit_pd){reference-type="ref" reference="thm:limit_pd"} follows verbatim from the proofs in [@hiraoka_limit_2018], provided that we prove the following lemma. **Lemma 11**. *$$\left| \beta_0^{r,s}(\tilde G^{(k)}) - \beta_0^{r,s}(G^{(k)})\right| \leq 1.$$* In other words, if we add a single edge to the filtration, the number of persistent clusters can change by at most $1$. Note that the proof here is not a straightforward application of Lemma 2.10 in [@hiraoka_limit_2018], since in our case, when a single edge is added to the filtration, the filtration values of other vertices and edges might be affected. *Proof.* Let $e_0=(u,v)$ with $W(e_0)=w_0$. Let $C_u$ and $C_v$ denote the components of the end points of $e_0$ at $w_0$ in the original filtration $\{G_t^{(k)}\}$. There are three possible cases which can occur. Both $|C_u|<k$ and $|C_v| < k$. Note that in this case $\tau_k(u),\tau_k(v) > w_0$. Let $C'_u$ be the cluster of $u$ at $\tau_k(u)$, so that it is the component of $u$ when it first appears in $G^{(k)}_t$. Similarly define $C'_v$. Note that aside from $C'_u\cup C'_v$ the filtration value of all other vertices remains unchanged by adding $e_0$. Without loss of generality, suppose that $w_0 < \tau_k(u) < \tau_k(v)$. Then comparing the persistence diagram for $G_t^{(k)}$ and $\tilde G_t^{(k)}$, only two differences can occur: 1. The point representing $C'_v$ in $G_t^{(k)}$ is removed, since $C'_v$ is no longer a connected component in $\tilde G_t^{(k)}$ (as it is merged with $C'_u$). 2. The point representing $C'_u$ in $G_t^{(k)}$ may get an earlier birth time in $\tilde G_t^{(k)}$, in the interval $[w_0, \tau_k(u))$. For a given $r,s$, the first change might decrease $\beta_k^{r,s}$ by $1$, while the second change might increase it by $1$. In any case, the total difference between $\beta_0^{r,s}(\tilde G^{(k)})$ and $\beta_0^{r,s}(G^{(k)})$ is no more than one. $|C_u|\ge k$, and $|C_v| < k$. Defining $C'_v$ the same as above, note in this case the filtration value of all points outside $C'_v$ will remain unchanged by adding $e_0$. The only change that will occur in this case is that the point in the diagram of $G_t^{(k)}$ corresponding to $C'_v$ will be removed in $\tilde G_t^{(k)}$, since it is now merged with $C_u$. Therefore, the difference in the persistent Betti numbers is at most 1. Both $|C_u|\ge k$ and $|C_v|\ge k$. If $C_u=C_v$ then adding $e_0$ creates a $1$-cycle (loop) and does not affect the $k$-cluster persistence diagram. If $C_u\ne C_v$, then both $C_u$ and $C_v$ are represented by different points in the persistence diagram of $G_t^{(k)}$. Adding $e_0$ will cause one of these components to die earlier. In this case $\beta_0^{r,s}$ may be decreased by $1$ (if $s>w_0$). ◻

arxiv_math

--- abstract: | It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$, isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $\mathbf{A}$ they have an amalgamated free product structure, for type $\mathbf{D}$ they are subgroups of the groups of Dynkin diagram automorphisms. For type $\mathbf{D}$ we also compute all the possible affine isomorphisms between deformations; this was not known before. author: - Simone Castellan bibliography: - R/myreferences.bib title: "**AUTOMORPHISM GROUPS OF DEFORMATIONS AND QUANTIZATIONS OF KLEINIAN SINGULARITIES**" --- # Introduction Consider a non-commutative, associative filtered algebra $B$ and its semi-classical limit ${\mathcal B}$, which is a Poisson algebra. The Poisson structure on ${\mathcal B}$ can be thought of as a first-order approximation of the algebra structure on $B$. It is then natural to ask how much does ${\mathcal B}$ recover of the properties of $B$. This question has a rich history; there is no general theory, but a lot of work has been done in this direction for specific algebras. In [@goodearlSemiclassicalLimitsQuantized2010], it is conjectured that the prime and primitive spectra of the quantized coordinate rings are respectively homeomorphic to the Poisson prime and Poisson primitive spectra of their corresponding semi-classical limits when the base field is algebraically closed and of characteristic zero. This conjecture has since been verified in a number of examples [@fryerPrimeSpectrumQuantum2017; @goodearlSemiclassicalLimitsQuantum2009; @goodearlSemiclassicalLimitsQuantized2010]. In [@launoisPoissonDerivationsSemiclassical2023; @launoisDerivationsFamilyQuantum2023], the authors studied the derivations of certain deformations $A_{{\alpha},{\beta}}$ of the second Weyl algebra and the Poisson derivations of their semi-classical limit ${\mathcal A}_{{\alpha},{\beta}}$, proving that the first Hochschild cohomology group $HH^1(A_{{\alpha},{\beta}})$ is isomorphic to the first Poisson cohomology group $HP^1({\mathcal A}_{{\alpha},{\beta}})$. In [@choSemiclassicalLimitsOre2016], it is proved that the endomorphisms of the generalized Weyl algebras are the same as the Poisson endomorphisms of the Poisson generalized Weyl algebras. The connection between the $0$-th Poisson homology $HP_0({\mathcal B})$ and the $0$-th Hochschild homology $HH_0(B)$ was studied by Etingof and Schedler in a series of papers [@etingofPoissonTracesDModules2010; @etingofTracesFiniteMathcal2010; @etingofZerothPoissonHomology2012; @etingofPoissonTracesSymmetric2014], proving that $HP_0({\mathcal B})\cong HH_0(B)$ in many examples, including symmetric powers of isolated quasi-homogeneous surface singularities and finite $W$-algebras. A particularly interesting problem concerns the relation between automorphisms. Belov-Kanel and Kontsevich made the following remarkable conjecture: **Conjecture 1** ([@belov-kanelAutomorphismsWeylAlgebra2005]). *The automorphism group of the Weyl algebra of index $n$ over ${\mathbb C}$ is isomorphic to the group of Poisson automorphisms of ${\mathbb C}[x_1,\dots,x_n,y_1,\dots,y_{2n}]$, with the standard symplectic structure.* The conjecture is verified for $n=1$ and for the subgroups of tame automorphisms [@belov-kanelAutomorphismsWeylAlgebra2005]. The importance of Conjecture [Conjecture 1](#conj kontsevich){reference-type="ref" reference="conj kontsevich"} relies on its connection to other important open problems, like the Jacobian conjecture and the Dixmier conjecture [@belov-kanelJacobianConjectureStably2007; @belov-kanelPolynomialAutomorphismsQuantization2022]. It is a hard problem, and it motivates the study of the relation between the automorphism groups of Poisson algebras and their quantizations in other cases. For a graded Poisson algebra ${\mathcal B}$, a filtered deformation is a filtered Poisson algebra ${\mathcal C}$ such that $\mathop{\mathrm{gr}}({\mathcal C})\cong{\mathcal B}$ as graded Poisson algebras. Suppose that the deformation and quantization theory of ${\mathcal B}$ has the desirable property that deformations and quantizations are parameterized by the same moduli space $\mathfrak{M}$ (see [@ambrosioUniversalFilteredQuantizations2022]). We denote by $B_{0,c}$ and $B_{1,c}$ respectively the deformation and quantization corresponding to the parameter $c\in\mathfrak{M}$. Consider all possible associative algebra isomorphisms between the various quantizations and all Poisson isomorphisms between the various deformations. They form two groupoids, where the objects are the algebras $B_{1,c}$ (respectively $B_{0,c}$) and the morphisms are the (Poisson) isomorphisms; we denote them by $\mathop{\mathrm{Iso}}(B)$ (respectively $\mathop{\mathrm{PIso}}(B)$). In this paper, we shall investigate the relation between $\mathop{\mathrm{Iso}}(B)$ and $\mathop{\mathrm{PIso}}(B)$, through the case study of the algebra of functions on a Kleinian singularity. In particular, understanding the groupoid of isomorphisms will give us information about the automorphism groups of all the deformations and quantizations. Recall that a Kleinian singularity is a quotient space $V/\Gamma$, where $V$ is a two dimensional complex vector space, and $\Gamma$ is a finite subgroup of $SL(2,{\mathbb C})$. Kleinian singularities are isomorphic to surfaces in ${\mathbb A}^3$, so $${\mathbb C}[V/\Gamma]={\mathbb C}[V]^\Gamma\cong{\mathbb C}[x,y,z]/(F), \quad \text{with $F\in{\mathbb C}[x,y,z]$.}$$ Moreover, ${\mathbb C}[V]^\Gamma$ carries the structure of a Poisson algebra, induced by the standard symplectic form on $V$. Kleinian singularities, and their corresponding algebras of functions, are classified by simply laced Dynkin diagrams (type $\mathbf{A_n} \ n\geq1, \mathbf{D_n} \ n\geq4, \mathbf{E_6},\mathbf{E_7},\mathbf{E_8}$) (see [@mckayGraphsSingularitiesFinite1980]). The study of deformations and quantizations of Kleinian singularities has a long and rich history. In both cases the moduli space is ${\mathfrak h}/W$, where ${\mathfrak h}$ and $W$ are, respectively, the Cartan subalgebra and the Weyl group of the Lie algebra of corresponding Dynkin type. The construction of the semi-universal deformation is due to the work of Grothendieck, Brieskorn, Kronheimer and Slodowy [@slodowySimpleSingularitiesSimple1980; @kronheimerConstructionALESpaces1989; @katzGorensteinThreefoldSingularities1992]. The quantizations of a Kleinian singularity were constructed by Crawley-Boevey and Holland in [@crawley-boeveyNoncommutativeDeformationsKleinian1998] as a family of algebras that they denote ${\mathcal O}^{\lambda}$. In [@crawley-boeveyNoncommutativeDeformationsKleinian1998] it is also proved that ${\mathcal O}^{\lambda}$ is isomorphic to $e_0\Pi^{\lambda}(Q) e_0$, the spherical subalgebra of the deformed preprojective algebra of the McKay quiver $Q$ associated to the singularity (see [@mckayGraphsSingularitiesFinite1980]). Explicit presentation with generators and relations for the quantizations in type $\mathbf{A}$ appeared in earlier works by Hodges [@hodgesNoncommutativeDeformationsTypeA1993], Smith [@smithClassAlgebrasSimilar1990], and Bavula and Jordan, who studied them as special cases of generalized Weyl algebras [@bavulaGeneralizedWeylAlgebras1992; @bavulaIsomorphismProblemsGroups2001]. For type $\mathbf{D}$, an explicit construction of the quantizations was given by Levy in [@levyIsomorphismProblemsNoncommutative2009] and, using different methods, by Boddington [@boddingtonDeformationsTypeKleinian2007]. We formulate the following conjecture. **Conjecture 2**. *Let $V/\Gamma$ be a Kleinian singularity. Then there is an isomorphism of groupoids $$\mathop{\mathrm{PIso}}({\mathbb C}[V/\Gamma])\cong\mathop{\mathrm{Iso}}({\mathbb C}[V/\Gamma]).$$* Since the only filtered quantization of the polynomial algebra ${\mathbb C}[x_1,\dots,x_n,y_1,\dots,y_{2n}]$ is the Weyl algebra, Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} can be seen as a direct analog of Belov-Kanel-Kontsevich Conjecture for the algebra of function of a Kleinian singularity. The main result of this paper is the following theorem (see Theorems [Theorem 39](#teo iso type A){reference-type="ref" reference="teo iso type A"} and [Theorem 53](#teo iso type D){reference-type="ref" reference="teo iso type D"}) **Theorem 1**. *Let $X$ be a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$. Then Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} holds. In particular, the group of automorphisms and Poisson automorphisms of the quantization and deformation corresponding to the same parameter are isomorphic.* The isomorphism problem for the quantizations was already solved in [@bavulaIsomorphismProblemsGroups2001], for type $\mathbf{A}$, and in [@levyIsomorphismProblemsNoncommutative2009], for type $\mathbf{D}$, while the Poisson side is less studied in the literature. In type $\mathbf{A}$, the automorphism groups of the deformations (considered as affine varieties) were computed by Makar-Limanov [@makar-limanovGroupsAutomorphismsClass1990] and further studied by Blanc [@blancAutomorphismsPreFibered2011]. Naurazbekova and Umirbaev studied in [@naurazbekovaAutomorphismsSimpleQuotients2021] the groups of Poisson automorphisms of deformed Kleinian singularities in type $\mathbf{A_1}$. They proved that the groups are independent of the deformation parameter and that they have an amalgamated free product structure. In the case $n=1$, the deformation and quantizations can be identified with quotients of the symmetric algebra and universal enveloping algebra of ${\mathfrak s}{\mathfrak l}_2$, respectively. The automorphism groups of these quotients of $U({\mathfrak s}{\mathfrak l}_2)$ are known to have the same amalgamated free product structure [@dixmierQuotientsSimplesAlgebre1973; @fleurySousgroupesFinisAut1998], which implies that the two groups are isomorphic [@naurazbekovaAutomorphismsSimpleQuotients2021 Theorem 5]. In the present paper, we show that these results can be generalized to $n>1$, even if we no longer have the ${\mathfrak s}{\mathfrak l}_2$ structure in higher degree. The main difference is that, for $n>1$, the (Poisson) automorphism groups depend on the deformation parameter. **Theorem 2**. *Let $n>2$. The group $G$ of Poisson automorphisms of a deformation of a $\mathbf{A_{n-1}}$ Kleinian singularity is isomorphic to the group of automorphisms of the corresponding quantization. The dependence of $G$ on the deformation parameter splits into two classes, one for generic and one for special parameter (see Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"} for more details):* (i) *for special deformation parameter and $n$ even, $G\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} ({\mathbb C}^\times\rtimes{\mathbb Z}/2{\mathbb Z});$* (ii) *for special deformation parameter and $n$ odd, $G\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} H$, where\ $H=\langle {\mathbb C}^\times, \Omega \, | \, \Omega^2=-1, \,{\lambda}\cdot\Omega=\Omega\cdot{\lambda}^{-1} \ \forall{\lambda}\in{\mathbb C}^\times \rangle$;* (iii) *for generic deformation parameter, $G\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} ({\mathbb C}[x]\rtimes{\mathbb C}^\times).$* In type $\mathbf{D}$, the affine automorphism group of the undeformed Kleinian singularity was computed by Blanc in [@blancNonrationalityFibrationsAssociated2015]. To our knowledge, the affine isomorphisms between deformations were unknown; we compute them in Theorem [Theorem 51](#teo iso Dn){reference-type="ref" reference="teo iso Dn"}. We are then able to check which ones are Poisson, directly confirming Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"}. In particular, we can give an explicit presentation of the groups of Poisson automorphisms of the deformations. **Theorem 3**. *Let $n\geq4$. The group $G$ of Poisson automorphisms of a deformation of a $\mathbf{D_{n}}$-Kleinian singularity is isomorphic to the group of automorphisms of the corresponding quantization. The dependence of $G$ on the deformation parameter splits into three classes, one for generic, one for sub-generic and one for special parameter (see Theorem [Theorem 40](#teo iso quantizations type D){reference-type="ref" reference="teo iso quantizations type D"} for more details):* (i) *for generic deformation parameter, $G=\langle \mathop{\mathrm{id}}\rangle$;* (ii) *for $n>4$ and "nice" deformation parameter, $G={\mathbb Z}/2{\mathbb Z}$;* (iii) *for $n=4$ and "nice" deformation parameter, $G=S_3$, the symmetric group on $3$ elements;* (iv) *for $n=4$ and "almost nice" deformation parameter, $G={\mathbb Z}/2{\mathbb Z}$.* Even though the results are uniform, the methods used to prove them are quite different: algebraic for type $\mathbf{A}$, geometric for type $\mathbf{D}$. Additionally, in type $\mathbf{A}$ we computed the automorphism groups first, and then used that to get the whole isomorphism groupoid, while in type $\mathbf{D}$ we were not able to compute the automorphism groups directly, but we needed to compute the whole isomorphism groupoid first. Recall that a Kleinian singularity $V/\Gamma$ is a symplectic quotient, thus the results of the present paper can be seen in the more general context of symplectic singularities (in the sense of Beauville [@beauvilleSymplecticSingularities2000]). In [@namikawaPoissonDeformationsAffine2010], Namikawa constructed a Cartan space $\mathfrak{P}$ and Weyl group $W$ associated to a conic symplectic singularity $X$, with $W$ acting on $\mathfrak{P}$ by crystallographic reflections, and proved that the deformations of ${\mathbb C}[X]$ are parameterized by $\mathfrak{P}/W$. Recently, Losev proved that the same is true for the quantizations of ${\mathbb C}[X]$ (see [@losevDeformationsSymplecticSingularities2022 Theorem 3.4]). The reductive part of the group of graded Poisson algebra isomorphisms of ${\mathbb C}[X]$ acts on the space of deformations and quantizations as filtered (Poisson) algebra isomorphisms. By a general result of Losev [@losevDeformationsSymplecticSingularities2022 Proposition 3.21 and Corollary 3.22], these two actions coincide and induce all the possible filtered (Poisson) algebra isomorphisms between different deformations or quantizations. Theorem [Theorem 1](#teo main){reference-type="ref" reference="teo main"} can be restated as saying that, for the special case of type $\mathbf{A}$ and $\mathbf{D}$ Kleinian singularities, all (Poisson) isomorphisms come from the action of an automorphism of ${\mathbb C}[X]$ on the moduli space. What we show also in the present paper is that in type $\mathbf{D}$ all (Poisson) isomorphisms are filtered, while in type $\mathbf{A}$ the non-filtered (Poisson) isomorphisms come from the exponentiation of nilpotent inner derivations. Notice that the associated graded of each of the quantizations of a Kleinian singularity $X$ is not the corresponding deformation, but the algebra ${\mathbb C}[X]$ itself. Nonetheless, there is a different way to realize a deformation as the semi-classical limit of the corresponding quantization. For each symplectic quotient, Etingof and Ginzburg [@etingofSymplecticReflectionAlgebras2002] constructed a symplectic reflection algebra $H_{t,c}$, for all $t\in{\mathbb C}$ and all $c$ in a certain vector space of dimension $\dim \mathfrak{P}$. They are defined as quotients of the smash-product ${\mathcal T}(V^*)\rtimes\Gamma$, where ${\mathcal T}(V^*)$ is the tensor algebra of $V^*$. Let $e$ be the idempotent element in $\Gamma$; the algebra $eH_{t,c}e$ is called the spherical subalgebra of $H_{t,c}$. A result by Bellamy [@bellamyCountingResolutionsSymplectic2016] shows that every filtered Poisson deformation of ${\mathbb C}[V]^\Gamma$ is of the form $eH_{0,c}e$, for some $c$. In [@losevDeformationsSymplecticSingularities2022], it is proved that every filtered quantization ${\mathbb C}[V]^\Gamma$ is of the form $eH_{1,c}e$, for some $c$; we can thus regard every deformation as the semi-classical limit of $eH_{t,c}e$, as $t$ goes to $0$. Unfortunately, this does not seem to allow us to directly compare the automorphisms of the two objects. The paper is structured as follows. In Section [2](#section 2){reference-type="ref" reference="section 2"} we study the case of type $\mathbf{A}$ singularities. We prove that the group of automorphisms of every quantization admits an amalgamated free product structure (Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"}). We compute the group of Poisson automorphisms of every deformation, using techniques similar to [@naurazbekovaAutomorphismsSimpleQuotients2021], and prove that it is isomorphic to the automorphism group of the corresponding deformation (Theorem [Theorem 31](#teo poisson automorphisms type A){reference-type="ref" reference="teo poisson automorphisms type A"}). Using these results we are then able to confirm Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} (Theorem [Theorem 39](#teo iso type A){reference-type="ref" reference="teo iso type A"}) in type $\mathbf{A}$. In Section [3](#section 3){reference-type="ref" reference="section 3"} we study the case of type $\mathbf{D}$ singularities. Using techniques similar to [@blancNonrationalityFibrationsAssociated2015], we manage to compute all the isomorphisms between deformations as affine varieties (Theorem [Theorem 51](#teo iso Dn){reference-type="ref" reference="teo iso Dn"}); we then compute the subgroupoid of Poisson isomorphisms and confirm Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} (Theorem [Theorem 53](#teo iso type D){reference-type="ref" reference="teo iso type D"}). #### Acknowledgments. I am grateful for the constant encouragement, guidance and support provided by my supervisors Daniele Valeri and Gwyn Bellamy, which made this work possible. I would also like to thank Franco Rota for the many helpful conversations, and for patiently answering my questions. This paper was supported by the EPSRC (Engineering and Physical Sciences Research Council) via a postgraduate scholarship. # Type A {#section 2} Let $V$ be a complex vector space of dimension $2$. Choose a basis for $V$, and let $X,Y$ be the corresponding coordinate functions. We then identify $SL(V)$ with $SL_2({\mathbb C})$. Take $\Gamma=C_n\subset SL_2({\mathbb C})$ the cyclic group of order $n$, with $n\geq 2$, acting on $V$ via matrices of the form $$\begin{pmatrix} e^{2k\pi i/n } & 0\\ 0 & e^{-2k\pi i/n } \end{pmatrix}, \hspace{4mm} \text{with $1\leq k\leq n$}\, .$$ The quotient $V/\Gamma$ is the Kleinian singularity of type $\mathbf{A_{n-1}}$. It is well-known that the algebra of functions ${\mathbb C}[V/\Gamma]={\mathbb C}[V]^\Gamma$ is generated by the monomials $X^n,Y^n$ and $XY$, so we have $${\mathbb C}[\mathbf{A_{n-1}}]={\mathbb C}[x,y,z]/(xy-z^n)\, ,$$ for all $n\geq 2$. Considering $X,Y$ to be of degree $1$, we have a grading on ${\mathbb C}[\mathbf{A_{n-1}}]$ given by $\deg x=\deg y=n$ and $\deg z=2$. We also have a Poisson structure on ${\mathbb C}[\mathbf{A_{n-1}}]$, induced by the symplectic structure on $V$; we choose here the normalization that gives the bracket $\{Y,X\}=1/n$ on ${\mathbb C}[V]$. The induced Poisson structure on ${\mathbb C}[\mathbf{A_{n-1}}]$ is then $$\{x,y\}=-nz^{n-1}\, , \hspace{5mm} \{z,x\}=x \hspace{5mm} \{z,y\}=-y\, .$$ This gives ${\mathbb C}[\mathbf{A_{n-1}}]$ the structure of a graded Poisson algebra, with Poisson bracket of degree $-2$. **Remark 4**. Let $\psi$ be a polynomial in $x,y,z$. It induces a Poisson structure on ${\mathbb C}[x,y,z]$, defined by $$\{x,y\}=\pdv{\psi}{z}, \ \ \ \{x,z\}=-\pdv{\psi}{y}, \ \ \ \{y,z\}=\pdv{\psi}{x}.$$ Since the ideal $(\psi)\subset{\mathbb C}[x,y,z]$ is Poisson, $\psi$ induces a Poisson structure on the quotient ${\mathbb C}[x,y,z]/(\psi)$. If additionally $\psi$ is homogeneous with respect to some grading of ${\mathbb C}[x,y,z]$, then ${\mathbb C}[x,y,z]$ and ${\mathbb C}[x,y,z]/(\psi)$ are graded Poisson algebras, and the Poisson bracket has degree $(\deg \psi-\deg x-\deg y-\deg z)$. The Poisson structure on ${\mathbb C}[\mathbf{A_{n-1}}]$ is of this type, where $\psi=xy-z^n$. ## Deformations and quantizations We have explicit presentations for both deformations and quantizations of the algebra ${\mathbb C}[\mathbf{A_{n-1}}]$. In both cases, they are parameterised by a monic polynomial $P\in{\mathbb C}[z]$ of degree $n$, with no degree $n-1$ term. Fix $n\geq2$ and denote by ${\mathcal A}(P)$ the deformation associated to the parameter $P$ of the algebra ${\mathbb C}[\mathbf{A_{n-1}}]$. Explicitly, we have $${\mathcal A}(P)={\mathbb C}[x,y,z]/(xy-P(z)).$$ These algebras come from specializing the semi universal deformation of the Kleinian singularity (see for example [@katzGorensteinThreefoldSingularities1992 Table 3]). We give ${\mathcal A}(P)$ the structure of a Poisson algebra as in Remark [Remark 4](#oss generale poisson structure on C[x,y,z]){reference-type="ref" reference="oss generale poisson structure on C[x,y,z]"} $$\{x,y\}=\pdv{\psi}{z}=-P'(z), \ \ \ \{z,x\}=\pdv{\psi}{y}=x, \ \ \ \{z,y\}=-\pdv{\psi}{x}=-y.$$ The algebra ${\mathcal A}(P)$ is a filtered Poisson algebra, with filtration induced by the degree on the generators $\deg x=\deg y=n$ and $\deg z=2$, with Poisson bracket of degree $-2$, i.e. $\{F_k,F_m\}\subset F_{k+m-2}$ for all filtration terms $F_k,F_m$. Quantizations of ${\mathbb C}[\mathbf{A_{n-1}}]$ are examples of generalized Weyl algebras [@bavulaIsomorphismProblemsGroups2001]. We recall here the general definition. **Definition 5**. Let $D$ be a ring, $\sigma$ an automorphism of $D$, and $a$ a central element of $D$. The generalized Weyl algebra $D(\sigma,a)$ is the ring extension of $D$ generated by two indeterminates $x,y$ subject to the relations 1. $xd=\sigma(d)x$ and $yd=\sigma^{-1}(d)y$ for all $d\in D$, 2. $xy=\sigma(a)$ and $yx=a$. In our case, the quantization associated to the parameter $P$ is the generalized Weyl algebra $$A(P):={\mathbb C}[z](\sigma,P),$$ with $\sigma$ defined by $z\mapsto z-1$. Explicitly, $A(P)$ is the ${\mathbb C}$-algebra generated by $x,y,z$ subject to the relations $$\label{eq defining A(P)} xz=(z-1)x, \ \ yz=(z+1)y, \ \ xy=P(z-1), \ \ yx=P(z).$$ The algebra $A(P)$ is a filtered associative algebra, with filtration induced by $\deg x=\deg y=n$ and $\deg z=2$. The commutator is of degree $-2$, i.e. $[F_k,F_m]\subset F_{k+m-2}$ for all filtration terms $F_k,F_m$. We can give an explicit construction of the algebra ${\mathcal A}(P)$ as the semi-classical limit of $A(P)$. Define $$A_t(P):={\mathbb C}[t,z](\sigma_t,P),$$ with $\sigma_t$ defined by $z\mapsto z-t$, $t\mapsto t$. Explicitly, $A_t(P)$ is the ${\mathbb C}[t]$-algebra generated by $x,y,z$ subject to the following relations: $$\label{eq defining A_t(P)} xz=(z-t)x, \ \ yz=(z+t)y, \ \ xy=P(z-t), \ \ yx=P(z).$$ **Remark 6**. The defining relations of $A(P)$ and $A_t(P)$ imply respectively that $$=(\sigma-1)(P), \ \ \ \ \ \ [x,y]=(\sigma_t-1)(P).$$ Define for all $m\in{\mathbb N}$ the operators $\delta_m:=(\sigma^m-1)$ and $\delta_{m,t}:=(\sigma_t^m-1)$, so that $$=\delta_1(P) \ \ \ \ \ \ [x,y]=\delta_{1,t}(P),$$ respectively in $A(P)$ and $A_t(P)$. Denote by $A_1(P):=A_t(P)/((t-1)A_t(P))$ and by $A_0(P):=A_t(P)/(tA_t(P))$. **Proposition 7**. *The algebras $A_1(P)$ and $A(P)$ are isomorphic as filtered associative algebras. The algebra $A_0$ has the structure of a filtered Poisson algebra, with Poisson bracket given by $\{\cdot,\cdot\}=\frac{1}{t}[\cdot, \cdot]$ mod $(t)$. Moreover, the algebras $A_0(P)$ and ${\mathcal A}(P)$ are isomorphic as filtered Poisson algebras.* *Proof.* The isomorphism $A_1(P)\cong A(P)$ is trivial. Notice that $[A_t(P),A_t(P)]\subset tA_t(P)$ and $A_t(P)$ is free over ${\mathbb C}[t]$, so $A_0(P)$ is commutative and $\{\cdot,\cdot\}=\frac{1}{t}[\cdot, \cdot]$ mod $(t)$ defines a Poisson bracket. We can see that it coincides with the one of ${\mathcal A}(P)$: the only non-trivial check is $$\frac{1}{t}[x,y] \! \! \mod (t)=\frac{P(z-t)-P(z)}{t}\!\!\mod (t)=-P'(z) \!\!\mod (t).$$ ◻ **Remark 8**. If we instead take the associated graded of either ${\mathcal A}(P)$ or $A(P)$, we get back the algebra of functions on the Kleinian singularity ${\mathbb C}[\mathbf{A_{n-1}}]$. We introduce here the following notation, that will be used throughout the rest of the paper. If $\phi$ is an endomorphism of an algebra with three generators $x,y,z$, we will identify $\phi$ with the triple $(\phi(x),\phi(y),\phi(z))$ of its value on the generators $(x,y,z)$. **Theorem 9** ([@bavulaIsomorphismProblemsGroups2001 Theorem 3.9]). *The group of automorphisms of the algebra $A(P)$ has the following generators:* (a) *$$\Phi_{\lambda,m}=\left(x+\sum_{i=1}^n{\frac{(-\lambda)^i}{i!}y^{im-1}\delta_m^i(P)}, y, z+m\lambda y^m \right),$$ for every ${\lambda}\in{\mathbb C}$ and $m\in{\mathbb N}$;* (b) *$$\Psi_{\lambda,m}=\left(x, y+\sum_{i=1}^n{\frac{\lambda^i}{i!}\delta_m^i(P)x^{im-1}}, z-m\lambda x^m\right),$$ for every ${\lambda}\in{\mathbb C}$ and $m\in{\mathbb N}$;* (c) *$$\Theta_\nu=(\nu x, \nu^{-1} y, z),$$ for all $\nu\in{\mathbb C}^\times$;* (d) *$$\Omega=(y, (-1)^n x, 1-z),$$ only if the polynomial $P$ is either odd or even, i.e. if $P(-z)=\pm P(z)$.* **Remark 10**. A polynomial of degree $n$ is called reflective if there exists a $\rho\in{\mathbb C}$ such that $P(\rho-x)=(-1)^nP(x)$. If $P$ is reflective, then $A(P)$ has an automorphism of the form $\Omega_\rho=(y, (-1)^n x, 1+\rho-z)$ [@bavulaIsomorphismProblemsGroups2001 Lemma 3.8]. Notice though that if we restrict ourselves to polynomials with no term of degree $n-1$, $P$ can be reflective only if $\rho=0$. In that case, the polynomial $P$ is either odd or even and $\Omega_\rho=\Omega$. For brevity and to stick with the original notation of Bavula and Jordan, we will write "$P$ is reflective" instead of "$P$ is either odd or even". **Remark 11**. We can define the automorphisms of type $\Phi$ and $\Psi$ in a different way. For all $m\geq0$, consider $\mathop{\mathrm{ad}}(x^m)$ and $\mathop{\mathrm{ad}}(y^m)$, the adjoint actions of $x^m$ and $y^m$. These are nilpotent the derivations of $A(P)$ (see [@bavulaIsomorphismProblemsGroups2001 Lemma 3.4]). If we exponentiate them, we get automorphisms of $A(P)$. In fact, $$e^{{\lambda}\mathop{\mathrm{ad}}(x^m)}=\Phi_{{\lambda},m} \text{ and } e^{{\lambda}\mathop{\mathrm{ad}}(y^m)}=\Psi_{{\lambda},m}$$ (see [@bavulaIsomorphismProblemsGroups2001 Lemma 3.4]). We can lift the generators of $\mathop{\mathrm{Aut}}(A(P))$ to the algebra $A_t(P)$. Define the following ${\mathbb C}[t]$-linear automorphisms of $A_t(P)$: (a) $$\Phi_{{\lambda},m,t}:=e^{{\lambda}/t\mathop{\mathrm{ad}}(x^m)}=\left(x+\sum_{i=1}^n{\frac{(-\lambda)^i}{t^ii!}y^{im-1}\delta_{m,t}^i(P)}, y, z+m\lambda y^m \right),$$ for all ${\lambda}\in{\mathbb C}$ and $m\geq0$; (b) $$\Psi_{{\lambda},m,t}:=e^{{\lambda}/t\mathop{\mathrm{ad}}(y^m)}=\left(x, y+\sum_{i=1}^n{\frac{\lambda^i}{t^ii!}\delta_{m,t}^i(P)x^{im-1}}, z-m\lambda x^m\right),$$ for all ${\lambda}\in{\mathbb C}$ and $m\geq0$; (c) $\Theta_{\nu,t}=(\nu x, \nu^{-1} y, z)$ for all $\nu\in{\mathbb C}^\times$; (d) $\Omega_t=(y, (-1)^n x, t-z)$, only if $P$ is reflective. We can also consider how they act on the quotient $A_0$. Using that $\{\cdot,\cdot\}=\frac{1}{t}[\cdot, \cdot] \mod (t)$, we have $$e^{{\lambda}\mathop{\mathrm{ad}}(x^m)}=\Phi_{{\lambda},m,0} \text{ and } e^{{\lambda}\mathop{\mathrm{ad}}(y^m)}=\Psi_{{\lambda},m.0},$$ where now "$\mathop{\mathrm{ad}}$" denotes the adjoint action with respect to the Poisson bracket. Denote by $G_t$ the group generated by the automorphisms (a)-(d) of $A_t(P)$. Since the automorphisms in $G_t$ are $t$-linear, $G_t$ acts on the algebras $A_0(P)$ and $A_1(P)$. Denote the images of $G_t$ in these representations as $G_0$ and $G_1$ respectively. By Theorem [Theorem 9](#teo automorfismi quantizzazione type A){reference-type="ref" reference="teo automorfismi quantizzazione type A"}, $$\label{eq G_1=Aut} G_1=\mathop{\mathrm{Aut}}(A(P)).$$ We also have $$\label{eq G_0 in PAut} G_0\leq \mathop{\mathrm{PAut}}({\mathcal A}(P)).$$ In fact, let $\phi\in G_0$ and $a,b\in{\mathcal A}(P)\cong A_0(P)$. Then $$\phi(\{a,b\})=\frac{1}{t}\phi([\hat{a},\hat{b}]) \!\! \mod (t)=\frac{1}{t}[\phi(\hat{a}),\phi(\hat{b})] \!\! \mod (t)= \{\phi(a),\phi(b)\},$$ where $\hat{a},\hat{b}$ are some lifts of $a,b$ in $A_t(P)$. ## Amalgamated free product group structure {#section amalgamated free product} The goal of this section is to prove Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"}, which gives us the explicit group structure of $G_t(P)$ as an abstract group. The following proposition gives us some useful identities. **Lemma 12**. *The following relations hold in $G_t$. $$\begin{aligned} &\Theta_{\nu,t}\circ\Theta_{\mu,t}=\Theta_{\mu+\nu,t}, \label{eq lemma commuting relations G_t -1} \\ &\Omega_t^2=\Theta_{(-1)^n,t},\label{eq lemma commuting relations G_t 0}\\ &\Theta_{\nu,t}\circ\Psi_{\lambda,m,t}=\Psi_{{\lambda}\nu^m,m,t}\circ\Theta_{\nu,t},\label{eq lemma commuting relations G_t 1}\\ &\Theta_{\nu,t}\circ\Phi_{\lambda,m,t}=\Phi_{{\lambda}\nu^{-m},m,t}\circ\Theta_{\nu,t},\label{eq lemma commuting relations G_t 2} \end{aligned}$$ for all $\nu\in{\mathbb C}^\times$, ${\lambda}\in{\mathbb C}$ and $m\in{\mathbb N}$.* *If $P$ is reflective, we also have $$\begin{aligned} &\Omega_t\circ \Theta_{\nu,t}=\Theta_{\nu^{-1},t}\circ\Omega_t \label{eq lemma commuting relations G_t 3} \\ &\Theta_{(-1)^n,t}\circ \Omega_t\circ\Phi_{\lambda,m,t}\circ\Omega_t=\Psi_{\lambda,m,t}, \label{eq lemma commuting relations G_t 4} \end{aligned}$$ for all $\nu\in{\mathbb C}^\times$, ${\lambda}\in{\mathbb C}$ and $m\in{\mathbb N}$.* *In particular, if $P$ is reflective, $G_t$ can be generated by just the automorphisms $\Theta_{\nu,t}$, $\Phi_{{\lambda},m,t}$ and $\Omega_t$.* *Proof.* Equations [\[eq lemma commuting relations G_t -1\]](#eq lemma commuting relations G_t -1){reference-type="eqref" reference="eq lemma commuting relations G_t -1"} and [\[eq lemma commuting relations G_t 0\]](#eq lemma commuting relations G_t 0){reference-type="eqref" reference="eq lemma commuting relations G_t 0"} follow from a direct check. To prove [\[eq lemma commuting relations G_t 1\]](#eq lemma commuting relations G_t 1){reference-type="eqref" reference="eq lemma commuting relations G_t 1"}, compute $$\begin{aligned} \Theta_{\nu,t}\circ\Psi_{\lambda,m,t}&=\left(\nu x,\nu^{-1} y+\sum_{i=1}^n\frac{{\lambda}^i}{t^ii!}\nu^{im-1}\delta_{m,t}^i(P)x^{im-1}, z-\nu^m m{\lambda}x^m\right)\\ &=\Psi_{{\lambda}\nu^m,m,t}\circ\Theta_{\nu,t}. \end{aligned}$$ Similarly, for equation [\[eq lemma commuting relations G_t 2\]](#eq lemma commuting relations G_t 2){reference-type="eqref" reference="eq lemma commuting relations G_t 2"} $$\begin{aligned} \Theta_{\nu,t}\circ\Phi_{\lambda,m,t}&=\left(\nu x+\sum_{i=1}^n\frac{(-{\lambda})^i}{t^ii!}\nu^{1-im}x^{im-1}\delta_{m,t}^i(P),\nu^{-1} y, z+\nu^{-m} m{\lambda}y^m\right)\\ &=\Phi_{{\lambda}\nu^{-m},m,t}\circ\Theta_{\nu,t}. \end{aligned}$$ Equation [\[eq lemma commuting relations G_t 3\]](#eq lemma commuting relations G_t 3){reference-type="eqref" reference="eq lemma commuting relations G_t 3"} follows from a direct check $$\begin{aligned} \Omega_t\circ\Theta_{\nu,t}=(\nu y, (-1)^n\nu^{-1} x,t-z)=\Theta_{\nu^{-1},t}\circ\Omega_t.\end{aligned}$$ Equation [\[eq lemma commuting relations G_t 4\]](#eq lemma commuting relations G_t 4){reference-type="eqref" reference="eq lemma commuting relations G_t 4"} is more involved. Let us check it separately on each of the three generators. For $x$ and $z$ we have $$x\mapsto y\mapsto y\mapsto (-1)^nx\mapsto x,$$ $$z\mapsto t-z\mapsto t-z-m\lambda y^m\mapsto z-(-1)^{n\cdot m}{\lambda}mx^m\mapsto z-m{\lambda}x^m.$$ Let us now check it for $y$: $$\label{eq lemma commuting relations G_t check 4 } \begin{aligned} y&\mapsto (-1)^nx \mapsto (-1)^n\left[x+\sum_{i=1}^n{\frac{(-\lambda)^i}{t^ii!}y^{im-1}\delta_{m,t}^i(P)}\right] \\ &\mapsto (-1)^n\left[y+\sum_{i=1}^n{\frac{(-\lambda)^i}{t^ii!}(-1)^{(im-1)n}x^{im-1}(\delta_{m,t}^iP)(t-z)}\right] \\ &\mapsto y+(-1)^n\sum_{i=1}^n{\frac{(-\lambda)^i}{t^ii!}x^{im-1}(\delta_{m,t}^iP)(t-z)}. \end{aligned}$$ The final term in [\[eq lemma commuting relations G_t check 4 \]](#eq lemma commuting relations G_t check 4 ){reference-type="eqref" reference="eq lemma commuting relations G_t check 4 "} is similar to $\Psi_{{\lambda},m,t}(y)$, but we need to move $x^{im-1}$ to the right of $(\delta_{m,t}^iP)(t-z)$. By relations [\[eq defining A_t(P)\]](#eq defining A_t(P)){reference-type="eqref" reference="eq defining A_t(P)"}, we can do that by applying $\sigma^{im-1}_t$ to $(\delta_{m,t}^iP)(t-z)$. Notice that $$(\delta_{m,t}^iP)(t-z)=\sigma_t\circ\gamma\circ(\sigma^m_t-1)^i(P),$$ where $\gamma$ is the map $z\mapsto -z$. Hence $$(\delta_{m,t}^iP)(t-z)=(-1)^n\sigma_t\circ(\sigma^{-m}_t-1)^i(P),$$ because $P$ is reflective. It follows that $$\begin{aligned} \sigma^{im-1}_t((\delta_{m,t}^iP)(t-z))&=(-1)^n\sigma_t^{im}\circ(\sigma^{-m}_t-1)^i(P)\\ &=(-1)^{n}(1-\sigma^m)^i(P)\\ &=(-1)^{n+i}\delta^i_{m,t}(P). \end{aligned}$$ Putting this together with [\[eq lemma commuting relations G_t check 4 \]](#eq lemma commuting relations G_t check 4 ){reference-type="eqref" reference="eq lemma commuting relations G_t check 4 "} we get $$\Theta_{(-1)^n,t}\circ \Omega_t\circ\Phi_{\lambda,m,t}\circ\Omega_t(y)=y+\sum_{i=1}^n{\frac{\lambda^i}{t^ii!}\delta_{m,t}^i(P)x^{im-1}}=\Psi_{{\lambda},m,t}(y)$$ This completes the proof. ◻ Let us recall the definition of the amalgamated free product. Let $G$ be a group and $H,K$ two subgroups of $G$, and let $L:=H\cap K$. The group $G$ is the free product of the subgroups $H$ and $K$ with the amalgamated subgroup $L$, and is denoted by $G=H\ast_{L}K$, if (a) $G$ is generated by the subgroups $H$ and $K$; (b) the defining relations of $G$ consist only of the defining relations of the subgroups $H$ and $K$. **Theorem 13**. *Let $H,K$ be subgroups of $G$, and $L=H\cap K$. If $S_1$ is a set of coset representatives for $L$ in $H$ and $S_2$ is a set of left coset representatives for $L$ in $K$, then $G=H\ast_{L} K$ if and only if every element $g\in G$ can be written uniquely as $$g=g_1\dotso g_k\alpha,$$ where $\alpha\in L$, $g_i\in S_1\cup S_2$ and $g_i,g_{i+1}$ do not belong in $S_1$ and $S_2$ at the same time, for all $i=1,\dotso,k$.* A proof of Theorem [Theorem 13](#teo tecnico amalgamated free product){reference-type="ref" reference="teo tecnico amalgamated free product"} can be found in [@solitarCombinatorialGroupTheory1976 Corollary 4.4.1]. We want to show that $G_t, G_0$ and $G_1$ have an amalgamated free product structure. We introduce the following notation. (i) $\Phi:=\langle \Phi_{{\lambda},m,t} \,| \,{\lambda}\in{\mathbb C}, m\in{\mathbb N}\rangle$ (ii) $\Psi:=\langle \Psi_{{\lambda},m,t} \,| \,{\lambda}\in{\mathbb C}, m\in{\mathbb N}\rangle$ (iii) $\Theta:=\langle \Theta_{\nu,t} \,| \,\nu\in{\mathbb C}^\times\rangle$ Let us first consider the case when $P$ is reflective. Define (a) $T:=\langle \Phi, \Theta \rangle$; (b) $J:=\langle \Omega_t, \Theta \rangle$. Lemma [Lemma 12](#lemma commuting relations G_t){reference-type="ref" reference="lemma commuting relations G_t"} implies that $G_t$ is generated by $T$ and $J$. Clearly, $T\cap J=\Theta$. **Remark 14**. We have that $T=\Phi\rtimes\Theta$. This follows at once from relation [\[eq lemma commuting relations G_t 1\]](#eq lemma commuting relations G_t 1){reference-type="eqref" reference="eq lemma commuting relations G_t 1"}. Thus, the elements of $\Phi$ form a set of left coset representatives for $\Theta$ in $T$. Notice that, due to the properties of the exponential, each automorphism in $\Phi$ has the form $\exp^{1/t\mathop{\mathrm{ad}}(g(y))}$, for some $g\in{\mathbb C}[y]$. For consistency with the notation used in Section [2.3](#section Poisson automorphisms type A){reference-type="ref" reference="section Poisson automorphisms type A"}, we define $$\label{def Phi_g} \Phi_{g,t}:=\exp^{1/t\mathop{\mathrm{ad}}(\hat{g}(y))},$$ where $\hat{g}:=\int^y_0g(t)dt$ denotes the antiderivative of $g$. From the properties of the exponential it follows that $$\Phi_{g,t}\circ\Phi_{h,t}=\Phi_{g+h,t} \hspace{8mm} \Phi_{0,t}=\mathop{\mathrm{id}},$$ so $\Phi\cong{\mathbb C}[y]$ as an additive group via the identification $g\mapsto\Phi_{g,t}$. From relation [\[eq lemma commuting relations G_t 3\]](#eq lemma commuting relations G_t 3){reference-type="eqref" reference="eq lemma commuting relations G_t 3"} we can also see that $\{\Omega_t,\mathop{\mathrm{id}}\}$ is a set of left coset representatives for $\Theta$ in $J$. **Proposition 15**. *If $P$ is reflective, all $\phi\in G_t$ can be written in the form $$\label{eq decomposition G_t reflective} \phi=\Phi_{g_1,t}\circ\Omega_t\circ\dots\circ\Omega_t\circ\Phi_{g_s,t}\circ \Theta_{\nu,t},$$ where $s\geq0$, $g_i\in{\mathbb C}[y]$ for all $i$ and $0\neq g_i$ for $1<i<s$ , and $\nu\in{\mathbb C}^\times$.* *Proof.* By definition of $G_t$ we have $$\label{eq proof decomposition G_t reflective} \phi=\phi_1\circ\dots\circ\phi_k,$$ with $\phi_i$ either $\Phi_{g,t},\Omega_t$ or $\Theta_{\nu,t}$. From relations [\[eq lemma commuting relations G_t 1\]](#eq lemma commuting relations G_t 1){reference-type="eqref" reference="eq lemma commuting relations G_t 1"} and [\[eq lemma commuting relations G_t 3\]](#eq lemma commuting relations G_t 3){reference-type="eqref" reference="eq lemma commuting relations G_t 3"}, we can take every automorphism of type $\Theta_{\nu,t}$ to the right. Since $\Theta_{\nu,t}\circ \Theta_{\mu,t}=\Theta_{\nu\mu,t}$, $\Omega_t^2=\Theta_{(-1)^n,t}$ and $\Phi_g\circ\Phi_h=\Phi_{g+h}$, we can always rewrite [\[eq proof decomposition G_t reflective\]](#eq proof decomposition G_t reflective){reference-type="eqref" reference="eq proof decomposition G_t reflective"} as $$\phi=\Phi_{g_1,t}\circ\Omega_t\circ\dots\circ\Omega_t\circ\Phi_{g_s,t}\circ \Theta_{\nu,t},$$ with $s\geq0$, $g_i\in{\mathbb C}[y]$ for all $i$ and $0\neq g_i$ for $1<i<s$ , and $\nu\in{\mathbb C}^\times$. ◻ Let us now consider the case where $P$ is not reflective. Define (a) $Q_1=\langle \Phi, \Theta\rangle$; (b) $Q_2=\langle \Psi, \Theta\rangle$. By definition, $G_t$ is generated by $Q_1$ and $Q_2$. Clearly, $Q_1\cap Q_2=\Theta$. **Remark 16**. We have that $Q_1=\Phi\rtimes\Theta$ and $Q_2=\Phi\rtimes\Theta$. This follows at once from relation [\[eq lemma commuting relations G_t 2\]](#eq lemma commuting relations G_t 2){reference-type="eqref" reference="eq lemma commuting relations G_t 2"}. Thus, the elements of $\Phi$ (respectively $\Psi$) form a set of left coset representatives for $\Theta$ in $Q_1$ (respectively for $\Theta$ in $Q_2$). As in [\[def Phi_g\]](#def Phi_g){reference-type="eqref" reference="def Phi_g"} we can define $$\Psi_{g,t}:=\exp^{1/t\mathop{\mathrm{ad}}(\hat{g}(x))}.$$ This defines an isomorphism $\Psi\cong{\mathbb C}[x]$ as additive groups via the map $g\mapsto \Psi_{g,t}$. **Proposition 17**. *If $P$ is not reflective, every $\phi\in G_t$ can be written in the form $$\phi=\Phi_{g_1,t}\circ\Psi_{h_1,t}\circ\dots\circ\Phi_{g_{s-1},t}\circ\Psi_{h_{s-1},t}\circ\Phi_{g_{s},t}\circ \Theta_{\nu,t},$$ with $s\geq1$, $g_i\in{\mathbb C}[y]$ for all $i$ with $0\neq g_i$ for $i=2,\dotso,s-1$, $0\neq h_i\in{\mathbb C}[x]$ for all $i$, and $\nu\in{\mathbb C}^\times$.* *Proof.* The proof is analogous to that of Proposition [Proposition 15](#prop decomposition G_t reflective){reference-type="ref" reference="prop decomposition G_t reflective"}, and it is a simple application of the relations [\[eq lemma commuting relations G_t -1\]](#eq lemma commuting relations G_t -1){reference-type="eqref" reference="eq lemma commuting relations G_t -1"}, [\[eq lemma commuting relations G_t 1\]](#eq lemma commuting relations G_t 1){reference-type="eqref" reference="eq lemma commuting relations G_t 1"} and [\[eq lemma commuting relations G_t 2\]](#eq lemma commuting relations G_t 2){reference-type="eqref" reference="eq lemma commuting relations G_t 2"}. ◻ We are left to prove that the decompositions of Propositions [Proposition 15](#prop decomposition G_t reflective){reference-type="ref" reference="prop decomposition G_t reflective"} and [Proposition 17](#prop decomposition G_t non reflective){reference-type="ref" reference="prop decomposition G_t non reflective"} are unique. To do that, we introduce the notion of multidegree of an automorphism. For all $a\in A_t(P)$, we define $\deg(a)$ as the smallest natural number $i$ such that $a\in F_i$, with $\deg(0)=-\infty$. For every automorphism $\phi$ in $G_t$, we define its multidegree to be: $$\mathop{\mathrm{mdeg}}(\phi)=(\deg(\phi(x)),\deg(\phi(y)),\deg(\phi(z))).$$ **Lemma 18**. *For all $a,b\in A_t(P)$, $\deg(ab)=\deg(a)+\deg(b)$.* *Proof.* For $a\in A_t(P)$, let $\mathop{\mathrm{gr}}(a):=a+F_{i-1}\in\mathop{\mathrm{gr}}(A_t(P))$, where $i=\deg(a)$. Clearly, $i=\deg(a)=\deg(\mathop{\mathrm{gr}}(a))$. Since $\mathop{\mathrm{gr}}(A_t(P))\cong{\mathbb C}[t,x,y,z]/(xy-z^n)$ is a domain, $$\deg((\mathop{\mathrm{gr}}(a)\mathop{\mathrm{gr}}(b))=\deg(\mathop{\mathrm{gr}}(a))+\deg(\mathop{\mathrm{gr}}(b))$$ for all $a,b\in A_t(P)$. Thus $$\deg(ab)=\deg(\mathop{\mathrm{gr}}(ab))=\deg((\mathop{\mathrm{gr}}(a)\mathop{\mathrm{gr}}(b))=\deg(a)+\deg(b).$$ ◻ **Lemma 19**. *Let $g\in{\mathbb C}[x]$ be a polynomial of degree $k$, and $\phi=(v_1,v_2,v_3)$ be an automorphism in $G_t$, such that $deg(v_2)>\deg(v_3)$ and $\deg(v_2)\geq\deg(v_1)$. Then $\phi\circ\Phi_{g,t}$ has multidegree $$((nk+n-1)\deg(v_2), \deg(v_2), (k+1)\deg(v_2) ).$$* *Similarly, if $\deg(v_1)>\deg(v_3)$ and $\deg(v_1)\geq\deg(v_2)$, then $\phi\circ\Psi_{g,t}$ has multidegree $$(\deg(v_1), (nk+n-1)\deg(v_1), (k+1)\deg(v_1)).$$* *Proof.* Recall that $\Phi_{g,t}=e^{1/t\mathop{\mathrm{ad}}(\hat{g}(y))}$ [\[def Phi_g\]](#def Phi_g){reference-type="eqref" reference="def Phi_g"}. We have that $$\label{eq adjoint action y^m} \mathop{\mathrm{ad}}(y^m): x\mapsto -y^{m-1}\delta_{m,t}(P), \ f(z)\mapsto -y^m\delta_{m,t}(f), \ y\mapsto 0$$ for all $f\in{\mathbb C}[z]$ (see [@bavulaIsomorphismProblemsGroups2001 Equation 8]). This implies that $\Phi_{g,t}(y)=y$, hence $\deg(\phi\circ\Phi_{g,t}(y))=\deg(v_2)$. Let $a y^{k+1}$ be the leading term of $\hat{g}$. From [\[eq adjoint action y\^m\]](#eq adjoint action y^m){reference-type="eqref" reference="eq adjoint action y^m"} $$\phi\circ\Phi_{g,t}(z)=v_3+(k+1)av_2^{k+1}+\text{lower terms in {$v_2$}. }$$ Since $\deg(v_2)\geq\deg(v_i)$ for $i=1,2,3$, from Lemma [Lemma 18](#lemma degree){reference-type="ref" reference="lemma degree"} $$\deg(\phi\circ\Phi_{g,t}(z))=(k+1)\deg(v_2).$$ From [\[eq adjoint action y\^m\]](#eq adjoint action y^m){reference-type="eqref" reference="eq adjoint action y^m"} it follows that, for all $m,r\in{\mathbb N}$ $$\label{eq leading term 1} \relax [\hat{g}(y),y^mz^r]=\alpha t y^{k+m+1}z^{r-1}+\text{monomials proportional to $y^iz^j$,}$$ with $i\leq k+m+1$, $j\leq r-1$ and $(i,j)\neq (k+m+1,r-1)$, for some $\alpha\in{\mathbb C}^\times$. It also follows that $$\label{eq leading term 2} \relax [\hat{g}(y),x]=\beta t y^{k}z^{n-1}+\text{monomials proportional to $y^iz^j$,}$$ with $i\leq k$, $j\leq n-1$ and $(i,j)\neq (k,n-1)$, for some $\beta\in{\mathbb C}^\times$. We now want to prove, by induction on $s$, that $$\label{eq lemma multidegree induction} \mathop{\mathrm{ad}}^s(\hat{g}(y))(x)=\alpha t^s y^{s(k+1)-1}z^{n-s}+\text{monomials proportional to $y^iz^j$,}$$ with $i\leq s(k+1)-1$, $j\leq n-s$ and $(i,j)\neq (s(k+1)-1,n-s)$, for some $\alpha\in{\mathbb C}^\times$. The base step is [\[eq leading term 1\]](#eq leading term 1){reference-type="eqref" reference="eq leading term 1"}. Consider now $$\mathop{\mathrm{ad}}^{s+1}(\hat{g}(y))(x)=[\hat{g}(y),\alpha t^s y^{s(k+1)-1}z^{n-s}+\text{lower terms}],$$ for some $\alpha\in{\mathbb C}^\times$, true by inductive hypothesis. Applying [\[eq leading term 1\]](#eq leading term 1){reference-type="eqref" reference="eq leading term 1"} we get [\[eq lemma multidegree induction\]](#eq lemma multidegree induction){reference-type="eqref" reference="eq lemma multidegree induction"}. It is then clear that $$\deg{(\phi\circ\mathop{\mathrm{ad}}^s(\hat{g}(y))(x))}=\deg{(\alpha t^s v_2^{s(k+1)-1}v_3^{n-s})}.$$ Since $\deg(v_2)>\deg(v_3)$, we get that $\deg ({\phi\circ\Phi_{g,t}(x)})=\deg ({\alpha v_2^{nk+n-1}})$, hence from Lemma [Lemma 18](#lemma degree){reference-type="ref" reference="lemma degree"} $$\deg(\phi\circ\Phi_{g,t})=(nk+n-1)\deg(v_2).$$ The proof for $\phi\circ\Psi_{g,t}$ is analogous, using the relations $$\label{eq adjoint action x^m} \mathop{\mathrm{ad}}(x^m): x\mapsto 0, \ f(z)\mapsto \delta_{m,t}(f)x^m, \ y\mapsto \delta_{m,t}(P)x^{m-1}$$ [@bavulaIsomorphismProblemsGroups2001 Equation 6] instead of [\[eq adjoint action y\^m\]](#eq adjoint action y^m){reference-type="eqref" reference="eq adjoint action y^m"}. ◻ **Proposition 20**. *Let $n>2$. An automorphism of the form $$\phi=\Phi_{g_1,t}\circ\Omega_t\circ\Phi_{g_2,t}\dots\circ\Omega_t\circ\Phi_{g_s,t},$$ with $s\geq1$, $0\neq g_i\in{\mathbb C}[y]$ has multidegree $$\left(n\prod_{i=1}^s(nk_i+n-1),n\prod_{i=1}^{s-1}(nk_i+n-1),n(k_s+1)\prod_{i=1}^{s-1}(nk_i+n-1)\right),$$ where $k_i=\deg(g_i)$.* *Proof.* We will prove this by induction on $s$. If $s=1$, then we have $\phi=\Phi_{g_1,t}$; since $\deg(y)=\deg(x)>\deg(z)$, we can apply Lemma [Lemma 19](#lemma multidegree){reference-type="ref" reference="lemma multidegree"} on $\mathop{\mathrm{id}}\circ\Phi_{g_1,t}$. Thus $$\mathop{\mathrm{mdeg}}(\phi)=(n(nk+n-1),n,n(k+1)).$$ When $s>1$, we have $$\phi=\psi\circ\Omega_t\circ\Phi_{g_s,t},$$ with $$\psi=\Phi_{g_1,t}\circ\Omega_t\circ\Phi_{g_2,t}\dots\circ\Omega_t\circ\Phi_{g_{s-1},t}.$$ We know by the induction hypothesis that $\psi=(u_1,u_2,u_3)$ has multidegree $$\left(n\prod_{i=1}^{s-1}(nk_i+n-1),n\prod_{i=1}^{s-2}(nk_i+n-1),n(k_{s-1}+1)\prod_{i=1}^{s-2}(nk_i+n-1)\right).$$ Notice that $\deg(u_2)\leq\deg(u_3)<\deg(u_1)$, because $(nk+n-1)>(k+1)\geq1$ for all $k\geq0$, when $n>2$. But now $\psi\circ\Omega_t=(u_2,(-1)^nu_1,t-u_3)$, hence we can once again apply lemma [\[lemma multidegree\]](#lemma multidegree){reference-type="eqref" reference="lemma multidegree"} to $(\psi\circ\Omega_t)\circ\Phi_{g_s,t}$ and complete the proof. ◻ **Proposition 21**. *Let $n>2$. An automorphism of the form $$\phi=\Phi_{g_1,t}\circ\Psi_{h_1,t}\circ\dots\circ\Psi_{h_{s-1},t}\circ\Phi_{g_s,t},$$ with $s\geq1$, $0\neq g_i\in{\mathbb C}[y]$ and $0\neq h_i\in{\mathbb C}[x]$ has multidegree $$\begin{pmatrix} n(nk_s+n-1)\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1) \\ \\ n\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1) \\ \\ n(k_s+1)\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1) \end{pmatrix}$$ where $k_i=\deg(g_i)$ and $l_i=\deg(h_i)$.* *Proof.* We will prove this by induction on $s$. When $s=1$ we have $\phi=\Phi_{g_1,t}$, which has multidegree $(n(nk_1+n-1),n,n(k_1+1)).$ Let us now consider the case with $s>1$. Take $$\psi=\Phi_{g_1,t}\circ\Psi_{h_1,t}\circ\dots\circ\Phi_{g_{s-1},t}=(v_1,v_2,v_3),$$ so that $\phi=\psi\circ\Psi_{h_{s-1},t}\circ\Phi_{h_s,t}$. By induction hypothesis, we know that $\psi$ has multidegree $$\begin{pmatrix} n(nk_{s-1}+n-1)\prod_{i=1}^{s-2}(nk_i+n-1)(nl_i+n-1) \\ \\ n\prod_{i=1}^{s-2}(nk_i+n-1)(nl_i+n-1) \\ \\ n(k_{s-1}+1)\prod_{i=1}^{s-2}(nk_i+n-1)(nl_i+n-1) \end{pmatrix}$$ Since $n>2$, we have $nk_{s-1}+n-1>k_{s-1}+1\geq 1$, which implies that $\deg(v_2)\leq\deg(v_3)<\deg(v_1)$. We can apply again Lemma $\ref{lemma multidegree}$ to $\psi\circ\Psi_{h_{s-1},t}$ to get $\mathop{\mathrm{mdeg}}(\psi\circ\Psi_{h_{s-1},t})=$ $$\begin{pmatrix} n(nk_{s-1}+n-1)\prod_{i=1}^{s-2}(nk_i+n-1)(nl_i+n-1) \\ \\ n\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1) \\ \\ n(l_{s-1}+1)(nk_{s-1}+n-1)\prod_{i=1}^{s-2}(nk_i+n-1)(nl_i+n-1) \end{pmatrix}$$ Let $\psi\circ\Psi_{h_{s-1},t}=(u_1,u_2,u_3)$. Then $\deg(u_2)>\deg(u_3)\geq\deg(u_1)$, and we can apply Lemma [Lemma 19](#lemma multidegree){reference-type="ref" reference="lemma multidegree"} to $(\psi\circ\Psi_{h_{s-1},t})\circ\Phi_{g_s,t}$ to complete the proof. ◻ **Theorem 22**. *Let $n>2$.* (i) *If $P$ is reflective, the group $G_t$ is a free product with amalgamation $$G_t=T\ast_\Theta J.$$ As an abstract group, if $n$ is even $$G_t\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} ({\mathbb C}^\times\rtimes{\mathbb Z}/2{\mathbb Z}).$$ If $n$ is odd, then $$G_t\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} H,$$ where $H=\langle {\mathbb C}^\times, \Omega \, | \, \Omega^2=-1, \,{\lambda}\cdot\Omega_t=\Omega_t\cdot{\lambda}^{-1} \ \forall{\lambda}\in{\mathbb C}^\times \rangle$.* (ii) *If $P$ is not reflective, the group $G_t$ is a free product with amalgamation $$G_t=Q_1\ast_\Theta Q_2.$$ As an abstract group, $$G_t\cong ({\mathbb C}[y]\rtimes{\mathbb C}^\times)\ast_{{\mathbb C}^\times} ({\mathbb C}[x]\rtimes{\mathbb C}^\times).$$* *Proof.* Thanks to Theorem [Theorem 13](#teo tecnico amalgamated free product){reference-type="ref" reference="teo tecnico amalgamated free product"}, we only need to prove that the decompositions in Propositions [Proposition 15](#prop decomposition G_t reflective){reference-type="ref" reference="prop decomposition G_t reflective"} and [Proposition 17](#prop decomposition G_t non reflective){reference-type="ref" reference="prop decomposition G_t non reflective"} are unique. Consider first the case where $P$ is reflective. Assume, for contradiction, that $$\phi=\Phi_{g_1,t}\circ\Omega_t\circ\dots\circ\Omega_t\circ\Phi_{g_s,t}=\Theta_{\nu,t},$$ for some $g_i\in{\mathbb C}[y]$ such that $0\neq g_i$ for $1<i<s$, with $s\geq1$, and for some $\nu\in{\mathbb C}^\times$. In particular, $\phi$ is a linear automorphism, since $\Theta_{\nu,t}$ is linear. Assume first that $g_1,g_s\neq0$. By Proposition [Proposition 20](#prop multidegree 2a){reference-type="ref" reference="prop multidegree 2a"} we have that $$\deg(\phi(x))=n\prod_{i=1}^s(nk_i+n-1),$$ where $k_i=\deg(g_i)$. So if $n>2$ then $\deg(\phi(x))>n$, which leads to a contradiction. If $g_1$ is $0$, we can move $\Omega_t$ to the right to get $$\phi'=\Phi_{g_2,t}\circ\Omega_t\circ\dots\circ\Omega_t\circ\Phi_{g_s,t}=\Omega_t^{-1}\circ\Theta_{\nu,t}.$$ The right-hand side is still a linear automorphism, so we still get a contradiction. We can reason similarly if $g_s=0$. Let us assume now that $P$ is not reflective. Assume, by contradiction, that $$\label{eq decomposizione unica} \phi=\Phi_{g_1,t}\circ\Psi_{h_1,t}\circ\dots\circ\Phi_{g_{s-1},t}\circ\Psi_{h_{s-1},t}\circ\Phi_{g_{s},t} =\Theta_{\nu,t}$$ for some $g_i\in{\mathbb C}[y]$ such that $0\neq g_i$ for $i=2,\dotso,s-1$, $0\neq h_i\in{\mathbb C}[x]$, with $s\geq1$, and for some $\nu\in{\mathbb C}^\times$. Assume first that $g_1,g_s\neq0$. Then by Proposition [Proposition 21](#prop multidegree 2b){reference-type="ref" reference="prop multidegree 2b"} we have $$\deg(\phi(x))=n(nk_s+n-1)\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1),$$ where $k_i=\deg(g_i)$ and $l_i=\deg(h_i)$. So if $n>2$ then $\deg(\phi(x))>n$, which leads to a contradiction, because $\Theta_{\nu^{-1},t}$ is linear. If one or both of $g_1$ and $g_s$ are zero, then we can multiply both sides of [\[eq decomposizione unica\]](#eq decomposizione unica){reference-type="eqref" reference="eq decomposizione unica"} on the left and/or on the right by some $\Phi_{s_i,t}$, for some $0\neq s_i\in{\mathbb C}[y]$. We get an equation of the form $$\phi'=\Phi_{g_1,t}\circ\Psi_{h_1,t}\circ\dots\circ\Phi_{g_{s-1},t}\circ\Psi_{h_{s-1},t}\circ\Phi_{g_{s},t} =\Theta_{\nu,t}\circ\Phi_{q,t},$$ where $q$ is either $g_1, g_s$ or $g_1+g_s$, and $0\neq g_i$ for all $i$. We can now apply Proposition [Proposition 21](#prop multidegree 2b){reference-type="ref" reference="prop multidegree 2b"} and get that $\deg(\phi' (y))=n\prod_{i=1}^{s-1}(nk_i+n-1)(nl_i+n-1)$, while $\deg(\Theta_{\nu^{-1},t}\circ\Phi_{q,t}(y))=n$. This leads to a contradiction, unless $s=1$. In that case though, equation [\[eq decomposizione unica\]](#eq decomposizione unica){reference-type="eqref" reference="eq decomposizione unica"} is either of the form $\mathop{\mathrm{id}}=\Theta_{\nu,t}$ or $\Phi_{g_1,t}=\Theta_{\nu^{-1},t}$, which again leads to contradiction. The decomposition of Proposition [Proposition 17](#prop decomposition G_t non reflective){reference-type="ref" reference="prop decomposition G_t non reflective"} is thus unique. The abstract group structures follow from the isomorphisms $$\begin{aligned} &{\mathbb C}[y]\cong \Phi \hspace{15mm} &g(y)\mapsto\Phi_{g,t} \\ &{\mathbb C}[x]\cong \Psi \hspace{15mm} &g(x)\mapsto\Psi_{g,t} \\ &{\mathbb C}^\times\cong \Theta \hspace{15mm} &\nu\mapsto\Theta_{\nu,t}, \end{aligned}$$ and from the relations in Lemma [Lemma 12](#lemma commuting relations G_t){reference-type="ref" reference="lemma commuting relations G_t"}. ◻ **Corollary 23**. *Let $n>2$. The groups $G_0$ and $G_1$ have the same amalgamated free product structure described in Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"}. In particular, $$G_0\cong G_1=\mathop{\mathrm{Aut}}(A(P)).$$* *Proof.* Notice that the relations in Lemma [Lemma 12](#lemma commuting relations G_t){reference-type="ref" reference="lemma commuting relations G_t"} still hold in $G_0$ and $G_1$, so we get decompositions analogous to those of Propositions [Proposition 15](#prop decomposition G_t reflective){reference-type="ref" reference="prop decomposition G_t reflective"} and [Proposition 17](#prop decomposition G_t non reflective){reference-type="ref" reference="prop decomposition G_t non reflective"}. We can define a notion of degree for basis monomials in $A(P)$ and ${\mathcal A}(P)$ just as for $A_t(P)$. Notice also that in the proof of Lemma [Lemma 19](#lemma multidegree){reference-type="ref" reference="lemma multidegree"}, no $t$ appears in the coefficients of the leading words, so the result holds in $G_0$ and $G_1$ too. These two lemmas were the only ingredient used in the proofs of Propositions [Proposition 20](#prop multidegree 2a){reference-type="ref" reference="prop multidegree 2a"} and [Proposition 21](#prop multidegree 2b){reference-type="ref" reference="prop multidegree 2b"}, and subsequently of Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"}. So, $G_0$ and $G_1$ have the same amalgamated free product structure as of $G_t$. In particular, they are isomorphic. ◻ ## The group of Poisson automorphisms of ${\mathcal A}(P)$ {#section Poisson automorphisms type A} We know that $G_0\leq \mathop{\mathrm{PAut}}({\mathcal A}(P))$ by [\[eq G_0 in PAut\]](#eq G_0 in PAut){reference-type="eqref" reference="eq G_0 in PAut"}. In this section we show that this is an equality. The group of automorphisms of ${\mathcal A}(P)$ as an affine variety is well known. Makar-Limanov computed its generators [@makar-limanovGroupsAutomorphismsClass1990], and Blanc and Dubouloz [@blancAutomorphismsPreFibered2011] proved that it has an amalgamated free product structure. **Theorem 24** ([@makar-limanovGroupsAutomorphismsClass1990 Theorem 1]). *Let $R$ be the quotient algebra $$R={\mathbb C}[x,y,z]/(xy-P(z)),$$ with $P(z)\in{\mathbb C}[z]$. Then the group $Aut(R)$ is generated by the following automorphisms:* (a) *Hyperbolic rotations: $\Theta_{\nu,0}=(\nu x,\nu^{-1} y, z)$, for all $\nu\in{\mathbb C}^\times$;* (b) *Involution: $V=(y,x,z)$;* (c) *Triangular automorphisms: $$\Delta_g=(z+[P(z+yg(y))-P(z)]y^{-1}, y, z+yg(y))$$ for all $g(y)\in{\mathbb C}[y]$;* (d) **(If $P(z)=c(z+a)^n$)* Rescalings $R_\nu=(\nu^n x, y,\nu z+(\nu-1)a)$, for all $\nu\in{\mathbb C}^\times$;* (e) **(If $P(z)=(z+a)^iQ((z+a)^d)$)* Symmetries: $S_\mu=(\mu^ix, y, \mu z+(\mu-1)a)$, for all $\mu^d=1$.* **Lemma 25**. *If $P$ is a monic polynomial of degree $n$ with no term of degree $n-1$, we can substitute (d) and (e) from the list of generators in Theorem [Theorem 24](#teo automorphism affine type A){reference-type="ref" reference="teo automorphism affine type A"} with* (d) **(If $P(z)=z^n$)* Rescalings $R_\nu=(\nu^n x, y,\nu z)$, for all $\nu\in{\mathbb C}^\times$;* (e) **(If $P(z)=z^iQ(z^d)$)* Symmetries: $S_\mu=(\mu^ix, y, \mu z)$, for all $\mu^d=1$.* *Proof.* For (d), $P(z)=c(z+a)^n$ only if $c=1$ and $a=0$, since $P$ is monic and has no term of degree $n-1$. For (e), first notice that the polynomial $Q$ must be monic, since $P$ is. For degree reason, $n=i+dk$, where $k$ is the degree of $Q$. We can ignore the case $d=1$, since the only corresponding automorphism is $S_1=\mathop{\mathrm{id}}$. Expanding $P(z)$ we get $$(z+a)^{i+dk}+\alpha(z+a)^{i+d(k-1)}+\text{terms of lower degree}$$ for some $\alpha\in{\mathbb C}$. Since $d>1$, the term of degree $n-1=i+dk-1$ of $P(z)$ comes only from the expansion of $(z+a)^{i+dk}$. Since $P(z)$ has no term of degree $n-1$, we must have $a=0$. ◻ **Remark 26**. Notice that if $P(z)=z^iQ(z^d)$ is of degree $n$, and $\mu^d=1$, then $\mu^i=\mu^n$. So, if they are both defined, the automorphisms of type (e) are actually a special type of automorphisms of type (d), i.e. $R_\mu=S_\mu$ for all $\mu^d=1$. Let us introduce the affine automorphism $$\begin{aligned} &\nabla_g=V\circ\Delta_{-g}\circ V\\ &=(x, y+[P(z-xg(x))-P(z)]x^{-1}, z-xg(x)).\end{aligned}$$ **Proposition 27**. *Let $g\in{\mathbb C}[x]$. Then $$\Delta_g=e^{\mathop{\mathrm{ad}}(\hat{g}(y))}=\Phi_{g,0},$$ $$\nabla_g=e^{\mathop{\mathrm{ad}}(\hat{g}(x))}=\Psi_{g,0},$$ where $\hat{g}:=\int^x_0gdx$ denotes the antiderivative of $g$. In particular, $\Delta_g$ and $\nabla_g$ are Poisson automorphisms and elements of $G_0$.* *Proof.* We have that $$\label{eq poisson adjoint action y^n} \{y^k, f(z)\}=kf'(z)y^{k},\hspace{8pt} \{y^k,x\}=kP'(z)y^{k-1},$$ and $$\label{eq poisson adjoint action x^n} \{x^k, f(z)\}=kx^kf'(z), \hspace{8pt}\{x^k,y\}=-kx^{k-1}P'(x),$$ for all $k\geq0$. These relations can be checked directly or derived from [\[eq adjoint action y\^m\]](#eq adjoint action y^m){reference-type="eqref" reference="eq adjoint action y^m"} and [\[eq adjoint action x\^m\]](#eq adjoint action x^m){reference-type="eqref" reference="eq adjoint action x^m"} using the isomorphism ${\mathcal A}(P)\cong A_0(P)$. By [\[eq poisson adjoint action y\^n\]](#eq poisson adjoint action y^n){reference-type="eqref" reference="eq poisson adjoint action y^n"}: $$\begin{aligned} &\{\hat{g}(y),z\}=yg(y), \\ &\{\hat{g}(y),x\}=P'(z)g(y),\\ &\{\hat{g}(y), P^{(i)}(z)y^{i-1}g^i(y)\}=P^{(i+1)}(z)y^ig^{i+1}(y),\end{aligned}$$ where $P^{(i)}$ denotes the $i$-th derivative of $P$. Hence, $$e^{\mathop{\mathrm{ad}}(\hat{g}(y))}=(x+\sum_{i=1}^n{\frac{1}{i!}P^{(i)}(z)y^{i-1}g^i(y)},y, z+yg(y)).$$ To complete the proof, we need to show that $\Phi_{g,0}$ acts on $x$ in the same way as $\Delta_g$. Let $P(z)=\sum_{i=0}^n{a_iz^i}$, with $a_n=1$ and $a_{n-1}=0$. $$\begin{aligned} &P(z+yg(y))-P(z)=\sum_{i=0}^n{a_i(z+yg(y))^i}-\sum_{i=0}^n{a_iz^i}= \\ &\sum_{i=1}^n{\binom{n}{i}z^{n-i}y^ig^i(y)}+a_{n-2}\sum_{i=1}^{n-2}{\binom{n}{i}z^{n-2-i}y^ig^i(y)}+\dotso+a_1yg(y)= \\ &\sum_{i=1}^n{y^ig^i(y)\left[\binom{n}{i}z^{n-i}+a_{n-2}\binom{n-2}{i}z^{n-2-i}+\dotso+a_i\right]}=\\ &\sum_{i=1}^n{\frac{1}{i!}P^{(i)}(z)y^ig^i(y)}.\end{aligned}$$ A similar computation using [\[eq poisson adjoint action x\^n\]](#eq poisson adjoint action x^n){reference-type="eqref" reference="eq poisson adjoint action x^n"} shows that $$\nabla_g=e^{\mathop{\mathrm{ad}}(\hat{g}(x))}=(x, y+\sum_{j=1}^n{\frac{(-1)^j}{j!}P^{(j)}(z)x^{j-1}g^j(A)}, z-xg(x)).$$ ◻ **Remark 28**. Consider the automorphism $S_{-1}=((-1)^nx,y,-z)$. This is in $\mathop{\mathrm{Aut}}({\mathcal A})$ if and only if $P(z)$ is of the form $z^iQ(z^2)$, by Proposition [Lemma 25](#lemma automorphism affine type A){reference-type="ref" reference="lemma automorphism affine type A"}, i.e. if and only if $P(z)$ is reflective. If $P$ is reflective, then $$\Omega_0=S_{-1}\circ V=(y,(-1)^n x,-z).$$ **Theorem 29**. *The group $\mathop{\mathrm{PAut}}({\mathcal A}(P))$ of Poisson automorphisms of ${\mathcal A}(P)$ is equal to $G_0$.* *Proof.* It is sufficient to prove that $\mathop{\mathrm{PAut}}({\mathcal A}(P))$ is generated by $\Theta_{\nu,0}$, $\Delta_g$ and $\Omega_0$ when $P$ is reflective, and by $\Theta_{\nu,0}$, $\Delta_g$ and $\nabla_h$ when $P$ is not reflective, for all $\nu\in{\mathbb C}^\times$ and $g,h\in{\mathbb C}[x]$. Let us compute some relations between the generators in $\mathop{\mathrm{Aut}}({\mathcal A}(P))$. The following relations are true whenever the automorphisms are defined. 1. $V^2=\mathop{\mathrm{id}}$ 2. $\Theta_{\nu,0}\circ V=V\circ \Theta_{\nu^{-1},0}$ 3. $\Delta_g\circ V=V\circ\nabla_{-g}$ and $\nabla_g\circ V=V\circ\Delta_{-g}$ 4. $R_{{\lambda}}\circ R_{\mu}=R_{{\lambda}\cdot\mu}$ and $S_{\lambda}\circ S_\mu=S_{{\lambda}\cdot\mu}$ 5. $\Theta_{\nu,0}\circ R_\mu=R_\mu\circ \Theta_{\nu,0}$ and $\Theta_{\nu,0}\circ S_\mu=S_\mu\circ \Theta_{\nu,0}$ 6. $V\circ R_\nu=R_\nu\circ V\circ \Theta_{\nu^n,0}$ and $V\circ S_\mu=S_{\mu}\circ V\circ \Theta_{\mu^n,0}$. Suppose now that $P$ is of the form $P(z)=z^iQ(z^d)$. We have: $$\Delta_g\circ S_\mu =\left(\mu^ix+\mu^i\sum_{j=1}^n{\frac{1}{j!}P^{(j)}(z)C^{j-1}g^j(y)},y, \mu z+\mu yg(y)\right)$$ $$\label{eq composition aut 1} S_\mu\circ \Delta_{\mu g}=\left(\mu^i x+\sum_{j=1}^n{\frac{\mu^j}{j!}P^{(j)}(\mu z)y^{j-1}g^j(y)},y, \mu z+\mu yg(y)\right).$$ Since $P(z)=z^iQ(z^d)$, the $j$-th derivative $P^{(j)}$ has only terms of degree equal to $d(k-l)+i-j$, where $k$ is the degree of $Q$, for some $l\geq0$. Hence each term of the sum in [\[eq composition aut 1\]](#eq composition aut 1){reference-type="eqref" reference="eq composition aut 1"} has a factor of $\mu^{d(k-l)+i}=\mu^i$. Thus 7. $\Delta_g\circ S_\mu=S_\mu\circ\Delta_{\mu g}$. Similarly, we get $$\begin{aligned} & \nabla_g\circ S_\mu=(\mu^i x, y+\sum_{j=1}^n{\frac{(-1)^j}{j!}P^{(j)}(z)x^{j-1}g^j(x)}, \mu z-\mu xg(x))\\ &S_\mu\circ \nabla_{\mu^{1-i} g(\mu^{-i}x)}=(\mu^i x, y+\sum_{j=1}^n{\frac{(-1)^j}{j!}P^{(j)}(\mu z)\mu^{j-i}x^{j-1}g^j(x)},\mu z-\mu xg(x)).\end{aligned}$$ Again, from $P^{(j)}(\mu z)$ we can take out $\mu$, getting factors of the form $\mu^{d(k-l)+i-j}$, which is equal to $\mu^{i-j}$. Thus 8. $\nabla_g\circ S_\mu=S_\mu\circ \nabla_{\mu^{1-i} g(\mu^{-i}x)}.$ Suppose now that $P(z)=z^n$. We have: 9. $\Delta_g\circ R_\nu=R_\nu\circ \Delta_{\nu g}$, since they are both equal to $$(\nu^n x+\nu^n\sum_{i=1}^n{\binom{n}{i}z^{n-i}y^{i-1}g^i(y)},y, \nu z+\nu yg(y))$$ 10. $\nabla_g\circ R_\nu=R_\nu\circ\nabla_{\nu^{1-n} g(\nu^{-n}x)}$, since they are both equal to $$(\nu^n x, y+\sum_{j=1}^n{(-1)^j\binom{n}{j}z^{n-j}x^{j-1}g^j(x)}, \nu z-\nu xg(x) ).$$ Take any automorphism $\psi\in\mathop{\mathrm{Aut}}({\mathcal A}(P))$. From Theorem [Theorem 24](#teo automorphism affine type A){reference-type="ref" reference="teo automorphism affine type A"} and Proposition [Lemma 25](#lemma automorphism affine type A){reference-type="ref" reference="lemma automorphism affine type A"} we know that we can write it as a composition of automorphisms of the form $V,\Theta_{\nu,0},\Delta_g,S_\mu$ or $R_\mu$. Using relations (1) to (10) we can rewrite $\psi$ as $\psi=\omega\circ\phi$, where $\phi$ is in $G_0$ and $\omega$ is one of the following automorphisms: $\mathop{\mathrm{id}}, V, R_\nu, S_\mu, R_\nu\circ V$ or $S_\mu\circ V$. If $\omega=\mathop{\mathrm{id}}$ then $\phi\in G_0$. If $\nu$ or $\mu$ are equal to $(-1)$, then $R_{-1}\circ V=S_{-1}\circ V=\Omega_0$. Since $\Omega_0\in G_0$ then $\psi\in G_0$. We want to show that in all the other cases, $\omega$ is not a Poisson automorphism. It follows from a direct check: $$\{R_\nu(z),R_\nu(y)\}=-\nu y\neq -y= R_\nu (\{z,y\}),$$ for all $\nu\neq 1$, $$\{ S_\mu(z),S_\mu(y)\} =-\mu y\neq -y=S_\mu(\{ z,y\} ),$$ for all $\mu\neq 1$, $$\{S_\mu\circ V(z),S_\mu\circ V(x)\}=-\mu y\neq y=R_\mu\circ V(\{z,x\}),$$ for $\mu\neq-1$, $$\{R_\nu\circ V(z),R_\nu\circ V(x)\}=-\nu y\neq y=R_\nu\circ V(\{z,x\}),$$ for $\nu\neq-1$. Hence, for all $\psi\in\mathop{\mathrm{Aut}}({\mathcal A}(P))$, either $\psi\in G_0$ or $\psi$ is a composition of a Poisson automorphism and of a non-Poisson automorphism, hence $\psi$ is non-Poisson. ◻ From the relations in the proof of Theorem [Theorem 29](#teo generatori poisson type a){reference-type="ref" reference="teo generatori poisson type a"} we also get: **Corollary 30**. *The subgroup $G$ of Poisson automorphisms is normal in $Aut({\mathcal A})$.* **Theorem 31**. *Let $n>2$. Then $\mathop{\mathrm{Aut}}(A(P))$ and $\mathop{\mathrm{PAut}}({\mathcal A}(P))$ admit the amalgamated free product structure described in Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"} and $$\mathop{\mathrm{Aut}}(A(P))\cong\mathop{\mathrm{PAut}}({\mathcal A}(P)),$$ for every deformation parameter $P$.* *Proof.* It follows directly from Theorem [Theorem 22](#teo amalgamated free product){reference-type="ref" reference="teo amalgamated free product"} and Theorem [Theorem 29](#teo generatori poisson type a){reference-type="ref" reference="teo generatori poisson type a"}. ◻ The case $n=2$ is summarised in the following theorem, due to the work of Dixmier [@dixmierQuotientsSimplesAlgebre1973], O'Fleury [@fleurySousgroupesFinisAut1998] and Naurazbekova and Umirbaev [@naurazbekovaAutomorphismsSimpleQuotients2021]. **Theorem 32**. *For $n=2$ and for every deformation parameter $P$, we have a group isomorphism $$\mathop{\mathrm{Aut}}(A(P))\cong\mathop{\mathrm{PAut}}({\mathcal A}(P)),$$ and they have an amalgamated free product structure given by $$L\ast_{L\cap T}T,$$ where $L$ is the subgroup of linear (Poisson) automorphisms and $T$ is the subgroup generated by triangular (Poisson) automorphisms and hyperbolic rotations.* **Remark 33**. The main difference for $n=2$ is that triangular automorphisms $\Delta_g$ can be linear (it is if and only if $g$ is a constant). This is a consequence of the fact that the case $n=2$ is the only one where the Poisson algebra structure is linear on the generators $x,y,z$, inducing a Lie algebra isomorphic to ${\mathfrak s}{\mathfrak l}_2$. ## Isomorphism groupoids (type $\mathbf{A}$) In this section we compute $\mathop{\mathrm{Iso}}(A)$ and $\mathop{\mathrm{PIso}}({\mathcal A})$, the groupoid of (Poisson) isomorphisms between quantizations and deformations of ${\mathbb C}[\mathbf{A_{n-1}}]$, respectively, and verify Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} for type $\mathbf{A}$. Let us first look at the quantization case. **Theorem 34** ([@bavulaIsomorphismProblemsGroups2001 Theorem 3.28]). *Let $P_1,P_2\in{\mathbb C}[z]$. Then the generalized Weyl algebras $A(P_1),A(P_2)$ are isomorphic if and only if there exist some $\eta,{\alpha}\in{\mathbb C}$, with $\eta\neq0$, such that $P_2(z)=\eta P_1(\pm z+{\alpha})$.* **Corollary 35**. *For all $n\geq2$, the groupoid $\mathop{\mathrm{Iso}}(A)$ is generated by the groups $\mathop{\mathrm{Aut}}(A(P))$, for all deformation parameters $P$, and by the isomorphism $$\Omega:A(P(z))\rightarrow A((-1)^nP(-z)), \quad \Omega:=(y,(-1)^n x,1-z).$$* *Proof.* Let $P_1$ be a deformation parameter, and $\eta,{\alpha}\in{\mathbb C}$, with $\eta\neq0$. Suppose that $P_2=\eta P_1(\pm z+{\alpha})$ is a deformation parameter. Then ${\alpha}=0$ because $P_1$ and $P_2$ have no term of degree $n-1$, and $\eta=(\pm 1)^n$ because $P_2$ and $P_1$ are monic. The claim now follows from Theorem [Theorem 34](#teo iso generalized weyl algebras){reference-type="ref" reference="teo iso generalized weyl algebras"}. ◻ Let us consider the deformation case. We will first compute a set of generators for $\mathop{\mathrm{Iso}}({\mathcal A})$, the groupoid of affine isomorphisms between deformations. In [@blancAutomorphismsPreFibered2011] the authors study the isomorphisms between ${\mathbb A}^1$-fibered surfaces. An affine surface $X$ is ${\mathbb A}^1$-fibered if there exists a surjective morphism of affine varieties $\pi_X:X\rightarrow {\mathbb A}^1$, with general fibres isomorphic to ${\mathbb A}^1$, where ${\mathbb A}^1$ denotes the complex affine line. Two ${\mathbb A}^1$-fibered surfaces $(X,\pi)$ and $(X',\pi')$ are isomorphic as ${\mathbb A}^1$-fibered surfaces if there are affine isomorphisms $f:X\rightarrow X'$ and $g:{\mathbb A}^1\rightarrow{\mathbb A}^1$ such that $$\pi'\circ f=g\circ \pi.$$ Let $P\in{\mathbb C}[z]$ be a polynomial of degree $n$. The algebra ${\mathcal A}(P)$ is ${\mathbb A}^1$-fibered by ${\pi}_P:{\mathcal A}(P)\rightarrow {\mathbb C}[x]$, the projection on the variable $x$. **Lemma 36**. *Let $P,Q\in{\mathbb C}[z]$ be of degree $n$.* (i) *Every fibration ${\pi}':{\mathcal A}(P)\rightarrow{\mathbb A}^1$ is of the form ${\pi}_P\circ{\gamma}$, where ${\gamma}$ is an affine automorphism of ${\mathcal A}(P)$;* (ii) *$({\mathcal A}(P),{\pi}_P)\cong({\mathcal A}(Q),{\pi}_Q)$ if and only if $Q(z)=\eta P({\alpha}z+\beta)$, for some $\eta,{\alpha},{\beta}\in{\mathbb C}$, with $\eta\neq0$.* *Proof.* A proof can be found in [@blancAutomorphismsPreFibered2011 Theorem 5.4.5]. Point $(i)$ was first proved by Daigle in [@daigleLocallyNilpotentDerivations2003]. ◻ **Proposition 37**. *For all $n\geq2$, the groupoid $Iso(A)$ is generated by the groups $\mathop{\mathrm{Aut}}({\mathcal A}(P))$, for all deformation parameters $P$, and by the isomorphisms $$R_{\alpha}:{\mathcal A}(P(z))\rightarrow {\mathcal A}({\alpha}^{-n}P({\alpha}z)), \quad R_{\alpha}:=({\alpha}^{n} x,y,{\alpha}z), \quad \forall{\alpha}\in{\mathbb C}^\times.$$* *Proof.* Let $P,Q$ be two deformation parameters, and $\phi:{\mathcal A}(P)\rightarrow{\mathcal A}(Q)$ be an affine isomorphism. Then $\phi$ is a fibered isomorphism between $$\phi:({\mathcal A}(P),\pi_Q\circ\phi)\rightarrow({\mathcal A}(Q),\pi_Q).$$ From Lemma [Lemma 36](#lemma iso fibered surfaces){reference-type="ref" reference="lemma iso fibered surfaces"} there exists ${\gamma}\in\mathop{\mathrm{Aut}}({\mathcal A}(P))$ such that $\pi_Q\circ\phi=\pi_P\circ{\gamma}$. Thus $\psi:=\phi\circ{\gamma}^{-1}:({\mathcal A}(P),\pi_P)\rightarrow({\mathcal A}(Q),\pi_Q)$ is a fibered isomorphism. By Lemma [Lemma 36](#lemma iso fibered surfaces){reference-type="ref" reference="lemma iso fibered surfaces"}, there exist $\eta,{\alpha},{\beta}\in{\mathbb C}$, with $\eta\neq0$, such that $Q(z)=\eta P({\alpha}z+{\beta})$. Since $P$ and $Q$ are deformation parameters, ${\beta}=0$ and ${\alpha}^n=\eta$. Consider the fibered isomorphism $$R_{\alpha}:({\mathcal A}(P),{\pi}_P)\rightarrow ({\mathcal A}(Q),{\pi}_Q) \quad R_{\alpha}:=({\alpha}^{n} x,y,{\alpha}z).$$ Clearly $R_{\alpha}^{-1}\circ \psi\in\mathop{\mathrm{Aut}}({\mathcal A}(P))$, so $\phi$ can be written as composition of $R_{\alpha}$ and an automorphism of ${\mathcal A}(P)$. ◻ **Theorem 38**. *For all $n\geq2$, the groupoid $\mathop{\mathrm{PIso}}({\mathcal A})$ is generated by the groups $\mathop{\mathrm{PAut}}({\mathcal A}(P))$, for all deformation parameters $P$, and by the isomorphism $${\tau}:{\mathcal A}(P(z))\rightarrow {\mathcal A}((-1)^nP(-z)) \quad {\tau}:=(y,(-1)^n x,-z).$$* *Proof.* Let $P,Q$ be two deformation parameters, and $\phi:{\mathcal A}(P)\rightarrow{\mathcal A}(Q)$ be a Poisson isomorphism. In particular, $\phi$ is an affine isomorphism, so, by Proposition [Proposition 37](#prop affine iso deformation type A){reference-type="ref" reference="prop affine iso deformation type A"}, $\phi=R_{\alpha}\circ{\gamma}$, for some ${\gamma}\in\mathop{\mathrm{Aut}}({\mathcal A}(P))$. By the proof of Theorem [Theorem 29](#teo generatori poisson type a){reference-type="ref" reference="teo generatori poisson type a"}, ${\gamma}$ can be written as $\omega\circ\xi$, where $\xi\in\mathop{\mathrm{PAut}}({\mathcal A}(P))$ and $\omega$ is one of the following affine automorphisms: $\mathop{\mathrm{id}}, V, R_\nu, S_\mu, R_\nu\circ V$ or $S_\mu\circ V$. Since $\phi$ is Poisson, $R_{\alpha}\circ\omega$ needs to be too. With the same computations as in the proof of Theorem [Theorem 29](#teo generatori poisson type a){reference-type="ref" reference="teo generatori poisson type a"}, we can check that the only case where $R_{\alpha}\circ\omega$ is Poisson is ${\alpha}=-1$ and $\omega=V$. In that case, $R_{-1}\circ V=\tau$. ◻ Putting together Corollary [Corollary 35](#cor iso quantizzazion type A){reference-type="ref" reference="cor iso quantizzazion type A"} and Theorem [Theorem 38](#teo Poisson iso type A){reference-type="ref" reference="teo Poisson iso type A"} we verify Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} for type $\mathbf{A}$. **Theorem 39**. *Let $n\geq 4$. We have an isomorphism of groupoids $$\mathop{\mathrm{Iso}}(A)\cong\mathop{\mathrm{PIso}}({\mathcal A}).$$* # Type D {#section 3} Once again, let $V$ be a complex vector space of dimension $2$. Choose a basis for $V$, and let $X,Y$ be the corresponding coordinate functions. We then identify $SL(V)$ with $SL_2({\mathbb C})$. Take $\Gamma=BD_{n-2}\subset SL_2({\mathbb C})$ the binary dihedral group of order $4(n-2)$, with $n\geq 4$. The group $\Gamma$ is generated by $$\sigma:=\begin{pmatrix} e^{\pi i/(n-2) } & 0\\ 0 & e^{-\pi i/(n-2) } \end{pmatrix}, \hspace{4mm} \tau:=\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}.$$ The quotient $V/\Gamma$ is the Kleinian singularity of type $\mathbf{D_{n}}$. The algebra of functions ${\mathbb C}[V/\Gamma]={\mathbb C}[V]^\Gamma$ is generated by the monomials $X^2Y^2,(X^{2(n-2)}+Y^{2(n-2)})$ and $XY(X^{2(n-2)}-Y^{2(n-2)})$. We have $${\mathbb C}[\mathbf{D_{n}}]\cong{\mathbb C}[x,y,z]/(x^{n-1}+xy^2+z^2)\, ,$$ for all $n\geq 4$. Considering $X,Y$ to be of degree $1$, we have a grading on ${\mathbb C}[\mathbf{D_{n}}]$ given by $\deg x=4$, $\deg y=2(n-2)$ and $\deg z=2(n-1)$. We give ${\mathbb C}[\mathbf{D_{n}}]$ the structure of a graded Poisson algebra as described in Remark [Remark 4](#oss generale poisson structure on C[x,y,z]){reference-type="ref" reference="oss generale poisson structure on C[x,y,z]"}, with Poisson bracket of degree $-2$. This structure is equivalent to the one induced by the standard symplectic structure on $V$, after a suitable renormalization. Explicitly, we have $$\{x,y\}=2z\, , \hspace{5mm} \{x,z\}=-2xy \hspace{5mm} \{y,z\}=(n-1)x^{n-2}+y^2\, .$$ ## Deformations and quantizations We have explicit presentations for both deformations and quantizations of the algebra ${\mathbb C}[\mathbf{D_{n}}]$. In both cases, they are parameterised by a pair $(Q,\gamma)$, where $Q\in{\mathbb C}[x]$ is a monic polynomial of degree $n-1$, and $\gamma\in{\mathbb C}$. Fix $n\geq4$ and denote by ${\mathcal D}_n(Q,\gamma)$ the deformation associated to the parameter $(Q,\gamma)$ of the algebra ${\mathbb C}[\mathbf{D_{n}}]$. Explicitly, we have $${\mathcal D}_n(Q,\gamma)={\mathbb C}[x,y,z]/(Q(x)+xy^2+z^2-\gamma y).$$ These algebras come from specializing the semi-universal deformation of the Kleinian singularity (see for example [@katzGorensteinThreefoldSingularities1992 Table 3]). We give ${\mathcal D}_n(Q,\gamma)$ the structure of a Poisson algebra as in Remark [Remark 4](#oss generale poisson structure on C[x,y,z]){reference-type="ref" reference="oss generale poisson structure on C[x,y,z]"}, $$\{x,y\}=\pdv{\psi}{z}=2z, \ \ \ \{x,z\}=-\pdv{\psi}{y}=-2xy+\gamma, \ \ \ \{y,z\}=\pdv{\psi}{x}=Q' (x)+y^2.$$ The algebra ${\mathcal D}_n(Q,\gamma)$ is a filtered Poisson algebra, with filtration induced by $\deg x=4$, $\deg y=2(n-2)$ and $\deg z=2(n-1)$, with Poisson bracket of degree $-2$, i.e. $\{F_k,F_m\}\subset F_{m+k-2}$ for all filtration terms $F_k,F_m$. We denote by $D_n(Q,{\gamma})$ the quantization associated to the parameter $(Q,{\gamma})$. The algebras $D_n(Q,{\gamma})$ were studied by Levy in [@levyIsomorphismProblemsNoncommutative2009]. The algebra $D_n(Q,{\gamma})$ is the ${\mathbb C}$-algebra generated by $x,y,z$, subject to the relations $$=2z, \quad [x,z]=-2xy+2z+{\gamma}, \quad [y,z]=y^2+P(x)-n,$$ $$Q(x)+x(y^2-n)+z^2-2yz-{\gamma}y=0,$$ where $P(x)$ is the unique degree $n-2$ polynomial satisfying $$Q(-x(x-1))-Q(-x(x+1))=(x-1)P(-x(x-1))+(x+1)P(-x(x+1)).$$ **Theorem 40** ([@levyIsomorphismProblemsNoncommutative2009 Theorem 2.22 & 3.6] ). *For all $n\geq 4$, there is an isomorphism $\Sigma:D_n(Q,{\gamma})\rightarrow D_n(Q,-{\gamma})$ given by $\Sigma:=(x,-y,-z)$. If $n=4$, write $Q(x)=x^3+ax^2+bx+c$. Set $$\begin{aligned} &b'=\frac{1}{8}(3a^2-4b-12 i \gamma), \\ &c'=c+\frac{1}{16}(a^3-4ab-4ia\gamma), \\ &\gamma'=\frac{i}{8}(a^2-4b+4i\gamma), \end{aligned}$$ and let $Q'(x)=x^3+ax^2+b'x+c'$. There is an isomorphism $T:D_n(Q,{\gamma})\rightarrow D_n(Q',{\gamma}')$ given by $$T:=\left(-\frac{1}{2}x+\frac{i}{2}y-\Big(1+\frac{1}{4}a\Big),\frac{3i}{2}x-\frac{1}{2}y+i\Big(1+\frac{1}{4}a\Big),z\right),$$* (i) *$\Sigma$ and $T$ satisfy the $S_3$ relations, whenever the compositions make sense. In particular $\Sigma^2=\mathop{\mathrm{id}}$ and $T^3=\mathop{\mathrm{id}}$;* (ii) *The only possible isomorphisms between quantizations $D(Q,{\gamma})$ are $\mathop{\mathrm{id}}, \Sigma, T$, $T^{-1}, (\Sigma\circ T)$ and $(\Sigma\circ T^{-1})$.* *In particular, the group $\mathop{\mathrm{Aut}}(D_n(Q,\gamma))$ has the following description. If $n>4$:* - *if $\gamma\neq0$, then $\mathop{\mathrm{Aut}}(D_n(Q,\gamma))=\langle\mathop{\mathrm{id}}\rangle$;* - *if $\gamma=0$, then $\mathop{\mathrm{Aut}}(D_n(Q,\gamma))={\mathbb Z}/2{\mathbb Z}$, generated by $\Sigma$.* *If $n=4$:* - *if $\gamma=0$ and $b=a^2/4$, then $\mathop{\mathrm{Aut}}(D_4(Q,\gamma))=S_3$, generated by $T$ and $\Sigma$;* - *if $\gamma\neq0$ and $b=a^2/4-i\gamma$ (respectively $b=a^2/4+i\gamma$), then $\mathop{\mathrm{Aut}}(D_4)={\mathbb Z}/2{\mathbb Z}$, generated by $\Sigma\circ T$ (respectively $\Sigma\circ T^{-1}$);* - *if $\gamma=0$ and $b\neq a^2/4$, then $\mathop{\mathrm{Aut}}(D_4(Q,\gamma))={\mathbb Z}/2{\mathbb Z}$, but this time it is generated by $\Sigma$;* - *if $\gamma\neq0$ and $b\neq a^2/4\pm i\gamma$, then $\mathop{\mathrm{Aut}}(D_4(Q,\gamma))=\langle\mathop{\mathrm{id}}\rangle$.* **Remark 41**. Our parametrization of the quantizations $D_n(Q,{\gamma})$ differs slightly from Levy's. The algebra defined in [@levyIsomorphismProblemsNoncommutative2009 Definition 1.5] coincides with our $D_n(Q+nx,{\gamma})$. ## Isomorphisms of deformations In this section we compute all the Poisson isomorphisms between the deformations ${\mathcal D}_n(Q,\gamma)$ and confirm Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"}. Let us first compute the groupoid $\mathop{\mathrm{Iso}}({\mathcal D}_n)$ of isomorphisms between the deformations ${\mathcal D}_n(Q,\gamma)$ as affine varieties. We follow a similar method to [@blancNonrationalityFibrationsAssociated2015]. We embed each ${\mathcal D}_n(Q,\gamma)$ into a projective normal surface $X_n(Q,\gamma)$, such that every point in $X_n(Q,\gamma)\setminus{\mathcal D}_n(Q,\gamma)$ is smooth in $X_n(Q,\gamma)$. We then prove that every isomorphism ${\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$ extends to an isomorphism of $X_n(Q,\gamma)\rightarrow X_n(Q',\gamma')$. This way we reduce ourselves to study the groupoid $\mathop{\mathrm{Iso}}(X_n, {\mathcal D}_n)$ of isomorphisms $X_n(Q,\gamma)\rightarrow X_n(Q',\gamma')$ that restrict to ${\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$, which is simpler to compute. We construct $X_n(Q,\gamma)$ as a hypersurfaces of $F_{a,b}$, a ${\mathbb P}^2$-bundle over ${\mathbb P}^1$. This ${\mathbb P}^2$-bundle is ${\mathbb P}({\mathcal O}_{{\mathbb P}^1}\oplus{\mathcal O}_{{\mathbb P}^1} (a)\oplus {\mathcal O}_{{\mathbb P}^1} (b))$, and can be viewed as the gluing of $U_{a,b,0} = {\mathbb P}^2\times{\mathbb C}$ and $U_{a,b,\infty} = {\mathbb P}^2 \times{\mathbb C}$ along ${\mathbb P}^2 \times{\mathbb C}^*$, where the identification map is given by the involution $$((w : y : z), x)\longrightarrow \left((w : x^{-a}y : x^{-b}z), \ \frac{1}{x}\right).$$ The ${\mathbb P}^2$-bundle is given by the map $F_{a,b}\rightarrow{\mathbb P}^1$ corresponding to $((w : y : z), x) \mapsto (x : 1)$ in the first chart and $((w : y : z), x) \mapsto (1 : x)$ in the second one. If $n = 2k$, we take the ${\mathbb P}^2$-bundle $F_{k-1,k-1}$, and denote by $X_n=X_n(Q,\gamma)$ the projective surface that restricts to the following surfaces on each chart: $$\begin{aligned} &\{((w : y : z), x) \in U_{k-1,k-1,0} \, | \,Q(x)w^2 + xy^2 + z^2-\gamma yw=0\},\\ &\{((w : y : z), x) \in U_{k-1,k-1,\infty} \,| \, Q^{r}(x)w^2 + y^2 + xz^2-\gamma x^{k}yw=0\}.\end{aligned}$$ Here by $Q^{r}(x)$ we mean the reciprocal polynomial of $Q$, i.e. $Q^r=x^{n-1}Q(x^{-1})$. If $n = 2k + 1$, we take the ${\mathbb P}^2$-bundle $F_{k-1,k}$, and denote by $X_n=X_n(Q,\gamma)$ the projective surface that restrict to the following surfaces on each chart: $$\begin{aligned} &\{((w : y : z), x) \in U_{k-1,k,0}\, | \, Q(x)w^2 + xy^2 + z^2-\gamma yw=0\},\\ &\{((w : y : z), x) \in U_{k-1,k,\infty} \, | \, Q^{r}(x)w^2 + xy^2 + z^2-\gamma x^{k+1}yw=0\}.\end{aligned}$$ In both cases, we embed the surface ${\mathcal D}_n$ in the first affine chart of $X_n$, via the embedding $$(x, y, z)\mapsto ((1 : y : z), x),$$ so that $X_n$ is the closure of ${\mathcal D}_n$ in $F_{a,b}$. Geometrically the situation is similar to the case considered in [@blancNonrationalityFibrationsAssociated2015] and we can make similar remarks. All the singular points of $X_n$ are in the image, under the above embedding, of a singular point in ${\mathcal D}_n$; in particular, for generic $Q$ and $\gamma$, the surface ${\mathcal D}_n$ is smooth and so is $X_n$. The ${\mathbb P}^2$-bundles $F_{k-1,k-1}\rightarrow {\mathbb P}^1$ and $F_{k-1,k}\rightarrow {\mathbb P}^1$ restrict to a morphism $\rho : X_n \rightarrow {\mathbb P}^1$. The fibres are conics in ${\mathbb P}^2$, which are smooth for generic values of $x$. The generic fibre is thus isomorphic to ${\mathbb P}^1$. The fibre $F_\infty$ over $(1 : 0)$ is always degenerate. It is a union of two transversal lines in the second chart, given by the equations $x=0, \, w=\pm iy$ (if $n$ is even) or $x=0, \, w=\pm iz$ (if $n$ is odd). Denote the two lines in $F_\infty$ by $F_+$ and $F_-$, respectively. The other degenerate fibres are the ones over the solutions of the equation $$\det \begin{pmatrix} Q(x) & -\gamma/2 & 0\\ -\gamma/2 & x & 0 \\ 0 & 0 & 1\\ \end{pmatrix}=0.$$ Counting with multiplicity, we have $n+1$ points in ${\mathbb P}^1$ with degenerate fibre. For every $Q$ and $\gamma$, the complement $X_n(Q,\gamma)\setminus{\mathcal D}_n(Q,\gamma)$ consists of the curve $C_n$, given by the equation $w = 0$ in each chart, and the curve $F_\infty=F_+\cup F_-$, with equation $x = 0$ in the second chart, corresponding to the fibre over $(1:0)$. We do not specify the deformation parameters when referring to the curves $C_n, F_+, F_-$, since they are defined by the same equation in $F_{k-1,k-1}$ or $F_{k-1,k}$. The geometric description of the boundary is the same for every deformation parameter. We have the following (compare with [@blancNonrationalityFibrationsAssociated2015 Lemma 4.3]). **Lemma 42**. *For every monic polynomial $Q$ of degree $n-1$ and for all $\gamma\in{\mathbb C}$, the complement of ${\mathcal D}_n(Q,\gamma)$ in $X_n(Q,\gamma)$ is the union of the three curves $C_n, F_+, F_-$, all isomorphic to ${\mathbb P}^1$. Any two of them intersect transversally, in exactly one point, which is $C_n\cap F_+\cap F_-$. Moreover, $C_n^2=3-n$, and $F_+^2=F_-^2=-1$.* *Proof.* The only thing that doesn't follow from the discussion above are the self intersection numbers. Let $P_n$ be the curve given by the equation $y=0$ in both charts. If $n=2k$, $P_n$ and $C_n$ intersect only in the point $((0:0:1),0)$ of the second chart. This happens along the distinct directions $w=0$ and $y=0$, so $C_n\cdot P_n=1$. If $n=2k+1$, the two curves are disjoint, so $C_n\cdot P_n=0$. In both cases, $C_n\cdot P_n=2k+1-n$. Consider the rational map $g\in{\mathbb C}(X_n)^*$ given by $w/y$ on the second chart, and by $wx^{k-1}/y$ on the first chart. The associated principal divisor is $C_n$ $+(k-1)F_0-P_n$, where $F_0$ is given by the equation $x=0$ in the first chart, i.e. it is the fibre over $(0:1)$. Computing the principal divisor associated to the rational function $x$, it is clear that $F_0$ is linearly equivalent to $F_\infty$. The intersection of $C_n$ and $F_\infty$ has multiplicity 2, since $$C_n\cdot F_\infty=C_n\cdot(F_++ F_-)=2.$$ Thus $$C_n^2=C_n\cdot(P_n-(k-1)F_{\infty})=(2k+1-n)-(2k-2)=3-n.$$ Clearly, $F_0$ and $F_\infty$ are disjoint, so $F_0\cdot F_\infty=0$. Since they are linearly equivalent, this means $$0=F_0\cdot F_\infty=F_\infty^2=(F_++F_-)^2=F_+^2+F_-^2+2F_+\cdot F_-=F_+^2+F_-^2+2.$$ The linear equivalence of $F_0$ and $F_\infty$ implies $F_+=F_0-F_-$, so $F_+^2=F_-^2+F_0^2.$ Since $F_0^2=F_0\cdot F_\infty=0$, this implies that $F_+^2=F_-^2=-1$. ◻ Consider now $\phi\in\mathop{\mathrm{Iso}}({\mathcal D}_n)$, an isomorphism ${\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$. This extends to a birational map $\phi:X_n(Q,\gamma)\dashrightarrow X_n(Q',\gamma')$, that is biregular between ${\mathcal D}_n(Q,\gamma)\subset X_n(Q, \gamma)$ and ${\mathcal D}_n(Q',\gamma')\subset X_n(Q', \gamma')$. We recall the following results about birational maps and blow-ups. **Lemma 43**. *Let $X, X'$ be projective complex surfaces and $\phi:X\dashrightarrow X'$ a birational map. Assume that $X$ and $X'$ are smooth outside of the open sets $S$ and $S'$, and that $\phi:S\rightarrow S'$ is biregular. Then there exists a surface $Z$ and a commutative diagram $$\xymatrix{ &Z\ar[ld]_\eta \ar[rd]^\pi \\ X\ar@{.>}[rr]_\phi & &X' }$$ where the morphisms $\eta,\pi$ are composite of blow-ups.* *Proof.* If $X,X'$ are smooth, this is Theorem II.11 of [@beauvilleComplexAlgebraicSurfaces1996]. If $X$ or $X'$ are not smooth, $\phi$ induces a birational map $\hat{\phi}$ between the resolutions $\hat{X}$ and $\hat{X'}$. We are now in the smooth case, so we get the following commutative diagram $$\xymatrix{ &Z'\ar[ld]_{\eta'} \ar[rd]^{\pi'} \\ \hat{X}\ar@{.>}[rr]_{\hat{\phi}} \ar[d]_f & &\hat{X'}\ar[d]^g \\ X\ar@{.>}[rr]_\phi & &X' }$$ where both $f$ and $g$ are composition of blow-ups. Now, the singularities of $X$ and $X'$ are inside $S$ and $S'$ respectively, and $\phi$ is biregular between $S$ and $S'$. We can thus contract all the curves blown up when resolving the singularities. Call $Z$ the image of $Z'$ under these contractions. We end up with the following diagram $$\xymatrix{ &Z\ar[ld]_\eta \ar[rd]^\pi \\ X\ar@{.>}[rr]_\phi & &X' }$$ with $\eta=f\circ\eta'\circ f^{-1}$ and $\pi=g\circ\pi'\circ g^{-1}$. ◻ **Lemma 44** ([@beauvilleComplexAlgebraicSurfaces1996 Proposition II.2-3]). *Let $S$ be a smooth surface, $\pi:\hat{S}\rightarrow S$ the blow-up of a point $p\in S$ and $E\subset \hat{S}$ the exceptional divisor. Then* (i) *Let $C\subset S$ be an irreducible curve, then $\pi^*C=\overline{C}+mE$, where $m$ is the multiplicity of $p$ in $C$, and the curve $\overline{C}$ is the strict transform of $C$;* (ii) *Let $D,D'$ be divisors on $S$. Then $(\pi^*D)\cdot(\pi^*D')=D\cdot D'$, $E\cdot\pi^*D=0$ and $E^2=-1$.* **Corollary 45**. *In the setting of Lemma [Lemma 44](#lemma technical properties blow-ups){reference-type="ref" reference="lemma technical properties blow-ups"}, let $C\subset S$ be an irreducible curve. Then* (i) *the intersection number $E\cdot\overline{C}$ is the multiplicity of $p$ in $C$.* (ii) *$C^2\geq \overline{C}^2$, and $C^2=\overline{C}^2$ if and only if $p\notin C$;* *Proof.* From point (i) of Lemma [Lemma 44](#lemma technical properties blow-ups){reference-type="ref" reference="lemma technical properties blow-ups"}, $\pi^*C=\overline{C}+mE$. Multiply now both sides by $E$. Using point (ii) of Lemma [Lemma 43](#lemma birational map blow-ups diagram){reference-type="ref" reference="lemma birational map blow-ups diagram"}, this becomes $0=\overline{C}\cdot E-m$. We compute the self intersection of $C$. Using again point (ii), we get $$C^2=(\pi^*C)^2=(\overline{C}+mE)^2=\overline{C}^2+2m\overline{C}\cdot E-m^2E^2=\overline{C}^2+m^2\geq \overline{C}^2.$$ ◻ We can apply Lemma [Lemma 43](#lemma birational map blow-ups diagram){reference-type="ref" reference="lemma birational map blow-ups diagram"} to our birational map $\phi$ and get the diagram $$\label{diagram blow-ups} \xymatrix{ &Z\ar[ld]_\eta \ar[rd]^\pi \\ X_n(Q,\gamma)\ar@{.>}[rr]_\phi & &X_n(Q',\gamma') }$$ where the morphisms $\eta,\pi$ are compositions of blow-ups of points of $X_n(Q,\gamma)\setminus {\mathcal D}_n(Q,\gamma)$ and $X_n(Q',\gamma')\setminus{\mathcal D}_n(Q',\gamma')$ respectively. Denote by $\overline{F_+},\overline{F_-}$ and $\overline{C_n}$ the strict transforms via $\eta^{-1}$ of $F_+,F_-$ and $C_n$. Note that since we are blowing-up away from the singularities, we can assume to be in the setting of Lemma [Lemma 44](#lemma technical properties blow-ups){reference-type="ref" reference="lemma technical properties blow-ups"}. Since $\phi$ is biregular between ${\mathcal D}_n(Q,\gamma)$ and ${\mathcal D}_n(Q',\gamma')$, it sends the curves $F_+,F_-$ and $C_n$ into themselves. Suppose now that $\phi$ is not an isomorphism $X_n(Q,\gamma)\xrightarrow{\sim}X_n(Q',\gamma')$. Then $\pi$ must contract one of the curves $\overline{F_+},\overline{F_-}$ and $\overline{C_n}$ to a point. **Remark 46**. Without loss of generality, we can assume that $\pi$ decomposes into a sequence of blow-ups of points $\pi_s\circ\cdots\circ\pi_1$, where $\pi_1$ contracts one of the curves $\overline{F_+},\overline{F_-}$ or $\overline{C_n}$. To see why, let us decompose $\eta=\eta_k\circ\dotso\circ\eta_1$ into blow-ups of points. Each $\eta_i$ adds an exceptional divisor, which we denote $E_i$. Denote by $\overline{E_i}$ their strict transforms. If $\pi_1$ contracts $E_k$, then we can remove $\pi_1$ and $\eta_k$ and get another diagram of the form [\[diagram blow-ups\]](#diagram blow-ups){reference-type="eqref" reference="diagram blow-ups"}. Suppose now that $\pi_1$ contracts $\overline{E_i}$ for some $i<k$. This means that $\overline{E_i}^2=-1$, so the self intersection number of $E_i$ does not change under the strict transform. By point (ii) of Corollary [Corollary 45](#cor technical properties blow-ups){reference-type="ref" reference="cor technical properties blow-ups"}, the maps $\eta_j$ with $j>i$ blow-up points that are not in $E_i$. So $\eta_i$ commutes with all $\eta_j$ with $j>i$ and we reduce ourselves to the case $E_k$. We have the following proposition (compare with [@blancNonrationalityFibrationsAssociated2015 Proposition 4.4]). **Proposition 47**. *Let $n\geq4$, $Q,Q'$ monic polynomials of degree $n-1$ and $\gamma,\gamma'\in{\mathbb C}$. Every isomorphism $\phi:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$ extends to an isomorphism $X_n(Q,\gamma)\rightarrow X_n(Q',\gamma')$.* *Proof.* Suppose that $\phi:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$ does not extend to an isomorphism $X_n(Q,\gamma)\rightarrow X_n(Q',\gamma')$. We are in the situation described in diagram [\[diagram blow-ups\]](#diagram blow-ups){reference-type="eqref" reference="diagram blow-ups"}. By Remark [Remark 46](#oss order blow-ups){reference-type="ref" reference="oss order blow-ups"}, we can assume that the first curve contracted by $\pi$ is either $\overline{F_+},\overline{F_-}$ or $\overline{C_n}$. Since this curve is a $(-1)$-curve, it is either $\overline{F_+}$ or $\overline{F_-}$ (or $\overline{C_n}$ if $n=4$), by (ii) of Corollary [Corollary 45](#cor technical properties blow-ups){reference-type="ref" reference="cor technical properties blow-ups"}; say it is $\overline{F_+}$. Since taking the strict transform via $\eta^{-1}$ does not change the self intersection number of $F_+$, $\eta$ does not blow-up any point of $F_+$ by (ii) of Corollary [Corollary 45](#cor technical properties blow-ups){reference-type="ref" reference="cor technical properties blow-ups"}. In particular, it does not blow-up the triple intersection point of the components of the boundary. So $\overline{F_+},\overline{F_-}$ and $\overline{C_n}$ still intersect transversely in one point inside $Z$. Denote $F^{(1)}:=\pi_1(\overline{F_-})$ and $C^{(1)}=\pi_1(\overline{C_n})$. Then $$F^{(1)}\cdot C^{(1)}=(\overline{F_-}+\overline{F_+})\cdot (\overline{C_n}+\overline{F_+})=1+1+1-1=2,$$ by applying both points of Lemma [Lemma 44](#lemma technical properties blow-ups){reference-type="ref" reference="lemma technical properties blow-ups"} and the fact that $m=\overline{F_-}\cdot\overline{F_+}=1$ (by (i) of Corollary [Corollary 45](#cor technical properties blow-ups){reference-type="ref" reference="cor technical properties blow-ups"}). Thus $F^{(1)}$ and $C^{(1)}$ are tangent. This leads to a contradiction, because it is not possible to recover the original boundary via other contractions. In fact, if $\pi$ contracts $F^{(1)}$ or $C^{(1)}$, say $F^{(1)}$, then $\pi(C^{(1)})$ is a curve whose strict transform $C^{(1)}$ intersects the exceptional divisor with multiplicity $2$. This means, by (i) of Corollary [Corollary 45](#cor technical properties blow-ups){reference-type="ref" reference="cor technical properties blow-ups"}, that it contains a point with multiplicity $2$, so it is not smooth. This is a contradiction, because the curves of the boundary are all smooth, and contracting does not resolve singularities. If instead $\pi$ never contracts $F^{(1)}$ or $C^{(1)}$, then it does not modify their intersection point, and it is impossible to recover the original boundary. The same reasoning works if the curve contracted by $\pi_1$ is $\overline{F_-}$ (or $\overline{C_n}$ if $n=4$). ◻ Using Proposition [Proposition 47](#Prop Blackbox){reference-type="ref" reference="Prop Blackbox"}, we can compute the groupoid $\mathop{\mathrm{Iso}}({\mathcal D}_n)$ for all $n\geq4$ (compare with [@blancNonrationalityFibrationsAssociated2015 Corollary 4.5]). First, let us recall some facts that will be used in the proof. **Lemma 48**. *Every isomorphism between two non-degenerate conics in ${\mathbb P}^2$ can be extended to a projective transformation of ${\mathbb P}^2$.* *Proof.* Since two non degenerate conics are always isomorphic via a projective transformation of ${\mathbb P}^2$, it is sufficient to show that every automorphism of a specific non degenerate conic extends to a projective transformation of ${\mathbb P}^2$. Consider the conic $C: xy-z^2=0$. Since it is non degenerate, it is isomorphic to ${\mathbb P}^1$. Explicitily, this is given by $$(s:t)\mapsto (s^2:st:t^2).$$ Thus, every automorphism of $C$ will be induced by an automorphism of ${\mathbb P}^1$, which has the form $(s:t)\mapsto (as+bt:cs+dt)$, with $a,b,c,d\in{\mathbb C}$ with $ad-bc\neq 0$. The induced automorphism on the conic is $$(x:y:z)\mapsto (a^2x+2aby+b^2z: acx+(ad+bc)y+bdz: c^2+2cdy+d^2z),$$ which extends to an element of $PGL(3,{\mathbb C})$, because the determinant of the associated matrix is\ $(ad-bc)^3$. ◻ **Proposition 49**. *Consider the algebra $A:={\mathbb C}[x,y,z]/(xy^2-z^2)$, graded with $\deg x=0, \deg y=1, \deg z=1$. The group of graded automorphisms of $A$ is: $$\{(\alpha^2\beta^{-2} x, \beta y, \alpha z) \, | \, \alpha,\beta \in {\mathbb C}^*\} .$$* *Proof.* It is shown in [@makar-limanovGroupAutomorphismsSurface2001 Theorem 1] that the group of automorphisms of $A$ is generated by: 1. Hyperbolic rotations: $H_{\nu}=(\nu^{-2} x,\nu y, z)$, for all $\nu\in{\mathbb C}^*$; 2. Rescalings: $R_\mu=(\mu^2 x, y, \mu z)$, for all $\mu\in{\mathbb C}^*$: 3. Triangular automorphisms: $$\Delta_g=(x+[(z+y^2g(y))^2-z^2]y^{-2}, y, z+y^2 g(y)),$$ for all $g(y)\in{\mathbb C}[y]$. The triangular automorphisms form a normal subgroup $\Delta$ isomorphic to the additive group ${\mathbb C}[x]$ via $g\mapsto\Delta_g$, and the group of automorphisms is the semidirect product of $\Delta$ and of the subgroup generated by the automorphisms of type (1) and (2) ([@makar-limanovGroupAutomorphismsSurface2001 Final Remark]). The proposition follows by noticing that automorphisms in $\Delta$ don't preserve the grading (except for the identity), and that $H_\mu H_{\nu}=H_{\mu\nu}$, $R_\mu R_\nu=R_{\mu \nu}$ and $R_\mu \Theta_{\nu,0}= H_{\nu} R_\mu$. ◻ **Lemma 50**. *Suppose $\rho_1: X_1\rightarrow {\mathbb P}^1$ and $\rho_2: X_2\rightarrow {\mathbb P}^1$ are bundles with compact, connected fibres, and let $\phi$ be an isomorphism $X_1\rightarrow X_2$. If there exists a fibre in $X_1$ that $\phi$ sends to a fibre in $X_2$, then $\phi$ sends all fibres to fibres.* *Proof.* Suppose there exists $x_0\in{\mathbb P}^1$ such that $\phi(\rho_1^{-1}(x_0))=\rho_2^{-1}(x_1)$ for some $x_1\in{\mathbb P}^1$. Take any other fibre $\rho_1^{-1}(z)$, with $z\in{\mathbb P}^1$. Assume $\phi(\rho_1^{-1}(z))$ is not a fibre. Then $\rho_2(\phi(\rho_1^{-1}(z)))={\mathbb P}^1$, which means that there is a point in $\phi(\rho_1^{-1}(z))$ that gets mapped to $x_1$. So, $\phi(\rho^{-1}(x_0))\cap \phi(\rho^{-1}(z))\neq\emptyset$, which is absurd because $\phi$ is an isomorphism. ◻ Define the following morphisms in $\mathop{\mathrm{Iso}}({\mathcal D}_n)$, for $n\geq 4$. 1. $$R^\pm_{\lambda}:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma'),$$ $$R^\pm_\lambda=(\lambda^2 x, \mu\lambda^{-2} y, \pm\mu\lambda^{-1} z),$$ with $\mu=\gamma/\gamma'$, for all $Q,Q'$ such that $Q(\lambda^2 x)=\lambda^{2(n-1)}Q'(x)$ and for all $\gamma,\gamma'\neq0$ such that $\lambda^n=\pm\mu$; 2. $$P^\pm_{\lambda}:{\mathcal D}_n(Q,0)\rightarrow{\mathcal D}_n(Q',0),$$ $$P^\pm_\lambda=(\lambda^2 x, \pm\lambda^{n-2} y, \pm\lambda^{n-1} z),$$ for all $Q,Q'$ such that $Q(\lambda^2 x)=\lambda^{2(n-1)}Q'(x)$ and for all ${\lambda}\in{\mathbb C}^\times$. **Theorem 51**. *Let $Q,Q'$ monic polynomials of degree $n-1$ and $\gamma,\gamma'\in {\mathbb C}$.* (i) *If $n>4$, the only isomorphisms in $\mathop{\mathrm{Iso}}({\mathcal D}_n)$ are of the form $R_{\lambda}$ and $P_{\lambda}$.* (ii) *If $n=4$, let $Q(x)=x^3+ax^2+bx+c$ and $Q'(x)=x^3+a'x^2+b'x+c'$ . Then we have an isomorphism $\tau:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$ given by $$\label{def tau} \tau=\left(-\frac{1}{2}x+\frac{i}{2}y-\frac{1}{4}a,\frac{3i}{2}x-\frac{1}{2}y+\frac{i}{4}a,z\right),$$ every time $(Q',\gamma')$ satisfies the following condition:* *$$\label{eq condition iso order 3} \begin{aligned} &a'=a, \\ &b'=\frac{1}{8}(3a^2-4b-12 i \gamma), \\ &c'=c+\frac{1}{16}(a^3-4ab-4ia\gamma), \\ &\gamma'=\frac{i}{8}(a^2-4b+4i\gamma). \end{aligned}$$ The groupoid $\mathop{\mathrm{Iso}}({\mathcal D}_4)$ is generated by all isomorphisms of the form $R_{\lambda}$, $P_{\lambda}$ and $\tau$.* *Proof.* It is easy to check that the listed actions are well defined isomorphisms. Thus, we only need to prove that they generate the whole of $\mathop{\mathrm{Iso}}({\mathcal D}_n)$. We know from Proposition [Proposition 47](#Prop Blackbox){reference-type="ref" reference="Prop Blackbox"} that every element of $\mathop{\mathrm{Iso}}({\mathcal D}_n)$ lifts to an element of $\mathop{\mathrm{Iso}}(X_n)$. We can thus consider the groupoid $K:=\mathop{\mathrm{Iso}}(X_n, {\mathcal D}_n)$ of isomorphisms $X_n(Q,\gamma)\rightarrow X_n(Q',\gamma')$ that restrict to ${\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$. First, let us compute the subgroupoid $K_0\leq K$ of isomorphisms that preserve the conic bundle structure, i.e. that send a fibre to another fibre. Every isomorphism $g\in K_0$ induces an automorphism of ${\mathbb P}^1$. In addition, it needs to preserve the boundary $F_\infty \cup C_n$, so it must preserve the fibre over $(1:0)$. Thus, the action on ${\mathbb P}^1$ is given by a degree one polynomial $x\mapsto ax+b$. Since $g$ sends conics to isomorphic conics, by Lemma [Lemma 48](#Prop iso conic){reference-type="ref" reference="Prop iso conic"}, it extends to a projective transformation of the projective plane that contains them (which is the fibre of the ${\mathbb P}^2$ bundle $F_{a,b}$). Since it also preserves the intersection of the conics with $C_n$ (i.e. with $w=0$) we can write $g$ (as an isomorphism ${\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$) as $$(x,y,z)\mapsto (ax+b, cy+dz+e, fy+hz+k),$$ with $a,b\in{\mathbb C}$ and $c,d,e,f,h,k\in{\mathbb C}[x]$ satisfying $ch-df\neq0$. Let us consider on ${\mathcal D}_n(Q,\gamma)$ the filtration ${\mathcal F}$ induced by $\deg x=0$ and $\deg y=\deg z=1$. Then $$\mathop{\mathrm{gr}}_{\mathcal F}({\mathcal D}_n(Q,\gamma))\cong A={\mathbb C}[x,y,z]/(xy^2-z^2)$$ for every deformation parameter $(Q,\gamma)$. Since $g$ preserves the filtration ${\mathcal F}$, it induces a graded automorphism $\mathop{\mathrm{gr}}(g)$ of $A$. By Proposition [Proposition 49](#Prop Makar-Limanov){reference-type="ref" reference="Prop Makar-Limanov"}, $\mathop{\mathrm{gr}}(g)=(\alpha^2\beta^{-2} x, \beta y, \alpha z)$ for some $\alpha,\beta\in{\mathbb C}^*$, so $$\label{eq isomorphism g} g=(\alpha^2\beta^{-2} x + b, \beta y+c, \alpha z+d),$$ with $b\in{\mathbb C}$ and $c,d\in{\mathbb C}[x]$. Plugging [\[eq isomorphism g\]](#eq isomorphism g){reference-type="eqref" reference="eq isomorphism g"} into the defining equation of ${\mathcal D}_n(Q,\gamma)$ we get the following condition: $$Q(\alpha^2\beta^{-2} x + b)+(\alpha^2\beta^{-2} x + b)(\beta y+c)^2+(\alpha z+d)^2-\gamma (\beta y+c)= k(Q'(x)+xy^2+z^2-\gamma' y),$$ for some $k\in{\mathbb C}[x,y,z]$. Comparing the coefficients of $z^2,z,y^2,y$ and the constant term we get, respectively $$\begin{aligned} &\alpha^2=k, \label{z^2}\\ &2\alpha d=0,\label{z}\\ &(\alpha^2\beta^{-2}x+b)\beta^2=kx, \label{y^2}\\ &2(\alpha^2\beta^{-2}x+b)\beta c-\gamma\beta=-k\gamma', \label{y}\\ &Q(\alpha^2\beta^{-2}x+b)=kQ'(x). \label{cost} \end{aligned}$$ From [\[z\^2\]](#z^2){reference-type="eqref" reference="z^2"}, [\[z\]](#z){reference-type="eqref" reference="z"} and [\[y\^2\]](#y^2){reference-type="eqref" reference="y^2"} we get that $k\in{\mathbb C}^*$, $d=0$ and $b=0$. From [\[y\]](#y){reference-type="eqref" reference="y"}, since $k$ is a constant, we get that the coefficient of $x$ is $0$, so $c=0$. Combining [\[y\]](#y){reference-type="eqref" reference="y"},[\[z\^2\]](#z^2){reference-type="eqref" reference="z^2"} and [\[y\^2\]](#y^2){reference-type="eqref" reference="y^2"} we also get $\gamma=\gamma'=0 \wedge \gamma\beta=\alpha^2\gamma'$. From [\[cost\]](#cost){reference-type="eqref" reference="cost"} and [\[z\^2\]](#z^2){reference-type="eqref" reference="z^2"} we have $Q(\alpha^2\beta^{-2} x)=\alpha^2 Q'(x)$. In particular, from comparing the $x^{n-1}$ coefficients, we get $\alpha^{2n-4}=\beta^{2n-2}$, so $$\label{eq rel alpha beta} \alpha^{n-2}=\pm\beta^{n-1}.$$ Consider $\lambda:=\alpha\beta^{-1}$. Then from [\[eq rel alpha beta\]](#eq rel alpha beta){reference-type="eqref" reference="eq rel alpha beta"} we get that $\lambda^{n-1}=\pm\alpha$ and $\lambda^{n-2}=\pm\beta$, with the same sign. We can thus rewrite the relation on $Q$ as $$\label{eq Q Q'} Q(\lambda^2 x)=\lambda^{2(n-1)}Q'(x).$$ If we write $Q(x)=x^{n-1}+a_{n-2}x^{n-2}+\dots+a_0$, then by [\[eq Q Q\'\]](#eq Q Q'){reference-type="eqref" reference="eq Q Q'"} we have $Q'(x)=x^{n-1}+{\lambda}^{-2}a_{n-2}x^{n-2}+\dots+{\lambda}^{-2(n-1)}a_0$. Let us first consider the case with $\gamma,\gamma'\neq0$. Denote by $\mu:=\gamma/\gamma'$. We have the condition $\mu\beta=\alpha^2$. So $\pm\mu\lambda^{n-2}=\lambda^{2n-2}$, i.e. $\lambda^{n}=\pm\mu$. Thus, $K_0$ contains an isomorphism $R^\pm_{\lambda}:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q',\gamma')$ of the form $$R^\pm_\lambda=(\lambda^2 x, \lambda^{n-2} y, \pm\lambda^{n-1} z)=(\lambda^2 x, \mu\lambda^{-2} y, \pm\mu\lambda^{-1} z),$$ for all $\lambda$ such that $\lambda^{n}=\pm\mu$ and for all $Q,Q'$ such that [\[eq Q Q\'\]](#eq Q Q'){reference-type="eqref" reference="eq Q Q'"} holds. When $\gamma=\gamma'=0$ we have no extra condition. Thus, $K_0$ contains an isomorphism $P^\pm_{\lambda}:{\mathcal D}_n(Q,0)\rightarrow{\mathcal D}_n(Q',0)$ of the form $$P^\pm_\lambda=(\lambda^2 x, \pm\lambda^{n-2} y, \pm\lambda^{n-1} z),$$ for all $\lambda\in{\mathbb C}^*$ and for all $Q,Q'$ such that [\[eq Q Q\'\]](#eq Q Q'){reference-type="eqref" reference="eq Q Q'"} holds. By equations [\[z\^2\]](#z^2){reference-type="eqref" reference="z^2"} to [\[cost\]](#cost){reference-type="eqref" reference="cost"}, the only elements in $K_0$ are of the form $R_{\lambda}, P_{\lambda}$. As we noted before, the automorphisms in $K_0$ preserve the fibre $F_\infty$. On the other hand, if an isomorphism in $K$ preserves $F_\infty$, then it is in $K_0$, by Proposition [Lemma 50](#prop bundle isomorphism){reference-type="ref" reference="prop bundle isomorphism"}. Thus an isomorphism in $K\setminus K_0$ does not preserve $F_\infty$ or, equivalently, $C_n$. Since the self intersection numbers for $C_n, F_+$ and $F_-$ are $3-n, -1, -1$ we see that $\mathop{\mathrm{Iso}}({\mathcal D}_n)=K_0$ for $n>4$. For $n=4$, one directly checks that we have an isomorphism $\tau: {\mathcal D}_4(Q,\gamma)\rightarrow{\mathcal D}_4(Q',\gamma')$ of order $3$ defined by [\[def tau\]](#def tau){reference-type="eqref" reference="def tau"}, with $Q',\gamma'$ as in [\[eq condition iso order 3\]](#eq condition iso order 3){reference-type="eqref" reference="eq condition iso order 3"}. The isomorphism $\tau$ does not preserve the conic bundle, so it cyclically permutes the curves $F_+,F_-$ and $C_4$. Together with the isomorphism $\sigma_y:{\mathcal D}_4(Q,\gamma)\rightarrow{\mathcal D}_4(Q,-\gamma)$ defined by $\sigma_y=(x,-y,z)$, we have a full action of $S_3$ on set of the three curves $F_+, F_-, C_4$. Take now any isomorphism $\phi:{\mathcal D}_4(Q,\gamma)\rightarrow{\mathcal D}_4(Q',\gamma')$. If $\phi$ fixes the curves $F_+,F_-$ and $C_4$, then it is in $K_0$. Otherwise, we can compose $\phi$ with the appropriate permutation generated by $\tau$ and $\sigma_y$ to get an isomorphism that fixes the three curves, i.e. it is again in $K_0$. Since $\sigma_y\in K_0$, it follows that $\tau$ and $K_0$ generate the whole groupoid $K$. ◻ **Theorem 52**. *Let $Q,Q'$ monic polynomials of degree $n-1$ and $\gamma,\gamma'\in {\mathbb C}$.* - *If $n>4$ there are only two classes of Poisson isomorphisms, $\mathop{\mathrm{id}}$ and $\sigma$, where $$\label{def sigma} \sigma:{\mathcal D}_n(Q,\gamma)\rightarrow{\mathcal D}_n(Q,-\gamma), \hspace{8mm} \sigma=(x,-y,-z).$$* - *If $n=4$ the groupoid $\mathop{\mathrm{PIso}}({\mathcal D}_4)$ is generated by the isomorphism $\sigma$ defined in [\[def sigma\]](#def sigma){reference-type="eqref" reference="def sigma"} and by the isomorphism $\tau$ defined in [\[def tau\]](#def tau){reference-type="eqref" reference="def tau"}. Moreover, every time the composition makes sense, $\sigma$ and $\tau$ satisfy the $S_3$ relations $\tau^3=\mathop{\mathrm{id}}$, $\sigma^2=\mathop{\mathrm{id}}$ and $\sigma\circ\tau\circ\sigma=\tau^2$.* *Proof.* It is sufficient to check when the isomorphisms of Theorem [Theorem 51](#teo iso Dn){reference-type="ref" reference="teo iso Dn"} satisfy the Poisson relations. In particular, by imposing $\{x,y\}=2z$, we get that $R_{\lambda}$ and $P_{\lambda}$ are Poisson isomorphisms only if $\lambda=1$. It is then straightforward to check that $\sigma$ is a Poisson isomorphism. For the case $n=4$, one can directly check that $\tau$ is a Poisson isomorphism and that $\tau$ and $\sigma$ satisfy the $S_3$ relations. ◻ Putting together Theorem [Theorem 40](#teo iso quantizations type D){reference-type="ref" reference="teo iso quantizations type D"} and [Theorem 52](#teo Poisson iso Dn){reference-type="ref" reference="teo Poisson iso Dn"} we can confirm Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} for type $\mathbf{D}$. **Theorem 53**. *Let $n\geq 4$. We have an isomorphisms of groupoids $$\mathop{\mathrm{Iso}}(D_n)\cong\mathop{\mathrm{PIso}}({\mathcal D}_n).$$ In particular, for every deformation parameter $(Q,{\gamma})$ $$\mathop{\mathrm{Aut}}(D(Q,{\gamma}))\cong\mathop{\mathrm{PAut}}({\mathcal D}(Q,{\gamma})).$$* We can also describe the groups of affine automorphisms of ${\mathcal D}_n(Q,\gamma)$ by checking under which deformation parameter conditions the isomorphisms in Theorem [Theorem 51](#teo iso Dn){reference-type="ref" reference="teo iso Dn"} are automorphisms. **Theorem 54**. *Let $n\geq4$, $Q$ monic of degree $n-1$ and $\gamma\in{\mathbb C}$. Write $Q(x)=x^dP(x^m)$, with $d\in{\mathbb Z}_{\geq 0}$ and $m\in{\mathbb Z}_{\geq 1}$, with $m$ maximal possible.* - *If $\gamma\neq0$, the group $$G=\{(\lambda^2 x, \lambda^{-2} y, \pm\lambda^{-1} z) \, | \, \lambda^{2n}=1, \, \lambda^{2m}=1 \}$$ acts on ${\mathcal D}_n(Q,\gamma)$ via automorphism.* - *If $\gamma=0$, the group $$G=\{(\lambda^2 x, \pm\lambda^{n-2} y, \pm\lambda^{n-1} z) \, | \, \lambda^{2m}=1 \}$$ acts on ${\mathcal D}_n(Q,\gamma)$.* *If $n>4$, then $\mathop{\mathrm{Aut}}({\mathcal D}_n(Q,\gamma))=G$.* *In particular, for generic $(Q,\gamma)$, $$\mathop{\mathrm{Aut}}({\mathcal D}_n(Q,\gamma))={\mathbb Z}/2{\mathbb Z},$$ generated by $\sigma_z=(x,y,-z)$.* *For generic $Q$ and $\gamma=0$, $$\mathop{\mathrm{Aut}}({\mathcal D}_n(Q,\gamma))=({\mathbb Z}/2{\mathbb Z})\times({\mathbb Z}/2{\mathbb Z}),$$ with $\sigma_y=(x,-y,z)$ added to the generators.* *If $n=4$, write $Q(x)=x^3+ax^2+bx+c$. Then:* - *if $\gamma=0$ and $b=a^2/4$, then $\mathop{\mathrm{Aut}}({\mathcal D}_4(Q,\gamma))$ is generated by $G$ and $\tau$;* - *if $\gamma\neq0$ and $b=a^2/4-i\gamma$ (respectively $b=a^2/4+i\gamma$), then $Aut({\mathcal D}_4(Q,\gamma))$ is generated by $G$ and $\sigma\circ\tau$ (respectively $\sigma\circ\tau^{-1}$);* - *otherwise, $Aut({\mathcal D}_4(Q,\gamma))=G$.* *Proof.* Theorem [Theorem 54](#teo aut Dn){reference-type="ref" reference="teo aut Dn"} follows at once from Theorem [Theorem 51](#teo iso Dn){reference-type="ref" reference="teo iso Dn"}, by noticing that $Q({\lambda}^2 x)={\lambda}^{2(n-1)}Q(x)$ if and only if $Q(x)=x^dP(x^m)$, with ${\lambda}^{2m}=1$. ◻ **Remark 55**. Notice that from Theorem [Theorem 54](#teo aut Dn){reference-type="ref" reference="teo aut Dn"} we recover the result of [@blancNonrationalityFibrationsAssociated2015 Corollary 4.5] for the undeformed case, i.e. when $Q(x)=x^{n-1}$ and $\gamma=0$. ## Final remarks {#final-remarks .unnumbered} Let $X$ be a conic symplectic singularity. The moduli space of filtered quantization and deformations of $X$ is the quotient of a Cartan space $\mathfrak{P}$ by the action of a Weyl group $W$ [@namikawaPoissonDeformationsAffine2010; @losevDeformationsSymplecticSingularities2022]. All the filtered (Poisson) isomorphisms between deformations and quantizations are induced by a graded Poisson automorphism of ${\mathbb C}[X]$ [@losevDeformationsSymplecticSingularities2022 Proposition 3.21 and Corollary 3.22]. If $X=V/\Gamma$ is a symplectic quotient, then the group of graded Poisson automorphisms of ${\mathbb C}[X]$ is equal to $\Theta:=N_{Sp(V)}(\Gamma)/\Gamma$ [@losevDeformationsSymplecticSingularities2022 Lemma 3.20]. From the results of this paper, we know that, for Kleinian singularities of type $\mathbf{A}$ and $\mathbf{D}$, there are no "wild" non-filtered isomorphisms in $\mathop{\mathrm{Iso}}$ and $\mathop{\mathrm{PIso}}$. In fact, in type $\mathbf{D}$ there exist only filtered isomorphisms, while in type $\mathbf{A}$ the only non-filtered isomorphisms are inner, that is they are the exponentiation of an inner nilpotent derivation. For a Kleinian singularity $X$ with Dynkin diagram $\Delta$, the group of diagram automorphisms $\mathop{\mathrm{Aut}}(\Delta)$ acts on ${\mathbb C}[X]$ by graded Poisson automorphisms, and can be identified with a subgroup of $\Theta$. In every type except type $\mathbf{A}$ the diagram automorphisms coincide with $\Theta$, while in type $\mathbf{A}$ they are a proper subgroup. In this case, the hyperbolic rotations are not in $\mathop{\mathrm{Aut}}(\Delta)$. Interestingly, the only isomorphisms in $\mathop{\mathrm{PIso}}({\mathcal A})$ that are not automorphisms come from $\mathop{\mathrm{Aut}}(\Delta)$. In other words, the action of the hyperbolic rotations and the inner automorphisms on the moduli space $\mathfrak{P}/W$ is trivial. For type $\mathbf{E}$, we expect no non-filtered isomorphisms to exist. Thus $\mathop{\mathrm{PIso}}({\mathcal E})$ and $\mathop{\mathrm{Iso}}(E)$ should be trivial, except for type $\mathbf{E_6}$, where $\mathop{\mathrm{Aut}}(\Delta)={\mathbb Z}/2{\mathbb Z}$. Unfortunately, an explicit presentations of the quantizations in type $\mathbf{E}$ with generators and relations is not known. It would be interesting to understand if the absence of "wild" non-filtered isomorphisms for Kleinian singularities is incidental or if Conjecture [Conjecture 2](#conj main){reference-type="ref" reference="conj main"} holds in the more general setting of conical symplectic singularities. [School of Mathematics & Statistics, University of Glasgow, G12 8QQ Glasgow, UK]{.smallcaps} *E-mail address*: `Simone.Castellan@glasgow.ac.uk`

arxiv_math

--- abstract: | We provide explicit faithful re-embeddings for all hyperelliptic curves of genus at most three and an algorithmic way to construct them. Both in the faithful tropicalization algorithm and the proofs of correctness, we showcase [Oscar]{.smallcaps}-methods for commutative algebra, polyhedral and tropical geometry. This article is submitted as a bookchapter in the upcoming [Oscar]{.smallcaps} book. address: - | Hannah Markwig, Universität Tübingen\ Auf der Morgenstelle 10\ 72076 Tübingen\ Germany - | Lukas Ristau, Fraunhofer Institute for Industrial Mathematics\ Fraunhofer-Platz 1\ 67663 Kaiserslautern\ Germany - | Victoria Schleis, Universität Tübingen\ Auf der Morgenstelle 10\ 72076 Tübingen\ Germany author: - Hannah Markwig - Lukas Ristau - Victoria Schleis bibliography: - references.bib title: Faithful tropicalization of hyperelliptic curves --- # Introduction Tropical geometry allows a fruitful exchange of methods between algebraic and convex geometry, as tropicalization preserves important properties of algebraic varieties such as dimension. The question which properties can be preserved under tropicalization --- and how --- is of foundational importance in order to make use of the exchange of methods. In this article, we concretely answer this question for the case of hyperelliptic curves of genus $3$. We assume that the reader is familiar with varieties defined over the Puiseux series e.g., their tropicalizations, in particular tropical hypersurfaces and duality, especially tropical plane curves and their dual Newton subdivisions, see for instance [@MS2015]. Consider a plane curve and its tropicalization. As dimension is preserved, the latter is a $1$-dimensional polyhedral complex, more precisely a balanced graph in $\mathbb{R}^2$ that is dual to its Newton subdivision, as described in [@MS2015]. Is the genus of the curve reflected in its tropicalization? Examples of a tropical line and a tropical conic, as in [@MS2015], suggest that the tropicalization of a rational (i.e. genus $0$) curve should be a rational graph, i.e. a tree. **Example 1**. What about the elliptic curve defined by the Weierstraß equation $$y^2=x^3+4\cdot x^2+8t^4\cdot x,$$ which is of genus $1$? We expect the tropicalization to be a graph containing a cycle, thus reflecting the genus. To compute its tropicalization, we use the following [Oscar]{.smallcaps} [@Oscar] commands: ``` {.jlcon breaklines=""} julia> Kt,t = RationalFunctionField(QQ,"t") (Rational function field over QQ, t) julia> Kxy, (x,y) = Kt["x", "y"] julia> f=-x^3-4*x^2+y^2+(-8*t^4)*x; julia> val_t = TropicalSemiringMap(Kt,t) The t-adic valuation on Rational function field over QQ julia> ftrop=tropical_polynomial(f,val_t) x^3 + x^2 + (4)*x + y^2 julia> Tf=TropicalHypersurface(ftrop) min tropical hypersurface embedded in 2-dimensional Euclidean space julia> PC = underlying_polyhedral_complex(Tf) Polyhedral complex in ambient dimension 2 julia> visualize(PC) ``` {#fig-ellipticrational width="17cm"} Figure [1](#fig-ellipticrational){reference-type="ref" reference="fig-ellipticrational"} shows the output of the computation. It turns out that the tropicalization is again a rational graph, i.e. the genus was somehow lost in the process of tropicalization. We should not let this discourage ourselves, as we can see that this loss was happening merely because of an unfavorable choice of embedding: Let us change coordinates and define $$g=f(x,y-2x)=-x^3-4\cdot xy+y^2-8t^4\cdot x.$$ Repeating the computation from above with $g$, we obtain the tropicalization depicted in Figure [2](#fig-ellipticrationalrepaired){reference-type="ref" reference="fig-ellipticrationalrepaired"}, which contains a cycle as desired. {#fig-ellipticrationalrepaired width="17cm"} The genus is only a topological invariant: every elliptic curve is of genus $1$, but their various complex structures may be distinguished using the $j$-invariant. Is the $j$-invariant of an elliptic curve reflected in the tropicalization? As suggested by Mikhalkin in [@Mik06 Example 3.15], the tropicalization of an elliptic curve defined over a field with a non-Archimedean valuation should contain a cycle whose length is equal to the valuation of the $j$-invariant. **Example 2**. Consider the elliptic cubic curve given by the equation $$\begin{aligned} f=&(-t^2)\cdot x^3+(t^{20})\cdot x^2y+(t^2)\cdot xy^2+(t^{14})\cdot y^3+(-3t^3)\cdot x^2+xy\\ &+(t^3+t^5-t^6)\cdot y^2+(-3t^4)\cdot x+(t+t^2)\cdot y+(2t^2+t^5).\end{aligned}$$ Repeating the computation from Example [Example 1](#ex-tropicalweierstrass){reference-type="ref" reference="ex-tropicalweierstrass"} with $f$ we obtain the picture in Figure [3](#fig-wrongcyclelength){reference-type="ref" reference="fig-wrongcyclelength"}. {reference-type="ref" reference="ex-wrongcyclelength"} has a cycle, but its length does not equal the valuation of its j-invariant.](wrongcyclelength.pdf){#fig-wrongcyclelength width="8cm"} After a coordinate change $$\begin{aligned} g=f(x-t,y)&=(-t^2)\cdot x^3+(t^{20})\cdot x^2y+(t^2)\cdot xy^2+(t^{14})\cdot y^3+(1-2t^{21})\cdot xy\\&+(t^5-t^6)\cdot y^2+(t^2+t^{22})\cdot y+(2t^2+2t^5)\end{aligned}$$ we obtain the picture from Figure [4](#fig-cyclelengthrepaired){reference-type="ref" reference="fig-cyclelengthrepaired"}. Using the command `vertices(PC2)` for the underlying polyhedral complex `PC2` of our tropical plane curve we obtain the coordinates of the vertices: ``` {.jlcon} julia> vertices(PC2) 6-element SubObjectIterator{PointVector{QQFieldElem}}: [3, -9] [-2, -2] [2, -2] [3, -3] [2, 0] [0, 2] ``` Before the coordinate change, we had the vertices: ``` {.jlcon} julia> vertices(PC1) 5-element SubObjectIterator{PointVector{QQFieldElem}}: [1, -11] [1, -2] [-2, -2] [0, 2] [1, 1] ``` Thus, before the coordinate change we obtained as cycle length 9, while we obtain 10 after the coordinate change. {#fig-cyclelengthrepaired width="12cm"} We compute the j-invariant in [Oscar]{.smallcaps} [@Oscar] using the function `j_invariant_cubic(f)` as supplied at <https://victoriaschleis.github.io/bookchapter>. It turns out that 10 is the valuation of the j-invariant and thus only the second embedding, after the coordinate change, reflects the j-invariant. Both examples illustrate that the tropicalization of a curve depends on the chosen embedding. We can see that for some, but not all, choices of embeddings, many properties of an elliptic curve are reflected in its tropicalization: dimension, genus, and $j$-invariant. This motivates the following definition: *Definition 3* (Faithful tropicalizations of elliptic curves). An embedding of an elliptic curve $E$ with $j$-invariant $j(E)$ is called *faithful* if its tropicalization is a graph containing a cycle of length $\mathop{\mathrm{val}}(j(E))$.[\[def-faithfulell\]]{#def-faithfulell label="def-faithfulell"} A faithful embedding of an elliptic curve can be viewed as optimal from the point of view of tropical geometry, as all important features are preserved to first order. The fact that some, but not all, embeddings are faithful motivates the study of the limit of all tropicalizations, where tropicalization depends on the chosen embedding. *Definition 4* (Analytification is the limit of all tropicalizations, [@P2009], Theorem 1.1). Let $\chi$ be an affine variety over $K$. Let $e$ be an affine embedding of $\chi$. Then we define the *Berkovich space* (or *analytification*) $\chi^{\mathop{\mathrm{an}}} := \varprojlim \mathop{\mathrm{Trop}}(e(\chi))$, i.e. analytification is the limit of all tropicalizations. Historically, the analytification is defined differently (in terms of extensions of valuations) and the statement above is a theorem. For the purpose of this text however, we can take it as a definition. The statement holds analogously for projective curves after compactification, resp. by gluing affine charts. **Example 5**. The tropicalization of a $\mathbb{P}^1$ can be viewed as the compactification of $\mathbb{R}$. If we embed our line into the (torus of a) plane, e.g. as $V(x+y+1)$ and tropicalize, we obtain the well-known picture of a tripod, which projects onto $\mathbb{R}$. Next, we embed our line into threespace. After tropicalization, several combinatorial types may occur. In general, we obtain a tropical curve with 2 vertices and one bounded edge as in the middle of Figure [\[fig-line\]](#fig-line){reference-type="ref" reference="fig-line"}. Continuing this procedure and taking the limit of all tropicalizations, we obtain as analytification of a line an infinite tree as sketched in the right of Figure [\[fig-line\]](#fig-line){reference-type="ref" reference="fig-line"} which is often called the witch's broom. Analytifications satisfy many desirable properties. They are infinitely branched metric graphs, [@BPR2014 Section 2.4]. In some sense, analytifications can be viewed as generalizations of the $j$-invariant for elliptic curves: the analytification of an elliptic curve is an infinitely branched graph containing precisely one cycle of length $\mathop{\mathrm{val}}(j)$. More generally, the analytification of a curve $\chi$ of genus $g$ contains a finite metric subgraph of genus $g$ as a deformation retract, which is called the *minimal Berkovich skeleton* of $\chi$. In the case of an elliptic curve, the minimal Berkovich skeleton is a loop of length $\mathop{\mathrm{val}}(j)$. Using the analytification, we can extend our notion of faithful tropicalization to arbitrary curves: *Definition 6* (Faithful tropicalization). Let $\chi$ be a projective curve and $e:\chi\rightarrow \mathbb{P}^n$ an embedding of $\chi$ into projective space. The embedding $e$ of $\chi$ is called *faithful* on a subgraph $\Gamma$ of the analytification $\chi^{\mathop{\mathrm{an}}}$ if $\mathop{\mathrm{Trop}}(e(\chi))$ contains an isometric copy of $\Gamma$. If we fix the cycle as the subgraph $\Gamma$ of the analytification of an elliptic curve, we recover Definition [\[def-faithfulell\]](#def-faithfulell){reference-type="ref" reference="def-faithfulell"}. Faithful tropicalizations are the best tropicalizations we can hope for, in the sense that they reflect as much of the geometry of the algebraic curve as possible. Analytifications also reflect the geometry, but they are infinite objects which are hard to control. A tropicalization which is chosen arbitrarily is accessible for computations, but may not reflect the features we want to observe. A faithful tropicalization combines the advantages of both: it is a finite object which is easier to control, but it still reflects the geometric features we want to observe. In [@BPR2014 Theorem 5.20], it is shown that faithful tropicalizations exist for every curve. But how can we find them concretely? In this paper, we answer this question for hyperelliptic curves of genus $3$. The technique we use is called re-embedding into tropical modifications and described in detail in Section [2](#sec_reembeddings_and_blocks){reference-type="ref" reference="sec_reembeddings_and_blocks"}. **Theorem 7**. *Let $\chi$ be a hyperelliptic curve of genus $3$. Then Table [7](#table_reembedding){reference-type="ref" reference="table_reembedding"} gives re-embedding polynomials such that the re-embedding into the tropical modification induced by the polynomials is a faithful tropicalization of $\chi$.* Faithful tropicalizations of curves of genus $2$ have been studied in [@CM2019]. The concrete construction of a faithful tropicalization depends on a case-by-case analysis, considering different possibilities for equalities and inequalities among the valuations of the coefficients. As the Newton polygon of such a curve is sufficiently small, it was possible to perform this case-by-case analysis by hand for curves of genus $2$. Generalizing to hyperelliptic curves of genus $3$ enlarges the amount of possibilities for inequalities among the valuations of the coefficients and the Newton subdivisions to such an extend that a case-by-case analysis by hand is out of reach. We have obtained Theorem [Theorem 7](#thm-main){reference-type="ref" reference="thm-main"} with the aid of [Oscar]{.smallcaps} [@Oscar]. # Techniques for faithful re-embeddings {#sec_reembeddings_and_blocks} In the following, we analyze hyperelliptic curves over a valued field $K$ of characteristic 0 given by a defining equation $$\label{eq_defining} g: \hspace{0.3cm} y^2 = x\cdot\prod_{i=1}^{8}(x_i-\alpha_i);$$ and we denote the valuations of coefficients by $\omega_i = \mathop{\mathrm{val}}(\alpha_i)$. In the introduction we showed an example of a coordinate change that resulted in a faithful tropicalization. In this section, we will introduce techniques to construct these coordinate changes methodically, by using *re-embeddings* into *tropical modifications*. The faithful re-embedding construction we describe follows the divide-and-conquer principle. Given a hyperelliptic curve $\chi$ of genus $g\leq 3$ defined by an equation $g$, see [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"}, we *separate* the tropical hyperelliptic curve into subgraphs according to conditions on the initial forms of $g$ using Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. For each subgraph identified this way, we change coordinates by *finding a correct re-embedding polynomial $f_i$* using Table [7](#table_reembedding){reference-type="ref" reference="table_reembedding"}. For all subgraphs connected by bridges, we further *reduce the dimension* of the ambient space of the re-embedding via Theorem [Theorem 28](#thm_mainthm_bridges){reference-type="ref" reference="thm_mainthm_bridges"}. In this section we show how the Berkovich skeleton can be decomposed into subgraphs that can be considered independently when constructing the re-embedding. Further, we provide details on the re-embeddings themselves. Finally, we describe how to to check faithfulness on a tropical curve. Throughout this section, we will follow the example of the elliptic curve $V(y^2 -x^3-4x^2-8t^4x)$ from Figure [1](#fig-ellipticrational){reference-type="ref" reference="fig-ellipticrational"} and construct its faithful tropicalization using the methods we introduce. ## Classifying hyperelliptic curves via their Berkovich skeleton. Berkovich skeleta of hyperelliptic curves can be glued from a small collection of possible subgraphs, called *building blocks*. In our construction of faithful re-embeddings, these subgraphs serve a vital role: It is sufficient to determine the faithful re-embeddings of building blocks as re-embeddings can be glued with methods we present in Section [4](#sec_globalizing){reference-type="ref" reference="sec_globalizing"}. **Lemma 8**. *The minimal Berkovich skeleton of a hyperelliptic curve is built by gluing building blocks of the types given in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}, and the coefficient conditions in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"} hold.* *Proof.* The types of subgraphs that can appear in the Berkovich skeleton are a special case of [@corey2021hyperelliptictype Theorem 1.1]. The coefficient conditions for cycles, 2-thetas, points and bridges are due to (or straightforward generalizations of) the conditions given in [@CM2019 Table 5.1], and the correctness of the conditions for 3-thetas will be shown later in Theorem [Theorem 18](#thm_3-thetas_faithful){reference-type="ref" reference="thm_3-thetas_faithful"}. ◻ -------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- **at** $x^i$ **Berkovich** **skeleton** **Newton subdivision** **conditions** $\omega_3 = \omega_4,\omega_{5} = \omega_{6}$ $\mathop{\mathrm{in}}(\alpha_{3}) = \mathop{\mathrm{in}}(\alpha_{4}), \mathop{\mathrm{in}}(\alpha_{5}) = \mathop{\mathrm{in}}(\alpha_{6})$ $i$ odd; $\omega_{i+2} = \omega_{i+3};$ $\mathop{\mathrm{in}}(\alpha_{i+2}) = \mathop{\mathrm{in}}(\alpha_{i+3}$) $i$ odd $\omega_{i+1} = \dots = \omega_{i+2k}$ $\mathop{\mathrm{in}}(\alpha_l) \neq \mathop{\mathrm{in}}(\alpha_j)$ for $i < l < j < 2k+2$ Bridge $i$ odd connector $i$ odd, $1\leq j \leq k+1$; $\omega_{i+2l} = \omega_{i+2l+1};$ $\mathop{\mathrm{in}}(\alpha_{i+2l}) \neq \mathop{\mathrm{in}}(\alpha_{i+2l+1});$ $1 \leq l \leq k-1;$ -------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- : The different building blocks appearing for hyperelliptic curves and their coefficient conditions. While point connectors and thetas have the same local Newton subdivision, their coefficient conditions and thus their behaviour under modification are distinguishable. [\[table_standard_trop_coeff_cond\]]{#table_standard_trop_coeff_cond label="table_standard_trop_coeff_cond"} ## Re-embeddings. By re-embedding a plane curve, we can lift its tropicalization in $\mathbb{R}^2$ to a tropical modification of $\mathbb{R}^2$, hoping to make new features visible in the newly attached cells and thus getting closer to the Berkovich skeleton (see, for instance, Examples [Example 10](#modifying_along_tropical_line){reference-type="ref" reference="modifying_along_tropical_line"} and [Example 30](#ex_1-thetas_reduced-dim){reference-type="ref" reference="ex_1-thetas_reduced-dim"}). Formally, they are defined as follows: *Definition 9* (Tropical modifications). Let $F$ be a polynomial over the tropical semiring $\mathbb{T}$. To each cell $\sigma$ of the graph of $F$ that does not have codimension 0 we attach a new cell $\varsigma$ spanned by $\sigma$ and $e_{n+1} := (0, \dots , 0, 1)$. We obtain a pure rational polyhedral complex in $\mathbb{R}^{n+1}$, the *modification* of $\mathbb{R}^n$ along $F$. **Example 10**. We modify $\mathbb{R}^2$ along $F = \min\{x+2,y\}$. ----- ------------------------------------------ $F$ Modification of $\mathbb{R}^2$ along $F$ ----- ------------------------------------------ We can write the modification of $F$ as a tropicalization by finding appropriate *polynomial lifts* $f$ of $F$, i.e. polynomials $f$ whose tropicalization is $F$: $f(x) = \sum_{\beta \in \text{supp(F)}} c_{\beta}x^{\beta}.$ *Definition 11*. In the remainder of the text, we will use the following notation: - $\chi$ denotes a hyperelliptic curve and $g$ its defining equation of the form given in [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"}; - uppercase letters will be used to denote objects in the tropical world, in particular, we will use $F$ (or $F_i$) to denote the tropical polynomials in the tropical polynomial ring $\mathbb{T}[X,Y]$ along which we modify $\mathbb{R}^2$, and call them *modification polynomials*; - lowercase letters will be used to denote objects over the valued field $K$. In particular, $f$ (or $f_i$) will denote a polynomial lift of a modification polynomial. We will call these polynomials *re-embedding polynomials*. *Definition 12* ([@CM2019]). Let $F_1, \dots F_n \in \mathbb{T}[X,Y]$ and $f_1 \dots, f_n$ be lifts of the respective $F_i$. Let $g$ be a defining equation for $\chi$ (see [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"}). Then the tropicalization of the variety $V(I_{g,f})$ of the ideal $I_{g,f} = \langle g, z_1-f_1, \dots, z_n -f_n \rangle \subset K[x^{\pm}, y^{\pm}, z_1^{\pm}, \dots, z_n^{\pm}]$ is a tropical curve in the modification of $\mathbb{R}^2$ along the $F_i$. *Definition 13*. In the following, we will denote by $I_{g,f}$ the ideal corresponding to the re-embedding of the hyperelliptic curve $\chi$ given by a defining equation $g$ (see [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"}) into the tropical modification induced by the re-embedding polynomials $f = (f_1, \dots, f_n)$. ## Analyzing re-embeddings. Re-embeddings into tropical modifications only make hidden parts of the tropical curve visible if the modification curve passes through the parts we want to investigate. Then, we can uncover the hidden parts of the curve by investigating what happens on the new cells. As hidden cycles occur on edges with multiplicity greater than one where two edges get mapped onto each other by the embedding, we restrict our attention to re-embedding polynomials of the form $f(x,y) = y - a_1x - a_2x^2 - \dots a_nx^n \in K[x,y]$. Their tropicalizations pass through the multiplicity two edges of the tropical hyperelliptic curve before re-embedding, see Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. An example of this can be found in Example [Example 15](#ex-running-inside-modification){reference-type="ref" reference="ex-running-inside-modification"}. After re-embedding the tropical curve into modification of $\mathbb{R}^2$, we can check for faithfulness. Since re-embedding is compatible with projection, i.e. projections of curves inside a tropical modification are tropical curves under projection [@CM2019; @S2020], re-embeddings can be analyzed by investigating projections. **Lemma 14** ([@CM2019], Lemma 3.3; and [@S2020], Lemma 1.1). *Given an irreducible curve $\chi \subset (K^*)^2$ defined by polynomials $g \in K[x,y]$ and $f(x,y) = y - a_1x - a_2x^2 - \dots a_nx^n \in K[x,y]$, the tropicalization induced by the ideal $I_{g,f} = \langle g, z-f \rangle \in K[x^{\pm}, y^{\pm}, z^{\pm}]$ is completely determined by the tropical plane curves $\mathop{\mathrm{trop}}(V(g))$, $\mathop{\mathrm{trop}}(V(I_{g,f}\cap K[x^{\pm}, z^{\pm}]))$, $\mathop{\mathrm{trop}}(V(I_{g,f}\cap K[y^{\pm}, z^{\pm}]))$ and one further projection onto a rational plane $P$ in $\mathbb{R}^3$ that is not parallel to any top-dimensional cell in the tropicalization of $z-f$ and not parallel to any of the coordinate planes.* **Example 15**. Now, let us investigate the example $\chi = V(y^2-x^3-4x^2-8t^4x)$ in Figure [1](#fig-ellipticrational){reference-type="ref" reference="fig-ellipticrational"} inside of the modification constructed in Example [Example 10](#modifying_along_tropical_line){reference-type="ref" reference="modifying_along_tropical_line"}. We have actually already seen its $xz$-projection - it is $\mathop{\mathrm{trop}}(-x^3-4 xy+y^2-8t^4x)$ and is Figure [2](#fig-ellipticrationalrepaired){reference-type="ref" reference="fig-ellipticrationalrepaired"}. Finally, if we have identified a re-embedding polynomial constructed the appropriate tropical curve inside the modification, how can we be sure that the resulting tropical curve is actually faithful? The answer lies in the multiplicities of the tropical curve [@MS2015]. **Lemma 16** ([@BPR2014], Theorem 5.24). *An embedded tropical curve is faithful on a subgraph $\Sigma$ of the Berkovich space $\chi^{an}$ if it contains a copy of $\Sigma$, $c_{\Sigma}$, where all edges of $c_{\Sigma}$ have tropical multiplicity one and all vertices are three-valent, i.e. have three incident edges or legs.* # Faithful tropicalizations of building blocks {#sec_local_faithful} In this section we find polynomials for faithful re-embeddings for curves that have genus at least one, i.e. the upper part of Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. We proceed by carefully analyzing the tropical curve obtained in the re-embedding by analyzing its coordinate projections. **Remark 17** (Re-embeddings for cycles and 2-thetas). In [@CM2016], re-embedding polynomials (see Definition [Definition 11](#notation){reference-type="ref" reference="notation"}) for cycles have been constructed as $f(x,y) = y+ \sqrt{-\alpha_3}x$ and in [@CM2019], re-embedding polynomials for 2-thetas were constructed as $f(x,y) = y- \sqrt{-\alpha_3\alpha_4\alpha_5}x + \sqrt{-\alpha_5}x^2$, where the $\alpha_i$ are the coefficients of the defining polynomial of the hyperelliptic curve. **Theorem 18** (Faithful re-embeddings of 3-thetas). *Let $\chi$ be a hyperelliptic curve given by a defining equation $g$ satisfying the coefficient conditions of being a 3-theta as in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. Then, the embedding of $\mathop{\mathrm{trop}}(\chi)$ into the modification of $\mathbb{R}^2$ at $y - (\sqrt{(\prod_{i = 3}^{7} \alpha_i)} x + \sqrt{(\alpha_5\alpha_6\alpha_7)} x^2 - \sqrt{\alpha_{7}}x^3)$ given by $I_{g,f}$ is faithful on the minimal Berkovich skeleton.* Depending on the valuations of the coefficients, several cases have to be considered. The case-by-case analysis is summed up in the following proof. The tables, code and subprocedures providing further details of this proof will be given throughout the section. *Proof.* For the remainder of this section, let $\alpha_i = \beta_i^2$ for $i = 3, ..., 6$ and $\alpha_{7} = -\beta_{7}^2$, so that the re-embedding may be equivalently given as $y - (\prod_3^7\beta_ix+\beta_5\beta_6\beta_7x^2-\beta_7x^3).$ We have $\mathop{\mathrm{in}}(\alpha_3) = \mathop{\mathrm{in}}(\alpha_4)$ and $\mathop{\mathrm{in}}(\alpha_5) = \mathop{\mathrm{in}}(\alpha_6)$ by Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. Let $d_{34} = \mathop{\mathrm{val}}(\alpha_3-\alpha_4)$ and $d_{56}=\mathop{\mathrm{val}}(\alpha_5-\alpha_6)$. We control these differences by defining new variables $\beta_{34} := \beta_3-\beta_4$ and $\beta_{56} := \beta_5-\beta_6$, and substituting $\alpha_3:= (\beta_4 + \beta_{34})^2$ and $\alpha_5:= (\beta_6 + \beta_{56})^2$. We obtain $-\mathop{\mathrm{val}}(\beta_{34}) = d_{34}+\mathop{\mathrm{val}}(\alpha_4)/2= -\mathop{\mathrm{val}}(\alpha_3-\alpha_4)+\mathop{\mathrm{val}}(\beta_4)=\omega_{34}$ and $-\mathop{\mathrm{val}}(\beta_{56}) = d_{56}+\mathop{\mathrm{val}}(\alpha_6)/2 = -\mathop{\mathrm{val}}(\alpha_5-\alpha_6)+\mathop{\mathrm{val}}(\beta_6)=\omega_{56}.$ Substituting the new variables into the equation [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"} of the hyperelliptic curve, we obtain $f(x,y) = y - \beta_4(\beta_{34}+\beta_4)\beta_6(\beta_6+\beta_{56})\beta_7x - \beta_6(\beta_6+\beta_{56})\beta_7x^2 + \beta_7x^3.$ The weight vector $\underline{u}\in\mathbb{R}^6$ encoding the valuation of the six parameters equals $$\underline{u}=\mathop{\mathrm{val}}(\beta_7,\beta_6,\beta_{56},\beta_4,\beta_{34},\beta_2)=(\frac{\omega_7}{2},\frac{\omega_6}{2}, d_{56}-\frac{\omega_6}{2},\frac{\omega_4}{2}, d_{34}-\frac{\omega_4}{2},\frac{\omega_2}{2}).$$ Using the valuation conditions from Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}, the valuations satisfy the following inequalities: $$\begin{aligned} \label{eq_defining_inequalities_3theta} u_7 > u_6, \hspace{0.2cm} u_6 > u_{56}, \hspace{0.2cm} u_6 > u_4, \hspace{0.2cm} u_4 > u_{34}, \hspace{0.2cm}\mathrm{ and } \hspace{0.2cm} u_4 > u_2.\end{aligned}$$ These inequaities span a polyhedral cone $\mathcal{C} \subset \mathbb{R}^6$. Its proper faces correspond to other types of curves. We subdivide the cone further using the data of the $xz$- and the $yz$-projections of $I_{g,f}$, computed in [Oscar]{.smallcaps} [@Oscar] via elimination. For the sake of brevity, we only supply the full computations for the maximal cones of this subdivision here. Supplementary material and further analysis of the lower dimensional cones can be found at <https://victoriaschleis.github.io/bookchapter>. For both projections, we list the leading terms that can appear for all monomials in the projection. For the $xz$-projection, this gives rise to 81 cones, of which 16 are maximal, defined by the inequalities in Table [3](#defining_conditions_xz){reference-type="ref" reference="defining_conditions_xz"}, and give the corresponding leading terms of the $xz$-projection in Table [2](#table_xz_analysis){reference-type="ref" reference="table_xz_analysis"}. Using the $yz$-projection, we subdivide our cones further as given in Remark [Remark 20](#lem tropical dominance conditions){reference-type="ref" reference="lem tropical dominance conditions"}. We do this computationally by taking the common refinement of the two fans spanned by these respective collections of cones. We list the leading terms of the $yz$-projection in Table [5](#leading_terms_coneI){reference-type="ref" reference="leading_terms_coneI"}. We proceed by giving a precise analysis of the Newton subdivision inside of each cone in Lemma [Lemma 19](#3lem_existence_cones_xz){reference-type="ref" reference="3lem_existence_cones_xz"} for the $xz$-projection and in Proposition [Proposition 22](#prop: faithful_yz){reference-type="ref" reference="prop: faithful_yz"} for the $yz$-projection and list them in Table [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} and Table [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"} respectively. Combining the information of these two projections with the naive tropicalization of the curve from Section [2](#sec_reembeddings_and_blocks){reference-type="ref" reference="sec_reembeddings_and_blocks"} and with Lemma [Lemma 14](#lem_proj_lemma_genus_n){reference-type="ref" reference="lem_proj_lemma_genus_n"}, we obtain multiplicity one on all edges of the $3$-theta and thus, using Lemma [Lemma 16](#lem: cartify_faithful){reference-type="ref" reference="lem: cartify_faithful"}, faithfulness on the minimal Berkovich skeleton. ◻ ## Analysis of the $xz$- projection {#sec_xz-projection} We begin our analysis of the re-embedding $I_{g,f}$ by investigating the $xz$-projection of Trop$(V(I_{g,f}))$. Computationally, this is done by elimination of $y$, using the `eliminate`-command in [OSCAR]{.smallcaps} [@Oscar]: ``` {.jlcon breaklines=""} julia> S,(b2,b34,b4,b56,b6,b7)=QQ["b2", "b34", "b4", "b56", "b6", "b7"] (Multivariate polynomial ring in 6 variables over QQ, QQMPolyRingElem[b2, b34, b4, b56, b6, b7]) julia> R,(x, y, z)=S["x", "y", "z"] (Multivariate polynomial ring in 3 variables over multivariate polynomial ring, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z]) julia> g = y^2-x*(x-b2^2)*(x-(b34+b4)^2)*(x-(b4)^2)*(x-(b6+b56)^2) *(x-(b6)^2)*(x+(b7)^2); julia> f= y - b4*(b34 + b4)*b6*(b6 + b56)*b7*x + b6*(b6 + b56)*b7*x^2 - b7*x^3; julia> Igf = ideal(R,[g,f]); julia> I_xz = eliminate(Igf,[y]); ``` The resulting projection is alternatively given by the polynomial $g_{xz} (x,z) := g(x, z + (\prod_{3}^{7} \beta_i) x + \beta_5\beta_6\beta_7 x^2 - \beta_{7}x^3).$ **Lemma 19**. *The conditions on the valuations of the coefficients for 3-thetas (see [\[eq_defining_inequalities_3theta\]](#eq_defining_inequalities_3theta){reference-type="eqref" reference="eq_defining_inequalities_3theta"}) describe a cone $\mathcal{C}$ which is subdivided into 81 cones by inequalities of the coefficients of the $xz$-projection. 16 of these are maximal corresponding to the rows in Table [3](#defining_conditions_xz){reference-type="ref" reference="defining_conditions_xz"}, although only seven distinct Newton subdivisions arise from them.* *Proof.* To analyze the Newton subdivision of $g_{xz}$, we compute all possible leading terms of all monomials in the $xz$-projection. Except for $x^5$ and $x^3$, all monomials admit the same leading terms for any admissible choice of ordering on the $\beta_i$ using [\[eq_defining_inequalities_3theta\]](#eq_defining_inequalities_3theta){reference-type="eqref" reference="eq_defining_inequalities_3theta"}. Omitting constant coefficients, for $x^5$, the possible leading terms are $\beta_6^4$, $-\beta_{56}^2\beta_7^2$, and $-\beta_4^2\beta_7^2$. For $x^3$, the possible leading terms are $\beta_2^2\beta_6^4\beta_7^2$, $\beta_{34}^2\beta_6^4\beta_7^2$, and $\beta_4^4\beta_6^2\beta_7^2$. For the remaining coefficients, we refer to Table [2](#table_xz_analysis){reference-type="ref" reference="table_xz_analysis"}. $x^7, z^2$ $x^6$ $x^4$ $x^2$ $x$ $x^3z$ $x^2z$ $xz$ ------------ ------------- ------------------------------- ---------------------------------------- ---------------------------------------- ----------- -------------------- ----------------------------- -- $1$ $\beta_6^2$ $\beta_4^2\beta_6^2\beta_7^2$ $\beta_2^2\beta_4^2\beta_6^4\beta_7^2$ $\beta_2^2\beta_4^4\beta_6^4\beta_7^2$ $\beta_7$ $\beta_6^2\beta_7$ $\beta_4^2\beta_6^2\beta_7$ : Valuations of the $xz$-projection of the modification ideal $I_{g,f}$ From the computed initial forms we can determine the induced subdivision of the cone $\mathcal{C}$ by the occurring possible leading terms. The inequalities are as follows: ----------------- ------------------------------------ -------------------------- ----------------- ------------------------------------ -------------------------- **Cone** **Condition 1** **Condition 2** **Cone** **Condition 3** **Condition 4** $\mathcal{C}_1$ $2\omega_6 < \omega_{56}+\omega_7$ $\omega_{56}<\omega_{4}$ $\mathcal{C}_A$ $2\omega_4 < \omega_{34}+\omega_6$ $\omega_{34}<\omega_2$ $\mathcal{C}_2$ $2\omega_6 < \omega_{4}+\omega_7$ $\omega_{4}<\omega_{56}$ $\mathcal{C}_B$ $2\omega_4 < \omega_{2}+\omega_6$ $\omega_{2}<\omega_{34}$ $\mathcal{C}_3$ $\omega_{56}+\omega_7<2\omega_6$ $\omega_{56}<\omega_{4}$ $\mathcal{C}_C$ $\omega_{34}+\omega_6<2\omega_4$ $\omega_{34}<\omega_{2}$ $\mathcal{C}_4$ $\omega_{4}+\omega_7<2\omega_6$ $\omega_{4}<\omega_{56}$ $\mathcal{C}_D$ $\omega_{2}+\omega_6<2\omega_4$ $\omega_{2}<\omega_{34}$ ----------------- ------------------------------------ -------------------------- ----------------- ------------------------------------ -------------------------- : Coefficient conditions for cones in the $xz$-projection Now we can assign each of the possible leading terms of $x^5$ and $x^3$ we computed earlier to a cone: for $x^5$, $\beta_6^4$ leads for points in $\mathcal{C}_1$ and $\mathcal{C}_2$, $-\beta_{56}^2\beta_7^2$ for $\mathcal{C}_3$ and $-\beta_4^2\beta_7^2$ for $\mathcal{C}_4$. For $x^3$, the leading terms are $\beta_2^2\beta_6^4\beta_7^2$ for points in $\mathcal{C}_D$, $\beta_{34}^2\beta_6^4\beta_7^2$ for $\mathcal{C}_C$, and $\beta_4^4\beta_6^2\beta_7^2$ for $\mathcal{C}_A$ and $\mathcal{C}_B$. There are seven possible combinatorial types of tropical curves that can arise in the $xz$-projection of the re-embedding, corresponding to the cones determined above. We give them and their corresponding Newton subdivisions in Table [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"}. All Newton subdivisions share a common part, given in Figure [5](#rem_general_xz_subdiv){reference-type="ref" reference="rem_general_xz_subdiv"}, and one that is distinct for each type, given as a subdivision of $\mathcal{P}$ in Table [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"}. ◻ {#rem_general_xz_subdiv} Source codes, examples and material for the analysis of the remaining nonmaximal cones can be found at <https://victoriaschleis.github.io/bookchapter>. Further, an example for each maximal combinatorial type is given there. The following Table [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} is color-coded in accordance with the later analysis for the $yz$-projections. ------------------------------------------- ------------------------------------------------------------- ------------------------ $\mathcal{C}_{1C}, \mathcal{C}_{1D}$ and $\mathcal{C}_{2C}$ $\mathcal{C}_{2D}$       $\mathcal{C}_{3A}$ and $\mathcal{C}_{3B}$ $\mathcal{C}_{3C}$ $\mathcal{C}_{3D}$       $\mathcal{C}_4$   ------------------------------------------- ------------------------------------------------------------- ------------------------ : Tropical curves associated to cones in the $xz$-projection. The associated Newton subdivisions given here are subdivisions of the trapeze $\mathcal{P}$ from Figure [5](#rem_general_xz_subdiv){reference-type="ref" reference="rem_general_xz_subdiv"}. ## Analysis of the $yz$-projection Since the analogous elimination computation for the $yz$-projection does not terminate within reasonable time (2 weeks), we proceed as follows. First, we investigate the structure of the possible coefficients and monomials in the $yz$-projection. The $yz$-projection is given as the variety of a single defining polynomial of degree $7$ in $y$ and $z$. The coefficients of the monomials $y^iz^j$ are homogeneous polynomials $p_{y^iz^j}$ in $\beta^2,\beta^{34},\beta^{4},\beta^{56},\beta^{6}$ and $\beta^{7}$. The homogeneous degree of the $p_{y^iz^j}$ is 1 for monomials of degree $7$ and $(7-(i+j))\cdot 7$ otherwise. Using the defining equation of the cone of $3$-thetas [\[eq_defining_inequalities_3theta\]](#eq_defining_inequalities_3theta){reference-type="eqref" reference="eq_defining_inequalities_3theta"}, we obtain the following technical *tropical dominance conditions* for coefficients: **Remark 20** (Tropical dominance conditions). We note that by the structure of the equations, with the exception of $\beta_7$ all exponents in leading monomials are even, and if the exponent of $\beta_7$ is odd for one possible leading monomial in $p_{y^iz^j}$, they are odd in $\beta_7$ for all leading monomials. Thus, we can assume that inequalities on the valuations between leading monomials are integral. Further, there are minimal inequalities generating all inequalities, that are of the following form: $\omega_{i}<\omega_{j}$ and $\omega_{i}>\omega_{j}$ for pairs $(i,j)\in\{(2,34),(4,56)\}$, $2\omega_i < \omega_{j}+\omega_{k}$ and $2\omega_i > \omega_{j}+\omega_{k}$ for combinations $i,(j,k) \in \{(4,(2,6)),(4,(34,6)),(56,(2,6)),(56,(34,6)),(6,(4,7)),(6,(56,7))\}$. We call a cone $\mathcal{C}_i$ in the subdivision provided by Table [3](#defining_conditions_xz){reference-type="ref" reference="defining_conditions_xz"} *$\beta_{56}$-dominant* if $\omega_{56}<\omega_4$. In this case, only the coefficient conditions we already determined contribute. Alternatively, we call $\mathcal{C}_i$ *$\beta_4$-dominant* (i.e. $\omega_4<\omega_{56}$). In this case, we obtain an additional subdivision of the $yz$-projection, given by inequalities on $\omega_{56}$ in relation to $\omega_{34}$ and $\omega_2$, as now, potentially $\min(\omega_{34},\omega_2)<\omega_{56}$. This is not possible for $\beta_{56}$-dominant terms, as here, $\omega_{56}<\omega_4<\min(\omega_{34},\omega_2)$. In particular, in addition to the conditions considered in [3](#defining_conditions_xz){reference-type="ref" reference="defining_conditions_xz"}, the following five inequalities contribute: ------------ ---------------------------------------------------------------------------- ------------ ---------------------------------------------------------------------------- $\alpha_<$ $2\beta_4 < \min(\beta_2,\beta_{34})+\beta_{56}$ $\alpha_>$ $2\beta_4 > \min(\beta_2,\beta_{34})+\beta_{56}$ $\beta_<$ $\max(\beta_2,\beta_{34})+\beta_{56} < \min(\beta_2,\beta_{34})+\beta_{6}$ $\beta_>$ $\max(\beta_2,\beta_{34})+\beta_{56} > \min(\beta_2,\beta_{34})+\beta_{6}$ $\gamma_<$ $\min(\beta_2,\beta_{34})+\beta_{4} < \max(\beta_2,\beta_{34})+\beta_{56}$ $\gamma_>$ $\min(\beta_2,\beta_{34})+\beta_{4} > \max(\beta_2,\beta_{34})+\beta_{56}$ $\delta_<$ $\min(\beta_2,\beta_{34})+2\beta_{6} < 2\beta_4+\beta_{7}$ $\delta_>$ $\min(\beta_2,\beta_{34})+2\beta_{6} > 2\beta_4+\beta_{7}$ $\phi_<$ $2\beta_4 < \beta_{56} + \beta_6$ $\phi_>$ $2\beta_4 > \beta_{56} + \beta_6$ ------------ ---------------------------------------------------------------------------- ------------ ---------------------------------------------------------------------------- **Proposition 21** (Leading terms). *The leading terms for the coefficients of monomials given in Table [5](#leading_terms_coneI){reference-type="ref" reference="leading_terms_coneI"} are correct and complete.* *Proof.* We proceed in two steps. First, we consider monomials of degree $\geq 5$. For monomials $y^iz^j$ of degree $\geq 5$, the $p_{y^iz^j}$ are of homogeneous degree $\leq 14$. By the structure of the equations, we then expect $p_{y^iz^j}$ to contain monomials that are only in terms of $\beta_{4},\beta_{56},\beta_{6}$ and $\beta_{7}$, and where the exponents in terms of $\beta_4$ and $\beta_{56}$ sum to less than or equal 4. Using the tropical dominance conditions from Remark [Remark 20](#lem tropical dominance conditions){reference-type="ref" reference="lem tropical dominance conditions"}, this implies that these are indeed the only possible leading terms for the monomial satisfying the defining inequality [\[eq_defining_inequalities_3theta\]](#eq_defining_inequalities_3theta){reference-type="eqref" reference="eq_defining_inequalities_3theta"}. Thus, we can substitute integers for the coefficients $\beta_2$ and $\beta_{34}$ and compute the elimination only with respect to the remaining coefficients. This can be computed in [Oscar]{.smallcaps} [@Oscar] using ``` {.jlcon breaklines=""} julia> S,(b4,b56,b6,b7)=QQ["b4", "b56", "b6", "b7"] (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[b4,b56, b6, b7]) julia> R,(x, y, z)=S["x", "y", "z"] (Multivariate polynomial ring in 3 variables over multivariate polynomial ring, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z]) julia> g = y^2-x*(x-1000^2)*(x-(1000000+b4)^2)*(x-(b4)^2)*(x-(b6+b56)^2)*(x-(b6)^2) *(x+(b7)^2); julia> f = y - b4*(1000000 + b4)*b6*(b6 + b56)*b7*x + b6*(b6 + b56)*b7*x^2 - b7*x^3; julia> Igf = ideal(R,[g,f]); julia> I_xy = eliminate(Igf,[x]); ``` which terminates within two hours. This procedure turns the previously homogeneous polynomials into non-homogeneous ones. The leading terms are now in the coefficients of highest degree. Here, we can identify the actual leading terms using the tropical dominance conditions from Remark [Remark 20](#lem tropical dominance conditions){reference-type="ref" reference="lem tropical dominance conditions"} encoded in the following algorithm: [\[alg_initials\]]{#alg_initials label="alg_initials"} **Input:**A polynomial $p_{y^iz^j}$ obtained from the elimination **Output:**A list of potential leading terms $lt$ satisfying the inequalities in [\[eq_defining_inequalities_3theta\]](#eq_defining_inequalities_3theta){reference-type="eqref" reference="eq_defining_inequalities_3theta"} $h_{b_7}$ is the maximal exponent of $\beta_7$ in $p_{y^iz^j}$ potential_monomials = list of monomials containing $\beta_7^{h_{b_7}}$ $h_{b_6}$ is the maximal exponent of $\beta_6$ in potential_monomials curr_monomial = monomial of maximal exponent $h_{b_4}$ of $\beta_4$ (resp. $h_{b_{56}}$ of $\beta_{56}$) among monomials containing $\beta_6^{h_{b_6}}\beta_7^{h_{b_7}}$ $lt$ = $lt\cup$ current_monomial $lt=lt\cup$cross_verification(current_monomial) Adjust $h_{b_7}$, $h_{b_6}$, $h_{b_{56}}$, and $h_{b_4}$ according to Remark [Remark 20](#lem tropical dominance conditions){reference-type="ref" reference="lem tropical dominance conditions"} Now, we proceed with monomials of lower degree. Here, knowing the elimination results in coefficients $\beta_{4},\beta_{56},\beta_{6}$ and $\beta_{7}$ is not sufficient, as the leading terms contain coefficients $\beta_2$ and $\beta_{34}$. To tackle this, we now compute the analogous eliminations, where instead of substituting integers for $\beta_2$ and $\beta_{34}$, we substitute integers for one of $\beta_{2}$ and $\beta_{34}$ and one of $\beta_{4}$ and $\beta_{56}$ respectively, and call the outcomes of these elimination computations *cross-verification polynomials*. Since $\beta_4,\beta_{56},\beta_6$ and $\beta_7$ remain the dominant coefficients, we first identify the potential monomials again as in Algorithm [\[alg_initials\]](#alg_initials){reference-type="ref" reference="alg_initials"}, remembering the gap $k$ in exponents. As we are in a maximal cone, we can assume $\omega_{34}>\omega_2$ or vice versa. Thus, in the cross-verification polynomials, we search for a term containing the correct amount of $\beta_4$ or $\beta_{56}$ respectively, and the maximal exponent of $\beta_2$ or $\beta_{34}$ less than or equal to $k$. We verify the existence of such a term using the remaining cross-verification polynomials. For $\beta_4$-dominant terms, in accordance with Remark [Remark 20](#lem tropical dominance conditions){reference-type="ref" reference="lem tropical dominance conditions"} we repeat the procedure and identify further leading terms by searching for terms containing maximal amounts of $\beta_2$ and $\beta_{34}$ instead of $\beta_{56}$. For mixed terms, we again use the cross-verification process. This computes all leading terms in five or less variables. For the final case of leading terms involving all six variables, we observe that the cross-verification algorithm yields two symmetric outputs of a degree smaller than the expected degree. We combine the two leading terms computed into a single leading term missing no monomials. We supply code computing all possible leading terms at <https://victoriaschleis.github.io/bookchapter>. ◻ **Monomial** **Leading Terms** **Weights** --------------------------------------------------------------------------- ---------------------- ------------------------ ----- $y^4z^3,y^3z^4,$ $y^2z^5, yz^6,z^7$ $0$ all $y^6$ $\beta_{56}^2\beta_6^2\beta_7^3$ $\omega_{56}+\omega_6+3\omega_7/2$ $(1,3)$ $y^5z$ $\beta_{56}^2\beta_7^5$ $\omega_{56} + 5 \omega_7/2$ $(1,3)$ $y^4z^2$ $\beta_6^2\beta_7^5$ $\omega_6+5\omega_7/2$ all $yz^5,z^6$ $7\omega_7/2$ all $y^5$ $\beta_4^6\beta_7^8$ $\beta_{56}^6\beta_7^8$ $3\omega_4+4\omega_7$ $3\omega_{56}+4\omega_7$ $(4)$ $(3)$ $y^4z$ $\beta_{4}^4\beta_6^2\beta_7^8$ $\beta_{56}^4\beta_6^2\beta_7^8$ $2\omega_4+\omega_6+4\omega_7$ $2\omega_{56}+\omega_6+4\omega_7$ ($4$) ($3$) $yz^4, z^5$ $\beta_{56}^2\beta_6^4\beta_7^8$ $\omega_{56}+2\omega_6+4\omega_7$ $(1,3)$ $y^4$ $\beta_2^2\beta_{56}^2\beta_6^{10}\beta_7^7$ $\beta_{34}^2\beta_{56}^4\beta_6^6\beta_7^9$ $\beta_2^2\beta_{56}^4\beta_6^6\beta_7^9$ $\beta_4^4\beta_{56}^2\beta_6^6\beta_7^9$ $\omega_2+\omega_{56}+5\omega_6+7/2\omega_7$ $\omega_{34}+2\omega_{56}+3\omega_6+9/2\omega_7$ $\omega_2+2\omega_{56}+3\omega_6+9/2\omega_7$ $2\omega_4+\omega_{56}+3\omega_6+9/2\omega_7$ ($1B_{\delta_{<}}$,$1D_{\delta_{<}}$,$2B_{\delta_{<}}$,$2D_{\delta_{<}}$) ($3A_{\alpha_{>}}$,$3C_{\alpha_{>}}$,$4A_{\alpha_{>}}$,$4C_{\alpha_{>}}$) ($3B_{\alpha_{>}}$,$3D_{\alpha_{>}}$,$4B_{\alpha_{>}}$,$4D_{\alpha_{>}}$) ($\alpha_{<},\delta_{>}$) $y^3z$ $y^2z^2, yz^3, z^4$ $y^3$ $\beta_{34}^4\beta_{56}^2\beta_6^{12}\beta_7^{10}$ $\beta_2^2\beta_4^4\beta_{56}^2\beta_6^{10}\beta_7^{10}$ $\beta_{34}^2\beta_4^4\beta_{56}^2\beta_6^{10}\beta_7^{10}$ $2\omega_{34}+\omega_{56}+6\omega_6+5\omega_7$ $\omega_2+2\omega_4+\omega_{56}+5\omega_6+5\omega_7$ $\omega_{34}+2\omega_4+\omega_{56}+5\omega_6+5\omega_7$ ($C$) ($B$) ($A$) $y^2z, yz^2,z^3$ $\beta_{34}^2\beta_4^2\beta_{56}^2\beta_6^{12}\beta_7^{10}$ $\omega_{34}+\omega_4+\omega_{56}+6\omega_6+5\omega_7$ ($A,C$) $y^2$ $\beta_2^4\beta_4^4\beta_{56}^4\beta_6^{12}\beta_7^{11}$ $\beta_{34}^4\beta_4^4\beta_{56}^4\beta_6^{12}\beta_7^{11}$ $\beta_2^2\beta_4^8\beta_{56}^2\beta_6^{12}\beta_7^{11}$ $\beta_{34}^2\beta_4^8\beta_{56}^2\beta_6^{12}\beta_7^{11}$ $2\omega_2+2\omega_4+2\omega_{56}+6\omega_6+11/2\omega_7$ $2\omega_{34}+2\omega_4+2\omega_{56}+6\omega_6+11/2\omega_7$ $\omega_2+4\omega_4+\omega_{56}+6\omega_6+11/2\omega_7$ $\omega_{34}+4\omega_4+\omega_{56}+6\omega_6+11/2\omega_7$ ($B_{\alpha_{>}}$,$D_{\beta_{<}}$) ($A_{\alpha_{>}}$,$C_{\beta_{<}}$) ($B_{\alpha_{<}}$) ($A_{\alpha_{<}}$) $yz, z^2$ $y,z$ $\beta_2^4\beta_4^8\beta_{56}^4\beta_6^{14}\beta_7^{12}$ $\beta_{34}^4\beta_4^8\beta_{56}^4\beta_6^{14}\beta_7^{12}$ $\beta_2^2\beta_{34}^2\beta_4^{12}\beta_6^{14}\beta_7^{12}$ $\beta_2^4\beta_4^{10}\beta_{56}^2\beta_6^{14}\beta_7^{12}$ $\beta_{34}^4\beta_4^{10}\beta_{56}^2\beta_6^{14}\beta_7^{12}$ $2\omega_2+4\omega_4+2\omega_{56}+7\omega_6+6\omega_7$ $2\omega_{34}+4\omega_4+2\omega_{56}+7\omega_6+6\omega_7$ $\omega_2+\omega_{34}+6\omega_4+7\omega_6+6\omega_7$ $2\omega_2+5\omega_4+\omega_{56}+7\omega_6+6\omega_7$ $2\omega_{34}+5\omega_4+\omega_{56}+7\omega_6+6\omega_7$ ($1B,1D,3B,3D$) ($1A,1C,3A,3C$) ($2_{\phi_<\gamma_>}$, $4_{\phi_<\gamma_>}$) ($2B_{\gamma_{<}}$,$2D_{\gamma_{<}}$,$4B_{\gamma_{<}}$,$4D_{\gamma_{<}}$) ($2A_{\gamma_{<}}$,$2C_{\gamma_{<}}$,$4A_{\gamma_{<}}$,$4C_{\gamma_{<}}$) : Leading terms for monomials in the $yz$-projection. r6.5cm[\[picture_reembedding\]]{#picture_reembedding label="picture_reembedding"}  To analyze the projections, we to provide a geometric description of the re-embedding, as parallel cells need to be carefully analyzed. For the re-embedding $f$ given in Theorem [Theorem 18](#thm_3-thetas_faithful){reference-type="ref" reference="thm_3-thetas_faithful"}, the tropical hypersurface Trop($V(z-f)$) consists of nine 2-dimensional cells, defined by the following systems of inequalities: $\sigma_1 = \{ Z = X+A \geq Y, X \leq A-B \}$, $\sigma_2 = \{ Z = 2X + B \geq Y, B-C \geq X \geq A-B\}$, $\sigma_3 = \{ Z = 3X+C \geq Y, X \geq B-C \}$, $\sigma_4 = \{ Z, Y \leq 2A-B, X = A-B\}$, $\sigma_5 = \{ Z, Y \leq 3B -2C, X = B-C \}$, $\sigma_6 = \{ Y = X+A \geq Z, X \leq A-B \}$, $\sigma_7 = \{ Y = 2X+B \geq Z, A-B \leq X \leq B-C\}$, $\sigma_8 = \{ Y = 3X+C \geq Z, B-C \leq X \}$, $\sigma_9 = \{ Z = Y \geq X+A,2X+B,3X+C \}$. **Proposition 22**. *Using the re-embedding of Theorem [Theorem 18](#thm_3-thetas_faithful){reference-type="ref" reference="thm_3-thetas_faithful"}, all edges of the re-embedded tropical curve $\mathop{\mathrm{Trop}}(g)$ have tropical multiplicity one, i.e. the re-embedding yields a faithful tropicalization.* *Proof.* We observe that in the $yz$-projection, the pairs of cells $\sigma_1$ and $\sigma_4$, $\sigma_1$ and $\sigma_5$, $\sigma_2$ and $\sigma_5$, and $\sigma_4$ and $\sigma_5$ overlap in parts. Further, the behaviour on these cells cannot be directly observed on the $xz$-projection. Thus, we expect some of the edges in the projections to have higher multiplicities, and need to carefully analyze them to understand the multiplicities and structure of the re-embedded curve. All in all, there are four types of such irregularities that can occur: 1. Some multiplicity one edges or legs inherit higher multiplicities in the $yz$-tropicalization due to the push-forward formula for multiplicities, see [@Tev07]. Multiplicities inherited in this way are the black multiplicities in Table [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. On the images of the overlapping cells $\sigma_1$, $\sigma_2$, $\sigma_4$ or $\sigma_5$, the following further behaviour can occur: 1. Parallelograms inside the Newton subdivision may indicate either a vertex (red in the subdivision) or not (marked in blue), depending on whether two edges or legs of $I_{g,f}$ that intersect in the $yz$-projection intersect in $I_{g,f}$ or not. 2. Two edges or legs overlap in the $yz$-tropicalization and their multiplicities get added accordingly, indicated by dashed edges and sums of multiplicities on the curve in Table [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. 3. Vertices lie in the relative interiors of edges or on another vertex. The vertices for which this can happen are the purple and the orange vertices in Tables [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} and [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. We resolve these issues by investigating the corresponding $xz$-projection for each case and combining the information from both projections. A color-coded analysis of each individual representative is given in Table [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} for the $xz$-projection and for one representative in Table [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"} for the $yz$-projection. Edges that are the same color in the projections correspond to each other. In Example [Example 23](#ex: refined modification){reference-type="ref" reference="ex: refined modification"}, a three dimensional reconstruction is carried out in full for the cone $\mathcal{C}_0$, and analogous constructions can be obtained using the subdivisions and tropical curves in Tables [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} and [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. ◻ ---------------------------- ---------------------------------   ---------------------------- --------------------------------- : Newton subdivisions and tropical curves associated to cones in the $yz$-projection. **Example 23**. We give an example of a re-embedded curve in Cone $\mathcal{C}_{1A_{\alpha_<\delta_>}}$ in Table [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. The colors of edges and vertices coincide with the colors in the $xz$- and $yz$-projections in Tables [4](#table_tropical_curves_xz){reference-type="ref" reference="table_tropical_curves_xz"} and [6](#table_newton_subdivs_yz){reference-type="ref" reference="table_newton_subdivs_yz"}. # Gluing re-embeddings {#sec_globalizing} In the previous section we have seen how to locally modify the tropical hyperelliptic curve according to specific building blocks of the associated Berkovich skeleton. This allows us to faithfully re-embed in the case of 3-thetas and 3-points. But what happens if the Berkovich skeleton consists of multiple building blocks? In this section, we show how to glue the constructed re-embeddings of building blocks to a re-embedding that works for the whole curve. **Remark 24**. First, let us assume that we are considering two building blocks connected by point connector $p$. Then, the union is already faithful on the minimal Berkovich skeleton. Thus, we assume that both building blocks are connected by a bridge $e_i$. The coefficient condition of bridges gives rise to a dual edge of length one, as can be observed in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. Therefore, $e_i$ has tropical multiplicity one and is faithful on every subgraph of the Berkovich skeleton that contains it. Thus, if the tropicalizations on both sides of the $e_i$ are faithful on respective subgraphs of the Berkovich skeleton connected by $e_i$ in the Berkovich skeleton, the whole curve is faithful on the union. The second piece of the gluing puzzle is the ability to shift re-embeddings: The modifications given in Section [3](#sec_local_faithful){reference-type="ref" reference="sec_local_faithful"} work for the first monomial in the building block being $x$, but multiple building blocks will necessarily start at different monomials. The next proposition solves just that: multiplying the modification polynomial by the correct monomial at which the building block starts preserves faithfulness. **Lemma 25**. *Let $\chi$ be an irreducible hyperelliptic curve given by the defining equation $g(x,y) := y^2 - h(x) = 0$ and let $\tilde{\chi}$ be another curve given by the defining equation $\tilde{g}(x,y):= y^2 - x^{2l+1}h(x) = 0.$ Further, let $f \in K[x,y]$ be a modification polynomial such that the tropicalization induced by the ideal $I_{g,f} = \langle g, z-f \rangle \in K[x^{\pm}, y^{\pm}, z^{\pm}]$ is faithful. Then, for the modification polynomial $\tilde{f}(x,y) = y - x^{l-1}f \in K[x,y]$, the tropicalization induced by the ideal $I_{\tilde{g},\tilde{f}} = \langle \tilde{g}, z-\tilde{f} \rangle \in K[x^{\pm}, y^{\pm}, z^{\pm}]$ is faithful.* *Proof.* Let $\mathcal{P}$ be the Newton polytope of $f$ with the negative valuation of every monomial corresponding to a point attached as a last coordinate. Then, for every vertex $\tilde{v}$ in $\tilde{\mathcal{P}}$ corresponding to the monomial $m$ there is a vertex $v$ in $\mathcal{P}$ corresponding to $mx^{-l}$ with the same valuation. Thus, $\mathcal{P}$ and $\tilde{\mathcal{P}}$ yield congruent subdivisions on their respective Newton polytopes $\mathcal{N}$ and $\tilde{\mathcal{N}}$ under the coordinate shift $x^i \mapsto x^{i+l}$. Analogous computations hold for the projections of their respective re-embeddings. Further, lattice lengths stay constant under this shift. Thus, $\tilde{g}$ only has multiplicity one edges in the re-embedding along $\tilde{f}$ by construction and is thus faithful by Lemma [Lemma 16](#lem: cartify_faithful){reference-type="ref" reference="lem: cartify_faithful"}. ◻ Combining the results from Section [3](#sec_local_faithful){reference-type="ref" reference="sec_local_faithful"} and the shifting from Lemma [Lemma 25](#prop_shifting_building_blocks){reference-type="ref" reference="prop_shifting_building_blocks"}, we obtain the following re-embedding polynomials for building blocks: --------------- -- -- **at** $x^i$ **Berkovich** **skeleton** --------------- -- -- : Re-embedding polynomials for the building blocks. The $\alpha_i$ are the coefficients in the defining equation of the hyperelliptic curve $\chi$, see [\[eq_defining\]](#eq_defining){reference-type="eqref" reference="eq_defining"}, and the $i$ is the exponent of the first monomial contributing to the cycle or 2-theta, as in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. [\[table_reembedding\]]{#table_reembedding label="table_reembedding"} Finally, we can combine all the pieces constructed in the last Sections to obtain our main result, Theorem [Theorem 7](#thm-main){reference-type="ref" reference="thm-main"}. **Theorem 26**. *Let $\chi$ be an irreducible hyperelliptic curve of genus three defined by an equation $g$ as in Equation [\[eq_defining\]](#eq_defining){reference-type="ref" reference="eq_defining"}. The embeddding of $\chi$ in the ideal given by modifications along hyperplanes generated by $z_i - f_i(x,y)$, where $f_i(x,y) = y - x^l(a_1x - a_2x^2 - \dots - a_{k_i}x^{k_i})$ for suitable $l$ and $a_i$ given in Table [7](#table_reembedding){reference-type="ref" reference="table_reembedding"} by considering $I_{g,f} = \langle g, z_1-f_1, \dots , z_m - f_m \rangle \in K[x^{\pm}, y^{\pm}, z_1^{\pm}, \dots, z_m^{\pm}]$ is faithful.* *Proof.* First, we separate the defining equation $g$ into its different components that identify the different (possibly shifted) building blocks using the coefficient conditions in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}. In Section [3](#sec_local_faithful){reference-type="ref" reference="sec_local_faithful"}, we showed that for the first building block of the Berkovich skeleton of $\chi$, this is a suitable faithful re-embedding regardless of the type of the building block. By applying Proposition [Lemma 25](#prop_shifting_building_blocks){reference-type="ref" reference="prop_shifting_building_blocks"}, we see that this is true for all building blocks contained in the tropical curve. For two connected adjacent building blocks carrying genus, we proceed pairwise by elimination of variables: We denote by $I_{g,f_i, k_1}$ the re-embedding along $f_i$, by $I_{g,f_{i+1}, k_2}$ the re-embedding along $f_{i+1}$ and by $I_{g,f_i,f_{i+1}, k_1, k_2}$ the re-embedding along both $f_i$ and $f_{i+1}$. Since the two sets of variables of the re-embedding are disjoint, eliminating the variables $z_1, \dots, z_m$ we obtain $I_{g,f_i,f_{i+1}, k_1, k_2} \cap K[x^{\pm}, y^{\pm}, w_1^{\pm}, \dots, w_{\tilde{m}}^{\pm}] = I_{g,f_{i+1}, k_2}.$ Analogously, we obtain that $I_{g,f_i,f_{i+1},k_1, k_2} \cap K[x^{\pm}, y^{\pm}, z_1^{\pm}, \dots, z_{m}^{\pm}] = I_{g,f_i, k_1}$. Since $I_{g,f_i, k_1} \subset I_{g,f_i, f_{i+1}, k_1, k_2}$ and $I_{g,f_{i+1}, k_2} \subset I_{g,f_i,f_{i+1}, k_1, k_2}$, all edges that appear in $I_{g,f_i, k_1}$ and $I_{g,f_{i+1}, k_2}$ can be identified with an edge of the same length in $I_{g,f_i,f_{i+1}, k_1, k_2}$. Thus, as $I_{g,f_i, k_1}$ and $I_{g,f_{i+1}, k_2}$ contain isometric copies of their respective building blocks, $I_{g,f_i,f_{i+1}, k_1, k_2}$ does as well. By assumption, all multiplicities of edges in $I_{g,f_i, k_1}$ and $I_{g,f_{i+1}, k_2}$ corresponding to isometric copies of the building blocks of genus $k_1$ and $k_2$ are one. Thus, all multiplicities of edges in $I_{g,f_i,f_{i+1}, k_1, k_2}$ corresponding to isometric copies of the building blocks of genus $k_1$ and $k_2$ are one. Then, by Remark [Remark 24](#lem_bridges_faithful){reference-type="ref" reference="lem_bridges_faithful"}, the tropicalization induced by $I_{g,f_i,f_{i+1}, k_1, k_2}$ is faithful on both building blocks and their connector. Thus, we obtain a faithful tropicalization of $\chi$. ◻ **Example 27** (Partial modification of two cycles). Consider the curve $g(x,y) = y^2 - x(x-t^2)(x-t^4)(x-t^6)(x-t^8).$ All valuations of the monomials in $x$ are distinct. Using the coefficient conditions in Table [1](#table_standard_trop_coeff_cond){reference-type="ref" reference="table_standard_trop_coeff_cond"}, we obtain that $g$ consists of two cycles, separated by a bridge. First, we modify at the lower edge in direction $z$ with $f = y - \sqrt{t^{2}t^4t^6}x = y - t^6x$. Then, we modify at the upper edge using $\tilde{f} = y - \sqrt{t^2}x^2 = y-tx^2$, and obtain a faithful tropicalization. The resulting modification is depicted on the left in Example [Example 30](#ex_1-thetas_reduced-dim){reference-type="ref" reference="ex_1-thetas_reduced-dim"}. ## Reducing dimensions {#sec_shrinking_dimensions} We have a faithful re-embedding now, but we can do better, for curves whose Berkovich skeleton contains bridges: we can systematically cut down ambient dimension by combining multiple re-embedding polynomials into one. **Theorem 28**. *Let $\chi$ be an irreducible hyperelliptic curve given by a defining equation $g$ as in Equation ([\[eq_defining\]](#eq_defining){reference-type="ref" reference="eq_defining"}) consisting of two $k$-thetas of genus $k_1$ and $k_2$ respectively, connected by a bridge. Let $I_{g,\hat{f}, k_1} = \langle g, z_1-\hat{f} \rangle \in K[x^{\pm}, y^{\pm}, z_1^{\pm}]$ and $I_{g,\tilde{f}, k_2} = \langle g, w_1-\tilde{f}_1 \rangle \in K[x^{\pm}, y^{\pm}, w_1^{\pm}]$ be the ideals determined in Section [3](#sec_local_faithful){reference-type="ref" reference="sec_local_faithful"} for which the embedding is faithful, where $\hat{f} = y - \hat{h}(x)$ and $\tilde{f}= y-x^{k_1 + 1}\tilde{h}(x)$, obtained from Proposition [Lemma 25](#prop_shifting_building_blocks){reference-type="ref" reference="prop_shifting_building_blocks"}. Let $f = y -\hat{h}(x) - x^{k_1 + 1}\tilde{h}(x)$.* *Then, the ideal $I_{g,f} = \langle g, z_1-f \rangle \in K[x^{\pm}, y^{\pm}, z_1^{\pm}]$ induces a faithful tropicalization.* *Proof.* By Theorem [Theorem 26](#thm_mainthm_point_connectors_minimal_berkovich){reference-type="ref" reference="thm_mainthm_point_connectors_minimal_berkovich"}, $I_{g,f, k_1, k_2} = \langle g, z_1-\hat{f}, w_1-\tilde{f} \rangle \in K[x^{\pm}, y^{\pm}, z_1^{\pm}, w_1^{\pm}]$ induces a faithful tropicalization on the whole Berkovich skeleton. Let $F=\mathop{\mathrm{trop}}(f)$. As $\hat{F}= \mathop{\mathrm{trop}}(\hat{f})$ and $\tilde{F}= \mathop{\mathrm{trop}}(\tilde{f})$ pass through the respective multiplicity two edges of the thetas in the standard $xy$-tropicalization of $g$ by Section [2](#sec_reembeddings_and_blocks){reference-type="ref" reference="sec_reembeddings_and_blocks"}, so does $F$ by its construction. Let $C$ be the additional cell in $F$ that is not contained in any cell of $\hat{F}$ or $\tilde{F}$. (For an example, see Example [Example 30](#ex_1-thetas_reduced-dim){reference-type="ref" reference="ex_1-thetas_reduced-dim"}) Outside of $C$, the modification agrees with $I_{g,\hat{f}, k_1}$ before the bridge and $I_{g,\tilde{f}, k_2}$ after the bridge respectively. They are thus faithful on the two thetas as they lie on distinct cells. Hence, by Remark [Remark 24](#lem_bridges_faithful){reference-type="ref" reference="lem_bridges_faithful"}, the modification is faithful on the minimal Berkovich skeleton. By construction, there is no edge or leg on $C$. Further, all legs and edges of the projected curve around $C$ have multiplicity one by Remark [Remark 24](#lem_bridges_faithful){reference-type="ref" reference="lem_bridges_faithful"}. By Lemma [Lemma 14](#lem_proj_lemma_genus_n){reference-type="ref" reference="lem_proj_lemma_genus_n"}, this extends to the re-embedding. Thus, the modification is faithful on $C$ as well. ◻ **Remark 29**. If the Berkovich skeleton of $\chi$ consists only of thetas and bridges (i.e. $\chi_{trop}$ is in a *top-dimensional cone of the moduli space $M_{3,n}^{trop}$*), the tropicalization of $\chi$ can be faithfully modified in dimension three by re-embedding Trop($\chi$) into a tropical hypersurface obtained as the tropicalization of $V(z- (y - a_1x - a_2x^2 - \dots + a_7x^7) \in K[x,y]$. **Example 30** (Global tropical modification of two cycles). Consider the curve $g(x,y) = y^2 - x(x-t^2)(x-t^4)(x-t^6)(x-t^8)$ from Example [Example 27](#ex_partial_mod_2_1_thetas){reference-type="ref" reference="ex_partial_mod_2_1_thetas"}. By using reducing dimension using Theorem [Remark 29](#thm_mainthm_topdim_cones){reference-type="ref" reference="thm_mainthm_topdim_cones"}, we can further simplify Example [Example 27](#ex_partial_mod_2_1_thetas){reference-type="ref" reference="ex_partial_mod_2_1_thetas"} while preserving faithfulness. By modifying $\mathbb{R}^2$ at $f = y- t^6x - tx^2$, we obtain the following re-embedding that is faithful on the tropical curve. ------------------------------------------------ --------------------------------------------   ------------------------------------------------ -------------------------------------------- The picture on the left shows the modification of Example [Example 27](#ex_partial_mod_2_1_thetas){reference-type="ref" reference="ex_partial_mod_2_1_thetas"}, and the picture on the right shows the combined modification. The purple cell on the right is $C$ referred to in the proof of Theorem [Theorem 28](#thm_mainthm_bridges){reference-type="ref" reference="thm_mainthm_bridges"}.

arxiv_math

--- abstract: | We study semilinear elliptic equations with the fractional Laplacian in $\mathbb{R}$. The equations with single power nonlinearities have been observed by Weinstein(1987), Frank--Lenzmann(2013) and so on. We focus on the equations with double power nonlinearities and consider the existence of ground states. author: - "Kaito Kokubu[^1]" bibliography: - reference_list.bib title: On solitary wave solutions to dispersive equations with double power nonlinearities --- Department of Mathematics, Graduate School of Science, Tokyo University of Science,\ 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan # Introduction We consider the following stationary problem: $$\begin{aligned} D_{x}^{\sigma} \phi + c \phi - f(\phi) = 0, \quad x\in\mathbb{R}, \label{eq:SP0}\end{aligned}$$ where $\phi$ is a real-valued unknown function, $c$ is a positive constant and $f$ is some given nonlinearity. Furthermore, for $0 < \sigma < 2$, we define the operator $D_{x}^{\sigma}$ as $D_{x}^{\sigma}:=\mathscr{F}^{-1}|\xi|^{\sigma}\mathscr{F}$, where $\mathscr{F}$ is the Fourier transform. This operator is often written as $(-\Delta)^{\sigma/2}$. It is well known that the solutions of [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} provide solitary wave solutions of some dispersive equations. For example, suppose that $\phi$ is a solution to [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"}, then $u(t,x)=\phi(x-ct)$ is a travelling wave solution to Benjamin--Ono type equations $$\begin{aligned} \partial_{t}u + \partial_{x}f(u) + \partial_{x}D_{x}^{\sigma}u = 0, \quad x\in \mathbb{R}, \ t\in \mathbb{R},\end{aligned}$$ with the wave speed $c>0$, and $u(t,x)=e^{i \omega t}\phi(x)$ is a standing wave solution to fractional Schrödinger equations $$\begin{aligned} i\partial_{t}u - D_{x}^{\sigma}u + f(u) = 0, \quad x\in \mathbb{R}, \ t\in \mathbb{R},\end{aligned}$$ with the frequency of oscillation $\omega>0$. (Here we replace $c$ with $\omega$.) One of the important solutions is *a ground state*. A ground state is a nontrivial solution to [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} which minimizes the value of *the action functional* corresponding to [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"}: $$\begin{aligned} S(u) = \frac{1}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} - \int_{\mathbb{R}} F(u) \, dx,\end{aligned}$$ where $F(s):=\int_{0}^{s} \, f(t) dt$ for $s\in\mathbb{R}$. The stationary problems [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with single power nonlinearities have been studied well so far. First, in case $\sigma=1$, Benjamin[@Benjamin] obtained the explicit solution to [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with $f(s) = s^{2}$: $\phi(x) = \frac{2c}{1+c^{2}x^{2}}$. Furthermore, Weinstein[@Weinstein] found a positive and even ground state of [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with $f(s)=s^{p} \ (p\in\mathbb{N}, \ p\geq2.)$ With the same method as [@Weinstein], we can obtain the existence of a positive and even ground state of more general problems $$\begin{aligned} D_{x}^{\sigma} \phi + c \phi - |\phi|^{p-1}\phi = 0,\end{aligned}$$ where $0 <\sigma < 2, \ 1 < p < 2^{\ast}_{\sigma}-1$, and $2^{\ast}_{\sigma}:=2/(1-\sigma)_{+}$ (see Remark (1) of Proposition 1.1 in [@Frank-Lenzmann].) The uniqueness of ground states of these equations is shown by Frank--Lenzmann[@Frank-Lenzmann]. Recently, the stability and instability of solitary wave solutions of dispersive equations with double power nonlinearities have been studied: for example, Ohta[@Ohta_1995_S], Fukaya-Hayashi[@Fukaya-Hayashi-2011], and Lewin--Nodari [@Lewin-Nodari] for the Schrödinger equation. Then our interest is the stability and instability of travelling wave solutions to the Benjamin--Ono equation $$\begin{aligned} \partial_{t} u + \partial_{x}(-\phi^{p} + \phi^{q}) + \partial_{x}D_{x}u = 0, \quad x\in\mathbb{R}, \ t\in\mathbb{R}, \tag{GBO} \label{eq:GBO}\end{aligned}$$ where $p, \ q \in\mathbb{N}, \ 2 \leq p < q$. To observe this, we need to study the existence of solutions, especially ground states, of the following stationary problem derived from [\[eq:GBO\]](#eq:GBO){reference-type="eqref" reference="eq:GBO"}: $$\begin{aligned} D_{x}\phi + c\phi + \phi^{p} - \phi^{q} = 0, \quad x\in\mathbb{R}. \tag{SP} \label{eq:SP00}\end{aligned}$$ The results on the existence depend on the parities of $p$ and $q$. Therefore, the aim of this paper is to classify the existence and properties of solutions to [\[eq:SP00\]](#eq:SP00){reference-type="eqref" reference="eq:SP00"} with respect to the parities. To consider the classification, we first observe [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with the following double power nonlinearities: $$\begin{aligned} & f_{1}(s) := -|s|^{p-1}s + |s|^{q-1}s, \\ & f_{2}(s) := |s|^{p-1}s + |s|^{q-1}s, \\ & f_{3}(s) := |s|^{p-1}s - |s|^{q-1}s,\end{aligned}$$ for $s\in\mathbb{R}$, where $1 < p < q < 2^{\ast}_{\sigma} -1$, and apply these results to the classification of solutions of [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}. # Main Results {#section:main_results} First, for convenience, we name equations [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} as $$\begin{aligned} D_{x}^{\sigma} \phi + c \phi - f_{j}(\phi) = 0, \quad x\in\mathbb{R}. \tag{SP$j$} \label{eq:SPj}\end{aligned}$$ Moreover, we put $$\begin{aligned} 2^{\ast}_{\sigma} := \frac{2}{(1-\sigma)_{+}} = \left\{ \begin{aligned} & \frac{2}{1-\sigma}, \ &\text{for} \ 0 < \sigma < 1, \\ & \infty, \ &\text{for} \ 1 \leq \sigma < 2. \end{aligned} \right.\end{aligned}$$ Before stating the main results, we define a ground state. First, we write the action funcional $S_{j}\colon H^{\sigma/2}(\mathbb{R}) \rightarrow \mathbb{R}$ corresponding to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} as $$\begin{aligned} S_{j}(u) := \frac{1}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} - \int_{\mathbb{R}} F_{j}(u) \, dx,\end{aligned}$$ where $$\begin{aligned} F_{j}(s):=\int_{0}^{s} f_{j}(t) \, dt\end{aligned}$$ for $s\in\mathbb{R}$. We can see that $S_{j}\in C^{1}(H^{\sigma/2}(\mathbb{R}),\mathbb{R})$ and $$\begin{aligned} S_{j}'(u) = D_{x}^{\sigma}u + cu - f_{j}(u),\end{aligned}$$ which implies that $u\in H^{\sigma/2}(\mathbb{R})$ is a solution to [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} if and only if $S_{j}'(u)=0$. Here we put the set of nontrivial solutions of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} as $$\begin{aligned} \mathcal{N}_{j}:= \{ v\in H^{\sigma/2}(\mathbb{R}) \setminus \{ 0 \} : S_{j}'(v)=0 \} .\end{aligned}$$ ****Definition** 1**. By *a ground state*, we mean $\phi\in\mathcal{N}_{j}$ which minimizes the value of $S_{j}$ in $\mathcal{N}_{j}$. We put the set of ground states of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} as $$\begin{aligned} \mathcal{G}_{j} := \{ \phi\in\mathcal{N}_{j} : S_{j}(\phi) \leq S_{j}(v) \ \text{for all} \ v\in\mathcal{N}_{j} \} . \end{aligned}$$ Here we state the results for [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. ****Theorem** 1**. *Set $j=1, \ 2$ and let $0 < \sigma < 2$. For any $c>0$, there exists a ground state of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} which belongs to $H^{\sigma + 1}(\mathbb{R})$. Moreover, it can be taken as positive, even, and decreasing in $|x|$. In addition, there do not exist any ground states of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} with sign changes. [\[Thm:SP1_2\]]{#Thm:SP1_2 label="Thm:SP1_2"}* ****Theorem** 2**. *Let $0 < \sigma \leq 1$ and set $$\begin{aligned} \alpha := \frac{q-1}{q-p}, \quad \beta := \frac{p-1}{q-p}, \quad c_{0} := \frac{2(q-p)(p-1)^{\beta}(q+1)^{\beta}}{(p+1)^{\alpha}(q-1)^{\alpha}}. \end{aligned}$$ If $c\in(0,c_{0})$, then there exists a ground state of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"} which belongs to $H^{\sigma+1}(\mathbb{R})$ and is positive, even, and decreasing in $|x|$. [\[Thm:SP3\]]{#Thm:SP3 label="Thm:SP3"}* ****Remark** 3**. In Theorem [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"}, we do not deny the existence of a ground state of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"} which has sign changes. ****Remark** 4**. We can also obtain a negative ground state of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} if it has a positive ground state. Indeed, let $\phi\in\mathcal{G}_{j}$ be positive. Then we obtain $-\phi\in\mathcal{N}_{j}$ and $S_{j}(-\phi)=S_{j}(\phi)$, which means $-\phi\in\mathcal{G}_{j}$. Applying Theorems [\[Thm:SP1_2\]](#Thm:SP1_2){reference-type="ref" reference="Thm:SP1_2"} and [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"}, we obtain the classification results of [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}: ****Theorem** 5**. *Let $c>0, \ p,\ q \in \mathbb{N}, \ 2 \leq p < q$. Then the followings hold for [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}:* 1. *Assume that $p$ is odd and $q$ is odd. Then there exists a ground state which is positive, even, and decreasing in $|x|$. There also exists a ground state which is negative, even, and increasing in $|x|$. [\[case:odd_odd\]]{#case:odd_odd label="case:odd_odd"}* 2. *Assume that $p$ is odd and $q$ is even. Then there exists a ground state which is positive, even, and decreasing in $|x|$, while there do not exist any negative solutions. [\[case:odd_even\]]{#case:odd_even label="case:odd_even"}* 3. *Assume that $p$ is even and $q$ is odd. Then there exists a solution which is positive, even, and decreasing in $|x|$. Moreover, there exists a ground state which is negative, even, and increasing in $|x|$. In addition, none of the positive solutions is a ground state. [\[case:even_odd\]]{#case:even_odd label="case:even_odd"}* 4. *Assume that $p$ is even and $q$ is even. Then there exists a solution which is positive, even, and decreasing in $|x|$. Moreover, there exists a solution which is negative, even, and increasing in $|x|$ if $c \in (0,c_{0})$, where $c_{0}$ is defined in Theorem [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"}. [\[case:even_even\]]{#case:even_even label="case:even_even"}* *[\[thm:classification\]]{#thm:classification label="thm:classification"}* ****Remark** 6**. In Case ([\[case:even_even\]](#case:even_even){reference-type="ref" reference="case:even_even"}) of Theorem [\[thm:classification\]](#thm:classification){reference-type="ref" reference="thm:classification"}, ground states are not identified yet. In the following, we will split the proof of Theorems [\[Thm:SP1_2\]](#Thm:SP1_2){reference-type="ref" reference="Thm:SP1_2"} and [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"} into two parts. We prove the existence of ground states of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} in §[3](#Section:existence_of_GS){reference-type="ref" reference="Section:existence_of_GS"}, and the regularity and positivity of solutions of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} in §[4](#section:properties){reference-type="ref" reference="section:properties"}. Finally, the proof of Theorem [\[thm:classification\]](#thm:classification){reference-type="ref" reference="thm:classification"} will be given in §[5](#section:application){reference-type="ref" reference="section:application"}. # Existence of ground states {#Section:existence_of_GS} In this section, we consider the existence of ground states of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. Before considering the existence, we recall the symmetric decreasing rearrangement of a nonnegative function, which plays an important role to obtain a solution which is positive, even, and decreasing in $|x|$. Let $u \in H^{\sigma/2}(\mathbb{R})$. Then we obtain $|u| \in H^{\sigma/2}(\mathbb{R})$ and the following inequality holds: $$\begin{aligned} \| D_{x}^{\sigma/2}|u| \|_{L^{2}} \leq \| D_{x}^{\sigma/2}u \|_{L^{2}} . \label{eq:A_101}\end{aligned}$$ Therefore, we can define the symmetric decreasing rearrangement $|u|^{\ast}$ of $|u|$, and we obtain $$\begin{aligned} \| D_{x}^{\sigma/2}|u|^{\ast} \|_{L^{2}} \leq \| D_{x}^{\sigma/2}|u| \|_{L^{2}} . \label{eq:A_102}\end{aligned}$$ We can prove these properties similarly to [@Angulo Lemma 8.15]. In the following, we simply denote $u^{\ast}:=|u|^{\ast}$ for $u\in H^{\sigma/2}(\mathbb{R})$. ## Case $j=1,2$ {#subsection:Nehari} In this subsection, we prove the following proposition. ****Proposition** 7**. *For $j=1,2$, there exists $\phi\in\mathcal{G}_{j}$ which is nonpositive, even, and decreasing in $|x|$. [\[prop:existense_of_gs_12\]]{#prop:existense_of_gs_12 label="prop:existense_of_gs_12"}* To simplify the discussion of Proposition [\[prop:existense_of_gs_12\]](#prop:existense_of_gs_12){reference-type="ref" reference="prop:existense_of_gs_12"}, we focus on the equation $$\begin{aligned} D_{x}^{\sigma}\phi + c\phi + |\phi|^{p-1}\phi - |\phi|^{q-1}\phi = 0, \quad x\in\mathbb{R}. \tag{SP1} \label{eq:SP1}\end{aligned}$$ We can prove the existence for (SP2) in the same way. In this case, the action functional $S_{1}$ is written as $$\begin{aligned} S_{1}(u) = \frac{1}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} + \frac{1}{p+1} \| u \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u \|_{L^{q+1}} ^{q+1}. \label{eq:def_action_SP1}\end{aligned}$$ Then using this functional, we define the Nehari functional $K_{1}$ as $$\begin{aligned} K_{1}(u) &:= \langle S_{1}'(u) , u \rangle \\ &= \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + c \| u \|_{L^{2}} ^{2} + \| u \|_{L^{p+1}} ^{p+1} - \| u \|_{L^{q+1}} ^{q+1}, \quad u\in H^{\sigma/2}(\mathbb{R}), \label{eq:def_nehari_SP1}\end{aligned}$$ and set $$\begin{aligned} \mathcal{K}_{1} := \{ v\in H^{\sigma/2}(\mathbb{R}) \setminus \{ 0 \} : K_{1}(v)=0 \} .\end{aligned}$$ By the definition of $K_{1}$, the inclusion $\mathcal{N}_{1}\subset\mathcal{K}_{1}$ obviously holds. First, show that $\mathcal{G}_{1}$ is not empty. To prove this, we introduce the following value and set: $$\begin{aligned} d_{1}:= \inf_{v\in\mathcal{K}_{1}} S_{1}(v), \quad \mathcal{M}_{1} := \{ v\in\mathcal{K}_{1} : S_{1}(v)=d_{1} \} .\end{aligned}$$ ****Lemma** 8**. *If $\mathcal{M}_{1}$ is not empty, then $\mathcal{M}_{1}=\mathcal{G}_{1}$ holds. [\[lem:relation_G\_and_M\]]{#lem:relation_G_and_M label="lem:relation_G_and_M"}* *Proof.* First, we show $\mathcal{M}_{1}\subset\mathcal{G}_{1}$. Let $\phi \in \mathcal{M}_{1}$. Then we can see that $K_{1}(\phi) = 0$. Here we let $\lambda > 0$ and consider the function $$\begin{aligned} \lambda \mapsto K_{1}(\lambda \phi) = \lambda^{2} \left( \| D_{x}^{\sigma/2}\phi \|_{L^{2}} ^{2} + c \| \phi \|_{L^{2}} ^{2} \right) + \lambda^{p+1} \| \phi \|_{L^{p+1}} ^{p+1} - \lambda^{q+1} \| \phi \|_{L^{q+1}} ^{q+1}. \label{function:K1} \end{aligned}$$ Considering the graph of [\[function:K1\]](#function:K1){reference-type="eqref" reference="function:K1"}, we obtain $$\begin{aligned} \left. \partial_{\lambda} K_{1}(\lambda\phi) \right|_{\lambda = 1} = \langle K_{1}'(\phi) , \phi \rangle < 0, \label{eq:A_2} \end{aligned}$$ which implies $K_{1}'(\phi) \neq 0$. Then there exists a Lagrange multiplier $\mu \in \mathbb{R}$ such that $S_{1}'(\phi) = \mu K_{1}'(\phi)$. From [\[eq:A_2\]](#eq:A_2){reference-type="eqref" reference="eq:A_2"} and $0 = K_{1}(\phi) = \langle S_{1}'(\phi) , \phi \rangle = \mu \langle K_{1}'(\phi) , \phi \rangle$, we see that $\mu = 0$, which implies $\phi\in\mathcal{N}_{1}$. Furthermore, we have $K_{1}(v) = 0$ for any $v\in\mathcal{N}_{1}$ and, by the definition of $\mathcal{M}_{1}$, we obtain $S_{1}(\phi) = d_{1} \leq S_{1}(v)$, which means $\phi\in\mathcal{G}_{1}$. Next, we show $\mathcal{G}_{1}\subset\mathcal{M}_{1}$. By the assumption of this lemma, we can take some $v\in\mathcal{M}_{1}$, and we have shown $\mathcal{M}_{1}\subset\mathcal{G}_{1}$ above. Then we have $S_{1}(\phi) \leq S_{1}(v) = d_{1}$ for any $\phi\in\mathcal{G}_{1}$. On the other hand, from $\phi\in\mathcal{K}_{1}$ and the definition of $d_{1}$, we can see that $d_{1} \leq S_{1}(\phi)$. This implies $\mathcal{G}_{1}\subset\mathcal{M}_{1}$. ◻ ****Remark** 9**. It is difficult to verify the same claim as Lemma [\[lem:relation_G\_and_M\]](#lem:relation_G_and_M){reference-type="ref" reference="lem:relation_G_and_M"} for the equation $$\begin{aligned} D_{x}^{\sigma}\phi + c \phi - |\phi|^{p-1}\phi + |\phi|^{q-1}\phi = 0 \tag{SP3} \label{eq:SP3} \end{aligned}$$ because we hardly obtain the condition independent of $\phi\in\mathcal{M}_{3}$ that gives the same shape graph of $K_{3}(\lambda \phi)$ as $K_{1}(\lambda\phi)$ (The definitions of $K_{3}$ and $\mathcal{M}_{3}$ are similar to those of $K_{1}$ and $\mathcal{M}_{1}$, respectively.) Therefore, we do not apply the same method to the existence of a ground state of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}. [\[rem:difficulty\]]{#rem:difficulty label="rem:difficulty"} Thanks to Lemma [\[lem:relation_G\_and_M\]](#lem:relation_G_and_M){reference-type="ref" reference="lem:relation_G_and_M"}, we shall show that $\mathcal{M}_{1}$ is not empty to obtain the existence of a ground state of [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"}. However, we claim a stronger statement. ****Lemma** 10**. *Let $(u_{n})_{n}$ be a minimizing sequence of $d_{1}$, that is, $(u_{n})_{n}\subset H^{\sigma/2}(\mathbb{R})$ satisfies $S_{1}(u_{n})\rightarrow d_{1}$ and $K_{1}(u_{n})\rightarrow 0$. Then there exists a sequence $(z_{n})_{n}\subset\mathbb{R}$ such that, by taking a subsequence, we find $v\in\mathcal{M}_{1}$ which satisfies $u_{n}(\cdot + z_{n}) \rightarrow v$ in $H^{\sigma/2}(\mathbb{R})$. [\[prop:existence_of_M1\]]{#prop:existence_of_M1 label="prop:existence_of_M1"}* To prove Proposition [\[prop:existence_of_M1\]](#prop:existence_of_M1){reference-type="ref" reference="prop:existence_of_M1"}, we need the following two lemmas. ****Lemma** 11** (Lieb[@Lieb]). *Let $(u_{n})_{n}$ be a bounded sequence in $H^{\sigma/2}(\mathbb{R})$. Moreover, we assume that there exists $r\in(2,\infty)$ such that $\inf_{n\in\mathbb{N}} \| u_{n} \|_{L^{r}} > 0$. Then there exists $(z_{n})_{n}\subset\mathbb{R}$ such that, by taking a subsequence, we find $v\in H^{\sigma/2}(\mathbb{R})\setminus \{ 0 \}$ which satisfies $u_{n}(\cdot + z_{n}) \rightharpoonup v$ weakly in $H^{\sigma/2}(\mathbb{R})$. [\[lem:Lieb\]]{#lem:Lieb label="lem:Lieb"}* Originally, Lieb[@Lieb] showed the same statement for a sequence in $H^{1}(\mathbb{R})$. For the proof, see Appendix [6](#appendix:proof_of_Lieb){reference-type="ref" reference="appendix:proof_of_Lieb"}. ****Lemma** 12** (Brezis--Lieb[@Brezis-Lieb]). *Let $r\in(1,\infty)$ and $(u_{n})_{n}$ be a bounded sequence in $L^{r}(\mathbb{R})$. Moreover, we assume that there exists a measurable function $u\colon \mathbb{R}\rightarrow \mathbb{R}$ such that $u_{n} \rightarrow u$ a.e. in $\mathbb{R}$. Then $u\in L^{r}(\mathbb{R})$ and we have $$\begin{aligned} \lim_{n\rightarrow\infty} \left( \int_{\mathbb{R}}|u_{n}|^{r} \, dx - \int_{\mathbb{R}}|u_{n}-u|^{r} \, dx \right) = \int_{\mathbb{R}} |u|^{r} \, dx. \end{aligned}$$ [\[lem:Brezis-Lieb\]]{#lem:Brezis-Lieb label="lem:Brezis-Lieb"}* Before proceeding with the proof of Proposition [\[prop:existence_of_M1\]](#prop:existence_of_M1){reference-type="ref" reference="prop:existence_of_M1"}, we define the functional $I_{1}\colon H^{\sigma/2}(\mathbb{R})\rightarrow \mathbb{R}$ as $$\begin{aligned} I_{1}(u) := \left( \frac{1}{2} - \frac{1}{q+1} \right) \| u \|_{H^{\sigma/2}_{c}} ^{2} + \left( \frac{1}{p+1} - \frac{1}{q+1} \right) \| u \|_{L^{p+1}} ^{p+1}, \label{eq:A_3}\end{aligned}$$ where $\| u \|_{H^{\sigma/2}_{c}} ^{2}:= \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + c \| u \|_{L^{2}} ^{2}$ for $u\in H^{\sigma/2}(\mathbb{R})$ and $c$ is the positive constant included in [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. Using [\[eq:A_3\]](#eq:A_3){reference-type="eqref" reference="eq:A_3"}, we can write $$\begin{aligned} S_{1}(u) = \frac{1}{q+1} K_{1}(u) + I_{1}(u), \label{eq:A_4}\end{aligned}$$ which implies that $$\begin{aligned} d_{1} = \inf_{v\in\mathcal{K}_{1}}S_{1}(v) = \inf_{v\in\mathcal{K}_{1}}I_{1}(v). \label{eq:A_5}\end{aligned}$$ Here we obtain the following two lemmas. ****Lemma** 13**. *Let $v\in H^{\sigma/2}(\mathbb{R})$ satisfy $K_{1}(v)<0$. Then $I_{1}(v) > d_{1}$ holds. [\[lem:negativeness_of_I1\]]{#lem:negativeness_of_I1 label="lem:negativeness_of_I1"}* *Proof.* Considering the shape of the graph of $(0,\infty) \ni \lambda \mapsto K_{1}(\lambda v)$, there exists $\lambda_{0}\in(0,1)$ such that $K_{1}(\lambda_{0} v)=0$, which means $\lambda_{0} v \in \mathcal{K}_{1}$. Finally, by the definition of $d_{1}$, we obtain $d_{1} \leq I_{1}(\lambda_{0}v) < I_{1}(v)$. ◻ ****Lemma** 14**. *$d_{1}>0$. [\[lem:positiveness_of_d1\]]{#lem:positiveness_of_d1 label="lem:positiveness_of_d1"}* *Proof.* By [\[eq:A_5\]](#eq:A_5){reference-type="eqref" reference="eq:A_5"}, we shall show that there exists $C_{0}>0$ such that $I_{1}(v) \geq C$ holds for any $v\in\mathcal{K}_{1}$. Let $v\in\mathcal{K}_{1}$. By the Sobolev embedding, we have $$\begin{aligned} 0 = K_{1}(v) &= \| v \|_{H^{\sigma/2}_{c}} ^{2} + \| v \|_{L^{p+1}} ^{p+1} - \| v \|_{L^{q+1}} ^{q+1} \\ &\geq \| v \|_{H^{\sigma/2}_{c}} ^{2}\left( 1 - C_{1} \| v \|_{H^{\sigma/2}_{c}} ^{p-1} - C_{2} \| v \|_{H^{\sigma/2}_{c}} ^{q-1} \right), \end{aligned}$$ which implies $1 \leq C_{1} \| v \|_{H^{\sigma/2}_{c}} ^{p-1}$ or $1 \leq C_{2} \| v \|_{H^{\sigma/2}_{c}} ^{q-1}$. Here we put $$\begin{aligned} C_{0} := \left( \frac{1}{2} - \frac{1}{q+1} \right) \min \left\{ C_{1}^{-\frac{2}{p-1}}, \ C_{2}^{-\frac{2}{q-1}} \right\}, \end{aligned}$$ and then we obtain $$\begin{aligned} I_{1}(v) \geq \left( \frac{1}{2} - \frac{1}{q+1} \right) \| v \|_{H^{\sigma/2}} ^{2} \geq C_{0}, \end{aligned}$$ which is the desired inequality. ◻ Now, we prove Proposition [\[prop:existence_of_M1\]](#prop:existence_of_M1){reference-type="ref" reference="prop:existence_of_M1"}. *Proof of Lemma [\[prop:existence_of_M1\]](#prop:existence_of_M1){reference-type="ref" reference="prop:existence_of_M1"}.* First, we remark that, by using the Nehari functional $K_{1}$, we can write $S_{1}$ as $$\begin{aligned} S_{1}(u) = \frac{1}{2} K_{1}(u) - \left( \frac{1}{2} - \frac{1}{p+1} \right) \| u \|_{L^{p+1}} ^{p+1} + \left( \frac{1}{2} - \frac{1}{q+1} \right) \| u \|_{L^{q+1}} ^{q+1}. \label{eq:A_6} \end{aligned}$$ Let $(u_{n})_{n}$ be a sequence in $H^{\sigma/2}(\mathbb{R})$ which satisfies $S_{1}(u_{n}) \rightarrow d_{1}$ and $K_{1}(u_{n}) \rightarrow 0$. From [\[eq:A_4\]](#eq:A_4){reference-type="eqref" reference="eq:A_4"}, [\[eq:A_6\]](#eq:A_6){reference-type="eqref" reference="eq:A_6"} and the assumption, we obtain $$\begin{aligned} & I_{1}(u_{n}) \rightarrow d_{1}, \label{eq:A_7} \\ & -\left( \frac{1}{2} - \frac{1}{p+1} \right) \| u_{n} \|_{L^{p+1}} ^{p+1} + \left( \frac{1}{2} - \frac{1}{q+1} \right) \| u_{n} \|_{L^{q+1}} ^{q+1} \rightarrow d_{1}. \label{eq:A_8} \end{aligned}$$ Then [\[eq:A_7\]](#eq:A_7){reference-type="eqref" reference="eq:A_7"} implies that $(u_{n})_{n}$ is bounded in $H^{\sigma/2}(\mathbb{R})$. Furthermore, by [\[eq:A_8\]](#eq:A_8){reference-type="eqref" reference="eq:A_8"}, we can see that $\inf_{n\in\mathbb{N}} \| u_{n} \|_{L^{p+1}} >0$. Indeed, if $\inf_{n\in\mathbb{N}} \| u_{n} \|_{L^{p+1}} =0$, it contradicts to the positivity of $d_{1}$. Therefore, we can apply Lemma [\[lem:Lieb\]](#lem:Lieb){reference-type="ref" reference="lem:Lieb"} to $(u_{n})_{n}$ and then there exists $(z_{n})_{n}\subset \mathbb{R}$ and $v\in H^{\sigma/2}(\mathbb{R})\setminus \{0\}$ which satisfies $u_{n}(\cdot + z_{n}) \rightharpoonup v$ weakly in $H^{\sigma/2}(\mathbb{R})$ by taking a subsequence. Here we denote $v_{n}:=u_{n}(\cdot + z_{n})$. By the Rellich compact embedding $H^{\sigma/2}(\mathbb{R}) \hookrightarrow L^{r}_{\textrm{loc}}(\mathbb{R})$ for $r\in[2,2^{\ast}_{\sigma})$, we may assume that $v_{n} \rightarrow v$ a.e. in $\mathbb{R}$. Then by using Lemma [\[lem:Brezis-Lieb\]](#lem:Brezis-Lieb){reference-type="ref" reference="lem:Brezis-Lieb"}, we have the following convergences: $$\begin{aligned} & I_{1}(v_{n}) - I_{1}(v_{n}-v) \rightarrow I_{1}(v), \label{eq:A_9} \\ & K_{1}(v_{n}) - K_{1}(v_{n}-v) \rightarrow K_{1}(v). \label{eq:A_10} \end{aligned}$$ From [\[eq:A_9\]](#eq:A_9){reference-type="eqref" reference="eq:A_9"}, we can see that $$\begin{aligned} \lim_{n\rightarrow\infty} I_{1}(v_{n}-v) &= \lim_{n\rightarrow\infty} I_{1}(v_{n}) - I_{1}(v) \\ &< \lim_{n\rightarrow\infty} I_{c}(v_{n}) = \lim_{n\rightarrow\infty} I_{c}(u_{n}) = d_{1}, \label{eq:A_11} \end{aligned}$$ which implies that $I_{1}(v_{n}-v) \leq d_{1}$ holds for large $n\in\mathbb{N}$ and then we have $K_{1}(v_{n}-v) \geq 0$ by Lemma [\[lem:negativeness_of_I1\]](#lem:negativeness_of_I1){reference-type="ref" reference="lem:negativeness_of_I1"}. Moreover, from [\[eq:A_10\]](#eq:A_10){reference-type="eqref" reference="eq:A_10"}, we obtain $$\begin{aligned} K_{1}(v) = \lim_{n\rightarrow\infty}K_{1}(v_{n}) - \lim_{n\rightarrow\infty}K_{1}(v_{n}-v) \leq 0. \label{eq:A_12} \end{aligned}$$ Therefore, applying Lemma [\[lem:negativeness_of_I1\]](#lem:negativeness_of_I1){reference-type="ref" reference="lem:negativeness_of_I1"} again, we have $$\begin{aligned} d_{1} \leq I_{1}(v) \leq \liminf_{n\rightarrow\infty}I_{1}(v_{n}) = d_{1}, \end{aligned}$$ that is, $$\begin{aligned} I_{1}(v) = d_{1}. \label{eq:A_13} \end{aligned}$$ Using Lemma [\[lem:negativeness_of_I1\]](#lem:negativeness_of_I1){reference-type="ref" reference="lem:negativeness_of_I1"} again, we see that $K_{1}(v) \geq 0$. Combining this inequality and [\[eq:A_12\]](#eq:A_12){reference-type="eqref" reference="eq:A_12"} gives $$\begin{aligned} K_{1}(v)=0. \label{eq:A_15} \end{aligned}$$ Therefore, $v\in\mathcal{M}_{1}$ follows from [\[eq:A_5\]](#eq:A_5){reference-type="eqref" reference="eq:A_5"}, [\[eq:A_13\]](#eq:A_13){reference-type="eqref" reference="eq:A_13"} and [\[eq:A_15\]](#eq:A_15){reference-type="eqref" reference="eq:A_15"}. Finally, we can obtain the strong convergence by [\[eq:A_9\]](#eq:A_9){reference-type="eqref" reference="eq:A_9"}. This completes the proof. ◻ ****Lemma** 15**. *Let $\phi\in\mathcal{G}_{1}$. Then we obtain $\phi^{\ast}\in\mathcal{G}_{1}$. [\[prop:even_ground_state_of_SP1_2\]]{#prop:even_ground_state_of_SP1_2 label="prop:even_ground_state_of_SP1_2"}* *Proof.* Thanks to Lemma [\[lem:relation_G\_and_M\]](#lem:relation_G_and_M){reference-type="ref" reference="lem:relation_G_and_M"}, it suffices to show $\phi^{\ast}\in\mathcal{M}_{1}$. From $\phi\in\mathcal{G}_{1} = \mathcal{M}_{1}$, we have $S_{1}(\phi) = d_{1}, \ K_{1}(\phi) = 0$. Using the inequalities [\[eq:A_101\]](#eq:A_101){reference-type="eqref" reference="eq:A_101"} and [\[eq:A_102\]](#eq:A_102){reference-type="eqref" reference="eq:A_102"} yields that $$\begin{aligned} & S_{1}(\phi^{\ast}) \leq S_{1}(\phi) = d_{1}, \label{eq:A_21} \\ & K_{1}(\phi^{\ast}) \leq 0, \label{eq:A_22} \\ & I_{1}(\phi^{\ast}) \leq I_{1}(\phi) = d_{1}. \label{eq:A_23} \end{aligned}$$ Moreover, we can see that $K_{1}(\phi^{\ast}) \geq 0$ from [\[eq:A_23\]](#eq:A_23){reference-type="eqref" reference="eq:A_23"} and Lemma [\[lem:negativeness_of_I1\]](#lem:negativeness_of_I1){reference-type="ref" reference="lem:negativeness_of_I1"}. Then combining this and [\[eq:A_22\]](#eq:A_22){reference-type="eqref" reference="eq:A_22"}, we obtain $K_{1}(\phi^{\ast}) = 0$. Therefore, $\phi^{\ast}\in\mathcal{M}_{1}$ follows from this identity and [\[eq:A_21\]](#eq:A_21){reference-type="eqref" reference="eq:A_21"}. This concludes the proof. ◻ Finally, Proposition [\[prop:existense_of_gs_12\]](#prop:existense_of_gs_12){reference-type="ref" reference="prop:existense_of_gs_12"} follows from Lemmas [\[lem:relation_G\_and_M\]](#lem:relation_G_and_M){reference-type="ref" reference="lem:relation_G_and_M"}, [\[prop:existence_of_M1\]](#prop:existence_of_M1){reference-type="ref" reference="prop:existence_of_M1"}, and [\[prop:even_ground_state_of_SP1_2\]](#prop:even_ground_state_of_SP1_2){reference-type="ref" reference="prop:even_ground_state_of_SP1_2"}. ****Remark** 16**. 1. This method is also valid for the case that $\sigma = 2$. 2. We can also characterize the ground state of [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with $f(s)=|s|^{p-1}s \ (1 < p < 2^{\ast}_{\sigma}-1)$ with the same method as Proposition [\[prop:existense_of_gs_12\]](#prop:existense_of_gs_12){reference-type="ref" reference="prop:existense_of_gs_12"}. 3. Here we define $f_{4}:=-|s|^{p-1}s -|s|^{q-1}s$ and consider (SP4). The Nehari functional $K_{4}$ corresponding to (SP4) is given as [\[rem:SP4\]]{#rem:SP4 label="rem:SP4"} $$\begin{aligned} K_{4}(u) = \langle S_{4}'(u) , u \rangle = \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + c \| u \|_{L^{2}} ^{2} + \| u \|_{L^{p+1}} ^{p+1} + \| u \|_{L^{q+1}} ^{q+1} \geq 0. \end{aligned}$$ Here we let $v\in H^{\sigma/2}(\mathbb{R})$ be a soluion to (SP4). Then we obtain $v=0$ by the nonnegativity of the value of $K_{4}$. [\[rem:end_of_1\_2\]]{#rem:end_of_1_2 label="rem:end_of_1_2"} ## Case $j=3$ {#subsection:Pohizaev} In this subsection, we prove the following proposition. ****Proposition** 17**. *Let $0 < \sigma \leq 1$. If $c\in(0,c_{0})$, then there exists $\phi\in\mathcal{G}_{3}$ which is nonpositive, even, and decreasing in $|x|$. [\[prop:existence_of_ground_state_of_SP3\]]{#prop:existence_of_ground_state_of_SP3 label="prop:existence_of_ground_state_of_SP3"}* As mentioned in Remark [\[rem:difficulty\]](#rem:difficulty){reference-type="ref" reference="rem:difficulty"}, it is difficult to find any solutions to [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"} with the same method as in Proposition [\[prop:existense_of_gs_12\]](#prop:existense_of_gs_12){reference-type="ref" reference="prop:existense_of_gs_12"}. So, we prove the existence of ground states of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"} with the Pohozaev identity, which is the method introduced by Berestycki--Lions [@Berestycki-Lions-1] or Berestycki--Gallouët--Kavian [@BGK]. ****Lemma** 18** (The Pohozaev identity). *The following identity holds for any solution $u\in H^{\sigma/2}(\mathbb{R})$ to [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}: $$\begin{aligned} \frac{1-\sigma}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} - \frac{1}{p+1} \| u \|_{L^{p+1}} ^{p+1} + \frac{1}{q+1} \| u \|_{L^{q+1}} ^{q+1} = 0. \end{aligned}$$ [\[lem:Pohozaev\]]{#lem:Pohozaev label="lem:Pohozaev"}* *Proof.* See [@Chang-Wang Proposition 4.1] and the references therein. ◻ By Lemma [\[lem:Pohozaev\]](#lem:Pohozaev){reference-type="ref" reference="lem:Pohozaev"}, we obtain $$\begin{aligned} S_{3}(u) = \frac{1}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} - \frac{1}{p+1} \| u \|_{L^{p+1}} ^{p+1} + \frac{1}{q+1} \| u \|_{L^{q+1}} ^{q+1} = \frac{\sigma}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2}\end{aligned}$$ for $u\in\mathcal{N}_{3}$. Here we define some functionals to use for the proof. Let $c>0$. Then we define $$\begin{aligned} P_{c}(u) := -\frac{c}{2} \| u \|_{L^{2}} ^{2} + \frac{1}{p+1} \| u \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u \|_{L^{q+1}} ^{q+1} \end{aligned}$$ for $u\in H^{\sigma/2}(\mathbb{R})$, and $$\begin{aligned} \mathcal{P}_{c} &:= \left\{ \begin{aligned} & \{ v\in H^{\sigma/2}(\mathbb{R})\setminus \{0\} : P_{c}(v)=0 \} , & & \text{for} \ \sigma = 1, \label{eq:B_3} \\ & \{ v\in H^{\sigma/2}(\mathbb{R})\setminus \{0\} : P_{c}(v)=1 \} , & & \text{for} \ 0 < \sigma < 1. \end{aligned} \right.\end{aligned}$$ To find a solution to [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}, we consider the following minimizing problem: $$\begin{aligned} j_{3} := \inf_{v\in\mathcal{P}_{c}} J_{3}(v), \label{eq:minimizing_problem}\end{aligned}$$ where $$\begin{aligned} J_{3}(u):= \frac{\sigma}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2}\end{aligned}$$ for $u\in H^{\sigma/2}(\mathbb{R})$. ****Lemma** 19**. *Let $c\in(0,c_{0})$. Then $\mathcal{P}_{c} \neq \emptyset$ holds. [\[lem:nonempty_of_constraint\]]{#lem:nonempty_of_constraint label="lem:nonempty_of_constraint"}* *Proof.* First, we show that there exists $v\in H^{1}(\mathbb{R})$ such that $P_{c}(v)<0$. We put $$\begin{aligned} G_{c}(s) := \frac{c}{2}s^{2} - \frac{1}{p+1}|s|^{p+1} + \frac{1}{q+1}|s|^{q+1}, \ s\in\mathbb{R}. \end{aligned}$$ If $c\in(0,c_{0})$, we can see that there exists $s_{0}>0$ such that $G_{c}(s_{0})<0$. Under this condition, we can find $v_{0}\in H^{1}(\mathbb{R})$ satisfying $P_{c}(v_{0}) < 0$ (see the proof of Theorem 2 of [@Berestycki-Lions-1].) For case $\sigma=1$, considering the graph of the function $(0,\infty) \ni \lambda \mapsto I_{c}(\lambda v_{0})$ yields that there exists $\lambda_{0} > 0$ such that $P_{c}(\lambda_{0} v_{0}) = 0$, which implies that $\lambda_{0} v_{0}$ is a desired function to find. If $0 < \sigma < 1$, there exist $v_{0}\in H^{1}(\mathbb{R})$ and $\alpha>0$ such that $P_{c}(v_{0}) = \alpha$, from the discussion in the first half. Here we set $v_{0}^{\alpha}(x) := v_{0}(\alpha x)$ for $x\in\mathbb{R}$ so that $P_{c}(v_{0}^{\alpha}) = 1$. Therefore, $v_{0}^{\alpha}$ is a desired function. ◻ Now, we state the existence of a minimizer of [\[eq:minimizing_problem\]](#eq:minimizing_problem){reference-type="eqref" reference="eq:minimizing_problem"}. ****Proposition** 20**. *Let $0 < \sigma \leq 1$. If $c\in(0,c_{0})$, then there exists a minimizer of [\[eq:minimizing_problem\]](#eq:minimizing_problem){reference-type="eqref" reference="eq:minimizing_problem"} which is nonnegative, even, and decreasing in $|x|$. [\[prop:existense_of_minimizer_of_j\]]{#prop:existense_of_minimizer_of_j label="prop:existense_of_minimizer_of_j"}* Here we introduce two lemmas related to the compactness used in the proof. ****Lemma** 21**. *Let $r\in[1,\infty)$ and $u\in L^{r}(\mathbb{R})$ be nonnegative, even, and decreasing in $|x|$. Then $$\begin{aligned} u(x) \leq 2^{-1/r} |x|^{-1/r} \| u \|_{L^{r}} \end{aligned}$$ holds for almost all $x\in \mathbb{R}$. [\[lem:Radial\]]{#lem:Radial label="lem:Radial"}* *Proof.* Without loss of generality, we may assume that $x>0$. A direct calculation yields $$\begin{aligned} \| u \|_{L^{r}} ^{r} = 2 \int_{0}^{\infty} u(y)^{r} \ dy \geq 2\int_{0}^{x} u(y)^{r} \ dy \geq 2u(x)^{r} x, \end{aligned}$$ which implies the desired inequality. ◻ ****Lemma** 22** (Strauss[@Strauss]). *Assume that $P, \ Q\colon \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions satisfying $$\begin{aligned} \frac{P(s)}{Q(s)} \rightarrow 0 \ \text{\rm as} \ |s| \rightarrow \infty \ \text{\rm and} \ |s| \rightarrow 0. \end{aligned}$$ Moreover, let $(u_{n})_{n}$ be a sequence of measurable functions in $\mathbb{R}$ which satisfies $$\begin{aligned} \sup_{n\in\mathbb{N}}\int_{\mathbb{R}}|Q(u_{n})| \, dx < \infty \end{aligned}$$ and assume that there exists a measurable function $v\colon \mathbb{R}\rightarrow \mathbb{R}$ such that $P(u_{n}(x))\rightarrow v(x)$ a.e. in $\mathbb{R}$, and $u_{n}(x)\rightarrow 0$ as $|x|\rightarrow 0$ uniformly respect to $n\in\mathbb{N}$. Then we obtain $P(u_{n})\rightarrow v$ in $L^{1}(\mathbb{R})$. [\[lem:Strauss\]]{#lem:Strauss label="lem:Strauss"}* Here we prove Proposition [\[prop:existense_of_minimizer_of_j\]](#prop:existense_of_minimizer_of_j){reference-type="ref" reference="prop:existense_of_minimizer_of_j"}. *Proof of Proposition [\[prop:existense_of_minimizer_of_j\]](#prop:existense_of_minimizer_of_j){reference-type="ref" reference="prop:existense_of_minimizer_of_j"}.* We split the proof into two cases with respect to $\sigma$. [Case 1. $\sigma=1$]{.ul}: Let $(u_{n})_{n}\subset H^{1/2}(\mathbb{R})$ be a minimizing sequence of $j_{3}$ so that we have $S_{3}(u_{n}) \rightarrow j_{3}$ and $P_{c}(u_{n})=0$ for all $n\in\mathbb{N}$. By a certain scaling, we may assume that $\| u_{n} \|_{L^{2}} = 1$ for all $n\in\mathbb{N}$. Therefore, $(u_{n})_{n}$ is bounded in $H^{1/2}(\mathbb{R})$. Here we take the symmetric decreasing rearrangement of $(u_{n})_{n}$, denoting $(u^{\ast}_{n})_{n}$. Then $(u^{\ast}_{n})_{n}$ is also bounded in $H^{1/2}(\mathbb{R})$ and we have $J_{3}(u^{\ast}_{n}) \leq J_{3}(u_{n}), \ P_{c}(u^{\ast}_{n}) = 0, \ \| u^{\ast}_{n} \|_{L^{2}} = 1$ for all $n\in\mathbb{N}$. Taking a subsequence of $(u^{\ast}_{n})_n$ and the Rellich compact embedding yield that there exists $v_{0}\in H^{1/2}(\mathbb{R})$ such that $u^{\ast}_{n} \rightharpoonup v_{0}$ weakly in $H^{1/2}(\mathbb{R})$ and $u^{\ast} \rightarrow v_{0}$ a.e. in $\mathbb{R}$. The second convergence means that we can take the weak limit $v_{0}$ as even, nonnegative, and decreasing in $|x|$. In the following, we show that $v_{0}$ attains [\[eq:minimizing_problem\]](#eq:minimizing_problem){reference-type="eqref" reference="eq:minimizing_problem"}. First, we obtain $$\begin{aligned} J_{3}(v_{0}) \leq \liminf_{n\rightarrow\infty} J_{3}(u^{\ast}_{n}) \leq \liminf_{n\rightarrow\infty} J_{3}(u_{n}) = j_{3}, \label{eq:B_4} \end{aligned}$$ by the lower semicontinuity of the norms. Next, by applying Lemma [\[lem:Radial\]](#lem:Radial){reference-type="ref" reference="lem:Radial"} to $(u^{\ast}_{n})_{n}$, there exists $C > 0$ such that $|u^{\ast}_{n}(x)| \leq C |x|^{-1/(q+1)}$ for all $n\in\mathbb{N}$, which means that $u^{\ast}_{n} \rightarrow 0$ as $|x|\rightarrow\infty$ uniformly respect to $n\in\mathbb{N}$. Here we let $r \in (1,\infty)$ and put $P(s):=|s|^{r+1}, \ Q(s):=s^{2} + |s|^{r+2}$ for $s\in\mathbb{R}$ and $w_{n}:=u^{\ast}_{n}-v_{0}$ for $n\in\mathbb{N}$. Then we can see that the functions $P$ and $Q$ and the sequence $(w_{n})_{n}$ satisfy the assumptions of the Lemma [\[lem:Strauss\]](#lem:Strauss){reference-type="ref" reference="lem:Strauss"}. Therefore, we obtain $\| w_{n} \|_{L^{r+1}} ^{r+1} \rightarrow 0$. Furthermore, by $P_{c}(u^{\ast}_{n})=0$ and $\| u^{\ast}_{n} \|_{L^{2}} =1$, we have $$\begin{aligned} \frac{c}{2} = P_{c}(u^{\ast}_{n}) + \frac{c}{2} \| u^{\ast}_{n} \|_{L^{2}} ^{2} = \frac{1}{p+1} \| u^{\ast}_{n} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u^{\ast}_{n} \|_{L^{q+1}} ^{q+1}. \label{eq:B_5} \end{aligned}$$ Combining [\[eq:B_5\]](#eq:B_5){reference-type="eqref" reference="eq:B_5"} with the convergence yields $$\begin{aligned} \frac{c}{2} = \frac{1}{p+1} \| v_{0} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| v_{0} \|_{L^{q+1}} ^{q+1} = P_{c}(v_{0}) + \frac{c}{2} \| v_{0} \|_{L^{2}} ^{2}, \end{aligned}$$ which implies $v_{0}\neq 0$ and this concludes $j_{3}>0$. Moreover, by the Fatou lemma, we have $P_{c}(v_{0}) \geq 0$. Here we see that $P_{c}(v_{0})=0$ by contradiction. Assume $P_{c}(v_{0}) > 0$ and we consider the function $$\begin{aligned} (0,\infty) \ni \lambda \mapsto P_{c}(\lambda v_{0}) = -\frac{c}{2}\lambda^{2} \| v_{0} \|_{L^{2}} ^{2} + \frac{\lambda^{p+1}}{p+1} \| v_{0} \|_{L^{p+1}} ^{p+1} - \frac{\lambda^{q+1}}{q+1} \| v_{0} \|_{L^{q+1}} ^{q+1}. \end{aligned}$$ The shape of the graph gives $\lambda_{0}\in(0,1)$ such that $P_{c}(\lambda_{0} v_{0}) = 0$. However, $J_{3}(\lambda_{0} v_{0}) = \lambda_{0}^{2}J_{3}(v_{0}) \leq \lambda_{0} j_{3} < j_{3}$, which contradicts to the definition of $j_{3}$. Hence, we have $P_{c}(v_{0})=0$. Thus, we finally obtain $J_{3}(v_{0})=j_{3}$, which implies that $v_{0}$ is a desired minimizer. [Case 2. $0 < \sigma < 1$]{.ul}: Let $(u_{n})_{n}\subset H^{\sigma/2}(\mathbb{R})$ be a minimizing sequence of $j_{3}$, that is, $S_{3}(u_{n})\rightarrow j_{3}$, $P_{c}(u_{n})=0$ for all $n\in\mathbb{N}$. First, we show that $(u_{n})_{n}$ is bounded in $H^{\sigma/2}(\mathbb{R})$. We can see that there exists $C_{0}>0$ such that $$\begin{aligned} s^{p} - s^{q} \leq \frac{c}{2}s + C_{0}s^{2^{\ast}_{\sigma}-1} \end{aligned}$$ for $s>0$. By integration, we have $$\begin{aligned} \frac{1}{p+1}s^{p+1} - \frac{1}{q+1}s^{q+1} \leq \frac{c}{4}s^{2} + Cs^{2^{\ast}_{\sigma}} \label{eq:B_9} \end{aligned}$$ for $s>0$. From $P_{c}(u_{n})=0$, we have $$\begin{aligned} 1 + \frac{c}{2} \| u_{n} \|_{L^{2}} ^{2} = \frac{1}{p+1} \| u_{n} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u_{n} \|_{L^{q+1}} ^{q+1}. \label{eq:B_10} \end{aligned}$$ Applying [\[eq:B_9\]](#eq:B_9){reference-type="eqref" reference="eq:B_9"} and the Sobolev inequality, we obtain $$\begin{aligned} \frac{1}{p+1} \| u_{n} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u_{n} \|_{L^{q+1}} ^{q+1} &\leq \frac{c}{4} \| u_{n} \|_{L^{2}} ^{2} + C \| D_{x}^{\sigma/2}u_{n} \|_{L^{2}} ^{2} \\ &\leq \frac{c}{4} \| u_{n} \|_{L^{2}} ^{2} + C. \label{eq:B_11} \end{aligned}$$ Combining [\[eq:B_10\]](#eq:B_10){reference-type="eqref" reference="eq:B_10"} and [\[eq:B_11\]](#eq:B_11){reference-type="eqref" reference="eq:B_11"} yields $$\begin{aligned} \frac{c}{4} \| u_{n} \|_{L^{2}} ^{2} \leq C, \end{aligned}$$ which implies that $(u_{n})_{n}$ is bounded in $H^{\sigma/2}(\mathbb{R})$. Therefore, the symmetric decreasing rearrangement $(u^{\ast}_{n})_{n}$ of $(u_{n})_{n}$ is also bounded in $H^{\sigma/2}(\mathbb{R})$. By taking a subsequence, we obtain $v_{0}\in H^{\sigma/2}(\mathbb{R})$ such that $u^{\ast}_{n} \rightharpoonup v_{0}$ weakly in $H^{\sigma/2}(\mathbb{R})$ and $u^{\ast}_{n} \rightarrow v_{0}$ a.e. in $\mathbb{R}$. Same as Case 1, $v_{0}$ can be taken as even, nonnegative, and decreasing in $|x|$. Next, we prove that $v_{0}$ is a desired minimizer. By the Fatou lemma and Lemmas [\[lem:Radial\]](#lem:Radial){reference-type="ref" reference="lem:Radial"}, [\[lem:Strauss\]](#lem:Strauss){reference-type="ref" reference="lem:Strauss"}, we obtain $$\begin{aligned} \frac{c}{2} \| v_{0} \|_{L^{2}} ^{2} &\leq \liminf_{n\rightarrow\infty} \left( \frac{c}{2} \| u^{\ast}_{n} \|_{L^{2}} ^{2} \right) \\ &= \liminf_{n\rightarrow\infty} \left( \frac{1}{p+1} \| u^{\ast}_{n} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| u^{\ast}_{n} \|_{L^{q+1}} ^{q+1} -1 \right) \\ &= \frac{1}{p+1} \| v_{0} \|_{L^{p+1}} ^{p+1} - \frac{1}{q+1} \| v_{0} \|_{L^{q+1}} ^{q+1} -1, \end{aligned}$$ which implies $P_{c}(v_{0}) \geq 1$. Furthermore, we have $J_{3}(v_{0}) \leq j_{3}$ by the lower semicontinuity of the norms. Therefore, we can see $P_{c}(v_{0}) = 1$. Indeed, assume that $P_{c}(v_{0}) > 1$. Then there exists $\alpha\in(0,1)$ such that $P_{c}(v_{0}^{\alpha}) = 1$, where $v_{0}^{\alpha}(x) := v_{0}(x/\alpha)$. Therefore, we have $J_{3}(v_{0}^{\alpha}) = \alpha^{1-\sigma} J_{3}(v_{0}) < j_{3}$, which contradicts to the definition of $j_{3}$. Hence, $v_{0}$ is a desired minimizer. ◻ Now, we show Proposition [\[prop:existence_of_ground_state_of_SP3\]](#prop:existence_of_ground_state_of_SP3){reference-type="ref" reference="prop:existence_of_ground_state_of_SP3"}. *Proof of Propositon [\[prop:existence_of_ground_state_of_SP3\]](#prop:existence_of_ground_state_of_SP3){reference-type="ref" reference="prop:existence_of_ground_state_of_SP3"}.* Let $v_{0}\in H^{\sigma/2}(\mathbb{R})$ be a minimizer of [\[eq:minimizing_problem\]](#eq:minimizing_problem){reference-type="eqref" reference="eq:minimizing_problem"} which is nonnegative, even, and decreasing in $|x|$. Then there exists a Lagrange multiplier $\mu\in\mathbb{R}$ such that $J_{3}'(v_{0}) = \mu P_{c}'(v_{0})$, that is $$\begin{aligned} D_{x}^{\sigma}(v_{0}) = \mu (-cv_{0} + |v_{0}|^{p-1}v_{0} - |v_{0}|^{q-1}v_{0}), \label{eq:B_14} \end{aligned}$$ Here we show $\mu > 0$. Multiplying $v_{0}$ to both hand sides of [\[eq:B_14\]](#eq:B_14){reference-type="eqref" reference="eq:B_14"} yields $$\begin{aligned} 0 < \| D_{x}^{\sigma/2}v_{0} \|_{L^{2}} ^{2} = \mu \langle P_{c}'(v_{0}) , v_{0} \rangle , \label{eq:B_16} \end{aligned}$$ which implies that $\mu \neq 0$ and $$\begin{aligned} \langle P_{c}'(v_{0}) , v_{0} \rangle \neq 0. \label{eq:B_15} \end{aligned}$$ Now, we consider the graph of the function $(0,\infty) \ni \lambda \mapsto P_{c}(\lambda v_{0})$. Thanks to [\[eq:B_15\]](#eq:B_15){reference-type="eqref" reference="eq:B_15"}, we can find two points $0 < \lambda_{1} < \lambda_{2} < \infty$ which satisfy $P_{c}(\lambda_{j} v_{0})= 0$ for $j=1, \ 2$ and either of whom equals to $1$. Then we can see that $\lambda_{1}=1$. Indeed, if we assume that $\lambda_{2}=1$, the same contradiction occurs as in the proof of Case 1 of Proposition [\[prop:existense_of_minimizer_of_j\]](#prop:existense_of_minimizer_of_j){reference-type="ref" reference="prop:existense_of_minimizer_of_j"}. Therefore, we obtain $\langle P_{c}'(v_{0}) , v_{0} \rangle > 0$, which implies $\mu > 0$ by [\[eq:B_16\]](#eq:B_16){reference-type="eqref" reference="eq:B_16"}. Then we set $\phi(x) := v_{0}(\mu^{1/\sigma}x)$ for $x\in\mathbb{R}$ so that $\phi$ is a solution to [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}. In case $\sigma = 1$, we can see that $\| D_{x}^{1/2}\phi \|_{L^{2}} ^{2} = \| D_{x}^{1/2}v_{0} \|_{L^{2}} ^{2}$ and $P_{c}(\phi)=P_{c}(v_{0})=0$, which means $S_{3}(\phi) = S_{3}(v_{0}) = J_{3}(v_{0}) = j_{3}$ and then $\phi$ is a ground state of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}. If $0 < \sigma <1$, we use the Pohozaev identity to find a ground state. First, since $\phi\in\mathcal{N}_{3}$, we obtain $$\begin{aligned} \frac{1-\sigma}{2} \| D_{x}^{\sigma/2}\phi \|_{L^{2}} ^{2} = P_{c}(\phi). \label{eq:B_51} \end{aligned}$$ by Lemma [\[lem:Pohozaev\]](#lem:Pohozaev){reference-type="ref" reference="lem:Pohozaev"}. Moreover, we have $$\begin{aligned} & \frac{1-\sigma}{2} \| D_{x}^{\sigma/2}\phi \|_{L^{2}} ^{2} = \frac{1-\sigma}{2} \mu^{(\sigma - 1) / \sigma} \| D_{x}^{\sigma/2}v_{0} \|_{L^{2}} ^{2} = \frac{1-\sigma}{\sigma}\mu^{(\sigma-1) / \sigma}j_{3}, \label{eq:B_52} \\ & P_{c}(\phi) = \mu^{-1/\sigma} P_{c}(v_{0}) = \mu^{-1/\sigma}. \label{eq:B_53} \end{aligned}$$ Combining [\[eq:B_51\]](#eq:B_51){reference-type="eqref" reference="eq:B_51"}, [\[eq:B_52\]](#eq:B_52){reference-type="eqref" reference="eq:B_52"}, and [\[eq:B_53\]](#eq:B_53){reference-type="eqref" reference="eq:B_53"} yields $$\begin{aligned} \mu = \left( \frac{1-\sigma}{\sigma} \right)^{-1} j_{3}^{-1}. \label{eq:B_54} \end{aligned}$$ Using [\[eq:B_54\]](#eq:B_54){reference-type="eqref" reference="eq:B_54"}, we obtain $$\begin{aligned} S_{3}(\phi) = J_{3}(\phi) = \mu^{(\sigma-1) / \sigma} j_{3} = \left( \frac{1-\sigma}{\sigma} \right)^{(1-\sigma) / \sigma} j_{3} ^{1/\sigma}. \label{eq:B_55} \end{aligned}$$ Next, let $u\in\mathcal{N}_{3}$. By the Pohozaev identity, we can see that $$\begin{aligned} P_{c}(u) = \frac{1-\sigma}{2} \| D_{x}^{\sigma/2}u \|_{L^{2}} ^{2} = \frac{1-\sigma}{\sigma} J_{3}(u) \end{aligned}$$ and $P_{c}(u^{\alpha}) = 1$, where $\alpha := P_{c}(u)^{-1} > 0$ and $u^{\alpha}(x) := u(x/\alpha)$ for $x\in\mathbb{R}$, which means $u^{\alpha}\in\mathcal{P}_{c}$. Moreover, we have $$\begin{aligned} J_{3}(u^{\alpha}) = \alpha^{1-\sigma} J_{3}(u) = P_{c}(u)^{\sigma-1} J_{3}(u) = \left( \frac{1-\sigma}{\sigma} \right)^{\sigma -1} J_{3}(u)^{\sigma}, \end{aligned}$$ which implies $$\begin{aligned} S_{3}(u) = J_{3}(u) = \left( \frac{1-\sigma}{\sigma} \right)^{(1-\sigma) / \sigma} J_{3}(u^{\alpha})^{1/\sigma}. \label{eq:B_56} \end{aligned}$$ Finally, considering [\[eq:B_55\]](#eq:B_55){reference-type="eqref" reference="eq:B_55"}, [\[eq:B_56\]](#eq:B_56){reference-type="eqref" reference="eq:B_56"}, and the definition of $j_{3}$ yield $S_{3}(\phi) \leq S_{3}(u)$ for all $u\in\mathcal{N}_{3}$. This means that $\phi$ is a ground state of [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"}. ◻ # Properties of solutions {#section:properties} In this section, we consider the regularity and positivity of solutions to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. First, we introduce a function which plays an important role in this section. For $0<\sigma<2$ and $\nu>0$, we define the function $N_{\nu}^{\sigma}\colon \mathbb{R}\rightarrow \mathbb{R}$ as $$\begin{aligned} N_{\nu}^{\sigma}(x) := \frac{1}{2\pi} \int_{\mathbb{R}} \frac{1}{|\xi|^{\sigma} + \nu} e^{i\xi x} \, d\xi.\end{aligned}$$ Properties of $N_{\nu}^{\sigma}$ have been investigated in [@Frank-Lenzmann Appendix A]. Here we recall them. ****Lemma** 23** (Frank--Lenzmann[@Frank-Lenzmann]). *The following properties hold:* 1. *$N_{\nu}^{\sigma} \in L^{r}(\mathbb{R})$ for $r \in \left( 1, 1/(1-\sigma)_{+} \right)$.* 2. *$N_{\nu}^{\sigma}$ is positive, even, and strictly decreasing in $|x|$.* *[\[lem:properties_of_integral_kernel\]]{#lem:properties_of_integral_kernel label="lem:properties_of_integral_kernel"}* ## Regularity of solutions Here we consider the regularity results of solutions to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. ****Proposition** 24**. *Let $\phi\in H^{\sigma/2}(\mathbb{R})$ be a solution to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. Then we obtain $\phi\in H^{\sigma +1}(\mathbb{R})$. [\[prop:regularity_of_solution\]]{#prop:regularity_of_solution label="prop:regularity_of_solution"}* The same statement for [\[eq:SP0\]](#eq:SP0){reference-type="eqref" reference="eq:SP0"} with single power nonlinearities is proven by Frank--Lenzmann[@Frank-Lenzmann Appendix B]. We can show Proposition [\[prop:regularity_of_solution\]](#prop:regularity_of_solution){reference-type="ref" reference="prop:regularity_of_solution"} with some modifications. *Proof.* Before proceeding, we remark that taking the Fourier transform to both sides of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}, we have $$\begin{aligned} \hat{\phi}(\xi) = \frac{C_{0}}{|\xi|^{\sigma} + c} \mathscr{F}[f_{j}(\phi)](\xi) \label{eq:AppB_02} \end{aligned}$$ with some constant $C_{0}>0$. First, we show $\phi\in H^{1}(\mathbb{R})$. The proof of this statement depends on the value of $\sigma$. [Case 1. $1 < \sigma <2$:]{.ul} First, we easily see that $\phi\in L^{\infty}(\mathbb{R})$ by the Sobolev embedding. Then using [\[eq:AppB_02\]](#eq:AppB_02){reference-type="eqref" reference="eq:AppB_02"}, we have $$\begin{aligned} \| D_{x}^{\sigma}\phi \|_{L^{2}} = \| |\xi|^{\sigma}\hat{\phi} \|_{L^{2}} &= C_{0} \left\| \frac{|\xi|^{\sigma}}{|\xi|^{\sigma}+c} \mathscr{F}[f_{j}(\phi)] \right\|_{L^{2}} \\ &\leq C_{0} \left( \| \mathscr{F}[ |\phi|^{p-1}\phi ] \|_{L^{2}} + \| \mathscr{F}[|\phi|^{q-1}\phi] \|_{L^{2}} \right) \\ &\leq C_{0} \left( \| \phi \|_{L^{\infty}} ^{(p-1)/2} + \| \phi \|_{L^{\infty}} ^{(q-1)/2} \right) \| \phi \|_{H^{\sigma/2}} , \end{aligned}$$ which implies $\phi\in H^{\sigma}(\mathbb{R}) \hookrightarrow H^{1}(\mathbb{R})$. [Case 2. $\sigma =1$:]{.ul} By [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} and the Sobolev embedding $H^{1/2}(\mathbb{R}) \hookrightarrow L^{r}(\mathbb{R})$ for $r\in[2,\infty)$, we have $$\begin{aligned} \| |\xi|\phi \|_{L^{2}} = \| D_{x}\phi \|_{L^{2}} &= \| -c \phi + f_{j}(\phi) \|_{L^{2}} \leq c \| \phi \|_{L^{2}} + \| \phi \|_{L^{2p}} ^{p} + \| \phi \|_{L^{2q}} ^{q} \\ &\leq C (1 + \| \phi \|_{H^{1/2}} ^{p-1} + \| \phi \|_{H^{1/2}} ^{q-1}) \| \phi \|_{H^{1/2}} , \end{aligned}$$ where $C>0$ is a constant dependent on $c, \ p$, and $q$. This concludes $\phi\in H^{1}(\mathbb{R})$. [Case 3. $0 < \sigma < 1$:]{.ul} First, we claim that there exists some $r_{1}, r_{2}\in (1,1/(1-\sigma))$ which satisfy $$\begin{aligned} \frac{1}{r_{1}} + \frac{p}{2^{\ast}_{\sigma}} = \frac{1}{r_{1}} + \frac{p(1-\sigma)}{2} = 1, \label{eq:AppB_03} \\ \frac{1}{r_{2}} + \frac{q}{2^{\ast}_{\sigma}} = \frac{1}{r_{2}} + \frac{q(1-\sigma)}{2} = 1. \label{eq:AppB_04} \end{aligned}$$ Indeed, since $1 < p < q < 2^{\ast}_{\sigma}-1$, we can see that $$\begin{aligned} 1 < \frac{2}{p(1-\sigma)}, \ \frac{2}{q(1-\sigma)} < \frac{2}{1-\sigma} = 2^{\ast}_{\sigma}. \end{aligned}$$ Now, we put $$\begin{aligned} r_{1} := \left( 1 - \frac{p(1-\sigma)}{2} \right)^{-1} = \frac{2}{2-p(1-\sigma)}. \end{aligned}$$ Then by $0 < \sigma < 1$ and $p < 2^{\ast}_{\sigma} -1$, we can see that $r_{1} \in (1,1/(1-\sigma))$. We can take some $r_{2}$ in the same way as taking $r_{1}$. By the way, we can write a solution $\phi$ of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} as $\phi = C_{0} N^{\sigma}_{c} \ast f_{j}(\phi)$ with some constant $C_{0}>0$. Therefore, combining this and [\[eq:AppB_03\]](#eq:AppB_03){reference-type="eqref" reference="eq:AppB_03"}, [\[eq:AppB_04\]](#eq:AppB_04){reference-type="eqref" reference="eq:AppB_04"}, we obtain $$\begin{aligned} \| \phi \|_{L^{\infty}} = C_{0} \| N_{c}^{\sigma} \ast f_{j}(\phi) \|_{L^{\infty}} &\leq C_{0} \left( \| N_{c}^{\sigma} \ast |\phi|^{p-1}|\phi| \|_{L^{\infty}} + \| N_{c}^{\sigma} \ast |\phi|^{q-1}|\phi| \|_{L^{\infty}} \right) \\ &\leq C_{0} \left( \| N_{c}^{\sigma} \|_{L^{r_{1}}} + \| N_{c}^{\sigma} \|_{L^{r_{2}}} \right) \| \phi \|_{L^{2^{\ast}_{\sigma}}} \\ &\leq C \left( \| N_{c}^{\sigma} \|_{L^{r_{1}}} + \| N_{c}^{\sigma} \|_{L^{r_{2}}} \right) \| \phi \|_{H^{\sigma/2}} , \end{aligned}$$ where $C>0$ depends on $\sigma$. Hence, $\phi\in L^{\infty}(\mathbb{R})$ follows from this inequality and Lemma [\[lem:properties_of_integral_kernel\]](#lem:properties_of_integral_kernel){reference-type="ref" reference="lem:properties_of_integral_kernel"}. Next, we show $\phi\in H^{1}(\mathbb{R})$. Using [\[eq:AppB_02\]](#eq:AppB_02){reference-type="eqref" reference="eq:AppB_02"}, we obtain $$\begin{aligned} \| D_{x}^{(k+1)\sigma/2}\phi \|_{L^{2}} &= C_{0} \left\| \frac{|\xi|^{(k+1)\sigma/2}}{|\xi|^{\sigma}+c} \mathscr{F}[f_{j}(\phi)] \right\|_{L^{2}} \\ &\leq C \left( \left\| |\xi|^{k\sigma/2} \mathscr{F}[|\phi|^{p-1}\phi] \right\|_{L^{2}} + \left\| |\xi|^{k\sigma/2} \mathscr{F}[|\phi|^{q-1}\phi] \right\|_{L^{2}} \right) \\ &\leq C \left( \| \phi \|_{L^{\infty}} ^{(p-1)/2} + \| \phi \|_{L^{\infty}} ^{(q-1)/2} \right) \| |\xi|^{k\sigma/2}\hat{\phi} \|_{L^{2}} \\ &\leq C \| D_{x}^{k\sigma/2}\phi \|_{L^{2}} \label{eq:AppB_05} \end{aligned}$$ for all $k\in\mathbb{N}$, where $C>0$ depends on $\sigma, \ p$, and $q$. Here, we take some $n\in\mathbb{N}$ such that $1/(n+1)\leq \sigma/2 < 1/n$. Then using [\[eq:AppB_05\]](#eq:AppB_05){reference-type="eqref" reference="eq:AppB_05"} infinitely many times, we have $\| D_{x}^{(n+1)\sigma/2} \|_{L^{2}} \leq C \| D_{x}^{\sigma/2}\phi \|_{L^{2}} \leq C \| \phi \|_{H^{\sigma/2}}$, which implies $\phi\in H^{(n+1)\sigma/2}(\mathbb{R}) \hookrightarrow H^{1}(\mathbb{R})$. Finally, we show that $\phi\in H^{\sigma + 1}(\mathbb{R})$. Using [\[eq:AppB_02\]](#eq:AppB_02){reference-type="eqref" reference="eq:AppB_02"}, we have $$\begin{aligned} \| D_{x}^{\sigma+1}\phi \|_{L^{2}} = \| |\xi|^{\sigma+1}\phi \|_{L^{2}} &\leq \| |\xi|\mathscr{F}[|\phi|^{p-1}\phi] \|_{L^{2}} + \| |\xi|\mathscr{F}[|\phi|^{q-1}\phi] \|_{L^{2}} \\ &= \| \partial_{x}(|\phi|^{p-1}\phi) \|_{L^{2}} + \| \partial_{x}(|\phi|^{p-1}\phi) \|_{L^{2}} \\ &\leq \left( p \| \phi \|_{L^{\infty}} ^{(p-1)/2} + q \| \phi \|_{L^{\infty}} ^{(q-1)/2} \right) \| \phi \|_{H^{1}} , \end{aligned}$$ which means $\phi \in H^{\sigma+1}(\mathbb{R})$. This completes the proof. ◻ ## Positivity of solutions In this subsection, we consider the positivity of solutions to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. ****Proposition** 25**. *Let $\phi\in H^{\sigma/2}(\mathbb{R})$ be a nonnegative solution to [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"}. Then $\phi$ is strictly positive. [\[prop:positiveness_of_GS\]]{#prop:positiveness_of_GS label="prop:positiveness_of_GS"}* *Proof.* For simplicity, we only consider [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"}. We can prove this for the remaining cases in almost the same way. Let $\phi\in H^{\sigma/2}(\mathbb{R})$ be a nonnegative solution of [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"}, and we can see that $\phi\in H^{\sigma + 1}(\mathbb{R})$ from Proposition [\[prop:regularity_of_solution\]](#prop:regularity_of_solution){reference-type="ref" reference="prop:regularity_of_solution"}, which implies $\phi\in L^{\infty}(\mathbb{R})$ and $\phi$ is continuous in $\mathbb{R}$. Therefore, we put $$\begin{aligned} \lambda_{1} := -\min_{x\in\mathbb{R}} (|\phi(x)|^{p-1}-|\phi(x)|^{q-1}) + 1, \end{aligned}$$ so that we obtain $\lambda_{1} + |\phi(y)|^{p-1} - |\phi(y)|^{q-1} > 0$ for all $y\in\mathbb{R}$. Then by adding $\lambda_{1}\phi$ to both sides of [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"} and transforming, we obtain $$\begin{aligned} \phi(x) = C \int_{\mathbb{R}} N_{c+\lambda_{1}}^{\sigma}(x-y)(\lambda_{1} + |\phi(y)|^{p-1} - |\phi(y)|^{q-1})\phi(y) \ dy. \label{eq:C_02} \end{aligned}$$ Then we can see that $\phi>0$ in $\mathbb{R}$. Indeed, if there exists $x_{0}\in\mathbb{R}$ such that $\phi(x_{0})=0$, then, by [\[eq:C_02\]](#eq:C_02){reference-type="eqref" reference="eq:C_02"}, we have $$\begin{aligned} \int_{\mathbb{R}} N_{c+\lambda_{1}}^{\sigma}(x_{0}-y)(\lambda_{1} + |\phi(y)|^{p-1} - |\phi(y)|^{q-1})\phi(y) \ dy = 0. \label{eq:C_03} \end{aligned}$$ Since $N_{c+\lambda_{1}}^{\sigma}(x_{0}-y)(\lambda_{1} + |\phi(y)|^{p-1} - |\phi(y)|^{q-1}) > 0$ for all $y\in\mathbb{R}$, we obtain $\phi\equiv 0$ in $\mathbb{R}$ from [\[eq:C_03\]](#eq:C_03){reference-type="eqref" reference="eq:C_03"}, which contradicts to that $\phi$ is nontrivial. This concludes the proof. ◻ By Proposition [\[prop:positiveness_of_GS\]](#prop:positiveness_of_GS){reference-type="ref" reference="prop:positiveness_of_GS"}, we can obtain more properties of ground states. ****Proposition** 26**. *Set $j=1, \ 2$. Then there do not exist any ground states of [\[eq:SPj\]](#eq:SPj){reference-type="eqref" reference="eq:SPj"} with sign changes. [\[prop:no_sign_changes\]]{#prop:no_sign_changes label="prop:no_sign_changes"}* *Proof.* For simplicity, we only consider the case $j=1$. We show this corollary with contradiction. Assume that there exists $\phi\in\mathcal{G}_{1}$ with sign changes. Then we can see that $|\phi|\in\mathcal{G}_{1}$ similarly to Proposition [\[prop:even_ground_state_of_SP1_2\]](#prop:even_ground_state_of_SP1_2){reference-type="ref" reference="prop:even_ground_state_of_SP1_2"} by using [\[eq:A_101\]](#eq:A_101){reference-type="eqref" reference="eq:A_101"}. Therefore, by Proposition [\[prop:positiveness_of_GS\]](#prop:positiveness_of_GS){reference-type="ref" reference="prop:positiveness_of_GS"}, $|\phi|$ is strictly positive. However, by the assumption of $\phi$, there exists $x_{0}\in\mathbb{R}$ such that $|\phi(x_{0})|=0$. This is contradiction. ◻ Here we conclude the proof of Theorems [\[Thm:SP1_2\]](#Thm:SP1_2){reference-type="ref" reference="Thm:SP1_2"} and [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"}. Theorem [\[Thm:SP1_2\]](#Thm:SP1_2){reference-type="ref" reference="Thm:SP1_2"} follows from Propositions [\[prop:existense_of_gs_12\]](#prop:existense_of_gs_12){reference-type="ref" reference="prop:existense_of_gs_12"}, [\[prop:regularity_of_solution\]](#prop:regularity_of_solution){reference-type="ref" reference="prop:regularity_of_solution"}, [\[prop:positiveness_of_GS\]](#prop:positiveness_of_GS){reference-type="ref" reference="prop:positiveness_of_GS"}, and [\[prop:no_sign_changes\]](#prop:no_sign_changes){reference-type="ref" reference="prop:no_sign_changes"}. Theorem [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"} follows from Propositions [\[prop:existence_of_ground_state_of_SP3\]](#prop:existence_of_ground_state_of_SP3){reference-type="ref" reference="prop:existence_of_ground_state_of_SP3"}, [\[prop:regularity_of_solution\]](#prop:regularity_of_solution){reference-type="ref" reference="prop:regularity_of_solution"}, and [\[prop:positiveness_of_GS\]](#prop:positiveness_of_GS){reference-type="ref" reference="prop:positiveness_of_GS"}. # Application and specific phenomenon {#section:application} In this section, we consider Theorem [\[thm:classification\]](#thm:classification){reference-type="ref" reference="thm:classification"}. Here we recall the equation [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}: $$\begin{aligned} D_{x}\phi + c\phi + \phi^{p} - \phi^{q} = 0, \quad x\in\mathbb{R}, \tag{SP} \label{eq:SP}\end{aligned}$$ where $c>0, \ p, \ q \in \mathbb{N}, \ 2 \leq p < q$. The action functional corresponding to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"} is $$\begin{aligned} S(u) = \frac{1}{2} \| D_{x}^{1/2}u \|_{L^{2}} ^{2} + \frac{c}{2} \| u \|_{L^{2}} ^{2} + \frac{1}{p+1}\int_{\mathbb{R}} u^{p+1} \, dx - \frac{1}{q+1} \int_{\mathbb{R}} u^{q+1} \, dx.\end{aligned}$$ The existence of ground states is obtained by the method in §[3.1](#subsection:Nehari){reference-type="ref" reference="subsection:Nehari"} with certain corrections. The remaining statements of the non/existence come from the main results of this paper. For example, in Case ([\[case:even_even\]](#case:even_even){reference-type="ref" reference="case:even_even"}), a positive ground state of [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"} becomes also a positive solution to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}. Furthermore, if the assumptions of Theorem [\[Thm:SP3\]](#Thm:SP3){reference-type="ref" reference="Thm:SP3"} are satisfied, we can find a negative solution to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"} as below; let $v\in H^{1/2}(\mathbb{R})$ be a positive solution to [\[eq:SP3\]](#eq:SP3){reference-type="eqref" reference="eq:SP3"} and put $u := -v$. Then $u$ becomes a negative solution to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}. In the following, we prove the statement in Case ([\[case:even_odd\]](#case:even_odd){reference-type="ref" reference="case:even_odd"}) that none of the positive solutions is a ground state. *Proof.* First, we recall that $\phi_{1}\in\mathcal{G}_{1}$ attains $d_{1} = \inf_{v\in\mathcal{K}_{1}}S_{1}(v)$. Moreover, we define notations corresponding to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"} which are similar to those in §[3.1](#subsection:Nehari){reference-type="ref" reference="subsection:Nehari"}: $$\begin{aligned} &\mathcal{N} := \{ v\in H^{1/2}(\mathbb{R}) \setminus \{ 0 \} : S'(v)=0 \} , \\ &K(u) := \langle S'(u) , u \rangle = \| D_{x}^{1/2}u \|_{L^{2}} ^{2} + c \| u \|_{L^{2}} ^{2} + \int_{\mathbb{R}} u^{p+1} \, dx - \int_{\mathbb{R}} u^{q+1} \, dx, \\ &\mathcal{K} := \{ v\in H^{1/2}(\mathbb{R})\setminus \{ 0 \} : K(v) = 0 \} , \quad d := \inf_{v\in\mathcal{K}}S(v), \\ &\mathcal{M} := \{ v\in H^{1/2}(\mathbb{R}) : S(v) = d \} . \end{aligned}$$ Then as in Lemma [\[lem:relation_G\_and_M\]](#lem:relation_G_and_M){reference-type="ref" reference="lem:relation_G_and_M"}, we can see that $\mathcal{M}$ coincides with the set of ground states of [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}, which we denote $\mathcal{G}$ in the following. Next, we put $$\begin{aligned} \mathcal{A} := \{ v\in\mathcal{N} : v>0 \ \text{in} \ \mathbb{R} \} , \quad a := \inf_{v\in\mathcal{A}}S(v). \end{aligned}$$ To prove the statement, it suffices to see that $d < a$. First, we show that $d < d_{1}$. Here, we let $\phi\in\mathcal{G}$ and take some $\phi_{1}\in\mathcal{G}_{1}$ which is positive. Moreover, we put $\psi_{1}:=-\phi_{1}$ so that $\psi_{1}$ becomes a negative solution to [\[eq:SP\]](#eq:SP){reference-type="eqref" reference="eq:SP"}. Therefore, by the definitions of $d$ and $d_{1}$, we obtain $$\begin{aligned} d = S(\phi) \leq S(\psi_{1}) < S_{1}(\psi_{1}) = S_{1}(\phi_{1}) = d_{1}. \end{aligned}$$ Next, we prove that $d_{1} \leq a$ holds. Here we take some $\phi_{1}\in\mathcal{G}_{1}$. In addition, for each $v\in\mathcal{A}$, we put $w:=-v$ so that $w$ becomes a solution to [\[eq:SP1\]](#eq:SP1){reference-type="eqref" reference="eq:SP1"}. Therefore, $$\begin{aligned} d_{1} = S_{1}(\phi_{1}) \leq S_{1}(w) = S_{1}(v) = S(v). \end{aligned}$$ holds for all $v\in\mathcal{A}$, which implies $d_{1} \leq a$. This completes the proof. ◻ # Proof of Lemma [\[lem:Lieb\]](#lem:Lieb){reference-type="ref" reference="lem:Lieb"} {#appendix:proof_of_Lieb} Before starting the proof, we introduce the Sobolev space $\tilde{H}^{\sigma/2}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}$. $\tilde{H}^{\sigma/2}(\Omega)$ is defined as $$\begin{aligned} \tilde{H}^{\sigma/2}(\Omega) := \left\{ v\in L^{2}(\Omega) : \frac{|v(x)-v(y)|}{|x-y|^{\frac{1}{2} + \frac{\sigma}{2}}}\in L^{2}(\Omega) \right\}\end{aligned}$$ with a norm $$\begin{aligned} \| u \|_{\tilde{H}^{\sigma/2}(\Omega)} ^{2} := \| u \|_{L^{2}(\Omega)} ^{2} + \iint_{\Omega\times\Omega} \frac{|u(x)-u(y)|^{2}}{|x-y|^{1+\sigma}} \, dxdy.\end{aligned}$$ Then we can see that $\tilde{H}^{\sigma/2}(\mathbb{R}) = H^{\sigma/2}(\mathbb{R})$ and $\| \cdot \|_{\tilde{H}^{\sigma/2}(\mathbb{R})} \sim \| \cdot \|_{H^{\sigma/2}}$. For details, see [@HitchhikersOfFractionalSobolev]. *Proof of Lemma [\[lem:Lieb\]](#lem:Lieb){reference-type="ref" reference="lem:Lieb"}.* First, we put $I_{z}:=(z,z+1)$ for $z\in\mathbb{Z}$. Then we see that $$\begin{aligned} \| u \|_{H^{\sigma/2}} ^{2} \geq \sum_{z\in\mathbb{Z}} \| u \|_{\tilde{H}^{\sigma/2}(I_{z})} ^{2} \label{eq:A_16} \end{aligned}$$ holds for any $u\in H^{\sigma/2}(\mathbb{R})$. In addition, we set $$\begin{aligned} M:=\sup_{n\in\mathbb{N}} \| u_{n} \|_{H^{\sigma/2}} , \quad m:= \inf_{n\in\mathbb{N}} \| u_{n} \|_{L^{r}(\mathbb{R})} , \quad C_{0}:=\frac{M^{2}+1}{m^{r}}. \end{aligned}$$ Then we claim the following statement [\[eq:A_17\]](#eq:A_17){reference-type="eqref" reference="eq:A_17"}: $$\begin{aligned} \text{For all} \ n\in\mathbb{N}, \ \text{there exists} \ z_{n}\in\mathbb{Z} \ \text{which satisfies} \ \| u_{n} \|_{\tilde{H}^{\sigma/2}(I_{z})} ^{2} < C_{0} \| u_{n} \|_{L^{r}(I_{z})} ^{r}. \label{eq:A_17} \end{aligned}$$ Indeed, if [\[eq:A_17\]](#eq:A_17){reference-type="eqref" reference="eq:A_17"} does not hold, there exists $n_{0}\in\mathbb{N}$ which satisfies $\| u_{n_{0}} \|_{\tilde{H}^{\sigma/2}(I_{z})} ^{2} \geq C_{0} \| u_{n_{0}} \|_{L^{r}(I_{z})} ^{r}$ for any $z\in\mathbb{Z}$. Then using [\[eq:A_16\]](#eq:A_16){reference-type="eqref" reference="eq:A_16"} yields $$\begin{aligned} M^{2} \geq \| u_{n_{0}} \|_{H^{\sigma/2}} ^{2} &\geq \sum_{z\in\mathbb{Z}} \| u_{n_{0}} \|_{\tilde{H}^{\sigma/2}(I_{z})} ^{2} \\ &\geq C_{0} \sum_{z\in\mathbb{Z}} \| u_{n_{0}} \|_{L^{r}(I_{z})} ^{r} = C_{0} \| u_{n_{0}} \|_{L^{r}(\mathbb{R})} ^{r} \geq C_{0}m^{r} = M^{2} + 1, \end{aligned}$$ which is impossible to occur. Then from [\[eq:A_17\]](#eq:A_17){reference-type="eqref" reference="eq:A_17"}, we can choose a sequence $(z_{n})_{n}\subset\mathbb{R}$. Here we set $v_{n}:=u_{n}(\cdot + z_{n})$ and then $(v_{n})_{n}$ is also bounded in $H^{\sigma/2}(\mathbb{R})$. Therefore, by taking a subsequence, we obtain $v\in H^{\sigma/2}(\mathbb{R})$ which satisfies $v_{n} \rightharpoonup v$ weakly in $H^{\sigma/2}(\mathbb{R})$. To complete the proof, we shall see that $v\neq 0$. By the Sobolev embedding and [\[eq:A_17\]](#eq:A_17){reference-type="eqref" reference="eq:A_17"}, we have $$\begin{aligned} \| v_{n} \|_{L^{r}(I_{0})} ^{2} \leq C \| v_{n} \|_{\tilde{H}^{\sigma/2}(I_{0})} ^{2} < C \| v_{n} \|_{L^{r}(I_{0})} ^{r}, \end{aligned}$$ which means $\| v_{n} \|_{L^{r}(I_{0})} ^{r-2} > C$. Furthermore, we obtain $v_{n} \rightarrow v$ in $L^{r}(I_{0})$ by the Rellich compact embedding. Thus, we have $\| v \|_{L^{r}(I_{0})} >0$, which means $v\neq 0$. ◻ # Acknowledgements {#acknowledgements .unnumbered} The author would like to thank Professor Masahito Ohta, Dr. Noriyoshi Fukaya, and Dr. Yoshinori Nishii for their support and helpful discussions for this study. [^1]: Email: `1123701@ed.tus.ac.jp`

arxiv_math

--- abstract: | Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$. The Gallai-Ramsey multiplicity $\operatorname{GM}_k(G:H)$ is defined as the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{\operatorname{gr}_k(G:H)}$. In this paper, we get some exact values of the Gallai-Ramsey multiplicity for rainbow small trees versus general monochromatic graphs under a sufficiently large number of colors. We also discuss the bipartite Gallai-Ramsey multiplicity.\ **Keywords:** Ramsey theory; Gallai-Ramsey number; Gallai-Ramsey multiplicity\ **AMS subject classification 2020:** 05C15, 05C30, 05C55. author: - | Xueliang Li, Yuan Si\ Center for Combinatorics, Nankai University, Tianjin 300071, China\ `lxl@nankai.edu.cn, yuan_si@aliyun.com` title: "**Gallai-Ramsey multiplicity for rainbow small trees[^1]** " --- # Introduction In this paper, the graphs we consider are finite, undirected, simple and without isolated vertices. Let $V(G)$ and $E(G)$ denote the vertex set and edge set of a graph $G$, respectively. An *edge-coloring* of $G$ is a function $c:E(G)\rightarrow \{1,2,\ldots,k\}$, where $\{1,2,\ldots,k\}$ is called the set of colors. We can also use red, blue or other specific names to represent these colors. An edge-colored graph is called *rainbow* if all its edges have distinct colors and *monochromatic* if all its edges have the same color. A $k$-edge-coloring of a graph is *exact* if all the $k$ colors are used at least once. In this paper, we only consider exact edge-colorings of graphs. The *union* $G\cup H$ of two graphs $G$ and $H$ is the graph with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)$. The *degree*, $\deg_G(v)$ or $\deg(v)$ for short, of a vertex $v$ of $G$ is the number of edges incident to $v$ in $G$. We usually say that a vertex with degree $1$ is a *leaf vertex*, and an edge incident to a leaf vertex is called a *pendent edge*. A path with $n$ vertices from $v_1$ to $v_n$ is denoted as $P_n=v_1v_2\ldots v_n$ or $P_n=e_1e_2\ldots e_{n-1}$, which is a vertex-edge alternative sequence $v_1,e_1,v_2,e_2,\ldots,v_{n-1},e_{n-1},v_n$ such that $v_1,v_2,\ldots,v_n$ are distinct vertices, $e_1,e_2,\ldots,e_{n-1}$ are distinct edges and $v_iv_{i+1}= e_i$ for each $i\in\{1,2,\ldots,n-1\}$. $S_3^{+}$ is a the graph consisting of a triangle with one pendent edge and $P_4^{+}$ is a the graph consisting of $P_4$ with one pendent edge incident with an inner vertex of $P_4$. $P_4^{+}$ can also be seen as the graph by adding an extra pendent edge to a leaf vertex of $K_{1,3}$. For convenience, we call the newly added pendent edge at $K_{1,3}$ the *tail edge* of $P_4^{+}$. The *automorphism group* of a graph $G$ is denoted as $\operatorname{Aut}(G)$. For the automorphism groups of some special graphs, we give the following conclusions: $\operatorname{Aut}(K_n)\cong S_n, \operatorname{Aut}(K_{m,n})\cong S_m\times S_n$ for $m\ne n$, and $\operatorname{Aut}(P_{n})\cong S_2$, where $S_n$ is the $n$-order symmetric group. For more notation and terminology not defined here, we refer to [@BondyMurty]. ## Gallai-Ramsey number and multiplicity In 1930, Ramsey problems started to be studied first; see [@Ramsey]. Given two graphs $G$ and $H$, the *Ramsey number* $r(G,H)$ is defined as the minimum positive integer $n$ such that every red/blue-edge-coloring of $K_{n}$ contains either a red subgraph $G$ or a blue subgraph $H$. If $G=H$, then we simply denote $r(G, H)$ as $r(G)$. More generally, the definition of Ramsey number has been extended to multicolor and hypergraph, and there are currently many research results available. Determining the exact value of the Ramsey numbers or improving the known upper or lower bounds of the number has always been a hot research topic in graph theory. For more results on Ramsey numbers, we refer to [@Radziszowski]. In 2010, Faudree, Gould, Jacobson and Magnant in [@FGJM10] provided a definition of a rainbow version of the Ramsey number, called the *Gallai-Ramsey number*. **Definition 1**. *[@FGJM10][\[Def:GR\]]{#Def:GR label="Def:GR"} Given two non-empty graphs $G,H$ and a positive integer $k$, define the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ to be the minimum integer $N$ such that for all $n\ge N$, every $k$-edge-coloring of $K_n$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$.* In the past decade, there has been a wealth of research on the Gallai-Ramsey numbers. In terms of current research progress, six types of rainbow graphs have been studied, which are $K_3, S_3^{+}, K_{1,3}, P_4, P_5$ and $P_4^{+}$. For more results about Gallai-Ramsey numbers, we refer to [@MagnantNowbandegani]. Considering the Gallai-Ramsey number on $k$-edge-colored balanced complete bipartite graph $K_{n,n}$ is another research direction. In 2019, Li, Wang and Liu in [@LWL19] gave the definition of *bipartite Gallai-Ramsey number*. **Definition 2**. *[@LWL19][\[Def:bGR\]]{#Def:bGR label="Def:bGR"} Given two non-empty bipartite graphs $G, H$ and a positive integer $k$, define the bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ to be the minimum integer $N$ such that for all $n\ge N$, every $k$-edge-coloring of $K_{n,n}$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$.* Recently, a counting problem related to the Gallai-Ramsey number has been studied. Li, Broersma and Wang in [@LiBroersmaWangDiscrete] studied the minimum number of subgraphs $H$ in all $k$-edge-colored complete graphs without rainbow triangles (also known as Gallai-coloring). Later, Mao in [@Mao] proposed the definition of *Gallai-Ramsey multiplicity*. **Definition 3**. *[@Mao][\[Def:GM\]]{#Def:GM label="Def:GM"} Given two non-empty graphs $G, H$ and a positive integer $k$, define the Gallai-Ramsey multiplicity $\operatorname{GM}_k(G:H)$ to be the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{\operatorname{gr}_k(G:H)}$.* Based on the definitions of bipartite Gallai-Ramsey number and Gallai-Ramsey multiplicity, we give the definition of *bipartite Gallai Ramsey multiplicity*. **Definition 4**. *Given two non-empty bipartite graphs $G, H$ and a positive integer $k$, define the bipartite Gallai-Ramsey multiplicity $\operatorname{bi-GM}_k(G:H)$ to be the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{\operatorname{bgr}_k(G:H),\operatorname{bgr}_k(G:H)}$.* ## Structural theorems under rainbow-tree-free colorings The five $k$-edge-colored structures of complete graphs or complete bipartite graphs given below is for convenience of describing several structural theorems. **Colored Structure 1:** Let $(V_1,V_2,\ldots,V_k)$ be a partition of $V(K_n)$ such that for each $i$, all the edges connecting two vertices in $V_i$ are colored by either $1$ or $i$ and all the edges between $V_i$ and $V_j$ with $i\ne j$ are colored by $1$. **Colored Structure 2:** Let $K_n$ be a $k$-edge-colored complete graph such that $K_n-v$ is monochromatic for some vertex $v$. **Colored Structure 3:** Let $(U,V)$ be the bipartition of complete bipartite graph $K_{n,n}$. $U$ can be partitioned into $k$ non-empty parts $U_1, U_2,\ldots,U_k$ such that all the edges between $U_i$ and $V$ have color $i$ for $i\in\{1,2,\ldots,k\}$. **Colored Structure 4:** Let $(U,V)$ be the bipartition of complete bipartite graph $K_{n,n}$. $U$ can be partitioned into two parts $U_1$ and $U_2$ with $|U_1|\ge 1, |U_2|\ge 0$, and $V$ can be partitioned into $k$ parts $V_1,V_2,\ldots,V_k$ with $|V_1|\ge 0$ and $|V_j|\ge 1$, $j\in\{2,3,\ldots,k\}$, such that all the edges between $V_i$ and $U_1$ have color $i$ and all the edges between $V_i$ and $U_2$ have color $1$ for $i\in\{1,2,\ldots,k\}$. **Colored Structure 5:** Let $(U,V)$ be the bipartition of complete bipartite graph $K_{n,n}$. $U$ can be partitioned into $k$ parts $U_1,U_2,\ldots,U_k$ with $|U_1|\ge 0, |U_j|\ge 1$ and $V$ can be partitioned into $k$ parts $V_1,V_2,\ldots,V_k$ with $|V_1|\ge 0, |V_j|\ge 1$, $j\in\{2,3,\ldots,k\}$, such that only colors $1$ and $i$ can be used on the edges between $U_i$ and $V_i$ for $i\in\{1,2,\ldots,k\}$, and all the other edges have color $1$. Thomason and Wagner in [@ThomasonWagner2007] obtained the following results. **Theorem 5**. *[@ThomasonWagner2007][\[th-3-path-Structure\]]{#th-3-path-Structure label="th-3-path-Structure"} For an integer $n\geq 4$, let $K_n$ be an edge-colored complete graph with at least three colors so that it contains no rainbow $P_4$ if and only if $n=4$ and three colors are used, each color forming a perfect matching.* **Theorem 6**. *[@ThomasonWagner2007][\[th-4-path-Structure\]]{#th-4-path-Structure label="th-4-path-Structure"} For integers $k\ge 5$ and $n\ge 5$, let $K_n$ be a $k$-edge-colored complete graph so that it contains no rainbow $P_5$ if and only if Colored Structure 1 or Colored Structure 2 occurs.* Bass, Magnant, Ozeki and Pyron in [@BassMagnantOzekiPyron] obtained the following results. The study of this structural theorem can be traced back to the study of local $k$-coloring Ramsey numbers by Gyárfás, Lehel, Schelp and Tuza in [@GLST1987]. **Theorem 7**. *[@BassMagnantOzekiPyron; @GLST1987][\[th-Star-Structure\]]{#th-Star-Structure label="th-Star-Structure"} For integers $k\ge 4$ and $n\ge 4$, let $K_n$ be a $k$-edge-colored complete graph so that it contains no rainbow $K_{1,3}$ if and only if Colored Structure 1 occurs.* Schlage-Puchta and Wagner in [@Schlage-PuchtaWagner] first described the colored structure of a complete graph without rainbow $P_{4}^{+}$ using a local $2$-coloring. Later, Bass, Magnant, Ozeki and Pyron in [@BassMagnantOzekiPyron] once again described this structural theorem. **Theorem 8**. *[@BassMagnantOzekiPyron; @Schlage-PuchtaWagner][\[th-P4+-Structure\]]{#th-P4+-Structure label="th-P4+-Structure"} For integers $k\ge 5$ and $n\ge 5$, let $K_n$ be a $k$-edge-colored complete graph so that it contains no rainbow $P_{4}^{+}$ if and only if Colored Structure 1 occurs.* In terms of edge-colorings of complete bipartite graphs, Li, Wang and Liu in [@LWL19] first obtained the following results. **Theorem 9**. *[@LWL19][\[th-P4-bGR-Structure\]]{#th-P4-bGR-Structure label="th-P4-bGR-Structure"} Let $(U,V)$ be the bipartition of a complete bipartite graph. For integers $k\ge 3$ and $n\ge 2$, let $K_{n,n}$ be a $k$-edge-colored complete bipartite graph so that it contains no rainbow $P_4$ if and only if Colored Structure 3 occurs.* **Theorem 10**. *[@LWL19][\[th-P5-bGR-Structure\]]{#th-P5-bGR-Structure label="th-P5-bGR-Structure"} Let $(U,V)$ be the bipartition of a complete bipartite graph. For integers $k\ge 5$ and $n\ge 3$, let $K_{n,n}$ be a $k$-edge-colored complete bipartite graph so that it contains no rainbow $P_5$ if and only if Colored Structure 4 or Colored Structure 5 occurs.* Recently, Chen, Ji, Mao and Wei in [@CJMW23] provided the colored structure for complete bipartite graphs without rainbow $K_{1,3}$. **Theorem 11**. *[@CJMW23][\[th-K13-bGR-Structure\]]{#th-K13-bGR-Structure label="th-K13-bGR-Structure"} Let $(U,V)$ be the bipartition of a complete bipartite graph. For integers $k\ge 5$ and $n\ge 3$, let $K_{n,n}$ be a $k$-edge-colored complete bipartite graph so that it contains no rainbow $K_{1,3}$ if and only if Colored Structure 5 occurs.* It should be noted that the structural theorems cited above are only a part of what is needed in this paper. For a complete survey of the structural theorems, we refer to the original references. ## Main results Due to the fact that the research in this paper is based on the exact $k$-edge-colorings, it is naturally required that the number of edges in the graph be not less than the number of colors used. From the above structural theorems, it can be seen that if there are no rainbow subgraphs $G\in\{P_4,P_5,K_{1,3},P_4^{+}\}$ for $k$-edge-colored complete graphs or complete bipartite graphs, then there must be some monochromatic graphs under this colored structure. For example, a $k$-edge-colored complete graph without rainbow $K_{1,3}$ must contain a monochromatic $K_{\underbrace{2,2,\ldots,2}_{k-1}}$; a $k$-edge-colored complete bipartite graph without rainbow $P_4$ must contain a monochromatic $K_{1,k}$; a $k$-edge-colored complete bipartite graph without rainbow $P_5$ or $K_{1,3}$ must contain a monochromatic $K_{1,\left\lceil\frac{k-1}{2}\right\rceil}$. Based on these properties, we know that when the color number $k$ is sufficiently large with respect to the subgraph $H$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ (also bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$) does not depend on the subgraph $H$, but only on $k$. In the third section of this paper, we give some exact values of the Gallai-Ramsey number and multiplicity when the number of colors $k=\binom{t}{2}$ is sufficiently large with respect to the subgraph $H$. The main results for the Gallai-Ramsey multiplicity are shown in the following table. Main results for Gallai-Ramsey multiplicity ------------------------------------------------------------------ -------------------------------- -------------------------------- $\operatorname{GM}_{k-1}(G:H)$ $\operatorname{GM}_{k-2}(G:H)$ $G=K_{1,3}$ $18 \ (t=5)$ $(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2} \ (t\ge 6)$ $14\ (t=5)$ $(t-3)\binom{t-1}{3}+3\binom{t-3}{3}+6\binom{t-3}{2} \ (t\ge 6)$ $G=P_{4}^{+}$ $G=P_4$ $12\binom{t}{4}-2(t-3) \ (t\ge 5)$ $12\binom{t}{4}-6(t-3) \ (t\ge 5)$ $G=P_5$ $60\binom{t}{5}-3(t-3)(t-4) \ (t\ge 7)$ $288\ (t=6)$ $60\binom{t}{5}-9(t-3)(t-4) \ (t\ge 7)$ In the fourth section of this paper, similarly, we give some exact values of the bipartite Gallai-Ramsey number and multiplicity when the number of colors $k=t^2$ is sufficiently large with respect to the subgraph $H$. The main results for the bipartite Gallai-Ramsey multiplicity are shown in the following table. Main results for bipartite Gallai-Ramsey multiplicity ---------------------------------------------------------------- ----------------------------------- ----------------------------------- $\operatorname{bi-GM}_{k-1}(G:H)$ $\operatorname{bi-GM}_{k-2}(G:H)$ $G=K_{1,3}$ $30 \ (t=4)$ $(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2} \ (t\ge 5)$ $28\ (t=4)$ $93\ (t=5)$ $(2t-1)\binom{t}{3}+\binom{t-3}{3}+3\binom{t-3}{2} \ (t\ge 6)$ $G=P_4$ $G=P_5$ # Preliminaries Some propositions and lemmas presented in section are very helpful for the proof in the third and fourth sections of this paper. In 2008, Fox in [@Fox] provided the total number of different subgraphs $G$ in $K_n$. **Proposition 12**. *[@Fox][\[Prop-counting\]]{#Prop-counting label="Prop-counting"} The total number of different subgraphs $G$ in $K_n$ is $\frac{|V(G)|!\binom{n}{|V(G)|}}{|\operatorname{Aut}(G)|}$.* Li, Wang and Liu in [@LWL19] determined the sharp bound of $k$ such that any $k$-edge-colored $K_n$ always has a rainbow subgraph $P_5$. Bass, Magnant, Ozeki and Pyron in [@BassMagnantOzekiPyron] obtained the sharp bound of $k$ such that any $k$-edge-colored $K_n$ always has a rainbow subgraph $K_{1,3}$ by studying anti-Ramsey numbers. **Proposition 13**. *[@LWL19][\[k-color-have-rainbowP5\]]{#k-color-have-rainbowP5 label="k-color-have-rainbowP5"} For integers $n\ge 5$ and $k$ with $n+1 \le k\le \binom{n}{2}$, there is always a rainbow subgraph $P_5$ under any $k$-edge-colored $K_n$.* **Proposition 14**. *[@BassMagnantOzekiPyron][\[k-color-have-rainbowK13\]]{#k-color-have-rainbowK13 label="k-color-have-rainbowK13"} For integers $n\ge 4$ and $k$ with $\lceil \frac{n+3}{2}\rceil \le k\le \binom{n}{2}$, there is always a rainbow subgraph $K_{1,3}$ under any $k$-edge-colored $K_n$.* Also, Li, Wang and Liu in [@LWL19] determined the sharp bound of $k$ such that any $k$-edge-coloring of $K_{n,n}$ always has a rainbow subgraph $P_4$ or $P_5$. **Proposition 15**. *[@LWL19][\[k-color-have-rainbowP4-bipartite\]]{#k-color-have-rainbowP4-bipartite label="k-color-have-rainbowP4-bipartite"} For integers $n\ge 2$ and $k$ with $n+1 \le k\le n^2$, there is always a rainbow subgraph $P_4$ under any $k$-edge-colored $K_{n,n}$.* **Proposition 16**. *[@LWL19][\[k-color-have-rainbowP5-bipartite\]]{#k-color-have-rainbowP5-bipartite label="k-color-have-rainbowP5-bipartite"} For integers $n\ge 3$ and $k$ with $n+2 \le k\le n^2$, there is always a rainbow subgraph $P_5$ under any $k$-edge-colored $K_{n,n}$.* Similarly, we can directly obtain the following result through Theorem [\[th-K13-bGR-Structure\]](#th-K13-bGR-Structure){reference-type="ref" reference="th-K13-bGR-Structure"}. **Proposition 17**. *For integers $n\ge 3$ and $k$ with $n+2 \le k\le n^2$, there is always a rainbow subgraph $K_{1,3}$ under any $k$-edge-colored $K_{n,n}$.* The following lemmas are very useful in the proof of the third section. **Lemma 18**. *Let integer $t\ge 4$ and edges $e_1, e_2$ in $E(K_t)$. If $e_1$ and $e_2$ are not adjacent, then the number of different $P_4$ containing $e_1$ and $e_2$ is $4$. If $e_1$ and $e_2$ are adjacent, then the number of different $P_4$ containing $e_1$ and $e_2$ is $2(t-3)$.* *Proof.* Assume that the edges $e_1$ and $e_2$ are not adjacent, and let $e_1=v_1v_2$, $e_2=v_3v_4$. Consider $P_4$ with edges $e_1$ and $e_2$. Since $P_4$ is a connected graph, there is an edge that connects $e_1$ and $e_2$. In this case, there are four different $P_4$, which are $v_1v_2v_3v_4$, $v_1v_2v_4v_3$, $v_2v_1v_3v_4$ and $v_2v_1v_4v_3$. Assume that the edges $e_1$ and $e_2$ are adjacent. From the structure of $P_4$, it can be seen that $e_1$ and $e_2$ cannot be both pendent edges of $P_4$, and one of $e_1$ and $e_2$ must be the pendent edge of $P_4$. If $e_1$ is the pendent edge of $P_4$, then there are $t-3$ different $P_4$. By symmetry, if $e_2$ is the pendent edge of $P_4$, then there are $t-3$ different $P_4$. Therefore, in this case, there are $2(t-3)$ different $P_4$. ◻ **Lemma 19**. *Let integer $t\ge 5$ and edges $e_1, e_2$ in $E(K_t)$. If $e_1$ and $e_2$ are not adjacent, then the number of different $P_5$ containing $e_1$ and $e_2$ is $12(t-4)$. If $e_1$ and $e_2$ are adjacent, then the number of different $P_5$ containing $e_1$ and $e_2$ is $3(t-3)(t-4)$.* *Proof.* Assume that the edges $e_1$ and $e_2$ are not adjacent. From the structure of $P_5$, it can be seen that one of $e_1$ and $e_2$ must be the pendent edge of $P_5$. If $e_1$ is the pendent edge of $P_5$ but $e_2$ is not the pendent edge of $P_5$, then there are $4(t-4)$ different $P_5$. By symmetry, if $e_2$ is the pendent edge of $P_5$ but $e_1$ is not the pendent edge of $P_5$, then there are $4(t-4)$ different $P_5$. If the $e_1$ and $e_2$ are both pendent edges of $P_5$, then there are $4(t-4)$ different $P_5$. Therefore, in this case, there are $12(t-4)$ different $P_5$. Assume that the edges $e_1$ and $e_2$ are adjacent. From the structure of $P_5$, it can be seen that $e_1$ and $e_2$ cannot be both pendent edges of $P_5$. If $e_1$ is the pendent edge of $P_5$ but $e_2$ is not the pendent edge of $P_5$, then there are $(t-3)(t-4)$ different $P_5$. By symmetry, if $e_2$ is the pendent edge of $P_5$ but $e_1$ is not the pendent edge of $P_5$, then there are $(t-3)(t-4)$ different $P_5$. If neither $e_1$ nor $e_2$ are the pendent edges of $P_5$, then there are $(t-3)(t-4)$ different $P_5$. Therefore, in this case, there are $3(t-3)(t-4)$ different $P_5$. ◻ **Lemma 20**. *Let integer $t\ge 5$ and edges $e_1, e_2$ in $E(K_t)$. If $e_1$ and $e_2$ are not adjacent, then the number of different $P_4^{+}$ containing $e_1$ and $e_2$ is $8(t-4)$. If $e_1$ and $e_2$ are adjacent, then the number of different $P_4^{+}$ containing $e_1$ and $e_2$ is $5(t-3)(t-4)$.* *Proof.* Assume that the edges $e_1$ and $e_2$ are not adjacent, and let $e_1=v_1v_2$, $e_2=v_3v_4$. Consider $P_{4}^{+}$ with edges $e_1$ and $e_2$. From the proof of Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"}, we know that there are four different $P_4$, which are $v_1v_2v_3v_4$, $v_1v_2v_4v_3$, $v_2v_1v_3v_4$ and $v_2v_1v_4v_3$. Since $P_{4}^{+}$ is a graph by adding an extra pendent edge to an inner vertex of $P_4$, it follows that there are $4\cdot 2(t-4)=8(t-4)$ different $P_{4}^{+}$. Assume that the edges $e_1$ and $e_2$ are adjacent. If $e_1$ is the tail edge of $P_{4}^{+}$, then there are $(t-3)(t-4)$ different $P_{4}^{+}$. By symmetry, if $e_2$ is the tail edge of $P_{4}^{+}$, then there are $(t-3)(t-4)$ different $P_{4}^{+}$. If neither $e_1$ nor $e_2$ are the tail edges of $P_{4}^{+}$, then there are $3(t-3)(t-4)$ different $P_{4}^{+}$. Therefore, in this case, there are $5(t-3)(t-4)$ different $P_{4}^{+}$. ◻ Also, the following lemmas are very useful in the proof of the fourth section. **Lemma 21**. *Let integer $t\ge 3$ and edges $e_1, e_2$ in $E(K_{t,t})$. If $e_1$ and $e_2$ are not adjacent, then the number of different $P_4$ containing $e_1$ and $e_2$ is $2$. If $e_1$ and $e_2$ are adjacent, then the number of different $P_4$ containing $e_1$ and $e_2$ is $2(t-1)$.* *Proof.* Let $(X,Y)$ be the bipartition of $K_{t,t}$, $e_1=v_1v_2$, $e_2=v_3v_4$ and $v_1,v_3\in X$, $v_2,v_4\in Y$. Consider $P_4$ with edges $e_1$ and $e_2$. Since $P_4$ is a connected graph, there is an edge that connects $e_1$ and $e_2$. In this case, there are two different $P_4$, which are $v_1v_2v_3v_4$ and $v_2v_1v_4v_3$. Assume that the edges $e_1$ and $e_2$ are adjacent. From the structure of $P_4$, it can be seen that $e_1$ and $e_2$ cannot be both pendent edges of $P_4$, and one of $e_1$ and $e_2$ must be the pendent edge of $P_4$. If $e_1$ is the pendent edge of $P_4$, then there are $t-1$ different $P_4$. By symmetry, if $e_2$ is the pendent edge of $P_4$, then there are $t-1$ different $P_4$. Therefore, in this case, there are $2(t-1)$ different $P_4$. ◻ **Lemma 22**. *Let integer $t\ge 3$ and edges $e_1, e_2$ in $E(K_{t,t})$. If $e_1$ and $e_2$ are not adjacent, then the number of different $P_5$ containing $e_1$ and $e_2$ is $6(t-2)$. If $e_1$ and $e_2$ are adjacent, then the number of different $P_5$ containing $e_1$ and $e_2$ is $3(t-1)(t-2)$.* *Proof.* Assume that the edges $e_1$ and $e_2$ are not adjacent. From the structure of $P_5$, it can be seen that one of $e_1$ and $e_2$ must be the pendent edge of $P_5$. If $e_1$ is the pendent edge of $P_5$ but $e_2$ is not the pendent edge of $P_5$, then there are $2(t-2)$ different $P_5$. By symmetry, if $e_2$ is the pendent edge of $P_5$ but $e_1$ is not the pendent edge of $P_5$, then there are $2(t-2)$ different $P_5$. If $e_1$ and $e_2$ are both pendent edges of $P_5$, then there are $2(t-2)$ different $P_5$. Therefore, in this case, there are $6(t-2)$ different $P_5$. Assume that the edges $e_1$ and $e_2$ are adjacent. From the structure of $P_5$, it can be seen that $e_1$ and $e_2$ cannot be both pendent edges of $P_5$. If $e_1$ is the pendent edge of $P_5$ but $e_2$ is not the pendent edge of $P_5$, then there are $(t-1)(t-2)$ different $P_5$. By symmetry, if $e_2$ is the pendent edge of $P_5$ but $e_1$ is not the pendent edge of $P_5$, then there are $(t-1)(t-2)$ different $P_5$. If neither $e_1$ nor $e_2$ are the pendent edges of $P_5$, then there are $(t-1)(t-2)$ different $P_5$. Therefore, in this case, there are $3(t-1)(t-2)$ different $P_5$. ◻ Recall that the $k$-edge-coloring we studied in this paper is exact, meaning that each color is used at least once. Based on this, a basic principle is that the number of colors does not exceed the total number of edges in an edge-colored graph. So by solving the equations $$k\le \binom{n}{2}=|E(K_n)| \mbox{ and } k\le n^2=|E(K_{n,n})|,$$ we obtain $n\ge \frac{1+\sqrt{1+8k}}{2}$ and $n\ge \sqrt{k}$, respectively. Therefore, we directly get the following basic lower bound lemma. **Lemma 23**. *For integer $k\ge 4$, $G\in\{P_4,K_{1,3}\}$ and any graph $H$, we have $$\operatorname{gr}_k(G:H)\ge \left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil.$$ For integer $k\ge 5$, $G\in\{P_4,P_5,K_{1,3}\}$ and any bipartite graph $H$, we have $$\operatorname{bgr}_k(G:H)\ge \left\lceil \sqrt{k}\right\rceil,$$ in particular, this lower bound also holds when $3\le k\le 4$ and $G=P_4$.* # Results for Gallai-Ramsey multiplicity We consider four kinds of rainbow graphs $K_{1,3}$, $P_{4}^{+}$, $P_{4}$, $P_{5}$, respectively, by the following four subsections. ## For rainbow $K_{1,3}$ **Theorem 24**. *Let integer $k\ge 4$. If $H$ is a subgraph of the balanced complete $(k-1)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-1}}$. Then $$\operatorname{gr}_k(K_{1,3}:H)=\left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil.$$* *Proof.* The lower bound follows from Lemma [Lemma 23](#basic-lower-bound-lemma){reference-type="ref" reference="basic-lower-bound-lemma"}. Let $N_k=\left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil$. For the upper bound, we consider any $k$-edge-coloring of $K_N \ (N\ge N_k)$. Noticing that $N_k<2k-2$ for $k\geq 4$. If $N_k\le N\le 2k-3$, then it follows from Proposition [\[k-color-have-rainbowK13\]](#k-color-have-rainbowK13){reference-type="ref" reference="k-color-have-rainbowK13"} that there is always a rainbow $K_{1,3}$, the result thus follows. Next we assume $N\ge 2k-2$. Suppose to the contrary that $K_N$ contains neither a rainbow subgraph $K_{1,3}$ nor a monochromatic subgraph $H$. It follows from Theorem [\[th-Star-Structure\]](#th-Star-Structure){reference-type="ref" reference="th-Star-Structure"} that the Colored Structure 1 occurs. From exact $k$-edge-coloring, we have $|V_i|\ge 2$ for each $i \in\{2,3,\ldots,k\}$. Since $H$ is a subgraph of the balanced complete $(k-1)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-1}}$, it follows that there is a monochromatic $H$, a contradiction. The result thus follows. ◻ According to Theorem [Theorem 24](#k-color-Star-H-general){reference-type="ref" reference="k-color-Star-H-general"} and Proposition [\[Prop-counting\]](#Prop-counting){reference-type="ref" reference="Prop-counting"}, the following corollary can be directly deduced. **Corollary 25**. For integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, and a subgraph $H$ of the balanced complete $(k-1)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-1}}$ with $|E(H)|\ge 2$, we have $$\operatorname{GM}_{k}(K_{1,3}:H)=\frac{4!\binom{t}{4}}{|\operatorname{Aut}(K_{1,3})|}=t\binom{t-1}{3}.$$ **Theorem 26**. *For integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, and a subgraph $H$ of the balanced complete $(k-2)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-2}}$ with $|E(H)|\ge 3$, we have $$\operatorname{GM}_{k-1}(K_{1,3}:H)=\left\{ \begin{array}{ll} 3, & t=4;\\ 18, & t=5;\\ (t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}, & t\ge 6. \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 24](#k-color-Star-H-general){reference-type="ref" reference="k-color-Star-H-general"} that $\operatorname{gr}_{k-1}(K_{1,3}:H)=t$. Consider any $(k-1)$-edge-coloring of $K_t$. Since $|E(K_t)|=\binom{t}{2}$ and each color is used at least once, it follows that there are only two edges with the same color in $K_t$. Without loss of generality, we assume that there are two red edges $e_1$ and $e_2$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. If the edges $e_1$ and $e_2$ are not adjacent, then this case is equivalent to Corollary [Corollary 25](#Cor-GM-k-color-star){reference-type="ref" reference="Cor-GM-k-color-star"}. Therefore, $$\operatorname{GM}_{k-1}(K_{1,3}:H)\le t\binom{t-1}{3}.$$ If the edges $e_1$ and $e_2$ are adjacent, then $e_1$ and $e_2$ form a red $P_3$. Let vertex $v$ be incident to the edges $e_1$ and $e_2$. We first investigate the number of rainbow $K_{1,3}$ with center $v$ for $t\ge 6$. Noticing that $\deg(v)=t-1$, the number of rainbow $K_{1,3}$ with center $v$ and without red edges is $\binom{t-3}{3}$, and number of rainbow $K_{1,3}$ with center $v$ and with a red edge is $2\binom{t-3}{2}$. In $K_t$, there are $(t-1)\binom{t-1}{3}$ rainbow $K_{1,3}$ with center in $V(K_t)\setminus \{v\}$. Therefore, $$\operatorname{GM}_{k-1}(K_{1,3}:H)\le (t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}.$$ Next, we compare the sizes of $t\binom{t-1}{3}$ and $(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}$, which is equivalent to comparing the sizes of $\binom{t-1}{3}$ and $\binom{t-3}{3}+2\binom{t-3}{2}$. Noticing that $$\binom{t-1}{3}=\frac{(t-1)(t-2)(t-3)}{6},$$ $$\binom{t-3}{3}+2\binom{t-3}{2}=\frac{(t-3)(t-4)(t-5)}{6}+(t-3)(t-4).$$ Thus we only need to compare the sizes of $(t-1)(t-2)$ and $(t+1)(t-4)$. Since $$(t-1)(t-2)=t^2-3t+2>t^2-3t-4=(t+1)(t-4),$$ it follows that $$\operatorname{GM}_{k-1}(K_{1,3}:H)=(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}.$$ When $t=4$, there is no rainbow $K_{1,3}$ with center $v$. In $K_4$, each other vertex has one rainbow $K_{1,3}$. So there are three rainbow $K_{1,3}$ in $K_4$. Since $3<4\binom{4-1}{3}$, it follows that $\operatorname{GM}_{5}(K_{1,3}:H)=3$. When $t=5$, the number of rainbow $K_{1,3}$ with center $v$ and without red edges is $0$, and number of rainbow $K_{1,3}$ with center $v$ and with a red edge is $2$. In $K_5$, there are $4\binom{4}{3}=16$ rainbow $K_{1,3}$ with center in $V(K_5)\setminus \{v\}$. Therefore, $\operatorname{GM}_{9}(K_{1,3}:H)\le 18$. Since $18<5\binom{5-1}{3}$. It follows that $\operatorname{GM}_{9}(K_{1,3}:H)=18$. ◻ **Theorem 27**. *For integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, and a subgraph $H$ of the balanced complete $(k-3)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-3}}$ with $|E(H)|\ge 4$, we have $$\operatorname{GM}_{k-2}(K_{1,3}:H)=\left\{ \begin{array}{ll} 1, & t=4;\\ 14, & t=5;\\ (t-3)\binom{t-1}{3}+3\binom{t-3}{3}+6\binom{t-3}{2}, & t\ge 6.\\ \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 24](#k-color-Star-H-general){reference-type="ref" reference="k-color-Star-H-general"} that $\operatorname{gr}_{k-2}(K_{1,3}:H)=t$. Consider a $(k-2)$-edge-coloring of $K_t$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. Noticing that each color needs to be used at least once. We first color any $k-2$ edges in $K_t$ with $k-2$ colors, and the remaining two edges are temporarily not colored, denoted as $e_1$ and $e_2$. Next, we discuss the edges $e_1$ and $e_2$ in two cases. **Case 1**. The edges $e_1$ and $e_2$ have the same color. Without loss of generality, we assume that these two edges are red. According to the structure of $K_t$, it is easy to calculate that if the red edges form a $3P_2$, then there are $$f_1(t)=t\binom{t-1}{3}$$ rainbow $K_{1,3}$ in $K_t$; if the red edges form a $P_3\cup P_2$, then there are $$f_2(t)=(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the red edges form a $P_4$, then there are $$f_3(t)=(t-2)\binom{t-1}{3}+2\binom{t-3}{3}+4\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the red edges form a $K_3$, then there are $$f_4(t)=(t-3)\binom{t-1}{3}+3\binom{t-3}{3}+6\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the red edges form a $K_{1,3}$, then there are $$f_5(t)=(t-1)\binom{t-1}{3}+\binom{t-4}{3}+3\binom{t-4}{2}$$ rainbow $K_{1,3}$ in $K_t$. **Case 2**. The edges $e_1$ and $e_2$ have different colors. When the $P_3$ is formed by the edges $e_1$ and $e_2$ in $K_t$, without loss of generality, we assume that $e_1$ is red and $e_2$ is blue. Let $V(P_3)=\{u,v,w\}$ and vertex $v$ is incident to edges $e_1$ and $e_2$. According to the structure of $K_t$, it is easy to calculate that if the other red edge is not incident to vertex $u$ or $v$, and the other blue edge is not incident to vertex $v$ or $w$, then there are $$f_1(t)=t\binom{t-1}{3}$$ rainbow $K_{1,3}$ in $K_t$; if the other red edge is incident to vertex $u$ or $v$, and the other blue edge is not incident to vertex $v$ or $w$, then there are $$f_2(t)=(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the other red edge is incident to vertex $u$, and the other blue edge is incident to vertex $w$, then there are $$f_3(t)=(t-2)\binom{t-1}{3}+2\binom{t-3}{3}+4\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the other red edge is incident to vertex $v$, and the other blue edge is also incident to vertex $v$, then there are $$f_6(t)=(t-1)\binom{t-1}{3}+\binom{t-5}{3}+4\binom{t-5}{2}+4(t-5)$$ rainbow $K_{1,3}$ in $K_t$. When the $2P_2$ is formed by the edges $e_1$ and $e_2$ in $K_t$, without loss of generality, we assume that $e_1$ is red and $e_2$ is blue. According to the structure of $K_t$, it is easy to calculate that if the other red edge is not adjacent to $e_1$, and the other blue edge is not adjacent to $e_2$, then there are $$f_1(t)=t\binom{t-1}{3}$$ rainbow $K_{1,3}$ in $K_t$; if the other red edge is adjacent to $e_1$, and the other blue edge is not adjacent to $e_2$, then there are $$f_2(t)=(t-1)\binom{t-1}{3}+\binom{t-3}{3}+2\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$; if the other red edge is adjacent to $e_1$, and the other blue edge is adjacent to $e_2$, then there are $$f_3(t)=(t-2)\binom{t-1}{3}+2\binom{t-3}{3}+4\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_t$. Next, we compare the sizes of $f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)$ and $f_6(t)$. Based on the practical significance of counting in this paper, we only define in the operations of expressions for $t$ in $f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)$ and $f_6(t)$ that when integers $a<b$, we have $\binom{a}{b}\equiv 0$ and $a-b\equiv 0$. For $t=4$, we have $$f_1(4)=4, f_2(4)=3, f_3(4)=2, f_4(4)=1, f_5(4)=3, f_6(4)=3.$$ Thus, $\min\{f_1(4), f_2(4), f_3(4), f_4(4), f_5(4), f_6(4)\}=1.$ For $t=5$, we have $$f_1(5)=20, f_2(5)=18, f_3(5)=16, f_4(5)=14, f_5(5)=16, f_6(5)=16.$$ Thus, $\min\{f_1(5), f_2(5), f_3(5), f_4(5), f_5(5), f_6(5)\}=14.$ For $t=6$, we have $$f_1(6)=60, f_2(6)=57, f_3(6)=54, f_4(6)=51, f_5(6)=53, f_6(6)=54.$$ Thus, $\min\{f_1(6), f_2(6), f_3(6), f_4(6), f_5(6), f_6(6)\}=51.$ For $t=7$, we have $$f_1(7)=140, f_2(7)=136, f_3(7)=132, f_4(7)=128, f_5(7)=130, f_6(7)=132.$$ Thus, $\min\{f_1(7), f_2(7), f_3(7), f_4(7), f_5(7), f_6(7)\}=128.$ For $t\ge 8$ and $1\le i\le 6$, let $f_{ii}(t)=f_i(t)-(t-3)\binom{t-1}{3}-\frac{1}{2}t^3+3t^2-\frac{5}{2}t$, then $$f_{11}(t)=3t-3, f_{22}(t)=2t, f_{33}(t)=t+3, f_{44}(t)=6, f_{55}(t)=8, f_{66}(t)=t+3.$$ Therefore, for $t\ge 8$, $$\min\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t), f_{66}(t)\}=f_{44}(t)=6,$$ and thus, $$\min\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t), f_6(t)\}=f_4(t)=(t-3)\binom{t-1}{3}+3\binom{t-3}{3}+6\binom{t-3}{2}.$$ Based on the above discussion, we have $$\min\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t), f_6(t)\}=\left\{ \begin{array}{ll} (t-3)\binom{t-1}{3}, & t=4;\\ (t-3)\binom{t-1}{3}+6\binom{t-3}{2}, & t=5;\\ (t-3)\binom{t-1}{3}+3\binom{t-3}{3}+6\binom{t-3}{2}, & t\ge 6.\\ \end{array}\right.$$ The result thus follows. ◻ ## For rainbow $P_{4}^{+}$ According to Theorems [\[th-Star-Structure\]](#th-Star-Structure){reference-type="ref" reference="th-Star-Structure"} and [\[th-P4+-Structure\]](#th-P4+-Structure){reference-type="ref" reference="th-P4+-Structure"}, we directly give the following observation. **Observation 1**. For integers $k\ge 5$ and $\operatorname{gr}_k(K_{1,3}:H)\ge 5$, we have $$\operatorname{gr}_k(P_{4}^{+}:H)=\operatorname{gr}_k(K_{1,3}:H).$$ From Observation [Observation 1](#obv){reference-type="ref" reference="obv"} and Theorem [Theorem 24](#k-color-Star-H-general){reference-type="ref" reference="k-color-Star-H-general"}, the following theorem can be directly deduced. **Theorem 28**. *Let integer $k\ge 5$. If $H$ is a subgraph of the balanced complete $(k-1)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-1}}$, then $$\operatorname{gr}_k(P_4^{+}:H)= \begin{cases} 5, & 5\le k\le 6;\\ \left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil, & k\ge 7. \end{cases}$$* Noticing that $P_4^{+}$ is obtained by adding a pendent edge to a leaf vertex at $K_{1,3}$, there are $t\binom{t-1}{3}\cdot 3(t-4)$ different $P_4^{+}$ in $K_t$. We can also calculate $|\operatorname{Aut}(P_4^{+})|=2$ from Proposition [\[Prop-counting\]](#Prop-counting){reference-type="ref" reference="Prop-counting"}. Therefore, we directly provide the following theorem. **Theorem 29**. *For integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 10$, and a subgraph $H$ of the balanced complete $(k-1)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-1}}$ with $|E(H)|\ge 2$, we have $$\operatorname{GM}_{k}(P_4^{+}:H)=\frac{5!\binom{t}{5}}{|\operatorname{Aut}(P_{4}^{+})|}=60\binom{t}{5}.$$* **Theorem 30**. *For integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 10$, and a subgraph $H$ of the balanced complete $(k-2)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-2}}$ with $|E(H)|\ge 3$, we have $$\operatorname{GM}_{k-1}(P_4^{+}:H)=60\binom{t}{5}-5(t-3)(t-4).$$* *Proof.* It follows from Theorem [Theorem 28](#k-color-P4+-H-general){reference-type="ref" reference="k-color-P4+-H-general"} that $\operatorname{gr}_{k-1}(P_4^{+}:H)=t$. Consider any $(k-1)$-edge-coloring of $K_t$. Since $|E(K_t)|=\binom{t}{2}$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_t$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. According to Theorem [Theorem 29](#THM-GM-k-color-P4+){reference-type="ref" reference="THM-GM-k-color-P4+"}, there are $60\binom{t}{5}$ different $P_4^{+}$ in $K_t$, and we only need to find the number of different $P_4^{+}$ containing the edges $e_1$ and $e_2$. This is because only $P_4^{+}$ containing edges $e_1$ and $e_2$ are not rainbow, and all other $P_4^{+}$ are rainbow. If $e_1$ and $e_2$ are not adjacent, then according to Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"}, there are $8(t-4)$ different $P_4^{+}$ in $K_t$ that contain edges $e_1$ and $e_2$. If $e_1$ and $e_2$ are adjacent, then according to Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"}, there are $5(t-3)(t-4)$ different $P_4^{+}$ in $K_t$ that contain edges $e_1$ and $e_2$. Noticing that $8(t-4)<5(t-3)(t-4)$ for $t\ge 5$, the result thus follows. ◻ **Theorem 31**. *For integers $k$ and $t$ satisfy $k=\binom{t}{2}\ge 10$, if $H$ is a subgraph of the balanced complete $(k-3)$-partite graph $K_{\underbrace{2,2,\ldots,2}_{k-3}}$ with $|E(H)|\ge 4$, then we have $$\operatorname{GM}_{k-2}(P_4^{+}:H)=60\binom{t}{5}-3(5t-14)(t-4).$$* *Proof.* It follows from Theorem [Theorem 28](#k-color-P4+-H-general){reference-type="ref" reference="k-color-P4+-H-general"} that $\operatorname{gr}_{k-2}(P_4^{+}:H)=t$. Consider $(k-2)$-edge-coloring of $K_t$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. Since each color is used at least once, there are only the following two cases. Due to the arbitrariness of colors, we can describe them using specific color names such as red and blue. Next, we calculate the number of different $P_{4}^{+}$ containing two or more edges with the same color. There are three red edges $e_1$, $e_2$ and $e_3$. The remaining edges are not red and the colors of any two remaining edges are not the same. The edges $e_1$, $e_2$ and $e_3$ form a red $3P_2$. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $24(t-4)$ different $P_{4}^{+}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_3\cup P_2$. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $5(t-3)(t-4)$ different $P_{4}^{+}$ containing red $P_3$ and $2(8(t-4)-2)$ different $P_{4}^{+}$ without red $P_3$. So there are a total of $(5t+1)(t-4)-4$ different $P_{4}^{+}$ containing two or more red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{1,3}$. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $15(t-3)(t-4)$ different $P_{4}^{+}$ containing two red edges and $3(t-4)$ different $P_{4}^{+}$ containing three red edges. So there are a total of $3(5t-14)(t-4)$ different $P_{4}^{+}$ containing two or more red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{3}$. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $15(t-3)(t-4)$ different $P_{4}^{+}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_4$. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $2(5(t-3)(t-4)-2(t-4))+3\cdot 2(t-4)=2(5t-14)(t-4)$ different $P_{4}^{+}$ containing two red edges and $2(t-4)$ different $P_{4}^{+}$ containing three red edges. So there are a total of $2(5t-13)(t-4)$ different $P_{4}^{+}$ containing two or more red edges. There are two red edges $e_1, e_2$ and two blue edges $e_3, e_4$. The remaining edges are not red or blue and the colors of any two remaining edges are not the same. The edges $e_1$ and $e_2$ are not adjacent, and the edges $e_3$ and $e_4$ are also not adjacent. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are $16(t-4)$ different $P_{4}^{+}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, but the edges $e_3$ and $e_4$ are not adjacent. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are at most $8(t-4)+5(t-3)(t-4)=(5t-7)(t-4)$ different $P_{4}^{+}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, and the edges $e_3$ and $e_4$ are also adjacent. In this subcase, it follows from Lemma [Lemma 20](#Lem: Conut P4+ in complete graph){reference-type="ref" reference="Lem: Conut P4+ in complete graph"} that there are at most $10(t-3)(t-4)$ different $P_{4}^{+}$ containing two edges with the same color. Let $$f_1(t)=24(t-4), f_2(t)=(5t+1)(t-4)-4,$$ $$f_3(t)=3(5t-14)(t-4), f_4(t)=2(5t-13)(t-4), f_5(t)=(5t-7)(t-4).$$ Based on the data calculated from the eight subcases above, we need to compare the sizes of $f_1(t), f_2(t), f_3(t), f_4(t)$ and $f_5(t)$. For $1\le i\le 5$, let $f_{ii}(t)=\frac{f_i(t)}{t-4}$. Then $$f_{11}(t)=24, f_{22}(t)=5t+1-\frac{4}{t-4}, f_{33}(t)=3(5t-14), f_{44}(t)=2(5t-13), f_{55}(t)=5t-7.$$ Therefore, for $t\ge 5$ $$\max\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{33}(t)=3(5t-14),$$ and thus, $$\max\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=f_3(t)=3(5t-14)(t-4).$$ The result thus follows. ◻ ## For rainbow $P_{4}$ From Theorem [\[th-3-path-Structure\]](#th-3-path-Structure){reference-type="ref" reference="th-3-path-Structure"}, we directly obtain the following theorem. **Theorem 32**. *For a graph $H$ and integer $k\ge 4$, we have $$\operatorname{gr}_k(P_4:H)=\left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil.$$* According to Theorem [Theorem 32](#k-color-P4-H-general){reference-type="ref" reference="k-color-P4-H-general"} and Proposition [\[Prop-counting\]](#Prop-counting){reference-type="ref" reference="Prop-counting"}, the following corollary can be directly deduced. **Corollary 33**. For a graph $H$ with $|E(H)|\ge 2$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, we have $$\operatorname{GM}_{k}(P_4:H)=\frac{4!\binom{t}{4}}{|\operatorname{Aut}(P_4)|}=12\binom{t}{4}.$$ **Theorem 34**. *For a graph $H$ with $|E(H)|\ge 3$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, we have $$\operatorname{GM}_{k-1}(P_4:H)=\left\{ \begin{array}{ll} 8, & t= 4;\\ 12\binom{t}{4}-2(t-3), & t\ge 5. \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 32](#k-color-P4-H-general){reference-type="ref" reference="k-color-P4-H-general"} that $\operatorname{gr}_{k-1}(P_4:H)=t$. Consider any $(k-1)$-edge-coloring of $K_t$. Since $|E(K_t)|=\binom{t}{2}$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_t$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. According to Corollary [Corollary 33](#Cor-GM-k-color-P4){reference-type="ref" reference="Cor-GM-k-color-P4"}, there are $12\binom{t}{4}$ different $P_4$ in $K_t$, and we only need to find the number of different $P_4$ containing the edges $e_1$ and $e_2$. This is because only $P_4$ containing edges $e_1$ and $e_2$ are not rainbow, and all other $P_4$ are rainbow. If $e_1$ and $e_2$ are not adjacent, then according to Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"}, there are $4$ different $P_4$ in $K_t$ that contain edges $e_1$ and $e_2$. If $e_1$ and $e_2$ are adjacent, then according to Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"}, there are $2(t-3)$ different $P_4$ in $K_t$ that contain edges $e_1$ and $e_2$. Noticing that $4>2(t-3)$ for $t=4$ and $4\le 2(t-3)$ for $t\ge 5$, the result thus follows. ◻ **Theorem 35**. *For a graph $H$ with $|E(H)|\ge 4$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge 6$, we have $$\operatorname{GM}_{k-2}(P_4:H)=\left\{ \begin{array}{ll} 4, & t= 4;\\ 12\binom{t}{4}-6(t-3), & t\ge 5. \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 32](#k-color-P4-H-general){reference-type="ref" reference="k-color-P4-H-general"} that $\operatorname{gr}_{k-2}(P_4:H)=t$. Consider $(k-2)$-edge-coloring of $K_t$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. Since each color is used at least once, there are only the following two cases. Due to the arbitrariness of colors, we can describe them using specific color names such as red and blue. Next, we calculate the number of different $P_{4}$ containing two or more edges with the same color. There are three red edges $e_1$, $e_2$ and $e_3$. The remaining edges are not red and the colors of any two remaining edges are not the same. The edges $e_1$, $e_2$ and $e_3$ form a red $3P_2$. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $12$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_3\cup P_2$. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $8+2(t-3)=2(t+1)$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{1,3}$. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $6(t-3)$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{3}$. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $6(t-3)$ different $P_{4}$ containing two red edges. Thew edges $e_1$, $e_2$ and $e_3$ form a red $P_4$. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $4+2(t-3)+2(t-4)=2(2t-5)$ different $P_{4}$ containing two or more red edges. There are two red edges $e_1, e_2$ and two blue edges $e_3, e_4$. The remaining edges are not red or blue and the colors of any two remaining edges are not the same. The edges $e_1$ and $e_2$ are not adjacent, and the edges $e_3$ and $e_4$ are also not adjacent. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $8$ different $P_{4}$ containing two edges with the same color. the dges $e_1$ and $e_2$ are adjacent, but the edges $e_3$ and $e_4$ are not adjacent. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are $4+2(t-3)=2(t-1)$ different $P_{4}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, and edges $e_3$ and $e_4$ are also adjacent. In this subcase, it follows from Lemma [Lemma 18](#Lem: Conut P4 in complete graph){reference-type="ref" reference="Lem: Conut P4 in complete graph"} that there are at most $4(t-3)$ different $P_{4}$ containing two edges with the same color. We first consider the result when $t=4$. Noticing that there are no red $3P_2$ or $P_3\cup P_2$ in a $4$-edge-colored $K_4$. Thus there are at most $8$ different $P_4$ in a $4$-edge-colored $K_4$ that contain two or more edges of the same color. Let $f_1(t)=12, f_2(t)=2(t+1), f_3(t)=6(t-3), f_4(t)=2(2t-5)$. Based on the data calculated from the eight subcases above, we need to compare the sizes of $f_1(t), f_2(t), f_3(t)$ and $f_4(t)$. For $t\ge 5$, we have $$\max\{f_1(t), f_2(t), f_3(t), f_4(t)\}=f_3(t)=6(t-3).$$ The result thus follows. ◻ ## For rainbow $P_{5}$ In 2023, Zou, Wang, Lai and Mao provided results on the Gallai-Ramsey number for rainbow $P_5$. **Theorem 36**. *[@ZWLM2023][\[k-color-P5-H-general\]]{#k-color-P5-H-general label="k-color-P5-H-general"} For a graph $H$ and an integer $k\ge 5$, we have $$\operatorname{gr}_k(P_5:H)= \begin{cases} \max\left\{\left\lceil \frac{1+\sqrt{1+8k}}{2} \right\rceil,5\right\}, & k\ge |V(H)|+1;\\ |V(H)|+1, & \text{$k=|V(H)|$ and $H$ is not a complete graph;}\\ (|V(H)|-1)^2+1, & \text{$k=|V(H)|$ and $H$ is a complete graph.} \end{cases}$$* According to Theorem [\[k-color-P5-H-general\]](#k-color-P5-H-general){reference-type="ref" reference="k-color-P5-H-general"} and Proposition [\[Prop-counting\]](#Prop-counting){reference-type="ref" reference="Prop-counting"}, the following corollary can be directly deduced. **Corollary 37**. For a graph $H$ with $|E(H)|\ge 2$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge \max\{|V(H)|+1,10\}$, we have $$\operatorname{GM}_{k}(P_5:H)=\frac{5!\binom{t}{5}}{|\operatorname{Aut}(P_5)|}=60\binom{t}{5}.$$ **Theorem 38**. *For a graph $H$ with $|E(H)|\ge 3$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge \max\{|V(H)|+2,10\}$, we have $$\operatorname{GM}_{k-1}(P_5:H)=\left\{ \begin{array}{ll} 60\binom{t}{5}-12(t-4), & 5\le t\le 6;\\ 60\binom{t}{5}-3(t-3)(t-4), & t\ge 7. \end{array}\right.$$* *Proof.* It follows from Theorem [\[k-color-P5-H-general\]](#k-color-P5-H-general){reference-type="ref" reference="k-color-P5-H-general"} that $\operatorname{gr}_{k-1}(P_5:H)=t$. Consider any $(k-1)$-edge-coloring of $K_t$. Since $|E(K_t)|=\binom{t}{2}$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_t$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. According to Corollary [Corollary 37](#Cor-GM-k-color-P5){reference-type="ref" reference="Cor-GM-k-color-P5"}, there are $60\binom{t}{5}$ different $P_5$ in $K_t$, and we only need to find the number of different $P_5$ containing the edges $e_1$ and $e_2$. This is because only $P_5$ containing edges $e_1$ and $e_2$ are not rainbow, and all other $P_5$ are rainbow. If $e_1$ and $e_2$ are not adjacent, then according to Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"}, there are $12(t-4)$ different $P_5$ in $K_t$ that contain edges $e_1$ and $e_2$. If $e_1$ and $e_2$ are adjacent, then according to Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"}, there are $3(t-3)(t-4)$ different $P_5$ in $K_t$ that contain edges $e_1$ and $e_2$. Noticing that $12(t-4)>3(t-3)(t-4)$ for $5 \le t\le 6$ and $12(t-4)\le 3(t-3)(t-4)$ for $t\ge 7$, the result thus follows. ◻ **Theorem 39**. *For a graph $H$ with $|E(H)|\ge 4$ and integers $k$ and $t$ satisfying $k=\binom{t}{2}\ge \max\{|V(H)|+3,10\}$, we have $$\operatorname{GM}_{k-2}(P_5:H)=\left\{ \begin{array}{ll} 38, & t=5;\\ 288, & t=6;\\ 60\binom{t}{5}-9(t-3)(t-4), & t\ge 7. \end{array}\right.$$* *Proof.* It follows from Theorem [\[k-color-P5-H-general\]](#k-color-P5-H-general){reference-type="ref" reference="k-color-P5-H-general"} that $\operatorname{gr}_{k-2}(P_5:H)=t$. Consider $(k-2)$-edge-coloring of $K_t$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_t$. Since each color is used at least once, there are only the following two cases. Due to the arbitrariness of colors, we can describe them using specific color names such as red and blue. Next, we calculate the number of different $P_5$ containing two or more edges with the same color. There are three red edges $e_1$, $e_2$ and $e_3$. The remaining edges are not red and the colors of any two remaining edges are not the same. The edges $e_1$, $e_2$ and $e_3$ form a red $3P_2$. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $36(t-4)$ different $P_{5}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_3\cup P_2$. Let $P_3=e_1e_2$ and $P_2=e_3$. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $3(t-3)(t-4)$ different $P_{5}$ containing red edges $e_1$ and $e_2$, $12(t-4)-4$ different $P_{5}$ only containing red edges $e_1$ and $e_3$, and symmetrically $12(t-4)-4$ different $P_{5}$ only containing red edges $e_2$ and $e_3$. So there are a total of $3(t-3)(t-4)+2(12(t-4)-4)=3(t+5)(t-4)-8$ different $P_{5}$ containing two or more red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{1,3}$. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $9(t-3)(t-4)$ different $P_{5}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{3}$. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $9(t-3)(t-4)$ different $P_{5}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_4$. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $12(t-4)$ different $P_{5}$ containing red $2P_2$ and $2(3(t-3)(t-4)-2(t-4))=2(3t-11)(t-4)$ different $P_{5}$ without red $2P_2$. So there are a total of $12(t-4)+2(3t-11)(t-4)=2(3t-5)(t-4)$ different $P_{5}$ containing two or more red edges. There are two red edges $e_1, e_2$ and two blue edges $e_3, e_4$. The remaining edges are not red or blue and the colors of any two remaining edges are not the same. The edges $e_1$ and $e_2$ are not adjacent, and edges $e_3$ and $e_4$ are also not adjacent. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are $24(t-4)$ different $P_{5}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, but edges $e_3$ and $e_4$ are not adjacent. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are at most $12(t-4)+3(t-3)(t-4)=3(t+1)(t-4)$ different $P_{5}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, and edges $e_3$ and $e_4$ are also adjacent. In this subcase, it follows from Lemma [Lemma 19](#Lem: Conut P5 in complete graph){reference-type="ref" reference="Lem: Conut P5 in complete graph"} that there are at most $6(t-3)(t-4)$ different $P_{5}$ containing two edges with the same color. Let $$f_1(t)=36(t-4), f_2(t)=3(t+5)(t-4)-8,$$ $$f_3(t)=9(t-3)(t-4), f_4(t)=2(3t-5)(t-4), f_5(t)=3(t+1)(t-4).$$ Based on the data calculated from the eight subcases above, we need to compare the sizes of $f_1(t), f_2(t), f_3(t), f_4(t)$ and $f_5(t)$. For $1\le i\le 5$, let $f_{ii}(t)=\frac{f_i(t)}{t-4}$. Then $$f_{11}(t)=36, f_{22}(t)=3(t+5)-\frac{8}{t-4}, f_{33}(t)=9(t-3), f_{44}(t)=2(3t-5), f_{55}=3(t+1).$$ For $t=5$, note that there are no red $3P_2$ in a $8$-edge-colored $K_5$. Thus $$\max\{f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{22}(t)=3(t+5)-\frac{8}{t-4}=22,$$ and thus $$\max\{f_2(t), f_3(t), f_4(t), f_5(t)\}=f_3(t)=3(t+5)(t-4)-8=22.$$ For $t=6$, we have $$\max\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{11}(t)=36,$$ and thus $$\max\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=f_1(t)=36(t-4)=72.$$ For $t\ge 7$, we have $$\max\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{33}(t)=9(t-3),$$ and thus $$\max\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=f_3(t)=9(t-3)(t-4).$$ The result thus follows. ◻ # Results for bipartite Gallai-Ramsey multiplicity We consider four kinds of rainbow graphs $P_{4}$, $P_{5}$ and $K_{1,3}$, respectively, by the following two subsections. ## For rainbow $P_{4}$ **Theorem 40**. *Let integer $k\ge 3$. If $H$ is a subgraph of $K_{1,k}$, then $$\operatorname{bgr}_k(P_4:H)=\left\lceil\sqrt{k}\right\rceil.$$* *Proof.* The lower bound follows from Lemma [Lemma 23](#basic-lower-bound-lemma){reference-type="ref" reference="basic-lower-bound-lemma"}. For the upper bound, we consider any $k$-edge-coloring of $K_{N,N} \ (N\ge \lceil\sqrt{k}\rceil)$. Let $(U,V)$ be the bipartition of $K_{N,N}$ and suppose to the contrary that $K_{N,N}$ contains neither a rainbow subgraph $P_4$ nor a monochromatic subgraph $H$. Noticing that $\lceil\sqrt{k}\rceil\le k-1$ for $k\geq 3$. If $\lceil\sqrt{k}\rceil\le N\le k-1$, then it follows from Proposition [\[k-color-have-rainbowP4-bipartite\]](#k-color-have-rainbowP4-bipartite){reference-type="ref" reference="k-color-have-rainbowP4-bipartite"} that there is always a rainbow $P_4$, and the result thus follows. Next we assume $N\ge k$. It follows from Theorem [\[th-P4-bGR-Structure\]](#th-P4-bGR-Structure){reference-type="ref" reference="th-P4-bGR-Structure"} that the Colored Structure 3 occurs. Thus $U$ can be partitioned into $k$ non-empty parts $U_1, U_2,\ldots,U_k$ such that all the edges between $U_i$ and $V$ have color $i$, $i\in\{1,2,\ldots,k\}$. Since $H$ is a subgraph of $K_{1,k}$ and $|V|=N \ge k$, it follows that there is a monochromatic $H$, a contradiction. The result thus follows. ◻ It is easy to calculate that when $t\ge 2$, there are $t^2(t-1)^2$ different $P_4$ in $K_{t,t}$. Therefore, we directly provide the following theorem. **Theorem 41**. *For integers $k$ and $t$ satisfying $k=t^2\ge 4$, and a subgraph $H$ of $K_{1,k}$ with $|E(H)|\ge 2$, we have $$\operatorname{bi-GM}_{k}(P_4:H)=t^2(t-1)^2.$$* **Theorem 42**. *For integers $k$ and $t$ satisfying $k=t^2\ge 4$, and a subgraph $H$ of $K_{1,k-1}$ with $|E(H)|\ge 3$, we have $$\operatorname{bi-GM}_{k-1}(P_4:H)=t^2(t-1)^2-2(t-1).$$* *Proof.* It follows from Theorem [Theorem 40](#k-color-P4-H-general-bGR){reference-type="ref" reference="k-color-P4-H-general-bGR"} that $\operatorname{bgr}_{k-1}(P_4:H)=t$. Consider any $(k-1)$-edge-coloring of $K_{t,t}$. Since $|E(K_{t,t})|=t^2$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_{t,t}$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. According to Theorem [Theorem 41](#THM-biGM-k-color-P4){reference-type="ref" reference="THM-biGM-k-color-P4"}, there are $t^2(t-1)^2$ different $P_4$ in $K_{t,t}$, and we only need to find the number of different $P_4$ containing the edges $e_1$ and $e_2$. This is because only $P_4$ containing edges $e_1$ and $e_2$ are not rainbow, and all other $P_4$ are rainbow. If $e_1$ and $e_2$ are not adjacent, then according to Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"}, there are $2$ different $P_4$ in $K_{t,t}$ that contain edges $e_1$ and $e_2$. If $e_1$ and $e_2$ are adjacent, then according to Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"}, there are $2(t-1)$ different $P_4$ in $K_{t,t}$ that contain edges $e_1$ and $e_2$. Noticing that $2\le 2(t-1)$ for $t\ge 2$, the result thus follows. ◻ **Theorem 43**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,k-2}$ with $|E(H)|\ge 4$, we have $$\operatorname{bi-GM}_{k-2}(P_4:H)=t^2(t-1)^2-6(t-1).$$* *Proof.* It follows from Theorem [Theorem 40](#k-color-P4-H-general-bGR){reference-type="ref" reference="k-color-P4-H-general-bGR"} that $\operatorname{bgr}_{k-2}(P_4:H)=t$. Consider $(k-2)$-edge-coloring of $K_{t,t}$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. Since each color is used at least once, there are only the following two cases. Due to the arbitrariness of colors, we can describe them using specific color names such as red and blue. Next, we calculate the number of different $P_{4}$ containing two or more edges with the same color. There are three red edges $e_1$, $e_2$ and $e_3$. The remaining edges are not red and the colors of any two remaining edges are not the same. The edges $e_1$, $e_2$ and $e_3$ form a red $3P_2$. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $6$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_3\cup P_2$. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $4+2(t-1)=2(t+1)$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{1,3}$. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $6(t-1)$ different $P_{4}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_4$. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $2+2(t-1)+2(t-2)=4(t-1)$ different $P_{4}$ containing two or more red edges. There are two red edges $e_1, e_2$ and two blue edges $e_3, e_4$. The remaining edges are not red or blue and the colors of any two remaining edges are not the same. The edges $e_1$ and $e_2$ are not adjacent, and edges $e_3$ and $e_4$ are also not adjacent. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $4$ different $P_{4}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, but edges $e_3$ and $e_4$ are not adjacent. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $2+2(t-1)=2t$ different $P_{4}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, and edges $e_3$ and $e_4$ are also adjacent. In this subcase, it follows from Lemma [Lemma 21](#Lem: Conut P4 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P4 in complete bipartite graph"} that there are $2(t-1)$ different $P_{4}$ containing two edges with the same color. Let $f_1(t)=6, f_2(t)=2(t+1), f_3(t)=6(t-1)$. Based on the data calculated from the seven subcases above, we need to compare the sizes of $f_1(t), f_2(t)$ and $f_3(t)$. For $t\ge 3$, we have $$\max\{f_1(t), f_2(t), f_3(t)\}=f_3(t)=6(t-1).$$ The result thus follows. ◻ ## For rainbow $P_{5}$ and $K_{1,3}$ **Theorem 44**. *Let integer $k\ge 5$. If $H$ is a subgraph of $K_{1,\left\lceil\frac{k-1}{2}\right\rceil}$, then $$\operatorname{bgr}_k(P_5:H)=\operatorname{bgr}_k(K_{1,3}:H)=\left\lceil\sqrt{k}\right\rceil.$$* *Proof.* The lower bound follows from Lemma [Lemma 23](#basic-lower-bound-lemma){reference-type="ref" reference="basic-lower-bound-lemma"}. For the upper bound, we consider any $k$-edge-coloring of $K_{N,N} \ (N\ge \lceil\sqrt{k}\rceil)$. Let $(U,V)$ be the bipartition of $K_{N,N}$ and suppose to the contrary that $K_{N,N}$ contains neither a rainbow subgraph $P_5$ nor a monochromatic subgraph $H$. Noticing that $\lceil\sqrt{k}\rceil\le k-2$ for $k\geq 5$. If $\lceil\sqrt{k}\rceil\le N\le k-2$, then it follows from Proposition [\[k-color-have-rainbowP5-bipartite\]](#k-color-have-rainbowP5-bipartite){reference-type="ref" reference="k-color-have-rainbowP5-bipartite"} that there is always a rainbow $P_5$, and the result thus follows. Next we assume $N\ge k-1$. It follows from Theorems [\[th-P5-bGR-Structure\]](#th-P5-bGR-Structure){reference-type="ref" reference="th-P5-bGR-Structure"} and [\[th-K13-bGR-Structure\]](#th-K13-bGR-Structure){reference-type="ref" reference="th-K13-bGR-Structure"} that either the Colored Structure 4 or Colored Structure 5 occurs. If Colored Structure 4 occurs, then $U$ can be partitioned into two parts $U_1$ and $U_2$ with $|U_1|\ge 1, |U_2|\ge 0$, and $V$ can be partitioned into $k$ parts $V_1,V_2,\ldots,V_k$ with $|V_1|\ge 0$ and $|V_j|\ge 1$, $j\in\{2,3,\ldots,k\}$. Since $N\ge k-1$, it follows from pigeonhole principle that $|U_1|\ge \left\lceil\frac{k-1}{2}\right\rceil$ or $|U_2|\ge \left\lceil\frac{k-1}{2}\right\rceil$. Without loss of generality, we assume that $|U_1|\ge \left\lceil\frac{k-1}{2}\right\rceil$. Noticing that $|V_2|\ge 1$ and all the edges between $V_2$ and $U_1$ have color $2$, there is a monochromatic $H$ with color $2$, a contradiction. If the Colored Structure 5 occurs, then $U$ can be partitioned into $k$ parts $U_1,U_2,\ldots,U_k$ with $|U_1|\ge 0, |U_j|\ge 1$ and $V$ can be partitioned into $k$ parts $V_1,V_2,\ldots,V_k$ with $|V_1|\ge 0, |V_j|\ge 1$, $j\in\{2,3,\ldots,k\}$. Noticing that $\left\lceil\frac{k-1}{2}\right\rceil< k-2$ for $k\ge 5$, $|V_2|\ge 1$ and all the edges between $V_2$ and $U_3\cup U_4\cup\ldots\cup U_k$ have color $1$, there is a monochromatic $H$ with color $1$, a contradiction. The result thus follows. ◻ It is easy to calculate that when $t\ge 3$, there are $t^2(t-1)^2(t-2)$ different $P_5$ and $2t\binom{t}{3}$ different $K_{1,3}$ in $K_{t,t}$. Therefore, we directly provide the following theorem. **Theorem 45**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,\left\lceil\frac{k-1}{2}\right\rceil}$ with $|E(H)|\ge 2$, we have $$\operatorname{bi-GM}_{k}(G:H)=\left\{ \begin{array}{ll} t^2(t-1)^2(t-2), & G=P_5;\\ 2t\binom{t}{3}, & G=K_{1,3}. \end{array}\right.$$* **Theorem 46**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,\left\lceil\frac{k-2}{2}\right\rceil}$ with $|E(H)|\ge 3$, we have $$\operatorname{bi-GM}_{k-1}(P_5:H)=t^2(t-1)^2(t-2)-3(t-1)(t-2).$$* *Proof.* It follows from Theorem [Theorem 44](#k-color-P5&K13-H-general-bGR){reference-type="ref" reference="k-color-P5&K13-H-general-bGR"} that $\operatorname{bgr}_{k-1}(P_5:H)=t$. Consider any $(k-1)$-edge-coloring of $K_{t,t}$. Since $|E(K_{t,t})|=t^2$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_{t,t}$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. According to Theorem [Theorem 45](#THM-biGM-k-color-P5&K13){reference-type="ref" reference="THM-biGM-k-color-P5&K13"}, there are $t^2(t-1)^2(t-2)$ different $P_5$ in $K_{t,t}$, and we only need to find the number of different $P_5$ containing the edges $e_1$ and $e_2$. This is because only $P_5$ containing edges $e_1$ and $e_2$ are not rainbow, and all other $P_5$ are rainbow. If $e_1$ and $e_2$ are not adjacent, then according to Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"}, there are $6(t-2)$ different $P_5$ in $K_{t,t}$ that contain edges $e_1$ and $e_2$. If $e_1$ and $e_2$ are adjacent, then according to Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"}, there are $3(t-1)(t-2)$ different $P_5$ in $K_{t,t}$ that contain edges $e_1$ and $e_2$. Noticing that $6(t-2)\le 3(t-1)(t-2)$ for $t\ge 3$, the result thus follows. ◻ **Theorem 47**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,\left\lceil\frac{k-3}{2}\right\rceil}$ with $|E(H)|\ge 4$, we have $$\operatorname{bi-GM}_{k-2}(P_5:H)=t^2(t-1)^2(t-2)-9(t-1)(t-2).$$* *Proof.* It follows from Theorem [Theorem 44](#k-color-P5&K13-H-general-bGR){reference-type="ref" reference="k-color-P5&K13-H-general-bGR"} that $\operatorname{bgr}_{k-2}(P_5:H)=t$. Consider $(k-2)$-edge-coloring of $K_{t,t}$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. Since each color is used at least once, there are only the following two cases. Due to the arbitrariness of colors, we can describe them using specific color names such as red and blue. Next, we calculate the number of different $P_5$ containing two or more edges with the same color. There are three red edges $e_1$, $e_2$ and $e_3$. The remaining edges are not red and the colors of any two remaining edges are not the same. The edges $e_1$, $e_2$ and $e_3$ form a red $3P_2$. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are $18(t-2)$ different $P_{5}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_3\cup P_2$. Let $P_3=e_1e_2$ and $P_2=e_3$. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are $3(t-1)(t-2)$ different $P_{5}$ containing red edges $e_1$ and $e_2$, $6(t-2)-2$ different $P_{5}$ only containing red edges $e_1$ and $e_3$, and symmetrically $6(t-2)-2$ different $P_{5}$ only containing red edges $e_2$ and $e_3$. So there are a total of $3(t-1)(t-2)+2(6(t-2)-2)=3(t+3)(t-2)-4$ different $P_{5}$ containing two or more red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $K_{1,3}$. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are $9(t-1)(t-2)$ different $P_{5}$ containing two red edges. The edges $e_1$, $e_2$ and $e_3$ form a red $P_4$. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are $6(t-2)$ different $P_{5}$ containing red $2P_2$ and $2(3(t-1)(t-2)-2(t-2))=2(3t-5)(t-2)$ different $P_{5}$ without red $2P_2$. So there are a total of $6(t-2)+2(3t-5)(t-2)=2(3t-2)(t-2)$ different $P_{5}$ containing two or more red edges. There are two red edges $e_1, e_2$ and two blue edges $e_3, e_4$. The remaining edges are not red or blue and the colors of any two remaining edges are not the same. The edges $e_1$ and $e_2$ are not adjacent, and edges $e_3$ and $e_4$ are also not adjacent. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are $12(t-2)$ different $P_{5}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, but edges $e_3$ and $e_4$ are not adjacent. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are at most $6(t-2)+3(t-1)(t-2)=3(t+1)(t-2)$ different $P_{5}$ containing two edges with the same color. The edges $e_1$ and $e_2$ are adjacent, and edges $e_3$ and $e_4$ are also adjacent. In this subcase, it follows from Lemma [Lemma 22](#Lem: Conut P5 in complete bipartite graph){reference-type="ref" reference="Lem: Conut P5 in complete bipartite graph"} that there are at most $6(t-1)(t-2)$ different $P_{5}$ containing two edges with the same color. Let $$f_1(t)=18(t-2), f_2(t)=3(t+3)(t-2)-4,$$ $$f_3(t)=9(t-1)(t-2), f_4(t)=2(3t-2)(t-2), f_5(t)=3(t+1)(t-2).$$ Based on the data calculated from the seven subcases above, we need to compare the sizes of $f_1(t), f_2(t), f_3(t), f_4(t)$ and $f_5(t)$. For $1\le i\le 5$, let $f_{ii}(t)=\frac{f_i(t)}{t-2}$. Then $$f_{11}(t)=18, f_{22}(t)=3(t+3)-\frac{4}{t-2}, f_{33}(t)=9(t-1), f_{44}(t)=2(3t-2), f_{55}=3(t+1).$$ For $t\ge 3$, we have $$\max\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{33}(t)=9(t-1),$$ and thus $$\max\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=f_3(t)=9(t-1)(t-2).$$ The result thus follows. ◻ **Theorem 48**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,\left\lceil\frac{k-2}{2}\right\rceil}$ with $|E(H)|\ge 3$, we have $$\operatorname{bi-GM}_{k-1}(K_{1,3}:H)=\left\{ \begin{array}{ll} 5, & t=3;\\ 30, & t=4;\\ (2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}, & t\ge 5. \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 44](#k-color-P5&K13-H-general-bGR){reference-type="ref" reference="k-color-P5&K13-H-general-bGR"} that $\operatorname{bgr}_{k-1}(K_{1,3}:H)=t$. Consider any $(k-1)$-edge-coloring of $K_{t,t}$. Since $|E(K_{t,t})|=t^2$ and each color is used at least once, it follows that there are only two edges, say $e_1$ and $e_2$, with the same color in $K_{t,t}$. Since $|E(H)|\ge 3$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. Next, we calculate the number of rainbow $K_{1,3}$ in $K_{t,t}$. If the edges $e_1$ and $e_2$ are not adjacent, then this case is equivalent to Theorem [Theorem 45](#THM-biGM-k-color-P5&K13){reference-type="ref" reference="THM-biGM-k-color-P5&K13"}. Hence there are $2t\binom{t}{3}$ rainbow $K_{1,3}$ in $K_{t,t}$. If the edges $e_1$ and $e_2$ are adjacent, then $e_1$ and $e_2$ form a monochromatic $P_3$. Without loss of generality, we assume that the edges $e_1$ and $e_2$ are red and vertex $v$ is incident with the edges $e_1$ and $e_2$. We first investigate the number of rainbow $K_{1,3}$ with center $v$ for $t\ge 5$. Noticing that $\deg(v)=t$, the number of rainbow $K_{1,3}$ with center $v$ and without red edges is $\binom{t-2}{3}$, and number of rainbow $K_{1,3}$ with center $v$ and with a red edge is $2\binom{t-2}{2}$. In $K_{t,t}$, there are $(2t-1)\binom{t}{3}$ rainbow $K_{1,3}$ with center in $V(K_{t,t})\setminus \{v\}$. Therefore, the total number of rainbow $K_{1,3}$ in $K_{t,t}$ is $(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}$. Then what we need to do is to compare the sizes of $2t\binom{t}{3}$ and $(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}$, which is equivalent to comparing the sizes of $\binom{t}{3}$ and $\binom{t-2}{3}+2\binom{t-2}{2}$. Noticing that $$\binom{t}{3}=\frac{1}{6}t^3-\frac{1}{2}t^2+\frac{1}{3}t,$$ $$\binom{t-2}{3}+2\binom{t-2}{2}=\frac{1}{6}t^3-\frac{1}{2}t^2-\frac{2}{3}t+2.$$ Thus for $t\ge 5$, $$\min\left\{2t\binom{t}{3},(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}\right\}=(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}.$$ When $t=3$, the number of rainbow $K_{1,3}$ with center $v$ and without red edges is $0$, and number of rainbow $K_{1,3}$ with center $v$ and with a red edge is $0$. In $K_{3,3}$, there are $5\binom{3}{3}=5$ rainbow $K_{1,3}$ with center in $V(K_{3,3})\setminus \{v\}$, therefore $\operatorname{bi-GM}_{8}(K_{1,3}:H)\le 5$. Since $5<6\binom{5}{3}$, it follows that $\operatorname{bi-GM}_{8}(K_{1,3}:H)=5$. When $t=4$, the number of rainbow $K_{1,3}$ with center $v$ and without red edges is $0$, and number of rainbow $K_{1,3}$ with center $v$ and with a red edge is $2$. In $K_{4,4}$, there are $7\binom{4}{3}=28$ rainbow $K_{1,3}$ with center in $V(K_{4,4})\setminus \{v\}$, and therefore $\operatorname{bi-GM}_{15}(K_{1,3}:H)\le 30$. Since $30<8\binom{4}{3}$, it follows that $\operatorname{bi-GM}_{15}(K_{1,3}:H)=30$. ◻ **Theorem 49**. *For integers $k$ and $t$ satisfying $k=t^2\ge 9$, and a subgraph $H$ of $K_{1,\left\lceil\frac{k-3}{2}\right\rceil}$ with $|E(H)|\ge 4$, we have $$\operatorname{bi-GM}_{k-2}(K_{1,3}:H)=\left\{ \begin{array}{ll} 4, & t=3;\\ 28, & t=4;\\ 93, & t=5;\\ (2t-1)\binom{t}{3}+\binom{t-3}{3}+3\binom{t-3}{2}, & t\ge 6.\\ \end{array}\right.$$* *Proof.* It follows from Theorem [Theorem 44](#k-color-P5&K13-H-general-bGR){reference-type="ref" reference="k-color-P5&K13-H-general-bGR"} that $\operatorname{bgr}_{k-2}(K_{1,3}:H)=t$. Consider $(k-2)$-edge-coloring of $K_{t,t}$. Since $|E(H)|\ge 4$, it follows that we do not need to consider the number of monochromatic $H$ in $K_{t,t}$. Noticing that each color needs to be used at least once. We first color any $k-2$ edges in $K_{t,t}$ with $k-2$ colors, and the remaining two edges are temporarily not colored, denoted as $e_1$ and $e_2$. Next, we discuss the edges $e_1$ and $e_2$ in two cases. **Case 3**. The edges $e_1$ and $e_2$ have the same color. Without loss of generality, we assume that these two edges are red. According to the structure of $K_{t,t}$, it is easy to calculate that if the red edges form a $3P_2$, then there are $$f_1(t)=2t\binom{t}{3}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the red edges form a $P_3\cup P_2$, then there are $$f_2(t)=(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the red edges form a $P_4$, then there are $$f_3(t)=(2t-2)\binom{t}{3}+2\binom{t-2}{3}+4\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the red edges form a $K_{1,3}$, then there are $$f_4(t)=(2t-1)\binom{t}{3}+\binom{t-3}{3}+3\binom{t-3}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$. **Case 4**. The edges $e_1$ and $e_2$ have different colors. When the $P_3$ formed by the edges $e_1$ and $e_2$ in $K_{t,t}$, without loss of generality, we assume that $e_1$ is red and $e_2$ is blue. Let $V(P_3)=\{u,v,w\}$ and vertex $v$ be incident with the edges $e_1$ and $e_2$. According to the structure of $K_{t,t}$, it is easy to calculate that if the other red edge is not incident with vertex $u$ or $v$, and the other blue edge is not incident with vertex $v$ or $w$, then there are $$f_1(t)=2t\binom{t}{3}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the other red edge is incident with vertex $u$ or $v$, and the other blue edge is not incident with vertex $v$ or $w$, then there are $$f_2(t)=(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the other red edge is incident with vertex $u$, and the other blue edge is incident with vertex $w$, then there are $$f_3(t)=(2t-2)\binom{t}{3}+2\binom{t-2}{3}+4\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the other red edge is incident with vertex $v$, and the other blue edge is also incident with vertex $v$, then there are $$f_5(t)=(2t-1)\binom{t}{3}+\binom{t-4}{3}+4\binom{t-4}{2}+4(t-4)$$ rainbow $K_{1,3}$ in $K_{t,t}$. When the $2P_2$ formed by the edges $e_1$ and $e_2$ in $K_{t,t}$, without loss of generality, we assume that $e_1$ is red and $e_2$ is blue. According to the structure of $K_{t,t}$, it is easy to calculate that if the other red edge is not adjacent to $e_1$, and the other blue edge is not adjacent to $e_2$, then there are $$f_1(t)=2t\binom{t}{3}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the other red edge is adjacent to $e_1$, and the other blue edge is not adjacent to $e_2$, then there are $$f_2(t)=(2t-1)\binom{t}{3}+\binom{t-2}{3}+2\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$; if the other red edge is adjacent to $e_1$, and the other blue edge is adjacent to $e_2$, then there are $$f_3(t)=(2t-2)\binom{t}{3}+2\binom{t-2}{3}+4\binom{t-2}{2}$$ rainbow $K_{1,3}$ in $K_{t,t}$. Next, we compare the sizes of $f_1(t), f_2(t), f_3(t), f_4(t)$ and $f_5(t)$. Based on the practical significance of counting in this paper, we only define in the operations of expressions for $t$ in $f_1(t), f_2(t), f_3(t), f_4(t)$ and $f_5(t)$ that when integers $a<b$, we have $\binom{a}{b}\equiv 0$ and $a-b\equiv 0$. For $t=3$, we have $$f_1(3)=6, f_2(3)=5, f_3(3)=4, f_4(3)=5, f_5(3)=5.$$ Thus, $\min\{f_1(3), f_2(3), f_3(3), f_4(3), f_5(3)\}=4.$ For $t=4$, we have $$f_1(4)=32, f_2(4)=30, f_3(4)=28, f_4(4)=28, f_5(4)=28.$$ Thus, $\min\{f_1(4), f_2(4), f_3(4), f_4(4), f_5(4)\}=28.$ For $t=5$, we have $$f_1(5)=100, f_2(5)=97, f_3(5)=94, f_4(5)=93, f_5(5)=94.$$ Thus, $\min\{f_1(5), f_2(5), f_3(5), f_4(5), f_5(5)\}=93.$ For $t=6$, we have $$f_1(6)=240, f_2(6)=236, f_3(6)=232, f_4(6)=230, f_5(6)=232.$$ Thus, $\min\{f_1(6), f_2(6), f_3(6), f_4(6), f_5(6)\}=230.$ For $t\ge 7$ and $1\le i\le 5$, let $f_{ii}(t)=f_i(t)-(2t-2)\binom{t}{3}-\frac{1}{3}t^3+t^2+\frac{1}{3}t$, then $$f_{11}(t)=t, f_{22}(t)=2, f_{33}(t)=-t+4, f_{44}(t)=-2t+8, f_{55}(t)=-t+4.$$ Therefore, when $t\ge 7$ we have $$\min\{f_{11}(t), f_{22}(t), f_{33}(t), f_{44}(t), f_{55}(t)\}=f_{44}(t)=-2t+8,$$ and thus $$\min\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=f_4(t)=(2t-1)\binom{t}{3}+\binom{t-3}{3}+3\binom{t-3}{2}.$$ Based on the above discussion, we have $$\min\{f_1(t), f_2(t), f_3(t), f_4(t), f_5(t)\}=\left\{ \begin{array}{ll} (2t-2)\binom{t}{3}, & t=3;\\ (2t-1)\binom{t}{3}, & t=4;\\ (2t-1)\binom{t}{3}+3\binom{t-3}{2}, & t=5;\\ (2t-1)\binom{t}{3}+\binom{t-3}{3}+3\binom{t-3}{2}, & t\ge 6.\\ \end{array}\right.$$ The result thus follows. ◻ 1 R. Bass, C. Magnant, K. Ozeki, B. Pyron, Characterizations of edge-colorings of complete graphs that forbid certain rainbow subgraphs, manuscript. J. A. Bondy, U. S. R. Murty, Graph Theory, *Springer GTM 244*, 2008. W. Chen, M. Ji, Y. Mao, M. Wei, Complete bipartite graphs without small rainbow stars, *Discrete Appl. Math.* 340 (2023), 14--20. R. J. Faudree, R. Gould, M. Jacobson, C. Magnant, Ramsey numbers in rainbow triangle free colorings, *Australas. J. Combin.* 46 (2010), 269--284. J. Fox, There exist graphs with super-exponential Ramsey multiplicity constant, *J. Graph Theory* 57(2) (2008), 89--98. A. Gyárfás, J. Lehel, R. H. Schelp, Z. Tuza, Ramsey numbers for local colorings, *Graphs Combin.* 3 (1987), 267--277. X. H. Li, H. Broersma, L. Wang, Extremal problems and results related to Gallai-colorings, *Discrete Math.* 344 (2021), 112567. X. H. Li, L. Wang, X. Liu, Complete graphs and complete bipartite graphs without rainbow path, *Discrete Math.* 342 (2019), 2116--2126. C. Magnant, P. S. Nowbandegani, Topics in Gallai-Ramsey Theory, *Springer*, 2020. Y. Mao, Gallai-Ramsey multiplicity. arXiv:2302.10770. S. P. Radziszowski, Small Ramsey numbers, *Electron. J. Combin.* 1 (1996). F. P. Ramsey, On a problem of formal logic, *Proc. London Math. Soc.* 30 (1930), 264--286. J. C. Schlage-Puchta, P. Wagner, Complete graphs with no rainbow tree, *J. Graph Theory* 93(2) (2020), 157--167. A. Thomason, P. Wagner, Complete graphs with no rainbow path, *J. Graph Theory* 54(3) (2007), 261--266. J. Zou, Z. Wang, H. Lai, Y. Mao, Gallai-Ramsey numbers involving a rainbow $4$-path, *Graphs Combin.* 39 (2023), 54. [^1]: Supported by NSFC No.12131013 and 12161141006.

arxiv_math

--- abstract: | Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation. author: - "Immanuel Bomze[^1]" - "Bo Peng[^2]" - "Yuzhou Qiu[^3]" - "E. Alper Yıldırım[^4]" bibliography: - references.bib date: - - September 28, 2023 title: On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints --- ***Keywords---*** Standard quadratic optimization problems, sparsity, mixed-integer quadratic optimization, reformulation-linearization technique, Shor relaxation **AMS Subject Classification:** 90C11, 90C20, 90C22 # Introduction The Standard Quadratic optimization Problem (StQP) consists of minimizing a quadratic form over the standard simplex (all vectors with no negative coordinates that sum up to one). Since no assumptions on the definiteness of quadratic form are made, this problem class is NP-hard. Indeed, the maximum-clique problem can be reduced to [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} [@motzkin_straus_1965]. Therefore, we can view the class of StQP as the simplest of the hard problems: the simplest non-convex objective functions are generated by indefinite Hessians, and the feasible set is the simplest bounded polyhedron (polytope) with a very obvious structure of faces comprised of standard simplices in lower dimensions when some variables are fixed to zero. Despite its simplicity, the class of StQPs provides a quite versatile modelling tool (see, e.g., [@Bomze98]). Applications are numerous, ranging from the famous Markowitz portfolio problem in finance, evolutionary game theory in economics and quadratic resource allocation problems, through machine learning (background--foreground clustering in image analysis), to the life sciences ----- e.g., in population genetics (selection models) and ecology (replicator dynamics). StQPs appear also quite naturally as subproblems in copositive-conic relaxations of mixed-integer or combinatorial optimization problems of all sorts. Finally, using barycentric coordinates, every quadratic optimization problem over a polytope with known (and not too many) vertices can be rephrased as an StQP. The aforementioned structural simplicity does not preclude coexistence of an exponential number of (local or global) solutions to some StQPs. Some of these solutions may be sparse (and will be so with high probability in the average case, see below), others may have many positive coordinates. However, in important applications like some variants of sparse portfolio optimization problems where one is interested in investments with a limited number of assets (see, e.g., [@MDA2019] and the references therein), sparsity of a solution must be enforced by an additional, explicit hard constraint on the number of positive coordinates. Introducing this sparsity constraint can render StQPs NP-hard even if the Hessian is positive-definite. This paper deals with such problems and investigates the structural properties of tractable linear and semidefinite relaxations which scale well with the dimension. # Background, Motivation, and Layout of Contribution In this section, we provide some background on standard quadratic optimization problems. We present our motivation for studying the variant with a hard sparsity constraint. We introduce our notation and give an outline of the paper. ## The Combinatorial Nature of Standard Quadratic Optimization -- Coexistence of Solutions and Role of Active Sets The well-studied Standard Quadratic optimization Problem (StQP) is given by $$\tag{StQP}\label{stqp} \ell(Q) := \min\limits_{x \in \mathbb{R}^n} \left\{x^\top Q x: x \in F\right\},$$ where $Q \in {\cal S}^n$ is the problem data, $x \in \mathbb{R}^n$ is the decision variable, and $F \subset \mathbb{R}^n$ denotes the standard simplex given by $$\label{def_F} F := \left\{x \in \mathbb{R}^n: e^\top x = 1, \quad x \geq 0\right\},$$ where $e \in \mathbb{R}^n$ denotes the vector of all ones. There is an exponential number, namely $2^n-1$, of faces of $F$, which form the "combinatorial" reason for NP-hardness. Indeed, if the active set $\{ i : x_i^* =0 \}$ at the global solution $x^*$ is known exactly, locating the solution (i.e., determining $x^*$ or a value-equivalent alternative with the same set of zero coordinates) reduces to solving an $n\times n$ linear equation system. The same holds true for locating local solutions and even first-order critical (KKT) points. This phenomenon may be the reason why recently iterative first-order methods were proposed, which can achieve identification of the correct active set in finite time [@Bomz19a]. For any instance of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}, not all faces of $F$ can contain an isolated (local or global) solution in their relative interior, as there is an upper bound on their cardinality given by Sperner's theorem on the maximal antichain (and Stirling's asymptotics), namely $$\label{sperstir} \binom{n}{\lfloor \frac n2\rfloor} \sim \sqrt{\frac 2{\pi n}}\, 2^n\quad\mbox{as } n\to \infty\, .$$ Scozzari and Tardella [@Scoz08] show that solutions can occur only in the relative interior of a face restricted to which the objective function is strictly convex. Nevertheless, recent research [@Bomz16a] has shown an exponential behavior regarding the number of local (or global) solutions: in the worst case, an instance of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} of order $n$ can have at least $$\label{lowbd} (15120)^{n/4} \approx (1.4933)^n$$ coexisting optimal solutions, a lower bound that currently seems to be the largest one known. The other bad news is that rounding on the standard simplex is, from the asymptotic point of view, also not always successful [@Bomz14c]. In spite of all this, [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} admits a polynomial-time approximation scheme (PTAS) [@Bomz02c]. ## Worst-case versus Average-Case Behavior -- Expected Sparsity All of the above observations refer to the worst case, of course. Several researchers turned to the average case, modelled by randomly chosen instances. Already in 1988, Kingman [@King61a] observed that very large polymorphisms (i.e., solutions $x^*$ with more than $C\sqrt n$ positive coordinates) are atypical. More recently, in a series of papers Kontogiannis and Spirakis [@Kont05; @Kont09; @Kont10] looked at models with several independent and identically distributed (e.g., Gaussian or uniform) entries of $Q \in {\cal S}^n$ and proved, among other results, that the expected number of (local) solutions does not grow faster than $\exp(0.138n)\approx (1.148)^n$, way smaller than the worst-case lower bound in [\[lowbd\]](#lowbd){reference-type="eqref" reference="lowbd"}. Based upon more recent research by Chen and coauthors [@chen2015new; @chen2013sparse], under quite reasonable distributional assumptions modeling the random average case, the probability that the global solution has more than 2 positive coordinates (i.e., that it does not lie on an edge of $F$) is asymptotically vanishing faster than $$K \frac {(\log n)^2}n\quad\mbox{with } n\to \infty\, ,$$ where $K>0$ is a universal constant [@Bomz16a Proposition 1]. ## StQPs with a Hard Sparsity Constraint However, if the instances are somehow structured, we cannot rely on our "luck" that $Q$ exhibits an average behavior in the above sense, and still, we may prefer a sparse solution to ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}). So, in pursuit of these sparse solutions, we introduce the following variant under a cardinality constraint, referred to as the *sparse StQP*: $$\ell_\rho (Q) := \min\limits_{x \in \mathbb{R}^n} \left\{x^\top Q x: x \in F_\rho\right\},$$ where $$\label{def_F_rho} F_\rho := \left\{x \in F : \quad \|x\|_0 \leq \rho\right\}\, .$$ Here, $\|x\|_0$ denotes the number of nonzero components of a vector $x$ and $\rho \in \{1,\ldots,n\}$ is the sparsity parameter. The elements of $F_\rho$ will be referred to as $\rho$-sparse. When $\rho$ is fixed independently of $n$, $F_\rho$ is the union of ${\mathfrak O} (n^\rho)$ faces of $F$, a number polynomial in $n$. In each of these faces, due to [\[sperstir\]](#sperstir){reference-type="eqref" reference="sperstir"}, at most $\binom{\rho}{\lfloor \frac \rho 2\rfloor}$ local solutions to [\[def_F\_rho\]](#def_F_rho){reference-type="eqref" reference="def_F_rho"} can coexist, so we end up with a polynomial set of candidates which makes problem [\[def_F\_rho\]](#def_F_rho){reference-type="eqref" reference="def_F_rho"} solvable in polynomial time, again for universally fixed $\rho$. However, if $\rho$ may increase with $n$, e.g. $\rho= \lfloor \frac n2\rfloor$, or even $\rho =n$, the above observations show that the sparse StQP is NP-hard even when $Q$ is positive semidefinite due to the combinatorial nature of the sparsity term $\|x\|_0$. Evidently, any (feasible or optimal) solution of the sparse StQP is a feasible solution to [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} with guaranteed $\rho$-sparsity, which can be crucial. Even if $\rho$ is fixed to a moderate number, say to 6, and for medium-scale dimensions, say $n=100$, polynomial worst-case behavior would not help much in practical optimization since $n^\rho=10^{12}$. This emphasizes the need for tractable relaxations of the sparse StQP. We start with some simple observations. **Lemma 1**. *The following relations hold: $$\label{rels1} \ell(Q) = \ell_n(Q) \leq \ell_{n-1}(Q) \leq \ldots \leq \ell_2(Q)\leq\ell_1(Q) \, ,$$ with $$\label{rho1} \ell_1(Q) = \min\limits_{1\le k \le n} Q_{kk} \, ,$$ and $$\label{rho2} \ell_2(Q) = \min\left\{\min \left\{ \textstyle{\frac{Q_{ii}Q_{jj}-Q_{ij}^2}{Q_{ii}+Q_{jj}-2Q_{ij}}}: Q_{ij} < \min\{Q_{ii},Q_{jj}\}, {1\le i<j\le n} \right \}, \, \ell_1(Q)\right\}\, .$$ Furthermore, we have $\ell(Q) = \ell_\rho(Q)$ if and only if ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) has a $\rho$-sparse optimal solution.* *Proof.* The relations [\[rels1\]](#rels1){reference-type="eqref" reference="rels1"} and [\[rho1\]](#rho1){reference-type="eqref" reference="rho1"} follow from $F_1 = \{e^1,e^2,\ldots,e^n\} \subset F_2 \subset \cdots \subset F_{n-1} \subset F_n = F$, where $F_\rho$ and $F$ are given by [\[def_F\]](#def_F){reference-type="eqref" reference="def_F"} and [\[def_F\_rho\]](#def_F_rho){reference-type="eqref" reference="def_F_rho"}, respectively. For $\rho=2$, a straightforward discussion of univariate quadratics over the edges $\mathrm{conv} \left(\left\{e^i,e^j\right\}\right),~1 \leq i < j \leq n$ (in case these are strictly convex functions yielding a minimizer in the relative interior of the edge) is sufficient to establish [\[rho2\]](#rho2){reference-type="eqref" reference="rho2"}. The last assertion is trivial. ◻ The condition $\ell(Q)=\ell_2(Q)$ is related to edge-convexity of the instance of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} as discussed in [@Scoz08 Theorem 1] but we will not dive into details here. Rather observe that the effort to calculate $\ell_2(Q)$, obviously an upper bound of $\ell(Q)$, is the same as for the closed-form lower bound $\ell^{\rm ref}(Q)\leq \ell(Q)$ proposed in [@Bomz08b]. The bracket $$\ell^{\rm ref}(Q)\leq \ell(Q) \leq \ell_2(Q)$$ shrinks to a singleton (i.e. the discussed bounds are exact) if and only if all off-diagonal entries of $Q$ are equal, in which case, an optimal solution $x^*$ to [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} must satisfy $\|x^*\|_0 \leq 2$ (see [@Bomz08b Theorem 2] and [\[rho2\]](#rho2){reference-type="eqref" reference="rho2"}). ## Mixed-Binary Quadratic Formulation of Sparse StQPs and Contributions By introducing binary variables, the sparse StQP can be reformulated as a mixed-binary QP: $$\begin{array}{llrcl} \tag{StQP($\rho$)}\label{sstqp} & \ell_\rho (Q) =\min\limits_{x \in \mathbb{R}^n} & x^\top Q x & & \\ & \textrm{s.t.} & & & \\ & & e^\top x & = & 1 \\ & & e^\top u & = & \rho \\ & & x & \leq & u \\ & & u & \in & \{0,1\}^n \\ & & x & \geq & 0. \end{array}$$ In this paper, we focus on various convex relaxations of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"}, all more tractable than the conic ones presented in [@Bomz23a Section 3.2] for general quadratic optimization problems. In particular, we establish several structural properties of these relaxations and shed light on the relations between each relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} and the corresponding relaxation of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}. We then draw several conclusions about the relations between different relaxations as well as the strength of each relaxation. While it turns out that all relaxations behave as expected for the case of $\rho = n$, already for the cases $\rho=1$ and $\rho=2$ (which cannot be excluded with a high probability in the random average case models) and other moderate sparsity values, there is a sharp contrast between the relaxations, which contributes to the motivation of this study. Typically, applications would require models with sparsity (significantly) less than half of the dimension, for which we obtain more interesting results. We will also pay particular attention to the case of rank-one solutions to the relaxations (all of them use matrix variables by lifting), in particular, because they certify optimality if optimal to the relaxed problems, and also because in algorithmic frameworks, we may (warm-)start with some (good) feasible solutions to the original problem of larger sparsity than desired. ## Notation and Organisation of the Paper We use $\mathbb{R}^n$, $\mathbb{R}^n_+$, $\mathbb{R}^{m \times n}$, and ${\cal S}^n$ to denote the $n$-dimensional Euclidean space, the nonnegative orthant, the set of $m \times n$ real matrices, and the space of $n \times n$ real symmetric matrices, respectively. We use 0 to denote the real number 0, the vector of all zeroes, as well as the matrix of all zeroes, which should always be clear from the context. We denote by $e \in \mathbb{R}^n$ and $e^i \in \mathbb{R}^n,~1 \leq i \leq n$, the vector of all ones and the $i$th unit vector, respectively. All inequalities on vectors or matrices are understood to be applied componentwise. For $A \in {\cal S}^n$ and $B \in {\cal S}^n$, we use $A \succeq B$ to denote that $A - B$ is positive semidefinite. For $x \in \mathbb{R}^n$ and an index set $\mathbf{K} \subseteq \{1,\ldots,n\}$, we denote by $x_{\mathbf{K}} \in \mathbb{R}^{\lvert\mathbf{K}\rvert}$ the subvector of $x$ restricted to the indices in $\mathbf{K}$, where $\lvert\cdot\rvert$ denotes the cardinality of a finite set. For singleton index sets, we simply use $x_j$ and $A_{ij}$ to denote the components of $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. For $B \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$, we denote by $\textrm{diag}(B) \in \mathbb{R}^n$ and $\textrm{Diag}(b) \in {\cal S}^n$ the vector given by the diagonal entries of $B$ and the diagonal matrix whose diagonal entries are given by $b$, respectively. The convex hull of a set is denoted by $\textrm{conv}(\cdot)$. For any $u \in \mathbb{R}^n$ and $v \in \mathbb{R}^n$, $u^\top v$ denotes the Euclidean inner product. Similarly, for any $U \in \mathbb{R}^{m \times n}$ and $V \in \mathbb{R}^{m \times n}$, the trace inner product is denoted by $$\langle U, V \rangle = \textrm{trace}(U^T V) = \sum\limits_{i=1}^m \sum\limits_{j = 1}^n U_{ij} V_{ij}.$$ The paper is organized as follows. In Section [3](#Sec2){reference-type="ref" reference="Sec2"}, we consider several convex relaxations of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}). Section [3.1](#RLT-Relaxation){reference-type="ref" reference="RLT-Relaxation"} focuses on the RLT (reformulation-linearization technique) relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} and presents several results in comparison with the RLT relaxation of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}). The Shor relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} is treated in Section [3.2](#SDP-Relaxation){reference-type="ref" reference="SDP-Relaxation"} and compared with that of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}. In Section [4](#SDP-RLT-Relaxation){reference-type="ref" reference="SDP-RLT-Relaxation"}, we then study the convex relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} given by combining the RLT and Shor relaxations and compare it with that of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}. We conclude the paper in Section [5](#conc){reference-type="ref" reference="conc"}. # Convex Relaxations: RLT and Shor {#Sec2} In this section, we consider several well-known convex relaxations of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}), which use LP (linear programming) and SDP (semidefinite programming) methods. We study their properties and establish relations between each relaxation of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}) and the corresponding relaxation of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}. ## RLT Relaxation {#RLT-Relaxation} In this section, we consider the RLT (reformulation-linearization technique) relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} and compare it with the RLT relaxation of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}). RLT relaxations of optimization problems with a quadratic objective function and a mix of linear and quadratic constraints are obtained by a two-stage process (see, e.g., [@Sherali1999]). The first stage, referred to as reformulation, consists of generating (additional) valid quadratic constraints from linear constraints by multiplying each pair of linear inequality constraints as well as each linear equality constraint by each variable. In the second stage, referred to as linearization, all of the original and additional quadratic functions are linearized by replacing the quadratic terms $x_i x_j$ by a lifted variable $X_{ij},~1 \leq i \leq j \leq n$. Together with the original linear constraints, this gives rise to the RLT relaxation. We first start with the RLT relaxation of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}): $$\tag{R1}\label{R1} \quad \ell^{R1}(Q) := \min\limits_{x \in \mathbb{R}^n, X \in {\cal S}^n} \left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R1}\right\},$$ where $$\label{def_F_R1} {\cal F}^{R1} := \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \quad X e = x, \quad x \geq 0, \quad X \geq 0\right\}.$$ Note that $x \geq 0$ is a redundant constraint in ${\cal F}^{R1}$ since it is implied by $X e = x$ and $X \geq 0$. Furthermore, it is easy to see that ${\cal F}^{R1}$ is a polytope. We first recall the following result about [\[R1\]](#R1){reference-type="eqref" reference="R1"}. **Proposition 1** (Qiu and Yıldırım (2023) [@QiuY23a]). *The set of vertices of ${\cal F}^{R1}$ is given by $$\label{rlt-vertices} \left\{(e^i, e^i (e^i)^\top): i = 1,\ldots,n\right\} \cup \left\{\left(\frac{1}{2}(e^i+e^j), \frac{1}{2}(e^i (e^j)^\top+ e^j (e^i)^\top)\right): 1 \leq i < j \leq n\right\}.$$ Therefore, $$\ell^{R1} (Q) = \min\limits_{1\leq i \leq j \leq n} Q_{ij} \leq \ell(Q).$$ Furthermore, [\[R1\]](#R1){reference-type="eqref" reference="R1"} is exact (i.e., $\ell^{R1} (Q) = \ell(Q)$) if and only if $$\min\limits_{1\leq i \leq j \leq n} Q_{ij} = \min\limits_{1\le k \le n} Q_{kk}.$$* Proposition [Proposition 1](#rlt-closed-form){reference-type="ref" reference="rlt-closed-form"} implies that [\[R1\]](#R1){reference-type="eqref" reference="R1"} is exact if and only if the minimum entry of $Q$ is on the diagonal. In this case, [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} has a 1-sparse optimal solution, i.e., the optimal solution of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} without any sparsity constraint is already the sparsest possible solution. Furthermore, by Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}, we immediately obtain $$\label{min-diag} \ell(Q) = \ell_n(Q) = \ell_{n-1}(Q) = \ldots = \ell_1(Q) = \min\limits_{1\le k \le n} Q_{kk}.$$ By reformulating the binarity constraint $u_j \in \{0,1\}$ with $u_j^2 = u_j,~j = 1,\ldots,n$ in ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}), we obtain the following RLT relaxation: $$\begin{array}{llrcl} \tag{R1($\rho$)}\label{R1rho} & \ell^{R1}_\rho (Q) := \min\limits_{x \in \mathbb{R}^n, u \in \mathbb{R}^n, X\in {\cal S}^n,U \in {\cal S}^n, R \in \mathbb{R}^{n \times n}} & \langle Q,X \rangle & & \\ & \textrm{s.t.} & & & \\ & & e^\top x &= &1\\ & & e^\top u &=& \rho \\ & & x & \leq & u \\ & & x & \geq & 0 \\ & & \textrm{diag} (U) & = & u\\ & & Xe & = & x \\ & & R^\top e & = & u \\ & & Re & = & \rho \, x \\ & & Ue & = & \rho \, u \\ & & X - R^\top- R + U & \geq & 0 \\ & & X - R^\top& \leq & 0 \\ & & R - U & \leq & 0 \\ & & X , R, U & \geq & 0. \\ \end{array}$$ Before we continue, let us remark that the constraints $x \leq u$ and $x \geq 0$ are redundant in (R1($\rho$)) since they are implied by the constraints $Xe = x$, $X \geq 0$, $R^\top e = u$, and $X - R^\top\leq 0$. Likewise, they imply $u\geq 0$ and $R \geq 0$. Furthermore, it is easy to verify that $R - U \leq 0$ and $U \geq 0$ are implied by the constraints $X - R^\top\leq 0$ and $X - R^\top- R + U\geq 0$. Note that $u\le e$ is not implied in this formulation. Let us denote the projection of the feasible region of [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} onto $(x,X)$ by $$\label{def_F_R1_rho} {\cal F}^{R1}_\rho := \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: (x,u,X,U,R) \textrm{ is \eqref{R1rho}-feasible for some } (u,U,R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}\right\}\, .$$ Note that $$\label{ell_rho_R1_alt} \ell^{R1}_\rho (Q) = \min\limits_{(x,X) \in \mathbb{R}^n \times {\cal S}^n}\left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R1}_\rho\right\}.$$ Clearly, we have ${\cal F}^{R1}_\rho \subseteq {\cal F}^{R1}$ if $1\le\rho \le n$, where ${\cal F}^{R1}$ is given by [\[def_F\_R1\]](#def_F_R1){reference-type="eqref" reference="def_F_R1"}. Our next result gives a description of ${\cal F}^{R1}_\rho$ in closed form for each $\rho \in \{1,\ldots,n\}$. **Lemma 2**. 1. *${\cal F}^{R1}_1 = \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \quad X = \textrm{Diag}(x), \quad x \geq 0 \right\}$.* 2. *For each $\rho \in \{2,3,\ldots,n\}$, we have ${\cal F}^{R1}_\rho = {\cal F}^{R1}$, where ${\cal F}^{R1}$ is given by [\[def_F\_R1\]](#def_F_R1){reference-type="eqref" reference="def_F_R1"}.* *Proof.* (i) Let $( x, X) \in {\cal F}^{R1}_1$. Then $e^\top x=1$ and $x\geq 0$. Moreover, there exists $( u, U, R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}$ such that $( x, u, X, U, R)$ is [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"}-feasible with $\rho=1$. Since $U \geq 0, ~\textrm{diag}( U) = u$, and $U e = u$, we obtain $U = \textrm{Diag}( u)$. By $R - U \leq 0$ and $R \geq 0$, we obtain that $R$ is a diagonal matrix. Similarly, using $X - R^\top\leq 0$, we conclude that $X$ is a diagonal matrix. Since $X e = x$, we obtain that $X = \textrm{Diag}( x)$. Conversely, if $e^\top x = 1$, $X = \textrm{Diag}( x)$, and $x \geq 0$, then it is easy to verify that $( x, u, X, U, R) = ( x, x, X, X, X)$ is [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"}-feasible. It follows that $( x, X) \in {\cal F}^{R1}_1$.\ (ii) Let $\rho \in \{2,3,\ldots,n\}$. We clearly have ${\cal F}^{R1}_\rho \subseteq {\cal F}^{R1}$. Evidently, ${\cal F}^{R1}$ is a bounded polyhedron/polytope, so for the reverse inclusion, it suffices to show that each vertex of ${\cal F}^{R1}$ belongs to ${\cal F}^{R1}_\rho$. By Proposition [Proposition 1](#rlt-closed-form){reference-type="ref" reference="rlt-closed-form"}, the set of vertices of ${\cal F}^{R1}$ is given by [\[rlt-vertices\]](#rlt-vertices){reference-type="eqref" reference="rlt-vertices"}. If $( x, X) = (e^i, e^i (e^i)^\top)$ for some $i = 1,\ldots,n$, then choose an arbitrary $u \in \{0,1\}^n$ such that $u_i = 1$ and $e^\top u = \rho$. If, on the other hand, $( x, X) = (\frac{1}{2}(e^i + e^j), \frac{1}{2}(e^i (e^j)^\top+e^j (e^i)^\top))$ for some $1 \leq i < j \leq n$, then choose an arbitrary $u \in \{0,1\}^n$ such that $u_i = 1$, $u_j = 1$, and $e^\top u = \rho$. In both cases then, define $R = x u^\top$ and $U = u u^\top$. It is easy to verify that $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}$ is [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"}-feasible, which implies that each vertex of ${\cal F}^{R1}$ belongs to ${\cal F}^{R1}_\rho$. We conclude that ${\cal F}^{R1}_\rho = {\cal F}^{R1}$. ◻ Lemma [Lemma 2](#F_R1_rho_desc){reference-type="ref" reference="F_R1_rho_desc"} immediately gives rise to the following results. **Corollary 1**. 1. *For $\rho = 1$, [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact, i.e., $\ell^{R1}_1 (Q) = \ell_1(Q)$.* 2. *For each $\rho \in \{2,3,\ldots,n\}$, we have $\ell^{R1} (Q) = \ell^{R1}_\rho (Q) = \min\limits_{1\leq i \leq j \leq n} Q_{ij}$.* *Proof.* Both assertions follow from Lemma [Lemma 2](#F_R1_rho_desc){reference-type="ref" reference="F_R1_rho_desc"}, Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}, and [\[ell_rho_R1_alt\]](#ell_rho_R1_alt){reference-type="eqref" reference="ell_rho_R1_alt"}. ◻ We arrive at the following exactness result for the classical RLT relaxation of sparse StQPs: **Theorem 1**. *[\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact (i.e., $\ell^{R1}_\rho (Q) = \ell_\rho(Q)$) if and only if $\rho = 1$ or $\min\limits_{1\leq i \leq j \leq n} Q_{ij} = \min\limits_{1\le k \le n} Q_{kk}$.* *Proof.* By Corollary [Corollary 1](#rlt-all-results){reference-type="ref" reference="rlt-all-results"}(i), [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact for $\rho = 1$. Let $\rho \in \{2,3,\ldots,n\}$. If [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact, then Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"} and Corollary [Corollary 1](#rlt-all-results){reference-type="ref" reference="rlt-all-results"}(ii) imply that $\ell^{R1}_\rho (Q) = \min\limits_{1\leq i \leq j \leq n} Q_{ij} = \ell^{R1} (Q) \leq \ell(Q) \leq \ell_\rho (Q) = \ell^{R1}_\rho (Q) = \ell^{R1} (Q)$. The claim follows from Proposition [Proposition 1](#rlt-closed-form){reference-type="ref" reference="rlt-closed-form"}. Conversely, if $\min\limits_{1\leq i \leq j \leq n} Q_{ij} = \min\limits_{1\le k \le n} Q_{kk}$, then $\ell^{R1}_\rho (Q) = \ell^{R1} (Q) = \min\limits_{1\leq i \leq j \leq n} Q_{ij} = \min\limits_{1\le k \le n} Q_{kk} = \ell(Q) = \ell_\rho (Q)$ by Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}, Corollary [Corollary 1](#rlt-all-results){reference-type="ref" reference="rlt-all-results"}(ii), and Proposition [Proposition 1](#rlt-closed-form){reference-type="ref" reference="rlt-closed-form"}. Therefore, [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact. ◻ By Theorem [Theorem 1](#exact-rlt){reference-type="ref" reference="exact-rlt"}, [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is exact if and only if $\rho = 1$ or ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) itself already has a 1-sparse optimal solution. Otherwise, in view of Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"} and the relation $\ell^{R1} (Q) \leq \ell(Q)$, it follows from Corollary [Corollary 1](#rlt-all-results){reference-type="ref" reference="rlt-all-results"} that, for each $\rho \geq 2$, the lower bound $\ell^{R1}_\rho (Q)$ arising from [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} is, in general, quite weak as it already agrees with the lower bound $\ell^{R1} (Q)$ obtained from the RLT relaxation [\[R1\]](#R1){reference-type="eqref" reference="R1"} of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}). ## SDP Relaxation {#SDP-Relaxation} In this section, we consider the standard Shor relaxation of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}) in relation to that of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"}. The Shor relaxation of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) is given by $$\tag{R2} \label{R2} \quad \ell^{R2}(Q) := \inf\limits_{x \in \mathbb{R}^n, X \in {\cal S}^n} \left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R2}\right\},$$ where $$\label{def_F_R2} {\cal F}^{R2} := \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \quad x \geq 0, \quad X \succeq x x^\top\right\}\, ,$$ a closed convex set not necessarily bounded, which necessitates the use of '$\inf$' in [\[R2\]](#R2){reference-type="eqref" reference="R2"}. Indeed, we have the following well-known result about (R2); we include a short proof for the sake of completeness. **Lemma 3**. *If $Q \succeq 0$, then (R2) is exact (i.e., $\ell^{R2}(Q) = \ell (Q)$). If $Q \not \succeq 0$, then $\ell^{R2}(Q) = -\infty$.* *Proof.* If $Q \succeq 0$, then, for any (R2)-feasible solution $( x, X) \in \mathbb{R}^n \times {\cal S}^n$, we have $\langle Q, X \rangle \geq x^\top Q x$ since $X \succeq x x^\top$, which implies that $\ell(Q) \geq \ell^{R2}(Q) \geq \ell (Q)$. If $Q \not \succeq 0$, then there exists $d \in \mathbb{R}^n$ such that $d^\top Q d < 0$. Let $x \in \mathbb{R}^n$ be any feasible solution of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) and let $X (\lambda) = x x^\top+ \lambda d d^\top$, where $\lambda \geq 0$. The assertion follows by observing that $( x, X (\lambda)) \in {\cal F}^{R2}$ for each $\lambda \geq 0$ and that the objective function of (R2) evaluated at $( x, X (\lambda))$ tends to $-\infty$ as $\lambda \to \infty$. ◻ The Shor relaxation of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}) is given by $$\begin{array}{llrcl} \tag{R2($\rho$)}\label{R2rho} & \ell^{R2}_\rho(Q) := \min\limits_{x \in \mathbb{R}^n, u \in \mathbb{R}^n, X \in {\cal S}^n, U \in {\cal S}^n, R \in \mathbb{R}^{n\times n}} & \langle Q,X \rangle & & \\ & \textrm{s.t.} & & & \\ & & e^\top x & = &1\\ & & e^\top u & = & \rho\\ & & \textrm{diag}(U) & = & u\\ & & x & \leq & u \\ & & x & \geq & 0 \\ & & \begin{bmatrix} 1 & x^\top& u^\top\\ x & X & R \\ u & R^\top& U \\ \end{bmatrix} & \succeq & 0. \end{array}$$ Note that the constraint $u \leq e$ is implied by $\textrm{diag}(U) = u$ and the semidefiniteness constraint. Similar to the RLT relaxation of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"}, let us introduce the following projection of the feasible region of [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"} onto $(x,X)$: $$\label{def_F_R2_rho} {\cal F}^{R2}_\rho := \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: (x,u,X,U,R) \textrm{ is \eqref{R2rho}-feasible for some } (u,U,R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}\right\}\, .$$ We again observe that $$\label{ell_rho_R2_alt} \ell^{R2}_\rho (Q) = \min\limits_{(x,X) \in \mathbb{R}^n \times {\cal S}^n}\left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R2}_\rho\right\}.$$ Our next result gives a complete description of ${\cal F}^{R2}_\rho$ for each $\rho = 1,2,\ldots,n$. **Lemma 4**. *For each $\rho \in \{1,2,\ldots,n\}$, we have ${\cal F}^{R2}_\rho = {\cal F}^{R2}$, where ${\cal F}^{R2}$ is given by [\[def_F\_R2\]](#def_F_R2){reference-type="eqref" reference="def_F_R2"}.* *Proof.* We clearly have ${\cal F}^{R2}_\rho \subseteq {\cal F}^{R2}$. For the reverse inclusion, let $( x, X) \in {\cal F}^{R2}$ so that $e^\top x=1$ and $x\geq 0$, so also $x\leq e$. Furthermore $X = x x^\top+ M$ for some $M \succeq 0$. Define $u = x + \left( \frac{\rho - 1}{n-1}\right) (e - x)$ so that $e^\top u = \rho$ and $0 \leq x \leq u \leq e$. Let $R = x u^\top$ and $U = u u^\top+ D$, where $D \in {\cal S}^n$ is a diagonal matrix such that $D_{jj} = u_j - ( u_j)^2 \geq 0,~j = 1,\ldots,n$. Note that $\textrm{diag}( U) = u$ and $$\begin{bmatrix} X & R \\ R^\top& U \end{bmatrix} - \begin{bmatrix} x \\ u \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix}^\top= \begin{bmatrix} M & 0 \\ 0 & D \end{bmatrix} \succeq 0.$$ By Schur complementation, it follows that $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R2($\rho$))-feasible. Therefore $( x, X) \in {\cal F}^{R2}_\rho$. ◻ Lemma [Lemma 4](#F_R2_rho_desc){reference-type="ref" reference="F_R2_rho_desc"} reveals that none of the feasible solutions of ${\cal F}^{R2}$ is cut off in the projection of the feasible region of [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"} for any choice of $\rho \in \{1,2,\ldots,n\}$. In view of [\[ell_rho_R2_alt\]](#ell_rho_R2_alt){reference-type="eqref" reference="ell_rho_R2_alt"}, we obtain the following corollary. **Corollary 2**. *For any $\rho \in \{1,\ldots,n\}$, we have $\ell^{R2}_\rho(Q) = \ell^{R2}(Q)$.* *Proof.* The assertion follows from [\[R2\]](#R2){reference-type="eqref" reference="R2"}, [\[ell_rho_R2_alt\]](#ell_rho_R2_alt){reference-type="eqref" reference="ell_rho_R2_alt"}, and Lemma [Lemma 4](#F_R2_rho_desc){reference-type="ref" reference="F_R2_rho_desc"}. ◻ Now we obtain the following exactness result for the Shor relaxation of the sparse StQP: **Theorem 2**. *[\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"} is exact (i.e., $\ell^{R2}_\rho(Q) = \ell_\rho(Q)$) if and only if $Q \succeq 0$ and ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) has a $\rho$-sparse optimal solution.* *Proof.* The assertion follows from Lemma [Lemma 3](#shor-stqp){reference-type="ref" reference="shor-stqp"}, Corollary [Corollary 2](#shor-sstqp){reference-type="ref" reference="shor-sstqp"}, and Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}. ◻ Theorem [Theorem 2](#exact-shor-sstqp){reference-type="ref" reference="exact-shor-sstqp"} shows that [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"} provides a finite lower bound if and only if $Q \succeq 0$. Furthermore, in this case, the bound is tight if and only if the problem [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} without any sparsity constraint already has a $\rho$-sparse optimal solution. It follows that [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"}, in general, is a weak relaxation. We close this section by specializing Theorem [Theorem 2](#exact-shor-sstqp){reference-type="ref" reference="exact-shor-sstqp"} to the particular case with a rank-one $Q \in {\cal S}^n$. **Corollary 3**. *Let $Q = vv^\top$, where $v \in \mathbb{R}^n$. If $v \in \mathbb{R}^n_+$ or $-v \in \mathbb{R}^n_+$ or $v_i = 0$ for some $i \in \{1,\ldots,n\}$, then $\ell^{R2}_\rho(Q) = \ell_\rho(Q)$ for each $\rho\in \{1,\ldots,n\}$. Otherwise, $\ell^{R2}_1(Q) < \ell_1(Q)$ and $\ell^{R2}_\rho(Q) = \ell_\rho(Q)$ for each $\rho \in \{ 2,\ldots,n\}$.* *Proof.* Let $Q = vv^\top$, where $v \in \mathbb{R}^n$. Note that $x^\top Q x = (v^\top x)^2 \geq 0$ for each $x \in \mathbb{R}^n$. If $v \in \mathbb{R}^n_+$ (resp., $-v \in \mathbb{R}^n_+$ or $v_i = 0$ for some $i \in \{1,\ldots,n\}$), then ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) has a 1-sparse optimal solution given by $e^j \in \mathbb{R}^n$, where $j = \arg \min\limits_{1\leq i \leq n} v_i$ (resp., $j = \arg \min\limits_{1\leq i \leq n}(-v_i)$ or $i = j$). The assertion follows from Theorem [Theorem 2](#exact-shor-sstqp){reference-type="ref" reference="exact-shor-sstqp"}. Otherwise, there exist $i \in \{1,\ldots,n\}$ and $j \in \{1,\ldots,n\}$ such that $v_i < 0 < v_j$. Therefore, setting $x= \frac{v_j}{v_j-v_i}\, e^i - \frac{v_i}{v_j-v_i}\, e^j$, we obtain $x \in F$ and $x^\top Q x = (v^\top x)^2 = 0 = \ell(Q)$. On the other hand $\ell_1 (Q) = \min\limits_{1\le k \le n}Q_{kk} = \min\limits_{1\le k \le n}v_{k}^2 > 0$ by Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}. ◻ A comparison of Corollary [Corollary 3](#rank-one-shor){reference-type="ref" reference="rank-one-shor"} and Theorem [Theorem 1](#exact-rlt){reference-type="ref" reference="exact-rlt"} reveals that the Shor relaxation [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"} can be strictly weaker than the RLT relaxation [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} for $\rho = 1$, even when $Q \succeq 0$. # SDP-RLT Relaxation {#SDP-RLT-Relaxation} In this section, we consider the SDP-RLT relaxations of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}) and [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} obtained by combining the corresponding RLT relaxations and SDP relaxations presented in Section [3.1](#RLT-Relaxation){reference-type="ref" reference="RLT-Relaxation"} and Section [3.2](#SDP-Relaxation){reference-type="ref" reference="SDP-Relaxation"}, respectively. In particular, our objective is to shed light on the properties of the combined relaxation in relation to those of the two individual relaxations. The SDP-RLT relaxation of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) is given by $$\tag{R3} \label{R3} \quad \ell^{R3}(Q) := \min\limits_{x \in \mathbb{R}^n, X \in {\cal S}^n} \left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R3}\right\},$$ where $$\label{def_F_R3} {\cal F}^{R3} := \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \quad X e = x, \quad x \geq 0, \quad X \geq 0, \quad X \succeq x x^\top\right\}.$$ A complete description of instances of [\[stqp\]](#stqp){reference-type="eqref" reference="stqp"} that admit exact SDP-RLT relaxations is given below. **Theorem 3** (Gökmen and Yıldırım [@Goek22]). *[\[R3\]](#R3){reference-type="eqref" reference="R3"} is exact (i.e., $\ell^{R3}(Q) = \ell(Q)$) if and only if (i) $n \leq 4$; or (ii) $n \geq 5$ and there exist $x \in F$, $P \succeq 0$, $N \in {\cal S}^n$, $N \geq 0$, $\lambda \in \mathbb{R}$ such that $P x = 0$, $x^\top N x = 0$, and $Q = P + N + \lambda E$. Furthermore, for any such decomposition, $x \in F$ is an optimal solution of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) and $\ell^{R3}(Q) = \ell(Q) = \lambda$.* We next consider the SDP-RLT relaxation of ([\[sstqp\]](#sstqp){reference-type="ref" reference="sstqp"}): $$\begin{array}{llrcl} \tag{R3($\rho$)} \label{R3rho} &\ell^{R3}_\rho(Q) := \min\limits_{x \in \mathbb{R}^n, u \in \mathbb{R}^n, X\in {\cal S}^n,U \in {\cal S}^n, R \in \mathbb{R}^n} & \langle Q,X \rangle & & \\ & \textrm{s.t.} & & & \\ & & e^\top x &= &1\\ & & e^\top u &=& \rho \\ & & x & \leq & u \\ & & x & \geq & 0 \\ & & \textrm{diag}(U) & = & u \\ & & Xe & = & x \\ & & R^\top e & = & u \\ & & Re & = & \rho \, x \\ & & Ue & = & \rho \, u \\ & & X - R^\top- R + U & \geq & 0 \\ & & X - R^\top& \leq & 0 \\ & & R - U & \leq & 0 \\ & & X, R, U & \geq & 0 \\ & & \begin{bmatrix}1 & x^\top& u^\top\\ x& X&R \\ u&R^\top&U\end{bmatrix}& \succeq & 0. \\ \end{array}$$ Similar to the RLT and SDP relaxations, consider the projection of the feasible region of [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} onto $(x,X)$ given by $$\label{def_F_R3_rho} {\cal F}^{R3}_\rho = \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: (x,u,X,U,R) \textrm{ is \eqref{R3rho}-feasible for some }(u,U,R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n} \right\}.$$ In a similar manner as above, we have $$\label{ell_rho_R3_alt} \ell^{R3}_\rho (Q) = \min\limits_{(x,X) \in \mathbb{R}^n \times {\cal S}^n}\left\{\langle Q, X \rangle: (x,X) \in {\cal F}^{R3}_\rho\right\}\, .$$ It is also easy to see that $$\label{proj_rels_1} {\cal F}^{R3}_\rho \subseteq {\cal F}^{R1}_\rho \cap {\cal F}^{R2}_\rho\, ,$$ where ${\cal F}^{R1}_\rho$ and ${\cal F}^{R2}_\rho$ are given by [\[def_F\_R1_rho\]](#def_F_R1_rho){reference-type="eqref" reference="def_F_R1_rho"} and [\[def_F\_R2_rho\]](#def_F_R2_rho){reference-type="eqref" reference="def_F_R2_rho"}, respectively. Therefore, $$\label{rels_R1_R2_R3} \max\{\ell^{R1}_\rho(Q), \ell^{R2}_\rho(Q)\} \leq \ell^{R3}_\rho(Q) \leq \ell_\rho(Q) \quad \textrm{for all }\rho \in \{ 1,\ldots,n\}\, ,$$ which implies that [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} is at least as tight as each of [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} and [\[R2rho\]](#R2rho){reference-type="eqref" reference="R2rho"}. Our first result follows from the previous results on weaker relaxations. **Corollary 4**. *If (i) $\rho = 1$, or (ii) $\min\limits_{1\leq i \leq j \leq n} Q_{ij} = \min\limits_{1\le k \le n} Q_{kk}$, or (iii) $Q \succeq 0$ and ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) has a $\rho$-sparse optimal solution, then [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} is exact.* *Proof.* The assertion follows from Theorem [Theorem 1](#exact-rlt){reference-type="ref" reference="exact-rlt"}, Theorem [Theorem 2](#exact-shor-sstqp){reference-type="ref" reference="exact-shor-sstqp"}, and [\[rels_R1_R2_R3\]](#rels_R1_R2_R3){reference-type="eqref" reference="rels_R1_R2_R3"}. ◻ ## Projected Feasible Sets and Their Inner Approximations We now focus on the sets ${\cal F}^{R3}_\rho$, $\rho \in \{ 1,\ldots,n\}$. By Lemma [Lemma 2](#F_R1_rho_desc){reference-type="ref" reference="F_R1_rho_desc"}, Lemma [Lemma 4](#F_R2_rho_desc){reference-type="ref" reference="F_R2_rho_desc"}, [\[def_F\_R3\]](#def_F_R3){reference-type="eqref" reference="def_F_R3"}, and [\[proj_rels_1\]](#proj_rels_1){reference-type="eqref" reference="proj_rels_1"}, $$\begin{aligned} \label{proj_rels_2} {\cal F}^{R3}_1 &\subseteq &\left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \, X = \textrm{Diag}(x), \, X \succeq x x^\top, \, x \geq 0\right\} \subseteq {\cal F}^{R3}, \label{proj_rels_2a}\\ {\cal F}^{R3}_\rho &\subseteq &\left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \, Xe = x, \, X \succeq x x^\top, \, X \geq 0, \,x \geq 0\right\} = {\cal F}^{R3}, \;\rho \geq 2\,. \label{proj_rels_2b}\end{aligned}$$ Next, we consider inner approximations of the sets ${\cal F}^{R3}_\rho$, where $\rho \in \{1,2,\dots,n\}$. **Proposition 2**. *For any fixed $\rho\in \{ 1,\ldots,n\}$, consider the corresponding formulation [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"}. Then, we have $$\label{proj_rels_4} \textrm{conv}\left\{(x,x x^\top): x \in F_\rho\right\} \subseteq {\cal F}^{R3}_\rho \, ,$$ where $F_\rho$ and ${\cal F}^{R3}_\rho$ are given by [\[def_F\_rho\]](#def_F_rho){reference-type="eqref" reference="def_F_rho"} and [\[def_F\_R3_rho\]](#def_F_R3_rho){reference-type="eqref" reference="def_F_R3_rho"}, respectively.* *Proof.* For any [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"}-feasible solution $( x, u) \in \mathbb{R}^n \times \mathbb{R}^n$, we define $X = x x^\top$, $R = x u^\top$, and $U = u u^\top$. Then obviously $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times\mathbb{R}^{n \times n}$ is [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"}-feasible. The claim now follows by [\[def_F\_R3_rho\]](#def_F_R3_rho){reference-type="eqref" reference="def_F_R3_rho"} and the convexity of ${\cal F}^{R3}_\rho$. ◻ In the remainder of this section, we identify further properties of the sets ${\cal F}^{R3}_\rho$, where $\rho \in \{1,2,\dots,n\}$ and their implications on the tightness of the lower bound $\ell^{R3}_\rho(Q)$. ## The Extremely Sparse Case $\rho=1$ In this section, we give an exact description of the set ${\cal F}^{R3}_1$ and discuss its implications. We start with a technical lemma. **Lemma 5**. *For any $a \in \mathbb{R}^n_+$ such that $e^\top a \leq 1$, we have $\textrm{Diag}(a) - aa^\top\succeq 0$.* *Proof.* Let $a_\mathbf{P}$ denote the subvector of $a$ with strictly positive components. Note that $\textrm{Diag}(a) - aa^\top\succeq 0$ if and only if $\textrm{Diag}(a_\mathbf{P}) - a_\mathbf{P}a_\mathbf{P}^\top\succeq 0$. Therefore, without loss of generality, we may and do assume that $a = a_\mathbf{P}$. We have $\textrm{Diag}(a) - aa^\top\succeq 0$ if and only if $\textrm{Diag}(\sqrt{a})^{-1} (\textrm{Diag}(a) - aa^\top) \textrm{Diag}(\sqrt{a})^{-1} = I - \sqrt{a}\sqrt{a}^\top\succeq 0$, where $\sqrt{a} := [\sqrt{a_1},\sqrt{a_2},\ldots,\sqrt{a_n}]^\top$. The only nonzero eigenvalue of the rank-one matrix $\sqrt{a}\sqrt{a}^\top$ is $\sqrt{a}^\top\sqrt{a} = e^\top a \leq 1$. Therefore, $I - \sqrt{a}\sqrt{a}^\top\succeq 0$, which implies $\textrm{Diag}(a) - aa^\top\succeq 0$. ◻ By Lemma [Lemma 5](#psd-condition){reference-type="ref" reference="psd-condition"}, it is easy to see that the constraint $X - x x^\top\succeq 0$ on the right-hand side of [\[proj_rels_2a\]](#proj_rels_2a){reference-type="eqref" reference="proj_rels_2a"} is redundant. Therefore, by Lemma [Lemma 2](#F_R1_rho_desc){reference-type="ref" reference="F_R1_rho_desc"}, we obtain $$\label{proj_rels_3} {\cal F}^{R3}_1 \subseteq \left\{(x,X) \in \mathbb{R}^n \times {\cal S}^n: e^\top x = 1, \quad X = \textrm{Diag}(x), \quad x \geq 0\right\} = {\cal F}^{R1}_1.$$ Our next result shows that the inclusion in [\[proj_rels_3\]](#proj_rels_3){reference-type="eqref" reference="proj_rels_3"} actually holds with equality, thereby yielding an exact description of ${\cal F}^{R3}_1$. **Lemma 6**. *We have $$\label{proj_rels_5} {\cal F}^{R3}_1 = {\cal F}^{R1}_1 = \textrm{conv}\left\{(x,x x^\top): x \in F_1 \right\} = \textrm{conv}\left\{(e^j,e^j (e^j)^\top): j \in \{ 1,\ldots,n \}\right\},$$ where ${\cal F}^{R3}_1$ and ${\cal F}^{R1}_1$ are defined as in [\[def_F\_R3_rho\]](#def_F_R3_rho){reference-type="eqref" reference="def_F_R3_rho"} and [\[def_F\_R1_rho\]](#def_F_R1_rho){reference-type="eqref" reference="def_F_R1_rho"}, respectively.* *Proof.* The assertion follows from the observation that ${\cal F}_1^{R1} = \textrm{conv}\left\{(e^j,e^j (e^j)^\top): 1\le j \le n \right\}$ in conjunction with Proposition [Proposition 2](#inner-rank-one){reference-type="ref" reference="inner-rank-one"} and [\[proj_rels_3\]](#proj_rels_3){reference-type="eqref" reference="proj_rels_3"}. ◻ Lemma [Lemma 6](#F_R3_1_desc){reference-type="ref" reference="F_R3_1_desc"} reveals that the SDP-RLT relaxation [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} is identical to the RLT relaxation [\[R1rho\]](#R1rho){reference-type="eqref" reference="R1rho"} for $\rho = 1$: semidefinite constraints in [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} are redundant. ## Case of Larger Sparsity $\rho \geq 2$ In this section, we focus on the sets ${\cal F}^{R3}_\rho$, where $\rho \in \{2,3,\ldots,n\}$, and establish several properties and relations. Our first result strengthens the inner approximation of ${\cal F}^{R3}_\rho$ given by Proposition [Proposition 2](#inner-rank-one){reference-type="ref" reference="inner-rank-one"}. **Lemma 7**. *We have $$\left\{(x,X) \in {\cal F}^{R3}: x \in F_\rho\right\} \subseteq {\cal F}^{R3}_\rho, \quad \textrm{all }\rho \in \{ 2,\ldots,n\}\, ,$$ where $F_\rho$, ${\cal F}^{R3}$, and ${\cal F}^{R3}_\rho$ are given by [\[def_F\_rho\]](#def_F_rho){reference-type="eqref" reference="def_F_rho"}, [\[def_F\_R3\]](#def_F_R3){reference-type="eqref" reference="def_F_R3"} and [\[def_F\_R3_rho\]](#def_F_R3_rho){reference-type="eqref" reference="def_F_R3_rho"}, respectively.* *Proof.* Fix $\rho \in \{2,\ldots,n\}$ and let $( x, X) \in {\cal F}^{R3}$ with $\| x \|_0 \leq \rho$. Choose $u \in \{0,1\}^n$ such that $x \leq u$ and $e^\top u = \rho$. Define $R = x u^\top$ and $U = u u^\top$. Clearly, $\textrm{diag}( U) = u$, $R^\top e = u, R e = \rho x$, $U e = \rho u$, $R - U \leq 0$, $R \geq 0$, and $U \geq 0$. Since $X = x x^\top+ M$ for some $M \succeq 0$, we obtain $$\begin{bmatrix} X & R \\ R^\top& U \end{bmatrix} - \begin{bmatrix} x \\ u \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix}^\top= \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} \succeq 0.$$ Next, we consider the constraint $X - R^\top\leq 0$. Since $X \geq 0$ and $X e = x$, we obtain $0 \leq X_{ij} \leq \min\{ x_i, x_j\}$ for each $1 \leq i \leq j \leq n$. Therefore, if $\min\{ x_i, x_j\} = 0$, then $X_{ij} - u_i x_j = - u_i x_j \leq 0$. On the other hand, if $\min\{ x_i, x_j\} > 0$, then $u_i = 1$, which implies that $X_{ij} - u_i x_j = X_{ij} - x_j \leq 0$. It follows that $X - R^\top\leq 0$. Finally, we need to show that $X - R - R^\top+ U \geq 0$. For each $1 \leq i \leq j \leq n$, if $\min\{ x_i, x_j\} = 0$, then $X_{ij} = 0$ and $\min\{ R_{ij}, R_{ji}\} = \min\{ x_i u_j, x_j u_i\} = 0$. Therefore, $$X_{ij} - R_{ij} - R_{ji} + U_{ij} = 0-\max\{ R_{ij}, R_{ji}\}-0 + u_iu_j = - \max\{ x_i u_j, x_j u_i\} + u_i u_j \geq 0\, ,$$ since $x \leq u$. Here, we used the lattice identity $v+w=\min \{ v,w\} + \max\{v,w\}$. On the other hand, if $\min\{ x_i, x_j\} > 0$, then $u_i = u_j = 1$, which implies that $X_{ij} - R_{ij} - R_{ji} + U_{ij} = X_{ij} - x_i - x_j + 1$. For any $1 \leq i < j \leq n$, since $x_i + x_j \leq 1$, we clearly have $X_{ij} - x_i - x_j + 1 \geq 0$ since $X \geq 0$. Finally, if $i = j$, since $X \succeq 0$ and $X e = x$, we obtain $$X_{ii} - 2 x_i + 1 = ( e^i - e)^\top X ( e^i - e) \geq 0, \quad i = 1,\ldots,n,$$ which completes the proof. ◻ By Lemma [Lemma 7](#comp1-R3-R3rho){reference-type="ref" reference="comp1-R3-R3rho"}, none of the solutions in $(x,X) \in {\cal F}^{R3}$ with $x \in F_\rho$ is cut off by the projection ${\cal F}^{R3}_\rho$. This observation gives rise to the following corollary. **Corollary 5**. 1. *For each $\rho \in \{2,\ldots,n\}$, if there exists an optimal solution $( x, X) \in \mathbb{R}^n \times {\cal S}^n$ of [\[R3\]](#R3){reference-type="eqref" reference="R3"} such that $\| x \|_0 \leq \rho$, then $\ell^{R3}(Q) = \ell^{R3}_\rho(Q)$.* 2. *We have ${\cal F}^{R3}_n = {\cal F}^{R3}$ and $\ell^{R3}(Q) = \ell^{R3}_n(Q)$.* *Proof.* 1. We clearly have $\ell^{R3}(Q) \leq \ell^{R3}_\rho(Q)$ by [\[R3\]](#R3){reference-type="eqref" reference="R3"}, [\[ell_rho_R3_alt\]](#ell_rho_R3_alt){reference-type="eqref" reference="ell_rho_R3_alt"}, and [\[proj_rels_2b\]](#proj_rels_2b){reference-type="eqref" reference="proj_rels_2b"}. The reverse inequality follows from Lemma [Lemma 7](#comp1-R3-R3rho){reference-type="ref" reference="comp1-R3-R3rho"}. 2. As $F_n=F$, the first equality follows from [\[proj_rels_2b\]](#proj_rels_2b){reference-type="eqref" reference="proj_rels_2b"} and Lemma [Lemma 7](#comp1-R3-R3rho){reference-type="ref" reference="comp1-R3-R3rho"}, and the second one from the first assertion (i).  ◻ By Corollary [Corollary 5](#stqp_sparse_x){reference-type="ref" reference="stqp_sparse_x"}, we can identify a particular set of instances of [\[sstqp\]](#sstqp){reference-type="eqref" reference="sstqp"} that admit an exact SDP-RLT relaxation. **Corollary 6**. *Let $\rho \in \{2,\ldots,n\}$. For any $x \in F_\rho$, any $P \succeq 0$ such that $P x = 0$, any $N \in {\cal S}^n$ such that $N \geq 0$ and $x^\top N x = 0$, and any $\lambda \in \mathbb{R}$, if $Q = P + N + \lambda E$, then the SDP-RLT relaxation [\[R3rho\]](#R3rho){reference-type="eqref" reference="R3rho"} is exact, i.e., $\ell^{R3}_\rho(Q) = \ell_\rho(Q)$.* *Proof.* Under the hypotheses, Theorem [Theorem 3](#exact_dnn_stqp){reference-type="ref" reference="exact_dnn_stqp"} implies that $x \in F$ is an optimal solution of ([\[stqp\]](#stqp){reference-type="ref" reference="stqp"}) and $\ell^{R3}(Q) = \ell(Q) = \lambda$. The assertion follows from Corollary [Corollary 5](#stqp_sparse_x){reference-type="ref" reference="stqp_sparse_x"}(i) and Lemma [Lemma 1](#simple-obs1){reference-type="ref" reference="simple-obs1"}. ◻ ## Rank-One Elements of ${\cal F}^{R3}_\rho$ Recall that each solution $(x,X) \in {\cal F}^{R3}$, where $x \in F_\rho$, is retained in the projection ${\cal F}^{R3}_\rho,~\rho = 1,\ldots,n$ by Lemma [Lemma 7](#comp1-R3-R3rho){reference-type="ref" reference="comp1-R3-R3rho"}. In this section, our goal is to shed light on the relations between ${\cal F}^{R3}_\rho$ and the set of solutions $(x,X) \in {\cal F}^{R3}$, where $\|x\|_0 > \rho$. First, it follows from Proposition [Proposition 2](#inner-rank-one){reference-type="ref" reference="inner-rank-one"} and Lemma [Lemma 6](#F_R3_1_desc){reference-type="ref" reference="F_R3_1_desc"} that $$\label{proj_rels_6} {\cal F}^{R3}_1 \subseteq {\cal F}^{R3}_\rho \quad\mbox{for all } \rho \in \{ 2,\ldots,n\}\, ,$$ which, in turn, implies that $(x,X) = (\frac{1}{n} e, \frac{1}{n} I) \in {\cal F}^{R3}_\rho$ for each $\rho \in \{1,\ldots,n\}$ by Lemma [Lemma 6](#F_R3_1_desc){reference-type="ref" reference="F_R3_1_desc"}. Therefore, for each $\rho \in \{1,\ldots,n\}$, there exists $(x,X) \in {\cal F}^{R3}_\rho$ such that $\|x\|_0 > \rho$. Let us restrict our attention to the subset of "rank-one solutions" $(x,X) \in {\cal F}^{R3}$, i.e., those with $\|x\|_0 = \nu > \rho$ and $X = x x^\top$. Note that $\langle Q, X \rangle = x^\top Q x$ for each rank-one solution. This, in turn, enables us to compare $\ell^{R3}_\rho(Q)$ and $\ell_{\nu}(Q)$ for some $\nu > \rho$. We start with the following result for $\rho = 1$. **Corollary 7**. *$( x, x x^\top) \in {\cal F}^{R3}_1$ if and only if $x \in F_1$.* *Proof.* The claim follows from Lemma [Lemma 6](#F_R3_1_desc){reference-type="ref" reference="F_R3_1_desc"}. ◻ By Corollary [Corollary 7](#rank-one-rho-1){reference-type="ref" reference="rank-one-rho-1"}, each rank-one solution $( x, x x^\top) \in {\cal F}^{R3}$, where $\| x \|_0 > 1$, is cut off by ${\cal F}^{R3}_1$. We next focus on ${\cal F}^{R3}_\rho$ for $\rho \geq 2$. To that end, we first state a technical result about the feasible region of (R3($\rho$)). **Lemma 8**. *Let $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ be (R3($\rho$))-feasible, where $\rho \in \{1,\ldots,n\}$. Then, $$\label{u_lb_1} \left(\rho - 2\right) u_i + 2 R_{ii} + (1 - \rho) x_i - X_{ii} \geq 0, \quad \mbox{for all }i \in \{1,\ldots,n\}\, .$$* *Proof.* Suppose that $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R3($\rho$))-feasible. Let us fix $i \in \{1,\ldots,n\}$. For each $j \in \{1,\ldots,n\}$ such that $j \neq i$, we have $$U_{ij} - R_{ij} - R_{ji} + X_{ij} \geq 0.$$ Therefore, $$\begin{aligned} 0 & \leq & \sum\limits_{j \in \{1,\ldots,n\} \backslash \{i\}} \left( U_{ij} - R_{ij} - R_{ji} + X_{ij}\right) \\ & = & \left(\rho u_i - u_i \right) - \left( \rho x_i - R_{ii} \right) - \left( u_i - R_{ii}\right) + \left( x_i - X_{ii}\right)\\ & = & \left(\rho - 2\right) u_i + 2 R_{ii} + (1 - \rho) x_i - X_{ii}, \end{aligned}$$ where we used $\textrm{diag}( U) = u$, $X e = x$, $R^\top e = u$, $R e = \rho \, x$, and $U e = \rho \, u$ in the second line. The assertion follows. ◻ Using this technical result, we can establish the following result about rank-one solutions for $\rho = 2$. **Corollary 8**. *For each $x \in F$ such that $\| x\|_0 \geq 4$, we have $( x, x x^\top) \not \in {\cal F}^{R3}_2$.* *Proof.* We prove the contrapositive. Let $\rho = 2$ and let $( x, x x^\top) \in {\cal F}^{R3}_\rho$. Then, there exists $( u, U, R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}$ such that $( x, u, X, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R3($\rho$))-feasible, where $X = x x^\top$. Since $X = x x^\top$, it follows from the positive semidefiniteness constraint that $R = x u^\top$. By Lemma [Lemma 8](#u_lower_bound){reference-type="ref" reference="u_lower_bound"}, we obtain $$\left(\rho - 2\right) u_i + 2 x_i u_i + (1 - \rho) x_i - x_i^2 \geq 0 \quad\mbox{for all } i \in \{ 1,\ldots,n\}\, .$$ Using $\rho = 2$, for each $i \in \{1,\ldots,n\}$ such that $x_i > 0$, we obtain $$u_i \geq \frac{1 + x_i}{2}\, .$$ Summing over each $i \in \{1,\ldots,n\}$ such that $x_i > 0$, and observing $\sum\limits_{i:x_i>0} x_i= e^\top x= 1$, we arrive at $$2 = \sum_i u_i \ge\sum_{i: x_i>0} u_i \geq \frac{\| x\|_0 + 1}{2}\, ,$$ which implies that $\| x \|_0 \leq 3$. The assertion follows. ◻ By Corollary [Corollary 8](#ub-rankone-rho-2){reference-type="ref" reference="ub-rankone-rho-2"}, each rank-one solution $( x, x x^\top) \in {\cal F}^{R3}$, where $\| x \|_0 > 3$, is cut off by ${\cal F}^{R3}_2$. Furthermore, for each $x \in F$ such that $\| x\|_0 = 3$, the proof of Corollary [Corollary 8](#ub-rankone-rho-2){reference-type="ref" reference="ub-rankone-rho-2"} implies that there exists a unique $u \in \mathbb{R}^n$ given by $u = \frac{1}{2}( x + e) = x + \frac{1}{2}(e - x)$ such that $( x, u, x x^\top, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R3($\rho$))-feasible. Our next result establishes that the choice of $u$ can be generalized to larger values of $\rho$. **Theorem 4**. *We have $$\begin{aligned} \label{incls1-3} \left\{(x, x x^\top) \in {\cal F}^{R3}: x \in F_{2 \rho - 1}\right\} & \subseteq & {\cal F}^{R3}_\rho \quad \mbox{for all }\rho\in \left\{ 2,\ldots,\left \lfloor \textstyle{\frac{n+1}{2}} \right \rfloor\right\}\, , \label{incl1} \\ \left\{(x, x x^\top) \in {\cal F}^{R3}: x \in F\right\} & \subseteq & {\cal F}^{R3}_\rho \quad \mbox{for all }\rho \in \left\{ \left \lfloor \textstyle{\frac{n+1}{2}} \right \rfloor + 1, \ldots,n\right\}\, , \label{incl2} \\ \left\{(x, x x^\top) \in {\cal F}^{R3}: x \in G_\rho \right\} & \subseteq & {\cal F}^{R3}_\rho \quad \mbox{for all }\rho \in \left\{ 2, \ldots, \left \lfloor \textstyle{\frac{n}{2}} \right \rfloor\right\}\, , \label{incl3}\end{aligned}$$ where we define for $\rho \in \left\{ 2, \ldots, \left \lfloor \textstyle{\frac{n}{2}} \right \rfloor\right\}$ $$\label{suff_cond_set1} G_\rho := \left\{x \in F: \|x\|_0 > 2 \rho - 1, \quad \max\limits_{1 \leq i < j \leq n: x_i x_j > 0} \frac{x_i x_j}{1 - x_i - x_j} \leq \frac{(\rho - 1)(\rho - 2)}{\left(\|x\|_0 - 2\right) \left(\|x\|_0 - 2 \rho + 1\right)}\right\}\, .$$* *Proof.* By Corollary [Corollary 5](#stqp_sparse_x){reference-type="ref" reference="stqp_sparse_x"}(ii), we have ${\cal F}^{R3}_n = {\cal F}^{R3}$, which implies [\[incl2\]](#incl2){reference-type="eqref" reference="incl2"} for $\rho = n$. Therefore, let $\rho \in \{2,\ldots,n-1\}$. By Lemma [Lemma 7](#comp1-R3-R3rho){reference-type="ref" reference="comp1-R3-R3rho"}, it suffices to focus on rank-one solutions $(x, x x^\top)$, where $x\in F$ with $\| x\|_0 \geq \rho + 1$. We abbreviate $\nu:=\| x\|_0$ to ease notation. Our proof is constructive. Let us define $u \in \mathbb{R}^n$ as follows: $$u_i = \begin{cases} x_i + \lambda (1 - x_i)\, , & \textrm{if}~ x_i > 0\, , \\ 0\, , & \textrm{otherwise,} \end{cases}$$ where $$\label{def_lambda} \lambda := \frac{\rho - 1}{\nu - 1} \in (0,1).$$ Note that $0 \leq x \leq u \leq e$ and $e^\top u = \rho$. Let us define $X = x x^\top$, $R = x u^\top$, and $$U = u u^\top+ U^1 + U^2,$$ where $$\begin{aligned} U^1 & := & \alpha\left(\textrm{Diag}( x) - x x^\top\right),\\ U^2 & := & \beta \left(\textrm{Diag}(a) - a a^\top\right),\end{aligned}$$ and $\alpha$, $\beta$, and $a \in \mathbb{R}^n$ are given by $$\begin{aligned} \label{def_params} \alpha & := & \frac{(\nu - \rho)(\nu - \rho - 1)}{(\nu - 1)(\nu - 2)} \geq 0, \label{def_alpha} \\ \beta & := & \frac{(\nu - \rho)(\rho - 1)}{\nu - 2} > 0, \label{def_beta} \\ a_i & := &\begin{cases} \frac{1 - x_i}{\nu - 1}, & \textrm{if}~ x_i > 0, \\ 0, & \textrm{otherwise.} \end{cases} \label{def_a}\end{aligned}$$ Note that $a \in \mathbb{R}^n_+$, $e^\top a = 1$, and $U e = \rho u$, because $U^1 e =U^2 e=0$. Furthermore, $$\begin{bmatrix} X & R \\ R^\top& U \end{bmatrix} - \begin{bmatrix} x \\ u \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix}^\top= \begin{bmatrix} 0 & 0 \\ 0 & U^1 + U^2 \end{bmatrix} \succeq 0,$$ where we used Lemma [Lemma 5](#psd-condition){reference-type="ref" reference="psd-condition"}. In addition, if $x_i = 0$, then $U_{ii} = 0 = u_i$. If $x_i > 0$, then it follows as well that $U_{ii}=u_i$, along the following lines: $$\label{crux}\begin{array}{rcl} U_{ii} & = & u_i^2 + \alpha ( x_i - x_i^2) + \beta (a_i - a_i^2) \\ & = & \left((1 - \lambda) x_i + \lambda \right)^2 + \alpha ( x_i - x_i^2) + \frac{\beta}{\nu - 1} (1 - x_i) - \frac{\beta}{(\nu - 1)^2} (1 - x_i)^2\\ & = & \left((1 - \lambda)^2 - \alpha - \frac{\beta}{(\nu - 1)^2}\right) x_i^2 + \left(2 \lambda (1 - \lambda) + \alpha - \frac{\beta}{\nu - 1} + \frac{2 \beta}{(\nu - 1)^2}\right) x_i \\ & & \quad + \lambda^2 + \frac{\beta}{\nu - 1} - \frac{\beta}{(\nu - 1)^2} \, .\end{array}$$ We claim the last expression of [\[crux\]](#crux){reference-type="eqref" reference="crux"} equals $(1 - \lambda) x_i + \lambda = u_i$, which follows by equating the coefficients of $x_i^2$, $x_i$, and $1$, in above expression and re-arranging all terms with $\lambda$ to the right-hand side: $$\label{crux2}\left\{ .\begin{array}{rcccl} \alpha + \frac{\beta}{(\nu - 1)^2} & = & (1 - \lambda)^2 & = & \frac{(\nu - \rho)^2}{(\nu - 1)^2}\\ \alpha + \frac{\beta (3-\nu)}{(\nu - 1)^2} & = &(1 - \lambda) (1- 2 \lambda) & = & \frac{(\nu - \rho)(\nu - 2 \rho + 1)}{(\nu - 1)^2} \\ 0 + \frac{\beta (\nu-2)}{(\nu - 1)^2} &=& \lambda (1-\lambda) & = & \frac{(\nu - \rho)(\rho - 1)}{(\nu - 1)^2} \end{array} \right.\, .$$ Observe that the system [\[crux2\]](#crux2){reference-type="eqref" reference="crux2"} has a unique solution given by [\[def_alpha\]](#def_alpha){reference-type="eqref" reference="def_alpha"} and [\[def_beta\]](#def_beta){reference-type="eqref" reference="def_beta"} since subtracting the second equation from the first one yields the third equation. Therefore, we obtain that $\textrm{diag}( U) = u$. We clearly have $X \geq 0$, $R \geq 0$, and $X - R^\top= ( x - u) x^\top\leq 0$. Finally, we focus on $X - R - R^\top+ U \geq 0$ since each of $R - U \leq 0$ and $U \geq 0$ is implied by these constraints. If $x_i = 0$, then $U_{ii} - 2 R_{ii} + X_{ii} = 0 \geq 0$. On the other hand, if $x_i > 0$, we have $$U_{ii} - 2 R_{ii} + X_{ii} = u_i^2 + U^1_{ii} + U^2_{ii} - 2 x_i u_i + x_i^2 = ( u_i - x_i)^2 + U^1_{ii} + U^2_{ii} \geq 0,$$ where we used $U^1 \succeq 0$ and $U^2 \succeq 0$. Similarly, $U_{ij} - R_{ij} - R_{ji} + X_{ij} = 0 \geq 0$ whenever $1 \leq i < j \leq n$ and $x_i x_j = 0$. On the other hand, if $1 \leq i < j \leq n$ and $x_i x_j > 0$, we obtain $$\begin{aligned} U_{ij} - R_{ij} - R_{ji} + X_{ij} & = & u_i u_j - \alpha x_i x_j - \beta a_i a_j - x_i u_j - x_j u_i + x_i x_j \\ & = & ( u_i - x_i)( u_j - x_j) - \alpha x_i x_j - \beta a_i a_j \\ & = & \lambda^2 (1 - x_i) (1 - x_j) - \alpha x_i x_j - {\textstyle\frac{\beta}{(\nu - 1)^2}}\, (1 - x_i) (1 - x_j) \\ & = & \left(\lambda^2 - {\textstyle\frac{\beta}{(\nu - 1)^2}}\right) (1 - x_i - x_j) + \left(\lambda^2 - {\textstyle\frac{\beta}{(\nu - 1)^2}} - \alpha\right) x_i x_j\\ & = & {\textstyle\frac{(\rho - 1)(\rho - 2)}{(\nu - 1)(\nu - 2)}}\left(1 - x_i - x_j\right) + {\textstyle\frac{ 2 \rho - 1 - \nu }{\nu - 1}} \, x_i x_j \, ,\end{aligned}$$ where we used [\[def_lambda\]](#def_lambda){reference-type="eqref" reference="def_lambda"}, [\[def_alpha\]](#def_alpha){reference-type="eqref" reference="def_alpha"}, and [\[def_beta\]](#def_beta){reference-type="eqref" reference="def_beta"} to derive the last equation. Since $\rho \geq 2$, $\nu \geq 3$, $1 - x_i - x_j\ge 0$ and $x_i x_j > 0$, it follows that $U_{ij} - R_{ij} - R_{ji} + X_{ij} \geq 0$ if $\nu \leq 2 \rho - 1$, which establishes [\[incl1\]](#incl1){reference-type="eqref" reference="incl1"} and [\[incl2\]](#incl2){reference-type="eqref" reference="incl2"}. If, on the other hand, $\nu > 2 \rho - 1$, then $U_{ij} - R_{ij} - R_{ji} + X_{ij} \geq 0$ by [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"}, giving rise to [\[incl3\]](#incl3){reference-type="eqref" reference="incl3"}. This completes the proof. ◻ Before we proceed to the important consequences of the above result, let us motivate the construction in its proof, in particular the choice of $\lambda$ and the other constants. **Observation 1**. *Let $\rho\in \{ 2,\ldots,n\}$ and let ${x}\in F$. Assume that ${u}_i = \tau {x}_i +b$ if $x_i>0$ with $0<\tau<1$, while $u_i=0$ if $x_i=0$. Furthermore, assume that ${U}_{ij} = c {x}_i + c {x}_j +d$ if $x_ix_j>0$ while $U_{ij}=0$ if $x_ix_j=0$ for $1 \leq i < j \leq n$. It is easy to verify that the choices of $u$ and $U$ in the proof of Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} are in this form. Then, the best choice of $\tau$, $b$, $c$ and $d$ ensuring that $(x, u, X, U, R) = ( {x} , {u}, {x} {x}^\top, {U}, {x} {u}^\top)$ is R3($\rho$)-feasible, is the choice in the proof of Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"}.* *Proof.* Let $\rho\in \{ 2,\ldots,n\}$ and let ${x}\in F$. Again, abbreviate $\nu=\| {x}\|_0$. From $e^\top(\tau {x} +b) = e^\top u= \rho$, we derive $b = \frac{\rho-\tau}{\nu}\in (0,1)$ as $\rho>1>\tau$ and $\rho-\tau< \rho < \nu$. Furthermore, the constraints $x\leq u \leq e$ become $$x_i \le \min \left \{ {\frac{1-b}{\tau}, \frac{b}{1-\tau}}\right \} = \min \left \{{\frac{ \nu-\rho+\tau}{ \nu \tau}, \frac{\rho-\tau}{ \nu(1-\tau)}}\right \} \quad\mbox{for all }i = 1,\ldots,n\, .$$ Since $g(\tau):=\frac{ \nu-\rho+\tau}{ \nu \tau}$ decreases and $h(\tau):=\frac{\rho-\tau}{ \nu(1-\tau)}$ increases with $\tau\in (0,1)$, the maximum of $\min \left \{ g(\tau), h(\tau)\right \}$ is attained at $\tau^*$ satisfying $g(\tau^*)=h(\tau^*)$, and this value ensures that the formulation covers as many $x\in F$ as possible. Hence the best choice of $\tau$ would be the solution $\tau^*$ of $g(\tau^*)=h(\tau^*)$, namely $\tau^*=\frac{ \nu-\rho}{ \nu-1}$, which is exactly our choice in the proof of Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} with $\lambda = 1 - \tau^*= \frac{\rho-1}{\nu-\rho}$. Since $U_{ii} = u_i$ and $Ue = \rho u$, we have for $x_i>0$ $$\begin{aligned} \tau^* x _i + b + \sum_{j\neq i:x_j>0}(c x _i + c x _j +d) &=\rho(\tau^* x _i + b)\quad\mbox{or} \\[0.3em] ( \nu -1)d + c + \frac{\rho - \tau^*}{ \nu} + ( \nu -1)c x _i + (\tau^* - c) x _i &= \rho\tau^* x _i + \frac{\rho(\rho - \tau^*)}{ \nu}, \end{aligned}$$ which implies, comparing coefficients of $x_i$ and 1, that $$\begin{cases} ( \nu -1)c + \tau^* - c = \rho \tau^* \quad\mbox{and}\\[0.3em] ( \nu -1)d + c + \frac{\rho - \tau^*}{ \nu} = \frac{\rho(\rho - \tau^*)}{ \nu} \, , \end{cases}$$ so that $c = \frac{(\rho-1)\tau^*}{ \nu-2}= \frac{(\rho-1)(\nu-\rho)}{(\nu-1)(\nu-2)}$ and $d = \frac{\rho-1}{\nu(\nu-1)(\nu-2)}[(\nu-2)\rho-2\tau^*( \nu-1)] =\frac{(\rho-1)(\rho-2)}{ ( \nu-1)( \nu-2)}$, substituting $\tau^* = \frac{ \nu-\rho}{ \nu-1}$. This justifies our choice of $c$ and $d$ in the proof of Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"}. ◻ **Example 1**. *The condition [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"} is sufficient but not necessary. For $n = 6$ and $\rho = 3$, the point $$x = [0.6,0.2,0.05,0.05,0.05,0.05]^\top\in F$$ violates [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"} since $$0.6 =\frac{(0.6)\, (0.2)}{1 - 0.6 - 0.2} = \max\limits_{1 \leq i < j \leq n: x_i x_j > 0} \frac{x_i x_j}{1 - x_i - x_j}>\frac{(\rho - 1)(\rho - 2)}{\left(\|x\|_0 - 2\right) \left(\|x\|_0 - 2 \rho + 1\right)} = 0.5\, ,$$ while there exists $(u, U, R) \in \mathbb{R}^6 \times {\cal S}^6 \times \mathbb{R}^{6 \times 6}$ such that $(x, u, X, U, R) = ( {x} , {u}, {x} {x}^\top, {U}, {x} {u}^\top)$ is (SDP-RLT(3))-feasible. One choice of $u$ and $U$ is $u = [0.8866,0.5512,0.3905,0.3906,0.3906,0.3905]^\top$ and $$U = \begin{bmatrix} 0.8866 &0.4674 &0.3264 &0.3265 &0.3265 &0.3264\\ 0.4674 &0.5512 &0.1588 &0.1588 &0.1588 &0.1588\\ 0.3264 &0.1588 &0.3905 &0.0986 &0.0986 &0.0986\\ 0.3265 &0.1588 &0.0986 &0.3906 &0.0986 &0.0986\\ 0.3265 &0.1588 &0.0986 &0.0986 &0.3906 &0.0986\\ 0.3264 &0.1588 &0.0986 &0.0986 &0.0986 &0.3905\\ \end{bmatrix}\, .$$ Note that $u$ is not given by an affine function of $x$ in the sense of Observation [Observation 1](#obs){reference-type="ref" reference="obs"}.* Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} reveals that an increasingly larger and nontrivial set of rank-one solutions is contained in the sets ${\cal F}^{R3}_\rho$ as $\rho$ increases. Note that $G_\rho$ given by [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"} is a nonconvex set. Our next result gives further insight into this set by providing a piecewise convex inner approximation. **Lemma 9**. *We have $G_2 = \emptyset$ and $$\label{def_Hrho} H_\rho := \bigcup\limits_{\nu = 2 \rho}^n \left\{x \in F: \| x \|_0 = \nu, \quad x_i + x_j \leq \delta_{\rho,\nu},~1 \leq i < j \leq n\right\} \subseteq G_\rho \quad\mbox{if } \rho \in \left\{3,\ldots,\left \lfloor \textstyle{\frac{n}{2}} \right \rfloor \right\},$$ where $G_\rho$ is defined as in [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"} and $$\label{def_delta} \delta_{\rho,\nu} := 2 \left[\left(\tau_{\rho,\nu}^2 + \tau_{\rho,\nu}\right)^{1/2} - \tau_{\rho,\nu}\right],$$ with $$\label{def_tau} \tau_{\rho,\nu} := \frac{(\rho - 1)(\rho - 2)}{\left(\nu - 2\right) \left(\nu - 2 \rho + 1\right)}.$$ Furthermore, $$\label{incl4} \left\{(x, x x^\top) \in {\cal F}^{R3}: x \in H_\rho \right\} \subseteq {\cal F}^{R3}_\rho \quad \mbox{for all }\rho \in \left\{ 3, \ldots, \left \lfloor \textstyle{\frac{n}{2}} \right \rfloor\right\}\, .$$* *Proof.* For $\rho = 2$, the upper bound in [\[suff_cond_set1\]](#suff_cond_set1){reference-type="eqref" reference="suff_cond_set1"} equals zero, which implies that $G_2 = \emptyset$. Let us fix $\rho \in \left\{3,\ldots,\left \lfloor \textstyle{\frac{n}{2}} \right \rfloor \right\}$ and let $x \in H_\rho$. Then, $x \in F$, $\| x \|_0 = \nu > 2 \rho - 1$, and it is easy to verify that $$\max\limits_{1 \leq i < j \leq n: x_i x_j > 0} \frac{x_i x_j}{1 - x_i - x_j} \leq \frac{\delta_{\rho,\nu}^2}{4 \left( 1 - \delta_{\rho,\nu} \right)} = \tau_{\rho,\nu} \, ,$$ where the last equality follows from [\[def_delta\]](#def_delta){reference-type="eqref" reference="def_delta"} and [\[def_tau\]](#def_tau){reference-type="eqref" reference="def_tau"}. Both inclusions [\[def_Hrho\]](#def_Hrho){reference-type="eqref" reference="def_Hrho"} and [\[incl4\]](#incl4){reference-type="eqref" reference="incl4"} now follow from Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} by observing that $\| x \|_0 = \nu$. ◻ For fixed $\rho \in \left\{3,\ldots,\left \lfloor \textstyle{\frac{n}{2}} \right \rfloor \right\}$, it is worth noticing that $\tau_{\rho,\nu}$ given by [\[def_tau\]](#def_tau){reference-type="eqref" reference="def_tau"} is a decreasing function of $\nu$, which, in turn, implies that $\delta_{\rho,\nu}$ given by [\[def_delta\]](#def_delta){reference-type="eqref" reference="def_delta"} is a decreasing function of $\nu$. Therefore, the positive components of the elements of $H_\rho$ given by Lemma [Lemma 9](#inner_approx_Grho){reference-type="ref" reference="inner_approx_Grho"} tend to get closer to each other as $\nu$ increases. For instance, if $\rho = 3$, then $\delta_{\rho,\nu}$ equals $0.7321$, $0.5798$, and $0.4805$ for $\nu = 6$, $\nu = 7$, and $\nu = 8$, respectively. Note that the point $x$ of Example [Example 1](#exle){reference-type="ref" reference="exle"} satisfies $x\notin G_3$, readily certifying $x\notin H_3$ since $x_1+x_2=0.8>0.7321$. Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} gives rise to several results about rank-one solutions of ${\cal F}^{R3}_\rho$. Our next result gives a complete description of such solutions for $\rho = 2$. **Corollary 9**. *We have $( x, x x^\top) \in {\cal F}^{R3}_2$ if and only if $x \in F_3$.* *Proof.* By Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"}, for any $x \in F_3$, we have $( x, x x^\top) \in {\cal F}^{R3}_2$ by [\[incl1\]](#incl1){reference-type="eqref" reference="incl1"}. The assertion follows from Corollary [Corollary 8](#ub-rankone-rho-2){reference-type="ref" reference="ub-rankone-rho-2"}. ◻ For $\rho = 1$ and $\rho = 2$, it follows from Corollary [Corollary 7](#rank-one-rho-1){reference-type="ref" reference="rank-one-rho-1"} and Corollary [Corollary 9](#rank-one-rho0-3-rho-2){reference-type="ref" reference="rank-one-rho0-3-rho-2"} that $( x, x x^\top) \in {\cal F}^{R3}_\rho$ if and only if $x \in F_1$ and $x \in F_3$, respectively. On the other hand, for $\rho \geq 3$, Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"} gives rise to our next result, which reveals that such a nontrivial upper bound on $\| x \|_0$ concerning rank-one solutions of ${\cal F}^{R3}_\rho$ does not exist. **Lemma 10**. *Let $\rho\geq 3$. Then, for any $\nu \in \{\rho + 1,\ldots,n\}$, there exists $x \in F_{\nu}$ such that $( x, x x^\top) \in {\cal F}^{R3}_\rho$.* *Proof.* Let $\rho \geq 3$ and $\nu \geq \rho + 1$. By Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"}, the assertion clearly holds for any $x \in F_{\nu}$ such that $\| x \|_0 = \nu \leq 2 \rho - 1$. Suppose that $\nu > 2 \rho - 1$. By Lemma [Lemma 9](#inner_approx_Grho){reference-type="ref" reference="inner_approx_Grho"}, it suffices to construct an $x \in F_{\nu}$ such that $\| x \|_0 = \nu$ and $x \in H_\rho$, where $H_\rho$ is defined as in [\[def_Hrho\]](#def_Hrho){reference-type="eqref" reference="def_Hrho"}. Let $x \in F_{\nu}$ be given by $$x_i = \begin{cases} \frac{1}{\nu}\,, & \textrm{if } i \in\{ 1,\ldots,\nu\}\, ,\\ 0\, , & \textrm{otherwise.} \end{cases}$$ We therefore need to verify that $$\frac{2}{\nu} \leq 2 \left[\left(\tau_{\rho,\nu}^2 + \tau_{\rho,\nu}\right)^{1/2} - \tau_{\rho,\nu}\right]\, ,$$ where $\tau_{\rho,\nu}$ is given by [\[def_tau\]](#def_tau){reference-type="eqref" reference="def_tau"}. Rearranging and simplifying the terms, the above inequality reduces to $$\frac 1{\nu(\nu - 2)} \leq \tau_{\rho,\nu}\, .$$ By [\[def_tau\]](#def_tau){reference-type="eqref" reference="def_tau"}, this inequality holds if $$\frac{\nu - 2 \rho + 1}{\nu} \leq (\rho - 1)(\rho - 2)\, .$$ Since $\rho \geq 3$ (and thus $2\rho-1>0$), we even have $$\frac{\nu - 2 \rho + 1}{\nu} \leq 1 \leq (\rho - 1)(\rho - 2)\, ,$$ which establishes the assertion. ◻ Following our earlier discussion about the positive components of the elements of the set $H_\rho$, we remark that all such components of the solution constructed in the proof of Lemma [Lemma 10](#no-upper-bound){reference-type="ref" reference="no-upper-bound"} are equal. Our next result establishes another useful property of the rank-one solutions of ${\cal F}^{R3}_\rho$. **Theorem 5**. *For each $\rho \in \{1,\ldots,n-1\}$, if $(x, xx^\top) \in {\cal F}^{R3}_\rho$, then $(x, xx^\top) \in {\cal F}^{R3}_{\rho+1}$.* *Proof.* Let $\rho \in \{1,\ldots,n-1\}$ and let $(x, xx^\top) \in {\cal F}^{R3}_\rho$. Let us define $\nu = \| x \|_0$. If $\rho\in \left\{ 2,\ldots,\left \lfloor \textstyle{\frac{n+1}{2}}\right \rfloor - 1\right\}$ and $\nu \leq 2 (\rho + 1) - 1 = 2 \rho + 1$; or if $\rho \in \left\{ \left \lfloor \textstyle{\frac{n+1}{2}} \right \rfloor, \ldots,n\right\}$, then the assertion follows from Theorem [Theorem 4](#general-construct){reference-type="ref" reference="general-construct"}. Therefore, let us assume that $\rho\in \left\{ 2,\ldots,\left \lfloor \textstyle{\frac{n+1}{2}}\right \rfloor - 1\right\}$ and $\nu > 2 \rho + 1$. For each $\rho \geq 3$, we remark that the set of rank-one solutions with this property is nonempty by Lemma [Lemma 10](#no-upper-bound){reference-type="ref" reference="no-upper-bound"}. Since $(x, xx^\top) \in {\cal F}^{R3}_\rho$, there exists $(u,U,R) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}$ such that $( x, u, x x^\top, U, R) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R3($\rho$))-feasible. Since $X = x x^\top$, we have $R = x u^\top$ and $U - u u^\top\succeq 0$ by Schur complementation. We will construct $(u^\prime, U^\prime, R^\prime) \in \mathbb{R}^n \times {\cal S}^n \times \mathbb{R}^{n \times n}$ such that $( x, u^\prime, x x^\top, U^\prime, R^\prime) \in \mathbb{R}^n \times \mathbb{R}^n \times {\cal S}^n \times {\cal S}^n \times \mathbb{R}^{n\times n}$ is (R3($\rho + 1$))-feasible. Let $u^\prime = u + s$, where $s \in \mathbb{R}^n$ is given by $$s_i = \begin{cases} \frac{1 - u_i}{\Vert u \Vert_0 - \rho}, & \text{if $u_i > 0$},\\ 0, & \text{otherwise.} \end{cases}$$ Since $0 \leq u \leq e$, we have $s \in \mathbb{R}^n_+$ since $\Vert u \Vert_0 \geq \nu > 2 \rho + 1$, which implies that $$\mu := \Vert u \Vert_0 - \rho > \rho + 1 \geq 3.$$ Therefore, we obtain $0 \leq x \leq u \leq u^\prime \leq e$. Furthermore, $e^\top s = 1$, which implies that $e^\top u^\prime = \rho + 1$. Since $X = x x^\top$, we define $R^\prime = x (u^\prime)^T = R + x s^\top$. Finally, we define $$U^\prime = u^\prime (u^\prime)^T + \frac{\mu - 2}{\mu} \left( U - uu^T \right) + \textrm{Diag}(s) - ss^T.$$ By Schur complementation, $$\begin{bmatrix} X & R^\prime \\ (R^\prime)^\top& U^\prime \end{bmatrix} - \begin{bmatrix} x \\ u^\prime \end{bmatrix} \begin{bmatrix} x \\ u^\prime \end{bmatrix}^\top= \begin{bmatrix} 0 & 0 \\ 0 & \frac{\mu - 2}{\mu} \left( U - uu^T \right) + \textrm{Diag}(s) - ss^T \end{bmatrix} \succeq 0,$$ where we used Lemma [Lemma 5](#psd-condition){reference-type="ref" reference="psd-condition"}, $\mu > 3$, and $U - u u^\top\succeq 0$. Therefore, the semidefiniteness constraint is satisfied. We clearly have $X e = x$, $R^\prime e = (\rho + 1) x$, $(R^\prime)^\top e = u^\prime$, and $U^\prime e = (\rho + 1) u^\prime$. We next focus on the constraint $\textrm{diag}(U^\prime) = u^\prime$. If $u_i = 0$, then $u^\prime_i = u_i = U_{ii} = U^\prime_{ii} = 0$ since $s_i = 0$. If $u_i > 0$, then $$\begin{aligned} U^\prime_{ii} & = & (u^\prime_i)^2 + \frac{\mu - 2}{\mu} \left( U_{ii} - u_i^2 \right) + s_i - s_i^2 \\ & = & \left( u_i + s_i \right)^2 + \frac{\mu - 2}{\mu} \left( u_i - u_i^2 \right) + s_i - s_i^2 \\ & = & \frac{2}{\mu} u_i^2 + \frac{\mu - 2}{\mu} u_i + s_i + 2 u_i s_i \\ & = & \frac{1}{\mu} \left(2 u_i^2 + (\mu - 2) u_i + 1 - u_i + 2 u_i (1 - u_i) \right) \\ & = & \frac{(\mu - 1) u_i + 1}{\mu} \\ & = & u^\prime_i,\end{aligned}$$ where we used $\textrm{diag}(U) = u$ in the second line and the definition of $s$ in the fourth line. This establishes $\textrm{diag}(U^\prime) = u^\prime$. Furthermore, we have $X \geq 0$, $R^\prime =R + x s^\top\geq 0$, and $X - (R^\prime)^\top= X - R^\top- s x^\top\leq 0$ since $X - R^\top\leq 0$, $x \geq 0$, and $s \geq 0$. We next verify $X - (R^\prime)^\top- R^\prime + U^\prime \geq 0$. Recall again that the remaining inequality constraints are implied. For the diagonal components, we have $$U^\prime_{ii} - 2 R^\prime_{ii} + X_{ii} = u^\prime_i - 2 x_i u^\prime_i + x_i^2 \geq (u^\prime_i)^2 - 2 x_i u^\prime_i + x_i^2 = ( u_i - x_i)^2 \geq 0, \quad i = 1,\ldots,n,$$ where we used $\textrm{diag}(U^\prime) = u^\prime$ and $0 \leq x \leq u \leq e$. If $1 \leq i < j \leq n$, then $$\begin{aligned} U^\prime_{ij} - R^\prime_{ij} - R^\prime_{ji} + X_{ij} & = & u^\prime_i u^\prime_j + \frac{\mu - 2}{\mu} \left( U_{ij} - u_i u_j \right) - s_i s_j - x_i u^\prime_j - x_j u^\prime_i + x_i x_j \\ & = & (u_i + s_i) (u_j + s_j) + \frac{\mu - 2}{\mu} \left( U_{ij} - u_i u_j \right) - s_i s_j \\ & & \quad - x_i (u_j + s_j) - x_j (u_i + s_i) + x_i x_j \\ & = & u_i s_j + s_i u_j - x_i u_j - x_i s_j - x_j u_i - x_j s_i + x_i x_j + \frac{2}{\mu} u_i u_j + \frac{\mu - 2}{\mu} U_{ij} \\ & = & \frac{\mu - 2}{\mu} \left( U_{ij} - x_i u_j - x_j u_i + x_i x_j \right) + \frac{2}{\mu} \left( u_i u_j - x_i u_j - x_j u_i + x_i x_j \right) \\ & & \quad + u_i s_j + s_i u_j - x_i s_j - x_j s_i \\ & \geq & \frac{2}{\mu} \left( (u_i - x_i) (u_j - x_j) \right) + \left( s_j (u_i - x_i) + s_i (u_j - x_j) \right) \\ & \geq & 0,\end{aligned}$$ where we used $\mu > 3$ and $U_{ij} - R_{ij} - R_{ji} + X_{ij} = U_{ij} - x_i u_j - x_j u_i + x_i x_j \geq 0$ to derive the first inequality, and $0 \leq x \leq u$ together with $s \geq 0$ to arrive at the final one. This completes the proof. ◻ Theorem [Theorem 5](#nested-rank-one){reference-type="ref" reference="nested-rank-one"} establishes the nested behavior of the set of rank-one solutions of ${\cal F}^{R3}_\rho$ with respect to $\rho$. We close this section with the following result about the tightness of the lower bound $\ell^{R3}_\rho(Q)$ arising from [\[R3\]](#R3){reference-type="eqref" reference="R3"}. **Corollary 10**. *We have $$\begin{aligned} \label{quality_lb_rels} \ell^{R3}_\rho(Q) & \leq & \ell_{2 \rho - 1}(Q)\, , \quad \textrm{if } \rho\in \left\{ 2, \ldots, \left \lfloor \textstyle{\frac{n+1}{2}} \right \rfloor\right\} \, , \textrm{ while} \label{quality_lb_rel1} \\ \ell^{R3}_\rho(Q) & \leq & \ell(Q)\, , \quad \quad \textrm{if }\rho \in \left\{ \left \lfloor \textstyle{\frac{n+1}{2}} \right \rfloor + 1, \ldots,n\right\} \, .\label{quality_lb_rel2} \end{aligned}$$* *Proof.* The relations follow from [\[ell_rho_R3_alt\]](#ell_rho_R3_alt){reference-type="eqref" reference="ell_rho_R3_alt"} and from [\[incl1\]](#incl1){reference-type="eqref" reference="incl1"} and [\[incl2\]](#incl2){reference-type="eqref" reference="incl2"}, respectively. ◻ Corollary [Corollary 10](#quality_lb){reference-type="ref" reference="quality_lb"} reveals that the lower bound $\ell^{R3}_\rho(Q)$ can be potentially quite weak especially for larger values of $\rho$. # Concluding Remarks {#conc} A Standard Quadratic optimization Problem with hard sparsity constraints can be exactly reformulated as a mixed-binary QP. Therefore, it is tempting to use tractable LP- or SDP-based relaxations, either in a straightforward/vanilla way or by suitable combinations as we did in Section [4](#SDP-RLT-Relaxation){reference-type="ref" reference="SDP-RLT-Relaxation"}. The aim is to achieve tight rigorous bounds with a computational effort that scales well with the problem size. However, our analysis reveals that some caveats are in place when following this approach. In unfavorable circumstances (e.g., if the sparsity constraints are not stringent enough), the resulting bounds are quite weak. We characterized the exactness of the bounds and studied the behavior of rank-one solutions to the relaxations. The findings of this article definitely call for more investigation, either in the direction of refined RLT models, or equally importantly, tighter conic-based relaxations which still offer some tractability. While these avenues are beyond the scope of the present work, they remain on our research agenda for the near future. [^1]: VCOR and Research Network Data Science, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. ORCID ID: 0000-0002-6288-9226 E-mail: `immanuel.bomze@univie.ac.at` [^2]: VGSCO and ISOR, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. ORCID ID: 0000-0002-2650-0295 E-mail: `bo.peng@univie.ac.at` [^3]: School of Mathematics, Peter Guthrie Tait Road, The University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom. E-mail: `y.qiu-16@sms.ed.ac.uk` [^4]: School of Mathematics, Peter Guthrie Tait Road, The University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom. ORCID ID: 0000-0003-4141-3189 E-mail: `E.A.Yildirim@ed.ac.uk`

arxiv_math

--- abstract: | In this paper, we study the absolute continuity of radial projections of missing digits measures. We show that for large enough missing digits measures $\lambda$ on $\mathbb{R}^n,n\geq 2,$ for all $x\in\mathbb{R}^n\setminus \mathrm{supp}(\lambda),$ $\Pi_x(\lambda)$ is absolutely continuous with a density function in $L^2(S^{n-1}).$ Our method applies to linear projections as well. In particular, we show that for $\lambda$ as above, the linearly projected measure $P_\theta(\lambda)$ is absolutely continuous with a continuous density function for almost all directions $\theta\in S^{n-1}.$ This implies a version of Palis' conjecture for missing digits sets. address: Han Yu, Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK author: - Han Yu title: On the absolute continuity of radial and linear projections of missing digits measures --- # Introduction ## Radial Projection Let $n\geq 2$ be an integer. Let $x\in\mathbb{R}^n.$ Define the function $$\Pi_x: y\in\mathbb{R}^n\setminus\{x\}\to \Pi_x(y)=\frac{y-x}{|y-x|}\in \mathbb{S}^{n-1}.$$ This function $\Pi_x$ is the radial projection centred at $x$. Let $A\subset\mathbb{R}^n.$ The set $\Pi_x(A\setminus\{x\})$ is the radial projection of $A$ from the observing position $x$. Let $\lambda$ be a compactly supported Borel probability measure on $\mathbb{R}^n.$ Suppose that $x$ is not in $\mathrm{supp}(\lambda).$ Then $\Pi_x(\lambda)$ is the pushed forward measure. See Section [3.4](#sec: rad){reference-type="ref" reference="sec: rad"}. The following result was proved in [@O19 Theorem 1.11]. **Theorem 1**. *Let $n\geq 2$ be an integer. Let $\lambda$ be a compactly supported Borel probability measure on $\mathbb{R}^n.$ Suppose that $$I_s(\lambda)=\int\int\frac{d\lambda(x)d\lambda(y)}{|x-y|^s}<\infty$$ for some $s>n-1.$ Then $\Pi_x(\lambda)$ is absolutely continuous for all $x\in\mathbb{R}^n\setminus\mathrm{supp}(\lambda)$ outside of an exceptional set with Hausdorff dimension at most $2(n-1)-s.$* For example, if $\lambda$ is $AD$-regular (see Section [3.3](#sec: AD){reference-type="ref" reference="sec: AD"}), then the finiteness of the integral happens when $s>\dim_{\mathrm{H}}\lambda.$ Roughly speaking, this result tells us that for thick enough measures, the radial projections are absolutely continuous for most of the 'observing positions'. The projected measures can have more regularities than just being absolutely continuous. In fact, the following finer result holds. See [@O19 Theorem 1.13]. **Theorem 2**. *Let $n\geq 2$ be an integer. Let $\lambda$ be a compactly supported Borel probability measure on $\mathbb{R}^n.$ Suppose that $$I_s(\lambda)=\int\int\frac{d\lambda(x)d\lambda(y)}{|x-y|^s}<\infty$$ for some $s>n-1.$ Let $p>1$. Then $\Pi_x(\lambda)$ is absolutely continuous with a density function in $L^p$ for all $x\in\mathbb{R}^n\setminus\mathrm{supp}(\lambda)$ outside of an exceptional set with Hausdorff dimension at most $2(n-1)-s+\delta(p)$ where $\delta(p)>0$ and moreover $\delta(p)\downarrow 0$ as $p\downarrow 1.$* For general $\lambda$, it is very difficult, if at all possible, to obtain more information about the exceptional sets in Theorem [Theorem 1](#thm: Or1){reference-type="ref" reference="thm: Or1"} and Theorem [Theorem 2](#thm: Or2){reference-type="ref" reference="thm: Or2"}. However, if $\lambda$ satisfies some other restrictions, e.g. being self-similar, then it is likely that the exceptional set for $\lambda$ is empty. In this paper, we consider radial projections of missing digits measures. See Section [3.2](#sec: missing){reference-type="ref" reference="sec: missing"}. In this case, it is plausible that the following conjecture holds. **Conjecture 3**. *Let $n\geq 2$. Let $\lambda$ be a missing digits measure (or a Cartesian product of missing digits measures). Suppose that $\dim_{\mathrm{H}}\lambda>n-1.$ Then* - *Difficulty level 1: $\Pi_x(\lambda)$ is absolutely continuous for all $x\in \mathbb{R}^n\setminus \mathrm{supp}(\lambda).$* - *Difficulty level 2: $\Pi_x(\lambda)$ is in $L^2(S^{n-1})$ for all $x\in \mathbb{R}^n\setminus \mathrm{supp}(\lambda).$* - *Difficulty level 3: $\Pi_x(\lambda)$ is in $L^p(S^{n-1})$ for all $p>1$ and all $x\in \mathbb{R}^n\setminus \mathrm{supp}(\lambda).$* - *Difficulty level 4: $\Pi_x(\lambda)$ is absolutely continuous with a upper semi continuous density function for all $x\in \mathbb{R}^n\setminus \mathrm{supp}(\lambda).$* **Remark 4**. *It might be the case that the difficulty level 1,2,3 statements hold with $\lambda$ being a general self-similar measure with the open set condition. However, this seems to be too strong to hold. We are not able to find counterexamples either.* **Remark 5**. *If instead of \"for all $x\in\mathbb{R}^n\setminus\mathrm{supp}(\lambda)$\", we require only \"most of $x\in\mathbb{R}^n\setminus\mathrm{supp}(\lambda)$\" then the situation is much more clear. In fact, Theorems [Theorem 1](#thm: Or1){reference-type="ref" reference="thm: Or1"}, [Theorem 2](#thm: Or2){reference-type="ref" reference="thm: Or2"} already give such a result. Stronger results establishing the nullity of the dimension of the exceptional sets can be found in [@KO; @ShmerkinSolomyak].* Currently, even the difficulty level 1 problem is open. In this paper, we partially solve the above conjecture up to the difficulty level 3. For clearness, we state our result for a particularly chosen $\lambda.$ In the next section, we will provide general results. **Theorem 1**. *Let $\lambda=\lambda_{p,D}$ be the missing digits measure on $\mathbb{R}$ with base $p=10^{10000}$ and the digit set $$D=\{1,2,\dots,10^{8000}\}.$$ Then for each $x\notin \mathrm{supp}(\lambda)\times\mathrm{supp}(\lambda),$ $\Pi_x(\lambda\times \lambda)$ is absolutely continuous with a density function in $L^2(S^1).$* Let $n\geq 2$ be an integer. Let $\lambda$ be a compactly supported Borel measure on $\mathbb{R}^n.$ Suppose that $x\in\mathbb{R}^n\setminus \mathrm{supp}(\lambda).$ We see that if $\Pi_x(\lambda)$ is absolutely continuous, then $\Pi_x(\mathrm{supp}(\lambda))\subset S^{n-1}$ has positive Lebesgue measure, or equivalently, positive $\mathcal{H}^{n-1}$ measure. If $\Pi_x(\lambda)$ is absolutely continuous with a upper semi continuous density function, then $\Pi_x(\mathrm{supp}(\lambda))$ has non-empty interior. For more applications, see Section [8](#sec: Integers){reference-type="ref" reference="sec: Integers"}. ## Linear Projection Let $n\geq 2$ be an integer. Let $\theta\in S^{n-1}$. Let $H_\theta$ be the ($n-1$ dimensional) linear subspace normal to $\theta.$ For each $y\in\mathbb{R}^n,$ there is a point $h_y\in H_\theta$ so that $y\in h_y+\theta\mathbb{R}.$ This $h_y$ is uniquely determined with respect to $y.$ Define the function $$P_{\theta}:y\in \mathbb{R}^n\to h_y\in H_\theta.$$ Thus $P_\theta$ is the linear projection that maps $\mathbb{R}^n$ onto $H_\theta.$ The fibres of this map $P_\theta$ are affine lines with direction $\theta.$ Notice that $P_\theta$ is the same as $P_{-\theta}.$ More generally, it is possible to define linear projections from $\mathbb{R}^n$ onto $k$ subspaces as long as $1\leq k\leq n-1.$ We only treat $k=n-1$ in this paper. In this setting, one has the general Marstrand projection theorem [@BP Section 3.5]. **Theorem 6**. *Let $n\geq 2$ be an integer. Let $\lambda$ be a Borel probability measure on $\mathbb{R}^n.$ Suppose that $$I_s(\lambda)=\int\int \frac{d\lambda(x)d\lambda(y)}{|x-y|^s}<\infty$$ for some $s>n-1.$ Then for Lebesgue almost all $\theta\in S^{n-1},$ $P_\theta(\lambda)$ is absolutely continuous with respect to the Lebesgue measure on $H_\theta$. In particular, $P_\theta(supp(\lambda))$ has positive Lebesgue measure for almost all $\theta\in S^{n-1}.$* As for radial projection, here we also have the following conjecture. **Conjecture 7**. *Let $n\geq 2.$ Let $\lambda$ be a missing digits measure (or a Cartesian product of missing digits measures). Suppose that $\dim_{\mathrm{H}}\lambda>n-1,$ then $P_\theta(\lambda)$ is absolutely continuous with a continuous density function for Lebesgue almost all $\theta\in S^{n-1}$.* **Remark 8**. *Here, if the reader would like to have a more challenging problem to solve, $\lambda$ can be replaced with a general self-similar measure with the open set condition.* Our method treating radial projections provides us with the following by-product which partially solves Conjecture [Conjecture 7](#conj: linear projection){reference-type="ref" reference="conj: linear projection"}. **Theorem 2**. *Let $\lambda_1$ be the missing digits measure on $\mathbb{R}$ with base $p=10^{10000}$ and the digit set $$D=\{1,2,\dots,10^{5005}\}.$$ Let $\lambda_2$ be the missing digits measure on $\mathbb{R}$ with base $p=11^{10000}$ and the digit set $$D=\{1,2,\dots,11^{5005}\}.$$ Then for almost all $\theta\in S^1,$ $P_\theta(\lambda_1\times\lambda_2),$ $P_\theta(\lambda_1\times\lambda_1)$ are continuous functions.* This implies, in particular, that $\mathrm{supp}{(\lambda_1)}+T_x\mathrm{supp}(\lambda_2)$ contains non-trivial interior for Lebesgue almost all $x\in\mathbb{R}.$ Here, $T_x$ is the map $y\in\mathbb{R}\to xy\in\mathbb{R}.$ In this direction, we recall the following verison of Palis' conjecture. **Conjecture 9**. *Let $A,B\subset [0,1]$ be two missing digits sets. Suppose that $$\dim_{\mathrm{H}}A+\dim_{\mathrm{H}}B>1.$$ Then 'generically', $A+B$ contains non-trivial interior.* **Remark 10**. *This is not a very honest (and not even precise) version of Palis' conjecture. In fact, to be honest, we should say that, 'generically', if $A+B$ has a positive Lebesgue measure then it also has a non-trivial interior. For Cantor sets defined via non-linear dynamics, (the honest version of) Palis' conjecture was proved by Moreira and Yoccoz in [@MY]. See [@PS], [@T1], [@T2] for more discussions on (the honest and precise version of) Palis' conjecture for (linear) self-similar sets.* In different settings, the meaning of the word 'generically' can differ from each other. In our situation with missing digits sets, we require that $A+T_x(B)$ contains a non-trivial interior for Lebesgue almost all $x\in\mathbb{R}.$ In this way, we see that Theorem [Theorem 2](#thm: B){reference-type="ref" reference="thm: B"} provides us with a special answer. We also suspect that the following much stronger result holds. This will provide us with a very definitive notion of 'genericity'. **Conjecture 11**. *Let $A,B\subset [0,1]$ be two missing digits sets. Suppose that the base $p_A$ of $A$ and $p_B$ of $B$ are such that $$\frac{\log p_A}{\log p_B}\notin\mathbb{Q},$$ and that $$\dim_{\mathrm{H}}A+\dim_{\mathrm{H}}B>1.$$ Then $A+T_x(B)$ contains non-trivial interior for all $x\in\mathbb{R}\setminus\{0\}.$ If $\log p_A/\log p_B\in\mathbb{Q},$ then for each $x\notin\mathbb{Q},$ $A+T_x(B)$ contains non-trivial interior.* Although we are nowhere near this conjecture, the following result gives us hope. **Theorem 12** (Furstenberg's conjecture [@Fu2]). *Let $A,B\subset [0,1]$ be two missing digits sets. Suppose that the base $p_A$ of $A$ and $p_B$ of $B$ are such that $$\frac{\log p_A}{\log p_B}\notin\mathbb{Q},$$ and that $$\dim_{\mathrm{H}}A+\dim_{\mathrm{H}}B>1.$$ Then $A+T_x(B)$ has full Hausdorff dimension for all $x\in\mathbb{R}\setminus\{0\}.$ If $\log p_A/\log p_B\in\mathbb{Q},$ then for each $x\notin\mathbb{Q},$ $A+T_x(B)$ has full Hausdorff dimension.* For the case when $\log p_A/\log p_B\notin\mathbb{Q},$ the result was proved in [@HS12]. For the other case, the result was proved in [@H14] as well as in [@Sh]. # General result The general result depend on the notion of $\dim_{l^1}.$ See Section [3.5](#sec: L1){reference-type="ref" reference="sec: L1"}. For now, we remark that for missing digits measures $\lambda$ with large bases and simple digits sets, $\dim_{l^1} \lambda$ is almost equal to $\dim_{\mathrm{H}}\lambda.$ **Theorem 13**. *Let $n\geq 2$ be an integer. Let $\lambda$ be a Borel probability measure on $\mathbb{R}^n.$* *Suppose that $$\dim_{l^1}\lambda>n-1.$$ For each $x\in\mathbb{R}^n\setminus \mathrm{supp}(\lambda),$ if $\dim_{\mathrm{H}}\Pi_x(\lambda)=n-1$ then $\Pi_x(\lambda)$ is absolutely continuous.* *Suppose that for an integer $p>1,$ $$\dim_{l^1}\lambda>n-p^{-1}.$$ Then for each $x\in\mathbb{R}^n\setminus \mathrm{supp}(\lambda),$ $\Pi_x(\lambda)$ is absolutely continuous with a density function in $L^{p}(S^{n-1}).$ In this case, the condition that $\dim_{\mathrm{H}}\Pi_x(\lambda)=n-1$ is not required to draw the conclusion.* **Remark 14**. *In some cases, the condition that $\dim_{\mathrm{H}}\Pi_x(\lambda)=n-1$ is automatic. For example, let $\lambda$ be a missing digits measure on $\mathbb{R}^2$ with $\dim_{\mathrm{H}}\lambda>1.$ Then [@Sh Theorem 6.2] implies that for each line $l\subset\mathbb{R}^2$ with irrational slope, for each $\epsilon>0,$ $\lambda(l^\delta)\ll \delta^{1-\epsilon}.$ This implies that for each $x\notin\mathrm{supp}(\lambda),$ each irrational $\theta\in S^1$ ($\theta=(\theta_1,\theta_2)$ with $\theta_1\theta_2\neq 0, ,\theta_1/\theta_2\notin\mathbb{Q}$), $\Pi_x(\lambda)(B_\delta(\theta))\ll \delta^{1-\epsilon}.$ Since $\dim_{\mathrm{H}}\lambda>1,$ $\lambda$ does not give positive measures for lines. This implies that $\Pi_x(\lambda)$ does not give positive measure for points. This implies further that the set of rational points in $S^1$ has zero $\Pi_x(\lambda)$ measure. This implies that $\dim_{\mathrm{H}}\Pi_x(\lambda)\geq 1-\epsilon$. Since $\epsilon>0$ can be arbitrarily chosen we see that $\dim_{\mathrm{H}}\Pi_x(\lambda)=1.$* *Similarly ([@Sh Theorem 7.2]), suppose that $\lambda$ is a Cartesian product of two missing digits measures on $\mathbb{R}$ with bases $b_1,b_2$ such that $\log b_1/\log b_2\notin\mathbb{Q}.$ If $\dim_{\mathrm{H}}\lambda>1,$ then for each line $l\subset\mathbb{R}^2$ with non-trivial slope, i.e. not being parallel with the coordinate axis, then for each $\epsilon>0,$ $\lambda(l^\delta)\ll \delta^{1-\epsilon}.$ From here we again have $\dim_{\mathrm{H}}\Pi_x(\lambda)=1$ for all $x\notin\mathrm{supp}(\lambda).$* *It is very likely that the aforementioned results in [@Sh] can be generalised to $\mathbb{R}^n$ for $n\geq 3.$ If this would be the case, then the condition $\dim_{\mathrm{H}}\Pi_x(\lambda)=n-1$ would be automatic if $\lambda$ is a missing digits measure or a Cartesian product of missing digits measures in lower-dimensional Euclidean spaces.* It is not easy to obtain $\dim_{l^1}\lambda$ for general measures $\lambda.$ However, for missing digits measures, it is possible to obtain useful estimates. See Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"}. Theorem [Theorem 1](#thm: A){reference-type="ref" reference="thm: A"} follows from Theorems [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"} and the above theorem. On the other hand, if $\lambda$ is a missing digits measure on $\mathbb{R}^n$ with $\dim_{\mathrm{H}}\lambda<n-1.$ Then $\Pi_x(\lambda)$ cannot be absolutely continuous for each $x\notin\mathrm{supp}(\lambda).$ Thus Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"} is almost sharp. In general, we have $\dim_{l^1}\lambda\leq \dim_{\mathrm{H}}\lambda$. In some cases, they can be almost the same. See Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"}. **Theorem 15**. *Let $n\geq 2$ be an integer. Let $\lambda$ be a Borel probability measure on $\mathbb{R}^n.$ Suppose that $$\dim_{l^1}\lambda>n-1.$$ Then for almost all $\theta\in S^{n-1},$ $P_\theta(\lambda)$ is absolutely continuous with a continuous density function.* Theorem [Theorem 2](#thm: B){reference-type="ref" reference="thm: B"} follows as a corollary of this Theorem and Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"}. # Preliminaries For more background materials in fractal geometry, see [@Fa], [@Ma1], [@Ma2]. ## Hausdorff dimension/measure Let $n\geq 1$ be an integer. Let $F\subset\mathbb{R}^n$ be a Borel set. Let $g: [0,1)\to [0,\infty)$ be a continuous function such that $g(0)=0$. Then for all $\delta>0$ we define the quantity $$\mathcal{H}^g_\delta(F)=\inf\left\{\sum_{i=1}^{\infty}g(\mathrm{diam} (U_i)): \bigcup_i U_i\supset F, \mathrm{diam}(U_i)<\delta\right\}.$$ The $g$-Hausdorff measure of $F$ is $$\mathcal{H}^g(F)=\lim_{\delta\to 0} \mathcal{H}^g_{\delta}(F).$$ When $g(x)=x^s$ then $\mathcal{H}^g=\mathcal{H}^s$ is the $s$-Hausdorff measure and Hausdorff dimension of $F$ is $$\dim_{\mathrm{H}}F=\inf\{s\geq 0:\mathcal{H}^s(F)=0\}=\sup\{s\geq 0: \mathcal{H}^s(F)=\infty \}.$$ Let $\mu$ be a Borel probability measure on $\mathbb{R}^n.$ We define the upper/lower Hausdorff dimension of $\mu$ as follows, $$\overline{\dim_{\mathrm{H}}} \mu=\inf\{\dim_{\mathrm{H}}A: \mu(A)=1\}, \underline{\dim_{\mathrm{H}}} \mu=\inf\{\dim_{\mathrm{H}}A: \mu(A)>0\}.$$ It is known that $$\underline{\dim_{\mathrm{H}}}\mu=\mathrm{essinf}_{x\sim \mu} \liminf_{\delta\to 0}\log \mu(B_\delta(x))/\log \delta.$$ In particular, if $\underline{\dim_{\mathrm{H}}}\mu=s>0$ then $\mu(A)=0$ for all $A$ with $\dim_{\mathrm{H}}A<s.$ In this paper, we only consider the lower Hausdorff dimension. For this reason, we will write $$\dim_{\mathrm{H}}\mu=\underline{\dim_{\mathrm{H}}} \mu.$$ ## Missing digits sets/measures {#sec: missing} We introduce the notion of missing digits sets. Let $n\geq 1$ be an integer. Let $p\geq 3$ be an integer. Let $D\subset\{0,\dots,p-1\}^{n}.$ Consider the set $$K_{p,D}=\mathrm{cl}\{x\in\mathbb{R}^n: [p\{p^kx\}]\in D,k\geq 0\},$$ where $\{x\},[x]$ are the component wise fractional part, integer part respectively of $x\in\mathbb{R}^n.$ Let $p_1,\dots,p_{\#D}$ be a probability vector, i.e. they are non-negative and sum to one. We can then assign each element in $D$ a probability weight. To be specific, one can first introduce an ordering on $D$ and assign the probabilities accordingly. We can now construct the random sum $$S=\sum_{i\geq 1} p^{-i} \mathbf{d}_i$$ where $\mathbf{d}_i\in D,i\geq 1$ are randomly and independently chosen from the set $D$ with the assigned probabilities. If $p_1=\dots=p_{\#D}=1/\#D,$ the distribution of $S$ is a Borel probability measure supported on $[0,1]^n.$ We call this measure to be $\lambda_{p,D}.$ It is a Borel probability measure supported on $K_{p,D}\cap [0,1]^n.$ Moreover, it is AD-regular with exponent $\dim_{\mathrm{H}}K_{p,D}.$ We also write $$\dim_{l^1}K_{p,D}=\dim_{l^1}\lambda_{p,D}.$$ ## AD-regularity {#sec: AD} Let $n\geq 1$ be an integer. Let $\mu$ be a Borel measure. Let $s>0$ be a number. We say that $\mu$ is $s$-regular, or AD-regular with exponent $s$ if there is a constant $C>1$ such that for all $x\in \mathrm{supp}(\mu)$ and all small enough $r>0$ $$C^{-1} r^s\leq \mu(B_r(x))\leq C r^s,$$ where $B_r(x)$ is the Euclidean ball of radius $r$ and centre $x$. For an AD-regular measure $\mu$, the exponent can be seen as $$s=\dim_{\mathrm{H}}\mathrm{supp}(\mu)=\dim_{\mathrm{H}}\mu.$$ Missing digits measure $\lambda_{p,D}$ in $\mathbb{R}^n$ are AD-regular measures with exponent $$s=\dim_{\mathrm{H}}\lambda_{p,D}=\dim_{\mathrm{H}}K_{p,D}=\frac{\log \#D}{\log p^n}.$$ ## Radial projections of measures {#sec: rad} Let $n\geq 2$ be an integer. Let $\lambda$ is a compactly supported Borel probability measure on $\mathbb{R}^n.$ For each $x\in \mathbb{R}^n,$ recall the radial projection function $$y\in\mathbb{R}^n\setminus\{x\}\to \Pi_x(y)=\frac{y-x}{|x-y|}\in S^{n-1}.$$ Assume that $x$ is not in the support of $\lambda.$ We now define the measure $\Pi_x(\lambda).$ Let $h$ be a continuous and real valued function on $S^{n-1}.$ Let $Ph$ be the function on $\mathbb{R}^n\setminus \{x\}$ such that $Ph(y)=h(\theta)$ if and only if $\Pi_{x}(y)=\theta.$ Then $\Pi_{x}(\lambda)$ can be defined via the equality $$\int_{S^{n-1}} h(\theta)d\Pi_{x}(\lambda)(\theta)=\int Ph(y)d\lambda(y)$$ for all continuous function $h$ on $S^{n-1}.$ The measure $\Pi_{x}(\lambda)$ is a compactly supported Borel probability measure on $S^{n-1}$. Alternatively, it is possible to define $\Pi_x(\lambda)$ by differentiation. Let $\theta\in S^{n-1}.$ Let $\delta>0.$ Let $B_\delta(\theta)\subset S^{n-1}$ be the metric ball (Euclidean metric) of radius $\delta.$ Define the function $$\theta\in S^{n-1}\to r_\delta(\theta)=\frac{\lambda(\Pi_x^{-1}(B_\delta(\theta)))}{L(B_\delta(\theta))}\in [0,\infty),$$ where $L$ is the Lebesgue probability measure on $S^{n-1}$. Then $r_\delta$ determines a compactly supported Borel probability measure on $S^{n-1}.$ It is possible to check that $\Pi_x(\lambda)$ is equal to the weak limit $\lim_{\delta\to 0} r_\delta$. ## Fourier norm dimensions {#sec: L1} Let $n\geq 1$ be an integer. Let $\lambda$ be a compactly supported Borel probability measure on $\mathbb{R}^n.$ Consider the Fourier transform $$\hat{\lambda}(\xi)=\int_{\mathbb{R}^n} e^{-2\pi i (x,\xi)}d\lambda(x),$$ where $(.,.)$ is the standard Euclidean bilinear form. **Definition 16**. *Let $p>0.$ We define $$\dim_{l^p}\lambda=\sup\left\{s>0: \sup_{\theta\in [0,1]^n}\sum_{|\xi|\leq R,\xi\in\mathbb{Z}^n} |\hat{\lambda}(\xi+\theta)|^p \ll R^{n-s}\right\}.$$* With the help of the Cauchy-Schwarz inequality, it is possible to show that $$\frac{\dim_{l^1}\lambda}{2}\leq \dim_{l^2}\lambda.$$ Moreover, we have for each AD-regular measure $\lambda$ $$\dim_{l^2}\lambda=\dim_{\mathrm{H}}\mu=\dim_{\mathrm{H}}\mathrm{supp}(\lambda).$$ Furthermore, let $n\geq 1$ be an integer. Let $\lambda_1,\dots,\lambda_n$ be a Borel probability measure on $\mathbb{R}.$ The $n$-fold Cartesian product $\lambda'=\lambda_1\times\dots\times \lambda_n$ satisfies $$\dim_{l^1}\lambda'\geq \dim_{l^1}\lambda_1+\dots+\dim_{l^1}\lambda_n.$$ We prove the following lemma. **Lemma 17**. *Let $n\geq 1$ be an integer. Let $\lambda$ be a Borel probability measure on $\mathbb{R}^{n}$ with $\dim_{l^1}\lambda=s\in (0,n).$ Then for each $\sigma>n-s,$ we have $$\int_{\mathbb{R}^n} \frac{|\hat{\lambda}(\xi)|}{|\xi|^{\sigma}+1}d\xi<\infty.$$* *Proof.* From the definition of $\dim_{l^1}\lambda$ we see that for each $\epsilon>0,$ $$\int_{2^{k-1}\leq |\xi|\leq 2^k}|\hat{\lambda}(\xi)|d\xi\ll 2^{k(n-s+\epsilon)}$$ as $k\to\infty.$ Thus we see that $$\int_{2^{k-1}\leq |\xi|\leq 2^k} 2^{-k(n-s+2\epsilon)}|\hat{\lambda}(\xi)|d\xi\ll 2^{-k\epsilon}.$$ This implies that $$\int_{\mathbb{R}^n} \frac{|\hat{\lambda}(\xi)|}{|\xi|^{n-s+2\epsilon}+1}d\xi<\infty.$$ Since $\epsilon>0$ can be arbitrarily chosen, the result concludes. ◻ The next result was proved in [@ACVY] (for $n=1$) and [@Y22] (for $n\geq 1$). **Theorem 18**. *Let $n\geq 1$ be an integer. The following results hold.* - *1: Let $t\geq 1$ be an integer. We have $$\liminf_{p\to\infty,\#D\geq p^n-t} \dim_{l^1} \lambda_{p,D}=n.$$ In particular, for each number $\epsilon>0,$ as long as $p$ is large enough, $\dim_{l^1}\lambda_{p,D}>n-\epsilon$ holds for each $D$ with $\#D=p^n-1.$* - *2: For each integer $p\geq 4,$ let $D\subset\{0,\dots,p-1\}^n$ be a 'rectangle', i.e. a set of form $[a_1,b_1]\times [a_2,b_2]\dots [a_n,b_n]\cap \{0,\dots,p-1\}^n.$ Then we have [^1] $$\begin{aligned} \label{T} \dim_{l^1}\lambda_{p,D}\geq \dim_{\mathrm{H}}\lambda_{p,D}-\frac{n\log\log p^2}{\log p}. \end{aligned}$$* This result only provides crude bounds of $\dim_{l^1}\lambda$ for missing digits measures $\lambda.$ See [@ACVY], [@Y21], [@Y22] for more detailed estimates and more applications. ## Asymptotic symbols We use the Vinogradov ($\ll,\gg,\asymp$) notations: Let $f(\delta), g(\delta)$ be two real valued quantities depending on $\delta>0.$ Then - $f\ll g$ if $|f(\delta)|\leq C|g(\delta)|$ for a constant $C>0$ and all $\delta>0.$ - $f\asymp g$ if $f\ll g$ and $g\ll f$. # Bump functions on rectangles {#sec: bump} Let $\phi: \mathbb{R}\to[0,\infty)$ be a non-zero Schwartz function such that $\hat{\phi}$ is supported on $[-1,1]$ and $0\leq \hat{\phi}(\xi)\leq 1$ for $\xi\in [-1,1]$. Then there is a constant $c=c_\phi>0$ such that for all $x\in\mathbb{R},$ $$0\leq \phi(x)\leq c.$$ Moreover, we have $\phi(0)>0.$ Thus, there are numbers $c',c''>0$ such that whenever $x\in [-c',c']$, $$\phi(x)>c''.$$ To find such a non-zero function $\phi,$ we can first choose a function $\phi'$ supported on $[-1/2,1/2]$ and $0\leq \phi'(\xi)\leq 1$ for $\xi\in [-1/2,1/2].$ We can also assume that $\phi'$ is an even function. Then we can choose $\phi=\hat{\phi'*\phi'}.$ Let $\delta>0.$ Let $\phi_\delta(x)=\delta^{-1}\phi(x/\delta).$ Then $\hat{\phi}_\delta$ is supported on $[-1/\delta,1/\delta]$ and $0\leq \hat{\phi}(\xi)\leq 1$ for all $\xi\in\mathbb{R}$. Of course, we also have $$0\leq \phi(x)\leq c/\delta$$ for all $x\in\mathbb{R}$. Next, for $x\in [-c'\delta,c'\delta],$ we have $$\phi(x)\geq \delta^{-1}c''.$$ Let $n>1$ be an integer. Let $r_1,\dots,r_n>0$ be positive numbers. Let $R_{r_1,\dots,r_n}$ be the rectangle $$R_{r_1,\dots,r_n}=[-r_1,r_1]\times\dots\times [-r_n,r_n].$$ Consider the function $\phi_{r_1,\dots,r_n},$ $$x=(x_1,\dots,x_n)\to \phi_{r_1,\dots,r_n}(x)=\prod_{i=1}^n \phi_{r_i}(x_i).$$ Then $\hat{\phi}_{r_1,\dots,r_n}$ is supported and with norm at most one on the rectangle $$\tilde{R}_{r_1,\dots,r_n}=[-r^{-1}_1,r^{-1}_1]\times\dots\times [-r^{-1}_n,r^{-1}_n].$$ Moreover, we have that for all $x\in\mathbb{R}^n,$ $$0\leq \phi_{r_1,\dots,r_n}(x)\leq c^n r^{-1}_1r^{-1}_2\times r^{-1}_n$$ and for all $x\in c'R_{r_1,\dots,r_n}= [-c'r_1,c'r_1]\times\dots\times [-c'r_n,c'r_n],$ we have $$\phi(x)\geq {c''}^{n}/(r_1\dots r_n).$$ Let $R\subset\mathbb{R}^n$ be a rectangle with side lengths $2r_1,\dots,2r_n.$ Let $\tilde{R}$ be the rectangle centred at the origin with side lengths $2r^{-1}_1,\dots,2r^{-1}_n$ and for each $i\in\{1,\dots,n\},$ the side of $\tilde{R}$ with length $r^{-1}_i$ is parallel with the side of $R$ with length $r_i.$ It is possible to find a rotation $g\in\mathbb{O}(n)$ and a translation $t\in\mathbb{R}^n$ such that $g(R_{r_1,\dots,r_n})+t=R.$ In this case, the function $\phi_R(x)=\phi_{r_1,\dots,r_n}(gx+t)$ has the property that that $\hat{\phi}_R$ is supported and with norm at most one on the rectangle $\tilde{R}.$ Moreover, we have that for all $x\in\mathbb{R},$ $$0\leq \phi_R(x)\leq c^nr^{-1}_1\dots r^{-1}_n$$ and for $x\in c'R,$ we have $$\phi_R(x)\geq {c''}^{n}/(r_1\dots r_n).$$ We note that $\hat{\phi}_R$ may not be real valued. # Proof of Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"}: the $L^p, p\geq 2$ case {#proof-of-theorem-thm-n1-the-lp-pgeq-2-case} We prove the second conclusion of Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"}. The proof of the first conclusion (case $p=1$) is of a different nature and will be provided in the next section. The proof in this section is divided into 4 steps. In the first two steps. We provide the proof for the case $n=p=2.$ In this way, it is easier to picture the underlying ideas. In Step 3, we will provide the proof for larger $n.$ Finally, in Step 4, we extend the proof to deal with the cases when $p\geq 2.$ ## Step 1 {#step-1 .unnumbered} Let $\lambda$ be an $AD$-regular probability measure with $\dim_{l^1}\lambda>3/2.$ Let $K$ be the support of $\lambda$. Consider the radial projection $\Pi_x(\lambda)$ of the measure $\lambda$ at $x\in\mathbb{R}^2.$ Without loss of generality, we can assume that $x=(0,0)$ and that $K$ is contained in the ball centred at $(0,0)$ with radius one. Moreover, $(0,0)$ is not in the support of $\lambda.$ We now write $\Pi(\lambda)$ instead of $\Pi_x(\lambda).$ Let $\delta>0.$ For each $\theta\in S^1,$ let $T^\delta_\theta$ be the rectangle centred at the origin of side lengths $2,2\delta$ whose long side is pointing towards the direction $\theta$, i.e. the long side has a direction vector parallel with $\theta.$ Consider the function $f_\delta$ $$\theta\in S^1\to f_\delta(\theta)=\frac{\lambda(T^\delta_\theta)}{\delta}\in [0,\infty).$$ Since $\lambda$ is $AD$-regular with dimension larger than one, we see that $\lambda$ does not give positive measures to lines. This implies that $f_\delta$ is continuous. Our goal is to show that $$\begin{aligned} \label{eqn: toshow} \int_{S^1} f^2_\delta(\theta)d\theta\ll 1.\end{aligned}$$ This implies that there is a sequence $\delta_i\to 0$ and a function $f\in L^2(S^1)$ such that $f_{\delta_1}\to f$ weakly (in the separable Hilbert space $L^2(S^1)$). The function $f_\delta$ is not the same as $r_\delta$ defined in Section [3.4](#sec: rad){reference-type="ref" reference="sec: rad"}. Since $\lambda$ is compactly supported and $(0,0)$ is not in the support of $\lambda,$ we see that $\lambda(T^\delta_\theta)$ is almost the same as $\lambda(\Pi^{-1}(B_\delta(\theta)))$. More precisely, there is a constant $0<C<1$ depending on $\lambda$ such that for all $\theta\in S^1,$ $$\Pi^{-1}(B_{C\delta}(\theta))\subset T^\delta_\theta,$$ and on the other direction, we have (as the support of $\lambda$ omit a ball centered at $(0,0)$) $$\lambda(T^\delta_\theta)\leq \lambda(\Pi^{-1}(B_{C^{-1}\delta}(\theta))).$$ This implies that $$f_\delta(\theta)\ll r_\delta(\theta)\ll f_\delta(\theta)$$ for all $\theta$ and the implicit constants in $\ll$ depend only on $\lambda.$ If ([\[eqn: toshow\]](#eqn: toshow){reference-type="ref" reference="eqn: toshow"}) holds, then $$\int_{S^1} r^2_\delta(\theta) d\theta\ll 1$$ as well. As above, we see that the limit $\lim_{\delta}r_\delta$ exists both in distribution and $L^2.$ This implies that $\Pi(\lambda)$ is absolutely continuous with respect to the Lebesgue measure and has an $L^2$ density function. ## Step 2 {#step-2 .unnumbered} Recall the bump function $\phi_{{c'}^{-1}T^\delta_\theta}.$ We see that $$\lambda(T^\delta_\theta)\ll \delta\int \phi_{{c'}^{-1}T^\delta_\theta}(x) d\lambda(x).$$ This is because $\phi_{{c'}^{-1}T^\delta_\theta}$ takes value in $[0,(c/c')^2 /\delta]$ and for $x\in c' ({c'}^{-1}T^\delta_\theta)=T^\delta_\theta,$ we have $$\phi_{{c'}^{-1}T^\delta_\theta}(x)\geq {(c''/c')}^2 /\delta.$$ Recall that $c,c',c''>0$ are constants depending on the choice of the bump function $\phi$ in Section [4](#sec: bump){reference-type="ref" reference="sec: bump"}. By using Parseval's theorem, we see that $$\int \phi_{{c'}^{-1}T^\delta_\theta}(x) d\lambda(x)=\int \hat{\phi}_{{c'}^{-1}T^\delta_\theta}(\xi)\hat{\lambda}(-\xi)d\xi.$$ Notice that $\hat{\phi}_{{c'}^{-1}T^\delta_\theta}$ is supported and with norm at most one on the rectangle $\tilde{{c'}^{-1}T^\delta_\theta}.$ For convenience, we write $R^\delta_\theta=\tilde{{c'}^{-1}T^\delta_\theta}.$ The rectangle $R^\delta_\theta$ has side lengths $2c'$ and $2c'/\delta.$ The side with length $2c'$ is pointing at the direction $\theta.$ From here we see that $$\int \hat{\phi}_{{c'}^{-1}T^\delta_\theta}(\xi)\hat{\lambda}(-\xi)d\xi\leq \int_{R^\delta_\theta} |\hat{\lambda}(\xi)|d\xi.$$ Then we see that $$f_\delta(\theta)\ll\int_{R^\delta_\theta} |\hat{\lambda}(\xi)|d\xi,$$ where the implicit constant in $\ll$ depends on $\phi$ only. Thus we have $$\int_{S^1} f^2_\delta(\theta)d\theta\ll \int_{S^1}\int\int |\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi' d\theta.$$ We now consider the triple integral above. By using Fubini's theorem, we see that $$\int_{S^1}\int_{R^\delta_\theta}\int_{R^\delta_\theta} |\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi' d\theta=\int_{R^\delta_\theta}\int_{R^\delta_\theta} h(\xi,\xi') |\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi',$$ where the function $h(\xi,\xi')$ is defined via $$h(\xi,\xi')=|\{\theta\in S^1: \xi,\xi'\in R^\delta_\theta\}|.$$ Here $|A|$ for a measurable set $A\subset S^1$ is the Lebesgue measure of $A.$ It is possible to see that for each $\xi\in\mathbb{R}^2$ $$|\{\theta\in S^1: \xi\in R^\delta_\theta\}|\ll \frac{1}{|\xi|},$$ where the implicit constant in $\ll$ depends on $\phi.$ Thus we see that $$h(\xi,\xi')\ll \frac{1}{\max\{|\xi|,|\xi'|\}}\leq \frac{1}{|\xi|^{1/2}|\xi'|^{1/2}}.$$ This implies that $$\int\int h(\xi,\xi') |\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi'\ll \left(\int |\hat{\lambda}(\xi)||\xi|^{-1/2}d\xi\right)^2.$$ Since $\dim_{l^1}\lambda>3/2,$ Lemma [Lemma 17](#lma: l1 integral){reference-type="ref" reference="lma: l1 integral"} implies that $$\int |\hat{\lambda}(\xi)||\xi|^{-1/2}d\xi<\infty.$$ From here we see that $$\int_{S^1} f^2_\delta(\theta)d\theta\ll 1.$$ This is what we wanted to show in ([\[eqn: toshow\]](#eqn: toshow){reference-type="ref" reference="eqn: toshow"}). ## Step 3 {#step-3 .unnumbered} The general case follows with similar arguments although it may not be easy to picture the ideas. Now let $n\geq 3$ be an integer and $\lambda$ be a compactly supported Borel probability measure on the unit ball centred at the origin with $\dim_{l^1}\lambda>n-(1/2).$ For the radial projection $\Pi_x$, we assume that $x=(0,\dots,0).$ Moreover, we assume that $(0,\dots,0)$ is not in the support of $\lambda$. We again write $\Pi$ instead of $\Pi_x.$ Let $1/2>\delta>0$ and $\theta\in S^{n-1}.$ Now, let $T^\delta_\theta$ be the rectangle of side lengths $2,2\delta,\dots,2\delta$ centred at the origin with the side of length $2$ being parallel with $\theta.$ Consider the function $$\theta\in S^{n-1}\to f_\delta(\theta)=\frac{\lambda(T^\delta_\theta)}{\delta^{n-1}}\in [0,\infty).$$ We want to show that $$\int_{S^{n-1}}f^2_\delta(\theta)d\theta\ll 1.$$ This will conclude the result just as in the case when $n=2$. Consider the bump function $\phi_{{c'}^{-1}T^\delta_\theta}.$ The Fourier transform of $\phi_{{c'}^{-1}T^\delta_\theta}$ is essentially supported on a rectangle $R^\delta_\theta$. The rectangle $R^\delta_\theta$ has side lengths $$2c', 2c'/\delta, \dots, 2c'/\delta$$ and it is centred at the origin. Moreover, the side with length $2c'$ is parallel with $\theta.$ Intuitively speaking, the rectangle $R^\delta_\theta$ is a 'thin plate' normal to $\theta.$ Next, with the same arguments as in the previous steps, it is possible to see that $$\int_{S^{n-1}}f^2_\delta(\theta)d\theta\ll \int_{R^\delta_\theta}\int_{R^\delta_\theta}h(\xi,\xi')|\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi'$$ where $$h(\xi,\xi')=|\{\theta\in S^{n-1}: \xi,\xi'\in R^\delta_\theta\}|.$$ Similar to the case when $n=2,$ it is possible to check that $$|\{\theta\in S^{n-1}: \xi\in R^\delta_\theta\}|\ll\frac{1}{|\xi|}.$$ Loosely speaking, this is because that $$\{\theta\in S^{n-1}: \xi\in R^\delta_\theta\}\subset S^{n-1}$$ is a stripe of width roughly $1/|\xi|.$ Thus we see that[^2] $$h(\xi,\xi')\ll \frac{1}{\max\{|\xi|,|\xi'|\}}.$$ This implies that $$\int\int h(\xi,\xi') |\hat{\lambda}(\xi)||\hat{\lambda}(\xi')|d\xi d\xi'\ll \left(\int |\hat{\lambda}(\xi)||\xi|^{-1/2}d\xi\right)^2.$$ Since $\dim_{l^1}\lambda>n-(1/2),$ the result concludes as in the previous step. ## Step 4 {#step-4 .unnumbered} We now consider the case when $p>2.$ As before, the goal is to show that as $\delta\to 0,$ $$\int_{S^{n-1}}f^p_{\delta}(\theta)d\theta\ll 1.$$ The arguments in Step 3 (or Step 2) can be borrowed without problems. Notice that $$\int_{S^{n-1}}f^{p}_\delta(\theta)d\theta\ll \int_{R^\delta_\theta}\int_{R^\delta_\theta}\dots \int_{R^\delta_\theta} h(\xi_1,\xi_2,\dots,\xi_p)|\hat{\lambda}(\xi_1)||\hat{\lambda}(\xi_2)|\dots |\hat{\lambda}(\xi_p)|d\xi_1\dots d\xi_p,$$ where $$h(\xi_1,\dots,\xi_p)=|\{\theta\in S^{n-1}: \xi_1,\dots,\xi_p\in R^{\delta}_\theta\}|\ll \frac{1}{\max\{|\xi_1|,\dots,|\xi_p|\}}.$$ Since $\dim_{l^1}\lambda>n-p^{-1},$ we see that $$\int_{S^{n-1}}f^{p}_\delta(\theta)d\theta\ll \left(\int \frac{1}{|\xi|^{1/p}}|\hat{\lambda}(\xi)|d\xi\right)^p\ll 1.$$ This is what we wanted to show. # Proof of Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"}: the $L^1$ case {#sec: planar} We prove the first conclusion of Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"}. We no longer separate the proof of the special case $n=2$. However, with $n=2,$ the geometrical idea is more transparent. Let $\lambda$ be a Borel probability measure on $\mathbb{R}^n$ with $s_1=\dim_{l^1}\lambda>n-1.$ We assume that $(0,\dots,0)\notin\mathrm{supp}(\lambda)$ and consider $\Pi(\lambda)=\Pi_x(\lambda)$ with $x=(0,\dots,0).$ Assume that $\dim_{\mathrm{H}}\Pi(\lambda)=n-1.$ The result for other points $x\in\mathbb{R}^n$ can be considered in a similar way. Let $R>1.$ Consider the annulus $$A_R=\{R\leq |\xi|\leq 2R\}.$$ We see that for each $\epsilon>0,$ as $R\to\infty,$ $$\int_{A_R} |\hat{\lambda}(\xi)|d\xi\ll R^{n-s_1+\epsilon}.$$ For each $\theta\in S^{n-1},$ let $S_{1/R}(\theta)\subset S^{n-1}$ be the stripe of width $1/R$ 'normal' to $\theta.$ More precisely, it is the set $$S_{1/R}(\theta)=\{\theta'\in S^{n-1}: |(\theta,\theta')|\leq 1/R\}.$$ Consider the set $$Q_{R}(\epsilon)=\left\{\theta\in S^{n-1}: \int_{\xi\in A_R, \xi/|\xi|\in S_{1/R}(\theta)}|\hat{\lambda}{(\xi)}|d\xi\geq R^{n-1-s_1+2\epsilon} \right\}.$$ We claim that $Q_R(\epsilon)$ can be covered with $\ll R^{n-1-\epsilon}$ many balls of radius $4/R.$ To see this, cover $S^{n-1}$ with $\asymp R^{n-1}$ many balls of radius $2/R$ whose centres at least $1/R$-separated from each other. Furthermore, it is possible to arrange that the maximum multiplicity of those balls is less than $C_n>1,$ a constant which depends on $n$ only. Let $\theta_i,i\geq 1$ be a numeration of the centres of those balls. Consider the sum $$\sum_{i\geq 1} \int_{S_{1/R}(\theta_i)}|\hat{\lambda}(\xi)|d\xi=\int |\{i\geq 1: \xi\in S_{1/R}(\theta_i)\}||\hat{\lambda}(\xi)|d\xi.$$ Observe that $$|\{i\geq 1: \xi\in S_{1/R}(\theta_i)\}|\ll R^{n-2}.$$ To see this, we temporally assume (without loss of generality) that $\xi=(1,0,\dots,0).$ In this case, we need $\theta_i$ to have $$|(\xi,\theta_i)|\leq R^{-1}.$$ Namely, we write this condition in coordinates ($\theta_i=(\theta_{i,1},\dots,\theta_{i,n})$) as $$|\theta_{i,1}|\leq R^{-1}$$ and $$\sum_{j=1}^n \theta^2_{i,j}=1.$$ We require that $\theta_i's$ are $1/R$-separated. From here, we can confirm the observation. Therefore we see that $$\sum_{i\geq 1} \int_{S_{1/R}(\theta_i)}|\hat{\lambda}(\xi)|d\xi\ll R^{n-2}R^{n-s_1+\epsilon}.$$ Thus there are at most $$\ll R^{n-1-\epsilon}$$ many $i$ with $$\int_{S_{1/R}(\theta_i)}|\hat{\lambda}(\xi)|d\xi\geq R^{n-1-s_1+2\epsilon}.$$ This proves the claim. Let $k\geq 1$ be an integer. From the claim, we see that $Q_{2^k}(\epsilon)$ can be covered with $\ll 2^{k(n-1-\epsilon)}$ many balls of radius $\asymp 1/2^{k}.$ Let $1>\rho_\epsilon>(n-1-\epsilon)/(n-1)$ be a positive number. We see that $$\sum_{k\geq 1} 2^{k(n-1-\epsilon)} (1/2^{k})^{(n-1)\rho_\epsilon}<\infty.$$ From the convergence Hausdorff-Borel-Cantelli lemma ([@HS18]), we see that $$\limsup_{k\to\infty} Q_{2^k}(\epsilon)\subset S^{n-1}$$ has zero $(n-1)\rho_\epsilon$-Hausdorff measure. This implies that $$\dim_{\mathrm{H}}\limsup_{k\to\infty} Q_{2^k}(\epsilon)\leq (n-1)\rho_\epsilon<n-1.$$ Since $\dim_{\mathrm{H}}\Pi(\lambda)=n-1,$ we see that $\Pi(\lambda)(\limsup_{k\to\infty} Q_{2^k}(\epsilon))=0.$ In other words, for $\Pi(\lambda)$ almost all $\theta\in S^{n-1},$ there are at most finitely many $k\geq 1$ such that $\theta\in Q_{2^k}(\epsilon).$ Since $s_1>n-1,$ it is possible to choose $\epsilon>0$ such that $n-1-s_1+2\epsilon<0.$ Fix such a choice of $\epsilon$. Let $\theta\in S^{n-1}$ be such that $\theta\in Q_{2^k}(\epsilon)$ for at most finitely many $k.$ Let $P^{1/100}_\theta$ be the slab normal to $\theta,$ centred at the origin, with width $1/100,$ $$P^{1/100}_\theta=\{x\in\mathbb{R}^n: |(\theta,x)|\leq 1/200\}.$$ Then we see that $$\begin{aligned} \label{eqn: L^1 Fourier} \int_{P^{1/100}_\theta} |\hat{\lambda}(\xi)|d\xi=\int_{P^{1/100}_\theta\cap B_1(0)} |\hat{\lambda}(\xi)|d\xi+\sum_{k\geq 0}\int_{P^{1/100}_\theta\cap A_{2^k}} |\hat{\lambda}(\xi)|d\xi<\infty.\end{aligned}$$ This is because for all large enough $k,$ $$dir(P^{1/100}_\theta\cap A_{2^k})\subset S_{1/{2^k}}(\theta),$$ where $dir(x)=x/|x|$ for $x\in\mathbb{R}^n\setminus\{0\}$ and $dir(A)=\{dir(x): x\in A\}\subset \mathbb{S}^{n-1}.$ Thus there is a constant $c'''>0$ (depending on the choice of $\phi$ in Section [4](#sec: bump){reference-type="ref" reference="sec: bump"}) such that $$\lambda(T^{c'''\delta}_\theta)/\delta^{n-1}\ll \int_{P^{1/100}_\theta}|\hat{\lambda}(\xi)|d\xi<\infty.$$ In other words (rescale $\delta$ if necessary), we have $$\limsup_{\delta\to 0}\lambda(T^{\delta}_\theta)/\delta^{n-1}<\infty.$$ Recall the function $f_\delta(\theta)=\lambda(T^\delta_\theta)/(\delta^{n-1}).$ Consider the limit $\lim_{\delta\to 0}f_\delta$ as a measure. This limit measure is equivalent to the measure $\Pi(\lambda).$ We have seen that for $\Pi(\lambda)$ almost all $\theta\in S^{n-1},$ $$\liminf_{\delta\to 0}f_\delta(\theta)\leq \limsup_{\delta\to 0} f_\delta(\theta)<\infty.$$ Thus the limit measure $\lim_{\delta\to 0}f_\delta$ and $\Pi(\lambda)$ are absolutely continuous with respect to the Lebesgue measure on $S^{n-1}.$ See [@Ma2 Theorem 2.11]. # Proof of Theorem [Theorem 15](#thm: n2){reference-type="ref" reference="thm: n2"} {#proof-of-theorem-thm-n2} Let $\theta\in S^{n-1}.$ By performing a rotation, let us assume that $\theta=(0,\dots,0,1).$ We consider the projection $P_{\theta}(\lambda).$ For doing this, observe that for each $\xi\in\mathbb{R}^{n-1},$ $$\hat{P_\theta(\lambda)}(\xi)=\int_{\mathbb{R}^{n-1}} e^{-2 \pi i (\xi,x)}dP_\theta(\lambda)(x)=\int_{\mathbb{R}^{n}} e^{-2 \pi i (\xi',x')}d\lambda(x')=\hat{\lambda}(\xi'),$$ where $\xi'$ is the unique element in $\{\xi\}\times\{0\}\subset\mathbb{R}^n.$ Let $\xi'_t$ be the unique element in $\{\xi\}\times\{t\}$ for $t\in\mathbb{R}.$ Consider the function $$J_{\xi}:t\to \hat{\lambda}(\xi'_t).$$ Then we see that (write $x'=(x'_1,\dots,x'_{n})$) $$J'_\xi(t)=\int_{\mathbb{R}^{n}} e^{-2\pi i (\xi'_t,x')} (-2\pi i x'_{n})d\lambda(x').$$ The equality holds because $\lambda$ is compactly supported. Thus we see that $$|J'_\xi(t)|\leq 2\pi \int_{\mathbb{R}^{n}} |x'_n| d\lambda(x')<\infty.$$ Notice that the RHS above is independent with respect to $\xi\in\mathbb{R}^{n-1}.$ This implies that ($d_{n-1}$ is the Lebesgue measure on $\mathbb{R}^{n-1}\times \{0\}$, $d_n$ is the Lebesgue measure on $\mathbb{R}^n$) $$\int_{\mathbb{R}^{n-1}\times\{0\}}|\hat{\lambda}(\xi')|d_{n-1} (\xi')\ll \int_{\mathbb{R}^{n-1}\times [-1/100,1/100]} |\hat{\lambda}(\xi')|d_n(\xi').$$ Thus as long as $$\int_{\mathbb{R}^{n-1}\times [-1/100,1/100]} |\hat{\lambda}(\xi')|d_n(\xi')<\infty,$$ we have, for $\theta=(0,\dots,1),$ $$\int_{\mathbb{R}^{n-1}}|\hat{P_\theta(\lambda)}(\xi)|d\xi<\infty.$$ It follows that $P_\theta(\lambda)$ has a absolutely integrable Fourier transform. This implies that the measure $P_\theta(\lambda)$ has a continuous density function. From ([\[eqn: L\^1 Fourier\]](#eqn: L^1 Fourier){reference-type="ref" reference="eqn: L^1 Fourier"}), we see that there is a set $E\subset S^{n-1}$ with Hausdorff dimension smaller than $n-1$ such that as long as $\theta\notin E,$ after performing a suitable rotation so that $\theta$ becomes $(0,\dots,1),$ the rotated measure (still written as $\lambda$) satisfies, $$\int_{\mathbb{R}^{n-1}\times [-1/100,1/100]} |\hat{\lambda}(\xi')|d_n(\xi')<\infty.$$ From here we conclude that $P_\theta(\lambda)$ has a continuous density function for all such $\theta\notin E.$ Since $E$ does not have full Hausdorff dimension in $S^{n-1},$ it also has zero Lebesgue measure. From here, the proof finishes. # Integers with restricted digits: Graham's problem {#sec: Integers} The difficulty level 4 part of Conjecture [Conjecture 3](#conj){reference-type="ref" reference="conj"} has a consequence in number theory. Let $k\geq 1$ be an integer. Let $2<b_1<b_2<\dots<b_k$ be integers. Let $D_1\subset\{0,\dots,b_1\},\dots,D_k\subset\{0,\dots,b_k\}$ be digit sets. Let $N^{D_1,\dots,D_k}_{b_1,\dots,b_k}$ be the set of positive integers whose base $b_1$ expansion contains digits only in $D_1,$ ..., base $b_k$ expansion contains digits only in $D_k.$ We have the following result. See [@Y20]. **Theorem 19**. *Assume the difficulty level 4 part of Conjecture [Conjecture 3](#conj){reference-type="ref" reference="conj"}. Suppose that $$1,\frac{\log b_1}{\log b_2},\dots,\frac{\log b_1}{\log b_k}$$ are $\mathbb{Q}$-linearly independent. If $$\sum_{i=1}^{k} \frac{\log |D_i|}{\log b_i}>k-1,$$ then $N^{D_1,\dots,D_k}_{b_1,\dots,b_k}$ is infinite.* The above theorem would answer a question of Graham which asks whether or not $N^{\{0,1\},\{0,1,2\},\{0,1,2,3\}}_{3,5,7}$ is infinite. In addition to the challenging difficulty level 4 problem, the $\mathbb{Q}$-linear independence of $$1,\frac{\log b_1}{\log b_2},\dots,\frac{\log b_1}{\log b_k}$$ is also not easy to be checked if $k\geq 3$. Those are the two heavy boulders in front of Graham's problem.[^3] From Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"} and Theorem [Theorem 13](#thm: n1){reference-type="ref" reference="thm: n1"}, it is possible to draw some weaker conclusions. Let $t_1,\dots,t_k$ be numbers in $(0,1].$ Let $N^{D_1,\dots,D_k}_{b_1,\dots,b_k}(t_1,\dots,t_k)$ be the set of positive integers such that the base $b_1$ expansion of $[t_1n]$ contains digits only in $D_1,$ ..., the base $b_k$ expansion of $[t_k n]$ contains digits only in $D_k.$ **Theorem 20**. *Let $4<b_1<b_2$ be integers with $\log b_1/\log b_2\notin\mathbb{Q}.$ Let $D_1, D_2$ be consecutive digit sets. If $$\sum_{i=1}^{2} \left(\frac{\log |D_i|}{\log b_i}-\frac{\log(\log b_i^2)}{\log b_i}\right)>1,$$ then $N^{D_1,D_2}_{b_1,b_2}(t_1,t_2)$ is infinite for Lebesgue almost all $(t_1,t_2)\in (0,1]^2$.* It is not difficult to find explicit examples. For instance, let $b_1=10^{10000}$ and $b_2=11^{10000}.$ Consider the digit sets $$D_1=\{1,\dots,10^{5005}\}, D_2=\{1,\dots,11^{5005}\}.$$ Without the above theorem, it is not clear whether or not there exist $t,t'\in (0,1]$ such that $$N^{D_1,D_2}_{b_1,b_2}(t,t')$$ is infinite. Of course, Conjecture [Conjecture 3](#conj){reference-type="ref" reference="conj"} implies that the above set is infinite for all $t,t'\in (0,1]$ including the most interesting case $t=t'=1.$ # Acknowledgement HY was financially supported by the University of Cambridge and the Corpus Christi College, Cambridge. 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Yu, *Rational points near self-similar sets*, preprint, arXiv:2101.05910, (2021). H. Yu, *Missing digits points near manifolds*, arXiv:2309.00130 # Appendix {#appendix .unnumbered} For being self-contained, we now prove Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"}. ## Step 1 {#step-1-1 .unnumbered} Let $n\geq 1$ be an integer. Let $p>2$ be an integer and $D\subset\{0,\dots,p-1\}^n.$ Consider the missing digit measure $\lambda_{p,D}.$ In this case, $\hat{\lambda}(\xi)$ can be computed with the help of the formula, $$\hat{\lambda}_{p,D}(\xi)=\prod_{j\geq 0} \frac{1}{\#D}\sum_{d\in D} e^{-2\pi i (d,\xi)/p^j}.$$ For convenience, let $$g(\xi)=\frac{1}{\#D}\sum_{d\in D} e^{-2\pi i (d,\xi)}.$$ Then we have $$\hat{\lambda}_{p,D}(\xi)=\prod_{j\geq 0} g(\xi/p^j).$$ We want to estimate for large integers $k\geq 1,$ $$S_k=\sup_{\theta\in [0,1]^n}\sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^k}|\hat{\lambda}_{p,D}(\xi+\theta)|.$$ Notice that in the sum, we conditioned on the max norm $|\xi|_\infty=\max\{|\xi_1|,\dots,|\xi_n|\}.$ We now estimate $S_k.$ Let $\theta\in (0,1)^n$ be a vector. Consider the function $$f(\theta)=\sum_{\mathbf{i}\in\{0,\dots,p-1\}^n} |g((\mathbf{i}+\theta)/p)|.$$ Clearly we have for all $\theta,$ $$0\leq f(\theta)\leq p^n.$$ Observe that for each $\theta\in [0,1]^n$ $$\begin{aligned} &S_{k}(\theta)=\sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k}}|\hat{\lambda}_{p,D}(\xi+\theta)|\\ &=\sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k}}\left|\prod_{j\geq 0}g((\xi+\theta)/p^j)\right|.\end{aligned}$$ Let $\xi'=\xi+\mathbf{i}$ for some $\mathbf{i}\in p^{k-1}\mathbb{Z}^n.$ Then we have $$g((\xi'+\theta)/p^j)=g((\xi+\theta)/p^j)$$ for all $j=0,1,\dots,k-1.$ From here we see that (recall that $|g|\leq 1$) $$\begin{aligned} S_k(\theta)&=\sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k}}\left|\prod_{j\geq 0}g((\xi+\theta)/p^j)\right|\\ &\leq \sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k}}\left|\prod^{k}_{j= 0}g((\xi+\theta)/p^j)\right| \\ &= \sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty<p^{k-1}}\sum_{\mathbf{i}\in\{0,\dots,p-1\}^n p^{k-1}} \left|\prod^{k}_{j= 0}g((\xi+\mathbf{i}+\theta)/p^j)\right| \\ &= \sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k-1}}\left|\prod_{j=0}^{k-1} g((\xi+\theta)/p^j)\right|\sum_{\mathbf{i}\in \{0,\dots,p-1\}^n }\left|g(\mathbf{i}p^{-1}+\theta p^{-k}+\xi p^{-k})\right|\\ &\leq \sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k-1}}\left|\prod_{j=0}^{k-1} g((\xi+\theta)/p^j)\right|\sup_{\theta'} f(\theta')\\ &\overset{\text{Continue inductively}}{\leq} \sum_{\xi\in\mathbb{Z}^n,|\xi|_\infty< p^{k-2}}\left|\prod_{j=0}^{k-2} g((\xi+\theta)/p^j)\right|(\sup_{\theta'} f(\theta'))^2\\ &\dots\\ &\leq (\sup_{\theta'} f(\theta'))^k.\end{aligned}$$ Thus we have $$S_k(\theta)\leq (\sup_{\theta'}f(\theta'))^{k}.$$ Therefore we see that $$S_k\leq (\sup_{\theta'}f(\theta'))^{k}.$$ This implies that (we take the Euclidean norm $|\xi|$) $$\sup_{\theta\in [0,1]^n}\sum_{\xi\in\mathbb{Z}^n,|\xi|\leq p^k}|\hat{\lambda}_{p,D}(\xi+\theta)|\leq S_k\leq (\sup_{\theta'}f(\theta'))^{k}.$$ From here one can see that $$n-\frac{\log \sup_{\theta}f(\theta)}{\log p}\leq \dim_{l^1}\lambda_{p,D}.$$ ## Step 2: Proof of Part 1 {#step-2-proof-of-part-1 .unnumbered} We now a upper bound for the value $$\sup_{\theta} f(\theta)$$ when $D$ is a large set. This will give us a lower bound for $\dim_{l^1}\lambda_{p,D}.$ First, observe that $$\#D g(\xi)=\prod_{j=1}^n\frac{1-e^{2\pi i p\xi_j}}{1-e^{2\pi i \xi_j}}-\sum_{d\notin D}e^{-2\pi i (d,\xi)}.$$ Let $\# D=p^n-t$ for some integer $t>0.$ Then we have $$-t\leq | (p^n-t) g(\xi)|-\left|\prod_{j=1}^n\frac{1-e^{2\pi i p\xi_j}}{1-e^{2\pi i \xi_j}}\right|\leq t.$$ Now we want to consider the sum $$f_1(\theta)=\sum_{\mathbf{i}\in\{0,p-1\}^n} |g_1((\mathbf{i}+\theta)/p)|,$$ where $$g_1(\xi)=\prod_{j=1}^n\frac{1-e^{2\pi i p\xi_j}}{1-e^{2\pi i \xi_j}}.$$ To do this, consider the function $h:\mathbb{R}\to\mathbb{R},$ $$h(x)=\left|\frac{1-e^{2\pi i px}}{1-e^{2\pi i x}}\right|.$$ We want to provide an estimate for $$H(\theta)=\sum_{j\in\{0,\dots,p-1\}}h((j+\theta)/p).$$ Notice that ($|e^{ix}-1|\leq 2, \forall x\in\mathbb{R}$) $$\begin{aligned} \label{eqn: *} H(\theta)\leq 2 \sum_{j=0, \{(j+\theta)/p\}\geq 1/p}^{p-1}\frac{1}{|1-e^{2\pi i (j/p)}e^{2\pi i (\theta/p)}|}+\sum_{j: \{(j+\theta)/p\}<1/p} h((j+\theta)/p).\tag{*} \end{aligned}$$ For the first sum in ([\[eqn: \*\]](#eqn: *){reference-type="ref" reference="eqn: *"}), we see that ($|1-e^{2\pi i x}|^2=2(1-\cos(2\pi x))\geq 16x^2$, for $x\in [0,1/2]$) $$\begin{aligned} &2\sum_{j=0, \{(j+\theta)/p\}\geq 1/p}^{p-1}\frac{1}{|1-e^{2\pi i (j/p)}e^{2\pi i (\theta/p)}|}\leq 8 \sum_{k=1}^{[(p-1)/2]} \frac{1}{|1-e^{2\pi i (k/p)}|}\\ &\leq \frac{1}{\pi }\sum_{k=1}^{[(p-1)/2]} \frac{k}{p} \leq \frac{p}{\pi} (\log p+1). \end{aligned}$$ For the second sum in ([\[eqn: \*\]](#eqn: *){reference-type="ref" reference="eqn: *"}), notice that there are at most two $j$'s in the sum. As $|h|\leq p$, we have $$\sum_{j: \{(j+\theta)/p\}<1/p} h((j+\theta)/p)\leq 2p.$$ From here we see that for $p\geq 4,$ $$H(\theta)\leq 2p\log p.$$ We can then use this estimate to see that $$\sup_{\theta} f_1(\theta)\leq (2p\log p)^n.$$ Thus we have $$(p^n-t)\sup_{\theta}f(\theta)\leq tp^n+(2p\log p)^n.$$ This implies that $$\begin{aligned} \label{Crude bound} \dim_{l^1}\lambda_{p,D}\geq n-\frac{\log \frac{tp^n+(2p\log p)^n}{p^n-t} }{\log p}.\tag{Crude bound} \end{aligned}$$ This ([\[Crude bound\]](#Crude bound){reference-type="ref" reference="Crude bound"}) tells us, for example, if we fix $t>0,$ then as long as $p$ is large enough, $\dim_{l^1}\lambda_{p,D}$ can be arbitrarily close to $n.$ This finishes the proof of Part 1 of Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"} ## Step 3: Proof of Part 2 {#step-3-proof-of-part-2 .unnumbered} Let $n\geq 1$ be an integer. Let $p>1$ be an integer. For a missing digits set, we choose a subset $D\subset\{0,\dots,p-1\}^n$ and construct $K_{p,D}.$ We found a lower bound for $\dim_{l^1}$ with the help of ([\[Crude bound\]](#Crude bound){reference-type="ref" reference="Crude bound"}). It turns out that if the digit sets are chosen to be well structured, then we can have a much better estimate than the ([\[Crude bound\]](#Crude bound){reference-type="ref" reference="Crude bound"}). Let $D\subset \{0,\dots,p-1\}^n$ be an rectangle, i.e. it is of form $$[a_1,b_1]\times\dots\times [a_n,b_n]\cap \{0,\dots,p-1\}^n$$ with integers $a_1\leq b_1,\dots,a_n\leq b_n.$ With this special digit set $D$, we see that the corresponding function $g_{b,D}$ is of form $$g_{b,D}(\theta)=(\#D)^{-1}\sum_{z\in D} e^{-2\pi i((z,\theta))}=(\#D)^{-1}\prod_{j=1}^n e^{-2\pi i a_j \theta_j}\frac{1-e^{-2\pi i (b_j-a_j+1)\theta_j}}{1-e^{-2\pi i\theta_j}}.$$ Next, we estimate the sum $$(\#D) f_{b,D}(\theta)=(\#D)\sum_{\mathbf{i}\in\{0,\dots,p-1\}^n}|g_{p,D}(\mathbf{i}+\theta)/p|.$$ For each $j\in \{1,\dots,n\},$ define $$S_j(\theta_j)=\left|\frac{1-e^{-2\pi i (b_j-a_j+1)\theta_j}}{1-e^{-2\pi i\theta_j}}\right|.$$ Then we see that $$\begin{aligned} (\#D)\sum_{\mathbf{i}\in\{0,\dots,p-1\}^n}|g_{p,D}(\mathbf{i}+\theta)/p|\\ =\sum_{\mathbf{i}\in\{0,\dots,p-1\}^n}\prod_{j=1}^n S_j((\mathbf{i}_j+\theta_j)/p)\\ =\prod_{j=1}^n \sum_{i\in\{0,\dots,p-1\}} S_j((i+\theta_j)/p). \end{aligned}$$ Now we need to estimate for each $j\in\{1,\dots,n\},$ $$\sum_{i=0}^{p-1} S_j((i+\theta_j)/p).$$ We have already considered this type of sums before, see ([\[eqn: \*\]](#eqn: *){reference-type="ref" reference="eqn: *"}). Notice that $S_j(\theta_j)\leq b_j-a_j+1.$ As a result, we have for $p\geq 4,$ $$\sup_{\theta_j} \sum_{i=0}^{p-1} S_j((i+\theta_j)/p)\leq \frac{p}{\pi}(\log p+1)+2(b_j-a_j+1)\leq 2p\log p.$$ Thus we see that $$\sup_{\theta} f_{b,D}(\theta)\leq (\# D)^{-1} (2p\log p)^n.$$ From here we see that $$\begin{aligned} \dim_{l^1}\lambda_{p,D}\geq& n-\frac{\log ((\# D)^{-1} (2p\log p)^n)}{\log p}\\ &= n+\frac{\log \# D}{\log p}-n-\frac{n\log\log p^2}{\log p}\\ &= \dim_{\mathrm{H}}K_{p,D}-\frac{n\log\log p^2}{\log p}, \end{aligned}$$ where have used the fact that $\dim_{\mathrm{H}}K_{p,D}=\log \#D/\log p.$ This finishes the Part 2 of Theorem [Theorem 18](#thm: l1 bound){reference-type="ref" reference="thm: l1 bound"}. [^1]: *The base of $\log$ in this paper is $e$.* [^2]: We remark that the inequality here is rather crude. In fact, for "generic' $\xi,\xi'$ we have $$h(\xi,\xi')\ll \frac{1}{|\xi||\xi'|}.$$ Our upper bound corresponds to the worst-case scenario when the two stripes (one for $\xi$ and one for $\xi'$) have a large intersection, e.g. they can be the same stripe. [^3]: Of course, there might be a completely different approach to Graham's question that could bypass the two boulders.

arxiv_math

--- abstract: | Praeger-Xu graphs are connected, symmetric, 4-regular graphs that are unusual both in that their automorphism groups are large, and in that vertex stabilizer subgroups are also large. Determining number and distinguishing number are parameters that measure the symmetry of a graph by investigating additional conditions that can be imposed on a graph to eliminate its nontrivial automorphisms. In this paper, we compute the values of these parameters for Praeger-Xu graphs. Most Praeger-Xu graphs are 2-distinguishable; for these graphs we also proved the cost of 2-distinguishing. author: - Sally Cockburn - "Max Klivans [^1]" bibliography: - PXBibliography.bib title: Symmetry Parameters of Praeger-Xu Graphs --- **Keywords**: Praeger-Xu graphs; determining number; distinguishing number; cost of 2-distinguishing. **Subject Classification:** 05C15, 05C25, 05C69 # Introduction A finite simple graph $G = (V, E)$ consists of a finite, nonempty set $V$ of vertices, and a set of $2$-subsets of $V$, called edges. An appealing feature of graphs is that they can be represented geometrically, with dots corresponding to vertices and lines between dots corresponding to edges. Certain graphs have the property that when the positions of the dots are carefully chosen, this geometric representation displays visual symmetry. However, a graph is an set-theoretic object, not a drawing. There are various ways to give a rigorous mathematical characterization of symmetry. An *automorphism* $\alpha$ of a graph $G = (V,E)$ is a permutation of $V$ such that for all $u, v \in V$, $\{u, v\} \in E$ if and only if $\{\alpha(u), \alpha(v)\} \in E$. The set of automorphisms of $G$, denoted $\operatorname{Aut}(G)$, is a group under composition. For example, the automorphism group of the complete graph on $n$ vertices is the entire group of permutations on $n$ elements; that is, $\operatorname{Aut}(K_n) = S_n$. For $n\ge 3$, the automorphism group of the cycle $C_n$ is the dihedral group $D_n$, consisting of rotations and reflections. One way of characterizing the symmetry of a graph is to determine whether the vertices and/or edges play the same role, in the following sense. A graph $G$ is *vertex-transitive* if for all $u, v \in V$, there is $\alpha \in \operatorname{Aut}(G)$ such that $\alpha(v) = u$. Similarly, $G$ is *edge-transitive* if for all $\{u, v\}, \{x, y\} \in E$, there is $\alpha \in \operatorname{Aut}(G)$ such that $\alpha(\{u, v\}) = \{\alpha(u), \alpha(v)\} = \{x, y\}$. More stringently, a graph is *arc-transitive* if for all $\{u, v\}, \{x, y\} \in E$, there is $\alpha \in \operatorname{Aut}(G)$ such that $\alpha(u) = x$ and $\alpha(v) = y$. Connected, arc-transitive graphs are automatically both vertex-transitive and edge-transitive, and are simply called *symmetric* graphs. Complete graphs $K_n$ and cycles $C_n$ are examples of symmetric graphs. Another way of characterizing the symmetry of a graph $G$ is to quantify extra measures that can taken to prevent the existence of nontrivial automorphisms of $G$. As one example, we could require that automorphisms of $G$ fix point-wise a subset $S$ of vertices. If the only automorphism doing so is the identity, then $S$ is called a *determining set* of $G$. The *determining number* of $G$, denoted $\operatorname{Det}(G)$, is the minimum size of a determining set of $G$. (Some authors use the term *fixing* instead of determining, for both sets and numbers.) Graphs with no nontrivial automorphisms, sometimes called asymmetric or rigid graphs, have determining number $0$; at the opposite end of the spectrum, $\operatorname{Det}(K_n) = n-1$. A minimum determining set for $C_n$ is any set of two non-antipodal vertices, so $\operatorname{Det}(C_n) = 2$. As another example, we could paint the vertices with different colors and require that automorphisms preserve set-wise the color classes. Graph $G$ is *d-distinguishable* if the vertices can be colored with $d$ colors in such a way that the only automorphism preserving the color classes is the identity. The *distinguishing number* of $G$, denoted $\operatorname{Dist}(G)$, is the minimum number of colors required for a distinguishing coloring. For a discussion of elementary properties of determining numbers and distinguishing numbers, as well as the connections between them, see [@AlBo2007]. Remarkably, many infinite families of symmetric graphs have been found to have distinguishing number $2$, including hypercubes  [@BoCo2004], Cartesian powers $G^{\Box n}$ of a connected graph where $G\ne K_2,K_3$ and $n\geq 2$ [@Al2005; @ImKl2006; @KlZh2007] and Kneser graphs $K_{n:k}$ with $n\geq 6, k\geq 2$ [@AlBo2007]. Boutin [@B2008] introduced an additional invariant in such cases; the *cost of 2-distinguishing* $G$, denoted $\rho(G)$, is the minimum size of a color class in a 2-distinguishing coloring of $G$. In this paper, we find these symmetry parameters for a family of symmetric graphs called Praeger-Xu graphs. They are remarkable among all connected, symmetric, 4-regular graphs for having very large automorphism groups. Moroever, there is an infinite family of Praeger-Xu graphs with the property that the smallest subgroup of automorphisms that acts transitively on the vertices has an arbitrarily large vertex stabilizer. For these and more results on Praeger-Xu graphs, see [@OTCOPXG], [@PSV2010] and [@JPW2019]. The large automorphism group suggest that they might have large determining and distinguishing numbers; the large vertex stabilizers suggest the opposite. This paper is organized as follows. In Section [2](#sec:PXIntro){reference-type="ref" reference="sec:PXIntro"}, we provide a definition of the Praeger-Xu graphs and facts about their automorphism groups. In Section [3](#sec:PXn1){reference-type="ref" reference="sec:PXn1"}, we show that most Praeger-Xu graphs are twin-free; for those with twins, we use a quotient graph construction to find the determining and distinguishing number. In Section [4](#sec:DetNoTwins){reference-type="ref" reference="sec:DetNoTwins"}, we find the determining number for twin-free Praeger-Xu graphs. As a tool for computing distinguishing number, in Section [5](#sec:Interchangeable){reference-type="ref" reference="sec:Interchangeable"} we characterize pairs of vertices in twin-free Praeger-Xu graphs that are interchangeable via an automorphism. Finally, in Section [6](#sec:DistNoTwins){reference-type="ref" reference="sec:DistNoTwins"} we show that all twin-free Praeger-Xu graphs are $2$-distinguishable and compute the cost of $2$-distinguishing. Our results are summarized in Table [1](#tab:summary){reference-type="ref" reference="tab:summary"}. Parameter Value Condition(s) ----------------------------------------------- --------------------------------- --------------------------------------------------------- -- -- $\operatorname{Det}(\operatorname{PX}(n,k))$ 6 $(n,k) = (4,1)$ $\lceil\frac{n}{k}\rceil$ $k \neq \frac{n}{2}$ but $(n,k) \neq (4,1)$ $\lceil\frac{n}{k}\rceil+1 = 3$ $k = \frac{n}{2}$ $\operatorname{Dist}(\operatorname{PX}(n,k))$ 5 $(n,k) = (4,1)$ $3$ $n \neq 4$, $k=1$ $2$ $k \ge 2$ $\rho(\operatorname{PX}(n,k))$ $5$ $(n,k) = (4,2)$ $(k \ge 2)$ $\lceil\frac{n}{k}\rceil$ $5 \le n < 2k$, or $2k<n \text{ and } n \notin \{0 \bmod k, -1 \bmod k\}$ $\lceil\frac{n}{k}\rceil + 1$ otherwise : Summary of Symmetry Parameters ($n \ge 3$) # Praeger-Xu Graphs, $\operatorname{PX}(n,k)$ {#sec:PXIntro} In 1989, Praeger and Xu [@SGOTPV] introduced a family of connected graphs they denoted by $C(m,r,s)$, where $m\ge 2$, $r \ge 3$ and $s \ge 1$, that are vertex-transitive for $r \ge s$ and arc-transitive, hence symmetric, for $r \ge s+1$. This was part of an investigation into connected, symmetric graphs whose automorphism groups have the property that for any vertex $v$, the subgroup of automorphisms fixing $v$ (the stabilizer of $v$) does not act primitively on the set of neighbors of $v$. The Praeger-Xu graphs are those where $p=2$; the notation $PX(n,k)$ denotes $C(2, n, k)$. There are several ways of describing Praeger-Xu graphs (see [@GP1994]); we use what is called the bitstring model. **Definition 1**. *Let $n\geq3$ and $1\leq k<n$. The corresponding the Praeger-Xu graph is $\operatorname{PX}(n,k)=(V,E)$, where $V$ is the set of all ordered pairs $(i,x)$, where $i\in\mathbb Z_n$ and $x= x_0x_1 \cdots x_{k-1}$ is a bitstring of length $k$, and $\{(i,x),(j,y)\}\in E$ if and only if $j=i+1$ and $x=az_1z_2\cdots z_{k-1}$ and $y=z_1z_2\cdots z_{k-1}b$ for some $z_1,\dots,z_{k-1},a,b\in \mathbb Z_2$.* Throughout this paper, subscripts on bits will be considered elements of $\mathbb Z_k$. We say that the bit $x_j$ in $x$ is *flipped* if it is switched to $x_j+1$ in $\mathbb Z_2$. There is a natural partition of $V$ into *fibres* $\mathcal F_i=\{(i,x): x\in \mathbb Z_2^k\}$ for each $i\in\mathbb Z_n$. Each fibre is an independent set of $2^k$ vertices; every vertex in $\mathcal F_i$ is adjacent to exactly two vertices in each of $\mathcal F_{i+1}$ and $\mathcal F_{i-1}$, so $\operatorname{PX}(n,k)$ is $4$-regular, or tetravalent. Two fibres $\mathcal F_i$ and $\mathcal F_j$ are *antipodal* if and only if $n$ is even and $i-j = \frac{n}{2} \bmod n$. Two Praeger-Xu graphs are illustrated in Figure [\[fig:EZsamples\]](#fig:EZsamples){reference-type="ref" reference="fig:EZsamples"}. Figure [1](#fig:px32){reference-type="ref" reference="fig:px32"} shows the smallest Praeger-Xu graph having $k>1$, namely $\operatorname{PX}(3,2)$, of order $3 \cdot 2^2 = 12$. Figure [2](#fig:px205){reference-type="ref" reference="fig:px205"} shows the larger Praeger-Xu graph $\operatorname{PX}(20,5)$ of order $20\cdot 2^5 = 640$. In all our diagrams of Praeger-Xu graphs, $\mathcal F_0$ is the fibre in the $12$ o'clock position, with remaining fibres labeled consecutively clockwise. The vertices in $\mathcal F_0$ on $\operatorname{PX}(3,2)$ have been labeled with their bitstring components; the bitstring components of vertices in $\mathcal F_1$ and $\mathcal F_2$ follow the same pattern. More generally, the bitstring components are the binary representations of the integers $0$ to $2^k$, starting with the innermost vertex. Throughout this paper, we will be assuming that $n \ge 3$ and $1 \le k < n$, unless otherwise explicitly indicated. {#fig:px32} {#fig:px205} ## Automorphisms of $\operatorname{PX}(n,k)$ In [@SGOTPV], Praeger and Xu described the automorphism groups of all graphs in the family $C(p, r, s)$. We will adopt the notation used in [@OTCOPXG] for automorphisms of the Praeger-Xu graphs, $PX(n,k)$. The automorphism group is generated by three different types of automorphisms. The first is the rotation $\rho$, defined by $\rho\cdot(i,x)=(i+1,x).$ Composing $\rho$ with itself $s$ times corresponds to a rotation by $s$ fibres: $\rho^s\cdot(i,x)=(i+s,x)$. If $s$ is a multiple of $n$, then the resulting map is the identity, and so we can interpret $s$ as an element of $\mathbb Z_n$. The second automorphism is the reflection defined by $\mu\cdot(i,x) = (-i, x^-),$ where $x^- = (x_0x_1 \cdots x_{k-1})^- = x_{k-1} \cdots x_1x_0$. It is easily verified that $\mu^2 =$ id and $\mu \rho \mu = \rho^{-1}$, so the subgroup $\langle \rho, \mu\rangle$ of $\operatorname{Aut}(\operatorname{PX}(n,k))$ is the dihedral group $D_n$. Following [@OTCOPXG], for each $s\in\mathbb Z_n$ we let $\mu_s = \rho^{s+1-k} \mu \in \langle \rho, \mu\rangle$, so that $\mu_s\cdot(i,x)=(s+1-k-i,x^-).$ With this notation, $\mu=\mu_{k-1}$; in particular, note that $\rho^0=\text{id}$ but $\mu_0\neq\text{id}$. We collect some elementary facts about the reflections $\mu_s$ in the following lemma. **Lemma 1**. *Let $s,i,j\in\mathbb Z_n$.* (1) *The reflection $\mu_s$ interchanges fibres $\mathcal F_i$ and $\mathcal F_{s+1-k-i}$; equivalently, fibres $\mathcal F_i$ and $\mathcal F_j$ are interchanged by $\mu_{i+j+k-1}$.* (2) *If $n$ is odd, then each $\mu_s$ preserves exactly one fibre. If $n$ is even and $s= k \bmod 2$, then $\mu_s$ does not preserves any fibre, and if $s \neq k \bmod 2$, then $\mu_s$ preserves exactly two antipodal fibres.* *Proof.* The proof is straightforward and left to the reader. ◻ The third type of automorphism is, for each $s\in \mathbb Z_n$, defined by $$\tau_s\cdot(i,x)=\begin{cases} (i,x^{s-i}), \quad & \text{if } i\in \{s, s-1, s-2, \dots, s-k+1\},\\ (i,x), & \text{otherwise,}\end{cases}$$ where $x^j$ denotes the bitstring $x$ with bit $x_j$ flipped. Thus $\tau_s$ flips bit $x_{s-i}$ of the bitstring component of every vertex in $\mathcal F_i$ if $i\leq s\leq i+k-1$, and acts trivially on $\mathcal F_i$ otherwise. Equivalently, vertices in $\mathcal F_i$ have their bitstring components altered only by $\tau_i, \tau_{i+1}, \dots, \tau_{i+k-1}$. Clearly each $\tau_s$ has order $2$ and $\tau_s,\tau_t$ commute for all $s, t \in \mathbb Z_n$. Hence the subgroup of $\operatorname{Aut}(\operatorname{PX}(n,k))$ generated by these automorphisms satisfies $K=\langle \tau_0,\tau_1,\tau_2,\dots,\tau_{n-1}\rangle\simeq \mathbb Z_2^n$. Each $\tau\in K$ can be represented by $$\tau=\tau_0^{u_0}\tau_1^{u_1}\tau_2^{u_2}\cdots\tau_{n-1}^{u_{n-1}},$$ where $u_m\in\{0,1\}$ for each $m\in\mathbb Z_n$. It is easy to verify that $\rho^{-1}\tau_s \rho = \tau_{s+1}$ and $\mu \tau_s \mu = \tau_{k-1-s}$, so $K$ is a normal subgroup of the group generated by $\rho, \mu$ and $\tau_0, \dots \tau_{n-1}$. Let $\mathcal{A}= K \rtimes \langle\rho,\mu\rangle = K \rtimes D_n$. Then if $\alpha\in\mathcal{A}$, $\alpha=\tau\delta$ for some $\tau\in K$ and $\delta\in\langle\rho,\mu\rangle = D_n$. Praeger and Xu showed in [@SGOTPV] that for all $n\neq 4$, $\mathcal{A}=\operatorname{Aut}(\operatorname{PX}(n,k))$, while for $n=4$, $\mathcal{A}$ is a proper subgroup of $\operatorname{Aut}(\operatorname{PX}(4,k))$. Note that under any $\alpha \in \mathcal A$, vertices in the same fibre will be mapped to vertices in the same fibre. In other words, the fibres form a block system for the action of $\mathcal A$ on $\operatorname{PX}(n,k)$. From [@OTCOPXG], the induced action of $\alpha = \tau\delta\in \mathcal A$ on the fibres of $\operatorname{PX}(n,k)$ is $\alpha(\mathcal F_i) = \mathcal F_{\delta(i)}$, where $\delta(i)$ is simply the action of the dihedral group element $\delta \in D_n$ on $i \in V(C_n) = \mathbb Z_n$. Since any $\tau=\tau_0^{u_0}\tau_1^{u_1}\tau_2^{u_2}\cdots\tau_{n-1}^{u_{n-1}} \in K$ preserves fibres, for any $\alpha=\tau\delta\in\mathcal{A}$ and $(i,x)\in V$, we have $$\alpha\cdot (i,x)=\tau\cdot(\delta\cdot (i,x))=\tau\cdot(\delta(i),y)=(\delta(i),z),$$ where $y = x$ if $\delta$ is a rotation $\rho^s$, $y = x^-$ if $\delta$ is a reflection $u_s$, and for all $j \in \mathbb Z_k$, $z_j = y_j + 1$ if $u_{\delta(i) -j}=1$ and $z_j = y_j$ otherwise. # Determining and Distinguishing $\operatorname{PX}(n,1)$ {#sec:PXn1} For any vertex $v$ in a graph $G=(V,E)$, the (open) *neighborhood* of $v$ is $N(v) = \{u : \{u,v\} \in E\}$. Distinct vertices $x$ and $y$ are (nonadjacent) *twins* if and only if $N(x)=N(y)$. Twins are relevant to notions of graph symmetry because if $x$ and $y$ are twins, then the map that interchanges $x$ and $y$ and fixes all other vertices is a graph automorphism. **Theorem 1**. *For $k=1$, two distinct vertices in $\operatorname{PX}(n,1)$ are twins if and only if either they are in the same fibre, or $n=4$ and they are in antipodal fibres. For $k\geq2$, $\text{PX}(n,k)$ is twin-free.* *Proof.* First assume $k=1$. For any $i\in\mathbb Z_n$, $\mathcal F_i=\{(i,0),(i,1)\}$. By definition, $$N((i,0))=\{(i+1,0),(i+1,1),(i-1,0),(i-1,1)\} = \mathcal F_{i+1} \cup \mathcal F_{i-1} = N((i,1)).$$ Hence $(i,0)$ and $(i,1)$ are twins. More generally, $(i,x)$ and $(j,y)$ are twins if and only if $\mathcal F_{i+1} \cup \mathcal F_{i-1} = \mathcal F_{j+1} \cup \mathcal F_{j-1}$, or equivalently, as subsets of $\mathbb Z_n$, $\{i+1, i-1\} = \{j+1, j-1\}$. If $i \neq j$, then $i+1 = j-1$ and $i-1 = j+1$. This implies $i-j = 2 = -2$in $\mathbb Z_n$. Since $n \ge 3$, we can conclude $n=4$ and $\mathcal F_i$ and $\mathcal F_j$ are antipodal fibres. Next assume $k\ge 2$. Let $u$ and $v$ be distinct vertices in $\operatorname{PX}(n,k)$ such that $N(u)=N(v)$. Let $u=(i,axb)$ and $v=(j,cyd)$ for some $i, j \in \mathbb Z_n$, $a, b, c, d \in \{0, 1\}$, and $y, x \in \mathbb Z_2^{k-2}$ (where $y$ and $x$ are empty strings if $k=2$). By definition, $N(u)=\{(i+1,xb0),(i+1,xb1),(i-1,0ax),(i-1,1ax)\}$ and $N(v)=\{(j+1,yd0),(j+1,yd1),(j-1,0cy),(j-1,1cy)\}.$ Since $N(u)$ consists of two vertices in each of $\mathcal F_{i+1}$ and $\mathcal F_{i-1}$, and $N(v)$ consists of two vertices in each of $\mathcal F_{j+1}$ and $\mathcal F_{j-1}$, $\{i+1, i-1\} = \{j+1, j-1\}$. Suppose $i= j \bmod n$. Comparing neighbors in $\mathcal F_{i+1} = \mathcal F_{j+1}$ with the same final bit gives $xb0=yd0$ and $xb1=yd1$. Hence $xb=yd$ in $\mathbb Z_2^{k-1}$. An analogous argument can be used in $\mathcal F_{i-1}$ to show that $ax=cy$ in $\mathbb Z_2^{k-1}$. Thus $axb=cyd$ in $\mathbb Z_2^k$. Since $i= j$ in $\mathbb Z_n$, $(i,axb)=(j,cyd)$ and so $u=v$, contradicting the assumption that $u$ and $v$ are distinct. Alternatively, if $i\neq j \bmod n$, then as argued earlier in this proof, $n=4$ and $i-1 =j+1 \bmod n$. Hence $N(u)\cap\mathcal F_{i-1}= N(v) \cap \mathcal F_{j+1}$, so $$\{(i-1,0ax),(i-1,1ax)\} = \{(j+1, yd0), (j+1, yd1)\}.$$ Since $x$ and $y$ cannot be simultaneously both $0$ and $1$, this is impossible. ◻ Figure [\[fig:EZsamples\]](#fig:EZsamples){reference-type="ref" reference="fig:EZsamples"} depicts two Praeger-Xu graph with twins. Note that every vertex of $\operatorname{PX}(4,1)$ is in a set of $t=4$ mutual twins, while for any $n\ge 3, n\neq 4$, every vertex of $\operatorname{PX}(n,1)$ is in a set of $t=2$ mutual twins. {#fig:px61} {#fig:px41} For any graph $G= (V,E)$, can define an equivalence relation on $V$ by $x\sim y$ if and only if $x$ and $y$ are twins. The corresponding *twin quotient graph* $\widetilde{G}$ has as its vertex set the set of equivalence classes $[x]=\{y\in V(G):x\sim y\}$, with $\{[x],[z]\}\in E(\widetilde{G})$ if and only if there exist $p\in[x]$ and $q\in[z]$ such that $\{p,q\}\in E(G)$. (Note that by definition of the twin relation, $\{[x],[z]\}\in E(\widetilde{G})$ if and only if for all $p\in[x]$ and $q\in[z]$, $\{p,q\}\in E(G)$.) The symmetry parameters of the twin quotient graph $\widetilde G$ can be used to give information about the symmetry parameters of $G$. A subset $T\subseteq V(G)$ that contains all but one of the vertices from each equivalence class of twin vertices is called a *minimum twin cover* of $G$. In 2020, Boutin *et al.* proved the following result. **Theorem 2**. *[@BCKLPR2020b Theorem 19] Let $G$ be a graph with twin quotient graph $\widetilde G$. Let $T$ be a minimum twin cover of $G$ and let $\widetilde T = \{[u] : u \in T\} \subseteq V(\widetilde G)$. Let $\widetilde S$ be a determining set for $\widetilde G$ containing $\widetilde T$. Let $R=\{x\in V(G) : [x]\in \widetilde S\setminus \widetilde T\}$. Then $S=T \cup R$ is a determining set for $G$. Furthermore, if $\widetilde S$ is of minimum size among determining sets for $\widetilde G$ that contain $\widetilde T$, then $S$ is a minimum determining set for $G$.* For distinguishing number, we have the following result from Cockburn and Loeb. **Theorem 3**. *[@CL2023 Theorem 2] Let $G$ be a graph in which every vertex is in a set of $t$ mutual twins. If $\operatorname{Dist}(\widetilde G) = \widetilde d$, then $\operatorname{Dist}(G) = d$, where $d$ is the smallest positive integer such that $\binom{d}{t} \ge \widetilde d$.* These theorems can be used to find the symmetry parameters of all Praeger-Xu graphs with twins, $\operatorname{PX}(n,1)$. **Theorem 4**. *For $n=4$, $\operatorname{Det}(\operatorname{PX}(4,1)) = 6$ and $\operatorname{Dist}(\operatorname{PX}(4,1)) = 5$. For all $n \neq 4$, $\operatorname{Det}(\operatorname{PX}(n,1)) = n$ and $\operatorname{Dist}(\operatorname{PX}(n,1)) = 3$.* *Proof.* For $n=4$, there are only two equivalence class under the twin relation, namely $[(0,0)]=\mathcal F_0 \cup \mathcal F_2$ and $[(1,0)] = \mathcal F_1 \cup \mathcal F_3$, so the twin quotient graph is $\widetilde{\operatorname{PX}}(4,1) = K_2$. In this case, $T = \{(0,0), (0,1), (2,0), (1,0), (1,1), (3,0)\}$ is a minimum twin cover with $\widetilde T = V(\widetilde{\operatorname{PX}}(4,1))$. The only determining set $\widetilde S$ of the twin quotient graph containing $\widetilde T$ is the entire vertex set. Hence by Theorem [Theorem 2](#thm:DettG){reference-type="ref" reference="thm:DettG"}, $T$ is a minimum determining set and so $\operatorname{Det}(\operatorname{PX}(4,1)) = 6$. Since every vertex of $\operatorname{PX}(4,1)$ is in a set of $t=4$ mutual twins and $\operatorname{Dist}(K_2)= 2$, by Theorem [Theorem 3](#thm:DistTwins){reference-type="ref" reference="thm:DistTwins"}, $\operatorname{Dist}(\operatorname{PX}(4,1)) = 5$. Next assume $n \neq 4$. In this case, the equivalence classes under the twin relation are the fibres of $\operatorname{PX}(n,1)$ and so $T=\{(i,0):i\in\mathbb Z_n\}$ is a minimum twin cover of $\operatorname{PX}(n,1)$. Additionally, every vertex in $V(\operatorname{PX}(n,1))$ belongs to one of these fibres, so $V(\widetilde{\operatorname{PX}}(n,1))=\widetilde{T}$. Again the only determining set $\widetilde S$ of the twin quotient graph containing $\widetilde T$ is the entire vertex set and so by Theorem [Theorem 2](#thm:DettG){reference-type="ref" reference="thm:DettG"}, $T$ is a minimum determining set. Thus $\operatorname{Det}(\operatorname{PX}(n,1)) = |T| = n$. Since every fibre in $\operatorname{PX}(n,1)$ is a vertex in $\widetilde{\operatorname{PX}}(n,i)$, and vertices in $\mathcal{F}_i$ are adjacent only to vertices in $\mathcal{F}_{i+1}$ and $\mathcal{F}_{i-1}$, $\widetilde{\operatorname{PX}}(n,1)= C_n$. Hence, $\widetilde{d}=\text{Dist}(\widetilde{\operatorname{PX}}(n,1))=\text{Dist}(C_n)=3$ if $n \in \{3, 5\}$, and $\widetilde{d}=2$ if $n\geq 6$. Since each vertex is in a set of $t=2$ twins, by Theorem [Theorem 3](#thm:DistTwins){reference-type="ref" reference="thm:DistTwins"}, $d$ is the smallest integer such that ${d\choose2} \geq3$ if $n \in \{3, 5\}$, and $d$ is the smallest integer such that ${d\choose2}\geq2$ if $n \ge 6$. In both cases, $d=3$. ◻ # Determining $\operatorname{PX}(n,k)$, $k \ge 2$ {#sec:DetNoTwins} In this section, we find the determining number for twin-free Praeger-Xu graphs. Recall from Section [2](#sec:PXIntro){reference-type="ref" reference="sec:PXIntro"} that for $n \neq 4$, $\mathcal A = K \rtimes \langle \rho, \mu \rangle=\operatorname{Aut}(\operatorname{PX}(n,k))$, whereas for $n=4$, $\mathcal A$ is a proper subgroup of $\operatorname{Aut}(\operatorname{PX}(4,k))$. We begin with a lemma that applies to all Praeger-Xu graphs and apply it to the general case $n \neq 4$. We then consider the exceptional cases $\operatorname{PX}(4,2)$ and $\operatorname{PX}(4,3)$. **Lemma 2**. *Let $i\in\mathbb Z_n$, $S_i\subseteq \mathcal F_i$ and $\tau = \tau_0^{u_0} \cdots \tau_{n-1}^{u_{n-1}}\in K$. If $\tau(S_i)=S_i$ and $|S_i|$ is odd, then $\tau$ acts trivially on $\mathcal F_i$; equivalently, $$u_i = u_{i+1} = \dots = u_{i+k-1} = 0.$$* *Proof.* Assume $\tau(S_i)=S_i$ and that $\tau$ acts nontrivially on $\mathcal F_i$. Let $s\in S \subseteq \mathcal F_i$, so by assumption, $\tau\cdot s\in S_i$. Since every element in $K\simeq \mathbb Z_2^n$ has order $2$, $\tau\cdot(\tau\cdot s)=\tau^{-1}\cdot(\tau\cdot s) = s.$ Additionally, since $\tau$ acts nontrivially on every vertex of $\mathcal F_i$, $s\neq \tau\cdot s$. Thus, $S$ can be partitioned into pairs of the form $\{s, \tau \cdot s\}$, implying that $S_i$ has an even number of vertices in total. ◻ **Theorem 5**. *For $n\neq4$, $$\operatorname{Det}(\operatorname{PX}(n,k))=\begin{cases}\lceil \frac{n}{k}\rceil, \quad & \text{ if } k\neq\frac{n}{2},\\ \lceil \frac{n}{k}\rceil + 1 = 3, & \text{ if } k=\frac{n}{2}.\end{cases}$$* *Proof.* First suppose $k\neq\frac{n}{2}$. Let $S\subset V(\operatorname{PX}(n,k))$ such that $|S|=\lceil\frac{n}{k}\rceil-1$, and assume $S$ is a determining set for $\operatorname{PX}(n,k)$. The set of indices of the fibres containing elements of $S$ is $$I_S=\{i\in\mathbb Z_n\mid S\cap\mathcal F_i\neq\emptyset\}=\{i_1,i_2,\dots,i_s\},$$ where $0\le i_1 < i_2 \dots < i_s \le n-1$. Then the number of fibres in the gaps between these fibres are $i_2-i_1-1,i_3-i_2-1,\dots,n+i_1-i_s-1$. If $i_{p+1}-i_p-1\geq k$ for some $i_p, i_{p+1} \in S$ , then $\tau_{i_{p+1}-1}$ is a nontrivial automorphism that fixes $S$, contradicting the assumption that $S$ is a determining set. Thus each gap contains at most $k-1$ fibres. Since every fibre either contains a vertex in $S$ or is in a gap between two such fibres, the total number of fibres satisfies $$|I_S|+|I_S|(k-1)=|I_S|k\ \le |S| k =(\lceil\tfrac{n}{k}\rceil-1)k<n,$$ a contradiction. Thus, $\operatorname{Det}(\operatorname{PX}(n,k))>\lceil\frac{n}{k}\rceil-1$. We claim $S=\big\{v_{ik}\in\mathcal F_{ik}:i\in\{0,1,\dots,\lceil\tfrac{n}{k}\rceil-1\}\big\},$ where $v_{ik}$ is any vertex in $\mathcal F_{ik}$, is a determining set for $\text{PX}(n,k)$. Let $\alpha=\tau\delta\in\mathcal A=\operatorname{Aut}(\operatorname{PX}(n,k))$ such that $\alpha$ fixes every vertex in $S$. Since $k\neq \tfrac{n}{2}$, $\mathcal F_0$ and $\mathcal F_k$ are non-antipodal fibres. Since the induced action of $\alpha$ on the fibres is an element of $D_n$ that fixes non-antipodal vertices $0$ and $k$ in $C_n$, $\delta=\text{id}$. Next we show that $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}=\text{id}$. Let $S_0 = S\cap \mathcal F_0$; note that $|S_0|=1$ is odd. Since $\tau$ fixes every vertex in $S$, $\tau(S_0) = S_0,$ and so by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_0 = u_1 = \dots = u_{k+1} = 0$. Applying the same logic to $S_k = S\cap \mathcal F_k$, we get $u_k = u_{k+1} = \dots = u_{2k-1} = 0$. Iterating this argument for $S_{ik}$ for all $i \in \{0, 1, \dots \lceil\frac{n}{k}\rceil - 1\}$, we conclude $u_0 =\dots= u_{n-1}=0$. Thus $\tau = \text{id}$. By definition, $S$ is a determining set and so $\operatorname{Det}(\operatorname{PX}(n,k))\leq |S|=\lceil\frac{n}{k}\rceil$. Now suppose $k=\frac{n}{2}$. Assume $S$ is a determining set of cardinality $2$. Since $\operatorname{PX}(n,k)$ is vertex-transitive, we can assume without loss of generality that $S=\{z=(0,00\cdots 0),v= (i,x)\}$. There are three cases: $i=0$, $i=k=\frac{n}{2}$, or $i\neq 0$ and $i\neq k=\frac{n}{2}$. For the first case, assume $i=0$, so both $z,v\in\mathcal F_0$. Then since $\tau_{n-1}$ affects $\mathcal F_i$ if and only if $i\in\{k=\frac{n}{2},\frac{n}{2}+1,\dots,n-1\}$, and $0$ is not in that set, $\tau_{n-1}$ is a nontrivial automorphism that fixes $S$, a contradiction. For the second case, assume $i=k=\frac{n}{2}$. Then $z$ and $v$ are in antipodal fibres. If we we apply the reflection $\mu$ to $S$, we get $$\mu\big(S\big)=\big\{(0,(00\cdots0)^-),(-k,x^-)\big\}=\big\{(0,00\cdots0),(k,x^-)\big\}.$$ For each $m$ such that $x_m\neq (x^-)_m$, the automorphism $\tau_{k+m}\in K$ flips this bit in the bitstring component of every vertex in $\mathcal F_k$, but has no effect on the vertices in $\mathcal F_0$. Let $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}$ where $u_{k+m} = 1$ if $x_m \neq (x^-)_m$ and $0$ otherwise. Then $\zeta = \tau \mu$ fixes both $z$ and $v$, contradicting our assumption that $S$ is a determining set. In the third case, $i\neq0$ and $i\neq k =\frac{n}{2}$, so we can assume that in $\mathbb Z$, either $0<i<k$ or $k < i < n$. In the first case, $\tau_{n-1}$ fixes both $z$ and $v$, and in the second, $\tau_{i-1}$ fixes both $z$ and $v$. Hence, $S$ is not a determining set for $\operatorname{PX}(n,k)$. As we have covered all possible cases, we conclude $\operatorname{Det}(\operatorname{PX}(n,k))>2$. Finally, let $S = \{v_0, v_1, v_k\}$ where $v_0\in \mathcal F_0$, $v_1 \in \mathcal F_1$ and $v_k \in \mathcal F_k$ and assume $\alpha = \tau \delta$ fixes $S$. The induced action of $\alpha$ on the fibres corresponds to an element of the dihedral group that fixes non-antipodal vertices $0$ and $1$ in $C_n$, so $\delta = \text{id}$. Next, $\tau = \tau_0^{u_0} \cdots \tau_{n-1}^{u_{n-1}}$ fixes one vertex in each of $\mathcal F_0$ and $\mathcal F_k$, so by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_0 = \dots = u_{k-1} = u_k = \dots = u_{2k-1}= u_{n-1} = 0$. Thus $\tau = \text{id}$. By definition, $S$ is a determining set for $\operatorname{PX}(n,k)$ so $\operatorname{Det}(\operatorname{PX}(n,k))\leq|S|=3$. Thus $\operatorname{Det}(\operatorname{PX}(n,k))=3$. ◻ We now turn our attention to the exceptional cases $\operatorname{PX}(4,2)$ and $\operatorname{PX}(4,3)$. It is stated without proof in [@SGOTPV] that $Q_4\cong\operatorname{PX}(4,2)$; we provide an explicit isomorphism. A bitstring $x$ is *even* if $x$ has an even number of ones, and $x$ is *odd* otherwise. Define $\varphi:V(Q_4)\rightarrow V(\operatorname{PX}(4,2))$ by $$\varphi(x_0x_1x_2x_3)=\begin{cases}(j,x_1x_3), \quad &\text{ if } j\text{ is odd},\\ (j,x_3x_1), &\text{ if } j\text{ is even},\end{cases}$$ where $$j=\begin{cases} 0, \quad & \text{ if } x_0x_1\text{ and } x_2x_3 \text{ are both odd},\\ 1, & \text{ if } x_0x_1\text{ is odd and } x_2x_3\text{ is even},\\ 2, & \text{ if } x_0x_1\text{ and } x_2x_3\text{ are both even},\\ 3, & \text{ if } x_0x_1\text{ is even and } x_2x_3\text{ is odd}. \end{cases}$$ Figure [5](#fig:Q4PX42){reference-type="ref" reference="fig:Q4PX42"} is a drawing of $Q_4$, with vertices positioned as they would be in a canonical drawing of $\operatorname{PX}(4,2)$, as explained at the beginning of Section [2](#sec:PXIntro){reference-type="ref" reference="sec:PXIntro"}. {#fig:Q4PX42} **Proposition 1**. *$\operatorname{Det}(\operatorname{PX}(4,2))=3, \operatorname{Dist}(\operatorname{PX}(4,2))=2 \text{ and }\rho(\operatorname{PX}(4,2))=5.$* *Proof.* This follows immediately from previous work on the symmetry parameters of $Q_4$. By Theorem 3 from [@Bo2009a], $$\operatorname{Det}(\operatorname{PX}(4,2))=\operatorname{Det}(Q_4)=\lceil\log_24\rceil+1=2+1=3.$$ Notably, this expression agrees with the formula given in Theorem [Theorem 5](#thm:DetnNot4){reference-type="ref" reference="thm:DetnNot4"} because $\lceil\frac{4}{2}\rceil+1=2+1=3$. By Theorem 5 from [@BoCo2004], $\operatorname{Dist}(\operatorname{PX}(4,2))=\operatorname{Dist}(Q_4)=2$; by Theorem 11 from [@Bo2021], $\rho(\operatorname{PX}(4,2))=\rho(Q_4)=5$. ◻ **Proposition 2**. *$\operatorname{Det}(\operatorname{PX}(4,3)) =2 = \lceil \frac{4}{3} \rceil.$* *Proof.* From [@SGOTPV], $\mathcal A =K\rtimes \langle\rho,\mu\rangle$ is a proper subgroup of $\operatorname{Aut}(\operatorname{PX}(4,3))$ of index $2$. The proof of Theorem [Theorem 5](#thm:DetnNot4){reference-type="ref" reference="thm:DetnNot4"} can be used when $(n,k) = (4,3)$ to show that the only $\alpha \in \mathcal A$ that fixes two vertices from non-antipodal fibres is the identity. However, more care must be taken when choosing the two vertices. For example, there is a nontrivial automorphism $\xi \in \operatorname{Aut}(\operatorname{PX}(4,3))$ that fixes both elements of $S = \{(0,000), (3,000)\}$. More precisely, as a permutation $\xi$ is the product of the disjoint $2$-cycles in Table [2](#tab:2CyclesInXi){reference-type="ref" reference="tab:2CyclesInXi"}. Unlike for any $\alpha \in \mathcal A$, the fibres do not constitute a block system for $\xi$. However, if we partition each fibre into vertices whose bitstrings are palindromic ($x = x^-$) and vertices whose bitstrings are nonpalindromic ($x \neq x^-$), these half-fibres constitute a block system for $\xi$. within $\mathcal F_0$, $x = x^-$ ((0,010), (0,101)) -------------------------------------------------------------------- ----------------------- -------------------- within $\mathcal F_2$, $x \neq x^-$ ((2,001), (2,110)) between $\mathcal F_0$, $x \neq x^-$ and $\mathcal F_2$, $x = x^-$ ((0,001), (2,000)),   ((0,100), (2,010)) ((0,011), (2,101)), ((0,110), (2,111)) within $\mathcal F_1$, $x \neq x^-$ ((1,100), (1,011)) within $\mathcal F_3$, $x = x^-$ ((3,010), (3,101)) between $\mathcal F_1$, $x = x^-$ and $\mathcal F_3$, $x \neq x^-$ ((1,000), (3,100)), ((1,010), (3,001)) ((1,101), (3,110)), ((1,111), (3,011)) : Disjoint $2$-cycles of $\xi \in \operatorname{Aut}(\operatorname{PX}(4,3))$. Implicitly $\xi$ has eight fixed points, with two in each fibre, namely $(0,000)$, $(0,111)$, $(1,001)$, $(1,110)$, $(2,100)$, $(2,001)$, $(3,000)$ and $(3,111)$. Since $\mathcal A$ has index $2$ in $\operatorname{Aut}(\operatorname{PX}(4,3))$, every automorphism of $\operatorname{PX}(4,3)$ is in one of the two cosets, $\mathcal A$ and $\mathcal A \xi$. Next we show that $S' = \{(0,000), (3,001)\}$ is a determining set. The two vertices in $S'$ are from non-antipodal fibres, so no nontrivial automorphism in $\mathcal A$ fixes both vertices in $S'$. Table [2](#tab:2CyclesInXi){reference-type="ref" reference="tab:2CyclesInXi"} shows that $\xi$ clearly does not fix $(3,001)$; we must also show that no other automorphism in the coset $\mathcal A \xi$ fixes $S'$. Assume there exists $\beta \in \mathcal A$ such that $\beta \circ \xi$ fixes $S'$. Then $(0,000) = \beta \circ \xi \cdot (0,000) = \beta\cdot (0,000)$ and $(3,001) = \beta \circ \xi \cdot (3,001) = \beta \cdot (1, 010)$. The induced action of $\beta$ on the fibres fixes $\mathcal F_0$ and takes $\mathcal F_1$ to $\mathcal F_3$, so by Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, $\beta = \tau \circ \mu = \tau_0^{u_0} \tau_1^{u_1} \tau_2^{u_2} \tau_3^{u_3} \mu$. Since $\beta$ and $\mu$ both fix $(0,000)$, so must $\tau$ and by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_0=u_1=u_2 = 0$. However, because $\tau_3$ can only affect the $0$-th bit of the bitstring component of a vertex in $\mathcal F_3$, no value of $u_3$ satisfies $(3,001) = \beta \cdot (1,010) = \tau_3^{u_3} \mu \cdot (1,010) = \tau_3^{u_3} \cdot (3,010)$. ◻ The following theorem summarizes our results on the determining number of twin-free Praeger-Xu graphs. **Theorem 6**. *For all $n \ge 3$ and $2 \le k <n$, $$\operatorname{Det}(\operatorname{PX}(n,k))= \begin{cases} \lceil\frac{n}{k}\rceil, \quad &\text{ if } k\neq \frac{n}{2},\\ \lceil\frac{n}{k}\rceil+1=3, &\text{ if } k=\frac{n}{2}. \end{cases}$$* # Interchangeable Vertices in $\operatorname{PX}(n,k)$ {#sec:Interchangeable} As mentioned in Section [3](#sec:PXn1){reference-type="ref" reference="sec:PXn1"}, if two vertices in a graph are twins, then the map that interchanges them and leaves all vertices fixed is a graph automorphism. By Theorem [Theorem 1](#thm:NoTwin){reference-type="ref" reference="thm:NoTwin"}, if $k \ge 2$, then $\operatorname{PX}(n,k)$ is twin-free, but we will find it useful to identify when two vertices can be interchanged by an automorphism, regardless of its action on other vertices. **Definition 2**. *Distinct vertices $u, v$ in a graph $G$ are interchangeable if and only if there exists $\alpha \in \operatorname{Aut}(G)$ such that $\alpha\cdot u = v$ and $\alpha \cdot v = u$.* There are some situations where it is easy to find an automorphism interchanging vertices $u=(i,x)$ and $v=(j,y)$ in $\operatorname{PX}(n,k)$. If $i=j$, there exists $\tau=\tau_0^{u_0} \cdots \tau_{n-1}^{u_{n-1}}\in K$ that flips exactly the right bits in bitstring components of vertices in $\mathcal F_i$. More precisely, for each $t \in \{0, 1, \dots, k-1\}$, if $x_t \neq y_t$, set $u_{i+t}=1$, and otherwise set $u_{i+t}=0$. The values of $u_m$ for any $m\in \mathbb Z_n$ not of the form $i+t$ do not affect the action of $\tau$ on $u$ and $v$. If $i\neq j$, then we can find $\delta \in \langle \rho, \mu\rangle$ such that the induced action of $\delta$ on the fibres interchanges $\mathcal F_i$ and $\mathcal F_j$; we can then look for $\tau=\tau_0^{u_0} \cdots \tau_{n-1}^{u_{n-1}}\in K$ so that $\tau$ flips exactly the right bits in both $\mathcal F_i$ and $\mathcal F_j$ to ensure that $\tau\delta$ interchanges $u$ and $v$. If $\mathcal F_i$ and $\mathcal F_j$ are far enough apart, then we can set the values $u_i, u_{i+1}, \dots u_{i+k-1}$ and $u_j, u_{j+1}, \dots u_{j+k-1}$ independently. However, if $M =\big\{i,i+1,i+2,\dots,i+k-1\big\}\cap\big\{j,j+1,j+2,\dots,j+k-1\big\} \neq \emptyset$, then for any $m \in M$, $\tau_m$ affects both vertices in $\mathcal F_i$ and $\mathcal F_j$ and there is potential for conflict. **Lemma 3**. *Let $u=(i,x),v=(j,y)\in V(\operatorname{PX}(n,k))$ and let $$M =\big\{i,i+1,i+2,\dots,i+k-1\big\}\cap\big\{j,j+1,j+2,\dots,j+k-1\big\} \subseteq \mathbb Z_n.$$ Then $u$ and $v$ are interchangeable by some $\alpha \in \mathcal A$ if and only if one of the following holds:* (1) *$j=i$,* (2) *$j \neq i$ and for all $m\in M$, $(x^-)_{m-j}=y_{m-j}$ if and only if $(y^-)_{m-i}=x_{m-i}$,* (3) *$j=i+\frac{n}{2}$ and for all $m\in M$, $x_{m-j}=y_{m-j}$ if and only if $y_{m-i}=x_{m-i}$.* *Proof.* Assume $u$ and $v$ are interchangeable by $\alpha = \tau\delta \in\mathcal{A}$, but neither (1) nor (2) holds. Then $j\neq i$ and for some $m\in M$, either $(x^-)_{m-j}=y_{m-j}$ but $(y^-)_{m-i}\neq x_{m-i}$, or $(x^-)_{m-j}\neq y_{m-j}$ but $(y^-)_{m-i}= x_{m-i}$. As usual, either $\delta = \rho^s$ or $\delta=\mu_s$ for some $s\in\mathbb Z_n$. If $\delta=\mu_s$, then $\mu_s\cdot u=(j,x^-)$ and $\mu_s\cdot v=(i,y^-)$, and hence $\tau\cdot(j,x^-)=(j,y)$ and $\tau\cdot(i,y^-)=(i,x)$, where $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}$. If $(x^-)_{m-j}=y_{m-j}$ and $(y^-)_{m-i} \neq x_{m-i}$, then since $\tau_m$ flips the $(m-j)$-th bit of the bitstrings in $\mathcal F_j$, $u_m=0$. However, $\tau_m$ flips the $(m-i)$-th bit of bitstrings in $\mathcal F_i$, so $u_m=1$, a contradiction. A completely analogous argument works if $(x^-)_{m-j}\neq y_{m-j}$ and $(y^-)_{m-i}= x_{m-i}$. Thus, $\delta=\rho^s$. Since $\rho^s\cdot u=(i+s,x)$ and $\rho^s\cdot v=(j+s,y)$, and $\tau$ fixes every fibre, $i+s=j$ and $j+s=i$. Since we are assuming $i\neq j$, $s=\frac{n}{2}$, so $j=i+\frac{n}{2}$. Let $m\in M$, and assume $x_{m-j}=y_{m-j}$. Since $\tau\cdot(j,x)=(j,y)$, $\tau$ must not flip the $(m-j)$-th bit of the bitstrings of $\mathcal F_{j}$, so $u_m=0$. That means that $\tau$ must also not flip the $(m-i)$-th bit of the bitstrings of $\mathcal F_i$, so since $\tau\cdot(i,y)=(i,x)$, $y_{m-i}=x_{m-i}$. A completely analogous argument works if we assume $y_{m-i}=x_{m-i}$. Thus condition (3) holds. Conversely, we will show that if one of (1), (2) or (3) holds, then $u$ and $v$ are interchangeable by some $\alpha\in\mathcal{A}$. First, assume (1) holds, so $j=i$. As noted in the paragraph before the statement of this lemma, there is some $\tau \in K \subset \mathcal A$ that interchanges $u$ and $v$. Next, assume (2) holds, so $j\neq i$ and for all $m\in M$, $(x^-)_{m-j}=y_{m-j}$ if and only if $(y^-)_{m-i}=x_{m-i}$. Let $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}\in K$, where for each $s\in\mathbb Z_n$, $u_s=1$ if $(x^-)_{s-j}\neq y_{s-j}$ and $(y^-)_{s-i}\ne x_{s-i}$ and $u_s=0$ otherwise. Hence, in $\mathcal F_j$, $\tau$ flips the bits in every position that $x^-$ and $y$ differ and no others, and in $\mathcal F_i$, $\tau$ flips the bits in every position that $y^-$ and $x$ differ and no others. By Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, there exists $\mu_s\in\langle\rho,\mu\rangle$ such that $\mu_s\cdot u=(j,x^-)$ and $\mu_s\cdot v=(i,y^-)$. Let $\alpha=\tau\mu_s\in\mathcal{A}$. Then $\alpha\cdot u = \tau\cdot(\mu_s\cdot u)= \tau\cdot(j,x^-)=(j,y),$ and $\alpha\cdot v = \tau\cdot(\mu_s\cdot v)=\tau\cdot(i,y^-)=(i,x),$ so $u$ and $v$ are interchangeable by $\alpha\in\mathcal{A}$. Lastly, assume (3) holds, so $j=i+\frac{n}{2}$ and for all $m\in M$, $x_{m-j}=y_{m-j}$ if and only if $y_{m-i}=x_{m-i}$. Let $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}\in K$, where for each $s\in\mathbb Z_n$, $u_s=1$ if $x_{s-j}\neq y_{s-j}$ and $u_s=0$ otherwise. By assumption, that also means that $u_s=1$ if $y_{s-i}\neq x_{s-i}$ and $u_s=0$ otherwise. Hence, in both $\mathcal F_i$ and $\mathcal F_j$, $\tau$ flips the bits in every position that $x$ and $y$ differ, and no others. It is straighforward to verify that $\alpha=\tau\rho^{n/2}\in\mathcal{A}$ interchanges $u$ and $v$. ◻ For example, let $u=(i,x)=(0,101),v=(j,y)=(1,001)\in V(\operatorname{PX}(5,3))$. Then $M=\{0,1,2\}\cap\{1,2,3\}=\{1,2\}$. Since $x^-=101$ and $y^-=100$, for $m=1$, $(x^-)_{m-j}=1\neq 0=y_{m-j}$ and $(y^-)_{m-i}=0=x_{m-i}$. Hence, by Lemma [Lemma 3](#lem:interchangeable){reference-type="ref" reference="lem:interchangeable"}, $u$ and $v$ are not interchangeable. Note that it is possible for a pair of vertices to be interchangeable by two different automorphisms. For example, $z = (0, 000)$ and $v=(5,000)$ in $\operatorname{PX}(10,3)$ satisfy both conditions (2) and (3), and so can be interchanged using either a rotation or a reflection. We will find it useful in the next section to identify which vertices of $\operatorname{PX}(n,k)$ are interchangeable with $z=(0, 00\cdots 0)$. **Corollary 1**. *Let $v=(j,y)\in V(\operatorname{PX}(n,k))$ and let $$M = \big\{0,1,2,\dots,k-1\big\}\cap\big\{j,j+1,j+2,\dots,j+k-1\big\}.$$ Then $z=(0,00\cdots0)$ and $v$ are interchangeable by some $\alpha \in \mathcal A$ if and only if one of the following holds:* (1) *$j=0$,* (2) *$j \neq 0$ and for all $m\in M$, $y_{m-j}=(y^-)_m= y_{k-1-m}$,* (3) *$j=\frac{n}{2}$ and for all $m\in M$, $y_{m-j}= y_m$.* To illustrate how vertex interchangeability can be used to compute symmetry parameters, we consider the smallest twin-free Praeger-Xu graph. **Lemma 4**. *Any two distinct vertices of $\operatorname{PX}(3,2)$ are interchangeable.* *Proof.* By vertex-transitivity, it suffices to show that $z = (0,00)$ is interchangeable with any vertex $v = (j,y)$. Since $n = 3$ is odd, we need only check that condition (2) of Corollary [Corollary 1](#cor:zInterchangeable){reference-type="ref" reference="cor:zInterchangeable"} holds. If $j = 1$, then $M=\{1\}$ and $y_{m-j} = y_{k-1-m} = y_0$. If $j = 2$, then $M =\{0\}$ and $y_{m-j} = y_{k-1-m} = y_1$. ◻ **Theorem 7**. *$\operatorname{Dist}(\operatorname{PX}(3,2))=2$ and $\rho(\operatorname{PX}(3,2))=3$.* *Proof.* Color the vertices in $R=\{(0,00), (1,01), (2,00)\}$ red and every other vertex blue. Assume $\alpha=\tau\delta\in\mathcal{A}=\operatorname{Aut}(\operatorname{PX}(3,2))$ preserves colors, where $\tau=\tau_0^{u_0}\tau_1^{u_1}\tau_2^{u_2}$. Suppose $\delta=\rho^s$ for some $s\in\{1,2\}$. If $s=1$, then $\delta\cdot(1,01) =(2,01)$. Since $\alpha$ preserves colors, $\tau\cdot (2,01)=(2,00)$, which implies $u_0=1$ and $u_2 = 0$. However, then $\tau\delta\cdot(2,00)= \tau\cdot(0,00)=(0,1a)$ for some $a\in\mathbb Z_2$. This contradicts our assumption that $\alpha$ preserves colors. If $s=2$, then $\delta\cdot(1,01)=(0,01)$, and hence $u_1=1$ and $u_0=0$. A similar contradiction arises because $\delta\cdot(2,00)=(1,00)$, and $u_1=1$ means $\tau\cdot(1,00)=(1,1a)$ for some $a\in\mathbb Z_2$. Thus $\delta$ cannot be a nontrivial rotation. Suppose $\delta=\mu_s$ for some $s \in \mathbb Z_3$. By Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, $\delta$ preserves one fibre and interchanges the other two. If $\delta$ preserves $\mathcal F_0$ and interchanges $\mathcal F_1$ and $\mathcal F_2$, then $\delta\cdot (1,01)=(2,10)$. Since $\alpha$ preserves colors, $\tau\cdot(2,10)=(2,00)$, so $u_2=1$. However, $\delta\cdot(2,00)=(1,00)$, which implies $\tau\cdot(1,00)=(1,1a)$ for some $a\in\mathbb Z_2$, a contradiction. Similar arguments apply to the remaining two cases. Since $\delta$ is neither a nontrivial rotation nor a reflection, $\delta = \text{id}$. Since $\alpha$ preserves colors and fibres, $\tau$ preserves the one red vertex in each fibre, so by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_0=u_1=u_2 = 0$. Hence, $\alpha=\tau=\text{id}$, so this is a $2$-distinguishing coloring with smaller color class of size $3$. Thus, $\operatorname{Dist}(\operatorname{PX}(3,2))=2$ and $\rho(\operatorname{PX}(3,2))\leq3$. To show $\rho(\operatorname{PX}(3,2)> 2$, let $R=\{u,v\}\subset V(\operatorname{PX}(3,2))$. Color the vertices in $R$ red and every other vertex blue. By Lemma [Lemma 4](#lem:3,2interchange){reference-type="ref" reference="lem:3,2interchange"}, $u$ and $v$ are interchangeable; any automorphism interchanging them is a nontrivial color-preserving automorphism. Thus, $\rho(\operatorname{PX}(3,2))> 2$, so $\rho(\operatorname{PX}(3,2))=3$. ◻ # Distinguishing $\operatorname{PX}(n,k)$, $k \ge 2$ {#sec:DistNoTwins} We have already found the distinguishing parameters for a number of Praeger-Xu graphs. Theorem [Theorem 4](#thm:DetkIsOne){reference-type="ref" reference="thm:DetkIsOne"} covers the case $k=1$; Theorem [Theorem 7](#thm:dist3,2){reference-type="ref" reference="thm:dist3,2"} covers $\operatorname{PX}(3,2)$ and Proposition [Proposition 1](#prop:PX42){reference-type="ref" reference="prop:PX42"} covers $\operatorname{PX}(4,2)$. The next result covers the exceptional case $\operatorname{PX}(4,3)$. The remainder of this section covers the case $n \ge 5$ and $k \ge 2$. **Theorem 8**. *$\operatorname{Dist}(\operatorname{PX}(4,3)) = 2$ and $\rho(\operatorname{PX}(4,3))= 3 = \lceil \frac{4}{3}\rceil +1$.* *Proof.* Color the vertices in $R = \{(0,000), (2,000), (3,001)\}$ red and all other vertices blue. Suppose $\beta \in \operatorname{Aut}(\operatorname{PX}(4,3))$ preserves this coloring. Recall that $\operatorname{Aut}(\operatorname{PX}(4,3))$ can be partitioned into the cosets $\mathcal A$ and $\mathcal A \xi$. First assume $\beta = \alpha \xi$ for some $\alpha \in \mathcal A$. Then by assumption, $$R = \beta\big ( \{ (0,000), (2,000), (3,001)\}\big ) = \alpha(\{ (0,000), (0,001), (1,010)\}).$$ Note that $R$ contains vertices in three different fibres, but since the fibres form a block system for any $\alpha \in \mathcal A$, $\alpha(\{(0,000), (0,001), (1,010)\})$ contains two vertices in one fibre and a third vertex in a different fibre. So these two sets cannot be equal. Hence $\beta \notin \mathcal A \xi$. Thus $\beta \in \mathcal A$, so $\beta = \tau \delta$ for some $\delta \in \langle \rho, \mu\rangle$. Note that $(2,000)$ and $(3,001)$ are adjacent, but neither is adjacent to $(0,000)$. Thus $\beta$ fixes $(0,000)$. If the induced action of $\beta$ on the fibres fixes $\mathcal F_0$, then either $\delta =$ id or $\delta = \mu$. Since $\mu$ does not interchange fibres $\mathcal F_2$ and $\mathcal F_3$, $\delta =$ id. Thus $\beta$ fixes every vertex in $R$, which contains $\{(0,000), (3,001)\}$, the determining set for $\operatorname{PX}(4,3)$ found in Proposition [Proposition 2](#prop:detFourThree){reference-type="ref" reference="prop:detFourThree"}. Hence $\beta = \text{id}$. Thus this is a 2-distinguishing coloring, proving that $\operatorname{Dist}(\operatorname{PX}(4,3)) = 2$. Next we show that we cannot create a 2-distinguishing coloring with fewer red vertices. If $R = \{(i, x)\}$, then $\tau_{i-1}$ is a nontrivial automorphism preserving the coloring. To show that no two-element set of red vertices provides a distinguishing coloring, it suffices, by vertex transitivity, to show that every vertex in $\operatorname{PX}(4.3)$ is interchangeable with $z=(0,000)$. Corollary [Corollary 1](#cor:zInterchangeable){reference-type="ref" reference="cor:zInterchangeable"} shows that $z=(0,000)$ is interchangeable with every vertex in $\operatorname{PX}(4,3)$ by some $\alpha \in \mathcal A$ except those listed below: $$(1,010), (1,011), (1,100), (1,101), (3,010),(3,110),(3,001), (3,101). \tag{*}$$ For each vertex (\*), we can find $\alpha \in \mathcal A$ such that $\alpha \xi$ interchanges it with $(0,000)$. For $(1,010)$, we seek $\alpha \in \mathcal A$ that satisfies $\alpha (\xi \cdot (0,000)) = (1,010)$ and $\alpha (\xi \cdot (1,010)) = (0,000)$. Referring to Table [2](#tab:2CyclesInXi){reference-type="ref" reference="tab:2CyclesInXi"} for the action of $\xi$, we seek $\alpha \in \mathcal A$ such that $\alpha \cdot (0,000) = (1,010)$ and $\alpha \cdot (3,001)) = (0,000)$. It is easy to verify that $\alpha = \tau_2 \rho$ satisfies this condition. For each vertex $v$ in (\*), Table [3](#tab:interchangingAlphas){reference-type="ref" reference="tab:interchangingAlphas"} gives an $\alpha$ satisfying $\alpha\xi \cdot (0,000) =\alpha\cdot (0,000)= v$ and $\alpha\xi\cdot v = (0,000)$. ◻ $v$ $\xi \cdot v$ $\alpha$ $v$ $\xi \cdot v$ $\alpha$ ----------- --------------- ------------------------------ ----------- --------------- ------------------------------- $(1,010)$ $(3,001)$ $\tau_2 \rho$ $(3,010)$ $(3,101)$ $\tau_0 \tau_2 \mu_1$ $(1,011)$ $(1,100)$ $\tau_2 \tau_3 \mu_3$ $(3,110)$ $(1,101)$ $\tau_0 \tau_2 \tau_3 \rho^3$ $(1,100)$ $(1,011)$ $\tau_0 \tau_1 \mu_3$ $(3,001)$ $(1,010)$ $\tau_1 \rho^3$ $(1,101)$ $(3,110)$ $\tau_0 \tau_1 \tau_3 \rho$ $(3,101)$ $(3,010)$ $\tau_1 \tau_3 \mu_1$ : $\alpha \in \mathcal A$ such that $\alpha\xi$ interchanges $(0,000)$ and $v$ in $(*)$. **Theorem 9**. *Let $n\geq5$ and $k \ge 2$. Then $\operatorname{Dist}(\operatorname{PX}(n,k))=2$ and\ $\lceil \frac{n}{k}\rceil \le \rho(\operatorname{PX}(n,k))\leq\lceil\frac{n}{k}\rceil+1$.* *Proof.* Let $x = 00\cdots 0, y = 11\cdots 1 \in \mathbb Z_2^k$. Then let $$R = \big\{(ik, x) : i \in \{0, 1, \dots, \lceil\tfrac{n}{k} \rceil -1\}\big\} \cup \big\{(1, y)\big\}.$$ Color the vertices in $R$ red and all other vertices blue. Assume $\alpha=\tau\delta \in\operatorname{Aut}(\operatorname{PX}(n,k))$ preserves these color classes. Then the induced action of $\delta$ on the fibres must preserve the set $I=\{0, 1, k, 2k, \dots, (\lceil\tfrac{n}{k} \rceil -1)k\} \subset \mathbb Z_n$. Note that $|R|= |I| = \lceil\tfrac{n}{k} \rceil +1 < n$. Interpreting $\mathbb Z_n$ as the vertex set of the cycle $C_n$, the (non-spanning) subgraph of $C_n$ induced by $I$ consists of a path containing at least the vertices $0$ and $1$, and possibly some isolated vertices. Let $F\subset \mathbb Z_n$ denote the set of vertices in the path; these will be the indices corresponding to a set of adjacent fibres of $\operatorname{PX}(n,k)$ containing red vertices. Note that the action of $\delta$ on $C_n$ must preserve $F$. Since no nontrivial rotation preserves a proper subpath of $C_n$, $\delta \neq \rho^s$ for any $0 \neq s\in\mathbb Z_n$. So assume $\delta=\mu_s$ for some $s\in\mathbb Z_n$, and as usual, $\tau= \tau_0^{u_o} \cdots \tau_{n-1}^{u_{n-1}}$. First assume $k=2$. If $n \ge 5$ is odd, then $I= \{0, 1, 2, 4, \dots, n-1\}$ and $F=\{n-1,0,1,2\}$. For $F$ to be preserved under reflection, $\mu_s$ must interchange the vertex pairs $\{0, 1\}$ and $\{n-1, 2\}$ in $C_n$. By Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, $s = 2$. In $\operatorname{PX}(n,2)$, $\alpha = \tau \mu_2$ must interchange the vertex pairs $\{(0,00),(1,11)\}$ and $\{(n-1,00), (2,00)\}$. Thus $\tau\cdot(0,11)=(0,00)$ and $\tau\cdot(1,11)=(1,00)$, which implies $u_0=u_1=u_2=1$. However, it must also be the case that $\tau\cdot(n-1,00)=(n-1,00)$, so $u_0=0$, a contradiction. If instead $n\ge 5$ is even, then $I = \{0, 1, 2, 4, \dots, n-2\}$ and $F=\{0,1,2\}$. In this case, $\mu_s$ must fix $1$ and interchange $0$ and $2$, so by Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, $s=3$. In this case, as an element of $\operatorname{Aut}(\operatorname{PX}(n,2))$, $\mu_3$ is a nontrivial automorphism preserving $R$, so this does not define a $2$-distinguishing coloring. However, let $$R' = \{(0, y)\} \cup \big\{(ik, x) : i \in \{1, \dots, \lceil\tfrac{n}{k} \rceil -1\}\big\} \cup \big\{(1, x)\big\}.$$ Then $I'=I = \{0, 1, 2, 4, \dots, n-2\}$ and $F'=F=\{0, 1, 2\}$. The only reflection preserving $F'$ is still $\mu_3$. If $\alpha = \tau\mu_3$ preserves $R'$, then $\tau\mu_3\cdot (0,11) = \tau\cdot(2,11) = (2, 00)$, which means $u_2 = u_3 = 1$. Also, $\tau\mu_3\cdot (2,00) = \tau\cdot(0,00) = (0, 11)$, so $u_0=u_1 = 1$. This creates a contradiction because $\tau \mu_3\cdot(1,00) = \tau(1,00) = (1,00)$, which implies $u_1 = u_2 = 0$. Now assume $k>2$. If $n\neq 1\bmod k$, then $F= \{0,1\}$. Then $\mu_s$ must interchange $0$ and $1$, so by Lemma [Lemma 1](#lem:ijReflec){reference-type="ref" reference="lem:ijReflec"}, $s = k$. Since $\alpha = \tau \mu_k$ preserves $R$, $\tau\cdot(0,y)=(0,x)$ and $\tau\cdot(1,x)=(1,y)$. Then $u_0=u_1=\cdots=u_k=1$. Since $k \in I$ and $\mu_k$ preserves $I$, $\mu_k(j) = k$ for some $j \in I\setminus \{0, 1\}$. Then $\alpha(j,x) = \tau\mu_s(j,x) = \tau\cdot (k,x)=(k,x)$, so $u_k=0$, a contradiction. If instead $n=1\bmod k$, then $F=\{n-1,0,1\}$. Then $\mu_s$ fixes $0$ and interchanges $n-1$ and $1$, so $s=k-1$. Then $\tau\cdot(n-1,y)=(n-1,x)$ and $\tau\cdot(1,x)=(1,y)$. Hence $u_{n-1}=u_0=\cdots=u_k=1$. However, $\tau\cdot(0,x)=(0,x)$, so $u_0=u_1=\cdots=u_{k-1}=0$, a contradiction. Thus $\delta\neq \mu_s$ for any $s \in \mathbb Z_n$, so $\delta=\text{id}$ and hence $\alpha=\tau \in K$. For every $0 \le t \le \lceil{\frac{n}k}\rceil - 1$, $\mathcal F_{tk}$ contains exactly one red vertex that is fixed by $\tau$, so by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_{tk} = u_{tk+1} = \dots = u_{tk+k-1} = 0$. Hence $u_0 = \dots = u_{n-1} = 0$ and so $\tau = \text{id}$. Thus, this is a $2$-distinguishing coloring of $\operatorname{PX}(n,k)$ with a color class of size $\lceil\frac{n}{k}\rceil+1$, so $\operatorname{Dist}(\operatorname{PX}(n,k))=2$ and $\rho(\operatorname{PX}(n,k))\leq \lceil\tfrac{n}{k}\rceil+1$. To establish the lower bound on cost, assume there exists a set of vertices $R=\{u_1,u_2,\dots,u_r\}$ with $r<\lceil\frac{n}{k}\rceil$ such that coloring the vertices of $R$ red and all other vertices blue defines a $2$-distinguishing coloring of $\operatorname{PX}(n,k)$. If $\alpha\in\operatorname{Aut}(\operatorname{PX}(n,k))$ fixes every vertex in $R$, then certainly $\alpha$ preserves the color classes and so by assumption $\alpha = \text{id}$. Hence, $R$ is a determining set of size $r<\lceil\frac{n}{k}\rceil$, a contradiction of Theorem [Theorem 6](#thm:DET){reference-type="ref" reference="thm:DET"}. Thus, $\rho(\operatorname{PX}(n,k))\geq \lceil\tfrac{n}{k}\rceil$. ◻ {#fig:PX(13, 4)} The remaining theorems indicate which Praeger-Xu graphs (for $n\ge 5$ and $k \ge 2$) have cost $\lceil\frac{n}{k}\rceil$ and which have cost $\lceil\frac{n}{k}\rceil+1$. **Theorem 10**. *Let $n \ge 5$ and $2\leq k<n$. If $k$ divides $n$, then $\rho(\operatorname{PX}(n,k))=\lceil\frac{n}{k}\rceil+1=\frac{n}{k}+1$.* *Proof.* Let $R\subset V$ be any set of $\frac{n}{k}$ vertices. Color every vertex in $R$ red and every other vertex blue. It suffices to show we can always find a nontrivial automorphism preserving $R$. Let $I=\{i_1,i_2,\dots,i_r\} \subseteq \mathbb Z_n$ be the set of indices of fibres containing red vertices, where we assume that as integers, $0\le i_1 < i_2 < \dots < i_r <n$. Then the gaps between these fibres contain $i_2-i_1-1,i_3-i_2-1,\dots, n+i_1-i_r-1$ fibres, respectively. If there exists $i_p\in I$ such that the gap between $i_p$ and $i_{p+1}$ contains at least $k$ fibres, then $\tau_{i_{p+1}-1}$ is a nontrivial automorphism that preserves colors. So assume that for all $i_p\in I$, the gap between $i_p$ and $i_{p+1}$ contains fewer than $k$ fibres. Suppose there exists $i_p\in I$ such that the gap between $i_p$ and $i_{p+1}$ contains fewer than $k-1$ fibres. Since $r=|I|\leq |R|=\frac{n}{k}$, the total number of fibres is strictly less than $r+(k-1)r = kr \le k \cdot \frac{n}{k} = n$, a contradiction. Thus for all $i\in\mathbb Z_n$, $i\in I$ if and only if $i+k\in I$, so the induced action of $\rho^k$ preserves $I$ as a subset of $V(C_n)$. Since the fibres containing red vertices are separated by $k-1$ fibres, we can define $\tau\in K$ such that $\tau$ adjusts the bitstring components of vertices in these fibres independently. More precisely, let $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}$, where $u_m=1$ if and only if there exist $(i_p,x), (i_{p+1},y)\in R$ such that $x_{m-i_p}\neq y_{m-i_{p+1}}$. Then for all $(i_p,x)\in R$, $\tau\rho^k\cdot(i_p,x))=\tau\cdot(i_{p+1},x)=(i_{p+1},y)\in R.$ Thus, $\tau\rho^k$ is a nontrivial automorphism that preserves colors. ◻ **Theorem 11**. *If $5 \le n < 2k$, then $\rho(\operatorname{PX}(n,k)) = \lceil \frac{n}{k} \rceil = 2$ .* *Proof.* Let $j = \lfloor \tfrac{n}{2} \rfloor - 1$. Then $5 \le n < 2k$ implies $0 < j < k-1$. Next, let $R = \{z, v\}$ where $z = (0, 000\cdots 0)$ and $v = (j,y) = (j, 011\cdots 1))$. Color vertices in $R$ red and all other vertices blue; assume $\alpha \in \operatorname{Aut}(\operatorname{PX}(n,k))$ preserves these color classes. Let $M = \big\{0,1,\dots,k-1\big\}\cap\big\{j,j+1,\dots,j+k-1\big\}.$ Since $0< j < k-1$, $j \in M$. For $m = j$, $y_{m-j} = y_0 = 0$, but $y_{k-1-m} = y_{k-1-j} = 1.$ By Corollary [Corollary 1](#cor:zInterchangeable){reference-type="ref" reference="cor:zInterchangeable"}, $z$ and $v$ are not interchangeable, so $\alpha$ can only preserve $R$ by fixing $z$ and $v$. Because fibres $\mathcal F_0$ and $\mathcal F_j$ are not antipodal, $R$ is a determining set by Theorem [Theorem 5](#thm:DetnNot4){reference-type="ref" reference="thm:DetnNot4"}. By definition, $\alpha$ is the identity. Thus we have defined a $2$-distinguishing coloring in which the smaller color class has size $2$. ◻ **Theorem 12**. *Let $k\geq 2$ and $n>2k$ such that $k$ does not divide $n$. Then $$\rho(\operatorname{PX}(n,k)) = \begin{cases} \lceil\tfrac{n}{k}\rceil+1, &\text{if } n = -1 \bmod k,\\ \lceil\tfrac{n}{k}\rceil, \quad &\text{if } n \neq -1 \bmod k. \end{cases}$$* *Proof.* First assume $n= -1\bmod k$, so $n = \lceil \frac{n}{k} \rceil k - 1$. Let $R$ be any set of $\lceil\frac{n}{k}\rceil$ vertices. Color every vertex in $R$ red and every other vertex blue. Let $I=\{i_1,i_2,\dots,i_r\} \subset \mathbb Z_n$ be the set of indices of the fibres containing red vertices, where as integers, $0\leq i_1 < i_2<\cdots <i_r<n$. We will show that there is a nontrivial automorphism preserving $R$. If there exists $i_p\in I$ such that the gap between $i_p$ and $i_{p+1}$ contains at least $k$ fibres, then $\tau_{i_p+k}$ is a nontrivial automorphism that preserves colors. So assume that every gap has at most $k-1$ fibres. Suppose there exist at least two gaps that contain at most $k-2$ fibres. Then the total number of fibres is at most $$r+2(k-2)+(r-2)(k-1)= rk-2 \leq \lceil\tfrac{n}{k}\rceil k -2 <\lceil \tfrac{n}{k} \rceil k - 1 = n,$$ a contradiction. Thus at most one gap contains at most $k-2$ fibres and the others contain exactly $k-1$ fibres. If two vertices $u,v\in R$ are in the same fibre, then $r<\lceil\frac{n}{k}\rceil$ and then the total number of fibres is $$r+(r-1)(k-1)+k-2 = rk-1< \lceil\tfrac{n}{k}\rceil k -1 =n,$$ a contradiction. Thus $r=\lceil\frac{n}{k}\rceil$, every gap except one contains $k-1$ fibres, and the remaining gap contains $k-2$ fibres. By vertex-transitivity, we can assume $I=\{0,k,2k,\dots,(\lceil\tfrac{n}{k}\rceil-1)k\}$. Let $j=(\lceil\tfrac{n}{k}\rceil-1)k= n-(k-1)$. The gap between $\mathcal F_j$ and $\mathcal F_0$ is the one containing exactly $k-2$ fibres; all other gaps contain $k-1$ fibres. Let $u=(0,x),v=(j,y)\in R$ be the red vertices in $\mathcal F_0$ and $\mathcal F_j$, respectively. Then, as defined in Lemma [Lemma 3](#lem:interchangeable){reference-type="ref" reference="lem:interchangeable"}, let $$M = \big\{0,1,\dots,k-1\big\}\cap\big\{j, j+1, \dots, j+k-1\} = \big\{0\big\}.$$ For the only $m\in M$, $m-j=k-1$ and $m-i=0.$ Then $(x^-)_{m-j}=x_0$, $y_{m-j}=y_{k-1}$, $(y^-)_{m-i}=(y^-)_{0}=y_{k-1}$, and $x_{m-i}=x_0$. Of course, $x_0=y_{k-1}$ if and only if $y_{k-1}=x_0$. By Lemma [Lemma 3](#lem:interchangeable){reference-type="ref" reference="lem:interchangeable"} $u$ and $v$ are interchangeable by an automorphism of the form by $\alpha=\tau\mu_s\in\mathcal{A}$, where $s=n=0 \bmod n$ and $\tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}}$ is designed to flip exactly the right bits of the bitstring components of vertices in $\mathcal F_0$ and $\mathcal F_j$. More precisely, let $t \in \{0, 1, \dots, k-1\}$. If $y_t \neq (x^-)_t = x_{k-1-t}$, then $u_{j+t}=u_{k-1-t} = 1$; if $y_t = (x^-)_t = x_{k-1-t}$, then $u_{j+t}=u_{k-1-t} = 0$. Note that this prescribes the value of $u_m$ for all $m \in \{0, 1, \dots k-1\} \cup \{j, j+1, \dots, j+k-1\}$; vertices in $\mathcal F_0$ and $\mathcal F_j$ are unaffected by the value $u_\ell$ for any $\ell\in \{k, k+1, \dots, j-1\}$. We claim that we can set the value of $u_\ell$ for all $\ell\in \{k, k+1, \dots, j-1\}$ in such a way that $\tau\mu_0$ preserves $R$. Note that $$\{k, k+1, \dots, j-1\} = \bigsqcup_{a=1}^ {\lceil\frac{n}{k}\rceil-2} \{ak, ak+1, ak+2, \dots, ak+k-1\}.$$ Let $b \in \{1, \dots,\lceil\frac{n}{k}\rceil-2 \}$ and let $(bk, w)$ be the red vertex in $\mathcal F_{bk}$. Then $\tau\mu_n \cdot (bk, w) = \tau\cdot (ak, w^-)$, for some $a \in \{1, \dots,\lceil\frac{n}{k}\rceil-2 \} \setminus \{b\}$. We can arrange to have $\tau(ak, w^-)$ equal the red vertex in $\mathcal F_{ak}$ by flipping bits in $w^-$ as necessary; this can be achieved by appropriately setting the values of $u_{ak}, u_{ak+1}, \dots , u_{ak+k-1}$. These values won't affect vertices in any of the other fibres containing red vertices. Now assume $n\neq -1\bmod k$; because we are also assuming that $k \nmid n$, the remainder after $n$ is divided by $k$ satisfies $0<n-(\lceil\frac{n}{k}\rceil-1)k< k-1$. Again, let $I=\{0,k,2k,\dots,(\lceil\frac{n}{k}\rceil-1)k\}$. To simplify notation, again let $j =(\lceil\frac{n}{k}\rceil-1)k$. Then let $$R = \{(i, 00\cdots 00) : i\in I \setminus \{j\}\} \cup \{(j, 00\cdots 01)\}.$$ Note that $|R| = |I| = \lceil\frac{n}{k}\rceil$. Color every vertex in $R$ red and all other vertices blue. Let $\alpha=\tau\delta\in\mathcal{A}=\operatorname{Aut}(\operatorname{PX}(n,k))$ such that $\alpha$ preserves these color classes. The induced action of $\alpha$ on the fibres must preserve the set $I$, interpreted as a subset of $V(C_n)$. The distance between $0$ and $j=(\lceil\frac{n}{k}\rceil-1)k$ in $C_n$ is strictly less than $k+1$, whereas the distance between any other two consecutive elements of $I$ in $C_n$ is exactly $k+1$. So no nontrivial rotation preserves $I$. Thus $\delta=\mu_s$, where $\mu_s$ interchanges $0$ and $j$. Then $\tau\mu_s$ interchanges the red vertices in $\mathcal F_0$ and $\mathcal F_j$, so $\tau\delta\cdot(0,00\cdots0)=\tau (j,00\cdots0) = (j,00\cdots01)$, which implies $u_j = u_{j+1} = \dots =u_{j+k-2} = 0$ and $u_{j+k-1} =1$. Additionally, $\tau\delta\cdot(j,00\cdots01)=\tau \cdot(0,10\cdots00) = (0, 00\cdots00)$, which implies $u_0=1$ and $u_2 = u_3 = \dots = u_{k-1} = 0$. A contradiction arises because $0<n-j< k-1$ implies that in $\mathbb Z_n$, $0 =n = j+m$ for some $m \in \{1, 2, \dots, k-2\}$. Thus, $\delta=\text{id}$ and so $\alpha = \tau=\tau_0^{u_0}\tau_1^{u_1}\cdots\tau_{n-1}^{u_{n-1}} \in K$. For every $0 \le t \le \lceil{\frac{n}k}\rceil - 1$, $\mathcal F_{tk}$ contains exactly one red vertex that is fixed by $\tau$, so by Lemma [Lemma 2](#lem:tauPreservesEven){reference-type="ref" reference="lem:tauPreservesEven"}, $u_{tk} = u_{tk+1} = \dots = u_{tk+k-1} = 0$. Hence $u_0 = \dots = u_{n-1} = 0$ and so $\tau = \text{id}$. Thus, this is a $2$-distinguishing coloring of $\operatorname{PX}(n,k)$ of with a color class of size $\lceil\frac{n}{k}\rceil$. By Theorem [Theorem 9](#thm:2DistCostBounds){reference-type="ref" reference="thm:2DistCostBounds"}, $\rho(\operatorname{PX}(n,k))=\lceil\frac{n}{k}\rceil$. ◻ Our results on distinguishing number and cost are summarized below. **Theorem 13**. *Let $n\geq3$ and $2\leq k<n$. Then $\operatorname{Dist}(\operatorname{PX}(n,k))=2$ and $$\rho(\operatorname{PX}(n,k))= \begin{cases} 5, \, & \text{ if } (n,k) = (4,2),\\ \lceil\tfrac{n}{k}\rceil, &\text{ if } 5\leq n<2k \text{ or }\\ & \text{ if } 2k<n \text{ and } n \notin \{0\bmod k,-1\bmod{k}\},\\ \lceil\tfrac{n}{k}\rceil +1, &\text{ otherwise.} \end{cases}$$* [^1]: This research was supported by The Monica Odening '05 Student Internship and Research Fund in Mathematics.

arxiv_math

--- abstract: | This paper studies the effect of perturbations on the gradient flow of a general constrained nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectory of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. Interestingly, we show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow. author: - "Leilei Cui[^1], Zhong-Ping Jiang, and Eduardo D. Sontag[^2]" bibliography: - sample.bib title: "Small-Disturbance Input-to-State Stability of Perturbed Gradient Flows: Applications to LQR Problem [^3]" --- # Introduction Gradient-based optimization of loss functions constitutes a key tool in contemporary machine learning. Thus, the theoretical analysis of convergence to the minima of loss functions, in gradient-like iterations and/or in their continuous analogue, gradient-like flows (viewed as the limit of discrete-time gradient descent algorithms with infinitesimally small step size) have attracted considerable attention from both academic and industrial researchers. Besides convergence under ideal no-noise situations, a useful optimization algorithm should be capable of finding a near-optimal solution while degrading elegantly in the face of perturbations that might arise from noisy measurements of experimental data, arithmetic rounding errors due to numerical computation, numerically approximating the gradient from data through two-point estimates, discretization error when solving ODE's, or even early stopping when estimating gradients in a hierarchical learning setup [@ieee_tac_2018_cherukuri_et_al_convexity_saddle_point_dynamics; @2021_arxiv_iss_gradient_suttner_dashkovskiy; @2020arxiv_bianchin_poveda_dallanese_gradient_iss_switched_systems; @Mohammadi2022; @Sontag2022]. Especially in the setting of data-driven optimization, the analytical form of the gradient is typically unknown, and consequently the gradient has to be numerically approximated through sampling and experiments, which unavoidably introduces perturbations to the gradient iteration or flow. In computer science foundations of optimization theory, similarly, noisy or error-prone operations such as inexact or stochastic gradient computations have led to the introduction of the concept of "reproducibility in optimization" which is concerned conceptually with the same issues [@reproducibility2022]. One could also view adversarial attacks on neural network training as affecting gradient computations in "backpropagation" algorithms, and the effect of disturbances in that context has been the subject of recent work [@23cdc_brancodeoliveira_siami_sontag]. In summary, both the convergence and robustness properties of gradient descent should be theoretically analyzed in the presence of perturbations. Mathematically, gradient flows are more amenable to mathematical analysis than discrete iterations, so they are the main object of study in this paper. In order to formulate precisely the effect of perturbations on gradient flows, we employ as in [@Sontag2022] the formalism of input-to-state stability (ISS) introduced originally in [@Sontag1989] (see for example [@Sontag2008] for an exposition). The key to proving ISS for perturbed gradient flows is to verify a *Polyak-Łojasiewicz (PL)* type of condition [@Polyak1963; @Lojasiewicz1963] on the loss function to be optimized, meaning that the gradient of the loss function should not be "too small" compared to the loss. Roughly (precise definitions to be given) if we wish to minimize a continuously differentiable (but not necessarily convex) function $\mathcal{J}(z)$ on a domain $\mathcal{Z}$ and if a global minimizer $z^*$ exists, then we would like that, for some continuous function $\kappa: {\mathbb R}_{\geq0} \rightarrow {\mathbb R}_{\geq 0}$ which satisfies that $\kappa(0)=0$ and $\kappa(r)>0$ for all $r>0$ (a "positive definite" function), there should hold an estimate of the form $\lVert\nabla \mathcal{J}(z)\rVert \ge \kappa(\mathcal{J}(z) - \mathcal{J}(z^*))$ valid for all $z$ in $\mathcal{Z}$. The classical PL condition is often stated in a slightly different "semi-global" form, by requiring the existence, for each $r$, of a constant $c_r$ such that $\lVert\nabla \mathcal{J}(z)\rVert \ge c_r (\mathcal{J}(z) - \mathcal{J}(z^*))$ for every $z$ in the sublevel set $\{z\,|\,\mathcal{J}(z) \leq r\}$. For a coercive function $\mathcal{J}$, this is equivalent to the above estimate using positive definite functions. Sometimes, the PL condition is stated globally, that is, with a constant $c$ which is independent of $r$ (in other words, $\kappa$ can be picked as a linear function), but such a global condition is too strong for many applications, including the one to be pursued here. On the other hand, one could think of stronger forms of the PL condition, weaker than the existence of a linear function but stronger than merely requiring $\kappa$ to be positive definite. One particularly useful strengthening is to ask that $\kappa$ be a function of class ${\cal K}$, that is, that it be a strictly increasing function. For example, one could take $\kappa(r) = \frac{ar}{b+r}$: note that this function saturates, in the sense that it approaches a finite limit as $r\rightarrow+\infty$. This is stronger than asking that $\kappa$ be only positive definite, as illustrated by $\kappa(r) = \frac{a}{b+r}$ which is positive definite but is not of class ${\cal K}$. An even further strengthening would be to ask that $\kappa$ be of class ${\cal K_{\infty}}$, meaning that $\kappa$ does not saturate, $\kappa(r)\rightarrow\infty$ as $r\rightarrow\infty$, as for example when $\kappa$ is linear. Estimates with $\kappa$ of class ${\cal K_{\infty}}$ lead to ISS estimates for perturbed gradient flows, as discussed in [@Sontag2022], and a similar proof to that in [@Sontag2022] can be used to show that estimates with $\kappa(r)$ only positive definite lead to the weaker property of "integral ISS (iISS)" [@MR1629012; @angeli_sontag_wang_iISS_TAC00] for perturbed gradient flows. We may call the intermediate type of PL estimate, in which $\kappa$ is required to be of class ${\cal K}$ (a stronger property than positive definiteness, but not as strong as class ${\cal K}_{\infty}$) a *CJS-PL* ("comparison just saturated") estimate. It turns out that CJS-PL estimates are exactly what is required in order to establish "small-disturbance" ISS as studied in [@Pang_Jiang_2021; @Pang2022]. To be precise, we will show that when the objective function is coercive (the value of the objective function blows up when the decision variable approaches the boundary of ${\cal Z}$) and the CJS-PL condition holds, the perturbed gradient flow is small-disturbance ISS. This implies that the trajectory of the perturbed gradient flow will eventually enter a small neighborhood of the optimal solution, as long as the perturbation is sufficiently small. In addition, the size of the neighborhood is (in a nonlinear manner) proportional to the magnitude of the perturbation. In the application that motivated this work, the linear regulator problem (see below), CJS-PL is the correct notion to use, and we believe that this notion might be of more general applicability in optimization problems as well. We should remark that generalizations of the PL condition, and relations to ISS types of properties, can be found in other recent work. This includes [@2021_poveda_krstic_fixedtime_iss_extremum_seeking], which studies the gradient minimization of a function ${\cal J}_q$ on Euclidean space, where the parameter $q$ represents time-varying uncertainty. In that paper, an ISS property is established with respect to the rate of change of the parameter $q$, essentially showing differential ISS (DISS) [@DiffISS_IJRNC03]). Extremum-seeking controllers based on gradient flows and an ISS property with respect to disturbances, specifically for an integrator and a kinematic unicycle, are designed and analyzed in [@2021_arxiv_iss_gradient_suttner_dashkovskiy]; in that paper the domain is a closed submanifold of an Euclidean space. In [@ieee_tac_2018_cherukuri_et_al_convexity_saddle_point_dynamics] one finds results on gradient flows that are ISS with respect to additive errors, but assuming a "convex-concave" property for the loss function, and in  [@2020arxiv_bianchin_poveda_dallanese_gradient_iss_switched_systems] the authors solve an output regulation problem for switched linear systems, and show an ISS property for gradient flows with respect to unknown disturbances acting on the plant. We now turn to the main motivation for this work. Reinforcement learning (RL) is an active research field in which gradient-based optimization plays a pivotal role [@book_sutton Chapter 13]. In the setting of RL, an agent interacts continuously with an unknown environment, and iteratively optimizes a performance index by collecting data from the environment. By adopting gradient-based optimization methods, various policy optimization (PO) algorithms have been developed, such as actor-critic methods [@konda1999actor], deep deterministic policy gradient [@Lillicrap2015], and trust region policy optimization [@Schulman15]. The critical strategy of the policy optimization methods is to parameterize the policy by universal approximations and update the parameters of the policy along the gradient descent direction of the performance index. Starting in the early 1960s with the work of Kalman, the linear quadratic regulator (LQR) problem was shown to be theoretically tractable, and has become a widely utilized tool for optimal control and feedback design in engineering applications. In the classical approach, the (infinite-horizon) LQR problem relies upon the solution of a Riccati equation. In 1970, Athans and Levine [@1970levine_athans] introduced the idea of a direct gradient descent computation of optimal feedback gains, a procedure which can be interpreted as a form of RL. Thus, the LQR problem offers an ideal benchmark for better understanding policy optimization methods in the RL field, as one can compare solutions to the known optimal solution, and analysis of gradient methods can take advantage of theory developed for LQR. For policy optimization in the LQR problem, the objective function is a cumulative quadratic function of the state and control inputs, the control policy is parameterized as a linear function (feedback) of the state, and the admissible set, consisting of all the stabilizing control gains, is an open subset of an Euclidean space. As investigated for example in [@bu2020policy; @Mohammadi2022; @Hu2023Review], the gradient of the objective function can be computed by using a Lyapunov equation that depends on the system matrices. Nevertheless, if precise system knowledge is unavailable, as in the setting of model-free RL, the gradient has to be numerically approximated through sampling and experiments. For example, by utilizing the approximate dynamic programming technique [@books_Bertsekas; @book_Powell], the Lyapunov equation was solved by data-driven methods in [@book_Jiang; @Book_Lewis2013; @tutorial_Jiang]. In [@Mohammadi2022; @fazel2018global; @Li2021], the gradient is directly calculated by the finite differences method [@book_Nocedal Section 7.1], based on the change in function values in response to small perturbations near a given point. For these data-driven methods, a gradient estimation error is inevitable due to noisy data and insufficient samples. Therefore, the robustness analysis of the policy optimization algorithm in the presence of perturbations is critical for efficient learning, and lays the foundations for better understanding RL algorithms. Our main result will be that, for the LQR problem, the loss function is coercive and satisfies the CJS-PL property, and therefore, by the results in the first part of the paper, we conclude that the perturbed standard gradient flow is small-disturbance ISS. We also show that two variants of gradient flows, natural gradient flows and Newton gradient flows, are small-disturbance ISS. The new contribution is to establish the CJS-PL property for the LQR problem. This considerably extends previous work [@Mohammadi2022; @bu2020policy] that only showed a semiglobal estimate (and thus would imply merely iISS). In [@Sontag2022], it was mistakenly stated that the magnitude of the gradient is lower bounded by a $\mathcal{K}_\infty$-function, which is a stronger property. This is incorrect. Indeed, take a one-dimensional linear system with scalar inputs and assume that all constants in the system and cost function are equal to one. Then the loss function is ${\cal J}(z) = \frac{z^2 + 1}{2(z - 1)}$, so that its gradient is ${\cal J'}(z) = \frac{z^2 - 2z - 1}{2(z - 1)^2}$. The domain of ${\cal J}$ is the open set $(1,\infty)$. We claim that there is no function $\kappa$ of class $\mathcal{K}_\infty$ such that $\left|\mathcal{J}'(z)\right| \ge \kappa(\mathcal{J}(z) - \mathcal{J}(z^*))$, where $z^*$ is the global minimizer of $\mathcal{J}$. Indeed, as $z\rightarrow\infty$ we would have that $\mathcal{J}(z)\rightarrow\infty$, so, as $\kappa$ is of class $\mathcal{K}_\infty$, also $\kappa(\mathcal{J}(z) - \mathcal{J}(z^*))\rightarrow\infty$. However, the left-hand side $\left|\mathcal{J}'(z)\right|$ is bounded, and in fact converges to $1/2$, showing a contradiction. To summarize, the contributions of the paper are as follows. First, we provide a Lyapunov-like necessary and sufficient condition for small-disturbance ISS. Second, under assumptions of coercivity and the CJS-PL property, we use the Lyapunov characterization to show that the perturbed gradient flow for a general constrained nonlinear programming problem is small-disturbance ISS. Finally, we show the CJS-PL property for the LQR loss function, which in turn then implies that the standard gradient flow, natural gradient flow, and Newton gradient flow, are all small-disturbance ISS. The remaining contents of the paper are organized as follows. In Section 2, the concept of small-disturbance ISS is reviewed, followed by a necessary and sufficient condition. Section 3 introduces the perturbed gradient flow for a general constrained nonlinear programming problem over an open admissible set, and it is shown that the perturbed gradient flow is small-disturbance ISS under appropriate conditions on the loss function. In Section 4, we study the CSJ-PL property for the LQR problem, and three different kinds of the perturbed gradient flows for LQR are shown to be small-disturbance ISS. *Notations:* In this paper, $\mathbb{R}$ ($\mathbb{R}_+$) denotes the set of (nonnegative) real numbers. $\mathbb{P}^n$ denotes the set of $n$-dimensional real symmetric and positive semi-definite matrices. $\lambda_{\mathrm{min}}\left(\cdot\right)$ and $\lambda_{\mathrm{max}}\left(\cdot\right)$ denote the minimal and maximal eigenvalues of a real symmetric matrix, respectively. $\mathrm{Tr}\left(\cdot\right)$ denotes the trace of a square matrix. $\lVert\cdot\rVert$ denotes the spectral norm of a matrix or Euclidean norm of a vector, and $\lVert\cdot\rVert_F$ denotes the Frobenius norm of a matrix. For a continuous-time signal $w(\cdot) : \mathbb{R}_+ \to \mathbb{R}^{n}$, $\lVert w\rVert_\infty$ denotes its essential supremum norm, i.e. $\lVert w\rVert_\infty = \mathop{\mathrm{ess\,sup}}_{s \in \mathbb{R}_+}\lVert w(s)\rVert$. For a matrix-valued signal $K(\cdot) : \mathbb{R}_+ \to \mathbb{R}^{m \times n}$, $\lVert K\rVert_\infty = \mathop{\mathrm{ess\,sup}}_{s \in \mathbb{R}_+}\lVert K(s)\rVert_F$. $w_t(\cdot)$ denotes the truncation of $w(\cdot)$ at $t$, that is, $w_t(s) = w(s)$ if $s \le t$, and $w_t(s) = 0$ if $s > t$. For any $K_1, K_2 \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{P}^n$, define the inner product as $\langle K_1, K_2 \rangle_Y = \mathrm{Tr}\left(K_1^T K_2 Y\right)$. # Small-Disturbance Input-to-State Stability Let $\mathcal{S}$ denote an open subset of $\mathbb{R}^n$, which will be called the admissible set of states. Consider the following nonlinear system $$\begin{aligned} \label{eq:nonLinearSys} \dot{\chi}(t) = f(\chi(t), w(t)),\end{aligned}$$ where $f: \mathcal{S} \times \mathbb{R}^m \to \mathcal{S}$ is continuously differentiable, and inputs $w(\cdot): \mathbb{R}_+ \to \mathbb{R}^m$ are locally measurable essentially bounded functions. We assume given an equilibrium $\chi^*$ of the unforced system, that is $f(\chi^*,0)=0$. **Definition 1**. *[@book_Hahn] A function $\alpha(\cdot): \mathbb{R}_+ \to \mathbb{R}_+$ is a $\mathcal{K}$-function if it is continuous, strictly increasing, and vanishes at zero. For any $d >0$, a function $\alpha(\cdot): [0,d) \to \mathbb{R}_+$ is a $\mathcal{K}_{[0,d)}$-function if it is continuous, strictly increasing, and vanishes at zero. A function $\alpha(\cdot): \mathbb{R}_+ \to \mathbb{R}_+$ is a $\mathcal{K}_\infty$-function if it is a $\mathcal{K}$-function and also satisfies $\alpha(r) \to \infty$ as $r \to \infty$. A function $\beta(\cdot,\cdot): \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ is a $\mathcal{KL}$-function if for any fixed $t \geq 0$, $\beta(\cdot,t)$ is a $\mathcal{K}$-function, and for any fixed $r \geq 0$, $\beta(r,\cdot)$ is decreasing and $\beta(r,t) \to 0$ as $t \to \infty$.* **Definition 2**. *[@Sontag2022][\[def:sizeFunc\]]{#def:sizeFunc label="def:sizeFunc"} A function $\mathcal{V}: \mathcal{S} \to \mathbb{R}$ is a size function if $\mathcal{V}(\cdot)$ is* 1. *continuous;* 2. *positive definite with respect to $\chi^*$, i.e. $\mathcal{V}(\chi^*)=0$ and $\mathcal{V}(\chi)>0$ for all $\chi \neq \chi^*$, $\chi \in \mathcal{S}$;* 3. *coercive, i.e. for any sequence $\{\chi_k\}_{k=0}^\infty$, $\chi_k \to \partial \mathcal{S}$ or $\lVert\chi_k\rVert \to \infty$, it holds that $\mathcal{V}(\chi_k) \to \infty$.* **Definition 3**. *[@Pang_Jiang_2021; @Pang2022][\[def:smallISS\]]{#def:smallISS label="def:smallISS"} System [\[eq:nonLinearSys\]](#eq:nonLinearSys){reference-type="eqref" reference="eq:nonLinearSys"} is small-disturbance input-to-state stable (ISS) if there exist a size function $\mathcal{V}(\cdot)$, a constant $d > 0$, a $\mathcal{KL}$-function $\beta(\cdot,\cdot)$, and a $\mathcal{K}_{[0,d)}$-function $\gamma(\cdot)$, such that for all inputs $w$ locally bounded by $d$ (i.e. $\lVert w_t\rVert_\infty<d$, $\forall t\ge 0$), and all initial states $\chi(0) \in \mathcal{S}$, $\chi(t)$ remains in $\mathcal{S}$ and satisfies $$\begin{aligned} \label{eq:ISS} \mathcal{V}(\chi(t)) \leq \beta(\mathcal{V}(\chi(0)),t) + \gamma(\lVert w\rVert_\infty), \quad \forall t \ge 0.\end{aligned}$$* The notation $w_t$ means the restriction of $w$ to the interval $[0,t]$. By causality of the dynamical system, the same definition would result if $\lVert w\rVert_\infty$ is replaced by $\lVert w_t\rVert_\infty$ in [\[eq:ISS\]](#eq:ISS){reference-type="eqref" reference="eq:ISS"}. **Definition 4**. *A continuously differentiable function $\mathcal{V}:\mathcal{S} \to \mathbb{R}$ is a small-disturbance ISS-Lyapunov function for system [\[eq:nonLinearSys\]](#eq:nonLinearSys){reference-type="eqref" reference="eq:nonLinearSys"} if* 1. *$\mathcal{V}$ is a size function;* 2. *there exist $\mathcal{K}$-functions $\alpha_1(\cdot)$ and $\alpha_2(\cdot)$ such that if $\lVert\mu\rVert \leq \alpha_1(\mathcal{V}(\chi))$, $$\begin{aligned} \dot{\mathcal{V}}(\chi,\mu) = \langle\nabla\mathcal{V}(\chi), f(\chi,\mu) \rangle \leq - \alpha_2(\mathcal{V}(\chi)). \end{aligned}$$* **Theorem 1**. *System [\[eq:nonLinearSys\]](#eq:nonLinearSys){reference-type="eqref" reference="eq:nonLinearSys"} is small-disturbance ISS if and only if it admits a small-disturbance ISS-Lyapunov function.* *Proof.* **Sufficiency:** This is an adaptation of the proof of the analogous result for the ISS property [@Sontag1989]. Let $d= \sup_{r \in \mathbb{R}_+} \alpha_1(r)$. For any inputs $w$ with $\lVert w\rVert_\infty<d$, define the sublevel set $\mathcal{S}_c = \{\chi \in \mathcal{S}| \mathcal{V}(\chi) \leq c \}$, where $c = \alpha_1^{-1}(\lVert w\rVert_\infty)$. Borrowing techniques similar to those in the proof of [@Sontag1989 Theorem 1], we can show that $\mathcal{S}_c$ is forward invariant, i.e. if $\chi(t_0) \in \mathcal{S}_c$ for some $t_0 \ge 0$, then $\chi(t) \in \mathcal{S}_c$ for all $t \ge t_0$. Now, let $t_1 = \inf\{t \in \mathbb{R}_+| \chi(t) \in \mathcal{S}_c \} \leq \infty$. Therefore, for any $t \ge t_1$, we have $$\begin{aligned} \label{eq:sizefunctionUpperBound1} \mathcal{V}(\chi(t)) \le \alpha_1^{-1}(\lVert w\rVert_\infty).\end{aligned}$$ For $t < t_1$, $\alpha_1(\mathcal{V}(\chi(t))) \ge \lVert w\rVert_\infty$, which implies that $$\begin{aligned} \frac{\mathrm{d}\mathcal{V}(\chi(t))}{\mathrm{d}t} \le -\alpha_2(\mathcal{V}(\chi(t))) , \quad \forall t < t_1.\end{aligned}$$ By the comparison lemma [@book_Khalil Lemmas 3.4 and 4.4], there exists a $\mathcal{KL}$-function $\beta(\cdot,\cdot)$ such that $$\begin{aligned} \label{eq:sizefunctionUpperBound2} \mathcal{V}(\chi(t)) \le \beta(\mathcal{V}(\chi(0)),t), \quad \forall t < t_1.\end{aligned}$$ Combining [\[eq:sizefunctionUpperBound1\]](#eq:sizefunctionUpperBound1){reference-type="eqref" reference="eq:sizefunctionUpperBound1"} and [\[eq:sizefunctionUpperBound2\]](#eq:sizefunctionUpperBound2){reference-type="eqref" reference="eq:sizefunctionUpperBound2"}, the small-disturbance ISS property [\[eq:ISS\]](#eq:ISS){reference-type="eqref" reference="eq:ISS"} follows readily with $\gamma(\cdot) = \alpha_1^{-1}(\cdot)$. **Necessity:** We first prove the case when $\mathcal{S} = \mathbb{R}^n$. Reparameterize the input as $$\begin{aligned} w = \frac{d}{1+\lVert v\rVert}v =: h(v).\end{aligned}$$ It is shown in Lemma [Lemma 12](#lm:homomorphism){reference-type="ref" reference="lm:homomorphism"} that $h(\cdot)$ is a homeomorphism from $\mathbb{R}^m$ to $\mathcal{W} := \{w \in \mathbb{R}^m|\lVert w\rVert <d\}$. With the input change, we have $$\begin{aligned} \label{eq:nonlinearSysRepara} \dot{\chi}(t) = f\left(\chi(t), h(v(t))\right) =: f_1\left(\chi(t), v(t)\right).\end{aligned}$$ Since $\gamma(\cdot)$ is a $\mathcal{K}_{[0,d)}$-function and $\gamma_1(\cdot)$ is a $\mathcal{K}$-function with the range $[0,d)$, $\gamma_2(\cdot) = \gamma(\gamma_1(\cdot))$ is a $\mathcal{K}$-function. According to [\[eq:ISS\]](#eq:ISS){reference-type="eqref" reference="eq:ISS"}, it holds $$\begin{aligned} \label{eq:ISSreparameter} \mathcal{V}(\chi(t)) \leq \beta(\mathcal{V}(\chi(0)),t) + \gamma_2(\lVert v\rVert_\infty).\end{aligned}$$ Since $\mathcal{V}(\cdot)$ is a size function over $\mathbb{R}^n$, according to Lemma [Lemma 10](#lm:radicallyUnbounded){reference-type="ref" reference="lm:radicallyUnbounded"}, there exist $\mathcal{K}_\infty$-functions $\rho_1(\cdot)$ and $\rho_2(\cdot)$, such that $$\begin{aligned} \label{eq:sizeBound} \rho_1\left(\lVert\chi-\chi^*\rVert\right) \le \mathcal{V}(\chi) \le \rho_2\left(\lVert\chi-\chi^*\rVert\right), \quad \forall \chi \in \mathbb{R}^n.\end{aligned}$$ Plugging [\[eq:sizeBound\]](#eq:sizeBound){reference-type="eqref" reference="eq:sizeBound"} into [\[eq:ISSreparameter\]](#eq:ISSreparameter){reference-type="eqref" reference="eq:ISSreparameter"} yields $$\begin{aligned} \lVert\chi(t)-\chi^*\rVert \le \rho_1^{-1}\left[\beta\left(\rho_2(\lVert\chi(0)-\chi^*\rVert),t\right) + \gamma_2(\lVert v\rVert_\infty) \right].\end{aligned}$$ By the weak triangular inequality in [@Jiang1994 Equation (6)], there exist a $\mathcal{KL}$-function $\beta_1(\cdot,\cdot)$ and a $\mathcal{K}$-function $\gamma_3(\cdot)$, such that $$\begin{aligned} \lVert\chi(t)-\chi^*\rVert \le \beta_1\left(\lVert\chi(0)-\chi^*\rVert),t\right) + \gamma_3(\lVert v\rVert_\infty).\end{aligned}$$ Hence, system [\[eq:nonlinearSysRepara\]](#eq:nonlinearSysRepara){reference-type="eqref" reference="eq:nonlinearSysRepara"} is ISS with respect to $v$. According to [@sontag1995characterizations Theorem 1], there exists an ISS-Lyapunov function $\mathcal{V}_1(\cdot)$, such that $$\begin{aligned} \label{eq:V1bound} \rho_3(\lVert\chi-\chi^*\rVert) \le \mathcal{V}_1(\chi) \le \rho_4(\lVert\chi-\chi^*\rVert), \quad \forall \chi \in \mathbb{R}^n,\end{aligned}$$ and $$\begin{aligned} \label{eq:V1ISS} \langle\nabla \mathcal{V}_1(\chi), f_1(\chi,v) \rangle \le - \rho_6(\lVert\chi-\chi^*\rVert),\end{aligned}$$ for any $\chi \in \mathbb{R}^n$ and any $v \in \mathbb{R}^m$ satisfying $\lVert v\rVert \le \rho_5(\lVert\chi-\chi^*\rVert)$, where $\rho_{i}(\cdot)$ ($i$=3,4,5) are $\mathcal{K}_\infty$-functions, and $\rho_{6}(\cdot)$ is a $\mathcal{K}$-function. This, in turn, implies that if $\lVert w\rVert \le \gamma_1 \circ \rho_5 \circ \rho_4^{-1} (\mathcal{V}_1(\chi))$, equation [\[eq:V1ISS\]](#eq:V1ISS){reference-type="eqref" reference="eq:V1ISS"} holds. Since $\gamma_1\circ \rho_5 \circ \rho_4^{-1}(\cdot)$ is a $\mathcal{K}$-function with the range $[0,d)$, we obtain that $\mathcal{V}_1(\cdot)$ is a small-disturbance ISS-Lyapunov function. Next, we prove the general case when $\mathcal{S}$ is an open subset of $\mathbb{R}^n$. For the unforced system $\dot{\chi} = f(\chi,0)$, $\chi^*$ is an asymptotically stable point and the domain of stability is $\mathcal{S}$. By [@WILSON1967323 Theorem 2.2], $\mathcal{S}$ is diffeomorphic to $\mathbb{R}^n$. Therefore, the necessity holds. ◻ # Robustness Analysis of Perturbed Gradient Flows Consider the following constrained nonlinear programming problem $$\begin{aligned} \label{eq:optiPro} &\min_{z \in \mathcal{Z}} \mathcal{J}(z) \end{aligned}$$ where $\mathcal{Z}$ is an open subset of $\mathbb{R}^n$, which is called an admissible set of variables; $\mathcal{J}(\cdot): \mathcal{Z} \to \mathbb{R}$ is an objective function with a global minimizer $z^*$. **Definition 5**. *A function $\mathcal{J}(\cdot): \mathcal{Z} \to \mathbb{R}$ is a proper objective function if it is* 1. *continuously differentiable and its gradient $\nabla \mathcal{J}(\cdot)$ is locally Lipschitz continuous;* 2. *$\mathcal{J}(z) - \mathcal{J}(z^*)$ is a size function;* 3. *there exists a $\mathcal{K}$-function $\alpha_3(\cdot)$, such that $\lVert\nabla \mathcal{J}(z)\rVert \ge \alpha_3(\mathcal{J}(z) - \mathcal{J}(z^*))$ (CJS-PL estimate).* **Remark 1**. *As discussed in the introduction, the gradient dominance condition given by the CJS-PL estimate is weaker than its counterpart in [@Sontag2022 Definition 4.1], where $\alpha_3(\cdot)$ is required to be a $\mathcal{K}_\infty$-function. Note also that this condition implies that over any compact set $\{z \in \mathcal{Z}|(\mathcal{J}(z) - \mathcal{J}(z^*) \le r\}$, there exists a constant $\alpha_r > 0$, such that $\lVert\nabla \mathcal{J}(z)\rVert \ge \alpha_r(\mathcal{J}(z) - \mathcal{J}(z^*))$. That is, the regular "PL inequality" holds over any compact sets.* The perturbed gradient flow for [\[eq:optiPro\]](#eq:optiPro){reference-type="eqref" reference="eq:optiPro"} is $$\begin{aligned} \label{eq:gradientflowopti} \dot{z}(t) = -\eta \nabla \mathcal{J}(z(t)) + \Delta z(t),\end{aligned}$$ where $\eta > 0$ is a constant step size, and $\Delta z(\cdot)$ denotes the perturbation to the gradient flow. The perturbation $\Delta z(t)$ may arise from inaccurate gradient estimation for data-driven optimization or arithmetic rounding errors of numerical computation. **Theorem 2**. *If $\mathcal{J}(\cdot)$ is a proper objective function, then, system [\[eq:gradientflowopti\]](#eq:gradientflowopti){reference-type="eqref" reference="eq:gradientflowopti"} is small-disturbance ISS.* *Proof.* We will prove that $\mathcal{V}_2(z) = \mathcal{J}(z) - \mathcal{J}(z^*)$ is a small-disturbance ISS-Lyapunov function. Firstly, notice that $\mathcal{V}_2(\cdot)$ is a size function with respect to $z^*$. Then, differentiating $\mathcal{V}_2(z)$ along the trajectories of system [\[eq:gradientflowopti\]](#eq:gradientflowopti){reference-type="eqref" reference="eq:gradientflowopti"} yields $$\begin{aligned} \label{eq:derivativeVo1} \begin{split} \dot{\mathcal{V}}_2(z, \Delta z) &= - \eta \langle\nabla \mathcal{J}(z), \nabla \mathcal{J}(z) \rangle + \langle\nabla \mathcal{J}(z), \Delta z \rangle \\ & \leq - \eta \lVert \nabla \mathcal{J}(z)\rVert^2 + \lVert\nabla \mathcal{J}(z)\rVert \lVert\Delta z\rVert \\ &\leq - \frac{\eta}{2} \lVert \nabla \mathcal{J}(z)\rVert^2 + \frac{1}{2\eta}\lVert\Delta z\rVert^2 \\ &\leq - \frac{\eta}{2} \alpha_3^2(\mathcal{V}_2(z)) + \frac{1}{2\eta}\lVert\Delta z\rVert^2, \end{split} \end{aligned}$$ where the second line follows from the Cauchy-Schwarz inequality, the third line is from Young's inequality, and the fourth line is a direct consequence of the CJS-PL property. Hence, if $\lVert\Delta z\rVert \leq \frac{\eta}{\sqrt{2}}\alpha_3(\mathcal{V}_2(z))$, it follows from [\[eq:derivativeVo1\]](#eq:derivativeVo1){reference-type="eqref" reference="eq:derivativeVo1"} that $$\begin{aligned} \dot{\mathcal{V}}_2(z, \Delta z) \leq -\frac{\eta}{4}\alpha_3^2(\mathcal{V}_2(z)). \end{aligned}$$ Since $\alpha_3(\cdot)$ is a $\mathcal{K}$-function, $\frac{\eta}{\sqrt{2}}\alpha_3(\cdot)$ is also a $\mathcal{K}$-function. Therefore, by Definition [Definition 4](#def:smallISSLyap){reference-type="ref" reference="def:smallISSLyap"}, $\mathcal{V}_2(z) = \mathcal{J}(z) - \mathcal{J}(z^*)$ is a small-disturbance ISS-Lyapunov function. According to Theorem [Theorem 1](#thm:smallISSSufficient){reference-type="ref" reference="thm:smallISSSufficient"}, we conclude that system [\[eq:gradientflowopti\]](#eq:gradientflowopti){reference-type="eqref" reference="eq:gradientflowopti"} is small-disturbance ISS. ◻ # Application to LQR Problem ## Preliminaries Consider the following linear time-invariant system $$\begin{aligned} \dot{x} = Ax + Bu, \,\, x(0) = x_0,\end{aligned}$$ where $x \in \mathbb{R}^n$ is the state; $x_0$ is the initial state; $u \in \mathbb{R}^{m}$ is the control input; $A$ and $B$ are constant matrices with compatible dimensions. The continuous-time LQR aims at finding a state-feedback controller by solving the following optimal control problem $$\begin{aligned} \label{eq:costContinuous} \min_{u}\mathcal{J}_{1}(x_0, u) = \int_{0}^{\infty} x^T(t)Qx(t) + u^T(t) R u(t) \mathrm{d}t , \end{aligned}$$ with $Q = Q^T \succeq 0$, and $R=R^T \succ 0$. Under the assumption that $(A,B)$ is stabilizable and $(A,\sqrt{Q})$ is observable, as shown in [@book_sontag Section 8.4], the optimal controller is $$\begin{aligned} \label{eq:Koptexpression} u^*(x(t)) = -{K^*}x(t), \quad K^* = R^{-1}B^TP^*,\end{aligned}$$ and $P^* = (P^*)^T \succ 0$ is the unique solution of the following algebraic Riccati equation (ARE) $$\begin{aligned} \label{eq:continuousARE} A^T P^* + P^* A + Q - P^*BR^{-1}B^TP^* = 0.\end{aligned}$$ Let $\mathcal{G} = \{K \in \mathbb{R}^{m \times n}| A-BK \text{ is Hurwitz} \}$ denote the admissible set of all stabilizing control gains. For any stabilizing controller $u(t) = -Kx(t)$, where $K \in \mathcal{G}$, and any nonzero initial state $x_0 \in \mathbb{R}^n$, the corresponding cost is $$\begin{aligned} \mathcal{J}_{1}(x_0, K) = \int_{0}^\infty x_0^T e^{(A-BK)^Tt} (Q + K^TRK) e^{(A-BK) t} x_0 \mathrm{d}t = x_0^T P_K x_0,\end{aligned}$$ where $P_K = P_K^T \succ 0$ is the solution of the following Lyapunov equation $$\begin{aligned} \label{eq:continuousLyapunov} (A-BK)^{T}P_K + P_K (A-BK) + Q + K^T R K = 0.\end{aligned}$$ Since $K^*$ is the optimal control gain and $P^* = P_{K^*}$ is the minimal cost matrix, by [@book_sontag page 382], it holds $$\begin{aligned} \mathcal{J}_1(x_0,K) = x^T_0 P_K x_0 \ge x^T_0 P^* x_0 = \mathcal{J}_1(x_0,K^*), \quad \forall x_0 \in \mathbb{R}^n.\end{aligned}$$ This implies that $P_K \succeq P^*$. Given an arbitrary nonzero initial state $x_0$, the LQR problem can be interpreted as optimizing the control gain over the admissible set $\mathcal{G}$, i.e. $$\begin{aligned} \label{eq:LQRoptimization} \min_{K \in \mathcal{G}}\mathcal{J}_1(x_0, K).\end{aligned}$$ Since the objective function $\mathcal{J}_1$ depends on the initial condition, we are motivated to develop an initial condition-independent formulation for the LQR. For any initial state, an upper bound for $\mathcal{J}_1(x_0,K)$ is $$\begin{aligned} \mathcal{J}_1(x_0,K) \le \lVert x_0\rVert^2\mathrm{Tr}\left(P_K\right) = \lVert x_0\rVert^2\mathcal{J}_2(K),\end{aligned}$$ where $$\begin{aligned} \label{eq:costJc_closedform} \mathcal{J}_2(K) = \mathrm{Tr}\left(P_K\right),\end{aligned}$$ which is independent of $x_0$. Notice that, for any nonzero $x_0$, $\mathcal{J}_1(x_0,K)$ and $\mathcal{J}_2(K)$ have the same minimum at $K^*$. Thus, [\[eq:LQRoptimization\]](#eq:LQRoptimization){reference-type="eqref" reference="eq:LQRoptimization"} is equivalent to the following policy optimization problem [@book_Boyd Section 4.1.3] $$\begin{aligned} \min_{K \in \mathcal{G}}\mathcal{J}_2(K). \end{aligned}$$ Before calculating $\nabla \mathcal{J}_2(K)$, which is defined from the first-order term in Taylor expansion, let us define the matrix $Y_K \in \mathbb{P}^n$ as the solution of $$\begin{aligned} \label{eq:YKDef} (A-BK)Y_K + Y_K (A-BK)^T + I_n = 0.\end{aligned}$$ It is noticed that according to [@book_Zhou Lemma 3.18], $Y_K \succ 0$ for any $K \in \mathcal{G}$. In addition, $Y^*$ is defined as the solution of [\[eq:YKDef\]](#eq:YKDef){reference-type="eqref" reference="eq:YKDef"} with $K$ replaced by $K^*$. Since $A-BK$ is Hurwitz, by [@book_chen], $Y_K$ can be expressed as $$\begin{aligned} \label{eq:YKexpress} Y_K = \int_{0}^{\infty} e^{(A-BK)t}e^{(A-BK)^Tt} \mathrm{d}t.\end{aligned}$$ **Lemma 1**. *For any $K \in \mathcal{G}$, when it is perturbed by $E$ with $K+E\in \mathcal{G}$, the second-order approximation of $\mathcal{J}_2(K + E)$ is $$\begin{aligned} \label{eq:J2Expansion} \begin{split} \mathcal{J}_2(K+E ) &= \mathcal{J}_2(K )+ 2\mathrm{Tr}\left(E^T (RK - B^TP_K)Y_K \right) + \mathrm{Tr}\left(E^T R EY_K\right) \\ &+ 2\mathrm{Tr}\left(E^T (RK - B^TP_K) \Delta Y_K \right)+O(\lVert E\rVert_F^3) , \end{split} \end{aligned}$$ where $\Delta Y_K$ is defined in [\[eq:DeltaYK\]](#eq:DeltaYK){reference-type="eqref" reference="eq:DeltaYK"}, and $O(\lVert E\rVert_F^3)$ is the remainder of the approximation.* *Proof.* Firstly, we calculate $Y_{K+E}$ when $K$ is perturbed to $K+ E$. Using [\[eq:YKDef\]](#eq:YKDef){reference-type="eqref" reference="eq:YKDef"}, we have $$\begin{aligned} \label{eq:YKDeltaK} (A-BK - BE)Y_{K+E} + Y_{K+E}(A-BK -BE)^T + I_n = 0. \end{aligned}$$ Subtracting [\[eq:YKDef\]](#eq:YKDef){reference-type="eqref" reference="eq:YKDef"} from [\[eq:YKDeltaK\]](#eq:YKDeltaK){reference-type="eqref" reference="eq:YKDeltaK"}, we have $$\begin{aligned} (A-BK -BE) (Y_{K+E} - Y_K) + (Y_{K+E} - Y_K) (A-BK -BE )^T - BE Y_K - Y_K E^T B^T = 0. \end{aligned}$$ Hence, the first-order approximation of $(Y_{K+E} - Y_K)$, denoted as $\Delta Y_K$, satisfies $$\begin{aligned} (A-BK) \Delta Y_K + \Delta Y_K (A-BK )^T - BE Y_K - Y_K E^T B^T = 0, \end{aligned}$$ which, in turn, implies $$\begin{aligned} \label{eq:DeltaYK} \Delta Y_K = -\int_{0}^{\infty} e^{(A-BK )t} ( BE Y_K + Y_K E^T B^T) e^{(A-BK )^{T}t}. \end{aligned}$$ Then, we will calculate $P_{K + E}$ for the perturbed control gain $K+E$. By [\[eq:continuousLyapunov\]](#eq:continuousLyapunov){reference-type="eqref" reference="eq:continuousLyapunov"}, we have $$\begin{aligned} \label{eq:PKDeltaK} \begin{split} &(A-BK-BE)^{T}P_{K + E} + P_{K + E} (A-BK-BE) + Q + (K+E)^T R (K+E) = 0. \end{split} \end{aligned}$$ Subtracting [\[eq:continuousLyapunov\]](#eq:continuousLyapunov){reference-type="eqref" reference="eq:continuousLyapunov"} from [\[eq:PKDeltaK\]](#eq:PKDeltaK){reference-type="eqref" reference="eq:PKDeltaK"} yields $$\begin{aligned} \label{eq:PKDeltaK1} \begin{split} &(A-BK-BE)^{T}(P_{K + E}-P_K) + (P_{K + E}-P_K) (A-BK-BE)\\ &+ E^T (RK - B^TP_K) + (RK - B^TP_K)^T E + E^T R E= 0, \end{split} \end{aligned}$$ which is equivalent to $$\begin{aligned} \label{eq:PKE-PK} P_{K + E}-P_K = \int_{0}^{\infty} e^{(A-BK-BE)^Tt} [ E^T (RK - B^TP_K) + (RK - B^TP_K)^T E + E^T R E] e^{(A-BK-BE)t}\mathrm{d}t. \end{aligned}$$ Taking the trace of [\[eq:PKE-PK\]](#eq:PKE-PK){reference-type="eqref" reference="eq:PKE-PK"}, considering $\mathcal{J}_2(K+E) -\mathcal{J}_2(K) = \mathrm{Tr}\left(P_{K + E}-P_K\right)$, and using the cyclic property of the trace [@book_Petersen Equation (16)], we can obtain $$\begin{aligned} \label{eq:JKdiff} \begin{split} &\mathcal{J}_2(K+E ) -\mathcal{J}_2(K) \\ &= \mathrm{Tr}\left( [ E^T (RK - B^TP_K) + (RK - B^TP_K)^T E + E^T R E]\int_{0}^{\infty} e^{(A-BK-BE)t} e^{(A-BK-BE)^Tt} \mathrm{d}t\right) \\ &=2\mathrm{Tr}\left(E^T (RK - B^TP_K)Y_{K+E}\right) + \mathrm{Tr}\left(E^T R EY_{K+E}\right). \end{split} \end{aligned}$$ Since the first-order approximation of $Y_{K+E}$ is $Y_K + \Delta Y_K$, plugging it into [\[eq:JKdiff\]](#eq:JKdiff){reference-type="eqref" reference="eq:JKdiff"}, we can obtain [\[eq:J2Expansion\]](#eq:J2Expansion){reference-type="eqref" reference="eq:J2Expansion"}. ◻ In [\[eq:J2Expansion\]](#eq:J2Expansion){reference-type="eqref" reference="eq:J2Expansion"}, the first-order term is $2\mathrm{Tr}\left(E^T (RK - B^TP_K)Y_K \right) = \langle E, 2(RK - B^TP_K)Y_K \rangle$, which should be equal to $\langle E, \nabla \mathcal{J}_2(K) \rangle$ by Taylor expansion. Hence, the gradient of the objective function $\mathcal{J}_2(K)$ is $$\begin{aligned} \label{eq:JcGradient} \nabla \mathcal{J}_2(K) = 2(RK - B^TP_K)Y_K,\end{aligned}$$ We claim that $\mathcal{J}_2(K)$ is a proper objective function (Definition [Definition 5](#def:ProperLossFun){reference-type="ref" reference="def:ProperLossFun"}). To this end, we will first prove several intermediate lemmas providing bounds on $Y_K$ and $P_K$. The following lemma gives a lower bound of $\lambda_{\mathrm{min}}\left(Y_K\right)$. **Lemma 2**. *For any $K \in \mathcal{G}$, we have $$\begin{aligned} \lambda_{\mathrm{min}}\left(Y_K\right) \geq \frac{1}{2\lVert A-BK^*\rVert_F + 2\lVert B\rVert\lVert K-K^*\rVert_F}.\end{aligned}$$* *Proof.* Let $q \in \mathbb{R}^n$ denote a unit eigenvector of $Y_K$ associated with the eigenvalue $\lambda_{\mathrm{min}}\left(Y_K\right)$. Pre and post multiplying [\[eq:YKDef\]](#eq:YKDef){reference-type="eqref" reference="eq:YKDef"} by $q^T$ and $q$, respectively, we have $$\begin{aligned} \label{eq:A-BKeig1} \lambda_{\mathrm{min}}\left(Y_K\right)q^T[(A-BK) + (A-BK)^T]q = -1. \end{aligned}$$ Since $Y_K \succ 0$, it follows from [\[eq:A-BKeig1\]](#eq:A-BKeig1){reference-type="eqref" reference="eq:A-BKeig1"} that $q^T[(A-BK) + (A-BK)^T]q<0$. Consequently, we have $$\begin{aligned} \label{eq:A-BKeig2} \begin{split} &\lambda_{\mathrm{min}}\left((A-BK) + (A-BK)^T\right) = \min_{\lVert\nu\rVert=1} \nu^T[(A-BK) + (A-BK)^T]\nu \\ &\leq q^T[(A-BK) + (A-BK)^T]q<0. \end{split} \end{aligned}$$ By [\[eq:A-BKeig1\]](#eq:A-BKeig1){reference-type="eqref" reference="eq:A-BKeig1"} and [\[eq:A-BKeig2\]](#eq:A-BKeig2){reference-type="eqref" reference="eq:A-BKeig2"}, it holds $$\begin{aligned} \begin{split} &\lambda_{\mathrm{min}}\left(Y_K\right) = \frac{1}{-q^T[(A-BK) + (A-BK)^T]q} \geq \frac{1}{- \lambda_{\mathrm{min}}\left((A-BK) + (A-BK)^T\right)} \\ &= \frac{1}{- \lambda_{\mathrm{min}}\left((A-BK^*) + (A-BK^*)^T + B(K^*-K) + (K^*-K)^TB^T\right)} \\ &\geq \frac{1}{2\lVert A-BK^*\rVert_F + 2 \lVert B\rVert \lVert K-K^*\rVert_F}, \end{split} \end{aligned}$$ where the last line uses the relation $\lVert B(K^* - K)\rVert \le \lVert B\rVert\lVert(K^* - K)\rVert\le \lVert B\rVert\lVert(K^* - K)\rVert_F$. Thus, the proof is completed. ◻ The following lemma gives the bounds of $\mathrm{Tr}\left(Y_K\right)$. **Lemma 3**. *Given $K \in \mathcal{G}$ and $Q \succ 0$, we have $$\begin{aligned} \label{eq:YKBound} \frac{\mathrm{Tr}\left(P_K-P^*\right)}{\lVert R\rVert\lVert K-K^*\rVert_F^2} \le \mathrm{Tr}\left(Y_{K}\right) \le \frac{\mathrm{Tr}\left(P_K\right)}{\lambda_{\mathrm{min}}\left(Q\right)} \end{aligned}$$* *Proof.* Considering $K^* = R^{-1}B^TP^*$, we can rewrite [\[eq:continuousARE\]](#eq:continuousARE){reference-type="eqref" reference="eq:continuousARE"} as $$\begin{aligned} \label{eq:AoptPopt} (A-BK^*)^T P^* + P^* (A-BK^*) + Q + (K^*)^T R K^* = 0. \end{aligned}$$ It follows from [\[eq:Koptexpression\]](#eq:Koptexpression){reference-type="eqref" reference="eq:Koptexpression"} and [\[eq:AoptPopt\]](#eq:AoptPopt){reference-type="eqref" reference="eq:AoptPopt"} that $$\begin{aligned} \label{eq:AoptPoptREwrite} (A-BK)^TP^* + P^*(A-BK) + Q + (K^*)^TRK^* + (K-K^*)^TRK^* + (K^*)^TR(K-K^*) = 0. \end{aligned}$$ Subtracting [\[eq:AoptPoptREwrite\]](#eq:AoptPoptREwrite){reference-type="eqref" reference="eq:AoptPoptREwrite"} from [\[eq:continuousLyapunov\]](#eq:continuousLyapunov){reference-type="eqref" reference="eq:continuousLyapunov"} and completing the squares yield $$\begin{aligned} \label{eq:ABKPdiff} (A-BK)^T(P_K - P^*) +(P_K - P^*)(A-BK) + (K-K^*)^TR(K-K^*) = 0. \end{aligned}$$ Since $A-BK$ is Hurwitz, by [@book_chen Equation (5.18)], we have $$\begin{aligned} \label{eq:PKdiff1} P_K - P^* = \int_{0}^{\infty} e^{(A-BK)^{T}t} (K-K^*)^TR(K-K^*) e^{(A-BK)t} \mathrm{d}t. \end{aligned}$$ Taking the trace of [\[eq:PKdiff1\]](#eq:PKdiff1){reference-type="eqref" reference="eq:PKdiff1"} and using the cyclic property of the trace in [@book_Petersen Equation (16)] and [\[eq:YKexpress\]](#eq:YKexpress){reference-type="eqref" reference="eq:YKexpress"}, we obtain $$\begin{aligned} \label{eq:TrPKPopt} \begin{split} \mathrm{Tr}\left(P_K - P^*\right) &= \mathrm{Tr}\left(\int_{0}^{\infty}e^{(A-BK)t} e^{(A-BK)^{T}t} \mathrm{d}t (K-K^*)^TR(K-K^*) \right) \\ &= \mathrm{Tr}\left(Y_K(K-K^*)^TR(K-K^*)\right). \end{split} \end{aligned}$$ By the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"} and considering the following relation $$\begin{aligned} \lVert(K-K^*)^TR(K-K^*)\rVert \le \mathrm{Tr}\left((K-K^*)^TR(K-K^*)\right) \le \lVert R\rVert \lVert K-K^*\rVert_F^2, \end{aligned}$$ we have $$\begin{aligned} \mathrm{Tr}\left(P_K - P^*\right) \le \lVert R\rVert\lVert K-K^*\rVert_F^2 \mathrm{Tr}\left(Y_K\right). \end{aligned}$$ Hence, the lower bound of $\mathrm{Tr}\left(Y_K\right)$ in [\[eq:YKBound\]](#eq:YKBound){reference-type="eqref" reference="eq:YKBound"} is obtained. Since $A-BK$ is Hurwitz, it follows from [@book_chen Equation (5.18)] and [\[eq:continuousLyapunov\]](#eq:continuousLyapunov){reference-type="eqref" reference="eq:continuousLyapunov"} that $$\begin{aligned} \label{eq:PKIntofABK} P_K = \int_{0}^{\infty} e^{(A-BK)^Tt}(Q + K^T R K) e^{(A-BK)t} \mathrm{d}t. \end{aligned}$$ Taking the trace of [\[eq:PKIntofABK\]](#eq:PKIntofABK){reference-type="eqref" reference="eq:PKIntofABK"}, and again using the cyclic property of the trace in [@book_Petersen Equation (16)] and the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}, we have $$\begin{aligned} \begin{split} \mathrm{Tr}\left(P_K\right) = \mathrm{Tr}\left((Q + K^T R K)\int_{0}^{\infty} e^{(A-BK)t}e^{(A-BK)^Tt} \mathrm{d}t \right)\ge \lambda_{\mathrm{min}}\left(Q\right)\mathrm{Tr}\left(Y_K\right) . \end{split} \end{aligned}$$ Hence, the right inequality in [\[eq:YKBound\]](#eq:YKBound){reference-type="eqref" reference="eq:YKBound"} follows readily. ◻ **Lemma 4**. *For any $K \in \mathcal{G}$, $\mathrm{Tr}\left(P_K - P^*\right) \geq \alpha_4(\lVert K-K^*\rVert_F)$, where $\alpha_4(\cdot)$ is a $\mathcal{K}_\infty$-function defined as $$\begin{aligned} \alpha_4(r) = \frac{\lambda_{\mathrm{min}}\left(R\right)r^2}{2\lVert A-BK^*\rVert_F + 2 \lVert B\rVert r}, \quad \forall r \ge 0. \end{aligned}$$* *Proof.* Taking the trace of [\[eq:PKdiff1\]](#eq:PKdiff1){reference-type="eqref" reference="eq:PKdiff1"}, and using the cyclic property of the trace in [@book_Petersen Equation (16)], we have $$\begin{aligned} \mathrm{Tr}\left(P_K - P^*\right) = \mathrm{Tr}\left((K-K^*)^T R (K - K^*) Y_K\right) \ge \lambda_{\mathrm{min}}\left(Y_K\right) \lambda_{\mathrm{min}}\left(R\right) \lVert K-K^*\rVert_F^2.\end{aligned}$$ Considering the lower bound of $\lambda_{\mathrm{min}}\left(Y_K\right)$ in Lemma [Lemma 2](#lm:eigminYKLowerBound){reference-type="ref" reference="lm:eigminYKLowerBound"}, we can obtain $$\begin{aligned} \mathrm{Tr}\left(P_K - P^*\right) \ge \frac{\lambda_{\mathrm{min}}\left(R\right)\lVert K-K^*\rVert_F^2}{2\lVert A-BK^*\rVert_F + 2 \lVert B\rVert \lVert K-K^*\rVert_F} = \alpha_4(\lVert K-K^*\rVert_F).\end{aligned}$$ ◻ The following lemma shows that $\mathcal{J}_2(K)$ is coercive. This result is known, but we provide a proof in order to make this paper self-contained. **Lemma 5**. *The objective function $\mathcal{J}_2(K)$ is coercive, i.e. for any sequence $\{K_k\}_{k=0}^\infty$, $K_k \to \partial \mathcal{G}$ or $\lVert K_k\rVert_F \to \infty$, it holds $\mathcal{J}_2(K_k) \to \infty$.* *Proof.* Recall that $\mathcal{J}_2(K) = \mathrm{Tr}\left(P_K\right)$. It follows from Lemma [Lemma 4](#lm:PKPoptLowerBound){reference-type="ref" reference="lm:PKPoptLowerBound"} that $\mathcal{J}_2(K_k) \to \infty$ when $\lVert K_k\rVert_F \to \infty$. Next, suppose that $K_k \to \partial \mathcal{G}$ and the corresponding sequence of cost matrices are $\{ P_k\}_{k=1}^\infty = \{ P_{K_k}\}_{k=1}^\infty$. We will prove that $\lim_{k \to \infty}\mathrm{Tr}\left(P_k\right) = \infty$ by contradiction. Assume that there exists a bounded subsequence, denoted as $\{ \mathrm{Tr}\left(P_{k}\right)\}_{k=1}^\infty$ by slightly abusing the notations. By Bolzano--Weierstrass theorem, there exists a convergent subsequence $\{ P_{k_j}\}_{j=1}^\infty$, i.e. $\lim_{j \to \infty}P_{k_j} = \bar{P}$. Since $P_K$ is continuous with respect to $K$, the limit $\bar{P}$ satisfies $$\begin{aligned} \label{eq:LapunovBoundLim} (A-B\bar{K})^T\bar{P} + \bar{P}(A-B\bar{K}) + Q + \bar{K}^T R \bar{K} = 0, \end{aligned}$$ where $\bar{K} \in \partial \mathcal{G}$, which implies that there are some imaginary eigenvalues for $A-B\bar{K}$. Let $\nu$ denote the eigenvector of $A-B\bar{K}$ associated with the imaginary eigenvalue $\lambda = i\theta$. Pre and post multiplying [\[eq:LapunovBoundLim\]](#eq:LapunovBoundLim){reference-type="eqref" reference="eq:LapunovBoundLim"} with $\nu$, we have $$\begin{aligned} 2i\theta \nu^T\bar{P}\nu + \nu^T(Q + \bar{K}^T R \bar{K})\nu = 0. \end{aligned}$$ This implies that $Q\nu=0$, $\bar{K} \nu = 0$ and $A\nu = \lambda \nu$, which contradicts with the observability of $(A,Q)$. Therefore, $\lim_{k \to \infty}\mathrm{Tr}\left(P_k\right) = \infty$ is obtained. ◻ **Lemma 6**. *For any $K \in \mathcal{G}$, let $K' = R^{-1}B^TP_K$ and $M_K = (K - K')^T R (K - K')$. Then, $$\begin{aligned} \label{eq:KK'geqKdiff} \mathrm{Tr}\left(M_K\right) \geq a \lVert K - K^*\rVert_F^2 + a' \mathrm{Tr}\left(P_K-P^*\right), \end{aligned}$$ where $a$ and $a'$ are constants defined as $$\begin{aligned} \label{eq:aa'def} a = \frac{\lambda_{\mathrm{min}}\left(R\right)\lambda_{\mathrm{min}}\left(Y^*\right)}{2\lambda_{\mathrm{min}}\left(Y^*\right)+2\lambda_{\mathrm{max}}\left(Y^*\right)}, \quad a' = \frac{1}{\lambda_{\mathrm{min}}\left(Y^*\right) + \lambda_{\mathrm{max}}\left(Y^*\right)}. \end{aligned}$$* *Proof.* We can rewrite [\[eq:continuousLyapunov\]](#eq:continuousLyapunov){reference-type="eqref" reference="eq:continuousLyapunov"} as $$\begin{aligned} \begin{split} &(A-BK^*)^TP_K + P_K(A-BK^*) + Q + K^TRK + (K^* - K)^TB^TP_K + P_KB(K^* - K) = 0. \end{split} \end{aligned}$$ Considering $K' = R^{-1}B^T P_K$ and completing the squares, we have $$\begin{aligned} \label{eq:AoptPK} \begin{split} &(A-BK^*)^TP_K + P_K(A-BK^*) + Q + (K^*)^T R K^* + (K - K')^T R (K - K') \\ & - (K' - K^*)^T R (K' - K^*) = 0. \end{split} \end{aligned}$$ Subtracting [\[eq:AoptPopt\]](#eq:AoptPopt){reference-type="eqref" reference="eq:AoptPopt"} from [\[eq:AoptPK\]](#eq:AoptPK){reference-type="eqref" reference="eq:AoptPK"} yields $$\begin{aligned} \label{eq:AoptPKPoptDiff} \begin{split} &(A-BK^*)^T(P_K - P^*) + (P_K - P^*)(A-BK^*) + (K - K')^T R (K - K') \\ & - (K' - K^*)^T R (K' - K^*) = 0. \end{split} \end{aligned}$$ Since $A-BK$ is Hurwitz, according to [@book_chen Equation (5.18)], we have $$\begin{aligned} \label{eq:PKPoptYopt} \begin{split} P_K - P^* &= \int_{0}^{\infty} e^{(A-BK^*)^{T}t} [ (K - K')^T R (K - K') - (K' - K^*)^T R (K' - K^*) ] e^{(A-BK^*)t} \mathrm{d}t. \end{split} \end{aligned}$$ Taking the trace of [\[eq:PKPoptYopt\]](#eq:PKPoptYopt){reference-type="eqref" reference="eq:PKPoptYopt"}, and using the cyclic property of the trace in [@book_Petersen Equation (16)] yield $$\begin{aligned} \label{eq:tracePKPopt} \mathrm{Tr}\left(P_K - P^*\right) % \Tr{ \int_{0}^{\infty} e^{(A-BK^*)t} e^{(A-BK^*)^{T}t} \de t [ (K - K')^T R (K - K') - (K' - K^*)^T R (K' - K^*) ]} \nonumber \\ = \mathrm{Tr}\left(Y^* M_K\right) - \mathrm{Tr}\left(Y^* (K' - K^*)^TR(K'-K^*)\right) \end{aligned}$$ By the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}, we have $$\begin{aligned} \label{eq:MKupperbound1} \begin{split} \lambda_{\mathrm{max}}\left(Y^*\right)\mathrm{Tr}\left(M_K\right) \geq \lambda_{\mathrm{min}}\left(Y^*\right)\langle(K'-K^*), R(K'-K^*) \rangle + \mathrm{Tr}\left(P_K - P^*\right). \end{split} \end{aligned}$$ Using Lemma [Lemma 11](#lm:CSInequality){reference-type="ref" reference="lm:CSInequality"} and Young's inequality, we can obtain $$\begin{aligned} \label{eq:MKupperbound2} \begin{split} \langle K-K^*, R(K-K^*) \rangle &= \langle K-K', R(K-K') \rangle + \langle K'-K^*, R(K'-K^*) \rangle + 2\langle K-K', R(K'-K^*) \rangle \\ & \le 2\langle K-K', R(K-K') \rangle + 2\langle K'-K^*, R(K'-K^*) \rangle \end{split} \end{aligned}$$ Noticing that $\mathrm{Tr}\left(M_K\right) = \langle K-K', R(K-K') \rangle$. Plugging [\[eq:MKupperbound1\]](#eq:MKupperbound1){reference-type="eqref" reference="eq:MKupperbound1"} into [\[eq:MKupperbound2\]](#eq:MKupperbound2){reference-type="eqref" reference="eq:MKupperbound2"} and using the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"} yield $$\begin{aligned} \label{eq:MKupperbound3} \lambda_{\mathrm{min}}\left(R\right) \lVert K-K^*\rVert_F^2 \le \left(2 + 2\frac{\lambda_{\mathrm{max}}\left(Y^*\right)}{\lambda_{\mathrm{min}}\left(Y^*\right)}\right)\mathrm{Tr}\left(M_K\right) - \frac{2}{\lambda_{\mathrm{min}}\left(Y^*\right)}\mathrm{Tr}\left(P_K - P^*\right). \end{aligned}$$ Hence, [\[eq:KK\'geqKdiff\]](#eq:KK'geqKdiff){reference-type="eqref" reference="eq:KK'geqKdiff"} follows from [\[eq:MKupperbound3\]](#eq:MKupperbound3){reference-type="eqref" reference="eq:MKupperbound3"}. ◻ ## Perturbed Standard Gradient Flow At any position $K \in \mathcal{G}$, the steepest-descent direction of $\mathcal{J}_2(K)$ is the solution to the problem [@book_Nocedal] $$\begin{aligned} \label{eq:steepest-descent} \min_{E} \langle E, \nabla \mathcal{J}_2(K) \rangle, \quad \text{subject to } \langle E, E \rangle = 1.\end{aligned}$$ Therefore, by the method of Lagrange multipliers, the steepest-descent direction of $\mathcal{J}_2(K)$ is $$-\nabla \mathcal{J}_2(K)/\lVert\nabla \mathcal{J}_2(K)\rVert_F \,.$$ Consequently, along the the steepest-descent direction and considering the expression of $\nabla \mathcal{J}_2(K)$ in [\[eq:JcGradient\]](#eq:JcGradient){reference-type="eqref" reference="eq:JcGradient"}, the standard gradient flow is [@bu2020policy; @Mohammadi2022] $$\begin{aligned} \label{eq:gradientflow} \frac{\mathrm{d}K(s)}{\mathrm{d}{s}} = -\eta \nabla \mathcal{J}_2(K(s)) = -2\eta(RK(s) - B^TP(s))Y(s), \end{aligned}$$ where $\eta >0$ is a constant step size, $P(s) := P_{K(s)}$, and $Y(s) := Y_{K(s)}$. It is seen from [\[eq:JcGradient\]](#eq:JcGradient){reference-type="eqref" reference="eq:JcGradient"} that the calculation of the gradient $\nabla \mathcal{J}_2(K(s))$ relies on the system matrices, which are unknown in the setting of model-free RL. Many data-driven methods, e.g. approximate dynamic programming [@tutorial_Jiang] and finite-difference algorithms [@fazel2018global], are proposed to numerically approximate the gradient. Therefore, in practice, $\widehat{\nabla \mathcal{J}_2}(K(s))$, instead of ${\nabla \mathcal{J}_2}(K(s))$, is utilized to optimize the control gain, and the perturbation $\Delta K(s) = \eta[{\nabla \mathcal{J}_2}(K(s)) -\widehat{\nabla \mathcal{J}_2}(K(s))]$ is unavoidable for the gradient flow. Hence, the perturbed gradient flow of the LQR is $$\begin{aligned} \label{eq:gradientflow_con} \frac{\mathrm{d}K(s)}{\mathrm{d}{s}} = -2\eta(RK(s) - B^TP(s))Y(s) + \Delta K(s).\end{aligned}$$ The following lemma shows that the gradient of $\mathcal{J}_2(K)$ is lower bounded by a $\mathcal{K}$-function of the deviation from the optimal value (gradient dominance condition), which is an important property of a proper objective function in Definition [Definition 5](#def:ProperLossFun){reference-type="ref" reference="def:ProperLossFun"}. **Lemma 7**. *There exists a $\mathcal{K}$-function $\xi_1(\cdot)$, such that for any $K \in \mathcal{G}$, $$\begin{aligned} \label{eq:gradientLowerBound} \lVert\nabla \mathcal{J}_2(K)\rVert_F \geq \xi_1(\mathcal{J}_2(K) - \mathcal{J}_2(K^*)). \end{aligned}$$* *Proof.* Considering the expression of $\nabla \mathcal{J}_2(K)$ in [\[eq:JcGradient\]](#eq:JcGradient){reference-type="eqref" reference="eq:JcGradient"}, the expression of $K'$ in Lemma [Lemma 6](#lm:MKlowerbound){reference-type="ref" reference="lm:MKlowerbound"}, and the cyclic property of the trace [@book_Petersen Equation 16], we have $$\begin{aligned} \mathrm{Tr}\left(\nabla \mathcal{J}_2^T(K)\nabla \mathcal{J}_2(K)\right) = 4\mathrm{Tr}\left(Y_K^2(K-K')^TR^2(K-K')\right). \end{aligned}$$ By the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}, it holds $$\begin{aligned} \begin{split} &\mathrm{Tr}\left(\nabla \mathcal{J}_2^T(K) \nabla \mathcal{J}_2(K)\right) \ge 4 \lambda_{\mathrm{min}}\left(Y_K\right)^2\mathrm{Tr}\left((K-K')^T\sqrt{R}R\sqrt{R}(K-K')\right) \\ &= 4 \lambda_{\mathrm{min}}\left(Y_K\right)^2\mathrm{Tr}\left(\sqrt{R}(K-K')(K-K')^T\sqrt{R}R\right) \\ &\ge 4 \lambda_{\mathrm{min}}\left(Y_K\right)^2 \lambda_{\mathrm{min}}\left(R\right)\mathrm{Tr}\left(\sqrt{R}(K-K')(K-K')^T\sqrt{R}\right) = 4 \lambda_{\mathrm{min}}\left(Y_K\right)^2 \lambda_{\mathrm{min}}\left(R\right)\mathrm{Tr}\left(M_K\right). \end{split} \end{aligned}$$ By Lemmas [Lemma 2](#lm:eigminYKLowerBound){reference-type="ref" reference="lm:eigminYKLowerBound"} and [Lemma 6](#lm:MKlowerbound){reference-type="ref" reference="lm:MKlowerbound"}, it follows that $$\begin{aligned} \label{eq:gradientDominant1} \begin{split} \mathrm{Tr}\left(\nabla \mathcal{J}_2^T(K) \nabla \mathcal{J}_2(K)\right) &\geq \lambda_{\mathrm{min}}\left(R\right) \frac{a\lVert K-K^*\rVert_F^2 + a'\mathrm{Tr}\left(P_K - P^*\right)} {(\lVert A-BK^*\rVert_F + \lVert B\rVert \lVert K-K^*\rVert_F)^2}. \end{split} \end{aligned}$$ To simplify notations, let $$\begin{aligned} \label{eq:defa1234} a_1 = \lambda_{\mathrm{min}}\left(R\right)a, \quad a_2 = \lambda_{\mathrm{min}}\left(R\right)a', \quad a_3 = \lVert A-BK^*\rVert_F, \quad a_4 = \lVert B\rVert, \end{aligned}$$ and $$\begin{aligned} \sigma(r, p) = \frac{a_1 r^2 + a_2 p} {(a_3 + a_4 r)^2}. \end{aligned}$$ It is clear that $\sigma(r,p) \ge \sigma(r,0)$. Taking the derivative of $\sigma(r, p)$ with respect to $r$, we have $$\begin{aligned} \frac{\mathrm{d}\sigma(r,p)}{\mathrm{d}r} = \frac{2a_1a_3r - 2a_2a_4p}{(a_3 + a_4 r)^3}. \end{aligned}$$ When $p \ge \frac{a_1a_3 }{a_2 a_4} r$, $\sigma(r,p)$ is strictly decreasing in $r$. When $p \le \frac{a_1a_3 }{a_2 a_4} r$, $\sigma(r,p)$ is strictly increasing in $r$. Hence, for each $p$, $r=\frac{a_2a_4}{a_1a_3} p$ is the minimum point of $\sigma(r,p)$, and $$\begin{aligned} \label{eq:gradientDominant2} \sigma(r,p) \geq \sigma(\frac{a_2a_4}{a_1a_3} p,p) \geq \sigma(\frac{a_2a_4}{a_1a_3} p,0)=: \xi^2_{1}(p), \end{aligned}$$ where $$\begin{aligned} \xi_{1}(p) = \frac{a_5 p}{a_3 + a_6p}, \end{aligned}$$ and $$\begin{aligned} \label{eq:a56def} a_5 = \frac{a_2a_4}{\sqrt{a_1}a_3}, \quad a_6 = \frac{a_2a_4^2}{a_1a_3}. \end{aligned}$$ Since $\frac{\mathrm{d}\xi_1(p)}{\mathrm{d}p} = \frac{a_3 a_5}{(a_3 + a_6p)^2} > 0$, $\xi_1(\cdot)$ is a $\mathcal{K}$-function with the range $[0,\frac{\sqrt{a_1}}{a_4})$. Plugging [\[eq:gradientDominant2\]](#eq:gradientDominant2){reference-type="eqref" reference="eq:gradientDominant2"} into [\[eq:gradientDominant1\]](#eq:gradientDominant1){reference-type="eqref" reference="eq:gradientDominant1"} yields $$\begin{aligned} \lVert\nabla \mathcal{J}_2(K)\rVert_F = \mathrm{Tr}\left(\nabla \mathcal{J}_2^T(K) \nabla \mathcal{J}_2(K)\right)^{\frac{1}{2}} \ge \xi_1\left(\mathrm{Tr}\left(P_K - P^*\right)\right). \end{aligned}$$ ◻ **Remark 2**. *As remarked in the introduction, the gradient dominance condition in Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"}, the CJS-PL condition, can be considered as a generalization of the well-known Polyak-Łojasiewicz (PL) condition which only holds on a compact set of stabilizing control gains [@Mohammadi2022; @bu2020policy]. The gradient dominance condition in Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"} removes the restriction to compact sets.* We next revisit the one-dimensional system mentioned in the introduction, in order to illustrate Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"}. Suppose that $m=n=1$ and $A=B=Q=R=1$. In this case, the admissible set is $\mathcal{G}=\{K|K > 1\}$, and one obtains $$\begin{aligned} \label{eq:oneDExample} \begin{split} &P^* = 1+\sqrt{2}, \quad K^* = 1+\sqrt{2}, \quad Y_K = \frac{1}{2(K-1)}, \quad Y^* = \frac{\sqrt{2}}{4} , \\ & \mathcal{J}_2(K) = P_K = \frac{1 + K^2}{2(K-1)}, \quad \mathcal{J}_2(K) - \mathcal{J}_2(K^*) = \frac{(K-K^*)^2}{2(K-1)} \,. \end{split}\end{aligned}$$ The constants in [\[eq:aa\'def\]](#eq:aa'def){reference-type="eqref" reference="eq:aa'def"}, [\[eq:defa1234\]](#eq:defa1234){reference-type="eqref" reference="eq:defa1234"} and [\[eq:a56def\]](#eq:a56def){reference-type="eqref" reference="eq:a56def"} can be computed as $$\begin{aligned} a = \frac{1}{4}, \quad a' = \sqrt{2}, \quad a_1 = \frac{1}{4}, \quad a_2 = \sqrt{2}, \quad a_3 = \sqrt{2}, \quad a_4 = 1, \quad a_5 = 2, \quad a_6 = 4.\end{aligned}$$ Consequently, we have $$\begin{aligned} \xi_1(p) = \frac{2p}{\sqrt{2} + 4p}, \quad \xi_1\left(\mathcal{J}_2(K) - \mathcal{J}_2(K^*)\right) = \frac{(K-K^*)^2}{\sqrt{2}(K-1) + 2(K-K^*)^2}.\end{aligned}$$ The gradient of $\mathcal{J}_2(K)$ is $$\begin{aligned} \nabla \mathcal{J}_2(K) = \frac{K^2 - 2K - 1}{2(K-1)^2} = \frac{(K-K^*)^2 + 2\sqrt{2}(K-K^*)}{2(K-1)^2}.\end{aligned}$$ When $K \ge K^*$, $\nabla \mathcal{J}_2(K) \ge 0$, and $$\begin{aligned} \frac{\nabla \mathcal{J}_2(K)}{\xi_1\left(\mathcal{J}_2(K) - \mathcal{J}_2(K^*)\right)} = \frac{2(K-K^*)^3 + 5\sqrt{2}(K-K^*)^2 + 6(K-K^*) + 4\sqrt{2}}{2(K-K^*)^3 + 4\sqrt{2}(K-K^*)^2 + 4(K-K^*)} \ge 1.\end{aligned}$$ When $K^* \ge K > 1$, $\nabla \mathcal{J}_2(K) \le 0$, and $$\begin{aligned} \begin{split} &{-\nabla \mathcal{J}_2(K)} - {\xi_1\left(\mathcal{J}_2(K) - \mathcal{J}_2(K^*)\right)} \\ &= \frac{-4(K-1)^4 + 7\sqrt{2}(K-1)^3 -4(K-1)^2 -6\sqrt{2}(K-1) + 8}{2\sqrt{2}(K-1)^3 + 4 (K-1)^2(K-K^*)^2} \ge 0, \quad \sqrt{2} \ge (K-1) > 0 \end{split}\end{aligned}$$ Therefore, $\lVert\nabla \mathcal{J}_2(K)\rVert \ge \xi_1\left(\mathcal{J}_2(K) - \mathcal{J}_2(K^*)\right)$, which is consistent with Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"}. Based on Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"}, we are ready to state the main result on the small-disturbance ISS property of the perturbed standard gradient flow [\[eq:gradientflow_con\]](#eq:gradientflow_con){reference-type="eqref" reference="eq:gradientflow_con"}. **Theorem 3**. *System [\[eq:gradientflow_con\]](#eq:gradientflow_con){reference-type="eqref" reference="eq:gradientflow_con"} is small-disturbance ISS with respect to $\Delta K$.* *Proof.* Define $\mathcal{V}_3(K) := \mathcal{J}_2(K) - \mathcal{J}_2(K^*)$. It follows from [\[eq:costJc_closedform\]](#eq:costJc_closedform){reference-type="eqref" reference="eq:costJc_closedform"} that $\mathcal{V}_3(K) = \mathrm{Tr}\left(P_K - P^*\right)$. Clearly, $\mathcal{V}_3(K)$ is continuous in $K$. Since $K^*$ is the unique minimum of $\mathcal{J}_2(K)$, $\mathcal{V}_3(K)$ is a positive definite function with respect to $K^*$. The coercivity of $\mathcal{V}_3(K)$ can be obtained by Lemma [Lemma 5](#lm:coercive){reference-type="ref" reference="lm:coercive"}. Therefore, by Definition [\[def:sizeFunc\]](#def:sizeFunc){reference-type="ref" reference="def:sizeFunc"}, $\mathcal{V}_3(K)$ is a size function. In addition, since $P_K$ and $Y_K$ are continuously differentiable with respect to $K \in \mathcal{G}$, $\nabla \mathcal{J}_2(K) = 2(RK-B^TP_K)Y_K$ is locally Lipchitz continuous. Hence, by Lemma [Lemma 7](#lm:gradientClassK){reference-type="ref" reference="lm:gradientClassK"} and Definition [Definition 5](#def:ProperLossFun){reference-type="ref" reference="def:ProperLossFun"}, $\mathcal{J}_2(\cdot)$ is a proper objective function. According to Theorem [Theorem 2](#eq:gradFlowISS){reference-type="ref" reference="eq:gradFlowISS"}, the proof of Theorem [Theorem 3](#thm:smallISSGradientFlow){reference-type="ref" reference="thm:smallISSGradientFlow"} is completed. ◻ As a direct consequence of Theorem [Theorem 3](#thm:smallISSGradientFlow){reference-type="ref" reference="thm:smallISSGradientFlow"}, an estimate of $\lVert K(s) - K^*\rVert_F$ is provided in the following corollary. **Corollary 1**. *There exist a constant $d_1 > 0$, a $\mathcal{KL}$-function $\beta_2(\cdot,\cdot)$, and a $\mathcal{K}_{[0,d_1)}$-function $\gamma_4(\cdot)$, such that for all perturbations $\Delta K$ bounded by $d_1$ (i.e. $\lVert\Delta K\rVert_\infty < d_1$), and all initial conditions $K(0) \in \mathcal{G}$, $K(s)$ satisfies $$\begin{aligned} \label{eq:KsmallISS} \lVert K(s) - K^*\rVert_F \le \beta_2(\mathcal{V}_2(K(0)), s) + \gamma_4(\lVert\Delta K\rVert_\infty), \quad \forall s\ge 0. \end{aligned}$$* *Proof.* Since system [\[eq:gradientflow_con\]](#eq:gradientflow_con){reference-type="eqref" reference="eq:gradientflow_con"} is small-disturbance ISS, when $\lVert\Delta K\rVert_\infty < d_1$, it holds $$\begin{aligned} \mathcal{V}_3(K(s)) \le \beta_2'(\mathcal{V}_3(K(0)), s) + \gamma_4'(\lVert\Delta K\rVert_\infty), \quad \forall s \ge 0, \end{aligned}$$ where $\beta_2'(\cdot, \cdot)$ is a $\mathcal{KL}$-function and $\gamma_4'(\cdot)$ is a $\mathcal{K}_{[0,d_1)}$-function. According to Lemma [Lemma 4](#lm:PKPoptLowerBound){reference-type="ref" reference="lm:PKPoptLowerBound"} and recalling that $\mathcal{V}_3(K(s)) = \mathrm{Tr}\left(P(s) - P^*\right)$, we have $$\begin{aligned} \lVert K(s) - K^*\rVert_F \le \alpha_4^{-1}\left(\beta_2'(\mathcal{V}_3(K(0)), s) + \gamma_4'(\lVert\Delta K\rVert_\infty)\right). \end{aligned}$$ Using [@Jiang1994 Equation (6)], we can obtain [\[eq:KsmallISS\]](#eq:KsmallISS){reference-type="eqref" reference="eq:KsmallISS"}. ◻ ## Perturbed Natural Gradient Flow It follows from [\[eq:TrPKPopt\]](#eq:TrPKPopt){reference-type="eqref" reference="eq:TrPKPopt"} that $$\begin{aligned} \mathcal{J}_2(K) - \mathcal{J}_2(K^*) = \langle K-K^*, R(K-K^*) \rangle_{Y_K}.\end{aligned}$$ Hence, the objective objective function can be viewed as a quadratic function over the Riemannian manifold $(\mathcal{G}, \langle\cdot, \cdot \rangle_{Y_K})$. As seen in the expression of $\nabla \mathcal{J}_2(K)$, the magnitude of the gradient dependents on $Y_K$, and $Y_K$ may tend to infinity when $K \to \partial \mathcal{G}$, and tend to zero when $\lVert K\rVert_F \to \infty$ (see the illustrative one-dimensional system in [\[eq:oneDExample\]](#eq:oneDExample){reference-type="eqref" reference="eq:oneDExample"}). The non-isotropic property of the magnitude of the gradient may slow down the convergence of the gradient flow. To handle the non-isotropic property, in [@Amari1998Why; @Amari1998], the natural gradient was proposed as a way to modify the standard gradient search direction according to the Riemannian structure of the parameter space. Over the Riemannian manifold $(\mathcal{G}, \langle\cdot, \cdot \rangle_{Y_K})$, the steepest-descent direction can be obtained by solving $$\begin{aligned} \label{eq:Naturalsteepest-descent} \min_{E} \langle E, \nabla \mathcal{J}_2(K) \rangle, \quad \text{subject to } \langle E, E \rangle_{Y_K} = 1.\end{aligned}$$ By the method of Lagrange multipliers, the solution of [\[eq:Naturalsteepest-descent\]](#eq:Naturalsteepest-descent){reference-type="eqref" reference="eq:Naturalsteepest-descent"} is $$\begin{aligned} E = -\nabla \mathcal{J}_2(K)Y_K^{-1}/ \langle\nabla \mathcal{J}_2(K)Y_K^{-1}, \nabla \mathcal{J}_2(K)Y_K^{-1} \rangle_{Y_K}^{1/2}.\end{aligned}$$ The natural gradient over the Riemannian manifold $(\mathcal{G}, \langle\cdot, \cdot \rangle_{Y_K})$ is $$\begin{aligned} \label{eq:naturalflow} \mathrm{grad}\left(\mathcal{J}_2(K)\right) = \nabla \mathcal{J}_2(K) Y_K^{-1} = 2(RK - B^TP_K).\end{aligned}$$ Considering the perturbation, the natural gradient flow is $$\begin{aligned} \label{eq:GradientFlow_NaturalCont} \frac{\mathrm{d}K(s)}{\mathrm{d}s} = - 2\eta (RK(s) - B^TP(s)) + \Delta K(s).\end{aligned}$$ The following lemma is introduced to pave the foundation for the proof of small-disturbance ISS property of system [\[eq:GradientFlow_NaturalCont\]](#eq:GradientFlow_NaturalCont){reference-type="eqref" reference="eq:GradientFlow_NaturalCont"}. **Lemma 8**. *For any $K\in\mathcal{G}$, we have $$\begin{aligned} \label{eq:PdiffNatural} 2\langle K-K^*, R(K-K') \rangle_{Y^*} = \mathrm{Tr}\left(P_K - P^*\right) + \langle K-K^*, R(K-K^*) \rangle_{Y^*}. \end{aligned}$$* *Proof.* By completing the squares, we have $$\begin{aligned} \begin{split} &(K - K')^T R (K - K') - (K' - K^*)^T R (K' - K^*) \\ % &=(K - K')^T R (K - K') + (K - K')^T R (K' - K^*) \\ % &+ (K - K')^T R (K - K') + (K' - K^*)^T R (K - K') \\ % & - (K - K')^T R (K' - K^*) - (K - K')^T R (K - K') \\ % &- (K' - K^*)^T R (K - K')- (K' - K^*)^T R (K' - K^*) \\ &= (K - K')^T R (K - K^*) + (K - K^*)^T R (K - K') - (K - K^*)^T R (K - K^*). \end{split} \end{aligned}$$ Therefore, we can rewrite [\[eq:AoptPKPoptDiff\]](#eq:AoptPKPoptDiff){reference-type="eqref" reference="eq:AoptPKPoptDiff"} as $$\begin{aligned} \begin{split} &(A-BK^*)^T(P_K - P^*) + (P_K - P^*)(A-BK^*) + (K - K')^T R (K - K^*) \\ &+(K - K^*)^T R (K - K') - (K - K^*)^T R (K - K^*) = 0. \end{split} \end{aligned}$$ Since $A-BK^*$ is Hurwitz, by [@book_chen Equation (5.18)] we have $$\begin{aligned} \label{eq:PKdiffRewrite} \begin{split} P_K - P^* &= \int_{0}^{\infty} e^{(A-BK^*)^{T}t } [(K - K')^T R (K - K^*) \\ &+(K - K^*)^T R (K - K') - (K - K^*)^T R (K - K^*)]e^{(A-BK^*)t } \mathrm{d}t. \end{split} \end{aligned}$$ Taking the trace of [\[eq:PKdiffRewrite\]](#eq:PKdiffRewrite){reference-type="eqref" reference="eq:PKdiffRewrite"} and using the cyclic property of trace in [@book_Petersen Equation (16)], we have $$\begin{aligned} \begin{split} \mathrm{Tr}\left(P_K - P^*\right) &= 2\langle K-K^*, R(K-K') \rangle_{Y^*} - \langle K-K^*, R(K-K^*) \rangle_{Y^*}. \end{split} \end{aligned}$$ Thus, the proof is completed. ◻ Next, we will prove the small-disturbance ISS property of the perturbed natural gradient flow. **Theorem 4**. *Given $Q \succ 0$, system [\[eq:GradientFlow_NaturalCont\]](#eq:GradientFlow_NaturalCont){reference-type="eqref" reference="eq:GradientFlow_NaturalCont"} is small-disturbance ISS with respect to $\Delta K$.* *Proof.* Let $\mathcal{V}_4(K) = \frac{1}{2} \langle K - K^*, K - K^* \rangle_{Y^*}$. Clearly, by Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}, $\mathcal{V}_4(K)$ is bounded by $$\begin{aligned} \label{eq:VKbounds} \frac{1}{2}\lambda_{\mathrm{min}}\left(Y^*\right) \lVert K - K^*\rVert_F^2 \leq \mathcal{V}_4(K) \leq \frac{1}{2}\lVert Y^*\rVert \lVert K - K^*\rVert_F^2.\end{aligned}$$ Taking the derivative of $\mathcal{V}_4(K(s))$ and plugging [\[eq:GradientFlow_NaturalCont\]](#eq:GradientFlow_NaturalCont){reference-type="eqref" reference="eq:GradientFlow_NaturalCont"} into it yield $$\begin{aligned} \label{eq:Vnnature1} \begin{split} \frac{\mathrm{d}\mathcal{V}_4(K(s))}{\mathrm{d}s} &= -2\eta \langle K(s) - K^*, R(K(s) - K'(s)) \rangle_{Y^*} + \langle K(s) - K^*, \Delta K(s) \rangle_{Y^*}. \end{split}\end{aligned}$$ Recall that $K'(s) = R^{-1}B^T P(s)$. According to Lemma [Lemma 8](#lm:PdiffNatural){reference-type="ref" reference="lm:PdiffNatural"}, we obtain $$\begin{aligned} \label{eq:Vnnature2} \begin{split} &\frac{\mathrm{d}\mathcal{V}_4(K(s))}{\mathrm{d}s} = - \eta\mathrm{Tr}\left(P(s) - P^*\right) - \eta\langle K(s) - K^*, R(K(s) - K^*) \rangle_{Y^*} + \langle K(s) - K^*, \Delta K(s) \rangle_{Y^*}\\ &\le - \eta\mathrm{Tr}\left(P(s) - P^*\right) - \eta\langle K(s) - K^*, R(K(s) - K^*) \rangle_{Y^*} \\ &\quad + \langle K(s) - K^*, K(s) - K^* \rangle_{Y^*}^{\frac{1}{2}}\langle\Delta K(s), \Delta K(s) \rangle_{Y^*}^{\frac{1}{2}}\\ &\le - \eta\mathrm{Tr}\left(P(s) - P^*\right) - \frac{\eta\lambda_{\mathrm{min}}\left(R\right)}{2}\langle K(s) - K^*, K(s) - K^* \rangle_{Y^*} + \frac{\lVert Y^*\rVert}{2\eta\lambda_{\mathrm{min}}\left(R\right)} \lVert\Delta K(s)\rVert_F^2. \end{split}\end{aligned}$$ where the second line follows from Lemma [Lemma 11](#lm:CSInequality){reference-type="ref" reference="lm:CSInequality"}, and the last inequality is according to Young's inequality and the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}. Differentiating $\mathcal{V}_3(K(s)) = \mathcal{J}_2(K(s)) - \mathcal{J}_2(K^*)$ along the trajectories of [\[eq:GradientFlow_NaturalCont\]](#eq:GradientFlow_NaturalCont){reference-type="eqref" reference="eq:GradientFlow_NaturalCont"} results in $$\begin{aligned} \label{eq:VvNaturalFlow1} \begin{split} \frac{\mathrm{d}\mathcal{V}_3(K(s))}{\mathrm{d}s} &= -2 \eta \langle\nabla \mathcal{J}_2(K(s)), RK(s) - B^TP(s) \rangle + \langle\nabla \mathcal{J}_2(K(s)), \Delta K(s) \rangle \\ &= -4 \eta \langle RK - B^TP(s), RK(s) - B^TP(s) \rangle_{Y(s)} + 2\langle RK(s) - B^TP(s), \Delta K(s) \rangle_{Y(s)} \\ &\le -3 \eta \langle RK - B^TP(s), RK(s) - B^TP(s) \rangle_{Y(s)} + \frac{1}{\eta} \langle\Delta K(s), \Delta K(s) \rangle_{Y(s)}, \end{split}\end{aligned}$$ where the second line follows from [\[eq:JcGradient\]](#eq:JcGradient){reference-type="eqref" reference="eq:JcGradient"} and the last line is obtained by Lemma [Lemma 11](#lm:CSInequality){reference-type="ref" reference="lm:CSInequality"} and Young's inequality. Using Lemma [Lemma 3](#lm:YKBound){reference-type="ref" reference="lm:YKBound"} and the trace inequality in Lemma [Lemma 9](#lm:traceIneq){reference-type="ref" reference="lm:traceIneq"}, we have $$\begin{aligned} \label{eq:VvNaturalFlow2} \frac{\mathrm{d}\mathcal{V}_3(K(s))}{\mathrm{d}s} \le -3 \eta \langle RK - B^TP(s), RK(s) - B^TP(s) \rangle_{Y(s)} + \frac{\mathcal{V}_3(K(s)) + \mathrm{Tr}\left(P^*\right)}{\eta \lambda_{\mathrm{min}}\left(Q\right)} \lVert\Delta K(s)\rVert_F^2.\end{aligned}$$ Let $\mathcal{V}_5(K) = \mathcal{V}_3(K) + \mathcal{V}_4(K)$. Since $\mathcal{V}_3(\cdot)$ is a size function and $\mathcal{V}_4(\cdot)$ is positive definite respect to $K^*$, $\mathcal{V}_5(\cdot)$ is also a size function. It follows from [\[eq:Vnnature2\]](#eq:Vnnature2){reference-type="eqref" reference="eq:Vnnature2"} and [\[eq:VvNaturalFlow2\]](#eq:VvNaturalFlow2){reference-type="eqref" reference="eq:VvNaturalFlow2"} that $$\begin{aligned} \label{eq:Vcnaturalflow1} \begin{split} \frac{\mathrm{d}\mathcal{V}_5(K(s))}{\mathrm{d}s} &\le -\eta \mathcal{V}_3(K(s)) - {\eta \lambda_{\mathrm{min}}\left(R\right)} \mathcal{V}_4(K(s)) + \frac{\mathcal{V}_3(K(s)) + b_1}{\eta b_2} \lVert\Delta K(s)\rVert_F^2. \end{split} \end{aligned}$$ where $$\begin{aligned} b_1 = \frac{\lVert Y^*\rVert\lambda_{\mathrm{min}}\left(Q\right)}{2\lambda_{\mathrm{min}}\left(R\right)} + \mathrm{Tr}\left(P^*\right), \quad b_2 = \lambda_{\mathrm{min}}\left(Q\right), \quad b_3 = \eta^2\lambda_{\mathrm{min}}\left(R\right)\lambda_{\mathrm{min}}\left(Q\right).\end{aligned}$$ Without losing generality, assume that $\lambda_{\mathrm{min}}\left(R\right)\le1$. Then, it follows from [\[eq:Vcnaturalflow1\]](#eq:Vcnaturalflow1){reference-type="eqref" reference="eq:Vcnaturalflow1"} that $$\begin{aligned} \label{eq:Vcnaturalflow2} \begin{split} \frac{\mathrm{d}\mathcal{V}_5(K(s))}{\mathrm{d}s} &\le -\eta\lambda_{\mathrm{min}}\left(R\right) \mathcal{V}_5(K(s)) + \frac{\mathcal{V}_5(K(s)) + b_1}{\eta b_2} \lVert\Delta K(s)\rVert_F^2. \end{split} \end{aligned}$$ Thus, if $$\begin{aligned} \lVert\Delta K(s)\rVert_F \le \left(\frac{b_3 \mathcal{V}_5(K(s))}{2\mathcal{V}_5(K(s))+2b_1} \right)^{\frac{1}{2}} =: \xi_2(\mathcal{V}_5(K(s))), \end{aligned}$$ it is guaranteed that $$\begin{aligned} \frac{\mathrm{d}\mathcal{V}_5(K(s))}{\mathrm{d}s} &\le -\frac{\eta\lambda_{\mathrm{min}}\left(R\right)}{2} \mathcal{V}_5(K(s)).\end{aligned}$$ Since $\frac{\mathrm{d}\xi_2(r)}{\mathrm{d}r} = \frac{1}{2}\xi_2^{-1}(r) \frac{ 2b_1 b_3}{(2r+2b_1)^2} > 0, \quad \forall r>0$, $\xi_2(\cdot)$ is a $\mathcal{K}$-function, and its range is $[0, \sqrt{\frac{b_3}{2}})$. Consequently, $\mathcal{V}_5(\cdot)$ is a small-disturbance ISS-Lyapunov function. According to Theorem [Theorem 1](#thm:smallISSSufficient){reference-type="ref" reference="thm:smallISSSufficient"}, we conclude that system [\[eq:GradientFlow_NaturalCont\]](#eq:GradientFlow_NaturalCont){reference-type="eqref" reference="eq:GradientFlow_NaturalCont"} is small-disturbance ISS. ◻ ## Perturbed Newton Gradient Flow The Newton gradient descent method was first adopted in [@Kleinman1968] for solving the LQR problem, and it converges to the optimum at a quadratic convergence rate. The Newton direction is derived from the second-order Taylor series approximation of $\mathcal{J}_2(K+E)$, which, according to Lemma [Lemma 1](#lm:J2Expansion){reference-type="ref" reference="lm:J2Expansion"}, can be expressed as $$\begin{aligned} \label{eq:SecondOrderAppro} \mathcal{J}_2(K+E) = \mathcal{J}_2(K) + \langle E, 2(RK-B^TP_K) \rangle_{Y_K} + \langle E, RE \rangle_{Y_K} + O(\lVert E\rVert_F^2).\end{aligned}$$ By minimizing the second-order approximation of $\mathcal{J}_2(K)$ over $E$, the Newton direction is obtained as $-(K - R^{-1}B^T P_K)$. Considering the perturbation, the Newton gradient flow is $$\begin{aligned} \label{eq:GradientFlow_Newton} \frac{\mathrm{d}K(s)}{\mathrm{d}s} = -\eta(K(s) - R^{-1}B^TP(s)) + \Delta K(s).\end{aligned}$$ **Theorem 5**. *Given $Q \succ 0$, system [\[eq:GradientFlow_Newton\]](#eq:GradientFlow_Newton){reference-type="eqref" reference="eq:GradientFlow_Newton"} is small-disturbance ISS with respect to $\Delta K(s)$.* *Proof.* The proof follows from the proof of Theorem [Theorem 4](#thm:ISSnature){reference-type="ref" reference="thm:ISSnature"} by defining $\mathcal{V}_6(K) = \mathcal{V}_3(K) + \frac{1}{2} \langle K - K^*, R(K - K^*) \rangle_{Y^*}$. ◻ # Conclusion In this paper, we studied the small-disturbance ISS property of continuous-time gradient flows on an open subset of certain Euclidean space. In the framework of small-disturbance ISS, the transient behavior, the convergence speed, and the robustness to the perturbations of gradient flows can be well quantified. As a by-product, a Lyapunov characterization of small-disturbance ISS is given. Upon specification to the policy optimization of the LQR problem, three kinds of perturbed gradient flows, including standard gradient flow, natural gradient flow, and Newton gradient flow, were studied in greater details. In particular, they are all small-disturbance ISS. # Auxiliary Results **Lemma 9** (Trace Inequality [@Wang1986]). *Let $S=S^T \in \mathbb{R}^{n \times n}$ and $P \in \mathbb{P}^{n}$. Then, $$\begin{aligned} \label{eq:} \lambda_{\mathrm{min}}\left(S\right) \mathrm{Tr}\left(P\right) \le \mathrm{Tr}\left(SP\right) \le \lambda_{\mathrm{max}}\left(S\right) \mathrm{Tr}\left(P\right). \end{aligned}$$* **Lemma 10**. *Suppose $\omega_1(\cdot), \, \omega_2(\cdot): \mathbb{R}^n \to \mathbb{R}$ are continuous, positive definite with respect to $\chi^*$, and radically unbounded. Then, there exist $\mathcal{K}_\infty$-functions $\rho_1(\cdot)$ and $\rho_2(\cdot)$ such that $$\begin{aligned} \rho_1(\omega_2(\chi)) \le \omega_1(\chi) \le \rho_2(\omega_2(\chi)), \quad \forall \chi \in \mathbb{R}^n. \end{aligned}$$* *Proof.* The proof follows from [@Sontag2022 Proposition 2.6] by considering the open subset as $\mathbb{R}^n$ and the compact set as $\{\chi^*\}$. ◻ **Lemma 11** (Cauchy-Schwarz Inequality). *For any $K_1,K_2 \in \mathbb{R}^{m\times n}$, $R \in \mathbb{P}^{m}$, and $Y \in \mathbb{P}^n$, we have $$\begin{aligned} \label{eq:CSInequality} \langle K_1, RK_2 \rangle_Y \leq \sqrt{ \langle K_1, RK_1 \rangle_Y}\sqrt{ \langle K_2, RK_2 \rangle_Y}. \end{aligned}$$* *Proof.* Define $K_3$ as $$\begin{aligned} \label{eq:orthoDecom} K_3 = K_1 - \frac{\langle K_1, RK_2 \rangle_Y}{\langle K_2, RK_2 \rangle_Y}K_2. \end{aligned}$$ It is clear that $\langle K_3, RK_2 \rangle_Y=0$. Therefore, by plugging [\[eq:orthoDecom\]](#eq:orthoDecom){reference-type="eqref" reference="eq:orthoDecom"} into $\langle K_1, RK_1 \rangle_Y$, we can obtain $$\begin{aligned} \label{eq:CSInequality1} \langle K_1, RK_1 \rangle_Y = \langle K_3, RK_3 \rangle_Y + \frac{\langle K_1, RK_2 \rangle_Y^2}{\langle K_2, RK_2 \rangle_Y^2} \langle K_2, RK_2 \rangle_Y\ge \frac{\langle K_1, RK_2 \rangle_Y^2}{\langle K_2, RK_2 \rangle_Y}. \end{aligned}$$ Hence, [\[eq:CSInequality\]](#eq:CSInequality){reference-type="eqref" reference="eq:CSInequality"} readily follows from [\[eq:CSInequality1\]](#eq:CSInequality1){reference-type="eqref" reference="eq:CSInequality1"}. ◻ **Lemma 12**. *The map $h(v) = \frac{1}{1+\lVert v\rVert}v: \mathbb{R}^m \to \mathcal{W}:=\{w\in \mathbb{R}^m| \lVert w\rVert < 1\}$ is a homeomorphism.* *Proof.* For any $w \in \mathcal{W}$, let $g(w) = \frac{1}{1-\lVert w\rVert}w$. Clearly, for any $v \in \mathbb{R}^m$, $g(h(v)) = v$ and for any $w \in \mathcal{W}$, $h(g(w)) = w$. Hence, $g(\cdot)$ is the inverse function of $h(\cdot)$, i.e. $g(\cdot) = h^{-1}(\cdot)$. Since both $h(\cdot)$ and $g(\cdot)$ are continuous, $h(\cdot)$ is a homeomorphism. ◻ [^1]: Department of Electrical and Computer Engineering, New York University, Brooklyn, NY, USA [^2]: Department of Electrical and Computer Engineering and Department of BioEngineering, Northeastern University, Boston, MA, USA [^3]: This work has been supported in part by the NSF grants CNS-2227153 and ECCS-2210320 (L.C. and Z.P.J.) and AFOSR grant FA9550-21-1-0289 and ONR grant N00014-21-1-2431 (E.D.S.)

arxiv_math

--- author: - | Ce Xu${}^{a,}$[^1] and Jianqiang Zhao${}^{b,}$[^2]\ a. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, P.R. China\ b. Department of Mathematics, The Bishop's School, La Jolla, CA 92037, USA title: "**On Some Unramified Families of Motivic Euler Sums**" --- It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as $\mathbb{Q}$-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for a motivic Euler sums (MES) to be unramified, namely, expressible as $\mathbb{Q}$-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified MES in two groups. In one such group we can further prove the concrete identities relating the MES to the motivic MZVs, determined up to rational multiple of a motivic Riemann zeta value by a result of Brown. : (motivic) multiple zeta values, (motivic) Euler sums. 11M32; 11M99. # Introduction The ubiquitous nature of multiple zeta values (MZVs) has attracted many mathematicians and theoretical physicists in recent years after the seminal works of Zagier [@DZ1994] and Hoffman [@H1992]. Its higher level generalization is given by the *colored multiple zeta values* (CMZVs) of level $N$ defined as follows. Let $\mathbb{N}$ be the set of positive integers and $\mathbb{N}_0=\mathbb{N}\cup\{0\}$. For any composition $(s_1,\dots,s_d)\in\mathbb{N}^d$ and $N$-th roots of unity $({\varepsilon}_1,\dots,{\varepsilon}_d)$ we define $$\zeta\left(s_1,\dots,s_d \atop {\varepsilon}_1,\dots,{\varepsilon}_d\right):=\sum_{0<k_1<\dots<k_d} \frac{{\varepsilon}_1^{k_1}\cdots{\varepsilon}_d^{k_d}}{k_1^{s_1}\cdots k_d^{s_d}}.$$ To guarantee convergence we impose the condition that $(s_d,{\varepsilon}_d)\ne (1,1)$. The CMZVs have played a pivotal role in the theory of mixed Tate motives over $\mathbb{Z}[\mu_N][1/N]$ (resp. $\mathbb{Z}[\mu_N]$), where $\mu_N=\exp(2\pi i/N)$, for $N=1,2,4,6,8$ (resp. $N=6$) as manifested by the works [@Brown2012; @Deligne2010; @Glanois2015]. In fact, they first appeared unexpectedly in the computation of Feynman integrals in the 1990s. In particular, the level two MZVs, sometimes also called *Euler sums*, have been studied quite intensively in [@BlumleinBrVe2010; @Broadhurst1996a; @Glanois2016; @JinLi2018]. To save space, if ${\varepsilon}_j=-1$ then we conventionally put a bar on top of $s_j$. If such an Euler sum can be expressed in terms of the MZVs then we say it is an *unramified* Euler sum because the corresponding motivic version is unramified. These values are also called honorary MZVs by Broadhurst. For example, we have the following beautiful unramified family of Euler sums discovered numerically in [@BorweinBrBr1997] and proved by the second author [@Zhao2010a] $$\label{equ:12bar} 8^\ell \zeta(\{1,\bar2\}_\ell)=\zeta(3_\ell)$$ for all $\ell\in\mathbb{N}$. Here and in the rest of the paper, ${\boldsymbol{\sl{s}}}_n$ means the string ${\boldsymbol{\sl{s}}}$ is repeated $n$ times and if the string has only one number then we remove the curly brackets to save space. Our goal in this paper is to lift [\[equ:12bar\]](#equ:12bar){reference-type="eqref" reference="equ:12bar"} to its motivic version and then provide a few more families of unramified motivic Euler sums (MES) in two groups. Denote by $\zeta^{\mathfrak m}({\boldsymbol{\sl{s}}})$ the motivic version of the Euler sum $\zeta({\boldsymbol{\sl{s}}})$ (see next section for the precise definition). Then the first group is presented below. **Theorem 1**. *For all $\ell\in\mathbb{N}$, there are some rational numbers $c_\ell$, $d_\ell$ and $e_\ell$ such that $$\begin{aligned} \label{equ:1bar2Motivic} 2^{3\ell} \zeta^{\mathfrak m}(\{1,\bar2\}_\ell)=&\, \zeta^{\mathfrak m}(3_\ell),\\ 2^{3\ell+1}\zeta^{\mathfrak m}(\{\bar2,1\}_\ell,\bar2)=&\, c_\ell\zeta^{\mathfrak m}(3\ell+2)-\sum_{{\alpha}+{\beta}=\ell} (-1)^{\alpha}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}),\label{equ:bar21bar2Motivic}\\ 2^{3\ell}\zeta^{\mathfrak m}(\{\bar2,1\}_\ell)=&\, d_\ell\zeta^{\mathfrak m}(3\ell)- \zeta^{\mathfrak m}(3_\ell)+2\sum_{{\alpha}+{\beta}=\ell-1} (-1)^{a} \zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}), \label{equ:bar21Motivic}\\ 2^{3\ell-1}\zeta^{\mathfrak m}(\{1,\bar2\}_\ell,1)=&\, e_\ell\zeta^{\mathfrak m}(3\ell+1)-3\sum_{{\alpha}+{\beta}=\ell, 2\nmid b}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}}).\label{equ:12bar1Motivic}\end{aligned}$$* After applying the period map [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"} we see that [\[equ:bar21bar2Motivic\]](#equ:bar21bar2Motivic){reference-type="eqref" reference="equ:bar21bar2Motivic"} and [\[equ:12bar1Motivic\]](#equ:12bar1Motivic){reference-type="eqref" reference="equ:12bar1Motivic"} partially confirm a conjecture by M. Hirose and N. Sato [@SH2019]. Based on this conjecture and some further numerical evidence, we believe that all the undetermined constants in the above theorem must vanish. **Conjecture 2**. *For all $\ell\in\mathbb{N}$, we have $c_\ell=d_\ell=e_\ell=0$ in Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"}.* Note that the corresponding Euler sum for $\zeta^{\mathfrak m}(\{1,\bar2\}_\ell,1)$ diverges. When applying the period map [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"} we need to use its shuffle or stuffle regularized version (see the explanation after [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"}). We will further prove the following seond group of families of unramified Euler sums. Here $\zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})$ corresponds to some shuffle regularized Euler sum for all $a\in\mathbb{N}$ (see (I3) in the next section for precise definition). **Theorem 3**. *For all integers $a, \ell\ge 0$ the MES $$\begin{aligned} \label{equ:2bar3original} &\zeta_a^{\mathfrak m}(\{\bar2,3\}_\ell), \qquad \zeta_a^{\mathfrak m}(\{\bar2,3\}_\ell,\bar2), \\ &\zeta_a^{\mathfrak m}(\{3,\bar2\}_\ell), \qquad \zeta_a^{\mathfrak m}(\{3,\bar2\}_\ell,3), \label{equ:13bar2original}\end{aligned}$$ are all unramified.* Using exactly the same approach we may generalized the first group of unramified families to the the following form. We leave the details to the interested reader. **Theorem 4**. *For all integers $a, \ell\ge 0$ the MES $$\zeta_a^{\mathfrak m}(\{\bar2,1\}_\ell), \quad \zeta_a^{\mathfrak m}(\{\bar2,1\}_\ell,\bar2), \quad\zeta_a^{\mathfrak m}(\{1,\bar2\}_\ell), \quad \zeta_a^{\mathfrak m}(\{1,\bar2\}_\ell,1),$$ are all unramified.* **Remark 5**. In [@Glanois2015; @Glanois2016] Glanois proved that an interpolated version of MES denoted by $\zeta^{\sharp,{\mathfrak m}}({\boldsymbol{\sl{s}}})$ are all unramified if ${\boldsymbol{\sl{s}}}=(\{\overline{\text{even}},\text{odd}\}_\ell)$ where the even components can differ and so can the odd ones. This is unlikely true for the MES in general. Numerial compuatation using the LLL algorithm shows that it is unlikely that $\zeta(\bar2,3,\bar2,5)$ is in the weight 12 piece of the MZV space. The main results obtained in this paper can be regarded as some good evidence for the following conjecture. **Conjecture 6**. *Let $a\in\mathbb{N}_0$, $\ell\in\mathbb{N}$, and $u_j,v_j\in X:=\{1,\bar2,3,\bar4,\dots\}$ for $j=1,\dots,\ell$ such that $1\not\in \{u_j: 1\le j\le \ell\}\cap \{v_j: 1\le j\le \ell\}$. Then $$\sum_{{\sigma},\tau\in{\mathfrak S}_\ell} \zeta_a^{\mathfrak m}(u_{{\sigma}(1)},v_{\tau(1)},\dots,u_{{\sigma}(\ell)},v_{\tau(\ell)})$$ and $$\sum_{{\sigma}\in{\mathfrak S}_\ell,\tau\in{\mathfrak S}_{\ell-1}} \zeta_a^{\mathfrak m}(u_{{\sigma}(1)},v_{\tau(1)},\dots,v_{\tau(\ell-1)},u_{{\sigma}(\ell)}).$$ are both unramified, where ${\mathfrak S}_\ell$ is the symmetry group of $\ell$ letters.* When $a=0$, after taking the period map this conjecture is reduced to a conjecture first proposed by M. Hirose and N. Sato [@SH2019]. # Motivic set-up We adopt the motivic set-up defined by [@Glanois2016]. Let $\Gamma_{N}$ be the group of $N$-th unity for any $N\in\mathbb{N}$. Let $w\in\mathbb{N}$ and suppose $a_j=0$ or $a_j\in\Gamma_{N}$ for all $0\le j\le w+1$. The most important property of the motivic integrals is that there is a period map $\mathop{\mathrm{dch}}$ such that $$\label{equ:periodMap} \mathop{\mathrm{dch}}\big(I^{\mathfrak m}(a_0;a_1,\dots,a_w;a_{w+1})\big)=(-1)^{\mathop{\mathrm{dep}}(a_1,\dots,a_w)} \int_{a_0}^{a_{w+1}} \frac{dt}{t-a_1} \cdots \frac{dt}{t-a_w}$$ as an iterated integral (we always integrate from left to right in this paper) whenever it converges. Here $\mathop{\mathrm{dep}}(a_1,\dots,a_w):=\sharp\{1\le j\le w: a_j\ne 0\}$ is called the *depth* and $w$ the *weight* of the integral (and its corresponding MZV). When the integral diverges then there is a way to regularize it so that it becomes a polynomial in $\mathsf {CMZV}[T]$, where $\mathsf {CMZV}$ is the $\mathbb{Q}$-span of all CMZVs of level $N$. By setting $T=0$ we can define the so-called shuffle regularized CMZVs (see [@Zhao2016 Ch. 13] for more details). We now list all the important properties of the motivic integrals as follows (cf. [@Brown2012 §2.4], [@Glanois2016 p. 9], and [@Murakami2021 §2, (I1)-(I6)]): - Empty word: $I^{\mathfrak m}(a_{0}; a_{1})=1$. - Trivial path: $\forall n\ge1$, $I^{\mathfrak m}(a_{0}; a_{1}, \dots, a_{n}; a_{n+1})=0$ if $a_{0}=a_{n+1}$. - Shuffle product: for any $s_{1}, \dots, s_n\in\mathbb N$ and ${\varepsilon}_{1}, \dots,{\varepsilon}_n\in\Gamma_N$, define $$\zeta_k^{\mathfrak m}\left(s_{1}, \dots, s_n \atop {\varepsilon}_{1}, \dots,{\varepsilon}_n \right) :=I^{\mathfrak m}\big(0; 0^k,\eta_1, 0^{s_1-1},\eta_2, 0^{s_2-1}, \dots, \eta_n, 0^{s_n-1}; 1 \big)$$ where $\eta_j=1/{\varepsilon}_j\cdots {\varepsilon}_d$ for all $j=1,\dots,d$. Then $$\zeta_{k}^{\mathfrak m}\left( {s_{1}, \cdots , s_{p} \atop {\varepsilon}_{1}, \cdots ,{\varepsilon}_{p} }\right)= (-1)^{k}\sum_{\substack{i_{1}+ \cdots + i_{p}=k\\ i_{1},\dots,i_{p}\ge0}} \left(\prod_{j=1}^p\binom {s_j+i_j-1} {i_j}\right) \zeta^{\mathfrak m}\left( {s_{1}+i_{1}, \cdots , s_{p}+i_{p} \atop {\varepsilon}_{1}\ \ , \cdots ,\ \ {\varepsilon}_{p} }\right).$$ - Regularization: If $a_{1}=\cdots=a_n\in \{0,1\}$, then $I^{\mathfrak m}(0; a_{1}, \cdots, a_{n}; 1)=0.$ - Path reversal: $I^{\mathfrak m}(a_{0}; a_{1}, \cdots, a_{n}; a_{n+1})= (-1)^n I^{\mathfrak m}(a_{n+1}; a_{n}, \cdots, a_{1}; a_{0}).$ - Homothety: $\forall \alpha \in \Gamma_{N}, I^{\mathfrak m}(0; \alpha a_{1}, \cdots, \alpha a_{n}; \alpha a_{n+1}) = I^{\mathfrak m}(0; a_{1}, \cdots, a_{n}; a_{n+1})$. - Change of variable $t\to 1-t: \forall a_{1}, \cdots, a_{n}\in\{0, 1\},$ $$I^{\mathfrak m}(0; a_{1}, \cdots, a_{n}; 1)= I^{\mathfrak m}(0;1-a_n, \cdots, 1-a_1; 1).$$ - Path composition: $\forall a,b, x\in \Gamma_{N} \cup \left\{0\right\}$, $$I^{\mathfrak m}(a; a_{1}, \cdots, a_{n}; b)=\sum_{i=0}^{n} I^{\mathfrak m}(a; a_{1}, \cdots, a_{i}; x) I^{\mathfrak m}(x; a_{i+1}, \cdots, a_{n}; b) .$$ Let ${\mathcal H}^N$ be $\mathbb{Q}$-vector space spanned by the motivic CMZVs of the form $$\zeta^{\mathfrak m}_a\left(n_1,\dots,n_d \atop {\varepsilon}_1,\dots,{\varepsilon}_d\right):= I^{\mathfrak m}(0;0_a,\eta_1,0_{n_1-1},\dots,\eta_d,0_{n_d-1};1), \quad a\in\mathbb{N}_0,$$ where $n_j\in\mathbb{N}, {\varepsilon}_j\in\Gamma_{N}$, and $\eta_j$'s are defined in (I3) for all $j$. Note that $a+n_1+\dots+n_d$ is its weight which is denoted by $|{\boldsymbol{\sl{n}}}|$ when $a=0$. If ${\varepsilon}_j=1$ (resp. ${\varepsilon}_j=\pm1$) for all $j$ then we get the motivic MZV (resp. MES) $\zeta^{\mathfrak m}_a(n_1,\dots,n_d)$. Set ${\mathcal A}^N={\mathcal H}^N/\zeta^{\mathfrak m}(2){\mathcal H}^N$. Denote by ${\mathcal H}^N_w$ and ${\mathcal A}_w^N$ the weight $w$ part for all $w\ge 0$. Let ${\mathcal L}^N={\mathcal A}^N_{>0}/{\mathcal A}^N_{>0}\cdot {\mathcal A}^N_{>0}$. For any weight $w$ and odd $r$ such that $r<w$ we can define a derivation as part of a coaction $$D_r: {\mathcal H}_w^N \to {\mathcal L}_r^N {\otimes}{\mathcal H}_{w-r}^N$$ by sending $I^{\mathfrak m}(a_0;a_1,\dots,a_w;a_{w+1})$ to $$\sum_{p=0}^{w-r} I^{\mathfrak l}(a_p; a_{p+1},\dots,a_{p+r};a_{p+r+1}){\otimes}I^{\mathfrak m}(a_0;a_1,\dots,a_p,a_{p+r+1},\dots,a_w;a_{w+1}).$$ The sequence in the left motivic integral is called a *subsequence* of $(a_0;a_1,\dots,a_w;a_{w+1})$ while that in the right factor is called a *quotient sequence*. Each such a choice is called a *cut*. Now we set $N=2$ and recall the following theorem which combines Glanois's result [@Glanois2016 Cor. 2.4] and Brown's [@Brown2012 Thm. 3.3]. Set ${\mathcal H}={\mathcal H}^1$ and ${\mathcal L}={\mathcal L}^1$. **Theorem 7**. *Let $a\in\mathbb{N}_0$ and ${\boldsymbol{\sl{s}}}$ be a composition of positive integers such that $a+|{\boldsymbol{\sl{s}}}|=k$. Then the weight $k$ MES $\zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})\in{\mathcal H}_k$ if and only $D_1 \zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})=0$ and $D_r \zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})\in {\mathcal L}_r{\otimes}{\mathcal H}_{k-r}$ for all odd $r<k$. Moreover, if $D_r \zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})=0$ for all odd $r<k$ then $\zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})=c \, \zeta^{\mathfrak m}(k)$ for some rational number $c$.* Set $\eta_j=\prod_{i=j}^d{\varepsilon}_i$ for all $j=1,\dots,d$ as above (note that $\eta_j=1/\eta_j$) and define $$\rho\left(n_1,\dots,n_d \atop {\varepsilon}_1,\dots,{\varepsilon}_d\right):=(0;\eta_1,0_{n_1-1},\dots,\eta_d,0_{n_d-1};1).$$ We often use the bar notation instead of the two-row one for Euler sums. The following simple fact was already noticed by Glanois [@Glanois2015] without proof. We provide one here for completeness. **Lemma 8**. *If $\bar1$ does not appear in ${\boldsymbol{\sl{s}}}$ then $D_1\zeta_a^{\mathfrak m}({\boldsymbol{\sl{s}}})=0$ for all $a\in\mathbb{N}_0$.* *Proof.* If $|s_j|>1$ for all components of ${\boldsymbol{\sl{s}}}$ then no two consecutive nonzero terms can appear in $\rho({\boldsymbol{\sl{s}}})$ so that $D_1=0$ clearly. If $s_j=1$ then the corresponding sign ${\varepsilon}_j=1$ and therefore $\eta_j=\eta_{j+1}$. Namely, whenever two consecutive nonzero terms appear they must have the same sign. Thus if there are more than one nonzero numbers in the left factor $I^{\mathfrak l}(a;b;c)$ of $D_1$ then it must look like $I^{\mathfrak l}(0;\eta;\eta)$, or $I^{\mathfrak l}(\eta;\eta;0)$,or $I^{\mathfrak l}(\eta;0;\eta)$ or $I^{\mathfrak l}(\eta;\eta;\eta)$ all of which must be 0. All the other cases are clearly 0, too. ◻ Recall that the Bernoulli numbers $B_n$ are defined by the generating function $$\frac{e^t}{e^t-1}=\sum_{n=0}^\infty B_n\frac{t^n}{n!}.$$ **Proposition 9**. *For any positive integers $m\ge 2$ and $d\ge 1$ we have $$\label{equ:zbarmd} \zeta^{\mathfrak m}(\{\bar m\}_d)\in {\mathcal H}_{md}.$$ Moreover, if $m=2s$ is even we have $$\label{equ:2sd} \zeta^{\mathfrak m}(\{\overline{2s}\}_d)= c\, \zeta^{\mathfrak m}(2sd)$$ where $$c= \frac{-2 (2sd)!}{16^{sd} B_{2sd} } \sum_{\substack{\Sigma_{j=1}^s n_j=sd\\ n_j \in\mathbb{N}_0\ \forall j}} \left(\prod_{j=1}^s \frac{\big(1+e^{\tfrac{\pi i}{2s}}\big)^{2n_j+1}+\big(1-e^{\tfrac{\pi i}{2s}}\big)^{2n_j+1} }{2(2n_j+1)!} e^{\tfrac{\pi i}{s}\underset{\scriptscriptstyle j=1}{\overset{\scriptscriptstyle m}{\Sigma}} (2j-1)n_j} \right)\in\mathbb{Q}.$$* *Proof.* When $d=1$ we see that $D_r\zeta^{\mathfrak m}(\bar m)=0$ for all $r<m$ since all cuts for $D_r$ either have 0 on both ends or have all 0's inside. Thus [\[equ:2sd\]](#equ:2sd){reference-type="eqref" reference="equ:2sd"} follows directly from Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. Suppose now $d>1$. To save space we let $m=n+1$ and use $\bar1$ to denote $-1$ . Then $$\zeta^{\mathfrak m}(\{\bar m\}_{2\ell})=I^{\mathfrak m}(0;\{1,0_n,\bar1,0_n\}_\ell;1),\quad \zeta^{\mathfrak m}(\{\bar m\}_{2\ell+1})=I^{\mathfrak m}(0;\bar1,0_n,\{1,0_n,\bar1,0_n\}_\ell;1).$$ For every odd integer $r< m$ we clearly have $D_r\zeta^{\mathfrak m}(\{\bar m\}_d)=0$. Let $r\ge m$ be an odd integer. Suppose first that $m=2s$ is an even integer. If $d=2\ell$ is even let $r=m+i+2mp<md$ for some integer $i,p\ge 0$. Then all nonzero terms of $D_r$ can be paired as shown below: Strictly speaking there are also cut pairs such as ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 2$}}}\hskip.25ex{}}{}$  for which 1 and $\bar1$ are exchanged inside ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ in the above graph. Every such pair have the same quotient sequence but reversed subsequences of each other. Thus the pair cancel each other by path reversal (I5). The argument for odd $d=2\ell+1$ is exactly the same. Thus, by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} we see that $\zeta^{\mathfrak m}(\{\overline{2s}\}_d)=c\zeta^{\mathfrak m}(2sd)$ for some rational number $c$. Applying the period map [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"} we can verify [\[equ:2sd\]](#equ:2sd){reference-type="eqref" reference="equ:2sd"} by [@ShenHe2021 Thm. 2]. Suppose now that $m$ is an odd integer. Then the above proof still works unless $r=mp$ for some odd $p<d$. Now, if a cut starts at a 0 immediately before a $\pm 1$ then the term will be canceled by the cut starting from the next $\pm1$, unless the cut itself ends with the last 1 which yields the following single nonzero term $$D_r \zeta^{\mathfrak m}(\{\bar m\}_d)=\zeta^{\mathfrak m}(\{\bar m\}_p){\otimes}\zeta^{\mathfrak m}(\{\bar m\}_{d-p})\in{\mathcal L}_{mp}{\otimes}{\mathcal H}_{m(d-p)}$$ by induction. Hence [\[equ:zbarmd\]](#equ:zbarmd){reference-type="eqref" reference="equ:zbarmd"} follows by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. This completes the proof of the proposition. ◻ # Proof of [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"} in Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"} {#proof-of-equ1bar2motivic-in-theorem-thmbar21bar2motivic} In this section, we first prove [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"} due to its simplicity and repeated use throughout the paper. The proof technique is also very typical for this kind of claims on unramified MES. As pointed out by S. Charlton, the equation [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"} follows from the original proof of [\[equ:12bar\]](#equ:12bar){reference-type="eqref" reference="equ:12bar"} in [@Zhao2010a] which uses the double shuffle relations and the distribution relations all of which are motivic. For convenience we restate [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"} as follows: for all $\ell\in\mathbb{N}$ $$\label{equ:1bar2MotivicAgain} 8^\ell \zeta^{\mathfrak m}(\{1,\bar2\}_\ell)=\zeta^{\mathfrak m}(3_\ell).$$ We now prove by [\[equ:1bar2MotivicAgain\]](#equ:1bar2MotivicAgain){reference-type="eqref" reference="equ:1bar2MotivicAgain"} by induction on $\ell$. ## The base case If $\ell=1$ then we see easily that $D_1 \zeta^{\mathfrak m}(1,\bar2)=0$ by Lemma [Lemma 8](#lem:D1){reference-type="ref" reference="lem:D1"}. Thus by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} $\zeta^{\mathfrak m}(1,\bar2)=c\zeta^{\mathfrak m}(3)$ for some $c\in \mathbb{Q}$. Applying the period map [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"} and comparing with [\[equ:12bar\]](#equ:12bar){reference-type="eqref" reference="equ:12bar"} we get $c=1/8$ immediately. We now assume that [\[equ:1bar2MotivicAgain\]](#equ:1bar2MotivicAgain){reference-type="eqref" reference="equ:1bar2MotivicAgain"} holds for all positive integers $\ell<k$. We need to show that $$\label{equ:D_r1bar2} 8^k D_r \zeta^{\mathfrak m}(\{1,\bar2\}_k)=D_r \zeta^{\mathfrak m}(3_k)$$ for all positive integers $r=6n+3, 6n+5, 6n+7<3k$ where $n\in\mathbb{Z}$. ## Inductive step ### $r=6n+3$ We have the following picture when $k$ is even (odd $k$ case can be dealt with similarly). Suppose a cut starts in a 011-block. It is easy to see that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$ since they have opposite subsequences but the same quotient one. Further, the subsequence is anti-symmetric so that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$ by path reversal (I5) and homothety with $\alpha=-1$ in (I6). If a cut starts in a $0\bar1\bar1$-block then the same argument works. However, if a cut of type ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 1$}}}\hskip.25ex{}}$ ends at the last 1 then it cannot be canceled. Thus $$8^k D_{6n+3} \zeta^{\mathfrak m}(\{1,\bar2\}_k)= 8^k \zeta^{\mathfrak l}(\{1,\bar2\}_{2n+1}){\otimes}\zeta^{\mathfrak m}(\{1,\bar2\}_{k-2n-1}) =\zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(3_{k-2n-1})$$ by induction. Similarly, by the picture we see that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$ unless the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ ends at the last 1 so that it cannot be canceled. Thus [\[equ:D_r1bar2\]](#equ:D_r1bar2){reference-type="eqref" reference="equ:D_r1bar2"} holds for $r=6n+3$. ### $r=6n+5$ It is obvious that $D_{6n+5}=0$ on both sides of [\[equ:D_r1bar2\]](#equ:D_r1bar2){reference-type="eqref" reference="equ:D_r1bar2"} since both $\rho(3_k)$ and $\rho(\{1,\bar2\}_k)$ have period 6 so that every cut of $D_{6n+5}$ starts and ends with the same number. ### $r=6n+7$ We have the following pictures of possible cuts. $$\label{fig:r=6n+7-bar21} \text{ \begin{tikzpicture}[scale=0.9] \node (A1) at (-2.6,0) {$0;$}; \node (A2) at (-2.2,0) {$1,$}; \node (A3) at (-1.8,0) {$1,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$0,$}; \node (A5) at (-1,0) {$\bar1,$}; \node (A6) at (-0.6,0) {$\bar1,$}; \draw[dashed] (-0.3,0.7) to (-0.3,-0.7); \node (A7) at (0.1,0) {$\cdots f,$}; \draw[dashed] (0.5,0.7) to (0.5,-0.7); \node (A7) at (0.75,0) {$0,$}; \node (A7) at (1.15,0) {$1,$}; \node (A9) at (1.55,0) {$1,$}; \node (A9) at (2.8,0.04) {$\{0\bar1\bar1011\}_n$}; \node (A10) at (4.05,0) {$0,$}; \node (A11) at (4.45,0) {$\bar1,$}; \node (A11) at (4.85,0) {$\bar1,$}; \node (A12) at (5.25,0) {$0,$}; \node (A13) at (5.65,0) {$1,$}; \node (A10) at (6.05,0) {$1,$}; \node (A10) at (6.45,0) {$0,$}; \node (A11) at (6.85,0) {$\bar1,$}; \node (A13) at (7.65,0) {$\bar1,\cdots f,$}; \node (A14) at (8.55,0) {$0,$}; \node (A14) at (8.95,0) {$\bar1,$}; \node (A14) at (9.35,0) {$\bar1,$}; \node (A15) at (9.75,0) {$0;$}; \node (A16) at (10.15,0.03) {$1$}; \node (D3) at (3.4,-0.4) {${}$}; \draw (.7,-0.25) to (.7,-0.4) to (D3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.0,-0.4) to (6.0,-0.25); \node (C1) at (4.2,-0.6) {${}$}; \node (D4) at (3.95,0.6) {${}$}; \draw (1.5,-0.25) to (1.5,-0.6) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (6.8,-0.6) to (6.8,-0.25); \draw (1.1,0.25) to (1.1,0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (6.4,0.6) to (6.4,0.25); \draw[dashed] (1.8,0.7) to (1.8,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (5.05,0.7) to (5.05,-0.7); \draw[dashed] (6.25,0.7) to (6.25,-0.7); \draw[dashed] (7.45,0.7) to (7.45,-0.7); \draw[dashed] (8.3,0.7) to (8.3,-0.7); \draw[dashed] (9.55,0.7) to (9.55,-0.7); \node (lab) at (4,-1.2) {Possible cuts of $D_{6n+7} \zeta^{\mathfrak m}(\{1,\bar2\}_k)$}; \end{tikzpicture} }$$ $$\label{fig:r=6n+7-333} \text{ \begin{tikzpicture}[scale=0.9] \node (A4) at (-1.4,0) {$0,$}; \node (A5) at (-1,0) {$1,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.3,0.7) to (-0.3,-0.7); \node (A7) at (0.1,0) {$\cdots f,$}; \draw[dashed] (0.5,0.7) to (0.5,-0.7); \node (A7) at (0.75,0) {$0,$}; \node (A7) at (1.15,0) {$1,$}; \node (A9) at (1.55,0) {$0,$}; \node (A9) at (2.8,0.04) {$\{010010\}_n$}; \node (A10) at (4.05,0) {$0,$}; \node (A11) at (4.45,0) {$1,$}; \node (A11) at (4.85,0) {$0,$}; \node (A12) at (5.25,0) {$0,$}; \node (A13) at (5.65,0) {$1,$}; \node (A10) at (6.05,0) {$0,$}; \node (A10) at (6.45,0) {$0,$}; \node (A11) at (6.85,0) {$1,$}; \node (A13) at (7.65,0) {$0,\cdots f,$}; \node (A14) at (8.55,0) {$0,$}; \node (A14) at (8.95,0) {$1,$}; \node (A14) at (9.35,0) {$0,$}; \node (A15) at (9.75,0) {$0;$}; \node (A16) at (10.15,0.03) {$1$}; \node (C1) at (4.2,-0.6) {${}$}; \node (D4) at (3.95,0.6) {${}$}; \draw (1.5,-0.25) to (1.5,-0.6) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (6.8,-0.6) to (6.8,-0.25); \draw (1.1,0.25) to (1.1,0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.4,0.6) to (6.4,0.25); \draw[dashed] (1.8,0.7) to (1.8,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (5.05,0.7) to (5.05,-0.7); \draw[dashed] (6.25,0.7) to (6.25,-0.7); \draw[dashed] (7.45,0.7) to (7.45,-0.7); \draw[dashed] (8.3,0.7) to (8.3,-0.7); \draw[dashed] (9.55,0.7) to (9.55,-0.7); \node (lab) at (5,-1.2) {Possible cuts of $D_{6n+7} \zeta^{\mathfrak m}(3_k)$}; \end{tikzpicture} }$$ It is clear that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$ has an anti-symmetric subsequence so that it disappears and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$ in both pictures. This implies that [\[equ:D_r1bar2\]](#equ:D_r1bar2){reference-type="eqref" reference="equ:D_r1bar2"} holds for $r=6n+7$. By combining all the cases for different $r$'s above, we see that [\[equ:D_r1bar2\]](#equ:D_r1bar2){reference-type="eqref" reference="equ:D_r1bar2"} holds for all odd $r<k$. By Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} there is some $c\in\mathbb{Q}$ such that $8^k \zeta^{\mathfrak m}(\{1,\bar2\}_k)=\zeta^{\mathfrak m}(3_k)+c\,\zeta^{\mathfrak m}(3k)$. Applying the period map [\[equ:periodMap\]](#equ:periodMap){reference-type="eqref" reference="equ:periodMap"} and using formula [\[equ:12bar\]](#equ:12bar){reference-type="eqref" reference="equ:12bar"} we finally see that $c=0$ which implies [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"} (or its restatement [\[equ:1bar2MotivicAgain\]](#equ:1bar2MotivicAgain){reference-type="eqref" reference="equ:1bar2MotivicAgain"}) immediately. ## Proof of base cases of Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} {#proof-of-base-cases-of-theorem-thm2bar3motivic} We first prove Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} when $a=0$ and $a=1$ as part of the base cases. We will then prove the full Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} by induction in the next section. **Lemma 10**. *For all integers $\ell\ge 0$ the MES $$\begin{aligned} \label{equ:2bar3} &\zeta^{\mathfrak m}(\{\bar2,3\}_\ell), \qquad \zeta^{\mathfrak m}(\{\bar2,3\}_\ell,\bar2), \\ &\zeta_1^{\mathfrak m}(\{\bar2,3\}_\ell), \qquad \zeta_1^{\mathfrak m}(\{\bar2,3\}_\ell,\bar2),\label{equ:1-2bar3} \\ &\zeta^{\mathfrak m}(\{3,\bar2\}_\ell), \qquad \zeta^{\mathfrak m}(\{3,\bar2\}_\ell,3), \label{equ:3bar2}\\ &\zeta_1^{\mathfrak m}(\{3,\bar2\}_\ell), \qquad \zeta_1^{\mathfrak m}(\{3,\bar2\}_\ell,3) \label{equ:13bar2}\end{aligned}$$ are all unramified.* We originally only planned to prove the pair of families in [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"} are both unramified. When we tried the induction we found that we had to use a "quadruple" induction to prove all the four pairs in Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"} are unramified. This is what will do below. By simple computation or using the datamine [@BlumleinBrVe2010] we get $$\begin{aligned} \zeta(\bar2,3)=&\, -\tfrac{1}{8}\zeta(3,2)-\tfrac{3}{32}\zeta(5),\\ \zeta(\bar2,3,\bar2)=&\, \tfrac{7}{640}\zeta(7)-\tfrac{21}{640}\zeta(3,4),\\ \zeta_1(\bar2,3)=&\, - \big(2\zeta(\bar3,3)+3 \zeta(\bar2,4) \big)=\tfrac{3}{8}\zeta(3)^2+\tfrac{5}{64}\zeta(6),\\ \zeta_1(\bar2,3,\bar2)=&\,-\big(2\zeta(\bar3,3,\bar2)+3 \zeta(\bar2,4,\bar2)+2\zeta(\bar2,3,\bar3) \big)\\ =&\, \tfrac{41}{32}\zeta(6,2)-\tfrac{185}{64}\zeta(1,7)-\frac{1}{8}\zeta(2,3,3)-\tfrac{215}{256}\zeta(8),\\ \zeta(3,\bar2)=&\, \tfrac{7}{32}\zeta(5)-\tfrac{1}{4}\zeta(2,3),\\ \zeta(3,\bar2,3)=&\,\tfrac{3}{64}\zeta(6,2)-\tfrac{3}{16}\zeta(3,5)+\tfrac{3}{16}\zeta(3,3,2)-\tfrac{1}{48}\zeta(8),\\ \zeta_1(3,\bar2)=&\,- \big(2\zeta(3,\bar3)+3 \zeta(4,\bar2) \big)= \tfrac{147}{64}\zeta(6)-\tfrac{9}{8}\zeta(3)^2,\\ \zeta_1(3,\bar2,3)=&\,-\big(3\zeta(4,\bar2,3)+2\zeta(3,\bar3,3)+3\zeta(3,\bar2,4) \big)\\ =&\,\tfrac{153}{64}\zeta(7,2)-\tfrac{21}{4}\zeta(6,3)-\tfrac{387}{64}\zeta(3,6)+\tfrac{11}{8}\zeta(9) .\end{aligned}$$ This convinced us that the base cases are all correct even though we need to do this on the motivic level. First, we can see easily that $D_1=0$ for all the MES in Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} by Lemma [Lemma 8](#lem:D1){reference-type="ref" reference="lem:D1"}. ### The base cases Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"} clearly holds for $\ell=0$. To see the basic ideas we now prove $\ell=1$ case. To save space, we often suppress the commas in the sub- and quotient sequence in the rest of the paper. Using the left picture below we can check that $$\begin{aligned} D_3\zeta^{\mathfrak m}(\bar2,3)=&\,I^{\mathfrak l}(\bar1;010;0){\otimes}I^{\mathfrak m}(0;\bar10;1)+I^{\mathfrak l}(0;100;1){\otimes}I^{\mathfrak m}(0;\bar10;1)\\ =&\,(\zeta^{\mathfrak l}(3)+\zeta_1^{\mathfrak l}(\bar2)){\otimes}\zeta^{\mathfrak m}(\bar2) =(\zeta^{\mathfrak l}(3)-2\zeta^{\mathfrak l}(\bar3)){\otimes}\zeta^{\mathfrak m}(\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_2,\end{aligned}$$ by (I3) and Prop. [Proposition 9](#prop-singleES){reference-type="ref" reference="prop-singleES"}, which shows that $\zeta^{\mathfrak m}(\bar2,3)\in{\mathcal H}_5$ by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. To compute $D_3\zeta^{\mathfrak m}(\bar2,3,\bar2)$ we can use the middle picture above to see that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}=0$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$ by path reversal. Hence $D_3\zeta^{\mathfrak m}(\bar2,3,\bar2)=0$. Further, using the right picture above we get $$\begin{aligned} D_5\zeta^{\mathfrak m}(\bar2,3,\bar2)=&\,I^{\mathfrak l}(0;10\bar100;\bar1){\otimes}I^{\mathfrak m}(0;\bar10;1) +\big(I^{\mathfrak l}(1;0\bar100\bar1;0) +I^{\mathfrak l}(0;\bar100\bar10;1) \big){\otimes}I^{\mathfrak m}(0;10;1)\\ =&\,I^{\mathfrak l}(0;\bar10100;1){\otimes}I^{\mathfrak m}(0;\bar10;1)=\zeta^{\mathfrak l}(\bar2, 3){\otimes}\zeta^{\mathfrak m}(\bar2)\end{aligned}$$ by homothety and path reversal. Thus by the above proof that $\zeta^{\mathfrak m}(\bar2, 3)\in {\mathcal H}_5$ and Prop. [Proposition 9](#prop-singleES){reference-type="ref" reference="prop-singleES"} we get $D_5\zeta^{\mathfrak m}(\bar2,3,\bar2)\in {\mathcal L}_5{\otimes}{\mathcal H}_2.$ The above computation shows that both MES in [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"} are unramified if $\ell=1$ by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. One can similarly check that $\zeta_1^{\mathfrak m}(\bar2,3), \zeta_1^{\mathfrak m}(\bar2,3,\bar2), \zeta^{\mathfrak m}(3,\bar2), \zeta^{\mathfrak m}(3,\bar2,3)$ are all unramified. Indeed, by similar computation as above, we get $$\begin{aligned} D_3\zeta_1^{\mathfrak m}(\bar2,3)=&\, I^{\mathfrak l}(0;0\bar10;1){\otimes}I^{\mathfrak m}(0;100;1)+ \big( I^{\mathfrak l}(\bar1;010;0)+I^{\mathfrak l}(0;100;1)\big){\otimes}I^{\mathfrak m}(0;0\bar10;1)\\ =&\, \zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta^{\mathfrak m}(3)+\big(\zeta^{\mathfrak l}(3)-\zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta_1^{\mathfrak m}(\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_3, \\ D_3\zeta_1^{\mathfrak m}(\bar2,3,\bar2)=&\, I^{\mathfrak l}(0;010;\bar1){\otimes}I^{\mathfrak m}(0;\bar100\bar10;1) +\big( I^{\mathfrak l}(1;0\bar10;0)+I^{\mathfrak l}(0;\bar100;\bar1)\big){\otimes}I^{\mathfrak m}(0;010\bar10;1) \\ &\, + \big(I^{\mathfrak l}(\bar1;00\bar1;0) + I^{\mathfrak l}(0;0\bar10;1) \big){\otimes}I^{\mathfrak m}(0;010\bar10;1) \\ =&\, \zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta^{\mathfrak m}(3,\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_5, \\ D_5\zeta_1^{\mathfrak m}(\bar2,3,\bar2)=&\,I^{\mathfrak l}(0;10\bar100;\bar1){\otimes}I^{\mathfrak m}(0;0\bar10;1) +\big(I^{\mathfrak l}(1;0\bar100\bar1;0) +I^{\mathfrak l}(0;\bar100\bar10;1) \big){\otimes}I^{\mathfrak m}(0;010;1)\\ =&\, \zeta^{\mathfrak l}(\bar2, 3){\otimes}\zeta_1^{\mathfrak m}(\bar2) \in {\mathcal L}_5{\otimes}{\mathcal H}_3\end{aligned}$$ by essentially the same proof of unramification of $\zeta^{\mathfrak m}(\bar2,3)$ above. Moreover, using the above pictures (${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ always cancel) one checks easily that $$\begin{aligned} D_3\zeta_1^{\mathfrak m}(3,\bar2)=&\, I^{\mathfrak l}(0;0\bar10;1){\otimes}I^{\mathfrak m}(0;0\bar10;1) =\zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta_1^{\mathfrak m}(\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_3, \\ D_5\zeta_1^{\mathfrak m}(3,\bar2)=&\, 0, \\ D_3\zeta_1^{\mathfrak m}(3,\bar2,3)=&\, I^{\mathfrak l}(0;0\bar10;1){\otimes}I^{\mathfrak m}(0;0\bar10100;1) + \big( I^{\mathfrak l}(\bar1;010;0)+I^{\mathfrak l}(0;100;1)\big){\otimes}I^{\mathfrak m}(0;0\bar100\bar10;1) \\ =&\, \zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta_1^{\mathfrak m}(\bar2,3)+\big(\zeta^{\mathfrak l}(3)- \zeta_1^{\mathfrak l}(\bar2)\big){\otimes}\zeta_1^{\mathfrak m}(3,\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_6, \\ D_5\zeta_1^{\mathfrak m}(3,\bar2,3)=&\, I^{\mathfrak l}(0;\bar10100;1){\otimes}I^{\mathfrak m}(0;0\bar100;1) =\zeta_1^{\mathfrak l}(\bar2,3){\otimes}\zeta_1^{\mathfrak m}(\bar3) \in {\mathcal L}_5{\otimes}{\mathcal H}_4, \\ D_7\zeta_1^{\mathfrak m}(3,\bar2,3)=&0,\end{aligned}$$ using the known unramified MES. Similarly, $$\begin{aligned} D_3\zeta^{\mathfrak m}(3,\bar2)=&\, I^{\mathfrak l}(0;0\bar10;1){\otimes}I^{\mathfrak m}(0;\bar10;1) =\zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta^{\mathfrak m}(\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_2, \\ D_3\zeta^{\mathfrak m}(3,\bar2,3)=&\, I^{\mathfrak l}(0;0\bar10;1){\otimes}I^{\mathfrak m}(0;\bar10100;1) + \big( I^{\mathfrak l}(\bar1;010;0)+I^{\mathfrak l}(0;100;1)\big){\otimes}I^{\mathfrak m}(0;\bar100\bar10;1) \\ =&\,\zeta_1^{\mathfrak l}(\bar2){\otimes}\zeta^{\mathfrak m}(\bar2,3)+\big(\zeta^{\mathfrak l}(3)- \zeta_1^{\mathfrak l}(\bar2)\big){\otimes}\zeta^{\mathfrak m}(3,\bar2) \in {\mathcal L}_3{\otimes}{\mathcal H}_5, \\ D_5\zeta^{\mathfrak m}(3,\bar2,3)=&\, I^{\mathfrak l}(0;\bar10100;1){\otimes}I^{\mathfrak m}(0;\bar100;1) =\zeta^{\mathfrak l}(\bar2,3){\otimes}\zeta^{\mathfrak m}(\bar3) \in {\mathcal L}_5{\otimes}{\mathcal H}_3.\end{aligned}$$ The above computation shows that all the MES in Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"} are unrmified when $\ell=1$. We now assume that $k\ge 2$ and that the MES in [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"} are unramified for all $\ell<k$. ## The inductive step for $\zeta^{\mathfrak m}(\{\bar2,3\}_k)$ As the sequence $\rho(\{\bar2,3\}_k)$ has a period of 10, it is clear that $D_9=0$ and the pattern of $D_{10+r}$ is basically the same as that of $D_r$ for any $r>1$, with one more of $\{\bar2,3\}_2$ in the subsequence and one less of $\{\bar2,3\}_2$ in the quotient. Thus it suffices for us to show that $D_r\zeta^{\mathfrak m}(\{\bar2,3\}_k)$ are stable derivations for $r=10n+3,10n+5,10n+7,10n+11$, meaning that all its factors are of similar forms which are then unramified by induction. ### $r=10n+3$ {#subsec:2bar3caser=3} We have the following picture when $k$ is odd and we only need to modify the left end by exchange 1 and $\bar1$ when $k$ is even. We call a subsequence $S_{\bar1}=(0\bar1010)$ a $S_{\bar1}$-block, and a subsequence $S_{1}=(010\bar10)$ a $S_{1}$-block. For general $k$, the sequence $\rho(\{\bar2,3\}_k)$ always starts with a $S_{(-1)^k}$-block (here we identify $-1$ with $\bar 1$) and ends with a $S_{\bar1}$-block followed by $0,1$. If a subsequence starts in a $S_1$-block (as shown in the picture above) then there are only four possible nonzero cuts with the same quotient sequence: $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(1;0,\bar1,0,\{S_{\bar1},S_{1}\}_n;0){\otimes}I^{\mathfrak m}(0;\bar1,\dots,0;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;\bar1,0,\{S_{\bar1},S_{1}\}_n,0;\bar1){\otimes}I^{\mathfrak m}(0;\bar1,\dots,0;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(\bar1;0,\{S_{\bar1},S_{1}\}_n,0,\bar1;0){\otimes}I^{\mathfrak m}(0;\bar1,\dots,0;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;\{S_{\bar1},S_{1}\}_n,0,\bar1,0;1){\otimes}I^{\mathfrak m}(0;\bar1,\dots,0;1). \endaligned$$ By path reversal and homothety, we see immediately that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}= {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$. If a subsequence starts in a $S_{\bar1}$-block, then the argument is completely similar. We only need to exchange $1$ and $\bar1$ and therefore the four terms still cancel each other. However, when a subsequence starts in the final $S_{\bar1}$-block, we only get two terms $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 1$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(\bar1;0,1,0,\{S_{1},S_{\bar1}\}_n;0){\otimes}I^{\mathfrak m}(0;\{S_{\bar1},S_1\}_{k/2-n},\bar1,0;1),\\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 2$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;1,0,\{S_{1},S_{\bar1}\}_n,0;1){\otimes}I^{\mathfrak m}(0;\{S_{\bar1},S_1\}_{k/2-n},\bar1,0;1) \endaligned$$ when $k$ is odd and $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(\bar1;0,1,0,\{S_{1},S_{\bar1}\}_n;0){\otimes}I^{\mathfrak m}(0;S_1,\{S_{\bar1},S_1\}_{k/2-n-1},\bar1,0;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;1,0,\{S_{1},S_{\bar1}\}_n,0;1){\otimes}I^{\mathfrak m}(0;S_1,\{S_{\bar1},S_1\}_{k/2-n-1},\bar1,0;1) \endaligned$$ when $k$ is even. In both cases $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}}=&\, -\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n},\bar2){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2n-1},\bar2)\in{\mathcal L}_{10n+3}{\otimes}{\mathcal H}_{5k-10n-3}, \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}}=&\,\zeta^{\mathfrak l}(\{3,\bar2\}_{2n},3){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2n-1},\bar2)\in{\mathcal L}_{10n+3}{\otimes}{\mathcal H}_{5k-10n-3} \endaligned$$ by induction. ### $r=10n+5$ {#subsec:2bar3caser=5} We have the following picture when $k$ is odd and we only need to modify the left end by exchange 1 and $\bar1$ when $k$ is even. If a subsequence starts in a $S_{1}$-block as shown above then there are only four possible nonzero terms with the same quotient sequence in ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}$: $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 0$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;1,0,\bar1,0,\{S_{\bar1},S_{1}\}_n,0;\bar1){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},0\bar1010,S_1,\dots;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(1;0,\bar1,0,\{S_{\bar1},S_{1}\}_n,0,\bar1;0){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},01010,S_1,\dots;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;\bar1,0,\{S_{\bar1},S_{1}\}_n,0,\bar1,0;1){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},01010,S_1,\dots;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(\bar1;0,\{S_{\bar1},S_{1}\}_n,0,\bar1,0,1;0){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},010\bar10,S_1,\dots;1). \endaligned$$ By path reversal and homothety, we see easily that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$. But we don't see the immediate cancelation of ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 0$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$. However, all possible cuts of ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 0$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 0$}}}\hskip.25ex{}}\,$ together yield $$\zeta^{\mathfrak l}(\{\bar2,3\}_{2n+1}){\otimes}\sum_{j=0}^{k-2n-1}\zeta^{\mathfrak m}(\{2,\bar3\}_{k-2n-1-j},\{\bar2,3\}_{j})$$ while all possible cuts of ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 3$}}}\hskip.25ex{}}\,$ together yield $$-\zeta^{\mathfrak l}(\{\bar2,3\}_{2n+1}){\otimes}\sum_{j=0}^{k-2n-2}\zeta^{\mathfrak m}(\{2,\bar3\}_{k-2n-1-j},\{\bar2,3\}_{j})$$ by path reversal. Note that there is one extra term for ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 0$}}}\hskip.25ex{}}\,$ when a subsequence ends at the last 1. Therefore $$D_5\zeta^{\mathfrak m}(\{\bar2,3\}_k)=\zeta^{\mathfrak l}(\{\bar2,3\}_{2n+1}){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2n-1}) \in {\mathcal L}_{10n+5}{\otimes}{\mathcal H}_{5k-10n-5}.$$ ### $r=10n+7$ {#subsec:2bar3caser=7} We have the following picture when $k$ is odd. If a subsequence starts in a $S_1$-block then there are only three possible nonzero terms: $$\aligned { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(0;10\bar10,\{S_{\bar1},S_1\}_n,0\bar10;1){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},0,1,0,S_1,\dots;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(1;0\bar10,\{S_{\bar1},S_1\}_n,0\bar101;0){\otimes}I^{\mathfrak m}(0;\dots,S_{\bar1},0,1,0,S_1,\dots;1), \\ { {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}}=&\, I^{\mathfrak l}(\bar1;0,\{S_{\bar1},S_1\}_n,0\bar10100;1){\otimes}I^{\mathfrak m}(0;\cdots;1). \endaligned$$ Then ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$ since its subsequence is anti-symmetric. The quotient sequence in ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ are the same. By path reversal, we see immediately that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$. If a subsequence starts in a $S_{\bar1}$-block, then the argument is completely similar so that all the terms will be canceled. In conclusion we see that $D_{10n+7}\zeta^{\mathfrak m}(\{\bar2,3\}_k)=0$. ### $r=10n+11$ {#subsec:2bar3caser=11} We notice that any such cut would produce the same type of quotient sequence with $n+1$ blocks of $(S_1,S_{\bar1})$ missing. So the right factor is unramified by induction. The left factor must have one of the following forms: $$\begin{aligned} I^{\mathfrak l}(0; \{S_1,S_{\bar1}\}_{n+1},0;1)=I^{\mathfrak l}(0;\{S_{\bar1},S_1\}_{n+1},0;\bar1)= &\, \zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n+2})\in{\mathcal L}_{10n+11},\\ I^{\mathfrak l}(1;0,\{S_1,S_{\bar1}\}_{n+1};0)=I^{\mathfrak l}(\bar1;0,\{S_{\bar1},S_1\}_{n+1};0)= &\, -\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n+2})\in{\mathcal L}_{10n+11}.\end{aligned}$$ by inductive assumption. Combining all the findings in §[3.4.1](#subsec:2bar3caser=3){reference-type="ref" reference="subsec:2bar3caser=3"}-§[3.4.4](#subsec:2bar3caser=11){reference-type="ref" reference="subsec:2bar3caser=11"} we see that $D_1\zeta^{\mathfrak m}(\{\bar2,3\}_k)=0$ and $D_r\zeta^{\mathfrak m}(\{\bar2,3\}_k)\in{\mathcal L}_r{\otimes}{\mathcal H}_{5k-r}$ (i.e., $D_r$ is a stable derivation) for all odd $r<5k$. Consequently, $\zeta^{\mathfrak m}(\{\bar2,3\}_k)$ is unramified by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. ## The inductive step for $\zeta^{\mathfrak m}(\{\bar2,3\}_k,\bar2)$ {#subsec:bar23bar2} The computation of $D_r\zeta^{\mathfrak m}(\{\bar2,3\}_\ell,\bar2)$ can be carried out in a similar manner as that for $\zeta^{\mathfrak m}(\{\bar2,3\}_\ell)$. We can attach $0;\bar1$ to the right end of the picture in §[3.4.1](#subsec:2bar3caser=3){reference-type="ref" reference="subsec:2bar3caser=3"} and then exchange $\bar1$ and 1. By the same argument there, $D_{10n+r}$ are all stable for $r=3,5,7,9,11$. For brevity, we only show the computation when $r=7$ in which case the induction assumption for MES $\zeta^{\mathfrak m}(\{\bar2,3\}_\ell)$ is essentially used. In this case, all the computation will be the same as that of $D_7\zeta^{\mathfrak m}(\{\bar2,3\}_\ell)$ except for the only one extra possible cut starting at the beginning of the last $S_{\bar1}$-block for which we have the following picture. which produces $${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \text{$\bar 1$}$}}}\hskip.25ex{}}=I^{\mathfrak l}(0;10\bar100\bar10;1){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-1}) =\zeta^{\mathfrak l}(\bar2,3,\bar2){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-1})\in{\mathcal L}_7{\otimes}{\mathcal H}_{5k-5}$$ by the initial step and the inductive assumption. By Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} we have now proved that both families of MES in [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"} are unramified when $\ell=k$. ## The inductive step for $\zeta^{\mathfrak m}(\{3,\bar2\}_k)$ and $\zeta^{\mathfrak m}(\{3,\bar2\}_k,3)$ We now prove that the pair of families in [\[equ:3bar2\]](#equ:3bar2){reference-type="eqref" reference="equ:3bar2"} are both unramified. It is mostly the same as that of [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"} except that in the inductive step we need to use the validity of unramification of [\[equ:2bar3\]](#equ:2bar3){reference-type="eqref" reference="equ:2bar3"}. When $r=3$ (or similarly for $r=10n+3<5k$) we can modify the picture in [3.4.1](#subsec:2bar3caser=3){reference-type="ref" reference="subsec:2bar3caser=3"} at the two ends if $k$ is odd. We just add $0;\bar10$ to the left end and remove $0,0;1$ from the right end (no right-end change needed for $\zeta^{\mathfrak m}(\{3,\bar2\}_k,3)$). So we consider the following picture for even $k>1$: Similar to what we have shown previously we see easily that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 2$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 3$}}}\hskip.25ex{}}=0$. But ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 4$}}}\hskip.25ex{}}\,$ from the first block cannot be canceled by the missing ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 1$}}}\hskip.25ex{}}$ in front of it. All the other cuts in $D_3\zeta^{\mathfrak m}(\{3,\bar2\}_k)$ will either cancel each other or vanish themselves (see §[3.4.1](#subsec:2bar3caser=3){reference-type="ref" reference="subsec:2bar3caser=3"}). For general $D_{10n+3}$ the argument is the same and therefore $$D_{10n+3}\zeta^{\mathfrak m}(\{3,\bar2\}_k) =\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n},\bar2){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2n-1},\bar2) \in {\mathcal L}_{10n+3}{\otimes}{\mathcal H}_{5k-10n-3}$$ by inductive assumption. Similarly, we see that $$D_{10n+3}\zeta^{\mathfrak m}(\{3,\bar2\}_k,3)=\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n},\bar2){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2n}) +\zeta^{\mathfrak l}(\{3,\bar2\}_{2n},3){\otimes}\zeta^{\mathfrak m}(\{3,\bar2\}_{k-2n})$$ also lies in ${\mathcal L}_{10n+3}{\otimes}{\mathcal H}_{5k-10n}$ by induction, where the second term comes from the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ ending at the last 1. Applying the same idea we find that all of $D_{10n+5}$, $D_{10n+7}$ and $D_{10n+11}$ are stable derivations. By Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} we know $\zeta^{\mathfrak m}(\{3,\bar2\}_k)$ and $\zeta^{\mathfrak m}(\{3,\bar2\}_k,3)$ are both unramified. We leave the details to the interested reader. ## The inductive step for $\zeta_1^{\mathfrak m}(\{3,\bar2\}_k)$ and $\zeta_1^{\mathfrak m}(\{3,\bar2\}_k,3)$ We first consider the case $r=10n+9<5k$. There is only one cut in $D_r$ that is nonzero. This occurs when the subsequence starts at the very beginning which produces the terms $$\begin{aligned} D_r\zeta_1^{\mathfrak m}(\{3,\bar2\}_k)= {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 9$}}}\hskip.25ex{}}=&\, \zeta_1(\{3,\bar2\}_{2n+1},3){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2-2n},\bar2)\in {\mathcal L}_{10n+9}{\otimes}{\mathcal H}_{5k-10n-8}, \\ D_r\zeta_1^{\mathfrak m}(\{3,\bar2\}_k,3)= {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 9$}}}\hskip.25ex{}}=&\, \zeta_1(\{3,\bar2\}_{2n+1},3){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-1-2n})\in {\mathcal L}_{10n+9}{\otimes}{\mathcal H}_{5k-10n-5}\end{aligned}$$ by induction and the proceeding section. Similarly, if $r=10n+11<5k$ then the only nontrivial cut in $D_r$ occurs when the subsequence starts at the very beginning, which produces the terms $$\begin{aligned} D_r\zeta_1^{\mathfrak m}(\{3,\bar2\}_k)= {\bigcirc\hskip-2ex{\raisebox{-0.0cm}{\text{$\scriptstyle 11$}}}\hskip.25ex{}}=&\, \zeta_1(\{3,\bar2\}_{2n+2}){\otimes}\zeta^{\mathfrak m}(\{3,\bar2\}_{k-2-2n})\in {\mathcal L}_{10n+11}{\otimes}{\mathcal H}_{5k-10n-10},\\ D_r\zeta_1^{\mathfrak m}(\{3,\bar2\}_k,3)= {\bigcirc\hskip-2ex{\raisebox{-0.0cm}{\text{$\scriptstyle 11$}}}\hskip.25ex{}}=&\, \zeta_1(\{3,\bar2\}_{2n+2}){\otimes}\zeta^{\mathfrak m}(\{3,\bar2\}_{k-2-2n},3)\in {\mathcal L}_{10n+11}{\otimes}{\mathcal H}_{5k-10n-7}\end{aligned}$$ by induction and the proceeding section. For all other odd $r$'s the proof is essentially the same as the starting $0$ does not contribute to $D_r$. By Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} we see that $\zeta_1^{\mathfrak m}(\{3,\bar2\}_k)$ is unramified. ## The inductive step for $\zeta_1^{\mathfrak m}(\{\bar2,3\}_k)$ and $\zeta_1^{\mathfrak m}(\{\bar2,3\}_k,\bar2)$ We now consider the pair of families of MES in [\[equ:1-2bar3\]](#equ:1-2bar3){reference-type="eqref" reference="equ:1-2bar3"}. We first consider the case $r=10n+3$. Adding one 0 to the left of the picture in §[3.4.1](#subsec:2bar3caser=3){reference-type="ref" reference="subsec:2bar3caser=3"} we see that there is only one extra cut in $D_r$ compared to the [\[equ:3bar2\]](#equ:3bar2){reference-type="eqref" reference="equ:3bar2"} case when the subsequence starts at the very beginning. This yields the terms $${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle \bar 4$}}}\hskip.25ex{}}=\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n},\bar2){\otimes}\zeta^{\mathfrak m}(\{3,\bar2\}_{k-1-2n},3) \quad\text{or} \quad \zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n},\bar2){\otimes}\zeta^{\mathfrak m}(\{3,\bar2\}_{k-2n})$$ By induction and the proceeding section (if $n=0$) we see that all the factors are unramified. Similarly, if $r=10n+11$ then the only nontrivial cut in $D_r$ occurs when the subsequence starts at the very beginning which produces the terms $$\zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n+2}){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2-2n}) \quad\text{or} \quad \zeta_1^{\mathfrak l}(\{\bar2,3\}_{2n+2}){\otimes}\zeta^{\mathfrak m}(\{\bar2,3\}_{k-2-2n},\bar2).$$ Again, all the factors are unramified by induction. For other cuts when $r=10n+3$ as well as for other odd $r$'s the proof is essentially the same as that for $\zeta^{\mathfrak m}(\{\bar2,3\}_k)$ and $\zeta^{\mathfrak m}(\{\bar2,3\}_k,\bar2)$ since the starting $0$ does not contribute to $D_r$. By Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} we see that both $\zeta_1^{\mathfrak m}(\{\bar2,3\}_k)$ and $\zeta_1^{\mathfrak m}(\{\bar2,3\}_k,\bar2)$ are unramified. We have now completed the proof of Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"}. # Proof of Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} {#proof-of-theorem-thm2bar3motivic} We first handle another set of base cases for arbitrary $a$. **Lemma 11**. *For any integer $a\ge 0$ the MES $$\zeta_a^{\mathfrak m}(\bar2), \quad \zeta_a^{\mathfrak m}(\bar2,3), \quad \zeta_a^{\mathfrak m}(\bar2,3,\bar2), \quad \zeta_a^{\mathfrak m}(3,\bar2), \quad \zeta_a^{\mathfrak m}(3,\bar2,3)$$ are all unramified.* *Proof.* The proof is again a standard application of Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} using the stable derivations $D_r$. We leave the details to the interested reader. ◻ We now use induction to prove Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"}. Assume $a\ge 2$, $\ell\ge 2$ and Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} holds if $a$ or $k$ or both are replaced by any smaller nonnegative integers. For any odd integer $r\ge 3$ if a cut of $D_r$ starting after the first nonzero $\pm 1$ is not canceled by another cut nor does it vanish by itself then by the proof of Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"} the left factor must have the form treated in Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"} and the right factor one of the forms in Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} but with smaller $\ell$. By induction assumption $D_r$ must be stable. If a cut starts somewhere within the first block of $a+1$ 0's then its left factor must have one of the forms in Theorem [Theorem 3](#thm:2bar3Motivic){reference-type="ref" reference="thm:2bar3Motivic"} but with $\ell<k$ and its right factor must have the form treated in Lemma [Lemma 10](#lem:2bar3Motivic){reference-type="ref" reference="lem:2bar3Motivic"}. Thus $D_r$ must be stable by induction assumption again. This completes the proof of the theorem. # Proof of [\[equ:bar21bar2Motivic\]](#equ:bar21bar2Motivic){reference-type="eqref" reference="equ:bar21bar2Motivic"} and [\[equ:bar21Motivic\]](#equ:bar21Motivic){reference-type="eqref" reference="equ:bar21Motivic"} in Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"} {#proof-of-equbar21bar2motivic-and-equbar21motivic-in-theorem-thmbar21bar2motivic} We now show that for all $\ell\in\mathbb{N}$, there are some rational numbers $c_\ell$ and $d_\ell$ such that $$\begin{aligned} 2^{3\ell+1}\zeta^{\mathfrak m}(\{\bar2,1\}_\ell,\bar2)=&\, c_\ell\zeta^{\mathfrak m}(3\ell+2)-\sum_{{\alpha}+{\beta}=\ell} (-1)^{\alpha}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}),\label{equ:bar21bar2MotivicN}\\ 2^{3\ell}\zeta^{\mathfrak m}(\{\bar2,1\}_\ell)=&\, d_\ell\zeta^{\mathfrak m}(3\ell)- \zeta^{\mathfrak m}(3_\ell)+2\sum_{{\alpha}+{\beta}=\ell-1} (-1)^{a} \zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}), \label{equ:bar21MotivicN}\end{aligned}$$ by induction on $\ell$. ## Proof of base cases of [\[equ:bar21bar2MotivicN\]](#equ:bar21bar2MotivicN){reference-type="eqref" reference="equ:bar21bar2MotivicN"} and [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} If $\ell=1$ then we see easily that $D_1=0$ for all MES involved by Lemma [Lemma 8](#lem:D1){reference-type="ref" reference="lem:D1"}. Then by the regularized stuffle relation of Euler sums $$\zeta_*(\bar2,1)=-\zeta(1,\bar2)-\zeta(\bar3)=-\frac18\zeta(3)+\frac34\zeta(3)=\frac58\zeta(3)$$ using Prop. [Proposition 9](#prop-singleES){reference-type="ref" reference="prop-singleES"} and [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"}. Thus [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} follows from Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} when $\ell=1$: $$\label{equ:ell=1bar21MotivicN} 8\zeta^{\mathfrak m}(\bar2,1)=-\zeta^{\mathfrak m}(3)+2\zeta_1^{\mathfrak m}(2)=5\zeta^{\mathfrak m}(3).$$ Using the pictures [\[fig:bar21bar2Base\]](#fig:bar21bar2Base){reference-type="eqref" reference="fig:bar21bar2Base"} $$\label{fig:bar21bar2Base} \text{ \begin{tikzpicture}[scale=0.9] \node (RA0) at (5.05,0) {$0;$}; \node (RA1) at (5.45,0) {$1,$}; \node (RA3) at (5.85,0) {$0,$}; \node (RA3) at (6.25,0) {$\bar1,$}; \node (RB3) at (6.2,0.4) {${}$}; \node (RA4) at (6.65,0) {$\bar1,$}; \node (RD4) at (6.6,0.8) {${}$}; \node (RA5) at (7.05,0) {$0;$}; \node (RA6) at (7.45,0) {$1\phantom{,}$}; \node (RC5) at (5.8,-0.5) {${}$}; \draw (5.4,0.25) to (5.4,0.4) to (RB3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (7.0,0.4) to (7.0,0.25); \draw (5.8,0.25) to (5.8,0.8) to (RD4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (7.4,0.8) to (7.4,0.25); \draw (5,-0.25) to (5,-0.5) to (RC5) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.6,-0.5) to (6.6,-0.325); \node (label) at (6.25,-1.2) {$D_3 \zeta^{\mathfrak m}(\bar2,1,\bar2)$}; \end{tikzpicture} \ \begin{tikzpicture}[scale=0.9] \node (RA0) at (5.05,0) {$0;$}; \node (RA1) at (5.45,0) {$1,$}; \node (RA3) at (5.85,0) {$0,$}; \node (RA3) at (6.25,0) {$0,$}; \node (RB3) at (6.2,0.4) {${}$}; \node (RA4) at (6.65,0) {$1,$}; \node (RD4) at (6.6,0.8) {${}$}; \node (RA5) at (7.05,0) {$0;$}; \node (RA6) at (7.45,0) {$1\phantom{,}$}; \node (RC5) at (5.8,-0.5) {${}$}; \draw (5.4,0.25) to (5.4,0.4) to (RB3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (7.0,0.4) to (7.0,0.25); \draw (5.8,0.25) to (5.8,0.8) to (RD4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (7.4,0.8) to (7.4,0.25); \draw (5,-0.25) to (5,-0.5) to (RC5) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.6,-0.5) to (6.6,-0.325); \node (label) at (6.25,-1.2) {$D_3 \zeta^{\mathfrak m}(3,2)$}; \end{tikzpicture} \ \begin{tikzpicture}[scale=0.9] \node (RA0) at (5.05,0) {$0;$}; \node (RA1) at (5.45,0) {$1,$}; \node (RA3) at (5.85,0) {$0,$}; \node (RA3) at (6.25,0) {$1,$}; \node (RB3) at (6.2,0.4) {${}$}; \node (RA4) at (6.65,0) {$0,$}; \node (RD4) at (6.6,0.8) {${}$}; \node (RA5) at (7.05,0) {$0;$}; \node (RA6) at (7.45,0) {$1\phantom{,}$}; \node (RC5) at (5.8,-0.5) {${}$}; \draw (5.4,0.25) to (5.4,0.4) to (RB3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (7.0,0.4) to (7.0,0.25); \draw (5.8,0.25) to (5.8,0.8) to (RD4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (7.4,0.8) to (7.4,0.25); \draw (5,-0.25) to (5,-0.5) to (RC5) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.6,-0.5) to (6.6,-0.325); \node (label) at (6.25,-1.2) {$D_3 \zeta^{\mathfrak m}(2,3)$}; \end{tikzpicture} \ \begin{tikzpicture}[scale=0.9] \node (RA1) at (5.45,0) {$0,$}; \node (RA3) at (5.85,0) {$\bar1,$}; \node (RA3) at (6.25,0) {$\bar1,$}; \node (RB3) at (6.2,-0.4) {${}$}; \node (RA4) at (6.65,0) {$0,$}; \node (RD4) at (6.6,0.4) {${}$}; \node (RA5) at (7.05,0) {$1;$}; \node (RA6) at (7.45,0) {$1\phantom{,}$}; \draw (5.8,0.25) to (5.8,0.4) to (RD4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (7.4,0.4) to (7.4,0.25); \draw (5.4,-0.25) to (5.4,-0.4) to (RB3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (7.0,-0.4) to (7.0,-0.25); \node (label) at (6.45,-1.2) {$D_3 \zeta^{\mathfrak m}(1,\bar2,1)$}; \end{tikzpicture} \ \begin{tikzpicture}[scale=0.9] \node (RA1) at (5.45,0) {$0,$}; \node (RA3) at (5.85,0) {$1,$}; \node (RA3) at (6.25,0) {$1,$}; \node (RB3) at (6.2,-0.4) {${}$}; \node (RA4) at (6.65,0) {$0,$}; \node (RD4) at (6.6,0.4) {${}$}; \node (RA5) at (7.05,0) {$0;$}; \node (RA6) at (7.45,0) {$1\phantom{,}$}; \draw (5.8,0.25) to (5.8,0.4) to (RD4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (7.4,0.4) to (7.4,0.25); \draw (5.4,-0.25) to (5.4,-0.4) to (RB3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (7.0,-0.4) to (7.0,-0.25); \node (label) at (6.45,-1.2) {$D_3 \zeta^{\mathfrak m}(1,3)$}; \end{tikzpicture} }$$ together with the fact that $\zeta^{\mathfrak m}(\bar2)=-\zeta^{\mathfrak m}(2)/2$ by Prop. [Proposition 9](#prop-singleES){reference-type="ref" reference="prop-singleES"} and [\[equ:ell=1bar21MotivicN\]](#equ:ell=1bar21MotivicN){reference-type="eqref" reference="equ:ell=1bar21MotivicN"}, we have $$\begin{aligned} D_3 \zeta^{\mathfrak m}(\bar2,1,\bar2)=&\, I^{\mathfrak l}(0;10\bar1;\bar1){\otimes}I^{\mathfrak m}(0;\bar10;1) =-\frac12\zeta^{\mathfrak l}(\bar2,1){\otimes}\zeta^{\mathfrak m}(2)=-\frac5{16} \zeta^{\mathfrak l}(3){\otimes}\zeta^{\mathfrak m}(2),\\ D_3 \zeta^{\mathfrak m}(3,2)=&\, I^{\mathfrak l}(0;010;1){\otimes}I^{\mathfrak m}(0;10;1)=\zeta_1^{\mathfrak l}(2){\otimes}\zeta^{\mathfrak m}(2)=-2\zeta^{\mathfrak l}(3){\otimes}\zeta^{\mathfrak m}(2),\\ D_3 \zeta^{\mathfrak m}(2,3)=&\, \big(I^{\mathfrak l}(1;010;0)+I^{\mathfrak l}(0;100;1)\big){\otimes}I^{\mathfrak m}(0;10;1)=3\zeta^{\mathfrak l}(3){\otimes}\zeta^{\mathfrak m}(2).\end{aligned}$$ Hence $$16 D_3 \zeta^{\mathfrak m}(\bar2,1,\bar2)=D_3\big( \zeta^{\mathfrak m}(3,2)- \zeta^{\mathfrak m}(2,3) \big).$$ For [\[equ:12bar1MotivicN\]](#equ:12bar1MotivicN){reference-type="eqref" reference="equ:12bar1MotivicN"} it is easy to see from the last two pictures in [\[fig:bar21bar2Base\]](#fig:bar21bar2Base){reference-type="eqref" reference="fig:bar21bar2Base"} that $$D_r \zeta^{\mathfrak m}(1,\bar2,1)=D_r \zeta^{\mathfrak m}(1,3)=0 \quad \text{for } r=1,3.$$ Hence, the $\ell=1$ case of [\[equ:bar21bar2MotivicN\]](#equ:bar21bar2MotivicN){reference-type="eqref" reference="equ:bar21bar2MotivicN"} and [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} follows from Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. We now assume [\[equ:bar21bar2MotivicN\]](#equ:bar21bar2MotivicN){reference-type="eqref" reference="equ:bar21bar2MotivicN"} and [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} holds for all positive integers $\ell<k$. ## Inductive proof of [\[equ:bar21bar2MotivicN\]](#equ:bar21bar2MotivicN){reference-type="eqref" reference="equ:bar21bar2MotivicN"} We first show that $$\label{equ:D_rbar21bar2} 2^{3\ell+1}D_r \zeta^{\mathfrak m}(\{\bar2,1\}_k,\bar2)=-\sum_{{\alpha}+{\beta}=k} (-1)^{\alpha}D_r \zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$$ for all positive integers $r=6n+3, 6n+5, 6n+7<3k+2$. ### $r=6n+3$ {#sec:D_6n+3-3a23b} For $r=6n+3$ we have the following picture when $k$ is even (the case of odd $k$ can be dealt with similarly). $$\label{fig:D_6n+3-bar21bar2} \text{ \begin{tikzpicture}[scale=0.9] \node (A1) at (-3.8,0) {$0;$}; \node (A12) at (-3.4,0) {$\bar1,$}; \node (A13) at (-3,0) {$0,$}; \node (A4) at (-2.6,0) {$1,$}; \node (A2) at (-2.2,0) {$1,$}; \node (A3) at (-1.8,0) {$0,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$\bar1,$}; \node (A5) at (-1,0) {$\bar1,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.2,0.7) to (-0.2,-0.7); \node (A7) at (0.2,0) {$\cdots f,$}; \draw[dashed] (0.6,0.7) to (0.6,-0.7); \node (A8) at (0.85,0) {$1,$}; \node (A8) at (1.25,0) {$1,$}; \node (A9) at (1.65,0) {$0,$}; \node (A9) at (2.8,0.04) {$\{\bar1\bar10110\}_n$}; \node (A10) at (4.05,0) {$\bar1,$}; \node (A11) at (4.45,0) {$\bar1,$}; \node (A11) at (4.85,0) {$0,$}; \node (A12) at (5.25,0) {$1,$}; \node (A13) at (5.65,0) {$1,$}; \node (A13) at (6.45,0) {$0,\cdots f,$}; \node (A14) at (7.35,0) {$\bar1,$}; \node (A14) at (7.75,0) {$\bar1,$}; \node (A14) at (8.15,0) {$0;$}; \node (A15) at (8.55,0) {$1\phantom{,}$}; \node (C1) at (2.6,-0.4) {${}$}; \node (D4) at (3,0.6) {${}$}; \node (D3) at (3.4,-0.6) {${}$}; \draw (0.8,-0.25) to (0.8,-0.4) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (4.4,-0.4) to (4.4,-0.325); \draw (1.2,0.25) to (1.2,0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (4.8,0.6) to (4.8,0.25); \draw (1.6,-0.25) to (1.6,-0.6) to (D3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (5.2,-0.6) to (5.2,-0.25); \draw[dashed] (1.8,0.7) to (1.8,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (6.2,0.7) to (6.2,-0.7); \draw[dashed] (5,0.7) to (5,-0.7); \draw[dashed] (7.1,0.7) to (7.1,-0.7); \draw[dashed] (8.3,0.7) to (8.3,-0.7); \node (C1) at (2.5,-1.2) {Possible cuts of $D_{6n+3}\zeta^{\mathfrak m}(\{\bar2,1\}_k,\bar2)$}; \end{tikzpicture}}$$ Note that in general ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ is antisymmetric so that they all disappear, except for the following special case that cannot be canceled: a type ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ cut starts at the initial $0$. Thus $$\begin{aligned} D_{6n+3}\zeta^{\mathfrak m}(\{\bar2,1\}_k,\bar2) =&\, I^{\mathfrak l}(0;10\{\bar1\bar10110\}_n\bar1;\bar1){\otimes}I^{\mathfrak m}(0;10\bar1\bar10\{110\bar1\bar10\}_{k/2-n-1};1) \notag\\ =&\, \zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\zeta^{\mathfrak m}(\{\bar2,1\}_{k-2n-1},\bar2). \label{equ:inductUsebar21bar2}\end{aligned}$$ We note that there are two kinds of cuts for $D_r \zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$: (i) those that don't involve the component $\rho(2)$ and (ii) those that do. $$\label{fig:D_6n+3-3a23bType(i)} \text{ \begin{tikzpicture}[scale=0.9] \node (A13) at (-3,0) {$0;$}; \node (A4) at (-2.6,0) {$1,$}; \node (A2) at (-2.2,0) {$0,$}; \node (A3) at (-1.8,0) {$0,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$1,$}; \node (A5) at (-1,0) {$0,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.3,0.7) to (-0.3,-0.7); \node (A7) at (0.1,0) {$\cdots f,$}; \draw[dashed] (0.5,0.7) to (0.5,-0.7); \node (A8) at (0.7,0) {$1,$}; \node (A8) at (1.1,0) {$0,$}; \node (A9) at (1.5,0) {$0,$}; \node (A9) at (2.7,0.04) {$\{100100\}_n$}; \node (A10) at (4.05,0) {$1,$}; \node (A11) at (4.45,0) {$0,$}; \node (A11) at (4.85,0) {$0,$}; \node (A12) at (5.25,0) {$1,$}; \node (A13) at (5.65,0) {$0,$}; \node (A13) at (6.45,0) {$0,\cdots f,$}; \node (A14) at (7.35,0) {$1,$}; \node (A14) at (7.75,0) {$0,$}; \node (A14) at (8.15,0) {$0;$}; \node (A15) at (8.55,0) {$1\phantom{,}$}; \node (C1) at (3.6,0.4) {${}$}; \node (D2) at (3.2,-0.4) {${}$}; \draw (2,0.25) to (2,0.4) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (5.6,0.4) to (5.6,0.25); \draw (1.5,-0.25) to (1.5,-0.4) to (D2) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (5.2,-0.4) to (5.2,-0.25); \draw[dashed] (1.7,0.7) to (1.7,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (6.25,0.7) to (6.25,-0.7); \draw[dashed] (5.05,0.7) to (5.05,-0.7); \draw[dashed] (7.1,0.7) to (7.1,-0.7); \draw[dashed] (8.35,0.7) to (8.35,-0.7); \node (C1) at (2.5,-1.2) {Type (i) cuts of $D_{6n+3}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$}; \end{tikzpicture}}$$ Consider the picture above for cuts of type (i). Then ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ is always canceled by ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ except when ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ ends at the last 1. Type (i) has only one nontrivial contribution $$\label{equ:type(i)Contr} \zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},2,3_{{\beta}}).$$ For type (ii) cuts shown below $$\label{fig:D_6n+3-3a23b} \text{ \begin{tikzpicture}[scale=0.9] \node (A13) at (-3,0) {$0;$}; \node (A4) at (-2.6,0) {$1,$}; \node (A2) at (-2.2,0) {$0,$}; \node (A3) at (-1.8,0) {$0,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$1,$}; \node (A5) at (-1,0) {$0,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.2,0.7) to (-0.2,-0.7); \node (A7) at (0.2,0) {$\cdots f,$}; \draw[dashed] (0.6,0.7) to (0.6,-0.7); \node (A8) at (0.85,0) {$1,$}; \node (A8) at (1.25,0) {$0,$}; \node (A9) at (1.65,0) {$0,$}; \node (A9) at (2.8,0.04) {$\{1,0,0\}_i,$}; \node (A10) at (4.05,0) {$1,$}; \node (A11) at (4.45,0) {$0,$}; \node (A9) at (5.65,0.04) {$\{1,0,0\}_j,$}; \node (A12) at (6.85,0) {$1,$}; \node (A13) at (7.25,0) {$0,$}; \node (A13) at (8.05,0) {$0,\cdots f,$}; \node (A14) at (8.85,0) {$1,$}; \node (A14) at (9.25,0) {$0,$}; \node (A14) at (9.65,0) {$0;$}; \node (A15) at (10.05,0) {$1\phantom{,}$}; \node (C1) at (2.9,-0.4) {${}$}; \node (D2) at (3.5,0.4) {${}$}; \node (D3) at (2.8,0.6) {${}$}; \node (D4) at (5.0,-0.6) {${}$}; \node (D5) at (5.6,0.6) {${}$}; \draw (0.8,-0.25) to (0.8,-0.4) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (6.0,-0.4) to (6.0,-0.325); \draw (1.2,0.25) to (1.2,0.4) to (D2) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (6.8,0.4) to (6.8,0.25); %\draw (1.6,0.25) to (1.6,0.6) to (D3) node {$\cic{3}$} to (4,0.6) to (4,0.25); \draw (4,-0.25) to (4,-0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (6.0,-0.6) to (6.0,-0.25); \draw (4.4,0.25) to (4.4,0.6) to (D5) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 4$}}}\hskip.25ex{}}$} to (6.8,0.6) to (6.8,0.25); \draw[dashed] (1.8,0.7) to (1.8,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (4.6,0.7) to (4.6,-0.7); \draw[dashed] (6.6,0.7) to (6.6,-0.7); \draw[dashed] (7.9,0.7) to (7.9,-0.7); \draw[dashed] (8.7,0.7) to (8.7,-0.7); \draw[dashed] (9.84,0.7) to (9.84,-0.7); \node (C1) at (3.5,-1.2) {Type (ii) cuts $D_{6n+3}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}), i+j=2n$, starting before $\rho(2)$}; \end{tikzpicture}}$$ we see that ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ contribute to $$\sum_{i+j=2n} \big(\zeta_1^{\mathfrak l}(3_i,2,3_j)-\zeta_1^{\mathfrak l}(3_{j-1},2,3_{i+1}) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-i-1},2,3_{{\beta}-j}).$$ When $j=0$ the second term inside the left factor means $$-\zeta_1^{\mathfrak l}(3_{-1},2,3_{2n+1}) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},2,3_{{\beta}}):=-\zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},2,3_{{\beta}}),$$ which is canceled by [\[equ:type(i)Contr\]](#equ:type(i)Contr){reference-type="eqref" reference="equ:type(i)Contr"}. There are two more special cases when a type (ii) cut starts in $\rho(2)$ (consider $j=2n+1$ in [\[fig:D_6n+3-3a23b\]](#fig:D_6n+3-3a23b){reference-type="eqref" reference="fig:D_6n+3-3a23b"}) $${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}= \big(\zeta^{\mathfrak l}(3_{2n+1})-\zeta_1^{\mathfrak l}(3_{2n},2) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}-2n-1}).$$ Putting all the above together we obtain $$\begin{aligned} D_{6n+3}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})=&\, \sum_{\substack{i+j=2n\\ a-1\ge i\ge 0, b\ge j\ge 1}} \big(\zeta_1^{\mathfrak l}(3_i,2,3_j)-\zeta_1^{\mathfrak l}(3_{j-1},2,3_{i+1}) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-i-1},2,3_{{\beta}-j}) \\ &\, + {\delta}_{{\alpha}\ge 2n+1} \zeta_1^{\mathfrak l}(3_{2n},2) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},2,3_{{\beta}}) \\ &\, +{\delta}_{{\beta}\ge 2n+1} \big(\zeta^{\mathfrak l}(3_{2n+1})-\zeta_1^{\mathfrak l}(3_{2n},2) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}-2n-1}).\end{aligned}$$ Therefore $$\begin{aligned} &\, \sum_{{\alpha}+{\beta}=k} (-1)^{\alpha}D_{6n+3}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ =&\, \sum_{\substack{{\alpha}+{\beta}=k,i+j=2n\\ {\alpha}-1\ge i\ge 0, {\beta}\ge j\ge 1}} (-1)^{\alpha}\big(\zeta_1^{\mathfrak l}(3_i,2,3_j)-\zeta_1^{\mathfrak l}(3_{j-1},2,3_{i+1}) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-i-1},2,3_{{\beta}-j}) \\ &\, +\sum_{{\alpha}+{\beta}=k} (-1)^{\alpha}{\delta}_{{\alpha}\ge 2n+1} \zeta_1^{\mathfrak l}(3_{2n},2) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},2,3_{{\beta}}) \\ &\, +\sum_{{\alpha}+{\beta}=k} (-1)^{\alpha}{\delta}_{{\beta}\ge 2n+1} \big(\zeta^{\mathfrak l}(3_{2n+1})-\zeta_1^{\mathfrak l}(3_{2n},2) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}-2n-1})\\ =&\, \sum_{\substack{i+j=2n,i\ge0,j\ge1\\ {\alpha}+{\beta}=k-2n-1}} (-1)^{a+i+1} \big(\zeta_1^{\mathfrak l}(3_i,2,3_j)-\zeta_1^{\mathfrak l}(3_{j-1},2,3_{i}) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ &\, +\sum_{{\alpha}+{\beta}=k-2n-1}(-1)^{\alpha}\big(\zeta^{\mathfrak l}(3_{2n+1})-2\zeta_1^{\mathfrak l}(3_{2n},2) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})\\ =&\, -2\sum_{\substack{i+j=2n,i\ge0,j\ge1\\ {\alpha}+{\beta}=k-2n-1}} (-1)^{a+i} \zeta_1^{\mathfrak l}(3_i,2,3_j) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ &\, +\sum_{{\alpha}+{\beta}=k-2n-1}(-1)^{\alpha}\big(\zeta^{\mathfrak l}(3_{2n+1})-2\zeta_1^{\mathfrak l}(3_{2n},2) \big) {\otimes}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ =&\,\left(\zeta^{\mathfrak l}(3_{2n+1})-2\sum_{i+j=2n} (-1)^{i} \zeta_1^{\mathfrak l}(3_i,2,3_j) \right) {\otimes}\sum_{{\alpha}+{\beta}=k-2n-1} (-1)^{\alpha}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ =&\,\zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\zeta^{\mathfrak m}(\{\bar2,1\}_{k-2n-1},\bar2)\end{aligned}$$ by inductive assumption. By comparing with [\[equ:inductUsebar21bar2\]](#equ:inductUsebar21bar2){reference-type="eqref" reference="equ:inductUsebar21bar2"} we see that [\[equ:D_rbar21bar2\]](#equ:D_rbar21bar2){reference-type="eqref" reference="equ:D_rbar21bar2"} holds for $r=6n+3$. ### $r=6n+5$ {#sec:D_6n+5-bar21bar2} It is obvious that if $r=6n+5$ then the left-hand side of [\[equ:D_rbar21bar2\]](#equ:D_rbar21bar2){reference-type="eqref" reference="equ:D_rbar21bar2"} vanishes since $\rho(\{\bar2,1\}_k,\bar2)$ has period 6 so that every cut of $D_{6n+5}$ starts and ends with the same number. Similar to the case $r=6n+3$, there are two kinds of cuts for $D_{6n+5}\zeta^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$: (i) those involve the component $\rho(2)$ and (ii) those don't. All cuts in (ii) together with the cut starting at the 0 in $\rho(2)$ clearly vanish by periodicity. For type (ii) cuts we consider the following picture Then we see immediately that cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ with index $(i,j)$ shown as in the picture is canceled by the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ with index $(j,i)$ by path reversal. Note that we allow ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ to start at the number 1 in $\rho(2)=10$ but not after. To summarize, [\[equ:D_rbar21bar2\]](#equ:D_rbar21bar2){reference-type="eqref" reference="equ:D_rbar21bar2"} holds (actually vanishes on both sides) for $r=6n+5$. ### $r=6n+7$ By periodicity, each cut of $D_{6n+7}$ straddling over $\rho(2)$ must start and end with the same number and must vanish. For all the other cut we may use the pictures [\[fig:r=6n+7-bar21\]](#fig:r=6n+7-bar21){reference-type="eqref" reference="fig:r=6n+7-bar21"} and [\[fig:r=6n+7-333\]](#fig:r=6n+7-333){reference-type="eqref" reference="fig:r=6n+7-333"} to conclude that all terms in [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} vanish under the derivation $D_{6n+7}$. In summary, the equation [\[equ:D_rbar21bar2\]](#equ:D_rbar21bar2){reference-type="eqref" reference="equ:D_rbar21bar2"} holds for all odd $r<3\ell$. This completes our inductive proof of [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. ## Inductive proof of [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} We now prove that $$\label{equ:D_rbar21k} 2^{3\ell}D_r \zeta^{\mathfrak m}(\{\bar2,1\}_k)= - D_r \zeta^{\mathfrak m}(3_k)+2\sum_{{\alpha}+{\beta}=k-1} (-1)^{a} D_r \zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$$ for all positive integers $r=6n+3, 6n+5, 6n+7<3k$. ### $r=6n+3$ {#sec:D_6n+3-bar21} For $r=6n+3$ we get the following picture when $k$ is even (the case of odd $k$ can be dealt with similarly) by inserting another 1 to the left of the last 1 and removing the $\bar101$ to the right of the initial 0 in picture [\[fig:D_6n+3-bar21bar2\]](#fig:D_6n+3-bar21bar2){reference-type="eqref" reference="fig:D_6n+3-bar21bar2"}: $$\label{fig:D_6n+3-bar21} \text{ \begin{tikzpicture}[scale=0.9] \node (A4) at (-2.6,0) {$0;$}; \node (A2) at (-2.2,0) {$1,$}; \node (A3) at (-1.8,0) {$0,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$\bar1,$}; \node (A5) at (-1,0) {$\bar1,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.3,0.7) to (-0.3,-0.7); \node (A7) at (0.1,0) {$\cdots f,$}; \draw[dashed] (0.5,0.7) to (0.5,-0.7); \node (A8) at (0.7,0) {$1,$}; \node (A8) at (1.1,0) {$1,$}; \node (A9) at (1.5,0) {$0,$}; \node (A9) at (2.75,0.04) {$\{\bar1\bar10110\}_n$}; \node (A10) at (4.05,0) {$\bar1,$}; \node (A11) at (4.45,0) {$\bar1,$}; \node (A11) at (4.85,0) {$0,$}; \node (A12) at (5.25,0) {$1,$}; \node (A13) at (5.65,0) {$1,$}; \node (A13) at (6.45,0) {$0,\cdots f,$}; \node (A14) at (7.35,0) {$\bar1,$}; \node (A14) at (7.75,0) {$\bar1,$}; \node (A14) at (8.15,0) {$0,$}; \node (A15) at (8.55,0) {$1;$}; \node (A15) at (8.95,0) {$1\phantom{,}$}; \node (C1) at (2.6,-0.4) {${}$}; \node (D4) at (3,0.6) {${}$}; \node (D3) at (3.4,-0.6) {${}$}; \draw (0.65,-0.25) to (0.65,-0.4) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (4.4,-0.4) to (4.4,-0.325); \draw (1.05,0.25) to (1.05,0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (4.8,0.6) to (4.8,0.25); \draw (1.45,-0.25) to (1.45,-0.6) to (D3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (5.2,-0.6) to (5.2,-0.25); \draw[dashed] (1.75,0.7) to (1.75,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (6.25,0.7) to (6.25,-0.7); \draw[dashed] (5.05,0.7) to (5.05,-0.7); \draw[dashed] (7.1,0.7) to (7.1,-0.7); \draw[dashed] (8.35,0.7) to (8.35,-0.7); \node (C1) at (2.25,-1.2) {Possible cuts of $D_{6n+3}\zeta^{\mathfrak m}(\{\bar2,1\}_k)$}; \end{tikzpicture}}$$ Using the same argument as for [\[fig:D_6n+3-bar21bar2\]](#fig:D_6n+3-bar21bar2){reference-type="eqref" reference="fig:D_6n+3-bar21bar2"} we get $$\label{equ:D_6n+3-bar21k} D_{6n+3}\zeta^{\mathfrak m}(\{\bar2,1\}_k) =\zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\zeta^{\mathfrak m}(\{\bar2,1\}_{k-2n-1}).$$ On the other hand, by [\[equ:D_r1bar2\]](#equ:D_r1bar2){reference-type="eqref" reference="equ:D_r1bar2"} $$D_{6n+3} \zeta^{\mathfrak m}(3_k)= \zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(3_{k-2n-1}).$$ Further, for any fixed ${\alpha}+{\beta}=k-1$, $D_{6n+3}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})$ can be computed similarly as in §[5.2.1](#sec:D_6n+3-3a23b){reference-type="ref" reference="sec:D_6n+3-3a23b"} according to whether the cuts involve the component $\rho(2)$ or not. By adding one more 0 at the beginning of pictures [\[fig:D_6n+3-3a23bType(i)\]](#fig:D_6n+3-3a23bType(i)){reference-type="eqref" reference="fig:D_6n+3-3a23bType(i)"} and [\[fig:D_6n+3-3a23b\]](#fig:D_6n+3-3a23b){reference-type="eqref" reference="fig:D_6n+3-3a23b"} we find that there will be one more term of cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ starting at the very first 0 giving rise to the additional term $$\zeta_1^{\mathfrak l}(3_{{\alpha}},2,3_{2n-a}){\otimes}\zeta^{\mathfrak m}(3_{k-2n-1}).$$ Therefore, by induction assumption. $$\begin{aligned} &\, \sum_{{\alpha}+{\beta}=k-1} (-1)^{\alpha}D_{6n+3}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ = &\, \left(\zeta^{\mathfrak l}(3_{2n+1})-2\sum_{i+j=2n} (-1)^{i} \zeta_1^{\mathfrak l}(3_i,2,3_j) \right) {\otimes}\sum_{{\alpha}+{\beta}=k-2n-2} (-1)^{\alpha}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ &\,+\sum_{{\alpha}+{\beta}=2n} (-1)^{\alpha}\zeta_1^{\mathfrak l}(3_{{\alpha}},2,3_{{\beta}}){\otimes}\zeta^{\mathfrak m}(3_{k-2n-1}) \\ =&\, -8^{2n+1}\zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\sum_{{\alpha}+{\beta}=k-2n-2} (-1)^{\alpha}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) +\sum_{{\alpha}+{\beta}=2n} (-1)^{\alpha}\zeta_1^{\mathfrak l}(3_{{\alpha}},2,3_{{\beta}}){\otimes}\zeta^{\mathfrak m}(3_{k-2n-1}).\end{aligned}$$ Thus by induction again $$\begin{aligned} &\, -D_{6n+3}\zeta^{\mathfrak m}(3_k)+2\sum_{{\alpha}+{\beta}=k-1} (-1)^{\alpha}D_{6n+3}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}}) \\ = &\, 8^{2n+1} \zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\left(-2\sum_{{\alpha}+{\beta}=k-2n-2} (-1)^{\alpha}\zeta_1^{\mathfrak m}(3_{{\alpha}},2,3_{{\beta}})\right) +8^{2n+1} \zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\zeta^{\mathfrak m}(3_{k-2n-1}) \\ =&\,8^k \zeta^{\mathfrak l}(\{\bar2,1\}_{2n+1}) {\otimes}\zeta_1^{\mathfrak m}(\{\bar2,1\}_{k-2n-1}).\end{aligned}$$ Comparing to [\[equ:D_6n+3-bar21k\]](#equ:D_6n+3-bar21k){reference-type="eqref" reference="equ:D_6n+3-bar21k"} we see immediately that [\[equ:D_rbar21k\]](#equ:D_rbar21k){reference-type="eqref" reference="equ:D_rbar21k"} holds for $r=6n+3$. ### $r=6n+5$ {#r6n5-1} Exactly the same argument as in §[5.2.2](#sec:D_6n+5-bar21bar2){reference-type="ref" reference="sec:D_6n+5-bar21bar2"} works almost word-for-word, implying that both sides of [\[equ:D_rbar21k\]](#equ:D_rbar21k){reference-type="eqref" reference="equ:D_rbar21k"} vanish if $r=6n+5$. ### $r=6n+7$ {#sec:D_6n+7-bar21} By modifying the pictures [\[fig:r=6n+7-bar21\]](#fig:r=6n+7-bar21){reference-type="eqref" reference="fig:r=6n+7-bar21"} and [\[fig:r=6n+7-333\]](#fig:r=6n+7-333){reference-type="eqref" reference="fig:r=6n+7-333"} we find that [\[equ:D_rbar21k\]](#equ:D_rbar21k){reference-type="eqref" reference="equ:D_rbar21k"} holds for $r=6n+7$. Combining all the results in §[5.3.1](#sec:D_6n+3-bar21){reference-type="ref" reference="sec:D_6n+3-bar21"}-§[5.3.3](#sec:D_6n+7-bar21){reference-type="ref" reference="sec:D_6n+7-bar21"} we see that [\[equ:bar21MotivicN\]](#equ:bar21MotivicN){reference-type="eqref" reference="equ:bar21MotivicN"} holds by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"}. # Proof of [\[equ:12bar1Motivic\]](#equ:12bar1Motivic){reference-type="eqref" reference="equ:12bar1Motivic"} in Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"} {#proof-of-equ12bar1motivic-in-theorem-thmbar21bar2motivic} We finally show that for all $\ell\in\mathbb{N}$, there is some rational number $e_\ell$ such that $$\label{equ:12bar1MotivicN} 2^{3\ell-1}\zeta^{\mathfrak m}(\{1,\bar2\}_\ell,1)= e_\ell\zeta^{\mathfrak m}(3\ell+1)-3\sum_{{\alpha}+{\beta}=\ell, 2\nmid {\beta}}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$$ by using induction on $\ell$. ## The base case If $\ell=1$ then clearly $D_1 \zeta^{\mathfrak m}(1,\bar2,1)=D_1 \zeta^{\mathfrak m}(1,3)=0$ by Lemma [Lemma 8](#lem:D1){reference-type="ref" reference="lem:D1"}. To consider the action of the derivation $D_3$ on [\[equ:12bar1MotivicN\]](#equ:12bar1MotivicN){reference-type="eqref" reference="equ:12bar1MotivicN"} we may look at the following pictures: We find that $D_3\zeta^{\mathfrak m}(1,3)=0$ since there is no nonzero cut, ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}=0$ by anti-symmetry and $$D_3\zeta^{\mathfrak m}(1,\bar2,1)= {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}=I^{\mathfrak l}(0;\bar1\bar10;1){\otimes}I^{\mathfrak m}(0;1;1)=0.$$ Hence by Theorem [Theorem 7](#thm-Glanois){reference-type="ref" reference="thm-Glanois"} there is some $e_1\in\mathbb{Q}$ such that $4\zeta^{\mathfrak m}(1,\bar2,1)=e_1\zeta^{\mathfrak m}(4)-3\zeta^{\mathfrak m}(1,3)$ and thus [\[equ:12bar1MotivicN\]](#equ:12bar1MotivicN){reference-type="eqref" reference="equ:12bar1MotivicN"} holds for $\ell=1$. In fact, by the table of values contained in [@Zhao2016 Appendix E.1], $\zeta(1,3)=-16A-4B$ where $A=\zeta(1,1,\bar2)$ and $B=\zeta(2,\bar2)$. Further, the stuffle regularized value $$\zeta_*(1,\bar2,1)=-2\zeta(1,1,\bar2)-\zeta(2,\bar2)-\zeta(1,\bar3)=-12A-3B.$$ Hence $$\label{equ:1bar21} 4\zeta_*(1,\bar2,1)=-3\zeta(1,3).$$ By applying the period and using [\[equ:1bar21\]](#equ:1bar21){reference-type="eqref" reference="equ:1bar21"} we get $e_1=0$ . ## The inductive step We now assume that [\[equ:12bar1MotivicN\]](#equ:12bar1MotivicN){reference-type="eqref" reference="equ:12bar1MotivicN"} holds for $\ell<k$ for some $k\ge 2$. We will now prove $$\label{equ:D_r1bar2k1} 2^{3\ell-1}D_r\zeta^{\mathfrak m}(\{1,\bar2\}_k,1)=-3\sum_{{\alpha}+{\beta}=k, 2\nmid {\beta}}D_r\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$$ for all positive integers $r=6n+3, 6n+5, 6n+7<3k$. ### $r=6n+3$ {#sec:D_6n+3-1bar2k1} For $r=6n+3$ we get the following picture when $k$ is even (the case of odd $k$ can be dealt with similarly) by inserting another 1 to the right of the initial 0 in picture [\[fig:D_6n+3-bar21\]](#fig:D_6n+3-bar21){reference-type="eqref" reference="fig:D_6n+3-bar21"}: $$\label{fig:D_6n+3bar21} \text{ \begin{tikzpicture}[scale=0.9] \node (A4) at (-3,0) {$0;$}; \node (A4) at (-2.6,0) {$1,$}; \node (A2) at (-2.2,0) {$1,$}; \node (A3) at (-1.8,0) {$0,$}; \draw[dashed] (-1.6,0.7) to (-1.6,-0.7); \node (A4) at (-1.4,0) {$\bar1,$}; \node (A5) at (-1,0) {$\bar1,$}; \node (A6) at (-0.6,0) {$0,$}; \draw[dashed] (-0.3,0.7) to (-0.3,-0.7); \node (A7) at (0.1,0) {$\cdots f,$}; \draw[dashed] (0.45,0.7) to (0.45,-0.7); \node (A8) at (0.7,0) {$1,$}; \node (A8) at (1.1,0) {$1,$}; \node (A9) at (1.5,0) {$0,$}; \node (A9) at (2.75,0.04) {$\{\bar1\bar10110\}_n$}; \node (A10) at (4.05,0) {$\bar1,$}; \node (A11) at (4.45,0) {$\bar1,$}; \node (A11) at (4.85,0) {$0,$}; \node (A12) at (5.25,0) {$1,$}; \node (A13) at (5.65,0) {$1,$}; \node (A13) at (6.45,0) {$0,\cdots f,$}; \node (A14) at (7.35,0) {$\bar1,$}; \node (A14) at (7.75,0) {$\bar1,$}; \node (A14) at (8.15,0) {$0,$}; \node (A15) at (8.55,0) {$1;$}; \node (A15) at (8.95,0) {$1\phantom{,}$}; \node (C1) at (2.6,-0.4) {${}$}; \node (D4) at (2.9,0.6) {${}$}; \node (D3) at (3.3,-0.6) {${}$}; \draw (0.65,-0.25) to (0.65,-0.4) to (C1) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 1$}}}\hskip.25ex{}}$} to (4.4,-0.4) to (4.4,-0.325); \draw (1.05,0.25) to (1.05,0.6) to (D4) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 2$}}}\hskip.25ex{}}$} to (4.8,0.6) to (4.8,0.25); \draw (1.45,-0.25) to (1.45,-0.6) to (D3) node {${}^{\bigcirc\hskip-1.15ex{\raisebox{-0.015cm}{\text{$\scriptscriptstyle 3$}}}\hskip.25ex{}}$} to (5.2,-0.6) to (5.2,-0.25); \draw[dashed] (1.7,0.7) to (1.7,-0.7); \draw[dashed] (3.8,0.7) to (3.8,-0.7); \draw[dashed] (6.25,0.7) to (6.25,-0.7); \draw[dashed] (5.05,0.7) to (5.05,-0.7); \draw[dashed] (7.1,0.7) to (7.1,-0.7); \draw[dashed] (8.35,0.7) to (8.35,-0.7); \node (C1) at (2.25,-1.2) {Possible cuts of $D_{6n+3}\zeta^{\mathfrak m}(\{1,\bar2\}_k,1)$}; \end{tikzpicture}}$$ By the same argument as for [\[fig:D_6n+3-bar21bar2\]](#fig:D_6n+3-bar21bar2){reference-type="eqref" reference="fig:D_6n+3-bar21bar2"} ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}=0$ by anti-symmetry and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}+ {\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}=0$ by path reversal, except for the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$ starting at the initial 0. Hence $$\label{equ:D_6n+3-1bar2k} D_{6n+3}\zeta^{\mathfrak m}(\{1,\bar2\}_k,1) =\zeta^{\mathfrak l}(\{1,\bar2\}_{2n+1}) {\otimes}\zeta^{\mathfrak m}(\{1,\bar2\}_{k-2n-1},1).$$ We note that there are two kinds of cuts for $D_r \zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$: (i) those that don't involve the component $\rho(1)$ and (ii) those that do. By periodicity, each type (ii) cut starts and ends with the same number so that it vanishes. Further, as shown in [\[fig:D_6n+3-3a23bType(i)\]](#fig:D_6n+3-3a23bType(i)){reference-type="eqref" reference="fig:D_6n+3-3a23bType(i)"} type (i) has only one nontrivial contribution $$\label{equ:type(i)Contr313} \zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},1,3_{{\beta}}).$$ Thus $$\begin{aligned} -3\sum_{{\alpha}+{\beta}=k, 2\nmid {\beta}} D_{6n+3}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}}) =&\, \zeta^{\mathfrak l}(3_{2n+1}){\otimes}\left(-3\sum_{{\alpha}+{\beta}=k, 2\nmid {\beta}}\zeta^{\mathfrak m}(3_{{\alpha}-2n-1},1,3_{{\beta}})\right)\\ =&\,2^{3(k-2n-1)-1} \zeta^{\mathfrak l}(3_{2n+1}){\otimes}\zeta^{\mathfrak m}(\{1,\bar2\}_{k-2n-1},1) \\ =&\,2^{3k-1} \zeta^{\mathfrak l}(\{1,\bar2\}_{2n+1}){\otimes}\zeta^{\mathfrak m}(\{1,\bar2\}_{k-2n-1},1)\end{aligned}$$ by inductive assumption and [\[equ:1bar2Motivic\]](#equ:1bar2Motivic){reference-type="eqref" reference="equ:1bar2Motivic"}. Comparing to [\[equ:D_6n+3-1bar2k\]](#equ:D_6n+3-1bar2k){reference-type="eqref" reference="equ:D_6n+3-1bar2k"} we see that [\[equ:D_r1bar2k1\]](#equ:D_r1bar2k1){reference-type="eqref" reference="equ:D_r1bar2k1"} holds for $r=6n+3$. ### $r=6n+5$ {#sec:D_6n+5-1bar2k1} It is obvious that the left-hand side of [\[equ:D_r1bar2k1\]](#equ:D_r1bar2k1){reference-type="eqref" reference="equ:D_r1bar2k1"} vanishes for $r=6n+5$ since $\rho(\{1,\bar2\}_k,1)$ have period 6 so that every cut of $D_{6n+5}$ starts and ends with the same number. Similar to the case $r=6n+3$, there are two kinds of cuts for $D_{6n+5}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$: (i) those involve the component $\rho(1)$ and (ii) those don't. All cuts in (ii) clearly vanish by periodicity. For type (ii) cuts we consider the following picture: We see immediately that cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ with index $(i,j)$ shown as in the picture is canceled by the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ with index $(j,i)$ by path reversal. The two special cases denoted by ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}\,$ with $i=j=2n$ cancel each other by path reversal. Thus both sides of [\[equ:D_r1bar2k1\]](#equ:D_r1bar2k1){reference-type="eqref" reference="equ:D_r1bar2k1"} vanish if $r=6n+5$. ### $r=6n+7$ {#sec:D_6n+7-1bar2k1} By attaching one more 1 to the right end of the picture [\[fig:r=6n+7-bar21\]](#fig:r=6n+7-bar21){reference-type="eqref" reference="fig:r=6n+7-bar21"} and [\[fig:r=6n+7-333\]](#fig:r=6n+7-333){reference-type="eqref" reference="fig:r=6n+7-333"} exactly the same argument shows that $D_{6n+7}\zeta^{\mathfrak m}(\{1,\bar2\}_k,1)=0$ and all cuts for $D_{6n+5}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$ that do not involve $\rho(1)$ cancel each other. For cuts of $D_{6n+5}\zeta^{\mathfrak m}(3_{{\alpha}},1,3_{{\beta}})$ that involve the component $\rho(1)$ we consider the following picture: We see immediately that cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 1$}}}\hskip.25ex{}}\,$ with index $(i,j)$ shown as in the picture is canceled by the cut ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 2$}}}\hskip.25ex{}}\,$ with index $(j,i)$ by path reversal. The two special cases denoted by ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 3$}}}\hskip.25ex{}}\,$ and ${\bigcirc\hskip-1.6ex{\raisebox{-0.0cm}{\text{$\scriptstyle 4$}}}\hskip.25ex{}}\,$ with $i=j=2n$ cancel each other by path reversal. Thus both sides of [\[equ:D_r1bar2k1\]](#equ:D_r1bar2k1){reference-type="eqref" reference="equ:D_r1bar2k1"} vanish if $r=6n+7$. Combining all the results in §[6.2.1](#sec:D_6n+3-1bar2k1){reference-type="ref" reference="sec:D_6n+3-1bar2k1"}-§[6.2.3](#sec:D_6n+7-1bar2k1){reference-type="ref" reference="sec:D_6n+7-1bar2k1"} we see that [\[equ:12bar1MotivicN\]](#equ:12bar1MotivicN){reference-type="eqref" reference="equ:12bar1MotivicN"} holds for $\ell=k$ which is the last statement in Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"}. Therefore we have completed the proof of Theorem [Theorem 1](#thm:bar21bar2Motivic){reference-type="ref" reference="thm:bar21bar2Motivic"}. Ce Xu is supported by the National Natural Science Foundation of China (Grant No. 12101008), the Natural Science Foundation of Anhui Province (Grant No. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057). Jianqiang Zhao is supported by the Jacobs Prize from The Bishop's School. Both authors would like to thank Prof. F. Xu at the Capital Normal University and Prof. C. Bai at the Chern Institute of Mathematics for their warm hospitality. The authors have no competing interests to declare. 99 J. Blümlein, D.J. Broadhurst and J.A.M. Vermaseren, The multiple zeta value data mine, *Comput. Phys. Commun.* **181** (2010), pp. 582--625, J.M. Borwein, D.M. Bradley and D.J. Broadhurst, Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$, *Electron. J. Combin.* **4**(2) (1997), R5. D.J. Broadhurst, On the enumeration of irreducible $k$-fold Euler sums and their roles in knot theory and field theory. arXiv:hep-th/9604128 F. Brown, Mixed Tate motives over Spec($\mathbb{Z}$), *Ann. Math.* **175**(2) (2012), pp. 949--976. P. Deligne, Le groupe fondamental de la $\mathbb{G}_m-{\boldsymbol \mu}_N$, pour $N =$ 2, 3, 4, 6 ou 8 (in French), *Publ. Math. Inst. Hautes Etudes Sci.* **112** (2010), pp. 101--141. C. Glanois, Periods of the motivic fundamental groupoid of, Ph.D. thesis, Université Pierre et Marie Curie, 2016. C. Glanois, Unramified Euler sums and Hoffman $\star$ basis, arXiv:1603.05178 M.E. Hoffman, Multiple harmonic series, *Pacific J. Math.* **152**(1992), pp. 275--290. Z. Jin and J. Li, Motivic multiple zeta values reletive to $\mu_2$, arxiv:1805.02126. T. Murakami, On Hoffman's $t$-values of maximal height and generators of multiple zeta values, *Math. Ann.*, **382** (2022), pp. 421-458. N. Sato and M. Hirose, A memo for an analogous formulas to Zhao's 1-$\bar2$ formula, private email dated June 26, 2019. Z. Shen and H. He, Some identities for multiple alternating zeta values, *J. Number Thy.* **228**(2021), pp. 8--21. D. Zagier, *Values of zeta functions and their applications*, in: First European Congress of Mathematics, Vol. II, pp. 497--512, Birkhauser, Boston, 1994. J. Zhao, On a conjecture of Borwein, Bradley and Broadhurst, *J. Reine Angew. Math.* **639** (2010), pp. 223--233. J. Zhao, *Multiple zeta functions, multiple polylogarithms and their special values*, Series on Number Theory and its Applications, Vol. 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. [^1]: Email: cexu2020\@ahnu.edu.cn [^2]: Email: zhaoj\@ihes.fr

arxiv_math

--- abstract: | With rectangular doubling weight, a generalized Hardy-Littlewood-Sobolev inequality for rectangular fractional integral operators is verified. The result is a nice application of $M$-linear embedding theorem for dyadic rectangles. address: Research and Support Center on Higher Education for the Hearing and Visually Impaired, National University Corporation Tsukuba University of Technology, Kasuga 4-12-7, Tsukuba 05-8521, Japan author: - Hitoshi Tanaka title: The rectangular fractional integral operators --- [^1] # Introduction {#sec1} The purpose of this paper is to demonstrate a nice application of $M$-linear embedding theorem for dyadic rectangles obtained in [@TY] and [@Ta4], and to study a generalized Hardy-Littlewood-Sobolev inequality for rectangular fractional integral operators. For a positive integer $N$, let $0<\alpha<N$. For the rectangular doubling weight $\mu$[^2] on ${\mathbb R}^{N}$, define the rectangular the fractional integral operator $R_{\alpha}^{\mu}$ by $$R_{\alpha}^{\mu}f(x) := \int_{{\mathbb R}^{N}} \mu(R(x,y))^{\frac{\alpha}{N}-1} f(y)\,{\rm d}\mu(y), \quad x\in{\mathbb R}^{N},$$ where $R(x,y)$ stands for the minimal rectangle, with respect to inculusion, which contains two deferent points, up to coordinates, $x$ and $y$ and has their sides parallel to the coordinate axes. We have the following theorem the proof of which is our goal. **Theorem 1**. *For $1<p<q<\infty$ with $\frac1q=\frac1p-\frac{\alpha}{N}$, the generalized Hardy-Littlewood-Sobolev inequality $$\|R_{\alpha}^{\mu}f\|_{L^q(\mu)} \lesssim \|f\|_{L^p(\mu)}$$ holds for all $f\in L^p(\mu)$.* In the case $\mu \equiv 1$ and when we restrict rectangles to cubes, Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is just the Hardy-Littlewood-Sobolev inequality which is one of the most fundamental norm inequality of real variable harmonic analysis. In the case $\mu$ is a doubling weight and when we restrict rectangles to cubes, Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} was studied in Stein's book [@St]. Thanks to the fact that $\mu$ is rectangular doubling weight, the operator $R_{\alpha}^{\mu}$ almost commutes with a multi-parameter family of dilations. The weight theory of such product operators, commuting with a multi-parameter family of dilations, is not developed so much up to now despite a number of pioneering works in the 1980's due to Robert Fefferman and Elias Stein (see [@Fe1; @Fe2; @FS]). The area remains largely open for product fractional integrals (see also [@CXY; @KM; @SZ; @Wa]). The letter $C$ will be used for constants that may change from one occurrence to another. Constants with subscripts, such as $C_1$, $C_2$, do not change in different occurrences. By $A\approx B$ we mean that $c^{-1}B\le A\le cB$ with some positive finite constant $c$ independent of appropriate quantities. We write $X\lesssim Y$, $Y\gtrsim X$ if there is a independent constant $c$ such that $X \le cY$. # $M$-linear embedding theorem for product dyadic cubes {#sec2} In what follows we recall $M$-linear embedding theorem for dyadic rectangles obtained in [@TY] and [@Ta4]. Let $(N_1,N_2,\ldots,N_n)$ be an $n$-tuple of positive integers and let $N=\sum_{i=1}^nN_i$. We decompose the $N$ dimensional Euclidean space ${\mathbb R}^{N}$ by $${\mathbb R}^{N} = \prod_{i=1}^n{\mathbb R}^{N_i}.$$ By $x\in{\mathbb R}^{N}$ we will denote $$x=(x_1,x_2,\ldots,x_n), \quad x_i\in{\mathbb R}^{N_i}.$$ By a *product cube* we will always mean a rectangle $R\subset{\mathbb R}^{N}$ of the form $$R = \prod_{i=1}^nQ_i, \quad Q_i\in{\mathcal Q}({\mathbb R}^{N_i}),$$ where ${\mathcal Q}({\mathbb R}^d)$ stands for the set of all cubes in ${\mathbb R}^d$ with sides parallel to the coordinate axes. We will denote by ${\mathcal R}({\mathbb R}^{N})$ the family of all such product cubes. That is, $${\mathcal R}({\mathbb R}^{N}) := \prod_{i=1}^n{\mathcal Q}({\mathbb R}^{N_i}).$$ We denote by ${\mathcal D\!\mathcal Q}({\mathbb R}^d)$ the family of all dyadic cubes $Q=2^{-k}(m+[0,1)^d)$, $k\in{\mathbb Z},\,m\in{\mathbb Z}^d$. We denote by ${\mathcal D\!\mathcal R}({\mathbb R}^{N})$ the family of all product dyadic cubes on the product space ${\mathbb R}^{N}$, that is, $${\mathcal D\!\mathcal R}({\mathbb R}^{N}) := \prod_{i=1}^n{\mathcal D\!\mathcal Q}({\mathbb R}^{N_i}).$$ By a *weight* we will always mean a nonnegative, locally integrable function on the product space ${\mathbb R}^{N}$ which is positive on a set of positive measure. Given a measurable set $E$ and a weight $\omega$, we will use the following notation: - Denote by $|E|$ the volume of $E$; - The symbol $\omega(E)$ denotes the quantity $\int_{E}\omega(x)\,{\rm d}x$; - The symbol ${\bf 1}_{E}$ stands for the characteristic function of $E$; - The measure of weight ${\rm d}\omega$ is defined by ${\rm d}\omega:=\omega(x)\,{\rm d}x$. Let $1\le p<\infty$ and $\omega$ be a weight. We define the weighted Lebesgue space $L^p(\omega)$ to be a Banach space equipped with the norm $$\|f\|_{L^p(\omega)} := \left(\int_{{\mathbb R}^{N}}|f|^p\,{\rm d}\omega\right)^{\frac1p}.$$ Given $1<p<\infty$, $p'=\frac{p}{p-1}$ denotes the conjugate exponent of $p$. **The condition $({\rm D})$:** We denote by $P_i$, $i=1,2,\ldots,n$, the projection onto the coordinate subspace ${\mathbb R}^{N_i}$. For a product cube $R\in{\mathcal R}({\mathbb R}^{N})$, an integer $j=1,2,\ldots,n$ and a cube $Q\in{\mathcal Q}({\mathbb R}^{N_j})$, we define the product cube $$\langle R;\,Q,j\rangle := \left(\prod_{i=1}^{j-1}P_i(R)\right) \times Q\times \left(\prod_{i=j+1}^nP_i(R)\right),$$ which simply replaces $P_j(R)$ by $Q$. For a cube $Q\in{\mathcal Q}({\mathbb R}^d)$, let ${\mathcal D}(Q)$ be the collection of all dyadic subcubes of $Q$, that is, all those cubes obtained by dividing $Q$ into $2^d$ congruent cubes of half its length, dividing each of those into $2^d$ congruent cubes, and so on. By convention, $Q$ itself belongs to ${\mathcal D}(Q)$. > For all positive number $\varepsilon>0$, we assume that the weight $\sigma$ satisfies *the condition $({\rm D})$*: $$\tag{{\rm D}} > \sum_{Q\in{\mathcal D}(P_j(R))} > \sigma(\langle R;\,Q,j\rangle)^{1+\varepsilon} > \le C_{\sigma,\varepsilon} > \sigma(R)^{1+\varepsilon}$$ holds for all product cubes $R\in{\mathcal R}({\mathbb R}^{N})$ and all integers $j=1,2,\ldots,n$, where the constant $C_{\sigma,\varepsilon}$ depends only on the weight $\sigma$ and the parameter $\varepsilon$. The following $M$-linear embedding theorem for product dyadic cubes is our crucial tool. For the case of dyadic cubes, we refer to a series of works [@HHL; @Hy; @LSU; @NTV; @Ta1; @Ta2; @Ta3; @Tr]. **Theorem 2** ([@Ta4 Theorem 3.1]). *Let ${\mathfrak K}:\,{\mathcal D\!\mathcal R}({\mathbb R}^{N})\to[0,\infty)$ be a map. For $k=1,2,\ldots,M$, let $\sigma_k$ be a weight on ${\mathbb R}^{N}$ that satisfies the condition $({\rm D})$ and let $1<p_k<\infty$ with $\sum_{k=1}^{M}\frac{1}{p_k}>1$. The following statements are equivalent* - *The $M$-linear embedding inequality for product dyadic cubes $$\sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} {\mathfrak K}(R)\prod_{k=1}^{M} \left|\int_{R}f_k\,{\rm d}\sigma_k\right| \le c_1 \prod_{k=1}^{M} \|f_k\|_{L^{p_k}(\sigma_k)}$$ holds for all $f_k\in L^{p_k}(\sigma_k)$, $k=1,2,\ldots,M$;* - *The Fefferman--Phong-type condition $${\mathfrak K}(R) \prod_{k=1}^{M} \sigma_k(R)^{\frac{1}{p_k'}} \le c_2$$ holds for all product dyadic cubes $R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})$.* *Moreover, the least possible constants $c_1$ and $c_2$ are equivalent.* By a simple duality argument for the case $M=2$ we have the following corollary. **Corollary 3**. *Let ${\mathfrak K}:\,{\mathcal D\!\mathcal R}({\mathbb R}^{N})\to[0,\infty)$ be a map. Let $\omega$ and $\sigma$ be the weights on ${\mathbb R}^{N}$ that satisfy the condition $({\rm D})$ and let $1<p<q<\infty$. The following statements are equivalent* - *The weighted norm inequality for rectangular dyadic positive operator $T_{{\mathfrak K}}^{\sigma}$ $$\|T_{{\mathfrak K}}^{\sigma}f\|_{L^q(\omega)} \le c_1 \|f\|_{L^p(\sigma)}$$ holds for all $f\in L^p(\sigma)$. Here, $$T_{{\mathfrak K}}^{\sigma}f(x) := \sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} {\mathfrak K}(R){\bf 1}_{R}(x) \int_{R}f\,{\rm d}\sigma, \quad x\in{\mathbb R}^{N}.$$* - *The Fefferman--Phong-type condition $${\mathfrak K}(R) \omega(R)^{\frac1q} \sigma(R)^{\frac{1}{p'}} \le c_2$$ holds for all product dyadic cubes $R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})$.* *Moreover, the least possible constants $c_1$ and $c_2$ are equivalent.* We have the following corollary too. **Corollary 4**. *Let $0<\alpha<N$. Let $\mu$ be a weight on ${\mathbb R}^{N}$ that satisfies the condition $({\rm D})$ and let $1<p<q<\infty$ with $\frac1q=\frac1p-\frac{\alpha}{N}$. Then, the Hardy-Littlewood-Sobolev inequality for the rectangular dyadic positive operator $T_{\alpha}^{\mu}$ $$\|T_{\alpha}^{\mu}f\|_{L^q(\mu)} \lesssim \|f\|_{L^p(\mu)}$$ holds for all $f\in L^p(\mu)$[^3]. Here, $$T_{\alpha}^{\mu}f(x) := \sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} \mu(R)^{\frac{\alpha}{N}-1}{\bf 1}_{R}(x) \int_{R}f\,{\rm d}\mu, \quad x\in{\mathbb R}^{N}.$$* # Proof of Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} {#sec3} In what follows we will prove Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. ## Rectangular doubling weight {#ssec3.1} We need some definitions and remarks. **Definition 5**. For the cube $Q\in{\mathcal Q}({\mathbb R}^d)$, let $${\mathcal D}^{(1)}(Q) := \{Q'\in{\mathcal D}(Q):\,2\ell(Q')=\ell(Q)\},$$ where, by $\ell(Q)$, we denote the side-lengths of cube $Q$. - We say that a weight $\sigma$ on the product space ${\mathbb R}^{N}$ is a *doubling weight* if there is a constant $\delta>0$ such that $$\sigma(R)\le\delta\sigma(\langle R;\,Q,j\rangle)$$ holds for all product cubes $R\in{\mathcal R}({\mathbb R}^{N})$, all integers $j=1,2,\ldots,n$ and all dyadic cubes $Q\in{\mathcal D}^{(1)}(P_j(R))$. - We say that a weight $\sigma$ on the product space ${\mathbb R}^{N}$ is a *reverse doubling weight* if there is a constant $\gamma>1$ such that $$\gamma\sigma(\langle R;\,Q,j\rangle)\le\sigma(R)$$ holds for all product cubes $R\in{\mathcal R}({\mathbb R}^{N})$, all integers $j=1,2,\ldots,n$ and all dyadic cubes $Q\in{\mathcal D}^{(1)}(P_j(R))$. **Remark 6**. If the weight $\sigma$ is doubling, then it is reverse doubling. Indeed, for any $R\in{\mathcal R}({\mathbb R}^{N})$, any $j=1,2,\ldots,n$ and any $Q\in{\mathcal D}^{(1)}(P_j(R))$, $$\sigma(R) = \sum_{Q'\in{\mathcal D}^{(1)}(P_j(R))} \sigma(\langle R;\,Q',j\rangle) \ge \left(1+\frac{2^{N_j}-1}{\delta}\right) \sigma(\langle R;\,Q,j\rangle) \ge \left(1+\frac{1}{\delta}\right) \sigma(\langle R;\,Q,j\rangle),$$ where we have used $$\sigma(\langle R;\,Q',j\rangle) \ge \frac{\sigma(R)}{\delta} \ge \frac{\sigma(Q)}{\delta}.$$ Conversely, if the weight $\sigma$ is reverse doubling and $\gamma>2^{\max_iN_i}-1$, then it is doubling. Indeed, for any $R\in{\mathcal R}({\mathbb R}^{N})$, any $j=1,2,\ldots,n$ and any $Q\in{\mathcal D}^{(1)}(P_j(R))$, $$\begin{aligned} \sigma(R) &= \sum_{Q'\in{\mathcal D}^{(1)}(P_j(R))} \sigma(\langle R;\,Q',j\rangle) \le \frac{2^{N_j}-1}{\gamma}\sigma(R) + \sigma(\langle R;\,Q,j\rangle) \\ &\le \frac{2^{\max_iN_i}-1}{\gamma}\sigma(R) + \sigma(\langle R;\,Q,j\rangle).\end{aligned}$$ Hence, $$\sigma(R) \le \frac{\gamma}{\gamma+1-2^{\max_iN_i}} \sigma(\langle R;\,Q,j\rangle).$$ **Remark 7**. If $\sigma$ is a reverse doubling weight on the product space ${\mathbb R}^{N}$, then it satisfies condition $({\rm D})$. Indeed, for any product cube $R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})$, any integer $j=1,2,\ldots,n$ and any positive number $\varepsilon>0$, $$\begin{aligned} \sum_{Q\in{\mathcal D}(P_j(R))} \sigma(\langle R;\,Q,j\rangle)^{1+\varepsilon} &= \sum_{k=0}^{\infty} \sum_{\substack{ Q\in{\mathcal D}(P_j(R)) \\ \ell(Q)=2^{-k}\ell(P_j(R)) }} \sigma(\langle R;\,Q,j\rangle)^{1+\varepsilon} \\ &= \sum_{k=0}^{\infty} \sum_{\substack{ Q\in{\mathcal D}(P_j(R)) \\ \ell(Q)=2^{-k}\ell(P_j(R)) }} \sigma(\langle R;\,Q,j\rangle)^{\varepsilon} \sigma(\langle R;\,Q,j\rangle) \\ &\le \sum_{k=0}^{\infty} \left(\frac{1}{\gamma^k}\right)^{\varepsilon} \sigma(R)^{\varepsilon} \sum_{\substack{ Q\in{\mathcal D}(P_j(R)) \\ \ell(Q)=2^{-k}\ell(P_j(R)) }} \sigma(\langle R;\,Q,j\rangle) \\ &= \sigma(R)^{1+\varepsilon} \sum_{k=0}^{\infty} \left(\frac{1}{\gamma^k}\right)^{\varepsilon} \\ &= C_{\gamma,\varepsilon} \sigma(R)^{1+\varepsilon}.\end{aligned}$$ ## The Pérez representation {#ssec3.2} For a number $c>0$ and a product cube $R\in{\mathcal R}({\mathbb R}^{N})$, we will use $cR$ to denote the product cube with the same center as $R$ but with $c$ times the side-lengths of $R$. We define a C. Pérez type representation of fractional integrals (see [@Pe]) by $$\overline{R}_{\alpha}^{\mu}f(x) := \sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} \mu(R)^{\frac{\alpha}{N}-1}{\bf 1}_{R}(x) \int_{3R}f\,{\rm d}\mu, \quad x\in{\mathbb R}^{N}.$$ We now verify the point-wise equivalence $$\label{3.1} R_{\alpha}^{\mu}f(x) \approx \overline{R}_{\alpha}^{\mu}f(x).$$ We first observe that, for $u,v\in{\mathbb R}^d$ with $u \neq v$, the minimal dyadic cube $Q\in{\mathcal D\!\mathcal Q}({\mathbb R}^d)$ such that $Q\ni u$ and $3Q\ni v$ satisfies $$\frac{\ell(Q)}{2}<|u-v|<2\sqrt{d}\ell(Q).$$ We will refer to such a dyadic cube as $Q(u,v)$. This observation, together with the fact that $\mu$ is a doubling weight, enable us to deduce that $$\mu(R(x,y)) \approx \mu(R_0(x,y)),$$ where, $x=(x_1,x_2,\ldots,x_n)\in{\mathbb R}^{N}$, $y=(y_1,y_2,\ldots,y_n)\in{\mathbb R}^{N}$, $x_i \neq y_i$, and $$R_0(x,y) := \prod_{i=1}^nQ(x_i,y_i).$$ The fact that $\mu$ is a reverse doubling weight and a calculus of geometric series also enable us to deduce that $$\sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} \mu(R)^{\frac{\alpha}{N}-1} {\bf 1}_{R}(x){\bf 1}_{3R}(y) \approx \mu(R_0(x,y))^{\frac{\alpha}{N}-1} \approx \mu(R(x,y))^{\frac{\alpha}{N}-1}.$$ This equation and Fubini's theorem yield [\[3.1\]](#3.1){reference-type="eqref" reference="3.1"}. ## The dyadic grid argument {#ssec3.3} For $\tau\in\{0,\pm\frac13\}^d$, the dyadic grid ${\mathcal D\!\mathcal Q}^{\tau}({\mathbb R}^d)$ is defined by $${\mathcal D\!\mathcal Q}^{\tau}({\mathbb R}^d) := \{2^{-k}(m+\tau+[0,1)^d):\, k\in{\mathbb Z},m\in{\mathbb Z}^d\}.$$ **Claim 8**. *We claim that, for any dyadic cube $Q\in{\mathcal D\!\mathcal Q}({\mathbb R}^d)$, there exist $\tau\in\{0,\pm\frac13\}^d$ and $\tau$-shifted dyadic cube $P\in{\mathcal D\!\mathcal Q}^{\tau}({\mathbb R}^d)$ such that $3Q\subset P$ and $\ell(P)=8\ell(Q)$.* *Proof.* We need only verify the one-dimensional case $d=1$. (The claim for $d>1$ holds after $d$ steps.) We may assume further $k=0$. Let $Q=[m,m+1)$, $m\in{\mathbb Z}$. Then $3Q=[m-1,m+2)$. We cover $3Q$ by disjoint dyadic intervals of ${\mathcal D}({\mathbb R})$ with the same length $8$. If $3Q$ is covered by such an interval $P$, then we choose $\tau=0$ and have $P\in{\mathcal D}^{\tau}({\mathbb R})$. We assume that $3Q$ is covered by such two intervals as $P_1\ni(m-1)$ and $P_2\ni(m+2)$. If $|Q\cap P_1|\ge 1.5$, then we choose $\tau=\frac13$ and let $P=\frac83+P_1$. If $|Q\cap P_2|>1.5$, then we choose $\tau=-\frac13$ and let $P=-\frac83+P_2$. This proves the claim. ◻ ## Proof of Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} {#ssec3.4} Using Claim [Claim 8](#clm3.4){reference-type="ref" reference="clm3.4"}, we see that for any product dyadic cube $R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})$ there exist $\tau\in\{0,\pm\frac13\}^{N}$ and $R'\in{\mathcal D\!\mathcal R}^{\tau}({\mathbb R}^{N})$, which is defined by $${\mathcal D\!\mathcal R}^{\tau}({\mathbb R}^{N}) := \prod_{i=1}^n {\mathcal D\!\mathcal Q}^{P_i(\tau)}({\mathbb R}^{N_i}),$$ such that $3R\subset R'$ and $\ell(P_i(R'))=8\ell(P_i(R))$, $i=1,2,\ldots,n$. Moreover, we notice that this correspondence $R \mapsto R'$ becomes *almost bijective*. By letting $$T_{\alpha}^{\mu,\tau}f(x) := \sum_{R\in{\mathcal D\!\mathcal R}^{\tau}({\mathbb R}^{N})} \mu(R)^{\frac{\alpha}{N}-1}{\bf 1}_{R}(x) \int_{R}f\,{\rm d}\mu, \quad x\in{\mathbb R}^{N},$$ since $\mu$ is doubling, $$\begin{aligned} \overline{R}_{\alpha}^{\mu}f(x) &= \sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} \mu(R)^{\frac{\alpha}{N}-1}{\bf 1}_{R}(x) \int_{3R}f\,{\rm d}\mu \\ &\lesssim \sum_{\tau\in\{0,\pm\frac13\}^{N}} T_{\alpha}^{\mu,\tau}f(x).\end{aligned}$$ Applying Corollary [Corollary 4](#cor2.3){reference-type="ref" reference="cor2.3"} to the right-hand side of this inequality, we obtain Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. # Appendix {#appendix .unnumbered} As an appendix, we state the Carleson-type embedding theorem for product dyadi c cubes. The importance of this theorem is that the Fefferman--Phong-type condition simply links the Carleson-type embedding theorem to the $M$-linear embedding theorem. **Lemma 9** ([@Ta4 Lemma 2.2]). *Given a weight $\sigma$ in the product space ${\mathbb R}^{N}$ and $1<p<q<\infty$, the following statements are equivalent* - *The Carleson-type embedding inequality for product dyadic cubes $$\sum_{R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})} \sigma(R)^{\frac{q}{p}} \left(\frac{1}{\sigma(R)}\int_{R}f\,{\rm d}\sigma\right)^q \le c_1 \left(\int_{{\mathbb R}^{N}}f^p\,{\rm d}\sigma\right)^{\frac{q}{p}}$$ holds for all nonnegative functions $f\in L^p(\sigma)$;* - *The testing condition $$\sum_{Q\in{\mathcal D}(P_j(R))} \sigma(\langle R;\,Q,j\rangle)^{\frac{q}{p}} \le c_2 \sigma(R)^{\frac{q}{p}}$$ holds for all product dyadic cubes $R\in{\mathcal D\!\mathcal R}({\mathbb R}^{N})$ and all integers $j=1,2,\ldots,n$.* *Moreover, the least possible constants $c_1$ and $c_2$ enjoy $c_1\le C c_2^n$ and $c_2\le c_1$.* 999 M. Cao, Q. Xue and K. Yabuta, *On multilinear fractional strong maximal operator associated with rectangles and multiple weights*, Rev. 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V. Kokilashvili and A. Meskhi, *Two-weight estimates for strong fractional maximal functions and potentials with multiple kernels*, J. Korean Math. Soc., **46** (2009), no. 3, 523--550. M. Lacey, E. Sawyer and I. Uriarte-Tuero, *Two weight inequalities for discrete positive operators*, Preprint, 2009, arXiv:0911.3437 \[math.CA\]. F. Nazarov, S. Treil and A. Volberg, *The Bellman functions and two-weight inequalities for Haar multipliers*, J. of Amer. Math. Soc., **12** (1999), no. 4, 909--928. C. Pérez, *Sharp $L^p$-weighted Sobolev inequalities*, Ann. Inst. Fourier (Grenoble) **45** (1995), 809--824. Y. Sawano, S. Sugano and H. Tanaka, *Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces*, Trans. Amer. Math. Soc., **363** (2011), no. 12, 6481--6503. E. T. Sawyer and R. L. Wheeden, *Weighted inequalities for fractional integrals on euclidean and homogeneous spaces*, Amer. J. Math., **114** (1992), 813--874. E. T. Sawyer and Z. Wang, *The $theta$-bump theorem for product fractional integrals*, Studia Math., **253** (2020), 109--127. E. M. Stein, *Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals*, Princeton University Press, 1993. H. Tanaka, *A characterization of two-weight trace inequalities for positive dyadic operators in the upper triangle case*, Potential Anal., **41** (2014), no. 2, 487--499. , *The trilinear embedding theorem*, Studia Math., **227** (2015), no. 3, 238--249. , *The $n$ linear embedding theorem*, Potential Anal., **44** (2016), no. 4, 793--809. , *Two-weight norm inequalities for product fractional integral operators*, Bull. Sci Math., **166** (2021) 102940, 1--18. H. Tanaka and K. Yabuta, *The $n$-linear embedding theorem for dyadic rectangles*, Ann. Acad. Sci. Fenn. Math, **44** (2019), 29--39. S. Treil, *A remark on two weight estimates for positive dyadic operators*, In Operator-related function theory and time-frequency analysis, volume 9 of Abel Symp., pages 185--195, Springer, Cham, 2015. arXiv:1201.1455 \[math.CA\]. Z. Wang, *Stein-Weiss inequality on product spaces*, Rev. Mat. Iberoamericana, **37** (2021), no.5, 1641--1667. [^1]: The author was supported by Grant-in-Aid for Scientific Research (C) (19K03510), the Japan Society for the Promotion of Science. Part of this work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. [^2]: For precise definition, see Subsection [3.1](#ssec3.1){reference-type="ref" reference="ssec3.1"} [^3]: *We merely check that $\frac{\alpha}{N}-1+\frac1q+1-\frac1p = \frac{\alpha}{N}+\frac1q-\frac1p =0$.*

arxiv_math

--- abstract: | Renewable energy sources (RES) has become common in modern power systems, helping to address decarbonization and energy security goals. Despite being attractive, RES such as solar and have low inertia and high uncertainty, thus compromising power grid stability and increasing the risk of energy blackouts. Stochastic (chance-constrained) optimization and other state-of-the-art algorithms to optimize and control power generation under uncertainty either explicitly assume the distribution of renewables, or use data-driven approximations. The latter becomes time-consuming and inaccurate, esp. when optimizing over multiple time steps. This paper considers a discrete-time chance-constraint direct current optimal power flow control problem for minimizing power generation costs subjected to power balance and security constraints. We propose an importance-sampling-based data-driven approximation for the optimal automated generation control, which allows to improve accuracy and reduce data requirements compared to state-of-the-art methods. We support the proposed approach theoretically and empirically. The results demonstrate the approach superior performance in handling generation uncertainty, enhancing the stability of renewable-integrated power systems, and facilitating the transition to clean energy. author: - Aleksander Lukashevich, Aleksander Bulkin, Roland Grinis, Ilya Makarov, and Yury Maximov bibliography: - biblio.bib title: Importance Sampling Approach for Dynamic Stochastic Optimal Power Flow Control --- power system control, importance sampling, uncertainty, optimal power flow # Introduction {#sec:introduction} renewable energy sources (RES) aligns with the United Nations sustainable development goals, promoting affordable and clean energy while enhancing energy security and resilience. But the integration of RES introduces significant uncertainty in power systems generation, posing challenges to existing grid optimization and control policies. The Optimal Power Flow (OPF) [@stott2012optimal] is a fundamental power optimization problem that aims to achieve economically optimal generation while adhering to grid security and power balance constraints. To account for generation uncertainty, the (joint) chance-constrained extension considers an unknown (joint) distribution of renewable energy sources [@geng2019data; @bienstock2014chance]. Alternatively, a robust approach, assuming bounded uncertainty, offers a more conservative solution in practice [@sousa2010robust; @ben2002robust; @ding2016adjustable]. The discrete-time dynamic chance-constrained optimal power flow problem [@lou2019multi; @capitanescu2007improving; @monticelli1987security] models optimal generation set-points for sequential timestamps, temporarily binding generator states through ramp-up and ramp-down constraints. For fast and efficient power dispatch in bulk power systems, the automatic generation control (AGC) is commonly employed [@xu2017real]. While the chance-constrained extension enhances flexibility in modeling uncertainty, solving it for an arbitrary distribution becomes computationally infeasible [@nemirovski2012safe; @jia2021iterative]. To overcome this, data-driven approximations such as Scenario Approximation (SA) [@calafiore2006scenario] and Sample Average Approximation (SAA) [@ahmed2008solving; @sen1992relaxations] have proven successful. Chance-constrained formulations offer a more flexible approach to model uncertainty, allowing control over the confidence level of solution feasibility. These formulations can be represented as Joint Chance Constraints (JCC) or Single Chance Constrained (SCC) forms. While analytical solutions exist for the SCC type, they are often conservative due to the sub-additivity of the probability measure. In contrast, JCC does not allow for an analytical formulation, necessitating the use of approximations like Scenario Approximation (SA) [@calafiore2006scenario] or Stochastic Average Approximation (SAA) [@ahmed2008solving; @sen1992relaxations]. Unfortunately, these approaches become computationally prohibitive when higher accuracy approximations are required. This compromises power system operation security practices and hinders the clean energy transition. To address this challenge, this paper proposes an importance-sampling-based scenario approximation for the discrete-time optimal power flow problem. The approach significantly reduces the sample requirements compared to baseline methods, while improving accuracy. The contribution of this paper is two-fold: - propose an importance-sampling-based scenario approximation for the discrete-time optimal power flow problem, requiring nearly half the samples of the baseline method. - demonstrate the state-of-the-art empirical performance of our approach across various IEEE test cases. The rest of this paper is organized as follows. Section [2](#sec:setup){reference-type="ref" reference="sec:setup"} provides background, notation, and problem setup. Section [3](#sec:prob){reference-type="ref" reference="sec:prob"} presents the mathematical background of the problem. In Section [4](#sec:emp){reference-type="ref" reference="sec:emp"}, we conduct an empirical study, and finally, Section [5](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} concludes the paper. # Background and Problem Setup {#sec:setup} The higher-voltage direct current (DC) model is a widely used load flow model in power systems. Its simplicity for analysis arises from the linear relationships between power injections and phase angles. To describe the model, we consider a power grid given by a graph $G = (V, E)$, where $V$ is the set of $n$ nodes (buses) and $E$ is the set of $m$ lines (edges). Let $p_g \in \mathbb{R}^{n_g}$, $p_d \in \mathbb{R}^{n_d}$, and $\theta\in\mathbb{R}^{n}$ be vectors representing power generations, power demands, and phase angles, respectively. Additionally, we define the vector of power injections as $p_{inj} \in \mathbb{R}^{n}$, where $p_{inj}^i$ denotes the power injection at bus $i$, calculated as the difference between the power generation $p^i_g$ and the power demand $p^i_d$ at that bus. If there is no generation or demand at a bus $i$, either $p^i_g$ or $p^i_d$ is zero, respectively. The power system is balanced, so the sum of all power injections is zero, $\sum_{i \in V} p^i_{inj} = 0$. To maintain clarity, we designate one bus as the slack bus, with its phase angle set as $\theta_s = 0$. The power injection vector $p_{inj}$ is related to the phase angle vector $\theta$ through an admittance matrix $B \in \mathbb{R}^{n \times n}$ as $p_{inj} = B\theta$. The components of $B$, denoted as $B_{ij}$, are non-zero if there is a line between buses $i$ and $j$. For each node $i$, $B_{ii}$ is defined as the negative sum of the off-diagonal elements $B_{ij}$ with $j \neq i$, resulting in a Laplacian matrix structure. The DC power flow equations and the associated constraints are expressed as follows: [\[eq:DC-PF\]]{#eq:DC-PF label="eq:DC-PF"} $$p_{inj} = B \theta \label{eq:DC-PF-a}$$ $$\underline{p}^i_g \leq p^i_g \leq \overline{p}^i_g, i \in V \text{ and } |\theta^i - \theta^j| \leq \bar{\theta}^{ij}, \; (i,j)\in E. \label{eq:DC-PF-b}$$ $$\sum_{i=1}^{n_g}p_g^i = \sum_{i=1}^{n_d}p^i_d \label{eq:DC-PF-c}$$ These equations represent the DC power flow and enforce generation and reliability constraints. The reliability constraints [\[eq:DC-PF-b\]](#eq:DC-PF-b){reference-type="eqref" reference="eq:DC-PF-b"} define a polytope $P$ in the space of power generations, where $P$ is given by $P = {p_g: W p_g \leq b}$. Violations of reliability constraints occur when power injections $p_g$ do not belong to this polytope $P$. The matrices $W \in \mathbb{R}^{J \times n_g}$ and vector $b \in \mathbb{R}^J$ can be derived from the set of DC power flow equations. We leave out relevant bulky equations to meet the paper lenght requirements. The objective is to solve the power flow problem by minimizing a convex cost function $c(p)$ subject to the constraints specified in Eqs. [\[eq:DC-PF-a\]](#eq:DC-PF-a){reference-type="eqref" reference="eq:DC-PF-a"} and [\[eq:DC-PF-b\]](#eq:DC-PF-b){reference-type="eqref" reference="eq:DC-PF-b"}. The function $c(p)$ captures the cost associated with the power flow generation in the system. Table [1](#tab:notation){reference-type="ref" reference="tab:notation"} summarizes the paper notation. We use upper indices for elements of vectors and matrices, lower-case letters for probability density functions (PDFs), and upper-case letters for cumulative distribution functions (CDFs). Once it does not lead to confusion, we use $\mathbb{P}$, $\mathbb{E}$, $\mathbb{V}$ for probability, expectation, and variance without explicitly mentioning a distribution. ------------------------------------------ -------------------------------------------------------------------------------------------------------------------------- ------------------------------------------ ------------------------------------------------------------------------- $\mathcal{F}$ Feas. set of CC Prob. [\[eq:optimal_control\]](#eq:optimal_control){reference-type="ref" reference="eq:optimal_control"} $n_g$ number of generators ${P}$ reliability set, $p\!:\! W p \!\le\! b\!\!\!\!$ $n_d$ number of demand nodes $E$ set of lines, $|E| = m$ $D_i$ $p\!\sim\!{\cal N}(\!\mu, \!\Sigma)$ s.t. $p^\top\! \omega^i \!> \!b_i$ $V$ set of buses, $|V| = n$ $\pi$ feasibility prob., $\!p\in\! P$ $B$ $n\times n$ admittance matrix $\chi$ DC-OPF variables, $(\theta, p_g)$ $\!p_g^i,p^i$ power gen. at bus $i$ in the mixture, $x\!\!\in\!\!X\!\!\subseteq\! \mathbb{R}^J$ $\!\!\underline{p}^i_g,\overline{p}^i_g$ lower/upper gen. limit $N$ number of samples $D$ mixture distribution $\mathbb{P},\mathbb{E}, \mathbb{V}$ prob., expectation, variance $R$ ramp-up/down limits ${\cal N}\!(\mu, \!\Sigma)\hspace{-5mm}$ Gauss. distribution with $\xi^t$ total uncertainty at time $t$ mean $\mu$, covariance $\Sigma$ $\theta_i$ phase angle at bus $i$ $f_D$ density of the mixture $\theta_{ij}$ phase angle difference $\Phi$ ${\cal N}(0,1)$ distribution CDF $\bar\theta_{ij}$ angle difference limits $U(0,1)$ uniform $(0,1)$ distribution $I_n$ $n\times n$ identity matrix $B$ imaginary of admitt. mat. $J$ number of constraints $c(p)$ convex cost function $D_i$ Gaussian dist. $w_i$ weight of $D_i$ conditioned on plane $i$ $\alpha$ gen. participation factors ------------------------------------------ -------------------------------------------------------------------------------------------------------------------------- ------------------------------------------ ------------------------------------------------------------------------- : Paper notation. [\[tab:notation\]]{#tab:notation label="tab:notation"} # Optimal control under uncertainty {#sec:prob} In this section, we formulate the multistage chance-constrained DC-OPF. We begin by describing the fluctuations that introduce uncertainty into the system, their origins, and their significance in control and optimization. Next, we address the concept of Automated Generation Control (AGC). Further, we present the chance-constrained optimization problem. Accordingly, we introduce and analyze the Scenario Approximation (SA) for the chance constrainted control. Finally, we address and define the redundant scenarios for this problem and present a procedure to efficiently sample them. ## Fluctuations and AGC {#sec:fluctuations} Power systems fluctuations typically occur both on the generation and demand side and arise from the intermittency of renewable energy generation, unstable demand from the customers, and intra-day electricity trading. Often the distribution of such fluctuations can be recovered based on historical data [@roald2017chance; @owen2019importance]. Below we assume that the fluctuations are Gaussian with the mean and covariance recoverable from historical data [@anvari2016short; @roald2017chance]. Next, we address the fluctuations impact the feasibility set of the DC-OPF, see Eq. [\[eq:DC-PF\]](#eq:DC-PF){reference-type="eqref" reference="eq:DC-PF"}. These fluctuations are involved the power balance equation [\[eq:DC-PF-c\]](#eq:DC-PF-c){reference-type="eqref" reference="eq:DC-PF-c"} and typically managed through primary and secondary control [@machowski2020power]. In this paper, we consider linear Automatic Generation Control (AGC). In our model, the generation and demand fluctuate as $(p_g^i)^t + (\xi^i_g)^t$ at bus $i=1, \dots, n_g$ and $p_d^j + (\xi^i_d)^t$ at node $j=1,\dots, n_d$ resp. As we assume that the initial generation is balanced, i.e., $\sum_{i=1}^{n_g} (p_g^i)^0 = \sum_{i=1}^{n_d} p_d^i$, the purpose of AGC is to balance the aggregated uncertainty term $\xi^t = \sum_{i=1}^{n_g}(\xi^i_g)^t-\sum_{i=1}^{n_d}(\xi^i_d)^t,$ where $\xi^t$ is distributed as the sum of all the nodes' uncertainties. For example, if the uncertainties are independent and identically distributed (i.i.d) Gaussians with mean $\mu = 0$ and some variance $\sigma^2$, then $\xi^t$ follows a Gaussian distribution $\mathcal{N}\left(0, (n_g+n_d) \cdot \sigma^2 \right)$. Following [@roald2017chance; @baros2021examining; @mezghani2020stochastic], the AGC recourse brings the generation to a new setpoint $p^{t+1}_g = p^t_g + \alpha \xi^t$. In the latter the participation factors $\alpha \in \mathbb{R}^{n_g}$ satisfy $~\alpha \geq 0, ~ \boldsymbol{1}^\top \alpha = 1$. It is easy to check that if the system is initially balanced, then the new generation set-point also ensures a balanced system. The power mismatch at timestamp $t=1, \dots, T$ is given by: $$\begin{gathered} \label{eq:power-balance} {\mathbf 1}^\top p^t - {\mathbf 1}^\top p_d - \xi^t = {\bold 1}^\top p^{t-1} - {\mathbf 1}^\top p_d + \xi^t - \xi^t = 0.\end{gathered}$$ i.e., this control strategy keeps the system balanced. Notice that $\xi^t$ represents the overall fluctuation, including load and generation uncertainties, and $p_d$ remains constant in time. ## Chance constrained multi-stage control We consider an uncertain discrete-time dynamical system with $T$, $T<\infty$, timestamps, and $\xi^t$, $1 \le t\le T$, total power mismatch due to uncertainties. Below, we use $p$ instead of $p_g$ to simplify the notation. We assume that $\xi^t$ are distributed normally and represent the total power mismatch in the system at timestamp $t$. Below we consider the linear Automatic Generation Control (AGC) [@de1983automatic]: $p^t = p^{t-1} + \alpha \xi^t$, where $\alpha \in \mathbb{R}^{n_g}$ is a vector of the participation factors $\alpha_k~\ge~0$, $\sum_{1\le k \le n} \alpha_k = 1$, and $\xi^t$ is a one dimensional random variable representing total disbalance at time step $t$. We assume that for balancing the system generator $i$ contributes proportionally to its participation factor $\alpha_i$ for each $t, 1\le t\le T$. We would use vector and scalar notation interchangeably later in the paper if it does not lead to confusion. We assume the power system is balanced at the initial time step $t=0$, i.e., $\mathbf{1}^\top p^0 = \mathbf{1}^\top p_d$. Thus, with the linear AGC the system remains balanced at each timestamp, as it was shown at the end of Section [3.1](#sec:fluctuations){reference-type="ref" reference="sec:fluctuations"}. We assume uncertainties being uncorrelated in time and follow the Gaussian distribution, $\xi^t \sim \mathcal{N}(0, (\sigma^t)^2)$, $1\le t\le T$. In practice, the ramp rates of generators are limited as $$\begin{gathered} |p^t_i - p^{t-1}_i| \leq R_i, \qquad R_i > 0.\label{eq:ramp_up-down}\end{gathered}$$ The discrete-time dynamic chance-constrained optimization problem is as follows: $$\begin{aligned} & \qquad \min_{p^t, \alpha} \mathbb{E} \sum_{t=1}^T c(p^t) \\ & \texttt{subject to:} \\ & \; \mathbb{P} \begin{pmatrix} Wp^t \leq b, 1\le t\le T \\ p^t = p^{t-1} + \alpha\xi^t, 1\le t\le T \\ |p_k^t - p_k^{t-1}| \leq R_k, 1 \leq k \leq n_g \end{pmatrix} \geq 1 - \eta. \end{aligned} \label{eq:optimal_control}$$ where $0 < \eta < 1$, $\mathbb{P}$ is a joint measure induced by the uncertainty distribution, and $\alpha$ is a set of participating factors. An equivalent formulation to the problem above is $$\begin{aligned} & \qquad \min_{p^t, \alpha} \mathbb{E} \sum_{t=1}^T c(p^t) \\ & \texttt{subject to:} \\ & \; \mathbb{P} \begin{pmatrix} Wp_0 + W \alpha\sum_{t\le \tau} (\xi^t)\leq b, 0 \le \tau\le T \\ %p^t = p^{t-1} + \xi^t + D(\alpha) \delta p^t, 1\le t\le T \\ |\alpha_k \xi^t| \leq R_k, \le 1\le k\le n_g \end{pmatrix} \geq 1 - \eta. \end{aligned} \label{eq:optimal_control_2}$$ where for $t = 0$ we assume no uncertainty, i.e, $\xi^0 = 0$. ## Scenario approximation of chance constrained control A (convex) Scenario Approximation to the problem above is $$\begin{aligned} & \qquad \min_{p^0t, \alpha} c(p^0) \\ & \hspace{-4mm}\texttt{subject to:} \\ & \; Wp_0 + W \alpha\sum\limits_{t\le \tau} \xi^t(j)\leq b, 0 \le \tau \le T \\ %p^t = p^{t-1} + \xi^t + D(\alpha) \delta p^t, 1\le t\le T \\ & |\alpha_k\xi^t(j)| \leq R_k, \le 1\le k\le n_g %\end{pmatrix} \geq 1 - \eta. \end{aligned} \label{eq:optimal_control_sampling_02}$$ with scenarios $\xi^t(j) \sim {\cal N}(0, (\sigma^t)^2)$, $1\le j \le N$. We omit expectation in the cost function since for typical cost functions in power systems (quadratic and linear, with diagonal matrix in quadratic term) the cost remains the same (in terms of $p^0$) after taking expectation, provided the uncertainty has zero mean. Scenario approximation (SA) is very attractive from the practical perspective but requires an extreme number of samples to get reasonable accuracy [@calafiore2006scenario]. Importance sampling helps to reduce the SA computational burden [@lukashevich2021importance; @lukashevich2021power; @mitrovic2023gp] in chance-constrained optimization; however, it has not yet been studied for dynamic power flow optimization problems. The importance sampling approach we leverage in this paper consists of two steps. First, we derive a conservative outer approximation to the set of optimal solutions of Problem [\[eq:optimal_control_2\]](#eq:optimal_control_2){reference-type="eqref" reference="eq:optimal_control_2"}. Based on this, we add a sequence of deterministic constraints to eliminate redundant scenarios from the scenario approximation. Finally, instead of using vanilla Monte-Carlo, we sample from a proxy (importance) distribution having less redundant scenarios. This approach improves the sample complexity and reliability [@lukashevich2021importance]. The paper extends the importance sampling based scenario approximation to discrete-time dynamic chance-constrained optimization and verifies the empirical performance of this approach over the optimal power flow problem. ## Redundant scenarios We now formulate the polytope that contains the samples that are not able to drive the system out of the safe operating zone during $T$ snapshots. We will use the notion of a *redundant* scenario. **Definition 1**. *Let $\vec{\xi}^t(j), ~j=1, \dots, N$, be scenarios drawn from $\mathcal{N}(0, \texttt{diag}((\sigma^1)^2, \dots, (\sigma^T)^2))$. A scenario $\vec{\xi}(j)$ is called redundant if the solution of a scenario approximation remains the same even when it is excluded from the latter.* Further, we develop an approximation of the set of redundant scenarios for the current optimal control problem. The participation factors $\alpha$ for secondary control are often fixed, enabling long-term grid stability and fast corrective secondary control [@machowski2020power]. However, in practice, small deviations from the long-term optimal point $\alpha^0$, where $|\alpha - \alpha^0|_2 \leq \delta\alpha$, are economically advantageous. Based on that assumption, we derive the proxy distribution for efficient scenario generation. **Lemma 1**. *Let AGC control be $p^t = p^{t-1} + \alpha\xi^{t}$. Then $p^t - p^{t-1}$ is either in the positive or negative orthant.* *Proof.* Recall that $\{\alpha: \sum_{k=1}^{n_g} \alpha^\top \mathbb{1} = 1, \alpha \geq 0\}$. Thus, since $\xi^t$ is a scalar, $\alpha \xi^t$ has either all positive or all negative components. ◻ The generation adjustments $\alpha \xi^t$ can only either decrease or increase all current powers $p^{t-1}$ as illustrated in Fig. [1](#fig:importance_sampled_vs_mc){reference-type="ref" reference="fig:importance_sampled_vs_mc"}. Moreover, since the variance of $\sum_{\tau\le t}\xi^\tau$ grows in time as shown in Fig. [\[fig:alpha_and_poly_snapshots\]](#fig:alpha_and_poly_snapshots){reference-type="ref" reference="fig:alpha_and_poly_snapshots"}, the probability of failure for each operating point is increasing. {width=".6\\textwidth"} {width=".6\\textwidth"} {#fig:importance_sampled_vs_mc width=".6\\textwidth"} {#fig:t=1 width=".8\\linewidth"} {#fig:t=2 width=".8\\linewidth"} {#fig:t=3 width=".8\\linewidth"} {#fig:t=4 width=".8\\linewidth"}       Our next step is to derive an upper bound for the failure probability of a control strategy ${(\xi^1, \dots, \xi^T), \alpha}$ with a given starting point $p^0$. To achieve this, we utilize the following bounds on the probability of $\mathbb{P}(S_1 \cup \dots \cup S_k)$ for some arbitrary sets $S_1, \dots, S_k:$ $\max_{i\le k} \mathbb{P}(S_i) \le \mathbb{P}(S_1 \cup \dots \cup S_k) \le \sum_{i\le k} \mathbb{P}(S_i).$ Let us consider the probability that all system states are feasible with a prescribed probability at least $1-\eta$: $$\begin{aligned} \pi = \mathbb{P}\left(\forall k, t: \omega_k^\top p^t \leq \beta_k, \; k\le n, t\le T\right), %1 - \mathbb{P}(\omega^\top\sum_{t\le T}\sum_{k\le n} \xi_k^t (1-\alpha_k) < \beta - \omega^\top p^0)\\\end{aligned}$$ note that $\pi \le 1 - \max_{k\le n,t\le T}\mathbb{P}(\omega_k^\top p^t > \beta_k)$ and $\mathbb{P}\left(\forall i, t: \bigcup_i \omega_i^\top p^t > \beta_i\right) \le 1 - \max_{i, t}\mathbb{P}(\omega_i^\top p^t > \beta_i)$ with $p^t\sim {\cal N}(p^0, \texttt{diag}((\sigma^1)^2, \dots, \sum_{t=1}^{T-1}(\sigma^\tau)^2))$. Now, we have the upper bound $\pi$ greater or equal than $1 - \eta$ we get $$\max_{i, t}\mathbb{P}(\omega_i^\top p^t > \beta_i) \leq \eta. \label{eq:necessary_JCC}$$ We use this necessary feasibility condition to get the outer approximation of the Joint Chance Constraint (JCC) in Eq. [\[eq:optimal_control_2\]](#eq:optimal_control_2){reference-type="eqref" reference="eq:optimal_control_2"}. We will now derive the *outer* approximation of the JCC feasibility set based on the necessary condition. It is important to note that if there exists a timestamp $t$ in the range of $1$ to $T$ and a plane $\omega_i$ in the range of $1$ to $J$, such that the necessary feasibility condition [\[eq:necessary_JCC\]](#eq:necessary_JCC){reference-type="eqref" reference="eq:necessary_JCC"} does not hold, it implies that the entire JCC, Prob. [\[eq:optimal_control_2\]](#eq:optimal_control_2){reference-type="eqref" reference="eq:optimal_control_2"}, cannot be satisfied. Thus, assuming that for some $t, i$ [\[eq:necessary_JCC\]](#eq:necessary_JCC){reference-type="eqref" reference="eq:necessary_JCC"} is violated and recalling that $p^{t} = p^0 + \alpha \zeta^{t}, ~ \zeta^t = \sum_{\tau=1, \dots t-1} \xi^\tau, ~ \zeta^t \sim \mathcal{N}(0, (\tilde{\sigma}^t)^2), \textup{ with } (\tilde{\sigma}^t)^2 = \sum_{\tau=1}^{t-1} (\sigma^{\tau})^2$, we get for a plane with $\omega_i^\top \alpha \neq 0$: $\mathbb{P}\left( \omega_i^\top p^0 + \omega_i^\top \alpha \zeta^{t} > \beta_i\right) > \eta, \Longrightarrow \Phi\left(\frac{\omega_i^\top p^0 - \beta_i}{\tilde{\sigma}^t \| \omega_i^\top \alpha \|}\right) > \eta.$ The latter expression defines a complement of the polytope $\mathcal{P}_{out} = \left\{ p: \omega_i^\top p \leq \beta_i - \Delta_i^t \| \omega_i^\top \alpha \|\right\}$, where $\Delta_i^t = \Phi^{-1}(1 - \eta) \tilde{\sigma}^t$. Essentially, $\mathcal{P}_{out}$ defines a necessary feasibility polytope for the original chance-constrained feasibility set. Note that those hyperplanes that are orthogonal to the AGC participation factors vector $\alpha$, $\omega_i^\top \alpha = 0$, cannot be violated as a result of the AGC action. Moreover, right-hand sides of the inequalities decrease from deterministic $\beta_i$ as $\tilde{\sigma}^t = \sqrt{\sum_{\tau=1}^{t-1} (\sigma^\tau)^2}$ grows. Figure [1](#fig:importance_sampled_vs_mc){reference-type="ref" reference="fig:importance_sampled_vs_mc"} illustrates the mutual geometry between initial feasibility polytope and necessary feasibility polytope at $t > 1$. **Remark 1**. *Additionally, the same reasoning applies to the ramp-up and ramp-down constraints $|\alpha_k \xi^t| \leq R_k$. In this case, the logic remains unchanged for the hyperplanes $\alpha_i\xi^t \leq R_i$. It's important to note that $|\alpha_k \xi^t| \leq R_k$ is equivalent to the simultaneous satisfaction of $\alpha_k \xi^t \leq R_k$ and $-\alpha_k \xi^t \leq R_k$.* To derive efficient samples for scenario approximation considering AGC recourse and constraint satisfaction, we need to impose conditions on sample redundancy. At timestamp $t$, we have the condition $\omega_t^\top p^0 + \omega_i^\top \alpha \zeta^t \leq \beta_i$. If a scenario $\zeta^t(j)$ satisfies $\omega_i^\top \alpha \zeta^t(j) \leq \Phi^{-1}(1 - \eta) \| \omega_i^\top \alpha \| \tilde{\sigma}^t$, it implies that the initial generation should satisfy $\omega_i^\top p^0 \leq \beta_i - \omega_i^\top \alpha \zeta^t(j)$. However, this right-hand side is looser than $\beta_i - \Phi^{-1}(1 - \eta) \| \omega_i^\top \alpha \| \tilde{\sigma}^t$, meaning that even the necessary feasibility condition would not be satisfied. Thus, the scenario $\zeta^t(j)$ with $\omega_i^\top \alpha \zeta^t(j) \leq \Phi^{-1}(1 - \eta) \| \omega_i^\top \alpha \| \tilde{\sigma}^t$ is redundant. To get efficient samples for scenario approximation, one must sample $\zeta^t(j)$ outside of the polytope $\mathcal{P}_{r} = \left\{ (\zeta^1, \dots, \zeta^T)^\top: ~ \omega_i^\top \alpha \zeta > \Phi^{-1}(1 - \eta) \| \omega_i^\top \| \tilde{\sigma}^t , ~ \forall~ t,~ \forall~ i\right\}$. The polytope can equally be described as $\mathcal{P}_{r} = \left\{ \mathbf{\zeta}: ~ \mathcal{W} \mathbf{\zeta} \leq \Phi^{-1}(1-\eta) \mathcal{S}\right\}$. Here $\mathcal{W} \in \mathbb{R}^{(J \cdot T) + (2 \cdot n_g \cdot T)\times T}$ is a block diagonal matrix. The $j^{\textup{th}}$ block of the matrix is of a shape $T \times T$ and $\mathcal{W}_j = \mathop{\mathrm{sign}}(\omega_k^\top \alpha) \cdot I$ where $j \in 1 + (k - 1) \cdot T, \dots, k \cdot T$, $k=1, \dots, J$. The other blocks correspond to the ramp-up/down constraints. They are as follows: $\mathcal{W}_j=I$ for $j\in J\cdot T + 1 + (k-1) \cdot T, \dots J\cdot T + 1 + k \cdot T$ and $k=1, \dots, n_g$. The remaining blocks of the matrix are the same except the sign and correspond to the ramp-down constraints. The vector on right-hand side is $\mathcal{S} = (\mathcal{S}_{\mathcal{W}}, \mathcal{S}_{ramp})$, where $\mathcal{S}_{\mathcal{W}}= (\tilde{\sigma}^1, \dots, \tilde{\sigma}^T, \dots, \tilde{\sigma}^1, \dots, \tilde{\sigma}^T)$ and $\mathcal{S}_{ramp} =\left[\frac{R_1}{\alpha_1 \sigma^1}, \dots, \frac{R_{n_g}}{\alpha_{n_g} \sigma^1}, \dots, \frac{R_1}{\alpha_1 \sigma^\top}, \dots, \frac{R_{n_g}}{\alpha_{n_g} \sigma^\top}\right],$ $\mathcal{S} \in \mathbb{R}^{J \cdot T + 2 \cdot n_g \cdot T}$. ## Sampling non-redundant scenarios {#sec:sampling} In order to sample non-redundant scenarios, we approximate their distribution with a mixtre of distributions $D$ that has the density $f_D(x) = \sum_{i=1}^{|\mathcal{S}|}w_i f_{D_i} (x).$ Here $|\mathcal{S}| = J \cdot T + 2 \cdot n_g \cdot T$, mixture components' weights are positive and $\sum_{i} w_i = 1$, $f_{D_i}$ are mixture components densities. The mixture components are Gaussians conditioned outsides of the plane $i$: $x \sim D_i \Longleftrightarrow \mathcal{N}(0, \Sigma_{\zeta}), ~ \mathcal{W}_i^\top x > \mathcal{S}_i$, where $\Sigma_{\zeta}=\texttt{diag}((\tilde{\sigma^1}))^2, \dots, (\tilde{\sigma^T}))^2$. The exact expression for the density is $f_{D_i} = \textbf{1} (\mathcal{W}_i^\top x > \mathcal{S}_i) f(x) / \mathbb{P}(\mathcal{W}_i^\top x > \mathcal{S}_i)$, where the probability also has explicit expression: $\mathbb{P}(\mathcal{W}_i^\top x > \mathcal{S}_i) = \Phi(-\mathcal{S}_i / \| \Sigma^{1/2}_{\zeta} \mathcal{W}_{i} \|)$. Such mixture has proven to be useful in approximating the distribution of interest [@lukashevich2021importance], [@lukashevich2021power], [@owen2019importance]. For this study, we use $w_1 = \dots = w_{|\mathcal{S}|}$. To sample from this mixture, one must pick a plane with probability $w_i$ and then sample outside of this plane. The Algorithm [\[alg:sample1d\]](#alg:sample1d){reference-type="ref" reference="alg:sample1d"} summarizes the procedure to produce samples outside of a plane of $\mathcal{P}_r$. With the obtained samples $\zeta^t(j)$ where $j=1, \dots, N$ and $t=1, \dots, T$, we construct the scenario approximation [\[eq:optimal_control_sampling_02\]](#eq:optimal_control_sampling_02){reference-type="ref" reference="eq:optimal_control_sampling_02"}. Fig. [1](#fig:importance_sampled_vs_mc){reference-type="ref" reference="fig:importance_sampled_vs_mc"} illustrates the difference between the samples obtained from the importance sampling procedure described above and the classical Monte Carlo (MC) sampling. Mean $\mu$, covariance $\Sigma$, and a constraint $p^\top \omega^i \le b_i$. $p\sim{\cal N}(\mu,\Sigma)$ s.t. $p^\top \omega^i \ge b_i$ Sample $z \sim {\cal N}(0, I_n)$ and $u \sim U(0,1)$ Compute $y = \Phi^{-1}(\Phi(\tau) + u(1 - \Phi(\tau)))$ Set $\phi = \bar\phi y + (I_n - \bar\phi\phi^\top) z$, with $\bar\phi = \Sigma^{1/2} \omega^i / \|\Sigma^{1/2} \omega^i\|_2$ $p = \Sigma^{1/2} (\phi+\mu)$ # Empirical Study {#sec:emp} ## Algorithms and Implementation Details We compare the performance of scenario approximations based on different sampling strategies: classical Monte-Carlo (SA) and the proposed Importance Sampling (SA-IS). For our test cases, we consider power systems from MATPOWER [@zimmerman2010matpower] (Washington-14, IEEE-30, and IEEE-57). We implemented[^1] the methods using Python 3.9.13 and PandaPower 2.8.0 [@pandapower.2018] on a MacBook Pro (M1 MAX, 64 GB RAM). We use CVX [@diamond2016cvxpy] and HiGHs [@huangfu2018parallelizing] optimization solvers. ## Test Cases and Numerical Results We studied the solutions of the scenario-based chance-constrained optimal power flow problem, considering Gaussian fluctuations, for both the classical Monte-Carlo strategy (SA) and the proposed Importance Sampling-based scenario generation (SA-IS). The comparisons were made for three different test cases: Washington-14, IEEE-30, and IEEE-57 bus systems, which have 14, 30, and 57 buses/nodes, respectively. In all cases, we assumed that the power generation and consumption levels fluctuated with a standard deviation of 0.01 of their nominal values. Thus, $\zeta^t \sim \mathcal{N}(0, (\tilde{\sigma_t}^2)$, where $\tilde{\sigma^t} = 0.01 \cdot t \cdot n_g$. We simulated for $T=5$ snapshots. The main point of comparison between SA-IS and classical SA is the number of samples $N$ required to obtain a solution with a reliability of $\delta$. The value of $1 - \eta$ stands for the required confidence threshold for linear constraints, $1-\delta$ for the reliability of the SA solution; see Sec. [3](#sec:prob){reference-type="ref" reference="sec:prob"} for details. To evaluate the empirical estimation of the solution reliability $\hat{\delta}$, we independently construct $L=200$ different scenario approximations for both SA and SA-IS with $N$ samples. For each approximation, we obtain $L$ different solutions: $(x^*_N)_l, ~ l=1, \dots, L$. Next, we estimate the confidence of each obtained solution over $10^4$ Monte-Carlo samplesto get $(\hat{\mathbb{P}}_N)_l$, $l=1, \dots, L$. Finally, the reliability of a solution is estimated by $1 - \hat{\delta}$, which represents the fraction of $L$ solutions $(x^*_N)_l$ such that $(\hat{\mathbb{P}}_N)_l \geq 1 - \eta$. See Alg. [\[alg:estimate_delta\]](#alg:estimate_delta){reference-type="ref" reference="alg:estimate_delta"} for details. $L$ -- number of trials, DC-OPF problem parameters, $\eta$ -- confidence level, $N_0$ -- initial size of scenario approximation, $N_{\max}$ -- maximal size of scenario approximation $N \gets N_0$ $\hat{ \mathbf{\delta}}$ -- storage for $\hat{\delta}_N$ $C_N \gets 0$ -- feasibility counter $l \gets 1$ Get $(x^*_N)_l$ with SA-IS or SA using Eq. [\[eq:optimal_control_sampling_02\]](#eq:optimal_control_sampling_02){reference-type="eqref" reference="eq:optimal_control_sampling_02"} Get constraint satis. prob. $(\hat{\mathbb{P}}_N)_l$ using Monte Carlo [\[alg:estimate_delta:phat_N\_l\]]{#alg:estimate_delta:phat_N_l label="alg:estimate_delta:phat_N_l"} $C_N \gets C_N +1$ $1-\hat{\delta}_N \gets C_N / L$ -- fraction of trials is feasible Append $\hat{\delta}_N$ to $\hat{ \mathbf{\delta}}$ $n \gets n + N_{\max}/ 10$ $\hat{ \mathbf{\delta}}$ We provide the number of samples required to reach confidence thresholds of $1-\eta = 0.95$ and $0.99$ with reliability of $0.99$ for SA-IS and SA in Table [\[tab:summary_results\]](#tab:summary_results){reference-type="ref" reference="tab:summary_results"}: the number of samples required is 30-50% less for SA-IS compared to classical SA. Furthermore, the advantage of SA-IS becomes more substantial as the increase of $1-\eta$. {#fig:ieee57conservatism width="\\linewidth"} {#fig:ieee30conservatism width="\\linewidth"} {#fig:washington14conservatism width="\\linewidth"} {#fig:ieee57reliability width="\\linewidth"}       {#fig:ieee30reliability width="\\linewidth"}       {#fig:washington14reliability width="\\linewidth"}       To better illustrate the dependency between number of samples in the approximation $N$ and relibaility estimates $1 - \hat{\delta}$, we demonstrate Fig. [9](#fig:ieee57reliability){reference-type="ref" reference="fig:ieee57reliability"}. We illustrate the dependence between empirical reliability $1-\hat{\delta}$ and $N$ over a range of values for the IEEE 118 bus system in Fig.  [9](#fig:ieee57reliability){reference-type="ref" reference="fig:ieee57reliability"}. Here, we keep a confidence threshold of joint chance constraint feasibility $1-\eta =0.99$. In addition to SA-IS and SA. Moreover, we present box-plots for the spread of $(\hat{\mathbb{P}}_N)_l$ for different $N$, in Fig. [6](#fig:ieee57conservatism){reference-type="ref" reference="fig:ieee57conservatism"}. We denote $(\hat{\mathbb{P}}_N)_l$ as the probability of constraint satisfaction, empirically computed using $10^4$ Monte Carlo samples of Algorithm [\[alg:estimate_delta\]](#alg:estimate_delta){reference-type="ref" reference="alg:estimate_delta"}, see Eq. [\[alg:estimate_delta:phat_N\_l\]](#alg:estimate_delta:phat_N_l){reference-type="ref" reference="alg:estimate_delta:phat_N_l"}. Note that SA-IS reaches a higher reliability level ($1-\hat{\delta}$) with a substantially fewer number of samples $N$. These boxplots indicate that the variance in the obtained solution's chance-constraint feasibility reduces faster for SA-IS, noted by thinner boxplots and lack of outliers, compared to SA. # Conclusion {#sec:conclusion} Importance sampling emerges as a valuable tool for real-time reliability assessment in power grids. In this study, we present an algorithm that, firstly, constructs a physics-informed mixture distribution for importance sampling, and, secondly, employs convex optimization to fine-tune the weights of this mixture. The proposed method surpasses existing state-of-the-art algorithms in terms of both accuracy and efficiency for reliability assessment. This approach can be extended and applied to various uncertain optimization and control problems further enhancing their stability and performance. # Appendix {#sec:appendix_Wb} First, we get an explicit expression for matrix $W$, $P = \left\{ W p_g \leq b \right\}$. Let $\chi \in \mathbb{R}^{n_{\chi}}$, where $n_{\chi} = n + n_g$, be the vector of variables describes the power system. For DC-OPF, $\chi = (\theta, p_g)^\top$. Let $A_{eq} \in \mathbb{R}^{m_{eq} \times n_{\chi}}$ and $b_{eq} \in \mathbb{R}^{m_{eq}}$ be a matrix and a right-hand side vector that define equality constraints: $A_{eq} \chi = b_{eq}$. These equality constraints include generation-demand balance, i.e., the slack bus balancing equation and the relation between power injections and phase angles [\[eq:DC-PF-a\]](#eq:DC-PF-a){reference-type="eqref" reference="eq:DC-PF-a"}. Matrix $A_{eq}$ is not a full-rank matrix because of the slack bus balancing equation. We start with eliminating $\theta$. Without loss of generality, we assume that the balancing equation is the first equation in $A_{eq}\chi = b_{eq}$. First, we drop the balancing equation and $\theta^s$ variable. Also, assume that $s=1$, i.e., the slack bus has index $1$. In this case, the resulting $\tilde{A}_{eq} \tilde{\chi} = \tilde{b}_{eq}$ has a full-rank matrix. We will enforce the dropped constraint later. Let $\tilde{\chi}_{\theta} = (\theta^2, \dots, \theta^n), ~ \tilde{\chi}_{p_g} = (p^1_g, \dots, p^{n_g}_g)$. Thus $$\tilde{A}_{eq} \tilde{\chi} = \tilde{b}_{eq} \Longleftrightarrow \left[ \tilde{A}_{eq,\theta}, \tilde{A}_{eq,p_g} \right] (\tilde{\chi}_{\theta}^\top, \tilde{\chi}_{p_g}^\top)^{\top} = \tilde{b}_{eq},$$ where $[\cdot, \cdot ]$ is a horizontal stack, i.e., $\tilde{A}_{eq,\theta}$ contains only those columns that are related to $\theta$ solely, the similar applies for $\tilde{A}_{eq,p_g}$. Having said that, using the fact that $\tilde{A}_{eq, \theta}$ is a full-rank square matrix, we can eliminate phase angles: $\tilde{\chi}_{\theta} = \left( \tilde{A}_{eq, \theta} \right)^{-1}\left( b_{eq} - \tilde{A}_{eq, p_g} p_g \right).$ The power balance equation is as follows: $$\left[ A^0_{eq, \theta}, A^0_{eq, p_g}\right] \left( \tilde{\chi}_{\theta}^\top, \tilde{\chi}_{p_g}^\top \right)^\top = b^0_{eq}$$ Substituting the $\tilde{\chi}_{\theta}$, we get $$\begin{aligned} \biggl(-A^0_{eq, \theta} \bigl(\tilde{A}_{eq, \theta} \bigr)^{-1} & \tilde{A}_{eq, p_g} + A^0_{eq, p_g} \biggr) \tilde{\chi}_{p_g} = \\ & b_{eq}^0 - (A^0_{eq, \theta})^\top \tilde{A}_{eq, \theta} b_{eq}.\end{aligned}$$ Denote $a_{ref}^\top=\left( -A^0_{eq, \theta} \left( \tilde{A}_{eq, \theta} \right)^{-1} \tilde{A}_{eq, p_g} + A^0_{eq, p_g} \right) \in \mathbb{R}^{1 \times n_g}$ and $b_{ref} = b_{eq}^0 - (A^0_{eq, \theta})^\top \tilde{A}_{eq, \theta} b_{eq}$, thus, one can define slack bus generation from the rest: $$p^0_g = \left( - (a_{ref}^{1:n_g})^\top p_g[1:]+ b_{ref}\right) / a_{ref}^0,$$ where $x^{k_1:k_2}$ denotes the elements of vector $x$ staring from $k_1$ ending on $k_2-1$, in the initial order. Next, we substitute the expressions obtained into inequalities to get $Wp_g \leq b$ formulation. Here we use superscript slices and indices for matrices, e.g., $(A'_{ineq})^{0}$ denote corresponding indexes for *columns*. Let $A_{ineq} \tilde{\chi} \leq b_{ineq}$ represent [\[eq:DC-PF-b\]](#eq:DC-PF-b){reference-type="eqref" reference="eq:DC-PF-b"}. Then, substitution of the obtained expression for $\theta$ leads to an equivalent system of inequalities $A'_{ineq} p_g \leq b'_{ineq}$, where $$A'_{ineq} = A_{ineq} - A_{ineq, \theta} \left( \tilde{A}_{eq, \theta} \right)^{-1} \tilde{A}_{eq, p_g}$$ and $b'_{ineq} = b'_{ineq} - A_{ineq, \theta} \left( \tilde{A}_{eq, \theta} \right)^{-1} \tilde{b}_{eq}$. Finally, we substitute the expression for the slack bus generation from the balancing equation. This will lead to the final formulation $Wp_g \leq b$ with $$W = (A'_{ineq})^{0} a_{ref}^\top / a_{ref}^0 + (A'_{ineq})^{1:n_g},$$ and $b = b'_{ineq} - (b_{ref} / a_{ref}^0) \cdot (A'_{ineq})^{0}$. The last step is the transformation of the linear cost function. Let $c(p_g) = c_1^\top p_g$. Then, the cost function after the transformation is $c^\top p_g^{1:n_g} + ct$ with $c = c_1^{1:n_g} - (c_1^0 / a_{ref}^0) a_{ref}^{1:n_g}$ and the cost correction term $ct = c_1^0 \cdot (b_{ref} / a_{ref}^0)$. [^1]: <https://urlr.me/crdtR>

arxiv_math

--- abstract: | We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system $$\partial_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \textup{in}\;\; \mathbb{R}^d,\; d = 3,4,$$ and derive the final blowup profile $$u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \textup{as}\;\; r \to 0, \;\; c_d > 0.$$ To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner *et al* in [@BCKSVnon99]. title: Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$. --- Van Tien Nguyen$^{(1)}$, Nejla Nouaili$^{(2)}$ and Hatem Zaag$^{(3)}$ $^{(1)}$ Department of Mathematics, Institute of Applied Mathematical Sciences, National Taiwan University. $^{(2)}$ CEREMADE, Université Paris Dauphine, Paris Sciences et Lettres, France $^{(3)}$Université Sorbonne Paris Nord, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France. [^1] # Introduction. We are interested in the existence of blowup solutions to the Keller-Segel system $$\label{sys:KS} \left\{ \begin{array}{rl} \partial_t u = &\Delta u - \nabla \mathcal{K}_u \cdot\nabla u + u^2, \\ 0 = &\Delta \mathcal{K}_u + u, \end{array} \right. \quad x \in \mathbb{R}^d,$$ where $u(t): x \in \mathbb{R}^d \to \mathbb{R}$ subject to initial data $u(0) = u_0$. The system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} appears in many biological and astrophysical contexts. Here, $u(x,t)$ stands for the density of particles or cells and $\mathcal{K}_u$ is a self-interaction potential. In the two-dimensional case, it is used to model the so-called *chemotaxis* phenomena in biology first introduced by Patlak [@PATbmb53] and Keller-Segel in [@KSjtb70] (see also [@KSjtb71a] [@KSjtb71b] for a derivation of a general model). In higher dimensional cases, the system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} appears as a simplified model for self-gravitating matter in stellar dynamics, see for example [@Warma92], [@Wjam92], [@SCPRSN02], [@CSPRE11] and [@DLMN13]. We refer the paper [@Hjdmv03] where the author gives a nice survey of mathematical derivation of [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} and related models. It's worth mentioning that the system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} is a special case belonging to a much wider class of nonlocal aggregation equations including those with degenerate diffusion read as $$\label{eq:generalequation} \partial_t u = \Delta A(u) + \nabla \cdot(B(u) \nabla \mathcal{K}\ast u), \quad (x,t) \in \mathbb{R}^d \times [0, \infty),$$ where $A(u)$ and $B(u)$ can be nonlinear functions and $\mathcal{K}$ is an arbitrary local integrable function. In [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}, we have $A(u) = B(u) = u$ and $\mathcal{K}$ is the Newtonian potential. Beside covering a wide range of application, equation [\[eq:generalequation\]](#eq:generalequation){reference-type="eqref" reference="eq:generalequation"} posses an interesting mathematical phenomena already observed in [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}: the competition between the diffusion and the nonlocal aggregation for which we may have global existence or finite time singularity of solutions. The local Cauchy problem for [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} can be solved in $L^\infty(\mathbb{R}^d)$ for the class of radially symmetric solutions, see for example Karch-Suzuki [@KSAM11], Souplet-Winkler [@SWCMP19], Winkler [@Wom23]. For the study of Cauchy problem for the general model [\[eq:generalequation\]](#eq:generalequation){reference-type="eqref" reference="eq:generalequation"}, we refer the works of Masmoudi-Bedrossian [@BMARMA14] (see also [@BRBN11], [@BRDCDS14], [@BCPDE15]), Biler-Karch-Pilarczyk [@BKPJDE19] and references therein. Due to the singularity of the Newtonian potential at the origin, the existence of finite blowup solutions and their mechanism are delicate, and this is precisely our focus in this paper. Neglecting the drift term in [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} gives the classical semilinear heat equation $$\label{eq:NLH} \partial_t u = \Delta u + |u|^{p-1}u \quad \textup{with} \;\; p = 2,$$ where the existence of global-in-time or blowup solutions has received lots of attention in the last five decades, starting from the work of Fujita [@FUJsut66]. We refer the book of Quittner-Souplet [@QSbook07] as a nice source of references on this subject. Up to now, a fairly completed picture on the singularity formation for [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} has been established, especially for the existence with a detailed description of the mechanism near the singularity. A very first result in this direction based on a numerical rescaling algorithm by Berger-Kohn [@BKcpam88] suggested that a stable (generic) blowup solution to [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} is given by $$\label{eq:profileNLH} u(x,t) \sim \frac{1}{T- t}P\left( \frac{x}{\sqrt{(T-t)\log(T-t)}}\right), \quad P(\xi) = \frac{1}{1 + |\xi^2|/8}.$$ A rigorous construction was later done by Bricmont-Kupiainen [@BKnon94], Merle-Zaag [@MZdm97] and a refined description by Nguyen-Zaag [@NZens17]. A completed classification of all Type I blowup behaviors was established by Fillipas-Kohn [@FKcpam92], Filippas-Liu [@FLaihn93], Velázquez [@VELcpde92], Herrero-Velázquez [@HVcpde92] where we see unstable blowup profiles. Here, a blowup solution to [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} is of Type I if it satisfies $$\limsup_{t \to T} (T-t)\|u(t)\|_{L^\infty} < +\infty,$$ otherwise, the blowup is called Type II. The existence of Type II blowup solutions to [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} was established in some special range of $p = p(d)$, see for example [@FHVslps00], [@Sjfa12], [@PMWZdcds20], [@Hihp20], [@PMWjfa21], [@PMWams19], [@PMWZZarx20] [@HVcras94],[@Cmams18], [@CMRjams19] and references therein.\ We wonder whether the presence of the drift term in [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} would create a different blowup mechanism. Indeed, this was the case $d =2$ where the system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} is said to be $L^1$-critical in the sense that the rescaling function $$\forall \lambda > 0, \quad u_\lambda(x,t) = \frac{1}{\lambda^2} u\Big(\frac{x}{\lambda}, \frac{t}{\lambda^2}\Big)$$ solves the same system and preserves the same $L^1$-norm, $$\| u\|_{L^1(\mathbb{R}^2)} = \|u_\lambda\|_{L^1(\mathbb{R}^2)}.$$ The blowup is completely different from the nonlinear heat equation [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} where only Type I blowup may occur (see [@GMSiumj04]). In fact, for the two dimensional Keller-Segel system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}, the blowup rate is quantized in [@CGMNcpam21; @CGMNapde22] (see also [@HVma96], [@Vsiam02], [@Sna07] and [@RSma14] for earlier results) with $$\|u(t)\|_{L^\infty(\mathbb{R}^2)} \sim \left[ \begin{array}{ll} c_0(T-t)^{-1} \exp (\sqrt{2 |\ln(T-t)|}), \\ c_\ell(u_0) (T-t)^{-\ell} |\ln (T-t)|^\frac{\ell + 1}{ \ell - 1}, \quad \ell \in \mathbb{N}, \ell\geq 2. \end{array} \right.$$ We also have the existence of blowup solutions in higher dimensional cases $d \geq 3$ established in [@CGMNjfa23] where the authors rigorously constructed a so-called collapsing-ring blowup solutions in the spirit of blowup for the nonlinear Schrödinger equation [@MRSduke14] and found the blowup rate $$\|u(t)\|_{L^\infty(\mathbb{R}^d)} \sim (T-t)^{-\frac{2d}{d-1}}.$$ This result was formally derived in [@HMVnonl97] for $d = 3$ and reestablished by Brenner *et al* in [@BCKSVnon99] for $d \geq 3$ (with a formal analysis too). The authors of [@BCKSVnon99] also predicted many other blowup patterns for [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} in the higher dimensional cases. There are also self-similar solutions having the blowup speed $\|u(t)\|_{L^\infty} \sim (T-t)^{-1}$ described in [@HMVjcam98], [@Sfe05], [@GMSarma11] and [@SWCMP19].\ In this paper we exhibit a new type of blowup solutions to [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} that haven't been observed in the literature to our knowledge (the possible occurrence of this solution was briefly mentioned in Section 4.3 of [@BCKSVnon99]). Consider the space dimension $$d = 3,\; 4,$$ and restrict to the case of radially symmetric solutions. Then, for any smooth radial function $u \in L^\infty(\mathbb{R}^d)$, the potential term is defined as $$\partial_r\mathcal{K}_u(r) = -\frac{1}{r^{d-1}}\int_0^r u(\zeta) \zeta^{d-1} d\zeta, \quad r = |x|,$$ and the system [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} is written as a nonlocal semilinear heat equation, $$\label{eq:urt} \partial_t u = \Delta_d u +\Big( \frac{1}{r^{d-1}}\int_0^r u(\zeta) \zeta^{d-1}d\zeta\Big) \partial_r u + u^2,$$ where $u(t): \mathbb{R}_+ \to \mathbb{R}$ and $\Delta_d$ is the Laplacian acting on radial functions in $\mathbb{R}^d$, i.e. $$\Delta_d = \partial_r^2 + \frac{d-1}{r}\partial_r.$$ Our main result is the following. **Theorem 1** (Existence of finite-time blowup solutions to [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}). *Consider $d = 3, 4$ and let $\ell = \frac{d}{d-2}$ and $\alpha = \frac{d-2}{2d}$. There exists radially symmetric initial data $u_0 \in L^{\infty}(\mathbb{R}_+)$ such that the corresponding solution to System [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} blows up in finite-time $T < \infty$ only at the origin and admits the following asymptotic dynamics.* - *[(Inner expansion)]{.upright} $$\label{exp:innerIntro} u\big(y \sqrt{T-t}, t\big) = \frac{1}{T-t}\Big[ 1 - \frac{1}{B_\ell} \frac{\phi_{2\ell} (y) }{|\log (T-t)|} + o \big( \frac{1}{|\log (T-t)|}\big) \Big] \quad \textup{as} \quad t \to T,$$ where the convergence holds on any compact sets $\{ y \leq C\}$, the function $\phi_{2\ell}(y)$ is the polynomial of degree $2\ell$ satisfying $\Delta \phi_{2\ell} - \frac{1}{2\ell} y \partial_y \phi_{2\ell} + \phi_{2\ell} = 0$, $$\label{eq:B3B2Intro} B_3 = 39360 \;\; \textup{for}\;\; (d, \ell) = (3,3) \quad \textup{and} \quad B_2 = 576 \;\;\textup{for}\;\; (d, \ell) = (4,2).$$* - *[(Intermediate profile)]{.upright} Let $Q$ be a positive function defined by $$\label{def:QIntro} \forall \xi \in \mathbb{R}_+, \quad \frac{1 - dQ}{Q^\ell} = c_\ell \xi^{2\ell} \quad \textup{with} \quad c_\ell = \frac{d^{\ell + 1}}{B_\ell(d + 2\ell)\ell^\ell} > 0.$$ Let $F(\xi) = dQ(\xi) + \xi Q'(\xi)$, we have $$\label{exp:interProfileIntro} \sup_{|y| \in \mathbb{R}^d} \left|(T-t) u\big(y\sqrt{T-t}, t\big) - F\Big( \frac{y}{|\log (T-t)|^\frac{1}{2\ell}} \Big) \right| \to 0 \quad \textup{as} \;\; t \to T.$$* - *[(Final profile)]{.upright} There exists $u^* \in \mathcal{C}(\mathbb{R}_+ \setminus \{0\}, \mathbb{R})$ such that $u(r,t) \to u^*(r)$ as $t \to T$ uniformly on compact subsets of $\mathbb{R}_+\setminus \{0\}$, where $$\label{exp:finalprofileIntro} u^*(r) \sim (d- 2) \left(\frac{2}{c_\ell} \right)^\frac{1}{\ell} \frac{|\log r| ^ \frac{1}{\ell}}{r^2} \quad \textup{as}\;\; r \to 0.$$* **Remark 2** (New blowup profile). One of the significant contribution of this work is the construction of a *new* blowup profile [\[def:QIntro\]](#def:QIntro){reference-type="eqref" reference="def:QIntro"} with a log correction to the blowup variable $$\label{def:xiIntro} \xi = \frac{r}{\sqrt{T-t} \; |\log(T-t)|^\frac{1}{2\ell}} \quad \textup{with} \quad \ell \geq 2.$$ Let us mention that Brenner *et al* [@BCKSVnon99] wondered whether blowup solutions exist with this particular scaling for any dimension $d \geq 3$, not necessarily with the same power we get here for the log correction (see Section 4.3 in that paper). In particular, they did not provide the exact power of the log correction (i.e. $\frac{1}{2\ell}$), as we found here for $d = 3$ or $d = 4$. The appearance of the *new* blowup scale given in [\[def:xiIntro\]](#def:xiIntro){reference-type="eqref" reference="def:xiIntro"} shows a strong influence of the drift-term to the blowup dynamic of [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}. Recall from [\[eq:profileNLH\]](#eq:profileNLH){reference-type="eqref" reference="eq:profileNLH"} that the stable blowup scale for [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} is given with $\ell = 1$ and the intermediate profile is explicitly given. As a consequence, the existing analysis developed for [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} can not be straightforward implemented to [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"}, although the general framework for the construction remains the same once the blowup is concerned. It's worth remarking that the final blowup profile derived in [@SWCMP19] for the class of radially symmetric *decreasing* solutions satisfies $$c_1r^{-2} \leq u(r,T) \leq c_2 r^{-2} \quad \textup{and} \quad u(x,T) \leq C(T-t + r^2)^{-1}.$$ Recall that our constructed solution is radially symmetric, but not a decreasing function. Indeed, the inner expansion [\[exp:interProfileIntro\]](#exp:interProfileIntro){reference-type="eqref" reference="exp:interProfileIntro"} involves a Hermite type polynomial of degree $2\ell$ which changes signs on the set $y \in (0, y_0)$ for $y_0\gg 1$. Hence, our intermediate and final blowup profiles [\[exp:interProfileIntro\]](#exp:interProfileIntro){reference-type="eqref" reference="exp:interProfileIntro"} and [\[exp:finalprofileIntro\]](#exp:finalprofileIntro){reference-type="eqref" reference="exp:finalprofileIntro"} are excluded from what described in [@SWCMP19] and in agreement with the description of [@GMSarma11] asserting that all Type I blowup solutions are asymptotically backward self-similar. **Remark 3** (Co-dimensional stability). The initial data we consider in the construction depends on $\ell$ parameters $(d_i)_{0 \leq i \leq \ell-1}$ (see [\[def:intitialdata_q\]](#def:intitialdata_q){reference-type="eqref" reference="def:intitialdata_q"} for a proper definition) to control growing eigenmodes of the linearized operator. One parameter can be eliminated by the translation in time invariance of the problem, so it remain $(\ell - 1)$ eigenmodes to be handled. Roughly speaking, our constructed solution is $(\ell - 1)$ co-dimension stable in the sense that if we fix those $(\ell-1)$ unstable directions and perturb only the remaining components of the solution (see Definition [\[Definition-shrinking -set\]](#Definition-shrinking -set){reference-type="eqref" reference="Definition-shrinking -set"} for a definition of solution decomposition and a bootstrap regime to control them), the solution still admits the same behavior as described in Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}. We also remark the connection of our constructed solution to the backward self-similar solutions to [\[eq:urt\]](#eq:urt){reference-type="eqref" reference="eq:urt"} which is of the form $$u(r,t) = \frac{1}{T-t}\Phi(y), \quad y = \frac{r}{\sqrt{T-t}},$$ where $\Phi$ solves $$0 = \Delta_d \Phi + \Big( \frac{1}{y^{d-1}}\int_0^y \Phi(\zeta) \zeta^{d-1} d\zeta \Big) \partial_y \Phi + \Phi^2 - \frac{1}{2}y\partial_y \Phi - \Phi.$$ There are four explicit solutions $$\Phi_1 \equiv 0, \quad \Phi_2 \equiv 1, \quad \Phi_3 = \frac{2(d-2)}{y^2}, \quad \Phi_4 = \frac{1}{y^{d-1}}\partial_y\Big[ \frac{4(d-2)(2d + y^2) y^d}{(2(d-2) + y^2)^2} \Big].$$ The blowup profile introduced in Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"} is asymptotically like the constant solution $\Phi_2$ in compact sets. In a recent work [@GSarma23], Glogic and Schörkhuber showed the stability of the solution with $\Phi_4$ profile in three dimensions in $H^3(\mathbb{R}^3)$. **Remark 4** (Extensions and related problems). From our analysis, we suspect that the blowup scale with a log correction [\[def:xiIntro\]](#def:xiIntro){reference-type="eqref" reference="def:xiIntro"} can only occur in dimensions $d = 3$ and $d = 4$. In fact, the appearance of such a log correction is strongly related from our point of view to the presence of a zero eigenvalue for the linearized operator defined below in [\[def:Ls_alp\]](#def:Ls_alp){reference-type="eqref" reference="def:Ls_alp"}, which occurs only for $d = 3,4$ (see Remark [\[remark:specofLs\]](#remark:specofLs){reference-type="ref" reference="remark:specofLs"}). Other blowup scales without a log correction are suspected to exist similarly as in the nonlinear heat equation [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} where a negative eigenmode of the linearized operator is assumed to be dominant in the inner expansion [\[exp:innerIntro\]](#exp:innerIntro){reference-type="eqref" reference="exp:innerIntro"}. The suppression of blowup in [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} by modifying the nonlinearity has been an interesting direction recently, see for an example [@KXarma16], [@BHsiam17], [@HTarma19], [@IXZtams21]. We remark that our construction actually works for the nonlinear perturbation problem where we consider the nonlinearity $u^2$ added a perturbation $f(u)$, namely the problem $$\partial_t u= \Delta u - \nabla \mathcal{K}_u \cdot \nabla u + u^2 + f(u),$$ where $f$ satisfies the growth condition $$|f(u)| \leq C (1 + |u|^q)\;\; \textup{for}\;\; 0 \leq q < 2, \quad \textup{or} \quad f(u) = \epsilon u^2 \;\; \textup{for}\;\; 1 + \epsilon > 0.$$ The first assumption turns to be exponentially small in the self-similarity setting [\[def:selfsimilarity\]](#def:selfsimilarity){reference-type="eqref" reference="def:selfsimilarity"} and the contribution from this small nonlinear term is neglectable. The later assumption ensures that there associated ODE $u' = (1 + \epsilon)u^2$ still blows up in finite time, hence the intermediate blowup profile [\[exp:finalprofileIntro\]](#exp:finalprofileIntro){reference-type="eqref" reference="exp:finalprofileIntro"} and the final profile [\[exp:finalprofileIntro\]](#exp:finalprofileIntro){reference-type="eqref" reference="exp:finalprofileIntro"} are modified with the factor $\frac{1}{1 + \epsilon}$. We suspect the case $\epsilon \leq -1$ would prevent blowup to happen, or if blowup does occur, its dynamic would be completely different from what established in this paper. The analysis presented in this work is expectedly applicable to the general equation [\[eq:generalequation\]](#eq:generalequation){reference-type="eqref" reference="eq:generalequation"} with $A(u) = u$ and $B(u) = u^{p-1}$ for $p > 1$ or $A(u) = u^m$ for $m > 0$ and $B(u) = u$ up to some technicalities. The later case is an interesting model (*porous medium* type equation) used to describe gravitational collapse phenomena (see [@CSPRE11]). #### Strategy of the construction: We briefly describe the idea of the proof of Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}.\ - *Change of variables*: We consider the change of variables $$u(r,t) = \frac{1}{T-t}w(y,s), \;\; y = \frac{r}{\sqrt{T-t}}, \;\; s = -\log(T-t),$$ and introduce the partial mass setting $$v(y,s) = \frac{1}{y^d}\int_0^y w(\zeta, s)\zeta^{d-1}d\zeta, \qquad w = \frac{1}{y^{d-1}}\partial_y(y^d v).$$ where $v$ solves the semilinear heat equation $$\partial_s v = \Delta_{d+2} v -\frac{1}{2}y\partial_y v + dv^2 + yv \partial_y v.$$ We note that such a transformation is just to simplify the analysis, and emphasize that the strategy and main idea remain the same once we work with the equation satisfied by $w$.\ - *Linearization*: Through a formal computation of the blowup profile $Q$ given in Section [2](#sec:blowupprofile){reference-type="ref" reference="sec:blowupprofile"}, we introduce the linearization $$v(y,s) = Q(\xi) + \varepsilon(y,s), \quad \xi = \frac{y}{s^\frac{1}{2\ell}},$$ where $\varepsilon$ solves the linearized problem $$\partial_s \varepsilon= \mathscr{H}\varepsilon+ NL(\varepsilon) + E,$$ where $NL$ is a quadratic nonlinear term, $E$ is a generated error and $\mathscr{H}$ is the linearized operator $$\mathscr{H}= \Delta_{d+2} - \Big(\frac{1}{2} - Q(\xi)\Big)y\partial_y + \big(2dQ -1 + \xi \partial_\xi Q(\xi)\big).$$ We observe that $\mathscr{H}$ behaves differently depending on the behavior of the profile $Q(\xi)$:\ - For $y \gg s^\frac{1}{2\ell} (\xi \gg 1)$ , we have by the decaying property $|Q(\xi)| + |\xi \partial_\xi Q(\xi)| = \mathcal{O}(\xi^{-2})$, the linear operator $\mathscr{H}$ behaves like $\Delta_{d+2} - \frac{1}{2}y\partial_y - \textup{Id}$, which has fully negative spectrum.\ - For $y \ll s^\frac{1}{2\ell} (\xi \ll 1)$, we have by the asymptotic behavior $\big|Q(\xi) - \frac{1}{d}\big| + |\xi \partial_\xi Q(\xi)| = \mathcal{O}(\xi^2)$, the linear operator $\mathscr{H}$ behaves like $\Delta_{d+2} - \frac{1}{2\ell }y\partial_y + \textup{Id}$, which has $\ell$ positive eigenvalues, a zero eigenvalue and infinity many negative ones.\ - For $y \sim s^\frac{1}{2\ell} (\xi \sim 1)$, this is the transition region (intermediate zone) that we don't have any asymptotic simplification of $\mathscr{H}$. This is one of the major difficulties of the paper. Indeed, it makes one of the main differences with respect to the analysis for the nonlinear heat equation [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} that results in a different approach presented in this paper.\ - *Decomposition:* Based on the behavior of the linearized operator $\mathscr{H}$, we split the control of $\varepsilon$ into three regions: for a fixed large constant $K \gg 1$,\ - The outer region $y \geq K s^\frac{1}{2\ell} (\xi \geq K)$: Since $\mathscr{H}$ behaves like the one with fully negative spectrum, the estimate of $\varepsilon$ in this region is straightforward by using the semigroup associated to the linear operator $\Delta_{d+2} - \frac{1}{2}y \partial_y$ (see Section [4.6](#sec:outer){reference-type="ref" reference="sec:outer"}), $$j =0, 1, \quad \| (y \partial_y)^j \varepsilon(y,s) \mathbf{1}_{\{\xi \geq K\}}\|_{L^\infty} \lesssim \|(y \partial_y)^jE(s)\|_{L^\infty} + \| (y \partial_y)^j\varepsilon(y,s) \mathbf{1}_{\{\xi \sim K\}}\|_{L^\infty},$$ where $\|(y \partial_y)^jE(s)\|_{L^\infty} \lesssim s^{-\frac{1}{\ell}}$ is the typical size of the generated error. We need the information from the intermediate region for the boundary term located on $\xi \sim K$ to completely close the estimate in the outer region.\ - The intermediate region $K \leq y \leq 2K s^\frac{1}{2\ell}$, we control the solution in the weighted $L^2$ norm, $$\|\varepsilon(s)\|_\flat^2 = \int_K^\infty \frac{|\varepsilon(y,s)|^2}{y^{4\ell + 2}} \frac{dy}{y}.$$ (we can replace the weight $y^{4\ell + 2}$ by $y^{2k}$ for any $k \geq 2\ell +1$ with an improved refinement of the generated error). Thanks to the monotone property of the profile $Q$ and the dissipative structure of the parabolic equation, we are able to arrive at the monotonicity formula (see Lemma [Lemma 14](#lemm:mid){reference-type="ref" reference="lemm:mid"}) $$j = 0, 1, 2, \quad \frac{d}{ds} \|(y\partial_y)^j \varepsilon\|_\flat^2 \leq -\delta_0 \|(y\partial_y)^j \varepsilon\|_\flat^2 + \|(y\partial_y)^j E\|_\flat^2 + \|(y\partial_y)^j \varepsilon\mathbf{1}_{ y \sim K}\|_\flat^2,$$ where $\|(y\partial_y)^j E\|_\flat^2 \lesssim s^{-2 - \frac{3}{\ell}}$ is the size of the error term. By Sobolev inequality, we obtain a pointwise estimate $|\varepsilon(y,s)| \lesssim s^{-1 - \frac{3}{2\ell}} (|y|^{2\ell + 1} + 1)$ that provides the necessary information of $\varepsilon$ at the boundary $y \sim s^\frac{1}{2\ell}$ to complete the estimate in the outer region. It's worth mentioning that in the case of the nonlinear heat equation [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"}, this kind of pointwise estimate can be directly achieved by using a semigroup approach. However, we are not able to follow that approach for the Keller-Segel equation [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} due to the lack of knowledge on semigroup theory associated to $\mathscr{H}$. Here, we still need the information of $\varepsilon$ at the boundary $y \sim K$ to completely close the estimate of $\|(y\partial_y)^j \varepsilon\|_\flat$ after a forward integration in time.\ - The inner region $y \leq 2K$, the linearized operator $\mathscr{H}$ is regarded as a perturbation of $$\mathscr{H}_\frac{1}{2\ell} + \textup{Id} \quad \textup{with} \quad \mathscr{H}_\frac{1}{2\ell}:=\Delta_{d+2} - \frac{1}{2\ell} y\partial_y.$$ The operator $\mathscr{H}_\frac{1}{2\ell}$ is self-adjoint in $L^2_\rho$ with the exponential weight $\rho = \exp( - \frac{y^2}{4\ell}) y^{d+1}$. Since $\mathscr{H}_\frac{1}{2\ell}$ posses $\ell$ positive eigenmodes and a zero one, we further decompose $$\varepsilon(y,s) = \varepsilon_\natural(y,s) + \tilde \varepsilon(y,s), \quad \varepsilon_\natural(y,s) = \sum_{k = 0}^{2\ell-1} \varepsilon_k(s) \varphi_{2k}(y), \quad \langle \tilde{\varepsilon},\varphi_{2k}\rangle_{L^2_\rho} =0 \; \textup{for} \; k = 0, \cdots, 2\ell-1,$$ where $\varphi_{2k}$ is the eigenfunction of $\mathscr{H}_\frac{1}{2\ell}$ corresponding to the eigenvalue $- \frac{k}{\ell}$ and $\tilde{\varepsilon}$ solves the equation $$\partial_s \tilde{\varepsilon} = \mathscr{H}_\frac{1}{2\ell}\tilde{\varepsilon} + \tilde{\varepsilon} + \sum_{k = 0}^{2\ell -1} \big[-\varepsilon_k' + \big( 1 - \frac{k}{\ell}\big) \varepsilon_k \big] \varphi_{2k}(y) + R + \tilde{\mathcal{V}}(\tilde \varepsilon) + NL(\tilde \varepsilon),$$ where $\tilde \mathcal{V}$ and $NL$ are small linear and nonlinear terms. Using the spectral gap $\langle \mathscr{H}_\frac{1}{2\ell} \tilde \varepsilon, \tilde{\varepsilon} \rangle_{L^2_\rho} \leq -2 \| \tilde{ \varepsilon}\|^2_{L^2_\rho}$ and a standard energy estimate, we end up with (see Lemma [Lemma 13](#lemm:L2rho){reference-type="ref" reference="lemm:L2rho"}) $$\frac{d}{ds}\| \tilde{ \varepsilon}\|^2_{L^2_\rho} \leq - \delta \| \tilde{ \varepsilon}\|^2_{L^2_\rho} + \| R \|^2_{L^2_\rho}, \quad \textup{with} \quad \| R\|_{L^2_\rho} \lesssim s^{-3}.$$ As for the finite dimensional part, we simply obtain by a projection onto the eigenmode $\varphi_{2k}$, $$\big|-\varepsilon_k' + \big( 1 - \frac{k}{\ell}\big) \varepsilon_k \big| \lesssim \| \tilde{\varepsilon}\|_{L^2_\rho} + s^{-2}, \quad \big|\varepsilon_\ell' + \frac{2}{s}\varepsilon_\ell \big| \lesssim \| \tilde{\varepsilon}\|_{L^2_\rho} + s^{-3}.$$ The equation of $\varepsilon_\ell$ is delicate as it's related to the projection onto the null mode where the contribution from the small potential term must be taken into account to produce the factor $\frac{2}{s}$ as well as an algebraic cancellation in the projection of the error term onto the null mode to reach $\mathcal{O}(s^{-3})$. Those calculations get used of the precise value $B_\ell$ given in [\[eq:B3B2Intro\]](#eq:B3B2Intro){reference-type="eqref" reference="eq:B3B2Intro"} (see Lemma [Lemma 12](#lemm:finitepart){reference-type="ref" reference="lemm:finitepart"}). A forward integration in time yields the estimates $$\|\tilde \varepsilon\|_{L^2_\rho} \lesssim s^{-3}, \quad |\varepsilon_k(s)| \lesssim s^{-2} \; \textup{for} \;\ell + 1 \leq 2\ell -1, \quad |\varepsilon_\ell(s)| \lesssim \frac{\log s}{s^2}.$$ The remaining growing modes $\big(\varepsilon_k(s)\big)_{0 \leq k \leq \ell-1}$ are then controlled by a topological argument where we need to construct initial data for which these components converges to zeros as $s \to \infty$. The $L^2_\rho$ estimate provides information on compact sets of $y$, where we can get estimate of $\varepsilon(y,s)$ for $y \sim K$ to close the estimate for the intermediate region. This decomposition is detailed in the definition of bootstrap regime [\[Definition-shrinking -set\]](#Definition-shrinking -set){reference-type="eqref" reference="Definition-shrinking -set"} in which we successfully construct solutions admitting the behavior as described in Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}.\ We organize the rest of the paper as follows: In Section [2](#sec:blowupprofile){reference-type="ref" reference="sec:blowupprofile"} we perform a formal spectral analysis to obtain the blowup profile. We formulate the linearized problem in Section [3](#sec:linear){reference-type="ref" reference="sec:linear"} and define a bootstrap regime to control the remainder. In Section [4](#sec:ControlBs){reference-type="ref" reference="sec:ControlBs"} we control the remainder in the bootstrap regime and prove that the solution of the linearized problem is trapped in this regime for all time from which we conclude the proof of the main theorem.\ #### Acknowlegments: V.T. Nguyen is supported by the National Science and Technology Council of Taiwan (ref. 111-2115-M-002-012 and ref. 112-2628-M-002-006). A part of this work was done when V.T. Nguyen visited the Université Paris Dauphine and he wants to thank the institution for their hospitality and support during his visit. H. Zaag is supported by the ERC Advanced Grant LFAG/266 "Singularities for waves and fluids". # Formal derivation of the blowup profile {#sec:blowupprofile} In this section we formally derive the blowup profile through a spectral approach that will be rigorously implemented in the next section. As you will see, the profile with a logarithmic correction to the blowup scale only appears in the case of dimension $d = 3$ or $d = 4$ for which the linearized operator poses zero eigenvalues. We work with the self-similar variables $$\label{def:selfsimilarity} u(r,t) = \frac{1}{(T-t)} w(y,s), \quad \mathcal{K}_u(x,t) = \mathcal{K}_w(y,s), \quad y = \frac{r}{\sqrt{T-t}}, \quad s = - \log(T-t),$$ where $w$ solves the equation for $(y,s) \in \mathbb{R}_+ \times [-\log T, +\infty)$, $$\label{eq:wys} \partial_s w = \Delta_d w - \left( \partial_y\mathcal{K}_w + \frac{1}{2} y \right)\partial_y w - w + w^2,$$ with $\Delta_d$ being the Laplacian in $\mathbb{R}^d$ acting on radial functions, i.e. $$\Delta_d = \partial_y^2 + \frac{d-1}{y}\partial_y,$$ and the potential term is defined by $$\label{eq:exprePhiy} \partial_y \mathcal{K}_w(y,s) = - \frac{1}{y^{d-1}}\int_0^y w(\zeta,s) \zeta^{d-1} d\zeta.$$ A linearization of $w$ around the nonzero constant solution $$\bar w = 1, \qquad -\partial_y \mathcal{K}_{\bar w}= \frac{y}{d},$$ leads to the linearized equation for $q = w - \bar w, \quad \mathcal{K}_q = \mathcal{K}_w - \mathcal{K}_{\bar w}$: $$\label{eq:qys} \partial_s q = \mathscr{L}_\alpha q - \partial_y \mathcal{K}_q \partial_y q + q^2, \quad (y,s) \in \mathbb{R}_+ \times [-\log T, +\infty),$$ where $$\label{def:Ls_alp} \mathscr{L}_\alpha = \Delta_d - \alpha\; y \partial_y +1 \quad \textup{with} \quad \alpha = \frac{d-2}{2d}.$$ #### Spectral properties of $\mathscr{L}_\alpha$: The linear operator $\mathscr{L}_\alpha$ is formally a self-adjoint operator in $L^2_\omega(\mathbb{R}_+)$ with the weight function $$\label{def:omega} \omega(y) = c_d y^{d-1} e^{-\frac{\alpha |y|^2}{2}},\qquad \int_0^\infty \omega (y) dy=1,$$ where $c_d = 2^{\frac d2-1}\alpha^{-\frac d2}\Gamma \left (\frac d2\right )$ to normalize $\|\omega\|_{L^1} = 1$, $\Gamma$ is the Gamma function. The spectrum of $\mathscr{L}_\alpha$ is discrete $$\label{def:specLsalpha} \textup{spec}(\mathscr{L}_\alpha) = \{ \lambda_{2n} = 1 - 2 n \alpha, \; n \in \mathbb{N}\},$$ and the corresponding eigenfunction $\phi_{2n}$ is a polynomial of degree $2n$ that satisfies $$\label{eq:phi2n} \Delta_d \phi_{2n} - \alpha y \partial_y \phi_{2n}= -2n \alpha \phi_{2n}.$$ In particular, we have the close form of $\phi_{2n}$ given by $$\label{def:phi2ell} \phi_{2n}(y) = H_n(z) = \sum_{k = 0}^n A_{n, k} z^{k} \quad \textup{with} \quad z =2\alpha y^2,$$ where $H_n$ is the regular solution to the Kummer-type ODE $$\label{eq:Hz} 4zH_n'' + \big(2d- z\big) H_n' + n H_n = 0.$$ and $A_{n,k}$'s satisfy the recurrence relation $$\label{def:Ank} A_{n,n} = 1, \quad A_{n, k - 1} = - \frac{2k (2k + d - 2)}{n - k + 1} \; A_{n, k} \;\; \textup{for}\;\; k = 1, ..., n.$$ We note the orthogonality identity $$\label{eq:ortho} \int_0^\infty \phi_{2n}(y) \phi_{2m}(y) \omega(y) dy = (2\alpha)^{-\frac{d-2}{2}}\int_0^\infty H_n(z)H_m(z) z^{\frac{d-1}{2}} e^{-\frac{z}{4}} dz = a_{n}\delta_{n,m},$$ where $\delta_{n,m} = 0$ if $n \ne m$ and $\delta_{n,m} = 1$ if $n = m$. [\[remark:specofLs\]]{#remark:specofLs label="remark:specofLs"} Consider $\ell \in \mathbb{N}$ such that $\lambda_{2\ell} = 1 - 2 \alpha \ell = 0$, which gives $\ell = \frac{1}{2\alpha} = \frac{d}{d -2} \in (1,3]$. We see that there are only two cases for which $\ell$ is an integer number: $$\ell = 3 \;\; \textup{for}\;\; d = 3 \quad \textup{and} \quad \ell = 2 \;\; \textup{for}\;\; d = 4.$$ We list here the first few eigenfunctions (generated by Matlab symbolic) served for our computation later:\ - for $d = 3$, $$\begin{aligned} &H_0(z) = 1, \qquad \quad H_1(z) = z - 6, \qquad \quad H_2(z) = z^2 - 20 z + 60,\\ & H_3(z) = z^3 - 42z^2 + 420 z - 840,\\ &H_4(z) = z^4 - 72z^3 + 1512z^2 - 10080z + 15120,\\ & H_5(z) = z^5 - 110z^4 + 3960z^3 - 55440z^2 + 277200z - 332640,\\ & H_6(z) = z^6 - 156z^5 + 8580z^4 - 205920 z^3 + 2162160 z^2 - 8648640z + 8648640,\end{aligned}$$ - for $d = 4$, $$\begin{aligned} &H_0(z) = 1, \quad H_1(z) = z - 8, \quad H_2(z) = z^2 - 24 z + 96, \\ &H_3(z) = z^3 - 48z^2 + 576z - 1536,\\ &H_4(z) = z^4 - 80z^3 + 1920z^2 - 15360z + 30720.\end{aligned}$$ We can expand an arbitrary polynomial $P_n(z)$ in terms of $\sum_{k = 0}^nH_k(z)$ through the inverse $$\begin{pmatrix} 1\\ z \\ z^2\\ \vdots \\ z^n \end{pmatrix} = \mathcal{D}_{n}^{-1} \begin{pmatrix} H_0\\ H_1 \\ H_2 \\ \vdots \\ H_n \end{pmatrix} \quad \textup{with} \quad \mathcal{D}_{n} = \begin{pmatrix} 1\\ A_{1,0} & 1\\ A_{2,0}& A_{2,1} & 1\\ \vdots & \vdots & & \ddots\\ A_{n,0} & A_{n, 1} & A_{n, 2}& \cdots & 1 \end{pmatrix},$$ where $A_{i,j}$ is given by [\[def:Ank\]](#def:Ank){reference-type="eqref" reference="def:Ank"}. A direct check yields $$\label{re:zntoHn} \mathcal{D}_n^{-1} =\{ |A_{i,j}|\}_{1 \leq i,j \leq n}, \quad z^n = \sum_{k = 0}^n |A_{n,k}| H_k(z),$$ from which and the orthogonality [\[eq:ortho\]](#eq:ortho){reference-type="eqref" reference="eq:ortho"} imply $$\int_0^\infty y^{2n} \phi_{2m}(y) \omega(y) dy = 0 \quad \textup{for}\;\; m \geq n+1.$$ #### Appearance of Type I log-profile: Consider $$(d, \ell) = (3,3) \quad \textup{and} \quad (d, \ell) = (4,2).$$ We decompose $q(y,s)$ according to the eigenspace of $\mathscr{L}_\alpha$, namely that $$\begin{aligned} q(y,s) & = \sum_{k \in \mathbb{N} } a_{k}(s)\phi_{2k}(y) \equiv \sum_{k \in \mathbb{N}} a_k(s) H_k(z), \quad z = 2\alpha y^2.\end{aligned}$$ Since $\sum_{k \geq \ell + 1} a_{k}(s) \phi_{2k}(y)$ is the projection of $q(y,s)$ on the negative mode of $\mathscr{L}_\alpha$, we may ignore it in this formal derivation, meaning that we only consider the ansatz $$q(y,s) = \sum_{k \leq \ell}a_{k}(s) \phi_{2k}(y).$$ By assuming the zero mode is dominant in the sense that $$\label{eq:assumall} |a_{k} (s)| \ll |a_{\ell}(s)| \;\; \textup{for}\;\; k \leq \ell -1,$$ we plugin this ansatz into [\[eq:qys\]](#eq:qys){reference-type="eqref" reference="eq:qys"} and project it onto the eigenmode $\phi_{2j}$ for $j = 0,..., \ell$: $$\begin{aligned} a_{j}' = (1 - 2j \alpha) a_{j} + \| \phi_{2j} \|_{L^2_\omega}^{-2} \left \langle NL, \phi_{2j}\right\rangle_{L^2_\omega},\end{aligned}$$ where $$NL = -\Big(\sum_{k \leq \ell} a_{k} \partial_y \phi_{2k}\Big)\Big(\sum_{k \leq \ell}a_{k}\partial_y\mathcal{K}_{\phi_{2k}} \Big) + \Big(\sum_{k \leq \ell}a_{k}\phi_{2k}\Big)^2.$$ From [\[eq:assumall\]](#eq:assumall){reference-type="eqref" reference="eq:assumall"} and the fact the $\int_0^\infty P_n(y) \omega(y)dy = \mathcal{O}(1)$, we see that $$\| \phi_{2j} \|_{L^2_\omega}^{-2}\left \langle NL, \phi_{2j}\right\rangle_{L^2_\omega} = \mathcal{O}\big(a_{\ell}^2\big) \quad \textup{and} \quad \| \phi_{2\ell} \|_{L^2_\omega}^{-2}\left \langle NL, \phi_{2\ell}\right\rangle_{L^2_\omega} = a_{\ell}^2 B_{\ell} + o\big(a_{\ell}^2\big),$$ where $$B_{\ell} = \| \phi_{2\ell} \|_{L^2_\omega}^{-2}\left \langle - \partial_y\phi_{2\ell} \partial_y \mathcal{K}_{\phi_{2\ell}} + \phi_{2\ell}^2, \phi_{2\ell}\right\rangle_{L^2_\omega}.$$ To compute $B_{\ell}$, we simply expand $$\label{eq:expNL} - \partial_y\phi_{2\ell} \partial_y \mathcal{K}_{\phi_{2\ell}} + \phi_{2\ell}^2 = \sum_{k = 0}^{2\ell} B_{k} \phi_{2k}(r) \equiv \sum_{k = 0}^{2\ell} B_{k} H_k(z),$$ from which we directly obtain the constant $B_\ell$ by the orthogonality [\[eq:ortho\]](#eq:ortho){reference-type="eqref" reference="eq:ortho"}.\ [\[sec:computeBell\]]{#sec:computeBell label="sec:computeBell"} #### Compute the constant $B_\ell$ in [\[eq:expNL\]](#eq:expNL){reference-type="eqref" reference="eq:expNL"}: From the definition $\phi_{2\ell}(y) = H_\ell(z)$ with $z = 2\alpha r^2$, we have $$8\alpha z^{\frac{d}{2}} \partial_z \mathcal{K}_{H_\ell} = - \int_0^z H_\ell(\xi) \xi^{\frac{d-2}{2}} d\xi.$$ We then write $$\begin{aligned} - \partial_y\phi_{2\ell} \partial_y \mathcal{K}_{\phi_{2\ell}} + \phi_{2\ell}^2 &= \frac{1}{y^{d-1}} \partial_y \big( y^{d-1} \phi_{2\ell}\partial_y \mathcal{K}_{\phi_{2\ell}} \big) = \frac{1}{ z^{\frac{d-2}{2}}}\partial_z \left(H_\ell \int_0^z H_\ell(\xi) \xi^{\frac{d-2}{2}} d\xi \right).\end{aligned}$$ For the case $(d =3, \ell = 3)$, we have $$\begin{aligned} - \partial_y\phi_{2\ell} \partial_y \mathcal{K}_{\phi_{2\ell}} + \phi_{2\ell}^2 &= \frac{1}{\sqrt{z}} \partial_z \left( H_3 \int_0^z H_3(\xi) \sqrt{\xi} d\xi\right) \nonumber\\ & = \frac 53 z^6 - \frac{416}{3}z^5 + \frac{12628}{3}z^4 - 57792z^3 + 364560 z^2 - 940800 z + 705600 \nonumber \\ & = \sum_{k = 0}^6 B_k H_6(z),\label{eq:B3ex}\end{aligned}$$ where the second line is computed by Matlab symbolic, and we have from [\[re:zntoHn\]](#re:zntoHn){reference-type="eqref" reference="re:zntoHn"} the value of $B_3$ is given by $$\label{def:valueB3} B_3 = \frac{5}{3}|A_{6,3}| - \frac{416}{3}|A_{5,3}| + \frac{12628}{3}|A_{4,3}| - 57792 = 39360.$$ Thus, the ODE satisfied by $a_3$ is $$a_3' \sim B_3 a_3^2, \qquad \textup{hence,}\quad a_3(s) \sim -\frac{1}{B_3 s}.$$ For the case $(d = 4, \ell = 2)$, we similarly compute $$\begin{aligned} - \partial_y\phi_{2\ell} \partial_y \mathcal{K}_{\phi_{2\ell}} + \phi_{2\ell}^2 &= \frac{1}{z} \partial_z \left( H_2 \int_0^z H_2(\xi) \xi d\xi\right) \nonumber\\ & = \frac 32 z^4 - 70 z^3 + 1056 z^2 - 5760 z + 9216 \nonumber \\ & = \frac{3}{2}H_4 + B_3 H_3 + B_2 H_2 + \cdots, \label{eq:B2ex}\end{aligned}$$ where we use [\[re:zntoHn\]](#re:zntoHn){reference-type="eqref" reference="re:zntoHn"} to compute the value of $B_2$ which is given by $$\label{def:valueB2} B_2 = \frac{3}{2} |A_{4,2}| - 70 |A_{3,2}| + 1056= 576.$$ Hence, $$a_2' \sim 576 a_2^2 \qquad \textup{hence,}\quad a_2(s) \sim -\frac{1}{576 s}.$$ In summary, we have found the asymptotic expansion in $L^2_\omega$ as follows: $$\begin{aligned} \label{eq:expinn} w(y,s) = 1 - \frac{1}{B_\ell s} \phi_{2\ell}(y) + \cdots = 1 - \frac{1}{B_\ell} \frac{(2\alpha y^2)^{\ell}}{s} + \cdots,\end{aligned}$$ where $B_\ell$ is given in [\[def:valueB3\]](#def:valueB3){reference-type="eqref" reference="def:valueB3"} and [\[def:valueB2\]](#def:valueB2){reference-type="eqref" reference="def:valueB2"} for the case $(d = 3, \ell =3)$ and $d = 2, \ell = 2$ respectively.\ #### Matching asymptotic expansions and the profile: The inner expansion [\[eq:expinn\]](#eq:expinn){reference-type="eqref" reference="eq:expinn"} suggests to look for a profile of the form $$w(y,s) \sim F(\xi) \quad \textup{with} \quad \xi = \frac{y}{s^\frac{1}{2\ell}} = \frac{|x|} {\sqrt{T-t} |\log (T-t)|^\frac{1}{2\ell}}.$$ Plugging this form to [\[eq:wys\]](#eq:wys){reference-type="eqref" reference="eq:wys"}, we obtain at the leading order the nonlocal ODE satisfied by $F$: $$\label{eq:odeF} \frac{1}{\xi^{d-1}} \partial_\xi \Big( F \int_0^\xi F(\zeta) \zeta^{d-1} d\zeta \Big) - \frac{1}{2}\xi F' - F = 0,$$ subjected to the initial condition $$\quad F(0) = 1.$$ One looks for solution of the form $$F(\xi) = \frac{1}{\xi^{d-1}} \partial_\xi (\xi^d Q),$$ and convert [\[eq:odeF\]](#eq:odeF){reference-type="eqref" reference="eq:odeF"} to the ODE $$\begin{aligned} \label{equation-Q} dQ\big(Q - 1/d \big) + \big(Q - 1/2 \big) \xi Q' = 0, \quad Q(0) = \frac{1}{d}.\end{aligned}$$ The two trivial solutions are $Q = 0$ and $Q = \frac{1}{d} < 1/2$ for $d \geq 3$. We observe from [\[equation-Q\]](#equation-Q){reference-type="eqref" reference="equation-Q"} that $Q(\xi)$ is a positive decreasing function on $(0,\infty)$, namely that $$\begin{aligned} Q(\xi)>0, \; Q'(\xi)<0\mbox{ for }\xi \in (0,\infty).\end{aligned}$$ As for the positivity, we assume there exists a first point $\xi^*\in (0,\infty)$ such that $Q(\xi^*)=0$ and $Q(\xi)>0$ for $\xi \in (0,\xi^*)$, then $Q'(\xi^*)<0$. From [\[equation-Q\]](#equation-Q){reference-type="eqref" reference="equation-Q"}, we have $-\frac{1}{2} Q'(\xi^*)=0$, which is a contradiction. As for the decreasing, we assume there exist a first point $\tilde \xi$ such that $Q'(\tilde\xi=0)$ and $Q'(\xi)<0$ for all $\xi\in (0,\tilde\xi)$, then equation [\[equation-Q\]](#equation-Q){reference-type="eqref" reference="equation-Q"} gives either $Q(\tilde\xi)=0$ or $Q(\tilde\xi)=\frac 1d$. The first case is not possible because of the strict positivity of $Q$ , the second case gives $Q(0)=Q(\tilde\xi)=\frac 1d$ for all $\xi\in (0,\tilde \xi)$ consequently $Q'(\xi)=0$ for all $\xi\in (0,\tilde \xi)$, which is a contradiction. We conclude that $Q$ is decreasing and connects $Q(0) = 1/d$ to $Q(\infty) = 0$. By solving $\xi = \xi(Q)$, we obtain the autonomous ODE $$\begin{aligned} \frac{\partial \xi}{\partial Q} + \frac{Q - 1/2}{dQ(Q - 1/d)} \xi = 0 \quad \mbox{with }\xi(0)=\frac 1d\end{aligned}$$ whose solution is implicitly given by $$\label{eq:formQ} c_\ell \xi^{2\ell} = \frac{1 - d Q}{Q^\ell}, \quad c_\ell \in \mathbb{R}_+.$$ From the initial condition $Q(0) = 1/d$, we look for an expansion near $\xi = 0$, $$\label{est:TaylorQat0} Q(\xi) = \frac{1}{d} - \frac{c_\ell}{d^{\ell + 1}} \xi^{2\ell} + \frac{\ell c_\ell^2}{d^{2\ell + 1}} \xi^{4\ell} + \mathcal{O}(\xi^{6\ell}) \quad \textup{for} \quad \xi \to 0.$$ As for $\xi$ large, the decay condition $Q(\xi) \to 0$ as $\xi \to \infty$, then from [\[eq:formQ\]](#eq:formQ){reference-type="eqref" reference="eq:formQ"} we see that $$\begin{aligned} Q(\xi) \sim c_\ell^{-\frac 1\ell} \xi^{-2}+\mathcal{O}(\xi^{-4}) \quad \textup{for} \quad \xi \to \infty. \label{est:QxiInf}\end{aligned}$$ From the relation $F(\xi) = dQ + \xi Q'$, we end up with $$\begin{aligned} F(\xi) = 1 - \frac{c_\ell (d + 2\ell)}{d^{\ell+1}} \xi^{2\ell} + \frac{c_\ell^2 \ell(d + 4\ell)}{d^{2\ell + 1}} \xi^{4\ell}+ \mathcal{O}(\xi^{6\ell}) \quad \textup{as}\;\; \xi \to 0.\end{aligned}$$ Matching this expansion with [\[eq:expinn\]](#eq:expinn){reference-type="eqref" reference="eq:expinn"} yields the value of $c_\ell$: $$\label{eq:con_cell} \frac{c_\ell (d + 2\ell)}{d^{\ell+1}} = \frac{(2\alpha)^\ell}{B_\ell}, \quad \textup{hence}, \quad c_\ell = \frac{(2\alpha)^\ell d^{\ell + 1}}{B_\ell (d + 2\ell)}.$$ The correct value of $c_\ell$ is crucial in many algebra cancellation in the rigorous analysis later (for example, improved estimates of the projection onto the null mode in Lemmas [Lemma 10](#lemm:EEhatys){reference-type="ref" reference="lemm:EEhatys"} and [\[lemm:finitepart\]](#lemm:finitepart){reference-type="eqref" reference="lemm:finitepart"}). # Linearized problem and bootstrap regime {#sec:linear} In this section, the constants $\ell$ and $\alpha$ are fixed as $$\ell = 3 \;\; \textup{for}\;\; d = 3 \quad \textup{and} \quad \ell = 2 \;\; \textup{for}\;\; d = 4, \quad \alpha = \frac{d-2}{2d} = \frac{1}{2\ell}.$$ We formulate the problem to show that there exist initial data so that the problem [\[eq:wys\]](#eq:wys){reference-type="eqref" reference="eq:wys"} has a global in time solution that satisfies $$\sup_{y \geq 0} \Big| w(y,s) - F\big(y s^{-\frac{1}{2\ell}} \big) \Big| \to 0 \quad \textup{as} \quad s \to \infty,$$ where $F(\xi) = \frac{1}{\xi^{d-1}}\partial_\xi(\xi^d Q(\xi)) = dQ(\xi) + \xi \partial_\xi Q(\xi)$ and $Q(\xi)$ is defined by [\[eq:formQ\]](#eq:formQ){reference-type="eqref" reference="eq:formQ"}. ## The partial mass setting Since we are working in the radial setting, it's convenient to work in the partial mass to simplify the analysis, namely we introduce $$\begin{aligned} \label{def:mw} m_w(y,s) = \int_0^y w(\zeta, s) \zeta^{d-1} d\zeta, \quad w(y,s) = \frac{\partial_y m_w}{y^{d-1}}, \quad \partial_y \mathcal{K}_w = -\frac{m_w}{y^{d-1}}.\end{aligned}$$ We write from [\[eq:wys\]](#eq:wys){reference-type="eqref" reference="eq:wys"} the equation for $m_w$, $$\label{eq:mw} \partial_s m_w = \partial_y^2m_w - \frac{d-1}{y}\partial_y m_w - \frac{1}{2}y \partial_y m_w + \frac{d-2}{2}m_w + \frac{m_w \partial_y m_w}{y^{d-1}}.$$ To keep the same scaling invariance of the original solution $w(y,s)$, we further introduce $$\begin{aligned} \label{def:vfrommw} v(y,s) = \frac{m_w(y,s)}{y^d}, \quad w = d v + y\partial_y v = \frac{1}{y^{d-1}}\partial_y (y^d v),\end{aligned}$$ where we write from [\[eq:mw\]](#eq:mw){reference-type="eqref" reference="eq:mw"} the equation for $v$: $$\label{eq:vys} \partial_s v =\Delta_{d+2} v - \frac{1}{2}y\partial_y v - v + dv^2 + y v\partial_y v,$$ where $\Delta_{d+2}$ stands for the Laplacian in dimension $d+2$ acting in the radial functions, i.e. $$\Delta_{d+2} = \partial_y^2 + \frac{d+1}{y}\partial_y.$$ We have found in the previous section through a formal spectral analysis and matching asymptotic expansions the following approximate blowup profile to [\[eq:vys\]](#eq:vys){reference-type="eqref" reference="eq:vys"}, $$Q = Q(\xi), \quad \forall \xi = ys^{-\frac{1}{2\ell}} \geq 0,$$ that solves [\[equation-Q\]](#equation-Q){reference-type="eqref" reference="equation-Q"} and is defined by [\[eq:formQ\]](#eq:formQ){reference-type="eqref" reference="eq:formQ"}. We recall that the profile $Q$ is strictly monotone and positive, $$\label{est:propQ} Q(0) = \frac{1}{d}, \quad \lim_{\xi \to \infty}Q(\xi) = 0, \quad Q(\xi) > 0, \quad Q'(\xi) < 0, \quad \forall \xi > 0.$$ The monotonicity of $Q$ makes the perturbative analysis simpler for the associated linearized problem from [\[eq:vys\]](#eq:vys){reference-type="eqref" reference="eq:vys"}. In contrary, a linearization from [\[eq:wys\]](#eq:wys){reference-type="eqref" reference="eq:wys"} around $F$ would make the analysis more complicated in terms of technicalities due to the lack of monotone property of $F$. This is one of the reasons we choose to work with the partial mass setting [\[def:vfrommw\]](#def:vfrommw){reference-type="eqref" reference="def:vfrommw"}. Nevertheless, there is always equivalent between the analysis with [\[def:vfrommw\]](#def:vfrommw){reference-type="eqref" reference="def:vfrommw"} and the one with the original variable $w(y,s)$ upto some technicalities. ## Linearization A linearization of [\[eq:vys\]](#eq:vys){reference-type="eqref" reference="eq:vys"} around the profile $Q$, namely $$v(y,s) = Q(\xi) + \varepsilon(y,s),$$ leads to the linearized problem $$\label{eq:vepys} \partial_s \varepsilon= \mathscr{H}\varepsilon+ NL(\varepsilon) + E,$$ where $\mathscr{H}$ is the second order linear operator $$\label{def:Hs} \mathscr{H}= \Delta_{d+2} - V_1 y \partial_y +V_2 ,$$ with $$\label{def:V12} V_1(\xi)= \frac{1}{2} - Q(\xi), \quad V_2(\xi) = 2dQ - 1 + \xi \partial_\xi Q,$$ and $E$ is the generated error, $$\label{def:E} E = \Delta_{d+2} Q - \partial_s Q.$$ and $NL$ is the nonlinear quadratic term $$\label{def:NLq} NL(\varepsilon) = d\varepsilon^2 + y \varepsilon\partial_y \varepsilon.$$ From the asymptotic behavior of the profile $Q(\xi)$ given in [\[est:TaylorQat0\]](#est:TaylorQat0){reference-type="eqref" reference="est:TaylorQat0"} and [\[est:QxiInf\]](#est:QxiInf){reference-type="eqref" reference="est:QxiInf"}, we observe $$V_1(\xi) = \frac{1}{2\ell} + \mathcal{O}(\xi^{2\ell}), \quad V_2(\xi) = 1 + \mathcal{O}(\xi^{2\ell}) , \quad \xi \ll 1,$$ and $$V_1(\xi) = \frac{1}{2} + \mathcal{O}(\xi^{-2}), \quad V_2(\xi) = -1 + \mathcal{O}(\xi^{-2}) , \quad \xi \gg 1,$$ Thus, the full linearized operator $\mathscr{H}$ behaves differently depending on the region: $$\label{def:Hxell} \mathscr{H}\sim \mathscr{H}_{\frac{1}{2\ell}} + \textup{Id}, \quad \textup{where}\quad \mathscr{H}_{\frac{1}{2\ell}} = \Delta_{d+2} - \frac{1}{2\ell} y\partial_y, \quad \textup{for}\;\; y \ll s^{-\frac{1}{2\ell}},$$ and $$\label{def:Hs12} \mathscr{H}\sim \mathscr{H}_{\frac{1}{2}} - \textup{Id}, \quad \textup{where}\quad \mathscr{H}_{\frac 12} = \Delta_{d+2} - \frac{1}{2} y\partial_y, \quad \textup{for}\;\; y \gg s^{-\frac{1}{2\ell}}.$$ We note that in *the outer region* $y \gg s^{-\frac{1}{2\ell}}$, the operator $\mathscr{L}$ behaves the same as for the one considered for the classical semilinear heat equation [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"}. However, *the inner region* $y \ll s^{-\frac{1}{2\ell}}$ and *the intermediate region* $y \sim s^{-\frac{1}{2\ell}}$, the operator behaves differently in comparison with what is known in the analysis for [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"}. To our knowledge, there is no a complete spectral theory for the full linear operator $\mathscr{H}$. We thus use a different approach to avoid this missed piece in the analysis, especially in the intermediate region. We recall here the spectral properties of $\mathscr{H}_{\frac{1}{2\ell}}$, which play an important role in the analysis when the inner region is concerned. The linear operator $\mathscr{H}_{\frac{1}{2\ell}}: H^2_{\rho}(\mathbb{R}_+) \to L^2_{\rho}(\mathbb{R}_+)$ is self-adjoint, where the weight function $$\rho(y) = e^{- \frac{|y|^2}{4\ell}} y^{d+1}.$$ In particular, we have from [\[eq:phi2n\]](#eq:phi2n){reference-type="eqref" reference="eq:phi2n"} and [\[def:vfrommw\]](#def:vfrommw){reference-type="eqref" reference="def:vfrommw"}, $$\mathscr{H}_{\frac{1}{2\ell}} \varphi_{2n} = - \frac{n}{\ell} \varphi_{2n}, \quad n \in \mathbb{N},$$ where the following relation holds $$\varphi_{2n}(y) = \frac{1}{y^{d}}\int_0^y \phi_{2n}(\zeta) \zeta^{d-1}d\zeta,$$ with $\phi_{2n}$ being defined explicitly in [\[def:phi2ell\]](#def:phi2ell){reference-type="eqref" reference="def:phi2ell"}. We use the even index $2n$ instead of $n$ to infer that we are in the radial setting and the eigenfunctions are only polynomials of even degrees. Note that $\varphi_{2n}$ has the same form as $\phi_{2n}$. We also have the orthogonality $$\int_0^\infty \varphi_{2n}(y) \varphi_{2m}(y) \rho(y) dy = c_{n} \delta_{n,m},$$ and the family of the eigenfunctions $\{ \varphi_{2n}\}_{n \in \mathbb{N}}$ forms a complete orthogonal basis in $L^2_\rho(\mathbb{R}_+)$ in the sense that for any $g \in L^2_\rho(\mathbb{R}_+)$ we can decompose it as $$g(y) = \sum_{n \in \mathbb{N}} g_n \varphi_{2n}(y), \quad g_n = \langle g, \varphi_{2n}\rangle_\rho = \int_0^\infty g \varphi_{2n} \rho dy.$$ ## Bootstrap regime Our aim is to construct a global in time solution $\varepsilon(y,s)$ to [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} that satisfies $$\|\varepsilon(s)\|_{L^\infty(\mathbb{R}_+)} \to 0 \quad \textup{as} \quad s \to \infty.$$ This requirement implies that the dynamics of [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} mainly relies on the linear part $\mathscr{H}$ since the nonlinear term is roughly quadratic. The construction is based on the following observation: #### - the outer region $y \gg s^\frac{1}{2\ell}$ ($\xi \gg 1$): thanks to the decay of $Q(\xi)$, the linear part $\mathscr{H}\sim \mathscr{H}_{\frac 12} - \textup{Id}$ has a fully negative spectrum. Thus, we can control the solution in this region without difficulties. In particular, let $K \gg 1$ be a large fixed constant and define $$\label{def:vep_out} \varepsilon^\textup{ex}(y,s) = \varepsilon(y,s) \big(1 - \chi_{_{K}}(\xi)\big),$$ where $$\label{def:chiK} \chi_{_K}(\xi) = \chi_{_0}\Big( \frac{\xi}{ K}\Big),$$ and $\chi_{_0}$ is the cut-off function $$\chi_{_0} \in \mathcal{C}^\infty\big(\mathbb{R}_+, [0,1]\big), \quad \chi_{_0}(\xi) = 1 \; \textup{if}\;\xi \in [0,1] \quad \textup{and} \quad \chi_{_0}(\xi) = 0\; \textup{if}\;\xi \geq 2.$$ From [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}, we write the equation for $\varepsilon^\textup{ex}(y,s)$, $$\label{eq:vepout} \partial_s \varepsilon^\textup{ex}= \big(\mathscr{H}_\frac 12 - \textup{Id} \big) \varepsilon^\textup{ex}+ \big(1 - \chi_{_{K}} \big)\big[ Q y\partial_y \varepsilon+ (2dQ + \xi \partial_\xi Q) \varepsilon+ NL(\varepsilon) + E\big] + \mathcal{E}^{bd}(\varepsilon),$$ and $\mathcal{E}^{bd}$ is the boundary term due to the cut-off defined by $$\label{def:EM} \mathcal{E}^{bd}(\varepsilon) = \Big(-\partial_s \chi_{_K} + \Delta_{d+2} \chi_{_K} - \frac 1{2\ell} y \partial_y \chi_{_K}\Big) \varepsilon+ 2\partial_y \chi_{_K}\partial_y \varepsilon.$$ We need here the information of $\varepsilon$ at the boundary $K s^{\frac{1}{2\ell}} \leq y \leq 2K s^\frac{1}{2\ell}$ ($K \leq \xi \leq 2K$) that we retrieve from the estimate of $\varepsilon$ in the intermediate region to close the estimate for $\varepsilon^\textup{ex}$.\ #### - the intermediate region $y \sim s^\frac{1}{2\ell}$ ($\xi \sim 1$): As mentioned earlier, there is a lack of a complete description of spectral property of $\mathscr{H}$, we are not able to use the semigroup estimate as for the nonlinear heat [\[eq:NLH\]](#eq:NLH){reference-type="eqref" reference="eq:NLH"} (see for example, [@MZdm97], [@NZens17]). Thanks to the monotone property of $Q$ and the dissipation, we can achieve the control of $\varepsilon$ in this region through a standard energy estimate from [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}. To obtain a small enough estimate, we need to refine the approximate solution by introducing $$\label{def:PsiApp} \Psi(y,s) = Q(\xi) + \hat{\Psi}(y,s), \quad \hat{\Psi}(y,s) = - \frac{1}{B_\ell s} \Big( \varphi_{2\ell}(y) - \frac{(2\alpha y^2)^\ell}{2\ell + d} \Big)\chi_{_0}(\xi),$$ where $B_\ell$ is the constant defined in [\[def:valueB2\]](#def:valueB2){reference-type="eqref" reference="def:valueB2"} and [\[def:valueB3\]](#def:valueB3){reference-type="eqref" reference="def:valueB3"} and $2\alpha = \frac{1}{\ell}$. Recall from [\[eq:expinn\]](#eq:expinn){reference-type="eqref" reference="eq:expinn"} and [\[def:vfrommw\]](#def:vfrommw){reference-type="eqref" reference="def:vfrommw"}, we have an equivalent inner expansion of $v(y,s)$ in $L^2_\rho$: $$\label{eq:InnerExpv} v(y,s) = \frac{1}{d} - \frac{1}{B_\ell s} \varphi_{2\ell}(y) + \cdots,$$ and from [\[est:TaylorQat0\]](#est:TaylorQat0){reference-type="eqref" reference="est:TaylorQat0"} and [\[eq:con_cell\]](#eq:con_cell){reference-type="eqref" reference="eq:con_cell"}, $$\label{eq:Qat0} Q(\xi) = \frac{1}{d} - \frac{1}{B_\ell(2\ell + d)} (2\alpha \xi^2)^\ell + \frac{\ell d }{B_\ell^2(2\ell + d)^2} (2\alpha \xi^2)^{2\ell} + \mathcal{O}(\xi^{6\ell}) \quad \textup{as} \quad \xi \ll 1.$$ Hence, we have for $y < s^\frac{1}{2\ell}$, $$\begin{aligned} \Psi(y,s) &= \frac{1}{d} - \frac{\varphi_{2\ell}(y) }{B_\ell s} + \frac{\ell d }{B_\ell^2(2\ell + d)^2} \frac{(2\alpha y^2)^{2\ell}}{s^2} + \mathcal{O}\Big( \frac{\langle y\rangle^{6\ell}}{ s^{3}} \Big), \label{eq:Psiat0}\end{aligned}$$ which agrees with the expansion [\[eq:InnerExpv\]](#eq:InnerExpv){reference-type="eqref" reference="eq:InnerExpv"}. We then linearize $$\begin{aligned} v(y,s) = \Psi(y,s) + \hat \varepsilon(y,s),\label{def:vephat}\end{aligned}$$ and write from [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} and the relation $\varepsilon= \hat{\Psi} + \hat \varepsilon$, $$\label{eq:vephat} \partial_s \hat \varepsilon= \mathscr{H}\hat \varepsilon+ \hat\mathcal{V}\hat\varepsilon+ NL(\hat \varepsilon) + \hat E,$$ where $\mathscr{H}$, $NL$ are defined in [\[def:Hs\]](#def:Hs){reference-type="eqref" reference="def:Hs"}, [\[def:NLq\]](#def:NLq){reference-type="eqref" reference="def:NLq"}, and $\hat \mathcal{V}$ is small linear operator $$\label{def:hatVc} \hat \mathcal{V}\hat \varepsilon= -\hat \Psi y \partial_y \hat \varepsilon+ (2d \hat \Psi + y\partial_y \hat \Psi) \hat \varepsilon,$$ and $\hat E$ is the generated error $$\label{def:Ehat} \hat E(y,s) = -\partial_s (Q(\xi) + \hat \Psi) + \Delta_{d+2}Q(\xi) + \mathscr{H}\hat \Psi + NL(\hat \Psi).$$ We introduce the following norm: for a fixed large constant $K\gg 1$, $$\label{def:normvepmid} \|\hat \varepsilon(s)\|_{\flat}^2 = \int_{0}^\infty (1 - \chi_{_K}(y)) \Big(\frac{|\hat \varepsilon|^2}{|y|^{4\ell + 2}}\Big)\frac{dy}{y}.$$ In fact, we can replace the power $4\ell + 2$ by any integer number $2k$ with $k \geq 2\ell + 1$, so that after some integration by parts, we get an estimate of the form $$\frac{d}{ds} \|\hat \varepsilon(s)\|_{\flat}^2 \leq -\delta(k) \|\hat \varepsilon(s)\|_{\flat}^2 + \|\hat E(s)\|_{\flat}^2 + \textup{"boundary terms $y \sim K$"},$$ where $\delta(k)$ is strictly positive for $k \geq 2\ell + 1$. Due to the cut-off $(1 - \chi_{_K}(y))$, we need information of $\hat \varepsilon$ for $K \leq y \leq 2K$ to estimate the boundary term and to complete the estimate of $\|\hat \varepsilon(s)\|_{\flat}$. This information is retrieved from the control in the inner region.\ #### - the inner region $y \ll s^\frac{1}{2\ell}$ ($\xi \ll 1$): The linearized operator $\mathscr{H}$ behaves like $\mathscr{H}_{\frac{1}{2\ell}} + \textup{Id}$ that has $\ell - 1$ positive eigenvalues, a zero mode and infinite many negative ones. We need further refinement to achieve the control of $\hat \varepsilon$ in this region. More precisely, we decompose $$\begin{aligned} \hat \varepsilon(y,s) = \Psi(y,s) + \hat{\varepsilon}_\natural(y,s) + \tilde \varepsilon(y,s), \label{decomp:vepys}\end{aligned}$$ where $$\label{def:vepnatural} \hat{\varepsilon}_\natural(y,s) = \sum_{k = 0}^{2\ell - 1} \hat \varepsilon_k(s) \varphi_{2k}(y), \quad \hat \varepsilon_k(s) = \langle \hat \varepsilon, \varphi_{2k}\rangle_\rho. \$$ The introduction of $\hat \varepsilon_\natural$ is to obtain a spectral gap estimate once we perform $L^2_\rho$ estimate for $\tilde \varepsilon$. In fact, we have the orthogonality condition $$\label{eq:orhtogonalvep} \langle \tilde \varepsilon, \varphi_{2k}\rangle_\rho = 0 \quad \textup{for} \quad k = 0, \cdots, \ell,$$ where $\varphi_{2k}$ is the eigenfunction associated to the linear operator $\mathscr{H}_{\frac{1}{2\ell}}$, from which we have the estimate $$\label{est:spectralgap} \langle \mathscr{H}_{\frac{1}{2\ell}} \tilde{ \varepsilon}, \tilde{\varepsilon} \rangle_\rho + \langle \tilde{\varepsilon}, \tilde{\varepsilon}\rangle_\rho \leq - \langle \tilde{\varepsilon}, \tilde{\varepsilon}\rangle_\rho.$$ From [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}, we write the equation for $\tilde{\varepsilon}$, $$\label{eq:veptil} \partial_s \tilde \varepsilon= \big(\mathscr{H}_\frac{1}{2\ell} + \textup{Id}\big) \tilde \varepsilon+ \tilde \mathcal{V}\tilde \varepsilon+ NL(\tilde{\varepsilon}) + \tilde E(y,s),$$ where $\mathscr{H}$ is the linearized operator around $Q$ introduced in [\[def:Hs\]](#def:Hs){reference-type="eqref" reference="def:Hs"}, $\mathcal{V}$ is a small first order linear term, $$\label{def:Vcqtil} \tilde \mathcal{V}= - \tilde{V}_1 y\partial_y + \tilde V_2 ,$$ with $$\label{def:V12til} \tilde{V}_1 = \frac{1}{d} - \Psi - \hat{\varepsilon}_\natural, \quad \tilde V_2 = 2d \Psi -2 + y \partial_y \Psi + 2d \hat{\varepsilon}_\natural + y\partial_y \hat{\varepsilon}_\natural,$$ the nonlinear term $NL(\tilde{\varepsilon})$ is defined by [\[def:NLq\]](#def:NLq){reference-type="eqref" reference="def:NLq"} and $E$ is the total error term, $$\label{def:Etil} \tilde E(y,s) = -\partial_s (\Psi + \hat{\varepsilon}_\natural) + \Delta_{d+2}Q(\xi) + \mathscr{H}(\hat \Psi+ \hat{\varepsilon}_\natural) + NL( \hat \Psi + \hat{\varepsilon}_\natural).$$ We now define the bootstrap regime to fully control the solution to [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}. **Definition 5** (Bootstrap regime). Let $s > 1$ and $A > 1$, we define $\mathcal{S}_A(s)$ the set of all functions $\varepsilon(s) \in W^{1,\infty}(\mathbb{R}_+)$ such that $$\label{est:vepk} |\hat \varepsilon_k(s)| \leq A s^{-2} \quad \textup{for} \quad 0 \leq k \ne \ell \leq 2\ell- 1,$$ $$\label{est:vepell} |\hat \varepsilon_\ell(s)| \leq A^2 s^{-2}\log s,$$ $$\label{est:vepL2rho} \| \tilde{\varepsilon}(s) \|_{L^2_\rho(\mathbb{R}_+)} \leq A s^{-3},$$ $$\label{est:vepmid} j =0,1,2, \quad \|(y\partial_y)^j \hat \varepsilon(s)\|_{\flat} \leq A^{1+j} s^{-1 - \frac{3}{2\ell}},$$ $$\label{est:vepout} j =0,1,\quad \|(y \partial_y)^j\varepsilon^\textup{ex}(s)\|_{L^\infty(\mathbb{R}_+)} \leq A^{4+j} s^{-\frac{1}{\ell}}, \quad \|y\varepsilon^\textup{ex}(s)\|_{L^\infty(\mathbb{R}_+)} \leq A^{4} s^{-\frac{1}{2\ell}},$$ where $\hat \varepsilon$, $\tilde \varepsilon$, $\varepsilon^\textup{ex}$ and $\|\cdot\|_{\flat}$ are introduced in [\[decomp:vepys\]](#decomp:vepys){reference-type="eqref" reference="decomp:vepys"}, [\[def:vep_out\]](#def:vep_out){reference-type="eqref" reference="def:vep_out"} and [\[def:normvepmid\]](#def:normvepmid){reference-type="eqref" reference="def:normvepmid"}. **Remark 6** (Order of estimates). The bootstrap estimates defined in the shrinking set $\mathcal{S}_A$ shows the prioprities in order to achieve the control of $\varepsilon$ in the whole $\mathbb{R}_+$ as follows: we first obtain $L^2_\rho$-estimate for $\tilde{\varepsilon}$ which directly gives $L^2_{loc}$-estimate, then a standard parabolic regularity yields $L^\infty(y \lesssim K)$ bound for $\hat \varepsilon$ for any $K > 0$; this $L^\infty_{loc}$-estimate is then used in the energy estimate of $\|\hat \varepsilon(s)\|_{\flat}$ to bound boundary terms (having the support $y \in [K, 2K]$) due to the cut-off $\chi_{_K}(y)$ (see [\[def:normvepmid\]](#def:normvepmid){reference-type="eqref" reference="def:normvepmid"}). A parabolic regularity type argument yields a similar estimate for $\|y\partial_y\hat \varepsilon(s)\|_{\flat}$, from which and Sobolev we get $L^\infty$-esimates for $y \lesssim K s^{\frac{1}{s^{2\ell}}}$. This $L^\infty$-bound for $y \sim K s^{\frac{1}{s^{2\ell}}}$ enters the estimate of $\|\varepsilon^\textup{ex}\|_{L^{\infty}}$ due to the cut-off $\chi_{_K}(\xi)$. We thus can obtain an $L^\infty$ bound for $\varepsilon$ in the whole $\mathbb{R}_+$. Since the nonlinear is quadratic, we just need a rough bound of $\|\varepsilon\|_{L^\infty(\mathbb{R})}$ to handle this nonlinear term in all estimates. ## Existence of solutions in the bootstrap regime The strategy is to show that if we start with initial data $\varepsilon(s_0) \in \mathcal{S}_A(s_0)$, then the corresponding solution $\varepsilon(s)$ to the equation [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} stays in $\mathcal{S}_A(s)$ for all $s \geq s_0$. In particular, we consider the initial data of the form $$\label{def:intitialdata_q} \varepsilon(y,s_0) = \hat{\Psi}(y,s_0) + \hat \varepsilon(y,s_0), \quad \hat \varepsilon(y,s_0) \equiv \hat \psi[\mathbf{d},A, s_0](y)= \frac{A}{s_0^2}\left(\sum_{i=0}^{\ell-1}d_i \varphi_{2i}\right) \chi_{_0}(\xi)$$ where $\hat{\Psi}$ is defined in [\[def:PsiApp\]](#def:PsiApp){reference-type="eqref" reference="def:PsiApp"}, $\varphi_{2i}$'s are the eigenfunctions of $\mathscr{H}_\frac{1}{2\ell}$, $\chi_{_0}$ is the cut-off function introduced right after [\[def:chiK\]](#def:chiK){reference-type="eqref" reference="def:chiK"}, $$s_0 \gg 1, \;\; A \gg 1, \; \; \mathbf{d} = (d_0, \cdots, d_{\ell - 1}) \in B_1(\mathbb{R}^{\ell}),$$ are real parameters to be determined for which the corresponding solution $\varepsilon(s)$ is trapped in $\mathcal{S}_A(s)$ for all $s \geq s_0$. Precisely, we aim at proving the following proposition which is the central of our construction leading to the conclusion of Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}. **Proposition 7** (Existence of solutions in $\mathcal{S}_A$). *There are $s_0 \gg 1 , A \gg 1$ and $\mathbf{d} \in B_1(\mathbb{R}^\ell)$ such that the solution $\varepsilon(s)$ to [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} with the initial data $\varepsilon(s_0) = \psi[\mathbf{d}, A, s_0]$ defined in [\[def:intitialdata_q\]](#def:intitialdata_q){reference-type="eqref" reference="def:intitialdata_q"} is trapped in $\mathcal{S}_A(s)$ for all $s \geq s_0$.* *Proof.* By the definition of $\hat \psi[\mathbf{d}, A, s_0]$ and the projection of $\hat\psi[\mathbf{d}, A, s_0]$ onto $\varphi_{2k}$, we obtain by a direct computation and the exponential decay of the weight function $\rho$, $$\hat \psi_k = \frac{A d_k}{s_0^2} +\mathcal{O}(s_0^{-2}e^{-\kappa s_0^{1/\ell}}) \quad \textup{for} \;\; k = 1, \cdots, \ell - 1,$$ and $$|\hat \psi_k | = \mathcal{O}( A s_0^{-2}e^{-\kappa s_0^{1/\ell}}) \quad \textup{for}\;\; k \geq \ell,$$ for some $\kappa > 0$, and $$\begin{aligned} |\tilde \psi(y)| &= \big| \hat \psi(y) - \sum_{k = 0}^{2\ell-1} \hat \psi_k \varphi_{2k}(y)\big| = \Big|\sum_{k = 0}^{\ell - 1} \hat \psi_k \varphi_{2k}(y) \big(1 - \chi_{_0}(\xi)\big) - \sum_{k = \ell}^{2\ell - 1}\hat \psi_k \varphi_{2k}(y) \Big|\\ & \lesssim \frac{A}{s_0^2} \langle y \rangle^{2\ell -2} \mathbf{1}_{\{\xi \geq 1\}} + A s_0^{-2}e^{-\kappa s_0^{1/\ell}} \langle y \rangle^{4\ell - 2}, \quad \forall y > 0. \end{aligned}$$ This yields the bounds $$\|\tilde \psi\|_{L^2_\rho} \lesssim A s_0^{-2}e^{-\kappa s_0^{1/\ell}}, \quad \sum_{j = 0}^2 \| (y\partial_y)^j \hat \psi\|_{\flat} \lesssim A s_0^{-2}e^{-\kappa s_0^{1/\ell}}.$$ By the definition [\[def:vep_out\]](#def:vep_out){reference-type="eqref" reference="def:vep_out"}, we have by $\chi_{_0}(\xi) \big(1 - \chi_{_K}(\xi) \big) = 0$ for $K \geq 2$, thus, $\varepsilon^\textup{ex}(s_0) \equiv 0$. We then conclude that for $A \gg 1$ and $s_0 \gg 1$, the initial data $\varepsilon(y,s_0) \in \mathcal{S}_A(s_0)$ with strictly inequalities, except for the first $\ell$ components $\hat \psi_k$ with $k = 0, \cdots, \ell - 1$.\ From the local Cauchy problem of [\[sys:KS\]](#sys:KS){reference-type="eqref" reference="sys:KS"} in the radial setting in $L^\infty(\mathbb{R}^d)$, for each initial data $\varepsilon( s_0) = \tilde{\Psi}(s_0) + \hat \psi_{\mathbf{d}, A, s_0} \in \mathcal{S}_A(s_0)$, there is a unique solution $\varepsilon(s) \in \mathcal{S}_A(s)$ for $s \in [s_0, s_*)$. If $s_* = +\infty$, we are done. If $s_* < \infty$, we have $\varepsilon(s_*) \in \partial \mathcal{S}_A(s_*)$. We claim that $\varepsilon(s_*)$ touches the boundary $\partial \mathcal{S}_A(s_*)$ only for the first $\ell$ components $\hat \varepsilon_k(s_*)$ with $k = 0, \cdots, \ell-1$. In particular, we claim the following: **Proposition 8** (Reduction to finite dimensional problem). *For $A \gg 1$, $s_0 = s_0(A) \gg 1$, there exists $\mathbf{d} = (d_0, \cdots, d_{\ell-1}) \in B_1(\mathbb{R}^{\ell})$ such that if the solution $\varepsilon(s)$ to [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"} with the initial data $\varepsilon(y,s_0) = \tilde{\Psi}(y,s_0) + \hat\psi[\mathbf{d}, A, s_0](y)$ defined as in [\[def:intitialdata_q\]](#def:intitialdata_q){reference-type="eqref" reference="def:intitialdata_q"} satisfies $\varepsilon(s) \in \mathcal{S}_A(s)$ for $s \in [s_0, s_*]$ and $\varepsilon(s_*) \in \partial \mathcal{S}_A(s_*)$, then it holds\ $$\label{est:qks*} \big(\hat \varepsilon_0, \cdots, \hat \varepsilon_{\ell-1}\big)(s_*) \in \partial \Big(\Big[ -\frac{A}{s_*^2}, \frac{A}{s_*^2}\Big]\Big)^{\ell}.$$ Moreover, we have $$\label{transversecross} \frac{d}{ds} \hat \varepsilon_k^2(s_*) > 0 \quad \textup{for}\;\; k = 0, \cdots, \ell-1.$$* Assuming Proposition [\[prop:3\]](#prop:3){reference-type="eqref" reference="prop:3"}, we see from [\[est:qks\*\]](#est:qks*){reference-type="eqref" reference="est:qks*"} that the map $$\begin{aligned} \Theta: [-1,1]^\ell & \to \partial \big([-1, 1]^{\ell} \big),\\ (d_0, \cdots, d_{\ell -1}) &\mapsto \frac{s_*^2}{A}\big(\hat \varepsilon_0, \cdots, \hat \varepsilon_{\ell-1} \big)(s_*)\end{aligned}$$ is well defined. From the transverse crossing [\[transversecross\]](#transversecross){reference-type="eqref" reference="transversecross"}, we see that $(\hat \varepsilon_0, \cdots, \hat \varepsilon_{\ell-1})(s)$ actually crosses it boundary at $s = s_*$, hence, $(\hat \varepsilon_0, \cdots, \hat \varepsilon_{\ell-1})(s)$ only leaves $\mathcal{S}_A(s)$ at $s = s_0$. This is a contradiction since $\Theta$ is the identity map on the boundary sphere and it can not be a continuous retraction of the unit ball. This concludes the proof of Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}, assuming Proposition [Proposition 8](#prop:3){reference-type="ref" reference="prop:3"}. ◻ # Control the solution in the bootstrap regime {#sec:ControlBs} ## Properties of the shrinking set We claim the following. **Lemma 9** (Properties of the shrinking set). *Let $A \gg 1$ and $s \geq s_0 \gg 1$ and $\varepsilon(s) \in \mathcal{S}_A(s)$ be a solution to [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}. We have\ i) (Local $L^\infty$-estimate) For all $M > 0$ and $j = 0,1$, $$\label{est:vepLinfloc} \|\partial_y^j \tilde \varepsilon(s)\|_{L^\infty(y \leq M)} \lesssim C(M) As^{-3}, \quad \|\partial_y^j \hat \varepsilon(s)\|_{L^\infty(y \leq M)} \lesssim C(M) A^2 s^{-2}|\log s|.$$ ii) (Pointwise estimate) $$\label{est:vephatpointswise} \forall y > 0, \quad |\hat \varepsilon(y,s)| + |y\partial_y \hat \varepsilon(y,s)| \lesssim A^3 s^{-1 - \frac{3}{2\ell}} \langle y \rangle^{2\ell + 1}.$$ iii) (Global $L^\infty$-estimate) $$\label{est:vepLinfbound} \| \hat \varepsilon(s)\|_{L^\infty(\mathbb{R}_+)} + \| (y\partial_y) \hat \varepsilon(s)\|_{L^\infty(\mathbb{R}_+)} \lesssim A^5s^{-\frac{1}{\ell}}.$$* *Proof.* (i) From the $L^2_\rho$ bound [\[est:vepL2rho\]](#est:vepL2rho){reference-type="eqref" reference="est:vepL2rho"} of $\tilde{\varepsilon}$, we get $$\| \tilde{\varepsilon}(s)\|_{L^2(y \leq 2M)} \lesssim e^{\frac{M^2}{2\ell}}\| \tilde \varepsilon(s)\|_{L^2_\rho} \lesssim C(M) A s^{-3}.$$ A standard parabolic regularity then yields the estimate $$\| \tilde{\varepsilon}(s)\|_{L^{\infty}(y \leq M)} + \| \partial_y\tilde{\varepsilon}(s)\|_{L^{\infty}(y \leq M)} \lesssim C(M)A s^{-3}.$$ We recall from the decomposition [\[decomp:vepys\]](#decomp:vepys){reference-type="eqref" reference="decomp:vepys"}, $$\label{def:vephat} \hat \varepsilon(y,s) = \sum_{k = 0}^{2\ell - 1} \hat \varepsilon_k(s) \varphi_{2k}(y) + \tilde{\varepsilon}(y,s).$$ Using the bootstrap bounds [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"}, [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"} and the $L^\infty$ local bound of $\tilde \varepsilon$ yields the desired estimates.\ (ii) We first claim that if $\int_{y \geq 1} (u^2 + |u\partial_y u|^2) y^{-1}dy < +\infty$, then $$\label{est:sobolev} \| u \|_{L^\infty(y \geq 1)} \lesssim \int_{y \geq 1} (u^2 + |y\partial_y u|^2) y^{-1}dy.$$ By making a change of variable $v(z) = u(M z)$, we write $$\begin{aligned} \| u\|^2_{L^\infty([M, 2M])} &= \| v\|^2_{L^\infty([1,2])} \lesssim \int_1^2 v^2(z) dz + \int_1^2 |\partial_z v(z)|^2 dz\\ & \lesssim \int_M^{2M} v^2(y) \frac{dy}{M} + \int_{M}^{2M} M|\partial_y v(y)|^2 dy \\ & \lesssim \int_M^{2M} v^2(y) y^{-1} dy + \int_{M}^{2M} |y \partial_y v(y)|^2 y^{-1} dy. \end{aligned}$$ Taking the supremum with respect to $M$ yields the inequality [\[est:sobolev\]](#est:sobolev){reference-type="eqref" reference="est:sobolev"}. We then apply [\[est:sobolev\]](#est:sobolev){reference-type="eqref" reference="est:sobolev"} with $u = \frac{\hat \varepsilon}{y^{2\ell + 1}}$ and $u = \frac{y\partial_y\hat \varepsilon}{ y^{2\ell + 1}}$ to obtain from [\[est:vepmid\]](#est:vepmid){reference-type="eqref" reference="est:vepmid"} $$\forall y \geq 1, \quad | \hat \varepsilon(y,s)| + |y\partial_y \hat \varepsilon(y,s)| \lesssim A^{3} s^{-1 - \frac{3}{2\ell}} \langle y \rangle^{2\ell + 1},$$ from which and [\[est:vepLinfloc\]](#est:vepLinfloc){reference-type="eqref" reference="est:vepLinfloc"}, we obtain [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"}.\ (iii) The estimate [\[est:vepLinfbound\]](#est:vepLinfbound){reference-type="eqref" reference="est:vepLinfbound"} follows from [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"} and the bootstrap bound [\[est:vepout\]](#est:vepout){reference-type="eqref" reference="est:vepout"}. ◻ ## Decomposition of the error We claim the following. **Lemma 10** (Estimate of the generated error). *We have\ (i) [($L^\infty$-bound of $E$ and $\hat E$)]{.upright} $$\label{est:ELinf} \|E(s)\|_{L^\infty(\mathbb{R}_+)} + \|y\partial_y E(s)\|_{L^\infty(\mathbb{R}_+)}+ \|\hat E(s)\|_{L^\infty(\mathbb{R}_+)} + \|y\partial_y \hat E(s)\|_{L^\infty(\mathbb{R}_+)} \lesssim s^{-\frac 1{\ell}}.$$ (ii) [(Decomposition of $\hat E$)]{.upright} $$\label{dec:Ehat} \hat E(y,s) = \sum_{k = 0}^{2\ell - 1} \hat E_k(s) \varphi_{2k}(y) + \hat R(y,s),$$ where $$\label{est:Ehatkandell} \sum_{k = 0, k \ne \ell}^{2\ell-1}|\hat E_k(s)| \lesssim s^{-2}, \quad \quad |\hat E_\ell(s)| \lesssim s^{-3}, \quad \|\hat R(s)\|_{L^2_\rho} \lesssim s^{-3},$$ and $$\label{est:Rhatpointwise} \forall y \lesssim s^\frac{1}{2\ell}, \quad \sum_{j = 0}^2 |(\langle y \rangle\partial_y)^j\hat R(y,s)| \lesssim s^{-3}\langle y \rangle^{6\ell - 2}.$$ In particular, we have $$\label{est:Ehatmid} \sum_{j = 0}^2\int_{1}^\infty \frac{|(y\partial_y)^j \hat E(y,s)|^2}{y^{4\ell +2}} \frac{dy}{y} \lesssim s^{-2 - \frac{3}{\ell}}.$$* **Remark 11**. The improved estimate for $\hat E_\ell$ reaching the order $s^{-3}$ comes from a crucial algebraic cancellation to remove the term of order $s^{-2}$ thanks to the precise choice of [\[eq:con_cell\]](#eq:con_cell){reference-type="eqref" reference="eq:con_cell"}. *Proof.* (i) From [\[def:E\]](#def:E){reference-type="eqref" reference="def:E"}, we have the estimate for all $y \geq 0,$ $$\begin{aligned} |E(y,s)| &= |E(\xi, s)| = \big|-\partial_s Q(\xi) + \Delta_{d+2}Q(\xi) \big|\\ & \lesssim s^{-1}\Big|\xi Q'(\xi)\Big| + s^{-\frac{1}{\ell}} \Big|Q''(\xi) + \xi^{-1}Q'(\xi) \Big| \lesssim s^{-\frac{1}{\ell}},\end{aligned}$$ and $$\begin{aligned} |y\partial_yE(y,s)| = |\xi \partial_\xi E(\xi, s)| = s^{-1}\Big| (\xi \partial_\xi)^2Q(\xi)\Big| + s^{-\frac{1}{\ell}} \Big| \xi \partial_\xi Q''(\xi) + \xi \partial_\xi (\xi^{-1}Q'(\xi))\Big| \lesssim s^{-\frac{1}{\ell}}.\end{aligned}$$ As for $\hat E$, we have by the definitions of $\varphi_{2\ell}$ and $\hat \Psi$, $$j = 0, \cdots, 3, \quad |(\langle y \rangle\partial_y)^j\hat \Psi(y,s)| \lesssim s^{-\frac{1}{\ell}} , \quad |\partial_s (\langle y \rangle\partial_y)^j\hat \Psi(y,s)| \lesssim s^{-1 - \frac{1}{\ell}}, \quad \forall y \geq 0.$$ Then, we have from [\[def:Ehat\]](#def:Ehat){reference-type="eqref" reference="def:Ehat"}, [\[def:Hs\]](#def:Hs){reference-type="eqref" reference="def:Hs"} and the bound $Q + |\xi \partial_\xi Q| \lesssim 1$, $$\begin{aligned} |\hat E(y,s)| &= | E(y,s) - \partial_s \hat \Psi + \mathscr{H}\hat \Psi + NL(\hat \Psi)| \\ & \lesssim |E(y,s)| + |\partial_s \hat \Psi| + |\Delta_{d+2} \hat \Psi| + |y\partial_y \hat \Psi| + |\hat \Psi| + |y \partial_y\hat \Psi \hat \Psi | + |\hat \Psi|^2 \lesssim s^{-\frac{1}{\ell}},\end{aligned}$$ and a similar bound for $|y\partial_y \hat E|$, which concludes the proof of $(i)$.\ (ii) From [\[def:Ehat\]](#def:Ehat){reference-type="eqref" reference="def:Ehat"}, [\[def:Hs\]](#def:Hs){reference-type="eqref" reference="def:Hs"} and [\[def:Hxell\]](#def:Hxell){reference-type="eqref" reference="def:Hxell"}, we rewrite $\hat E$ as $$\begin{aligned} \hat E(y,s) &= -\partial_s \Psi + \Delta_{d+1} Q(\xi) + \mathscr{H}_{\frac 1{2\ell}} \hat \Psi + \hat \Psi \\ & - \Big( \frac{1}{d} - Q(\xi) \Big) y\partial_y \hat \Psi + \big( 2d Q - 2 + \xi \partial_\xi Q\big) \hat \Psi + NL(\hat \Psi). \end{aligned}$$ We use the expansions [\[eq:Psiat0\]](#eq:Psiat0){reference-type="eqref" reference="eq:Psiat0"} and [\[eq:Qat0\]](#eq:Qat0){reference-type="eqref" reference="eq:Qat0"} of $\Psi$ and $Q$, the cancellations $\big(\mathscr{H}_{\frac{1}{2\ell}} + \textup{Id}\big) \varphi_{2\ell} = 0$ and $\big(- \frac{1}{2\ell} y\partial_y + \textup{Id}\big) y^{2\ell} = 0$ to write for $y \leq s^\frac{1}{2\ell}$ ($\xi \leq 1$), $$\begin{aligned} -\partial_s \Psi + &\Delta_{d+2} Q(\xi) + \mathscr{H}_{\frac 1{2\ell}} \hat \Psi + \hat \Psi \\ & \qquad = \frac{1}{s^2} \left[ - \frac{\varphi_{2\ell}}{B_\ell} + \frac{\ell d}{B_\ell^2(2\ell + d)^2} \Delta_{d + 2} (2\alpha y^2)^{2\ell}\right] + \mathcal{O}\Big( \frac{\langle y \rangle^{6\ell - 2}}{s^3} \Big).\end{aligned}$$ We notice that $$\tilde \varphi_{2\ell}(y) =\varphi_{2\ell}(y) - \frac{(2\alpha y^2)^\ell}{2\ell + d} = \mathcal{O}(\langle y \rangle^{2\ell - 2}), \quad \hat \Psi(y,s) = -\frac{1}{B_\ell s} \tilde \varphi_{2\ell} \chi_{_0}(\xi) = \mathcal{O}\Big( \frac{\langle y \rangle^{2\ell - 2}}{s}\Big),$$ and use again [\[eq:Qat0\]](#eq:Qat0){reference-type="eqref" reference="eq:Qat0"} to expand (keep track the terms of order $\mathcal{O}(s^{-2})$ only), $$\begin{aligned} &- \Big( \frac{1}{d} - Q(\xi) \Big) y\partial_y \hat \Psi = \frac{1}{s^2} \frac{(2\alpha y^2)^\ell }{B_\ell^2(2\ell + d)} y\partial_y \tilde \varphi_{2\ell} + \mathcal{O}\Big( \frac{\langle y \rangle^{6\ell - 2}}{s^3} \Big), \\ &\big( 2d Q - 2 + \xi \partial_\xi Q\big) \hat \Psi = \frac{1}{s^2} \frac{(2d + 2\ell) (2\alpha y^2)^\ell}{B_\ell^2 (2\ell + d)} \tilde \varphi_{2\ell} + \mathcal{O}\Big( \frac{\langle y \rangle^{6\ell - 2}}{s^3} \Big),\\ & NL(\hat \Psi) = \frac{1}{B_\ell^2 s^2} \left[ d \tilde \varphi_{2\ell}^2 + \frac{1}{2}y\partial_y \tilde \varphi_{2\ell}^2 \right] + \mathcal{O}\Big( \frac{\langle y \rangle^{6\ell - 2}}{s^3} \Big).\end{aligned}$$ A collection of these expansions yields $$\begin{aligned} \label{dec:Ehattmp} \hat E(y,s) = \frac{1}{B_\ell^2 s^2}P_{4\ell - 2}(y) + \hat{R}(y,s) \quad \textup{with} \quad \sum_{j=0}^2 | (\langle y \rangle\partial_y)^j \hat R(y,s)| = \mathcal{O}\Big( \frac{\langle y \rangle^{6\ell - 2}}{s^3} \Big),\end{aligned}$$ where $$\begin{aligned} P_{4\ell - 2}(y) &= -B_\ell \varphi_{2\ell} + \frac{\ell d}{(2\ell + d)^2} \Delta_{d + 2} (2\alpha y^2)^{2\ell} + \frac{(2\alpha y^2)^\ell }{(2\ell + d)} y\partial_y \tilde \varphi_{2\ell} \nonumber \\ & \qquad + \frac{(2d + 2\ell) (2\alpha y^2)^\ell}{(2\ell + d)} \tilde \varphi_{2\ell} + d \tilde \varphi_{2\ell}^2 + \frac{1}{2}y\partial_y \tilde \varphi_{2\ell}^2. \label{def:P4ell2}\end{aligned}$$ A projection of $\hat E$ onto $\varphi_{2k}$ immediately gives $$\hat E_k(s) = \langle \hat E, \varphi_{2k} \rangle_\rho = \mathcal{O}(s^{-2}), \quad k \in \mathbb{N}.$$ We claim that the projection of $P_{4\ell - 2}$ onto $\varphi_{2\ell}$ is identically zero, i.e. $$\label{est:ProjectionP4ell2} \langle P_{4\ell - 2}, \varphi_{2\ell} \rangle_\rho = 0,$$ from which we get the improved bound $$\hat E_\ell(s) = \langle \hat E, \varphi_{2\ell} \rangle_\rho = \langle P_{4\ell - 2}, \varphi_{2\ell} \rangle_\rho + \langle \hat R, \varphi_{2\ell} \rangle_\rho = \mathcal{O}(s^{-3}).$$ This concludes the proofs of [\[dec:Ehat\]](#dec:Ehat){reference-type="eqref" reference="dec:Ehat"}, [\[est:Ehatkandell\]](#est:Ehatkandell){reference-type="eqref" reference="est:Ehatkandell"} and [\[est:Rhatpointwise\]](#est:Rhatpointwise){reference-type="eqref" reference="est:Rhatpointwise"}. The estimate [\[est:Ehatmid\]](#est:Ehatmid){reference-type="eqref" reference="est:Ehatmid"} is straightforward from [\[dec:Ehattmp\]](#dec:Ehattmp){reference-type="eqref" reference="dec:Ehattmp"} and [\[est:ELinf\]](#est:ELinf){reference-type="eqref" reference="est:ELinf"}. Indeed, we have the estimate for $j = 0,1,2$, $$\begin{aligned} \int_{1}^\infty \frac{| (y \partial_y)^j \hat E |^2}{y^{4\ell + 2}} \frac{dy}{y} &\lesssim \int_1^{s^\frac{1}{2\ell}} \Big( \frac{| (y \partial_y)^j P_{4\ell - 2}|^2}{s^4 y^{4\ell + 2}} + \frac{| (y \partial_y)^j \hat R|^2}{ y^{4\ell + 2}}\Big) \frac{dy}{y} + \int_{s^\frac{1}{2\ell}}^\infty \frac{|(y \partial_y)^j \hat E|^2}{y^{4\ell +2}} \frac{dy}{y}\\ & \lesssim \int_1^{s^\frac{1}{2\ell}} \Big(\frac{y^{4\ell - 7}}{s^4} +\frac{y^{6\ell - 7}}{s^6} \Big) dy + s^{-\frac{2}{\ell}}\int_{s^\frac{1}{2\ell}}^\infty \frac{dy}{y^{4\ell +3}} \lesssim s^{-2 - \frac{3}{\ell}}.\end{aligned}$$ It remains to prove [\[est:ProjectionP4ell2\]](#est:ProjectionP4ell2){reference-type="eqref" reference="est:ProjectionP4ell2"} to complete the proof of Lemma [Lemma 10](#lemm:EEhatys){reference-type="ref" reference="lemm:EEhatys"}. ◻ *Proof of [\[est:ProjectionP4ell2\]](#est:ProjectionP4ell2){reference-type="eqref" reference="est:ProjectionP4ell2"}.* We use the exact value $(d, \ell)$ to simplify the computation which we can easily implement with Matlab symbolic. Recall from [\[def:phi2ell\]](#def:phi2ell){reference-type="eqref" reference="def:phi2ell"} and the relation $$\varphi_{2\ell}(y) = \frac{1}{y^d}\int_0^y \phi_{2\ell}(\zeta) \zeta^{d-1} d\zeta,$$ we have for $(d, \ell) = (3,3)$: $$\begin{aligned} \varphi_6(y) &= \frac{y^6}{243} - \frac{2}{3}y^4 + 28y^2 - 280, \quad \tilde{\varphi}_6(y) = - \frac{2}{3}y^4 + 28y^2 - 280,\\ P_{10}(y) &= -B_3 \varphi_6(y) -\frac{4\,y^{10} }{243}+\frac{1148\,y^8 }{243}-\frac{19264\,y^6 }{81}+\frac{17360\,y^4 }{3}-62720\,y^2 +235200,\end{aligned}$$ and recall from [\[def:valueB3\]](#def:valueB3){reference-type="eqref" reference="def:valueB3"} that $B_3 = 39360$ to get $$\frac{1}{\| \varphi_{6}\|^2_\rho} \langle P_{10}, \varphi_{6} \rangle_\rho = -B_3 + 39360 = 0.$$ Similarly, we have for $(d, \ell) = (4,2)$: $$\begin{aligned} \varphi_4(y) &= \frac{y^4}{32} - 2 y^2 + 24, \quad \tilde{\varphi}_4(y) = - 2 y^2 + 24,\\ P_{6}(y) &= -B_2 \varphi_4(y) - \frac{y^6}{8} + 33y^4 - 480y^2 + 2304,\end{aligned}$$ and recall from [\[def:valueB2\]](#def:valueB2){reference-type="eqref" reference="def:valueB2"} that $B_2 = 576$ to get $$\frac{1}{\| \varphi_{4}\|^2_\rho} \langle P_{6}, \varphi_{4} \rangle_\rho = -B_2 + 576 = 0.$$ This concludes the proof of [\[est:ProjectionP4ell2\]](#est:ProjectionP4ell2){reference-type="eqref" reference="est:ProjectionP4ell2"} and completes the proof of Lemma [Lemma 10](#lemm:EEhatys){reference-type="ref" reference="lemm:EEhatys"}. ◻ ## Dynamics of the finite dimensional part We obtain in this section the ODE satisfied by the finite dimensional part $\hat \varepsilon_\natural$. We claim the following. **Lemma 12** (Dynamics of the finite dimensional part $\hat \varepsilon_\natural$). *Let $\varepsilon(s) \in \mathcal{S}_A(s)$, we have\ (i) [(Decomposition of $\tilde E$)]{.upright} The function $\tilde{E}$ defined by [\[def:Etil\]](#def:Etil){reference-type="eqref" reference="def:Etil"} can be decomposed as $$\label{est:decompEys} \tilde E(y,s) = \sum_{k = 0, k\ne \ell}^{2\ell-1} \Big[ \hat E_k -\hat \varepsilon_k' + \big(1 - \frac{k}{\ell}\big)\hat \varepsilon_k \Big] \varphi_{2k}(y) + \big(\hat E_\ell - \hat \varepsilon_\ell' - \frac{2}{s} \hat \varepsilon_\ell\big) \varphi_{2\ell}+ \tilde R(y,s),$$ where $\hat E_k$ is introduced in [\[dec:Ehat\]](#dec:Ehat){reference-type="eqref" reference="dec:Ehat"} and satisfies the estimate [\[est:Ehatkandell\]](#est:Ehatkandell){reference-type="eqref" reference="est:Ehatkandell"}, $$\label{est:RtilL2rho} \|\tilde R(s)\|_{L^2_\rho} \lesssim s^{-3}.$$ (ii) [(Dynamics of $\hat \varepsilon_\natural$)]{.upright} $$\begin{aligned} \label{eq:ODEvepk} \sum_{k = 0, k \ne \ell}^{2\ell - 1}\left|\hat \varepsilon_k' + \big(1 - \frac{k}{\ell}\big)\hat \varepsilon_k \right| \lesssim s^{-2}, \quad \left| \hat \varepsilon_\ell' + \frac{2}{s} \hat \varepsilon_\ell \right| \lesssim s^{-3}. \end{aligned}$$* *Proof.* (i) From the definitions [\[def:Etil\]](#def:Etil){reference-type="eqref" reference="def:Etil"} and [\[def:Ehat\]](#def:Ehat){reference-type="eqref" reference="def:Ehat"} of $\tilde{E}$ and $\hat E$, the decomposition [\[dec:Ehat\]](#dec:Ehat){reference-type="eqref" reference="dec:Ehat"} and $\mathscr{H}_{\frac{1}{2\ell}} \varphi_{2k} = -\frac{k}{\ell} \varphi_{2k}$, we have by $$\begin{aligned} \tilde{E}(y,s) &= \hat E(y,s) - \partial_s \hat \varepsilon_\natural + (\mathscr{H}_\frac{1}{2\ell} + \textup{Id}) \hat \varepsilon_\natural - \tilde P_1 y \partial_y \hat \varepsilon_\natural + \tilde P_2 \hat \varepsilon_\natural + NL(\hat \varepsilon_\natural)\\ & = \sum_{k = 0}^{2\ell - 1}\Big[ \hat E_k - \hat \varepsilon_k' + \big(1 - \frac{k}{\ell} \big) \Big] \varphi_{2k} + \hat R - \tilde P_1 y \partial_y \hat \varepsilon_\natural + \tilde P_2 \hat \varepsilon_\natural + NL(\hat \varepsilon_\natural),\end{aligned}$$ where $NL(\hat \varepsilon_\natural) = d \hat \varepsilon_\natural^2 + y \hat \varepsilon_\natural\partial_y \hat \varepsilon_\natural$, $\tilde{P}_1$ and $\tilde{P}_2$ are defined by $$\tilde{P}_1 = \frac{1}{d} - \Psi, \quad \tilde{P}_2 = 2d \Psi - 2 + \xi \partial_\xi \Psi.$$ From the expansion [\[eq:Psiat0\]](#eq:Psiat0){reference-type="eqref" reference="eq:Psiat0"} of $\Psi$, we have the rough estimate $$\forall y \geq 0, \quad |\tilde{P}_1(y,s)| + |y \partial_y \tilde{P}_1(y,s)| + |\tilde{P}_2(y,s)| \lesssim \frac{\langle y \rangle^{2\ell}}{s}.$$ From the bootstrap bounds [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"} and [\[est:vepell\]](#est:vepell){reference-type="eqref" reference="est:vepell"}, we have $$\forall y \geq 0, \quad |\hat \varepsilon_\natural(y,s) - \hat \varepsilon_\ell \varphi_{2\ell}| \lesssim \frac{1}{s^2} \langle y \rangle^{4\ell - 2}, \quad |\hat \varepsilon_\natural(y,s)| \lesssim \frac{\log s}{s^2} \langle y \rangle^{4\ell - 2}.$$ Using these estimates, [\[est:Ehatkandell\]](#est:Ehatkandell){reference-type="eqref" reference="est:Ehatkandell"} and Cauchy-Schwarz inequality, we end up with $$\|\hat R - \tilde P_1 y \partial_y (\hat \varepsilon_\natural(y,s) - \hat \varepsilon_\ell \varphi_{2\ell}) + \tilde P_2 (\hat \varepsilon_\natural(y,s) - \hat \varepsilon_\ell \varphi_{2\ell}) + NL(\hat \varepsilon_\natural)\|_{L^2_\rho} \lesssim s^{-3}.$$ We claim the following: $$\label{est:tmpP12onvarp2ell} \langle - \tilde P_1 y \partial_y \varphi_{2\ell} + \tilde{P}_2 \varphi_{2\ell}, \varphi_{2\ell} \rangle_\rho \; \hat \varepsilon_\ell(s)= -\frac{2}{s} \hat \varepsilon_\ell(s) + \mathcal{O}(s^{-4}\log s ).$$ We recall from [\[eq:Psiat0\]](#eq:Psiat0){reference-type="eqref" reference="eq:Psiat0"} the expansion $\Psi(y,s) = \frac{1}{d} - \frac{\varphi_{2\ell}}{B_{\ell}s} + \mathcal{O}(s^{-2} \langle y \rangle^{2\ell})$, and write (keep track only terms of order $\mathcal{O}(s^{-1})$) $$\begin{aligned} - \tilde P_1 y \partial_y \varphi_{2\ell} + \tilde{P}_2 \varphi_{2\ell} = -\frac{2}{B_\ell s} \Big(d\varphi_{2\ell}^2 + y \varphi_{2\ell}\partial_y \varphi_{2\ell} \Big) + \mathcal{O}(s^{-2} \langle y \rangle^{6\ell}).\end{aligned}$$ A direct computation (Matlab symbolic) yields $$\frac{1}{\| \varphi_{2\ell}\|^2_\rho} \langle d\varphi_{2\ell}^2 + y \varphi_{2\ell}\partial_y \varphi_{2\ell}, \varphi_{2\ell} \rangle_\rho = \left\{ \begin{array}{ll} 39360 \; &\textup{if} \; (d, \ell) = (3,3)\\ 576 \; &\textup{if} \; (d, \ell) = (4,2) \end{array} \right. \equiv B_\ell,$$ which agrees with the formal computation given at page where the constant $B_\ell$ is the projection of the nonlinear term (in the original setting) onto the eigenmode $\phi_{2\ell}$. This proves [\[est:tmpP12onvarp2ell\]](#est:tmpP12onvarp2ell){reference-type="eqref" reference="est:tmpP12onvarp2ell"} and concludes the proof of [\[est:decompEys\]](#est:decompEys){reference-type="eqref" reference="est:decompEys"}.\ (ii) We project the equation [\[eq:veptil\]](#eq:veptil){reference-type="eqref" reference="eq:veptil"} onto the eigenmode $\varphi_{2k}$ and use the orthogonality [\[eq:orhtogonalvep\]](#eq:orhtogonalvep){reference-type="eqref" reference="eq:orhtogonalvep"} to get $$0 = \langle -\tilde V_1 y\partial_y \tilde{ \varepsilon} + \tilde V_2 \tilde{\varepsilon} + NL(\tilde \varepsilon) + \tilde{E}, \varphi_{2k} \rangle_\rho.$$ From [\[def:V12til\]](#def:V12til){reference-type="eqref" reference="def:V12til"}, [\[eq:Psiat0\]](#eq:Psiat0){reference-type="eqref" reference="eq:Psiat0"} and the bootstrap bounds [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"}, [\[est:vepell\]](#est:vepell){reference-type="eqref" reference="est:vepell"}, we have the rough bound $$\forall y \geq 0, \quad |\tilde{V}_1(y,s)| + |y\partial_y \tilde{V}_1(y,s)| + |\tilde{V}_2(y,s)| \lesssim \frac{\langle y \rangle^{4\ell - 2}}{s}.$$ We use Cauchy-Schwarz inequality, integration by parts and the fact that $\rho$ is exponential decay to estimate $$\big|\langle -\tilde V_1 y\partial_y \tilde{ \varepsilon} + \tilde V_2 \tilde{\varepsilon}, \varphi_{2k}\rangle_\rho \big| \lesssim s^{-1}\| \tilde{\epsilon}(s)\|_{L^2_\rho}.$$ For the nonlinear term, we use the relation $\tilde{\varepsilon} = \hat \varepsilon- \hat \varepsilon_\natural$, the pointwise estimate [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"} and the bootstrap bounds [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"}, [\[est:vepell\]](#est:vepell){reference-type="eqref" reference="est:vepell"} to get $$\forall y \geq 0, \quad |\tilde{\varepsilon}(y,s)| + |y \partial_y \tilde{\varepsilon}(y,s)| \lesssim A^6 s^{-1 - \frac{3}{2\ell}} \langle y \rangle^{4\ell - 2}.$$ Then, using Cauchy-Schwarz inequality and the exponential decay of $\rho$ yields $$\big| \langle NL(\tilde \varepsilon), \varphi_{2k} \rangle_\rho \big| \lesssim A^6 s^{-1 - \frac{3}{2\ell}} \| \tilde{\varepsilon}(s)\|_{L^2_\rho}.$$ Putting all these estimates together with [\[est:decompEys\]](#est:decompEys){reference-type="eqref" reference="est:decompEys"} and the bootstrap bound [\[est:vepL2rho\]](#est:vepL2rho){reference-type="eqref" reference="est:vepL2rho"} yield [\[eq:ODEvepk\]](#eq:ODEvepk){reference-type="eqref" reference="eq:ODEvepk"} and completes the proof of Lemma [Lemma 12](#lemm:finitepart){reference-type="ref" reference="lemm:finitepart"}. ◻ ## $L^2_\rho$-estimate We give the formulation to control $L^2_\rho$ of $\tilde{\varepsilon}$. The orthogonality [\[eq:orhtogonalvep\]](#eq:orhtogonalvep){reference-type="eqref" reference="eq:orhtogonalvep"}, which provides the spectral gap [\[est:spectralgap\]](#est:spectralgap){reference-type="eqref" reference="est:spectralgap"}, plays a crucial role in the improvement of $L^2_\rho$ bootstrap estimate [\[est:vepL2rho\]](#est:vepL2rho){reference-type="eqref" reference="est:vepL2rho"}. We claim the following. **Lemma 13** (Energy estimate in $L^2_\rho$). *Let $A \geq 1$ and $s \geq s_0 = s_0(A) \gg 1$ and $\varepsilon(s) \in \mathcal{S}_A(s)$, there is $\delta > 0$ such that $$\frac{d}{ds}\|\tilde \varepsilon\|_{L^2_\rho}^2 \leq - \frac{1}{2} \|\tilde \varepsilon\|_{L^2_\rho}^2 + Cs^{-6},$$ where $C$ is independent of $A$.* *Proof.* The proof is just a standard energy estimate in $L^2_\rho$ from the equation [\[eq:veptil\]](#eq:veptil){reference-type="eqref" reference="eq:veptil"}. We take the scalar product of [\[eq:veptil\]](#eq:veptil){reference-type="eqref" reference="eq:veptil"} with $\tilde{\varepsilon}$ in $L^2_\rho$ and use the spectral gap [\[est:spectralgap\]](#est:spectralgap){reference-type="eqref" reference="est:spectralgap"} to get $$\begin{aligned} \frac{1}{2}\frac{d}{ds}\|\tilde \varepsilon\|_{L^2_\rho}^2 \leq -\|\tilde \varepsilon\|_{L^2_\rho}^2 + \Big| \langle \tilde{\mathcal{V}}\tilde{\varepsilon} + NL(\tilde{\varepsilon})+ \tilde E, \tilde{\varepsilon} \rangle_\rho\big|. \end{aligned}$$ From the definition [\[def:Vcqtil\]](#def:Vcqtil){reference-type="eqref" reference="def:Vcqtil"} of $\tilde \mathcal{V}$ and integration by part, we get $$\begin{aligned} \Big|\langle \tilde{\mathcal{V}}\tilde{\varepsilon}, \tilde{\varepsilon}\rangle_\rho\Big| &= \Big|-\frac{1}{2}\int_0^\infty \tilde{V_1}y \partial_y \tilde{\varepsilon}^2 \rho dy + \int_0^\infty \tilde{V}_2 \tilde{\varepsilon}^2 \rho dy\Big| \lesssim \int_0^\infty \big(|y\partial_y \tilde V_1| + \langle y \rangle^2 |\tilde V_1| + |\tilde V_2| \big) \tilde{\varepsilon}^2 \rho dy. \end{aligned}$$ From the definition [\[def:V12til\]](#def:V12til){reference-type="eqref" reference="def:V12til"} of $\tilde{V}_1$ and $\tilde{V}_2$, we have the rough bound $$\label{est:V12tilpointwise} \forall y \geq 0, \quad |y\partial_y \tilde V_1| + \langle y \rangle^2 |\tilde V_1| + |\tilde V_2| \lesssim s^{-1} \langle y \rangle^C,$$ for some constant $C > 0$. Let $0< \kappa \ll 1$ be a small constant such that $$\forall y \leq s^\kappa, \quad s^{-1} \langle y \rangle^C \leq s^{-\kappa}.$$ We also get from [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"} and the definition [\[def:vepnatural\]](#def:vepnatural){reference-type="eqref" reference="def:vepnatural"} of $\hat \varepsilon_\natural$ to have the pointwise bound $$\label{est:veptilpointwise} \forall y \geq 0, \quad |\tilde \varepsilon(y,s)| \lesssim |\hat \varepsilon(y,s)| + |\hat \varepsilon_\natural(y,s)| \lesssim A^3 s^{-1 - \frac{3}{2\ell}} \langle y \rangle^C.$$ By splitting the integral and using [\[est:V12tilpointwise\]](#est:V12tilpointwise){reference-type="eqref" reference="est:V12tilpointwise"}, [\[est:veptilpointwise\]](#est:veptilpointwise){reference-type="eqref" reference="est:veptilpointwise"}, we obtain $$\begin{aligned} \Big|\langle \tilde{\mathcal{V}}\tilde{\varepsilon}, \tilde{\varepsilon}\rangle_\rho\Big| &\lesssim s^{-1}\int_0^{s^\kappa} \langle y \rangle^C \tilde{\varepsilon}^2 \rho dy + A^{6}s^{-3 - \frac{3}{\ell}}\int_{s^\kappa}^\infty \langle y \rangle^C e^{-\frac{|y|^2}{2\ell}} dy\\ & \lesssim s^{-\kappa} \|\tilde{\varepsilon}\|^2_{L^2_\rho} + A^{6}e^{-\eta s^{2\kappa}},\end{aligned}$$ for some $\eta > 0$. Arguing in a similar way to estimate the nonlinear term by using [\[est:veptilpointwise\]](#est:veptilpointwise){reference-type="eqref" reference="est:veptilpointwise"} and integration by parts, we obtain $$\begin{aligned} \big| \langle NL(\tilde\varepsilon), \tilde{\varepsilon}\rangle_\rho\big| &\lesssim \Big| \int_0^\infty d \tilde{\varepsilon}^3 \rho dy + \frac{1}{3}\int_0^\infty y\partial_y \tilde{\varepsilon}^3 \rho dy \Big| \lesssim \int_0^\infty \langle y \rangle^2 |\tilde{\varepsilon}|^3 \rho dy\\ &\lesssim A^3 s^{-1 - \frac{3}{2\ell}} \int_0^{s^\kappa} \langle y \rangle^C |\tilde \varepsilon|^2 \rho dy + A^{9} s^{-3 - \frac{9}{2\ell}}\int_{s^\kappa}^\infty \langle y \rangle^{3C + d+1} e^{ -\frac{|y|^2}{2\ell}}dy\\ &\lesssim A^3s^{-\kappa} \|\tilde{\varepsilon}\|^2_{L^2_\rho} + A^{9}e^{-\eta s^{2\kappa}}. \end{aligned}$$ For the error term, we use the decomposition [\[est:decompEys\]](#est:decompEys){reference-type="eqref" reference="est:decompEys"}, the orthogonality [\[eq:orhtogonalvep\]](#eq:orhtogonalvep){reference-type="eqref" reference="eq:orhtogonalvep"}, Cauchy-Schwarz inequality and the estimate [\[est:RtilL2rho\]](#est:RtilL2rho){reference-type="eqref" reference="est:RtilL2rho"} to obtain $$\begin{aligned} \big| \langle \tilde{E}, \tilde \varepsilon\rangle_\rho \big| = \big| \langle \tilde{R}, \tilde \varepsilon\rangle_\rho \big| \leq \frac{1}{4} \|\tilde{\varepsilon}\|^2_{L^2_\rho} + C\|\tilde{R}\|^2_{L^2_\rho} \leq \frac{1}{4} \|\tilde{\varepsilon}\|^2_{L^2_\rho} + Cs^{-6}. \end{aligned}$$ Putting together all the estimates and take $s_0 = s_0(A) \gg 1$ yields the desired formulation and concludes the proof of Lemma [Lemma 13](#lemm:L2rho){reference-type="ref" reference="lemm:L2rho"}. ◻ ## Estimate for the intermediate region We perform an energy estimate to control the solution in the intermediate region $y \lesssim s^\frac{1}{2\ell} (\xi \lesssim 1)$. We claim the following. **Lemma 14** (Energy estimate in the intermediate region). *Let $A \geq 1$ and $s \geq s_0 = s_0(A) \gg 1$ and $\varepsilon(s) \in \mathcal{S}_A(s)$. We have $$\begin{aligned} \frac{d}{ds} \|\hat \varepsilon(s)\|^2_{\flat}& \leq -\delta \|\hat \varepsilon(s)\|^2_{\flat} + C s^{-2 - \frac{3}{\ell}}, \label{est:vepmid0}\\ \frac{d}{ds} \|y\partial_y\hat \varepsilon(s)\|^2_{\flat} &\leq -\delta \|y\partial_y\hat \varepsilon(s)\|^2_{\flat} + C \big(\|\hat \varepsilon(s)\|^2_{\flat} + s^{-2 - \frac{3}{\ell}}\big),\label{est:vepmid1}\\ \frac{d}{ds} \|(y\partial_y)^2\hat \varepsilon(s)\|^2_{\flat} &\leq -\delta \|(y\partial_y)^2\hat \varepsilon(s)\|^2_{\flat} + C \big( \|y\partial_y\hat \varepsilon(s)\|^2_{\flat} + \|\hat \varepsilon(s)\|^2_{\flat} + s^{-2 - \frac{3}{\ell}}\big), \label{est:vepmid2}\end{aligned}$$ where $\delta > 0$ and $C = C(K) > 0$ is independent of $A$.* *Proof.* We begin with [\[est:vepmid0\]](#est:vepmid0){reference-type="eqref" reference="est:vepmid0"}. To ease the notation, we write in this proof $$\chi_{_K}(y) = \chi(y), \quad \int_0^\infty \square dy = \int \square dy.$$ From the equation [\[eq:vephat\]](#eq:vephat){reference-type="eqref" reference="eq:vephat"}, we have $$\begin{aligned} \frac{1}{2} \frac{d}{ds} \|\hat \varepsilon(s)\|^2_{\flat} &= \int \big(1 - \chi\big) \big(\mathscr{H}\hat \varepsilon+ \hat \mathcal{V}\hat \varepsilon+ NL(\hat \varepsilon) + \hat E \big) \frac{\hat \varepsilon}{y^{4\ell + 3}} dy.\end{aligned}$$ We rewrite from the definition [\[def:Hs\]](#def:Hs){reference-type="eqref" reference="def:Hs"} of $\mathscr{H}$ and $\Psi = Q + \hat \Psi$, $$\mathscr{H}+ \hat \mathcal{V}= \Delta_{d+2} + \Big( \Psi - \frac{1}{2} \Big)y\partial_y + \Big( 2d \Psi - 1 + y\partial_y \Psi \Big).$$ We compute by integration by parts and use the fact that the compact support of $\partial_y \chi$ and $\partial_y^2 \chi$ is in $(K, 2K)$, $$\begin{aligned} \int(1 - \chi) \frac{ \hat \varepsilon\Delta_{d+2} \hat \varepsilon}{y^{4\ell + 3}} dy &\leq - \int (1 - \chi) \frac{|\partial_y \hat \varepsilon|^2}{y^{4\ell + 3}} dy + C\int \hat\varepsilon^2 \Big( \frac{|\partial_y \chi|}{y^{4\ell + 4}} + \frac{|\partial_y^2 \chi|}{y^{4\ell +3}} \Big)dy + C \int \frac{\hat \varepsilon^2(1 - \chi)}{y^{4\ell + 5}} dy\\ &\leq - \int (1 - \chi) \frac{|\partial_y \hat \varepsilon|^2}{y^{4\ell + 3}} dy + \frac{C}{K^{4\ell + 3}} \int_K^{2K} |\hat \varepsilon|^2 dy + \frac{C}{K^2}\| \hat\varepsilon\|^2_\flat. \end{aligned}$$ Using the relation $\hat \varepsilon= \tilde{\varepsilon} + \hat \varepsilon_\natural$ and the bootstrap bounds in Definition [Definition 5](#Definition-shrinking -set){reference-type="ref" reference="Definition-shrinking -set"}, we obtain $$\begin{aligned} \int_K^{2K} |\hat \varepsilon|^2 dy &\leq \frac{e^{\frac{K^2}{\ell}}}{ K^{d+1}} \|\hat \varepsilon\|_{L^2_\rho}^2 \leq \frac{e^{\frac{K^2}{\ell}}}{ K^{d+1}} \big( \|\tilde \varepsilon\|_{L^2_\rho}^2 + \| \hat \varepsilon_\natural\|^2_{L^2_\rho} \big) \nonumber \\ & \qquad \leq \frac{e^{\frac{K^2}{\ell}}}{ K^{d+1}}\Big(\frac{A^{6}}{ s^{6}} + \frac{A^4 \log^2s}{ s^{4}}\Big) \leq s^{-2 - \frac{3}{\ell}}. \label{est:vephapK}\end{aligned}$$ Using integration by parts , we derive $$\begin{aligned} &\int \frac{(1 - \chi)\hat \varepsilon}{y^{4\ell + 3}} \left[\Big( \Psi - \frac{1}{2} \Big)y\partial_y \hat \varepsilon+ \Big( 2d \Psi - 1 + y\partial_y \Psi \Big) \hat \varepsilon\right]dy\\ & \quad =-\int \frac{(1 - \chi)\hat \varepsilon^2}{y^{4\ell + 3}}\Big[(2\ell +1) \big( \frac 12 - \Psi \big) - \frac{1}{2} y\partial_y\Psi + 1 - 2d \Psi\Big] dy + \int \frac{\hat \varepsilon^2 \partial_y \chi}{2 y^{4\ell +2}} dy. \end{aligned}$$ We now use the monotonicity of $Q$ stated in [\[est:propQ\]](#est:propQ){reference-type="eqref" reference="est:propQ"} and the fact that $\|\hat \Psi(s)\|_\infty = \mathcal{O}(s^{-\frac{1}{2\ell}})$ to have $$\forall y \geq 0, \quad \frac{1}{2} - \Psi(y,s) \geq \frac{1}{2\ell} - Cs^{-\frac{1}{2\ell}}, \quad \Psi(y,s) \leq \frac{1}{d} + Cs^{-\frac{1}{2\ell}}, \quad y\partial_y \Psi(y,s) \leq Cs^{-\frac{1}{2\ell}},$$ hence, $$\forall y \geq 0, \quad (2\ell +1) \big( \frac 12 - \Psi \big) - \frac{1}{2} y\partial_y\Psi + 1 - 2d \Psi \geq \frac{2\ell + 1}{2\ell} - 1 - Cs^{-\frac{1}{2\ell}} = \frac{1}{2\ell} - Cs^{-\frac{1}{2\ell}} \geq \frac{1}{4\ell}.$$ The term with cutoff $\partial_y \chi$ is simply estimated as in [\[est:vephapK\]](#est:vephapK){reference-type="eqref" reference="est:vephapK"}, $$\Big|\int \frac{\hat \varepsilon^2 \partial_y \chi}{2 y^{4\ell +2}} dy \Big| \lesssim K^{-4\ell -2}\int_K^{2K} |\hat \varepsilon|^2 dy \lesssim s^{-2 - \frac{3}{\ell}}.$$ Hence, by taking $K \gg 1$ large, the full linear term is estimate by $$\label{est:linearestimate} \int (1 - \chi) \frac{\hat \varepsilon(\mathscr{H}+ \hat \mathcal{V}) \hat \varepsilon}{y^{4\ell +3}} dy \leq - \int (1 - \chi) \frac{|\partial_y \hat \varepsilon|^2}{y^{4\ell + 3}} dy -\frac{1}{6\ell} \| \hat \varepsilon(s)\|^2_\flat + Cs^{-2 - \frac{3}{\ell}}.$$ As for the nonlinear term, we estimate by using [\[est:vepLinfbound\]](#est:vepLinfbound){reference-type="eqref" reference="est:vepLinfbound"}, $$\begin{aligned} \Big| \int (1 - \chi) \frac{\hat \varepsilon NL(\hat \varepsilon)}{y^{4\ell + 3}}dy \Big| &= \Big| \int (1 - \chi) \frac{\hat \varepsilon^2 (d\hat \varepsilon+ y \partial_y \hat \varepsilon)}{y^{4\ell + 3}}dy \Big|\\ & \qquad \leq (\|\hat \varepsilon(s)\|_\infty + \| y\partial_y \hat \varepsilon(s)\|_\infty) \|\hat \varepsilon(s)\|^2_\flat \lesssim A^8s^{-\frac{1}{\ell}} \|\hat \varepsilon(s)\|^2_\flat. \end{aligned}$$ For the error term, we use Cauchy-Schwarz inequality and [\[est:Ehatmid\]](#est:Ehatmid){reference-type="eqref" reference="est:Ehatmid"}, $$\begin{aligned} \int (1 - \chi) \frac{|\hat \varepsilon| |\hat E|}{y^{4\ell +3}} dy \leq \frac{1}{8\ell} \|\hat \varepsilon(s)\|^2_\flat + C\| \hat E\|^2_\flat \leq \frac{1}{8\ell} \|\hat \varepsilon(s)\|^2_\flat + Cs^{-2 - \frac{3}{\ell}}. \end{aligned}$$ Collecting all the above bounds and taking $K \gg 1$ and $s_0(A) \gg 1$, we end up with $$\begin{aligned} \frac{1}{2} \frac{d}{ds}\|\hat \varepsilon(s)\|^2_\flat &\leq \Big( - \frac{1}{6\ell} + \frac{1}{8\ell} + \frac{CA^8}{s^{\frac{1}{\ell}}}\Big) \|\hat \varepsilon(s)\|^2_\flat + Cs^{-2 - \frac{3}{\ell}} \leq - \delta \|\hat \varepsilon(s)\|^2_\flat + Cs^{-2 - \frac{3}{\ell}}, \end{aligned}$$ for some $\delta > 0$, which concludes the proof of [\[est:vepmid0\]](#est:vepmid0){reference-type="eqref" reference="est:vepmid0"}. The derivation of [\[est:vepmid1\]](#est:vepmid1){reference-type="eqref" reference="est:vepmid1"} and [\[est:vepmid2\]](#est:vepmid2){reference-type="eqref" reference="est:vepmid2"} is similar as for [\[est:vepmid0\]](#est:vepmid0){reference-type="eqref" reference="est:vepmid0"}. Indeed, the equations satisfied by $$g_1 = y\partial_y \hat \varepsilon, \quad g_2 = y\partial_y g_1,$$ have the same forms as for $\hat \varepsilon$ with an extra commutator, $$\begin{aligned} \partial_s g_1 &= (\mathscr{H}+ \hat\mathcal{V})g_1 + [y\partial_y, \mathscr{H}+ \hat \mathcal{V}] \hat \varepsilon+ y\partial_y (NL(\hat \varepsilon) + \hat E), \\ \partial_s g_2 &= (\mathscr{H}+ \hat\mathcal{V})g_2 + [y\partial_y, \mathscr{H}+ \hat \mathcal{V}] g_1 + y\partial_y ( [\mathscr{H}+ \hat \mathcal{V}, y\partial_y] \hat \varepsilon) + (y\partial_y)^2(NL(\hat \varepsilon) + \hat E).\end{aligned}$$ The linear part is estimated as in [\[est:linearestimate\]](#est:linearestimate){reference-type="eqref" reference="est:linearestimate"} that provides a dissipative term and a coercive estimate with the constant $- \frac{1}{6\ell}$. The commutator term $[y\partial_y, \mathscr{H}+ \hat \mathcal{V}] \hat \varepsilon$ is then controlled either by the dissipation or by the norm $\|\hat \varepsilon(s)\|^2_\flat$, and similarly for the term $[y\partial_y, \mathscr{H}+ \hat \mathcal{V}] g_1$. The nonlinear term and the error term are estimated by integration by parts, Cauchy-Schwarz inequality and the dissipation with the provided estimate [\[est:Ehatmid\]](#est:Ehatmid){reference-type="eqref" reference="est:Ehatmid"} of the error term that we omit the detail here. This concludes the proof of Lemma [Lemma 14](#lemm:mid){reference-type="ref" reference="lemm:mid"}. ◻ ## Estimate for the outer region {#sec:outer} This section is devoted to the control of the remainder $\varepsilon$ in the outer region $y \gg s^\frac{1}{2\ell} (\xi \gg 1)$ based on the well-known semigroup properties of the Hermite operator $\mathscr{L}_\eta = \Delta - \eta z\cdot \nabla$ with $\eta = \frac{1}{2}$. Let $\chi_{_K}$ be the cut-off function defined by [\[def:chiK\]](#def:chiK){reference-type="eqref" reference="def:chiK"} and recall the definition of $\varepsilon^\textup{ex}$, $$\varepsilon^\textup{ex}(y,s) = \varepsilon(y,s) (1 - \chi_{_K}(\xi)), \quad \xi = ys^{-\frac{1}{2\ell}}.$$ In what follows, we write without distinguishing $$\varepsilon= \varepsilon(z, s) \equiv \varepsilon(y, s), \quad \varepsilon^\textup{ex}= \varepsilon^\textup{ex}(z, s) \equiv \varepsilon^\textup{ex}(y, s), \quad y = |z|, \;\; z \in \mathbb{R}^d,$$ and notice that $$z\cdot\nabla_z \varepsilon\equiv y\partial_y \varepsilon, \quad \nabla \cdot(z \varepsilon) = z\cdot\nabla \varepsilon+ d \varepsilon\equiv y\partial_y \varepsilon+ d \varepsilon,$$ From [\[eq:vepys\]](#eq:vepys){reference-type="eqref" reference="eq:vepys"}, we have the equation satisfied by $\varepsilon^\textup{ex}$, $$\label{eq:vepout1} \partial_s \varepsilon^\textup{ex}= \big(\mathscr{L}_\eta - \textup{Id}\big)\varepsilon^\textup{ex}+ F + \mathcal{E}^{bd}, \quad \eta = \frac{1}{2},$$ where $$\begin{aligned} F &= \big(1 - \chi_{_{K}} \big)\big[ \big( 2y^{-2} + Q\big) y\partial_y \varepsilon+ (2dQ + y \partial_yQ) \varepsilon+ NL(\varepsilon) + E\big], \nonumber \\ & = \big(1 - \chi_{_{K}} \big)\big[ P_1 y\partial_y \varepsilon+ P_2 \varepsilon+ NL(\varepsilon) + E\big] = \big(1 - \chi_{_{K}} \big) \hat F, \label{def:F} \\ \mathcal{E}^{bd} & = \Big(-\partial_s \chi_{_K} + \Delta \chi_{_K} - \frac 1{2\ell} y \partial_y \chi_{_K}\Big) \varepsilon+ 2\partial_y \chi_{_K}\partial_y \varepsilon. \label{def:EcbdK}\end{aligned}$$ We restate some well-known semigroup properties of the Hermite operator $\mathscr{L}_\eta = \Delta - \eta z.\nabla$ acting on general functions (not necessary radially symmetric) defined from $\mathbb{R}^{d}$ to $\mathbb{R}$ **Lemma 15** (Properties of the semigroup $e^{s \mathscr{L}_\eta}$). *The kernel of the semigroup $e^{s \mathscr{H}_\eta}$ is given by $$\label{def:kernelLalp} e^{s\mathscr{L}_\eta}(z,\xi) = \frac{1}{[2\pi \eta (1 - e^{-s})]^\frac{d}{2}} \exp\Big( - \frac{\eta}{2} \frac{ |z e^{-s/2} - \xi|^2}{(1 - e^{-s})} \Big).$$ The action of $e^{s \mathscr{L}_\eta}$ on the function $g: \mathbb{R}^{d} \to \mathbb{R}$ is defined by $$e^{s \mathscr{L}_\eta} g(z) = \int_{\mathbb{R}^{d}} e^{s\mathscr{L}_\eta} (z,\xi) g(\xi ) d\xi.$$ We have the following properties:\ (i) $\big\| e^{s \mathscr{L}_\eta} g\big\|_\infty \leq \|g\|_\infty$ for all $g \in L^\infty(\mathbb{R}^{d})$.\ (ii) $\big\| e^{s \mathscr{L}_\eta}\nabla g \big\|_\infty \leq \frac{C}{\sqrt{1 - e^{-s}}} \|g\|_\infty$ for all $g \in L^\infty(\mathbb{R}^{d})$.\ * *Proof.* The formulation [\[def:kernelLalp\]](#def:kernelLalp){reference-type="eqref" reference="def:kernelLalp"} can be verified by a direct check after a simple change of variable, thanks to the fact that the function $\rho_0(z) = e^{-\frac{\eta |z|^2}{2}}$ satisfies $\Delta \rho_0 + \eta z \cdot \nabla \rho_0 + d\eta\rho_0 = 0$ for all $z \in \mathbb{R}^{d}$. The estimates in (i)-(ii) are straightforward from [\[def:kernelLalp\]](#def:kernelLalp){reference-type="eqref" reference="def:kernelLalp"}. ◻ **Lemma 16** (Estimates in the outer region). *For $A \geq 1$ and $s_0 = s_0(A) \gg 1$ and $\varepsilon(s) \in \mathcal{S}_A(s)$, we have for all $\tau \in [s_0, s]$, $$\label{est:vepoutDuh} j = 0,1, \quad \|(y\partial_y)^j \varepsilon^\textup{ex}(s)\|_{L^\infty} \leq e^{-(s - \tau)} \|(y\partial_y)^j\varepsilon^\textup{ex}(\tau)\|_{L^\infty} + \frac{C(K) A^{3+j}}{\tau^{\frac{1}{\ell}}}(1 + s - \tau),$$ and $$\label{est:vepoutDy1} \|y\varepsilon^\textup{ex}(s)\|_{L^\infty} \leq e^{-(s - \tau)} \|y\varepsilon^\textup{ex}(\tau)\|_{L^\infty} + \frac{C(K) A^{3}}{\tau^{\frac{1}{2\ell}}}(1 + s - \tau),$$* *Proof.* We use Duhamel's formula and item (i) of Lemma [Lemma 15](#lemm:semigroupH){reference-type="ref" reference="lemm:semigroupH"} to write from [\[eq:vepout1\]](#eq:vepout1){reference-type="eqref" reference="eq:vepout1"} for all $\tau \in [s_0, s]$, $$\begin{aligned} \|\varepsilon^\textup{ex}(s)\|_{L^\infty} \leq e^{-(s - \tau)}\|\varepsilon^\textup{ex}(\tau)\|_{L^\infty} + \int_\tau^s e^{-(s - s')}\big(\| F(s')\|_{L^\infty} + \|\mathcal{E}^{bd}(s')\|_{L^\infty}\big) ds',\end{aligned}$$ Due to the the cut-off $\chi_{_K}(\xi)$, the boundary term $\mathcal{E}^{bd}(\varepsilon)$ is located in the zone $K s^{\frac{1}{2\ell}} \leq y \leq 2K s^{\frac{1}{2\ell}} (K \leq \xi \leq 2K)$ and it is bounded by using the estimate from the intermediate region. In particular, we have from [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"} and the bootstrap bounds [\[est:vepk\]](#est:vepk){reference-type="eqref" reference="est:vepk"} and [\[est:vepell\]](#est:vepell){reference-type="eqref" reference="est:vepell"} the following estimate for $j = 0,1$, $$K \leq \xi \leq 2K, \quad | (y\partial_y)^j\varepsilon(y,s)| \leq |(y\partial_y)^j\hat \varepsilon(y,s)| + |(y\partial_y)^j\hat \varepsilon_\natural(y,s)| \leq C(K)A^3 s^{-\frac{1}{\ell}}.$$ Hence, from the definition [\[def:EcbdK\]](#def:EcbdK){reference-type="eqref" reference="def:EcbdK"}, we obtain $$\| \mathcal{E}^{bd} (s)\|_{L^\infty} \lesssim \| \varepsilon(s)\|_{L^\infty(K \leq \xi \leq 2K)} + \|y \partial_y\varepsilon(s)\|_{L^\infty(K \leq \xi \leq 2K)} \lesssim C(K) A^3 s^{-\frac{1}{\ell}}.$$ For the term $F$, we use the decay of $Q$ that is $$\forall y \geq K s^\frac{1}{2\ell}, \quad |Q(\xi)| + |\xi \partial_\xi Q(\xi)| \lesssim \xi^{-2} \lesssim K^{-2}s^{-\frac{1}{\ell}},$$ and from the definition [\[def:NLq\]](#def:NLq){reference-type="eqref" reference="def:NLq"} of $NL(\varepsilon)$ and the bootstrap bound [\[est:vepout\]](#est:vepout){reference-type="eqref" reference="est:vepout"} and the bound [\[est:ELinf\]](#est:ELinf){reference-type="eqref" reference="est:ELinf"}, we get $$\begin{aligned} \|F\|_{L^\infty} &\lesssim s^{-\frac{1}{\ell}} \big(\| \varepsilon^\textup{ex}\|_{L^\infty} + \| y\partial_y \varepsilon^\textup{ex}\|_{L^\infty}\big) + \|\varepsilon^\textup{ex}\|^2_{L^\infty} + \|\varepsilon^\textup{ex}\|_{L^\infty}\| y\partial_y \varepsilon^\textup{ex}\|_{L^\infty} + \|E\|_{L^\infty}\\ & \lesssim A^{9}s^{-\frac{2}{\ell}} + s^{-\frac{1}{\ell}} \lesssim s^{-\frac{1}{\ell}},\end{aligned}$$ for $s_0(A) \gg 1$ so that $A^9 s_0^{-\frac{1}{\ell}} \lesssim 1$. We gather all these estimates and simply bound $\int_\tau^s e^{-(s - s')} s'^{-\frac{1}{\ell}} ds' \lesssim \tau^{-\frac{1}{\ell}} (1 + s - \tau)$ to conclude the estimate [\[est:vepoutDuh\]](#est:vepoutDuh){reference-type="eqref" reference="est:vepoutDuh"} for $j = 0$.\ The proof of [\[est:vepout\]](#est:vepout){reference-type="eqref" reference="est:vepout"} for the case $j =1$ is similar as for $j = 0$ by using (ii) of Lemma [Lemma 15](#lemm:semigroupH){reference-type="ref" reference="lemm:semigroupH"}. The only difference is due to the extra commutator term in the equation satisfied by $$g^\textup{ex}= z \cdot \nabla \varepsilon^\textup{ex}\equiv y\partial_y \varepsilon^\textup{ex},$$ which reads as $$\label{eq:gout1} \partial_s g^\textup{ex}= \big(\mathscr{L}_\eta - \textup{Id}\big)g^\textup{ex}+ [z\cdot \nabla ,\Delta_{d+2}] \varepsilon^\textup{ex}+ z\cdot \nabla (F + \mathcal{E}^{bd}),$$ where $$[z\cdot \nabla , \Delta_{d+2}] \varepsilon^\textup{ex}= - 2\Delta \varepsilon^\textup{ex}= -2\nabla \cdot \Big( \frac{z g^\textup{ex}}{y^2}\Big).$$ Let $$g = z\cdot \nabla \varepsilon\equiv y\partial_y \varepsilon,$$ we write from the definition [\[def:F\]](#def:F){reference-type="eqref" reference="def:F"} of $F$, $$\begin{aligned} z\cdot \nabla F & = -y\partial_y \chi_{_K} (\hat F + yP_1 g ) + \partial_y(yP_1 g (1 - \chi_{_K})) \\ & \quad \qquad + (1 - \chi_{_K})\Big[ (y\partial_y P_1 + P_2 - yP_1)g + y\partial_y P_2 \varepsilon+ E + NL(\varepsilon)\Big] ,\end{aligned}$$ and from the definition [\[def:NLq\]](#def:NLq){reference-type="eqref" reference="def:NLq"} of $NL$, $$(1 - \chi_{_K})y\partial_y NL(\varepsilon) = (1 - \chi_{_K})\Big[ (2d-1) \varepsilon g + g^2\Big] - y\partial_y\chi_{_K} \varepsilon g + \partial_y(y\varepsilon^\textup{ex}g).$$ Hence, $$z\cdot \nabla F = \partial_y \big(yP_1 g (1 - \chi_{_K}) + y\varepsilon^\textup{ex}g) + G,$$ where we can bound $G$ in $L^\infty$ from the bootstrap estimates [\[est:vepout\]](#est:vepout){reference-type="eqref" reference="est:vepout"}, the decay of $Q$, the support of $\chi_{_K}$ and its derivatives, $$\begin{aligned} \|G(s)\|_{L^\infty} \lesssim s^{-\frac{1}{\ell}}, \quad \| yP_1 g(1 - \chi_{_K}) + y \varepsilon^\textup{ex}g)\|_{L^\infty} \lesssim A^9 s^{-\frac{3}{2\ell}} \lesssim s^{-\frac{1}{\ell}}. \end{aligned}$$ Similar, we have $$\begin{aligned} z\cdot \nabla \mathcal{E}^{bd} = 2\partial_y(\partial_y\chi_{_K} g) + G^{bd}, \end{aligned}$$ where $G^{bd}$ and $\partial_y\chi_{_K} g$ have supports on $\{Ks^{\frac{1}{2\ell}} \leq y \leq 2Ks^{\frac{1}{2\ell}}\}$ that can be bounded using the estimate [\[est:vephatpointswise\]](#est:vephatpointswise){reference-type="eqref" reference="est:vephatpointswise"}, $$\|G^{bd}(s)\|_{L^\infty} + \|\partial_y\chi_{_K} g\|_{L^\infty} \lesssim A^3 s^{-\frac{1}{\ell}}.$$ We now use the Duhamel's formula applied to [\[eq:gout1\]](#eq:gout1){reference-type="eqref" reference="eq:gout1"}, Lemma [Lemma 15](#lemm:semigroupH){reference-type="ref" reference="lemm:semigroupH"} and [\[est:ELinf\]](#est:ELinf){reference-type="eqref" reference="est:ELinf"} to get $$\begin{aligned} \|g^\textup{ex}(s)\|_{L^\infty} &\leq e^{-(s - \tau)}\|g^\textup{ex}(\tau)\|_{L^\infty} + \int_\tau^s \frac{e^{-(s - s')}}{\sqrt{1 - e^{-(s - s')}}} \Big[ \| y^{-1}g^\textup{ex}\|_{L^\infty} + \|\partial_y\chi_{_K} g\|_{L^\infty} + \|\partial_y\chi_{_K} g\|_{L^\infty}\Big] \\ & \qquad \qquad + \int_\tau^s e^{-(s - s')} \Big[ \| G(s')\|_{L^\infty} + \| G^{bd}(s')\|_{L^\infty} + \|y\partial_y E(s')\|_{L^\infty}\Big] ds'\\ & \lesssim e^{-(s - \tau)}\|g^\textup{ex}(\tau)\|_{L^\infty} + A^3\int_\tau^s \frac{e^{-(s - s')}}{\sqrt{1 - e^{-(s - s')}}} (s')^{-\frac{1}{ \ell}} ds' + \int_\tau^s e^{-(s - s')}(s')^{-\frac{1}{ \ell}}ds'\\ & \lesssim e^{-(s - \tau)}\|g^\textup{ex}(\tau)\|_{L^\infty} + A^3 \tau^{-\frac{1}{\ell}} (1 + s - \tau). \end{aligned}$$ This concludes the proof of [\[est:vepoutDuh\]](#est:vepoutDuh){reference-type="eqref" reference="est:vepoutDuh"} for $j = 1$. The estimate [\[est:vepoutDy1\]](#est:vepoutDy1){reference-type="eqref" reference="est:vepoutDy1"} for $\|y \varepsilon^\textup{ex}\|_{L^\infty}$ follows the same proof as for [\[est:vepoutDuh\]](#est:vepoutDuh){reference-type="eqref" reference="est:vepoutDuh"}, except that the bound on the error $\| yE (1 - \chi_{_K})\|_{L^\infty} \lesssim s^{-\frac{1}{2\ell}}$. This completes the proof of Lemma [Lemma 16](#lemm:outer){reference-type="ref" reference="lemm:outer"}. ◻ ## Proof of Proposition [Proposition 8](#prop:3){reference-type="ref" reference="prop:3"} and Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"} {#proof-of-proposition-prop3-and-theorem-theo1} In this section we give the proof of Proposition [Proposition 8](#prop:3){reference-type="ref" reference="prop:3"} to complete the proof of Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}. Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"} is a direct consequence of Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}. *Proof of Proposition [Proposition 8](#prop:3){reference-type="ref" reference="prop:3"}.* The basic idea is to improve the bootstrap estimates given in Definition [Definition 5](#Definition-shrinking -set){reference-type="ref" reference="Definition-shrinking -set"}, except for the first $\ell$ modes $(\hat \varepsilon_k)_{0 \leq k \leq \ell-1}$. Regarding the constants, we fix them in the following order: we fix $K \gg 1$ a large constant independent of $A$, then $A = A(K) \gg 1$, then $s_0 = s_0(A) \gg 1$. We recall from the assumption that $$\varepsilon(s) \in \mathcal{S}_A(s) \quad \forall s \in [s_0, s_1] \quad \textup{and} \quad \varepsilon(s_1) \in \partial\mathcal{S}_A(s_1).$$ (i) (*Improve bootstrap estimates*) Let's begin with $\hat \varepsilon_\ell$ and argue by contradiction that there is $\bar s \in [s_0, s_1]$ such that $$|\hat \varepsilon_\ell(s) | < \frac{A^2 \log s}{s^2} \; \forall s\in [s_0, \bar s), \quad \hat \varepsilon_\ell(\bar s) = \pm\frac{A^2 \log \bar s}{\bar s^2},$$ then, we have by equation [\[eq:ODEvepk\]](#eq:ODEvepk){reference-type="eqref" reference="eq:ODEvepk"} (consider that case $\hat \varepsilon(\bar s) > 0$, similar for the negative case) $$-\frac{2A^2 \log \bar s}{\bar s^3} + \frac{C}{\bar s^3} \hat \varepsilon_\ell'(\bar s) \geq A^2 \frac{d}{ds} \frac{A^2 \log s}{s^2}\Big\vert_{\bar s} = \frac{A^2}{\bar s^3} - \frac{2A^2 \log \bar s}{\bar s^3},$$ which can not happen for $A$ large enough. Therefore, $\hat \varepsilon_\ell(s)$ never touches its boundary, $$|\hat \varepsilon_\ell(s_1) | < \frac{A^2 \log s_1}{s_1^2}.$$ As for the modes $\hat \varepsilon_k$ with $k = \ell + 1, \cdots, 2\ell - 1$, we integrate the ODE [\[eq:ODEvepk\]](#eq:ODEvepk){reference-type="eqref" reference="eq:ODEvepk"} forward in time and use the fact that the eigenvalue is negative to conclude that $\hat \varepsilon_k(s)$ can not touch its boundary as well. The same way for $\|\tilde \varepsilon(s)\|_{L^2_\rho}$ and $\|(y\partial_y)^j \hat \varepsilon\|_\flat$ thanks to the energy estimates derived in Lemmas [Lemma 13](#lemm:L2rho){reference-type="ref" reference="lemm:L2rho"} and [\[lemm:mid\]](#lemm:mid){reference-type="eqref" reference="lemm:mid"}. The improvement of $\| (y\partial_y)^j\varepsilon^\textup{ex}\|_{L^\infty}$ and $\|y \varepsilon^\textup{ex}\|_{L^\infty}$ follows from Lemma [Lemma 16](#lemm:outer){reference-type="ref" reference="lemm:outer"} by taking $\lambda = \log A \gg 1$ and $s_0 \geq \lambda$ such that for all $\tau \geq s_0$ and $s \in [\tau, \tau + \lambda]$, we have $$\tau \leq s \leq \tau +\lambda \leq \tau + s_0 \leq 2\tau, \quad \textup{hence}, \quad \frac{1}{2\tau}\leq \frac{1}{s} \leq \frac{1}{\tau}\leq \frac{2}{s}.$$ This give us the bound $$\frac{C(K) A^{3+j}}{\tau^{\frac{1}{\ell}}}(1 + s - \tau) \lesssim \frac{C(K) A^{3+j} \log A}{s^{\frac{1}{\ell}}} < \frac{A^{4+j}}{s^\frac{1}{\ell}},$$ for $A$ large enough. This concludes that $\varepsilon(s_1)$ can only touch its boundary $\partial\mathcal{S}_A(s_1)$ at the first $\ell$ modes $(\hat \varepsilon_k)_{0 \leq k \leq \ell-1}$.\ (ii) (*Transverse crossing*) The estimate [\[transversecross\]](#transversecross){reference-type="eqref" reference="transversecross"} follows from a direct computation thanks to [\[eq:ODEvepk\]](#eq:ODEvepk){reference-type="eqref" reference="eq:ODEvepk"}, $$\frac{1}{2}\frac{d}{ds}\sum_{k = 0}^{\ell - 1} \hat \varepsilon^2_k(s_1) = \sum_{k = 0}^{\ell - 1} \Big[ (1 - k/\ell) \hat \varepsilon^2(s_1) + \mathcal{O}(s^{-2} |\hat \varepsilon(s_1)|)\Big] \geq \frac{A^4 - CA^2}{s_1^4} > 0,$$ for $A$ large enough. This completes the proof of Proposition [Proposition 8](#prop:3){reference-type="ref" reference="prop:3"} as well as Proposition [Proposition 7](#prop:2){reference-type="ref" reference="prop:2"}. ◻ *Proof of Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}.* (i) and (ii) follows from the definition of the shrinking set [Definition 5](#Definition-shrinking -set){reference-type="ref" reference="Definition-shrinking -set"} and the relation $w = d v + y\partial_y v$ and $\phi_{2\ell} = d \varphi_{2\ell} + y\partial_y \varphi_{2\ell}$. As for (iii), we use the same argument as in Herrero-Velázquez [@HVaihn93] for the classical nonlinear heat equation (see also Bebernes-Bricher [@BBsima92], Zaag [@ZAAihn98], [@GNZana20] for a similar approach), we only sketch the computation for the reader convenience. We introduce the auxiliary function $$g(x_0, \xi, \tau) = (T - t_0)u(x,t), \quad x = x_0 + \xi \sqrt{T- t_0}, \quad t = t_0 + \tau(T-t_0),$$ where $t_0 = t_0(x_0)$ is uniquely determined by $$|x_0| = K_0\sqrt{T - t_0} | \log(T - t_0)| ^\frac{1}{2\ell}, \quad K_0 \gg 1.$$ We have the relation $$\log (T - t_0) \sim 2\log |x_0|, \quad T - t_0 \sim \frac{|x_0|^2}{K_0^2 \big(2 |\log |x_0|| \big)^ \frac{1}{\ell} }.$$ From [\[exp:innerIntro\]](#exp:innerIntro){reference-type="eqref" reference="exp:innerIntro"}, we have $$\begin{aligned} u^*(x_0) = \lim_{t \to T} u(x,t) = (T - t_0)^{-1} \lim_{\tau \to 1} g(x_0,0, \tau) = (T-t_0)^{-1}\hat g_{K_0}(1).\end{aligned}$$ We compute from [\[est:QxiInf\]](#est:QxiInf){reference-type="eqref" reference="est:QxiInf"}, $$\hat g_{K_0}(1) = F(K_0) \sim (d - 2) c_\ell^{-\frac{1}{\ell}} K_0^{-2},$$ which gives $$u^*(x_0) \sim (d- 2) \left(\frac{2}{c_\ell} \right)^\frac{1}{\ell} \frac{|\log |x_0|| ^ \frac{1}{\ell}}{|x_0|^2} \quad \textup{as}\;\; |x_0| \to 0.$$ This completes the proof of Theorem [Theorem 1](#theo:1){reference-type="ref" reference="theo:1"}. ◻ 60 urlstyle J. Bebernes and S. Bricher. 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arxiv_math

--- abstract: | Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1,\ldots ,x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$, where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$. This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K)$ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K)$. address: - "D.Efimov: Department of Algebra and Computer Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrska street, 01033 Kyiv, Ukraine " - "M.Sydorov: Department of Algebra and Computer Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrska street, 01033 Kyiv, Ukraine" - "K.Sysak: Department of Higher and Applied Mathematics, Education and Research Institute of Energetics, Automatics and Energy saving, National University of Life and Environmental Sciences of Ukraine, 15, Heroiv Oborony street, 03041 Kyiv, Ukraine" author: - D.Efimov, M.Sydorov, K.Sysak title: | On maximality of some solvable and locally\ nilpotent subalgebras of the Lie algebra $W_n(K)$ --- # Introduction Let $K$ be an algebraically closed field of characteristic zero and $P_n=K[x_1,\ldots ,x_n]$ the polynomial ring in $n$ variables. Recall that a $K$-linear map $D: P_n \to P_n$ is called a $K$-derivation (or simply a derivation if the field $K$ is fixed) if it satisfies the Leibniz rule: $D(fg)=D(f)g+fD(g)$ for any $f, g \in P_n$. For any $f_1,\ldots ,f_n \in P_n$ there exists a unique $K$-derivation ${D} \in W_n(K)$ of the form ${D} = f_1\partial_1+\cdots +f_n\partial_n$ such that ${D}(x_i)=f_i, i=1,\ldots ,n,$ where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$. The vector space $W_n(K)$ of all $K$-derivations on $P_n$ is a Lie algebra over the field $K$ with respect to the Lie bracket $[{D}_1, {D}_2]={D}_1{D}_2-{D}_2{D}_1$, ${D}_1, {D}_2 \in W_n(K)$. This Lie algebra is of great interest in many areas of mathematics and physics because in geometric language any derivation can be considered as a vector field on $K^n$ with polynomial coefficients. Nilpotent, locally nilpotent and solvable subalgebras of $W_n(K)$ were studied by many authors, started from [@Lie1] (see, for example, [@Bavula; @Olver2; @MP1]). One of the most important subalgebras of $W_n(K)$ is the triangular Lie algebra $$u_n(K)=P_0\partial_1+\cdots +P_{n-1}\partial_n,$$ which consists of locally nilpotent derivations on $K[x_1,\ldots ,x_n].$ This Lie algebra is locally nilpotent but not nilpotent, its structure and properties were studied in [@Bavula]. We consider its embedding in $W_n(K)$ and prove that $u_n(K)$ is a maximal locally nilpotent subalgebra of $W_n(K)$ (Theorem 1). Another maximality property of $u_n(K)$ was considered in [@Skutin], where it was proved that $u_n(K)$ is a maximal subalgebra contained in the set of locally nilpotent derivations on $P_n$ (note that this set is not a Lie subalgebra of $W_n(K)$). In [@MP1], it was proved that the derived length of solvable subalgebras in $W_n(K)$ does not exceed $2n$. The known example of solvable subalgebras that reaches this bound was pointed out in [@Martello], this is the subalgebra $$s_n(K)=(P_0+x_1P_0)\partial_1+\dots+(P_{n-1}+x_nP_{n-1})\partial_n.$$ It is clear that the subalgebra $u_n(K)$ is properly contained in $s_n(K)$. The subalgebra $s_n(K)$ has also a maximality property: we prove that $s_n(K)$ is a maximal solvable subalgebra of $W_n(K)$ (Theorem 2). Note that $s_n(K)$ appears in a natural way while studying Lie algebras of vector fields on $\mathbb{C}^n$ (see [@Martello]). In general, maximal subalgebras of the Lie algebra $W_n(K)$ are not described, but some types of such subalgebras are known (see, for example, [@Amemiya]). Note that the structure of maximal subalgebras of semisimple Lie algebras was described in [@Dynkin]. We use standard notations. Recall that a derivation $D \in W_n(K)$ is called locally nilpotent if for any $f \in P_n$ there exists a positive integer $k=k(f)$ such that $D^k(f)=0$. Let a derivation $D \in W_n(K)$ be written in the form $D = f_1\partial_1+\dots +f_n\partial_n$, $f_i \in P_n,$ $i=1,\dots, n.$ Then we say that $D$ has an index $k$ if $f_k \neq 0$ and $f_m= 0$ for all $m>k$. Let $f=f(x_1,\ldots ,x_n)\in P_n$ be a polynomial. Then we say that $f$ has an index $s$ if $\frac{\partial f}{\partial x_s}\neq 0$ but $\frac{\partial f}{\partial x_i}=0$ for $i>s$. If $D_1, \ D_2 \in W_n(K)$ then we write $[{D}_1^k, {D}_2]=\underbrace{[{D}_1,[ \ldots [{D}_1 }_{k},{D}_2]\ldots ]\ldots ]$. We denote as usual by $K^*$ the multiplicative group of the field $K$, that is $K^*=K\setminus{\{0\}}.$ # Maximality of $u_n(K)$ We need some technical lemmas to prove Theorem 1, the main result of this section. Lemma [Lemma 2](#diff){reference-type="ref" reference="diff"} seems to be known but having no exact references we point out its proof for completeness. **Lemma 1** ([@MP1], Lemma 1). *Let $D_{1}, D_{2}\in W_n(K)$ and $a, b\in K[x_1, \ldots , x_n].$ Then it holds:* *(1) $[aD_{1}, bD_{2}]=ab[ D_1, D_2]+aD_1(b)D_2-bD_2(a)D_{1}.$* *(2) If $[D_{1}, D_{2}]=0,$ then $[aD_{1}, bD_{2}]=aD_1(b)D_2-bD_2(a)D_{1}.$* **Lemma 2**. *Let $f \in P_n = K[x_1, \ldots, x_n]$, $\deg f\geq 1$. Then:* *(1) there exist nonnegative integers $\alpha_1,\ldots ,\alpha_n$ such that $\partial_1^{\alpha_1}\dots \partial_n^{\alpha_n}(f)$ is a nonzero constant;* *(2) if $\deg _{x_i}f \geq 1$, then there exist nonnegative integers $\beta_1,\ldots,\beta_n$ (depending on $i$) such that $\partial_1^{\beta_1}\dots\partial_n^{\beta_n}(f)=\lambda_i x_i+g_i(x_1, \ldots , x_{i-1}, x_{i+1}, \ldots ,x_n),$ where $\lambda_i \in K^*$.* *Proof.* (1) Let $d=\deg f$ and $f=f_0+\cdots +f_d$ be the sum of homogeneous components of $f$. Choose any monomial $ax_1^{\alpha _1}\dots x_n^{\alpha _n}$ of the homogeneous polynomial $f_d$. Then $\partial_1^{\alpha_1}\dots \partial_i^{\alpha _n}(a x_1^{\alpha_1}\dots x_n^{\alpha_n})=\gamma$ for some $\gamma \in K^*$. If there exists another monomial $bx_1^{\beta_1}\dots x_n^{\beta_n}$ of the polynomial $f_d$ then the polynomial $\partial_1^{\alpha_1}\dots \partial_i^{\alpha _n}(bx_1^{\beta_1}\dots x_n^{\beta_n})$ is a constant. This constant is nonzero only in the case when $\beta _1=\alpha _1, \ldots , \beta _n=\alpha _n.$ The latter is impossible because of the choice of the monomial $bx_1^{\beta_1}\cdots x_n^{\beta_n}$. So $\partial_1^{\alpha_1}\dots \partial_i^{\alpha _n}(f_d)=\gamma,$ and since the equalities $\partial_1^{\alpha_1}\dots \partial_i^{\alpha _n}(f_i)=0$ hold for all $i<d,$ we get $\partial_1^{\alpha_1}\dots \partial_i^{\alpha _n}(f)=\gamma, \ \gamma \in K^*.$ \(2\) Let $\deg _{x_{i}} (f)=d\geq 1.$ Expand $f$ in powers of $x_i:$ $f=h_0+h_1x_i+\cdots + h_dx_i^d$, where $\deg _{x_{i}} (h_j)=0, j=1, \ldots ,d.$ Then $$\partial _i^{d-1}(f)=t_0(x_1, \ldots x_{i-1}, x_{i+1}, \ldots , x_n)+t_1(x_1, \ldots x_{i-1}, x_{i+1}, \ldots , x_n)x_i.$$ If the polynomial $t_1( x_1, \ldots x_{i-1}, x_{i+1}, \ldots , x_n)$ is nonconstant then by the first part of this lemma there exist nonnegative integers $\beta _1, \ldots \beta _{i-1}, \beta _{i+1}, \ldots ,\beta _n$ such that $$\partial_1^{\beta_1}\dots \partial _{i-1}^{\beta _{i-1}} \partial_{i+1}^{\beta _{i+1}}\dots\partial_{n}^{\beta _{n}}(t_1)=\lambda _i, \ \lambda _i \in K^*.$$ Denoting $\beta _i=d-1$ we get $$\partial_1^{\beta_1}\dots \partial _{n}^{\beta _{n}}(f)=\lambda _ix_i+g_i(x_1, \ldots x_{i-1},x_{i+1}, \ldots ,x_n),$$ where $g_i= \partial_1^{\beta_1}\dots \partial _{i-1}^{\beta _{i-1}} \partial_{i+1}^{\beta _{i+1}}\dots \partial_{n}^{\beta _{n}}(t_0),$ $\lambda _i\in K^*.$ ◻ **Lemma 3**. *If there exists a locally nilpotent subalgebra $S$ of the Lie algebra $W_n(K)$ that properly contains $u_n(K)$, then there exists a (nonzero) linear derivation $D \in S \setminus u_n(K)$ of the form $D = \sum_{i, j = 1}^n \lambda_{ij}x_j\partial_i$, where $\lambda_{ij} = 0$ for all $i>j$.* *Proof.* Suppose the statement of the lemma is false and the set $S \setminus u_n({K})$ does not contain any nonzero linear derivation. Let us choose a derivation $D\in S \setminus u_n(K)$ of minimum degree and write it in the form $D=f_1\partial _1+\cdots +f_n\partial _n$, $f_i\in K[x_1, \ldots , x_n].$ Since $\partial _i \in S, i=1,\ldots ,n,$ we have that $$[\partial _i, D]=\partial _i(f_1)\partial _1+\cdots +\partial _i(f_n)\partial _n\in S$$ and $\deg [\partial _i, D]<\deg D.$ By the choice of $D,$ we see that $[\partial _i, D]\in u_n(K), i=1, \ldots ,n.$ Let us show that $\deg _{x_j}f_i\leq 1$ for $j\geq i.$ Indeed, if $\deg _{x_j}f_i\geq 2$, then $\deg _{x_j}\partial _j(f_i)\geq 1$, which contradicts the inclusion $[\partial _j, D]\in u_n(K)$ mentioned above. Write the polynomial $f_i$ in powers of $x_j\colon$ $f_i=f_{i0}+f_{i1}x_j$ for some $f_{i0}, f_{i1}\in K[x_1, \ldots ,x_{j-1}, x_{j+1}, \ldots , x_n]$. If the polynomial $f_{i1}$ is nonconstant, then $\partial _k(f_{i1})\not =0$ for some $k, 1\leq k\leq n.$ But then $\deg _{x_j}\partial _k(f_i)=1$ and therefore $[\partial _k, D]\not\in u_n(K).$ The latter contradicts the proven above. So, $f_i=f_{i0}+\lambda _{ij}x_j$ for some $\lambda _{ij}\in K.$ Repeating these considerations for every $j\geq i$ we see that $f_i$ can be chosen in the form $f_i=\sum_{i, j = 1}^n \lambda_{ij}x_j+\overline{f_{i}},$ where $\lambda_{ij} = 0$ for all $i>j$ and $\overline{f_{i}}$ does not depend on $x_j$, $j\geq i,$ i.e., $\overline{f_{i}} \in K[x_1, \ldots , x_{i-1}].$ But then $\overline{f_{i}}\partial _i\in u_n(K)$ and we can subtract $\overline{f_{i}}\partial _i$ from $D.$ Applying such considerations to all $f_i,$ we get the statement of the lemma. ◻ **Theorem 1**. *The triangular subalgebra $u_n(K)$ is a maximal locally nilpotent subalgebra of the Lie algebra $W_n(K)$.* *Proof.* Suppose to the contrary that $u_n(K)$ is properly contained in a locally nilpotent subalgebra $S$ of $W_n(K)$. By Lemma [Lemma 3](#lem2){reference-type="ref" reference="lem2"}, there exists an element $D \in S \setminus u_n(K)$ of the form $$D = \sum_{i=1}^n \sum_{j=i}^n \lambda_{ij} x_j \partial_i, \ \lambda_{ij} \in K,$$ i.e. $D = f_1 \partial_1 + \dots + f_n \partial_n$, where $f_i = \lambda_{ii}x_i + \lambda_{i, i+1}x_{i+1} + \dots + \lambda_{in}x_n$. Firstly, let us prove that every linear derivation $D\in S \setminus u_n(K)$ is diagonal, i.e. that the matrix $(\lambda_{ij})_{i,j=1}^n$ is diagonal. Let it be not the case and choose any linear (non-diagonal) derivation $D = \sum_{i=1}^n \sum_{j=i}^n \lambda_{ij} x_j \partial_i, \ \lambda_{ij} \in K$ from the set $S \setminus u_n(K)$. Note that the derivation $x_i \partial_j \in u_n(K)$ for $i<j$ and consider the product $$\begin{gathered} \label{eq2} [x_i\partial _j, D]=[x_i\partial_j, \sum_{s=1}^n \lambda _{1s}x_s \partial_1+\cdots +\sum_{s=k}^n \lambda _{ks}x_s \partial_k+ \cdots +\lambda _{nn}x_n\partial _n] = \\ = x_i \lambda_{1j} \partial_1 +\cdots + x_i\lambda_{ij}\partial _i + \dots + x_i\lambda_{jj}\partial_j - ( \lambda _{1i}x_i+ \dots + \lambda_{ji}x_i)\partial_j.\end{gathered}$$ Denote this product $[x_i\partial _j, D]$ by $D_0$. Using ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) one can easily see that $[D_0, x_i\partial _j]=\lambda_{ij}x_i\partial _j,$ i.e. $x_i\partial _j$ is an eigenvector for the linear operator $\mathop{\mathrm{ad}}D_0$ with the eigenvalue $\lambda_{ij}.$ Since $D$ is non-diagonal (by our assumption) there exists a nonzero coefficient $\lambda _{ij}, i<j.$ The latter is impossible because the subalgebra $S$ is locally nilpotent. The obtained contradiction shows that all linear derivations from $S \setminus u_n(K)$ are diagonal. Take any linear derivation $D\in S \setminus u_n(K)$, $D = \mu_1 x_1 \partial_1 + \dots + \mu_n x_n \partial_n.$ Obviously, $D \neq 0$. Let us show that $S \setminus u_n(K)$ contains a derivation $D_1 = \mu E_n$, where $E_n = x_1 \partial_1 + \dots + x_n \partial_n$ is the Euler derivation. If $D = \sum \mu_i x_i \partial_i$ is not proportional to $E_n$, then there exist $\mu_i, \ \mu_j,$ such that $\mu_i \neq \mu_j, \ i < j$. Then $S$ contains the product $$[\sum_{i=1}^n \mu_i x_i \partial_i, x_i\partial_j] = (\mu_i - \mu_j)x_i \partial_j, \ \mu_i - \mu_j \neq 0.$$ The latter means that $x_i \partial_j$ is an eigenvector for the linear operator $\mathop{\mathrm{ad}}D$ with the (nonzero) eigenvalue $\mu_i - \mu_j$, which is impossible because $S$ is a locally nilpotent subalgebra of $W_n({K})$. Therefore, we have that $D = \mu E_n$ for some $\mu \in K^*$. But $[\mu E_n, x_1^2\partial_2] = \mu x_1^2 \partial_2$ for the element $x^2_1 \partial_2 \in u_n({K}) \subset S$. The latter is impossible as it was mentioned above. The obtained contradiction shows that $S = u_n({K})$ and $u_n({K})$ is a maximal locally nilpotent subalgebra of the Lie algebra $W_n({K})$. ◻ # Maximality of $s_n(K)$ Recall that we denote by $P_i=K[x_1, \dots, x_i]$ the polynomial ring over $K.$ We also denote for convenience $P_0= K.$ It is easy to see that the $K$-subspace $$s_n( K)=(P_0+x_1P_0)\partial_1+\dots+(P_{n-1}+x_nP_{n-1})\partial_n$$ is a subalgebra of $W_n( K)$ and $u_n( K)\subset s_n(K).$ This subalgebra is solvable of the derived length $2n.$ Some properties of $s_n(K)$ were pointed out in [@Martello]. Since the derived length of solvable subalgebras of $W_n(K)$ does not exceed $2n$ (see [@MP1], [@Martello]), the subalgebra $s_n(K)$ has the maximum possible derived length. Here we prove that $s_n(K)$ is a maximal solvable subalgebra of $W_n(K)$ (Theorem [Theorem 2](#th2){reference-type="ref" reference="th2"}). Let $D\in W_n( K)$ be a derivation of the form $D=f_1\partial_1+\dots+f_n\partial_n,$ where $f_i \in P_n,$ $i=1,\ldots ,n.$ Recall that we say that $D$ has an index $k$ if $f_k\neq 0$ and $f_m=0$ for all $m>k.$ We also say that a polynomial $f\in P_n$ has an index $s$ if $\frac{\partial f}{\partial x_s}\neq 0$ and $\frac{\partial f}{\partial x_i}=0$ for all $i>s.$ **Lemma 4**. *Let $T_1=\sum _{i=1}^{k-1}g_i\partial _i+\partial _k$, $T_2=\sum _{i=1}^{k-1}h_i\partial _i-x_k^2\partial _k$, $T_3=\sum _{i=1}^{k-1}f_i\partial _i-2x_k\partial _k$ be derivations from $W_n(K)$ for some $k\leq n$. Then $T_1, T_2, T_3$ generate a non-solvable subalgebra of the Lie algebra $W_n(K)$.* *Proof.* Direct calculations show that $[T_1, T_2]=\sum _{i=1}^{k-1}a_i\partial _i-2x_k\partial _k$, $[T_3, T_1]=\sum _{i=1}^{k-1}b_i\partial _i+2\partial _k$, $[T_3, T_2]=\sum _{i=1}^{k-1}c_i\partial _i+2x_k^2\partial _k$ for some polynomials $a_i, b_i, c_i\in K[x_1, \ldots , x_n] .$ Denote by $L$ the subalgebra of the Lie algebra $W_n(K)$ generated by the elements $T_1, T_2, T_3$ and by $L_1$ the subalgebra generated by $\partial _k, -x_k^2\partial _k, -2x_k\partial _k.$ Define a map $\varphi$ from the set $\{ T_1, T_2, T_3\}$ onto the set $\{ \partial _k, -x_k^2\partial _k, -2x_k\partial _k \}$ by the rule: $$\varphi (T_1)= \partial _k, \ \varphi (T_2)= -x_k^2\partial _k, \ \varphi (T_3)=-2x_k\partial _k.$$ The rule of commutation of generators $T_1, T_2, T_3$ shows that $\varphi$ can be extended to a homomorphism of the Lie algebra $L$ onto $L_1$. Since $L_1$ is isomorphic to $sl_2(K)$ we conclude that $L$ is non-solvable. ◻ **Theorem 2**. *The subalgebra $s_n( K)$ is a maximal solvable subalgebra of the Lie algebra $W_n( K).$* *Proof.* Suppose to the contrary that there exists a solvable subalgebra $S\subset W_n( K)$ such that $s_n(K)$ is properly contained in $S.$ Denote by $k$ the smallest index of derivations from the set $S\setminus s_n( K)$ and consider the set $\mathfrak D_k$ of all derivations $D\in S\setminus s_n( K)$ that have the index $k.$ Let us choose a derivation $D\in \mathfrak D_k$ in such a way that its (nonzero) polynomial coefficient $f_k$ (by the partial derivative $\partial _k$) has the smallest index $s.$ Then we have $$D=f_1\partial _1+\dots +f_k\partial _k,$$ where $f_i\in P_n, \ i=1, \ldots,k,$ $\frac {\partial f_k}{\partial x_s}\not =0$ and $\frac {\partial f_k}{\partial x_i}=0$ for all $i>s.$ Firstly, let us show that $s\geq k$ and if $s=k$ then $\deg_{x_s}f_k\geq 2.$ Indeed, if $s<k$ then $f_k\partial_k\in u_n( K),$ since $f_k\in P_{s}$ and $s<k.$ By our assumption, $u_n( K)\subset S.$ Then $D-f_k\partial_k \in S\setminus s_n( K)$ and this derivation has an index less than $k,$ which contradicts our choice of $D.$ Therefore, $s\geq k.$ Let $s=k$. Then $x_s=x_k,$ $\frac{\partial f_k}{\partial x_k}\neq 0$ and $f_k\in P_k.$ Let us expand the polynomial $f_k$ in powers of $x_k:$ $$f_k(x_1,\dots, x_k)=h_0(x_1, \dots, x_{k-1})+h_1(x_1, \dots, x_{k-1})x_k+\dots +h_t(x_1, \dots, x_{k-1})x_k^t$$ for some $t\geq 1$ and polynomials $h_i\in P_{k-1},$ $i=1\ldots, t.$ If $t=1$ then $$f_k\partial_k=(h_0(x_1, \dots, x_{k-1})+h_1(x_1, \dots, x_{k-1})x_k)\partial_k\in s_n( K).$$ It is obvious that $D-f_k\partial_k\in S\setminus s_n( K)$ and the derivation $D-f_k\partial_k$ has the index less than $k.$ The latter contradicts the choice of $D$. Thus, if $s=k$ then $\deg_{x_k}f_k\geq 2.$ Therefore, we can write $D\in \mathfrak D_k$ in the form $$\label{th3_eq1} D=f_1\partial_1+\dots+f_k\partial_k,$$ where $\deg_{x_s}f_k\neq 0,$ $\deg_{x_i}f_k=0$ for all $i>s$ and $s\geq k.$ We investigate the possible two cases: $s>k$ and $s=k.$ [*Case 1.*]{.ul} Let us begin with the case $s>k.$ Let us expand the polynomial $f_k\in P_{s}$ from the derivation $D$ (written in the form  ([\[th3_eq1\]](#th3_eq1){reference-type="ref" reference="th3_eq1"})) in powers of $x_s:$ $$f_k(x_1, \dots, x_s)=g_0(x_1, \dots, x_{s-1})+g_1(x_1, \dots, x_{s-1})x_s+\dots+g_l(x_1, \dots, x_{s-1})x_s^l,$$ for some $l\geq 1$, where $g_i\in P_{s-1}, \ g_l\neq 0.$ Then the product $D_0=[\underbrace{\partial_s, \dots, [\partial_s,}_{l-1~\text{times}} D]\dots ]\in S$ can be written in the form $$D_0=\alpha_1 \partial_1+\dots +\alpha_{k-1}\partial_{k-1}+(u_0(x_1, \dots, x_{s-1})+u_1(x_1, \dots, x_{s-1})x_s)\partial_k,$$ for some $\alpha_i\in P_n, \ i=1,\ldots ,k-1,$ $u_0, u_1\in P_{s-1}$ and $u_1\neq 0$ by the choice of $s.$ By Lemma [Lemma 2](#diff){reference-type="ref" reference="diff"}, there exists a differential operator $\partial_1^{\beta_1}\dots\partial_{s-1}^{\beta_{s-1}},$ $\beta_i\geq 0$ such that $$\partial_1^{\beta_1}\dots\partial_{s-1}^{\beta_{s-1}}(u_1)=\lambda\in \ K^*,$$ where $K^*$ is the group of units of the field $K.$ Applying this operator to the derivation $D_0$ we obtain a derivation $D_1$ of the form $$D_1=\gamma_1 \partial_1+\dots +\gamma_{k-1}\partial_{k-1}+(v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k,$$ for some $\gamma_i\in P_n, \ i=1, \ldots ,k-1,$ $v_0 \in P_{s-1}$ and $\lambda \neq 0.$ The derivation $D_1\in S\setminus s_n( K),$ since $\lambda\neq 0$ and $s>k.$ Consider the subcase $s-1>k.$ Then it holds $$\begin{gathered} \label{th3_eq2} [x_{s-1}\partial_s; (v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k]=\\ =x_{s-1}\partial_s(v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k-(v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k(x_{s-1})\partial_s=\lambda x_{s-1}\partial_k, \end{gathered}$$ since $\partial_k(x_{s-1})=0$ for $s-1>k.$ Moreover, for all $i=1, \ldots ,k-1$ we get $$\label{th3_eq3} [x_{s-1}\partial_s; \gamma_i \partial_i]=x_{s-1}\partial_s(\gamma_i)\partial_i - \gamma_i \partial_i(x_{s-1})\partial_s=x_{s-1}\partial_s(\gamma_i)\partial_i.$$ Taking into account the relations ([\[th3_eq2\]](#th3_eq2){reference-type="ref" reference="th3_eq2"}) and ([\[th3_eq3\]](#th3_eq3){reference-type="ref" reference="th3_eq3"}), we obtain $$[x_{s-1}\partial_s; D_1]=\sum_{i=1}^{k-1} x_{s-1}\partial_s(\gamma_i)\partial_i+ \lambda x_{s-1}\partial_k.$$ Since $s-1>k$ we have that $[x_{s-1}\partial_s; D_1]\not \in s_n(K).$ Therefore, $[x_{s-1}\partial_s; D_1]\in S\setminus s_n(K),$ and we get a contradiction, since the polynomial $\lambda x_{s-1}$ has the index less than $s.$ Now let $s-1=k.$ Then $\partial_k(x_{s})=0,$ so we get $$\begin{gathered} \label{th3_eq4} [x_{s}\partial_s; (v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k]=\\ =x_{s}\partial_s(v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k-(v_0(x_1, \dots, x_{s-1})+\lambda x_s)\partial_k(x_{s})\partial_s=\lambda x_{s}\partial_k. \end{gathered}$$ Since $\partial_i(x_s)=0$ for all $i=1, \ldots , k-1$ one can easily show (using the relation ([\[th3_eq4\]](#th3_eq4){reference-type="ref" reference="th3_eq4"})) that $$[(1/{\lambda })x_{s}\partial_s; D_1]=(1/{\lambda })\sum_{i=1}^{k-1} x_{s}\partial_s(\gamma_i)\partial_i+ x_{s}\partial_k.$$ Denote $D_2=(1/{\lambda })\sum_{i=1}^{k-1} x_{s}\partial_s(\gamma_i)\partial_i+ x_{s}\partial_k.$ It is obvious that $D_2\in S\setminus s_n(K)$ because $s=k+1.$ Note that $x_k^2\partial _{k+1}\in s_n(K)$ and therefore $$[x_k^2\partial _{k+1}, D_2]=[x_k^2\partial _{k+1}, (1/{\lambda })\sum _{i=1}^{k-1}x_s\partial _s(\gamma _i)\partial _i+x_{k+1}\partial _k]=\sum _{i=1}^{k-1}\alpha _i\partial _i+[x_k^2\partial _{k+1}, x_{k+1}\partial _k]$$ for some $\alpha _i\in K[x_1, \ldots ,x_n]$ is an element of subalgebra $S.$ But $[x_k^2\partial _{ k+1}, x_{k+1}\partial _k]=x_k^{2}\partial _k-2x_kx_{k+1}\partial _{k+1}.$ Therefore $$[x_k^2\partial _{k+1}, D_2]=\sum _{i=1}^{k-1}\alpha _i\partial _i+x_k^{2}\partial _k-2x_kx_{k+1}\partial _{k+1}.$$ But $2x_kx_{k+1}\partial _{k+1}\in s_n(K)$ and therefore $\sum _{i=1}^{k-1}\alpha _i\partial _i+x_k^{2}\partial _k\in S\setminus s_n(K).$ Besides, $\partial _k, x_k\partial _k\in s_n(K)$ and denoting $$T_1=\partial _k, T_2=-\sum _{i=1}^{k-1}\alpha _i\partial _i-x_k^{2}\partial _k, T_3=-2x_k\partial _k$$ we see by Lemma [Lemma 4](#sl2){reference-type="ref" reference="sl2"} that $S$ is non-solvable. The latter contradicts the choice of $S$ and this contradiction shows that the case $s>k$ is impossible. [*Case 2.*]{.ul} Now let us consider the case $s=k.$ As shown above, in this case $\deg_{x_s}f_k=\deg_{x_k}f_k\geq 2.$ Then the chosen derivation $D\in \mathcal D_k$ is of the form $$D=f_1\partial_1+\dots+f_k\partial_k,$$ where $\deg_{x_k}f_k\geq 2,$ $\deg_{x_i}f_k=0$ for all $i>k.$ Let us expand the polynomial $f_k\in P_{k}$ by powers of $x_k:$ $$f_k(x_1, \dots, x_k)=g_0(x_1, \dots, x_{k-1})+g_1(x_1, \dots, x_{k-1})x_k+\dots+g_l(x_1, \dots, x_{k-1})x_k^l,$$ where $g_l\neq 0, \ l\geq 2, \ g_i\in P_{k-1}, i=1,\ldots ,l.$ As in the previous case, let us consider the derivation $$\begin{gathered} D_0=[\underbrace{\partial_k, \dots, [\partial_k,}_{l-2~\text{times}} D]\dots ]=\alpha_1 \partial_1+\dots +\alpha_{k-1}\partial_{k-1}+\\+(u_0(x_1, \dots, x_{k-1})+u_1(x_1, \dots, x_{k-1})x_k+u_2(x_1, \dots, x_{k-1})x_k^2)\partial_k \in S, \end{gathered}$$ where $\alpha_i\in P_n, \ i=1, \ldots k-1,$ $u_0, u_1, u_2\in P_{k-1}$ and $u_2\neq 0.$ Using a differential operator $\partial_1^{\gamma_1}\dots\partial_{k-1}^{\gamma_{k-1}}$ with appropriate $\gamma_i\geq 0, \ i=1,\ldots ,k-1,$ we can assume without loss of generality that $u_2(x_1, \dots, x_{k-1})=\lambda_k \in K^*$. We obtain the derivation $$D_1=\mu_1 \partial_1+\dots +\mu_{k-1}\partial_{k-1}+(v_0(x_1, \dots, x_{k-1})+v_1(x_1, \dots, x_{k-1})x_k+\lambda_kx_k^2)\partial_k \in S\setminus s_n(K),$$ where $\mu_i\in P_n, \ i=1,\ldots ,k-1,$ and $v_0, \ v_1\in P_{k-1}.$ Since $$(v_0(x_1, \dots, x_{k-1})+v_1(x_1, \dots, x_{k-1})x_k)\partial_k\in s_n(K),$$ we have the derivation $$D_2=\mu_1 \partial_1+\dots +\mu_{k-1}\partial_{k-1}+\lambda_kx_k^2\partial_k\in S\setminus s_n(K).$$ Denote by $L_1$ the subalgebra of $S$ generated by $T_1=\partial _k, T_2=-D_2,$ and $T_3=-2x_k\partial _k$. By Lemma [Lemma 4](#sl2){reference-type="ref" reference="sl2"} the subalgebra $L_1$ is non-solvable which contradicts to its choice. The obtained contradiction shows that case $s=k$ is also impossible. Therefore our assumption about $S$ is false and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K).$ ◻ 99 I. Amemiya, K. Masuda and Shiga. Lie algebras of differential operators. Osaka J. Math. 1212 (1975), 139--172. V.V. Bavula, Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras, Izv. Math. 77 (2013) 1067--1104. A. González-López, N. Kamran and P.J. Olver. Lie algebras of vector fields in the real plane. Proc. London Math. Soc. (3) 64 (1992), no. 2, 339--368. E.B. Dynkin. Maximal subgroups of classical groups. Trudy Moskovskogo Matem. obshchestva 1 (1952), (Russian), English transl. in: Moscow Math. Soc. Translations Ser. 2, v.6, 245--378. S. Lie, Theorie der Transformationsgruppen, Bd. 3. Teubner, Leipzig, 1893. Ie. O. Makedonskyi, A.P. Petravchuk. On nilpotent and solvable Lie algebras of derivations. Journal of Algebra, 2014, **401**, 245--257. M.Martello, J. Ribon. Derived length of solvable groups of local diffeomorphisms. Mathematische Annalen (2014), **358**, 701--728. A. Skutin. Maximal Lie algebras among locally nilpotent derivations. Mat. Sb. **212** (2021), no.2, 138--146.

arxiv_math

--- abstract: | We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, decide whether the intersection $\langle \mathcal{G} \rangle \cap \langle \mathcal{H} \rangle$ is trivial. The second problem is Coset Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, as well as elements $g, h \in G$, decide whether the intersection of the two cosets $g \langle \mathcal{G} \rangle \cap h \langle \mathcal{H} \rangle$ is empty. We show that both problems are decidable in finitely generated abelian-by-cyclic groups. In particular, we reduce them to the Shifted Monomial Membership problem (whether an ideal of the Laurent polynomial ring over integers contains any element of the form $X^z - f,\; z \in \mathbb{Z} \setminus \{0\}$). We also point out some obstacles for generalizing these results from abelian-by-cyclic groups to arbitrary metabelian groups. author: - Ruiwen Dong bibliography: - intermeta.bib title: Subgroup and Coset Intersection in abelian-by-cyclic groups --- # Introduction #### Algorithmic problems in groups {#algorithmic-problems-in-groups .unnumbered} Computational group theory is one of the oldest and most well-developed parts of computational algebra. Dating back to the first half of the twentieth century, the area provided some of the first undecidability results in the theory of computing. Among the most prominent problems is the Subgroup Membership problem, proposed by Mikhailova [@mikhailova1966occurrence] in the 1960s. For this problem, we work in a group $G$ that is typically infinite but finitely generated (such as the additive group of $\mathbb{Z}$, or a matrix group over integers). For a finite subset $\mathcal{G}$ of $G$, denote by $\langle {\mathcal{G}} \rangle$ the subgroup of $G$ generated by $\mathcal{G}$. The Subgroup Membership problem for the group $G$ is defined as follows. (i) *(Subgroup Membership)* Given a finite set of elements $\mathcal{G}\subseteq G$ and $h \in G$, decide whether $h \in \langle {\mathcal{G}} \rangle$. Denote by $e$ the neutral element of $G$. Another widely studied problem is the Subgroup Intersection problem: (i) *(Subgroup Intersection)* Given two finite sets of elements $\mathcal{G}, \mathcal{H}\subseteq G$, decide whether $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ is the trivial group $\{e\}$. While the intersection $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ is always a group, it is usually not clear how to compute its effective representation. In fact, the group $G$ might not satisfy the *Howson property*, meaning the intersection of two finitely generated subgroups $\langle {\mathcal{G}} \rangle, \langle {\mathcal{H}} \rangle$ of $G$ might not be finitely generated [@Moldavanskii1968IntersectionOF]. Deciding triviality of $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ can be considered as a first step towards understanding intersection of subgroups in the given group $G$. For an element $g \in G$ and a subgroup $S \leq G$, define the coset $g S \coloneqq \{gs \mid s \in S\}$. We also consider the Coset Intersection problem: (i) *(Coset Intersection)* Given two finite sets of elements $\mathcal{G}, \mathcal{H}\subseteq G$ and $g, h \in G$, decide whether $g \langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$ is empty. Coset Intersection as well as Subgroup Intersection has been extensively studied in various contexts such as permutation groups [@babai2010coset], abelian and nilpotent groups [@babai1996multiplicative; @macdonald2019low], and right-angled Artin groups [@delgado2018intersection]. They are intimately related to problems from numerous other areas such as Graph Isomorphism [@luks1982isomorphism], vector reachability [@potapov2019vector] and automata theory [@delgado2018intersection]. Since the intersection $g \langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$ is empty if and only if $\langle {\mathcal{G}} \rangle \cap g^{-1} h \langle {\mathcal{H}} \rangle$ is empty, we can without loss of generality suppose $g = e$ in the definition of Coset Intersection. That is, we want to decide whether $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. We may note that by setting $\mathcal{H}= \{e\}$, Coset Intersection subsumes Subgroup Membership. However, Coset Intersection does not subsume Subgroup Intersection, despite their obvious connection. It is not surprising that for general groups, all three problems are undecidable. A classic result of Mikhailova [@mikhailova1966occurrence] shows that Subgroup Membership is undecidable for the direct product $F_2 \times F_2$ of two free groups $F_2$ over two generators. Moreover, Mikhailova's construction also implies that Subgroup Intersection is undecidable for $F_2 \times F_2$ [@199499]. Nevertheless, for finitely generated free groups and abelian groups, Subgroup Membership, Subgroup Intersection and Coset Intersection have been shown to be decidable. For free groups, these decidability results were obtained using the classic construction of *Stallings foldings* [@bogopolski2010orbit Proposition 6.1] [@KAPOVICH2002608 Proposition 7.2, Corollary 9.5] [@stallings1991foldings]. This automata-inspired construction has now become the standard tool for describing subgroups of free groups. For abelian matrix groups, Babai, Beals, Cai, Ivanyos and Luks [@babai1996multiplicative] famously reduced computational problems for commutative matrices to computation over lattices. Therefore Subgroup Membership, Subgroup Intersection and Coset Intersection reduce to linear algebra over $\mathbb{Z}$ and are decidable in polynomial time. #### Metabelian and abelian-by-cyclic groups {#metabelian-and-abelian-by-cyclic-groups .unnumbered} As most algorithmic problems for abelian groups are well-understood due to their relatively simple structure, much effort has focused on relaxations of the commutativity requirement. For example, the aforementioned decidability results have been successfully extended to the class of nilpotent groups [@macdonald2019low]. Among the simplest and most well-studied extensions to abelian groups is the class of *metabelian groups*. A group $G$ is called metabelian if it admits an abelian normal subgroup $A$ such that the quotient group $G/A$ is abelian. Developing a complete algorithmic theory for finitely generated metabelian groups has been the focus of intense research since the 1950s [@baumslag1994algorithmic; @hall1954finiteness]. Despite their simple definition, many problems in finitely generated metabelian groups are still far from being well understood. Unlike free groups, abelian groups and nilpotent groups, metabelian groups do not satisfy the Howson property [@baumslag2010subgroups; @howson1954intersection]. This makes solving intersection-type problems in metabelian groups much more difficult. Among the three problems introduced above, only the decidability of Subgroup Membership is known, thanks to a classic result of Romanovskii [@romanovskii1974some]. Subgroup Intersection has been solved only for *free* metabelian groups [@baumslag2010subgroups] and the wreath products $\mathbb{Z}^m \wr \mathbb{Z}^n, m, n \geq 1$. Unfortunately, this solution does not generalize to arbitrary metabelian groups, as explicitly stated after [@baumslag2010subgroups Corollary C]. Despite Subgroup Intersection and Coset Intersection being currently out of reach for arbitrary metabelian groups, various results have been obtained for specific classes of metabelian groups. Recent results by Lohrey, Steinberg and Zetzsche [@lohrey2015rational] showed decidability of the *Rational Subset Membership* problem (which subsumes Coset Intersection) in the wreath products $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z},\; p \geq 2$. This result has been extended to the *Baumslag-Solitar groups* $\mathsf{BS}(1, p),\; p \geq 2,$ by Cadilhac, Chistikov and Zetzsche [@DBLP:conf/icalp/CadilhacCZ20]. The groups $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ and $\mathsf{BS}(1, p)$ can be respectively represented as groups of $2 \times 2$ matrices over the Laurent polynomial ring $\left(\mathbb{Z}/ p \mathbb{Z}\right)[X, X^{-1}]$ and over the ring $\mathbb{Z}[1/p] = \{\frac{a}{p^n} \mid a \in \mathbb{Z}, n \in \mathbb{N}\}$: $$\begin{aligned} \left(\mathbb{Z}/ p \mathbb{Z}\right) \wr \mathbb{Z}& \cong \left\{ \begin{pmatrix} X^{b} & f \\ 0 & 1 \end{pmatrix} \;\middle|\; f \in \left(\mathbb{Z}/ p \mathbb{Z}\right)[X, X^{-1}], b \in \mathbb{Z} \right\}, \label{eq:defwr} \\ \mathsf{BS}(1, p) & \cong \left\{ \begin{pmatrix} p^{b} & f \\ 0 & 1 \end{pmatrix} \;\middle|\; f \in \mathbb{Z}[1/p], b \in \mathbb{Z} \right\}. \label{eq:defbs}\end{aligned}$$ Alternatively, the element $\begin{pmatrix} X^{b} & f \\ 0 & 1 \end{pmatrix} \in (\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ can be thought of as a Turing machine configuration whose tape cells contain letters in $\{0, 1, \ldots, p-1\}$ which correspond to the coefficients of the polynomial $f$, while the head of the machine is positioned at the cell $b$. Multiplication in $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ corresponds to operating the machine by moving the head and adding integers to the cells modulo $p$. (See [@lohrey2015rational] for a complete description.) Similarly, $\mathsf{BS}(1, p)$ can be considered as a version of $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ with "carrying". The element $\begin{pmatrix} p^{b} & f \\ 0 & 1 \end{pmatrix} \in \mathsf{BS}(1, p)$ can be seen the base-$p$ expansion of the rational number $f \in \mathbb{Z}[1/p]$, along with a cursor at the $b$-th position. Multiplication in $\mathsf{BS}(1, p)$ corresponds to aligning the cursors of the two elements and adding up the numbers $f$. (See [@DBLP:conf/icalp/CadilhacCZ20] for a complete description.) The Turing machine-like structure of $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ and $\mathsf{BS}(1, p)$ can be explained by the following fact. Both groups belong to the much broader class of groups called *abelian-by-cyclic* groups. A group is called abelian-by-cyclic if it admits an abelian normal subgroup $A$ such that the quotient group $G/A$ is isomorphic to $\mathbb{Z}$. Intuitively, this isomorphism to $\mathbb{Z}$ gives them the Turing machine-like structure described above, as $\mathbb{Z}$ represents the indices of the tape. Abelian-by-cyclic groups have been extensively studied from the point of view of geometry and growth [@farb2000asymptotic; @hurtado2021global], algorithmic problems [@boler1976conjugacy], random walks [@pittet2003random], and group algebra isomorphism [@baginski1999isomorphism]. They also serve as a first step towards understanding general metabelian groups, whose definition is obtained by replacing $\mathbb{Z}$ with an arbitrary abelian group. Figure [\[fig:metabelian\]](#fig:metabelian){reference-type="ref" reference="fig:metabelian"} illustrates the relations between the classes of groups introduced above, as well as their known decidability results. In this paper, we show decidability of Subgroup Intersection and Coset Intersection in finitely generated abelian-by-cyclic groups. Our approach is different from the automata-based methods [@DBLP:conf/icalp/CadilhacCZ20; @lohrey2015rational] used for $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$ and $\mathsf{BS}(1, p)$. We reduce both Subgroup and Coset Intersection to the problem of finding an element of the form $X^z - f,\; z \in \mathbb{Z}\setminus \{0\}$ in a given ideal of the Laurent polynomial ring $\mathbb{Z}[X, X^{-1}]$. This problem has already been solved by Noskov [@noskov1982conjugacy]. However, Noskov's solution relies on a series of intricate arguments in commutative algebra. We propose a more direct solution using a combination of computational algebraic geometry and number theory. A natural follow-up to our work would be trying to generalize our results to arbitrary metabelian groups. This boils down to generalizing several arguments in this paper to *multivariate* polynomial rings, which become significantly more difficult. # Preliminaries #### Laurent polynomial ring and modules {#laurent-polynomial-ring-and-modules .unnumbered} A (univariate) *Laurent polynomial* with coefficients over $\mathbb{Z}$ is an expression of the form $$f = \sum_{i = p}^q a_i X^i, \quad \text{where $p, q \in \mathbb{Z}$ and $a_i \in \mathbb{Z}, i = p, p+1, \ldots, q$.}$$ The set of all Laurent polynomials with coefficients over $\mathbb{Z}$ forms a ring and is denoted by $\mathbb{Z}[X^{\pm}]$. On the other hand, we denote by $\mathbb{Z}[X]$ the *usual* univariate polynomial ring over $\mathbb{Z}$: it contains elements whose monomials have non-negative degree. Let $d \geq 1$ be a positive integer. One can similarly define the Laurent polynomial ring $$\mathbb{Z}[X^{\pm d}] \coloneqq \left\{\sum_{i = p}^q a_{di} X^{di} \in \mathbb{Z}[X^{\pm}] \;\middle|\; p, q \in \mathbb{Z}, a_{dp}, \ldots, a_{dq} \in \mathbb{Z}\right\}.$$ Its elements are Laurent polynomials whose monomials have degrees divisible by $d$. Let $R$ be a commutative ring. An $R$-module is defined as an abelian group $(M, +)$ along with an operation $\cdot \;\colon R \times M \rightarrow M$ satisfying $f \cdot (a+b) = f \cdot a + f \cdot b$, $(f + g) \cdot a = f \cdot a + g \cdot a$, $fg \cdot a = f \cdot (g \cdot a)$ and $1 \cdot a = a$. We will denote by $\boldsymbol{0}$ the neutral element of an $R$-module $M$. For example, for any $d \in \mathbb{N}$, the group $\mathbb{Z}[X^{\pm}]$ can be seen as a $\mathbb{Z}[X^{\pm d}]$-module by $f \cdot g \coloneqq fg, \; f \in \mathbb{Z}[X^{\pm d}], g \in \mathbb{Z}[X^{\pm}]$. In general, in order to define a $\mathbb{Z}[X^{\pm d}]$-module structure on an abelian group $M$, it suffices to define $X^d \cdot m$ and $X^{-d} \cdot m$ for all $m \in M$. The value of $f \cdot m, \; f \in \mathbb{Z}[X^{\pm d}], m \in M,$ would then follow from the linearity of the operation $\cdot$. An *ideal* of $R$ is a subset of $R$ that is an $R$-module. If $M$ is an $R$-module and $m \in M$, then $R \cdot m \coloneqq \{r \cdot m \mid r \in R\}$ is again an $R$-module. If $N$ and $N'$ are $R$-submodule of $M$, then $N + N' \coloneqq \{n + n' \mid n \in N, n' \in N'\}$ is again an $R$-submodule of $M$. #### Finite presentation of modules {#finite-presentation-of-modules .unnumbered} For any $D \in \mathbb{N}$, $\mathbb{Z}[X^{\pm}]^D$ is a $\mathbb{Z}[X^{\pm}]$-module by $f \cdot (g_1, \ldots, g_D) \coloneqq (fg_1, \ldots, fg_D)$. Throughout this paper, we use the bold symbol $\boldsymbol{f}$ to denote a vector $(f_1, \ldots, f_d) \in \mathbb{Z}[X^{\pm}]^D$. Given $\boldsymbol{g}_1, \ldots, \boldsymbol{g}_m \in \mathbb{Z}[X^{\pm}]^D$, we say they *generate* the $\mathbb{Z}[X^{\pm}]$-module $\sum_{i=1}^D \mathbb{Z}[X^{\pm}] \cdot \boldsymbol{g}_i$. A module is called *finitely generated* if it can be generated by a finite number of elements. Given two finitely generated $\mathbb{Z}[X^{\pm}]$-submodules $N, M$ of $\mathbb{Z}[X^{\pm}]^D$ such that $N \subseteq M$, we can define the quotient $M/N \coloneqq \{\overline{\boldsymbol{m}} \mid \boldsymbol{m}\in M\}$ where $\overline{\boldsymbol{m}_1} = \overline{\boldsymbol{m}_2}$ if and only if $\boldsymbol{m}_1 - \boldsymbol{m}_2 \in N$. This quotient is also an $\mathbb{Z}[X^{\pm}]$-module. We say that an $\mathbb{Z}[X^{\pm}]$-module $\mathcal{A}$ is *finitely presented* if it can be written as a quotient $M/N$ for two finitely generated submodules $N \subseteq M$ of $\mathbb{Z}[X^{\pm}]^D$ for some $D \in \mathbb{N}$. Such a pair $(M, N)$, given by their respective generators, is called a *finite presentation* of $\mathcal{A}$. The element $\overline{\boldsymbol{m}}$ of $\mathcal{A}$ is effectively represented by $\boldsymbol{m}\in \mathbb{Z}[X^{\pm}]^D$, this representation is unique modulo $N$. Effective computation in finitely presented modules over polynomial rings is a well-studied area, with numerous algorithms developed to solve a wide range of computation problems. In particular, these algorithms have been applied to solve various other decision problems in metabelian groups, see the paper [@baumslag1981computable] by Baumslag, Cannonito and Miller for a comprehensive account on this subject. The following are some classic computational problems with effective algorithms that we will make use of. **Lemma 1** ([@baumslag1981computable Lemma 2.1, 2.2]). *Let $\mathcal{A}$ be a $\mathbb{Z}[X^{\pm}]$-module with a given finite presentation. The following problems are effectively solvable:* (i) **(Submodule Membership)* Given elements $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k, \boldsymbol{a}\in \mathcal{A}$, decide whether $\boldsymbol{a}$ is in the submodule generated by $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k$.* (ii) **(Computing Syzygies)* Given elements $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k \in \mathcal{A}$, compute a finite set of generators for the *Syzygy module* $S \subseteq \mathbb{Z}[X^{\pm}]^k$: $$S \coloneqq \left\{(f_1, \ldots, f_k) \in \mathbb{Z}[X^{\pm}]^k \;\middle|\; f_1 \cdot \boldsymbol{a}_1 + \cdots + f_k \cdot \boldsymbol{a}_k = \boldsymbol{0}\right\}.$$* (iii) **(Computing Intersection)* Given the generators $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k$ of a submodule $A \subseteq \mathcal{A}$ and the generators $\boldsymbol{b}_1, \ldots, \boldsymbol{b}_m$ of a submodule $B \subseteq \mathcal{A}$, compute a finite set of generators for the submodule $A \cap B$.* *This effectiveness still holds if we replace the Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ with the regular polynomial ring $\mathbb{Z}[X]$.* In particular, taking $\mathcal{A}\coloneqq \mathbb{Z}[X^{\pm}]$, Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(i) becomes the well-known *Ideal Membership* problem: given elements $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k, \boldsymbol{a}\in \mathbb{Z}[X^{\pm}]$, decide whether $\boldsymbol{a}$ is in the ideal generated by $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_k$. For Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii), it states that one can compute the generators for the solution set of any homogeneous linear equation. Alternatively, Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii) can be understood as a procedure to compute the finite presentation $\mathbb{Z}[X^{\pm}]^k/S$ of the module $\sum_{i = 1}^k \mathbb{Z}[X^{\pm}] \cdot \boldsymbol{a}_i$. The following lemma shows we can effectively compute the intersection of a submodule of $\mathbb{Z}[X^{\pm}]^k$ with $\mathbb{Z}^k$. **Lemma 1** ([@baumslag1981computable Corollary 2.5(2)]). *Suppose we are given $k \in \mathbb{N}$ and elements $\boldsymbol{g}_1, \ldots, \boldsymbol{g}_n$ of the $\mathbb{Z}[X^{\pm}]$-module $\mathbb{Z}[X^{\pm}]^k$. Let $\mathcal{M}$ denote the $\mathbb{Z}[X^{\pm}]$-module generated by $\boldsymbol{g}_1, \ldots, \boldsymbol{g}_n$, and define $\Lambda \coloneqq \mathcal{M}\cap \mathbb{Z}^k$. Then $\Lambda \subseteq \mathbb{Z}^k$ is a $\mathbb{Z}$-module, and a finite set of generators for $\Lambda$ can be effectively computed.* Recall that for any $d \geq 1$, a $\mathbb{Z}[X^{\pm}]$-module is naturally a $\mathbb{Z}[X^{\pm d}]$-module. In particular, $\mathbb{Z}[X^{\pm}]^D$ is isomorphic as a $\mathbb{Z}[X^{\pm d}]$-module to $\mathbb{Z}[X^{\pm d}]^{Dd}$, and any finitely presented $\mathbb{Z}[X^{\pm}]$-module can be considered as a finitely presented $\mathbb{Z}[X^{\pm d}]$-module: lemlemchangebase[\[lem:changebase\]]{#lem:changebase label="lem:changebase"} Let $d \geq 2$. Given a finite presentation of a $\mathbb{Z}[X^{\pm}]$-module $\mathcal{A}$, one can compute a finite presentation of $\mathcal{A}$ as a $\mathbb{Z}[X^{\pm d}]$-module. Furthermore, let $\boldsymbol{a}\in \mathcal{A}$ be given in the finite presentation of $\mathcal{A}$ as $\mathbb{Z}[X^{\pm}]$-module, then one can compute the representation of $\boldsymbol{a}$ in $\mathcal{A}$ considered as a $\mathbb{Z}[X^{\pm d}]$-module. #### Abelian-by-cyclic groups {#abelian-by-cyclic-groups .unnumbered} We now formally define abelian-by-cyclic groups, the main object of study in this paper. **Definition 1**. A group $G$ is called *abelian-by-cyclic* if it admits an abelian normal subgroup $A$ such that $G/A \cong \mathbb{Z}$. It is a classic result [@boler1976conjugacy p.17] that every finitely generated abelian-by-cyclic group $G$ can be written as a *semidirect product* $\mathcal{A}\rtimes \mathbb{Z}$: $$\mathcal{A}\rtimes \mathbb{Z}\coloneqq \left\{ (\boldsymbol{a}, z) \;\middle|\; \boldsymbol{a}\in \mathcal{A}, z \in \mathbb{Z}\right\},$$ where $\mathcal{A}$ is a finitely presented $\mathbb{Z}[X^{\pm}]$-module. The group law in $\mathcal{A}\rtimes \mathbb{Z}$ is defined by $$(\boldsymbol{a}, z) \cdot (\boldsymbol{a}', z') = (\boldsymbol{a}+ X^z \cdot \boldsymbol{a}', z + z'), \quad (\boldsymbol{a}, z)^{-1} = (- X^{-z} \cdot \boldsymbol{a}, -z).$$ The neutral element of $\mathcal{A}\rtimes \mathbb{Z}$ is $(\boldsymbol{0}, 0)$. Intuitively, the element $(\boldsymbol{a}, z)$ is analogous to a $2 \times 2$ matrix $\begin{pmatrix} X^{z} & \boldsymbol{a}\\ 0 & 1 \\ \end{pmatrix}$, where group multiplication is represented by matrix multiplication. By direct computation, for all $m \in \mathbb{Z}$, we have $$\begin{aligned} (\boldsymbol{a}, z)^m = \begin{cases} \left(\frac{X^{mz} - 1}{X^z - 1} \cdot \boldsymbol{a}, mz\right), \quad & z \neq 0, \\ \left(m \cdot \boldsymbol{a}, 0\right), \quad & z = 0. \\ \end{cases}\end{aligned}$$ We naturally identify $\mathcal{A}$ with the subgroup $\left\{ (\boldsymbol{a}, 0) \;\middle|\; \boldsymbol{a}\in \mathcal{A}\right\}$ of $\mathcal{A}\rtimes \mathbb{Z}$. In particular, the quotient $\left( \mathcal{A}\rtimes \mathbb{Z}\right)/\mathcal{A}$ is isomorphic to $\mathbb{Z}$, so $\mathcal{A}\rtimes \mathbb{Z}$ is indeed abelian-by-cyclic. If we take $\mathcal{A}\coloneqq (\mathbb{Z}/p\mathbb{Z})[X, X^{-1}] = \mathbb{Z}[X^{\pm}]/\left(\mathbb{Z}[X^{\pm}] \cdot p\right)$, then we recover the definition [\[eq:defwr\]](#eq:defwr){reference-type="eqref" reference="eq:defwr"} of the group $(\mathbb{Z}/p\mathbb{Z}) \wr \mathbb{Z}$. If we take $\mathcal{A}\coloneqq \mathbb{Z}[1/p] = \mathbb{Z}[X^{\pm}]/\left(\mathbb{Z}[X^{\pm}] \cdot (X - p)\right)$, then we recover the definition [\[eq:defbs\]](#eq:defbs){reference-type="eqref" reference="eq:defbs"} of the group $\mathsf{BS}(1, p)$. Throughout this paper, a finitely generated abelian-by-cyclic group $G$ is always represented as the semidirect product $\mathcal{A}\rtimes \mathbb{Z}$, where $\mathcal{A}$ is a $\mathbb{Z}[X^{\pm}]$-module given by a finite presentation. # Main results and overview The main result of this paper is the following. [\[thm:main\]]{#thm:main label="thm:main"} Subgroup Intersection and Coset Intersection are decidable for finitely generated abelian-by-cyclic groups. Our proof of Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} is divided into two parts. The first part is to reduce both Subgroup Intersection and Coset Intersection to the *Shifted Monomial Membership* problem: **Definition 1**. *Shifted Monomial Membership* is the following decision problem. Given as input a finite set of generators of an ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$, as well as a Laurent polynomial $f \in \mathbb{Z}[X^{\pm}]$, decide if there exists $z \in \mathbb{Z}\setminus \{\boldsymbol{0}\}$ such that $X^z - f \in \mathcal{I}$. The reduction from Subgroup Intersection and Coset Intersection to Shifted Monomial Membership combines various classic techniques from effective computation of finitely presented $\mathbb{Z}[X^{\pm}]$-modules, and will be shown in Section [4](#sec:reduction){reference-type="ref" reference="sec:reduction"}. The main difficulty in this part is the interaction of modules over different base rings. Our key is to adaptively change the base rings when combining different modules. The second part is to prove decidability of Shifted Monomial Membership: this will be shown in Section [5](#sec:smm){reference-type="ref" reference="sec:smm"}. As written in the introduction, we provide a more direct proof than that of Noskov [@noskov1982conjugacy]. We will use structural theorems to classify ideals of $\mathbb{Z}[X]$ and consider each case separately. In some cases, we employ arguments of *height of algebraic numbers* to produce a bound on $|z|$ whenever $X^z - f \in \mathcal{I}$. In other cases, we will show certain periodicity stemming from the finiteness of quotients or from roots of unity. As a result, in all cases it suffices to verify whether $X^z - f \in \mathcal{I}$ for a finite number of $z$. Omitted proofs can be found in Appendix [6](#app:proof){reference-type="ref" reference="app:proof"}. A summary of the algorithms for deciding Subgroup Intersection, Coset Intersection, and Shifted Monomial Membership is given in Appendix [7](#app:alg){reference-type="ref" reference="app:alg"}. # Reduction to Shifted Monomial Membership {#sec:reduction} Let $\mathcal{A}$ be a $\mathbb{Z}[X^{\pm}]$-module given by a finite presentation. Recall that we naturally identify $\mathcal{A}$ with the subgroup $\left\{ (\boldsymbol{a}, 0) \;\middle|\; \boldsymbol{a}\in \mathcal{A}\right\}$ of $\mathcal{A}\rtimes \mathbb{Z}$; that is, we will sometimes write $\boldsymbol{a}$ instead of $(\boldsymbol{a}, 0)$ when the context is clear. We start with a lemma that effectively describes finitely generated subgroups of $\mathcal{A}\rtimes \mathbb{Z}$. Such a description follows from the general description of subgroups of finitely generated metabelian groups [@romanovskii1974some proof of Theorem 1]. Here we give a systematic reformulation in the context of abelian-by-cyclic groups. lemlemstruct[\[lem:struct\]]{#lem:struct label="lem:struct"} Let $\langle {\mathcal{G}} \rangle$ be a subgroup of $\mathcal{A}\rtimes \mathbb{Z}$ generated by the elements $g_1 \coloneqq (\boldsymbol{a}_1, z_1), \ldots, g_K \coloneqq (\boldsymbol{a}_K, z_K)$. Then (i) If $z_1 = \cdots = z_K = 0$, then $\langle {\mathcal{G}} \rangle$ is contained in $\mathcal{A}$ and it is the $\mathbb{Z}$-module generated by $\boldsymbol{a}_1, \ldots, \boldsymbol{a}_K$. (ii) If $z_1, \ldots, z_K$ are not all zero, then $\langle {\mathcal{G}} \rangle \not\subset \mathcal{A}$. Let $d \in \mathbb{N}$ denote the greatest common divisor of $z_1, \ldots, z_K$. Consider the lattice $$\Lambda \coloneqq \left\{(s_1, \ldots, s_K) \in \mathbb{Z}^K \;\middle|\; s_1 z_1 + \cdots + s_K z_K = 0\right\}.$$ Let $(s_{11}, \ldots, s_{1K}), \ldots, (s_{T1}, \ldots, s_{TK})$ be a finite set of generators for $\Lambda$. Then $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is a $\mathbb{Z}[X^{\pm d}]$-submodule of $\mathcal{A}$, generated by the set of elements $$\label{eq:genmod} S \coloneqq \left\{g_i g_j g_i^{-1} g_j^{-1} \;\middle|\; 1 \leq i < j \leq K \right\} \cup \left\{g_1^{s_{i1}} \cdots g_K^{s_{iK}} \;\middle|\; i \in [1, T] \right\}.$$ (iii) In case (ii), let $\boldsymbol{a}\in \mathcal{A}$ be any element such that $(\boldsymbol{a}, d) \in \langle {\mathcal{G}} \rangle$. Then $\langle {\mathcal{G}} \rangle$ is generated by $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and $(\boldsymbol{a}, d)$ as a group. In other words, every element of $\langle {\mathcal{G}} \rangle$ can be written as $(\boldsymbol{b}, 0) \cdot (\boldsymbol{a}, d)^m$ for some $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and $m \in \mathbb{Z}$. We point out that in case (ii), the subgroup $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is finitely generated as a $\mathbb{Z}[X^{\pm d}]$-module; but it is not necessarily finitely generated as a group. Let $\mathcal{A}= \mathbb{Z}[X^{\pm}]$, considered as a $\mathbb{Z}[X^{\pm}]$-module. Let $\langle {\mathcal{G}} \rangle$ be the subgroup of $\mathcal{A}\rtimes \mathbb{Z}$ generated by the elements $g_1 = (X, 4), g_2 = (1+X, -6)$. Then $d = 2$, and $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is the $\mathbb{Z}[X^{\pm 2}]$-module generated by the elements $$g_1 g_2 g_1^{-1} g_2^{-1} = (X^5 + X^4 - 1 - X^{-5}, 0), \quad g_1^3 g_2^2 = (X^{13} + X^{12} + X^{9} + X^{7} + X^{6} + X^{5} + X, 0).$$ For example, consider the element $g_1^2 g_2 g_1 g_2 \in \langle {\mathcal{G}} \rangle$. By direct computation, its second entry is zero, therefore $g_1^2 g_2 g_1 g_2 \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$. Furthermore, $g_1^2 g_2 g_1 g_2$ can be written as $$\begin{gathered} g_1^2 g_2 g_1 g_2 = g_1^2 (g_2 g_1) g_2 = g_1^2 (g_2 g_1 g_2^{-1} g_1^{-1}) (g_1 g_2) g_2 = g_1^2 (g_1 g_2 g_1^{-1} g_2^{-1})^{-1} g_1 g_2^2 \\ = g_1^2 (g_1 g_2^2) (g_1 g_2^2)^{-1} (g_1 g_2 g_1^{-1} g_2^{-1})^{-1} (g_1 g_2^2) = g_1^3 g_2^2 \cdot (g_1 g_2^2)^{-1} (g_1 g_2 g_1^{-1} g_2^{-1})^{-1} (g_1 g_2^2) \\ = (X^{13} + X^{12} + X^{9} + X^{7} + X^{6} + X^{5} + X, 0) \cdot (g_1 g_2^2)^{-1} (X^5 + X^4 - 1 - X^{-5}, 0)^{-1} (g_1 g_2^2) \\ = (X^{13} + X^{12} + X^{9} + X^{7} + X^{6} + X^{5} + X) + X^{-8} \cdot (-1) \cdot (X^5 + X^4 - 1 - X^{-5}). \end{gathered}$$ It is therefore indeed in the $\mathbb{Z}[X^{\pm 2}]$-module generated by $g_1 g_2 g_1^{-1} g_2^{-1} = X^5 + X^4 - 1 - X^{-5}$ and $g_1^3 g_2^2 = X^{13} + X^{12} + X^{9} + X^{7} + X^{6} + X^{5} + X$. Intuitively, modulo the generator $g_1 g_2 g_1^{-1} g_2^{-1}$, one can permute letters in any word over $\mathcal{G}$ (in the above example, $g_1^2 g_2 g_1 g_2$ is congruent to $g_1^3 g_2^2$). Whereas the generator $g_1^3 g_2^2$ guarantees the second entry of the product to be zero. Let $\langle {\mathcal{G}} \rangle, \langle {\mathcal{H}} \rangle$ be finitely generated subgroups of $\mathcal{A}\rtimes \mathbb{Z}$ given by their respective generators, let $h$ be an element of $\mathcal{A}\rtimes \mathbb{Z}$. We now consider Subgroup and Coset Intersection for $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$. We split into three cases according to whether $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ are contained in the subgroup $\mathcal{A}$. If at least one of $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$ (Case 1 and 2 below), then the solutions to Subgroup and Coset Intersection are relatively straightforward using the standard tools introduced in Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}. They do not need to be reduced to Shifted Monomial Membership. If neither $\langle {\mathcal{G}} \rangle$ nor $\langle {\mathcal{H}} \rangle$ is contained in the subgroup $\mathcal{A}$ (Case 3 below), then the solution is more complicated and we reduce both Subgroup and Coset Intersection to Shifted Monomial Membership. ## Case 1: $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ are both contained in $\mathcal{A}$ Suppose $\langle {\mathcal{G}} \rangle$ is generated by the elements $g_1 = (\boldsymbol{a}_1, 0), \ldots, g_K = (\boldsymbol{a}_K, 0)$, and $\langle {\mathcal{H}} \rangle$ is generated by the elements $h_1 = (\boldsymbol{a}'_1, 0), \ldots, h_M = (\boldsymbol{a}'_M, 0)$. #### Subgroup Intersection {#subgroup-intersection .unnumbered} In this case, we have $$\langle {\mathcal{G}} \rangle = \{y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K \mid y_1, \ldots, y_K \in \mathbb{Z}\}, \quad \langle {\mathcal{H}} \rangle = \{z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M \mid z_1, \ldots, z_M \in \mathbb{Z}\}.$$ Then $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ if and only if every element of $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ is equal to the neutral element. This means that every solution of $y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M, \; y_1, \ldots, y_K, z_1, \ldots, z_M \in \mathbb{Z}$ is also a solution of $y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = \boldsymbol{0}$. Let $\mathcal{M}$ denote the $\mathbb{Z}[X^{\pm}]$-module $$\begin{gathered} \label{eq:M1s} \mathcal{M}\coloneqq \Big\{ (y_1, \ldots, y_K, z_1, \ldots, z_M) \in \mathbb{Z}[X^{\pm}]^{K+M} \;\Big|\; \\ y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K - z_1 \cdot \boldsymbol{a}'_1 - \cdots - z_M \cdot \boldsymbol{a}'_M = \boldsymbol{0}\Big\},\end{gathered}$$ and let $\mathcal{Z}$ denote the $\mathbb{Z}[X^{\pm}]$-module $$\label{eq:Z1s} \mathcal{Z}\coloneqq \Big\{ (y_1, \ldots, y_K, z_1, \ldots, z_M) \in \mathbb{Z}[X^{\pm}]^{K+M} \;\Big|\; y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = \boldsymbol{0}\Big\}.$$ Then the statement above can be summarized as follows. **Observation 1**. We have $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ if and only if $\mathcal{M}\cap \mathbb{Z}^{K+M} = (\mathcal{M}\cap \mathcal{Z}) \cap \mathbb{Z}^{K+M}$. Indeed, the left hand side $\mathcal{M}\cap \mathbb{Z}^{K+M}$ denotes all the integer solutions of the equation $y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M$, while the right hand side $(\mathcal{M}\cap \mathcal{Z}) \cap \mathbb{Z}^{K+M}$ denotes all integer solutions of $y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M = \boldsymbol{0}$. By Observation [Observation 1](#obs:C1SI){reference-type="ref" reference="obs:C1SI"}, we can decide Subgroup Intersection in this case. By Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii) we can compute the generators of $\mathcal{M}$ and $\mathcal{Z}$, then by Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(iii) we can compute the generators of $\mathcal{M}\cap \mathcal{Z}$. Next, by Lemma [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"} we can compute the generators of $\mathcal{M}\cap \mathbb{Z}^{K+M}$ and $(\mathcal{M}\cap \mathcal{Z}) \cap \mathbb{Z}^{K+M}$. Since these are subgroups of $\mathbb{Z}^{K+M}$, their equality can be decided by checking whether all generators of one subgroup belong to the other subgroup. #### Coset Intersection {#coset-intersection .unnumbered} Let $h = (\boldsymbol{a}_h, z_h)$. If $z_h \neq 0$ then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. Therefore we only need to consider the case where $z_h = 0$. Then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ if and only if there is no solution for $y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K = z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M + z \cdot \boldsymbol{a}_h, \; y_1, \ldots, y_K, z_1, \ldots, z_M \in \mathbb{Z}, z = 1$. Let $\mathcal{M}'$ denote the $\mathbb{Z}[X^{\pm}]$-module $$\begin{gathered} \label{eq:M1c} \mathcal{M}' \coloneqq \Big\{ (y_1, \ldots, y_K, z_1, \ldots, z_M, z) \in \mathbb{Z}[X^{\pm}]^{K+M+1} \;\Big|\; \\ y_1 \cdot \boldsymbol{a}_1 + \cdots + y_K \cdot \boldsymbol{a}_K - z_1 \cdot \boldsymbol{a}'_1 - \cdots - z_M \cdot \boldsymbol{a}'_M - z \cdot \boldsymbol{a}_h = \boldsymbol{0}\Big\}.\end{gathered}$$ **Observation 1**. We have $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ if and only if $\big(\mathcal{M}' \cap \mathbb{Z}^{K+M+1}\big) \cap \big(\mathbb{Z}^{K+M} \times \{1\}\big) = \emptyset$. Again, $\mathcal{M}' \cap \mathbb{Z}^{K+M+1}$ can be computed by Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii)(iii). So Coset Intersection in this case can be decided using linear algebra over $\mathbb{Z}$. ## Case 2: one of $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$ This case is similar to the previous case, we leave the detailed proofs in the appendix and summarize the result by the following proposition. proppropCtwo[\[prop:C2\]]{#prop:C2 label="prop:C2"} Suppose exactly one of $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$. Let $h \in \mathcal{A}\rtimes \mathbb{Z}$. Given finite sets of generators of $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ as groups, it is decidable whether $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ and whether $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. ## Case 3: neither $\langle {\mathcal{G}} \rangle$ nor $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$ By Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}, suppose $\langle {\mathcal{G}} \rangle$ is generated by $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and an element $(\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})$; and $\langle {\mathcal{H}} \rangle$ is generated by $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ and an element $(\boldsymbol{a}_{\mathcal{H}}, d_{\mathcal{H}})$. The elements $(\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})$ and $(\boldsymbol{a}_{\mathcal{H}}, d_{\mathcal{H}})$ can be effectively computed from the generating sets $\mathcal{G}, \mathcal{H}$ by performing the Euclidean algorithm. Furthermore, $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is a $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module whose generators are explicitly given (by Equation [\[eq:genmod\]](#eq:genmod){reference-type="eqref" reference="eq:genmod"}), and $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ is a $\mathbb{Z}[X^{\pm d_{\mathcal{H}}}]$-module whose generators are explicitly given. #### Subgroup Intersection {#subgroup-intersection-1 .unnumbered} We have $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ if and only if the equation $(\boldsymbol{b}, 0) \cdot (\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})^m = (\boldsymbol{c}, 0) \cdot (\boldsymbol{a}_{\mathcal{H}}, d_{\mathcal{H}})^n$ has non-trivial solutions $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A},\; \boldsymbol{c}\in \langle {\mathcal{H}} \rangle \cap \mathcal{A},\; m, n \in \mathbb{Z}$. Here, non-trivial means $\boldsymbol{b}, \boldsymbol{c}, m, n$ are not all zero. By direct computation, this equation is equivalent to the system $$\label{eq:intersys} \boldsymbol{b}+ \frac{X^{m d_{\mathcal{G}}} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} = \boldsymbol{c}+ \frac{X^{n d_{\mathcal{H}}} - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}}, \quad m d_{\mathcal{G}} = n d_{\mathcal{H}}.$$ An obstacle here is that $\boldsymbol{b}$ and $\boldsymbol{c}$ take values respectively in the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and the $\mathbb{Z}[X^{\pm d_{\mathcal{H}}}]$-module $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$. This difference makes solving [\[eq:intersys\]](#eq:intersys){reference-type="eqref" reference="eq:intersys"} complicated. To overcome this obstacle we define $d \coloneqq \operatorname{lcm}(d_{\mathcal{G}}, d_{\mathcal{H}})$; that is, $d$ is the smallest positive integer such that $d_{\mathcal{G}} \mid d, \; d_{\mathcal{H}} \mid d$. We can thus consider both $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ as $\mathbb{Z}[X^{\pm d}]$-modules (see Lemma [\[lem:changebase\]](#lem:changebase){reference-type="ref" reference="lem:changebase"}). One can compute a finite set of generators $S_{\mathcal{G}}$ for $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ as a $\mathbb{Z}[X^{\pm d}]$-module. Similarly, one can compute a finite set of generators $S_{\mathcal{H}}$ for $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ as a $\mathbb{Z}[X^{\pm d}]$-module. Define the $\mathbb{Z}[X^{\pm d}]$-module $\mathcal{M}\coloneqq (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + (\langle {\mathcal{H}} \rangle \cap \mathcal{A})$, it is generated by the set $S_{\mathcal{G}} \cup S_{\mathcal{H}}$. lemlemintertoeq[\[lem:intertoeq\]]{#lem:intertoeq label="lem:intertoeq"} The intersection $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ is non-trivial if and only if at least one of the following two conditions is satisfied: (i) $(\langle {\mathcal{G}} \rangle \cap \mathcal{A}) \cap (\langle {\mathcal{H}} \rangle \cap \mathcal{A}) \neq \{\boldsymbol{0}\}$. (ii) The equation $$(X^{zd} - 1) \cdot \boldsymbol{a}_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}$$ has solution $z \in \mathbb{Z}\setminus \{\boldsymbol{0}\}$. Here, $$\boldsymbol{a}_{\mathcal{G}, \mathcal{H}} \coloneqq \frac{X^d - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - \frac{X^d - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}}.$$ *Proof.* Suppose $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$ is non-trivial. Let $(\boldsymbol{a}, z') \in \langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle$, then $d_{\mathcal{G}} \mid z'$ and $d_{\mathcal{H}} \mid z'$. Therefore $d \mid z'$ and we can write $z' = zd$ for some $z \in \mathbb{Z}$. If $z = 0$ then $\boldsymbol{a}\in (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) \cap (\langle {\mathcal{H}} \rangle \cap \mathcal{A})$, so $(\langle {\mathcal{G}} \rangle \cap \mathcal{A}) \cap (\langle {\mathcal{H}} \rangle \cap \mathcal{A}) \neq \{\boldsymbol{0}\}$. If $z \neq 0$ then Equation [\[eq:intersys\]](#eq:intersys){reference-type="eqref" reference="eq:intersys"} has solution with $m d_{\mathcal{G}} = n d_{\mathcal{H}} = z d$, meaning $$\frac{X^{zd} - 1}{X^d - 1} \cdot \left(\frac{X^d - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - \frac{X^d - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}}\right) = \boldsymbol{c}- \boldsymbol{b}.$$ In particular, we have $\boldsymbol{c}- \boldsymbol{b}\in (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + (\langle {\mathcal{H}} \rangle \cap \mathcal{A}) = \mathcal{M}$. Multiplying both sides by $X^d - 1$ yields $(X^{zd} - 1) \cdot \boldsymbol{a}_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}$. Suppose either (i) or (ii) is satisfied. In case (i), we have $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle \supseteq (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) \cap (\langle {\mathcal{H}} \rangle \cap \mathcal{A}) \neq \{\boldsymbol{0}\}$. In case (ii), we have $$\frac{X^{zd} - 1}{X^d - 1} \cdot \boldsymbol{a}_{\mathcal{G}, \mathcal{H}} \in \mathcal{M}= (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + (\langle {\mathcal{H}} \rangle \cap \mathcal{A}),$$ so it can be written as $\boldsymbol{c}- \boldsymbol{b}$ for some $\boldsymbol{c}\in \langle {\mathcal{H}} \rangle \cap \mathcal{A}, \; \boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$. Therefore Equation [\[eq:intersys\]](#eq:intersys){reference-type="eqref" reference="eq:intersys"} has non-trivial solutions by taking $m \coloneqq zd/d_{\mathcal{G}}, \, n \coloneqq zd/d_{\mathcal{H}}$. ◻ **Proposition 1**. *Suppose neither $\langle {\mathcal{G}} \rangle$ nor $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$. Then deciding whether $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ reduces to Shifted Monomial Membership (see Definition [Definition 1](#def:SMM){reference-type="ref" reference="def:SMM"}).* *Proof.* (See Algorithm [\[alg:groupinter\]](#alg:groupinter){reference-type="ref" reference="alg:groupinter"} for a summary.) We use Lemma [\[lem:intertoeq\]](#lem:intertoeq){reference-type="ref" reference="lem:intertoeq"}. Condition (i) can be decided using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(iii), so it suffices to decide condition (ii). The set $$\label{eq:defI} \mathcal{I}\coloneqq \left\{f \in \mathbb{Z}[X^{\pm d}] \;\middle|\; f \cdot \boldsymbol{a}_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}\right\}$$ is an ideal of $\mathbb{Z}[X^{\pm d}]$. Furthermore, given a finite set of generators for $\mathcal{M}$, one can compute a finite set of generators for $\mathcal{I}$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii). Condition (ii) in Lemma [\[lem:intertoeq\]](#lem:intertoeq){reference-type="ref" reference="lem:intertoeq"} is equivalent to "$\mathcal{I}$ contains an element $X^{zd} - 1$ for some $z \in \mathbb{Z}\setminus \{0\}$". Performing the variable change $X^d \rightarrow X$, we can consider $\mathcal{I}$ as an ideal of $\mathbb{Z}[X^{\pm}]$ and the condition becomes "$\mathcal{I}$ contains an element $X^{z} - 1$ for some $z \in \mathbb{Z}\setminus \{0\}$". This is exactly the Shifted Monomial Membership problem (Definition [Definition 1](#def:SMM){reference-type="ref" reference="def:SMM"}) with $f = 1$. ◻ **Remark 1**. The fact that $\mathcal{M}\coloneqq (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + (\langle {\mathcal{H}} \rangle \cap \mathcal{A})$ is a finitely generated $\mathbb{Z}[X^{\pm d}]$-module is crucial to the reduction in Proposition [Proposition 1](#prop:interredsmm){reference-type="ref" reference="prop:interredsmm"}. This argument is specific to abelian-by-cyclic groups and no longer holds in arbitrary metabelian groups. For example, let $\mathcal{A}$ now be a finitely presented module over the bivariate Laurent polynomial ring $\mathbb{Z}[X^{\pm}, Y^{\pm}]$. We can similarly define the semidirect product $\mathcal{A}\rtimes \mathbb{Z}^2$, which is metabelian but not abelian-by-cyclic. We can find subgroups $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$ such that $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is a finitely generated $\mathbb{Z}[X^{\pm}]$-module, while $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ is a finitely generated $\mathbb{Z}[Y^{\pm}]$-module. In this case, if we define the sum $\mathcal{M}\coloneqq (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + (\langle {\mathcal{H}} \rangle \cap \mathcal{A})$, then $\mathcal{M}$ is not a $\mathbb{Z}[X^{\pm d}]$-module, a $\mathbb{Z}[Y^{\pm d}]$-module, or a $\mathbb{Z}[X^{\pm d}, Y^{\pm d}]$-module for any $d \geq 1$. While $\mathcal{M}$ is still a $\mathbb{Z}$-module (since both $\mathbb{Z}[X^{\pm}]$-modules and $\mathbb{Z}[Y^{\pm}]$-modules can be seen as $\mathbb{Z}$-modules), it is not finitely generated as a $\mathbb{Z}$-module. Since being finitely generated is essential to the effectiveness results in Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}-[\[lem:changebase\]](#lem:changebase){reference-type="ref" reference="lem:changebase"}, this constitutes the key difficulty in generalizing our results from abelian-by-cyclic groups to arbitrary metabelian groups. The same difficulty also appears in Coset Intersection. #### Coset Intersection {#coset-intersection-1 .unnumbered} Let $h = (\boldsymbol{a}_h, z_h)$. Then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ if and only if the equation $(\boldsymbol{b}, 0) \cdot (\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})^m = (\boldsymbol{a}_h, z_h) \cdot (\boldsymbol{c}, 0) \cdot (\boldsymbol{a}_{\mathcal{H}}, d_{\mathcal{H}})^n$ has solutions $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}, \boldsymbol{c}\in \langle {\mathcal{H}} \rangle \cap \mathcal{A}, m, n \in \mathbb{Z}$. By direct computation, this is equivalent to the system $$\label{eq:Cosetsys} \boldsymbol{b}+ \frac{X^{m d_{\mathcal{G}}} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} = \boldsymbol{a}_h + X^{z_h} \cdot \boldsymbol{c}+ X^{z_h} \cdot \frac{X^{n d_{\mathcal{H}}} - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}}, \quad m d_{\mathcal{G}} = n d_{\mathcal{H}} + z_h.$$ Again we define $d \coloneqq \operatorname{lcm}(d_{\mathcal{G}}, d_{\mathcal{H}})$ and consider both $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ as $\mathbb{Z}[X^{\pm d}]$-modules, respectively generated by the sets $S_{\mathcal{G}}$ and $S_{\mathcal{H}}$. This time, we define the $\mathbb{Z}[X^{\pm d}]$-module $\mathcal{M}' \coloneqq (\langle {\mathcal{G}} \rangle \cap \mathcal{A}) + X^{z_h} \cdot (\langle {\mathcal{H}} \rangle \cap \mathcal{A})$, it is generated by the set $S_{\mathcal{G}} \cup (X^{z_h} \cdot S_{\mathcal{H}})$. If $m d_{\mathcal{G}} = n d_{\mathcal{H}} + z_h$ has no integer solutions $m, n$, then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. Otherwise, there exist $z_{\mathcal{G}} \coloneqq m d_{\mathcal{G}}, z_{\mathcal{H}} \coloneqq n d_{\mathcal{H}} \in \mathbb{Z}$ such that $d_{\mathcal{G}} \mid z_{\mathcal{G}},\; d_{\mathcal{H}} \mid z_{\mathcal{H}}$ and $z_{\mathcal{G}} = z_{\mathcal{H}} + z_h$. Then, every solution $(m, n) \in \mathbb{Z}^2$ of the equation $m d_{\mathcal{G}} = n d_{\mathcal{H}} + z_h$ is of the form $$m = (z_{\mathcal{G}} + z d)/d_{\mathcal{G}}, \quad n = (z_{\mathcal{H}} + z d)/d_{\mathcal{H}}, \quad z \in \mathbb{Z}.$$ Similar to Lemma [\[lem:intertoeq\]](#lem:intertoeq){reference-type="ref" reference="lem:intertoeq"}, we can show the following: lemlemcosettoeq[\[lem:cosettoeq\]]{#lem:cosettoeq label="lem:cosettoeq"} Let $z_{\mathcal{G}}, z_{\mathcal{H}}$ be integers such that $d_{\mathcal{G}} \mid z_{\mathcal{G}},\; d_{\mathcal{H}} \mid z_{\mathcal{H}}$ and $z_{\mathcal{G}} = z_{\mathcal{H}} + z_h$. The intersection $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$ is non-empty if and only if the equation $$\label{eq:Coseteqz} X^{z d} \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}'$$ has solution $z \in \mathbb{Z}$. Here, $$\boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} \coloneqq X^{z_{\mathcal{G}}} \cdot \frac{X^d - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - X^{z_{\mathcal{H}}} \cdot \frac{X^d - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}},$$ and $$\boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \coloneqq \frac{X^d - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - \frac{X^d - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}} + (X^d - 1) \cdot \boldsymbol{a}_h.$$ We can decide if $$\label{eq:faainM} f \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}'$$ has solution $f \in \mathbb{Z}[X^{\pm d}]$ by deciding membership of $\boldsymbol{a}''_{\mathcal{G}, \mathcal{H}}$ in the $\mathbb{Z}[X^{\pm d}]$-module generated by $\boldsymbol{a}'_{\mathcal{G}, \mathcal{H}}$ and $\mathcal{M}'$ (see Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(i)). If Equation [\[eq:faainM\]](#eq:faainM){reference-type="eqref" reference="eq:faainM"} does not have a solution $f \in \mathbb{Z}[X^{\pm d}]$, then [\[eq:Coseteqz\]](#eq:Coseteqz){reference-type="eqref" reference="eq:Coseteqz"} cannot have a solution $z \in \mathbb{Z}$; otherwise, we can compute a solution $f = f_0$ of Equation [\[eq:faainM\]](#eq:faainM){reference-type="eqref" reference="eq:faainM"}. For example, $f_0$ can be computed by enumerating all elements $f \in \mathbb{Z}[X^{\pm d}]$ (there are countably many), and test for each one whether it satisfies Equation [\[eq:faainM\]](#eq:faainM){reference-type="eqref" reference="eq:faainM"}. Since Equation [\[eq:faainM\]](#eq:faainM){reference-type="eqref" reference="eq:faainM"} has a solution, this procedure must terminate. Consider the ideal $$\label{eq:defIp} \mathcal{I}' \coloneqq \left\{f \in \mathbb{Z}[X^{\pm d}] \;\middle|\; f \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}' \right\}$$ of $\mathbb{Z}[X^{\pm d}]$, then a finite set of generators for $\mathcal{I}'$ can be computed by Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii). lemlemctoi[\[lem:Cosettoideal\]]{#lem:Cosettoideal label="lem:Cosettoideal"} The solution set $$\left\{f \in \mathbb{Z}[X^{\pm d}] \;\middle|\; f \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}' \right\}$$ is equal to $f_0 + \mathcal{I}' \coloneqq \{f_0 + g \mid g \in \mathcal{I}'\}$. **Proposition 1**. *Suppose neither $\langle {\mathcal{G}} \rangle$ nor $\langle {\mathcal{H}} \rangle$ is contained in $\mathcal{A}$. Then deciding whether $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ reduces to Shifted Monomial Membership.* *Proof.* (See Algorithm [\[alg:cosetinter\]](#alg:cosetinter){reference-type="ref" reference="alg:cosetinter"} for a summary.) Suppose there exist $z_{\mathcal{G}}, z_{\mathcal{H}} \in \mathbb{Z}$ such that $d_{\mathcal{G}} \mid z_{\mathcal{G}},\; d_{\mathcal{H}} \mid z_{\mathcal{H}}$ and $z_{\mathcal{G}} = z_{\mathcal{H}} + z_h$, otherwise $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. By Lemma [\[lem:cosettoeq\]](#lem:cosettoeq){reference-type="ref" reference="lem:cosettoeq"} and [\[lem:Cosettoideal\]](#lem:Cosettoideal){reference-type="ref" reference="lem:Cosettoideal"}, it suffices to decide whether there exists $z \in \mathbb{Z}$ such that $X^z \in f_0 + \mathcal{I}'$. This is equivalent to $X^z - f_0 \in \mathcal{I}'$. We can decide whether $X^0 - f_0 \in \mathcal{I}'$ using ideal membership of $1 - f_0$ in $\mathcal{I}'$. Then we use Shifted Monomial Membership to decide whether there exists $z \in \mathbb{Z}\setminus \{\boldsymbol{0}\}$ such that $X^z - f_0 \in \mathcal{I}'$. ◻ # Deciding Shifted Monomial Membership {#sec:smm} In this section we show that Shifted Monomial Membership is decidable. Recall that for this problem, we are given a finite set of generators of an ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$, as well as a Laurent polynomial $f \in \mathbb{Z}[X^{\pm}]$. We want to decide if there exists $z \in \mathbb{Z}\setminus \{\boldsymbol{0}\}$ such that $X^z - f \in \mathcal{I}$. The outline of the proof is as follows. In Subsection [5.1](#subsec:redzx){reference-type="ref" reference="subsec:redzx"} we first simplify the problem by reducing to ideals $\widetilde{\mathcal{I}}$ over the ring $\mathbb{Z}[X]$ instead of $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$. We then consider the greatest common divisor $\varphi$ of the elements in $\widetilde{\mathcal{I}}$, and divide into five cases according to $\varphi$. Each of Subsections [5.2](#subsec:caseone){reference-type="ref" reference="subsec:caseone"}-[5.6](#subsec:casefive){reference-type="ref" reference="subsec:casefive"} treats a separate case. A common idea in each case is to give a bound on the absolute value of $z$ whenever Shifted Monomial Membership has positive answer. See Algorithm [\[alg:smm\]](#alg:smm){reference-type="ref" reference="alg:smm"} for a summary. ## Reduction to ideals of $\mathbb{Z}[X]$ {#subsec:redzx} Let $g_1, \ldots, g_m$ be the given generators of the ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$. Without loss of generality suppose none of the $g_i$ is zero. Multiplying any $g_i$ with any power of $X$ does not change the ideal they generate, because $X$ is invertible in $\mathbb{Z}[X^{\pm}]$. Therefore we can multiply each $g_i$ with a suitable power of $X$, and without loss of generality suppose $g_1, \ldots, g_m$ are polynomials in $\mathbb{Z}[X]$ instead of $\mathbb{Z}[X^{\pm}]$, and that they are not divisible by $X$. Let $\widetilde{\mathcal{I}}$ denote the ideal of $\mathbb{Z}[X]$ generated by $g_1, \ldots, g_m$. lemlemlaurenttoreg[\[lem:laurenttoreg\]]{#lem:laurenttoreg label="lem:laurenttoreg"} Let $g$ be a polynomial in $\mathbb{Z}[X^{\pm}]$. Then $g \in \mathcal{I}$ if and only if for some $c \in \mathbb{N}$, $X^c \cdot g \in \widetilde{\mathcal{I}}$. By Lemma [\[lem:laurenttoreg\]](#lem:laurenttoreg){reference-type="ref" reference="lem:laurenttoreg"}, the ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$ contains an element $X^z - f$ for some $z \neq 0$, if and only if $\widetilde{\mathcal{I}}$ contains an element $X^a - X^b f$ for some $a, b \in \mathbb{N}, a \neq b$. Furthermore, in this case, we have $z = a - b$. Hence, Shifted Monomial Membership reduces to the following problem: **Problem 1**. Given the generators of an ideal $\widetilde{\mathcal{I}}\subseteq \mathbb{Z}[X]$, decide whether $\widetilde{\mathcal{I}}$ contains any element of the form $X^a - X^b f$, $a, b \in \mathbb{N}, a \neq b$. For a non-zero polynomial $g \in \mathbb{Z}[X]$, its *leading coefficient* is defined as the coefficient of its monomial of largest degree. For example, the leading coefficient of $3X^2 + 4$ is $3$. A *common divisor* of a set $S \subseteq \mathbb{Z}[X]$ is a polynomial $g$ with positive leading coefficient, such that $g \mid s$ for all $s \in S$. The *greatest common divisor* of $S$, denoted by $\gcd(S)$, is a polynomial that has the largest degree and largest leading coefficient among all common divisors of $S$. The greatest common divisor is well-defined over $\mathbb{Z}[X]$ because it is a Unique Factorization Domain [@sharpe1987rings]. In particular, as $\widetilde{\mathcal{I}}\subseteq \mathbb{Z}[X]$ is the ideal generated by $g_1, \ldots, g_m$, the greatest common divisor $\gcd(\widetilde{\mathcal{I}})$ is equal to $\gcd(\{g_1, \ldots, g_m\})$. Denote $\varphi \coloneqq \gcd(\widetilde{\mathcal{I}})$. Then $X \nmid \varphi$ because $X \nmid g_1$. We say that a polynomial $g \in \mathbb{Z}[X]$ is *primitive* if there is no integer $d \geq 2$ such that $d \mid g$. A complex number $x$ is called a *root of unity* if $x^p = 1$ for some $p \geq 1$. We say that a polynomial $g \in \mathbb{Z}[X]$ has a *square divisor* if $\phi^2 \mid g$ for some polynomial $\phi \in \mathbb{Z}[X]$ with degree at most one. A polynomial is called *square-free* if it does not have a square divisor. Since $\varphi \neq 0$, there are only five cases regarding $\varphi$: (i) $\varphi = 1$, (ii) $\varphi$ is not primitive, (iii) $\varphi$ is primitive and has a root that is not a root of unity, (iv) $\varphi$ is primitive, all roots of $\varphi$ are roots of unity, and $\varphi$ has a square divisor, (v) $\varphi$ is primitive, all roots of $\varphi$ are roots of unity, and $\varphi$ is square-free. Each of the following subsections deals with one case. ## Case (i): trivial GCD {#subsec:caseone} In this case, $\varphi = 1$. The following lemma gives the structure of the ideal $\widetilde{\mathcal{I}}$ in this case. A polynomial in $\mathbb{Z}[X]$ is called *monic* if its leading coefficient is one. **Lemma 1** ([@szekeres1952canonical p.384-385]). *Let $\widetilde{\mathcal{I}}$ be an ideal of $\mathbb{Z}[X]$ such that $\gcd(\widetilde{\mathcal{I}}) = 1$. Then there are only two possible cases for $\widetilde{\mathcal{I}}$:* (i) *either $\widetilde{\mathcal{I}}= \mathbb{Z}[X]$,* (ii) *or $\widetilde{\mathcal{I}}$ contains an integer $c \geq 2$, as well as a monic polynomial $g$ of degree at least one.* *Furthermore, given a finite set of generators for $\widetilde{\mathcal{I}}$, one can decide which case is true. In case (ii), one can explicitly compute such $c$ and $g$.* If $\widetilde{\mathcal{I}}= \mathbb{Z}[X]$ then obviously it contains an element $X^a - X^b f, \; a \neq b$. Suppose now that $\widetilde{\mathcal{I}}$ contains an integer $c \geq 2$ and a monic polynomial $g$ of degree at least one. In particular, $\widetilde{\mathcal{I}}\subseteq (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$. **Lemma 1**. *The quotient $\mathbb{Z}[X] / (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$ is finite.* *Proof.* Let $\deg g$ denote the degree of $g$. Since $g$ is monic, every $f \in \mathbb{Z}[X]$ can be written as $f = gh + r$ where $g, h, r \in \mathbb{Z}[X]$ and $\deg r < \deg g$. Therefore, every element in $\mathbb{Z}[X]$ is equivalent modulo $g$ to a polynomial with degree at most $\deg g - 1$. But there are only finitely many polynomials modulo $c$ with degree at most $\deg g - 1$. Therefore, the quotient $\mathbb{Z}[X] / (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$ is finite. ◻ Let $f \mapsto \overline{f}$ denote the canonical projection $\mathbb{Z}[X] \rightarrow\mathbb{Z}[X] / (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$. Consider the sequence $\overline{1}, \overline{X}, \overline{X^2}, \cdots \in \mathbb{Z}[X] / (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$. Since $\mathbb{Z}[X] / (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$ is finite, there exists $0 \leq p < q$ such that $\overline{X^p} = \overline{X^q}$. Furthermore, such integers $p, q$ can be effectively found by incrementally testing whether $X^q - X^p \in (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$ (see Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}). Then $X^q - X^p \in (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$, so $\overline{X^r} = \overline{X^{r - (q-p)}}$ for every $r \geq q$. From this, we easily obtain the following result. lemlemgcdone[\[lem:gcdone\]]{#lem:gcdone label="lem:gcdone"} Suppose $\widetilde{\mathcal{I}}$ contains an integer $c \geq 2$, as well as a monic polynomial $g$ of degree at least one. Then $\widetilde{\mathcal{I}}$ contains an element of the form $X^a - X^b f$, $a, b \in \mathbb{N}, a \neq b$, if and only if $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, q-1]$. *Proof.* Since $\overline{X^r} = \overline{X^{r - (q-p)}}$ for every $r \geq q$, every $X^r, \; r \in \mathbb{N}$ is equivalent modulo $(\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$ to $X^{r'}$ for some $r' \in [0, q-1]$. Since $(\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c) \subseteq \widetilde{\mathcal{I}}$, every $X^r$ is also equivalent modulo $\widetilde{\mathcal{I}}$ to $X^{r'}$ for some $r' \in [0, q-1]$. Therefore, if $\widetilde{\mathcal{I}}$ contains an element of the form $X^a - X^b f$, $a, b \in \mathbb{N}, a \neq b$, then $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, q-1]$. And if $\widetilde{\mathcal{I}}$ contains some $X^{a'} - X^{b'} f$, $a', b' \in [0, q-1]$, then it also contains $X^{a' + q(q - p)} - X^{b'} f$. Taking $a \coloneqq a' + q(q - p), b \coloneqq b'$ we have $X^a - X^b f \in \widetilde{\mathcal{I}}$ and $a \neq b$. ◻ Since there are only finitely many integers in $[0, q-1]$, one can decide whether $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, q-1]$ by enumerating all such $a', b'$. ## Case (ii): non-primitive GCD In this case, $\varphi$ is not primitive. Suppose $d \mid \varphi$ with $d \geq 2$. Then $d$ divides every element in $\widetilde{\mathcal{I}}$. We show that there is an effectively computable bound on $a - b$. **Lemma 1**. *Let $d \geq 2$. If $d \mid X^a - X^b f$, then $0 \leq a - b \leq \deg f$.* *Proof.* If $a > b + \deg f$ then $\deg X^a > \deg X^b f$, so the leading coefficient of $X^a - X^b f$ is $1$, a contradiction to $d \mid X^a - X^b f$. Similarly if $a < b$ then the coefficient of the monomial $X^a$ in $X^a - X^b f$ is one, a contradiction to $d \mid X^a - X^b f$. Therefore $0 \leq a - b \leq \deg f$. ◻ Therefore if $X^a - X^b f \in \widetilde{\mathcal{I}}, \; a \neq b,$ then $d \mid X^a - X^b f$, and we must have $a - b \in [1, \deg f]$. By Lemma [\[lem:laurenttoreg\]](#lem:laurenttoreg){reference-type="ref" reference="lem:laurenttoreg"}, we have $X^a - X^b f \in \widetilde{\mathcal{I}}$ if and only if $X^{a - b} - f \in \mathcal{I}$. Therefore in this case, it suffices to decide for each $r \in [1, \deg f]$ whether $X^r - f \in \mathcal{I}$. ## Case (iii): non-root of unity In this case, $\varphi$ has a root $x$ that is not a root of unity. Since $X \nmid g_1$ we have $X \nmid \varphi$, so $x \neq 0$. Let $\mathbb{K}$ be an algebraic number field that contains $x$. The key idea in this case is to use the *height function* over $\mathbb{K}$ to give a bound on $|a - b|$. For an exact construction of the height function, see [@waldschmidt2013diophantine Section 3.2]. In this paper we will only make use of its properties listed in the following lemma. **Lemma 1** (Height of algebraic numbers [@waldschmidt2013diophantine Property 3.3 and Section 3.6]). *Let $\mathbb{K}$ be an algebraic number field and denote $\mathbb{K}^* \coloneqq \mathbb{K}\setminus \{0\}$. There exists a map $H \colon \mathbb{K}^* \rightarrow \mathbb{R}_{\geq 0}$ that satisfies to following properties.* (i) *For any $n \in \mathbb{Z}$ and $y \in \mathbb{K}^*$, we have $H(y^n) = H(y)^{|n|}$.* (ii) *For all $y \in \mathbb{K}^*$, we have $H(y) \geq 1$. And $H(y) = 1$ if and only if $y$ is a root of unity.* *For any $y \in \mathbb{K}^*$, the value $H(y)$ is called the *height* of $y$, it is an algebraic number that can be effectively computed.* Since $x$ is not a root of unity, we have $H(x) > 1$. **Lemma 1**. *Let $x \neq 0$ be a root of $\varphi$ that is not a root of unity. If $\varphi \mid X^a - X^b f$, then $f(x) \neq 0$ and $|a - b| = \frac{\log H(f(x))}{\log H(x)}$.* *Proof.* Since $\varphi \mid X^a - X^b f$ and $x$ is a root of $\varphi$, we have $x^a - x^b f(x) = 0$. Therefore $x^{a - b} = f(x)$. Since $x \neq 0$ we have $f(x) \neq 0$. Taking the height function on both sides of $x^{a - b} = f(x)$ yields $H(x)^{|a - b|} = H(x^{a - b}) = H(f(x))$, so $|a - b| = \frac{\log H(f(x))}{\log H(x)}$. ◻ If $X^a - X^b f \in \widetilde{\mathcal{I}}$ then we must have $\varphi \mid X^a - X^b f$. By Lemma [\[lem:laurenttoreg\]](#lem:laurenttoreg){reference-type="ref" reference="lem:laurenttoreg"}, we have $X^a - X^b f \in \widetilde{\mathcal{I}}$ if and only if $X^{a - b} - f \in \mathcal{I}$. Therefore by Lemma [Lemma 1](#lem:ht){reference-type="ref" reference="lem:ht"}, it suffices to decide whether $r \coloneqq \frac{\log H(f(x))}{\log H(x)}$ is a non-zero integer, and then decide whether one of $X^r - f$ and $X^{-r} - f$ is in $\mathcal{I}$. ## Case (iv): square divisor In this case, $\varphi$ has a square divisor. Suppose $\phi^2 \mid \varphi$ where $\deg \phi \geq 1$. Let $x$ be a root of $\phi$, then $x \neq 0$ since $X \nmid \varphi$. The key here is that if $\phi^2 \mid g$, then $\phi \mid g'$ where $g'$ denotes the derivative of $g$. Indeed, writing $g = \phi^2 h$, then $g' = 2 \phi \phi' h + \phi^2 h'$ is divisible by $\phi$. **Lemma 1**. *Let $x \neq 0$ be any root of $\phi$. If $\phi^2 \mid X^a - X^b f$ where $a \neq b$, then $a - b = \frac{x f'(x)}{f(x)}$.* *Proof.* If $a > b$ then $\phi^2 \mid X^{a-b} - f$. Taking the derivative of $X^{a-b} - f$ yields $\phi \mid (a-b) X^{a-b-1} - f'$. Since $\phi(x) = 0$ this yields $(a - b) x^{a-b-1} = f'(x)$. On the other hand, $\phi^2 \mid X^a - X^b f$ yields $x^{a-b} = f(x)$. Combining these two equations, we obtain $a - b = \frac{x f'(x)}{f(x)}$. If $a < b$ then $\phi^2 \mid 1 - X^{b - a} f$. Taking the derivative of $1 - X^{b - a} f$ yields $\phi \mid (b - a) X^{b - a - 1} f + X^{b - a} f'$. Since $\phi(x) = 0$ this yields $(b - a) x^{b - a - 1} f(x) + x^{b - a} f'(x) = 0$. Since $\phi^2 \mid X^a - X^b f$ we have $x^a = x^b f(x)$, so $f(x) \neq 0$. Therefore $a - b = \frac{x f'(x)}{f(x)}$. ◻ As in the previous cases we have $X^a - X^b f \in \widetilde{\mathcal{I}}$ if and only if $X^{a - b} - f \in \mathcal{I}$. Therefore, by Lemma [Lemma 1](#lem:sqfindiff){reference-type="ref" reference="lem:sqfindiff"}, it suffices to decide whether $r \coloneqq \frac{x f'(x)}{f(x)}$ is a non-zero integer, and then decide whether $X^r - f \in \mathcal{I}$. ## Case (v): only roots of unity {#subsec:casefive} In this case, $\varphi$ is primitive, square-free, and all its roots are roots of unity. lemlemdiv[\[lem:div\]]{#lem:div label="lem:div"} Let $\varphi \in \mathbb{Z}[X]$ be a primitive, square-free polynomial such that all its roots are roots of unity. Then there exists an effectively computable integer $p \geq 1$ such that $\varphi \mid X^p - 1$. *Proof.* Since $\varphi$ is square-free, it has no repeated roots over the complex numbers [@yun1976square]. Recall that roots of unity are of the form $e^{\frac{2\pi i r}{s}}, \; r, s \in \mathbb{N}$. Let $e^{\frac{2\pi i q_1}{p_1}}, \ldots, e^{\frac{2\pi i q_d}{p_d}}$ be all the roots of $\varphi$. Let $p \geq 1$ be a common multiplier of $p_1, \ldots, p_d$, then these roots can be written as $e^{\frac{2\pi i Q_1}{p}}, \ldots, e^{\frac{2\pi i Q_d}{p}}$ where $Q_1, \ldots, Q_d \in [0, p-1]$ are pairwise distinct. Therefore $\varphi$ divides $X^p - 1 = \prod_{q = 0}^{p-1} (X - e^{\frac{2\pi i q}{p}})$ in the ring $\mathbb{Q}[X]$. Hence $\varphi$ divides $c(X^p - 1)$ in the ring $\mathbb{Z}[X]$ for some $c \in \mathbb{Z}$. Without loss of generality suppose $\varphi \neq 1$. Since $\varphi$ is primitive, it does not divide $c$, so it must divide $X^p - 1$ in the ring $\mathbb{Z}[X]$. ◻ Let $p \geq 1$ be such that $\varphi \mid X^p - 1$. Write $\widetilde{\mathcal{I}}= \varphi \cdot \widetilde{\mathcal{J}}$ where $\widetilde{\mathcal{J}}$ is an ideal of $\mathbb{Z}[X]$ with $\gcd(\widetilde{\mathcal{J}}) = 1$. In particular, the generators of $\widetilde{\mathcal{J}}$ are $\frac{g_1}{\varphi}, \ldots, \frac{g_m}{\varphi}$. We apply Lemma [Lemma 1](#lem:idealstr){reference-type="ref" reference="lem:idealstr"} for $\widetilde{\mathcal{J}}$. If $\widetilde{\mathcal{J}}= \mathbb{Z}[X]$, then $\widetilde{\mathcal{I}}$ is simply the ideal generated by $\varphi$. Then $X^a - X^b f \in \widetilde{\mathcal{I}}$ if and only if $\varphi \mid X^a - X^b f$. Since $\varphi \mid X^p - 1$, there exist $a \neq b$ such that $\varphi \mid X^a - X^b f$, if and only if there exist $a', b' \in [0, p-1]$ (not necessarily distinct), such that $\varphi \mid X^{a'} - X^{b'} f$. If $\widetilde{\mathcal{J}}\neq \mathbb{Z}[X]$, then by Lemma [Lemma 1](#lem:idealstr){reference-type="ref" reference="lem:idealstr"}, $\widetilde{\mathcal{J}}$ contains an integer $c \geq 2$, as well as a monic polynomial $g$ of degree at least one. Similar to Case (i), consider the equivalent class of the elements $\frac{X^p - 1}{\varphi}, \frac{X^{2p} - 1}{\varphi}, \ldots,$ in the quotient $\mathbb{Z}[X]/(\mathbb{Z}[X] \cdot c + \mathbb{Z}[X] \cdot g)$. Since $\mathbb{Z}[X]/(\mathbb{Z}[X] \cdot c + \mathbb{Z}[X] \cdot g)$ is finite by Lemma [Lemma 1](#lem:finring){reference-type="ref" reference="lem:finring"}, there exist $0 \leq q' < q$ such that $$\label{eq:diffin} \frac{X^{qp} - 1}{\varphi} - \frac{X^{q'p} - 1}{\varphi} \in \mathbb{Z}[X] \cdot c + \mathbb{Z}[X] \cdot g \subseteq \widetilde{\mathcal{J}}.$$ and we have the following result which is analogous to Lemma [\[lem:gcdone\]](#lem:gcdone){reference-type="ref" reference="lem:gcdone"}. lemlemgcdcyc[\[lem:gcdcyc\]]{#lem:gcdcyc label="lem:gcdcyc"} Suppose $\widetilde{\mathcal{J}}$ contains an integer $c \geq 2$, as well as a monic polynomial $g$ of degree at least one. Then $\widetilde{\mathcal{I}}= \varphi \cdot \widetilde{\mathcal{J}}$ contains an element of the form $X^a - X^b f$, $a, b \in \mathbb{N}, a \neq b$, if and only if $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, pq-1]$. Since there are only finitely many integers in $[0, pq-1]$, one can decide whether $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, pq-1]$ by enumerating all such $a', b'$. # Omitted proofs {#app:proof} *Proof.* Every element $\boldsymbol{f}\in \mathbb{Z}[X^{\pm}]^D$ can be uniquely written as $\boldsymbol{f}= \boldsymbol{f}_0 + X \cdot \boldsymbol{f}_1 + \cdots + X^{d-1} \cdot \boldsymbol{f}_{d-1}$ where $\bf_0, \bf_1, \ldots, \bf_{d-1}$ are in $\mathbb{Z}[X^{\pm d}]$. This gives an effective isomorphism $\varphi \colon \mathbb{Z}[X^{\pm}]^D \rightarrow \mathbb{Z}[X^{\pm d}]^{Dd}$. Write $\mathcal{A}= M/N$ where $M, N \subseteq \mathbb{Z}[X^{\pm}]^D$. The generators of $\varphi(M)$ can be obtained by simply applying $\varphi$ to the generators of $M$, similarly for $\varphi(N)$. Hence $\mathcal{A}= \varphi(M)/\varphi(N)$ is a finite presentation of $\mathcal{A}$ as a $\mathbb{Z}[X^{\pm d}]$-module. ◻ *Proof.* (i) is obvious. For (ii), suppose $z_1, \ldots, z_K$ are not all zero and let $d \in \mathbb{N}$ be their greatest common divisor. Let $n_1, \ldots, n_K \in \mathbb{Z}$ be such that $n_1 z_1 + \cdots + n_K z_K = d$, then $g \coloneqq g_1^{n_1} \cdots g_K^{n_K}$ is of the form $(\boldsymbol{b}, d), \boldsymbol{b}\in \mathcal{A}$. Then for any $\boldsymbol{a}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$, we have $\langle {\mathcal{G}} \rangle \cap \mathcal{A}\ni g^{-1} \boldsymbol{a}g = (- X^{-d} \cdot \boldsymbol{b}, -d) (\boldsymbol{a}, 0) (\boldsymbol{b}, d) = (X^d \cdot \boldsymbol{a}, 0) = X^d \cdot \boldsymbol{a}$. Similarly, $\langle {\mathcal{G}} \rangle \cap \mathcal{A}\ni g \boldsymbol{a}g^{-1} = X^{-d} \cdot \boldsymbol{a}$. Therefore, $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ is a $\mathbb{Z}[X^{\pm d}]$-module. On one hand, the elements in $S$ are obviously in $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$, so the $\mathbb{Z}[X^{\pm d}]$-module generated by $S$ is a submodule of $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$. On the other hand, we show that the quotient $(\langle {\mathcal{G}} \rangle \cap \mathcal{A})/S$ is trivial. Notice that since $(g_i g_j g_i^{-1} g_j^{-1})^{-1} = g_j g_i g_j^{-1} g_i^{-1}$, we have $\left\{g_i g_j g_i^{-1} g_j^{-1} \;\middle|\; i, j \in [1, K] \right\} \subseteq S$. Therefore, the quotient by $S$ allows one to permute elements in any product $g_{i_1}^{\epsilon_1} \cdots g_{i_n}^{\epsilon_n} \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$ without changing their class in $(\langle {\mathcal{G}} \rangle \cap \mathcal{A})/S$. More precisely, for $g, g' \in \langle {\mathcal{G}} \rangle$ and $i, j \in [1, K]$, if $g g_i g_j g' \in \mathcal{A}$ then we have $g g_i g_j g' + S = g g_j g_i g' + S$. Indeed, we have $g g_i g_j g' = g (g_i g_j g_i^{-1} g_j^{-1}) g_j g_i g' = g (g_i g_j g_i^{-1} g_j^{-1}) g^{-1} + g g_j g_i g'$ and $g (g_i g_j g_i^{-1} g_j^{-1}) g^{-1}$ is in the $\mathbb{Z}[X^{\pm d}]$-module generated by $S$. For every product $g_{i_1}^{\epsilon_1} \cdots g_{i_n}^{\epsilon_n} \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$, we must have $$\left(\sum_{j \in [1, n], i_j = 1} \epsilon_j, \ldots, \sum_{j \in [1, n], i_j = K} \epsilon_j\right) \in \Lambda$$ by looking at the second component. Since $(s_{11}, \ldots, s_{1K}), \ldots, (s_{T1}, \ldots, s_{TK})$ be are the generators for $\Lambda$, by permuting the elements in the product we can rewrite $g_{i_1}^{\epsilon_1} \cdots g_{i_n}^{\epsilon_n}$ as $\left(g_1^{s_{1 1}} \cdots g_K^{s_{1 K}}\right)^{j_1} \cdots \left(g_1^{s_{T 1}} \cdots g_K^{s_{T K}}\right)^{j_T}$ where $j_1, \ldots, j_T \in \mathbb{Z}$. Therefore, $g_{i_1}^{\epsilon_1} \cdots g_{i_n}^{\epsilon_n}$ is in the $\mathbb{Z}[X^{\pm d}]$-module generated by $S$. For (iii), $\boldsymbol{a}\in \mathcal{A}$ be any element such that $(\boldsymbol{a}, d) \in \langle {\mathcal{G}} \rangle$. Since $d \in \mathbb{N}$ is the greatest common divisor for $z_1, \ldots, z_K$, every element $g$ of $\langle {\mathcal{G}} \rangle$ must be of the form $(\boldsymbol{c}, md), \; \boldsymbol{c}\in \mathcal{A}, m \in \mathbb{Z}$. Then $g \cdot (\boldsymbol{a}, d)^{-m} \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$. Let $(\boldsymbol{b}, 0) \coloneqq g \cdot (\boldsymbol{a}, d)^{-m}$, then $\boldsymbol{a}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and $g = (\boldsymbol{b}, 0) \cdot (\boldsymbol{a}, d)^m$. ◻ *Proof.* We can without loss of generality suppose $\langle {\mathcal{H}} \rangle \subseteq \mathcal{A}$ and $\langle {\mathcal{G}} \rangle \not\subset \mathcal{A}$. Otherwise notice that $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ if and only if $h^{-1} \langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \emptyset$, so we can exchange the role of $\langle {\mathcal{G}} \rangle$ and $\langle {\mathcal{H}} \rangle$. By Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}, suppose $\langle {\mathcal{G}} \rangle$ is generated by the element $(\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})$ and the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$, and $\langle {\mathcal{H}} \rangle$ is generated by the elements $h_1 = (\boldsymbol{a}'_1, 0), \ldots, h_M = (\boldsymbol{a}'_M, 0)$. Recall that the generators of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ can be effectively computed. Also, by Lemma [\[lem:changebase\]](#lem:changebase){reference-type="ref" reference="lem:changebase"}, we can consider $\mathcal{A}$ as a finitely presented $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module instead of a $\mathbb{Z}[X^{\pm}]$-module, and suppose the generators of $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ as well as $\boldsymbol{a}'_1, \ldots, \boldsymbol{a}'_M$ are given as elements of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module. **Subgroup Intersection.** In this case, $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ if and only if every solution of $z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}, \; z_1, \ldots, z_M \in \mathbb{Z}$ is also a solution of $z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M = \boldsymbol{0}$. Let $\mathcal{M}$ denote the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $$\label{eq:M2s} \mathcal{M}\coloneqq \Big\{ (z_1, \ldots, z_M) \in \mathbb{Z}[X^{\pm d_{\mathcal{G}}}]^{M} \;\Big|\; z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}\Big\}, \\$$ and $\mathcal{Z}$ denote the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $$\label{eq:Z2s} \mathcal{Z}\coloneqq \Big\{ (z_1, \ldots, z_M) \in \mathbb{Z}[X^{\pm d_{\mathcal{G}}}]^{M} \;\Big|\; z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M = \boldsymbol{0}\Big\}.$$ Then we have $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$ if and only if $\mathcal{M}\cap \mathbb{Z}^M \neq \mathcal{Z}\cap \mathbb{Z}^M$. Note that $z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}$ can be rewritten as a linear equation $$\label{eq:equiveq} z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M + x_1 \cdot \boldsymbol{g}_1 + \cdots + x_m \cdot \boldsymbol{g}_m = \boldsymbol{0}, \quad x_1, \ldots, x_m \in \mathbb{Z}[X^{\pm d_{\mathcal{G}}}],$$ where $\boldsymbol{g}_1, \ldots, \boldsymbol{g}_m$ are the generators of $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$. Therefore the generators of $\mathcal{M}$ can be computed by projecting the solution set of [\[eq:equiveq\]](#eq:equiveq){reference-type="eqref" reference="eq:equiveq"} to the coordinates $(z_1, \ldots, z_M)$. Similar to the previous case, Subgroup Intersection can be decided by computing the generators of $\mathcal{M}\cap \mathbb{Z}^M$ and $\mathcal{Z}\cap \mathbb{Z}^M$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}, and deciding equality using linear algebra over $\mathbb{Z}$. **Coset Intersection.** Let $h = (\boldsymbol{a}_h, z_h)$. If $d_{\mathcal{G}} \nmid z_h$ then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$. Therefore we only need to consider the case where $z_h = z d_{\mathcal{G}}$ for some $z \in \mathbb{Z}$. Then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle \neq \emptyset$ if and only if the equation $(\boldsymbol{b}, 0) \cdot (\boldsymbol{a}_{\mathcal{G}}, d_{\mathcal{G}})^z = (\boldsymbol{a}_h, z_h) \cdot (\boldsymbol{c}, 0)$ has solutions $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A},\; \boldsymbol{c}\in \sum_{i = 1}^M \mathbb{Z}\cdot \boldsymbol{a}'_i$. Direct computation shows this is equivalent to $$X^{z_h} \cdot \boldsymbol{c}+ \left(\boldsymbol{a}_h - \frac{X^{z d_{\mathcal{G}}} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} \right) = \boldsymbol{b}.$$ Let $\mathcal{M}'$ denote the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $$\begin{gathered} \label{eq:M2c} \mathcal{M}' \coloneqq \Bigg\{ (z_1, \ldots, z_M, z) \in \mathbb{Z}[X^{\pm d_{\mathcal{G}}}]^{M+1} \;\Bigg|\; \\ X^{z_h} \cdot \left(z_1 \cdot \boldsymbol{a}'_1 + \cdots + z_M \cdot \boldsymbol{a}'_M\right) + z \cdot \left(\boldsymbol{a}_h - \frac{X^{z_h} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} \right) \in \langle {\mathcal{G}} \rangle \cap \mathcal{A}\Bigg\}. \end{gathered}$$ Then $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$ if and only if $d_{\mathcal{G}} \nmid z_h$ and $\big(\mathcal{M}' \cap \mathbb{Z}^{M+1}\big) \cap \big(\mathbb{Z}^{M} \times \{1\}\big) = \emptyset$. Similarly, Coset Intersection is decidable in this case. ◻ *Proof.* Suppose the $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$ is non-empty. Let $(\boldsymbol{a}, z') \in \langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$, then $d_{\mathcal{G}} \mid z'$ and $d_{\mathcal{H}} \mid (z' - z_h)$. Hence $z' = z_{\mathcal{G}} + zd = z_{\mathcal{H}} + zd + z_h$ for some $z \in \mathbb{Z}$. Since $(\boldsymbol{a}, z') \in \langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle$, the Equation [\[eq:Cosetsys\]](#eq:Cosetsys){reference-type="eqref" reference="eq:Cosetsys"} has solution with $m d_{\mathcal{G}} = z'$, meaning $$\begin{gathered} \frac{X^{z d} \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}}}{X^d - 1} = \frac{X^{z_{\mathcal{G}} + zd}}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - \frac{X^{z_{\mathcal{H}} + zd}}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}} - \frac{1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} + \frac{1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}} - \boldsymbol{a}_h \\ = \frac{X^{z_{\mathcal{G}} + zd} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - X^{z_h} \cdot \frac{X^{z_{\mathcal{H}} + zd} - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}} - \boldsymbol{a}_h = X^{z_h} \cdot \boldsymbol{c}- \boldsymbol{b}\in \mathcal{M}'. \end{gathered}$$ Therefore [\[eq:Coseteqz\]](#eq:Coseteqz){reference-type="eqref" reference="eq:Coseteqz"} is satisfied. On the other hand, suppose Equation [\[eq:Coseteqz\]](#eq:Coseteqz){reference-type="eqref" reference="eq:Coseteqz"} is satisfied. Then we have $$\frac{X^{z_{\mathcal{G}} + zd} - 1}{X^{d_{\mathcal{G}}} - 1} \cdot \boldsymbol{a}_{\mathcal{G}} - X^{z_h} \cdot \frac{X^{z_{\mathcal{H}} + zd} - 1}{X^{d_{\mathcal{H}}} - 1} \cdot \boldsymbol{a}_{\mathcal{H}} - \boldsymbol{a}_h = \frac{X^{z d} \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}}}{X^d - 1} \in \mathcal{M}',$$ so it can be written as $X^{z_h} \cdot \boldsymbol{c}- \boldsymbol{b}$ for some $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A}, \; \boldsymbol{c}\in \langle {\mathcal{H}} \rangle \cap \mathcal{A}$. Hence the system [\[eq:Cosetsys\]](#eq:Cosetsys){reference-type="eqref" reference="eq:Cosetsys"} has solutions $\boldsymbol{b}\in \langle {\mathcal{G}} \rangle \cap \mathcal{A},\; \boldsymbol{c}\in \langle {\mathcal{H}} \rangle \cap \mathcal{A},\; m = \frac{z_{\mathcal{G}} + z d}{d_{\mathcal{G}}},\; n = \frac{z_{\mathcal{H}} + z d}{d_{\mathcal{H}}}$. ◻ *Proof.* Since $f_0 \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}'$, we have $f \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} - \boldsymbol{a}''_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}'$ if and only if $(f - f_0) \cdot \boldsymbol{a}'_{\mathcal{G}, \mathcal{H}} \in (X^d - 1) \cdot \mathcal{M}'$. This is equivalent to $f - f_0 \in \mathcal{I}'$, in other words $f \in f_0 + \mathcal{I}'$. ◻ *Proof.* Suppose $g \in \mathcal{I}$, then $g = f_1 \cdot g_1 + \cdots + f_m \cdot g_m$ for some $f_1, \ldots, f_m \in \mathbb{Z}[X^{\pm}]$. Let $c \in \mathbb{N}$ be large enough so that $X^c f_1, \ldots, X^c f_m$ are in $\mathbb{Z}[X]$, then $X^c g = X^c f_1 \cdot g_1 + \cdots + X^c f_m \cdot g_m \in \widetilde{\mathcal{I}}$. On the other hand, suppose $X^c g \in \widetilde{\mathcal{I}}$ for some $c \in \mathbb{N}$, then $X^c g = F_1 \cdot g_1 + \cdots + F_m \cdot g_m$ for some $F_1, \ldots, F_m \in \mathbb{Z}[X]$. Then $g = X^{-c} F_1 \cdot g_1 + \cdots + X^{-c} F_m \cdot g_m \in \mathcal{I}$. ◻ *Proof.* By Equation [\[eq:diffin\]](#eq:diffin){reference-type="eqref" reference="eq:diffin"}, we have $\frac{X^{qp} - X^{q'p}}{\varphi} \in \widetilde{\mathcal{J}}$, so $X^{qp} - X^{q'p} \in \varphi \cdot \widetilde{\mathcal{J}}= \widetilde{\mathcal{I}}$. Therefore, every $X^r, \; r \in \mathbb{N}$ is equivalent modulo $\widetilde{\mathcal{I}}$ to $X^{r'}$ for some $r' \in [0, pq-1]$. Therefore similar to Lemma [\[lem:gcdone\]](#lem:gcdone){reference-type="ref" reference="lem:gcdone"}, $\widetilde{\mathcal{I}}$ contains an element of the form $X^a - X^b f$, $a, b \in \mathbb{N}, a \neq b$, if and only if $\widetilde{\mathcal{I}}$ contains an element of the form $X^{a'} - X^{b'} f$, $a', b' \in [0, pq-1]$. ◻ # Summary of algorithms {#app:alg} Input: : a finite presentation of the $\mathbb{Z}[X^{\pm}]$-module $\mathcal{A}$, two finite sets of elements $\mathcal{G}= \{(\boldsymbol{a}_1, z_1), \ldots, (\boldsymbol{a}_K, z_K)\},\; \mathcal{H}= \{(\boldsymbol{a}'_1, z'_1), \ldots, (\boldsymbol{a}'_M, z'_M)\}$ in the group $\mathcal{A}\rtimes \mathbb{Z}$. Output: : **True** (when $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle = \{e\}$) or **False** (when $\langle {\mathcal{G}} \rangle \cap \langle {\mathcal{H}} \rangle \neq \{e\}$). 1. **If $z_1, \ldots, z_K$ and $z'_1, \ldots, z'_M$ are all zero.** Compute generators of the modules $\mathcal{M}$ and $\mathcal{Z}$ defined in Equations [\[eq:M1s\]](#eq:M1s){reference-type="eqref" reference="eq:M1s"} and [\[eq:Z1s\]](#eq:Z1s){reference-type="eqref" reference="eq:Z1s"}. Decide whether $\mathcal{M}\cap \mathbb{Z}^{K+M} = (\mathcal{M}\cap \mathcal{Z}) \cap \mathbb{Z}^{K+M}$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}. If yes, return **True**, otherwise return **False**. 2. **If one of the sets $\{z_1, \ldots, z_K\}$ and $\{z'_1, \ldots, z'_M\}$ is all zero.** Without loss of generality suppose $z'_1 = \cdots = z'_M = 0$, otherwise swap the sets $\mathcal{G}, \mathcal{H}$. (i) Compute $d_{\mathcal{G}} \coloneqq \gcd(z_1, \ldots, z_K)$, and compute the generators of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ using Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}. (ii) Compute generators of the modules $\mathcal{M}$ and $\mathcal{Z}$ in Equations [\[eq:M2s\]](#eq:M2s){reference-type="eqref" reference="eq:M2s"} and [\[eq:Z2s\]](#eq:Z2s){reference-type="eqref" reference="eq:Z2s"}. Decide whether $\mathcal{M}\cap \mathbb{Z}^M \neq \mathcal{Z}\cap \mathbb{Z}^M$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}. If yes, return **True**, otherwise return **False**. 3. **If none of the sets $\{z_1, \ldots, z_K\}$ and $\{z'_1, \ldots, z'_M\}$ is all zero.** (i) Compute $d_{\mathcal{G}} \coloneqq \gcd(z_1, \ldots, z_K), d_{\mathcal{H}} \coloneqq \gcd(z'_1, \ldots, z'_M), d \coloneqq \gcd(d_{\mathcal{G}}, d_{\mathcal{H}})$. (ii) Compute generators of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and the $\mathbb{Z}[X^{\pm d_{\mathcal{H}}}]$-module $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ using Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}. Compute their respective generators $S_{\mathcal{G}}, S_{\mathcal{H}}$ as $\mathbb{Z}[X^{\pm d}]$-modules using Lemma [\[lem:changebase\]](#lem:changebase){reference-type="ref" reference="lem:changebase"}. Let $\mathcal{M}$ be the $\mathbb{Z}[X^{\pm d}]$-module generated by $S_{\mathcal{G}} \cup S_{\mathcal{H}}$. (iii) Decide whether $(\langle {\mathcal{G}} \rangle \cap \mathcal{A}) \cap (\langle {\mathcal{H}} \rangle \cap \mathcal{A}) = \{\boldsymbol{0}\}$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(iii). If yes, continue, otherwise return **False**. (iv) Compute the generators of the ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm d}]$ defined in Equation [\[eq:defI\]](#eq:defI){reference-type="eqref" reference="eq:defI"} using the generators of $\mathcal{M}$ and Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii). (v) Using Algorithm [\[alg:smm\]](#alg:smm){reference-type="ref" reference="alg:smm"}, decide whether $\mathcal{I}$ contains an element $X^{zd} - 1$ for some $z \in \mathbb{Z}\setminus \{0\}$. If yes, return **False**, otherwise return **True**. Input: : a finite presentation of the $\mathbb{Z}[X^{\pm}]$-module $\mathcal{A}$, two finite sets of elements $\mathcal{G}= \{(\boldsymbol{a}_1, z_1), \ldots, (\boldsymbol{a}_K, z_K)\},\; \mathcal{H}= \{(\boldsymbol{a}'_1, z'_1), \ldots, (\boldsymbol{a}'_M, z'_M)\}$ in the group $\mathcal{A}\rtimes \mathbb{Z}$, an element $h = (\boldsymbol{a}_h, z_h)$. Output: : **True** (when $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle = \emptyset$) or **False** (when $\langle {\mathcal{G}} \rangle \cap h \langle {\mathcal{H}} \rangle \neq \emptyset$). 1. **If $z_1, \ldots, z_K$ and $z'_1, \ldots, z'_M$ are all zero.** Compute generators of the module $\mathcal{M}'$ defined in Equation [\[eq:M1c\]](#eq:M1c){reference-type="eqref" reference="eq:M1c"}. Decide whether $\big(\mathcal{M}' \cap \mathbb{Z}^{K+M+1}\big) \cap \big(\mathbb{Z}^{K+M} \times \{1\}\big) = \emptyset$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}. If yes, return **True**, otherwise return **False**. 2. **If one of the sets $\{z_1, \ldots, z_K\}$ and $\{z'_1, \ldots, z'_M\}$ is all zero.** Without loss of generality suppose $z'_1 = \cdots = z'_M = 0$, otherwise swap the sets $\mathcal{G}, \mathcal{H}$ and replace $h$ with $h^{-1}$. (i) Compute $d_{\mathcal{G}} \coloneqq \gcd(z_1, \ldots, z_K)$, and compute generators of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ using Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}. (ii) If $d_{\mathcal{G}} \mid z_h$, continue, otherwise return **False**. (iii) Compute generators of the modules $\mathcal{M}'$ defined in Equation [\[eq:M2c\]](#eq:M2c){reference-type="eqref" reference="eq:M2c"}. Decide whether $\big(\mathcal{M}' \cap \mathbb{Z}^{M+1}\big) \cap \big(\mathbb{Z}^{M} \times \{1\}\big) = \emptyset$ using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"} and [Lemma 1](#lem:decinterZ){reference-type="ref" reference="lem:decinterZ"}. If yes, return **True**, otherwise return **False**. 3. **If none of the sets $\{z_1, \ldots, z_K\}$ and $\{z'_1, \ldots, z'_M\}$ is all zero.** (i) Compute $d_{\mathcal{G}} \coloneqq \gcd(z_1, \ldots, z_K), d_{\mathcal{H}} \coloneqq \gcd(z'_1, \ldots, z'_M), d \coloneqq \gcd(d_{\mathcal{G}}, d_{\mathcal{H}})$. (ii) Decide whether the equation $m d_{\mathcal{G}} = n d_{\mathcal{H}} + z_h$ has solutions $(m, n) \in \mathbb{Z}^2$. If yes, take any solution $(m, n)$ and let $z_{\mathcal{G}} \coloneqq m d_{\mathcal{G}},\; z_{\mathcal{H}} \coloneqq n d_{\mathcal{H}}$; otherwise return **True**. (iii) Compute generators of the $\mathbb{Z}[X^{\pm d_{\mathcal{G}}}]$-module $\langle {\mathcal{G}} \rangle \cap \mathcal{A}$ and the $\mathbb{Z}[X^{\pm d_{\mathcal{H}}}]$-module $\langle {\mathcal{H}} \rangle \cap \mathcal{A}$ using Lemma [\[lem:struct\]](#lem:struct){reference-type="ref" reference="lem:struct"}. Compute their respective generators $S_{\mathcal{G}}, S_{\mathcal{H}}$ as $\mathbb{Z}[X^{\pm d}]$-modules using Lemma [\[lem:changebase\]](#lem:changebase){reference-type="ref" reference="lem:changebase"}. Let $\mathcal{M}'$ be the $\mathbb{Z}[X^{\pm d}]$-module generated by $S_{\mathcal{G}} \cup (X^{z_h} \cdot S_{\mathcal{H}})$. (iv) Using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(i), decide whether Equation [\[eq:faainM\]](#eq:faainM){reference-type="eqref" reference="eq:faainM"} has a solution $f \in \mathbb{Z}[X^{\pm d}]$.\ If yes, compute (by enumeration) such a solution $f_0$; otherwise return **True**. (v) Compute the generators of the ideal $\mathcal{I}' \subseteq \mathbb{Z}[X^{\pm d}]$ defined in Equation [\[eq:defIp\]](#eq:defIp){reference-type="eqref" reference="eq:defIp"} using the generators of $\mathcal{M}'$ and Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(ii). (vi) Using Lemma [Lemma 1](#lem:classicdec){reference-type="ref" reference="lem:classicdec"}(i), decide whether $1 - f_0 \in \mathcal{I}'$. If yes, return **False**, otherwise continue. (vii) Using Algorithm [\[alg:smm\]](#alg:smm){reference-type="ref" reference="alg:smm"}, decide whether $\mathcal{I}'$ contains an element $X^{zd} - f_0$ for some $z \in \mathbb{Z}\setminus \{0\}$. If yes, return **False**, otherwise return **True**. Input: : The generators $g_1, \ldots, g_m$ of an ideal $\mathcal{I}\subseteq \mathbb{Z}[X^{\pm}]$ and an element $f \in \mathbb{Z}[X^{\pm}]$. Output: : **True** (when there exists $z \in \mathbb{Z}\setminus \{0\}$ such that $X^{z} - f \in \mathcal{I}$) or **False**. 1. Multiply each $g_i, i = 1, \ldots, m,$ by a suitable power of $X$ so that $g_i \in \mathbb{Z}[X]$ and $X \nmid g_i$. Denote by $\widetilde{\mathcal{I}}$ the ideal of $\mathbb{Z}[X]$ generated by $g_1, \ldots, g_m$. 2. Compute $\varphi \coloneqq \gcd(g_1, \ldots, g_m)$. 3. **If $\varphi = 1$.** (i) If $\widetilde{\mathcal{I}}= \mathbb{Z}[X]$ (equivalent to $1 \in \widetilde{\mathcal{I}}$), return **True**; otherwise compute $c, g \in \widetilde{\mathcal{I}}$ defined in Lemma [Lemma 1](#lem:idealstr){reference-type="ref" reference="lem:idealstr"}. (ii) Enumerate all pairs $(p, q) \in \mathbb{N}^2,\; p < q,$ and test whether $X^q - X^p \in (\mathbb{Z}[X] \cdot g + \mathbb{Z}[X] \cdot c)$. Stop when we find such a pair $p, q$. (iii) For all pairs of integers $a', b' \in [0, q-1]$, decide whether $X^{a'} - X^{b'} f \in \widetilde{\mathcal{I}}$. If such a pair exists, return **True**; otherwise return **False**. 4. **If $\varphi$ is not primitive.** For each integer $r \in [1, \deg f]$, decide whether $X^r - f \in \mathcal{I}$. If such $r$ exists, return **True**; otherwise return **False**. 5. **If $\varphi$ is primitive and has a root $x$ that is not a root of unity.** Let $\mathbb{K}$ be an algebraic number field that contains $x$ and let $H$ be the height function over $\mathbb{K}^*$ (Lemma [Lemma 1](#lem:height){reference-type="ref" reference="lem:height"}). Let $r \coloneqq \frac{\log H(f(x))}{\log H(x)}$. If $r = 0$ or $r$ is not an integer, return **False**. Otherwise decide whether one of $X^{r} - f$ and $X^{-r} - f$ is in $\mathcal{I}$. If yes, return **True**; otherwise return **False**. 6. **If $\varphi$ is primitive, all roots of $\varphi$ are roots of unity, and $\varphi$ has a square divisor $\phi$.** Let $x$ be a root of $\phi$. Let $r \coloneqq \frac{x f'(x)}{f(x)}$. If $r$ is a non-zero integer and $X^r - f \in \mathcal{I}$, return **True**; otherwise return **False**. 7. **If $\varphi$ is primitive, all roots of $\varphi$ are roots of unity, and $\varphi$ is square-free.** (i) Compute $p \geq 1$ such that $\varphi \mid X^p - 1$ (Lemma [\[lem:div\]](#lem:div){reference-type="ref" reference="lem:div"}). (ii) Compute the generators $g'_i \coloneqq \frac{g_i}{\varphi}, i = 1, \ldots, m$ of the ideal $\widetilde{\mathcal{J}}= \frac{\widetilde{\mathcal{I}}}{\varphi}$. (iii) If $\widetilde{\mathcal{J}}= \mathbb{Z}[X]$ (equivalent to $1 \in \widetilde{\mathcal{J}}$). For each pair of integers $a', b' \in [0, p-1]$, decide whether $\varphi \mid X^{a'} - X^{b'} f$. If such a pair exists return **True**, otherwise return **False**. (iv) If $\widetilde{\mathcal{J}}\neq \mathbb{Z}[X]$, compute $c, g \in \widetilde{\mathcal{J}}$ defined in Lemma [Lemma 1](#lem:idealstr){reference-type="ref" reference="lem:idealstr"}. (a) Enumerate all pairs $(q', q) \in \mathbb{N}^2,\; q' < q,$ and test whether $\frac{X^{pq} - 1}{\varphi} - \frac{X^{pq'} - 1}{\varphi} \in \mathbb{Z}[X] \cdot c + \mathbb{Z}[X] \cdot g$. Stop when we find such a pair $q', q$. (b) For all pairs of integers $a', b' \in [0, pq-1]$, decide whether $X^{a'} - X^{b'} f \in \widetilde{\mathcal{I}}$. If such a pair exists, return **True**; otherwise return **False**.

arxiv_math

--- abstract: | In this paper, we give the Jacobi polynomials for first-order generalized Reed--Muller codes. We show as a corollary the nonexistence of combinatorial $3$-designs in these codes. address: School of Fundamental Science and Engineering, Waseda University, Tokyo 169--8555, Japan author: - Ryosuke Yamaguchi\* title: Jacobi polynomials for first-order generalized Reed--Muller codes --- # Introduction Let $m$ be a positive integer and $q$ be a prime power, and set $V = \mathbb{F}_q^m$. The first-order generalized Reed--Muller (GRM) code $RM_q(1, m)$ is defined as the subspace of $\mathbb{F}_q^V$ consisting of affine linear functions: $$RM_q(1, m) = \left\{(\lambda(x) + b)_{x \in V} \in \mathbb{F}_q^V \mid \lambda \in V^*, b \in \mathbb{F}_q\right\},$$ where $V^* = \mathrm{Hom}(V, \mathbb{F}_q)$. We remark that the weight enumerator of $RM_q(1, m)$ is $$\label{weight} x^{q^m} + (q^{m + 1} - q)x^{q^{m - 1}}y^{(q - 1)q^{m - 1}} + (q - 1)y^{q^m}.$$ Let $C = RM_q(1, m)$ and $C_\ell := \{ c \in C \mid \mathrm{wt}(c) = \ell \}$. In this paper, we use Jacobi polynomials to show that the shells of $RM_q(1, m)$ support combinatorial $2$-designs. More precisely, the set $\mathcal{B}(C_\ell) := \{ \mathrm{supp}(x) \mid x \in C_\ell\}$ supports a combinatorial $2$-design. Herein, we always assume that a combinatorial $t$-design allows the existence of repeated blocks, and we exclude the trivial design $\mathcal{D} = (\Omega, \mathcal{B})$ where $\Omega = \{1,\dots,n\}$ and $\mathcal{B} = \{\Omega,\dots,\Omega\}$. **Theorem 1**. *Let $C=RM_q(1, m)$ and $T = \{0, u\} \in \binom{V}{2}$. Then, $$\begin{aligned} J_{C, T}(w, z, x, y) =& w^2x^{q^m - 2} + (q^{m - 1} - 1)w^2x^{q^{m-1} - 2}y^{(q-1)q^{m-1}} \\ &+ 2(q-1)q^{m-1}wzx^{q^{m-1}-1}y^{(q-1)q^{m-1}-1}\\ &+ (q-1)(q^m-q^{m-1}-1)z^2x^{q^{m-1}}y^{(q-1)q^{m-1}-2} \\ &+ (q-1)z^2y^{q^m - 2}. \end{aligned}$$* Using this theorem, we show that the shells of $RM_q(1, m)$ and $RM_q(1, m)^\perp$ support $2$-designs. **Corollary 2**. *Let $C=RM_q(1,m)$. Then for any $\ell \in \mathbb{N}$, $C_\ell$ is a combinatorial $2$-design. Similarly, $(C^\perp)_\ell$ is a combinatorial $2$-design.* Miezaki and Munemasa [@MM] showed that the shells of $RM_2(1, m)$ support $3$-designs but do not support $4$-designs. As an analogy, we show that the shells of $RM_q(1, m)$ do not support combinatorial $3$-designs, where $q \geq 3$. **Theorem 3**. *Let $C=RM_q(1, m)$, $T=\{0, u_1, u_2\} \in \binom{V}{3}$, and $A = {}^t[u_1\ u_2]$.* 1. *If $\mathrm{rank}\, A = 2$, then $$\begin{aligned} J_{C, T}(w, z, x, y) =& w^3x^{q^m - 3} + (q^{m - 2} - 1) w^3x^{q^{m - 1} - 3}y^{(q - 1)q^{m - 1}} \\ &+ 3q^{m - 2}(q - 1) w^2zx^{q^{m - 1} - 2}y^{(q - 1)q^{m - 1} - 1} \\ &+ 3q^{m - 2}(q - 1)^2 wz^2x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 2} \\ &+ (q - 1)(q^m - 2q^{m - 1} + q^{m - 2} - 1) z^3x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)z^3y^{q^m - 3}. \end{aligned}$$* 2. *If $\mathrm{rank}\, A = 1$, then $$\begin{aligned} J_{C, T}(w, z, x, y) =& w^3x^{q^m - 3} + (q^{m - 1} - 1) w^3x^{q^{m - 1} - 3}y^{(q - 1)q^{m - 1}} \\ &+ 3q^{m - 1}(q - 1) wz^2x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 2} \\ &+ (q - 1)(q^m - 2q^{m - 1} - 1) z^3x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)z^3y^{q^m - 3}. \end{aligned}$$* Using this theorem, we show that $C_\ell$ does not support a $3$-design. In addition, we show that $(C^\perp)_\ell$ is not a combinatorial $3$-design. **Corollary 4**. *Let $C=RM_q(1,m)$. If $q \geq 3$, then for any $\ell \in \mathbb{N}$, $C_\ell$ is not a combinatorial $3$-design. Similarly, $(C^\perp)_\ell$ is not a combinatorial $3$-design.* Using Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}, we show that $C_\ell$ supports a $3$-$(v, k, (\lambda_1, \lambda_2))$-design. **Corollary 5**. *Let $C = RM_q(1, m)$ and $\ell = (q - 1)q^{m-1}$. Then, $C_\ell$ is a combinatorial $3$-$(v, k, (\lambda_1, \lambda_2))$-design, where $$\begin{aligned} v &= (q - 1)q^{m-1},\ k = q^{m + 1} - q,\\ \lambda_1 &= (q - 1)(q^m - 2q^{m - 1} + q^{m - 2} - 1),\\ \lambda_2 &= (q - 1)(q^m - 3q^{m - 1} - 1). \end{aligned}$$* Also, we use Jacobi polynomials to show that the shells of $RM_q(1, m)$ support combinatorial $4$-$(v,k,(\lambda_1,\lambda_2,\lambda_3,\lambda_4))$-designs. First, we give the Jacobi polynomials $J_{C, T}$ with $|T| = 4$. **Theorem 6**. *Let $C=RM_q(1, m)$, $T=\{0, u_1, u_2, u_3\} \in \binom{V}{4}$, and $A = {}^t[u_1\ u_2\ u_3]$.* 1. *If $\mathrm{rank}\, A = 3$ then, $$\begin{aligned} J_{C, T}(w, z, x, y) = &w^4 x^{q^m - 4} + (q^{m-3} - 1)w^4x^{q^{m - 1} - 4}y^{(q - 1)q^{m - 1}} \\ &+ 4q^{m - 3}(q - 1)w^3zx^{q^{m - 1} - 3}y^{(q - 1)q^{m - 1} - 1} \\ &+ 6(q - 1)^2q^{m - 3}w^2z^2x^{q^{m - 1} - 2}y^{(q - 1)q^{m - 1} - 2} \\ &+ 4q^{m - 3}(q - 1)^3wz^3x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} - q^{m - 3} - 1)z^4x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 4} \\ &+ (q - 1)z^4y^{q^m - 4}. \end{aligned}$$* 2. *If $\mathrm{rank}\, A = 2$ then, $$\begin{aligned} J_{C, T}(w, z, x, y) = &w^4 x^{q^m - 4} + (q^{m - 2} - 1)w^4x^{q^{m - 1} - 4}y^{(q - 1)q^{m - 1}} \\ &+ q^{m - 2}(q - 1) w^3zx^{q^{m - 1} - 3}y^{(q - 1)q^{m - 1} - 1} \\ &+ 3q^{m - 2} (q - 1) w^2z^2x^{q^{m - 1} - 2}y^{(q - 1)q^{m - 1} - 2} \\ &+ q^{m - 2}(q - 1)(4q - 5) wz^3x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)(q^m - 3q^{m - 1} + 2q^{m - 2} - 1) z^4x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 4} \\ &+ (q - 1)z^4y^{q^m - 4} \end{aligned}$$ or $$\begin{aligned} J_{C, T}(w, z, x, y) = &w^4 x^{q^m - 4} + (q^{m - 2} - 1)w^4x^{q^{m - 1} - 4}y^{(q - 1)q^{m - 1}} \\ &+ 6q^{m - 2}(q - 1)w^2z^2x^{q^{m - 1} - 2}y^{(q - 1)q^{m - 1} - 2} \\ &+ q^{m - 2}(q - 1)(4q - 8) wz^3x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} + 1) z^4x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 4} \\ &+ (q - 1)z^4y^{q^m - 4}. \end{aligned}$$* 3. *If $\mathrm{rank}\, A = 1$ then, $$\begin{aligned} J_{C, T}(w, z, x, y) = &w^4 x^{q^m - 4} + (q^{m - 1} - 1)w^4x^{q^{m - 1} - 4}y^{(q - 1)q^{m - 1}} \\ &+ 4q^{m - 1}(q - 1)wz^3x^{q^{m - 1} - 1}y^{(q - 1)q^{m - 1} - 3} \\ &+ (q - 1)(q^m - 3q^{m - 1} - 1) z^4x^{q^{m - 1}}y^{(q - 1)q^{m - 1} - 4} \\ &+ (q - 1)z^4y^{q^m - 4}. \end{aligned}$$* By this theorem, we show that $C_\ell$ supports a combinatorial $4$-$(v,k,(\lambda_1,\lambda_2,\lambda_3,\lambda_4))$-design. **Corollary 7**. *Let $C = RM_q(1, m)$ and $\ell = (q-1)q^{m-1}$. Then $C_\ell$ is a $4$-$(v, k, (\lambda_1,\lambda_2,\lambda_3,\lambda_4))$-design, where $$\begin{aligned} v &= (q - 1)q^{m-1}, \ k = q^{m + 1} - 1,\\ \lambda_1 &= (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} - q^{m - 3} - 1), \\ \lambda_2 &= (q - 1)(q^m - 3q^{m - 1} + 2q^{m - 2} - 1), \\ \lambda_3 &= (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} + 1),\\ \lambda_4 &= (q - 1)(q^m - 3q^{m - 1} - 1). \end{aligned}$$* This paper is organized as follows. In Section [2](#sec:pre){reference-type="ref" reference="sec:pre"}, we define and give some basic properties of the codes, combinatorial $t$-designs, and Jacobi polynomials used in this paper. In Sections [3](#sec:2-design){reference-type="ref" reference="sec:2-design"}, [4](#sec:3-design){reference-type="ref" reference="sec:3-design"}, and [5](#sec:4-design){reference-type="ref" reference="sec:4-design"}, we show Theorems [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"}, [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}, and [Theorem 6](#thm:4-design){reference-type="ref" reference="thm:4-design"}, respectively. All computer calculations reported in this paper were carried out using Magma [@Magma] and Mathematica [@Mathematica]. # Preliminaries {#sec:pre} In this section, we give definitions and some properties of codes, combinatorial designs, and Jacobi polynomials. We mainly refer to [@MM] and [@CMOT]. ## Codes and combinatorial $t$-designs. Let $q$ be a prime power. A $q$-ary linear code $C$ of length $n$ is a linear subspace of $\mathbb{F}_q^n$. For $c \in \mathbb{F}_q^n$, the weight $\mathrm{wt}(c)$ is the number of its nonzero components. The shell of a weight-$\ell$ codeword is the set of codewords whose weight is $\ell$: $C_\ell := \{ c \in C \mid \mathrm{wt}(c) = \ell \}$. A combinatorial $t$-design is a pair $\mathcal{D} = (\Omega, \mathcal{B})$, where $\Omega$ is a set of points of cardinality $v$, and $\mathcal{B}$ is a collection of $k$-element subsets of $\Omega$ called blocks, with the property that any $t$-element subset of $\Omega$ is contained in precisely $\lambda$ blocks. Recall [@BMP] for the definitions of various types of designs. A pair $\mathcal{D} = (\Omega, \mathcal{B})$ is a combinatorial design with parameters $t$-$(v, k, (\lambda_1,\dots,\lambda_N))$ if $\mathcal{B}$ is a collection of $k$-element subsets of $\Omega$ called blocks and every $t$-element subset of $\Omega$ is contained in $\lambda_i$ blocks. Note that for $N=1$, the design coincides with an exactly $t$-design. The support of a vector $x := (x_1,\dots,x_n)$, $x_i \in \mathbb{F}_q$, is the set of indices of its nonzero coordinates: $\mathrm{supp}(x) = \{ (c) \mid x_i \neq 0\}$. Let $\Omega := \{1,\dots,n\}$ and $\mathcal{B}(C_\ell):= \{\mathrm{supp}(x)\mid x \in C_\ell\}$. Then for a code $C$ of length $n$, we say that the shell $C_\ell$ is a combinatorial $t$-design if $(\Omega, \mathcal{B}(C_\ell))$ is a combinatorial $t$-design. Similarly, we say that the shell $C_\ell$ is a combinatorial $t$-$(v, k, (\lambda_1,\dots,\lambda_N))$-design if $(\Omega, \mathcal{B}(C_\ell))$ is a combinatorial $t$-$(v, k, (\lambda_1,\dots, \lambda_N))$-design. ## Jacobi poynomials. Let $C$ be a $q$-ary code of length $n$ and $T \subset [n] := \{1,\dots,n\}$. Then the Jacobi polynomial of $C$ with $T$ is defined as follows [@Ozeki]: $$J_{C, T}(w, z, x, y) := \sum_{c \in C} w^{m_0(c)}z^{m_1(c)}{n_0(c)}y^{n_1(c)},$$ where for $c = (c_1,\dots,c_n)$, $$\begin{aligned} m_0(c) &= |\{ j \in T \mid c_j = 0\}|,\\ m_1(c) &= |\{ j \in T \mid c_j \ne(c) \}|,\\ n_0(c) &= |\{ j \in [n] \setminus T \mid (c) = 0\}|,\\ n_1(c) &= |\{ j \in [n] \setminus T \mid c_j \neq 0\}|.\end{aligned}$$ The Jacobi polynomial of the dual code is written as follows. **Theorem 8** (Theorem 4 of [@Ozeki]). *Let $C$ be a $q$-ary code of length $n$ and $T \subset [n]$. Then we have $$J_{C^\perp, T}(w, z, x, y) = \frac{1}{|C|}J_{C, T}(w + (q - 1)z, w - z, x + (q - 1)y, x - y).$$* Clearly, we have the following relation between Jacobi polynomials and combinatorial designs. **Proposition 9**. *Let $C$ be a linear code. $C_\ell$ is a combinatorial $t$-design if and only if the coefficient of $z^tx^{n-\ell}y^{\ell-t}$ in $J_{C, T}$ is independent of the choice of $T$ with $|T| = t$.* We remark that the Jacobi polynomials are invariant under the automorphisms of codes. In particular, for $C = RM_q(1, m)$, a map $\varphi$ such that $\varphi(\lambda(x) + b') = \lambda(x) + b + b'$ is an automorphism on $C$. Then, we have the following result. **Proposition 10**. *Let $C = RM_q(1, m), T \subset V$, and $T' = T + v$, where $v \in V$. Then we ha'e $J_{C, T} = J_{C, T'}$.* This fact is useful because it suffices to consider the case that an element of $T$ is zero. ## Notation {#subsec:notations} We next introduce some notation. Let $C = RM_q(1, m)$, $T \subset V$, and $t = |T|$. For $c \in C$, $i \in \{0,1,\dots,t\}$, $j \in \mathbb{F}_q$, we define $$\begin{gathered} n_{j, T}(c) := |\{u \in T \mid c(u) = j\}|, \\ a_{i, j} := | \{c \in C \setminus \mathbb{F}_q\mathbf{1} \mid n_{j, T} (c) = i\} |, \\ b_{i, j} := | \{c \in V^* \mid n_{j, T}(c) = i\} |\end{gathered}$$ and $$\begin{gathered} a_i := | \{c \in C \setminus \mathbb{F}_q\mathbf{1} \mid \mathrm{wt}(c|_T) = i\} | = a_{t-i, 0}, \\ b_i := \sum_{j \in \mathbb{F}_q} b_{i, j} = \sum_{j \in \mathbb{F}_q} | \{c \in V^* \mid n_{j, T}(c) = i\} |.\end{gathered}$$ Then, the Jacobi polynomial of $C$ with $T$ is written as $$\begin{aligned} \label{eq:Jacobi} J_{C, T}(w, z, x, y) &= w^tx^{q^m-t} \\ &+ \sum_{i = 0}^t a_i w^{t-i}z^ix^{q^{m-1}-(t-i)}y^{(q-1)q^{m-1}-i} \notag\\ &+ (q-1)z^ty^{q^m-4}. \notag\end{aligned}$$ We have the following relation between $a_i$ and $b_i$. **Lemma 11**. *We have $$a_i = b_{t - i} - \delta_{i, 0} - (q - 1)\delta_{i, t}.$$* *Proof.* Since $$\begin{aligned} a_{i, 0} &= |\{c \in C \setminus \mathbb{F}_q \mathbf{1} \mid n_{0, T}(c) = i\}| \\ &= \sum_{j \in \mathbb{F}_q } |\{c \in (V^* + j\mathbf{1}) \setminus \{j\mathbf{1}\} \mid n_{0, T}(c) = i\}| \\ &= \sum_{j \in \mathbb{F}_q } |\{c \in V^* \setminus\{0\} \mid n_{0, T}(c + j\mathbf{1}) = i \}| \\ &= \sum_{j \in \mathbb{F}_q } |\{c \in V^* \setminus\{0\} \mid n_{-j, T}(c) = i \}| \\ &= |\{ c \in V^* \setminus \{0\} \mid n_{0, T}(c) = i\} | + \sum_{j \in \mathbb{F}_q \setminus\{0\}} | \{ c \in V^* \setminus\{0\} \mid n_{-j, T}(c) = i \} | \\ &= b_{i, 0} - \delta_{i, t} + \sum_{j \in \mathbb{F}_q \setminus \{0 \} } (b_{i, -j} - \delta_{i, 0}) \\ &= \sum_{j \in \mathbb{F}_q} b_{i, -j} - \delta_{i, t} - (q - 1)\delta_{i, 0} \\ &= \sum_{j \in \mathbb{F}_q} b_{i, j} - \delta_{i, t} - (q - 1)\delta_{i, 0} \\ &= b_i - \delta_{i, t} - (q - 1)\delta_{i, 0}, \end{aligned}$$ we obtain $$\begin{aligned} a_i &= a_{t - i, 0} = b_{t - i} - \delta_{t - i, t} - (q - 1)\delta_{t - i, 0} \\ &= b_{t - i} - \delta_{i, 0} - (q - 1)\delta_{i, t}. \\ \end{aligned}$$ ◻ Using this lemma, we obtain $a_i$ by calculating $b_i$. # Proofs of Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"} and Corollary [Corollary 2](#cor:2-design){reference-type="ref" reference="cor:2-design"} {#sec:2-design} In this section, we give proofs of Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"} and Corollary [Corollary 2](#cor:2-design){reference-type="ref" reference="cor:2-design"} using the notation introduced in Section [2.3](#subsec:notations){reference-type="ref" reference="subsec:notations"}. First, we give a lemma to show Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"}. Let $T = \{ 0, u \} \in \binom{V}{2}$. **Lemma 12**. *We have $$\begin{aligned} b_0 &= q^{m - 1}(q - 1)^2, \\ b_1 &= 2q^{m - 1}(q - 1), \\ b_2 &= q^{m - 1}. \end{aligned}$$* *Proof.* Considering $u \in T$ as an element of $V^{**}$, $u$ is a surjective linear map from $V^*$ to $\mathbb{F}_q$ because $u \neq 0$. Then, for $j \in \mathbb{F}_q \setminus \{0\}$, $$\begin{gathered} b_{0, 0} = 0,\ b_{0, j} = \left| \bigcup_{a\in\mathbb{F}_q \setminus\{j\}}u^{-1}(a) \right| = (q-1)q^{m-1},\\ b_{1, 0} = b_{0, j} = (q-1)q^{m-1},\ b_{1, j} = \left| u^{-1}(j) \right| = q^{m-1},\\ b_{2, 0} = b_{1, j} = q^{m-1},\ b_{2, j} = 0.\end{gathered}$$ Therefore, $$\begin{aligned} b_0 &= b_{0,0} + \sum_{j\in\mathbb{F}_q\setminus\{0\}}b_{0,j} = (q-1)^2q^{m-1},\\ b_1 &= b_{1,0} + \sum_{j\in\mathbb{F}_q\setminus\{0\}}b_{1,j} = 2(q-1)q^{m-1},\\ b_2 &= b_{2,0} + \sum_{j\in\mathbb{F}_q\setminus\{0\}}b_{2,j} = q^{m-1}.\end{aligned}$$ ◻ Using this Lemma, we show Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"}. *Proof of Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"}.* Using Lemmas [Lemma 11](#lem:b-to-a){reference-type="ref" reference="lem:b-to-a"} and [Lemma 12](#lem:2-design){reference-type="ref" reference="lem:2-design"}, we obtain $$\begin{aligned} a_0 &= b_2 - 1 = q^{m-1} - 1,\\ a_1 &= b_1 = 2(q-1)q^{m-1},\\ a_2 &= b_0 - (q-1) = (q-1)(q^m - q^{m-1} - 1). \end{aligned}$$ Thus, we give coefficients of the Jacobi polynomial by ([\[eq:Jacobi\]](#eq:Jacobi){reference-type="ref" reference="eq:Jacobi"}). ◻ Finally, we give a proof of Corollary [Corollary 2](#cor:2-design){reference-type="ref" reference="cor:2-design"}. *Proof of Corollary [Corollary 2](#cor:2-design){reference-type="ref" reference="cor:2-design"}.* By Proposition [Proposition 10](#prop:Jacobi-inv){reference-type="ref" reference="prop:Jacobi-inv"}, it suffices to show that for any $T = \{ 0, u\} \subset \binom{V}{2}$, the coefficient of $z^2x^{q^m - \ell}y^{\ell-2}$ is the same value. This is true by using Theorem [Theorem 1](#thm:2-design){reference-type="ref" reference="thm:2-design"}. In addition, we have the Jacobi polynomial of $RM_q(1, m)^\perp$ by using Theorem [Theorem 8](#thm:dual-code){reference-type="ref" reference="thm:dual-code"}. Thus, we obtain the desired results. ◻ # Proofs of Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"} and Corollary [Corollary 4](#cor:3-design){reference-type="ref" reference="cor:3-design"} {#sec:3-design} In this section, we give proofs of Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"} and Corollaries [Corollary 4](#cor:3-design){reference-type="ref" reference="cor:3-design"} and [Corollary 5](#cor:3-gen-design){reference-type="ref" reference="cor:3-gen-design"} using the notation introduced in Section [2.3](#subsec:notations){reference-type="ref" reference="subsec:notations"}. First, we give two lemmas to show Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}. Let $T = \{0, u_1, u_2\}\in \binom{V}{3}$, $A = {}^t[u_1\ u_2]$. **Lemma 13**. *If $\mathrm{rank}\, A = 2$, then $$\begin{aligned} b_0 &= q^{m - 2}(q - 1)^3, \\ b_1 &= 3q^{m - 2}(q - 1)^2, \\ b_2 &= 3q^{m - 2}(q - 1), \\ b_3 &= q^{m - 2}.\end{aligned}$$* *Proof.* We remark that $A$ is a surjective map from $V$ to $\mathbb{F}_q^2$, and for all $a, b \in \mathbb{F}_q, |A^{-1}({}^t[a\ b])| = |\mathrm{Ker}\, A| = q^{m - 2}$. Then, for $j \in \mathbb{F}_q \setminus \{0\}$, $$\begin{aligned} b_{0, j} &= \left|\bigcup_{a,b\neq j} A^{-1}({}^t[a\ b])\right| =\sum_{a, b\in \mathbb{F}_q \setminus \{j\}} \left| A^{-1}({}^t [a\ b]) \right| = (q - 1)^2q^{m - 2} ,\\ b_{1, j} &= 2\times \left|\bigcup_{a\neq j} A^{-1} ({}^t[j\ a])\right| = 2(q - 1)q^{m - 2},\\ b_{2, j} &= \left| A^{-1} ({}^t[j\ j]) \right| = q^{m - 2},\\ b_{3, j} &= 0\end{aligned}$$ and $$\begin{aligned} b_{0,0} &= 0,\\ b_{1,0} &= \left|\bigcup_{a,b\neq 0} A^{-1}({}^t[a\ b])\right| =\sum_{a, b\in \mathbb{F}_q \setminus \{0\}} \left| A^{-1}({}^t [a\ b]) \right| = (q - 1)^2q^{m - 2},\\ b_{2,0} &= 2\times \left|\bigcup_{a\neq 0} A^{-1} ({}^t[j\ a])\right| = 2(q - 1)q^{m - 2},\\ b_{3,0} &= \left| \mathrm{Ker}\, A \right| = q^{m - 2}.\end{aligned}$$ Since $$b_i = b_{i, 0} + \sum_{j \in \mathbb{F}_q\setminus \{0\}}b_{i, j},$$ we obtain the desired results. ◻ **Lemma 14**. *If $\mathrm{rank}\, A = 1$, then $$\begin{aligned} b_0 &= q^{m - 1}(q - 1)(q - 2), \\ b_1 &= 3q^{m - 1}(q - 1), \\ b_2 &= 0, \\ b_3 &= q^{m - 1}.\end{aligned}$$* *Proof.* Because $u_1$ and $u_2$ are not equal to $0$ or each other, there exist $a, b \in \mathbb{F}_q \setminus \{0\}$ such that $a \neq b$ and $\{{}^t[a\ b] \}$ is a basis of $\mathrm{Im}\, A$. Hence, $\mathrm{rank}\, A = 1$ and for any $v \in \mathrm{Im}\, A$, $$|A^{-1}(v)| = |\mathrm{Ker}\, A| = q^{\dim{\mathrm{Ker}\, A}} = q^{m - \mathrm{rank}\, A} = q^{m - 1}.$$ Then, for $j \in \mathbb{F}_q \setminus \{0\}$, we have $$\begin{aligned} b_{1, j} &= |A^{-1}({}^t[j\ ja^{-1}b])| + |A^{-1}({}^t[jb^{-1}a\ j])| = 2q^{m - 1},\\ b_{2, j} &= b_{3, j} = 0, \\ b_{0, j} &= |V| - (b_{1, j} + b_{2, j} + b_{3, j}) = q^m - 2q^{m - 1}\end{aligned}$$ and $$\begin{aligned} b_{1, 0} &= (q - 1)q^{m - 1},\\ b_{0, 0} &= b_{2, 0} = 0, \\ b_{3, 0} &= |\mathrm{Ker}\, A| = q^{m - 1}.\end{aligned}$$ Thus, since $b_i = \sum_{j \in \mathbb{F}_q} b_{i, j}$, we obtain the desired results. ◻ Using these lemmas, we give a proof of Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}. *Proof of Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}.* By Lemmas [Lemma 11](#lem:b-to-a){reference-type="ref" reference="lem:b-to-a"} and ([\[eq:Jacobi\]](#eq:Jacobi){reference-type="ref" reference="eq:Jacobi"}), (1) follows from Lemma [Lemma 13](#lem:3-design-rank2){reference-type="ref" reference="lem:3-design-rank2"}, and (2) follows from Lemma [Lemma 14](#lem:3-design-rank1){reference-type="ref" reference="lem:3-design-rank1"}. ◻ Then, we show that the shells of $RM_q(1, m)$ and $RM_q(1, m)^\perp$ do not support $3$-designs if $q \geq 3$. *Proof of Corollary [Corollary 4](#cor:3-design){reference-type="ref" reference="cor:3-design"}.* Let $C = RM_q(1, m)$. We give a proof relying on the properties of Jacobi polynomials. Let $T_1 = \{ 0, u_1, u_2 \} \in \binom{V}{3}$, $T_2 = \{ 0, v_1, v_2 \} \in \binom{V}{3}$, $A_1 = {}^t[u_1\ u_2]$, and $A_2 = {}^t[v_1 \ v_2]$. We assume that $\mathrm{rank}\, A_1 = 2$ and $\mathrm{rank}\, A_2 = 1$. Indeed, if $q \geq 3$, there exist such $T_1, T_2$. By Theorem [Theorem 3](#thm:3-design){reference-type="ref" reference="thm:3-design"}, $$J_{C,T_1} - J_{C,T_2} = -q^{m-2}(q-1)x^{q^{m-1}-3}y^{(q-1)q^{m-1}-3}(wy - xz)^3.$$ Since the coefficient of $z^3x^{q^{m-1}-\ell}y^{\ell-3}$ in $J_{C, T_1}, J_{C, T_2}$ is non-zero whenever $C_\ell$ is non-empty, $C_\ell$ is not a $3$-design. By using Theorem [Theorem 8](#thm:dual-code){reference-type="ref" reference="thm:dual-code"}, we have $$\begin{aligned} J_{C^\perp, T_1} - J_{C^\perp, T_2} = (q - 1)\{x + (q - 1)y\}^{q^{m-1}-3}(x - y)^{(q-1)q^{m-1}-3}(wy - xz)^3. \end{aligned}$$ Thus, $(C^\perp)_\ell$ is not a $3$-design. ◻ Finally, we claim that the shells of $RM_q(1, m)$ support $3$-$(v,k,(\lambda_1, \lambda_2))$-designs. *Proof of Corollary [Corollary 5](#cor:3-gen-design){reference-type="ref" reference="cor:3-gen-design"}.* It is clear from Theorem [Corollary 4](#cor:3-design){reference-type="ref" reference="cor:3-design"}. ◻ # Proof of Theorem [Theorem 6](#thm:4-design){reference-type="ref" reference="thm:4-design"} {#sec:4-design} In this section, we give proofs of Theorem [Theorem 6](#thm:4-design){reference-type="ref" reference="thm:4-design"} and Corollary [Corollary 7](#cor:4-gen-design){reference-type="ref" reference="cor:4-gen-design"} using the notation introduced in Section [2.3](#subsec:notations){reference-type="ref" reference="subsec:notations"}. Let $T = \{0, u_1, u_2, u_3\} \in \binom{V}{4}$, $A = {}^t[u_1\ u_2\ u_3]$. Considering $A$ as a linear map from $V$ to $\mathbb{F}_q^3$, for all $j \in \mathbb{F}_q\setminus \{0\}$, we have $$\begin{aligned} b_{i, j} &= |\{ c \in V \mid i \text{ elements of $Ac$ are equal to }j\}| \\ &= q^{m - \mathrm{rank}\, A} \times |\{ v \in \mathrm{Im}\, A \mid i \text{ elements of $v$ are equal to } j\}|,\end{aligned}$$ and for $j = 0$, we have $$\begin{aligned} b_{i, 0} &= |\{ c \in V \mid i - 1 \text{ elements of $Ac$ are equal to } 0\}|\notag \\ &= q^{m - \mathrm{rank}\, A} \times |\{ v \in \mathrm{Im}\, A \mid i - 1 \text{ elements of $v$ are equal to } 0\}|. \end{aligned}$$ Next, we prepare three lemmas for proving Theorem [Theorem 6](#thm:4-design){reference-type="ref" reference="thm:4-design"}. **Lemma 15**. *If $\mathrm{rank}\, A = 3$, $$\begin{gathered} a_0 = q^{m - 3} - 1,\ a_1 = 4(q - 1)q^{m - 3} ,\ a_2 = 6(q - 1)^2q^{m - 3},\\ a_3 = 4(q - 1)^3q^{m - 3},\ a_4 = (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} - q^{m - 3} - 1). \end{gathered}$$* *Proof.* If $\mathrm{rank}\, A = 3$, $A$ is a surjective linear map from $V$ to $\mathbb{F}_q^3$. Then, for all $v \in \mathbb{F}_q^3$, $|A^{-1}(v)| = q^{m-3}$. Thus, $$\begin{aligned} b_0 &= b_{0, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{0, j} = (q - 1) \times (q - 1)^3 \times q^{m - 3} = (q - 1)^4q^{m - 3}, \\ b_1 &= b_{1, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{1, j} = (q - 1)^3q^{m - 3} + 3(q - 1)^3q^{m - 3} = 4(q - 1)^3q^{m - 3}, \\ b_2 &= b_{2, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{2, j} = 3(q - 1)^2q^{m - 3} + 3(q - 1)^2q^{m - 3} = 6(q - 1)^2q^{m - 3}, \\ b_3 &= b_{3, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{3, j} = 3(q - 1)q^{m - 3} + (q - 1)q^{m - 3} = 4(q - 1)q^{m - 3}, \\ b_4 &= b_{4, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{4, j} = q^{m - 3}. \end{aligned}$$ Therefore, we obtain the desired results by Lemma [Lemma 11](#lem:b-to-a){reference-type="ref" reference="lem:b-to-a"}. ◻ **Lemma 16**. *If $\mathrm{rank}\, A = 1$, $$\begin{gathered} a_0 = q^{m - 1} - 1,\ a_1 = 0,\ a_2 = 0,\\ a_3 = 4(q - 1)q^{m - 1} ,\ a_4 = (q - 1)(q^m - 3q^{m - 1} - 1). \end{gathered}$$* *Proof.* Since $u_1, u_2, u_3$ are different from each other and not equal to $0$, we take a basis $\{{}^t[a\ b\ c]\}$ of $\mathrm{Im}\, A$, where $a, b, c \in \mathbb{F}_q\setminus\{0\}$ and are different from each other. If $\mathrm{rank}\, A = 1$, then for all $v \in \mathrm{Im}\, A$, $|A^{-1}(v)| = q^{m-1}$. Then, for all $j \in \mathbb{F}_q\setminus\{0\}$, $$\begin{aligned} &b_{1, j} = 3 \times q^{m - 1} = 3q^{m - 1},\\ &b_{2, j} = b_{3, j} = b_{4, j} = 0,\\ &b_{0, j} = q^m - (b_{1, j} + b_{2, j} + b_{3, j} + b_{4, j}) = q^m - 3q^{m - 1} \end{aligned}$$ and $$\begin{aligned} &b_{1, 0} = (q - 1) \times q^{m - 1}, \\ &b_{0, 0} = b_{2, 0} = b_{3, 0} = 0, \\ &b_{4, 0} = q^{m - 1}. \end{aligned}$$ Therefore, $b_0,\dots,b_4$ are written as follows: $$\begin{aligned} b_0 &= b_{0, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{0, j} = (q - 1) \times (q^m - 3q^{m - 1}) = (q - 1)(q - 3)q^{m - 1}, \\ b_1 &= b_{1, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{1, j} = (q - 1)q^{m - 1} + (q - 1) \times 3q^{m - 1} = 4(q - 1)q^{m - 1}, \\ b_2 &= b_{2, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{2, j} = 0, \\ b_3 &= b_{3, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{3, j} = 0, \\ b_4 &= b_{4, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{4, j} = q^{m - 1}. \end{aligned}$$ Thus, we obtain $a_0,\dots,a_4$ by using Lemma [Lemma 11](#lem:b-to-a){reference-type="ref" reference="lem:b-to-a"}. ◻ Before claiming the lemma under the condition of $\mathrm{rank}\, A = 2$, we give a basis of $\mathrm{Im}\, A$. Since $\mathrm{Im}\, A$ is invariant under the right multiplication of an invertible matrix to $A$, we confine a basis of $\mathrm{Im}\, A$ to the following: $$\left\{\begin{bmatrix}1\\0\\a\end{bmatrix}, \begin{bmatrix}0\\1\\b\end{bmatrix}\right\}, \left\{\begin{bmatrix}1\\a\\0\end{bmatrix}, \begin{bmatrix}0\\0\\1\end{bmatrix}\right\}, \left\{\begin{bmatrix}0\\1\\0\end{bmatrix}, \begin{bmatrix}0\\0\\1\end{bmatrix}\right\},$$ where $a, b \in \mathbb{F}_q$. However, since $u_1, u_2, u_3$ are not equal to $0$, the last one is excluded. Without loss of generality, we take the first one because $b_{i, j}$ is invariant under a permutation on indices of codewords. Note that we have $(a, b) \neq (0,0), (1,0), (0,1)$ because $u_1, u_2, u_3 \neq 0$. **Lemma 17**. *If $\mathrm{rank}\, A = 2$, we have the following equations.* 1. *If $a + b = 1$ or $ab = 0$, then $$\begin{gathered} % b_0 = q^{m - 2}(q - 1)^2(q - 2),\ b_1 = q^{m - 2}(q - 1)(4q - 5),\ b_3 = q^{m - 2}(q - 1),\ b_4 = q^{m - 2} a_0 = q^{m - 2} - 1,\ a_1 = q^{m - 2}(q - 1),\ a_2 = 3q^{m - 2}(q - 1),\\ a_3 = q^{m - 2}(q - 1)(4q - 5),\ a_4 = (q - 1)(q^m - 3q^{m - 1} + 2q^{m - 2} - 1). \end{gathered}$$* 2. *If $a + b \neq 1$ and $ab \neq 0$, then $$\begin{gathered} a_0 = q^{m - 2} - 1,\ a_1 = 0,\ a_2 = 6q^{m - 2}(q - 1),\\ a_3 = q^{m - 2}(q - 1)(4q - 8),\ a_4 = (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} - 1). \end{gathered}$$* *Proof.* First, we give $b_0$. Clearly, $b_{0, 0} = 0$. For all $j \in\mathbb{F}_q\setminus\{0\}$, $$\begin{aligned} b_{0, j} &= q^{m - \mathrm{rank}\, A} \times |\{ v \in \mathrm{Im}\, A \mid \text{ zero elements of $v$ are equal to } j\}|\\ &= q^{m-2} \times |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 \neq j,\, c_2 \neq j,\, c_1a+c_2b\neq j\}|.\end{aligned}$$ We have $$\begin{aligned} |\{(c_1, c_2)&\in\mathbb{F}_q^2\mid c_1 \neq j, c_2 \neq j, c_1a+c_2b \neq j\}| \\ = (q - 1)^2 &- |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 \neq j, c_2 \neq j, c_1a+c_2b = j\}|\\ = (q - 1)^2 &- ( |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1a+c_2b = j\}| \\ &- |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 = j, c_1a+c_2b = j\}| \\ &- |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_2 = j, c_1a+c_2b = j\}| \\ &+ |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 = j, c_2 = j, c_1a+c_2b = j\}| ).\end{aligned}$$ If $a = 0$, then $b \neq 0, 1$, and if $b = 0$, then $a \neq 0, 1$. Thus, $$\begin{aligned} |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1a+c_2b = j\}| &= q,\\ |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 = j, c_1a+c_2b = j\}| &= \left\{\begin{array}{ll} 1 & \text{if } b \neq 0 \\ 0 & \text{if } b = 0 \end{array}\right.,\\ |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_2 = j, c_1a+c_2b = j\}| &= \left\{\begin{array}{ll} 1 & \text{if } a \neq 0 \\ 0 & \text{if } a = 0 \end{array}\right.,\\ |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 = j, c_2 = j, c_1a+c_2b = j\}| &= \left\{\begin{array}{ll} 1 & \text{if } a + b = 1 \\ 0 & \text{otherwise} \end{array}\right..\end{aligned}$$ Therefore, $$\begin{aligned} |\{(c_1, c_2)\in\mathbb{F}_q^2\mid c_1 \neq j, c_2 \neq j, c_1a+c_2b \neq j\}|\\ = \left\{\begin{array}{ll} (q - 1)^2 - (q - 1) & \text{if } ab = 0 \text{ or } a + b = 1 ,\\ (q - 1)^2 - (q - 2) & \text{otherwise}. \end{array}\right.\end{aligned}$$ Hence, $$\begin{aligned} b_0 &= b_{0, 0} + \sum_{j \in \mathbb{F}_q \setminus\{0\}} b_{0, j} \\ &= \left\{\begin{array}{ll} \sum_{j \in \mathbb{F}_q \setminus\{0\}} q^{m - 2}(q - 1)(q - 2) & \text{if } a + b = 1 \text{ or } ab = 0 \\ \sum_{j \in \mathbb{F}_q \setminus\{0\}} q^{m - 2}(q^2 - 3q + 3) & \text{otherwise} \end{array}\right.\\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)^2(q - 2) & \text{if } a + b = 1 \text{ or } ab = 0, \\ q^{m - 2}(q - 1)(q^2 - 3q + 3) & \text{otherwise}. \end{array}\right.\end{aligned}$$ Second, we give $b_1$. Similarly, $$b_{1, 0} = \left\{\begin{array}{ll} (q - 1)^2 & \text{if } ab = 0,\\ (q - 1)(q - 2) & \text{otherwise}. \end{array}\right.$$ Fix $j \in \mathbb{F}_q\setminus\{0\}$, and let $I_1$, $I_2$, and $I_3$ be non-negative numbers such that $$\begin{gathered} I_1 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 = j, c_2 \neq j, ac_1 + bc_2 \neq j\}|, \\ I_2 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 \neq j, c_2 = j, ac_1 + bc_2 \neq j\}|,\\ I_3 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 \neq j, c_2 \neq j, ac_1 + bc_2 = j\}|. \end{gathered}$$ Then, $b_{1, j} = q^{m - 2} \times (I_1 + I_2 + I_3)$. We have $$\begin{aligned} I_1 &= \left\{\begin{array}{lll} q - 1 & \text{if } b = 0 \text{ or } a + b = 1,\\ q - 2 & \text{otherwise}, \end{array}\right.\\ I_2 &= \left\{\begin{array}{lll} q - 1 & \text{if } a = 0 \text{ or } a + b = 1,\\ q - 2 & \text{otherwise}, \end{array}\right.\\ I_3 &= \left\{\begin{array}{lll} q - 1 & \text{if } ab = 0 \text{ or } a + b = 1,\\ q - 2 & \text{otherwise}, \end{array}\right.\\ I_1 + I_2 + I_3 &= \left\{\begin{array}{ll} 3q - 3 & \text{if } a + b = 1,\\ 3q - 4 & \text{if } ab = 0,\\ 3q - 6 & \text{otherwise}. \end{array}\right.\end{aligned}$$ Therefore, $$\begin{aligned} b_1 &= b_{1, 0} + \sum_{j \in \mathbb{F}_q\setminus\{0\}} b_{1, j} \\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)(q - 2) + \sum_{j}q^{m - 2}(3q - 3) & \text{if } a + b = 1, \\ q^{m - 2}(q - 1)^2 + \sum_{j}q^{m - 2}(3q - 4) & \text{if } ab = 0, \\ q^{m - 2}(q - 1)(q - 2) + \sum_{j}q^{m - 2}(3q - 6) & \text{otherwise,} \end{array}\right.\\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)(q - 2) + (q - 1)q^{m - 2}(3q - 3) & \text{if } a + b = 1, \\ q^{m - 2}(q - 1)^2 + (q - 1)q^{m - 2}(3q - 4) & \text{if } ab = 0, \\ q^{m - 2}(q - 1)(q - 2) + (q - 1)q^{m - 2}(3q - 6) & \text{otherwise,} \end{array}\right.\\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)(4q - 5) & \text{if } a + b = 1 \text{ or } ab = 0, \\ q^{m - 2}(q - 1)(4q - 8) & \text{otherwise}. \end{array}\right.\end{aligned}$$ Similarly, we give $b_3$. For all $j \in \mathbb{F}_q\setminus\{0\}$, we have $$b_{3, j} = \left\{\begin{array}{ll} q^{m - 2} & \text{if } a + b = 1,\\ 0 & \text{otherwise}. \end{array}\right.$$ Let $J_1$, $J_2$, and $J_3$ be non-negative integers such that $$\begin{gathered} J_1 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 = 0, c_2 \neq 0, ac_1 + bc_2 \neq 0\}|, \\ J_2 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 \neq 0, c_2 = 0, ac_1 + bc_2 \neq 0\}|, \\ J_3 = |\{ (c_1, c_2) \in \mathbb{F}_q^2 \mid c_1 \neq 0, c_2 \neq 0, ac_1 + bc_2 = 0\}|. \end{gathered}$$ Clearly, $J_3 = 0$. We have $$\begin{aligned} J_1 &= \left\{\begin{array}{ll} q - 1 & \text{if } a = 0 ,\\ 0 & \text{otherwise}, \end{array}\right.\\ J_2 &= \left\{\begin{array}{ll} q - 1 & \text{if } b = 0 ,\\ 0 & \text{otherwise}. \end{array}\right.\end{aligned}$$ Thus, $$\begin{aligned} b_{3, 0} = q^{m - 2}\times(J_1 + J_2 + J_3) = \left\{\begin{array}{ll} q - 1 & \text{if } ab = 0, \\ 0 & \text{otherwise}. \end{array}\right.\end{aligned}$$ Therefore, $$\begin{aligned} b_3 &= b_{3, 0} + \sum_{j \in \mathbb{F}_q \setminus \{0\}} b_{3, j} \\ &= \left\{\begin{array}{ll} q^{m - 2} (q - 1) & \text{if } a + b = 1 \text{ or } ab = 0,\\ 0 & \text{otherwise} , \end{array}\right.\end{aligned}$$ and we have $b_{4, j} = 0$, where $j \in \mathbb{F}_q\setminus\{0\}$, and $b_{4, 0} = |\mathrm{Ker}\, A| = q^{m-2}$. Then we have $$b_4 = q^{m-2}.$$ Finally, by using Lemma [Lemma 11](#lem:b-to-a){reference-type="ref" reference="lem:b-to-a"}, we give $a_0, \dots, a_4$ as follows: $$\begin{aligned} a_0 &= b_4 - 1 \\ &= q^{m - 2} - 1, \\ a_1 &= b_3 \\ &= \left\{\begin{array}{ll} q^{m - 2} (q - 1) & \text{if } a + b = 1 \text{ or } ab = 0,\\ 0 & \text{otherwise}, \end{array}\right. \\ a_3 &= b_1 \\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)(4q - 5) & \text{if } a + b = 1 \text{ or } ab = 0, \\ q^{m - 2}(q - 1)(4q - 8) & \text{otherwise}, \end{array}\right.\\ a_4 &= b_0 - (q - 1) \\ &= \left\{\begin{array}{ll} q^{m - 2}(q - 1)^2(q - 2) - (q - 1) & \text{if } a + b = 1 \text{ or } ab = 0, \\ q^{m - 2}(q - 1)(q^2 - 3q + 3) - (q - 1) & \text{otherwise}, \end{array}\right.\\ &= \left\{\begin{array}{ll} (q - 1)(q^m - 3q^{m - 1} + 2q^{m - 2} - 1) & \text{if } a + b = 1 \text{ or } ab = 0, \\ (q - 1)(q^m - 3q^{m - 1} + 3q^{m - 2} - 1) & \text{otherwise}, \end{array}\right.\\ a_2 &= |C \setminus \mathbb{F}_q\mathbf{1}| - (a_0 + a_1 + a_3 + a_4) \\ &= q^{m + 1} - q - (a_0 + a_1 + a_3 + a_4)\\ &= \left\{\begin{array}{ll} 3q^{m - 2}(q - 1) & \text{if } a + b = 1 \text{ or } ab = 0, \\ 6q^{m - 2}(q - 1) & \text{otherwise}. \end{array}\right.\end{aligned}$$ ◻ *Proof of Theorem [Theorem 6](#thm:4-design){reference-type="ref" reference="thm:4-design"} and Corollary [Corollary 7](#cor:4-gen-design){reference-type="ref" reference="cor:4-gen-design"}.* Using Lemmas [Lemma 15](#lem:4-design-rank3){reference-type="ref" reference="lem:4-design-rank3"}, [Lemma 16](#lem:4-design-rank1){reference-type="ref" reference="lem:4-design-rank1"}, and [Lemma 17](#lem:4-design-rank2){reference-type="ref" reference="lem:4-design-rank2"}, we obtain the desired results. ◻ # Acknowledgments {#acknowledgments .unnumbered} We are grateful to Professors Tsuyoshi Miezaki and Akihiro Munemasa for helpful discussions and comments on the manuscript. 999 E.F. Assmus, Jr. and H.F. Mattson, Jr., New $5$-designs, *J. Combin. Theory* **6** (1969), 122--151. T. Miezaki and A. Munemasa, Jacobi polnomials and harmonic weight enumerators of the first-order Reed-Muller codes and the extended hamming codes, *submitted* (2023). T. Miezaki, A. Munemasa, and H. Nakasora, A note on Assmus--Mattson type theorems, *Des. Codes Cryptogr.*, **89** (2021), 843--858. A. Bonnecaze, B. Mourrain, P. Solé, Jacobi polynomials, type II codes, and designs, *Des. Codes Cryptogr.* **16** (1999), no. 3, 215-234. H.S. Chakraborty, T. Miezaki, M. Oura, and Y. Tanaka, Jacobi polynomials and design theory I, *Discrete Mathematics* **346** (2023), 113339. M. Ozeki, On the notion of Jacobi polynomials for codes. *Math. Proc. Cambridge Philos. Soc.* **121** (1997), no. 1, 15--30. W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language, *J. Symb. Comp.* **24** (1997), 235--265. Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).

arxiv_math

--- author: - "Diego Chamorro[^1]" - "Gastón Vergara-Hermosilla[^2]" title: "**Lebesgue spaces with variable exponent: some applications to the Navier-Stokes equations**" --- \ **Keywords: Lebesgue spaces of variable exponent; Navier-Stokes equations; mild solutions.**\ **Mathematics Subject Classification: 35Q30; 76D05.** # Introduction We consider here the classical incompressible Navier-Stokes equations defined in the whole space $\mathbb{R}^3$: $$\label{NS_Intro} \begin{cases} \partial_t\vec{u}=\Delta \vec{u}-(\vec{u}\cdot \vec{\nabla}) \vec{u}-\vec{\nabla}P + \vec{f}, \qquad div(\vec{u})=0,\\[5pt] \vec{u}(0,x)=\vec{u}_0(x),\quad div(\vec{u}_0)=0, \qquad x\in \mathbb{R}^3, \end{cases}$$ where $\vec{u}:[0, +\infty[\times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ is the velocity field, $P:[0, +\infty[\times\mathbb{R}^3 \longrightarrow \mathbb{R}$ denotes the pressure and $\vec{u}_0:\mathbb{R}^3 \longrightarrow \mathbb{R}^3$, $\vec{f}:[0, +\infty[\times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ are a given initial data and a given exterior force, respectively.\ From the point of view of the existence of solutions we have at our disposal (at least) two main theories: *mild* solutions which are local in time (for any generic initial data $\vec{u}_0$) and *weak* (Leray) solutions which satisfy an energy inequality and are global in time.\ In this article we are concerned by the existence of *mild* solutions which are obtained with the help of a fixed-point theorem. Of course, in this theory the choice of a *good* functional setting is crucial: indeed, from the seminal work of Fujita-Kato [@fujita1964navier] (where classical Lebesgue spaces and Sobolev spaces were considered), many other functional spaces were used such as Fourier-Herz spaces [@cannone2012global; @le1997cascades; @lei2011global], Besov spaces [@cannone1994ondelettes; @lemarie2002recent], Morrey spaces [@giga1989navier; @kato1992strong; @taylor1992analysis], the $BMO^{-1}$ space [@KochTataru], etc. For a more complete review of possible functional spaces, see the book [@lemarie2018navier].\ The main feature of this work is to explore some existence results for equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) using as base spaces the Lebesgue spaces of variable exponent $L^{p(\cdot)}$, which -to the best of our knowledge- were not used before in the analysis of the Navier-Stokes equations. These spaces are quite different from the usual Lebesgue spaces $L^p$. Indeed, the parameter $p$ now is a function $p(\cdot):\mathbb{R}^3\longrightarrow [1,+\infty[$. To define the spaces $L^{p(\cdot)}$ we will proceed as follows: for a measurable function $\vec{f}:\mathbb{R}^3\longrightarrow \mathbb{R}^3$, we consider the *modular function* $\varrho_{p(\cdot)}$ associated to $p(\cdot)$, which is given by the expression $$\label{Def_Modular_Intro} \varrho_{p(\cdot)}(\vec{f})=\int_{\mathbb{R}^3}|\vec{f}(x)|^{p(x)}dx.$$ We note that if the function $p(\cdot)$ is constant (*i.e.* if $p(\cdot)\equiv p\in [1,+\infty[$) we obtain the classical Lebesgue spaces and we can derive from the modular function $\varrho_p$ a norm defined by $$\|\vec{f}\|_{L^{p}}=\left(\int_{\mathbb{R}^3}|\vec{f}(x)|^{p}dx\right)^{\frac1p}.$$ However, in the general case where $p(\cdot)$ is a measurable function, it is not possible replace in the previous formula the constant exponent $\frac1p$ outside the integral by $\frac{1}{p(\cdot)}$. To overcome this issue it is classical (see the books [@Cruz_Libro] and [@Diening_Libro]) to consider the *Luxemburg norm* $\|\cdot\|_{L^{p(\cdot)}}$ associated to the modular function $\varrho_{p(\cdot)}$, which is given by: $$\label{Def_LuxNormLebesgue} \|\vec{f}\|_{L^{p(\cdot)}}=\inf\{\lambda > 0: \, \varrho_{p(\cdot)}(\vec{f}/\lambda)\leq1\}.$$ We then define the Lebesgue spaces of variable exponent $L^{p(\cdot)}(\mathbb{R}^3)$ as the set of all the measurable functions such that the quantity $\|\cdot\|_{L^{p(\cdot)}}$ given above is finite (for more details we refer the reader to the previously cited books or to the Section [2](#Secc_Notaciones_Presentaciones){reference-type="ref" reference="Secc_Notaciones_Presentaciones"} below).\ In our results will thus study mild solutions for the Navier-Stokes system in the framework of the Lebesgue spaces of variable exponent. Indeed, in Theorem [Theorem 1](#Theoreme_1){reference-type="ref" reference="Theoreme_1"} below we will consider a particular variant of these functional spaces for the space variable $x\in \mathbb{R}^3$ and we will consider a classical $L^\infty$ space for the time variable $t>0$. In Theorem [Theorem 2](#Theoreme_2){reference-type="ref" reference="Theoreme_2"} we will work with a Lebesgue space of variable exponent in the time variable with a usual $L^q$ space in the space variable. In these two different cases we will be able to close the fixed point argument and to obtain mild solutions: these results give a first idea of what can be done in this framework of Lebesgue spaces of variable exponent. Moreover, since the $L^{p(\cdot)}$ spaces can not be easily related to the usual Lebesgue spaces, we expect that these results could offer a different point of view in some applications.\ The complete analysis of the Navier-Stokes equations is a deep and complex field and our aim here is to present a first application of the Lebesgue spaces of variable exponent to this topic. We thus hope that this work will help to attract the attention to these functional spaces.\ The outline of the article is the following. In Section [2](#Secc_Notaciones_Presentaciones){reference-type="ref" reference="Secc_Notaciones_Presentaciones"} we first present a small review of the main properties of the spaces $L^{p(\cdot)}$ and then we state our results. Section [3](#Secc_Proof_Existence){reference-type="ref" reference="Secc_Proof_Existence"} is devoted to the proof of the theorems. # Preliminaries and presentation of results {#Secc_Notaciones_Presentaciones} For $n\geq 1$, let us first consider a function $p:\mathbb{R}^n\longrightarrow [1,+\infty[$, we will say that $p\in \mathcal{P}(\mathbb{R}^n)$ if $p(\cdot)$ is a measurable function and we define $p^-=\underset{x\in \mathbb{R}^n}{\mbox{inf ess}} \; \{p(x)\}$ and $p^+=\underset{x\in \mathbb{R}^n}{\mbox{sup ess}} \; \{p(x)\}$. In order to distinguish between variable and constant exponents, we will always denote exponent functions by $p(\cdot)$, moreover, for the sake of simplicity and to avoid technicalities (see [@Diening_Libro Chapter 3]), we will always assume here that we have $$\label{BornePminPmax} 1<p^-\leq p^+<+\infty.$$ With these exponents we can consider the Luxemburg norm $\|\cdot\|_{L^{p(\cdot)}}$ as defined in ([\[Def_LuxNormLebesgue\]](#Def_LuxNormLebesgue){reference-type="ref" reference="Def_LuxNormLebesgue"}). These functional spaces $L^{p(\cdot)}(\mathbb{R}^n)$ possess some of the structural properties of normed spaces (they are moreover Banach function spaces) but they also present some very special features.\ In this setting, the Hölder inequalities have the following version: let $p(\cdot),\,q(\cdot),\,r(\cdot)\in \mathcal{P}(\mathbb{R}^n)$ be functions such that we have the pointwise relationship $\frac{1}{p(x)}=\frac{1}{q(x)}+\frac{1}{r(x)}$, $x\in \mathbb{R}^n$. Then there exists a constant $C>0$ such that for all $f\in L^{q(\cdot)}(\mathbb{R}^n)$ and $g \in L^{r(\cdot)}(\mathbb{R}^n)$, the pointwise product $fg$ belongs to the space $L^{p(\cdot)}(\mathbb{R}^n)$ and we have the estimate $$\label{Holder_LebesgueVar} \|f g\|_{L^{p(\cdot)}} \leq C\|f\|_{L^{q(\cdot)}}\|g\|_{L^{r(\cdot)}},$$ see [@Cruz_Libro Section 2.4] or [@Diening_Libro Section 3.2] for a proof of this fact. This estimate can be easily generalized to vector fields $\vec{f}, \vec{g}:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ and to the product $\vec{f}\cdot \vec{g}$.\ Note that the quantity $\|\cdot\|_{L^{p(\cdot)}}$ satisfies the *Norm conjugate formula* given in [@Diening_Libro Corollary 3.2.14]: $$\label{Norm_conjugate_formula} \|f\|_{L^{p(\cdot)}}\leq \underset{\|g\|_{L^{p'(\cdot)}}\leq 1}{\sup}\int_{\mathbb{R}^n}|f(x)||g(x)|dx\qquad \mbox{with } \frac{1}{p(\cdot)}+\frac{1}{p'(\cdot)}=1.$$ **Remark 1**. *Note that in the previous notions the space $\mathbb{R}^n$ can be replaced by an interval $[0,T]$.* It is crucial to remark now that the convolution product $f\ast g$ is not well adapted to the structure of the $L^{p(\cdot)}$ spaces, in particular the Young inequalities for convolution are not valid anymore (see [@Diening_Libro Section 3.6]) and thus many of the usual operators that appear in PDEs must be treated very carefully. Note also that Fourier-based methods are not so easy to use as we lack of an alternative for the Plancherel formula.\ To study the boundedness of such operators we first need to impose some conditions over the functions $p(\cdot)\in \mathcal{P}(\mathbb{R}^n)$: indeed, following [@Diening_Libro Section 4.1], we will say that a measurable function $p(\cdot)\in \mathcal{P}(\mathbb{R}^n)$ belongs to the class $\mathcal{P}^{log}(\mathbb{R}^n)$ if we have $$\left|\frac{1}{p(x)}-\frac{1}{p(y)}\right|\leq \frac{C}{\log(e+1/|x-y|)}\qquad \mbox{for all } x,y\in \mathbb{R}^n,$$ and if $$\left|\frac{1}{p(x)}-\frac{1}{p_\infty}\right| \leq \frac{C}{\log(e+|x|)}\qquad \mbox{for all } x\in \mathbb{R}^n,$$ where $$\frac{1}{p_\infty}=\underset{|x|\to +\infty}{\lim}\frac{1}{p(x)}.$$ With the condition $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^n)$ we have the following results. - For $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ a locally integrable function, the Hardy-Littlewood maximal function $\mathcal{M}$ is given by $\displaystyle{\mathcal{M}(f)(x)=\underset{B \ni x}{\sup } \;\frac{1}{|B|}\int_{B }|f(y)|dy}$ where $B$ is an open ball of $\mathbb{R}^n$. Thus, if $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^n)$ we have the estimate $$\label{MaximalFunc_LebesgueVar} \|\mathcal{M}(f)\|_{L^{p(\cdot)}}\leq C \|f\|_{L^{p(\cdot)}}.$$ See [@Diening_Libro Section 4.3] for a proof of this fact.\ - Note also that the usual Riesz transforms $(\mathcal{R}_j)_{1\leq j\leq n}$ defined formally in the Fourier level by $\widehat{\mathcal{R}_j(f)}(\xi)=-\frac{i\xi_j}{|\xi|}\widehat{f}(\xi)$ are also bounded in Lebesgue spaces of variable exponent and we have the inequality $$\label{Riesz_LebesgueVar} \|\mathcal{R}_j(f)\|_{L^{p(\cdot)}}\leq C \|f\|_{L^{p(\cdot)}},$$ with the conditions $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^n)$ and $1<p^-\leq p^+<+\infty$. See [@Diening_Libro Section 6.3].\ - Let us recall now that for $0<\sigma<n$, the Riesz potentials $\mathcal{I}_\sigma$ are defined by $$\label{Definition_RieszPotential} \mathcal{I}_\sigma(f)(x)=\int_{\mathbb{R}^n}\frac{|f(y)|}{|x-y|^{n-\sigma}}dy.$$ If $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^n)$ and if $0<\sigma<n/p^+$, then, following [@Diening_Libro Section 6.1], we have the inequality $$\label{PotentialRieszVariable0} \|\mathcal{I}_\sigma(f)\|_{L^{q(\cdot)}}\leq C\|f\|_{L^{p(\cdot)}},\qquad \mbox{with} \quad \frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\sigma}{n}.$$ This estimate introduces a very strong relationship between the parameters $p(\cdot)$ and $q(\cdot)$. In order to obtain some more freedom in the parameters (see Remark [Remark 3](#Rem_Riesz_MixedLebesgue){reference-type="ref" reference="Rem_Riesz_MixedLebesgue"} below) we will consider the mixed Lebesgue spaces $\mathcal{L}^{p(\cdot)}_\mathfrak{p}(\mathbb{R}^n)=L^{p(\cdot)}(\mathbb{R}^n)\cap L^{\mathfrak{p}}(\mathbb{R}^n)$ introduced in [@chamorro_paper_lpvar], where $1<\mathfrak{p}<+\infty$ is a constant exponent. These spaces that can be normed by the quantity $$\label{MixedLebesgue} \|\cdot\|_{\mathcal{L}^{p(\cdot)}_\mathfrak{p}}=\max\{\|\cdot\|_{L^{p(\cdot)}}, \|\cdot\|_{L^{\mathfrak{p}}}\}.$$ With the help of these spaces we have the following result. **Proposition 1**. *Let $1<\mathfrak{p}<+\infty$ be a constant exponent, $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^n)$ a variable exponent and fix a parameter $0<\sigma<\min\{n/p^+, n/\mathfrak{p}\}$. If $f\in \mathcal{L}^{p(\cdot)}_\mathfrak{p}(\mathbb{R}^n)$, then we have the inequality $$\label{Inegalite3} \|\mathcal{I}_\sigma(f)\|_{L^{\rho(\cdot)}}\leq C \|f\|_{\mathcal{L}^{p(\cdot)}_\mathfrak{p}},$$ where the function $\rho(\cdot)$ satisfies the following condition $$\label{Inegalite2Condition} \rho(\cdot)=\frac{np(\cdot)}{n-s\mathfrak{p}}.$$* See [@chamorro_paper_lpvar] for a proof. Note in particular that the index $\mathfrak{p}$ is not to related to $p^-$ or $p^+$ nor to $p(\cdot)$ and this inequality gives more flexibility in the indexes than the conditions of the estimate ([\[PotentialRieszVariable0\]](#PotentialRieszVariable0){reference-type="ref" reference="PotentialRieszVariable0"}). **Remark 2**. *Note that, by construction, the mixed spaces $\mathcal{L}^{p(\cdot)}_\mathfrak{p}$ inherit the properties of the spaces $L^{p(\cdot)}$ and $L^{\mathfrak{p}}$. In particular we have the Hölder inequality $\|\varphi\|_{\mathcal{L}^{p(\cdot)}_\mathfrak{p}}\leq \|\varphi\|_{\mathcal{L}^{q(\cdot)}_\mathfrak{q}}\|\varphi\|_{\mathcal{L}^{r(\cdot)}_\mathfrak{r}}$ with $\frac{1}{p(\cdot)}=\frac{1}{q(\cdot)}+\frac{1}{r(\cdot)}$ and $\frac{1}{\mathfrak{p}}=\frac{1}{\mathfrak{q}}+\frac{1}{\mathfrak{r}}$ and of course the Riesz transforms are also bounded in these spaces.\ * For more details on the Lebesgue spaces of variable exponent, on their inner structure as well as many other properties, see the books [@Cruz_Libro] and [@Diening_Libro].\ With these preliminaries, which were presented for the sake of generality in $\mathbb{R}^n$, we can state our first result about the existence of mild solutions for the 3D Navier-Stokes system: **Theorem 1** (Global Mild Solutions). *Consider a variable exponent $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^3)$, a divergence free initial data $\vec{u}_0\in \mathcal{L}^{p(\cdot)}_{3}(\mathbb{R}^3)$ (as defined in ([\[MixedLebesgue\]](#MixedLebesgue){reference-type="ref" reference="MixedLebesgue"}) above) and let $\vec{f}$ be a divergence free external force such that $\vec{f}=div(\mathbb{F})$ where $\mathbb{F}$ is a tensor such that $\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2}}(\mathbb{R}^3, L^\infty([0,T[))$. If the quantity $\|\vec{u}_0\|_{\mathcal{L}^{p(\cdot)}_{3}}+\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}$ is small, then the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) admits a unique, global mild solution in the space $\mathcal{L}^{p(\cdot)}_{3}\left(\mathbb{R}^3,L^\infty([0,T[)\right)$.* Some remarks are in order here. First note that as long as the initial data and the external force are small enough, then we can obtain a unique, global solution for the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) in the framework of Lebesgue spaces of variable exponent. This seems to be, to the best of our knowledge, the first application of this type of spaces in the analysis of these equations. Let us mention for the sake of completness that steady versions of some PDEs from fluid mechanics were studied in the Section 14.4 of the book [@Diening_Libro] but evolution problems, as the one considered here, require a different treatment. Remark also that global mild solution are frequently associated to small data, however the study of large initial data that could generate global mild solution is a completely (and hard) open problem. Finally, let us point out that the use of the mixed Lebesgue spaces of variable exponent $\mathcal{L}^{p(\cdot)}_{\mathfrak{p}}$ is essentially technical and it is driven by the lack of flexibility of the indexes that intervene in the boundedness of the Riesz transforms. See the Remark [Remark 3](#Rem_Riesz_MixedLebesgue){reference-type="ref" reference="Rem_Riesz_MixedLebesgue"} below for more details in this particular point.\ In the previous result, we have first analyzed the behavior of the solutions in the time variable (in a $L^\infty$ norm) and then we studied the information in the space variable in a mixed Lebesgue space of variable exponent. In our second theorem we will proceed in a different fashion: **Theorem 2** (Local Mild Solutions ). *Let $p(\cdot )\in \mathcal{P}^{\text{log}}(\mathbb{R}^3)$ with $p^->2$ and fix an index $q>3$ by the relationship $\frac{2}{p(\cdot)} + \frac{3}{q} < 1$. If $\vec{f}\in L^{p(\cdot)} \left( [0,+\infty[, L^{q} (\mathbb{R}^3) \right)$ is an exterior force and if $\vec{u}_0\in L^{q} (\mathbb{R}^3)$ is an initial data such that $div(\vec{u}_0)=0$, then there exist a time $0<T<+\infty$ and an unique local in time mild solution of the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$.* Note that, if we compare this result with Theorem [Theorem 1](#Theoreme_1){reference-type="ref" reference="Theoreme_1"}, we changed here the order of the variables: we first measure the information in the space variable and then we consider the information in the time variable. Although it is a slightly more popular way to construct mild solutions for evolutive PDEs, we face here the inexorable problem of the time of existence of such solutions: large initial conditions can be considered but then the time of existence will be very small.\ Note moreover that, if the exponent $p$ is constant, the condition $\frac{2}{p} + \frac{3}{q} < 1$ is common in the analysis of the Navier-Stokes system, however we should expect that the additional freedom given by the variable exponent $p(\cdot)$ would be used to deepen the study of these equations. # Mild solutions in Lebesgue Spaces of variable exponent {#Secc_Proof_Existence} We present here a first general approach to mild solutions for the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) in the setting of Lebesgue spaces of variable exponent. These mild solutions are obtained via the following classical result: **Theorem 3** (Banach-Picard contraction principle). *Let $(E,\|\cdot \|_E )$ a Banach space and consider $B: E \times E \longrightarrow E$ a bounded bilinear application: $$\|B(e,e)\|_{E}\leq C_B\|e\|_E\|e\|_E.$$ Given $e_0\in E$ such that $\|e_0\|_E \leq \delta$ with $0<\delta < \frac{1}{4C_B}$, then the equation $$e = e_0 - B(e,e),$$ admits an unique solution $e\in E$ which satsifies $\| e \|_E \leq 2 \delta$.* In order to apply this result to the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}) we need to get rid of the pressure $P$ and for this we apply to this system the Leray projector $\mathbb{P}$ defined by[^3] $\mathbb{P}(\vec{\varphi})=\vec{\varphi}+\vec{\nabla}\frac{1}{(-\Delta)}(\vec{\nabla}\cdot \vec{\varphi})$. Recall that we thus have $\mathbb{P}(\vec{\nabla}P)\equiv 0$ and if a vector field is divergence free we have the identity $\mathbb{P}(\vec{\varphi})= \vec{\varphi}$. Since $\vec{u}$ and $\vec{f}$ are divergence free, we obtain the equation $$\begin{cases} \partial_t\vec{u}=\Delta \vec{u}-\mathbb{P}(div(\vec{u}\otimes \vec{u}))+\vec{f}, \quad div(\vec{u})=0,\\[3mm] \vec{u}(0,x)=\vec{u}_0(x), \qquad x\in \mathbb{R}^3. \end{cases}$$ Now, due to the Dumahel formula, we can write this equation in the following form $$\label{NS_Integral} \vec{u}(t,x)=\mathfrak{g}_t\ast \vec{u}_0(x)+\int_{0}^t\mathfrak{g}_{t-s}\ast\vec{f}(s, x)ds-\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, x)ds,$$ where $\mathfrak{g}_t$ is the usual gaussian heat kernel.\ ## Proof of Theorem [Theorem 1](#Theoreme_1){reference-type="ref" reference="Theoreme_1"} {#proof-of-theorem-theoreme_1} The integral equation ([\[NS_Integral\]](#NS_Integral){reference-type="ref" reference="NS_Integral"}) above is now of the form considered in Theorem [Theorem 3](#BP_principle){reference-type="ref" reference="BP_principle"}: indeed, it is enough to set the quantity $$\mathfrak{g}_t\ast \vec{u}_0(x)+\int_{0}^t\mathfrak{g}_{t-s}\ast\vec{f}(s, x)ds,$$ as the term $e_0$ and to analyse the bilinear application $$B(\vec{u}, \vec{u})=\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, x)ds.$$ We will thus look for a mild solution of equation [\[NS_Integral\]](#NS_Integral){reference-type="eqref" reference="NS_Integral"} in the functional space $\mathcal{E}=\mathcal{L}^{p(\cdot)}_{3}(\mathbb{R}^3,L^\infty([0,T[))$ endowed with the norm $$\label{Norm_PointFixe} \|\cdot\|_{\mathcal{E}}=\max\{\|\cdot\|_{L^{p(\cdot)}_x(L^\infty_t)}, \|\cdot\|_{L^{3}_x(L^\infty_t)}\},$$ where $p(\cdot)\in \mathcal{P}^{\log}(\mathbb{R}^3)$ and $1<p^-\leq p^+<+\infty$. Now, following the argument presented in Theorem [Theorem 3](#BP_principle){reference-type="ref" reference="BP_principle"}, we will prove the following estimates $$\label{Control_Uo} \|\mathfrak{g}_t\ast \vec{u}_0\|_{\mathcal{E}}\leq C\|\vec{u}_0\|_{\mathcal{L}^{p(\cdot)}_{3}},$$ $$\label{Control_f_preambulo} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(\cdot, \cdot)ds\right\|_{\mathcal{E}} \leq C\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}.$$ and $$\label{Control_Nolineal} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(\cdot, \cdot)ds\right\|_{\mathcal{E}}\leq C_B\|\vec{u}\|_{\mathcal{E}}\|\vec{v}\|_{\mathcal{E}}.$$ Thus, if we have the condition $$C\big(\|\vec{u}_0\|_{\mathcal{L}^{p(\cdot)}_{3}}+\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}\big)<\frac{1}{4C_B},$$ then we can obtain a unique mild solution for the system ([\[NS_Integral\]](#NS_Integral){reference-type="ref" reference="NS_Integral"}).\ Now, we will deduce each one of the previous estimates ([\[Control_Uo\]](#Control_Uo){reference-type="ref" reference="Control_Uo"}), ([\[Control_f\_preambulo\]](#Control_f_preambulo){reference-type="ref" reference="Control_f_preambulo"}) and ([\[Control_Nolineal\]](#Control_Nolineal){reference-type="ref" reference="Control_Nolineal"}). - First we study the quantity ([\[Control_Uo\]](#Control_Uo){reference-type="ref" reference="Control_Uo"}) and for this we recall a classical lemma (see [@lemarie2018navier Lemma 7.4, Section 7.7]): **Lemma 1**. *If $\varphi$ is a radially decreasing function on $\mathbb{R}^3$ and $\vec{f}$ is a locally integrable function, then $$|(\varphi\ast\vec{f})(x)| \leq \|\varphi\|_{L^1} \mathcal{M} (\vec{f})(x),$$ where $\mathcal{M}$ is the Hardy-Littlewood maximal function.* Since the heat kernel $\mathfrak{g}_t$ is a radially decreasing function and since $\vec{u}_0$ is a locally integrable function, using the previous lemma we obtain the control $$\|\mathfrak{g}_t\ast \vec{u}_0(x)\|_{L^\infty_t}\leq C\mathcal{M}(\vec{u}_0)(x),$$ and we can write $$\begin{aligned} \|\mathfrak{g}_t\ast \vec{u}_0\|_{\mathcal{E}}&=&\max\{\|\mathfrak{g}_t\ast \vec{u}_0\|_{L^{p(\cdot)}_x(L^\infty_t)}, \|\mathfrak{g}_t\ast \vec{u}_0\|_{L^{3}_x(L^\infty_t)}\}\\ &\leq & C\max\{\|\mathcal{M}(\vec{u}_0)\|_{L^{p(\cdot)}}, \|\mathcal{M}(\vec{u}_0)\|_{L^{3}}\}.\end{aligned}$$ Since $p(\cdot)\in \mathcal{P}^{log}(\mathbb{R}^3)$, the estimate ([\[MaximalFunc_LebesgueVar\]](#MaximalFunc_LebesgueVar){reference-type="ref" reference="MaximalFunc_LebesgueVar"}) implies that the maximal function $\mathcal{M}$ is bounded in the Lebesgue space $L^{p(\cdot)}(\mathbb{R}^3)$ (this operator is also bounded in $L^3$). Then we obtain $$\|\mathfrak{g}_t\ast \vec{u}_0\|_{\mathcal{E}}\leq C\max\big\{\|\vec{u}_0\|_{L^{p(\cdot)}},\|\vec{u}_0\|_{L^{3}}\big\}\leq C\|\vec{u}_0\|_{\mathcal{L}^{p(\cdot)}_3},$$ which is the announced control ([\[Control_Uo\]](#Control_Uo){reference-type="ref" reference="Control_Uo"}). - To study the inequality [\[Control_f\_preambulo\]](#Control_f_preambulo){reference-type="eqref" reference="Control_f_preambulo"} we proceed as follows: since $\vec{f}=div(\mathbb{F})$ we can write $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right|\leq C\int_{0}^t\int_{\mathbb{R}^3}|\vec{\nabla}\mathfrak{g}_{t-s}(x-y)| |\mathbb{F}(s,y)|dyds,$$ and due to the decay properties of the heat kernel we obtain $$\begin{aligned} \left|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right|&\leq &C\int_{0}^t\int_{\mathbb{R}^3}\frac{1}{|t-s|^2+|x-y|^4} |\mathbb{F}(s,y)|dyds\\ &\leq &C\int_{\mathbb{R}^3}\int_{0}^t\frac{1}{|t-s|^2+|x-y|^4} |\mathbb{F}(s,y)|dsdy,\end{aligned}$$ where we have applied the Fubini Theorem. Moreover, since $\mathbb{F}\in \mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)$, one has $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right|\leq C\int_{\mathbb{R}^3}\int_{0}^t\frac{1}{|t-s|^2+|x-y|^4} ds \|\mathbb{F}(\cdot,y)\|_{L^\infty_t}dy,$$ and after an integration with respect to the time variable, it comes $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right|\leq C\int_{\mathbb{R}^3}\frac{1}{|x-y|^2} \|\mathbb{F}(\cdot,y)\|_{L^\infty_t}dy=\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(y),$$ where $\mathcal{I}_1$ is the Riesz potential defined in ([\[Definition_RieszPotential\]](#Definition_RieszPotential){reference-type="ref" reference="Definition_RieszPotential"}). We thus obtain $$\left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right\|_{L^\infty_t}\leq C\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(y).$$ Now, in order to reconstruct the $\mathcal{L}^{p(\cdot)}_3$ norm given in ([\[Norm_PointFixe\]](#Norm_PointFixe){reference-type="ref" reference="Norm_PointFixe"}), from the previous estimate we write $$\begin{aligned} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right\|_{L^{p(\cdot)}_x(L^\infty_t)}&\leq &C\left\|\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(\cdot)\right\|_{L^{p(\cdot)}_x}\\ \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right\|_{L^{3}_x(L^\infty_t)}&\leq &C\left\|\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(\cdot)\right\|_{L^{3}_x}.\end{aligned}$$ Thus by the Proposition [Proposition 1](#Proposition_RieszPotential){reference-type="ref" reference="Proposition_RieszPotential"} with $\sigma=1$, $\mathfrak{p}=\frac{3}{2}$ and $\rho(\cdot)=p(\cdot)$, we have the following estimate: $$\left\|\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(\cdot)\right\|_{L^{p(\cdot)}_x}\leq C\left\|\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t}\right\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}}=\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}.$$ Moreover, by the boundedness properties of the Riesz potentials in the usual Lebesgue spaces $L^3$ we obtain $$\left\|\mathcal{I}_1(\|\mathbb{F}(\cdot,\cdot)\|_{L^\infty_t})(\cdot)\right\|_{L^{3}_x}\leq C\left\|\|\mathbb{F}\|_{L^\infty_t}\right\|_{L^{\frac{3}{2}}_x}=\|\mathbb{F}\|_{L^{\frac{3}{2}}_x(L^\infty_t)}.$$ With these two estimates at hand, and following the definition of the norm $\|\cdot\|_{\mathcal{E}_T}$ given in ([\[Norm_PointFixe\]](#Norm_PointFixe){reference-type="ref" reference="Norm_PointFixe"}), we finally obtain $$\left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right\|_{\mathcal{E}}=\left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \vec{f}(s,x)ds\right\|_{\mathcal{L}^{p(\cdot)}_{3,x}(L^\infty_t)}\leq C\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}<+\infty.$$ - In order to establish the estimate ([\[Control_Nolineal\]](#Control_Nolineal){reference-type="ref" reference="Control_Nolineal"}), we first remark that, due to the properties of the Leray projector, we have the identities $$\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{v}))ds=\int_{0}^t\mathbb{P}\big(\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})\big)ds=\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right),$$ and as before, by the decay properties of the heat kernel, we obtain $$\begin{aligned} \left|\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})(s,x)ds\right|&\leq &C\int_{0}^t\int_{\mathbb{R}^3}|\vec{\nabla}\mathfrak{g}_{t-s}(x-y)|\vec{u}(s,y)| |\vec{u}(s,y)|dyds\\ &\leq & C\int_{\mathbb{R}^3}\int_{0}^t\frac{1}{|t-s|^2+|x-y|^4} |\vec{u}(s,y)| |\vec{u}(s,y)|dyds.\end{aligned}$$ Note that we have $|\vec{u}(t,x)|\leq \|\vec{u}(\cdot,x)\|_{L^\infty_t}$ and we can thus write $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})(s,x)ds\right|\leq C\int_{\mathbb{R}^3}\left(\int_{0}^t \frac{1}{|t-s|^2+|x-y|^4}ds\right)\|\vec{u}(\cdot,x)\|_{L^\infty_t}\|\vec{u}(\cdot,x)\|_{L^\infty_t}dy,$$ from which we deduce the estimate $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})(s,x)ds\right|\leq C\int_{\mathbb{R}^3}\frac{1}{|x-y|^2}\|\vec{u}(\cdot,x)\|_{L^\infty_t}\|\vec{u}(\cdot,x)\|_{L^\infty_t}dy.$$ Remark now that, using the definition of the Riesz potentials given in ([\[Definition_RieszPotential\]](#Definition_RieszPotential){reference-type="ref" reference="Definition_RieszPotential"}), the last term above can be written in the following form: $$\left|\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})(s,x)ds\right|\leq C\mathcal{I}_1\big(\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t}\big)(x),$$ and we obtain the estimate $$\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})(s,x)ds\right)\leq C\mathbb{P}\left(\mathcal{I}_1\big(\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t}\big)(x)\right).$$ In order to reconstruct the norm $\|\cdot\|_{\mathcal{E}_T}$ given in ([\[Norm_PointFixe\]](#Norm_PointFixe){reference-type="ref" reference="Norm_PointFixe"}) we write: $$\begin{aligned} \left\|\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right)\right\|_{L^{p(\cdot)}_x(L^\infty_t)}&\leq &C\|\mathcal{I}_1(\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t})\|_{L^{p(\cdot)}_x(L^\infty_t)}\\ \left\|\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right)\right\|_{L^{3}_x(L^\infty_t)}&\leq &C\|\mathcal{I}_1(\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t})\|_{L^{3}_x(L^\infty_t)},\end{aligned}$$ where we have used the fact that the Leray projector $\mathbb{P}$ is bounded in the Lebesgue space $L^3$ as well as in the Lebesgue space of variable exponent $L^{p(\cdot)}$ (since the Riesz transforms are bounded in such spaces). At this point we recall that the Riesz potential $\mathcal{I}_1$ satisfies $\|\mathcal{I}_1(\varphi)\|_{L^3}\leq C\|\varphi\|_{L^{\frac{3}{2}}}$ and following the Proposition [Proposition 1](#Proposition_RieszPotential){reference-type="ref" reference="Proposition_RieszPotential"} we have $\|\mathcal{I}_1(\varphi)\|_{L^{p(\cdot)}}\leq C \|\varphi\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2}}}$, so we can write, using the Hölder inequalities (see the Remark [Remark 2](#Rem_Holder_Mixed_Lebesgue_Var){reference-type="ref" reference="Rem_Holder_Mixed_Lebesgue_Var"} above): $$\begin{aligned} \left\|\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right)\right\|_{L^{p(\cdot)}_x(L^\infty_t)}&\leq & C \left\|\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t}\right\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2}}}\leq C\|\vec{u}\|_{\mathcal{L}^{p(\cdot)}_{3,x}(L^\infty_x)}\|\vec{u}\|_{\mathcal{L}^{p(\cdot)}_{3,x}(L^\infty_x)} \\ \left\|\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right)\right\|_{L^{3}_x(L^\infty_t)}&\leq &C\left\|\|\vec{u}\|_{L^\infty_t}\|\vec{u}\|_{L^\infty_t}\right\|_{L^{\frac{3}{2}}_x}\leq C\|\vec{u}\|_{L^{3}_x(L^\infty_t)}\|\vec{u}\|_{L^{3}_x(L^\infty_t)}.\end{aligned}$$ **Remark 3**. *It is worth noting here that it is the use of the mixed Lebesgue spaces of variable exponent $\mathcal{L}^{p(\cdot)}_{3}$ that allows us to obtain the inequality $\|\mathcal{I}_1(\varphi)\|_{L^{p(\cdot)}}\leq \|\varphi\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2}}}$ from which we will deduce the wished control by applying the Hölder inequalities. Indeed, if we only use the spaces $L^{p(\cdot)}$ and we look for an estimate of the form $\|\mathcal{I}_1(\varphi)\|_{L^{p(\cdot)}}\leq \|\varphi\|_{L^{\frac{p(\cdot)}{2}}}$, then the relationship ([\[PotentialRieszVariable0\]](#PotentialRieszVariable0){reference-type="ref" reference="PotentialRieszVariable0"}) will force us to have $p(\cdot)\equiv 3$ and this will cancel the interest of using Lebesgue spaces of variable exponent.* With all these estimates, and using the definition of the norm $\|\cdot\|_{\mathcal{E}}$ given in ([\[Norm_PointFixe\]](#Norm_PointFixe){reference-type="ref" reference="Norm_PointFixe"}) we finally can write $$\left\|\mathbb{P}\left(\int_{0}^t\mathfrak{g}_{t-s}\ast div(\vec{u}\otimes \vec{u})ds\right)\right\|_{\mathcal{E}}\leq C_B\|\vec{u}\|_{\mathcal{E}}\|\vec{u}\|_{\mathcal{E}}.$$ We have established the estimates ([\[Control_Uo\]](#Control_Uo){reference-type="ref" reference="Control_Uo"}), ([\[Control_f\_preambulo\]](#Control_f_preambulo){reference-type="ref" reference="Control_f_preambulo"}) and ([\[Control_Nolineal\]](#Control_Nolineal){reference-type="ref" reference="Control_Nolineal"}), thus in order to apply the fixed-point Theorem [Theorem 3](#BP_principle){reference-type="ref" reference="BP_principle"} we need the condition $$C\big(\|\vec{u}_0\|_{\mathcal{L}^{p(\cdot)}_{3}}+\|\mathbb{F}\|_{\mathcal{L}^{\frac{p(\cdot)}{2}}_{\frac{3}{2},x}(L^\infty_t)}\big)<\frac{1}{4C_B},$$ which is fulfilled by hypotheses as the initial data and the external force are assumed to be small, thus the Theorem [Theorem 1](#Theoreme_1){reference-type="ref" reference="Theoreme_1"} is proven. $\blacksquare$ ## Proof of Theorem [Theorem 2](#Theoreme_2){reference-type="ref" reference="Theoreme_2"} {#proof-of-theorem-theoreme_2} As we are interested in mild solutions for the integral problem ([\[NS_Integral\]](#NS_Integral){reference-type="ref" reference="NS_Integral"}), we will follow the general steps given in the Theorem [Theorem 3](#BP_principle){reference-type="ref" reference="BP_principle"} and here we will consider the following functional space $$E_{T}=L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right),$$ with $p(\cdot)\in \mathcal{P}^{log}([0,+\infty[)$, for some $0<T<+\infty$ to be determined later. This space is endowed with the following Luxemburg norm $$\label{Norm_PointFixe_2} \| \vec{\varphi}\|_{E_T}=\inf\left\{\lambda > 0: \,\int_0^{T}\left|\frac{ \| \vec{\varphi} (t,\cdot)\|_{L^{q}}}{\lambda}\right|^{p(t)} dt \leq1\right\}.$$\ Thus, in order to apply Theorem [Theorem 3](#BP_principle){reference-type="ref" reference="BP_principle"}, just as before we first need to establish the following estimates: $$\label{Control_Uo_Lplq} \|\mathfrak{g}_t * \vec{u}_0 \|_{ E_T}\leq C_1\| \vec{u}_0 \|_{L^{q}},$$ $$\label{Control_force_LpLq} \left\|\int_0^t\mathfrak{g}_{t-s} *\vec{f}(s, \cdot) ds\right\|_{E_T} \leq C_2\|\vec{f}\|_{L^1_t(L^q_x)},$$ and $$\label{Control_Nolineal_LpLq} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, \cdot)ds\right\|_{E_T} \leq C_B\|\vec{u}\|_{E_T}\|\vec{u}\|_{E_T}.$$ Note that the constants $C_1, C_2, C_B$ will depend on the time variable as we shall see later on. Each one of these estimates will be studied separately. - To deduce the inequality [\[Control_Uo_Lplq\]](#Control_Uo_Lplq){reference-type="eqref" reference="Control_Uo_Lplq"}, we simply write $\|\mathfrak{g}_t * \vec{u}_0 \|_{L^{q} (\mathbb{R}^3)} \leq \|\mathfrak{g}_t \|_{L^{1} (\mathbb{R}^3)}\| \vec{u}_0 \|_{L^{q} (\mathbb{R}^3)} = \| \vec{u}_0 \|_{L^{q} (\mathbb{R}^3)}$, and to continue, we will need the following result borrowed from the book [@Diening_Libro] (Lemma 3.2.12, Section 3.2). **Lemma 2**. *Let $p(\cdot)\in \mathcal{P}([0,+\infty[)$ with $1<p^-\leq p^+<+\infty$. Then $$\frac{1}{C} \min\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}\leq\|1\|_{L^{p(\cdot)}([0,T])} \leq C \max\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}.$$* Thus, with this result at hand, taking the $L^{p(\cdot)}$ norm in the time variable we have $$\label{Estimation_DonneeInitiale} \|\mathfrak{g}_t * \vec{u}_0 \|_{L^{p(\cdot)} _tL^{q}_x }\leq C\|\vec{u}_0\|_{L^{q} (\mathbb{R}^3)}\|1\|_{L^{p(\cdot)}([0,T])} \leq C \|\vec{u}_0\|_{L^{q} (\mathbb{R}^3)}\max\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}.$$ - Let us now prove the estimate [\[Control_force_LpLq\]](#Control_force_LpLq){reference-type="eqref" reference="Control_force_LpLq"}. We start by the usual $L^q$-norm in the space variable and we obtain $$\left\|\int_0^t\mathfrak{g}_{t-s} \ast \vec{f}(s, \cdot)ds\right\|_{L^{q}} \leq \int_0^t\|\mathfrak{g}_{t-s}\|_{L^1} \| \vec{f}(s, \cdot) \|_{L^{q}}ds \leq C\| \vec{f}\|_{L^1_t(L^{q}_x)}.$$ Proceeding as in the previous lines, *i.e.* taking the $L^{p(\cdot)}$-norm in the time variable and using the Lemma [Lemma 2](#Lemma_subset){reference-type="ref" reference="Lemma_subset"} above we can write $$\begin{aligned} \left\|\int_0^t\mathfrak{g}_{t-s} \ast \vec{f}(s, \cdot) ds\right\|_{L^{p(\cdot)}_t(L^{q}_x)}&\leq &C\left\|\| \vec{f}\|_{L^1_t(L^{q}_x)}\right\|_{L^{p(\cdot)}_t}\leq C\| \vec{f}\|_{L^1_t(L^{q}_x)}\|1\|_{L^{p(\cdot)}([0,T])}\notag\\ &\leq &C\| \vec{f}\|_{L^1_t(L^{q}_x)}\max\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}.\label{Estimation_ForceExterieure}\end{aligned}$$ - Let us study now the inequality [\[Control_Nolineal_LpLq\]](#Control_Nolineal_LpLq){reference-type="eqref" reference="Control_Nolineal_LpLq"}. We first take the $L^q$-norm in the space variable to obtain $$\left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, \cdot)ds \right\|_{L^q}\leq C\int_{0}^t\left\|\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, \cdot) \right\|_{L^q}ds,$$ by the properties of the Leray projector $\mathbb{P}$ and since it is a bounded operator in Lebesgue spaces $L^q$ for $1<q<+\infty$, we can write $$\left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, \cdot)ds \right\|_{L^q}\leq C\int_0^t \left\|\vec{\nabla}\mathfrak{g}_{t-s} * \vec{u}\otimes\vec{u}(s, \cdot)\right\|_{L^q} ds.$$ By the Young inequalities with $1+\frac{1}{q}=\frac{q-1}{q}+\frac{2}{q}$ we obtain $$\begin{aligned} \int_0^t \left\|\vec{\nabla}\mathfrak{g}_{t-s} * \vec{u}\otimes\vec{u}(s, \cdot)\right\|_{L^q} ds&\leq &\int_0^t \|\vec{\nabla}\mathfrak{g}_{t-s} \|_{L^{\frac{q}{q-1}}}\left\|\vec{u}\otimes\vec{u}(s, \cdot)\right\|_{L^{\frac{q}{2}}} ds\\ &\leq&C\int_0^t \frac{1}{(t-s)^{\frac{1}{2}+\frac{3}{2q}}}\|\vec{u}(s,\cdot)\|_{L^{q}}\|\vec{u}(s,\cdot)\|_{L^{q}} ds.\end{aligned}$$ We will take now the $L^{p(\cdot)}$-norm in the time variable and for this, we will proceed by duality. Thus, for $p'(\cdot)$ the associated conjugate exponent (*i.e.* $\frac{1}{p(\cdot)}+\frac{1}{p'(\cdot)}=1$) we have, by the norm conjugate formula ([\[Norm_conjugate_formula\]](#Norm_conjugate_formula){reference-type="ref" reference="Norm_conjugate_formula"}), the inequalities: $$\begin{aligned} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(s, \cdot)ds \right\|_{L_t^{p(\cdot)}(L_x^q)}&\leq&\left\|\int_0^t \frac{1}{(t-s)^{\frac{1}{2}+\frac{3}{2q}}}\left\|\vec{u}(s,\cdot)\right\|_{L^{q}}^2ds\right\|_{L_t^{p(\cdot)}([0,T])}\label{Identite_NormeDualite}\\ &\leq &\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1} \int_0^{T} \int_0^t \frac{|\psi(t)|}{|t-s|^{\frac{1}{2}+\frac{3}{2q}}}\left\|\vec{u}(s,\cdot)\right\|_{L^{q}}^2ds \,dt.\notag\end{aligned}$$ We write now, by the Fubini Theorem: $$\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^{T} \int_0^t \frac{|\psi(t)|}{|t-s|^{\frac{1}{2}+\frac{3}{2q}}}\left\|\vec{u}(s,\cdot)\right\|_{L^{q}}^2ds \,dt =\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^T\int_0^T\frac{1_{ \{ 0<s<t \} } |\psi(t)| }{|t-s|^{\frac{1}{2} + \frac{3}{2q} }}dt\|\vec{u}(s,\cdot) \|^2_{L^q}ds,$$ and extending the function $\psi(t)$ by zero if $t<0$ and if $t>T$, we can write $$\begin{aligned} \sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^{T} \int_0^t \frac{|\psi(t)|}{|t-s|^{\frac{1}{2}+\frac{3}{2q}}}\left\|\vec{u}(s,\cdot)\right\|_{L^{q}}^2ds \,dt =\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^T\left(\int_{-\infty}^{+\infty}\frac{|\psi(t)| }{|t-s|^{\frac{1}{2} + \frac{3}{2q} }}dt\right)\|\vec{u}(s,\cdot) \|^2_{L^q}ds\\ =\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^T\mathcal{I}_\sigma(|\psi|)(s)\|\vec{u}(s,\cdot)\|^2_{L^q}ds,\end{aligned}$$ where $\mathcal{I}_\sigma$ is the 1D Riesz potential given in ([\[Definition_RieszPotential\]](#Definition_RieszPotential){reference-type="ref" reference="Definition_RieszPotential"}) with $\sigma=\frac{1}{2}-\frac{3}{2q}<1$. **Remark 4**. *Note that the condition $0<\frac{1}{2}-\frac{3}{2q}<1$ which is needed here in order to use the Riesz potentials implies the constraint $3<q$.* We apply now the Hölder inequality with $1=\frac{1}{p(\cdot)}+\frac{1}{p(\cdot)}+\frac{1}{ \tilde{p}(\cdot)}$ to obtain the estimate $$\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\int_0^T\mathcal{I}_\sigma(|\psi|)(s)\|\vec{u}(s,\cdot)\|^2_{L^q}ds\leq C\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\|\mathcal{I}_\sigma(|\psi|)\|_{L_t^{\tilde{p}(\cdot)}}\Big\|\|\vec{u}(\cdot,\cdot)\|_{L_x^q}\Big\|_{L_t^{p(\cdot)}}\Big\|\|\vec{u}(\cdot,\cdot)\|_{L_x^q}\Big\|_{L_t^{p(\cdot)}}.$$ **Remark 5**. *The relationship $1=\frac{2}{p(\cdot)}+\frac{1}{ \tilde{p}(\cdot)}$ imposes the condition $p^->2$.* By the boundedness of the Riesz potentials in the Lebesgue spaces of variable exponent (see ([\[PotentialRieszVariable0\]](#PotentialRieszVariable0){reference-type="ref" reference="PotentialRieszVariable0"})) we obtain, $$\begin{aligned} \sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\|\mathcal{I}_\sigma(|\psi|)\|_{L_t^{\tilde{p}(\cdot)}}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\qquad\qquad\qquad\qquad\qquad\notag\\ \leq C\sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\|\psi\|_{L_t^{r(\cdot)}}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)},\label{Estimation_AvantInclusion}\end{aligned}$$ where $$\label{Indices_Riesz} \frac{1}{\tilde{p}(\cdot)}=\frac{1}{r(\cdot)}-\big(\frac{1}{2}-\frac{3}{2q}\big).$$ Note now that if $\frac{2}{p(\cdot)}+\frac{3}{q}<1$ (which is the case by the hypothesis of Theorem [Theorem 2](#Theoreme_2){reference-type="ref" reference="Theoreme_2"}), then by the previous identity since we have $\frac{1}{\tilde{p}(\cdot)}=1-\frac{2}{p(\cdot)}$ and $\frac{1}{p'(\cdot)}=1-\frac{1}{p(\cdot)}$, we easily deduce that $r(\cdot)<p'(\cdot)$. This particular relationship will help us to apply the following result **Lemma 3**. *For $n\geq 1$ and for a given bounded domain $X\subset \mathbb{R}^n$ consider two functions $p_1(\cdot), p_2(\cdot)\in \mathcal{P}(X)$ such that $1< p_1^+, \ p_2^+ <+\infty$. Then, $L^{p_2(\cdot)}(X)\subset L^{p_1(\cdot)}(X)$ if and only if $p_1(x)\leq p_2(x)$ almost everywhere. Furthermore, in this case we have that $$\|f\|_{L^{p_1(\cdot)} } \leq\left(1+\left| X \right|\right)\|f\|_{L^{p_2(\cdot)}}.$$* A proof of this lemma can be found in [@Cruz_Libro Corollary 2.48].\ Thus, if we apply this result to the case $r(\cdot)<p'(\cdot)$ in ([\[Estimation_AvantInclusion\]](#Estimation_AvantInclusion){reference-type="ref" reference="Estimation_AvantInclusion"}) we obtain (recall that we are working in the time interval $[0,T]$): $$\begin{aligned} \sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}\|\psi\|_{L_t^{r(\cdot)}}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\qquad\qquad\qquad\qquad\qquad\qquad\\ \leq \sup_{\| \psi \|_{L^{p'(\cdot)}}\leq 1}(1+T)\|\psi\|_{L_t^{p'(\cdot)}}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\\ \leq (1+T)\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}.\end{aligned}$$ Now, with all these estimates, getting back to ([\[Identite_NormeDualite\]](#Identite_NormeDualite){reference-type="ref" reference="Identite_NormeDualite"}) we obtain the following control $$\label{Estimation_ApplicationBilineaire} \left\|\int_{0}^t\mathfrak{g}_{t-s}\ast \mathbb{P}(div(\vec{u}\otimes \vec{u}))(\cdot, \cdot)ds \right\|_{L_t^{p(\cdot)}(L_x^q)}\leq C(1+T)\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}\|\vec{u}(\cdot,\cdot)\|_{L_t^{p(\cdot)}(L_x^q)}.$$ With the controls ([\[Estimation_DonneeInitiale\]](#Estimation_DonneeInitiale){reference-type="ref" reference="Estimation_DonneeInitiale"}), ([\[Estimation_ForceExterieure\]](#Estimation_ForceExterieure){reference-type="ref" reference="Estimation_ForceExterieure"}) and ([\[Estimation_ApplicationBilineaire\]](#Estimation_ApplicationBilineaire){reference-type="ref" reference="Estimation_ApplicationBilineaire"}) we have proven the estimates [\[Control_Uo_Lplq\]](#Control_Uo_Lplq){reference-type="eqref" reference="Control_Uo_Lplq"}, ([\[Control_force_LpLq\]](#Control_force_LpLq){reference-type="ref" reference="Control_force_LpLq"}) and [\[Control_Nolineal_LpLq\]](#Control_Nolineal_LpLq){reference-type="eqref" reference="Control_Nolineal_LpLq"}. Now, in order to close the fixed point argument we need the condition $$C \big(\|\vec{u}_0\|_{L^{q} (\mathbb{R}^3)}+\| \vec{f}\|_{L^1_t(L^{q}_x)}\big)\max\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}\leq\frac{C}{(1+T)},$$ which can be rewritten in the following manner: $$\|\vec{u}_0\|_{L^{q} (\mathbb{R}^3)}+\| \vec{f}\|_{L^1_t(L^{q}_x)}\leq\frac{C}{(1+T)\max\left\{T^{\frac{1}{p^{-}}}, T^{\frac{1}{p^{+}}}\right\}},$$ thus, if the initial data and the external force satisfy this estimate, we can obtain a unique (local in time) solution of the Navier-Stokes equations ([\[NS_Intro\]](#NS_Intro){reference-type="ref" reference="NS_Intro"}). This concludes the proof of Theorem [Theorem 2](#Theoreme_2){reference-type="ref" reference="Theoreme_2"}. $\blacksquare$ **Remark 6**. *Note that if in the estimate [\[Estimation_AvantInclusion\]](#Estimation_AvantInclusion){reference-type="eqref" reference="Estimation_AvantInclusion"} we aim to obtain $r(\cdot)=p'(\cdot)$, then by the relationship [\[Indices_Riesz\]](#Indices_Riesz){reference-type="eqref" reference="Indices_Riesz"} we will obtain the condition $\frac{2}{p(\cdot)} +\frac{3}{q} =1$ that forces the parameter $p(\cdot)$ to be constant.* #### **Conflict of interest.** On behalf of all authors, the corresponding author states that there is no conflict of interest.\ #### **Acknowledgements.** The second author is supported by the ANID postdoctoral program BCH 2022 grant No. 74220003. 10 M. Cannone, *Ondelettes, paraproduits et Navier-Stokes*, PhD thesis, Paris 9, 1994. M. Cannone and G. Wu, *Global well-posedness for Navier-Stokes equations in critical Fourier--Herz spaces*, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), pp. 3754--3760. D. Chamorro, *Mixed Sobolev-like inequalities in Lebesgue spaces of variable exponents and in Orlicz spaces*, Positivity, 26 (2022), pp. Paper No. 5, 21. D. V. Cruz-Uribe and A. Fiorenza, *Variable Lebesgue spaces: Foundations and harmonic analysis*, Springer Science & Business Media, 2013. L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, *Lebesgue and Sobolev spaces with variable exponents*, Springer, 2011. H. Fujita and T. Kato, *On the Navier-Stokes initial value problem I*, Archive for Rational Mechanics and Analysis, 16 (1964), pp. 269--315. Y. Giga and T. Miyakawa, *Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces*, Communications in partial differential equations, 14 (1989), pp. 577--618. L. Grafakos, *Classical and Modern Fourier Analysis*, Prentice Hall, 2004. T. Kato, *Strong solutions of the Navier-Stokes equation in morrey spaces*, Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society, 22 (1992), pp. 127--155. H. [Koch]{.smallcaps}, D. [Tataru]{.smallcaps}. *Well-posedness for the Navier--Stokes equations.* Adv. Math., 157:22--35, (2001). Y. Le Jan and A. S. Sznitman, *Cascades aléatoires et équations de Navier-Stokes*, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 324 (1997), pp. 823--826. Z. Lei and F.-H. Lin, *Global mild solutions of Navier-Stokes equations*, Communications on Pure and Applied Mathematics, 64 (2011), pp. 1297--1304. P. G. Lemarié-Rieusset, *Recent developments in the Navier-Stokes problem*, CRC press, 2002. P. G. Lemarié-Rieusset, *The Navier-Stokes problem in the 21st century*, CRC press, 2018. M. E. Taylor, *Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations*, Communications in Partial Differential Equations, 17 (1992), pp. 1407--1456. [^1]: *diego.chamorro\@univ-evry.fr* (corresponding author) [^2]: *gaston.vergarahermosilla\@univ-evry.fr* [^3]: Recall that the Leray projector $\mathbb{P}$ can also be defined in terms of the Riesz transforms: $\mathbb{P}(\vec{\varphi})=(Id_{3\times 3}-\vec{R}\otimes\vec{R})(\vec{\varphi})$ where $\vec{R}=(R_1, R_2, R_3)$ and $R_j$ is the $j$-th Riesz transform. Thus, as long as the Riesz transforms are bounded in a functional space $E$, then the Leray projector $\mathbb{P}$ is bounded in $E$ and we have $\|\mathbb{P}(\vec{\varphi})\|_E\leq C\|\vec{\varphi}\|_E$.

arxiv_math

--- abstract: | The author has shown that the category of analytic contravariant functors on $\mathbf{gr}$, the category of finitely-generated free groups, is equivalent to the category of left modules over the PROP associated to the Lie operad, working over $\mathbb{Q}$. This exploited properties of the polynomial filtration of the category of contravariant functors on $\mathbf{gr}$. The first purpose of this paper is to strengthen the corresponding result for *covariant* functors on $\mathbf{gr}$. This involves introducing the appropriate analogue of the category of analytic contravariant functors, namely a certain category of towers of polynomial functors on $\mathbf{gr}$. This category is abelian and has a natural symmetric monoidal structure induced by the usual tensor product of functors. Moreover, the projective generators of this category are described in terms of the *Mal'cev* functors that are introduced here. It follows that this category is equivalent to the category of right modules over the PROP associated to the Lie operad. As a fundamental example, the Passi functors arising from the group ring functors are described explicitly. The theory is applied to consider bifunctors on $\mathbf{gr}$. This allows the $\mathbb{Q}$-linearization of the category of free groups to be described, up to polynomial filtration. As a stronger application of the theory, this is generalized to the Casimir PROP associated to the Lie operad, as studied by Hinich and Vaintrob. Up to polynomial filtration, this recovers the category $\mathbf{A}$ introduced by Habiro and Massuyeau in their study of bottom tangles in handlebodies. author: - Geoffrey Powell title: On the Passi and the Mal'cev functors --- # Introduction {#sect:intro} For $G$ a discrete group and $\mathbbm{k}$ a unital commutative ring, one can consider the group ring $\mathbbm{k}G$ and its augmentation ideal $\mathcal{I}G$. This leads to the filtration of $\mathbbm{k}G$ by powers of the augmentation ideal $\mathcal{I}^n G$, for $n \in \mathbb{N}$, and the quotient rings $\mathbbm{k}G / \mathcal{I}^n G$; their underlying functors to $\mathbbm{k}$-modules are sometimes termed the Passi functors in reference to [@MR537126], for example. These constructions are natural with respect to the group $G$; here we usually restrict to the full subcategory $\mathbf{gr}$ of the category of groups with objects the finite-rank free groups. Forgetting the multiplicative structure, it is still interesting to ask what is the structure of these functors from $\mathbf{gr}$ to $\mathbbm{k}$-modules, denoted $\mathcal{F}(\mathbf{gr})$. For example, the associated graded of the filtration of $G \mapsto \mathbbm{k}G$ by powers of the augmentation ideal is $$\bigoplus_{n \in \mathbb{N}} \mathfrak{a}^{\otimes n}$$ where $\mathfrak{a}$ is the functor $G \mapsto \mathbbm{k}\otimes _\mathbb{Z}G_\mathsf{ab}$, $G_\mathsf{ab}$ the abelianization of $G$. This filtration does not split, and understanding its structure is a key to analysing the structure of the category of functors from $\mathbf{gr}$ to $\mathtt{Mod}_\mathbbm{k}$. Moreover, the filtration can be constructed without using the multiplicative structure of $\mathbbm{k}G$ by using the *polynomial filtration*, based on a generalization of the notion of polynomial functor due to Eilenberg and Mac Lane [@MR65162]. Indeed, the functor $\mathfrak{a}^{\otimes n}$ is a basic example of a polynomial functor of degree $n$. One can go further, replacing the group ring functor by $P_\Gamma : G \mapsto \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(\Gamma, G)$, where $\Gamma$ is a finite rank free group. The notation $P_\Gamma$ reflects the fact that it is the projective that corepresents evaluation on $\Gamma$. Moreover, $P_\Gamma$ is contravariantly functorial with respect to $\Gamma$, leading to the consideration of the bifunctor $P_\bullet$, a functor from $\mathbf{gr}^\mathrm{op}\times \mathbf{gr}$ to $\mathbbm{k}$-modules. One of the original motivations for this work was to analyse this bifunctor by exploiting the polynomial filtration. This is related to studying the $\mathbbm{k}$-linearization $\mathbbm{k}\mathbf{gr}$ of the category $\mathbf{gr}$, since $P_\bullet$ encodes the morphisms of this category. Specializing to $\mathbbm{k}= \mathbb{Q}$, the question can be rephrased using the results of [@2021arXiv211001934P], which give a model for the category $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ of *analytic* functors on $\mathbf{gr}^\mathrm{op}$ in terms of the 'infinitesimal structure', namely the action of the $\mathbb{Q}$-linear category $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ associated to the Lie operad. Namely, $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ is equivalent to the category of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules (this theory is reviewed and revisited here in Section [9](#sect:catlie){reference-type="ref" reference="sect:catlie"}). Here the case of *covariant* functors is developed, improving upon the result established in [@2021arXiv211001934P], which only treated polynomial functors with a finite composition series. The solution is to work with pro-polynomial functors, exploiting properties of the polynomial filtration: the appropriate analogue of $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ is the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ introduced in Section [4](#sect:propoly){reference-type="ref" reference="sect:propoly"}. This has objects given by towers $$\ldots \twoheadrightarrow F_d \twoheadrightarrow F_{d-1} \twoheadrightarrow \ldots \twoheadrightarrow F_0 \twoheadrightarrow F_{-1} =0,$$ where $F_d$ has polynomial degree $d$ and the structure map $F_d \rightarrow F_{d-1}$ induces an isomorphism $\mathsf{q}^\mathbf{gr}_{d-1} F_d \cong F_{d-1}$, where $\mathsf{q}^\mathbf{gr}_{d-1}$ is the $(d-1)$st Taylor functor that yields the universal quotient of polynomial degree $d-1$. Moreover, ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is shown to be abelian and is equipped with a symmetric monoidal structure $\overline{\otimes}$ induced by the tensor product on $\mathcal{F}(\mathbf{gr})$. The category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ can be considered as being 'dual' to $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$; in particular, vector space duality induces an adjunction: $${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightleftarrows \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}).$$ The category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is related to $\mathcal{F}(\mathbf{gr})$ by the completion functor $\mathfrak{c}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ given by passage to the inverse limit. This is right adjoint to the functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ that sends a functor $F$ to the tower provided by its polynomial filtration, $(\mathsf{q}^\mathbf{gr}_\bullet F)$. For example, taking $P_\mathbb{Z}: G \mapsto \mathbb{Q}G$ as above, $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ encodes the tower of Passi functors; applying the completion functor yields the completed group ring functor $G \mapsto \widehat{\mathbb{Q}G}$. The adjunction unit is the natural map $\mathbb{Q}G \rightarrow \widehat{\mathbb{Q}G}$, which is injective when $G$ is a finite rank free group. For current purposes, it is more natural to work at the level of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, without passing to the completion, since one needs to retain the information on the polynomial filtration, anyway. The key ingredient is that ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ has enough projectives; these are introduced in Section [6](#sect:malcev){reference-type="ref" reference="sect:malcev"}. The fundamental object is $\mathfrak{malcev}\in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, which is shown to be the projective cover of $\mathfrak{a}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ (see Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}). The functor $\mathfrak{malcev}$ is the counterpart in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ of $\mathscr{P}\widehat{\mathbb{Q}G}$, the primitives of the completed group ring, as considered by Quillen [@MR258031 Appendix A]. In particular, $\mathfrak{malcev}$ is a Lie algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. For each $s \in \mathbb{N}$, one has $\mathfrak{malcev}^{\overline{\otimes}s}\in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. By Corollary [Corollary 105](#cor:malcev_s_proj_generators){reference-type="ref" reference="cor:malcev_s_proj_generators"}, $\{ \mathfrak{malcev}^{\overline{\otimes}s} \ | \ s\in \mathbb{N}\}$ is a set of projective generators for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Moreover, using the Lie algebra structure of $\mathfrak{malcev}$, these assemble to a left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module, denoted by $\underline{\mathfrak{malcev}}$. This structure encodes *all* the morphisms between the above projective generators, by Proposition [Proposition 132](#prop:malcev_full_subcat){reference-type="ref" reference="prop:malcev_full_subcat"}. Then, denoting the category of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules by $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, one has the following: **Theorem 1**. *(Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}.) The functor $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -)$ induces an equivalence of categories $$\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -) \ : \ {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\stackrel{\cong}{\rightarrow} \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}.$$* *The inverse equivalence is given by $- \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \ : \ \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* This is the counterpart of the result of [@2021arXiv211001934P] establishing the equivalence between $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ and the category ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$ of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. The latter result is reformulated here, based on the following observation: starting from the Lie operad $\mathfrak{Lie}$, one can form the universal enveloping algebra $U \mathfrak{Lie}$, which is a unital associative algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. This has the structure of a cocommutative Hopf algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules, so that one can form the exponential functor $\Phi U\mathfrak{Lie}$, a functor from $\mathbf{gr}^\mathrm{op}$ to $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. The equivalence can be restated as: **Theorem 2**. *(Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"}.) The functor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} -$ induces an equivalence of categories $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} - : {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod} \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}).$$* These results are then combined to study bifunctors on $\mathbf{gr}$, i.e., functors from $\mathbf{gr}^\mathrm{op}\times \mathbf{gr}$ to $\mathbb{Q}$-vector spaces. The category of bifunctors is denoted $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$; it is equivalent to $\mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$, the category of functors from $\mathbf{gr}^\mathrm{op}$ to $\mathcal{F}(\mathbf{gr})$. Clearly one cannot expect to recover *all* bifunctors by the current techniques derived from polynomial approximation. Instead, one uses $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$, the category of analytic functors from $\mathbf{gr}^\mathrm{op}$ to ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. This category will (abusively) be referred to as the category of (pro)bifunctors; it contains as a full subcategory the category of bifunctors that are polynomial in the appropriate sense. Moreover, it comes equipped with a symmetric monoidal structure induced by $\overline{\otimes}$ on ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The category of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules is written ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$; this is equipped with a symmetric monoidal structure based on the convolution product for $\mathbb{Q}{\bm{\Sigma}}$-bimodules (where ${\bm{\Sigma}}$ is the category of finite sets and bijections). The main result of Section [11](#sect:bifunctors){reference-type="ref" reference="sect:bifunctors"} is: **Theorem 3**. *(Theorem [Theorem 178](#thm:bifunctors_sym_monoidal){reference-type="ref" reference="thm:bifunctors_sym_monoidal"}.) The functor $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}( - )\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \ : \ {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$$ is a symmetric monoidal equivalence.* This is applied in Section [12](#sect:model_qgr_pbif){reference-type="ref" reference="sect:model_qgr_pbif"} to study the $\mathbb{Q}$-linearization $\mathbb{Q}\mathbf{gr}$ of $\mathbf{gr}$. More precisely, in Section [3](#sect:polyq){reference-type="ref" reference="sect:polyq"}, it is shown that the polynomial filtration induces a tower of categories under $\mathbb{Q}\mathbf{gr}$ (see Theorem [Theorem 45](#thm:tower){reference-type="ref" reference="thm:tower"}): $$\xymatrix{ \mathbb{Q}\mathbf{gr} \ar@{.>}[d] \ar[rd] \ar[rrd] \ar@{.>}[rrrd] \ar[rrrrd] \\ \ldots \ar[r] & \mathsf{q}^\mathbf{gr}_d \mathbb{Q}\mathbf{gr} \ar[r] & \mathsf{q}^\mathbf{gr}_{d-1} \mathbb{Q}\mathbf{gr} \ar[r] & \ldots \ar[r] & \mathsf{q}^\mathbf{gr}_0 \mathbb{Q}\mathbf{gr}. }$$ Likewise, there is a tower of $\mathbb{Q}$-linear categories under $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$: $$\xymatrix{ \mathbf{Cat}\hspace{1pt}\mathfrak{Lie} \ar@{.>}[d] \ar[rd] \ar[rrd] \ar@{.>}[rrrd] \ar[rrrrd] \\ \ldots \ar[r] & \mathbf{Cat}\hspace{1pt}^{\leq d+1} \mathfrak{Lie} \ar[r] & \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie} \ar[r] &\ldots \ar[r] & \mathbf{Cat}\hspace{1pt}^{\leq 0} \mathfrak{Lie}. }$$ These are related: **Theorem 4**. *(Theorem [Theorem 186](#thm:tower_isomorphism){reference-type="ref" reference="thm:tower_isomorphism"}.) Under the equivalence of Theorem [Theorem 3](#THM:bimod){reference-type="ref" reference="THM:bimod"}, the tower $$\ldots \rightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\rightarrow \mathbf{Cat}\hspace{1pt}^{\leq d-1} \mathfrak{Lie}\rightarrow \ldots \rightarrow \mathbf{Cat}\hspace{1pt}^{\leq 0} \mathfrak{Lie}$$ corresponds to the tower $$\ldots \rightarrow \mathsf{q}^\mathbf{gr}_{d+1}\mathbb{Q}\mathbf{gr}\rightarrow \mathsf{q}^\mathbf{gr}_d \mathbb{Q}\mathbf{gr}\rightarrow \ldots \rightarrow \mathsf{q}^\mathbf{gr}_0 \mathbb{Q}\mathbf{gr}.$$* This is the starting point for analysing the more general situation in which $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ is replaced by the PROP $\mathbb{P}$ that encodes Casimir Lie algebras, as described in the work of Hinich and Vaintrob [@MR1913297]; this relies on the fact that the operad $\mathfrak{Lie}$ has a cyclic structure. The PROP $\mathbb{P}$ is $\mathbb{N}$-graded and $\mathbb{P}_0$ identifies with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$; this means that each $\mathbb{P}_n$ has a natural $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule structure and the composition in $\mathbb{P}$ is compatible with these bimodule structures. One can thus form the associated (pro)bifunctors: $$\mathfrak{A}(n) := \Phi U \mathfrak{Lie}\otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbb{P}_n \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$$ in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$, for each $n \in\mathbb{N}$. Then, applying the general theory developed here, the composition in $\mathbb{P}$ induces natural transformations: $$\mathfrak{A}(n) \otimes_\mathbf{gr}\mathfrak{A}(m) \rightarrow \mathfrak{A}(m+n)$$ for $m,n \in \mathbb{N}$. Unlike the situation of Theorem [Theorem 4](#THM:tower_iso){reference-type="ref" reference="THM:tower_iso"}, this cannot be analysed one polynomial degree at a time; the more general framework proposed here is necessary. These structures are the basis of the following result: **Theorem 5**. *(Theorem [Theorem 209](#thm:ajac){reference-type="ref" reference="thm:ajac"}.) The (pro)bifunctors $\mathfrak{A}(*)$ have the structure of a unital, $\mathbb{N}$-graded associative monoid in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$, with $\mathfrak{A}(0)$ isomorphic to $(\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr})^\mathrm{op}$ considered as a monoid.* Part of the interest of this result is that $\mathfrak{A}(*)$ is related to Habiro and Massuyeau's category $\mathbf{A}$ [@MR4321214], introduced in their work on bottom tangles in handlebodies. By definition, $\mathrm{Ob}\hspace{2pt}\mathbf{A}= \mathbb{N}$ and $\mathbf{A}$ is $\mathbb{N}$-graded. The category $\mathbf{A}$ is equipped with an embedding $\mathbb{Q}\mathbf{gr}^\mathrm{op}\hookrightarrow \mathbf{A}$ that sends the free group $\mathbb{Z}^{\star r}$ to the object $r \in \mathbb{N}$ and induces an equivalence $\mathbb{Q}\mathbf{gr}^\mathrm{op}\cong \mathbf{A}_0$. In particular, for each grading $n \in \mathbb{N}$, $\mathbf{A}_n(-,-)$ is a bifunctor in $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$. Applying the functor $\mathsf{q}^\mathbf{gr}_\bullet$ gives $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_n (-, -) \in \mathcal{F}_\omega(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$ The composition of $\mathbf{A}$ induces an associative composition operation: $$\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A} \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A} \rightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}$$ making $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}$ into an associative monoid with unit $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_0 \cong (\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr})^\mathrm{op}$. A proof of the following is sketched in Section [13.5](#subsect:sketchy){reference-type="ref" reference="subsect:sketchy"}: **Theorem 6**. *(Theorem [Theorem 210](#thm:ajac_versus_A){reference-type="ref" reference="thm:ajac_versus_A"}.) [\[THM:ajac_HM\]]{#THM:ajac_HM label="THM:ajac_HM"} There is an isomorphism $\mathfrak{A}(*) \cong \mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}$ of $\mathbb{N}$-graded associative monoids in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$.* The significance of this result is the following: the construction of $\mathfrak{A}(*)$ is direct, with the composition following directly from that of $\mathbb{P}$. However, the construction does not give natural 'models' for the $\mathfrak{A}(*)$ as bifunctors (without passing to completion). By contrast, the construction of the composition in $\mathbf{A}$ given in [@MR4321214 Section 4] may appear somewhat complex on first acquaintance. However, that construction has the advantage of furnishing *genuine* bifunctors, rather than (pro)bifunctors. ## Notation and conventions {#subsect:nota_conv} Throughout $\mathbbm{k}$ is a unital commutative ring (for the general theory, this can be taken to be $\mathbb{Z}$; for the applications it will be the field $\mathbb{Q}$ of rational numbers); $\mathtt{Mod}_\mathbbm{k}$ denotes the category of $\mathbbm{k}$-modules, considered as symmetric monoidal with respect to $\otimes_\mathbbm{k}$ (this will be written $\otimes$ when no confusion can result). *Notation 1*. Denote by - $\mathbf{gr}$ the category of finitely-generated free groups, considered as a full subcategory of the category of groups; - ${\bm{\Sigma}}$ the category of finite sets and bijections; - $\mathbf{n}$, for $n \in \mathbb{N}$, the finite set $\{1, \ldots , n\}$ (with $\mathbf{0} = \emptyset$, by convention), so that ${\bm{\Sigma}}$ has small skeleton with objects $\{ \mathbf{n}\ | \ n \in \mathbb{N}\}$; - $\mathfrak{S}_n$ the symmetric group on $n$ letters (aka. $\mathrm{Aut}(\mathbf{n})$), for $n \in \mathbb{N}$ . For an essentially small category $\mathcal{C}$, $\mathcal{F}(\mathcal{C})$ denotes the category of functors from $\mathcal{C}$ to $\mathtt{Mod}_\mathbbm{k}$. This is abelian, equipped with the pointwise tensor product induced by $\otimes_\mathbbm{k}$. (The category $\mathcal{F}(\mathcal{C})$ is sometimes also referred to as the category of $\mathcal{C}$-modules.) More generally, for any category $\mathcal{E}$, $\mathcal{F}(\mathcal{C}; \mathcal{E})$ denotes the category of functors from $\mathcal{C}$ to $\mathcal{E}$, so that $\mathcal{F}(\mathcal{C})$ is shorthand for $\mathcal{F}(\mathcal{C}; \mathtt{Mod}_\mathbbm{k})$. If $\mathcal{E}$ is abelian, then so is $\mathcal{F}(\mathcal{C}; \mathcal{E})$; moreover, if $\mathcal{E}$ is symmetric monoidal, then so is $\mathcal{F}(\mathcal{C}; \mathcal{E})$ for the induced pointwise structure. For small categories $\mathcal{C}, \mathcal{D}$, the exterior tensor product is denoted $\boxtimes : \mathcal{F}(\mathcal{C}) \times \mathcal{F}(\mathcal{D}) \rightarrow \mathcal{F}(\mathcal{C}\times \mathcal{D})$, given for $F_1 \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathcal{C})$ and $F_2 \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathcal{D})$ by $F_1 \boxtimes F_2 : (X,Y) \mapsto F_1 (X) \otimes F_2 (Y)$. Taking $\mathcal{C}= \mathcal{D}^\mathrm{op}$, the composite of $\boxtimes$ with the coend construction is denoted: $$\otimes_\mathcal{D}: \mathcal{F}(\mathcal{D}^\mathrm{op}) \times \mathcal{F}(\mathcal{D}) \rightarrow \mathtt{Mod}_\mathbbm{k}.$$ *Remark 2*. The category $\mathcal{F}(\mathcal{C})$ will also be referred to here as the category of $\mathbbm{k}\mathcal{C}$-modules, where $\mathbbm{k}\mathcal{C}$ is the $\mathbbm{k}$-linear category constructed from $\mathcal{C}$. In the literature, the terminology $\mathcal{C}$-modules is also used for this; this is avoided here since we also work with modules over $\mathbbm{k}$-linear categories which are not of the form $\mathbbm{k}\mathcal{C}$. **Example 3**. The category $\mathcal{F}({\bm{\Sigma}})$ (or $\mathbbm{k}{\bm{\Sigma}}$-modules) is equivalent to the category of sequences $M(n)$ of left $\mathbbm{k}\mathfrak{S}_n$-modules, for $n \in \mathbb{N}$, and equivariant maps. Likewise, the category $\mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ is equivalent to the category of sequences of right $\mathbbm{k}\mathfrak{S}_n$-modules. The isomorphism ${\bm{\Sigma}}\cong {\bm{\Sigma}}^\mathrm{op}$ of categories (provided by the passage to inverse morphisms for the groupoid ${\bm{\Sigma}}$) induces the isomorphism $\mathcal{F}({\bm{\Sigma}}) \cong \mathcal{F}({\bm{\Sigma}}^\mathrm{op})$. This corresponds (for each $n \in \mathbb{N}$) to the isomorphism of categories between left $\mathbbm{k}\mathfrak{S}_n$-modules and right $\mathbbm{k}\mathfrak{S}_n$-modules obtained using the group isomorphism $\mathfrak{S}_n \cong \mathfrak{S}_n^\mathrm{op}$ provided by the inverse $g \mapsto g^{-1}$. Given two ${\bm{\Sigma}}$-modules $M_1$, $M_2$, by the above, one can consider $M_1$ as a ${\bm{\Sigma}}^\mathrm{op}$-module and then form $M_1 \otimes_{\bm{\Sigma}}M_2$. This is given explicitly by $$M_1 \otimes_{\bm{\Sigma}}M_2 = \bigoplus_{n \in \mathbb{N}} M_1 (n) \otimes_{\mathfrak{S}_n} M_2 (n).$$ *Remark 4*. The passage between $\mathcal{F}({\bm{\Sigma}})$ and $\mathcal{F}({\bm{\Sigma}}^\mathrm{op})$-modules as above will be used freely. Post-composing with the $\mathbbm{k}$-linear duality functor $(-)^\sharp := \mathrm{Hom}_{\mathtt{Mod}_\mathbbm{k}} (-, \mathbbm{k})$ yields the functors $\mathcal{F}(\mathcal{C}) ^\mathrm{op}\rightarrow \mathcal{F}(\mathcal{C}^\mathrm{op})$ and $\mathcal{F}(\mathcal{C}^\mathrm{op}) ^\mathrm{op}\rightarrow \mathcal{F}(\mathcal{C})$, both denoted by $D$, since context should make clear what is intended. If $\mathbbm{k}$ is a field, these are adjoint and restrict to equivalences between the respective full subcategories of functors taking finite-dimensional values. ## Organization of the paper An overview of functors on $\mathbf{gr}$ and $\mathbf{gr}^\mathrm{op}$ is provided in Section [2](#sect:gr){reference-type="ref" reference="sect:gr"} and this is illustrated in Section [3](#sect:polyq){reference-type="ref" reference="sect:polyq"}, where the polynomial filtration of the category $\mathbbm{k}\mathbf{gr}$ is introduced. Section [4](#sect:propoly){reference-type="ref" reference="sect:propoly"} introduces the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ that serves to encode the polynomial filtration of functors on $\mathbf{gr}$. This is applied in Section [5](#sect:group-ring){reference-type="ref" reference="sect:group-ring"}, where the group ring is considered as a functor on $\mathbf{gr}$. This feeds into the definition of the *Mal'cev* functors in Section [6](#sect:malcev){reference-type="ref" reference="sect:malcev"} and their tensor products. Section [7](#sect:opd){reference-type="ref" reference="sect:opd"} is an interlude to introduce background on operadic structures. This is used in Section [8](#sect:endo_malcev){reference-type="ref" reference="sect:endo_malcev"} to relate ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. Section [9](#sect:catlie){reference-type="ref" reference="sect:catlie"} revisits the relationship between left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and analytic functors on $\mathbf{gr}^\mathrm{op}$. Section [10](#sect:compare){reference-type="ref" reference="sect:compare"} then shows how this is related to the framework for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ introduced here. These results are put together in Section [11](#sect:bifunctors){reference-type="ref" reference="sect:bifunctors"} to treat bifunctors. Finally, Section [12](#sect:model_qgr_pbif){reference-type="ref" reference="sect:model_qgr_pbif"} and Section [13](#sect:modular){reference-type="ref" reference="sect:modular"} apply the theory to the two fundamental examples. ## Acknowledgement The author is grateful to Christine Vespa, who drew Habiro and Massuyeau's work [@MR4321214] and Katada's work [@2021arXiv210206382K] to his attention, in relation to the study of functors on $\mathbf{gr}$. This formed one of the original motivations for studying the relationship between bifunctors and $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules. Moreover, the basic results on the polynomial filtration of functors on $\mathbf{gr}$ were first considered in the joint work with Vespa, [@PV]. # Background on functors on $\mathbf{gr}$ {#sect:gr} This section surveys various flavours of functors on $\mathbf{gr}$, so as to make this paper reasonably self-contained. Most of this material is covered in [@PV] and elsewhere; in particular, no claim to originality is made. Throughout $\mathbbm{k}$ is a unital commutative ring; for some arguments this will be taken to be a field and, for the analysis of the polynomial filtration, $\mathbbm{k}$ is taken to be $\mathbb{Q}$, in which case the theory is particularly powerful. $\mathcal{F}(\mathbf{gr})$ denotes the category of functors from $\mathbf{gr}$ to $\mathtt{Mod}_\mathbbm{k}$ (respectively $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ for contravariant functors). As in Section [1.1](#subsect:nota_conv){reference-type="ref" reference="subsect:nota_conv"}, the tensor product on $\mathtt{Mod}_\mathbbm{k}$ induces symmetric monoidal structures on $\mathcal{F}(\mathbf{gr})$ and on $\mathcal{F}(\mathbf{gr}^\mathrm{op})$, denoted simply $\otimes$. **Example 5**. The abelianization functor $\mathfrak{a}$ in $\mathcal{F}(\mathbf{gr})$ is given by $\mathfrak{a}(G) = G_\mathsf{ab}\otimes_\mathbb{Z}\mathbbm{k}$. The linear dual $\mathfrak{a}^\sharp$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ is given by $G \mapsto \mathrm{Hom}(G, \mathbbm{k}) \cong \mathrm{Hom}(G_\mathsf{ab}, \mathbbm{k})$, where $\mathrm{Hom}$ denotes homomorphisms of groups, using the underlying additive structure of $\mathbbm{k}$. The free product defines a symmetric monoidal structure on $\mathbf{gr}$ and on $\mathbf{gr}^\mathrm{op}$. This gives the respective shift functors $\tau_\mathbb{Z}^{\mathbf{gr}}$ and $\tau_\mathbb{Z}^{\mathbf{gr}^\mathrm{op}}$ defined by precomposition with $- \star \mathbb{Z}$; for instance, for $F\in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, one has $\tau_\mathbb{Z}^{\mathbf{gr}}F (\mathbb{Z}^{\star r}) = F (\mathbb{Z}^{\star r} \star \mathbb{Z}) \cong F (\mathbb{Z}^{\star r+1})$. The associated difference functors $\delta^{\mathbf{gr}}$ and $\delta^{\mathbf{gr}^\mathrm{op}}$ are defined so that there are natural isomorphisms: $$\begin{aligned} \tau_\mathbb{Z}^{\mathbf{gr}}& \cong & \delta^{\mathbf{gr}}\oplus \mathrm{Id}_{\mathcal{F}(\mathbf{gr})} \\ \tau_\mathbb{Z}^{\mathbf{gr}^\mathrm{op}}& \cong & \delta^{\mathbf{gr}^\mathrm{op}}\oplus \mathrm{Id}_{\mathcal{F}(\mathbf{gr}^\mathrm{op})}.\end{aligned}$$ *Notation 6*. Let $P_\mathbb{Z}$ denote the functor $H \mapsto \mathbbm{k}H$ (the $\mathbbm{k}$-module generated by the underlying set of $H$). This splits canonically in $\mathcal{F}(\mathbf{gr})$ as $P_\mathbb{Z}\cong \mathbbm{k}\oplus \overline{P}$, where $\overline{P}$ is the reduced part (a functor $F$ on $\mathbf{gr}$ is reduced if $F(\mathbb{Z}^{\star 0}) =0$). *Remark 7*.   1. The functor $P_\mathbb{Z}$ is projective in $\mathcal{F}(\mathbf{gr})$; it corepresents the evaluation functor $F \mapsto F (\mathbb{Z})$. 2. For $H \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, $P_\mathbb{Z}(H)$ is the $\mathbbm{k}$-module underlying the group ring $\mathbbm{k}H$; as explained in Section [5](#sect:group-ring){reference-type="ref" reference="sect:group-ring"}, this implies that $P_\mathbb{Z}$ is a cocommutative Hopf algebra in $\mathcal{F}(\mathbf{gr})$. 3. The functor $\overline{P}$ identifies as the functor to $\mathbbm{k}$-modules underlying the augmentation ideal functor $H \mapsto \mathcal{I}H$, where $\mathcal{I}H := \ker \big ( \mathbbm{k}H \rightarrow \mathbbm{k}\big).$ ## Exponential functors on $\mathbf{gr}^\mathrm{op}$ {#subsect:expo} Suppose that $(\mathcal{M}, \odot, \mathbbm{1})$ is a symmetric monoidal category, then one can consider Hopf algebras in $\mathcal{M}$ and, in particular, cocommutative Hopf algebras in $\mathcal{M}$. Recall (see [@PV], for example) that an exponential functor on $\mathbf{gr}^\mathrm{op}$ with values in $\mathcal{M}$ is a symmetric monoidal functor from $\mathbf{gr}^\mathrm{op}$ to $\mathcal{M}$. *Notation 8*. For $H$ a cocommutative Hopf algebra in the symmetric monoidal category $\mathcal{M}$, $\Phi H : \mathbf{gr}^\mathrm{op}\rightarrow \mathcal{M}$ denotes the associated exponential functor. This has values given by $\mathbb{Z}^{\star n} \mapsto H^{\odot n}$, and with morphisms of $\mathbf{gr}^\mathrm{op}$ acting via the Hopf algebra structure of $H$. The construction $\Phi$ is natural: it defines a functor from cocommutative Hopf algebras in $\mathcal{M}$ to functors from $\mathbf{gr}^\mathrm{op}$ to $\mathcal{M}$. Moreover, it is natural with respect to the symmetric monoidal category $\mathcal{M}$: namely, if $\alpha: \mathcal{M}_1 \rightarrow \mathcal{M}_2$ is a symmetric monoidal functor and $H$ is a cocommutative Hopf algebra in $\mathcal{M}_1$ then $\alpha H$ is a cocommutative Hopf algebra in $\mathcal{M}_2$ and there is a natural isomorphism $\alpha (\Phi H) \cong \Phi (\alpha H)$, where $\Phi H$ is constructed in $\mathcal{M}_1$ and $\Phi (\alpha H)$ in $\mathcal{M}_2$. ## Bifunctors {#subsect:bifunctors} We consider the category of bifunctors on $\mathbf{gr}$, by which we mean $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. The exterior tensor product $\boxtimes : \mathcal{F}(\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ gives one way of constructing bifunctors. Another fundamental example of a bifunctor is provided by the standard projectives $P_\Gamma$ of the category $\mathcal{F}(\mathbf{gr})$: *Notation 9*. Denote by $P_{(-)}$ the bifunctor $P_{(-)}: (\Gamma,H) \mapsto P_\Gamma (H) := \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(\Gamma, H).$ Since $\star$ is the coproduct in $\mathbf{gr}$, there is a natural isomorphism: $$\begin{aligned} \label{eqn:P_bullet_star} \quad \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(G_1 \star G_2, H) \cong \mathbbm{k}( \mathrm{Hom}_\mathbf{gr}(G_1 , H) \times \mathrm{Hom}_\mathbf{gr}(G_2 , H)) \cong \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(G_1 , H) \otimes \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(G_2 , H).\end{aligned}$$ This property can be restated as: **Proposition 10**. *Considered as a functor from $\mathbf{gr}^\mathrm{op}$ to $\mathcal{F}(\mathbf{gr})$, $P_{(-)}$ is exponential. Explicitly, $P_{(-)}\cong \Phi P_\mathbb{Z}$, where $P_\mathbb{Z}$ is considered as a cocommutative Hopf algebra in $\mathcal{F}(\mathbf{gr})$.* In particular, one has the following: **Lemma 11**. *There are natural isomorphisms $\tau_\mathbb{Z}^{\mathbf{gr}^\mathrm{op}}P_{(-)}\cong P_{(-)}\otimes P_\mathbb{Z}$ and $\delta^{\mathbf{gr}^\mathrm{op}}P_{(-)}\cong P_{(-)}\otimes \overline{P}$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$, where $P_\mathbb{Z}$ and $\overline{P}$ are constant with respect to $\mathbf{gr}^\mathrm{op}$.* *Proof.* Using ([\[eqn:P_bullet_star\]](#eqn:P_bullet_star){reference-type="ref" reference="eqn:P_bullet_star"}), it is clear that $\tau_\mathbb{Z}^{\mathbf{gr}^\mathrm{op}}P_{(-)}\cong P_{(-)}\otimes P_\mathbb{Z}$. The splitting $\tau_\mathbb{Z}^{\mathbf{gr}^\mathrm{op}}\cong \delta^{\mathbf{gr}^\mathrm{op}}\oplus \mathrm{Id}$ then leads to the result for $\delta^{\mathbf{gr}^\mathrm{op}}$. ◻ This is the main ingredient in the proof of the following standard result: **Proposition 12**. *As endofunctors of $\mathcal{F}(\mathbf{gr})$, the shift functor $\tau_\mathbb{Z}^{\mathbf{gr}}$ is right adjoint to $-\otimes P_\mathbb{Z}$ and the difference functor $\delta^{\mathbf{gr}}$ is right adjoint to $- \otimes \overline{P}$.* ## Polynomial functors The functors $\delta^{\mathbf{gr}}$ and $\delta^{\mathbf{gr}^\mathrm{op}}$ allow the following definition of polynomiality: **Definition 13**.   1. A functor $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$ has polynomial degree $d \in \mathbb{N}$ if $(\delta^{\mathbf{gr}})^{d+1} F = 0$. 2. A functor $F' \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$ has polynomial degree $d \in \mathbb{N}$ if $(\delta^{\mathbf{gr}^\mathrm{op}})^{d+1} F' = 0$. The full subcategories of polynomial functors of degree at most $d$ are denoted $\mathcal{F}_d (\mathbf{gr})$ and $\mathcal{F}_d (\mathbf{gr}^\mathrm{op})$ respectively. *Remark 14*.   1. For $d \in \mathbb{N}$, $\mathcal{F}_d (\mathbf{gr})$ is an abelian subcategory of $\mathcal{F}(\mathbf{gr})$ and $\mathcal{F}_d(\mathbf{gr}^\mathrm{op})$ an abelian subcategory of $\mathcal{F}(\mathbf{gr}^\mathrm{op})$. 2. An alternative, equivalent definition uses the *cross-effect functors*, as in [@MR3340364], generalizing the approach of Eilenberg and Mac Lane for functors on additive categories. Not all functors considered here are polynomial: **Example 15**.   1. The functor $P_\mathbb{Z}$ is not polynomial in $\mathcal{F}(\mathbf{gr})$: for $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, the set map $G \rightarrow G \star \mathbb{Z}$ given by $g \mapsto g x$, where $x$ denotes the generator of $\mathbb{Z}$, is injective. On linearization, this induces an injective natural transformation $P_\mathbb{Z}\hookrightarrow \delta^\mathbf{gr}P_\mathbb{Z}$. Since $\delta ^\mathbf{gr}$ is exact, a straightforward induction shows that $(\delta^\mathbf{gr})^{d+1}P_\mathbb{Z}$ is non-zero, for all $d \in \mathbb{N}$. More generally, for $\Gamma \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}^\mathrm{op}$ of positive rank, $P_\Gamma$ is not polynomial in $\mathcal{F}(\mathbf{gr})$. 2. For $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$ of positive rank, the functor $\Gamma \mapsto P_\Gamma (G)$ is not polynomial in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$. Indeed, by Lemma [Lemma 11](#lem:dgrop_P_bullet){reference-type="ref" reference="lem:dgrop_P_bullet"}, $\delta^{\mathbf{gr}^\mathrm{op}}P_{(-)}(G) \cong P_{(-)}(G) \otimes \overline{P}(G)$. Since $G$ has positive rank, $\overline{P}(G)$ is a non-zero, free $\mathbbm{k}$-module. Iterating this shows that $(\delta^{\mathbf{gr}^\mathrm{op}})^{d+1} P_{(-)}(G)$ is non-zero for all $d \in \mathbb{N}$. *Notation 16*. Denote by 1. $\mathcal{F}_{<\infty}(\mathbf{gr})$ the full subcategory of $\mathcal{F}(\mathbf{gr})$ given by $\bigcup_{d \in \mathbb{N}} \mathcal{F}_d (\mathbf{gr})$; 2. $\mathcal{F}_{< \infty}(\mathbf{gr}^\mathrm{op})$ the full subcategory of $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ given by $\bigcup_{d \in \mathbb{N}} \mathcal{F}_d (\mathbf{gr}^\mathrm{op})$. There are inclusions $$\begin{aligned} 0 = \mathcal{F}_{-1} (\mathbf{gr}) \subset \mathcal{F}_0 (\mathbf{gr}) \subset \mathcal{F}_1 (\mathbf{gr}) \subset \ldots \subset \mathcal{F}_d (\mathbf{gr}) \subset \mathcal{F}_{d+1}(\mathbf{gr}) \subset \ldots \subset \mathcal{F}_{<\infty}(\mathbf{gr}) \subset \mathcal{F}(\mathbf{gr})\end{aligned}$$ and likewise for $\mathcal{F}(\mathbf{gr}^\mathrm{op})$. **Example 17**. Let $F$ be a functor in $\mathcal{F}(\mathbf{gr})$. 1. $F$ has polynomial degree $0$ if and only if it is constant. 2. If $F$ is reduced (i.e., $F(\mathbb{Z}^{\star 0})=0$), then $F$ has polynomial degree $1$ if and only if $F$ is additive (i.e., the obvious morphisms induce an isomorphism $F (\mathbb{Z}^{\star r} \star \mathbb{Z}^{\star s}) \cong F(\mathbb{Z}^{\star r} ) \oplus F(\mathbb{Z}^{\star s})$). The abelianization functor $\mathfrak{a}$ in $\mathcal{F}(\mathbf{gr})$ is reduced and additive; in particular it has polynomial degree one. Likewise, the linear dual $\mathfrak{a}^\sharp$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ is reduced and additive, thus has polynomial degree one. The tensor product $\otimes$ is compatible with the polynomial filtration. Namely, for $d, e \in \mathbb{N}$ the tensor product on $\mathcal{F}(\mathbf{gr})$ restricts to $\mathcal{F}_d (\mathbf{gr}) \times \mathcal{F}_e (\mathbf{gr}) \stackrel{\otimes}{\longrightarrow} \mathcal{F}_{d+e} (\mathbf{gr})$. The corresponding statement holds for functors on $\mathbf{gr}^\mathrm{op}$. **Example 18**. For $d \in \mathbb{N}$, the iterated tensor product $\mathfrak{a}^{\otimes d}$ is polynomial of degree $d$, i.e., belongs to $\mathcal{F}_d (\mathbf{gr})$. This is the fundamental example of a degree $d$ polynomial functor. Likewise, in the contravariant case, $(\mathfrak{a}^\sharp)^{\otimes d}$ belongs to $\mathcal{F}_d (\mathbf{gr}^\mathrm{op})$. For $d \in \mathbb{N}$, the inclusion $\mathcal{F}_d (\mathbf{gr}) \hookrightarrow \mathcal{F}(\mathbf{gr})$ has both a left and a right adjoint. Here it is the right adjoint $$\mathsf{q}^\mathbf{gr}_d : \mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}_d (\mathbf{gr})$$ that is exploited; this will usually be considered as a functor $\mathsf{q}^\mathbf{gr}_d : \mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}(\mathbf{gr})$. With this interpretation, for a functor $F$, $\mathsf{q}^\mathbf{gr}_d F$ is the largest quotient of $F$ that has polynomial degree $d$, so that there is a canonical surjection $F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d F$, corresponding to the adjunction unit. The inclusion $\mathcal{F}_d (\mathbf{gr}) \subset \mathcal{F}_{d+1} (\mathbf{gr})$ then gives the commutative (up to natural isomorphism) diagram of natural surjections $$\xymatrix{ \mathrm{Id}_{\mathcal{F}(\mathbf{gr})} \ar@{->>}[r] \ar@/^1pc/@{->>}[rr] & \mathsf{q}^\mathbf{gr}_{d+1} \ar@{->>}[r] & \mathsf{q}^\mathbf{gr}_d . }$$ **Example 19**. Applied to the standard projective $P_\mathbb{Z}$, one has $\mathsf{q}^\mathbf{gr}_0 P_\mathbb{Z}\cong \mathbbm{k}$ and $\mathsf{q}^\mathbf{gr}_1 P_\mathbb{Z}\cong \mathfrak{a}\oplus \mathbbm{k}$. Hence $\mathsf{q}^\mathbf{gr}_0 \overline{P}=0$ and $\mathsf{q}^\mathbf{gr}_1 \overline{P}=\mathfrak{a}$. The corresponding surjection $P_\mathbb{Z}\twoheadrightarrow \mathfrak{a}$ evaluated on $G\in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$ is the map $\mathbbm{k}G \twoheadrightarrow \mathbbm{k}\otimes_\mathbb{Z}G_\mathsf{ab}$; this is the $\mathbbm{k}$-linearization of the map $G \rightarrow \mathbbm{k}\otimes_\mathbb{Z}G_\mathsf{ab}$, $g \mapsto 1 \otimes \overline{g}$, where $\overline{g}$ denotes the image of $g$ in $G_\mathsf{ab}$. This restricts to the surjection $\overline{P}\twoheadrightarrow \mathfrak{a}$. An analysis of the functors $\mathsf{q}^\mathbf{gr}_d \overline{P}$, for $d \in \mathbb{N}$, is given in Section [5](#sect:group-ring){reference-type="ref" reference="sect:group-ring"}. The behaviour of $\mathsf{q}^\mathbf{gr}_d$ is illustrated by the following: **Lemma 20**. *For $d < n \in \mathbb{N}$, $\mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n}) =0 = \mathsf{q}^\mathbf{gr}_d (\overline{P}^{\otimes n})$.* *Proof.* Consider the composite surjection $\overline{P}^{\otimes n} \twoheadrightarrow \mathfrak{a}^{\otimes n} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n})$, where the first map is the $n$-fold tensor product of the surjection $\overline{P}\twoheadrightarrow \mathfrak{a}$ of Example [Example 19](#exam:qgr_PZ){reference-type="ref" reference="exam:qgr_PZ"}. To show that $\mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n})=0$, it suffices to show that the composite is zero. By the adjunction of Proposition [Proposition 12](#prop:dgr_adjoint){reference-type="ref" reference="prop:dgr_adjoint"}, this is adjoint to a map $\mathbbm{k}\rightarrow (\delta^{\mathbf{gr}})^n \mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n})$. By construction, $\mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n})$ has polynomial degree $d$ hence, since $d<n$, by the definition of polynomiality, $(\delta^{\mathbf{gr}})^n \mathsf{q}^\mathbf{gr}_d (\mathfrak{a}^{\otimes n})=0$. The result follows in this case; the argument for $\mathsf{q}^\mathbf{gr}_d (\overline{P}^{\otimes n})$ is similar. ◻ This leads to the following: **Proposition 21**. *For $d \in \mathbb{N}$, the functors $\mathsf{q}^\mathbf{gr}_d (\overline{P}^{\otimes n})$, $0 \leq n \leq d$, form a set of projective generators of $\mathcal{F}_d (\mathbf{gr})$.* *Proof.* Yoneda's lemma implies that the set of functors $P_{\mathbb{Z}^n}$, for $n \in \mathbb{N}$, is a set of projective generators of $\mathcal{F}(\mathbf{gr})$; from this, one deduces readily that $\overline{P}^{\otimes n}$, $n \in \mathbb{N}$, also forms a set of projective generators. The functor $\mathsf{q}^\mathbf{gr}_d :\mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}_d (\mathbf{gr})$ is left adjoint to an exact functor, hence preserves projectives. Using Lemma [Lemma 20](#lem:qgr_tensor){reference-type="ref" reference="lem:qgr_tensor"}, it follows that $\{\mathsf{q}^\mathbf{gr}_d (\overline{P}^{\otimes n}) \ | \ 0 \leq n \leq d\}$ is a set of projective generators of $\mathcal{F}_d (\mathbf{gr})$. ◻ We record the following, which is clear from the constructions: **Proposition 22**. *For $d \in \mathbb{N}$ $\mathcal{F}_d (\mathbf{gr})$ and $\mathcal{F}_d(\mathbf{gr}^\mathrm{op})$ are both complete and cocomplete.* ## The tensor product over $\mathbf{gr}$ and duality {#subsect:tensor_gr} The tensor product $\otimes_\mathbf{gr}: \mathcal{F}(\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr}) \rightarrow \mathtt{Mod}_\mathbbm{k}$ is defined as usual: this can be taken to be the composite of the external tensor product $\boxtimes : \mathcal{F}(\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ followed by the coend $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \rightarrow \mathtt{Mod}_\mathbbm{k}$. **Example 23**. Consider the standard projective $P_\Gamma$, for $\Gamma \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}^\mathrm{op}$. Then $- \otimes _\mathbf{gr}P_\Gamma : \mathcal{F}(\mathbf{gr}^\mathrm{op}) \rightarrow \mathtt{Mod}_\mathbbm{k}$ is naturally isomorphic to the evaluation functor $F \mapsto F(\Gamma)$, for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$. Likewise, for $H \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, consider $P_{(-)}(H) = \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-, H)$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$, which is projective. The functor $P_{(-)}(H) \otimes_\mathbf{gr}- : \mathcal{F}(\mathbf{gr}) \rightarrow \mathtt{Mod}_\mathbbm{k}$ is naturally isomorphic to the evaluation functor $G \mapsto G(H)$, for $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$. *Remark 24*. The bifunctor $\otimes_\mathbf{gr}$ commutes with colimits with respect to both variables. For current purposes it is colimits in $\mathcal{F}(\mathbf{gr})$ that are of most interest. For the rest of this subsection, we suppose that $\mathbbm{k}$ is a field. Post-composing with vector space duality yields $D: \mathcal{F}(\mathbf{gr})^\mathrm{op}\rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$ and $D: \mathcal{F}(\mathbf{gr}^\mathrm{op}) ^\mathrm{op}\rightarrow \mathcal{F}(\mathbf{gr})$. These are exact and are adjoint: for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$ and $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, there is a natural isomorphism: $$\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})}(G, DF) \cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}^\mathrm{op}) } (F, DG).$$ Moreover, the duality functors restrict to equivalences between the respective full subcategories of functors taking finite-dimensional values. Duality restricts to the full subcategories of polynomial functors: **Lemma 25**. *For $d \in \mathbb{N}$, the duality adjunction restricts to $D : \mathcal{F}_d (\mathbf{gr}) ^\mathrm{op}\rightleftarrows \mathcal{F}_d (\mathbf{gr}^\mathrm{op}) : D$.* The tensor product $\otimes_\mathbf{gr}$ relates to duality as follows: **Proposition 26**. *For $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$ and $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, there are natural isomorphisms: $$\mathrm{Hom}_{\mathtt{Mod}_\mathbbm{k}} (F \otimes _\mathbf{gr}G, \mathbbm{k}) \cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})}(G, DF) \cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}^\mathrm{op}) } (F, DG).$$* Now consider bifunctors, i.e., the category $\mathcal{F}(\mathbf{gr}\times \mathbf{gr}^\mathrm{op})$. For two bifunctors $F_1$, $F_2$, one can form $F_1 \otimes_\mathbf{gr}F_2$, using the $\mathbf{gr}$-structure of $F_2$ and the $\mathbf{gr}^\mathrm{op}$-structure of $F_1$. This has a natural bifunctor structure, with $\mathbf{gr}$ acting via $F_1$ and $\mathbf{gr}^\mathrm{op}$ via $F_2$. As usual, one has: **Proposition 27**. *The functor $\otimes_\mathbf{gr}$ yields a monoidal structure $(\mathcal{F}(\mathbf{gr}\times \mathbf{gr}^\mathrm{op}) , \otimes_\mathbf{gr}, P_{(-)})$.* ## The polynomial filtration for functors on $\mathbf{gr}$ {#subsect:poly_Q} For $d \in \mathbb{N}$, the relationship between $\mathcal{F}_d(\mathbf{gr})$ and the category $\mathbbm{k}\mathfrak{S}_d \hspace{-2pt}-\hspace{-2pt}\mathrm{mod}$ of $\mathbbm{k}\mathfrak{S}_d$-modules is important. This uses the functors introduced below, exploiting the action of $\mathfrak{S}_d$ by place permutations on $\mathfrak{a}^{\otimes d}$. *Notation 28*. For $d \in \mathbb{N}$, denote by 1. $\alpha_d : \mathbbm{k}\mathfrak{S}_d \hspace{-2pt}-\hspace{-2pt}\mathrm{mod}\rightarrow \mathcal{F}_d (\mathbf{gr})$ the functor $\mathfrak{a}^{\otimes _d} \otimes_{\mathfrak{S}_d } -$; 2. $\mathrm{cr}_d: \mathcal{F}_d (\mathbf{gr}) \rightarrow \mathbbm{k}\mathfrak{S}_d \hspace{-2pt}-\hspace{-2pt}\mathrm{mod}$ the functor $\mathrm{Hom}_{\mathcal{F}_d (\mathbf{gr})} (\mathfrak{a}^{\otimes d}, -)$. The functor $\mathrm{cr}_d$ is sometimes referred to as the *cross-effect* functor (more precisely, it is the restriction to $\mathcal{F}_d (\mathbf{gr})$ of the $d$th cross-effect functor as in [@MR3340364]); it is exact. The functor $\alpha_d$ is left adjoint to $\mathrm{cr}_d$ and is right exact. The composite $\mathrm{cr}_d \alpha_d$ is naturally isomorphic to the identity on $\mathbbm{k}\mathfrak{S}_d$-modules via the unit of the adjunction; for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr})$, the adjunction counit $\alpha_d \mathrm{cr}_d F \rightarrow F$ has kernel and cokernel that are both polynomial of degree $d-1$. *Remark 29*. The cross-effect functor also has a right adjoint, $\beta_d : \mathbbm{k}\mathfrak{S}_d\hspace{-2pt}-\hspace{-2pt}\mathrm{mod}\rightarrow \mathcal{F}_d (\mathbf{gr})$. This is of significant importance in the theory of polynomial functors, as exploited in [@PV] and [@2021arXiv211001934P], for example. We now specialize to $\mathbbm{k}= \mathbb{Q}$. **Proposition 30**. *[@PV] For $\mathbbm{k}= \mathbb{Q}$,* 1. *the functor $\alpha_d$ is exact;* 2. *the adjunction counit $\alpha_d \mathrm{cr}_d \rightarrow \mathrm{Id}_{\mathcal{F}_d(\mathbf{gr})}$ is injective and fits into the natural short exact sequence (for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr})$): $$0 \rightarrow \alpha_d \mathrm{cr}_d F \rightarrow F \rightarrow \mathsf{q}^\mathbf{gr}_{d-1} F \rightarrow 0;$$* 3. *$\alpha_d \mathrm{cr}_d F$ is semisimple and all of its composition factors have polynomial degree exactly $d$.* *Remark 31*. A polynomial functor of the form $\alpha_d M$ for $M$ a $\mathbb{Q}\mathfrak{S}_d$-module will be referred to as a *homogeneous polynomial functor of degree $d$*. (Thus a functor $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d(\mathbf{gr})$ is homogeneous of degree $d$ if and only if the adjunction counit $\alpha_d \mathrm{cr}_d F \rightarrow F$ is an isomorphism.) Now, for any $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, one has the natural tower under $F$: $$\ldots \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d+1}F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d F \twoheadrightarrow \ldots \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{-1} F=0.$$ By considering the kernels of the natural surjections $F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d F$, this is equivalent to giving a decreasing filtration of $F$, the *polynomial filtration*. This filtration stabilizes if and only if $F$ is polynomial. We introduce the following notation for the subquotients of the polynomial filtration: *Notation 32*. For $d\in \mathbb{N}$, denote by $\widehat{\mathsf{q}_{d}^\mathbf{gr}}$ the kernel of the natural surjection $\mathsf{q}^\mathbf{gr}_d \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d-1}$. The associated graded to the polynomial filtration of $F\in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$ is thus $\bigoplus_{d \in \mathbb{N}} \widehat{\mathsf{q}_{d}^\mathbf{gr}} F$. The following is used in Corollary [Corollary 35](#cor:fbcr){reference-type="ref" reference="cor:fbcr"} to encode this as a ${\bm{\Sigma}}$-module: **Proposition 33**. *[@PV] For $\mathbbm{k}=\mathbb{Q}$ and $d \in \mathbb{N}$,* 1. *the functors $\mathsf{q}^\mathbf{gr}_d$ and $\widehat{\mathsf{q}_{d}^\mathbf{gr}}$ are exact on $\mathcal{F}_{< \infty} (\mathbf{gr})$;* 2. *for $F \in \mathcal{F}(\mathbf{gr})$, there is a natural isomorphism $\widehat{\mathsf{q}_{d}^\mathbf{gr}} F \cong \alpha_d \mathrm{cr}_d \mathsf{q}^\mathbf{gr}_d F$; in particular, $\widehat{\mathsf{q}_{d}^\mathbf{gr}} F$ is a homogeneous polynomial functor of degree $d$.* The category $\mathcal{F}({\bm{\Sigma}})$ used below was introduced in Example [Example 3](#exam:fb-modules){reference-type="ref" reference="exam:fb-modules"}. *Notation 34*. Let $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}: \mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}({\bm{\Sigma}})$ be the functor that sends $F$ to the ${\bm{\Sigma}}$-module $\mathbf{t} \mapsto \mathrm{cr}_t \mathsf{q}^\mathbf{gr}_t F = \mathrm{cr}_t \widehat{\mathsf{q}_{t}^\mathbf{gr}} F$, for $\mathbf{t} = \{ 1, \ldots , t \} \in {\bm{\Sigma}}$. The following consequence of Proposition [Proposition 33](#prop:qhat){reference-type="ref" reference="prop:qhat"} shows that $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ determines the associated graded of the polynomial filtration. **Corollary 35**. *Suppose $\mathbbm{k}= \mathbb{Q}$.* 1. *For $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, the associated graded of the polynomial filtration is naturally isomorphic to $$\bigoplus_{t \in \mathbb{N}} \alpha_t \big((\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}F) (\mathbf{t})\big).$$* 2. *The restriction of $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ to $\mathcal{F}_{< \infty}(\mathbf{gr})$ is exact.* ## The polynomial filtration for $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ and analytic functors {#subsect:analytic_grop} In [@2021arXiv211001934P] it is the polynomial filtration for functors in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ that is used. When working with $\mathcal{F}(\mathbf{gr}^\mathrm{op})$, the relevant counterpart of $\mathsf{q}^\mathbf{gr}_d$ is the *right* adjoint to the inclusion $\mathcal{F}_d (\mathbf{gr}^\mathrm{op}) \hookrightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$. We also introduce the left adjoint, since this occurs when considering the behaviour of $\otimes _\mathbf{gr}$. *Notation 36*. For $d \in \mathbb{N}$, denote by 1. $\mathsf{p}_d : \mathcal{F}(\mathbf{gr}^\mathrm{op}) \rightarrow \mathcal{F}_ d (\mathbf{gr}^\mathrm{op})$ the right adjoint to the inclusion $\mathcal{F}_d (\mathbf{gr}^\mathrm{op}) \hookrightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$; 2. $\mathsf{q}_d : \mathcal{F}(\mathbf{gr}^\mathrm{op}) \rightarrow \mathcal{F}_ d (\mathbf{gr}^\mathrm{op})$ the left adjoint to the inclusion $\mathcal{F}_d (\mathbf{gr}^\mathrm{op}) \hookrightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$. These may also be considered as functors $\mathcal{F}(\mathbf{gr}^\mathrm{op}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$. Every functor $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$ has a canonical polynomial filtration $$0 = \mathsf{p}_{-1} G \subset \mathsf{p}_0 G \subset \mathsf{p}_1 G \subset \ldots \subset \mathsf{p}_d G \subset \ldots \subset G.$$ The functor $G$ is *analytic* if the canonical morphism $\lim_{\substack{\rightarrow\\ d} } \mathsf{p}_d G \rightarrow G$ is an isomorphism. **Definition 37**. The category $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \subset \mathcal{F}(\mathbf{gr}^\mathrm{op})$ is the full subcategory of analytic functors. *Remark 38*. Properties of the polynomial filtration for $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ are studied in [@2021arXiv211001934P]; these are analogous (via 'duality') to those of the polynomial filtration of $\mathcal{F}(\mathbf{gr})$. # The tower of categories $\mathsf{q}^\mathbf{gr}_\bullet \mathbbm{k}\mathbf{gr}$ {#sect:polyq} This section considers the bifunctor $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ (for $d \in \mathbb{N}$) that is constructed from $P_{(-)}= \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-,-)$. These bifunctors lead to the tower of categories $\mathsf{q}^\mathbf{gr}_\bullet \mathbbm{k}\mathbf{gr}$, providing 'polynomial approximations' to $\mathbbm{k}\mathbf{gr}$. Here $\mathbbm{k}$ is taken to be a field, so as to simplify the exposition. For the main applications, it will be taken to be $\mathbb{Q}$. ## Polynomial functors and $\otimes_\mathbf{gr}$ Proposition [Proposition 26](#prop:otimes_gr_duality){reference-type="ref" reference="prop:otimes_gr_duality"} allows a quick proof of the following: **Proposition 39**. *For $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op})$ and $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$,* 1. *if $F$ has polynomial degree $d$, then the canonical surjection $G \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d G$ induces an isomorphism $F \otimes _\mathbf{gr}G \stackrel{\cong}{\rightarrow} F \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d G$;* 2. *if $G$ has polynomial degree $d$, then the canonical surjection $F \twoheadrightarrow \mathsf{q}_d F$ induces an isomorphism $F \otimes_\mathbf{gr}G \stackrel{\cong}{\rightarrow} \mathsf{q}_d F \otimes_\mathbf{gr}G$.* *Hence, in general, there are natural isomorphisms: $$\mathsf{q}_d F \otimes_\mathbf{gr}G \cong \mathsf{q}_d F \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d G \cong F \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d G.$$* *Proof.* Consider the first statement. By Proposition [Proposition 26](#prop:otimes_gr_duality){reference-type="ref" reference="prop:otimes_gr_duality"}, there is a natural isomorphism $\mathrm{Hom}_{\mathbbm{k}} (F \otimes_\mathbf{gr}G, \mathbbm{k}) \cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (G, DF)$. Moreover, $DF$ has polynomial degree $d$ in $\mathcal{F}(\mathbf{gr})$, by Lemma [Lemma 25](#lem:duality_polynomial){reference-type="ref" reference="lem:duality_polynomial"}. Hence, there is a natural isomorphism $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (G, DF)\cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (\mathsf{q}^\mathbf{gr}_d G, DF)$. Applying the isomorphism of Proposition [Proposition 26](#prop:otimes_gr_duality){reference-type="ref" reference="prop:otimes_gr_duality"} again gives $\mathrm{Hom}_{\mathbbm{k}} (\mathsf{q}^\mathbf{gr}_d F \otimes_\mathbf{gr}G, \mathbbm{k})$. More precisely, the induced map $F \otimes _\mathbf{gr}G \stackrel{\cong}{\rightarrow} F \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d G$ gives an isomorphism on applying $\mathrm{Hom}_\mathbbm{k}(-, \mathbbm{k})$, hence is an isomorphism. The proof of the second statement is categorically dual. The final statement then follows. ◻ We now seek to apply this to bifunctors (i.e., working with $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$). The key example is the bifunctor $P_{(-)}$ that sends $(\Gamma, H) \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}^\mathrm{op}\times \mathbf{gr}$ to $\mathbbm{k}\mathrm{Hom}_{\mathbf{gr}} (\Gamma, H)$. When working with bifunctors, we have the following, which follows directly from naturality: **Lemma 40**. *The functors $\mathsf{q}^\mathbf{gr}_d$ and $\mathsf{q}_d$, for $d \in \mathbb{N}$, induce endofunctors of $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. Moreover, the natural surjections $\mathrm{Id}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d+1} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d$ are natural transformations of endofunctors of $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* In particular, a bifunctor $B$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ is said to be polynomial of degree $d$ with respect to $\mathbf{gr}$ (respectively $\mathbf{gr}^\mathrm{op}$) if the natural surjection $B \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d B$ (resp. $B \twoheadrightarrow \mathsf{q}_d B$) is an isomorphism. Equivalently, $B$ is polynomial of degree $d$ with respect to $\mathbf{gr}$ if, for every $\Gamma \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}^\mathrm{op}$, $B (\Gamma, -)$ (considered as an object of $\mathcal{F}(\mathbf{gr})$) is polynomial of degree $d$. The corresponding statement holds for $\mathbf{gr}^\mathrm{op}$. **Proposition 41**. *For $d \in \mathbb{N}$, there is an isomorphism $\mathsf{q}^\mathbf{gr}_d P_{(-)}\cong \mathsf{q}_d P_{(-)}$. In particular, $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ is polynomial with respect to $\mathbf{gr}^\mathrm{op}$.* *Moreover, for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr}^\mathrm{op})$ and $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr})$, there are natural isomorphisms: $$\begin{aligned} F \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d P_{(-)}& \cong & F \\ \mathsf{q}^\mathbf{gr}_d P_{(-)}\otimes _\mathbf{gr}G & \cong & G.\end{aligned}$$* *Proof.* By Proposition [Proposition 27](#prop:gr_bifunctors_monoidal){reference-type="ref" reference="prop:gr_bifunctors_monoidal"}, $P_{(-)}$ is the unit for the monoidal structure on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ given by $\otimes_\mathbf{gr}$. In particular, this gives the isomorphisms $P_{(-)}\otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_d P_{(-)}\cong \mathsf{q}^\mathbf{gr}_d P_{(-)}$ and $\mathsf{q}_d P_{(-)}\otimes_\mathbf{gr}P_{(-)}\cong \mathsf{q}_d P_{(-)}$. Proposition [Proposition 39](#prop:otimes_gr_poly){reference-type="ref" reference="prop:otimes_gr_poly"} gives that both of the left hand expressions of the above isomorphisms are isomorphic to $\mathsf{q}_d P_{(-)}\otimes _\mathbf{gr}\mathsf{q}^\mathbf{gr}_d P_{(-)}$. Hence the first statement follows by composing the isomorphisms of bifunctors. The remaining statements are proved similarly. ◻ *Remark 42*.   1. The fact that $\mathsf{q}^\mathbf{gr}_d$, $d \in \mathbb{N}$ applied to $P_{(-)}$ (with respect to $\mathbf{gr}^\mathrm{op}$) yields an interesting tower of surjections may appear surprising, since it is usually the increasing filtration associated to the functors $\mathsf{p}_d$ that is considered when dealing with $\mathbf{gr}^\mathrm{op}$. 2. Proposition [Proposition 41](#prop:pbif_bipolynomiality){reference-type="ref" reference="prop:pbif_bipolynomiality"} implies that, restricting the monoidal structure $(\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}), \otimes_\mathbf{gr}, P_{(-)})$ to the full subcategory of $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ with objects bifunctors that are polynomial degree $d$ with respect to both $\mathbf{gr}$ and $\mathbf{gr}^\mathrm{op}$ gives a monoidal structure with unit $\mathsf{q}^\mathbf{gr}_d P_{(-)}$. **Example 43**.   1. The bifunctor $\mathsf{q}^\mathbf{gr}_0 P_{(-)}$ is the constant bifunctor $\mathbbm{k}$. 2. The bifunctor $\mathsf{q}^\mathbf{gr}_1 P_{(-)}$ is isomorphic to $\mathbbm{k}\oplus (\mathfrak{a}^\sharp \boxtimes \mathfrak{a})$. This can also be viewed as $(\Gamma , H) \mapsto \mathbbm{k}\otimes \mathrm{Hom}(\Gamma_\mathsf{ab}, H_\mathsf{ab})$, where $(-)_\mathsf{ab}: \mathbf{gr}\rightarrow \mathbf{ab}$ is the abelianization functor. The surjection to $\mathfrak{a}^\sharp \boxtimes \mathfrak{a}$ is then induced by the $\mathbbm{k}$-linear extension of $\mathrm{Hom}_\mathbf{gr}(\Gamma, H) \rightarrow \mathrm{Hom}(\Gamma_\mathsf{ab}, H_\mathsf{ab}) \rightarrow \mathbbm{k}\otimes \mathrm{Hom}(\Gamma_\mathsf{ab}, H_\mathsf{ab})$, where the first map is given by the abelianization functor $(-)_\mathsf{ab}$ and the second is induced by the unit of $\mathbbm{k}$. Proposition [Proposition 41](#prop:pbif_bipolynomiality){reference-type="ref" reference="prop:pbif_bipolynomiality"} provides the tower of quotients in $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$: $$\xymatrix{ P_{(-)} \ar@{.>>}[d] \ar@{->>}[rd] \ar@{.>>}[rrd] \ar@{->>}[rrrd] \ar@{->>}[rrrrd] \ar@<.3ex>@{->>}[rrrrrd] \\ \ldots \ar@{->>}[r] & \mathsf{q}^\mathbf{gr}_d P_{(-)} \ar@{->>}[r] & \ldots \ar@{->>}[r] & \mathsf{q}^\mathbf{gr}_2 P_{(-)} \ar@{->>}[r] & \mathsf{q}^\mathbf{gr}_1 P_{(-)}= \mathbbm{k}\oplus (\mathfrak{a}^\sharp \boxtimes \mathfrak{a}) \ar@{->>}[r] & \mathsf{q}^\mathbf{gr}_0 P_{(-)}= \mathbbm{k}, }$$ in which $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ is polynomial of degree $d$ with respect to both $\mathbf{gr}$ and $\mathbf{gr}^\mathrm{op}$. We note the following: **Proposition 44**. *For $d \in \mathbb{N}$, the bifunctor $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ is finite (i.e., has a finite composition series). In particular, it takes finite-dimensional values.* *Proof.* This can be proved by using the explicit analysis of the polynomial filtration carried out in [@MR3505136]. ◻ ## The tower {#subsect:tower_kgr} Composition of morphisms in $\mathbf{gr}$ induces 'composition maps' $P_{(-)}\otimes P_{(-)}\rightarrow P_{(-)}$ for $P_{(-)}$ by $\mathbbm{k}$-linearizing. Explicitly, for $H \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$ one has the morphism of bifunctors $$\mathbbm{k}\mathrm{Hom}_\mathbf{gr}(H, -) \otimes \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-, H) \rightarrow \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-,-).$$ Now, there is a natural morphism $\mathbbm{k}\mathrm{Hom}_\mathbf{gr}(H, -) \otimes \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-, H) \rightarrow \mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-,-) \otimes_\mathbf{gr}\mathbbm{k}\mathrm{Hom}_\mathbf{gr}(-,-) = P_{(-)}\otimes_\mathbf{gr}P_{(-)}$ and the latter is isomorphic to $P_{(-)}$. For $d \in \mathbb{N}$, this can be repeated, replacing $P_{(-)}$ by $\mathsf{q}^\mathbf{gr}_d P_{(-)}$. This gives the following commutative diagram: $$\begin{aligned} \label{eqn:diag_comp} \xymatrix{ P_{(-)}(H, -) \otimes P_{(-)}(-,H) \ar[r] \ar@{->>}[d] & P_{(-)} \ar@{->>}[d] \\ \mathsf{q}^\mathbf{gr}_d P_{(-)}(H,-) \otimes \mathsf{q}^\mathbf{gr}_d P_{(-)}(-,H) \ar[r] & \mathsf{q}^\mathbf{gr}_d P_{(-)} }\end{aligned}$$ where the vertical surjections are induced by the canonical map $P_{(-)}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_d P_{(-)}$. (Note that $P_{(-)}(H,-)$ is the same as $P_H (-)$ and $P_{(-)}(-,H)$ is the same as $P_{(-)}(H)$.) This diagram determines the lower horizontal 'composition map'. **Theorem 45**. *For $d \in \mathbb{N}$,* 1. *there is a $\mathbbm{k}$-linear category $\mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}$ with objects $\mathrm{Ob}\hspace{2pt}\mathbf{gr}$ and such that, for $\Gamma, H \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, $\mathrm{Hom}_{\mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}} (\Gamma, H) = \mathsf{q}^\mathbf{gr}_d P_{(-)}(\Gamma, H)$ and composition of morphisms is given by diagram ([\[eqn:diag_comp\]](#eqn:diag_comp){reference-type="ref" reference="eqn:diag_comp"});* 2. *there is a full $\mathbbm{k}$-linear functor $\mathbbm{k}\mathbf{gr}\rightarrow \mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}$ that is the identity on objects and, on morphisms, is the $\mathbbm{k}$-linear surjection $P_{(-)}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_d P_{(-)}$.* *These categories form a tower under $\mathbbm{k}\mathbf{gr}$, with projection $\mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}\rightarrow \mathsf{q}^\mathbf{gr}_{d-1} \mathbbm{k}\mathbf{gr}$ that is the identity on objects and, on morphisms, is induced by the natural surjection $\mathsf{q}^\mathbf{gr}_d \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d-1}$: $$\xymatrix{ \mathbbm{k}\mathbf{gr} \ar@{.>}[d] \ar[rd] \ar@{.>}[rrd] \ar[rrrd] \ar[rrrrd] \\ \ldots \ar[r] & \mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr} \ar[r] & \ldots \ar[r] & \mathsf{q}^\mathbf{gr}_1 \mathbbm{k}\mathbf{gr} \ar[r] & \mathsf{q}^\mathbf{gr}_0 \mathbbm{k}\mathbf{gr}. }$$* *Proof.* That $\mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}$ forms a category as stated is essentially a restatement of the assertion of Remark [Remark 42](#rem:monoidal_polynomial_bifunctors){reference-type="ref" reference="rem:monoidal_polynomial_bifunctors"} that $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ forms the unit for $\otimes_\mathbf{gr}$ in the appropriate full subcategory of bifunctors. This provides the composition of morphisms in $\mathsf{q}^\mathbf{gr}_d \mathbbm{k}\mathbf{gr}$. The fact that composition is unital and associative follows from the corresponding properties for $\mathbbm{k}\mathbf{gr}$. That these constructions, for varying $d$, form a tower is clear. ◻ *Remark 46*. This result can be proved directly by exploiting the definition of $\mathsf{q}^\mathbf{gr}_d$ as a left adjoint to construct the composition maps. The above approach has been preferred, since it explains why $\mathsf{q}^\mathbf{gr}_d P_{(-)}$ is the natural object to consider. # Pro-polynomial functors on $\mathbf{gr}$ {#sect:propoly} We introduce a framework for studying functors on $\mathbf{gr}$ that are *not* polynomial. This is based on using appropriate towers of polynomial functors on $\mathbf{gr}$, such as those arising from the polynomial filtration of an object of $\mathcal{F}(\mathbf{gr})$. Throughout, $\mathbbm{k}$ is taken to be $\mathbb{Q}$, so that the results of Section [2.5](#subsect:poly_Q){reference-type="ref" reference="subsect:poly_Q"} apply. ## Introducing ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ Consider $(\mathbb{N}, \leq)$ as a poset and hence as a category. Thus $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$ is the category of towers in $\mathcal{F}(\mathbf{gr})$ and this inherits an abelian structure from $\mathcal{F}(\mathbf{gr})$. We use this to define the following category of pro-polynomial functors: **Definition 47**. Let ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ be the full subcategory of $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$ with objects $F_\bullet : \mathbb{N}^\mathrm{op}\rightarrow \mathcal{F}(\mathbf{gr})$, $n \mapsto F_n$, such that, for all $d \in \mathbb{N}$, 1. $F_d \in \mathcal{F}_d (\mathbf{gr})$; 2. the morphism $\mathsf{q}^\mathbf{gr}_{d} F_{d+1} {\rightarrow} F_{d}$ induced by $F_{d+1} \rightarrow F_{d}$ is an isomorphism. *Remark 48*. The hypothesis implies that, for each $d \in \mathbb{N}$, the structure morphism $F_{d+1} \rightarrow F_{d}$ is surjective. In particular, the endofunctor of $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$ that sends a tower $F_\bullet$ to the tower $(\mathsf{q}^\mathbf{gr}_d F_d)$ (with the induced structure morphisms) does not take values in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, since this surjectivity property is not in general satisfied. **Proposition 49**. *The category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is an abelian, $\mathbb{Q}$-linear full subcategory of $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$. Moreover, ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is cocomplete.* *Proof.* For the first statement, it suffices to show that the kernel and cokernel of a morphism in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ both lie in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The condition on the polynomiality is immediate, hence it suffices to check that the formation of the kernel and cokernel of a morphism between polynomial functors preserves the polynomial filtration in the appropriate sense. This follows since the functors $\mathsf{q}^\mathbf{gr}_d$ restricted to $\mathcal{F}_{< \infty}(\mathbf{gr})$ are exact, by Proposition [Proposition 33](#prop:qhat){reference-type="ref" reference="prop:qhat"}. That ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is cocomplete is a consequence of the fact that $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$ is cocomplete (since $\mathcal{F}(\mathbf{gr})$ is) and that the functor $\mathsf{q}^\mathbf{gr}_d$, $d \in \mathbb{N}$, commutes with colimits, since it is a left adjoint. ◻ **Proposition 50**. * * 1. *For $d \in \mathbb{N}$, the functor ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}_d (\mathbf{gr})$ given by $F_\bullet \mapsto F_d$ is exact.* 2. *The functors $\mathsf{q}^\mathbf{gr}_n$, $n \in \mathbb{N}$, induce a $\mathbb{Q}$-linear functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ given by $F \mapsto (\mathsf{q}^\mathbf{gr}_n F \ | \ n \in \mathbb{N})$. This is exact when restricted to $\mathcal{F}_{< \infty} (\mathbf{gr})$.* *Proof.* The first statement is clear, as is the fact that the functors $\mathsf{q}^\mathbf{gr}_n$ give rise to a $\mathbb{Q}$-linear functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The exactness statement follows from that for the functors $\mathsf{q}^\mathbf{gr}_n$ given by Proposition [Proposition 33](#prop:qhat){reference-type="ref" reference="prop:qhat"}, which requires the restriction to $\mathcal{F}_{< \infty} (\mathbf{gr})$. ◻ Restricting to polynomial functors, one has: **Proposition 51**. *For $d \in \mathbb{N}$, the category $\mathcal{F}_d (\mathbf{gr})$ is equivalent to the full subcategory of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ with objects $F_\bullet$ such that the structure morphism $F_{n+1} \rightarrow F_n$ is the identity for all $n \geq d$.* *Hence, $\mathcal{F}_{< \infty} (\mathbf{gr})$ is equivalent to the full subcategory of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ with objects $F_\bullet$ such that there exists $d \in \mathbb{N}$ such that the structure morphism $F_{n+1} \rightarrow F_n$ is the identity for all $n \geq d$.* *Proof.* The restriction of the functor of Proposition [Proposition 50](#prop:d_evaluate_exact_f_to_propoly){reference-type="ref" reference="prop:d_evaluate_exact_f_to_propoly"}, $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, to $\mathcal{F}_d (\mathbf{gr})$ clearly takes values in the given full subcategory. We claim that this has quasi-inverse given by the restriction of the functor of Proposition [Proposition 50](#prop:d_evaluate_exact_f_to_propoly){reference-type="ref" reference="prop:d_evaluate_exact_f_to_propoly"}. This is verified as follows: for $F$ a functor of polynomial degree $d$, one has the natural isomorphism $(\mathsf{q}^\mathbf{gr}_\bullet F) _d \cong F$; for $F_\bullet \in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ lying in the given full subcategory, the definition of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ implies that there is a unique morphism $\mathsf{q}^\mathbf{gr}_\bullet (F_d) \rightarrow F_\bullet$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ that is the identity on $F_d$. The second statement follows from the first. In particular, the quasi-inverse sends an object $(F_\bullet)$ of the given subcategory to $\lim_{\substack{\leftarrow \\ n }} F_n$, which identifies with $F_N$ for some $N \gg 0$. ◻ ## The symmetric monoidal structure The category $\mathcal{F}(\mathbf{gr})^{\mathbb{N}^\mathrm{op}}$ comes equipped with the 'pointwise' tensor product; namely, for $F_\bullet$ and $G_\bullet$ two such functors, one has $F_\bullet \otimes G_\bullet$ such that, for all $n \in \mathbb{N}$, $(F_\bullet \otimes G_\bullet)_n = F_n \otimes G_n$. However, ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is not stable under this, so one introduces the following truncation, designed so that Proposition [Proposition 55](#prop:propoly_sym_mon){reference-type="ref" reference="prop:propoly_sym_mon"} below holds: **Definition 52**. Let $\overline{\otimes}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\times {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ be the functor defined on objects by $(F_\bullet \overline{\otimes}G_\bullet)_n:= \mathsf{q}^\mathbf{gr}_n (F_n \otimes G_n)$ and with structure morphism (for $n>0$) determined by the commutative diagram: $$\xymatrix{ F_n \otimes G_n \ar@{->>}[d] \ar[r] & F_{n-1} \otimes G_{n-1} \ar@{->>}[d] \\ \mathsf{q}^\mathbf{gr}_n(F_n \otimes G_n) \ar@{.>}[r] & \mathsf{q}^\mathbf{gr}_{n-1}(F_{n-1} \otimes G_{n-1}) }$$ in which the vertical maps are the canonical surjections, the top map is the tensor product of the respective structures maps and the dotted arrow is provided by the adjunction defining $\mathsf{q}^\mathbf{gr}_n$. *Remark 53*. One must check that $\overline{\otimes}$ does indeed take values in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, namely that, for $0< d \in \mathbb{N}$, the canonical surjections $F_d \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d-1} F_d \cong F_{d-1}$ and $G_d \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{d-1} G_d \cong G_{d-1}$ induce an isomorphism $$\mathsf{q}^\mathbf{gr}_{d-1} (F_d \otimes G_d) \cong \mathsf{q}^\mathbf{gr}_{d-1} (F_{d-1} \otimes G_{d-1}).$$ This is established using the properties of the polynomial filtration presented in Section [2.5](#subsect:poly_Q){reference-type="ref" reference="subsect:poly_Q"}, as follows. By hypothesis one has the short exact sequence $0 \rightarrow \widehat{\mathsf{q}_{d}^\mathbf{gr}} F_d \rightarrow F_{d} \rightarrow F_{d-1} \rightarrow 0$, where $F_d \rightarrow F_{d-1}$ is the structure morphism. Forming the tensor product with $G_d$ gives the short exact sequence in $\mathcal{F}_{< \infty}(\mathbf{gr})$ $$0 \rightarrow (\widehat{\mathsf{q}_{d}^\mathbf{gr}} F_d) \otimes G_d \rightarrow F_{d} \otimes G_d \rightarrow F_{d-1} \otimes G_d \rightarrow 0.$$ Since $\widehat{\mathsf{q}_{d}^\mathbf{gr}} F_d$ is homogeneous polynomial of degree $d$ and $G_d$ is polynomial, it is straightforward to see that $\mathsf{q}^\mathbf{gr}_{d-1} ( (\widehat{\mathsf{q}_{d}^\mathbf{gr}} F_d) \otimes G_d) =0$. Since $\mathsf{q}^\mathbf{gr}_{d-1}$ is exact (by Proposition [Proposition 33](#prop:qhat){reference-type="ref" reference="prop:qhat"}), the structure morphism $F_d \rightarrow F_{d-1}$ induces an isomorphism $\mathsf{q}^\mathbf{gr}_{d-1} (F_d \otimes G_d) \cong \mathsf{q}^\mathbf{gr}_{d-1} (F_{d-1} \otimes G_d)$. Repeating the argument with the rôles of $F$ and $G$ reversed gives the required result. *Remark 54*. The definition of $\overline{\otimes}$ given in Definition [Definition 52](#defn:obar){reference-type="ref" reference="defn:obar"} adapts to give the natural symmetric monoidal structure on $\mathcal{F}_d (\mathbf{gr})$, for each $d \in \mathbb{N}$. Namely, for $F, G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr})$, $F\overline{\otimes}_d G := \mathsf{q}^\mathbf{gr}_d (F \otimes G).$ In the following $\mathbb{Q}\in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$ is considered as an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ via the functor $\mathsf{q}^\mathbf{gr}_\bullet$ of Proposition [Proposition 50](#prop:d_evaluate_exact_f_to_propoly){reference-type="ref" reference="prop:d_evaluate_exact_f_to_propoly"}. Moreover, $\mathcal{F}_{< \infty}(\mathbf{gr})$ is considered as symmetric monoidal for the structure inherited from $(\mathcal{F}(\mathbf{gr}), \otimes, \mathbb{Q})$. **Proposition 55**. *The category $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes},\mathbb{Q})$ is symmetric monoidal. Moreover, the functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}_{< \infty}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is symmetric monoidal with respect to this structure.* *Proof.* The first statement is clear. The fact that $\mathsf{q}^\mathbf{gr}_\bullet$ is symmetric monoidal follows by using the argument employed in Remark [Remark 53](#rem:obar_well_defined){reference-type="ref" reference="rem:obar_well_defined"}. ◻ ## Completion and projectives A further relationship between ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and $\mathcal{F}(\mathbf{gr})$ is provided by completion: *Notation 56*. Denote by $\mathfrak{c}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ the completion functor given by $F_\bullet \mapsto \lim_{\substack{\leftarrow \\ n }} F_n$. **Proposition 57**. *The completion functor $\mathfrak{c}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ is right adjoint to $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* *Proof.* By definition of the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, for any two objects $G'_\bullet$ and $G_\bullet$, $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(G'_\bullet, G_\bullet)$ is given by the equalizer of the usual diagram: $$\prod_{n\in \mathbb{N}} \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (G'_n, G_n) \rightrightarrows \prod_{j\in \mathbb{N}} \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (G'_{j+1}, G_j).$$ Taking $G'_\bullet = \mathsf{q}^\mathbf{gr}_\bullet F$, for $F \in \mathcal{F}(\mathbf{gr})$, using the definition of $\mathsf{q}^\mathbf{gr}_n$ as the left adjoint to the inclusion $\mathcal{F}_n (\mathbf{gr}) \hookrightarrow \mathcal{F}(\mathbf{gr})$, this diagram can be rewritten naturally as: $$\prod_{n\in \mathbb{N}} \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (F, G_n) \rightrightarrows \prod_{j\in \mathbb{N}} \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (F, G_j),$$ with structure morphisms induced by the inverse system $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (F, G_\bullet)$. The equalizer thus identifies with $\lim_\leftarrow \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (F, G_\bullet)$. By the universal property of the inverse limit defining $\mathfrak{c}G_\bullet$, the latter is naturally isomorphic to $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (F, \mathfrak{c}G_\bullet)$, as required. ◻ *Remark 58*. Composing the functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ with the completion functor gives the composite $\mathcal{F}(\mathbf{gr}) \rightarrow \mathcal{F}(\mathbf{gr})$ that sends a functor $F$ to the inverse limit $\lim_{\substack{\leftarrow \\ n }} \mathsf{q}^\mathbf{gr}_n F$ of the polynomial filtration of $F$. The adjunction unit is the natural map $F \rightarrow \mathfrak{c}(\mathsf{q}^\mathbf{gr}_\bullet F)$ induced by the canonical surjections $F \twoheadrightarrow q_n F$. (The behaviour of this morphism in the case $F= P_\mathbb{Z}$ is explained in Corollary [Corollary 87](#cor:compl_qgr_unit){reference-type="ref" reference="cor:compl_qgr_unit"}.) **Proposition 59**. *The functor $\mathfrak{c}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ is exact.* *Proof.* This follows from the Mittag-Leffler condition by using the fact (cf. Remark [Remark 48](#rem:propoly_surj){reference-type="ref" reference="rem:propoly_surj"}) that the structure morphisms in the tower of an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ are surjective. This implies that $\lim_\leftarrow^1$ vanishes, as required. ◻ We note the following consequence: **Lemma 60**. *For $G_\bullet$ an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and $n \in \mathbb{N}$, the canonical morphism $\mathfrak{c}G_\bullet \rightarrow G_n$ is surjective.* *Proof.* For the purposes of this proof, write $\overline{G}_\bullet$ for the quotient of $G_\bullet$ with $\overline{G}_t$ equal to $G_t$ for $t \leq n$ and $G_n$ for $t \geq n$, with the obvious structure morphisms. Applying $\mathfrak{c}$ (which is exact by Proposition [Proposition 59](#prop:compl_exact){reference-type="ref" reference="prop:compl_exact"}) gives the surjection $\mathfrak{c}G_\bullet \twoheadrightarrow \mathfrak{c}\overline{G}_\bullet$. The codomain is canonically isomorphic to $G_n$, by construction of $\overline{G}_\bullet$, and the corresponding map identifies with that of the statement, which is thus surjective, as required. ◻ Combining Proposition [Proposition 57](#prop:compl_right_adjoint){reference-type="ref" reference="prop:compl_right_adjoint"} with Proposition [Proposition 59](#prop:compl_exact){reference-type="ref" reference="prop:compl_exact"} allows the identification of a set of projective generators of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$: **Corollary 61**. *For $\Gamma$ a finite rank free group, $\mathsf{q}^\mathbf{gr}_\bullet P_\Gamma$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$; it corepresents the functor $G_\bullet \mapsto (\mathfrak{c}G_\bullet) (\Gamma)$.* *The set of functors $\mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star n}}$, for $n \in \mathbb{N}$, is a set of projective generators for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* *Proof.* The first statement is an immediate consequence of Propositions [Proposition 57](#prop:compl_right_adjoint){reference-type="ref" reference="prop:compl_right_adjoint"} and [Proposition 59](#prop:compl_exact){reference-type="ref" reference="prop:compl_exact"}. For the second statement, consider an object $G_\bullet$ of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Since $\mathcal{F}(\mathbf{gr})$ has set of projective generators $\{ P_{\mathbb{Z}^{\star n}}\ | \ n \in \mathbb{N}\}$, there exists a set of finite rank free groups $\{ \Gamma_i \ | \ i \in \mathcal{I} \}$ and a surjection $$\begin{aligned} \label{eqn:proj_compl} \bigoplus_{i \in \mathcal{I}} P_{\Gamma_i} \twoheadrightarrow \mathfrak{c}G_\bullet.\end{aligned}$$ In particular, for each $n \in \mathbb{N}$, composing with the canonical surjection $\mathfrak{c}G_\bullet \twoheadrightarrow G_n$ given by Lemma [Lemma 60](#lem:surj_compl_eval){reference-type="ref" reference="lem:surj_compl_eval"}, this induces a surjection $\bigoplus_{i \in \mathcal{I}} P_{\Gamma_i} \twoheadrightarrow G_n$. Using the adjunction (which implies that $\mathsf{q}^\mathbf{gr}_\bullet$ commutes with coproducts), ([\[eqn:proj_compl\]](#eqn:proj_compl){reference-type="ref" reference="eqn:proj_compl"}) yields: $\bigoplus_{i \in \mathcal{I}} \mathsf{q}^\mathbf{gr}_\bullet P_{\Gamma_i} \rightarrow G_\bullet$. This is surjective since, by the above observation, the composite $\bigoplus_{i \in \mathcal{I}} P_{\Gamma_i} \twoheadrightarrow \bigoplus_{i \in \mathcal{I}} \mathsf{q}^\mathbf{gr}_n P_{\Gamma_i} \twoheadrightarrow G_n$ is surjective (using the canonical factorization) for each $n \in \mathbb{N}$. ◻ ## Relating to analytic functors on $\mathbf{gr}^\mathrm{op}$ That the category $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ of analytic functors on $\mathbf{gr}^\mathrm{op}$ (introduced in Section [2.6](#subsect:analytic_grop){reference-type="ref" reference="subsect:analytic_grop"}) is related to ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ via duality $D$ is manifest. This is made explicit by the following statement: **Lemma 62**. * * 1. *The duality functor $D :\mathcal{F}(\mathbf{gr}) ^\mathrm{op}\rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op})$ induces an exact functor $D: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$, given on objects by $D(G_\bullet):= \lim_{\substack{\rightarrow \\ n}} (DG_n)$.* 2. *The duality functor $D: \mathcal{F}(\mathbf{gr}^\mathrm{op}) ^\mathrm{op}\rightarrow \mathcal{F}(\mathbf{gr})$ induces an exact functor $D : \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) ^\mathrm{op}\rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, given on objects by $(DF)_n := D (p_n F)$.* *Proof.* Consider the functor $D : \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) ^\mathrm{op}\rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. For an analytic functor $F$ on $\mathbf{gr}^\mathrm{op}$, we require to show that $DF$ as defined is an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. By exactness of $D$ on $\mathcal{F}(\mathbf{gr}^\mathrm{op})$, for any $n \in \mathbb{N}$, it is clear that $\mathsf{p}_{n-1} F \hookrightarrow \mathsf{p}_n F$ induces a surjection $D (\mathsf{p}_n F)\twoheadrightarrow D(\mathsf{p}_{n-1}F)$, with the domain of polynomial degree $n$ and codomain of degree $n-1$. It remains to show that this induces an isomorphism $\mathsf{q}^\mathbf{gr}_{n-1} D(\mathsf{p}_n F) \cong D(\mathsf{p}_{n-1}F)$. For this, consider the short exact sequence $0 \rightarrow \mathsf{p}_{n-1} F \rightarrow \mathsf{p}_n F \rightarrow (\mathsf{p}_n F) / (\mathsf{p}_{n-1} F) \rightarrow 0$. By construction of the polynomial filtration, $(\mathsf{p}_n F) / (\mathsf{p}_{n-1} F)$ is a *homogeneous* polynomial functor of degree $n$. On applying $D$, which is exact, the short exact sequence gives: $$0 \rightarrow D ((\mathsf{p}_n F) / (\mathsf{p}_{n-1} F)) \rightarrow D (\mathsf{p}_n F) \rightarrow D(\mathsf{p}_{n-1}F) \rightarrow 0.$$ By the above, $D ((\mathsf{p}_n F) / (\mathsf{p}_{n-1} F))$ is a a *homogeneous* polynomial functor of degree $n$, in particular applying $\mathsf{q}^\mathbf{gr}_{n-1}$ to this gives $0$. The result follows. ◻ *Remark 63*. The functor $D :{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ may be thought of as a 'continuous dual'. **Proposition 64**. *The duality functors of Lemma [Lemma 62](#lem:propoly_analytic_duality){reference-type="ref" reference="lem:propoly_analytic_duality"} are adjoint: $$D : {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightleftarrows \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) : D.$$* *Proof.* This adjunction extends the duality adjunctions $D : \mathcal{F}_d (\mathbf{gr})^\mathrm{op}\rightleftarrows \mathcal{F}_d (\mathbf{gr}^\mathrm{op}) :D$ between the categories of polynomial functors (for $d \in \mathbb{N}$) as follows. Consider $G_\bullet \in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$, then by definition, $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (G_\bullet , DF)$ is the equalizer of the associated diagram: $$\prod _n \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (G_n, D (\mathsf{p}_n F)) \rightrightarrows \prod _j \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (G_{j+1}, D (\mathsf{p}_jF)).$$ By the duality adjunction, this can be rewritten as $$\prod _n \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (\mathsf{p}_n F, D G_n) \rightrightarrows \prod _j \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (\mathsf{p}_j F, DG_{j+1}).$$ Since $D(G_\bullet)$ is (by construction) analytic on $\mathbf{gr}^\mathrm{op}$, with polynomial filtration such that $\mathsf{p}_n (DG_\bullet) = D G_n$, using the defining property of the functors $\mathsf{p}_n$ this can be rewritten as: $$\prod _n \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (\mathsf{p}_n F, D (G_\bullet)) \rightrightarrows \prod _j \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (\mathsf{p}_j F, D(G_\bullet)).$$ This has equalizer $\lim_{\substack{\leftarrow \\ n}} \mathrm{Hom}_{\mathcal{F}(\mathbf{gr}) } (\mathsf{p}_n F, D (G_\bullet))$, which is isomorphic to $\mathrm{Hom}_{\mathcal{F}_\omega (\mathbf{gr})} (F, D (G_\bullet))$, since $F$ is analytic, by hypothesis. This provides the required natural isomorphism $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (G_\bullet , DF) \cong \mathrm{Hom}_{\mathcal{F}_\omega (\mathbf{gr})} (F, D (G_\bullet))$. ◻ This leads to the following version of $\otimes_\mathbf{gr}$ for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and $\mathcal{F}_\omega(\mathbf{gr}^\mathrm{op})$: **Proposition 65**. *The tensor product $\otimes_\mathbf{gr}: \mathcal{F}(\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr}) \rightarrow \mathtt{Mod}_\mathbbm{k}$ induces $$\ {\widetilde{\otimes}_\mathbf{gr}}: \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \times {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathtt{Mod}_\mathbbm{k}$$ given explicitly for $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ and $G_\bullet \in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ by: $$F \ {\widetilde{\otimes}_\mathbf{gr}}G_\bullet := \lim_{\substack{\rightarrow \\ n}} (\mathsf{p}_n F \otimes _\mathbf{gr}G_n),$$ where the morphisms of the direct system are induced by the structure morphisms.* *There are natural isomorphisms: $$\mathrm{Hom}_{\mathtt{Mod}_\mathbbm{k}} (F \ {\widetilde{\otimes}_\mathbf{gr}}G_\bullet, \mathbbm{k}) \cong \mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (G_\bullet , DF) \cong \mathrm{Hom}_{\mathcal{F}_\omega (\mathbf{gr})} (F, D (G_\bullet)).$$* *Proof.* This is a direct consequence of the isomorphism given by Proposition [Proposition 39](#prop:otimes_gr_poly){reference-type="ref" reference="prop:otimes_gr_poly"}. To illustrate this, consider the structure morphisms of the direct system defining $F \otimes_\mathbf{gr}G_\bullet$. For $n \in\mathbb{N}$, the natural inclusion $\mathsf{p}_n F \hookrightarrow \mathsf{p}_{n+1} F$ and the natural surjection $G_{n+1} \twoheadrightarrow G_n \cong \mathsf{q}^\mathbf{gr}_n (G_{n+1})$ give the (natural) solid arrows in $$\xymatrix{ \mathsf{p}_n F \otimes_\mathbf{gr}G_{n+1} \ar[r] \ar[d]_\cong & \mathsf{p}_{n+1} F \otimes_\mathbf{gr}G_{n+1} \\ \mathsf{p}_n F \otimes_\mathbf{gr}G_{n}, \ar@{.>}[ur] }$$ where the isomorphism is provided by Proposition [Proposition 39](#prop:otimes_gr_poly){reference-type="ref" reference="prop:otimes_gr_poly"}. This yields the required structure morphism, indicated by the dotted arrow making the diagram commute. The given natural isomorphisms extend those of Corollary [Proposition 26](#prop:otimes_gr_duality){reference-type="ref" reference="prop:otimes_gr_duality"}, arguing as in the proof of Proposition [Proposition 64](#prop:D_propoly_analytic_adjoint){reference-type="ref" reference="prop:D_propoly_analytic_adjoint"}. ◻ One may also consider $\otimes_\mathbf{gr}$ restricted to $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr})$. This is compatible with the functor of Proposition [Proposition 65](#prop:otimes_gr_propoly_analytic){reference-type="ref" reference="prop:otimes_gr_propoly_analytic"}: **Proposition 66**. *For $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ and $G \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr})$, there is a natural isomorphism: $$F \otimes_\mathbf{gr}G \cong F \ {\widetilde{\otimes}_\mathbf{gr}}\mathsf{q}^\mathbf{gr}_\bullet G.$$* ## The associated graded There is a functor ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ that generalizes the construction of the associated graded of the polynomial filtration. This is given by $$G_\bullet \mapsto \mathfrak{gr}(G_\bullet):= \bigoplus_{n \in \mathbb{N}} \ker \big( G_n \rightarrow G_{n-1} \big).$$ This can be rewritten $\mathfrak{gr}(G_\bullet):= \bigoplus_{n \in \mathbb{N}} \widehat{\mathsf{q}_{n}^\mathbf{gr}} G_n.$ As in Notation [Notation 34](#nota:fbcr){reference-type="ref" reference="nota:fbcr"}, by composing with the cross-effect functors, one obtains a functor with values in $\mathbb{Q}{\bm{\Sigma}}$-modules (aka. $\mathcal{F}({\bm{\Sigma}})$): *Notation 67*. Denote by $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}({\bm{\Sigma}})$ the functor that associates to $G_\bullet \in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ the ${\bm{\Sigma}}$-module $\mathbf{t} \mapsto \mathrm{cr}_t \widehat{\mathsf{q}_{t}^\mathbf{gr}} (G_t) \cong \mathrm{cr}_t \mathsf{q}^\mathbf{gr}_t (G_t) \cong \mathrm{cr}_t G_t$. *Remark 68*. Using the same notation $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ as in Notation [Notation 34](#nota:fbcr){reference-type="ref" reference="nota:fbcr"} should cause no confusion, since these functors are compatible via $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr}) \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The category of $\mathbb{Q}{\bm{\Sigma}}$-modules is symmetric monoidal with respect to the Day convolution product $\odot$. Here it is useful to recall its construction in terms of representations of the symmetric groups: a $\mathbb{Q}{\bm{\Sigma}}$-module can be considered as as sequence of representations of the symmetric groups $M(n)$, $n \in \mathbb{N}$. For two such, $M(-)$ and $N(-)$, the Day convolution $M\odot N$ has $n$th term $$\bigoplus _{s+t=n} M(s) \otimes N(t) \uparrow_{\mathfrak{S}_s \times \mathfrak{S}_t}^{\mathfrak{S}_n}.$$ The unit is the $\mathbb{Q}{\bm{\Sigma}}$-module $\mathbbm{k}$ (with $\mathbbm{k}(0) = \mathbbm{k}$ and $\mathbbm{k}(s)=0$ otherwise). **Proposition 69**. *The functor $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}({\bm{\Sigma}})$ is exact. Moreover, it is symmetric monoidal with respect to the structures $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes}, \mathbbm{k})$ and $(\mathcal{F}({\bm{\Sigma}}), \odot , \mathbbm{k})$.* *Proof.* That $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ is exact follows from the fact that $\mathrm{cr}_d$ is exact (as recalled in Section [2.5](#subsect:poly_Q){reference-type="ref" reference="subsect:poly_Q"}). It remains to show that it is symmetric monoidal. First consider the behaviour of the cross-effect functor $\mathrm{cr}_d$ on the tensor product of two homogeneous polynomial functors $F:= \alpha_s M(s) =\mathfrak{a}^{\otimes s} \otimes_{\mathfrak{S}_s} M(s)$ and $G:= \alpha _t N(t) = \mathfrak{a}^{\otimes t} \otimes_{\mathfrak{S}_t} N(t)$, for a $\mathfrak{S}_s$-module $M(s)$ and $\mathfrak{S}_t$-module $N(t)$. By construction, $\mathrm{cr}_j \widehat{\mathsf{q}_{j}^\mathbf{gr}} F$ is zero unless $j=s$, when one recovers $M(s)$; similarly for $G$. Now, $F \otimes G = (\mathfrak{a}^{\otimes s} \otimes_{\mathfrak{S}_s} M(s)) \otimes (\mathfrak{a}^{\otimes t} \otimes_{\mathfrak{S}_t} N(t) \cong (\mathfrak{a}^{\otimes s+t} \otimes_{\mathfrak{S}_s \times \mathfrak{S}_t} (M(s)\otimes N(t))$, so that $F \otimes G \cong \mathfrak{a}^{\otimes s+t} \otimes_{\mathfrak{S}_{s+t}} (M(s)\otimes N(t))\uparrow_{\mathfrak{S}_s \times \mathfrak{S}_t} ^{\mathfrak{S}_{s+t}}$. Hence, for $d \in \mathbb{N}$, and the above homogeneous polynomial functors $F$, $G$: $$\mathrm{cr}_d (F \otimes G) \cong \left\{ \begin{array}{ll} (M(s) \otimes N(t))\uparrow_{\mathfrak{S}_s \times \mathfrak{S}_t} ^{\mathfrak{S}_{d}} & d = s+t \\ 0 & \mbox{otherwise}. \end{array} \right.$$ From this, one deduces the result in the homogeneous case. Now suppose that $F$ and $G$ are polynomial. Using the finite length polynomial filtrations (as in Section [2.5](#subsect:poly_Q){reference-type="ref" reference="subsect:poly_Q"}) of $F$ and $G$ respectively and the fact that $\mathrm{cr}_d$ is exact, the above extends to give the natural isomorphism $$\mathrm{cr}_d \mathsf{q}^\mathbf{gr}_d(F \otimes G) \cong \bigoplus_{s+t=d} (\mathrm{cr}_s \mathsf{q}^\mathbf{gr}_s F \otimes \mathrm{cr}_t \mathsf{q}^\mathbf{gr}_t G)\uparrow_{\mathfrak{S}_s \times \mathfrak{S}_t} ^{\mathfrak{S}_{d}}.$$ From this, it is straightforward to deduce the required natural isomorphism for $F$ and $G$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$: $$\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}(F \overline{\otimes}G) \cong \mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}F \odot \mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}G$$ and that this makes $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ a symmetric monoidal functor. ◻ It is a useful fact that $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ is conservative: **Proposition 70**. *An object $G_\bullet$ of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is zero if and only if $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}(G_\bullet) =0$.* *Hence a morphism $f : G_\bullet \rightarrow G'_\bullet$ is an isomorphism (resp. surjective, resp. injective) if and only if $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}f$ is an isomorphism (resp. surjective, resp. injective).* *Proof.* Clearly, if $G_\bullet=0$, then $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}(G_\bullet)=0$. Otherwise, choose $\ell$ minimal such that $G_\ell \neq 0$. Then, by minimality of $\ell$, $\widehat{\mathsf{q}_{\ell}^\mathbf{gr}} G_{\ell} \neq 0$ so that $\mathrm{cr}_\ell G_{\ell} \neq 0$. It follows that $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}(G_\bullet)$ is non-zero. The remaining statements follow using that $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ is exact, by Proposition [Proposition 69](#prop:properties_fbcr_propoly){reference-type="ref" reference="prop:properties_fbcr_propoly"}. ◻ # The group ring functor {#sect:group-ring} The purpose of this section is to review the structure of the group ring functors $G \mapsto \mathbb{Q}G$ when restricted to $\mathbf{gr}$, by using the polynomial filtration and the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ introduced in Section [4](#sect:propoly){reference-type="ref" reference="sect:propoly"}. In particular, this introduces the Passi functors. ## Recollections For $G$ a group, the group ring $\mathbbm{k}G$ is equipped with a natural cocommutative Hopf algebra structure. In particular, the unit is induced by the inclusion $\{ e \} \subset G$ and the augmentation (or counit) by the canonical surjection of groups $G \twoheadrightarrow \{ e\}$. The product $\mathbbm{k}G \otimes \mathbbm{k}G \rightarrow \mathbbm{k}G$ is induced by the multiplication of the group $G$ and the coproduct makes the generators 'grouplike' (i.e., $[g] \mapsto [g] \otimes [g]$). This structure gives the augmentation ideal $\mathcal{I}G \lhd \mathbbm{k}G$; this has basis given by the elements $[g] - [e]$, for $g \in G \backslash \{ e \}$. One has the (natural) filtration of $\mathbbm{k}G$ by powers of the augmentation ideal: $$\ldots \subset \mathcal{I}^n G \subset \mathcal{I}^{n-1} G \subset \ldots \subset \mathcal{I}G \subset \mathbbm{k}G.$$ The associated graded ring $\mathfrak{gr}(\mathbbm{k}G)$ is $\bigoplus_{n \in \mathbb{N}} \mathcal{I}^n G/ \mathcal{I}^{n+1} G$; this is natural in $G$ and it is convenient to write $\mathcal{I}^n/ \mathcal{I}^{n+1} (G)$ for $\mathcal{I}^n G/ \mathcal{I}^{n+1} G$. *Remark 71*. For $d \in \mathbb{N}$, $G \mapsto \mathbbm{k}G / \mathcal{I}^{d+1} G$ is sometimes termed the $d$th Passi functor, in reference to Passi's work [@MR537126]. *Remark 72*. Restricting to $G$ in $\mathbf{gr}$, this structure transposes to the functor category $\mathcal{F}(\mathbf{gr})$. The underlying functor is the projective $P_\mathbb{Z}$. This is equipped with a cocommutative Hopf algebra structure in $\mathcal{F}(\mathbf{gr})$: the unit and counit induce (and are determined by) the natural splitting $P_\mathbb{Z}\cong \mathbbm{k}\oplus \overline{P}$, the product is a natural transformation $P_\mathbb{Z}\otimes P_\mathbb{Z}\rightarrow P_\mathbb{Z}$ (unital and associative), and the coproduct a natural transformation $P_\mathbb{Z}\rightarrow P_\mathbb{Z}\otimes P_\mathbb{Z}$ (counital, cocommutative and coassociative); conjugation is the natural transformation $P_\mathbb{Z}\rightarrow P_\mathbb{Z}$ corepresented by $[-x]$, for $x$ the chosen generator of $\mathbb{Z}$. The product restricts to $\overline{P}\otimes \overline{P}\rightarrow \overline{P}$ and, under the above identifications, the functor $G \mapsto \mathcal{I}^n G$ corresponds to the image of the iterated multiplication $\overline{P}^{\otimes n} \rightarrow \overline{P} \subset P_\mathbb{Z}.$ The quotient $\mathcal{I}/ \mathcal{I}^2 (G)$ is naturally isomorphic to $\mathbbm{k}\otimes _\mathbb{Z}G_\mathsf{ab}$, where $G_\mathsf{ab}$ is the abelianization of $G$. Restricted to $\mathbf{gr}$, this has the following interpretation: **Lemma 73**. *The product of $P_\mathbb{Z}$ gives the exact sequence $\overline{P}\otimes \overline{P}\rightarrow \overline{P}\rightarrow \mathfrak{a}\rightarrow 0$ which induces the isomorphism: $\mathcal{I}/ \mathcal{I}^2 (G) \cong \mathbbm{k}\otimes _\mathbb{Z}G_\mathsf{ab}$.* ## The associated graded and more We use the following notation for the lower central series of a group (which is functorial): *Notation 74*. For $G$ a group, let $(\gamma_t G)$ denote the lower central series of $G$, defined recursively by $\gamma_1 G = G$ and $\gamma_{i+1} G = [\gamma_i G, G]$. Recall the definition of the dimension subgroups of $G$ (with respect to $\mathbbm{k}$): the $n$th dimension subgroup is $$D_{n,\mathbbm{k}} (G) := G \cap (1 + \mathcal{I}^n G) = \{ g \in G \ | \ [g] - [e] \in \mathcal{I}^n G \}.$$ This is a normal subgroup of $G$ and $D_{n+1,\mathbbm{k}}(G) \subseteq D_{n,\mathbbm{k}} (G)$; moreover, $[D_{i,\mathbbm{k}} (G) , D_{j,\mathbbm{k}} (G)] \subseteq D_{i+j,\mathbbm{k}} (G)$ so that there is a natural inclusion $\gamma_t G \subseteq D_{t,\mathbbm{k}} G,$ for all $0< t \in \mathbb{N}$. *Remark 75*. Jennings [@MR68540] identified the dimension subgroups over $\mathbbm{k}= \mathbb{Q}$ as $D_{n , \mathbb{Q}} (G) = \sqrt {\gamma_n G},$ where, for a subgroup $H \subseteq G$, $\sqrt H = \{ x \in G \ |\ \exists 0<n \in \mathbb{N}, x^n \in H \}$. **Example 76**. For $G$ a finite rank free group, Magnus [@MR1581549] showed that $D_{t,\mathbb{Z}}G =\gamma_t G$, $\forall 0< t \in \mathbb{N}$. One also has $D_{t,\mathbb{Q}}G =\gamma_t G$, $\forall 0< t \in \mathbb{N}$. The associated Lie algebra $\mathfrak{l}_\mathbbm{k}G$ (defined over $\mathbb{Z}$) is the associated graded: $$\mathfrak{l}_\mathbbm{k}G := \bigoplus _{0< t \in \mathbb{N}} D_{t,\mathbbm{k}} G / D_{t+1,\mathbbm{k}} G$$ with Lie bracket induced by the group commutator. By construction, there is a map $D_{t, \mathbbm{k}} G \rightarrow \mathcal{I}^t G$ given by $g \mapsto [g] -[e]$ and this induces a morphism of Lie algebras $\mathfrak{l}_\mathbbm{k}G \rightarrow \mathfrak{gr}(\mathbbm{k}G)$, where the algebra $\mathfrak{gr}(\mathbbm{k}G)$ is considered as a Lie algebra for the commutator Lie bracket. Quillen proved: **Theorem 77**. *[@MR231919] For $\mathbbm{k}$ a field of characteristic zero, the induced morphism of Hopf algebras $U (\mathbbm{k}\otimes_\mathbb{Z}\mathfrak{l}_\mathbbm{k}G ) \rightarrow \mathfrak{gr}(\mathbbm{k}G)$ is an isomorphism.* **Example 78**. For $G$ a finite rank free group, by the work of Magnus, $\mathfrak{l}G$ is isomorphic to the free Lie algebra (over $\mathbb{Z}$) on the abelianization $G_\mathsf{ab}$ of $G$ (see, for example, [@MR2109550 Theorem 5.12]). Thus, for $\mathbbm{k}$ a field of characteristic zero, Quillen's theorem gives the isomorphism of Hopf algebras $$U (\mathrm{Lie}(G_\mathsf{ab}\otimes \mathbbm{k}) ) \stackrel{\cong}{\rightarrow} \mathfrak{gr}(\mathbbm{k}G),$$ where $\mathrm{Lie}(G_\mathsf{ab}\otimes \mathbbm{k})$ is the free Lie algebra (over $\mathbbm{k}$) on $G_\mathsf{ab}\otimes \mathbbm{k}$. In particular, this shows that $\mathfrak{gr}(\mathbbm{k}G)$ is isomorphic to the tensor Hopf algebra on $G_\mathsf{ab}\otimes \mathbbm{k}$. These isomorphisms are natural with respect to $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$. Essentially by construction of the dimension subgroups, one has the following (the proof is left to the reader): **Proposition 79**. *For $1 \leq t \leq n \in \mathbb{N}$ and a group $G$, the canonical surjection $G \twoheadrightarrow G/ D_{n,\mathbbm{k}} G$ induces a natural isomorphism of $\mathbbm{k}$-algebras: $$\mathbbm{k}G / \mathcal{I}^t G \stackrel{\cong}{\rightarrow} \mathbbm{k}(G/ D_{n,\mathbbm{k}} G) / \mathcal{I}^t (G/D_{n,\mathbbm{k}} G).$$* *Hence, the canonical projection $G \twoheadrightarrow G/ \gamma_n G$ induces a natural isomorphism of $\mathbbm{k}$-algebras: $$\mathbbm{k}G / \mathcal{I}^t G \stackrel{\cong}{\rightarrow} \mathbbm{k}(G/ \gamma_n G) / \mathcal{I}^t (G/\gamma_n G).$$* *Remark 80*. Since $\gamma_t G \subseteq D_{t, \mathbbm{k}} G$, Proposition [Proposition 79](#prop:passi_nilpotent){reference-type="ref" reference="prop:passi_nilpotent"} means that, when studying the Passi functor $G \mapsto \mathbbm{k}G / \mathcal{I}^t G$, one can reduce to working with nilpotent groups of class $t-1$. In particular, for $G$ a finite rank free group and $\mathbbm{k}= \mathbb{Q}$, since $\gamma_t G = D_{t, \mathbb{Q}} G$, one can replace $G$ by $G / \gamma_t G$. The group $G/ \gamma_t G$ is a finitely-generated free nilpotent group of class $t-1$; it is torsion-free. ## Completion of the group ring Quillen [@MR258031 Appendix A] showed the interest of studying the completion of the group ring $\mathbbm{k}G$ with respect to the $\mathcal{I}G$-adic filtration: $$\widehat{\mathbbm{k}G} = \lim_{\substack{\leftarrow\\ t}}\mathbbm{k}G/ \mathcal{I}^t G,$$ with augmentation ideal $\widehat{\mathcal{I}G}$. This has the structure of a complete Hopf algebra, with coproduct $\Delta : \widehat{\mathbbm{k}G} \rightarrow \widehat{\mathbbm{k}G} \hat{\otimes} \widehat{\mathbbm{k}G}$. One has the primitive elements $\mathscr{P}\widehat{\mathbbm{k}G}$, defined as $\{ x \in \widehat{\mathcal{I}G} \ | \ \Delta x = x \hat{\otimes} 1 + 1 \hat{\otimes} x\}$ and the grouplike elements $\mathscr{G}\widehat{\mathbbm{k}G}$, defined as $\{ x \in 1 + \widehat{\mathcal{I}G} \ | \ \Delta x = x \hat {\otimes } x \}$. The primitives yield a Lie subalgebra of $\widehat{\mathbbm{k}G}$ and the grouplike elements form a subgroup of $\widehat{\mathbbm{k}G} ^{\times}$. There is a natural map $G \rightarrow \mathscr{G}\widehat{\mathbbm{k}G}$. When $\mathbbm{k}$ is a field of characteristic zero, one has $\log$ and $\exp$ defined by the usual power series; Quillen [@MR258031 Proposition A.2.6] showed that these induce filtration-preserving isomorphisms: $$\log : \mathscr{G}\widehat{\mathbbm{k}G} \stackrel{\cong}{\leftrightarrow} \mathscr{P}\widehat {\mathbbm{k}G} : \exp.$$ *Remark 81*. Quillen explained the relationship between the passage from $G$ to $\mathscr{G}\widehat{\mathbb{Q}G}$ and Mal'cev completion of nilpotent groups. (The functor $G \mapsto \mathscr{G}\widehat{\mathbb{Q}G}$ is presented as Mal'cev completion in [@MR3643404 Chapter 8].) Recall that a nilpotent group is uniquely divisible if the map $x \mapsto x^n$ is bijective for each $0 \neq n \in \mathbb{Z}$. In [@MR258031 Corollary A.3.8], Quillen established that, for $G$ a nilpotent group and $j : G \rightarrow \mathscr{G}\widehat{\mathbb{Q}G}$ the canonical map, $j$ is the universal map to a nilpotent, uniquely divisible group and $j$ is characterized up to canonical isomorphism by the following conditions: 1. $\mathscr{G}\widehat{\mathbb{Q}G}$ is nilpotent, uniquely divisible; 2. $\ker (j)$ is the torsion subgroup of $G$; 3. for $x \in \mathscr{G}\widehat{\mathbb{Q}G}$, there exists $0 \neq n \in \mathbb{N}$ such that $x^n \in \mathrm{image} j$. Hence $\mathscr{G}\widehat{\mathbb{Q}G}$ is the Mal'cev completion of the nilpotent group $G$. For example, if $G$ is abelian, then $\mathscr{G}\widehat{\mathbb{Q}G}$ is isomorphic to $G \otimes_\mathbb{Z}\mathbb{Q}$. For a finite rank free group and working over a field of characteristic zero, these structures are well understood: **Example 82**. [@MR258031 Example A.2.11] For $G$ a finite rank free group, $\widehat{\mathbb{Q}G}$ is isomorphic as a complete Hopf algebra to the completed tensor Hopf algebra $\widehat{T} (G_\mathsf{ab}\otimes \mathbb{Q})$. Under this isomorphism, $\mathscr{P}\widehat{\mathbb{Q}G}$ is isomorphic to the completed free Lie algebra $\widehat{\mathrm{Lie}} (G_\mathsf{ab}\otimes \mathbb{Q})$ and there is an isomorphism of complete Hopf algebras $$\widehat{U} (\mathrm{Lie}(G_\mathsf{ab}\otimes \mathbb{Q})) \stackrel{\cong}{\rightarrow} \widehat{\mathbb{Q}G},$$ where $\widehat{U}$ denotes the completion of the universal enveloping algebra with respect to the augmentation ideal filtration. **Proposition 83**. *Let $G$ be a finitely-generated, torsion-free nilpotent group. Then $\mathscr{P}\widehat{\mathbb{Q}G}$ is generated as a $\mathbb{Q}$-vector space by the elements $\{ \log (g) \ | \ g \in G \}$.* *Proof.* The canonical map $G \rightarrow \mathscr{G}\widehat{\mathbb{Q}G}$ is a model for the Mal'cev completion of the finitely-generated, torsion-free nilpotent group $G$ (see Remark [Remark 81](#rem:quillen_case_G_nilpotent){reference-type="ref" reference="rem:quillen_case_G_nilpotent"}). In particular, for each $x \in \mathscr{G}\widehat{\mathbb{Q}G}$, there exists $0<n \in \mathbb{N}$ such that $x^n \in G$. Since $\log (x^n)= n \log (x)$, it follows that $\log (x)$ is in the $\mathbb{Q}$-vector space generated by $\{ \log (g) \ | \ g \in G \}$. Since $\log$ induces an isomorphism between $\mathscr{G}\widehat{\mathbb{Q}G}$ and $\mathscr{P}\widehat{\mathbb{Q}G}$, the result follows. ◻ ## The Passi functors restricted to $\mathbf{gr}$ On restricting to the category of functors on $\mathbf{gr}$, we can exploit the polynomial filtration to study the underlying functor of the group ring functor $G \mapsto \mathbbm{k}G$, which identifies as above with $P_\mathbb{Z}$. The Passi functors are closely related to the polynomial filtration of $P_\mathbb{Z}$, as explained in [@MR3505136]: **Proposition 84**. *For $d \in \mathbb{N}$, the Passi functor $G \mapsto \mathbbm{k}G / \mathcal{I}^{d+1} G$ on $\mathbf{gr}$ is isomorphic to $\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$ and the tower of Passi functors under $\mathbbm{k}G$ identifies with the polynomial tower $$\ldots \twoheadrightarrow \mathsf{q}^\mathbf{gr}_n P_\mathbb{Z} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{n-1} P_\mathbb{Z} \twoheadrightarrow \ldots$$ under $P_\mathbb{Z}$.* *The associated graded, $G \mapsto \mathfrak{gr}(\mathbbm{k}G)$, is naturally isomorphic to the Tensor algebra $T (\mathfrak{a}):= \bigoplus_{n \in \mathbb{N}} \mathfrak{a}^{\otimes n}$, considered as a cocommutative Hopf algebra.* By Proposition [Proposition 55](#prop:propoly_sym_mon){reference-type="ref" reference="prop:propoly_sym_mon"}, $\mathsf{q}^\mathbf{gr}_\bullet$ is symmetric monoidal when restricted to $\mathcal{F}_{<\infty}(\mathbf{gr})$. We would like to be able to apply this to tensor products of $P_\mathbb{Z}$, but this does not belong to $\mathcal{F}_{<\infty} (\mathbf{gr})$. Instead, we proceed directly: **Proposition 85**. *For $d, t \in \mathbb{N}$, the natural projection $P_\mathbb{Z}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$ induces an isomorphism: $$\mathsf{q}^\mathbf{gr}_d (P_\mathbb{Z}^{\otimes t}) \stackrel{\cong}{\rightarrow} \mathsf{q}^\mathbf{gr}_d ((\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z})^{\otimes t} ).$$* *Hence, there is a natural isomorphism $\mathsf{q}^\mathbf{gr}_\bullet (P_\mathbb{Z}^{\otimes t} ) \cong (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t}$ and this is $\mathfrak{S}_t$-equivariant.* *Proof.* The final conclusion follows directly from the first statement, so we concentrate on this. By construction, the surjection $P_\mathbb{Z}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$ induces a surjection $$\mathsf{q}^\mathbf{gr}_d (P_\mathbb{Z}^{\otimes t}) \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d ((\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z})^{\otimes t} ).$$ We require to show that this is an isomorphism. For this, it is sufficient to show that the canonical projection $P_\mathbb{Z}^{\otimes t} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_d (P_\mathbb{Z}^{\otimes t})$ factors across the surjection $P_\mathbb{Z}^{\otimes t} \twoheadrightarrow (\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z})^{\otimes t}$. Since $\mathsf{q}^\mathbf{gr}_d (P_\mathbb{Z}^{\otimes t})$ is polynomial of degree $d$, it suffices to show that any map $P_{\mathbb{Z}}^{\otimes t} \rightarrow F$, where $F$ has polynomial degree $d$, factors across $(\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z})^{\otimes t}$ By Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}, there is an exact sequence in $\mathcal{F}(\mathbf{gr})$ $$\overline{P}^{\otimes d+1} \rightarrow P_\mathbb{Z}\rightarrow \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}\rightarrow 0,$$ where the first map is induced by the product of $P_\mathbb{Z}$. This induces the following presentation of $(\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}) ^{\otimes t}$: $$\bigoplus _{i=1}^t P_\mathbb{Z}^{\otimes i-1} \otimes (\overline{P}^{\otimes d+1}) \otimes P_\mathbb{Z}^{\otimes t-i} \rightarrow P_\mathbb{Z}^{\otimes t} \rightarrow (\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z})^{\otimes t} \rightarrow 0,$$ To prove the result, it suffices to show that, for $F$ polynomial of degree $d$ and any $i \in \{1, \ldots , t\}$: $$\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (P_\mathbb{Z}^{\otimes i-1} \otimes (\overline{P}^{\otimes d+1}) \otimes P_\mathbb{Z}^{\otimes t-i}, F ) =0.$$ By Proposition [Proposition 12](#prop:dgr_adjoint){reference-type="ref" reference="prop:dgr_adjoint"}, one has $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (P_\mathbb{Z}^{\otimes i-1} \otimes (\overline{P}^{\otimes d+1}) \otimes P_\mathbb{Z}^{\otimes t-i}, F ) \cong \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (P_\mathbb{Z}^{\otimes t-1} , (\delta^{\mathbf{gr}})^{d+1} F )$. Since $F$ has polynomial degree $d$ by hypothesis, $(\delta^{\mathbf{gr}})^{d+1} F=0$ and the result follows. ◻ The category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ provides a way of modelling the completed group ring functor (restricted to $\mathbf{gr}$): **Proposition 86**. *The cocommutative Hopf algebra structure of $P_\mathbb{Z}$ in $\mathcal{F}(\mathbf{gr})$ induces a cocommutative Hopf algebra structure on $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ (with respect to $\overline{\otimes}$).* *In particular, the coproduct induces the reduced coproduct: $$\overline{\Delta} : \mathsf{q}^\mathbf{gr}_\bullet \overline{P}\rightarrow \mathsf{q}^\mathbf{gr}_\bullet \overline{P}\overline{\otimes}\mathsf{q}^\mathbf{gr}_\bullet \overline{P}.$$* *Moreover, there is a natural isomorphism $\widehat{\mathbb{Q}G} \cong \mathfrak{c}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}) (G)$ of complete Hopf algebras, for $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$.* *Proof.* The first statement follows by using the properties of $\mathsf{q}^\mathbf{gr}_\bullet$, in particular Proposition [Proposition 85](#prop:qgr_tensor_Pzed){reference-type="ref" reference="prop:qgr_tensor_Pzed"}. The final statement is an immediate consequence of the identification of the Passi filtration given in Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}. ◻ **Corollary 87**. *The adjunction unit $P_\mathbb{Z}\rightarrow \mathfrak{c}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ evaluated on $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$ identifies with the canonical morphism $\mathbb{Q}G \rightarrow \widehat{\mathbb{Q}G}.$ In particular, $P_\mathbb{Z}\rightarrow \mathfrak{c}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ is injective.* *Proof.* The first statement follows from Proposition [Proposition 86](#prop:P_zed_propoly){reference-type="ref" reference="prop:P_zed_propoly"}, together with the fact that the adjunction unit is induced by the canonical surjections $P_\mathbb{Z}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$. The injectivity follows from the fact that finitely-generated free groups are residually nilpotent, as established by Magnus. ◻ ## Dualizing the group ring functor One can dualize the functor $G \mapsto \mathbbm{k}G$ (here $G$ can be any discrete group and $\mathbbm{k}$ an arbitrary commutative unital ring); this gives the functor $G \mapsto \mathbbm{k}^G$, the set of maps from $G$, since $\mathrm{Hom}_\mathbbm{k}(\mathbbm{k}G, \mathbbm{k}) \cong \mathbbm{k}^G$. This gives a contravariant functor of $G$. However, for current purposes, the functor $G \mapsto \mathbbm{k}^G$ is too large. Instead, we use a 'continuous dual' with respect to the filtration by powers of the augmentation ideal. **Definition 88**. For $s \in \mathbb{N}$, let 1. $\mathbf{pmap}_s (G, \mathbbm{k})$ be the submodule of $\mathbbm{k}^G$ given by $\mathrm{Hom}_\mathbbm{k}(\mathbbm{k}G / \mathcal{I}^{s+1} G, \mathbbm{k})$; 2. $\mathbf{pmap}_\infty (G, \mathbbm{k})$ be the colimit $\lim_{\substack{\rightarrow \\ s} } \mathbf{pmap}_s (G, \mathbbm{k})$, for the direct system induced by the filtration by powers of the augmentation ideal. *Remark 89*. The notation $\mathbf{pmap}_s (G, \mathbbm{k})$ reflects the fact that this can be viewed as the module of 'polynomial maps of degree $s$' from $G$ to $\mathbbm{k}$. By construction, $G \mapsto \mathbf{pmap}_s (G, \mathbbm{k})$ and $G \mapsto \mathbf{pmap}_\infty (G, \mathbbm{k})$ are contravariant functors of $G$. Moreover, there are natural inclusions: $$\mathbf{pmap}_s (G, \mathbbm{k}) \hookrightarrow \mathbf{pmap}_\infty (G, \mathbbm{k}) \hookrightarrow \mathbbm{k}^G.$$ Restricting to $\mathbf{gr}$ and taking $\mathbbm{k}= \mathbb{Q}$, one identifies: **Proposition 90**. *For $s\in \mathbb{N}$, there are natural isomorphisms: $$\begin{aligned} \mathbf{pmap}_s (-, \mathbb{Q}) & \cong & D (\mathsf{q}^\mathbf{gr}_s P_\mathbb{Z}) \\ \mathbf{pmap}_\infty (-, \mathbb{Q}) & \cong & D (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}),\end{aligned}$$ where for the second isomorphism, $D$ is the duality functor ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$.* *In particular, $\mathbf{pmap}_s (-, \mathbb{Q})$ is a polynomial functor of degree $s$ and $\mathbf{pmap}_\infty (-, \mathbb{Q})$ is analytic and is not polynomial.* *Remark 91*. Proposition [Proposition 90](#prop:dualize_passi){reference-type="ref" reference="prop:dualize_passi"} generalizes to consider the duals of $\mathsf{q}^\mathbf{gr}_s P_{(-)}$ and $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}$ respectively, carrying over the bifunctoriality. If one neglects the contravariant functoriality of $P_{(-)}$, for $n \in \mathbb{N}$, the functor $D (\mathsf{q}^\mathbf{gr}_s P_{\mathbb{Z}^{\star n}})$ is isomorphic to $G \mapsto \mathbf{pmap}_s (G^{\times n}, \mathbb{Q})$ and $D (\mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star n}})$ is isomorphic to $G \mapsto \mathbf{pmap}_\infty (G^{\times n}, \mathbb{Q})$. **Corollary 92**. *For $n \in \mathbb{N}$, the functor $G \mapsto \mathbf{pmap}_\infty (G^{\times n}, \mathbb{Q})$ is injective in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op};\mathbb{Q})$.* *Proof.* Since $\mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star n}}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, by Corollary [Corollary 61](#cor:proj_gen_propoly){reference-type="ref" reference="cor:proj_gen_propoly"}, this follows from the duality adjunction of Proposition [Proposition 64](#prop:D_propoly_analytic_adjoint){reference-type="ref" reference="prop:D_propoly_analytic_adjoint"}, together with the exactness of the duality functors given by Lemma [Lemma 62](#lem:propoly_analytic_duality){reference-type="ref" reference="lem:propoly_analytic_duality"}. ◻ ## An explicit description of $\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$ {#subsect:describe_qgr_d_PZ} Fix $0< d \in \mathbb{N}$ and work over $\mathbb{Q}$, for convenience. By Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}, the associated graded of the polynomial filtration of $\mathsf{q}^\mathbf{gr}_d \overline{P}$ is: $$\mathfrak{gr}( \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}) \cong \bigoplus_{i=0}^d \mathfrak{a}^{\otimes i}.$$ The right hand side is well-understood as a functor on $\mathbf{gr}$. However, this is not sufficient to describe the full structure of $\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$. Recall that $\overline{P}(G) \cong \mathcal{I}G$, for $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, and has basis $\{[g]- [e] \ | \ g \in G \backslash \{ e \} \}$. Moreover, one has the following standard relation between the product in $\mathcal{I}G$ and that in $G$ (this has already been used implicitly in identifying the associated graded in Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}): **Lemma 93**. *For $g, h \in G$, one has $$([g]- [e]) ([h]- [e]) = ([gh]- [e]) -([g]- [e]) -([h]- [e]).$$ In particular, $$([g]- [e]) ([g^{-1}]- [e]) = -([g]- [e]) -([g^{-1}]- [e]) = ([g^{-1}]- [e])([g]- [e]).$$ Hence, for any $1 \leq t \in \mathbb{N}$: $$[g_i^{-1}] - [e] = \sum_{s=1}^t (-1)^s ([g_i] - [e]) ^s + (-1)^t ([g_i] - [e]) ^t ([g^{-1}] - [e]) .$$* To proceed, we need to recall the structure of $\mathbf{gr}$ (cf. [@2021arXiv211001934P Appendix A] for example). Consider the free group functor $\mathrm{Free} : \mathbf{Fin}\rightarrow \mathbf{gr}$, where $\mathbf{Fin}$ is the category of finite sets. This preserves coproducts, is faithful and is essentially surjective on objects. However, it is not full. Using the symmetric monoidal structure of $\mathbf{gr}$, the remaining morphisms can be obtained from: $$\begin{aligned} p : \mathbb{Z}& \rightarrow & \{e \} \\ \chi : \mathbb{Z}& \rightarrow & \mathbb{Z}\\ \nabla : \mathbb{Z}& \rightarrow & \mathbb{Z}\star \mathbb{Z}\end{aligned}$$ where $\chi$ is the passage to the inverse $x \mapsto x^{-1}$ and $\nabla$ is the cogroup structure of $\mathbb{Z}$ given by sending the generator $x \in \mathbb{Z}$ to $x_1 x_2$, where $x_i$ are the generators of the copies of $\mathbb{Z}$ in the free product. For a $\mathbb{Q}$-vector space $V$, write $T^{\leq d} (V)$ for the quotient of the tensor algebra $T(V)$ obtained by killing the two-sided ideal generated by $V^{\otimes d+1}$, so that $V \mapsto T^{\leq d}(V)$ is a functor that takes values in associative algebras. **Proposition 94**. *For $1<d \in \mathbb{N}$, $\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$ is isomorphic to the functor with values in associative algebras $$\mathrm{Free}(S) \mapsto T^{\leq d} (\mathbb{Q}S),$$ where $\mathbb{Q}S$ is the $\mathbb{Q}$-vector space generated by the finite set $S$, such that:* 1. *morphisms in the image of $\mathrm{free} : \mathbf{Fin}\rightarrow \mathbf{gr}$ act via $\mathbb{Q}( -) \ : \ \mathbf{Fin}\rightarrow \mathrm{mod}_\mathbb{Q}$ and the naturality of $T^{\leq d} (-)$;* 2. *$p : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \{ e \} = \mathrm{Free} (\emptyset)$ sends $x \in T^{\leq d} (\mathbb{Q}\{x\})$ to zero;* 3. *$\chi : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \mathbb{Z}= \mathrm{Free}(x)$ sends $x \in T^{\leq d} (\mathbb{Q}\{x\})$ to $\sum_{s=1}^d (-1)^s x^s$;* 4. *$\nabla : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \mathbb{Z}\star \mathbb{Z}= \mathrm{Free} (x_1, x_2)$ acts via $x \mapsto x_1 + x_2 + x_1 x_2$.* *Proof.* For a finite set $S$, the universal property of the truncated tensor algebra $T^{\leq d}(-)$ functor induces $$T^{\leq d} (\mathbb{Q}S) \rightarrow \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}(\mathrm{Free}(S))$$ by sending a generator corresponding to $x \in S$ to the class of $[x]-[e]$. This is clearly a natural transformation with respect to $\mathbf{Fin}$ and is an isomorphism by Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}. It remains to check that it is a natural transformation with respect to $\mathbf{gr}$, using the structure on the domain defined in the statement. This follows immediately from the relations given in Lemma [Lemma 93](#lem:aug_G_relations){reference-type="ref" reference="lem:aug_G_relations"}. ◻ # The Mal'cev functors {#sect:malcev} This section serves to introduce a convenient set of projective generators of the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, working over $\mathbbm{k}= \mathbb{Q}$. Namely, $\mathfrak{malcev}$ is introduced as a Lie algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$; the underlying object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is projective and the tensor products $\mathfrak{malcev}^{\overline{\otimes}s}$, $s \in \mathbb{N}$, provide a family of projective generators. ## Introducing $\mathfrak{malcev}$ There are various ways of introducing the Mal'cev Lie algebras; here we adapt the approach of Quillen [@MR258031 Appendix A] (as reviewed in Section [5](#sect:group-ring){reference-type="ref" reference="sect:group-ring"}), who defined the Mal'cev Lie algebra of a group $G$ as $\mathscr{P}\widehat{\mathbb{Q}G}$, the primitives of the completed group ring. We restrict to the category $\mathbf{gr}$ and work with the category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ (equipped with its symmetric monoidal structure $\overline{\otimes}$) rather than completing. Recall from Proposition [Proposition 86](#prop:P_zed_propoly){reference-type="ref" reference="prop:P_zed_propoly"} that $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is a cocommutative Hopf algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. **Definition 95**. The Mal'cev functor $\mathfrak{malcev}$ is the kernel (in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$) of the reduced coproduct: $$\overline{\Delta} : \mathsf{q}^\mathbf{gr}_\bullet \overline{P}\rightarrow \mathsf{q}^\mathbf{gr}_\bullet \overline{P}\overline{\otimes}\mathsf{q}^\mathbf{gr}_\bullet \overline{P}.$$ For $d \in \mathbb{N}$, the $d$th Mal'cev functor is $\mathfrak{malcev}_d \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_d (\mathbf{gr})$ (i.e., the $d$th polynomial component of $\mathfrak{malcev}$). There are natural inclusions $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet\overline{P}\subset \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ and $\mathfrak{malcev}_d \hookrightarrow \mathsf{q}^\mathbf{gr}_d \overline{P}\subset \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$, for $d \in \mathbb{N}$. **Proposition 96**. * * 1. *The Mal'cev functor $\mathfrak{malcev}$ is a Lie algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* 2. *The inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is a morphism of Lie algebras, where $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is equipped with the commutator Lie structure arising from its associative algebra structure given by Proposition [Proposition 86](#prop:P_zed_propoly){reference-type="ref" reference="prop:P_zed_propoly"}.* 3. *The completion $\mathfrak{c}(\mathfrak{malcev})$ is naturally isomorphic to the underlying functor of $G \mapsto \mathscr{P}\widehat{\mathbb{Q}G}$.* *Proof.* The first two statements follow from the fact that $\mathfrak{malcev}$ is defined as the primitives of the Hopf algebra structure of $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ given by Proposition [Proposition 86](#prop:P_zed_propoly){reference-type="ref" reference="prop:P_zed_propoly"}. On applying the completion functor, one recovers $\mathscr{P}\widehat{\mathbb{Q}G}$, using that $\mathfrak{c}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\overline{\otimes}\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ yields the functor $G \mapsto \widehat{\mathbb{Q}G} \hat{\otimes } \widehat {\mathbb{Q}G}$. ◻ The above structure is described at the level of the associated graded objects as follows (compare Proposition [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"}): **Proposition 97**. * * 1. *The associated graded of $\mathfrak{malcev}$ is isomorphic (as a Lie algebra in $\mathcal{F}(\mathbf{gr})$) to $\mathrm{Lie}(\mathfrak{a})$, the functor $\mathfrak{a}$ composed with the free Lie algebra functor $\mathrm{Lie}(-)$.* 2. *The associated graded of the inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ identifies as the canonical inclusion $\mathrm{Lie}(\mathfrak{a}) \hookrightarrow T (\mathfrak{a})$ of Lie algebras in $\mathcal{F}(\mathbf{gr})$.* *Proof.* Proposition [Proposition 69](#prop:properties_fbcr_propoly){reference-type="ref" reference="prop:properties_fbcr_propoly"} implies that the passage to the associated graded is an exact, symmetric monoidal functor from ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ to $\mathcal{F}(\mathbf{gr})$. Now, the associated graded of the Hopf algebra $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is the tensor algebra $T (\mathfrak{a})$ (considered as a primitively-generated Hopf algebra), by Propositions [Proposition 84](#prop:passi_filt_polynomial){reference-type="ref" reference="prop:passi_filt_polynomial"} and [Proposition 86](#prop:P_zed_propoly){reference-type="ref" reference="prop:P_zed_propoly"}. The result thus follows from the definition of $\mathfrak{malcev}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. ◻ The universal enveloping algebra construction applies in the symmetric monoidal category ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$; it is defined as the left adjoint to the restriction from unital associative algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ to Lie algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ given by the commutator bracket. This is analogous to the classical case. Hence, one can construct $U \mathfrak{malcev}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, which is a cocommutative Hopf algebra with primitives $\mathfrak{malcev}$. **Proposition 98**. *The inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ induces an isomorphism $U \mathfrak{malcev}\stackrel{\cong}{\rightarrow} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ of cocommutative Hopf algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* *Proof.* The morphism is constructed using the universal property of $U(-)$ from the inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ and this is a morphism of cocommutative Hopf algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. It remains to show that this is an isomorphism. By Proposition [Proposition 70](#prop:fbcr_conservative){reference-type="ref" reference="prop:fbcr_conservative"}, it is sufficient to check this after passage to the associated graded, when it identifies with the isomorphism $U (\mathrm{Lie}(\mathfrak{a})) \stackrel{\cong}{\rightarrow} T (\mathfrak{a})$ induced by the inclusion $\mathrm{Lie}(\mathfrak{a}) \rightarrow T(\mathfrak{a})$ given in Proposition [Proposition 97](#prop:assoc_graded_malcev){reference-type="ref" reference="prop:assoc_graded_malcev"}. ◻ ## The splitting Recall from Proposition [Proposition 96](#prop:malcev_first_properties){reference-type="ref" reference="prop:malcev_first_properties"} that $\mathfrak{c}(\mathfrak{malcev})$ is isomorphic to the underlying functor of $G \mapsto \mathscr{P}\widehat{\mathbb{Q}G}$. **Lemma 99**. *There is an isomorphism $\mathfrak{c}(\mathfrak{malcev})(\mathbb{Z}) \cong \mathbb{Q}$, with generator given by $\log (x)$, for $x$ a generator of $\mathbb{Z}$.* *Proof.* Since $\mathbb{Z}$ is a torsion-free nilpotent group, Proposition [Proposition 83](#prop:torsion-free_nilp_prim){reference-type="ref" reference="prop:torsion-free_nilp_prim"} applies: hence $\mathfrak{c}(\mathfrak{malcev})(\mathbb{Z})$ is generated as a $\mathbb{Q}$-vector space by the elements $\log (g)$, for $g \in \mathbb{Z}$. Since $\log (g)$ is always a scalar multiple of $\log (x)$, the result follows. ◻ Now, using the adjunction of Proposition [Proposition 57](#prop:compl_right_adjoint){reference-type="ref" reference="prop:compl_right_adjoint"}, the element $\log (x) \in \mathfrak{c}(\mathfrak{malcev}) (\mathbb{Z})$ gives the map $\pi : \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\rightarrow \mathfrak{malcev}$. **Lemma 100**. *The map $\pi : \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\rightarrow \mathfrak{malcev}$ is surjective.* *Proof.* We require to show that, for each $d\in \mathbb{N}$, the corresponding map $\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}\rightarrow \mathfrak{malcev}_d$ is surjective. By adjunction, this map corresponds to a map $P_\mathbb{Z}\rightarrow \mathfrak{malcev}_d$ and hence to an element of $\mathfrak{malcev}_d (\mathbb{Z})$. By construction, this element is the image of $\log (x)$ under the surjection $\mathfrak{c}(\mathfrak{malcev}) (\mathbb{Z}) \twoheadrightarrow \mathfrak{malcev}_d (\mathbb{Z})$. Hence we require to prove that, for any $G \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, $\mathfrak{malcev}_d (G)$ is generated as a $\mathbb{Q}$-vector space by the images of the elements $\log (g)$, for $g \in G$. By Quillen's results (reviewed in Section [5](#sect:group-ring){reference-type="ref" reference="sect:group-ring"}), $\mathfrak{c}(\mathfrak{malcev}) (G)$ is given by the set of elements $\log (\tilde{g})$, for $\tilde{g}$ in $\mathscr{G}\widehat{\mathbb{Q}G}$. Hence $\mathfrak{malcev}_d (G)$ is certainly generated by the set of images of these elements. By Proposition [Proposition 79](#prop:passi_nilpotent){reference-type="ref" reference="prop:passi_nilpotent"}, for $G$ a finitely-generated free group, the quotient $G \twoheadrightarrow G/ \gamma_{d+1} G$ induces an isomorphism: $$\mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}(G) \cong \mathbb{Q}(G/\gamma_{d+1} G) / \mathcal{I}^{d+1}( G/\gamma_{d+1} G).$$ Now, Proposition [Proposition 83](#prop:torsion-free_nilp_prim){reference-type="ref" reference="prop:torsion-free_nilp_prim"} implies that, for $\tilde{g}$ in $\mathscr{G}\widehat{\mathbb{Q}G}$ the image of $\log (\tilde{g})$ in $\widehat{\mathbb{Q}(G/\gamma_{d+1} G)}$ is a $\mathbb{Q}$-linear combination of images of elements of the form $\log (g)$ for $g \in G$. Hence, by using the above isomorphism, the result follows. ◻ Recall that there is a splitting $P_\mathbb{Z}\cong \mathbb{Q}\oplus \overline{P}$ and an isomorphism $\mathsf{q}^\mathbf{gr}_1 \overline{P}\cong \mathfrak{a}$. The composite surjection $P_\mathbb{Z}\twoheadrightarrow \mathfrak{a}$ corresponds (under the corepresenting property of $P_\mathbb{Z}$) to the generator of $\mathfrak{a}(\mathbb{Z})$ given by the chosen generator of $\mathbb{Z}$. *Notation 101*. Write $\rho : \mathfrak{malcev}\rightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}$ for the morphism of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ given by the composite of the canonical inclusion $\mathfrak{malcev}\subset \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ with the morphism induced by the surjection $P_\mathbb{Z}\twoheadrightarrow \mathfrak{a}$. On completion, $\rho$ gives $\mathfrak{c}(\rho) : \mathfrak{c}(\mathfrak{malcev}) \rightarrow \mathfrak{a}$ in $\mathcal{F}(\mathbf{gr})$. **Lemma 102**. *Evaluating $\mathfrak{c}(\rho) : \mathfrak{c}(\mathfrak{malcev}) \rightarrow \mathfrak{a}$ on $\mathbb{Z}$ gives an isomorphism $\mathbb{Q}\cong \mathfrak{c}(\mathfrak{malcev})(\mathbb{Z}) \stackrel{\cong}{\rightarrow} \mathfrak{a}(\mathbb{Z}) \cong \mathbb{Q}.$* *Proof.* By Lemma [Lemma 99](#lem:compl_malcev_zed){reference-type="ref" reference="lem:compl_malcev_zed"}, a generator of $\mathfrak{c}(\mathfrak{malcev})(\mathbb{Z})$ is given by $\log (x)$ (for the chosen generator $x$ of $\mathbb{Z}$). Under the map $\mathfrak{c}(\rho)$, it is straightforward to check that this maps to the generator of $\mathfrak{a}(\mathbb{Z})$ given by the class of $x$. ◻ Recall that $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, by Corollary [Corollary 61](#cor:proj_gen_propoly){reference-type="ref" reference="cor:proj_gen_propoly"}. **Proposition 103**. * * 1. *The surjection $\pi : \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$ is a retract of the defining inclusion $\mathfrak{malcev}\subset \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$.* 2. *$\mathfrak{malcev}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$; it corepresents the functor $G_\bullet \mapsto \mathrm{cr}_1 G_1$.* *Proof.* For the first statement, we require to show that the composite $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$ is the identity. Since $\pi : \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$ is surjective by Lemma [Lemma 100](#lem:pi_surjective){reference-type="ref" reference="lem:pi_surjective"}, it suffices to show that the composite $$\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\stackrel{\pi}{\twoheadrightarrow} \mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$$ coincides with $\pi : \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$. By adjunction, a map $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev}$ corresponds to a map $P_\mathbb{Z}\rightarrow \mathfrak{c}(\mathfrak{malcev})$ and hence to an element of $\mathfrak{c}(\mathfrak{malcev}) (\mathbb{Z})$. Now, by Lemma [Lemma 102](#lem:compl_rho_iso){reference-type="ref" reference="lem:compl_rho_iso"}, $\mathfrak{c}(\rho)$ evaluated on $\mathbb{Z}$ is an isomorphism. Using this, one reduces to showing that the composite $$\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev} \stackrel{\rho}\rightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}$$ coincides with $\rho$. Now, the composite $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\twoheadrightarrow \mathfrak{malcev} \stackrel{\rho}\rightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}$ can easily be seen to identify with $\mathsf{q}^\mathbf{gr}_\bullet (P_\mathbb{Z}\rightarrow \mathfrak{a})$, where the morphism $P_\mathbb{Z}\rightarrow \mathfrak{a}$ is the surjection used in the construction of $\rho$. The result follows. ◻ ## A convenient set of projective generators of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ Proposition [Proposition 103](#prop:malcev_s=1){reference-type="ref" reference="prop:malcev_s=1"} allows the construction of a useful set of projective generators for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. **Theorem 104**. *For $s \in \mathbb{N}$, $\mathfrak{malcev}^{\overline{\otimes}s}$* 1. *is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$;* 2. *is the projective cover of $\mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$;* 3. *corepresents the functor $G_\bullet \mapsto \mathrm{cr}_s G_s$, where the $\mathfrak{S}_s$-action on $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\mathfrak{malcev}^{\overline{\otimes}s},G_\bullet )$ is given by place permutations of the factors $\mathfrak{malcev}$ with respect to $\overline{\otimes}$.* *Proof.* The case $s=0$ is straightforward, so we suppose that $s>0$. Since $\mathfrak{malcev}^{\overline{\otimes}s}$ is a direct summand of $(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}s}$, to establish projectivity, it suffices to show that the latter is projective. The exponential property of $P_{(-)}$ gives the isomorphism $P_\mathbb{Z}^{\otimes s} \cong P_{\mathbb{Z}^{\star s}}$. Proposition [Proposition 85](#prop:qgr_tensor_Pzed){reference-type="ref" reference="prop:qgr_tensor_Pzed"} thus gives the isomorphism $\mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star s}} \cong (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}s}$, hence the latter is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, by Corollary [Corollary 61](#cor:proj_gen_propoly){reference-type="ref" reference="cor:proj_gen_propoly"}. The surjection $\rho : \mathfrak{malcev}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}$ induces $\mathfrak{malcev}^{\overline{\otimes}s} \twoheadrightarrow (\mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a})^{\overline{\otimes}s} \cong \mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$, using the symmetric monoidal property of $\mathsf{q}^\mathbf{gr}_\bullet$ for the isomorphism. Moreover, by construction, the kernel $K$ of this surjection in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ has the property $K_t =0$ for $t \leq s$. To show that the surjection exhibits $\mathfrak{malcev}^{\overline{\otimes}s}$ as the projective cover of $\mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$, we require to show that, if $Y \subset \mathfrak{malcev}^{\overline{\otimes}s}$ such that the composite surjects to $\mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$, then $Y = \mathfrak{malcev}^{\overline{\otimes}s}$. Suppose otherwise, so that $X:= \mathfrak{malcev}^{\overline{\otimes}s} / Y \neq 0$. This has the property $X_t = 0$ for $t \leq s$ and there exists a minimal $\ell >s$ such that $X_\ell \neq 0$. Minimality implies that $\mathsf{q}^\mathbf{gr}_{\ell -1} X_\ell =0$, so that $X_\ell$ is homogeneous polynomial degree of $\ell$, so that there is a non-zero map $X_\ell \rightarrow \mathfrak{a}^{\otimes \ell}$. This gives a non zero-map $\mathfrak{malcev}^{\overline{\otimes}s} \rightarrow \mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes \ell})$, where $\ell > s$. We show that this is not possible, hence establishing a contradiction. One checks that it suffices to show that $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} ((\mathfrak{malcev}_\ell) ^{\otimes s}, \mathfrak{a}^{\otimes \ell})$ is zero. Now $\mathfrak{malcev}_\ell$ is a quotient of $\overline{P}$, hence it suffices to show that $\mathrm{Hom}_{\mathcal{F}(\mathbf{gr})} (\overline{P}^{\otimes s}, \mathfrak{a}^{\otimes \ell})$ is zero; this follows from Lemma [Lemma 20](#lem:qgr_tensor){reference-type="ref" reference="lem:qgr_tensor"}. Finally, we show that that $\mathfrak{malcev}^{\overline{\otimes}s}$ corepresents the functor $G_\bullet \mapsto \mathrm{cr}_s G_s$. As in the proof of Lemma [Lemma 60](#lem:surj_compl_eval){reference-type="ref" reference="lem:surj_compl_eval"}, write $\overline{G}_\bullet$ for the quotient of $G_\bullet$ with $\overline{G}_t$ equal to $G_t$ for $t \leq s$ and $G_s$ for $t \geq s$, with the obvious structure morphisms. Consider the associated short exact sequence: $$0 \rightarrow K \rightarrow G_\bullet \rightarrow \overline{G}_\bullet \rightarrow 0$$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. By construction, $K_t =0$ for $t \leq s$. Arguing as above, one shows that $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\mathfrak{malcev}^{\overline{\otimes}s}, \overline{G}_\bullet)$ is isomorphic to $\mathrm{cr}_s G_s$. Hence, to prove the result, it suffices to show that $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\mathfrak{malcev}^{\overline{\otimes}s}, K)=0$. Suppose that such a non-zero map exists and let $X$ be its image, a non-zero quotient of $\mathfrak{malcev}^{\overline{\otimes}s}$ with $X_t =0$ for $t \leq s$. As above, this is not possible, establishing the contradiction. ◻ **Corollary 105**. *The set $\{ \mathfrak{malcev}^{\overline{\otimes}s} \ | \ s\in \mathbb{N}\}$ is a set of projective generators for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* *Proof.* Consider $G_\bullet \in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. For each $s \in \mathbb{N}$, using the corepresenting property of $\mathfrak{malcev}^{\overline{\otimes}s}$, one has the map $$\varphi_s : \mathrm{cr}_s G_s \otimes \mathfrak{malcev}^{\overline{\otimes}s} \rightarrow G_\bullet$$ that is adjoint to the identity on $\mathrm{cr}_s G_s$ (considered as a vector space). Here $\mathrm{cr}_s G_s \otimes \mathfrak{malcev}^{\overline{\otimes}s}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Summing over $s$, the maps $\varphi_s$ yield the map $(\varphi_s) : \bigoplus_{s \in \mathbb{N}} \big( \mathrm{cr}_s G_s \otimes \mathfrak{malcev}^{\overline{\otimes}s} \big) \rightarrow G_\bullet$. Again the domain of this map is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Hence it suffices to show that the map is surjective. Suppose that $(\varphi_s)$ is not surjective and write $C$ for its cokernel, so that there exists a minimal $\ell \in \mathbb{N}$ such that $C_\ell \neq 0$. Minimality implies that $\mathrm{cr}_\ell C_\ell \neq 0$, so that the surjection $G_\ell \twoheadrightarrow C_\ell$ induces a non-zero surjection $\mathrm{cr}_\ell G_\ell \twoheadrightarrow \mathrm{cr}_\ell C_\ell$. This is in contradiction with the construction of $\varphi_\ell$. ◻ *Remark 106*. For $d \in \mathbb{N}$, $\{ (\mathfrak{malcev}^{\overline{\otimes}s})_d \ |\ 0 \leq s \leq d \}$ is a set of projective generators of $\mathcal{F}_d (\mathbf{gr})$. Heuristically, the set of projective generators $\{ \mathfrak{malcev}^{\overline{\otimes}s} \ | \ s\in \mathbb{N}\}$ is what one obtains 'on passage to the limit' as $d \rightarrow \infty$. ## Decomposing $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ By Corollary [Corollary 61](#cor:proj_gen_propoly){reference-type="ref" reference="cor:proj_gen_propoly"}, $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The aim of this subsection is to give a decomposition of this functor using the projective generators $\mathfrak{malcev}^{\otimes s}$, for $s \in \mathbb{N}$. This can be interpreted as a functorial form of (a weak version of) the Poincaré-Birkhoff-Witt theorem. By construction, there is a monomorphism $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Moreover, $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ has an associative algebra structure in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, so that one can form the morphisms $\mu_s$ (for each $s \in \mathbb{N}$), where $\mu_s$ is given by the composite $$\mathfrak{malcev}^{\overline{\otimes}s} \hookrightarrow (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}s} \rightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$$ where the second map is the product. (For $s =0$, this is just the map induced by the canonical inclusion $\mathbb{Q}\hookrightarrow P_\mathbb{Z}$.) As usual, these induce an increasing filtration of $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$, $\mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$, $n \in \mathbb{N}$, where $\mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ is the image of $$\bigoplus_{s=0}^n \mathfrak{malcev}^{\overline{\otimes}s} \rightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$$ given by the morphisms $\mu_0, \ldots, \mu_n$. (By convention one takes $\mathfrak{f}_{-1} (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}) =0$.) **Lemma 107**. *The filtration $\mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ is exhaustive, i.e., $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}\cong \lim_\rightarrow \mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$. Moreover, for each $n \in \mathbb{N}$, there is an isomorphism in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$: $$\mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})/\mathfrak{f}_{n-1} (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}) \cong \mathrm{triv}_n \otimes_{\mathfrak{S}_n} \mathfrak{malcev}^{\overline{\otimes}n},$$ where $\mathrm{triv}_n$ is the trivial representation of $\mathfrak{S}_n$.* *Proof.* Proposition [Proposition 98](#prop:U_malcev){reference-type="ref" reference="prop:U_malcev"} gives that the inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ induces an isomorphism $U \mathfrak{malcev}\stackrel{\cong}{\rightarrow} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ of Hopf algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. The filtration $\mathfrak{f}_n (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ corresponds under this isomorphism to the filtration of $U \mathfrak{malcev}$ induced by the length filtration of the tensor algebra on $\mathfrak{malcev}$. Using this, the result follows by using the classical Poincaré-Birkhoff-Witt theorem. ◻ *Remark 108*.   1. The functor $\mathrm{triv}_n \otimes_{\mathfrak{S}_n} \mathfrak{malcev}^{\overline{\otimes}n}$ can be considered as the $n$th symmetric power functor $S^n$ applied to $\mathfrak{malcev}$ in $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes}, \mathbb{Q})$, written $S^n (\mathfrak{malcev})$. 2. $S^n (\mathfrak{malcev})$ is a direct summand of $\mathfrak{malcev}^{\overline{\otimes}n}$ (since we are working over $\mathbb{Q}$), hence is projective. More precisely, it is the projective cover of $\mathsf{q}^\mathbf{gr}_\bullet S^n (\mathfrak{a})$ ($S^n (\mathfrak{a})$ identifies with $\mathrm{triv}_n \otimes_{\mathfrak{S}_n} \mathfrak{a}^{\otimes n}$). In particular, it is indecomposable. 3. $S^n (\mathfrak{malcev})$ has the property that $(S^n (\mathfrak{malcev}))_\ell =0$ for $\ell <n$ and it is isomorphic to $S^n (\mathfrak{a})$ for $\ell=n$. The canonical surjection $T^n \twoheadrightarrow S^n$ from the $n$th tensor power functor to the $n$th tensor power functor admits a unique section; this is given by $\prod_{i=1}^n x_i \mapsto \frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_n} x_{\sigma(1)} \otimes \ldots \otimes x_{\sigma(n)}$. Hence, the projection $\mathfrak{malcev}^{\overline{\otimes}n} \twoheadrightarrow S^n (\mathfrak{malcev})$ has an induced section $S^n (\mathfrak{malcev}) \hookrightarrow \mathfrak{malcev}^{\overline{\otimes}n}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. *Notation 109*. For $n \in \mathbb{N}$, write $\tilde{\mu}_n : S^n (\mathfrak{malcev}) \rightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ for the composite $$S^n (\mathfrak{malcev}) \hookrightarrow \mathfrak{malcev}^{\overline{\otimes}n} \stackrel{\mu_n}{\rightarrow} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}.$$ **Theorem 110**. *The morphisms $\tilde{\mu}_n$, $n \in \mathbb{N}$, yield an isomorphism in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ $$\bigoplus_{n \in \mathbb{N}} S^n (\mathfrak{malcev}) \stackrel{\cong}{\rightarrow} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}.$$ This expresses $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ as a direct sum of indecomposable projectives in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$.* *Proof.* To establish the isomorphism, it suffices to prove that, for each $N \in \mathbb{N}$, the map induced by $\tilde{\mu}_n$, for $n \in \{ 0, \ldots , N\}$ $$\bigoplus_{n=0}^N S^n (\mathfrak{malcev}) {\rightarrow} \mathfrak{f}_N \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$$ is an isomorphism. This is proved by induction upon $N$, the cases $N \in \{ 0, 1\}$ being clear. For the inductive step, by the five-lemma, it suffices to show that the composite map $$S^N (\mathfrak{malcev}) \stackrel{\tilde{\mu}_N}{\rightarrow} \mathfrak{f}_N \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z} \twoheadrightarrow \mathfrak{f}_N \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}/\mathfrak{f}_{N-1} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$$ is an isomorphism. Since $S^N (\mathfrak{malcev})$ is the projective cover of $S^N (\mathfrak{a})$, it suffices to check behaviour after composing with the projection $S^N (\mathfrak{malcev}) \twoheadrightarrow S^N (\mathfrak{a})$. By construction of $\tilde{\mu}_N$, one sees that the following diagram commutes: $$\xymatrix{ S^N (\mathfrak{malcev}) \ar[r]^{\tilde{\mu}_N} \ar@{->>}[d] & \mathfrak{f}_N \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z} \ar@{->>}[r] & \mathfrak{f}_N \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}/\mathfrak{f}_{N-1} \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z} \ar[r]^(.6)\cong & S^N (\mathfrak{malcev}) \ar@{->>}[d] \\ S^N(\mathfrak{a}) \ar[rrr]_= &&& S^N(\mathfrak{a}), }$$ in which the isomorphism in the top right is given by Lemma [Lemma 107](#lem:filt_qgr_bullet_PZ){reference-type="ref" reference="lem:filt_qgr_bullet_PZ"}. This establishes the inductive step. Since $S^n (\mathfrak{malcev})$ is an indecomposable projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, the isomorphism exhibits a decomposition of $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ as a direct sum of indecomposable projectives. ◻ ## The dual Mal'cev functors Recall from Proposition [Proposition 64](#prop:D_propoly_analytic_adjoint){reference-type="ref" reference="prop:D_propoly_analytic_adjoint"} that duality induces an adjunction: $$D : {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}^\mathrm{op}\rightleftarrows \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) : D.$$ Hence, one has the dual Mal'cev functor $D \mathfrak{malcev}\in \mathrm{Ob}\hspace{2pt}\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$, i.e. an analytic functor on $\mathbf{gr}^\mathrm{op}$. Explicitly, this is constructed as the colimit $\lim_{\substack{\rightarrow \\ d}} D \mathfrak{malcev}_d$, where $D \mathfrak{malcev}_d$ is a *finite* polynomial functor of degree $d$. Likewise, for $s \in \mathbb{N}$, one has the dual functor $D (\mathfrak{malcev}^{\overline{\otimes}s})$. **Lemma 111**. *For $s \in \mathbb{N}$, there is a $\mathfrak{S}_s$-equivariant isomorphism: $$D (\mathfrak{malcev}^{\overline{\otimes}s}) \cong (D \mathfrak{malcev})^{\otimes s}.$$* *Proof.* This is a straightforward verification using the definition of $\overline{\otimes}$ on ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. ◻ **Theorem 112**. *For $s \in \mathbb{N}$, $D (\mathfrak{malcev}^{\overline{\otimes}s})$ is injective in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$. It is the injective envelope of the functor $(\mathfrak{a}^\sharp)^{\otimes s}$.* *Proof.* Since $\mathfrak{malcev}^{\overline{\otimes}s}$ is projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, the injectivity follows from the duality adjunction of Proposition [Proposition 64](#prop:D_propoly_analytic_adjoint){reference-type="ref" reference="prop:D_propoly_analytic_adjoint"} together with the fact that $D : \mathcal{F}_\omega (\mathbf{gr})\rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is exact, by Lemma [Lemma 62](#lem:propoly_analytic_duality){reference-type="ref" reference="lem:propoly_analytic_duality"}. Moreover, the surjection $\mathfrak{malcev}^{\overline{\otimes}s} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$ induces the monomorphism $$(\mathfrak{a}^\sharp)^{\otimes s} \hookrightarrow D(\mathfrak{malcev}^{\overline{\otimes}s}).$$ It remains to show that this exhibits $D (\mathfrak{malcev}^{\overline{\otimes}s})$ as the injective envelope of $(\mathfrak{a}^\sharp)^{\otimes s}$. Once again, this follows from the duality adjunction together with the fact that $\mathfrak{malcev}^{\overline{\otimes}s}$ is the projective cover of $\mathsf{q}^\mathbf{gr}_\bullet (\mathfrak{a}^{\otimes s})$. Details are left to the reader. ◻ From this one deduces: **Corollary 113**. *The set $\{ D (\mathfrak{malcev}^{\overline{\otimes}s}) \ | \ s \in \mathbb{N}\}$ is a set of injective cogenerators of $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$.* ## Understanding the structure of $\mathfrak{malcev}$ As in Section [5.6](#subsect:describe_qgr_d_PZ){reference-type="ref" reference="subsect:describe_qgr_d_PZ"}, one can describe the structure of the functors $\mathfrak{malcev}_d$, for $d \in \mathbb{N}$, and hence of $\mathfrak{malcev}$. This uses a similar approach to that in Proposition [Proposition 94](#prop:describe_qgr_Pzed){reference-type="ref" reference="prop:describe_qgr_Pzed"}, recalling that there is a canonical inclusion $\mathfrak{malcev}_d \subset \mathsf{q}^\mathbf{gr}_d P_\mathbb{Z}$, by construction. Here, rather than working with the truncated tensor algebra $T^{\leq d}(-)$, one works with the truncated Lie algebra $\mathrm{Lie}_{\leq d}( -)$; this embeds naturally $\mathrm{Lie}_{\leq d} (-) \hookrightarrow T^{\leq d} (-)$ as a functor to Lie algebras. *Notation 114*. Write $\mathsf{bch}_d(x_1, x_2)$ for the truncated Baker-Campbell-Hausdorff element of $\mathrm{Lie}_{\leq d} (\mathbb{Q}\{ x_1, x_2 \})$ corresponding to the truncation of $\log (\exp (x_1)\exp(x_2))$. In the following statement, $p$, $\chi$ and $\nabla$ are as in Section [5.6](#subsect:describe_qgr_d_PZ){reference-type="ref" reference="subsect:describe_qgr_d_PZ"}. **Proposition 115**. *For $1<d \in \mathbb{N}$, $\mathfrak{malcev}_d$ is isomorphic to the functor with values in Lie algebras $$\mathrm{Free}(S) \mapsto \mathrm{Lie}_{\leq d} (\mathbb{Q}S),$$ where $\mathbb{Q}S$ is the $\mathbb{Q}$-vector space generated by the finite set $S$, such that:* 1. *morphisms in the image of $\mathrm{free} : \mathbf{Fin}\rightarrow \mathbf{gr}$ act via $\mathbb{Q}( -) \ : \ \mathbf{Fin}\rightarrow \mathrm{mod}_\mathbb{Q}$ and the naturality of $\mathrm{Lie}_{\leq d} (-)$;* 2. *$p : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \{ e \} = \mathrm{Free} (\emptyset)$ sends $x \in \mathrm{Lie}_{\leq d}(\mathbb{Q}\{x\})$ to zero;* 3. *$\chi : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \mathbb{Z}= \mathrm{Free}(x)$ sends the generator $x \in \mathrm{Lie}_{\leq d} (\mathbb{Q}\{x\})$ to $-x$;* 4. *$\nabla : \mathbb{Z}= \mathrm{Free} (x) \rightarrow \mathbb{Z}\star \mathbb{Z}= \mathrm{Free} (x_1, x_2)$ acts via $x \mapsto \mathsf{bch}_d (x_1, x_2)\in \mathrm{Lie}_{\leq d} (\mathbb{Q}\{ x_1, x_2\})$.* *Proof.* Using the universal property of $\mathrm{Lie}_{\leq d} (-)$, there is a natural transformation of functors from $\mathbf{Fin}$ to Lie algebras defined on a finite set $S$ $$\mathrm{Lie}_{\leq d} (\mathbb{Q}S) \rightarrow \mathfrak{malcev}_d (S)$$ by sending a generator $y \in S$ to the image of $\log(y)$ in $\mathfrak{malcev}_d (S)$. This is easily seen to be an isomorphism. As in the proof of Proposition [Proposition 94](#prop:describe_qgr_Pzed){reference-type="ref" reference="prop:describe_qgr_Pzed"}, it remains to check that this is a natural transformation of functors on $\mathbf{gr}$, using the specified structure on the domain. The behaviour of $\chi$ follows from the identity (for any $g \in G$), $\log (g^{-1}) = - \log (g)$. The behaviour of $\nabla$ is a consequence of the *definition* of the Baker-Campbell-Hausdorff series. Here it is essential that the natural transformation above has been constructed as a natural transformation of Lie algebras. ◻ # Operadic structures {#sect:opd} This section reviews the relationship between operads in $\mathbbm{k}$-modules and $\mathbbm{k}$-linear categories. Here, $\mathbbm{k}$ can usually be taken to be an arbitrary commutative, unital ring; for all applications it will be $\mathbb{Q}$ and the reader may prefer to assume this throughout. The most important example here is the operad $\mathcal{O}= \mathfrak{Lie}$, which encodes Lie algebras in $\mathtt{Mod}_\mathbbm{k}$. Again, the reader may prefer to focus upon this case. This material is well-known, and is mostly treated in [@MR1854112], for example. (Most of this background material is also covered in [@2021arXiv211001934P]; it is included here so as to make the exposition self-contained.) ## The category associated to an operad The category $\mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ (aka. $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-modules) underlies the category of $\mathbbm{k}$-linear operads (i.e., operads in the symmetric monoidal category $\mathtt{Mod}_\mathbbm{k}$): an operad $\mathcal{O}$ has underlying $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-module given by the family (for $n\in \mathbb{N}$) of $\mathfrak{S}_n^\mathrm{op}$-modules $\mathcal{O}(n)$ (or $\mathcal{O}(\mathbf{n})$ in the notation for $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-modules), the term of *arity* $n$. The identity operad $I$ has $I(n)$ that is zero, except for $n=1$, when $I(1) = \mathbbm{k}$. Each operad $I$ is equipped with the canonical unit morphism $I \rightarrow \mathcal{O}$. *Notation 116*. For $\mathcal{O}$ an operad, the associated $\mathbbm{k}$-linear category is denoted by $\mathbf{Cat}\hspace{1pt}\mathcal{O}$, so that $\mathcal{O}(n)= \mathbf{Cat}\hspace{1pt}\mathcal{O}(n ,1)$. This has set of objects $\mathbb{N}$. The morphisms are given explicitly by: $$\mathbf{Cat}\hspace{1pt}\mathcal{O}(s,t) = \bigoplus_{f : \mathbf{s} \rightarrow \mathbf{t}} \bigotimes_{i=1}^t \mathcal{O}(f^{-1} (i)),$$ where the sum is over set maps $f : \mathbf{s} \rightarrow \mathbf{t}$. Composition is defined using the composition of the operad. *Remark 117*. There is more structure: $\mathbf{Cat}\hspace{1pt}\mathcal{O}$ is naturally a PROP, i.e., it is symmetric monoidal with the structure corresponding to $(\mathbb{N}, +)$ on objects. The construction $\mathcal{O}\mapsto \mathbf{Cat}\hspace{1pt}\mathcal{O}$ is functorial: a morphism of operads $\varphi: \mathcal{O}\rightarrow \mathcal{P}$ induces a $\mathbbm{k}$-linear functor $\mathbf{Cat}\hspace{1pt}\varphi: \mathbf{Cat}\hspace{1pt}\mathcal{O}\rightarrow \mathbf{Cat}\hspace{1pt}\mathcal{P}$. **Example 118**. For $I$ the identity operad, $\mathbf{Cat}\hspace{1pt}I$ is equivalent to $\mathbbm{k}{\bm{\Sigma}}$, the $\mathbbm{k}$-linearization of ${\bm{\Sigma}}$. Explicitly, $\mathbf{Cat}\hspace{1pt}I (m,n)$ is zero unless $m=n$, when $\mathbf{Cat}\hspace{1pt}I (n,n) \cong \mathbbm{k}\mathfrak{S}_n$ as associative $\mathbbm{k}$-algebras. For any operad $\mathcal{O}$, the unit $I \rightarrow \mathcal{O}$ induces $\mathbf{Cat}\hspace{1pt}I \rightarrow \mathbf{Cat}\hspace{1pt}\mathcal{O}$. In particular, for $n\in \mathbb{N}$, this induces a morphism of $\mathbbm{k}$-algebras $\mathbbm{k}\mathfrak{S}_n \rightarrow \mathbf{Cat}\hspace{1pt}\mathcal{O}(n,n).$ In the cases of interest here, the following applies: **Lemma 119**. *Suppose that $\mathcal{O}(0)=0$ and $\mathcal{O}(1)=\mathbbm{k}$, corresponding to the identity of the operad. Then, for $m, n \in \mathbb{N}$:* 1. *$\mathbf{Cat}\hspace{1pt}\mathcal{O}(n,n) \cong \mathbbm{k}\mathfrak{S}_n$ as associative algebras;* 2. *$\mathbf{Cat}\hspace{1pt}\mathcal{O}(m,n) =0$ if $n>m$.* *In particular, there is a natural $\mathbb{N}$-grading of the morphism modules of $\mathbf{Cat}\hspace{1pt}\mathcal{O}$, where $\mathbf{Cat}\hspace{1pt}\mathcal{O}(m,n)$ is placed in degree $m-n$; this is compatible with composition in the obvious sense.* For any operad $\mathcal{O}$, one has the categories of left (respectively right) modules over $\mathbf{Cat}\hspace{1pt}\mathcal{O}$: 1. a left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module is a $\mathbbm{k}$-linear functor $\mathbf{Cat}\hspace{1pt}\mathcal{O}\rightarrow \mathtt{Mod}_\mathbbm{k}$; 2. a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module is a $\mathbbm{k}$-linear functor $(\mathbf{Cat}\hspace{1pt}\mathcal{O})^\mathrm{op}\rightarrow \mathtt{Mod}_\mathbbm{k}$; morphisms are $\mathbbm{k}$-linear natural transformations. *Notation 120*. For an operad $\mathcal{O}$, 1. the category of right (respectively left) $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules is denoted $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$ (resp. ${}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$). 2. $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}^{< \infty} \subset \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$ (respectively ${}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}^{< \infty} \subset {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$) denote the full subcategory of modules such that the underlying $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-module (resp. $\mathbbm{k}{\bm{\Sigma}}$-module) has finite support ($M$ has finite support if $M(n)=0$ for $n\gg 0$). *Remark 121*. Using restriction along $\mathbf{Cat}\hspace{1pt}I \rightarrow \mathbf{Cat}\hspace{1pt}\mathcal{O}$ (see Example [Example 118](#exam:catI){reference-type="ref" reference="exam:catI"}), a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module has an underlying right $\mathbbm{k}{\bm{\Sigma}}$-module structure. This allows the following alternative formulation of the definition of a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module (and analogously for left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules). A right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module is a sequence $M(n)$ of right $\mathbbm{k}\mathfrak{S}_n$-modules, for $n \in \mathbb{N}$ (i.e., a $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-module), equipped with structure morphisms: $$\begin{aligned} \label{eqn:M_right_cat_opd} M (n) \otimes_{\mathfrak{S}_n} \mathbf{Cat}\hspace{1pt}\mathcal{O}(m,n) \rightarrow M(m)\end{aligned}$$ that are unital and associative in the obvious sense; in particular the morphism is $\mathfrak{S}_m^\mathrm{op}$-equivariant. Natural transformations are identified similarly. Clearly one has: **Proposition 122**. *The categories ${}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$ and $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$ are abelian $\mathbbm{k}$-linear categories.* **Example 123**. Using the identification of Example [Example 118](#exam:catI){reference-type="ref" reference="exam:catI"}, the category of left $\mathbf{Cat}\hspace{1pt}I$-modules is equivalent to $\mathcal{F}({\bm{\Sigma}})$ and that of right $\mathbf{Cat}\hspace{1pt}I$-modules is equivalent to $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}) \cong \mathcal{F}( {\bm{\Sigma}})$. Suppose that $\mathcal{O}(0)=0$ and $\mathcal{O}(1)=\mathbbm{k}$, then $\mathcal{O}$ is canonically augmented: there is a morphism of operads $\mathcal{O}\rightarrow I$ that is an isomorphism in arity $1$. This induces the 'augmentation' of $\mathbbm{k}$-linear categories $\mathbf{Cat}\hspace{1pt}\mathcal{O}\rightarrow \mathbf{Cat}\hspace{1pt}I$. Hence any $\mathbbm{k}{\bm{\Sigma}}$-module has a natural left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module structure (by restriction) and, likewise, any $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-module a natural right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module structure. In particular, for $n\in \mathbb{N}$, a $\mathbbm{k}\mathfrak{S}_n$-module can be considered as an object of $\mathcal{F}({\bm{\Sigma}})\cong \mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ supported on $\mathbf{n}$ and hence as a left (respectively right) $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module. **Example 124**. The $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-module underlying an operad $\mathcal{O}$ has a canonical *right* $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module, with structure morphism $\mathcal{O}(n) \otimes_{\mathfrak{S}_n} \mathbf{Cat}\hspace{1pt}\mathcal{O}(m, n) \rightarrow \mathcal{O}(m)$ given by the composition operation of the operad $\mathcal{O}$. This structure is natural in the following sense: for $\varphi: \mathcal{O}\rightarrow \mathcal{P}$ a morphism of operads, $\mathcal{O}$ is a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module and $\mathcal{P}$ is a right $\mathbf{Cat}\hspace{1pt}\mathcal{P}$-module, hence a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module by restriction along $\mathbf{Cat}\hspace{1pt}\varphi: \mathbf{Cat}\hspace{1pt}\mathcal{O}\rightarrow \mathbf{Cat}\hspace{1pt}\mathcal{P}$. With respect to these structures, the underlying morphism of $\varphi: \mathcal{O}\rightarrow \mathcal{P}$ is a morphism of right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules. ## Canonical filtrations {#subsect:canon_filt_catopd_mod} Throughout this section, we suppose that $\mathcal{O}(0)=0$ and $\mathcal{O}(1) = \mathbbm{k}$, so that Lemma [Lemma 119](#lem:cat_opd_connexe){reference-type="ref" reference="lem:cat_opd_connexe"} applies. The following is then clear: **Lemma 125**. *Let $d \in \mathbb{N}$.* 1. *There is an exact functor $(-)_{\leq d} : {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod} \rightarrow {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$ equipped with a canonical inclusion $(-)_{\leq d} \hookrightarrow \mathrm{Id}$, where for a left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module $M$, $M_{\leq d}(n)=M(n)$ for $n \leq d$ and $0$ if $n>d$.* 2. *There is an exact functor $(-)^{\leq d} : \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} \rightarrow \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$ equipped with a canonical surjection $\mathrm{Id}\twoheadrightarrow (-)^{\leq d}$, where for a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module $N$, $N^{\leq d}(t)=N(t)$ for $t \leq d$ and $0$ if $t>d$.* The functor $(-)_{\leq d} : {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod} \rightarrow {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$ can be constructed as the right adjoint to the inclusion of the full subcategory of left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules $M$ for which $M (t) =0$ for $t>d$. In particular, for any left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module $M$, there is a canonical sequence of inclusions of subobjects of $M$ in ${}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$: $$M_{\leq 0} \subset M_{\leq 1} \subset \ldots \subset M_{\leq d} \subset M_{\leq d+1} \subset \ldots \subset M.$$ Likewise, for a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module $N$, there is a canonical tower under $N$ in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$: $$N \twoheadrightarrow \ldots \twoheadrightarrow N^{\leq d+1} \twoheadrightarrow N^{\leq d} \twoheadrightarrow \ldots \twoheadrightarrow N^{\leq 1} \twoheadrightarrow N^{\leq 0}.$$ Moreover, the following is clear: **Proposition 126**. * * 1. *For $M \in \mathrm{Ob}\hspace{2pt}{}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$, the canonical morphism $\lim_{\substack{\rightarrow\\d}} M_{\leq d} \rightarrow M$ is an isomorphism.* 2. *For $N \in \mathrm{Ob}\hspace{2pt}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$, the canonical morphism $N \rightarrow \lim_{\substack{\leftarrow \\ d}} N^{\leq d}$ is an isomorphism.* As well as the functor $(-)_{\leq d} : {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod} \rightarrow {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$, we shall also require the following: *Notation 127*. For $d \in \mathbb{N}$, write $(-)_{[\leq d]} : {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod} \rightarrow {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$ for the functor induced by the left adjoint to the inclusion of the full subcategory of left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules $M$ for which $M (t) =0$ for $t>d$. *Remark 128*. Let $d \in \mathbb{N}$. 1. For $N \in \mathrm{Ob}\hspace{2pt}{}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod}$, the canonical surjection $N \twoheadrightarrow N_{[\leq d]}$ is the universal morphism to an object in the full subcategory of left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules $M$ for which $M (t) =0$ for $t>d$. 2. Explicitly, $N_{[\leq d]}(t)$ is the cokernel of $$\bigoplus_{s> d} \bigoplus_{f \in \mathbf{Cat}\hspace{1pt}\mathcal{O}(s,t)} N (s) \rightarrow N(t),$$ where the morphisms are given by the $N(f)$. 3. The composite morphism $N_{\leq d} \hookrightarrow N \twoheadrightarrow N_{[\leq d]}$ is a surjection but is not in general an isomorphism. For instance, taking $\mathcal{O}= \mathfrak{Lie}$, $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s, -)_{\leq d}$ is non-zero if $d>0$, whereas $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s, -)_{[\leq d]}$ is zero whenever $s>d$, as follows from the Yoneda lemma. ## The tensor product over $\mathbf{Cat}\hspace{1pt}\mathcal{O}$ One has the tensor product $$-\otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} - \ : \ \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} \times {}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}\mathtt{Mod} \rightarrow \mathtt{Mod}_\mathbbm{k}.$$ For $M$ a left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module and $N$ a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module, $N \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M$ is defined as the coequalizer of the diagram $$\bigoplus_{s,t} \bigoplus_{f \in \mathbf{Cat}\hspace{1pt}\mathcal{O}(s,t)} N(t) \otimes M(s) \rightrightarrows \bigoplus_u N(u) \otimes M(u),$$ where the maps are given by the evident structure morphisms $N(s) \otimes M(s) \leftarrow N(t) \otimes M(s) \rightarrow N(t)\otimes M(t)$ induced by a morphism $f \in \mathbf{Cat}\hspace{1pt}\mathcal{O}(s,t)$. We note the following basic properties: **Proposition 129**. *For $M$ a left $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module and $N$ a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module:* 1. *The filtration $(M_{\leq d})$ gives rise to a canonical isomorphism $$\lim_{\substack{\rightarrow \\ d}} \big( N \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M_{\leq d}\big) \stackrel{\cong}{\rightarrow } N \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M.$$* 2. *For $d\in \mathbb{N}$, the canonical surjections $N \twoheadrightarrow N^{\leq d}$ and $M \twoheadrightarrow M_{[\leq d]}$ induce isomorphisms: $$N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M \cong N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M_{[\leq d]} \cong N \otimes _{\mathbf{Cat}\hspace{1pt}\mathcal{O}} M_{[\leq d]}.$$* *Proof.* The first statement follows from the fact that $N \otimes_{\mathbf{Cat}\hspace{1pt}\mathcal{O}} -$ commutes with colimits. The second statement is the analogue of Proposition [Proposition 39](#prop:otimes_gr_poly){reference-type="ref" reference="prop:otimes_gr_poly"}; it can be proved directly from the definition of $\otimes _{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$. ◻ ## The convolution product for right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules {#subsect:convolution_cat_opd} Recall that Day convolution defines a symmetric monoidal structure $(\mathcal{F}({\bm{\Sigma}}^\mathrm{op}), \odot, \mathbbm{k})$, where $\mathbbm{k}$ is concentrated in arity $0$. Here, for two $\mathbbm{k}{\bm{\Sigma}}^\mathrm{op}$-modules, $M \odot N$ is defined on objects by: $$(M \odot N) (S) = \bigoplus_{S_1 \amalg S_2 = S} M(S_1) \otimes N(S_2),$$ where $S$ is a finite set and the sum is indexed by ordered decompositions of $S$ into two subsets. This extends to $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}$: by [@MR1854112 Proposition 1.6.3] (which references [@MR1617616]), $\odot$ makes the category of right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules into a symmetric monoidal category $(\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}},\odot, \mathbbm{k})$. (See also [@MR2494775 Section 6.1], where the structure is defined for the category of right modules over the operad (defined using $\circ$), which is equivalent to the category of right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules.) *Remark 130*. Analogously, the convolution product on $\mathcal{F}({\bm{\Sigma}})$ extends to a convolution product on *left* $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules (see [@2021arXiv211001934P]). # The $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\underline{\mathfrak{malcev}}$ and modelling ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ {#sect:endo_malcev} The purpose of this section is to prove that ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is equivalent to the category $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ (see Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}), working over $\mathbbm{k}= \mathbb{Q}$. This follows from the fact that the projective generators $\mathfrak{malcev}^{\overline{\otimes}s}$, for $s \in \mathbb{N}$, assemble to form a left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module, and these structure morphisms provide all possible morphisms. ## The equivalence of categories By construction, $\mathfrak{malcev}$ is a Lie algebra in $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes}, \mathbb{Q})$. *Notation 131*. Denote by $\underline{\mathfrak{malcev}}$ the left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\underline{\mathfrak{malcev}}$ given by $\underline{\mathfrak{malcev}} : s \mapsto \mathfrak{malcev}^{\overline{\otimes}s}$, with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ acting via the Lie algebra structure. In particular, for $s, t \in \mathbb{N}$, this gives the $\mathfrak{S}_s^\mathrm{op}\times \mathfrak{S}_t$-equivariant morphism encoding this structure: $$\begin{aligned} \label{eqn:cat_lie_malcev} \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s, t) \rightarrow \mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\mathfrak{malcev}^{\overline{\otimes}s}, \mathfrak{malcev}^{\overline{\otimes}t}).\end{aligned}$$ Here, $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t) =0$ if $s<t$, so we may assume without loss of generality that $s\geq t$. **Proposition 132**. *The full subcategory of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ with objects $\mathfrak{malcev}^{\overline{\otimes}s}$, for $s\in \mathbb{N}$, is equivalent to the $\mathbb{Q}$-linear category $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$.* *Proof.* We require to prove that, for all $s, t \in \mathbb{N}$, the structure morphism ([\[eqn:cat_lie_malcev\]](#eqn:cat_lie_malcev){reference-type="ref" reference="eqn:cat_lie_malcev"}) is an isomorphism. We first establish that ([\[eqn:cat_lie_malcev\]](#eqn:cat_lie_malcev){reference-type="ref" reference="eqn:cat_lie_malcev"}) is injective. The associated graded of $\mathfrak{malcev}$ is $\mathrm{Lie}(\mathfrak{a})$, by Proposition [Proposition 97](#prop:assoc_graded_malcev){reference-type="ref" reference="prop:assoc_graded_malcev"}, as a Lie algebra. Since passage to the associated graded is symmetric monoidal (cf. Proposition [Proposition 69](#prop:properties_fbcr_propoly){reference-type="ref" reference="prop:properties_fbcr_propoly"}), the associated graded of $\mathfrak{malcev}^{\overline{\otimes}s}$ is $\mathrm{Lie}(\mathfrak{a}) ^{\otimes s}$. Hence, passage to the associated graded gives the $\mathfrak{S}_s^\mathrm{op}\times \mathfrak{S}_t$-equivariant morphism $$\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\mathfrak{malcev}^{\overline{\otimes}s}, \mathfrak{malcev}^{\overline{\otimes}t}) \rightarrow \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})}(\mathrm{Lie}(\mathfrak{a})^{\otimes s}, \mathrm{Lie}(\mathfrak{a}) ^{\otimes t}).$$ Moreover, the composite $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\rightarrow \mathrm{Hom}_{\mathcal{F}(\mathbf{gr})}(\mathrm{Lie}(\mathfrak{a})^{\otimes s}, \mathrm{Lie}(\mathfrak{a}) ^{\otimes t})$ is the structure morphism for the left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ module $\underline{\mathrm{Lie}(\mathfrak{a})}$ associated to the Lie algebra $\mathrm{Lie}(\mathfrak{a})$ in $\mathcal{F}(\mathbf{gr})$. Since $\mathrm{Lie}(-)$ is the free Lie algebra functor, it is essentially tautological that this composite is injective. This gives the required injectivity of ([\[eqn:cat_lie_malcev\]](#eqn:cat_lie_malcev){reference-type="ref" reference="eqn:cat_lie_malcev"}). It remains to establish surjectivity, refining the above argument. By Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}, there is a natural isomorphism: $$\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\mathfrak{malcev}^{\overline{\otimes}s}, \mathfrak{malcev}^{\overline{\otimes}t}) \cong \mathrm{cr}_s (\mathfrak{malcev}^{\overline{\otimes}t})_s.$$ The right hand side is the $\mathbb{Q}\mathfrak{S}_s$-module that encodes the homogeneous degree $s$ part of $\mathfrak{malcev}^{\overline{\otimes}t}$. Equivalently, it encodes the homogeneous degree $s$ part of the associated graded of $\mathfrak{malcev}^{\overline{\otimes}t}$, which identifies as $\mathrm{Lie}(\mathfrak{a})^{\otimes t}$, as above. We claim that $\mathrm{cr}_s (\mathrm{Lie}(\mathfrak{a})^{\otimes t})$ is isomorphic to $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t)$ as a vector space. To see this, consider the Schur functor $V \mapsto \bigoplus_s \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s, t) \otimes_{\mathfrak{S}_s} V^{\otimes s}$. By the construction of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$, this is isomorphic to $\mathrm{Lie}(V) ^{\otimes t}$ and $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t)$ is recovered from the component of polynomial degree $s$ in $V$ by applying the appropriate cross-effect functor. This calculation is equivalent to that of $\mathrm{cr}_s (\mathrm{Lie}(\mathfrak{a})^{\otimes t})$. Now $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t)$ has finite dimension; the morphism considered is a linear monomorphism between vector spaces of the same (finite) dimension, hence is an isomorphism, as required. ◻ This has the immediate consequence: **Theorem 133**. *The functor $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -)$ induces an equivalence of categories $$\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -) \ : \ {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\stackrel{\cong}{\rightarrow} \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}.$$* *The inverse equivalence is given by: $$- \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \ : \ \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}.$$* *Proof.* By Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}, for $s \in \mathbb{N}$, $\mathfrak{malcev}^{\overline{\otimes}s}$ corepresents the functor $G_\bullet \mapsto \mathrm{cr}_s G_s$. This functor commutes with colimits, hence $\mathfrak{malcev}^{\otimes s}$ is a small projective in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Thus $\{ \mathfrak{malcev}^{\otimes s} \ | \ s\in \mathbb{N}\}$ is a generating set of small projectives. The result thus follows from Freyd's theorem (see [@MR294454 Theorem 3.1] for a version over $\mathbb{Z}$). ◻ *Remark 134*. Recall that restriction along $\mathbf{Cat}\hspace{1pt}I \rightarrow \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ induces the forgetful functor $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow \mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ and that $\mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ is isomorphic to $\mathcal{F}({\bm{\Sigma}})$. Hence this gives the functor $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow \mathcal{F}({\bm{\Sigma}})$. Composing $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -)$ with the restriction functor gives ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}({\bm{\Sigma}})$. Using Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}, this identifies with the functor $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$ (cf. Notation [Notation 67](#nota:fbcr_propoly){reference-type="ref" reference="nota:fbcr_propoly"}) that encodes the associated graded of an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Thus Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} refines $\mathsf{grad}^\mathbf{gr}_{\bm{\Sigma}}$, retaining the necessary additional structure, namely the right action of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$. The equivalence of Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} restricts to the full subcategory of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ corresponding to $\mathcal{F}_{<\infty}(\mathbf{gr})$ (see Proposition [Proposition 51](#prop:poly_propoly){reference-type="ref" reference="prop:poly_propoly"}) to give the following result, which strengthens the corresponding result of [@2021arXiv211001934P] (which restricted to functors in $\mathcal{F}_{< \infty}(\mathbf{gr})$ with a finite composition series). **Corollary 135**. *The category $\mathcal{F}_{< \infty} (\mathbf{gr})$ is equivalent to the full subcategory $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{< \infty} \subset \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules $M$ such that $M(n) =0$ for $n \gg 0$.* *Proof.* It suffices to check that the two functors given in Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} restrict as required. A functor $F\in \mathrm{Ob}\hspace{2pt}{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is in $\mathcal{F}_{< \infty} (\mathbf{gr})\subset {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ if and only if $\mathrm{cr}_s F_s =0$ for $s \gg 0$. This is equivalent to $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\mathfrak{malcev}^{\overline{\otimes}s}, F) =0$ for $s \gg 0$. The result follows. ◻ This can be made more precise by comparing the canonical filtration of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules with the canonical filtration of objects of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Recall that, for $d \in \mathbb{N}$ and $F_\bullet$ an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, $F_d$ denotes the $d$th term of $F_\bullet$, which has polynomial degree $d$. There is a canonical surjection in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ $$F_\bullet \twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet (F_d)$$ to an object arising from a functor of $\mathcal{F}_d (\mathbf{gr})$. **Proposition 136**. *For $d \in \mathbb{N}$ and $N \in \mathrm{Ob}\hspace{2pt}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, the canonical surjection $N \twoheadrightarrow N^{\leq d}$ induces an isomorphism: $$\mathsf{q}^\mathbf{gr}_\bullet \big((N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}})_d\big) \cong N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}.$$* *In particular, there is an isomorphism in $\mathcal{F}_d (\mathbf{gr})$: $$(N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}})_d \cong (N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}})_d.$$* *Proof.* The argument employed in the proof of Corollary [Corollary 135](#cor:finite_functors_equiv_rmod){reference-type="ref" reference="cor:finite_functors_equiv_rmod"} implies that $N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ arises from a functor of $\mathcal{F}_d (\mathbf{gr})$; explicitly: $$N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \cong \mathsf{q}^\mathbf{gr}_\bullet \big((N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} )_d \big).$$ This yields the surjection $\mathsf{q}^\mathbf{gr}_\bullet \big((N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}})_d\big) \twoheadrightarrow N^{\leq d} \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$. Comparing the polynomial filtrations of both sides shows that this is an isomorphism. ◻ *Remark 137*. It is instructive to consider the universal example for Proposition [Proposition 136](#prop:compare_filtrations){reference-type="ref" reference="prop:compare_filtrations"}, namely taking $N = \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$, so that $N^{\leq d}$ is $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$. Then Proposition [Proposition 129](#prop:properties_otimes_cat_opd){reference-type="ref" reference="prop:properties_otimes_cat_opd"} gives the isomorphism $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \cong \underline{\mathfrak{malcev}}_{[\leq d]}$. Proposition [Proposition 136](#prop:compare_filtrations){reference-type="ref" reference="prop:compare_filtrations"} asserts that the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\underline{\mathfrak{malcev}}_{[\leq d]}$ is given in arity $s$ by $$\underline{\mathfrak{malcev}}_{[\leq d]} (s) = \mathsf{q}^\mathbf{gr}_\bullet \big ( (\mathfrak{malcev}^{\overline{\otimes}s})_d \big).$$ That this holds can be seen by analysing the construction of $\underline{\mathfrak{malcev}}_{[\leq d]}$. In arity $s$, this is given by the cokernel of $$\bigoplus_{u >d} \bigoplus _{f \in \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(u, s) } \mathfrak{malcev}^{\overline{\otimes}u} \rightarrow \mathfrak{malcev}^{\overline{\otimes}s},$$ as in Remark [Remark 128](#rem:left_adjoint_left_cat_opd){reference-type="ref" reference="rem:left_adjoint_left_cat_opd"}, where the action of $f$ on $\underline{\mathfrak{malcev}}$ is induced by the Lie algebra structure of $\mathfrak{malcev}$. Combining Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"} with Proposition [Proposition 132](#prop:malcev_full_subcat){reference-type="ref" reference="prop:malcev_full_subcat"}, one sees that the cokernel identifies as claimed. ## The equivalence is symmetric monoidal Recall from Section [7.4](#subsect:convolution_cat_opd){reference-type="ref" reference="subsect:convolution_cat_opd"} that $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ is symmetric monoidal for the convolution product $\odot$ and that we have the symmetric monoidal structure $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes}, \mathbb{Q})$. **Theorem 138**. *The functor $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -) : {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\stackrel{\cong}{\rightarrow} \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ is symmetric monoidal.* *Proof.* Using the identification given in Remark [Remark 134](#rem:identify_w_fbcr){reference-type="ref" reference="rem:identify_w_fbcr"}, if we only retain the underlying $\mathbb{Q}{\bm{\Sigma}}^\mathrm{op}$-module structure, the result follows from Proposition [Proposition 69](#prop:properties_fbcr_propoly){reference-type="ref" reference="prop:properties_fbcr_propoly"}. It remains to check that this isomorphism is compatible with the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structures. This follows by analysing the above construction at the level of morphisms in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Consider two morphisms $f: \mathfrak{malcev}^{\overline{\otimes}s} \rightarrow G_\bullet$ and $g :\mathfrak{malcev}^{\overline{\otimes}t} \rightarrow G'_\bullet$. One has the tensor product $$f \overline{\otimes}g : \mathfrak{malcev}^{\overline{\otimes}s+t} \cong \mathfrak{malcev}^{\overline{\otimes}s} \overline{\otimes}\mathfrak{malcev}^{\overline{\otimes}t} \rightarrow G_\bullet \overline{\otimes}G'_\bullet.$$ This represents the element in $\mathrm{cr}_{s+t} (G_\bullet \overline{\otimes}G'_\bullet)_s$ given by forming the appropriate convolution product of $f \in \mathrm{cr}_s G_s$ and $g \in \mathrm{cr}_t G'_t$. To understand the action of the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-morphisms, it suffices to consider the precomposition with morphisms of the form $$\begin{aligned} \mathfrak{malcev}^{\overline{\otimes}u} \overline{\otimes}\mathfrak{malcev}^{\overline{\otimes}t} & \rightarrow & \mathfrak{malcev}^{\overline{\otimes}s} \overline{\otimes}\mathfrak{malcev}^{\overline{\otimes}t} \\ \mathfrak{malcev}^{\overline{\otimes}s} \overline{\otimes}\mathfrak{malcev}^{\overline{\otimes}v} & \rightarrow & \mathfrak{malcev}^{\overline{\otimes}s} \overline{\otimes}\mathfrak{malcev}^{\overline{\otimes}t}\end{aligned}$$ induced by morphisms in $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\mathfrak{malcev}^{\overline{\otimes}u}, \mathfrak{malcev}^{\overline{\otimes}s})$ and $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\mathfrak{malcev}^{\overline{\otimes}v}, \mathfrak{malcev}^{\overline{\otimes}t})$ respectively, where we may take $u \geq s$ and $v \geq t$. These composites are determined by the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structures of $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\underline{\mathfrak{malcev}}, G_\bullet)$ and of $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\underline{\mathfrak{malcev}}, G'_\bullet)$ respectively. One checks that the composites correspond to the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure on the convolution product $\mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\underline{\mathfrak{malcev}}, G_\bullet) \odot \mathrm{Hom}_{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}(\underline{\mathfrak{malcev}}, G'_\bullet)$. ◻ # Relating left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules to functors on $\mathbf{gr}^\mathrm{op}$ {#sect:catlie} This section revisits the relationship between left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ that was established in [@2021arXiv211001934P]. The equivalence of categories is restated here as Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"}, using the exponential functor $\Phi U \mathfrak{Lie}$ in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules that is introduced below. *Remark 139*. In Section [10](#sect:compare){reference-type="ref" reference="sect:compare"} we analyse the relationship between this result and the analogous result for ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}. The operad $\mathfrak{Ass}^u$ encodes unital, associative algebras; it is related to the Lie operad $\mathfrak{Lie}$ by the morphism of operads $\mathsf{lie}: \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$ that encodes the underlying commutator algebra of an associative algebra (this factorizes across the associative operad $\mathfrak{Ass}$ as $\mathfrak{Lie}\rightarrow \mathfrak{Ass}\rightarrow \mathfrak{Ass}^u$, where $\mathfrak{Ass}\rightarrow \mathfrak{Ass}^u$ corresponds to forgetting the unit of an associative, unital algebra). The operad $\mathfrak{Com}^u$ encodes unital, commutative associative algebras, so that there is a morphism of operads $\mathfrak{Ass}^u\rightarrow \mathfrak{Com}^u$ that corresponds to forgetting about commutativity. The results of [@2021arXiv211001934P] are based on the fact that the structure of the projective generators of the category $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ are encoded in the functor $\mathbf{gr}^\mathrm{op}\rightarrow \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ given on objects by $$\mathbb{Z}^{\star n} \mapsto \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u(-, n),$$ where $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ is considered as a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module via the restriction of the regular right module structure along $\mathbf{Cat}\hspace{1pt}\mathsf{lie}: \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\rightarrow \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$. In particular, this asserts that the naturality with respect to $\mathbf{gr}^\mathrm{op}$ commutes with the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure. A further crucial ingredient is that $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ is free as a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module, a consequence of the Poincaré-Birkhoff-Witt theorem. Here we provide a model for $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$, together with its right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure, exploiting the symmetric monoidal structure $(\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathcal{O}}, \odot, \mathbb{Q})$ for an operad $\mathcal{O}$. **Proposition 140**. *The operad $\mathcal{O}$ is an $\mathcal{O}$-algebra in the category of right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules.* *If $\varphi: \mathcal{O}\rightarrow \mathcal{P}$ is a morphism of operads, then $\varphi$ is a morphism of $\mathcal{O}$-algebras in right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules, where $\mathcal{P}$ is equipped with the respective restricted structures: i.e., considered as a right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-module by restriction of the canonical right $\mathbf{Cat}\hspace{1pt}\mathcal{P}$-module structure; considered as a $\mathcal{O}$-algebra by restriction of the canonical $\mathcal{P}$-algebra structure.* *Proof.* This result is stated in [@MR2494775 Observation 9.1.3] (in terms of right $\mathcal{O}$-modules, defined in the operadic sense). For the convenience of the reader, a proof is outlined here. For $n \in \mathbb{N}$, by construction, there is an isomorphism of right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules: $$\mathbf{Cat}\hspace{1pt}\mathcal{O}(-, n) \cong \mathcal{O}(-)^{\odot n}$$ and this is $\mathfrak{S}_n$-equivariant with respect to the canonical action on the left hand side and the 'place permutation' action on the right hand side. The operad structure then induces the structure map: $$\mathcal{O}(n) \otimes_{\mathfrak{S}_n} \mathcal{O}(-)^{\odot n} \rightarrow \mathcal{O}(-)$$ in right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-modules. This gives the natural $\mathcal{O}$-algebra structure in right $\mathbf{Cat}\hspace{1pt}\mathcal{O}$-algebras, as required. ◻ **Example 141**. By Proposition [Proposition 140](#prop:opd_alg_right_mod){reference-type="ref" reference="prop:opd_alg_right_mod"}, $\mathfrak{Lie}$ is a Lie algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and $\mathfrak{Ass}^u$ is a unital, associative algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$-modules. Thus, by restriction along $\mathsf{lie}: \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$ and its associated functor $\mathbf{Cat}\hspace{1pt}\mathsf{lie}$, $\mathfrak{Ass}^u$ is a Lie algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and, with respect to this structure, $\mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$ is a morphism of Lie algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. The universal enveloping algebra functor $U$ from Lie algebras to unital associative algebras can be applied in the category of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules with respect to the structure $(\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}, \odot, \mathbb{Q})$. It is the left adjoint to the forgetful functor corresponding to restriction along $\mathsf{lie}: \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$. Applied to $\mathfrak{Lie}$, considered as a Lie algebra in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, this yields $U \mathfrak{Lie}$, which is an $\mathfrak{Ass}^u$-algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. **Proposition 142**. *The morphism $\mathsf{lie}: \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$ induces $U \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$ that is an isomorphism of $\mathfrak{Ass}^u$-algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules, where $\mathfrak{Ass}^u$ is equipped with the restricted module structure.* *Proof.* The morphism is given by the universal property of the universal enveloping algebra construction. To prove that it is an isomorphism, it suffices to establish that the underlying morphism of $\mathbb{Q}{\bm{\Sigma}}^\mathrm{op}$-modules is an isomorphism. Since we are working over $\mathbb{Q}$, as exploited in [@2021arXiv211001934P], it is sufficient to show that it induces an isomorphism on the associated Schur functors. We use the following notation for the Schur functor construction, following [@2021arXiv211001934P]. For $V$ a finite-dimensional vector space, let $\underline{V}$ be the $\mathbb{Q}{\bm{\Sigma}}$-module with $\underline{V}(\mathbf{n}) = V^{\otimes n}$, the $\mathfrak{S}_n$-action given by place permutations of the tensor factors; this construction is natural in $V$. The Schur functor construction then identifies as $- \otimes_{\bm{\Sigma}}\underline{V}$ (naturally in $V$). This is an exact functor from $\mathbb{Q}{\bm{\Sigma}}^\mathrm{op}$-modules to functors from finite-dimensional $\mathbb{Q}$-vector spaces to $\mathbb{Q}$-vector spaces. Now, $\mathfrak{Ass}^u\otimes_{\bm{\Sigma}}\underline{V}$ is naturally isomorphic to the tensor algebra functor $T(V)$, since the latter is the free unital associative algebra on $V$. Likewise, $U \mathfrak{Lie}\otimes_{\bm{\Sigma}}\underline{V}$ is naturally isomorphic to the universal enveloping algebra $U \mathrm{Lie}(V)$ on the free Lie algebra on $V$ and the map $(U \mathfrak{Lie}\rightarrow \mathfrak{Ass}^u) \otimes_{\bm{\Sigma}}\underline{V}$ identifies with the canonical isomorphism $U \mathrm{Lie}(V) \stackrel{\cong}{\rightarrow} T(V)$. This concludes the proof. ◻ More is true: **Proposition 143**. *The right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $U \mathfrak{Lie}$ has a unique cocommutative Hopf algebra structure extending the associative, unital algebra structure, for which $\mathfrak{Lie}\subset U \mathfrak{Lie}$ lies in the primitives.* *Proof.* This generalizes the classical construction of the Hopf algebra structure on the universal enveloping algebra of a Lie algebra. Namely, restricted to $\mathfrak{Lie}\subset U \mathfrak{Lie}$, the coproduct is defined to be $$\mathfrak{Lie}\rightarrow \mathfrak{Lie}\oplus \mathfrak{Lie}\cong \mathfrak{Lie}\odot \mathbb{Q}\ \oplus \ \mathbb{Q}\odot \mathfrak{Lie} \rightarrow U \mathfrak{Lie}\odot U \mathfrak{Lie},$$ where the first map is the diagonal and the final map is induced by the canonical inclusions of $\mathfrak{Lie}$ and $\mathbb{Q}$ in $U\mathfrak{Lie}$. This is clearly a morphism of Lie algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules, where $U \mathfrak{Lie}\odot U \mathfrak{Lie}$ is considered as the Lie algebra associated to the $\odot$-product associative algebra structure. Hence, this induces the coproduct $U \mathfrak{Lie}\rightarrow U\mathfrak{Lie}\odot U \mathfrak{Lie}$, a morphism of $\mathfrak{Ass}^u$-algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. Similarly, the conjugation is obtained as the anti-automorphism of algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules extending $\mathfrak{Lie}\stackrel{ (-1)\times}{\rightarrow} \mathfrak{Lie}\subset U \mathfrak{Lie}$. This gives the required cocommutative Hopf algebra structure on $U \mathfrak{Lie}$ in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. ◻ *Remark 144*.   1. Upon passing to the associated Schur functor $U \mathfrak{Lie}\otimes_{\bm{\Sigma}}\underline{V}$ as in the proof of Proposition [Proposition 142](#prop:U_right_cat_Lie_modules){reference-type="ref" reference="prop:U_right_cat_Lie_modules"}, this recovers the primitively-generated Hopf algebra structure on $U \mathrm{Lie}(V)$ with primitives $\mathrm{Lie}(V) \subset U \mathrm{Lie}(V)$. 2. The augmentation of the Lie operad $\mathfrak{Lie}\rightarrow I$ can be considered as a morphism of Lie algebras in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, where the structure of $I$ is induced by restriction along the augmentation. On applying the universal enveloping algebra functor this gives a morphism of cocommutative Hopf algebras in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules $U \mathfrak{Lie}\rightarrow U I$. Here, the underlying right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module of $U I$ is isomorphic to $\mathfrak{Com}^u$ (with 'trivial' right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure) and, via the isomorphism of Proposition [Proposition 142](#prop:U_right_cat_Lie_modules){reference-type="ref" reference="prop:U_right_cat_Lie_modules"}, the underlying morphism of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules is the morphism $\mathfrak{Ass}^u\rightarrow \mathfrak{Com}^u$. On passage to Schur functors, this corresponds to the symmetrization map $T (V) \rightarrow S(V)$, where $S(V)$ is the symmetric algebra, which is a morphism of primitively-generated Hopf algebras (naturally with respect to $V$). The exponential functor construction $\Phi$ can be applied (working in $(\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}, \odot, \mathbb{Q})$) to $U\mathfrak{Lie}$. Essentially by construction, one has: **Proposition 145**. *The exponential functor $\Phi U \mathfrak{Lie}$ is a functor from $\mathbf{gr}^\mathrm{op}$ to right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. This is isomorphic to $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$, equipped with the structure of [@2021arXiv211001934P].* *Remark 146*. In arity $s \in \mathbb{N}$, the functor $(\Phi U \mathfrak{Lie})(s)$ (evaluated with respect to the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure) is isomorphic to the functor on $\mathbf{gr}^\mathrm{op}$: $$\mathbb{Z}^{\star t} \mapsto \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u(s, t),$$ with the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure of $\Phi U \mathfrak{Lie}$ corresponding to the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-action on $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$. Moreover, by the results of [@2021arXiv211001934P], $(\Phi U \mathfrak{Lie})(s)$ is polynomial on $\mathbf{gr}^\mathrm{op}$ of degree $s$ and there is a surjection $$(\Phi U \mathfrak{Lie}) (s) \twoheadrightarrow (\mathfrak{a}^{\sharp})^{\otimes s}$$ that exhibits $(\Phi U \mathfrak{Lie})(s)$ as the projective cover of $(\mathfrak{a}^{\sharp})^{\otimes s}$ in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$. It follows that the dual $D (\Phi U \mathfrak{Lie})(s)$ is the injective envelope of $\mathfrak{a}^{\otimes s}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Hence, the main result of [@2021arXiv211001934P] can be rewritten as follows, where $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ is the full subcategory of analytic functors: **Theorem 147**. *The functor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} -$ induces an equivalence of categories $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} - : {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod} \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}).$$* *This restricts to an equivalence of categories: $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} - : {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}^{< \infty} \rightarrow \mathcal{F}_{<\infty} (\mathbf{gr}^\mathrm{op}),$$ where ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}^{< \infty} \subset {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$ is the full subcategory of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules $M$ such that $M (n)=0$ for $n \gg 0$.* **Example 148**. For a Lie algebra $\mathfrak{g}$, one has the associated left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\underline{\mathfrak{g}}$ given by $\mathbf{n} \mapsto \mathfrak{g}^{\otimes n}$, with morphisms of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ acting via the Lie algebra structure of $\mathfrak{g}$. As proved in [@2021arXiv211001934P], one has the natural isomorphism $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{g}} \cong \Phi (U \mathfrak{g}).$ as functors in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$. One can also consider $\mathfrak{Lie}$ as a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module and form $\mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{g}}$. Since $\mathfrak{Lie}$ identifies with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(-, 1)$ as a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module, which is projective, using a Yoneda lemma type argument, the above isomorphism 'restricts' to $\mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{g}} \cong \mathfrak{g}$, and this is an isomorphism of Lie algebras, where the Lie algebra structure on the left is induced by considering $\mathfrak{Lie}$ as a Lie algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. The naturality of these isomorphisms implies that they apply to $\mathfrak{malcev}$, considered as a Lie algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Hence there are natural isomorphisms: $$\begin{aligned} \Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} &\cong & \Phi (U \mathfrak{malcev}) \\ \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} & \cong & \mathfrak{malcev}.\end{aligned}$$ # Relating the covariant and contravariant cases {#sect:compare} The purpose of this section is to make explicit the relationship between Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} and Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"}. This is based on the duality between ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ and $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$. This involves relating $\underline{\mathfrak{malcev}}$, which is a left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, and $\Phi U\mathfrak{Lie}$, a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$. Theorem [Theorem 158](#thm:otimes_gr_PhiUL_malcev){reference-type="ref" reference="thm:otimes_gr_PhiUL_malcev"} gives a concrete 'duality' statement. Theorem [Theorem 151](#thm:equiv_Umalcev_PhiUlie){reference-type="ref" reference="thm:equiv_Umalcev_PhiUlie"} explains *why* $\Phi U \mathfrak{Lie}$ arises naturally in the theory. Essentially this section serves to verify that the theories behave as they should. ## Analysing $\Phi U \mathfrak{Lie}$ via ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} gives the equivalence of categories: $$\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, - ) : {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}.$$ Underlying this equivalence is the fact that $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, \mathfrak{malcev})$ is naturally isomorphic to $\mathfrak{Lie}$ as a Lie algebra in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, by Proposition [Proposition 132](#prop:malcev_full_subcat){reference-type="ref" reference="prop:malcev_full_subcat"}. Now, the functor $P_{(-)}: \Gamma \mapsto P_\Gamma$ is an exponential functor, via the natural equivalence $P_{\mathbb{Z}^{\star s} } \cong P_\mathbb{Z}^{\otimes s}$. This translates to ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$: **Lemma 149**. *The functor $\Gamma \mapsto \mathsf{q}^\mathbf{gr}_\bullet P_\Gamma$ from $\mathbf{gr}^\mathrm{op}$ to ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is exponential.* *Namely, $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}\cong \Phi (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$, where the exponential functor construction $\Phi$ is applied in $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \overline{\otimes}, \mathbb{Q})$.* *Proof.* The result follows from Proposition [Proposition 85](#prop:qgr_tensor_Pzed){reference-type="ref" reference="prop:qgr_tensor_Pzed"}. ◻ *Remark 150*. One would have liked to prove Lemma [Lemma 149](#lem:expo_propoly){reference-type="ref" reference="lem:expo_propoly"} by appealing to the 'symmetric monoidality' of $\mathsf{q}^\mathbf{gr}_\bullet$ given by Proposition [Proposition 55](#prop:propoly_sym_mon){reference-type="ref" reference="prop:propoly_sym_mon"}; unfortunately the latter is only established restricted to $\mathcal{F}_{<\infty} (\mathbf{gr})$, considered as a full subcategory of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Proposition [Proposition 85](#prop:qgr_tensor_Pzed){reference-type="ref" reference="prop:qgr_tensor_Pzed"} allows this problem to be circumvented. **Theorem 151**. *Under the equivalence of Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}, $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}$ corresponds to $\Phi U \mathfrak{Lie}$, considered as a functor from $\mathbf{gr}^\mathrm{op}$ to $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$.* *Proof.* The functor $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, -)$ is symmetric monoidal, by Theorem [Theorem 138](#thm:hom_underline_malcev_sym_monoidal){reference-type="ref" reference="thm:hom_underline_malcev_sym_monoidal"}. In particular, it commutes with the exponential functor construction $\Phi$, up to natural isomorphism. Since $\Gamma \mapsto \mathsf{q}^\mathbf{gr}_\bullet P_\Gamma$ is naturally isomorphic to $\Phi (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$ by Lemma [Lemma 149](#lem:expo_propoly){reference-type="ref" reference="lem:expo_propoly"}, it suffices to show that there is an isomorphism $$\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}) \cong U \mathfrak{Lie}$$ as Hopf algebras in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, where the Hopf algebra structure of the domain is induced by that of $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Proposition [Proposition 98](#prop:U_malcev){reference-type="ref" reference="prop:U_malcev"} gives the isomorphism $U \mathfrak{malcev}\cong \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ of Hopf algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Thus, we require to establish the isomorphism of Hopf algebras $$\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, U \mathfrak{malcev}) \cong U \mathfrak{Lie}.$$ Now, $\mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, \mathfrak{malcev}) \cong \mathfrak{Lie}$ as Lie algebras in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, by Proposition [Proposition 132](#prop:malcev_full_subcat){reference-type="ref" reference="prop:malcev_full_subcat"} and the inclusion $\mathfrak{malcev}\hookrightarrow U \mathfrak{malcev}$ induces an inclusion of Lie algebras $\mathfrak{Lie}\hookrightarrow \mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, U \mathfrak{malcev})$ in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. This induces a morphism of Hopf algebras in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$: $$U \mathfrak{Lie}\rightarrow \mathrm{Hom}_{{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}} (\underline{\mathfrak{malcev}}, U \mathfrak{malcev}).$$ To check that this is an isomorphism, one proceeds as in the proof of Proposition [Proposition 98](#prop:U_malcev){reference-type="ref" reference="prop:U_malcev"}, i.e., by reducing to considering the associated graded of $U \mathfrak{malcev}$ and forgetting the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure. ◻ By construction, the Passi functors are encoded in $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$, considered as an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Combining Proposition [Proposition 142](#prop:U_right_cat_Lie_modules){reference-type="ref" reference="prop:U_right_cat_Lie_modules"} with the identification established above in the proof of Theorem [Theorem 151](#thm:equiv_Umalcev_PhiUlie){reference-type="ref" reference="thm:equiv_Umalcev_PhiUlie"}, one has: **Corollary 152**. *The Passi functors $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$, considered as forming a unital associative algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, are the image under the equivalence of Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} of $\mathfrak{Ass}^u$, considered as a unital associative algebra in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules.* ## Relating $\underline{\mathfrak{malcev}}$ and $\Phi U \mathfrak{Lie}$ The functor $\mathfrak{malcev}^{\overline{\otimes}s}$ in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ is the projective cover of $\mathfrak{a}^{\otimes s}$, for $s \in \mathbb{N}$, by Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}. By Proposition [Proposition 132](#prop:malcev_full_subcat){reference-type="ref" reference="prop:malcev_full_subcat"}, these assemble to form the left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\underline{\mathfrak{malcev}}$, where the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure encodes the morphisms between such functors. In the contravariant setting, as explained in Remark [Remark 146](#rem:identifications_Phi_ULie){reference-type="ref" reference="rem:identifications_Phi_ULie"}, the functor $(\Phi U \mathfrak{Lie})(s)$ is the projective cover in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ of $(\mathfrak{a}^{\sharp})^{\otimes s}$ and these assemble to give the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module $\Phi U \mathfrak{Lie}$. The purpose of this section is to relate these structures, showing that they are 'dual' by using $\otimes_\mathbf{gr}$. Recall from Proposition [Proposition 85](#prop:qgr_tensor_Pzed){reference-type="ref" reference="prop:qgr_tensor_Pzed"} that there is a $\mathfrak{S}_t$-equivariant isomorphism $\mathsf{q}^\mathbf{gr}_\bullet (P_{\mathbb{Z}^{\star t}}) \cong (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t}$, for $t \in \mathbb{N}$. In particular, via this isomorphism one deduces that $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is a unital, associative algebra in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. This gives: **Lemma 153**. *The association $t \mapsto \mathsf{q}^\mathbf{gr}_\bullet (P_{\mathbb{Z}^{\star t}}) \cong (\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t}$ defines a left $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$-module in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, denoted $(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$.* On forming the tensor product over $\mathbf{gr}$ (using $\ {\widetilde{\otimes}_\mathbf{gr}}$ of Proposition [Proposition 65](#prop:otimes_gr_propoly_analytic){reference-type="ref" reference="prop:otimes_gr_propoly_analytic"}), one has $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$, which has the structure of a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$, left $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ bimodule, with the actions derived respectively from $\Phi U \mathfrak{Lie}$ and $(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$. Explicitly, in bi-arity $(s,t)$ (where $s$ corresponds to the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-structure and $t$ to the $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$-structure), this identifies as: $$(\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*})(s,t) = \Phi U \mathfrak{Lie}(s) \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t}.$$ *Remark 154*. Since $\Phi U \mathfrak{Lie}(s)$ is polynomial, in particular lies in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$, $\Phi U \mathfrak{Lie}(s) \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t}$ is naturally isomorphic to $\Phi U \mathfrak{Lie}(s) \otimes_\mathbf{gr}P_\mathbb{Z}^{\otimes t} \cong \Phi U \mathfrak{Lie}(s) \otimes_\mathbf{gr}P_{\mathbb{Z}^{\star t}}$, by Proposition [Proposition 66](#prop:compatibility_otimes_gr){reference-type="ref" reference="prop:compatibility_otimes_gr"}. Recall that $P_\mathbb{Z}$ splits as $\overline{P}\oplus \mathbb{Q}$, hence $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$ contains $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}*}$ as a direct summand. The canonical surjection $\overline{P}\twoheadrightarrow \mathfrak{a}$ induces $(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}s} \twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}^{\otimes s}$, for $s \in \mathbb{N}$. Together with the surjection $(\Phi U \mathfrak{Lie}) (s) \twoheadrightarrow (\mathfrak{a}^\sharp)^{\otimes s}$ in $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op})$, this induces $$(\Phi U \mathfrak{Lie}) (s) \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}s} \twoheadrightarrow (\mathfrak{a}^\sharp)^{\otimes s} \otimes_\mathbf{gr}\mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}^{\otimes s} \cong \mathrm{Hom}_{\mathcal{F}_{<\infty} (\mathbf{gr})} (\mathfrak{a}^{\otimes s} , \mathfrak{a}^{\otimes s}).$$ **Lemma 155**. *For $s \in \mathbb{N}$, the map $(\Phi U \mathfrak{Lie}) (s) \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}s} \twoheadrightarrow \mathrm{Hom}_{\mathcal{F}_{<\infty} (\mathbf{gr})} (\mathfrak{a}^{\otimes s} , \mathfrak{a}^{\otimes s})$ is an isomorphism of $\mathfrak{S}_s$-bimodules.* *Proof.* By construction, the map is a morphism of $\mathfrak{S}_s$-bimodules, hence it suffices to show that it is an isomorphism. Now, $(\Phi U \mathfrak{Lie}) (s)$ has polynomial degree $s$ with respect to $\mathbf{gr}^\mathrm{op}$ and the kernel of $(\Phi U \mathfrak{Lie}) (s) \twoheadrightarrow (\mathfrak{a}^\sharp)^{\otimes s}$ has polynomial degree less than $s$. The functor $- \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}s}$ is exact and vanishes on functors of polynomial degree less than $s$; the result follows. ◻ *Notation 156*. For $s \in \mathbb{N}$, let $\widetilde{\mathrm{Id}_s} \in (\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*})(s,s)$ be the element corresponding to the identity morphism of $\mathrm{Hom}_{\mathcal{F}_{<\infty} (\mathbf{gr})} (\mathfrak{a}^{\otimes s} , \mathfrak{a}^{\otimes s})$ via the isomorphism of Lemma [Lemma 155](#lem:iso_pbar){reference-type="ref" reference="lem:iso_pbar"} and the inclusion $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}*} \subset \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$. Restricting along $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\rightarrow \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$, induced by $\mathfrak{Lie}\rightarrow \mathfrak{Ass}^u$, $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ can be considered as a right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$, left $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ bimodule. Proposition [Proposition 145](#prop:Phi_U_L_model){reference-type="ref" reference="prop:Phi_U_L_model"} then leads to: **Proposition 157**. *There is an isomorphism of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$, left $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ bimodules $$\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u\stackrel{\cong}{\rightarrow} \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}.$$* *This is determined by the property that, for $s \in \mathbb{N}$, the identity morphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u(s,s)$ is sent to $\widetilde{\mathrm{Id}_s} \in (\Phi U \mathfrak{Lie}\otimes_\mathbf{gr}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*})(s,s)$.* *Proof.* For $t \in \mathbb{N}$, one has the isomorphism $(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}t} \cong \mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star t}}$. Hence, for $s \in \mathbb{N}$: $$(\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*})(s,t) \cong \Phi U \mathfrak{Lie}(s) \ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_{\mathbb{Z}^{\star t}}) \cong \Phi U \mathfrak{Lie}(\mathbb{Z}^{\star t})(s) ,$$ where the second isomorphism follows from the Yoneda lemma. From the construction, one identifies $\Phi U \mathfrak{Lie}(\mathbb{Z}^{\star t}) \cong (U \mathfrak{Lie})^{\odot t}$, where the convolution product is formed in right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. In bi-arity $(s,t)$, this is isomorphic to $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u(s,t)$, by construction of the latter, using the isomorphism $U \mathfrak{Lie}\cong \mathfrak{Ass}^u$ of Proposition [Proposition 142](#prop:U_right_cat_Lie_modules){reference-type="ref" reference="prop:U_right_cat_Lie_modules"}, as in Proposition [Proposition 145](#prop:Phi_U_L_model){reference-type="ref" reference="prop:Phi_U_L_model"}. To complete the proof of the first statement it remains to check that this corresponds to an isomorphism of bimodules. This is deduced from Proposition [Proposition 142](#prop:U_right_cat_Lie_modules){reference-type="ref" reference="prop:U_right_cat_Lie_modules"} together with the properties of the exponential functor construction. Since the isomorphism is one of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$-modules, by Yoneda it is uniquely determined by the images of the identity elements in $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u(s,s)$, for $s \in \mathbb{N}$. To conclude, one checks from the above identifications that these images are the elements $\widetilde{\mathrm{Id}_s}$, as required. ◻ This Proposition is the key input in the following result that relates $\underline{\mathfrak{malcev}}$ and $\Phi U \mathfrak{Lie}$. **Theorem 158**. *There is a unique isomorphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules $$\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\stackrel{\cong}{\rightarrow} \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}}$$ that fits into the commutative diagram: $$\begin{aligned} \label{eqn:diag_PhiULie_malcev} \xymatrix{ \mathbf{Cat}\hspace{1pt}\mathfrak{Lie} \ar[r]^(.4)\cong \ar@{^(->}[d] & \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}} \ar@{^(->}[d] \\ \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u \ar[r]_(.4)\cong & \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}, }\end{aligned}$$ in which the left hand vertical map is induced by $\mathfrak{Lie}\hookrightarrow \mathfrak{Ass}^u$, the right hand vertical map by the canonical inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$, and the lower horizontal isomorphism is given by Proposition [Proposition 157](#prop:iso_cat_uass){reference-type="ref" reference="prop:iso_cat_uass"}.* *Proof.* The inclusion of $\mathbb{Q}$-linear categories $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\hookrightarrow \mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$ can be considered as a morphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules, using the restricted structure on $\mathbf{Cat}\hspace{1pt}\mathfrak{Ass}^u$. Likewise, since $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$ is a morphism of Lie algebras in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ (for the commutator bracket structure on $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$), the map $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}} \rightarrow \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$ is a morphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules. It is injective since, as an object of ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, $\mathfrak{malcev}$ is a direct summand of $\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$. Hence, to construct the morphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules, it suffices to show that the composite around the bottom of the diagram under construction ([\[eqn:diag_PhiULie_malcev\]](#eqn:diag_PhiULie_malcev){reference-type="ref" reference="eqn:diag_PhiULie_malcev"}) factors across $\Phi U \mathfrak{Lie}\otimes_\mathbf{gr}\underline{\mathfrak{malcev}}$. By the characterization given in Proposition [Proposition 157](#prop:iso_cat_uass){reference-type="ref" reference="prop:iso_cat_uass"}, this reduces to showing that, for each $s \in \mathbb{N}$, the element $\widetilde{\mathrm{Id}_s}$, lies in the image of $\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}} \rightarrow \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})^{\overline{\otimes}*}$ in bi-arity $(s,s)$. Now, $\mathfrak{malcev}$ is a subobject of $\mathsf{q}^\mathbf{gr}_\bullet \overline{P}\subset \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z}$. The inclusion $\mathfrak{malcev}\hookrightarrow \mathsf{q}^\mathbf{gr}_\bullet \overline{P}$ induces an isomorphism $$(\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}}) (s,s) \stackrel{\cong}{\rightarrow} (\Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}(\mathsf{q}^\mathbf{gr}_\bullet \overline{P})^{\overline{\otimes}*}) (s,s)$$ since the projection $\mathsf{q}^\mathbf{gr}_\bullet \overline{P}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathfrak{a}$ factors across the retract $\mathsf{q}^\mathbf{gr}_\bullet \overline{P}\twoheadrightarrow \mathfrak{malcev}$. Hence, by the construction of the elements $\widetilde{\mathrm{Id}_s}$, one obtains the required factorization. The resulting map $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\rightarrow \Phi U \mathfrak{Lie}\ {\widetilde{\otimes}_\mathbf{gr}}\underline{\mathfrak{malcev}}$ is clearly injective. To conclude, it suffices to observe that the underlying bigraded vector spaces are of finite type (i.e., finite dimensional in each bi-arity) and isomorphic. This follows from Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} together with the fact that $- \ {\widetilde{\otimes}_\mathbf{gr}}\mathfrak{malcev}^{\overline{\otimes}t}$ is equivalent to the $t$th cross-effect functor when restricted to objects of $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op})$ that take finite-dimensional values. The latter fact follows by duality from the corepresenting property of $\mathfrak{malcev}^{\overline{\otimes}t}$ given in Theorem [Theorem 104](#thm:proj_cover_malcev_s){reference-type="ref" reference="thm:proj_cover_malcev_s"}. ◻ Theorem [Theorem 158](#thm:otimes_gr_PhiUL_malcev){reference-type="ref" reference="thm:otimes_gr_PhiUL_malcev"} has the following useful Corollary: **Corollary 159**. *For $M \in \mathrm{Ob}\hspace{2pt}{}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$ and $N \in \mathrm{Ob}\hspace{2pt}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, there is a canonical isomorphism: $$(\Phi U \mathfrak{Lie}\otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M) \ {\widetilde{\otimes}_\mathbf{gr}}(N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}) \cong N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M.$$* *Proof.* The left hand side is naturally isomorphic to $$(N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}) \widetilde{\otimes}_{\mathbf{gr}^\mathrm{op}} (\Phi U \mathfrak{Lie}\otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M) \cong N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (\underline{\mathfrak{malcev}} \otimes_{\mathbf{gr}^\mathrm{op}} \Phi U \mathfrak{Lie}) \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M,$$ reversing the order of the tensor factors and then using associativity. (Here the tensor product $\widetilde{\otimes}_{\mathbf{gr}^\mathrm{op}} : {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\times \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \rightarrow \mathtt{Mod}_\mathbbm{k}$ is as in Proposition [Proposition 65](#prop:otimes_gr_propoly_analytic){reference-type="ref" reference="prop:otimes_gr_propoly_analytic"}, up to the change of order of the factors.) Theorem [Theorem 158](#thm:otimes_gr_PhiUL_malcev){reference-type="ref" reference="thm:otimes_gr_PhiUL_malcev"} gives that $\underline{\mathfrak{malcev}} \widetilde{\otimes}_{\mathbf{gr}^\mathrm{op}} \Phi U \mathfrak{Lie}$ is isomorphic to $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule. The result follows, since $N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M$ is naturally isomorphic to $N \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M$. ◻ *Remark 160*. In the isomorphism of Corollary [Corollary 159](#cor:otimes_compatibility){reference-type="ref" reference="cor:otimes_compatibility"}, on the left hand side $M$ appears on the left of $\ {\widetilde{\otimes}_\mathbf{gr}}$ whereas, on the right hand side, it is on the right of $\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. This reflects the fact that left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules correspond to analytic functors on $\mathbf{gr}^\mathrm{op}$, which can be considered as *right* $\mathbb{Q}\mathbf{gr}$-modules, whereas right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules are related to *left* $\mathbb{Q}\mathbf{gr}$-modules. # Bimodules and Bifunctors {#sect:bifunctors} The purpose of this section is to put together the relations between left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ and right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules and $\mathcal{F}(\mathbf{gr})$ so as to explain the relationship between $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules and bifunctors, i.e., objects of $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. Due to the *analyticity* arising in considering functors on $\mathbf{gr}^\mathrm{op}$ and the *pro-polynomiality* for functors of $\mathbf{gr}$, the general case has an unavoidable technical aspect. If one restricts to *polynomial bifunctors*, then the situation is much simpler. On first reading, the reader is encouraged to focus upon this case, for which the results are immediate consequences of the earlier ones. The general case can be modelled by passing to the appropriate category of ind-pro objects. Rather than follow this formal approach, we phrase the construction in terms of analytic functors on $\mathbf{gr}^\mathrm{op}$ with values in ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Throughout, $\mathbbm{k}= \mathbb{Q}$. ## $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules {#subsect:catlie_bimodules} By definition, a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule is an object of $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}\times {\bm{\Sigma}})$ equipped with commuting left and right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-actions. Explicitly, a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule is a bigraded $\mathbb{Q}$-module, given by $M (s,t)$, for $s , t \in \mathbb{N}$, equipped with structure morphisms: $$\begin{aligned} &&\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(t ,u) \otimes M (s,t) \rightarrow M (s,u) \\ && M (s,t) \otimes \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(r ,s)\rightarrow M (r,t)\end{aligned}$$ that commute and satisfy the unital and associativity constraints. (This fixes our conventions for denoting the bi-arities: for fixed $s$, $M(s,-)$ is a *left* $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module and, for fixed $t$, $M(-,t)$ is a *right* $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module.) Then, extending Notation [Notation 120](#nota:catopd_modules){reference-type="ref" reference="nota:catopd_modules"}: *Notation 161*. Denote by 1. ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ the category of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules; 2. ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{<\infty} \subset {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ the full subcategory of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules with finite support (i.e., such that $M(s,t)=0$ for $s+t \gg 0$). One has the following standard identification: **Lemma 162**. *The category ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ is equivalent to both the following* 1. *the category of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$;* 2. *the category of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules in ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$.* The canonical filtrations of Section [7.2](#subsect:canon_filt_catopd_mod){reference-type="ref" reference="subsect:canon_filt_catopd_mod"} for left (respectively right) $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules induce the increasing filtration corresponding to the subobjects $M_{\leq d} \hookrightarrow M$ and the decreasing filtration corresponding to the quotients $M \twoheadrightarrow M^{\leq e}$ in ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. As usual, the tensor product $\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ induces a monoidal structure on bimodules: **Proposition 163**. *There is a monoidal structure $({}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}, \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} , \mathbf{Cat}\hspace{1pt}\mathfrak{Lie})$.* ## The convolution product on $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules The Day convolution $\odot$ extends to a symmetric monoidal structure on ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$ and on $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, as recalled in Section [7.4](#subsect:convolution_cat_opd){reference-type="ref" reference="subsect:convolution_cat_opd"}. To consider bimodules, we start with the Day convolution product for $\mathbb{Q}{\bm{\Sigma}}$-bimodules: **Lemma 164**. *There is a symmetric monoidal structure $(\mathcal{F}({\bm{\Sigma}}^\mathrm{op}\times {\bm{\Sigma}}), \circledcirc , \mathbb{Q})$, where $\mathbb{Q}$ is considered as a $\mathbb{Q}{\bm{\Sigma}}$-bimodule concentrated in arity $(0,0)$. The convolution product $\circledcirc$ is given on objects $B_1$, $B_2$ by $$(B_1 \circledcirc B_2) (m,n) := \bigoplus_{i+j =m} \bigoplus_{s+t=n} \big (B_1 (i,s) \otimes B_2 (j,t)\big) \uparrow_{\mathfrak{S}_i^\mathrm{op}\times \mathfrak{S}_j^\mathrm{op}}^{\mathfrak{S}_m^\mathrm{op}} \uparrow _{\mathfrak{S}_s \times \mathfrak{S}_t}^{\mathfrak{S}_n}.$$* *Remark 165*. This is related to the Day convolution product for $\mathcal{F}({\bm{\Sigma}}^\mathrm{op})$ and for $\mathcal{F}({\bm{\Sigma}})$ as follows. The category $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}\times {\bm{\Sigma}})$ is equivalent to $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}; \mathcal{F}({\bm{\Sigma}}))$. Here $\mathcal{F}({\bm{\Sigma}})$ is equipped with the Day convolution product $\odot$. Then $$(B_1 (-,*) \circledcirc B_2 (-,*)) (m) = \bigoplus_{i+j =m} B_1 (i, *) \odot B_2 (j,*) \uparrow_{\mathfrak{S}_i^\mathrm{op}\times \mathfrak{S}_j^\mathrm{op}}^{\mathfrak{S}_m^\mathrm{op}}$$ and the right hand side is simply the convolution product in $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}; \mathcal{F}({\bm{\Sigma}}))$ with respect to ${\bm{\Sigma}}^\mathrm{op}$ (using the convolution symmetric monoidal structure $\odot$ on $\mathcal{F}({\bm{\Sigma}})$). **Proposition 166**. *The convolution product provides a symmetric monoidal structure $({}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}, \circledcirc , \mathbb{Q})$.* *Proof.* The proof generalizes the analysis of $\circledcirc$ on $\mathcal{F}({\bm{\Sigma}}^\mathrm{op}\times {\bm{\Sigma}})$ given in Remark [Remark 165](#rem:biconvolution){reference-type="ref" reference="rem:biconvolution"}, using that ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ is equivalent to the category of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$, by Lemma [Lemma 162](#lem:bimodules_left_right){reference-type="ref" reference="lem:bimodules_left_right"}. One forms the convolution product in left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules with respect to the symmetric monoidal structure on $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ given by the convolution product. The details are left to the reader. ◻ ## Bifunctors {#subsect:bimod_functors} We now consider bifunctors, i.e., the category $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. For a bifunctor $F$ and $d,e \in \mathbb{N}$, there are canonical inclusions $\mathsf{p}_d F \hookrightarrow \mathsf{p}_{d+1} F \hookrightarrow F$ and canonical surjections $F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_{e+1} F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_e F,$ given by using the respective polynomial filtrations with respect to $\mathbf{gr}^\mathrm{op}$ and $\mathbf{gr}$ respectively. **Lemma 167**. *For a bifunctor $F$ and $d, e \in \mathbb{N}$, there is a natural morphism $\mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F \rightarrow \mathsf{p}_d \mathsf{q}^\mathbf{gr}_e F$.* *Proof.* Applying the functor $\mathsf{p}_d$ to the canonical surjection $F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_e F$ gives $\mathsf{p}_d F \rightarrow \mathsf{p}_d \mathsf{q}^\mathbf{gr}_e F$. Since the codomain has polynomial degree $e$ with respect to $\mathbf{gr}$, this factorizes naturally as required. ◻ *Remark 168*. This gives the notion of a bifunctor that is analytic with respect to $\mathbf{gr}^\mathrm{op}$: namely, a bifunctor $F$ for which the canonical morphism $\lim_{\substack {\rightarrow \\ d} } \mathsf{p}_d F \rightarrow F$ is an isomorphism. This is equivalent to the condition that, for each $\Gamma \in \mathrm{Ob}\hspace{2pt}\mathbf{gr}$, the functor $F (- \times \Gamma)$ in $\mathcal{F}(\mathbf{gr}^\mathrm{op})$ is analytic. This can be placed in a more general framework: for any abelian category $\mathscr{A}$, the above definition generalizes to give the full subcategory $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; \mathscr{A}) \subset \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathscr{A})$ of analytic functors. Then, using the natural equivalence $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$, the category of analytic bifunctors introduced above is the full subcategory $$\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr})) \subset \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))\cong \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}).$$ The difference functors $\delta^{\mathbf{gr}}$ and $\delta^{\mathbf{gr}^\mathrm{op}}$ act on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ and commute, so that one can adopt the following definition of polynomiality for bifunctors: **Definition 169**. A functor $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ has polynomial degree $d$ if $(\delta^{\mathbf{gr}})^i (\delta^{\mathbf{gr}^\mathrm{op}})^j F =0$ for all $i, j \in \mathbb{N}$ such that $i+j>d$. *Notation 170*. The full subcategory of polynomial bifunctors of degree $d$ is denoted $\mathcal{F}_d (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ and $\mathcal{F}_{<\infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ denotes $\bigcup_d \mathcal{F}_d (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. Basic examples of polynomial bifunctors are given by using the exterior tensor product: **Lemma 171**. *For $i, j\in \mathbb{N}$, the exterior tensor product $\mathcal{F}(\mathbf{gr}^\mathrm{op}) \times \mathcal{F}(\mathbf{gr}) \stackrel{\boxtimes}{\rightarrow} \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ restricts to $\mathcal{F}_i (\mathbf{gr}^\mathrm{op}) \times \mathcal{F}_j (\mathbf{gr}) \stackrel{\boxtimes}{\rightarrow} \mathcal{F}_{i+j} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* Under the hypothesis of polynomiality, one has the following, which ensures that forming the respective polynomial filtrations behaves as expected. **Proposition 172**. *For $F \in \mathrm{Ob}\hspace{2pt}\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ and $d, e \in \mathbb{N}$, the natural morphism $\mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F \rightarrow \mathsf{p}_d \mathsf{q}^\mathbf{gr}_e F$ is an isomorphism.* *Proof.* Restricted to $\mathcal{F}_{< \infty}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$, the functors $\mathsf{p}_d$ and $\mathsf{q}^\mathbf{gr}_e$ are exact. Using this, one checks that the natural transformation of Lemma [Lemma 167](#lem:compare_pq){reference-type="ref" reference="lem:compare_pq"} is an isomorphism, as follows. Write $F'$ for the kernel of the natural surjection $F \twoheadrightarrow \mathsf{q}^\mathbf{gr}_e F$. Then, applying the natural surjection $\mathsf{p}_d \twoheadrightarrow \mathsf{q}^\mathbf{gr}_e \mathsf{p}_d$ gives a commutative diagram: $$\xymatrix{ \mathsf{p}_d F' \ar@{->>}[d] \ar[r] & \mathsf{p}_d F \ar@{->>}[d] \ar[r] & \mathsf{p}_d \mathsf{q}^\mathbf{gr}_e F \ar@{->>}[d]|{\cong} \\ \mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F' \ar[r] & \mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F \ar[r] & \mathsf{q}^\mathbf{gr}_e\mathsf{p}_d \mathsf{q}^\mathbf{gr}_e F }$$ in which the rows are short exact. The result is equivalent to showing that the bottom right horizontal map is an isomorphism. This is equivalent to the vanishing of $\mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F'$. Now $\mathsf{q}^\mathbf{gr}_e F'=0$, by construction; applying $\mathsf{q}^\mathbf{gr}_e$ to the inclusion $\mathsf{p}_d F' \hookrightarrow F'$ therefore gives $\mathsf{q}^\mathbf{gr}_e \mathsf{p}_d F' =0$, by exactness of $\mathsf{q}^\mathbf{gr}_e$, as required. ◻ ## Mixing analyticity and pro-polynomiality Our aim is to use ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ to model bifunctors. Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} leads to an *analyticity* condition with respect to $\mathbf{gr}^\mathrm{op}$, hence one should expect to restrict to analytic bifunctors, $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$, as in Remark [Remark 168](#rem:analytic_bifunctor){reference-type="ref" reference="rem:analytic_bifunctor"}. The relationship between $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ and $\mathcal{F}(\mathbf{gr})$ as made precise by Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}, makes clear that one should not expect to model all analytic bifunctors on the nose. Namely, the target category should be replaced by ${\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, so that one can consider the abelian category $\mathcal{F}(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ and its full subcategory $\mathcal{F}_\omega(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ of analytic functors (defined as in Remark [Remark 168](#rem:analytic_bifunctor){reference-type="ref" reference="rem:analytic_bifunctor"}). *Remark 173*. The above reflects the choice to work with $\mathcal{F}_\omega(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$, based upon the equivalence $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$. One could equally well have used the equivalence $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}; \mathcal{F}(\mathbf{gr}^\mathrm{op}))$. In this case, one is lead to study the appropriate category of pro-polynomial functors on $\mathbf{gr}$ with values in analytic functors on $\mathbf{gr}^\mathrm{op}$. **Lemma 174**. *The functor $\mathsf{q}^\mathbf{gr}_\bullet : \mathcal{F}(\mathbf{gr})\rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$ induces a functor $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr})) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op};{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$.* *This induces an exact functor $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ that is the inclusion of a full subcategory.* *Proof.* The first statement is immediate. The second follows using the fact that the restriction of $\mathsf{q}^\mathbf{gr}_\bullet$ to $\mathcal{F}_{<\infty} (\mathbf{gr})$ is exact; one checks easily that the given functor takes values in analytic functors with respect to $\mathbf{gr}^\mathrm{op}$. The identification of the essential image is straightforward. ◻ **Proposition 175**. * * 1. *The category $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ is abelian and the inclusion functor $$\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \hookrightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$$ is exact.* 2. *The completion functor $\mathfrak{c}: {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathcal{F}(\mathbf{gr})$ induces an exact functor $\mathcal{F}(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr})) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* 3. *The composite gives an exact functor $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* 4. *The composite of $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ with $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ of Lemma [Lemma 174](#lem:include_finite_bifunctors){reference-type="ref" reference="lem:include_finite_bifunctors"} is naturally equivalent to the inclusion $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \subset \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* *Proof.* The first statement is clear; the second follows from the exactness of the completion functor given by Proposition [Proposition 59](#prop:compl_exact){reference-type="ref" reference="prop:compl_exact"}. The restriction to polynomial bifunctors is analysed directly from the definitions. ◻ The following is immediate: **Proposition 176**. *The symmetric monoidal structure $({\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}, \odot, \mathbb{Q})$ induces a symmetric monoidal structure on $\mathcal{F}(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$. This restricts to a symmetric monoidal structure on the full subcategory $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$.* *Restricted to the subcategory $\mathcal{F}_{<\infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})\hookrightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$, this symmetric monoidal structure is equivalent to that induced by $\otimes$ on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* ## Modelling bifunctors We combine the equivalence of Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"} with that of Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} to treat bifunctors. **Theorem 177**. *The functor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (-) \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ induces an equivalence of categories: $${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$$* *This restricts to an equivalence of categories: $${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{<\infty} \stackrel{\cong}{\rightarrow} \mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$$ fitting into the commutative (up to natural isomorphism) diagram: $$\xymatrix{ {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{<\infty} \ar[r]^\cong \ar@{^(->}[d] & \mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \ar@{^(->}[d] \\ {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \ar[r]_(.4)\cong & \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}), }$$ in which the right hand vertical functor is as in Proposition [Proposition 175](#prop:complete_bifunctors){reference-type="ref" reference="prop:complete_bifunctors"}.* *Proof.* The category ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ is equivalent to the category of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules in $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ by Lemma [Lemma 162](#lem:bimodules_left_right){reference-type="ref" reference="lem:bimodules_left_right"}. One checks that Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} implies that $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} -$ induces an equivalence of categories $${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}).$$ Indeed, $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} -$ clearly induces an exact functor ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}})$ and this takes values in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}})$. Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} implies that this yields the stated equivalence. Post-composing with the equivalence $- \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} : \mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \rightarrow {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$, this gives the equivalence $${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$$ On restriction to ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{<\infty}$, one checks that the equivalence restricts as stated. ◻ The category ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ has the symmetric monoidal structure $({}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}, \circledcirc, \mathbb{Q})$ by Proposition [Proposition 166](#prop:convolution_bimodules){reference-type="ref" reference="prop:convolution_bimodules"} and $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ the symmetric monoidal structure of Proposition [Proposition 176](#prop:sym_mon_odot_bifunctors){reference-type="ref" reference="prop:sym_mon_odot_bifunctors"} (which is induced by the tensor product on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$). The equivalence of Theorem [Theorem 177](#thm:equivalence_bimodules){reference-type="ref" reference="thm:equivalence_bimodules"} is symmetric monoidal with respect to these: **Theorem 178**. *The functor $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}( - )\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \ : \ {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$$ is a symmetric monoidal equivalence.* *Proof.* This follows from the fact that $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} -$ is symmetric monoidal for the convolution product $\odot$ on ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}$ and the usual tensor product on $\mathcal{F}_{\omega}(\mathbf{gr}^\mathrm{op})$, by [@2021arXiv211001934P] and the corresponding result for $- \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$, which follows from Theorem [Theorem 138](#thm:hom_underline_malcev_sym_monoidal){reference-type="ref" reference="thm:hom_underline_malcev_sym_monoidal"}. ◻ **Example 179**. Under the equivalence of Theorem [Theorem 177](#thm:equivalence_bimodules){reference-type="ref" reference="thm:equivalence_bimodules"}, the bifunctors $\mathbb{Q}\boxtimes \mathfrak{a}$ and $\mathfrak{a}^{\sharp} \boxtimes \mathbb{Q}$ in $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ correspond respectively to the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules $\mathbb{Q}(1,0)$ and $\mathbb{Q}(0,1)$ given by $\mathbb{Q}$ concentrated in the indicated bi-arities. Using the convolution $\circledcirc$ for bimodules, $\mathbb{Q}(1,0) ^{\circledcirc s} \circledcirc \mathbb{Q}(0,1) ^{\circledcirc t}$ identifies as the bimodule $\mathbb{Q}\mathfrak{S}_t \boxtimes \mathbb{Q}\mathfrak{S}_s$ concentrated in bi-arity $(s,t$), with left action of $\mathfrak{S}_t$ and right action of $\mathfrak{S}_s$ (morphisms of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ that do not preserve the arity necessarily act by zero). The associated bifunctor is $(\mathbb{Q}\boxtimes \mathfrak{a})^{\otimes s} \otimes (\mathfrak{a}^{\sharp} \boxtimes \mathbb{Q})^{\otimes t}$, which identifies as $(\mathfrak{a}^\sharp) ^{\otimes t} \boxtimes \mathfrak{a}^{\otimes s}$, which is isomorphic to the bifunctor obtained by applying $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (-) \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ to $\mathbb{Q}\mathfrak{S}_t \boxtimes \mathbb{Q}\mathfrak{S}_s$. This analysis can be extended to treat all 'homogeneous polynomial bifunctors' (i.e., those corresponding to a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule supported on a single bi-arity). ## Comparing $\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ and $\otimes_\mathbf{gr}$ By Proposition [Proposition 27](#prop:gr_bifunctors_monoidal){reference-type="ref" reference="prop:gr_bifunctors_monoidal"}, $\otimes_\mathbf{gr}$ gives a monoidal structure on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. Analogously to the induced $$\ {\widetilde{\otimes}_\mathbf{gr}}: \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \times {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathtt{Mod}_\mathbbm{k}$$ given in Proposition [Proposition 65](#prop:otimes_gr_propoly_analytic){reference-type="ref" reference="prop:otimes_gr_propoly_analytic"}, there is an induced monoidal structure on $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$. **Proposition 180**. *The functor $\ {\widetilde{\otimes}_\mathbf{gr}}$ induces a monoidal structure $(\mathcal{F}_\omega(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}), \ {\widetilde{\otimes}_\mathbf{gr}}, \mathsf{q}^\mathbf{gr}_\bullet P_{(-)})$.* *Proof.* One first checks that $\otimes_\mathbf{gr}: \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \times \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \rightarrow \mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ induces $$\ {\widetilde{\otimes}_\mathbf{gr}}: \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \times \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}) \rightarrow \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}),$$ as in Proposition [Proposition 65](#prop:otimes_gr_propoly_analytic){reference-type="ref" reference="prop:otimes_gr_propoly_analytic"} for $\ {\widetilde{\otimes}_\mathbf{gr}}: \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}) \times {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}\rightarrow \mathtt{Mod}_\mathbbm{k}$. This is associative, by construction. It remains to check that $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}$ is the unit. This corresponds to the fact that $P_{(-)}$ is the unit for $\otimes_\mathbf{gr}$ on $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$, which follows from the behaviour exhibited in Example [Example 23](#exam:otimes_gr){reference-type="ref" reference="exam:otimes_gr"}. ◻ We seek to compare this structure with the monoidal structure on ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ induced by $\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$. Due to the behaviour stressed in Remark [Remark 160](#rem:order_otimes_catlie){reference-type="ref" reference="rem:order_otimes_catlie"}, the *opposite* monoidal structure is used, defined for bimodules $M_1$, $M_2$ by $$M_1 \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^\mathrm{op}M_2 := M_2 \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M_1.$$ **Theorem 181**. *The functor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (-) \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} : {}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ is monoidal for the structures $({}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}, \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^\mathrm{op}, \mathbb{Q})$ and $(\mathcal{F}_\omega(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}), \ {\widetilde{\otimes}_\mathbf{gr}}, \mathsf{q}^\mathbf{gr}_\bullet P_{(-)})$.* *In particular, for bimodules $M_1$ and $M_2$, there is a canonical isomorphism $$\begin{aligned} &&\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (M_1 \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M_2) \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}} \cong \\ && \quad \quad (\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M_2 \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}) \ {\widetilde{\otimes}_\mathbf{gr}} (\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} M_1 \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}).\end{aligned}$$* *Proof.* This follows from Corollary [Corollary 159](#cor:otimes_compatibility){reference-type="ref" reference="cor:otimes_compatibility"}. ◻ # Modelling the tower of categories $\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr}$ {#sect:model_qgr_pbif} As a first application of the relationship between $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules and bifunctors, in this section we exhibit a model for the tower of categories $\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr}$. ## The tower of categories $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ For $d \in \mathbb{N}$, as in Section [11.1](#subsect:catlie_bimodules){reference-type="ref" reference="subsect:catlie_bimodules"}, let $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ be the quotient $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule obtained by applying the truncation functor $(-)^{\leq d}$ with respect to the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure. Explicitly, $$\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}(s, t) = \left\{ \begin{array}{ll} \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t) & s \leq d \\ 0 &\mbox{otherwise.} \end{array} \right.$$ *Remark 182*. Since $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t) =0$ if $s <t$, one has $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}(s,t)=0$ if $t>d$ or if $s>d$; it follows that the bimodule $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ belongs to ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{< \infty}$. We have the following analogue of Proposition [Proposition 41](#prop:pbif_bipolynomiality){reference-type="ref" reference="prop:pbif_bipolynomiality"}, using the functor $(-)_{[\leq d]}$ on left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules introduced in Notation [Notation 127](#nota:left_adjoint_left_cat_opd){reference-type="ref" reference="nota:left_adjoint_left_cat_opd"}: **Proposition 183**. *For $d \in \mathbb{N}$ the morphism $(\mathbf{Cat}\hspace{1pt}\mathfrak{Lie})_{[\leq d]} \rightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ induced by the surjection $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ is an isomorphism.* *Proof.* This can be proved by the same formal argument as used for Proposition [Proposition 41](#prop:pbif_bipolynomiality){reference-type="ref" reference="prop:pbif_bipolynomiality"}. It can also be proved directly, by exhibiting an inverse as follows. By definition, the canonical surjection $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow (\mathbf{Cat}\hspace{1pt}\mathfrak{Lie})_{[\leq d]}$ sends $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s, t)$ to zero if $t >d$. In particular, if $s >d$, it sends $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,s)$ to zero. It follows, by the Yoneda Lemma, that it sends $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,t)$ to zero for all $s>d$ (and any $t$). This shows that this canonical surjection factorizes over $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$, whence the result. ◻ This helps motivate the following counterpart of Theorem [Theorem 45](#thm:tower){reference-type="ref" reference="thm:tower"}. **Proposition 184**. *For $d \in\mathbb{N}$, the composition in $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ induces a $\mathbb{Q}$-linear category structure on $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ such that the canonical surjection $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ gives a $\mathbb{Q}$-linear functor that is the identity on objects and the canonical surjection on morphisms.* *Moreover, these form a tower of $\mathbb{Q}$-linear categories under $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$: $$\xymatrix{ \mathbf{Cat}\hspace{1pt}\mathfrak{Lie} \ar@{.>}[d] \ar[rd] \ar[rrd] \\ \ar@{.>}[r] & \mathbf{Cat}\hspace{1pt}^{\leq d+1} \mathfrak{Lie} \ar[r] & \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie} \ar[r] &\ldots }$$* *Proof.* This can be proved by the reasoning applied for Theorem [Theorem 45](#thm:tower){reference-type="ref" reference="thm:tower"}. It can also be seen directly, as follows. The key is that $\mathbf{Cat}\hspace{1pt}^{\leq d}\mathfrak{Lie}(s,t) =0$ if $t > d$ (this is encoded in Proposition [Proposition 183](#prop:catlie_left_adjoint){reference-type="ref" reference="prop:catlie_left_adjoint"}). Thus non-trivial compositions are understood by restricting to $s\leq d$ and $t \leq d$, where it is given by that of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$. ◻ ## Passage to bifunctors By Remark [Remark 182](#rem:catlie_trunc_finite){reference-type="ref" reference="rem:catlie_trunc_finite"}, for $d \in \mathbb{N}$, the bimodule $\mathbf{Cat}\hspace{1pt}^{\leq d}\mathfrak{Lie}$ belongs to ${}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}^{< \infty}$. It follows that applying the functor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} (-) \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ of Theorem [Theorem 177](#thm:equivalence_bimodules){reference-type="ref" reference="thm:equivalence_bimodules"} yields a bifunctor in $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$. One has the key identification: **Proposition 185**. *For $d \in \mathbb{N}$, the bifunctor $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ is isomorphic to $\mathsf{q}^\mathbf{gr}_d P_{(-)}$.* *Proof.* By Proposition [Proposition 129](#prop:properties_otimes_cat_opd){reference-type="ref" reference="prop:properties_otimes_cat_opd"}, $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie} \cong (\Phi U \mathfrak{Lie})^{\leq d} \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$$ since $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}(s,t)=0$ for $t>d$. Now, Proposition [Proposition 183](#prop:catlie_left_adjoint){reference-type="ref" reference="prop:catlie_left_adjoint"} gives the isomorphism $(\mathbf{Cat}\hspace{1pt}\mathfrak{Lie})_{[\leq d]}\cong \mathbf{Cat}\hspace{1pt}^{\leq d}\mathfrak{Lie}$. Hence, Proposition [Proposition 129](#prop:properties_otimes_cat_opd){reference-type="ref" reference="prop:properties_otimes_cat_opd"} implies that the surjection $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ induces an isomorphism $$(\Phi U \mathfrak{Lie})^{\leq d} \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}\mathfrak{Lie} \stackrel{\cong}{\rightarrow} (\Phi U \mathfrak{Lie})^{\leq d} \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}.$$ Putting these together gives the isomorphisms: $$\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie} \cong (\Phi U \mathfrak{Lie})^{\leq d} \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}\mathfrak{Lie} \cong (\Phi U \mathfrak{Lie})^{\leq d} .$$ Hence $\Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ is isomorphic to $(\Phi U \mathfrak{Lie})^{\leq d} \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathfrak{malcev}.$ By Theorem [Theorem 133](#thm:equiv_propoly_modcatlie){reference-type="ref" reference="thm:equiv_propoly_modcatlie"}, $- \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$ induces an equivalence of categories $\mathtt{Mod}_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \stackrel{\cong}{\rightarrow} {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}$. Under this equivalence, $\Phi U \mathfrak{Lie}$ corresponds to $\Phi U \mathfrak{malcev}$, arguing as in Theorem [Theorem 151](#thm:equiv_Umalcev_PhiUlie){reference-type="ref" reference="thm:equiv_Umalcev_PhiUlie"}, and the latter is isomorphic to $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}$, by the Theorem. Proposition [Proposition 136](#prop:compare_filtrations){reference-type="ref" reference="prop:compare_filtrations"} shows that applying the truncation $(-)^{\leq d}$ corresponds to forming the quotient $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}\twoheadrightarrow \mathsf{q}^\mathbf{gr}_\bullet ( \mathsf{q}^\mathbf{gr}_d P_{(-)})$. The result follows. ◻ Composition in $\mathbf{Cat}\hspace{1pt}^{\leq d } \mathfrak{Lie}$ gives the morphism of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules: $$\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}\rightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}.$$ This is unital and associative in the obvious sense. (More precisely, by Proposition [Proposition 129](#prop:properties_otimes_cat_opd){reference-type="ref" reference="prop:properties_otimes_cat_opd"}, it is an isomorphism, corresponding to the fact that $\mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ is the unit for the restriction of $\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}}$ to the full subcategory of bimodules that vanish whenever one of the arities is greater than $d$.) **Theorem 186**. *For $d\in \mathbb{N}$, under the equivalence of categories of Theorem [Theorem 177](#thm:equivalence_bimodules){reference-type="ref" reference="thm:equivalence_bimodules"}, the category $\mathbf{Cat}\hspace{1pt}^{\leq d}\mathfrak{Lie}$ corresponds to $\big( \mathsf{q}^\mathbf{gr}_d \mathbb{Q}\mathbf{gr}\big)^\mathrm{op}$.* *Under this identification, the $\mathbb{Q}$-linear functor $\mathbf{Cat}\hspace{1pt}^{\leq d+1} \mathfrak{Lie}\rightarrow \mathbf{Cat}\hspace{1pt}^{\leq d} \mathfrak{Lie}$ of Proposition [Proposition 184](#prop:tower_cat_lie){reference-type="ref" reference="prop:tower_cat_lie"} corresponds to the functor $\big( \mathsf{q}^\mathbf{gr}_{d+1} \mathbb{Q}\mathbf{gr}\big)^\mathrm{op} \rightarrow \big( \mathsf{q}^\mathbf{gr}_d \mathbb{Q}\mathbf{gr}\big)^\mathrm{op}$ of Theorem [Theorem 45](#thm:tower){reference-type="ref" reference="thm:tower"}.* *Proof.* This follows from Proposition [Proposition 185](#prop:catlie_trunc_d){reference-type="ref" reference="prop:catlie_trunc_d"} combined with Theorem [Theorem 181](#thm:monoidal_otimes_catlie_gr){reference-type="ref" reference="thm:monoidal_otimes_catlie_gr"}, paying attention to the variance. ◻ # The Casimir PROP associated to $\mathfrak{Lie}$ viewed as a cyclic operad {#sect:modular} The purpose of this section is to extend the analysis of the tower of categories $\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr}$ in terms of the tower $\mathbf{Cat}\hspace{1pt}^{\leq \bullet } \mathfrak{Lie}$ to the case of the modular (or Casimir) PROP associated to the Lie operad. This is inspired by the work of Habiro and Massuyeau [@MR4321214], who introduced a category $\mathbf{A}$ (constructed from Jacobi diagrams in handlebodies) in their study of the Kontsevich integral for bottom tangles in handlebodies. The category $\mathbf{A}$ can be defined over $\mathbb{Q}$; it is $\mathbb{N}$-graded and $\mathbf{A}_0$ is equivalent to $\mathbb{Q}\mathbf{gr}^\mathrm{op}$. The positive part of $\mathbf{A}$ arises due to the *Casimir* structure. Part of the structure of $\mathbf{A}$ has been analysed by Katada [@2021arXiv210206382K]; she considers the structure of $\mathbf{A}(\mathbf{0}, -)$, restricting to the action of automorphism groups of finite rank free groups. Vespa [@2022arXiv220210907V] has analysed part of the structure of $\mathbf{A}$ by using *beaded* Jacobi diagrams. ## From $\mathfrak{Lie}$ to the modular operad $\mathfrak{MLie}$ and the PROP $\mathbb{P}$ The material of this subsection is inspired (and covered) by the work of Hinich and Vaintrob [@MR1913297]; the reader is referred to their paper for further details. The starting point is the fact that the operad $\mathfrak{Lie}$ underlies a cyclic operad, $\mathfrak{CLie}$. In particular, the cyclic operad $\mathfrak{CLie}$ has underlying $\mathbb{Q}{\bm{\Sigma}}^\mathrm{op}$-module such that $\mathfrak{CLie}(\mathbf{0}) = \mathfrak{CLie}(\mathbf{1})=0$ and $\mathfrak{CLie}(\mathbf{n+1}) \downarrow ^{\mathfrak{S}_{n+1}}_{\mathfrak{S}_n}\cong \mathfrak{Lie}(\mathbf{n})$ as a $\mathfrak{S}_n$-module. One can go further and consider the associated *modular operad* $\mathfrak{MLie}$ and then the PROP $\mathbb{P}$ constructed from $\mathfrak{MLie}$; these are constructed from *(open) Jacobi diagrams*. Recall that a Jacobi diagram is a vertex-oriented, uni-trivalent graph (we will always suppose that the graph is finite); it is *open* if each connected component has at least one univalent vertex (see [@MR2962302 Chapter 5] for details on this and other standard material, such as the AS and IHX relations). The vertex orientation is a cyclic order of the half edges at a trivalent vertex; equivalently, this can be specified by giving a planar embedding, so that one has the following two possibilities at a trivalent vertex: indexing the half edges by $\{1, 2, 3 \}$ and taking the convention that the trivalent vertex inherits the standard orientation from the plane. 1. For a (non-empty) finite set $X$, $\mathfrak{MLie}(X)$ is the $\mathbb{Q}$-vector space generated by connected Jacobi diagrams equipped with a bijection between the univalent vertices and $X$, modulo the following relations: 1. a generator corresponding to a closed (i.e., not open) Jacobi diagram is equivalent to zero; 2. the AS (antisymmetry) and IHX (Jacobi) relations. The automorphisms $\mathrm{Aut} (X)$ act by relabelling. 2. The underlying cyclic operad has composition operation defined as follows: for finite sets $X$, $Y$ and elements $x \in X$, $y \in Y$, the composition $\circ_{x,y} : \mathfrak{MLie}(X) \otimes \mathfrak{MLie}(Y)\rightarrow \mathfrak{MLie}((X \amalg Y) \backslash \{x, y\})$ is induced by the glueing operation on uni-trivalent graphs that glues together the univalent vertices labelled by $x$ and $y$ respectively. 3. The contraction operation for the modular structure is defined similarly: for a subset $\{x_1, x_2 \}\subset X$, the contraction $c_{x_1, x_2} : \mathfrak{MLie}(X) \rightarrow \mathfrak{MLie}(X \backslash \{ x_1, x_2 \})$ is given by glueing together the univalent vertices labelled by $x_1$ and $x_2$ respectively. *Remark 187*.   1. The composition and contraction operations on uni-trivalent graphs may give rise to *closed* diagrams; these are then identified to zero. 2. There is an inclusion of cyclic operads $\mathfrak{CLie}\hookrightarrow \mathfrak{MLie}$ (forgetting the modular structure of $\mathfrak{MLie}$); the image corresponds to the sub cyclic operad represented by trivalent planar trees (i.e. connected, vertex-oriented uni-trivalent graphs that are simply-connected). The *degree* of a uni-trivalent graph is defined as $\frac{1}{2} (\sharp \mathrm{vertices})$, where $\sharp \mathrm{vertices}$ is the total number of vertices; the degree is a positive integer. This induces an $\mathbb{N}_+$-grading of the underlying ${\bm{\Sigma}}^\mathrm{op}$-module of $\mathfrak{MLie}$ (however, composition is not additive with respect to this grading). The PROP $\mathbb{P}$ is constructed from $\mathfrak{MLie}$ as follows, requiring that each term from $\mathfrak{MLie}$ contributes at least one exit: 1. For finite sets $X$, $Y$: $$\mathbb{P}(X,Y) = \bigoplus_{\substack{X = \amalg_{i \in \mathcal{I}}X_i \\ Y = \amalg_{i \in \mathcal{I}} Y_i \\ Y_i \neq \emptyset}} \bigotimes _{i \in \mathcal{I}} \mathfrak{MLie}(X_i \amalg Y_i),$$ where the sum is over all possible pairs of decompositions (indexed by the same set $\mathcal{I}$) $(X = \amalg_{i \in \mathcal{I}}X_i , Y = \amalg_{i \in \mathcal{I}} Y)$, subject to the condition that each $Y_i$ is non-empty. In particular, $\mathbb{P}(X, \emptyset) =0$ if $X\neq \emptyset$, whereas $\mathbb{P}(\emptyset, \emptyset) =\mathbb{Q}$, by convention. 2. Composition is induced by the modular operad structure of $\mathfrak{MLie}$, giving $$\mathbb{P}(Y, Z) \otimes_{\mathrm{Aut} (Y)} \mathbb{P}(X, Y) \rightarrow \mathbb{P}(X, Z).$$ *Remark 188*.   1. One can take $X= \emptyset$ in $\mathbb{P}(X,Y)$; there is a non-trivial composition from the left $$\mathbb{P}(Y, Z) \otimes_{\mathrm{Aut} (Y)} \mathbb{P}(\emptyset, Y) \rightarrow \mathbb{P}(\emptyset, Z).$$ 2. There is an inclusion of PROPs $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\hookrightarrow \mathbb{P}$ induced by the inclusion of cyclic operads $\mathfrak{CLie}\hookrightarrow \mathfrak{MLie}$: namely, $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(X,Y)$ identifies as the following subspace of $\mathbb{P}(X,Y)$: $$\bigoplus_{\substack{X = \amalg_{i \in \mathcal{I}}X_i \\ Y = \amalg_{i \in \mathcal{I}} Y_i \\ |Y_i|=1 }} \bigotimes _{i \in \mathcal{I}} \mathfrak{CLie}(X_i \amalg Y_i) \subset \bigoplus_{\substack{X = \amalg_{i \in \mathcal{I}}X_i \\ Y = \amalg_{i \in \mathcal{I}} Y_i \\ Y_i \neq \emptyset}} \bigotimes _{i \in \mathcal{I}} \mathfrak{MLie}(X_i \amalg Y_i),$$ (note the condition $|Y_i|=1$ on the left hand side). *Remark 189*. Hinich and Vaintrob [@MR1913297] place the above construction in a general framework. Namely they exhibit a functor $(-)^C$ from cyclic operads to PROPs that sends the cyclic operad $\mathcal{O}$ to the PROP $\mathcal{O}^C$ for Casimir $\mathcal{O}$-algebras. Applied to the cyclic operad $\mathfrak{CLie}$, this yields $\mathbb{P}$. Recall (see [@MR4321214 Section 7] or [@MR1913297]) that a Casimir Lie algebra is a Lie algebra $\mathfrak{g}$ equipped with a Casimir element, i.e., $c \in \Gamma^2 (\mathfrak{g})\subset \mathfrak{g}^{\otimes 2}$ that is $\mathrm{ad}$-invariant (explicitly, writing $c = c_i \otimes c'_i$ with implicit summation, then $c_i \otimes c'_i= c'_i \otimes c_i$ and, for any $x \in \mathfrak{g}$, $[x, c_i] \otimes c'_i + c_i \otimes [x, c'_i]=0$). Casimir Lie algebras form a category with morphisms those morphisms of Lie algebras that are compatible with the Casimir elements. There is an obvious forgetful functor from Casimir Lie algebras to Lie algebras. Hinich and Vaintrob observed (see [@MR1913297 Lemma 3.1.7]) that the category of Casimir Lie algebras is precisely the category of algebras over the PROP $\mathbb{P}$; the Casimir element is encoded by the action of the element of $\mathbb{P}(\mathbf{0} , \mathbf{2})$ that is given by the canonical generator of $\mathfrak{CLie}(\mathbf{2})$. *Remark 190*. Habiro and Massuyeau observed in [@MR4321214 Section 7] that the universal enveloping algebra $U \mathfrak{g}$ of a Casimir Lie algebra is a (cocommutative) Casimir Hopf algebra, as defined in [@MR4321214 Section 5]. This is at the heart of the relationship between $\mathbb{P}$ and the category $\mathbf{A}$ introduced in [@MR4321214 Section 4] and described in terms of Casimir Hopf algebras in [@MR4321214 Theorem 5.11]. This is also related to the results of Hinich and Vaintrob, notably [@MR1913297 Theorem 7.1.1]. ## First properties of $\mathbb{P}$ Using the explicit construction of $\mathbb{P}$, one deduces the following, which gives rise to a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule structure. **Proposition 191**. *The PROP $\mathbb{P}$ is $\mathbb{N}$-graded, i.e., $\mathbb{P}(-,-)= \bigoplus_{n \in \mathbb{N}} \mathbb{P}_n (-,-)$ and this grading is compatible with the PROP structure. This grading is determined by placing an element of $\mathfrak{MLie}(X \amalg Y)$ (with $Y \neq \emptyset$), represented by a connected, vertex-oriented uni-trivalent graph of degree $d$ in grading $d - |X|$.* *Moreover, $\mathbb{P}_0$ identifies with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ as a PROP. Hence, for each $n \in \mathbb{N}$, $\mathbb{P}_n$ has the structure of a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule.* *Proof.* Using the PROP structure, it is clear that specifying the grading on terms of $\mathfrak{MLie}(X \amalg Y)$ determines the grading, by using the 'horizontal' composition of the PROP given by the symmetric monoidal structure of the PROP. One first checks that the given grading yields an $\mathbb{N}$-grading (i.e., there are no terms in negative grading). The identification of the degree zero part is proved by induction on the number of trivalent vertices. A direct verification shows that the grading is compatible with the 'vertical' composition; indeed, the correction of the degree by subtracting the number of incoming univalent vertices is designed to ensure the appropriate additivity. The above analysis also yields the identification of $\mathbb{P}_0$ with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ as a PROP. The composition of $\mathbb{P}$ therefore yields a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule structure. ◻ *Remark 192*. If the condition $Y_i \neq \emptyset$ were not imposed, one would not obtain an $\mathbb{N}$-grading above. For instance, considering a generator of $\mathfrak{CLie}(\mathbf{3})$ (represented by the connected uni-trivalent graph with one trivalent vertex), as having three entries and no exits, the degree is $2$ so the grading would be $-1$. The grading gives the following useful finiteness property: **Lemma 193**. *For $n, s,t \in \mathbb{N}$:* 1. *$\mathbb{P}_n (\mathbf{s}, \mathbf{t})$ has finite dimension;* 2. *$\mathbb{P}_n (\mathbf{s}, \mathbf{t})=0$ if $t =0$ or $t > 2n+s$.* *In particular $\sum_{t \in \mathbb{N}} \dim_\mathbb{Q}\mathbb{P}_n (\mathbf{s}, \mathbf{t}) <\infty$.* *Proof.* For fixed degree $d$, there are only finitely-many uni-trivalent graphs of degree $d$. The finite-dimensionality of $\mathbb{P}_n (\mathbf{s}, \mathbf{t})$ follows. For the second statement, elements of $\mathbb{P}_n (\mathbf{s}, \mathbf{t})$ are represented by linear combinations of uni-trivalent graphs with $2(n+s)$ vertices, hence with at most $2n+s$ exit vertices. Putting these results together gives the total finite-dimensionality. ◻ **Example 194**. By Proposition, [Proposition 191](#prop:nat_grading_ojac){reference-type="ref" reference="prop:nat_grading_ojac"}, for any $n \in \mathbb{N}$, $\mathbb{P}_n (\mathbf{0}, -)$ has the structure of a left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module. Moreover, the grading $n$ coincides with the degree. One has $\mathbb{P}_n (\mathbf{0}, \mathbf{t})=0$ for $t > 2n$ and $\mathbb{P}_n (\mathbf{0}, \mathbf{t})$ is always finite-dimensional. The associated polynomial functor $J_n := \Phi U \mathfrak{Lie}\otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbb{P}_n (\mathbf{0}, -)$ provided by the equivalence of Theorem [Theorem 147](#thm:analytic_grop){reference-type="ref" reference="thm:analytic_grop"} lies in $\mathcal{F}_{2n} (\mathbf{gr}^\mathrm{op})$ and has a finite composition series. This underlies the structures studied by Katada in [@2021arXiv210206382K]. (Katada does not work with functors on $\mathbf{gr}^\mathrm{op}$, but restricts to the underlying representations of automorphism groups of free groups.) The symmetric monoidal structure of the PROP provides the 'horizontal' compositions: $$\mathbb{P}_m (\mathbf{s}, \mathbf{t}) \otimes \mathbb{P}_n (\mathbf{u}, \mathbf{v}) \rightarrow \mathbb{P}_{m+n} (\mathbf{s}\amalg \mathbf{u}, \mathbf{t}\amalg \mathbf{v}).$$ These assemble to form a map of $\mathbb{Q}{\bm{\Sigma}}^\mathrm{op}\otimes \mathbb{Q}{\bm{\Sigma}}$-modules, using the convolution product for bimodules: $$\mathbb{P}_m \circledcirc \mathbb{P}_n \rightarrow \mathbb{P}_{m+n}.$$ This is associative and unital in the evident sense (noting that $\mathbb{P}(\emptyset, \emptyset) = \mathbb{P}_0 (\emptyset, \emptyset) =\mathbb{Q}$). For each $t \in \mathbb{N}$, we can consider $\mathbb{P}_t$ as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule. Hence, using the convolution product structure of Proposition [Proposition 166](#prop:convolution_bimodules){reference-type="ref" reference="prop:convolution_bimodules"}, one can consider $\mathbb{P}_m \circledcirc \mathbb{P}_n$ as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule. **Proposition 195**. *For $m,n \in \mathbb{N}$, the horizontal composition $\mathbb{P}_m \circledcirc \mathbb{P}_n \rightarrow \mathbb{P}_{m+n}$ is a morphism of right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules. If both $m$ and $n$ are positive, it is not a morphism of left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-modules.* *Proof.* One can restrict to considering partial compositions, and use the identification $\mathfrak{Lie}(s) = \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(s,1)$, which underlines the fact that there is a single 'exit'. Hence, precomposing an element in the image of the horizontal composition with this corresponds either to precomposition in $\mathbb{P}_m$ or precomposition in $\mathbb{P}_n$. Unwinding the definitions, this implies that the horizontal composition respects the right $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure. This argument breaks down for the left $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-module structure: if $m>0$ and $n>0$, a post-composition with $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(2,1)$ can always be found that mixes the contributions from $\mathbb{P}_m$ and $\mathbb{P}_n$. ◻ ## Generating $\mathbb{P}$ as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule Unfortunately, a full understanding of the $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule structure of $\mathbb{P}$ (or even the underlying $\mathbb{Q}{\bm{\Sigma}}$-bimodule) is out of reach; this stems from the fact that the $\mathfrak{S}_n$-representations $\mathfrak{MLie}(\mathbf{n})$, for $n \in \mathbb{N}$, are not known for large $n$. Nevertheless, there is an explicit $\mathbb{Q}{\bm{\Sigma}}$-bimodule that generates $\mathbb{P}$ as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule. This is based on the following, which is immediate from the definitions: **Lemma 196**. * * 1. *$\mathbb{P}_0 (\mathbf{1}, \mathbf{1}) \cong \mathbb{Q}$ (with trivial action of $\mathfrak{S}_1^\mathrm{op}\times \mathfrak{S}_1$), with generator $\iota$ corresponding to the identity of $\mathfrak{Lie}(\mathbf{1})= \mathfrak{CLie}(\mathbf{2})$;* 2. *$\mathbb{P}_1 (\mathbf{0}, \mathbf{2}) \cong \mathbb{Q}$, with trivial action of $\mathfrak{S}_0^\mathrm{op}\times \mathfrak{S}_2$, with generator $c$ corresponding to the Casimir element.* *Moreover, $c$ and $\iota$ generate the subspace of $\mathbb{P}$ generated by connected open Jacobi diagrams with no trivalent vertex.* *Proof.* There is a unique connected open Jacobi diagram with no trivalent vertex. After distinguishing the exit vertices (denoted by $\circ$) and the entrance vertices (denoted $\bullet$), since there must be at least one $\circ$ vertex, there are the following two possibilities 1. representing the generator $\iota$; 2. representing the generator $c$. (We use the convention that entries are at the top and exits the bottom.) ◻ The symmetric monoidal structure of $\mathbb{P}$ allows one to generate a sub $\mathbb{Q}{\bm{\Sigma}}$-bimodule of $\mathbb{P}$ from the elements $c$ and $\iota$. This can be expressed by using the convolution product $\circledcirc$ for $\mathbb{Q}{\bm{\Sigma}}$-bimodules; here a more concrete description is given: **Definition 197**. Let $\mathsf{Chord}\subset \mathbb{P}$ be the sub ${\bm{\Sigma}}$-bimodule generated by open Jacobi diagrams with no trivalent vertices and let $\mathsf{Chord}_n \subset \mathbb{P}_n$ be the subspace of grading $n$. By convention, $\mathsf{Chord}(\mathbf{0}, \mathbf{0}) = \mathbb{P}(\mathbf{0}, \mathbf{0}) = \mathbb{Q}$, in grading $0$. *Remark 198*. As suggested by the notation, $\mathsf{Chord}$ is related to *open* chord diagrams appearing in studying finite type invariants (see [@MR2962302 Chapter 4], for example). Via the symmetric monoidal structure of $\mathbb{P}$, $\mathsf{Chord}$ is generated by $c$ and $\iota$ of Lemma [Lemma 196](#lem:no_trivalent){reference-type="ref" reference="lem:no_trivalent"}. In particular one has: **Lemma 199**. *For $n,s \in \mathbb{N}$,* 1. *$\mathsf{Chord}_n (\mathbf{s}, \mathbf{t})=0$ unless $t = 2n +s$;* 2. *the vector space $\mathsf{Chord}_n (\mathbf{s}, \mathbf{2n+s})$ has basis given by pairs given by an injection $\alpha : \mathbf{s} \hookrightarrow \mathbf{2n+s}$ together with a decomposition of $\mathbf{2n+s} \backslash \alpha (\mathbf{s})$ into $n$ subsets of cardinal two.* 3. *The group $\mathfrak{S}_s ^\mathrm{op}\times \mathfrak{S}_{2n+s}$ acts in the obvious way on this basis.* *In particular, $\mathsf{Chord}_n (\mathbf{s}, \mathbf{2n+s})$ is a permutation representation of $\mathfrak{S}_s ^\mathrm{op}\times \mathfrak{S}_{2n+s}$ .* *Proof.* This follows by identifying the uni-trivalent graphs with no trivalent vertex and applying the condition that each connected component must have at least one exit vertex (see the following example). The associated permutation representation can be written down explicitly. ◻ The ${\bm{\Sigma}}$-bimodule $\mathsf{Chord}$ gives generators for $\mathbb{P}$, as follows. (This result is implicit in the work of Habiro and Massuyeau [@MR4321214].) **Proposition 200**. *For $n \in \mathbb{N}$, $\mathbb{P}_n$ is generated as a $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodule by $\mathsf{Chord}_n$, i.e., there is a surjection of $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules: $$\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\otimes_{\bm{\Sigma}}\mathsf{Chord}_n \otimes_{\bm{\Sigma}}\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbb{P}_n.$$* *Proof.* This is proved by a straightforward induction on the number of vertices. ◻ **Example 201**. The surjection of Proposition [Proposition 200](#prop:chord_generates){reference-type="ref" reference="prop:chord_generates"} gives, in particular, the surjection $$\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}(2n, -) \otimes_{\mathfrak{S}_{2n}} \mathsf{Chord}_n (\mathbf{0}, \mathbf{2n}) \twoheadrightarrow \mathbb{P}_n (\mathbf{0}, -).$$ This has been used by Katada (with a very different formulation) in [@2021arXiv210206382K], based on the results of [@MR4321214]. *Remark 202*. The surjection of Proposition [Proposition 200](#prop:chord_generates){reference-type="ref" reference="prop:chord_generates"} is far from being an isomorphism. 1. For $n=0$, $\mathsf{Chord}_0$ identifies as the ${\bm{\Sigma}}$-bimodule with $\mathsf{Chord}_0 (\mathbf{s}, \mathbf{s}) \cong \mathbb{Q}\mathfrak{S}_s$. The surjection is $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\otimes_{\bm{\Sigma}}\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}\twoheadrightarrow \mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$ induced by composition. 2. Moreover, for $n>0$, the surjection does not taken into account: 1. the 4T-relation for chord diagrams (see [@MR2962302], for example); 2. analogous commutation relations between $c$ and $\iota$ (using the notation of Lemma [Lemma 196](#lem:no_trivalent){reference-type="ref" reference="lem:no_trivalent"}). ## Passage to bifunctors The general theory of Section [11](#sect:bifunctors){reference-type="ref" reference="sect:bifunctors"} applies, generalizing the considerations of Section [12](#sect:model_qgr_pbif){reference-type="ref" reference="sect:model_qgr_pbif"}, where $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}= \mathbb{P}_0$ was treated. *Notation 203*. For $n \in \mathbb{N}$, let $\mathfrak{A}(n)$ denote the image of $\mathbb{P}_n$ in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ under the equivalence of Theorem [Theorem 178](#thm:bifunctors_sym_monoidal){reference-type="ref" reference="thm:bifunctors_sym_monoidal"}, $\mathfrak{A}(n):= \Phi U \mathfrak{Lie}\otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbb{P}_n \otimes _{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \underline{\mathfrak{malcev}}$. Proposition [Proposition 200](#prop:chord_generates){reference-type="ref" reference="prop:chord_generates"} immediately gives the following, which provides some intuition when considering $\mathfrak{A}(*)$: **Proposition 204**. *For $n \in \mathbb{N}$, there is a surjection in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$: $$\Phi U \mathfrak{Lie}\otimes _{{\bm{\Sigma}}} \mathsf{Chord}_n \otimes _{\bm{\Sigma}}\underline{\mathfrak{malcev}} \twoheadrightarrow \mathfrak{A}(n).$$* By construction, $\mathfrak{A}(n)$ is given by the tower of bifunctors: $$\begin{aligned} \label{eqn:tower_ajac} \ldots \twoheadrightarrow \mathfrak{A}(n)_d \twoheadrightarrow \mathfrak{A}(n)_{d-1} \twoheadrightarrow \ldots \mathfrak{A}(n)_1 \twoheadrightarrow \mathfrak{A}(n)_0,\end{aligned}$$ where $\mathfrak{A}(n)_d$ is polynomial of degree $d$ with respect to $\mathbf{gr}$. **Example 205**. The tower $\mathfrak{A}(0)_\bullet$ coincides with the tower $\mathsf{q}^\mathbf{gr}_\bullet P_{(-)}$. Lemma [Lemma 193](#lem:finiteness_pjac){reference-type="ref" reference="lem:finiteness_pjac"} implies the following: **Proposition 206**. *For $d, n \in \mathbb{N}$, $\mathfrak{A}(n)_d$ lies in $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$ and has a finite composition series. More precisely, $\mathfrak{A}(n)_d$ has polynomial degree $d$ with respect to $\mathbf{gr}$ and polynomial degree $\leq 2n+ d$ with respect to $\mathbf{gr}^\mathrm{op}$.* *Hence, ([\[eqn:tower_ajac\]](#eqn:tower_ajac){reference-type="ref" reference="eqn:tower_ajac"}) is a tower of surjections between finite objects in $\mathcal{F}_{< \infty} (\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$.* The (vertical) composition in the PROP $\mathbb{P}_*$ induces composition operations between the bifunctors $\mathfrak{A}(*)$. This requires using the symmetric monoidal structure $\ {\widetilde{\otimes}_\mathbf{gr}}$ on $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ given in Proposition [Proposition 180](#prop:otimes_gr_analytic_coanalytic){reference-type="ref" reference="prop:otimes_gr_analytic_coanalytic"}. **Lemma 207**. *For $m,n \in \mathbb{N}$, the composition operation in $\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}$-bimodules $$\mathbb{P}_m \otimes_{\mathbf{Cat}\hspace{1pt}\mathfrak{Lie}} \mathbb{P}_n \rightarrow \mathbb{P}_{m+n}$$ induces the natural transformation $\mathfrak{A}(n) \ {\widetilde{\otimes}_\mathbf{gr}}\mathfrak{A}(m) \rightarrow \mathfrak{A}(m+n)$ in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op};{\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$.* *Via the natural projections in the respective towers, in polynomial degree $d$ with respect to $\mathbf{gr}$, this is determined by the morphism of polynomial bifunctors: $$\mathfrak{A}(n)_d \otimes_\mathbf{gr}\mathfrak{A}(m)_{2n+d} \rightarrow \mathfrak{A}(m+n)_d.$$* *Proof.* The first statement follows from Theorem [Theorem 181](#thm:monoidal_otimes_catlie_gr){reference-type="ref" reference="thm:monoidal_otimes_catlie_gr"}. The second statement follows from Proposition [Proposition 39](#prop:otimes_gr_poly){reference-type="ref" reference="prop:otimes_gr_poly"} by using the fact that $\mathfrak{A}(n)_d$ is polynomial of degree $2n+d$ with respect to $\mathbf{gr}^\mathrm{op}$, so that there is a natural isomorphism $\mathfrak{A}(n)_d \otimes_\mathbf{gr}\mathfrak{A}(m) \cong \mathfrak{A}(n)_d \otimes_\mathbf{gr}\mathfrak{A}(m)_{2n+d}$. ◻ *Remark 208*. Contrary to the behaviour exhibited in Theorem [Theorem 186](#thm:tower_isomorphism){reference-type="ref" reference="thm:tower_isomorphism"}, composition on $\mathfrak{A}(*)$ does not correspond to a tower of monoid structures on $\mathfrak{A}(*)_d$, for $d \in \mathbb{N}$. This justifies the additional work in Section [11](#sect:bifunctors){reference-type="ref" reference="sect:bifunctors"} in developing the framework based upon $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$. Putting these facts together, one has: **Theorem 209**. *The (pro)bifunctors $\mathfrak{A}(*)$ have the structure of a unital, $\mathbb{N}$-graded associative monoid in the monoidal category $(\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}), \ {\widetilde{\otimes}_\mathbf{gr}}, \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$, with $\mathfrak{A}(0)$ isomorphic to $(\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr})^\mathrm{op}$ considered as a monoid.* ## Relating to $\mathbf{A}$ {#subsect:sketchy} This section sketches how $\mathfrak{A}(*)$ is related to Habiro and Massuyeau's category $\mathbf{A}$ [@MR4321214]. By definition, $\mathrm{Ob}\hspace{2pt}\mathbf{A}= \mathbb{N}$ and $\mathbf{A}$ is $\mathbb{N}$-graded. It is equipped with an embedding $\mathbb{Q}\mathbf{gr}^\mathrm{op}\hookrightarrow \mathbf{A}$ that sends the free group $\mathbb{Z}^{\star r}$ to the object $r \in \mathbb{N}$ and induces an equivalence $\mathbb{Q}\mathbf{gr}^\mathrm{op}\cong \mathbf{A}_0$. In particular, for each grading $n \in \mathbb{N}$, $\mathbf{A}_n(-,-)$ is a bifunctor in $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr}) \cong \mathcal{F}(\mathbf{gr}^\mathrm{op}; \mathcal{F}(\mathbf{gr}))$. Post-composing with $\mathsf{q}^\mathbf{gr}_\bullet$ gives $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_n (-, -) \in \mathcal{F}(\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$ More precisely, one has the analogue of Proposition [Proposition 206](#prop:properties_ajac){reference-type="ref" reference="prop:properties_ajac"}, namely that $\mathsf{q}^\mathbf{gr}_d \mathbf{A}_n$ has polynomial degree at most $2n +d$ with respect to $\mathbf{gr}^\mathrm{op}$. It follows in particular that $$\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_n (-, -) \in \mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}).$$ Moreover, the composition of $\mathbf{A}$ induces an associative composition operation in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$: $$\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A} \ {\widetilde{\otimes}_\mathbf{gr}}\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A} \rightarrow \mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}$$ making $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}$ into an associative monoid with unit $\mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_0 \cong (\mathsf{q}^\mathbf{gr}_\bullet \mathbb{Q}\mathbf{gr})^\mathrm{op}$. **Theorem 210**. *There is an isomorphism of $\mathbb{N}$-graded associative monoids in $(\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})}), \ {\widetilde{\otimes}_\mathbf{gr}}, \mathsf{q}^\mathbf{gr}_\bullet P_\mathbb{Z})$: $$\mathfrak{A}(*) \cong \mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}.$$* *Proof.* (Indications.) The proof that there is a natural isomorphism $\mathfrak{A}(n) \cong \mathsf{q}^\mathbf{gr}_\bullet \mathbf{A}_n$ in $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$ uses the methods employed in the proof of Theorem [Theorem 186](#thm:tower_isomorphism){reference-type="ref" reference="thm:tower_isomorphism"}. (A partial result in this sense is given in [@2022arXiv220210907V Theorem 6.7].) A careful analysis of the definition of $\mathbf{A}$ in [@MR4321214 Section 4] shows that these natural isomorphisms respect the respective monoid structures. (Heuristically, this follows from the presentation of the category $\mathbf{A}$ that is given in [@MR4321214 Theorem 5.11] and is related to [@MR1913297 Theorem 7.1.1].) ◻ *Remark 211*. Using the constructions of this paper, the monoid $\mathfrak{A}(*)$ arises directly from the Casimir PROP $\mathbb{P}$ constructed from the Lie operad. The significance of Theorem [Theorem 210](#thm:ajac_versus_A){reference-type="ref" reference="thm:ajac_versus_A"} is that it shows that the objects $\mathfrak{A}(n)$ arise from honest bifunctors of $\mathcal{F}(\mathbf{gr}^\mathrm{op}\times \mathbf{gr})$, rather than just $\mathcal{F}_\omega (\mathbf{gr}^\mathrm{op}; {\mathcal{F}_{\mathrm{pro}<\infty}(\mathbf{gr})})$. Pow21 S. Chmutov, S. Duzhin, and J. Mostovoy, *Introduction to Vassiliev knot invariants*, Cambridge University Press, Cambridge, 2012. MR 2962302 Aurélien Djament, Teimuraz Pirashvili, and Christine Vespa, *Cohomologie des foncteurs polynomiaux sur les groupes libres*, Doc. Math. **21** (2016), 205--222. MR 3505136 Samuel Eilenberg and Saunders Mac Lane, *On the groups $H(\Pi,n)$. II. Methods of computation*, Ann. of Math. (2) **60** (1954), 49--139. MR 65162 Benoit Fresse, *Lie theory of formal groups over an operad*, J. Algebra **202** (1998), no. 2, 455--511. 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--- author: - title: Discrepancy estimates related to the fractional parts of $b^n/n$. --- # Introduction {#sec1} Let $b\in\mathbb{N}, b\ge2$. In 2013, Cilleruelo et al. [@CKLRS] proved that $$\label{Seq} \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}$$ is dense in $[0,1]$. (See also [@D2] for a number of interesting related results.) For $A\subset\mathbb{N}$, we set $$\nonumber \mathcal{S}_b(A)=\left\{\frac{b^n\pmod{n}}{n}:n\in A\right\}.$$ Note in particular that the set ([\[Seq\]](#Seq){reference-type="ref" reference="Seq"}) is simply $\mathcal{S}_b(\mathbb{N})$. Let $\mathbb{P}=\{2,3,\ldots\}$ denote the prime numbers and set $$\nonumber \mathcal{A}=\left\{pq:p,q\in\mathbb{P}, p>b^q\right\}.$$ The main result of [@CKLRS] is an estimate of the discrepancy of $\mathcal{S}_b(\mathcal{A})$. Denote $\mathcal{A}_N=\mathcal{A}\cap[1,N]$, then $$\label{discrEstimate1} D(\mathcal{S}_b(\mathcal{A}_N))=\mathcal{O}\left(\frac{\log\log\log\log(N)}{\log\log\log(N)}\right).$$ where $D(\mathcal{S}_b(\mathcal{A}_N))$ denotes the *discrepancy* of $\mathcal{S}_b(\mathcal{A}_N)$ (see Section [2](#sec2){reference-type="ref" reference="sec2"} below). In particular, it follows from ([\[discrEstimate1\]](#discrEstimate1){reference-type="ref" reference="discrEstimate1"}) that $\mathcal{S}_b(\mathcal{A})$ is uniformly distributed modulo 1 and this implies the density of $\mathcal{S}_b(\mathcal{A})$ in $[0,1]$. Unaware of the work [@CKLRS], the author studied properties of $\mathcal{S}_b(\mathcal{A})$ from a different point of view in [@M2]. When informed of the paper [@CKLRS], we found that some observations from [@M2] could be used to improve on ([\[discrEstimate1\]](#discrEstimate1){reference-type="ref" reference="discrEstimate1"}). The main result of this note is the following. **Theorem 1**. *There holds $$\label{discrEstimate2} D(\mathcal{S}_b(\mathcal{A}_N))=\mathcal{O}\left(\frac{1}{\log\log\log(N)}\right).$$* The improvement ([\[discrEstimate2\]](#discrEstimate2){reference-type="ref" reference="discrEstimate2"}) is not due to any sharper number-theoretic inequalities. In fact, we use the same estimates as in [@CKLRS]. Rather, we employ a different strategy to estimate the discrepancy. Instead of using the Erdős-Turán inequality and exponential sums as in [@CKLRS], we use a sort of triangle inequality (Lemma [Lemma 3](#triangleIneq){reference-type="ref" reference="triangleIneq"}) to decompose $\mathcal{S}_b(\mathcal{A}_N)$ into well-structured subsequences. By combining a number of basic facts about discrepancy with some observations from [@M2] (Proposition [Proposition 5](#countProp){reference-type="ref" reference="countProp"} in particular) and a variant of the Siegel-Walfisz theorem, we obtain in Lemma [\[discrepancyLemma\]](#discrepancyLemma){reference-type="ref" reference="discrepancyLemma"} an estimate of the discrepancy of each subsequence and these estimates allow us to establish Theorem [Theorem 1](#discrepancyThm){reference-type="ref" reference="discrepancyThm"}. In connection with this, we mention our previous work [@M1] where a similar strategy based on Lemma [Lemma 3](#triangleIneq){reference-type="ref" reference="triangleIneq"} was used to find optimal discrepancy decay rates. # Auxiliary results {#sec2} ## Discrepancy For a finite set $A$, we denote by $|A|$ the cardinality of $A$. Let $S=\{x_1,x_2,\ldots,x_M\}\subset[0,1]$ be a finite sequence. The *extreme discrepancy* of $S$ is defined by $$\nonumber D(S)=\sup_{J\subseteq[0,1]}\left|\frac{A_S(J)}{M}-\lambda(J)\right|,$$ where $A_S(J)=|\{n:x_n\in J\}|$ and $\lambda$ is the linear Lebesgue measure. Similarly, the *star discrepancy* of $S$ is defined by $$\nonumber D^*(S)=\sup_{r>0}\left|\frac{A_S([0,r])}{M}-r\right|.$$ It is well-known (see e.g. [@DP], Chapter 3) that $$\nonumber D^*(S)\le D(S)\le 2D^*(S),$$ hence it is sufficient to only consider $D^*(S)$. **Lemma 2**. *Let $R\in\mathbb{N}$ and $S=\{x_1,x_2,\ldots,x_M\}$ be a finite sequence such that the elements of $S$ only attain values in the set $\{k/R: k=0,1,\ldots, R-1\}$. Assume that $$\nonumber |\{n:x_n=k/R\}|=\alpha_kM+\epsilon_k$$ where $\alpha_k\ge0~~(k=0,1,\ldots,R-1)$, $\alpha_1=\alpha_2=\ldots=\alpha_{R-1}$ and $$\nonumber \sum_{k=0}^{R-1}\alpha_k=1.$$ Then there exists an absolute constant $C>0$ such that $$\nonumber MD^*(S)\le\max\left\{\alpha_0M,\frac{M}{R}\right\}+\sum_{k=0}^{R-1}|\epsilon_k|.$$* *Proof.* Take any $J_r=[0,r]$ and let $j=\lfloor rR\rfloor$, so that $M\mu(J_r)=jM/R+M\delta$ for some $\delta\in[0,1/R]$. We have $$\nonumber A_S(J_r)=M\sum_{k=0}^j\alpha_k+\sum_{k=0}^j\epsilon_k$$ so $$\nonumber \left|A_S(J_r)-M\mu(J_r)\right|\le\left|M\alpha_0+M\sum_{k=1}^j\left(\alpha_k-\frac{1}{R}\right)-M\delta\right|+\sum_{k=1}^j|\epsilon_k|$$ The first term of the expression at the right-hand side above is either increasing or decreasing in $\delta$, hence we have $$\nonumber MD^*(S)\le\max_{j=0,\ldots, R-1}\left|M\alpha_0+M\sum_{k=1}^j\left(\alpha_k-\frac{1}{R}\right)\right|+\sum_{j=0}^{R-1}|\epsilon_k|$$ where $j=0$ means that the sum is 0. The maximum of the first term is attained either at $k=0$ or $k=R-1$, since the terms of the sum have the same sign (due to the fact that $\alpha_1=\ldots=\alpha_{R-1}$. Further, $\alpha_0+(R-1)\alpha_1=1$ so $$\begin{aligned} \nonumber \max_{j=0,\ldots, R-1}\left|M\alpha_0+M\sum_{k=1}^j\left(\alpha_k-\frac{1}{R}\right)\right|&=&\max\left\{M\alpha_0,\left|M\alpha_0+M\sum_{k=1}^{R-1}\left(\alpha_k-\frac{1}{R}\right)\right|\right\}\\ \nonumber &=&\max\left\{M\alpha_0,\left|M-\frac{M(R-1)}{R}\right|\right\}\\ \nonumber &=&\max\left\{M\alpha_0,\frac{M}{R}\right\} \end{aligned}$$ Consequently, $$MD^*(S)\le\max\left\{M\alpha_0,\frac{M}{R}\right\}+\sum_{k=0}^{R-1}|\epsilon_k|.$$ ◻ **Lemma 3** ([@DP], Chapter 3). *Assume that $S=\cup_j S_j$ where $S_i\cap S_j=\emptyset$. Denote $M_j=|S_j|$ and $M=|S|=\sum_jM_j$. Then $$\nonumber MD^*(S)\le\sum_{j=1}^KM_jD^*(S_j).$$* **Lemma 4** ([@NW], Chapter 4). *Let $S'=\{x_1,x_2,\ldots,x_M\}$ and $S''=\{y_1,y_2,\ldots,y_M\}$ such that $$\nonumber |x_j-y_j|\le\epsilon$$ for $j=1,2,\ldots,M$. Then $$\nonumber |D^*(S')-D^*(S'')|<\epsilon.$$* ## Primes in arithmetic progressions Denote $$\nonumber Z_k=\{r\in\mathbb{Z}^*_{q(q-1)}: b^r\equiv kr+b\pmod{q}\}.$$ For any $p\in\mathbb{P}$ with $p>b^q$ and $p\equiv r\pmod{q(q-1)}$ for some $r\in Z_k$, there holds $$\label{pqEstimate} \left|\frac{b^{pq}\pmod{pq}}{pq}-\frac{k}{q}\right|<\frac{1}{q},$$ see [@M2]. Let ${\rm ord}_q(b)=|\langle b\rangle|$ where $\langle b\rangle$ is the subgroup of $\mathbb{Z}^*_q$ generated by $b$. In [@M2], we proved the following proposition. **Proposition 5**. *For $q\in\mathbb{P}$ there holds $$\nonumber |Z_k|=\varphi(q-1)-m_b(q)\quad(k=1,2,\ldots,q-1),$$ and $$\nonumber |Z_0|=(q-1)m_b(q)$$ where $$\nonumber m_b(q)=|\{r\in\mathbb{Z}^*_{q-1}: r\equiv 1\pmod{{\rm ord}_q(b)}\}|.$$* We shall need to estimate the number of primes in certain arithmetic progressions. Denote $$\pi(N;q(q-1),r)=|\{p\in\mathbb{P}: p\le N, p\equiv r\pmod{q(q-1)}\}.$$ We use the following consequence of the Siegel-Walfisz theorem (see [@CKLRS] and the reference given there): $$\label{siegel} \pi(N;q(q-1),r)=\frac{\pi(N)}{\varphi(q(q-1))}+\mathcal{O}\left(\frac{N}{(\log(N))^A}\right)$$ for any $A>0$ and $N\ge2$. (The implied constant in ([\[siegel\]](#siegel){reference-type="ref" reference="siegel"}) depends on $A$.) Here, $\pi(N)$ is the prime counting function. In particular, there is asymptotically the same amount of primes in the progression $r+nq(q-1)$ for each $r\in\mathbb{Z}^*_{q(q-1)}$. More precisely, ([\[siegel\]](#siegel){reference-type="ref" reference="siegel"}) and Proposition [Proposition 5](#countProp){reference-type="ref" reference="countProp"} imply that $$\begin{aligned} \nonumber |\{p\in\mathbb{P}:p\le N, \exists r\in Z_k\text{ such that } p\equiv r\pmod{q(q-1)}\}|= \\ \nonumber =\frac{|Z_k|}{\varphi(q(q-1))}\pi(N)+\mathcal{O}\left(\frac{N|Z_k|}{(\log(N))^A}\right)\end{aligned}$$ # Proof of Theorem [Theorem 1](#discrepancyThm){reference-type="ref" reference="discrepancyThm"} {#sec3} Denote $$\nonumber F_{q,N}=\{p\in\mathbb{P}:b^q<p\le N/q\}\quad\text{and}\quad \mathcal{F}_{q,N}=\{pq:p\in F_{q,N}\},$$ then $$\nonumber \mathcal{A}_N=\bigcup_{q\in\mathbb{P}}\mathcal{F}_{q,N}.$$ (Note that $F_{q,N}=\emptyset$ for $q$ sufficiently large.) Define $$\nonumber F_k=\{p\in F_{q,N}: \exists r\in Z_k\text{ such that } p\equiv r\pmod{q(q-1)}\}.$$ for $k=0,1,\ldots,q-1$. **Lemma 6**. *For each $k\in\{0,1,\ldots,q-1\}$, there holds $$\nonumber |F_k|=\frac{|Z_k|}{\varphi(q(q-1))}|F_{q,N}|+\epsilon_k$$ where $$\nonumber \sum_{k=0}^{q-1}|\epsilon_k|\le \frac{CN}{q^2\log(N)}.$$* *Proof.* Observe that $$|F_k|=|Z_k|\left(\pi(N;q(q-1),r)-\pi(b^q,q(q-1),r)\right).$$ Taking $A=4$ in ([\[siegel\]](#siegel){reference-type="ref" reference="siegel"}), we obtain $$\begin{aligned} \nonumber |F_k|&=&\frac{|Z_k|}{\varphi(q(q-1))}\left(\pi(N/q)-\pi(b^q)\right)+\epsilon_k\\ \nonumber &=&\frac{|Z_k|}{\varphi(q(q-1))}|F_{q,N}|+\epsilon_k,\end{aligned}$$ where $$\begin{aligned} \nonumber |\epsilon_k|&\le& C|Z_k|\left(\frac{N}{q(\log(N/q))^4}+\frac{b^q}{q^4}\right)\le 2C|Z_k|\frac{N}{q(\log(N/q))^4}\end{aligned}$$ since $x\mapsto x/(\log(x))^4$ is increasing. Note that $$\nonumber \sum_{k=0}^{q-1}|\epsilon_k|\le\frac{2CN}{q(\log(N/q))^4}\sum_{k=0}^{q-1}|Z_k|=\frac{2CN}{q(\log(N/q))^4}\varphi(q(q-1))\le\frac{2CNq^2}{q(\log(N/q))^4}$$ Since $b^q<N/q$, we have $q\le\log(N/q)$. Furthermore, $q^2< qb^q< N$, so $N/q>\sqrt{N}$. Consequently, $$\frac{1}{\log(N/q)}\le\frac{1}{q}\quad\text{and}\quad \frac{\log(N)}{2}\le\log(N/q)\le\log(N)$$ and therefore $$\nonumber \sum_{k=0}^{q-1}|\epsilon_k|\le\frac{2CNq^2}{q(\log(N/q))^4}\le\frac{4CN}{q^2\log(N)}.$$ ◻ Denote by $$\nonumber n(q,N)=|F_{q,N}|$$ **Lemma 7**. *There exists an absolute constant $C$ such that for any $q\in\mathbb{P}$, $N>b^q$, there holds [\[discrepancyLemma\]]{#discrepancyLemma label="discrepancyLemma"} $$\label{mainDiscrEstimate} n(q,N)D^*(\mathcal{S}_b(\mathcal{F}_{q,N}))\le C\left(\frac{\log\log(q)}{\log(q)}n(q,N)+\frac{N}{q^2\log(N)}\right).$$* *Proof.* By ([\[pqEstimate\]](#pqEstimate){reference-type="ref" reference="pqEstimate"}), for any $p>b^q$ there is a $k\in\{0,1,\ldots,q-1\}$ such that $$\label{diff} \left|\frac{b^{pq}\pmod{pq}}{pq}-\frac{k}{q}\right|<\frac{1}{q}$$ holds. Furthermore, for a specific $k$ the estimate ([\[diff\]](#diff){reference-type="ref" reference="diff"}) holds if and only if $p\equiv r\pmod{q(q-1)}$ where $r\in Z_k$. For $p\in F_{q,N}$ we define $a_p=k/q$ if $p\in F_k$. Set $S'=\{a_p:p\in F_{q,N}\}$. Then $|S'|=|F_{q,N}|$ and $$\nonumber \left|\frac{b^{pq}\pmod{pq}}{pq}-a_p\right|<\frac{1}{q}$$ for each $p\in F_{q,N}$. By Lemma [Lemma 4](#continuity){reference-type="ref" reference="continuity"}, there holds $$\label{mainLemmaEq1} D^*(S')-\frac{1}{q}<D^*(\mathcal{S}_b(\mathcal{F}_{q,N})<D^*(S')+\frac{1}{q}$$ We shall now use Lemma [Lemma 2](#discreteLemma){reference-type="ref" reference="discreteLemma"} compute $D^*(S')$. Set $\alpha_k=|Z_k|/\varphi(q(q-1))$, so $\sum\alpha_k=1$ and this, together with Lemma [Lemma 6](#numberLemma){reference-type="ref" reference="numberLemma"}, implies that we may apply Lemma [Lemma 2](#discreteLemma){reference-type="ref" reference="discreteLemma"} to conclude $$\nonumber n(q,N)D^*(S')\le\max\left\{\frac{|Z_0|n(q,N)}{\varphi(q(q-1))},\frac{n(q,N)}{q}\right\}+\frac{CN}{q^2\log(N)}$$ Further, we have $$\nonumber \frac{|Z_0|}{\varphi(q(q-1))}=\frac{(q-1)m_b(q)}{(q-1)\varphi(q-1)}=\frac{|\mathcal{N}|}{\varphi(q-1)}.$$ Clearly, $$|\mathcal{N}|\le\frac{q-1}{{\rm ord}_q(b)}$$ and it is well-known that $$\varphi(q-1)\ge\frac{C(q-1)}{\log\log(q-1)}$$ Taking into consideration ${\rm ord}_q(b)\ge C\log(q)$, we get $$\begin{aligned} \nonumber n(q,N)D^*(S')&\le&\max\left\{\frac{Cn(q,N)\log\log(q)}{\log(q)},\frac{n(q,N)}{q}\right\}+\frac{CN}{q^2\log(N)}\\ \label{mainLemmaEq2} &=&\frac{Cn(q,N)\log\log(q)}{\log(q)}+\frac{CN}{q^2\log(N)}\end{aligned}$$ By ([\[mainLemmaEq1\]](#mainLemmaEq1){reference-type="ref" reference="mainLemmaEq1"}) and ([\[mainLemmaEq2\]](#mainLemmaEq2){reference-type="ref" reference="mainLemmaEq2"}), we get $$\nonumber n(q,N)D^*(\mathcal{S}_b(\mathcal{F}_{q,N})\le \frac{Cn(q,N)\log\log(q)}{\log(q)}+\frac{CN}{q^2\log(N)},$$ concluding the proof of ([\[mainDiscrEstimate\]](#mainDiscrEstimate){reference-type="ref" reference="mainDiscrEstimate"}). ◻ *Proof of Theorem [Theorem 1](#discrepancyThm){reference-type="ref" reference="discrepancyThm"}.* Fix $N>N_0$ and set $M=|\mathcal{A}_N|$, it was shown in [@CKLRS] that $$\label{Msize} M\sim\frac{N\log\log\log(N)}{\log(N)}.$$ (We write $A\sim B$ if $c_1A\le B\le c_2A$ for absolute constants $c_1,c_2$.) Using Lemma [Lemma 3](#triangleIneq){reference-type="ref" reference="triangleIneq"} and Lemma [\[discrepancyLemma\]](#discrepancyLemma){reference-type="ref" reference="discrepancyLemma"}, we obtain $$\begin{aligned} \nonumber MD^*(\mathcal{S}_b(\mathcal{A}_N))&\le&\sum_{q\in\mathbb{P}}n(q,N)D^*(\mathcal{S}_b(\mathcal{F}_{q,N}))\\ \label{discrEstProof1} &\le&C\sum_{q\in\mathbb{P}}\left(\frac{\log\log(q)}{\log(q)}n(q,N)+\frac{N}{q^2\log(N)}\right), \end{aligned}$$ where $n(q,N)=0$ if $F_{q,N}=\emptyset$. By the prime number theorem $$\label{discrEstProof2} n(q,N)\le\pi(N/q)=\frac{N}{q\log(N/q)}+\mathcal{O}\left(\frac{N}{q(\log(N/q))^2}\right)$$ Using ([\[discrEstProof1\]](#discrEstProof1){reference-type="ref" reference="discrEstProof1"}), ([\[discrEstProof2\]](#discrEstProof2){reference-type="ref" reference="discrEstProof2"}) and the fact that $\log(N)/2\le \log(N/q)\le\log(N)$ for every $q\in\mathbb{P}$ with $n(q,N)>0$, we get $$\begin{aligned} \label{discrEstProof3} MD^*(\mathcal{S}_b(\mathcal{A}_N))&\le& \frac{CN}{\log(N)}\sum_{q\in\mathbb{P}}\left(\frac{\log\log(q)}{q\log(q)}\left(1+\mathcal{O}\left(\frac{1}{\log(N)}\right)\right)+\frac{1}{q^2}\right). \end{aligned}$$ Since the series $\sum_{q\in\mathbb{P}}\log\log(q)/(q\log(q))$ and $\sum_{q\in\mathbb{P}} 1/q^2$ both are convergent, it follows from ([\[discrEstProof3\]](#discrEstProof3){reference-type="ref" reference="discrEstProof3"}) that $$\label{discrEstProof4} MD^*(\mathcal{S}_b(\mathcal{A}_N))\le\frac{CN}{\log(N)}\sum_{q\in\mathbb{P}}\left(\frac{\log\log(q)}{q\log(q)}+\frac{1}{q^2}\right)\le\frac{CN}{\log(N)}.$$ From ([\[discrEstProof4\]](#discrEstProof4){reference-type="ref" reference="discrEstProof4"}) and ([\[Msize\]](#Msize){reference-type="ref" reference="Msize"}), we have $$D^*(\mathcal{S}_b(\mathcal{A}_N))=\mathcal{O}\left(\frac{1}{\log\log\log(N)}\right).$$ ◻ **Acknowledgements** The author is grateful to Professor A. Dubickas (Vilnius) for pointing out the references [@CKLRS; @D2]. 0 J. Cilleruelo, A. Kumchev, F. Luca, J. Rué and I. E. Shparlinski, \"On the fractional parts of $a^n/n$\", Bull. London Math. Soc. **45** (2013), 249-256 J. Dick and F. Pillichshammer, *Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte-Carlo Integration*, Cambridge University Press, 2010 A. Dubickas, \"Density of some special sequences modulo 1\", Mathematics **11** (2023), 1727 M. Lind, \"A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes\", Int. J. Number Theory **18** (2022), 1409-1416 M. Lind, \"Some remarks related to the density of $\{b^n\pmod{n}/n:n\in\mathbb{N}\}$\", preprint 2023, arXiv:2308.14354 \[NT\] H. Niederreiter and A. Winterhof, *Applied Number Theory* Springer, 2015.

arxiv_math

--- abstract: | In this paper we define a continuous version of multiple zeta functions with double variables. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the first shuffle product and the second shuffle product. We proved that the dimension of the $\mathbb{Q}-$linear spaces generated by continuous multiple zeta values with given weight are finite. By using a theorem of C.Glanois, we proved that continuous multiple zeta values include all cyclotomic multiple zeta values of level 2. We will give a detail analysis about the two different shuffle products. Furthermore, we will discuss the extension of the two different products, we proved a theorem about comparing the two different shuffle product, this is an analogy of Ihara-Kaneko-Zagier's comparison theorem in the case of continuous multiple zeta values. As an application, we will give a new method to proof some Ramanujan's identities. Finally, we will provide some conjectures. address: | Jia Li\ School of Mathematical Sciences, Peking University, Beijing, China author: - Jia Li title: A continuous version of multiple zeta values with double variables --- UTF8gbsn # **Introduction** As a generalization of the famous Riemann zeta function, we have the following definition of classical multiple zeta function. For $r\in\mathbb{Z}_{>0}$, the classical multiple zeta function is defined by $$\zeta(s_1, s_2,\cdots s_r)=\sum_{0<n_1<n_2<\cdots<n_r}\frac{1}{n_1^{s_1}n_2^{s_2}\cdots n_r^{s_r}}.$$ Where $(s_1,s_2.\cdots, s_r)\in\mathbb{C}^r$. If $(s_1,s_2.\cdots, s_r)=(k_1,k_2,\cdots,k_r)\in(\mathbb{Z}_{>0})^r, k_r>1$, the values $$\zeta(k_1,k_2,\cdots,k_r)$$ are called multiple zeta values and write MZVs for short. Let $\mathcal{Z}$ denote the $\mathbb{Q}-$subvector space of $\mathbb{R}$ which spanned by all multiple zeta values. We know that multiple zeta values satisfy the stuffle product and the shuffle product [@bj], thus $\mathcal{Z}$ has an algebra structure. As a result, there are many relations among multiple zeta values. In recent decades, a host of mathematicians, including Brown, Cartier, Deligne, Drinfeld, Ecalle, Goncharov, Hain, Hoffman, Kontsevich, Terasoma, Zagier, and many others have made a huge progress in this regard. But fundamental questions still open, such as the irrationality of $\zeta(2n+1)$. As an analogy of classical situations, Jiangtao Li [@li] recently defined a continuous version of multiple zeta function: $$\zeta^{\mathcal{C}}(s_1,\cdots,s_r):=\idotsint\limits_{[1,+\infty)^r}\frac{dx_1dx_2\cdots dx_r}{x_1^{s_1}(x_1+x_2)^{s_2}\cdots(x_1+\cdots+x_r)^{s_r}}.$$ Where $(s_1,\cdots,s_r)\in\mathbb{C}^r$. In particular, he considered that the variables are positive integers, that is, $(s_1,s_2.\cdots, s_r)=(k_1,k_2,\cdots,k_r)\in(\mathbb{Z}_{>0})^r, k_r>1$, the values which denoted by $$\zeta^{\mathcal{C}}(k_1,\cdots,k_r)$$ are called continuous multiple zeta values . Let $\mathcal{Z}^{\mathcal{C}}$ denote the $\mathbb{Q}-$subvector space of $\mathbb{R}$ which spanned by all continuous multiple zeta values. He proved that there is a shuffle product structure on $\mathcal{Z}^{\mathcal{C}}$ and a series of important results. We know that there are two different product structures on $\mathcal{Z}$, and then given a rich algebraic relationship. However, there is only one product structure on $\mathcal{Z}^{\mathcal{C}}$, in order to better analogy with $\zeta(s_1,\cdots,s_r)$, we consider a further generalization of $\zeta^{\mathcal{C}}(s_1,\cdots,s_r)$. If we consider the following variable substitution $$\begin{cases} y_1=x_1-1\\ y_2=x_1+x_2-2\\ \quad\vdots\\ y_r=x_1+x_2+\cdots+x_r-r \end{cases}$$ Then, we have $$\begin{aligned} \zeta^{\mathcal{C}}(s_1,\cdots,s_r)&=\idotsint\limits_{[1,+\infty)^r}\frac{dx_1dx_2\cdots dx_r}{x_1^{s_1}(x_1+x_2)^{s_2}\cdots(x_1+\cdots+x_r)^{s_r}}\\ &=\idotsint\limits_{0<y_1<y_2<\cdots<y_r}\frac{dy_1dy_2\cdots dy_r}{(y_1+1)^{s_1}(y_2+2)^{s_2}\cdots(y_r+r)^{s_r}} \end{aligned}$$ Inspired by the above results, we define a further generalized continuous multiple zeta function $$\zeta^{\mathcal{C}}(s_1,s_2,\cdots,s_r;t_1,t_2,\cdots,t_r):=\idotsint\limits_{0<x_1<x_2<\cdots<x_r}\frac{dx_1dx_2\cdots dx_r}{(x_1+t_1)^{s_1}(x_2+t_2)^{s_2}\cdots(x_r+t_r)^{s_r}}.$$ Where $s_1,\cdots,s_r\in\mathbb{C}$ and $t_1,\cdots,t_r\in\mathbb{R}$. We call it a **continuous multiple zeta function with double variables**. For convenience, we omit "double variables\". We have: **Theorem 1**. *For any given $t_1,\cdots,t_r\in\mathbb{R}_{>0}$, the continuous multiple zeta function $\zeta^{\mathcal{C}}(s_1,\cdots, s_r;t_1,\cdots,t_r)$ is convergent for $$\mathrm{Re}(s_1+s_2+\cdots+s_r)>r,\; \mathrm{Re}(s_2+\cdots+s_r)>r-1,\;\cdots, \mathrm{Re}(s_r)>1.$$ Moreover, it can be analytically continued to a meromorphic function on $\mathbb{C}^r$ with possible poles at some special hyperplanes.* In this article, we mainly consider that the variables are positive integers, that is, $$\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)$$ where $(k_1,\cdots,k_r;m_1,\cdots,m_r)\in(\mathbb{Z}_{>0})^r\times(\mathbb{Z}_{>0})^r, k_r>1$. For the simplicity of notations, we introduce the concept of multi-index. **Definition 1**. *A multi-index $$(\boldsymbol{k};\boldsymbol{m}):=(k_1,\cdots,k_r;m_1,\cdots,m_r)\in\mathbb{Z}^r\times\mathbb{Z}^r$$ is called positive if $k_1,\cdots,k_r>0;m_1,\cdots,m_r>0$, and called admissible if it is positive and, in addition, satisfies $k_r>1$. By convention, the empty multi-index ($r=0$) will also be considered to be admissible.* By the above notation, if $(\boldsymbol{k};\boldsymbol{m})$ is admissible, the value $\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})$ is called **continuous multiple zeta value with double variables**. Sometimes, we omit "double variables\", and we write CMZVs(don't confuse with cyclotomic multiple zeta values) for short. let $\mathcal{Z}^{\text{Dou},\mathcal{C}}$ denote the $\mathbb{Q}-$subvector space of $\mathbb{R}$ which spanned by all multiple zeta values, that is, $$\mathcal{Z}^{\text{Dou},\mathcal{C}}=\langle\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})|(\boldsymbol{k};\boldsymbol{m})\ \text{is admissible}\rangle_{\mathbb{Q}}$$ Let $(\boldsymbol{k};\boldsymbol{m}):=(k_1,\cdots,k_r;m_1,\cdots,m_r)\in\mathbb{Z}^r\times\mathbb{Z}^r$ be a multi-index, the number $r$ is called the **length** of $(\boldsymbol{k};\boldsymbol{m})$, denoted by $$\ell(\boldsymbol{k};\boldsymbol{m})=r$$ furthermore, we write $$\begin{aligned} \textbf{wt}(\boldsymbol{k})&:=k_1+\cdots+k_r\\ \textbf{wt}(\boldsymbol{m})&:=m_1+\cdots+m_r \end{aligned}$$ where $\textbf{wt}(\boldsymbol{k})(\text{resp.}\textbf{wt}(\boldsymbol{m}))$ called the **weight** of $\boldsymbol{k}(\text{resp.}\boldsymbol{m})$. Now we consider the following $\mathbb{Q}-$linear spaces $$\begin{aligned} \mathcal{Z}^{\text{Dou},\mathcal{C}}_m&:=\langle\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\big|\textbf{wt}(\boldsymbol{m})=m\rangle_{\mathbb{Q}}\\ F_{r}\mathcal{Z}^{\text{Dou},\mathcal{C}}&:=\langle\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\big|\ell(\boldsymbol{k};\boldsymbol{m})= r\rangle_{\mathbb{Q}}\\ F_{r}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m&:=\langle\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\big|\textbf{wt}(\boldsymbol{m})=m,\ell(\boldsymbol{k};\boldsymbol{m})=r\rangle_{\mathbb{Q}} \end{aligned}$$ clearly, we have $\mathcal{Z}^{\text{Dou},\mathcal{C}}_0=\{0\},\mathcal{Z}^{\text{Dou},\mathcal{C}}_1=\mathbb{Q}$. Now we have the following result: **Theorem 1**. *The subspace $F_{r}\mathcal{Z}^{\text{Dou},\mathcal{C}}$ define an increasing filtration of $\mathcal{Z}^{\text{Dou},\mathcal{C}}$, that is, we have $$\{0\}\subset \mathbb{Q}=F_{0}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset F_{1}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset F_{2}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset\cdots$$* Furthermore, we also have: **Theorem 1**. *The subspace $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$ define an increasing filtration of $\mathcal{Z}^{\text{Dou},\mathcal{C}}$, that is, we have $$0=\mathcal{Z}^{\text{Dou},\mathcal{C}}_0\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_1\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_3\subset\cdots$$ and $$\text{dim}_{\mathbb{Q}}F_r\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant(2^r-r)\binom{m-1}{r-1}$$ $$\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant 2\cdot3^{m-1}-2^{m-2}(m+1)$$.* **Remark 1**. *This is a very roughly estimation, far from the sharper bound. For example $$\mathcal{Z}^{\text{Dou},\mathcal{C}}_0=\{0\}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_1=\mathbb{Q}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_2=\mathbb{Q}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_3=\mathbb{Q}+\mathbb{Q}\ln(2)$$ $$\begin{aligned} \mathcal{Z}^{\text{Dou},\mathcal{C}}_4&=\mathbb{Q}+\mathbb{Q}\pi^2+\mathbb{Q}\ln(2)+\mathbb{Q}\ln(3)\\ \mathcal{Z}^{\text{Dou},\mathcal{C}}_5&=\mathbb{Q}+\mathbb{Q}\pi^2+\mathbb{Q}\ln(2)+\mathbb{Q}\ln(3)+\mathbb{Q}\zeta(3)+\mathbb{Q}\text{Li}_2\left(-\frac{1}{2}\right) \end{aligned}$$ where $\text{Li}_2(x):=\sum\limits_{n=1}^{\infty}\frac{x^n}{n^2}$.* Let $N$ be a positive integer and $\mu_N:=\left\{x\in\mathbb{C}|x^N=1\right\}$. Recall that the cyclotomic multiple zeta values of level $N$ are defined by $$\zeta\left(\begin{matrix} k_1,\cdots,k_r\\ \varepsilon_1,\cdots,\varepsilon_r \end{matrix}\right):=\sum_{0<n_1<\cdots<n_r}\frac{\varepsilon_1^{n_1}\cdots\varepsilon_r^{n_r}}{n_1^{k_1}\cdots n_r^{k_r}}\qquad\varepsilon_1,\cdots,\varepsilon_r\in\mu_N,(k_r,\varepsilon_r)\neq(1,1)$$ Denote by $\mathcal{Z}^N$ the $\mathbb{Q}-$vector space spanned by these cyclotomic multiple zeta values. Obviously, we have $\mathcal{Z}=\mathcal{Z}^1\subset\mathcal{Z}^2$. Using C.Glanois's theorem which proved in 2016, we have **Theorem 1**. *The cyclotomic multiple zeta values of level $2$ are continuous multiple zeta values with double variables, that is $$\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$$ especially, for all riemann zeta valus $\zeta(n)$, we have $\zeta(n)\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$.* **Remark 1**. *Notice that $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$ is only an including of $\mathbb{Q}-$vector space, but not an embedding of $\mathbb{Q}-$algebra.* For the classical multiple zeta values, we have an integral representation, known as the Kontsevich's integral representation [@bj], as follows: $$\zeta(\boldsymbol{k})=\zeta(k_1,\cdots,k_r)=\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}}\omega_{\boldsymbol{k}}$$ and using the integral representation, we obtain another product on $\mathcal{Z}$ called shuffle product. For the continuous multiple zeta values with double variables, We have the similar results: **Theorem 1**. *(Integral representation) Let $(\boldsymbol{k};\boldsymbol{m})$ be an admissible multi-index, then we have the following integral representation $$\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$* For details of the above theorem, refer to Section 4.\ Using the integral representation, we also obtain another product on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$. And we have **Theorem 1**. *Their are two different product on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$, and satisfy $$F_{r_1}\mathcal{Z}^{\text{Dou},\mathcal{C}}\cdot F_{r_2}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset F_{r_1+r_2}\mathcal{Z}^{\text{Dou},\mathcal{C}}$$* We know that the multiplication rules on $\mathcal{Z}$ obtained by stuffle product and shuffle product are not the same, the equality of the products are very important that they give will be our main tool for obtaining linear dependences among MZVs. Ihara, Kaneko and Zagier [@ikz] proved the comparison theorem among the two different products. For the continuous multiple zeta values with double variables, we have the similar comparison theorem **Theorem 1**. *(**The first extension homomorphism**) For any given $m\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{Q}-$linear map $$\zeta_{\shuffle_1,m}^{\mathcal{C},T}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ satisfy $$\begin{cases} \zeta_{\shuffle_1,m}^{\mathcal{C},T}(w)=\zeta_{\shuffle_1}^{\mathcal{C}}(w)\qquad&\forall w\in\widetilde{\mathbb{Q}\langle Y\rangle^0}\\ \zeta_{\shuffle_1,m}^{\mathcal{C},T}(y_{1,m})=T\\ \zeta_{\shuffle_1,m}^{\mathcal{C},T}(v\shuffle_1w)=\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v)\cdot\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w)&\forall v,w\in\mathbb{Q}\langle Y\rangle \end{cases}$$* For details of the above theorem, refer to Section 5.\ **Theorem 1**. *(**The second extension homomorphism**) For any given $m\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{Q}-$linear map $$\zeta_{\shuffle_2,m}^{\mathcal{C},T}:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ satisfy $$\begin{cases} \zeta_{\shuffle_2,m}^{\mathcal{C},T}(w)=\zeta_{\shuffle_2}^{\mathcal{C}}(w)\qquad&\forall w\in\widetilde{\mathbb{Q}\langle X\rangle^0}\\ \zeta_{\shuffle_2,m}^{\mathcal{C},T}(y_{m})=T\\ \zeta_{\shuffle_2,m}^{\mathcal{C},T}(v\shuffle_2w)=\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v)\cdot\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w)&\forall v,w\in\mathbb{Q}\langle X\rangle^1 \end{cases}$$* For details of the above theorem, refer to Section 5.\ Now by the Theorem 1.10 and theorem 1.11, we have the following comparison theorem. **Theorem 1**. *(The comparison theorem) We have the following results:* *1. For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\begin{aligned} \rho_{m,m'}^{\shuffle_1}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$, called **the first comparison map** satisfy $$\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_1}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\text{Id}}VV@VV{\rho_{m,m'}^{\shuffle_1}}V\\ (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$\ 2. For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\begin{aligned} \rho^{\shuffle_2}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$, called **the second comparison map** satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_2}(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle X\rangle^1$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\text{Id}}VV@VV{\rho_{m,m'}^{\shuffle_2}}V\\ (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$\ 3. For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\rho_{m,m'}:\mathbb{R}[T]\longrightarrow\mathbb{R}[T]$$ called **the mixed comparison map** satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\varphi}VV@VV{\rho_{m,m'}}V\\ (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$* For details of the above theorem, refer to Section 5.\ From the theorem 1.12, we also have the following result\ **Theorem 1**. *We have the following commutative diagram $$\begin{CD} (\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]@>\rho^{\shuffle_1}_{m_1,m_1'}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\rho_{m_1,m_2}}VV@VV{\rho_{m_1',m_2'}}V\\ (\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]@>\rho^{\shuffle_2}_{m_2,m_2'} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$* *that is, we have a network which composed of comparative mappings. We denoted this network by the following diagram: $$\begin{tikzpicture} \draw (-6,3) -- (6,3); \draw (-6,-3) -- (6,-3); \draw (-6,1) -- (6,1); \draw (-6,-1) -- (6,-1); \draw (4,-3) -- (4,3); \draw (2,-3) -- (2,3); \draw (0,-3) -- (0,3); \draw (-2,-3) -- (-2,3); \draw (-4,-3) -- (-4,3); \draw (4,-1) -- (2,-3); \draw (4,1) -- (0,-3); \draw (4,3) -- (-2,-3); \draw (2,3) -- (-4,-3); \draw (0,3) -- (-4,-1); \draw (-2,3) -- (-4,1); \draw (2,3) -- (4,1); \draw (0,3) -- (4,-1); \draw (-2,3) -- (4,-3); \draw (-4,3) -- (2,-3); \draw (-4,1) -- (0,-3); \draw (-4,-1) -- (-2,-3); \draw (-4,3) -- (0,1); \draw (-4,3) -- (2,1); \draw (-4,3) -- (4,1); \draw (-2,3) -- (2,1); \draw (-2,3) -- (4,1); \draw (0,3) -- (-4,1); \draw (0,3) -- (4,1); \draw (2,3) -- (-4,1); \draw (2,3) -- (-2,1); \draw (4,3) -- (-4,1); \draw (4,3) -- (-2,1); \draw (4,3) -- (0,1); \draw (4,-1) -- (4,1); \draw (4,-1) -- (2,1); \draw (4,-1) -- (0,1); \draw (4,-1) -- (-2,1); \draw (4,-1) -- (-4,1); \draw (2,-1) -- (4,1); \draw (2,-1) -- (2,1); \draw (2,-1) -- (0,1); \draw (2,-1) -- (-2,1); \draw (2,-1) -- (-4,1); \draw (0,-1) -- (4,1); \draw (0,-1) -- (2,1); \draw (0,-1) -- (0,1); \draw (0,-1) -- (-2,1); \draw (0,-1) -- (-4,1); \draw (-2,-1) -- (4,1); \draw (-2,-1) -- (2,1); \draw (-2,-1) -- (0,1); \draw (-2,-1) -- (-2,1); \draw (-2,-1) -- (-4,1); \draw (-4,-1) -- (4,1); \draw (-4,-1) -- (2,1); \draw (-4,-1) -- (0,1); \draw (-4,-1) -- (-2,1); \draw (-4,-1) -- (-4,1); \draw (-4,-3) -- (0,-1); \draw (-4,-3) -- (2,-1); \draw (-4,-3) -- (4,-1); \draw (-2,-3) -- (2,-1); \draw (-2,-3) -- (4,-1); \draw (0,-3) -- (-4,-1); \draw (0,-3) -- (4,-1); \draw (2,-3) -- (-4,-1); \draw (2,-3) -- (-2,-1); \draw (4,-3) -- (-4,-1); \draw (4,-3) -- (-2,-1); \draw (4,-3) -- (0,-1); \node [above] at (-4,3) {$1$}; \node [above] at (-2,3) {$2$}; \node [above] at (-0,3) {$3$}; \node [above] at (2,3) {$4$}; \node [above] at (4,3) {$5$}; \node [below] at (-4,-3) {$1$}; \node [below] at (-2,-3) {$2$}; \node [below] at (-0,-3) {$3$}; \node [below] at (2,-3) {$4$}; \node [below] at (4,-3) {$5$}; \node [above] at (5,3) {$\cdots$}; \node at (5,0) {$\cdots$}; \node at (5,2) {$\cdots$}; \node at (5,-2) {$\cdots$}; \node [below] at (5,-3) {$\cdots$}; \node [left] at (-6,1) {$y_{1,m_1}$}; \node [left] at (-6,3) {$y_{1,m_1'}$}; \node [left] at (-6,-1) {$y_{m_2}$}; \node [left] at (-6,-3) {$y_{m_2'}$}; \node at (-5,2) {$\rho_{m_1',m_1}^{\shuffle_1}$}; \node at (-5,0) {$\rho_{m_1,m_2}$}; \node at (-5,-2) {$\rho_{m_2,m_2'}^{\shuffle_2}$}; \draw [->] (-4.3,2.75) -- (-4.3,1.25); \draw [->] (-4.3,0.75) -- (-4.3,-0.75); \draw [->] (-4.3,-1.25) -- (-4.3,-2.75); \draw [->] (-6.8,2.7) arc [radius=8, start angle=160, end angle= 199]; \node [left] at (-7.3,0) {$\rho_{m_1',m_2'}$}; \end{tikzpicture}$$* By the theorem 1.13, if we want to give a detail description of the map $\rho_{m_1,m_2}$, we only need to consider the map $\rho_{1,1}$. Then **Theorem 1**. **Remark 1**. *This is a very mysterious conclusion, It makes me very very excited ! We know that, for MZVs, Ihara, Kaneko and Zagier [@ikz] proved that:\ The identity $$\zeta^T_{\shuffle}(w)=\rho(\zeta_*^T(w))$$ holds for all words $w\in\mathfrak{H}^1$. Where $$\begin{aligned} \rho:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^n&\longmapsto n!\sum_{k=0}^n\frac{\gamma_k}{(n-k)!}\cdot T^{n-k} \end{aligned}$$ and $$\sum_{k=0}^{\infty}\gamma_ku^k=e^{\gamma u}\Gamma(1+u)$$ we obtained the same results from two different definitions, that is "All roads lead to Rome".* As an application, we will give a new method to proof the following identities, which know as Ramanujan's identity $$\begin{aligned} \sum_{i=1}^{\infty}\frac{1}{i^32^i}&=\frac{7}{8}\zeta(3)-\frac{\pi^2\ln(2)}{12}+\frac{\ln^3(2)}{6}\\ \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{(-1)^{i-1}}{i^2}\frac{1}{j2^j}&=\frac{13}{24}\zeta(3)\\ \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i}\frac{1}{j^22^j}&=\zeta(3)-\frac{\pi^2\ln(2)}{12}\\ \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i^2}\frac{1}{j2^j}&=\frac{5}{8}\zeta(3) \end{aligned}$$ furthermore, we also discovered a new identity $$\sum_{i=0}^{ \infty}\sum_{j=0}^{\infty}\frac{1}{(2i+1)^2}\frac{1}{(2i+1+j)2^{2i+1+j}}=\frac{7}{12}\zeta(3)$$ In the end, We will also propose some conjectures that these conjectures are similar with the classical MZVs. This article is just the beginning of a new story, and subsequent research work is underway. # **Analytic continuation of continuous multiple zeta functions** {#acc} In this section we show that the continuous multiple zeta functions are convergent under some natural conditions. Furthermore, they can be analytically continued to meromorphic functions with only simple poles at some special hyperplanes. **Theorem 1**. *For any given $t_1,\cdots,t_r\in\mathbb{R}_{>0}$, the continuous multiple zeta function $\zeta^{\mathcal{C}}(s_1,\cdots, s_r;t_1,\cdots,t_r)$ is convergence for $$\mathrm{Re}(s_1+s_2+\cdots+s_r)>r,\; \mathrm{Re}(s_2+\cdots+s_r)>r-1,\;\cdots, \mathrm{Re}(s_r)>1.$$ Moreover, it can be analytically continued to a meromorphic function on $\mathbb{C}^r$ with possible poles at some special hyperplanes.* We will prove the above theorem in two steps.\ **Step 1: The convergence**. Let $t:=\min\{t_1,\cdots,t_r\}$, then $t>0$ and $$\begin{aligned} |\zeta^{\mathcal{C}}(s_1,\cdots,s_r;t_1,\cdots,t_r)|&=\left|\quad\idotsint\limits_{0<x_1<x_2<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+t_1)^{s_1}\cdots(x_r+t_r)^{s_r}}\right|\\ &\leqslant\idotsint\limits_{0<x_1<x_2<\cdots<x_r}\left|\frac{dx_1\cdots dx_r}{(x_1+t_1)^{s_1}\cdots(x_r+t_r)^{s_r}}\right|\\ &=\idotsint\limits_{0<x_1<x_2<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+t_1)^{\text{Re}(s_1)}\cdots(x_r+t_r)^{\text{Re}(s_r)}}\\ &\leqslant\idotsint\limits_{0<x_1<x_2<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+t)^{\text{Re}(s_1)}\cdots(x_r+t)^{\text{Re}(s_r)}}\\ &=\frac{1}{\text{Re}(s_r)-1}\idotsint\limits_{0<x_1<x_2<\cdots<x_{r-1}}\frac{dx_1\cdots dx_{r-1}}{(x_1+t)^{\text{Re}(s_1)}\cdots(x_{r-1}+t)^{\text{Re}(s_{r-1}+s_r)-1}}\\ &=\frac{1}{(\text{Re}(s_r)-1)(\text{Re}(s_{r-1}+s_r)-2)\cdots(\text{Re}(s_1+\cdots+s_r)-r)} \end{aligned}$$ As a result, the continuous multiple zeta function $$\zeta^{\mathcal{C}}(s_1,\cdots, s_r;t_1,\cdots,t_r)$$ is convergent. $\hfill\Box$\ **Step 2: Analytic continuation**. The following lemma will be useful in the analytic continuation of continuous multiple zeta function. **Lemma 1**. *If $\varphi(t)$ is an infinitely differentiable bounded function on $(-\epsilon, 1+\epsilon)$ for some $\epsilon>0$, then $$I_{\varphi}(s)=\int_0^1\varphi(t)t^{s-1}dt$$ can be analytically continued to a meromorphic function on $\mathbb{C}$ with only simple poles at $s=0,-1,\cdots,-n,\cdots$ and $\mathrm{Res}_{s=-n}I_{\varphi}(s)=\frac{ \varphi^{(n)}(0)}{n!}$.* **Proof**. One can see [@li]. But for the convenience of readers, we will restate the proof of this lemma. For $k\geqslant 1$, $\mathrm{Re}(s)>0$, we have $$I_{\varphi}(s)=\int^1_0\left(\varphi(t)-\sum_{n=0}^k\frac{\varphi^{(n)}(0)}{n!}t^n \right)t^{s-1}dt+\sum_{n=0}^k\frac{\varphi^{(n)}(0)}{n!}\frac{1}{s+n}.$$ Denote by $R_{\varphi}(t)=\varphi(t)-\sum\limits_{n=0}^k\frac{\varphi^{(n)}(0)}{n!}t^n$. As $$R_{\varphi}(t)=O(t^k),\,t\to 0,$$ it follows that $$\int^1_0 R_{\varphi}(t) t^{s-1}dt$$ is a holomorphic function for $\mathrm{Re}(s)>-k$. Since the above analysis holds for any $k\geqslant 1$. The lemma is proved. $\hfill\Box$\ Now we return to the analytic continuation. For given $t_1,\cdots,t_r\in\mathbb{R}_{>0}$, let $t=\frac{\min\{t_1,\cdots,t_r\}}{2}$, and $y_{r+1-i}=\frac{t}{x_i+t},i=1,\cdots,r$, we have $$\begin{aligned} \zeta^{\mathcal{C}}(s_1,\cdots,s_r;t_1,\cdots,t_r)&=\int\limits_{0<x_1<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+t_1)^{s_1}\cdots(x_r+t_r)^{s_r}}\\ &=t^{r-(s_1+\cdots+s_r)}\cdot\int\limits_{0<y_1<\cdots<y_r<1}\frac{y_r^{s_1-2}\cdots y_1^{s_r-2}}{(1+t_1'y_r)^{s_1}\cdots(1+t'_ry_1)^{s_r}}dy_1\cdots dy_r \end{aligned}$$ where $t_i'=t_i/t-1,i=1,\cdots,r$. Next,we will induction on $r$. If $r=1$, by the above lemma, we are done. We assume this result for $r-1$. Now, we consider the case for $r$, we let $$\varphi(\tau)=\frac{1}{1+t_1'\tau}\int\limits_{0<y_1<\cdots<y_{r-1}<\tau}\frac{y_{r-1}^{s_1-2}\cdots y_1^{s_r-2}}{(1+t_1'y_{r-1})^{s_1}\cdots(1+t'_ry_1)^{s_r}}dy_1\cdots dy_r$$ one can verify that $\varphi(\tau)$ is a infinitely differentiable function on $(-\varepsilon,1+\varepsilon)$ for some $\varepsilon>0$. By using the above lemma, we are done. $\hfill\Box$ **Remark 1**. *Zhao [@zhao] proved that the multiple zeta function $$\zeta(s_1, s_2, \cdots, s_r)$$ can be analytically continued to a meromorphic function on $\mathbb{C}^r$ with possible poles at some special hyperplanes. theorem 2.1 shows that the structure of the poles of the continuous multiple zeta function $$\zeta^{\mathcal{C}}(s_1,s_2,\cdots, s_r;t_1,\cdots,t_r)$$ is more complicated than that of the multiple zeta function $$\zeta(s_1, s_2, \cdots, s_r).$$* # **Continuous multiple zeta values with double variables** In this section, we will use the notations in the above sections. Firstly, we will prove that there is an increasing filtration on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$ and the dimension of the subspace $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$ is finite. Secondly, we will give some actual calculation examples and put forward some problems. Lastly, we will discuss the relations among the cyclotomic multiple zeta values and the continuous multiple zeta values. ## **Filtration structure and finiteness of dimension** **Theorem 1**. *The subspace $F_{r}\mathcal{Z}^{\text{Dou},\mathcal{C}}$ define an increasing filtration of $\mathcal{Z}^{\text{Dou},\mathcal{C}}$, that is, we have $$\{0\}\subset \mathbb{Q}=F_{0}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset F_{1}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset F_{2}\mathcal{Z}^{\text{Dou},\mathcal{C}}\subset\cdots$$* **Proof**: If $\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\in F_{r}\mathcal{Z}^{\text{Dou},\mathcal{C}}$, then $$\begin{aligned} \zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})&=\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)\\ &=\idotsint\limits_{0<x_1<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}\\ &=\idotsint\limits_{0<x_1<\cdots<x_r<x_{r+1}}\frac{dx_1\cdots dx_rdx_{r+1}}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r-1}(x_{r+1}+m_r)^2}\\ &=\zeta^{\mathcal{C}}(k_1,\cdots,k_r-1,2;m_1,\cdots,m_r,m_r)\in F_{r+1}\mathcal{Z}^{\text{Dou},\mathcal{C}} \end{aligned}$$ $\hfill\Box$\ Next, we discuss the dimension of $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$. The proof of this result is relatively complicated, we are divided into the following parts. Firstly, we need some useful lemma and theorem. **Lemma 1**. *Let $m,n\in\mathbb{Q},m\neq n$ and $a,b\in\mathbb{Z}_{>0}$, we have $$\frac{1}{(x+m)^a(x+n)^b}=\sum_{i=1}^a\frac{a_i}{(x+m)^i}+\sum_{j=1}^{b}\frac{b_j}{(x+n)^j}$$ where $a_i,b_j\in\mathbb{Q}$ and $a_1+b_1=0$.* **Proof**: We only need to show $a_1+b_1=0$. We multiply both sides of the equation by $x+m$, we have $$\frac{1}{(x+m)^{a-1}(x+n)^b}=a_1+b_1\frac{x+m}{x+n}+\sum_{i=2}^a\frac{a_i}{(x+m)^{i-1}}+\sum_{j=2}^{b}\frac{b_j(x+m)}{(x+n)^j}$$ Let $x\to\infty$, then $a_1+b_1=0$. $\hfill\Box$\ **Theorem 1**. *We have the following formula* *$$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_i,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_r)\\ &=\frac{1}{k_i-1}\sum_{a=1}^{k_{i-1}}A_a\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},a,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i-1},m_{i+1},\cdots,m_r)\\ &+\frac{1}{k_i-1}\sum_{b=1}^{k_{i}-1}B_b\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},b,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i},m_{i+1},\cdots,m_r)\\ &-\frac{1}{k_i-1}\sum_{l=i+1}^rk_l\zeta^{\mathcal{C}}(k_1,\cdots,k_i-1,\cdots,k_l+1,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_l,\cdots,m_r) \end{aligned}$$ where $A_a,B_b\in\mathbb{Q},k_i\in\mathbb{Z}_{>1}(i=1,\cdots,r-1)$, and $m_{i-1}\neq m_i$. If $m_{i-1}=m_i$, then $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_i,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_r)\\ &=\frac{1}{k_i-1}\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},k_{i-1}+k_i-1,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i-1},m_{i+1},\cdots,m_r)\\ &-\frac{1}{k_i-1}\sum_{l=i+1}^rk_l\zeta^{\mathcal{C}}(k_1,\cdots,k_i-1,\cdots,k_l+1,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_l,\cdots,m_r) \end{aligned}$$ If $i=r$, then $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},k_r;m_1,\cdots,m_{r-1},m_r)\\ &=\frac{1}{k_r-1}\sum_{a=2}^{k_{r-1}}A_a\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},a;m_1,\cdots,m_{r-2},m_{r-1})\\ &+\frac{1}{k_r-1}\sum_{b=2}^{k_{r}-1}B_b\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},b;m_1,\cdots,m_{r-2},m_{r})\\ &+\frac{A_1}{(k_r-1)(m_r-m_{r-1})}\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},1,2;m_1,\cdots,m_{r-2},m_{r-1},m_{r}) \end{aligned}$$ where $k_r\in\mathbb{Z}_{>2},m_r\neq m_{r-1}$. If $m_r=m_{r-1}$, then $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},k_r;m_1,\cdots,m_{r-1},m_r)=\frac{1}{k_r-1}\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},k_{r-1}+k_r-1;m_1,\cdots,m_{r-2},m_{r-1}) \end{aligned}$$* **Proof**: If $i=1,\cdots,r-1$, Let $$\begin{aligned} F(x_i)&=\idotsint\limits_{x_i<x_{i+1}<\cdots<x_r}\frac{dx_{i+1}\cdots dx_r}{(x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_r+m_r)^{k_r}}\\ &=\idotsint\limits_{0<x_{i+1}<\cdots<x_r}\frac{dx_{i+1}\cdots dx_r}{(x_i+x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_i+x_r+m_r)^{k_r}} \end{aligned}$$ we have $$\begin{aligned} \zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})&=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_i}\frac{F(x_i)dx_{1}\cdots dx_i}{(x_{1}+m_{1})^{k_{1}}\cdots(x_i+m_i)^{k_i}}\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_{i-1}}\frac{1}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-1}+m_{i-1})^{k_{i-1}}}\left(\int_{x_{i-1}}^{+\infty}\frac{F(x_i)dx_i}{(x_i+m_i)^{k_i}}\right)dx_{1}\cdots dx_{i-1}\\ \end{aligned}$$ since $$\begin{aligned} \int_{x_{i-1}}^{+\infty}\frac{F(x_i)}{(x_i+m_i)^{k_i}}dx_i&=\frac{1}{1-k_i}\frac{F(x_i)}{(x_i+m_i)^{k_i-1}}\bigg|_{x_{i-1}}^{+\infty}-\frac{1}{1-k_i}\int_{x_{i-1}}^{+\infty}\frac{F'(x_i)}{(x_i+m_i)^{k_i-1}}dx_i\\ &=\frac{1}{k_i-1}\frac{F(x_{i-1})}{(x_{i-1}+m_i)^{k_i-1}}+\frac{1}{k_i-1}\int_{x_{i-1}}^{+\infty}\frac{F'(x_i)}{(x_i+m_i)^{k_i-1}}dx_i \end{aligned}$$ notice that $$\begin{aligned} F'(x_i)&=\sum_{l=i+1}^r\ \idotsint\limits_{0<x_{i+1}<\cdots<x_r}\frac{-k_ldx_{i+1}\cdots dx_r}{(x_i+x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_i+x_{l}+m_{l})^{k_{l}+1}\cdots(x_i+x_r+m_r)^{k_r}}\\ &=\sum_{l=i+1}^r\ \idotsint\limits_{x_i<x_{i+1}<\cdots<x_r}\frac{-k_ldx_{i+1}\cdots dx_r}{(x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_{l}+m_{l})^{k_{l}+1}\cdots(x_r+m_r)^{k_r}} \end{aligned}$$ thus, we have $$\begin{aligned} &\idotsint\limits_{0<x_1<x_{2}<\cdots<x_{i-1}}\frac{1}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-1}+m_{i-1})^{k_{i-1}}}\left(\frac{1}{k_i-1}\int_{x_{i-1}}^{+\infty}\frac{F'(x_i)}{(x_i+m_i)^{k_i-1}}dx_i\right)dx_{1}\cdots dx_{i-1}\\ &=-\sum_{l=i+1}^r\frac{k_l}{k_i-1}\idotsint\limits_{0<x_1<<\cdots<x_r}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i}+m_{i})^{k_{i}+1}\cdots(x_{l}+m_{l})^{k_{l}+1}\cdots(x_r+m_r)^{k_r}}\\ &=-\frac{1}{k_i-1}\sum_{l=i+1}^rk_l\zeta^{\mathcal{C}}(k_1,\cdots,k_i-1,\cdots,k_l+1,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_l,\cdots,m_r) \end{aligned}$$ if $m_{i-1}\neq m_{i}$, then $$\begin{aligned} &\idotsint\limits_{0<x_1<\cdots<x_{i-1}}\frac{1}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-1}+m_{i-1})^{k_{i-1}}}\left(\frac{1}{k_i-1}\frac{F(x_{i-1})}{(x_{i-1}+m_i)^{k_i-1}}\right)dx_{1}\cdots dx_{i-1}\\ &=\frac{1}{k_i-1}\idotsint\limits_{0<x_1<\cdots<x_{i-1}}\frac{F(x_{i-1})}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-2}+m_{i-2})^{k_{i-2}}}\left(\sum_{a=1}^{k_{i-1}}\frac{A_{a}}{(x_{i-1}+m_{i-1})^a}+\sum_{b=1}^{k_{i}-1}\frac{B_{b}}{(x_{i-1}+m_{i})^b}\right)dx_{1}\cdots dx_{i-1}\\ &=\frac{1}{k_i-1}\sum_{a=1}^{k_{i-1}}A_a\idotsint\limits_{0<x_1<\cdots<x_{i-1}<x_{i+1}<\cdots<x_r}\frac{dx_1\cdots dx_{i-1}dx_i\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-2}+m_{i-2})^{k_{i-2}}(x_{i-1}+m_{i-1})^a(x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_{r}+m_{r})^{k_r}}\\ &+\frac{1}{k_i-1}\sum_{b=1}^{k_{i}-1}B_b\idotsint\limits_{0<x_1<\cdots<x_{i-1}<x_{i+1}<\cdots<x_r}\frac{dx_1\cdots dx_{i-1}dx_i\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{i-2}+m_{i-2})^{k_{i-2}}(x_{i-1}+m_{i})^b(x_{i+1}+m_{i+1})^{k_{i+1}}\cdots(x_{r}+m_{r})^{k_r}}\\ &=\frac{1}{k_i-1}\sum_{a=1}^{k_{i-1}}A_a\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},a,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i-1},m_{i+1},\cdots,m_r)\\ &+\frac{1}{k_i-1}\sum_{b=1}^{k_{i}-1}B_b\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},b,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i},m_{i+1},\cdots,m_r) \end{aligned}$$ we obtain $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_i,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_r)\\ &=\frac{1}{k_i-1}\sum_{a=1}^{k_{i-1}}A_a\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},a,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i-1},m_{i+1},\cdots,m_r)\\ &+\frac{1}{k_i-1}\sum_{b=1}^{k_{i}-1}B_b\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},b,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i},m_{i+1},\cdots,m_r)\\ &-\frac{1}{k_i-1}\sum_{l=i+1}^rk_l\zeta^{\mathcal{C}}(k_1,\cdots,k_i-1,\cdots,k_l+1,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_l,\cdots,m_r) \end{aligned}$$ if $m_{i-1}=m_i$, then $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_i,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_r)\\ &=\frac{1}{k_i-1}\zeta^{\mathcal{C}}(k_1,\cdots,k_{i-2},k_{i-1}+k_i-1,k_{i+1},\cdots,k_r;m_1,\cdots,m_{i-2},m_{i-1},m_{i+1},\cdots,m_r)\\ &-\frac{1}{k_i-1}\sum_{l=i+1}^rk_l\zeta^{\mathcal{C}}(k_1,\cdots,k_i-1,\cdots,k_l+1,\cdots,k_r;m_1,\cdots,m_i,\cdots,m_l,\cdots,m_r) \end{aligned}$$ if $i=r$, using the same techniques, we have $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},k_r;m_1,\cdots,m_{r-1},m_r)\\ &=\frac{1}{k_r-1}\sum_{a=2}^{k_{r-1}}A_a\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},a;m_1,\cdots,m_{r-2},m_{r-1})\\ &+\frac{1}{k_r-1}\sum_{b=2}^{k_{r}-1}B_b\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},b;m_1,\cdots,m_{r-2},m_{r})\\ &+\frac{A_1}{(k_r-1)(m_r-m_{r-1})}\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},1,2;m_1,\cdots,m_{r-2},m_{r-1},m_{r}) \end{aligned}$$ where $k_r\in\mathbb{Z}_{>2},m_r\neq m_{r-1}$. If $m_r=m_{r-1}$, then $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},k_r;m_1,\cdots,m_{r-1},m_r)=\frac{1}{k_r-1}\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-2},k_{r-1}+k_r-1;m_1,\cdots,m_{r-2},m_{r-1}) \end{aligned}$$ $\hfill\Box$\ **Corollary 1**. *Using the above theorem, we have $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)\in&\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,m_{r-1},m_r)\\ &+\sum_{i=1}^r\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,m_r)\\ &+\sum_{1\leqslant i<j\leqslant r}\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,\widehat{m}_j,\cdots,m_r)+\cdots\\ &+\sum_{1\leqslant i<j\leqslant r}\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_i,m_j)\\ &+\sum_{i=1}^r\mathbb{Q}\zeta^{\mathcal{C}}(2;m_i) \end{aligned}$$* **Example 1**. *We list some examples where $r$ is relatively small.\ 1. If $r=1$, then $$\zeta^{\mathcal{C}}(k_1;m_1)\in\mathbb{Q}\zeta^{\mathcal{C}}(2;m_1)=\mathbb{Q}\cdot\frac{1}{m_1}=\mathbb{Q}$$\ 2. If $r=2$, then $$\zeta^{\mathcal{C}}(k_1,k_2;m_1,m_2)\in\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_2)+\mathbb{Q}\zeta^{\mathcal{C}}(2;m_1)+\mathbb{Q}\zeta^{\mathcal{C}}(2;m_2)=\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_2)+\mathbb{Q}$$\ 3. If $r=3$, then $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,k_2,k_3;m_1,m_2,m_3)\in&\mathbb{Q}\zeta^{\mathcal{C}}(1,1,2;m_1,m_2,m_3)\\ &+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_2)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_3)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_2,m_3)\\ &+\mathbb{Q} \end{aligned}$$\ 4. If $r=4$, then $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,k_2,k_3,k_4;m_1,m_2,m_3,m_4)\in&\mathbb{Q}\zeta^{\mathcal{C}}(1,1,1,2;m_1,m_2,m_3,m_4)\\ &+\mathbb{Q}\zeta^{\mathcal{C}}(1,1,2;m_1,m_2,m_3)+\mathbb{Q}\zeta^{\mathcal{C}}(1,1,2;m_1,m_2,m_4)\\ &+\mathbb{Q}\zeta^{\mathcal{C}}(1,1,2;m_1,m_3,m_4)+\mathbb{Q}\zeta^{\mathcal{C}}(1,1,2;m_2,m_3,m_4)\\ &+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_2)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_3)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_1,m_4)\\ &+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_2,m_3)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_2,m_4)+\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_3,m_4)\\ &+\mathbb{Q} \end{aligned}$$* Now Using the theorem 3.3 and corollary 3.1, we obtain the following result and the finiteness of the dimension of $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$.\ **Theorem 1**. *The subspace $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$ define an increasing filtration of $\mathcal{Z}^{\text{Dou},\mathcal{C}}$, that is, we have $$0=\mathcal{Z}^{\text{Dou},\mathcal{C}}_0\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_1\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_3\subset\cdots$$* **Proof**. Let $(\boldsymbol{k};\boldsymbol{m})=(1,k_2,\cdots,k_r;1,m_2,\cdots,m_r)$ be a admissible multi-index, and $$1+m_2+\cdots+m_r=m\Longrightarrow m_2+\cdots+m_r=m-1$$ by the theorem 3.3, we have $$\begin{aligned} \zeta^{\mathcal{C}}(2,k_2,\cdots,k_r;1,m_2,\cdots,m_r)&=\zeta^{\mathcal{C}}(k_2,\cdots,k_r;m_2,\cdots,m_r)\\ &\quad-\sum_{i=2}^rk_i\zeta^{\mathcal{C}}(1,k_2,\cdots,k_i,\cdots,k_r;1,m_2,\cdots,m_i,\cdots,m_r) \end{aligned}$$ hence, we conclude that $$\zeta^{\mathcal{C}}(k_2,\cdots,k_r;m_2,\cdots,m_r)\in\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\Longrightarrow\mathcal{Z}^{\text{Dou},\mathcal{C}}_{m-1}\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$$ $\hfill\Box$ **Corollary 2**. *Fix $(m_1,\cdots,m_r)$, then we have $$\begin{aligned} &\langle\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)|\forall k_1,\cdots,k_r\in\mathbb{Z}_{>0},k_r>1\rangle_{\mathbb{Q}}\\ &=\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,m_{r-1},m_r)\\ &\quad+\sum_{i=1}^r\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,m_r)\\ &\quad+\sum_{1\leqslant i<j\leqslant r}\mathbb{Q}\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,\widehat{m}_j,\cdots,m_r)+\cdots\\ &\quad+\sum_{1\leqslant i<j\leqslant r}\mathbb{Q}\zeta^{\mathcal{C}}(1,2;m_i,m_j)\\ &\quad+\sum_{i=1}^r\mathbb{Q}\zeta^{\mathcal{C}}(2;m_i) \end{aligned}$$* **Proof**. Using corollary 3.1 and theorem 3.5, we conclude this result. $\hfill\Box$\ Next, we will proof the finiteness of $\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$. **Theorem 1**. *The dimension of $\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$ is finite and satisfy $$\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant 2\cdot3^{m-1}-2^{m-2}(m+1)$$ furthermore, we also have $$\text{dim}_{\mathbb{Q}}F_r\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant(2^r-r)\binom{m-1}{r-1}$$* **Proof**: We know that $$\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m=\langle\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\big|\textbf{wt}(\boldsymbol{m})=m\rangle_{\mathbb{Q}}$$ for any give $m,r\in\mathbb{Z}_{>0},r\leqslant m$, we consider the Diophantine equation $$m_1+\cdots+m_r=m$$ we know that the number of the solutions of the above equation is $$N(r,m):=\left|\left\{(m_1,\cdots,m_r)\in(\mathbb{Z}_{>0})^r\big|m_1+\cdots+m_r=m\right\}\right|=\binom{m-1}{r-1}$$ By corollary 3.2, for a give solution $\boldsymbol{m}=(m_1,\cdots,m_r)$, the number of the value $\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,m_{r-1},m_r)$ of length $r-1$ is $\binom{r}{r}$; the number of the value\ $\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,m_r)$ of length $r-1$ is $\binom{r}{r-1}$; the number of the value $\zeta^{\mathcal{C}}(1,\cdots,1,2;m_1,\cdots,\widehat{m}_i,\cdots,\widehat{m}_j,\cdots,m_r)$ of length $r-2$ is $\binom{r}{r-2}$, by induction, we obtain the total number $$\binom{r}{1}+\binom{r}{2}+\cdots+\binom{r}{r-1}+\binom{r}{r}=2^r-1$$ notice that if $\ell(\boldsymbol{k};\boldsymbol{m})=1$, we always have $\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})\in\mathbb{Q}$. Thus for a given solution $(m_1,\cdots,m_r)$, we have $$\begin{aligned} &\text{dim}_{\mathbb{Q}}\langle\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)|\forall k_1,\cdots,k_r\in\mathbb{Z}_{>0},k_r>1\rangle_{\mathbb{Q}}\\ &\leqslant1+\binom{r}{2}+\cdots+\binom{r}{r-1}+\binom{r}{r}\\ &=2^r-r \end{aligned}$$ finally, we obtain $$\text{dim}_{\mathbb{Q}}F_r\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant(2^r-r)\binom{m-1}{r-1}$$ and $$\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m\leqslant\sum_{r=1}^m(2^r-r)\binom{m-1}{r-1}= 2\cdot3^{m-1}-2^{m-2}(m+1)$$ $\hfill\Box$\ **Remark 1**. *This is a very roughly estimation, far from the sharper bound. For example $$\mathcal{Z}^{\text{Dou},\mathcal{C}}_0=\{0\}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_1=\mathbb{Q}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_2=\mathbb{Q}\qquad\mathcal{Z}^{\text{Dou},\mathcal{C}}_3=\mathbb{Q}+\mathbb{Q}\ln(2)$$ $$\begin{aligned} \mathcal{Z}^{\text{Dou},\mathcal{C}}_4&=\mathbb{Q}+\mathbb{Q}\pi^2+\mathbb{Q}\ln(2)+\mathbb{Q}\ln(3)\\ \mathcal{Z}^{\text{Dou},\mathcal{C}}_5&=\mathbb{Q}+\mathbb{Q}\pi^2+\mathbb{Q}\ln(2)+\mathbb{Q}\ln(3)+\mathbb{Q}\zeta(3)+\mathbb{Q}\text{Li}_2\left(-\frac{1}{2}\right) \end{aligned}$$ where $\text{Li}_2(x):=\sum\limits_{n=1}^{\infty}\frac{x^n}{n^2}$.* ## **Some examples and problems** .\ We give some interesting and useful examples. $$\begin{aligned} \zeta^{\mathcal{C}}(1,2;1,2)&=\ln(2)\\ \zeta^{\mathcal{C}}(1,2;1,3)&=\ln(3)\\ \zeta^{\mathcal{C}}(1,1,2;1,1,2)&=\frac{\pi^2}{12}\\ \zeta^{\mathcal{C}}(1,1,3;1,1,2)&=\frac{\pi^2}{24}-\frac{\ln(2)}{2}\\ \zeta^{\mathcal{C}}(1,2,2;1,1,2)&=1-\frac{\pi^2}{12}\\ \zeta^{\mathcal{C}}(2,1,2;1,1,2)&=2\ln(2)-1\\ \zeta^{\mathcal{C}}(1,1,2;2,1,1)&=\ln(2)\\ \zeta^{\mathcal{C}}(1,1,2;2,1,2)&=\frac{\pi^2}{12}-\frac{\ln^2(2)}{2}\\ \zeta^{\mathcal{C}}(2,2,3;2,1,2)&=\ln(2)-\frac{11}{16}\\ \zeta^{\mathcal{C}}(1,1,2;1,2,3)&=\frac{\text{Li}_2(1/4)}{2}+\ln^2(2) \end{aligned}\qquad\begin{aligned} \zeta^{\mathcal{C}}(1,1,4;1,1,2)&=\frac{\pi^2}{36}-\frac{\ln(2)}{2}+\frac{1}{12}\\ \zeta^{\mathcal{C}}(1,1,5;1,1,2)&=\frac{\pi^2}{48}-\frac{11}{24}\ln(2)+\frac{11}{96}\\ \zeta^{\mathcal{C}}(1,2,2;1,2,3)&=-\frac{\text{Li}_2(1/4)}{2}-\ln^2(2)+\ln(2)\\ \zeta^{\mathcal{C}}(1,2,3;1,2,3)&=-\frac{\text{Li}_2(1/4)}{2}-\ln^2(2)+\frac{\ln(2)}{2}+\frac{\ln(3)}{4}\\ \zeta^{\mathcal{C}}(1,1,1,2;2,1,1,2)&=\zeta(3)-\frac{\pi^2\ln(2)}{12}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,2)&=\frac{5}{8}\zeta(3)\\ \zeta^{\mathcal{C}}(1,1,1,2;2,2,1,2)&=\frac{7}{8}\zeta(3)+\frac{\ln^3(2)}{6}-\frac{\pi^2\ln(2)}{12}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)&=\frac{3}{4}\zeta(3)\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)&=\frac{13}{24}\zeta(3) \end{aligned}$$ $$\begin{aligned} \zeta^{\mathcal{C}}(1,2;m_1,m_2)&=\frac{\ln(m_2)-\ln(m_1)}{m_1-m_2}(m_1\neq m_2)\\ \zeta^{\mathcal{C}}(1,1,2;1,1,m)&=\frac{1}{m-1}\left(\text{Li}_2\left(\frac{1}{1-m}\right)+\frac{\pi^2}{6}+\frac{\ln^2(m-1)}{2}\right)(m>1)\\ \zeta^{\mathcal{C}}(1,1,2;1,2,m)&=\frac{1}{m-2}\left(\text{Li}_2\left(\frac{1}{1-m}\right)+\frac{\pi^2}{12}+\frac{\ln^2(m-1)}{2}\right)(m>2)\\ \end{aligned}$$ where $\text{Li}_2(x):=\sum\limits_{n=1}^{\infty}\frac{x^n}{n^2}$.\ **Question**: Find out the dimension $d_m:=\text{dim}_{\mathbb{Q}}\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$ and $d_{m,r}:=\text{dim}_{\mathbb{Q}}F_r\mathcal{Z}^{\text{Dou},\mathcal{C}}_m$.\ ## Cyclotomic multiple zeta values and continuous multiple zeta values {#de} In this subsection, we will talk about the relation among the cyclotomic multiple zeta values of level 2 and continuous multiple zeta values with double variables. Using C.Glanois's theorem [@CG] which proved in 2016, we will proof that all cyclotomic multiple zeta values of level 2 are continuous multiple zeta values with double variables, that is $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$. **Theorem 1**. *Let $k_1,\cdots,k_s\in\mathbb{Z}_{>0}$, we have $$\zeta^{\mathcal{C}}(1^{\{k-1\}},2;1^{\{k_1-1\}},2,1^{\{k_2-1\}},2,\cdots,1^{\{k_s-1\}},2)=-\sum_{0<n_1\leqslant n_2\leqslant\cdots\leqslant n_s}\frac{(-1)^{n_s}}{n_1^{k_s-k_{s-1}-1}n_2^{k_{s-1}-k_{s-2}}\cdots n_s^{k_1}}$$ where $1^{\{r\}}$ means that the entry 1 is repeated $r$ times, and $k=k_1+\cdots+k_s$.* **Proof**. The idea of the proof of ​​this theorem is easy, but the process of narrative proves is very cumbersome. In order to facilitate readers to understand the proof, we only calculate two specific examples. Through these two examples, readers can immediately obtain the theorem's proof. The first example: $$\begin{aligned} \zeta^{\mathcal{C}}(1^{\{r-1\}},2;1^{\{r-1\}},2)&=\idotsint\limits_{0<x_1<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+2)^2}\\ &=\idotsint\limits_{0<x_1<\cdots<x_{r-1}}\frac{dx_1\cdots dx_{r-1}}{(x_1+1)\cdots(x_{r-1}+1)(x_{r-1}+2)}\\ &=\idotsint\limits_{0<x_1<\cdots<x_{r-1}}\sum_{n=0}^{\infty}\frac{(-1)^n}{(x_1+1)\cdots(x_{r-2}+1)(x_{r-1}+1)^{n+2}}dx_1\cdots dx_{r-1}\\ &=\idotsint\limits_{0<x_1<\cdots<x_{r-2}}\sum_{n=0}^{\infty}\frac{(-1)^n}{n+1}\frac{dx_1\cdots dx_{r-2}}{(x_1+1)\cdots(x_{r-3}+1)(x_{r-2}+1)^{n+2}}\\ &=\sum_{n=0}^{\infty}\frac{(-1)^n}{(n+1)^{r-1}}\\ &=-\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{r-1}} \end{aligned}$$\ The second example: $$\begin{aligned} &\zeta^{\mathcal{C}}(1^{\{r+s-1\}},2;1^{\{r-1\}},2,1^{s-1},2)\\ =&\idotsint\limits_{0<x_1<\cdots<x_{r+s}}\frac{dx_1\cdots dx_{r+s}}{(x_1+1)\cdots(x_{r-1}+1)(x_r+2)(x_{r+1}+1)\cdots(x_{r+s-1}+1)(x_{r+s}+2)^2}\\ =&\idotsint\limits_{0<x_1<\cdots<x_{r+s-1}}\frac{dx_1\cdots dx_{r+s-1}}{(x_1+1)\cdots(x_{r-1}+1)(x_r+2)(x_{r+1}+1)\cdots(x_{r+s-1}+1)(x_{r+s-1}+2)}\\ =&\idotsint\limits_{0<x_1<\cdots<x_{r+s-1}}\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{m+n}dx_1\cdots dx_{r+s-1}}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^{m+1}(x_{r+1}+1)\cdots(x_{r+s-1}+1)^{n+2}} \end{aligned}$$ $$\begin{aligned} =&\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\ \idotsint\limits_{0<x_1<\cdots<x_{r+s-1}}\frac{(-1)^{m+n}dx_1\cdots dx_{r+s-1}}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^{m+1}(x_{r+1}+1)\cdots(x_{r+s-1}+1)^{n+2}}\\ =&\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(-1)^{m+n}}{(n+1)^{s-1}(m+n+1)^r}\qquad(n_1=n+1,n_2=m+n+1)\\ =&-\sum_{0<n_1\leqslant n_2}\frac{(-1)^{n_2}}{n_1^{s-1}n_2^r} \end{aligned}$$ if we repeat the above steps, we get the proof of this theorem. $\hfill\Box$ Below, we will cite the C.Glanois's theorem which proved in 2016. **Theorem 1**. *(C.Glanois,2016). The set $$\left\{\zeta\left(\begin{matrix} k_1,\cdots,k_{r-1},k_r\\ 1,\cdots,1,-1 \end{matrix}\right)\Bigg|k_1,\cdots,k_r\in\mathbb{Z}_{>0}\right\}$$ is a set of generators for $\mathcal{Z}^2$.* **Proof**. See [@CG].$\hfill\Box$\ Using the theorem 3.8 and theorem 3.9, we conclude that **Theorem 1**. *All cyclotomic multiple zeta values of level 2 are continuous multiple zeta values with double variables, that is $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$.* **Proof**. We will proof this theorem by induction on the length $r$. If $r=1$, then $$\zeta\left(\begin{matrix} k\\ -1 \end{matrix}\right)=\sum_{n=1}^{\infty}\frac{(-1)^k}{n^k}\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$$ if $r=2$, then $$\zeta\left(\begin{matrix} k_1,k_2\\ 1,-1 \end{matrix}\right)=\sum_{0<n_1<n_2}\frac{(-1)^{n_2}}{n_1^{k_1}n_2^{k_2}}=\sum_{0<n_1\leqslant n_2}\frac{(-1)^{n_2}}{n_1^{k_1}n_2^{k_2}}-\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{k_1+k_2}}\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$$ repeat the above steps, we have $$\zeta\left(\begin{matrix} k_1,\cdots,k_{r-1},k_r\\ 1,\cdots,1,-1 \end{matrix}\right)\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$$ hence, we conclude that $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$.$\hfill\Box$\ **Remark 1**. *The result $\zeta\left(\begin{matrix} k\\ -1 \end{matrix}\right)=\sum\limits_{n=1}^{\infty}\frac{(-1)^k}{n^k}\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$ was first observed by myself, and then I told Jiangtao Li about this result. Thanks to his profound knowledge, he used C.Glanois's theorem to generalize it to $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$. Notice that $\mathcal{Z}^2\subset\mathcal{Z}^{\text{Dou},\mathcal{C}}$ is only an including of $\mathbb{Q}-$vector space, but not an embedding of $\mathbb{Q}-$algebra.* **Question**: Is there exists a canonical embedding of $\mathbb{Q}-$algebra $\mathcal{Z}^2\hookrightarrow\mathcal{Z}^{\text{Dou},\mathcal{C}}$ ? # **The algebraic structure on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$** In the previous section, we only regard $\mathcal{Z}^{\text{Dou},\mathcal{C}}$ as a $\mathbb{Q}-$linear space. In this section, We will proved that the product of two continuous multiple zeta values is a linear combination of continuous multiple zeta values with coefficients given by the first multiplicities. and we will give another integral representation of these values which similar with Kontsevich's integral representation [@bj]. This way of writing continuous multiple zeta values is central to the theory. From a combinatorial point of view, it yields the second shuffle product, a new structure from which many other linear relations are obtained in a systematic way.\ ## **The first product on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$** Firstly, we introduce some necessary notations.\ **Definition 1**. *Let $r,s\in\mathbb{Z}_{\geqslant0}$. A permutation of the set $\{1,2,\cdots,r+s\}$ is called a shuffle of type $(r, s)$ if the following two conditions are satisfied: $$\sigma(1)<\sigma(2)<\cdots<\sigma(r)\qquad\sigma(r+1)<\sigma(r+2)<\cdots<\sigma(r+s)$$ We denote the set of all shuffles of type $(r,s)$ by $\shuffle(r, s)$. That is $$\shuffle(r,s):=\left\{\sigma\in S_{r+s}\Bigg|\ \begin{aligned} &\sigma(1)<\sigma(2)<\cdots<\sigma(r),\\ &\sigma(r+1)<\sigma(r+2)<\cdots<\sigma(r+s) \end{aligned}\right\}$$* Notice that the subset $\shuffle(r,s)$ is not a subgroup of permutation group $S_{r+s}$.\ **Example 1**. *If $(r,s)=(2,2)$, then $$\shuffle(2,2)=\{(1),(23),(243),(123),(1243),(13)(24)\}\subset S_4$$* **Definition 1**. *Let $M$ be a positive real number, let $$D_r(M):=\left\{(x_1,\cdots,x_r)\in\mathbb{R}^r\Big|0<x_1<\cdots<x_r<M\right\}$$ especially, we write $D_r:=D_r(+\infty)$. * Using the above notations, we have $$\begin{aligned} \zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)&=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}\\ &=\int_{D_r}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}} \end{aligned}$$ similarly, we have $$\begin{aligned} \zeta^{\mathcal{C}}_M(k_1,\cdots,k_r;m_1,\cdots,m_r):&=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r<M}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}\\ &=\int_{D_r(M)}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}} \end{aligned}$$ furthermore, if $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ is a admissible multi-index($k_r>1$), then $$\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\lim_{M\to+\infty}\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})$$\ **Definition 1**. *Let $$\begin{aligned} (\boldsymbol{k}_1;\boldsymbol{m}_1)&=(k_1,\cdots,k_r;m_1,\cdots,m_r)\\ (\boldsymbol{k}_2;\boldsymbol{m}_2)&=(k_{r+1},\cdots,k_{r+s};m_{r+1},\cdots,m_{r+s}) \end{aligned}$$ we define $$\begin{aligned} &\shuffle_1((\boldsymbol{k}_1;\boldsymbol{m}_1),(\boldsymbol{k}_2;\boldsymbol{m}_2);(\boldsymbol{k};\boldsymbol{m}))\\ &:=|\{\sigma\in\shuffle(r,s)|(\boldsymbol{k};\boldsymbol{m})=(k_{\sigma^{-1}(1)},\cdots,k_{\sigma^{-1}(r+s)};m_{\sigma^{-1}(1)},\cdots,m_{\sigma^{-1}(r+s)})\}| \end{aligned}$$* With the previous preparations, we will provide the first product on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$.\ **Theorem 1**. *Let $(\boldsymbol{k}_1;\boldsymbol{m}_1)=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ and $(\boldsymbol{k}_2;\boldsymbol{m}_2)=$\ $(k_{r+1},\cdots,k_{r+s};m_{r+1},\cdots,m_{r+s})$ are positive multi-index, and $M$ be a positive real number, then we have $$\begin{aligned} \zeta^{\mathcal{C}}_M(\boldsymbol{k}_1;\boldsymbol{m}_1)\cdot\zeta^{\mathcal{C}}_M(\boldsymbol{k}_2;\boldsymbol{m}_2)=\sum_{(\boldsymbol{k};\boldsymbol{m})}\shuffle_1((\boldsymbol{k}_1;\boldsymbol{m}_1),(\boldsymbol{k}_2;\boldsymbol{m}_2);(\boldsymbol{k};\boldsymbol{m}))\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m}) \end{aligned}$$* **Proof**: We consider the decomposition $$D_r(M)\times D_s(M)=\bigcup_{\sigma\in\shuffle(r,s)}\left\{(x_1,\cdots,x_{r+s})\in\mathbb{R}^{r+s}\big|0<t_{\sigma^{-1}(1)}<\cdots<t_{\sigma^{-1}(r+s)}<M\right\}$$ then we conclude that $$\begin{aligned} &\zeta^{\mathcal{C}}_M(\boldsymbol{k}_1;\boldsymbol{m}_1)\cdot\zeta^{\mathcal{C}}_M(\boldsymbol{k}_1;\boldsymbol{m}_1)\\ &=\zeta^{\mathcal{C}}_M(k_1,\cdots,k_r;m_1,\cdots,m_r)\cdot\zeta^{\mathcal{C}}_M(k_{r+1},\cdots,k_{r+s};m_{r+1},\cdots,m_{r+s})\\ &=\int_{D_r(M)}\frac{dx_{1}\cdots dx_r}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}\cdot\int_{D_s(M)}\frac{dx_{r+1}\cdots dx_{r+s}}{(x_{r+1}+m_{r+1})^{k_{r+1}}\cdots(x_{r+s}+m_{r+s})^{k_{r+s}}}\\ &=\int_{D_r(M)\times D_s(M)}\frac{dx_{1}\cdots dx_{r+s}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{r+s}+m_{r+s})^{k_{r+s}}}\\ &=\sum_{\sigma\in\shuffle(r,s)}\quad\idotsint\limits_{0<x_{\sigma^{-1}(1)}<\cdots<x_{\sigma^{-1}(r+s)}<M}\frac{dx_{1}\cdots dx_{r+s}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_{r+s}+m_{r+s})^{k_{r+s}}}\\ &=\sum_{\sigma\in\shuffle(r,s)}\quad\int_{D_{r+s}(M)}\frac{dx_{\sigma^{-1}(1)}\cdots dx_{\sigma^{-1}(r+s)}}{(x_{\sigma^{-1}(1)}+m_{\sigma^{-1}(1)})^{k_{\sigma^{-1}(1)}}\cdots(x_{\sigma^{-1}(r+s)}+m_{\sigma^{-1}(r+s)})^{k_{\sigma^{-1}(r+s)}}}\\ &=\sum_{\sigma\in\shuffle(r,s)}\zeta^{\mathcal{C}}_M(k_{\sigma^{-1}(1)},\cdots,k_{\sigma^{-1}(r+s)};m_{\sigma^{-1}(1)},\cdots,m_{\sigma^{-1}(r+s)})\\ &=\sum_{(\boldsymbol{k};\boldsymbol{m})}\shuffle_1((\boldsymbol{k}_1;\boldsymbol{m}_1),(\boldsymbol{k}_2;\boldsymbol{m}_2);(\boldsymbol{k};\boldsymbol{m}))\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m}) \end{aligned}$$ $\hfill\Box$ **Corollary 3**. *If $(\boldsymbol{k}_1;\boldsymbol{m}_1)$ and $(\boldsymbol{k}_2;\boldsymbol{m}_2)$ are admissible multi-index, by taking the limit on $M$, we obtain $$\zeta^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)\cdot\zeta^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)=\sum_{(\boldsymbol{k};\boldsymbol{m})}\shuffle_1((\boldsymbol{k}_1;\boldsymbol{m}_1),(\boldsymbol{k}_2;\boldsymbol{m}_2);(\boldsymbol{k};\boldsymbol{m}))\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})$$* As a result, the linear spaces $\mathcal{Z}^{\text{Dou},\mathcal{C}}$ form a $\mathbb{Q}-$algebra under the above multiplicative rule, we denoted by $(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$.\ **Example 1**. *We have the following equations: $$\begin{aligned} &\zeta^{\mathcal{C}}(k_1,k_2;m_1,m_2)\cdot\zeta^{\mathcal{C}}(k_3,k_4;m_3,m_4)\\ &=\int\limits_{0<x_1<x_2}\frac{dx_1dx_2}{(x_1+m_1)^{k_1}(x_2+m_2)^{k_2}}\cdot\int\limits_{0<x_3<x_4}\frac{dx_3dx_4}{(x_3+m_3)^{k_3}(x_4+m_4)^{k_4}}\\ &=\int_U\frac{dx_1dx_2dx_3dx_4}{(x_1+m_1)^{k_1}(x_2+m_2)^{k_2}(x_3+m_3)^{k_3}(x_4+m_4)^{k_4}}\\ &=\sum_{i=1}^6\int_{U_i}\frac{dx_1dx_2dx_3dx_4}{(x_1+m_1)^{k_1}(x_2+m_2)^{k_2}(x_3+m_3)^{k_3}(x_4+m_4)^{k_4}}\\ &=\zeta^{\mathcal{C}}(k_1,k_2,k_3,k_4;m_1,m_2,m_3,m_4)+\zeta^{\mathcal{C}}(k_1,k_3,k_2,k_4;m_1,m_3,m_2,m_4)\\ &+\zeta^{\mathcal{C}}(k_1,k_3,k_4,k_2;m_1,m_3,m_4,m_2)+\zeta^{\mathcal{C}}(k_3,k_1,k_2,k_4;m_3,m_1,m_2,m_4)\\ &+\zeta^{\mathcal{C}}(k_3,k_1,k_4,k_2;m_3,m_1,m_4,m_2)+\zeta^{\mathcal{C}}(k_3,k_4,k_1,k_2;m_3,m_4,m_1,m_2) \end{aligned}$$ where $U$ and $U_i(i=1,\cdots,6)$ are defined by $$U=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4|0<x_1<x_2,0<x_3<x_4\right\}$$ and $$\begin{aligned} U_1&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_1<x_2<x_3<x_4\right\}\\ U_2&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_1<x_3<x_2<x_4\right\}\\ U_3&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_1<x_3<x_4<x_2\right\}\\ U_4&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_3<x_1<x_2<x_4\right\}\\ U_5&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_3<x_1<x_4<x_2\right\}\\ U_6&=\left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\big|0<x_3<x_4<x_1<x_2\right\} \end{aligned}$$* ## **Integral representation and the second product on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$** For the classical MZVs, Kontsevich and Drinfel'd discovered the following integral representation **Theorem 1**. *Let $\boldsymbol{k}=(k_1,\cdots,k_r)$ be a admissible multi-index($k_r>1$), we have $$\zeta(\boldsymbol{k})=\int_{\Delta^{\text{wt}(\boldsymbol{k})}}\omega_{\boldsymbol{k}}$$* **Proof**: See [@bj] chapter 1.$\hfill\Box$\ We return to the continuous multiple zeta value $\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})$, we have the similar result. Before starting this story, we define the "continuous multiple polylogarithm function". **Definition 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a positive multi-index, we define $$\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t):=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{t^{x_r}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dx_{1}\cdots dx_r\qquad0\leqslant t<1$$ called the **continuous multiple polylogarithm function**.* Obviously, if $(\boldsymbol{k};\boldsymbol{m})$ is a admissible multi-index, then $$\lim_{t\to1^{-}}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)=\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})$$ An important property of polylogarithms is that they satisfy many functional equations, the easiest being **Lemma 1**. *The following identities hold for all $0\leqslant t<1$. $$\begin{aligned} t^{m_r}\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r+1;m_1,\cdots,m_r)(t)&=\int_0^t\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_r}\frac{dt_1}{t_1}\\ t^{m_{r+1}}\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r,1;m_1,\cdots,m_r,m_{r+1})(t)&=\int_0^t\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_{r+1}}\frac{dt_1}{-t_1\ln(t_1)} \end{aligned}$$* **Proof**. The first identity $$\begin{aligned} &\int_0^t\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_r}\frac{dt_1}{t_1}\\ &=\int_0^t\left(\ \ \idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{t_1^{x_r}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dx_{1}\cdots dx_r\right)t_1^{m_r}\frac{dt_1}{t_1}\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\left(\int_0^t\frac{t_1^{x_r+m_r-1}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r-1)^{k_r}}dt_1\right)dx_1\cdots dx_r\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{t^{x_r+m_r}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r+1}}dx_{1}\cdots dx_r\\ &=t^{m_r}\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r+1;m_1,\cdots,m_r)(t) \end{aligned}$$ The second identity $$\begin{aligned} &\int_0^t\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_{r+1}}\frac{dt_1}{-t_1\ln(t_1)}\\ &=\int_0^t\left(\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{t_1^{x_r}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dx_{1}\cdots dx_r\right)t_1^{m_{r+1}}\frac{dt_1}{-t_1\ln(t_1)}\\ &=\int_0^t\left(\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\frac{t_1^{x_r+m_{r+1}-1}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}\cdot\frac{1}{-\ln(t_1)}dx_{1}\cdots dx_r\right)dt_1\\ &=\int_0^t\left(\idotsint\limits_{0<x_1<x_{2}<\cdots<x_r}\int_{x_r}^{\infty}\frac{t_1^{x_{r+1}+m_{r+1}-1}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dx_{r+1}dx_{1}\cdots dx_r\right)dt_1\\ &=\int_0^t\left(\idotsint\limits_{0<x_1<x_{2}<\cdots<x_{r+1}}\frac{t_1^{x_{r+1}+m_{r+1}-1}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dx_{1}\cdots dx_{r+1}\right)dt_1\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_{r+1}}\left(\int_0^t\frac{t_1^{x_{r+1}+m_{r+1}-1}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}}dt_1\right)dx_1\cdots dx_{r+1}\\ &=\idotsint\limits_{0<x_1<x_{2}<\cdots<x_{r+1}}\frac{t^{x_{r+1}+m_{r+1}}}{(x_{1}+m_{1})^{k_{1}}\cdots(x_r+m_r)^{k_r}(x_{r+1}+m_{r+1})}dx_{1}\cdots dx_r\\ &=t^{m_{r+1}}\text{Li}^{\mathcal{C}}(k_1,\cdots,k_r,1;m_1,\cdots,m_r,m_{r+1})(t) \end{aligned}$$$\hfill\Box$\ A piece of notation is needed to describe the general integral representation of multiple zeta values. **Definition 1**. *Given a real number $0<t\leqslant1$, we define $$\Delta^r(t):=\left\{(x_1,\cdots,x_r)\in\mathbb{R}^r\big|0<x_1<\cdots<x_r<t\right\}\qquad0<t\leqslant1$$* when $t=1$, we will simply write $\Delta^r:=\Delta^r(1)$. Furthermore, consider the measures on the open interval $(0,1)$ $$\omega_0(t):=\frac{dt}{t}\qquad\omega_1^{(m)}(t):=\frac{t^mdt}{-t\ln(t)}$$ where $m\in\mathbb{Z}$. Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)\in(\mathbb{Z}_{>0})^r\times(\mathbb{Z}_{>0})^r$ is a positive multi-index, we denote $s_i=k_1+\cdots+k_i(i=1,\cdots,r)$, for convenience, we write $s_0=0,m_0=0$. Let $\omega_{(\boldsymbol{k};\boldsymbol{m})}$ be the measure on the interior of the simplex $\Delta^{\textbf{wt}(\boldsymbol{k})}$ given by $$\omega_{(\boldsymbol{k};\boldsymbol{m})}=\prod_{i=1}^r\omega_1^{(m_i-m_{i-1})}(t_{s_{r+1-i}})\omega_0(t_{s_{r+1-i}-1})\cdots\omega_0(t_{s_{r-i}+1})$$ For example, one has: $$\begin{aligned} \omega_{(2;m_1)}&=\frac{t_2^{m_1}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ &=\omega_1^{(m_1)}(t_2)\omega_0(t_1)\\ \omega_{(2,2;m_1,m_2)}&=\frac{t_4^{m_1}dt_4}{-t_4\ln(t_4)}\ \frac{dt_3}{t_3}\ \frac{t_2^{m_2-m_1}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ &=\omega_1^{(m_1)}(t_4)\omega_0(t_3)\omega_1^{(m_2-m_1)}(t_2)\omega_0(t_1)\\ \omega_{(1,3;m_1,m_2)}&=\frac{t_4^{m_1}dt_4}{-t_4\ln(t_4)}\ \frac{t_3^{m_2-m_1}dt_3}{-t_3\ln(t_3)}\ \frac{dt_2}{t_2}\ \frac{dt_1}{t_1}\\ &=\omega_1^{(m_1)}(t_4)\omega_1^{(m_2-m_1)}(t_3)\omega_0(t_2)\omega_0(t_1)\\ \omega_{(2,1,2;m_1,m_2,m_3)}&=\frac{t_5^{m_1}dt_5}{-t_5\ln(t_5)}\ \frac{dt_4}{t_4}\ \frac{t_3^{m_2-m_1}dt_3}{-t_3\ln(t_3)}\ \frac{t_2^{m_3-m_2}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ &=\omega_1^{(m_1)}(t_5)\omega_0(t_4)\omega_1^{(m_2-m_1)}(t_3)\omega_1^{(m_3-m_2)}(t_2)\omega_0(t_1) \end{aligned}$$\ Now the integral representation is a particular case of the next result **Theorem 1**. *If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ is a positive multi-index and $0<t<1$ a real number, then the following identity holds: $$t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)=\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$* **Proof**. The proof is by induction on the weight of $\boldsymbol{k}$. If $\textbf{wt}(\boldsymbol{k})=1$, then $(\boldsymbol{k};\boldsymbol{m})=(1;m_1)$ and the statement is just the second identity in the above lemma. The inductive step follows from the functional equations in the above lemma. Indeed, let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a positive multi-index and assume that the result is true for all multi-indices of lower weight. If $k_r>1$, we write $$(\boldsymbol{k}';\boldsymbol{m})=(k_1,\cdots,k_r-1;m_1,\cdots,m_r)$$ then, by the first identity in the above lemma and induction, we have $$t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)=\int_0^tt_1^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k}';\boldsymbol{m})(t_1)\frac{dt_1}{t_1}=\int_0^t\int_{\Delta^{\text{wt}(\boldsymbol{k}')}(t_1)}\omega_{(\boldsymbol{k}';\boldsymbol{m})}\frac{dt_1}{t_1}=\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$ If $k_r=1$, we write $$(\boldsymbol{k}';\boldsymbol{m}')=(k_1,\cdots,k_{r-1};m_1,\cdots,m_{r-1})$$ then, by the second identity in the above lemma and induction, we have $$\begin{aligned} t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)&=\int_0^tt_1^{m_{r-1}}\text{Li}^{\mathcal{C}}(\boldsymbol{k}';\boldsymbol{m}')(t_1)\frac{t_1^{m_r-m_{r-1}}dt_1}{-t_1\ln(t_1)}\\ &=\int_0^t\int_{\Delta^{\text{wt}(\boldsymbol{k}')}(t_1)}\omega_{(\boldsymbol{k}';\boldsymbol{m}')}\frac{t_1^{m_r-m_{r-1}}dt_1}{-t_1\ln(t_1)}\\ &=\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})} \end{aligned}$$ Now we finished this proof.$\hfill\Box$\ **Corollary 4**. *If $(\boldsymbol{k};\boldsymbol{m})$ is a admissible multi-index, then the following identity holds: $$\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\int_{\Delta^{\text{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$* **Proof**. By assumption, $(\boldsymbol{k};\boldsymbol{m})$ is a admissible multi-index, then $$\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\lim_{t\to1^{-}}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)=\lim_{t\to1^{-}}\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=\int_{\Delta^{\text{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$ $\hfill\Box$ **Example 1**. *$$\begin{aligned} \zeta^{\mathcal{C}}(2;m_1)&=\int\limits_{0<t_2<t_1<1}\frac{t_2^{m_1}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ \zeta^{\mathcal{C}}(2,2;m_1,m_2)&=\int\limits_{0<t_4<t_3<t_2<t_1<1}\frac{t_4^{m_1}dt_4}{-t_4\ln(t_4)}\ \frac{dt_3}{t_3}\ \frac{t_2^{m_2-m_1}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ \zeta^{\mathcal{C}}(1,3;m_1,m_2)&=\int\limits_{0<t_4<t_3<t_2<_1<1}\frac{t_4^{m_1}dt_4}{-t_4\ln(t_4)}\ \frac{t_3^{m_2-m_1}dt_3}{-t_3\ln(t_3)}\ \frac{dt_2}{t_2}\ \frac{dt_1}{t_1}\\ \zeta^{\mathcal{C}}(2,1,2;m_1,m_2,m_3)&=\int\limits_{0<t_5<t_4<t_3<t_2<t_1<1}\frac{t_5^{m_1}dt_5}{-t_5\ln(t_5)}\ \frac{dt_4}{t_4}\ \frac{t_3^{m_2-m_1}dt_3}{-t_3\ln(t_3)}\ \frac{t_2^{m_3-m_2}dt_2}{-t_2\ln(t_2)}\ \frac{dt_1}{t_1}\\ \end{aligned}$$* **Remark 1**. *If we define $$\text{Li}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t):=\sum\limits_{0<n_1<\cdots<n_r}\frac{t^{n_r}}{(n_1+m_1)^{k_1}\cdots(n_r+m_r)^{k_r}}\qquad |t|<1$$ where $k_1,\cdots,k_r\in\mathbb{Z}_{>0};m_1\cdots,m_r\in\mathbb{Z}_{\geqslant0}$, we also have the similar result $$\begin{aligned} t^{m_r}\text{Li}(k_1,\cdots,k_r+1;m_1,\cdots,m_r)(t)&=\int_0^t\text{Li}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_r}\frac{dt_1}{t_1}\\ t^{m_{r+1}}\text{Li}(k_1,\cdots,k_r,1;m_1,\cdots,m_r,m_{r+1})(t)&=\int_0^t\text{Li}(k_1,\cdots,k_r;m_1,\cdots,m_r)(t_1)t_1^{m_{r+1}}\frac{dt_1}{1-t_1} \end{aligned}$$ and $$t^{m_r}\text{Li}(\boldsymbol{k};\boldsymbol{m})(t)=\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$ where $$\omega_{(\boldsymbol{k};\boldsymbol{m})}=\prod_{i=1}^r\omega_1^{(m_i-m_{i-1})}(t_{s_{r+1-i}})\omega_0(t_{s_{r+1-i}-1})\cdots\omega_0(t_{s_{r-i}+1})$$ and $s_i=k_1+\cdots+k_i(i=1,\cdots,r),s_0=0,m_0=0$, moreover $$\omega_0(t):=\frac{dt}{t}\qquad\omega_1^{(m)}(t):=\frac{t^mdt}{1-t}$$ as a result, we also have $$\zeta(\boldsymbol{k};\boldsymbol{m})=\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}$$ where $$\zeta(\boldsymbol{k};\boldsymbol{m})=\zeta(k_1,\cdots,k_r;m_1,\cdots,m_r):=\sum\limits_{0<n_1<\cdots<n_r}\frac{1}{(n_1+m_1)^{k_1}\cdots(n_r+m_r)^{k_r}}$$* ## **Multi-indices and binary sequences** To exploit the preceding proposition to derive relations continuous multiple among polylogarithms, and in particular among continuous multiple zeta values, we need a new notation. We will use the following notation to go from multi-indices to binary sequences and the other way around. **Definition 1**. *To each positive multi-index $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ we attach the positive binary sequence $$\text{bs}(\boldsymbol{k};\boldsymbol{m})=(1,0^{\{k_1-1\}},1,0^{\{k_2-1\}},\cdots,1,0^{\{k_r-1\}};m_1,0^{\{k_1-1\}},m_2-m_1,0^{\{k_2-1\}},\cdots,m_r-m_{r-1},0^{\{k_r-1\}})$$ where $0^{\{k\}}$ means that the entry zero is repeated $k$ times.* Clearly, we have a inverse map $$\text{bs}^{-1}(1,0^{\{k_1\}},1,0^{\{k_2\}},\cdots,1,0^{\{k_r\}};m_1,0^{\{k_1\}},m_2,0^{\{k_2\}},\cdots,m_r,0^{\{k_r\}})=(k_1',\cdots,k_r';m_1',\cdots,m_r')$$ where $k_i'=k_i+1,m_i'=m_1+\cdots+m_i,i=1,\cdots,r$.\ **Example 1**. *$$\begin{aligned} \text{bs}(2;m_1)&=(1,0;m_1,0)\\ \text{bs}(2,2;m_1,m_2)&=(1,0,1,0;m_1,0,m_2-m_1,0)\\ \text{bs}(1,3;m_1,m_2)&=(1,1,0,0;m_1,m_2-m_1,0,0)\\ \text{bs}(2,1,2;m_1,m_2,m_3)&=(1,0,1,1,0;m_1,0,m_2-m_1,m_3-m_2,0) \end{aligned}$$* **Lemma 1**. *Let $r,s\geqslant0$ be integers, $0<t<1$ a real number and $\mu_i(i=1,\cdots,r+s)$ be differential 1-forms on $(0,1)$. Then $$\begin{aligned} &\int_{\Delta^r(t)}\mu_1(t_1)\cdots\mu_r(t_r)\cdot\int_{\Delta^r(t)}\mu_1(t_1)\cdots\mu_r(t_r)\\ &=\sum_{\sigma\in\shuffle(r,s)}\int_{\Delta^{r+s}(t)}\mu_{\sigma^{-1}}(t_1)\cdots\mu_{\sigma^{-1}(r+s)}(t_{r+s}) \end{aligned}$$* **Proof**. Using the decomposition $$\Delta^r(t)\times\Delta^s(t)=\bigcup_{\sigma\in\shuffle(r,s)}\left\{(t_1,\cdots,t_{r+s})\big|0<t_{\sigma^{-1}(r+s)}<\cdots<t_{\sigma^{-1}(1)}<t\right\}$$ together with the fact that the intersection of two simplices on the right-hand side is a set of measure zero we obtain $$\begin{aligned} &\int_{\Delta^r(t)}\mu_1(t_1)\cdots\mu_r(t_r)\cdot\int_{\Delta^r(t)}\mu_1(t_1)\cdots\mu_r(t_r)\\ &=\int_{\Delta^r(t)\times\Delta^s(t)}\mu_1(t_1)\cdots\mu_{r+s}(t_{r+s})\\ &=\sum_{\sigma\in\shuffle(r,s)}\ \int\limits_{0<t_{\sigma^{-1}(r+s)}<\cdots<t_{\sigma^{-1}(1)}<t}\mu_1(t_1)\cdots\mu_{r+s}(t_{r+s})\\ &=\sum_{\sigma\in\shuffle(r,s)}\int_{\Delta^{r+s}(t)}\mu_{\sigma^{-1}}(t_1)\cdots\mu_{\sigma^{-1}(r+s)}(t_{r+s}) \end{aligned}$$ where, in the last equality we have made the change of variables $t_i = t_{\sigma^{-1}(i)}$ to put the set $0<t_{\sigma^{-1}(r+s)}<\cdots<t_{\sigma^{-1}(1)}<t$ as $\Delta^{r+s}(t)$. $\hfill\Box$ **Definition 1**. *Let $(\boldsymbol{s}_1;\boldsymbol{\tau}_1)=(s_1,\cdots,s_r;\tau_1,\cdots,\tau_r)$,$(\boldsymbol{s}_2;\boldsymbol{\tau}_2)=(s_{r+1},\cdots,s_{r+s};\tau_{r+1},\cdots,\tau_{r+s})$ are binary sequences, we define $$\begin{aligned} &\shuffle_2((\boldsymbol{s}_1;\boldsymbol{\tau}_1),(\boldsymbol{s}_2;\boldsymbol{\tau}_2);(\boldsymbol{s};\boldsymbol{\tau}))\\ &:=|\{\sigma\in\shuffle(r,s)|(\boldsymbol{s};\boldsymbol{\tau})=(s_{\sigma^{-1}(1)},\cdots,s_{\sigma^{-1}(r+s)};\tau_{\sigma^{-1}(1)},\cdots,\tau_{\sigma^{-1}(r+s)})\}| \end{aligned}$$* Now, we have the following important result. **Theorem 1**. *Let $(\boldsymbol{k}_1;\boldsymbol{m}_1)=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ and $(\boldsymbol{k}_2;\boldsymbol{m}_2)=$\ $(k_{r+1},\cdots,k_{r+s};m_{r+1},\cdots,m_{r+s})$ are positive multi-index, and $0<t<1$ be a positive real number, then we have $$\begin{aligned} \text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\cdot\text{Li}^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)(t)=\sum_{(\boldsymbol{k};\boldsymbol{m})}\shuffle_2(\text{bs}(\boldsymbol{k}_1;\boldsymbol{m}_1),\text{bs}(\boldsymbol{k}_2;\boldsymbol{m}_2);\text{bs}(\boldsymbol{k};\boldsymbol{m}))\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t) \end{aligned}$$* **Proof**. This result comes from Lemma 4.16.$\hfill\Box$\ **Corollary 5**. *If $(\boldsymbol{k}_1;\boldsymbol{m}_1)$ and $(\boldsymbol{k}_2;\boldsymbol{m}_2)$ are admissible multi-index, by taking the limit on $t\to1^-$, we obtain $$\zeta^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)\cdot\zeta^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)=\sum_{(\boldsymbol{k};\boldsymbol{m})}\shuffle_2(\text{bs}(\boldsymbol{k}_1;\boldsymbol{m}_1),\text{bs}(\boldsymbol{k}_2;\boldsymbol{m}_2);\text{bs}(\boldsymbol{k};\boldsymbol{m}))\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})$$* By the corollary 4.3, we obtain another multiplicative rule on $\mathcal{Z}^{\text{Dou},\mathcal{C}}$.\ **Example 1**. *We have $$\begin{aligned} \zeta^{\mathcal{C}}(2;m_1)\cdot\zeta^{\mathcal{C}}(2;m_2)&=\zeta^{\mathcal{C}}(2,2;m_1,m_1+m_2)+\zeta^{\mathcal{C}}(2,2;m_2,m_1+m_2)\\ &\quad+2\zeta^{\mathcal{C}}(1,3;m_1,m_1+m_2)+2\zeta^{\mathcal{C}}(1,3;m_2,m_1+m_2) \end{aligned}$$ if we take $m_1=m_2=m$, then $$\zeta^{\mathcal{C}}(2;m)\cdot\zeta^{\mathcal{C}}(2;m)=2\zeta^{\mathcal{C}}(2,2;m,2m)+4\zeta^{\mathcal{C}}(1,3;m,2m)$$ this is a similar result with $$\zeta(2)^2=2\zeta(2,2)+4\zeta(1,3)$$* # **The algebra of continuous multiple zeta values and the comparison theorem** In the previous sections, we saw two methods to express a product of multiple zeta values as a linear combination of MZVs. As we saw in examples 4.6 and example 4.19, both methods may give different linear combinations for the same product of multiple zeta values, thus leading to linear relations among them. The first shuffle multiplicities are easily expressed in terms of multi-indices as in theorem 4.5, while the second shuffle multiplicities are expressed more conveniently using binary sequences as in theorem 4.18. We now want to put a little order to make the combinatorial structure of multiple zeta values clearer. We will define the first shuffle product and the second shuffle product as products in certain formal algebras that encode the first and the second shuffle multiplicities respectively. To this end, we discuss how to extend multiple zeta values to non-admissible words and use this extension to derive relations among them. We will derive the comparison theorem among the two different products which is an analogy of Ihara-Kaneko-Zagier's comparison theorem. As an application, we will give a new method to proof some Ramanujan's identities. Conjecturally, all relations can be obtained in this way. ## **The algebra of continuous multiple zeta values** Let $Y=\{y_{i,j}|i,j\in\mathbb{Z}_{>0}\}$ be a countable set. The elements of $Y$ will be called **letters** and $Y$ is called an **alphabet**. Let $\mathbb{Q}\langle Y\rangle$ be the non-commutative polynomial $\mathbb{Q}-$algebra over $Y$. The first shuffle product $\shuffle_1$ on $\mathbb{Q}\langle Y\rangle$ is defined by $$1\shuffle_1 w=w\shuffle_11=w\qquad \forall w\in\mathbb{Q}\langle Y\rangle$$ $$uw_1\shuffle_1vw_2=u(w_1\shuffle_1vw_2)+v(uw_1\shuffle_1w_2)\qquad \forall u,v\in Y,\forall w_1,w_2\in\mathbb{Q}\langle Y\rangle$$ inductively. We have the following properties **Proposition 1**. *$(\mathbb{Q}\langle Y\rangle,\shuffle_1)$ is a commutative $\mathbb{Q}-$algebra with unit.* **Proof**. One can check this result directly from the definition of the product $\shuffle_1$. $\hfill\Box$ **Definition 1**. *A word $w=y_{k_1,m_1}\cdots y_{k_r,m_r}$ is called admissible if $k_r>1$,i.e. if it corresponds to an admissible multi-index. By convention, the word 1 will also be considered to be admissible. We will denote by $\mathbb{Q}\langle Y\rangle^0$ the subspace of $\mathbb{Q}\langle Y\rangle$ generated by admissible words.* **Proposition 1**. *We have the following results:* *(1) The subspace $(\mathbb{Q}\langle Y\rangle^0,\shuffle_1)$ is a subalgebra of $(\mathbb{Q}\langle Y\rangle,\shuffle_1)$\ (2) We have a morphism of $\mathbb{Q}-$algebra $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_1}:(\mathbb{Q}\langle Y\rangle^0,\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot) \end{aligned}$$ determined by the assignment $$y_{k_1,m_1}\cdots y_{k_r,m_r}\longmapsto\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)$$ i.e. we have $$\zeta^{\mathcal{C}}_{\shuffle_1}(w\shuffle_1v)=\zeta^{\mathcal{C}}_{\shuffle_1}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_1}(v)\qquad\forall w,v\in\mathbb{Q}\langle Y\rangle^0$$* **Proof**. The first statement can be checked directly from the definition of the product $\shuffle_1$. The second statement follows from Corollary 4.1.$\hfill\Box$\ **Definition 1**. *Let $X=\{x,y_i|i\in\mathbb{Z}_{>0}\}$ be a countable set. The elements of $X$ will be also called **letters** and $X$ is called an **alphabet**. Let $\mathbb{Q}\langle X\rangle$ be the non-commutative polynomial $\mathbb{Q}-$algebra over $X$. The second shuffle product $\shuffle_2$ on $\mathbb{Q}\langle X\rangle$ is defined by $$1\shuffle_2 w=w\shuffle_21=w\qquad \forall w\in\mathbb{Q}\langle X\rangle$$ $$uw_1\shuffle_2vw_2=u(w_1\shuffle_2vw_2)+v(uw_1\shuffle_2w_2)\qquad \forall u,v\in X,\forall w_1,w_2\in\mathbb{Q}\langle X\rangle$$ inductively.* We have the following properties **Proposition 1**. *$(\mathbb{Q}\langle X\rangle,\shuffle_2)$ is a commutative $\mathbb{Q}-$algebra with unit.* **Proof**. One can check this result directly from the definition of the product $\shuffle_2$.$\hfill\Box$ **Definition 1**. *A word $w=y_{l_1}x^{t_1}y_{l_2}x^{t_2}\cdots y_{l_r}x^{t_r}$ is said to be positive if $l_1\geqslant1,l_1+l_2\geqslant1,\cdots,l_1+\cdots+l_r\geqslant1$ and said to be admissible if it is positive and, in addition, $t_r>1$. By convention, the word 1 will also be considered to be admissible. We will denote by $\mathbb{Q}\langle X\rangle^0$ the subspace of $\mathbb{Q}\langle X\rangle$ generated by admissible words, and $\mathbb{Q}\langle X\rangle^1$ the subspace of $\mathbb{Q}\langle X\rangle$ generated by positive words, i.e. $$\begin{aligned} \mathbb{Q}\langle X\rangle^0&:=\langle y_{l_1}x^{t_1}y_{l_2}x^{t_2}\cdots y_{l_r}x^{t_r}|l_1\geqslant1,l_1+l_2\geqslant1,\cdots,l_1+\cdots+l_r\geqslant1,t_r\geqslant1\rangle_{\mathbb{Q}}\\ \mathbb{Q}\langle X\rangle^1&:=\langle y_{l_1}x^{t_1}y_{l_2}x^{t_2}\cdots y_{l_r}x^{t_r}|l_1\geqslant1,l_1+l_2\geqslant1,\cdots,l_1+\cdots+l_r\geqslant1\rangle_{\mathbb{Q}} \end{aligned}$$* **Proposition 1**. *We have the following results:* *(1) The subspace $(\mathbb{Q}\langle X\rangle^0,\shuffle_2)$ and $(\mathbb{Q}\langle X\rangle^1,\shuffle_2)$ are subalgebra of $(\mathbb{Q}\langle X\rangle,\shuffle_2)$, i.e. we have $$(\mathbb{Q}\langle X\rangle^0,\shuffle_2)\subset(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\subset(\mathbb{Q}\langle X\rangle,\shuffle_2)$$* *(2) We have a morphism of $\mathbb{Q}-$algebra $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_2}:(\mathbb{Q}\langle X\rangle^0,\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot) \end{aligned}$$ determined by the assignment $$y_{m_1}x^{k_1-1}y_{m_2-m_1}x^{k_2-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}\longmapsto\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)$$ i.e. we have $$\zeta^{\mathcal{C}}_{\shuffle_2}(w\shuffle_2v)=\zeta^{\mathcal{C}}_{\shuffle_2}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_2}(v)\qquad\forall w,v\in\mathbb{Q}\langle X\rangle^0$$* **Proof**. The first statement can be checked directly from the definition of the product $\shuffle_2$. The second statement follows from Corollary 4.3.$\hfill\Box$\ In the previous sections, we saw that positive multi-indices can be translated into binary sequences, hence, there is a natural map between $\mathbb{Q}\langle Y\rangle$ and $\mathbb{Q}\langle X\rangle^1$. To remedy this, we define a $\mathbb{Q}-$linear map between them. **Definition 1**. *There is a natural $\mathbb{Q}-$linear map between $\mathbb{Q}\langle Y\rangle$ and $\mathbb{Q}\langle X\rangle^1$ which defined as follows $$\begin{aligned} \varphi:\mathbb{Q}\langle Y\rangle&\longrightarrow\mathbb{Q}\langle X\rangle^1\\ y_{k_1,m_1}\cdots y_{k_r,m_r}&\longmapsto y_{m_1}x^{k_1-1}y_{m_2-m_1}x^{k_2-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1} \end{aligned}$$* Notice that, this map does not transform the the first shuffle product on $\mathbb{Q}\langle Y\rangle$ into the second shuffle product on $\mathbb{Q}\langle X\rangle^1$. Obviously, $\varphi$ is a bijection and the inverse of $\varphi$ is given by $$\begin{aligned} \varphi^{-1}:\mathbb{Q}\langle X\rangle^1&\longrightarrow\mathbb{Q}\langle Y\rangle\\ y_{l_1}x^{t_1}y_{l_2}x^{t_2}\cdots y_{l_r}x^{t_r}&\longmapsto y_{l_1,t_1+1}y_{l_1+l_2,t_2+1}\cdots y_{l_1+\cdots+l_r,t_r+1} \end{aligned}$$ furthermore, if we restrict the map $\varphi$ to $\mathbb{Q}\langle Y\rangle^0$, then $\varphi$ induced a bijection between $\mathbb{Q}\langle Y\rangle^0$ and $\mathbb{Q}\langle X\rangle^0$. **Theorem 1**. *Let $\zeta^{\mathcal{C}}_{\shuffle_1}:(\mathbb{Q}\langle Y\rangle^0,\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$ and $\zeta^{\mathcal{C}}_{\shuffle_2}:(\mathbb{Q}\langle X\rangle^0,\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$ be as before. Then we have $$\zeta^{\mathcal{C}}_{\shuffle_1}(w)=\zeta^{\mathcal{C}}_{\shuffle_2}(\varphi(w))\qquad\forall w\in\mathbb{Q}\langle Y\rangle^0$$ i.e. we have the following commutative diagram $$\begin{CD} (\mathbb{Q}\langle Y\rangle^0,\shuffle_1)@>\zeta_{\shuffle_1}^{\mathcal{C}}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)\\ @V{\varphi}VV@VV{\text{Id}}V\\ (\mathbb{Q}\langle X\rangle^0,\shuffle_2)@>\zeta_{\shuffle_2}^{\mathcal{C}} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)\end{CD}$$* **Proof**. This result comes from directly the definition of $\zeta_{\shuffle_1}^{\mathcal{C}}$ and $\zeta_{\shuffle_2}^{\mathcal{C}}$.$\hfill\Box$\ ## **Extension homomorphism** In this section, we discuss how to extend continuous multiple zeta values to non-admissible words.\ Firstly, we considering the extension of $\zeta_{\shuffle_1}^{\mathcal{C}}$. We know that a positive word $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1.\cdots,m_r)$ defines a word $$y_{(\boldsymbol{k};\boldsymbol{m})}:=y_{k_1.m_1}\cdots y_{k_r,m_r}$$ In fact, the set of positive multi-indices and the set of words in the alphabet $Y$ are bijection. We will use this bijection to identify both sets. Let $M$ be a positive real number, we defines the $\mathbb{Q}-$linear map $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_1,M}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)&\longrightarrow\mathbb{R}\\ y_{(\boldsymbol{k};\boldsymbol{m})}&\longmapsto\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m}) \end{aligned}$$ **Proposition 1**. *The $\mathbb{Q}-$linear map $\zeta^{\mathcal{C}}_{\shuffle_1,M}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)\longrightarrow\mathbb{R}$ is a $\mathbb{Q}-$algebra homomorphism, i.e. we have $$\zeta^{\mathcal{C}}_{\shuffle_1,M}(w\shuffle_1v)=\zeta^{\mathcal{C}}_{\shuffle_1,M}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_1,M}(v)\qquad\forall w,v\in\mathbb{Q}\langle Y\rangle$$* **Proof**. By theorem 4.5.$\hfill\Box$\ Next, we need some useful lemma. **Lemma 1**. *Let $$U_{r,i}(M):=\left\{(x_1,\cdots,x_r)\in\mathbb{R}^r\big|0<x_1<\cdots<x_i<M<x_{i+1}<\cdots<x_r\right\}$$ by convention, $x_0=0$. Notice that $U_{r,r}(M)=D_r(M)$. Then we have $$\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}=\frac{\ln^i(M+1)}{i!(M+1)}\qquad i=0,\cdots,r-1$$ and $$\int_{U_{r,r}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}=1-\sum_{i=0}^{r-1}\frac{\ln^i(M+1)}{i!(M+1)}$$* **Proof**. If $i=0,\cdots,r-1$, We have $$\begin{aligned} &\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=\int\limits_{0<x_1<\cdots<x_i<M<x_{i+1}<\cdots<x_r}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=\frac{1}{M+1}\int\limits_{0<x_1<\cdots<x_i<M}\frac{dx_1\cdots dx_i}{(x_1+1)\cdots(x_{i}+1)}\\ &=\frac{1}{i!(M+1)}\int\limits_{[0,M]^i}\frac{dx_1\cdots dx_i}{(x_1+1)\cdots(x_{i}+1)}\\ &=\frac{\ln^i(M+1)}{i!(M+1)} \end{aligned}$$ if $i=r$, then $$\begin{aligned} &\int_{U_{r,r}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=\int_{D_r}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}-\sum_{i=0}^{r-1}\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=1-\sum_{i=0}^{r-1}\frac{\ln^i(M+1)}{i!(M+1)} \end{aligned}$$$\hfill\Box$\ **Lemma 1**. *Let $M$ be a positive real number, then* *1. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a positive multi-index, then $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=O(\ln^r(M))\qquad M\to +\infty$$* *2. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a admissible multi-index(i.e. $k_r>1$), then $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})+O\left(\frac{\ln^{r-1}(M)}{M}\right)\qquad M\to+\infty$$ and $$\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}=O\left(\frac{\ln^i(M)}{M}\right)\qquad M\to+\infty$$ where $i=0,\cdots,r-1$* **Proof**. 1. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a positive multi-index, then $$\begin{aligned} \zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})&=\int_{D_r(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}\\ &\leqslant\int_{D_r(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_r+1)}\\ &=\frac{\ln^r(M+1)}{r!}\\ &=O(\ln^r(M))\qquad M\to+\infty \end{aligned}$$ 2\. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a admissible multi-index, we have $$\begin{aligned} &|\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})-\zeta^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})|\\ &=\int_{D_r-D_r(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}\\ &=\sum_{i=0}^{r-1}\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}\\ &\leqslant\sum_{i=0}^{r-1}\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=\sum_{i=0}^{r-1}\frac{\ln^i(M+1)}{i!(M+1)}\qquad\text{By\ Lemma\ 5.11}\\ &=O\left(\frac{\ln^{r-1}(M)}{M}\right)\qquad M\to+\infty \end{aligned}$$ if $i=0,\cdots,r-1$, we have $$\begin{aligned} &\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}\\ &\leqslant\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}\\ &=\frac{\ln^i(M+1)}{i!(M+1)}\qquad\text{By\ Lemma\ 5.11}\\ &=O\left(\frac{\ln^{i}(M)}{M}\right)\qquad M\to+\infty \end{aligned}$$ $\hfill\Box$ **Lemma 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r)$ and $(\boldsymbol{k}_1;\boldsymbol{m}_1)=$\ $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r')$ are positive multi-index, let $$\begin{aligned} (\boldsymbol{k}_2;\boldsymbol{m}_2)&=(k_1,\cdots,k_{r-1},1,2;m_1,\cdots,m_{r-1},m_r,m_r') \end{aligned}$$ then $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=\zeta_M^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)+(m_r'-m_r)\zeta^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)+O\left(\frac{\ln^r(M)}{M}\right)$$* **Proof**. we have $$\begin{aligned} &\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})-\zeta_M^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)\\ &=(m_r'-m_r)\int_{D_r(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_{r-1}+m_{r-1})^{k_{r-1}}(x_r+m_r)(x_r+m_r')}\\ &=(m_r'-m_r)\int_{D_r}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_{r-1}+m_{r-1})^{k_{r-1}}(x_r+m_r)(x_r+m_r')}\\ &-(m_r'-m_r)\sum_{i=0}^{r-1}\int_{U_{r,i}(M)}\frac{dx_1\cdots dx_r}{(x_1+m_1)^{k_1}\cdots(x_{r-1}+m_{r-1})^{k_{r-1}}(x_r+m_r)(x_r+m_r')}\\ &=(m_r'-m_r)\int_{D_{r+1}}\frac{dx_1\cdots dx_rdx_{r+1}}{(x_1+m_1)^{k_1}\cdots(x_{r-1}+m_{r-1})^{k_{r-1}}(x_r+m_r)(x_{r+1}+m_r')^2}\\ &-(m_r'-m_r)\sum_{i=0}^{r-1}O\left(\frac{\ln^i(M)}{M}\right)\qquad(\text{By\ lemma\ 5.12})\\ &=(m_r'-m_r)\zeta^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)+O\left(\frac{\ln^{r-1}(M)}{M}\right) \end{aligned}$$$\hfill\Box$ **Definition 1**. *Let $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r)$ and $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r')$ are positive multi-index, the element $$y_{k_1,m_1}\cdots y_{k_{r-1},m_{r-1}}(y_{1,m_{r}}-y_{1,m_r'})$$ is called quasi-admissible word. The subspace generated by all admissible words and all quasi-admissible words over $\mathbb{Q}$ denoted by $\widetilde{\mathbb{Q}\langle Y\rangle^0}$.* **Proposition 1**. *We have the following results:* *(1) The subspace $(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)$ is a subalgebra of $(\mathbb{Q}\langle Y\rangle,\shuffle_1)$ and $$(\mathbb{Q}\langle Y\rangle^0,\shuffle_1)\subset(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)\subset(\mathbb{Q}\langle Y\rangle,\shuffle_1)$$* *(2) We have a morphism of $\mathbb{Q}-$algebra $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_1}:(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot) \end{aligned}$$ determined by the assignment $$\begin{aligned} y_{k_1,m_1}\cdots y_{k_r,m_r}&\longmapsto\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)\\ y_{k_1,m_1}\cdots y_{k_{r-1},m_{r-1}}(y_{1,m_r}-y_{1,m_r'})&\longmapsto(m_r'-m_r)\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},1,2;m_1,\cdots,m_{r-1},m_r,m_r') \end{aligned}$$ i.e. we have $$\zeta^{\mathcal{C}}_{\shuffle_1}(w\shuffle_1v)=\zeta^{\mathcal{C}}_{\shuffle_1}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_1}(v)\qquad\forall w,v\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$$* **Proof**. The first statement can be checked directly from the definition of the product $\shuffle_1$. Next, we proof the second statement. By lemma 5.12 and lemma 5.13, we conclude that $$\zeta_{\shuffle_1,M}^{\mathcal{C}}(w)=\zeta_{\shuffle_1}^{\mathcal{C}}(w)+O\left(\frac{\ln^j(M)}{M}\right)\qquad\forall w\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$$ and we also have $$\zeta_{\shuffle_1,M}^{\mathcal{C}}(w\shuffle_1v)=\zeta_{\shuffle_1,M}^{\mathcal{C}}(w)\cdot\zeta_{\shuffle_1,M}^{\mathcal{C}}(v)$$ hence, we obtain $$\zeta^{\mathcal{C}}_{\shuffle_1}(w\shuffle_1v)=\zeta^{\mathcal{C}}_{\shuffle_1}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_1}(v)\qquad\forall w,v\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$$ $\hfill\Box$ **Theorem 1**. *The map of $(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)-$algebra $$\begin{aligned} \psi_{\shuffle_1,m}:(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)[T]&\longrightarrow(\mathbb{Q}\langle Y\rangle,\shuffle_1)\\ T&\longmapsto y_{1,m} \end{aligned}$$ is an isomorphism.* **Proof**. Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$, and $$y_{(\boldsymbol{k};\boldsymbol{m})}=y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m_{r+1}}\cdots y_{1,m_{r+s}}$$ we will proof this theorem by induction on $s$. If $s=0$, then $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a admissible multi-index, we are done. Now, we assume this conclusion for $s-1$, we considering the case $s$. We write $$\begin{aligned} y_{(\boldsymbol{k};\boldsymbol{m})}&=y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m_{r+1}}\cdots y_{1,m_{r+s}}\\ &=y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m}^s+y_{k_1,m_1}\cdots y_{k_r,m_r}(y_{1,m_{r+1}}-y_{1,m})y_{1,m}^{s-1}\\ &+y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m_{r+1}}(y_{1,m_{r+2}}-y_{1,m})y_{1,m}^{s-2}+\cdots\\ &+y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m_{r+1}}\cdots y_{1,m_{r+s-1}}(y_{1,m_{r+s}}-y_{1,m}) \end{aligned}$$ by compute, we have $$\begin{aligned} y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,m}^s=w_s\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s+w_{s-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-1}+\cdots+w_0 \end{aligned}$$ where $w_0,\cdots,w_s\in\mathbb{Q}\langle Y\rangle^0$. For $i=1,\cdots,s$, we have $$\begin{aligned} y_{k_1,m_1}\cdots y_{k_r,m_r}y_{1,r+1}\cdots y_{1,r+i-1}(y_{1,r+i}-y_{1,m})y_{1,m}^{s-i}&=w^{(i)}_{s-i}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-i}\\ &+w^{(i)}_{s-i-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-i-1}+\cdots+w^{(i)}_0 \end{aligned}$$ where $w^{(i)}_0,\cdots,w^{(i)}_{s-i}\in\widetilde{\mathbb{Q}\langle Y\rangle}$. Hence ,we conclude that $$y_{(\boldsymbol{k};\boldsymbol{m})}=v_s\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s+v_{s-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-1}+\cdots+v_0$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle Y\rangle}$. $\hfill\Box$\ Using the previous results, we describe one of the most important theorem. **Theorem 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$ be a positive multi-index, and $m\in\mathbb{Z}_{>0}$, then $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=a_s(\zeta_M^{\mathcal{C}}(1;m))^s+a_{s-1}(\zeta_M^{\mathcal{C}}(1;m))^{s-1}+\cdots+a_0+O\left(\frac{\ln^{r+s-1}(M)}{M}\right)$$ where $a_0,\cdots,a_s\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$.* **Proof**. By theorem 5.16, we have $$y_{(\boldsymbol{k};\boldsymbol{m})}=v_s\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s+v_{s-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-1}+\cdots+v_0$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle Y\rangle}$. hence, we conclude that $$\begin{aligned} \zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})&=\zeta^{\mathcal{C}}_{\shuffle_1,M}(y_{(\boldsymbol{k};\boldsymbol{m})})\\ &=\sum_{i=0}^s\zeta^{\mathcal{C}}_{\shuffle_1,M}(v_i\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^i)\\ &=\sum_{i=0}^s\zeta^{\mathcal{C}}_{\shuffle_1,M}(v_i)\cdot(\zeta^{\mathcal{C}}_{\shuffle_1,M}(y_{1,m}))^i\\ &=\sum_{i=0}^s\left(a_i+O\left(\frac{\ln^{r+s-1}(M)}{M}\right)\right)\cdot(\zeta^{\mathcal{C}}_{M}(1;m))^i\\ &=\sum_{i=0}^sa_i\cdot(\zeta^{\mathcal{C}}_{M}(1;m))^i+O\left(\frac{\ln^{r+s-1}(M)}{M}\right) \end{aligned}$$ $\hfill\Box$ **Example 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,1,1;m_1,m_2,m_3)$ be a positive multi-index and $k_1>1$, $m\in\mathbb{Z}_{>0}$, then we have $$\begin{aligned} y_{k_1,m_1}y_{1,m_2}y_{1,m_3}&=y_{k_1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m})+y_{k_1,m_1}(y_{1,m_2}-y_{1,m})y_{1,1}+y_{k_1,m_1}y_{1,m}^2\\ &=\frac{1}{2}y_{k_1,m_1}\shuffle_1 y_{1,m}\shuffle_1 y_{1,m}+(y_{k_1,m_1}(y_{1,m_2}-y_{1,m})-y_{1,m}y_{k_1,m_1})\shuffle_1 y_{1,m}\\ &+y_{k_1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m})-y_{1,m}y_{k_1,m_1}(y_{1,m_2}-y_{1,m})\\ &-y_{k_1,m_1}y_{1,m}(y_{1,m_2}-y_{1,m})+y_{1,m}^2y_{k_1,m_1} \end{aligned}$$ hence $$\begin{aligned} \zeta_M^{\mathcal{C}}(k_1,1,1;m_1,m_2,m_3)&=\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}y_{1,m_2}y_{1,m_3})\\ &=\frac{1}{2}\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}\shuffle_1 y_{1,m}\shuffle_1 y_{1,m})\\ &\quad+\zeta_{\shuffle_1,M}^{\mathcal{C}}((y_{k_1,m_1}(y_{1,m_2}-y_{1,m})-y_{1,m}y_{k_1,m_1})\shuffle_1 y_{1,m})\\ &\quad+\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m}))-\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m}y_{k_1,m_1}(y_{1,m_2}-y_{1,m}))\\ &\quad-\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}y_{1,m}(y_{1,m_2}-y_{1,m}))+\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m}^2y_{k_1,m_1})\\ &=\frac{1}{2}\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1})\cdot(\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m}))^2\\ &\quad+\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}(y_{1,m_2}-y_{1,m})-y_{1,m}y_{k_1,m_1})\cdot\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m})\\ &\quad+\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m}))-\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m}y_{k_1,m_1}(y_{1,m_2}-y_{1,m}))\\ &\quad-\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{k_1,m_1}y_{1,m}(y_{1,m_2}-y_{1,m}))+\zeta_{\shuffle_1,M}^{\mathcal{C}}(y_{1,m}^2y_{k_1,m_1})\\ &=\frac{\zeta^{\mathcal{C}}(k_1;m_1)}{2}\cdot(\zeta_M^{\mathcal{C}}(1;m))^2\\ &\quad+((m-m_2)\zeta^{\mathcal{C}}(k_1,1,2;m_1,m_2,m)-\zeta^{\mathcal{C}}(1,k_1;m,m_1))\cdot\zeta^{\mathcal{C}}_M(1;m)\\ &\quad+(m-m_3)\zeta^{\mathcal{C}}(k_1,1,1,2;m_1,m_2,m_3,m)\\ &\quad-(m-m_2)\zeta^{\mathcal{C}}(1,k_1,1,2;m,m_1,m_2,m)\\ &\quad-(m-m_2)\zeta^{\mathcal{C}}(k_1,1,1,2;m_1,m,m_2,m)+\zeta^{\mathcal{C}}(1,1,k_1;m,m,m_1)\\ &\quad+O\left(\frac{\ln^2(M)}{M}\right) \end{aligned}$$* The theorem 5.17 is very important for us to define the extension of $\zeta^{\mathcal{C}}_{\shuffle_1}$. **Definition 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$ be a positive multi-index, and $m\in\mathbb{Z}_{>0}$, then by the theorem 5.17, we have $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=a_s(\zeta_M^{\mathcal{C}}(1;m))^s+a_{s-1}(\zeta_M^{\mathcal{C}}(1;m))^{s-1}+\cdots+a_0+O\left(\frac{\ln^{r+s-1}(M)}{M}\right)$$ where $a_0,\cdots,a_s\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$. We define **The first extending homomrphism** as follows $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_1,m}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)&\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ y_{(\boldsymbol{k};\boldsymbol{m})}&\longmapsto\sum_{i=0}^sa_iT^i \end{aligned}$$* **Theorem 1**. *The map $\zeta^{\mathcal{C},T}_{\shuffle_1,m}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$ is a $\mathbb{Q}-$algebra homomorphism.* **Proof**. By the definition of $\zeta^{\mathcal{C}}_{\shuffle_1,M}$, we know that $$\zeta^{\mathcal{C}}_{\shuffle_1,M}(w\shuffle_1v)=\zeta^{\mathcal{C}}_{\shuffle_1,M}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_1,M}(v)\qquad\forall w,v\in\mathbb{Q}\langle Y\rangle$$ by the definition of $\zeta_{\shuffle_1,m}^{\mathcal{C},T}$, we have $$\zeta^{\mathcal{C}}_{\shuffle_1,M}(w)=\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w)+O\left(\frac{\ln^r(M)}{M}\right)$$ where $T=\zeta^{\mathcal{C}}_{M}(1;m)$. Then we have $$\zeta^{\mathcal{C},T}_{\shuffle_1,m}(w\shuffle_1v)=\zeta^{\mathcal{C},T}_{\shuffle_1,m}(w)\cdot\zeta^{\mathcal{C},T}_{\shuffle_1,m}(v)\qquad\forall w,v\in\mathbb{Q}\langle Y\rangle$$ $\hfill\Box$ Furthermore, The map $\zeta^{\mathcal{C},T}_{\shuffle_1,m}$ is characterized by the following conditions, that is we have **Theorem 1**. *For any given $m\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{Q}-$linear map $$\zeta_{\shuffle_1,m}^{\mathcal{C},T}:(\mathbb{Q}\langle Y\rangle,\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ satisfy $$\begin{cases} \zeta_{\shuffle_1,m}^{\mathcal{C},T}(w)=\zeta_{\shuffle_1}^{\mathcal{C}}(w)\qquad&\forall w\in\widetilde{\mathbb{Q}\langle Y\rangle^0}\\ \zeta_{\shuffle_1,m}^{\mathcal{C},T}(y_{1,m})=T\\ \zeta_{\shuffle_1,m}^{\mathcal{C},T}(v\shuffle_1w)=\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v)\cdot\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w)&\forall v,w\in\mathbb{Q}\langle Y\rangle \end{cases}$$* **Proof**. The existence of $\zeta_{\shuffle_1,m}^{\mathcal{C},T}$ is the conclusion of theorem 5.20. Now, we only need to proof the uniqueness. By the proof of theorem 5.16, we have $$w=v_s\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s+v_{s-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-1}+\cdots+v_0$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$. If $f$ is another map satisfy the above properties, then we have $$f(w)=\sum_{i=0}^sf(v_i)\cdot(f(y_{1,m}))^i=\sum_{i=0}^sf(v_i)\cdot T^i$$ notice that, $v_i$ is the $\mathbb{Q}-$linear combination of admissible words and quasi-admissible words, hence $$f(v_i)=\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v_i)$$ $\hfill\Box$ **Example 1**. *$$\begin{aligned} \zeta_{\shuffle_1,m}^{\mathcal{C},T}(y_{k_1,m_1}y_{1,m_2}y_{1,m_3})&=\frac{\zeta^{\mathcal{C}}(k_1;m_1)}{2}\cdot T^2+((m-m_2)\zeta^{\mathcal{C}}(k_1,1,2;m_1,m_2,m)-\zeta^{\mathcal{C}}(1,k_1;m,m_1))\cdot T\\ &\quad+(m-m_3)\zeta^{\mathcal{C}}(k_1,1,1,2;m_1,m_2,m_3,m)\\ &\quad-(m-m_2)\zeta^{\mathcal{C}}(1,k_1,1,2;m,m_1,m_2,m)\\ &\quad-(m-m_2)\zeta^{\mathcal{C}}(k_1,1,1,2;m_1,m,m_2,m)+\zeta^{\mathcal{C}}(1,1,k_1;m,m,m_1) \end{aligned}$$* Next, we considering the extension of $\zeta_{\shuffle_2}^{\mathcal{C}}$. We know that a positive multi-index $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1.\cdots,m_r)$ defines a word $$v_{(\boldsymbol{k};\boldsymbol{m})}:=y_{m_1}x^{k_1-1}y_{m_2-m_1}x^{k_2-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}$$ In fact, the set of positive multi-indices and the set of words in $\mathbb{Q}\langle X\rangle^1$ are bijection. We will use this bijection to identify both sets. Let $0<t<1$, we defines the $\mathbb{Q}-$linear map $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_2,t}:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)&\longrightarrow\mathbb{R}\\ y_{m_1}x^{k_1-1}y_{m_2-m_1}x^{k_2-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}&\longmapsto\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})} \end{aligned}$$ **Lemma 1**. *Let $0<t<1$, we have the following result* *1. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a positive multi-index, then $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=O(\ln^r(1/(1-t)))\qquad t\to 1^-$$* *2. If $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a admissible multi-index, then $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=\int_{\Delta^{\text{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}+O((1-t)\ln^{r}(1/(1-t)))\qquad t\to 1^-$$* **Proof**. The first statement. By theorem 4.11, we have $$\begin{aligned} \int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}&=t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)\\ &=\int_{D_r}\frac{t^{x_r+m_r}}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdots dx_r\\ &\leqslant\int_{D_r}\frac{t^{x_r+1}}{(x_1+1)\cdots(x_r+1)}dx_1\cdots dx_r\\ &=\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{-\ln(t_r)}\frac{dt_{r-1}}{-t_{r-1}\ln(t_{r-1})}\frac{dt_{r-2}}{-t_{r-2}\ln(t_{r-2})}\cdots\frac{dt_{1}}{-t_{1}\ln(t_{1})}\\ &<\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{1-t_r}\frac{dt_{r-1}}{t_{r-1}(1-t_{r-1})}\frac{dt_{r-2}}{t_{r-2}(1-t_{r-2})}\cdots\frac{dt_{1}}{t_{1}(1-t_1)}\\ &=\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{1-t_r}\left(\frac{dt_{r-1}}{t_{r-1}}+\frac{dt_{r-1}}{1-t_{r-1}}\right)\left(\frac{dt_{r-2}}{t_{r-2}}+\frac{dt_{r-2}}{1-t_{r-1}}\right)\cdots\left(\frac{dt_{1}}{t_{1}}+\frac{dt_{1}}{1-t_{1}}\right)\\ &=\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{1-t_r}\frac{dt_{r-1}}{1-t_{r-1}}\cdots\frac{dt_{1}}{1-t_{1}}\\ &+\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{1-t_r}\frac{dt_{r-1}}{t_{r-1}}\cdots\frac{dt_{1}}{1-t_{1}}\\ &+\cdots\\ &+\int\limits_{0<t_r<\cdots<t_1<t}\frac{dt_r}{1-t_r}\frac{dt_{r-1}}{t_{r-1}}\cdots\frac{dt_{1}}{t_{1}}\\ &=\frac{\ln^r(1/(1-t))}{r!}+O(\ln^{r-1}(1/(1-t)))+\cdots+O(1)\\ &=O(\ln^r(1/(1-t))) \end{aligned}$$ The second statement. For $0<t<1$ and $x_r>0$, we have the inequalities $$0<1-t^{x_r+m_r}<1\qquad 0<1-t^{x_r+m_r}<(x_r+m_r)(1-t)$$ therefore $$\begin{aligned} 0&<\int_{\Delta^{\text{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k};\boldsymbol{m})}-\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}\\ &=\int_{D_r}\frac{1-t^{x_r+m_r}}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdots dx_r\\ &=\sum_{i=0}^r\int_{U_{r,i}\left(\frac{1}{1-t}-1\right)}\frac{1-t^{x_r+m_r}}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdots dx_r\\ &\leqslant\int_{U_{r,r}\left(\frac{1}{1-t}-1\right)}\frac{(x_r+m_r)(1-t)}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdot dx_r\\ &+\sum_{i=0}^{r-1}\int_{U_{r,i}\left(\frac{1}{1-t}-1\right)}\frac{1}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdots dx_r\\ &\leqslant\int_{U_{r,r}\left(\frac{1}{1-t}-1\right)}\frac{1-t}{(x_1+1)\cdots(x_r+1)}dx_1\cdot dx_r\\ &+\sum_{i=0}^{r-1}\int_{U_{r,i}\left(\frac{1}{1-t}-1\right)}\frac{1}{(x_1+1)\cdots(x_{r-1}+1)(x_r+1)^2}dx_1\cdots dx_r\\ &=\frac{(1-t)\ln^r(1/(1-t))}{r!}+\sum_{i=0}^{r-1}\frac{(1-t)\ln^i(1/(1-t))}{i!}\\ &=\sum_{i=0}^{r}\frac{(1-t)\ln^i(1/(1-t))}{i!}\\ &=O((1-t)\ln^r(1/(1-t)))\qquad t\to 1^- \end{aligned}$$$\hfill\Box$ **Lemma 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r)$ and $(\boldsymbol{k}_1;\boldsymbol{m}_1)=$\ $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r')$ are positive multi-index, let $$\begin{aligned} (\boldsymbol{k}_2;\boldsymbol{m}_2)&=(k_1,\cdots,k_{r-1},1,2;m_1,\cdots,m_{r-1},m_r,m_r') \end{aligned}$$ then, we have $$\begin{aligned} \int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=\int_{\Delta^{\text{wt}(\boldsymbol{k}_1)}(t)}\omega_{(\boldsymbol{k}_1;\boldsymbol{m}_1)}+(m_r'-m_r)\int_{\Delta^{\text{wt}(\boldsymbol{k}_2)}}\omega_{(\boldsymbol{k}_2;\boldsymbol{m}_2)}+O((1-t)\ln^r(1-t)) \end{aligned}$$* **Proof**. We consider the equality $$\begin{aligned} &\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}-\int_{\Delta^{\text{wt}(\boldsymbol{k}_1)}(t)}\omega_{(\boldsymbol{k}_1;\boldsymbol{m}_1)}\\ &=t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-t^{m_r'}\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &=t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)\\ &+\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &+\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)-t^{m_r'}\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &=(1-t^{-m_r})\cdot t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)+(t^{-m_r'}-1)\cdot t^{m_r'}\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &+\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &=(1-t^{-m_r})\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}+(t^{m_r'}-1)\int_{\Delta^{\textbf{wt}(\boldsymbol{k}_1)}(t)}\omega_{(\boldsymbol{k}_1;\boldsymbol{m}_1)}\\ &+\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &=O((1-t)\ln^r(1/(1-t)))+\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)-\text{Li}^{\mathcal{C}}(\boldsymbol{k}_1;\boldsymbol{m}_1)(t)\\ &=O((1-t)\ln^r(1/(1-t)))+(m_r'-m_r)\text{Li}^{\mathcal{C}}(\boldsymbol{k}_2;\boldsymbol{m}_2)(1)+O((1-t)\ln^r(1/(1-t)))\\ &=(m_r'-m_r)\int_{\Delta^{\textbf{wt}(\boldsymbol{k})}}\omega_{(\boldsymbol{k}_2;\boldsymbol{m}_2)}+O((1-t)\ln^r(1/(1-t))) \end{aligned}$$$\hfill\Box$\ **Definition 1**. *Let $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r)$ and $(k_1,\cdots,k_{r-1},1;m_1,\cdots,m_{r-1},m_r')$, are positive multi-index, the element $$y_{m_1}x^{k_1-1}\cdots y_{m_{r-1}-m_{r-2}}x^{k_{r-1}-1}(y_{m_{r}-m_{r-1}}-y_{m_r'-m_{r-1}})$$ is called quasi-admissible word. The subspace generated by all admissible words and all quasi-admissible words over $\mathbb{Q}$ denoted by $\widetilde{\mathbb{Q}\langle X\rangle^0}$.* **Proposition 1**. *We have the following results:* *(1) The subspace $(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)$ is a subalgebra of $(\mathbb{Q}\langle X\rangle^1,\shuffle_2)$ and $$(\mathbb{Q}\langle X\rangle^0,\shuffle_2)\subset(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)\subset(\mathbb{Q}\langle X\rangle^1,\shuffle_2)$$* *(2) We have a morphism of $\mathbb{Q}-$algebra $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_2}:(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot) \end{aligned}$$ determined by the assignment $$y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}\\ \longmapsto\zeta^{\mathcal{C}}(k_1,\cdots,k_r;m_1,\cdots,m_r)\\$$ $$\begin{aligned} &y_{m_1}x^{k_1-1}\cdots y_{m_{r-1}-m_{r-2}}x^{k_{r-1}-1}(y_{m_{r}-m_{r-1}}-y_{m_r'-m_{r-1}})\\ &\longmapsto(m_r'-m_r)\zeta^{\mathcal{C}}(k_1,\cdots,k_{r-1},1,2;m_1,\cdots,m_{r-1},m_r,m_r') \end{aligned}$$ i.e. we have $$\zeta^{\mathcal{C}}_{\shuffle_2}(w\shuffle_2v)=\zeta^{\mathcal{C}}_{\shuffle_2}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_2}(v)\qquad\forall w,v\in\widetilde{\mathbb{Q}\langle X\rangle^0}$$* **Proof**. The first statement can be checked directly from the definition of the product $\shuffle_2$. Next, we proof the second statement. By lemma 5.23 and lemma 5.24, we conclude that $$\zeta_{\shuffle_2,t}^{\mathcal{C}}(w)=\zeta_{\shuffle_2}^{\mathcal{C}}(w)+O((1-t)\ln^j(1/(1-t)))\qquad\forall w\in\widetilde{\mathbb{Q}\langle X\rangle^0}$$ and we also have $$\zeta_{\shuffle_2,t}^{\mathcal{C}}(w\shuffle_2v)=\zeta_{\shuffle_2,t}^{\mathcal{C}}(w)\cdot\zeta_{\shuffle_2,t}^{\mathcal{C}}(v)$$ hence, we obtain $$\zeta^{\mathcal{C}}_{\shuffle_2}(w\shuffle_2v)=\zeta^{\mathcal{C}}_{\shuffle_2}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_2}(v)\qquad\forall w,v\in\widetilde{\mathbb{Q}\langle X\rangle^0}$$ $\hfill\Box$ **Theorem 1**. *The map of $(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)-$algebra $$\begin{aligned} \psi_{\shuffle_2,m}:(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)[T]&\longrightarrow(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\\ T&\longmapsto y_{m} \end{aligned}$$ is an isomorphism.* **Proof**. Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$, and $$v_{(\boldsymbol{k};\boldsymbol{m})}=y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_{m_{r+1}-m_r}\cdots y_{m_{r+s}-m_{r+s-1}}$$ we will proof this theorem by induction on $s$. If $s=0$, then $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r;m_1,\cdots,m_r)$ be a admissible multi-index, we are done. Now, we assume this conclusion for $s-1$, we considering the case $s$. We write $$\begin{aligned} v_{(\boldsymbol{k};\boldsymbol{m})}&=y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_{m_{r+1}-m_r}\cdots y_{m_{r+s}-m_{r+s-1}}\\ &=y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_m^s\\ &+y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}(y_{m_{r+1}-m_r}-y_m)y_m^{s-1}\\ &+\cdots\\ &+y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_{m_{r+1}-m_r}\cdots y_{m_{r+s-1}-m_{r+s-2}}(y_{m_{r+s}-m_{r+s-1}}-y_m) \end{aligned}$$ by compute, we have $$\begin{aligned} y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_m^s=w_s\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^s+w_{s-1}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-1}+\cdots+w_0 \end{aligned}$$ where $w_0,\cdots,w_0\in\mathbb{Q}\langle X\rangle^0$. For $i=1,\cdots,s$, we have $$\begin{aligned} &y_{m_1}x^{k_1-1}\cdots y_{m_r-m_{r-1}}x^{k_r-1}y_{m_{r+1}-m_r}\cdots y_{m_{r+i-1}-m_{r+i-2}}(y_{m_{r+i}-m_{r+i-1}}-y_m)y_m^{s-i}\\ &=w^{(i)}_{s-i}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-i}+w^{(i)}_{s-i-1}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-i-1}+\cdots+w^{(i)}_0 \end{aligned}$$ where $w^{(i)}_0,\cdots,w^{(i)}_{s-i}\in\widetilde{\mathbb{Q}\langle X\rangle^0}$. Hence ,we conclude that $$v_{(\boldsymbol{k};\boldsymbol{m})}=u_s\shuffle_1\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^s+u_{s-1}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-1}+\cdots+u_0$$ where $u_0,\cdots,u_s\in\widetilde{\mathbb{Q}\langle X\rangle^0}$. $\hfill\Box$\ Using the previous results, we describe one of the most important theorem. **Theorem 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$, and $m\in\mathbb{Z}_{>0}$, then $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=a_s\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^s+a_{s-1}\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^{s-1}+\cdots+a_0+O\left((1-t)\ln^{r+s}(1/(1-t))\right)$$ where $a_0,\cdots,a_s\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$.* **Proof**. By the theorem 5.27, we conclude that $$v_{(\boldsymbol{k};\boldsymbol{m})}=u_s\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^s+u_{s-1}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-1}+\cdots+u_0$$ where $u_0,\cdots,u_s\in\widetilde{\mathbb{Q}\langle X\rangle^0}$. then we have $$\begin{aligned} \int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}&=\zeta^{\mathcal{C}}_{\shuffle_2,t}(v_{(\boldsymbol{k};\boldsymbol{m})})\\ &=\sum_{i=0}^s\zeta^{\mathcal{C}}_{\shuffle_2,t}(u_i\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^i)\\ &=\sum_{i=0}^s\zeta^{\mathcal{C}}_{\shuffle_2,t}(u_i)\cdot(\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_m))^i\\ &=\sum_{i=0}^s\left(a_i+O\left((1-t)\ln^{r+s}(1/(1-t))\right)\right)\cdot\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^i\\ &=\sum_{i=0}^sa_i\cdot\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^i+O\left((1-t)\ln^{r+s}(1/(1-t))\right) \end{aligned}$$ $\hfill\Box$ **Example 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(2,1;m_1,m_2)$ be a positive multi-index, $m\in\mathbb{Z}_{>0}$, then we have $$\begin{aligned} v_{(\boldsymbol{k};\boldsymbol{m})}&=v_{(2,1;m_1,m_2)}\\ &=y_{m_1}xy_{m_2-m_1}\\ &=y_{m_1}xy_m+y_{m_1}x(y_{m_2-m_1}-y_m)\\ &=y_{m_1}x\shuffle_2y_m+y_{m_1}x(y_{m_2-m_1}-y_m)-y_my_{m_1}x-y_{m_1}y_mx \end{aligned}$$ hence $$\begin{aligned} \zeta^{\mathcal{C}}_{\shuffle_2,t}(v_{(\boldsymbol{k};\boldsymbol{m})})&=\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}xy_{m_2-m_1})\\ &=\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}x\shuffle_2y_m)+\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}x(y_{m_2-m_1}-y_m))-\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_my_{m_1}x)-\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}y_mx)\\ &=\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}x)\cdot\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_m)+\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}x(y_{m_2-m_1}-y_m))-\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_my_{m_1}x)-\zeta^{\mathcal{C}}_{\shuffle_2,t}(y_{m_1}y_mx)\\ &=\zeta^{\mathcal{C}}(2;m_1)\cdot\int_{\Delta^1(t)}\omega_{(1;m)}+(m+m_1-m_2)\zeta^{\mathcal{C}}(2,1,2;m_1,m_2,m+m_1)\\ &-\zeta^{\mathcal{C}}(1,2;m,m+m_1)-\zeta^{\mathcal{C}}(1,2;m_1,m_1+m)+O((1-t)\ln^2(1/(1-t))) \end{aligned}$$* The theorem 5.28 is very important for us to define the second extending homomorphism. **Definition 1**. *Let $(\boldsymbol{k};\boldsymbol{m})=(k_1,\cdots,k_r,1,\cdots,1;m_1,\cdots,m_r,m_{r+1},\cdots,m_{r+s})$ be a positive multi-index, and $m\in\mathbb{Z}_{>0}$, then by the theorem 5.28, we have $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=a_s\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^s+a_{s-1}\left(\int_{\Delta^1(t)}\omega_{(1;m)}\right)^{s-1}+\cdots+a_0+O\left((1-t)\ln^{r+s}(1/(1-t))\right)$$ where $a_0,\cdots,a_s\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$. We define **The second extending homomrphism** as follows $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_2,m}:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)&\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ v_{(\boldsymbol{k};\boldsymbol{m})}&\longmapsto\sum_{i=0}^sa_iT^i \end{aligned}$$* **Theorem 1**. *The map $\zeta^{\mathcal{C},T}_{\shuffle_2,m}:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$ is a $\mathbb{Q}-$algebra homomorphism.* **Proof**. By the definition of $\zeta^{\mathcal{C}}_{\shuffle_2,t}$, we know that $$\zeta^{\mathcal{C}}_{\shuffle_2,t}(w\shuffle_2v)=\zeta^{\mathcal{C}}_{\shuffle_2,t}(w)\cdot\zeta^{\mathcal{C}}_{\shuffle_2,t}(v)$$ by the definition of $\zeta_{\shuffle_2,m}^{\mathcal{C},T}$, we have $$\zeta^{\mathcal{C}}_{\shuffle_2,t}(w)=\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w)+O((1-t)\ln^j(1/(1-t)))$$ where $T=\int_{\Delta^1(t)}\omega_{(1;m)}$. Then we have $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_2,m}(w\shuffle_2v)=\zeta^{\mathcal{C},T}_{\shuffle_2,m}(w)\cdot\zeta^{\mathcal{C},T}_{\shuffle_2,m}(v) \end{aligned}$$ $\hfill\Box$\ Furthermore, The map $\zeta^{\mathcal{C},T}_{\shuffle_1,m}$ is characterized by the following conditions, that is we have **Theorem 1**. *For any given $m\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{Q}-$linear map $$\zeta_{\shuffle_2,m}^{\mathcal{C},T}:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ satisfy $$\begin{cases} \zeta_{\shuffle_2,m}^{\mathcal{C},T}(w)=\zeta_{\shuffle_2}^{\mathcal{C}}(w)\qquad&\forall w\in\widetilde{\mathbb{Q}\langle X\rangle^0}\\ \zeta_{\shuffle_2,m}^{\mathcal{C},T}(y_m)=T\\ \zeta_{\shuffle_2,m}^{\mathcal{C},T}(v\shuffle_2w)=\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v)\cdot\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w)&\forall v,w\in\mathbb{Q}\langle X\rangle^1 \end{cases}$$* **Proof**. The existence of $\zeta_{\shuffle_2,m}^{\mathcal{C},T}$ is the conclusion of theorem 5.31. Now, we only need to proof the uniqueness. By the theorem 5.27, we have $$v_{(\boldsymbol{k};\boldsymbol{m})}=v_s\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^s+v_{s-1}\shuffle_2\overbrace{y_m\shuffle_2\cdots\shuffle_2y_m}^{s-1}+\cdots+v_0$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle X\rangle^0}$. If $f$ is another map satisfy the above properties, then we have $$f(v_{(\boldsymbol{k};\boldsymbol{m})})=\sum_{i=0}^sf(v_i)\cdot(f(y_m))^i=\sum_{i=0}^sf(v_i)\cdot T^i$$ notice that, $v_i$ is the $\mathbb{Q}-$linear combination of admissible words and quasi-admissible words, hence $$f(v_i)=\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v_i)$$ **Example 1**. *$$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_2,m}(y_{m_1}xy_{m_2-m_1}) &=\zeta^{\mathcal{C}}(2;m_1)\cdot T+(m+m_1-m_2)\zeta^{\mathcal{C}}(2,1,2;m_1,m_2,m+m_1)\\ &-\zeta^{\mathcal{C}}(1,2;m,m+m_1)-\zeta^{\mathcal{C}}(1,2;m_1,m_1+m) \end{aligned}$$ * Notice that, the $\mathbb{Q}-$linear map $\varphi:\mathbb{Q}\langle Y\rangle\longrightarrow\mathbb{Q}\langle X\rangle^1$ also induced a bijection $$\varphi:\widetilde{\mathbb{Q}\langle Y\rangle^0}\longrightarrow\widetilde{\mathbb{Q}\langle X\rangle^0}$$ that is, **Proposition 1**. *We have the following commutative diagram $$\begin{CD} (\mathbb{Q}\langle Y\rangle^0,\shuffle_1)@>>>(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)@>>>(\mathbb{Q}\langle Y\rangle,\shuffle_1)\\ @V{\varphi}VV@V{\varphi}VV@V{\varphi}VV\\ (\mathbb{Q}\langle X\rangle^0,\shuffle_2)@>>>(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)@>>>(\mathbb{Q}\langle X\rangle^1,\shuffle_2) \end{CD}$$* **Proof**. One can check this result by the definition of $\varphi$.\ If we combine the map $\varphi, \zeta_{\shuffle_1}^{\mathcal{C}}$ and $\zeta_{\shuffle_2}^{\mathcal{C}}$, then we have **Theorem 1**. *Let $\zeta^{\mathcal{C}}_{\shuffle_1}:(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$ and $\zeta^{\mathcal{C}}_{\shuffle_2}:(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$ be as before. Then we have $$\zeta^{\mathcal{C}}_{\shuffle_1}(w)=\zeta^{\mathcal{C}}_{\shuffle_2}(\varphi(w))\qquad\forall w\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$$ i.e. we have the following commutative diagram $$\begin{CD} (\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)@>\zeta_{\shuffle_1}^{\mathcal{C}}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)\\ @V{\varphi}VV@VV{\text{Id}}V\\ (\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)@>\zeta_{\shuffle_2}^{\mathcal{C}} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)\end{CD}$$* **Proof**. This result comes from directly the definition of $\zeta_{\shuffle_1}^{\mathcal{C}}$ and $\zeta_{\shuffle_2}^{\mathcal{C}}$.$\hfill\Box$\ ## **The comparison theorem** In the previous sections, we have put a lot of effort into constructing the first extend homomorphism $\zeta_{\shuffle_1,m}^{\mathcal{C},T}$ and the second extend homomorphism $\zeta_{\shuffle_2,m}^{\mathcal{C},T}$. As we just saw in the previous example, the extend homomorphism $\zeta_{\shuffle_1,m}^{\mathcal{C},T}$ and $\zeta_{\shuffle_2,m}^{\mathcal{C},T}$ are in general different from each other. In this section, we will discuss how to compare them. **Theorem 1**. *For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\begin{aligned} \rho^{\shuffle_1}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$, called **the first comparison map** satisfy $$\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_1}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\text{Id}}VV@VV{\rho_{m,m'}^{\shuffle_1}}V\\ (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$\ * **Proof**. Firstly, we proof the existence of $\rho_{m,m'}^{\shuffle_1}$. We construct the $\mathbb{R}-$linear map $$\begin{aligned} \rho_{m,m'}^{\shuffle_1}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s\\ \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$. Next, we will proof that the map $\rho_{m,m'}^{\shuffle_1}$ satisfy $$\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_1}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ Let $w\in\mathbb{Q}\langle Y\rangle$ be any word, by the theorem 5.16, we write $$\begin{aligned} w=v_s\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s+v_{s-1}\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^{s-1}+\cdots+v_0 \end{aligned}$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle Y\rangle^0}$, then we have $$\begin{aligned} \rho_{m,m'}^{\shuffle_1}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))&=\rho_{m,m'}^{\shuffle_1}\left(\sum_{i=0}^s\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(y_{1,m}))^i\right)\\ &=\rho_{m,m'}^{\shuffle_1}\left(\sum_{i=0}^s\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v_i)\cdot T^i\right)\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m}^{\mathcal{C},T}(v_i)\cdot\rho_{m,m'}^{\shuffle_1}(T^i)\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(v_i)\cdot(T+\ln(m'/m))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m'})+\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m}-y_{1,m'}))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m}))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(v_i)\cdot\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^i)\\ &=\sum_{i=0}^s\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(v_i\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^i)\\ &=\zeta_{\shuffle_1,m'}^{\mathcal{C},T}\left(\sum_{i=0}^sv_i\shuffle_1\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^i\right)\\ &=\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w) \end{aligned}$$ Next, we proof the uniqueness of $\rho_{m,m'}^{\shuffle_1}$. If there exists another $\mathbb{R}-$linear map $f:\mathbb{R}[T]\longrightarrow\mathbb{R}[T]$ satisfy $$\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w)=f(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ especially, we take $$w=\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s$$ then, we have $$\begin{aligned} f(T^s)&=f((\zeta_{\shuffle_1,m}^{\mathcal{C},T}(y_{1,m}))^s)\\ &=f(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s))\\ &=f(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\\ &=\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(w)\\ &=\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s)\\ &=(\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m}))^s\\ &=(\zeta_{\shuffle_1,m'}^{\mathcal{C},T}((y_{1,m}-y_{1,m'})+y_{1,m'}))^s\\ &=(\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m}-y_{1,m'})+\zeta_{\shuffle_1,m'}^{\mathcal{C},T}(y_{1,m'}))^s\\ &=(T+\ln(m'/m))^s\\ &=\rho_{m,m'}^{\shuffle_1}(T^s) \end{aligned}$$$\hfill\Box$\ Next, we will compare the map $\zeta_{\shuffle_2,m}^{\mathcal{C},T}$ and $\zeta_{\shuffle_2,m'}^{\mathcal{C},T}$. We have\ **Theorem 1**. *For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\begin{aligned} \rho^{\shuffle_2}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$, called **the second comparison map** satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_2}(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\text{Id}}VV@VV{\rho_{m,m'}^{\shuffle_2}}V\\ (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$\ * **Proof**. Firstly, we proof the existence of $\rho_{m,m'}^{\shuffle_2}$. We construct the $\mathbb{R}-$linear map $$\begin{aligned} \rho_{m,m'}^{\shuffle_2}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+\ln(m'/m))^s\\ \end{aligned}$$ where $s\in\mathbb{Z}_{\geqslant0}$. Next, we will proof that the map $\rho_{m,m'}^{\shuffle_2}$ satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w)=\rho_{m,m'}^{\shuffle_2}(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle X\rangle^1$$ Let $w\in\mathbb{Q}\langle X\rangle^1$ be any word, by the theorem 5.27, we write $$\begin{aligned} w=v_s\shuffle_2\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^s+v_{s-1}\shuffle_2\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^{s-1}+\cdots+v_0 \end{aligned}$$ where $v_0,\cdots,v_s\in\widetilde{\mathbb{Q}\langle X\rangle^0}$, then we have $$\begin{aligned} \rho_{m,m'}^{\shuffle_2}(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))&=\rho_{m,m'}^{\shuffle_2}\left(\sum_{i=0}^s\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(y_m))^i\right)\\ &=\rho_{m,m'}^{\shuffle_2}\left(\sum_{i=0}^s\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v_i)\cdot T^i\right)\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m}^{\mathcal{C},T}(v_i)\cdot\rho_{m,m'}^{\shuffle_2}(T^i)\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(v_i)\cdot(T+\ln(m'/m))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m'})+\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_m-y_{m'}))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(v_i)\cdot(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m}))^i\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(v_i)\cdot\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^i)\\ &=\sum_{i=0}^s\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(v_i\shuffle_2\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^i)\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}\left(\sum_{i=0}^sv_i\shuffle_2\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^i\right)\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w) \end{aligned}$$ Next, we proof the uniqueness of $\rho_{m,m'}^{\shuffle_2}$. If there exists another $\mathbb{R}-$linear map $f:\mathbb{R}[T]\longrightarrow\mathbb{R}[T]$ satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w)=f(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle X\rangle^1$$ especially, we take $$w=\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^s$$ then, we have $$\begin{aligned} f(T^s)&=f((\zeta_{\shuffle_2,m}^{\mathcal{C},T}(y_m))^s)\\ &=f(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^s))\\ &=f(\zeta_{\shuffle_2,m}^{\mathcal{C},T}(w))\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w)\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\overbrace{y_{m}\shuffle_2\cdots\shuffle_2y_{m}}^s)\\ &=(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m}))^s\\ &=(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}((y_m-y_{m'})+y_{m'}))^s\\ &=(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_m-y_{m'})+\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m'}))^s\\ &=(T+\ln(m'/m))^s\\ &=\rho_{m,m'}^{\shuffle_2}(T^s) \end{aligned}$$$\hfill\Box$\ Finally, we will compare the map $\zeta_{\shuffle_1,m}^{\mathcal{C},T}$ and $\zeta_{\shuffle_2,m'}^{\mathcal{C},T}$. This is the most important and interesting case. We will give a detail analysis about this.\ **Lemma 1**. *Let $r\in\mathbb{Z}_{\geqslant0}$ and $0<t<1$, then we have $$\int_0^{+\infty}\frac{\ln^r(M+1)}{M+1}t^{M+1}dM=O(\ln^{r+1}(1/(1-t)))\qquad t\to1^-$$* **Proof**. By explicit calculations, we have $$\begin{aligned} \int_0^{+\infty}\frac{\ln^r(M+1)}{M+1}t^{M+1}dM&=\int_0^{+\infty}\int\limits_{0<x_1<\cdots<x_r<M}\frac{r!dx_1\cdots dx_r}{(x_1+1)\cdots(x_r+1)}\cdot\frac{t^{M+1}}{M+1}dM\\ &=r!\int\limits_{0<x_1<\cdots<x_r<x_{r+1}}\frac{t^{x_{r+1}+1}}{(x_1+1)\cdots(x_r+1)(x_{r+1}+1)}dx_1\cdots dx_rdx_{r+1}\\ &=O(\ln^{r+1}(1/(1-t)))\qquad(\text{by\ lemma}\ 5.23) \end{aligned}$$$\hfill\Box$\ **Theorem 1**. *For any given $m,m'\in\mathbb{Z}_{>0}$, their exists an unique $\mathbb{R}-$linear map $$\rho_{m,m'}:\mathbb{R}[T]\longrightarrow\mathbb{R}[T]$$ called **the mixed comparison map** satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ That is, we have the following commutative diagram: $$\begin{CD} (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta_{\shuffle_1,m}^{\mathcal{C},T}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\varphi}VV@VV{\rho_{m,m'}}V\\ (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta_{\shuffle_2,m'}^{\mathcal{C},T} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$\ * **Proof**. Firstly, we proof the existence of $\rho_{m,m'}$. We consider the word $$\begin{aligned} \varphi(\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s)&=\varphi(s!\cdot y_{1,m}^s)\\ &=s!\cdot y_{m}y_0^{s-1}\\ &=s!\cdot y_{m}y_0^{s-2}(y_{0}-y_{m'})+s!\cdot y_{m}y_0^{s-3}(y_{0}-y_{m'})y_{m'}+\cdots\\ &\quad+s!\cdot y_{m}(y_{0}-y_{m'})y_{m'}^{s-2}+s!\cdot(y_{m}-y_{m'})y_{m'}^{s-1}+s!\cdot y_{m'}^s\\ &=\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^s+w_{s-1}^{(s)}\shuffle_2\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^{s-1}+\cdots\\ &\quad+w_1^{(s)}\shuffle_2y_{m'}+w_0^{(s)} \end{aligned}$$ where $w_0,\cdots,w_{s-1}\in\widetilde{\mathbb{Q}\langle X\rangle^0}$. Let $$a_i^{(s)}:=\zeta_{\shuffle_1,m}^{\mathcal{C},T}(\varphi^{-1}(w_i^{(s)}))=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w_i^{(s)})\in(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$$ we define $\rho_{m,m'}$ as follows $$\rho_{m,m'}(T^s)=T^s+a_{s-1}^{(s)}T^{s-1}+\cdots+a_0^{(s)}\in(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ notice that $\rho_{m,m'}$ is a $\mathbb{R}-$linar map, hence we obtain a map $$\rho_{m,m'}:(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\longrightarrow(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]$$ by linear expansion. Next, we will proof that the map $\rho_{m,m'}$ satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ Let $w=y_{(\boldsymbol{k};\boldsymbol{m})}\in\mathbb{Q}\langle Y\rangle$, where $(\boldsymbol{k};\boldsymbol{m})$ be a positive multi-index. By the theorem 5.28, we have $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=b_s\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^s+b_{s-1}\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^{s-1}+\cdots+b_0+O\left((1-t)\ln^{r}(1/(1-t))\right)$$ hence $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=b_sT^s+b_{s-1}T^{s-1}+\cdots+b_0$$ and $$\int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))+O\left((1-t)\ln^{r}(1/(1-t))\right)$$ where $T=\int_{\Delta^1(t)}\omega_{(1;m')}$. By explicit calculations, we have $$\begin{aligned} \int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}&=t^{m_r}\text{Li}^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})(t)\\ &=\int\limits_{0<x_1<\cdots<x_r}\frac{t^{x_r+m_r}}{(x_1+m_1)^{k_1}\cdots(x_r+m_r)^{k_r}}dx_1\cdots dx_r\\ &=\int_0^{+\infty}\int\limits_{0<x_1<\cdots<x_{r-1}<M}\frac{dx_1\cdots dx_{r-1}}{(x_1+m_1)^{k_1}\cdots(x_{r-1}+m_{r-1})^{k_{r-1}}}\cdot \frac{t^{M+m_r}}{(M+m_r)^{k_r}}dM\\ &=\int_0^{+\infty}\frac{d}{dM}\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})\cdot t^{M+m_r}dM\\ &=\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})\cdot t^{M+m_r}\bigg|_0^{+\infty}-\ln(t)\int_0^{+\infty}\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})\cdot t^{M+m_r}dM\\ &=-\ln(t)\int_0^{+\infty}\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})\cdot t^{M+m_r}dM \end{aligned}$$ by theorem 5.17, we have $$\zeta_M^{\mathcal{C}}(\boldsymbol{k};\boldsymbol{m})=a_s(\zeta_M^{\mathcal{C}}(1;m))^s+a_{s-1}(\zeta_M^{\mathcal{C}}(1;m))^{s-1}+\cdots+a_0+O\left(\frac{\ln^{r-1}(M+1)}{M+1}\right)$$ where $a_0,\cdots,a_s\in\mathcal{Z}^{\text{Dou},\mathcal{C}}$. Then we have $$\begin{aligned} &-\ln(t)\int_0^{+\infty}\zeta^{\mathcal{C}}_M(\boldsymbol{k};\boldsymbol{m})\cdot t^{M+m_r}dM\\ &=-\ln(t)\int_0^{+\infty}\left(\sum_{i=0}^sa_i(\zeta^{\mathcal{C}}_M(1;m))^i+O\left(\frac{\ln^{r-1}(M+1)}{M+1}\right)\right)\cdot t^{M+m_r}dM\\ &=-\ln(t)t^{m_r-m}\sum_{i=0}^sa_i\int_0^{+\infty}(\zeta^{\mathcal{C}}_M(1;m))^i\cdot t^{M+m}dM\\ &\quad+\ln(t)t^{m_r-1}\int_0^{+\infty}O\left(\frac{\ln^{r-1}(M+1)}{M+1}\right)\cdot t^{M+1}dM\\ &=-\ln(t)t^{m_r-m}\sum_{i=0}^sa_i\int_0^{+\infty}(\zeta^{\mathcal{C}}_M(1;m))^i\cdot t^{M+m}dM\\ &\quad+O((1-t)\ln^{r}(1/(1-t)))\qquad(\text{By\ Lemma 5.38}) \end{aligned}$$ notice that $$\begin{aligned} &\int_0^{+\infty}(\zeta^{\mathcal{C}}_M(1;m))^i\cdot t^{M+m}dM\\ &=i!\int_0^{+\infty}\int\limits_{0<x_1<\cdots<x_i<M}\frac{dx_1\cdots dx_i}{(x_1+m)\cdots(x_i+m)}\cdot t^{M+m}dM\\ &=i!\int\limits_{0<x_1<\cdots<x_i<M}\frac{dx_1\cdots dx_i}{(x_1+m)\cdots(x_i+m)}\cdot\frac{t^{M+m}}{\ln(t)}\bigg|_0^{+\infty}\\ &\quad-\frac{i!}{\ln(t)}\int_0^{+\infty}\int\limits_{0<x_1<\cdots<x_{i-1}<M}\frac{dx_1\cdots dx_i}{(x_1+m)\cdots(x_{i-1}+m)}\cdot\frac{t^{M+m}}{M+m}dM\\ &=-\frac{i!}{\ln(t)}\int\limits_{0<x_1<\cdots<x_i}\frac{t^{x_i+m}}{(x_1+m)\cdots(x_i+m)}dx_1\cdots dx_i\\ &=-\frac{i!}{\ln(t)}\int_{\Delta^i(t)}\omega_{(1,\cdots,1;m,\cdots,m)}\\ &=-\frac{1}{\ln(t)}\left(\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^i+a_{i-1}^{(i)}\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^{i-1}+\cdots+a_{0}^{(i)}+O((1-t)\ln^i(1/(1-t)))\right)\\ &=-\frac{1}{\ln(t)}\left(\rho_{m,m'}\left(\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^i\right)+O((1-t)\ln^i(1/(1-t)))\right) \end{aligned}$$ hence,we conclude that $$\begin{aligned} \int_{\Delta^{\text{wt}(\boldsymbol{k})}(t)}\omega_{(\boldsymbol{k};\boldsymbol{m})}&=\rho_{m,m'}\left(\sum_{i=0}^sa_i\cdot\left(\int_{\Delta^1(t)}\omega_{(1;m')}\right)^i\right)+O((1-t)\ln^r(1/(1-t)))\\ &=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))+O((1-t)\ln^r(1/(1-t))) \end{aligned}$$ where $T=\int_{\Delta^1(t)}\omega_{(1;m')}$. Now we obtain the following result $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))+O\left((1-t)\ln^{r}(1/(1-t))\right)=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))+O((1-t)\ln^r(1/(1-t)))$$ hence we have $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=\rho_{m,m'}(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))$$ Next, we proof the uniqueness of $\rho_{m,m'}$. If there exists another $\mathbb{Q}-$linear map $f:\mathbb{R}[T]\longrightarrow\mathbb{R}[T]$ satisfy $$\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))=f(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\qquad\forall w\in \mathbb{Q}\langle Y\rangle$$ especially, we take $$w=\overbrace{y_{1,m}\shuffle_1\cdots\shuffle_1y_{1,m}}^s$$ then, we have $$\begin{aligned} f(T^s)&=f(\zeta_{\shuffle_1,m}^{\mathcal{C},T}(w))\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\varphi(w))\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(s!\cdot y_my_0^{s-1})\\ &=\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^s+w_{s-1}^{(s)}\shuffle_2\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^{s-1}+\cdots+w_0^{(s)})\\ &=(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m'}))^s+\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w_{s-1}^{(s)})(\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(y_{m'}))^{s-1}+\cdots+\zeta_{\shuffle_2,m'}^{\mathcal{C},T}(w_0^{(s)})\\ &=T^s+a_{s-1}^{(s)}T^{s-1}+\cdots+a_{0}^{s}\\ &=\rho_{m,m'}(T^s) \end{aligned}$$ $\hfill\Box$ **Theorem 1**. *We have the following commutative diagram $$\begin{CD} (\mathbb{Q}\langle (\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]@>\rho^{\shuffle_1}_{m_1,m_1'}>>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\\ @V{\rho_{m_1,m_2}}VV@VV{\rho_{m_1',m_2'}}V\\ (\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]@>\rho^{\shuffle_2}_{m_2,m_2'} >>(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)[T]\end{CD}$$ that is, we have a network which composed of comparative mappings. We denoted this network by the following diagram $$\begin{tikzpicture} \draw (-6,3) -- (6,3); \draw (-6,-3) -- (6,-3); \draw (-6,1) -- (6,1); \draw (-6,-1) -- (6,-1); \draw (4,-3) -- (4,3); \draw (2,-3) -- (2,3); \draw (0,-3) -- (0,3); \draw (-2,-3) -- (-2,3); \draw (-4,-3) -- (-4,3); \draw (4,-1) -- (2,-3); \draw (4,1) -- (0,-3); \draw (4,3) -- (-2,-3); \draw (2,3) -- (-4,-3); \draw (0,3) -- (-4,-1); \draw (-2,3) -- (-4,1); \draw (2,3) -- (4,1); \draw (0,3) -- (4,-1); \draw (-2,3) -- (4,-3); \draw (-4,3) -- (2,-3); \draw (-4,1) -- (0,-3); \draw (-4,-1) -- (-2,-3); \draw (-4,3) -- (0,1); \draw (-4,3) -- (2,1); \draw (-4,3) -- (4,1); \draw (-2,3) -- (2,1); \draw (-2,3) -- (4,1); \draw (0,3) -- (-4,1); \draw (0,3) -- (4,1); \draw (2,3) -- (-4,1); \draw (2,3) -- (-2,1); \draw (4,3) -- (-4,1); \draw (4,3) -- (-2,1); \draw (4,3) -- (0,1); \draw (4,-1) -- (4,1); \draw (4,-1) -- (2,1); \draw (4,-1) -- (0,1); \draw (4,-1) -- (-2,1); \draw (4,-1) -- (-4,1); \draw (2,-1) -- (4,1); \draw (2,-1) -- (2,1); \draw (2,-1) -- (0,1); \draw (2,-1) -- (-2,1); \draw (2,-1) -- (-4,1); \draw (0,-1) -- (4,1); \draw (0,-1) -- (2,1); \draw (0,-1) -- (0,1); \draw (0,-1) -- (-2,1); \draw (0,-1) -- (-4,1); \draw (-2,-1) -- (4,1); \draw (-2,-1) -- (2,1); \draw (-2,-1) -- (0,1); \draw (-2,-1) -- (-2,1); \draw (-2,-1) -- (-4,1); \draw (-4,-1) -- (4,1); \draw (-4,-1) -- (2,1); \draw (-4,-1) -- (0,1); \draw (-4,-1) -- (-2,1); \draw (-4,-1) -- (-4,1); \draw (-4,-3) -- (0,-1); \draw (-4,-3) -- (2,-1); \draw (-4,-3) -- (4,-1); \draw (-2,-3) -- (2,-1); \draw (-2,-3) -- (4,-1); \draw (0,-3) -- (-4,-1); \draw (0,-3) -- (4,-1); \draw (2,-3) -- (-4,-1); \draw (2,-3) -- (-2,-1); \draw (4,-3) -- (-4,-1); \draw (4,-3) -- (-2,-1); \draw (4,-3) -- (0,-1); \node [above] at (-4,3) {$1$}; \node [above] at (-2,3) {$2$}; \node [above] at (-0,3) {$3$}; \node [above] at (2,3) {$4$}; \node [above] at (4,3) {$5$}; \node [below] at (-4,-3) {$1$}; \node [below] at (-2,-3) {$2$}; \node [below] at (-0,-3) {$3$}; \node [below] at (2,-3) {$4$}; \node [below] at (4,-3) {$5$}; \node [above] at (5,3) {$\cdots$}; \node at (5,0) {$\cdots$}; \node at (5,2) {$\cdots$}; \node at (5,-2) {$\cdots$}; \node [below] at (5,-3) {$\cdots$}; \node [left] at (-6,1) {$y_{1,m_1}$}; \node [left] at (-6,3) {$y_{1,m_1'}$}; \node [left] at (-6,-1) {$y_{m_2}$}; \node [left] at (-6,-3) {$y_{m_2'}$}; \node at (-5,2) {$\rho_{m_1',m_1}^{\shuffle_1}$}; \node at (-5,0) {$\rho_{m_1,m_2}$}; \node at (-5,-2) {$\rho_{m_2,m_2'}^{\shuffle_2}$}; \draw [->] (-4.3,2.75) -- (-4.3,1.25); \draw [->] (-4.3,0.75) -- (-4.3,-0.75); \draw [->] (-4.3,-1.25) -- (-4.3,-2.75); \draw [->] (-6.8,2.7) arc [radius=8, start angle=160, end angle= 199]; \node [left] at (-7.3,0) {$\rho_{m_1',m_2'}$}; \end{tikzpicture}$$* **Proof**. This result comes from theorem 5.36, theorem 5.37 and theorem 5.39. $\hfill\Box$\ Next, we will give a detail description of the map $\rho_{m,m'}$. By the theorem 5.40, we only need to consider the map $\rho_{1,1}$. Firstly, we will give some useful lemma.\ **Lemma 1**. *As $t\to1^-$, we have $$\sum_{m=0}^{\infty}\frac{\ln^j(m+1)}{m+1}t^m=O\left(\ln^{j+1}\left(\frac{1}{1-t}\right)\right)$$* **Proof**. One can see \[2\] page 61. $\hfill\Box$\ **Lemma 1**. *As $t\to1^-$, we have $$\int_0^t\frac{1}{-\ln(\tau)}d\tau=\ln\left(\frac{1}{1-t}\right)-\gamma+\frac{1-t}{2}+\frac{(1-t)^2}{24}+\frac{(1-t)^3}{72}+O((1-t)^4)$$* **Proof**. This result comes from https://www.wolframalpha.com/input?i=-li(x). $\hfill\Box$\ If we let $$\psi(t):=\int_0^t\frac{1}{-\ln(\tau)}d\tau+\gamma-\ln\left(\frac{1}{1-t}\right)\qquad0<t<1$$ then by the above lemma we have $$\lim_{t\to1^-}\frac{\psi(t)}{1-t}=\frac{1}{2}$$ **Lemma 1**. *If $0<t<1$, we have $$\sum_{m=0}^{\infty}\ln^j(m+1)t^m=\int_0^{\infty}\ln^j(x+1)t^xdx+O(1)$$* **Proof**. Using the Abel-Plana formular: $$\sum_{m=0}^{\infty}f(n)=\frac{f(0)}{2}+\int_0^{\infty}f(x)dx+i\int_0^{\infty}\frac{f(ix)-f(-ix)}{e^{2\pi x}-1}dx$$ if we take $f(x)=\ln^j(x+1)t^x$, then we have $$\sum_{m=0}^{\infty}\ln^j(m+1)t^m=\int_0^{\infty}\ln^j(x+1)t^xdx+i\int_0^{\infty}\frac{\ln^j(1+ix)t^{ix}-\ln^j(1-ix)t^{-ix}}{e^{2\pi x}-1}dx$$ notice that $$\begin{aligned} &\left|i\int_0^{\infty}\frac{\ln^j(1+ix)t^{ix}-\ln^j(1-ix)t^{-ix}}{e^{2\pi x}-1}dx\right|\\ &\leqslant\int_0^{\infty}\frac{|\ln(1+ix)|^j+|\ln(1-ix)|^j}{e^{2\pi x}-1}dx\\ &=\int_0^{1}\frac{|\ln(1+ix)|^j+|\ln(1-ix)|^j}{e^{2\pi x}-1}dx+\int_1^{\infty}\frac{|\ln(1+ix)|^j+|\ln(1-ix)|^j}{e^{2\pi x}-1}dx\\ &\leqslant\int_0^{1}\frac{|\ln(1+ix)|^j+|\ln(1-ix)|^j}{e^{2\pi x}-1}dx+2\int_1^{\infty}\left(\left(\frac{\ln(1+x^2)}{2}\right)^2+\left(\frac{\pi}{2}\right)^2\right)^\frac{j}{2}\frac{dx}{e^{2\pi x}-1}\\ &<+\infty \end{aligned}$$$\hfill\Box$ **Lemma 1**. *Let $j$ be a positive integer, we have $$\sum_{m=0}^{\infty}\frac{\Gamma^{(j)}(m+1)}{\Gamma(m+1)}t^m=\frac{1}{1-t}O\left(\ln^j\left(1/(1-t)\right)\right)\qquad t\to 1^-$$* **Proof**. We use that, for $m\to\infty$ and all $j$, we have the estimate $$\frac{\Gamma^{(j)}(m+1)}{\Gamma(m+1)}=\ln^j(m+1)+O\left(\frac{\ln^{j-1}(m+1)}{m+1}\right)$$ hence $$\begin{aligned} \sum_{m=0}^{\infty}\frac{\Gamma^{(j)}(m+1)}{\Gamma(m+1)}t^m&=\sum_{m=0}^{\infty}\ln^j(m+1)t^m+\sum_{m=0}^{\infty}O\left(\frac{\ln^{j-1}(m+1)}{m+1}\right)t^m \end{aligned}$$ by lemma 5.41 we have $$\sum_{m=0}^{\infty}O\left(\frac{\ln^{j-1}(m+1)}{m+1}\right)t^m=O\left(\ln^j\left(1/(1-t)\right)\right)\qquad t\to1^-$$ by lemma 5.43 we have $$\sum_{m=0}^{\infty}\ln^j(m+1)t^m=\int_0^{\infty}\ln^j(x+1)t^xdx+O(1)\qquad t\to1^-$$ notice that $$\begin{aligned} \int_0^{\infty}\ln^j(x+1)t^xdx&=j!\int_0^{\infty}\int\limits_{0<x_1<\cdots<x_j<x}\frac{dx_1\cdots dx_j}{(x_1+1)\cdots(x_j+1)}\cdot t^xdx\\ &=\frac{j!}{-\ln(t)}\int_{0<x_1<\cdots<x_j}\frac{t^{x_j}}{(x_1+1)\cdots(x_j+1)}dx_1\cdots dx_j\\ &=\frac{1}{1-t}O(\ln^j(1/(1-t)))\qquad(\text{by\ lemma\ 5.23}) \end{aligned}$$ hence wa are done.$\hfill\Box$.\ With the previous preparation work, we now prove the following theorem.\ **Theorem 1**. *The map $\rho_{1,1}$ is characterized by $$\begin{aligned} \rho_{1,1}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^n&\longmapsto n!\sum_{k=0}^n\frac{\gamma_k}{(n-k)!}\cdot T^{n-k} \end{aligned}$$ where $$\sum_{k=0}^{\infty}\gamma_ku^k=e^{\gamma u}\Gamma(1+u)=\exp\left(\sum_{n=2}^{\infty}\frac{(-1)^n}{n}\zeta(n)u^n\right)$$* **Proof**. Let $$Q(T):=\frac{d^n}{du^n}\left(e^{\gamma u}\Gamma(1+u)\cdot e^{Tu}\right)=\frac{d^n}{du^n}\left(\Gamma(1+u) e^{(T+\gamma)u}\right)\bigg|_{u=0}$$ Let $$T=\int_{\Delta^1(t)}\omega_{(1;1)}=\int_0^t\frac{1}{-\ln(\tau)}d\tau$$ we have $$\begin{aligned} \frac{1}{1-t}Q\left(\int_0^t\frac{1}{-\ln(\tau)}d\tau\right)&=\frac{1}{1-t}Q\left(\ln\left(\frac{1}{1-t}\right)-\gamma+\psi(t)\right)\qquad(\text{see\ lemma\ 5.42})\\ &=\frac{d^n}{du^n}\left(\frac{\Gamma(1+u)}{(1-t)^{1+u}}\cdot e^{\psi(t)u}\right)\bigg|_{u=0}\\ &=\sum_{k=0}^n\binom{n}{k}\left(\frac{d^{n-k}}{du^{n-k}}\frac{\Gamma(1+u)}{(1-t)^{1+u}}\right)\cdot(\psi(t))^ke^{\psi(t)u}\bigg|_{u=0}\\ &=\sum_{k=0}^n\binom{n}{k}\left(\frac{d^{n-k}}{du^{n-k}}\sum_{m=0}^{\infty}\frac{\Gamma(m+1+u)}{\Gamma(m+1)}t^m\right)\cdot(\psi(t))^ke^{\psi(t)u}\bigg|_{u=0}\\ &=\sum_{k=0}^n\binom{n}{k}\left(\sum_{m=0}^{\infty}\frac{\Gamma^{(n-k)}(m+1+u)}{\Gamma(m+1)}t^m\right)\cdot(\psi(t))^ke^{\psi(t)u}\bigg|_{u=0}\\ &=\sum_{k=0}^n\binom{n}{k}\left(\sum_{m=0}^{\infty}\frac{\Gamma^{(n-k)}(m+1)}{\Gamma(m+1)}t^m\right)\cdot(\psi(t))^k\\ &=\sum_{m=0}^{\infty}\frac{\Gamma^{(n)}(m+1)}{\Gamma(m+1)}t^m+\sum_{k=1}^n\binom{n}{k}\left(\sum_{m=0}^{\infty}\frac{\Gamma^{(n-k)}(m+1)}{\Gamma(m+1)}t^m\right)\cdot(\psi(t))^k\\ (\text{By\ Lemma\ 5.44})&=\sum_{m=0}^{\infty}\frac{\Gamma^{(n)}(m+1)}{\Gamma(m+1)}t^m+\sum_{k=1}^n\binom{n}{k}\left(\frac{1}{1-t}O(\ln^{n-k}(1/(1-t)))\right)\cdot(\psi(t))^k\\ (\text{By\ Lemma\ 5.42})&=\sum_{m=0}^{\infty}\frac{\Gamma^{(n)}(m+1)}{\Gamma(m+1)}t^m+O(\ln^{n-1}(1/(1-t)))\\ &=\sum_{m=0}^{\infty}\ln^n(m+1)t^m+\sum_{m=0}^{\infty}O\left(\frac{\ln^{n-1}(m+1)}{m+1}\right)t^m+O(\ln^{n-1}(1/(1-t)))\\ (\text{By\ Lemma\ 5.41\ and\ 5.43})&=\int_0^{\infty}\ln^n(x+1)t^xdx+O(1)+O(\ln^{n}(1/(1-t)))\\ &=\int_0^{\infty}\ln^n(x+1)t^xdx+O(\ln^{n}(1/(1-t)))\\ (\text{By\ the\ proof\ of\ Lemma\ 5.44})&=\frac{n!}{-t\ln(t)}\int\limits_{0<x_1<\cdots<x_n}\frac{t^{x_n+1}}{(x_1+1)\cdots(x_n+1)}dx_1\cdots dx_n+O(\ln^{n}(1/(1-t)))\\ &=\frac{n!}{1-t}\int\limits_{0<x_1<\cdots<x_n}\frac{t^{x_n+1}}{(x_1+1)\cdots(x_n+1)}dx_1\cdots dx_n+O(\ln^{n}(1/(1-t)))\\ &=\frac{1}{1-t}\zeta_{\shuffle_2,1}^{\mathcal{C},T}(n!\cdot y_1y_0^{n-1})+O(\ln^{n}(1/(1-t)))\qquad T=\int_{\Delta^1(t)}\omega_{(1;1)}\\ &=\frac{1}{1-t}\zeta_{\shuffle_2,1}^{\mathcal{C},T}(\varphi(n!\cdot y_{1,1}^n))+O(\ln^{n}(1/(1-t)))\\ \end{aligned}$$ $$\begin{aligned} &=\frac{1}{1-t}\rho_{1,1}(\zeta_{\shuffle_1,1}^{\mathcal{C},T}(n!\cdot y_{1,1}^n))+O(\ln^{n}(1/(1-t)))\\ &=\frac{1}{1-t}\rho_{1,1}(T^n)+O(\ln^{n}(1/(1-t))) \end{aligned}$$ we conclude that $$Q(T)=\rho_{1,1}(T^n)+O((1-t)\ln^{n}(1/(1-t)))\qquad t\to 1^-$$ hence $$\rho_{1,1}(T^n)=Q(T)=n!\sum_{k=0}^n\frac{\gamma_k}{(n-k)!}T^{n-k}$$ $\hfill\Box$ Some specific values for $\gamma_k$. $$\begin{tikzpicture} \draw (-4.5,1)--(6,1); \draw (-4.5,0)--(6,0); \draw (-4.5,-1)--(6,-1); \draw (-4.5,1)--(-4.5,-1); \draw (-3,1)--(-3,-1); \draw (-1.5,1)--(-1.5,-1); \draw (-0,1)--(-0,-1); \draw (1.5,1)--(1.5,-1); \draw (3,1)--(3,-1); \draw (4.5,1)--(4.5,-1); \node at (-3.75,0.5) {$k$}; \node at (-3.75,-0.5) {$\gamma_k$}; \node at (-2.25,0.5) {$0$}; \node at (-2.25,-0.5) {$1$}; \node at (-0.75,0.5) {$1$}; \node at (-0.75,-0.5) {$0$}; \node at (0.75,0.5) {$2$}; \node at (0.75,-0.5) {$\frac{\zeta(2)}{2}$}; \node at (2.25,0.5) {$3$}; \node at (2.25,-0.5) {$-\frac{\zeta(3)}{3}$}; \node at (3.75,0.5) {$4$}; \node at (3.75,-0.5) {$\frac{9\zeta(4)}{16}$}; \node at (5.25,0.5) {$\cdots$}; \node at (5.25,-0.5) {$\cdots$}; \end{tikzpicture}$$ hence we have $$\begin{aligned} \rho_{1,1}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ 1&\longmapsto1\\ T&\longrightarrow T\\ T^2&\longmapsto T^2+\zeta(2)\\ T^3&\longmapsto T^3+3\zeta(2)T-2\zeta(3)\\ T^4&\longmapsto T^4+6\zeta(2)T^2-8\zeta(3)T+\frac{27}{2}\zeta(4)\\ &\vdots \end{aligned}$$ **Corollary 6**. *The map $\rho_{m,m'}$ is characterized by $$\begin{aligned} \rho_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^n&\longmapsto\sum_{k=0}^n\binom{n}{k}(-\ln(m))^{n-k}\left(k!\sum_{l=0}^k\frac{\gamma_l}{(k-l)!}(T+\ln(m'))^{k-l}\right) \end{aligned}$$* **Proof**. Notice that $$\rho_{m,m'}=\rho_{1,m'}^{\shuffle_2}\circ\rho_{1,1}\circ\rho_{m,1}^{\shuffle_1}$$ hence $$\begin{aligned} \rho_{m,m'}(T^n)&=(\rho_{1,m'}^{\shuffle_2}\circ\rho_{1,1}\circ\rho_{m,1}^{\shuffle_1})(T^n)\\ &=(\rho_{1,m'}^{\shuffle_2}\circ\rho_{1,1})((T-\ln(m))^n)\\ &=(\rho_{1,m'}^{\shuffle_2}\circ\rho_{1,1})\left(\sum_{k=0}^n\binom{n}{k}(-\ln(m))^{n-k}T^k\right)\\ &=\rho_{1,m'}^{\shuffle_2}\left(\sum_{k=0}^n\binom{n}{k}(-\ln(m))^{n-k}\left(k!\sum_{l=0}^k\frac{\gamma_l}{(k-l)!}T^{k-l}\right)\right)\\ &=\sum_{k=0}^n\binom{n}{k}(-\ln(m))^{n-k}\left(k!\sum_{l=0}^k\frac{\gamma_l}{(k-l)!}(T+\ln(m'))^{k-l}\right) \end{aligned}$$ $\hfill\Box$\ next we will give some applications of the map $\rho_{m,m'}$.\ **Example 1**. *We have $$\zeta^{\mathcal{C}}(1,1,2;1,1,2)=\frac{\zeta(2)}{2}\qquad\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)=\frac{13}{24}\zeta(3)$$* **Proof**. Notice that $$\begin{aligned} \varphi(3!\cdot y_{1,1}^3)&=3!\cdot y_1y_0^2\\ &=6y_1y_0(y_0-y_1)+6y_1(y_0-y_1)\shuffle_2y_1-12y_1^2(y_0-y_1)+y_1\shuffle_2y_1\shuffle_2y_1 \end{aligned}$$ hence we have $$\begin{aligned} T^3+3\zeta(2)T-2\zeta(3)&=\rho_{1,1}(T^3)\\ &=\rho_{1,1}(\zeta_{\shuffle_1,1}^{\mathcal{C},T}(3!\cdot y_{1,1}^3))\\ &=\zeta_{\shuffle_2,1}^{\mathcal{C},T}(\varphi(3!\cdot y_{1,1}^3))\\ &=\zeta_{\shuffle_2,1}^{\mathcal{C},T}(3!\cdot y_1y_0^2)\\ &=\zeta_{\shuffle_2,1}^{\mathcal{C},T}(6y_1y_0(y_0-y_1)+6y_1(y_0-y_1)\shuffle_2y_1-12y_1^2(y_0-y_1)+y_1\shuffle_2y_1\shuffle_2y_1)\\ &=T^3+6\zeta^{\mathcal{C}}(1,1,2;1,1,2)T+6(\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)-2\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)) \end{aligned}$$ wo obtain $$\begin{aligned} 3\zeta(2)&=6\zeta^{\mathcal{C}}(1,1,2;1,1,2)\\ -2\zeta(3)&=6(\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)-2\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)) \end{aligned}$$ by the theorem 3.8, we know that $$\zeta^{\mathcal{C}}(1^{\{r-1\}},2;1^{\{r-1\}},2)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{r-1}}=(1-2^{2-r})\zeta(r-1)$$ hence we have $$\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)=\frac{3}{4}\zeta(3)\Longrightarrow\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)=\frac{13}{24}\zeta(3)$$ $\hfill\Box$ **Example 1**. *we have $$\int_1^{+\infty}\int_0^1\int_0^1\frac{dzdydx}{x(x+y)(x+y+z)}=\frac{5}{24}\zeta(3)$$* **Proof**. We consider $$\begin{cases} x_1=x\\ y_1=x+y\\ z_1=x+y+z \end{cases}\Longrightarrow\begin{cases} x_1=x\in(1,+\infty)\\ y_1-x_1=y\in(0,1)\\ z_1-y_1=z\in(0,1) \end{cases}$$ then we have $$\begin{aligned} \int_1^{+\infty}\int_0^1\int_0^1\frac{dzdydx}{x(x+y)(x+y+z)}&=\int_1^{+\infty}\int_{x_1}^{x_1+1}\int_{y_1}^{y_1+1}\frac{dz_1dy_1dx_1}{x_1y_1z_1}\\ &=\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{x_1+1}\frac{1}{y_1}\left(\int_{y_1}^{y_1+1}\frac{dz_1}{z_1}\right)dy_1\right)dx_1\\ &=\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{x_1+1}\frac{1}{y_1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &=\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{+\infty}\frac{1}{y_1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &\quad-\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1+1}^{+\infty}\frac{1}{y_1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &=\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{+\infty}\frac{1}{y_1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &\quad-\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{+\infty}\frac{1}{y_1+1}\left(\int_{y_1+1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &=\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{+\infty}\frac{1}{y_1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{z_1(z_1+1)}\right)dy_1\right)dx_1\\ &\quad-\int_1^{+\infty}\frac{1}{x_1}\left(\int_{x_1}^{+\infty}\frac{1}{y_1+1}\left(\int_{y_1}^{+\infty}\frac{dz_1}{(z_1+1)(z_1+2)}\right)dy_1\right)dx_1\\ &=\int\limits_{0<x_1<y_1<z_1}\frac{dz_1dy_1dx_1}{(x_1+1)(y_1+1)(z_1+1)(z_1+2)}\\ &\quad-\int\limits_{0<x_1<y_1<z_1}\frac{dz_1dy_1dx_1}{(x_1+1)(y_1+2)(z_1+2)(z_1+3)}\\ &=\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)-\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)\\ &=\frac{5}{24}\zeta(3) \end{aligned}$$ $\hfill\Box$ **Remark 1**. *The example 5.47 was first proved by Tewodros Amdeberhan, Victor H. Moll, Armin Straub and Christophe Vignat in 2020, one can see [@THAC], using the map $\rho_{1,1}$, we give a new proof about that.* **Example 1**. *We have $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;2,2,1,2)&=\sum_{i=1}^{\infty}\frac{1}{i^32^i}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)&=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{(-1)^{i-1}}{i^2}\frac{1}{j2^j}\\ \zeta^{\mathcal{C}}(1,1,1,2;2,1,1,2)&=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i}\frac{1}{j^22^j}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,2)&=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i^2}\frac{1}{j2^j}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,3)&=\sum_{i=0}^{ \infty}\sum_{j=0}^{\infty}\frac{1}{(2i+1)^2}\frac{1}{(2i+1+j)2^{2i+1+j}} \end{aligned}$$* **Proof**. The first identity $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;2,2,1,2)&=\int\limits_{0<x<y<z<w}\frac{dxdydzdw}{(x+2)(y+2)(z+1)(w+2)^2}\\ &=\int\limits_{0<x<y<z}\frac{dxdydz}{(x+2)(y+2)(z+1)(z+2)}\\ &=\sum_{n=0}^{\infty}\int\limits_{0<x<y<z}\frac{dxdydz}{(x+2)(y+2)(z+2)^{n+2}}\\ &=\sum_{i=1}^{\infty}\frac{1}{i^32^i} \end{aligned}$$ The second identity $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3)&=\int\limits_{0<x<y<z<w}\frac{dxdydzdw}{(x+1)(y+2)(z+2)(w+3)^2}\\ &=\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+2)(z+3)}\\ &=\sum_{n=0}^{\infty}(-1)^n\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+2)^{n+2}}\\ &=\sum_{n=0}^{\infty}\frac{(-1)^n}{(n+1)^2}\int_0^{\infty}\frac{dx}{(x+1)(x+2)^{n+1}}\\ &=\sum_{n=0}^{\infty}\frac{(-1)^n}{(n+1)^2}\sum_{m=0}^{\infty}\int_0^{\infty}\frac{dx}{(x+2)^{m+n+2}} \end{aligned}$$ $$\begin{aligned} &=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{(-1)^n}{(n+1)^2}\frac{1}{(m+n+1)2^{m+n+1}}\\ &=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{(-1)^{i-1}}{i^2}\frac{1}{j2^j} \end{aligned}$$ The third identity $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;2,1,1,2)&=\int\limits_{0<x<y<z<w}\frac{dxdydzdw}{(x+2)(y+1)(z+1)(w+2)^2}\\ &=\int\limits_{0<x<y<z}\frac{dxdydz}{(x+2)(y+1)(z+1)(z+2)}\\ &=\sum_{n=0}^{\infty}\int\limits_{0<x<y<z}\frac{dxdydz}{(x+2)(y+1)(z+2)^{n+2}}\\ &=\sum_{n=0}^{\infty}\frac{1}{n+1}\int_{0<x<y}\frac{dxdy}{(x+2)(y+1)(y+2)^{n+1}}\\ &=\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{m=0}^{\infty}\int_{0<x<y}\frac{dxdy}{(x+2)(y+2)^{m+n+2}}\\ &=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{n+1}\frac{1}{(m+n+1)^22^{m+n+1}}\\ &=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i}\frac{1}{j^22^j} \end{aligned}$$ The fourth identity $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,2)&=\int\limits_{0<x<y<z<w}\frac{dxdydzdw}{(x+1)(y+2)(z+1)(w+2)^2}\\ &=\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+1)(z+2)}\\ &=\sum_{n=0}^{\infty}\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+2)^{n+2}}\\ &=\sum_{n=0}^{\infty}\frac{1}{(n+1)^2}\int_{0}^{\infty}\frac{dx}{(x+1)(x+2)^{n+1}}\\ &=\sum_{n=0}^{\infty}\frac{1}{(n+1)^2}\sum_{m=0}^{\infty}\int_{0}^{\infty}\frac{dx}{(x+2)^{m+n+2}} \end{aligned}$$ $$\begin{aligned} &=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{(n+1)^2}\frac{1}{(m+n+1)2^{m+n+1}}\\ &=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\frac{1}{i^2}\frac{1}{j2^j} \end{aligned}$$ The fifth identity $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,3)&=\int\limits_{0<x<y<z<w}\frac{dxdydzdw}{(x+1)(y+2)(z+1)(w+3)^2}\\ &=\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+1)(z+3)}\\ &=\sum_{n=0}^{\infty}\int\limits_{0<x<y<z}\frac{dxdydz}{(x+1)(y+2)(z+2)^{2n+2}}\\ &=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\int_{0}^{\infty}\frac{dx}{(x+1)(x+2)^{2n+1}}\\ &=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\sum_{m=0}^{\infty}\int_{0}^{\infty}\frac{dx}{(x+2)^{m+2n+2}}\\ &=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{(2n+1)^2}\frac{1}{(m+2n+1)2^{m+2n+1}}\\ &=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{(2i+1)^2}\frac{1}{(2i+1+j)2^{2i+1+j}} \end{aligned}$$ Using the same method, We can obtain many similar identities. $\hfill\Box$\ **Example 1**. *(Ramanujan's identity. [@BB1], page 259;[@BB4], page 324). We have $$\begin{aligned} \zeta^{\mathcal{C}}(1,1,1,2;2,2,1,2)&=\frac{7}{8}\zeta(3)+\frac{\ln^3(2)}{6}-\frac{\pi^2\ln(2)}{12}\\ \zeta^{\mathcal{C}}(1,1,1,2;2,1,1,2)&=\zeta(3)-\frac{\pi^2\ln(2)}{12}\\ \zeta^{\mathcal{C}}(1,1,1,2;1,2,1,2)&=\frac{5}{8}\zeta(3) \end{aligned}$$ Furthermore, we discovered a new identity $$\zeta^{\mathcal{C}}(1,1,1,2;1,2,1,3)=\frac{7}{12}\zeta(3)\\$$* **Proof**. We consider the word $$w=y_{1,m_1}y_{1,m_2}y_{1,m_3}$$ then we have $$\begin{aligned} y_{1,m_1}y_{1,m_2}y_{1,m_3}&=y_{1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m})+y_{1,m_1}(y_{1,m_2}-y_{1,m})y_{1,m}+(y_{1,m_1}-y_{1,m})y_{1,m}^2+y_{1,m}^3\\ &=y_{1,m_1}y_{1,m_2}(y_{1,m_3}-y_{1,m})\\ &\quad+y_{1,m_1}(y_{1,m_2}-y_{1,m})\shuffle_1y_{1,m}-y_{1,m}y_{1,m_1}(y_{1,m_2}-y_{1,m})-y_{1,m_1}y_{1,m}(y_{1,m_2}-y_{1,m})\\ &\quad+\frac{(y_{1,m_1}-y_{1,m})\shuffle_1y_{1,m}\shuffle_1y_{1,m}}{2}-y_{1,m}(y_{1,m_1}-y_{1,m})\shuffle_1y_{1,m}+y_{1,m}^2(y_{1,m_1}-y_{1,m})\\ &\quad+y_{1,m}^3 \end{aligned}$$ hence $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_1,m}(y_{1,m_1}y_{1,m_2}y_{1,m_3})&=\frac{T^3}{3!}+\frac{(m-m_1)\zeta^{\mathcal{C}}(1,2;m_1,m)}{2}T^2\\ &\quad+((m-m_2)\zeta^{\mathcal{C}}(1,1,2;m_1,m_2,m)+(m-m_1)\zeta^{\mathcal{C}}(1,1,2;m,m_1,m))T\\ &\quad+(m-m_3)\zeta^{\mathcal{C}}(1,1,1,2;m_1,m_2,m_3,m)-(m-m_2)\zeta^{\mathcal{C}}(1,1,1,2;m,m_1,m_2,m)\\ &\quad-(m-m_2)\zeta^{\mathcal{C}}(1,1,1,2;m_1,m,m_2,m)+(m-m_1)\zeta^{\mathcal{C}}(1,1,1,2;m,m,m_1,m) \end{aligned}$$ we also have $$\begin{aligned} \varphi(y_{1,m_1}y_{1,m_2}y_{1,m_3})&=y_{m_1}y_{m_2-m_1}y_{m_3-m_2}\\ &=y_{m_1}y_{m_2-m_1}(y_{m_3-m_2}-y_{m'})+y_{m_1}(y_{m_2-m_1}-y_{m'})y_{m'}+(y_{m_1}-y_{m'})y_{m'}^2+y_{m'}^3\\ &=y_{m_1}y_{m_2-m_1}(y_{m_3-m_2}-y_{m'})\\ &\quad+y_{m_1}(y_{m_2-m_1}-y_{m'})\shuffle_2y_{m'}-y_{m'}y_{m_1}(y_{m_2-m_1}-y_{m'})-y_{m_1}y_{m'}(y_{m_2-m_1}-y_{m'})\\ &\quad+\frac{(y_{m_1}-y_{m'})\shuffle_2y_{m'}\shuffle_2y_{m'}}{2}-y_{m'}(y_{m_1}-y_{m'})\shuffle_2y_{m'}+y_{m'}^2(y_{m_1}-y_{m'})\\ &\quad+y_{m'}^3 \end{aligned}$$ we obtain $$\begin{aligned} &\zeta^{\mathcal{C},T}_{\shuffle_2,m'}(y_{m_1}y_{m_2-m_1}y_{m_3-m_2})\\ &=\frac{T^3}{3!}+\frac{(m'-m_1)\zeta^{\mathcal{C}}(1,2;m_1,m')}{2}T^2\\ &\quad+((m'+m_1-m_2)\zeta^{\mathcal{C}}(1,1,2;m_1,m_2,m'+m_1)+(m'-m_1)\zeta^{\mathcal{C}}(1,1,2;m',m'+m_1,2m'))T\\ &\quad+(m'+m_2-m_3)\zeta^{\mathcal{C}}(1,1,1,2;m_1,m_2,m_3,m'+m_2)\\ &\quad-(m'+m_1-m_2)\zeta^{\mathcal{C}}(1,1,1,2;m',m'+m_1,m'+m_2,2m'+m_1)\\ &\quad-(m'+m_1-m_2)\zeta^{\mathcal{C}}(1,1,1,2;m_1,m'+m_1,m'+m_2,2m'+m_1)\\ &\quad+(m'-m_1)\zeta^{\mathcal{C}}(1,1,1,2;m',2m',2m'+m_1,3m') \end{aligned}$$ if we let $(m_1,m_2,m_3;m,m')=(1,2,1;1,1)$, using $\rho_{1,1}$, we have $$\zeta^{\mathcal{C}}(1,1,1,2;1,2,1,3)=\frac{7}{12}\zeta(3)$$ if we let $(m_1,m_2,m_3;m,m')=(1,2,2;2,1)$, using $\rho_{2,1}$, we have $$\zeta^{\mathcal{C}}(1,1,1,2;2,2,1,2)=\frac{7}{8}\zeta(3)+\frac{\ln^3(2)}{6}-\frac{\pi^2\ln(2)}{12}$$ if we let $(m_1,m_2,m_3;m,m')=(1,2,1;2,1)$, using $\rho_{2,1}$, we have $$\zeta^{\mathcal{C}}(1,1,1,2;1,2,1,2)=\frac{5}{8}\zeta(3)$$ if we let $(m_1,m_2,m_3;m,m')=(1,1,2;2,1)$, using $\rho_{2,1}$, we have $$\zeta^{\mathcal{C}}(1,1,1,2;2,1,1,2)=\zeta(3)-\frac{\pi^2\ln(2)}{12}$$ Using the same technique, we can proof many identities which appearing in Ramanujan's Notebook. $\hfill\Box$ **Example 1**. *We have $$2\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,2,2,3)+\zeta^{\mathcal{C}}(1,1,1,1,2;1,1,2,2,3)-3\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,3,3,4)=\frac{5}{16}\zeta(4)$$* **Proof**. We repeat the same steps in Example 5.46, then $$\begin{aligned} \varphi(4!\cdot y_{1,1}^4)&=4!\cdot y_1y_0^3\\ &=24y_1y_0y_0(y_0-y_1)+24(y_1y_0(y_0-y_1)\shuffle_2y_1-2y_1^2y_0(y_0-y_1)-y_1y_0y_1(y_0-y_1))\\ &\quad+12(y_1(y_0-y_1)\shuffle_2y_1\shuffle_2y_1-4y_1^2(y_0-y_1)\shuffle_2y_1+6y_1^3(y_0-y_1))\\ &\quad+y_1\shuffle_2y_1\shuffle_2y_1\shuffle_2y_1 \end{aligned}$$ hence we have $$\begin{aligned} \rho_{1,1}(T^4)&=T^4+12\zeta^{\mathcal{C}}(1,1,2;1,1,2)T^2+24(\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)-2\zeta^{\mathcal{C}}(1,1,1,2;1,2,2,3))T\\ &\quad+24\zeta^{\mathcal{C}}(1,1,1,1,2;1,1,1,1,2)-48\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,2,2,3)-24\zeta^{\mathcal{C}}(1,1,1,1,2;1,1,2,2,3)\\ &\quad+72\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,3,3,4) \end{aligned}$$ by the result $$\rho_{1,1}(T^4)=T^4+6\zeta(2)T^2-8\zeta(3)T+\frac{27}{2}\zeta(4)$$ notice that $$\zeta^{\mathcal{C}}(1,1,1,2;1,1,1,2)=\frac{7}{8}\zeta(4)$$ we conclude that $$2\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,2,2,3)+\zeta^{\mathcal{C}}(1,1,1,1,2;1,1,2,2,3)-3\zeta^{\mathcal{C}}(1,1,1,1,2;1,2,3,3,4)=\frac{5}{16}\zeta(4)$$ $\hfill\Box$ **Example 1**. *If we take $w=y_{2,1}y_{1,2}$, then we have $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_1,1}(y_{2,1}y_{1,2})=\zeta^{\mathcal{C}}(2;1)T-\zeta^{\mathcal{C}}(2,1,2;1,2,1)-\zeta^{\mathcal{C}}(1,2;1,1) \end{aligned}$$ and $$\begin{aligned} \zeta^{\mathcal{C},T}_{\shuffle_2,1}(\varphi(y_{2,1}y_{1,2}))=\zeta^{\mathcal{C},T}_{\shuffle_2}(y_{1}xy_{1})=\zeta^{\mathcal{C}}(2;1)T-2\zeta^{\mathcal{C}}(1,2;1,2) \end{aligned}$$ notice that $\rho_{1,1}(1)=1, \rho_{1,1}(T)=T$, hence $$\zeta^{\mathcal{C}}(2,1,2;1,2,1)+\zeta^{\mathcal{C}}(1,2;1,1)=2\zeta^{\mathcal{C}}(1,2;1,2)$$* **Example 1**. *If we take $w=y_{1,2}y_{1,4}$, then we have $$\zeta^{\mathcal{C},T}_{\shuffle_1,1}(y_{1,2}y_{1,4})=\frac{T^2}{2}-\zeta^{\mathcal{C}}(1,2;2,1)T-3\zeta^{\mathcal{C}}(1,1,2;2,4,1)+\zeta^{\mathcal{C}}(1,1,2;1,2,1)$$ and $$\zeta^{\mathcal{C},T}_{\shuffle_2,1}(\varphi(y_{1,2}y_{1,4}))=\zeta^{\mathcal{C},T}_{\shuffle_2,1}(y_2^2)=\frac{T^2}{2}-\zeta^{\mathcal{C}}(1,2;2,1)T-\zeta^{\mathcal{C}}(1,1,2;2,4,3)+\zeta^{\mathcal{C}}(1,1,2;1,3,2)$$ notice that $\rho_{1,1}(1)=1, \rho_{1,1}(T)=T,\rho_{1,1}(T^2)=T^2+\zeta(2)$, hence $$\zeta^{\mathcal{C}}(1,1,2;1,2,1)-3\zeta^{\mathcal{C}}(1,1,2;2,4,1)+\frac{\zeta(2)}{2}=\zeta^{\mathcal{C}}(1,1,2;1,3,2)-\zeta^{\mathcal{C}}(1,1,2;2,4,3)$$* # **The universal algebra and some conjectures** We now introduce the extended double shuffle relations. We first recall the two commutative diagrams $$\begin{CD} (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\psi_{\shuffle_1,m}^{-1}>>(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)[T]\\ @V{\text{Id}}VV@VV{w\mapsto\zeta^{\mathcal{C}}_{\shuffle_1}(w),T\mapsto T}V\\ (\mathbb{Q}\langle Y\rangle,\shuffle_1)@>\zeta^{\mathcal{C},T}_{\shuffle_1,m}>>\mathbb{R}[T] \end{CD}$$ and $$\begin{CD} (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\psi_{\shuffle_2,m}^{-1}>>(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)[T]\\ @V{\text{Id}}VV@VV{w\mapsto\zeta^{\mathcal{C}}_{\shuffle_2}(w),T\mapsto T}V\\ (\mathbb{Q}\langle X\rangle^1,\shuffle_2)@>\zeta^{\mathcal{C},T}_{\shuffle_2,m}>>\mathbb{R}[T] \end{CD}$$ **Definition 1**. *Let $(R,\cdot)$ be a $\mathbb{Q}-$algebra, $Z^R_{\shuffle_1}:\widetilde{\mathbb{Q}\langle Y\rangle^0}\longrightarrow R$ and $Z^R_{\shuffle_2}:\widetilde{\mathbb{Q}\langle X\rangle^0}\longrightarrow R$ are map. We say that $(R,Z^R_{\shuffle_1,}Z^R_{\shuffle_2})$ satisfy the finite double shuffle relations if $Z^R_{\shuffle_1}$ is an algebra homomoephism $Z^R_{\shuffle_1}:(\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)\longrightarrow(R,\cdot)$, $Z^R_{\shuffle_2}$ is an algebra homomorphism $Z^R_{\shuffle_2}:(\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)\longrightarrow(R,\cdot)$, and the following commutative diagram $$\begin{CD} (\widetilde{\mathbb{Q}\langle Y\rangle^0},\shuffle_1)@>Z^R_{\shuffle_1}>>(R,\cdot)\\ @V{\varphi}VV@VV{\text{Id}}V\\ (\widetilde{\mathbb{Q}\langle X\rangle^0},\shuffle_2)@>Z^R_{\shuffle_2} >>(R,\cdot)\end{CD}$$* Composing $Z^R_{\shuffle_1}(\text{resp}.Z^R_{\shuffle_2})$ with the map $\psi^{-1}_{\shuffle_1,m}(\text{resp}.\psi^{-1}_{\shuffle_2,m})$(see section 5.2), we obtain extensions $$\begin{aligned} Z^{R,T}_{\shuffle_1,m}&:(\mathbb{Q}\langle Y\rangle,\shuffle_1)\longrightarrow (R,\cdot)[T]\\ Z^{R,T}_{\shuffle_2,m}&:(\mathbb{Q}\langle X\rangle^1,\shuffle_2)\longrightarrow (R,\cdot)[T] \end{aligned}$$ Next, we define the comparing maps $\rho^{R,\shuffle_1}_{m,m'},\rho^{R,\shuffle_2}_{m,m'}$ and $\rho^{R}_{m,m'}$. $$\begin{aligned} \rho^{R,\shuffle_1}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+Z^R_{\shuffle_1}(y_{1,m}-y_{1,m'}))^s \end{aligned}$$ $$\begin{aligned} \rho^{R,\shuffle_2}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto(T+Z^R_{\shuffle_2}(y_{m}-y_{m'}))^s \end{aligned}$$ and $$\begin{aligned} \rho^{R}_{m,m'}:\mathbb{R}[T]&\longrightarrow\mathbb{R}[T]\\ T^s&\longmapsto\sum_{i=0}^sa^{(s)}_iT^i \end{aligned}$$ where $$s!\cdot y_my_0^{s-1}=w_{s}^{(s)}\shuffle_2\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^s+w_{s-1}^{(s)}\shuffle_2\overbrace{y_{m'}\shuffle_2\cdots\shuffle_2y_{m'}}^{s-1}+\cdots+w_1^{(s)}\shuffle_2y_{m'}+w_0^{(s)}$$ and $a_i^{(s)}=Z_{\shuffle_2}^R(w_{i}^{(s)}),i=0,\cdots,s$.\ **Definition 1**. *Assume that $(R,Z_{\shuffle_1}^R,Z_{\shuffle_2}^R)$ satisfies the finite double shuffle relations. We say $(R,Z_{\shuffle_1}^R,Z_{\shuffle_2}^R)$ satisfies the extended double shuffle relations ,if in addition, for all $w\in\mathbb{Q}\langle Y\rangle$, one has $$\begin{aligned} Z^{R,T}_{\shuffle_1,m'}(w)&=\rho^{R,\shuffle_1}_{m,m'}(Z^{R,T}_{\shuffle_1,m}(w))\\ Z^{R,T}_{\shuffle_2,m'}(\varphi(w))&=\rho^{R,\shuffle_2}_{m,m'}(Z^{R,T}_{\shuffle_2,m}(\varphi(w)))\\ Z^{R,T}_{\shuffle_2,m'}(\varphi(w))&=\rho^{R}_{m,m'}(Z^{R,T}_{\shuffle_1,m}(w)) \end{aligned}$$* Combining theorem 5.36, theorem 5.37 and theorem 5.39, we obtain the main result of this section **Theorem 1**. *The pair $(\mathbb{R},\zeta^{\mathcal{C}}_{\shuffle_1},\zeta^{\mathcal{C}}_{\shuffle_2})$ satisfies the extend double shuffle relations.* Let $(R_{\text{EDS}},Z_{\shuffle_1}^{\text{EDS}},Z_{\shuffle_2}^{\text{EDS}})$ be the universal algebra satisfying the extended double shuffle relations. This means that, for any $(R,Z_{\shuffle_1}^R,Z_{\shuffle_2}^R)$ satisfy the extend double shuffle relations, there exists a pair of unique map $(\varphi^R_{\shuffle_1},\varphi^R_{\shuffle_2})$ such that the following diagram commutes $$\begin{CD} \mathbb{Q}\langle Y\rangle@>Z^{\text{EDS}}_{\shuffle_1}>>R_{\text{EDS}}\\ @V{\text{Id}}VV@VV{\varphi^R_{\shuffle_1}}V\\ \mathbb{Q}\langle Y\rangle@>Z^{R}_{\shuffle_1}>>R_{\text{EDS}}\end{CD}\qquad\qquad\begin{CD} \mathbb{Q}\langle X\rangle^1@>Z^{\text{EDS}}_{\shuffle_2}>>R_{\text{EDS}}\\ @V{\text{Id}}VV@VV{\varphi^R_{\shuffle_2}}V\\ \mathbb{Q}\langle X\rangle^1@>Z^{R}_{\shuffle_2}>>R_{\text{EDS}}\end{CD}$$ The following conjecture describes the combinatorial structure of the algebra of continuous multiple zeta values. **Conjecture 1**. *The map $(\varphi_{\shuffle_1}^{\mathbb{R}},\varphi_{\shuffle_2}^{\mathbb{R}})$ is injective. Equivalently the algebra $(\mathcal{Z}^{\text{Dou},\mathcal{C}},\cdot)$ of CMZVs is isomorphic to $(R_{\text{EDS}},\cdot)$.* This article is just the beginning of a new story, and subsequent research work is underway. # Acknowledgements {#acknowledgements .unnumbered} The author wants to thank the anonymous referee for his/her helpful comments to improve this paper. The author is supported by the School of Mathematical Sciences, Peking University, Beijing, China. 1 Jiangtao Li. *A continuous version of multiple zeta functions and multiple zeta values*. https://arxiv.org/abs/math/2111.15062v2,2023. Jose Ignacio Burgos Gil, Javier Fresan, *Multiple zeta values: from number to motives*, chapter 1. http://javier.fresan.perso.math.cnrs.fr/mzv.pdf T. Amdeberhan, H. Moll, A. Straub, C. Vignat, *A triple integral analog of a multiple zeta value*. https://arXiv:2004.06232v1 \[math.NT\] C. Glanois. *Motivic unipotent fundamental groupoid of $\mathbb{G}_m\backslash\mu_N$ for $N=2,3,4,6,8$ and Galois descents*. Journal of Number Theory(2016), Volume 160, pages 334--384. B. Berndt. *Ramanujan's Notebooks, Part I*. Springer-Verlag, New York, 1985. B. Berndt. *Ramanujan's Notebooks, Part IV*. Springer-Verlag, New York, 1994. D. Broadhurst, D. Kreimer, *Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops*, Phys. Lett. B 393 (1997), no. 3-4, 303-412. K. Chen, *Iterated path integrals*, Bull. Amer. Math. Soc.,(1977), 83, 831-879. H. Gangl, M. Kaneko, D. Zagier, *Double zeta values and modular forms*, Automorphic forms and zeta functions, In: Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific (2006), 71-106. A. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197-318. A. Goncharov, D. Rudenko, *Motivic correlators, cluster varieties, and Zagier's conjecture on $\zeta_F(4)$*, arXiv: 1803.08585. A. Goncharov, *The dihedral Lie algebras and Galois symmetries of $\pi_1^{(l)}(\mathbb{P}^1-\{0,\infty\}\cup \mu_{N})$*, Duke Math. J., 110 (3) (2001), 397-487. K. Ihara, M. Kaneko, D. Zagier, *Derivation and double shuffle relations for multiple zeta values*, Compositio Math. 142 (2006), 307-338. M. Kontsevich, D. Zagier, *Periods*, Mathematics Unlimited 2001 and Beyond, Springer, Berlin, (2001), 771-808. S. Yamamoto, *Multiple zeta-star values and multiple integrals*, arXiv: 1405.6499. D. Zagier, *Hyperbolic manifolds and special values of Dedekind zeta functions*, Inventiones Math., 83 (1986), 285-301. D. Zagier, *Polylogarithms, zeta-functions, and algebraic K-theory of fields*, Progress Math Vol 89. Birkhauser, Boston, MA, (1991), 392-430. J. Zhao, *Analytic continuation of multiple zeta functions* , Proceedings of the American Mathematical Society, 128 (5) (1999), 1275-1283.

arxiv_math

--- abstract: | We consider the class numbers of imaginary quadratic extensions $F(\sqrt{-p})$, for certain primes $p$, of totally real quadratic fields $F$ which have class number one. Using seminal work of Shintani, we obtain two elementary class number formulas for many such fields. The first expresses the class number as an alternating sum of terms that we generate from the coefficients of the power series expansions of two simple rational functions that depend on the arithmetic of $F$ and $p$. The second makes use of expansions of $1/p$, where $p$ is a prime such that $p \equiv 3 \pmod{4}$ and $p$ remains inert in $F$. More precisely, for a generator $\varepsilon_F$ of the totally positive unit group of $\mathcal{O}_F$, the base-$\varepsilon_{F}$ expansion of $1/p$ has period length $\ell_{F,p}$, and our second class number formula expresses the class number as a finite sum over disjoint cosets of size $\ell_{F,p}$. address: - Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 - Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138 - Department of Mathematics, Stanford University, 450 Jane Stanford Way, Stanford, CA 94305 author: - Elizabeth Athaide - Emma Cardwell - Christina Thompson bibliography: - bibliography.bib title: Class Number Formulas for Certain Biquadratic Fields --- # Introduction The theory of class numbers has a rich history, beginning with Gauss's effort to understand how primes could be represented by positive definite binary quadratic forms [@cox]. Gauss recognized that $\textrm{SL}_2(\mathbb{Z})$ acts naturally on positive definite integral binary quadratic forms $f(X, Y) = aX^{2} + bXY + cY^{2}$ with fixed discriminant $-d = b^{2} - 4ac$. He proved that the set of equivalence classes under this action is a finite abelian group; the order of this group is known as the *class number* $h(-d)$. The class group for quadratic forms of discriminant $d$ is also isomorphic to the ideal class group for the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{d})$. Therefore, it is natural to ask whether results about Gauss's class numbers are glimpses of results for the class numbers $h_K$ of more general number fields $K$. In this spirit, we recall two surprising results for Gauss's class numbers. [In the 1970s, Hirzebruch [@hirzebruch_sqrt_p] and Zagier [@zagier_sqrt_p] found an elegant formula for $h(-p)$]{style="color: black"}, when $7\leq p\equiv 3 \pmod{4}$ is prime and $h(4p) = 1$. [If t]{style="color: black"}he simple continued fraction for $\sqrt{p}$ is written as $$\begin{aligned} \sqrt{p} = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{\ddots}}} = [a_0,\overline{a_1,\ldots, a_{2t}}], \end{aligned}$$ where the repeating period begins with $a_1$ and has minimal even length $2t$, they proved that $$\begin{aligned} \label{hirzebruch_zagier_formula} h(-p) = \dfrac{1}{3} \sum_{k=1}^{2t}(-1)^{k}a_{k}.\end{aligned}$$ More recently in the 1990s, Girstmair [@girstmair_g_expansions] found another elegant formula as an alternating sum of numbers that are even simpler to describe. Namely, if $g$ is a primitive root modulo $p$, he examines the base $g$ expansion of $1/p$, which is eventually periodic with period length $p-1$ (see [@hardy], Section $9.6$). If this period is $\overline{x_{1}x_{2}...x_{p-1}}$, where $0 \leq x_i\leq g-1$, then he proved that $$\begin{aligned} \label{GirstmairFormula} h(-p) = \dfrac{1}{g+1}\sum_{k = 1}^{p -1} (-1)^{k}x_{k}. \end{aligned}$$ A priori, these results are unexpected relationships between combinatorial sums and class numbers of binary quadratic forms with discriminant $-p$. Since class numbers of binary quadratic forms are also examples of class numbers of number fields, it is natural to ask whether [\[thm:1.1\]](#thm:1.1){reference-type="eqref" reference="thm:1.1"} and [\[thm 1.2\]](#thm 1.2){reference-type="eqref" reference="thm 1.2"} are glimpses of a more general theory where class numbers of number fields can be described as alternating sums of combinatorial numbers. We show that this is indeed the case for a large class of imaginary quadratic extensions of real quadratic fields $F$. To make this precise, suppose that $F = \mathbb{Q}(\sqrt{d})$, where $d > 1$ is square-free. Throughout, we assume that its ring of integers $\mathcal{O}_F$ has class number 1. We note that $\mathcal{O}_F = \mathbb{Z}[\theta_F]$, where we let $$\begin{aligned} \theta_F \coloneqq \begin{cases} \sqrt{d} &\quad \textrm{if }d\equiv 3\pmod{4}\\ \frac{1 + \sqrt{d}}{2} &\quad \textrm{if }d\equiv 1\pmod{4}. \end{cases}\end{aligned}$$ The imaginary quadratic extensions of $F$ that we consider are of the form $F(\sqrt{-p})$, where $p$ is a prime for which $7\leq p \equiv 3 \pmod{4}$ and $\genfrac{(}{)}{}{}{d}{p} = -1$. These conditions imply that the relative discriminant ideal is the prime ideal $p\mathcal{O}_F$ (see Lemma [Lemma 5](#pof){reference-type="ref" reference="pof"}). Moreover, for convenience, we fix a generator $\rho_{F,p} \coloneqq a + b \theta_F \in \mathcal{O}_F$ such that $(\mathcal{O}_F/p\mathcal{O}_F)^{\times} = \langle \rho_{F,p} +p\mathcal{O}_F\rangle\cong \mathbb{F}_{p^2}^{\times}$. In this setting, we derive a class number formula for $F(\sqrt{-p})$ as an alternating sum that arises from $p$ and invariants of $F$. Our key observation is that the combinatorial structure that underlies [\[hirzebruch_zagier_formula\]](#hirzebruch_zagier_formula){reference-type="eqref" reference="hirzebruch_zagier_formula"} and [\[GirstmairFormula\]](#GirstmairFormula){reference-type="eqref" reference="GirstmairFormula"} can be reformulated in terms of recurrence relations that can be captured by the coefficients of distinguished rational functions. Therefore, our goal is to define two rational functions (reflecting that $F$ [has degree 2 over $\mathbb{Q}$]{style="color: black"}) whose coefficients can be incorporated into an alternating sum that yields the class number $h_{F(\sqrt{-p})}$. To this end, we use $\rho_{F,p} = a+b\theta_F$ to define integers $$\begin{aligned} \label{def:c_defns} C_{F,p} &\coloneqq a^{2} + ab\cdot \textrm{Tr}_{F/\mathbb{Q}}(\theta_F) + \textrm{Norm}_{F/\mathbb{Q}}(\theta_F)b^{2}, \\ \label{def:d_defns} D_{F,p} &\coloneqq 2a + b\cdot \textrm{Tr}_{F/\mathbb{Q}}({\theta_F}), \end{aligned}$$ and in turn, to define the rational functions as $$\begin{aligned} \label{eqn:x1defns} X_{F,p}(z) &= \sum_{m \geq 1} x(m) z^m \coloneqq \dfrac{az -C_{F,p}z^2}{C_{F,p}z^2 - D_{F,p}z + 1}, \\ \label{eqn:x2defns} Y_{F,p}(z) &= \sum_{m\geq 1} y(m) z^m \coloneqq \dfrac{bz}{C_{F,p}z^{2} - D_{F,p}z + 1}. \end{aligned}$$ Moreover, we must delicately take into account the presence of nontrivial units as they inform class number calculations. To make this precise, we recall that Dirichlet's Unit Theorem implies that $\mathcal{O}_F^\times = \{\pm \varepsilon_F^j, j\in \mathbb{Z}\}$, where $\varepsilon_F = s+t\theta_F$ is the totally positive fundamental unit. We then define $t$ pairs of sequences, say $\left\{({x}_i(m), {y}_i(m))\;:\; \;m\geq 1\right\}$, where $t$ is the coefficient of $\theta_F$ in $\varepsilon_F$, that encode the action of $\varepsilon_F$ by means of expressions involving $x(m)$ and $y(m)$ (see [\[tilde_defns\]](#tilde_defns){reference-type="eqref" reference="tilde_defns"}). Finally, we find that the analogues of the right hand side of [\[GirstmairFormula\]](#GirstmairFormula){reference-type="eqref" reference="GirstmairFormula"} turn out to be obtained from the quadratic form $$\begin{aligned} Q_{F}(Y_1, Y_2)\coloneqq \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)Y_1^2+4Y_1Y_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)Y_2^2. \end{aligned}$$ In terms of this data, we obtain the following theorem, which [gives a formula for the class number]{style="color: black"} $h_{F(\sqrt{-p})}$. **Theorem 1**. *Assuming the notation and hypotheses above, we have $$\begin{aligned} h_{F(\sqrt{-p})}=\frac{1}{16t^2 p^2}\sum_{\substack{1\leq m\leq p^2-1\\ 1\leq i\leq t}}(-1)^mQ_{F}\left({x}_i(m),{y}_i(m)\right). \end{aligned}$$* **Remark 1**. For real quadratic fields $F$ with $h_F=1$, Theorem [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} applies for one-fourth of the primes. This follows from the strong version of Dirichlet's Theorem on primes in arithmetic progressions, which implies that the primes $p$ such that $p \equiv 3 \pmod 4$ and $\genfrac{(}{)}{}{}{d}{p} = -1$ have density 1/4. **Example 1**. *Here we illustrate Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} with $F=\mathbb{Q}(\sqrt{3})$ and $p = 7$. The field $F(\sqrt{-7})$ has class number $h_{F(\sqrt{-7})} = 2$. Note that $F$ has class number $1$, and its totally positive fundamental unit is $\varepsilon_F= 2+\sqrt{3}$, and so $t=1$. The prime $p = 7$ satisfies the required conditions that $p\equiv 3\pmod 4$ and $\genfrac{(}{)}{}{}{3}{7} = -1$. Therefore, we have that the principal ideal $7\mathcal{O}_F\subset\mathcal{O}_F= \mathbb{Z}[\sqrt{3}]$ is prime, and so we have that $\mathcal{O}_F/7\mathcal{O}_F\cong \mathbb{F}_{49}$. One can check that $\rho_{F,p} = 6+\sqrt{3}$ generates the multiplicative cyclic group $(\mathcal{O}_F/7\mathcal{O}_F)^\times\cong \mathbb{F}_{49}^\times$. Thus we have $a=6, b=1,$ and using [\[def:c_defns\]](#def:c_defns){reference-type="eqref" reference="def:c_defns"} and [\[def:d_defns\]](#def:d_defns){reference-type="eqref" reference="def:d_defns"}, we find that $C_{F,p}=33,$ and $D_{F,p}=-12,$ which in turn by [\[eqn:x1defns\]](#eqn:x1defns){reference-type="eqref" reference="eqn:x1defns"} and [\[eqn:x2defns\]](#eqn:x2defns){reference-type="eqref" reference="eqn:x2defns"} give $$\begin{aligned} X_{F,p}(z) &= \sum_{m\ge 1} x(m)z^m = 6z + 39z^2 + 270 z^3 + 1953z^4 + \ldots = \dfrac{6z - 33z^2}{33z^2 +12z+ 1} ,\\ Y_{F,p}(z) &= \sum_{m\ge 1} y(m)z^m = z+12z^2+111z^3 + 936z^4 + \ldots= \dfrac{z}{33z^{2} +12z + 1} . \end{aligned}$$ Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} offers a formula for $h_{F(\sqrt{-7})}$ as an alternating sum of $7^2-1=48$ terms that are assembled from the first $48$ coefficients of $X_{F,p}(z)$ and $Y_{F,p}(z)$. Furthermore, because $t=1$, the relevant pairs $\{x_1(m), y_1(m)\}$ are merely reductions of the pairs of coefficients $\{x(m),y(m)\}$ to a specific *fundamental domain*, as given in [\[tilde_defns\]](#tilde_defns){reference-type="eqref" reference="tilde_defns"}. One finds that* --------------------- ---------------------- ------------- ------------------------ *${x}_{1}(1) = 1,$* *${x}_{1}(2) = -5,$* *$\ldots,$* *${x}_{1}(48) = -5,$* *$y_1(1) = -5,$* *${y}_{1}(2) = -4$,* *$\ldots,$* *${y}_{1}(48) = -7.$ * --------------------- ---------------------- ------------- ------------------------ *[We now use Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} to calculate $h_{F(\sqrt{-7})}$:]{style="color: black"} $$\begin{aligned} h_{F(\sqrt{-7})}&=\frac{1}{784}\sum_{1\leq m\leq 48}(-1)^m\bigg[4x_1(m)^2+4x_1(m)y_1(m)+4y_1(m)^2\bigg]\\ &= \frac{1}{784}(-84+76-300+52-28+\cdots + 436)=2.\end{aligned}$$* We circle back to the fact that the class number formula in [\[GirstmairFormula\]](#GirstmairFormula){reference-type="eqref" reference="GirstmairFormula"} makes use of the base $g$ expansion of $1/p$. We stress that the number of terms in the sum, which is $p-1$, is the length of the repeating period of this expansion. Therefore, we ask whether the expression in Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} can be reformulated so that the number of terms in the sum equals the period length of an analogous expansion of $1/p$. We find, indeed, that this is the case. In the setting of Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}, it is natural to consider the *base-$\varepsilon_{F}$ expansion* of elements $\alpha \in F$. To be precise, there is a unique sequence of integers $a_n, a_{n-1}, \ldots, a_{0}, a_{-1}, a_{-2}, \ldots$, with $0 \le a_{i} \le \lfloor \varepsilon_{F} \rfloor$, for which $$\begin{aligned} \alpha = a_{n}\varepsilon_F^n + a_{n-1}\varepsilon_{F}^{n-1} + \ldots + a_0 + a_{-1}\varepsilon_{F}^{-1} + a_{-2}\varepsilon_{F}^{-2} + \ldots .\end{aligned}$$ The above expression is called the *base*-$\varepsilon_{F}$ *expansion* of $\alpha$, and it is well-known that such expansions are eventually periodic (see, for example, [@pisot]). To recast Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} in terms of these expansions, we require the following finite set: $$\begin{aligned} R_{F,p} := \left\{r_1+r_2\varepsilon_F\in\frac{1}{p}\mathcal{O}_F : r_1\in\mathbb{Q}\cap(0,1],r_2\in\mathbb{Q}\cap[0,1)\right\},\end{aligned}$$ which is known as the *Shintani set* for $F$ at $p$, when $p\equiv 3\pmod 4$ and $\genfrac{(}{)}{}{}{d}{p}=-1$. The totally positive units define a group action of $\mathcal{O}_F^{\times,+}\coloneqq\langle\varepsilon_F\rangle$ onto $R_{F,p}$ as follows. $$\varepsilon_F \ast (r_1+r_2\varepsilon_F) \coloneqq (1-r_2) + \{r_1+r_2 \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)\}\varepsilon_F,$$ where $\{x\}\coloneqq x - \lfloor x\rfloor$ is the fractional part of $x$. Under this action, the set $R_{F,p}$ is a finite disjoint union of orbits, say $$R_{F,p} = \bigsqcup_{r\in \mathcal{O}_F^{\times,+}\backslash R_{F,p}} \mathcal{O}_F^{\times, +}\ast r.$$ For $r\in R_{F,p} \backslash \mathcal{O}_F$, we prove (see Lemma  [Lemma 19](#orbit_length_is_cycle_length){reference-type="ref" reference="orbit_length_is_cycle_length"}) that the number of elements in the orbit of $r$ under $\varepsilon_F$ is equal to the period length of $1/p$ in base $\varepsilon_F,$ which we denote $\ell_{F,p}.$ This allows us to now state the desired class number formula as a sum over $\ell_{F,p}$ terms, where we make the following abuse of notation: $$\begin{aligned} {Q}_F(r_1 + r_2\varepsilon_{F}) = Q_{F}(r_1, r_2). \end{aligned}$$ **Theorem 2**. *Assuming the notation and hypotheses from Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}, we have $$h_{F(\sqrt{-p})}=\frac{1}{4}\sum_{i=1}^{\ell_{F,p}}\hspace{+0.23cm}\sum_{r\in \mathcal{O}_F^{\times,+}\backslash R_{F,p}} \chi_{F(\sqrt{-p})/F} \left({rp}\mathcal{O}_{F}\right) Q_F(\varepsilon_{F}^i \ast r),$$ where $\chi_{F(\sqrt{-p})/F}$ is the unique quadratic Hecke character of conductor $p\mathcal{O}_{F}$.* **Example 2**. *Now we illustrate Theorem  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} with $F=\mathbb{Q}(\sqrt{3})$ and $p=7$, where $h_F=1$ and $\varepsilon_F=2+\sqrt{3}$ (so $t=1$). One can check (for example, using `SageMath`) that the base-$\varepsilon_F$ expansion of $1/7$ is $$\begin{aligned} \frac{1}{7}&=\varepsilon_F^{-2} +\sum_{i=0}^\infty \left({3\varepsilon_F^{-8i-3}+2\varepsilon_F^{-8i-4}+2\varepsilon_F^{-8i-5}+2\varepsilon_F^{-8i-7}+2\varepsilon_F^{-8i-8}+3\varepsilon_F^{-8i-9}}\right)\\ &=0.01\overline{32202230}. \end{aligned}$$ Thus, we see that the base $\varepsilon_F$ expansion of $1/7$ has period length $\ell_{F,7}=8$. Since $|R_{F,7}-\mathcal{O}_F|=tp^2-t=48$ (see Lemmas  [Lemma 8](#shin_set_construction){reference-type="ref" reference="shin_set_construction"} and  [Lemma 11](#explicit_kernel){reference-type="ref" reference="explicit_kernel"}), we deduce that $\mathcal{O}_F^{\times,+}\backslash(R_{F,7}-\mathcal{O}_F)$ contains $48/\ell_{F,7}=6$ disjoint orbits. One can also verify that the set $$\begin{aligned} \left\{\frac{1}{7}+\frac{1}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{1}{7},\hspace{+0.1 cm}\frac{1}{7}+\frac{4}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{1}{7}+\frac{5}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{2}{7}+\frac{2}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{3}{7}\right\}. \end{aligned}$$ is a complete set of orbit representatives for $\mathcal{O}_{F}^{\times,+}\backslash (R_{F,7}-\mathcal{O}_F)$. Equipped with these values, Theorem   [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} states that $$\begin{aligned} h_{F(\sqrt{-7})}&=\frac{1}{4}\sum_{i=1}^{8}\hspace{+0.23cm}\sum_{r\in R_{F,p}\backslash \mathcal{O}_F^{\times,+}} \chi_{F(\sqrt{-7})/F} \left(r {p}\mathcal{O}_{F}\right) Q_F(\varepsilon_{F}^i \ast r)\\ &=\frac{1}{4}\bigg(-\frac{220}{7} + \frac{228}{7} - \frac{188}{7} + \frac{212}{7} - \frac{180}{7} + \frac{204}{7}\bigg)=2. \end{aligned}$$* Theorems  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} and  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} are generalizations of the results from Hirzebruch-Zagier and Girstmair to the setting of imaginary quadratic extensions of real quadratic fields $F$ with $h_F=1$. Within this new setting, we prove our theorems by working with a class number formula analogous to the one used in the quadratic setting by Hirzebruch, Zagier, and Girstmair. Both [\[hirzebruch_zagier_formula\]](#hirzebruch_zagier_formula){reference-type="eqref" reference="hirzebruch_zagier_formula"} and [\[GirstmairFormula\]](#GirstmairFormula){reference-type="eqref" reference="GirstmairFormula"} arise from a finite version of Dirichlet's class number formula, which relates the Dirichlet $L$-function, an infinite series, to the class number of $\mathbb{Q}(\sqrt{-d})$: $$\begin{aligned} L(1, \chi_{d}) = \dfrac{2\pi}{\omega\sqrt{d}}h(-d),\end{aligned}$$ where $\omega$ represents the number of roots of unity in $\mathbb{Q}(\sqrt{-d})$, and $\chi_d$ is a primitive Dirichlet character of conductor $d$. Using the functional equation of this $L$-function, the above equation can be written in terms of $L(0, \chi_{d})$, which in turn allows us to use the Hurwitz $\zeta$-function and the periodicity of $\chi_d$ to rewrite this class number formula as a finite sum of Bernoulli polynomials evaluated at integer points. Our work uses an analogous formula of Shintani [@shintani_thm], which expresses the class numbers of totally imaginary quadratic extensions of totally real fields as finite sums assembled from Bernoulli numbers. In Section  [2](#sec2){reference-type="ref" reference="sec2"}, we review the background needed to state Shintani's class number formula for imaginary quadratic extensions of real quadratic fields $F$ with $h_F=1$. These formulae involve which are something like fundamental domains for the action of the totally positive units on $\frac{1}{p}\mathcal{O}_F$. The crux of our work relies on combinatorial properties of these sets, which we derive in Section  [2](#sec2){reference-type="ref" reference="sec2"}. Then, we prove Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} in Section  [3](#section_1.1_proof){reference-type="ref" reference="section_1.1_proof"} and Theorem  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} in Section  [4](#section4){reference-type="ref" reference="section4"}. Finally, in Section  [5](#example_section){reference-type="ref" reference="example_section"}, we use Theorems  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} and  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} to calculate class numbers of $\mathbb{Q}(\sqrt{3},\sqrt{-p})$, where $p\equiv 3 \pmod{4}$ is prime, $\genfrac{(}{)}{}{}{d}{p} = -1$, and $p<100$. # Acknowledgements {#sec: intro .unnumbered} The authors were participants in the 2023 UVA REU in Number Theory. They are grateful for the support of grants from Jane Street Capital, the National Science Foundation (DMS-2002265 and DMS- 2147273), the National Security Agency (H98230-23-1-0016), and the Templeton World Charity Foundation. The authors thank Ken Ono, Wei-Lun Tsai, Alejandro De Las Penas Castano, and Eleanor McSpirit for suggesting the problem and for their mentorship and support. They would also like to thank Marie-Hélène Tomé and the other participants of the 2023 UVA REU for many thoughtful discussions. # Shintani's Class Number Formula and Properties of Shintani Sets {#sec2} In this section, we discuss the background needed to state Shintani's class number formula. While Shintani's theorem is true for totally imaginary quadratic extensions of a totally real field of arbitrary degree, we restrict the following commentary and definitions to the case that $F$ is quadratic with class number $1.$ Throughout this section, we fix a real quadratic field $F$ of class number is 1 and a totally imaginary quadratic extension of $F$, which we denote $K = F(\sqrt{-p})$, where $p \equiv 3\pmod{4}$. ## Algebraic Background {#alg-background} Shintani's formula can be used to calculate the relative class number $h_K/h_F$ in terms of invariants of $F$, $K$, and the extension $K/F$ itself. Before stating the formula, we review the definitions of these invariants. The *regulator* $R_L$ of a number field $L$ measures the density of units in the ring of integers. The regulator can be determined by considering the matrix $$[N_j\log(\sigma_j(u_i))],$$ where each $u_i$ is a fundamental unit from the set $u_1, \cdots, u_{k}$ generating the unit group in $\mathcal{O}_L$, each $\sigma_j$ is a unique Archimedian place of $L$, and $N_j$ is defined to be $1$ if $\sigma_j$ is real, and $2$ if $\sigma_j$ is complex. If we define $r_1$ and $r_2$ respectively to be the number of real and complex embeddings of $L$, by Dirichlet's unit theorem, we see that this matrix has dimension $(r_1+r_2-1)\times (r_1+r_2)$. The regulator $R_L$ is the determinant of the square submatrix which is formed by deleting any single column of this matrix. Since the sum of the entries in each row of this matrix is $0$, this determinant is independent of which column is deleted. If we consider the rows of this matrix as forming a lattice in $\mathbb{R}^{r_1+r_2-1},$ then the regulator is directly proportional to the volume of the fundamental domain associated to this lattice. Next, we examine the unit groups of $\mathcal{O}_K$ and $\mathcal{O}_F.$ Since $F$ is a real quadratic field, and $K$ is a totally imaginary quadratic extension of $F$, Dirichlet's unit theorem implies that $\mathcal{O}_F$ and $\mathcal{O}_K$ are both $\mathbb{Z}$-modules of rank $1$. More precisely, if we let $\mu_F$ and $\mu_K$ represent the groups of roots of unity in $F$ and $K$ respectively, there exists $\varepsilon_F\in\mathcal{O}_F$ and $\varepsilon_K\in\mathcal{O}_K$ such that $\mathcal{O}_F^\times=\mu_F\times\langle\varepsilon_F\rangle$ and $\mathcal{O}_K^\times=\mu_K\times\langle\varepsilon_K\rangle.$ Since $F$ is real quadratic and any of $\varepsilon_F,-\varepsilon_F,\varepsilon_F^{-1},-\varepsilon_F^{-1}$ can generate the free part of $\mathcal{O}_F,$ we can choose $\varepsilon_F$ to be totally positive and greater than $1.$ **Lemma 3**. *We have $\mathcal{O}_K^\times=\mathcal{O}_F^\times.$ In particular, we may choose $\varepsilon_K=\varepsilon_F.$* *Proof.* A theorem by Frölich and Taylor shows that $[\mathcal{O}_K^\times:\mathcal{O}_F^\times\mu_K]=1\text{ or }2$ (see Theorem 42 in [@taylor]). Since $K=\mathbb{Q}(\sqrt{d},\sqrt{-p})$ for $p\geq 7,$ $\mu_K=\{\pm 1\}.$ Thus $\mu_K=\mu_F,$ so $[\mathcal{O}_K^\times:\mathcal{O}_F^\times]=1\text{ or }2.$ Now, assume for the sake of contradiction that $[\mathcal{O}_K^\times:\mathcal{O}_F^\times]=2,$ so $\varepsilon_K \notin \mathcal{O}_F^{\times},$ and $\varepsilon_K^2\in\mathcal{O}_F^\times$. Since $K\cap\mathbb{R}=F,$ we see that $\varepsilon_K\not\in\mathbb{R}.$ However, we know that $\varepsilon_K^2 \in \mathcal{O}_F \subset \mathbb{R}$. Observe that both $\varepsilon_K \in \mathbb{C}-\mathbb{R}$ and $\varepsilon_K^2 \in \mathbb{R}$ if and only if $\mathrm{Re}(\varepsilon_K) = 0$. Additionally, $\textrm{Norm}_{K/\mathbb{Q}}(\varepsilon_K) = \pm 1$, which implies that $\varepsilon_K = \pm i.$ However, this is a contradiction since $\pm i \notin K,$ so we see that $[\mathcal{O}_K^\times:\mathcal{O}_F^\times]=1$, and hence $\mathcal{O}_K^\times=\mu_F\times\langle\varepsilon_F\rangle=\mu_K\times\langle\varepsilon_F\rangle,$ so we can choose $\varepsilon_K = \varepsilon_F$. ◻ Equipped with the fact that $\varepsilon_K = \varepsilon_F,$ we may now relate the regulators $R_K$ and $R_F$ of $K$ and $F,$ which we do in the lemma which follows. **Lemma 4**. *For $F$ and $K$ as defined in the beginning of this section, we have that $R_K=2R_F.$* *Proof.* Since $\varepsilon_F=\varepsilon_K$ by Lemma   [Lemma 3](#ep_F_is_ep_K){reference-type="ref" reference="ep_F_is_ep_K"}, the regulators of the fields $F$ and $K$ as previously defined are determined using the following matrices: $$\begin{aligned} R_F: \begin{bmatrix} \log|\varepsilon_F| && \log|-\varepsilon_F| \end{bmatrix}\\ R_K: \begin{bmatrix} 2\log|\varepsilon_F| && 2\log|-\varepsilon_F| \end{bmatrix}\end{aligned}$$ Thus $R_F=\log|\varepsilon_F|$ and $R_K=2\log|\varepsilon_F|=2R_F.$ ◻ Next, we review the definition of the *relative discriminant ideal* $D_{K/F}$ for our fields $K$ and $F$. Recall that $F$ has class number $1$, so $D_{K/F}$ is principal. Since $K/F$ is quadratic, it is Galois, and its Galois group consists of two elements: the identity and complex conjugation. In this setting, $D_{K/F}$ is given by $$\begin{aligned} D_{K/F}\coloneqq \left(\det \begin{bmatrix} \omega_1 && \omega_2\\ \overline{\omega_1} && \overline{\omega_2} \end{bmatrix}\right)^2\mathcal{O}_F,\end{aligned}$$ where $\{\omega_1,\omega_2\}$ is an integral basis of $K/F$. We know that an integral basis will exist in our case by the following argument. From the structure theorem for finitely generated modules over a Dedekind domain, we have that $\mathcal{O}_K\cong\mathcal{O}_F^n\oplus\mathfrak{a},$ where $\mathfrak{a}$ is an ideal of $\mathcal{O}_F$ and $n\in\mathbb{Z}_{\geq 0}$ (see Theorem $1.32$ of [@narkiewicz]). Since $h_F=1,$ implying $\mathcal{O}_F$ is a principal ideal domain, $\mathcal{O}_K$ must be a free $\mathcal{O}_F$-module of rank $2=[K:F].$ **Lemma 5**. *The set $\{1, \frac{1+\sqrt{-p}}{2}\}$ is an integral basis of $K/F,$ and thus we have $D_{K/F}=p\mathcal{O}_F.$* *Proof.* Let $A$ be the change-of-basis matrix from the integral basis $\{\omega_1,\omega_2\}$ to the $F$-basis $\{1, \frac{1+\sqrt{-p}}{2}\}$. We see that $$\begin{aligned} p\mathcal{O}_F = \left(\det \begin{bmatrix} 1 && \frac{1+\sqrt{-p}}{2}\\ 1 && \frac{1-\sqrt{-p}}{2} \end{bmatrix}\right)^2\mathcal{O}_F = (\det A)^2 D_{K/F}.\end{aligned}$$ Since $\mathcal{O}_F$ is a Dedekind domain, ideals in $\mathcal{O}_F$ factor uniquely. Therefore since $p\mathcal{O}_F$ is prime by assumption, $\det A$ must be a unit in $\mathcal{O}_F$, so $A \in \mathrm{GL}_2(\mathcal{O}_F)$. Thus, $\{1, \frac{1+\sqrt{-p}}{2}\}$ is an integral basis of $K/F$. Using this integral basis, we see that $$\begin{aligned} D_{K/F}=\left(\frac{1-\sqrt{-p}}{2}-\frac{1+\sqrt{-p}}{2}\right)^2\mathcal{O}_F=p\mathcal{O}_F. \end{aligned}$$ ◻ Finally, since $\mathrm{Gal}(K/F)\cong \mathbb{Z}/2\mathbb{Z}$, there is a unique nontrivial character $\chi:\mathrm{Gal}(K/F)\to\mathbb{C}^\times$. By class field theory, we can consider the precomposition of $\chi$ with the Artin symbol to obtain a character $\chi_{K/F}$ of the group of fractional ideals that are relatively prime to $D_{K/F}$. This is known as the *Hecke character of $K/F$ with conductor $D_{K/F}$.* By definition of the Artin symbol (see, for example, [@cox] page 106), we can explicitly compute the value of $\chi_{K/F}$ for any prime ideal $\mathfrak{p}$: $$\begin{aligned} \chi_{K/F}(\mathfrak{p})= \begin{cases} 1 & \mathfrak{p} \text{ splits in } \mathcal{O}_{K}\\ -1 & \mathfrak{p} \text{ remains inert in } \mathcal{O}_{K}\\ 0 & \mathfrak{p} \text{ ramifies in } \mathcal{O}_{K}. \end{cases} \end{aligned}$$ **Remark 1**. *Shintani's class number formula relies on the narrow ideal class group character with conductor $D_{K/F}$ evaluated at fractional ideals. This corresponds to a primitive Grössencharakter with modulus $D_{K/F}$ (see Prop. 6.9 in [@neukirch]). Since $\mathrm{Gal}(K/F)\cong \mathbb{Z}/2\mathbb{Z}$, the nontrivial character $\chi: \mathrm{Gal}(K/F)\to\mathbb{C}^\times$ is unique and injective, so class field theory implies that the primitive Grössencharacter with modulus $D_{K/F}$ is unique and corresponds to $\chi$. Hence, we can see that the character used in Shintani's formula is exactly the Grössencharakter. For more details, see Sections 6 and 10 in [@neukirch].* **Remark 2**. *For any unit $u \in K$ and any ideal $\mathfrak{a}\subset\mathcal{O}_{K}$, $u\cdot\mathfrak{a}=\mathfrak{a},$ and hence $\chi_{K/F}(u\cdot\mathfrak{a})=\chi_{K/F}(\mathfrak{a}).$* ## Shintani's Class Number Formula In this section, we prove a simplified version of Shintani's formula for real quadratic base fields $F$ with $h_F = 1$. **Proposition 6**. *For a totally real quadratic extension $F$ of $\mathbb{Q}$ with $h_{F} = 1$ and $K = F(\sqrt{-p})$ a totally imaginary quadratic extension of $F$ where $7\leq p \equiv 3\pmod{4}$ remains inert in $\mathcal{O}_F$, Shintani's formula simplifies to the following: $$\begin{aligned} h_{K} = \frac{1}{2}\sum_{r\in R_{F,p}}{\chi_{K/F}\bigg((r_1+r_2\varepsilon_F) D_{K/F}\bigg)\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(r_1)}{l_1!}\frac{B_{l_2}(r_2)}{l_2!}\mathrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)^{l_2-1}}},\end{aligned}$$ where $$R_{F,p} = \left\{r=r_1+r_2\varepsilon_F\;:\; 0<r_1\leq 1, 0\leq r_2< 1, r\in \tfrac{1}{p}\mathcal{O}_F\right\},$$ and $B_n(x)$ is the degree $n$ Bernoulli polynomial. As $[F:\mathbb{Q}] = 2$, Shintani's formula only requires the following Bernoulli polynomials: $$\begin{aligned} B_0(x)=1,\quad B_1(x)=x-\frac{1}{2},\quad B_2(x)=x^2-x+\frac{1}{6}.\end{aligned}$$* *Proof.* We follow [@shintani_thm] by first considering the embedding $F\to\mathbb{R}^2$ via $$\begin{aligned} F\hookrightarrow \mathbb{R}^2 \quad \alpha \mapsto \left(\alpha, \alpha'\right),\end{aligned}$$ where $\alpha\mapsto\alpha'$ is the nontrivial automorphism in $\mathrm{Gal}(F/\mathbb{Q})$. Shintani shows that the first quadrant $\mathbb{R}^2_+\coloneqq\{(x,y)\in \mathbb{R}^2\;:\; x,y>0\}$ can be decomposed as the following disjoint union: $$\begin{aligned} \mathbb{R}_+^2 &= \bigcup_{\eta\in \mathcal{O}_F^{\times,+}}\eta C_1\sqcup \bigcup_{\eta\in \mathcal{O}_{F}^{\times,+}}\eta C_2\end{aligned}$$ where $C_1$ is generated by the images of $1,\varepsilon_F$ in $\mathbb{R}^2$ and $C_2$ is generated by the image of $1$: $$\begin{aligned} C_1= \{\lambda_1(1,1) + \lambda_2 (\varepsilon_F, \varepsilon_F') \in \mathbb{R}^2 \;:\; \lambda_1,\lambda_2>0\},\quad C_2 = \{\lambda (1,1) \in \mathbb{R}^2 \;:\; \lambda>0 \},\end{aligned}$$ and $\eta\in \mathcal{O}_F^{\times,+}$ acts by component-wise multiplication. Next, for each cone $C_i$, Shintani defines the set $R(i,\frac{1}{p}\mathcal{O}_F)$ as the following vectors with components in $\mathbb{Q}\cap (0,1]$: $$\begin{aligned} R\left(1,\tfrac{1}{p}\mathcal{O}_F\right) &\coloneqq \left\{(r_1,r_2)\in \mathbb{Q}^2 \;:\; 0< r_1,r_2\leq 1, \; r_1+r_2\varepsilon_F\in\tfrac{1}{p}\mathcal{O}_F\right\}\\ R\left(2, \tfrac{1}{p}\mathcal{O}_F\right) &\coloneqq \left\{r_3 \in \mathbb{Q}\;:\; 0< r_3 \leq 1, \; r_3\in \tfrac{1}{p}\mathcal{O}_F\right\}.\end{aligned}$$ Let $\chi_{K/F}$ be the unique quadratic character of the narrow ideal class group of $F$ with conductor $p\mathcal{O}_F$, associated to $K$. Then, assuming the notation above, we have the class number formula $$\begin{gathered} h_{K} = \frac{2\omega_K R_F}{R_K \left[\mathcal{O}_F^\times:\mathcal{O}_F^{\times,+}\right]} \left(\sum_{r\in R\left(1,\tfrac{1}{p}\mathcal{O}_F\right)} \chi_{K/F} \bigg(\left(r_1 + r_2\varepsilon_F\right)p\mathcal{O}_F\bigg)\sum_{\substack{(l_1,l_2)\in \mathbb{Z}_{\geq 0}^2\\ l_1+l_2 = 2}} \frac{B_{l_1}(r_1)B_{l_2}(r_2)}{2\cdot l_1!l_2!} \mathrm{Tr}_{F/\mathbb{Q}}\left(\varepsilon_F^{l_2-1} \right)\right.\\ - \left.\sum_{r_3\in R\left(2,\tfrac{1}{p}\mathcal{O}_F\right)} \chi_{K/F}(r_3 p\mathcal{O}_F)B_1(r_3)\right), \end{gathered}$$ where $\omega_K$ is the number of roots of unity in $K$ ([@shintani_thm], Theorem 2). We first simplify the coefficient term in this formula. Recall from Lemma   [Lemma 4](#regulator_lemma){reference-type="ref" reference="regulator_lemma"} that $R_F/R_K ~=~ 1/2$. Since $K = \mathbb{Q}(\sqrt{d},\sqrt{-p})$ for $p\geq 7,$ we have $\omega_K = 2.$ Furthermore, since we may choose the fundamental unit of $F$ to be totally positive, we see that $\mathcal{O}_F^\times = \{\pm 1\} \times \mathcal{O}_F^{\times,+}$, so we get $[\mathcal{O}_F^\times:\mathcal{O}_{F}^{\times,+}] = 2$. Then, $$\begin{aligned} \frac{2 \omega_K R_F}{R_K \left[\mathcal{O}_F^\times:\mathcal{O}_F^{\times,+}\right]} &= 1.\end{aligned}$$ Next, we reindex the sum. First we split the set $R(1, \frac{1}{p}\mathcal{O}_F)$ into two parts. Consider the sets $R_1,R_2$ given by $$\begin{aligned} R_1 &\coloneqq \left\{(r_1,r_2)\in \mathbb{Q}^2\;:\; 0<r_1\leq 1, 0< r_2< 1, r_1+r_2\varepsilon_F\in\tfrac{1}{p}\mathcal{O}_F\right\}\\ R_2 &\coloneqq \left\{(r_1,r_2)\in \mathbb{Q}^2\;:\; 0<r_1\leq 1, r_2=1, r_1+r_2\varepsilon_F\in\tfrac{1}{p}\mathcal{O}_F\right\}.\end{aligned}$$ Additionally, for simplicity we denote the inner sum of Shintani's formula by: $$\begin{aligned} \mathcal{B}(r_1+r_2\varepsilon_F) \coloneqq \sum_{\substack{(l_1,l_2)\in \mathbb{Z}_{\geq 0}^2\\ l_1+l_2 = 2}} \frac{B_{l_1}(r_1)B_{l_2}(r_2)}{2\cdot l_1!l_2!} \textrm{Tr}_{F/\mathbb{Q}}\left(\varepsilon_F^{l_2-1} \right).\end{aligned}$$ By splitting the sum with $R(1,\tfrac{1}{p}\mathcal{O}_F) = R_1\bigsqcup R_2$, we see that $$\begin{aligned} \notag &h_{K} = \sum_{r\in R_1} \chi_{K/F} \bigg(\left(r_1 + r_2\varepsilon_F\right)p\mathcal{O}_F\bigg)\mathcal{B}(r_1+r_2\varepsilon_F) \\\label{merging_Rs} &\hspace{2 cm} + \sum_{r\in R_2} \chi_{K/F} \bigg(\left(r_1 + \varepsilon_F\right)p\mathcal{O}_F\bigg)\mathcal{B}(r_1+\varepsilon_F)- \sum_{r\in R\left(2,\tfrac{1}{p}\mathcal{O}_F\right)} \chi_{K/F}(r p\mathcal{O}_F)(r-1/2).\end{aligned}$$ Since $\varepsilon_F\in\mathcal{O}_F^\times$ and $\chi_{K/F}$ has conductor $p\mathcal{O}_F,$ we have $\chi_{K/F}\big(r_1p\mathcal{O}_F\big)=\chi_{K/F}\Big((r_1 + \varepsilon_F)p\mathcal{O}_F\big)$. Moreover, comparing $\mathcal{B}(r_1)$ and $\mathcal{B}(r_1+\varepsilon_F),$ we see that $$\begin{aligned} \mathcal{B}(r_1)&=\frac{r_1^2 - r_1 + 1/3}{4}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)- \frac{r_1-1/2}{2} \\ \mathcal{B}(r_1+\varepsilon_F) &=\frac{r_1^2 - r_1 + 1/3}{4}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)+ \frac{r_1-1/2}{2}=\mathcal{B}(r_1)+r_1-1/2.\end{aligned}$$Thus [\[merging_Rs\]](#merging_Rs){reference-type="eqref" reference="merging_Rs"} simplifies to $$\begin{aligned} h_{K} = \frac{1}{2}\sum_{r\in R_{F,p}} \chi_{K/F} \bigg(\left(r_1 + r_2\varepsilon_F\right)p\mathcal{O}_F\bigg)\sum_{\substack{(l_1,l_2)\in \mathbb{Z}_{\geq 0}^2\\ l_1+l_2 = 2}} \frac{B_{l_1}(r_1)B_{l_2}(r_2)}{ l_1!l_2!} \textrm{Tr}_{F/\mathbb{Q}}\left(\varepsilon_F^{l_2-1} \right)\end{aligned}$$ where $R_{F,p}$ is given by $$\begin{aligned} R_{F,p} & = \left\{r=r_1+r_2\varepsilon_F\;:\; 0<r_1\leq 1, 0\leq r_2< 1, r\in \tfrac{1}{p}\mathcal{O}_F\right\}.\end{aligned}$$ ◻ **Definition 7**. *We call $R_{F,p}$ the **Shintani set** associated to $F$ and $p$.* ## Properties of Shintani Sets In this subsection, we identify a correspondence between $R_{F,p}$ and the finite field $\mathbb{F}_{p^2}$, which will play an important role in our proof of Theorem   [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}. Namely, we make use of this correspondence and the cyclic structure of the multiplicative group $\mathbb{F}_{p^2}^{\times}$ to enumerate the elements of $R_{F,p}-\mathcal{O}_F$ using the powers of a generator of $\mathbb{F}_{p^2}^{\times}$. Throughout this subsection, we fix a totally real quadratic field $F$ and an imaginary quadratic extension $K = F(\sqrt{-p})$, where $p \equiv 3\pmod{4}$ and $p$ remains inert in $\mathcal{O}_F$. We let $\varepsilon_F=s+t\theta_F,$ and to simplify notation, we denote $$\begin{aligned} \varepsilon\coloneqq \varepsilon_F \quad \text{and}\quad R \coloneqq R_{F,p}.\end{aligned}$$ We begin by giving an explicit construction of the Shintani set: **Lemma 8**. *The Shintani set $R$ can be written as: $$\begin{aligned} R &= %R(D_{K/F}) = %\Bigg\{ \frac{A}{tp} + \frac{B}{tp}\ep \;\ : \;A + s B \equiv 0 \pmod{t}, \; A,B\in \Z,\; A\in (0,tp], B\in [0, tp ) \Bigg\}.\\ \Bigg\{ \frac{A}{tp} + \frac{B}{tp}\varepsilon\;\ : \;A + s B \equiv 0 \pmod{t}, \; A\in (0,tp] \cap \mathbb{Z}, B\in [0, tp ) \cap \mathbb{Z}\Bigg\}. \end{aligned}$$* *Proof.* By Lemma [Lemma 5](#pof){reference-type="ref" reference="pof"}, we have $D_{K/F} = p\mathcal{O}_F.$ As such, for any element $r_1+r_2\varepsilon\in \frac{1}{p}\mathcal{O}_F,$ we have $$\begin{aligned} r_1 + r_2 \varepsilon= (r_1 + s r_2) + t r_2\theta_F \in \frac{1}{p}\mathcal{O}_F. % \implies r_1+s r_2 = \frac{m}{p} \text{ and } t r_2 = \frac{B}{p}\quad\textrm{ for some }m,B\in \Z. \end{aligned}$$ The set $\{1, \theta_F\}$ constitutes an integral basis of $\mathcal{O}_F$, meaning we can write any element of the Shintani set as $r_1+r_2\varepsilon= \frac{A'}{p} + \frac{B}{p}\theta_F \in \frac{1}{p}\mathcal{O}_F$, for some $A', B \in \mathbb{Z}$. Note that $$\begin{aligned} \frac{A'}{p} = {r_1+sr_2}\quad \text{and}\quad \frac{B}{p} = tr_2. \end{aligned}$$ In particular, we have $$\begin{aligned} r_2 = \frac{B}{tp}, \end{aligned}$$ and since $r_2 \in [0,1)$, we see that $B \in [0, tp)$. Additionally, we see that $$\begin{aligned} r_1 = \frac{A'}{p}-sr_2 = \frac{tA'-sB}{tp} = \frac{A}{tp} \end{aligned}$$ where $A\coloneqq tA'-sB.$ We know that $r_1 \in (0, 1]$, so $A \in (0, tp]$. Moreover, since $$\frac{A}{tp}+\frac{B}{tp}\varepsilon=\frac{A+sB}{tp}+\frac{B}{p}\theta_F\in\frac{1}{p}\mathcal{O}_F,$$ we must also have $A+sB\equiv 0\pmod{t}.$ From the expression above, we can see that every element of the form $\frac{A}{tp}+\frac{B}{tp}\varepsilon$ with $A,B\in\mathbb{Z}$, $A\in(0,tp]$, $B\in[0,tp)$, $A+sB\equiv 0\pmod{t}$ is in the Shintani set. This finishes the proof. ◻ Next, we want to identify $R$ with the finite field $\mathbb{F}_{p^2}$. We begin with the work of Barquero-Sanchez, Masri, and Tsai, who proved that $R$ is a finite abelian group with respect to the following operation: $$\begin{aligned} r \oplus r' \coloneqq r + r' + \mathbb{Z}[\varepsilon]\end{aligned}$$ (see Proposition 4.3 in [@wei_lun_stark_units]). This allows us to prove the following proposition relating $R$ to $\mathbb{F}_{p^2}$, a property that is central to our proof of Theorem [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}. **Proposition 9**. *The Shintani set $R$ has a structure as a $\mathbb{Z}[\varepsilon]$-module. This structure admits a surjective $\mathbb{Z}[\varepsilon]$-module homomorphism $\pi: R\to \mathbb{F}_{p^2}.$* *Proof.* We begin with the $\mathbb{Z}[\varepsilon]$-module structure on $R$. By definition, the fractional ideal $\frac{1}{p}\mathcal{O}_F$ is an $\mathcal{O}_F$-module, and since $\mathbb{Z}[\varepsilon]$ is a subring of $\mathcal{O}_F$, we observe that $\frac{1}{p}\mathcal{O}_F$ is a $\mathbb{Z}[\varepsilon]$-module by restriction of scalars. Furthermore, since $\mathbb{Z}[\varepsilon]$ is a $\mathbb{Z}[\varepsilon]$-submodule of $\frac{1}{p}\mathcal{O}_F$, we have that $\tfrac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$ is a $\mathbb{Z}[\varepsilon]$-module. Moreover, $R\subset\tfrac{1}{p}\mathcal{O}_F$ is a complete reduced set of coset representatices for $\tfrac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$ (see Proposition 4.1 in [@wei_lun_stark_units]). Thus $R$ has the structure of a $\mathbb{Z}[\varepsilon]$-module and can be identified with $\tfrac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon].$ Since $R \subseteq \frac{1}{p}\mathcal{O}_F$, multiplication by $p$ defines an injective $\mathbb{Z}[\varepsilon]$-module homomorphism $$R \longrightarrow \mathcal{O}_F, \hspace{0.5cm} r \longmapsto pr+p\mathcal{O}_F.$$ If we compose this map with the projection $\mathcal{O}_F \longrightarrow \mathcal{O}_F/p\mathcal{O}_F,$ we obtain the map $$\pi: R \longrightarrow \mathcal{O}_F/p\mathcal{O}_F, \hspace{0.5cm} r \longmapsto pr,$$ which is surjective, as shown in Proposition 4.4 of [@wei_lun_stark_units]. If $p$ remains inert in $F$, then $p\mathcal{O}_F$ is prime and thus maximal, so $\mathcal{O}_F/p\mathcal{O}_F$ is a finite field. Then, since $F$ is quadratic and $\mathcal{O}_F=\mathbb{Z}[\theta_F]$, we know that $$\begin{aligned} \mathcal{O}_F/p\mathcal{O}_F \cong \mathbb{Z}[\theta_F]/p\mathbb{Z}[\theta_F] \cong \mathbb{F}_{p}[\theta]\cong\mathbb{F}_{p^2}.\end{aligned}$$ ◻ **Remark 2**. By the First Isomorphism Theorem, $$R/\ker(\pi) \cong \mathcal{O}_F/p\mathcal{O}_F.$$ Note that this is an isomorphism of groups, and therefore pertains only to the structure of $R$ as an additive abelian group. We do not require a multiplicative structure within $R$ here; rather we point out that the map $$\pi: R/\ker(\pi) \to \mathbb{F}_{p^2}$$ is bijective. We will make use of this bijective correspondence in the proof of Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}. **Lemma 10**. *The elements of $\ker(\pi)$ are exactly those elements of $R$ which are in $\mathcal{O}_F.$* *Proof.* Assume $r \in \ker(\pi)$. Then, since $\mathrm{Im}(\pi) = \mathcal{O}_F/p\mathcal{O}_F$, we have $$\begin{aligned} \pi(r) =0 \iff pr \in p\mathcal{O}_{F} \iff r\in \mathcal{O}_{F}. \end{aligned}$$ ◻ Now we are in a position to explicitly describe the elements in $\ker(\pi)$. **Lemma 11**. *The kernel of the map $\pi$ is given by $$\begin{aligned} \ker(\pi) = \left\{1 - \bigg\{\frac{si}{t}\bigg\}_{[0,1)}+ \frac{i}{t}\varepsilon\;|\; 0\leq i \leq t-1\right\} \end{aligned}$$ In particular, we have that $|\ker(\pi)| = t$.* *Proof.* Consider $r\in\ker(\pi).$ Using Lemma [Lemma 8](#shin_set_construction){reference-type="ref" reference="shin_set_construction"} and noting that $\varepsilon=s+t\theta_F,$ we see that $r$ has the form $$\begin{aligned} r=\frac{A}{tp}+\frac{B}{tp}\varepsilon=\frac{A+sB}{tp}+\frac{B}{p}\theta_F. \end{aligned}$$ By Lemma [Lemma 10](#kernel_of_phi_lemma){reference-type="ref" reference="kernel_of_phi_lemma"}, $r\in\ker(\pi)\iff r\in R\cap\mathcal{O}_F,$ so we have that $\tfrac{A+sB}{tp}\in\mathbb{Z}$ and $\tfrac{B}{p}\in\mathbb{Z}.$ The second condition implies $p|B,$ and since $B\in[0,tp)\cap\mathbb{Z}$ by Lemma [Lemma 8](#shin_set_construction){reference-type="ref" reference="shin_set_construction"}, we see that $B=pi$ for $i\in[0,t)\cap\mathbb{Z}.$ The condition that $\tfrac{A+sB}{tp}\in\mathbb{Z}$ implies that $A\equiv -sB=spi\pmod{tp}.$ Since $A\in(0,tp]\cap\mathbb{Z}$ by Lemma [Lemma 8](#shin_set_construction){reference-type="ref" reference="shin_set_construction"}, $A$ is uniquely determined by $B.$ More precisely, $$\begin{aligned} A=tp-(spi\pmod{tp}) \end{aligned}$$ where $spi\pmod{tp}$ is the least positive residue of $spi\in\mathbb{Z}$ modulo $tp.$ We can further simplify this expression; since $$spi \pmod{tp} = spi - tp\bigg\lfloor \frac{spi}{tp}\bigg\rfloor,$$ we have that $$\begin{aligned} \ker(\pi)\subseteq\left\{1 - \bigg\{\frac{si}{t}\bigg\}_{[0,1)}+ \frac{i}{t}\varepsilon\;:\; i\in[0,t)\cap\mathbb{Z}\right\}, \end{aligned}$$ where $\{\cdot\}$ denotes the fractional part function $\{x\}_I$, defined as the unique element of $I$ satisfying $x-\{ x\}_I \in \mathbb{Z}$. The converse containment is seen immediately from the fact that that $\varepsilon=s+t\theta_F$ and the definition of $spi\pmod{tp}.$ Thus, $\ker(\pi)$ has size exactly $t.$ ◻ We will denote the elements of $\ker(\pi)$ as $$\kappa_i \coloneqq 1 - \bigg\{\frac{si}{t}\bigg\}_{[0,1)}+ \frac{i}{t}\varepsilon\quad\text{for}\quad i\in\{0,1,\ldots,t-1\}.$$ # Proof of Theorem 1.1 {#section_1.1_proof} Equipped with these facts about the Shintani set described in the previous section, we now prove Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}. Our proof relies on features of the structure of the Shintani set which come from from the bijection between $R_{F,p}/\ker(\pi)$ and $\mathbb{F}_{p^2}$, as well as some properties we derive of the Hecke character across the Shintani set. Again, to simplify notation, we let $\varepsilon= \varepsilon_F$, $R =R_{F,p}$, and $\rho = \rho_{F,p}.$ We are now able to describe the Shintani set using the multiplicative structure of $\mathbb{F}_{p^2}^\times = \langle \rho +p\mathcal{O}_F\rangle$. Using the bijection from $R/\ker(\pi)$ to $\mathbb{F}_{p^2}$, we have $$\begin{aligned} R = \ker(\pi) \sqcup \left(\bigsqcup_{m=1}^{p^2-1} \pi^{-1}(\rho^m+p\mathcal{O}_F)\right).\end{aligned}$$ For each $m$ between $1$ and $p^2-1$, choose one element in the coset $\pi^{-1}(\rho^m+p\mathcal{O}_F)$, which we denote $\tilde{x}(m) + \tilde{y}(m)\varepsilon\in R$. Next, we explicitly calculate each $\tilde{x}(m)$ and $\tilde{y}(m)$ in terms of $\rho^m$. Note that $\{1, \theta_F\}$ is a $\mathbb{F}_{p}$-basis of $\mathbb{F}_{p^2}$, so we can write $\rho^m+p\mathcal{O}_F \coloneqq x(m) + y(m)\theta_F$ for some integers $x(m), y(m)$. Observe that, since $\varepsilon= s + t\theta_F$, we have $$\begin{aligned} \rho^m+p\mathcal{O}_F = x(m) - \frac{s\cdot y(m)}{t} + \frac{y(m)}{t}\varepsilon.\end{aligned}$$ Under multiplication by $p$ and reduction modulo $p$, the point $$\frac{x(m)}{p} - \frac{s\cdot y(m)}{tp} + \frac{y(m)}{tp}\varepsilon\in \frac{1}{p}\mathcal{O}_F$$ maps to $\rho^m+p\mathcal{O}_F$. Thus, if we subtract a suitable element of $\mathbb{Z}[\varepsilon]$ from this point, we obtain a point $\tilde{x}(m) + \tilde{y}(m)\varepsilon\in R$ that is a preimage of $\pi^{-1}(\rho^m+p\mathcal{O}_F)$. In particular, we see that $$\begin{aligned} \tilde{x}(m) = \left\{\frac{{x(m)}}{p} - \frac{s\cdot y(m)}{tp}\right\}_{(0,1]}, \hspace{0.5cm} \tilde{y}(m) = \left\{\frac{y(m)}{tp}\right\}_{[0,1)}.\end{aligned}$$ Since $\pi(\tilde{x}(m) + \tilde{y}(m)\varepsilon) = \rho^m+p\mathcal{O}_F$, we can construct the entire coset from this element: $$\begin{aligned} \pi^{-1}(\rho^m+p\mathcal{O}_F) = \bigg\{(\tilde{x}(m) + \tilde{y}(m)\varepsilon) \oplus \kappa_i : 1 \le i \le t\bigg\}.\end{aligned}$$ For simplicity, we write $$\begin{aligned} \tilde{x}_i(m) + \tilde{y}_i(m)\varepsilon\coloneqq (\tilde{x}(m) + \tilde{y}(m)\varepsilon) \oplus \kappa_i.\end{aligned}$$ Using our explicit construction of $\ker(\pi)$ given in Lemma  [Lemma 11](#explicit_kernel){reference-type="ref" reference="explicit_kernel"}, we can similarly explicitly construct each $\tilde{x}_i(m)$, $\tilde{y}_i(m)$. We see that $$\begin{aligned} \tilde{x}_i(m) &= \left\{\frac{x(m)}{p} - \frac{s\cdot y(m)}{tp} + 1 - \bigg\{\frac{si}{t}\bigg\}_{[0,1)} \right\}_{(0,1]}\\ \tilde{y}_i(m) &= \left\{\frac{y(m)}{tp}+ \frac{i}{t}\right\}_{[0,1)}.\end{aligned}$$ Thus, we can write $R$ as the following disjoint union: $$\begin{aligned} R = \ker(\pi) \sqcup \left(\bigsqcup_{m=1}^{p^2-1}\bigg\{\tilde{x}_i(m) + \tilde{y}_i(m)\varepsilon: 1 \le i \le t\bigg\}\right).\end{aligned}$$ By Proposition [Proposition 6](#shintani_simplified){reference-type="ref" reference="shintani_simplified"}, we simplify Shintani's class number formula to obtain $$\begin{aligned} \label{thm1_unsimplified} h_{K} = \frac{1}{2}\sum_{\substack{1\leq m\leq p^2-1\\ 1\leq i \leq t}} \chi_{K/F}\bigg((\tilde{x}_i(m) + \tilde{y}_i(m)\varepsilon) \cdot p\mathcal{O}_F\bigg) \cdot\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(\tilde{x}_i(m))}{l_1!}\frac{B_{l_2}(\tilde{y}_{i}(m))}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)^{l_2-1}}.\end{aligned}$$ We can also simplify the Hecke character term. Consider any element $r_1+r_2\varepsilon\in R$ and any element $k_1+k_2\varepsilon\in \ker(\pi)$. By Lemma  [Lemma 10](#kernel_of_phi_lemma){reference-type="ref" reference="kernel_of_phi_lemma"}, we have $k_1+k_2\varepsilon\in\mathcal{O}_F.$ Since the Hecke character has conductor $p\mathcal{O}_F,$ we have $$\begin{aligned} \chi_{K/F}\bigg((r_1+r_2\varepsilon+k_1+k_2\varepsilon)p\mathcal{O}_F\bigg)=\chi_{K/F}\bigg(\left((r_1+r_2\varepsilon)p\mathcal{O}_F\right)+p\mathcal{O}_F\bigg) =\chi_{K/F}\bigg((r_1+r_2\varepsilon)p\mathcal{O}_F\bigg).\end{aligned}$$ Thus, the value of $\chi_{K/F}(r_1 + r_2\varepsilon)$ depends only on the coset of $r_1 + r_2\varepsilon$ in $R/\ker(\pi)$. Therefore, we have that $$\begin{aligned} \chi_{K/F}\bigg((\tilde{x}_i(m)+\tilde{y}_i(m)\varepsilon)p\mathcal{O}_F\bigg) &= \chi_{K/F}\bigg((\tilde{x}(m)+\tilde{y}(m)\varepsilon)p\mathcal{O}_F\bigg).\end{aligned}$$ By definition, $$p(\tilde{x}(m)+\tilde{y}(m)\varepsilon)-\rho^m \in p\mathcal{O}_F.$$ Using this and the multiplicativity of the Hecke character, we get $$\begin{aligned} \chi_{K/F}((\tilde{x}_i(m)+\tilde{y}_i(m)\varepsilon)p\mathcal{O}_F) = \chi_{K/F}((\rho^m+p\mathcal{O}_F)\mathcal{O}_F) = \chi_{K/F}((\rho+p\mathcal{O}_F)\mathcal{O}_F)^m.\end{aligned}$$ If $\chi_{K/F}((\rho+p\mathcal{O}_F)\mathcal{O}_F) = 0$, then since $(\mathcal{O}_F/p\mathcal{O}_F)^{\times} = \langle \rho+p\mathcal{O}_F \rangle$, we would have that $\chi_{K/F}(rp\mathcal{O}_F)=0$ for all $r\in R.$ However, this contradicts the definition of $\chi_{K/F}$. Moreover, $\chi_{K/F}((\rho+p\mathcal{O}_F)\mathcal{O}_F) \neq 1,$ since we would then similarly have that $\chi_{K/F}(rp\mathcal{O}_F)=1$ for all $r\in R-\mathcal{O}_F,$ but $\chi_{K/F}$ is a non-trivial character by construction. Thus, we see that $\chi_{K/F}((\rho+p\mathcal{O}_F)\mathcal{O}_F) = -1,$ which implies $$\begin{aligned} \chi_{K/F}\bigg((\tilde{x}_i(m)+\tilde{y}_i(m)\varepsilon)p\mathcal{O}_F\bigg) = \chi_{K/F}((\rho+p\mathcal{O}_F)\mathcal{O}_F)^m=(-1)^m.\end{aligned}$$ Thus, Equation [\[thm1_unsimplified\]](#thm1_unsimplified){reference-type="ref" reference="thm1_unsimplified"} simplifies further: $$\begin{aligned} h_{K} = \frac{1}{2}\sum_{\substack{1\leq m\leq p^2-1\\1\leq i \leq t}}(-1)^{m}\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(\tilde{x}_i(m))}{l_1!}\frac{B_{l_2}(\tilde{y}_i(m))}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)^{l_2-1}}.\end{aligned}$$ Next, we simplify the Bernoulli polynomial part of the class number formula. We consider $$\begin{aligned} &\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(\tilde{x}_i(m))}{l_1!}\frac{B_{l_2}(\tilde{y}_i(m))}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)^{l_2-1}} \\& \hspace{1 cm}= \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\frac{\tilde{x}_i(m)^2-\tilde{x}_i(m)+1/6}{2} + 2\left(\tilde{x}_i(m)-\frac{1}{2}\right)\left(\tilde{y}_i(m)-\frac{1}{2}\right) +\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\frac{\tilde{y}_i(m)^2-\tilde{y}_i(m)+1/6}{2}\\ &\hspace{1 cm}= \frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2} \left(\tilde{x}_i(m) - \frac{1}{2}\right)^2 + 4\left(\tilde{x}_i(m) - \frac{1}{2}\right)\left(\tilde{y}_i(m) - \frac{1}{2}\right) + \frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2} \left(\tilde{y}_i(m) - \frac{1}{2}\right)^2 + c_0,\end{aligned}$$ for some constant $c_0$. Since for any constant $c$, $$\sum_{\substack{1\leq m \leq p^2-1\\ 1\leq i \leq t}}(-1)^m \cdot c =0,$$ we can ignore the constant term $c_0$ that arises in the inner sum of Bernoulli polynomials. We can write the class number $h_K$ as $$\label{longscaryeqn} h_{K} = \frac{1}{4}\sum_{\substack{1\leq m\leq p^2-1\\1\leq i \leq t}}(-1)^{m} \Bigg[\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\left(\tilde{x}_i(m) - \frac{1}{2}\right)^2 + 4\left(\tilde{x}_i(m) - \frac{1}{2}\right)\left(\tilde{y}_i(m) - \frac{1}{2}\right) + \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\left(\tilde{y}_i(m) - \frac{1}{2}\right)^2\Bigg].$$ If we define $$\label{tilde_defns} x_i(m) \coloneqq tp(2\tilde{x}_i(m)-1) \quad\text{and}\quad y_i(m) \coloneqq tp(2\tilde{y}_i(m)-1),$$ we can then rewrite the above equation as $$\begin{aligned} \label{actualscaryeqn} h_{K} &= \frac{1}{16t^2p^2}\sum_{\substack{1\leq m \leq p^2-1\\ 1\leq i\leq t}}(-1)^m \bigg[\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\Big({x}_i(m)\Big)^2 + 4\Big({x}_i(m)\Big)\Big({y}_i(m)\Big) + \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\Big({y}_i(m)\Big)^2\bigg].\end{aligned}$$ Finally, by defining the quadratic form $$Q_{F}(Y_1, Y_2)\coloneqq \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)Y_1^2+4Y_1Y_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)Y_2^2,$$ we rewrite [\[actualscaryeqn\]](#actualscaryeqn){reference-type="eqref" reference="actualscaryeqn"} as $$\begin{aligned} h_K=\frac{1}{16t^2 p^2}\sum_{\substack{1\leq m\leq p^2-1\\ 1\leq i\leq t}}(-1)^mQ_{F}\left({x}_i(m),{y}_i(m)\right). \end{aligned}$$ The last step is to derive recurrence relations for $x(m), y(m)$, the coefficients of $\rho^m +p\mathcal{O}_F = x(m) + y(m)\theta_F$. The minimal polynomial of $\theta_F$ is $x^2 - \textrm{Tr}(\theta_F)x + \textrm{Norm}_{F/\mathbb{Q}}(\theta_F)$, which implies $$\theta_F^2 = \textrm{Tr}_{F/\mathbb{Q}}(\theta_F)\theta_F - \textrm{Norm}_{F/\mathbb{Q}}(\theta_F).$$ To simplify notation, let $T=\textrm{Tr}_{F/\mathbb{Q}}(\theta_F)$ and $N=\textrm{Norm}_{F/\mathbb{Q}}(\theta_F)$. Since $\rho\coloneqq a+b\varepsilon,$ we have the initial conditions $x(1) = a$ and $y(1) = b$. Then, we get $$\begin{aligned} \rho^{m+1} = x(m+1) + y(m+1)\theta_F &= \big(x(m) + y(m)\theta_F\big)\cdot \big(a+b\theta_F\big)\\ &= a\cdot x(m)-Nb\cdot y(m)+\bigg(b\cdot x(m)+\bigg(a+Tb\bigg)\cdot y(m)\bigg)\theta_F.\end{aligned}$$ This implies the following recurrence relations: $$\begin{aligned} x(m+1) &= a\cdot x(m)-Nb\cdot y(m)\\ y(m+1) &= b\cdot x(m)+(a+Tb)\cdot y(m). \end{aligned}$$ Then, consider functions $X(z)$, $Y(z)$ given by $$\begin{aligned} X(z) = \sum_{m=1}^\infty x(m)\cdot z^m,\quad Y(z) = \sum_{m=1}^\infty y(m)\cdot z^m. \end{aligned}$$ Using our recurrence relations, we can set up a system of equations to find explicit expressions for $X(z), Y(z)$ as rational functions determined by $x(1)$ and $y(1)$. We see that $$\begin{aligned} \notag X(z) &= z\cdot \left[a\cdot X(z) - N b\cdot Y(z)\right] + az\\\notag Y(z) &= z\cdot \left[ b\cdot X(z) + \bigg(a + Tb\bigg)\cdot Y(z)\right] + bz,\end{aligned}$$ which gives $$\begin{aligned} \notag X(z) &= \dfrac{az - (a^2 + abT + Nb^2)z^2}{(a^2 + abT + Nb^2)z^2 - (2a + bT)z + 1} \\\notag Y(z) &= \dfrac{bz}{(a^{2} + abT + b^{2}N)z^{2} - (2a + bT)z + 1} .\end{aligned}$$ We simplify these by letting $C_{F,p} \coloneqq a^{2} + abT + Nb^{2}$ and $D_{F,p} \coloneqq 2a + bT$ to get $$\begin{aligned} \notag X(z) = \dfrac{az - C_{F,p}z^2}{C_{F,p}z^2 - D_{F,p}z + 1},\quad Y(z) = \dfrac{bz}{C_{F,p}z^{2} - D_{F,p}z + 1}.\end{aligned}$$ Note that the coefficients $x(m), y(m)$ of the power series of these rational functions correspond to those $x(m), y(m)$ which we use to generate each $x_i(m), y_i(m)$ using the formulas $$\begin{aligned} x_i(m) \coloneqq tp(2\tilde{x}_i(m)-1) \quad\text{and}\quad y_i(m) \coloneqq tp(2\tilde{y}_i(m)-1). \end{aligned}$$This concludes the proof of Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"}. # Proof of Theorem 1.2 {#section4} Here we prove Theorem $~\ref{thm 1.2},$ which relies heavily on the structure of the Shintani set as a $\mathbb{Z}[\varepsilon_F]$-module and the related attributes of the base-$\varepsilon_F$ expansions of its elements. Through a series of preliminary lemmas, we set up the proof of Theorem  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} by relating the base-$\varepsilon_F$ expansion of $1/p$ to the orbit of elements in $R_{F,p}-\ker(\pi)$ under the action of $\varepsilon_F$. This allows us to derive a finite sum analogous to Girstmair's [\[GirstmairFormula\]](#GirstmairFormula){reference-type="eqref" reference="GirstmairFormula"}, in which the number of summands is equal to the period length of the base $\varepsilon_F$ expansion of $1/p$. Thoughout this section, we fix a totally real quadratic field $F$ and an imaginary quadratic extension $K \coloneqq F(\sqrt{-p})$, where $p \equiv 3\pmod{4}$ and $p$ remains inert in $\mathcal{O}_F$. To simplify notation, we also let $$\begin{aligned} \varepsilon\coloneqq \varepsilon_F \quad\text{and}\quad R \coloneqq R_{F,p}.\end{aligned}$$ Additionally, we denote $r \in R$ as $r \coloneqq r_1 + r_2\varepsilon$. ## Shintani Cycles Recall from Section  [2](#sec2){reference-type="ref" reference="sec2"} that we can identify $R$ with $\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$ to make it into a $\mathbb{Z}[\varepsilon]$ module. In particular, the multiplicative group $\langle \varepsilon\rangle$ acts on $\frac{1}{p}\mathcal{O}_F$ via scalar multiplication. If we denote the map for this group action by $$\mu: \langle \varepsilon\rangle \times \frac{1}{p}\mathcal{O}_F \longrightarrow \frac{1}{p}\mathcal{O}_F,$$ we can compose $\mu$ with the projection map $$\nu: \frac{1}{p}\mathcal{O}_F \longrightarrow \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$$ to yield $$\mu' \coloneqq \nu\circ\mu: \langle \varepsilon\rangle \times \frac{1}{p}\mathcal{O}_F \longrightarrow \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon].$$ Note that, since $\nu$ is a $\mathbb{Z}[\varepsilon]$-module homomorphism, $\mu'$ constitutes a group action of $\langle\varepsilon\rangle$ on $\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon].$ **Lemma 12**. *The map $$\begin{aligned} \overline{\mu}: \langle \varepsilon\rangle \times \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon] \longrightarrow \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon], \hspace{0.3 cm} (\varepsilon, \alpha + \mathbb{Z}[\varepsilon]) \longmapsto \mu'(\varepsilon, \alpha) \end{aligned}$$is a well-defined group action.* *Proof.* Since $\mu'$ is a group action, it is sufficient to show that $\overline{\mu}$ is well-defined. Take $\alpha, \alpha' \in \frac{1}{p}\mathcal{O}_F$ such that $\alpha + \mathbb{Z}[\varepsilon] = \alpha' + \mathbb{Z}[\varepsilon]$. Thus, for any $n \in \mathbb{Z}$, we have $$\begin{aligned} \overline{\mu}(\varepsilon^n,\alpha + \mathbb{Z}[\varepsilon]) &= \mu'(\varepsilon^n,\alpha + \mathbb{Z}[\varepsilon])\\ &= \varepsilon^n\alpha + \mathbb{Z}[\varepsilon] = \varepsilon^n\alpha' + \mathbb{Z}[\varepsilon]\\ &= \overline{\mu}(\varepsilon^n, \alpha' + \mathbb{Z}[\varepsilon]). \end{aligned}$$ Thus, $\overline{\mu}$ is well-defined, so $\overline{\mu}$ constitutes a group action of $\langle \varepsilon\rangle$ on $\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$. ◻ We know from Proposition 4.1 in [@wei_lun_stark_units] that $R$ is a complete and reduced set of representatives of $\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$. Thus, we obtain a group action $\langle\varepsilon\rangle \curvearrowright R$ given by $(\varepsilon,r) \mapsto \varepsilon\ast r,$ where $\varepsilon\ast r = \varepsilon r + z,$ and $z$ is the unique element of $\mathbb{Z}[\varepsilon]$ such that $\varepsilon r + z \in R$. We define the *Shintani cycle* of any element $r\in R$ to be the orbit of $r$ under this action, and we denote this set as $C_r \coloneqq \langle \varepsilon\rangle \ast r.$ **Remark 3**. Note that $\varepsilon\ast r \in \mathcal{O}_F \iff r \in \mathcal{O}_F$. Thus, for any $r \in R \cap \mathcal{O}_F$, every element in the Shintani cycle of $r$ is an element of $\mathcal{O}_F$. We will call Shintani cycles containing elements in $R -\mathcal{O}_F$ the *nontrivial Shintani cycles* of $R$. We refer to Shintani cycles of elements in $R\cap\mathcal{O}_F$ as *trivial Shintani cycles* because the elements in these cycles are weighted by a factor of $0$ in Shintani's class number formula (see the remark in Section [4.3](#thm1.2proof_subsection){reference-type="ref" reference="thm1.2proof_subsection"}), and hence for our purposes are "trivial.\" ## Epsilon Expansions {#ep_exp_section} A base-$\varepsilon$ expansion is an analogue to the usual decimal expansion. The base-$\varepsilon$ expansion of any element $\alpha\in F$ is computed in the following way. Let $n\coloneqq\lfloor\log_{\varepsilon}(\alpha)\rfloor.$ Then we have $$\alpha=a_n\varepsilon^n+a_{n-1}\varepsilon^{n-1}+\ldots+a_0+a_{-1}\varepsilon^{-1}+\ldots,$$where $$\begin{aligned} a_n\coloneqq\lfloor\alpha/\varepsilon^n\rfloor, a_{n-1}\coloneqq\lfloor(\alpha-a_n\varepsilon^n)/\varepsilon^{n-1}\rfloor,\ldots, a_i\coloneqq\lfloor(\alpha-a_n\varepsilon^n-\ldots-a_{i+1}\varepsilon^{i+1})/\varepsilon^{i}\rfloor, \ldots\end{aligned}$$ We observe that $\varepsilon$ is an algebraic integer which is real since $F$ is real quadratic, and that $\varepsilon$ must be $>1$ since it is a totally positive fundamental unit. Moreover, $\varepsilon$ must have Galois conjugate with absolute value $<1$ since $F$ is a real quadratic field and $\varepsilon$ has norm $1$. Thus, $\varepsilon$ is a Pisot number by definition, and by consequence, Theorem $3.1$ in [@pisot] shows that any element of $R$ has an eventually periodic base-$\varepsilon$ expansion. For some $\alpha \in F$ whose base $\varepsilon$ expansion can be written as $$\alpha = a_n\varepsilon^n+\ldots+a_0+a_{-1}\varepsilon^{-1}+\ldots+a_{-k}\varepsilon^{-k}+\overline{a_{-k-1}\varepsilon^{-k-1} + \ldots + a_{-k-P_\alpha}\varepsilon^{-k-P_\alpha}},$$ we call $P_\alpha$ the *period length* of the base-$\varepsilon$ expansion of $\alpha$. Additionally, we will call the ordered set $$\{a_{-k-1}, \ldots, a_{-k-P_\alpha}\}$$ the *period set* of the base-$\varepsilon$ expansion of $\alpha$. We can further observe that any element of $F$ whose base-$\varepsilon$ expansion is finite is an element of $\mathbb{Z}[\varepsilon]$, by the following argument. **Lemma 13**. *If $\alpha \in F$ has a finite base-$\varepsilon$ expansion, then $\alpha \in \mathbb{Z}[\varepsilon]$.* *Proof.* If $\gamma$ has a finite base-$\varepsilon$ expansion, we can express it as $$\begin{aligned} \gamma=\sum_{i=K_1}^{K_2}m_i\varepsilon^i \end{aligned}$$ where $K_1, K_2$ are integers. Using that $\varepsilon^2=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\varepsilon-1$ and that $\varepsilon^{-1}=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)-\varepsilon$, we can perform the following replacement on any term $m_k\varepsilon^k$ where $k\neq 0$ or $1$: $$\begin{aligned} m_k\varepsilon^k = \begin{cases} m_k(\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\varepsilon-1)^{k/2} & \text{if }k \text{ is an even positive integer}\\ m_k(\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\varepsilon-1)^{(k-1)/2}\varepsilon& \text{if }k \text{ is an odd positive integer}\\ m_k(\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)-\varepsilon)^{-k} & \text{if }k \text{ is a negative integer}. \end{cases} \end{aligned}$$ The third equality implies that any negative integer power of $\varepsilon$ can be converted to a linear combination of positive integer powers of $\varepsilon$. Therefore, it suffices to show that a linear combination of positive integer powers of $\varepsilon$ can be expressed as an element of $\mathbb{Z}[\varepsilon]$. The first two equalities guarantee that any positive power of $\varepsilon$ can be expressed a strictly lower positive power of $\varepsilon$. Thus by induction, any finite linear combination of (possibly negative) powers of $\varepsilon$ can be expressed as an element of $\mathbb{Z}[\varepsilon]$. ◻ **Proposition 14**. *The repeating part in the base-$\varepsilon$ expansion of any two elements in the same Shintani cycle is the same.* To prove this lemma, we require some preliminaries. Consider some $r \in R$, where $r = r_1 + r_2\varepsilon$, and recall that $0 < r_1 \leq 1$ and $0 \leq r_2 < 1$, $r_1, r_2 \in \mathbb{Q}$. Hence the action of $\varepsilon$ on $R$ amounts to: $$\begin{aligned} \varepsilon\ast r &= \varepsilon\cdot(r_1+r_2\varepsilon)+z_1+z_2\varepsilon\end{aligned}$$ where $z_1+z_2\varepsilon$ is the unique element in $\mathbb{Z}[\varepsilon]$ such that $\varepsilon\cdot(r_1+r_2\varepsilon)+z_1+z_2\varepsilon\in R$. We can explicitly compute bounds for $z_1$ and $z_2$: **Lemma 15**. *If $\varepsilon\ast(r_1+r_2\varepsilon)=\varepsilon\cdot(r_1+r_2\varepsilon)+z_1+z_2\varepsilon$, then $$\begin{aligned} z_1 = 1,\quad\text{and}\quad z_2 = -\lfloor r_1+r_2\emph{Tr}_{F/\mathbb{Q}}(\varepsilon)\rfloor. \end{aligned}$$* *Proof.* The minimal polynomial of $\varepsilon$ is $$x^2 - \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)x + \textrm{Norm}_{F/\mathbb{Q}}(\varepsilon) = x^2 - \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)x + 1,$$ and thus $$\varepsilon^2 = \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\varepsilon- 1.$$ Consider $$\begin{aligned} \varepsilon(r_1 + r_2\varepsilon) = r_1\varepsilon+ r_2\varepsilon^2 = -r_2 + (r_1+r_2\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon))\varepsilon. \end{aligned}$$ To find $\varepsilon\ast(r_1+r_2\varepsilon)$, we must shift $\varepsilon(r_1+r_2\varepsilon)$ by some $z_1+z_2\varepsilon\in \mathbb{Z}[\varepsilon]$ such that $$\begin{aligned} -r_2 + z_1 \in (0,1],\quad\text{and}\quad r_1+r_2\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)+ z_2 \in [0,1). \end{aligned}$$ It is immediately apparent that $z_2 = -\lfloor r_1+r_2\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\rfloor.$ Further, since $r_1+r_2\varepsilon\in R$, we have that $r_2 \in [0,1)$, so we see that $z_1 = 1$. ◻ Note that the above proposition and our bounds on $r_1$ and $r_2$ imply that $-\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq z_2 \leq 0$. Before we show that the repeating part of the base $\varepsilon$ of elements in the same Shintani cycle is the same, we require one more fact about $\varepsilon$, which we now prove. **Lemma 16**. *We have that $\left\lceil\varepsilon\right \rceil=\emph{Tr}_{F/\mathbb{Q}}(\varepsilon)$.* *Proof.* Let $\varepsilon=s+t\sqrt{d}$. We start by showing $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\geq\left\lceil \varepsilon\right\rceil$. Observe that $$\begin{aligned} \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)= \varepsilon+\frac{1}{\varepsilon} \implies \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)>\varepsilon \implies \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\geq\left\lceil \varepsilon\right\rceil,\end{aligned}$$ because the trace of an algebraic integer is always an element of $\mathbb{Z}$. Now we will show that $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq \lceil \varepsilon\rceil$. Assume for the sake of contradiction that $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\ge \lceil \varepsilon\rceil + 1$. Then we have that $$\begin{aligned} \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\ge \lceil \varepsilon\rceil + 1 \implies \frac{\varepsilon^2+1}{\varepsilon} \ge \lceil\varepsilon\rceil + 1 \implies 1 - \varepsilon\ge \varepsilon\lceil\varepsilon\rceil - \varepsilon^2 \ge 0 \implies 1 \ge \varepsilon.\end{aligned}$$ However, by definition, $\varepsilon> 1$, so we see that $$\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)< \lceil\varepsilon\rceil + 1 \implies \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\le \lceil\varepsilon\rceil.$$ ◻ Now we proceed to prove that the repeating part in the base-$\varepsilon$ expansion of any two elements in the same Shintani cycle is the same. *Proof of Proposition [Proposition 14](#same_repeating_part){reference-type="ref" reference="same_repeating_part"}.* Consider some element $r\in R,$ with base-$\varepsilon$ expansion $$\begin{aligned} r &= a_1\varepsilon+ a_0+ a_{-1} \varepsilon^{-1} + a_{-2} \varepsilon^{-2} + a_{-3}\varepsilon^{-3} + \ldots \end{aligned}$$ Note that since $r\in R,$ $\lfloor\log_{\varepsilon}(r)\rfloor = 0\text{ or } 1,$ so the highest power of $\varepsilon$ appearing in the base-$\varepsilon$ expansion of $r$ is at most $1.$ Given this base-$\varepsilon$ expansion of $r,$ we have that $$\label{ep_star_r} \varepsilon\ast r = \varepsilon\cdot r + z_2\varepsilon+1 = a_{1}\varepsilon^2 + (a_0 + z_2)\varepsilon^1 +( a_{-1}+1) \varepsilon^0 + a_{-2}\varepsilon^{-1} + a_{-3}\varepsilon^{-2} + \ldots$$ Recall that in a base-$\varepsilon$ expansion, each digit (in this case $a_i$ for $i\in\mathbb{Z}$) must be an element of the set $A\coloneqq\{0,1,\ldots, \lfloor\varepsilon\rfloor\}.$ We now consider the following two cases: in Case $1,$ both $a_0 + z_2$ and $a_{-1}+1$ are in $A$; in Case $2,$ one or both of $a_0+z_2$ and $a_{-1}+1$ is not in $A.$ **Case 1.** In Case $1,$ the expression in [\[ep_star_r\]](#ep_star_r){reference-type="eqref" reference="ep_star_r"} is already a valid base-$\varepsilon$ expansion of $\varepsilon\ast r.$ We can see that only a finite number of digits differ between the base-$\varepsilon$ expansion of $\varepsilon\ast r$ and the base-$\varepsilon$ expansion of $r,$ so in this case the repeating part of $\varepsilon\ast r$ must be the same as $r.$ **Case 2.** Now we address Case $2,$ which we can split into Case $2.1$ and Case $2.2.$ In Case $2.1,$ $a_0+z_2\not\in A;$ in Case $2.2,$ $a_{-1}+1\not\in A.$ **Case 2.1.** Assume that $a_0+z_2\not\in A.$ Since $a_0$ is a digit in the base-$\varepsilon$ expansion of $r$, $0\leq a_0\leq\lfloor\varepsilon\rfloor$ by definition. Additionally, $-\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq z_2\leq 0$ by Lemma  [Lemma 15](#prop:abbounds){reference-type="ref" reference="prop:abbounds"}. Thus it always true that $-\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq a_0+z_2\leq \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon).$ Therefore if $a_0+z_2\not\in A,$ it must be that $-\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq a_0+z_2\leq -1.$ Then we have that $0\leq a_0+z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\leq\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)-1,$ so $a_0+z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\in A$ and is hence an acceptable digit. Since $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)=\varepsilon+\varepsilon^{-1},$ we can rewrite [\[ep_star_r\]](#ep_star_r){reference-type="eqref" reference="ep_star_r"} as $$\begin{aligned} \notag \varepsilon\ast r &= (a_{1}-1)\varepsilon^2 + (a_0 + z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon))\varepsilon+ (a_{-1}+1-1) + a_{-2}\varepsilon^{-1} + a_{-3}\varepsilon^{-2} +\ldots\\ \label{2.1_shift} &= (a_{1}-1)\varepsilon^2 + (a_0 + z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon))\varepsilon+ a_{-1} + a_{-2}\varepsilon^{-1} + a_{-3}\varepsilon^{-2} +\ldots \end{aligned}$$ Since $0\leq a_0+z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)<\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon),$ the above base-$\varepsilon$ expansion is valid as long as $0\leq a_{1}-1\leq\lfloor\varepsilon\rfloor.$ Since $0\leq a_1\leq\lfloor\varepsilon\rfloor,$ we know $-1\leq a_1-1\leq\lfloor\varepsilon\rfloor-1.$ Thus unless $a_1-1=-1,$ it must be true that $0\leq a_{1}-1\leq\lfloor\varepsilon\rfloor.$ Let us assume for the sake of contradiction that $a_1-1=-1.$ If we let $$\begin{aligned} \alpha&=\varepsilon^2\text{ and }\\ \beta&=(a_0 + z_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon))\varepsilon+ a_{-1} + a_{-2}\varepsilon^{-1} + a_{-3}\varepsilon^{-2} +\ldots, \end{aligned}$$ then [\[2.1_shift\]](#2.1_shift){reference-type="eqref" reference="2.1_shift"} implies that $\varepsilon\ast r=-\alpha+\beta.$ However, it follows directly from the definition of a base-$\varepsilon$ expansion and the fact that $\varepsilon$ is a Pisot number that $\alpha>\beta.$ Thus $\varepsilon\ast r =-\alpha+\beta<0,$ which contradicts the fact that $\varepsilon\ast r\in R.$ Thus we have that $0\leq a_{1}-1\leq\lfloor\varepsilon\rfloor,$ so [\[2.1_shift\]](#2.1_shift){reference-type="eqref" reference="2.1_shift"} is a valid base-$\varepsilon$ expansion of $\varepsilon\ast r.$ We can see that only a finite number of digits differ between the base-$\varepsilon$ expansion of $\varepsilon\ast r$ and the base-$\varepsilon$ expansion of $r.$ Therefore the repeating part of $\varepsilon\ast r$ must be the same as $r.$ **Case 2.2** Assume $a_{-1}+1\not\in A.$ By Case $2.1,$ we may assume without loss of generality that $a_0+z_2\in A.$ Since $0\leq a_{-1}\leq\lfloor\varepsilon\rfloor,$ we know that $1\leq a_{-1}+1\leq\lfloor\varepsilon\rfloor+1.$ Thus if $a_{-1}+1\not\in A,$ it must be that $a_{-1}+1=\lfloor\varepsilon\rfloor+1,$ so $a_{-1}=\lfloor\varepsilon\rfloor.$ Since $\lceil\varepsilon\rceil=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)$ by Lemma  [Lemma 16](#prop:eps=trace){reference-type="ref" reference="prop:eps=trace"}, we have $a_{-1}+1=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon).$ Again using that $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)=\varepsilon+\varepsilon^{-1},$ we can rewrite [\[ep_star_r\]](#ep_star_r){reference-type="eqref" reference="ep_star_r"} as $$\label{2.2_shift} \varepsilon\ast r = a_{1}\varepsilon^2 + (a_0 + z_2+1)\varepsilon+ (a_{-2}+1)\varepsilon^{-1} + a_{-3}\varepsilon^{-2} + \ldots$$ Since $0\leq a_0 + z_2\leq\lfloor\varepsilon\rfloor$ by assumption, if $a_0 + z_2+1\not\in A,$ then $a_0+z_2+1=\lfloor\varepsilon\rfloor+1.$ This would imply that $\varepsilon\ast r>\varepsilon+1,$ which contradicts the fact that $\varepsilon\ast r \in R.$ Thus it must be that $0\leq a_0 + z_2+1 \leq\lfloor\varepsilon\rfloor.$ With this, we see that if $a_{-2}+1\in A,$ then [\[2.2_shift\]](#2.2_shift){reference-type="ref" reference="2.2_shift"} is a valid base-$\varepsilon$ expansion of $\varepsilon\ast r.$ Otherwise, if $a_{-2}+1\not\in A,$ then since $a_{-2}\in A,$ it must be that $a_{-2}=\lfloor\varepsilon\rfloor,$ so $a_{-2}+1=\lceil\varepsilon\rceil=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon).$ Using that $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)=\varepsilon+\varepsilon^{-1},$ we can rewrite [\[2.2_shift\]](#2.2_shift){reference-type="ref" reference="2.2_shift"} as $$\varepsilon\ast r = a_{1}\varepsilon^2 + (a_0 + z_2+1)\varepsilon+ 1 + (a_{-3}+1)\varepsilon^{-2} + \ldots$$ We note that by the same argument used before, if $a_{i}+1\not\in A$ for any $i\in\mathbb{Z},$ then $a_i=\lfloor\varepsilon\rfloor=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)- 1.$ Thus if we let $j$ be the smallest positive integer such that $a_{-j}\neq\lfloor\varepsilon\rfloor,$ then continuing in the same manner, we see that $$\varepsilon\ast r = a_{1}\varepsilon^2 + (a_0 + z_2+1)\varepsilon+ 1 + \varepsilon^{-1} + \varepsilon^{-2} + \ldots + \varepsilon^{-j+3}+(a_j+1)\varepsilon^{-j+1}+a_{j+1}\varepsilon^{-j}\ldots$$ Since the base-$\varepsilon$ expansion of $r$ must be finite or periodic, it is certainly possible to choose such an index $j.$ We can assume that $r$ does not have repeating part $\overline{\lfloor\varepsilon\rfloor},$ since $\lfloor\varepsilon\rfloor\varepsilon^{i}+\lfloor\varepsilon\rfloor\varepsilon^{i-1}+\lfloor\varepsilon\rfloor\varepsilon^{i-2}+\ldots=\varepsilon^{i+1}$ for any $i\in\mathbb{Z}.$ Thus in Case $2.2,$ we see that only a finite number of digits differ between the base-$\varepsilon$ expansion of $\varepsilon\ast r$ and the base-$\varepsilon$ expansion of $r.$ Therefore the repeating part of $\varepsilon\ast r$ must be the same as $r$ in this case. Now we have seen that in all cases, the repeating part of the base-$\varepsilon$ expansion of $\varepsilon\ast r$ is the same as that of $r,$ which finishes the proof. ◻ In many of the results which follow, it will prove useful for us to note the following fact about the map $\pi$ as defined in Proposition [Proposition 9](#pi_prop){reference-type="ref" reference="pi_prop"}. **Lemma 17**. *The map $\pi$ is equivariant under the action of $\langle\varepsilon\rangle.$* *Proof.* Since $\frac{1}{p}\mathcal{O}_F$ is an ideal of $\mathcal{O}_F,$ it is also a $\mathcal{O}_F$-module. Moreover, $\mathcal{O}_F$ is trivially an $\mathcal{O}_F$-module, and $p\mathcal{O}_F$ is a submodule of $\mathcal{O}_F$ since $p\mathcal{O}_F$ is an ideal of $\mathcal{O}_F$. Thus the map $$\phi:\frac{1}{p}\mathcal{O}_F\to\mathcal{O}_F/p\mathcal{O}_F$$ defined by multiplication by $p$ is an $\mathcal{O}_F$-module homomorphism. The kernel of this map is $\mathcal{O}_F,$ so by the first isomorphism theorem, $\frac{1}{p}\mathcal{O}_F/\mathcal{O}_F\xrightarrow{\sim}\mathcal{O}_F/p\mathcal{O}_F$ is an isomorphism of $\mathcal{O}_F$-modules. Since $\mathbb{Z}[\varepsilon]$ is a subring of $\mathcal{O}_F,$ by restriction of scalars, $\frac{1}{p}\mathcal{O}_F/\mathcal{O}_F\cong\mathcal{O}_F/p\mathcal{O}_F$ is also an isomorphism of $\mathbb{Z}[\varepsilon]$-modules. We can also observe that since $\mathcal{O}_F$ is a submodule of $\frac{1}{p}\mathcal{O}_F,$ the projection map $$\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]\to\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]\bigg/\mathcal{O}_F/\mathbb{Z}[\varepsilon]$$ is a surjective $\mathbb{Z}[\varepsilon]$-module homomorphism. By the third isomorphism theorem, we further have that $$\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]\bigg/\mathcal{O}_F/\mathbb{Z}[\varepsilon]\cong\frac{1}{p}\mathcal{O}_F/\mathcal{O}_F,$$ which implies $$\psi: \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]\to\frac{1}{p}\mathcal{O}_F/\mathcal{O}_F$$ is a surjective $\mathbb{Z}[\varepsilon]$-module homomorphism. Since $R\subset \frac{1}{p}\mathcal{O}_F$ and $R$ constitutes a complete set of coset representatives for $\frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$ (see [@wei_lun_stark_units], Proposition $4.1$), the identity map $$\iota: R\to \frac{1}{p}\mathcal{O}_F/\mathbb{Z}[\varepsilon]$$ is a $\mathbb{Z}[\varepsilon]$-module isomorphism. Now we can see that since $\pi=\phi\circ\psi\circ\iota,$ $\pi$ is a surjective $\mathbb{Z}[\varepsilon]$-module homomorphism. Thus $\pi$ is equivariant under the action of $\langle\varepsilon\rangle.$ ◻ With this result, we may now examine more closely the action of $\varepsilon$ on $R.$ Namely, we can deduce the following fact about nontrivial Shintani cycles. **Lemma 18**. *All nontrivial Shintani cycles have length equal to the multiplicative order of $\varepsilon+p\mathcal{O}_F$ in $(\mathcal{O}_F/p\mathcal{O}_F)$.* *Proof.* Let $M$ denote the multiplicative order of $\varepsilon+p\mathcal{O}_F$ in $\mathcal{O}_F/p\mathcal{O}_F$, and consider $r\in R-\mathcal{O}_F.$ We will show that $|C_r| = M.$ Suppose that $\varepsilon^m\ast r =r$ for some $m\in\mathbb{Z}^+.$ Then by Lemma  [Lemma 17](#equivariance){reference-type="ref" reference="equivariance"}, $$\begin{aligned} \pi(r)=\pi(\varepsilon^m\ast r)=(\varepsilon^m+p\mathcal{O}_F)\pi(r) \implies (\varepsilon^m-1+p\mathcal{O}_F)\pi(r)=0.\end{aligned}$$ By assumption $r\not\in\mathcal{O}_F,$ so $\pi(r)\neq 0$ by Lemma [Lemma 10](#kernel_of_phi_lemma){reference-type="ref" reference="kernel_of_phi_lemma"}. Since $\mathcal{O}_F/p\mathcal{O}_F$ is a field, it must be that $(\varepsilon^m-1+p\mathcal{O}_F)=0.$ Therefore $\varepsilon^m\equiv 1 \mod{p\mathcal{O}_F},$ so $M | m.$ As a consequence, if we denote the stabilizer subgroup associated to $r$ under the action of $\varepsilon$ as $\langle\varepsilon\rangle_r,$ then $\langle\varepsilon\rangle_r\subset\langle\varepsilon^M\rangle.$ Moreover, we note that since $\langle\varepsilon\rangle_r$ is a subgroup of $\langle\varepsilon^M\rangle$, it must be that $\langle\varepsilon\rangle_r=\langle\varepsilon^{M'}\rangle$ for some $M'$ such that $M|M'.$ Moreover, we also have that $$\begin{aligned} \varepsilon^{M'}\ast r=r\implies (\varepsilon^{M'}-1+p\mathcal{O}_F)\pi(r)=0\end{aligned}$$ where $\pi(r)\neq 0,$ so that $\varepsilon^{M'}\equiv 1 \mod{p\mathcal{O}_F}.$ Thus it must also be that $M'|M,$ so $M'=M,$ and hence $\langle\varepsilon\rangle_r=\langle\varepsilon^M\rangle.$ In other words, $\langle\varepsilon^M\rangle$ is the stabilizer subgroup of $r$ for all $r\in R-\mathcal{O}_F$. By the orbit-stabilizer theorem, we then have that $|C_r|=[\langle\varepsilon\rangle:\langle\varepsilon^M\rangle]=M.$ ◻ In the lemma which follows, we equate $M$ with the minimal period length of the base-$\varepsilon$ expansion of $r$ for all $r\in R-\mathcal{O}_F.$ This fact, combined with Lemma [Lemma 18](#same_cycle_length){reference-type="ref" reference="same_cycle_length"}, will then imply that for all $r\in R-\mathcal{O}_F$, $|C_r|=P_r,$ where $P_r$ denotes the minimal period length of the base-$\varepsilon$ expansion of $r$. **Lemma 19**. * For any $r\in R-\mathcal{O}_F$, the minimal period length of the base-$\varepsilon$ expansion of $r$ is equal to the multiplicative order of $\varepsilon+p\mathcal{O}_F$ in $(\mathcal{O}_F/p\mathcal{O}_F)$.* *Proof.* Consider an element $r = r_1+r_2\varepsilon\in R-\mathcal{O}_F.$ We start by showing that $P_r | M$. As mentioned at the beginning of Section [4.2](#ep_exp_section){reference-type="ref" reference="ep_exp_section"}, the base-$\varepsilon$ expansion of $r$ is always eventually periodic, say $$\begin{aligned} \label{ep_exp_sum} r=\sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i}}+\varepsilon^{-N}\sum_{j=0}^\infty {\left(a_1\varepsilon^{-jP_r}+a_2\varepsilon^{-jP_r-1}+\ldots+a_{P_r}\varepsilon^{-jP_r-P_r+1}\right)}.\end{aligned}$$ We remind the reader that since $r\in R, \lfloor\log_{\varepsilon}(r)\rfloor = 0\text{ or } 1,$ so the highest power of $\varepsilon$ in [\[ep_exp_sum\]](#ep_exp_sum){reference-type="eqref" reference="ep_exp_sum"} is $1.$ Multiplying [\[ep_exp_sum\]](#ep_exp_sum){reference-type="eqref" reference="ep_exp_sum"} by $\varepsilon^{P_r}$, we obtain $$\begin{aligned} \varepsilon^{P_r}r&=\sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i+P_r}}+\varepsilon^{-N}\sum_{j=0}^\infty {\left(a_1\varepsilon^{-(j-1)P_r}+a_2\varepsilon^{-(j-1)P_r-1}+\ldots+a_{P_r}\varepsilon^{-(j-1)P_r-P_r+1}\right)}.\end{aligned}$$ Reindexing [\[ep_exp_sum\]](#ep_exp_sum){reference-type="eqref" reference="ep_exp_sum"}, we get $$\begin{aligned} r=\sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i}}+\varepsilon^{-N}\sum_{j=1}^\infty {\left(a_1\varepsilon^{-(j-1)P_r}+a_2\varepsilon^{-(j-1)P_r-1}+\ldots+a_{P_r}\varepsilon^{-(j-1)P_r-P_r+1}\right)}.\end{aligned}$$ And thus $$\begin{aligned} \varepsilon^{P_r}r-r=\left(\sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i+P_r}-a'_i\varepsilon^{-i}}\right)+\varepsilon^{-N}\left(a_1\varepsilon^{P_r}+a_2\varepsilon^{P_r-1}+\ldots+a_{P_r}\varepsilon\right).\end{aligned}$$ Let $$\begin{aligned} \alpha = \sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i+P_r}}, \quad\text{and}\quad\beta = -\sum_{i=-1}^{N-1}{a'_i\varepsilon^{-i}}, \quad\text{and}\quad\gamma = \varepsilon^{-N}\left(a_1\varepsilon^{P_r}+a_2\varepsilon^{P_r-1}+\ldots+a_{P_r}\varepsilon\right).\end{aligned}$$ Note that, because $\alpha$, $\beta$, and $\gamma$ have finite $\varepsilon$ expansions, we have that $\alpha, \beta, \gamma \in \mathbb{Z}[\varepsilon]$, and thus $$\alpha + \beta + \gamma = \varepsilon^{P_r}r-r \in \mathbb{Z}[\varepsilon].$$ By definition, we know that $$\varepsilon^{P_r}\ast r = \varepsilon^{P_r}r + z$$ for some $z \in \mathbb{Z}[\varepsilon]$. Thus, using Lemmas  [Lemma 10](#kernel_of_phi_lemma){reference-type="ref" reference="kernel_of_phi_lemma"} and  [Lemma 17](#equivariance){reference-type="ref" reference="equivariance"}, we have $$\begin{aligned} \varepsilon^{P_r}\ast r-r-z= \varepsilon^{P_r}r - r \implies \pi(\varepsilon^{P_r}\ast r-r-z) = \pi(\varepsilon^{P_r}r - r) \implies (\varepsilon^{P_r} + p\mathcal{O}_F - 1)\pi(r) = 0.\end{aligned}$$ Since $F$ is a field in which $\pi(r)\neq 0$ since $r\not\in\mathcal{O}_F,$ we have that $$\begin{aligned} \varepsilon^{P_r} - 1 + p\mathcal{O}_F = 0 \implies \varepsilon^{P_r} \equiv 1 \pmod{p\mathcal{O}_F}.\end{aligned}$$ for any $r \in R-\mathcal{O}_F$. Recall that $M$ is the multiplicative order of $\varepsilon$ in $\mathcal{O}_F/p\mathcal{O}_F$, so we see that $M|P_r$. Next, we show that $P_{r} | M$. Since both $r$ and $\varepsilon\ast r$ have periodic base-$\varepsilon$ expansions, we let $N_1$ represent the smallest integer such that the repeating part of the base-$\varepsilon$ expansion of $r$ begins in the $\varepsilon^{-N_{1}}$ place. Similarly, let $N_2$ represent the smallest integer such that the repeating part of the base-$\varepsilon$ expansion of $\varepsilon\ast r$ begins in the $\varepsilon^{-N_{2}}$ place. Let $S = \max(N_1, N_2)$ be the smallest integer such that the base-$\varepsilon$ expansion of both $r$ and $\varepsilon\ast r$ is periodic for all indices greater than $S$. Thus, the digits in the $\varepsilon^{-S}, \varepsilon^{-S-1}, \ldots, \varepsilon^{-S-P_r}$ place of the base-$\varepsilon$ expansion of $r$ constitute a full period, and we let the ordered set $$\begin{aligned} \{x_1, x_2, \ldots, x_{P_{r}}\}\end{aligned}$$ represent the period set of $r$. As shown in Proposition  [Proposition 14](#same_repeating_part){reference-type="ref" reference="same_repeating_part"}, the operation $\varepsilon\ast r$ shifts the digits within the repeating part of the base-$\varepsilon$ of $r$ to the left by one index. In other words, the period set of $\varepsilon\ast r$ is the ordered set $$\begin{aligned} \{x_2, \ldots, x_{P_{r}}, x_1\}.\end{aligned}$$ Note that moving between the period set of $r$ and the period set of $\varepsilon\ast r$ can be represented by applying the permutation $$\begin{aligned} \tau = (1 \hspace{0.2 cm} 2 \hspace{0.2 cm} \cdots \hspace{0.2 cm} P_r) \in S_{P_r}\end{aligned}$$ to the period set of $r.$ Additionally, we have that $r = \varepsilon^{M}\ast r$, so the period sets of $r$ and $\varepsilon^{M}\ast r$ must be equal. Thus, $$\begin{aligned} \tau^{M} \{x_1, x_2, \ldots, x_{P_{r}}\} = \{x_1, x_2, \ldots, x_{P_{r}}\}, \end{aligned}$$ which implies that $\tau^{M}$ is the identity permutation. Since the order of $\tau \in S_{P_r}$ is $P_r$, we have that $P_r | M$. So, we see that $P_r = M$. ◻ In our final lemma before we prove Theorem [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"}, we show that for any nontrivial Shintani cycle, the sum of the coefficients $r_1$ and $r_2$ where $r=r_1+r_2\varepsilon$ of all the elements $r$ in the Shintani cycle is a constant. In fact, these coefficients sum to $M.$ **Lemma 20**. *For any $r\in R-\mathcal{O}_F,$ let $r'\coloneqq r_1'+r_2'\varepsilon$. Then, $$\begin{aligned} \sum_{r'\in C_r}{(r_1'+r_2')}=M. \end{aligned}$$* *Proof.* Let $\varepsilon^i\ast r \coloneqq r_1(i)+r_2(i)\varepsilon,$ so $r_1(i+1)+r_2(i+1)\varepsilon=\varepsilon\ast (r_1(i)+r_2(i)\varepsilon).$ Recall that $$\begin{aligned} \varepsilon\ast (r_1(i)+r_2(i)\varepsilon)=(1-r_2(i))+\{r_1(i)+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_2(i)\}\varepsilon \end{aligned}$$ by Lemma [Lemma 15](#prop:abbounds){reference-type="ref" reference="prop:abbounds"}. Comparing coefficients, we see that $r_1(i+1)+r_2(i)=1$ for all $i\in\mathbb{Z}$. Moreover, since $M=|C_r|$ for all $r\in R-\mathcal{O}_F$ by Lemma  [Lemma 18](#same_cycle_length){reference-type="ref" reference="same_cycle_length"}, we have that $r_1(m)=r_1(m+M)$ for any integer $m.$ Using these facts, we see that $$\begin{aligned} \sum_{r\in C_r}{(r_1(i)+r_2(i))}&=\sum_{i=1}^M {(r_1(i)+r_2(i))}=r_1(1)+r_2(M)+\sum_{i=1}^{M-1} {r_1(i+1)}+\sum_{j=1}^{M-1}{r_2(j)} \\ &=r_1(M+1)+r_2(M)+\sum_{i=1}^{M-1} {\bigg(r_1(i+1)}+r_2(i)\bigg)=1+(M-1)=M. \end{aligned}$$ ◻ ## Proof of Theorem 1.2 {#thm1.2proof_subsection} Since $\langle \varepsilon\rangle$ acts on $R$, $R$ decomposes into a disjoint union of Shintani cycles, under this action. Letting $\mathcal{L}$ denote a complete reduced set of Shintani cycle representatives for $R,$ and recalling that $C_r$ denotes the Shintani cycle of $r,$ we can rewrite Shintani's formula as follows: $$\begin{aligned} \notag h_K &= \frac{1}{2}\sum_{r\in R}{\chi_{K/F}(rp\mathcal{O}_F)\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(r_1)}{l_1!}\frac{B_{l_2}(r_2)}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)^{l_2-1}}}\\ \label{eqn:begin_thm_1.2} &= \frac{1}{2}\sum_{i=1}^{|C_r|}\sum_{r\in\mathcal{L}}{\chi_{K/F}(\varepsilon_{F}^i\ast r \cdot p\mathcal{O}_F)\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(r_1)}{l_1!}\frac{B_{l_2}(r_2)}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)^{l_2-1}}}\end{aligned}$$ First, we show that $\chi_{K/F}(r' \cdot p\mathcal{O}_F)$ is constant for all $r'\in C_r$. By definition of $\varepsilon\ast r$, we see that $\varepsilon\ast r=\varepsilon r +z$ for some $z \in \mathcal{O}_F.$ Thus $$\begin{aligned} \chi_{K/F}(\varepsilon^i\ast r \cdot p\mathcal{O}_F) &= \chi_{K/F}( (\varepsilon^ir + z) \cdot p\mathcal{O}_F)= \chi_{K/F}(\varepsilon^irp\mathcal{O}_F + zp\mathcal{O}_F ).\end{aligned}$$ Since $zp\mathcal{O}_F\subset p\mathcal{O}_F,$ $p\mathcal{O}_F | zp\mathcal{O}_F,$ and since $p\mathcal{O}_F$ is the conductor of this Hecke character, we see that $$\begin{aligned} \chi_{K/F}( \varepsilon^i\ast r \cdot p\mathcal{O}_F) &= \chi_{K/F}(\varepsilon^irp\mathcal{O}_F ).\end{aligned}$$ Additionally, $\varepsilon$ is a unit, so we know $$r\mathcal{O}_F = \varepsilon^ir \mathcal{O}_F$$ for any integer $i.$ Therefore $$\chi_{K/F}(\varepsilon^i\ast r \cdot p\mathcal{O}_F)= \chi_{K/F}(\varepsilon^ir p\mathcal{O}_F)= \chi_{K/F}( rp \mathcal{O}_F),$$ and thus the Hecke character value in [\[eqn:begin_thm_1.2\]](#eqn:begin_thm_1.2){reference-type="eqref" reference="eqn:begin_thm_1.2"} is constant throughout each Shintani cycle. By Lemma  [Lemma 18](#same_cycle_length){reference-type="ref" reference="same_cycle_length"}, all nontrivial Shintani cycles in $R$ contain the same number of elements. Since $1/p$ is an element of $R-\mathcal{O}_F$ by Lemma  [Lemma 19](#orbit_length_is_cycle_length){reference-type="ref" reference="orbit_length_is_cycle_length"}, the period length $\ell_{F,p}$ of the base-$\varepsilon$ expansion of $1/p$ is equal to the length of each nontrivial cycle. **Remark 4**. Note that, for all $r\in R\cap\mathcal{O}_F,$ the Hecke character $\chi_{K/F}(rp\mathcal{O}_F)$ evaluates to 0, so elements $r\in R\cap\mathcal{O}_F$ are all weighted by a factor of $0$ in [\[eqn:begin_thm_1.2\]](#eqn:begin_thm_1.2){reference-type="eqref" reference="eqn:begin_thm_1.2"}. Hence, we can ignore them in our calculations. Using these facts, we obtain $$\label{thm1.2_simplification_intermediate_step} h_K=\frac{1}{2}\sum_{i=1}^{\ell_{F,p}}\hspace{+.1 cm}\sum_{r\in\mathcal{L}}{\chi_{K/F}(rp\mathcal{O}_F)\sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}}{\frac{B_{l_1}(r_1)}{l_1!}\frac{B_{l_2}(r_2)}{l_2!}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)^{l_2-1}}}.$$ As shown in [@wei_lun_stark_units], we have that $$\begin{aligned} \sum_{r \in R} \chi_{K/F}( r p\mathcal{O}_F) = \sum_{r \in R\cap\mathcal{O}_F} \chi_{K/F}( r p\mathcal{O}_F) + \sum_{r \in R -\mathcal{O}_F} \chi_{K/F}( r p\mathcal{O}_F) = 0.\end{aligned}$$ For all $r \in \mathcal{O}_F,$ we have already seen that $\chi_{K/F}( rp\mathcal{O}_F) = 0.$ Thus, $$\begin{aligned} 0 &= \sum_{r \in R-\mathcal{O}_F} \chi_{K/F}( r p\mathcal{O}_F) = \sum_{{\color{black} r \in \mathcal{L}-\mathcal{O}_F}} \chi_{K/F}( r p\mathcal{O}_F)\cdot\ell_{F,p} = \ell_{F,p}\sum_{{\color{black} r \in \mathcal{L}-\mathcal{O}_F}} \chi_{K/F}(r p\mathcal{O}_F), \end{aligned}$$ which yields $$\sum_{r \in \mathcal{L}-\mathcal{O}_F} \chi_{K/F}( r p\mathcal{O}_F)=0.$$ In other words, we have character orthogonality across the elements $r \in \mathcal{L}-\mathcal{O}_F$. With this, we consider the sum over Bernoulli polynomials within this formula. Letting $$\begin{aligned} \mathcal{B}(r_1+r_2\varepsilon) \coloneqq \sum_{\substack{0\leq l_1,l_2\leq 2\\ l_1+l_2=2}} \frac{B_{l_1}(r_1)B_{l_2}(r_2)}{l_1!l_2!} \textrm{Tr}_{F/\mathbb{Q}}\left(\varepsilon^{l_2-1} \right),\end{aligned}$$we see that $$\begin{aligned} \notag \mathcal{B}(r_1+r_2\varepsilon) &= \dfrac{r_1^2-r_1+\frac{1}{6}}{2}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)+ 2\bigg(r_1 - \frac{1}{2}\bigg)\bigg(r_2 - \frac{1}{2}\bigg) + \dfrac{r_2^2-r_2+\frac{1}{6}}{2}\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)\\\label{eqn:thm_1.2_middle}~~ &= \frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2}\bigg(r_1^2 + r_2^2 - (r_1+r_2) + \frac{1}{3}\bigg) + 2r_1r_2 - (r_1+r_2) + \frac{1}{2}.\end{aligned}$$ Recall that by Lemmas  [Lemma 20](#sum_of_a_cycle_is_l){reference-type="ref" reference="sum_of_a_cycle_is_l"}, for all $r\in\mathcal{L}-\mathcal{O}_F,$ $$\sum_{r'\in C_r}r_1'+r_2'=M.$$ Thus, we can further simplify [\[eqn:thm_1.2_middle\]](#eqn:thm_1.2_middle){reference-type="eqref" reference="eqn:thm_1.2_middle"} to $$\label{thm1.2_simplification_second_intermediate_step} \mathcal{B}(r_1+r_2\varepsilon)=\frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2}\bigg(r_1^2 + r_2^2 - M + \frac{1}{3}\bigg) + 2r_1r_2 - M - \frac{1}{2}.$$ Because we have character orthogonality over $\mathcal{L}-\mathcal{O}_F,$ we can add a constant to the inner Bernoulli sum of [\[thm1.2_simplification_intermediate_step\]](#thm1.2_simplification_intermediate_step){reference-type="eqref" reference="thm1.2_simplification_intermediate_step"} without changing the value of the whole expression. In particular, if we let $$\begin{aligned} c\coloneqq\frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2}\bigg(-M+\frac{1}{3}\bigg) - M-\frac{1}{2},\end{aligned}$$ we see that [\[thm1.2_simplification_second_intermediate_step\]](#thm1.2_simplification_second_intermediate_step){reference-type="eqref" reference="thm1.2_simplification_second_intermediate_step"} can be rewritten as $$\begin{aligned} \mathcal{B}(r_1+r_2\varepsilon)&= \frac{\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)}{2}\bigg(r_1^2 + r_2^2\bigg) + 2r_1r_2 + c\\ &=\frac{1}{2}\bigg(\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_1^2 + 4r_1r_2 +\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_2^2\bigg) + c.\end{aligned}$$ Using these results and letting $\varepsilon\ast r\coloneqq r_1(i)+r_2(i)\varepsilon,$ we obtain $$\begin{aligned} \label{eqn:1.2stinky} h_K = \frac{1}{4}\sum_{i=1}^{\ell_{F,p}}\sum_{r\in \mathcal{L}-\mathcal{O}_F}{\chi_{K/F}}\bigg(r p\mathcal{O}_F\bigg)\bigg(\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_1(i)^2 + 4r_1(i)r_2(i) + \textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_2(i)^2 \bigg).\end{aligned}$$ Recall that $Q_F(Y_1,Y_2)\coloneqq\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)Y_1^2+4Y_1Y_2+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)Y_2^2.$ Thus if we make a slight abuse of notation by letting $Q_F(\varepsilon^i\ast r)=\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)(r_1(i)^2+4r_1(i)r_2(i)+\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon)r_2(i)^2,$ then we can express $h_K$ as $$\begin{aligned} h_K = \frac{1}{4}\sum_{i=1}^{\ell_{F,p}}\sum_{r\in \mathcal{L}-\mathcal{O}_F}{\chi_{K/F}}(r p\mathcal{O}_F)Q(\varepsilon^i\ast r).\end{aligned}$$ # Examples {#example_section} Here we illustrate Theorems [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} and [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} for $\mathbb{Q}(\sqrt{3},\sqrt{-p})$, where $p$ is prime. Note that the ring of integers of $F=\mathbb{Q}(\sqrt{3})$ is given by $\mathbb{Z}[\sqrt{3}]$, and its totally positive unit group $\mathcal{O}_{F}^{\times,+}$ is generated by $\varepsilon_{F} = 2 + \sqrt{3}$. We require that $p\equiv 3\pmod{4}, \genfrac{(}{)}{}{}{3}{p} = -1,$ and $7\leq p.$ The first two conditions imply that the relative discriminant ideal is the prime ideal $p\mathbb{Z}[\sqrt{3}]$. Consequently, $\mathbb{Z}[\sqrt{3}]/p\mathbb{Z}[\sqrt{3}] ~\cong ~\mathbb{F}_p[\sqrt{3}]$. ## Theorem 1.1 with $F=\mathbb{Q}(\sqrt{3})$ Let $\rho_{F,p} = a + b\sqrt{3}$ be a generator of $\mathbb{F}_p[\sqrt{3}]$. Table [1](#table1){reference-type="ref" reference="table1"} lists values of $\rho_{F,p}$ as computed with `SageMath`. Using these values, we use [\[def:c_defns\]](#def:c_defns){reference-type="eqref" reference="def:c_defns"} and [\[def:d_defns\]](#def:d_defns){reference-type="eqref" reference="def:d_defns"} to calculate $C_{F,p}$ and $D_{F,p}$, then use [\[eqn:x1defns\]](#eqn:x1defns){reference-type="eqref" reference="eqn:x1defns"} and [\[eqn:x2defns\]](#eqn:x2defns){reference-type="eqref" reference="eqn:x2defns"} to find the corresponding rational functions $X_{F,p}(z)$ and $Y_{F,p}(z)$, which are also displayed in Table [1](#table1){reference-type="ref" reference="table1"}. We extract the first $p^2-1$ coefficients from our rational functions by taking the $k^{th}$ derivative of $X(z)$ and $Y(z)$, evaluating each function at $z=0$, and dividing by $k!$. Note that in this case, since $t=1$, we obtain only $1$ sequence $x_1(m),$ and $y_1(m),$ from each of $X_{F,p}(z)$ and $Y_{F,p}(z)$ respectively. Since $\textrm{Tr}_{F/Q}(\varepsilon_F)=4$, we have $$\begin{aligned} Q_{F}(Y_1, Y_2)=4Y_1^2+4Y_1Y_2+4Y_2^2.\end{aligned}$$ Now we may apply Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} to obtain $$\begin{aligned} h_{F(\sqrt{-p})}=\frac{1}{16p^2}\sum_{1\leq m\leq p^2-1}(-1)^mQ_{F}\bigg(x_1(m),y_1(m)\bigg).\end{aligned}$$ The smallest suitable prime for which we can apply Theorem  [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} here is $p=7,$ for which we calculate $$\begin{aligned} h_{F(\sqrt{-7})}=\frac{1}{784}(&-84+76-300+52-28+436-100+148-196+52-108+124-84+148\\ &-36+172-28+124-12+76-196+172-4+156-84+76-300+52-28\\ &+156-100+316-196+52-108+124-84+316-36+228-28+124-12\\ &+76-196+228-4+436)=2.\end{aligned}$$In Table [1](#table1){reference-type="ref" reference="table1"}, we list some terms of our alternating sum for the class numbers of all such primes less than $100$, along with the corresponding class numbers calculated using Theorem [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} and verified using `SageMath`. $\hspace{0.25 cm} p \hspace{0.25 cm}$ $\hspace{0.25 cm}\rho_{F,p} \hspace{0.25 cm}$ $\hspace{0.4 cm}X_{F,p}(z) \hspace{0.4 cm}$ $\hspace{0.4 cm}Y_{F,p}(z) \hspace{0.4 cm}$ $\hspace{1.0 cm}$$h_{F_(\sqrt{-p})}$ Calculation $\hspace{1.0 cm}$ --------------------------------------- ----------------------------------------------- --------------------------------------------- --------------------------------------------- -------------------------------------------------------------------- 7 $6+\sqrt{3}$ $\dfrac{6z - 33z^2}{33z^2 - 12z + 1}$ $\dfrac{z}{33z^{2} - 12z + 1}$ $\frac{1}{784}\bigg( -84 + 76 - \ldots + 436\bigg) = 2$ 19 $1+4\sqrt{3}$ $\dfrac{6z +47z^2}{-47z^2 - 12z + 1}$ $\dfrac{4z}{-47z^{2} - 12z + 1}$ $\frac{1}{5776} \bigg(-364 + 252 - \ldots + 3892 \bigg) = 2$ 31 $1+6\sqrt{3}$ $\dfrac{z +107 z^2}{-107z^2 - 2z + 1}$ $\dfrac{6z}{-107z^{2} - 2z + 1}$ $\frac{1}{15376} \bigg(-1084 + 676 - \ldots + 10804 \bigg) = 6$ 43 $1+5\sqrt{3}$ $\dfrac{z +74z^2}{-74z^2 - 2z + 1}$ $\dfrac{5z}{-74z^{2} - 2z + 1}$ $\frac{1}{29584} \bigg(-3556 + 4836 - \ldots + 21172 \bigg) = 6$ 67 $2+5\sqrt{3}$ $\dfrac{2z + 71z^2}{-71z^2 - 4z + 1}$ $\dfrac{5z}{-71z^{2} - 4z + 1}$ $\frac{1}{71824}\bigg(-11772 + 2212 - \ldots + 52276\bigg) = 6$ 79 $2+6\sqrt{3}$ $\dfrac{2z + 104z^2}{-104z^2 - 4z + 1}$ $\dfrac{6z}{-104z^{2} - 4z + 1}$ $\frac{1}{99856} \bigg(-16068 + 7372 - \ldots + 73012\bigg) = 30$ : Theorem [Theorem 1](#thm:1.1){reference-type="ref" reference="thm:1.1"} for primes $p<100.$ ## Theorem 1.2 with $F = \mathbb{Q}(\sqrt{3})$ We illustrate Theorem  [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} in the same setting. Letting $F=\mathbb{Q}(\sqrt{3})$, we calculate $h_{K}$ for $p\equiv 3 \pmod{4}$ where $7\leq p$ and $\genfrac{(}{)}{}{}{3}{p}=-1$. We remind the reader that $\varepsilon_F=2+\sqrt{3}$, so $t=1$. Thus by Lemma  [Lemma 11](#explicit_kernel){reference-type="ref" reference="explicit_kernel"}, $\ker(\pi)=R_{F,p}\cap\mathcal{O}_F=\{1\}$. In the case that $p=7,$ we first calculate the base-$\varepsilon_F$ expansion of $1/7,$ $$\begin{aligned} \frac{1}{7}&=\varepsilon_F^{-2}+3\varepsilon_F^{-3}+2\varepsilon_F^{-4}+2\varepsilon_F^{-6}+2\varepsilon_F^{-7}+3\varepsilon_F^{-8}+3\varepsilon_F^{-11}+2\varepsilon_F^{-12}+2\varepsilon_F^{-14}+2\varepsilon_F^{-15}+3\varepsilon_F^{-16}+\ldots\\ &=0.01\overline{32202230}.\end{aligned}$$ Noticing that $1/7$ has period length $\ell_{F,7}=8$, by Lemma $~\ref{orbit_length_is_cycle_length}$, we can then deduce that there are $$\begin{aligned} \frac{|R_{F,7}-\mathcal{O}_F|}{\ell_{F,7}}=\frac{1\cdot 7^2-1}{8}=6\end{aligned}$$ disjoint Shintani cycles which comprise $R_{F,7}-\mathcal{O}_F$. We can generate these Shintani cycles explicitly, by calculating $\varepsilon_F^i \ast r$ for $0\leq i < 8$ for $r\in R_{F,7}-\mathcal{O}_F$. One can verify that $$\begin{aligned} \mathcal{L}=\left\{\frac{1}{7}+\frac{1}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{1}{7},\hspace{+0.1 cm}\frac{1}{7}+\frac{4}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{1}{7}+\frac{5}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{2}{7}+\frac{2}{7}\varepsilon_F,\hspace{+0.1 cm}\frac{3}{7}\right\}\end{aligned}$$ is a complete reduced set of representatives for all $6$ distinct nontrivial cycles in $R_{F,7}.$ With these values, we now calculate $h_{F(\sqrt{-7})}$ using Theorem $~\ref{thm 1.2}$. Noting that $\textrm{Tr}_{F/\mathbb{Q}}(\varepsilon_F)=4$ so $Q_F(Y_1,Y_2)=4Y_1^2+4Y_1Y_2+4Y_2^2,$ we compute $$\begin{aligned} h_{F(\sqrt{-7})}&=\frac{1}{4}\sum_{i=1}^{8}\hspace{+0.23cm}\sum_{r\in \mathcal{L}} \chi_{F(\sqrt{-7})/F} \left(r {p}\mathcal{O}_{F}\right) \bigg(4 r_1(i)^2 + 4r_1(i)r_2(i) + 4r_2(i)^2 \bigg)\\ &=\frac{1}{4}\bigg(-\frac{220}{7} + \frac{228}{7} - \frac{188}{7} + \frac{212}{7} - \frac{180}{7} + \frac{204}{7}\bigg)=2.\end{aligned}$$ In Table [2](#table_thm1.2){reference-type="ref" reference="table_thm1.2"}, we carry out the same procedure for all suitable primes less than $100.$ $\hspace{0.25 cm}p \hspace{0.25 cm}$ $\ell_{F,p}$ $h_{F_(\sqrt{-p})}$ Calculation -------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------ -------------- --------------------------------------------------------------------------------------------------------------------------- $7$ $0.01\hspace{+0.07 cm}\overline{32202230}$ $8$ $\frac{1}{4}\bigg(-\frac{220}{7} + \frac{228}{7} - \frac{188}{7} + \frac{212}{7} - \frac{180}{7} + \frac{204}{7}\bigg)=2$ $19$ $0.002\hspace{+0.07 cm}\overline{22231}$ $5$ $\frac{1}{4}\bigg(\frac{396}{19}+\frac{400}{19}-\frac{360}{19}+\ldots+\frac{332}{19}+\frac{328}{19}\bigg)=2$ $31$ $0.001\hspace{+0.07 cm}\overline{2132023120322221002303200122}$$\overline{2230}$ $32$ $\frac{1}{4}\bigg(\frac{3876}{31}-\frac{3788}{31}-\frac{3764}{31}-\ldots -\frac{3444}{31}+\frac{3420}{31}\bigg)=6$ $43$ $0.001\hspace{+0.07 cm}\overline{02311222230}$ $11$ $\frac{1}{4}\bigg(\frac{1856}{43}+\frac{1848}{43}+\frac{1940}{43}+\ldots+\frac{1656}{43}+\frac{1624}{43}\bigg)=6$ $67$ $0.0002\hspace{+0.07 cm}\overline{3110011313222221320122102312}$$\overline{222231}$ $34$ $\frac{1}{4}\bigg(\frac{8908}{67} +\frac{9228}{67}-\frac{8604}{67}+\ldots+\frac{7492}{67}+\frac{7860}{67}\bigg)=6$ $79$ $0.0002\hspace{+0.07 cm}\overline{122101031011213031211013010}$$\overline{122113222222010012113021100303001}$$\overline{12031121001022222231}$ $80$ $\frac{1}{4}\bigg(\frac{22740}{79} -\frac{22364}{79}+\ldots-\frac{22372}{79}+\frac{22500}{79}\bigg)=30$ : Theorem [Theorem 2](#thm 1.2){reference-type="ref" reference="thm 1.2"} for primes $p<100.$

arxiv_math

**LOCALLY NILPOTENT POLYNOMIALS OVER $\mathbb{Z}$** **Sayak Sengupta**\ Department of Mathematics and Statistics, Binghamton University - SUNY, Binghamton, New York, USA\ `sengupta@math.binghamton.edu`\ **Abstract** For a polynomial $u=u(x)$ in $\mathbb{Z}[x]$ and $r\in\mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by $\mathcal{O}_u(r):=\{u(r),u(u(r)),\ldots\}$. We ask two questions here: (i) what are the polynomials $u$ for which $0\in \mathcal{O}_u(r)$, and (ii) what are the polynomials for which $0\not\in \mathcal{O}_u(r)$ but, modulo every prime $p$, $0\in \mathcal{O}_u(r)$? In this paper, we give a complete classification of the polynomials for which (ii) holds for a given $r$. We also present some results for some special values of $r$ where (i) can be answered. -30pt -1ex -.2ex 2.3ex .2ex Introduction In Example 1 of [@B13], A. Borisov set up a polynomial map, called the *additive trap*, $F_{at}:\mathbb{A}_\mathbb{Z}^2\to \mathbb{A}_\mathbb{Z}^2$, which maps $(x,y)\mapsto (x^2y,x^2y+xy^2)$. This polynomial map satisfies many interesting properties. In particular, $F_{at}^{(p)}(x,y)\equiv (0,0)\pmod p$ for every $(x,y)\in \mathbb{A}^2_{\mathbb{F}_p}$ and for all primes $p$, where $F_{at}^{(p)}$ is the $p$th iteration of $F_{at}$. To prove this, suppose that $p$ is a prime. Note that all points $(x,y)\in\mathbb{A}_{\mathbb{F}_p}^2$ with either $x=0$ or $y=0$ are taken to (0,0) by $F_{at}$. Let $x\in \mathbb{F}_p^*$. Then for any $y\in\mathbb{F}_p^*$ we get $$\frac{x^2y+xy^2}{x^2y}=\frac{y}{x}+1.$$ So, after at most $p-1$ iterations, the second coordinate becomes 0 and thus, applying $F_{at}$ once more, we reach (0,0). Since $p$ is arbitrary, the proof follows. For more details see [@B13]. One can see from the discussion that the $p$th iteration of $F_{at}$ modulo $p$ is the zero map, which follows from the fact that the polynomial $u(x)=x+1$ has the following property: for every $n\in \mathbb{N}$, $u^{(n)}(x)=x+n$, so that, in particular, for every prime $p$, $u^{(p-1)}(1)=p\equiv 0\pmod p$. Throughout this paper, by $\mathbb{N}$ we will mean the set of all positive integers, and by $u^{(n)}$, we will mean $\underbrace{u\circ u\circ \cdots\circ u}_{n\text{ times}}$, the $n$th iteration of $u$. We can write our first definition now which is motivated by the behavior of the polynomial $u(x)=x+1$. Suppose $r\in\mathbb{Z}$ and $A$ is a finite subset of the set of all prime numbers. If, for every prime $p$ not contained in $A$, there exists an $m\in\mathbb{N}$ such that $u^{(m)}(r)\equiv 0\pmod p$, we will say that *$u$ is weakly locally nilpotent at $r$ outside $A$*. The set of all weakly locally nilpotent polynomials at $r$ outside $A$ of degree $d$ will be denoted by $L_{r,A}^d$ and $L_{r,A}$ is the union of $L_{r,A}^d$'s, where the union is taken over all degrees $d\in\mathbb{N}$. When $A=\emptyset$, i.e., for all primes $p$, $m$ exists satisfying $u^{(m)}(r)\equiv 0\pmod p$, we say that $u$ is locally nilpotent at $r$. Thus $u(x)=x+1$ is locally nilpotent at $1$, i.e., $x+1\in L_{1,\emptyset}^1$. If, in particular, a polynomial $u$ is such that $u^{(n)}(r)=0,$ for some $n\in\mathbb{N}$, then we will say that $u$ is *nilpotent at $r$* and the *nilpotency index* is the least of such $n's$. We denote the set of all nilpotent polynomials at $r$ of degree $d$ and nilpotency index $i$ by $N_{r,i}^d$. Also, $N_r$ is the union of all such $N_{r,i}^d$, where the union is taken over $i,d\in\mathbb{N}$. Thus $u(x)=x-1$ is nilpotent at 1 of nilpotency index 1, i.e., $x-1\in N_{1,1}^1$. One of the principal goals of this paper is to understand the polynomials that are locally nilpotent without being nilpotent at $r$, i.e., $0$ is not in the orbit of a polynomial $u$ at $r$ but modulo every prime $p$, some iteration of $u$ at $r$ hits $0$. This is an interesting question and it has been answered completely here. To classify these polynomials we have used Theorem 5 of [@TS13]. In particular, if we take $K=\mathbb{Q},g=1,A_1=\{0\},T_1=\{r\},\varphi_1=u\text{ polynomial}$ in this theorem, we get Fact 1.1 in our paper which says: **Fact 1**. If $u$ is a polynomial that is of degree at least 2 and it is weakly locally nilpotent at $r$ outside some finite set of primes $A$, then it must be nilpotent at $r$. Equivalently, if there is a polynomial $u$ such that it is weakly locally nilpotent at $r$ outside $A$ but not nilpotent at $r$, then it must be a linear polynomial. It should be noted that Theorem 5 of [@TS13] was built upon the work of Silverman in [@S93], and the works of Benedetto-Ghioca-Kurlberg-Tucker-Zannier in Lemma 4.1 of [@TT12]. It should also be noted that the question of when an orbit passes through a given point modulo a prime came naturally from the study of the Dynamical Mordell-Lang (DML) Conjecture, which is a major open question in arithmetic dynamics. The paper [@TS13] was developed to prove certain special cases of the DML Conjecture. Interested readers can also look at [@TT16]. From Fact 1.1 it is imperative to study the behavior of linear polynomials in order to understand locally nilpotent polynomials which are not nilpotent. This is where we use Theorem 1 of [@CS97] and derive a Lemma, which we call CRS Lemma (Lemma 3.2). This Lemma has been stated and proved in section 3. The paper contains four main results: 1. Complete classification of all polynomials in $L_{r,\emptyset}$, when $r\in\{0,-1,1\}$. This can be found in Theorems 4.1, 4.4 and Corollary 4.2. 2. Complete classification of all polynomials in $L_{1,A}^1$ for any given finite subset $A$ of the set of prime numbers. This can be found in Theorem 5.1. To establish this we have used Lemma 3.2. 3. Complete classification of all polynomials in $S_r$ which can only be linear polynomials by Fact 1.1, where $S_r:=L_{r,\emptyset}\setminus N_r$. This can be found in Corollaries 4.3, 4.5, 5.4 and Theorem 5.3. Lemma 3.2 has also been used to establish this. The main tools that we have used here are the following (see section 3 for details): 1. Facts 1.1 and 3.1, 2. Lemma 3.2, 3. The reduction of polynomials, a technique described after Remark 3.3. This paper has 6 sections in total. Section 1 is the introduction. In sections 2 and 3, respectively, we formalize the definitions and introduce the main tools. Section 4 contains the main results listed in (1) above. Section 5 is dedicated to the classification of polynomials in $S_r$. The last section has some open questions and discussions that arose from the study of the polynomials in this paper. Interested reader can also look at the works of Shallit, Vasiga in [@VS04] and Odoni in [@O85]. -30pt -1ex -.2ex 2.3ex .2ex Definitions, notation and terminology We will start by formally defining the polynomials mentioned in the introduction and fixing some basic terminology that we will use throughout this paper. Let $\mathcal{P}$ be the set of all positive primes in $\mathbb{Z}$. For a finite subset $A$ of $\mathcal{P}$ and for $a\in \mathbb{Z}$, we define $$\mathcal{P}_A:=\mathcal{P}\setminus A\;\;\textup{and}\;\; P_A(a):=\{p\in \mathcal{P}_A~|~p \text{ divides }a\}\;\; \textup{and}\;\; P(a):=P_{\emptyset}(a).$$ So $P(a)$ is the set of all positive primes that divides $a$. For $u=u(x)\in\mathbb{Z}[x]$ of degree at least 1, we define the polynomials $u^{(1)}(x):=u(x)$ and $u^{(n+1)}(x):=u(u^{(n)}(x))$, $n\in\mathbb{N}$. Having fixed $r$ in $\mathbb{Z}$, $A$ a finite subset of $\mathcal{P}$, $d$ in $\mathbb{N}$ a degree and $i$ in $\mathbb{N}$ an index, we define the following: 1. We will say that $u(x)$ is a *weakly locally nilpotent polynomial* at $r$ outside $A$ if for each $p\in \mathcal{P}_A$, there exists $m\in\mathbb{N}$ (possibly depending on $p$) such that $u^{(m)}(r)\equiv 0\pmod p$. For each $p\in \mathcal{P}_A$, we let $m_p$ be the least of all such $m'$s. We fix the following notation for weakly locally nilpotent polynomials at $r$ outside $A$:\ $L_{r,A}^d:=\{u~|~u\textup{, of degree } d,\textup{ is weakly locally nilpotent at }r \textup{ outside }A\},$\ $L_{r,A}:=\sqcup_{d=1}^\infty L_{r,A}^d~$. 2. If $A=\emptyset$ in (1), then we will just drop the terms "weakly\" and "outside $A$\". 3. We will say that $u(x)$ is a *nilpotent polynomial* at $r$ if $\text{there exists an}~ n\in\mathbb{N}$ such that $u^{(n)}(r)=0$. We will call the smallest of all such $n'$s the *nilpotency index/index of nilpotency* of $u(x)$ at $r$. If $u^{(n)}(r)\neq 0$ for all $n\in\mathbb{N}$, we will say that $u$ is *non-nilpotent* at $r$. We fix the following notations for nilpotent polynomials at $r$:\ $N_{r,i}^d:=\{u~|~u\textup{ is nilpotent at }r \textup{ of nilpotency index }i\textup{ and degree }d\},$\ $N_{r,i}:=\sqcup_{d=1}^\infty N_{r,i}^d~,$\ $N_r:=\sqcup_{i=1}^\infty N_{r,i}~.$ 4. The rest of the notation are as follows:\ $S_r:=L_r\setminus N_r$. For integers $a,b,c~(c\neq 0)$, we will write $a\equiv_c b$ to mean $a\equiv b\pmod c$. **Remark 2**. It is clear that $N_r\subset L_{r,\emptyset}$. But it turns out that, for every given $r\in\mathbb{Z}$, $S_r$ is non-empty (see Corollaries 4.3, 4.5 and 5.4, and Theorem 5.3 below). -3.25ex-1ex -.2ex 1.5ex .2ex Some examples 1. Let $r\in\mathbb{Z}$. For each non-zero $q(x)\in\mathbb{Z}[x]$, $(x-r)q(x)\in N_{r,1}$. 2. If $u(x)=-2x-4$, then $u(-1)=-2,~u(-2)=0$. So $u(x)\in N_{-1,2}^1$. If $r\in\mathbb{Z}\setminus\{-1\}$, $u_r(x):=-(r+1)x+(r+1)^2\in N_{r,2}^1$, and if $r\in\mathbb{Z}\setminus\{0\}$, $u_r(x):=-2x+4r\in N_{r,2}^1$. Also if $r\in\mathbb{Z}\setminus\{0\}$, $u(x)=x\pm 1\in N_{r,r}^1$, where we use the negative sign when $r$ is positive and positive sign otherwise. 3. Let $u(x)=-2x^2+7x-3$. Then $u(1)=2,~u(2)=3$ and $u(3)=0$. So $u(x)\in N_{1,3}^2$. From this and Fact 3.1 (stated and proved below) it follows that $v(x):=2x^2+7x+3\in N_{-1,3}^2$. 4. The polynomial $u(x)=-x^3+9x^2-25x+25\in N_{2,4}^3$. 5. **This example shows the existence of non-nilpotent, locally nilpotent polynomials at 1.** Let $u(x)=x+1$. Then by induction it is easy to see that $u^{(n)}(1)=n+1$, for every $n\in\mathbb{N}$, and hence $u\notin N_{1}$. For each $p\in \mathcal{P}$, $u^{(p-1)}(1)=p\equiv_p 0$. Thus $u(x)\in S_1$. In Corollary 4.3 we will see that $S_1=\{x+1\}$. 6. For every $a\in\mathbb{Z}\setminus\{0\}$, let $u_a=u_a(x):=x+a$. By induction, we get $u_a^{(n)}(0)=na$. So it is clear that $u_a\notin N_0$. For each prime $p$, $u_a^{(p)}(0)=pa\equiv_p 0$. Thus $u_a\in S_0$. 7. Let $u(x)=4x-2$. Then $u(1)=2$ and $u(2)=6\equiv_5 1$. This means that $u^{(n)}(1)$ is either 1 or 2$\pmod 5$, for every $n\in\mathbb{N}$. This shows that $u(x)\notin L_{1,A}$, for every finite subset $A\subset \mathcal{P}_{\{5\}}$. 8. Let $u(x)$ be as in *example* (g). Then by induction we have $u^{(n)}(0)=\frac{2}{3}(1-4^n)$, which cannot be zero for any $n\in \mathbb{N}$ and so the above polynomial is not in $N_0$. Clearly $m_2=1, ~m_3=3$. For every prime $p\in \mathcal{P}_{\{2,3\}}$, we have that $u^{(p-1)}(0)\equiv_p 0$, by *Fermat's little theorem* and so $u(x)\in S_0$. **Remark 3**. The computation of polynomial iterations are very complicated. But the linear polynomials have a nice and easy-to-understand iteration formula: let $u(x)=ax+b$ be a linear polynomial, i.e., $a\in\mathbb{Z}\setminus\{0\}$. Then by induction it follows that for every $n\geq 1$, $$u^{(n)}(x)=a^nx+b\left(\sum\limits_{i=0}^{n-1}a^i\right),~~n\in\mathbb{N}.$$ So $u^{(n)}(r)=a^nr+b\left(\sum\limits_{i=0}^{n-1}a^i\right)$, for each $n\in\mathbb{N}$. Throughout this paper we will refer to this formula as the ***linear iteration formula.*** -30pt -1ex -.2ex 2.3ex .2ex The main tools *In this section we will develop the necessary tools. We begin with the proof of Fact 3.1, which indicates that for our purposes it is enough to study the polynomials at non-negative values of $r$. In particular, it shows that there is a one-to-one correspondence between $S_r$ and $S_{-r}$.* **Fact 4**. Let $u(x)$ be a polynomial of degree $d$ and let $r\in\mathbb{Z}\setminus\{0\}$. Define $v(x):=-u(-x)$. Then $u(x)\in L_{r,\emptyset}^d \iff v(x)\in L_{-r,\emptyset}^d$. Similarly, $u(x)\in N_{r,n}^d \iff v(x)\in N_{-r,n}^d$, and $u(x)\in S_r\iff v(x)\in S_{-r}$. *Proof.* Since $v(-x)=-u(x)$, by induction one sees that $v^{(n)}(-r)=-u^{(n)}(r)$, from which the fact follows. ◻ We will now state and prove the CRS Lemma which was mentioned in the introduction. We will also justify its importance in understanding linear locally nilpotent polynomials. **Lemma 5** (CRS Lemma). *Let $\alpha,\beta,\gamma\in\mathbb{Z}\setminus \{0\}$ be such that there is no $k\in\mathbb{Z}$ such that $\frac{\beta}{\gamma}=\alpha^k$. Then $\mathcal{P}\setminus \cup_{n\in\mathbb{N}}P(\gamma \alpha^n-\beta)$ is an infinite set.* *Proof.* Suppose, if possible, that $\mathcal{P}\setminus \cup_{n\in\mathbb{N}}P(\gamma \alpha^n-\beta)$ is a finite set. This means that the set $\cup_{n\in\mathbb{N}}P(\gamma \alpha^n-\beta)$ contains all but finitely many primes. Then $\cup_{n\in\mathbb{N}}P(\gamma \alpha^n-\beta)\setminus P(\gamma)$ also contains all but finitely many primes. So, for almost all $p\in \mathcal{P}_{P(\gamma)}, ~\alpha^{n_p}\equiv_p \beta\gamma^{-1}$ for some $n_p\in \mathbb{N}$ (choice of $n_p$ possibly depends on $p$). So, if $k\in\mathbb{N}$ is such that $\alpha^k\equiv_p 1$, then $(\beta \gamma^{-1})^k\equiv_p (\alpha^{n_p})^k\equiv_p 1$. Thus taking $\alpha=x$, $\beta\gamma^{-1}=y$ and $F=\mathbb{Q}$ in Theorem 1 of [@CS97], we arrive at a contradiction! Thus $\mathcal{P}\setminus \cup_{n\in\mathbb{N}}P(\gamma \alpha^n-\beta)$ is an infinite set. ◻ **Remark 6** (Importance of the CRS lemma). Let $r\in \mathbb{Z}\setminus\{0\}$ and $u=u(x)=ax+b\in L_{r,\emptyset}^1$ with $a\neq \pm 1$. By the *linear iteration formula*, we get $$u^{(n)}(r)=\frac{a^n(r-ar-b)+b}{1-a}.$$ Since $u\in L_{r,\emptyset}^1$, we can say that $\mathcal{P}\setminus \cup_{n\in\mathbb{N}} P(\gamma \alpha^n-\beta)$ is a finite set (in fact, it is an empty set), where $\alpha=a,\beta=-b$ and $\gamma=r-ar-b$. Then it follows from the Lemma 3.2 that either $\frac{\beta}{\gamma}$ or $\frac{\gamma}{\beta}$ is a power of $\alpha$, i.e., $b=-a^m(r-ar-b),$ for some $m\in\mathbb{Z}$. Moreover, if $m\in\mathbb{N}$ we can say that $u\in N_r$. So, to summarize, if $u$ is in $S_r$ (with $a\notin\{ \pm 1\}$), then $\text{there exists}~m\in\mathbb{N}\cup \{0\}$ such that $a^mb=b+ar-r$. ### Reduction of polynomials. {#reduction-of-polynomials. .unnumbered} Let $r\in \mathbb{N}$ and $u=u(x)\in\mathbb{Z}[x]$ such that $r|u(0)$. Define $v=v(x):=\frac{1}{r}u(rx)$. Note that $v$ is a polynomial over $\mathbb{Z}$ of the same degree as $u$ and $rv(1)=u(r)$. Using induction one can see that $rv^{(n)}(1)=u^{(n)}(r),\text{ for all} ~n\in\mathbb{N}$. Then it follows that $u(x)$ is weakly locally nilpotent at $r$ outside some $A$ iff $v(x)$ is weakly locally nilpotent at 1 outside $A\cup P(r)$, and also $u(x)$ is nilpotent at $r$ iff $v(x)$ is nilpotent at 1. Thus we can reduce any polynomial $u(x)$ in $L_{r,\emptyset}^d$ with the extra condition that $r|u(0)$ to a polynomial $v(x)$ in $L_{1,P(r)}^d$. We will call this the *reduction of $u(x)$ to $v(x)$*, where, of course, $u$ and $v$ are as above. -30pt -1ex -.2ex 2.3ex .2ex Arbitrary $d$ and $r\in\{0,1,-1\}$ In this section, we state and prove two theorems that provide the classification of all locally nilpotent polynomials at $r$ when $r\in\{0,\pm 1\}$. We start with the $r=1$ case. **Theorem 7**. *The following is the list of all polynomials in $L_{1,\emptyset}$[:]{.upright}* 1. *$(x-1)p(x)$, with $p(x)\in\mathbb{Z}[x]\setminus\{0\}$ (*Nilpotent of nilpotency index 1*).* 2. *$-2x+4+p(x)(x-1)(x-2)$, with $p(x)\in\mathbb{Z}[x]$ (*Nilpotent of nilpotency index 2*).* 3. *$-2x^2+7x-3+p(x)(x-1)(x-2)(x-3)$, with $p(x)\in\mathbb{Z}[x]$ (*Nilpotent of nilpotency index 3*).* 4. *$x+1$ (*Locally nilpotent but not nilpotent*).* *Proof.* Let $u=u(x)\in L_{1,\emptyset}^d$. We will consider the following three cases: ### Case 1. $u(1)-1\not \in\{\pm 1\}$. {#case-1.-u1-1not-inpm-1. .unnumbered} Then $P(u(1)-1)\neq\emptyset$ and for each $p\in P(u(1)-1)$ we have $u(1)\equiv_p 1$, i.e., for each $p\in P(u(1)-1)$, $m_p$ does not exist. This is a contradiction to the hypothesis that $u\in L_{1,\emptyset}^d$! ### Case 2. $u(1)-1=-1$. {#case-2.-u1-1-1. .unnumbered} These are just the polynomials listed in (1). ### Case 3. $u(1)-1=1\textup{ or }u(1)=2$. {#case-3.-u1-11textup-or-u12. .unnumbered} We now explore the possibilities for $u(2)$. If $u(2)=0$, then $u(x)$ is of the form listed in (2). So suppose that $u(2)\neq 0$. Of course $u(2)\notin\{1,2\}$ as otherwise we get $u^{(n)}(1)=$ 1 or 2, for every $n\in\mathbb{N}$ and hence it cannot be in $L_{1,\emptyset}^d$. Thus $u(2)$ is either $\le -1$ or $\ge 3$, i.e., $|u(2)-1|\ge 2$. In other words, $P(u(2)-1)\neq \emptyset$. Let $p\in P(u(2)-1)$. Then $u(2)\equiv_p 1$. As $u$ is locally nilpotent at $1$, $p$ must be 2 so that $u(2)-1$ must be of the form $\pm 2^t$, for some $t\in\mathbb{N}$. To arrive at a contradiction suppose that $u(2)\neq 3$. That means $u(2)$ is either $\geq 4$ or $\leq -1$. So we consider these two possibilities one by one. **Possibility 1**. $u(2)\ge 4$. We know that $u(2)-1=2^t$, i.e., $u(2)$ is odd. So, in fact, $u(2)\ge 5$. Thus there exists $p\in \mathcal{P}_{\{2\}}$ such that $p\in P(u(2)-2)$ and $u^{(n)}(2)\equiv_p 2$, for every $n\in\mathbb{N}$, a contradiction to the hypothesis that $u\in L_{1,\emptyset}^d$! **Possibility 2**. $u(2)\le -1$. We know that $u(2)-1=-2^t$, i.e., $u(2)$ is odd so that $u(2)-2$ is odd as well and less or equal to $-3$. Using the same argument as in possibility 1 we get a contradiction! Thus $u(2)$ must be 3. Next, we look at $u(3)$. If $u(3)=0$, then $u(x)$ is of the form listed in (3). So suppose that $u(3)\neq 0$. For the same reason as above $u(3)\notin\{0,1,2,3\}$. Thus $u(3)$ is either $\le -1$ or $\ge 4$. To get to a contradiction suppose that $u(3)\neq 4$. Then either $u(3)-3\le -4$ or $\geq 2$. In any case, $P(u(3)-3)\neq\emptyset$. Let $p\in P(u(3)-3)$. Then $u(3)\equiv_p 3$ so that $p\in \{2,3\}$, which follows because $u\in L_{1,\emptyset}^d$. If $p=2$ then $u(1)\equiv_p 0$. Then $u(3)\equiv_p 3\equiv_p 1\not\equiv_p u(1)$, which is an impossibility as $3\equiv_p 1$ must imply $u(3)\equiv_p u(1)$! This means $p=3$ and $u(3)-3=\pm 3^s$, for some $s\in\mathbb{N}$. For similar reason, $P(u(3)-1)\neq \emptyset$ and for each $q\in P(u(3)-1)$, $u(3)\equiv_q 1$ which implies $q\in\{2,3\}$. But $q|u(3)-1=2\pm 3^s$, which is a contradiction to the fact that $q\in\{2,3\}$! Thus $u(3)=4$. Next, we look at $u(4)$. We claim that no further iteration of $u$ at $1$ can be zero and we would like to prove this by showing that $u(n-1)=n$, $\text{ for all} ~n\ge 4$ and that would mean $u(x)=x+1$. We use *mathematical induction* to prove this claim. Let $u(j-1)=j$, for every $2\le j\le n$, for some $n\ge 4$ and we want to show that $u(n)=n+1$. Since $u(1)=2, ~u(2)=3, ~u(3)=4,~\ldots, u(n-1)=n$, there is a polynomial $p(x)$ such that $u(x)=x+1+p(x)(x-1)(x-2)(x-3)\cdots (x-n+1)$. So $u(n)=n+1+p(n)\cdot (n-1)!$, which must be different from 0 as $n\ge 4$. If $u(n)=i$ for some $i\in\{1,\ldots,n\}$, then the iterations $u^{(m)}(1)\in\{1,\ldots,n\}$, for every $m\in\mathbb{N}$ and that means for only finitely many primes $p$, $m_p$ can exist. Thus $u$ cannot be locally nilpotent at $1$ and $u(n)\not \in \{0,\ldots,n\}$. This means $u(n)$ is either $\ge n+1$ or $\le -1$. For a contradiction, suppose that $u(n)\neq n+1$. Then, either $u(n)-n\ge 2$ or $u(n)-n\le -(n+1)$. In any case, we get $P(u(n)-n)\neq\emptyset$. For each $p\in P(u(n)-n)$ we have $u(n)\equiv_p n$, which is an impossibility unless $p\le n$. Suppose, if possible, $p<n$. Then $n\equiv_p a$ for some $a\in \{1,\ldots,p-1\}$. Note that $a$ cannot be zero as otherwise $n\equiv_p 0$, so that $u(0)\equiv_p u(n)\equiv_p n\equiv_p 0$, i.e., $p|u(0)$ and so $p|u(p)=p+1$, an impossibility! Now by the induction hypothesis we have $u(a)=a+1$ and also $a\equiv_p n\equiv_p u(n)\equiv_p u(a)$, i.e., $u(a)\equiv_p a$, which is absurd as this means $m_p$ does not exist! So $p=n$, i.e., $n$ is prime and $u(n)=n\pm n^s$, for some $s\in\mathbb{N}$. For similar reason, $P(u(n)-1)\neq \emptyset$. So for every $q\in P(u(n)-1)$, $u(n)\equiv_q 1$ and it follows that $q$ is less than or equal to $n$. But if $q=n$, then $n=q|u(n)-1=(n-1)\pm n^s$ and so $n|1$, an impossibility! So, in fact, we have $q\le n-1$. We can choose $b\in\{0,\ldots,q-1\}$ such that $n\equiv_q b+1$. By the induction hypothesis $u(b+1)=b+2$ and also $u(b+1)\equiv_q u(n)\equiv_q 1$. These two relations together imply $b+1\equiv_q 0$, i.e., $q|n$. But, since $n$ is a prime, $n=q$, which is an impossibility, and hence $u(n)=n+1$. ◻ **Corollary 8**. *It follows from Fact 3.1 and Theorem 4.1 that the following is the list of all polynomials in $L_{-1,\emptyset}$:* 1. *$(x+1)p(x)$, with $p(x)\in\mathbb{Z}[x]\setminus\{0\}$ (Nilpotent of nilpotency index 1).* 2. *$-2x-4+p(x)(x+1)(x+2)$, with $p(x)\in\mathbb{Z}[x]$ (Nilpotent of nilpotency index 2).* 3. *$2x^2+7x+3+p(x)(x+1)(x+2)(x+3)$, with $p(x)\in\mathbb{Z}[x]$ (Nilpotent of nilpotency index 3).* 4. *$x-1$ (Locally nilpotent but not nilpotent).* **Corollary 9**. *The sets $S_1$ and $S_{-1}$ are singleton sets.* *Proof.* Let $u(x)\in S_1$. Then by Theorem 4.1, $u(x)$ must be $x+1$ as all the other polynomials in the list (1)-(4) in there are in $N_1$. Now by Fact 3.1, it follows that $S_{-1}=\{x-1\}$. ◻ We end this section by looking at the case when $r=0.$ **Theorem 10**. *The following is the list of all polynomials in $L_{0,\emptyset}$[:]{.upright}* 1. *$x+b$, with $b\in\mathbb{Z}\setminus \{0\}$ (*Locally nilpotent but not nilpotent*).* 2. *$ax+b$, with $P(b)\supseteq P(a)\neq \emptyset$ and $b\neq 0$ (*Locally nilpotent but not nilpotent*).* 3. *$xp(x)$, with $p(x)\in\mathbb{Z}[x]\setminus\{0\}$ (*Nilpotent of nilpotency index 1*).* 4. *$(x-a)p(x)$, with $a\in\mathbb{Z}\setminus\{0\}$ and $p(x)\in \mathbb{Z}[x]$ s.t $p(0)=-1$ (*Nilpotent of nilpotency index 2*).* *Proof.* First suppose that $u$ is nilpotent of nilpotency index $m$, for some $m\in\mathbb{N}$. If $u(0)=0$, then $m=1$ and $u(x)=xp(x),$ for some non-zero $p(x)\in\mathbb{Z}[x]$, which is (3) in the list. So suppose that $u(0)\neq 0$. Define $$u_0:=u(0),~u_n:=u^{(n+1)}(0)-u^{(n)}(0),~n\in\mathbb{N}.$$ Then $u_{n+1}=u^{(n+2)}(0)-u^{(n+1)}(0)= u(u^{(n+1)}(0))-u(u^{(n)}(0)).$ That means $u_n$ divides $u_{n+1},\text{ for all}~n\in\mathbb{Z}_{\ge 0}$. We also have $u^{(m)}(0)=0$ so that $u_m=u^{(m+1)}(0)-u^{(m)}(0)=u^{(m+1)}(0)=u_0$. As $u_0|u_1|\ldots|u_m=u_0,$ it follows that $u_n=\pm u_0,$ for all $n.$ Also note that $u_0+\cdots+u_{m-1}=u^{(m)}(0)=0$. This means $m$ must be even and half of these integers is positive and the other half is negative (since $|u_n|=|u_0|,\text{ for all}~n$). So there exists $k\in\{1,\ldots,m-1\}$ such that $u_{k-1}=-u_k,$ i.e., $u^{(k)}(0)-u^{(k-1)}(0)=u^{(k)}(0)-u^{(k+1)}(0),$ i.e., $u^{(k+1)}(0)=u^{(k-1)}(0)$. Thus $u^{(n+2)}(0)=u^{(n)}(0),\text{ for all} ~n\ge k-1$ and so, in particular, $0=u^{(m)}(0)=u^{(m+2)}(0)=u^{(2)}(0)$. Hence, $m=2$ and $u(x)=(x-a)p(x)$, with $a\in\mathbb{Z}\setminus\{0\}$ and $p(x)\in\mathbb{Z}[x]$ with $p(0)=-1$; here $a=u(1),$ which is (4) in the list. Now suppose that $u\in S_0$. Then by Fact 1.1, it must be linear. Let $u(x):=ax+b, a\neq 0$. Note that if $b\neq0$ as otherwise $u$ would be nilpotent. When $a=1$, every $u(x)\in S_0$: in fact if $u(x)=x+b$, then by the *linear iteration formula*, $u^{(n)}(0)=b(1+\cdots+1)=bn$, which is non-zero $\text{ for all}~n\in\mathbb{N}$ and for each prime $p$, $u^{(p)}(0)=bp\equiv_p 0$. When $a=-1$, $u^{(2)}(0)=0$. So $a$ must be different from $-1$. Thus we can assume that $|a|\geq 2$, i.e., $P(a)$ is a non-empty, finite set. Again, using the *linear iteration formula*, we get $u^{(n)}(0)=b(1+\cdots+a^{n-1}),~n\in\mathbb{N}$. Suppose, if possible, there exists some prime $p$ in $P_{P(b)}(a)$, i.e., there is a prime $p$ such that $p|a$ but $p\nmid b$. Then $u^{(n)}(0)\equiv_p b$, for every $n\in\mathbb{N}$ and that means $u$ cannot be locally nilpotent. Thus $P(b)\supseteq P(a)\neq\emptyset$. If $p\in P(b)$, then $m_p=1$.\ If $p\notin P(b)\cup P(a-1)$, then $u^{(p-1)}(0)=\frac{b}{a-1}(a^{p-1}-1)\equiv_p 0$, which follows from *Fermat's little theorem*.\ Finally if $p\in P(a-1)$, then $u^{(p)}(0)=b(1+\cdots+a^{p-1})\equiv_p b(1+\cdots+1)\equiv_p 0$. Thus $m_p$ exists for every $p\in\mathcal{P}$. ◻ **Corollary 11**. *The following is the list of all polynomials in $S_{0,\emptyset}$[:]{.upright}* 1. *$x+b$, with $b\in\mathbb{Z}\setminus \{0\}$.* 2. *$ax+b$, with $P(b)\supseteq P(a)\neq \emptyset$ and $b\neq 0$.* -30pt -1ex -.2ex 2.3ex .2ex Linear case $d=1$ In this section we classify all polynomials that are locally nilpotent but non-nilpotent at $r$. Since the cases $r=0$ and $r=\pm 1$ have been covered in the previous section, here our focus would be on the values of $r$ that lie in $\mathbb{Z}\setminus\{0,\pm 1\}$. First, we state and prove a classification theorem for polynomials which are weakly locally nilpotent at 1 outside some given finite subset $A$ of $\mathcal{P}$. We will use this theorem to prove our final main result, Theorem 5.3. **Theorem 12**. *Let $A=\{q_1,\ldots, q_k\}$, where $q_1,\ldots, q_k$ are $k$ distinct primes. Then the following is the list of all the polynomials in $L_{1,A}^1$[:]{.upright}* 1. *$x\pm q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$.* 2. *$\alpha(x-1)$, $\alpha\in\mathbb{Z}\setminus\{0\}$.* 3. *$\pm q_1^{s_1}\cdots q_k^{s_k} x+1$, where $s_i\in\mathbb{N}\cup\{0\}$ such that $\sum s_i\ge 1$.* 4. *$-2x-1$ (only when $2\in A$).* 5. *$-2x+4$.* *Proof.* Let $u=u(x)\in L_{1,A}^1$. Then by Fact 1.1 it must be linear, say $ax+b$. It is clear that $b\neq 0$ as otherwise $u^{(n)}(1)$ would just be $a^n$ and it cannot be divisible by any prime $p\in \mathcal{P}_{P(a)}$. By the *linear iteration formula*, $u^{(n)}(1)=a^n+b(1+\cdots+a^{n-1})$, for every $n\in\mathbb{N}$. $$\begin{aligned} \text{Note that if } a=1, ~u^{(m_p)}(1)& = 1+bm_p\equiv_p 0, \text{ for every prime }p\notin A,\\ \implies & bm_p\equiv_p -1, \text{ for every prime }p\notin A,\\ \implies & b\text{ is invertible in }\mathbb{F}_p, \text{ for every prime }p\notin A,\\ \implies & b=\pm q_1^{s_1}\cdots q_k^{s_k}, \text{ for some }s_i's \textup{ in }\mathbb{N}\cup \{0\}.\hspace{57pt}\end{aligned}$$ One can check that the polynomials $x\pm q_1^{s_1}\cdots q_k^{s_k}$ are indeed in $L_{1,A}^1$, for $s_i\in\mathbb{N}\cup\{0\}$. If $a=-1$, $u(x)=-x+b$ and $u^{(2)}(x)=x$. So $u$ cannot be in $L_{1,A}^1$ unless $b=1$, and in that case, it is in fact in $N_{1,1}^1$. Thus we can assume that $|a|\ge 2$. Similar to the proof of Theorem 4.1, we can break down these polynomials into the following three cases: ### Case 1. $u(1)-1\notin \{\pm 1\}$. {#case-1.-u1-1notin-pm-1. .unnumbered} This means that $P(u(1)-1)\neq \emptyset$. So $a+b=1 \pm q_1^{s_1}\cdots q_k^{s_k}$, i.e., $b=1-a \pm q_1^{s_1}\cdots q_k^{s_k}$, for some $s_i\in\mathbb{N}\cup\{0\}$ with $\sum s_i\ge 1$. Then by the *linear iteration formula*, we have $$u^{(n)}(1)=\frac{b\pm a^n(1-a-b)}{1-a}$$ and it follows from Remark 3.3 that $\text{there exists an }m\in\mathbb{Z}$ such that $b=\pm a^m(1-a-b)$. If $m=0$, then $b=\pm(1-a-b)$, i.e., $a+2b=1$ or $a=1$. Since $|a|\ge 2$, we deduce that $a+2b=1$.We also have $a+b=1\pm q_1^{s_1}\cdots q_k^{s_k}$. Solving $a$ and $b$ from these two equations we get $a=1\pm 2q_1^{s_1}\cdots q_k^{s_k},~b=\mp q_1^{s_1}\cdots q_k^{s_k}$. So $u(x)=(1\pm 2q_1^{s_1}\cdots q_k^{s_k})x\mp q_1^{s_1}\cdots q_k^{s_k}$ and so $u^{(n)}(1)=\frac{1+(1\pm 2q_1^{s_1}\cdots q_k^{s_k})^n}{2},~n\in\mathbb{N}$. Letting $\alpha=1\pm 2q_1^{s_1}\cdots q_k^{s_k},\beta=-1$ and $\gamma =1$, it is clear that neither $\frac{\beta}{\gamma}$ nor $\frac{\gamma}{\beta}$ is a power of $\alpha$. So it follows from the Lemma 3.2 that $(1\pm 2q_1^{s_1}\cdots q_k^{s_k})x\mp q_1^{s_1}\cdots q_k^{s_k}\notin L_{1,A}^1$. Thus $m$ must be a non-zero integer. We consider the following four possibilities: [If $m\in\mathbb{N}$ and $b=a^m(1-a-b)$]{.ul}, then $b(1+a^m)=a^m(1-a)$. Since $\gcd(a^m,a^m+1)=1$, we must have $a^m+1|1-a$ which is only possible if $m=1$. [If $m\in\mathbb{N}$ and $b=-a^m(1-a-b)$]{.ul}, then $b(1-a^m)=-a^m(1-a)$, i.e., $b(1+\cdots+a^{m-1})=-a^m$. Since $\gcd(1+\cdots+a^{m-1},a^m)=1$, we must have $1+\cdots+a^{m-1}=\pm 1$ which is only possible if $m\in\{1,2\}$. [If $m=-n,\text{ with }n\in\mathbb{N}$ and $b=a^m(1-a-b)$]{.ul}, then $ba^n=1-a-b$, i.e., $b(a^n+1)=1-a$. It follows from above that this is only possible if $n=1$. [If $m=-n,\text{ with }n\in\mathbb{N}$ and $b=-a^m(1-a-b)$]{.ul}, then $ba^n=-(1-a-b)$, i.e., $b(a^n-1)=a-1$. Again using the same logic as above, we conclude that $n\in\{1,2\}$. Thus we only need to look at $m=\pm 1,\pm 2$, which we investigate in the following four subcases: **Subcase 1.** $m=-1$.\ Here we have $ba=\pm(1-a-b)$. First suppose that $ba=1-a-b$, i.e., $b(a+1)=1-a$. This means that $a+1|a-1$ and this is only possible if $a=-2$ and $a=-3$. These values generate the polynomials $u(x)=-2x-3$ and $u(x)=-3x-2$, respectively. When $u(x)=-2x-3$, the *linear iteration formula* gives $$u^{(n)}(1)=2(-2)^n-1.$$ Letting $\alpha=-2,\beta=1$ and $\gamma=2$, it is clear that neither $\frac{\beta}{\gamma}$ nor $\frac{\gamma}{\beta}$ is a power of $\alpha$. So, by the Lemma 3.2, $-2x-3\notin L_{1,A}^1$. Similarly we can show that $-3x-2\notin L_{1,A}^1.$ Now suppose $ba=-(1-a-b)$. This gives $b=1$ and hence $a=\pm q_1^{s_1}\cdots q_k^{s_k}$. Thus $u(x)=\pm q_1^{s_1}\cdots q_k^{s_k}x+1$ and it follows from the *linear iteration formula* that $$u^{(n)}(1)=(\pm q_1^{s_1}\cdots q_k^{s_k})^n+[1+\cdots+(\pm q_1^{s_1}\cdots q_k^{s_k})^{n-1}]=\frac{1-(\pm q_1^{s_1}\cdots q_k^{s_k})^{n+1}}{1-(\pm q_1^{s_1}\cdots q_k^{s_k})}, ~n\in\mathbb{N}.$$ If $p\in P_A(1-(\pm q_1^{s_1}\cdots q_k^{s_k}))$, then $u^{(p)}(1)\equiv_p p\equiv_p 0$. So suppose that $p\notin P_A(1-(\pm q_1^{s_1}\cdots q_k^{s_k}))$. Now if $2\in A$, then the existence of $m_2$ is not a concern and if $2\notin A$, then $2\in P_A(1-(\pm q_1^{s_1}\cdots q_k^{s_k}))$ which was covered above. Finally, if $p$ is in ${A\cup \{2\}}$, then $u^{(p-2)}(1)\equiv_p 0$, by *Fermat's little theorem*. So $u(x)=\pm q_1^{s_1}\cdots q_k^{s_k}x+1$ is in $L_{1,A}^1$. **Subcase 2.** $m=1$.\ Here we have $b=\pm a(1-a-b)$. First suppose that $b=a(1-a-b)$, i.e., $b(a+1)=a(1-a)$. The same reasoning as above implies $a+1|a-1$ so that the only possibilities we get are $a=-2,~b=6$ or $a=-3,~b=6$. These values produce the polynomials $u(x)=-2x+6$ and $u(x)=-3x+6$, respectively. When $u(x)=-2x+6$, the *linear iteration formula* gives $$u^{(n)}(1)=-(-2)^n+2.$$ Letting $\alpha=-2,\beta=-2$ and $\gamma=-1$, it is clear that neither $\frac{\beta}{\gamma}$ nor $\frac{\gamma}{\beta}$ is a power of $\alpha$. So, by the Lemma 3.2, $-2x+6\notin L_{1,A}^1$. Similarly, we can show that $-3x+6\notin L_{1,A}^1$. **Subcase 3.** $m=-2$.\ Here we have $ba^2=\pm (1-a-b)$. First suppose $ba^2=1-a-b$, i.e., $b(a^2+1)=1-a$. This means that $a^2+1|a-1$ which is not possible as $|1-a|\le 1+|a|<1+a^2$. Thus $ba^2=-(1-a-b)$, i.e., $b(a+1)=1$, i.e., $b=a+1=\pm 1$. So $u(x)=-2x-1$. It follows from the *linear iteration formula* that $$u^{(n)}(1)=(-2)^n-[1+\cdots+(-2)^{n-1}]=\frac{(-2)^{n+2}-1}{3},~n\in\mathbb{N}.$$ It is easy to see that $m_2$ does not exist, $m_3=1$ and for all $p\in \mathcal{P}_{\{2,3\}}$, $u^{(p-3)}(1)\equiv_p 0$. So $-2x-1$ is in $L_{1,A}^1$ iff $2\in A$. **Subcase 4.** $m=2$.\ Here we have $b=\pm a^2(1-a-b)$. First suppose that $b=a^2(1-a-b)$, i.e., $b(1+a^2)=a^2(1-a)$. Since $\gcd(1+a^2,a^2)=1$, we must have $1+a^2|1-a$, which is impossible (see the subcase 3 above). So $b=-a^2(1-a-b)$, i.e., $b(a+1)=-a^2$ which means $a+1=\pm 1$ and $b=\mp a^2$. Since $|a|\ge 2$, this means $a=-2$ and $b=4$. But then $u(1)-1=1\in \{\pm 1\},$ an impossibility in this *case*! This wraps us case 1. Now we look at the remaining two cases. ### Case 2. $u(1)-1=-1$, i.e., $u(1)=0$. {#case-2.-u1-1-1-i.e.-u10. .unnumbered} These are the polynomials in $N_{1,1}^1$. ### Case 3. $u(1)-1=1$, i.e., $u(1)=2$. {#case-3.-u1-11-i.e.-u12. .unnumbered} If $u(2)=0$, then $u(x)=-2x+4$. So we can suppose that $u(2)\notin\{0,1,2\}$, i.e., $u(2)$ is either $\le -1$ or $\ge 3$, i.e., $|u(2)-1|\ge 2,$ i.e., $P(u(2)-1)\neq \emptyset$. If $u(2)=3$, then $u(x)=x+1\in S_1$. So we can further assume that $u(2)\neq 3$. Since $u(1)=2,~b=2-a$ and so $u(x)=ax+(2-a)$. Then by the *linear iteration formula*, we get $$u^{(n)}(1)=\frac{2-a-a^n}{1-a},~~n\in\mathbb{N}.$$ Since $u\in L_{1,A}^1$ it follows from Remark 3.3 that $2-a=a^m$, for some $m\in\mathbb{Z}$. Note that $m$ cannot be a non-positive integer as otherwise it would follow that $a$ must be equal to $\pm 1$. Thus $m\in\mathbb{N}$ and $2=a(1+ a^{m-1})$. Therefore $a=\pm 2$ and $1+a^{m-1}=\pm 1,$ i.e., $a^{m-1}=-2$, i.e., $a=-2$. But $a=-2$ implies that $b=4$ and hence $u(2)=2a+b=0$, which cannot happen as we have already said that $u(2)\not\in\{0,1,2,3\}$. Thus $ax+(2-a)\notin L_{1,A}^1$. ◻ The next corollary follows directly from the computations in the proof of Theorem 5.1: **Corollary 13**. *Let $A=\{q_1,\ldots,q_k\}$ and $q_1,\ldots,q_k$ be $k$ distinct primes. Then the following is the list of all polynomials in $L_{1,A}^1\setminus N_1$[:]{.upright}* 1. *$x+ q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$.* 2. *$x- q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$ such that $\sum s_i\ge 1$.* 3. *$\pm q_1^{s_1}\cdots q_k^{s_k} x+1$, where $s_i\in\mathbb{N}\cup\{0\}$ such that $\sum s_i\ge 1$.* 4. *$-2x-1$ (only when $2\in A$).* Finally we state and prove the last (main) result of this paper. **Theorem 14**. *Let $r$ be a natural number greater than or equal to 2 and $r=q_1^{a_1}\cdots q_k^{a_k}$ be the prime decomposition. Then the following is the list of all polynomials in $S_r$[:]{.upright}* 1. *$x+ q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$.* 2. *$x-q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$ with at least one $j\in\{1,\ldots, k\}$ s.t. $s_j>a_j$.* 3. *$\pm q_1^{s_1}\cdots q_k^{s_k}x+r$, where $s_i\in\mathbb{N}\cup\{0\}$ with $\sum\limits_{i} s_i\ge 1$.* 4. *$-2x-r$ (only when $r$ is even).* *Proof.* Let $u=u(x)\in S_r$ and $A:=P(r)$. Then by Fact 1.1 $u$ must be linear, say $ax+b$. First we will look at the instances when $a=\pm 1$. $$\begin{aligned} \text{Note that if } a=1, ~u^{(m_p)}(1)=&1+bm_p\equiv_p 0, \text{ for every prime }p\notin A,\\ \implies & bm_p\equiv_p -1, \text{ for every prime }p\notin A,\\ \implies & b\text{ is invertible in }\mathbb{F}_p, \text{ for every prime }p\notin A,\\ \implies & b=\pm q_1^{s_1}\cdots q_k^{s_k}, \text{ for some }s_i's \textup{ in }\mathbb{N}\cup \{0\}.\hspace{57pt}\end{aligned}$$ So, when $a=1$, we expect $u(x)$ to be of the form $x \pm q_1^{s_1}\cdots q_k^{s_k}$. First suppose that $u(x)=x+q_1^{s_1}\cdots q_k^{s_k}$. Then by the *linear iteration formula*, $u^{(n)}(r)=q_1^{a_1}\cdots q_k^{a_k}+n\cdot q_1^{s_1}\cdots q_k^{s_k}$, which is always non-zero for every $n\in\mathbb{N}$. One can check that these polynomials are locally nilpotent and they are in (1) in the list above. Now suppose that $u(x)=x-q_1^{s_1}\cdots q_k^{s_k}$. Then by the *linear iteration formula* $u^{(n)}(r)=q_1^{a_1}\cdots q_k^{a_k}-n\cdot q_1^{s_1}\cdots q_k^{s_k},~n\in\mathbb{N}$. If, for all $i\in\{1,\ldots,k\}$, $s_i\le a_i$, then $u^{(q_1^{a_1-s_1}\cdots q_k^{a_k-s_k})}(r)=0$, which is a contradiction as $u$ is non-nilpotent! That means we must have at least one $j\in\{1,\ldots,k\}$ such that $a_j<s_j$. One can now check that $u$ is non-nilpotent but locally nilpotent and so we get the polynomials in (2) in the list above. If $a=-1$, $u(x)=-x+b$ and $u^{(2)}(x)=x$. So $u$ cannot be in $L_{r,\emptyset}^1$ unless $b=r$, and in that case, it is in fact in $N_{r,1}^1$, a contradiction! So $a\neq -1$. Thus we suppose $|a|\ge 2$. It follows from Remark 3.3 that $\text{there exists an }~m\in\mathbb{N}\cup \{0\}$ such that $a^mb=b+ar-r$. If $m=0$ then $r(1-a)=0$ which is an impossibility as $r\neq 0$ and $|a|\ge 2$. That means that $m\in\mathbb{N}$ and $b(a^m-1)=r(a-1),$ i.e., $b(1+\cdots+a^{m-1})=r$ so that $b|r$. We want to show that $u(r)-r\notin \{\pm 1\}$. Suppose otherwise, i.e., $u(r)=r\pm 1$. This means that $b=r-ar\pm 1$, i.e., $u(x)=ax+(r-ar\pm 1)$. We will only consider the possibility $b=r-ar-1$ as the other possibility can be rejected using the same argument. Applying the *linear iteration formula*, we get $$u^{(n)}(r)=\frac{a^n+r-ar-1}{1-a}=\frac{a^n+b}{1-a},~\text{ for all }n\in \mathbb{N}.$$ From Remark 3.3 it follows that $r-ar-1=-a^t$, for some $t\in\mathbb{Z}$. It is clear that $t\neq 0$, as otherwise $r-ar=0,$ i.e., $r(1-a)=0,$ i.e., either $r=0$ or $a=1$, which is not true! If $t=-n$ for some $n\in\mathbb{N}$, then $a^n(r-ar-1)=-1,$ again an impossibility as $|a|\ge 2$! Thus $t\in\mathbb{N}$ and $r(1-a)=1-a^t$, i.e., $r=1+\cdots+a^{t-1}$. So $t\ge 2$, $a|r-1$ and $u^{(t)}(r)=0$, i.e., $u$ is nilpotent at $r$, a contradiction! Thus $u(r)-r$ cannot be a unit and $u(r)=r\pm q_1^{s_1}\cdots q_k^{s_k}$, for a suitable collection of $s_i's$ in $\mathbb{N}\cup\{0\}$ with $\sum\limits_i s_i\ge 1$. So $b=r-ar\pm q_1^{s_1}\cdots q_k^{s_k}$. But then $b|r$ implies that $b|q_1^{s_1}\cdots q_k^{s_k}$, i.e., $\text{there exists}~t_i\in\mathbb{N}\cup \{0\}$, with $t_i\le s_i$ for every $i$, such that $b=\pm q_1^{t_1}\cdots q_k^{t_k}$. From $b=r-ar\pm q_1^{s_1}\cdots q_k^{s_k}$ we get $ra=r-b\pm q_1^{s_1}\cdots q_k^{s_k}=r-b(\pm 1\pm q_1^{s_1-t_1}\cdots q_k^{s_k-t_k})$, i.e., $r|b(\pm 1\pm q_1^{s_1-t_1}\cdots q_k^{s_k-t_k})$.\ Suppose, if possible, all the $t_i's$ are zero. Then $b=\pm 1$ and so $r$ must divide $\pm 1\pm q_1^{s_1}\cdots q_k^{s_k}$, which is clearly absurd as $\gcd(r,\pm 1\pm q_1^{s_1}\cdots q_k^{s_k})=\gcd(q_1^{a_1}\cdots q_k^{a_k},\pm 1\pm q_1^{s_1}\cdots q_k^{s_k})=1$. Thus $\sum\limits_i t_i\ge 1$. All of these now boil down to the following two cases: ### Case 1. $There~ exists~j\in\{1,\ldots,k\}~ s.t.~s_j>t_j$. {#case-1.-there-existsjin1ldotsk-s.t.s_jt_j. .unnumbered} Since $\gcd(r,\pm 1\pm q_1^{s_1-t_1}\cdots q_k^{s_k-t_k})=1$, $r$ must divide $b$ so that we can deduce $r=\pm b$. So $a_i=t_i\le s_i,~\text{ for all}~i\in\{1,\ldots,k\}$. We use the technique of reduction of polynomials here. Define $v=v(x):=\frac{1}{r}u(rx)=ax\pm 1.$ Then $$v(1)=\frac{1}{r}u(r)=\frac{1}{r}(r\pm q_1^{s_1}\cdots q_k^{s_k})=1\pm q_1^{s_1-a_1}\cdots q_k^{s_k-a_k}$$ and $v\in L_{1,A}^1\setminus N_1$. It follows now from the list in Corollary 5.2 that we have two possibilities for $v$ (since $|a|\ge 2$): 1. $v(x)=\pm q_1^{s_1-a_1}\cdots q_k^{s_k-a_k}x+1$, in which case $u(x)=\pm q_1^{s_1-a_1}\cdots q_k^{s_k-a_k}x+r$, or 2. $v(x)=-2x-1$ (only when $2\in A$), in which case $u(x)=-2x-r$. One can check that both (i) and (ii) are indeed in $S_r$. ### Case 2. $For~each~i\in\{1,\ldots, k\},~s_i=t_i$. {#case-2.-foreachiin1ldots-ks_it_i. .unnumbered} Then $\pm q_1^{s_1}\cdots q_k^{s_k}=b|r$, i.e., $a_i\ge s_i$ for each $i$. From $b=r-ar\pm q_1^{s_1}\cdots q_k^{s_k}$ we get $r(1-a)=\pm 2q_1^{s_1}\cdots q_k^{s_k}=\pm 2b$. Thus either $r=\pm b$ or $r=\pm 2b$. The first possibility has been taken care of in *Case* 1. So we can assume that $r=2q_1^{s_1}\cdots q_k^{s_k}=\pm 2b$. That would mean $a=2$ and that $2\in A$. Without loss of generality, let $q_1=2$ so that $r=2^{s_1+1}\cdots q_k^{s_k}$. Rewriting $b=r-ar\pm q_1^{s_1}\cdots q_k^{s_k}$ gives us $2r=r-2b$, i.e., $r=-2b$. This means that $u(x)=2x-\frac{r}{2}$. It follows from the *linear iteration formula* that $$u^{(n)}(r)=\frac{r}{2}\cdot (2^n+1),~n\in\mathbb{N}.$$ Letting $\alpha=2,\beta =-1, \gamma=1$, we can see that neither $\frac{\beta}{\gamma}$ nor $\frac{\gamma}{\beta}$ is a power $\alpha$. Thus from the Lemma 3.2, $\mathcal{P}\setminus P(2^n+1)$ is an infinite set so that $\mathcal{P}\setminus P(\frac{r}{2}\cdot (2^n+1))$ is also an infinite set. So $2x-\frac{r}{2}\not\in S_r$. This completes the proof. ◻ It follows directly from Fact 3.1 and Theorem 5.3 that: **Corollary 15**. *If $r=-q_1^{a_1}\cdots q_k^{a_k}$ is the prime decomposition of an integer $r\le -2$, then the following is the list of all polynomials in $S_r$[:]{.upright}* 1. *$x- q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$.* 2. *$x+q_1^{s_1}\cdots q_k^{s_k}$, where $s_i\in\mathbb{N}\cup\{0\}$ with at least one $j\in\{1,\ldots, k\}$ s.t. $s_j>a_j$.* 3. *$\pm q_1^{s_1}\cdots q_k^{s_k}x-r$, where $s_i\in\mathbb{N}\cup\{0\}$ with $\sum\limits_{i} s_i\ge 1$.* 4. *$-2x+r$ (only when $r$ is even).* -30pt -1ex -.2ex 2.3ex .2ex Some open problems For $u(x)$, a non-constant polynomial over $\mathbb{Z}$, let $$N(u):=\{r\in\mathbb{Z}~|~u\in N_r \} ,~~LN(u):=\{r\in\mathbb{Z}~|~u\in L_{r,\emptyset} \}.$$ Then one can look at the following questions: 1. Describe all $u's$ such that $N(u)$ is finite. 2. Describe all $u's$ such that $LN(u)$ is finite. 3. Given $r\in\mathbb{Z}$, describe all $u's$ such that $r\in LN(u)$. **Few words about the open problems.** The methods used in the paper are actually useful for linear polynomials and due to Fact 1.1 we were able to achieve our goal by studying the linear polynomials only. It is clear that if $u$ is assumed to be linear, then $N(u)$ is finite for every $u$ except for the polynomials of the form $\pm x\pm c$, $c\in\mathbb{Z}\setminus\{0\}$. This answers Q1 for linear polynomials $u(x)$. It should be noted that $LN(u)=N(u)\cup\{r\in\mathbb{Z}~|~u\in S_r\}$. So for Q2, due to Fact 1.1 again, if the degree of $u$ is greater than or equal to 2 then $LN(u)$ is finite so that $N(u)$ is finite. For linear polynomials of the form $\pm x\pm c$, $c\in\mathbb{Z}\setminus\{0\}$, $N(u)$ is infinite and so $LN(u)$ is infinite. If $u=ax+b$ with $a\neq 1$, then we know that $u^{(n)}(r)=\frac{a^n(r-a-b)+b}{1-a}$. Note that $\frac{b}{a+b-r}=a^k$, for some $k\in\mathbb{Z}$ can only be true for finitely many values of $r$. So it follows from Lemma 3.2 that $\mathcal{P}\setminus\cup_{n\in\mathbb{N}} P((r-a-b)a^n+b)$ is finite for only finitely many values of $r$. Thus for these polynomials, $LN(u)$ is indeed finite. In this paper we have fully classified $S_r$, for every integer $r$. Also given an $r$, using the *linear iteration formula* it should be easy enough to describe the linear $u's$ such that $r\in N(u)$. So we have a partial answer for Q3 here. Thus if one knows enough about the form of the nilpotent polynomials of degree higher than or equal to 2, one should potentially be able to answer all of the questions above. This is a work in progress and it is of course important enough to be considered as a separate work. The author gratefully acknowledges his advisors Prof. Alexander Borisov and Prof. Adrian Vasiu for their constant support, encouragement and very helpful suggestions. The author would also like to thank Prof. Jeremy Rouse for suggesting [@CS97] which was one of the main tools used to prove Theorems 5.1 and 5.3 and Prof. Kiran Kedlaya for maintaining a wonderful archive of William Lowell Putnam Mathematics Competition questions and answers here <https://kskedlaya.org/putnam-archive/>, which was very helpful in the proof of Theorem 4.4. Finally, the author would like to thank Prof. Thomas Tucker for his suggestion to use Theorem 5 of [@TS13], which led to Fact 1.1 and that was very useful to boil down the possible candidates for locally nilpotent polynomials. 1 A. Borisov. Geometrically nilpotent subvarieties. *Finite Fields Appl*., **50** (2018), 366-371 A. Borisov. Iterations of integer polynomial maps modulo primes. *J. Integer Seq*. Vol. **16** (2013), Article 13.8.3 C. Corrales-Rodrigáñez and R. Schoof. The support problem and its elliptic analogue. *J. Number Theory*, **64** (1997), Article No. NT972114, 276-290 R.L. Benedetto, D. Ghioca, B. Hutz, P. Kurlberg, T. Scanlon and T.J. Tucker. Periods of rational maps modulo primes. *Math. Ann*., **355** (2013), 637-660 R. Benedetto, D. Ghioca, P. Kurlberg, T. J. Tucker, U. Zannier. A case of the dynamical Mordell-Lang conjecture. (with an appendix by Umberto Zannier) *Math. Ann*. **352** (2012), 1-26 J. P. Bell, D. Ghioca, and T. J. Tucker. The dynamical Mordell-Lang conjecture. *Math. Surveys Monogr.*, **210** (2016), xiv+280 pp J. H. Silverman. Integer points, Diophantine approximation, and iteration of rational maps. *Duke Math. J.* **71**(3) (1993), 793-829 J.O. Shallit and M.J.T. Vasiga. On the iteration of certain quadratic maps over GF($p$). *Discrete Math*., **277** (2004), no. 1-3, 219-240 R.W.K. Odoni. The Galois theory of iterates and composites of polynomials. *Proc. Lond. Math. Soc. (3)* **51** (1985), 385-414

arxiv_math

--- abstract: | We show that for certain sequences escaping to infinity in the $\operatorname{SL}_3\mathbb{R}$ Hitchin component, growth rates of trace functions are described by natural Finsler metrics. More specifically, as the Labourie-Loftin cubic differential gets big, logarithms of trace functions are approximated by lengths in a Finsler metric which has triangular unit balls and is defined directly in terms of the cubic differential. This is equivalent to a conjecture of Loftin from 2006 [@loftin_flat_2007] which has recently been proven by Loftin, Tambourelli, and Wolf [@loftin_limits_2022], though phrasing the result in terms of Finsler metrics is new and leads to stronger results with simpler proofs. From our perspective, the result is a corollary of a more local theorem which may have other applications. The key ingredient of the proof is another asymmetric Finsler metric, defined on any convex projective surface, recently defined by Danciger and Stecker, in which lengths of loops are logarithms of eigenvalues. We imitate work of Nie [@nie_limit_2022] to show that, as the cubic differential gets big, Danciger and Stecker's metric converges to our Finsler metric with triangular unit balls. While [@loftin_limits_2022] addresses cubic differential rays, our methods also address sequences of representations which are asymptotic to cubic differential rays, giving us more insight into natural compactifications of the moduli space of convex projective surfaces. author: - Charlie Reid bibliography: - bib.bib title: Limits of Convex Projective Surfaces and Finsler Metrics --- # Introduction A projective structure on a closed manifold $M$ is an atlas of charts valued in $\mathbb{R}\mathbb{P}^n$ with transition functions in $\operatorname{PGL}_{n+1}\mathbb{R}$. Such a structure gives rise to a developing map $\tilde{M}\to \mathbb{R}\mathbb{P}^n$, and a holonomy representation $\pi_1(M)\to \operatorname{PGL}_{n+1}\mathbb{R}$. A projective structure is called convex if the developing map is a homeomorphism onto a properly convex domain in $\mathbb{R}\mathbb{P}^n$. Classification of convex projective manifolds in general is a largely open subject, but the case of surfaces is understood: the space of convex projective structures on an oriented closed surface of genus $g$, which we will denote $\operatorname{Conv}(S)$, is a ball of dimension $16g-16$ [@choi_convex_1993]. Taking holonomy representations identifies $\operatorname{Conv}(S)$ with the Hitchin component of the space of representations $\operatorname{Rep}(\pi_1 (S),\operatorname{SL}_3\mathbb{R})$. This is analogous to the fact that the space $\operatorname{Teich}(S)$ of hyperbolic structures on $S$, is a ball of dimension $6g-6$, and is in bijection with a component of $\operatorname{Rep}(\pi_1 (S),\operatorname{PSL}_2\mathbb{R})$. In fact the symmetric square $\operatorname{PSL}_2\mathbb{R}\to \operatorname{SL}_3\mathbb{R}$ embeds $\operatorname{Teich}(S)$ into $\operatorname{Conv}(S)$ as a submanifold, which we call the Fuchsian locus. This paper is largely motivated by the aspiration to extend Thurston's compactification of $\operatorname{Teich}(S)$ by measured foliations to a compactification of $\operatorname{Conv}(S)$. Features of Thurston's compactification which are desiderata for the convex projective case include the following. - Thurston's compactification is homeomorphic to a closed ball. - A boundary point records ratios of growth rates of lengths of closed curves. - The boundary points parametrize some kind of geometric structures on $S$. In this paper we show that certain sequences in $\operatorname{Conv}(S)$ converge to the following class of Finsler metrics in a natural way. **Definition 1**. *Let $\mu$ be a cubic differential on a Riemann surface $C$; that is, a holomorphic section of $(T^*C)^{\otimes 3}$. We define $F^\Delta_\mu$ to be the maximum of twice the real parts of the cube roots of $\mu$. $$F^\Delta_\mu(v) := \max_{\{\alpha\in T_x^*C:\alpha^3=\mu_x\}} 2 Re(\alpha(v))$$ Here $x\in C$ is a point, and $v\in T_x C$ is a tangent vector.* We conjecture that these Finsler metrics, considered up to scaling, comprise an open dense subset of a compactification which extends Thurston's compactification of Teichmüller space, and satisfies the above three desiterata. The sequences we consider are "orthogonal\" to $\operatorname{Teich}(S)$ in the following sense. In the early 2000's Labourie [@labourie_flat_2006] and Loftin [@loftin_affine_2001] found a beautiful, but non-explicit parametrization of the space of convex projective structures on a closed surface $S$ by pairs $(J,\mu)$ where $J$ is a complex structure, and $\mu$ is a holomorphic cubic differential. The Fuchsian locus corresponds to $\mu=0$. We will consider sequences $(J_i,\mu_i)$ where $J_i$ converges, and $\mu_i$ diverges. A natural measurement one can make on a convex projective structure is the logarithm of the top eigenvalue of the action of a group element $\gamma\in \pi_1(S)$. $$\log(\lambda_1(\rho(\gamma)))$$ Here, $\rho:\pi_1(S)\to \operatorname{SL}_3\mathbb{R}$ is the holonomy representation. If $\rho$ is Fuchsian, then this is the hyperbolic length of the geodesic representing the conjugacy class of $\gamma$, so we think of it as a notion of geodesic length for closed curves in convex projective manifolds. We will call $\log(\lambda_1(\rho(\gamma)))$ the asymmetric length, because it is a function of oriented loops, and to distinguish it from the more commonly used Hilbert length: $\log(\lambda_1(\rho(\gamma))/\lambda_3(\rho(\gamma)))$. Hilbert length is the symmetrization of the asymmetric length. **Theorem 1**. *Let $\mu_i$ be a sequence of cubic differentials on a smooth oriented surface $S$ of genus at least $2$, each holomorphic with respect to a complex structure $J_i$, such that $a_i^3\mu_i$ converges uniformally to $\mu$, for some sequence of positive real numbers $a_i$ tending to $0$. (It follows that $J_i$ converge to $J$.) Let $\gamma\in \pi_1(S)$. Let $F^{\Delta}_\mu(\gamma)$ denote the infimal length of loops representing $\gamma$ in the Finsler metric $F_\mu^{\Delta}$. $$\lim_{i \to \infty} a_i \log(\lambda_1(\rho(\gamma))) = F^{\Delta}_\mu(\gamma)$$* Part of the appeal of theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} is that the left hand side is complicated, necessitating both solution of a PDE and an ODE to compute directly from $\mu_i$, while the right hand side only involves integrals of cube roots of $\mu$. The cubic differential $\mu$ is equivalent to a $1/3$-translation structure on $S$ in which $F^\Delta_\mu$ geodesics can always be straightened to be concatenations of straight line segments which have angle at least $\pi$ on either side at zeros. To compute $F^{\Delta}_\mu(\gamma)$, it suffices to find such a geodesic representative and add up the $F^\Delta_\mu$ lengths of the constituent segments. Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} was proved in [@loftin_limits_2022] for the case of cubic differential rays $\mu_i = \mu/a_i^3$. We are not sure if one can directly deduce Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} from [@loftin_limits_2022], but in any case, theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} is a useful improvement. Our main new result is a more local version of this theorem. Danciger and Stecker have recently discovered an asymmetric Finsler metric $F^{DS}$, defined on any closed convex projective manifold, whose length function is the asymmetric length function. $$F^{DS}(\gamma) = \log(\lambda_1(\rho(\gamma)))$$ We call it the domain shape metric because its unit balls are all projectively equivalent to the developing image of the convex projective structure. Our main theorem says that when $\mu$ is large, the domain shape metric for the projective structure on $S$ specified by $(J,\mu)$, looks roughly like a much simpler Finsler metric $F^\Delta_\mu$. **Theorem 2**. *If $S, \mu_i$, $a_i$ and $\mu$ are as in theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"}, then $a_i F^{DS}_{\mu_i}$ converges uniformally to $F^{\Delta}_{\mu}$ on any compact set in the complement of the zeros of $\mu$.* Pointwise convergence would follow from [@nie_limit_2022] which describes the Gromov-Hausdorff limit of a sequence of pointed convex domains coming from a sequence of pointed Riemann surfaces with cubic differentials tending to infinity, but we will need uniformity to deduce theorem $\ref{evlimit}$ from theorem $\ref{metriclimit}$. To get uniform convergence away from zeros, we will retrace the steps of [@nie_limit_2022], with a slightly different setup, and make sure things work uniformly. In section 5 we show that uniform convergence away from zeros is sufficient for deducing convergence of length functions. ## What theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} says about compactification {#what-theorem-evlimit-says-about-compactification} Let $\mathcal{L}(S)$ denote the set of homotopy classes of closed loops in $S$. An attractive way to define a compactification of $\operatorname{Conv}(S)$, which goes by various names, including Morgan-Shalen compactification, tropical compactification, and spectral radius compactification, is to embed it into $\mathbb{P}(\mathbb{R}^{\mathcal{L}(S)})$ via taking projectivized marked asymmetric length spectrum, then take the closure. The map $\operatorname{Conv}(S)\to \mathbb{P}(\mathbb{R}^{\mathcal{L}(S)})$ is an embedding, but the topology of the closure is still unknown. Part of the difficulty is that we still don't have a good understanding of what geometric structures on $S$ the tropical boundary points might be parametrizing. Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} takes a step towards answering this last question. The Labourie-Loftin parametrization identifies $\operatorname{Conv}(S)$ with a vector bundle over Teichmuller space $Q(S)$ which we can compactify in the fiber directions by adding in a point for each $\mathbb{R}_+$ orbit. Let $\bar{Q}(S)$ denote this radial partial compactification. Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} implies that the projectivized marked length spectrum map $\operatorname{Conv}(S)\to \mathbb{P}(\mathbb{R}^{\mathcal{L}(S)})$ extends continuously to $\bar{Q}(S)$, and the boundary map is given by the projectivized length spectrum of $F^\Delta_{\mu}$. {width="8cm"} Let $\Delta Fins(S)$ denote the set of triangular Finsler metrics $\{F^\Delta_\mu:\mu\in Q(S)\backslash\underline{0}\}$. We conjecture that $\Delta Fins(S)/\mathbb{R}_+$, injects into $\mathbb{P}(\mathbb{R}^{\mathcal{L}(S)})$ and comprises an open dense subset of the boundary of $\operatorname{Conv}(S)$. ## Relation to other work, and future directions This work fits into a few different bigger stories. Firstly, Gaiotto Moore and Neitzke [@gaiotto_wall-crossing_2013] have a conjecture for the asymptotics of $tr(\rho(\gamma))$ along rays of Higgs bundles $\{(E,R\phi):R\in \mathbb{R}_+\}$ under the nonabelian hodge correspondence. Our paper, and [@loftin_limits_2022], can be seen as proving the leading order part of the conjecture for the subspace of the $\operatorname{SL}_3$ Hitchin section where the quadratic differential is zero. The methods of [@loftin_limits_2022] seem like they might work in higher rank, while our methods seem very specific to $\operatorname{SL}_3\mathbb{R}$. On the other hand, our method is less similar to ideas of [@gaiotto_wall-crossing_2013] in that Stokes lines don't make an appearence, so it might bring a new perspective to the conjectures. Our Theorem [Theorem 2](#metriclimit){reference-type="ref" reference="metriclimit"} is similar in flavor to work of Oyang-Tambourelli [@ouyang_limits_2021] where it is shown that when $\mu$ is big, the Blaschke metric is close to the singular flat Riemannian metric defined by $\mu$. They are also able to fully understand the closure of the space of projectivised length spectra of Blaschke metrics as mixed structures on $S$, which consist of a singular flat metric on part of the surface, and a measured lamination on the rest. We hope that the tropical compactification of $\operatorname{Conv}(S)$ has a similar description, but with singular flat triangular Finsler metrics instead of singular flat Riemannian metrics. In [@ouyang_limits_2021], it is shown that singular flat metrics comprise an open dense subset of mixed structures, motivating our analogous conjecture about singular, flat, triangular Finsler metrics. In the case of punctured surfaces, Fock and Goncharov [@fock_moduli_2006] have explained how to compactify Hitchin components by the projectivization of the tropical points of the character variety. In the $\operatorname{SL}_2$ case, these tropical points parametrize measured laminations, and the integral tropical points correspond to integral laminations. This has led many people to investigate what kind of objects the Fock-Goncharov tropical points are parametrizing in higher rank. In [@parreau_invariant_2015], Parreau shows that a certain cone of FG tropical points is parametrizing strucutres on the surface which are part $1/3$ translation structure, and part tree. She shows that asymptotics of Jordan projections of group elements are encoded in a Weyl cone valued metric. For the $1/3$ translation structure part, this is exactly equivalent to the way we encode asymptotics of top eigenvalues with the Finsler metric $F_\mu$. The present paper can be seen as accomplishing, for closed surfaces, something quite analogous to Parreau's work on punctured surfaces. In neither case are all tropical points covered, but the the limitations are not quite the same. Parreau has to restrict to a certain cone of tropical points which depends on the chosen triangulation, but this cone includes some cases which exhibit tree behavior. We, on the other hand, study exactly the cases which do not exhibit tree behavior. Douglous and Sun [@douglas_tropical_2021] have developed a different perspective on the Fock-Goncharov integral $\operatorname{SL}_3$ tropical points, showing that they parametrize certain bipartite trivalent graphs introduced by Sikora-Westbury [@sikora_confluence_2007] called non-elliptic webs. Webs and $1/3$ translation structures are related by a simple geometric construction (which has been contemplated by J. Farre. myself, and possibly other people.) Let $S$ be a closed surface. Suppose we have a filling non-elliptic web: a trivalent bipartite graph $W$ embedded in $S$ whose complementary regions are all disks with at least six sides. We can replace each black vertex of $W$ with the equilateral triangle $\operatorname{Conv}(0,1,1/2+i\sqrt{3}/2)\subset \mathbb{C}$ with cubic differential $dz^3$, and each white vertex with $\operatorname{Conv}(0,1,1/2-i\sqrt{3}/2)\subset \mathbb{C}$, then glue these triangles according to the edges of the web. This construction produces a surface with cubic differential, which is identified with $S$ up to isotopy. In the other direction, a cubic differential on $S$ with integral periods is the same as a singular $1/3$ translation structure with holonomy valued in $\mathbb{Z}/3 \ltimes \mathbb{Z}^2\subset \mathbb{Z}/3 \ltimes \mathbb{R}^2$. One can take the preimage of a standard hexagonal web on $\mathbb{R}^2$ via the developing map to get a filling non-elliptic web on $S$. Below is an example of a web on a genus 2 surface specifying a $1/3$ translation structure glued from $16$ equilateral triangles. {width="8cm"} Using this correspondence, we get a map from filling non-elliptic webs to integral tropical points of $Rep(\pi_1S, \operatorname{SL}_3\mathbb{R})$. It would great to extend the results of this paper to punctured surfaces so that they could be compared with work of Parreau, Douglas-Sun, and Fock-Goncharov. A final, much more wide open, future direction is to use ideas of this paper to understand degenerations of higher dimensional convex projective manifolds. As we will see in the next section, the domain shape metric is defined in any dimension. It is conceivable that one could make sense of limits of domain shape metrics in higher dimensions, and relate these to tropical points of the relevant character varieties. There are various instances throughout geometry where studying more degenerate, combinatorial versions of a class of objects is the key to understanding many important things. Maybe convex projective geometry will be another example of this trend. ## Structure of the paper In Section 2 we define domain shape metrics and prove their important properties. In section 3 we review the Labourie-Loftin correspondence, mostly to recall formulas and set notation. In section 4, the technical heart of the paper, we prove theorem [Theorem 2](#metriclimit){reference-type="ref" reference="metriclimit"}. In section 5 we prove that length functions of Finsler metrics are sufficiently continuous to deduce theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} from theorem [Theorem 2](#metriclimit){reference-type="ref" reference="metriclimit"}. In section 6 we explain how to apply theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} to triangle reflection groups, and present some numerical computations. Some readers may prefer to look at section 6 first. ## Acknowledgements This paper would not exist without many discussions with J. Danciger and F. Stecker. In particular, they told me about domain shape metrics, and F. Stecker did various computer experiments giving evidence for Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"}. This research was supported in part by NSF grant DMS-1937215, and NSF grant DMS-1945493. # Domain shape Finsler metrics In this section, we construct a Finsler metric $F^{DS}$ on any convex projective manifold (of any dimension) whose geodesics are projective lines, and whose length spectrum is the assymmetric length spectrum. The construction depends on a lift of the domain in $\mathbb{RP}^n$ to a convex hypersurface in $\mathbb{R}^{n+1}$. After this section we will specialize to the case where we use the affine sphere as our choice of lift. This metric, and its main properties, were shown to me by Danciger and Stecker, but have not yet appeared in the literature. **Definition 2**. *Let $v$ be a tangent vector to a point $x$ in a properly convex domain $\Omega\subset \mathbb{R}\mathbb{P}^n$. Let $\beta$ be a linear functional defining a supporting hyperplane at the point where the ray starting at $x$ and tangent to $-v$ intersects $\partial\Omega$. Let $S\subset \mathbb{R}^{n+1}$ be a convex, differentiable lift, which is asymptotic to the cone over $\Omega$ in the sense that that the line going through two points in $S$ is never in $\bar{\Omega}$. Let $\tilde{x}$ and $\tilde{v}$ be the lifts of $x$ and $v$ to $S$. $$F^{DS}(x,v):= \frac{\beta(\tilde{v})}{\beta(\tilde{x})}$$ We call $F^{DS}$ the domain shape metric of $\Omega$ for the hypersurface $S$.* The name is justified by the fact that each unit ball of $F^{DS}$ is projectively equivalent to $\Omega$. More specifically, the unit ball at $x\in \Omega$ is the antipodal image of the intersection of the tangent space of $S$ at $\tilde{x}$ with the cone over $\Omega$. {width="14cm"} **Remark 3**. *Domain shape metrics are a simple generalization of Funk metrics [@funk_uber_1930], which have been studied since Funk introduced them in 1929, because their geodesics are straight lines. An interpretation of Hilbert's 4th problem is to classify Finsler metrics on euclidean space whose geodesics are straight lines, so there has been interest in such metics for a long time.* It turns out that $F^{DS}$ integrates to the asymmetric path metric $d^{DS}$ which we now define. **Definition 4**. *Let $\Omega$ be a convex domain in $\mathbb{R}\mathbb{P}^n$, and let $S$ be a lift of $\Omega$ to $\mathbb{R}^{n+1}$ which is convex, such the line going through two points of $S$ is never in $\bar{\Omega}$. We define a metric on $\Omega$ as follows. Let $x,y\in \Omega$ be distinct points, and let $\tilde{x},\tilde{y}\in S$ be their lifts. Let $p$ be the point where the projective line starting at $y$ and passing through $x$ first hits the boundary of $\Omega$. Let $[\beta]$ be a supporting hyperplane to $\Omega$ at $p$, defined by a linear functional $\beta$. $$d^{DS}(x,y) := \log\frac{\beta(\tilde{y})}{\beta(\tilde{x})}$$* {width="8cm"} **Remark 5**. *In the case when $\Omega$ has a sharp corner at $p$, there are multiple choices of $[\beta]$, but they give the same value.* **Remark 6**. *Nicolas Tholozan pointed out to me an infinite dimensional case where $d^{DS}$ has been studied. Let $C$ be the cone of parametrizations of the geodisic foliation of the unit tangent bundle of a hyperbolic surface (up to weak conjugacy) and let $S$ be the hypersurface of positive, entropy 1 parametrizations. One can restrict the resulting domain shape metric on $\mathbb{P}(C)$ to spaces of Anosov representations, and get asymmetric Finsler metrics studied in [@tholozan_teichmuller_nodate] and [@carvajales_thurstons_2022].* Before we relate $d^{DS}$ to $F^{DS}$, we check that $d^{DS}$ satisfies the two axioms of an assymmetric metric: reflexivity, and the triangle inequality. Reflexivity is left to the reader. The triangle inequality will follow from the following lemma. **Lemma 7**. *Let $x,y,\Omega, S, \beta$, and $p$ be as above. If $[\beta']$ is any hyperplane which doesn't intersect $\Omega$, and doesn't pass through $p$, then: $$\frac{\beta'(\tilde{y})}{\beta'(\tilde{x})} < \frac{\beta(\tilde{y})}{\beta(\tilde{x})}$$* *Proof.* Let $p'$ be the point where the line through $x$ and $y$ hits the hyperplane $[\beta']$. We can find an affine hyperplane $A$ going through $\tilde{x}$ and $\tilde{y}$ whose intersection with the cone over $\Omega$ is as follows. By assumption, $[\tilde{x}-\tilde{y}]$ cannot be in $\bar{\Omega}$. This means that the dual hyperplane $[\tilde{x}-\tilde{y}]^*$ intersects the dual domain $\Omega^*$. Let $\alpha$ be a linear functional representing a point in the intersection. By construction, $\alpha(\tilde{x})=\alpha(\tilde{y})$, and we can scale $\alpha$ so that $\alpha(\tilde{x})=\alpha(\tilde{y})=1$. Let $A$ be the hyperplane defined by $\alpha=1$. This affine hyperplane $A$ is identified with an affine chart of projective space containing $\Omega$. Arbitrarily choose a euclidean metric $d_A$ on $A$ compatible with the affine structure. Since $\beta|_A$ and $\beta'|_A$ are affine linear functions vanishing $[\beta]$ and $[\beta']$, they are proportional to the affine linear functions which simply measure signed euclidean distance to $[\beta]$ and $[\beta']$. Our desired inequality is thus equated with an inequality involving euclidean distances which is visually clear. $$\frac{\beta'(\tilde{y})}{\beta'(\tilde{x})} = \frac{d_A(y,p')}{d_A(x,p')} < \frac{d_A(y,p)}{d_A(x,p)}=\frac{\beta(\tilde{y})}{\beta(\tilde{x})}$$ ◻ **Lemma 8**. 1. *$d^{DS}$ satisfies the triangle inequality, $d^{DS}(x,y) + d^{DS}(y,z) \geq d^{DS}(x,z)$.* 2. *If $x$, $y$, $z$ are collinear, then we have equality.* 3. *If the domain is strictly convex, and $x$, $y$, $z$ are not collinear, then the inequality is strict.* {width="8cm"} *Proof.* Let $\beta_1,\beta_2$, and $\beta_3$ be linear functionals defining hyperplanes tangent to the points on $\partial\Omega$ intersecting the rays $\overrightarrow{yx}$, $\overrightarrow{zy}$, and $\overrightarrow{zx}$ respectively. $$\frac{\beta_3(z)}{\beta_3(x)} = \frac{\beta_3(z)}{\beta_3(y)} \frac{\beta_3(y)}{\beta_3(x)} \leq \frac{\beta_2(z)}{\beta_2(y)} \frac{\beta_1(y)}{\beta_1(x)}$$ Taking logarithms gives the triangle inequality. If $x$, $y$, and $z$ are colinear, then we can choose $\beta_1=\beta_2=\beta_3$, so we have equality. If $x$, $y$, and $z$ are not colinear, and $\Omega$ is strictly convex, then $\beta_1$, $\beta_2$, and $\beta_3$ are supporting hyperplanes at unique, and distinct points, so the previous lemma gives us strict inequality. ◻ Part $2$ of the preceding lemma implies that projective lines are geodesics of $d^{DS}$, and that $d^{DS}$ is a path metric. Part $3$ implies that if $\Omega$ is strictly convex, projective lines are the only geodesics. Finally, we check that $d^{DS}$ and $F^{DS}$ agree. **Lemma 9**. *$d^{DS}$ differentiates to $F^{DS}$.* *Proof.* Let $v$ be a tangent vector at $x\in \Omega$. Let $\gamma:(-\epsilon,\epsilon)\to \Omega$ be a curve with $\gamma'(0)=v$ which we choose for convenience to be a projective line segment. Let $\beta$ be a linear functional as in the definition of $F^{DS}(x,v)$. $$\frac{d}{dt}d^{DS}(\tilde\gamma(0),\tilde\gamma(t))|_{t=0} = \frac{d}{dt}\log(\frac{\beta(\tilde\gamma(t))}{\beta(\tilde\gamma(0))})|_{t=0} = \frac{\beta(\tilde\gamma'(0))}{\beta(\tilde\gamma(0))}=F^{DS}(x,v)$$ ◻ ## Domain shape metrics for dual hypersurfaces In this section we show that, if we choose compatible lifts, domain shape metrics for projectively dual domains are pointwise dual, with respect to a natural Riemannian metric. In the case when the lift is an affine sphere, this metric is known as the Blaschke metric. This will be used in section [4.6](#lower bounds){reference-type="ref" reference="lower bounds"} to turn upper bounds for $F^{DS}$ into lower bounds. Let $V$ be a real vector space of dimension $n+1$. Let $\Omega\subset \mathbb{P}(V)$ be a properly convex domain. Let $S\subset V$ be a proper convex lift. Let $\Omega^* \subset \mathbb{P}(V^*)$ be the dual convex domain, and let $S^*\subset V^*$ be the dual convex lift: the set of linear functionals $\alpha$ such that $\inf \alpha(S)=1$. We say that $x\in S$ and $\alpha\in S^*$ are dual points if $\alpha(x) = 1$. If $S$ and $S^*$ are strictly convex, then each point in $S$ has exactly one dual point in $S^*$. Assume that $S$ is smooth, with everywhere positive hessian, so that the duality mapping between $S$ and $S^*$ is a diffeomorphism. In this setting, a natural Riemannian metric appears. **Lemma 10**. *Let $x:\mathbb{R}^n\to S$ and $\alpha:\mathbb{R}^n\to S^*$ be smooth, dual parametrizations. The matrix $$g_{ij} = -\langle \partial_i \alpha,\partial_j x\rangle = \langle \alpha, \partial_i \partial_j x\rangle = \langle \partial_i \partial_j \alpha, x\rangle$$ is a Riemannian metric which doesn't depend on the parametrization.* *Proof.* The first expression $-\langle \partial_i \alpha,\partial_j x\rangle$ is independant of parametrization because we can phrase it in parametrization independant language: if $\beta\in T_\alpha S^*$ and $v\in T_x S$, these both give tangent vectors to $S$, (or $S^*$,) and we define $g(\beta,v)=-\langle \beta, v\rangle$. The second expression $\langle \alpha, \partial_i \partial_j x\rangle$ is clearly symmetric and positive definite because partial derivatives commute, and $S$ is convex with positive Hessian. It remains to show that these expressions are indeed equal. That $x$ and $\alpha$ are dual means that $\langle \alpha, \partial_i x\rangle = \langle \partial_i\alpha, x\rangle = 0$ for all $i$. Applying $\partial_j$ to these equations gives the result. ◻ The Blaschke metric is $\langle \partial_i \partial_j \alpha, v\rangle$ where $v$ is a certain canonically defined normal vector called the 'affine normal'. An affine sphere (centered at $0$) is precisely a hypersurface satisfying $v=x$, so for affine spheres $g$ is the Blaschke metric. **Definition 11**. *Let $\Omega$ be a convex subset of a vector space $V$ containing the origin, and let $g$ be a metric on $V$. The dual of $\Omega$ with respect to $g$ is $\Omega^{*_g}:=\{v \in V : g(v,\Omega) < 1\}\subset V$* If $M$ is a manifold with Riemannian metric $g$, and Finsler metric $F$, let $F^{*_g}$ denote the Finsler metric whose unit balls are pointwise dual to unit balls of $F$ with respect to $g$. **Lemma 12**. *If $S$ is a smooth, properly embedded lift of a properly convex domain $\Omega$ with positive definite Hessian, and we identify $S$ with $S^*$ by the duality diffeomorphism, then $F^{DS}_{S^*} = (F^{DS}_S)^{*_g}$, where $g$ is the metric defined above.* *Proof.* Let $C$ denote the cone $\mathbb{R}_+ S$. Let $B$ denote the unit ball of $F^{DS}_S$ at $x\in S$. $$B = \{v \in T_x S : x-v\in C\}$$ The unit ball of $F^{DS}_{S^*}$ at $\alpha$ is $$B^* = \{\beta \in T_\alpha S^* : \langle \alpha - \beta, C \rangle > 0 \}$$ $C$ is the $\mathbb{R}_+$ span of $x-B$, so equivalently $$B^* = \{\beta \in T_\alpha S^* : \langle \alpha - \beta, x-v \rangle > 0 \;\;\;\forall v\in B\}$$ Recall $\langle \alpha,x\rangle=1$, $\alpha$ vanishes on $T_x S$ and $x$ vanishes on $T_\alpha S^*$, so we have $\langle \alpha - \beta, x-v \rangle = 1 + \langle \beta, v\rangle$. Finally we have $$B^* = \{\beta \in T_\alpha S^* : -\langle \beta, v \rangle < 1 \;\;\;\forall v\in B\}$$ If we identify $T_\alpha S^*$ and $T_x S$ by the differential of the duality map, then $-\langle \beta, v \rangle$ is $g$ evaluated on $\beta$ and $v$, so the unit balls of $F^{DS}_S$ and $F^{DS}_{S^*}$ are dual with respect to the $g$. ◻ # Review of the Labourie-Loftin correspondence The Labourie-Loftin correspondence [@loftin_affine_2001], [@labourie_flat_2006] provides a bijection between pairs $(J,\mu)$, where $J$ is a complex structure, and $\mu$ is a holomorphic cubic differential, and convex projective structures, on a surface of genus at least $1$. After quotienting both the space of pairs $(J,\mu)$, and the space of convex projective structures by $\operatorname{Diff}_0(S)$, the Labourie-Loftin correspondence becomes a diffeomorphism from the bundle of cubic differentials over Teichmuller space, to the Hitchin component. Here, we review how one gets a convex projective structure on $S$ from a pair $(J,\mu)$. Let $S$ be a Riemann surface with cubic differential $\mu$. It turns out that there is a unique hermitian metric $g$ on $S$ satisfying Wang's equation. $$\kappa_g = |\mu|^2_g - 1$$ From the data $(S,J,\mu,g)$ we will construct a convex projective structure. This means a developing-holonomy pair: a representation $\pi_1(S)\to \operatorname{SL}_3\mathbb{R}$, and an equivariant embedding $\tilde{S}\to \mathbb{P}(\mathbb{R}^3)$ whose image is a convex set. We will actually construct an equivariant map $\tilde{S}\to \mathbb{R}^3$, whose image is a strictly convex hypersurface, (in fact an affine sphere) which we can compose with the projection to $\mathbb{P}(\mathbb{R}^3)$ to get a developing map. The construction of the developing holonomy pair necessitates a choice of base point, and it is useful to pay attention to this choice. For each point $x\in S$, we will construct an affine sphere in the three dimensional vector space $T_x S\oplus \underline{\mathbb{R}}$ with $\pi_1(x,S)$ action, but different choices of $x$ will give isomorphic results. There is an explicit formula for a real flat connection $\nabla$ on $TS\oplus \underline{\mathbb{R}}$ in terms of $\mu$ and $g$. We write this formula in terms of the complexification which has a natural line decomposition $TS \oplus \overline{TS} \oplus \underline{\mathbb{C}}$: $$\nabla = \begin{bmatrix} D_{TS}^g & g^{-1}\bar{\mu} & 1 \\ g^{-1}\mu & D_{\overline{TS}}^g & 1 \\ g & g & 0 \\ \end{bmatrix}$$ Here, $D_{TS}^g$, and $D_{\overline{TS}}^g$ are the Chern connections, which both coincide with the Levi-Civita connection for $g$. The off-diagonal entries are maps of various line bundles. One checks that $\nabla$ is real, and that its flatness is equivalent to Wang's equation. We sketch this second computation here. **Lemma 13**. *$\nabla$ is flat if and only if $\kappa_g = |\mu|^2_g - 1$.* *Proof.* This statement can be checked locally. Choose a local holomorphic coordinate $z$ on $S$. Write $g=e^\phi dzd\bar{z}$, and $\mu=\mu_0 dz^3$. We get a frame $\partial_z,\partial_{\bar{z}},\underline{1}$ of $(TS\oplus\mathbb{R})\otimes \mathbb{C}$. In this frame, we can write $\nabla$ as the de Rahm differential plus a matrix valued $1$-form. $$\nabla = d + A_1 dz + A_2 d\bar{z}$$ $$\label{connection formula} A_1=\begin{bmatrix} \partial\phi & 0 & 1 \\ e^{-\phi}\mu_0 & 0 & 0 \\ 0 & e^\phi & 0 \\ \end{bmatrix}dz\;\;\;\;\;\;\;\;\;A_2 = \begin{bmatrix} 0 & e^{-\phi}\bar{\mu}_0 & 0 \\ 0 & \bar{\partial}\phi & 1 \\ e^\phi & 0 & 0 \\ \end{bmatrix}d\bar{z}$$ The curvature of $\nabla$ is $$F_\nabla= (-\partial_{\bar{z}} A_1 + \partial_z A_2 + \frac12[A_1,A_2])dz\wedge d\bar{z}$$ This comes out to be $$F_\nabla = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}(\partial_z\partial_{\bar{z}}\phi + \frac12(e^{-2\phi}|\mu_0|^2 - e^\phi))dz\wedge d\bar{z}$$ Recall that $\partial_z\partial_{\bar{z}}=\frac14\Delta$ where $\Delta=\partial_x^2+\partial_y^2$ is the laplacian. Vanishing of $F_\nabla$ becomes a PDE for $\phi$: $$\frac12\Delta\phi = -e^{-2\phi}|\mu_0|^2 + e^\phi$$ Recalling the formula $\kappa_g = -\frac12 e^{-\phi}\Delta \phi$ for the gauss curvature, we get the coordinate independant form of the equation. $$\kappa_g = |\mu|_g-1$$ ◻ The formula for $\nabla$ may seem a bit mysterious, so we mention two perspectives from which one could derive it. First, for a general strictly convex surface $S$ in three dimensional affine space $\mathbb{R}^3$ (endowed with a translation invariant volume form), there is a canonically defined normal vector field called the affine normal, and a metric called the Blaschke metric. The affine normal lets us identify $T\mathbb{R}^3|_S$ with $TS\oplus \mathbb{R}$, and the complex structure induced by the Blaschke metric gives a line decomposition $TS\otimes \mathbb{C}= TS\oplus \overline{TS}$. If we write the trivial connection on $T\mathbb{R}^3|_S$ with respect to this line decomposition, it takes a form quite similar to ([\[connection formula\]](#connection formula){reference-type="ref" reference="connection formula"}), but with non-trivial tensors in the last column. The fact that the connection takes the form ([\[connection formula\]](#connection formula){reference-type="ref" reference="connection formula"}) is equivalent to $S$ being a hyperbolic affine sphere. We refer to [@loftin_survey_2008] for a survey on affine spheres. From a completely different point of view, Wang's equation is really a special case of Hitchin's equation: if we use $g$ to identify $\overline{TS}$ with $TS^{*}$, we get a harmonic bundle solving the Hitchin equation for the rank 3 Higgs bundle in the Hitchin section corresponding to the cubic differential $\mu$, and the zero quadratic differential. $TS\oplus \underline{\mathbb{R}}$ has a natural section, namely $\underline{1}\in \Gamma(\underline{\mathbb{R}})$. Let $\tilde{S}_x$ denote the universal cover of $S$ based at $x\in S$, constructed explicitly as the collection of pairs $(y,[\gamma])$, where $y\in S$ and $[\gamma]$ is a homotopy class of path from $x$ to $y$. For each point $x\in S$, let $\xi_x:\tilde{S}_x\to T_xS\oplus \mathbb{R}$ denote the map which takes a point $(y,[\gamma])$ to the parallel transport of $\underline{1}(y)$ back to $x$, along $\gamma$ using the connection $\nabla$. We call $\xi_x$ the affine sphere developing map based at $x$. One can deduce, from the form of $\nabla$, that the image of $\xi_x$ is indeed an affine sphere: the affine normal at $x$ is a fixed scalar multiple of $\xi_x$. {width="12cm"} The fundamental group $\pi_1(S,x)$ acts on $\tilde{S}_x$ via deck transformations, and on $T_x S\oplus \mathbb{R}$ via the holonomy of $\nabla$. The map $\xi_x$ is equivariant for these two actions by construction. If we choose a different point $x'\in S$, then parallel transport along any path from $x$ to $x'$ will be a volume preserving linear map $T_x S\oplus \mathbb{R}\to T_{x'} S\oplus \mathbb{R}$ identifying the affine spheres $\operatorname{Im}(\xi_{x})$ and $\operatorname{Im}(\xi_{x'})$, which intertwines the $\pi_1(x, S)$ and $\pi_1(x', S)$ actions. # $F^{DS}$ is close to $F^\Delta$ far from zeros To make precise what we mean by "far from zeros\" we need a metric. A Riemann surface $S$ with cubic differential $\mu$, has a singular, flat Riemannian metric $h$ defined by the equation $|\mu|_{h} = 1$. We will call this singular flat metric $h_\mu$. **Theorem 3**. *There exists a function $\epsilon:\mathbb{R}_+ \to (0, \infty]$ with $\lim_{r\to\infty}\epsilon(r) = 0$ such that for any closed Riemann surface $S$ with cubic differential $\mu$, $$|\log\frac{F^{DS}_\mu(x,v)}{F^\Delta_\mu(x,v)}| < \epsilon(r(x))$$ for any non-zero tangent vector $v$ at any point $x$, where $r(x)$ denotes the distance from $x$ to the closest zero of $\mu$, with respect to the metric $h$.* This theorem is simply a more uniform version of the part of Nie's result [@nie_limit_2022] dealing with triangles, and we will prove it following his method. ## Proof of theorem [Theorem 2](#metriclimit){reference-type="ref" reference="metriclimit"} assuming theorem [Theorem 3](#metricestimate){reference-type="ref" reference="metricestimate"} {#proof-of-theorem-metriclimit-assuming-theorem-metricestimate} We will now prove Theorem [Theorem 2](#metriclimit){reference-type="ref" reference="metriclimit"} from the introduction as an easy consequence of Theorem [Theorem 3](#metricestimate){reference-type="ref" reference="metricestimate"}. *Proof.* Fix $R>0$. Let $S_R \subset S$ be the set of points which are at least distance $R$ away from all zeros of $\mu$, in the metric $h_\mu$. It will suffice to show that $F^{DS}_{\mu_i}/a^i$ converges uniformly to $F^\Delta_\mu$ on $S_R$. Uniform convergence of $a_i^3\mu_i$ to $\mu$ implies that for all $\epsilon>0$, there exists $N$ such that zeros of $\mu_i$ for $i>N$ are all in an $\epsilon$ neighborhood of the zeros of $\mu$ with respect to $h_\mu$. Consequently, the limit of the $h_{\mu}$ distance between the zeros of $\mu_i$ and $S_R$ is $R$. The $h_{\mu_i}$ distance between zeros of $\mu_i$ and $S_R$ must then go to infinity, because $a_i h_{\mu_i}$ converges uniformly to $h_\mu$. This means that the ratio between $F^{DS}_{\mu_i}$ and $F^\Delta_{\mu_i}$ limits to $1$ uniformly on $S_R$. The ratio of $a_i F^\Delta_{\mu_i}$ to $F^\Delta_{\mu}$ also goes to $1$ uniformally on $S_R$. It follows that the ratio of $a^iF^{DS}_{\mu_i}$ to $F^\Delta_\mu$ goes uniformally to $1$ on $S_R$. ◻ ## Blaschke metric estimate Note that the singular flat metric $h_\mu$, defined by $|\mu|_{h_\mu}=1$, is a solution to Wang's equation on the complement of the zeros of $\mu$. The first step in proving theorem [Theorem 3](#metricestimate){reference-type="ref" reference="metricestimate"} is to show that, far from zeros, the global solution to Wang's equation is close, in $C^1$ norm, to this singular flat solution. **Lemma 14**. *There exists a function $\epsilon_{C^1}:\mathbb{R}_+ \to (0,\infty]$ limiting to zero as the input goes to infinity, such that if $S$ is a Riemann surface with holomorphic cubic differential $\mu$ such that $h_\mu$ is complete, $g$ is the complete solution to Wang's equation, and $g=e^\phi h_\mu$, then $$|\phi(x)| + |d\phi_x|_{h_\mu} < \epsilon_{C^1}(r(x))$$ for all $x\in S$, where $r(x)$ denotes the $h_\mu$ distance from $x$ to the closest zero of $\mu$.* Wang's equation implies that $\phi$ satisfies the following PDE. $$\frac12\Delta_{h_\mu}\phi = e^\phi - e^{-2\phi}$$ The intuition is that this equation very much wants to force $\phi$ close to zero, and ellipticity can promote $C^0$ bounds to $C^k$ bounds for whatever $k$ we want. For us $C^1$ will be sufficient. *Proof.* In [@nie_poles_2023] it is shown that there are uniform constants $C,r_0$ such that $\phi(x)$ is bounded between $0$ and $\epsilon_{C^0}(r(x))$, where: $$\epsilon_{C^0}(r)=\begin{cases} C\sqrt{r}e^{-\sqrt {6}r}& \text{if } r\geq r_0\\ \infty & \text{otherwise} \end{cases}$$ This means in particular that on the disk $B_{r(x)/2}(x)$, $\phi$ is bounded between $0$ and $\epsilon_{C^0}(r(x)/2)$. We now use the following interior gradient estimate. **Lemma 15**. *If $u$ is a twice differentiable function on the closed disk $\overline{B}_x(R)\subset \mathbb{R}^2$, then we have the following estimate for its gradient at $x$. $$|\nabla u (x)|\leq R|\Delta u|_{C^0(B_x(R))} + \frac{2}{\pi R}|u|_{C^0(\partial B_x(R))}$$* This is proved by writing $u$ as a sum of a function which vanishes on the boundary of the disk and has the same laplacian as $u$, and a harmonic function which has the same boundary value as $u$. The gradient of the former can be estimated using its representation in terms of the Green's function for the disk, and the gradient of the latter can be bounded using the maximum principal, and mean value property for harmonic functions. Much more general estimates of this flavor are proved in PDE texts such as [@gilbarg_elliptic_2001]. Apply this to $\phi$ on $B_{x}(r(x)/2)$, recalling that $\Delta \phi = F(\phi)$ where $F(y)=2(e^y-e^{-2y})$. $$|\nabla\phi(x)| \leq \frac{r(x)}{2}F(\epsilon_{C^0}(r(x)/2)) + \frac{2n}{\pi r(x)} \epsilon_{C^0}(r(x)/2)$$ Note that this gradient estimate goes to zero as $r(x)$ goes to infinity. We simply add the pointwise estimate and the gradient estimate to get the desired $C^1$ estimate. $$\epsilon_{C^1}(r) := \epsilon_{C^0}(r) + \frac{r}{2}F(\epsilon_{C^0}(r/2)) + \frac{2n}{\pi r} \epsilon_{C^0}(r/2)$$ ◻ ## $F^{DS}$ and $F^\Delta$ coincide for Ţiţeica surfaces In this section, we review the fact that the affine sphere corresponding to the constant cubic differential $dz^3$ on the complex plane $\mathbb{C}$ is a Ţiţeica surface, and we show that the DS metric is exactly the $\Delta$ metric in this case. **Lemma 16**. *$F^{DS}_{dz^3}=F^\Delta_{dz^3}$* *Proof.* The standard hermitian metric $h=dzd\bar{z}$ is the complete solution to Wang's equation. By specializing formula ([\[connection formula\]](#connection formula){reference-type="ref" reference="connection formula"}) to the case $\phi=0$, we get the flat connection $$\label{nabla0 formula} \nabla_0 = d +\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix}dz +\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{bmatrix}d\bar{z}$$ on $T\mathbb{C}\oplus \underline{\mathbb{R}}$, expressed in terms of the frame $(\partial_z, \partial_{\bar{z}}, \underline{1})$ of the complexification $(T\mathbb{C}\oplus \mathbb{R})\otimes \mathbb{C}$. One directly checks that for each 3rd root of unity $\zeta$, $$s_\zeta=\begin{bmatrix} \zeta e^{-2Re(\zeta z)} \\ \bar{\zeta} e^{-2Re(\zeta z)} \\ e^{-2Re(\zeta z)} \\ \end{bmatrix}$$ is a flat section. We then express $\underline{1}$ in this flat frame. $$\label{titeica coordinates} \underline{1} = \frac{1}{3}\sum_{\zeta \in \sqrt[3]{1}} e^{2 Re(\zeta z)}s_\zeta$$ Identifying all fibers of $T\mathbb{C}\oplus \mathbb{R}$ via $\nabla_0$, the flat frame $\{s_\zeta\}$ collapses to the basis $\{(\zeta,1):\zeta\in \sqrt[3]{1}\}\subset \mathbb{C}\oplus \mathbb{R}$, and $\frac{1}{3} e^{2 Re(\zeta z)}$ are the coefficients of the affine sphere developing map in this basis. {width="8cm"} This particular affine sphere is called a Ţiţeica surface. All we need to notice about it is that its projection to $\mathbb{C}\times \{1\}$ is indeed the antipodal image of the unit ball for $F^\Delta_{dz^3}$ at $z=0$. Both $F^{DS}_{dz^3}$ and $F^\Delta_{dz^3}$ are translation invariant, so agreement at $z=0$ implies that they are the same everywhere. ◻ ## Upper bounds on $F^{DS}$ and $F^\Delta$ We won't be able to compare $F^{DS}$ and $F^\Delta$ directly. Instead, we will define a families of upper and lower bounds for $F^{DS}$ and $F^\Delta$, and compare these bounds. In this section we define upper bounds $F^{DS,d}>F^{DS}$, and $F^{\Delta,d}>F^\Delta$, and show they are close. In the next section we will use duality arguments to get lower bounds. The tangent bundle $TS$ embeds into the $\mathbb{RP}^2$ bundle $\mathbb{P}(TS\oplus \mathbb{R})$ as a bundle of affine charts via $v\mapsto [v:1]$. The projectivization of the affine sphere $Im(\xi_x)$, is a convex subset of the affine chart $T_xS \subset \mathbb{P}(T_xS\oplus \mathbb{R})$, and in fact is the antipodal image of the unit ball of $F^{DS}$ at $x$. This suggests a family of upper bounds defined by truncating affine spheres. **Definition 17**. *For $d>0$, let $F^{DS,d}$ denote the Finsler metric whose unit ball at $x\in S$ is $-\operatorname{Conv}(\pi(\xi_x(\tilde{B}_{\tilde{x}}(d)))$, where $\tilde{B}_{\tilde{x}}(d)$ denotes the $h_\mu$-ball of radius $d$, centered at $\tilde{x}$ in the universal cover $\tilde{S}_x$, and $\pi$ is the projection from $(T_x S \oplus \mathbb{R}) \backslash T_x S$ to the affine chart $T_x S\subset \mathbb{P}(T_x S \oplus \mathbb{R})$.* Now we do a similar thing for $F^\Delta$. We let $\mu_0$ and $h_0$ denote the constant cubic differential and hermitian metric on $T_x S$ which agree with $\mu_x$, and $h_x$. Let $\xi_{0,p}: T_x S\to T_x S\oplus \mathbb{R}$ denote the affine sphere developing map determined by $\mu_0$ and $h_0$. By lemma $\ref{titeica coincidence}$, the $DS$, and $\Delta$ metrics for the Ţiţeica surface $\xi_{0,x}(T_pS)$ coincide. At the point $x$, this means that the unit ball of $F^\Delta$ at $x$ is $-\pi(\xi_{0,x}(T_pS))$. **Definition 18**. *Let $F^{\Delta,d}$ denote the Finsler metric whose unit ball at $p$ is $-\operatorname{Conv}(\pi(\xi_{0,p}(T^{\leq d}_pS)))$ where $T^{\leq d}_pS$ denotes the ball of radius $d$ in $T_p S$.* Note that by construction, $F^{\Delta,d}$ converges to $F^\Delta$. We can thus find a function $\epsilon_{\Delta,d}$, which goes to zero as $d$ goes to infinity, such that $\log(F^{\Delta,d}_\mu/F^{\Delta}_\mu) < \epsilon_{\Delta,d}$ for all $d$, for any Riemann surface with cubic differential, on the complement of the zeros. ## Closeness of truncated affine spheres The next step is to show that, far from zeros, $F^{DS,d}$ is close to $F^{\Delta,d}$. This will follow from showing the two affine sphere developing maps $\xi_p$ and $\xi_{p,0}$ are close on the ball of radius $d$. This will give us a bound on the Hausdorff distance between unit balls of $F^{DS,d}$ and $F^{\Delta,d}$. We will then have to prove a simple lemma relating ratios between norms, and Hausdorff distances between their unit balls. **Lemma 19**. *There is a function $\epsilon_{d,Haus}:(0,\infty)\to (0,\infty]$ limiting to zero as the argument goes to infinity, such that for a Riemann surface $S$ with cubic differential $\mu$, the Hausdorff distance, with respect to $h_\mu$, between the unit balls of $F^{DS}_\mu$ and $F^\Delta_\mu$ at any point $p$, is bounded above by $\epsilon_{d,Haus}(r(p))$.* *Proof.* We define $\epsilon_{d,Haus}(r)$ to be $\infty$ for $r \leq d$, so we can assume that $p$ is distance at least $d$ from zeros. Let $B(p,d)$ denote the ball centered at $p$ of radius $d$ with respect to $h$. We have two solutions to Wang's equation on $B(p,d)$: $g$, and $h$. These give rise to two affine sphere developing maps $\xi_p, \xi_{0,p}: B(p,d) \to T_p S\oplus \mathbb{R}$ which are constructed by parallel transporting the section $\underline{1}$ back to the fiber over $p$, via two different connections $\nabla_0$, and $\nabla$. As we did for $\nabla_0$ in equation [\[nabla0 formula\]](#nabla0 formula){reference-type="ref" reference="nabla0 formula"}, we can write an explicit formula for $\nabla$ by choosing a coordinate $z$ on $B(p,d)$ which takes $\mu$ to $dz^3$, and using the frame $\partial_z, \bar{\partial}_z, \underline{1}$ of the complexification of $TS\oplus \mathbb{R}$. In this frame, we have the following formula for $\nabla$ in terms of $\phi = \log(g/h)$. $$\nabla = d+\begin{bmatrix} 0 & 0 & 1 \\ e^{-\phi} & \partial\phi & 0 \\ 0 & e^\phi & 0 \\ \end{bmatrix}dz +\begin{bmatrix} \bar{\partial}\phi & e^{-\phi} & 0 \\ 0 & 0 & 1 \\ e^\phi & 0 & 0 \\ \end{bmatrix}d\bar{z}$$ Let $A_0$ and $A$ be the matrix valued $1$ forms representing $\nabla_0$ and $\nabla$ in this frame: $\nabla_0=d+A_0$, $\nabla=d+A$. To conclude closeness of $\xi_p$ and $\xi_{0,p}$ from closeness of $\nabla$ and $\nabla_0$, we need the following standard consequence of Gronwall's inequality. **Lemma 20**. *Suppose $f' = Af$ and $g'= B g$ where $A,B\in C^0([0,t],\mathbb{R}^{n\times n})$ are matrix valued functions, and $f,g\in C^1([0,t],\mathbb{R}^n)$ are vector valued functions with $f(0)=g(0)$. Then $|g-f|$ has the following bound. $$|g(t)-f(t)|\leq |B-A|_{C^0}|f|_{C^0} t e^{t|B|_{C^0}}$$* Let $q\in B(p,d)$. Applying Gronwalls inequality to the restriction of $\nabla$ and $\nabla_0$ to the straight line segment from $q$ to $p$ gives the following. $$|\xi_p(q)-\xi_{0,p}(q)|\leq |A-A_0|_{C^0}|\xi_{0,p}|_{C^0} de^{d|A|_{C^0}}$$ $|A-A_0|$ is bounded by $C\epsilon_{C^1}(r)$ for some fixed constant $C$. By equation [\[titeica coordinates\]](#titeica coordinates){reference-type="ref" reference="titeica coordinates"}, $|\xi_{0,p}|$ is bounded by $\frac13 e^{4d}$. $|A|$ is bounded by $|A_0|+C\epsilon_{C^1}(r)$. We get $$|\xi_p(q)-\xi_{0,p}(q)|\leq C \epsilon_{C^1}(r) \frac13 e^{4d} d e^{d(|A_0|+C\epsilon_{C^1}(r))}$$ Define $\epsilon_d(r)$ to be the right hand side of this inequality, and note that it indeed goes to zero as $r$ goes to infinity for any fixed $d$ which is large enough that $\epsilon_{C^1}$ is finite. This bound persists after projecting to $T_p S$, and negating to get the unit balls of $F^{\Delta,d}$, and $F^{DS,d}$. This is because, by convexity, $\xi_0$ and $\xi$ are valued in $\{(v,y)\subset TS\oplus \mathbb{R}\;|\; y\geq 0\}$, a region in which the radial projection $\pi$ to $T_p S$ is contracting. This bound on $|\pi(\xi_p(q))-\pi(\xi_{0,p}(q))|$ gives the same bound on the Hausdorff distance between unit balls of $F^{DS,d}(p)$ and $F^{\Delta,d}(p)$. ◻ **Definition 21**. *Let $\Omega_1,\Omega_2\subset \mathbb{R}^n$ be convex sets containing the origin. We can describe $\Omega_1$ and $\Omega_2$ as the unit balls for norms $f_1$ and $f_2$ on $\mathbb{R}^n$. Let the radial distance, $d_R(\Omega_1,\Omega_2)$ denote the supremum of $|\log(f_1/f_2)|$.* Note that $f_1/f_2$ is the scaling factor which takes the boundary of $\Omega_1$ onto the boundary of $\Omega_2$. **Lemma 22**. *Let $\Omega_1,\Omega_2\subset \mathbb{R}^n$ be convex sets containing the origin. Let $d_H(\Omega_1,\Omega_2)$ denote the Hausdorff distance. We have $$rd_R(\Omega_1,\Omega_2) \leq d_H(\Omega_1,\Omega_2)$$ where $r$ is the radius of a ball centered at $0$ contained in both $\Omega_1$ and $\Omega_2$.* *Proof.* For $x\in \partial\Omega_1$, let $\alpha(x) := f_1(x)/f_2(x)$ be the scaling factor necessary such that $\alpha x\in \partial\Omega_2$. Let $x\in \partial{\Omega_1}$ be a point realizing the supremum of $|\log(\alpha(x))|$ and, without loss of generality, assume $\alpha(x)\geq 1$. Suppose we have a supporting hyperplane $H_x$ for $\Omega_1$ at $x$. We argue that $H_y=\alpha(x) H_x$ must be a supporting hyperplane for $\Omega_2$ at $y$. Indeed, suppose that there is $y'\in \Omega_2$ on the other side of $H_y$ from $0$. Let $x'$ be the point where the line from $0$ to $z$ intersects $\partial\Omega^1$, one sees that $\alpha(x')>\alpha(x)$, a contradiction. It follows that there is no $y'$ on the other side of $H_y$, so $H_y$ is a supporting hyperplane for $\Omega_2$. The Hausdorff distance is at least the euclidean distance from $y$ to $\Omega_1$ which is at least the distance between $H_y$ and $H_x$, which is at least $(\alpha-1)r$. $$d_H(\Omega_1,\Omega_2) \geq (\alpha-1)r \geq \log(\alpha)r \geq rd_R(\Omega_1,\Omega_2)$$ ◻ If we assume $d$ is above some threshold $d_0$, we then the unit balls of $F^\Delta_\mu$ will contain the unit balls of $r_0^{-1} h_\mu$ for some $r_0$. Setting $\epsilon_d(r) = \infty$ for $d\leq d_0$, and $\epsilon_d(r) = r_0^{-1}\epsilon_{d,Haus}(r)$ for $d>d_0$ we get our desired control over the ratio between $F^{\Delta,d}$ and $F^{DS,d}$. **Lemma 23**. *There is a function $\epsilon_d:(0,\infty)\to (0,\infty]$ limiting to zero as the argument goes to infinity, such that for a Riemann surface $S$ with cubic differential $\mu$, $$|\log \frac{F^{DS,d}_\mu}{F^{\Delta,d}_\mu}|\leq \epsilon_d(r(p))$$* ## Lower bounds on $F^{\Delta}$ and $F^{DS}$ {#lower bounds} Next, we will construct a family of lower bounds, using projective duality ideas. **Lemma 24**. *On a Riemann surface with cubic differential $\mu$, inducing flat metric $h$, we have $$(F^{\Delta}_\mu)^{*_{2h}} = F^{\Delta}_{-\mu}$$ on the complement of zeros.* *Proof.* Around any point, there is a local coordinate $z$ such that $\mu=dz^{3}$, so it suffices to treat the case of of $\mu=dz^3$ on the complex plane. This is an easy computation. Alternatively, it is a special case of the next lemma. ◻ When we have a Riemann surface with cubic differential $\mu$, and $g$ satisfying $\kappa_g = |\mu|_g^2 -1$, the Blaschke metric for the corresponding affine sphere is $2g$. The dual affine sphere is given by replacing $\mu$ with $-\mu$ in the formula for the connection. It follows from lemma [Lemma 12](#DS duality){reference-type="ref" reference="DS duality"} that negating the cubic differential corresponds to dualizing the domain shape metric with respect to $2g$. **Lemma 25**. *If $S$ is a Riemann surface with cubic differential $\mu$, and $g$ is a complete solution to Wang's equation, then $(F^{DS}_\mu)^{*_{2g}} = F^{DS}_{-\mu}$ where $g$ is the complete solution to Wang's equation.* Equivalently, we have $F^{DS}_\mu = (F^{DS}_{-\mu})^{*_{2g}}$. Since taking duals reverses containment of convex sets, $F^{DS}_{-\mu} < F^{DS,d}_{-\mu}$ implies $F^{DS}_{\mu} > (F^{DS,d}_{-\mu})^{*_{2g}}$. This is our desired lower bound. We will need to show that it is close to $F^{\Delta}_\mu$ just like our upper bound. In the last subsection we showed that $F^{DS,d}$ and $F^{\Delta,d}$ are close. The following lemma implies that their duals, with respect to $2g$, are the same amount close. **Lemma 26**. *Taking dual convex sets is an isometry for $d_R$. $$d_R(\Omega_1,\Omega_2) = d_R(\Omega_1^*,\Omega_2^*)$$* *Proof.* For $x\in \partial\Omega_1$, let $\alpha(x) := f_1(x)/f_2(x)$ be the scaling factor necessary such that $\alpha x\in \partial\Omega_2$. Let $x\in \partial{\Omega_1}$ realize the supremum of $\alpha$. By definition, $d_R(\Omega_1,\Omega_2) = |\log(\alpha)|$. Without loss of generality, assume $\alpha\geq 1$. As in the proof of Lemma [Lemma 22](#hausdorff-radial){reference-type="ref" reference="hausdorff-radial"}, if $H_x$ is a supporting hyperplane for $\Omega_1$ at $x$, then $H_y$ is a supporting hyperplane for $\Omega_2$. Note that $\partial\Omega_1^*$ is identified with the set of supporting hyperplanes of $\Omega_1$. If $H$ is a supporting hyperplane for $\Omega_1$, then let $\beta(H)$ be the positive number such that $\beta(H)H$ is a supporting hyperplane for $\Omega_2$. $d_R(\Omega_1^*,\Omega_2^*)$ is the supremum of $|\log(\beta)|$. We have shown that $d_R(\Omega_1,\Omega_2)\leq d_R(\Omega_1^*,\Omega_2^*)$. The reverse inequality follows from the fact that taking dual convex sets is an involution. ◻ ## End of proof of theorem C We have an upper bound on $F^{DS}_\mu$ which is close to $F^\Delta_\mu$ $$F^{DS}_\mu\: \leq \:F^{DS,d}_\mu\:\underset{\epsilon_{d}(r)}{\approx} \: F^{\Delta,d}_\mu\: \underset{\epsilon_{\Delta,d}}{\approx} \: F^\Delta_\mu$$ and a lower bound on $F^{DS}_\mu$ which is also close to $F^\Delta_\mu$ $$F^{DS}_\mu = (F_{-\mu}^{DS})^{*_{2g}} \:\geq \: (F^{DS,d}_{-\mu})^{*_{2g}}\:\underset{\epsilon_{d}(r)}{\approx} \: (F^{\Delta,d }_{-\mu})^{*_{2g}}\:\underset{\epsilon_{C^0}(r)}{\approx} \: (F^{\Delta,d }_{-\mu})^{*_{2h}}\:\underset{\epsilon_{\Delta,d}}{\approx} \: (F^\Delta_{-\mu})^{*_{2h}} = F^\Delta_\mu$$. Each '$\approx$' symbol means there is a bound, named in the subscript, on the absolute value of the log of the ratio. Combining upper and lower bounds gives bound on the distance between $F^{DS}_\mu$ and $F^{\Delta}_\mu$: $$|\log \frac{F^{DS}}{F^{\Delta}}| \leq \epsilon_{\Delta,d} + \epsilon_d(r) + \epsilon_{C^0}(r)$$ We have that $\epsilon_{\Delta,d}$ limits to zero as $d$ goes to infinity, and for each fixed $d$, $\epsilon_d(r)$ goes to zero as $r$ goes to infinity. We just need to choose a function $d(r)$ which limits to infinity, but slowly enough such that $\epsilon_{d(r)}(r)$ goes to zero. Setting $\epsilon(r) = \epsilon_{\Delta,d(r)} + \epsilon_{d(r)}(r) + \epsilon_{C^0}(r)$ finishes the proof. # Continuity of length functions of Finsler metrics Depending on context, there are minor variations on what one means by a Finsler metric. So far, we have discussed specific Finsler metrics, so we haven't needed to specify a class of Finsler metrics to work with, but now we need to prove a general fact about Finsler geometry, so we will specify exactly what we mean by Finsler metric. **Definition 27**. *A Finsler metric on a differentiable manifold $M$ is a continuous function $F:TM\to \mathbb{R}_{\geq 0 }$ satisfying* - *$F_x(\lambda v) = \lambda F_x(v)$ for all $\lambda \in \mathbb{R}_{\geq 0 }$* - *$F_x(v+v')\leq F_x(v) + F_x(v')$* - *$F_x(v) = 0$ iff $v=0$* Importantly, we don't assume $F_x(v) = F_x(-v)$, and we don't assume that $F$ is differentiable. If $F$ satisfies the first two conditions, but not the third, we call it a degenerate Finsler metric, and we call the subset of $M$ where $F$ is degenerate the degeneracy locus. Let $\mathcal{F}(M)$ denote the set of Finsler metrics on $M$, and let $\mathcal{F}^{\textrm{fd}}(M)$ denote the set of Finsler metrics with finite degeneracy locus. These definitions are useful to us because if $S$ is a convex projective surface, $F^{DS}$ is always in $\mathcal{F}(M)$, and $F^{\Delta}$ is always in $\mathcal{F}^{\textrm{fd}}(S)$. If $L$ is a free homotopy class of loop in $M$, and $F\in \mathcal{F}(M)$, then we denote by $F(L)$ the infimum of the lengths of paths in $L$ with respect to $F$. $$F(L) := \inf_{\gamma \in L}\int_{\gamma}F$$ We give $\mathcal{F}(M)$ the topology in which $F_i$ converge to $F$ if $F_i/F$ converge uniformly to $1$ as functions on $TM- \underline{0}$. It is easy to see that $F(\gamma)$ is a continuous function of $F$ with respect to this topology. We need to be a little more thoughtful in choosing a topology for $\mathcal{F}^{\textrm{fd}}(M)$ **Definition 28**. *We endow $\mathcal{F}^{\textrm{fd}}(M)$ with the topology in which $F_i$ converges to a degenerate Finsler metric $F$ with degeneracy locus $X$ if for every open neighborhood $U$ of $X$, $F_i/F$ converges uniformly to $1$ in $M\backslash U$.* **Theorem 4**. *If $F_i\in \mathcal{F}^{\textrm{fd}}(M)$ is a sequence converging to a Finsler metric $F$ with finite degeneracy locus $X$, and $Y$ is a free homotopy class of loop in $M$, then $$\lim_{i\to\infty} F_i(Y) = F(Y)$$* *Proof.* First we show $$\lim_{i\to \infty}F_i(Y) \leq F(Y)$$ Let $\epsilon>0$. We can choose $\gamma\in Y$ which doesn't hit $X$ such that $F(\gamma) \leq F(Y)+\epsilon$. This follows from continuity of $F$, because only very small deformations of a path are necessary to avoid $X$, and these will change the path's length by a small amount. Because $\gamma$ avoids the degeneracy locus, we get the following: $$\lim_{i\to \infty} F_i(Y)\leq \lim_{i\to \infty} F_i(\gamma) = F(\gamma) \leq F(Y)+\epsilon$$ Since this holds for all $\epsilon$, it follows that $\lim F_i(\gamma) \leq F(Y)$. Now we show the reverse inequality. $$F(Y) \leq \lim_{i\to \infty} F_i(Y)$$ For every $i\in \mathbb{N}$, let $\gamma_i\in Y$ be a path which nearly realizes minimal length in the metric $F_i$. $$F_i(\gamma_i) \leq F_i(Y) + \epsilon$$ Choose $\delta>0$ such that closed $\delta$-balls around points in $X$, for the metric $F$ are disjoint. Let $d_{\min}$ be shortest distance between two balls. If a path in the universal cover $\gamma:[0,1]\to \tilde{M}$ hits $N$ different balls, then $F(\gamma)\geq N d_{\min}$. Let $K>1$. For sufficiently large $i$, we have $K^{-1} < F_i/F < K$ on the complement of the balls. We get a bound on the number of balls a path can visit in terms of its $F_i$ length: $$N \leq K F_i(\gamma)/d_{\min}$$ For sufficiently large $i$, $F_i(Y)\leq F(Y) + \epsilon$, so $F_i(\gamma)\leq F(Y) + 2\epsilon$. Let $\tilde\gamma_i:[0,1]\to \tilde{M}$ be a lift of $\gamma_i$ to the universal cover. For simplicity, choose $\tilde\gamma$ so that that $\tilde\gamma(0)$ is not in the $\delta$ neighborhood of $X$. Putting things together, we get a bound on the number of balls $\tilde\gamma_i$ can visit which holds for all sufficiently large $i$. $$N_i\leq K (F(Y)+2\epsilon)/d_{\min}$$ We now define a path $\gamma_i'$ whose $F$ length we can estimate. Let $B_1$ be the first ball that $\gamma_i$ touches, and let $t_0,t_1\in [0,1]$ be the first and last points which are sent by $\tilde\gamma_i$ to $\overline{B_1}$. Replace $\tilde\gamma_i|_{[t_0,t_1]}$ with a path that goes straight to $p$, and straight out. Now apply the same procedure to the rest of the path $\tilde\gamma_i|_{[t_1,1]}$. Proceed inductively, and call the final result $\tilde\gamma_i'$. We have an upper bound on the $F$ length of $\tilde\gamma_i'$, thus an upper bound on $F(Y)$. $$F(Y) \leq F(\gamma') \leq K F_i(\gamma_i) + 2N_i\delta \leq K (F_i(Y)+\epsilon) + 2N_i\delta$$ This bound holds for sufficiently large $i$ for any choices of $K$ and $\delta$, and $N_i$ eventually has a uniform bound, so we have $F(Y) \leq \lim_{i\to \infty} F_i(Y)+\epsilon$. This holds for any $\epsilon$, so we get the desired inequality. ◻ {width="14cm"} # Application to a triangle reflection group The broad aspiration of this project is to understand what happens when we have a sequence of Hitchin representations of a surface group into $\operatorname{SL}_3\mathbb{R}$ going to infinity. Replacing the surface group with a triangle reflection group is a great way to probe this question, because the Hitchin component is diffeomorphic to $\mathbb{R}$, so there are only two ways to go to infinity. Before [@loftin_limits_2022] was published, and before we knew of Loftin's work [@loftin_flat_2007], it was triangle group computations which convinced us that theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} should hold. Consider a triangle group. $$\Gamma_{pqr} := \langle a,b,c \:|\: a^2 = b^2 = c^2 = (ab)^p = (bc)^q = (ca)^r = 1 \rangle$$ For $1/p + 1/q + 1/r < 1$, there is a unique conjugacy class of homomorphism $\rho_0:\Gamma_{pqr}\to SO(2,1)$ giving a proper discontinuous action on the hyperbolic plane $\mathbb{H}^2$, and the quotient $S=\Gamma\backslash\mathbb{H}^2$ is an orbifold. Let $\operatorname{Conv}(\Gamma)$ denote the component of $\operatorname{Rep}(\Gamma, \operatorname{SL}_3\mathbb{R})$ containing $\rho_0$. We can easily make an explicit algebraic parametrization of $\operatorname{Conv}(\Gamma_{pqr})$. Let $v_1,v_2,v_3$ and $\alpha_1,\alpha_2,\alpha_3$ be bases of $\mathbb{R}^3$ and $(\mathbb{R}^3)^*$ with the following matrix of pairings. $$\alpha_i(v_j) = \begin{bmatrix} 2 & -2\cos(\frac{\pi}{p})t & -2\cos(\frac{\pi}{r}) \\ -2\cos(\frac{\pi}{p})t^{-1} & 2 & -2\cos(\frac{\pi}{q}) \\ -2\cos(\frac{\pi}{r}) & -2\cos(\frac{\pi}{q}) & 2 \\ \end{bmatrix}$$ We define $\rho_t:\Gamma_{pqr}\to \operatorname{SL}_3\mathbb{R}$ to send the generators $a$, $b$, $c$ to the three reflections $I - v_i\otimes \alpha_i$. The parameter $t$ is the square root of the triple ratio of the three reflections. It has the following expression: $$t^2=\frac{\alpha_1(v_2)\alpha_2(v_3)\alpha_3(v_1)}{\alpha_1(v_3)\alpha_2(v_1)\alpha_3(v_2)}$$ which makes it clear that it is an invariant of the three reflections. By [@lee_anosov_2021], we know that $t$ gives a global parametrization of $\operatorname{Conv}(\Gamma_{pqr})$ by $\mathbb{R}_+$. A naive way to try to understand the representation $\rho_t$ is to plot the eigenvalues of $\rho_t(g)$ for a bunch of $g\in \Gamma$. Let $\phi:\operatorname{SL}_3\mathbb{R}\to \mathbb{R}^2$ be the function $(\log|\lambda_1|,-\log|\lambda_3|)$ where $\lambda_1$ is the top eigenvalue, and $\lambda_3$ is the bottom eigenvalue. $\phi$ is called the Jordan projection for $\operatorname{SL}_3\mathbb{R}$ and lands in the cone spanned by $(1,2)$ and $(2,1)$. In general, the Jordan projection of an element of a semi-simple lie group is the Weyl-chamber valued translation length of the the element acting on the associated symmetric space, so Jordan projections are a generalization of hyperbolic translation length. As an example, we plot here $\phi(g)$ for orientation preserving elements $g\in\Gamma_{444}$ of length at most 24, for triple ratio $t^2$ set to $1,10,1000$, and $10^{12}$. {width="8cm"} {width="8cm"} {width="8cm"} {width="8cm"} We see that the Jordan projections increasingly tend to lie on an integral lattice. Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} gives a way to compute which lattice point each group element converges to. From an algebraic perspective, it not so surprising that the Jordan projections collect onto a lattice. Note that when $\lambda_1(g)$ is big, $\log|\lambda_1(g)|$ is approximately $\log|tr(g)|$, and $-\log|\lambda_3(g)|$ is approximately $\log|tr(g^{-1})|$. Trace functions $tr(\rho_t(\gamma))$ for $\gamma\in \Gamma_{pqr}$ are Laurent polynomials in $t$. This means that when $t$ is big, $$\phi(g) \approx \log(t)(d_1,d_2)$$ where $d_1$ and $d_2$ are the highest powers of $t$ in $tr(\rho_t(\gamma))$ and $tr(\rho_t(\gamma^{-1}))$ respectively. From this perspective, theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} tells us the ratios between the highest power of $t$ in $tr(\rho_t(\gamma))$. The Labourie-Loftin parametrization can be modified to apply to our reflection orbifold $S$, by defining a holomorphic cubic differential on $S$ to be a holomorphic cubic differential $\tilde{\mu}$ on a universal cover which is preserved by orientation preserving deck transformations, and complex conjugated by orientation reversing deck transformations. The correspondence for quotient orbifolds (such as our triangle reflection orbifold) follows from the correspondence on a smooth, compact covering space, and the general fact that when solutions to PDE's are unique, they have to be invariant under the symmetry group of the input data. The correspondence between Higgs bundles and Hitchin components for orbifolds was worked out in generality in [@alessandrini_hitchin_2020]. Note that this definition of cubic differential for orbifolds forces the fixed loci of reflections to be real trajectories of $\tilde{\mu}$. As predicted by the Labourie-Loftin correspondence, the space of cubic differentials on $S$ is $1$ dimensional. We now construct a nonzero element of this space. Consider a euclidian equilateral triangle $T\subset \mathbb{C}$, whose sides are unit length, and such that the restriction of $dz^3$ to each side is real. Let $\bar{T}$ denote $T$ with the conjugate cubic differential $d\bar{z}^3$. We can glue copies of $T$ and $\bar{T}$ appropriately to get a Riemann surface with cubic differential which has an action of $\Gamma_{pqr}$. Call this cubic differential $\mu$. If we color copies of $T$ grey, and color copies of $\bar{T}$ white, and conformally map the resulting surface to the disk, we get the standard picture of the $pqr$ triangulation of the hyperbolic plane. Below is the picture for $p=q=r=4$. {width="7cm"} We now illustrate how to apply theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} with an example. The word $$w=cbcacbcacbcacbacbabcabab$$ was chosen arbitrarily from the words forming the cluster around the lattice point $(6,5)$ in the $t=10^{12}$ Jordan projection picture. We can compute the $F_\mu^\Delta$ translation lengths of $w$, and $w^{-1}$ as follows. First we construct a curve in the universal cover $\tilde{S}$ which is preserved by $w$ by concatinating straight line segemnts connecting centers of triangles, which cross reflection loci prescribed by the letters of $w$. Then pull this curve curve tight in the metric $h_\mu$. Then push it onto the reflection locus, while making sure not to change its $F_\mu^\Delta$ length. $F^\Delta_\mu$ assigns length $2$ to edges going counter clockwise around grey triangles, and length $1$ to edges going clockwise around grey triangles (and vice versa for white triangles) so we can just add up these numbers to compute translation length once we have a geodesic representative in the reflection locus. {width="13cm"} Here, the thin black line represents the easy to construct $w$ invariant curve, the pink line represents the geodesic for the singular flat metric $h_\mu$, which is also a geodesic for $F_\mu^\Delta$, and the red line represents a geodesic for $F_\mu^\Delta$ which lies in the reflection locus. We see that the translation length of $w$ is $18$, and the translation length $w^{-1}$ is $15$. Theorem [Theorem 1](#evlimit){reference-type="ref" reference="evlimit"} thus predicts $$\lim_{t\to\infty} \frac{\log(\lambda_1(\rho_t(w)))}{\log(\lambda_1(\rho_t(w^{-1})))} = \frac{6}{5}$$ which is what we observed in the Jordan projection picture. We can verify this limit rigorously by using computer algebra software to directly compute that the highest powers of $t$ in $tr(\rho_t(w))$ and $tr(\rho_t(w^{-1}))$ are $6$, and $5$ respectively. The reader can easily try these computations for other elements of triangle reflection groups.

arxiv_math

--- abstract: | We find Fano threefolds $X$ admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are ${\mathbb T}$-varieties of complexity two. More precisely, we show that the weighted K-stability of $(X,\xi_0)$ (where $\xi_0$ is the soliton candidate) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso's theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair $(V,\Delta_V)$ is equivalent to the weighted K-stability of a cone $(Y, \Delta_Y, \xi_0)$ over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of [@AZ22], which gives a lower bound of the weighted stability threshold $\delta^g_{{\mathbb T}}(X,\Delta)$. This is an effective way to check the weighted K-semistablity of a log Fano triple $(X,\Delta,\xi_0)$. Surprisingly, such an estimate is also useful in testing (weighted) K-polystability based on the work of [@BLXZ23]. address: - Department of Mathematics, Nanjing University, Nanjing 210093, China - Department of Mathematics, Nanjing University, Nanjing 210008, China author: - Minghao Miao - Linsheng Wang bibliography: - ref.bib date: - - title: Kähler-Ricci solitons on Fano threefolds with non-trivial moduli --- # Introduction The existence of canonical metrics on a given manifold is a fundamental problem in Kähler geometry. The so-called Yau-Tian-Donaldson (YTD) conjecture predicts that the existence of canonical metrics is equivalent to certain algebro-geometric stability conditions. The concept of K-stability introduced by Tian [@Tia97] detects the existence of the Kähler-Einstein (KE) metrics on a Fano manifold. It was proved in [@Tia15] that a Fano manifold admits a Kähler-Einstein metric if and only if it is K-polystable. Other proofs were also given in [@CDS15; @DS16; @CSW18; @BBJ18; @Zhang22]. But it is in general very hard to verify the K-stability of a given Fano manifold $X$. The first powerful tool in testing K-stability is Tian's criterion [@Tia87] via $\alpha$-invariant, with a notable application that every smooth Fano hypersurface of index one is K-stable [@Fuj19]. More recently, [@AZ22] introduced an inductive approach to give a lower bound estimate of the stability threshold $\delta(X)$, which proved that every smooth Fano hypersurface of index $r \ge 3$ and dimension $\ge r^3$ is K-stable [@AZ23]. The Abban-Zhuang theory is also widely used in the study of the K-stability of Fano threefolds, see [@ACC+]. Besides testing K-stability, the Abban-Zhuang theory is also useful in finding the minimizer of the local delta invariant. For example, [@AZ22 Theorem 4.6] found the minimizer of $\delta_p(X)$ for every cubic surface $X$ and every closed point $p\in X$. However, not every Fano manifold admits KE metric. The Kähler-Ricci soliton (KRS) metrics provide a natural generalization of the KE metrics on Fano manifolds, which is closely related to the limiting behavior of the Kähler-Ricci flow on Fano manifolds. The uniqueness of Kähler-Ricci solitons is established by [@TZ00; @TZ02], There are several results on the existence of Kähler-Ricci solitons on Fano varieties with large symmetry, for example, on toric varieties [@WZ04; @SZ12; @BB13], on spherical varieties [@LZZ18; @Del20; @LLW22] and on ${\mathbb T}$-varieties of complexity one [@IS17; @CS18; @HHS23]. In this paper, we find new examples of Fano threefolds admitting Kähler-Ricci solitons, which are ${\mathbb T}$-varieties of complexity two (Section [2.5](#Subsection:torus action){reference-type="ref" reference="Subsection:torus action"}). **Theorem 1** (Theorem [Theorem 28](#Theorem. stability of 2.28){reference-type="ref" reference="Theorem. stability of 2.28"} and [Theorem 33](#Theorem: stability of 3.14){reference-type="ref" reference="Theorem: stability of 3.14"}). *Every smooth Fano threefold in the families №2.28 and №3.14 of Mori and Mukai's list admits a Kähler-Ricci soliton.* By the celebrated works of [@HL23] and [@BLXZ23], a log Fano triple $(X,\Delta,\xi_0)$ admits a twisted Kähler-Ricci soliton if and only if it is weighted K-polystable (proved by [@DS16; @CSW18] for smooth $X$ and $\Delta$ = 0). It suffices to show that $(X,\xi_0)$ is weighted K-polystable, where $X$ is a Fano threefold in the above families in Theorem [Theorem 1](#Theorem: Intro №2.28 and No 3.14 soliton){reference-type="ref" reference="Theorem: Intro №2.28 and No 3.14 soliton"}, and the holomorphic vector field $\xi_0$ is the soliton candidate of $X$ (see Section [3.1](#Subsection: soliton candidate){reference-type="ref" reference="Subsection: soliton candidate"}). Since there is only one-dimensional torus action on $X$, all known methods do not apply. To overcome the difficulty, we establish the weighted Abban-Zhuang estimate to show that the weighted stability threshold $\delta^g_{{\mathbb T}}(X,\Delta)\geq 1$, which proves the weighted K-semistability of a log Fano triple $(X,\Delta,\xi_0)$. **Theorem 2** (Theorem [Theorem 20](#Theorem: weighted AZ){reference-type="ref" reference="Theorem: weighted AZ"}, weighted Abban-Zhuang estmate). *Let $F$ be a ${\mathbb T}$-invariant plt-type divisor over $X$, and $Z \subseteq X$ be a ${\mathbb T}$-invariant subvariety contained in $C_X(F)$. For any ${\mathbb T}$-invariant multi-graded linear series $V_\bullet$ on $X$ we have $$\begin{aligned} \delta^{g}_{Z, {\mathbb T}}(X,\Delta;V_\bullet) \ge {\rm min}\Big\{\frac{A_{X,\Delta}(F)}{S^{g}(V_\bullet; F)},\,\, \delta^{g}_{Z,{\mathbb T}}(F,\Delta_F;W_\bullet) \Big\}, \end{aligned}$$ where $W_\bullet$ is the refinement of $V_\bullet$ by $F$.* The theorem still holds if we replace ${\mathbb T}$ by an algebraic group $G$ acting on $X$ and $V_\bullet$ (see Lemma [Lemma 22](#Theorem: weighted AZ 2){reference-type="ref" reference="Theorem: weighted AZ 2"}), and it is an equivariant version of the Abban-Zhuang estimate [@AZ22 Theorem 3.2] when $g=1$. The proof is based on the $G$-equivariant K-stability theory developed by [@Zhu21]. Surprisingly, this estimate is not only useful in checking the (weighted) K-semistability but also in checking the (weighted) K-polystability, when the ${\mathbb T}$-action on $F$ is trivial and the second term on the right-hand side of the estimate is strictly greater than $1$, see the proof of Theorem [Theorem 28](#Theorem. stability of 2.28){reference-type="ref" reference="Theorem. stability of 2.28"}. The moduli spaces of Fano threefolds in the families №2.28 and №3.14 are non-trivial. Recall that every smooth Fano threefold in the family №2.28 is given by the blowup of ${\mathbb P}^3$ along a smooth plane cubic curve $C\subseteq H \cong {\mathbb P}^2$. If further blowing up a point outside $H$, we get a smooth Fano manifold in the family №3.14. Parallel with the proof of Theorem [Theorem 1](#Theorem: Intro №2.28 and No 3.14 soliton){reference-type="ref" reference="Theorem: Intro №2.28 and No 3.14 soliton"}, we will show that for any Fano threefold $X$ in the families №2.28 and №3.14, the weighted K-stability of $(X,\xi_0)$ is equivalent to the GIT-stability of the plane cubic curves. **Theorem 3** (Theorem [Theorem 34](#Theorem: equivalence of weighted K-stability of №2.28 and №3.14 to the GIT-stability of plane cubic){reference-type="ref" reference="Theorem: equivalence of weighted K-stability of №2.28 and №3.14 to the GIT-stability of plane cubic"}). *Let $X$ be the blowup of ${\mathbb P}^3$ along a plane cubic curve $C\subseteq H\cong {\mathbb P}^2$ or further blowing up a point outside $H$. Let $\xi_0$ be the soliton candidate of $X$. Then $(X, \xi_0)$ is weighted K-semistable (weighted K-polystable) if and only if $C\subseteq H$ is GIT-semistable (GIT-stable or polystable).* Hence the GIT-moduli space (stack) of plane cubic curves should be viewed as a weighted K-moduli space (stack) of Fano threefolds in the families №2.28 or №3.14. To the best of the authors' knowledge, these may be the first examples of KRS Fano manifolds with non-trivial moduli, which give an answer to [@Ino19 Question 5.6]. On the other hand, we have the following generalization of Koiso's theorem [@Koi90], which establishes the equivalence between the K-stability of a log Fano pair $(V,\Delta)$ and the weighted K-stability of the cone $(Y, \Delta_Y, \xi_0)$ (see Section [9](#Section: Weighted K-stability of cones){reference-type="ref" reference="Section: Weighted K-stability of cones"}) over $(V,\Delta_V)$. **Theorem 4** (Theorem [Theorem 38](#Theorem: soliton of cone){reference-type="ref" reference="Theorem: soliton of cone"}). *Let $(V, \Delta_V)$ be a $(n-1)$-dimensional log Fano pair such that $L=-\frac{1}{r}(K_V+\Delta_V)$ is an ample Cartier divisor for some $0<r\le 1$. Let $Y=\overline{{\mathcal C}}(V,L)$ be the projective cone over $V$ with polarization $L$, and $\Delta_Y$ be the closure of $\Delta_V\times {\mathbb C}^*$ in $Y$. Then $(Y, \Delta_Y, \xi_0)$ is weighted K-semistable (weighted K-polystable) if and only if $(V,\Delta_V)$ is K-semistable (K-stable or K-polystable).* By this theorem, the K-moduli space (stack) of log Fano pairs $(V,\Delta_V)$ should be viewed as the weighted K-moduli space (stack) of log Fano triples $(Y, \Delta_Y, \xi_0)$. This also gives us a series of examples of KRS Fano varieties with non-trivial moduli. **Remark 5**. *There are strictly semistable objects in the GIT-moduli space of plane cubic curves, and the K-moduli space of log Fano pairs $(V,\Delta_V)$. Theorem [Theorem 1](#Theorem: Intro №2.28 and No 3.14 soliton){reference-type="ref" reference="Theorem: Intro №2.28 and No 3.14 soliton"} and [Theorem 4](#Theorem: Intro stability of cones){reference-type="ref" reference="Theorem: Intro stability of cones"} give us examples of strictly weighted K-semistable Fano varieties.* We briefly state the proof of Theorem [Theorem 1](#Theorem: Intro №2.28 and No 3.14 soliton){reference-type="ref" reference="Theorem: Intro №2.28 and No 3.14 soliton"}. Recall that a Fano manifold $X$ in family №2.28 is the blowup of ${\mathbb P}^3$ along a plane cubic $C\subseteq H$, where $H$ is the plane containing $C$. We still denote by $H$ the strict transform of the plane. The Fano manifold $X$ is destabilized by $H$, hence is K-unstable and does not admit the KE metric. We take refinement of $R_\bullet=R(-K_X)$ by $H$ in the first step, and get a ${\mathbb N}^2$-graded linear series $W_\bullet$. From the definition of the soliton candidate, we know that $A_X(H)/S^g(R_\bullet; H)=1$. Explicit computation (see Section [7.2](#Subsection: computing S^g of №2.28){reference-type="ref" reference="Subsection: computing S^g of №2.28"}) shows that $\delta^g_{p,{\mathbb T}}(H; W_\bullet)>1$ for any point $p\in H$. By Theorem [Theorem 2](#Theorem: Intro weighted Abban-Zhuang){reference-type="ref" reference="Theorem: Intro weighted Abban-Zhuang"}, we deduce that $(X, \xi_0)$ is weighted K-semistable. To prove the weighted K-polystability, we need the existence of the delta-minimizer established in [@BLXZ23 Lemma 4.14]. Assume that $(X,\xi_0)$ is strictly K-semistable. By the above Lemma, there exists a ${\mathbb T}$-invariant quasi-monomial valuation $v$ which is not of the form ${\rm wt}_\xi$ for $\xi\in N({\mathbb T})_{\mathbb R}$ such that $A_X(v)/S^g(R_\bullet;v)=1$. We can find a valuation $v_0$ on $H$ whose extension on $X$ is just $v$, and it's not difficult to show that $A_X(v)=A_H(v_0)$ and $S^g(R_\bullet;v)=S^g(W_\bullet; v_0)$. Hence $$1=\frac{A_X(v)}{S^g(R_\bullet;v)}=\frac{A_{H}(v_0)}{S^g(W_\bullet;v_0)} \ge \delta^g_{p,{\mathbb T}}(H;W_\bullet)>1,$$ which is a contradiction. **Remark 6**. *There is a well-known example found in [@Fut83] similar to the one we introduced above, that is, the blowup of ${\mathbb P}^3$ along a plane conic and a point outside the plane. This is the first example of a Fano manifold with reductive automorphism group and does not admit a Kähler-Einstein metric. It admits a Kähler-Ricci soliton using the same argument as above, which was proved in [@CS18] based on the structure of ${\mathbb T}$-varieties of complexity one.* The paper is organized as follows. In Section [2](#Section: Preliminaries){reference-type="ref" reference="Section: Preliminaries"}, we recall the basic notions in K-stability, especially the ${\mathbb T}$-action on a graded linear series and the weight decomposition. The concepts of weighted K-stability can be found in Section [3](#Section: Weighted K-stability){reference-type="ref" reference="Section: Weighted K-stability"}. Various formulations of the weighted expected vanishing order of a multi-graded linear series will be stated in Section [4](#Section: Invariants associated to multi-graded linear series){reference-type="ref" reference="Section: Invariants associated to multi-graded linear series"}. In Section [5](#Section: Weighted Abban-Zhuang estmate){reference-type="ref" reference="Section: Weighted Abban-Zhuang estmate"}, we establish the weighted Abban-Zhuang estimate (Theorem [Theorem 2](#Theorem: Intro weighted Abban-Zhuang){reference-type="ref" reference="Theorem: Intro weighted Abban-Zhuang"}), which is the main technical results of this paper. As one application, we generalize the $G$-equivariant K-stability of [@Zhu21] to the weighted setting in Section [6](#Section: G-Equivariant weighted K-stability){reference-type="ref" reference="Section: G-Equivariant weighted K-stability"}. Finally, we prove the main theorems in the last three sections. In Section [7](#Section: Applications){reference-type="ref" reference="Section: Applications"} and [8](#Section: GIT and weighted K, 2.28 and 3.14){reference-type="ref" reference="Section: GIT and weighted K, 2.28 and 3.14"}, we prove Theorem [Theorem 3](#Theorem: Intro GIT and weighted K, 2.28 and 3.14){reference-type="ref" reference="Theorem: Intro GIT and weighted K, 2.28 and 3.14"}, the equivalence between the weighted K-stability of Fano threefolds in the families №2.28, №3.14 and the GIT-stability of plane cubic curves. In Section [9](#Section: Weighted K-stability of cones){reference-type="ref" reference="Section: Weighted K-stability of cones"}, we prove Theorem [Theorem 4](#Theorem: Intro stability of cones){reference-type="ref" reference="Theorem: Intro stability of cones"}, the equivalence between the K-stability of a log Fano pair $(V,\Delta)$ and the weighted K-stability of the log Fano cone $(Y, \Delta_Y, \xi_0)$ over it. . We would like to thank our advisor, Gang Tian, for his constant support and guidance. We thank Chenyang Xu and Yuchen Liu for their suggestions on the weighted K-stability and K-moduli. We thank Akito Futaki for telling us the example of ${\mathbb P}^3$ with a conic blowup. We also thank Lu Qi, Fei Si, and Shengxuan Zhou for helpful discussions. # Preliminaries {#Section: Preliminaries} ## Notations and Conventions We work over the field of complex numbers ${\mathbb C}$. A *variety* is a separated integral scheme of finite type over ${\mathbb C}$. A *pair* $(X, \Delta)$ consists of a normal variety $X$ and an effective divisor $\Delta$ on $X$ such that $K_X+\Delta$ is ${\mathbb Q}$-Cartier. The notions of klt and lc singularities are defined as in [@K13 Definition 2.8]. A pair $(X, \Delta)$ is called *log Fano* if it is klt and $-K_X-\Delta$ is ample. Let $(X,\Delta)$ be a $n$-dimensional log Fano pair. Fix an integer $l_0 > 0$ such that $-l_0(K_X+\Delta)$ is Cartier. We denote by $R:= \oplus_{m\in l_0{\mathbb N}} R_m=:R(X,\Delta)$ the anti-canonical ring of $(X,\Delta)$ where $R_m := H^0(X, -m(K_X+\Delta))$. The number $m$ is always assumed to be a multiple of $l_0$. ## Valuations Let $K$ be a field. An ${\mathbb R}$-*valuation* $v$ on $K$ is a function $v: K^\times \to {\mathbb R}$ such that $v(fg)=v(f) + v(g), v(f+g)\ge {\rm min}\{v(f), v(g)\}$ for all $f, g \in K^\times$. For convenience, we set $v(0)=+\infty$. The *trivial valuation* $v_{{\rm triv}}$ is defined as $v_{{\rm triv}}(f)=0$ for all $f\in K^\times$. A *valuation* $v$ on $X$ is an ${\mathbb R}$-valuation on the rational function field $K(X)$ with a center on $X$ and $v|_{{\mathbbm k}^\times}=0$. Recall that the *center* of $v$, denoted by $c_X(v)$, is a scheme-theoretic point $\zeta$ on $X$ such that $v\ge 0$ on ${\mathcal O}_{X,\zeta}$. We denote by $C_X(v)=\overline{c_X(v)}\subseteq X$ the corresponding closed irreducible subscheme on $X$. Since $X$ is proper, every valuation $v$ has a unique center on $X$. Let $\pi:Y\to X$ be a birational morphism. Any prime divisor $E\subseteq Y$ determines a *divisorial valuation* ${\rm ord}_E$ on $X$. The log discrepancy $A_{X,\Delta}({\rm ord}_E)=A_{X,\Delta}(E) = {\rm ord}_E(K_Y - \pi^*K_X) + 1 -{\rm ord}_E(\Delta)$ is defined for such a valuation. One may define the *log discrepancy* $A_X(v)$ for a general valuation $v$ on $X$, see [@jm12]. We denote by ${\rm Val}_X^\circ$ the set of non-trivial valuations on $X$ with finite log discrepancies. If $(X, \Delta)$ admits a torus ${\mathbb T}={\mathbb G}_m^r$ action, we denote by ${\rm Val}_X^{{\mathbb T}}$ the set of ${\mathbb T}$-invariant valuations on $X$, and ${\rm Val}_X^{{\mathbb T}, \circ}:= {\rm Val}_X^\circ\cap {\rm Val}_X^{\mathbb T}$. ## Filtrations {#Subsection: filtration} A *filtration* ${\mathcal F}$ on $R$ is collection of subspaces ${\mathcal F}^\lambda R_m \subseteq R_m$ for each $\lambda\in {\mathbb R}$ and $m\ge 0$ such that - *Decreasing.* ${\mathcal F}^\lambda R_m \supseteq {\mathcal F}^{\lambda'}R_m$ for $\lambda\le \lambda'$; - *Left-continuous.* ${\mathcal F}^\lambda R_m={\mathcal F}^{\lambda-\epsilon}R_m$ for $0<\epsilon \ll 1$; - *Bounded.* ${\mathcal F}^\lambda R_m = R_m$ for $\lambda\ll 0$ and ${\mathcal F}^\lambda R_m = 0$ for $\lambda\gg 0$; - *Multiplicative.* ${\mathcal F}^\lambda R_m \cdot {\mathcal F}^{\lambda'}R_{m'} \subseteq {\mathcal F}^{\lambda+\lambda'}R_{m+m'}$. Since $R$ is finitely generated and ${\mathcal F}$ is bounded and multiplicable, there is a constant $C>0$ such that ${\mathcal F}^{-mC}R_m=R_m$ for all $m$. A filtration ${\mathcal F}$ is called *linearly bounded* if there is a constant $C>0$ such that ${\mathcal F}^{mC}R_m=0$ for all $m$. For any valuation $v$ on $X$, there is a filtration ${\mathcal F}_v$ on $R$ defined by $${\mathcal F}_v^\lambda R_m := \{s\in R_m: v(s)\ge \lambda\}.$$ If $A_X(v)<+\infty$, then ${\mathcal F}_v$ is linearly bounded, see [@BJ20]. In particular, the trivial valuation induces the trivial filtration ${\mathcal F}_{{\rm triv}}^0 R_m = R_m,\,\,{\mathcal F}_{{\rm triv}}^{>0}R_m= 0.$ Let $L$ be a line bundle on $X$ and $V\subseteq H^0(X, L)$ be a subspace. A filtration ${\mathcal F}$ on $V$ is simply defined to be a decreasing set $\{{\mathcal F}^\lambda V\}$ of subspaces which is left-continuous in $\lambda$. For any $s \in V\setminus \{0\}$, we define ${\rm ord}_{\mathcal F}(s) := {\rm max}\{\lambda: s\in {\mathcal F}^\lambda R_m\}$. We also set ${\rm ord}_{\mathcal F}(0)=+\infty$ for convenience. **Definition 7**. *Let $N=\dim V$. A basis $\{s_1, \cdots, s_{N}\}$ of $V$ is said to be *compatible* with ${\mathcal F}$ if ${\mathcal F}^\lambda V$ is generated by those $s_j$ with ${\rm ord}_{\mathcal F}(s_j) \ge \lambda$ for any $\lambda\in {\mathbb R}$.* The lifting of any basis of ${\rm gr}_{\mathcal F}V=\oplus_\lambda{\mathcal F}^\lambda V / {\mathcal F}^{>\lambda}V$ to $V$ is a basis compatible with ${\mathcal F}$. Moreover, for any two filtrations, there exists a basis compatible with both the two filtrations, see [@AZ22]. **Lemma 8**. *For any two filtrations ${\mathcal F}, {\mathcal G}$ on a finite-dimensional vector space $V$, there exists a basis of $V$ that is compatible with both ${\mathcal F}$ and ${\mathcal G}$.* *Proof.* Consider the index set: $$\begin{aligned} \Gamma=\{(i,j): {\mathcal F}^i\cap{\mathcal G}^j\ne 0, {\mathcal F}^{>i}\cap{\mathcal G}^j=0, {\mathcal F}^i\cap{\mathcal G}^{>j}=0\}. \end{aligned}$$ Then ${\rm gr}_{\mathcal F}^i{\rm gr}_{\mathcal G}^j V={\mathcal F}^i\cap{\mathcal G}^j$ for $(i,j)\in \Gamma$. We first show that $\Sigma_{(i,j)\in \Gamma}{\mathcal F}^i\cap{\mathcal G}^j$ is a direct sum. If there exists $0\ne s_k \in {\mathcal F}^{i_k}\cap{\mathcal G}^{j_k}, 0\le k\le K$ with $(i_k,j_k) \in \Gamma$, such that they are linearly dependent. We may assume that $i_0>i_1>\cdots >i_K$, then the definition of $\Gamma$ implies that $j_0<j_1<\cdots <j_K$ (This is the key step that holds for two filtrations, which will be lost for more filtrations). Hence $s_0=\sum_{k=1}^K a_ks_k \in {\mathcal F}^{i_0}\cap{\mathcal G}^{j_1}=0$, a contradiction. We denote by $\Gamma_1=\Gamma$ and ${\mathcal H}^{-1}=\oplus_{(i,j)\in \Gamma_1} {\mathcal F}^i\cap {\mathcal G}^j$. We define $\Gamma_k$ and ${\mathcal H}^{-k}$ for $k\ge2$ inductively: $$\begin{aligned} \Gamma_k=\{(i,j): {\mathcal F}^i\cap{\mathcal G}^j\nsubseteq {\mathcal H}^{-k+1}, {\mathcal F}^{>i}\cap{\mathcal G}^j + {\mathcal F}^i\cap{\mathcal G}^{>j}\subseteq{\mathcal H}^{-k+1}\}, \end{aligned}$$ $$\begin{aligned} {\mathcal H}^{-k}=\sum_{(i,j)\in\Gamma_k} {\mathcal F}^i\cap{\mathcal G}^j. \end{aligned}$$ With the same argument as in the previous paragraph, $$\begin{aligned} {\mathcal H}^{-k}/{\mathcal H}^{-k+1}=\oplus_{(i,j)\in\Gamma_k} {\mathcal F}^i\cap{\mathcal G}^j/{\mathcal H}^{-k+1}\end{aligned}$$ is a direct sum. Hence for each $k$ and $(i,j)\in\Gamma_k$, we may choose a basis of ${\mathcal F}^i\cap{\mathcal G}^j/{\mathcal H}^{-k+1}$ and lift it to $V$ as a set of vectors $\Omega_{i,j}$. Then $\cup \Omega_{i,j}$ is a basis of $V$, which is compatible with both ${\mathcal F}$ and ${\mathcal G}$. ◻ **Definition 9**. *[\[Definition: basis type divisor and compatible\]]{#Definition: basis type divisor and compatible label="Definition: basis type divisor and compatible"} Let $V\subseteq H^0(X, L)$ be a subspace for some line bundle $L$ on $X$. A *basis type divisor* $D$ of $V$ is of the form $D=\sum_i\{s_i=0\}$ where $\{s_i\}$ is a basis of $V$. And $D$ is said to be *compatible* with a filtration ${\mathcal F}$ on $V$ if the associated basis is.* By the above lemma, for any two filtrations ${\mathcal F}, {\mathcal G}$ on $V$, there always exists a basis type divisor $D$ of $V$ which is compatible with both ${\mathcal F}$ and ${\mathcal G}$. ## Test configurations Let $(X,\Delta)$ be a projective klt pair, and $L$ be a ${\mathbb Q}$-Cartier ample divisor. A *test configuration (TC)* of $(X,\Delta,L)$ is a collection $({\mathcal X}, \Delta_{\mathcal X},{\mathcal L},\eta)$ consisting of - A variety ${\mathcal X}$ with a ${\mathbb G}_m$-action generated by a holomorphic vector field $\eta\in {\rm Hom}({\mathbb G}_m, {\rm Aut}({\mathcal X}))$; - A ${\mathbb G}_m$-equivariant morphism $\pi: {\mathcal X}\to {\mathbb A}^1$, where the ${\mathbb G}_m$-action on ${\mathbb A}^1$ is standard; - A ${\mathbb G}_m$-equivariant $\pi$-semiample ${\mathbb Q}$-Cartier divisor ${\mathcal L}$ on ${\mathcal X}$; - A ${\mathbb G}_m$-equivariant trivialization over the punctured plane $i_\eta:({\mathcal X},{\mathcal L})|_{\pi^{-1}({\mathbb G}_m)}\cong (X,L)\times {\mathbb G}_m$, which is compatible with $\pi$ and ${\rm pr}_1$. And $\Delta_{\mathcal X}$ is the closure of $i_\eta^{-1}(\Delta\times{\mathbb G}_m)$ in ${\mathcal X}$. If ${\mathcal X}$ is a normal variety, then $({\mathcal X},\Delta_{\mathcal X},{\mathcal L},\eta)$ is called a *normal test configuration*. In the log Fano case, we always choose $L=-K_X-\Delta$, and a normal TC $({\mathcal X},\Delta_{\mathcal X},{\mathcal L},\eta)$ is called *(weakly) special* if $({\mathcal X},{\mathcal X}_0+\Delta_{\mathcal X})$ is (lc) plt, and ${\mathcal L}=-K_{{\mathcal X}/{\mathbb A}^1}-\Delta_{\mathcal X}+ c{\mathcal X}_0$ for some $c\in{\mathbb Q}$. Note by adjunction that $({\mathcal X}, \Delta_{\mathcal X},{\mathcal L})$ being special is equivalent that the central fiber $({\mathcal X}_0, -K_{{\mathcal X}_0}-\Delta_{{\mathcal X},0})$ is log Fano. ## Torus actions and moment polytopes {#Subsection:torus action} Let $(X, \Delta)$ be a log Fano pair of dimension $n$ with a ${\mathbb T}$-action. We say that the ${\mathbb T}$-action is of *complexity* $k$ if the ${\mathbb T}$-orbit of a general point in $X$ is of codimension $k$. Assume that $(X,\Delta)$ admits a ${\mathbb T}={\mathbb G}^r_m$-action ($r=n-k$). Then the ${\mathbb T}$-action has a canonical lifting to $R=R(X,\Delta)$, and we have a weight decomposition $R_m=\oplus_{\alpha\in M}R_{m,\alpha}$ where $M=M({\mathbb T})={\rm Hom}({\mathbb T},{\mathbb G}_m)$ is the weight lattice of the ${\mathbb T}$-action. Let $$\begin{aligned} \rm{P}\Gamma(R_\bullet) &:=& \{(m,\alpha)\in {\mathbb N}\times M: R_{m,\alpha}\ne0\}, \\ \mathbf{P}_m(R_\bullet) &:=& (\{m\}\times M)\cap\rm{P}\Gamma(R_\bullet), \\ \mathbf{P}(R_\bullet) &:=&(\{1\}\times M_{\mathbb R}) \cap {\rm Cone}(\rm{P}\Gamma(R_\bullet)) = \overline{\bigcup_{\it{m}} \frac{1}{\it{m}} \mathbf{P}_{\it{m}}(R_\bullet)}. \end{aligned}$$ The closed convex body $\mathbf{P}=\mathbf{P}(R_\bullet)\subseteq M_{\mathbb R}$ is called the *moment polytope* of the ${\mathbb T}$-action on $R=R_\bullet$. We also denote by $N=N({\mathbb T}) = {\rm Hom}({\mathbb G}_m, {\mathbb T})$ the lattice of one-parameter subgroup (1-PS), or the coweight lattice of the ${\mathbb T}$-action. Any $\xi \in N_{\mathbb R}$ can be viewed as a holomorphic vector field on $(X, \Delta)$. ## The $\xi$-twist of a valuation We shortly recall the $\xi$-twist of a valuation introduced by [@Li19]. This is a fundamental notion in the reduced uniform K-stability which is equivalent to the K-polystability by [@LXZ22]. Let $(X,\Delta)$ be a log Fano pair with a ${\mathbb T}$-action. Then there exists a ${\mathbb T}$-equivariant dominant birational map $\pi: X \dashrightarrow Z$, where $Z$ is the Chow quotient of $X$ and ${\mathbb T}$ acts on $Z$ trivially. The function field $K(X)$ of $X$ is the fractional field of $K(Z)[M] = \oplus_{\alpha \in M} K(Z) \cdot 1^\alpha$. For any valuation $\mu$ on $Z$ and $\xi \in N_{\mathbb R}$ we define the ${\mathbb T}$-invariant valuation $v_{\mu, \xi}$ on $X$ such that $$v_{\mu, \xi}(f) = {\rm min}_\alpha\{\mu(f_\alpha)+\langle\alpha, \xi \rangle\},$$ for any $f=\sum_\alpha f_\alpha \cdot 1^\alpha \in K(Z)[M]$. By [@BHJ17] we know that any ${\mathbb T}$-invariant valuation on $X$ is obtained in this way, and we get a non-canonical isomorphism ${\rm Val}^{\mathbb T}_X \cong {\rm Val}_Z \times N_{\mathbb R}$. For any $v = v_{\mu, \xi_0}\in {\rm Val}^{\mathbb T}_X$ and $\xi \in N_{\mathbb R}$, we define the $\xi$-twist of $v$ by $v_\xi:= v_{\mu,\xi_0+\xi}$. One can check that the definition is not dependent on the choice of the birational map $X\dashrightarrow Z$. If $\mu$ is the trivial valuation, then we denote by ${\rm wt}_\xi = v_{\mu, \xi}$. ## Multi-graded linear series Fix integers $0\le l < n$. Let $X_l$ be a projective variety of dimension $n-l$, and $L,L_1, \cdots, L_l$ be a sequence of line bundles on $X_l$. A *${\mathbb N}\times {\mathbb N}^l$-graded linear series* $W_\bullet$ associated to those $L_i$ is a collection of finite dimensional subspaces $$\begin{aligned} W_{m,\beta} \subseteq H^0(X_l, mL+\beta_1L_1+\cdots+\beta_lL_l), \end{aligned}$$ for $(m,\beta)=(m,\beta_1,\cdots,\beta_l)\in {\mathbb N}\times {\mathbb N}^l$ such that $W_0 = {\mathbb C}$ and $W_{m,\beta}\cdot W_{m',\beta'} \subseteq W_{m+m',\beta+\beta'}$. For convenience we may write $W_{m,\bullet}=\oplus_{\beta}W_{m,\beta}$ and use $W_{(1,\beta)}$ to denote the graded linear series $\{W_{m(1,\beta)}=W_{m,m\beta}\}_m$. We define the semigroup $\rm{Q}\Gamma(W_\bullet)=\{(\it{m},\beta)\in {\mathbb N}\times{\mathbb N}^l: W_{\it{m},\beta} \ne 0\}$ and the $m$-slicing $\mathbf{Q}_m(W_\bullet)=(\{m\}\times {\mathbb N}^l)\cap\rm{Q}\Gamma(W_\bullet)$. The *base* of $W_\bullet$ is defined by $\mathbf{Q}(W_\bullet)=(\{1\}\times {\mathbb R}^l) \cap {\rm Cone}(\rm{Q}\Gamma(W_\bullet))$. We say that $W_\bullet$ has *bounded support* if $\mathbf{Q}(W_\bullet)$ is bounded. The series $W_\bullet$ *contains an ample series* if there exists $(1,\beta)\in{\mathbb Q}\times{\mathbb Q}^l$ in the interior of $\mathbf{Q}=\mathbf{Q}(W_\bullet)$ such that $L+\beta_1L_1+\cdots+\beta_lL_l = A+E$, where $A$ is an ample ${\mathbb Q}$-divisor and $E$ is an effective ${\mathbb Q}$-divisor, and $H^0(X_l,mA)\subseteq W_{m(1,\beta)}$ for sufficiently divisible $m$. We use the following definition that appeared in [@Xu23]. We say that $W_\bullet$ is *asymptotically complete* with respect to line bundles $L,L_1,\cdots,L_l$ if $$\begin{aligned} \mathop{{\rm lim}}_{m\to \infty} \frac{h^0(W_{m(1,\beta_1,\cdots,\beta_l)})}{h^0(X_l,mL+m_1\beta_1L_1+\cdots+m_l\beta_lL_l)} = 1, \end{aligned}$$ for all $\beta=(\beta_1,\cdots,\beta_l) \in {\rm int}(\mathbf{Q})_{\mathbb Q}$, that is, ${\mathbb Q}$-point in the interior of $\mathbf{Q}$. In the following, we will always assume that the multi-graded linear series $W_\bullet$ has bounded support and contains an ample series. The volume of $W_\bullet$ is defined as ${\rm vol}(W_\bullet)={\rm lim\,sup}_{m\to \infty}\frac{h^0(W_{m,\bullet})}{m^n/n!}.$ We also define the fiberwise volume ${\rm vol}(W_{(1,\beta)})={\rm lim\,sup}_{m\to \infty}\frac{h^0(W_{m(1,\beta)})}{m^{n-l}/(n-l)!}$ for any $\beta\in{\rm int}(\mathbf{Q})_{\mathbb Q}$. A filtration ${\mathcal F}$ on $W_\bullet$ is a descending sequence of subspaces $\{{\mathcal F}^\lambda W_{m,\beta}\}_{\lambda\in{\mathbb R}}$, which is bounded, left-continuous and multiplicative (see [2.3](#Subsection: filtration){reference-type="ref" reference="Subsection: filtration"}). We say that a filtration ${\mathcal F}$ is linearly bounded if there exists $C>0$ such that ${\mathcal F}^{-Cm}W_{m,\beta}=W_{m,\beta}$ and ${\mathcal F}^{Cm}W_{m,\beta}=0$ for all $m,\beta$. We usually use the Abban-Zhuang estimate succeedingly to deduce a lower bound of the weighted stability threshold of a $n$-dimensional Fano variety $X$. In the first step, we will take refinement of the anti-canonical ring $R$ of $X$, and the dimension of the Okounkov body of the multi-graded linear series in each step does not change. So we should consider the ${\mathbb N}\times{\mathbb N}^l$-graded linear series on a $(n-l)$-dimensional variety $X_l$ as the $l$-th step in the program. ## Okounkov bodies Equip ${\mathbb Z}^{n-l}$ with the lexicographic ordering. A valuation $\mathfrak{v}:K(X_l)^\times\to {\mathbb Z}^{n-l}$ is *faithful* if it is surjective. By [@LM09], one may pick an admissible flag over $X_l$ to define a faithful valuation on $X_l$. For a ${\mathbb N}\times{\mathbb N}^l$-graded linear series $W_\bullet$ on $X_l$ we define the semigroup $$\begin{aligned} \Gamma(W_\bullet)=\{(m,\beta,\mathfrak{v}(s))\in {\mathbb N}\times{\mathbb N}^l\times {\mathbb N}^{n-l}: 0\ne s\in W_{m,\beta}\}, \end{aligned}$$ and the $m$-slicing $\mathbf{O}_m(W_\bullet) = \Gamma(W_\bullet) \cap (\{m\} \times {\mathbb N}^n)=\mathfrak{v}(W_{m,\bullet}\setminus \{0\})$. We denote the natural projection by $q:\mathbf{O}_m\to \mathbf{Q}_m$. We know that $\sharp \Gamma_m(W_\bullet) = \dim W_{m,\bullet}$ (see, for example, [@Xu23 Lemma 1.6]). The *Okounkov body* of $W_\bullet$ is defined by $\mathbf{O}(W_\bullet)= (\{1\} \times {\mathbb R}^n)\cap {\rm Cone}(\Gamma(W_\bullet))$. In other word, $$\begin{aligned} \mathbf{O}(W_\bullet)=\overline{\bigcup_{m\in {\mathbb N}}\frac{1}{m}\mathbf{O}_m(W_\bullet)}. \end{aligned}$$ Let ${\mathcal F}$ be a linearly bounded filtration on $W_\bullet$. Then for each $t \in {\mathbb R}$ we have a ${\mathbb N}\times{\mathbb N}^l$-graded linear series ${\mathcal F}^{(t)} W_\bullet$ defined by $({\mathcal F}^{(t)} W)_{m,\beta}={\mathcal F}^{mt}W_{m,\beta}$. Note that ${\mathcal F}^{(t)} W_\bullet$ is linearly bounded and contains an ample series since $W_\bullet$ does. We denote the Okounkov body of ${\mathcal F}^{(t)} W_\bullet$ by $\mathbf{O}^{(t)}$, and let $\mathbf{O}=\mathbf{O}(W_\bullet)$. Then $\mathbf{O}^{(t)}\subseteq\mathbf{O}$ is a descending collection of convex bodies. The linear boundedness of ${\mathcal F}$ guarantees that $\mathbf{O}^{(-C)}=\mathbf{O}$ and $\mathbf{O}^{(C)}=0$. The *concave transform* of ${\mathcal F}$ is the function on $\mathbf{O}$ defined by $G^{\mathcal F}(y)={\rm sup}\{t: y\in\mathbf{O}^{(t)}\}$ (see [@BC11; @HL20]). Note that $G^{\mathcal F}$ is concave and continuous. If $(\beta,\gamma) \in \mathbf{O}_{\mathbb Q}$, then $$G^{\mathcal F}(\beta,\gamma)={\rm sup}\Big\{\frac{{\rm ord}_{\mathcal F}(s)}{m}: \frac{\mathfrak{v}(s)}{m} = \gamma, \exists m\in {\mathbb N},\,\exists s \in W_{m,m\beta} \Big\}.$$ ## Multi-graded linear series with a torus action {#Subsection: multi-graded linear series with a torus action} Let $X_l$ be a $(n-l)$-dimensional projective variety with a ${\mathbb T}={\mathbb G}^r_m$-action, and $V_\bullet$ be a ${\mathbb T}$-invarant ${\mathbb N}\times{\mathbb N}^l$-graded linear series on $X_l$, that is, $V_{m,\beta}$ is ${\mathbb T}$-invariant and $g(s_1s_2)=g(s_1)g(s_2)$ for any $g\in{\mathbb T}, s_i\in V_\bullet$. Hence each $V_{m,\beta}$ admits a weight decomposition $V_{m,\beta} = \oplus_{\alpha\in M} V_{m,\alpha,\beta}$, where the direct sum is taken over $\mathbf{P}_{m,\beta}=\{\alpha\in M: V_{m,\alpha,\beta}\ne 0\}$. We will write $V_{m,\bullet,\beta}$ instead of $V_{m,\beta}$. Hence $V_m=\oplus_{\beta\in \mathbf{Q}_m} V_{m,\bullet,\beta} =\oplus_{\beta\in \mathbf{Q}_m,\alpha\in \mathbf{P}_{m,\beta}} V_{m,\alpha,\beta}$. Let $\mathbf{P}_{m}=\cup_{\beta\in \mathbf{Q}_m} \mathbf{P}_{m,\beta}\subseteq M$. Then the moment polytope of the ${\mathbb T}$-action on $V_\bullet$ is $\mathbf{P}=\overline{\cup_m\frac{1}{m}\mathbf{P}_m}\subseteq M_{\mathbb R}$. For $\beta \in {\rm int}(\mathbf{Q})_{\mathbb Q}$, we also have the moment polytope $\mathbf{P}_{(1,\beta)}=\overline{\cup_m\frac{1}{m}\mathbf{P}_{m,m\beta}}\subseteq M_{\mathbb R}$ of the graded linear series $V_{(1,\bullet,\beta)}$, which is a slicing of $\mathbf{P}$. We also set $\mathbf{Q}_{m,\alpha}:=\{\beta\in{\mathbb N}^l:V_{m,\alpha,\beta}\ne 0\}$ and $V_{m,\alpha,\bullet}=\oplus_{\beta\in\mathbf{Q}_{m,\alpha}} V_{m,\alpha,\beta}$. So $V_m=\oplus_{\alpha\in \mathbf{P}_m} V_{m,\alpha,\bullet}$ is the weight decomposition of $V_\bullet$. **Remark 10**. *The ${\mathbb T}$-action on $V_\bullet$ is always assumed to be faithful. But the ${\mathbb T}$-action on $X$ may have a non-trivial stabilizer at general points. See Section [7](#Section: Applications){reference-type="ref" reference="Section: Applications"} for examples.* *We also remark that $\alpha$ and $\beta$ may not be independent, see Example [\[Example: formula when Q=P\]](#Example: formula when Q=P){reference-type="ref" reference="Example: formula when Q=P"}.* ## Refinements {#Subsection: Refinements and toric filtrations} For any ${\mathbb T}$-invariant ${\mathbb Z}$-filtration ${\mathcal F}$ on $V_\bullet$, we define the *refinement* of $V_\bullet$ along ${\mathcal F}$ by $W_\bullet:={\rm gr}_{\mathcal F}V_\bullet$, which is also ${\mathbb T}$-invariant. More precisely, $W_{m,\bullet,\beta, j}={\mathcal F}^jV_{m,\bullet,\beta}/{\mathcal F}^{j+1}V_{m,\bullet,\beta}$. For the weight decomposition, there are two different cases (1). The filtration ${\mathcal F}$ is horizontal (or toric), which means that each ${\mathcal F}^\lambda V_{m,\bullet,\beta}$ is equal to the sum of some $V_{m,\alpha,\beta}$. Up to shift ${\mathcal F}$, there exists a piecewise linear function $\Phi$ on ${\rm Cone}(\rm{P}\Gamma)$ such that ${\rm ord}_{\mathcal F}(s)=\Phi(\alpha)$ for any $s \in V_{m,\alpha,\bullet}\setminus \{0\}$. Hence $${\mathcal F}^j V_{m,\bullet,\beta} = \oplus_{\alpha\in \mathbf{P}_{m,\beta},\Phi(\alpha)\ge j} V_{m,\alpha,\beta}.$$ For $\alpha\in \mathbf{P}_{m,\beta}$ with $\Phi(\alpha)= j$, we denote by $$W_{m,\alpha,\beta,j}=V_{m,\alpha,\beta}/{\mathcal F}^{j+1}V_{m,\bullet,\beta} \subseteq W_{m,\beta, j},$$ (we may omit $j$ and write $W_{m,\alpha,\beta}$ instead of $W_{m,\alpha,\beta,j}$ if the multi-graded structure is not important). Hence $W_{m,\bullet,\beta, j} =\oplus_{\alpha\in \mathbf{P}_{m,\beta,j}(W_\bullet)} W_{m,\alpha,\beta,j}$ where $\mathbf{P}_{m,\beta,j}(W_\bullet)=\{\alpha\in \mathbf{P}_{m,\beta}:\Phi(\alpha)= j\}\subseteq\mathbf{P}_{m,\beta}(V_\bullet)$. On the other hand, $\mathbf{Q}_{m,\alpha}(W_\bullet) = \{(\beta,\Phi(\alpha))\in {\mathbb N}^{l+1}: \beta \in \mathbf{Q}_{m,\alpha}\} = \mathbf{Q}_{m,\alpha}(V_\bullet)\times \{\Phi(\alpha)\}\subseteq{\mathbb N}^{l+1}$. In conclusion $$\begin{aligned} \mathbf{P}(W_\bullet) &=&\overline{\bigcup_{m,\beta,j}\frac{1}{m} \mathbf{P}_{m,\beta,j}(W_\bullet)} =\overline{\bigcup_{m,\beta}\frac{1}{m} \mathbf{P}_{m,\beta}(V_\bullet)} =\mathbf{P}(V_\bullet), \\ \mathbf{Q}(W_\bullet) &=&\overline{\bigcup_{m,\alpha}\frac{1}{m} \mathbf{Q}_{m,\alpha}(W_\bullet)}\to \mathbf{Q}(V_\bullet), \quad (\beta, j) \mapsto \beta. \end{aligned}$$ The moment polytope does not change and the base polytope admits a one-dimensional fibration over the origional base polytope, whose fiber over $\beta \in \mathbf{Q}(V_\bullet)$ is $\Phi(\overline{\cup_m\frac{1}{m}\mathbf{P}_{m,\beta}})$. This process should be considered as a refinement of the ${\mathbb T}$-action, which means that it takes some information of the ${\mathbb T}$-action into the multi-graded structure. (2). The filtration ${\mathcal F}$ is vertical, that is, not horizontal. Then $W_{m,\alpha,\beta, j}={\mathcal F}^jV_{m,\alpha,\beta}/{\mathcal F}^{j+1}V_{m,\alpha,\beta}$. We also have $\cup_j\mathbf{P}_{m,\beta,j}(W_\bullet)=\mathbf{P}_{m,\beta}(V_\bullet)$. If $V_{m,\alpha,\beta}\ne0$, then there exists $j\in{\mathbb Z}$ such that $W_{m,\alpha,\beta,j}\ne 0$. We have a surjective map $\mathbf{Q}_{m,\alpha}(W_\bullet)\to \mathbf{Q}_{m,\alpha}(V_\bullet)$. Hence the moment polytope does not change, and the base polytope admits a one-dimensional fibration over the original base polytope $\mathbf{Q}(W_\bullet) \to \mathbf{Q}(V_\bullet)$ whose fiber over $\beta$ is $[\lambda^{\mathcal F}_{\rm min}(V_{(1,\beta)}),\lambda^{\mathcal F}_{\rm max}(V_{(1,\beta)})]\subseteq{\mathbb R}$, where $\lambda_{\rm min}^{\mathcal F}(V_\bullet) = {\rm inf}\{t\in{\mathbb R}: {\rm vol}({\mathcal F}^{(t)}V_\bullet) < {\rm vol}(V_\bullet)\}$ and $\lambda_{\rm max}^{\mathcal F}(V_\bullet)= {\rm inf}\{t\in{\mathbb R}: {\rm vol}({\mathcal F}^{(t)}V_\bullet) > 0\}$ for any graded linear series $V_\bullet$ and filtration ${\mathcal F}$. This process should be considered as an ordinary refinement, which takes the information of the linear series into the multi-graded structure. Using this program inductively, one can construct an Okounkov body. **Remark 11**. *The anti-canonical ring $R$ of a $n$-dimensional log Fano pair $(X,\Delta)$ with a ${\mathbb T}={\mathbb G}^r_m$-action admits a weight decomposition $R_m=\oplus_{\alpha}R_{m,\alpha}$, which should be viewed as a ${\mathbb N}\times{\mathbb N}^r$-graded linear series on a $(n-r)$-dimensional ${\mathbb T}$-invariant subvariety $Z$ over $X$ on which the ${\mathbb T}$-action is trivial. Such a subvariety $Z$ will be obtained by taking refinements by toric divisors succeedingly.* # Weighted K-stability {#Section: Weighted K-stability} In this section, we define the invariants about the *$g$-weighted K-stability*. The invariants are almost the same as those of the ordinary K-stability. The only difference is that we replace the measures in the integrals defining the invariants by the *$g$-weighted measures*. Let $(X, \Delta)$ be a log Fano pair with a ${\mathbb T}$-action. Then there is a canonical lifting of the ${\mathbb T}$-action on the anti-canonical ring $R_\bullet=R(X,\Delta)$. We denote the $1$-PS lattice by $N=N({\mathbb T})$ and the weight lattice by $M=M({\mathbb T})$. Let $\mathbf{P}\subseteq M_{\mathbb R}$ be the moment polytope of the ${\mathbb T}$-action on $R_\bullet$, and $\mathbf{O}\subseteq{\mathbb R}^n$ be the Okounkov body of $-(K_X+\Delta)$ which is compatible with the ${\mathbb T}$-action. Then there is a linear projection $p:{\mathbb R}^n\to M_{\mathbb R}$ mapping $\mathbf{O}$ onto $\mathbf{P}$. The ${\mathbb T}$-action on $R_\bullet$ gives the weight decomposition $R_m=\oplus_{\alpha \in M\cap m\mathbf{P}} R_{m,\alpha}$. ## Soliton candidates {#Subsection: soliton candidate} In many examples, the main reason that leads to a Fano manifold being K-unstable is that there exists a product test configuration destabilizing it. It's natural to ask how can we remove the defect of such a test configuration. The *modified Futaki invariant* ${\rm Fut}_g$ introduced by [@TZ02] is the natural candidate to give an affirmative answer to this question. We state an algebraic definition of ${\rm Fut}_g$ here. Let $\xi_0\in N_{\mathbb R}$ be a holomorphic vector field on $X$, and $g(\alpha)=e^{-\langle\alpha, \xi_0 \rangle}$ for $\alpha\in\mathbf{P}$. Then $${\rm Fut}_g(\xi)=-\frac{1}{\mathbf{v}^g} \int_\mathbf{P}\langle\alpha, \xi\rangle g(\alpha) {\rm DH}_\mathbf{P}({\rm d}\alpha), \,\,\xi\in N_{\mathbb R}$$ where $\mathbf{v}^g=\int_\mathbf{P}g(\alpha) {\rm DH}_\mathbf{P}({\rm d}\alpha)$ is the $g$-weighted volume of $-(K_X+\Delta)$ and ${\rm DH}_\mathbf{P}$ is the Duistermaat-Heckman measure of $-(K_X+\Delta)$ on $\mathbf{P}$ (see Section [3.3](#Subsection: Sg via Okounkov body){reference-type="ref" reference="Subsection: Sg via Okounkov body"}). Note that ${\rm Fut}_g$ is a rescaling of the first order derivative of the following function at $\xi=\xi_0$, $$H(\xi)= \int_\mathbf{P}e^{-\langle\alpha,\xi\rangle} {\rm DH}_\mathbf{P}({\rm d}\alpha), \,\,\xi\in N_{\mathbb R},$$ which is strictly convex function on $N_{\mathbb R}$ and tends to $+\infty$ when $|\xi|\to +\infty$. Hence there exists a unique holomorphic vector field minimizing $H$. We call such a vector field the *soliton candidate* of $(X,\Delta)$. If $(X,\Delta)$ is K-semistable, then $\xi_0=0$ and ${\rm Fut}_g$ is the ordinary Futaki invariant. In the following, we will choose $\xi_0$ to be the soliton candidate. So ${\rm Fut}_g(\xi)=0$ for all $\xi\in N_{\mathbb R}$. ## Weighted expected vanishing order via DH measures Let ${\mathcal F}$ be a ${\mathbb T}$-invariant linearly bounded filtration on $R_\bullet$. We simply denote by $$\begin{aligned} N_m &=& \dim R_m, \\ N_{m,\alpha} &=& \dim R_{m,\alpha}, \\ N^\lambda_m &=& \dim {\mathcal F}^\lambda R_m - \dim {\mathcal F}^{>\lambda} R_m \\ N^\lambda_{m,\alpha} &=& \dim {\mathcal F}^\lambda R_{m,\alpha} - \dim {\mathcal F}^{>\lambda} R_{m,\alpha}. \end{aligned}$$ Then we define the asymptotic invariant $$\begin{aligned} S^g_m(R_\bullet; {\mathcal F}) &:=& \frac{1}{N^g_m} \sum_{\lambda\in{\mathbb R}} \frac{\lambda}{m} \sum_{\alpha\in \mathbf{P}_m} g(\frac{\alpha}{m}) N^\lambda_{m,\alpha}, \end{aligned}$$ where $N^g_m := \sum_{\alpha\in \mathbf{P}_m} g(\frac{\alpha}{m}) N_{m,\alpha} =: m^n \mathbf{v}^g_m$. One may choose a basis $\{s_j\}_{j=1}^{N_m}$ of $R_m$ which is compatible with the weight decomposition and the filtration ${\mathcal F}$ (exists since ${\mathcal F}$ is ${\mathbb T}$-invariant) with $\lambda_j={\rm ord}_{\mathcal F}(s_j)$ such that $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_{N_m}.$ The collection $\{\lambda_j\}$ is called the *($m$-th) successive minima* of ${\mathcal F}$. We also set $\alpha_j=\alpha$ if $s_j\in R_{m,\alpha}$. Note that $\{\alpha_j\}\subseteq M, \{\lambda_j\}\subseteq{\mathbb R}$ are both independent of the compatible basis. Then $$\begin{aligned} S^g_m(R_\bullet; {\mathcal F}) &=& \frac{1}{N^g_m} \sum_{j} \frac{\lambda_j}{m}\cdot g(\frac{\alpha_j}{m}). \end{aligned}$$ Consider the following discrete measure on ${\mathbb R}$ determined by ${\mathcal F}$ $$\begin{aligned} {\rm DH}^g_{{\mathcal F},m} &=& \frac{1}{m^n} \sum_{\lambda\in{\mathbb R}} \sum_{\alpha\in \mathbf{P}_m} g(\frac{\alpha}{m}) N^\lambda_{m,\alpha} \cdot \delta_{\frac{\lambda}{m}}. \end{aligned}$$ Hence we have $\mathbf{v}^g_m=\int_{\mathbb R}{\rm DH}^g_{{\mathcal F},m}$ and $$\begin{aligned} S^g_m(R_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g_m} \int_{\mathbb R}t \cdot {\rm DH}^g_{{\mathcal F}, m}({\rm d}t). \end{aligned}$$ By [@BJ20 Lemma 2.8] we see that ${\rm DH}^g_{{\mathcal F},m}$ converges weakly to the *Duistermaat-Heckman (DH) measure* ${\rm DH}^g_{\mathcal F}$ on ${\mathbb R}$. We define the *$g$-weighted volume of $R_\bullet$* by $\mathbf{v}^g := \int_{\mathbb R}{\rm DH}^g_{\mathcal F}$, and the *$g$-weighted expected vanishing order* of ${\mathcal F}$ by $$\begin{aligned} S^g(R_\bullet; {\mathcal F}) &:=& \frac{1}{\mathbf{v}^g} \int_{\mathbb R}t \cdot {\rm DH}^g_{\mathcal F}({\rm d}t). \end{aligned}$$ We also denote it by $S^g(R;E)$ or $S^g(R;v)$ if ${\mathcal F}$ is induced by a prime divisor $E$ or a valuation $v$. It's clearly that $\mathbf{v}^g_m$ and $S^g_m(R_\bullet; {\mathcal F})$ converge to $\mathbf{v}^g, S^g(R_\bullet; {\mathcal F})$ respectively. **Remark 12**. *Our definition of ${\rm DH}^g_{\mathcal F}$, hence the $g$-weighted volume of $R_\bullet$, are $(n!)^{-1}$ multiple of those in the literature. This is convenient for us since the $g$-weighted volumes of $R_\bullet$ and it's Okounkov body are the same under this definition. We will not use the notion *$g$-weighted volume of $R_\bullet$* but use the $g$-weighted volume of the Okounkov body in the following sections.* ## Weighted expected vanishing order via Okounkov bodies {#Subsection: Sg via Okounkov body} We define the following discrete measures on the Okounkov body $\mathbf{O}$ and the moment polytope $\mathbf{P}$ respectively $$\begin{aligned} {\rm LE}_m &=&\frac{1}{m^n}\sum_{\gamma\in \mathbf{O}_m} \delta_\frac{\gamma}{m}, \\ {\rm DH}_{\mathbf{P},m} &=&\frac{1}{m^n}\sum_{\alpha\in \mathbf{P}_m} N_{m,\alpha} \cdot \delta_\frac{\alpha}{m}.\end{aligned}$$ It's clear that ${\rm DH}_{\mathbf{P},m}=p_*{\rm LE}_m$, and ${\rm LE}_m$ converges weakly to the Lebesgue measure ${\rm LE}$ on $\mathbf{O}$. Hence ${\rm DH}_{\mathbf{P},m}$ converges weakly to the DH measure ${\rm DH}_\mathbf{P}=p_*{\rm LE}$ on $\mathbf{P}$. We define the following *$g$-weighted measures* on $\mathbf{O}$ and $\mathbf{P}$ $$\begin{aligned} {\rm LE}^g=g\circ p\cdot {\rm LE},\quad {\rm DH}^g_\mathbf{P}=g\cdot {\rm DH}_\mathbf{P}, \end{aligned}$$ respectively. Then ${\rm DH}^g_\mathbf{P}=p_*{\rm LE}^g$. We similarly define ${\rm LE}^g_m=g\circ p\cdot {\rm LE}_m$ and ${\rm DH}^g_{\mathbf{P},m}=g\cdot {\rm DH}_{\mathbf{P},m}$. They converge weakly to ${\rm LE}^g$ and ${\rm DH}^g_\mathbf{P}$ respectively. Let $\mathbf{O}' \subseteq\mathbf{O}$ be a convex body. The *$g$-weighted volume* of $\mathbf{O}'$ is ${\rm vol}^g(\mathbf{O}'):=\int_{\mathbf{O}'}{\rm LE}^g$. Clearly we have $\int_\mathbf{O}{\rm LE}^g_m = N^g_m/m^n=\mathbf{v}^g_m$, and the $g$-weighted volume of $\mathbf{O}$ is just $$\begin{aligned} \mathbf{v}^g={\rm vol}^g(\mathbf{O}) =\int_{\mathbf{O}}{\rm LE}^g =\int_{\mathbf{P}}{\rm DH}^g_\mathbf{P}. \end{aligned}$$ For any ${\mathbb T}$-invarant linearly bounded filtration ${\mathcal F}$ on $R=R_\bullet$, let $G^{\mathcal F}$ be the concave transform of the filtration ${\mathcal F}$. One may show that $$\begin{aligned} {\rm DH}^g_{\mathcal F}({\rm d}t) &=& -{\rm d}{\rm vol}^g\{G^{\mathcal F}\ge t\}, \end{aligned}$$ where $\{G^{\mathcal F}\ge t\}\subseteq\mathbf{O}$ is the Okounkov body of the graded linear series ${\mathcal F}^{(t)}R_\bullet$. Integrating by part, we get $S^g(R_\bullet; {\mathcal F}) = \frac{1}{\mathbf{v}^g} \int_{\mathbb R} {\rm vol}^g\{G^{\mathcal F}\ge t\}{\rm d}t$. Hence $$\begin{aligned} S^g(R_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g} \int_\mathbf{O}G^{\mathcal F}\cdot {\rm LE}^g. \end{aligned}$$ If ${\mathcal F}$ is toric (see Section [2.10](#Subsection: Refinements and toric filtrations){reference-type="ref" reference="Subsection: Refinements and toric filtrations"}), then $G^{\mathcal F}$ is invariant on each fiber of $p$, hence descends to a function on $\mathbf{P}$ (still denoted by $G^{\mathcal F}$). Hence $$\begin{aligned} S^g(R_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g} \int_\mathbf{P}G^{\mathcal F} \cdot {\rm DH}^g_{\mathbf{P}}. \end{aligned}$$ With this formualtion of $S^g$, we may also define the asymptotic invariants $\tilde{S}^g_m$ analogous to $S^g_m$ by $$\begin{aligned} \tilde{S}^g_m(R_\bullet; {\mathcal F}) \,\,:=\,\, \frac{1}{\mathbf{v}^g_m} \int_\mathbf{O}G^{\mathcal F}\cdot{\rm LE}^g_m \,\,=\,\, \frac{1}{N^g_m} \sum_{j} G^{\mathcal F}(\frac{\mathfrak{v}(s_j)}{m}) \cdot g(\frac{\alpha_j}{m}). \end{aligned}$$ By definition of $G^{\mathcal F}$, we have $\frac{{\rm ord}_{\mathcal F}(s_j)}{m} \le G^{\mathcal F}(\frac{\mathfrak{v}(s_j)}{m})$. Hence $S^g_m(R_\bullet; {\mathcal F})\le \tilde{S}^g_m(R_\bullet; {\mathcal F})$. We have the following analog of [@BJ20 Corollary 2.10], see also [@BLXZ23]. For the convenience of the reader, we state a proof in the appendix. **Lemma 13**. *For any $\varepsilon>0$ there exists $m_0\in{\mathbb N}$ such that $S^g_m(R_\bullet; {\mathcal F}) \le (1+\varepsilon)S^g(R_\bullet; {\mathcal F})$ for any $m\ge m_0$ and any linearly bounded filtration ${\mathcal F}$ on $R_\bullet$.* ## Weighted stability thresholds We define $$\begin{aligned} \delta^g_{{\mathbb T},m}(X,\Delta) \,\,:=\,\, {\rm inf}_v \frac{A_{X,\Delta}(v)}{S^g_m(R_\bullet;v)}, \quad \delta^g_{\mathbb T}(X,\Delta) \,\,:=\,\, {\rm inf}_v \frac{A_{X,\Delta}(v)}{S^g(R_\bullet;v)},\end{aligned}$$ where the infimum runs over all the valuations $v\in {\rm Val}^{{\mathbb T},\circ}_X$. By Lemma [Lemma 13](#Lemma: uniform bound of S^g_m/S^g){reference-type="ref" reference="Lemma: uniform bound of S^g_m/S^g"} we have **Lemma 14**. *$\mathop{{\rm lim}}_{m\to \infty}\delta^g_{{\mathbb T},m}(X,\Delta) = \delta^g_{\mathbb T}(X,\Delta).$* **Definition 15**. *[\[Definition: T-equivariantly weighted K-semistable\]]{#Definition: T-equivariantly weighted K-semistable label="Definition: T-equivariantly weighted K-semistable"} A log Fano triple $(X,\Delta, \xi_0)$ with a ${\mathbb T}$-action is called *${\mathbb T}$-equivariantly $g$-weighted K-semistable* if $\mathbf{D}^g({\mathcal F}) = \mu({\mathcal F}) - S^g(R_\bullet;{\mathcal F}) \ge 0$ for any ${\mathbb T}$-invariant linearly bounded filtration ${\mathcal F}$ on $R_\bullet=R(X,\Delta)$, where $\mu({\mathcal F})$ is the *slope* of ${\mathcal F}$ defined in [@XZ19]. It is *${\mathbb T}$-equivariantly $g$-weighted K-polystable* if for any ${\mathbb T}$-equivariantly special test configuration $({\mathcal X}, \Delta_{\mathcal X})$ of it with $g$-weighted K-semistable central fiber $({\mathcal X}_0, \Delta_{{\mathcal X},0}, \xi_0)$, then $({\mathcal X}, \Delta_{\mathcal X})$ is a product test configuration, see [@LWX15; @XZ19; @HL20].* **Remark 16**. *Whenever we talk about the weighted K-semistability of a log Fano triple $(X,\Delta,\xi_0)$, a torus action containing the soliton candidate $\xi_0$ is fixed. We shall omit the words "${\mathbb T}$-equivariantly" if the ${\mathbb T}$-action is clear.* We have the following valuative criterion of $g$-weighted K-semistability, see [@HL23]. **Theorem 17**. *Let $(X, \Delta)$ be a log Fano pair with a ${\mathbb T}$-action, $\xi\in N({\mathbb T})_{\mathbb R}$ and $g(\alpha)=e^{-\langle\alpha,\xi\rangle}$. Then $(X, \Delta, \xi)$ is ${\mathbb T}$-equivariantly $g$-weighted $K$-semistable if and only if $\delta^g_{{\mathbb T}}(X,\Delta)\ge 1$.* # Invariants associated with multi-graded linear series {#Section: Invariants associated to multi-graded linear series} In this section, We deal with the invariants of weighted K-stability associated with multi-graded linear series, which will appear in the weighted Abban-Zhuang estimate. They are the straightforward generalizations of the invariants in the previous section. We mainly focus on the various formulations which are useful in explicit computations. ## Weighted expected vanishing order We work with the same assumption as Section [2.9](#Subsection: multi-graded linear series with a torus action){reference-type="ref" reference="Subsection: multi-graded linear series with a torus action"}. Let $X_l$ be a $(n-l)$-dimensional projective variety with a ${\mathbb T}={\mathbb G}^r_m$-action, and $V_\bullet$ be a ${\mathbb T}$-invarant ${\mathbb N}\times{\mathbb N}^l$-graded linear series on $X_l$. We simply denote by $N_*$ the dimension of $V_*$ where $*=m, (m,\alpha,\bullet), (m,\bullet,\beta)$ or $(m,\alpha,\beta)$. Let ${\mathcal F}$ be a ${\mathbb T}$-invariant linearly bounded filtration on $V_\bullet$. We also denote by $N^\lambda_*$ the dimension of ${\rm gr}^\lambda_{\mathcal F}V_*$ for $*$ as above. Then the definition of $S^g_m(V_\bullet;{\mathcal F})$ follows $$\begin{aligned} S^g_m(R_\bullet; {\mathcal F}) &:=& \frac{1}{\mathbf{v}^g_m} \int_{\mathbb R}t \cdot {\rm DH}^g_{{\mathcal F}, m}({\rm d}t), \end{aligned}$$ where $\mathbf{v}^g_m = \int_{\mathbb R}{\rm DH}^g_{{\mathcal F}, m}({\rm d}t)$ and $$\begin{aligned} {\rm DH}^g_{{\mathcal F},m} &:=& \frac{1}{m^n} \sum_{\alpha\in \mathbf{P}_m} \sum_{\lambda\in{\mathbb R}} g(\frac{\alpha}{m}) N^\lambda_{m,\alpha,\bullet} \cdot \delta_{\frac{\lambda}{m}}. \end{aligned}$$ Similarly ${\rm DH}^g_{{\mathcal F},m}$ converges weakly to the DH measure ${\rm DH}^g_{\mathcal F}$ on ${\mathbb R}$. We define $\mathbf{v}^g={\rm lim}_{m\to \infty} \mathbf{v}^g_m$ and $$\begin{aligned} S^g(R_\bullet; {\mathcal F}) &:=& \frac{1}{\mathbf{v}^g} \int_{\mathbb R}t \cdot {\rm DH}^g_{{\mathcal F}}({\rm d}t), \end{aligned}$$ Let $\mathbf{O}\subseteq{\mathbb R}^n, \mathbf{Q}\subseteq{\mathbb R}^l, \mathbf{P}\subseteq{\mathbb R}^r$ be the Okounkov body, base polytope, and moment polytope of $V_\bullet$ respectively. Denote by $q:\mathbf{O}\to \mathbf{Q}$ the natural linear projection. We may assume that the Okounkov body $\mathbf{O}$ is compatible with the ${\mathbb T}$-action, then there exists a natural linear projection $p:\mathbf{O}\to \mathbf{P}$. We have the discrete measures on $\mathbf{O}$ and $\mathbf{P}$ respectively $$\begin{aligned} {\rm LE}^g_m &=&\frac{1}{m^n}\sum_{\gamma\in \mathbf{O}_m} g(p(\frac{\gamma}{m}))\cdot \delta_\frac{\gamma}{m}, \\ {\rm DH}^g_{\mathbf{P},m} &=&\frac{1}{m^n}\sum_{\alpha\in \mathbf{P}_m} g(\frac{\alpha}{m}) h^0(V_{m,\alpha,\bullet}) \cdot\delta_\frac{\alpha}{m}.\end{aligned}$$ They converge weakly to ${\rm LE}^g$ and ${\rm DH}^g_\mathbf{P}$ respectively. We also have $\mathbf{v}^g=\int_\mathbf{O}{\rm LE}^g=\int_\mathbf{P}{\rm DH}^g_\mathbf{P}={\rm vol}^g(\mathbf{O})$, and $\mathbf{v}^g_m = \int_\mathbf{O}{\rm LE}^g_m=\int_\mathbf{P}{\rm DH}^g_{\mathbf{P},m}=N^g_m/m^n$. Let $G^{\mathcal F}$ be the concave transform of ${\mathcal F}$, which is a concave function on $\mathbf{O}$. We have $$\begin{aligned} \label{Formula: S^g 2} S^g(V_\bullet; {\mathcal F}) \,\,=\,\, \frac{1}{\mathbf{v}^g} \int_{\mathbb R}t \cdot {\rm DH}^g_{\mathcal F} \,\,=\,\, \frac{1}{\mathbf{v}^g} \int_{\mathbb R}{\rm vol}^g\{G\ge t\}{\rm d}t \,\,=\,\, \frac{1}{\mathbf{v}^g} \int_\mathbf{O}G^{\mathcal F}\cdot {\rm LE}^g. \end{aligned}$$ We may compute $S^g$ using the multi-graded structure. Let $\beta \in \mathbf{Q}_{\mathbb Q}$, and denote by $\mathbf{O}^{(t)}=\{G^{\mathcal F}\ge t\}$ the Okounkov body of ${\mathcal F}^{(t)}V_\bullet$. Then $q^{-1}(\beta)\cap \mathbf{O}^{(t)}$ is the Okounkov body of the ${\mathbb N}$-graded linear series ${\mathcal F}^{(t)}V_{(1,\bullet,\beta)}$, and we have $$\begin{aligned} {\rm vol}^g(\mathbf{O}^{(t)}) &=& \int_\mathbf{Q}{\rm vol}^g(q^{-1}(\beta)\cap \mathbf{O}^{(t)}) {\rm LE}_\mathbf{Q}({\rm d}\beta), \end{aligned}$$ where ${\rm LE}_\mathbf{Q}({\rm d}\beta)$ is the Lebesgue measure on $\mathbf{Q}$. Hence $$\begin{aligned} \label{Formula: S^g 3} S^g(V_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g} \int_{\mathbb R} \int_\mathbf{Q}{\rm vol}^g(q^{-1}(\beta)\cap \mathbf{O}^{(t)}) {\rm LE}_\mathbf{Q}({\rm d}\beta) {\rm d}t, \end{aligned}$$ **Example 18**. *[\[Example: formula when Q=P\]]{#Example: formula when Q=P label="Example: formula when Q=P"} Assume that the projection $p:\mathbf{O}\to \mathbf{P}$ factor through $q:\mathbf{O}\to \mathbf{Q}$, that is, there exists a linear projection $\bar{p}:\mathbf{Q}\to \mathbf{P}$ such that $p=\bar{p}\circ q$. The weighted volume is clear in this case $$\begin{aligned} {\rm vol}^g(q^{-1}(\beta)\cap \mathbf{O}^{(t)}) = g(\bar{p}(\beta)) \cdot {\rm vol}(q^{-1}(\beta)\cap \mathbf{O}^{(t)}). \end{aligned}$$ Changing the order of the integration, we get $$\begin{aligned} \label{Formula: S^g 4} S^g(V_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g} \int_\mathbf{P}g(\alpha) \Big( \int_{\mathbb R} \int_{\bar{p}^{-1}(\alpha)} {\rm vol}(q^{-1}(\beta)\cap \mathbf{O}^{(t)}) {\rm LE}_{\bar{p}^{-1}(\alpha)}({\rm d}\beta){\rm d}t \Big) {\rm LE}_\mathbf{P}({\rm d}\alpha) \end{aligned}$$ Recalling the definition of the $S$-invariant (formula ([\[Formula: S\^g 3\]](#Formula: S^g 3){reference-type="ref" reference="Formula: S^g 3"}) with $g=1$), we further have $$\begin{aligned} \label{Formula: S^g 5} S^g(V_\bullet; {\mathcal F}) &=& \frac{1}{\mathbf{v}^g} \int_\mathbf{P}g(\alpha) {\rm vol}(p^{-1}(\alpha))S(V_{(1,\alpha,\bullet)};{\mathcal F}) {\rm LE}_\mathbf{P}({\rm d}\alpha). \end{aligned}$$ This is the key formula in our proof of Theorem [Theorem 1](#Theorem: Intro №2.28 and No 3.14 soliton){reference-type="ref" reference="Theorem: Intro №2.28 and No 3.14 soliton"} and [Theorem 4](#Theorem: Intro stability of cones){reference-type="ref" reference="Theorem: Intro stability of cones"}.* ## Local weighted stability thresholds Let $(X,\Delta)$ be a projective pair with ${\mathbb T}$-action and a ${\mathbb T}$-invariant subvariety $Z\subseteq X$. Let $V_\bullet$ be a ${\mathbb T}$-invariant multi-graded linear series on $X$. We define $$\begin{aligned} \delta^g_{Z, {\mathbb T}, m}(X,\Delta;V_\bullet) = {\rm inf}_{v} \frac{A_{X,\Delta}(v)}{S^g_m(V_\bullet; v)}, \quad \delta^g_{Z, {\mathbb T}}(X,\Delta;V_\bullet) = {\rm inf}_{v} \frac{A_{X,\Delta}(v)}{S^g(V_\bullet; v)}, \end{aligned}$$ where the infimum runs over all the valuations $v\in {\rm Val}^{{\mathbb T},\circ}_X$ passing through $Z$, that is, $C_X(v)\supseteq Z$. **Remark 19**. *If there is a dominant morphism $f:X\to U$ with a subvariety $Z\subseteq U$, then we also denote by $\delta^g_{Z, {\mathbb T}}(X,\Delta;V_\bullet) = {\rm inf}_{v} \frac{A_{X,\Delta}(v)}{S^g(V_\bullet; v)},$ where the infimum runs over all the $v\in{\rm Val}^{{\mathbb T},\circ}_X$ passing through $Z$, that is, $f(C_X(v))\supseteq Z$.* For a ${\mathbb T}$-invariant valuation $v$ passing through $Z$, we have $S^g_m(V_\bullet; v)={\rm sup}_D v(D)$ where the supremum runs over all the ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ of $V_\bullet$. For such a divisor $D$, ${\rm lct}_Z(X,\Delta;D)$ is minimized by a ${\mathbb T}$-invariant valuation. Hence $$\begin{aligned} \label{formula: delta_m = lct} \delta^g_{Z, {\mathbb T}, m}(X,\Delta;V_\bullet) = {\rm inf}_{v} {\rm inf}_D \frac{A_{X,\Delta}(v)}{v(D)} ={\rm inf}_D {\rm lct}_Z(X,\Delta;D), \end{aligned}$$ where the infimum runs over all the ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ of $V_\bullet$. Similarly, we have $\delta^g_{Z, {\mathbb T}}(X,\Delta;V_\bullet) ={\rm inf}_D {\rm lct}_Z(X,\Delta;D)$, where the infimum runs over all the ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ of $V_\bullet$ for all $m\in {\mathbb N}$. # Weighted Abban-Zhuang estmate {#Section: Weighted Abban-Zhuang estmate} We establish the weighted Abban-Zhuang estimate in this section, which gives a lower bound of the $g$-weighted stability threshold. The main difference with the original Abban-Zhuang estimate is that we need to preserve the weight decomposition in each step of refinements. ## Weighted basis type divisors Let $(X_l,\Delta_l)$ be a $(n-l)$-dimensional projective klt pair with a ${\mathbb T}$-action, and $V_\bullet$ be a ${\mathbb T}$-invariant ${\mathbb N}\times{\mathbb N}^l$-graded linear series on $X_l$ as defined in Section [2.9](#Subsection: multi-graded linear series with a torus action){reference-type="ref" reference="Subsection: multi-graded linear series with a torus action"}. We denote by $\mathbf{O}\subseteq{\mathbb R}^n$ the Okounkov body of $V_\bullet$ and $\mathbf{Q}\subseteq{\mathbb R}^l$ the base polytope of $V_\bullet$ with the canonical linear projection $q:{\mathbb R}^n\to {\mathbb R}^l$ mapping $\mathbf{O}$ onto $\mathbf{Q}$. We have weight decomposition $V_m=\oplus_{\alpha \in \mathbf{P}_m} V_{m,\alpha,\bullet}$. A *${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor* $D$ of $V_\bullet$ is of the form $$\begin{aligned} \label{Formula: weighted basis type divisor} D= \frac{1}{N^g_m} \sum_{\alpha\in\mathbf{P}_m} g(\frac{\alpha}{m}) \frac{D_{m,\alpha,\bullet}}{m}, \end{aligned}$$ where $D_{m,\alpha,\bullet}= \sum_{\beta\in\mathbf{Q}_{m,\alpha}} D_{m,\alpha,\beta}$ and $D_{m,\alpha,\beta}$ is a basis type divisor of $V_{m,\alpha,\beta}$. We say that $D$ is *compatible* with a filtration ${\mathcal F}$ on $V_\bullet$ if every $D_{m, \alpha,\beta}$ is compatible with ${\mathcal F}$ on $V_{m,\alpha,\beta}$. From the definition we see that, if $D$ is compatible with ${\mathcal F}_v$ for some valuation $v$, then we have $$v(D)=S^g_m(V_\bullet; {\mathcal F}_v).$$ Let $F$ be a ${\mathbb T}$-invariant plt-type divisor over $X_l$ and $\pi: Y\to X_l$ be the associated plt-type blowup. We denote by $W_\bullet$ the refinement of $V_\bullet$ by ${\mathcal F}={\mathcal F}_F$. We consider a ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ which is compatible with $F$ and has decomposition as ([\[Formula: weighted basis type divisor\]](#Formula: weighted basis type divisor){reference-type="ref" reference="Formula: weighted basis type divisor"}). If $F$ is toric, then $W_{m,\alpha,\beta,\Phi(\alpha)} = V_{m,\alpha,\beta} \cdot s_F^{-\Phi(\alpha)} |_F$ locally (for $m$ sufficiently divisible such that $\Phi(\alpha) \in {\mathbb Z}$ for all $\alpha\in \mathbf{P}_m$), where $s_F\in {\mathcal O}_X$ is the local defining function of $F$. Globally we have $\pi^*D_{m,\alpha,\beta} = \Phi(\alpha) N_{m,\alpha,\beta} \cdot F + \Gamma_{m,\alpha,\beta,\Phi(\alpha)}$, where $\Gamma_{m,\alpha,\beta,\Phi(\alpha)}$ does not contain $F$ as a component, and $\Gamma_{m,\alpha,\beta,\Phi(\alpha)}|_F$ is a basis type divisor of $W_{m,\alpha,\beta,\Phi(\alpha)}$. Hence we have $$\begin{aligned} \Gamma &=& \frac{1}{N^g_m} \sum_{\alpha\in\mathbf{P}_m(V_\bullet)} g(\frac{\alpha}{m}) %\sum_{\beta\in\BQ_{m,\alpha}(V_\bu)} \frac{1}{m} \cdot \Gamma_{m,\alpha,\bullet,\Phi(\alpha)}, \\ \pi^*D &=& \Big( \frac{1}{N^g_m} \sum_{\alpha\in \mathbf{P}_m(V_\bullet)} g(\frac{\alpha}{m}) \frac{\Phi(\alpha)}{m} N_{m,\alpha,\bullet} \Big) \cdot F + \Gamma\\ &=& S^g_m(V_\bullet; F) \cdot F + \Gamma, \end{aligned}$$ where $\mathbf{P}_m(W_\bullet)=\mathbf{P}_m(V_\bullet), \mathbf{Q}_{m,\alpha}(W_\bullet) =\mathbf{Q}_{m,\alpha}(V_\bullet) \times \{\Phi(\alpha)\}.$ Hence $\Gamma|_F$ is a ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor of $W_\bullet$. If $F$ is vertical, then $W_{m,\alpha,\beta,j}={\mathcal F}^jV_{m,\alpha,\beta}\cdot s_F^{-j}|_F$ locally. Hence we have $D_{m,\alpha,\beta}=\sum_jD_{m,\alpha,\beta,j}$, where ${\rm ord}_F(D_{m,\alpha,\beta,j})=j$. Then $$D_{m,\alpha,\beta,j} =jN^j_{m,\alpha,\beta}\cdot F + \Gamma_{m,\alpha,\beta,j},$$ where $\Gamma_{m,\alpha,\beta,j}$ does not contain $F$ as a component. Summing up we also have $$\begin{aligned} \label{Formula: refinement of weighted basis type divisor} \pi^*D &=& S^g_m(V_\bullet; F) \cdot F + \Gamma, \\ \Gamma &=& \frac{1}{N^g_m} \sum_{\alpha\in\mathbf{P}_m(W_\bullet)} g(\frac{\alpha}{m}) \frac{1}{m} \cdot \Gamma_{m,\alpha,\bullet}, \\ \Gamma_{m,\alpha,\bullet} &=& \sum_{(\beta,j)\in\mathbf{Q}_{m,\alpha}(W_\bullet)} \Gamma_{m,\alpha,\beta,j}\end{aligned}$$ Hence $\Gamma|_F$ is a ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor of $W_\bullet$. ## Weighted Abban-Zhuang estmate {#weighted-abban-zhuang-estmate} Let $(X,\Delta)$ be a projective klt pair with a ${\mathbb T}$-action, $V_{\bullet}$ be a ${\mathbb T}$-invariant multi-graded linear series on $X$. Let $F$ be a plt-type divisor over $X$, with the associated plt-type blowup $\pi: Y \to X$. We denote by $W_{\bullet}$ the refinement of $V_\bullet$ by $F$, and by $\Delta_Y$ be the strict transform of $\Delta$ on $Y$. Then we have $$\begin{aligned} K_Y+\Delta_Y+(1-A_{X,\Delta}(F))F=\pi^*(K_X+\Delta).\end{aligned}$$ We denote by $\Delta_F$ the difference of $\Delta_Y$ on $F$, then $$\begin{aligned} (K_Y+\Delta_Y+F)|_F=K_F+\Delta_F. \end{aligned}$$ **Theorem 20**. *Let $Z \subseteq X$ be a ${\mathbb T}$-invariant subvariety contained in $C_X(F)$. Then we have $$\begin{aligned} \label{ineq.1} \delta^{g}_{Z, {\mathbb T}}(X,\Delta;V_\bullet) \ge {\rm min}\Big\{\frac{A_{X,\Delta}(F)}{S^{g}(V_\bullet; F)},\,\, \delta^{g}_{Z,{\mathbb T}}(F,\Delta_F;W_\bullet) \Big\}. \end{aligned}$$* **Remark 21**. *If $F$ is a divisor on $X$, we may set $Y=X, \Delta=\Delta_Y+{\rm ord}_F(\Delta)F$, and assume that $(Y,\Delta_Y+F)$ is lc in a neighbourhood of $F$. The theorem still holds in this case, with $\delta^g_Z(F,\Delta_F;W_\bullet)$ replaced by $\delta^g_Z(\hat{F},\Delta_{\hat{F}};W_\bullet)$, where $\hat{F}$ is the normalization of $F$.* *Proof of Theorem [Theorem 20](#Theorem: weighted AZ){reference-type="ref" reference="Theorem: weighted AZ"}.* We denote by $\lambda$ the right hand side of the inequality ([\[ineq.1\]](#ineq.1){reference-type="ref" reference="ineq.1"}), and let $$\begin{aligned} \lambda_m := {\rm min}\Big\{\frac{A_{X,\Delta}(F)}{S^{g}_m(V_\bullet; F)},\,\, \delta^{g}_{Z,{\mathbb T},m}(F,\Delta_F,W_\bullet) \Big\}. \end{aligned}$$ One may show that ${\rm lim}_{m \to \infty} \lambda_m = \lambda$. Hence it suffices to show $\delta^{g}_{Z, {\mathbb T}, m}(X,\Delta,V_\bullet) \ge \lambda_m$. For any ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ of $V_\bullet$ which is compatible with $F$, we have $\pi^*D = S^{g}_m(V_\bullet; F)\cdot F + \Gamma$ by ([\[Formula: refinement of weighted basis type divisor\]](#Formula: refinement of weighted basis type divisor){reference-type="ref" reference="Formula: refinement of weighted basis type divisor"}), where $\Gamma$ is the strict transform of $D$ on $Y$, and $F \nsubseteq {\rm Supp}\Gamma$. We see that $\Gamma|_F$ is a ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor of $W_\bullet$. Now we have $$\begin{aligned} K_Y+\Delta_Y+a_mF+\lambda_m \Gamma=\pi^*\big(K_X+\Delta+\lambda_m D\big), \end{aligned}$$ where $a_m=1-A_{X,\Delta}(F)+\lambda_m S^{g}_m(V_\bullet; F) \le1.$ By inversion of adjunction, we know that $(Y,\Delta_Y+F+\lambda_m \Gamma)$ is lc at the generic point of $Z$ is equivalent that $(F, \Delta_F+\lambda_m \Gamma|_F)$ is lc at the generic point of $Z$. Note that $\Gamma|_F$ is ${\mathbb T}$-invariant. By [@Zhu21], ${\rm lct}_Z(F,\Delta_F;\Gamma|_F)$ is minimized by a ${\mathbb T}$-invariant valuation, and $$\begin{aligned} A_{F,\Delta_F+\lambda_m \Gamma|_F}(w) &=& A_{F,\Delta_F}(w)-\lambda_m w(\Gamma|_F) \\ &=&\Big( \frac{A_{F,\Delta_F}(w)}{w(\Gamma|_F)}-\lambda_m \Big)w(\Gamma|_F)\\ &\ge& \Big(\delta^g_{Z,{\mathbb T},m}(F,\Delta_F;W_\bullet)-\lambda_m\Big)w(\Gamma|_F) \quad \ge \quad 0, \end{aligned}$$ for any valuation $w\in {\rm Val}^{{\mathbb T},\circ}_F$ passing through $Z$, where we use ([\[formula: delta_m = lct\]](#formula: delta_m = lct){reference-type="ref" reference="formula: delta_m = lct"}) in the first inequality. Hence $(F, \Delta_F+\lambda_m \Gamma|_F)$ is lc at the generic point of $Z$, so is $(Y,\Delta_Y+F+\lambda_m \Gamma)$. Then we have $$\begin{aligned} A_{X,\Delta+\lambda_m D}(v) &=& A_{Y,\Delta_Y+a_mF+\lambda_m\Gamma}(v) \\ &=& A_{Y,\Delta_Y+F+\lambda_m\Gamma}(v)+(1-a_m)v(F)\\ &\ge& (1-a_m)v(F) \quad\ge\quad 0, \end{aligned}$$ hence $A_{X,\Delta}(v)/v(D)\ge \lambda_m$ for any valuation $v\in {\rm Val}^{{\mathbb T},\circ}_X$ passing through $Z$. So ${\rm lct}_Z(X,\Delta;D)\ge \lambda_m$. We conclude that $$\begin{aligned} \delta^g_{Z,{\mathbb T},m}(X,\Delta;V_\bullet) = \delta^g_{Z,{\mathbb T},m}(X,\Delta;V_\bullet;{\mathcal F}_F) = {\rm inf}_D {\rm lct}_Z(X,\Delta;D) \ge \lambda_m, \end{aligned}$$ where the infimum runs over all the ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor $D$ of $V_\bullet$ which is compatible with $F$. ◻ # G-Equivariant weighted K-stability {#Section: G-Equivariant weighted K-stability} We prove the equivalence of the $G$-equivariant $g$-weighted K-stability and the ${\mathbb T}$-equivariant $g$-weighted K-stability for a log Fano triple $(X,\Delta,\xi_0)$ with $G$-action, where ${\mathbb T}\subseteq G$ is a subtorus containing $\xi_0$. This is an application of [@Zhu21] to the *$g$-weighted boundary* on $X$. Recall that a *boundary* $V$ on $X$ is a formal sum $V=\sum_i a_i V_i$, where $a_i \in {\mathbb R}_{>0}$ and $V_i\subseteq H^0(X,L_i)$ for some line bundle $L_i$ on $X$. We use the notion $h^0(V_i)=\dim V_i$ for convenience. A *basis type ${\mathbb R}$-divisor* of $V$ is of the form $D=\sum_i a_i D_i$, where $D_i=\frac{1}{h^0(V_i)}\sum_j\{s_j=0\}$, and $\{s_j\} \subseteq V_i$ is a basis. Then we define $c_1(V)$ by $\sum_i a_i L_i \sim_{\mathbb R}D$. For any filtration ${\mathcal F}$ on $V$, we have $S(V;{\mathcal F})= \sum_i a_i \sum_\lambda\lambda\cdot h^0({\rm gr}^\lambda_{\mathcal F}V_i)$. Assume that the line bundles $L_i$ admit $G$-linearization. We say that $V$ is $G$-invariant if $V_i\subseteq H^0(X,L_i)$ is $G$-invariant for each $i$. Let $F$ be a $G$-invariant plt-type divisor over $X$, the *refinement* of $V$ by $F$ is defined as $$W=\sum_i a_i \sum_\lambda\frac{h^0({\rm gr}^\lambda_{\mathcal F}V_i)}{h^0(V_i)}\cdot V_i(-\lambda F)|_F,$$ where $V_i(-\lambda F)|_F \cong {\rm gr}^\lambda_{\mathcal F}V_i$ is the image of the restriction map $V_i\to H^0(F, L_i|_F)$. For a $G$-equivariant morphism $f: X\to U$ and a $G$-invariant subvariety $Z\subseteq U$, we shall also define $\delta_{Z,G}(V)=\delta_{Z,G}(X,\Delta,V) = {\rm inf}_v \frac{A_{X,\Delta}(v)}{S(V;v)}$, where the infimum runs over all the $G$-invariant valuations $v$ on $X$ passing through $Z$. With the same argument in Theorem [Theorem 20](#Theorem: weighted AZ){reference-type="ref" reference="Theorem: weighted AZ"}, we have **Lemma 22**. *$\delta_{Z, G}(X,\Delta;V) \ge {\rm min}\Big\{\frac{A_{X,\Delta}(F)}{S(V; F)},\,\, \delta_{Z,G}(F,\Delta_F;W) \Big\}.$* Assume moreover that $H\subseteq G$ is an algebraic subgroup. Then we have the following minor strengthening of [@Zhu21 Theorem 4.4]. **Lemma 23**. *Assume that $\delta_{Z,H}(V)< \infty$ and $c_1(V)$, $-(K_X+\Delta+\delta_{Z,H}(V) c_1(V))$ are $f$-ample, then there exists a $G$-invariant prime divisor $E$ over $X$ with $f(C_X(E))\supseteq Z$ such that $$\delta_{Z,H}(V)=\frac{A_{X,\Delta}(E)}{S(V;E)}.$$* *Proof.* We follow the proof of [@Zhu21 Theorem 4.4]. Let $\dim X = n$. Assume that the lemma holds for $(n-1)$-dimensional varieties. With the same argument in [@Zhu21 Lemma 4.5] using Kollár-Shokurov's connectedness theorem, we have the following claim. **Claim 24**. *There exists a $G$-invariant subvariety $\widetilde{Z}\subseteq X$ with $f(\widetilde{Z})\supseteq Z$ such that $$\delta_{\widetilde{Z},H}(V)=\delta_{Z,H}(V).$$* By [@Zhu21 Lemma 4.7], there exists a $G$-invariant plt-type prime divisor $E$ over $X$ with $C_X(E)\supseteq \widetilde{Z}$ minimizing $\delta = \delta_{\widetilde{Z},G}(V)$. We will show that $\delta_{\widetilde{Z},H}(V)$ is also minimized by $E$. By Lemma [Lemma 22](#Theorem: weighted AZ 2){reference-type="ref" reference="Theorem: weighted AZ 2"}, it suffices to show $\delta_{\widetilde{Z}, H}(E,\Delta_E; W)\ge \delta$, where $W$ is the refinement of $V$ by $E$. We denote by $\pi:Y\to X$ the plt-type blowup associated with $E$. Then for any basis type ${\mathbb R}$-divisor $D$ of $V$ which is compatible with $E$, we have $\pi^*D=S(V;E)E+\Gamma$ where $\Gamma$ does not contain $E$ as a component. Note that $\Gamma|_E$ is a basis type ${\mathbb R}$-divisor of $W$, and any basis type ${\mathbb R}$-diviosr of $W$ is of this form. We also have $c_1(W)=(\pi^*c_1(V)-S(V;E)E)|_E$, which is $\pi$-ample. For any $G$-invariant prime divisor $F$ over $E$ with $\pi(F)\supseteq \widetilde{Z}$, let $D_0$ be a basis type ${\mathbb R}$-divisor of $W$ which is compatible with $F$. The filtration induced by $F$ on $W$ lifts to a $G$-invariant filtration ${\mathcal F}$ on $V$ which is a refinement of the filtration ${\mathcal F}_E$ induced by $E$. Then $D_0$ lifts to a basis type ${\mathbb R}$-divisor $D$ of $V$ which is compatible with ${\mathcal F}$ (hence with ${\mathcal F}_E$). We have $$\delta =\frac{A_{X,\Delta}(E)}{S(V;E)} =\frac{A_{X,\Delta}(E)}{{\rm ord}_E(D)} \ge {\rm lct}_{\widetilde{Z}}(X,\Delta;D) = {\rm lct}_{\widetilde{Z}}(X,\Delta;\mathfrak{a}({\mathcal F})) \ge \tilde{\delta}_{\widetilde{Z},G}(V) =\delta.$$ Hence ${\rm lct}_{\widetilde{Z}}(X,\Delta;D)$ is minimized by $E$. Write $\pi^*D=S(V;E)E+\Gamma$, then $\Gamma|_E=D_0$. We also have $\pi^*(K_X+\Delta+\delta D)=K_Y+\Delta_Y+E+\delta \Gamma$ and $(K_Y+\Delta_Y+E+\delta \Gamma)|_E = K_E+\Delta_E+\delta D_0$. Hence $A_{Y,\Delta_Y+E+\delta \Gamma}(v)=A_{X,\Delta+\delta D}(v)\ge (\frac{A_{X,\Delta}(v)}{v(D)}-\delta)v(\Gamma)\ge 0$ for all valuation $v$ on $Y$ passing through $\widetilde{Z}$. So $(Y,\Delta_Y+E+\delta \Gamma)$ is lc at the generic point of $\widetilde{Z}$. By inversion of adjunction, $(E,\Delta_E+\delta D_0)$ is lc at the generic point of $\widetilde{Z}$. Hence $\frac{A_{E,\Delta_E}(F)}{S(W;F)}= \frac{A_{E,\Delta_E}(F)}{{\rm ord}_F(D_0)} \ge \delta$. (Taking infimum we only get $\delta_{\widetilde{Z},G}(E,\Delta_E;W) \ge \delta$). Assume that $\delta_0=\delta_{\widetilde{Z},H}(E,\Delta_E;W) < \delta$, then $-(K_E+\Delta_E+\delta_0c_1(W))\sim_{\pi,{\mathbb R}} (\delta-\delta_0) c_1(W)$ is $\pi$-ample. By induction hypothesis, $\delta_{\widetilde{Z},H}(E,\Delta_E;W)$ is minimized by a $G$-invariant prime divisor $F$ over $E$ passing through $\widetilde{Z}$, that is, $\delta_0=\frac{A_{E,\Delta_E}(F)}{S(W;F)} \ge \delta$. We get a contradiction. We conclude that $\delta_{\widetilde{Z}, H}(E,\Delta_E; W)\ge \delta$, and $\delta_{\widetilde{Z},H}(X,\Delta;V) = \delta$ is minimized by $E$. ◻ For a multi-graded linear series $V_\bullet$ with a ${\mathbb T}$-action decending to $X$, we define the *$m$-th $g$-weighted boundary* of $V_\bullet$ by (recall that $N^g_m=\sum_{\alpha,\beta}g(\frac{\alpha}{m})N_{m,\alpha,\beta}$) $$\begin{aligned} V^g_m = \sum_{\alpha, \beta} g(\frac{\alpha}{m}) \frac{N_{m,\alpha,\beta}}{m\cdot N^g_m} \cdot V_{m,\alpha,\beta}. \end{aligned}$$ Then the ${\mathbb T}$-invariant $g$-weighted $m$-basis type ${\mathbb R}$-divisor of $V_\bullet$ is just the basis type ${\mathbb R}$-divisor of $V^g_m$, and $S^g_m(V_\bullet;{\mathcal F})=S(V^g_m; {\mathcal F})$ for any ${\mathbb T}$-invariant filtration ${\mathcal F}$ on $V_\bullet$. **Theorem 25**. *Let $(X,\Delta,\xi_0)$ be a log Fano triple admitting an algebraic group $G$-action, and ${\mathbb T}\subseteq G$ be a subtorus containing $\xi_0$. If $(X,\Delta,\xi_0)$ is not ${\mathbb T}$-equivariantly $g$-weighted K-semistable, then $\delta^g_{{\mathbb T}}(X,\Delta)$ is achieved by a sequence of $G$-invariant weakly special divisors over $X$.* *Proof.* We follow the proof of [@Zhu21 Theorem 4.1]. Let $V_\bullet=R(X,\Delta)$ be the anti-canonical ring of $(X,\Delta)$, and $\delta_m = \delta_{\mathbb T}(X,\Delta;V^g_m)$. Hence $c_1(V^g_m)\sim_{\mathbb R}-(K_X+\Delta)$ is ample. By Lemma [Lemma 13](#Lemma: uniform bound of S^g_m/S^g){reference-type="ref" reference="Lemma: uniform bound of S^g_m/S^g"}, for any $\varepsilon>0$ there exists $m_0=m_0(\varepsilon)\in{\mathbb N}$ such that $S(V^g_m;{\mathcal F})\le (1+\varepsilon) S^g(V_\bullet;{\mathcal F})$ holds for all $m\ge m_0$ and all the linearly bounded filtrations ${\mathcal F}$ on $V_\bullet$. On the other hand, we shall increase $m_0\in{\mathbb N}$ such that $\delta_m<1$ for all $m\ge m_0$ by Lemma [Lemma 14](#Lemma: convergence of delta_m){reference-type="ref" reference="Lemma: convergence of delta_m"}. Then $-(K_X+\Delta+\delta_mc_1(V^g_m))\sim_{{\mathbb R}} -(1-\delta_m)(K_X+D)$ is ample. By Lemma [Lemma 23](#Lemma: existence of G-invariant minimizer){reference-type="ref" reference="Lemma: existence of G-invariant minimizer"} (with $U=$ point and $V=V^g_m$), we have $G$-invariant prime divisor $E_m$ minimizing $\delta_m = \delta_{\mathbb T}(X,\Delta;V^g_m)=\delta^g_{{\mathbb T},m}(X,\Delta;V_\bullet)$. Hence $$(1+\varepsilon) \delta_m = (1+\varepsilon) \frac{A_{X,\Delta}(E_m)}{S(V^g_m;E_m)} \ge \frac{A_{X,\Delta}(E_m)}{S^g(V_\bullet;E_m)} \ge \delta^g_{{\mathbb T}}(X,\Delta;V_\bullet).$$ Pick series $\varepsilon_i \to 0$, we have constants $m_i=m_i(\varepsilon_i)\in{\mathbb N}$ as above. By Lemma [Lemma 14](#Lemma: convergence of delta_m){reference-type="ref" reference="Lemma: convergence of delta_m"}, we conclude that $$\frac{A_{X,\Delta}(E_{m_i})}{S^g(V_\bullet;E_{m_i})} \to \delta^g_{{\mathbb T}}(X,\Delta;V_\bullet),\, (i\to \infty).$$ On the other hand, for $m\ge m_0$, the divisor $E_m$ is an lc place of $\delta_m D + (1-\delta_m) H$ where $D$ is a basis type ${\mathbb R}$-divisor of $V^g_m$ compatible with $E_m$, and $H$ is a general effective divisor such that $H\sim_{\mathbb R}-(K_X+\Delta)$. Hence $E_m$ is weakly special. ◻ **Remark 26**. *Using the argument of [@LXZ22], one may further show that $\delta^g_{\mathbb T}(X,\Delta)<1$ is minimized by a $G$-invariant weakly special divisor over $X$.* **Corollary 27**. *If $\delta^g_G(X,\Delta)\ge 1$, then $(X,\Delta,\xi_0)$ is ${\mathbb T}$-equivariantly $g$-weighted K-semistable,* *Proof.* Otherwise, by the above theorem there exists a $G$-invariant prime divisor $E$ over $(X,\Delta)$ such that $\frac{A_{X,\Delta}(E)}{S^g(V_\bullet;E)} < 1$, which contradicts $\delta^g_G(X,\Delta)\ge 1$. ◻ # Kähler-Ricci solitons on Fano threefolds of №2.28 and №3.14 {#Section: Applications} Let $(X, \Delta)$ be a log Fano pair of dimension $n$ with a ${\mathbb T}$-action. We say that the ${\mathbb T}$-action is of *complexity* $k$ if the ${\mathbb T}$-orbit of a general point in $X$ is of codimension $k$. By [@ACC+], we have the following K-unstable Fano threefolds with torus actions of complexity two: - №2.26: blow up $V_5\subset {\mathbb P}^6$ along line. - №2.28: blow up ${\mathbb P}^3$ along plane cubic. - №3.14: blow up of ${\mathbb P}^3$ along plane cubic curve and point that are not coplanar. - №3.16: blow up of $V_7$ along proper transform via blow up $V_7 \rightarrow {\mathbb P}^3$ of twisted cubic passing through blown up point. Note that any member in these families is destabilized by a product TC. We show that every smooth Fano threefold in the family №2.28 and №3.14 admits a Kähler-Ricci soliton in this section. ## Weighted K-polystability of №2.28 {#Subsection: №2.28} Every smooth Fano threefold $X$ of family №2.28 is given by the blowup of ${\mathbb P}^3_{[u,x,y,z]}$ along a smooth plane cubic curve $C=\{u=0, y^2z=x(x-z)(x-\lambda z)\}$ for $\lambda \neq 0,1$. Denote by $\pi: X\rightarrow {\mathbb P}^3$ the blowup morphism. We will write $v:= y^2z-x(x-z)(x-\lambda z)$ for simplicity. Note that the ${\mathbb G}_m$-action $\lambda(t): [u,x,y,z] \mapsto [tu,x,y,z]$ of ${\mathbb P}^3$ fixes $C$, hence it lifts to an ${\mathbb G}_m$-action to $X$, which we also denote by $\lambda$. This ${\mathbb G}_m$-action generates the maximal torus of the automorphism group of $X$ since the connected component of the identity is given by ${\rm Aut}^0(X) = {\mathbb G}_m^3 \rtimes {\mathbb G}_m$, see [@ACC+]. We first take the affine chart $U_{z} = \{[u,x,y,z]\in {\mathbb P}^4: z\neq 0\}$, then $X|_{U_z} = \text{Proj}_{C}\mathbb{A}_{{\mathbb C}}^3 = \text{Proj}\Big({\mathbb C}[u,x,y][\zeta_0,\zeta_1]/(\zeta_0\cdot v-\zeta_1\cdot u) \Big)$. If we further restrict to the affine chart $\{\zeta_1 \neq 0\}$, then $u = \zeta_0\cdot v$. We denote $w:=\zeta_0$ for convenience. In this affine chart, the exceptional divisor $E_C=\{v=0\}$ and the strict transform of hyperplane $H_u$ is given by $\{w =0\}$, still denoted by $H_u$. We have $-K_X = \pi^* (-K_{{\mathbb P}^3})-E_C$. Hence $$\begin{aligned} H^0(X,-K_X) &=& w\cdot H^0({\mathbb P}^3,\mathcal{O}_{{\mathbb P}^3}(3)) \oplus H^0({\mathbb P}^3,\mathcal{O}_{{\mathbb P}^3}(1)) \\ &=& \langle w u^3 \rangle\oplus w u^2\cdot\langle x,y,z \rangle\oplus w u \cdot \langle x,y,z \rangle^2 \oplus w \cdot \langle x,y,z \rangle^3 \oplus \langle x,y,z \rangle\\ &=& \langle w^4v^3 \rangle\oplus w^3v^2\cdot\langle x,y,z \rangle\oplus w^2v\cdot \langle x,y,z \rangle^2 \oplus w \cdot \langle x,y,z \rangle^3 \oplus \langle x,y,z \rangle. \end{aligned}$$ It also generates the weight decomposition of $R$. Since $H_u$ is a toric divisor of log discrepancy one, the moment polytope is $[-1,3] \subseteq M_{\mathbb R}= {\mathbb R}$. We take the admissible flag: $\mathbb{A}^3_{w,y,x}\supseteq \{w=0\}\supseteq \{w=y=0\}\supseteq \{w=y=x=0\}$, which gives a faithful valuation $\mathfrak{v}: w\mapsto (1,0,0), y\mapsto (0,1,0), x\mapsto (0,0,1)$. We have $\mathfrak{v}(v) = (0,0,1)$. Thus, $$\begin{aligned} w^4\cdot v^3 &\mapsto &(4,0,3);\\ w^3\cdot v^2\cdot \langle x,y,z \rangle &\mapsto &(3,0,2), (3,1,2), (3,0,3);\\ w^2\cdot v\cdot \langle x,y,z \rangle^2 &\mapsto &(2,0,1),(2,1,1),(2,0,2),(2,2,1),(2,1,2),(2,0,3);\\ w\cdot \langle x,y,z \rangle^3&\mapsto &(1,0,0),(1,1,0),(1,0,1),(1,2,0),(1,1,1),(1,0,2),\\ &&(1,3,0),(1,2,1),(1,1,2),(1,0,3);\\ \langle x,y,z \rangle&\mapsto &(0,0,0),(0,1,0),(0,0,1).\end{aligned}$$ Let $\mathbf{O}_0\subseteq{\mathbb R}^3_{w,y,x}$ be the convex polytope generated by these points. Computing volume we see that $\mathbf{O}_0$ is an Okounkov body of $X$. In order to make the Okounkov body compatible with the ${\mathbb G}_m$-action, we may shift the first coordinate by one such that the body lives in $-1\le w\le3$. We denote this shifted body by $\mathbf{O}$, and the natural projection by $p:\mathbf{O}\to \mathbf{P}=[-1,3]$. Note that $\mathbf{O}$ has vertices $\{(3,0,3), (0,0,0),(0,0,3),(0,3,0),(-1,0,0),(-1,1,0),(-1,0,1)\}$. From the equation ${\rm Fut}_g(1)=0$, we derive $0=\int_{\mathbf{O}} w\cdot e^{-\xi_0\cdot w}{\rm d}w {\rm d}y {\rm d}x = \int_{-1}^0 w\cdot e^{-\xi_0\cdot w}\cdot \frac{1}{2}\cdot (3+2w)^2{\rm d}w + \int_0^3 w\cdot e^{-\xi_0\cdot w}\cdot \frac{1}{2}\cdot (3-w)^2 {\rm d}w$. We can numerically solve $\xi_0 \approx 0.9377815610300645$. **Theorem 28**. *Every smooth Fano threefold $X=X_{2.28}$ admits a Kähler-Ricci soliton.* *Proof.* By the work of [@HL23] and [@BLXZ23], we know that $X$ admits a Kähler-Ricci soliton if and only if $(X,\xi_0)$ is weighted K-polystable. We first show that it is weighted K-semistable. Note that $G={\rm Aut}^0(X)\cong {\mathbb G}^3_a \rtimes {\mathbb G}_m$, and the minimal $G$-invariant subvarieties of $X$ are exactly the subvarieties of $H_u$. By $G$-equivariant weighted K-stability, it suffices to show $\delta^g_{p,{\mathbb T}}(X)\ge 1$ for any closed point $p\in H_u$ (note that $\delta^g_{p,{\mathbb T}}(X)\le 1$ by the choice of $\xi_0$). In the first step, we take refinement of $R_\bullet$ by $H_u$ and denote it by $W_\bullet$. In the second step, let $C=H_u\cap E_C$. There are three different cases of $p\in H_u$. Case (A) if $p\notin C$, we choose refinement by a line $l\subseteq H_u$ passing through $p$; case (B), if $p\in C$ and the tangent line $\{X=0\}$ of $C$ at $p$ has multiplicity two at $p$ when restricting to $C$, we choose refinement by the exceptional line $E$ of the $(2,1)$-weighted blowup of $H_u$ at the point $p$, where ${\rm wt}(X)=2, {\rm wt}(Y)=1$ and $(X,Y)$ is the local coordinate of ${\mathbb P}^2$ at $p$; case (C), if $p\in C$ and the tangent line is of multiplicity three, we choose refinement by the exceptional line $E$ of the $(3,1)$-weighted blowup. Denote the second step refinement by $W^l_\bullet$. Let $\widetilde{C}$ be the strict transform of $C$ in the above blowup. Finally, we conclude by the weighted Abban-Zhuang estimate that $$\delta^g_{p,{\mathbb T}}(X)\ge {\rm min}\{1, \delta^g_p({\mathbb P}^2; W_\bullet)\},$$ where $\delta^g_p({\mathbb P}^2; W_\bullet) \ge \frac{4}{3.773902}>1$ for any $p\in {\mathbb P}^2 = H_u$, see Section [7.2](#Subsection: computing S^g of №2.28){reference-type="ref" reference="Subsection: computing S^g of №2.28"}. We next prove the weighted K-polystability of $(X,\xi_0)$. Let $Y\to X$ be the blowup of a point outside $H_u\cup E$. Then $Y$ admits a ${\mathbb T}$-equivariant morphism $\tau:Y\to H_u$, which is the composition of the blow-down map of $E$ and the canonical map of ${\mathbb P}^3$ with one point blowup to $H_u$. Hence $H_u$ can be viewed as a Chow quotient of $X$ by the ${\mathbb T}$-action. Hence ${\rm Val}^{{\mathbb T},\circ}_X\cong {\rm Val}^{\circ}_{H_u}\times N_{\mathbb R}$. Assume that $(X,\xi_0)$ is not weighted K-polystable. By [@BLXZ23 Lemma 4.14], there exist a ${\mathbb T}$-invarant quasi-monimial valuation $v$ on $X$ which is not of the form ${\rm wt}_\xi$ such that $\delta^g(v)=1$. Up to twisting by some $\xi\in N_{\mathbb R}$, we may assume that $v$ is the pull-back of some $v_0\in {\rm Val}^{\circ}_{H_u}$. One may show that $A_X(v)=A_{H_u}(v_0)$. On the other hand, the filtration induced by $v$ on $R_\bullet$ and by $v_0$ on $W_\bullet$ are the same. Hence $S^g(R_\bullet;v)=S^g(W_\bullet;v_0)$, and we get $$1=\frac{A_X(v)}{S^g(R_\bullet;v)}=\frac{A_{H_u}(v_0)}{S^g(W_\bullet;v_0)} \ge \delta^g_p({\mathbb P}^2; W_\bullet)>1,$$ which is a contradiction. ◻ ## Computing $S^g$ of №2.28 {#Subsection: computing S^g of №2.28} Recall that $H_u$ is the strict transform of $\{u=0\}\subseteq{\mathbb P}^3$ and $-K_X={\mathcal O}_{{\mathbb P}^3}(4)-E_C$. We have decomposition ${\mathcal O}_{{\mathbb P}^3}(1)=H_u+E_C$ on $X$. Hence $-K_X-(w+1) H_u$ is ample for $w\le 0$, and has fixed part $w E_C$ and movable part ${\mathcal O}_{{\mathbb P}^3}(3-w)$ when $0< w < 3$. Since ${\mathcal O}_{{\mathbb P}^3}(1)|_{H_u}={\mathcal O}_{{\mathbb P}^2}(1), E_C|_{H_u}={\mathcal O}_{{\mathbb P}^2}(3)$, we have $$\begin{aligned} W_{(1,w)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^2, {\mathcal O}(3+2w)\Big) & -1\le w < 0,\\ wC+H^0\Big({\mathbb P}^2, {\mathcal O}(3-w)\Big) & 0\le w\le 3. \end{array} \right. \end{aligned}$$ This is the case of Example [\[Example: formula when Q=P\]](#Example: formula when Q=P){reference-type="ref" reference="Example: formula when Q=P"}. Now the question is reduced to the plane $H_u\cong {\mathbb P}^2$. **Case (A).** If $p\notin C$ and $l$ is a line on ${\mathbb P}^2$. Then $$\begin{aligned} {\mathcal F}^{(y)}_l W_{(1,w)} = \left\{ \begin{array}{ll} yl+H^0\Big({\mathbb P}^2, {\mathcal O}(3+2w-y)\Big) & -1\le w < 0, 0\le y\le 3+2w\\ yl+wC+H^0\Big({\mathbb P}^2, {\mathcal O}(3-w-y)\Big) & 0\le w\le 3, 0\le y\le 3-w. \end{array} \right. \end{aligned}$$ By ([\[Formula: S\^g 4\]](#Formula: S^g 4){reference-type="ref" reference="Formula: S^g 4"}) we have $$\begin{aligned} S^g(W_\bullet; l) &=&\frac{1}{2!} \frac{1}{\mathbf{v}^g} \int_{\mathbb R} \int^3_{-1} g(w) {\rm vol}({\mathcal F}^{(y)}W_{(1,w)}) {\rm d}w {\rm d}y \\ &=& \frac{1}{2\mathbf{v}^g} \Big( \int_{-1}^0 g(w) \int_0^{3+2w} (3+2w-y)^2 {\rm d}y {\rm d}w + \int_0^3 g(w) \int_0^{3-w} (3-w-y)^2 {\rm d}y {\rm d}w \Big) \\ &\approx& 0.773902 \,\, <1, \end{aligned}$$ where $g(w) = e^{-\xi_0 w}$ and $\mathbf{v}^g \approx 5.61542$. Hence $A_X(l)/S^g(W_\bullet; l) >1$. Moreover, let $p_1,p_2,p_3\in C$ be the points that $l$ and $C$ intersect at (we just assume that $l$ intersects $C$ transversally). Then $$\begin{aligned} W^l_{(1,w,y)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^1, {\mathcal O}(3+2w-y)\Big) & -1\le w < 0, 0\le y\le 3+2w,\\ w(p_1+p_2+p_3)+H^0\Big({\mathbb P}^1, {\mathcal O}(3-w-y)\Big) & 0\le w\le 3, 0\le y\le 3-w. \end{array} \right. \end{aligned}$$ Since $p\notin C$, we have $$\begin{aligned} {\mathcal F}^{(x)}_p W^l_{(1,w,y)} = \left\{\begin{array}{ll} xp+H^0\Big({\mathbb P}^1, {\mathcal O}(3+2w-y-x)\Big) & (w,y,x)\in \Delta_-,\\ xp+w(p_1+p_2+p_3)+H^0\Big({\mathbb P}^1, {\mathcal O}(3-w-y-x)\Big) & (w,y,x)\in \Delta_+. \end{array} \right. \end{aligned}$$ where $\Delta_-=\{-1\le w<0, 0\le y\le 3+2w, 0\le x\le 3+2w-y\}$ and $\Delta_+ = \{0\le w\le 3, 0\le y\le 3-w, 0\le x\le 3-w-y\}$. Hence $$\begin{aligned} S^g(W^l_\bullet; p) &=&\frac{1}{1!} \frac{1}{\mathbf{v}^g} \int^3_{-1} g(w) \Big( \int_{\Delta_-\cup \Delta_+} {\rm vol}({\mathcal F}^{(x)}W_{1,w,y}) {\rm d}x {\rm d}y \Big) {\rm d}w \\ &=& \frac{1}{\mathbf{v}^g} \Big( \int_{-1}^0 g(w) \int_0^{3+2w} \int_0^{3+2w-y} (3+2w-y-x) {\rm d}x {\rm d}y {\rm d}w \\ &&\quad +\int_0^3 g(w) \int_0^{3-w} \int_0^{3-w-y} (3-w-y-x) {\rm d}x {\rm d}y {\rm d}w \Big) \\ &\approx& 0.773902 \,\, <1, \end{aligned}$$ We conclude that $A_{{\mathbb P}^1}(p)/S^g(W^l_\bullet; p)>1$. **Remark 29**. *We remark here that the two $S^g$ computed above are the same. One may show that $S^g(W_\bullet;{\mathcal F}_l)= \int_\mathbf{O}yg(w)\cdot{\rm LE}= S^g(W^l_\bullet, p)$.* **Case (B).** If $p\in C$, and the tangent line is of multiplicity $2$ at $p$, and $E$ is the exceptional line of the $(2,1)$-weighted blowup $\tilde{{\mathbb P}}^2\to {\mathbb P}^2$. We denote by $L=\{X=0\}$ where $(X,Y)$ is a local coordinate of ${\mathbb P}^2$ at $p$ such that ${\rm ord}_E(X)=2, {\rm ord}_E(Y)=1$. Let $P_0$ be the unique singular point of $\tilde{{\mathbb P}}^2$, and $P_1=E\cap \tilde{L}, P_2=E\cap\tilde{C}$. The three points $P_0, P_1, P_2$ are different. For each $w, t$, the linear system ${\mathcal F}^{(t)}_E W_{(1,w)}$ admits Zariski decomposition $${\mathcal F}^{(t)}_E W_{(1,w)} = N({\mathcal F}^{(t)}_E W_{(1,w)}) + H^0({\mathbb P}^2, P({\mathcal F}^{(t)}_E W_{(1,w)})).$$ For $-1\le w\le0$, we have $$\begin{aligned} N({\mathcal F}^{(t)}_E W_{(1,w)}) = \left\{ \begin{array}{ll} tl & 0\le t \le 3+2w, \\ tl+(t-3-2w)\tilde{L} & 3+2w\le t\le 6+4w, \end{array} \right. \end{aligned}$$ For $0\le w\le 3$, we have $$\begin{aligned} N({\mathcal F}^{(t)}_E W_{(1,w)}) = \left\{ \begin{array}{ll} 2wl+ w\tilde{C} & 0\le t \le 2w, \\ tl+w\tilde{C} & 2w\le t\le 3+w, \\ tl+w\tilde{C}+(t-3-w)\tilde{L} & 3+w\le t \le 6. \end{array} \right. \end{aligned}$$ Then the positive part follows as $$\begin{aligned} P({\mathcal F}^{(t)}_E W_{(1,w)}) = \left\{ \begin{array}{ll} {\mathcal O}(3+2w) - N({\mathcal F}^{(t)}_E W_{(1,w)}) & -1\le w \le 0, \\ {\mathcal O}(3-w)+w(2l+\tilde{C})-N({\mathcal F}^{(t)}_E W_{(1,w)}) & 0\le w\le 3. \end{array} \right. \end{aligned}$$ The volume ${\rm vol}({\mathcal F}^{(t)}_E W_{(1,w)}) = P({\mathcal F}^{(t)}_E W_{(1,w)})^2$. Hence for $-1\le w\le 0$ $$\begin{aligned} {\rm vol}({\mathcal F}^{(t)}_E W_{(1,w)}) = \left\{\begin{array}{ll} (3+2w)^2-\frac{1}{2}t^2 & 0\le t \le 3+2w, \\ (3+2w)^2-\frac{1}{2}t^2 + (t-3-2w)^2 & 3+2w\le t\le 6+4w, \end{array} \right. \end{aligned}$$ and for $0\le w\le 3$, we have $$\begin{aligned} {\rm vol}({\mathcal F}^{(t)}_E W_{(1,w)}) = \left\{ \begin{array}{ll} (3-w)^2 & 0\le t \le 2w, \\ (3-w)^2 - \frac{1}{2}(t-2w)^2 & 2w\le t\le 3+w, \\ (3-w)^2 - \frac{1}{2}(t-2w)^2 +(t-3-w)^2 & 3+w\le t \le 6. \end{array} \right. \end{aligned}$$ Hence we have $$\begin{aligned} S^g(W_\bullet; E) &=& \frac{1}{2\mathbf{v}^g} \Big( \int_{-1}^0 g(w) \int_0^{6+4w} {\rm vol}({\mathcal F}^{(t)}_EW_{(1,w)}) {\rm d}t {\rm d}w \\ && \quad + \int_{0}^3 g(w) \int_0^{6} {\rm vol}({\mathcal F}^{(t)}_EW_{(1,w)}) {\rm d}t {\rm d}w \Big)\\ &\approx& 2.773902 \,\, < 3 \,\, =\,\, A_{{\mathbb P}^2}(E), \end{aligned}$$ that is, $A_{{\mathbb P}^2}(E)/S^g(W_\bullet; E)>1$. **Remark 30**. *[\[Remark: 2.28 compute S\^g using G\]]{#Remark: 2.28 compute S^g using G label="Remark: 2.28 compute S^g using G"} There is another way to compute such $S^g$ using formula ([\[Formula: S\^g 2\]](#Formula: S^g 2){reference-type="ref" reference="Formula: S^g 2"}). The concave transform of ${\mathcal F}_E$ on $\mathbf{O}$ is $G(w,y,x)=2x+y$, then $S^g(W_\bullet;E)= \frac{1}{\mathbf{v}^g}\int_\mathbf{O}(2x+y){\rm LE}^g \approx 2.773902$. But we do not know how to get the further step refinement in this way.* With the above calculation, taking the quotient of ${\mathcal F}_E$ we get the second step refinement $$\begin{aligned} W^E_{(1,w,t)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{t}{2})\Big) & -1\le w < 0, 0\le t \le 3+2w, \\ (t-3-2w)P_1+H^0\Big({\mathbb P}^1, {\mathcal O}(3+2w-\frac{t}{2})\Big) & -1\le w < 0, 3+2w\le t \le 6+4w, \\ wP_2 +H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{t}{2} -w)\Big) & 0\le w\le 3, 2w\le t\le 3+w, \\ wP_2 +(t-3-w)P_1 +H^0\Big({\mathbb P}^1, {\mathcal O}(3-\frac{t}{2})\Big) & 0\le w\le 3, 3+w\le t\le 6. \end{array} \right. \end{aligned}$$ Note that ${\rm Diff}_l(0) = \frac{1}{2} P_0$. Hence we have $$\begin{aligned} \delta^g({\mathbb P}^1;W_\bullet^E) ={\rm min}\Big\{\frac{1}{2 S^g(W^E_\bullet;P_0)}, \frac{1}{S^g(W^E_\bullet;P_1)}, \frac{1}{S^g(W^E_\bullet;P_2)}\Big\}. \end{aligned}$$ We first consider the filtration of $W^E_\bullet$ induced by $P_0$. The fiberwise volume is clear in this case $$\begin{aligned} {\rm vol}({\mathcal F}^s_{P_0}W^E_{(1,w,t)}) = \left\{ \begin{array}{ll} \frac{t}{2}-s & -1\le w < 0, 0\le t \le 3+2w,\\ 3+2w-\frac{t}{2}-s & -1\le w < 0, 3+2w\le t \le 6+4w,\\ \frac{t}{2} -w-s & 0\le w\le 3, 2w\le t\le 3+w,\\ 3-\frac{t}{2}-s & 0\le w\le 3, 3+w\le t\le 6. \end{array} \right. \end{aligned}$$ Hence $$\begin{aligned} S^g(W^E_\bullet; P_0) &=& \frac{1}{\mathbf{v}^g} \Big[ \int_{-1}^0 g(w) \Big( \int_0^{3+2w}\frac{t^2}{8}{\rm d}t+ \int_{3+2w}^{6+4w}\frac{1}{2} (3+2w-\frac{t}{2})^2 {\rm d}t \Big) {\rm d}w \\ && \quad +\int_0^3 g(w) \Big( \int_{2w}^{3+w} \frac{1}{2} (\frac{t}{2}-w)^2 {\rm d}t+ \int_{3+w}^6 \frac{1}{2} (3-\frac{t}{2})^2 {\rm d}t \Big) {\rm d}w \Big]\\ &\approx& 0.386951 \,\, < \frac{1}{2} \,\, . \end{aligned}$$ The filtrations induced by $P_1$ and $P_2$ are slightly different from that of $P_0$, we conclude that $$\begin{aligned} S^g(W^E_\bullet; P_1) - S^g(W^E_\bullet; P_0) &=& \frac{1}{\mathbf{v}^g} \Big[ \int_{-1}^0 g(w) \int_{3+2w}^{6+4w} (3+2w-\frac{t}{2})(t-3-2w) {\rm d}t {\rm d}w \\ && \quad +\int_0^3 g(w) \int_{3+w}^6 (3-\frac{t}{2})(t-3-w) {\rm d}t {\rm d}w \Big] \,\, \approx\,\, 0.386951, \\ S^g(W^E_\bullet; P_2) - S^g(W^E_\bullet; P_0) &=& \frac{1}{\mathbf{v}^g} \int_0^3 g(w) w \Big( \int_{2w}^{3+w} (\frac{t}{2} - w) {\rm d}t \\ && \quad + \int_{3+w}^6 (3-\frac{t}{2}) {\rm d}t \Big) {\rm d}w \,\, \approx\,\, 0.226098. \end{aligned}$$ Hence $\delta^g({\mathbb P}^1;W_\bullet^E) \approx \frac{1}{0.773902}$. **Remark 31**. *We have $S^g(W^E_\bullet; P_1)=2 S^g(W^E_\bullet; P_0)$ and $S^g(W^E_\bullet; P_2)+S^g(W^E_\bullet; P_0)=1$.* **Case (C).** If $p\in C$ and $L=\{X=0\}$ tangent to $C$ of multiplicity $3$ at $p$, where $(X,Y)$ is a local coordinate of ${\mathbb P}^2$ at $p$. Let $E$ be the exceptional curve of $(3,1)$-weighted blowup. With the same argument in the previous case or Remark [\[Remark: 2.28 compute S\^g using G\]](#Remark: 2.28 compute S^g using G){reference-type="ref" reference="Remark: 2.28 compute S^g using G"}, we have $S^g(W_\bullet;E)=\int_{\mathbf{O}}(3x+y){\rm LE}^g \approx 3.773902$. Hence $A_{{\mathbb P}^2}(E)/S^g(W_\bullet;E)\approx\frac{4}{3.773902}$, which is less than all the $A/S^g$ computed above. We will see that $\delta^g(X)$ is minimized by $E$. Firstly we have $$\begin{aligned} W^E_{(1,w,t)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{t}{3})\Big) & -1\le w < 0, 0\le t \le 3+2w, \\ \frac{1}{2}(t-3-2w)P_1+ H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{1}{2}(3+2w-\frac{t}{3}))\Big) & -1\le w < 0, 3+2w\le t \le 9+6w, \\ wP_2 +H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{t}{3} - w)\Big) & 0\le w\le 3, 3w\le t\le 3+2w, \\ wP_2 +\frac{1}{2}(t-3-2w)P_1 + H^0\Big({\mathbb P}^1, {\mathcal O}(\frac{1}{2}(3-\frac{t}{3}))\Big) & 0\le w\le 3, 3+2w\le t\le 9. \end{array} \right. \end{aligned}$$ Then for any point $P\ne P_1$ or $P_2$, we have $$\begin{aligned} S^g(W^E_\bullet; P) &=& \frac{1}{\mathbf{v}^g} \Big[ \int_{-1}^0 g(w) \Big( \int_0^{3+2w}\frac{1}{2}(\frac{t}{3})^2{\rm d}t+ \int_{3+2w}^{9+6w}\frac{1}{2} (\frac{1}{2}(3+2w-\frac{t}{3}))^2 {\rm d}t \Big) {\rm d}w \\ && \quad +\int_0^3 g(w) \Big( \int_{3w}^{3+2w} \frac{1}{2} (\frac{t}{3}-w)^2 {\rm d}t+ \int_{3+2w}^9 \frac{1}{2} (\frac{1}{2}(3-\frac{t}{3}))^2 {\rm d}t \Big) {\rm d}w \Big] \\ &\approx& 0.257967, \end{aligned}$$ $$\begin{aligned} S^g(W^E_\bullet; P_1) - S^g(W^E_\bullet; P) &=& \frac{1}{\mathbf{v}^g} \Big[ \int_{-1}^0 g(w) \int_{3+2w}^{9+6w} \frac{1}{2}(3+2w-\frac{t}{3})\cdot \frac{1}{2}(t-3-2w) {\rm d}t {\rm d}w \\ && \quad +\int_0^3 g(w) \int_{3+2w}^9 \frac{1}{2}(3-\frac{t}{3}) \cdot \frac{1}{2}(t-3-2w) {\rm d}t {\rm d}w \Big]\\ &\approx& 0.515935,\\ S^g(W^E_\bullet; P_2) - S^g(W^E_\bullet; P) &=& \frac{1}{\mathbf{v}^g} \int_0^3 g(w) w \Big( \int_{3w}^{3+2w} (\frac{t}{3}-w) {\rm d}t+ \int_{3+2w}^9 \frac{1}{2}(3-\frac{t}{3}) {\rm d}t \Big) {\rm d}w \\ &\approx& 0.226098. \end{aligned}$$ **Remark 32**. *We have $S^g(W^E_\bullet; P_1)=3 S^g(W^E_\bullet; P)$ and $S^g(W^E_\bullet; P_2)+2S^g(W^E_\bullet; P)=1$.* Note that ${\rm Diff}_l(0) = \frac{2}{3} P_0$, where $P_0$ is the unique singular point of the $(3,1)$-weighted blowup of ${\mathbb P}^2$. Hence we have $$\begin{aligned} \delta^g({\mathbb P}^1;W_\bullet^E) ={\rm min}\Big\{\frac{1}{3 S^g(W^E_\bullet;P_0)}, \frac{1}{S^g(W^E_\bullet;P_1)}, \frac{1}{S^g(W^E_\bullet;P_2)}\Big\} \approx \frac{1}{0.773902}. \end{aligned}$$ We conclude that $\delta^g({\mathbb P}^2; W_\bullet) \approx \frac{4}{3.773902}$ is minimized by the exceptional line $E$ of the $(3,1)$-weighted blowup of ${\mathbb P}^2$ at $p\in C$ where the tangent line has multiplicity $3$. ## Weighted K-polystability of №3.14 The smooth Fano threefold $X$ of №3.14 is the ordinary blowup of the Fano threefold $X_0$ of №2.28 at one point outside the plane containing $C$. We use the same notations as Section [7.1](#Subsection: №2.28){reference-type="ref" reference="Subsection: №2.28"}. We denote by $CC=\{y^2z-x(x-z)(x-\lambda z)=0\}$ the cone over the curve $C$ in ${\mathbb P}^3$, by $CC_{0,C}$ the strict transform of $CC$ via the two step blowups, and by $E_C, E_0$ the exceptional divisor of blowing up $C$ and the point $[1,0,0,0]$ respectively. We compute $R_\bullet=R(X,-K_X)=R(X,{\mathcal O}(4)-E_C-2E_0)$ on the affine open chart $X\setminus (CC_{0,C}\cup H_z)$. Then $u=w\cdot v, x=x_0\cdot z, y=y_0\cdot z$, and we have $E_0=\{z=0\}, E_C=\{v=0\}$. Hence $$\begin{aligned} H^0(X,-K_X) &=& H^0(X, -K_{X_0}-2E_0) \\ &=& {\mathcal F}^2_{E_0}\big(z^3 H^0({\mathcal O}(1)) \oplus w H^0({\mathcal O}(3))\big)\cdot z^{-2} \\ &=& z^2 \langle 1,x_0,y_0\rangle\oplus wz\cdot \langle 1,x_0,y_0\rangle^3 \oplus w^2v\langle 1,x_0,y_0\rangle^2. \end{aligned}$$ We take the faithful valuation by $\mathfrak{v}:w\mapsto (1,0,0), y_0\mapsto (0,1,0), x_0\mapsto (0,0,1)$, and we get a convex body by these sections. To make it is compatible with the moment polytope, we shift the first coordinate by one and denote the body by $\mathbf{O}$, which is spanned by vertices $$(-1,0,0), (-1,0,1), (-1,1,0), (0,0,0), (0,0,3), (0,3,0), (1,0,1), (1,0,3), (1,2,1).$$ The volume of $\mathbf{O}$ is $\frac{16}{3}$. Hence $\mathbf{O}$ is a Okounkov body of $X$ since $(-K_X)^3=32$. Solving ${\rm Fut}_g(1)=0$, we get the soliton candidate numarically as $\xi_0 \approx 0.5265255550640977$. **Theorem 33**. *Every smooth Fano threefold $X=X_{3.14}$ admits a Kähler-Ricci soliton.* *Proof.* We prove using the same argument as Theorem [Theorem 28](#Theorem. stability of 2.28){reference-type="ref" reference="Theorem. stability of 2.28"}. Note that the one step more blowup kills the ${\mathbb G}_a$-part of the automorphism of $X_0$ and hence $G={\rm Aut}^0(X)={\mathbb G}_m$. The minimal $G$-orbits of $X$ are exactly the closed subvarieties of $H_u$ and $E_0$. We shall take refinements by the toric divisors $H_u$ and $E_0$ on $X$ respectively. Let $W_\bullet$ be the refinement of $R_\bullet=R(-K_X)$ by $H_u$. For any point $p\in H_u$, we take $l$ be a line passing through $p$ if $p\ne C$, the $(k,1)$ blowup of $p$ if $p\in C$ and the tangent line $L$ of $C$ at $p$ has multiplicity $k$ at $p$. It suffices to show $\delta_l = \frac{A_{{\mathbb P}^2}(l)}{S^g(W_\bullet;l)}>1,\delta_p = \delta^g({\mathbb P}^1, \Delta_l; W^E_\bullet) >1$, see Section [7.4](#Subsection: computing S^g of №3.14){reference-type="ref" reference="Subsection: computing S^g of №3.14"}. For points $p\in E_0$, the computation of $\delta_l$ and $\delta_p$ is totally the same as $p\in H_u$. The values of invariants are also the same. Hence the same argument of Theorem [Theorem 28](#Theorem. stability of 2.28){reference-type="ref" reference="Theorem. stability of 2.28"} shows that $X$ is weighted K-polystable. ◻ ## Computing $S^g$ of №3.14 {#Subsection: computing S^g of №3.14} Firstly we have $$\begin{aligned} W_{(1,w)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^2, {\mathcal O}(3+2w)\Big) & -1\le w < 0,\\ wC+H^0\Big({\mathbb P}^2, {\mathcal O}(3-w)\Big) & 0\le w\le 1. \end{array} \right. \end{aligned}$$ Note the formula of $S^g$ remains the same with the computation of №2.28 except we change the integration interval $0\le w \le 3$ into $0 \le w \le 1$. Here we list the results below: **(A).** If $p \notin C$, then we choose a general line $l$ on $H_u$ passing through $p$. Then $$S^g(W_{\bullet};l) = S^g(W^{E}_{\bullet};p) \approx 0.806338.$$ Hence $\delta^g_p({\mathbb P}^2, W_\bullet)\ge {\rm min}\{ \frac{1}{S^g(W_\bullet; l)}, \frac{1}{S^g(W^E_\bullet; p)} \} >1;$ **(B).** If $p\in C$ and the tangent line is of multiplicity $2$, let $E$ be the exceptional line of the $(2,1)$-weighted blowup $\tilde{{\mathbb P}}^2\to {\mathbb P}^2$. Denote by $P_0$ the singular point of $\tilde{{\mathbb P}}^2$ and $P_1=\tilde{L}\cap E, P_2=\tilde{C} \cap E$. We know that $P_0, P_1$, and $P_2$ are different points. Then $$\begin{aligned} S^g(W_{\bullet};E) \approx 2.806338, &&S^g(W^{E}_{\bullet};P_0) \approx 0.403169, \\ S^g(W^{E}_{\bullet};P_1) \approx 0.806338, &&S^g(W^{E}_{\bullet};P_2) \approx 0.596831. \end{aligned}$$ Hence $\delta^g_p({\mathbb P}^2, W_\bullet)\ge {\rm min}\{ \frac{3}{S^g(W_\bullet; E)}, \frac{1}{2S^g(W^E_\bullet; P_0)}, \frac{1}{S^g(W^E_\bullet; P_1)}, \frac{1}{S^g(W^E_\bullet; P_2)}\} >1;$ **(C).** If $p\in C$ and the tangent line is of multiplicity $3$, let $E$ be the exceptional line of the $(3,1)$-weighted blowup $\tilde{{\mathbb P}}^2\to {\mathbb P}^2$. Denote by $P_0$ the singular point of $\tilde{{\mathbb P}}^2$ and $P_1=\tilde{L}\cap E, P_2=\tilde{C} \cap E$. The points $P_0, P_1, P_2$ are different. Then $$\begin{aligned} S^g(W_{\bullet};E) \approx 3.806338, &&S^g(W^{E}_{\bullet};P_0) \approx 0.268799, \\ S^g(W^{E}_{\bullet};P_1) \approx 0.806338, &&S^g(W^{E}_{\bullet};P_2) \approx 0.462442. \end{aligned}$$ Hence $\delta^g_p({\mathbb P}^2, W_\bullet)\ge {\rm min}\{ \frac{4}{S^g(W_\bullet; E)}, \frac{1}{3S^g(W^E_\bullet; P_0)}, \frac{1}{S^g(W^E_\bullet; P_1)}, \frac{1}{S^g(W^E_\bullet; P_2)}\} >1.$ # GIT-stability and Weighted K-stability of Fano threefolds {#Section: GIT and weighted K, 2.28 and 3.14} In this section, we show that the weighted K-stability of $(X,\xi_0)$ is equivalent to the GIT-stability of plane cubic curves, where $X$ is a Fano threefold in the family №2.28 or №3.14. **Theorem 34**. *Let $X$ be the blowup of ${\mathbb P}^3$ along a plane cubic curve $C\subseteq H\cong {\mathbb P}^2$ or further blowing up a point outside $H$. Let $\xi_0$ be the soliton candidate of $X$. Then $(X, \xi_0)$ is weighted K-semistable (K-polystable) if and only if $C\subseteq H$ is GIT-semistable (GIT-stable or polystable).* The proof follows from the same calculation in the previous section, where the only difference is that we shall change the function $v$ of $C$ respectively. We will only do for №2.28. The computation for №3.14 is similar. It is well known that - the plane cubic curve $C$ is GIT-stable if and only if it is smooth ($v=y^2z-x(x-z)(x-\lambda z), \lambda\ne 0,1$); - it is strictly GIT-semistable if and only if it is a nodal cubic ($v=y^2z-x^2(x-z)$) or the union of a conic and a secant line ($v=xyz-x^3$); - it is GIT-polystable if and only if it is the union of three lines ($v=xyz$) that do not intersect at one point; - it is GIT unstable for other cases, that is, cuspidal cubic ($v=y^2z-x^3$) or the union of a conic and a tangent line ($v=y^2z-x^2y$) or the union of three lines with one common point ($v=y^3-x^2y$). In the first case, we have shown that $(X, \xi_0)$ is K-polystable in the previous section. **Lemma 35**. *In the second and the third cases, $(X,\xi_0)$ is K-semistable. In the fourth case, $(X,\xi_0)$ is K-unstable.* *Proof.* We have the ${\mathbb N}^2$-graded linear series on $H_u$ $$\begin{aligned} W_{(1,w)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^2, {\mathcal O}(3+2w)\Big) & -1\le w < 0,\\ wC+H^0\Big({\mathbb P}^2, {\mathcal O}(3-w)\Big) & 0\le w\le 3. \end{array} \right. \end{aligned}$$ In the fourth case, let $l$ be the exceptional line of $(2,3)$-blowup of the point $\{x=y=0\}$. Then ${\rm ord}_l(v)\ge6$ and we have $S^g(W_\bullet; l)\ge\frac{1}{\mathbf{v}^g}\int_\mathbf{O}(2y+3y+6(x-y)){\rm LE}^g\approx 5.226098>5=A_{{\mathbb P}^2}(l)$. Pulling-back $l$ to the cone $\tilde{l}$ over $l$ as a vertical divisor over $X$ we see that $$\frac{A_X(\tilde{l})}{S^g(R_\bullet;\tilde{l})} =\frac{A_{H_u}(l)}{S^g(W_\bullet;l)} <1,$$ hence $(X,\xi_0)$ is K-unstable. In the second and the third cases, $C$ has only an ordinary double point. Let $l$ be the exceptional line of the $(1,1)$-blowup of the ordinary double point. We have ${\rm ord}_l(v)=2, {\rm ord}_l(x)={\rm ord}_l(y)=1$. Hence we have $S^g(W_\bullet; l)=\frac{1}{\mathbf{v}^g}\int_\mathbf{O}(y+y+2(x-y)){\rm LE}^g=2=A_{{\mathbb P}^2}(l)$. Suppose that $\tilde{C}\cap l= \{P_1, P_2\}$, then the refinement of $W_\bullet$ by $l$ is $$\begin{aligned} W^l_{(1,w,t)} = \left\{ \begin{array}{ll} H^0\Big({\mathbb P}^1, {\mathcal O}(t)\Big) & -1\le w < 0, 0\le t\le 3+2w,\\ w(P_1+P_2)+H^0\Big({\mathbb P}^1, {\mathcal O}(t-2w)\Big) & 0\le w\le 3, 2w\le t\le 3+w. \end{array} \right. \end{aligned}$$ For $P\ne P_1, P_2$ we have $S^g(W^l_\bullet; P) \approx 0.587831$ and $S^g(W^l_\bullet; P_1) = S^g(W^l_\bullet; P_2) \approx 0.625755$. We conclude by the weighted Abban-Zhuang estimate that $(X, \xi_0)$ is K-semistable. ◻ Next, we show that $(X,\xi_0)$ is strictly K-semistable in the second case. Note that the nodal cubic $v=y^2z-x^2(x-z)$ is degenerated by $\lambda(t)\cdot [u,x,y,z]=[u,t^{-1}x, t^{-1}y,z]$ to the union of three lines with no common point $v=z(y-x)(y+x)$. The union of a conic with a secant line $v=xyz-x^3$ is degenerated by $\lambda(t)\cdot [u,x,y,z]=[u,t^{-1}x, y,z]$ to the union of three lines with no common point $v=xyz$. These degenerations induce the special degenerations of $(X,\xi_0)$ in the second case to the $(X, \xi_0)$ in the third case. It remains to show that $(X,\xi_0)$ is K-polystable in the third case. **Lemma 36**. *Let $C$ be the union of three lines with no common point, and $X$ the blowup of ${\mathbb P}^3$ along $C$. Then $(X,\xi_0)$ is K-polystable.* *Proof.* Let $X$ be the blowup of ${\mathbb P}^3$ along $\{u=xyz=0\}$. Note that $X$ is toric in this case, and it admits three ${\mathbb G}_m$-action: the horizontal one ${\mathbb T}_h=[t^{-1}u,x,y,z]$ and the vertical ones ${\mathbb T}_v={\mathbb G}_m^2$: $[u,t^{-1}x,y,z]$ and $[u, x, t^{-1}y,z]$. We denote by ${\mathbb T}={\mathbb T}_h\times{\mathbb T}_v$. One may compute directly that the $h$-minimizer $\xi_1\in N({\mathbb T})_{\mathbb R}$ (the soliton candidate of the toric variety $X$) indeed living in $N({\mathbb T}_h)_{\mathbb R}$ and is just $\xi_0$. Hence by [@BLXZ23 Theorem 5.11], $(X,\xi_0)$ is weighted K-polystable. ◻ *Proof of Theorem [Theorem 34](#Theorem: equivalence of weighted K-stability of №2.28 and №3.14 to the GIT-stability of plane cubic){reference-type="ref" reference="Theorem: equivalence of weighted K-stability of №2.28 and №3.14 to the GIT-stability of plane cubic"}.* It follows directly from Lemma [Lemma 35](#Lemma: K-moduli semistable and unstable){reference-type="ref" reference="Lemma: K-moduli semistable and unstable"} and Lemma [Lemma 36](#Lemma: K-moduli polystable){reference-type="ref" reference="Lemma: K-moduli polystable"}. ◻ # Weighted K-stability of cones and projective bundles {#Section: Weighted K-stability of cones} We establish the equivalence between the K-stability of a Fano manifold $V$ and the weighted K-stability of a projective cone $Y$ and bundle $\tilde{Y}$ over $V$ in this section, which generalizes the so-called Koiso's theorem [@Koi90]. We first recall some basic notions. ## Weighted K-stability of log Fano cones Let $(V, \Delta_V)$ be a $(n-1)$-dimensional log Fano pair such that $L=-\frac{1}{r}(K_V+\Delta_V)$ is an ample Cartier divisor for some $0<r\le n$. We define the projective cone over $V$ with polarization $L$ by $$Y=\overline{{\mathcal C}}(V,L):={\rm Proj}\Big(\bigoplus_{m\ge0}\bigoplus_{0\le\lambda\le m}H^0(V, (m-\lambda)L)s^\lambda\Big),$$ which is the union of the affine cone ${\mathcal C}(V,L)={\rm Spec}\big(\oplus_{m\ge0}H^0(V, m L)\big)$ and a divisor $V_\infty =\{s=0\}$ at infinity. It admits a ${\mathbb G}_m$-action along the cone direction, which we denote by ${\mathbb T}_c$. Let $\Delta_Y$ be the closure of $\Delta_V\times {\mathbb C}^*$ in $Y$. Then $-(K_{Y}+\Delta_Y)$ is Cartier and $$-(K_Y+\Delta_Y)\sim_{\mathbb Q}(1+r)V_\infty.$$ Hence $(Y, \Delta_Y)$ is also log Fano, see for example [@ZZ22 Lemma 2.1]. Let $\tilde{Y}\to Y$ be the blowup of the vertex, $V_0$ be the exceptional divisor and $\Delta_{\tilde{Y}}$ be the strict transform of $\Delta_Y$. Then $\tilde{Y}\cong {\mathbb P}_V({\mathcal O}_V\oplus L)$ is a ${\mathbb P}^1$-bundle over $V$. We have weight decomposition of $R=R(Y, \Delta_Y)$ via the ${\mathbb T}_c$-action, $$H^0(Y, -m(K_Y+\Delta_Y)) = \bigoplus_{0\le\lambda\le m(1+r)}H^0(V, (m(1+r)-\lambda)L)s^\lambda$$ The toric divisor of the ${\mathbb T}_c$-action is just $V_\infty$ and $V_0$. Since $A_{Y,\Delta_Y}(V_\infty)=1, A_{Y, \Delta_Y}(V_0)=r$, the moment polytope of this ${\mathbb T}_c$-action is just $\mathbf{P}=[-1, r]\subseteq M_{\mathbb R}= {\mathbb R}$ and the DH measure is ${\rm DH}_\mathbf{P}({\rm d}\alpha) = \frac{(r-\alpha)^{n-1}L^{n-1}}{(n-1)!} {\rm d}\alpha$. We set the soliton candidate $\xi_0$ to be the solution of ${\rm Fut}_g(1)=0$ where $g(\alpha)=e^{-\alpha\cdot\xi_0}$, that is $$\int_{-1}^r \alpha \cdot g(\alpha) (r-\alpha)^{n-1} {\rm d}\alpha = 0.$$ **Lemma 37**. *$\delta^{g}_{{\mathbb T}_c}(Y,\Delta_Y) = {\rm min}\big\{1, \,\, \delta(V,\Delta_V) \big\}.$* *Proof.* Let $W_\bullet$ be the refinement of $R_\bullet=R(Y, \Delta_Y)$ by $V_\infty$. We claim that $\delta^{g}_{p}(V,\Delta_V;W_\bullet)=\delta_{p}(V,\Delta_V)$ for any $p\in V_\infty$. It suffices to show that for any valuation $v$ on $V$, we have $S^g(W_\bullet; v)=rS(L;v)$. Since $W_{(1,\alpha)}=(r-\alpha)L$ for any $-1\le \alpha \le r$, we have $$\begin{aligned} S^g(W_\bullet; v) &=& \frac{1}{(n-1)!\mathbf{v}^g} \int_{-1}^r g(\alpha) \int_{0}^{\infty} {\rm vol}({\mathcal F}_v^{(t)}(r-\alpha)L) {\rm d}t {\rm d}\alpha \\ &=& \frac{1}{(n-1)!\mathbf{v}^g} \int_{-1}^r g(\alpha) (r-\alpha)^{n} {\rm d}\alpha \cdot \int_{0}^{\infty} {\rm vol}({\mathcal F}_v^{(t)}L) {\rm d}t. \end{aligned}$$ On the other hand $(n-1)!\mathbf{v}^g = \int_{-1}^r g(\alpha) (r-\alpha)^{n-1} {\rm d}\alpha \cdot {\rm vol}(L)$. Hence $$\begin{aligned} S^g(W_\bullet; v) = \frac{\int_{-1}^r g(\alpha) (r-\alpha)^{n} {\rm d}\alpha} {\int_{-1}^r g(\alpha) (r-\alpha)^{n-1} {\rm d}\alpha} \cdot S(L;v). \end{aligned}$$ Since $(r-\alpha)^n= (r-\alpha)\cdot (r-\alpha)^{n-1}$ and $\int_{-1}^r \alpha\cdot g(\alpha) (r-\alpha)^{n-1}{\rm d}\alpha=0$, we conclude that $$\int_{-1}^r g(\alpha) (r-\alpha)^{n}{\rm d}\alpha =r\int_{-1}^r g(\alpha) (r-\alpha)^{n-1}{\rm d}\alpha.$$ By Theorem [Theorem 20](#Theorem: weighted AZ){reference-type="ref" reference="Theorem: weighted AZ"}, for any point $p\in V_\infty$, we have $$\begin{aligned} \delta^{g}_{p, {\mathbb T}_c}(Y,\Delta_Y;R_\bullet) &\ge& {\rm min}\Big\{\frac{A_{Y,\Delta_Y}(V_\infty)}{ S^{g}(R_\bullet; V_\infty)}, \,\, \delta^{g}_{p}(V,\Delta_V;W_\bullet) \Big\}\\ &=& {\rm min}\Big\{1, \,\, \delta_{p}(V,\Delta_V) \Big\}. \end{aligned}$$ On the other hand, we should also consider the refinement $\tilde{W}_\bullet$ of $R_\bullet$ by $V_0$. One can show that $\tilde{W}_{(1,\lambda)}=W_{(1,r-\lambda)} (0\le \lambda\le 1+r)$ on $V$, hence they have the same $S^g$ for any valuation on $V$. We conclude that $\delta^{g}_{{\mathbb T}_c}(Y,\Delta_Y) \ge {\rm min}\big\{1, \,\, \delta(V,\Delta_V) \big\}.$ For the reverse inequality, it follows directly from $$\delta^g_{{\mathbb T}_c}(Y,\Delta_Y) \le \frac{A_{Y,\Delta_Y}(v)}{S^g(R_\bullet;v)} =\frac{A_{V,\Delta_V}(v_0)}{S^g(W_\bullet;v_0)} =\frac{A_{V,\Delta_V}(v_0)}{S(-K_V-\Delta_V;v_0)},$$ where $v_0$ is a valuation on $V$ and $v$ is the pull-back of $v_0$ to $Y$. ◻ We have the following generalization of Koiso's theorem [@Koi90]. **Theorem 38**. *The log Fano triple $(Y, \Delta_Y, \xi_0)$ is weighted K-semistable (weighted K-polystable) if and only if $(V,\Delta_V)$ is K-semistable (K-stable or K-polystable).* *Proof.* The equivalence of semistability follows from the above lemma. For the equivalence of polystability, we prove with the same argument of Theorem [Theorem 28](#Theorem. stability of 2.28){reference-type="ref" reference="Theorem. stability of 2.28"}. Let ${\mathbb T}_V = {\mathbb T}(V,\Delta_V)$ be a maximal torus of ${\rm Aut}(V,\Delta_V)$. Then the ${\mathbb T}_V$-action can be lifted to $(Y,\Delta_Y)$, and ${\mathbb T}={\mathbb T}_c\times {\mathbb T}_V$ is a maximal torus of $(Y, \Delta_Y)$. Then it also follows from $$\begin{aligned} \frac{A_{Y,\Delta_Y}(v)}{S^g(R_\bullet;v)} =\frac{A_{V,\Delta_V}(v_0)}{S(-K_V-\Delta_V;v_0)}, \end{aligned}$$ where $v_0$ is a valuation on $V$ and $v$ is the pull-back of $v_0$ to $Y$. Indeed, by [@BLXZ23] the log Fano triple $(Y,\Delta_Y,\xi_0)$ is strictly weighted K-semistable if and only if there exists a valuation $v\ne{\rm wt}_\xi, \xi\in N({\mathbb T})_{\mathbb R}$ such that $A_{Y,\Delta_Y}(v)=S^g(R_\bullet;v)$. By the above equality, this holds if and only if there exists a valuation $v_0$ on $V$ whose pull-back to $Y$ is $v$, such that $v_0\ne {\rm wt}_\xi, \xi\in N({\mathbb T}_V)_{\mathbb R}$ and $A_{V,\Delta_V}(v_0)=S(-K_V-\Delta_V;v_0)$. This is equivalent that $(V,\Delta_V)$ is strictly K-semistable by [@LXZ22]. ◻ The theorem will lead to new examples of weighted K-moduli spaces based on the existence of K-moduli spaces of log Fano pairs [@LXZ22]. In particular, it gives us examples of strictly weighted K-semistable Fano varieties. Let $\pi: ({\mathcal V}, {\mathcal L}) \to {\mathbb A}^1$ be a TC of $(V, L)$. We can also define the cone over $({\mathcal V}, {\mathcal L})$ $${\mathcal Y} = \overline{{\mathcal C}}({\mathcal V},{\mathcal L}) :={\rm Proj}_{{\mathbb A}^1} \Big( \bigoplus_{m\ge0}\bigoplus_{0\le\lambda\le m} \pi_*({\mathcal L}^{\otimes(m-\lambda)}) \cdot s^\lambda \Big),$$ which gives a TC of $Y$ whose central fiber is the cone ${\mathcal Y}_0=\overline{{\mathcal C}}({\mathcal V}_0, {\mathcal L}_0)$. The ${\mathbb T}_c$-action on $Y$ extends to the whole family ${\mathcal Y}$ and is compatible with the ${\mathbb T}_c$-action on the central fiber. Hence we have **Corollary 39**. *Let $V$ be a strictly K-semistable Fano manifold with $-K_V \sim_{\mathbb Q}r L$ for some ample Cartier divisor $L$ and $0<r \le 1$. Then the cone $(Y=\overline{{\mathcal C}}(V; L), \xi_0)$ is strictly weighted K-semistable.* **Example 40**. *It was shown in [@Tia97] that there exists a Fano manifold $V$ in the family №1.10 of Mori and Mukai's list such that there exists a non-product type special degeneration $({\mathcal V}, {\mathcal L})$ of $(V, L)$ whose central fiber $({\mathcal V}_0, {\mathcal L}_0)$ is the Mukai-Umemura manifold. Since ${\mathcal V}_0$ is K-polystable, we know that $V$ is strictly K-semistable. The above Corollary gives us a four-dimensional Fano variety $Y=\overline{{\mathcal C}}(V,L)$ with the vertex as the only singular point, which has a special degeneration to the cone ${\mathcal Y}_0=\overline{{\mathcal C}}({\mathcal V}_0,{\mathcal L}_0)$ over the Mukai-Umemura manifold. By Theorem [Theorem 38](#Theorem: soliton of cone){reference-type="ref" reference="Theorem: soliton of cone"} we see that $({\mathcal Y}_0, \xi_0)$ is weighted K-polystable. Hence $(Y,\xi_0)$ is strictly weighted K-semistable.* ## Weighted K-stability of projective bundles over log Fano pairs Let $r\in{\mathbb Q}_{>0}, (V, \Delta_V)$ and $(\tilde{Y}, \Delta_{\tilde{Y}})$ be the same as above. The pair $(\tilde{Y}, \Delta_{\tilde{Y}})$ is log Fano if $r>1$. For $0<r\le 1$, the anti-canonical divisor of $(\tilde{Y}, \Delta_{\tilde{Y}})$ is not ample. Instead, we consider the log Fano pair $(\tilde{Y}, \Delta_{\tilde{Y}}+aV_0)$ where $1-r<a<1$. The ${\mathbb T}_c$-action of $(Y,\Delta_Y)$ lifts to $(\tilde{Y}, \Delta_{\tilde{Y}}+V_0)$ naturally. If $r>1$, then $A_{\tilde{Y}, \Delta_{\tilde{Y}}}(V_0)=A_{\tilde{Y}, \Delta_{\tilde{Y}}}(V_\infty)=1$. We have weight decomposition $$H^0(\tilde{Y}, -m(K_{\tilde{Y}}+\Delta_{\tilde{Y}})) = \bigoplus_{-1\le\alpha\le 1, m\alpha \in {\mathbb N}} H^0(V, m(r-\alpha)L)s^{m(1+\alpha)}.$$ If $r\le 1$, then $A_{\tilde{Y}, \Delta_{\tilde{Y}}+aV_0}(V_0)=1-a$ and $A_{\tilde{Y}, \Delta_{\tilde{Y}}+aV_0}(V_\infty)=1$. We also have $$H^0(\tilde{Y}, -m(K_{\tilde{Y}}+\Delta_{\tilde{Y}}+aV_0)) = \bigoplus_{-1\le\alpha\le 1-a, m\alpha \in {\mathbb N}} H^0(V, m(r-\alpha)L)s^{m(1+\alpha)}.$$ Hence the moment polytope $\mathbf{P}=[-1,1]$ for $r>1$ and $\mathbf{P}=[-1,1-a]$ for $0<r\le 1$, and the DH measure is ${\rm DH}_\mathbf{P}({\rm d}\alpha) = \frac{(r-\alpha)^{n-1}L^{n-1}}{(n-1)!} {\rm d}\alpha$. Solving equation in $\xi_0$ $$\int_\mathbf{P}\alpha \cdot g(\alpha) (r-\alpha)^{n-1} {\rm d}\alpha = 0, \quad g(\alpha)=e^{-\alpha\cdot\xi_0},$$ we get the soliton candidate $\xi_0\in{\mathbb R}$. Following the same proof of Theorem [Theorem 38](#Theorem: soliton of cone){reference-type="ref" reference="Theorem: soliton of cone"}, we also have **Theorem 41**. *The log Fano triples $(\tilde{Y}, \Delta_{\tilde{Y}}, \xi_0)$ for $r>1$ and $(\tilde{Y}, \Delta_{\tilde{Y}}+aV_0, \xi_0)$ for $0<r\le 1$ are weighted K-semistable (weighted K-polystable) if and only if $(V,\Delta_V)$ is K-semistable (K-stable or K-polystable).* # Proof of Lemma [Lemma 13](#Lemma: uniform bound of S^g_m/S^g){reference-type="ref" reference="Lemma: uniform bound of S^g_m/S^g"} {#proof-of-lemma-lemma-uniform-bound-of-sg_msg} It follows directly from the same argument of [@BJ20 Corollary 2.10] based on the following Lemma, which is a minor strengthening of [@BJ20 Lemma 2.2]. **Lemma 42**. *For any $\varepsilon>0$ there exists $m_0=m_0(\varepsilon)$ such that $\int_\mathbf{O}G \cdot {\rm LE}^g_m \le \int_\mathbf{O}G\cdot{\rm LE}^g + \varepsilon$ for any $m\ge m_0$ and every concave function $G$ on $\mathbf{O}$ satisfying $0\le G\le 1$.* *Proof.* For any $t>0$, let $\mathbf{O}^t=\{\alpha\in{\mathbb R}^n:\alpha+[-t,t]\subseteq\mathbf{O}\}$. This is a closed subset of $\mathbf{O}$ and converges to $\mathbf{O}$ as $t\to 0$. Fix $\varepsilon>0$. There exists $t>0$ such that ${\rm LE}(\mathbf{O}\setminus\mathbf{O}^{2t})<\varepsilon/4$. Since ${\rm LE}_m\to {\rm LE}$ converges weakly, we have ${\rm lim\,sup}\, {\rm LE}_m(\mathbf{O}\setminus\mathbf{O}^{t})\le {\rm LE}(\mathbf{O}\setminus\mathbf{O}^{2t}) \le \varepsilon/4$. So there exists $m_1$ such that ${\rm LE}_m(\mathbf{O}\setminus\mathbf{O}^{t})\le \varepsilon/2$ for any $m\ge m_1$. Note that $g\circ p$ is uniformly continuous on $\mathbf{O}$. Hence there exists $m_2$ such that $|g(p(\alpha))-g(p(\beta))| < \varepsilon/(2{\rm LE}(\mathbf{O}))$ for any $\alpha, \beta\in\mathbf{O}$ with $|\alpha-\beta|\le 2/m_2$. We may choose $m_0$ such that $m_0 \ge m_1,m_2$ and $t^{-1}$. For any $m\ge m_0$, we set $$\begin{aligned} A'_m &:=& \{\alpha \in \frac{1}{m}{\mathbb Z}^n: \alpha + [0,\frac{1}{m}]^n \subseteq\mathbf{O}\}, \\ A_m &:=& \{\alpha \in \frac{1}{m}{\mathbb Z}^n: \alpha + [-\frac{1}{m},\frac{1}{m}]^n \subseteq\mathbf{O}\}.\end{aligned}$$ It's clear that $\mathbf{O}^t\cap \frac{1}{m}{\mathbb Z}^n \subseteq A_m \subseteq A'_m$. For any $\alpha\in A'_m$, we denote by $g^\alpha_{\rm max}, g^\alpha_{\rm min}$ the maximum and minimum value of $g\circ p$ on $\alpha+[0,\frac{1}{m}]^n$ respectively. Since $m\ge m_2$, we have $g^\alpha_{\rm max}-g^\alpha_{\rm min}\le \varepsilon/(2{\rm LE}(\mathbf{O}))$. We define $$\begin{aligned} f(\alpha) &=& \Big( \sum_{w\in \{0,\frac{1}{m}\}^n} 2^{-n}(G\cdot g\circ p)(\alpha +w) \Big) - \int_{[0,\frac{1}{m}]^n} (G\cdot g\circ p)(\alpha +w) {\rm d}w . \end{aligned}$$ Then we have $$\begin{aligned} f(\alpha) &\le& g^\alpha_{\rm max}\Big( \sum_{w\in \{0,\frac{1}{m}\}^n} 2^{-n}G(\alpha +w) - \int_{[0,\frac{1}{m}]^n} G(\alpha +w) {\rm d}w \Big) \\ && + (g^\alpha_{\rm max}-g^\alpha_{\rm min}) \int_{[0,\frac{1}{m}]^n} G(\alpha +w) {\rm d}w . \end{aligned}$$ The first term on the right-hand side of the inequality is non-positive by the concavity of $G$. Hence $$\begin{aligned} f(\alpha) \le \frac{\varepsilon}{2{\rm LE}(\mathbf{O})}\cdot \int_{\alpha + [0,\frac{1}{m}]^n} G\cdot {\rm LE}. \end{aligned}$$ Then $\sum_{\alpha\in A'_m} f(\alpha) \le \varepsilon/2$. We are ready to give the desired estimate. $$\begin{aligned} &&\int_\mathbf{O}G\cdot{\rm LE}^g \,\,=\,\, \int_\mathbf{O}G\cdot g\circ p\cdot {\rm LE}\\ &\ge& \sum_{\alpha\in A'_m} \int_{\alpha + [0,\frac{1}{m}]^n} G\cdot g\circ p\cdot {\rm LE} \,\,=\,\, \sum_{\alpha\in A'_m} \int_{[0,\frac{1}{m}]^n} (G\cdot g\circ p)(\alpha +w) {\rm d}w \\ &=& \sum_{\alpha\in A'_m} \Big( \sum_{w\in \{0,\frac{1}{m}\}^n} 2^{-n}(G\cdot g\circ p)(\alpha +w) -f(\alpha) \Big) \,\,\ge\,\, \sum_{\alpha\in A_m} (G\cdot g\circ p)(\alpha ) -\Big( \sum_{\alpha\in A'_m} f(\alpha) \Big) \\ &\ge& \int_{\mathbf{O}^t} (G\cdot g\circ p)\cdot {\rm LE}_m -\frac{\varepsilon}{2} \,\,\ge\,\, \int_{\mathbf{O}} (G\cdot g\circ p)\cdot {\rm LE}_m -{\rm LE}_m(\mathbf{O}\setminus \mathbf{O}^t) -\frac{\varepsilon}{2} \,\,\ge\,\, \int_{\mathbf{O}} G\cdot {\rm LE}^g_m -\varepsilon. \end{aligned}$$ ◻

arxiv_math

--- abstract: | Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers, which typically require significant effort, and the presence of nonlinearity in the equation makes such designs even more challenging. Emerging deep learning-based methods for solving PDEs, with the physics-informed neural networks (PINNs) method as a representative, still face significant challenges when directly used to solve PDEs on unbounded domains. Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs. In light of this, this paper proposes a novel and effective operator learning-based method for solving PDEs on unbounded domains. The key idea behind this method is to generate high-quality training data. Specifically, we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term. Then, using these constructed data comprising exact solutions, initial conditions, and source terms, we train an operator learning model called MIONet, which is capable of handling multiple inputs, to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest. Finally, we utilize the generalization ability of this model to predict the solution of the target PDE. The effectiveness of this method is exemplified by solving the wave equation and the Schrödinger equation defined on unbounded domains. More importantly, the proposed method can deal with nonlinear problems, which has been demonstrated by solving Burger's equation and Korteweg-de Vries (KdV) equation. This ability to handle nonlinearity is expected to bring significant benefits in solving diverse real-world problems involving complex systems. address: - Research Center for Applied Mathematics and Machine Intelligence, Zhejiang Lab, Hangzhou 311121, China - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China author: - Jihong Wang - Xin Wang - Jing Li - Bin Liu bibliography: - DNN.bib - ABM.bib title: "**Data Generation-based Operator Learning for Solving Partial Differential Equations on Unbounded Domains**" --- Scientific machine learning ,Operator learning,Unbounded domain,Nonlinear PDEs # Introduction Real-world wave propagation problems in various fields, such as acoustics, aerodynamics, solid geophysics, oceanography, meteorology, and electromagnetics, are commonly described by partial differential equations (PDEs) on unbounded (or very large) domains. However, solving these PDEs numerically poses challenges due to the infinite domains involved. Standard domain-based numerical methods like finite difference and finite element methods are not directly applicable to solving such unbounded PDEs. The main reason is that these methods rely on discretizing the domain into a finite set of points or elements, leading to a reduced algebraic system with a finite number of degrees of freedom. However, when dealing with unbounded domains, these approaches lead to an algebraic system with an infinite number of degrees of freedom that cannot be effectively solved. Two popular techniques to deal with the problems on unbounded domains are the artificial boundary method (ABM) [@han2013artificial; @antoine2008review; @givoli2004high] and perfectly matched layer (PML) [@berenger1994perfectly; @pled2022review]. The ABM involves designing suitable absorbing/artificial boundary conditions (ABCs) that are satisfied by the solution of the original problem on the artificial boundaries. This approach reduces the original unbounded problem to a well-posed boundary value problem on the bounded computational domains of interest. The key ingredient of ABM is the construction of the ABCs. Based on the Fourier series expansion, Laplace transform, $z$-transform, Padé approximation, continued fraction expansion, and other techniques, exact or approximated ABCs are designed for various linear PDEs. However, it remains challenging to construct suitable ABCs for many nonlinear equations, even for some fundamental equations like the nonlinear Schrödinger equation [@han2013artificial; @zheng2006exact]. The PML method is used as absorbing layers that effectively eliminate reflections for all incident waves, regardless of their frequency and angle. PMLs have gained widespread usage due to their computational efficiency, ease of implementation, applicability to complex geometries, and high absorption accuracy. However, a major drawback of the PML method is its numerical instabilities in time-domain simulations for some wave propagation problems in anisotropic and/or dispersive media [@becache2003stability; @becache2018analysis; @duru2012well; @loh2009fundamental]. Several methods have been proposed to avoid and remove growing waves and to improve the stability and accuracy of PML formulations. Nevertheless, for some specific situations, such as linear elastodynamic equations in arbitrary anisotropic elastic media, developing a stable PML formulation remains an open problem [@pled2022review]. Recently, the rapid progress in deep learning has driven the development of solution techniques for PDEs. Learning-based PDE approaches can fall into two categories in terms of the objects approximated by neural networks (NN), i.e., the solution and the solution mapping. The typical methods of the first approach include physics-informed neural networks (PINNs) [@karniadakis2021physics], Deep Galerkin Method (DGM) [@sirignano2018dgm], Deep Ritz Method [@yu2018deep], Weak Adversarial Network (WAN) [@zang2020weak], etc. These methods train neural networks by sampling points within bounded computational domains and minimizing the physical loss at these sample points. However, when dealing with PDEs on unbounded domains, the method becomes impractical as it would require sampling training data points across the entire unbounded domains. On the other hand, if data points are only sampled within a bounded domain without imposing boundary conditions, the problem will become ill-posed, leading to the failure of the neural network training process to converge. The second approach uses neural networks to learn the solution mapping between infinite-dimensional function spaces (such as the mapping from the initial function space to the solution space). The typical methods include deep operator network (DeepONet) [@lu2021learning], Fourier neural operator (FNO) [@li2021fourier], PDE-Nets [@long2018pde; @long2019pde], etc. A significant advantage of this approach is that once the neural network is trained, it can quickly solve a large number of in-distribution tasks. However, this approach requires a large amount of data to effectively train neural networks, which is costly and challenging in many cases. For the task of solving PDEs on bounded domains, it is generally necessary to use traditional numerical methods to extensively solve PDEs to obtain data for training. For PDEs defined on unbounded domains, the cost of data acquisition is higher because of the need for complicated techniques, such as ABM or PML. Further, for many complex nonlinear equations, there is even no effective numerical method to solve them, resulting in the lack of training data. An alternative method, physics-informed DeepONet (PI-DeepONet) [@wang2021learning], is proposed to train the neural networks only relying on the equations and given initial/boundary conditions without any paired input-output data. As the combination of PINNs and DeepONet, PI-DeepONet adopts the architecture of DeepONet while taking the residual of the equation as the loss function during training just like PINNs. Unfortunately, this strategy does not work for solving PDEs on unbounded domains due to the same reason as PINNs. At present, most deep learning-based solvers focus on problems defined on bounded domains, while there is few work addressing the complexities associated with solving PDEs on unbounded domains. Xia et al. [@xia2023spectrally] blend the adaptive spectral method and PINNs to optimize the traditional numerical spectral schemes for solving PDEs in unbounded domains, while an assumption on the asymptotic spatial behavior is needed. Lin et al. [@lin2023binet; @lin2023bi] propose BINet, a method that combines boundary integral equations with neural networks to solve PDEs with known fundamental solutions. By leveraging potential theory to transform the original problem into boundary integral equations, this approach can handle problems on both bounded and unbounded domains. Xie et al. [@xie2021machine] are the first to explore the application of DeepONet for solving the PDEs on unbounded domains. They train the neural network in a data-driven setting, where data is obtained by exact solutions or traditional numerical methods. The trained model can be used to predict the solution of interpolation problems. This direct application of the DeepONet is confronted with the issues of high data acquisition costs and ineffectiveness for many challenging problems that traditional numerical methods struggle to solve, as mentioned earlier. Therefore, efficiently solving problems on unbounded domains, especially with the assistance of powerful deep learning techniques, is an area that merits further exploration. This work aims to develop an effective operator learning-based approach to solve PDEs (including nonlinear cases) defined on unbounded domains. The basic idea of our approach is to generate training data at a low cost and leverage the generalization ability of DeepONet to predict the solution of the PDE on the domain of interest. To ensure that the PDE to be solved falls within the interpolation range of the training data, we generate paired input-output training data that closely approximates the target PDE's initial value and source term. The main operations of the approach are as follows: firstly, construct a family of analytical solutions that satisfy the target equation needed to be solved, with initial value and source term designed to closely 'approximate' those of the target equation. Then, utilize MIONet [@jin2022mionet], an extension to DeepONet for learning multiple-input operators, to learn the mapping from initial value and source term to the PDE solution. Finally, utilize the learned mapping to directly predict the solution of the target equation on a bounded domain of interest. We test the effectiveness of our method on extensive equations including the second-order wave equation, Burger's equation, Korteweg-de Vries (KdV) equation, and Schrödinger equation. This approach offers several advantages. Firstly, it involves low computational costs for generating training data to train operator models. Secondly, it demonstrates effectiveness in solving nonlinear PDEs on unbounded domains, which are challenging for traditional numerical methods. Lastly, this method is generally applicable to various types of PDEs verified by numerical experiments and only needs to generate corresponding data that meets specific requirements. This flexible approach eliminates the need for designing distinct techniques for different equations, as often required by traditional numerical methods. The paper is structured as follows. Section 2 provides a brief introduction to the architecture of DeepONet and MIONet. In Section 3, we present a general workflow of our proposed method for solving PDEs on unbounded domains. Next, in Section 4, we demonstrate the effectiveness of our method through extensive numerical examples including both linear and nonlinear cases. Finally, Section 5 concludes the paper with a discussion of our main findings, potential limitations of the proposed method, and future research directions stemming from this study. # Preliminaries {#NN width="1.0\\linewidth"} ## DeepONet Based on the universal approximation theorem of Chen $\&$ Chen [@chen1995universal], DeepONet [@lu2021learning] is proposed to learn a nonlinear operator $$\begin{aligned} \mathcal{G}: \mathcal{V}\rightarrow \mathcal{U}, \quad v \mapsto u, \end{aligned}$$ where $\mathcal{V}$ and $\mathcal{U}$ are two infinite-dimensional Banach spaces of functions, $v\in \mathcal{V}$ and $u=\mathcal{G}(v)$. The DeepONet learns $\mathcal{G}$ via a neural network denoted by $\mathcal{G}_{\theta}$, where $\theta$ represents all trainable parameters of the network. Take $v\in \mathcal{V}$ evaluated at a collection of fixed locations $\{y_j\}_{j=1}^{m}$, i.e., $\bm{v}=[v(y_1),v(y_2),\cdots,v(y_{m})]$ and another family of locations $y=\{y_j\}_{j=1}^{P}$ as inputs, the output of DeepONet is calculated as follows $$\begin{aligned} \mathcal{G}_{\theta}(\bm{v})(y) = S\left( B(\bm{v}) \odot T(y) \right) + b, \end{aligned}$$ where $B$ is called branch net, and $T$ called trunk net, $\odot$ is the Hadamard product (i.e., element-wise product), $S$ is the summation of all components of a vector, and $b$ is a trainable bias. The architecture of DeepONet is shown in the left plane of Fig.[1](#NN){reference-type="ref" reference="NN"} and its detailed dataset structure can be referred to [@wang2021learning Remark 2.1]. ## MIONet The vanilla DeepONet is defined for input functions on a single Banach space. MIONet [@jin2022mionet] extends DeepONet to multiple input Banach spaces. Specifically, MIONet aims to learn a continuous operator $$\begin{aligned} \mathcal{G}: \mathcal{V}_1\times \mathcal{V}_2\times \cdots \times \mathcal{V}_n \rightarrow \mathcal{U}, \quad (v_1, v_2,\cdots, v_n) \mapsto u, \end{aligned}$$ where $\mathcal{V}_1, \mathcal{V}_2, \cdots, \mathcal{V}_n$ are $n$ different input Banach spaces, $\mathcal{U}$ is the output Banach space, and $v_i\in \mathcal{V}_i$, $u=\mathcal{G}(v_1, \cdots, v_2)$. Similar to the vanilla DeepONet, MIONet approximates $\mathcal{G}$ by $$\begin{aligned} \mathcal{G}_{\theta}(\bm{v_1}, \bm{v_2}, \cdots, \bm{v_n})(y) = S\left( B_1(\bm{v_1}) \odot \cdots \odot B_n(\bm{v_n}) \odot T(y) \right) + b, \end{aligned}$$ where $\{B_i\}_{i=1}^n$ are $n$ different branch nets, each $\bm{v_i}=[v_i(y^i_1),v_i(y^i_2),\cdots,v_i(y^i_{m})]$ can be calculated at different locations $\{\bm{y^i}\}$. The architecture of MIONet is shown at the right plane of Fig.[1](#NN){reference-type="ref" reference="NN"}. # Proposed method In this section, we present the proposed data generation-based operator learning method for solving PDEs on unbounded domains. Consider the following time-dependent equation $$\begin{aligned} \mathcal{L}u(x,t) = f(x,t), \quad x\in\mathbb{R}^d,~t>0, \label{Eq}\\ \mathcal{I}_i u(x,0) = \varphi_i(x), \quad x\in \mathbb{R}^d,~i\in{I}, \label{EqI} \end{aligned}$$ where $\mathcal{L}$ and $\mathcal{I}_i$ are partial differential operators, and $f(x,t)$ is the source term, $\varphi_i(x)$ is the initial function, $I=\{0\}$ if Eq.[\[Eq\]](#Eq){reference-type="eqref" reference="Eq"} is the first-order time evolution equation and $I=\{0,1\}$ when Eq.[\[Eq\]](#Eq){reference-type="eqref" reference="Eq"} is the second-order time evolution equation. There are no extra constraints on the initial value and source term. For convenience of exposition, let us refer to the equation that needs to be solved as the target equation, and the corresponding initial value and source term will be referred to as target functions, denoted by $\varphi^{tar}$ and $f^{tar}$, respectively. The proposed method of solving Eq.[\[Eq\]](#Eq){reference-type="eqref" reference="Eq"}-[\[EqI\]](#EqI){reference-type="eqref" reference="EqI"} can be summarized into the following three steps (see also Fig.[2](#flow){reference-type="ref" reference="flow"} for the entire workflow). {#flow width="1.0\\linewidth"} **Step 1: Data generation.** To generate data for learning a mapping from the initial value and source term to the solution, we construct a family of analytical solutions that satisfy the target governing equation, for which corresponding initial values and source terms approximate the target initial value and the target source term, respectively. We describe how to construct analytical solutions and select training data in detail as follows. The core idea of constructing analytical solutions is leveraging the information of known target functions. For example, if the target initial value is zero, we can construct an analytical solution that inherently satisfies this condition, simplifying the model's task to learn the mapping solely from the source term to the solution. Similarly, if the target function is compactly supported, we can generate an analytical solution using the compactly supported basis function accordingly. On the basis of utilizing the information of target functions, the choice of analytical solution in form is still diverse. For instance, we can choose the following solution in the form of variable separation $$\begin{aligned} \label{u} u(x,t) = \sum_{k=0}^K \alpha_k(t) \phi_k(w x+b), \end{aligned}$$ where $\phi_k(x)$ is some basis function on $x$, such as Hermite function, $\alpha_k(t)$ is basis function on $t$, such as Fourier function. Parameters $K, w,b$ and parameters contained in $\alpha_k(t)$ are randomly generated according to certain rules, such as normal random distributions. After obtaining an analytical solution, substituting it into Eq.[\[Eq\]](#Eq){reference-type="eqref" reference="Eq"}-[\[EqI\]](#EqI){reference-type="eqref" reference="EqI"}, one obtains the corresponding initial value $\varphi$ and source term $f$. To assess the quality of the generated data $\varphi, f$ in comparison to the target function $\varphi^{tar}, f^{tar}$, we introduce the following evaluation metrics: - Metric-1 (relative $L^2$ error): $$\begin{aligned} \label{metric1} E_{1}(g,g^{tar}) :=\frac{\Vert g^{tar}-g\Vert_2}{\Vert g^{tar}\Vert_2}, \end{aligned}$$ where $\Vert\cdot\Vert_2$ represents the discrete $L^2$ norm. - Metric-2 (mean square error): $$\begin{aligned} \label{metric2} E_{2}(g,g^{tar}) := \left(\frac{1}{P}\sum_{j=1}^P |g^{tar}(y_j)-g(y_j)|^2\right)^{\frac12}, \end{aligned}$$ where $\{y_j\}_{j=1}^P$ are sample points in a fixed spatial-temporal domain related to the computational domain of interest. Generally, we utilize metric-1 to evaluate the generated data. However, when $g^{tar}=0$, metric-1 becomes ineffective, and in such cases, we rely on metric-2 for evaluation. If $E(g,g^{tar}) \le \epsilon$, then the corresponding data is selected to train the model, where $\epsilon$ is a parameter selected based on experience. We remark that the evaluation should better be conducted in a relatively large domain compared to the domain of interest. By ensuring that the initial values and source terms closely match the target functions across a broader domain, one can achieve higher-quality generated data. We investigate the impact of the size of the spatial domain where the data is generated on the final results, as detailed in Appendix B. **Step 2: MIONet training.** MIONet is utilized to learn the mapping from the initial value and source term to the solution of the PDE, i.e., $\mathcal{G_{\theta}}: (\varphi, f) \rightarrow u$. We train the network parameters by minimizing the following loss function: $$\begin{aligned} L(\theta) = \frac{1}{NP} \sum_{n=1}^N \sum_{j=1}^P|\mathcal{G}_{\theta}(\bm{\varphi}^{(n)},\bm{f}^{(n)})(y_j)-u^{(n)}(y_j)|^2, \end{aligned}$$ where $\bm{\varphi}^{(n)} = [\varphi^{(n)}(y_1),\varphi^{(n)}(y_2), \cdots, \varphi^{(n)}(y_{m_1})]$ represents the initial function evaluated at a collection of fixed locations $\{y_j\}_{j=1}^{m_1}$, $\bm{f}^{(n)} = [f^{(n)}(y_1),f^{(n)}(y_2), \cdots, f^{(n)}(y_{m_2})]$ represents the source term evaluated at another set of fixed locations $\{y_j\}_{j=1}^{m_2}$, and $u^{(n)}(y_j)$ denotes the solution at points $y_j$ of the problem [\[Eq\]](#Eq){reference-type="eqref" reference="Eq"}-[\[EqI\]](#EqI){reference-type="eqref" reference="EqI"} with $\varphi^{(n)}$ and $f^{(n)}$. $N$ is the number of input function pairs, and $P$ is the number of sampling points on the computational domain. The training of the model is purely data-driven. Experiments have shown that incorporating the residual of the equation into the loss function for joint training does not lead to a significant improvement in accuracy. On the contrary, it likely leads to reduced accuracy (refer to Appendix D), and considerably slows down the training process (see [@wang2021learning Appendix C]). **Step 3: Prediction.** Apply the learned mapping operator $\mathcal{G_{\theta}}$ to target functions $\varphi^{tar}$ and $f^{tar}$ to obtain the solution of the target PDE. **Remark 1**. *The success of this approach relies on the well-posedness of the equation, particularly its stability concerning the initial value and source term, i.e., there exists a constant $C$, s.t., $$\begin{aligned} \Vert u-u_{\epsilon}\Vert \le C\Vert f-f_{\epsilon}\Vert, \end{aligned}$$ where $f_{\epsilon}$ represents the initial function or source function $f$ after a small perturbation, and $u_{\epsilon}$ is the corresponding solution.* # Numerical results In this section, we evaluate the effectiveness of the proposed method across various types of PDEs, including 1D and 2D wave equations, Burger's equation, KdV equation and Schrödinger equation. All experiments are conducted on an NVIDIA A100 GPU using PyTorch (v1.12.1) framework. The activation function employed in all networks is set to Tanh, and the batch size is fixed at 8192. The networks are trained with the Adam optimizer \[12\] with default settings. The detailed hyper-parameters for all examples are listed in Appendix A. In our evaluation, we compare the model prediction solution, denoted as $u_h$, with a reference solution, denoted as $u_{ref}$. The reference solution can be obtained from an exact solution or numerically computed via methods such as finite difference or spectral methods. For each numerical experiment, we present the following three types of errors for the predicted solutions: $$\begin{aligned} && \text{Relative~} L^2 \text{~error} := \displaystyle \frac{\Vert u_h-u_{ref}\Vert_2}{\Vert u_{ref}\Vert_2}; \\ && \text{Relative~} L^1 \text{~error} := \displaystyle \frac{\Vert u_h-u_{ref}\Vert_1}{\Vert u_{ref}\Vert_1};\\ && \text{Max error} := \Vert u_h-u_{ref}\Vert_{\infty}. \end{aligned}$$ In addition to demonstrating the capability of the proposed method to solve various equations on unbounded domains, we also investigate the impact of important factors such as the size of the spatial domain where the data is generated, the number of input functions, and physics-informed training on the experimental results. The findings and results of these investigations are provided in Appendix B, C, and D, respectively. ## 1-D wave equation We consider the 1-D second-order wave equation defined on the whole space: $$\begin{aligned} \label{wave} \begin{aligned} & u_{tt}-u_{xx}=f(x,t), && x\in \mathbb{R},~ t\in(0,1], \\ & u(x, 0) = \varphi_0(x), && x\in \mathbb{R},\\ & u_t(x,0) = \varphi_{1}(x), && x\in \mathbb{R}. \end{aligned} \end{aligned}$$ We solve two representative examples: one with a known analytical solution, and the other without an analytical solution. For the latter, we compute a reference solution using the finite difference method on a sufficiently large domain. To generate training data for both examples, we construct various forms of analytical expressions. ### Case 1: The analytical solution exists We consider the following initial conditions and the source term: $$\begin{aligned} \label{wave_case1_target} \begin{aligned} %&u(x,t) = \exp\left(-|x|^2\right)\cos(t-x),\\ &\varphi_0(x) = \exp\left(-x^2\right)\cos(x),\\ &\varphi_{1}(x) = \exp\left(-x^2\right)\sin(x),\\ &f(x,t) = \exp(-x^2)\left(4x\sin(t-x)+(2-4x^2)\cos(t-x)\right), \end{aligned} \end{aligned}$$ with the exact solution $u(x,t) = \exp\left(-x^2\right)\cos(t-x)$. The computational domain is $(x,t)\in [-1,1]\times[0,1]$. We choose the following form of analytical solutions to generate the required data: $$\begin{aligned} \label{wave_case1_construct_sol} u(x,t) = \sum_{i=0}^K A_i\sin(k_it+a_i)H_i(wx+b), \end{aligned}$$ where $H_i(x)$ stands for the Hermite funtion [@shen2011spectral], $A_i, k_i, a_i, w, b$ are parameters randomly generated. We set $K=2$. The spatial domain of generating data is $[-3,3]$. We select 5000 input function pairs, i.e., $N=5000$. The training loss and three-type errors computed at a $101\times201$ spatio-temporal grid are presented in Fig.[4](#wave1d-1-loss){reference-type="ref" reference="wave1d-1-loss"}, with the error computed at every 1,000 iterations. It is observed that as the loss stabilizes during training, the testing error also tends to stabilize. This implies that monitoring the loss can provide a rough indication of whether the trained model is suitable for the target problem. The average values of the last 20 relative $L^2$ errors, relative $L^1$ errors and max errors shown in Fig.[4](#wave1d-1-loss){reference-type="ref" reference="wave1d-1-loss"} are 2.34e-03, 2.30e-03, and 3.60e-03, respectively. We also illustrate visual representations of the exact solution, the prediction solution and the absolute error in Fig.[5](#wave1d-1-sol){reference-type="ref" reference="wave1d-1-sol"}. {#wave1d-1-loss width="0.45\\linewidth"} {#wave1d-1-loss width="0.45\\linewidth"} {#wave1d-1-sol width="1.0\\linewidth"} ### Case 2: Point source and without analytical solution We consider zero initial values, i.e., $\varphi_0(x) = 0, \varphi_1(x) = 0$, and $f(x,t) = 5\exp(-25x^2)$, which is an approximation of point source function. The reference solution $u(x,t)$ of this case is computed using a second-order finite difference method on a large enough spatial domain. To generate the approximate source term $f$, we choose the following form of the analytical solutions: $$\begin{aligned} % && \alpha(t) = At\sin(kt)\exp(-(t-\mu)^2/\sigma_t^2), \\ % && \phi(x) = \exp(-x^2/\sigma_x^2). % u(x,t) = \sum_{i=0}^N A_it\sin(k_it)\exp(-(t-\mu_i)^2/\sigma_{t,i}^2)\exp(-x^2/\sigma_{x,i}^2). u(x,t) = \sum_{i=0}^K A_it^2\cos(a_it+b_i)\exp(-x^2/\sigma_{i}^2). \end{aligned}$$ The spatial domain of generating data is $[-2,2]$, and $K=2$. We select 5000 input function pairs, i.e., $N=5000$. The training loss and the error computed at a $101\times201$ spatio-temporal grid are presented in Fig.[7](#wave1d-2-loss){reference-type="ref" reference="wave1d-2-loss"}. The average values of the final 20 errors are 4.98e-02, 6.60e-02, and 2.84e-02 for the relative $L^2$ error, relative $L^1$ error, and max error, respectively. Compared to case 1, it is relatively difficult to generate the analytical expression of this case due to the special form of the function $f$. As a result, the prediction error is also greater. To achieve higher accuracy, it is crucial to construct analytical expressions of higher quality. Finally, we draw the reference solution, the prediction solution, and the absolute error in Fig.[8](#wave1d-2-sol){reference-type="ref" reference="wave1d-2-sol"}. {#wave1d-2-loss width="0.45\\linewidth"} {#wave1d-2-loss width="0.45\\linewidth"} {#wave1d-2-sol width="1.0\\linewidth"} ## 2D wave equation This example aims to solve the following 2D wave equation $$\begin{aligned} \label{wave2d} \begin{aligned} & u_{tt}-\Delta u=f(\bm{x},t), && \bm{x}\in \mathbb{R}^2,~ t\in(0,1], \\ & u(\bm{x}, 0) = \varphi_0(\bm{x}), && \bm{x}\in \mathbb{R}^2,\\ & u_t(\bm{x},0) = \varphi_{1}(\bm{x}), && \bm{x}\in \mathbb{R}^2. \end{aligned} \end{aligned}$$ The target initial functions and source are given as: $$\begin{aligned} && \varphi_0(\bm{x}) = \exp\left(-\frac{x_1^2+x_2^2}{2}\right),\\ && \varphi_1(\bm{x}) = \exp\left(-\frac{x_1^2+x_2^2}{2}\right)(x_1+x_2),\\ && f(\bm{x}, t) = \exp\left(-\frac{(x_1-t)^2+(x_2-t)^2}{2}\right)\left((x_1-t+x_2-t)^2\cos(kt)-2k(x_1-t+x_2-t)\sin(kt)\right. \\ && \quad\quad\quad\quad \left.-(k^2+(x_1-t)^2+(x_2-t)^2)\cos(kt)\right), \end{aligned}$$ which are computed from the solution $$u(\bm{x},t)=\exp\left(-\frac{(x_1-t)^2+(x_2-t)^2}{2}\right)\cos(kt).$$ The spatial domain of interest is selected as $[-1,1]$. Based on the property of the target function, we construct the following form of analytical solutions to generate training data: $$\begin{aligned} u(\bm{x},t) = A\exp\left(-\frac{(x_1-a_1t)^2+(x_2-a_2t)^2}{\sigma^2}\right)\cos(k_1x_1+k_2x_2-wt). \end{aligned}$$ We generate 2000 input function pairs on the spatial domain $[-5,5]\times[-5,5]$. The training loss and errors computed at a $21\times21\times51$ spatio-temporal grid are presented in Fig.[10](#wave2d-loss){reference-type="ref" reference="wave2d-loss"}. Fig.[11](#wave2d-sol){reference-type="ref" reference="wave2d-sol"} displays the reference solution, prediction solution and the absolute error at final time $t=1$. Furthermore, we calculated the average values of the final 20 relative $L^2$ errors, relative $L^1$ errors, and max errors as 9.27e-03, 1.00e-02, and 9.65e-03, respectively. {#wave2d-loss width="0.45\\linewidth"} {#wave2d-loss width="0.45\\linewidth"} {#wave2d-sol width="1.0\\linewidth"} ## Burger's equation To highlight the ability of the proposed method to handle nonlinear equations on unbounded domains, we solve the following 1D Burger's equation $$\begin{aligned} \label{Burgers} \begin{aligned} & u_{t}-uu_{x}-\nu u_{xx} = f(x,t), && x\in \mathbb{R},~ t\in(0,1], \\ & u(x, 0) = \varphi(x), && x\in \mathbb{R}. \end{aligned} \end{aligned}$$ In this section, we present two examples: solving a single equation and solving a family of parameterized equations. The latter case means that we solve multiple PDEs simultaneously using a single trained model. ### Case 1: Solve one single equation {#sec_Burgers1} In this example, we consider $\varphi(x) = 0,~ f(x,t) = \cos(\pi t)\exp(-x^2)$, the viscosity is set to $\nu = 0.2$. To evaluate the accuracy of the predicted solution, we solve the equation using the finite difference method on a large enough domain, where the spatial step size and temporal step size are both set to 0.01. The domain of interest is selected as $[-2,2]$. We construct the analytical solution to generate training data: $$\begin{aligned} \label{BurgersData} u(x,t) = \sum_{i=0}^K A_i(\sin(k_{1,i}\pi t)+t\cos(k_{2,i}\pi t))H_{i}(w_ix+b_i), \end{aligned}$$ which spontaneously satisfies the target initial value. Therefore, our focus is on adjusting the parameters of this constructed solution to make it closely match the target source term. The spatial domain for generating data is also set as $[-2,2]$. We select 5000 input function pairs, i.e., $N=5000$. We discuss the impact of the number of input functions on model training and prediction accuracy in Appendix C. Generally, increasing $N$ tends to improve the prediction accuracy of the model. The training loss and errors during the training process are presented in Fig.[13](#Burgers1-loss){reference-type="ref" reference="Burgers1-loss"}. Additionally, Fig.[14](#Burgers1-sol){reference-type="ref" reference="Burgers1-sol"} illustrates the comparison between the predicted and the exact solution. We evaluate the average values of the final 20 errors at a $101\times101$ spatio-temporal grid, yielding a relative $L^2$ error, relative $L^1$ error, and max error of 4.15e-03, 4.31e-03, and 1.66e-03, respectively. These results demonstrate that the proposed method can effectively tackle nonlinear equations on unbounded domains, which are often challenging for traditional numerical methods to solve. {#Burgers1-loss width="0.45\\linewidth"} {#Burgers1-loss width="0.45\\linewidth"} {#Burgers1-sol width="1.0\\linewidth"} ### Case 2: Solve multiple equations simultaneously In this example, we show the proposed method is also capable of solving multiple PDEs simultaneously. We consider Burger's equation [\[Burgers\]](#Burgers){reference-type="eqref" reference="Burgers"} with a zero initial value and a family of source terms parameterized by $\theta$: $$f(x,t) = \cos(\theta t)\exp(-x^2).$$ We uniformly select 20 values of $\theta$ from the interval $[-\pi, \pi]$. The goal is to train only one model to solve these parameterized PDEs. The training data is also generated from the analytical solutions given in [\[BurgersData\]](#BurgersData){reference-type="eqref" reference="BurgersData"}. The selected solutions satisfy the condition that the error between the corresponding source term and each target source term is below a given tolerance $\epsilon$. The data is generated over the spatial domain $[-4,4]$. In the left plane of Fig.[17](#Burgers2){reference-type="ref" reference="Burgers2"}, we plot the training loss over 150,000 iterations. The middle panel displays the average error of the predicted solutions for all 20 target PDEs. After the model convergence, the average values of the relative $L^2$ errors, relative $L^1$ errors, and max errors are measured as 8.04e-03, 7.64e-03, and 5.83e-03, respectively. Lastly, the right plane of Fig.[17](#Burgers2){reference-type="ref" reference="Burgers2"} shows the final error of the prediction solution for each target PDE. This example demonstrates that the proposed method still performs well in solving multiple similar PDEs simultaneously. {#Burgers2 width="0.32\\linewidth"} {#Burgers2 width="0.32\\linewidth"} {#Burgers2 width="0.32\\linewidth"} ## KdV equation {#sec_KdV} This example considers the KdV equation, which describes the propagation of waves in certain nonlinear dispersive media. It takes the following form: $$\begin{aligned} \label{KdV} \begin{aligned} & u_{t}+ 6uu_{x}+u_{xxx} = f(x,t), && x\in \mathbb{R},~ t\in(0,1], \\ & u(x, 0) = \varphi(x), && x\in \mathbb{R}. \end{aligned} \end{aligned}$$ Here the target initial value and source term are generated from the exact solution $u(x,t) = \exp(-(x-t)^2)$. Substituting it to the KdV equation [\[KdV\]](#KdV){reference-type="eqref" reference="KdV"}, one obtains $$\begin{aligned} \label{KdV_target} \begin{aligned} & \varphi(x) = \exp(-x^2),\\ & f(x,t) = \exp(-(x-t)^2)(12(t-x)\exp(-(t - x)^2)+14(x-t) + 24tx(x-t) + 8(t^3-x^3)). \end{aligned} \end{aligned}$$ To solve the equation [\[KdV\]](#KdV){reference-type="eqref" reference="KdV"} with the target initial value and source [\[KdV_target\]](#KdV_target){reference-type="eqref" reference="KdV_target"} on domain $[-1,1]\times[0,1]$, we construct the following analytical expression: $$u(x,t)=A\exp(a(x+c_1t+c_2)^2)\cos(kx-wt),$$ where $A, a, c_1, c_2, k, w$ are tunable parameters. We measure the generated data using the relative $L^2$ error [\[metric1\]](#metric1){reference-type="eqref" reference="metric1"} on the spatial domain $[-5,5]$. We select $N=1000$ input functions. The training loss and errors tested at a $101\times101$ spatio-temporal grid are presented in Fig.[19](#KdV_loss){reference-type="ref" reference="KdV_loss"}. The exact solution, the prediction solution, and the absolute error are drawn in Fig.[20](#KdV_sol){reference-type="ref" reference="KdV_sol"}. The average values of the relative $L^2$ error, relative $L^1$ error and max error for the last 20 predictions are 1.34e-03, 1.30e-03, and 3.04e-03, respectively. {#KdV_loss width="0.45\\linewidth"} {#KdV_loss width="0.45\\linewidth"} {#KdV_sol width="1.0\\linewidth"} ## Schrödinger equation In this example, we consider the classical Schrödinger equation, which is typically defined in an unbounded domain $$\begin{aligned} \label{Schr} \begin{aligned} & {\rm i}u_{t}= -u_{xx} + f(x,t), && x\in \mathbb{R},~ t\in(0,1], \\ & u(x, 0) = \varphi(x), && x\in \mathbb{R}, \end{aligned} \end{aligned}$$ where the solution $u$ is the complex-value function. To handle complex values in neural networks, certain modifications need to be made to the network structure since direct gradient propagation is not supported for complex values. The approach involves decomposing complex numbers into their real and imaginary parts, feeding them as inputs to the neural networks, and producing separate outputs for the real and imaginary parts. For each input function, there are two corresponding subnetworks: one responsible for outputting the real part and another for outputting the imaginary part. The trunk network encoded input $y$ is shared between these two subnetworks. Fig.[21](#NN_Schr){reference-type="ref" reference="NN_Schr"} shows in detail the structure of this network. {#NN_Schr width="1.0\\linewidth"} The target initial value and source term are given as $\varphi(x) = \exp(-x^2+{\rm i}x)\cos(x)$ and $f=0$. It is known that Eq.[\[Schr\]](#Schr){reference-type="eqref" reference="Schr"} with $f=0$ has the following analytical solutions: $$\begin{aligned} \begin{aligned} & u(x, t)=\frac{1}{\sqrt{\zeta+{\rm i} t}} \exp \left[{\rm i} k(x-k t)-\frac{(x-2 k t)^2}{4(\zeta+{\rm i} t)}\right], \end{aligned} \end{aligned}$$ where $k$ is a real parameter that controls the beam propagation speed, and $\zeta$ is a positive parameter that controls the beam width. The training data are generated based on this family of solutions, and we remark that the target function is not contained in the training data. The spatial domain of generating data is $[-2,2]$. We select 5000 input function pairs, i.e., $N=5000$. The training loss and errors computed at a $101\times101$ spatio-temporal grid in the training process are presented in Fig.[23](#Schr-loss){reference-type="ref" reference="Schr-loss"}. We evaluate the average values of the final 20 relative $L^2$ errors, relative $L^1$ errors, and max errors, which are found to be 8.90e-03, 7.90e-03, and 9.05e-03, respectively. In the upper part of Fig.[24](#Schr-sol){reference-type="ref" reference="Schr-sol"}, the real parts of the reference solution and the predicted solution are displayed, accompanied by their respective absolute error. The lower part of Fig.[24](#Schr-sol){reference-type="ref" reference="Schr-sol"} illustrates the imaginary parts of the reference solution and the predicted solution, along with their corresponding absolute error. {#Schr-loss width="0.45\\linewidth"} {#Schr-loss width="0.45\\linewidth"} {#Schr-sol width="1.0\\linewidth"} # Conclusion and discussion This paper presents a data generation-based operator learning method for effectively solving PDEs on unbounded domains. The key ingredient of this approach is to generate high-quality training data based on the target PDE. Then, the MIONet model is trained to learn the mapping from the initial value and source term to the solution of the PDE. The generalization ability of the model allows it to directly operate on the initial value and source term of the target PDE, producing a solution with a certain level of accuracy. Through extensive testing on different equations, including both linear and nonlinear cases, we have demonstrated the effectiveness of our proposed method. This study is an attempt to solve PDEs on unbounded domains entirely based on operator learning. Unlike classical numerical methods, this method is not sensitive to the specific form of the equation, making it suitable for solving challenging problems, such as nonlinear cases. In addition, the proposed method can not only solve a single PDE but also offers the capability to simultaneously solve multiple PDEs with different parameters. This versatility highlights the wide-ranging applicability and potential impact of our approach. Moreover, the cost of data generation in this method is significantly lower compared to the original DeepONet approach which typically relies on expensive traditional numerical methods to generate a large volume of data. The concept of generating approximate data not only applies to solving problems on unbounded domains but also offers valuable insights for addressing problems on bounded domains and even inverse problems. Despite the promising results, there are still numerous open questions that require further investigation. A key concern is the absence of theoretical guarantees for prediction accuracy. Even if the loss function is minimized to an extremely small level, it does not provide a definitive measure of prediction accuracy for the target problem. In situations where some observation results are available, they can provide valuable information for assessing the reliability of the model. By comparing the model's predictions with the observed data, we can gain insights into the model's performance and determine its reliability to some extent. Another important concern is that constructing satisfactory analytical solutions is not always straightforward. For instance, in cases where the source term of the target PDE is zero, it is challenging to construct analytical solutions such that the corresponding source terms approximate the target source term. In such situations, we must utilize all available information to carefully construct analytical solutions. The development of effective construction methods requires further exploration in the future. # Acknowledgments {#acknowledgments .unnumbered} This work is supported by Exploratory Research Project (No.2022RC0AN02) and Research Initiation Project (No. K2022RC0PI01) of Zhejiang Lab. # A  Hyper-parameter settings {#a-hyper-parameter-settings .unnumbered} This appendix provides the detailed parameters used in all examples, including the size of training data, MIONet architectures, learning rate, and the rule of generating training data. ## A.1  Training data. {#a.1-training-data. .unnumbered} The number of input function pairs for training is denoted as $N$. We uniformly select $m$ sensors distributed across the domain of input functions defined. The training locations for computing initial loss are randomly selected from the spatial domain and locations for computing interior loss are uniformly selected from the spatial-temporal domain. Table [1](#appendix_data){reference-type="ref" reference="appendix_data"} presents the number of input functions $N$, the number of sensors $m$ of input functions (initial function and source function), and the number of locations $P$ for computing initial loss and interior loss. ---------------------------- ------ ---------- ------------ Case $N$ (\# initial, \# source) (\# initial + \# interior) 1D wave Eq. (case 1) 5000 51,225 101+$51^2$ 1D wave Eq. (case 2) 5000 51,225 101+$51^2$ 2D wave Eq. 2000 100,1000 21+$21^3$ Burger's Eq. (case 1) 5000 51,225 101+$51^2$ Burger's Eq. (case 2) 5000 51,225 51+$51^2$ KdV Eq. 1000 51,225 51+$51^2$ Schrödinger Eq. 5000 51,225 101+$51^2$ ---------------------------- ------ ---------- ------------ : Training data. [\[appendix_data\]]{#appendix_data label="appendix_data"} ## A.2  MIONet architectures {#a.2-mionet-architectures .unnumbered} Table [2](#appendix_arch){reference-type="ref" reference="appendix_arch"} shows the MIONet architectures used in all examples. The dimension of the input layer is determined by the data size, while the size of all network output layers remains the same. This consistency is essential as it enables the Hadamard product of each network's output to be taken to obtain the final output. ----------------------------------- -------------- ---------------- ----------------- Case ($D_{in}$-width\*depth-$D_{out}$) ($D_{in}$-width\*depth-$D_{out}$) ($D_{in}$-width\*depth-$D_{out}$) 1D wave Eq. (case 1) 2-400\*4-100 51-100\*3-100 225-225\*3-100 1D wave Eq. (case 2) 2-400\*3-100 51-100\*3-100 225-225\*3-100 2D wave Eq. 3-200\*4-200 100-200\*3-200 1000-200\*3-200 Burger's Eq. (case 1) 2-400\*3-100 51-100\*3-100 225-225\*3-100 Burger's Eq. (case 2) 2-400\*3-100 51-100\*3-100 225-225\*3-100 KdV Eq. 2-400\*4-100 51-100\*3-100 225-225\*3-100 Schrödinger Eq. 2-400\*3-100 51-51\*3-100 225-225\*3-100 ----------------------------------- -------------- ---------------- ----------------- : MIONet architectures. The trunk net and branch nets are composed of fully connected networks. Branch-1 net is for the initial function $\varphi$, and Branch-2 net is for the source function $f$. $D_{in}$ and $D_{out}$ are the input dimension and output dimension, respectively. The "width\" and "depth\" refer to the size of the hidden layer. [\[appendix_arch\]]{#appendix_arch label="appendix_arch"} ## A.3  Learning rate {#a.3-learning-rate .unnumbered} The learning rate directly affects the effect of model training. MIONet models are trained via mini-batch gradient descent using the Adam optimizer with default settings and exponential learning rate decay with a decay rate of $\gamma$ every $M$ iterations. Table [3](#appendix_lr){reference-type="ref" reference="appendix_lr"} gives the base learning rate (lr) $\eta$, step size $M$ and decay rate $\gamma$ used in all examples. Case base lr $\eta$ step size $M$ decay rate $\gamma$ ----------------------- ---------------- --------------- --------------------- 1D wave Eq. (case 1) 0.001 500 0.96 1D wave Eq. (case 2) 0.001 500 0.97 2D wave Eq. 0.0005 1000 0.96 Burger's Eq. (case 1) 0.0005 1000 0.95 Burger's Eq. (case 2) 0.0005 1000 0.96 KdV Eq. 0.001 1000 0.96 Schrödinger Eq. 0.001 500 0.96 : Learning rate. [\[appendix_lr\]]{#appendix_lr label="appendix_lr"} ## A.4  Parameters of generating training data {#a.4-parameters-of-generating-training-data .unnumbered} The parameters in the analytical expression are randomly generated, with each parameter being selected from a normal distribution. The mean and variance of the normal distribution for each parameter in each calculation example are provided in Table [4](#par_gen_data_random){reference-type="ref" reference="par_gen_data_random"}. To select the data for training, we need to calculate the errors between the generated initial values, source terms and those of the target PDE, respectively. Therefore, we need to select the domain and sampling points for calculating the errors. Here the time domain is always consistent with the computational domain of the target PDE, while the spatial domain can be selected larger than the computational domain of interest to better capture the global information of the target functions. Table [5](#par_gen_data){reference-type="ref" reference="par_gen_data"} provides the spatial domain, number of grid points (uniformly), and tolerance $\epsilon$ for each example. Case Parameters ---------------------- ----------------------------- --------------------------------- --------------------------- ---------------------------------- 1D wave Eq. (case 1) $A\sim\mathcal{N}(1,1)$ $k\sim\mathcal{N}(0,1)$ $a\sim\mathcal{N}(0,1)$ $w\sim\mathcal{N}(-2,1)$ $b\sim\mathcal{N}(0,1)$ 1D wave Eq. (case 2) $A\sim\mathcal{N}(0.2,1)$ $a\sim\mathcal{N}(1,1)$ $b\sim\mathcal{N}(0,1)$ $\sigma\sim\mathcal{N}(0.2,0.5)$ $A\sim\mathcal{N}(1,1)$ $a_1\sim\mathcal{N}(2,1)$ $a_2\sim\mathcal{N}(1,1)$ $\sigma\sim\mathcal{N}(1,1)$ $k_1\sim\mathcal{N}(0,1)$ $k_2\sim\mathcal{N}(0,1)$ $w\sim\mathcal{N}(3,1)$ $A\sim\mathcal{N}(0.2,1)$ $k_1\sim\mathcal{N}(0.8,1)$ $k_2\sim\mathcal{N}(1,2)$ $w\sim\mathcal{N}(1,1)$ $b\sim\mathcal{N}(0,1)$ $A\sim\mathcal{N}(0.2,1)$ $k_1\sim\mathcal{N}(0.8,1)$ $k_2\sim\mathcal{N}(1,2)$ $w\sim\mathcal{N}(1,1)$ $b\sim\mathcal{N}(0,1)$ $A\sim\mathcal{N}(1,1)$ $a\sim\mathcal{N}(-1,1)$ $c_1\sim\mathcal{N}(0,1)$ $c_2\sim\mathcal{N}(-1,1)$ $k\sim\mathcal{N}(0,1)$ $w\sim\mathcal{N}(0,1)$ Schrödinger Eq. $A\sim\mathcal{N}(0.5,0.5)$ $\zeta\sim\mathcal{N}(0.3,0.5)$ $k\sim\mathcal{N}(1,0.5)$ : Random rule. [\[par_gen_data_random\]]{#par_gen_data_random label="par_gen_data_random"} Case spatial domain grid (initial, source) tolerance $\epsilon$ ----------------------- ---------------- ------------------------------------ -------------------------------------------------------------- 1D wave Eq. (case 1) $[-3,3]$ 51, $101\times101$ $(\varphi_0,\varphi_1,f):(0.5,1,1)$ 1D wave Eq. (case 2) $[-2,2]$ 51, $101\times101$ $(\varphi_0,\varphi_1,f):(0,0,0.6)$ 2D wave Eq. $[-5,5]^2$ $41\times41$, $41\times41\times41$ $(\varphi_0,\varphi_1,f):(0.8,0.8,0.8)$ Burger's Eq. (case 1) $[-2,2]$ 51, $51\times41$ $(\varphi,f): (0,1)$ Burger's Eq. (case 2) $[-4,4]$ 51, $51\times51$ $(\varphi,f): (0,1)$ KdV Eq. $[-5,5]$ 51, $51\times51$ $(\varphi,f): (0.8,0.8)$ Schrödinger Eq. $[-5,5]$ 51, $101\times101$ $(\Re(\varphi),\Im(\varphi), \Re(f), \Im(f)): (0.4,0.4,0,0)$ : Parameters of generating training data. [\[par_gen_data\]]{#par_gen_data label="par_gen_data"} # B  Impact of generating data domain {#b-impact-of-generating-data-domain .unnumbered} We investigate the influence of the spatial domain size on the results by varying the size of the domain where the data is generated. We specifically focus on case 1 of the 1D wave equation [\[wave\]](#wave){reference-type="eqref" reference="wave"} and case 1 of Burger's equation [\[Burgers\]](#Burgers){reference-type="eqref" reference="Burgers"} as examples for our analysis. To make the compact support domain of the target function larger, we introduce a slight modification to the target function [\[wave_case1_target\]](#wave_case1_target){reference-type="eqref" reference="wave_case1_target"} of the wave equation. The modified target functions are given as $$\begin{aligned} \begin{aligned} %&u(x,t) = \exp\left(-|x|^2\right)\cos(t-x),\\ &\varphi_0(x) = \exp\left(-x^2/4\right)\cos(x),\\ &\varphi_{1}(x) = \exp\left(-x^2/4\right)\sin(x),\\ &f(x,t) = \exp(-x/4^2)\left((-x^2/4+1/2)\cos(t-x)+x\sin(t-x)\right), \end{aligned} \end{aligned}$$ with the exact solution $u(x,t) = \exp\left(-x^2/4\right)\cos(t-x)$. The initial values and source function are depicted in the left and right planes of Fig.[26](#test_fun){reference-type="ref" reference="test_fun"}, respectively. The domain of interest is defined as $[-1,1]$. We conduct tests by generating data domains of $[-1,1], [-2,2], [-3,3], [-4,4]$. Within each domain, we uniformly sample points while maintaining the same grid density, i.e., using a fixed spatial grid step size of 0.02. The training process and prediction errors are shown in Fig.[34](#wave_domain){reference-type="ref" reference="wave_domain"}. The final errors are summarized in Table [6](#domain){reference-type="ref" reference="domain"}. It can be observed that the best performance is achieved when the data domain is set as $[-4,4]$. For the case of Burger's equation, we adopt the same settings as described in section [4.3.1](#sec_Burgers1){reference-type="ref" reference="sec_Burgers1"}. The initial value is zero, and the source function is depicted in the right plane of Fig.[26](#test_fun){reference-type="ref" reference="test_fun"}. We generate data on domains $[-2,2], [-3,3], [-4,4], [-5,5]$ to test the performance. We uniformly collect points within the domain and maintain the same grid density. That is, we fix the spatial grid step size as 0.02. The training process and prediction errors are shown in Fig.[42](#burgers_domain){reference-type="ref" reference="burgers_domain"}. The final errors are shown in Table [6](#domain){reference-type="ref" reference="domain"}, which indicates that the results are optimal when the computational domain is chosen as $[-3, 3]$. Combining these two examples, we can tentatively conclude that a larger computational domain does not necessarily guarantee better results. However, the selected computational domain should not be too small either. It is preferable to choose a domain that includes the compact support set of target functions, meaning that target functions decay toward zero within that domain. {#test_fun width="0.66\\linewidth"} {#test_fun width="0.33\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"}\ {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#wave_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"}\ {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} {#burgers_domain width="0.24\\linewidth"} 1D wave Eq. (case 1) Burger's Eq. ---------------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- domain $L^2$-error $L^1$-error max-error domain $L^2$-error $L^1$-error max-error $[-1,1]$ 2.08e-02 1.98e-02 3.49e-02 $[-2,2]$ 8.42e-03 8.24e-03 2.86e-03 $[-3,3]$ 1.40e-02 1.28e-02 2.76e-02 $[-3,3]$ **7.56e-03** **8.33e-03** **2.08e-03** $[-4,4]$ **7.40e-03** **6.08e-03** **1.40e-02** $[-4,4]$ 9.05e-03 9.80e-03 2.78e-03 $[-5,5]$ 8.64e-03 7.54e-03 2.37e-02 $[-5,5]$ 1.06e-02 1.05e-02 2.64e-03 : Impact of generating data domain. [\[domain\]]{#domain label="domain"} # C  Impact of the number of input functions $N$ {#c-impact-of-the-number-of-input-functions-n .unnumbered} To investigate the impact of the number of input functions $N$ on the final prediction, we choose different $N$ values to train the model for the example of Burger's equation and KdV equation in section [4.3.1](#sec_Burgers1){reference-type="ref" reference="sec_Burgers1"} and [4.4](#sec_KdV){reference-type="ref" reference="sec_KdV"}, respectively. The values of hyperparameters remain unchanged. The experimental results suggest that the prediction accuracy is not significantly influenced by $N$. However, as we only tested it on one example, the prediction accuracy is more sensitive to the model parameters. Different model initialization or learning rates can lead to substantial fluctuations in the final prediction. ------- -------------- -------------- -------------- -------------- -------------- -------------- Burger's Eq. KdV Eq. $N$ $L^2$-error $L^1$-error max-error $L^2$-error $L^1$-error max-error 500 7.60e-03 8.46e-03 2.02e-03 1.90e-03 1.82e-03 3.63e-03 1000 6.28e-03 6.64e-03 2.39e-03 1.34e-03 1.30e-03 3.04e-03 5000 7.17e-03 7.74e-03 2.39e-03 0.84e-03 0.83e-03 1.38e-03 10000 **4.88e-03** **5.35e-03** **1.75e-03** **0.41e-03** **0.37e-03** **1.14e-03** ------- -------------- -------------- -------------- -------------- -------------- -------------- : Impact of $N$. {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"}\ {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"}\ {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} {width="0.24\\linewidth"} # D  Effect of physics-informed training {#d-effect-of-physics-informed-training .unnumbered} Besides generated data, the physics information can also be used to train the model. However, based on our experimental findings in this section, we have found that physics-informed training does not yield a positive impact. We reconsider Burger's equation of case 1 in section [4.3.1](#sec_Burgers1){reference-type="ref" reference="sec_Burgers1"}. Building upon the previous experiment that only utilized data for training, we incorporated physical information into the training process alongside the data. We randomly sample $Q$ points within the computational domain to calculate the residual of the equation. The experimental results are presented in Fig.[\[PI\]](#PI){reference-type="ref" reference="PI"}, which indicates that incorporating equation information not only failed to improve accuracy but had the opposite effect. After adjusting hyperparameters such as the learning rate and the weight of loss functions, the model still struggled to converge during training. This could be attributed to the fact that integrating physical information made the energy landscape of the loss function more complex, rendering the model training more difficult. More discussion on this topic can refer to [@basir2023investigating; @wang2021learning]. {#fig:subfig1 width="\\linewidth"} {#fig:subfig2 width="\\linewidth"} {#fig:subfig3 width="\\linewidth"} {#fig:subfig4 width="\\linewidth"} {#fig:subfig5 width="\\linewidth"} {#fig:subfig6 width="\\linewidth"} {#fig:subfig7 width="\\linewidth"} {#fig:subfig8 width="\\linewidth"}

arxiv_math

--- abstract: | Let $S$ and $T$ be smooth projective varieties over an algebraically closed field. Suppose that $S$ is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of $k$, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational, and motivic functor. This generalizes Kahn's result on the torsion order of $S$. We also exhibit an example of $S$ over $\mathbb{C}$ for which $S \times S$ violates the integral Hodge conjecture. address: - Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan - Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan author: - Kanetomo Sato - Takao Yamazaki title: | Torsion birational motives of surfaces\ and unramified cohomology --- [^1] # Introduction Let $k$ be an algebraically closed field. In the introduction we assume the characteristic $p$ of $k$ is zero for simplicity, although most results remain valid away from $p$ if $p>0$. Recall from [@K1] that a contravariant functor $F$ defined on the category of smooth projective varieties over $k$ and with values in the category of abelian groups is called - *motivic* if $F$ factors through an additive functor on the category of Chow motives, - *birational* if $F(f)$ is an isomorphism for any birational morphism $f$, and - *normalized* if $F(\operatorname{Spec}k)=0$. (See §[2.4](#sect:mot-inv){reference-type="ref" reference="sect:mot-inv"} for details.) Classical examples of a normalized, birational, and motivic functors include $H^0(-, \Omega^i_{-/k})$ for $i>0$. Other important examples are provided by the unramified cohomology. Recall that, for a smooth $k$-scheme $X$ and $i \in \mathbb{Z}_{> 0}$, the *unramified cohomology* is defined by $$H_{\operatorname{ur}}^i(X):=H^0_{\operatorname{Zar}}(X, \mathscr{H}^i),$$ where $\mathscr{H}^i$ is the Zariski sheaf on $X$ associated to the presheaf $U \mapsto H^i_{\operatorname{\acute{e}t}}(U, \mathbb{Q}/\mathbb{Z}(i-1))$. In particular, we have $H^1_{\operatorname{ur}}(X) \cong H^1_{\operatorname{\acute{e}t}}(X, \mathbb{Q}/\mathbb{Z})$ and $H^2_{\operatorname{ur}}(X) \cong \operatorname{Br}(X)$, where $\operatorname{Br}(X):=H^2_{\operatorname{\acute{e}t}}(X, \mathbb{G}_m)$ is the Brauer group of $X$ (see §[2.5](#sect:unram-coh){reference-type="ref" reference="sect:unram-coh"} for details). We say a smooth projective variety $X$ has a *decomposition of the diagonal* if the degree map induces an isomorphism ${\operatorname{CH}}_0(X_{k(X)}) \otimes \mathbb{Q}\cong \mathbb{Q}$, where $k(X)$ denotes the total ring of fractions of $X$. This implies that $X$ is connected, and $H^0(X, \Omega_{X/k}^1)=H^0(X, \Omega_{X/k}^2)=0$. If $\dim X=2$, Bloch's conjecture predicts the converse (see §[2.6](#sect:dec-diag){reference-type="ref" reference="sect:dec-diag"} for details). The main result of this note is the following. **Theorem 1** (Theorem [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"}). *Let $S$ and $T$ be smooth projective varieties over $k$. Suppose that $S$ has a decomposition of the diagonal and $\dim S=2$. Let $f : T \to S$ be an algebraic correspondence such that $H^i_{\operatorname{ur}}(f) : H^i_{\operatorname{ur}}(S) \to H^i_{\operatorname{ur}}(T)$ vanishes for $i=1, 2$. Then $F(f) : F(S) \to F(T)$ vanishes for any normalized, birational, and motivic functor $F$.* **Remark 2**. Under the assumptions of Theorem [Theorem 1](#thm:main1){reference-type="ref" reference="thm:main1"} on $S$, we have isomorphisms $$H^1_{\operatorname{ur}}(S) \cong \operatorname{NS}(S)_{\operatorname{Tor}}, \qquad H^2_{\operatorname{ur}}(S) \cong \operatorname{Br}(S),$$ and they are finite abelian groups canonically dual to each other (see [\[eq:etcoh13-fincoef\]](#eq:etcoh13-fincoef){reference-type="eqref" reference="eq:etcoh13-fincoef"}). This result will be applied to the K3 cover $f : T \to S$ of an Enriques surface $S$ over $\mathbb{C}$ to interpret Beauville's result [@B] in Example [Example 55](#ex:Beauville){reference-type="ref" reference="ex:Beauville"}. Another application of Theorem [Theorem 1](#thm:main1){reference-type="ref" reference="thm:main1"} is an alternative proof of a result of Kahn [@K1], which computes the *torsion order* of $S$ (see Example [Example 56](#ex:Bruno){reference-type="ref" reference="ex:Bruno"}; see also Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"}). Theorem [Theorem 1](#thm:main1){reference-type="ref" reference="thm:main1"} is deduced by an easy Yoneda-type argument from the following theorem, which is a central technical result in this note. **Theorem 3** (Theorem [Theorem 51](#thm:main2-full){reference-type="ref" reference="thm:main2-full"}). *Let $S$ and $T$ be smooth projective varieties over $k$. Suppose that $S$ has a decomposition of the diagonal and $\dim S=2$. Then we have an exact sequence $$\label{eq:Vishik-ex-seq} 0 \to {\operatorname{CH}}_0(S_{k(T)})_{\operatorname{Tor}}\to \bigoplus_{i=1, 2} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(T)) \to H^3_{\operatorname{ur}}(S \times T) \to 0.$$* Theorem [Theorem 3](#thm:main2){reference-type="ref" reference="thm:main2"} is reminiscent of [@K1 Theorem 6.3] (see Remark [Remark 52](#ex:Bruno2){reference-type="ref" reference="ex:Bruno2"}). A key ingredient for the proof is a method of Vishik [@Vishik §4], where the classical Godeaux surface over $\mathbb{C}$ is investigated. Each term in [\[eq:Vishik-ex-seq\]](#eq:Vishik-ex-seq){reference-type="eqref" reference="eq:Vishik-ex-seq"} can be made explicit in some special cases. **Theorem 4** (Theorem [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"}). *Let $S$ be a smooth projective surface over $k$ having a decomposition of the diagonal. Suppose moreover that $H^1_{\operatorname{ur}}(S)$ is a cyclic group of prime order $\ell$. Then we have $$|{\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}}|=|H^3_{\operatorname{ur}}(S \times S)|=\ell.$$* Theorem [Theorem 4](#thm:main3){reference-type="ref" reference="thm:main3"} answers a question raised by Kahn [@K1 p. 840, footnote], which concerns the structure of ${\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}}$ for an Enriques surface $S$. See Example [Example 63](#ex:chow-computation){reference-type="ref" reference="ex:chow-computation"} for this point and for more examples. It also provides us with counter-examples for the integral Hodge conjecture (see Corollary [Corollary 61](#cor:integHC){reference-type="ref" reference="cor:integHC"}). **Remark 5**. Suppose now that $k$ has characteristic $p>0$. As alluded at the beginning of the introduction, most of our proof works over $k$ for the non-$p$-primary torsion part. However it completely collapses for the $p$-primary torsion part. (We essentially use an isomorphism $\mathbb{Z}/m\mathbb{Z}\cong \mu_m$ for $m \in \mathbb{Z}_{>0}$ invertible in $k$.) In this situation, it is standard to define the $p$-primary torsion part of the unramified cohomology of $X \in \operatorname{\mathbf{Sm}}$ as $$H^i_{\operatorname{ur}}(X)\{ p \} := \operatornamewithlimits{\varinjlim}_m H^1_{\operatorname{Zar}}(X, \mathscr{H}^i_{p^m}),$$ where $\mathscr{H}^i_{p^m}$ is the Zariski sheaf on $X$ associated to the presheaf $U \mapsto H^1_{\operatorname{\acute{e}t}}(U, W_m \Omega_{X, \log}^{i-1})$, and $W_m \Omega_{X, \log}^{i-1}$ is the logarithmic part of the de Rham-Witt complex (see [@I]). It can be deduced from a result of [@KOY] that $H^i_{\operatorname{ur}}(-)\{ p \}$ is a normalized, birational, and motivic functor. (This fact is kindly communicated to us by B. Kahn. See also [@BRS].) Hence it makes sense to ask if the results of this note can be extended to the $p$-primary torsion part. We leave it as a future problem. ## Organization of the paper {#organization-of-the-paper .unnumbered} §2 is a recollection on the Chow motives and birational motives. We then study a torsion direct summand of the Chow motive of a surface admitting a decomposition of the diagonal in §3. A key result is Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"}. In §4, we employ the method of Vishik [@Vishik] to study the motivic cohomology of a torsion motive constructed in §3. This result is then applied to deduce an exact sequence in §5, which relates the Chow group ${\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}}$ appearing in Theorems [Theorem 3](#thm:main2){reference-type="ref" reference="thm:main2"} and [Theorem 4](#thm:main3){reference-type="ref" reference="thm:main3"} with the unramified cohomology. The main results (Theorems [Theorem 51](#thm:main2-full){reference-type="ref" reference="thm:main2-full"}, [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"}, [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"}) are proved in §6, which also contains a discussion of examples and related topics. The last section §7 is an appendix where we prove elementary results on homological algebra that are used in the body of the paper. ## Notation and conventions {#notation-and-conventions .unnumbered} We use the following notation throughout this paper. - $k$ is a field, which will be assumed to be algebraically closed from §[3](#sect:surf){reference-type="ref" reference="sect:surf"} onward. - $p$ is the characteristic of $k$ if it is positive, and $p:=1$ otherwise. - $\Lambda$ is either $\mathbb{Z}, \mathbb{Z}[1/p]$ or $\mathbb{Q}$. From §[3](#sect:surf){reference-type="ref" reference="sect:surf"} onward, we assume $\Lambda=\mathbb{Z}[1/p]$. Notation relative to $k$. - $\operatorname{\mathbf{Fld}}$ is the category of fields over $k$ and $k$-homomorphisms. Denote by $\operatorname{\mathbf{Fld}}^{\operatorname{fg}}$ (resp. $\operatorname{\mathbf{Fld}}^{\operatorname{ac}}$) its full subcategory consisting of those which are finitely generated over $k$ (resp. algebraically closed). - $\operatorname{\mathbf{Sch}}$ is the category of separated $k$-schemes of finite type and $k$-morphisms. Its full subcategory consisting of smooth (resp. smooth and projective) $k$-schemes is denoted by $\operatorname{\mathbf{Sm}}$ (resp. $\operatorname{\mathbf{SmProj}}$). We write $\times$ for the product in $\operatorname{\mathbf{Sch}}$ (i.e., the fiber product over $\operatorname{Spec}k$ in the category of all schemes). Notation relative to $X \in \operatorname{\mathbf{Sch}}$. - $X_R := X \times_{\operatorname{Spec}k} \operatorname{Spec}R$ for a $k$-algebra $R$. - $K(X)$ is the total ring of fractions of $X_K$ for $K \in \operatorname{\mathbf{Fld}}$. - $X_{(i)}$ is the set of all points of $X$ of dimension $i$ for $i \in \mathbb{Z}$. - ${\operatorname{CH}}_i(X)$ is the Chow group of dimension $i$ cycles on $X$ for $i \in \mathbb{Z}$. - $\operatorname{Pic}(X)$ is the Picard group of $X$. - $\operatorname{NS}(X)$ is the Néron-Severi group if $X \in \operatorname{\mathbf{Sm}}$. Additional general notation, where $A$ is an abelian group: - $A[m]:=\{ a \in A \mid ma=0 \}$ for $m \in \mathbb{Z}_{>0}$, $A_{\operatorname{Tor}}:=\cup_{m \in \mathbb{Z}_{>0}} A[m]$, and $A_{\operatorname{fr}}:=A/A_{\operatorname{Tor}}$. - $\exp(A):=\inf\{ m \in \mathbb{Z}_{>0} \mid m A=0 \} \in \mathbb{Z}_{>0} \cup \{ \infty \}$. - $A_R:=A \otimes_\mathbb{Z}R$ for a commutative ring $R$. - The set of all morphisms from $X$ to $Y$ in a category $\mathscr{C}$ is written by $\mathscr{C}(X, Y)$. - $\operatorname{\mathbf{Mod}}_\Lambda$ is the category of all $\Lambda$-modules and $\Lambda$-homomorphisms. # Preliminaries In this section we recall some definitions and results from [@CT; @GG; @K1; @KS1; @T; @Vishik] that will be used later. ## Chow motives {#sect:cat-mot} We write $\operatorname{\mathbf{Chow}}(k)_\Lambda$ for the *covariant* category of Chow motives over $k$ with coefficients in $\Lambda$, defined e.g. in [@K1 §1.5, 1.6], [@T §4, p.2092]. (This is opposite of the more frequently used contravariant version, see e.g. [@S].) It is a $\Lambda$-linear rigid symmetric monoidal pseudo-abelian category. Any object of $\operatorname{\mathbf{Chow}}(k)_\Lambda$ can be written as $(X, \pi, r)$ for some equidimensional $X \in \operatorname{\mathbf{SmProj}}$, a projector $\pi$ of $X$, and $r \in \mathbb{Z}$. (By a projector of $X$ we mean $\pi \in {\operatorname{CH}}_{\dim X}(X \times X)_\Lambda$ such that $\pi \circ \pi = \pi$, where $\circ$ denotes the composition of algebraic correspondences.) We have $$\operatorname{\mathbf{Chow}}(k)_\Lambda((X, \pi, r), (Y, \rho, s)) = \rho \circ {\operatorname{CH}}_{\dim X + r - s}(X \times Y)_\Lambda \circ \pi,$$ where $X, Y \in \operatorname{\mathbf{SmProj}}$ (with $X$ equidimensional), $\pi, \rho$ projectors of $X, Y$, and $r, s \in \mathbb{Z}$. We write $\Lambda(r):=(\operatorname{Spec}k, {\operatorname{id}}_{\operatorname{Spec}k}, r)$ and $M(r) := M \otimes \Lambda(r)$ for $M \in \operatorname{\mathbf{Chow}}(k)_\Lambda$. Thus $\Lambda := \Lambda(0)$ is a unit object for the monoidal structure. We denote by $M^\vee$ the dual object of $M$. The category of effective Chow motives $\operatorname{\mathbf{Chow}}(k)^{\operatorname{eff}}_\Lambda$ is the full subcategory of $\operatorname{\mathbf{Chow}}(k)_\Lambda$ consisting of all objects isomorphic to those the form $(X, \pi, r)$ with $r \ge 0$. There is a covariant functor $$\label{eq:h-eff} h^{\operatorname{eff}}: \operatorname{\mathbf{SmProj}}\to \operatorname{\mathbf{Chow}}(k)_\Lambda^{\operatorname{eff}}, \qquad h^{\operatorname{eff}}(X)=(X, {\operatorname{id}}_X, 0).$$ We have $h^{\operatorname{eff}}(X) = h^{\operatorname{eff}}(X)^\vee(d)$ if $X \in \operatorname{\mathbf{SmProj}}$ is purely $d$-dimensional. For $M \in \operatorname{\mathbf{Chow}}(k)_\Lambda$ and $r \in \mathbb{Z}$ we write ${\operatorname{CH}}_r(M)_\Lambda := \operatorname{\mathbf{Chow}}(k)_\Lambda(\Lambda(r), M)$ so that we have ${\operatorname{CH}}_r(h^{\operatorname{eff}}(X))_\Lambda={\operatorname{CH}}_r(X)_\Lambda$ for any $X \in \operatorname{\mathbf{SmProj}}$. We abbreviate $\operatorname{\mathbf{Chow}}_\Lambda:=\operatorname{\mathbf{Chow}}(k)_\Lambda$ and $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda:=\operatorname{\mathbf{Chow}}(k)^{\operatorname{eff}}_\Lambda$. For any $K \in \operatorname{\mathbf{Fld}}$, there is a base change functor $\operatorname{\mathbf{Chow}}_\Lambda \to \operatorname{\mathbf{Chow}}(K)_\Lambda$ written by $M \mapsto M_K$. ## Torsion motives Vishik [@Vishik Definition 2.4] defines a torsion motive to be an object $M \in \operatorname{\mathbf{Chow}}_\Lambda$ such that $m \cdot {\operatorname{id}}_M=0$ for some $m \in \mathbb{Z}_{>0}$. Since we will need a similar notion considered in different categories, we introduce the following general terminology: **Definition 6**. We say an object $A$ of an additive category $\mathscr{C}$ is *torsion* if there exists $m \in \mathbb{Z}_{>0}$ such that $m \cdot {\operatorname{id}}_A=0$ in $\mathscr{C}(A, A)$. This is equivalent to saying that $\mathscr{C}(A, B)$ (or $\mathscr{C}(B, A)$) is a torsion abelian group for any $B \in \mathscr{C}$. The following is an obvious variant of a result of Gorchinskiy-Orlov [@GO Lemma 1]. **Lemma 7**. *For $M \in \operatorname{\mathbf{Chow}}_\Lambda$, the following conditions are equivalent.* 1. *$M$ is a torsion object of $\operatorname{\mathbf{Chow}}_\Lambda$.* 2. *${\operatorname{CH}}_n(M_K)_\Lambda$ is torsion for any $n \in \mathbb{Z}$ and for any $K \in \operatorname{\mathbf{Fld}}$.* 3. *${\operatorname{CH}}_n(M_K)_\Lambda$ is torsion for any $n \in \mathbb{Z}$ and for any $K \in \operatorname{\mathbf{Fld}}^{\operatorname{ac}}$.* 4. *${\operatorname{CH}}_n(M_K)_\Lambda$ is torsion for any $n \in \mathbb{Z}$ and for any $K \in \operatorname{\mathbf{Fld}}^{\operatorname{fg}}$.* *Proof.* (2) $\Rightarrow$ (3) and (2) $\Rightarrow$ (4) are obvious. (3) $\Rightarrow$ (2) holds because $\ker({\operatorname{CH}}_n(M_K)_\Lambda \to {\operatorname{CH}}_n(M_{\overline{K}})_\Lambda)$ is torsion, where $\overline{K}$ is an algebraic closure of $K \in \operatorname{\mathbf{Fld}}$. (4) $\Rightarrow$ (2) is seen by taking colimit. We have shown the equivalence (2) $\Leftrightarrow$ (3) $\Leftrightarrow$ (4). Let us show (1) $\Rightarrow$ (4). By the shown equivalence (3) $\Leftrightarrow$ (4), we are reduced to the case $k$ is algebraically closed (in particular $k$ is perfect). Take $K \in \operatorname{\mathbf{Fld}}^{\operatorname{fg}}$. By Nagata's compactification and de Jong's alteration (see [@Co Theorem 4.1], [@dJ Theorem 4.1]), we can find an integral proper $k$-scheme $X \in \operatorname{\mathbf{Sch}}$ with $K=k(X)$ and a proper surjective generically finite morphism $f : Y \to X$ with $Y \in \operatorname{\mathbf{SmProj}}$ integral. We then have a sequence of induced maps $${\operatorname{CH}}_{n+d_Y}(M \otimes Y)_\Lambda \twoheadrightarrow {\operatorname{CH}}_{n}(M_{k(Y)})_\Lambda \overset{f_*}{\to} {\operatorname{CH}}_{n}(M_{k(X)})_\Lambda,$$ where $d_Y:=\dim Y$. The first map is surjective, and the cokernel of the second map is annihilated by $[k(Y): k(X)]$. Since ${\operatorname{CH}}_{n+d_Y}(M \otimes Y)_\Lambda =\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(\Lambda(n+d_Y), M \otimes Y)$ is torsion by the assumption (1), we conclude that ${\operatorname{CH}}_{n}(M_{k(X)})_\Lambda$ is torsion as well. It remains to prove (2) $\Rightarrow$ (1), for which we follow [@GG Lemma 1]. Write $M=(X, \pi, r) \in \operatorname{\mathbf{Chow}}_\Lambda$ with $X$ equidimensional and put $d_X := \dim X$. We take $N \in \operatorname{\mathbf{Chow}}_\Lambda$ and show that $\operatorname{\mathbf{Chow}}_\Lambda(M, N)$ is torsion. We may assume $N=h^{\operatorname{eff}}(Y)$ for connected $Y \in \operatorname{\mathbf{SmProj}}$ (by replacing $r$ if necessary). Given $Z \in \operatorname{\mathbf{Sch}}$, we define ${\operatorname{CH}}_n(M \otimes Z)_\Lambda$ as the image of an idempotent operator $${\operatorname{CH}}_n(X \times Z)_\Lambda \to {\operatorname{CH}}_n(X \times Z)_\Lambda, \qquad \alpha \mapsto p_{23 *}(p_{13}^*(\alpha) \cdot_{p_{12}} \pi),$$ where $p_{ij}$ are respective projections on $X \times X \times Z$, and $\cdot_{p_{12}}$ is the global product along $p_{12}$ defined in [@F §8.1]; this product exists since $X \times X$ is smooth. We show that ${\operatorname{CH}}_n(M \otimes Z)_\Lambda$ is torsion for any integral $Z \in \operatorname{\mathbf{Sch}}$ and for any $n$ by induction on $d_Z:=\dim Z$. The case $d_Z=0$ is immediate from the assumption (2). If $d_Z>0$, from the localization sequence for $X \times Z$ we deduce an exact sequence $$\bigoplus_W {\operatorname{CH}}_n(M \otimes W)_\Lambda \to {\operatorname{CH}}_n(M \otimes Z)_\Lambda \to {\operatorname{CH}}_{n-d_Z}(M_{k(Z)})_\Lambda \to 0,$$ where $W$ runs through integral proper closed subschemes of $Z$. The claim now follows by induction. Applying this to $Z=Y$ and $n=d_X+r$, we conclude ${\operatorname{CH}}_{d_X+r}(M \otimes Y)_\Lambda =\operatorname{\mathbf{Chow}}_\Lambda(M, N)$ is torsion. ◻ ## Birational motives {#sect:birat-mot} We write $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{bir}}$ for the category of birational motives over $k$ with coefficients in $\Lambda$ from [@KS1 Definition 2.3.6]. (This is denoted by $\operatorname{\mathbf{Chow}}^\circ(k, \Lambda)$ in [@KS1].) This comes equipped with a localization functor $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}\to \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{bir}}$. We write the composition of it with $h^{\operatorname{eff}}$ by $$\label{eq:h-bir} h^{\operatorname{bir}}: \operatorname{\mathbf{SmProj}}\to \operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda.$$ We then have $$\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{bir}}(h^{\operatorname{bir}}(X), h^{\operatorname{bir}}(Y)) = {\operatorname{CH}}_0(Y_{k(X)})_\Lambda$$ for any $X, Y \in \operatorname{\mathbf{SmProj}}$ (see [@KS1 Lemma 2.3.7]). **Remark 8**. There are several variants of $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$. We recall two of them. 1. Denote by $\operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}, 1}$ the pseudo-abelian envelope of the category obtained from $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ by inverting all birational morphisms. 2. Denote by $\operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}, 2}$ the pseudo-abelian envelope of $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda/\mathbb{L}$, where $\mathbb{L}$ is the ideal of $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ consisting of all morphisms which factor through an object of the form $M(1)$ with $M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$. There are functors $$\operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}, 2} \overset{\cong}{\longrightarrow} \operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}, 1} \overset{}{\longrightarrow} \operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}}.$$ The first one is always an equivalence, and so is the second if $p$ is invertible in $\Lambda$ (see [@KS1 Proposition 2.2.9, Corollary 2.4.3]). As $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda \to \operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$ factors through $\operatorname{\mathbf{Chow}}_\Lambda^{{\operatorname{bir}}, 2}$, the image of $M(1)$ vanishes in $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$ for any $M\in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$. Finally, we write $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$ for the quotient category of $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$ by the ideal generated by $\Lambda=h^{\operatorname{bir}}(\operatorname{Spec}k)$, introduced in [@K1 Definition 2.4]. Denote by $$\label{eq:h-nor} h^{\operatorname{nor}}: \operatorname{\mathbf{SmProj}}\to \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$$ the composition of $h^{\operatorname{bir}}$ and the localization functor $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda \to \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$. We have $$\label{eq:hom-in-chow-nor} \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(h^{\operatorname{nor}}(X), h^{\operatorname{nor}}(Y)) = \operatorname{Coker}({\operatorname{CH}}_0(Y)_\Lambda \to {\operatorname{CH}}_0(Y_{k(X)})_\Lambda)$$ for any $X, Y \in \operatorname{\mathbf{SmProj}}$ (see loc. cit.). **Remark 9**. If no confusion is likely, we abbreviate $h^{\operatorname{eff}}(X),~ h^{\operatorname{bir}}(X)$, and $h^{\operatorname{nor}}(X)$ by $X$ for $X \in \operatorname{\mathbf{SmProj}}$. Similarly, for $M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ we use the same letter $M$ to denote its images in $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$ and $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$. For instance, the left hand side of [\[eq:hom-in-chow-nor\]](#eq:hom-in-chow-nor){reference-type="eqref" reference="eq:hom-in-chow-nor"} will be written by $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(X, Y)$. ## Motivic invariants {#sect:mot-inv} Denote by $\operatorname{\mathbf{Mod}}_\Lambda$ the category of $\Lambda$-modules. Following [@K1 Definition 2.1], we introduce some definitions. **Definition 10**. Let $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$ be a functor. 1. We say $F$ is *birational* if $F(f)$ is an isomorphism for any birational morphism $f$. 2. We say $F$ is *motivic* if $F$ factors through an additive functor $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda \to \operatorname{\mathbf{Mod}}_\Lambda$. 3. We say $F$ is *normalized* if $F(\operatorname{Spec}k)=0$. **Lemma 11**. *Suppose that $p$ is invertible in $\Lambda$. A functor $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$ is birational and motivic (resp. normalized, birational, and motivic) if and only if $F$ factors through an additive functor $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda \to \operatorname{\mathbf{Mod}}_\Lambda$ (resp. $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda \to \operatorname{\mathbf{Mod}}_\Lambda$).* *Proof.* This is immediate from what we recalled in §[2.3](#sect:birat-mot){reference-type="ref" reference="sect:birat-mot"}. ◻ **Remark 12**. Given a motivic (resp. birational and motivic, resp. normalized, birational, and motivic) functor $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$, its extension to $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ (resp. $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$, resp. $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$) is denoted by the same letter $F$. **Example 13**. 1. Suppose $p=1$ or $\Lambda=\mathbb{Z}$. It is a classical fact that $H^0(-, \Omega^i_{-/k})$ is birational and motivic for any $i \in \mathbb{Z}_{\ge0}$; it is also normalized if $i>0$. It is less classical that the same is true of $H^i(-, \mathscr{O})$ if $k$ is perfect (see [@CR]). 2. It is obvious from the definition that the functor $$\label{eq:CH0-S} \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(-, S) : T \mapsto \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S) = \operatorname{Coker}({\operatorname{CH}}_0(S)_\Lambda \to {\operatorname{CH}}_0(S_{k(T)})_\Lambda)$$ is birational, motivic, and normalized for any fixed $S \in \operatorname{\mathbf{SmProj}}$. 3. Let $M$ be a cycle module in the sense of Rost [@R]. Then its $0$-th cycle cohomology $A^0(-, M_n)$ is birational and motivic by [@KS1 Corollary 6.1.3]. We will only use a special case of unramified cohomology, which will be recalled in the next subsection. **Lemma 14**. *Suppose that $p$ is invertible in $\Lambda$. For $S, T \in \operatorname{\mathbf{SmProj}}$ and a morphism $f : T \to S$ in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$, the following are equivalent:* 1. *The induced map $F(f) : F(S) \to F(T)$ vanishes for any normalized, birational, and motivic functor $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$.* 2. *The induced map $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(f, S) : \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(S, S) \to \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S)$ vanishes.* *Proof.* (2) is a special case of (1) for $F=\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(-, S)$, whence (1) $\Rightarrow$ (2). To see the converse, note that $f$ is the image of ${\operatorname{id}}_S$ by the map in (2) and apply Lemma [Lemma 11](#lem:bir-funct-factor){reference-type="ref" reference="lem:bir-funct-factor"}. ◻ ## Unramified cohomology {#sect:unram-coh} A general reference for this subsection is [@CT]. Let $K \in \operatorname{\mathbf{Fld}}$ and $i \in \mathbb{Z}$. For $n \in \mathbb{Z}_{>0}$ invertible in $k$, the *unramified cohomology* of $K/k$ is defined by $$\label{eq:def-unramcoh1} H_{{\operatorname{ur}}, n}^i(K/k) := \ker\left( H_{\operatorname{Gal}}^i(K, \mu_n^{\otimes (i-1)}) \to \bigoplus_v H_{\operatorname{Gal}}^{i-1}(F_v, \mu_n^{\otimes (i-2)}) \right),$$ where $v$ ranges over all discrete valuations of $K$ that are trivial on $k$, and $F_v$ is the residue field of $v$. The maps appearing in the definition are the residue maps (see [@CT (3.6)]). We set $$\label{eq:def-unramcoh2} H_{\operatorname{ur}}^i(K/k) := \operatornamewithlimits{\varinjlim}_{(n, p)=1} H_{{\operatorname{ur}}, n}^i(K/k),$$ where $n$ ranges over all $n \in \mathbb{Z}_{>0}$ that is invertible in $k$. By Rost-Voevodsky's norm residue isomorphism theorem (which is the former Bloch-Kato conjecture, see [@HW Theorem A]), we may identify $H_{{\operatorname{ur}}, n}^i(K/k)$ with the $n$-torsion part of $H_{\operatorname{ur}}^i(K/k)$: $$\label{eq:def-unramcoh3} H_{{\operatorname{ur}}, n}^i(K/k) \cong H_{\operatorname{ur}}^i(K/k)[n].$$ Let $X \in \operatorname{\mathbf{Sm}}$ and $i \in \mathbb{Z}$. For $n \in \mathbb{Z}_{>0}$ invertible in $k$, the *unramified cohomology* of $X$ is defined as $$\label{eq:def-unramcoh4} H^i_{{\operatorname{ur}}, n}(X) :=H^0_{\operatorname{Zar}}(X, \mathscr{H}^i_n), \qquad H^i_{\operatorname{ur}}(X) := \operatornamewithlimits{\varinjlim}_{(n, p)=1} H^i_{{\operatorname{ur}}, n}(X),$$ where $\mathscr{H}^i_n$ is the Zariski sheaf on $X$ associated to the presheaf $U \mapsto H^i_{\operatorname{\acute{e}t}}(U, \mu_n^{\otimes (i-1)})$, and the colimit in the second formula is taken in the same way as [\[eq:def-unramcoh2\]](#eq:def-unramcoh2){reference-type="eqref" reference="eq:def-unramcoh2"}. We have canonical isomorphisms (see [@CT Propositions 4.2.1, 4.2.3]) $$\label{eq:ur-coh-12} H^1_{{\operatorname{ur}}, n}(X) \cong H^1_{\operatorname{\acute{e}t}}(X, \mathbb{Z}/n\mathbb{Z}), \qquad H^2_{{\operatorname{ur}}, n}(X) \cong \operatorname{Br}(X)[n],$$ where $\operatorname{Br}(X):=H^2_{\operatorname{\acute{e}t}}(X, \mathbb{G}_m)$ is the Brauer group of $X$. If further $X$ is integral and proper over $k$, we also have (see [@CT Theorem 4.1.1]) $$\label{eq:unramcoh-K-S} H^i_{{\operatorname{ur}}, n}(X) \cong H^i_{{\operatorname{ur}}, n}(k(X)/k), \qquad H^i_{{\operatorname{ur}}}(X) \cong H^i_{{\operatorname{ur}}}(k(X)/k).$$ The following well-known fact plays an essential role in this paper: **Proposition 15**. *Let $i, n \in \mathbb{Z}$ and suppose that $n$ is invertible in $k$. Then the functor $H_{{\operatorname{ur}}, n}^i : \operatorname{\mathbf{SmProj}}\to \operatorname{\mathbf{Mod}}_{\mathbb{Z}[1/p]}$ is birational and motivic. The same is true for $H_{\operatorname{ur}}^i$. They are also normalized if $i>0$ and $k$ is algebraically closed.* *Proof.* The first statement follows from [@CT Theorem 4.1.1] (see also [@R (2.5)]) and [@KS1 Corollary 6.1.3], and the second from the first. The third statement is obvious from the definition. ◻ ## Varieties admitting decompositions of the diagonals {#sect:dec-diag} **Proposition 16**. *The following conditions are equivalent for $X \in \operatorname{\mathbf{SmProj}}$:* 1. *The degree map induces an isomorphism ${\operatorname{CH}}_0(X_{k(X)})_\mathbb{Q}\cong \mathbb{Q}$.* 2. *The class of the generic point of $X$ in ${\operatorname{CH}}_0(X_{k(X)})_\mathbb{Q}$ belongs to $$\operatorname{Im}({\operatorname{CH}}_0(X)_\mathbb{Q}\to {\operatorname{CH}}_0(X_{k(X)})_\mathbb{Q}).$$* 3. *The structure map induces an isomorphism $h^{\operatorname{bir}}(X) \cong \mathbb{Q}$ in $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\mathbb{Q}$* 4. *The object $h^{\operatorname{nor}}(X)$ of $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\mathbb{Z}$ is torsion in the sense of Definition [Definition 6](#def:tor-mot){reference-type="ref" reference="def:tor-mot"}.* *Proof.* See [@KS1 Proposition 3.1.1] for (1)--(3). Equivalence of (2) and (4) is obvious from the definition and [\[eq:hom-in-chow-nor\]](#eq:hom-in-chow-nor){reference-type="eqref" reference="eq:hom-in-chow-nor"} (see also [@K1 §2.3]). ◻ **Remark 17**. If $k$ is an algebraically closed field with infinite transcendental degree over its prime subfield, then these conditions are also equivalent to the following: 1. The degree map induces an isomorphism ${\operatorname{CH}}_0(X)_\Lambda \cong \Lambda$ for either $\Lambda=\mathbb{Z}$ or $\mathbb{Q}$. (See [@KS1 Proposition 3.1.1].) **Definition 18**. We say $X \in \operatorname{\mathbf{SmProj}}$ admits a *decomposition of the diagonal* if the conditions of Proposition [Proposition 16](#prop:equiv-UTC){reference-type="ref" reference="prop:equiv-UTC"} are satisfied. This notion goes back to Bloch-Srinivas [@BS]. For such $X$, Kahn [@K1 Definition 2.5] and Chatzistamatiou-Levine [@CL Definition 1.1] defined a numerical invariant called the *torsion order*, which can be written as ${\operatorname{Tor}}_\mathbb{Z}^{\operatorname{nor}}(X)$ in terms of the following definition: **Definition 19**. 1. Let $A$ be an object of an additive category $\mathscr{C}$ that is torsion in the sense of Definition [Definition 6](#def:tor-mot){reference-type="ref" reference="def:tor-mot"}. The smallest $m \in \mathbb{Z}_{>0}$ such that $m \cdot {\operatorname{id}}_A=0$ is called the *torsion order* of $A$. 2. The torsion order of a torsion object $M$ of $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}$ (resp. $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{bir}}$, resp. $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{nor}}$) is denoted by ${\operatorname{Tor}}^{\operatorname{eff}}_\Lambda(M)$ (resp. ${\operatorname{Tor}}^{\operatorname{bir}}_\Lambda(M)$, resp. ${\operatorname{Tor}}^{\operatorname{nor}}_\Lambda(M)$). We write $b_i(X)$ and $\rho(X)$ for the Betti and Picard numbers of $X \in \operatorname{\mathbf{SmProj}}$: $$b_i(X) := \dim_{\mathbb{Q}_\ell} H^i_{\operatorname{\acute{e}t}}(X_{\overline{k}}, \mathbb{Q}_\ell), \qquad \rho(X):= \operatorname{rank}_\mathbb{Z}\operatorname{NS}(X_{\overline{k}})/\operatorname{NS}(X_{\overline{k}})_{\operatorname{Tor}},$$ where $\overline{k}$ is an algebraic closure of $k$, and $\ell$ is any prime number different from $p$. **Proposition 20**. *Suppose that $X \in \operatorname{\mathbf{SmProj}}$ admits a decomposition of the diagonal.* 1. *We have $b_1(X)=0$,  $b_2(X)=\rho(X)$ and $\operatorname{Pic}(X)=\operatorname{NS}(X)$.* 2. *Suppose that $p$ is invertible in $\Lambda$, and put $m:={\operatorname{Tor}}^{\operatorname{nor}}_\Lambda(X)$. Then we have $m F(X)=0$ for any normalized, birational, and motivic functor $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$.* *Proof.* See [@KS1 Proposition 3.1.4] and [@K1 Lemma 2.6]. ◻ **Remark 21**. 1. If $S \in \operatorname{\mathbf{SmProj}}$ is a surface such that $b_1(S)=0$ and $b_2(S)=\rho(S)$, then Bloch's conjecture predicts that $S$ should admit a decomposition of the diagonal (see [@KS1 Proposition 3.1.4]). 2. It is obvious that ${\operatorname{Tor}}^{\operatorname{nor}}_\Lambda(M) ~|~ {\operatorname{Tor}}^{\operatorname{bir}}_\Lambda(M) ~|~ {\operatorname{Tor}}^{\operatorname{eff}}_\Lambda(M)$ for torsion $M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$. The opposite divisibility does not hold in general. (For example, we have ${\operatorname{Tor}}^{\operatorname{eff}}_\Lambda(M)={\operatorname{Tor}}^{\operatorname{eff}}_\Lambda(M(1))$ but the image of $M(1)$ vanishes in $\operatorname{\mathbf{Chow}}^{\operatorname{bir}}_\Lambda$.) Yet, it can hold in some non-trivial cases, as seen in Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"} below. # Torsion motives of surfaces {#sect:surf} **Setting 22**. From now on we suppose $k$ is algebraically closed and $\Lambda=\mathbb{Z}[1/p]$. Fix $S \in \operatorname{\mathbf{SmProj}}$ admitting a decomposition of the diagonal and such that $\dim S=2$. ## Surfaces admitting decompositions of the diagonals {#sect:triv-bir-mot} **Lemma 23**. *For any prime number $\ell \not=p$, we have the following:* 1. *$b_0(S)=b_4(S)=1$, $b_2(S)=\rho(S)$, and $b_i(S)=0$ for any $i \not= 0, 2, 4$.* 2. *$H^0_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell) =H^4_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(2))=\mathbb{Z}_\ell$,  $H^1_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell)=0$, and $$\begin{aligned} &H^1_{\operatorname{ur}}(S)_{\mathbb{Z}_\ell} \cong \operatorname{NS}(S)_{{\operatorname{Tor}}, \mathbb{Z}_\ell} \cong H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{\operatorname{Tor}}, \\ &H^2_{\operatorname{ur}}(S)_{\mathbb{Z}_\ell} \cong \operatorname{Br}(S)_{\mathbb{Z}_\ell} \cong H^3_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1)).\end{aligned}$$* 3. *$\operatorname{Pic}(S)=\operatorname{NS}(S)$ is a finitely generated $\mathbb{Z}$-module; $\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda}$ and $\operatorname{Br}(S)_{\Lambda}$ are finite abelian groups canonically dual to each other.* 4. *${\operatorname{CH}}_1(S_K) \cong \operatorname{NS}(S)$ for any $K \in \operatorname{\mathbf{Fld}}$ and ${\operatorname{CH}}_2(S_{\overline{K}}) \cong \mathbb{Z}$ for any $\overline{K} \in \operatorname{\mathbf{Fld}}^{\operatorname{ac}}$.* *Proof.* (1)--(3) are standard consequences of Proposition [Proposition 20](#prop:b1-b2-rho){reference-type="ref" reference="prop:b1-b2-rho"}, using Poincaré duality, Kummer sequence, and [\[eq:ur-coh-12\]](#eq:ur-coh-12){reference-type="eqref" reference="eq:ur-coh-12"}. It also implies the vanishing of the Picard variety of $X$, whence the first statement of (4). The last statement of (4) follows from Roitman's theorem. ◻ **Lemma 24**. *Let $\rho:=\rho(S)$ and take $e_1, \dots, e_\rho \in \operatorname{NS}(S)$ such that their classes form a $\mathbb{Z}$-basis of $\operatorname{NS}(S)/\operatorname{NS}(S)_{\operatorname{Tor}}$. Let $a_{ij} := \langle e_i, e_j \rangle \in \mathbb{Z}$, where $\langle \cdot, \cdot \rangle$ denotes the intersection form on $S$. Then $\delta:=\det((a_{ij})_{i,j=1, \dots, \rho})$ is invertible in $\Lambda$.* *Proof.* It suffices to show that $\delta \in \mathbb{Z}_\ell^\times$ for any prime number $\ell \not= p$. By Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"}, we have an isomorphism $\operatorname{NS}(S)_{\mathbb{Z}_\ell} \cong H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))$ which is compatible with the intersection pairing and the cup product. Therefore it suffices to show that the cup product induces an isomorphism $$H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{\operatorname{fr}} \overset{\cong}{\longrightarrow} \operatorname{Hom}_{\mathbb{Z}_\ell}(H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{\operatorname{fr}}, \mathbb{Z}_\ell),$$ where we put $M_{\operatorname{fr}}:= M/M_{\operatorname{Tor}}$ for a $\mathbb{Z}_\ell$-module $M$. This follows from the Poincaré duality, see Proposition [Proposition 69](#prop:PD-app){reference-type="ref" reference="prop:PD-app"} below. ◻ **Proposition 25**. *There exists a direct summand $M$ of $h^{\operatorname{eff}}(S)$ in $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ satisfying the following conditions:* 1. *$h^{\operatorname{eff}}(S) \cong \Lambda \oplus \Lambda(1)^{\rho(S)} \oplus \Lambda(2) \oplus M$ in $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}$;* 2. *$M \cong M^\vee(2)$ in $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}$;* 3. *$M$ is torsion in $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}$ in the sense of Definition [Definition 6](#def:tor-mot){reference-type="ref" reference="def:tor-mot"}.* *Proof.* The statement without the condition (2) is shown by Gorchinskiy-Orlov in (the proof of) [@GO Proposition 2.3, Remark 2.5] when $k=\mathbb{C}$, and the full statement by Vishik in [@Vishik Proposition 4.1] when $S$ is the classical Godeaux surface. The same proof works without any essential change, but for the sake of completeness we give a brief account. Let $e_1, \dots, e_\rho \in \operatorname{NS}(S)$ and $A:=(a_{ij}) \in {\operatorname{GL}}_\rho(\Lambda)$ be as in Lemma [Lemma 24](#lem:delta){reference-type="ref" reference="lem:delta"}. Write $A^{-1}=(b_{ij}) \in {\operatorname{GL}}_\rho(\Lambda)$. Take also a closed point $x_0 \in S_{(0)}$. We then define orthogonal projectors $$\begin{aligned} & \pi_1 := \sum_{i, j} b_{ij} [e_i \times e_j], \quad \pi_2 := [S \times x_0] + [x_0 \times S] \quad \in \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}(S, S)={\operatorname{CH}}_2(S \times S)_\Lambda.\end{aligned}$$ Set $M_i:=(S, \pi_i, 0) \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ for $i=1, 2$. Then we have $M_1 \cong \Lambda(1)^\rho$, $M_2 \cong \Lambda \oplus \Lambda(2)$, and $M_i^\vee \cong M_i(2)$ for $i=1, 2$. It follows that $M:=(S, \pi, 0)$ with $\pi := {\operatorname{id}}_S - \pi_1 - \pi_2$ satisfies (1) and (2). Observe that (1) and Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} imply that for any $\overline{K} \in \operatorname{\mathbf{Fld}}^{\operatorname{ac}}$ $${\operatorname{CH}}_1(M_{\overline{K}})_\Lambda=\operatorname{NS}(S)_{\Lambda, {\operatorname{Tor}}} \quad \text{and}\quad {\operatorname{CH}}_i(M_{\overline{K}})_\Lambda=0 ~\text{for}~ i \not= 1.$$ It then follows by Lemma [Lemma 7](#lem:GG){reference-type="ref" reference="lem:GG"} that $M$ satisfies (3) too. We are done. ◻ The summand $M$ is not necessarily unique. We choose one and fix it. **Setting 26**. In what follows we denote by $M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ a Chow motive constructed in Proposition [Proposition 25](#prop:GG-V){reference-type="ref" reference="prop:GG-V"}. **Proposition 27**. 1. *Suppose that $T \in \operatorname{\mathbf{SmProj}}$ is also a surface admitting a decomposition of the diagonal. Let $N \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ be the motive obtained by applying Proposition [Proposition 25](#prop:GG-V){reference-type="ref" reference="prop:GG-V"} to $T$. We consider the maps $$\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(N, M) \overset{a}{\longrightarrow} \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(N, M) \overset{b}{\longrightarrow} \underset{i=1, 2}{\bigoplus} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(T)), %\Chow^\eff(N, M) %\overset{\overset{a}{\cong}}{\longrightarrow} %\Chow^\nor(N, M) %\overset{b}{\hookrightarrow} %\underset{i=1, 2}{\bigoplus} \Hom(H_\ur^i(S), H_\ur^i(T)),$$ where $a$ is induced by the localization functor $\operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}\to \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{bir}}$, and $b$ is induced by the functors $H^i_{\operatorname{ur}}$ for $i=1, 2$ using Lemma [Lemma 11](#lem:bir-funct-factor){reference-type="ref" reference="lem:bir-funct-factor"} and Proposition [Proposition 15](#prop:unram-coh){reference-type="ref" reference="prop:unram-coh"}. Then $a$ is bijective and $b$ is injective.* 2. *We have $$\label{eq:torsion-order-S} {\operatorname{Tor}}_\Lambda^{\operatorname{eff}}(M) = {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(M) = {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(S) = \exp(\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda}) = \exp(\operatorname{Br}(S)_\Lambda),$$ where $\exp(A):= \min \{ m \in \mathbb{Z}_{>0} ~|~ m A = 0 \}$ for an abelian group $A$.* *Proof.* (1) This proof is adopted from [@GO Proposition 2.3]. Recall from Remark [Remark 8](#rem:comparison){reference-type="ref" reference="rem:comparison"} that $\Lambda(r)$ vanishes in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$ for any $r \ge 0$ and hence we have $S=M, ~T=N$ in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$. We consider a commutative diagram $$\xymatrix{ \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(N, M) \ar[r]^-a \ar[rd]^e & \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(N, M) = \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S) \ar[d]^b \\ \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(T, S) \ar[u]^c \ar[r]^-d & \underset{i=1, 2}{\bigoplus} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(T)). }$$ The maps $a$ and $c$ are surjective by definition. Therefore it suffices to prove the injectivity of $e$. Take $f \in \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}(N, M)$ such that $e(f)=0$. By Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} this means that, for any prime number $\ell \not= p$, we have $$\label{eq:vanishing-23} f^*=0 : H^i_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}} \to H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}} \quad \text{for $i=2, 3$}.$$ On the other hand, we have a commutative diagram $$\xymatrix{ {\operatorname{CH}}_2(S \times T)_{{\operatorname{Tor}}, \mathbb{Z}_{\ell}} \ar@{^{(}-^{>}}[r] & H^3_{\operatorname{\acute{e}t}}(S \times T, \mathbb{Q}_\ell/\mathbb{Z}_\ell(2)) \ar@{^{(}-^{>}}[d] \\ \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(N, M)_{\mathbb{Z}_\ell} \ar@{^{(}-^{>}}[u] \ar[r] & H^4_{\operatorname{\acute{e}t}}(S \times T, \mathbb{Z}_\ell(2))_{\operatorname{Tor}}. }$$ The left vertical map comes from ${\operatorname{CH}}_2(S \times T)_\Lambda =\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(T, S)$. The upper horizontal injective map is the one constructed by Bloch (see [@CT2 Théorème 4.3]). The right vertical map is injective since we have $H^3_{\operatorname{\acute{e}t}}(S \times T, \mathbb{Q}_\ell(2))=0$ by Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} and the Künneth formula. We have shown the injectivity of the lower horizontal map. By the Künneth formula, Poincaré duality and Lemma [Lemma 67](#lem:homalg){reference-type="ref" reference="lem:homalg"} below, we have $$\begin{aligned} H^4_{\operatorname{\acute{e}t}}(S \times T, \mathbb{Z}_\ell(2))_{{\operatorname{Tor}}} &\cong \underset{i=2, 3}{\bigoplus} {\operatorname{Tor}}(H^{5-i}_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}, H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}) \\ &\cong \underset{i=2, 3}{\bigoplus} \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}, H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}).\end{aligned}$$ Now [\[eq:vanishing-23\]](#eq:vanishing-23){reference-type="eqref" reference="eq:vanishing-23"} shows that $f=0$ in $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(N, M)$. We are done. \(2\) The relations $$\exp(\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda})= \exp(\operatorname{Br}(S)_\Lambda) ~|~ {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(S) = {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(M) ~|~ {\operatorname{Tor}}_\Lambda^{\operatorname{eff}}(M)$$ are seen by Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} (3), Propositions [Proposition 15](#prop:unram-coh){reference-type="ref" reference="prop:unram-coh"} and [Proposition 20](#prop:b1-b2-rho){reference-type="ref" reference="prop:b1-b2-rho"} (2) applied to $F=\operatorname{Br}(-)_\Lambda$, the equality $S=M$ in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$ recalled above, and Remark [Remark 21](#rem:div-nor-eff){reference-type="ref" reference="rem:div-nor-eff"} (2), respectively. To prove ${\operatorname{Tor}}_\Lambda^{\operatorname{eff}}(M) ~|~ \exp(\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda})$, it suffices to apply (1) to $T=S$ and $f=m \cdot {\operatorname{id}}_S$ with $m \in \mathbb{Z}_{>0}$. ◻ **Corollary 28**. 1. *If $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$ is a motivic functor, then $F(M)$ is annihilated by the integer in [\[eq:torsion-order-S\]](#eq:torsion-order-S){reference-type="eqref" reference="eq:torsion-order-S"}. (We used the convention of Remark [Remark 12](#rem:extension-F){reference-type="ref" reference="rem:extension-F"}.)* 2. *We have $H^i_{\operatorname{\acute{e}t}}(M, \mathbb{Z}_\ell) \cong H^i_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell)_{\operatorname{Tor}}$ for any $i \in \mathbb{Z}$ and any prime $\ell \not= p$.* *Proof.* (1) and (2) follows from Propositions [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"} and [Proposition 25](#prop:GG-V){reference-type="ref" reference="prop:GG-V"} respectively. ◻ Before proceeding, we pose to mention two problems related to Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"}. **Problem 29**. *Does Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"} hold without the assumption $1/p \in \Lambda$? (See Remark [Remark 5](#rem:p-in-char-pos){reference-type="ref" reference="rem:p-in-char-pos"}.)* **Problem 30**. *Let $\mathscr{C}$ be the full subcategory of $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ consisting of torsion direct summands of the motives of surfaces. Is the functor $\mathscr{C}\to \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$ fully faithful?* To ease the notation, put $$\label{eq:N-B-S} N_S:=\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda} %=H^1_\ur(S), \qquad B_S:=\operatorname{Br}(S)_\Lambda.$$ We denote the Bockstein operator for $m$ by $$\label{eq:bockstein} Q : H^i_{\operatorname{\acute{e}t}}(-, \mu_m) \to H^{i+1}_{\operatorname{\acute{e}t}}(-, \mu_m),$$ i.e., the connecting map associated to the short exact sequence $0 \to \mu_m \to \mu_{m^2} \to \mu_m \to 0$. **Lemma 31**. *For any $m \in \mathbb{Z}_{>0}$ invertible in $k$, we have canonical isomorphisms $$\label{eq:etcoh-M-coeff-m} H^i_{\operatorname{\acute{e}t}}(M, \mu_m) \cong \begin{cases} 0 & (i \not= 1, 2, 3), \\ N_S[m] &(i=1), \\ B_S/m B_S & (i=3), \end{cases}$$ and an exact sequence $$\label{eq:etcoh} \xymatrix{ 0 \ar[r] & N_S/m N_S \ar[r] & H^2_{\operatorname{\acute{e}t}}(M, \mu_m) \ar[r] & B_S[m] \ar[r] & 0. }$$ If moreover $m N_S=0$ (so that we have $m B_S=0$ as well by [\[eq:torsion-order-S\]](#eq:torsion-order-S){reference-type="eqref" reference="eq:torsion-order-S"}), then we have a commutative diagram with exact rows $$\xymatrix{ 0 \ar[r] & H^1_{\operatorname{\acute{e}t}}(M, \mu_m) \ar[r]^Q \ar[d]^\cong & H^2_{\operatorname{\acute{e}t}}(M, \mu_m) \ar[r]^Q \ar@{=}[d] & H^3_{\operatorname{\acute{e}t}}(M, \mu_m) \ar[r] & 0 \\ 0 \ar[r] & N_S \ar[r] & H^2_{\operatorname{\acute{e}t}}(M, \mu_m) \ar[r] & B_S \ar[r] \ar[u]_\cong & 0, }$$ where the vertical isomorphisms are those in [\[eq:etcoh-M-coeff-m\]](#eq:etcoh-M-coeff-m){reference-type="eqref" reference="eq:etcoh-M-coeff-m"}, and the lower sequence is obtained from the exact sequence [\[eq:etcoh\]](#eq:etcoh){reference-type="eqref" reference="eq:etcoh"} with the identifications $N_S/m N_S=N_S, ~B_S[m]=B_S$.* *Proof.* The first statement follows from Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} and Corollary [Corollary 28](#cor:coh-S-M){reference-type="ref" reference="cor:coh-S-M"} (2), and the second from the definition of $Q$. ◻ **Lemma 32**. *Suppose that $m_0 \in \mathbb{Z}_{>0}$ is invertible in $k$ and $m_0 N_S=0$. Put $m:=m_0^2$ and let $Q$ be the Bockstein operator [\[eq:bockstein\]](#eq:bockstein){reference-type="eqref" reference="eq:bockstein"} for $m$. Then there exists a subgroup $\widetilde{B}_S$ of $H^2_{{\operatorname{\acute{e}t}}}(M, \mu_{m})$ fitting into a commutative diagram with exact row $$\label{eq:et-coh-M} \xymatrix{ & N_S \ar[d]_{\cong} & \widetilde{B}_S \ar@{^{(}-^{>}}[d] \ar[dr]^{\cong} & & \\ 0 \ar[r] & H^1_{{\operatorname{\acute{e}t}}}(M, \mu_{m}) \ar[r]_Q & H^2_{{\operatorname{\acute{e}t}}}(M, \mu_{m}) \ar[r]_Q & H^3_{{\operatorname{\acute{e}t}}}(M, \mu_{m}) \ar[r] & 0.}$$ In particular, we have an isomorphism $$\label{eq:etcoh2-fincoef} H^2_{{\operatorname{\acute{e}t}}}(M, \mu_{m}) \cong QN_S \oplus \widetilde{B}_S,$$ where we identified $N_S=H^1_{{\operatorname{\acute{e}t}}}(M, \mu_{m})$.* *Proof.* Put $H^i_{{\operatorname{\acute{e}t}}, n}(M):= H^i_{{\operatorname{\acute{e}t}}}(M, \mu_n)$. We consider a commutative diagram with exact rows and columns $$\xymatrix{ %0 \ar[r] & N_S/m_0 N_S \ar[r] \ar[d]^0 & & & H^2_{{\operatorname{\acute{e}t}}, m_0}(M) \ar[r] \ar[d] & B_S[m_0] \ar[r] \ar[d]^\cong & 0 \\ 0 \ar[r] & N_S/m N_S \ar[r] \ar[d]^\cong & H^2_{{\operatorname{\acute{e}t}}, m}(M) \ar[r] \ar[d] & B_S[m] \ar[r] & %\ar[d]^0 & 0 \\ 0 \ar[r] & N_S/m_0 N_S \ar[r] & H^2_{{\operatorname{\acute{e}t}}, m_0}(M). & %\ar[r] & & %B_S[m_0] \ar[r] & 0. }$$ All rows are from [\[eq:etcoh\]](#eq:etcoh){reference-type="eqref" reference="eq:etcoh"}. Due to our assumption on $m_0$ and $m$, the left and right vertical maps are bijective as indicated in the diagram. We now rewrite it using the latter half of Lemma [Lemma 31](#lem:etcoh-fincoef){reference-type="ref" reference="lem:etcoh-fincoef"}: $$\xymatrix{ %0 \ar[r] & H^1_{\et, m_0}(M) \ar[r]^{Q_0} \ar[d]^0 & & & H^2_{{\operatorname{\acute{e}t}}, m_0}(M) \ar[r]^{Q_0} \ar[d]^\iota & H^3_{{\operatorname{\acute{e}t}}, m_0}(M) \ar[r] \ar[d]^\cong & 0 \\ 0 \ar[r] & H^1_{{\operatorname{\acute{e}t}}, m}(M) \ar[r]^Q \ar[d]^\cong & H^2_{{\operatorname{\acute{e}t}}, m}(M) \ar[r]^Q \ar[d]^\pi & H^3_{{\operatorname{\acute{e}t}}, m}(M) \ar[r] & %\ar[d]^0 & 0 \\ 0 \ar[r] & H^1_{{\operatorname{\acute{e}t}}, m_0}(M) \ar[r]^{Q_0} & H^2_{{\operatorname{\acute{e}t}}, m_0}(M), & %\ar[r]^{Q_0} & %H^3_{\et, m_0}(M) \ar[r] & 0, }$$ where $Q_0$ denotes the Bockstein operator [\[eq:bockstein\]](#eq:bockstein){reference-type="eqref" reference="eq:bockstein"} for $m_0$. We then obtain the assertion from the middle horizontal exact row by putting $\widetilde{B}_S:=\operatorname{Im}(\iota)=\ker(\pi)$. ◻ # Vishik's method {#sect:vishik} In [@Vishik §4], Vishik obtained an exact sequence that computes the motivic cohomology with $\mathbb{Z}/5\mathbb{Z}$ coefficients of the classical Godeaux surface over $\mathbb{C}$. In this section we apply his method to a general surface having a decomposition of the diagonal over an arbitrary algebraically closed field. The main result of this section is Theorem [Theorem 34](#thm:vishik){reference-type="ref" reference="thm:vishik"} below. We retain the assumptions and notation introduced in Setting [Setting 22](#setting:surf){reference-type="ref" reference="setting:surf"} and [Setting 26](#setting:def-M){reference-type="ref" reference="setting:def-M"}. We also fix the following data: **Setting 33**. Fix $m_0 \in \mathbb{Z}_{>0}$ that is invertible in $k$ and divisible by [\[eq:torsion-order-S\]](#eq:torsion-order-S){reference-type="eqref" reference="eq:torsion-order-S"}. Put $m:=m_0^2$. We also fix an isomorphism $\mathbb{Z}/m\mathbb{Z}\cong \mu_m$ by which we will identify étale and Galois cohomology with different Tate twists. We write $$H^i_{{\operatorname{\acute{e}t}}}(-):=H^i_{\operatorname{\acute{e}t}}(-, \mathbb{Z}/m\mathbb{Z}), \qquad H^i_{{\operatorname{Gal}}}(-):=H^i_{\operatorname{Gal}}(-, \mathbb{Z}/m\mathbb{Z}).$$ Using the isomorphism from [\[eq:ur-coh-12\]](#eq:ur-coh-12){reference-type="eqref" reference="eq:ur-coh-12"}, [\[eq:N-B-S\]](#eq:N-B-S){reference-type="eqref" reference="eq:N-B-S"} and [\[eq:etcoh-M-coeff-m\]](#eq:etcoh-M-coeff-m){reference-type="eqref" reference="eq:etcoh-M-coeff-m"}, we identify $$\begin{aligned} \label{eq:etcoh13-fincoef} H^1_{\operatorname{ur}}(S) \cong H^1_{\operatorname{\acute{e}t}}(M) \cong N_S, \qquad H^2_{\operatorname{ur}}(S) \cong H^3_{\operatorname{\acute{e}t}}(M) \cong B_S,\end{aligned}$$ which are finite abelian groups dual to each other by Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} (3). ## Motivic cohomology For $X \in \operatorname{\mathbf{Sm}}$, $K \in \operatorname{\mathbf{Fld}}$, and $a, b \in \mathbb{Z}$ with $b \ge 0$, we write $$H^{a, b}_\mathscr{M}(X_K, \Lambda):= H^a_{\operatorname{Zar}}(X_K, \Lambda(b)), \qquad H^{a, b}_\mathscr{M}(X_K):= H^a_{\operatorname{Zar}}(X_K, \mathbb{Z}/m\mathbb{Z}(b)).$$ where $\Lambda(b)$ and $\mathbb{Z}/m\mathbb{Z}(b)$ are Voevodsky's motivic complex [@MVW Definition 3.1] with coefficients in $\Lambda$ and $\mathbb{Z}/m\mathbb{Z}$, respectively. By convention we put $H^{a, b}_\mathscr{M}(X_K, \Lambda) =H^{a, b}_\mathscr{M}(X_K)=0$ if $b<0$. We recall the following fundamental facts: $$\begin{aligned} & \label{eq:motcoh1} H^{a, b}_\mathscr{M}(X_K, \Lambda)= H^{a, b}_\mathscr{M}(X_K)=0 \quad \text{if $a>2b$ or $a>b+\dim X$}. \\ & \label{eq:motcoh2} H^{2b, b}_{\mathscr{M}}(X_K, \Lambda) \cong {\operatorname{CH}}^b(X_K)_\Lambda, \quad H^{2b, b}_{\mathscr{M}}(X_K) \cong {\operatorname{CH}}^b(X_K)/m {\operatorname{CH}}^b(X_K), \\ & \label{eq:motcoh3} H^{a, b}_{\mathscr{M}}(X_K) \cong H^a_{{\operatorname{\acute{e}t}}}(X_K) \quad \text{if $a \le b$}.\end{aligned}$$ The case $a>2b$ of [\[eq:motcoh1\]](#eq:motcoh1){reference-type="eqref" reference="eq:motcoh1"} and [\[eq:motcoh2\]](#eq:motcoh2){reference-type="eqref" reference="eq:motcoh2"} are consequences of Voevodsky's comparison theorem on the motivic cohomology with Bloch's higher Chow groups (see [@MVW Corollary 19.2, Theorem 19.3]). The second case of [\[eq:motcoh1\]](#eq:motcoh1){reference-type="eqref" reference="eq:motcoh1"} is immediate from the definition (see [@MVW Theorem 3.6]). For [\[eq:motcoh3\]](#eq:motcoh3){reference-type="eqref" reference="eq:motcoh3"} (which is the former Beilinson-Lichtenbaum conjecture), we refer to [@HW Theorem B]. These notation and results are extended to motives, cf. Remark [Remark 12](#rem:extension-F){reference-type="ref" reference="rem:extension-F"}. We now state the main result of this section. **Theorem 34**. *For any $a \in \mathbb{Z}$ and $K \in \operatorname{\mathbf{Fld}}$, we have an exact sequence $$0 \to H^{a, a-2}_{\mathscr{M}}(M_K) \to \bigoplus_{i=1, 2} H^{a-i-1}_{{\operatorname{Gal}}}(K) \otimes H^i_{{\operatorname{ur}}}(S) \overset{\Psi}{\to} H^{a-1}_{{\operatorname{ur}}}(K(S)/K) \to 0.$$ Here $\Psi$ is given by $\Psi(a \otimes b) = \operatorname{pr}_1^*(a) \cup \operatorname{pr}_2^*(b)$, where $\operatorname{pr}_i$ denotes the respective projectors on $\operatorname{Spec}(K) \times S$. (The last term is the unramified cohomology over $K$ and not over $k$.)* ## Étale cohomology **Proposition 35**. *For any $N \in \operatorname{\mathbf{Chow}}_\Lambda$ and $K \in \operatorname{\mathbf{Fld}}$, we have an isomorphism $$\label{eq:spec-decomp} H^*_{{\operatorname{Gal}}}(K) \otimes H^*_{{\operatorname{\acute{e}t}}}(N) \cong H^*_{{\operatorname{\acute{e}t}}}(N_K).$$* *Proof.* Vishik proved [\[eq:spec-decomp\]](#eq:spec-decomp){reference-type="eqref" reference="eq:spec-decomp"} in [@Vishik Proposition 4.2] assuming $k=\mathbb{C}$ and $m$ is a prime, although his proof did not use those assumptions. For the completeness sake we include a short proof. We may replace $N$ by $X \in \operatorname{\mathbf{Sm}}$. Consider the spectral sequence $$\label{eq:spec-seq} E_2^{a, b}= H^a_{{\operatorname{Gal}}}(K, H^b_{{\operatorname{\acute{e}t}}}(X_{\overline{K}})) \Rightarrow H^{a+b}_{{\operatorname{\acute{e}t}}}(X_K),$$ where $\overline{K}$ is a separable closure of $K$. By the smooth base change theorem we have $H^b_{{\operatorname{\acute{e}t}}}(X_{\overline{K}}) \cong H^b_{{\operatorname{\acute{e}t}}}(X)$ on which the absolute Galois group of $K$ acts trivially, and hence $$E_2^{a, b}= H^a_{{\operatorname{Gal}}}(K, H^b_{{\operatorname{\acute{e}t}}}(X_{\overline{K}})) \cong H^a_{{\operatorname{Gal}}}(K) \otimes H^b_{{\operatorname{\acute{e}t}}}(X).$$ Observe that $E_2^{*, *}$ is generated by $H^*_{\operatorname{\acute{e}t}}(X)$ as a $H^*_{\operatorname{Gal}}(K)$-module, and the differential maps $d_r^{*, *} : E_r^{*, *} \to E_r^{*+r, *-r+1}$ are $H^*_{\operatorname{Gal}}(K)$-linear. It follows from the commutative diagram $$\xymatrix{ H^j_{\operatorname{\acute{e}t}}(X_K) \ar[r] & E_2^{0, j}=H^0_{\operatorname{Gal}}(K, H^j_{\operatorname{\acute{e}t}}(X_{\overline{K}})) \ar@{^{(}-^{>}}[d] \\ H^j_{\operatorname{\acute{e}t}}(X) \ar[r]^{\cong} \ar[u] & H^j_{\operatorname{\acute{e}t}}(X_{\overline{K}}) }$$ that the edge maps $H^j_{\operatorname{\acute{e}t}}(X_K) \to E_2^{0, j}$ are surjective for all $j$, whence $E_2^{0, j}=E_\infty^{0, j}$. We conclude that [\[eq:spec-seq\]](#eq:spec-seq){reference-type="eqref" reference="eq:spec-seq"} degenerates at $E_2$-terms and induces the desired isomorphism. ◻ **Remark 36**. The proof shows that [\[eq:spec-decomp\]](#eq:spec-decomp){reference-type="eqref" reference="eq:spec-decomp"} remains valid when $N$ is replaced by any $X \in \operatorname{\mathbf{Sm}}$. **Corollary 37**. *For any $K \in \operatorname{\mathbf{Fld}}$ and $a \in \mathbb{Z}$, we have an isomorphism $$\begin{aligned} \label{eq:coniveau0} H^{a, a}_{\mathscr{M}}(M_K) \cong & (H^{a-1}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \\ \notag &\oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S).\end{aligned}$$* *Proof.* Apply Proposition [Proposition 35](#prop:etcoh-dec){reference-type="ref" reference="prop:etcoh-dec"} to $N=M$ and use [\[eq:etcoh2-fincoef\]](#eq:etcoh2-fincoef){reference-type="eqref" reference="eq:etcoh2-fincoef"}, [\[eq:etcoh13-fincoef\]](#eq:etcoh13-fincoef){reference-type="eqref" reference="eq:etcoh13-fincoef"} and [\[eq:motcoh3\]](#eq:motcoh3){reference-type="eqref" reference="eq:motcoh3"}. ◻ ## The first coniveau filtration The isomorphism $\mathbb{Z}/m\mathbb{Z}\cong \mu_m$ fixed in Setting [Setting 33](#setting:tate-twists){reference-type="ref" reference="setting:tate-twists"} yields a homomorphism $$\tau : H^{a, b}_{\mathscr{M}}(M_K) \to H^{a, b+1}_{\mathscr{M}}(M_K).$$ **Proposition 38**. *For any $K \in \operatorname{\mathbf{Fld}}$ and $a \in \mathbb{Z}$, the map $$\tau : H^{a, a-1}_{\mathscr{M}}(M_K) \to H^{a, a}_{\mathscr{M}}(M_K) \cong H^a_{{\operatorname{\acute{e}t}}}(M_K)$$ is injective and its image corresponds to the subgroup $$\begin{aligned} \label{eq:coniveau1} & (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S) \\ \notag &\oplus \ker[\alpha_a : (H^{a-1}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \to H^a_{{\operatorname{ur}}}(M_K)]\end{aligned}$$ under the isomorphism [\[eq:coniveau0\]](#eq:coniveau0){reference-type="eqref" reference="eq:coniveau0"} (see [\[eq:def-unramcoh4\]](#eq:def-unramcoh4){reference-type="eqref" reference="eq:def-unramcoh4"} for $H^a_{{\operatorname{ur}}}(M_K)$). Here $\alpha_a$ is given by the composition $$(H^{a-1}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \overset{\text{\eqref{eq:coniveau0}}}{\hookrightarrow} H_\mathscr{M}^{a, a}(M_K) \overset{\rho}{\longrightarrow} H_{{\operatorname{ur}}}^a(M_K),$$ where $\rho$ is given by Theorem [Theorem 40](#thm:TY){reference-type="ref" reference="thm:TY"} (1) below.* **Remark 39**. We will show that $\alpha_a$ is surjective in Proposition [Proposition 42](#prop:coniveau2){reference-type="ref" reference="prop:coniveau2"} below. For the proof, we recall an important result from [@TY]: **Theorem 40**. *Let $X \in \operatorname{\mathbf{Sm}}$, $K \in \operatorname{\mathbf{Fld}}$ and $a, b \in \mathbb{Z}$ with $b \ge 0$.* 1. *There exists a long exact sequence $$\cdots \to H^{a, b-1}_{\mathscr{M}}(X_K) \overset{\tau}{\to} H^{a, b}_{\mathscr{M}}(X_K) \overset{\rho}{\to} H^{a-b}_{{\operatorname{Zar}}}(X_K, \mathscr{H}^b_m) \to H^{a+1, b-1}_{\mathscr{M}}(X_K) \overset{\tau}{\to} \cdots,$$ where $\mathscr{H}^b_{m}$ is from [\[eq:def-unramcoh4\]](#eq:def-unramcoh4){reference-type="eqref" reference="eq:def-unramcoh4"}.* 2. *Let $E_1^{i, j}=H^{2i+j}_{\operatorname{Zar}}(X_K, \mathscr{H}^{-i}_m) \Rightarrow H^{i+j}_{\operatorname{\acute{e}t}}(X_K)$ be the $\tau$-Bockstein spectral sequence constructed in [@TY p. 4478] (using the long exact sequence in (1)). Let ${}^\dagger E_1^{i, j}=\oplus_{x \in (X_K)^{(i)}} H^{j-i}_{\operatorname{Gal}}(K(x)) \Rightarrow H^{i+j}_{\operatorname{\acute{e}t}}(X_K)$ be the coniveaux spectral sequence. Then we have $E_r^{i, j} \cong {}^\dagger E_{r+1}^{2i+j, -i}$ for all $r \ge 1$.* 3. *The composition $${\operatorname{CH}}^a(X_K)/m {\operatorname{CH}}^a(X_K) \cong H^{2a, a}_\mathscr{M}(X_K) \overset{\tau^a}{\longrightarrow} H^{2a, 2a}_\mathscr{M}(X_K) \cong H^{2a}_{\operatorname{\acute{e}t}}(X_K)$$ agrees with the cycle map.* *Proof.* This is taken from [@TY Lemma 2.1, Theorem 2.4]. Here we only recall that (1) is a consequence of [\[eq:motcoh3\]](#eq:motcoh3){reference-type="eqref" reference="eq:motcoh3"}, (2) is due to Deligne and Paranjape (see [@BO p.195, footnote], [@P Corollary 4.4]), and (3) is a consequence of (2). ◻ We need a simple lemma. **Lemma 41**. 1. *The following diagram is commutative: $$\xymatrix{ H^{a, b}_{\mathscr{M}}(M_K) \ar[r]^\tau & H^{a, b+1}_{\mathscr{M}}(M_K) \\ H^{a-1, b}_{\mathscr{M}}(M_K) \ar[r]_\tau \ar[u]^Q& H^{a-1, b+1}_{\mathscr{M}}(M_K). \ar[u]_Q }$$* 2. *We have $Q(H^a_{\operatorname{Gal}}(K) \otimes H^b_{\operatorname{\acute{e}t}}(M)) = H^a_{\operatorname{Gal}}(K) \otimes Q(H_{\operatorname{\acute{e}t}}^b(M))$.* *Proof.* We have $Q(\zeta)=0$ for any $\zeta \in \mu_m$ because of the surjectivity of the $m$-th power map $H^0_{\operatorname{Gal}}(k, \mu_{m^2}) \to H^0_{\operatorname{Gal}}(k, \mu_m)$. Thus (1) follows from a formal property of the Bockstein operator $Q(x \cup y)=Q(x) \cup y \pm x \cup Q(y)$. The same formal property reduces (2) to the surjectivity of $H_{\operatorname{Gal}}^a(K, \mu_{m^2}^{\otimes a}) \to H_{\operatorname{Gal}}^a(K, \mu_{m}^{\otimes a})$, which is a consequence of the norm residue isomorphism theorem (see [@HW Theorem A]). ◻ *Proof of Proposition [Proposition 38](#prop:coniveau1){reference-type="ref" reference="prop:coniveau1"}.* Since $H^{-1}_{\operatorname{Zar}}(S_K, \mathscr{H}^a_m)=0$ and $H^0_{\operatorname{Zar}}(S_K, \mathscr{H}^a_m)=H^a_{{\operatorname{ur}}, m}(S_K)$ by the definition [\[eq:def-unramcoh4\]](#eq:def-unramcoh4){reference-type="eqref" reference="eq:def-unramcoh4"}, we obtain from Theorem [Theorem 40](#thm:TY){reference-type="ref" reference="thm:TY"} (1) with $a=b$ an exact sequence sitting in the upper row of a diagram: $$\label{eq:diag-coniv1} \xymatrix{ 0 \ar[r] & H^{a, a-1}_{\mathscr{M}}(M_K) \ar[r]^\tau & H^{a, a}_{\mathscr{M}}(M_K) \ar[r]^\rho & H^a_{{\operatorname{ur}}, m}(M_K) \\ & H^{a-1, a-1}_{\mathscr{M}}(M_K) \ar[r]_\tau^\cong \ar[u]^Q & H^{a-1, a}_{\mathscr{M}}(M_K). \ar[u]_Q & }$$ This proves the desired injectivity. The square in [\[eq:diag-coniv1\]](#eq:diag-coniv1){reference-type="eqref" reference="eq:diag-coniv1"} is commutative by Lemma [Lemma 41](#lem:comm-bock){reference-type="ref" reference="lem:comm-bock"} (1). The lower horizontal arrow in the diagram is an isomorphism by [\[eq:motcoh3\]](#eq:motcoh3){reference-type="eqref" reference="eq:motcoh3"}. By [\[eq:coniveau0\]](#eq:coniveau0){reference-type="eqref" reference="eq:coniveau0"} we find that $H^{a-1, a-1}_{\mathscr{M}}(M_K)$ and $H^{a, a}_{\mathscr{M}}(M_K)$ are respectively decomposed as $$\begin{aligned} &(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \oplus (H^{a-4}_{{\operatorname{Gal}}}(K) \otimes B_S), \\ &(H^{a-1}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S).\end{aligned}$$ By Lemma [Lemma 41](#lem:comm-bock){reference-type="ref" reference="lem:comm-bock"} (2) and [\[eq:diag-coniv1\]](#eq:diag-coniv1){reference-type="eqref" reference="eq:diag-coniv1"}, we get $$\rho(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) = \rho(Q\tau(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S)) = \rho(\tau Q(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S))=0.$$ Similarly we obtain $\rho(H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S)=0$ since $B_S=Q\widetilde{B}_S$. To conclude [\[eq:coniveau1\]](#eq:coniveau1){reference-type="eqref" reference="eq:coniveau1"}, it suffices now to note that $H^a_{{\operatorname{ur}}, m}(M_K)=H^a_{{\operatorname{ur}}}(M_K)$ by [\[eq:def-unramcoh3\]](#eq:def-unramcoh3){reference-type="eqref" reference="eq:def-unramcoh3"} and use Corollary [Corollary 28](#cor:coh-S-M){reference-type="ref" reference="cor:coh-S-M"} (1). ◻ ## The second coniveau filtration **Proposition 42**. *For any $K \in \operatorname{\mathbf{Fld}}$ and $a \in \mathbb{Z}$, the map $$\tau : H^{a, a-2}_\mathscr{M}(M_K) \to H^{a, a-1}_\mathscr{M}(M_K)$$ is injective and its image corresponds to the subgroup $$\begin{aligned} \label{eq:coniveau2} \ker[\beta_a : (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S) \to H^{a-1}_{{\operatorname{ur}}}(M_K)]\end{aligned}$$ under the isomorphism [\[eq:coniveau1\]](#eq:coniveau1){reference-type="eqref" reference="eq:coniveau1"}. Here $\beta_a$ is defined by the commutativity of $$\xymatrix{ (H^{a-2}_{\operatorname{Gal}}(K) \otimes QN_S) \oplus (H^{a-3}_{\operatorname{Gal}}(K) \otimes B_S) \ar[rd]^-{\beta_a} & \\ (H^{a-2}_{\operatorname{Gal}}(K) \otimes N_S) \oplus (H^{a-3}_{\operatorname{Gal}}(K) \otimes \widetilde{B}_S) \ar[r]_-{\alpha_a} \ar[u]^Q_\cong & H_{\operatorname{ur}}^{a-1}(M_K). }$$ Moreover, the map $\alpha_a$ in [\[eq:coniveau1\]](#eq:coniveau1){reference-type="eqref" reference="eq:coniveau1"} is surjective.* *Proof.* Since $H^{a, b}_{\mathscr{M}}(M_K, \Lambda)$ is annihilated by $m$ for any $a, b \in \mathbb{Z}$, a commutative diagram with an exact row $$\xymatrix{ H_\mathscr{M}^{a-1, b}(M_K) \ar@{->>}[r] \ar[rd]_Q & H_\mathscr{M}^{a, b}(M_K, \Lambda) \ar[r]^{m=0} \ar@{^{(}-^{>}}[d] & H_\mathscr{M}^{a, b}(M_K, \Lambda) \ar@{^{(}-^{>}}[r] & H_\mathscr{M}^{a, b}(M_K) \\ & H^{a, b}_\mathscr{M}(M_K). & & }$$ shows that the complex $(H^{\bullet, b}_{\mathscr{M}}(M_K), Q)$ is exact. Consider a diagram $$\xymatrix{ H^{a, a-2}_{\mathscr{M}}(M_K) \ar[r]^\tau & H^{a, a-1}_{\mathscr{M}}(M_K) \\ H^{a-1, a-2}_{\mathscr{M}}(M_K) \ar[r]_\tau \ar@{->>}[u]^Q & H^{a-1, a-1}_{\mathscr{M}}(M_K), \ar[u]_Q }$$ which is commutative by Lemma [Lemma 41](#lem:comm-bock){reference-type="ref" reference="lem:comm-bock"} (1). Since $H^{a+1, a-2}_{\mathscr{M}}(M_K)=0$ by [\[eq:motcoh1\]](#eq:motcoh1){reference-type="eqref" reference="eq:motcoh1"}, the previous remark shows that the left vertical map in the diagram is surjective. The rest of the proof goes along the same line as Proposition [Proposition 38](#prop:coniveau1){reference-type="ref" reference="prop:coniveau1"}. We apply [\[eq:coniveau1\]](#eq:coniveau1){reference-type="eqref" reference="eq:coniveau1"} to obtain direct sum decompositions of $H^{a-1, a-2}_{\mathscr{M}}(M_K)$ and $H^{a, a-1}_{\mathscr{M}}(M_K)$ respectively as $$\begin{aligned} & (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-4}_{{\operatorname{Gal}}}(K) \otimes B_S) \oplus \ker(\alpha_{a-1}), \\ & (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S) \oplus \ker(\alpha_a).\end{aligned}$$ By Lemma [Lemma 41](#lem:comm-bock){reference-type="ref" reference="lem:comm-bock"} (2), the summand $(H^{a-3}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-4}_{{\operatorname{Gal}}}(K) \otimes B_S)$ of $H^{a-1, a-2}_{\mathscr{M}}(M_K)$ is killed by the left vertical map, because $Q^2=0$ and $B_S=Q\widetilde{B}_S$. On the other hand, $\tau \circ Q$ maps $\ker(\alpha_{a-1})$ injectively into the summand $(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S)$ of $H^{a, a-1}_{\mathscr{M}}(M_K)$, showing the first statement. The last statement follows from Theorem [Theorem 40](#thm:TY){reference-type="ref" reference="thm:TY"} and the injectivity of $\tau : H^{a+1, a-1}_\mathscr{M}(X_K) \to H^{a+1, a}_\mathscr{M}(X_K)$. ◻ *Proof of Theorem [Theorem 34](#thm:vishik){reference-type="ref" reference="thm:vishik"}.* As the unramified cohomology is normalized, birational, and motivic (Proposition [Proposition 15](#prop:unram-coh){reference-type="ref" reference="prop:unram-coh"}), we have $H^i_{{\operatorname{ur}}}(S)=H^i_{{\operatorname{ur}}}(M)$ and $H^i_{{\operatorname{ur}}}(K(S)/K)=H^i_{{\operatorname{ur}}}(M_K)$. Now Propositions [Proposition 38](#prop:coniveau1){reference-type="ref" reference="prop:coniveau1"} and [Proposition 42](#prop:coniveau2){reference-type="ref" reference="prop:coniveau2"} complete the proof. ◻ # Main exact sequence {#sect:proof} We keep the assumptions in Setting [Setting 22](#setting:surf){reference-type="ref" reference="setting:surf"}, [Setting 26](#setting:def-M){reference-type="ref" reference="setting:def-M"} and [Setting 33](#setting:tate-twists){reference-type="ref" reference="setting:tate-twists"}. ## Main exact sequence {#main-exact-sequence} The following is the main technical result of this paper. **Theorem 43**. *Suppose that $S \in \operatorname{\mathbf{SmProj}}$ admits a decomposition of the diagonal (see Definition [Definition 18](#def:dec-diag){reference-type="ref" reference="def:dec-diag"}) and $\dim S=2$. Then we have an exact sequence for any $K \in \operatorname{\mathbf{Fld}}$ $$0 \to {\operatorname{CH}}_0(S_K)_{{\operatorname{Tor}}, \Lambda} \to \bigoplus_{i=1, 2} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(K/k)) \to H^3_{\operatorname{ur}}(K(S)/k) \to 0.$$ (Unlike Theorem [Theorem 34](#thm:vishik){reference-type="ref" reference="thm:vishik"}, the last term is the unramified cohomology over $k$ and not over $K$.)* The proof of Theorem [Theorem 43](#thm:tech-main){reference-type="ref" reference="thm:tech-main"} will be complete in §[5.3](#sect:end){reference-type="ref" reference="sect:end"} below. **Remark 44**. In the situation of Theorem [Theorem 43](#thm:tech-main){reference-type="ref" reference="thm:tech-main"}, we have a canonical isomorphism $${\operatorname{CH}}_0(S_K)_{{\operatorname{Tor}}, \Lambda} \cong \operatorname{Coker}({\operatorname{CH}}_0(S)_\Lambda \to {\operatorname{CH}}_0(S_K)_\Lambda),$$ and this group is annihilated by the integer [\[eq:torsion-order-S\]](#eq:torsion-order-S){reference-type="eqref" reference="eq:torsion-order-S"}. To see this, it suffices to note that the degree map ${\operatorname{CH}}_0(S_K) \to \mathbb{Z}$ is split surjective (as $k$ is algebraically closed). As a special case where $K=k(T)$ for $T \in \operatorname{\mathbf{SmProj}}$, we also have (see [\[eq:CH0-S\]](#eq:CH0-S){reference-type="eqref" reference="eq:CH0-S"}) $$\label{eq:CH0S-CHtor} {\operatorname{CH}}_0(S_{k(T)})_{{\operatorname{Tor}}, \Lambda} \cong \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S).$$ ## Auxiliary lemmas **Lemma 45**. *Let $E$ be a field such that $m$ is invertible in $E$ and $\mu_{m^\infty} \subset E$. Then $H^j_{\operatorname{Gal}}(E)$ is a free $\mathbb{Z}/m\mathbb{Z}$-module for any $j \in \mathbb{Z}$.* *Proof.* We may assume $m=\ell^e$ for a prime number $\ell \not= p$ and $e \in \mathbb{Z}_{>0}$. Recall that a module over an Artin local ring is free if and only if it is flat (see, e.g. [@A Proposition 2.1.4]). By the norm residue isomorphism theorem (see [@HW Theorem A]), $K_{j-1}^M(E) \otimes \mu_{\ell^\infty}$ surjects onto $K_j^M(E)_{{\operatorname{Tor}}} \otimes \mathbb{Z}_{(\ell)}$, hence $K_j^M(E)_{{\operatorname{Tor}}}$ is divisible by $\ell$. It follows that $K_j^M(E)$ is the direct sum of an $\ell$-divisible group and a flat $\mathbb{Z}_{(\ell)}$-module. Thus $K_j^M(E) \otimes \mathbb{Z}/m\mathbb{Z}\cong H^j_{\operatorname{Gal}}(E)$ is a flat $\mathbb{Z}/m\mathbb{Z}$-module. ◻ By the Poincaré duallty, we have a perfect paring of finite abelian groups for any $i \in \mathbb{Z}$ $$\langle -, -\rangle : H^{4-i}_{\operatorname{\acute{e}t}}(S) \times H^i_{\operatorname{\acute{e}t}}(S) \to \mathbb{Z}/m\mathbb{Z}. %\quad %\langle -, -\rangle : %H^{4-i}_\et(M) \times H^i_\et(M) %\to \Z/m\Z.$$ For $i=1, 2$, we define the homomorphisms $$\begin{aligned} \label{eq:def-q-pi} Q_i' : H^{3-i}_{\operatorname{ur}}(S) \to H^{4-i}_{\operatorname{\acute{e}t}}(S), \qquad \pi_i : H^i_{\operatorname{\acute{e}t}}(S) \to H^i_{\operatorname{ur}}(S)\end{aligned}$$ as follows. For $i=1$, they are given by [\[eq:etcoh13-fincoef\]](#eq:etcoh13-fincoef){reference-type="eqref" reference="eq:etcoh13-fincoef"}. For $i=2$, $Q_2'$ and $\pi_2$ are the compositions $$H^1_{\operatorname{ur}}(S) \cong H^1_{\operatorname{\acute{e}t}}(S) \overset{Q}{\to} H^2_{\operatorname{\acute{e}t}}(S), \qquad H^2_{\operatorname{\acute{e}t}}(S) \overset{Q}{\to} H^3_{\operatorname{\acute{e}t}}(S) \cong H^3_{\operatorname{ur}}(S),$$ where $Q$ are the Bockstein operator [\[eq:bockstein\]](#eq:bockstein){reference-type="eqref" reference="eq:bockstein"}. (Hence $Q_1'$ and $\pi_1$ are bijective, and we have a split short exact sequence $0 \to H_{\operatorname{ur}}^1(S) \overset{Q_2'}{\to} H^2_{\operatorname{\acute{e}t}}(S) \overset{\pi_2}{\to} H_{\operatorname{ur}}^2(S) \to 0$.) **Lemma 46**. *We have a perfect paring of finite abelian groups for $i=1, 2$ $$\langle -, -\rangle : H^{3-i}_{\operatorname{ur}}(S) \times H^i_{\operatorname{ur}}(S) \to \mathbb{Z}/m\mathbb{Z}$$ characterized by the formula $$\label{eq:pairing} \langle Q_i'(a), b \rangle = \langle a, \pi_i(b) \rangle \qquad (a \in H^{3-i}_{\operatorname{ur}}(S), \, b \in H^i_{\operatorname{\acute{e}t}}(S)).$$* *Proof.* For $i=1$, [\[eq:pairing\]](#eq:pairing){reference-type="eqref" reference="eq:pairing"} is nothing other than the paring in Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} (3), whence the result. Assume now $i=2$. It suffices to show $\langle Q_2'(H_{\operatorname{ur}}^1(S), Q_2'(H_{\operatorname{ur}}^1(S) \rangle=0$. This follows from the computation $$Q(a) \cup Q(b) = Q(a \cup Q(b)) = 0$$ for $a, b \in H_{\operatorname{\acute{e}t}}^1(S)$. Here the first (resp. second) equality holds because $Q^2=0$ (resp. $Q : H^3_{\operatorname{\acute{e}t}}(S) \to H^4_{\operatorname{\acute{e}t}}(S)$ is the zero map, as $H^4_{\operatorname{\acute{e}t}}(S, \mathbb{Z}/m\mathbb{Z}) \to H^4_{\operatorname{\acute{e}t}}(S, \mathbb{Z}/m^2\mathbb{Z})$ is injective). ◻ **Lemma 47**. *Let $E$ be a field satisfying the assumption of Lemma [Lemma 45](#lem:freeness){reference-type="ref" reference="lem:freeness"}. Then for $i=1, 2$ and for any $j \in \mathbb{Z}$, we have isomorphisms $$\begin{aligned} &H^j_{\operatorname{Gal}}(E) \otimes H^{4-i}_{\operatorname{\acute{e}t}}(S) \cong \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S), H^j_{\operatorname{Gal}}(E)), \\ &H^j_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{ur}}(S) \cong \operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^j_{\operatorname{Gal}}(E)). %\quad a \otimes b \mapsto %[c \mapsto \langle b, c \rangle a].\end{aligned}$$* *Proof.* This follows from Lemmas [Lemma 45](#lem:freeness){reference-type="ref" reference="lem:freeness"} and [Lemma 65](#lem:homalg0){reference-type="ref" reference="lem:homalg0"} (2). ◻ **Lemma 48**. *The canonical map $H^2_{\operatorname{\acute{e}t}}(\operatorname{Spec}(E \otimes_k k(S))) \to H^2_{\operatorname{Gal}}(E(S))$ is injective for any $E \in \operatorname{\mathbf{Fld}}$.* *Proof.* We consider a commutative diagram with exact row: $$\xymatrix{ & & H^2_{\operatorname{\acute{e}t}}(\operatorname{Spec}(E \otimes_k k(S))) \ar[r] & H^2_{\operatorname{Gal}}(E(S)) & \\ 0 \ar[r] & \operatorname{Pic}(U_E)/m \operatorname{Pic}(U_E) \ar[r] & H^2_{\operatorname{\acute{e}t}}(U_E) \ar[r]_{\gamma_U} \ar[u] & H^2_{{\operatorname{ur}}, m}(U_E) \ar[r] \ar@{_{(}-{>}}[u]_{\iota_U}& 0, }$$ where $U$ is an open dense subscheme of $S$. Since the map in question is obtained as the colimit of $\iota_U \circ \gamma_U$ as $U$ ranges over such schemes, it suffices to show the vanishing of the lower left group for sufficiently small $U \subset S$. For this, we take a (possibly reducible) curve $C \subset S$ whose components generate $\operatorname{NS}(S)$. Then we find $\operatorname{Pic}(U_E)=0$ as soon as $U \subset S \setminus C$, because we have $\operatorname{Pic}(S_E)=\operatorname{NS}(S)$ by Lemma [Lemma 23](#lem:coh-surf){reference-type="ref" reference="lem:coh-surf"} (4). We are done. ◻ **Lemma 49**. *For any $E \in \operatorname{\mathbf{Fld}}$, the map $$\label{eq:phi} \bigoplus_{i=1, 2} H^{i-1}_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{ur}}(S) \to H^2_{\operatorname{Gal}}(E(S)), \quad a \otimes b \mapsto \operatorname{pr}_1^*(a) \cup \operatorname{pr}_2^*(b)$$ is injective, where $\operatorname{pr}_i$ denotes the respective projectors on $\operatorname{Spec}(E) \times S$.* *Proof.* We decompose [\[eq:phi\]](#eq:phi){reference-type="eqref" reference="eq:phi"} as follows: $$\begin{aligned} \bigoplus_{i=1, 2} H^{i-1}_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{ur}}(S) &\hookrightarrow \bigoplus_{i=1, 2} H^{i-1}_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{Gal}}(k(S)) \\ &\hookrightarrow H^2_{\operatorname{\acute{e}t}}(\operatorname{Spec}(E \otimes_k k(S))) \hookrightarrow H^2_{\operatorname{Gal}}(E(S)).\end{aligned}$$ The injectivity of the first map follows from Lemma [Lemma 45](#lem:freeness){reference-type="ref" reference="lem:freeness"}, since $H^i_{\operatorname{ur}}(S)=H^i_{\operatorname{ur}}(k(S)/k)$ is a subgroup of $H^i_{\operatorname{Gal}}(k(S))$ by definition (see [\[eq:def-unramcoh1\]](#eq:def-unramcoh1){reference-type="eqref" reference="eq:def-unramcoh1"}, [\[eq:unramcoh-K-S\]](#eq:unramcoh-K-S){reference-type="eqref" reference="eq:unramcoh-K-S"}). The second (resp. third) map is also injective by Remark [Remark 36](#rem:smooth-bc){reference-type="ref" reference="rem:smooth-bc"} (resp. Lemma [Lemma 48](#lem:inj){reference-type="ref" reference="lem:inj"}). ◻ ## End of the proof {#sect:end} We consider a commutative diagram $$\xymatrix{ 0 \ar[r] & {\operatorname{CH}}_0(S_K)_{{\operatorname{Tor}}, \Lambda} \ar[r] & \underset{i=1, 2}{\bigoplus} H^i_{{\operatorname{Gal}}}(K) \otimes H^{3-i}_{{\operatorname{ur}}}(S) \ar[r]^-\Psi \ar[d]^{\partial_1} & H^3_{{\operatorname{ur}}}(K(S)/K) \ar[r] \ar[d]^{\partial_2} & 0 \\ & & \underset{i=1, 2}{\bigoplus} \underset{v}{\bigoplus} H^{i-1}_{{\operatorname{Gal}}}(F_v) \otimes H^{3-i}_{{\operatorname{ur}}}(S) \ar[r]_-\psi & \underset{w}{\bigoplus} H^2_{\operatorname{Gal}}(F_w). & }$$ The upper row is an exact sequence obtained by setting $a=4$ and replacing $i$ with $3-i$ in Theorem [Theorem 34](#thm:vishik){reference-type="ref" reference="thm:vishik"}. In the lower row, $v$ (resp. $w$) ranges over all discrete valuations of $K$ (resp. $K(S)$) that are trivial on $k$, and $F_v$ (resp. $F_w$) denotes the residue field. For each $v$, let $w(v)$ be an extension of $v$ to $K(S)$. Then the $(v, w(v))$-component of $\psi$ is given by [\[eq:phi\]](#eq:phi){reference-type="eqref" reference="eq:phi"} for $E=F_v$, and the other components are zero. The two vertical maps are the residue maps recalled in §[2.5](#sect:unram-coh){reference-type="ref" reference="sect:unram-coh"}. Lemma [Lemma 49](#lem:inj2){reference-type="ref" reference="lem:inj2"} shows that $\psi$ is injective. By Lemma [Lemma 47](#lem:tensor-hom){reference-type="ref" reference="lem:tensor-hom"} we have isomorphisms $$\begin{aligned} &H^i_{{\operatorname{Gal}}}(K) \otimes H^{3-i}_{{\operatorname{ur}}}(S) \cong \operatorname{Hom}(H^i_{{\operatorname{ur}}}(S), H^i_{{\operatorname{Gal}}}(K)), \\ &H^{i-1}_{{\operatorname{Gal}}}(F_v) \otimes H^{3-i}_{{\operatorname{ur}}}(S) \cong \operatorname{Hom}(H^i_{{\operatorname{ur}}}(S), H^{i-1}_{{\operatorname{Gal}}}(F_v)).\end{aligned}$$ By [\[eq:def-unramcoh1\]](#eq:def-unramcoh1){reference-type="eqref" reference="eq:def-unramcoh1"} and the left exactness of $\operatorname{Hom}(H^i_{{\operatorname{ur}}}(S), -)$, we obtain $$\begin{aligned} \ker({\partial_1})= \underset{i=1, 2}{\bigoplus} \operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{ur}}(K/k)).\end{aligned}$$ On the other hand, since $H^3_{\operatorname{ur}}(K(S)/k) \subset H^3_{\operatorname{ur}}(K(S)/K) \subset H^3_{\operatorname{Gal}}(K(S))$ we have $$\begin{aligned} \ker({\partial_2}) &= H^3_{\operatorname{ur}}(K(S)/K) \cap \ker(H^3_{\operatorname{Gal}}(K(S)) \to \bigoplus_w H^2_{\operatorname{Gal}}(F_w)) \\ &= H^3_{\operatorname{ur}}(K(S)/K) \cap H^3_{\operatorname{ur}}(K(S)/k) =H^3_{\operatorname{ur}}(K(S)/k).\end{aligned}$$ Now a diagram chase completes the proof of Theorem [Theorem 43](#thm:tech-main){reference-type="ref" reference="thm:tech-main"}. 0◻ **Remark 50**. It is not always the case that $H^i_{\operatorname{ur}}(K/k) \otimes H^{3-i}_{{\operatorname{ur}}}(S) \cong \operatorname{Hom}(H^i_{{\operatorname{ur}}}(S), H^i_{\operatorname{ur}}(K/k))$. # Main results In this section, we suppose $k$ is algebraically closed and $\Lambda=\mathbb{Z}[1/p]$. ## An exact sequence **Theorem 51**. *Let $S, T \in \operatorname{\mathbf{SmProj}}$. Suppose that $S$ admits a decomposition of the diagonal and $\dim S=2$. Then we have an exact sequence $$\label{eq:Vishik-ex-seq-full} 0 \to {\operatorname{CH}}_0(S_{k(T)})_{{\operatorname{Tor}}, \Lambda} \overset{\Phi}{\to} \bigoplus_{i=1, 2} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(T)) \to H^3_{\operatorname{ur}}(S \times T) \to 0.$$* *Proof.* Apply Theorem [Theorem 43](#thm:tech-main){reference-type="ref" reference="thm:tech-main"} to $K=k(T)$ and use [\[eq:unramcoh-K-S\]](#eq:unramcoh-K-S){reference-type="eqref" reference="eq:unramcoh-K-S"}. ◻ **Remark 52**. Using Lemma [Lemma 67](#lem:homalg){reference-type="ref" reference="lem:homalg"}, we may rewrite [\[eq:Vishik-ex-seq-full\]](#eq:Vishik-ex-seq-full){reference-type="eqref" reference="eq:Vishik-ex-seq-full"} as follows: $$0 \to {\operatorname{CH}}_0(S_{k(T)})_{{\operatorname{Tor}}, \Lambda} \to \bigoplus_{i=1, 2} {\operatorname{Tor}}(H_{\operatorname{ur}}^{3-i}(S), H_{\operatorname{ur}}^i(T)) \to H^3_{\operatorname{ur}}(S \times T) \to 0.$$ This, together with [\[eq:etcoh13-fincoef\]](#eq:etcoh13-fincoef){reference-type="eqref" reference="eq:etcoh13-fincoef"}, recovers Kahn's exact sequence [@K1 Corollary 6.4] as a special case $T=S$. It also recovers [@K1 Corollary 6.5] as the case $\dim T=1$. The general case should be compared with [@K1 Theorem 6.3], where the map $${\operatorname{CH}}_0(S_{k(T)})_{{\operatorname{Tor}}, \Lambda} \to \bigoplus_{i=1, 2} \prod_{\ell \not= p} {\operatorname{Tor}}(H_{\operatorname{\acute{e}t}}^{3-i}(S, \mathbb{Z}_\ell), H_{\operatorname{\acute{e}t}}^i(T, \mathbb{Z}_\ell))$$ is studied. ## Faithful property of unramified cohomology **Theorem 53**. *Let $S, T \in \operatorname{\mathbf{SmProj}}$. Suppose that $S$ admits a decomposition of the diagonal and $\dim S=2$. Let $f : T \to S$ be a morphism in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$. Then the following are equivalent:* 1. *The map $F(f) : F(S) \to F(T)$ vanishes for any normalized, birational, and motivic functor $F : \operatorname{\mathbf{SmProj}}^{\operatorname{op}}\to \operatorname{\mathbf{Mod}}_\Lambda$.* 2. *The map $H^i_{\operatorname{ur}}(f) : H^i_{\operatorname{ur}}(S) \to H^i_{\operatorname{ur}}(T)$ vanishes for $i=1, 2$.* 3. *The map $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(f, S) : \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(S, S) \to \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S)$ of [\[eq:CH0-S\]](#eq:CH0-S){reference-type="eqref" reference="eq:CH0-S"} vanishes.* *Proof.* (2) and (3) are special cases of (1). (3) $\Rightarrow$ (1) follows from Lemma [Lemma 14](#lem:yoneda){reference-type="ref" reference="lem:yoneda"}, and (2) $\Rightarrow$ (3) follows from Theorem [Theorem 51](#thm:main2-full){reference-type="ref" reference="thm:main2-full"} and Lemma [Lemma 54](#lem:compatible){reference-type="ref" reference="lem:compatible"} below. ◻ **Lemma 54**. *Under the identification ${\operatorname{CH}}_0(S_{k(T)})_{{\operatorname{Tor}}, \Lambda} = \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(T, S)$ from [\[eq:CH0S-CHtor\]](#eq:CH0S-CHtor){reference-type="eqref" reference="eq:CH0S-CHtor"}, the map $\Phi$ in [\[eq:Vishik-ex-seq-full\]](#eq:Vishik-ex-seq-full){reference-type="eqref" reference="eq:Vishik-ex-seq-full"} is induced by the functors $H_{\operatorname{ur}}^i$ for $i=1, 2$.* *Proof.* Put $K:=k(T)$. We use a cartesian diagram $$\xymatrix{ S_K \ar[r]^-{\operatorname{pr}_2} \ar[d]_-{\operatorname{pr}_1} & \operatorname{Spec}K \ar[d]^-{s_2} \\ S \ar[r]^-{s_1} & \operatorname{Spec}k, }$$ where $\operatorname{pr}_i$ are the projections and $s_i$ are the structure maps. We first show, by a standard argument, the commutativity of the diagram $$\label{eq:comm-ku-pd} \xymatrix{ H^4_{\operatorname{\acute{e}t}}(S_K) \ar[r]^-{\operatorname{cr}}& \bigoplus_i \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S), H^i_{\operatorname{Gal}}(K)) \\ & \bigoplus_i H_{\operatorname{\acute{e}t}}^{4-i}(S) \otimes H_{\operatorname{Gal}}^i(K), \ar[lu]^-{\operatorname{k\ddot{u}}}_(0.4)\cong \ar[u]_-{\operatorname{pd}}^-\cong }$$ where ${\operatorname{cr}}$ is the correspondence action (that is, ${\operatorname{cr}}(\xi)(a)=\operatorname{pr}_{2*}(\operatorname{pr}_1^*(a) \cup \xi)$), ${\operatorname{k\ddot{u}}}$ is the Künneth isomorphism, and ${\operatorname{pd}}$ is the isomorphism from Lemma [Lemma 47](#lem:tensor-hom){reference-type="ref" reference="lem:tensor-hom"}. We take $a \in H^i_{\operatorname{\acute{e}t}}(S)$, $b \in H^{4-i}_{\operatorname{\acute{e}t}}(S)$ and $x \in H^i_{\operatorname{Gal}}(K)$, and compute $$\begin{aligned} ({\operatorname{cr}}\circ {\operatorname{k\ddot{u}}})(b \otimes x)(a) &= \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a) \cup \operatorname{pr}_1^*(b) \cup \operatorname{pr}_2^*(x)) \\ &= \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a \cup b) \cup \operatorname{pr}_2^*(x)) \overset{(1)}{=} \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a \cup b)) \cup x \\ & \overset{(2)}{=} s_2^* s_{1*}(a \cup b) \cup x ={\operatorname{pd}}(b \otimes x)(a).\end{aligned}$$ Here we have used the projection formula for étale cohomology and the base change property in [@SGA4 Exposé XVIII, Théorème 2.9] at $(1)$ and $(2)$, respectively. We have shown the commutativity of [\[eq:comm-ku-pd\]](#eq:comm-ku-pd){reference-type="eqref" reference="eq:comm-ku-pd"}. We now consider the following diagram $$\xymatrix{ {\operatorname{CH}}_0(S_K)_{{\operatorname{Tor}}, \Lambda} \ar[r]^-{\operatorname{cyc}}\ar[rrd]_-{(**)}& H^4_{\operatorname{\acute{e}t}}(S_K) \ar[r]^-{(*)} & \bigoplus_i \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S), H^i_{\operatorname{Gal}}(K)) \\ & & \bigoplus_i \operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{ur}}(T)). \ar@{_{(}-{>}}[u]_{\Pi} }$$ Here ${\operatorname{cyc}}$ is the cycle map, and $\Pi$ is the direct sum of the compositions $$\operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{ur}}(T)) \hookrightarrow \operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{Gal}}(K)) \overset{\pi_i^*}{\hookrightarrow} \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S), H^i_{\operatorname{Gal}}(K)),$$ where $\pi_i^*$ is induced by $\pi_i$ in [\[eq:def-q-pi\]](#eq:def-q-pi){reference-type="eqref" reference="eq:def-q-pi"} (which is split surjective). If we set ${\operatorname{pd}}\circ {\operatorname{k\ddot{u}}}^{-1}$ at $(*)$ and $\Phi$ at $(**)$, then the diagram commutes by Theorem [Theorem 40](#thm:TY){reference-type="ref" reference="thm:TY"} (3) and Lemma [Lemma 46](#lem:pi-q){reference-type="ref" reference="lem:pi-q"}. On the other hand, if we set ${\operatorname{cr}}$ at $(*)$ and the induced map by $H^*_{\operatorname{ur}}$ at $(**)$, then the diagram commutes by definition. Hence the assertion follows from the commutativity of [\[eq:comm-ku-pd\]](#eq:comm-ku-pd){reference-type="eqref" reference="eq:comm-ku-pd"}. ◻ **Example 55**. Let $S$ be an Enriques surface over $\mathbb{C}$ (so that $S$ admits a decomposition of the diagonal by [@BKL] and Remark [Remark 21](#rem:div-nor-eff){reference-type="ref" reference="rem:div-nor-eff"} (1)). Let $f : T \to S$ be its universal cover so that $\deg(f)=2$ and $T$ is a $K3$ surface. In [@B Corollary 5.7], Beauville showed that $H^2_{\operatorname{ur}}(f)$ vanishes if and only if there exists $L \in \operatorname{Pic}(T)$ such that $\sigma(L)=L^{-1}$ and $c_1(L)^2 \equiv 2 \bmod 4$, where $\sigma \in {\operatorname{Gal}}(f)$ is the non-trivial element. Moreover, it is shown that all the $S$ satisfying those conditions form an infinite countable union of hypersurfaces in the moduli space of Enriques surfaces [@B Corollary 6.5]. Explicit examples of $S$ satisfying those conditions can be found in [@GS; @HS]. As $H^1_{\operatorname{ur}}(f)=0$ by definition, Theorem [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"} shows that this condition implies $F(f)=0$ for any normalized, birational, and motivic functor $F$. **Example 56**. Let us apply Theorem [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"} to $T=S$ and $f=m \cdot {\operatorname{id}}_S$ with $m \in \mathbb{Z}_{>0}$. The minimal $m$ for which the condition (3) is satisfied is nothing other than the torsion order ${\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(S)$ in the sense of Definition [Definition 19](#def:tor-ord){reference-type="ref" reference="def:tor-ord"}. Thus Theorem [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"} (together with [\[eq:etcoh13-fincoef\]](#eq:etcoh13-fincoef){reference-type="eqref" reference="eq:etcoh13-fincoef"}) recovers a main result of [@K1 Corollary 6.4 (b)], which says ${\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(S) =\exp(\operatorname{NS}(S)_{\Lambda, {\operatorname{Tor}}})$. Theorem [Theorem 53](#thm:main1-full){reference-type="ref" reference="thm:main1-full"} suggests the following problem. **Problem 57**. *Is the functor $H^*_{\operatorname{ur}}$, viewed as a functor from the full subcategory of torsion objects in $\operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda$ to $\operatorname{\mathbf{Mod}}_\Lambda$, faithful? (Compare [@K1 Question 3.5].)* ## Explicit computation of the Chow group and unramified cohomology **Theorem 58**. *Suppose the characteristic of $k$ is zero. Let $S \in \operatorname{\mathbf{SmProj}}$ be a surface admitting a decomposition of the diagonal. If $H^1_{\operatorname{ur}}(S)$ is a cyclic group of prime order $\ell$, then so are ${\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}, \Lambda}$ and $H^3_{\operatorname{ur}}(S \times S)$.* *Proof.* Let $M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ be the Chow motive constructed in Proposition [Proposition 25](#prop:GG-V){reference-type="ref" reference="prop:GG-V"}. Since ${\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}, \Lambda} = \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(S, S) = \operatorname{\mathbf{Chow}}^{\operatorname{nor}}_\Lambda(M, M)$, Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"} (1) and [\[eq:Vishik-ex-seq-full\]](#eq:Vishik-ex-seq-full){reference-type="eqref" reference="eq:Vishik-ex-seq-full"} yields an exact sequence $$\label{eq:Vishik-ex-seq-full2} 0 \to \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}(M, M) \overset{\Phi}{\to} \bigoplus_{i=1, 2} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(S)) \to H^3_{\operatorname{ur}}(S \times S) \to 0.$$ We know ${\operatorname{id}}_M \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(S, S)$ has order $\ell$ by Proposition [Proposition 27](#prop:GG-V2){reference-type="ref" reference="prop:GG-V2"} (2). Thus it suffices to show $\Phi$ is not surjective. If it were surjective, then by [\[eq:Vishik-ex-seq-full2\]](#eq:Vishik-ex-seq-full2){reference-type="eqref" reference="eq:Vishik-ex-seq-full2"} there should be a projector $\pi : M \to M$ in $\operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ such that $N:=\operatorname{Im}(\pi) \subset M$ satisfies $\operatorname{Pic}(N)=0$ and $\operatorname{Br}(N) \cong \mathbb{Z}/\ell\mathbb{Z}$, but this would contradict the following result of Vishik. ◻ **Theorem 59** (Vishik). *Suppose that $k$ is of characteristic zero, and let $N \in \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda$ be a non-trivial direct summand of a motive of a surface such that $\ell \cdot {\operatorname{id}}_N=0$ for some prime number $\ell$. Then we have $\operatorname{Pic}(N) \not= 0$.* *Proof.* See [@Vishik Corollary 4.22]. ◻ **Remark 60**. The assumption on the characteristic is used only to invoke Vishik's result. It is likely to hold in any characteristic, as long as $\ell$ is invertible in $k$. **Corollary 61**. *In Theorem [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"}, suppose further that $k=\mathbb{C}$. Then we have $$\operatorname{Coker}({\operatorname{CH}}^2(S \times S) \to H^4(S \times S(\mathbb{C}), \mathbb{Z}(2)) \cap H^{2, 2}(S \times S)) \cong \mathbb{Z}/\ell \mathbb{Z}.$$ In particular, $S \times S$ violates the integral Hodge conjecture in codimension two.* *Proof.* Set $X:=S \times S$. We claim that ${\operatorname{CH}}_0(X) \cong \mathbb{Z}$. For this, it suffices to show that $\ker({\operatorname{CH}}_0(X) \to \mathbb{Z})$ is torsion by Roitman's theorem, but Proposition [Proposition 25](#prop:GG-V){reference-type="ref" reference="prop:GG-V"} implies that $$\ker({\operatorname{CH}}_0(X) \to \mathbb{Z}) \cong \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(\Lambda(4), M \otimes M),$$ which is obviously killed by $\ell$. Now the corollary is a consequence of Theorem [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"} and the following result of Colliot-Thélène and Voisin [@CTV]. ◻ **Theorem 62** (Colliot-Thélène, Voisin). *Suppose $k=\mathbb{C}$ and let $X \in \operatorname{\mathbf{SmProj}}$. Assume that there exist $Y \in \operatorname{\mathbf{SmProj}}$ and a morphism $f : Y \to X$ such that $\dim Y=2$ and $f_* : {\operatorname{CH}}_0(Y) \to {\operatorname{CH}}_0(X)$ is surjective. Then we have an isomorphism of finite abelian groups $$H^3_{\operatorname{ur}}(X) \cong \operatorname{Coker}({\operatorname{CH}}^2(X) \to H^4(X(\mathbb{C}), \mathbb{Z}(2)) \cap H^{2, 2}(X)).$$* *Proof.* See [@CTV Théorème 3.9]. ◻ **Example 63**. 1. By applying Theorem [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"} (2) to an Enriques surface $S$, we find that ${\operatorname{CH}}_0(S_{k(S)})_{\operatorname{Tor}}$ is of order two. This answers a question raised by Kahn [@K1 p. 840, footnote] (in case of characteristic zero). 2. Similarly, we may apply Theorem [Theorem 58](#thm:main3-full){reference-type="ref" reference="thm:main3-full"} (2) to a Godeaux surface $S$ over $\mathbb{C},$ as long as Bloch's conjecture holds for $S$ (see Remark [Remark 21](#rem:div-nor-eff){reference-type="ref" reference="rem:div-nor-eff"}). This is previously known for the classical Godeaux surface by Vishik (see a remark after Proposition 4.6 in [@Vishik]). Other Godeaux surfaces for which Bloch's conjecture is verified can be found in [@Voisin]. **Problem 64**. *Does the equality $$|{\operatorname{CH}}_0(S_{k(S)})_{{\operatorname{Tor}}}|=|H^3_{\operatorname{ur}}(S \times S)|$$ remain valid when $H^1_{\operatorname{ur}}(S) \cong \operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda}$ is not cyclic of prime order, e.g. for a Beauville surface (see [@GS2]) or for a Burniat surface (see [@AO]) over $\mathbb{C}$? Note that Bloch's conjecture is known for such surfaces, and we have $H^1_{\operatorname{ur}}(S) \cong \mathbb{Z}/5\mathbb{Z}\times \mathbb{Z}/5\mathbb{Z}$ or $H^1_{\operatorname{ur}}(S) \cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, respectively.* # Appendix : elementary homological algebra In this section we prove some elementary lemmas that have been used in the body of this paper. **Lemma 65**. 1. *Let $A, B$ be abelian groups. Suppose that $A$ is finitely generated and that $B$ is a free $\mathbb{Z}$-module. Then the canonical map $$\operatorname{Hom}(A, \mathbb{Q}/\mathbb{Z}) \otimes B \to \operatorname{Hom}(A, B \otimes \mathbb{Q}/\mathbb{Z}), \qquad \chi \otimes b \mapsto [a \mapsto b \otimes \chi(a)]$$ is an isomorphism.* 2. *Let $m \in \mathbb{Z}_{>0}$ and let $A, B$ be $\mathbb{Z}/m\mathbb{Z}$-modules. Suppose that $A$ is finite and that $B$ is a free $\mathbb{Z}/m\mathbb{Z}$-module. Then the canonical map $$\operatorname{Hom}(A, \mathbb{Z}/m\mathbb{Z}) \otimes B \to \operatorname{Hom}(A, B), \qquad \chi \otimes b \mapsto [a \mapsto \chi(a)b]$$ is an isomorphism.* *Proof.* (1) Write $B=\mathbb{Z}^{\oplus I}$ with some set $I$. Since tensor product commutes with arbitrary sums, we can identify $-\otimes B = (-)^{\oplus I}$. To conclude, it suffices to note that $\operatorname{Hom}(A, -)$ commutes with arbitrary sums because $A$ is finitely generated. The proof of (2) is identical. ◻ **Lemma 66**. *Let $A, B$ be abelian groups. Suppose that $A$ is finite and that $B$ is a free $\mathbb{Z}$-module. Then we have canonical isomorphisms $$\operatorname{Hom}(A, \mathbb{Q}/\mathbb{Z}) \otimes B \cong \operatorname{Hom}(A, B \otimes \mathbb{Q}/\mathbb{Z}) \cong \operatorname{Ext}(A, B).$$* *Proof.* The first isomorphism is from Lemma [Lemma 65](#lem:homalg0){reference-type="ref" reference="lem:homalg0"}. The second is seen by an exact sequence $0 \to B \to B \otimes \mathbb{Q}\to B \otimes \mathbb{Q}/\mathbb{Z}\to 0$, together with $\operatorname{Hom}(A, B \otimes \mathbb{Q})=\operatorname{Ext}(A, B \otimes \mathbb{Q})=0$ as $A$ is finite and $B \otimes \mathbb{Q}$ is injective. ◻ **Lemma 67**. *Let $A, B$ be abelian groups with $A$ finite. Then we have canonical isomorphisms $${\operatorname{Tor}}(\operatorname{Hom}(A, \mathbb{Q}/\mathbb{Z}), B) \cong \operatorname{Hom}(A, B), \qquad \operatorname{Hom}(A, \mathbb{Q}/\mathbb{Z}) \otimes B \cong \operatorname{Ext}(A, B).$$* *Proof.* Set $(-)^\vee := \operatorname{Hom}(-, \mathbb{Q}/\mathbb{Z})$. We take an exact sequence $0 \to B_1 \to B_0 \to B \to 0$ with free $\mathbb{Z}$-modules $B_i$. Applying the two functors $A^\vee \otimes -$ and $\operatorname{Hom}(A, -)$, we obtain a commutative diagram with exact rows $$\xymatrix{ 0 \ar[r] & {\operatorname{Tor}}(A^\vee, B) \ar[r] & A^\vee \otimes B_1 \ar[r] \ar[d]^\cong & A^\vee \otimes B_0 \ar[r] \ar[d]^\cong & A^\vee \otimes B \ar[r] & 0 \\ 0 \ar[r] & \operatorname{Hom}(A, B) \ar[r] & \operatorname{Ext}(A, B_1) \ar[r] & \operatorname{Ext}(A, B_0) \ar[r] & \operatorname{Ext}(A, B) \ar[r] & 0, }$$ where two middle vertical isomorphisms are from Lemma [Lemma 66](#lem:homalg2){reference-type="ref" reference="lem:homalg2"}. The lemma follows. ◻ **Lemma 68**. *Let $\ell$ be a prime number and $M$ a $\mathbb{Z}_\ell$-module. Then we have a canonical isomorphism $$\operatorname{Hom}_{\mathbb{Z}_\ell}(M_{\operatorname{fr}}, \mathbb{Z}_\ell) \cong \operatorname{Hom}_{\mathbb{Z}_\ell}(M \otimes \mathbb{Q}_\ell/\mathbb{Z}_\ell, \mathbb{Q}_\ell/\mathbb{Z}_\ell),$$ where $A_{\operatorname{fr}}:= A/A_{\operatorname{Tor}}$ for an abelian group $A$.* *Proof.* We claim that both sides are canonically isomorphic to $\operatorname{Hom}_{\mathbb{Z}_\ell}(M, \mathbb{Z}_\ell)$. For the left hand side it holds since $\mathbb{Z}_\ell$ is torsion free. For the right hand side, we note that $\mathbb{Z}_\ell=\operatorname{Hom}_{\mathbb{Z}_\ell}(\mathbb{Q}_\ell/\mathbb{Z}_\ell, \mathbb{Q}_\ell/\mathbb{Z}_\ell)$ and use the adjointness of $\operatorname{Hom}$ and $\otimes$. ◻ Finally, we prove a version of the Poincaré duality. **Proposition 69**. *Let $X$ be a smooth proper variety over an algebraically closed field $k$ of pure dimension $d$. Let $\ell$ be a prime number invertible in $k$ and let $i, j \in \mathbb{Z}$ be such that $i+j=2d$. Then for any $r \in \mathbb{Z}$ the cup product induces an isomorphism $$\label{eq:PD-app} H^i(\mathbb{Z}_\ell(d-r))_{\operatorname{fr}} \overset{\cong}{\longrightarrow} \operatorname{Hom}_{\mathbb{Z}_\ell}(H^j(\mathbb{Z}_\ell(r))_{\operatorname{fr}}, \mathbb{Z}_\ell),$$ where we write $H^i(M):=H^i_{\operatorname{\acute{e}t}}(X, M)$.* *Proof.* Set $(-)^\vee := \operatorname{Hom}_{\mathbb{Z}_\ell}(-, \mathbb{Q}_\ell/\mathbb{Z}_\ell)$. We claim that there is a commutative diagram with exact rows: $$\xymatrix{ 0 \ar[r] & H^i(\mathbb{Z}_\ell(d-r))_{\operatorname{Tor}}\ar[r] \ar[d]^\cong & H^i(\mathbb{Z}_\ell(d-r)) \ar[r] \ar[d]^\cong & H^i(\mathbb{Z}_\ell(d-r))_{\operatorname{fr}}\ar[r] \ar[d]^\cong & 0 \\ 0 \ar[r] & (H^{j+1}(\mathbb{Z}_\ell(r))_{\operatorname{Tor}})^\vee \ar[r] & H^j(\mathbb{Q}_\ell/\mathbb{Z}_\ell(r))^\vee \ar[r] & (H^j(\mathbb{Z}_\ell(r)) \otimes \mathbb{Q}_\ell/\mathbb{Z}_\ell)^\vee \ar[r] & 0. }$$ The upper row comes from the definition of $(-)_{\operatorname{fr}}$, while the lower one is obtained by taking $(-)^\vee$ of the colimit of an exact sequence $$0 \to H^j(\mathbb{Z}_\ell(r))/\ell^n H^j(\mathbb{Z}_\ell(r)) \to H^j(\mathbb{Z}/\ell^n \mathbb{Z}_\ell(r)) \to H^j(\mathbb{Z}_\ell(r))[\ell^n] \to 0.$$ The middle vertical map is an isomorphism as the limit of the Poincaré duality with finite coefficients. The left vertical map is also an isomorphism since $(H^{j+1}(\mathbb{Z}_\ell(r))_{\operatorname{Tor}})^\vee$ is precisely the torsion part of $H^j(\mathbb{Q}_\ell/\mathbb{Z}_\ell(r))^\vee$. It follows that the right vertical map is an isomorphism as well. We now apply Lemma [Lemma 68](#lem:dual-dual){reference-type="ref" reference="lem:dual-dual"} to conclude [\[eq:PD-app\]](#eq:PD-app){reference-type="eqref" reference="eq:PD-app"}. ◻ 99 V. Alexeev, D. Orlov. Derived categories of Burniat surfaces and exceptional collections. Math. Ann. **357** (2013), no. 2, 743--759. G. Anderson, F. Pablos Romo. Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring. Comm. Algebra **32** (2004), no. 1, 79--102. M. Artin, A. Grothendieck, J. Verdier, P. Deligne and B. Saint-Donat. *Théorie des topos et cohomologie étale des schémas, Tome 3*. Lecture Notes in Math. 305. Springer, Berlin, 1973. A. Beauville. 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[^1]: The first author is supported by JSPS KAKENHI Grant (JP20K03566). The second author is supported by JSPS KAKENHI Grant (JP21K03153).

arxiv_math

--- abstract: | Describing the equality conditions of the *Alexandrov--Fenchel inequality* .03cmhas been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem, and is a complexity counterpart of the recent result by Shenfeld and van Handel [@SvH-acta], which gave a geometric characterization of the equality conditions. The proof involves Stanley's *order polytopes* .03cmand employs poset theoretic technology. address: - Department of Mathematics, Rutgers University, Piscatway, NJ 08854. - Department of Mathematics, UCLA, Los Angeles, CA 90095. author: - Swee Hong Chan - Igor Pak title: | Equality cases of the Alexandrov--Fenchel inequality\ are not in the polynomial hierarchy --- -.5cm # Introduction {#s:intro} ## Foreword {#ss:intro-foreword} Geometric inequalities play a central role in convex geometry, probability and analysis, with numerous combinatorial and algorithmic applications. The *Alexandrov--Fenchel *(AF)* inequality* .03cmlies close to the heart of convex geometry. It is one of the deepest and most general results in the area, generalizing a host of simpler geometric inequalities such as the *isoperimetric inequality* .03cm and the *Brunn--Minkowski inequality*, see $\S$[3.1](#ss:hist-geom){reference-type="ref" reference="ss:hist-geom"}. The equality conditions for geometric inequalities are just as fundamental as the inequalities themselves, and are crucial for many applications. For simpler inequalities they tend to be straightforward and follow from the proof. As the inequalities become more complex, their proofs became more involved, and the equality cases become more numerous and cumbersome. This is especially true for the Alexandrov--Fenchel inequality, where the complete description of the equality cases remain open despite much effort and many proofs, see $\S$[3.2](#ss:hist-AF){reference-type="ref" reference="ss:hist-AF"}. We use the language and ideas from computational complexity and tools from poset theory, to prove that the equality cases of the Alexandrov--Fenchel inequality cannot be explicitly described for convex polytopes in a certain formal sense. We give several applications to stability in geometric inequalities and to combinatorial interpretation of the deficit of poset inequalities. We also raise multiple questions, both mathematical and philosophical, see Section [10](#s:finrem){reference-type="ref" reference="s:finrem"}. ## Alexandrov--Fenchel inequality {#ss:intro-main} Let .06cm $\textnormal{V}(\textnormal{Q}_1,\ldots, \textnormal{Q}_n)$ .06cm denote the *mixed volume* .03cm of convex bodies .06cm $\textnormal{Q}_1,\ldots,\textnormal{Q}_n$ .03cmin .03cm$\mathbb R^n$ (see below). The *Alexandrov--Fenchel inequality* .03cmstates that for convex bodies .03cm$\textnormal{K}, \textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}$ .03cmin $\mathbb R^{n}$, we have: $$\label{eq:AF} \tag{AF} \textnormal{V}\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big)^2 \, \geq \, \textnormal{V}\big(\textnormal{K},\textnormal{K},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \hskip.06cm\cdot \hskip.06cm \textnormal{V}\big(\textnormal{L},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big).$$ Let polytope .03cm$\textnormal{K}\subset \mathbb R^n$ .03cmbe defined by a system of inequalities .06cm $A \hskip.06cm \textbf{\textit{x}}\leqslant \textbf{{\textit{b}}}$. We say that .03cm$\textnormal{K}$ .03cmis a .03cm*$\text{{\rm TU}}$-polytope* .03cmif vector .06cm $\textbf{{\textit{b}}}\in \mathbb Z^n$, and matrix .03cm$A$ .03cmis *totally unimodular*, i.e. all its minors have determinants in .03cm$\{0,\pm 1\}$. Note that all vertices of TU-polytopes are integral. Denote by .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}$ .03cmthe *equality verification problem of the Alexandrov--Fenchel inequality*, defined as the decision problem whether [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} is an equality. **Theorem 1** (Main theorem). *Let .06cm $\textnormal{K}, \textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2} \subset \mathbb R^{n}$ .03cmbe .03cm$\text{{\rm TU}}$-polytopes. Then the equality verification problem of the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level: $$\mathop{\mathrm{\textsc{EqualityAF}}}\hskip.03cm\in \textup{\textup{\textsf{PH}}}\ \ \Longrightarrow \ \ \textup{\textup{\textsf{PH}}}=\ensuremath{\Sigma^{{\textup{p}}}}_m \quad \ \text{for some} \ \, m\hskip.03cm.$$* Informally, the theorem says that the equality cases of the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} are *unlikely to have a description in the polynomial hierarchy*.[^1] This is in sharp contrast with other geometric inequalities, including many special cases of [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}, when the equality cases have an explicit description, thus allowing an efficient verification (see $\S$[3.1](#ss:hist-geom){reference-type="ref" reference="ss:hist-geom"}). Let us emphasize that constraining to TU-polytopes makes the theorem stronger rather than weaker. Indeed, one would hope that the equality verification problem is easy at least in the case when both vertices and facets are integral (cf. $\S$[10.2](#ss:finrem-polytopes){reference-type="ref" reference="ss:finrem-polytopes"}). In fact, we chose the *smallest* .03cmnatural class of H-polytopes which contains all slices of the order polytopes (see below). Let us quickly unpack the very strong claim of Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"}. In particular, the theorem implies that given the polytopes, the equality in [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} cannot be decided in polynomial time: .03cm .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\notin \textup{\textup{\textsf{P}}}$, nor even in probabilistic polynomial time: .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\notin\textup{\textup{\textsf{BPP}}}$ .03cm(unless .03cm$\textup{\textup{\textsf{PH}}}$ .03cmcollapses). Moreover, there can be no polynomial size certificate which verifies that [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} is an equality: .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\hskip.03cm\notin \textup{\textup{\textsf{NP}}}$, or a strict inequality: .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\notin \textup{\textup{\textsf{coNP}}}$ .03cm(ditto). Our results can be viewed as a complexity theoretic counterpart of the *geometric description* .03cm of the Alexandrov--Fenchel inequality that was proved recently by Shenfeld and van Handel [@SvH-acta]. In this context, Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} says that this geometric description is not computationally effective, and cannot be made so under standard complexity assumptions. From this point of view, the results in [@SvH-acta] are optimal, at least for convex polytopes in the full generality (cf. $\S$[10.11](#ss:finrem-meaning){reference-type="ref" reference="ss:finrem-meaning"}). *Warning*: .03cmHere we only give statements of the results without a context. Our hands are tied by the interdisciplinary nature of the paper with an extensive background in both convex geometry, poset theory and computational complexity. We postpone the definitions until Section [2](#s:def){reference-type="ref" reference="s:def"}, and the review until Section [3](#s:hist){reference-type="ref" reference="s:hist"}. ## Stability {#ss:intro-stability} In particular, Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} prohibits certain *stability inequalities*. In the context of general inequalities, these results give quantitative measurements of how close are the objects of study (variables, surfaces, polytopes, lattice points, etc.) to the equality cases in some suitable sense, when the inequality is close to an equality, see e.g. [@Fig13]. In the context of geometric inequalities, many sharp stability results appear in the form of *Bonnesen type inequality*, see [@Oss79]. These are defined as the strengthening of a geometric inequality .06cm $f\geqslant g$ .06cm to .06cm $f-g\geqslant h$, such that .06cm $h\geqslant 0$, and .06cm $h=0$ .06cm [if and only if]{.ul} .06cm $f=g$.[^2] They are named after the celebrated extension of the *isoperimetric inequality* .03cmby Bonnesen (see $\S$[3.3](#ss:hist-stab){reference-type="ref" reference="ss:hist-stab"}). While there are numerous Bonnesen type inequalities of various strength for the Brunn--Minkowski inequalities and their relatives, the case of Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} remains unapproachable in full generality. Formally, define the *Alexandrov--Fenchel defect* .06cm as: $$\label{eq:AF-stab} % \tag{AF-stab} \delta\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \hskip.06cm := \hskip.06cm \textnormal{V}\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big)^2 \hskip.03cm- \hskip.03cm \textnormal{V}\big(\textnormal{K},\textnormal{K},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \cdot \hskip.03cm\textnormal{V}\big(\textnormal{L},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big).$$ One would want to find a bound of the form .06cm $\delta(\cdot) \geqslant \xi(\cdot)$, where .03cm $\xi$ .03cmis a nonnegative computable function of the polytopes. The following result is an easy corollary from the proof of Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"}. **Corollary 2**. *Suppose .06cm $\delta\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \geqslant \xi\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big)$ .06cm is a Bonnesen type inequality, such that .03cm$\xi$ .03cmis computable in polynomial time on all .03cm$\text{{\rm TU}}$-polytopes. Then .03cm$\textup{\textup{\textsf{PH}}}=\textup{\textup{\textsf{NP}}}$.* Informally, the corollary implies that for the stability of the AF inequality, one should either avoid polytopes altogether and require some regularity conditions for the convex bodies (as has been done in the past, see $\S$[3.3](#ss:hist-stab){reference-type="ref" reference="ss:hist-stab"}), or be content with functions .03cm$\xi$ .03cmwhich are hard to compute (such inequalities can still be very useful, of course). See $\S$[10.9](#ss:finrem-mass-transport){reference-type="ref" reference="ss:finrem-mass-transport"} for further implications. To understand how the corollary follows from the proof of Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"}, the Bonnesen condition in this case states that .06cm $\xi(\cdot) = 0$ .06cm [if and only if]{.ul} .06cm $\delta(\cdot) = 0$. Thus, the equality .06cm $\{\delta(\cdot) =^? 0\}$ .06cm can be decided in polynomial time on TU-polytopes, giving the assumption in the theorem. ## Stanley inequality {#ss:intro-Stanley} We restrict ourselves to a subset of TU-polytopes given by the slices of *order polytopes* .03cm(see $\S$[2.5](#ss:hist-poset-polytopes){reference-type="ref" reference="ss:hist-poset-polytopes"}). Famously, Stanley showed in [@Sta-AF], that the Alexandrov--Fenchel inequality applied to certain such polytopes gives the *Stanley inequality*, that the numbers of certain linear extensions of finite posets form a log-concave sequence. This inequality is of independent interest in order theory (see $\S$[3.4](#ss:hist-LE){reference-type="ref" reference="ss:hist-LE"}), and is the starting point of our investigation. Let .06cm $P=(X,\prec)$ .06cm be a poset with .06cm $|X|=n$ .06cm elements. Denote .06cm $[n]:=\{1,\ldots,n\}$. A *linear extension* of $P$ is a bijection .06cm $f: X \to [n]$, such that .06cm $f(x) < f(y)$ .06cm for all .06cm $x \prec y$. Denote by .03cm$\mathop{\mathrm{\mathcal{E}}}(P)$ .03cmthe set of linear extensions of $P$, and let .06cm $e(P):=|\mathop{\mathrm{\mathcal{E}}}(P)|$. Let .06cm $x,z_1,\ldots,z_k\in X$ .06cm and .06cm $a,c_1,\ldots,c_k\in [n]$; we write .06cm $\mathop{\mathrm{\mathbf{z}}}=(z_1,\ldots,z_k)$ .06cm and .06cm $\mathop{\mathrm{\mathbf{c}}}=(c_1,\ldots,c_k)$, and we assume without loss of generality that .06cm $c_1<\ldots< c_k$.06cm. Let .06cm $\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a)$ .06cm be the set of linear extensions .06cm $f\in \mathop{\mathrm{\mathcal{E}}}(P)$, such that .06cm $f(x)=a$ .06cm and .06cm $f(z_i)=c_i$ .06cm for all .06cm $1\le i \le k$. Denote by .06cm $\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a):=\bigl|\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a)\bigr|$ .06cm the number of such linear extensions. The *Stanley inequality* [@Sta-AF] states that the sequence .06cm $\big\{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a), 1\le a \le n\big\}$ .06cm is *log-concave*: $$\label{eq:Sta}\tag{Sta} \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)^2 \, \ge \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a+1) \hskip.06cm\cdot \hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\hskip.03cm\mathop{\mathrm{\mathbf{c}}}}(P, x,a-1).$$ The problem of finding the equality conditions for [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} was first asked by Stanley in the original paper [@Sta-AF]. Formally, for every .03cm$k\ge 0$, denote by .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_k$ .03cmthe *equality verification problem of the Stanley inequality .03cm with $k$ fixed elements*, defined as the decision problem whether [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is an equality. It was shown by Shenfeld and van Handel that .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_0 \in {\textup{\textup{\textsf{P}}}}$, see [@SvH-acta Thm 15.3]. **Theorem 3**. *Let .03cm$k\ge 6$. Then the equality verification problem of the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level: $$\mathop{\mathrm{\textsc{EqualityStanley}}}_k \hskip.03cm\in \textup{\textup{\textsf{PH}}}\ \ \Longrightarrow \ \ \textup{\textup{\textsf{PH}}}=\ensuremath{\Sigma^{{\textup{p}}}}_m \quad \ \text{for some} \ \, m\hskip.03cm.$$* In Section [5](#s:AF){reference-type="ref" reference="s:AF"}, we deduce Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} from Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}. For the proof, any $k$ in [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} suffices, but one can ask if .03cm$k\ge 6$ is really necessary. We prove the following extension of Shenfeld and van Handel's result above in Section [9](#s:Sta1){reference-type="ref" reference="s:Sta1"}, and conjecture that it is optimal (see $\S$[10.7](#ss:finrem-LE){reference-type="ref" reference="ss:finrem-LE"}). **Theorem 4**. *.06cm $\mathop{\mathrm{\textsc{EqualityStanley}}}_1 \hskip.03cm\in \hskip.03cm\textup{\textup{\textsf{P}}}$.* ## Combinatorial interpretation {#ss:intro-combin} The problem of finding a *combinatorial interpretation* .03cmis fundamental in both enumerative and algebraic combinatorics, and was the original motivation of this investigation (see $\S$[3.7](#ss:hist-combin-int){reference-type="ref" reference="ss:hist-combin-int"}). Although very different in appearance and technical details, there are certain natural parallels with the stability problems discussed above. Let .03cm$f\geqslant g$ .03cmbe an inequality between two counting functions .03cm$f,g \in \textup{\textup{\textsf{\#P}}}$. We say that .03cm$(f-g)$ .03cmhas a *combinatorial interpretation*, if .03cm$(f-g) \in \textup{\textup{\textsf{\#P}}}$. While many combinatorial inequalities have a combinatorial interpretation, for the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} this is an open problem. Formally, let $$%\label{eq:Sta-phi}%\tag{Sta} \Phi_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a) \, := \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)^2 \hskip.06cm - \hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a+1) \cdot \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a-1)$$ denote the deficit in [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}. Let .03cm$\phi_k: \big(P, X^{k+1}, [n]^{k+1}\big) \to \mathbb N$ .03cm be the function computing .03cm $\Phi_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)$. **Corollary 5**. *For all .03cm$k\ge 6$, function .03cm$\phi_k$ .03cmdoes not have a combinatorial interpretation unless the polynomial hierarchy collapses to the second level: $$\phi_k \in \textup{\textup{\textsf{\#P}}}\ \ \Longrightarrow \ \ \textup{\textup{\textsf{PH}}}=\ensuremath{\Sigma^{{\textup{p}}}}_2\hskip.03cm.$$* To see some context behind this result, note that .06cm $\Phi_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)\in \textup{\textup{\textsf{\#P}}}$ .06cm by definition, so .06cm $\phi_k \in \textup{\textup{\textsf{GapP}}}_{\ge 0}$, a class of nonnegative functions in .03cm $\textup{\textup{\textsf{GapP}}}:=\textup{\textup{\textsf{\#P}}}-\textup{\textup{\textsf{\#P}}}$. We currently know very few functions which are in .03cm$\textup{\textup{\textsf{GapP}}}_{\ge 0}$ .03cm but not in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$. The examples include $$\label{eq:maltese} \tag{\text{\small $\maltese$}} \big(\text{\#3SAT}(F) \hskip.03cm- \hskip.03cm1 \big)^2, \quad \big(\text{\#2SAT}(F) \hskip.03cm- \hskip.03cm\text{\#2SAT}(F')\big)^2 \quad \text{and} \quad \big(e(P)-e(P')\big)^2,$$ where .03cm$F,F'$ .03cmare CNF Boolean formulas and .03cm$P,P'$ .03cmare posets [@CP23; @IP22]. In other words, all three functions in [\[eq:maltese\]](#eq:maltese){reference-type="eqref" reference="eq:maltese"} *do not* .03cmhave a combinatorial interpretation (unless $\textup{\textup{\textsf{PH}}}$ collapses). The corollary provides the first *natural* .03cmexample of a defect function that is .03cm$\textup{\textup{\textsf{GapP}}}_{\ge 0}$ .03cmbut not in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$. The case .03cm$k=0$, whether .06cm $\phi_0 \in \textup{\textup{\textsf{\#P}}}$, is especially interesting and remains a challenging open problem, see [@CPP-effective $\S$`<!-- -->`{=html}9.12] and [@Pak-OPAC Conj. 6.3]. The corollary suggests that Stanley's inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is unlikely to have a direct combinatorial proof, see $\S$[10.8](#ss:finrem-injective){reference-type="ref" reference="ss:finrem-injective"}. To understand how the corollary follows from the proof of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}, note that .03cm$\phi_6 \in \textup{\textup{\textsf{\#P}}}$ .03cmimplies that there is a polynomial certificate for the Stanley inequality being strict. In other words, we have .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_6 \in \textup{\textup{\textsf{coNP}}}$, giving the assumption in the theorem. ## Structure of the paper {#structure-of-the-paper .unnumbered} We begin with definitions and notation in Section [2](#s:def){reference-type="ref" reference="s:def"}, followed by the lengthy background and literature review in Section [3](#s:hist){reference-type="ref" reference="s:hist"} (see also $\S$[10.1](#ss:finrem-hist){reference-type="ref" reference="ss:finrem-hist"}). In the key Section [4](#s:roadmap){reference-type="ref" reference="s:roadmap"}, we give proofs of Theorems [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} and [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}, followed by proofs of Corollaries [Corollary 2](#c:main-AF-stab){reference-type="ref" reference="c:main-AF-stab"} and [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"}. These results are reduced to several independent lemmas, which are proved one by one in Sections [5](#s:AF){reference-type="ref" reference="s:AF"}--[8](#s:verify){reference-type="ref" reference="s:verify"}. We prove Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"} in Section [9](#s:Sta1){reference-type="ref" reference="s:Sta1"}. This section is independent of the previous sections (except for notation in $\S$[6.1](#ss:main-MS){reference-type="ref" reference="ss:main-MS"}). We conclude with extensive final remarks and open problems in Section [10](#s:finrem){reference-type="ref" reference="s:finrem"}. # Definitions and notation {#s:def} ## Terminology {#ss:def-term} For functions .03cm$f,g: X\to \mathbb R$, we write .03cm *$f\geqslant g$*, if .03cm$f(x) \ge g(x)$ .03cmfor all .03cm$x\in X$. For an inequality .06cm $f \geqslant g$, the *defect* .03cmis a function .03cm$h:=f-g$. The *equality cases* .03cmto describe the set of .03cm$x\in X$ .03cmsuch that .03cm$f(x)=g(x)$. Denote by .03cm$X_h :=\{x\in X \hskip.06cm : h(x)=0\} \subseteq X$ .03cmthe subset of equality cases. We use .06cm E$_h$ .06cm to denote the *equality verification* .03cmof .06cm $f(x)=g(x)$, i.e. the decision problem $$\text{\sc E}_h \, := \, \big\{ \hskip.03cmf(x) \hskip.06cm =^? \hskip.06cm g(x) \big\},$$ where .03cm$x\in X$ .03cmis an input. In other words, this is a special case of the *inclusion problem*: .06cm E$_h = \big\{x\in^? X_h\}$. We use .06cm E$_h$ .06cm to denote the *verification* .03cmof .06cm $h(x)=a$, i.e. the decision problem $$\text{\sc V}_h \, := \, \big\{ \hskip.03cmf(x) \hskip.03cm- \hskip.03cmg(x) \hskip.06cm =^? \hskip.06cm a \big\},$$ where .03cm$a\in \mathbb R$ .03cmand .03cm$x\in X$ .03cmare the input. Clearly, V$_h$ .03cmis a more general problem than .03cmE$_h$.03cm. For a subset .03cm$Y\subseteq X$, we use *description* .03cmfor an equivalent condition for the inclusion problem .03cm$\big\{x\in^? Y\big\}$, where .03cm$x\in X$. We use *equality conditions* .03cmfor a description of .03cmE$_h$. We say that equality cases of .06cm $f\geqslant g$ .03cmhave a .03cm*description in the polynomial hierarchy* .03cm if .06cm E$_h \in \textup{\textup{\textsf{PH}}}$. In other words, there is a CNF Boolean formula .03cm$\Phi(y_1,y_2,y_3,\ldots,x)$, such that $$\forall \hskip.03cmx\in X \ : \ \text{\sc E}_h \ \Longleftrightarrow \ \exists y_1 \hskip.06cm \forall y_2 \hskip.06cm \exists y_3 \hskip.06cm \ldots \hskip.06cm \Phi(y_1,y_2,y_3,\ldots,x).$$ ## General notation {#s:def-gen} Let .03cm$[n]=\{1,\ldots,n\}$, .03cm$\mathbb N=\{0,1,2,\ldots\}$ .03cmand .03cm$\mathbb \mathbb Z_{\ge 1}=\{1,2,\ldots\}$. For a subset .03cm$S\subseteq X$ .03cmand element .03cm$x\in X$, we write .03cm$S+x:=S\cup \{x\}$ .03cmand .03cm$S-x:=S\smallsetminus\{x\}$. For a sequence .03cm$\mathop{\mathrm{\mathbf{a}}}=(a_1,\ldots,a_m)$, denote .03cm$|\mathop{\mathrm{\mathbf{a}}}| := a_1 + \ldots + a_m$.03cm. This sequence is *log-concave*, if .06cm $a_i^2\ge a_{i-1} \hskip.03cma_{i+1}$ .06cm for all .03cm$1< i < m$. ## Mixed volumes {#ss:def-mixed} Fix .03cm$n \geq 1$. For two sets .06cm $A, B \subset \mathop{\mathrm{\mathbb{R}}}^n$ .06cm and constants .03cm$\alpha,\beta>0$, denote by $$\alpha A \hskip.06cm + \hskip.06cm \beta B \, := \, \bigl\{ \hskip.03cm\alpha\mathop{\mathrm{\mathbf{x}}}\hskip.06cm + \hskip.06cm \beta\mathop{\mathrm{\mathbf{y}}}\, : \, \mathop{\mathrm{\mathbf{x}}}\in A, \hskip.06cm \mathop{\mathrm{\mathbf{y}}}\in B \hskip.03cm\bigr\}$$ the *Minkowski sum* of these sets. For a convex body .03cm$\textnormal{K}\subset \mathop{\mathrm{\mathbb{R}}}^n$ .03cmwith affine dimension $d$, denote by .03cm$\textnormal{Vol}_d(\textnormal{K})$ .03cmthe volume of $\textnormal{K}$. We drop the subscript when .03cm$d=n$. One of the basic result in convex geometry is *Minkowski's theorem*, see e.g. [@BZ-book $\S$`<!-- -->`{=html}19.1], that the volume of convex bodies with affine dimension $d$ behaves as a homogeneous polynomial of degree $d$ with nonnegative coefficients: **Theorem 6** (Minkowski). *For all convex bodies .06cm $\textnormal{K}_1, \ldots, \textnormal{K}_r \subset \mathop{\mathrm{\mathbb{R}}}^n$ .06cm and .06cm $\lambda_1,\ldots, \lambda_r > 0$, we have: $$\label{eq:mixed volume definition} \textnormal{Vol}_d(\lambda_1 \textnormal{K}_1+ \ldots + \lambda_r \textnormal{K}_r) \ = \ \sum_{1 \hskip.03cm\le \hskip.03cmi_1\hskip.03cm,\hskip.03cm\ldots \hskip.03cm, \hskip.03cmi_d\hskip.03cm\le \hskip.03cmr} \hskip.06cm \textnormal{V}\bigl(\textnormal{K}_{i_1},\ldots, \textnormal{K}_{i_d}\bigr) \, \lambda_{i_1} \hskip.03cm\cdots\hskip.03cm\lambda_{i_d}\hskip.06cm,$$ where the functions .03cm$\textnormal{V}(\cdot)$ .03cmare nonnegative and symmetric, and where $d$ is the affine dimension of .03cm$\lambda_1 \textnormal{K}_1+ \ldots + \lambda_r \textnormal{K}_r$ .03cm$($which does not depend on the choice of .03cm$\lambda_1,\ldots, \lambda_r)$.* The coefficients .06cm $\textnormal{V}(\textnormal{A}_{i_1},\ldots, \textnormal{A}_{i_d})$ .06cm are called *mixed volumes* of .06cm $\textnormal{A}_{i_1}, \ldots, \textnormal{A}_{i_d}$.06cm. We refer to [@HW; @Leicht; @Schn] for an accessible introduction to the subject. ## Posets {#ss:def-posets} For a poset .03cm$P=(X,\prec)$ .03cmand a subset .03cm$Y \subset X$, denote by .03cm$P_Y=(Y,\prec)$ .03cma *subposet* .03cmof $P$. We use .03cm $(P-z)$ .03cmto denote a subposet .03cm$P_{X-z}$.03cm, where .03cm$z\in X$. Element .03cm$x\in X$ .03cmis *minimal* .03cmin $\hskip.03cmP$, if there exists no element .03cm$y \in X-x$ .03cmsuch that .03cm$y \prec x$.03cm. Define *maximal* .03cm elements similarly. Denote by .03cm$\min(P)$ .03cmand .03cm$\max(P)$ .03cmthe set of minimal and maximal elements in $P$, respectively. In a poset .03cm$P=(X,\prec)$, elements .03cm$x,y\in X$ .03cmare called *parallel* or *incomparable* if .03cm$x\not\prec y$ .03cm and .03cm$y \not \prec x$. We write .06cm $x\parallel y$ .06cm in this case. *Comparability graph* .03cmis a graph on $X$, with edges .03cm$(x,y)$ .03cmwhere .03cm$x\prec y$. Element .03cm$x\in X$ .03cmis said to *cover* .03cm$y\in X$, if .03cm$y\prec x$ .03cmand there are no elements .03cm$z\in X$ .03cm such that .06cm $y\prec z \prec x$. A *chain* is a subset .03cm$C\subset X$ .03cmof pairwise comparable elements. The *height* of poset .03cm$P=(X,\prec)$ .03cmis the maximum size of a chain. An *antichain* is a subset .03cm$A\subset X$ .03cmof pairwise incomparable elements. The *width* of poset .03cm$P=(X,\prec)$ .03cmis the size of the maximal antichain. A *dual poset* .03cmis a poset .03cm$P^\ast=(X,\prec^\ast)$, where .03cm$x\prec^\ast y$ .03cmif and only if .03cm$y \prec x$. A *disjoint sum* .03cm$P+Q$ .03cmof posets .03cm$P=(X,\prec)$ .03cm and .03cm$Q=(Y,\prec')$ .06cm is a poset .03cm$(X\cup Y,\prec^{\small{\hskip.03cm\diamond\hskip.03cm}})$, where the relation $\prec^{\small{\hskip.03cm\diamond\hskip.03cm}}$ coincides with $\prec$ and $\prec'$ on $X$ and $Y$, and .06cm $x\hskip.06cm\|\hskip.06cm y$ .06cm for all .03cm$x\in X$, $y\in Y$. A *linear sum* .03cm$P\oplus Q$ .03cmof posets .03cm$P=(X,\prec)$ .03cm and .03cm$Q=(Y,\prec')$ .06cm is a poset .03cm$(X\cup Y,\prec^{\small{\hskip.03cm\diamond\hskip.03cm}})$, where the relation $\prec^{\small{\hskip.03cm\diamond\hskip.03cm}}$ coincides with $\prec$ and $\prec'$ on $X$ and $Y$, and .06cm $x\prec^{\small{\hskip.03cm\diamond\hskip.03cm}}y$ .06cm for all .03cm$x\in X$, $y\in Y$. Posets constructed from one-element posets by recursively taking disjoint and linear sums are called *series-parallel*. Both *$n$-chain* .03cm$C_n$ .03cmand *$n$-antichain* .03cm$A_n$ .03cmare examples of series-parallel posets. *Forest* .03cmis a series-parallel poset formed by recursively taking disjoint sums (as before), and linear sums with one element: .03cm$C_1 \oplus P$. We refer to [@Sta-EC Ch. 3] for an accessible introduction, and to surveys [@BrW; @Tro] for further definitions and standard results. ## Poset polytopes {#ss:hist-poset-polytopes} Let .03cm$P=(X,\prec)$ .03cmbe a poset with .03cm$|X|=n$ .03cmelements. The *order polytope* .03cm$\mathcal O_P\subset \mathbb R^n$ .03cmis defined as $$\label{eq:order-def} 0\le \alpha_x \le 1 \quad \text{for all} \quad x\in X\hskip.06cm, \qquad \alpha_x \le \alpha_y \quad \text{for all} \quad x\prec y, \ \ x,y \in X.$$ Similarly, the *chain polytope* (also known as the *stable set polytope*) .03cm$\mathcal S_P\subset \mathbb R^n$ .03cmis defined as $$\label{eq:chains-def} \beta_x \ge 0 \quad \text{for all} \quad x\in X\hskip.06cm, \qquad \beta_x + \beta_y + \ldots \le 1 \quad \text{for all} \quad x\prec y \prec \cdots\,, \ x,y,\ldots \in X.$$ In [@Sta-two], Stanley computed the volume of both polytopes: $$\label{eq:two-poset} \textnormal{Vol}\hskip.03cm(\mathcal O_P) \, = \, \textnormal{Vol}\hskip.03cm(\mathcal S_P) \, = \, \frac{e(P)}{n!}\hskip.06cm.$$ This connection is the key to many applications of geometry to poset theory and vice versa. ## Complexity {#ss:def-CS} We assume that the reader is familiar with basic notions and results in computational complexity and only recall a few definitions. We use standard complexity classes: .06cm $\textup{\textup{\textsf{P}}}$, .06cm $\textup{\textup{\textsf{FP}}}$, .06cm $\textup{\textup{\textsf{NP}}}$,.06cm $\textup{\textup{\textsf{coNP}}}$, .06cm $\textup{\textup{\textsf{\#P}}}$, .06cm $\ensuremath{\Sigma^{{\textup{p}}}}_m$ .06cm and .06cm $\textup{\textup{\textsf{PH}}}$. The notation .06cm $\{a =^? b\}$ .06cm is used to denote the decision problem whether .03cm$a=b$. We use the *oracle notation* .03cm R$^{\text{\sf S}}$ .03cmfor two complexity classes .03cmR, S $\subseteq \textup{\textup{\textsf{PH}}}$, and the polynomial closure .03cm$\langle$A$\rangle$ for a problem .03cmA $\in \textup{\textup{\textsf{PSPACE}}}$. We will also use less common classes .06cm $$\textup{\textup{\textsf{GapP}}}:= \{f-g \mid f,g\in \textup{\textup{\textsf{\#P}}}\} \quad \text{and} \quad {\textup{\textup{\textsf{C$_=$P}}}}:=\{f(x)=^?g(y) \mid f,g\in \textup{\textup{\textsf{\#P}}}\}.$$ Note that .03cm$\textup{\textup{\textsf{coNP}}}\subseteq {\textup{\textup{\textsf{C$_=$P}}}}$. We also assume that the reader is familiar with standard decision and counting problems: .03cm3SAT, .03cm\#3SAT .03cmand .03cmPERMANENT. Denote by .03cm\#LE .03cmthe problem of computing the number .03cm$e(P)$ .03cmof linear extensions. For a counting function .03cm$f\in \textup{\textup{\textsf{\#P}}}$, the *coincidence problem* .03cmis defined as: $$\text{\sc C}_f \ := \ \big\{\hskip.03cmf(x) \hskip.06cm = ^? \hskip.03cmf(y) \hskip.03cm\big\}.$$ Note the difference with the equality verification problem .03cmE$_{f-g}$ .03cm defined above. Clearly, we have both .03cm$\text{\sc E}_{f-g}\in {\textup{\textup{\textsf{C$_=$P}}}}$ .06cm and .06cm $\text{\sc C}_f \in {\textup{\textup{\textsf{C$_=$P}}}}$. Note also that .06cm$\text{\sc C}_\text{\#3SAT}$ .03cm is both .03cm${\textup{\textup{\textsf{C$_=$P}}}}$-complete .03cmand .03cm$\textup{\textup{\textsf{coNP}}}$-hard. The distinction between *binary* .03cmand *unary* .03cmpresentation will also be important. We refer to [@GJ78] and [@GJ79 $\S$`<!-- -->`{=html}4.2] for the corresponding notions of $\textup{\textup{\textsf{NP}}}$-completeness and *strong* .03cm$\textup{\textup{\textsf{NP}}}$-completeness. Unless stated otherwise, we use the word "*reduction*" .03cmto mean "polynomial Turing reduction". We refer to [@AB; @Gold; @Pap] for definitions and standard results in computational complexity. # Background and historical overview {#s:hist} ## Geometric inequalities {#ss:hist-geom} The history of equality conditions of geometric inequalities goes back to antiquity, see e.g. [@Bla; @Porter], when it was discovered that the *isoperimetric inequality* $$\label{eq:Isop}\tag{Isop} \ell(X)^2 \, \ge \, 4 \hskip.03cm\pi \hskip.03cma(X)$$ is an equality .06cm [if and only if]{.ul} .06cm $X$ .03cmis a circle. Here .03cm$\ell(X)$ .03cmis the perimeter and .03cm$a(X)$ .03cmis the area of a convex .03cm $X\subset\mathbb R^2$. This classical result led to numerous extensions and generalizations leading to the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}. We refer to [@BZ-book; @Schn] for a review of the literature. Below we highlight only the most important developments to emphasize how the equality conditions become more involved as one moves in the direction of the AF inequality (see also $\S$[10.3](#ss:finrem-discrete-isop){reference-type="ref" reference="ss:finrem-discrete-isop"} and $\S$[10.4](#ss:finrem-BM){reference-type="ref" reference="ss:finrem-BM"}). For the *Brunn--Minkowski inequality* $$\label{eq:BM}\tag{BM} \textnormal{Vol}(\textnormal{K}+\textnormal{L})^{1/d} \, \ge \, \textnormal{Vol}(\textnormal{K})^{1/d} \hskip.06cm + \hskip.06cm \textnormal{Vol}(\textnormal{L})^{1/d},$$ for all convex .03cm$\textnormal{K}, \textnormal{L}\subset \mathbb R^d$, the equality in [\[eq:BM\]](#eq:BM){reference-type="eqref" reference="eq:BM"} holds .06cm [if and only if]{.ul} .06cm $\textnormal{K}$ .03cmis an expansion of $\hskip.03cm\textnormal{L}$. For the *mean width inequality* $$\label{eq:Mink-mean} \tag{MWI} s(\textnormal{K})^2 \, \ge \, 6\hskip.03cm\pi \hskip.06cm w(\textnormal{K}) \hskip.06cm \textnormal{Vol}(\textnormal{K})\hskip.03cm,$$ for all convex $\hskip.03cm\textnormal{K}\subset \mathbb R^3$, Minkowski conjectured (1903) the equality cases are the *cap bodies* (balls with attached tangent cones). Here .03cm$s(\textnormal{K})$ .03cm is the surface area and .03cm$w(\textnormal{K})$ .03cmis the *mean width* .03cmof $\hskip.03cm\textnormal{K}$. Minkowski's conjecture was proved by Bol (1943), see e.g. [@BF; @BZ-book]. The *Minkowski's quadratic inequality* .03cmfor three convex bodies .03cm$\textnormal{K},\textnormal{L},\textnormal{M}\subset \mathbb R^3$, states: $$\label{eq:MQI} \tag{MQI} \textnormal{V}(\textnormal{K},\textnormal{L},\textnormal{M})^2 \, \ge \, \textnormal{V}(\textnormal{K},\textnormal{K},\textnormal{M}) \hskip.06cm \cdot \hskip.06cm \textnormal{V}(\textnormal{L},\textnormal{L},\textnormal{M})\hskip.03cm.$$ This is a special case of [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} for .03cm$n=d=3$. When .03cm$\textnormal{L}=\textnormal{B}_1$ .03cmis a unit ball and .03cm$\textnormal{K}=\textnormal{M}$, this gives [\[eq:Mink-mean\]](#eq:Mink-mean){reference-type="eqref" reference="eq:Mink-mean"}. Favard [@Fav p. 248] wrote that the equality conditions for [\[eq:MQI\]](#eq:MQI){reference-type="eqref" reference="eq:MQI"} *"parait difficile à énonce"* ("seem difficult to state"). There are even interesting families of convex polytopes that give equality cases (see e.g. [@SvH-acta Fig. 2.1]). Shenfeld and van Handel [@SvH-duke] gave a complete characterization of the equality cases of [\[eq:MQI\]](#eq:MQI){reference-type="eqref" reference="eq:MQI"} as triples of convex bodies that are similarly truncated in a certain formal sense. Notably, for the full-dimensional H-polytopes in $\mathbb R^3$, each with at most .03cm$n$ .03cmfacets, the equality conditions amount to checking .03cm$O(n)$ .03cmlinear relations for distances between facet inequalities. This can be easily done in polynomial time. ## Alexandrov--Fenchel inequality {#ss:hist-AF} For the AF inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}, the equality conditions have long believed to be out of reach, as they would generalize those for [\[eq:Mink-mean\]](#eq:Mink-mean){reference-type="eqref" reference="eq:Mink-mean"} and [\[eq:MQI\]](#eq:MQI){reference-type="eqref" reference="eq:MQI"}. Alexandrov made a point of this in his original 1937 paper: *"Serious difficulties occur in determining the conditions for equality to hold in the general inequalities just derived"* [@Ale37 $\S$`<!-- -->`{=html}4]. Half a century later, Burago and Zalgaller reviewed the literature and summarized: *"A conclusive study of all these situations when the equality sign holds has not been carried out, probably because they are too numerous"* [@BZ-book $\S$`<!-- -->`{=html}20.5]. Schneider made a case for perseverance: *"As [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} represents a classical inequality of fundamental importance and with many applications, the identification of the equality cases is a problem of intrinsic geometric interest. Without its solution, the Brunn--Minkowski theory of mixed volumes remains in an uncompleted state."* [@Schn94 p. 426]. The AF inequality has a number of proofs using ideas from convex geometry, analysis and algebraic geometry, going back to two proofs by Alexandrov (Fenchel's full proof never appeared). We refer to [@BZ-book; @Schn] for an overview of the older literature, especially [@Schn p. 398] for historical remarks, and to [@BL23; @CP2; @CEKMS; @KK; @SvH-pams; @Wang] for some notable recent proofs. All these proofs use a limit argument at the end, which can create new equality cases that do not hold for generic convex bodies. This partially explains the difficulty of the problem (cf. [@SvH-duke Rem. 3.7]). In [@Ale37], Alexandrov gave a description of equality cases for combinatorially isomorphic polytopes. This is a large family of full-dimensional polytopes, for which every convex body is a limit. In particular, he showed that for the full-dimensional axis-parallel boxes .03cm$[\ell_1\times \ldots \times \ell_n]$, the equality in [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} is equivalent to $\textnormal{K}$ and $\textnormal{L}$ being homothetic (cf. $\S$[10.5](#ss:finrem-vdW){reference-type="ref" reference="ss:finrem-vdW"}). In the pioneering work [@Schn85], Schneider published a conjectural description of the equality cases, corrected later by Ewald [@Ewald], see also [@Schn]. After many developments, this conjecture was confirmed for all .06cm smooth (full-dimensional) convex bodies .03cm$\textnormal{Q}_i$ .03cm[@Schn90a], and for all (not necessarily full-dimensional) convex bodies .03cm${\text {\rm Q} } _1=\ldots=\textnormal{Q}_{n-2}$.03cm, by Shenfeld and van Handel [@SvH-acta]. Closer to the subject of this paper, in a remarkable development, the authors gave a geometric description of the equality cases *for all* .03cm convex polytopes. They explain: *"Far from being esoteric, it is precisely the case of convex bodies with empty interior $($which is not covered by previous conjectures$)$ that arises in combinatorial applications"* [@SvH-acta $\S$`<!-- -->`{=html}1.3]. The geometric description of the equality cases in [@SvH-acta] is indirect, technically difficult to prove, and computationally hard in the degenerate cases.[^3] While we will not quote the full statement (Theorem 2.13 in [@SvH-acta]), let us mention the need to find witnesses polytopes .03cm$\textnormal{M}_i \hskip.03cm, \textnormal{N}_i \subset\mathbb R^n$ .03cmwhich must satisfy certain conditions (Def. 2.10, ibid.) The second of these conditions is an equality of certain mixed volumes (Eq. (2.4), ibid.) In [@SvH-acta $\S$`<!-- -->`{=html}2.2.3], the authors write: "Condition (2.4) should be viewed merely as a normalization".[^4] From the computational complexity point of view, asking for the equality of mixed volumes (known to be hard to compute, see $\S$[3.8](#ss:hist-CA){reference-type="ref" reference="ss:hist-CA"}), lifts the problem outside of the polynomial hierarchy, to a hard coincidence problem (see $\S$[2.6](#ss:def-CS){reference-type="ref" reference="ss:def-CS"}). This coincidence problem eventually percolated into [@MS22], see [\[eq:MS\]](#eq:MS){reference-type="eqref" reference="eq:MS"} below, which in turn led directly to this work. ## Stability {#ss:hist-stab} *Bonnesen's inequality* .03cmis an extension of the isoperimetric inequality [\[eq:Isop\]](#eq:Isop){reference-type="eqref" reference="eq:Isop"}, which states that for every convex .03cm $X\subset\mathbb R^2$, we have: $$\label{eq:Bon}\tag{Bon} \ell(X)^2 \hskip.06cm - \hskip.06cm a(X) \, \geq \, 4 \hskip.03cm\pi \hskip.03cm(R-r)^2,$$ where .03cm$R$ .03cmis the smallest radius of the circumscribed circle, and .03cm$r$ .03cmis the maximal radius of the inscribed circle.[^5] Moreover, Bonnesen proved [@Bon], that there is an *annulus* (thin shell) .03cm$U$ .03cmbetween concentric circles of radii .03cm$R\ge r$, such that .03cm$\partial X\subseteq U$ .03cmand [\[eq:Bon\]](#eq:Bon){reference-type="eqref" reference="eq:Bon"} holds. Note that the optimal such annulus can be computed in polynomial time, see [@AAHS]. Bonnesen's inequality [\[eq:Bon\]](#eq:Bon){reference-type="eqref" reference="eq:Bon"} was an inspiration for many Bonnesen type inequalities [@Oss78; @Oss79; @Gro90]. See also discrete versions in $\S$[10.3](#ss:finrem-discrete-isop){reference-type="ref" reference="ss:finrem-discrete-isop"}, and applications in computational geometry in [@KS99]. There is now a large literature on stability inequalities in geometric and more general context, see e.g. [@Fig13; @Gro93]. There is an especially large literature on the stability of the Brunn--Minkowski inequality [\[eq:BM\]](#eq:BM){reference-type="eqref" reference="eq:BM"}. For major early advances by Diskant (1973), Groemer (1988) and others, see e.g. [@Gro93] and references therein. We refer to [@Fig14] for an overview of more recent results, including [@FMP09; @FMP10]. See also [@EK14] for the thin shell .03cm type bounds, and [@FJ17] for the stability of [\[eq:BM\]](#eq:BM){reference-type="eqref" reference="eq:BM"} for *nonconvex sets*. For the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}, the are very few stability results, all for the full dimensional convex bodies with various regularity conditions, see e.g. [@Mar17; @Schn90]. ## Linear extensions {#ss:hist-LE} Linear extensions play a cental role in enumerative combinatorics and order theory. They appear in connection with saturated chains in *distributive lattices*, *standard Young tableaux* and *$P$-partitions*, see e.g. [@Sta-EC]. The world of inequalities for linear extensions has a number of remarkable results, some with highly nontrivial equality conditions. Notably, the *Björner--Wachs inequality* .03cmfor .03cm$e(P)$ .03cm is an equality if and only if .03cm$P$ .03cmis a forest [@BW89 Thm 6.3], see also [@CPP-effective]. On the other hand, the celebrated *XYZ inequality* .03cm established by Shepp in [@She-XYZ] (see also [@ASE $\S$`<!-- -->`{=html}6.4]), has no nontrivial equality cases [@Fish]. An especially interesting example is the *Sidorenko inequality* .03cm $$\label{eq:Sid} e(P) \cdot e(P^\circ) \hskip.06cm \ge \hskip.06cm n!$$ for posets .03cm$P, P^\circ$ .03cmon the same ground set with .03cm$n$ .03cmelements, which have complementary comparability graphs [@Sid] .03cm(other proofs are given in [@CPP-effective; @GG22]). Sidorenko also proved that the series-parallel posets are the only equality cases. This solves the equality verification problem of [\[eq:Sid\]](#eq:Sid){reference-type="eqref" reference="eq:Sid"}, since the recognition problem of series-parallel posets is in $\textup{\textup{\textsf{P}}}$, see [@VTL]. It was noticed in [@BBS], that the Sidorenko inequality follows from *Mahler's conjecture*, which states that for every convex centrally symmetric body .03cm$\textnormal{K}\subset \mathbb R^n$, we have: $$\label{eq:Mah} \textnormal{Vol}(\textnormal{K}) \cdot \textnormal{Vol}(\textnormal{K}^{o}) \, \ge \, \frac{4^n}{n!}\,.$$ To derive [\[eq:Sid\]](#eq:Sid){reference-type="eqref" reference="eq:Sid"} from [\[eq:Mah\]](#eq:Mah){reference-type="eqref" reference="eq:Mah"}, take .03cm$\textnormal{K}$ .03cmto be the union all axis reflections the chain polytope .03cm$\mathcal S_P$ .03cmdefined in [\[eq:chains-def\]](#eq:chains-def){reference-type="eqref" reference="eq:chains-def"}. Mahler's conjecture [\[eq:Mah\]](#eq:Mah){reference-type="eqref" reference="eq:Mah"} is known for all axis symmetric convex bodies [@StR], but remains open in full generality [@AASS], in part due to the many equality cases [@Tao $\S$`<!-- -->`{=html}1.3]. ## Stanley inequality {#ss:hist-Stanley} Stanley's inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is of independent interest in order theory, having inspired a large literature especially in the last few years. The case .03cm$k=0$ .03cmis especially interesting. The unimodality in this case was conjectured by Rivest, while the log-concavity was conjectured by Chung, Fishburn and Graham [@CFG], who established both conjectures for posets of width two. Stanley proved them in [@Sta-AF] in full generality.[^6] The authors of [@CFG] called Rivest's conjecture "tantalizing" and Stanley's proof "very ingenious". The *Kahn--Saks inequality* .03cmis a generalization of the .03cm$k=0$ .03cm case of [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}, and is also proved from the AF inequality. This inequality was used to obtain the first positive result in the direction of the .06cm *$\frac 13-\frac 23$ conjecture* [@KS]. For posets of width two, both the .03cm$k=0$ .03cmcase of the Stanley inequality, and the Kahn--Saks inequality have natural *$q$-analogues* [@CPP-KS]. A generalization of Stanley's inequality to *marked posets* .03cmwas given in [@LMS19]. For all .03cm$k\ge 0$, the *vanishing conditions* .03cm $\{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)=^?0\}$ .03cmare in .03cm$\textup{\textup{\textsf{P}}}$. This was shown by David and Jacqueline Daykin in [@DD Thm 8.2], via explicit necessary and sufficient conditions. Recently, this result was rediscovered in [@CPP-effective Thm 1.11] and [@MS22 Thm 5.3]. Similarly, the *uniqueness conditions* .03cm$\{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)=^?1\}$ .03cm are in ${\textup{\textup{\textsf{P}}}}$ .03cmby the result of Panova and the authors [@CPP-effective Thm 7.5], where we gave explicit necessary and sufficient conditions. Both the vanishing and the uniqueness conditions give examples of equality cases of the Stanley inequality, which remained a "major challenge" in full generality [@CPP-effective $\S$`<!-- -->`{=html}9.10]. As we mentioned in the introduction, Shenfeld and van Handel resolved the .03cm$k=0$ .03cmcase of Stanley equality conditions by giving explicit necessary and sufficient conditions, which can be verified in polynomial time, see [@SvH-acta]. Similar explicit necessary and sufficient conditions for the Kahn--Saks inequality were given in [@CPP-KS Conj. 8.7], where they were proved for posets of width two. Building on the technology in [@SvH-acta], van Handel, Yan and Zeng announced the proof of this conjecture in [@vHY]. In [@CP], we gave a new proof of the .03cm$k=0$ .03cm case of [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}, using a *combinatorial atlas* .03cm technology. This is an inductive self-contained linear algebraic approach; see [@CP2] for the introduction. We also gave a new proof of the Shenfeld and van Handel equality conditions, and generalized both results to *weighted linear extensions* (see $\S\S$`<!-- -->`{=html}1.16-18 in [@CP]). In an important development, Ma and Shenfeld [@MS22] advances the technology of [@SvH-acta], to give a clean albeit ineffective combinatorial description of the equality cases in full generality. In particular, they showed that [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} is an equality .03cm[if and only if]{.ul} $$\label{eq:MS} \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a-1)\, = \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)\, = \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a+1).$$ They proceeded to give explicit necessary and sufficient conditions for these equalities in some cases (see $\S$[10.10](#ss:finrem-eq){reference-type="ref" reference="ss:finrem-eq"}). About the remaining cases that they called *critical* (see $\S$[9.2](#ss:Sta1-critical){reference-type="ref" reference="ss:Sta1-critical"}), they write: "It is an interesting problem to find .03cm$[$an explicit description$]$ .03cmfor critical posets" [@MS22 Rem. 1.6]. Our Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} implies that such a description is unlikely, as it would imply a disproof of a major conjecture in computational complexity (see also $\S$[10.11](#ss:finrem-meaning){reference-type="ref" reference="ss:finrem-meaning"}). ## Complexity aspects {#ss:hist-CC} There are two standard presentations of polytopes: *H-polytopes* .03cmdescribed by the inequalities and *V-polytopes* .03cmdescribed by the vertices. These two presentation types have very different nature in higher dimensions, see e.g. [@DGH]. We refer to [@GK94; @GK97] for an overview of standard complexity problems in geometry, and to [@Schr86 $\S$`<!-- -->`{=html}19], .03cm[@Schr03 $\S$`<!-- -->`{=html}5.16], for the background on totally unimodular matrices and $\text{{\rm TU}}$-polytopes. Note also that testing whether matrix .03cm$A$ .03cmis totally unimodular can be done in polynomial time, see [@Sey]. When the dimension $n$ is bounded, H-polytopes and V-polytopes have the same complexity, so the volume and the mixed volumes are in .03cm$\textup{\textup{\textsf{FP}}}$. Thus, the dimension $n$ is unbounded throughout the paper. The volume of TU-polytopes is $\textup{\textup{\textsf{\#P}}}$-hard via reduction to .03cmKNAPSACK [@DF]. Note that for *rational* .03cmH-polytopes in .03cm$\mathbb R^n$, the volume denominators can be doubly exponential [@Law91], thus not in .03cm$\textup{\textup{\textsf{PSPACE}}}$. This is why we constrain ourselves to TU-polytopes which is a subclass of H-polytopes that includes slices of the order polytopes (see $\S$[5.1](#ss:AF-slices){reference-type="ref" reference="ss:AF-slices"}). The mixed volume .03cm$\textnormal{V}(\textnormal{Q}_1,\ldots,\textnormal{Q}_n)$ .03cmcoincides with the permanent when all .03cm$\textnormal{Q}_i$ .03cmare axis parallel boxes, see [@vL] and $\S$[10.5](#ss:finrem-vdW){reference-type="ref" reference="ss:finrem-vdW"}. Thus, computing the mixed volume is .03cm$\textup{\textup{\textsf{\#P}}}$-hard even for the boxes, see [@DGH]. For rational H-polytopes, the vanishing problem .03cm$\{\textnormal{V}(\cdot)=^?0\}$ .03cm can be described combinatorially, and is thus in .03cm$\textup{\textup{\textsf{NP}}}$, see [@DGH; @Est10]. It is equivalent to computing the rank of intersection of two geometric matroids (with a given realization), which is in $\textup{\textup{\textsf{P}}}$, see [@Schr03 $\S41$]. For TU-polytopes in $\mathbb R^n$, the uniqueness problem .03cm$\big\{\textnormal{V}(\cdot)=^?\frac{1}{n!}\big\}$ .03cmis in .03cm$\textup{\textup{\textsf{NP}}}$ .03cmby a result in [@EG15]. The problem .03cm\#LE .03cmis proved .03cm$\textup{\textup{\textsf{\#P}}}$-complete by Brightwell and Winkler [@BW Thm 1], and this holds even for posets of height two [@DP]. Linial noticed [@Lin], that this result and [\[eq:two-poset\]](#eq:two-poset){reference-type="eqref" reference="eq:two-poset"} together imply that the volume of H-polytopes is .03cm$\textup{\textup{\textsf{\#P}}}$-hard even when the input is in unary. Linial also observed that the number of vertices of order polytopes is $\textup{\textup{\textsf{\#P}}}$-complete (ibid.) Now, fix .03cm$k\ge 0$, .03cm$x\in X$ .03cmand .03cm$\mathop{\mathrm{\mathbf{z}}}\in X^k$. Clearly, we have: $$e(P) \ = \ \sum_{a\in [n]} \, \sum_{\mathop{\mathrm{\mathbf{c}}}\in [n]^k} \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)\hskip.03cm,$$ where the summation has size .03cm$O(n^{k+1})$. Thus, computing .06cm $\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)$ .06cm is also .03cm$\textup{\textup{\textsf{\#P}}}$-complete. Finally, it was proved in [@CP23], that .03cm$\text{\sc C}_\text{\#3SAT}$, .03cm$\text{\sc C}_\text{PERMANENT}$ .03cmand .03cm $\text{\sc C}_\text{\#LE}$ .03cmare not in .03cm$\textup{\textup{\textsf{PH}}}$, unless .03cm$\textup{\textup{\textsf{PH}}}$ .03cmcollapses to a finite level. The proof idea of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} is inspired by these results. ## Combinatorial interpretations {#ss:hist-combin-int} Finding a combinatorial interpretation is a standard problem throughout combinatorics, whenever a positivity phenomenon or an inequality emerges. Having a combinatorial interpretation allows one to deeper understand the underlying structures, give asymptotic and numerical estimates, as well as analyze certain algorithms. We refer to [@Huh; @Sta-log-concave; @Sta-pos] for an overview of inequalities in algebraic combinatorics and matroid theory, and to [@Pak-OPAC] for a recent survey from the complexity point of view. Recall that .03cm$\textup{\textup{\textsf{GapP}}}:=\textup{\textup{\textsf{\#P}}}-\textup{\textup{\textsf{\#P}}}$ .03cmis a the class of difference of two .03cm$\textup{\textup{\textsf{\#P}}}$ .03cmfunctions, and let .03cm$\textup{\textup{\textsf{GapP}}}_{\ge 0}$ .03cmbe a subclass of .03cm$\textup{\textup{\textsf{GapP}}}$ .03cmof nonnegative functions. Thus, for every inequality .03cm$f\geqslant g$ .03cmof counting functions .03cm$f,g\in \textup{\textup{\textsf{\#P}}}$, we have .03cm$(f-g) \in \textup{\textup{\textsf{GapP}}}_{\ge 0}$. It was shown in [@IP22 Prop. 2.3.1], that .03cm$\textup{\textup{\textsf{GapP}}}_{\ge 0}\ne \textup{\textup{\textsf{\#P}}}$, unless .03cm$\textup{\textup{\textsf{PH}}}= \ensuremath{\Sigma^{{\textup{p}}}}_2$. The key example is $$\big(\text{\#3SAT}(F) \hskip.03cm- \hskip.03cm\text{\#3SAT}(F')\big)^2,$$ see also the first function in [\[eq:maltese\]](#eq:maltese){reference-type="eqref" reference="eq:maltese"}. The other two functions in [\[eq:maltese\]](#eq:maltese){reference-type="eqref" reference="eq:maltese"} were given in [@CP23]. A *natural* .03cm$\textup{\textup{\textsf{GapP}}}_{\ge 0}$ .03cmproblem of computing $S_n$ character squared: .03cm$[\chi^\lambda(\mu)]^2$, was proved not in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$ .03cm(in unary), under the same assumptions [@IPP22]. The idea that some natural combinatorial inequalities can have no combinatorial interpretations appeared in [@Pak]. A number of interesting examples were given in [@IP22 $\S$`<!-- -->`{=html}7], including the *Cauchy*, *Minkowski*, *Hadamard*, *Karamata* and *Ahlswede--Daykin inequalities*, all proved not in .03cm$\textup{\textup{\textsf{\#P}}}$ .03cmunder varying complexity assumptions. Closer to the subject of this paper, Ikenmeyer and the second author showed that the AF defect .03cm$\delta(\cdot)$ .03cmis not in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$ (unless $\textup{\textup{\textsf{PH}}}=\ensuremath{\Sigma^{{\textup{p}}}}_2$), even for axis parallel rectangles in $\mathbb R^2$ whose edge length are given by .03cm\#3SAT .03cmformulas [@IP22 Thm 7.1.5]. This is a nonstandard model of computation. One can think of our Main Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} as a tradeoff: in exchange for needing a higher dimension, we now have unary input and the standard model of computation. ## Complexity assumptions {#ss:hist-CA} The results in the paper use different complexity assumptions, and navigating between them can be confusing. Here is short list of standard implications: $$\textup{\textup{\textsf{PH}}}\ne \ensuremath{\Sigma^{{\textup{p}}}}_m \ \ \text{for all} \ \ m\ge 2 \quad \Longrightarrow \quad \textup{\textup{\textsf{PH}}}\ne \ensuremath{\Sigma^{{\textup{p}}}}_2 \quad \Longrightarrow \quad \textup{\textup{\textsf{PH}}}\ne \textup{\textup{\textsf{NP}}} \quad \Longrightarrow \quad \textup{\textup{\textsf{P}}}\ne \textup{\textup{\textsf{NP}}}\hskip.03cm.$$ In other words, the assumption in Theorems [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} and [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} is the strongest, while .03cm$\textup{\textup{\textsf{P}}}\ne \textup{\textup{\textsf{NP}}}$ .03cmis the weakest. Proving either of these would be a major breakthrough in theoretical computer science. Disproving either of these would bring revolutionary changes to the way the computational complexity understands the nature of computation. We refer to [@Aar16; @Wig19] for an extensive discussion, philosophy and implications in mathematics and beyond. # Proof roadmap {#s:roadmap} The results in the paper follow from a series of largely independent polynomial reductions and several known results. In this section, we only state the reductions whose proofs will be given in the next few sections. We then deduce both theorems from these reductions. ## Around Stanley equality {#ss:roadmap-lemmas} First, we show that Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} follows from Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}. Recall the notation from the introduction. Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements. As before, let .03cm$x\in X$, .03cm$a\in [n]$, .03cm$\mathop{\mathrm{\mathbf{z}}}\in X^k$, and .03cm$\mathop{\mathrm{\mathbf{c}}}\in [n]^k$. Recall also $$%\label{eq:AF} \tag{AF} \mathop{\mathrm{\textsc{EqualityAF}}}\, := \, \big\{ \textnormal{V}\big(\textnormal{K},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big)^2 \, =^? \, \textnormal{V}\big(\textnormal{K},\textnormal{K},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \cdot \textnormal{V}\big(\textnormal{L},\textnormal{L},\textnormal{Q}_1,\ldots,\textnormal{Q}_{n-2}\big) \big\},$$ $$%\label{eq:AF} \tag{AF} \mathop{\mathrm{\textsc{EqualityStanley}}}_k \, := \, \big\{ \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a)^2 \, =^? \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a+1) \cdot \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a-1) \big\}.$$ **Proposition 7** (cf. [@Sta-AF $\S$`<!-- -->`{=html}3]). *For all .03cm$k\ge 0$, .06cm EqualityStanley$_k$ .06cm reduces to .06cm EqualityAF.* The proof of the proposition is given in Section [5](#s:AF){reference-type="ref" reference="s:AF"}, is very close to Stanley's original proof of the inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}. The key difference is the observation that slices of order polytopes are TU-polytopes. Next, we need a simple technical result. **Lemma 8**. *For all .03cm$k> \ell$, .06cm EqualityStanley$_\ell$ .06cm reduces to .06cm EqualityStanley$_k$.06cm.* *Proof.* Let .03cm$P=(X,\prec)$ .03cmbe a poset on $n$ elements, and let .03cm$\mathop{\mathrm{\mathbf{z}}}\in X^k$, .03cm$\mathop{\mathrm{\mathbf{c}}}\in [n]^k$, .03cm$x\in X$, .03cm$a\in [n]$ .03cmbe as in $\S$[1.4](#ss:intro-Stanley){reference-type="ref" reference="ss:intro-Stanley"}. Denote by .03cm$P':=P+A_{k-\ell}$ .03cma poset obtained by adding .03cm $(k-\ell)$ .03cmindependent elements $z'_1,\ldots,z_{k-\ell}'$.03cm. Let .03cm$c_i':=n+i$, for all .03cm$1\le i\le k-\ell$. For .03cm$\mathop{\mathrm{\mathbf{z}}}':=\big(z_1,\ldots,z_\ell,z_1',\ldots,z_{k-\ell}'\big)$ .03cmand .03cm$\mathop{\mathrm{\mathbf{c}}}':=\big(c_1,\ldots,c_\ell,c_1',\ldots,c_{k-\ell}'\big)$, we have: $$\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}'\mathop{\mathrm{\mathbf{c}}}'}(P', x,a) \, = \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a).$$ Varying $a$, we conclude that .06cm EqualityStanley$_k$ .06cm is equivalent to .06cm EqualityStanley$_\ell$ .06cm in this special case. This gives the desired reduction. ◻ Next, we simplify the Stanley equality problem to the following .03cm*flatness problem*: $$\text{\sc FlatLE}_k \ := \ \big\{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) =^? \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1)\big\},$$ where .03cm$\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a)$ .03cmare defined in $\S$[1.4](#ss:intro-Stanley){reference-type="ref" reference="ss:intro-Stanley"}. The idea is to ask whether .03cm$a$ .03cmis in the flat part of the distribution of .03cm$f(x)$ .03cm(cf. Figure 15.1 in [@SvH-acta]). **Lemma 9**. *For all .03cm$k \geq 0$, .06cm FlatLE$_k$ .06cm reduces to .06cm EqualityStanley$_{k+2}$.06cm.* We prove Lemma [Lemma 9](#l:Flat-Sta){reference-type="ref" reference="l:Flat-Sta"} in Section [6](#s:Flat-Sta){reference-type="ref" reference="s:Flat-Sta"}. ## Relative numbers of linear extensions {#ss:roadmap-relative} Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements, and let .03cm$\min(P)\subseteq X$ .03cmbe the set of minimal elements of $P$. For every .03cm$x \in \min(P)$, define the *relative number of linear extensions*: $$\label{eq:rho-def} \rho(P,x) \, := \, \frac{e(P)}{e(P-x)}\hskip.06cm.$$ In other words, .06cm $\rho(P,x) = \mathbb P[f(x)=1]^{-1}$, where .06cm $f\in \mathop{\mathrm{\mathcal{E}}}(P)$ .06cm is a uniform random linear extension of $P$. Denote by .06cm \#RLE .06cm the problem of computing .03cm$\rho(P,x)$. **Lemma 10**. *$\text{\sc \#RLE}$ .06cm is polynomial time equivalent to .06cm \#LE.* *Proof.* By definition, $\text{\sc \#RLE}$ .03cmreduces to .03cm$\text{\sc \#LE}$. In the opposite direction, let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements. Fix a linear extension .03cm$g\in \mathop{\mathrm{\mathcal{E}}}(P)$, and let .03cm$x_i:=g^{-1}(i)$, .03cm $1\le i \le n$. Denote by .03cm$P_i$ .03cma subposet of $P$ restricted to .03cm $x_i,\ldots,x_n$ .03cmand observe that .03cm$x_i \in \min(P_i)$. We have: $$e(P) \ = \ \frac{e(P_1)}{e(P_2)} \hskip.06cm \cdot \hskip.06cm \frac{e(P_2)}{e(P_3)} \, \cdots \ = \ \rho(P_1,x_1) \hskip.06cm \cdot \hskip.06cm \rho(P_2,x_2) \, \cdots \,,$$ which gives the desired reduction from .03cm$\text{\sc \#LE}$ .06cm to .03cm$\text{\sc \#RLE}$. ◻ Define the following decision problem: $$%\label{eq:AF} \tag{AF} \text{\sc QuadRLE} \ := \ \big\{ \hskip.06cm \rho(P_1,x_1) \cdot \rho(P_2,x_2) \, =^? \, \rho(P_3,x_3) \cdot \rho(P_4,x_4) \hskip.06cm \big\},$$ where .03cm$P_1,P_2,P_3,P_4$ .03cmare finite posets and .03cm$x_i\in \min(P_i)$, for all .06cm $1\le i \le 4$. **Lemma 11** (see Theorem [Theorem 17](#thm:gCF){reference-type="ref" reference="thm:gCF"}). *$\text{\sc QuadRLE}$ .06cm reduces to .06cm FlatLE$_4$.03cm.* ## Verification lemma {#ss:roadmap-TCS} Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements, and let .03cm$x\in \min(P)$. Consider $$\label{eq:VRLE} %\tag{AF} \text{\sc VerRLE} \ := \ \Big\{ \hskip.06cm \rho(P,x) \hskip.06cm =^? \, \frac{A}{B} \hskip.06cm \Big\},$$ where $\hskip.03cmA,B \hskip.03cm$ are coprime integers with .03cm$1\le B \le A\leq n!$.06cm. We need this lemma with the following: **Lemma 12** (Verification lemma). *.06cm $\textup{\textup{\textsf{NP}}}^{\langle\text{\sc VerRLE}\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc QuadRLE}\rangle}.$* Note that the opposite direction .06cm "$\supseteq$" .06cm is also true and easy to prove. Indeed, suppose you have an oracle .06cm . Guess the values .06cm $a_i:=\rho(P_i,x_i)\in \mathbb Q$, verify that they are correct, and check that .06cm $a_1\cdot a_2 = a_3\cdot a_4\hskip.03cm.$ This gives .06cm . We will only need the direction in the lemma which is highly nontrivial. ## Putting everything together {#ss:roadmap-proofs} We can now obtain all the results stated in the introduction, except for Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"} which uses different tools and is postponed until Section [9](#s:Sta1){reference-type="ref" reference="s:Sta1"}. *Proof of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}.* Recall that .03cm\#LE .03cmis .03cm$\textup{\textup{\textsf{\#P}}}$-complete [@BW] (see also $\S$[3.6](#ss:hist-CC){reference-type="ref" reference="ss:hist-CC"}). By Lemma [Lemma 10](#l:RLE){reference-type="ref" reference="l:RLE"}, we conclude that .03cm\#RLE .03cmis .03cm$\textup{\textup{\textsf{\#P}}}$-hard. We then have: $$\label{eq:together-LE} \textup{\textup{\textsf{PH}}} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{P}}}^{\textup{\textup{\textsf{\#P}}}} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{P}}}^{\langle\text{\sc \#RLE}\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc VerRLE}\rangle}$$ where the first inclusion is Toda's theorem [@Toda], the second inclusion is because .03cm\#RLE .03cm is $\textup{\textup{\textsf{\#P}}}$-hard, and the third inclusion is because one can simulate .03cm\#RLE .03cmby first guessing and then verifying the answer. Fix .03cm$k\ge 6$. Combining Lemma [Lemma 8](#l:Sta-Sta-more){reference-type="ref" reference="l:Sta-Sta-more"}, .03cmLemma [Lemma 9](#l:Flat-Sta){reference-type="ref" reference="l:Flat-Sta"} .03cmand .03cmLemma [Lemma 11](#l:Quart-Flat){reference-type="ref" reference="l:Quart-Flat"}, we conclude that .06cm QuadRLE .06cm reduces to .06cm EqualityStanley$_k$.03cm. We have: $$\label{eq:together-verify} \textup{\textup{\textsf{NP}}}^{\langle\text{\sc VerRLE}\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc QuadRLE} \rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc EqualityStanley}_k\rangle},$$ where the first inclusion is the Verification Lemma [Lemma 12](#l:Verify-Quart){reference-type="ref" reference="l:Verify-Quart"}. Now, suppose .03cm$\text{\sc EqualityStanley}_k \in \textup{\textup{\textsf{PH}}}$. Then .03cmEqualityStanley$_k \in \hskip.03cm\ensuremath{\Sigma^{{\textup{p}}}}_m$ .03cm for some $m$. Combining [\[eq:together-LE\]](#eq:together-LE){reference-type="eqref" reference="eq:together-LE"} and [\[eq:together-verify\]](#eq:together-verify){reference-type="eqref" reference="eq:together-verify"}, this implies: $$\label{eq:together-collapse} \textup{\textup{\textsf{PH}}}\hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc EqualityStanley}_k\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\ensuremath{\Sigma^{{\textup{p}}}}_m} \hskip.06cm \subseteq \hskip.06cm \ensuremath{\Sigma^{{\textup{p}}}}_{m+1}\,,$$ as desired. ◻ As a byproduct of the proof, we get the same conclusion for intermediate problems. This result is potentially of independent interest (cf. [@CP23]). **Corollary 13**. *Problems .06cm VerRLE, .06cm QuadRLE .06cm and .06cm FlatLE$_4$ .06cm are not in .03cm$\textup{\textup{\textsf{PH}}}$, unless .03cm$\textup{\textup{\textsf{PH}}}=\ensuremath{\Sigma^{{\textup{p}}}}_m$ .03cmfor some $m$.* *Proof of Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"}.* The result follows from Proposition [Proposition 7](#p:AF-Sta){reference-type="ref" reference="p:AF-Sta"} and Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}. ◻ *Proof of Corollary [Corollary 2](#c:main-AF-stab){reference-type="ref" reference="c:main-AF-stab"}.* By the "Bonnesen type" assumption, we have $$\big\{\xi(\cdot) =^?0\big\} \ \ \Longleftrightarrow \ \ \big\{\delta(\cdot) =^?0\big\} \, = \, \text{\sc EqualityAF}\hskip.03cm.$$ Since computing .03cm$\xi$ .03cmis in $\textup{\textup{\textsf{FP}}}$, we have .03cmEqualityAF $\in \textup{\textup{\textsf{P}}}$. Then [\[eq:together-collapse\]](#eq:together-collapse){reference-type="eqref" reference="eq:together-collapse"} for $k=6$, and Proposition [Proposition 7](#p:AF-Sta){reference-type="ref" reference="p:AF-Sta"} give: $$\label{eq:together-stab} \textup{\textup{\textsf{PH}}}\hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc EqualityStanley}_6\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc EqualityAF}\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\textup{\textup{\textsf{P}}}} \hskip.06cm = \hskip.06cm \textup{\textup{\textsf{NP}}},$$ as desired. ◻ *Proof of Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"}.* Suppose .03cm$\phi_k \in \textup{\textup{\textsf{\#P}}}$. By definition, we have: .03cm $$\big\{\hskip.06cm \Phi_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P, x,a) \, \ne^? \, 0\hskip.06cm\big\} \hskip.06cm \in \hskip.06cm \textup{\textup{\textsf{NP}}}.$$ In other words, we have .03cmEqualityStanley$_k \in \textup{\textup{\textsf{coNP}}}$. Then [\[eq:together-collapse\]](#eq:together-collapse){reference-type="eqref" reference="eq:together-collapse"} gives: $$\textup{\textup{\textsf{PH}}}\hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\langle\text{\sc EqualityStanley}_k\rangle} \hskip.06cm \subseteq \hskip.06cm \textup{\textup{\textsf{NP}}}^{\hskip.03cm\textup{\textup{\textsf{coNP}}}} \hskip.06cm = \hskip.06cm \ensuremath{\Sigma^{{\textup{p}}}}_{2}\,,$$ as desired. ◻ # AF equality from Stanley equality {#s:AF} ## Slices of order polytopes {#ss:AF-slices} Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements. Recall the construction of order polytopes .03cm$\mathcal O_P\subseteq [0,1]^n$ .03cm given in [\[eq:order-def\]](#eq:order-def){reference-type="eqref" reference="eq:order-def"}. Fix .06cm $z_1\prec \ldots \prec z_k$ .06cm and .06cm $1 \le c_1 < \ldots < c_k\le n$.03cm. Denote .03cm$Z:=\{z_1,\ldots,z_k\}$ .03cmand let .03cm$Y:=X\smallsetminus Z$. For all .06cm $0\le i \le k$, consider .03cmthe following *slices* .03cmof the order polytopes: $$\textnormal{S}_i \, := \, \mathcal O_P \hskip.06cm \cap \hskip.06cm \{\alpha_x=0 \, : \, x \preccurlyeq z_i, \hskip.06cm x \in X \} \hskip.06cm \cap \hskip.06cm \{\alpha_x=1 \, : \, x \succcurlyeq z_{i+1}, \hskip.06cm x \in X \}. % \. \cap \. \{\al_y=0 \, : \, y \in Z\}.$$ Here the conditions .03cm$x \preccurlyeq z_i$ .03cmand .03cm$x \succcurlyeq z_{i+1}$ .03cmare vacuous when .03cm$i=0$ .03cmand .03cm$i=k$, respectively. Note that .03cm$\dim \textnormal{S}_i \le n-k$ .03cm for all .03cm$0\le i \le k$, since .03cm$\alpha_x$ .03cmis a constant on .03cm$\textnormal{S}_i$ .03cm for all .03cm$x\in Z$.[^7] **Lemma 14**. *Slices .06cm $\textnormal{S}_i$ .03cmare .03cm$\text{{\rm TU}}$-polytopes.* *Proof.* Write .03cm$\textnormal{K}_i$ .03cmin the form .06cm $A \cdot (\alpha_y)_{y\in Y} \le \textbf{{\textit{b}}}$. Observe that .03cm$A$ .03cmhas .03cm$\{-1,0,1\}$ .03cmentries, and so does $\textbf{{\textit{b}}}$. Every square submatrix $B$ of $A$ corresponds to taking a subposet with added rows of $0$'s, or with rows of $0$'s and a single $\pm 1$. By definition of .03cm$\mathcal O_P$, we can rearrange columns in $B$ to make it upper triangular. Thus, $\det(B) \in \{-1,0,1\}$, as desired. ◻ ## Proof of Proposition [Proposition 7](#p:AF-Sta){reference-type="ref" reference="p:AF-Sta"} {#ss:AF-proofs} Denote by .03cm$\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P)$ .03cmthe set of all linear extensions .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P)$, such that .03cm$f(z_i)=c_i$ .03cmfor all $i$, and let .03cm$\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P):=|\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P)|$. Let .03cm$\textnormal{S}_0,\ldots,\textnormal{S}_k\subset\mathbb R^n$ .03cmbe the slices defined above, and note that .03cm $\dim\langle\textnormal{S}_0,\ldots,\textnormal{S}_k\rangle = n-k$. Stanley's original proof of [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is based on the following key observation: **Lemma 15** ([@Sta-AF Thm 3.2]). *Let .06cm $z_1\prec \ldots \prec z_k$ .06cm and .06cm $1 \le c_1 < \ldots < c_k\le n$. We have: $$\label{eq:Sta-pol} \textnormal{V}\big(\underbrace{\textnormal{S}_0 \hskip.06cm ,\hskip.06cm \ldots \hskip.06cm , \hskip.06cm \textnormal{S}_0}_{\text{$c_1-1$ times}}, \hskip.06cm \underbrace{\textnormal{S}_1 \hskip.06cm ,\hskip.06cm \ldots \hskip.06cm , \hskip.06cm \textnormal{S}_1}_{\text{$c_2-c_1-1$ times}}, \hskip.06cm \ldots \hskip.06cm , \underbrace{\textnormal{S}_k \hskip.06cm ,\hskip.06cm \ldots \hskip.06cm , \hskip.06cm \textnormal{S}_k}_{\text{$n-c_k$ times}}\big) \ = \ \frac{1}{(n-k)!} \, \textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P).$$* Now let .03cm$z_i \gets x$ .03cmand .03cm$c_i\gets a$ .03cmfor some $i$, such that .06cm $1 \le c_1 < \ldots < c_k\le n$. By Lemma [Lemma 15](#l:Sta-AF){reference-type="ref" reference="l:Sta-AF"}, the AF inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} becomes [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}. By Lemma [Lemma 14](#l:slices-TU){reference-type="ref" reference="l:slices-TU"}, slices .03cm$\textnormal{S}_i\subset\mathbb R^n$ .03cmare TU-polytopes defined by .03cm$O(n^2)$ .03cminequalities. This gives the desired reduction. 0◻ # Stanley equality from flatness {#s:Flat-Sta} ## Ma--Shenfeld poset notation {#ss:main-MS} Recall the following terminology from [@MS22]. For .06cm $s\in \{-1,0,1\}$ .06cm and .06cm $f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+s)$, the *companions* .03cmin $\hskip.03cmf$ .03cm are the elements in $${\text {\rm Com} } (f) \, := \, \big\{f^{-1}(a-1), \hskip.06cm f^{-1}(a), \hskip.06cm f^{-1}(a+1) \big\} \hskip.06cm - \hskip.06cm x.$$ Note that .06cm $| {\text {\rm Com} } (f)| =2$ .06cm for all $s$ as above. Let the *lower companion* .06cm ${\textup {\rm lc} } (f)\in {\text {\rm Com} } (f)$ .06cm be the companion with smaller of the two values in $f$. Similarly, let the *upper companion* .06cm .06cm ${\textup {\rm uc} } (f)\in {\text {\rm Com} } (f)$ .06cm be the companion with larger of the two values in $f$. Denote by .03cm $\mathcal C(x)\subset X$ .03cmthe set of elements .03cm$y\in X$ comparable to $x$, i.e. .06cm $\mathcal C(x) := \{y \in X \, : \, x \prec y \ \text{or} \ x \succ y\}$. ## Proof of Lemma [Lemma 9](#l:Flat-Sta){reference-type="ref" reference="l:Flat-Sta"} {#s:Flat-Sta-proof} Let .06cm $P=(X,\prec)$, and let .03cm$x$, .03cm$a$, .03cm$\mathop{\mathrm{\mathbf{z}}}=(z_1,\ldots,z_k)$ .03cmand .03cm$\mathop{\mathrm{\mathbf{c}}}=(c_1,\ldots,c_k)$ .06cm be an instance of .03cmFlatLE$_k$ .03cmas above. To prove the reduction in the lemma, we construct a poset .03cm$Q=(Y,\prec)$ .03cmfor which .03cm$P$ .03cm is a subposet, and .03cm$x$, .03cm$b$, .03cm$\mathop{\mathrm{\mathbf{y}}}$ .03cmand .03cm$\mathop{\mathrm{\mathbf{x}}}$, which give the desired instance .06cm EqualityStanley$_{k+2}$.03cm. Without loss of generality, we can assume that .03cm$\min(P)=\{z_0\}$ .03cm and .03cm$\max(P)=\{z_{k+1}\}$. In other words, assume that there are elements .06cm $z_0,z_{k+1} \in X$ .06cm such that .06cm $z_0 \preccurlyeq y \preccurlyeq z_{k+1}$ .06cm for all .03cm$y \in X$. Let .06cm $\textrm{M}_1,\textrm{M}_2, \textrm{M}_3$ .06cm be given by $$\begin{aligned} \textrm{M}_1 \ &:= \ \big|\big\{ f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) \, : \, f^{-1}(a+1) \succ x \big\} \big|\hskip.06cm, \\ \textrm{M}_2 \ &:= \ \big|\big\{ f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \, : \, f^{-1}(a) \prec x \big\} \big|\hskip.06cm,\\ \textrm{M}_3 \ &:= \ \big|\big\{ f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) \, : \, f^{-1}(a+1) \hskip.06cm \| \hskip.06cm x \big\} \big| \\ & \qquad = \ \big|\big\{ f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \, : \, f^{-1}(a) \hskip.06cm \| \hskip.06cm x \big\} \big|\hskip.06cm.\end{aligned}$$ Note that the two sets in the definition of .03cm$\textrm{M}_3$ .03cmare in bijection with each other via the map that swaps .06cm $f(a)$ .06cm with .06cm $f(a+1)$.06cm. It then follows from here that $$\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) \, = \, \textrm{M}_1 \hskip.06cm + \hskip.06cm \textrm{M}_3 \qquad \text{and} \qquad \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \, = \, \textrm{M}_2 \hskip.06cm + \hskip.06cm \textrm{M}_3\hskip.03cm.$$ This implies that $$\label{eq:cmkt-1} \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) \, = \, \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \qquad \Longleftrightarrow \qquad \textrm{M}_1 \ = \ \textrm{M}_2\hskip.03cm.$$ Now, let .06cm $Q=(Y,\prec)$ .06cm be the poset .03cm$P+C_3$.03cm, i.e. .06cm $Y:=X \cup \{u,v,w\}$ .06cm and with the additional relation .06cm $u \prec v \prec w$ .06cm and .06cm $\{u,v,w\}$ .06cm is incomparable to all elements in $X$. Let .06cm $\ell :=\max \{i \hskip.06cm : \hskip.06cm c_i< a\}$ .06cm be the maximal index such that the corresponding element in $\mathop{\mathrm{\mathbf{z}}}$ is less than $a$. Let .06cm $b := a+2$, and let $$\begin{aligned} &\mathop{\mathrm{\mathbf{y}}}\ := \ (z_1,\ldots, z_{\ell}, u, w, z_{\ell+1}, \ldots, z_{k}) \hskip.06cm \in \hskip.06cm Y^{k+2}, \\ & \mathop{\mathrm{\mathbf{b}}}\ := \ (c_1,\ldots, c_\ell, a, a+4, c_{\ell+1}+3, \ldots, c_{k}+3) \hskip.06cm \in \hskip.06cm \mathbb N^{k+2}.\end{aligned}$$ In the notation above, for .06cm $s\in \{-1,0,1\}$ .06cm and .06cm $f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b+s)$, the companions .03cmin $\hskip.03cmf$ .03cm are the elements in $${\text {\rm Com} } (f) \, := \, \big\{f^{-1}(b-1), \hskip.06cm f^{-1}(b), \hskip.06cm f^{-1}(b+1) \big\} \hskip.06cm - \hskip.06cm x\hskip.03cm.$$ Let[^8] $$\begin{aligned} \mathcal F(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ &:= \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \, : \, {\textup {\rm lc} } (f)\in \mathcal C(x), \hskip.06cm {\textup {\rm uc} } (f)\not\in \mathcal C(x) \big\},\\ \mathcal F(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}}) \ &:= \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \, : \, {\textup {\rm lc} } (f)\notin \mathcal C(x), \hskip.06cm {\textup {\rm uc} } (f)\in \mathcal C(x) \big\},\\ \mathcal F(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}}) \ &:= \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \, : \, {\textup {\rm lc} } (f)\in \mathcal C(x), \hskip.06cm {\textup {\rm uc} } (f)\in \mathcal C(x) \big\},\\ \mathcal F(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}}) \ &:= \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \, : \, {\textup {\rm lc} } (f)\not\in \mathcal C(x), \hskip.06cm {\textup {\rm uc} } (f)\not\in \mathcal C(x) \big\}, \end{aligned}$$ and we write .06cm $\mathop{\mathrm{\textnormal{F}}}(b,\cdot, \cdot) \hskip.06cm := \hskip.06cm |\mathcal F(b,\cdot,\cdot)|$.03cm. Note that by construction it follows that, for all .03cm$f \in \mathcal F(b,\cdot,\cdot)$, we have $$b-2 \ = \ f(u) \ < \ f(v) \ < \ f(w) \ = \ b+2,$$ so .03cm$f(v) \in \{b-1,b,b+1\}$.03cm, and thus $v$ will always be a companion in $f$. Sets .03cm$\mathcal F(b+1,\ast,\ast)$ .03cmand .03cm$\mathcal F(b-1,\ast,\ast)$ .03cmare defined analogously. **Claim 16**. *We have: $$\begin{aligned} {2}\label{eq:yskn-1} & \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ = \ \textrm{M}_2\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}}) \ = \ \textrm{M}_1\hskip.03cm,\\ & \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}}) \ = \ 0\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}}) \ = \ 2\hskip.03cm\textrm{M}_3\hskip.03cm,\\ & \mathop{\mathrm{\textnormal{F}}}(b+1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ = \ \textrm{M}_2\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b+1,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}}) \ = \ \textrm{M}_2\hskip.03cm,\\ & \mathop{\mathrm{\textnormal{F}}}(b+1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}}) \ = \ 0\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b+1,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}}) \ = \ 2\hskip.03cm\textrm{M}_3\hskip.03cm,\\ & \mathop{\mathrm{\textnormal{F}}}(b-1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ = \ \textrm{M}_1\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b-1,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}}) \ = \ \textrm{M}_1\hskip.03cm,\\ & \mathop{\mathrm{\textnormal{F}}}(b-1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}}) \ = \ 0\,, \qquad && \mathop{\mathrm{\textnormal{F}}}(b-1,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}}) \ = \ 2\hskip.03cm\textrm{M}_3\hskip.03cm.\end{aligned}$$* *Proof.* We only compute the values .03cm$\mathop{\mathrm{\textnormal{F}}}(b,\ast,\ast)$, as proof of the other cases is analogous. Denote by .03cm$\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P)$ .03cmthe set of all linear extensions .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P)$, such that .03cm$f(z_i)=c_i$ .03cmfor all $i$. Let .06cm $\psi :\mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q) \to \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P)$ .06cm be the map given by .03cm$\psi(f) = g$, where $$g(s) \ := \ \begin{cases} f(s) & \text{ if } \ \ f(s)<f(u),\\ f(s)-1 & \text{ if } \ \ f(u) < f(s) < f(v),\\ f(s)-2 & \text{ if } \ \ f(v) < f(s) < f(w),\\ f(s)-3 & \text{ if } \ \ f(s)>f(w), \end{cases}$$ for all .03cm$s\in X$. It follows from the definition of .03cm${\textup {\rm lc} } (f)$ .03cmand .03cm${\textup {\rm uc} } (f)$, that $$\mathcal F(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ = \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \, : \, f^{-1}(b-1) \prec x, \hskip.06cm f^{-1}(b+1)=v \big\} \hskip.06cm,$$ It then follows that .03cm$\varphi$ .03cmrestricted to .03cm$\mathcal F(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}})$ .03cmis a bijection onto $$\big\{g \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \, : \, g^{-1}(a+1) \prec x \big\},$$ which gives us $\mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}})=\textrm{M}_2$. Similar arguments gives .06cm $\mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}})=\textrm{M}_1$.03cm. Note that .06cm $\mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}})=0$, because $v$ is always a companion in $f$ but $v \hskip.06cm \| \hskip.06cm x$ by definition. Note also that $$\aligned \mathcal F(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}}) \ & = \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \ : \ f^{-1}(b-1) \hskip.06cm \| \hskip.06cm x, \hskip.06cm f^{-1}(b+1)=v \big\} \\ & \hskip1.cm \cup \ \big\{f \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b) \ : \ f^{-1}(b+1) \hskip.06cm \| \hskip.06cm x, \hskip.06cm f^{-1}(b-1)=v \big\}. \endaligned$$ It then follows that .03cm$\psi$ .03cmrestricted to .03cm$\mathcal F(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}})$ .03cmis a bijection onto $$\big\{g \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a+1) \, : \, g^{-1}(a) \hskip.06cm \| \hskip.06cm x \big\} \ \cup \ \big\{g \in \mathop{\mathrm{\mathcal{E}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(P,x,a) \, : \, g^{-1}(a+1) \hskip.06cm \| \hskip.06cm x \big\},$$ which gives .06cm $\mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}})=2\hskip.03cm\textrm{M}_3$. This finishes proof of the claim. ◻ By the claim, we have: $$\begin{split} \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,c,b) \ &= \ \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ + \ \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}}) \ + \ \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}}) \ + \ \mathop{\mathrm{\textnormal{F}}}(b,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{inc}}})\\ \ &= \ \textrm{M}_2 \ + \ \textrm{M}_1 \ + \ 2\textrm{M}_3. \end{split}$$ Similarly, we have: $$\begin{split} \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b+1) \ &= \ 2 \hskip.03cm\textrm{M}_2 \ + \ 2\hskip.03cm\textrm{M}_3\hskip.03cm, \\ \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b-1) \ &= \ 2 \hskip.03cm\textrm{M}_1 \ + \ 2\hskip.03cm\textrm{M}_3\hskip.03cm. \end{split}$$ We conclude: $$\begin{aligned} & \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b)^2 \hskip.06cm - \hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b+1) \cdot \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b-1) \\ & \hskip1.cm = \hskip.06cm (\textrm{M}_1+\textrm{M}_2+2\textrm{M}_3)^2 \hskip.06cm - \hskip.06cm 4(\textrm{M}_1+\textrm{M}_3)(\textrm{M}_2+\textrm{M}_3) \, = \, (\textrm{M}_1-\textrm{M}_2)^2.\end{aligned}$$ This implies that $$\label{eq:cmkt-2} \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b)^2 \, = \, \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b+1) \cdot \textrm{N}_{\mathop{\mathrm{\mathbf{y}}}\mathop{\mathrm{\mathbf{b}}}}(Q,x,b-1) \quad \Longleftrightarrow \quad \textrm{M}_1=\textrm{M}_2\hskip.03cm.$$ Lemma [Lemma 9](#l:Flat-Sta){reference-type="ref" reference="l:Flat-Sta"} now follows by combining [\[eq:cmkt-1\]](#eq:cmkt-1){reference-type="eqref" reference="eq:cmkt-1"} and [\[eq:cmkt-2\]](#eq:cmkt-2){reference-type="eqref" reference="eq:cmkt-2"}. 0◻ # Flatness from the quadruple relative ratio {#s:Quad-Fla} Recall from Section [4](#s:roadmap){reference-type="ref" reference="s:roadmap"} the key definitions: $$\label{eq:Flat-RLE} \text{\sc FlatLE}_4 \ := \ \big\{\hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{a}}}}(Q,y,b) \hskip.06cm =^? \hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{a}}}}(Q,y,b+1)\hskip.06cm\big\},$$ where .03cm$Q=(Y,\prec)$ is a finite poset on .03cm$[m]$ .03cmelements, .03cm$y\in Y$, .03cm$b \in [m]$, .03cm$\mathop{\mathrm{\mathbf{z}}}\in Y^4$, and .03cm$\mathop{\mathrm{\mathbf{a}}}\in [m]^4$. Also, $$\label{eq:Quart-RLE} \text{\sc QuadRLE} \ := \ \big\{ \hskip.06cm \rho(P_1,x_1) \cdot \rho(P_2,x_2) \, =^? \, \rho(P_3,x_3) \cdot \rho(P_4,x_4) \hskip.06cm \big\},$$ where .03cm$P_1,P_2,P_3,P_4$ .03cmare finite posets and .03cm$x_i\in \min(P_i)$, for all .06cm $1\le i \le 4$. The following result gives a quantitative version[^9] of Lemma [Lemma 11](#l:Quart-Flat){reference-type="ref" reference="l:Quart-Flat"}. **Theorem 17**. *$\text{\sc QuadRLE}$ .06cm reduces to .06cm FlatLE$_4$.03cm. More precisely, let .03cm$P_i=(X_i, \prec_i)$ .03cmbe posets on .03cm$n_i$ .03cmelements, and let .03cm$x_i \in \min(P_i)$, .03cm$1\le i \le 4$. Then there is a polynomial time construction of .03cm$Q=(Y,\prec)$, .03cm$y\in Y$, .03cm$b \in [m]$, .03cm$\mathop{\mathrm{\mathbf{z}}}\in Y^4$, .03cm$\mathop{\mathrm{\mathbf{a}}}\in [m]^4$, where .06cm $m:=|Y| \le 2(n_1+n_2+n_3+n_4)+4$, such that  [\[eq:Flat-RLE\]](#eq:Flat-RLE){reference-type="eqref" reference="eq:Flat-RLE"}  $\Leftrightarrow$  [\[eq:Quart-RLE\]](#eq:Quart-RLE){reference-type="eqref" reference="eq:Quart-RLE"}.* ## Two posets from four {#ss:Quad-Flat-ratio} We now build toward the proof of Theorem [Theorem 17](#thm:gCF){reference-type="ref" reference="thm:gCF"}, via a sequence of reductions which gradually embed four posets $P_i$ into one poset $Q$ as in the theorem. **Lemma 18**. *Let .03cm$P=(X,\prec)$, .03cm$Q=(Y,\prec')$ .03cmbe posets, let .03cm$n=|P|$, .03cm$m:=|Q|$, and let .03cm$x \in \min(P)$, .03cm$y \in \min(Q)$. Then there is a polynomial time construction of a poset .03cm$R=(Z,\prec^\circ)$ .03cm and .03cm$z\in \min(R)$, such that .06cm $|Z|=m+n$, and $$\frac{\textrm{\em N}(R,z,n+1)}{\textrm{\em N}(R,z,n)} \ = \ \frac{\rho(P,x)}{\rho(Q,y)}\,.$$* *Proof.* Let .03cm$P^\ast=(X,\prec^\ast)$ .03cmbe the dual poset of $P$. Define .03cm$R=(Z,\prec^\circ)$ .03cmto be a poset on $$Z \ := \ (X-x) \cup (Y-y) \cup \{w,z\},$$ where .03cm$w,z$ .03cmare two new elements. Let the partial order .06cm $\prec^\circ$ .06cm coincide with .03cm$\prec^\ast$ .03cmon .03cm$(X-x)$, and with .03cm$\prec'$ .03cmon .03cm$(Y-y)$, with additional relations $$\begin{aligned} \label{eq:koko-1} & p \hskip.06cm \prec^\circ \hskip.06cm z \hskip.06cm \prec^\circ \hskip.06cm q\hskip.06cm, \quad \text{ for all } \ \ p \in X-x, \ q \in Y-y,\\ \label{eq:koko-2} & p \hskip.06cm \prec^\circ w \ \ \text{ if and only if } \ \ x \prec p \hskip.06cm, \quad \text{ for all } \ \ p \in X-x, \\ \label{eq:koko-3} & w \hskip.06cm \prec^\circ q \ \ \text{ if and only if } \ \ y \prec' q\hskip.06cm, \quad \text{ for all } \ \ q \in Y-y. \end{aligned}$$ It then follows from a direct calculation that $$\textrm{N}(R,z,n+1) \, = \, e(P) \hskip.06cm \cdot \hskip.06cm e(Q-y).$$ Indeed, by [\[eq:koko-1\]](#eq:koko-1){reference-type="eqref" reference="eq:koko-1"}, for every .06cm $f \in \mathop{\mathrm{\mathcal{N}}}(R,z,n+1)$ .06cm we have: $$\big\{f^{-1}(1),\ldots, f^{-1}(n) \big\} \, = \, X-x +w, \qquad \big\{f^{-1}(n+2),\ldots, f^{-1}(m+n) \big\} \, = \, Y-y\hskip.03cm.$$ These two labelings define a linear extension of .03cm$P^*$ .03cmafter a substitution .06cm $w\gets x$ .06cm given by [\[eq:koko-2\]](#eq:koko-2){reference-type="eqref" reference="eq:koko-2"}, and a linear extension .03cm$Q-y$. By an analogous argument, we have: $$\textrm{N}(R,z,n) \, = \, e(P^*-x) \hskip.06cm \cdot \hskip.06cm e(Q),$$ and the result follows. ◻ We now reduce $\text{\sc QuadRLE}$ to the following intermediate decision problem: $$\label{eq:FlatRatio} %\tag{AF} \text{\sc FlatRatio} \ := \ \Big\{ \hskip.06cm \frac{\textrm{N}(P,x,a+1)}{\textrm{N}(P,x,a)} \, =^? \, \frac{\textrm{N}(Q,y,b+1)}{\textrm{N}(Q,y,b)} \hskip.06cm \Big\},$$ where .03cm$P=(X,\prec)$, .03cm$Q=(Y,\prec')$ .03cmare posets, .03cm$x\in X$, .03cm$y \in Y$, and .03cm$a,b \in \mathbb N$. **Lemma 19**. *$\text{\sc QuadRLE}$ .06cm reduces to .06cm FlatRatio. More precisely, for every .06cm $P_i = (X_i,\prec_i)$ .06cm posets on .06cm $n_i= |X_i|$ .06cm elements, and .03cm$x_i\in \min(P_i)$, .03cm$1\le i \le 4$, there exists a polynomial time construction of a poset .06cm $P=(X,\prec)$ .06cm on .06cm $n:=|X| \le n_1+n_3$ .06cm elements, of a poset .06cm $Q=(Y,\prec')$ .06cm on .06cm $m:=|Y|\le n_2+n_4$ .06cm elements, and of integers .06cm $a\in [n]$, .06cm $b\in [m]$, and such that  [\[eq:Quart-RLE\]](#eq:Quart-RLE){reference-type="eqref" reference="eq:Quart-RLE"}  $\Leftrightarrow$  [\[eq:FlatRatio\]](#eq:FlatRatio){reference-type="eqref" reference="eq:FlatRatio"}.* *Proof.* By Lemma [Lemma 18](#lem:recursion-1){reference-type="ref" reference="lem:recursion-1"}, there exists a poset .03cm$P=(X,\prec)$, .03cm$x\in X$, and .03cm$a \in [n]$, where .03cm$n=|X|=n_1+n_3$.03cm, such that $$\frac{\textrm{N}(P,x,a+1)}{\textrm{N}(P,x,a)} \ = \ \frac{\rho(P_1,x_1)}{\rho(P_3,x_3)}\,.$$ Similarly, there exists a poset $Q=(Y,\prec')$, .03cm$y\in Y$, and .03cm$b \in [m]$, where .03cm$m=|Y|=n_2+n_4$.03cm, such that $$\frac{\textrm{N}(Q,y,b+1)}{\textrm{N}(Q,y,b)} \ = \ \frac{\rho(P_4,x_4)}{\rho(P_2,x_2)}\,.$$ Comparing these, gives the desired reduction and proves the result. ◻ ## One poset from two {#ss:Flat-ratio-to-one} Our next reduction gets us closer to the flatness equality [\[eq:Flat-RLE\]](#eq:Flat-RLE){reference-type="eqref" reference="eq:Flat-RLE"}. **Lemma 20**. *Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$n=|X|$ .03cmelements, let .03cm$Q=(Y,\prec')$ be a poset on .03cm$m=|Y|$ .03cmelements, and let .03cm$x\in X$, .03cm$y \in Q$, .03cm$a\in [n]$, .03cm$b\in [m]$. Then there exists a polynomial time construction of a poset .06cm $R=(Z,\prec^\circ)$ .06cm on .06cm $\ell:=|Z| \le m+n+4$ .06cm elements, of elements .03cm$z\in Z$, .03cm$\mathop{\mathrm{\mathbf{z}}}\in Z^4$, and of integers .03cm$c \in [\ell]$, .03cm$\mathop{\mathrm{\mathbf{c}}}\in [\ell]^4$, such that $$\frac{\textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,n+3)}{\textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,n+2)} \ = \ \bigg(\frac{\textrm{\em N}(P,x,a+1)}{\textrm{\em N}(P,x,a)} + a\bigg) \ \bigg(\frac{\textrm{\em N}(Q,y,b+1)}{\textrm{\em N}(Q,y,b)} + b\bigg)^{-1}.$$* *Proof.* As before, let .03cm$P^\ast = (X, \prec^\ast)$ .03cmbe the poset dual to $P$. Define the poset .03cm$R=(Z,\prec^\circ)$ .03cmon elements $$Z \ := \ X \ \cup \ Y \ \cup \ \{w,z,x',y'\},$$ where .03cm$w,z,x',y'$ .03cmare four new elements. Let the partial order .06cm $\prec^\circ$ .06cm coincide with .03cm$\prec^\ast$ .03cmon .03cm$X$, coincide with .03cm$\prec'$ .03cmon .03cm$Y$, and with additional relations $$\begin{aligned} &p \hskip.06cm \prec^\circ \hskip.06cm z \hskip.06cm \prec^\circ \hskip.06cm q \qquad \text{ for all } \ \ p \in X, \, q \in Y,\\ &x' \hskip.06cm \prec^\circ \hskip.06cm w \hskip.06cm \prec^\circ y' \qquad \text{ and } \qquad x' \hskip.06cm \prec^\circ \hskip.06cm z \hskip.06cm \prec^\circ \hskip.06cm y', \\ &w \hskip.06cm \| \hskip.06cm u \quad \text{for all} \ \ u \in X \cup Y,\\ &x' \ \| \hskip.06cm v \quad \text{for all} \ \ v \in X, \quad \text{and} \quad y' \ \| \hskip.06cm v \quad \text{for all} \ \ v \in Y. \end{aligned}$$ We write $$\begin{aligned} c \, &:= \, n+2 , \quad \mathop{\mathrm{\mathbf{z}}}\hskip.06cm := \, (x',x,y,y'), \quad \text{and}\\ \mathop{\mathrm{\mathbf{c}}}\, &:= \, (c-a-2,c-a,c+b+1,c+b+3). \end{aligned}$$ **Claim 21**. *We have: $$\label{eq:lax-1} \textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c) \, = \, \textrm{\em N}(P,x,a) \cdot \Big( b \hskip.03cm\textrm{\em N}(Q,y,b) \hskip.06cm + \hskip.06cm \textrm{\em N}(Q,y,b+1)\Big).$$* *Proof.* Let .03cm$f \in \mathop{\mathrm{\mathcal{N}}}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c)$. Since .03cm$c=n+2$, we have: $$\big\{f^{-1}(1),\ldots, f^{-1}(c-1)\big\} \, = \, X +x'.$$ By construction, we have .03cm$f(x')=c-a-2=n-a$. Define .06cm $g:X \to [n]$ .06cm as follows: $$g(u) \ := \ \begin{cases} f(u) & \ \text{ if } \quad f(u)<f(x'),\\ f(u)-1 & \ \text{ if } \quad f(u) > f(x'). \end{cases}$$ It is clear that $g\in \mathop{\mathrm{\mathcal{E}}}(P^*)$ .03cmwith the property that .06cm $g(x)=n+1-a$. Thus there are .06cm $\textrm{N}(P^*,x,n+1-a)$ .06cm many such linear extensions $g$. By duality, we have .06cm $\textrm{N}(P^*,x,n+1-a)=\textrm{N}(P,x,a)$. This gives the first factor in [\[eq:lax-1\]](#eq:lax-1){reference-type="eqref" reference="eq:lax-1"}. Similarly, since .03cm$c=n+2$, we have: $$\big\{f^{-1}(c+1),\ldots, f^{-1}(m+n+4)\big\} \, = \, Y \cup \{w,y'\}.$$ There are two cases to consider. First, suppose that .06cm $f(w)<f(y)$. Since .03cm$w$ .03cmis incomparable to every element in $Q$, there are exactly .03cm$b$ .03cmmany choices for the value of .03cm$f(w)$. This gives the factor of $b$ in [\[eq:lax-1\]](#eq:lax-1){reference-type="eqref" reference="eq:lax-1"}. Because .03cm$y'$ .03cmis incomparable to every .03cm$q\in Q$, and since .03cm$f(y')=c+b+3$ is fixed by construction, this implies that .03cm$f$ .03cmcan be uniquely recovered from restrictions of .03cm$f$ .03cmto .03cm$X$, and to .03cm$Y+w$. We now define .06cm $h:Y \to [m]$ .06cm as follows: $$h(v) \ := \ \begin{cases} f(v)-c & \ \text{ if } \quad f(v)<f(w),\\ f(v)-c-1 & \ \text{ if } \quad f(w) < f(v) < f(y'),\\ f(v)-c-2 & \ \text{ if } \quad f(v)>f(y'). \end{cases}$$ Then .03cm$h$ .03cmis a linear extension in .03cm$\mathop{\mathrm{\mathcal{N}}}(Q,y,b)$, which gives the factor of .03cm$\textrm{N}(Q,y,b)$ .03cmin [\[eq:lax-1\]](#eq:lax-1){reference-type="eqref" reference="eq:lax-1"}. For the second case, suppose that $f(w)>f(y)$. Note that we also have $f(w)<f(y')$, since .06cm $w \prec^\circ y'$ .06cm by definition. Hence we have: $$c+b+1 \, = \, f(y) \ < \ f(w) \ < \ f(y') \ = \ c+b+3,$$ which implies $f(w)=c+b+2$. Now defined .06cm $h':Y \to [m]$ .06cm as $$h'(v) \ := \ \begin{cases} f(v) - c & \ \text{ if } \quad f(v)< f(w),\\ f(v) - c-2 & \ \text{ if } \quad f(v)> f(y'). \end{cases}$$ It then follows that .03cm$h'$ .03cm is a linear extension in .03cm$\mathop{\mathrm{\mathcal{N}}}(Q,y,b+1)$, which gives the factor of .03cm$\textrm{N}(Q,y,b+1)$ .03cmin [\[eq:lax-1\]](#eq:lax-1){reference-type="eqref" reference="eq:lax-1"}. This completes the proof of the claim. ◻ By the arguments analogous to the proof of the claim, we also have $$\label{eq:lax-2} \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c+1) \, = \, \textrm{N}(Q,y,b) \cdot \Big( a \hskip.03cm\textrm{N}(P,x,a) \hskip.06cm + \hskip.06cm \textrm{N}(P,x,a+1)\Big).$$ The lemma now follows by combining [\[eq:lax-1\]](#eq:lax-1){reference-type="eqref" reference="eq:lax-1"} and [\[eq:lax-2\]](#eq:lax-2){reference-type="eqref" reference="eq:lax-2"}. ◻ We slightly modify Lemma [Lemma 20](#lem:rec-duo){reference-type="ref" reference="lem:rec-duo"} as follows: **Lemma 22**. *In notation of Lemma [Lemma 20](#lem:rec-duo){reference-type="ref" reference="lem:rec-duo"}, there is a poset .06cm $R=(Z,\prec^\circ)$ .06cm on .06cm $\ell:=|Z| = m+n+ 4 +\max\{a,b\}-\min\{a,b\}$ .06cm elements, with .06cm $c, \hskip.03cmz, \hskip.03cm\mathop{\mathrm{\mathbf{c}}}, \hskip.03cm\mathop{\mathrm{\mathbf{z}}}$ .06cm as in the lemma, and such that $$\frac{\textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c+1)}{\textrm{\em N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c)} \ = \ \bigg(\frac{\textrm{\em N}(P,x,a+1)}{\textrm{\em N}(P,x,a)} \hskip.06cm + \hskip.06cm \max\{a,b\}\bigg) \ \bigg(\frac{\textrm{\em N}(Q,y,b+1)}{\textrm{\em N}(Q,y,b)} \hskip.06cm + \hskip.06cm \max\{a,b\}\bigg)^{-1}.$$* *Proof.* Without loss of generality, assume that .03cm$a \leq b$. Let .03cm$P':= C_{b-a} \oplus P$ .03cmbe the linear sum of .03cm$P$ .03cmwith a chain of .03cm $(b-a)$ .03cmelements, and let .06cm $a':= b$. Then we have: $$\textrm{N}(P',x,a'+1) \, = \, \textrm{N}(P,x,a+1) \quad \text{and} \quad \textrm{N}(P',x,a') \, = \, \textrm{N}(P,x,a).$$ By Lemma [Lemma 20](#lem:rec-duo){reference-type="ref" reference="lem:rec-duo"}, there exists exists a poset .06cm $R=(Z,\prec^\circ)$, .03cm$z \in R$, .03cm$c\in [\ell]$, .06cm $\mathop{\mathrm{\mathbf{z}}}\in Z^{4}$, .06cm $\mathop{\mathrm{\mathbf{c}}}\in [\ell]^{4}$, such that $$\frac{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c+1)}{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c)} \ = \ \bigg(\frac{\textrm{N}(P',x,a'+1)}{\textrm{N}(P',x,a')} \hskip.06cm + \hskip.06cm a'\bigg) \ \bigg(\frac{\textrm{N}(Q,y,b+1)}{\textrm{N}(Q,y,b)} \hskip.06cm + \hskip.06cm b\bigg)^{-1},$$ and where .06cm $\ell = |Z| = m+n+4+b-a$. This proves the lemma. ◻ ## Reduction to flatness {#ss:Flat-ratio-proof} We are now ready to prove Theorem [Theorem 17](#thm:gCF){reference-type="ref" reference="thm:gCF"} which extends Lemma [Lemma 11](#l:Quart-Flat){reference-type="ref" reference="l:Quart-Flat"}. **Lemma 23**. *.06cm FlatRatio .06cm reduces to .06cm FlatLE$_4$.03cm. More precisely, suppose we have a poset .03cm$P=(X,\prec)$ .03cmon .03cm$n=|X|$ .03cmelements, a poset .03cm$Q=(Y,\prec')$ .03cmon .03cm$m=|Y|$ .03cmelements, elements .03cm$x\in X$, .03cm$y\in Y$, and integers .03cm$a\in [n]$, .03cm$b\in [m]$. Then there exists a polynomial time construction of a poset .06cm $R=(Z,\prec^\circ)$ .06cm on .06cm $\ell:=|Z| \le 2(m+n)+4$ .06cm elements, of elements .03cm$z\in Z$, .03cm$\mathop{\mathrm{\mathbf{z}}}\in Z^4$, and of integers .03cm$c \in [\ell]$, .03cm$\mathop{\mathrm{\mathbf{c}}}\in [\ell]^4$, and such that  [\[eq:Flat-RLE\]](#eq:Flat-RLE){reference-type="eqref" reference="eq:Flat-RLE"}  $\Leftrightarrow$  [\[eq:FlatRatio\]](#eq:FlatRatio){reference-type="eqref" reference="eq:FlatRatio"}.* *Proof.* In notation of Lemma [Lemma 22](#lem:rec-duo-equal){reference-type="ref" reference="lem:rec-duo-equal"}, we have: $$\frac{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c+1)}{\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c)} \ = \ \bigg(\frac{\textrm{N}(P,x,a+1)}{\textrm{N}(P,x,a)} + \max\{a,b\}\bigg) \ \bigg(\frac{\textrm{N}(Q,y,b+1)}{\textrm{N}(Q,y,b)} + \max\{a,b\}\bigg)^{-1},$$ for a poset .03cm$R=(Z,\prec^\circ)$ .03cmon .03cm$\ell=|Z|$ .03cmelements, where $$\ell \, = \, m+n+ 4 +\max\{a,b\}-\min\{a,b\} \, \leq \, 2m \hskip.06cm + \hskip.06cm 2n \hskip.06cm + \hskip.06cm 4.$$ Observe that the equality $$\textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c+1) \hskip.06cm = \hskip.06cm \textrm{N}_{\mathop{\mathrm{\mathbf{z}}}\mathop{\mathrm{\mathbf{c}}}}(R,z,c)$$ is equivalent to $$\frac{\textrm{N}(P,x,a+1)}{\textrm{N}(P,x,a)} \ = \ \frac{\textrm{N}(Q,y,b+1)}{\textrm{N}(Q,y,b)}\,.$$ This proves the result. ◻ *Proof of Theorem [Theorem 17](#thm:gCF){reference-type="ref" reference="thm:gCF"}.* The result follows from Lemma [Lemma 19](#lem:reduction-1){reference-type="ref" reference="lem:reduction-1"} and Lemma [Lemma 23](#lem:reduction-2){reference-type="ref" reference="lem:reduction-2"}. ◻ # Verification lemma {#s:verify} The proof of the Verification Lemma [Lemma 12](#l:Verify-Quart){reference-type="ref" reference="l:Verify-Quart"} is different from other reductions which are given by parsimonious bijections. Before proceeding to the proof, we need several technical and seemingly unrelated results. ## Continuous fractions {#ss:verify-CF} Given .06cm $a_0\geq 0$.06cm, .06cm $a_1, \ldots, a_s \in \mathbb \mathbb Z_{\ge 1}$.06cm, where .03cm$s \geq 0$, the corresponding *continued fraction* .03cmis defined as follows: $$[a_0\hskip.03cm; \hskip.03cma_1,\ldots, a_s] \ := \ a_0 \hskip.06cm + \hskip.06cm \cfrac{1}{a_1 \hskip.06cm + \hskip.06cm \cfrac{1}{\ddots \ + \hskip.06cm \frac{1}{a_s}}}$$ Numbers .03cm$a_i$ .03cmare called .03cm*quotients*, see e.g. [@HardyWright $\S$`<!-- -->`{=html}10.1]. We refer to [@Knuth98 $\S$`<!-- -->`{=html}4.5.3] for a detailed asymptotic analysis of the quotients in connection with the Euclidean algorithm, and further references. The following technical result is key in the proof of the Verification Lemma. **Proposition 24** (cf. [@KS21 $\S$`<!-- -->`{=html}3]). *Let .06cm $a_0, \ldots, a_s \in \mathbb \mathbb Z_{\ge 1}$.06cm. Then there exists a poset .03cm$P=(X,\prec)$ .03cmof width two on .06cm $|X|=a_0+\ldots +a_s$ .06cm elements, and element .03cm$x \in \min(P)$, such that $$\rho(P,x) \ = \ [a_0 \hskip.03cm; \hskip.03cma_1,\ldots,a_s].$$* **Corollary 25**. *Let .06cm $a_1,\ldots a_s \in \mathbb \mathbb Z_{\ge 1}$. Then there exists a width two poset .03cm$P=(X,\prec)$ .03cm on .06cm $|X|=a_1+\ldots +a_s$ .06cm elements, and element .03cm$x \in \min(P)$, such that $$\frac{1}{\rho(P,x)} \ = \ [0\hskip.03cm; \hskip.03cma_1,\ldots,a_s].$$* *Proof.* This follows from .06cm $[a_1; \hskip.03cma_2,\ldots, a_s] \hskip.03cm= \hskip.03cm[0\hskip.03cm; \hskip.03cma_1,\ldots, a_s]^{-1}$. ◻ **Remark 26**. Proposition [Proposition 24](#prop:genheight-cf){reference-type="ref" reference="prop:genheight-cf"} was proved implicitly in [@KS21 $\S$`<!-- -->`{=html}3]. Unfortunately, the notation and applications in that paper are very different from ours, so we chose to include a self-contained proof for completeness. Note that the next two lemmas are also contained in the argument of [@KS21 $\S$`<!-- -->`{=html}3]. **Lemma 27**. *Let .03cm$P=(X,\prec)$ .03cmbe a width two poset on .03cm$n=|X|$ elements, and let .03cm$x\in \min(P)$. Then there exists a width two poset .03cm$Q=(Y,\prec')$ .03cmand .03cm$y \in \min(Q)$, such that .03cm$|Y|=n+1$ .03cmand $$\rho(Q,y) \ = \ 1 \, + \, \frac{1}{\rho(P,x)}\,.$$* *Proof.* Let .03cm$Y:=X+z$, and let .03cm$\prec'$ .03cmcoincide with .03cm$\prec$ .03cmon $P$, with added relations $$\begin{aligned} z \hskip.03cm\prec' \hskip.03cmu \quad \text{for all} \ \ u \in X-x \hskip.03cm, \quad \text{and} \quad z \hskip.03cm\| \hskip.03cmx. \end{aligned}$$ Note that .03cm$z\in \min(Q)$ .03cmand that .03cm$Q$ .03cmalso has width two. Note also that $$e(Q-z) \, = \, e(P) \quad \text{and} \quad e(Q) \, = \, e(P) \hskip.06cm + \hskip.06cm e(P-x),$$ since for every .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(Q)$ .03cmwe either have .03cm$f(z)=1$, or .03cm$f(z)=2$ and thus .03cm$f(x)=1$. We now take .06cm $y \gets z$, and observe that $$\rho(Q,y) \ = \ \frac{e(Q)}{e(Q-z)} \ = \ \frac{e(P) + e(P-x)}{e(P)} \ = \ 1 \, + \, \frac{1}{\rho(P,x)} \,,$$ as desired. ◻ **Lemma 28**. *Let .03cm$P=(X,\prec)$ .03cmbe a width two poset on .03cm$n=|X|$ elements, and let .03cm$x\in \min(P)$. Then there exists a width two poset .03cm$Q=(Y,\prec')$ .03cmand .03cm$y \in \min(Q)$, such that .03cm$|Y|=n+1$ .03cmand $$\rho(Q,y) \ = \ 1 + \rho(P,x).$$* *Proof.* Let $Q$ be as in the proof of Lemma [Lemma 27](#lem:KS-1){reference-type="ref" reference="lem:KS-1"}. Note that .03cm$x\in \min(Q)$, and that $$e(Q-x) \, = \, e(P-x),$$ since .03cm$z$ .03cmis the unique minimal element in .03cm$Q-x$. We now take .06cm $y \gets x$, and observe that $$\rho(Q,y) \ = \ \frac{e(Q)}{e(Q-x)} \ = \ \frac{e(P)+e(P-x)}{e(P-x)} \ = \ 1 \hskip.06cm + \hskip.06cm \rho(P,x),$$ as desired. ◻ **Corollary 29**. *Let .03cm$P=(X,\prec)$ .03cmbe a width two poset on .03cm$n=|X|$ elements, let .03cm$x\in \min(P)$, and let .03cm$a \in \mathbb \mathbb Z_{\ge 1}$. Then there exists a width two poset .03cm$Q=(Y,\prec')$ .03cmand .03cm$y \in \min(Q)$, such that .03cm$|Y|=n+a$ .03cmand $$\rho(Q,y) \ = \ a \hskip.06cm + \hskip.06cm \frac{1}{\rho(P,x)}.$$* *Proof.* Use Lemma [Lemma 27](#lem:KS-1){reference-type="ref" reference="lem:KS-1"} once, and Lemma [Lemma 28](#lem:KS-2){reference-type="ref" reference="lem:KS-2"} .06cm $(a-1)$ .06cm times. ◻ *Proof of Proposition [Proposition 24](#prop:genheight-cf){reference-type="ref" reference="prop:genheight-cf"}.* We use induction on .03cm$s$. For .03cm$s=0$, let .06cm $P := C_{a_0-1} + \{x\}$ .06cm be a disjoint sum of two chains, and observe that .06cm $\rho(P,x) = a_0$. Suppose the claim holds for .03cm$s-1$, i.e. there exists a poset .03cm$P_1$ .03cm on .06cm $n=a_1+\ldots+a_s$ .06cm elements and .03cm $x_1 \in \min(P_1)$, such that .06cm $\rho(P_1,x_1) \hskip.03cm= \hskip.03cm[a_1 \hskip.03cm; \hskip.03cma_2,\ldots, a_s]$, and with .06cm $|P_1|=a_1+\ldots+a_s$. By Corollary [Corollary 29](#cor:KS){reference-type="ref" reference="cor:KS"}, there exists a poset .03cm$Q$ .03cmon .03cm$a_0+n$ .03cmelements, and .03cm$x \in \min(P)$, such that $$\begin{aligned} \rho(P,x) \, = \, a_0 \, + \, \frac{1}{\rho(P_1,x_1)} \, = \, a_0 \, + \, \frac{1}{[a_1 \hskip.03cm; \hskip.03cma_2,\ldots, a_s]} \, = \, [a_0 \hskip.03cm; \hskip.03cma_1,\ldots,a_s]. \end{aligned}$$ This completes the proof. ◻ ## Number theoretic estimates {#ss:verify-NT} For .03cm$A\in \mathbb \mathbb Z_{\ge 1}$ .03cmand .03cm$m \in [A]$, consider the quotients in the continued fraction of .03cm$m/A$ .03cmand their sum: $$\frac{m}{A} \, = \, [0 \hskip.03cm; \hskip.03cma_1(m),\ldots, a_s(m)] \quad \text{and} \quad S_A(m) \, := \, \sum_{i=1}^s \hskip.06cm a_i(m)\hskip.06cm.$$ The following technical result will also be used in the proof of the Verification Lemma [Lemma 12](#l:Verify-Quart){reference-type="ref" reference="l:Verify-Quart"}. **Proposition 30**. *There exists a constant .03cm$C>0$, such that for all coprime integers .03cm$A,B$ .03cmwhich satisfy .06cm $C < B < A < 2B$, there exists an integer .06cm $m:=m(A,B)$ .06cm such that .06cm $m<B$, $$S_A(m) \ \leq \ (\log A)^3 \quad \text{ and } \quad S_B(m) \ \leq \ (\log B)^3.$$* We now build toward the proof of this result. For .03cm$n\in \mathbb \mathbb Z_{\ge 1}$, let $$\textmd{\em J}_n \, := \, \big\{\hskip.03cmk \ : \ (k,n)=1, \ 1\le k \le n \hskip.03cm\big\}$$ denote the set of integers smaller than $n$ and are coprime to $n$. Let .03cm $\varphi(n):=|\textmd{\em J}_n|$ .03cmdenote .03cm*Euler's totient function* of $n$. We need several technical results. **Lemma 31** (Yao--Knuth [@YK75]). *We have: $$\frac{1}{\varphi(n)} \hskip.06cm \sum_{m \hskip.03cm\in \hskip.03cm\textmd{\em J}_n} S_n(m) \ = \ \frac{6}{\pi^2} \hskip.06cm (\log n)^2 \hskip.06cm + \hskip.06cm O\big((\log n) (\log \log n)^2\big) \quad \text{as \ \ $n\to \infty$}.$$* By the Markov inequality, it follows from Lemma [Lemma 31](#lem:Knuth){reference-type="ref" reference="lem:Knuth"} that $$\label{eq:Knuth} \big|\big\{ m \in \textmd{J}_n \, : \, S_n(m) \hskip.06cm \geq \hskip.06cm (\log n)^3 \big\}\big| \, \leq \, O(\log n)^2 \hskip.06cm \frac{\varphi(n)}{( \log n)^3} \, = \, O\Big(\frac{\varphi(n)}{\log n}\Big).$$ **Lemma 32** (see e.g. [@HardyWright Thm 328]). *We have: $$\label{eq:phi-low-bound} \liminf_{n \to \infty} \hskip.06cm \frac{\varphi(n) \hskip.06cm \log \log n}{n} \,\hskip.06cm = \,\hskip.06cm e^{-\gamma}\hskip.06cm,$$ where .06cm $\gamma=-\Gamma'(1)\approx 0.5772156649$ .06cm is the *Euler--Mascheroni constant*.* **Lemma 33**. *Let .06cm $B < A<2B$. Then $$\label{eq:stex} \big| \textmd{J}_{AB} \cap [B]\big| \, \geq \, \frac{\varphi(AB)}{A} \hskip.06cm \big(1 - o(1)\big) \quad \ \text{as} \ \hskip.06cm \ A\to \infty.$$* *Proof.* Denote by .06cm $d(n) := \big|\big\{a \hskip.06cm:\hskip.06cm a\hskip.03cm| \hskip.03cmn\big\}\big|$ .06cm the number of divisors of $n$. By the Möbius inversion (see e.g. [@HardyWright $\S$`<!-- -->`{=html}16.4]), we have: $$\big| \textmd{J}_{AB} \cap [B]\big| \, = \, \sum_{a \hskip.03cm\mid \hskip.03cmAB} \hskip.06cm \mu(a) \hskip.03cm\left\lfloor \tfrac{B}{a} \right\rfloor \, \geq \, \sum_{a \hskip.03cm\mid \hskip.03cmAB} \hskip.06cm \mu(a) \hskip.06cm \tfrac{B}{a} \hskip.06cm - \hskip.06cm d(AB) \, = \, \tfrac{\varphi(AB)}{A} \hskip.06cm - \hskip.06cm d(AB),$$ where the inequality follows from .03cm$|\mu(a)|\le 1$. Recall that .06cm $d(n) = O(n^\delta)$ .06cm for all .03cm$\delta>0$, see e.g. [@HardyWright Thm 315]. Thus, .03cm $d(AB) = O(\sqrt{A})$. On the other hand, by [\[eq:phi-low-bound\]](#eq:phi-low-bound){reference-type="eqref" reference="eq:phi-low-bound"} we have .06cm $\tfrac{1}{A}\hskip.06cm\varphi(AB) \hskip.06cm = \hskip.06cm \Omega\big( \tfrac{A}{ \log \log A} \big)$. This implies the result. ◻ *Proof of Proposition [Proposition 30](#p:NTD){reference-type="ref" reference="p:NTD"}.* We will show that $$\begin{aligned} \vartheta(A,B) \, := \, \big|\{ m \in \textmd{J}_A \cap \textmd{J}_B \hskip.06cm \mid \hskip.06cm S_A(m) \leq (\log A)^3, \ S_B(m) \leq (\log B)^3 \ \} \big| \, = \, \Omega\big(\tfrac{A}{\log \log A}\big) \end{aligned}$$ as .03cm$A\to \infty$. This implies the result. Note that $$\begin{aligned} & \hskip3.cm \vartheta(A,B) \, \ge \, D \hskip.06cm - \hskip.06cm E \hskip.06cm - \hskip.06cm F, \quad \text{where} \quad D \hskip.06cm := \hskip.06cm \big|\textmd{J}_A \cap \textmd{J}_B\big|\hskip.03cm, \\ & E \, := \, \big|\{ m \in \textmd{J}_A \hskip.06cm \mid \hskip.06cm S_A(m) \geq (\log A)^3 \} \big| %\hskip1.cm \qquad \text{and} \qquad F \, := \, \big|\{ m \in \textmd{J}_B \hskip.06cm \mid \hskip.06cm S_B(m) \geq (\log B)^3 \} \big|\hskip.03cm.\end{aligned}$$ Since $(A,B)=1$, we have .03cm$\varphi(AB)=\varphi(A)\varphi(B)$ .03cmand .06cm $J_A \cap J_B = J_{AB} \cap [B]$. We have: $$%\label{eq:keihan-2} D \, = \, \big|\textmd{J}_{AB} \cap [B] \big| \, \geq_{\eqref{eq:stex}} \, \tfrac{\varphi(AB)}{A} (1-o(1)) \, = \, \tfrac{\varphi(A) \hskip.03cm\varphi(B)}{A} (1-o(1)) \, =_{\eqref{eq:phi-low-bound}} \, \Omega(\tfrac{\varphi(A)}{\log \log A}).$$ On the other hand, by [\[eq:Knuth\]](#eq:Knuth){reference-type="eqref" reference="eq:Knuth"}, we have .06cm $E = O\big(\tfrac{\varphi(A)}{\log A}\big) = o(D)$. Similarly, we have .03cm$F = o(D)$. This implies the result. ◻ **Remark 34**. The proof of Proposition [Proposition 30](#p:NTD){reference-type="ref" reference="p:NTD"} does not give a (deterministic) polynomial time algorithm to find the desired $m$, i.e. in .03cmpoly$\hskip.03cm(\log A)$ .03cmtime. There is, however, a relatively simple *probabilistic* .03cmpolynomial time algorithm, cf. [@CP23 Remark 5.31]. ## Bounds on relative numbers of linear extensions {#ss:verify-relative-bounds} The following simple bound is the final ingredient we need for the proof of the Verification Lemma. **Proposition 35** (see [@CPP-Quant-CPC; @EHS]). *Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$|X|=n$ .03cmelements, and let .03cm$x\in \min(X)$. Then .06cm $1 \leq \rho(P,x) \leq n$. Moreover, .03cm$\rho(P,x)=1$ .03cmif an only if .03cm$\min(P)=\{x\}$, i.e. .03cm$x$ .03cmis the unique minimal element.* The lower bound holds for all .03cm$x\in X$, see e.g. [@EHS]. The upper bound is a special case of .03cm[@CPP-Quant-CPC Lem. 5.1]. We include a short proof for completeness. *Proof.* The lower bound .06cm $e(P-x) \le e(P)$ .06cm follows from the injection .06cm $\mathop{\mathrm{\mathcal{E}}}(P-x) \to \mathop{\mathrm{\mathcal{E}}}(P)$ .06cm that maps .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P-x)$ .03cminto .03cm$g\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cmby letting .03cm$g(x)\gets 1$, .06cm $g(y)\gets f(x)+1$ .06cm for all .03cm$y\ne x$. For the second part, note that .06cm $e(P)-e(P-x)$ .06cm is the number of .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cm such that .03cm$f(x) >1$, so .06cm $e(P)-e(P-x)=0$ .06cm implies .06cm $\min(P)=\{x\}$. The upper bound .06cm $e(P) \le n \hskip.03cme(P-x)$ .06cm follows from the injection .06cm $\mathop{\mathrm{\mathcal{E}}}(P) \to \mathop{\mathrm{\mathcal{E}}}(P-x) \times [n]$ .06cm that that maps .03cm$g\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cminto a pair .03cm$\big(f, g(x)\big)$ .03cmwhere .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P-x)$ .03cm is defined as .03cm$f(y)\gets g(y)$ .03cmif .03cm$g(y)<g(x)$, .06cm $f(y)\gets g(y)-1$ .03cmif .03cm$g(y)>g(x)$. ◻ ## Proof of Verification Lemma [Lemma 12](#l:Verify-Quart){reference-type="ref" reference="l:Verify-Quart"} {#ss:verify-proof} Recall the decision problem $$%\label{eq:AF} \tag{AF} \text{\sc VerRLE} \ := \ \big\{ \hskip.06cm \rho(P,x) \hskip.06cm =^? \, \tfrac{A}{B} \hskip.06cm \big\},$$ where .03cm$P=(X,\prec)$ .03cmis a poset on .03cm$n=|X|$ .03cmelements, .03cm$x\in \min(P)$, and .03cm$A,B$ are coprime integers with .03cm$B< A\leq n!$.06cm. We simulate .06cm VerRLE .06cm with an oracle for .06cm QuadRLE .06cm as follows. By Proposition [Proposition 35](#p:g-bound){reference-type="ref" reference="p:g-bound"}, we need only to consider the cases .06cm $1 < \frac{A}{B} \leq n$. Indeed, when .03cm$\rho(P,x)<1$ .03cmor .03cm$\rho(P,x)>n!$, .06cm VerRLE .03cmdoes not hold. Additionally, when .03cm$\rho(P,x)=1$, .06cm VerRLE .03cmholds if and only if .03cm$P$ .03cmis a chain. Let .03cm$k := \left\lfloor \tfrac{A}{B} \right \rfloor$. As in the $s=0$ part of the proof of Proposition [Proposition 24](#prop:genheight-cf){reference-type="ref" reference="prop:genheight-cf"}, there exists a poset .03cm$P_3=(X_3,\prec_3)$ .03cmwith $|X_3| = k \le n$, and an element .03cm$x_3 \in \min(P_3)$, such that .06cm $\rho(P_3,x_3) = k$. Let $A',B'$ be coprime integers such that $$\frac{A}{B} \, = \, k \hskip.06cm \frac{A'}{B'}\,.$$ Then we have .06cm $B\le B'<A'<2B'$, .06cm $A'\le A$ .06cm and thus .06cm $\log A' = O(n \log n)$. By Proposition [Proposition 30](#p:NTD){reference-type="ref" reference="p:NTD"}, there is a positive integer .06cm $m \in [B']$, such that $$S_{A'}(m) \ \leq \ (\log A')^3 \quad \text{ and } \quad S_{B'}(m) \ \leq \ (\log B')^3.$$ At this point we *guess* .03cmsuch $m$. Since computing the quotients of .03cm$m/A'$ .03cm can be done in polynomial time, we can verify in polynomial time that .03cm$m$ .03cmsatisfies the inequalities above. By Corollary [Corollary 25](#cor:genheight-cf){reference-type="ref" reference="cor:genheight-cf"}, we can construct posets .03cm$P_2=(X_2,\prec_2)$, .03cm$P_4=(X_4,\prec_4)$ .03cmwith .03cm $x_2\in \min(P_2)$, .03cm$x_4\in \min(P_2)$, such that $$\rho(P_2,x_2) \, = \, \frac{B'}{m} \qquad \text{and} \qquad \rho(P_4,x_4) \, = \, \frac{A'}{m} \,.$$ The corollary also gives us $$|X_2| \, \leq \, S_{B'}(m) \, \leq \, (\log B')^3 \, = \, O\big(n^3 (\log n)^3\big),$$ and we similarly have .06cm $|X_4| = O\big(n^3 (\log n)^3\big)$. Since posets .03cm$P_2,P_3,P_4$ .03cmhave polynomial size, we can call .03cmQuadRLE .03cm to check $$%\label{eq:AF} \tag{AF} \big\{ \hskip.06cm \rho(P,x) \cdot \rho(P_2,x_2) \, =^? \, \rho(P_3,x_3) \cdot \rho(P_4,x_4) \hskip.06cm \big\}.$$ Observe that $$\frac{\rho(P_3,x_3) \cdot \rho(P_4,x_4)}{\rho(P_2,x_2)} \ = \ \frac{m}{B'} \hskip.06cm \cdot \hskip.06cm k \hskip.06cm \cdot \hskip.06cm \frac{A'}{m} \ = \ \frac{A}{B}\,.$$ Thus, in this case .03cmQuadRLE .03cmis equivalent to .03cmVerRLE, as desired. 0◻ # Fixing one element {#s:Sta1} In this section we prove Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"}. The proof relies heavily on [@MS22]. We also need the definition and basic properties of the *promotion* .03cmand *demotion* .03cmoperations on linear extensions, see e.g. [@Sta-promo] and [@Sta-EC $\S$`<!-- -->`{=html}3.20]. ## Explicit equality conditions {#ss:Sta1-equality} For .03cm$k=1$, the equality cases of Stanley's inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} are tuples .06cm $(P,x,z,a,c)$, .03cmwhere .03cm$P=(X,\prec)$ .03cmis a poset on .03cm$n=|X|$ .03cmelements, .03cm$x,z\in X$, .03cm$a,c \in [n]$, and the following holds: $$\label{eq:Sta1} \textrm{N}_{z\hskip.03cmc}(P, x,a)^2 \, = \, \textrm{N}_{z\hskip.03cmc}(P, x,a+1)\cdot \textrm{N}_{z\hskip.03cmc}(P, x,a-1).$$ The subscripts here and throughout this section are no longer bold, to emphasize that .03cm$k=1$. Recall also both the notation in $\S$[1.4](#ss:intro-Stanley){reference-type="ref" reference="ss:intro-Stanley"}, and the Ma--Shenfeld poset notation in $\S$[6.1](#ss:main-MS){reference-type="ref" reference="ss:main-MS"}. **Lemma 36**. *Let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$n=|X|$ .03cmelements, let .03cm$x,z\in X$, .03cm$a,c\in [n]$. Then the equality .03cm[\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} .03cmis equivalent to:* *$(\divideontimes)$ for every .03cm$f \in \mathop{\mathrm{\mathcal{E}}}_{z\hskip.03cmc}(P,x,a+s)$, .03cm$s \in \{0,\pm 1\}$, we have .06cm $x \hskip.06cm \| \, {\textup {\rm lc} } (f)$ .06cm and .06cm $x \hskip.06cm \| \, {\textup {\rm uc} } (f)$.* We prove Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"} later in this section. **Remark 37**. For the case $k=0$, the analogue of $(\divideontimes)$ that companions of $f$ are incomparable to $x$, was proved in [@SvH-acta Thm 15.3(c)]. However, $(\divideontimes)$ fails for .03cm$k= 2$, as shown in the "hope shattered" Example 1.4 in [@MS22]. Thus, Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"} closes the gap between these two results. See $\S$[10.7](#ss:finrem-LE){reference-type="ref" reference="ss:finrem-LE"} for potential complexity implications of this observation. Note also that condition $(\divideontimes)$ is in $\hskip.03cm\textup{\textup{\textsf{P}}}$ .03cmsince can be equivalently described in terms of explicit conditions on the partial order (rather than in terms of linear extensions of the poset). This is proved in [@SvH-acta Thm 15.3.03cm(d)] for .03cm$k=0$, and in [@MS22 Eq. (1.6)] for .03cm$k=1$. *Proof of Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"}.* As before, let .03cm$P=(X,\prec)$ .03cmbe a poset on .03cm$n=|X|$ .03cmelements, let .03cm$x,y,z\in X$ .03cmand .03cm$a,b,c\in [n]$. Denote by .06cm $\textrm{N}_{z c}(P,x,a,y,b)$ .06cm the number of linear extensions .06cm $f \in \mathop{\mathrm{\mathcal{E}}}_{z \hskip.03cmc}(P,x,a)$ .03cmthat additionally satisfy .03cm$f(y)=b$. Now, condition $(\divideontimes)$ in Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"}, can be rewritten as follows: $$\label{eq:sapporo-1} \textrm{N}_{z \hskip.03cmc}(P,x,a',y,b') \hskip.06cm = \hskip.06cm 0 \quad \text{for all} \quad y \in \mathcal C(x) \ \ \ \text{and} \ \ \ a',b' \in \{a-1,a,a+1\}.$$ Indeed, each vanishing condition in [\[eq:sapporo-1\]](#eq:sapporo-1){reference-type="eqref" reference="eq:sapporo-1"} is checking whether there exists a companion of $x$ in a linear extension that is comparable to $x$. Recall that each vanishing condition in [\[eq:sapporo-1\]](#eq:sapporo-1){reference-type="eqref" reference="eq:sapporo-1"} is in $\textup{\textup{\textsf{P}}}$, see references in $\S$[3.5](#ss:hist-Stanley){reference-type="ref" reference="ss:hist-Stanley"}. There are at most .03cm$6n$ .03cminstances to check, since for all .03cm$y\in X$ .03cmthere are at most $6$ choices of distinct .03cm$a',b'$ .03cmin .03cm$\{a-1,a,a+1\}$. Therefore, .03cmEqualityStanley$_{1} \in \textup{\textup{\textsf{P}}}$. ◻ ## Ma--Shenfeld theory {#ss:Sta1-critical} We now present several ingredients needed to prove Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"}. We follow closely the Ma--Shenfeld paper [@MS22], presenting several results from that paper. In [@MS22], Ma--Shenfeld defined the notions of *subcritical*, *critical*, and *supercritical* .03cmposets, which are directly analogous to the corresponding notions for polytopes given in [@SvH-acta], cf. $\S$[3.2](#ss:hist-AF){reference-type="ref" reference="ss:hist-AF"}. As the precise definitions are rather technical, we will not state them here while still including key properties of those families that are needed to prove Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"}. We start with the following hierarchical relationship between the three families: $$\{\text{subcritical posets}\} \quad \supseteq \quad \{\text{critical posets}\} \quad \supseteq \quad \{\text{supercritical posets}\}.$$ A poset that is subcritical but not critical is called *sharp subcritical*, and a poset that is critical but not super critical is called *sharp critical*. The equality conditions for [\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} are directly determined by the classes to which the poset $P$ belongs, as we explain below. We note that these families depend on the choices of .03cm$P,x,a,z,c$, which we omit from the notation to improve readability. Furthermore, without loss of generality we can assume that .06cm $z \notin \{a-1,a,a+1\}$, as otherwise one of the numbers in [\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} are equal to $0$, making the problem in $\textup{\textup{\textsf{P}}}$ (see above). We now state two other properties of these families, which require the following definitions. Following [@MS22], we add two elements .06cm $z_0,z_{k+1}$ .06cm into the poset such that .06cm $z_0 \preccurlyeq y \preccurlyeq z_{k+1}$ .06cm for all .03cm$y \in X$, and we define .06cm $c_0:=0$ .06cm and .06cm $c_{k+1}:=n+1$.06cm. A *splitting pair* .03cmis a pair of integers .03cm$(r,s)$ .03cmin .03cm$\{0,\ldots, k+1\}$, such that .06cm $(r,s)\neq (0,k+1)$.06cm.[^10] **Lemma 38** ([@MS22 Lemma 5.10]). *Let .03cm$P=(X,\prec)$ .03cmbe a sharp subcritical poset. Then there exists a splitting pair .03cm$(r,s)$ .03cmsuch that $$\label{eq:split} \big|\big\{ \hskip.03cmu \in X \, : \, z_r \hskip.03cm\prec \hskip.03cmu \hskip.03cm\prec \hskip.03cmz_s \hskip.03cm\big\}\big| \, = \, c_s\hskip.06cm - \hskip.06cm c_r \hskip.06cm - \hskip.06cm 1.$$* We say that poset .03cm$P$ .03cmis *split indecomposable* .03cmif, for every splitting pair $(r,s)$, $$\big|\big\{ \hskip.03cmu \in X \, : \, z_r \hskip.03cm\prec \hskip.03cmu \hskip.03cm\prec \hskip.03cmz_s \hskip.03cm\big\}\big| \, \leq \, c_s\hskip.06cm - \hskip.06cm c_r \hskip.06cm - \hskip.06cm 2.$$ In particular, by Lemma [Lemma 38](#lem:subcrit-prop){reference-type="ref" reference="lem:subcrit-prop"} every sharp subcritical poset is not split indecomposable. It was shown in [@MS22], that we can without loss of generality assume that poset .03cm$P$ .03cmis split indecomposable. Indeed, otherwise checking [\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} can be reduced to checking the same problem for a smaller poset: either restricting to the set in [\[eq:split\]](#eq:split){reference-type="eqref" reference="eq:split"}, or removing this set from the poset, see [@MS22 $\S$`<!-- -->`{=html}6] for details. Thus we can without loss of generality assume that .03cm$P$ .03cmis a critical poset. **Lemma 39** ([@MS22 Lemma 5.11]). *Let .03cm$P$ .03cmbe a split indecomposable sharp critical poset. Then there exists a splitting pair .03cm$(r,s)$ .03cmsuch that .06cm $c_r<a<c_s$ .06cm and $$\label{eq:split-2} \big|\big\{ \hskip.03cmu \in X \, : \, z_r \hskip.03cm\prec \hskip.03cmu \hskip.03cm\prec \hskip.03cm z_s \hskip.03cm\big\}\big| \, = \, c_s\hskip.06cm - \hskip.06cm c_r \hskip.06cm - \hskip.06cm 2.$$* **Remark 40**. Lemmas [Lemma 38](#lem:subcrit-prop){reference-type="ref" reference="lem:subcrit-prop"} and [Lemma 39](#lem:crit-prop){reference-type="ref" reference="lem:crit-prop"} can be modified to imply that deciding whether poset .03cm$P$ .03cmis subcritical, critical, or supercritical is in $\textup{\textup{\textsf{P}}}$. We do not need this result for the proof of Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"}, so we omit these changes to stay close to the presentation in [@MS22]. More generally, one can ask similar questions for H-polytopes. While we believe that for TU-polytopes these decision problems are still likely to be in .03cm$\textup{\textup{\textsf{P}}}$, proving that would already be an interesting challenge beyond the scope of this paper. Recall from §[6.2](#s:Flat-Sta-proof){reference-type="ref" reference="s:Flat-Sta-proof"} that .06cm $\mathcal F(a,\mathop{\mathrm{\text{com}}}, \mathop{\mathrm{\text{com}}})$ .06cm is the set of linear extensions in .06cm $\mathop{\mathrm{\mathcal{E}}}_{z \hskip.03cmc}(P,x,a)$, such that both the lower and upper companions of $x$ are incomparable to $x$. Next, .06cm $\mathcal F(a,\mathop{\mathrm{\text{com}}}, \mathop{\mathrm{\text{inc}}})$ .06cm is the set of linear extensions in .06cm $\mathop{\mathrm{\mathcal{E}}}_{z \hskip.03cmc}(P,x,a)$, such that the lower companion is comparable to $x$, but the upper companion is incomparable to $x$. Similarly, .06cm $\mathcal F(a,\mathop{\mathrm{\text{inc}}}, \mathop{\mathrm{\text{com}}})$ .06cm is the set of linear extensions in .06cm $\mathop{\mathrm{\mathcal{E}}}_{z \hskip.03cmc}(P,x,a)$, such that the lower companion is incomparable to $x$, but the upper companion is comparable to $x$. Let .06cm $\mathcal F(a-1,\cdot, \cdot)$ .06cm and .06cm $\mathcal F(a+1,\cdot, \cdot)$ .06cm be defined analogously. Finally, let .06cm $\mathop{\mathrm{\textnormal{F}}}(a+s,\cdot,\cdot) \hskip.06cm := \hskip.06cm |\mathcal F(a+s,\cdot,\cdot)|$ .06cm where .06cm $s\in \{0,\pm 1\}$, be the numbers of these linear extensions. **Lemma 41** ([@MS22 Thm 1.5]). *Let .03cm$P$ .03cmbe a critical poset. Then .03cm[\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} .03cmholds .06cm [if and only if]{.ul} $$\begin{aligned} & \textrm{\em F}(a-1,\textnormal{com},\textnormal{com})\ = \ \textrm{\em F}(a,\textnormal{com},\textnormal{com}) \ = \ \textrm{\em F}(a+1,\textnormal{com},\textnormal{com}) \, = \, 0 \quad \ \text{ and} \label{eq:crit-1}\\ & \aligned & \textrm{\em F}(a-1,\textnormal{com},\textnormal{inc}) \ = \ \textrm{\em F}(a-1,\textnormal{inc},\textnormal{com}) \ = \ \textrm{\em F}(a,\textnormal{com},\textnormal{inc}) \\ & \hskip1.cm = \ \textrm{\em F}(a,\textnormal{inc},\textnormal{com}) \ = \ \textrm{\em F}(a+1,\textnormal{com},\textnormal{inc}) \ = \ \textrm{\em F}(a+1,\textnormal{inc},\textnormal{com}). % \notag \endaligned \label{eq:crit-2} \end{aligned}$$* Now note that .06cm $\mathop{\mathrm{\textnormal{F}}}(a-1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}}) \ \leq \ \mathop{\mathrm{\textnormal{F}}}(a-1,\mathop{\mathrm{\text{inc}}},\mathop{\mathrm{\text{com}}})$, with the equality .06cm [if and only if]{.ul} .06cm every upper companion of $x$ is always incomparable to the lower companion of $x$. By an analogous arguments applied to .06cm $\mathop{\mathrm{\textnormal{F}}}(a,\cdot, \cdot)$ .06cm and .06cm $\mathop{\mathrm{\textnormal{F}}}(a+1,\cdot, \cdot)$, we get the following corollary. **Corollary 42**. *Let .03cm$P$ .03cmbe a critical poset. Suppose $$\textrm{\em N}_{z \hskip.03cmc}(P, x,a)^2 \, = \, \textrm{\em N}_{z \hskip.03cmc}(P, x,a+1) \hskip.06cm\cdot \hskip.06cm \textrm{\em N}_{z \hskip.03cmc}(P, x,a-1) \ \neq \ 0,$$ Then, for every linear extension .03cm$f\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cmcounted by [\[eq:crit-2\]](#eq:crit-2){reference-type="eqref" reference="eq:crit-2"}, the upper companion is incomparable to the lower companion: .06cm ${\textup {\rm uc} } (f) \hskip.06cm \| \, {} {\textup {\rm lc} } (f)$.* Finally, we have equality conditions for supercritical posets. **Lemma 43** ([@MS22 Thm 1.3]). *Let $P$ be a supercritical poset. Then .03cm[\[eq:Sta1\]](#eq:Sta1){reference-type="eqref" reference="eq:Sta1"} .03cmholds .06cm [if and only if]{.ul} .06cm equalities .03cm[\[eq:crit-1\]](#eq:crit-1){reference-type="eqref" reference="eq:crit-1"} .03cmand .03cm[\[eq:crit-2\]](#eq:crit-2){reference-type="eqref" reference="eq:crit-2"} .03cmhold, and additionally $$\label{eq:crit-3} \text{all numbers in \eqref{eq:crit-2} are equal to~$\hskip.03cm0$.}$$* ## Proof of Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"} {#ss:Sta1-proofi} Note that .03cm[\[eq:crit-1\]](#eq:crit-1){reference-type="eqref" reference="eq:crit-1"}, .03cm[\[eq:crit-2\]](#eq:crit-2){reference-type="eqref" reference="eq:crit-2"} .03cmand .03cm[\[eq:crit-3\]](#eq:crit-3){reference-type="eqref" reference="eq:crit-3"} are equivalent to requiring that .03cm$x$ .03cmis incomparable to both .06cm ${\textup {\rm lc} } (f)$ .06cm and .06cm ${\textup {\rm uc} } (f)$. Thus it suffices to show that, if .03cm$P$ .03cmis a critical poset, then [\[eq:crit-3\]](#eq:crit-3){reference-type="eqref" reference="eq:crit-3"} holds. Suppose to the contrary, that .03cm$P=(X,\prec)$ .03cmis a counterexample, and let .03cm$n:=|X|$. Then .03cm$P$ .03cmis a sharp critical poset. By taking the dual poset if necessary, we can assume, without loss of generality, that .03cm$c<a$. It then follows that the splitting pair .03cm$(r,s)$ .03cmin Lemma [Lemma 39](#lem:crit-prop){reference-type="ref" reference="lem:crit-prop"} is .03cm$(1,2)$. This means that .03cm$c_r=c$ .03cmand .03cm$c_s=n+1$, so we have from [\[eq:split-2\]](#eq:split-2){reference-type="eqref" reference="eq:split-2"} that $$\label{eq:split-3} \left|\{ u \in P \, : \, z \hskip.06cm \prec \hskip.06cm u \}\right| \ = \ n -c-1.$$ Since [\[eq:crit-3\]](#eq:crit-3){reference-type="eqref" reference="eq:crit-3"} does not hold, there exists .06cm $f \in \mathcal F(a,\mathop{\mathrm{\text{com}}}, \mathop{\mathrm{\text{inc}}})$ .06cm and .06cm $h \in \mathcal F(a-1,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}})$.06cm. Let .06cm $y_1:=f^{-1}(a-1)$ .06cm (i.e., the lower companion in $f$) and .06cm $y_2:=h^{-1}(a)$ .06cm (i.e., the lower companion in $h$). Note that we have .06cm $y_1 \hskip.06cm \prec \hskip.06cm x \hskip.06cm \prec \hskip.06cm y_2$.06cm. Let .06cm $m=f(y_2)$.06cm, and note that .06cm $m\geq a+2$ .06cm by definition. **We claim:** .06cm There exists a new linear extension .06cm $g\in \mathop{\mathrm{\mathcal{E}}}(P)$ .06cm such that $g(y_2)=m-1$, and such that .06cm $g \in \mathcal F(a,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{inc}}})$ .06cm if .03cm$m>a+2$, and .06cm $g \in \mathcal F(a,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}})$ .06cm if .03cm$m=a+2$. Note that this suffices to prove the lemma, as by replacing $f$ with $g$ and decreasing $m$ repeatedly, we get that .06cm $\mathop{\mathrm{\textnormal{F}}}(a,\mathop{\mathrm{\text{com}}},\mathop{\mathrm{\text{com}}})>0$, which contradicts [\[eq:crit-1\]](#eq:crit-1){reference-type="eqref" reference="eq:crit-1"}. ***We now prove the claim.*** Since $h(y_2)=a <m=f(y_2)$, there exists .06cm $w \in X$ .06cm such that .06cm $f(w) < m$ .06cm and .06cm $w \hskip.06cm \| \hskip.06cm y_2$.03cm. Suppose .03cm$w$ .03cmis such an element that maximizes .03cm$f(w)$. There are *three cases*: [First]{.ul}.03cm, .03cmsuppose that .06cm $f(w)> a$. By the maximality assumption, every element ordered between $w$ and $y_2$ according to $f$, is incomparable to $w$. Then we can promote .03cm$w$ .03cmto be larger than $\hskip.03cmy_2$. Note that the resulting linear extension .03cm$g\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cmsatisfies .06cm $g(y_2)=m-1$, .06cm $g(y_1)=a-1$ .06cm and .06cm $g(x)=a$, as desired. [Second]{.ul}.03cm, .03cmsuppose that .06cm $c<f(w)<a$. By the maximality assumption, every element ordered between .03cm$w$ .03cmand .03cm$y_2$ according to $f$, is incomparable to $\hskip.03cmw$. Then we can promote .03cm$w$ .03cmto be larger than $\hskip.03cmy_2$. The resulting linear extension .03cm$g'\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cmsatisfies .06cm $g'(y_2)=m-1$. Note, however, that we have .06cm $g'(y_1)=a-2$ .06cm and .06cm $g'(x)=a-1$. In order to fix this, let .06cm $v:=f^{-1}(a+1)$. It follows from Corollary [Corollary 42](#cor:crit-cond){reference-type="ref" reference="cor:crit-cond"}, that $v$ is incomparable to $y_1$ and $x$. Thus we can demote $v$ to be the smaller than $y_1$. We obtain a new linear extension .03cm$g\in \mathop{\mathrm{\mathcal{E}}}(P)$ .03cm that satisfies .06cm $g(y_1)=a-1$ .06cm and .06cm $g(x)=a$, as desired. [Third]{.ul}.03cm, .03cmsuppose that .06cm $f(w)<c$. Then, every element ordered between $z$ and $y_2$ according to $\hskip.03cmf$, is less than $\hskip.03cmy_2 \hskip.03cm.$ Note that there are .06cm $m-c-1$ .06cm many such elements. On the other hand, it follows from [\[eq:split-3\]](#eq:split-3){reference-type="eqref" reference="eq:split-3"}, that there is exactly one element in .06cm $\{f^{-1}(c+1), f^{-1}(c+2), \ldots, f^{-1}(n) \}$ .06cm that is incomparable to $\hskip.03cmz$. It then follows that there are at least .06cm $m-c-2$ .06cm elements that are greater than .03cm$z$ .03cmand less than $\hskip.03cmy_2$.03cm, i.e. $$\big|\big\{ u \in X \, : \, z \hskip.06cm \prec \hskip.06cm u \hskip.06cm \prec \hskip.06cm y_2 \big\}\big| \, \geq \, m-c-2.$$ On the other hand, the existence of .03cm$h$ .03cmimplies that $$\big|\big\{ u \in X \, : \, z \hskip.06cm \prec \hskip.06cm u \hskip.06cm \prec \hskip.06cm y_2 \big\}\big| \, \leq \, h(y_2)-c-1 \, = \, a-c-1 \, \leq \, m-c-3,$$ a contradiction. This finishes the proof of the claim. 0◻ # Final remarks {#s:finrem} ## The basis of our work {#ss:finrem-hist} Due to the multidisciplinary nature of this paper, we make a special effort to simplify the presentation. Namely, the proofs of our main results (Theorems [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} and [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}), are largely self-contained in a sense that we only use standard results in combinatorics (Stanley's theorem in $\S$[5.2](#ss:AF-proofs){reference-type="ref" reference="ss:AF-proofs"} and Brightwell--Winkler's theorem in $\S$[3.6](#ss:hist-CC){reference-type="ref" reference="ss:hist-CC"}), computational complexity (Toda's theorem in $\S$[4.4](#ss:roadmap-proofs){reference-type="ref" reference="ss:roadmap-proofs"}), and number theory (Yao--Knuth's theorem in $\S$[8.2](#ss:verify-NT){reference-type="ref" reference="ss:verify-NT"}). In reality, the paper freely uses tools and ideas from several recent results worth acknowledging. First, we heavily build on the recent paper by Shenfeld and van Handel [@SvH-acta], and the followup by Ma and Shenfeld [@MS22]. Without these results we would not know where to look for "bad posets" and "bad polytopes". Additionally, the proof in $\S$[6.2](#s:Flat-Sta-proof){reference-type="ref" reference="s:Flat-Sta-proof"} is a reworking and simplification of many technical results and ideas in [@MS22]. Second, in $\S$[8.1](#ss:verify-CF){reference-type="ref" reference="ss:verify-CF"} we use and largely rework the continued fraction approach by Kravitz and Sah [@KS21]. There, the authors employ the *Stern--Brocot* .03cmand *Calkin--Wilf tree* .03cmnotions, which we avoid in our presentation as we aim for different applications. Third, in the heart of our proof of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} in $\S$[4.4](#ss:roadmap-proofs){reference-type="ref" reference="ss:roadmap-proofs"}, we follow the complexity roadmap championed by Ikenmeyer, Panova and the second author in [@IP22; @IPP22]. Same for the heart of the proof of the Verification Lemma [Lemma 12](#l:Verify-Quart){reference-type="ref" reference="l:Verify-Quart"} in $\S$[8.4](#ss:verify-proof){reference-type="ref" reference="ss:verify-proof"}, which follows the approach in our companion paper [@CP23]. On the other hand, the proof of Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"} given in Section [9](#s:Sta1){reference-type="ref" reference="s:Sta1"}, is the opposite of self-contained, as we rely heavily on both results and ideas in [@MS22]. We also use properties the *promotion* .03cmand *demotion* .03cm operations on linear extensions, that were introduced by Schützenberger in the context of algebraic combinatorics, see [@Schu][^11]. Panova and the authors employed this approach in a closely related setting in [@CPP-KS; @CPP-effective; @CPP-Quant-CPC]. We emphasize once again that our proof of Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"} is independent of the rest of the paper and is the only part that uses results in [@MS22]. ## Polytopes {#ss:finrem-polytopes} The family of TU-polytopes that we chose is very special in that these H-polytopes have integral vertices (but not a description in $\textup{\textup{\textsf{P}}}$, as V-polytopes are defined to have). In [@CP23+], we consider a family of *axis-parallel boxes* .03cm which have similar properties. Clearly, for general convex bodies there is no natural way to set up a computational problem that would not be immediately intractable (unless one moves to a more powerful computational model, see e.g. [@BCSS98]). ## Discrete isoperimetric inequality {#ss:finrem-discrete-isop} For a discrete version of the isoperimetric inequality in the plane, one can consider convex polygons with given normals to edges. In this case, L'Huilier (1775) proved that the isoperimetric ratio is minimized for circumscribed polygons, see e.g. [@Fej $\S$I.4]. In the 1860s, Steiner and Lindelöf studied a natural generalization of this problem in $\mathbb R^3$, but were unable to solve it in full generality. At the turn of 20th century, Minkowski developed the *theory of mixed volumes*, motivated in part to resolve the Steiner--Lindelöf problem. He showed that among all polytopes with given normals, the isoperimetric ratio is minimized on circumscribed polytopes, see e.g. [@Fej $\S$V.7]. There are several Bonnesen type and stability versions of the discrete isoperimetric inequality, see e.g. [@FRS; @IN15; @Zhang98]. Let us single out a hexagon version used by Hales in his famous proof of the *honeycomb conjecture* [@Hales Thm 4]. ## Brunn--Minkowski inequality {#ss:finrem-BM} There are several proofs of the Brunn--Minkowski inequality [\[eq:BM\]](#eq:BM){reference-type="eqref" reference="eq:BM"}, but some of them do not imply the equality conditions, such as, e.g., the "brick-by-brick" inductive argument in [@Mat $\S$`<!-- -->`{=html}12.2]. Note also that Alexandrov's proof of the *Minkowski uniqueness theorem* .03cm(of polytopes with given facet volumes and normals) relies on the equality conditions for the Brunn--Minkowski inequality, see [@Ale-book]. This is essential for Alexandrov's "topological method", and is the basis for the *variational principle* .03cmapproach, see e.g. [@Pak-book]. ## Van der Waerden conjecture {#ss:finrem-vdW} The Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} came to prominence in combinatorics after Egorychev [@Egor] used it to prove the *van der Waerden conjecture*, that was proved earlier by Falikman [@Fal].[^12] See [@Knuth81; @vL] for friendly expositions. This development set the stage for Stanley's paper [@Sta-AF]. The conjecture states that for every bistochastic $n\times n$ matrix $A$, we have $$\label{eq:vdW}\tag{vdW} \mathrm{per}(A) \, \ge \, \frac{n!}{n^n}\hskip.06cm,$$ and the equality holds only if .03cm$A=(a_{ij})$ .03cmhas uniform entries: .03cm$a_{ij}=\frac{1}{n}$ .03cmfor all $1\le i,j\le n$. Note that Egorychev's proof of the equality conditions for [\[eq:vdW\]](#eq:vdW){reference-type="eqref" reference="eq:vdW"} used Alexandrov's equality conditions [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} for nondegenerate boxes, see $\S$[3.2](#ss:hist-AF){reference-type="ref" reference="ss:hist-AF"} (cf. [@Knuth81 p. 735] and [@vL $\S$`<!-- -->`{=html}7]). In a followup paper [@CP23+], we analyze the complexity of the Alexandrov--Fenchel equality condition for degenerate boxes. Note also that Knuth's exposition in [@Knuth81] is essentially self-contained, while Gurvits's proof of [\[eq:vdW\]](#eq:vdW){reference-type="eqref" reference="eq:vdW"} completely avoids [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}, see [@Gur; @LS10]. ## Matroid inequalities {#ss:finrem-matroid} Of the several log-concavity applications of the AF inequality given by Stanley in [@Sta-AF] (see also [@Sta-two $\S$`<!-- -->`{=html}6]), one stands out as a special case of a Mason's conjecture (Thm 2.9 in [@Sta-AF]). The strongest of the three Mason's conjectures states that the numbers .06cm ${\text {\rm I} } (M,k)/\binom{n}{k}$ .06cm are log-concave, where .03cm${\text {\rm I} } (M,k)$ .03cmis the number of independent sets of size $k$ in a matroid $M$ on $n$ elements. These Mason's conjectures were recently proved in a long series of spectacular papers culminating with [@AHK; @ALOV; @BH20], see also an overview in [@Huh; @Kalai]. Curiously, the equality cases for these inequalities are rather trivial and can be verified in polynomial time [@MNY] (see also [@CP $\S$`<!-- -->`{=html}1.6]). Here we assume that the matroid is given in a concise presentation (such presentations include graphical, bicircular and representable matroids). Curiously, for the weighted extension of Mason's third conjecture given in [@CP Thm 1.6], the equality cases are more involved. It follows from [@CP Thm 1.9], however, that this problem is in $\textup{\textup{\textsf{coNP}}}$. In other words, Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} shows that .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_6$ .03cmis likely much more powerful. Note that the defect .06cm $\psi(M,k):= {\text {\rm I} } (M,k)^2- {\text {\rm I} } (M,k+1)\cdot {\text {\rm I} } (M,k-1)$ .06cm is conjectured to be not in .03cm$\textup{\textup{\textsf{\#P}}}$, see [@Pak-OPAC Conj. 5.3]. Clearly, the argument in the proof of Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"} does not apply in this case. Thus, another approach is needed to prove this conjecture, just as another approach is need to prove that .03cm$\phi_0 \notin \textup{\textup{\textsf{\#P}}}$ .03cm(see $\S$[1.4](#ss:intro-Stanley){reference-type="ref" reference="ss:intro-Stanley"}). ## Complexity conjectures {#ss:finrem-LE} Note that Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} does not imply that .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}$ .03cm is .03cm$\textup{\textup{\textsf{NP}}}$-hard or .03cm$\textup{\textup{\textsf{coNP}}}$-hard, more traditional measures of computational hardness. This remains out of reach. Note also that .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_k$ .03cmis naturally in ${\textup{\textup{\textsf{C$_=$P}}}}$. Recall from $\S$[3.6](#ss:hist-CC){reference-type="ref" reference="ss:hist-CC"}, that .03cm${\textup{\textup{\textsf{C$_=$P}}}}$-complete problem .03cm $\text{\sc C}_\text{\#3SAT}$ .03cmis .03cm$\textup{\textup{\textsf{coNP}}}$-hard. It is not known whether .03cm$\text{\sc C}_\text{\#3SAT}$ .03cm is .03cm$\textup{\textup{\textsf{NP}}}$-hard under Turing reductions (this is unlikely). **Conjecture 44**. *.06cm $\mathop{\mathrm{\textsc{EqualityStanley}}}_k$ .03cmis .03cm${\textup{\textup{\textsf{C$_=$P}}}}$-complete for large enough $k$.* Note that our proof of Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"} does not extend to .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_2$.03cm. While it is possible that our proof of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} can extend to .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_4$ .03cmwith a considerable effort, it would not extend to .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_2$ .03cmwithout major changes and potentially some new tools. **Conjecture 45**. *.06cm $\mathop{\mathrm{\textsc{EqualityStanley}}}_2$ .03cmis not in .03cm$\textup{\textup{\textsf{PH}}}$, unless .03cm$\textup{\textup{\textsf{PH}}}$ .03cmcollapses.* There are two reasons to believe in this conjecture. First, the key to understand .03cm $\mathop{\mathrm{\textsc{EqualityStanley}}}_1\in \textup{\textup{\textsf{P}}}$ .03cmis the condition $(\divideontimes)$ in Lemma [Lemma 36](#lem:k=1){reference-type="ref" reference="lem:k=1"}. However, by [@MS22 Ex. 1.4] discussed in Remark [Remark 37](#r:Sta1-MS){reference-type="ref" reference="r:Sta1-MS"}, for .03cm$k=2$ .03cm there are critical equality cases that do not satisfy $(\divideontimes)$. This gives hope that the equality verification problem might not be in $\textup{\textup{\textsf{PH}}}$ in these cases. Second, it follows from our proof of Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}, that this conjecture can be derived from the following: **Conjecture 46**. *The coincidence problem .06cm C$_\rho := \big\{\rho(P,x)=^?\rho(Q,y)\big\}$ .06cm is not in .03cm$\textup{\textup{\textsf{PH}}}$, unless .03cm$\textup{\textup{\textsf{PH}}}$ .03cmcollapses.* To see the implication, recall that problem .03cmC$_\rho$ .03cmreduces to .03cmFlatLE$_0$ .03cm by Lemma [Lemma 18](#lem:recursion-1){reference-type="ref" reference="lem:recursion-1"}, which in turn reduces to .03cmEqualityStanley$_2$ .03cm by Lemma [Lemma 9](#l:Flat-Sta){reference-type="ref" reference="l:Flat-Sta"}. In fact, the coincidence problem .03cmC$_\rho$ .03cm is a more natural problem to study than .03cmQuadRLE, and [@CP23] was inspired by our unsuccessful effort to prove Conjecture [Conjecture 46](#conj:Sta-ver){reference-type="ref" reference="conj:Sta-ver"}. Finally, note that the proof of .03cm$\mathop{\mathrm{\textsc{EqualityStanley}}}_6\notin\textup{\textup{\textsf{PH}}}$ .03cmimplies that .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\notin \textup{\textup{\textsf{PH}}}$ .03cmeven when at most $8$ polytopes are allowed to be distinct. It would be interesting to decide if this number can be reduced down to $3$. It is known that $2$ distinct TU-polytopes are not enough.[^13] ## Injective proofs {#ss:finrem-injective} In enumerative combinatorics, whenever one has an equality between the numbers counting certain combinatorial objects, one is tempted to find a *direct bijection* .03cmbetween the sides, see e.g. [@Loe11; @Pak-part; @Sta-EC]. Similarly, when presented an inequality .03cm$f \geqslant g$, one is tempted to find a *direct injection*, see e.g. [@Pak; @Sta-log-concave]. In the context of linear extensions, such injections appear throughout the literature, see e.g. [@Bre; @BT02; @CPP-KS; @DD; @GG22; @LP07]. Typically, a direct injection and its inverse are given by simple polynomial time algorithms, thus giving a combinatorial interpretation for the defect .03cm$(f-g)$. Therefore, if a combinatorial inequality is not in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$, it is very unlikely that there is a proof by a direct injection. In particular, Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"} implies that the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} most likely cannot be proved by a direct injection. This confirm an old speculation: *"It appears unlikely that Stanley's Theorem for linear extensions quoted earlier can be proved using the kind of injection presented here."* [@DDP $\S$`<!-- -->`{=html}4]. Similarly, Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"} suggests that the strategy in [@CPP-effective $\S$`<!-- -->`{=html}9.12] is unlikely to succeed, at least for .03cm$k\ge 6$. Moreover, assuming Conjecture [Conjecture 45](#conj:Sta-ESta2){reference-type="ref" reference="conj:Sta-ESta2"} holds, we have the same conclusion for .03cm$k\ge 2$.[^14] To fully appreciate how delicate is Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"}, compare it with a closely related problem. It is known that for all .03cm$k\ge 0$, the analogue of the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} holds for the number of *order preserving maps* .06cm $X\to [t]$, for all .03cm$t \in \mathbb N$. This was conjectured by Graham in [@Gra82 p. 129] (see also [@Gra p. 233]), motivated by the proof of the XYZ inequality [@She-XYZ] (cf. $\S$[3.4](#ss:hist-LE){reference-type="ref" reference="ss:hist-LE"}). The result was proved in [@DDP Thm 4] by a direct injection (see also [@Day84 $\S$`<!-- -->`{=html}4.2] for additional details of the proof). In other words, in contrast with  $\hskip.03cm\phi_k\hskip.03cm$, the analogue of the defect for order preserving maps has a combinatorial interpretation. Note also that it is not known whether the defect of the XYZ inequality is in .03cm$\textup{\textup{\textsf{\#P}}}$, see [@Pak-OPAC Conj. 6.4]. ## Stability proofs {#ss:finrem-mass-transport} By analogy with the injective proofs, Corollary [Corollary 2](#c:main-AF-stab){reference-type="ref" reference="c:main-AF-stab"} suggests that certain proofs of the Alexandrov--Fenchel inequality are likely not possible. Here we are thinking of the mass transportation proof of characterization of the isoperimetric sets given in [@FMP10 App.], following Gromov's approach in [@Gromov]. It would be interesting to make this idea precise. ## Dichotomy of the equality cases {#ss:finrem-eq} As we discuss in $\S$[9.2](#ss:Sta1-critical){reference-type="ref" reference="ss:Sta1-critical"}, it follows from the results in [@MS22], that the equality verification of the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} can be decided in polynomial time for supercritical posets. In contrast, by Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"}, the problem is not in .03cm$\textup{\textup{\textsf{PH}}}$ .03cmfor critical posets. We believe that this dichotomy also holds for the equality cases of the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"} for classes of H-polytopes for which the scaled mixed volume is in $\hskip.03cm\textup{\textup{\textsf{\#P}}}$. ## The meaning of it all {#ss:finrem-meaning} Finding the equality conditions of an inequality may seem like a straightforward unambiguous problem, but the case of the Alexandrov--Fenchel inequality shows that it is nothing of the kind. Even the words "equality conditions" are much too vague for our taste. What the problem asks is a *description of the equality cases*. But since many geometric and combinatorial inequalities have large families of equalities cases, the word "description" becomes open-ended (cf. $\S$[2.1](#ss:def-term){reference-type="ref" reference="ss:def-term"}). How do you know when you are done? At what point are you satisfied with the solution and do not need further details? These are difficult questions which took many decades to settle, and the answers depend heavily on the area. In the context of geometric inequalities discussed in $\S$[3.1](#ss:hist-geom){reference-type="ref" reference="ss:hist-geom"}, the meaning of "description" starts out simple enough. There is nothing ambiguous about discs as equality cases of the isoperimetric inequality in the plane [\[eq:Isop\]](#eq:Isop){reference-type="eqref" reference="eq:Isop"}, or pairs of homothetic convex bodies for the Brunn--Minkowski inequality [\[eq:BM\]](#eq:BM){reference-type="eqref" reference="eq:BM"}, or circumscribed polygons with given normals for the discrete isoperimetric inequality (see $\S$[10.3](#ss:finrem-discrete-isop){reference-type="ref" reference="ss:finrem-discrete-isop"}). Arguably, Bol's equality cases of [\[eq:Mink-mean\]](#eq:Mink-mean){reference-type="eqref" reference="eq:Mink-mean"} are also unambiguous --- in $\mathbb R^3$, you literally know the cap bodies when you see them. However, when it comes to Minkowski's quadratic inequality [\[eq:MQI\]](#eq:MQI){reference-type="eqref" reference="eq:MQI"}, the exact meaning of "description" is no longer obvious. Shenfeld and van Handel write "The main results of this paper will provide a complete solution to this problem" [@SvH-duke]. Indeed, their description of $3$-dimensional triples of convex bodies cannot be easily improved upon, at least not in the case of convex polytopes (see $\S$[3.1](#ss:hist-geom){reference-type="ref" reference="ss:hist-geom"}). Some questions may still linger, but they are on the structure of the equality cases rather than on their recognition.[^15] What Shenfeld and van Handel did, is finished off the geometric approach going back to Brunn, Minkowski, Favard, Fenchel, Alexandrov and others, further formalized by Schneider. "Maybe a published conjecture will stimulate further study of this question", Schneider wrote in [@Schn85]. This was prophetic, but that conjecture was not the whole story, as it turned out. In [@SvH-acta], the authors write again: "We completely settle the extremals of the Alexandrov--Fenchel inequality for convex polytopes." Unfortunately, their description is extraordinary complicated in higher dimensions, so the problem of *recognizing* .03cmthe equality cases is no longer easy (see $\S$[3.2](#ss:hist-AF){reference-type="ref" reference="ss:hist-AF"}). And what good is a description if it cannot be used to recognize the equality cases? In combinatorics, the issue of "description" has also been a major problem for decades, until it was fully resolved with the advent of computational complexity. For example, consider the following misleadingly simple description: "Let $G$ be a planar cubic Hamiltonian graph." Is that good enough? How can you tell if a given graph $G$ is as you describe? We now know that the problem whether $G$ is planar, cubic and Hamiltonian is .03cm$\textup{\textup{\textsf{NP}}}$-complete [@GJT]. But if you only need the "planar" condition, the problem is computationally easy, while the "cubic" condition is trivial. Consequently, "planar cubic Hamiltonian" should not be viewed as a "good" description, but if one must consider the whole class of such graphs, this description is (most likely) the best one can do. Going over equality cases for various inequalities on the numbers of linear extensions, already gives an interesting picture. For the Björner--Wachs inequality (see $\S$[3.4](#ss:hist-LE){reference-type="ref" reference="ss:hist-LE"}), the recognition problem of forests is in $\textup{\textup{\textsf{P}}}$, of course. On the other hand, as we explain in $\S$[3.4](#ss:hist-LE){reference-type="ref" reference="ss:hist-LE"}, for the Sidorenko inequality [\[eq:Sid\]](#eq:Sid){reference-type="eqref" reference="eq:Sid"}, the recognition problem of series-parallel posets is in $\textup{\textup{\textsf{P}}}$ for a more involved reason. On the opposite end of the spectrum, for the (rather artificial) inequality .03cm$(e(P)-e(Q))^2\ge 0$, the equality verification is not in .03cm$\textup{\textup{\textsf{PH}}}$, unless $\textup{\textup{\textsf{PH}}}$ collapses, see $\S$[3.7](#ss:hist-combin-int){reference-type="ref" reference="ss:hist-combin-int"} and [@CP23 Thm 1.4]. In this language, for the .03cm$k=0$ .03cmcase of the Stanley inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"}, the description of equality cases given in [@SvH-acta] is trivially in $\textup{\textup{\textsf{P}}}$. Similarly, for the .03cm$k=1$ .03cmcase, the description of equality cases is also in $\textup{\textup{\textsf{P}}}$ by Theorem [Theorem 4](#t:ESta-1){reference-type="ref" reference="t:ESta-1"}. On the other hand, Theorem [Theorem 3](#t:main-Sta){reference-type="ref" reference="t:main-Sta"} shows that for .03cm$k\ge 6$, the description in [@MS22] is (very likely) not in $\textup{\textup{\textsf{P}}}$. Under standard complexity assumptions, there is no description of the equality cases in .03cm$\textup{\textup{\textsf{P}}}$ .03cmat all, or even in .03cm$\textup{\textup{\textsf{PH}}}$ .03cmfor that matter. Now, the problem of *counting* .03cmthe equality cases brings a host of new computational difficulties, making seemingly easy problems appear hard when formalized, see [@Pak-OPAC]. Even for counting non-isomorphic forest posets on $n$ elements, to show that this function in $\textup{\textup{\textsf{\#P}}}$ one needs to define a *canonical labeling* .03cmto be able to distinguish the forests, to make sure each is counted exactly once, see e.g. [@SW19]. In this language, Corollary [Corollary 5](#c:main-Stanley-not-SP){reference-type="ref" reference="c:main-Stanley-not-SP"} states that *there are no combinatorial objects* .03cmthat can be counted to give the number of non-equality cases of the Stanley inequality, neither the non-equality cases themselves nor anything else. The same applies to the equality cases. Fundamentally, this is because you should not be able to efficiently tell if the instances you are observing are the ones you should be counting. Back to the Alexandrov--Fenchel inequality [\[eq:AF\]](#eq:AF){reference-type="eqref" reference="eq:AF"}, the description of equality cases by Shenfeld and van Handel is a breakthrough in convex geometry, and gives a complete solution for a large family of ($n$-tuples of) convex polytopes (see $\S$[10.10](#ss:finrem-eq){reference-type="ref" reference="ss:finrem-eq"}). However, our Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} says that from the computational point of view, the equality cases are intractable in full generality. Colloquially, this says that *there is no good description* .03cmof the equality cases of the Alexandrov--Fenchel inequality, unless the world of computational complexity is not what we think it is. As negative as this may seem, this is what we call a complete solution indeed. .6cm ## Acknowledgements {#acknowledgements .unnumbered} We are grateful to Karim Adiprasito, Sasha Barvinok, Károly Böröczky, Christian Ikenmeyer, Jeff Kahn, Joe O'Rourke, Matvey Soloviev and Richard Stanley for useful remarks on the subject. Special thanks to Yair Shenfeld and Ramon van Handel for many very helpful comments on the first draft of the paper, and to Greta Panova for the numerous helpful discussions. This paper was finished when both authors were visiting the American Institute of Mathematics at their new location in Pasadena, CA. We are grateful to AIM for the hospitality. The first author was partially supported by the Simons Foundation. 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Xin-Min Zhang, Schur-convex functions and isoperimetric inequalities, *Proc. AMS* **126** (1998), 461--470. .7cm [^1]: The collapse in the theorem contradicts standard assumptions in computational complexity. A conjecture that the collapse does not happen is a strengthening of the .03cm$\textup{\textup{\textsf{P}}}\ne \textup{\textup{\textsf{NP}}}$ .03cmconjecture that remains out of reach, see $\S$[3.8](#ss:hist-CA){reference-type="ref" reference="ss:hist-CA"}. [^2]: Following [@Oss79], function $h$ should also have a (not formally defined) "geometric description". [^3]: It follows from our Theorem [Theorem 1](#t:main-AF){reference-type="ref" reference="t:main-AF"} that it has to be, see a discussion in $\S$[10.11](#ss:finrem-meaning){reference-type="ref" reference="ss:finrem-meaning"}. [^4]: By that the authors of [@SvH-acta] seem to mean that their description captured all the geometry in the problem, as opposed to the equality of mixed volumes which has no geometric content. [^5]: Note that when .03cm$X \subset\mathbb R$ .03cmis a nonzero interval, we have .03cm$r=0$ .03cmand .03cm$\ell(X)=4R$, so the inequality remains strict. [^6]: For .03cm$k \ge 1$, the inequality [\[eq:Sta\]](#eq:Sta){reference-type="eqref" reference="eq:Sta"} is sometimes called the *generalized Stanley inequality*, see [@CPP-effective]. [^7]: In geometric language, slices .03cm$\textnormal{S}_i$ .03cmare sections of the order polytope .03cm$\mathcal O_P$ .03cmwith a $k$-dimensional affine subspace. [^8]: We warn the reader that from this point on our notation is substantially different from that in [@MS22]. [^9]: We do not actually need the precise bounds below, other than the fact that they are polynomial. However, these bounds help to clarify the construction. [^10]: In [@MS22 Def 5.2], this pair is instead written as $(r+1,s)$. [^11]: These operations were rediscovered in [@DD; @Day84], where they are called *push up* .03cm and *push down*, respectively. [^12]: According to Vladimir Gurvich's [essay](https://tinyurl.com/47c6s9et), Egorychev was the referee of Falikman's article which was submitted prior to Egorychev's preprint. [^13]: This follows from a combination of our argument that for supercritical cases (in the sense of [@SvH-acta]), we have .03cm$\mathop{\mathrm{\textsc{EqualityAF}}}\in \textup{\textup{\textsf{coNP}}}$, and an argument by Ramon van Handel that for two polytopes the equality cases are supercritical (personal communication, April 2023). [^14]: In [@Gra p. 129], Graham asked if Stanley's inequality can be proved using the AD and FKG inequalities. This seems unlikely, but we don't know how to formalize this question. [^15]: For example, one can ask to characterize all possible triples of polytope graphs that arise as equality cases.

arxiv_math

# Main Theorem {#sec:main-thm} In this section, we prove our main theorem ([\[main th\]](#main th){reference-type="ref" reference="main th"}), characterizing closed symmetric monoidal structures on the category of graphs. For the remainder of the paper, we impose the following assumption: [\[assumption4\]]{#assumption4 label="assumption4"} Fix a closed symmetric monoidal structure $\otimes$ on $\Graph$, and set $F_\otimes := \otimes \circ (\yo \times \yo)$. We claim that the only two such functors $\otimes$ are the box and categorical products. In broad strokes, the argument can be summarized as: 1. $F_\otimes(E,E)$ cannot be a graph with more than four vertices ([\[at most 4\]](#at most 4){reference-type="ref" reference="at most 4"}). 2. $F_\otimes(E,E)$ cannot be a graph with less than four vertices ([\[cor:exactly 4\]](#cor:exactly 4){reference-type="ref" reference="cor:exactly 4"}). 3. If $F_\otimes(E,E)$ has exactly four vertices, and the resulting product is closed symmetric monoidal, then the resulting product is either the box or categorical product ([\[TH:monoidal structures\]](#TH:monoidal structures){reference-type="ref" reference="TH:monoidal structures"}). Recall that, by [\[Prop: monoidal as kan extensions\]](#Prop: monoidal as kan extensions){reference-type="ref" reference="Prop: monoidal as kan extensions"}, we know that $\otimes \iso \lan_{\yo \times \yo} F_\otimes$. By [\[cor:colimit formula\]](#cor:colimit formula){reference-type="ref" reference="cor:colimit formula"}, we only need to consider the action of $F_\otimes$ on $(E,E)$. [\[at most 4\]]{#at most 4 label="at most 4"} The product $I_1 \otimes I_1$ has at most four vertices. *Proof.* Consider the following pushout square in $\Graph$: $$\begin{tikzcd}[column sep = large] I_0 \sqcup I_0 \arrow[r,tail,"\partial"] \arrow[d,tail,swap,"\partial"] & I_1 \arrow[d,"\id"] \\ I_1 \arrow[r,"\id"] & I_1 \end{tikzcd}$$ Applying $I_1 \otimes -$ and using the fact that the monoidal structure is closed and has unit $I_0$, we get a pushout square $$\begin{tikzcd}[column sep = large] I_1 \sqcup I_1 \arrow[r,"\partial \otimes \id"] \arrow[d,swap,"\partial \otimes \id"] & I_1 \otimes I_1 \arrow[d,"\id"] \\ I_1 \otimes I_1 \arrow[r,"\id"] & I_1 \otimes I_1 \end{tikzcd}$$ (Note that we used here that functors preserve identity morphisms.) But a square of this form is a pushout if and only if $\partial \otimes \id \colon I_1 \sqcup I_1 \to I_1 \otimes I_1$ is an epimorphism, and hence in particular surjective on vertices. As its domain has four vertices, the codomain cannot have more than four vertices. ◻ The task of showing that $F_\otimes(E,E)$ must have at least four vertices is significantly more involved. We begin with the following proposition and definition, which are used throughout the remainder of this section: [\[image of I_0\]]{#image of I_0 label="image of I_0"} The functor $F_\otimes$ must satisfy: $F_\otimes(V,V)=I_0$ and $F_\otimes(V,E)=F_\otimes(E,V)=I_1$. *Proof.* We have: $$\begin{aligned} F_\otimes(V,V) &= \otimes \circ \yo \times \yo (V,V) \\ &= I_0 \otimes I_0 \\ &= I_0\end{aligned}$$ since $\otimes$ is monoidal and $I_0$ must be the unit by [\[cor: graph_unit\]](#cor: graph_unit){reference-type="ref" reference="cor: graph_unit"}. The other parts are analogous. ◻ [\[def:vertex representation\]]{#def:vertex representation label="def:vertex representation"} We define *labelled* vertices as follows: - We say a vertex $w$ in $F_\otimes(E,E)$ is *labelled* $(a,a')$ for $a, a' \in \{s,t\}$ if $F_\otimes(a,a')(I_0) = w$. - We say a vertex $w$ in $F_\otimes(E,V)$ is *labelled* $a$ for $a \in \{s,t\}$ if $F_\otimes(a,\id)(I_0) = w$, and the vertices in $F_\otimes(V,E)$ are labelled analogously. Consider the closed symmetric monoidal product $\square$. If $F_\square$ is such that $\square \iso \lan_{\yo \times \yo} F_\square$, then by [\[Prop: monoidal as kan extensions\]](#Prop: monoidal as kan extensions){reference-type="ref" reference="Prop: monoidal as kan extensions"} we must have $F_\square (V,V)=I_0$, $F_\square (E,V)=F_\square (V,E)=I_1$ and $F _\square(E,E)= C_4$. We also know $F_\square$ also acts on the morphisms $(s,s)$, $(s,t)$, $(t,s)$, and $(t,t)$ by sending them to distinct graph maps from $I_0$ to $C_4$. So, in this case, the four vertices of $F_\square (E,E)$ are each are labelled one of $(s,s)$, $(s,t)$, $(t,s)$, and $(t,t)$, based on where $F_\square (s,s)$, $F_\square (s,t)$, $F_\square (t,s)$, and $F_\square (t,t)$ map $I_0$, as shown below. Many of the following proofs also rely on using the pointwise formula and analyzing the colimit diagrams. As the diagrams can get rather complex, before proceeding it makes sense to first gain some familiarity with what these colimits can look like. We know that objects in $(\yo \times \yo \downarrow (X,X'))_{(E,E)}$ are of the form $((E,E), (f,f') \colon (I_1, I_1) \to (X,X'))$. For simplicity, we refer to such an object as $(f,f')$. [\[comma cat ex\]]{#comma cat ex label="comma cat ex"} Consider the category $(\yo \times \yo \downarrow (I_0,I_1))_{(E,E)}$. This category has four objects, each defined by a map $(I_1,I_1) \to (I_0,I_1)$. As there is only one map from $I_1$ to $I_0$, these objects are completely determined by maps $I_1$ to $I_1$. Call the four objects $\textsf{Idty}$, $\textsf{Flip}$, $\textsf{Const}_s$, and $\textsf{Const}_t$, based how they map $I_1$. Note that we label the vertices of $I_1$ as $s$ and $t$. The resulting diagram and morphisms can be seen below: Here each arrow represents multiple morphisms, as '-' is used to represent any morphism in $\mathbb{G}(E,E)$. The objects represent maps as follows: - The object $\textsf{Idty}$ corresponds to the identity map, sending the vertex $s$ to $s$ and $t$ to $t$. - The object $\textsf{Flip}$ corresponds to the map sending the $s$ vertex to $t$ and the $t$ vertex to $s$. - The object $\textsf{Const}_s$ corresponds to the constant map sending both vertices to the $s$ vertex of $I_1$. - The object $\textsf{Const}_t$ corresponds to the constant map sending both vertices to the $s$ vertex of $I_1$. If $F_\otimes$ is a functor $F_\otimes \colon \mathbb{G} \times \mathbb{G} \to \Graph$, each of these objects get sent to graphs in the colimit diagram by $F_\otimes \circ \pi$, and the morphisms between them to graph maps. With this out of the way, we now proceed with further restricting the possibilities for $F_\otimes$. Note that in the following proofs, for simplicity, when referring to maps such as $F_\otimes(sr,\sigma)$, we omit the $F_\otimes$ and simply write $(sr,\sigma)$. [\[double labels\]]{#double labels label="double labels"} Suppose $F_\otimes(E,E)$ has a vertex with more than one label. Then either $(s,s) = (t,s)$ and $(s,t) = (t,t)$, $(s,s) = (s,t)$ and $(t,s) = (t,t)$, or both. *Proof.* Suppose that $F_\otimes(E,E)$ contains a double-labelled vertex. Then at least one of the following must hold: 1. $(s,s) = (t,s)$ 2. $(s,t) = (t,t)$ 3. $(s,s) = (s,t)$ 4. $(t,s) = (t,t)$ 5. $(s,s)=(t,t)$ We show that (1) implies (2), (3) implies (4), and (5) implies both (1) and (3), and hence also (2) and (4). **(1) $\implies$ (2):** Suppose $(s,s) = (t,s)$. Then: $$\begin{aligned} (s,t) &= (\id, \sigma)(s,s)\\ &=(\id,\sigma)(t,s)\\ &=(t,t) \end{aligned}$$ **(3) $\implies$ (4):** Suppose $(s,s) = (s,t)$. Then: $$\begin{aligned} (t,s) &= (\sigma, \id)(s,s)\\ &=(\sigma,\id)(s,t)\\ &=(t,t) \end{aligned}$$ **(5) $\implies$ (1):** Suppose $(s,s) = (t,t)$. Then: $$\begin{aligned} (s,s) &= (\id, sr)(s,s)\\ &=(\id,sr)(t,t)\\ &=(t,s) \end{aligned}$$ **(5) $\implies$ (3):** Suppose $(s,s) = (t,t)$. Then: $$\begin{aligned} (s,s) &= (sr, \id)(s,s)\\ &=(sr,\id)(t,t)\\ &=(s,t) \end{aligned}$$ Thus, if $F_\otimes(E,E)$ has a double label, either $(s,s) = (t,s)$ and $(s,t) = (t,t)$, $(s,s) = (s,t)$ and $(t,s) = (t,t)$, or both. ◻ [\[no double labels\]]{#no double labels label="no double labels"} The graph $F_\otimes(E,E)$ cannot be such that any vertex has more than one label. *Proof.* Suppose $F_\otimes(E,E)$ contains a double labelled vertex. We show that $\otimes$ is not closed symmetric monoidal. By [\[double labels\]](#double labels){reference-type="ref" reference="double labels"}, we have two possibilities to consider. Either $(s,s) = (t,s)$ and $(s,t) = (t,t)$ or $(s,s) = (s,t)$ and $(t,s) = (t,t)$. For this proof, assume that $(s,s) = (s,t)$ and $(t,s) = (t,t)$. The proof where $(s,s) = (t,s)$ and $(s,t) = (t,t)$ is identical, the coordinates are just reversed. Also note that we are not assuming both are not the case, just that at least $(s,s) = (s,t)$ and $(t,s) = (t,t)$. To show that $\otimes$ is not closed symmetric monoidal, we show that $I_0$ is not the unit, which must be the case if $\otimes$ was closed symmetric monoidal by [\[cor: graph_unit\]](#cor: graph_unit){reference-type="ref" reference="cor: graph_unit"}. Before we begin, first note that the graph $F_\otimes(E,E)$ may have vertices that are unlabelled. These unlabelled vertices are vertices that are not mapped to by any map of the form $F_\otimes(a,b) \colon I_0 \to F_\otimes(E,E)$, where $a,b \in \{s,t\}$. We break the proof into two separate cases, based on how the map $(sr, \id): F_\otimes(E,E) \to F_\otimes(E,E)$ acts on any unlabelled vertices: Case 1: : The map $(sr, \id)$ is not such that $(sr, \id)(a) = b$ for some (not necessarily distinct) unlabelled vertices $a$ and $b$. Case 2: : The map $(sr, \id)$ is such that $(sr, \id)(a) = b$ for some (not necessarily distinct) unlabelled vertices $a$ and $b$. The proof goes roughly as follows: In case (1), we consider the product $I_0 \otimes I_1$. We show that $I_0 \otimes I_1 = I_0$, which contradicts unitality. This is roughly because, in the colimit diagram, all vertices get related through the double labelled vertices along with the maps of the form $(sr, \id)$. In case (2), we consider the product $I_0 \otimes I_3$. We show that $I_0 \otimes I_3 \neq I_3$, again contradicting unitality. This roughly happens because certain unlabelled vertices do not end up getting related to any other vertices, which causes there to be extra vertices in the colimit. More formally, we have: **Case 1:** Suppose the map $(sr, \id)$ sends any unlabelled vertices in $F_\otimes(E,E)$ to labelled ones (or that there are no unlabelled vertices). Consider the product $I_0 \otimes I_1$. Recall from [\[comma cat ex\]](#comma cat ex){reference-type="ref" reference="comma cat ex"} that the category $(\yo \times \yo \downarrow (I_0,I_1))_(E,E)$ looks like: Our goal is to show that all vertices in the colimit diagram get related, and thus belong to the same equivalence class. This would show that the colimit has only one vertex, and is thus $I_0$. First, we show that all labelled vertices are in the same equivalence class. Let $v$ be a vertex labelled $(s,s)$, and thus also $(s,t)$ in $F_\otimes(\pi(\mathsf{Idty}))$. Consider the map $(sr, sr) \colon F_\otimes(\pi(\textsf{Const}_s)) \to F_\otimes(\pi(\textsf{Idty}))$. This map is constant with a value of $v$, since for a vertex $w \in F_\otimes(\pi(\textsf{Const}))$: $$\begin{aligned} (sr,sr)(w) &= (s,s) \circ (r,r)(w)\\ &= (s,s)(I_0) \\ &= v\end{aligned}$$ since $v$ has a the label $(s,s)$. Likewise, there is a constant map $(sr, tr) \colon F_\otimes(\pi(\textsf{Const}_t)) \to F_\otimes(\pi(\textsf{Idty}))$ which is constant with a value of $v$. What we have just shown is that the vertex labelled both $(s,s)$ and $(s,t)$ in $F_\otimes(\pi(\textsf{Idty}))$ belongs to the same equivalence class as all vertices in both $F_\otimes(\pi(\textsf{Const}_s))$ and $F_\otimes(\pi(\textsf{Const}_t))$. We can use the same method, however, to show that this is the case for all labelled vertices in $F_\otimes(\pi(\textsf{Idty}))$ and $F_\otimes(\pi(\textsf{Flip}))$: - For the vertex labelled $(t,s)$ and $(t,t)$ in $F_\otimes(\pi(\textsf{Idty}))$, there are constant maps $(tr,sr)$ and $(tr,tr)$ from $F_\otimes(\pi(\textsf{Const}_s))$ and $F_\otimes(\pi(\textsf{Const}_t))$, respectively. - For the vertex labelled $(s,s)$ and $(s,t)$ in $F_\otimes(\pi(\textsf{Flip}))$, there are constant maps $(sr,tr)$ and $(sr,sr)$ from $F_\otimes(\pi(\textsf{Const}_s))$ and $F_\otimes(\pi(\textsf{Const}_t))$, respectively. - For the vertex labelled $(t,s)$ and $(t,t)$ in $F_\otimes(\pi(\textsf{Flip}))$, there are constant maps $(tr,tr)$ and $(tr,sr)$ from $F_\otimes(\pi(\textsf{Const}_s))$ and $F_\otimes(\pi(\textsf{Const}_t))$, respectively. Thus, all labelled vertices in both $F_\otimes(\pi(\textsf{Idty}))$ and $F_\otimes(\pi(\textsf{Flip}))$ belong to the same equivalence class as all vertices in both $F_\otimes(\pi(\textsf{Const}_s))$ and $F_\otimes(\pi(\textsf{Const}_t))$. The only vertices that remain are potentially unlabelled vertices in $F_\otimes(\pi(\textsf{Idty}))$ and $F_\otimes(\pi(\textsf{Flip}))$, if they exist. By assumption, however, the map $(sr,\id)$ must map any unlabelled vertex to labelled ones. Since there is a map $(sr,\id)$ from both $F_\otimes(\pi(\textsf{Idty}))$ to itself and $F_\otimes(\pi(\textsf{Flip}))$ to itself, we have that any unlabelled vertices in these two graphs must also belong to the same equivalence class as all the labelled vertices. Thus, since there is only one equivalence class of vertices in the colimit, we get $I_0 \otimes I_1 = I_0$, contradicting unitality, so $\otimes$ cannot be closed symmetric monoidal. **Case 2:** Suppose $(sr,\id)$ is such that $(sr, \id)(a) = b$ for some (not necessarily distinct) unlabelled vertices $a$ and $b$. In this case, the trick of considering $I_0 \otimes I_1$ does not work. Instead, we must consider $I_0 \otimes I_3$. This is more complicated, however, and requires some setup before proceeding. We first note that $(sr,\id)(b) = b$. This is because: $$\begin{aligned} (sr, \id)(b) &= (sr, \id)((sr,\id)(a))\\ &=(sr, \id) \circ (sr, \id)(a)\\ &= (sr, \id)(a)\\ &= b\end{aligned}$$ With this in mind, we construct a set of vertices in $F_\otimes(E,E)$ which we call $\mathsf{Inv}^\sigma$. The purpose of this construction, roughly, is to have a set of vertices that do not get related to other vertices in the colimit. First, add $a$ and $b$ into $\mathsf{Inv}^\sigma$. Next, if there exists a vertex $c$ such that $(sr, \id)(c) \in \mathsf{Inv}^\sigma$, add $c$ to $\mathsf{Inv}^\sigma$. Repeat this step until no such $c$ exists. Finally, for every vertex $a \in \mathsf{Inv}^\sigma$, add in $(\id, \sigma)(a)$ and $(\sigma,\id)(a)$. Note that we only need to do this once, since $(\id, \sigma)^2 = (\id, \id) = (\sigma,\id)^2$. Note that $\mathsf{Inv}^\sigma$ cannot contain any labelled vertices. This is because none of $(sr, \id)$, $(\sigma \id)$, or $(\id,\sigma)$ can map a labelled vertex to an unlabelled one. To see this, suppose $v$ is labelled $(x,y)$ for $x,y \in \{s,t\}$. This means that $(x,y)(I_0) = v$. Then by functoriality: $$\begin{aligned} (sr, \id)(v) &= (sr,\id) \circ (x,y)(I_0)\\ &= (srx, \id y)(I_0)\\ &= (s,y)(I_0)\end{aligned}$$ which is another labelled vertex by definition. An identical argument shows that $(\id, \sigma)(v)$ and $(\sigma, \id)(v)$ must also be labelled. So $\mathsf{Inv}^\sigma$ cannot contain any labelled vertex. We thus have a set of unlabelled vertices $\mathsf{Inv}^\sigma$, such that $(sr, \id)(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$, $(\id,\sigma)(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$, $(\sigma, \id)(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$, and for any vertex $v \notin \mathsf{Inv}^\sigma$, $(sr, \id)(v) \notin \mathsf{Inv}^\sigma$. In order to show that the vertices in $\mathsf{Inv}^\sigma$ don't get related to other vertices in the colimit, we have the following claim: **Claim:** - For any map $f$ of the form $(-,\id)$, $f(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$ - For any map $f$ of the form $(-, \sigma)$, $f(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$ - For any map $f$ of the form $(-,\id)$, and any any vertex $v \notin \mathsf{Inv}^\sigma$, $f(w) \notin \mathsf{Inv}^\sigma$ - For any map $f$ of the form $(-, \sigma)$, and any any vertex $v \notin \mathsf{Inv}^\sigma$, $f(w) \notin \mathsf{Inv}^\sigma$ **Proof of claim:** For (1), $(sr, \id)$ and $(\sigma, \id)$ are given by construction. Consider the map $(tr,\id)$. We have that $(sr,\id)(\mathsf{Inv}^\sigma)=(sr,\id) \circ (tr, \id)(\mathsf{Inv}^\sigma)$. But we know $(sr, \id)(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$ and for any $v \notin \mathsf{Inv}^\sigma$, $(sr,\id)(v) \notin \mathsf{Inv}^\sigma$. Thus $(tr, \id)(\mathsf{Inv}^\sigma) \subseteq{\mathsf{Inv}^\sigma}$. For (2), we have four possibilities: $(\id, \sigma)$, $(sr,\sigma)$, $(tr, \sigma)$, and $(\sigma, \sigma)$. By construction, we know that $(\id, \sigma)(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$. Then from (1), the remaining three can all be written as composites of maps $f \circ g$ such that $f(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$ and $g(\mathsf{Inv}^\sigma) \subseteq \mathsf{Inv}^\sigma$. For (3), there are three possibilities to consider: $(sr, \id)$, $(tr,\id)$, and $(\sigma, \id)$. $(sr,\id)$ is given by construction. Now, let $v \notin \mathsf{Inv}^\sigma$. We have that $(sr,\id)(v) = (sr, \id) \circ (tr,\id)(v)$. Thus $(tr, \id)(v) \notin \mathsf{Inv}^\sigma$, else we would have $(sr,\id)(v) \in \mathsf{Inv}^\sigma$, a contradiction. We also have that $v = (\sigma, \id) \circ (\sigma, \id)(v)$. So if $(\sigma, \id)(v) \in \mathsf{Inv}^\sigma$, we would get that $(\sigma, \id)(\mathsf{Inv}^\sigma) \not\subseteq \mathsf{Inv}^\sigma$, which is another contradiction. Finally, for (4), there are four more cases to consider: $(\id, \sigma)$, $(sr, \sigma)$, $(tr,\sigma)$, and $(\sigma, \sigma)$. But $(\id, \sigma)$ can be proven using an identical argument to $(\sigma, \id)$ in (3). Then from (3), the remaining three can be written as compositions of maps $f \circ g$, such that for any $v \notin \mathsf{Inv}^\sigma$ $f(v) \notin \mathsf{Inv}^\sigma$ and $g(v) \notin \mathsf{Inv}^\sigma$. This completes the proof of the claim. Note that in a colimit diagram, every graph will have one of these sets of vertices $\mathsf{Inv}^\sigma$. What we have just shown is that for a given graph, none of the vertices in $\mathsf{Inv}^\sigma$ get related to any other vertices by any maps of the form $(-,\id)$ or $(-, \sigma)$, except possibly vertices in $\mathsf{Inv}^\sigma$ for another graph. We use this fact to arrive at a contradiction when considering $I_0 \otimes I_3$. Consider two vertices $v$ and $w$ connected by an edge in $I_3$. We know there exists a graph in the colimit diagram corresponding to the edge from $v$ to $w$. Call this graph $\mathsf{Idty}_{vw}$. For a graph of this form, we refer to the set $\mathsf{Inv}^\sigma \subset (\mathsf{Idty}_{vw})_V$ by $\mathsf{Inv}^\sigma_{vw}$. There is also a graph $\mathsf{Idty}_{wv}$ corresponding to the edge going in the opposite direction, which is analogous to $\mathsf{Flip}$ in the case of $I_0 \otimes I_1$. There also exist two more objects, one corresponding to the loops on each vertex $v$ and $w$. Call these objects $\mathsf{Const}_v$ and $\mathsf{Const}_w$, respectively. We know that both $\mathsf{Idty}_{vw}$ and $\mathsf{Idty}_{wv}$ both a set of unlabelled vertices $\mathsf{Inv}^\sigma_{vw}$ and $\mathsf{Inv}^\sigma_{wv}$, respectively, which we constructed above. Our goal is to show that no vertex in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$ belongs to an equivalence class containing any vertex not in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$. We showed above that any maps from either $\mathsf{Idty}_{vw}$, or $\mathsf{Idty}_{wv}$, to themselves, or between the two, do not relate the vertices in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$ to any vertices not in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$. This is because maps between these graphs, or to themselves, are of the form $(-,\id)$ and $(-,\sigma)$, as was the case in [\[comma cat ex\]](#comma cat ex){reference-type="ref" reference="comma cat ex"}. There also do not exist any maps from either of these graphs to any other ones, and the only other maps to these graphs are from $\mathsf{Const}_v$ and $\mathsf{Const}_w$. These maps are of the form $(\id, sr)$, $(\id,tr)$, or $(ar,br)$ for $a,b \in \{s,t\}$. We show that these maps must not send any vertex in $\mathsf{Const}_v$ and $\mathsf{Const}_w$ to any vertex $a \in \mathsf{Inv}^\sigma_{vw}$, and the case of $\mathsf{Inv}^\sigma_{wv}$ is identical. The maps $(ar,br)$ are obvious since they're constant to the labelled vertices. For the other maps, we have $(sr, \id) \circ (\id,sr) = (sr,sr)$ which must map all vertices to the vertex $(s,s)$. Then if $(\id,sr)(v) \in \mathsf{Inv}^\sigma_{vw}$, we'd get $(sr,sr)(v) \in \mathsf{Inv}^\sigma_{vw}$ since $(sr,\id)(\mathsf{Inv}^\sigma_{vw}) \subseteq \mathsf{Inv}^\sigma_{wv}$. This is a contradiction, so this cannot be the case. The same can be said for $(tr, \id)$. Thus, for a pair of distinct vertices $v,w$ with an edge between them, no vertex in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$ belongs to an equivalence class containing any vertex not in $\mathsf{Inv}^\sigma_{vw} \cup \mathsf{Inv}^\sigma_{wv}$. Now, for every edge $e$ from vertices $v$ to $w$ in $I_3$, choose a vertex $a_e$ in $\mathsf{Inv}^\sigma_{vw}$, and let $[a_e]$ be the equivalence class in the colimit that $a_e$ belongs to. Then, we can choose edges $e_1, e_2$ and $e_3$, such that no pair of these edges is between the same two vertices. We then have three distinct equivalence classes $[a_{e_1}]$, $[a_{e_2}]$, and $[a_{e_3}]$, such that there is not an edge between any two of them. This is because if vertices $a$ and $b$ are in separate equivalence classes, they must be in separate graphs. Then, in the product $I_0 \otimes I_3$, there exists a set of three vertices, none of which are connected by edges. Since no such set exists in $I_3$, we have that $I_0 \otimes I_3 \neq I_3$, and thus $\otimes$ is not monoidal. ◻ [\[cor:exactly 4\]]{#cor:exactly 4 label="cor:exactly 4"} The graph $F_\otimes(E,E)$ must be a graph of four vertices, such that each vertex has exactly one label. *Proof.* By [\[at most 4\]](#at most 4){reference-type="ref" reference="at most 4"} we know that $F_\otimes(E,E)$ cannot have more than four vertices. We also know by [\[no double labels\]](#no double labels){reference-type="ref" reference="no double labels"} that $F_\otimes(E,E)$ cannot have any double labels. Thus, $F_\otimes(E,E)$ must be a graph with exactly four vertices, as if it had fewer we would have at least one double label. We also have that each vertex in $F_\otimes(E,E)$ must then have exactly one label, since if there was an unlabelled vertex, we would have to again have at least one double-labelled vertex. ◻ We are now ready to prove the main result of this paper: [\[TH:monoidal structures\]]{#TH:monoidal structures label="TH:monoidal structures"} Let $F_\otimes$ and $\otimes$ be as in [\[assumption4\]](#assumption4){reference-type="ref" reference="assumption4"}. Then either $\otimes = \boxtimes$ or $\otimes = \square$. *Proof.* We know by [\[cor:exactly 4\]](#cor:exactly 4){reference-type="ref" reference="cor:exactly 4"} that $F_\otimes(E,E)$ is a graph of four vertices all with distinct labels. Note that since there are no vertices in $F_\otimes(E,E)$ with double labels, then there also cannot be vertices in $F_\otimes(E,V)$ or $F_\otimes(V,E)$ with double labels. If this were the case, the maps from $F_\otimes(E,V)$ or $F_\otimes(V,E)$ would result in double labelled vertices in $F_\otimes(E,E)$. Thus, in both $F_\otimes(E,V)$ and $F_\otimes(V,E)$, one vertex must be labelled $s$ and the other $t$. We now show that $F_\otimes(E,E)$ must be either $C_4$ or $K_4$. As always, label the four vertices $(s,s),(s,t),(t,s)$ and $(t,t)$. We first show that we must have at least the condition that if $(a,a')$ and $(b,b')$ are such that $a=b$ or $a'=b'$ then $(a,a') \sim (b,b')$. First consider $(a,s)$ and $(a,t)$ (the other case is analogous). Suppose there is no edge between $(a,s)$ and $(a,t)$. We show that $F$ is not a functor. Consider the morphism $(a, \id) \colon (V,E) \to (E,E)$. Then the image of this morphism must send the $s$ vertex of $I_1$ to $(a,s)$ and the $t$ vertex to $(a,t)$. But this is not a graph map since $s$ and $t$ have an edge between them in $I_1$, but $(a,s)$ and $(a,t)$ don't in $F_\otimes(E,E)$. The case of $(s,a)$ and $(t,a)$ is identical. Thus $F_\otimes$ is not a functor. So, if $F_\otimes$ is a functor, we must have that $(a,a') \sim (b,b')$ whenever $a=a'$ or $b=b'$. Thus $C_4$ must be a subgraph of $F_\otimes(E,E)$. Now suppose $F_\otimes(E,E)$ has one of the diagonal edges. Assume that $(s,s) \sim (t,t)$. We know that $(\sigma, \id)$ must be a graph map. But this map sends $(s,s)$ to $(t,s)$ and $(t,t)$ to $(s,t)$. Since it's a graph map we get that $(s,t) \sim (t,s)$. Similarly if $(s,t) \sim (t,s)$ we must have $(s,s) \sim (t,t)$. So if $F_\otimes(E,E)$ is not $C_4$, then it must be $K_4$. Note that if $F_\otimes(E,E)$ is $K_4$ or $C_4$ with no unlabelled vertices, then the action of $F_\otimes$ on any morphism is fully determined by its action on $(s,s), (s,t),(t,s)$ and $(t,t)$ which are already defined. Finally, by [\[Prop: monoidal as kan extensions\]](#Prop: monoidal as kan extensions){reference-type="ref" reference="Prop: monoidal as kan extensions"}, the first case corresponds to the functor resulting in the product $\square$ and the second case corresponds to the functor resulting in the product $\boxtimes$. Thus, if $F_\otimes$ is a functor such that $\otimes$ is a closed symmetric monoidal structure, then $\otimes$ must be either $\boxtimes$ or $\square$. ◻ [\[main th\]]{#main th label="main th"} There are only two closed symmetric monoidal products on the category $\Graph$: the box and categorical products. *Proof.* By [\[TH:monoidal structures\]](#TH:monoidal structures){reference-type="ref" reference="TH:monoidal structures"}, we know that if a closed symmetric monoidal product $\otimes$ arises as a Kan extension $\otimes \iso \lan_{\yo \times \yo} F_\otimes$, then either $\otimes \iso \square$ or $\otimes \iso \boxtimes$. But, by [\[Prop: monoidal as kan extensions\]](#Prop: monoidal as kan extensions){reference-type="ref" reference="Prop: monoidal as kan extensions"}, every closed symmetric monoidal product must arise in this manner. ◻

arxiv_math

--- abstract: | We prove that the smallest elements of Shi parts and cone type parts exist and form Garside shadows. The latter resolves a conjecture of Parkinson and the second author as well as a conjecture of Hohlweg, Nadeau and Williams. address: - Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada - Department of Mathematics and Statistics, 1 University Heights, University of North Carolina Asheville, Asheville, NC 28804, USA author: - Piotr Przytycki$^{\dag}$ - Yeeka Yau$^\dag$ title: A Pair of Garside shadows --- [^1] [^2] # Introduction {#sec:introd} A *Coxeter group* $W$ is a group generated by a finite set $S$ subject only to relations $s^2=1$ for $s\in S$ and $(st)^{m_{st}}=1$ for $s\neq t\in S$, where $m_{st}=m_{ts}\in \{2,3,\ldots,\infty\}$. Here the convention is that $m_{st}=\infty$ means that we do not impose a relation between $s$ and $t$. By $X^1$ we denote the *Cayley graph* of $W$, that is, the graph with vertex set $X^0=W$ and with edges (of length $1$) joining each $g\in W$ with $gs$, for $s\in S$. For $g\in W$, let $\ell(g)$ denote the *word length* of $g$, that is, the distance in $X^1$ from $g$ to $\mathrm{id}$. We consider the action of $W$ on $X^0=W$ by left multiplication. This induces an action of $W$ on $X^1$. For $r\in W$ a conjugate of an element of $S$, the *wall* $\mathcal W_r$ of $r$ is the fixed point set of $r$ in $X^1$. We call $r$ the *reflection* in $\mathcal W_r$ (for fixed $\mathcal W_r$ such $r$ is unique). Each wall $\mathcal W$ separates $X^1$ into two components, called *half-spaces*, and a geodesic edge-path in $X^1$ intersects $\mathcal W$ at most once [@Ronan_2009 Lem 2.5]. Consequently, the distance in $X^1$ between $g,h\in W$ is the number of walls separating $g$ and $h$. We consider the partial order $\preceq$ on $W$ (called the 'weak order' in algebraic combinatorics), where $p\preceq g$ if $p$ lies on a geodesic in $X^1$ from $\mathrm{id}$ to $g$. Equivalently, there is no wall separating $p$ from both $\mathrm{id}$ and $g$. **Shi parts.** Let $\mathcal E$ be the set of walls $\mathcal W$ such that there is no wall separating $\mathcal W$ from $\mathrm{id}$ (these walls correspond to so-called 'elementary roots'). The components of $X^1\setminus \bigcup \mathcal E$ are *Shi components*. For a Shi component $Y$, we call $P = Y \cap X^0$ the corresponding *Shi part*. Our first result is the following. **Theorem 1**. *Let $P$ be a Shi part. Then $P$ has a smallest element with respect to $\preceq$.* Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"} was proved independently in a more general form by Dyer, Fishel, Hohlweg and Mark in [@DHFM Theorem 1.1(1)]). Here we give a short proof following the lines of the proof of a related result of the first author and Osajda [@OP Thm 2.1]. In [@Shi_1987], Shi proved Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"} for affine $W$. The family $\mathcal E$, which is finite by [@Brink1993], has been extensively studied ever since and has become an important object in algebraic combinatorics, geometric group theory and representation theory. See for example see the survey article [@fishel2020]. By [@Brink1993], Shi parts are in correspondence with the states of an automaton recognising the language of reduced words of the Coxeter group. This partition of a Coxeter group is thus one of the primary examples of 'regular' partitions, see [@Parkinson-Yau_2022]. For $g\in W$, let $m(g)$ be the smallest element in the Shi part containing $g$, guaranteed by Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"}. Let $M\subset W$ be the set of elements of the form $m(g)$ for $g\in W$. The $\emph{join}$ of $g,g'\in W$ is the smallest element $h$ (if it exists) satisfying $g\preceq h$ and $g'\preceq h$. A subset $B\subseteq W$ is a *Garside shadow* if it contains $S$, contains $g^{-1}h$ for every $h\in B$ and $g\preceq h$, and contains the join, if it exists, of every $g,g'\in B$. **Theorem 2**. *$M$ is a Garside shadow.* Theorem [Theorem 2](#thm:Shi2){reference-type="ref" reference="thm:Shi2"} was also obtained in [@DHFM Thm 1.1(2)], where the authors showed that $M$ is the set of so-called 'low elements' introduced in [@DH]. We give an alternative proof using 'bipodality', a notion introduced in [@DH] and rediscovered in [@OP]. **Cone type parts.** For each $g\in W$, let $T(g)=\{h\in W \ | \ \ell(gh)=\ell(g)+\ell(h)\}$. For $T\subset W$, the *cone type part* $Q(T)\subset W$ is the set of all $g^{-1}$ with $T(g)=T$. In other words, $Q(T)$ consists of $g$ such that $T$ is the set of vertices on geodesic edge-paths starting at $g$ and passing through $\mathrm{id}$ that appear after $\mathrm{id}$, including $\mathrm{id}$. We obtain a new proof of the following. **Theorem 3**. *[@Parkinson-Yau_2022 Thm 1] Let $Q$ be a cone type part. Then $Q$ has a smallest element with respect to $\preceq$.* For $g\in W$, let $\mu(g)$ be the smallest element in the cone type part containing $g$. Let $\Gamma\subset W$ be the set of elements of form $\mu(g)$ for $g\in W$ These elements are called the *gates* of the cone type partition in [@Parkinson-Yau_2022]. We also obtain the following new result, confirming in part [@Parkinson-Yau_2022 Conj 1]. **Theorem 4**. *For any $g,g'\in \Gamma$, if the join of $g$ and $g'$ exists, then it belongs to $\Gamma$.* By [@Parkinson-Yau_2022 Prop 4.27(i)], this implies that $\Gamma$ is a Garside shadow. Furthermore, $\Gamma$ is the set of states of a the minimal automaton (in terms of the number of states) recognising the language of reduced words of a Coxeter group. This verifies [@HNW Conj 1].\ The paper is organised as follows. In Section [2](#sec:shi_components){reference-type="ref" reference="sec:shi_components"} we discuss 'bipodality' and use it to prove Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"} and Theorem [Theorem 2](#thm:Shi2){reference-type="ref" reference="thm:Shi2"}. In Section [3](#sec:cone_type_components){reference-type="ref" reference="sec:cone_type_components"} we focus on the cone type parts and give the proofs of Theorem [Theorem 3](#thm:conetype){reference-type="ref" reference="thm:conetype"} and Theorem [Theorem 4](#thm:conetype2){reference-type="ref" reference="thm:conetype2"}. **Acknowledgements.** We thank Christophe Hohlweg and Damian Osajda for discussions and feedback. # Shi parts {#sec:shi_components} The following property was called *bipodality* in [@DH]. It was rediscovered in [@OP]. **Definition 5**. Let $r,q\in W$ be reflections. Distinct walls $\mathcal{W}_r,\mathcal{W}_{q}$ *intersect*, if $\mathcal{W}_r$ is not contained in a half-space for $\mathcal{W}_{q}$ (this relation is symmetric). Equivalently, $\langle r,q\rangle$ is a finite group. We say that such $r,q$ are *sharp-angled*, if $r$ and $q$ do not commute and $\{r,q\}$ is conjugate into $S$. In particular, there is a component of $X^1\setminus (\mathcal{W}_r\cup \mathcal{W}_q)$ whose intersection $F$ with $X^0$ is a fundamental domain for the action of $\langle r,q\rangle$ on $X^0$. We call such $F$ a *geometric fundamental domain for $\langle r,q\rangle$*. **Lemma 6** ( [@OP Lem 3.2], special case of [@DH Thm 4.18]). *Suppose that reflections $r,q\in W$ are sharp-angled, and that $g\in W$ lies in a geometric fundamental domain for $\langle r,q\rangle$. Assume that there is a wall $\mathcal{U}$ separating $g$ from $\mathcal{W}_r$ or from $\mathcal{W}_q$. Let $\mathcal{W}'$ be a wall distinct from $\mathcal{W}_r,\mathcal{W}_q$ that is the translate of $\mathcal{W}_r$ or $\mathcal{W}_q$ under an element of $\langle r,q\rangle$. Then there is a wall $\mathcal{U}'$ separating $g$ from $\mathcal{W}'$.* The following proof is surprisingly the same as that for a different result [@OP Thm 2.1]. *Proof of Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"}.* Let $P=Y\cap X^0$, where $Y$ is a Shi component. It suffices to show that for each $p_0,p_n\in P$ there is $p\in P$ satisfying $p_0\succeq p\preceq p_n$. Let $(p_0,p_1,\ldots,p_n)$ be the vertices of a geodesic edge-path $\pi$ in $X^1$ from $p_0$ to $p_n$, which lies in $Y$. Let $L=\max_{i=0}^n\ell(p_i)$. We will now modify $\pi$ and replace it by another embedded edge-path from $p_0$ to $p_n$ with vertices in $P$, so that there is no $p_i$ with $p_{i-1}\prec p_i\succ p_{i+1}$. Then we will be able to choose $p$ to be the smallest $p_i$ with respect to $\preceq$. If $p_{i-1}\prec p_i\succ p_{i+1}$, then let $\mathcal{W}_r,\mathcal{W}_q$ be the (intersecting) walls separating $p_i$ from $p_{i-1},p_{i+1}$, respectively. Moreover, if $r$ and $q$ do not commute, then $r,q$ are sharp-angled, with $\mathrm{id}$ in a geometric fundamental domain for $\langle r,q\rangle$. We claim that all the elements of the *residue* $R=\langle r,q\rangle (p_i)$ lie in $P$. Indeed, since $p_{i-1},p_{i+1}$ are both in $P$, we have that $\mathcal{W}_r,\mathcal{W}_q\notin \mathcal E$. It remains to justify that each wall $\mathcal{W}'\neq \mathcal{W}_r,\mathcal{W}_q$ that is the translate of $\mathcal{W}_r$ or $\mathcal{W}_q$ under an element of $\langle r,q\rangle$ does not belong to $\mathcal E$. We can thus assume that $r$ and $q$ do not commute, since otherwise there is no such $\mathcal{W}'$. Since $\mathcal{W}_r\notin \mathcal E$, there is a wall $\mathcal{U}$ separating $\mathrm{id}$ from $\mathcal{W}_r$. By Lemma [Lemma 6](#lem:key){reference-type="ref" reference="lem:key"}, there is a wall $\mathcal{U}'$ separating $\mathrm{id}$ from $\mathcal{W}'$, justifying the claim. We now replace the subpath $(p_{i-1},p_i,p_{i+1})$ of $\pi$ by the second embedded edge-path with vertices in the residue $R$ from $p_{i-1}$ to $p_{i+1}$. Since all the elements of $R$ are $\prec p_i$ [@Ronan_2009 Thm 2.9], this decreases the complexity of $\pi$ defined as the tuple $(n_L,\ldots,n_2,n_1)$, where $n_j$ is the number of $p_i$ in $\pi$ with $\ell(p_i)=j$, with lexicographic order. After possibly removing a subpath, we can assume that the new edge-path is embedded. After finitely many such modifications, we obtain the desired path. ◻ **Lemma 7**. *For $g\preceq h$, we have $m(g)\preceq m(h)$.* *Proof.* Let $k$ be the minimal number of distinct Shi components traversed by a geodesic edge-path $\gamma$ from $h$ to $g$. We proceed by induction on $k$, where for $k=1$ we have $m(g)=m(h)$. Suppose now $k>1$. If a neighbour $f$ of $h$ on $\gamma$ lies in the same Shi component as $h$, then we can replace $h$ by $f$. Thus we can assume that $f$ lies in a different Shi component than $h$. Consequently, the wall $\mathcal{W}_r$ separating $h$ from $f$ belongs to $\mathcal E$. Since $g\preceq f$, by the inductive assumption we have $m(g)\preceq m(f)$. Thus it suffices to prove $m(f)\preceq m(h)$. In the first case, where for every neighbour $h'$ of $h$ on a geodesic edge-path from $h$ to $\mathrm{id}$, the wall separating $h$ from $h'$ belongs to $\mathcal E$, we have $h=m(h)$ and we are done. Otherwise, let $\mathcal{W}_q$ be such a wall separating $h$ from $h'$ outside $\mathcal E$. If $r$ and $q$ do not commute, then $r,q$ are sharp-angled, with $\mathrm{id}$ in a geometric fundamental domain for $\langle r,q\rangle$. By Lemma [Lemma 6](#lem:key){reference-type="ref" reference="lem:key"}, among the walls in $\langle r,q\rangle\{\mathcal{W}_r,\mathcal{W}_q\}$ only $\mathcal{W}_r$ belongs to $\mathcal E$. Let $\bar h, \bar f$ be the vertices opposite to $f,h$ in the residue $\langle r,q\rangle h$. We have $m(\bar h)=m(h), m(\bar f)=m(f)$. Replacing $h,f$ by $\bar h,\bar f$, and possibly repeating this procedure finitely many times, we arrive at the first case. ◻ Lemma [Lemma 7](#lem:shadow){reference-type="ref" reference="lem:shadow"} has the following immediate consequence. **Corollary 8**. *For any $g,g'\in M$, if the join of $g$ and $g'$ exists, then it belongs to $M$.* For completeness, we include the proof of the following. **Lemma 9** ([@DH Prop 4.16]). *For any $h \in M$ and $g \preceq h$, we have $g^{-1}h \in M$.* *Proof.* For any neighbour $h'$ of $h$ on a geodesic edge-path from $h$ to $g$, the wall $\mathcal{W}$ separating $h$ from $h'$ belongs to $\mathcal E$. Consequently, we also have $g^{-1}\mathcal{W}\in \mathcal E$, and so $g^{-1}h\in M$. ◻ Also note that for each $s\in S$, we have $\mathcal W_s\in \mathcal E$ and so $m(s)=s$ implying $S\subset M$. Thus Corollary [Corollary 8](#cor:Shi){reference-type="ref" reference="cor:Shi"} and Lemma [Lemma 9](#lem:suffix_shi){reference-type="ref" reference="lem:suffix_shi"} imply Theorem [Theorem 2](#thm:Shi2){reference-type="ref" reference="thm:Shi2"}. # Cone type parts {#sec:cone_type_components} Let $T=T(g)$ for some $g\in W$. We denote by $\partial T$ the set of walls separating adjacent vertices $h\in T$ and $h'\notin T$. In particular, the walls in $\partial T$ separate $\mathrm{id}$ from $g^{-1}$. We note that one of the primary differences between the cone type parts and the Shi parts is that the cone type parts do not correspond to a 'hyperplane arrangement'. See for example Figure [1](#fig:conetype_arrangement334){reference-type="ref" reference="fig:conetype_arrangement334"}. {#fig:conetype_arrangement334} **Remark 10**. Note that for $g,g'\in Q(T)$ any geodesic edge-path from $g$ to $g'$ has all vertices $f$ in $Q(T)$. Indeed, for $h\in T$, any wall separating $\mathrm{id}$ from $f$ separates $\mathrm{id}$ from $g$ or $g'$ and so it does not separate $\mathrm{id}$ from $h$. Thus $h\in T(f^{-1})$ and so $T\subseteq T(f^{-1})$. Conversely, if we had $T\subsetneq T(f^{-1})$ then there would be a vertex $h\in T$ with a neighbour $h'\in T(f^{-1})\setminus T$ separated from $h$ by a wall $\mathcal{W}$ (in $\partial T$) that does not separate $h$ from $f$. The wall $\mathcal{W}$ would not separate $h'$ from $g$ or $g'$, contradicting $h'\notin T(g^{-1})$ or $h'\notin T(g'^{-1})$. See also [@Parkinson-Yau_2022 Thm 2.14] for a more general statement. *Proof of Theorem [Theorem 3](#thm:conetype){reference-type="ref" reference="thm:conetype"}.* The proof is identical to that of Theorem [Theorem 1](#thm:Shi){reference-type="ref" reference="thm:Shi"}, with $P$ replaced by $Q$. The vertices of a geodesic edge-path $\pi$ in $X^1$ from $p_0$ to $p_n$ belong to $Q$ by Remark [Remark 10](#rem:convex){reference-type="ref" reference="rem:convex"}. We also make the following change in the proof of the claim that all the elements of $R=\langle r,q\rangle (p_i)$ lie in $Q$. Namely, since $T=T(p^{-1}_i)$ equals $T(p^{-1}_{i-1})$, we have $\mathcal{W}_r\notin \partial T$. Analogously we obtain $\mathcal{W}_q\notin \partial T$. If $r$ and $q$ do not commute, we have that $T$ is contained in a geometric fundamental domain for $\langle r,q\rangle$, and so we also have $\mathcal{W}'\notin \partial T$ for any $\mathcal{W}'$ that is a translate of $\mathcal{W}_r$ or $\mathcal{W}_q$ under an element of $\langle r,q\rangle$. This justifies the claim. ◻ *Proof of Theorem [Theorem 4](#thm:conetype2){reference-type="ref" reference="thm:conetype2"}.* The proof structure is similar to that of Lemma [Lemma 7](#lem:shadow){reference-type="ref" reference="lem:shadow"}. We need to justify that for $g\preceq h$, we have $\mu(g)\preceq \mu(h)$, where we induct on the minimal number $k$ of distinct cone type components traversed by a geodesic edge-path $\gamma$ from $h$ to $g$. Suppose $k>1$, and let $Q=Q(T)$ be the cone type component containing $h$. If a neighbour $f$ of $h$ on $\gamma$ lies in $Q$, then we can replace $h$ by $f$. Thus we can assume $f\notin Q$. Consequently, the wall $\mathcal{W}_r$ separating $h$ from $f$ belongs to $\partial T$. Since $g\preceq f$, by the inductive assumption we have $\mu(g)\preceq \mu(f)$. Thus it suffices to prove $\mu(f)\preceq \mu(h)$. If for every neighbour $h'$ of $h$ on a geodesic edge-path from $h$ to $\mathrm{id}$, the wall separating $h$ from $h'$ belongs to $\partial T$, we have $h=\mu(h)$ and we are done. Otherwise, let $\mathcal{W}_q$ be such a wall separating $h$ from $h'$ outside $\partial T$. Let $\bar h, \bar f$ be the vertices opposite to $f,h$ in the residue $\langle r,q\rangle h$, and let $f'=rqh$. It suffices to prove $\mu(\bar h)=\mu(h), \mu(\bar f)=\mu(f)$. To justify $\mu(\bar h)=\mu(h)$, or, equivalently, $\bar h\in Q$, it suffices to observe that among the walls in $\langle r,q\rangle\{\mathcal{W}_r,\mathcal{W}_q\}$ only $\mathcal{W}_r$ belongs to $\partial T$: Indeed, if $r$ and $q$ do not commute, then $r,q$ are sharp-angled, with $T$ in the geometric fundamental domain $F$ for $\langle r,q\rangle$ containing $\mathrm{id}$. It remains to justify $\mu(\bar f)=\mu(f)$, or, equivalently, $T(\bar f^{-1})=\widetilde T$ for $\widetilde T=T(f^{-1})$. Since $\widetilde T\cap F=T$, to show, for example, $T(f'^{-1})=\widetilde T$, it suffices to show that the wall $\mathcal{W}=r\mathcal{W}_q$ does not belong to $\partial \widetilde T$. Otherwise, let $b\in \widetilde T$ be adjacent to $\mathcal{W}$. Then $rb\in F$ is adjacent to $\mathcal{W}_q$, which is outside $\partial T$. Consequently, $rb\notin T$. Thus there is a wall $\mathcal{W}'$ separating $\mathrm{id}$ from $h$ and $rb$. Note that $\mathcal{W}'\neq \mathcal{W}_r$ and so $\mathcal{W}'$ separates $\mathrm{id}$ from $f$. Since $\mathrm{id}$ lies on a geodesic edge-path from $f$ to $b$, we have that $\mathcal{W}'$ does not separate $\mathrm{id}$ from $b$. Thus $r\mathcal{W}'$ separates $r$ and $rb$ from $f,h,b$, and $\mathrm{id}$, since, again, $\mathrm{id}$ lies on a geodesic edge-path from $f$ to $b$. Consider the distinct connected components $\Lambda_1,\Lambda_2,\Lambda_3,\Lambda_4$ of $X^1\setminus (\mathcal{W}_r\cup r\mathcal{W}')$ with $\mathrm{id}\in \Lambda_1,b\in \Lambda_2,r\in \Lambda_3,rb\in \Lambda_4$. Since $\mathrm{id}$ and $r$ are interchanged by the reflection $r$ and they lie in the opposite connected components, we have $r\Lambda_2\subsetneq \Lambda_1$. On the other hand, since $b$ and $rb$ lie in the opposite connected components, we have $r\Lambda_1\subsetneq \Lambda_2$, which is a contradiction. This proves that the wall $\mathcal{W}$ does not belong to $\partial \widetilde T$, and hence neither does any other wall in $\langle r,q\rangle\{\mathcal{W}_r,\mathcal{W}_q\}$. Consequently $T(\bar f^{-1})=\widetilde T$, as desired. ◻ [^1]: $\dag$ Partially supported by NSERC and (Polish) Narodowe Centrum Nauki, UMO-2018/30/M/ST1/00668 [^2]: $\dag$ Partially supported by the National Science Foundation under Award No. 2316995.

arxiv_math

--- abstract: | In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passive structurally complete varieties of $\mathsf{MTL}$-algebras, i.e., bounded commutative integral residuated lattices generated by chains. address: - DIISM, Università di Siena, Siena, Italy - Artificial Intelligence Research Institute (IIIA), CSIC, Barcelona, Spain author: - Paolo Aglianò - Sara Ugolini title: | Structural and universal completeness\ in algebra and logic --- # Introduction The main aim of this paper is to explore some connections between algebra and logic; mainly, we try to produce some *bridge theorems*. A bridge theorem is a statement connecting logical (and mostly synctactical) features of deductive systems and properties of classes of algebras; this connection is usually performed using the tools of general algebra and the rich theory that is behind it. The main reason behind this kind of exploration is in the further understanding one can gain by connecting two apparently distant fields. In this way, we can explore logical properties in purely algebraic terms; at the same time statements can be imported from logic that have an important and often new algebraic meaning. The set of logical problems we want to explore is connected with the concept of *structural completeness* of a deductive system, in the different ways it can be declined. For a deductive system, being structurally complete means that each of its proper extensions admits new theorems. This notion can be formalized in a more rigorous way, using the concept of *admissible rule*. A rule is admissible in a logic if, whenever there is a substitution making its premises a theorem, such substitution also makes the conclusion a theorem. A logic is then structurally complete if all its admissible rules are derivable in the system. It is well-known that classical logic is structurally complete; intuitionistic logic is not but it satisfies a weaker although still interesting notion: it is *passively* structurally complete. We will see that this is not just a feature of intuitionism but it can be explained in a much more general framework, and it is connected to the way the contradictions of classical logic are treated. In more details, passive structural completeness means that all rules that do not apply to theorems are derivable. Naturally, the dual notion of *active* structural completeness also arises, which instead isolates the derivability of those rules for which there exists a substitution making their premises a theorem. The latter notion has been explored in generality in [@DzikStronkowski2016]. Structural completeness and its hereditary version have been deeply studied in the literature: e.g., in general algebraic terms in [@Bergman1991], in substructural logics in [@OlsonRafteryVanAlten2008], in fuzzy logics in [@CintulaMetcalfe2009], in intermediate logics in [@Citkin1978]. A natural extension of this kind of problems is to consider *clauses* instead of rules. A clause is a formal pair $\Sigma \Rightarrow \Delta$, where both $\Sigma$ and $\Delta$ are finite sets of formulas over a suitable language. A clause is then admissible if a substitution making all the formulas in $\Sigma$ into theorems makes at least one of the formulas in $\Delta$ a theorem. Likewise, a clause is derivable if at least one of the formulas in $\Delta$ is derivable from $\Sigma$. A logic is *universally complete* if every admissible clause is derivable in it. It is then also possible to investigate the situation in which admissible clauses are active or passive in a deductive system, and thus the corresponding notions of universal completeness. Universal completeness in connection to admissible clauses has been studied in [@CabrerMetcalfe2015a]. The way in which our bridge theorems will be created exploits the machinery of the so-called Blok-Pigozzi connection [@BlokPigozzi1989]. Without going into details, this machinery allows us to express purely logical concepts in an algebraic language. The advantage of doing so is evident: on one hand we can use the entire wealth of results about classes of algebras and various algebraic operators. On the other hand, very often by mean of this translation one ends up with algebraic results that are interesting in their own nature, irregardless of their logical origin. While not every logical system admits this translation, many interesting and/or classical systems do: classical and intuitionistic logic, relevance logics, substructural logics in general, many-valued logics, many modal logics and so on. In this framework, one can translate the previously described notions of structural and universal completeness into properties of the quasiequational or universal theory of a quasivariety of algebras. In this setting, we will rephrase the notions of interest not in terms of formulas, but in terms of equations in a suitable language. In this manuscript our aim is twofold; on one side we will try to describe in a complete and organic way (as much as it is possible) the phenomena mentioned above and the relations among them. In particular, we will recall the existing results trying to put them in a coherent perspective, which we believe is currently lacking, and we will provide many examples. On the other side, we will provide new results and novel characterizations of those notions that are missing an effective algebraic description. More specifically, we will first show how the characterization of active structural completeness in [@DzikStronkowski2016] can be extended to describe active universal completeness. Moreover, we will give algebraic descriptions of the notions of passive universal and structural completeness and the latter will result in an effective characterization. As a particularly interesting consequence, we show that a substructural logic satisfying the weakening rule is passively structurally complete if and only if every contradiction of classical logic is explosive in it. This generalizes and explains the passive structural completeness of intuitionistic logic. Moreover, it entails that all substructural logics (with weakening) with the *Glivenko property* with respect to classical logic are passively structurally complete. Further specializing the general result, we build on it to provide a clear characterization (and an axiomatization) of the minimal passive structurally complete logic that is an axiomatic extension of the t-norm based logic $\mathcal{MTL}$. From the algebraic side, this means that we characterize the passive structurally complete quasivarieties of bounded commutative integral residuated lattices generated by chains. The techniques we will employ in our study are the ones proper of general algebra. In particular, we will use the understanding of algebraic objects such as projective and exact algebras. The same objects are known to be relevant for the algebraic study of unification problems in algebraizable logics [@Ghilardi1997]. In fact, we will show how the notion of unifiability of a set of formulas (or, equivalently, a set of equations) plays a major role in our results. The structure of this manuscript is as follows. In the next section we will discuss the needed preliminary notions. In particular, the Blok-Pigozzi connection, projective and exact algebras, algebraic unification, and finally, we define the notions of structural and universal completeness. Section [3](#univquasi){reference-type="ref" reference="univquasi"} is devoted to universal completeness, and Section [4](#structprim){reference-type="ref" reference="structprim"} to structural completeness, both in their various declinations. The last section is devoted to a deeper understanding of some relevant examples from the realms of algebra and (algebraic) logic respectively. In particular, in Subsection [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"} we apply our results to the variety of (bounded) lattices; finally, in Subsection [5.2](#sec:FL){reference-type="ref" reference="sec:FL"}, we prove the aforementioned results and more about substructural logics. # Preliminaries {#sec: start} ## Universal algebra and the Blok-Pigozzi connection {#subsec:universal} Let $\mathsf {K}$ be a class of algebras; we denote by ${\mathbf I},{\mathbf H},{\mathbf P},{\mathbf S},{\mathbf P}_u$ the class operators sending $\mathsf {K}$ in the class of all isomorphic copies, homomorphic images, direct products, subalgebras and ultraproducts of members of $\mathsf {K}$. The operators can be composed in the obvious way; for instance ${\mathbf S}{\mathbf P}(\mathsf {K})$ denotes all algebras that are embeddable in a direct product of members of $\mathsf {K}$; moreover there are relations among the classes resulting from applying operators in a specific orders, for instance ${\mathbf P}{\mathbf S}(\mathsf {K}) \subseteq{\mathbf S}{\mathbf P}(\mathsf {K})$ and ${\mathbf H}{\mathbf S}{\mathbf P}(\mathsf {K})$ is the largest class we can obtain composing the operators. We will use all the known relations without further notice, but the reader can consult [@Pigozzi1972] or [@BurrisSanka] for a textbook treatment. If $\rho$ is a type of algebras, an *equation* is a pair $p,q$ of $\rho$-terms (i.e. elements of the absolutely free algebra ${\textbf{\upshape T}}_\rho(\omega)$) that we write suggestively as $p \approx q$; a *universal sentence* or *clause* in $\rho$ is a formal pair $(\Sigma, \Gamma)$ that we write as $\Sigma \Rightarrow \Gamma$, where $\Sigma,\Gamma$ are finite sets of equations; a universal sentence is a *quasiequation* if $|\Gamma| = 1$ and it is is *negative* if $\Delta= \emptyset$. Clearly an equation is a quasiequation in which $\Sigma = \emptyset$. Given any set of variables $X$, an assignment of $X$ into an algebra $A$ of type $\rho$ is a function $h$ mapping each variable $x \in X$ to an element of ${\textbf{\upshape A}}$, that extends (uniquely) to a homomorphism (that we shall also call $h$) from the term algebra ${\textbf{\upshape T}}_\rho(\omega)$ to ${\textbf{\upshape A}}$. An algebra ${\textbf{\upshape A}}$ satisfies an equation $p \approx q$ with an assignment $h$ (and we write ${\textbf{\upshape A}}, h \models p \approx q$) if $h(p) = h(q)$ in ${\textbf{\upshape A}}$. An equation $p \approx q$ is *valid* in ${\textbf{\upshape A}}$ (and we write ${\textbf{\upshape A}} \vDash p \approx q$) if for all assignments $h$ in ${\textbf{\upshape A}}$, ${\textbf{\upshape A}}, h \models p \approx q$; if $\Sigma$ is a set of equations then ${\textbf{\upshape A}} \vDash \Sigma$ if ${\textbf{\upshape A}} \vDash \sigma$ for all $\sigma \in \Sigma$. A universal sentence is *valid* in ${\textbf{\upshape A}}$ (and we write ${\textbf{\upshape A}} \vDash \Sigma \Rightarrow \Delta$) if for all assignments $h$ to ${\textbf{\upshape A}}$, $h(p) = h(q)$ for all $p \approx q \in \Sigma$ implies that there is an identity $s \approx t \in \Delta$ with $h(s) = h(t)$; in other words a universal sentence can be understood as the formula $\forall \mathbf x(\bigwedge \Sigma \rightarrow\bigvee \Delta)$. An equation or a universal sentence is *valid* in a class $\mathsf {K}$ if it is valid in all algebras in $\mathsf {K}$. A class of algebras is a variety if it is closed under ${\mathbf H}, {\mathbf S}$ and ${\mathbf P}$, a quasivariety if it is closed under ${\mathbf I}$,${\mathbf S}$,${\mathbf P}$ and ${\mathbf P}_u$ and a universal class if it is closed under ${\mathbf I}{\mathbf S}{\mathbf P}_u$. The following facts were essentially discovered by A. Tarski , J. Łòs and A. Lyndon in the pioneering phase of model theory; for proof of this and similar statements the reader can consult [@ChangKeisler]. **Lemma 1**. *Let $\mathsf {K}$ be any class of algebras. Then:* 1. *$\mathsf {K}$ is a universal class if and only if ${\mathbf I}{\mathbf S}{\mathbf P}_u(\mathsf {K}) = \mathsf {K}$ if and only if it is the class of algebras in which a set of universal sentences is valid;* 2. *$\mathsf {K}$ is a quasivariety if and only if ${\mathbf I}{\mathbf S}{\mathbf P}{\mathbf P}_u(\mathsf {K}) = \mathsf {K}$ if and only if it is the class of algebras in which a set of quasiequations is valid;* 3. *$\mathsf {K}$ is a variety if and only if ${\mathbf H}{\mathbf S}{\mathbf P}(\mathsf {K}) = \mathsf {K}$ if and only if it is the class of algebras in which a set of equations is valid.* **Notation 1**. We will often write ${\mathbf V}$ for ${\mathbf H}{\mathbf S}{\mathbf P}$ and ${\mathbf Q}$ for ${\mathbf I}{\mathbf S}{\mathbf P}{\mathbf P}_u$. For the definition of free algebras in a class $\mathsf {K}$ on a set $X$ of generators, in symbols ${\textbf{\upshape F}}_\mathsf {K}(X)$, we refer to [@BurrisSanka]; we merely observe that every free algebra on a class $\mathsf {K}$ belongs to ${\mathbf I}{\mathbf S}{\mathbf P}(\mathsf {K})$. It follows that every free algebra in $\mathsf {K}$ is free in ${\mathbf I}{\mathbf S}{\mathbf P}(\mathsf {K})$ and therefore for any quasivariety $\mathsf {Q}$, ${\textbf{\upshape F}}_\mathsf {Q}(X) = {\textbf{\upshape F}}_{{\mathbf V}(\mathsf {Q})}(X)$. There are two fundamental results that we will be using many times and deserve a spotlight. Let ${\textbf{\upshape B}}, ({\textbf{\upshape A}}_i)_{i \in I}$ be algebras in the same signature; we say that ${\textbf{\upshape B}}$ *embeds* in $\prod_{i \in I} {\textbf{\upshape A}}_i$ if ${\textbf{\upshape B}} \in {\mathbf I}{\mathbf S}(\prod_{i\in I} {\textbf{\upshape A}}_i)$. Let $p_i$ be the $i$-th projection, or better, the composition of the embedding and the $i$-th projection, from ${\textbf{\upshape B}}$ to ${\textbf{\upshape A}}_i$; the embedding is *subdirect* if for all $i \in I$, $p_i({\textbf{\upshape B}}) = {\textbf{\upshape A}}_i$ and in this case we will write $${\textbf{\upshape B}} \le_{sd} \prod_{i \in I} {\textbf{\upshape A}}_i.$$ An algebra ${\textbf{\upshape B}}$ is *subdirectly irreducible* if it is nontrivial and for any subdirect embedding $${\textbf{\upshape B}} \le_{sd} \prod_{i \in I} {\textbf{\upshape A}}_i$$ there is an $i \in I$ such that ${\textbf{\upshape B}}$ and ${\textbf{\upshape A}}_i$ are isomorphic. It can be shown that ${\textbf{\upshape A}}$ is *subdirectly irreducible* if and only if the congruence lattice $\operatorname{Con}({\textbf{\upshape A}})$ of ${\textbf{\upshape A}}$ has a unique minimal element different from the trivial congruence. If $\mathsf {V}$ is a variety we denote by $\mathsf {V}_{si}$ the class of subdirectly irreducible algebras in $\mathsf {V}$. **Theorem 2**. 1. *(Birkhoff [@Birkhoff1944]) Every algebra can be subdirectly embedded in a product of subdirectly irreducible algebras. So if ${\textbf{\upshape A}} \in \mathsf {V}$, then ${\textbf{\upshape A}}$ can be subdirectly embedded in a product of members of $\mathsf {V}_{si}$.* 2. *(Jónsson's Lemma [@Jonsson1967]) Suppose that $\mathsf {K}$ is a class of algebras such that $\mathsf {V}(\mathsf {K})$ is congruence distributive; then $\mathsf {V}_{si} \subseteq{\mathbf H}{\mathbf S}{\mathbf P}_u(\mathsf {K})$.* If $\mathsf {Q}$ is a quasivariety and ${\textbf{\upshape A}} \in \mathsf {Q}$, a *relative congruence* of ${\textbf{\upshape A}}$ is a congruence $\theta$ such that ${\textbf{\upshape A}}/\theta\in \mathsf {Q}$; relative congruences form an algebraic lattice $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$. Moreover, for an algebra ${\textbf{\upshape A}}$ and a set $H \subseteq A\times A$ there exists the least relative congruence $\theta_{\mathsf {Q}}(H)$ on ${\textbf{\upshape A}}$ containing $H$. When $H = \{(a,b)\}$, we just write $\theta_{\mathsf {Q}}(a,b)$. When $\mathsf {Q}$ is a variety we simplify the notation by dropping the subscript $\mathsf {Q}$. For any congruence lattice property $P$ we say that ${\textbf{\upshape A}} \in \mathsf {Q}$ is *relative $P$* if $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$ satisfies $P$. So for instance ${\textbf{\upshape A}}$ is *relative subdirectly irreducible* if $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$ has a unique minimal element; since clearly $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$ is a meet subsemilattice of $\operatorname{Con}({\textbf{\upshape A}})$, any subdirectly irreducible algebra is relative subdirectly irreducible for any quasivariety to which it belongs. For a quasivariety $\mathsf {Q}$ we denote by $\mathsf {Q}_{rsi}$ the class of relative subdirectly irreducible algebras in $\mathsf {Q}$. We have the equivalent of Birkhoff's and Jónsson's results for quasivarieties: **Theorem 3**. *Let $\mathsf {Q}$ be any quasivariety.* 1. *(Mal'cev [@Malcev1956]) Every ${\textbf{\upshape A}} \in \mathsf {Q}$ is subdirectly embeddable in a product of algebras in $\mathsf {Q}_{rsi}$.* 2. *(Czelakowski-Dziobiak [@CzelakowskiDziobiak1990]) If $\mathsf {Q} = {\mathbf Q}(\mathsf {K})$, then $\mathsf {Q}_{rsi} \subseteq{\mathbf I}{\mathbf S}{\mathbf P}_u(\mathsf {K})$.* The following fact will be used in the sequel. **Lemma 4**. *Let ${\textbf{\upshape A}}$ be an algebra, such that ${\mathbf V}({\textbf{\upshape A}})$ is congruence distributive. Then ${\mathbf Q}({\textbf{\upshape A}}) = {\mathbf V}({\textbf{\upshape A}})$ if and only if every subdirectly irreducible algebra in ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}})$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u{\textbf{\upshape A}}$.* *Proof.* Suppose first that ${\mathbf Q}({\textbf{\upshape A}}) = {\mathbf V}({\textbf{\upshape A}})$, and let ${\textbf{\upshape A}}$ be a subdirectly irreducible algebra in ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}})$. Thus ${\textbf{\upshape A}}$ is subdirectly irreducible in ${\mathbf V}({\textbf{\upshape A}}) = {\mathbf Q}({\textbf{\upshape A}})$, and by Theorem [Theorem 3](#quasivariety){reference-type="ref" reference="quasivariety"} ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}})$. Conversely assume that every subdirectly irreducible algebra in ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}})$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u{\textbf{\upshape A}}$. Since ${\mathbf V}({\textbf{\upshape A}})$ is congruence distributive, by Theorem [Theorem 2](#birkhoff){reference-type="ref" reference="birkhoff"}(2) every subdirectly irreducible algebra in ${\mathbf V}({\textbf{\upshape A}})$ is in ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}})$, thus in ${\mathbf I}{\mathbf S}{\mathbf P}_u{\textbf{\upshape A}}$. Now every algebra in ${\mathbf V}({\textbf{\upshape A}})$ is subdirectly embeddable in a product of subdirectly irreducible algebras in ${\mathbf V}({\textbf{\upshape A}})$ (Theorem [Theorem 2](#birkhoff){reference-type="ref" reference="birkhoff"}(1)). Therefore, ${\mathbf V}({\textbf{\upshape A}}) \subseteq{\mathbf I}{\mathbf S}{\mathbf P}{\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape A}}) \subseteq{\mathbf I}{\mathbf S}{\mathbf P}{\mathbf P}_u({\textbf{\upshape A}}) = {\mathbf Q}({\textbf{\upshape A}})$ and thus equality holds. ◻ In this work we are particularly interested in quasivarieties that are the equivalent algebraic semantics of a logic in the sense of Blok-Pigozzi [@BlokPigozzi1989]. We will spend some time illustrating the machinery of *Abstract Algebraic Logic* that establishes a Galois connection between *algebraizable logics* and *quasivarieties of logic*, since it is relevant to understand our results. For the omitted details we refer the reader to [@BlokPigozzi1989; @Font2016]. By a *logic* $\mathcal{L}$ in what follows we mean a substitution invariant consequence relation $\vdash$ on the set of terms ${\textbf{\upshape T}}_{\rho}(\omega)$ (also called *algebra of formulas*) of some algebraic language $\rho$. In loose terms, to establish the algebraizability of a logic $\mathcal{L}$ with respect to a quasivariety of algebras $\mathsf {Q}_{\mathcal{L}}$ over the same language $\rho$, one needs a finite set of one-variable equations $$\tau(x) =\{\delta_i(x) \approx\varepsilon_i(x): i = 1,\dots,n\}$$ over terms of type $\rho$ and a finite set of formulas of $\mathcal{L}$ in two variables $$\Delta(x,y)=\{\varphi_1(x,y),\dots,\varphi_m(x,y)\}$$ that allow to transform equations, quasiequations and universal sentences in $\mathsf {Q}_{\mathcal{L}}$ into formulas, rules and clauses of $\mathcal{L}$; moreover this transformation must respect both the consequence relation of the logic and the semantical consequence of the quasivariety. More precisely, for all sets of formulas $\Gamma$ of $\mathcal{L}$ and formulas $\varphi\in {\textbf{\upshape T}}_{\rho}(\omega)$ $$\Gamma \vdash_{\mathcal{L}} \varphi\quad\text{iff}\quad \tau(\Gamma) \vDash_{\mathsf {Q}_{\mathcal{L}}} \tau(\varphi)$$ where $\tau(\Gamma)$ is a shorthand for $\{\tau(\gamma): \gamma \in \Gamma\}$, and also $$(x \approx y) \Dashv \vDash_{\mathsf {Q}_{\mathcal{L}}}\tau(\Delta(x,y)).$$ A quasivariety $\mathsf {Q}$ is a *quasivariety of logic* if it is the equivalent algebraic semantics for some logic $\mathcal{L}_\mathsf {Q}$; the Galois connection between algebraizable logics and quasivarieties of logic is given by $$\mathcal{L}_{\mathsf {Q}_{\mathcal{L}}} = \mathcal{L} \qquad\qquad \mathsf {Q}_{\mathcal{L}_\mathsf {Q}} = \mathsf {Q}.$$ Not every quasivariety is a quasivariety of logic; for instance no *idempotent quasivariety*, such as any quasivariety of lattices, can be a quasivariety of logics. Nonetheless quasivarieties of logic are plentiful. In fact any ideal determined variety is such, as well as any quasivariety coming from a congruential variety with normal ideals (see [@OSV3] for details). Moreover, every quasivariety is *categorically equivalent* to a quasivariety of logic [@MoraschiniRaftery2019]. This means that if an algebraic concept is expressible through notions that are invariant under categorical equivalence, and it holds for a quasivariety $\mathsf {Q}$, then it holds for its categorically equivalent quasivariety of logic $\mathsf {Q}'$; and hence in can be transformed into a logical concept in $\mathcal{L}_{\mathsf {Q}'}$ using the Blok-Pigozzi connection. **Definition 5**. If $\mathsf {Q}$ is any quasivariety, with an abuse of notation, we will denote by $\mathcal{L}_\mathsf {Q}$ a logic whose equivalent algebraic semantics is categorically equivalent to $\mathsf {Q}$. The following result hints at what kind of properties can be transferred by categorical equivalence. **Theorem 6** ([@BankstonFox1983]). *Let $\mathsf {K}$ be a class closed under subalgebras and direct products; If $\mathsf {K}$ is categorically equivalent to a quasivariety $\mathsf {Q}$, then $\mathsf {K}$ is a quasivariety.* Suppose now that $\mathsf {Q}$ and $\mathsf {R}$ are quasivarieties and suppose that $F:\mathsf {Q} \longrightarrow \mathsf {R}$ is a functor between the two algebraic categories witnessing the categorical equivalence. Now, $F$ preserves all the so-called *categorical properties*, i.e., those notions that can be expressed as properties of morphisms. In particular, embeddings are mapped to embeddings (since in algebraic categories they are exactly the categorical monomorphisms), surjective homomorphisms are mapped to surjective homomorphisms (since they correspond to *regular* epimorphisms in the categories). Moreover, we observe that direct products are preserved as well, since they can be expressed via families of surjective homomorphisms (see e.g. [@BurrisSanka]). Therefore, if $\mathsf {Q}'$ is a subquasivariety of $\mathsf {Q}$, then the restriction of $F$ to $\mathsf {Q}'$ witnesses a categorical equivalence between $\mathsf {Q}'$ and $$\mathsf {R}'=\{{\textbf{\upshape B}} \in \mathsf {R}: {\textbf{\upshape B}} = F({\textbf{\upshape A}})\ \text{for some ${\textbf{\upshape A}} \in \mathsf {Q}'$}\}.$$ It follows from Theorem [Theorem 6](#thm:banks){reference-type="ref" reference="thm:banks"} that $\mathsf {R}'$ is a subquasivariety of $\mathsf {R}$, and that $\mathsf {R}'$ is a variety whenever $\mathsf {Q}'$ is such. Given a quasivariety $\mathsf {Q}$, we denote by $\Lambda_q(\mathsf {Q})$ the lattice of subquasivarieties of $\mathsf {Q}$. Hence the correspondence sending $\mathsf {Q}' \longmapsto \mathsf {R}'$ is a lattice isomorphism between $\Lambda_q(\mathsf {Q})$ in $\Lambda_q(\mathsf {R})$ that preserves all the categorical properties. Moreover, we observe that, since ultraproducts in an algebraic category admit a categorical definition which turns out to be equivalent to the algebraic one (see for instance [@Eklof1977]), the functor $F$ also map universal subclasses to universal subclasses; more precisely, $\mathsf {U} \subseteq\mathsf {Q}$ is a universal class if and only if $F(\mathsf {U}) \subseteq\mathsf {R}$ is a universal class. Let us show an example of how we can use these correspondences, that is also a preview of what we will see in the coming sections; if $\mathsf {Q}$ is a quasivariety, a subquasivariety $\mathsf {Q}'$ is *equational* in $\mathsf {Q}$ if $\mathsf {Q}'= {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q}$. A quasivariety is *primitive* if every subquasivariety of $\mathsf {Q}$ is equational in $\mathsf {Q}$. It is clear from the discussion above that this concept is preserved by categorical equivalence and that the lattice isomorphism described above sends primitive subquasivarieties in primitive subquasivarieties. ## Projectivity, weak projectivity and exactness We now introduce the algebraic notions that will be the key tools for our investigation: projective, weakly projective, exact, and finitely presented algebras. **Definition 7**. Given a class $\mathsf {K}$ of algebras, an algebra ${\textbf{\upshape A}} \in \mathsf {K}$ is *projective* in $\mathsf {K}$ if for all ${\textbf{\upshape B}},{\textbf{\upshape C}} \in \mathsf {K}$, any homomorphism $h:{\textbf{\upshape A}} \longmapsto {\textbf{\upshape C}}$, and any surjective homomorphism $g: {\textbf{\upshape B}}\longmapsto {\textbf{\upshape C}}$, there is a homomorphism $f: {\textbf{\upshape A}} \longmapsto {\textbf{\upshape B}}$ such that $h=gf$. Determining the projective algebras in a class is usually a challenging problem, especially in a general setting. If however $\mathsf {K}$ contains all the free algebras on $\mathsf {K}$ (in particular, if $\mathsf {K}$ is a quasivariety), projectivity admits a simpler formulation. We call an algebra ${\textbf{\upshape B}}$ a *retract* of an algebra ${\textbf{\upshape A}}$ if there is a homomorphism $g: {\textbf{\upshape A}} \longmapsto {\textbf{\upshape B}}$ and a homomorphism $f:{\textbf{\upshape B}} \longmapsto {\textbf{\upshape A}}$ with $gf= \textrm{\upshape id}_{\textbf{\upshape B}}$ (and thus, necessarily, $f$ is injective and $g$ is surjective). The following theorem was proved first by Whitman for lattices [@Whitman1941] but it is well-known to hold for any class of algebras. **Theorem 8**. *Let $\mathsf {Q}$ be a quasivariety. Then the following are equivalent:* 1. *${\textbf{\upshape A}}$ is projective in $\mathsf {Q}$;* 2. *${\textbf{\upshape A}}$ is a retract of a free algebra in $\mathsf {Q}$.* 3. *${\textbf{\upshape A}}$ is a retract of a projective algebra in $\mathsf {Q}$.* *In particular every free algebra in $\mathsf {Q}$ is projective in $\mathsf {Q}$.* **Definition 9**. Given a quasivariety $\mathsf {Q}$ we say that an algebra is *finitely presented in $\mathsf {Q}$* if there exists a finite set $X$ and a finite set $H$ of pairs of terms over $X$ such that ${\textbf{\upshape A}} \cong {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_{\mathsf {Q}}(H)$. The proof of the following theorem is standard (but see [@Ghilardi1997]). **Theorem 10**. *For a finitely presented algebra ${\textbf{\upshape A}}\in\mathsf {Q}$ the following are equivalent:* 1. *${\textbf{\upshape A}}$ is projective in $\mathsf {Q}$;* 2. *${\textbf{\upshape A}}$ is projective in the class of all finitely presented algebras in $\mathsf {Q}$;* 3. *${\textbf{\upshape A}}$ is a retract of a finitely generated free algebra in $\mathsf {Q}$.* As a consequence we stress that if $\mathsf {Q}$ is a quasivariety and $\mathsf {V} = {\mathbf V}(\mathsf {Q})$ then all the algebras that are projective in $\mathsf {Q}$ are also projective in $\mathsf {V}$ (and vice versa). Moreover, all the finitely generated projective algebras in $\mathsf {Q}$ lie inside ${\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$. **Definition 11**. An algebra ${\textbf{\upshape A}}$ is *weakly projective in an algebra ${\textbf{\upshape B}}$* if ${\textbf{\upshape A}} \in {\mathbf H}({\textbf{\upshape B}})$ implies ${\textbf{\upshape A}} \in {\mathbf S}({\textbf{\upshape B}})$; an algebra is *weakly projective in a class $\mathsf {K}$* if it is weakly projective in any algebra ${\textbf{\upshape B}} \in \mathsf {K}$. **Definition 12**. If $\mathsf {Q}$ is a quasivariety of algebras and ${\textbf{\upshape A}} \in \mathsf {Q}$, let $G_{\textbf{\upshape A}}$ be the set of generators of ${\textbf{\upshape A}}$; ${\textbf{\upshape A}}$ is *exact* in $\mathsf {Q}$ if it is weakly projective in some ${\textbf{\upshape F}}_\mathsf {Q}(X)$ with $|X| \ge |G_{\textbf{\upshape A}}|$. Clearly any projective algebra in $\mathsf {Q}$ is weakly projective in $\mathsf {Q}$ and any weakly projective algebra in $\mathsf {Q}$ is exact in $\mathsf {Q}$. Observe also the following consequence of the definition. **Lemma 13**. *Let $\mathsf {Q}$ be a quasivariety and let ${\textbf{\upshape A}}$ be a finitely generated algebra in $\mathsf {Q}$; then the following are equivalent:* 1. *${\textbf{\upshape A}}$ is exact in $\mathsf {Q}$;* 2. *${\textbf{\upshape A}} \in {\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* Therefore for finitely generated algebras our definition of exactness coincides with the one in [@CabrerMetcalfe2015]. We close this subsection with a couple of results connecting projectivity and weak projectivity. **Proposition 14**. *Let ${\textbf{\upshape A}}$ be a finite subdirectly irreducible algebra; if ${\textbf{\upshape A}}$ is weakly projective in ${\mathbf Q}({\textbf{\upshape A}})$, then it is projective in ${\mathbf Q}({\textbf{\upshape A}})$.* *Proof.* Let $\mathsf {Q}= {\mathbf Q}({\textbf{\upshape A}})$; since ${\textbf{\upshape A}}$ is finite, $\mathsf {Q}$ is locally finite. Let ${\textbf{\upshape F}}$ be a finitely generated (hence finite) free algebra in $\mathsf {Q}$ such that ${\textbf{\upshape A}} \in {\mathbf H}({\textbf{\upshape F}})$; since ${\textbf{\upshape A}}$ is weakly projective, ${\textbf{\upshape A}}$ is embeddable in ${\textbf{\upshape F}}$ and without loss of generality we may assume that ${\textbf{\upshape A}} \le {\textbf{\upshape F}}$. Consider the set $$V =\{\alpha\in \textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape F}}): \alpha\cap A^2 = 0_{\textbf{\upshape A}}\},$$ where we denote by $0_{{\textbf{\upshape A}}}$ the minimal congruence of ${\textbf{\upshape A}}$. It is easy to see that $V$ is an inductive poset so we may apply Zorn's Lemma to find a maximal congruence $\theta\in V$. Clearly $a \longmapsto a/\theta$ is an embedding of ${\textbf{\upshape A}}$ into ${\textbf{\upshape F}}/\theta$. We claim that ${\textbf{\upshape F}}/\theta$ is relative subdirectly irreducible and to prove so, since everything is finite, it is enough to show that $\theta$ is meet irreducible in $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape F}})$; so let $\alpha,\beta\in \textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$ such that $\alpha\wedge\beta = \theta$. Then $$0_ {\textbf{\upshape A}} = \theta\cap A^2 = (\alpha\wedge\beta) \cap A^2 = (\alpha\cap A^2) \wedge(\beta\cap A^2);$$ But ${\textbf{\upshape A}}$ is subdirectly irreducible, so $0_A$ is meet irreducible in $\textrm{\upshape Con}({\textbf{\upshape A}})$; hence either $\alpha\cap A^2 = 0_{\textbf{\upshape A}}$ or $\beta\cap A^2 = 0_{\textbf{\upshape A}}$, so either $\alpha\in V$ or $\beta\in V$. Since $\theta$ is maximal in $V$, either $\alpha= \theta$ or $\beta= \theta$, which proves that ${\textbf{\upshape F}}/\theta$ is relative subdirectly irreducible. Therefore, by Theorem [Theorem 3](#quasivariety){reference-type="ref" reference="quasivariety"}(2), ${\textbf{\upshape F}}/\theta\in {\mathbf I}{\mathbf S}({\textbf{\upshape A}})$; since ${\textbf{\upshape F}}/\theta$ and ${\textbf{\upshape A}}$ are both finite and each one is embeddable in the other, they are in fact isomorphic. Thus ${\textbf{\upshape A}} \le {\textbf{\upshape F}}$, and there is a homomorphism from ${\textbf{\upshape F}}$ onto ${\textbf{\upshape A}}$ that maps each $a \in A$ to itself. This shows that ${\textbf{\upshape A}}$ is a retract of ${\textbf{\upshape F}}$, and therefore ${\textbf{\upshape A}}$ is projective in ${\mathbf Q}({\textbf{\upshape A}})$. ◻ For varieties we have to add the hypothesis of congruence distributivity, since the use of Theorem [Theorem 2](#birkhoff){reference-type="ref" reference="birkhoff"}(2) is paramount; for the very similar proof see [@JipsenNation2022 Theorem 9]. **Proposition 15**. *Let ${\textbf{\upshape A}}$ be a finite subdirectly irreducible algebra such that ${\mathbf V}({\textbf{\upshape A}})$ is congruence distributive; if ${\textbf{\upshape A}}$ is weakly projective in ${\mathbf V}({\textbf{\upshape A}})$, then it is projective in ${\mathbf V}({\textbf{\upshape A}})$.* We observe that in algebraic categories projectivity is a property preserved by categorical equivalence and the same holds for weak projectivity and exactness. Finally by [@GabrielUllmer1971] being finitely presented and being finitely generated are also categorical properties preserved by equivalences. ## Algebraic unification The main objects of our study, i.e., the notions of universal and structural completeness, are closely related to unification problems. The classical syntactic unification problem given two term $s,t$ finds a *unifier* for them; that is, a uniform replacement of the variables occurring in $s$ and $t$ by other terms that makes $s$ and $t$ identical. When the latter syntactical identity is replaced by equality modulo a given equational theory $E$, one speaks of *$E$-unification*. S. Ghilardi [@Ghilardi1997] proved that there is a completely algebraic way of studying ($E$-)unification problems in varieties of logic, which makes use of finitely presented and projective algebras and thus is invariant under categorical equivalence. Let us discuss Ghilardi's idea in some detail showing how it can be applied to quasivarieties. If $\mathsf {Q}$ is a quasivariety and $\Sigma$ is a finite set of equations in the variables $X=\{x_1,\dots,x_{n}\}$ by a *substitution* $\sigma$ we mean an assignment from $X$ to ${\textbf{\upshape F}}_\mathsf {Q}(\omega)$, extending to a homomorphism from ${\textbf{\upshape F}}_\mathsf {Q}(X)$ to ${\textbf{\upshape F}}_\mathsf {Q}(\omega)$. **Definition 16**. A *unification problem* for a quasivariety $\mathsf {Q}$ is a finite set of identities $\Sigma$ in the language of $\mathsf {Q}$; $\Sigma$ is *unifiable* in $\mathsf {Q}$ if there is a substitution $\sigma$ such that $\mathsf {Q} \vDash \sigma(\Sigma)$, i.e. $$\mathsf {Q}\vDash p(\sigma(x_1),\dots,\sigma (x_n)) \approx q(\sigma(x_1),\dots,\sigma (x_n))$$ for all $p \approx q \in \Sigma$. The substitution $\sigma$ is called a *unifier* for $\Sigma$. Observe that $\Sigma$ is *unifiable* in $\mathsf {Q}$ if and only if it is unifiable in ${\mathbf V}(\mathsf {Q})$. Let us now present the algebraic approach, where a unification problem can be represented by a finitely presented algebra in $\mathsf {Q}$. **Definition 17**. If ${\textbf{\upshape A}}$ is in $\mathsf {Q}$, a *unifier* for ${\textbf{\upshape A}}$ is a homomorphism $u: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape P}}$ where ${\textbf{\upshape P}}$ is a projective algebra in $\mathsf {Q}$; we say that an algebra is *unifiable in $\mathsf {Q}$* if at least one such homomorphism exists. A quasivariety $\mathsf {Q}$ is *unifiable* if every finitely presented algebra in $\mathsf {Q}$ is unifiable. **Notation 2**. When we write ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$, $\theta_{\mathsf {Q}}(\Sigma)$ is the relative congruence generated in ${\textbf{\upshape F}}_\mathsf {Q}(X)$ by the set $\{(p,q): p \approx q \in \Sigma\}$. The following summarizes the needed results of [@Ghilardi1997] applied to quasivarieties. **Theorem 18**. *Let $\mathsf {Q}$ be a quasivariety, and let $\Sigma$ be a finite set of equations in the language of $\mathsf {Q}$ with variables in a (finite) set $X$; then:* 1. *if $\Sigma$ is unifiable via $\sigma: {\textbf{\upshape F}}_\mathsf {Q}(X) \to {\textbf{\upshape F}}_\mathsf {Q}(Y)$ then $u_\sigma: {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \to {\textbf{\upshape F}}_\mathsf {Q}(Y)$ defined by $$u_\sigma(t/\theta_{\mathsf {Q}}(\Sigma)) = \sigma(t)$$ is a unifier for ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$;* 2. *conversely let ${\textbf{\upshape A}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_{\mathsf {Q}}(\Sigma)$. If there is a unifier $u:{\textbf{\upshape A}} \to {\textbf{\upshape P}}$, where ${\textbf{\upshape P}}$ is projective and a retract of ${\textbf{\upshape F}}_\mathsf {Q}(Y)$ witnessed by an embedding $i: {\textbf{\upshape P}} \to {\textbf{\upshape F}}_\mathsf {Q}(Y)$, the substitution $$\sigma_u: x \longmapsto i(u (x/\theta_{\mathsf {Q}}(\Sigma)))$$ is a unifier for $\Sigma$ in $\mathsf {Q}$.* *Proof.* For the first claim, consider $\sigma: {\textbf{\upshape F}}_\mathsf {Q}(X) \to {\textbf{\upshape F}}_\mathsf {Q}(Y)$ and the natural epimorphism $\pi_\Sigma: {\textbf{\upshape F}}_\mathsf {Q}(X) \to {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_{\mathsf {Q}}(\Sigma)$. Since $\theta_{\mathsf {Q}}(\Sigma)$ is the least congruence of ${\textbf{\upshape F}}_\mathsf {Q}(X)$ containing the set of pairs $S = \{(p,q): p \approx q \in \Sigma\}$, and given that $S \subseteq\ker(\sigma)$, by the Second Homomorphism Theorem we can close the following diagram with exactly the homomorphism $u_\sigma$: The second claim is easily seen, since $\sigma_u$ is defined by a composition of homomorphism and as above the set of pairs $S = \{(p,q): p \approx q \in \Sigma\}$ is contained in its kernel, which yields that $\sigma_u$ is a unifier for $\Sigma$ in $\mathsf {Q}$. ◻ **Corollary 19**. *A finite set of identities $\Sigma$ is unifiable in $\mathsf {Q}$ if and only if the finitely presented algebra ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ is unifiable in $\mathsf {Q}$.* The following observation shows how to characterize unifiability in quasivarieties. **Definition 20**. For a quasivariety $\mathsf {Q}$, we let ${\textbf{\upshape F}}_\mathsf {Q}$ be the *smallest free algebra*, i.e. ${\textbf{\upshape F}}_\mathsf {Q}(\emptyset)$ (if there are constant operations) or else ${\textbf{\upshape F}}_\mathsf {Q}(x)$. We have the following : **Lemma 21**. *Let $\mathsf {Q}$ be a quasivariety and let ${\textbf{\upshape A}} \in \mathsf {Q}$. Then the following are equivalent:* 1. *${\textbf{\upshape A}}$ is unifiable in $\mathsf {Q}$;* 2. *there is a homomorphism from ${\textbf{\upshape A}}$ to ${\textbf{\upshape F}}_\mathsf {Q}$.* *Proof.* Note that (2) trivially implies (1), since ${\textbf{\upshape F}}_\mathsf {Q}$ is projective. Vice versa, if ${\textbf{\upshape A}}$ is unifiable, there is a homomorphism from ${\textbf{\upshape A}}$ to some projective algebra ${\textbf{\upshape P}}$. Since ${\textbf{\upshape P}}$ is a retract of some free algebra in $\mathsf {Q}$, and ${\textbf{\upshape F}}_{\mathsf {Q}}$ is a homomorphic image of every free algebra in $\mathsf {Q}$, the claim follows. ◻ The above lemma implies for instance that if ${\textbf{\upshape F}}_\mathsf {Q}$ is trivial, then $\mathsf {Q}$ is unifiable since every algebra admits a homomorphism onto a trivial algebra. Hence, examples of unifiable algebras include lattices, groups, lattice-ordered abelian groups, residuated lattices. On the other hand, both bounded lattices and bounded residuated lattices (explored in Subsection [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"} and [5.2](#sec:FL){reference-type="ref" reference="sec:FL"} respectively) are unifiable if and only if they admit a homomorphism onto the algebra (over the appropriate signature) with two elements $0$ and $1$. We observe in passing that if ${\textbf{\upshape A}} \cong {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}$ is finitely presented unifiable algebra in $\mathsf {Q}$, witnessed by a unifier $u: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape P}}$, then $u$ can be split into a homomorphism onto its image $u({\textbf{\upshape A}})$, and an embedding from $u({\textbf{\upshape A}})$ to ${\textbf{\upshape P}}$. By the Third Homomorphism Theorem there is a $\theta' \in \textrm{\upshape Con}({\textbf{\upshape F}}_\mathsf {Q}(X))$ corresponding to the kernel of the onto homomorphism $u: {\textbf{\upshape A}} \longrightarrow u({\textbf{\upshape A}})$, $\theta' \ge \theta$, such that ${\textbf{\upshape F}}_\mathsf {V}(X)/\theta'$ embeds in ${\textbf{\upshape P}}$; note that $\theta'\in \textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape F}}_\mathsf {Q}(X))$, since ${\textbf{\upshape P}} \in \mathsf {Q}$. The diagram in Figure [\[exact\]](#exact){reference-type="ref" reference="exact"} shows that indeed ${\textbf{\upshape F}}_\mathsf {V}(X)/\theta'$ is exact. Let us now introduce the usual notion of order among unifiers. Given two unifiers $u_1,u_2$ for ${\textbf{\upshape A}}$ we say that $u_1$ is *less general then* $u_2$ (and we write $u_1 \preceq u_2$), if there is a homomorphism $h$ that makes the following diagram commute. Clearly $\preceq$ is a preordering and so the equivalence classes of the associated equivalence relation (i.e. the unifiers that are *equally general*) form a poset $U_{\textbf{\upshape A}}$; using the maximal sets of that poset it is possible to define a hierarchy of unification types (see [@Ghilardi1997]). In particular, the unification type is *unitary* if there is one maximal element, that is called *the most general unifier* or *mgu*. **Definition 22**. We say that a quasivariety $\mathsf {Q}$ has *projective unifiers* if every finitely presented unifiable algebra in $\mathsf {Q}$ is projective, and that it has *exact unifiers* if every finitely presented unifiable algebra in $\mathsf {Q}$ is exact. If $\mathsf {Q}$ has projective unifiers, then (from the algebraic perspective) the identity map is a unifier, and it is also the most general unifier. Next we have a lemma whose proof is straightforward (modulo Lemma [Lemma 21](#free){reference-type="ref" reference="free"}). **Lemma 23**. *Let $\mathsf {Q}$ be a quasivariety; then the following are equivalent:* 1. *$\mathsf {Q}$ has projective (exact) unifiers;* 2. *for any finitely presented ${\textbf{\upshape A}}\in\mathsf {Q}$, ${\textbf{\upshape A}}$ has ${\textbf{\upshape F}}_\mathsf {Q}$ as a homomorphic image if and only if ${\textbf{\upshape A}}$ is projective (exact).* If $\mathsf {Q}$ is locally finite, then we have a necessary and sufficient condition. **Lemma 24**. *Let $\mathsf {Q}$ be a locally finite quasivariety of finite type, then the following are equivalent:* 1. *$\mathsf {Q}$ has projective unifiers;* 2. *every finite unifiable algebra in $\mathsf {Q}$ is projective in the class of finite algebras in $\mathsf {Q}$.* *Proof.* (1) implies (2) is obvious. Assume (2), let ${\textbf{\upshape A}}$ be unifiable and finite and let ${\textbf{\upshape B}} \in \mathsf {Q}$ such that $f: {\textbf{\upshape B}} \longrightarrow {\textbf{\upshape A}}$ is a onto homomorphism. Let $a_1,\dots,a_{n}$ be the generators of ${\textbf{\upshape A}}$ and let $b_1,\dots,b_{n} \in B$ with $f(b_i) = a_i$ for $i = 1 \ldots n$; if ${\textbf{\upshape B}}'$ is the subalgebra generated by $b_1,\dots,b_{n}$ then $f$ restricted to ${\textbf{\upshape B}}'$ is onto. Clearly ${\textbf{\upshape B}}'$ is finite. Hence by hypothesis there exists a $g: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape B}}$ such that $fg$ is the identity on ${\textbf{\upshape A}}$. This shows that ${\textbf{\upshape A}}$ is projective in ${\textbf{\upshape B}}$ and hence in $\mathsf {Q}$. Thus (1) holds. ◻ Having exact unifiers is weaker than having projective unifiers: **Example 25**. The variety $\mathsf {D}$ of distributive lattices is unifiable since it has no constants and it is idempotent; hence its least free algebra is trivial. But $\mathsf {D}$ does note have projective unifiers: a distributive lattice is projective if and only if the meet of join irreducible elements is again join irreducible [@Balbes1967], so there are finite non projective distributive lattices. However every finitely presented (i.e. finite) distributive lattice is exact [@CabrerMetcalfe2015a]. **Example 26**. A different example is the variety of $\mathsf{ST}$ of Stone algebras; a Stone algebra is a pseudocomplemented bounded distributive lattice in the signature $(\land, \lor, *, 0, 1)$ such that $x^* \lor x^{**} \approx 1$ holds. A Stone algebra is unifiable if and only if is has a homomorphism into the two element Boolean algebra if and only if it is nontrivial. While there are nontrivial Stone algebras that are not projective, any nontrivial Stone algebra is exact ([@CabrerMetcalfe2015a Lemma 17]). Hence $\mathsf{ST}$ has exact unifiers. Moreover, there are examples of varieties having a most general unifier that do not have projective unifiers. **Example 27**. From the results in [@Ghilardi1999], the variety $\mathsf{SH}$ of Stonean Heyting algebras (that is, Heyting algebras such that $\neg x \lor \neg\neg x \approx 1$ holds) is such that every unifiable algebra ${\textbf{\upshape A}} \in \mathsf{SH}$ has a most general unifier. However, $\mathsf{SH}$ does not have projective unifiers. The algebra ${\textbf{\upshape F}}_\mathsf{SH}(x,y,z)/\theta$, where $\theta$ is the congruence generated by the pair $( \neg x \rightarrow(y \vee z),1)$, is unifiable but not projective. We observe that Ghilardi's argument relies heavily on some properties of Heyting algebras and uses Kripke models, making it difficult to generalize. Trivial examples show that having projective or exact unifiers is not inherited in general by subvarieties (see for instance [@DzikStronkowski2016 Example 7.2]). The following lemma (that we extract from [@DzikStronkowski2016 Lemma 5.4]) gives a sufficient condition for having projective unifiers. We write a detailed proof for the reader's convenience. **Lemma 28** ([@DzikStronkowski2016]). *Let $\mathsf {Q}$ be a quasivariety and let $\mathsf {Q}'$ be a subquasivariety of $\mathsf {Q}$ such that if ${\textbf{\upshape B}} ={\textbf{\upshape F}}_{\mathsf {Q}'}(X)/\theta_{\mathsf {Q}'}(\Sigma)$ is finitely presented and unifiable in $\mathsf {Q}'$, then ${\textbf{\upshape A}} = {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_\mathsf {Q}(\Sigma)$ is unifiable in $\mathsf {Q}$. If $\mathsf {Q}$ has projective unifiers then $\mathsf {Q}'$ has projective unifiers.* *Proof.* It is an easy exercise in general algebra to show that if $\Theta = \bigcap\{\theta\in \textrm{\upshape Con} ({\textbf{\upshape F}}_\mathsf {Q}(X)): {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta\in \mathsf {Q}'\}$ then $${\textbf{\upshape F}}_{\mathsf {Q}'}(X)/\theta_{\mathsf {Q}'}(\Sigma) \cong {\textbf{\upshape F}}_\mathsf {Q}(X)/(\theta_\mathsf {Q}(\Sigma) \vee\Theta).$$ It follows that ${\textbf{\upshape B}}$ is a homomorphic image of ${\textbf{\upshape A}}$ via the natural surjection $$p:a/\theta_\mathsf {Q}(\Sigma) \longmapsto a/(\theta_\mathsf {Q}(\Sigma) \vee\Theta)$$ composed with the isomorphism. Moreover if $f: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape C}}$ is a homomorphism and ${\textbf{\upshape C}} \in \mathsf {Q}'$, then $\textrm{\upshape ker}(p) \le \textrm{\upshape ker}(f)$ and by the Second Homomorphism Theorem there is a $f': {\textbf{\upshape B}} \longrightarrow {\textbf{\upshape C}}$ with $f'p = f$. Now let ${\textbf{\upshape B}} = {\textbf{\upshape F}}_{\mathsf {Q}'}/\theta_{\mathsf {Q}'}(\Sigma)$ be finitely presented and unifiable and let ${\textbf{\upshape A}} = {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_\mathsf {Q}(\Sigma)$; then ${\textbf{\upshape A}}$ is finitely presented and unifiable as well, so, since $\mathsf {Q}$ has projective unifiers, ${\textbf{\upshape A}}$ is projective in $\mathsf {Q}$. We now show that ${\textbf{\upshape B}}$ is projective. Suppose there are algebras ${\textbf{\upshape C}}, {\textbf{\upshape D}} \in \mathsf {Q}' \subseteq\mathsf {Q}$ and homomorphisms $h: {\textbf{\upshape B}} \to {\textbf{\upshape D}}, g: {\textbf{\upshape C}} \to {\textbf{\upshape D}}$ with $g$ surjective. Then, there is a homomorphism $h p: {\textbf{\upshape A}} \to {\textbf{\upshape D}}$, and since ${\textbf{\upshape A}}$ is projective by the definition of projectivity there is a homomorphism $f: {\textbf{\upshape A}} \to {\textbf{\upshape C}}$ such that $gf = hp$. Factoring $f$ as above, there is $f'$ such that $f'p = f$. Therefore since $g f'p = gf = hp$ and $p$ is surjective, we get that $gf' = h$ which means that ${\textbf{\upshape B}}$ is projective in $\mathsf {Q}'$. ◻ We will see later in Section [3.2](#subsec: active universal){reference-type="ref" reference="subsec: active universal"} (Example [Example 67](#ex: De Morgan){reference-type="ref" reference="ex: De Morgan"}) that Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} does not hold with "projective unifiers" replaced by "exact unifiers". We can build on the previous lemma and obtain the following. **Lemma 29**. *Suppose that $\mathsf {Q}$ is a quasivariety such that ${\textbf{\upshape F}}_\mathsf {Q} = {\textbf{\upshape F}}_{\mathsf {Q}'}$ for all $\mathsf {Q}'\subseteq\mathsf {Q}$. If $\mathsf {Q}$ has projective unifiers, then every subquasivariety $\mathsf {Q}'$ has projective unifiers.* *Proof.* Let $\mathsf {Q}'$ be a subquasivariety of $\mathsf {Q}$, let ${\textbf{\upshape B}} = {\textbf{\upshape F}}_{\mathsf {Q}'}(X)/\theta(\Sigma)$ be finitely presented and unifiable in $\mathsf {Q}'$ and let ${\textbf{\upshape A}} ={\textbf{\upshape F}}_\mathsf {Q}(X)/\theta(\Sigma)$. Then ${\textbf{\upshape B}}$ is a homomorphic image of ${\textbf{\upshape A}}$ and, since ${\textbf{\upshape B}}$ is unifiable there is a homomorphism from ${\textbf{\upshape B}}$ to ${\textbf{\upshape F}}_{\mathsf {Q}'} = {\textbf{\upshape F}}_\mathsf {Q}$. Hence ${\textbf{\upshape A}}$ is unifiable as well; hence the hypothesis of Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} are satisfied, and so $\mathsf {Q}'$ has projective unifiers. ◻ We close this subsection with a corollary appearing also in [@DzikStronkowski2016] that is useful to some examples we will explore in what follows. We reproduce the easy proof for the reader's convenience. **Corollary 30**. *Let $\mathsf {Q}$ be a quasivariety and let ${\mathbf V}(\mathsf {Q})= \mathsf {V}$; if $\mathsf {V}$ has exact (projective) unifiers, then so does $\mathsf {Q}$.* *Proof.* First recall that $\mathsf {Q}$ and $\mathsf {V}$ have the same free algebras. Let ${\textbf{\upshape A}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {V}(\Sigma)$ and ${\textbf{\upshape B}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}(\Sigma)$; if ${\textbf{\upshape B}}$ is unifiable then, as ${\textbf{\upshape B}}$ is a homomorphic image of ${\textbf{\upshape A}}$ via the epimorphism $p$ described in the proof of Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"}, ${\textbf{\upshape A}}$ is unifiable as well hence it is exact. Therefore there is an embedding $u: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape F}}_\mathsf {Q}(\omega)$; then by (the proof of) Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} there is a $g:{\textbf{\upshape B}} \longrightarrow {\textbf{\upshape F}}_\mathsf {Q}(\omega)$ with $gp = u$. Since $u$ is injective, so is $p$ and hence ${\textbf{\upshape A}}$ and ${\textbf{\upshape B}}$ are isomorphic. This proves the thesis. ◻ ## Structural and universal completeness {#subsec:structuraluniversal} We now introduce the main notions of interest of this work, that is, structural and universal completeness. Let $\mathcal{L}$ be a logic with consequence relation $\vdash$. We call *clause* of $\mathcal{L}$ an ordered pair $(\Sigma,\Gamma)$ where $\Sigma,\Gamma$ are finite sets of formulas. We usually write a clause as $\Sigma \Rightarrow \Gamma$. A *rule* is a clause $\Sigma \Rightarrow \Gamma$ where $\Gamma = \{\gamma\}$. A rule is *admissible* in a logic if, when added to its calculus, it does not produce new theorems. More precisely: **Definition 31**. A clause $\Sigma \Rightarrow \Delta$ is *admissible* in a logic $\mathcal{L}$ if every substitution making all the formulas in $\Sigma$ a theorem, also makes at least one of the formulas in $\Delta$ a theorem. Moreover, we say that a clause $\Sigma \Rightarrow \Delta$ is *derivable* in a logic $\mathcal{L}$ if $\Sigma \vdash \delta$ for some $\delta \in \Delta$. An admissible clause is not necessarily derivable; a popular example is Harrop's rule for intuitionistic logic $$\{ \neg p \rightarrow(q \vee r)\} \ \Rightarrow \ \{(\neg p \rightarrow q) \vee(\neg p \rightarrow r)\}$$ which is admissible but not derivable. **Definition 32**. Let $\mathcal{L}$ be a logic. A clause $\Sigma \Rightarrow \Gamma$ is *passive* in $\mathsf {Q}$ if there is no substitution making the premises $\Sigma$ a theorem of $\mathcal{L}$; a clause is *active* otherwise. Finally, a clause $\Sigma \Rightarrow \Delta$ is *negative* if $\Delta = \emptyset$. We observe that every admissible negative clause is necessarily passive. **Definition 33**. A logic is said to be 1. *universally complete* if every admissible clause is derivable; 2. *structurally complete* if every admissible rule is derivable; 3. *actively universally complete* if every active admissible clause is derivable; 4. *actively structurally complete* if every active admissible rule is derivable[^1] 5. *passively universally complete* if every passive admissible clause is derivable; 6. *passively structurally complete* if every passive admissible rule is derivable; 7. *non negatively universally complete* if every non negative admissible clause is derivable. Modulo algebraizability, one obtains the corresponding notions for a quasivariety. In particular, we can express admissibility and derivability of clauses in $\mathcal{L}_\mathsf {Q}$ using the (quasi)equational logic of $\mathsf {Q}$; this is because the Blok-Pigozzi Galois connection transforms (sets of) formulas in $\mathcal{L}_\mathsf {Q}$ into (sets of) equations in $\mathsf {Q}$ in a uniform way. The obtained notions make sense for quasivarieties that do not necessarily correspond to a logic. **Definition 34**. Let $\mathsf {Q}$ be a quasivariety. A universal sentence $\Sigma \Rightarrow \Delta$ is *admissible* in $\mathsf {Q}$ if every substitution unifying all the identities in $\Sigma$ also unifies at least one of the identities in $\Delta$. A universal sentence is *passive* if there is no substitution unifying its premises, *active* otherwise. $\mathsf {Q}$ is *(active/passive) universally/structurally complete* if every (active/passive) admissible universal sentence/quasiequation is valid in $\mathsf {Q}$. If $P$ is one of those properties, then we say that a logic (or a quasivariety) is *hereditarily $P$* if the logic (or the quasivariety) and all its extensions have the property $P$. Some of these properties are well-known to be distinct: for instance classical logic is non-negative universally complete but not universally complete, while intuitionistic logic is not structurally complete (thanks to Harrop's example) but it is passively structurally complete (as reported by Wronski in 2005, see [@CintulaMetcalfe2009]). The following is a consequence of algebraizability. **Theorem 35**. *Let $\mathsf {Q}$ be a quasivariety of logic, $\Sigma,\Delta$ sets of equations in the language of $\mathsf {Q}$ and $\Sigma',\Delta'$ the corresponding sets of formulas in $\mathcal{L}_\mathsf {Q}$. Then:* 1. *$\Sigma' \Rightarrow \Delta'$ is admissible in $\mathcal{L}_\mathsf {Q}$ if and only if $\Sigma \Rightarrow \Delta$ is admissible in $\mathsf {Q}$;* 2. *$\Sigma' \Rightarrow \Delta'$ is derivable in $\mathcal{L}_\mathsf {Q}$ if and only if $\mathsf {Q} \vDash \Sigma \Rightarrow \Delta$.* Moreover, by Corollary [Corollary 19](#cor:unifiablesigma){reference-type="ref" reference="cor:unifiablesigma"} we get the following. **Proposition 36**. *Let $\mathsf {Q}$ be a quasivariety of logic, $\Sigma,\Delta$ sets of equations in the language of $\mathsf {Q}$ and $\Sigma',\Delta'$ the corresponding sets of formulas in $\mathcal{L}_{\mathsf {Q}}$. Then:* 1. *$\Sigma'\Rightarrow \Delta'$ is active in $\mathcal{L}_{\mathsf {Q}}$ if and only if ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_{\mathsf {Q}}(\Sigma)$ is unifiable in $\mathsf {Q}$;* 2. *$\Sigma'\Rightarrow \Delta'$ is passive in $\mathcal{L}_{\mathsf {Q}}$ if and only if ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_{\mathsf {Q}}(\Sigma)$ is not unifiable in $\mathsf {Q}$.* The next lemma (also derivable from [@CabrerMetcalfe2015a Theorem 2]) characterizes admissibility of universal sentences. **Lemma 37**. *Let $\mathsf {Q}$ be any quasivariety, let $\Sigma \Rightarrow \Delta$ be a clause in the language of $\mathsf {Q}$ and let $\mathsf {U}_{\Sigma\Rightarrow\Delta} = \{{\textbf{\upshape A}} \in \mathsf {Q}: {\textbf{\upshape A}} \vDash \Sigma\Rightarrow\Delta\}$. Then the following are equivalent:* 1. *$\Sigma \Rightarrow \Delta$ is admissible in $\mathsf {Q}$;* 2. *${\textbf{\upshape F}}_\mathsf {Q}(\omega) \vDash \Sigma \Rightarrow \Delta$;* 3. *${\mathbf H}(\mathsf {Q}) = {\mathbf H}(\mathsf {U}_{\Sigma \Rightarrow \Delta})$.* *Proof.* The equivalence between (1) and (2) follows directly from the definition of admissibility. Assume now ${\textbf{\upshape F}}_\mathsf {Q}(\omega) \vDash \Sigma \Rightarrow \Delta$, then ${\textbf{\upshape F}}_\mathsf {Q}(\omega) \in \mathsf {U}_{\Sigma \Rightarrow \Delta}$. Clearly ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega)) \subseteq{\mathbf H}(\mathsf {U}_{\Sigma \Rightarrow \Delta}) \subseteq{\mathbf H}(\mathsf {Q})$. Now every algebra is embeddable in an ultraproduct of its finitely generated subalgebras and every finitely generated algebra is a homomorphic image of ${\textbf{\upshape F}}_\mathsf {Q}(\omega)$. Therefore if ${\textbf{\upshape A}} \in \mathsf {Q}$, then ${\textbf{\upshape A}} \in {\mathbf S}{\mathbf P}_u{\mathbf H}({\textbf{\upshape F}}_\mathsf {Q}(\omega)) \subseteq{\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$. So ${\mathbf H}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q} (\omega)) = {\mathbf H}(\mathsf {Q})$ and thus (3) holds. Conversely assume (3). Since ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \in {\mathbf H}(\mathsf {Q}) = {\mathbf H}(\mathsf {U}_{\Sigma \Rightarrow \Delta})$, there is ${\textbf{\upshape A}} \in \mathsf {U}_{\Sigma \Rightarrow \Delta}$ such that ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \in {\mathbf H}({\textbf{\upshape A}})$. Since ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega)$ is projective in $\mathsf {Q}$, it follows that ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \in {\mathbf S}({\textbf{\upshape A}}) \subseteq{\mathbf S}(\mathsf {U}_{\Sigma \Rightarrow \Delta}) \subseteq\mathsf {U}_{\Sigma \Rightarrow \Delta}$. Therefore, ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \models \Sigma \Rightarrow \Delta$ and (2) holds. ◻ To conclude the preliminaries, we present the following lemma which will be particularly useful in our proofs. **Lemma 38**. *Let $\mathsf {Q}$ be a quasivariety, and $\Sigma, \Delta$ be finite sets of equations over variables in a finite set $X$. The following are equivalent:* 1. *$\mathsf {Q} \models \Sigma \Rightarrow \Delta$;* 2. *${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \models \Sigma \Rightarrow \Delta$;* 3. *there is $p \approx q \in \Delta$ such that $p / \theta_{\mathsf {Q}}(\Sigma) = q / \theta_{\mathsf {Q}}(\Sigma)$ in ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$.* *Proof.* It is clear that (1) implies (2) and (2) implies (3). We now show that (3) implies (1). Let ${\textbf{\upshape A}} \in \mathsf {Q}$. If there is no assignment of the variables in $X$ to ${\textbf{\upshape A}}$ that models $\Sigma$, then ${\textbf{\upshape A}} \models \Sigma \Rightarrow \Delta$. Otherwise, suppose there is an assignment $h$ such that ${\textbf{\upshape A}}, h \models \Sigma$. Then, since $\theta_{\mathsf {Q}}(\Sigma)$ is the smallest relative congruence of ${\textbf{\upshape F}}_{\mathsf {Q}}(X)$ containing the set of pairs $S = \{(p,q): p \approx q \in \Sigma\}$, by the Second Homomorphism Theorem we can close the following diagram: That is, there is a homomorphism $f: {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \to {\textbf{\upshape A}}$ such that $h = f \pi_{\Sigma}$, where $\pi_\Sigma$ is the natural epimorphism from ${\textbf{\upshape F}}_{\mathsf {Q}}(X)$ to ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$. Now by (3) there is at least an identity $p \approx q \in \Delta$ such that $(p,q) \in \ker(\pi_\Sigma)$. Since $h = f\pi_\Sigma$, $(p,q) \in \ker(h)$, which means that ${\textbf{\upshape A}}, h \models p \approx q$ and therefore ${\textbf{\upshape A}} \models \Sigma \Rightarrow \Delta$. Since ${\textbf{\upshape A}}$ is an arbitrary algebra of $\mathsf {Q}$ this shows that $\mathsf {Q} \models \Sigma \Rightarrow \Delta$. ◻ # Universal completeness {#univquasi} In this section we study from the algebraic perspective the notion of universal completeness and its variations: active, nonnegative, passive universal completeness, together with their hereditary versions. That is, we shall see which algebraic properties correspond to the notions coming from the logical perspective (detailed in the preliminaries Subsection [2.4](#subsec:structuraluniversal){reference-type="ref" reference="subsec:structuraluniversal"}). For each notion, we will present a characterization theorem and some examples. While the characterizations of active and passive universal completeness (to the best of our knowledge) are fully original, we build on existing ones for the other notions, presenting some new results and a coherent presentation in our framework. ## Universal quasivarieties {#univquasi1} We start with universal completeness. The following expands [@CabrerMetcalfe2015a Proposition 6]. **Theorem 39**. *For any quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *[\[prop:universal2\]]{#prop:universal2 label="prop:universal2"} $\mathsf {Q}$ is universally complete;* 2. *[\[prop:universal1\]]{#prop:universal1 label="prop:universal1"} for every universal class $\mathsf {U} \subseteq\mathsf {Q}$, ${\mathbf H}(\mathsf {U}) = {\mathbf H}(\mathsf {Q})$ implies $\mathsf {U} = \mathsf {Q}$.* 3. *[\[prop:universal3\]]{#prop:universal3 label="prop:universal3"} $\mathsf {Q} = {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$;* 4. *[\[prop:universal4\]]{#prop:universal4 label="prop:universal4"} every finitely presented algebra in $\mathsf {Q}$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* *Proof.* ([\[prop:universal1\]](#prop:universal1){reference-type="ref" reference="prop:universal1"}) implies ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}) via Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}. We show that ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}) implies ([\[prop:universal1\]](#prop:universal1){reference-type="ref" reference="prop:universal1"}). Let $\mathsf {U} \subseteq\mathsf {Q}$ be a universal class such that ${\mathbf H}(\mathsf {U}) = {\mathbf H}(\mathsf {Q})$ and suppose that $\mathsf {U} \vDash \Sigma \Rightarrow \Delta$; then $${\mathbf H}(\mathsf {Q}) = {\mathbf H}(\mathsf {U}) \subseteq{\mathbf H}(\mathsf {U}_{\Sigma \Rightarrow \Delta}) \subseteq{\mathbf H}(\mathsf {Q}).$$ So ${\mathbf H}(\mathsf {U}_{\Sigma \Rightarrow \Delta}) = {\mathbf H}(\mathsf {Q})$ and by Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"} $\Sigma \Rightarrow \Delta$ is admissible in $\mathsf {Q}$. By ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}), $\mathsf {Q} \vDash \Sigma \Rightarrow \Delta$; therefore $\mathsf {U}$ and $\mathsf {Q}$ are two universal classes in which exactly the same clauses are valid, thus they are equal. Hence ([\[prop:universal1\]](#prop:universal1){reference-type="ref" reference="prop:universal1"}) holds, and thus ([\[prop:universal1\]](#prop:universal1){reference-type="ref" reference="prop:universal1"}) and ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}) are equivalent. ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}) implies ([\[prop:universal3\]](#prop:universal3){reference-type="ref" reference="prop:universal3"}) follows by Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}. Moreover, ([\[prop:universal3\]](#prop:universal3){reference-type="ref" reference="prop:universal3"}) clearly implies ([\[prop:universal4\]](#prop:universal4){reference-type="ref" reference="prop:universal4"}). We now show that ([\[prop:universal4\]](#prop:universal4){reference-type="ref" reference="prop:universal4"}) implies ([\[prop:universal2\]](#prop:universal2){reference-type="ref" reference="prop:universal2"}), which completes the proof. Consider a universal sentence $\Sigma \Rightarrow \Delta$ that is admissible in $\mathsf {Q}$, or equivalently (by Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}), such that ${\textbf{\upshape F}}_\mathsf {Q}(\omega) \models \Sigma \Rightarrow \Delta$. The finitely presented algebra ${\textbf{\upshape F}}_\mathsf {Q} (X)/\theta_{\mathsf {Q}}(\Sigma) \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ by (4), and thus ${\textbf{\upshape F}}_\mathsf {Q} (X)/\theta_{\mathsf {Q}}(\Sigma) \models \Sigma \Rightarrow \Delta$. By Lemma [Lemma 38](#lemma:FThetaSigma){reference-type="ref" reference="lemma:FThetaSigma"}, $\mathsf {Q} \models \Sigma \Rightarrow \Delta$ and thus $\mathsf {Q}$ is universally complete. ◻ By algebraizability, and since the property of being universal (for the discussion in Subsection [2.1](#subsec:universal){reference-type="ref" reference="subsec:universal"}) is preserved by categorical equivalence, we get at once: **Corollary 40**. *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is universally complete;* 2. *$\mathcal{L}_\mathsf {Q}$ is universally complete.* The following theorem and lemma show a sufficient and a necessary condition respectively for a quasivariety to be universally complete. **Theorem 41**. *If every finitely presented algebra in $\mathsf {Q}$ is exact then $\mathsf {Q}$ is universally complete.* *Proof.* If every finitely presented algebra in $\mathsf {Q}$ is exact, it is in ${\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$, and thus also in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$. The claim then follows from Theorem [Theorem 39](#prop:universal){reference-type="ref" reference="prop:universal"}. ◻ **Theorem 42**. *If $\mathsf {Q}$ is universally complete, then $\mathsf {Q}$ is unifiable.* *Proof.* Suppose by counterpositive that there is a finite set of identities $\Sigma$ that is not unifiable in $\mathsf {Q}$. Then $\Sigma \Rightarrow \emptyset$ is (passively) admissible but not derivable; indeed it does not hold in the trivial algebra. This implies that $\mathsf {Q}$ is not universally complete, and the claim is proved. ◻ Since projectivity implies exactness, we observe the following immediate consequence of Theorem [Theorem 41](#thm: unifiable){reference-type="ref" reference="thm: unifiable"}. **Corollary 43**. *If every finitely presented algebra in $\mathsf {Q}$ is projective then $\mathsf {Q}$ is universally complete.* For locally finite varieties there is a stronger result, observed in [@CabrerMetcalfe2015a]. **Lemma 44**. *[@CabrerMetcalfe2015a] Let $\mathsf {Q}$ be a locally finite quasivariety; then ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ if and only if every finite subalgebra ${\textbf{\upshape B}}$ of ${\textbf{\upshape A}}$ is in ${\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* **Theorem 45** ([@CabrerMetcalfe2015a]). *Let $\mathsf {Q}$ be a locally finite variety of finite type. Then $\mathsf {Q}$ is universally complete if and only if $\mathsf {Q}$ is unifiable and has exact unifiers.* *Proof.* Suppose that $\mathsf {Q}$ is universally complete; then, by Theorem [Theorem 42](#lemma: FQ trivial){reference-type="ref" reference="lemma: FQ trivial"}, $\mathsf {Q}$ is unifiable. Since it is universally complete, every finite algebra in $\mathsf {Q}$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$, hence in ${\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ (by Lemma [Lemma 44](#lemma: techlemma){reference-type="ref" reference="lemma: techlemma"}). Thus every finite unifiable algebra in $\mathsf {Q}$ is exact and $\mathsf {Q}$ has exact unifiers. The converse claim follows from Theorem [Theorem 41](#thm: unifiable){reference-type="ref" reference="thm: unifiable"}. ◻ **Remark 46**. We observe that Theorem [Theorem 42](#lemma: FQ trivial){reference-type="ref" reference="lemma: FQ trivial"} limits greatly the examples of universally complete quasivarieties. In particular, in quasivarieties with finite type the trivial algebra is finitely presented, and thus if $\mathsf {Q}$ is universally complete, it must be unifiable. This means that a quasivariety with more than one constant in its finite type cannot be universally complete if there are nontrivial models where the constants are distinct; similarly if there is only one constant, then it must generate the trivial algebra in nontrivial models, or equivalently, in ${\textbf{\upshape F}}_\mathsf {Q}$. If there are no constants, then ${\textbf{\upshape F}}_\mathsf {Q} = {\textbf{\upshape F}}_\mathsf {Q}(x)$ and, in order to be able to embed the trivial algebra, there has to be an idempotent term. Let us now discuss some different examples of universally complete (quasi)varieties. **Example 47**. Let us consider *lattice-ordered abelian groups* (or abelian $\ell$-groups for short). These are algebras ${\textbf{\upshape G}} = (G, \land, \lor, \cdot, ^{-1}, 1)$ where $(G, \cdot, ^{-1}, 1)$ is an abelian group, $(G, \land, \lor)$ is a lattice, and the group operation distributes over the lattice operations. Every finitely presented abelian $\ell$-groups is projective [@Beynon1977]; thus, the variety of abelian $\ell$-groups is universally complete by Corollary [Corollary 43](#cor: unifiable){reference-type="ref" reference="cor: unifiable"}. The same holds for the variety of *negative cones* of abelian $\ell$-groups. Given an $\ell$-group ${\textbf{\upshape G}}$, the set of elements $G^-= \{x \in G: x \leq 1\}$ can be seen as a *residuated lattice* (see Section [5.2](#sec:FL){reference-type="ref" reference="sec:FL"}) ${\textbf{\upshape G}}^-=(G^-, \cdot, \to, \land, \lor, 1)$ where $(\cdot, \land, \lor, 1)$ are inherited by the group and $x \to y = x^{-1} \cdot y \land 1$. The algebraic category of negative cones of abelian $\ell$-groups is equivalent to the one of abelian $\ell$-groups [@GalatosTsinakis2005], thus every finitely presented algebra is projective and the variety of negative cones of $\ell$-groups $\mathsf {L} \mathsf {G}^-$ is universally complete. Observe that in all these cases the unique constant $1$ is absorbing w.r.t. any basic operation, and it generates the trivial algebra. **Example 48**. *Hoops* are a particular variety of residuated monoids related to logic which were defined in an unpublished manuscript by Büchi and Owens, inspired by the work of Bosbach on partially ordered monoids (see [@BlokFerr2000] for details on the theory of hoops). Hoops have a constant which is absorbing w.r.t. any basic operation; hence the least free algebra is trivial in any variety of hoops and any variety of hoops is unifiable. In [@AglianoUgolini2022] it was shown that every finite hoop is projective in the class of finite hoops which via Lemma [Lemma 24](#tame){reference-type="ref" reference="tame"} entails that every locally finite variety of hoops has projective unifiers. Since any locally finite quasivariety is contained in a locally finite variety, every locally finite quasivariety of hoops is universally complete. The same holds in the variety of $\rightarrow$-subreducts of hoops, usually denoted by $\mathsf{HBCK}$; again locally finite varieties of $\mathsf{HBCK}$-algebras have projective unifiers [@AglianoUgolini2022] and hence they are universally complete. For a non-locally finite example, we say that a hoop is *cancellative* if the underlying monoid is cancellative; cancellative hoops form a variety $\mathsf {C}$ that is categorically equivalent to the one of abelian $\ell$-groups [@BlokFerr2000]. Hence $\mathsf {C}$ is a non locally finite variety of hoops which is universally complete. The classes of algebras in the above examples all have projective unifiers. However: **Example 49**. In lattices there are no constants but any variety of lattices is idempotent; hence the least free algebra is trivial and every lattice is unifiable. Every finite distributive lattice is exact [@CabrerMetcalfe2015a] and distributive lattices are locally finite, so distributive lattices are universally complete by Theorem [Theorem 45](#thm: lf unifiable){reference-type="ref" reference="thm: lf unifiable"}. Moreover, as we have already observed in Example [Example 25](#ex: distlattice){reference-type="ref" reference="ex: distlattice"}, distributive lattices do not have projective unifiers. We now consider the hereditary version of universal completeness. **Definition 50**. A quasivariety $\mathsf {Q}$ is **primitive universal** if all its subquasivarieties are universally complete. All the above examples of universally complete varieties are primitive universal and this is not entirely coincidental. Distributive lattices are trivially primitive universal, since they do not have any trivial subquasivariety. For all the other examples, we have a general result. **Theorem 51**. *Let $\mathsf {Q}$ be a quasivariety with projective unifiers and such that ${\textbf{\upshape F}}_\mathsf {Q}$ is trivial; then $\mathsf {Q}$ is primitive universal.* *Proof.* Observe that for any subquasivariety $\mathsf {Q}'\subseteq\mathsf {Q}$, ${\textbf{\upshape F}}_{\mathsf {Q}'}$ is trivial as well. Hence every algebra in $\mathsf {Q}$ is unifiable in any subvariety to which it belongs. Let ${\textbf{\upshape B}}' = {\textbf{\upshape F}}_{\mathsf {Q}'}(X)/\theta_{\mathsf {Q}'}(\Sigma)$ be finitely presented in $\mathsf {Q}'$; then ${\textbf{\upshape A}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}(\Sigma)$ is finitely presented in $\mathsf {Q}$ and thus it is projective in $\mathsf {Q}$. But then Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} applies and ${\textbf{\upshape B}}$ is projective; thus $\mathsf {Q}'$ has projective unifiers and thus it is universally complete by Corollary [Corollary 43](#cor: unifiable){reference-type="ref" reference="cor: unifiable"}. ◻ Is the same conclusion true if we replace "projective unifiers" with "unifiable, locally finite with exact unifiers"? We do not know, but we know that we cannot use an improved version of Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} since it cannot be improved to account for exact unifiers (see Example [Example 67](#ex: De Morgan){reference-type="ref" reference="ex: De Morgan"}). ## Non-negative and active universal quasivarieties {#subsec: active universal} The situation in which universal completeness fails due *only* to the trivial algebras has been first investigated in [@CabrerMetcalfe2015a]; the following expands [@CabrerMetcalfe2015a Proposition 8]. **Theorem 52**. *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *[\[thm: non negative universal3\]]{#thm: non negative universal3 label="thm: non negative universal3"} $\mathsf {Q}$ is non-negative universally complete;* 2. *[\[thm: non negative universal2\]]{#thm: non negative universal2 label="thm: non negative universal2"} every admissible universal sentence is valid in $\mathsf {Q}^+$;* 3. *[\[thm: non negative universal1\]]{#thm: non negative universal1 label="thm: non negative universal1"} every nontrivial algebra is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* 4. *[\[thm: non negative universal4\]]{#thm: non negative universal4 label="thm: non negative universal4"} every nontrivial finitely presented algebra is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* *Proof.* The equivalence of the first three points is in [@CabrerMetcalfe2015a Proposition 8], and ([\[thm: non negative universal1\]](#thm: non negative universal1){reference-type="ref" reference="thm: non negative universal1"}) clearly implies ([\[thm: non negative universal4\]](#thm: non negative universal4){reference-type="ref" reference="thm: non negative universal4"}). Assume now that ([\[thm: non negative universal4\]](#thm: non negative universal4){reference-type="ref" reference="thm: non negative universal4"}) holds, we show ([\[thm: non negative universal3\]](#thm: non negative universal3){reference-type="ref" reference="thm: non negative universal3"}). Let $\Sigma \Rightarrow \Delta$ be a non-negative admissible universal sentence with variables in a finite set $X$, we show that ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}(\Sigma) \models \Sigma \Rightarrow \Delta$. If ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}(\Sigma)$ is trivial, then it models $\Sigma \Rightarrow \Delta$ (given that $\Delta$ is not $\emptyset$). Suppose now that ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta_\mathsf {Q}(\Sigma)$ is nontrivial, then it is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ by hypothesis and then it models $\Sigma \Rightarrow \Delta$ since the latter is admissible, and thus ${\textbf{\upshape F}}_\mathsf {Q}(\omega) \models \Sigma \Rightarrow \Delta$ by Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}. By Lemma [Lemma 38](#lemma:FThetaSigma){reference-type="ref" reference="lemma:FThetaSigma"}, $\mathsf {Q}$ models $\Sigma \Rightarrow \Delta$ and ([\[thm: non negative universal3\]](#thm: non negative universal3){reference-type="ref" reference="thm: non negative universal3"}) holds. ◻ Moreover: **Theorem 53**. *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is non-negative universally complete;* 2. *$\mathcal{L}_\mathsf {Q}$ is non-negative universally complete.* *Proof.* In a categorical equivalence between quasivarieties trivial algebras are mapped to trivial algebras, since the latter can be characterized as the algebras that are a homomorphic image of every algebra in a quasivariety. Thus nontrivial finitely presented algebras are mapped to nontrivial finitely presented algebras, and the result follows from the usual arguments. ◻ We can also obtain an analogue of Theorem [Theorem 41](#thm: unifiable){reference-type="ref" reference="thm: unifiable"}. **Theorem 54**. *If every nontrivial finitely presented algebra in $\mathsf {Q}$ is exact (or projective), then $\mathsf {Q}$ is non-negative universally complete.* *Proof.* If every nontrivial finitely presented algebra is exact (or projective), then it is in ${\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$, and therefore in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$. The claim then follows from Theorem [Theorem 52](#thm: non negative universal){reference-type="ref" reference="thm: non negative universal"}. ◻ Analogously to the case of universal completeness, we get a stronger result for locally finite quasivarieties. **Theorem 55**. *Let $\mathsf {Q}$ be a locally finite quasivariety. Then $\mathsf {Q}$ is non-negative universally complete if and only if every nontrivial finitely presented algebra is exact.* *Proof.* Suppose that $\mathsf {Q}$ is locally finite and there is a finite nontrivial algebra ${\textbf{\upshape A}} \in \mathsf {Q}$ that is not exact. Then ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ and thus, by Lemma [Lemma 44](#lemma: techlemma){reference-type="ref" reference="lemma: techlemma"}, ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_\mathsf {Q}(\omega))$. Therefore $\mathsf {Q}$ cannot be non-negative universally complete by Theorem [Theorem 52](#thm: non negative universal){reference-type="ref" reference="thm: non negative universal"}. The other direction follows from Theorem [Theorem 54](#thm:unifiable2){reference-type="ref" reference="thm:unifiable2"}. ◻ **Example 56**. Boolean algebras are an example of a non-negative universally complete variety that is not universally complete. It is easily seen that every nontrivial finite Boolean algebra is exact (indeed, projective), which shows that Boolean algebras are non-negative universally complete by Theorem [Theorem 55](#thm: nonnegative unifiable){reference-type="ref" reference="thm: nonnegative unifiable"}. However, there are negative admissible clauses: e.g., the ones with premises given by the presentation of the trivial algebra, which is finitely presented but not unifiable. Thus Boolean algebras are not universally complete. **Example 57**. Stone algebras are a different example; in [@CabrerMetcalfe2015a] the authors proved, using the duality between Stone algebras and particular Priestley spaces, that every finite nontrivial Stone algebra is exact; hence Stone algebras are non-negative universally complete. We now move on to describe active universal completeness from the algebraic perspective. **Theorem 58**. *Let $\mathsf {Q}$ be a quasivariety. The following are equivalent:* 1. *[\[thm: activeu main2\]]{#thm: activeu main2 label="thm: activeu main2"} $\mathsf {Q}$ is active universally complete;* 2. *[\[thm: activeu main5\]]{#thm: activeu main5 label="thm: activeu main5"} every unifiable algebra in $\mathsf {Q}$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$;* 3. *[\[thm: activeu main1\]]{#thm: activeu main1 label="thm: activeu main1"} every finitely presented and unifiable algebra in $\mathsf {Q}$ is in ${\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$;* 4. *[\[thm: activeu main3\]]{#thm: activeu main3 label="thm: activeu main3"} every universal sentence admissible in $\mathsf {Q}$ is satisfied by all finitely presented unifiable algebras in $\mathsf {Q}$;* 5. *[\[thm: activeu main4\]]{#thm: activeu main4 label="thm: activeu main4"} for every ${\textbf{\upshape A}} \in \mathsf {Q}$, ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}\in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$.* *Proof.* We start by showing that ([\[thm: activeu main2\]](#thm: activeu main2){reference-type="ref" reference="thm: activeu main2"}) implies ([\[thm: activeu main5\]](#thm: activeu main5){reference-type="ref" reference="thm: activeu main5"}). Assume ([\[thm: activeu main2\]](#thm: activeu main2){reference-type="ref" reference="thm: activeu main2"}), and let $\Sigma \Rightarrow \Delta$ be such that ${\textbf{\upshape F}}_\mathsf {Q}(\omega) \models \Sigma \Rightarrow \Delta$; equivalently, by Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}, $\Sigma \Rightarrow \Delta$ is an admissible universal sentence in $\mathsf {Q}$. If $\Sigma$ is unifiable, by hypothesis $\Sigma \Rightarrow \Delta$ is valid in $\mathsf {Q}$. Suppose now that $\Sigma$ has variables in a finite set $X$ and it is not unifiable, that is, via Corollary [Corollary 19](#cor:unifiablesigma){reference-type="ref" reference="cor:unifiablesigma"} there is no homomorphism from ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ to ${\textbf{\upshape F}}_{\mathsf {Q}}$. Let ${\textbf{\upshape A}}$ be a unifiable algebra in $\mathsf {Q}$; we argue that there is no assignment of the variables in $\Sigma$ that validates $\Sigma$ in ${\textbf{\upshape A}}$. Indeed otherwise the following diagram would commute and ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ would be unifiable, yielding a contradiction. Therefore, $\Sigma \Rightarrow \Delta$ is vacuously satisfied in ${\textbf{\upshape A}}$, which is any unifiable algebra in $\mathsf {Q}$, thus ([\[thm: activeu main5\]](#thm: activeu main5){reference-type="ref" reference="thm: activeu main5"}) holds. Now, clearly ([\[thm: activeu main5\]](#thm: activeu main5){reference-type="ref" reference="thm: activeu main5"}) implies ([\[thm: activeu main1\]](#thm: activeu main1){reference-type="ref" reference="thm: activeu main1"}), and ([\[thm: activeu main1\]](#thm: activeu main1){reference-type="ref" reference="thm: activeu main1"}) and ([\[thm: activeu main3\]](#thm: activeu main3){reference-type="ref" reference="thm: activeu main3"}) are equivalent by the definitions. Let us show that ([\[thm: activeu main3\]](#thm: activeu main3){reference-type="ref" reference="thm: activeu main3"}) implies ([\[thm: activeu main2\]](#thm: activeu main2){reference-type="ref" reference="thm: activeu main2"}). Let $\Sigma \Rightarrow \Delta$ be an active admissible universal sentence in $\mathsf {Q}$ with variables in a finite set $X$; we want to show that it is also valid in $\mathsf {Q}$. Since by hypothesis $\Sigma \Rightarrow \Delta$ is active admissible, $\Sigma$ is unifiable, and therefore so is ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ by Corollary [Corollary 19](#cor:unifiablesigma){reference-type="ref" reference="cor:unifiablesigma"}. Then by ([\[thm: activeu main3\]](#thm: activeu main3){reference-type="ref" reference="thm: activeu main3"}), ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \models \Sigma \Rightarrow \Delta$, which implies that $\mathsf {Q} \models \Sigma \Rightarrow \Delta$ by Lemma [Lemma 38](#lemma:FThetaSigma){reference-type="ref" reference="lemma:FThetaSigma"}. Therefore the first four points are equivalent. Finally, we show that ([\[thm: activeu main2\]](#thm: activeu main2){reference-type="ref" reference="thm: activeu main2"}) implies ([\[thm: activeu main4\]](#thm: activeu main4){reference-type="ref" reference="thm: activeu main4"}) and ([\[thm: activeu main4\]](#thm: activeu main4){reference-type="ref" reference="thm: activeu main4"}) implies ([\[thm: activeu main5\]](#thm: activeu main5){reference-type="ref" reference="thm: activeu main5"}), which completes the proof. We start with ([\[thm: activeu main2\]](#thm: activeu main2){reference-type="ref" reference="thm: activeu main2"}) $\Rightarrow$ ([\[thm: activeu main4\]](#thm: activeu main4){reference-type="ref" reference="thm: activeu main4"}). Let ${\textbf{\upshape A}} \in \mathsf {Q}$, and consider a clause $\Sigma \Rightarrow \Delta$ valid in ${\textbf{\upshape F}}_{\mathsf {Q}}(\omega)$. We show that ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \models \Sigma \Rightarrow \Delta$. Now, if $\mathsf {Q} \models \Sigma \Rightarrow \Delta$, in particular ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \models \Sigma \Rightarrow \Delta$. Suppose that $\mathsf {Q} \not\vDash \Sigma \Rightarrow \Delta$. Since $\mathsf {Q}$ is active universally complete, $\Sigma \Rightarrow \Delta$ must be a passive rule, thus $\Sigma$ is not unifiable. Equivalently, there is no assignment $h$ of the variables in $\Sigma$ such that ${\textbf{\upshape F}}_{\mathsf {Q}},h\models \Sigma$. Thus, there is also no assignment $h'$ of the variables in $\Sigma$ such that ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}},h'\models \Sigma$, thus ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}(\omega) \models \Sigma \Rightarrow \Delta$. It is left to prove ([\[thm: activeu main4\]](#thm: activeu main4){reference-type="ref" reference="thm: activeu main4"}) $\Rightarrow$ ([\[thm: activeu main5\]](#thm: activeu main5){reference-type="ref" reference="thm: activeu main5"}). Let ${\textbf{\upshape A}}$ be a unifiable algebra in $\mathsf {Q}$, then there is a homomorphism $h: {\textbf{\upshape A}} \to {\textbf{\upshape F}}_\mathsf {Q}$ (Lemma [Lemma 21](#free){reference-type="ref" reference="free"}). Consider the map $h': {\textbf{\upshape A}} \to {\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}$ be defined as $h'(a) = (a, h(a))$. Clearly, $h'$ is an embedding of ${\textbf{\upshape A}}$ into ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$ (by ([\[thm: activeu main4\]](#thm: activeu main4){reference-type="ref" reference="thm: activeu main4"})). Thus also ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$, which completes the proof. ◻ We observe that the previous characterization extends to universal sentences some of the results in [@DzikStronkowski2016] about active structural completeness. We also get the usual result. **Theorem 59**. *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is active universally complete;* 2. *$\mathcal{L}_\mathsf {Q}$ is active universally complete.* *Proof.* The result follows from the fact that embeddings, ultraproducts, being finitely presented and unifiable, are all categorical notions and thus preserved by categorical equivalence. ◻ Moreover, we have the following lemma whose proof is the same as the one of Theorems [Theorem 41](#thm: unifiable){reference-type="ref" reference="thm: unifiable"} and [Theorem 45](#thm: lf unifiable){reference-type="ref" reference="thm: lf unifiable"}. **Theorem 60**. *If $\mathsf {Q}$ has exact (or projective) unifiers, then $\mathsf {Q}$ is active universally complete. If $\mathsf {Q}$ is also locally finite then it is active universally complete if and only if it has exact unifiers.* **Example 61**. A *discriminator* on a set $A$ is a ternary operation $t$ on $A$ defined by $$t(a,b,c) =\left\{ \begin{array}{ll} a, & \hbox{if $a\ne b$;} \\ c, & \hbox{if $a=b$.} \end{array} \right.$$ A variety $\mathsf {V}$ is a *discriminator variety* [@Pixley1971] if there is a ternary term that is the discriminator on all the subdirectly irreducible members of $\mathsf {V}$. Discriminator varieties have many strong properties: for instance they are congruence permutable and congruence distributive. In [@Burris1992 Theorem 3.1] it has been essentially shown that discriminator varieties have projective unifiers, and therefore they are all active universally complete by Theorem [Theorem 60](#lemma: ex unif auc){reference-type="ref" reference="lemma: ex unif auc"}. **Example 62**. We now see some examples within the algebraic semantics of many-valued logics; in [@AglianoUgolini2022] it has been shown that in any locally finite variety of bounded hoops or $\mathsf {BL}$-algebras (the equivalent algebraic semantics of Hájek Basic Logic [@Hajek98]), the finite unifiable algebras are exactly the finite projective algebras. It follows that any of such varieties has projective unifiers and hence it is active universally complete. This holds also for any locally finite quasivariety of bounded hoops or $\mathsf {BL}$-algebras, or their reducts, i.e., bounded $\mathsf{HBCK}$-algebras. In contrast with the case of (unbounded) hoops, not all of them are non-negative universally complete, as we will now discuss. Let us call *chain* a totally orderef algebra. Every finite BL-chain is an ordinal sum of finite Wajsberg hoops, the first of which is an MV-algebra [@AglianoMontagna2003]. No finite MV-chain different from the 2-element Boolean algebra ${\textbf{\upshape 2}}$ is unifiable (they are all simple and the least free algebra is ${\textbf{\upshape 2}}$), and thus not exact. It follows by basic facts about ordinal sums that if a locally finite quasivariety $\mathsf {Q}$ of BL-algebras contains a chain whose first component is different from ${\textbf{\upshape 2}}$, $\mathsf {Q}$ is not non-negative universally complete. The same holds, mutatis mutandis, for bounded hoops and bounded $\mathsf{HBCK}$-algebras. In Section [5.2](#sec:FL){reference-type="ref" reference="sec:FL"} we shall see a different class of (discriminator) varieties coming from many-valued logics that are active universally complete. **Definition 63**. We call a quasivariety $\mathsf {Q}$ *active primitive universal* if every subquasivariety of $\mathsf {Q}$ is active universally complete. It is evident from the characterization theorem of active universally complete quasivarieties that a variety $\mathsf {Q}$ is active primitive universal if and only if $\mathcal{L}_\mathsf {Q}$ is hereditarily active universally complete. We have the following fact: **Theorem 64**. *Suppose that $\mathsf {Q}$ is a quasivariety such that ${\textbf{\upshape F}}_\mathsf {Q} = {\textbf{\upshape F}}_{\mathsf {Q}'}$ for all $\mathsf {Q}'\subseteq\mathsf {Q}$. If $\mathsf {Q}$ has projective unifiers then it is active primitive universal.* *Proof.* The proof follows from Theorem [Theorem 60](#lemma: ex unif auc){reference-type="ref" reference="lemma: ex unif auc"} and Lemma [Lemma 29](#lemma:dzik2){reference-type="ref" reference="lemma:dzik2"}. ◻ All varieties in Example [Example 62](#ex: BL-algebras au){reference-type="ref" reference="ex: BL-algebras au"} satisfy the hypotheses of Theorem [Theorem 64](#thm: activeprimitive){reference-type="ref" reference="thm: activeprimitive"} (as the reader can easily check). For discriminator varieties all the examples of lattice-based varieties in Section [5.2](#sec:FL){reference-type="ref" reference="sec:FL"} of this paper (but see also [@Burris1992] or [@Citkin2018a] for more examples) have the same property; hence they are all active primitive universal. Now, a variety is *q-minimal* if it does not have any proper nontrivial subquasivariety; so a q-minimal variety is necessarily equationally complete. We have this result by Bergman and McKenzie: **Theorem 65**. *[@BergmanMcKenzie1990] A locally finite equationally complete variety is q-minimal if and only if it has exactly one subdirectly irreducible algebra that is embeddable in any nontrivial member of the variety. Moreover, this is always the case if the variety is congruence modular.* It follows immediately that every active universally complete q-minimal variety is active primitive universal. **Example 66**. Discriminator varieties are active universally complete as seen in example [Example 61](#ex:discriminator){reference-type="ref" reference="ex:discriminator"}. Now, given a finitely generated discriminator variety $\mathsf {V}$, it is generated by a finite algebra ${\textbf{\upshape A}}$ having a discriminator term on it, also called a *quasi-primal* algebra. By [@Werner1970] $\mathsf {V}$ is equationally complete and, since it is congruence modular, it is q-minimal; hence $\mathsf {V}$ is active primitive universal. Finally, we observe that Lemma [Lemma 29](#lemma:dzik2){reference-type="ref" reference="lemma:dzik2"} cannot be improved to "having exact unifiers" and the counterexample is given by *De Morgan lattices*; we will see below that they form an active universally complete variety that is not active primitive universal. **Example 67**. A De Morgan lattice is a distributive lattice with a unary operation $\neg$ which is involutive and satisfies the De Morgan Laws. It is well-known that the variety $\mathsf{DM}$ of De Morgan lattices is locally finite and has exactly two proper non trivial subvarieties, i.e. the variety $\mathsf{BLa}$ of Boolean lattices (axiomatized by $x \le y \vee\neg y$) and the variety $\mathsf{KL}$ of Kleene lattices (axiomatized by $x \wedge\neg x \le y \vee\neg y$). It is easily seen that all these nontrivial varieties have the same one-generated free algebra whose universe is $\{x, \neg x, x \vee\neg x, x \wedge\neg x\}$. It follows that all the subquasivarieties of De Morgan lattices have the same least free algebra and $\mathsf{DM}$ satisfies the hypotheses of Theorem [Theorem 64](#thm: activeprimitive){reference-type="ref" reference="thm: activeprimitive"}. Admissibility in De Morgan lattices has been investigated in [@MetcalfeRothlisberger2012] and [@CabrerMetcalfe2015a]. Now for a finite algebra ${\textbf{\upshape A}} \in \mathsf{DM}$ the following are equivalent: 1. ${\textbf{\upshape A}}$ is unifiable; 2. the universal sentence $\{x \approx\neg x\} \Rightarrow \emptyset$ is valid in ${\textbf{\upshape A}}$; 3. ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {DM}(\omega))$. The equivalence of (2) and (3) has been proved in [@CabrerMetcalfe2015a Lemma 28], while (3) implies (1) trivially. If we assume that (2) does not hold for ${\textbf{\upshape A}}$, then there is an $a \in A$ with $\neg a = a$; so if $f:{\textbf{\upshape A}} \longrightarrow {\textbf{\upshape F}}_\mathsf {DM} (x)$ is a homomorphism and $f(a) = \varphi$, then $\varphi= \neg \varphi$. But there is no element in ${\textbf{\upshape F}}_\mathsf {DM} (x)$ with that property, so ${\textbf{\upshape A}}$ cannot be unifiable. This concludes the proof of the equivalence of the three statements. Therefore $\mathsf {DM}$ has exact unifiers and thus it is active universally complete by Theorem [Theorem 60](#lemma: ex unif auc){reference-type="ref" reference="lemma: ex unif auc"}. Now consider the subvariety of $\mathsf {DM}$ of Kleene lattices. In [@CabrerMetcalfe2015a] it is shown that the universal sentence $$\Phi := \{x \le \neg x, x \wedge\neg y \le \neg x \vee y\} \Rightarrow \neg y \le y$$ is admissible in $\mathsf{KL}$. It is also active, as the reader can easily check that the substitution $x \longmapsto z \land \neg z$, $y \longmapsto \neg z$ unifies the premises of $\Phi$. However it fails in the three element Kleene lattice ${\textbf{\upshape K}}_3$ in Figure [\[Kleene\]](#Kleene){reference-type="ref" reference="Kleene"}, with the assignment $x =a$, $y=\neg a$; hence $\mathsf{KL}$ is not active universally complete. So $\mathsf {DM}$ is a variety that is active universally complete but not active primitive universal. Note that in $\mathsf{KL}$ there must be a finite unifiable algebra that is not exact (since $\mathsf{KL}$ cannot have exact unifiers). Now a finite Kleene lattice ${\textbf{\upshape A}}$ is exact if and only if both $\{x \approx\neg x\} \Rightarrow \emptyset$ and $\Phi$ are valid in ${\textbf{\upshape A}}$ [@CabrerMetcalfe2015a Lemma 38]. Let ${\textbf{\upshape A}} = {\textbf{\upshape K}}_3 \times \mathbf 2$; the reader can easily check that ${\textbf{\upshape A}}$ is unifiable in $\mathsf{KL}$ (since it satisfies $\{x \approx\neg x\} \Rightarrow \emptyset$ and hence it is unifiable in $\mathsf{DM}$) but does not satisfy $\Phi$. This shows (as promised) that Lemma [Lemma 28](#lemma:DZlemma){reference-type="ref" reference="lemma:DZlemma"} cannot be improved. ## Passive universal quasivarieties We will now see that passive universal completeness in a quasivariety corresponds to an algebraic notion we have already introduced: unifiability. Moreover, we shall see that it corresponds to the apparently weaker notion of negative universal completeness, that is, every (passive) admissible negative universal sentence is derivable. We recall that a quasivariety $\mathsf {Q}$ is unifiable if every finitely presented algebra in $\mathsf {Q}$ is unifiable. **Theorem 68**. *For every quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is passive universally complete;* 2. *$\mathsf {Q}$ is negative universally complete;* 3. *$\mathsf {Q}$ is unifiable.* *Proof.* Assume (1) and let $\Sigma \Rightarrow \emptyset$ be a negative admissible universal sentence; then it is necessarily passive, since there is no substitution that unifies $\emptyset$. Thus, by (1), $\Sigma \Rightarrow \emptyset$ is valid in $\mathsf {Q}$. Assume now (2), we prove that it implies (3) by contrapositive. Suppose that $\mathsf {Q}$ is not unifiable, that is, there exists a finite set of identities $\Sigma$ that is not unifiable. Then the negative universal sentence $\Sigma \Rightarrow \emptyset$ is (passively) admissible, but it is not derivable (in particular, it fails in the trivial algebra). Finally, if (3) holds, then (1) trivially holds, since if every set of identities is unifiable there is no passive admissible clause. ◻ In some cases, we can improve the previous result. **Lemma 69**. *Let $\mathsf {Q}$ be a quasivariety such that ${\mathbf I}({\textbf{\upshape F}}_{\mathsf {Q}}) = {\mathbf I}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}})$, then the following are equivalent.* 1. *$\mathsf {Q}$ is unifiable;* 2. *every algebra in $\mathsf {Q}$ is unifiable.* *Proof.* We prove the nontrivial direction by contraposition. Consider an arbitrary algebra ${\textbf{\upshape A}} \in \mathsf {Q}$ and assume that it is not unifiable; without loss of generality we let ${\textbf{\upshape A}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta$ for some set $X$ and some relative congruence $\theta$. Since ${\textbf{\upshape A}}$ is not unifiable, there is no assignment $h: {\textbf{\upshape F}}_\mathsf {Q}(X) \to {\textbf{\upshape F}}_\mathsf {Q}$ such that ${\textbf{\upshape F}}_\mathsf {Q}, h \models \Sigma_\theta$, where $\Sigma_\theta = \{t \approx u : (t,u) \in \theta\}$. Equivalently, iff ${\textbf{\upshape F}}_\mathsf {Q} \models \Sigma_\theta \Rightarrow \emptyset$. Now, the equational consequence relation relative to a class of algebras $\mathsf {K}$ is finitary if and only if $\mathsf {K}$ is closed under ultraproducts (see for instance [@Raftery2006]); thus by hypothesis the equational consequence relation relative to ${\textbf{\upshape F}}_{\mathsf {Q}}$ is finitary, and we obtain that ${\textbf{\upshape F}}_\mathsf {Q} \models \Sigma'_\theta \Rightarrow \emptyset$, for $\Sigma'_\theta$ some finite subset of $\Sigma_\theta$. That is, $\Sigma'$ is finite and not unifiable, thus $\mathsf {Q}$ is not unifiable and the proof is complete. ◻ Observe that if a quasivariety $\mathsf {Q}$ is such that ${\textbf{\upshape F}}_\mathsf {Q}$ is finite, it satisfies the hypothesis of the previous lemma. **Corollary 70**. *Let $\mathsf {Q}$ be a quasivariety such that ${\mathbf I}({\textbf{\upshape F}}_{\mathsf {Q}}) = {\mathbf I}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}})$, then the following are equivalent.* 1. *$\mathsf {Q}$ is passive universally complete;* 2. *$\mathsf {Q}$ is negative universally complete;* 3. *$\mathsf {Q}$ is unifiable;* 4. *every algebra in $\mathsf {Q}$ is unifiable.* Since unifiability is preserved by categorical equivalence, we get the following. **Corollary 71**. *A quasivariety $\mathsf {Q}$ is passive universally complete if and only if $\mathcal{L}_\mathsf {Q}$ is passive universally complete.* # Structural completeness {#structprim} In this section we investigate the algebraic counterparts of structural completeness and its variations. The main new results are about the characterization of passive structurally complete quasivarieties; moreover, we also show a characterization of primitive quasivarieties grounding on the results in [@Gorbunov1998]. ## Structural quasivarieties The bridge theorems for structural completeness have been first established by Bergman [@Bergman1991]. We present the proof for the sake of the reader, expanding with point (6). **Theorem 72** ([@Bergman1991]). *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *[\[structural5\]]{#structural5 label="structural5"} $\mathsf {Q}$ is structurally complete;* 2. *[\[structural4\]]{#structural4 label="structural4"} $\mathsf {Q} = {\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$;* 3. *[\[structural1\]]{#structural1 label="structural1"} no proper subquasivariety of $\mathsf {Q}$ generates a proper subvariety of ${\mathbf H}(\mathsf {Q})$;* 4. *[\[structural2\]]{#structural2 label="structural2"} for all $\mathsf {Q}'\subseteq\mathsf {Q}$ if ${\mathbf H}(\mathsf {Q}') = {\mathbf H}(\mathsf {Q})$, then $\mathsf {Q} = \mathsf {Q}'$;* 5. *[\[structural3\]]{#structural3 label="structural3"} for all ${\textbf{\upshape A}} \in \mathsf {Q}$ if ${\mathbf V}({\textbf{\upshape A}}) = {\mathbf H}(\mathsf {Q})$, then ${\mathbf Q}({\textbf{\upshape A}}) = \mathsf {Q}$;* 6. *[\[structural6\]]{#structural6 label="structural6"} every finitely presented algebra in $\mathsf {Q}$ is in ${\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* *Proof.* First, ([\[structural5\]](#structural5){reference-type="ref" reference="structural5"}) is equivalent to ([\[structural4\]](#structural4){reference-type="ref" reference="structural4"}) via Lemma [Theorem 35](#admissiblederivable){reference-type="ref" reference="admissiblederivable"}. The implications ([\[structural1\]](#structural1){reference-type="ref" reference="structural1"}) $\Leftrightarrow$ ([\[structural2\]](#structural2){reference-type="ref" reference="structural2"}) $\Rightarrow$ ([\[structural3\]](#structural3){reference-type="ref" reference="structural3"}) $\Rightarrow$ ([\[structural4\]](#structural4){reference-type="ref" reference="structural4"}) are straightforward. ([\[structural4\]](#structural4){reference-type="ref" reference="structural4"}) implies ([\[structural2\]](#structural2){reference-type="ref" reference="structural2"}) since if $\mathsf {Q}'\subseteq\mathsf {Q}$ and ${\mathbf H}(\mathsf {Q}') = {\mathbf H}(\mathsf {Q})$, we get that ${\textbf{\upshape F}}_{\mathsf {Q}'}(\omega) = {\textbf{\upshape F}}_{{\mathbf H}(\mathsf {Q}')}(\omega) = {\textbf{\upshape F}}_{{\mathbf H}(\mathsf {Q})}(\omega) = {\textbf{\upshape F}}_{\mathsf {Q}}(\omega)$; thus $\mathsf {Q} = {\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega)) = {\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}'(\omega)) \subseteq\mathsf {Q}'$ and then equality holds. Thus the first five points are equivalent; Finally, clearly ([\[structural4\]](#structural4){reference-type="ref" reference="structural4"}) implies ([\[structural6\]](#structural6){reference-type="ref" reference="structural6"}), and ([\[structural6\]](#structural6){reference-type="ref" reference="structural6"}) implies ([\[structural4\]](#structural4){reference-type="ref" reference="structural4"}) since a quasivariety is generated by its finitely presented algebras ([@Gorbunov1998 Proposition 2.1.18]). ◻ **Corollary 73**. *A variety $\mathsf {V}$ is structurally complete if and only if every proper subquasivariety of $\mathsf {V}$ generates a proper subvariety; therefore if ${\textbf{\upshape A}}$ is such that ${\mathbf V}({\textbf{\upshape A}})$ is structurally complete, then ${\mathbf V}({\textbf{\upshape A}}) = {\mathbf Q}({\textbf{\upshape A}})$.* Since the definition of structural completeness is invariant under categorical equivalence we get also: **Corollary 74**. *Let $\mathsf {Q}$ be a quasivariety; then $\mathsf {Q}$ is structurally complete if and only if $\mathcal{L}_\mathsf {Q}$ is structurally complete.* Let us extract some sufficient conditions for structural completeness. **Lemma 75**. *Let $\mathsf {Q}$ be a quasivariety; if* 1. *every ${\textbf{\upshape A}}\in \mathsf {K}$ is exact in $\mathsf {Q} = {\mathbf Q}(\mathsf {K})$, or* 2. *every finitely generated algebra in $\mathsf {Q}$ is exact, or* 3. *every finitely presented algebra in $\mathsf {Q}$ is exact, or* 4. *every finitely generated relative subdirectly irreducible in $\mathsf {Q}$ is exact,* *then $\mathsf {Q}$ is structurally complete. Moreover if every ${\textbf{\upshape A}}\in \mathsf {K}$ is exact in ${\mathbf V}(\mathsf {K})$ and every subdirectly irreducible member of ${\mathbf V}(\mathsf {K})$ is in ${\mathbf I}{\mathbf S}(\mathsf {K})$, then ${\mathbf V}(\mathsf {K})$ is structurally complete.* *Proof.* If each algebra in $\mathsf {K}$ is exact in $\mathsf {Q} = {\mathbf Q}(\mathsf {K})$, then $\mathsf {K} \subseteq{\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$; therefore $\mathsf {Q} ={\mathbf Q}(\mathsf {K}) \subseteq{\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ and thus equality holds. Hence $\mathsf {Q}$ is structurally complete by the characterization theorem. The other points follow. For the last claim, every subdirectly irreducible member of ${\mathbf V}(\mathsf {K})$ lies in ${\mathbf I}{\mathbf S}(\mathsf {K})$ and thus is exact in ${\mathbf V}(\mathsf {K})$. Since any variety is generated as a quasivariety by its subdirectly irreducible members, ${\mathbf V}(\mathsf {K})$ is structurally complete. ◻ We observe that none of the previous conditions is necessary. For locally finite quasivarieties we have a necessary and sufficient condition for structural completeness because of the following: **Lemma 76** ([@CabrerMetcalfe2015a]). *Let $\mathsf {Q}$ be a locally finite quasivariety and ${\textbf{\upshape A}}$ a finite algebra in $\mathsf {Q}$. Then ${\textbf{\upshape A}} \in {\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ if and only if ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$.* The following theorem improves [@CabrerMetcalfe2015a Corollary 11]. **Theorem 77**. *For a locally finite quasivariety $\mathsf {Q}$ of finite type the following are equivalent:* 1. *$\mathsf {Q}$ is structurally complete;* 2. *each finite algebra in $\mathsf {Q}$ is in ${\mathbf I}{\mathbf S}{\mathbf P}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$;* 3. *every finite relative subdirectly irreducible in $\mathsf {Q}$ is exact.* *Proof.* Assume (1); then each finite algebra in $\mathsf {Q}$ is in ${\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ and thus, by Lemma [Lemma 76](#lemma: lf ISP){reference-type="ref" reference="lemma: lf ISP"}, is in ${\mathbf I}{\mathbf S}{\mathbf P}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$ and (2) holds. If (2) holds and ${\textbf{\upshape A}}$ is finite relative subdirectly irreducible, then it is in ${\mathbf I}{\mathbf S}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$, i.e. it is exact. Finally if (3) holds, then $\mathsf {Q}$ is structurally complete by Lemma [Lemma 75](#lemma: wpstructcomplete){reference-type="ref" reference="lemma: wpstructcomplete"}. ◻ ## Primitive quasivarieties {#subsec: prim quasivarieties} We now consider the hereditary notion of structural completeness. **Definition 78**. A class of algebras $\mathsf {K}$ in a quasivariety $\mathsf {Q}$ is *equational relative to* $\mathsf {Q}$ if $\mathsf {K} = {\mathbf V}(\mathsf {K})\cap \mathsf {Q}$. In particular, a subquasivariety $\mathsf {Q}'$ of $\mathsf {Q}$ is *equational relative to* $\mathsf {Q}$ if $\mathsf {Q}' = {\mathbf H}(\mathsf {Q}')\cap \mathsf {Q}$; a quasivariety $\mathsf {Q}$ is *primitive* if every subquasivariety of $\mathsf {Q}$ is equational relative to $\mathsf {Q}$. Clearly primitivity is downward hereditary and a variety $\mathsf {V}$ is primitive if and only if every subquasivariety of $\mathsf {V}$ is a variety. We can show the following. **Theorem 79**. *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is primitive;* 2. *every subquasivariety of $\mathsf {Q}$ is structurally complete;* 3. *for all subdirectly irreducible ${\textbf{\upshape A}} \in {\mathbf H}(\mathsf {Q})$ and for any ${\textbf{\upshape B}} \in \mathsf {Q}$, if ${\textbf{\upshape A}} \in {\mathbf H}({\textbf{\upshape B}})$, then ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape B}})$.* *Proof.* We first show the equivalence between (1) and (2). Suppose that $\mathsf {Q}$ is primitive and let $\mathsf {Q}'\subseteq\mathsf {Q}$; if $\mathsf {Q}'' \subseteq\mathsf {Q}'$ and ${\mathbf H}(\mathsf {Q}'') = {\mathbf H}(\mathsf {Q}')$ then $$\mathsf {Q}' = {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q} = {\mathbf H}(\mathsf {Q}'') \cap \mathsf {Q} = \mathsf {Q}''$$ so $\mathsf {Q}'$ is structurally complete by Theorem [Theorem 72](#structural){reference-type="ref" reference="structural"}. Conversely assume (2), let $\mathsf {Q}' \subseteq\mathsf {Q}$ and let $\mathsf {Q}'' = {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q}$ (it is clearly a quasivariety); then ${\mathbf H}(\mathsf {Q}'') = {\mathbf H}(\mathsf {Q}')$ and thus $\mathsf {Q}'' = \mathsf {Q}'$, again using the characterization of Theorem [Theorem 72](#structural){reference-type="ref" reference="structural"}. So $\mathsf {Q}'$ is equational in $\mathsf {Q}$ and $\mathsf {Q}$ is primitive. Assume (1) again, and let ${\textbf{\upshape A}},{\textbf{\upshape B}} \in \mathsf {Q}$ with ${\textbf{\upshape A}}$ subdirectly irreducible and ${\textbf{\upshape A}} \in {\mathbf H}({\textbf{\upshape B}})$. Since $\mathsf {Q}$ is primitive we have $${\mathbf Q}({\textbf{\upshape B}}) = {\mathbf H}({\mathbf Q}({\textbf{\upshape B}})) \cap \mathsf {Q}$$ and hence ${\textbf{\upshape A}} \in {\mathbf Q}({\textbf{\upshape B}})$. Since ${\textbf{\upshape A}}$ is subdirectly irreducible, ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape B}})$ by Theorem [Theorem 3](#quasivariety){reference-type="ref" reference="quasivariety"} and (3) holds. Conversely, assume (3) and let $\mathsf {Q}'$ be a subquasivariety of $\mathsf {Q}$; if ${\textbf{\upshape B}} \in {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q}$, observe that ${\textbf{\upshape B}} \in {\mathbf H}(\mathsf {Q})$ and hence ${\textbf{\upshape B}} \le_{sd} \prod {\textbf{\upshape A}}_i$ where the ${\textbf{\upshape A}}_i$ are subdirectly irreducible in ${\mathbf H}(\mathsf {Q}) \cap {\mathbf H}(\mathsf {Q}')$. Then for all $i$ there is ${\textbf{\upshape B}}_i \in \mathsf {Q}'$ such that ${\textbf{\upshape A}}_i \in {\mathbf H}({\textbf{\upshape B}}_i)$ and hence by hypothesis ${\textbf{\upshape A}}_i \in {\mathbf S}{\mathbf P}_u({\textbf{\upshape B}}_i)$ and so ${\textbf{\upshape A}}_i \in \mathsf {Q}'$ for all $i$. Therefore ${\textbf{\upshape B}} \in \mathsf {Q}'$, so ${\mathbf H}(\mathsf {Q}') \cap \mathsf {Q} =\mathsf {Q}'$ and $\mathsf {Q}'$ is equational in $\mathsf {Q}$. Therefore $\mathsf {Q}$ is primitive and (1) holds. ◻ As commented in the preliminary section (Subsection [2.1](#subsec:universal){reference-type="ref" reference="subsec:universal"}), primitivity is preserved under categorical equivalence, and therefore: **Corollary 80**. *A quasivariety is primitive if and only if $\mathcal{L}_\mathsf {Q}$ is hereditarily structurally complete.* We will see how Theorem [Theorem 79](#primitiveQ){reference-type="ref" reference="primitiveQ"} can be improved in the locally finite case. Let $\mathsf {Q}$ be a quasivariety and let ${\textbf{\upshape A}} \in \mathsf {Q}$; we define $$[\mathsf {Q}:{\textbf{\upshape A}}] =\{{\textbf{\upshape B}} \in \mathsf {Q}: {\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})\}.$$ The following lemma describes some properties of $[\mathsf {Q}:{\textbf{\upshape A}}]$; the proofs are quite standard with the exception of point (3). As a matter of fact a proof of the forward implication of (3) appears in [@Gorbunov1998 Corollary 2.1.17]. However the proof is somewhat buried into generality and it is not easy to follow; so we felt that a suitable translation would make it easier for the readers. **Lemma 81**. *Let $\mathsf {Q}$ be a quasivariety; then* 1. *if ${\textbf{\upshape A}} \in \mathsf {Q}$ is finite and $\mathsf {Q}$ has finite type, then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a universal class;* 2. *if ${\textbf{\upshape A}}$ is relative subdirectly irreducible and finitely presented, then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a quasivariety;* 3. *${\textbf{\upshape A}}$ is weakly projective in $\mathsf {Q}$ if and only if $[\mathsf {Q}:{\textbf{\upshape A}}]$ is closed under ${\mathbf H}$ if and only if $[\mathsf {Q}:{\textbf{\upshape A}}]$ is equational relative to $\mathsf {Q}$;* 4. *if ${\textbf{\upshape A}}$ is relative subdirectly irreducible, finitely presented and weakly projective in $\mathsf {Q}$, then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a variety.* *Moreover if $\mathsf {Q}$ is locally finite of finite type, the converse implications in (1),(2) and (4) hold.* *Proof.* For (1), if ${\textbf{\upshape A}}$ is finite, then there is a first order universal sentence $\Psi$ such that, for all ${\textbf{\upshape B}} \in \mathsf {Q}$, ${\textbf{\upshape B}} \vDash \Psi$ if and only if ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$. More in detail, if $|A|=n$, $$\Psi: = \exists x_1 \ldots \exists x_n (\&\{ x_i \neq x_j: i,j \leq n, i \neq j\} \,\,\&\,\, {\textbf{\upshape D}}({\textbf{\upshape A}})),$$ where ${\textbf{\upshape D}}({\textbf{\upshape A}})$ is the diagram of ${\textbf{\upshape A}}$, that is, a conjunction of universal sentences that describe the operation tables of ${\textbf{\upshape A}}$ (identifying each element of ${\textbf{\upshape A}}$ with a different $x_i$), and $\&$ is first order logic conjunction. Consider ${\textbf{\upshape B}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u([\mathsf {Q}:{\textbf{\upshape A}}])$, we show that ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$; if ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$, then ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u([\mathsf {Q}:{\textbf{\upshape A}}])$. Hence there exists a family $({\textbf{\upshape A}}_i)_{i\in I} \subseteq[\mathsf {Q}: {\textbf{\upshape A}}]$ and an ultrafilter $U$ on $I$ such that ${\textbf{\upshape C}} = \Pi_{i\in I}{\textbf{\upshape A}}/U$ and ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape C}})$. So ${\textbf{\upshape C}} \vDash \Psi$; but then by Łòs Lemma there is a (necessarily nonempty) set of indexes $I' \in U$ such that $\Psi$ is valid in each ${\textbf{\upshape A}}_i$ with $i \in I'$, which is clearly a contradiction, since each ${\textbf{\upshape A}}_i \in [\mathsf {Q}:{\textbf{\upshape A}}]$. Thus ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$ and ${\textbf{\upshape B}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$ and therefore ${\mathbf I}{\mathbf S}{\mathbf P}_u([\mathsf {Q}:{\textbf{\upshape A}}]) = [\mathsf {Q}:{\textbf{\upshape A}}]$ which is a universal class by Lemma [Lemma 1](#lemma:ISP){reference-type="ref" reference="lemma:ISP"} Conversely let $\mathsf {Q}$ be locally finite of finite type; every algebra in $\mathsf {Q}$ is embeddable in an ultraproduct of its finitely generated (i.e. finite) subalgebras, say ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}{\mathbf P}_u(\{{\textbf{\upshape B}}_i: i\in I\})$. If ${\textbf{\upshape A}}$ is not finite, then ${\textbf{\upshape A}} \notin {\mathbf S}({\textbf{\upshape B}}_i)$ for all $i$, so ${\textbf{\upshape B}}_i \in [\mathsf {Q}:{\textbf{\upshape A}}]$ for all $i$. Since $[\mathsf {Q}:{\textbf{\upshape A}}]$ is universal, we would have that ${\textbf{\upshape A}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$, a clear contradiction. So ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}}_i)$ for some $i$ and hence it is finite. For (2), suppose that ${\textbf{\upshape A}}$ is relative subdirectly irreducible and finitely presented, i.e. ${\textbf{\upshape A}} \cong {\textbf{\upshape F}}_\mathsf {Q}(\mathbf x)/\theta(\Sigma)$ where $\mathbf x = (x_1,\dots,x_{n})$ and $\Sigma = \{ p_i(\mathbf x) \approx q_i(\mathbf x): i = 1,\dots,m\}$. We set $a_i = x_i/\theta(\Sigma)$; since ${\textbf{\upshape A}}$ is relative subdirectly irreducible, it has a relative monolith $\mu$, i.e. a minimal non trivial relative congruence. Since $\mu$ is minimal, there are $c,d \in A$ such that $\mu$ is the relative congruence generated by the pair $(c,d)$. Now let $t_c,t_d$ terms in ${\textbf{\upshape F}}_\mathsf {Q}(\mathbf x)$ such that $t_c(a_1,\dots,a_{n}) = c$ and $t_d(a_1,\dots,a_{n})=d$ and let $\Phi$ be the quasiequation $$\bigwedge_{i=1}^m p_i(\mathbf x) \approx q_i(\mathbf x) \ \longrightarrow \ t_c(\mathbf x) \approx t_d(\mathbf x).$$ Then ${\textbf{\upshape A}} \not\vDash \Phi$; moreover if ${\textbf{\upshape C}} \in \mathsf {Q}$ is a homomorphic image of ${\textbf{\upshape A}}$ which is not isomorphic with ${\textbf{\upshape A}}$, then ${\textbf{\upshape C}} \vDash \Phi$. We claim that $[\mathsf {Q}:{\textbf{\upshape A}}] =\{{\textbf{\upshape B}} \in \mathsf {Q}: {\textbf{\upshape B}} \vDash \Phi\}$ and since $\Phi$ is a quasiequation this implies that $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a quasivariety. Clearly if ${\textbf{\upshape B}} \vDash \Phi$, then ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$; conversely assume that ${\textbf{\upshape B}} \not\vDash \Phi$. Then there are $b_1,\dots,b_{n} \in B$ such that $p_i(b_1,\dots,b_{n}) = q_i(b_1,\dots,b_{n})$ but $t_c(b_1,\dots,b_{n}) \ne t_d(b_1,\dots,b_{n})$. Let $g$ be the homomorphism extending the assignment $x_i \longmapsto b_i$; then $\theta(\Sigma) \subseteq\textrm{\upshape ker}(g)$ so by the Second Homomorphism Theorem there is a homomorphism $f: {\textbf{\upshape A}} \longrightarrow {\textbf{\upshape B}}$ such that $f(a_i) = b_i$. Observe that $f({\textbf{\upshape A}}) \in \mathsf {Q}$ (since it is a subalgebra of ${\textbf{\upshape B}} \in \mathsf {Q}$) and $f({\textbf{\upshape A}}) \not\vDash \Phi$, so by what we said above $f({\textbf{\upshape A}}) \cong {\textbf{\upshape A}}$; this clearly implies ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$, so ${\textbf{\upshape B}} \notin [\mathsf {Q}: {\textbf{\upshape A}}]$ as wished. For the converse, let $\mathsf {Q}$ be locally finite of finite type; by (1) ${\textbf{\upshape A}}$ is finite. Suppose that ${\textbf{\upshape A}} \le_{sd} \prod_{i \in I} {\textbf{\upshape B}}_i$ where each ${\textbf{\upshape B}}_i$ is relative subdirectly irreducible in $\mathsf {Q}$. Since ${\textbf{\upshape A}}$ is finite, each ${\textbf{\upshape B}}_i$ can be taken to be finite; if ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}}_i)$ for all $i$, then ${\textbf{\upshape B}}_i \in [\mathsf {Q}:{\textbf{\upshape A}}]$ for all $i$ and hence, being $[\mathsf {Q}:{\textbf{\upshape A}}]$ a quasivariety we have ${\textbf{\upshape A}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$ which is impossible. Hence there is an $i$ such that ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}}_i)$, so that $|A| \le |B_i|$; on the other hand ${\textbf{\upshape B}} \in {\mathbf H}({\textbf{\upshape A}})$, so $|B| \le |A|$. Since everything is finite we have ${\textbf{\upshape A}} \cong {\textbf{\upshape B}}_i$ and then ${\textbf{\upshape A}}$ is relative subdirectly irreducible. For the first forward direction of (3), suppose that ${\textbf{\upshape B}} \in {\mathbf H}([\mathsf {Q}:{\textbf{\upshape A}}])$. If ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$, then ${\textbf{\upshape A}} \in {\mathbf S}{\mathbf H}([\mathsf {Q}:{\textbf{\upshape A}}]) \subseteq{\mathbf H}{\mathbf S}([\mathsf {Q}:{\textbf{\upshape A}}])$. Now $[\mathsf {Q}: {\textbf{\upshape A}}]\subseteq\mathsf {Q}$ and ${\textbf{\upshape A}}$ is weakly projective in $\mathsf {Q}$; so ${\textbf{\upshape A}} \in {\mathbf S}([\mathsf {Q}:{\textbf{\upshape A}}])$ which is impossible. It follows that ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$ and ${\textbf{\upshape B}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$; thus $[\mathsf {Q}:{\textbf{\upshape A}}]$ is closed under ${\mathbf H}$. For the second forward direction, it is easy to see that if $[\mathsf {Q}:{\textbf{\upshape A}}]$ is closed under ${\mathbf H}$ then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is equational relative to $\mathsf {Q}$. Assume now that $[\mathsf {Q}:{\textbf{\upshape A}}]$ is closed under ${\mathbf H}$, we show that ${\textbf{\upshape A}}$ is weakly projective in $\mathsf {Q}$. Suppose that ${\textbf{\upshape A}} \in {\mathbf H}({\textbf{\upshape B}})$ for some ${\textbf{\upshape B}} \in \mathsf {Q}$; if ${\textbf{\upshape A}} \notin {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$, then ${\textbf{\upshape B}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$ and, since $[\mathsf {Q}:{\textbf{\upshape A}}]$ is closed under ${\mathbf H}$, ${\textbf{\upshape A}} \in [\mathsf {Q}:{\textbf{\upshape A}}]$, again a contradiction. Hence ${\textbf{\upshape A}} \in {\mathbf I}{\mathbf S}({\textbf{\upshape B}})$ and ${\textbf{\upshape A}}$ is weakly projective in $\mathsf {Q}$. A completely analogous proof shows that if $[\mathsf {Q}:{\textbf{\upshape A}}]$ is equational relative to $\mathsf {Q}$ then ${\textbf{\upshape A}}$ is weakly projective, which completes the proof of (3). \(4\) follows directly from (1), (2) and (3). ◻ Thus if ${\textbf{\upshape A}}$ is relative subdirectly irreducible and finitely presented, then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a quasivariety; this is the key to prove the following result, appearing in [@Gorbunov1998 Proposition 5.1.24]. We present a self-contained proof for the sake of the reader. **Theorem 82** ([@Gorbunov1998]). *If $\mathsf {Q}$ is a locally finite quasivariety of finite type, then the following are equivalent.* 1. *[\[mainstructural1\]]{#mainstructural1 label="mainstructural1"} $\mathsf {Q}$ is primitive;* 2. *[\[mainstructural2\]]{#mainstructural2 label="mainstructural2"} for all finite relative subdirectly irreducible ${\textbf{\upshape A}} \in \mathsf {Q}$, $[\mathsf {Q}:{\textbf{\upshape A}}]$ is equational relative to $\mathsf {Q}$;* 3. *[\[mainstructural3\]]{#mainstructural3 label="mainstructural3"} every finite relative subdirectly irreducible ${\textbf{\upshape A}} \in \mathsf {Q}$ is weakly projective in $\mathsf {Q}$;* 4. *[\[mainstructural4\]]{#mainstructural4 label="mainstructural4"} every finite relative subdirectly irreducible ${\textbf{\upshape A}} \in \mathsf {Q}$ is weakly projective in the class of finite algebras in $\mathsf {Q}$.* *Proof.* ([\[mainstructural2\]](#mainstructural2){reference-type="ref" reference="mainstructural2"}) and ([\[mainstructural3\]](#mainstructural3){reference-type="ref" reference="mainstructural3"}) are equivalent by Lemma [Lemma 81](#universal){reference-type="ref" reference="universal"}, and ([\[mainstructural3\]](#mainstructural3){reference-type="ref" reference="mainstructural3"}) and ([\[mainstructural4\]](#mainstructural4){reference-type="ref" reference="mainstructural4"}) are equivalent in locally finite quasivarieties. Now, ([\[mainstructural1\]](#mainstructural1){reference-type="ref" reference="mainstructural1"}) implies ([\[mainstructural2\]](#mainstructural2){reference-type="ref" reference="mainstructural2"}) by Lemma [Lemma 81](#universal){reference-type="ref" reference="universal"}, since if ${\textbf{\upshape A}}$ is a finite relative subdirectly irreducible algebra then $[\mathsf {Q}:{\textbf{\upshape A}}]$ is a quasivariety, and if $\mathsf {Q}$ is primitive every subquasivariety is equational relative to $\mathsf {Q}$ by definition. Finally, assume ([\[mainstructural3\]](#mainstructural3){reference-type="ref" reference="mainstructural3"}) and let $\mathsf {Q}'$ be a subquasivariety of $\mathsf {Q}$; consider a finite algebra ${\textbf{\upshape B}} \in {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q}$, then ${\textbf{\upshape B}}$ is a subdirect product of finite relative subdirectly irreducible algebras in $\mathsf {Q}$, that is, ${\textbf{\upshape B}} \le_{sd} \prod_{i \in I} {\textbf{\upshape A}}_i$ where each ${\textbf{\upshape A}}_i$ is finite relative subdirectly irreducible in $\mathsf {Q}$, and thus it is also weakly projective in $\mathsf {Q}$ by hypothesis. Since ${\textbf{\upshape B}} \in {\mathbf H}(\mathsf {Q}')$, there is ${\textbf{\upshape A}} \in \mathsf {Q}'$ such that ${\textbf{\upshape B}} \in {\mathbf H}({\textbf{\upshape A}})$. But then for each $i \in I$, ${\textbf{\upshape A}}_i \in {\mathbf H}({\textbf{\upshape A}})$; since each ${\textbf{\upshape A}}_i$ is weakly projective in $\mathsf {Q}$, it is also isomorphic to a subalgebra of ${\textbf{\upshape A}}$. Thus, ${\textbf{\upshape B}} \in {\mathbf I}{\mathbf S}{\mathbf P}({\textbf{\upshape A}}) \subseteq\mathsf {Q}'$, and therefore $\mathsf {Q}' = {\mathbf H}(\mathsf {Q}') \cap \mathsf {Q}$, which means that $\mathsf {Q}$ is primitive and ([\[mainstructural1\]](#mainstructural1){reference-type="ref" reference="mainstructural1"}) holds. ◻ Most results in the literature are about structurally complete and primitive *varieties* of algebras and the reason is quite obvious; first the two concepts are easier to formulate for varieties. Secondly being subdirectly irreducible is an absolute concept (every subdirectly irreducible algebra is relative subdirectly irreducible in any quasivariety to which it belongs) while being relative subdirectly irreducible depends on the subquasivariety we are considering. Of course when a quasivariety is generated by a "simple" class (e.g. by finitely many finite algebras), then Theorem [Theorem 3](#quasivariety){reference-type="ref" reference="quasivariety"}(2) gives a simple solution, but in general describing the relative subdirectly irreducible algebras in a quasivariety is not an easy task. Now, it is clear that if $\mathsf {Q}$ is non-negative universally complete, then it is structurally complete. Finding examples of (quasi)varieties that are structurally complete but not primitive is not easy; one idea is to find a finite algebra ${\textbf{\upshape A}}$ such that ${\textbf{\upshape A}}$ satisfies the hypotheses of Lemma [Lemma 75](#lemma: wpstructcomplete){reference-type="ref" reference="lemma: wpstructcomplete"}, but ${\mathbf V}({\textbf{\upshape A}})$ contains some strict (i.e. not a variety) subquasivariety. We will see an example of this in Section [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"}. Let us now show some different kinds of examples of primitive (quasi)varieties. **Example 83**. The variety of bounded distributive lattices is primitive (as we will discuss in Section [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"}), since it is equationally complete and congruence modular and so is q-minimal by Theorem [Theorem 65](#thm: mckenziebergman){reference-type="ref" reference="thm: mckenziebergman"}. It is well-known (and easy to check) that the variety of distributive lattices is a *dual discriminator variety*; a *dual discriminator* on a set $A$ is a ternary operation $d$ on $A$ defined by $$d(a,b,c) =\left\{ \begin{array}{ll} c, & \hbox{if $a\ne b$;} \\ a, & \hbox{if $a=b$.} \end{array} \right.$$ A variety $\mathsf {V}$ is a dual discriminator variety [@FriedPixley1979] if there is a ternary term that is the dual discriminator on all the subdirectly irreducible members of $\mathsf {V}$. Dual discriminator varieties, as opposed to discriminator varieties, do not necessarily have projective unifiers. However, recently in [@Caicedoetal2021] the authors have extended the results in [@BergmanMcKenzie1990] (such as Theorem [Theorem 65](#thm: mckenziebergman){reference-type="ref" reference="thm: mckenziebergman"}) in two directions: every minimal dual discriminator variety is q-minimal, hence primitive and, if the variety is also idempotent, then minimality can be dropped and the variety is primitive. This last fact gives raise to different examples of primitive varieties. **Example 84**. A *weakly associative lattice* is an algebra $\langle A,\vee,\wedge\rangle$ where $\vee$ and $\wedge$ are idempotent, commutative and satisfy the absorption laws but (as the name reveals) only a weak form of associativity. In [@FriedPixley1979] the authors proved that there is a largest dual discriminator variety $\mathsf {U}$ of weakly associative lattices; since weakly associative lattices are idempotent, $\mathsf {U}$ is the largest primitive variety of weakly associative lattices. **Example 85**. The *pure dual discriminator variety* $\mathsf {D}$ (see [@FriedPixley1979 Theorem 3.2]) is a variety with a single ternary operation $d(x,y,z)$ satisfying $$\begin{aligned} &d(x,y,y) \approx y\\ &d(x,y,x) \approx x \\ &d(x,x,y) \approx x\\ &d(x,y,d(x,y,z)) \approx d(x,y,z)\\ &d(u,v,d(x,y,z)) \approx d(d(u,v,x),d(u,v,y),d(u,v,z))\end{aligned}$$ which is enough to prove that $\mathsf {D}$ is a dual discriminator variety. Since $d$ is idempotent $\mathsf {D}$ is an idempotent dual discriminator variety and so it is primitive. A different example is given by the following. **Example 86**. A *modal algebra* is a Boolean algebra with a modal operator $\Box$, that we take as a basic unary operation, satisfying $\Box 1 \approx 1$ and $\Box (x \wedge y) \approx\Box x \wedge\Box y$; there is an extensive literature on modal algebras (see for instance [@Wolter1997] and the bibliography therein). A modal algebra is a *K4-algebra* if it satisfies $\Box x \le \Box\Box x$; in [@Rybakov1995] V.V. Rybakov classified all the primitive varieties of K4-algebras. However very recently [@Carr2022] Carr discovered a mistake in Rybakov's proof; namely Rybakov in his description missed some varieties that all have the properties of containing a unifiable weakly projective algebra that is not projective. So any of such varieties, though primitive, does not have projective unifiers. We now present some examples from (quasi)varieties that are the equivalent algebraic semantics of (fragments) of many-valued logics; in particular, of infinite-valued Łukasiewicz logic. **Example 87**. Wajsberg algebras are the equivalent algebraic semantics of infinite-valued Łukasiewicz logic in the signature of bounded commutative residuated lattices $(\cdot, \to, \land, \lor, 0, 1)$ and they are term-equivalent to the better known MV-algebras [@CDOM]; Wajsberg hoops are their $0$-free subreducts. About these algebras there are some recent results [@Agliano2022a]. In summary: 1. the only primitive variety of Wajsberg algebras is the variety of Boolean algebras, that is also non-negative universally complete, and it is the only non-negative universally complete variety of Wajsberg algebras; 2. there are nontrivial primitive quasivarieties of Wajsberg algebras; 3. a proper variety of Wajsberg hoops is structurally complete if and only if it is primitive if and only if every subdirectly irreducible is either finite or perfect. The third point above clearly implies that the variety of Wajsberg hoops is not primitive. Considering varieties of $\rightarrow$-subreducts, the $\rightarrow$-subreducts of Wajsberg hoops is a subvariety of $\mathsf{BCK}$-algebras usually denoted by $\mathsf{LBCK}$; every locally finite subvariety of $\mathsf{LBCK}$-algebras is a variety of $\mathsf{HBCK}$-algebras, so it is universally complete. However: 1. the only non locally finite subvariety is the entire variety $\mathsf{LBCK}$ [@Komori1978]; 2. $\mathsf{LBCK}$ it is generated as a quasivariety by its finite chains [@AFM]; 3. every infinite chain contains all the finite chains as subalgebras [@Komori1978]; 4. so if $\mathsf {Q}$ is a quasivariety which contains only finitely many chains, then ${\mathbf V}(\mathsf {Q})$ is locally finite, hence universally complete and so $\mathsf {Q} = {\mathbf V}(\mathsf {Q})$; 5. otherwise $\mathsf {Q}$ contains infinitely many chains and so ${\mathbf V}(\mathsf {Q}) = \mathsf {Q} = \mathsf{LBCK}$. Hence every subquasivariety of $\mathsf{LBCK}$ is a variety and $\mathsf{LBCK}$ is primitive. The status of non locally finite varieties of basic hoops and basic algebras is still unclear (except for the cases we mentioned) and it is under investigation. ## Active structurally complete quasivarieties The problem of active structural completeness has been tackled in [@DzikStronkowski2016]; it is an extensive and profound paper touching many aspects and there is no need to reproduce it here. We will only state the definition, the main result, and we will display an example. **Theorem 88** ([@DzikStronkowski2016]). *For a quasivariety $\mathsf {Q}$ the following are equivalent:* 1. *$\mathsf {Q}$ is active structurally complete;* 2. *every unifiable algebra of $\mathsf {Q}$ is in ${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$;* 3. *every finitely presented unifiable algebra in $\mathsf {Q}$ is in ${\mathbf Q}({\textbf{\upshape F}}_\mathsf {Q}(\omega))$;* 4. *every admissible quasiequation in $\mathsf {Q}$ is valid in all the finitely presented unifiable algebras in $\mathsf {Q}$;* 5. *for every ${\textbf{\upshape A}} \in \mathsf {Q}$, ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}\in {\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$.* 6. *for every ${\textbf{\upshape A}} \in \mathsf {Q}_{rsi}$, ${\textbf{\upshape A}} \times {\textbf{\upshape F}}_{\mathsf {Q}}\in {\mathbf I}{\mathbf S}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}}(\omega))$.* Given that, we have as usual: **Corollary 89**. *A quasivariety $\mathsf {Q}$ is active structurally completeif and only if $\mathcal{L}_\mathsf {Q}$ is actively structurally complete.* **Example 90**. An $S4$-algebra is a $K4$-algebra satisfying $\Box x \le x$; if we define $\Diamond x := \neg \Box \neg x$, then a *monadic* algebra is an $S4$-algebras satisfying $\Diamond x \le \Box\Diamond x$. Now let ${\textbf{\upshape A}}$, ${\textbf{\upshape B}}$ be the monadic algebra and the $S4$-algebra in Figure [\[monadic\]](#monadic){reference-type="ref" reference="monadic"} and let $\mathsf {V} = {\mathbf V}({\textbf{\upshape A}})$ and $\mathsf {W}= {\mathbf V}({\textbf{\upshape B}})$. Let $\mathsf {U} = \mathsf {V} \vee\mathsf {W}$ (the varietal join); from [@DzikStronkowski2016 Section 8] one can deduce that: 1. every finitely generated algebra in $\mathsf {U}$ is isomorphic to the direct product of an algebra in $\mathsf {V}$ and one in $\mathsf {W}$, hence $\mathsf {U}$ is locally finite; 2. $\mathsf {U}$ is active structurally complete but not structurally complete; 3. $\mathsf {U}$ does not have exact unifiers. Since $\mathsf {U}$ is locally finite, by Theorem [Theorem 60](#lemma: ex unif auc){reference-type="ref" reference="lemma: ex unif auc"}, it cannot be active universally complete; so $\mathsf {U}$ is an example of a variety that is active structurally complete but not active universally complete. ## Passive quasivarieties Passive structurally complete quasivarieties have been studied in [@MoraschiniWannenburg2020] in relation to the joint embedding property, while here we take a different path. We start with the following observation. **Proposition 91**. *A quasivariety $\mathsf {Q}$ is passive structurally complete if and only if every non-negative passive admissible universal sentence is derivable in $\mathsf {Q}$.* *Proof.* For the non-trivial direction, suppose $\mathsf {Q}$ is passive structurally complete, and let $\Sigma \Rightarrow \Delta$ be a non-negative ($\Delta \neq \emptyset$) passive admissible universal sentence. This means that $\Sigma$ is not unifiable, and thus, each quasiequation $\Sigma \Rightarrow \delta$, for any $\delta \in \Delta$, is passive admissible. By hypothesis, each such $\Sigma \Rightarrow \delta$ is valid in $\mathsf {Q}$, thus so is $\Sigma \Rightarrow \Delta$ and the conclusion holds. ◻ It is clear that a key concept to study passive clauses is understanding the unifiability of the premises. In order to do so, we introduce the following notion. **Definition 92**. We say that a finite set of identities $\Sigma$ is *trivializing* in a class of algebras $\mathsf {K}$ if the quasiequation $\Sigma \Rightarrow (x \approx y)$ is valid in $\mathsf {K}$, where the variables $x, y$ do not appear in $\Sigma$. Notice that such a quasiequation $\Sigma \Rightarrow (x \approx y)$ is valid in an algebra ${\textbf{\upshape A}}$ if and only if either ${\textbf{\upshape A}}$ is trivial, or there is no assignment of the variables of $\Sigma$ in ${\textbf{\upshape A}}$ that makes $\Sigma$ valid in ${\textbf{\upshape A}}$. **Lemma 93**. *Let $\mathsf {Q}$ be a quasivariety, and let $\Sigma$ be a finite set of equations in its language. The following are equivalent:* 1. *$\Sigma$ is not unifiable in $\mathsf {Q}$;* 2. *${\textbf{\upshape F}}_{\mathsf {Q}}$ is nontrivial and $\Sigma$ is trivializing in ${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}})$;* 3. *${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$.* *Proof.* It is easy to see that (2) and (3) are equivalent, modulo the fact that a set of identities is trivializing in ${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}})$ if and only if it is trivializing in ${\textbf{\upshape F}}_{\mathsf {Q}}$. Let us now assume that the identities in $\Sigma$ are on a (finite) set of variables $X$. Then, given Lemma [Lemma 21](#free){reference-type="ref" reference="free"}, $\Sigma$ is not unifiable in $\mathsf {Q}$ if and only if there is no homomorphism $h: {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \to {\textbf{\upshape F}}_{\mathsf {Q}}$. We show that the latter holds if and only if there is no homomorphism $k: {\textbf{\upshape F}}_{\mathsf {Q}}(X) \to {\textbf{\upshape F}}_{\mathsf {Q}}$ such that $k(t) = k(u)$ for each $t \approx u \in \Sigma$. Indeed, for the non-trivial direction, suppose that there is a homomorphism $k: {\textbf{\upshape F}}_{\mathsf {Q}}(X) \to {\textbf{\upshape F}}_{\mathsf {Q}}$ with the above property. Then the following diagram commutes, i.e., there is a homomorphism $h: {\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma) \to {\textbf{\upshape F}}_\mathsf {Q}$: Notice that there is no homomorphism $k: {\textbf{\upshape F}}_{\mathsf {Q}}(X) \to {\textbf{\upshape F}}_{\mathsf {Q}}$ such that $k(t) = k(u)$ for each $t \approx u \in \Sigma$ if and only if there is no assignment of variables in $X$ validating $\Sigma$ in ${\textbf{\upshape F}}_{\mathsf {Q}}$. The latter is equivalent to ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$. ◻ We are now ready to prove the characterization theorem. **Theorem 94**. *Let $\mathsf {Q}$ be a quasivariety, then the following are equivalent.* 1. *[\[thm:PUC1\]]{#thm:PUC1 label="thm:PUC1"} $\mathsf {Q}$ is passive structurally complete;* 2. *[\[thm:PUC4\]]{#thm:PUC4 label="thm:PUC4"} ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$ implies $\Sigma$ is trivializing in $\mathsf {Q}$;* 3. *[\[thm:PUC5\]]{#thm:PUC5 label="thm:PUC5"} either ${\textbf{\upshape F}}_{\mathsf {Q}}$ is trivial, or $\Sigma$ is trivializing in ${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}})$ implies $\Sigma$ is trivializing in $\mathsf {Q}$;* 4. *[\[thm:PUC2\]]{#thm:PUC2 label="thm:PUC2"} every nontrivial finitely presented algebra is unifiable.* *Proof.* We first show that ([\[thm:PUC1\]](#thm:PUC1){reference-type="ref" reference="thm:PUC1"}) and ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}) are equivalent. By definition, $\mathsf {Q}$ is passive structurally complete if and only if each quasiidentity $\Sigma \Rightarrow \delta$ where $\Sigma$ is not unifiable in $\mathsf {Q}$ is valid in $\mathsf {Q}$. That is, $\Sigma$ not unifiable in $\mathsf {Q}$ implies $\mathsf {Q} \models \Sigma \Rightarrow \delta$, for all identities $\delta$. By Proposition [Lemma 93](#prop:trivial){reference-type="ref" reference="prop:trivial"}, the latter is equivalent to: ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$ implies $\mathsf {Q} \models \Sigma \Rightarrow \delta$, for all identities $\delta$. From this it follows the particular case where $\delta = \{x \approx y\}$, with $x, y$ not appearing in $\Sigma$. In turn, if ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$ implies $\mathsf {Q} \models \Sigma \Rightarrow (x \approx y)$, then clearly $\mathsf {Q} \models \Sigma \Rightarrow \delta$ for any $\delta$, and thus ([\[thm:PUC1\]](#thm:PUC1){reference-type="ref" reference="thm:PUC1"}) $\Leftrightarrow$ ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}). Now, ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}) and ([\[thm:PUC5\]](#thm:PUC5){reference-type="ref" reference="thm:PUC5"}) are equivalent by Lemma [Lemma 93](#prop:trivial){reference-type="ref" reference="prop:trivial"}, thus the first three points are equivalent. Let us now assume ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}) and prove ([\[thm:PUC2\]](#thm:PUC2){reference-type="ref" reference="thm:PUC2"}). We consider a nontrivial finitely presented algebra in $\mathsf {Q}$, ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$. If it is not unifiable, by Lemma [Lemma 93](#prop:trivial){reference-type="ref" reference="prop:trivial"} ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$. By ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}) this implies that $\Sigma$ is trivializing in $\mathsf {Q}$, that is, $\mathsf {Q} \models \Sigma \Rightarrow (x \approx y)$ (with $x,y$ new variables). This clearly implies that ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ is trivial, a contradiction. Thus ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta_{\mathsf {Q}}(\Sigma)$ is unifiable and ([\[thm:PUC2\]](#thm:PUC2){reference-type="ref" reference="thm:PUC2"}) holds. Finally, we prove that ([\[thm:PUC2\]](#thm:PUC2){reference-type="ref" reference="thm:PUC2"}) implies ([\[thm:PUC1\]](#thm:PUC1){reference-type="ref" reference="thm:PUC1"}). Suppose $\Sigma \Rightarrow \delta$ is a passive quasiequation over variables in $X$, that is, $\Sigma$ is not unifiable in $\mathsf {Q}$. By Lemma [Lemma 93](#prop:trivial){reference-type="ref" reference="prop:trivial"} ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$. Let $x,y$ be variables not in $X$, and consider the finitely presented algebra ${\textbf{\upshape F}}_{\mathsf {Q}}(X')/\theta_{\mathsf {Q}}(\Sigma)$, where $X' = X \cup \{x, y\}$ and suppose by way of contradiction that it is not trivial. By ([\[thm:PUC2\]](#thm:PUC2){reference-type="ref" reference="thm:PUC2"}) it is unifiable, that is, there is a homomorphism $h: {\textbf{\upshape F}}_{\mathsf {Q}}(X')/\theta_{\mathsf {Q}}(\Sigma) \to {\textbf{\upshape F}}_{\mathsf {Q}}$. Then, considering the natural epimorphism $\pi_{\Sigma}: {\textbf{\upshape F}}_{\mathsf {Q}}(X') \to {\textbf{\upshape F}}_{\mathsf {Q}}(X')/\theta_{\mathsf {Q}}(\Sigma)$, the composition $h \pi_\Sigma$ is an assignment from $X'$ to ${\textbf{\upshape F}}_\mathsf {Q}$ satisfying $\Sigma$; but ${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$, a contradiction. Thus ${\textbf{\upshape F}}_{\mathsf {Q}}(X')/\theta_{\mathsf {Q}}(\Sigma)$ is trivial, and therefore $x / \theta_{\mathsf {Q}}(\Sigma) = y / \theta_{\mathsf {Q}}(\Sigma)$. By Lemma [Lemma 38](#lemma:FThetaSigma){reference-type="ref" reference="lemma:FThetaSigma"} $\mathsf {Q} \models \Sigma \Rightarrow (x \approx y)$, and thus $\mathsf {Q} \models \Sigma \Rightarrow \delta$ and ([\[thm:PUC1\]](#thm:PUC1){reference-type="ref" reference="thm:PUC1"}) holds. ◻ Analogously to the case of passive universal completeness, if the smallest free algebra is isomorphic to all its ultraproducts we can improve the previous result. **Lemma 95**. *Let $\mathsf {Q}$ be a quasivariety such that ${\mathbf I}({\textbf{\upshape F}}_{\mathsf {Q}}) = {\mathbf I}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}})$, then the following are equivalent.* 1. *every nontrivial finitely presented algebra in $\mathsf {Q}$ is unifiable;* 2. *every nontrivial algebra in $\mathsf {Q}$ is unifiable.* *Proof.* The proof is analogous to the one of Lemma [Lemma 69](#lemma:Qunifiable){reference-type="ref" reference="lemma:Qunifiable"}; we prove the nontrivial direction by contraposition. Consider an arbitrary algebra ${\textbf{\upshape A}} = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta \in \mathsf {Q}$ and assume that it is not unifiable. Then there is no assignment $h: {\textbf{\upshape F}}_\mathsf {Q}(X) \to {\textbf{\upshape F}}_\mathsf {Q}$ such that ${\textbf{\upshape F}}_\mathsf {Q}, h \models \Sigma_\theta$, where $\Sigma_\theta = \{t \approx u : (t,u) \in \theta\}$. Equivalently, iff ${\textbf{\upshape F}}_\mathsf {Q} \models \Sigma_\theta \Rightarrow \emptyset$. Now, the equational consequence relation relative to ${\textbf{\upshape F}}_{\mathsf {Q}}$ is finitary (since all ultraproducts of ${\textbf{\upshape F}}_{\mathsf {Q}}$ are isomorphic to ${\textbf{\upshape F}}_{\mathsf {Q}}$); thus we obtain that ${\textbf{\upshape F}}_\mathsf {Q} \models \Sigma'_\theta \Rightarrow \emptyset$, for $\Sigma'_\theta$ some finite subset of $\Sigma_\theta$. But ${\textbf{\upshape F}}_{\mathsf {Q}}(X)/\theta \not\vDash \Sigma'_\theta \Rightarrow (x \approx y)$ (with $x,y \notin X$), since it is nontrivial, which contradicts ([\[thm:PUC4\]](#thm:PUC4){reference-type="ref" reference="thm:PUC4"}) of Theorem [Theorem 94](#thm:PUC){reference-type="ref" reference="thm:PUC"}; equivalently it contradicts (1) and thus the proof is complete. ◻ **Corollary 96**. *Let $\mathsf {Q}$ be a quasivariety such that ${\mathbf I}({\textbf{\upshape F}}_{\mathsf {Q}}) = {\mathbf I}{\mathbf P}_u({\textbf{\upshape F}}_{\mathsf {Q}})$, then the following are equivalent.* 1. *$\mathsf {Q}$ is passively structurally complete;* 2. *${\textbf{\upshape F}}_{\mathsf {Q}} \models \Sigma \Rightarrow \emptyset$ implies $\Sigma$ is trivializing in $\mathsf {Q}$;* 3. *either ${\textbf{\upshape F}}_{\mathsf {Q}}$ is trivial, or $\Sigma$ is trivializing in ${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {Q}})$ implies $\Sigma$ is trivializing in $\mathsf {Q}$;* 4. *every nontrivial finitely presented algebra is unifiable;* 5. *every nontrivial algebra in $\mathsf {Q}$ is unifiable.* **Remark 97**. The previous corollary can be applied whenever ${\textbf{\upshape F}}_\mathsf {Q}$ is finite, therefore to all locally finite quasivarieties, but also to more complex classes of algebras, e.g., all subquasivarieties of $\mathsf{FL}_{w}$ (see Subsection [5.2](#sec:FL){reference-type="ref" reference="sec:FL"}). We will see an interesting application of Theorem [Theorem 94](#thm:PUC){reference-type="ref" reference="thm:PUC"} (or Corollary [Corollary 96](#cor:PUCPu){reference-type="ref" reference="cor:PUCPu"}) in substructural logics in Subsection [5.2](#sec:FL){reference-type="ref" reference="sec:FL"}; let us now show some other consequences. Given a quasivariety $\mathsf {Q}$ whose smallest free algebra ${\textbf{\upshape F}}_\mathsf {Q}$ is nontrivial, let us consider the following set: $$\mathcal{P}_{\mathsf {Q}} = \{\Sigma \Rightarrow \delta: \mathsf {Q}({\textbf{\upshape F}}_{\mathsf {Q}})\vDash \Sigma \Rightarrow \{x \approx y\}, \delta \mbox{ any identity}\}.$$ $\mathcal{P}_{\mathsf {Q}}$ axiomatizes a subquasivariety of $\mathsf {Q}$, that we denote with $\mathsf {P}_{\mathsf {Q}}$. From Theorem [Theorem 94](#thm:PUC){reference-type="ref" reference="thm:PUC"} we get the following. **Corollary 98**. *Let $\mathsf {Q}$ be a quasivariety such that ${\textbf{\upshape F}}_\mathsf {Q}$ is nontrivial. Every passive structurally complete subquasivariety of $\mathsf {Q}$ is contained in $\mathsf {P}_{\mathsf {Q}}$, which is the largest subquasivariety of $\mathsf {Q}$ that is passive structurally complete.* Moreover, for locally finite quasivarieties the characterization theorem reads as follows. **Corollary 99**. *Let $\mathsf {Q}$ be a locally finite quasivariety, then the following are equivalent.* 1. *$\mathsf {Q}$ is passive structurally complete;* 2. *every algebra in $\mathsf {Q}$ is unifiable;* 3. *every finite algebra in $\mathsf {Q}$ is unifiable;* 4. *every finite subdirectly irreducible in $\mathsf Q$ is unifiable.* A nontrivial algebra ${\textbf{\upshape A}}$ is *Kollár* if it has no trivial subalgebras, and a quasivariety $\mathsf {Q}$ is a *Kollár quasivariety* if all nontrivial algebras in $\mathsf {Q}$ are Kollár. By [@Kollar1979] if ${\textbf{\upshape A}}$ belongs to a Kollár quasivariety, $1_{\textbf{\upshape A}}$, the largest congruence of ${\textbf{\upshape A}}$, is compact in $\textrm{\upshape Con}_\mathsf {Q}({\textbf{\upshape A}})$; from there a straightforward application of Zorn's Lemma yields that if ${\textbf{\upshape A}}$ is nontrivial there is at least one maximal congruence $\theta\in \textrm{\upshape Con}_\mathsf {Q} ({\textbf{\upshape A}})$ below $1_{\textbf{\upshape A}}$ (i.e. ${\textbf{\upshape A}}/\theta$ is relative simple). **Theorem 100**. *If $\mathsf {Q}$ is a Kollár quasivariety and ${\textbf{\upshape F}}_\mathsf {Q}$ is the only finitely generated relative simple algebra in $\mathsf {Q}$, then $\mathsf {Q}$ is passive structurally complete.* *Proof.* Let ${\textbf{\upshape A}}$ be a nontrivial finitely presented algebra in $\mathsf {Q}$; since $\mathsf {Q}$ is a Kollár quasivariety, ${\textbf{\upshape A}}$ has a relative simple homomorphic image, that must be finitely generated. Hence it must be equal to ${\textbf{\upshape F}}_\mathsf {Q}$, so ${\textbf{\upshape A}}$ is unifiable; by Theorem [Theorem 94](#thm:PUC){reference-type="ref" reference="thm:PUC"} $\mathsf {Q}$ is passive structurally complete. ◻ **Corollary 101**. *For a locally finite Kollár quasivariety $\mathsf {Q}$ such that ${\textbf{\upshape F}}_{\mathsf {Q}}$ has no proper subalgebra the following are equivalent:* 1. *${\textbf{\upshape F}}_\mathsf {Q}$ is the only finite relative simple algebra in $\mathsf {Q}$;* 2. *$\mathsf {Q}$ is passive structurally complete.* *Proof.* If (1) holds, than (2) holds by Theorem [Theorem 100](#semideg){reference-type="ref" reference="semideg"}. Conversely assume (2); then every nontrivial finitely presented algebra in $\mathsf {Q}$ is unifiable. Since $\mathsf {Q}$ is locally finite ${\textbf{\upshape F}}_\mathsf {Q}$ is finite and nontrivial since $\mathsf {Q}$ is Kollár; now since ${\textbf{\upshape F}}_{\mathsf {Q}}$ has no proper subalgebra no finite relative simple algebra different from ${\textbf{\upshape F}}_\mathsf {Q}$ can be unifiable, but $\mathsf {Q}$ must contain at least a relative simple algebra [@Gorbunov1998 Theorem 3.1.8]. Hence ${\textbf{\upshape F}}_\mathsf {Q}$ must be relative simple and (1) holds. ◻ The next results will allow us to find interesting applications in varieties of bounded lattices, which we will explore in Section [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"}. We say that an algebra ${\textbf{\upshape A}}$ in a variety $\mathsf {V}$ is *flat* if ${\mathbf H}{\mathbf S}({\textbf{\upshape A}})$ does not contain any simple algebra different from ${\textbf{\upshape F}}_\mathsf {V}$. **Theorem 102**. *Let $\mathsf {V}$ be a Kollár variety; if every finitely generated algebra in $\mathsf {V}$ is flat then $\mathsf {V}$ is passive structurally complete. If $\mathsf {V}$ is locally finite, then the converse holds as well.* *Proof.* First, if ${\textbf{\upshape F}}_\mathsf {V}$ is trivial then $\mathsf {V}$ is vacuously passive structurally complete. If ${\textbf{\upshape F}}_\mathsf {V}$ is nontrivial and every finitely generated algebra is flat, then the only finitely generated simple lattice in $\mathsf {V}$ must be ${\textbf{\upshape F}}_{\mathsf {V}}$; since $\mathsf {V}$ is Kollár, $\mathsf {V}$ is passive structurally complete by Theorem [Theorem 100](#semideg){reference-type="ref" reference="semideg"}. If $\mathsf {V}$ is locally finite and passive structurally complete, then ${\textbf{\upshape F}}_\mathsf {V}$ is the only finite simple algebra in $\mathsf {V}$ by Corollary [Corollary 101](#semideglf){reference-type="ref" reference="semideglf"}. It follows that no finite simple algebra different from ${\textbf{\upshape F}}_\mathsf {V}$ can appear in ${\mathbf H}{\mathbf S}({\textbf{\upshape A}})$ for any finite ${\textbf{\upshape A}}\in\mathsf {V}$. So every finite algebra in $\mathsf {V}$ must be flat. ◻ **Theorem 103**. *Let $\mathsf {V}$ be a congruence distributive Kollár variety; a finitely generated variety $\mathsf {W} \subseteq\mathsf {V}$ is passive structurally complete if and only if each generating algebra is flat.* *Proof.* Suppose that $\mathsf {W} = {\mathbf V}(K)$ where $K$ is a finite set of finite algebras; by Jónsson Lemma any simple algebra in $\mathsf {V}$ is in ${\mathbf H}{\mathbf S}(K)$. If $K$ consists entirely of flat algebras, then there cannot be any simple algebra in $\mathsf {V}$ different from ${\textbf{\upshape F}}_\mathsf {W}$, so $\mathsf {W}$ is passive structurally complete. On the other hand if ${\textbf{\upshape A}} \in K$ is not flat, then there is an algebra ${\textbf{\upshape B}} \in {\mathbf H}{\mathbf S}(K)$ which is simple and different from ${\textbf{\upshape F}}_\mathsf {V}$. Clearly ${\textbf{\upshape B}} \in \mathsf {W}$, which is not passive structurally complete. ◻ # Applications to algebra and logic In this last section we will see some relevant examples and applications of our results in the realm of algebra and (algebraic) logic that deserve a deeper exploration than the examples already presented in the previous sections. We will start with focusing on varieties of lattices and bounded lattices, and then move to their expansions that are the equivalent algebraic semantics of subtructural logics: residuated lattices. As a main result, in the last subsection we present the logical counterpart of the characterization of passive structural completeness in substructural logics with weakening, that is, such a logic is passively structurally complete if and only if every classical contradiction is explosive in it; building on this, from the algebraic perspective, we are able to axiomatize the largest variety of representable bounded commutative integral residuated lattices that is passively structurally complete (and such that all of its quasivarieties have this property). Notice that this characterization establishes negative results as well: if a logic (or a quasivariety) is not passively structurally complete, a fortiori it is not structurally complete either. ## (Bounded) lattices {#sec:lattices} In this subsection we start with some results about primitive (quasi)varieties of lattices, and then move to bounded lattices, where in particular we obtain some new results about passive structurally complete varieties. ### Primitivity in lattices {#sec: primlattices} Many examples of quasivarieties that are primitive can be found in lattices satisfying *Whitman's condition* (W); Whitman's condition is a universal sentence that holds in free lattices: $$\{x \wedge y \le u \vee v\} \Rightarrow \{x \le u \vee v, y \le u \vee v, x \wedge y \le u, x \wedge y \le v\} \tag{W}.$$ Now a finite lattice is finitely projective in the variety of all lattices if and only if it satisfies (W) [@DaveySands1977], which implies: **Lemma 104**. *Let $\mathsf {K}$ be a finite set of finite lattices. If every lattice in $\mathsf {K}$ satisfies (W) then ${\mathbf Q}(\mathsf {K})$ is primitive.* *Proof.* ${\mathbf Q}(\mathsf {K})$ is locally finite and by Theorem [Theorem 3](#quasivariety){reference-type="ref" reference="quasivariety"}(2) every relative subdirectly irreducible lies in ${\mathbf I}{\mathbf S}(\mathsf {K})$; as (W) is a universal sentence it is preserved under subalgebras, thus they all satisfy (W) and hence they are all finitely projective in the variety of lattices and then also in ${\mathbf Q}(\mathsf {K})$. By Theorem [Theorem 82](#mainstructural){reference-type="ref" reference="mainstructural"}(4), ${\mathbf Q}(\mathsf {K})$ is primitive. ◻ Luckily finite lattices satisfying (W) abound, so there is no shortage of primitive quasivarieties of lattices. For varieties of lattices the situation is slightly different; in particular, because of Lemma [Lemma 75](#lemma: wpstructcomplete){reference-type="ref" reference="lemma: wpstructcomplete"} it is not enough that all lattices in $\mathsf {K}$ are weakly projective in ${\mathbf V}(\mathsf {K})$ to guarantee that ${\mathbf V}(\mathsf {K})$ is structurally complete. First we introduce some lattices: ${\textbf{\upshape M}}_n$ for $3\le n \le \omega$ are the modular lattices consisting of a top, a bottom, and $n$ atoms while the lattices ${\textbf{\upshape M}}_{3,3}$ and ${\textbf{\upshape M}}_{3,3}^+$ are displayed in Figure [\[m3lattices\]](#m3lattices){reference-type="ref" reference="m3lattices"}. Observe that all the above lattices, with the exception of ${\textbf{\upshape M}}_{3,3}$, satisfy (W). Now Gorbunov ([@Gorbunov1998], Theorem 5.1.29) showed that ${\textbf{\upshape M}}^+_{3,3}$ is *splitting* in the lattice of subquasivarieties of modular lattices. More in detail for any quasivariety $\mathsf {Q}$ of modular lattices, either ${\textbf{\upshape M}}^+_{3,3} \in \mathsf {Q}$ or else $\mathsf {Q}= {\mathbf Q}({\textbf{\upshape M}}_n)$ for some $n \le \omega$. Observe that, for $n <\omega$, $\mathsf {Q}({\textbf{\upshape M}}_n)$ is primitive by Lemma [Lemma 104](#whitman){reference-type="ref" reference="whitman"} and ${\mathbf V}({\textbf{\upshape M}}_n) = {\mathbf Q}({\textbf{\upshape M}}_n)$ by Lemma [Lemma 4](#lemma: Q(A) variety){reference-type="ref" reference="lemma: Q(A) variety"}; then the only thing left to show is that ${\mathbf V}({\textbf{\upshape M}}_\omega)$ is a primitive variety and Gorbunov did exactly that. On the other hand no variety $\mathsf {V}$ of lattices containing ${\textbf{\upshape M}}_{3,3}^+$ can be primitive; in fact ${\textbf{\upshape M}}_{3,3}$ is a simple homomorphic image of ${\textbf{\upshape M}}^+_{3,3}$ that cannot be embedded in ${\textbf{\upshape M}}^+_{3,3}$. By Lemma [Lemma 4](#lemma: Q(A) variety){reference-type="ref" reference="lemma: Q(A) variety"}, ${\mathbf Q}({\textbf{\upshape M}}_{3,3}^+) \subsetneq {\mathbf V}({\textbf{\upshape M}}_{3,3}^+)$, so $\mathsf {V}$ contains a strict (i.e. not a variety) subquasivariety and cannot be primitive. Thus Gorbunov's result can be formulated as: *a variety of modular lattices is primitive if and only if it does not contain ${\textbf{\upshape M}}^+_{3,3}$*. Note that it cannot be improved to quasivarieties: since ${\textbf{\upshape M}}_{3,3}^+$ satisfies (W), ${\mathbf Q}({\textbf{\upshape M}}^+_{3,3})$ is primitive by Lemma [Lemma 104](#whitman){reference-type="ref" reference="whitman"}. However we observe: **Lemma 105**. *If $\mathsf {Q}$ is a quasivariety of modular lattices and ${\textbf{\upshape M}}_{3,3} \in \mathsf {Q}$, then $\mathsf {Q}$ is not primitive.* *Proof.* Clearly the two element lattice $\mathbf 2 \in \mathsf {Q}$ and it is easy to check that ${\textbf{\upshape M}}_{3,3}^+ \le_{sd} \mathbf 2 \times {\textbf{\upshape M}}_{3,3}$ so ${\textbf{\upshape M}}_{3,3}^+ \in \mathsf {Q}$ and ${\textbf{\upshape M}}_{3,3} \in {\mathbf H}({\textbf{\upshape M}}_{3,3}^+)$. Since ${\textbf{\upshape M}}_{3,3}$ cannot be embedded in ${\textbf{\upshape M}}_{3,3}^+$, in $\mathsf {Q}$ there is a simple finite (so finitely presented, since lattices have finite type) algebra that is not weakly projective. By Theorem [Theorem 82](#mainstructural){reference-type="ref" reference="mainstructural"}, $\mathsf {Q}$ is not primitive. ◻ Therefore to find a variety of modular lattices that is structurally complete but not primitive it is enough to find a finite lattice ${\textbf{\upshape F}}$ such that ${\textbf{\upshape M}}_{3,3}^+ \in {\mathbf V}({\textbf{\upshape F}})$ but $\mathsf {K}=\{{\textbf{\upshape F}}\}$ satisfies the hypotheses of Lemma [Lemma 75](#lemma: wpstructcomplete){reference-type="ref" reference="lemma: wpstructcomplete"}. Bergman in [@Bergman1991] observed that the *Fano lattice* ${\textbf{\upshape F}}$ has exactly those characteristics; the Fano lattice is the (modular) lattice of subspaces of $(\mathbb Z_2)^3$ seen as a vector space on $\mathbb Z_2$ and it is displayed in Figure [\[fano\]](#fano){reference-type="ref" reference="fano"}. Now: 1. ${\textbf{\upshape F}}$ is projective in ${\mathbf V}({\textbf{\upshape F}})$ [@HermannHuhn1976]; 2. the subdirectly irreducible members of ${\mathbf V}({\textbf{\upshape F}})$ are exactly $\mathbf 2, {\textbf{\upshape M}}_3, {\textbf{\upshape M}}_{3,3}, {\textbf{\upshape F}}$ and they are all subalgebras of ${\textbf{\upshape F}}$. It follows that ${\textbf{\upshape F}}$ does not satisfies (W) (since ${\textbf{\upshape M}}_{3,3}$ does not), ${\mathbf V}({\textbf{\upshape F}})$ is structurally complete and (since ${\textbf{\upshape M}}_{3,3} \in {\mathbf V}({\textbf{\upshape F}})$) not primitive by Lemma [Lemma 105](#lemmaM33){reference-type="ref" reference="lemmaM33"}; also ${\mathbf Q}({\textbf{\upshape F}})$ is structurally complete but, since ${\textbf{\upshape M}}_{3,3} \in {\mathbf Q}({\textbf{\upshape F}})$, it cannot be primitive as well. Primitive varieties of lattices have been studied in depth in [@JipsenNation2022]; there the authors proved the following theorem that explains the behavior we have seen above. **Theorem 106** ([@JipsenNation2022]). *If ${\textbf{\upshape A}}$ is a lattice satisfying (W), then ${\mathbf V}({\textbf{\upshape A}})$ is primitive if and only if every subdirectly irreducible lattice in ${\mathbf H}{\mathbf S}({\textbf{\upshape A}})$ satisfies (W).* We believe that many of the techniques in [@JipsenNation2022] could be adapted to gain more understanding of primitive quasivarieties of lattices, but proceeding along this path would make this part too close to being a paper in lattice theory, and we have chosen a different focus. We only borrow an example from [@JipsenNation2022] that shows that Lemma [Lemma 104](#whitman){reference-type="ref" reference="whitman"} cannot be inverted for quasivarieties. Let ${\textbf{\upshape H}}^+, {\textbf{\upshape H}}$ be the lattices in Figure [\[H\]](#H){reference-type="ref" reference="H"}. It is easily seen that the pair ${\textbf{\upshape H}}^+, {\textbf{\upshape H}}$ behaves almost like the pair ${\textbf{\upshape M}}_{3,3}^+, {\textbf{\upshape M}}_{3,3}$: ${\textbf{\upshape H}}^+$ satisfies (W) (so ${\mathbf Q}({\textbf{\upshape H}}^+)$ is primitive), ${\textbf{\upshape H}}$ does not satisfy (W) and ${\textbf{\upshape H}}^+ \le_{sd} \mathbf 2 \times {\textbf{\upshape H}}$. As above we can conclude that ${\mathbf V}({\textbf{\upshape H}}^+)$ is not primitive. However ${\mathbf V}({\textbf{\upshape H}})$ is primitive [@JipsenNation2022] so ${\mathbf Q}({\textbf{\upshape H}})$ is a primitive quasivariety generated by a finite lattice not satisfying (W). ### Bounded lattices {#bounded-lattices} We now focus on applications of our results in varieties of bounded lattices. A *bounded* lattice is a lattice with two constants, $0$ and $1$, that represent the top and the bottom of the lattice. Bounded lattices form a variety $\mathsf {L}^b$ that shares many features with variety of lattices. In particular, let $\mathbf 2^b$ be the two element bounded lattice, then the variety of bounded distributive lattices is $\mathsf {D}^b = {\mathbf I}{\mathbf S}{\mathbf P}(\mathbf 2^b)$. Therefore $${\mathbf Q}({\textbf{\upshape F}}_{\mathsf {D}_b}(\omega)) \subseteq\mathsf {D}^b = {\mathbf I}{\mathbf S}{\mathbf P}(\mathbf 2^b) \subseteq{\mathbf Q}({\textbf{\upshape F}}_{\mathsf {D}_b}(\omega))$$ and by Theorem [Theorem 72](#structural){reference-type="ref" reference="structural"}, the variety of bounded distributive lattices $\mathsf {D}^b$ is structurally complete, as shown in [@DzikStronkowski2016]. In [@BergmanMcKenzie1990] it is shown that locally finite, congruence modular, minimal varieties are q-minimal; since these hypotheses apply to $\mathsf {D}^b$, the latter is also primitive. However, it is not non-negative universally complete; it is a nice exercise in general algebra to show that for any variety $\mathsf {V}$ of bounded lattices, $1$ is join irreducible in ${\textbf{\upshape F}}_{\mathsf {V}}(\omega)$. It follows that $$\{x \vee y \approx 1\} \Rightarrow \{x \approx 1, y \approx 1\}$$ is an active universal sentence that is admissible in $\mathsf {V}$. But it is clearly not derivable, since any nontrivial variety of bounded lattices contains $\mathbf 2^b \times \mathbf 2^b$ which does not satisfy the universal sentence. **Proposition 107**. *No nontrivial variety of bounded lattices is active universally complete.* Actually something more is true; if $\mathsf {V}$ is a variety of bounded lattices that is structurally complete, then by Theorem [Theorem 77](#thm: structexact){reference-type="ref" reference="thm: structexact"}, each finite subdirectly irreducible algebra ${\textbf{\upshape A}} \in \mathsf {V}$ must satisfy the above universal sentence, i.e. $1$ must be join irreducible in ${\textbf{\upshape A}}$. But the bounded lattices ${\textbf{\upshape N}}_5^b$ and ${\textbf{\upshape M}}_3^b$ do not satisfy that, so any structurally complete variety of bounded lattice must omit them both. As in the unbounded case, this means that the variety must be the variety of bounded distributive lattices. Thus: **Proposition 108** ([@DzikStronkowski2016]). *The variety of bounded distributive lattices is the only (active) structurally complete variety of bounded lattices.* We have seen that active structural completeness does not have much meaning in bounded lattices. Passive structural completeness has more content, as we are now going to show. Notice that any variety of bounded lattices is Kollár and ${\textbf{\upshape F}}_\mathsf {V}= \mathbf 2^b$ for any variety $\mathsf {V}$ of bounded lattices. Since $\mathbf 2^b$ is simple and has no proper subalgebras, any simple bounded lattice not isomorphic with $\mathbf 2^b$ is not unifiable; in particular if a variety $\mathsf {V}$ contains a finite simple lattice ${\textbf{\upshape L}}$ different from $\mathbf 2^b$, then ${\mathbf V}({\textbf{\upshape L}})$ cannot be passive structurally completeby Corollary [Corollary 101](#semideglf){reference-type="ref" reference="semideglf"}, and hence neither can $\mathsf {V}$. We will use this fact to show that the only variety of bounded modular lattices that is passive structurally complete is the one we already know to possess that property, i.e. the variety $\mathsf {D}^b$ of bounded distributive lattices. A key step is to show that ${\textbf{\upshape M}}^b_3$ is splitting in the variety of bounded modular lattices; in the unbounded case, this follows from the fact that ${\textbf{\upshape M}}_3$ is projective and subdirectly irreducible. However, ${\textbf{\upshape M}}_3^b$ is not projective in the variety of bounded modular lattices. Indeed, the lattice in Figure [\[notproj\]](#notproj){reference-type="ref" reference="notproj"} is a bounded modular lattice having ${\textbf{\upshape M}}^b_3$ as homomorphic image, but it has no subalgebra isomorphic with ${\textbf{\upshape M}}^b_3$, which hence cannot be a retract. [\[notproj\]]{#notproj label="notproj"} However we can use A. Day idea in [@Day1975]; a finite algebra ${\textbf{\upshape A}}$ is *finitely projected* in a variety $\mathsf {V}$ if for any ${\textbf{\upshape B}} \in \mathsf {V}$ if $f: {\textbf{\upshape B}} \longrightarrow {\textbf{\upshape A}}$ is surjective, then there is a finite subalgebra ${\textbf{\upshape C}}$ of ${\textbf{\upshape B}}$ with $f({\textbf{\upshape C}}) \cong {\textbf{\upshape A}}$. Clearly any finite projective lattice is finitely projected. A finite algebra ${\textbf{\upshape A}}$ *splitting* in a variety $\mathsf {V}$ if ${\textbf{\upshape A}} \in \mathsf {V}$ and there is a subvariety $\mathsf {W}_{\textbf{\upshape A}} \subseteq\mathsf {V}$ such that for any variety $\mathsf {U} \subseteq\mathsf {V}$ either ${\textbf{\upshape A}} \in \mathsf {U}$ or $\mathsf {U} \subseteq\mathsf {W}_{\textbf{\upshape A}}$. This simply means that the lattice of subvarieties of $\mathsf {V}$ is the disjoint union of the filter generated by ${\mathbf V}({\textbf{\upshape A}})$ and the ideal generated by $\mathsf {W}_{\textbf{\upshape A}}$. The key result is: **Theorem 109**. *([@Day1975], Theorem 3.7) If $\mathsf {V}$ is a congruence distributive variety, then any finitely projected subdirectly irreducible algebra in $\mathsf {V}$ is splitting in $\mathsf {V}$.* **Lemma 110**. *Let $\mathsf {V}^b$ be a variety of bounded lattices and let $\mathsf {V}$ be the variety of lattice subreducts of $\mathsf {V}^b$. If ${\textbf{\upshape L}}$ is finitely projected in $\mathsf {V}$, then ${\textbf{\upshape L}}^b$ is finitely projected in $\mathsf {V}^b$.* *Proof.* The fact that $\mathsf {V}$ is indeed a variety is easy to check. Let now ${\textbf{\upshape A}}^b \in \mathsf {V}^b$ and suppose that there is an onto homomorphism $f: {\textbf{\upshape A}}^b \longrightarrow {\textbf{\upshape L}}^b$; then $f$ is onto from ${\textbf{\upshape A}}$ to ${\textbf{\upshape L}}$ and since ${\textbf{\upshape L}}$ is finitely projected in $\mathsf {V}$ there is a subalgebra ${\textbf{\upshape B}}$ of ${\textbf{\upshape A}}$ with $f({\textbf{\upshape B}}) \cong {\textbf{\upshape L}}$. But $B\cup\{0,1\}$ is the universe of a finite subalgebra ${\textbf{\upshape C}}$ of ${\textbf{\upshape A}}^b$. Extend $f$ to $\hat f$ by setting $\hat f(0)=0$ and $\hat f(1)=1$; then $\hat f({\textbf{\upshape C}}) \cong {\textbf{\upshape L}}^b$ and so ${\textbf{\upshape L}}^b$ is finitely projected in $\mathsf {V}^b$. ◻ **Theorem 111**. *A variety of modular bounded lattices is passive structurally complete if and only if it is the variety of bounded distributive lattices.* *Proof.* $\mathsf {D}^b$ is structurally complete, hence passive structurally complete. Conversely observe that ${\textbf{\upshape M}}_3$ is projective in the variety of modular lattices, so ${\textbf{\upshape M}}_3^b$ is finitely projected in the variety of bounded modular lattices. Hence, by Theorem [Theorem 109](#day){reference-type="ref" reference="day"}, ${\textbf{\upshape M}}^b_3$ is splitting in the variety, which means that for any variety $\mathsf {V}$ of bounded modular lattices, either ${\textbf{\upshape M}}_3^b \in \mathsf {V}$ or $\mathsf {V}$ is $\mathsf {D}^b$. But if ${\textbf{\upshape M}}^b_3 \in \mathsf {V}$ then $\mathsf {V}$ cannot be passive universally complete, since ${\textbf{\upshape M}}_3^b$ is simple. The conclusion follows. ◻ In order to find other relevant varieties of bounded lattices that are passive structurally complete, we are going to take a closer look at flat lattices. Finding flat bounded lattices is not hard since the lattice of subvarieties of lattices has been studied thoroughly and a lot is known about it (an excellent survey is [@JipsenRose1992]). Clearly ${\textbf{\upshape N}}_5$ is flat and hence so is ${\textbf{\upshape N}}_5^b$; however we know exactly all the covers of the minimal nondistributive varieties of lattices (which is of course ${\mathbf V}({\textbf{\upshape N}}_5)$). There are 15 finite subdirectly irreducible nonsimple lattices, commonly called ${\textbf{\upshape L}}_1,\dots,{\textbf{\upshape L}}_{15}$ (some of them are in Figure [\[lattices\]](#lattices){reference-type="ref" reference="lattices"}) that generate all the join irreducible (in the lattice of subvarieties) covers of ${\mathbf V}({\textbf{\upshape N}}_5)$. It is easy to see their bounded versions all are join irreducible covers of ${\mathbf V}({\textbf{\upshape N}}^b_5)$ in the lattice of subvarieties of bounded lattices. We suspect that they are also the only join irreducible covers; one needs only to check that the (rather long) proof for lattices [@JonssonRival1979] goes through for bounded lattices but we leave this simple but tedious task to the reader. In any case for $i=1,\dots,15$ the subdirectly irreducible algebras in ${\mathbf V}({\textbf{\upshape L}}_i^b)$ are exactly $\mathbf 2^b, {\textbf{\upshape N}}_5^b$ and ${\textbf{\upshape L}}_i^b$ (via a straightforward application of Jónsson Lemma); so each ${\textbf{\upshape L}}_i^b$ is flat and each ${\mathbf V}({\textbf{\upshape L}}_i^b)$ is passively structurally complete (by Theorem [Theorem 103](#fgflat){reference-type="ref" reference="fgflat"}). Let's make more progress: consider the rules $$\begin{aligned} &x \wedge y \approx x \wedge z \quad\Rightarrow\quad x \wedge y \approx x \wedge(y \vee z)\tag{$SD_\wedge$}\\ &x \vee y \approx x \vee z \quad\Rightarrow\quad x\vee y \approx x \vee(y \wedge z)\tag{$SD_\vee$}.\end{aligned}$$ A lattice is *meet semidistributive* if it satisfies $SD_\wedge$, *join semidistributive* if it satisfies $SD_\wedge$ and *semidistributive* if it satisfies both. Clearly (meet/join) semidistributive lattices form quasivarieties called $\mathsf{SD}_\wedge$, $\mathsf{SD}_\vee$ and $\mathsf{SD}$ respectively, and so do their bounded versions. It is a standard exercise to show that homomorphic images of a finite (meet/join) semidistributive lattices are (meet/join) semidistributive. It is also possible to show none of the three quasivariety (and their bounded versions) is a variety (see [@JipsenRose1992] p. 82 for an easy argument); they are also not locally finite since for instance ${\textbf{\upshape F}} ={\textbf{\upshape F}}_\mathsf{SD}(x,y,z)$ is infinite; hence ${\textbf{\upshape F}}^b$ is a bounded infinite three-generated lattice and thus $\mathsf{SD}^b$ is not locally finite as well. A variety $\mathsf {V}$ of (bounded) lattices is (meet/join) semidistributive if $\mathsf {V} \subseteq\mathsf{SD}$ ($\mathsf {V} \subseteq\mathsf{SD}/\wedge$  / $\mathsf {V} \subseteq\mathsf{SD}/\vee$). We need a little bit of lattice theory. A filter of ${\textbf{\upshape L}}$ is an upset $F$ of ${\textbf{\upshape L}}$ that is closed under meet; a filter is *prime* if $a \vee b \in F$ implies $a \in F$ or $b \in F$. An *ideal* $I$ of ${\textbf{\upshape L}}$ is the dual concept, i.e. a downset that is closed under join; an ideal is *prime* if $a \wedge b \in I$ implies $a \in I$ or $b \in I$. The following lemma is straightforward. **Lemma 112**. *If $F$ is a prime filter of ${\textbf{\upshape L}}$ ($I$ is a prime ideal of ${\textbf{\upshape L}}$), then $L \setminus F$ is a prime ideal of ${\textbf{\upshape L}}$ (${\textbf{\upshape L}}\setminus I$ is a prime filter of ${\textbf{\upshape L}}$).* **Lemma 113**. *Any bounded (meet/join) semidistributive lattice is unifiable in the variety of bounded lattices.* *Proof.* Let ${\textbf{\upshape L}}$ be bounded and meet semidistributive. Since ${\textbf{\upshape L}}$ is lower bounded by $0$ a standard application of Zorn Lemma yields a maximal proper filter $F$ of ${\textbf{\upshape L}}$; we claim that $F$ is also prime. Let $a,b \notin F$; then the filter generated by $F \cup \{a\}$ must be the entire lattice. Hence there must be a $c \in F$ with $c \wedge a =0$; similarly there must be a $d \in F$ with $d \wedge b = 0$. Let $e = c \wedge d$; then $e \in F$ and $e\wedge a = e \wedge b = 0$ and by meet semidistributivity $e \wedge(a \vee b) = 0$. But if $a \vee b \in F$, then $0 \in F$, a clear contradiction. Hence $a \vee b \notin F$ and $F$ is prime. Let now $\varphi: {\textbf{\upshape L}}\Longrightarrow \mathbf 2^b$ defined by $$\varphi(x) = \left\{ \begin{array}{ll} 1, & \hbox{if $x \in F$;} \\ 0, & \hbox{if $x \notin F$.} \end{array} \right.$$ Using the fact that $F$ is prime and $L \setminus F$ is prime it is straightforward to check that $\varphi$ is a homomorphism. Therefore ${\textbf{\upshape L}}$ is unifiable. A dual proof shows that the conclusion holds for join semidistributivity and a fortiori for semidistributivity. ◻ **Proposition 114**. *Any bounded finite (meet/join) semidistributive lattice is flat.* *Proof.* If ${\textbf{\upshape L}}$ is finite and (meet/join) semidistributive, every lattice in ${\mathbf H}{\mathbf S}({\textbf{\upshape L}})$ is finite and (meet/join) semidistributive. So it is unifiable and, if simple, it must be equal to ${\textbf{\upshape 2}}^b$; therefore ${\textbf{\upshape L}}$ is flat. ◻ **Corollary 115**. *Every locally finite (meet/join) semidistributive variety of bounded lattices is passive structurally complete.* In [@Lee1985] several (complex) sets of equations implying semidistributivity are studied; one of them is useful to us, since it describes a class of locally finite varieties. The description is interesting in that involves some of the ${\textbf{\upshape L}}_i's$ we have introduced before. **Theorem 116**. *[@Lee1985] There exists a finite set $\Gamma$ of lattices equations such that, if $\mathsf {V}$ is any variety of lattices such that $\mathsf {V} \vDash \Gamma$, then the following hold:* 1. *$\mathsf {V}$ is semidistributive;* 2. *$\mathsf {V}$ is locally finite;* 3. *only ${\textbf{\upshape L}}_{13}, {\textbf{\upshape L}}_{14}, {\textbf{\upshape L}}_{15} \in \mathsf {V}$.* A variety satisfying $\Gamma$ is called *almost distributive* and it is straightforward to check that a similar result holds for varieties of bounded lattices. Therefore: **Proposition 117**. *Every almost distributive variety of bounded lattices is passive structurally complete.* We close this subsection with a couple of observations; first ${\mathbf V}({\textbf{\upshape L}}_1^b,{\textbf{\upshape L}}_2^b)$ is a variety of bounded lattices that is passive structurally complete (by Theorem [Theorem 103](#fgflat){reference-type="ref" reference="fgflat"}) but neither meet nor join semidistributive. Next, what about infinite flat (bounded) lattices? We stress that in [@McKenzie1994] there are several examples of this kind and we believe that a careful analysis of the proofs therein could give some insight on how to construct a non locally finite variety of bounded lattices that it is passive structurally complete. But again, this is not a paper in lattice theory; therefore we defer this investigation. ## Substructural logics and residuated lattices {#sec:FL} Originally, *substructural logics* were introduced as logics which, when formulated as Gentzen-style systems, lack some (including "none" as a special case) of the three basic *structural rules* (i.e. exchange, weakening and contraction) of classical logic. Nowadays, substructural logics are usually intended as those logics whose equivalent algebraic semantics are residuated structures, and they encompass most of the interesting non-classical logics: intuitionistic logic, basic logic, fuzzy logics, relevance logics and many other systems. Precisely, by substructural logics we mean here the axiomatic extensions of the Full Lambek calculus $\mathcal{FL}$ (see [@GJKO] for details and a survey on substructural logics). All these logics are *strongly algebraizable*: their equivalent algebraic semantics are all *varieties* of $\mathsf{FL}$-algebras, particular residuated lattices that we shall now define. A *residuated lattice* is an algebra ${\textbf{\upshape A}} = \langle A,\vee,\wedge,\cdot, {\slash}, {\backslash}, 1\rangle$ where 1. $\langle A, \vee, \wedge\rangle$ is a lattice; 2. $\langle A, \cdot,1\rangle$ is a monoid; 3. ${\slash}$ and ${\backslash}$ are the right and left divisions w.r.t. $\cdot$, i.e., $x \cdot y \leq z$ iff $y \leq x {\backslash}z$ iff $x \leq z {\slash}y$, where $\leq$ is given by the lattice ordering. Residuated lattices form a variety $\mathsf{RL}$ and an equational axiomatization, together with many equations holding in these very rich structures, can be found in [@BlountTsinakis2003]. A residuated lattice ${\textbf{\upshape A}}$ is *integral* if it satisfies the equation $x \le 1$; it is *commutative* if $\cdot$ is commutative, and in this case the divisions coincide: $x \backslash y = y / x$, and they are usually denoted with $x \to y$. The classes of residuated lattices that satisfy any combination of integrality and commutativity are subvarieties of $\mathsf{RL}$. We shall call the variety of integral residuated lattices $\mathsf{IRL}$, commutative residuated lattices $\mathsf{CRL}$, and their intersection $\mathsf{CIRL}$. Residuated lattices with an extra constant $0$ in the language are called $\mathsf{FL}$-algebras, since they are the equivalent algebraic semantics of the Full Lambek calculus $\mathcal{FL}$. Residuated lattices are then the equivalent algebraic semantics of $0$-free fragment of $\mathcal{FL}$, $\mathcal{FL}^{+}$. An $\mathsf{FL}$-algebra is *$0$-bounded* if it satisfies the inequality $0 \leq x$ and the variety of zero-bounded $\mathsf{FL}$-algebras is denoted by $\mathsf{FL}_o$; integral and $0$-bounded $\mathcal{FL}$-algebras are called $\mathcal {FL}_w$ algebras (since they are the equivalent algebraic semantics of the Full Lambek Calculus with weakening), and we call its commutative subvariety $\mathcal{FL}_{ew}$. Restricting ourselves to the commutative case there is another interesting equation: $$(x \rightarrow y) \vee(y \rightarrow x) \approx 1.$$ It can be shown (see [@BlountTsinakis2003] and [@JipsenTsinakis2002]) that a subvariety of $\mathsf{FL_{ew}}$ or $\mathsf{CIRL}$ satisfies the above equation if and only if any algebra therein is a subdirect product of totally ordered algebras, and this implies that all the subdirectly irreducible algebras are totally ordered. Such varieties are called *representable* and the subvariety axiomatized by that equation is the largest subvariety of $\mathsf{FL_{ew}}$ or $\mathsf{CIRL}$ that is representable. The representable subvariety of $\mathsf{FL}_{ew}$ is usually denoted by $\mathsf {M}\mathsf {T}\mathsf {L}$, since it is the equivalent algebraic semantics of Esteva-Godo's *Monoidal t-norm based logic* [@EstevaGodo2001]. ### Active universal completeness We have already seen examples of subvarieties of $\mathsf{FL}_{ew}$-algebras that are active universally complete, but those were all locally finite subvarieties of $\mathsf{BL}$-algebras, that is, $\mathsf{MTL}$-algebras satisfying the divisibility equation: $x \land y = x (x \to y)$. In this section we will display a different class of examples. If ${\textbf{\upshape A}}$ is any algebra a congruence $\theta\in \operatorname{Con}({\textbf{\upshape A}})$ is a **factor congruence** if there is a $\theta' \in \operatorname{Con}({\textbf{\upshape A}})$ such that $\theta\vee\theta' = 1_{\textbf{\upshape A}}$, $\theta\wedge\theta' = 0_{\textbf{\upshape A}}$ and $\theta,\theta'$ permute. It is an easy exercise in general algebra to show that in this case ${\textbf{\upshape A}} \cong {\textbf{\upshape A}}/\theta\times {\textbf{\upshape A}}/\theta'$; note that $1_{\textbf{\upshape A}}$ and $0_{\textbf{\upshape A}}$ are factor congruences that gives a trivial decomposition. A less known fact (that appears in [@Citkin2018a]) is: **Lemma 118**. *Let ${\textbf{\upshape A}}$ be any algebra and $\theta$ a factor congruence; then ${\textbf{\upshape A}}/\theta$ is a retract of ${\textbf{\upshape A}}$ if and only if there is a homomorphism $h: {\textbf{\upshape A}}/\theta\longrightarrow {\textbf{\upshape A}}/\theta'$.* *Proof.* Suppose first that there is a homomorphism $h: {\textbf{\upshape A}}/\theta\longrightarrow {\textbf{\upshape A}}/\theta'$. Since ${\textbf{\upshape A}} \cong {\textbf{\upshape A}}/\theta\times {\textbf{\upshape A}}/\theta'$ for $u \in A$, $u = (a/\theta,b/\theta')$, we set $f(u) = a/\theta$; then $f:{\textbf{\upshape A}} \longrightarrow {\textbf{\upshape A}}/\theta$ is clearly an epimorphism, since $(a/\theta,a/\theta') \in A$ for all $a \in A$. Let $$g(a/\theta) = (a/\theta,h(a/\theta)).$$ One can check that $g$ is a homomorphism with standard calculations and clearly $fg= id_{{\textbf{\upshape A}}/\theta}$. Hence ${\textbf{\upshape A}}/\theta$ is a retract of ${\textbf{\upshape A}}$. Conversely suppose that $f,g$ witness a retraction from ${\textbf{\upshape A}}/\theta$ in ${\textbf{\upshape A}}$; then if $g(a/\theta)= (u/\theta,v/\theta')$, set $h(a/\theta) = v/\theta'$. It is then easy to see that $h$ is a homomorphism and the thesis holds. ◻ Observe that in any $\mathsf{FL}$-algebra, every compact (i.e., finitely generated) congruence is principal; as a matter of fact if ${\textbf{\upshape A}}$ is in $\mathsf{FL}$, $X = \{(a_1,b_1),\dots,(a_n,b_n)\}$ is a finite set of pairs from $A$ and $p = \bigwedge_{i=1}^n [ (a_i \backslash b_i) \wedge(b_i \backslash a_i) \wedge 1]$ then $\vartheta_{\textbf{\upshape A}}(X) = \vartheta_{\textbf{\upshape A}}(p,1)$. **Theorem 119**. *Let $\mathsf {Q}$ be a quasivariety of $\mathsf{FL}_w$-algebras in which every principal congruence is a factor congruence; then $\mathsf {Q}$ has projective unifiers.* *Proof.* Let ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta$ be a finitely presented unifiable algebra in $\mathsf {Q}$; then there is an onto homomorphism from ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta(\Sigma)$ to ${\textbf{\upshape F}}_\mathsf {Q} = {\textbf{\upshape 2}}$. Now $\theta= \theta(\Sigma)$ is a principal congruence, hence it is a factor congruence with witness $\theta'$, i.e. ${\textbf{\upshape F}}_\mathsf {Q}(X) \cong {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta\times {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta'$. If $\theta' = 1_{\textbf{\upshape A}}$, then ${\textbf{\upshape F}}_\mathsf {Q}(X) = {\textbf{\upshape F}}_\mathsf {Q}(X)/\theta$ and so it is projective. Otherwise ${\textbf{\upshape F}}_\mathsf {Q} = {\textbf{\upshape 2}}$ is embeddable in ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta'$; hence there is a homomorphism from from ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta$ to ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta'$. By Lemma [Lemma 118](#factor){reference-type="ref" reference="factor"} ${\textbf{\upshape F}}_\mathsf {Q}(X)/\theta$ is a retract of ${\textbf{\upshape F}}_\mathsf {Q}(X)$, i.e. it is projective. ◻ So any quasivariety of $\mathsf{FL}_{w}$-algebras with the property that every principal congruence is a factor congruence is active universally complete (Theorem [Theorem 60](#lemma: ex unif auc){reference-type="ref" reference="lemma: ex unif auc"}); really it is active primitive universally complete, because ${\textbf{\upshape F}}_\mathsf {Q}$ is the two-element algebra for any quasivariety $\mathsf {Q}$ of $\mathsf{FL}_{w}$-algebras (Theorem [Theorem 64](#thm: activeprimitive){reference-type="ref" reference="thm: activeprimitive"}). We observe in passing that for any $\mathsf{FL}_{w}$ algebra every factor congruence is principal; this is because every variety of $\mathsf{FL}_{w}$-algebras is Kollár and congruence distributive. Discriminator varieties of $\mathsf{FL}_{ew}$-algebras have been completely described in [@Kowalski2005]; as a consequence we have: **Theorem 120**. *For a variety $\mathsf {V}$ of $\mathsf{FL}_{ew}$-algebras the following are equivalent:* 1. *$\mathsf {V}$ is a discriminator variety;* 2. *$\mathsf {V}$ is semisimple, i.e. all the subdirectly irreducible members of $\mathsf {V}$ are simple;* 3. *there is an $n \in \mathbb N$ such that $\mathsf {V} \vDash x \vee\neg x^n \approx 1$;* 4. *for any ${\textbf{\upshape A}} \in \mathsf {V}$ every compact (i.e. principal) congruence is a factor congruence.* *Proof.* The equivalence of (1), (2) and (3) has been proved in [@Kowalski2005]. Assume then (1); it is well-known that in every discriminator variety every principal congruence is a factor congruence. In fact if $\mathsf {V}$ is a discriminator variety with discriminator term $t(x,y,z)$ let for any ${\textbf{\upshape A}} \in \mathsf {V}$ and $a,b \in A$ $$\begin{aligned} &\theta_{\textbf{\upshape A}}(a,b) = \{(u,v): t(a,b,u)=t(a,b,v)\} \\ &\gamma_{\textbf{\upshape A}}(a,b)= \{(u,v): t(a,t(a,b,u),u) = t(a,t(a,b,v),v)\}.\end{aligned}$$ Using the properties of the discriminator term it is easy to verify that they are congruences and the complement of each other; since discriminator varieties are congruence permutable they are factor congruences and (4) holds. Conversely assume (4) and let ${\textbf{\upshape A}}$ be a subdirectly irreducible member of $\mathsf {V}$. Let $\mu_{\textbf{\upshape A}}$ be the minimal nontrivial congruence of ${\textbf{\upshape A}}$; then $\mu_{\textbf{\upshape A}}$ is principal, so it must be a factor congruence. This is possible if and only if $\mu_{\textbf{\upshape A}}=1_{\textbf{\upshape A}}$; therefore ${\textbf{\upshape A}}$ is simple, and $\mathsf {V}$ is semisimple. ◻ **Corollary 121**. *Every discriminator (or, equivalently, semisimple) variety $\mathsf {V}$ of $\mathsf{FL}_{ew}$-algebras is active primitive universal.* We observe that Theorem [Theorem 119](#thm: discr almost universal){reference-type="ref" reference="thm: discr almost universal"} does not add anything as far as $\mathsf{BL}$-algebras are concerned; in fact any discriminator variety of $\mathsf{FL}_{ew}$-algebras must satisfy $x^n \approx x^{n+1}$ for some $n$ [@Kowalski2005] and the varieties of $\mathsf{BL}$-algebras with that property are exactly the locally finite varieties, which we already pointed out are active universally complete. ### Passive structural completeness A particularly interesting application of our characterization of passive structurally complete varieties is in the subvariety of integral and $0$-bounded FL-algebras. Let us rephrase Theorem [Theorem 94](#thm:PUC){reference-type="ref" reference="thm:PUC"} in this setting. First, using residuation it is easy to see that every finite set of identities in $\mathsf{FL}$ is equivalent to a single identity. Moreover, in every subquasivariety $\mathsf {Q}$ of $\mathsf{FL}_w$, the smallest free algebra ${\textbf{\upshape F}}_{\mathsf {Q}}$ is the 2-element Boolean algebra ${\textbf{\upshape 2}}$, and its generated quasivariety is the variety of Boolean algebras. **Corollary 122**. *Let $\mathsf {Q}$ be a quasivariety of $\mathsf{FL}_w$-algebras, then the following are equivalent:* 1. *$\mathsf {Q}$ is passive structurally complete;* 2. *every trivializing identity in the variety of Boolean algebras is trivializing in $\mathsf {Q}$;* 3. *every nontrivial finitely presented algebra is unifiable.* 4. *every nontrivial algebra is unifiable.* The previous corollary has a possibly more transparent shape from the point of view of the logics. Let us call a formula $\varphi$ in the language of $\mathsf{FL}$-algebras *explosive in a logic $\mathcal{L}$*, with consequence relation $\vdash_{\mathcal{L}}$, if $\varphi \vdash_{\mathcal{L}} \delta$ for all formulas $\delta$ in the language of $\mathcal{L}$. Moreover, we call $\varphi$ a *contradiction in $\mathcal{L}$* if $\varphi \vdash_{\mathcal{L}} 0$. Since $\mathsf{FL}_w$-algebras are $0$-bounded, it is clear that contradictions coincide with explosive formulas in all axiomatic extensions of $\mathcal{F}\mathcal{L}_w$. **Corollary 123**. *Let $\mathcal{L}$ be an axiomatic extension of $\mathcal{FL}_w$, then the following are equivalent:* 1. *$\mathcal{L}$ is passively structurally complete.* 2. *Every contradiction of classical logic is explosive in $\mathcal{L}$.* 3. *Every passive rule of $\mathcal{L}$ has explosive premises.* Let us first explore the consequences of the equivalence between (1) and (2) in Corollary [Corollary 123](#cor:PUClogic){reference-type="ref" reference="cor:PUClogic"}. It is well known that intuitionistic logic is passively structurally complete (reported by Wronski at the 51st Conference on the History of Logic, Krakow, 2005). This is easily seen by Corollary [Corollary 123](#cor:PUClogic){reference-type="ref" reference="cor:PUClogic"}, indeed, observe that any contradiction of classical logic $\varphi$ is such that its negation $\neg \varphi$ is a theorem of classical logic. Using the Glivenko translation and the deduction theorem, we obtain that $\varphi$ is explosive in intuitionistic logic as well, which is then passively structurally complete. We will now show how this argument can be extended to a wide class of logics. Let us write the negations corresponding to the two divisions as $\neg x = x \backslash 0$ and $\sim x = 0 / x$. Following [@GalatosOno2006; @GalatosOno2009], we say that two logics $\mathcal{L}_1$ and $\mathcal{L}_2$ are *Glivenko equivalent* if for all formuals $\varphi$: $$\vdash_{\mathcal{L}_1} \neg \varphi \;\;\mbox{ iff }\;\; \vdash_{\mathcal{L}_2} \neg \varphi$$(equivalently, $\vdash_{\mathcal{L}_1} \sim \varphi \mbox{ iff } \vdash_{\mathcal{L}_2} \sim \varphi$). Given a logic $\mathcal{L}$, we call *Glivenko logic of $\mathcal{L}$* the smallest substructural logic that is Glivenko equivalent to $\mathcal{L}$ . Moreover, we call *Glivenko logic of $\mathcal{L}$ with respect to $\mathcal{L}'$*, and denote it with $\mathcal{G}_{\mathcal{L}'}(\mathcal{L})$ the smallest extension of $\mathcal{L}'$ that is Glivenko equivalent to $\mathcal{L}$ (all these notions make sense by the results in [@GalatosOno2006; @GalatosOno2009]). $\mathcal{G}_{\mathcal{L}'}(\mathcal{L})$ is axiomatized relatively to $\mathcal{L}'$ by the set of axioms $\{ \neg \sim \varphi : \vdash_{\mathcal{L}}\varphi\}$, or equivalently by the set $\{ \sim \neg \varphi : \vdash_{\mathcal{L}}\varphi\}$. Here we are interested in the Glivenko equivalent of classical logic with respect to $\mathcal{FL}_w$. From the algebraic perspective, this corresponds to the largest subvariety of $\mathsf{FL}_w$ that is Glivenko equivalent to Boolean algebras, $\mathsf{G}_{\mathsf{FL}_w}(\mathsf {B})$. The latter is axiomatized in [@GJKO Corollary 8.33] as the subvariety of $\mathsf{FL}_w$ satisfying: 1. $\sim (x \land y) = \sim (xy)$ 2. $\sim (x \backslash y) = \sim (\neg x \lor y)$ 3. $\neg (x \backslash y) = \neg (\sim x \lor y)$ 4. $\sim(x \backslash y) = \sim (\neg \sim x \backslash \neg \sim y)$ 5. $\sim(x / y) = \sim (\neg \sim x / \neg \sim y)$. **Theorem 124**. *Every axiomatic extension $\mathcal{L}$ of the Glivenko logic of classical logic with respect to $\mathcal{FL}_w$ is passively structurally complete.* *Proof.* Consider a contradiction of classical logic $\varphi$, by the deduction theorem $\vdash_{\mathcal{C}\mathcal{L}} \neg \varphi$ (where $\vdash_{\mathcal{C}\mathcal{L}}$ is the consequence relation of classical logic). Since $\mathcal{L}$ is Glivenko equivalent to classical logic, $\vdash_{\mathcal{L}} \neg \varphi$. It can be easily checked that this implies that $\varphi \vdash_{\mathcal{L}} 0$ (it is a consequence of the parametrized local deduction theorem which holds in every extension of $\mathcal{FL}$ [@GJKO], but it is also straightforward to see in models). Thus $\varphi$ is a contradiction of $\mathcal{L}$, or equivalently it is explosive in $\mathcal{L}$, which is then passively structurally complete by Corollary [Corollary 123](#cor:PUClogic){reference-type="ref" reference="cor:PUClogic"}. ◻ Thus, every subvariety of $\mathsf{G}_{\mathsf{FL}_w}(\mathsf {B})$ is passive structurally complete. In particular, the commutative subvariety $\mathsf{G}_{\mathsf{FL}_{ew}}(\mathsf {B})$ is the variety of pseudocomplemented $\mathsf{FL}_{ew}$-algebras ([@FazioStJohn2022]), axiomatized by $$x \land \neg x \approx 0.$$ Examples of passive structurally complete varieties then include Heyting algebras, Stonean MTL-algebras and as a consequence, e.g., product algebras and Gödel algebras. We observe that these are not all of the passive structural complete varieties of $\mathsf{FL}_{w}$ (nor of $\mathsf{FL}_{ew}$). Let us indeed obtain a different kind of examples. **Definition 125**. We say that a variety $\mathsf {V}$ has a *Boolean retraction term* if there exists a term $t$ in the language of residuated lattices (i.e., $0$-free) such that, for every ${\textbf{\upshape A}} \in \mathsf {V}$, $t$ defines an idempotent endomorphism on ${\textbf{\upshape A}}$ whose image is the Boolean skeleton of ${\textbf{\upshape A}}$, that is, the set of complemented elements of ${\textbf{\upshape A}}$. Varieties with a Boolean retraction term have been studied at length by Cignoli and Torrens in a series of papers, see in particular [@CignoliTorrens2012]. These are all varieties in which all nontrivial algebras retract onto a nontrivial Boolean algebra, thus they satisfy the hypotheses of Corollary [Corollary 122](#cor:psctriv){reference-type="ref" reference="cor:psctriv"} and they are passive structurally complete. Some of these varieties have been shown in [@AglianoUgolini2022a] to have projective unifiers, thus they satisfy Theorem [Theorem 54](#thm:unifiable2){reference-type="ref" reference="thm:unifiable2"} and they are non-negative universally complete. Among those we cite some varieties of interest in the realm of many-valued logics: the variety of product algebras, the variety generated by perfect MV-algebras, the variety $\mathsf{NM^-}$ of nilpotent minimum algebras without negation fixpoint and some varieties that have been called *nilpotent product* in [@AglianoUgolini2019a] or [@AglianoUgolini2019b]. We will see that in the representable variety of $\mathsf{FL}_{ew}$, $\mathsf{MTL}$, we can fully characterize passive structurally complete varieties as those with a Boolean retraction term. By [@CignoliTorrens2012], the largest subvariety of $\mathsf {M}\mathsf {T}\mathsf {L}$ with a Boolean retraction term is axiomatized relatively to $\mathsf {M}\mathsf {T}\mathsf {L}$ by the Di Nola-Lettieri equation: $$(x + x)^2 = x^2 + x^2 \tag{DL}$$ where $x + y = \neg(\neg x \cdot \neg y)$. The latter identity has been introduced by Di Nola and Lettieri to axiomatize within MV-algebras the variety generated by the Chang algebra. This variety is called $\mathsf{sDL}$ in [@Ugolini2018] ($\mathsf{BP}_0$ in [@Noguera2007; @AFU2017]), and it includes, for instance: pseudocomplemented MTL-algebras (also called SMTL-algebras), and thus Gödel algebras and product algebras; involutive $\mathsf{BP}_0$-algebras and thus the variety generated by perfect MV-algebras and nilpotent minimum algebras without negation fixpoint. Let us say that an element of an $\mathsf{FL}_{ew}$-algebra ${\textbf{\upshape A}}$ has *finite order $n$* if $x^n = 0$, and *infinite order* if there is no such $n$. We call *perfect* an algebra ${\textbf{\upshape A}} \in \mathsf{FL}_{ew}$ such that, for all $a \in A$, $a$ has finite order if and only if $\neg a$ has infinite order. Now, $\mathsf{sDL}$ turns out to be the variety generated by the perfect chains (see [@Ugolini2018; @AFU2017]). **Lemma 126**. *A chain ${\textbf{\upshape A}} \in \mathsf{FL}_{ew}$ is perfect if and only if there is no element with finite order $a \in A$ such that $a \geq \neg a$.* *Proof.* By order preservation, if there is an element $a \in A, a \geq \neg a, a^n = 0$, then both $a$ and its negation have finite order, thus the chain is not perfect. Suppose now a chain ${\textbf{\upshape A}}$ is not perfect. Observing that for every element $x \in A$ it cannot be that both $x$ and $\neg x$ gave infinite order, we get that there is an element $a \in A$ such that both $a$ and its negation $\neg a$ have finite order. If $a \not\geq \neg a$, since ${\textbf{\upshape A}}$ is a chain, $a < \neg a$. Then $\neg\neg a \leq \neg a$, and they both have finite order. ◻ **Theorem 127**. *For a subvariety $\mathsf {V}$ of $\mathsf{MTL}$ the following are equivalent:* 1. *$\mathsf {V}$ is passive structurally complete;* 2. *$\mathsf {V}$ is a subvariety of $\mathsf{sDL}$.* *Proof.* Since subvarieties of $\mathsf{sDL}$ have a Boolean retraction term (2) implies (1) by Corollary [Corollary 122](#cor:psctriv){reference-type="ref" reference="cor:psctriv"}. Suppose now that $\mathsf {V} \nsubseteq \mathsf{sDL}$. Then there is a chain ${\textbf{\upshape A}}$ in $\mathsf {V}$ that is not perfect. By Lemma [Lemma 126](#lemma:perfectchain){reference-type="ref" reference="lemma:perfectchain"}, there exists $a \in A, a \geq \neg a, a^n = 0 \mbox{ for some } n \in \mathbb{N}.$ Thus, $\neg(a \lor \neg a)^n = 1$. But the identity $\neg(x \lor \neg x)^n = 0$ holds in Boolean algebras. Thus $\neg(x \lor \neg x)^n \approx 1$ is trivializing in Boolean algebras but not in $\mathsf {V}$. By Corollary [Corollary 122](#cor:psctriv){reference-type="ref" reference="cor:psctriv"}, $\mathsf {V}$ is not passive structurally complete and thus (1) implies (2). ◻ Notice that the previous theorem also implies that a variety of MTL-algebras that is not a subvariety of $\mathsf{sDL}$ cannot be structurally complete. We mention that structural completeness in subvarieties of $\mathsf{MTL}$ (or their logical counterparts) has been studied by several authors: e.g., [@Wojtylak1976] and [@Gispert2016] for Łukasiewicz logics, [@DzikWronski1973] Gödel logic, and [@CintulaMetcalfe2009] for fuzzy logics in the MTL framework; in the latter the authors show for instance that all subvarieties of pseudocomplemented MTL-algebras ($\mathsf{SMTL}$) are passive structurally complete. This result is here obtained as a consequence of Theorem [Theorem 127](#thm: MTL collapse){reference-type="ref" reference="thm: MTL collapse"}, since $\mathsf{SMTL}$ is a subvariety of $\mathsf{sDL}$. From the results mentioned above and the characterzation theorem, it also follows that the only varieties of MV-algebras (the equivalent algebraic semantics of infinite-valued Łukasiewicz logic) that are structurally complete are Boolean algebras and the variety generated by perfect MV-algebras (this result has been obtained following a different path in [@Gispert2016]). We also remark that a variety of $\mathsf{FL}_{ew}$-algebras can be at most non-negative universally complete since trivial algebras are finitely presented and not unifiable (unifiability is a necessary condition for universal completeness by Theorem [Theorem 42](#lemma: FQ trivial){reference-type="ref" reference="lemma: FQ trivial"}); by Proposition [Proposition 91](#prop: nnu iff ps and au){reference-type="ref" reference="prop: nnu iff ps and au"} this happens if and only if the variety is active universally complete and passive structurally complete. Thus, for instance, a semisimple variety of $\mathsf{FL}_{ew}$-algebras satisfying the conditions in Corollary [Corollary 122](#cor:psctriv){reference-type="ref" reference="cor:psctriv"} would be non-negative universally complete. We stress that this observation is not of particular interest in MTL-algebras, since the only discriminator variety in $\mathsf{sDL}$ is the variety of Boolean algebras. Indeed, consider a chain ${\textbf{\upshape A}}$ in a discriminator variety $\mathsf {V}$ in $\mathsf{sDL}$. Then there is some $n \in \mathbb{N}$ such that $\mathsf {V} \models x \lor \neg x^n \approx 1$. Let now $a \in A$; either $a$ has finite order, and then from $a \lor \neg a^n$ we obtain that $a = 1$, or $a$ has infinite order, and then $\neg a$ has finite order. So by the analogous reasoning $\neg a = 1$. Therefore ${\textbf{\upshape A}}$ is the two-element chain, and $\mathsf {V}$ is the variety of Boolean algebras. # Conclusions {#sec: conclusions} In Figure [\[classes\]](#classes){reference-type="ref" reference="classes"} we display several classes of varieties that we have considered in this paper (and the labels should be self explanatory); we are dropping the hereditary subclasses to avoid clutter. Observe that this is really a meet semilattice under inclusion. Almost all the classes are provably distinct. 1. The variety of bounded distributive lattice is structurally complete (Proposition [Proposition 108](#proo: bdl structural){reference-type="ref" reference="proo: bdl structural"}) but it is neither passive universally complete, since it is Kollár and the least free algebra is not trivial, nor non-negative universally complete (Proposition [Proposition 107](#prop: bdl not nnu){reference-type="ref" reference="prop: bdl not nnu"}). Hence $S \ne NNU,S+PU$. 2. The variety of Boolean algebras is non-negative universally complete but not universally complete (Example [Example 56](#ex: boolean algebras){reference-type="ref" reference="ex: boolean algebras"}) so $NNU\ne U$. 3. Any locally finite variety of $\mathsf{BL}$-algebras is active universally complete and some of them are not non-negative universally complete (Example [Example 62](#ex: BL-algebras au){reference-type="ref" reference="ex: BL-algebras au"}), so $AU \ne NNU$. 4. The variety in Example [Example 90](#ex: as not au){reference-type="ref" reference="ex: as not au"} is active structurally complete but not active universally complete, hence $AS \ne AU$. 5. Any locally finite variety of bounded semidistributive lattices different from the distributive variety is passive structurally complete (Corollary [Corollary 115](#cor: sdbl is ps){reference-type="ref" reference="cor: sdbl is ps"}) but not structurally complete, since the only structurally complete variety of bounded distributive lattices is the distributive variety (Proposition [Proposition 108](#proo: bdl structural){reference-type="ref" reference="proo: bdl structural"}); as above it is also not passive universally complete. Hence $PS \ne S, PU$. 6. The variety ${\mathbf V}({\textbf{\upshape M}}_{3,3}^+)$ (Section [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"}) is passive universally complete, as any variety of lattices, but it is not structurally complete since ${\mathbf Q}({\textbf{\upshape M}}_{3,3}^+) \nsubseteq {\mathbf V}({\textbf{\upshape M}}_{3,3}^+)$; hence $PU \ne S+PU$. 7. Example 7.11 in [@DzikStronkowski2016] shows that $AS \ne S$. Moreover for the primitive counterparts: 1. the variety ${\mathbf V}({\textbf{\upshape F}})$ generated by the Fano lattice is structurally complete and passive universally complete but not primitive (Section [5.1](#sec:lattices){reference-type="ref" reference="sec:lattices"}). 2. the variety of De Morgan lattices (Example [Example 67](#ex: De Morgan){reference-type="ref" reference="ex: De Morgan"}) is active universally complete but not active primitive universal. 3. the variety of injective monounary algebras is active structurally complete but not active primitive structural (Example 7.2 in [@DzikStronkowski2016]). There are three examples that we were not able to find, which would guarantee total separation of all the classes we have considered: 1. A (quasi)variety that is structurally complete and passive universally complete, but not universally complete. 2. A non-negative universally complete (quasi)variety such that not all subquasivarieties are non-negative universally complete. 3. A universally complete variety which is not primitive universal. The natural example for (3) would be a locally finite variety with exact unifiers having a subvariety without exact unifiers. However we are stuck because of lack of examples: we have only one unifiable locally finite variety with exact (non projective) unifiers, i.e. the variety of distributive lattices, which is trivially primitive universal. A similar situation happens for (2); all the examples of non-negative universally complete varieties we have are either equationally complete and congruence distributive (so they do not have nontrivial subquasivarieties), or else are active universally complete just by consequence of their characterization (such as the subvarieties of $\mathsf{FL}_{ew}$ in Section [5.2](#sec:FL){reference-type="ref" reference="sec:FL"}). Then we have Stone algebras that are not equationally complete but the only nontrivial subvariety is the variety of Boolean algebras, that is non-negative universally complete. Now from Corollary [Corollary 30](#cor: VQ pu Q pu){reference-type="ref" reference="cor: VQ pu Q pu"} it is immediate that every subquasivariety of $\mathsf{ST}$ is non-negative universally complete. In conclusion a deeper investigation of universally complete and non-negative universally complete varieties is needed. For (1) the situation is (slightly) easier to tackle: any primitive variety of lattices that is not universally complete gives a counterexample. While it seems impossible that all the primitive varieties in Section [5.1.1](#sec: primlattices){reference-type="ref" reference="sec: primlattices"} are universally complete, actually proving that one it is not does not seem easy. 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arxiv_math

--- abstract: | We introduce anisotropic Hölder spaces useful for the study of the regularity theory for non local kinetic operators $\mathscr{L}$ whose prototypal example is $$\mathscr{L}u (t,x,v) = \int_{{\mathbb {R}}^d} \frac{C_{d,s}}{|v - v'|^{d+2s}} (u(t,x,v') - u(t,x,v)) \mathrm{d}v' + \langle v , \nabla_x \rangle + \partial_t, \quad (t,x,v)\in{\mathbb {R}}\times{\mathbb {R}}^{2d}.$$ The Hölder spaces are defined in terms of an anisotropic distance relevant to the Galilean geometric structure on ${\mathbb {R}}\times{\mathbb {R}}^{2d}$ the operator $\mathscr{L}$ is invariant with respect to. We prove an intrinsic Taylor-like formula, whose reminder is estimated in terms of the anisotropic distance of the Galilean structure. Our achievements naturally extend analogous known results for purely differential operators on Lie groups. author: - "Maria Manfredini [^1]" - "Stefano Pagliarani [^2]" - "Sergio Polidoro [^3]" bibliography: - Bibtex-Master-3.bib date: "This version: " title: Intrinsic Hölder spaces for fractional kinetic operators --- **Keywords**: Fractional kinetic operators, Kolmogorov operators, Hörmander's condition, Hölder spaces, Taylor formula. # Introduction {#intro} We consider Hölder spaces and Taylor-like formulas useful for the study of the regularity theory of the solutions to $\mathscr{L}u = f$, being $\mathscr{L}$ a non local kinetic operator of the form $$\label{e1} \mathscr{L}u (t,x,v) = \int_{{\mathbb {R}}^d} K(t,x,v,v') (u(t,x,v') - u(t,x,v)) \mathrm{d}v' + Yu,\qquad (t,x,v)\in{\mathbb {R}}\times{\mathbb {R}}^{2d}.% {\blue \qquad \text{($t$ non c'e' nell'operatore)}}$$ where $$\label{eq:Y} Y = \langle v , \nabla_x \rangle + \partial_t , \qquad v\in{\mathbb {R}}^d.$$ The integral part is of order $2s$, with $s\in ]0,1[$, in the following sense $$\frac{c^-}{|v - v'|^{d+2s}} \le K(t, x, v, v') \le \frac{c^+}{|v - v'|^{d+2s}}, \qquad (t,x,v,v')\in{\mathbb {R}}\times{\mathbb {R}}^{3d},$$ for some positive constants $c^-$ and $c^+$. One notable particular instance of $\mathscr{L}$ is the fractional kinetic Fokker--Planck operator $$\label{e1_prot} \mathscr{L}_s = (-\Delta_v)^s + Y,$$ which is the operator in [\[e1\]](#e1){reference-type="eqref" reference="e1"} with $$\label{eq:kernel_prot} K(t, x, v, v') = \frac{C_{d,s}}{|v - v'|^{d+2s}},$$ for a suitable positive constant $C_{d,s}$. Note that $\mathscr{L}_s = (-\Delta_v)^s + Y$ is also related to the infinitesimal generator of an $\alpha$-stable Lévy process, with $\alpha = 2s$. In the purely diffusive setting, which can be seen as the limiting case $s = 1$, $\mathscr{L}_1 = -\Delta_v + Y$ is a differential hypoelliptic operator. This means that every distributional solution $u$ to the equation $-\Delta_v u + Yu = f$ is a smooth function whenever $f$ is smooth. Indeed, setting the primary vector fields $$\label{eq:Z_fields} Z_i: = \partial_{v_i}, \qquad i=1,\cdots, d,$$ we obtain that $\mathscr{L}_1$ writes in the form $\mathscr{L}_1 = - \sum_{i=1}^d Z_i^2 + Y$ and the system $Z_1,\cdots,Z_d,Y$ satisfies the so-called Hörmander condition, namely $$\label{eq:hormander} \text{rank } \text{Lie}(Z_1,\dots,Z_d,Y) = 2d + 1. %, \qquad v\in \R^d.$$ This is a straightforward consequence of $$\label{eq:commutator} [\partial_{v_j} , Y] = \partial_{x_j} , \qquad j=1,\dots, d.$$ We emphasize that the regularity properties of the Hörmander's operators depend on a non-Euclidean underlying structure (see the survey [@anceschi2019survey]). In the setting of the kinetic operator $\mathscr{L}$, this structure agrees with the Galilean translation (see [\[eq:translation\]](#eq:translation){reference-type="eqref" reference="eq:translation"} below). In this work we rely on the geometric structure introduced for the differential operator $\mathscr{L}_1$, in order to study the fractional operator $\mathscr{L}_s$ for $0 < s < 1$. We give a definition of *intrinsic Hölder spaces*, which extends the one indroduced by [@PPP16], and we prove a Taylor polynomial approximation of a function $f$ belonging to this Hölder space. We conclude this section with some remarks about the intrinsic Hölder spaces and the applications of our main results. These spaces are *anisotropic*, as the variables $v$ and $x$ in the Galileian group have a different role. Moreover, the definition of *intrinsic Hölder spaces* is based on the non-Euclidean quasi-distance [\[eq:quasi_distance\]](#eq:quasi_distance){reference-type="eqref" reference="eq:quasi_distance"} of the underlying Galileian group. Remark [Remark 8](#rcaa){reference-type="ref" reference="rcaa"} contains a brief discussion on our definition of anisotropic Hölder spaces compared with other definition present in the literature. In the purely differential setting, such Hölder spaces were studied by several authors, and a characterization relating the regularity along the vector fields $\partial_{v_1},\dots,\partial_{v_d},Y$ to the existence of appropriate instrinsic Taylor formulas was given by [@PPP16]. We refer to the articles [@Bonfiglioli2009] and [@Arena] for similar Taylor formulas on homogeneous groups. The Taylor approximation of a solution to a PDE is a useful tool in the proof of Schauder estimates. We refer to [@PolidoroRebucciStroffolini2022], where the regularity of classical solutions to degenerate Kolmogorov equations is obtained by using the method introduced in [@Wang2006] for uniformly elliptic and parabolic equations. We also recall the article [@imbert2021schauder] where Taylor approxiamation results and Schauder estimates for kinetic equations are proved. In the fractional setting, Schauder estimates for the solutions to $\mathscr{L}u = 0$ have been recently proved by [@imbert2018schauder], in suitable Hölder spaces that take into account the intrinsic geometry of the Galilean group. We emphasize that the main results of this note do apply to the operators $\mathscr{L}$ consdered in the aforementioned reference. Our main results also apply to a non-linear non-local version of [\[e1_prot\]](#e1_prot){reference-type="eqref" reference="e1_prot"}, that is $$\label{e1_prot-p} \mathscr{L}_{s,p} = (-\Delta_v)^s_p + Y,$$ considered for $p \in ]1,\infty[$ and $s\in ]0,1[$ in [@AnceschiPiccinini2023]. In this case, the kernel $K$ in [\[e1\]](#e1){reference-type="eqref" reference="e1"} also depends on the unknown function $u$ and the term $(-\Delta_v)_p^s u$ takes the following form $$\label{eq:kernel_prot-p} (-\Delta_v)^s_p u(t, x, v) = \int_{{\mathbb {R}}^d} \frac{C_{d,p,s}}{|v - v'|^{d+ps}}|u(t,x,v) - u(t,x,v')|^{p-2}\big( u(t,x,v) - u(t,x,v')\big) \mathrm{d}v'.$$ We finally remark that not only anysotropic spaces of Hölder continuous functions have been considered in literature for the study of kinetic equations of the form $\mathscr{L}u = f$. Indeed, [@pascucci2022sobolev] prove intrinsic Taylor expansion for anisotropic Sobolev-Slobodeckij spaces, and prove continuous embeddings into Lorentz and intrinsic Hölder spaces. This article is organized as follows. Section 2 contains some recalls on the non-Euclidean geometry relevant to the kinetic operator $\mathscr{L}_1$. Note that the study of the non local operator $\mathscr{L}_s$ relies on the same geometric structure. In Section 2 the notion of intrinsic regularity and intrinsic Hölder spaces and the statement of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"}, which is the main results of this note, are given. Some examples illustrate the meaning of the definitions and the main results. Section 3 contains some preliminaries the the proof of the main results, which is given in Section 4. Section 5 contains a local version of our main results, which generalizes Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} in that it applies to the more general geometric framework of the *non-homogeneous Lie groups*. **Acknowldedgment.** We thank Luis Silvestre for advising us to investigate this subject. # Hölder spaces and Taylor polynomials In this part we fix some notation for the geometric strucutre on ${\mathbb {R}}^{2d+1}$ that will be used in this work. We recall that, remarkably, $\mathscr{L}_s$ in [\[e1_prot\]](#e1_prot){reference-type="eqref" reference="e1_prot"} has the property of being invariant with respect to left translations in the group $\left({\mathbb {R}}\times{\mathbb {R}}^{2d},\circ\right)$, where the non-commutative group law "$\circ$" is defined by $$\label{eq:translation} z_1\circ z_2 = \left(t_1+t_2, x_1 + x_2 + t_2 v_1 , v_1 + v_2 \right),\qquad z_1=(t_1,x_1,v_1),z_2=(t_2,x_2,v_2)\in \mathbb{R}\times\mathbb{R}^{2d}.$$ Precisely, we have $$\label{eq:translation_invariance} \big(\mathscr{L}u^{(z_1)}\big) (z_2)=(\mathscr{L}u)(z_1\circ z_2).%, \qquad (t,x),(s,\xi)\in\Rdd,$$ where $$u^{(z_1)} (z_2)=u(z_1\circ z_2).$$ This translation property is often referred to as "Galilean" change of coordinate and is very useful in kinetic theory. A systematic study of the PDEs theory on this group started in [@LanconelliPolidoro1994] in the limiting case $s=1$. Notice that $({\mathbb {R}}\times{\mathbb {R}}^{2d},\circ)$ is a group with the identity and the inverse elements $${\text{Id}=(0,0,0)}, \qquad (t,x,v)^{-1}=\left(-t, t v - x , -v \right).%, $(t,x)\in\Rdd$$$ Moreover, $\mathscr{L}_s$ *is homogeneous of degree ${\mathbf{\vartheta}}= 2s$* with respect to the dilations $\left(D_\lambda\right)_{\lambda>0}$ on ${\mathbb {R}}\times{\mathbb {R}}^{2d}$ given by $$\label{eq:dilation} D_\lambda =\textrm{diag}\big(\lambda^{{\mathbf{\vartheta}}},\lambda^{{\mathbf{\vartheta}}+1}I_{d},\lambda I_{d}\big),%(t,x), \qquad (t,x)\in\Rdd,$$ where $I_{d}$ is the $(d\times d)$ identity matrix, i.e. $$\label{eq:dilation_invariance} \big(Y u^{(\lambda)}\big)(z)=\lambda^{{\mathbf{\vartheta}}}(Y u)(D_\lambda z), \qquad z=(t,x,v)\in{\mathbb {R}}\times{\mathbb {R}}^{2d},\ \lambda>0 ,$$ where $$u^{(\lambda)} (z)=u\big(D_\lambda(z)\big).$$ Note that the exponent ${\mathbf{\vartheta}}$ in [\[eq:dilation\]](#eq:dilation){reference-type="eqref" reference="eq:dilation"} equals $ps$ in the case of non-local operator $\mathscr{L}_{s,p}$ in [\[e1_prot-p\]](#e1_prot-p){reference-type="eqref" reference="e1_prot-p"}, hence we don't impose any restrition to the choice of ${\mathbf{\vartheta}}\in ]0, + \infty[$. In the sequel we will say that $Y$ has formal degree ${\mathbf{\vartheta}}$ and that the vector fields $Z_1,\dots,Z_d$ in [\[eq:Z_fields\]](#eq:Z_fields){reference-type="eqref" reference="eq:Z_fields"} have formal degree $1$, in accordance with the terminology adopted in [@Bonfiglioli2009 Section 2.4]. Notice that the triplet $\left(\mathbb{R}\times\mathbb{R}^{2d},\circ,D_{\lambda}\right)$ forms a homogeneous group. Indeed, it is well defined the so-called homogeneous norm: $$\label{e7} \left\|{(t,x,v)}\right\|=|t|^{\frac{1}{{\mathbf{\vartheta}}}}+|x|^{\frac{1}{{\mathbf{\vartheta}}+1}} + |v|,%\qquad |x|_B=\sum_{j=1}^d |x_j|^{1/q_j},%{\blue \qquad (t,x)\in\R\times\R^d,}$$ and we consider the quasi-distance $$\label{eq:quasi_distance} d(z_1, z_2) := \left\|{z_2^{-1}\circ z_1}\right\|, \qquad z_1,z_2\in{\mathbb {R}}\times{\mathbb {R}}^{2d}.$$ The following properties directly follow from the definition of the quasi-distance $$\label{eq-dist-invariance} d(D_\lambda(z_1), D_\lambda(z_2)) = \lambda \, d(z_1, z_2), \qquad d(z \circ z_1, z \circ z_2) = d(z_1, z_2)$$ for every $z, z_1, z_2 \in \mathbb{R}\times\mathbb{R}^{2d}$ and for every $\lambda >0$. Note that $d$ is said *quasi-distance* as the following weaker form of triangular inequality holds for it: there exists a constant $\kappa \ge 1$ such that $$\label{eq:quasi_triangle} d(z_1, z_3) \le \kappa \left( d(z_1, z_2) + d(z_2, z_3) \right), \qquad z_1, z_2, z_3 \in {\mathbb {R}}\times{\mathbb {R}}^{2d}.$$ We recall that in [@imbert2018schauder] it is considered the *equivalent* distance $$\label{imbert2018distance} \min_{w\in\mathbb{R}^{d}} \big\{ \max\big( |t_1-t_2|^{\frac{1}{{\mathbf{\vartheta}}}} , |x_1-x_2- w (t_1-t_2)|^{\frac{1}{1+{\mathbf{\vartheta}}}} , |v_1-w| , | v_2 - w| \big) \big\} , \qquad z_1,z_2\in{\mathbb {R}}\times{\mathbb {R}}^{2d}.$$ We next introduce the notions of intrinsic regularity and intrinsic Hölder space. Let $X$ be a Lipschitz vector field on $\mathbb{R}\times\mathbb{R}^{2d}$. For any $z\in\mathbb{R}\times\mathbb{R}^{2d}$, we denote by ${\tau}\mapsto e^{{\tau}X}(z)$ the integral curve of $X$ defined as the unique solution to $$\begin{cases} \frac{d}{d{\tau}}e^{{\tau}X}(z)= X\left(e^{{\tau}X}(z)\right),\qquad &{\tau}\in{\mathbb {R}}, \\ e^{{\tau}X}(z)\vert_{{\tau}=0}= z. \end{cases}$$ In particular, for a vector $h \in {\mathbb {R}}^d$ we set $Z_h := h_1 Z_1 + \dots + h_d Z_d$ and we find we have $$\label{eq:def_curva_integrale_campo} e^{{\tau}%\partial_{v_i} Z_h }(t,x,v)=(t,x,v+{\tau}h),\qquad e^{{\tau}Y }(t,x,v)=(t+{\tau}, x + {\tau}v , v ), \qquad \tau\in{\mathbb {R}},$$ for any $(t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}$. Next we recall the general notion of Lie differentiability and Hölder regularity. **Definition 1**. Let $X$ be a Lipschitz vector field and $u$ be a real-valued function defined in a neighborhood of $z\in \mathbb{R}\times\mathbb{R}^{2d}$. We say that $u$ is *$X$-differentiable* in $z$ if the function ${\tau}\mapsto u\left(e^{{\tau}X}(z)\right)$ is differentiable in $0$. We will refer to the function $z \mapsto Xu (z) :=\frac{d}{d {\tau}} u\left(e^{{\tau}X}(z)\right)\big|_{{\tau}=0}$ as *$X$-Lie derivative of $u$*, or simply *Lie derivative of $u$* when the dependence on the field $X$ is clear from the context. **Definition 2**. Let $u:{\mathbb{R}\times\mathbb{R}^{2d}}\to {\mathbb {R}}$. Then, for $a\in]0,1]$, we say that $u\in C^{{\alpha}}_{Z_i}$, $i=1,\dots,d$, if $$%\label{eq:} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}}:=%\sup_{z\in\Rdd}|f(z)|+ \sup_{z\in\mathbb{R}\times\mathbb{R}^{2d}\atop {\tau}\in{\mathbb {R}}\setminus\{0\}} \frac{ \left|u\left(e^{{\tau}Z_i }(z)\right)- u(z)\right|}{|{\tau}|^{{\alpha}}} <\infty.$$ Moreover, for $\alpha\in]0,{\mathbf{\vartheta}}]$, we say that $u\in C^{{\alpha}}_{Y}$ if $$%\label{eq:} \left\|{u}\right\|_{C^{{\alpha}}_{Y}}:=%\sup_{z\in\Rdd}|f(z)|+ \sup_{z\in\mathbb{R}\times\mathbb{R}^{2d}\atop {\tau}\in{\mathbb {R}}\setminus\{0\}} \frac{ \left|u\left(e^{{\tau}Y }(z)\right)- u(z)\right|}{|{\tau}|^{\frac{{\alpha}}{{\mathbf{\vartheta}}}}} <\infty.$$ Let us introduce the set $$\label{index} \mathcal{I}= \{ \alpha \in {\mathbb {R}}\mid \alpha = k + j {\mathbf{\vartheta}}\, \text{ with } k,j \in \mathbb{N}_0 \}.$$ As we shall see in the sequel, $\alpha \in \mathcal{I}$ represents the regularity indices for which there is a *jump* in the regularity of a function $u$, meaning that new derivatives along $Z_i$ or/and $Y$ appear. In particular, the fllowing statements are true. - If $\alpha\in ]0,1 \wedge {\mathbf{\vartheta}}]$ there are no derivatives with respect to any of the vector fields $Z_1,\dots, Z_d$ or $Y$. - If $\alpha\in ]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}]$, there are two cases: if ${\mathbf{\vartheta}}<1$, then only derivatives along $Y$ appear, up to order $j$ with ${\mathbf{\vartheta}}j <\alpha$; if ${\mathbf{\vartheta}}>1$, then only derivatives along the vector fields $Z_1,\dots, Z_d$ appear, up to order $k$ with $k <\alpha$. Of course, if ${\mathbf{\vartheta}}=1$, the interval is empty and all the fields have the same formal degree (the gradation is the same as in the Heisenberg group). - If ${\alpha}> 1 \vee {\mathbf{\vartheta}}$, then there exist derivatives along $Z_1,\dots, Z_d$ and $Y$. - In Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} we show that, if ${\alpha}> 1 + {\mathbf{\vartheta}}$, then the derivatives $\partial_{x_j}$ also appear for $i=1,\dots,d$. Now we define the intrinsic Hölder spaces on the homogeneous group $\left(\mathbb{R}\times\mathbb{R}^{2d},\circ,D_{\lambda}\right)$, by extending the definitions of Hölder spaces given in [@PPP16]. Namely, our procedure is recursive with respect to the (ordered) indices in $\mathcal{I}$ . In particular, in the second step, which is for $\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}]$, the definition splits in two different cases depending on whether ${\mathbf{\vartheta}}<1$ or ${\mathbf{\vartheta}}>1$. If ${\mathbf{\vartheta}}=1$, $\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}]$ is empty and one moves on to the third step. **Definition 3**. Let $u:{{\mathbb {R}}\times}{\mathbb {R}}^{2d}\to {\mathbb {R}}$ and ${\alpha}>0$. Then: - if $\alpha\in]0,1 \wedge {\mathbf{\vartheta}}]$, $u\in C^{{\alpha}}$ if the semi-norm $$\label{e9} \left\|{u}\right\|_{C^{{\alpha}}}:=\left\|{u}\right\|_{C^{{\alpha}}_{Y}}+\sum_{i=1}^{d} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}}% < \infty.$$ is finite; - if $\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}]$, $u\in C^{{\alpha}}$ if the semi-norm $$\label{e10} \left\|{u}\right\|_{C^{{\alpha}}}:= \begin{cases} \left\|{Yu}\right\|_{C^{{\alpha}-{\mathbf{\vartheta}}}}+\sum_{i=1}^{d} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}}, \qquad \text {if }{\mathbf{\vartheta}}<1 \\ \left\|{u}\right\|_{C^{{\alpha}}_{Y}}+\sum_{i=1}^{d} \left\|{Z_i u}\right\|_{C^{{\alpha}-1}}, \qquad \text {if }{\mathbf{\vartheta}}>1 \end{cases}$$ is finite. If ${\mathbf{\vartheta}}=1$, $\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}]$ is empty and this case can be skipped; - if $\alpha>1 \vee {\mathbf{\vartheta}}$, $u\in C^{{\alpha}}$ if the semi-norm $$\label{e11} \left\|{u}\right\|_{C^{{\alpha}}}:=\left\|{Y u}\right\|_{C^{{\alpha}-{\mathbf{\vartheta}}}}+\sum_{i=1}^{d} \left\|{Z_i u}\right\|_{C^{{\alpha}-1}}$$ is finite. **Example 4**. Figure 1 below describes the pairs $(k, j {\mathbf{\vartheta}})$ with $k,j \in {\mathbb {N}}_0$ and ${\mathbf{\vartheta}}=1/3$. The filled dots form the set $\mathcal{I}$, the points ${\mathbf{\vartheta}}$ and $1 + {\mathbf{\vartheta}}$ are highlighted. Let us consider $u\in C^{\alpha}$. Then we have: - If $\alpha \in ]0,1/3]$, there are no derivatives with respect to either $Z_i$ or $Y$. - If $\alpha\in ]1/3, 2/3]$, the Lie derivative $Yu$ exists and belongs to $C^{\alpha-1/3}$. Furthermore, $u$ belongs to $C^{\alpha}_{Z_i}$. Note that the definition is well-posed. Indeed, the index $\alpha-1/3 \in]0,1/3]$ and thus the space $C^{\alpha-1/3}$ has already been defined; also $\alpha \le 1$ and thus $C^{\alpha}_{Z_i}$ is defined too. - If $\alpha\in ]2/3, 1]$, $Yu$ belongs to $C^{\alpha-1/3}$ where $\alpha - 1/3 \in ]1/3 , 2/3]$. In particular there exists $Y^2 u \in C^{\alpha-2/3}$. - If $\alpha\in ]1 , 4/3]$, there exist the derivatives along the fields $Z_i$, which are the Euclidean derivatives $\partial_{v_1}u, \cdots, \partial_{v_d}u$. Furthermore, such derivatives belong to $C^{\alpha-1}$ where $\alpha -1 \in ]0, 1/3]$. Also, the Lie derivatives $Yu, Y^2u$ and $Y^3u$ exist and belong to $C^{\alpha-1/3}, C^{\alpha-2/3}$ and $C^{\alpha-1}$, respectively. - If $\alpha\in ]4/3, 5/3]$, then $Y^4 u \in C^{\alpha-4/3}$, and $\partial_{v_i}u \in C^{\alpha-1}$. Moreover, the mixed derivatives $\partial_{v_i} Y u$ and $Y \partial_{v_i}u$ exist and belong to $C^{\alpha-4/3}$, and so do the commutators $[\partial_{v_i}, Y] u$. Indeed we have $Y u \in C^{\alpha-1/3}$ where $\alpha-1/3 \in ]1, 4/3]$, and $\partial_{v_i}u \in C^{\alpha-1}$ where $\alpha-1 \in ]1/3, 2/3]$. Once more, the last step is crucial as it can be proved that $[\partial_{v_i}, Y] u = \partial_{x_i} u$ (see Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} below). By induction, there exist the $n$-th order Euclidean derivatives with respect to $x$ so long as $\alpha > n ({\mathbf{\vartheta}}+1)$. **Example 5**. Figure 2 below describes the pairs $(k, j {\mathbf{\vartheta}})$ with $k,j \in {\mathbb {N}}_0$ and ${\mathbf{\vartheta}}=4/3$. The filled dots form the set $\mathcal{I}$. Let us consider $u\in C^{\alpha}$. Then we have: - If $\alpha \in ]0,1]$, there are no derivatives with respect to either $Z_i$ or $Y$. - If $\alpha\in ]1, 4/3]$, the $1$st order derivatives $Z_i u = \partial_{v_i} u$ exist and belong to $C^{\alpha-1}$. Plus, $u\in C^{\alpha}_Y$. - If $\alpha\in ]4/3 , 2 ]$, the Lie derivative $Yu$ exists and belongs to $C^{\alpha-4/3}$. Moreover, $\partial_{v_1}u, \cdots, \partial_{v_d}u\in C^{\alpha-1}$. - If $\alpha\in ]2, 7/3]$, the $2$nd order derivatives $\partial_{v_i v_j}u$, for $i,j=1,\cdots,d$, exist and belong to $C^{\alpha-2}$. Furthermore, $\partial_{v_i}u \in C^{\alpha-1}_Y$ and $Yu \in C^{\alpha-4/3}$. - If $\alpha\in ]7/3, 8/3]$, then the mixed derivatives $\partial_{v_i} Y u$ and $Y \partial_{v_i}u$ exist and belong to $C^{\alpha-7/3}$, and so do the commutators $[\partial_{v_i}, Y] u$. Also, $Y u \in C^{\alpha-4/3}$ and $\partial_{v_i v_j}u \in C^{\alpha-2}$. In the sequel, $\beta=(\beta_1,\cdots, \beta_d)\in \mathbb{N}^d_{0}$ will denote a multi-index. As usual $$|\beta|:=\sum_{j=1}^d \beta_j\quad \text{ and }\quad \beta!:=\prod_{j=1}^d \left(\beta_j !\right)$$ are called the length and the factorial of ${\beta}$ respectively. Moreover, for any $x\in\mathbb{R}^{d}$, we set $$x^{\beta}=x_1^{\beta_1}\cdots x_d^{\beta_d}\quad \text{ and }\quad \partial^{\beta}=\partial^{\beta}_x=\partial_{x_1}^{\beta_1}\cdots \partial_{x_d}^{\beta_d}.$$ Finally, for $u\in C^{{\alpha}}$ we let $T_{\alpha} u(z_0,\cdot)$ its intrinsic Taylor polynomial around $z_0=(t_0,x_0,v_0)$ defined as $$\label{eq:TAy_pol} T_{\alpha} u(z_0;z):=\!\! \sum_{{0\leq {\mathbf{\vartheta}}k + (1+{\mathbf{\vartheta}})|\gamma| + |\beta| < \alpha}} \!\! \frac{ Y^k {\partial_{v}^{\beta} \partial_{x}^{\gamma} }u(z_0) }{k!\, \gamma! \,\beta! } (t-t_0)^k \big(x-x_0-(t-t_0)v_0 \big)^{\gamma} (v-v_0)^{\beta}.$$ **Example 6**. For $d=1$ and ${\mathbf{\vartheta}}=4/3$ as in Example [Example 5](#ex:kappa_3_2){reference-type="ref" reference="ex:kappa_3_2"}, we have - If $\alpha \in ]0,1]$, then we set $\alpha_1 :=1$ and we have $T_{\alpha} u(z_0;z) = T_{\alpha_1} u(z_0;z) = u(z_0)$; - If $\alpha\in ]1, 4/3]$ ($\alpha_2 :=4/3$), then $T_{\alpha} u(z_0;z) = T_{\alpha_2} u(z_0;z) = T_{\alpha_1} u(z_0;z) + \big(\partial_v u(z_0)\big) (v-v_0)$; - If $\alpha\in ]4/3 , 2 ]$ ($\alpha_3 :=2$), then $T_{\alpha} u(z_0;z) = T_{\alpha_3} u(z_0;z) = T_{\alpha_2} u(z_0;z) + \big(Y u(z_0)\big) (t-t_0)$; - If $\alpha\in ]2, 7/3 ]$ ($\alpha_4 :=7/3$), then $T_{\alpha} u(z_0;z) = T_{\alpha_4} u(z_0;z) = T_{\alpha_3} u(z_0;z) + \frac{1}{2} \big(\partial^2_v u(z_0)\big) (v-v_0)^2$; - If $\alpha\in ]7/3, 8/3 ]$, then $T_{\alpha} u(z_0;z) %= T_{\alpha_5} u(z_0;z) = T_{\alpha_4} u(z_0;z) + \big(\partial_x u(z_0)\big) (x-x_0 - (t-t_0) v_0) + \big(Y \partial_v u(z_0)\big) (t-t_0)(v-v_0)$. **Theorem 7**. *For any $\alpha >0$ and for any $u\in C^{{\alpha}}$ the following statements are true:* Hölder spaces characterization. : *For any $k\in{\mathbb {N}}_0$ and $\gamma,\beta\in{\mathbb {N}}_0^d$ with ${0\leq {\mathbf{\vartheta}}k + (1+{\mathbf{\vartheta}})|\gamma| + |\beta| <\alpha}$, the derivatives $\ Y^k \partial_v^{\beta} \partial_x^{\gamma} u$ exist and $$\label{eq:maintheorem_part1_global} %\begin{split} % & \ Y^k \partial_x^{\gamma} \partial_v^{\beta}u \in C^{\alpha-\ddY k - (1+\ddY)|\gamma| - |\beta|} % \qquad \text{and} \\ %& \| Y^k \partial_v^{\beta} \partial_x^{\gamma}u \|_{C^{\alpha-{\mathbf{\vartheta}}k - (1+{\mathbf{\vartheta}})|\gamma| - |\beta|}} \leq \| u \|_{C^{\alpha}} . %\end{split}$$* Taylor formula. : *There exists a positive constant $c>0$, only dependent on $\alpha$, such that $$\label{eq:estim_tay_n} \left|u(z)-T_{\alpha} u(z_0;z)\right|\le c \|u\|_{C^{{\alpha}}} \|z_0^{-1}\circ z\|^{{\alpha}},\qquad z,z_0\in \mathbb{R}\times\mathbb{R}^{2d}.$$* **Remark 8**. If $\alpha\in]0, 1 \wedge {\mathbf{\vartheta}}]$, estimate [\[eq:estim_tay_n\]](#eq:estim_tay_n){reference-type="eqref" reference="eq:estim_tay_n"} restores the definition of Hölder continuous function given in Definition 1.2 in [@Polidoro94] $$\left|u(z)-u(z_0)\right|\le c \|u\|_{C^{{\alpha}}} \left\|{z_0^{-1}\circ z}\right\|^{{\alpha}},\qquad z,z_0\in\mathbb{R}\times\mathbb{R}^{2d}.$$ For a comparison between intrinsic and Euclidean Hölder continuity we refer to Proposition 2.1 in [@Polidoro94]. **Remark 9**. By [\[eq:maintheorem_part1_global\]](#eq:maintheorem_part1_global){reference-type="eqref" reference="eq:maintheorem_part1_global"}-[\[eq:estim_tay_n\]](#eq:estim_tay_n){reference-type="eqref" reference="eq:estim_tay_n"}, it is straightforward to see that $\partial_v^{\beta} \partial_x^{\gamma}u$ and $\partial_x^{\gamma} \partial_v^{\beta}u$ are both continuous on ${\mathbb {R}}\times{\mathbb {R}}^{2d}$ for any $\gamma,\beta\in{\mathbb {N}}_0^d$ with ${0\leq (1+{\mathbf{\vartheta}})|\gamma| + |\beta| <\alpha}$, hence they agree. By contrast, $Y^k$ does not commute with $\partial_v^{\beta}u$. # Preliminaries The method of the proof relies on the construction of a finite sequence of integral curves of the vector fields $Y, Z_1, \dots, Z_d$ which steer a point $z_0 = (t_0, x_0, v_0)$ to any other point $z = (t, x, v)$. We then rely on the usual Taylor expansion of the functions $\tau \mapsto u\left( e^{\tau Y}(z_0) \right)$ and $\tau \mapsto u\left( e^{\tau Z_h}(z_0) \right)$ introduced in [\[eq:def_curva_integrale_campo\]](#eq:def_curva_integrale_campo){reference-type="eqref" reference="eq:def_curva_integrale_campo"}, to obtain a good approximation of $u$ near $z_0$. The approximation of $u$ along the integral curves of the commutators $[Y, Z_i]$ is obtained by using a rather classical technique from control theory. We first consider the approximation along the integral curve $e^{\tau Y}(z)$. **Remark 10**. Let $n\in{\mathbb {N}}_0$, $\gamma \in ] 0, {\mathbf{\vartheta}}]$ and $u\in C^{{\mathbf{\vartheta}}n+\gamma}$. Then, by Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}, we have $Y^{j} u\in C^{{\mathbf{\vartheta}}(n-j)+\gamma}_{Y}$ with $j=1,\dots,n$. Therefore, by mean-value theorem along the vector field $Y$, for any $z=(t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}$ and $\tau \in{\mathbb {R}}$, there exists $\delta \in ]0,1[$ such that $$\label{eq:mean_value_Yn_loc} u\big(e^{\tau Y}(z)\big)-u(z)-\sum_{j=1}^{n} \frac{\tau^j}{j!}Y^j u(z)= \frac{\tau^{n}}{n!}\left( Y^{n} u\big(e^{{\delta \tau}Y}(z)\big)- Y^{n}u(z)\right),$$ and thus, Definition [Definition 2](#def:intrinsic_alpha_Holder){reference-type="ref" reference="def:intrinsic_alpha_Holder"} yields $$\begin{aligned} \label{eq:mean_value_Yn_B} \big| u \big( e^{\tau Y}(z) \big) - T_{{\mathbf{\vartheta}}n+\gamma} u\big(z;e^{\tau Y}(z)\big)\big| & = \Big| u\big(e^{\tau Y}(z)\big)-u(z)-\sum_{j=1}^{n} \frac{\tau^j}{j!}Y^iu(z) \Big| \\ & \leq \frac{1}{n!} %c_B \|u\|_{C^{{\mathbf{\vartheta}}n+\gamma}}|\tau|^{n+\frac{\gamma}{{\mathbf{\vartheta}}}},\qquad \tau\in{\mathbb {R}},\quad z\in {\mathbb {R}}\times{\mathbb {R}}^{2d}.\end{aligned}$$ Since the vector fields $Z_1, \dots, Z_d$ have constant coefficients, the usual Euclidean Taylor theorem with Lagrange reminder plainly gives the following result. **Remark 11**. Let $n\in{\mathbb {N}}_0$, $\gamma \in ]0,1]$ and $u\in C^{n+\gamma}$. Then, by Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}, we have $\partial_{v}^{\beta} u\in C^{\gamma + n - |\beta|}_{Z_i}$ for any $\beta\in{\mathbb {N}}_0^{d}$ with $|\beta|\leq n$ and $i=1,\cdots,d$. Therefore, recalling Definition [Definition 2](#def:intrinsic_alpha_Holder){reference-type="ref" reference="def:intrinsic_alpha_Holder"}, mean-value theorem yields $$\label{eq:mean_value_Zn_B} \big| u ( t, x, v + h ) - T_{n+\gamma} u\big(z; t, x, v + h \big)\big| \leq \frac{1}{n!} \|u\|_{C^{n+\gamma}}|h|^{n+ \gamma},\qquad z = (t,x,v)\in {\mathbb {R}}\times{\mathbb {R}}^{2d}, \quad h\in{\mathbb {R}}^d.$$ In view of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"}, we consider two points $z_0, z \in {\mathbb {R}}^{1+2d}$ and we note that Remarks [Remark 10](#rem:eq:mean_value_Yn){reference-type="ref" reference="rem:eq:mean_value_Yn"} and [Remark 11](#rem:eq:mean_value_Zn){reference-type="ref" reference="rem:eq:mean_value_Zn"} provide us with a bound of $|u(z_0) - u(z_1)|$ in terms of $\| z_0^{-1} \circ z \|$, where $z_1$ is a specific point of ${\mathbb {R}}^{1+2d}$ whose components $t$ and $v$ agree with the components of $z$. The following picture illustrates the situation in the case $d=1$ and $z = (0,0,0)$. We finally recover the regularity of the function $u$ with respect to the variable $x$ by the usual trajectory defined as concatenations of integral curves of $Z_1, \dots, Z_m$ and $Y$, as Figure 4 below shows. The following Proposition provides us with a quantitative estimate of the increment of a function $u\in C^{\alpha}$ with respect to the variable $x$. **Proposition 12**. *Let $u\in C^{\alpha}$ with $\alpha\in ]0, 1 + {\mathbf{\vartheta}}]$, then $$\label{eq:claim} |u(t,x,v) - u(t,x+h,v)| \leq c \, \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}},\qquad z=(t,x,v)\in \mathbb{R}\times\mathbb{R}^{2d}, \quad h\in\mathbb{R}^{d}.$$* *Proof.* Assume $h\neq 0$ and set $w:=h/|h|$, $\tau:= |h|^{\frac{1}{{\mathbf{\vartheta}}+1}}$. Recall the notation in [\[eq:def_curva_integrale_campo\]](#eq:def_curva_integrale_campo){reference-type="eqref" reference="eq:def_curva_integrale_campo"}, put $$\label{eq:gamma_k1_bis} z_2 = e^{\tau Z_w}(t,x,v), \qquad z_3 = e^{\tau^{{\mathbf{\vartheta}}}Y}(z_2) \qquad z_4 = e^{- \tau Z_w}(z_3),$$ and note that $$\label{eq:gamma_k1_ter} e^{-\tau^{{\mathbf{\vartheta}}}Y}(z_4) = \big(t, x + \tau^{{\mathbf{\vartheta}}+ 1} w , v \big) = \big(t, x + h , v \big).$$ With this notation, we express the left hand side of [\[eq:claim\]](#eq:claim){reference-type="eqref" reference="eq:claim"} as follows $$\label{eq-boxes} \begin{split} u(t,x+ h ,v) - u (t,x,v) &=\boxed{u(t, x+h,v) - u (z_4) - \sum_{{\mathbf{\vartheta}}\le {\mathbf{\vartheta}}j < \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} Y^j u(z_4)}_{(1)}\\ &+\boxed{u(z_4) - u (z_3) - \sum_{1 \le k < \alpha} \frac{(-\tau )^{k} Z_w^k u(z_3)}{k!}}_{(2)} \\ &+\boxed{u(z_3)-u(z_2) + \sum_{{\mathbf{\vartheta}}\le {\mathbf{\vartheta}}j < \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} Y^j u(z_3)}_{(3)} \\ &+\boxed{u (z_2) - u(t, x,v) + \sum_{1 \le k < \alpha} \frac{(-\tau )^{k}}{k!}Z_w^k u(z_2)}_{(4)} \\ &+\boxed{\sum_{{\mathbf{\vartheta}}\le {\mathbf{\vartheta}}j < \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} \left(Y^j u(z_4) - Y^j u(z_3) \right)}_{(5)}\\ &+\boxed{\sum_{1 \le k < \alpha} \frac{(-\tau )^{k}}{k!} \left(Z_w^k u(z_2) - Z_w^k u(z_3) \right)}_{(6)} =: I_1 + I_2 + I_3 + I_4 + I_5 + I_6, \end{split}$$ where the indices $j,k$ in the above summations are non-negative integers. Note that, if $\alpha \le {\mathbf{\vartheta}}$, then no derivatives $Y^j u$ apperar in $I_1, I_3$ and the term $I_5$ doesn't appear. Thus, in the following we consider separately this case and ${\mathbf{\vartheta}}< \alpha \le 1 + {\mathbf{\vartheta}}$. Analogously, if $\alpha \le 1$, no derivatives of the form $Z_w^k u$ apperar in $I_2, I_4$. We claim that, in every case, the terms $I_1, \dots, I_6$ are bounded by $\|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}$, up to multiplying by a positive constant. Consider first $I_1, I_3$ and $I_5$ with ${\mathbf{\vartheta}}< \alpha \le 1 + {\mathbf{\vartheta}}$. We apply Remark [Remark 10](#rem:eq:mean_value_Yn){reference-type="ref" reference="rem:eq:mean_value_Yn"} to $I_1$ and $I_3$, and we find $$\label{eq:estimate_I13} |I_1|, |I_3| \le \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}.$$ Moreover, $\alpha \le 1 + {\mathbf{\vartheta}}$ also yields $Y^j u \in C_{Z_i}^{\alpha - j {\mathbf{\vartheta}}}$, with $0 < \alpha - j {\mathbf{\vartheta}}\le 1$ for every $j$ such that ${\mathbf{\vartheta}}\le {\mathbf{\vartheta}}j < \alpha$. Then $$\label{eq:mean_value_Yn_prop_2} |\tau|^{j {\mathbf{\vartheta}}} \, \left| Y^j u(z_4) - Y^j u(z_3) \right| \le \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}, \qquad {\mathbf{\vartheta}}\le {\mathbf{\vartheta}}j < \alpha,$$ because of the very definition of $C^\alpha$ space. As a consequence, $$\label{eq:estimate_I5} |I_5| \le e \, \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}.$$ If $0 < \alpha \le {\mathbf{\vartheta}}$ instead, we set $I_5 = 0$, so that [\[eq:estimate_I5\]](#eq:estimate_I5){reference-type="eqref" reference="eq:estimate_I5"} trivially holds. Moreover, [\[eq:estimate_I13\]](#eq:estimate_I13){reference-type="eqref" reference="eq:estimate_I13"} follows again from the definiton of $C^\alpha$. In both cases [\[eq:estimate_I13\]](#eq:estimate_I13){reference-type="eqref" reference="eq:estimate_I13"} and [\[eq:estimate_I5\]](#eq:estimate_I5){reference-type="eqref" reference="eq:estimate_I5"} hold. The argument for $I_2, I_4$ and $I_6$ is analogous. If $1 < \alpha \le 1 + {\mathbf{\vartheta}}$, then Remark [Remark 11](#rem:eq:mean_value_Zn){reference-type="ref" reference="rem:eq:mean_value_Zn"} yields $$\label{eq:estimate_I24} |I_2|, |I_4| \le \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}},$$ and $\alpha \le {\mathbf{\vartheta}}+ 1$ implies $Z_w^k u \in C_{Y}^{\alpha - k}$, with $0 < \alpha - k \le {\mathbf{\vartheta}}$ for every $k \in {\mathbb {N}}$ such that $k < \alpha$. Then $$\label{eq:mean_value_Zn_prop_2} |\tau|^{k} \, \left| Z^k_w u(z_2) - Z^k_w u(z_3) \right| \le \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}, \qquad 1 \le k < \alpha,$$ and this inequality yields $$\label{eq:estimate_I6} |I_6| \le e \, \|u\|_{C^{{\alpha}}} | h |^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}.$$ If $0 < \alpha \le 1$ we set $I_6 = 0$ and the bound [\[eq:estimate_I24\]](#eq:estimate_I24){reference-type="eqref" reference="eq:estimate_I24"} holds by the definition of $C^\alpha$. The conclusion of the proof of [\[eq:claim\]](#eq:claim){reference-type="eqref" reference="eq:claim"} is then a direct consequence of [\[eq:estimate_I13\]](#eq:estimate_I13){reference-type="eqref" reference="eq:estimate_I13"}-[\[eq:estimate_I5\]](#eq:estimate_I5){reference-type="eqref" reference="eq:estimate_I5"} and [\[eq:estimate_I24\]](#eq:estimate_I24){reference-type="eqref" reference="eq:estimate_I24"}-[\[eq:estimate_I6\]](#eq:estimate_I6){reference-type="eqref" reference="eq:estimate_I6"}. ◻ # Proofs The proof of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} is divided into two steps: - (preliminary result) assuming true the first statement of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} for some $\alpha> 0$, we prove that also the second statement is true for the same $\alpha$; - we prove the first statement of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} for a general $\alpha> 0$, by means of a suitable induction procedure. ## The preliminary result We assume the first statement of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} to be true for fixed $\alpha>0$ and prove the second one. [Step I:]{.ul} We fix $$%\label{eq:} z_1 = (t_1, x_1, v_1)= e^{( t -t_0)Y}(z_0)= (t, x_0 +(t-t_0) v_0, v_0)$$ and we prove that $$\label{eq:estim_tay_n_space} \left| u(z)-T_{\alpha} u(z_1,z)\right|\le c \|u\|_{C^{{\alpha}}} \big(|x-x_1|^{\frac{1}{{\mathbf{\vartheta}}+1}} + |v-v_1|\big)^{{\alpha}} = c \|u\|_{C^{{\alpha}}} \, \|z_1^{-1}\circ z \|^{ {\alpha}}.$$ Set $z_2 := (t_1, x_1, v)$. We rearrange the first term as $$\begin{aligned} %\label{eq:} u(z)-T_{\alpha} u(z_1,z) &= \boxed{ u(z) - T_{\alpha} u(z_2,z)}_{(1)} + \boxed{ T_{\alpha} u(z_2,z) -T_{\alpha} u(z_1,z)}_{(2)} =: I_1 + I_2.\end{aligned}$$ We have $$%\label{eq:} I_1 = u(z) - \sum_{(1+ {\mathbf{\vartheta}}) |\gamma| < \alpha} \frac{1}{\gamma!} \partial^{\gamma}_x u(z_2) (x - x_1)^{\gamma}.$$ By assumption, for any multi-index $\gamma$ in the summation above with highest order, namely $|\gamma | = \lceil \alpha/(1+ {\mathbf{\vartheta}})\rceil - 1$, we have $\partial_x^{ \gamma} u \in C^{\alpha-(1+{\mathbf{\vartheta}})|\gamma|}$. Therefore, Proposition [Proposition 12](#prop:holder_x){reference-type="ref" reference="prop:holder_x"} together with Euclidean Taylor formula yield $$%\label{eq:} |I_1| \leq c\, \|u\|_{C^{{\alpha}}} \, |x - x_1|^{\frac{\alpha}{1+ {\mathbf{\vartheta}}}} \leq c\, \|u\|_{C^{{\alpha}}} \, \|z_1^{-1}\circ z \|^{ {\alpha}}.$$ For the second term, we have $$%\label{eq:} I_2 = \sum_{(1+ {\mathbf{\vartheta}}) |\gamma| < \alpha} \frac{1}{ \gamma!}(x - x_1)^{\gamma}\big( \partial_x^{\gamma} u(z_2) - T_{\alpha-(1+ {\mathbf{\vartheta}}) |\gamma|} (\partial_x^{\gamma} u)(z_1,z_2) \big).$$ By assumption $\partial_x^{\gamma} u \in C^{\alpha-(1+ {\mathbf{\vartheta}}) |\gamma|}$, and thus, by applying Remark [Remark 11](#rem:eq:mean_value_Zn){reference-type="ref" reference="rem:eq:mean_value_Zn"} with $h=v-v_0$ to each term in the summation we obtain $$%\label{eq:} |I_2| \leq c\, \|u\|_{C^{{\alpha}}} \!\!\! \sum_{(1+ {\mathbf{\vartheta}}) |\gamma| < \alpha} |x - x_1|^{\gamma} |v - v_0|^{\alpha-(1+ {\mathbf{\vartheta}}) |\gamma|} \leq c\, \|u\|_{C^{{\alpha}}} \, \|z_1^{-1}\circ z \|^{ {\alpha}}.$$ [Step II:]{.ul} we conclude the proof of [\[eq:estim_tay_n\]](#eq:estim_tay_n){reference-type="eqref" reference="eq:estim_tay_n"}. Note that $$%\label{eq:} z_1^{-1}\circ z= (0, x-x_0-(t- t_0)v_0,v- v_0), \quad z_0^{-1}\circ z= (t-t_0,x-x_0-(t- t_0)v_0,v- v_0),$$ so that $\|z_1^{-1}\circ z\| \leq \|z_0^{-1}\circ z\|$. Therefore, by [\[eq:estim_tay_n\_space\]](#eq:estim_tay_n_space){reference-type="eqref" reference="eq:estim_tay_n_space"}, we have $$%\label{eq:} |u(z)-T_{\alpha}(z_0, z)| = |u(z)-T_{\alpha}(z_1, z)| + |T_{\alpha}(z_1, z)-T_{\alpha}(z_0, z)| \le c\, \|u\|_{C^{\alpha}} \|z_0^{-1}\circ z\|^{\alpha}+ |T_{\alpha}(z_1, z)-T_{\alpha}(z_0, z)|.$$ Rearranging the Taylor polynomials we can write $$\begin{aligned} %\label{eq:} T_{\alpha}(z_1, z)-T_{\alpha}(z_0, z)= \sum_{{(1+{\mathbf{\vartheta}})|\gamma| + |\beta| < \alpha}} \!\! \frac 1 {\gamma! \,\beta! }& \Big( \partial_{x}^{\gamma}\partial_{v}^{\beta}u(z_1) -T_{{\alpha}-(1+{\mathbf{\vartheta}})|\gamma| - |\beta| } ( \partial_{x}^{\gamma}\partial_{v}^{\beta} u)(z_0,z_1)\Big)\\ &\times \big(x-x_0-(t-t_0)v_0 \big)^{\gamma} (v-v_0)^{\beta}.\end{aligned}$$ By Remarks [Remark 9](#rem-dx-dv){reference-type="ref" reference="rem-dx-dv"} and [Remark 10](#rem:eq:mean_value_Yn){reference-type="ref" reference="rem:eq:mean_value_Yn"} we obtain $$%\label{eq:} |T_{\alpha}(z_1, z)-T_{\alpha}(z_0, z)|\le c\, \|u\|_{C^{\alpha}} \|z_0^{-1}\circ z\|^{\alpha}.$$ This concludes the proof of the second statement of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"}, assuming that the first one holds true. ## Proof of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} {#proof-of-theorem-thmain} If $\alpha\leq 1+ {\mathbf{\vartheta}}$, then [\[eq:maintheorem_part1_global\]](#eq:maintheorem_part1_global){reference-type="eqref" reference="eq:maintheorem_part1_global"} does not contain derivatives with respect to $x$, thus it stems from Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}. We prove [\[eq:maintheorem_part1_global\]](#eq:maintheorem_part1_global){reference-type="eqref" reference="eq:maintheorem_part1_global"} for $\alpha > 1+ {\mathbf{\vartheta}}$. Notice that it is enough to show that, for any $i=1,\dots, d$, we have $$\label{eq:comm_equal_deriv_x} \partial_{x_i} u = [Z_i,Y] u. % \qquad i=1,\cdots, d.$$ Indeed, with [\[eq:comm_equal_deriv_x\]](#eq:comm_equal_deriv_x){reference-type="eqref" reference="eq:comm_equal_deriv_x"} at hand, $$\label{eq:der_x} \partial_{x_i} u \in C^{\alpha- ({\mathbf{\vartheta}}+ 1)}%, \qquad i=1,\cdots, d,$$ and the whole [\[eq:maintheorem_part1_global\]](#eq:maintheorem_part1_global){reference-type="eqref" reference="eq:maintheorem_part1_global"} follows, once more, by Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}, and Remark [Remark 9](#rem-dx-dv){reference-type="ref" reference="rem-dx-dv"}, combined with a plain induction argument. To show [\[eq:comm_equal_deriv_x\]](#eq:comm_equal_deriv_x){reference-type="eqref" reference="eq:comm_equal_deriv_x"}, we consider any $\alpha > 1+ {\mathbf{\vartheta}}$, we let $n:= \lceil \tfrac{\alpha}{{\mathbf{\vartheta}}+ 1} \rceil - 1$, so that $\alpha \in ] n(1+ {\mathbf{\vartheta}}), (n+1)(1+ {\mathbf{\vartheta}})]$, and we assume [\[eq:maintheorem_part1_global\]](#eq:maintheorem_part1_global){reference-type="eqref" reference="eq:maintheorem_part1_global"} true for $\bar \alpha := \alpha -({\mathbf{\vartheta}}+1)$. We claim that $$\label{eq:claim_step_2} \frac{u(t,x+{\delta}\, e_i ,v) - u (t,x,v)}{{\delta}} \to [Z_i,Y] u (t,x,v) \quad \text{as} \quad {\delta}\to 0, \qquad (t,x,v) \in\mathbb{R}\times\mathbb{R}^{2d}.$$ Here $e_i$ denotes the $i$-th element of the canonical basis of $\mathbb{R}^{d}$. To prove [\[eq:claim_step_2\]](#eq:claim_step_2){reference-type="eqref" reference="eq:claim_step_2"} we rely on an arugment similar to that used in the proof of Proposition [Proposition 12](#prop:holder_x){reference-type="ref" reference="prop:holder_x"}. We fix $(t,x,v) \in\mathbb{R}\times\mathbb{R}^{2d}, i \in \{1,\dots, d\}, {\delta}\ne 0$, and we recall the notation in [\[eq:def_curva_integrale_campo\]](#eq:def_curva_integrale_campo){reference-type="eqref" reference="eq:def_curva_integrale_campo"}. If $\delta >0$ we set $w := e_i$, otherwise we set $w := - e_i$. For simplicity we only consider the case $\delta>0$, the case $\delta<0$ being completely analogous. Set $\tau: = {\delta}^\frac{1}{1 + {\mathbf{\vartheta}}}$, $$\label{eq:gamma_k1_step_2} z_2 := {e^{\tau Z_w}(t,x,v) = e^{\tau Z_i}(t,x,v)}, \qquad z_3 := e^{\tau^{{\mathbf{\vartheta}}}Y}(z_2) \qquad {z_4 := e^{- \tau Z_w}(z_3) = e^{- \tau Z_i}(z_3),}$$ and note that we have $$\label{eq:gamma_k1_step_2+} e^{-\tau^{{\mathbf{\vartheta}}}Y}(z_4) = \big(t, x + \tau^{{\mathbf{\vartheta}}+ 1} e_i , v \big) = \big(t, x + {\delta}\, e_i , v \big).$$ We proceed as in the proof of Proposition [Proposition 12](#prop:holder_x){reference-type="ref" reference="prop:holder_x"}, the unique difference being that $\alpha > {\mathbf{\vartheta}}+ 1$ here. We have $$\label{eq-boxes-2} \begin{split} u(t,x+{\delta}\, e_i ,v) - u (t,x,v) %- [Z_i,Y] u (t,x,v) &=\boxed{u(t, x+{\delta}\, e_i,v) - u (z_4) - \sum_{{\mathbf{\vartheta}}\le j {\mathbf{\vartheta}}< \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} Y^j u(z_4)}_{(1)}\\ &+\boxed{u(z_4) - u (z_3) - \sum_{1 \le k < \alpha} \frac{(-\tau )^{k} Z_i^k u(z_3)}{k!}}_{(2)} \\ &+\boxed{u(z_3)-u(z_2) + \sum_{{\mathbf{\vartheta}}\le j {\mathbf{\vartheta}}< \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} Y^j u(z_3)}_{(3)} \\ &+\boxed{u (z_2) - u(t, x,v) + \sum_{1 \le k < \alpha} \frac{(-\tau )^{k}}{k!}Z_i^k u(z_2)}_{(4)} \\ &+ \boxed{\sum_{{\mathbf{\vartheta}}\le j {\mathbf{\vartheta}}< \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} \left(Y^j u(z_4) - Y^j u(z_3) \right)}_{(5)}\\ &+\boxed{\sum_{1 \le k < \alpha} \frac{(-\tau )^{k}}{k!} \left(Z_i^k u(z_3) - Z_i^k u(z_2) \right)}_{(6)} =: I_1 + I_2 + I_3 + I_4 + I_5 + I_6. \end{split}$$ Note that all the sums in the above boxes are not void, since $\alpha > {\mathbf{\vartheta}}+ 1$. By Remark [Remark 10](#rem:eq:mean_value_Yn){reference-type="ref" reference="rem:eq:mean_value_Yn"} and [Remark 11](#rem:eq:mean_value_Zn){reference-type="ref" reference="rem:eq:mean_value_Zn"} we find that $$\label{eq-boxes-1-4} |I_1| + |I_2| + |I_3| + |I_4| \le c_ \alpha \|u\|_{C^{{\alpha}}} | {\delta}|^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}},$$ for some positive constant $c_\alpha$. Concerning $I_5$ and $I_6$, we have $$\label{eq-boxes-5} \begin{split} I_5 + I_6 &=\boxed{\sum_{{\mathbf{\vartheta}}\le j {\mathbf{\vartheta}}< \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} \bigg(Y^j u(z_4) - Y^j u(z_3) - \sum_{0 < k < \alpha - j {\mathbf{\vartheta}}} \frac{(-\tau )^{k}}{k!}Z_i^k Y^j u(z_3)\bigg)}_{(5.1)}\\ &+\boxed{\sum_{1 \le k < \alpha} \frac{(-\tau )^{k}}{k!} \bigg(Z_i^k u(z_3) - Z_i^k u(z_2) + \sum_{0 < j {\mathbf{\vartheta}}< \alpha - k} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} Y^j Z_i^k u(z_3)\bigg)}_{(6.1)}\\ &+\boxed{\sum_{k >0,j>0 \atop j {\mathbf{\vartheta}}+ k < \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} \frac{(-\tau )^{k}}{k!} \big( v_{jk}(z_3) - T_{\alpha_{jk}}v_{jk}(z;z_3) \big)}_{(7)} \\ &+\boxed{\sum_{k >0,j>0 \atop j {\mathbf{\vartheta}}+ k < \alpha} \frac{(-\tau^{\mathbf{\vartheta}})^{j}}{j!} \frac{(-\tau )^{k}}{k!} T_{\alpha_{jk}}v_{jk}(z;z_3)}_{(8)}=: I_{5.1} + I_{6.1} + I_7 + I_8. \end{split}$$ where $$\alpha_{jk} := \alpha -(j {\mathbf{\vartheta}}+k), \qquad \text{and} \qquad v_{jk} := [Z_i^k, Y^j]u. %Z_i^k Y^j u - Y^j Z_i^k u.$$ Consider $I_{5.1}$ first. By Remark [Remark 11](#rem:eq:mean_value_Zn){reference-type="ref" reference="rem:eq:mean_value_Zn"}, we obtain $$\left| Y^j u(z_4) - Y^j u(z_3) - \sum_{0 < k < \alpha - j {\mathbf{\vartheta}}} \frac{(-\tau )^{k}}{k!}Z_i^k Y^j u(z_3) \right| \le \|u\|_{C^{{\alpha}}} |\tau|^{\alpha - j {\mathbf{\vartheta}}}.%, \qquad \ddY \le j \ddY < \alpha - 1.$$ The same argument applies to $I_{6.1}$, in this case we use Remark [Remark 10](#rem:eq:mean_value_Yn){reference-type="ref" reference="rem:eq:mean_value_Yn"}. By collecting the above inequalities we find that there exists a positive constant $c'_\alpha$ such that $$\label{eq-boxes-5-6.1} |I_{5.1}| + |I_{6.1}| \le c'_\alpha \|u\|_{C^{{\alpha}}} | {\delta}|^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}.$$ Concerning $I_{7}$, we first note that, by Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}, $v_{jk} \in C^{\alpha_{jk}}$. Moreover, only terms with $0 < \alpha_{jk} \le \bar \alpha = \alpha - ({\mathbf{\vartheta}}+ 1)$ appear there. Then, because of the induction hypothesis, in particular by [\[eq:estim_tay_n\]](#eq:estim_tay_n){reference-type="eqref" reference="eq:estim_tay_n"}, we have $$\label{eq:estim_vjk} \left|v_{jk}(z_3) - T_{\alpha_{jk}}v_{jk}(z;z_3) \right|\le c \|v_{jk}\|_{C^{{\alpha}_{jk}}} \|z^{-1}\circ z_3\|^{{\alpha}_{jk}}.$$ We then conclude that there exists a positive constant $c''_\alpha$ such that $$\label{eq-boxes-567} |I_7| \le c''_\alpha \|u\|_{C^\alpha} | {\delta}|^{\frac{{\alpha}}{{\mathbf{\vartheta}}+1}}.$$ We are left with the term $I_8$. We note that $$%\label{eq:boxes-8pre} I_8 = {\delta}\, [Z_i, Y]u(z) + \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \sum_{k >0,j>0 \atop {\mathbf{\vartheta}}+ 1 < m' ({\mathbf{\vartheta}}+1) + (j'+j) {\mathbf{\vartheta}}+ k' +k < \alpha} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\delta}^{\frac{m' ({\mathbf{\vartheta}}+1) + (j'+j) {\mathbf{\vartheta}}+ k' +k}{{\mathbf{\vartheta}}+1}} \big( \bar c_{m'j'k'jk} \partial_{x_i}^{m'}Y^{j'} Z_i^{k'+k} Y^j u + \widetilde c_{m'j'k'jk} \partial_{x_i}^{m'}Y^{j'} Z_i^{k'} Y^j Z_i^k u \big)(z),$$ where every constant $\bar c_{m'j'k'jk}$ and $\widetilde c_{m'j'k'jk}$ is obtained by collecting the coefficients of the Taylor polynomials of the $v_{jk}$ functions. As a consequence of the fact that $\frac{m' ({\mathbf{\vartheta}}+1) + (j'+j) {\mathbf{\vartheta}}+ k' +k}{{\mathbf{\vartheta}}+1} > 1$ for every term of the above sum, we finally obtain $$\label{eq-boxes-8} \frac{I_8}{{\delta}} \to [Z_i, Y]u(z) \qquad \text{as} \quad {\delta}\to 0.$$ Moreover, since $1 + {\mathbf{\vartheta}}< \alpha \le 2(1 + {\mathbf{\vartheta}})$, [\[eq-boxes-1-4\]](#eq-boxes-1-4){reference-type="eqref" reference="eq-boxes-1-4"}, [\[eq-boxes-5\]](#eq-boxes-5){reference-type="eqref" reference="eq-boxes-5"} and [\[eq-boxes-567\]](#eq-boxes-567){reference-type="eqref" reference="eq-boxes-567"} yield $$\label{eq-boxes-9} \frac{I_1 + I_2 + I_3 + I_4}{{\delta}} \to 0, \qquad \frac{I_{5.1} + I_{6.1}}{{\delta}} \to 0, \qquad \frac{I_7}{{\delta}} \to 0 \qquad \text{as} \quad {\delta}\to 0.$$ The proof of [\[eq:claim_step_2\]](#eq:claim_step_2){reference-type="eqref" reference="eq:claim_step_2"} follows from [\[eq-boxes-8\]](#eq-boxes-8){reference-type="eqref" reference="eq-boxes-8"} and [\[eq-boxes-9\]](#eq-boxes-9){reference-type="eqref" reference="eq-boxes-9"}. # Extensions: non-homogeneous and local case In some applications, the operator $Y$ in [\[e1\]](#e1){reference-type="eqref" reference="e1"} appears in a more general form than [\[eq:Y\]](#eq:Y){reference-type="eqref" reference="eq:Y"}, namely $$%\label{eq:} Y =\big\langle B (x,v)^\top , \nabla_{(x,v)} \big\rangle + \partial_t , \qquad (x,v)\in{\mathbb {R}}^{2d},$$ with $B$ being a $(2 d\times 2 d)$-matrix with real entries that admits the following block decomposition: $$%\label{eq:} B =\left( \begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix}\right),$$ where each block is a $(d\times d)$-matrix and $B_{12}$ has rank $d$. For instance, in the kinetic model originally introduced in ([@langevin1908theorie]), the term $B_{12}$ is the $d \times d$ identity matrix, while $B_{22}$ is non-null and depends on the viscosity of the liquid. In mathematical finance, $B_{22}$ may represent an interest rate in the pricing of path-dependent derivatives. Note that the Hörmander condition [\[eq:hormander\]](#eq:hormander){reference-type="eqref" reference="eq:hormander"} is still satisfied: in particular, [\[eq:commutator\]](#eq:commutator){reference-type="eqref" reference="eq:commutator"} becomes $$\label{eq:commutator2} B_{12} \nabla_{x} = [\nabla_{v} , Y] - B_{22} \nabla_{v}.%, \qquad j=1,\dots, d.$$ The integral curve of the vector field $Y$ now reads as $$\label{eq:def_curva_integrale_campo_non} e^{{\tau}Y }(t,x,v)=\big(t+{\tau}, e^{{\tau}B} (x,v)^\top \big), \qquad (t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d}, \quad {\tau}\in{\mathbb {R}}.$$ Accordingly, the relevant non-commutative group law "$\circ$" in [\[eq:translation_non\]](#eq:translation_non){reference-type="eqref" reference="eq:translation_non"} is replaced by $$\label{eq:translation_non} z_1\circ z_2 = \left(t_1+t_2, (x_2,v_2)^\top + e^{t_2 B} (x_1,v_1)^\top \right),\qquad z_1=(t_1,x_1,v_1),z_2=(t_2,x_2,v_2)\in \mathbb{R}\times\mathbb{R}^{2d}.$$ Notice that $({\mathbb {R}}\times{\mathbb {R}}^{2d},\circ)$ remains a group, with the identity and the inverse elements that now read as $${\text{Id}=(0,0,0)}, \qquad (t,x,v)^{-1}=\big(-t, - e^{-t B} (x,v)^\top \big).%, $(t,x)\in\Rdd$$$ In particular, we have $$\label{eq:translat_non} (t_0,x_0,v_0)^{-1} \circ (t,x,v) = \big(t- t_0, (x,v)^\top - e^{(t-t_0) B} (x_0,v_0)^\top \big) .$$ Despite the fact that $\mathscr{L}_s$ is no longer homogeneous with respect to the dilations $\left(D_\lambda\right)_{\lambda>0}$ in [\[eq:dilation\]](#eq:dilation){reference-type="eqref" reference="eq:dilation"}, the homogeneous norm [\[e7\]](#e7){reference-type="eqref" reference="e7"} remains well behaved with respect to the stratification induced by the Hörmander condition, in particular by [\[eq:commutator2\]](#eq:commutator2){reference-type="eqref" reference="eq:commutator2"}. A local version of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} can be proved in this more general setting, with the Hölder spaces $C^{{\alpha}}$ begin exactly as in Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"}. However, it is more natural to localize the definition of Hölder spaces as well. We follow the approach in [@PPP16], [@pagliarani2017intrinsic] for the case ${\mathbf{\vartheta}}=2$. Let ${\Omega}$ be a domain in $\mathbb{R}\times\mathbb{R}^{2d}$. For any $z\in{\Omega}$ we set $${\delta}_{z}:=\sup\left\{\bar{{\delta}}\in\,]0,1]\mid e^{{\delta}Z_1}(z),\cdots,e^{{\delta}Z_{d}}(z),e^{{\delta}Y}(z)\in{\Omega}\text{ for any }{\delta}\in [-\bar{{\delta}},\bar{{\delta}}]\right\}.$$ If ${\Omega}_{0}$ is a bounded domain with $\overline{{\Omega}}_{0}\subseteq{\Omega}$, we set ${\delta}_{{\Omega}_{0}}:=\min_{z\in \overline{{\Omega}}_{0}}{\delta}_{z}.$ Note that ${\delta}_{{\Omega}_{0}}\in\,]0,1]$. Now we first localize Definition [Definition 2](#def:intrinsic_alpha_Holder){reference-type="ref" reference="def:intrinsic_alpha_Holder"} of Hölder regularity along the Hörmander fields. **Definition 13**. Let $u:\mathbb{R}\times\mathbb{R}^{2d}\to {\mathbb {R}}$. Then, for $a\in]0,1]$, we say that $u\in C^{{\alpha}}_{Z_i}({\Omega})$, $i=1,\dots,d$ if, for any bounded domain ${\Omega}_{0}$ with $\overline{{\Omega}}_{0}\subseteq{\Omega}$, we have $$%\label{eq:} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}({\Omega}_0)}:=%\sup_{z\in\Rdd}|f(z)|+ \sup_{z\in {\Omega}_{0} \atop 0<|{\tau}|<{\delta}_{{\Omega}_{0}}} \frac{ \left|u\left(e^{{\tau}Z_i }(z)\right)- u(z)\right|}{|{\tau}|^{{\alpha}}} <\infty.$$ Moreover, for $\alpha\in]0,{\mathbf{\vartheta}}]$, we say that $u\in C^{{\alpha}}_{Y}(\Omega)$ if, for any bounded domain ${\Omega}_{0}$ with $\overline{{\Omega}}_{0}\subseteq{\Omega}$, we have $$%\label{eq:} \left\|{u}\right\|_{C^{{\alpha}}_{Y}({\Omega}_0)}:=%\sup_{z\in\Rdd}|f(z)|+ \sup_{z\in{\Omega}_{0}\atop 0<|{\tau}|<{\delta}_{{\Omega}_{0}}} \frac{ \left|u\left(e^{{\tau}Y }(z)\right)- u(z)\right|}{|{\tau}|^{\frac{{\alpha}}{{\mathbf{\vartheta}}}}} <\infty.$$ With Definition [Definition 13](#def:intrinsic_alpha_Holder_local){reference-type="ref" reference="def:intrinsic_alpha_Holder_local"} at hand, we can now localize the notion of intrinsic Hölder spaces $C^{{\alpha}}$ with the following definition, which is completely analogous to Definition [Definition 3](#def:C_alpha_spaces){reference-type="ref" reference="def:C_alpha_spaces"} and thus is written in a more compact form for sake of brevity. **Definition 14**. Let $u:{\Omega}\to {\mathbb {R}}$ and ${\alpha}>0$. Then $u\in C^{{\alpha}}({\Omega})$ if, for any bounded domain ${\Omega}_{0}$ with $\overline{{\Omega}}_{0}\subseteq{\Omega}$, the semi-norm defined recursively as $$%\label{eq:} \left\|{u}\right\|_{C^{{\alpha}}({\Omega}_0)}:= \begin{cases} \left\|{u}\right\|_{C^{{\alpha}}_{Y}({\Omega}_0)}+\sum_{i=1}^{d} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}({\Omega}_0)}& \quad \text {if } \alpha\in]0,1 \wedge {\mathbf{\vartheta}}],\\ \left\|{Yu}\right\|_{C^{{\alpha}-{\mathbf{\vartheta}}}({\Omega}_0)}+\sum_{i=1}^{d} \left\|{u}\right\|_{C^{{\alpha}}_{Z_i}({\Omega}_0)}& \quad \text {if }\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}] \text{ and }{\mathbf{\vartheta}}<1, \\ \left\|{u}\right\|_{C^{{\alpha}}_{Y}({\Omega}_0)}+\sum_{i=1}^{d} \left\|{Z_i u}\right\|_{C^{{\alpha}-1}({\Omega}_0)}& \quad \text {if }\alpha\in]1 \wedge {\mathbf{\vartheta}}, 1 \vee {\mathbf{\vartheta}}] \text{ and }{\mathbf{\vartheta}}>1, \\ \left\|{Y u}\right\|_{C^{{\alpha}-{\mathbf{\vartheta}}}({\Omega}_0)}+\sum_{i=1}^{d} \left\|{Z_i u}\right\|_{C^{{\alpha}-1}({\Omega}_0)}& \quad \text {if } \alpha>1 \vee {\mathbf{\vartheta}} \end{cases}$$ is finite. **Remark 15**. It is easy to see that $C^{{\alpha}}({\Omega})$ is a decreasing family in ${\alpha}$, meaning that $C^{{\alpha}}({\Omega}) \subset C^{{\alpha}'}({\Omega})$ for any ${\alpha}> {\alpha}'>0$. We now have the following local version of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"}. Before, for $u\in C^{{\alpha}}({\Omega})$, we define its intrinsic Taylor polynomial centered at $z_0=(t_0,x_0,v_0)$ as $$\label{eq:TAy_pol_non} T_{\alpha} u(z_0;z):=\!\! \sum_{{0\leq {\mathbf{\vartheta}}k + (1+{\mathbf{\vartheta}})|\gamma| + |\beta| < \alpha}} \!\! \frac{ Y^k {\partial_{v}^{\beta} \partial_{x}^{\gamma} } }{k!\, \gamma! \,\beta! } \big( (t_0,x_0,v_0)^{-1} \circ (t,x,v) \big)^{(k,\gamma,\beta)}, \qquad z=(t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d},$$ with $\circ$ as in [\[eq:translat_non\]](#eq:translat_non){reference-type="eqref" reference="eq:translat_non"}, and where we adopted the multi-product notation $$%\label{eq:} (s,h,y)^{(k,\gamma,\beta)} = s^{k} h_1^{\gamma_1} \cdots h_d^{\gamma_d} y_1^{\beta_1} \cdots y_d^{\beta_d}, \qquad (s,h,y)\in\mathbb{R}\times\mathbb{R}^{2d}.$$ Note that the term *polynomial* here is slightly abused, as $T_{\alpha} u(z_0;z)$ is not necessarily a polynomial in the time-increment. **Theorem 16**. *For any $\alpha >0$ and for any $u\in C^{{\alpha}}({\Omega})$ the following statements are true:* Hölder spaces characterization. : *For any $k\in{\mathbb {N}}_0$ and $\gamma,\beta\in{\mathbb {N}}_0^d$ with ${0\leq {\mathbf{\vartheta}}k + (1+{\mathbf{\vartheta}})|\gamma| + |\beta| <\alpha}$, the derivatives $\ Y^k \partial_v^{\beta} \partial_x^{\gamma} u$ exist on ${\Omega}$ and $$\label{eq:maintheorem_part1_loc} %\begin{split} % & \ Y^k \partial_x^{\gamma} \partial_v^{\beta}u \in C^{\alpha-\ddY k - (1+\ddY)|\gamma| - |\beta|} % \qquad \text{and} \\ %& \| Y^k \partial_v^{\beta} \partial_x^{\gamma}u \|_{C^{\alpha-{\mathbf{\vartheta}}k - (1+{\mathbf{\vartheta}})|\gamma| - |\beta|}({\Omega}_0)} \leq \| u \|_{C^{\alpha}({\Omega}_0)} % \qquad {\color{forestgreen} \big( \| u \|_{C^{\alpha}(\O)}? \big)} %\end{split}$$ for any bounded domain ${\Omega}_{0}$ with $\overline{{\Omega}}_{0}\subseteq{\Omega}$.* Taylor formula. : *For any ${\zeta}_0\in\Omega$, there exist two bounded domains $U,\Omega_0$ such that $\zeta_0\in \overline{U}\subset \overline{\Omega_0}\subset{\Omega}$, and a positive constant $c$, only dependent on $B$, $\alpha$ and $U$, such that $$\label{eq:estim_tay_n_loc} \left|u(z)-T_{\alpha} u(z_0;z)\right|\le c \|u\|_{C^{{\alpha}}(\Omega_0)} \|z_0^{-1}\circ z\|^{{\alpha}},\qquad z,z_0\in U.$$* In the homogeneous case, namely when the blocks $B_{11}$, $B_{21}$, $B_{22}$ are null, the proof of Theorem [Theorem 16](#th:main_loc){reference-type="ref" reference="th:main_loc"} is substantially identical to that of Theorem [Theorem 7](#th:main){reference-type="ref" reference="th:main"} with the only additional complexity being that $U$ and $\Omega_0$ have to be chosen in a way that all the integral curves employed to connect $z_0$ and $z$ are contained in $\Omega$. On the other hand, in the general non-homogeneous case, there is a substantial additional difficulty stemming from the fact that the discretization of [\[eq:commutator2\]](#eq:commutator2){reference-type="eqref" reference="eq:commutator2"} with the integral curves of $Z_1, \dots, Z_d$ and $Y$ is more involved than in the homogeneous case. In the rest of the section we give an account of this additional complexity and recall a result from [@pagliarani2017intrinsic] that allows to overcome it. Let $z=(t,x,v) \in \mathbb{R}\times\mathbb{R}^{2d}$, $w\in\mathbb{R}^{d}$ and recall the points $z_2$, $z_3$, $z_4$ as defined in [\[eq:gamma_k1_bis\]](#eq:gamma_k1_bis){reference-type="eqref" reference="eq:gamma_k1_bis"}, namely $$\label{eq:gamma_k1_bis_non} z_2 = e^{\tau Z_w}(t,x,v), \qquad z_3 = e^{\tau^{{\mathbf{\vartheta}}}Y}(z_2), \qquad z_4 = e^{- \tau Z_w}(z_3),$$ and set $$%\label{eq:} z_5 = e^{-\tau^{{\mathbf{\vartheta}}}Y}(z_4), \qquad z_6 = e^{- \tau^{{\mathbf{\vartheta}}+ 1} Z_{B_{22}w}}(z_5).$$ A direct computation shows that $$\label{eq:gamma_k1_ter_non} z_6 = \big(t, x + \tau^{{\mathbf{\vartheta}}+ 1} B_{12} w , v \big) - {\tau}^{2{\mathbf{\vartheta}}+1} \bigg(0 , \sum_{n=0}^{\infty} \frac{(-1)^{n}{\tau}^{{\mathbf{\vartheta}}n}}{(n+2)!} B^{n+2} (0,w)^\top \bigg), \qquad {\tau}\in{\mathbb {R}}. %= \big(t, x + h , v \big),$$ Assume that our aim is to move along the $x$ variable only, by an increment $h\in \mathbb{R}^{d}$. If we adjust the leading order increment in [\[eq:gamma_k1_ter_non\]](#eq:gamma_k1_ter_non){reference-type="eqref" reference="eq:gamma_k1_ter_non"} by choosing $w = B_{1,2}^{-1} h / |h|$ and ${\tau}= |h|^{\frac{1}{{\mathbf{\vartheta}}+1}}$ (recall that $B_{1,2}$ has maximum rank), then we are off by an error term of order ${\tau}^{2{\mathbf{\vartheta}}+1}$. Note that this error involves both the velocity and position variables as the blocks of $B$ are all non-null. Therefore, if we make a further correction in the velocity variables by considering $$\begin{aligned} \label{eq:def_g1} g_{w,{\tau}}(z) : = e^{\tau^{2 {\mathbf{\vartheta}}+ 1} Z_{w'}}(z_6), \qquad \text{with }\quad %\qquad v\in%\Rpz, \ %\d>0 %{\delta\in\R},% \d>0, \ (t,x,y)\in \R\times \R^2. %\end{align} %with %\begin{equation} %\label{eq:def_g2} %v_1 &= v, &\d_1 &= - \d^3 B_{1}, \\ w' = w'({\tau},w) = \sum_{n=0}^{\infty} \frac{(-1)^{n}{\tau}^{{\mathbf{\vartheta}}n}}{(n+2)!} B^{n+2}_{22} w, %, \quad \d' =\d'(\d) = \d^5,\end{aligned}$$ we fix the velocity variables but the increment in the position variables still differs from $h$ by an error term of order ${\delta}^{2{\mathbf{\vartheta}}+1}$. The next lemma allows to connect $z=(t,x,v)$ to $(t,x+h,v)$ moving along the curves above, while keeping $w$ bounded and $|{\tau}|$ controlled in terms of $|h|^{\frac{1}{{\mathbf{\vartheta}}+1}}$. It concides with [@pagliarani2017intrinsic Lemma 3.2], which has been proved for ${\mathbf{\vartheta}}=2$. We recall this result without proof, as dealing with a general ${\mathbf{\vartheta}}>0$ requires no modification. **Lemma 17**. *There exists ${\varepsilon}>0$, only dependent on $B$, such that: for any $h\in\mathbb{R}^{d}%\Rpo$ with $|h|\leq {\varepsilon}$, there exist $w\in\mathbb{R}^{d}%\Rpz$, $\tau\geq 0$ with $$%\label{eq:} |w|=1, \qquad |{\tau}| \leq \frac{2}{\left\|{B_{12}}\right\|}%_B |h|%_B% ^{\frac{1}{{\mathbf{\vartheta}}+ 1}},$$ such that $$\label{eq:g1_estim} g_{w,{\tau}}(z) = (t, x + h, v ), \qquad z=(t,x,v)\in\mathbb{R}\times\mathbb{R}^{2d},$$ where $g_{w,{\tau}}(z)$ is as in [\[eq:def_g1\]](#eq:def_g1){reference-type="eqref" reference="eq:def_g1"}.* With Lemma [Lemma 17](#lem:connect){reference-type="ref" reference="lem:connect"} at hand, which plays the role of identity [\[eq:gamma_k1_ter\]](#eq:gamma_k1_ter){reference-type="eqref" reference="eq:gamma_k1_ter"} that we had in the case $Y = \langle v , \nabla_x \rangle + \partial_t$, the proof of Theorem [Theorem 16](#th:main_loc){reference-type="ref" reference="th:main_loc"} is essentially the same as in the homogeneous case. [^1]: Dipartimento di Scienze Fisiche, Informatiche e Matematiche Università di Modena e Reggio Emilia, Modena, Italy. **e-mail**: maria.manfredini\@unimore.it [^2]: Dipartimento di Matematica, Universitá di Bologna, Bologna, Italy. **e-mail**: stefano.pagliarani9\@unibo.it [^3]: Dipartimento di Scienze Fisiche, Informatiche e Matematiche Università di Modena e Reggio Emilia, Modena, Italy. **e-mail**: sergio.polidoro\@unimore.it

arxiv_math

--- abstract: | In this paper, we present controlled finite continuous frames in a finite dimensional Hilbert space and we study some properties of them. Parseval controlled integral frames are presented and we characterize operators that construct controlled integral finite frames. address: - $^{1}$Department of Mathematics, Faculty Of Sciences, University of Ibn Tofail, Kenitra, Morocco - $^{2}$LaSMA Laboratory, Department of Mathematics, Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, Fes, Morocco - $^{3}$Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea author: - Hafida Massit$^{1}$, Mohamed Rossafi$^{2}$ and Choonkil Park$^{3*}$ date: " $^{*}$Corresponding author: Choonkil Park (email: baak\\@hanyang.ac.kr, fax: +82-2-2281-0019, orcid: 0000-0001-6329-8228)." title: Controlled finite continuous frames --- # Introduction and preliminaries The concept of frames in Hilbert spaces has been introduced by Duffin and Schaffer [@Duf] in 1952 to study some deep problems in nonharmonic Fourier series, after the fundamental paper [@DGM] by Daubechies, Grossman and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames. The majority of these applications requires frames in finite-dimensional spaces. For example, Jamali et al [@H] and Javanshiri et al [@JAN], were obtained results that are interesting in applications of frames. Recently, controlled frames were introduced by Balzas [@PDA], Antoine and Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces [@BAL], however they are used earlier in [@bo] for spherical wavelets. For more details, the reader can refer to [@PDA; @gx; @kvs; @Musazadeh]. The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaisar [@KAI] and independently by Ali, Antoine and Gazeau [@GAZ]. In this paper we try to give a generalization of the results given in [@Z] moving from the discrete case to the continuous case. For more information on frame theory and its applications, we refer the readers to [@FR1; @5; @r6; @r1; @RFDCA]. Throughout this paper, assume that $(\mathfrak{A},\mu)$ is a measure space with positive measure $\mu$, $\mathcal{H}$ and $\mathcal{H}^{N}$ are used for showing a Hilbert space and a finite-dimensional Hilbert space, respectively, $GL(H)$ denotes the set of all bounded linear operators with a bounded inverse, and $GL^{+}(H)$ is the set of positive operators in $GL(H)$. **Definition 1**. [@5] Let $\mathcal{H}^{N}$ be an $N$-dimentional Hilbert space, and $(\mathfrak{A},\mu)$ be a measure space. Then a map $F : \mathfrak{A} \rightarrow \mathcal{H}^{N}$ is called an integral frame in $\mathcal{H}^{N}$ if there exist $0<A\leq B < \infty$ such that $$\label{1} A\|f\|^{2}\leq \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle \langle F_{\varsigma},f \rangle d\mu(\varsigma)\leq B\|f\|^{2} \quad \forall f\in \mathcal{H}^{N} .$$ The elements $A$ and $B$ are called the integral frame bounds. If $A = B$, we call this an integral tight frame. If $A = B = 1$, it is called an integral Parseval frame. If only the right hand inequality of ([\[1\]](#1){reference-type="ref" reference="1"}) is satisfied, we call $F$ a controlled integral Bessel map with bound $B$. If $F$ is a Bessel map, then $T_{F}: L^{2}(\mathfrak{A},\mu) \rightarrow \mathcal{H}^{N}$, defined by $T_{F}(f)= \int_{\mathfrak{A}} \langle f, F(\varsigma) \rangle F(\varsigma) d\mu(\varsigma)$, is a bounded linear operator. $T_{F}$ is surjective and bounded if and only if $F$ is an integral frame. This operator is called the synthesis operator. The adjoint of $T_{F}$, which is called the analysis operator, is defined by $$T^{\ast}_{F}:\mathcal{H}^{N} \rightarrow L^{2}(\mathfrak{A}, \mu), \;\;\;\;T^{\ast}_{F}(f)(\varsigma)=\langle f,F(\varsigma)\rangle , \; \varsigma\in \mathfrak{A}.$$ The continuous frame operator is defined to be $S_{F}= T_{F}T^{\ast}_{F}$, it is invertible and positive. Recall that a Bessel map $F$ is a frame if and only if there exists a continuous Bessel mapping $G$ is a dual of F if for any $f,g\in \mathcal{H}^{N}$ $$\langle f,g\rangle= \int_{\mathfrak{A}} \langle f,G(\varsigma)\rangle \langle g,F(\varsigma)\rangle d\mu(\varsigma), \;f,g\in \mathcal{H}^{N},$$ $G$ is called a dual frame for $F$ and $S^{-1} _{F}F$ is a dual of $F$. # Main results We consider some properties of controlled continuous frames in finite Hilbert spaces. **Definition 2**. Let $\mathcal{H}^{N}$ be an $N$-dimensional Hilbert space and $(\mathfrak{A},\mu)$ be a measure space. Then a family $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is called a $V$-controlled integral frame for an invertible operator $V$ on $\mathcal{H}^{N}$ if there exist $0<A\leq B < \infty$ such that $$\label{d1eq1} A\|f\|^{2}\leq \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle \langle VF_{\varsigma},f \rangle d\mu(\varsigma)\leq B\|f\|^{2} \quad \forall f\in \mathcal{H}^{N}.$$ The elements $A$ and $B$ are called the $V$-controlled integral frame bounds. If $A = B$, we call this a $V$-controlled integral tight frame. If $A = B = 1$, it is called a $V$-controlled integral Parseval frame. If only the right hand inequality of ([\[d1eq1\]](#d1eq1){reference-type="ref" reference="d1eq1"}) is satisfied, we call $F$ a $V$-controlled integral Bessel map with bound $B$. Similar to ordinary frames, the controlled integral frame operator is defined for a controlled frame on $\mathcal{H}^{N}$ by $S_{VF}f= \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle VF_{\varsigma} d\mu(\varsigma)$.Wihch assumed in weak sense. The controlled synthesis operator $T^{\ast}_{VF}: L^{2}(\mathfrak{A},\mu) \rightarrow \mathcal{H}^{N}$ is defined by $T T^{\ast}_{VF}(f)= \int_{\mathfrak{A}} \langle f, F(\varsigma) \rangle VF(\varsigma) d\mu(\varsigma)$ and $S_{UF}= T^{\ast}_{VF}T_{F}$, where $T_{F}$ is the analysis operator of $\{F_{\varsigma}\}_{\varsigma}$. **Proposition 3**. *Let $F: \mathfrak{A} \rightarrow \mathcal{H}^{N}$ such that $\int_{\mathfrak{A}}\Vert VF((\varsigma)) \Vert^{2} d\mu(\varsigma) < \infty$. Then $V F$ is a Bessel map.* *Proof.* Using Cauchy-Schwarz inequality, we have $$\int_{\mathfrak{A}} \vert \langle f,VF(\varsigma)\rangle \vert^{2} d\mu(\varsigma)\leq \int_{\mathfrak{A}}\Vert f\Vert ^{2} \Vert VF(\varsigma) \Vert^{2}d\mu(\varsigma) \leq B \Vert f\Vert^{2}\; with\; B= \int_{\mathfrak{A}}\Vert VF((\varsigma)) \Vert^{2} d\mu(\varsigma).$$ This completes the proof. ◻ We prove that the converse of Proposition [Proposition 3](#p0){reference-type="ref" reference="p0"} holds if $\mathcal{H}^{N}$ is finite dimensional. **Proposition 4**. *Let $\mathcal{H}^{N}$ be an $N$-dimensional Hilbert space and $F: \mathfrak{A} \rightarrow \mathcal{H}^{N}$ be a Bessel map. Then $\int_{\mathfrak{A}}\Vert VF((\varsigma)) \Vert^{2} d\mu(\varsigma) < \infty$.* *Proof.* Let $\{e_{k}\}_{k\in\{1,2,...,n\}}$ be an orthonormal basis for $\mathcal{H}^{N}$. Then we have $\Vert VF(\varsigma) \Vert ^{2} =\sum_{k=1}^{ n}\vert \langle VF(\varsigma) , e_{k}\rangle \vert^{2}$. So $$\begin{aligned} \int_{\mathfrak{A}}\Vert VF(\varsigma) \Vert ^{2} d\mu(\varsigma)&=\sum_{k=1}^{ n}\int_{\mathfrak{A}}\vert \langle VF(\varsigma) , e_{k}\rangle \vert^{2}d\mu(\varsigma)\\ &\leq \sum_{k=1}^{ n} B \Vert e_{k} \Vert ^{2}= Bn<\infty. \end{aligned}$$ This completes the proof. ◻ We give a new identity for controlled integral frames in finite dimensional Hilbert spaces. **Proposition 5**. *Let $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a $V$-controlled integral frame where $V$ is an invertible operator on $\mathcal{H}^{N}$. Then the following statements are equivalent.* - *$\{F_{\varsigma}\}_{\varsigma \in \mathcal{A}}$ is a $V-$controlled integral frame with bounds $A$ and $B$.* - *$S_{VF}(f)= \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle VF_{\varsigma} d\mu(\varsigma)$ is an invertible and positive operator on $\mathcal{H}^{N}$.* *Proof.* $(1) \Rightarrow (2)$ is immediately from the definition of $V-$ controlled integral frame operator. $(2) \Rightarrow (1)$ for any $f\in \mathcal{H}^{N}$, suppose that $S_{VF}$ is positive and invertible. Then $$\langle S_{VF}f,f\rangle = \langle \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle VF_{\varsigma}f,f d\mu(\varsigma)\rangle= \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle VF_{\varsigma}f,f \rangle d\mu(\varsigma).$$ This implies that $$\Vert\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle VF_{\varsigma}f,f \rangle d\mu(\varsigma)\Vert= \Vert\langle S_{VF}f,f\rangle\Vert= \Vert S_{VF}^{\frac{1}{2}}f\Vert^{2} ,$$ there exists $0<m$ such that $$\label{2} m\langle f,f\rangle \leq \langle S_{VF}f,f\rangle .$$ On other hand, for all $f\in \mathcal{H}$, there exists $0<m'$ such that $$\label{1} \langle S_{VF}f,f\rangle \leq m' \langle f,f\rangle$$ From [\[2\]](#2){reference-type="ref" reference="2"} and [\[1\]](#1){reference-type="ref" reference="1"}, we conclude that $\{F_{\varsigma}\}_{\varsigma \in \mathcal{A}}$ is a $V-$controlled integral frame ◻ **Theorem 6**. *Let $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a continuous frame with the frame operator $S_{F}$. If $V \in GL^{+}(\mathcal{H}^{N})$ is self-adjoint operator with $V S_{F}=S_{F}V$, then $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a $V-$controlled integral frame.* *Proof.* For $f\in \mathcal{H}$, we have $$\langle S_{V}f,f\rangle =\langle \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle VF_{\varsigma}f,f d\mu(\varsigma)\rangle= \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle VF_{\varsigma}f,f \rangle d\mu(\varsigma).$$ So, we have $$A\Vert f\Vert^{2}\leq \langle S_{V}f,f\rangle.$$ Then, The operator $S_{VF}$ is positive, also it's selfadjoint. Let $S_{VF}=VS_{F}$. The operator $S_{VF}$ is invertible. By Proposition [Proposition 5](#p1){reference-type="ref" reference="p1"}, $\{F_{\varsigma} \}_{\varsigma}$ is a $V-$controlled integral frame. ◻ **Proposition 7**. *Let $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a $V-$controlled integral frame for $\mathcal{H}$ and $V \in GL(\mathcal{H})$. Then $\{F_{\varsigma}\}_{\varsigma}$ is a continuous frame and $V S_{F}=S_{F}V$, with $$\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle VF_{\varsigma} d\mu(\varsigma) =\int_{\mathfrak{A}}\langle f,VF_{\varsigma}\rangle F_{\varsigma} d\mu(\varsigma) .$$* *Proof.* Let $\{F_{\varsigma}\}_{\varsigma \in \mathcal{A}}$ is a $V-$controlled integral frame with bounds $A$ and $B$. We have $$A\langle f,f\rangle \leq \langle S_{V}f,f\rangle = \langle VS f,f\rangle =\langle V^{\frac{1}{2}} Sf,V^{\frac{1}{2}}f\rangle\leq \Vert V^{\frac{1}{2}} \Vert^{2}\langle Sf,f\rangle.$$ So, $$\label{3} A\Vert V^{\frac{1}{2}}\Vert^{-2} \langle f,f\rangle \leq \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle F_{\varsigma},f\rangle d\mu(\varsigma).$$ On other hand, for all $f\in\mathcal{H}$ we have $$\begin{aligned} \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle F_{\varsigma},f\rangle d\mu(\varsigma)&= \langle Sf,f\rangle\\ & = \langle V^{-1}V Sf,f\rangle \\ & =\langle (V^{-1}VS)^{\frac{1}{2}}f,(V^{-1}VS)^{\frac{1}{2}}f\rangle \\ & \leq \Vert V^{\frac{-1}{2}} \Vert^{2}\langle (VS)^{\frac{1}{2}}f,(VS)^{\frac{1}{2}}f\rangle \\ &=\Vert V^{\frac{-1}{2}} \Vert^{2}\langle (S_{V})^{\frac{1}{2}}f,(S_{V})^{\frac{1}{2}}f\rangle \\ &= \Vert V^{\frac{-1}{2}} \Vert^{2}\langle S_{V}f,f\rangle \\ &\leq \Vert V^{\frac{-1}{2}} \Vert^{2} B\langle f,f\rangle. \end{aligned}$$ Then, $$\label{4} \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle \langle F_{\varsigma},f\rangle d\mu(\varsigma)\leq \Vert V^{\frac{-1}{2}} \Vert^{2} B\langle f,f\rangle.$$ From [\[3\]](#3){reference-type="ref" reference="3"} and [\[4\]](#4){reference-type="ref" reference="4"} we conclude that $\{F_{\varsigma}\}$ is a continuous frame. ◻ We show that the condition $VS_{F}= S_{F}V$ is not given in a finite-dimensional real Hilbert space in the following example. **Example 8**. Consider the frame $\{F_{\varsigma}\}_{\varsigma} = \{\binom{1}{\varsigma}\}$ for $\mathbb{R}^{2}$ and $\mathfrak{A} = [0,1]$ endewed with the Lebesgue measure. With the operator $V= \begin{pmatrix} \begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array} \end{pmatrix},$ it is clear that $V$ is positive and invertible. By definition of frame operator, we have $S_{F}= \begin{pmatrix} \begin{array}{cc} 1 & 1/2 \\ -1/2 & 1/3 \end{array} \end{pmatrix}.$ For all $f =\binom{x}{y} \in \mathbb{R}^{2}$, we have $$\begin{aligned} \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle \langle VF_{\varsigma},f \rangle d\mu(\varsigma)&=\langle S_{VF}f,f\rangle\\ &=\langle VS_{F}\binom{x}{y},\binom{x}{y}\rangle\\ &= \dfrac{1}{2}x^{2}- \dfrac{y^{2}}{2}-\dfrac{2}{3}xy.\end{aligned}$$ We obtain that the frame $\{F_{\varsigma}\}$ is a $V-$controlled integral frame such that $VS_{F}\not= S_{F}V$. **Proposition 9**. *If $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a $V-$controlled integral frame for $\mathcal{H}^{N}$ with the frame operator $S_{VF}$, then $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a continuous frame for $\mathcal{H}$ with the frame operator $V^{-1}S_{VF}$.* *Proof.* Let $S_{F}f= \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle F_{\varsigma} d\mu(\varsigma), \forall f\in\mathcal{H}^{N}$ and $S_{VF}f= \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle VF_{\varsigma} d\mu(\varsigma) = VS_{F}f, \forall f\in\mathcal{H}^{N}$. Then $V^{-1}S_{VF}f= S_{F}f$. The operator $S_{F}$ is an injective operator on finite dimensional Hilbert space and $S_{F}$ is invertible. Therefore, for $f\in \mathcal{H}^{N}$ we have $$f= \int_{\mathfrak{A}} \langle S_{F}^{-1}f, F_{\varsigma}\rangle F_{\varsigma} d\mu(\varsigma)=\int_{\mathfrak{A}} \langle f,(S_{F}^{-1})^{\ast} F_{\varsigma}\rangle F_{\varsigma} d\mu(\varsigma).$$ This shows that $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a continuous frame generator for $\mathcal{H}^{N}$ with the frame operator $V^{-1}S_{VF}$ and $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a generator for $\mathcal{H}^{N}$. ◻ **Theorem 10**. *Let $F=\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a $V-$controlled integral frame for $\mathcal{H}^{N}$ where $V$ is an invertible operator and the controlled frame operator $S_{VF}$ be a normal operator with $VS_{VF}= S_{VF}V$. Then $V$ is a positive operator.* *Proof.* By Proposition [Proposition 9](#p9){reference-type="ref" reference="p9"}, $\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a continuous frame with the frame operator $S_{F}=V^{-1}S_{VF}$. We have $S_{VF}V=V S_{VF}$ and so $S_{VF}S_{F}= VS_{F}S_{F}= S_{F}VS_{F}= S_{F}S_{VF}$. There exists a set of common orthonormal eigenvectors of $S_{VF}$ and $S_{F}$ as $\{e_{k}\}_{k\in\{1,2,...,N\}}$. Let $\{\alpha_{k}\}_{k\in\{1,2,...,N\}}$ and $\{\beta_{k}\}_{k\in\{1,2,...,N\}}$ be eigenvalues of operators $S_{VF}$ and $S_{F}$, respectively. For $\varsigma\in\mathfrak{A}$, we have $$Ve_{k}= ( S_{VF} S_{F}^{-1}) (e_{k})= S_{VF}(\beta_{k}^{-1}e_{k})=\beta_{k}^{-1} \alpha_{k} e_{k}.$$ Then $$Vf= \sum_{k=1}^{N} \beta_{k}^{-1} \alpha_{k} \langle f,e_{k}\rangle$$ Which follows $V$ is a positive operator. ◻ **Proposition 11**. *Let $F=\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a continuous frame for $\mathcal{H}^{N}$ with the frame operator $S_{F}$. If $\{e_{k}\} _{k\in\{1,2,...,N\}}$ and $\{\beta_{k}\} _{k\in\{1,2,...,N\}}$ are the set of orthonormal eigenvectors and the set of eigenvalues of $S_{F}$, respectively, then for every set $\{\alpha_{k}\} _{k\in\{1,2,...,N\}} \subseteq (0,+\infty)$, $F=\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ is a $V-$controlled frame, where $V$ is defined by $V e_{k}= \alpha_{k}e_{k}$, for $k= 1,\cdots,N$.* *Proof.* Let $f\in \mathcal{H}^{N}$. Then we have $$\begin{aligned} VS_{F}f=VS_{F}(\sum_{k=1}^{N} \langle f,e_{k} \rangle e_{k}) )&= V(\sum_{k=1}^{N} \langle f,e_{k} \rangle S_{F}e_{k})\\ &=\sum_{k=1}^{N}\alpha_{k} \langle f,e_{k} \rangle Ve_{k}\\ &= \sum_{k=1}^{N}\beta_{k} \langle f,e_{k} \rangle \alpha_{k}e_{k}\\ &= \sum_{k=1}^{N}\alpha_{k} \langle f,e_{k} \rangle S_{F}e_{k}\\ &= S_{F}\sum_{k=1}^{N} \langle f,e_{k} \rangle \alpha_{k}e_{k}\\ &= S_{F}\sum_{k=1}^{N} \langle f,e_{k} \rangle Ve_{k}\\ &= S_{F}Vf. \end{aligned}$$ Since $\{\alpha_{k}\}\subset (0,\infty)$, so $V$ is positive and invertible and also $V$ and $VS_{F}$ commute with each other. So $VS_{F}$ is an invertible and positive operator with $$VS_{F}f=V (\sum_{k=1}^{N} \langle f,F_{k}\rangle F_{k} )=\sum_{k=1}^{N}\langle f,F_{k}\rangle VF_{k} .$$ Therefore, $\{F_{\varsigma}\} _{\varsigma}$ is a $V-$controlled integral frame with the frame operator $VS_{F}$. ◻ **Theorem 12**. *Let $F=\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a $V-$ controlled integral frame with frame operator $S_{VF}$ and $L\in GL(\mathcal{H}^{N})$ ($L$ is positive and so it is self -adjoint) such that $LV =VL$. Then $\{LF_{\varsigma}\}_{\varsigma \in\mathfrak{A}}$ is a $V-$controlled frame with frame operator $LS_{VF} L^{\ast}$. Moreover, $\{L^{k}F_{\varsigma}\}_{\varsigma \in\mathfrak{A}}$ is a $V-$ controlled integral frame for $k\in\mathbb{R}$ with frame operator $L^{k}S_{VF}(L^{k})^{\ast}$.* *Proof.* We have $$S_{VLF}(f)= \int_{\mathfrak{A}}\langle f,LF_{\varsigma}\rangle VLF_{\varsigma} d\mu(\varsigma)= \int_{\mathfrak{A}} \langle f,LF_{\varsigma}\rangle LVF_{\varsigma} d\mu(\varsigma)= LS_{VF}L^{\ast}f.$$ Thus $S_{VLF}= LS_{VF}L^{\ast}$ is invertible and $$\langle LS_{VF}L^{\ast}f,f\rangle = \langle S_{VF}L^{\ast},L^{\ast}f \geq0,\;\forall f\in\mathcal{H}^{N}, \; i.e.,\; S_{VF}\geq 0 .$$ This gives that $S_{VLF}$ is positive. Hence $\{LF_{\varsigma}\}_{\varsigma}$ is a $V-$ controlled integral frame. Also we have $L^k V = V l^k$. Thus $\{L^{k}F_{\varsigma}\}_{\varsigma}$ is a $V-$controlled integral frame with the frame operator $L^{k}S_{VF}(L^{k})^{\ast}$. ◻ **Corollary 13**. *If $\{F_{\varsigma}\}_{\varsigma}$ is a $V-$ controlled integral frame such that $VS_{F} =S_{F}V$, then $\{ S_{F}^{\frac{\beta-1}{2}}F_{\varsigma}\} _{\varsigma}$ is a $V-$controlled integral frame for any $\beta \in\mathbb{R}$, with frame operator $VS_{F}$.* Gramian operator or Gramian matrix for a $V-$controlled frame has been defined in [@Z] and we introduce Gramian operator for $V-$controlled frame in finite Hilbert spaces, and we consider its properties. **Definition 14**. Let $\{F_{\varsigma}\}_{\varsigma}$ be a $V-$controlled integral frame with analysis operator $T_{F}$ and synthesis operator $T^{\ast}_{VF}$. Then the operator $G_{VF}: = T_{F}T^{\ast}_{VF}$ is called a $V-$Gramian operator. The canonical matrix representation of Gramian operator of a $V-$controlled integral frame $\{F_{\varsigma}\}$ is obtained by $$G= (\langle V F_{i},F_{j}\rangle )_{i,j\in\mathfrak{A}}.$$ The following theorem investigates the Gramian matrix of the transferred $V-$controlled integral frames. **Theorem 15**. *Let $\{F_{\varsigma}\}_{\varsigma}$ be a $V-$controlled integral frame for $\mathcal{H}^{N}$ and $T$ be a linear operator that commutes with $V$. Then $T$ is unitary if and only if the $V-$Gramian matrix of $\{TF_{\varsigma}\}$ is equal to $G_{VF}$.* *Proof.* Suppose that $T$ is unitary. Then we have $$G_{V(TF)}=\{\langle TVF_{\beta},TF_{\alpha}\rangle \}_{\beta,\alpha}=\{\langle VF_{\beta},F_{\alpha} \rangle\}= G_{VF}.$$ Conversely, let $G_{VF}=G_{V(TF)}$. Then $$\langle VTF_{\beta},TF_{\alpha}\rangle = \langle VF_{\beta},F_{\alpha}\rangle$$ and $$\langle T^{\ast}VTF_{\beta} -VF_{\beta}, F_{\alpha}\rangle=0 .$$ For $f\in\mathcal{H}^{N}$, we have $$f=\int_{\mathfrak{A}} \langle f,S_{F}^{-1}F_{\varsigma} \rangle d\mu(\varsigma).$$ Then $$\begin{aligned} (T^{\ast}VT-V)f& = (T^{\ast}VT-V) \int_{\mathfrak{A}} \langle f,S_{F}^{-1}F_{\varsigma} \rangle F_{\varsigma} d\mu(\varsigma)\\ & = \int_{\mathfrak{A}} \langle f,S_{F}^{-1}F_{\varsigma} \rangle (T^{\ast}VT-V) F_{\varsigma} d\mu(\varsigma)\\ & =0 . \end{aligned}$$ Thus we have $T^{\ast}VT= V$ and $T^{\ast}T=I$. ◻ Parseval frames are the closest family to orthonormal bases. We present and study some properties of Parseval controlled integral frames in a finite-dimensional Hilbert space. **Lemma 16**. *[@GAV][\[l.12\]]{#l.12 label="l.12"} Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space, and $G, L: \mathfrak{A} \rightarrow \mathcal{H}^{N}$ be continuous Parseval frames and $K \in L( \mathcal{H}^{N})$ be self-adjoint. Then $$\int_{\mathfrak{A}} \Vert K G(\varsigma)\Vert ^{2}d\mu(\varsigma)= \int_{\mathfrak{A}} \Vert K L(\varsigma)\Vert ^{2}d\mu(\varsigma).$$* **Theorem 17**. *Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space, and $F$ be a frame for $\mathcal{H}^{N}$ and $G$ be a Parseval $V-$ controlled integral frame for $\mathcal{H}^{N}$. Then $$\begin{aligned} \int_{\mathfrak{A}}\Vert V (G(\varsigma)-F(\varsigma))\Vert^{2}d\mu(\varsigma) &=& \int_{\mathfrak{A}}\Vert VS^{-1/2}F(\varsigma)-VF(\varsigma)\Vert^{2}d\mu(\varsigma)\\ &+& \int_{\mathfrak{A}}\Vert V(S^{1/4}G(\varsigma)- S^{-1/4}F(\varsigma))\Vert^{2} d\mu(\varsigma). \end{aligned}$$* *Proof.* By Lemma [\[l.12\]](#l.12){reference-type="ref" reference="l.12"} with $L(\varsigma)= VS^{1/2}F(\varsigma)$, we have $$\int_{\mathfrak{A}}\Vert G(\varsigma)\Vert^{2} d\mu(\varsigma)= \int_{\mathfrak{A}}\Vert VS^{-1/2}F(\varsigma)\Vert^{2} d\mu(\varsigma)$$ and $$\int_{\mathfrak{A}}\Vert S^{1/4} G(\varsigma)\Vert^{2} d\mu(\varsigma)= \int_{\mathfrak{A}}\Vert S^{1/4}F(\varsigma)\Vert^{2} d\mu(\varsigma)=\int_{\mathfrak{A}} \Vert S ^{-1/4}F(\varsigma)\Vert^{2}d\mu(\varsigma).$$ Thus $$\begin{aligned} & \int_{\mathfrak{A}}\Vert G(\varsigma)-F(\varsigma)\Vert^{2}d\mu(\varsigma) -\int_{\mathfrak{A}}\Vert VS^{-1/2}F(\varsigma)-VF(\varsigma)\Vert^{2}d\mu(\varsigma)\\ &=-2Re \int_{\mathfrak{A}}\langle G(\varsigma),F(\varsigma)\rangle d\mu(\varsigma)+2\int_{\mathfrak{A}}\langle VS^{-1/2}F(\varsigma),F(\varsigma)\rangle d\mu(\varsigma)\\ &=-2Re\int_{\mathfrak{A}}\langle VS^{1/4}G(\varsigma),S^{-1/4}F(\varsigma)\rangle d\mu(\varsigma)+\int_{\mathfrak{A}} \Vert VS^{-1/4}F(\varsigma)\Vert^{2} d\mu(\varsigma)+\int_{\mathfrak{A}}\Vert VS^{-1/4}G(\varsigma)\Vert^{2} d\mu(\varsigma)\\ &= \int_{\mathfrak{A}}\Vert V(S^{1/4}G(\varsigma)- S^{-1/4}F(\varsigma))\Vert^{2} d\mu(\varsigma).\end{aligned}$$ This completes the proof. ◻ **Corollary 18**. *Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space and $F$ be a frame for $\mathcal{H}^{N}$ with frame operator $S$. For every Parseval $V-$controlled integral frame $G$ of $\mathcal{H}^{N}$, we have $$\int_{\mathfrak{A}}\Vert V (G(\varsigma)-F(\varsigma))\Vert^{2}d\mu(\varsigma)\geq \int_{\mathfrak{A}}\Vert VS^{-1/2}F(\varsigma)-VF(\varsigma)\Vert^{2}d\mu(\varsigma)$$ and we have equality if and only if $$G(\varsigma)=VS^{-1/2}F(\varsigma)\;,\varsigma\in\mathfrak{A}.$$* *Proof.* The first part follows immediately from Theorem [Theorem 17](#t){reference-type="ref" reference="t"}. We have equality if and only if $$\begin{aligned} & S^{1/4}G(\varsigma)= VS^{-1/4}F(\varsigma)\;,\varsigma\in\mathfrak{A}\\ & \Leftrightarrow G(\varsigma)=VS^{-1/2}F(\varsigma). \end{aligned}$$ This completes the proof. ◻ **Proposition 19**. *Let $F=\{F_{\varsigma}\}_{\varsigma\in \mathfrak{A}}$ be a $V-$ controlled integral frame for $\mathcal{H}^{N}$. Then $$\int_{\mathfrak{A}} \langle VF_{\varsigma},F_{\varsigma}\rangle d\mu(\varsigma)= N.$$* *Proof.* Let $\{e_{k}\}_{k=1}^{N}$be an orthonormal basis for $\mathcal{H}^{N}$. We have $$e_{k}=S_{VF}e_{k}= \int_{\mathfrak{A}}\langle e_{k},F_{\varsigma}\rangle VF_{\varsigma}d\mu(\varsigma).$$ Thus $$\begin{aligned} N= \sum_{k=1}^{N}\Vert e_{k}\Vert^{2}&=\sum_{k=1}^{N} \int_{\mathfrak{A}} \langle e_{k},F_{\varsigma}\rangle \langle VF_{\varsigma}, e_{k}\rangle d\mu(\varsigma)\\ &=\int_{\mathfrak{A}}\sum_{k=1}^{N}\langle e_{k},F_{\varsigma}\rangle \langle VF_{\varsigma}, e_{k}\rangle d\mu(\varsigma)\\ &=\int_{\mathfrak{A}} \langle VF_{\varsigma}, F_{\varsigma}\rangle d\mu(\varsigma). \end{aligned}$$ This completes the proof. ◻ The following proposition illustrates that the orthogonal projections can be preserved controlled frames in a finite-dimensional Hilbert space. **Proposition 20**. *Let $\{F_{\varsigma}\}_{\varsigma \in\mathfrak{A}}$ be a $V-$ controlled integral frame for $\mathcal{H}^{N}$, $E$ be a subspace of $\mathcal{H}^{N}$ and $U$ be an orthonogonal projection of $\mathcal{H}^{N}$ onto $E$ such that $VU = UV$. Then $\{UF_{\varsigma}\}_{\varsigma}$ is a $V-$controlled frame for $E$. If $\{F_{\varsigma}\}_{\varsigma \in\mathfrak{A}}$ is a Parseval $V-$controlled integral frame for $\mathcal{H}^{N}$, then $\{UF_{\varsigma} \}_{\varsigma}$ is a Parseval $V-$controlled integral frame for $E$.* *Proof.* For all $f\in E$, we have $$A\Vert f\Vert^{2}= A\Vert Uf\Vert^{2} \leq \int_{\mathfrak{A}}\langle Uf,F_{\varsigma}\rangle \langle VF_{\varsigma}, Uf \rangle d\mu(\varsigma) \leq B \Vert Uf\Vert^{2}= B\Vert f\Vert^{2}$$ and $$A\Vert f\Vert^{2} \leq \int_{\mathfrak{A}}\langle f,UF_{\varsigma}\rangle \langle UVF_{\varsigma}, f \rangle d\mu(\varsigma) \leq B\Vert f\Vert^{2},$$ which implies that $$A\Vert f\Vert^{2} \leq \int_{\mathfrak{A}}\langle f,UF_{\varsigma}\rangle \langle VUF_{\varsigma}, f \rangle d\mu(\varsigma) \leq B\Vert f\Vert^{2}.$$ Therefore, $\{UF_{\varsigma}\}_{\varsigma}$ is a $V-$ controlled integral frame for $E$. Suppose $\{F_{\varsigma}\}_{\varsigma}$ is a Parseval $V-$controlled integral frame. Then for every $f\in E$, $$\begin{aligned} S_{VUF}(f)&=\int_{\mathfrak{A}}\langle f,UF_{\varsigma}\rangle VUF_{\varsigma} d\mu(\varsigma)\\ &= \int_{\mathfrak{A}}\langle Uf,F_{\varsigma}\rangle UVF_{\varsigma} d\mu(\varsigma)\\ &=U\int_{\mathfrak{A}}\langle Uf,F_{\varsigma}\rangle VF_{\varsigma} d\mu(\varsigma)\\ &=U^{2}f\\ &=f.\end{aligned}$$ Therefore, $\{UF_{\varsigma}\} _{\varsigma}$ is a Parseval $V-$controlled integral frame for $E$. ◻ If $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled integral frame with the controlled frame operator $S_{VF}$, then $S _{VF}= VS_{F}$ and $f= \int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle (S_{VF}^{-1}V)fd\mu(\varsigma)$ for every $f\in \mathcal{H}^{N}$. This gives that $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ is a Parseval $S_{VF}^{-1}V-$controlled integral frame, and $\{S_{VF}^{-1}VF_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$. **Theorem 21**. *Let $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ be a continuous frame with the frame operator $S_{F}$. Then every tight controlled integral frame $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ is exactly an $\alpha-$tight $\alpha S_{F}^{-1}$ controlled integral frame for $\alpha \in \mathbb{C}$.* *Proof.* Let $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ be an $\alpha-$tight $V-$controlled integral frame, for $\alpha \in\mathbb{C}$. Then for $f\in \mathcal{H}^{N}$, $\alpha f=\int_{\mathfrak{A}} \langle f,F_{\varsigma}\rangle VF_{\varsigma}d\mu(\varsigma)$ and so $\alpha I= S_{VF} = VS_{F}$ and $V= \alpha S^{-1}_{F}$, i.e., $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ is a $\alpha-$tight $\alpha S_{F}^{-1}$ controlled integral frame. ◻ We need to recall properties of the trace of linear operators on $\mathcal{H}^{N}$ and then consider trace of an operator by controlled integral frames. The trace of a linear operator $L\in L(\mathcal{H}^{N})$ is defined by $$Tr(L)= \sum_{k=1}^{N}\langle Le_{k},e_{k}\rangle,$$ where $\{e_{k}\}_{k=1}^{N}$ is an orthonormal basis for $\mathcal{H}^{N}$. If $L _{1}$ and $L_{2}$ are self-adjoint positive operators, then $0\leq Tr(L_{1}L_{2}) \leq Tr(L_{1}) \cdot Tr(L_{2})$. **Proposition 22**. *Let $\{F_{\varsigma}\}_{\varsigma\in\mathfrak{A}}$ be a $V-$controlled integral frame such that $V\in GL(\mathcal{H}^{N})$ is a self-adjoint operator. Then $$Tr(S_{VF})\leq Tr(V)\int_{\mathfrak{A}} \Vert F_{\varsigma}\Vert ^{2}d\mu(\varsigma).$$* *Proof.* Let $\{\alpha_{k}\}_{k=1}^{N}$ be the set of eigenvalues of the operator frame $S_{F}$. Then $Tr(S_{F})= \sum_{k=1}^{N}\alpha_{k} = \int_{\mathfrak{A}}\Vert F_{\varsigma}\Vert ^{2} d\mu(\varsigma)$. Therefore, $$Tr(S_{VF})= Tr(VS_{F})\leq Tr(V)Tr(S_{F})= Tr(V)\sum_{k=1}^{N}\alpha_{k}= Tr(V)\int_{\mathfrak{A}}\Vert F_{\varsigma}\Vert ^{2}d\mu(\varsigma).$$ This completes the proof. ◻ **Proposition 23**. *Let $\{F_{\varsigma}\}_{\varsigma \in \mathfrak{A}}$ be a Parseval $V-$controlled integral frame for $\mathcal{H}^{N}$ and $G$ be a linear operator on $\mathcal{H}^{N}$. Then $Tr(G)= \int_{\mathfrak{A}} \langle GVF_{\varsigma},F_{\varsigma} \rangle d\mu(\varsigma)$.* *Proof.* Let $\{e_{k}\}_{k=1}^{N}$ be an orthonormal basis for $\mathcal{H}^{N}$. Then $Tr(G)= \sum_{k=1}^{N}\langle Ge_{k},e_{k}\rangle$. Since $\{F_{\varsigma}\}_{\varsigma}$ is a Parseval $V-$controlled integral frame, for $k\in\{1,2,3,...\}$, we have $$Ge_{k}= \int_{\mathfrak{A}} \langle Ge_{k},F_{\varsigma}\rangle V F_{\varsigma}d\mu(\varsigma)$$ and $$\begin{aligned} Tr(G)&= \sum_{k=1}^{N}\langle \int_{\mathfrak{A}} \langle Ge_{k},F_{\varsigma}\rangle VF_{\varsigma}, e_{k}\rangle d\mu(\varsigma)\\ &=\sum_{k=1}^{N} \int_{\mathfrak{A}} \langle e_{k},G^{\ast}F_{\varsigma}\rangle \langle VF_{\varsigma},e_{k}\rangle d\mu(\varsigma)\\ &= \int_{\mathfrak{A}} \langle \sum_{k=1}^{N}\langle VF_{\varsigma},e_{k}\rangle e_{k},G^{\ast}F_{\varsigma}\rangle d\mu(\varsigma)\\ &= \int_{\mathfrak{A}}\langle VF_{\varsigma},G^{\ast}F_{\varsigma}\rangle d\mu(\varsigma)\\ &=\int_{\mathfrak{A}}\langle GVF_{\varsigma},F_{\varsigma}\rangle d\mu(\varsigma). \end{aligned}$$ This completes the proof. ◻ *Remark 24*. Every $\alpha-$ tight $V-$controlled integral frame $\{ F_{\varsigma}\}_{\varsigma}$ induces a Parseval controlled integral frame. We have for every $f\in\mathcal{H}^{N}$ $$\alpha f= \int_{\mathfrak{A}} \langle f,F_{\varsigma}\rangle VF_{\varsigma}d\mu(\varsigma),$$ and then $$f= \int_{\mathfrak{A}} \langle f,F_{\varsigma}\rangle (\alpha^{-1}V)F_{\varsigma} d\mu(\varsigma).$$ This means that $\{F_{\varsigma}\}_{\varsigma}$ is a Parseval $\alpha^{-1}V -$ controlled frame. Also, $\{F_{\varsigma}\}$ is equivalent to $\{(\alpha^{-1}V)F_{\varsigma}\}_{\varsigma}$ and $\{(\alpha^{-1}V)F_{\varsigma}\}_{\varsigma}$ is a dual for $\{F_{\varsigma}\}$. We recall that a frame $\{F_{\varsigma} \}_{\varsigma}$ is equivalent to a frame $\{G_{\varsigma}\}_{\varsigma}$ if there exists an invertible operator $P \in B(\mathcal{H}^{N})$ such that $F_{\varsigma}=P G_{\varsigma}$. **Example 25**. Consider the Hilbert space $\L^{2}( \mathbb{R})$ and $\mathfrak{A} = [0,1]$ endewed with the Lebesgue measure. Let $F=\{F_{\varsigma}\}_{\varsigma}$ and $G=\{G_{\varsigma}\}_{\varsigma}$ with $F_{\varsigma}(x_{1},x_{2},x_{3},...)= \dfrac{x_{\varsigma}}{3}$ and $G_{\varsigma}(x_{1},x_{2},x_{3},...)= \dfrac{x_{\varsigma}}{2}$. Then we obtain $$\int_{\mathfrak{A}} \Vert F_{\varsigma}(x) \Vert^{2} d\mu(\varsigma) = \dfrac{1}{9}\Vert x\Vert^{2}$$ and $$\int_{\mathfrak{A}} \Vert G_{\varsigma}(x) \Vert^{2} d\mu(\varsigma) = \dfrac{1}{4}\Vert x\Vert^{2}.$$ Define $P: \mathcal{H}\rightarrow \mathcal{H}$ by $P(x_{1},x_{2},x_{3},...)= (\dfrac{2}{3}x_{1},\dfrac{2}{3}x_{2},\dfrac{2}{3}x_{3},...)$. Then $F_{\varsigma}P= G_{\varsigma}$ for all $\varsigma \in \mathfrak{A}$. **Theorem 26**. *Let $\{F_{\varsigma}\}_{\varsigma \in \mathfrak{A}}$ be a $V-$controlled integral frame. Then $\{F_{\varsigma}\}_{\varsigma \in \mathfrak{A}}$ has a dual frame that is equivalent to $\{F_{\varsigma}\}_{\varsigma \in \mathfrak{A}}$.* be *Proof.* Let $S_{VG}$ be the frame operator of $\{F_{\varsigma}\}_{\varsigma}$. Then for $f\in\mathcal{H}^{N}$, $$S_{VG}f= \int_{\mathfrak{A}} \langle f,F_{\varsigma}\rangle VF_{\varsigma}. d\mu(\varsigma)$$ Since $S_{VG}$ is invertible, we have $$f= \int_{\mathfrak{A}} \langle f, F_{\varsigma}\rangle (S^{-1}_{VG}V)F_{\varsigma}d\mu(\varsigma).$$ Thus $\{F_{\varsigma}\}_{\varsigma}$ is a Parseval controlled integral frame and the frame $\{S_{VF}^{-1}VF_{\varsigma}\}_{\varsigma}$ is a dual frame for $\{F_{\varsigma}\}_{\varsigma}$ such that it is equivalent to $\{F_{\varsigma}\}_{\varsigma}$. ◻ **Proposition 27**. *If $\{G_{\varsigma}\}_{\varsigma}$ is a dual of $\{F_{\varsigma}\}_{\varsigma}$ such that it is equivalent to $\{F_{\varsigma}\}_{\varsigma}$, then $\{G_{\varsigma}\}_{\varsigma}$ induces a Parseval controlled integral frame of $\{F_{\varsigma}\}_{\varsigma}$.* *Proof.* Let $\{G_{\varsigma}\}_{\varsigma}$ be a dual of $\{F_{\varsigma}\}_{\varsigma}$ such that it is equivalent to $\{F_{\varsigma}\}_{\varsigma}$. Then there exists an invertible operator $V$ such that $G_{\varsigma}=VF^{\varsigma}$ for every $\varsigma \in\mathfrak{A}$. We have $$f=\int_{\mathfrak{A}} \langle f,F_{\varsigma} \rangle G_{\varsigma} d\mu(\varsigma),\;\forall f\in\mathcal{H}^{N}.$$ Then $\{F_{\varsigma}\}_{\varsigma}$ is a Parseval $V-$controlled integral frame. ◻ # Conclusion In this manuscript we introduced and characterized controlled finite continuous frames particularly Parseval controlled finite continuous frames as a subset of dual frames and we reviewed some notions and properties of operators and frames in Hilbert spaces. Also, we defined controlled finite continuous frames and we gave their properties. Gramian matrix and its properties for controlled finite continuous frames are examined. In the end we studied controlled finite continuous frames as a proper subset of dual frames is presented by the equivalent frames. 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arxiv_math

--- abstract: | In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $TVB_n$. In particular, the twisted virtual pure braid group $TVP_n$ is the kernel of an epimorphism of $TVB_n$ onto the symmetric group $S_n$. We find the set of generators and defining relations for $TVP_n$ and show that $TVB_n = TVP_n \rtimes S_n$. Further we prove that $TVP_n$ is a semi-direct product of some subgroup and abelian group $\mathbb{Z}_2^n$. As corollary we get that the virtual pure braid group $VP_n$ is a subgroup of $TVP_n$. Also, we construct some other epimorphism of $TVB_n$ onto $S_n$. Its kernel, $TVH_n$ is an analogous of $TVP_n$. We find its set of generators and defining relations and construct its decomposition in a semi-direct product. address: - Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia. - Novosibirsk State Agrarian University, Dobrolyubova street, 160, Novosibirsk, 630039, Russia. - Regional Scientific and Educational Mathematical Center of Tomsk State University, 36 Lenin Ave., Tomsk, Russia. - Regional Scientific and Educational Mathematical Center of Tomsk State University, 36 Lenin Ave., Tomsk, Russia. - Indian Institute of Technology, Ropar-140001, Punjab, India - Indian Institute of Technology, Ropar-140001, Punjab, India author: - Valeriy G. Bardakov - Tatyana A. Kozlovskaya - Komal Negi - Madeti Prabhakar title: Twisted virtual braid group --- # Introduction M. O. Bourgoin [@Bour] took a pioneering role in developing the theory of twisted knots, which is a generalization the theory of classical knots and virtual knots. He introduced the twisted Jones polynomial and the twisted knot group for twisted knot. Following this, Naoko Kamada [@N] introduced polynomial invariants and specialized algebraic structures called quandles, designed for the analysis of twisted knots and links. Moreover, in [@NS], S. Kamada and N. Kamada explores the use of biquandles in the study of twisted knots. For the studying of twisted links in [@NPK] was introduced the twisted virtual braid group $TVB_n$ and was proved that the closing a twisted virtual braid leads to a twisted link (analogous of Alexander's theorem), also it was some analogous of the Markov theorem, explains that a twisted virtual braid achieves uniqueness up to modulo certain transformations called Markov moves. The group $TVB_n$ is some analogous of the the braid group $B_n$. During some last decades were introduced and intensively studied some generalizations of $B_n$ (see, for example, [@B; @Bir; @KL] and references therein). For example, virtual braid group $VB_n$, welded braid group $WB_n$, singular braid group $SB_n$ and some other. Any group of this type has a pure subgroup, which is the kernel of an epimorphism onto $S_n$. The pure braid group $P_n$ is the kernel of the epimorphism from braid group $B_n$ to the symmetric group $S_n$. The virtual pure braid group $VP_n$ is the kernel of the epimorphism of the virtual braid group $VB_n$ onto $S_n$ that maps, for each $i$, $\sigma_i$ and $\rho_i$ to the $\rho_i$. Here $\langle \rho_1, \rho_2, \ldots, \rho_{n-1} \rangle \cong S_n$. The twisted virtual braid group $TVB_n$ (see [@NPK]) is generated by three families of elements, $$\sigma_i, ~~\rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j,~~j=1, 2, \ldots, n.$$ One can define some epimorphisms of $TVB_n$ onto $S_n$. The first one is $$\varphi_P \colon TVB_n \to S_n,~~\sigma_i \mapsto \rho_i, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto e,~~j=1, 2, \ldots, n.$$ Its kernel $\ker(\varphi_P)$ is the twisted virtual pure braid group $TVP_n$. The second epimorphism is $$\varphi_H \colon TVB_n \to S_n,~~\sigma_i \mapsto e, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto e,~~j=1, 2, \ldots, n.$$ Its kernel $\ker(\varphi_H)$ is denoted by $TVH_n$. This group is some analogous of a subgroup of $VB_n$, which was introduced in [@R] and studied in [@BB]. The analogous of $TVH_n$ for the singular braid group $SB_n$ was defined in [@GKM], where it was found a presentation of this subgroup in the case $n=3$. In this paper we study some subgroups, their presentation and decomposition into semi-direct product of the twisted virtual braid group $TVB_n$. In particular, we proved that $$TVB_n = TVP_n \rtimes S_n = TVH_n \rtimes S_n.$$ These decompositions show that for the studying of $TVB_n$ it is need to study $TVP_n$ and $TVH_n$. The paper is organized as follows. In Section [2](#BD){reference-type="ref" reference="BD"}, we recall definitions and some known facts from classical and virtual braid theories, specifically concerning the pure braid group and pure virtual braid group. In Section [3](#pure){reference-type="ref" reference="pure"}, we delve into the group structures of $TVB_n, TVP_n, TVH_n$, and $TS_n$. Additionally, we investigate the decomposition of the groups $TVP_n$ and $TVH_n$, specifically, we establish that $TVP_n = PL_n \rtimes A_n$ and $TVH_n = HL_n \rtimes A_n$, where $A_n \cong \mathbb{Z}_2^n$. Furthermore, we demonstrate that the subgroups $TVP_n$ and $TVH_n$ are non-isomorphic for all $n\geq 3$. In Section [4](#PT){reference-type="ref" reference="PT"}, we define the endomorphisms $$\varphi_{PT} \colon TVB_n \to TS_n,~~\sigma_i \mapsto \rho_i, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto \gamma_j,~~j=1, 2, \ldots, n,$$ $$\varphi_{HT} \colon TVB_n \to TS_n,~~\sigma_i \mapsto e, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto \gamma_j,~~j=1, 2, \ldots, n,$$ where $$TS_n = \langle \rho_1, \rho_2, \ldots, \rho_{n-1}, \gamma_1, \gamma_2, \ldots, \gamma_n \rangle \leq TVB_n.$$ and find a set of generators and defining relations for the kernel $\ker(\varphi_{PT})$ and $\ker(\varphi_{HT})$ which we will denote by $PT_n$ and $HT_n$, respectively, and defining relations for the image $Im(\varphi_{PT})$ and $Im(\varphi_{HT})$ of these endomorphisms. It can be inferred from this result that the subgroup $TS_n$ is isomorphic to $A_n \rtimes S_n$. Also, we proved that $$TVB_n = PT_n \rtimes TS_n = HT_n \rtimes TS_n.$$ At the end of the paper we formulate some open problems and suggest directions for further research. # Acknowledgments {#acknowledgments .unnumbered} This work is supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2022-884). The third author would like to thank the University Grants Commission(UGC), India, for Research Fellowship with NTA Ref.No.191620008047. The fourth author acknowledges the support given by SERB research project(MATRICS) with F.No.MTR/2021/000394. This work is supported by the NBHM, Government of India under grant-in-aid with F.No.02011/2/20223NBHM(R.P.)/R&D II/970. # Basic definitions {#BD} In this section we recall some known definitions which can be found in [@Artin; @Bir1; @Mar]. The braid group $B_n$, $n\geq 2$, on $n$ strands can be defined as a group generated by $\sigma_1,\sigma_2,\ldots,\sigma_{n-1}$ with the defining relations $$\sigma_i \, \sigma_{i+1} \, \sigma_i = \sigma_{i+1} \, \sigma_i \, \sigma_{i+1},~~~ i=1,2,\ldots,n-2, \label{eq1}$$ $$\sigma_i \, \sigma_j = \sigma_j \, \sigma_i,~~~|i-j|\geq 2. \label{eq2}$$ The geometric interpretation of $\sigma_i$, its inverse $\sigma_{i}^{-1}$ and the unit $e$ of $B_n$ are depicted in the Figure [1](#figure1){reference-type="ref" reference="figure1"}. {#figure1} There exists a homomorphism of $B_n$ onto the symmetric group $S_n$ on $n$ symbols. This homomorphism maps $\sigma_i$ to the transposition $(i,i+1)$, $i=1,2,\ldots,n-1$. The kernel of this homomorphism is called the *pure braid group* and denoted by $P_n$. The group $P_n$ is generated by elements $a_{ij}$, $1\leq i < j\leq n$. These elements can be expressed by the generators of $B_n$ as follows $$a_{i,i+1}=\sigma_i^2,$$ $$a_{ij} = \sigma_{j-1} \, \sigma_{j-2} \ldots \sigma_{i+1} \, \sigma_i^2 \, \sigma_{i+1}^{-1} \ldots \sigma_{j-2}^{-1} \, \sigma_{j-1}^{-1},~~~i+1< j \leq n.$$ In these generators $P_n$ is defined by relations $$\begin{aligned} & a_{ik} a_{ij} a_{kj} = a_{kj} a_{ik} a_{ij}, \label{re2}\\ & a_{mj} a_{km} a_{kj} = a_{kj} a_{mj} a_{km}, ~\mbox{for}~m < j, \label{re3}\\ & (a_{km} a_{kj} a_{km}^{-1}) a_{im} = a_{im} (a_{km} a_{kj} a_{km}^{-1}), ~\mbox{for}~i < k < m < j, \label{re4}\\ & a_{kj} a_{im} = a_{im} a_{kj}, ~\mbox{for}~k < i < m < j ~\mbox{or}~m < k. \label{re1}\end{aligned}$$ Let $m_{kl} = \sigma_{k-1} \, \sigma_{k-2} \ldots \sigma_l$ for $l < k$ and $m_{kl} = 1$ in other cases. Then the set $$\Lambda_n = \left\{ \prod\limits_{k=2}^n m_{k,j_k}~ |~ 1 \leq j_k \leq k \right\}$$ is a Schreier set of coset representatives of $P_n$ in $B_n$. The subgroup $P_n$ is normal in $B_n$, and the quotient $B_n / P_n$ is isomorphic to $S_n$. The generators of $B_n$ act on the generator $a_{ij} \in P_n$ by the rules: $$\begin{aligned} & \sigma_k^{-1} a_{ij} \sigma_k = a_{ij}, ~\mbox{for}~k \not= i-1, i, j-1, j, \label{c1}\\ & \sigma_{i}^{-1} a_{i,i+1} \sigma_{i} = a_{i,i+1}, \label{c2}\\ & \sigma_{i-1}^{-1} a_{ij} \sigma_{i-1} = a_{i-1,j}, \label{c3}\\ & \sigma_{i}^{-1} a_{ij} \sigma_{i} = a_{i+1,j} [a_{i,i+1}^{-1}, a_{ij}^{-1}], ~\mbox{for}~j \not= i+1 \label{c4}\\ & \sigma_{j-1}^{-1} a_{ij} \sigma_{j-1} = a_{i,j-1}, \label{c5}\\ & \sigma_{j}^{-1} a_{ij} \sigma_{j} = a_{ij} a_{i,j+1} a_{ij}^{-1}, \label{c6}\end{aligned}$$ where $[a, b] = a^{-1} b^{-1} a b = a^{-1} a^b$. Denote by $$U_{i} = \langle a_{1i}, a_{2i}, \ldots, a_{i-1,i} \rangle,~~~i = 2, \ldots, n,$$ a subgroup of $P_n$. It is known that $U_i$ is a free group of rank $i-1$. One can rewrite the defining relations of $P_n$ as the following conjugation rules (for $\varepsilon = \pm 1$): $$\begin{aligned} & a_{ik}^{-\varepsilon} a_{kj} a_{ik}^{\varepsilon} = (a_{ij} a_{kj})^{\varepsilon} a_{kj} (a_{ij} a_{kj})^{-\varepsilon}, \label{co1}\\ & a_{km}^{-\varepsilon} a_{kj} a_{km}^{\varepsilon} = (a_{kj} a_{mj})^{\varepsilon} a_{kj} (a_{kj} a_{mj})^{-\varepsilon}, ~\mbox{for}~m < j, \label{co2}\\ & a_{im}^{-\varepsilon} a_{kj} a_{im}^{\varepsilon} = [a_{ij}^{-\varepsilon}, a_{mj}^{-\varepsilon}]^{\varepsilon} a_{kj} [a_{ij}^{-\varepsilon}, a_{mj}^{-\varepsilon}]^{-\varepsilon}, ~\mbox{for}~i < k < m, \label{co3}\\ & a_{im}^{-\varepsilon} a_{kj} a_{im}^{\varepsilon} = a_{kj}, ~\mbox{for}~k < i < m < j ~\mbox{or}~ m < k. \label{co4}\end{aligned}$$ The group $P_n$ is a semi--direct product of the normal subgroup $U_n$ and the group $P_{n-1}$. Similarly, $P_{n-1}$ is a semi--direct product of the free group $U_{n-1}$ and the group $P_{n-2},$ and so on. Therefore, $P_n$ is decomposable (see [@Mar]) into the following semi--direct product $$P_n=U_n\rtimes (U_{n-1}\rtimes (\ldots \rtimes (U_3\rtimes U_2))\ldots),~~~U_i\cong F_{i-1}, ~~~i=2,3,\ldots,n.$$ ## Virtual braid group The *virtual braid group* $VB_n$ was introduced in [@Ka]. This group is generated by the braid group $B_n = \langle \sigma_1, \sigma_2, \ldots, \sigma_{n-1} \rangle$ and the symmetric group $S_n=\langle \rho_1, \rho_2,\ldots, \rho_{n-1} \rangle$ with the following relations: $$\begin{aligned} \sigma_i \sigma_{i+1} \sigma_i&=\sigma_{i+1} \sigma_i \sigma_{i+1} & i=1, 2, \ldots, {n-2}, \\ \sigma_i \sigma_j&=\sigma_j \sigma_i & |i-j| \geq 2, \\ \rho_i^{2}&=1 & i=1, 2, \ldots, {n-1},\\ \rho_i \rho_j&= \rho_j \rho_i & |i-j| \geq 2,\\ \rho_i \rho_{i+1} \rho_i&= \rho_{i+1} \rho_{i} \rho_{i+1} & i=1, 2, \ldots, {n-2},\\ \sigma_i \rho_j&= \rho_j \sigma_i & |i-j| \geq 2 ,\\ \rho_i \rho_{i+1} \sigma_i&= \sigma_{i+1} \rho_i \rho_{i+1} & i=1, 2, \ldots, {n-2}.\end{aligned}$$ The virtual pure braid group $VP_n$, $n\geq 2$, was introduced in [@B] as the kernel of the homomorphism $VB_n \to S_n$, $\sigma_i \mapsto \rho_i$, $\rho_i \mapsto \rho_i$ for all $i=1, 2, \ldots, i-1$. $VP_n$ admits a presentation with the generators $\lambda_{ij},\ 1\leq i\neq j\leq n,$ and the following relations: $$\begin{aligned} & \lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij} \label{rel},\\ & \lambda_{ki}\lambda_{kj}\lambda_{ij}=\lambda_{ij}\lambda_{kj}\lambda_{ki} \label{relation},\end{aligned}$$ where distinct letters stand for distinct indices. The generators of $VP_n$ can be expressed in terms of the generators of $VB_n$ by the formulas $$\begin{aligned} \lambda_{i,i+1} &= \rho_i \, \sigma_i,\\ \lambda_{i+1,i} &= \rho_i \, \lambda_{i,i+1} \, \rho_i = \sigma_i \, \rho_i\end{aligned}$$ for $i=1, 2, \ldots, n-1$, and $$\begin{aligned} \lambda_{i,j} & = \rho_{j-1} \, \rho_{j-2} \ldots \rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots \rho_{j-2} \, \rho_{j-1}, \\ \lambda_{j,i} & = \rho_{j-1} \, \rho_{j-2} \ldots \rho_{i+1} \, \lambda_{i+1,i} \, \rho_{i+1} \ldots \rho_{j-2} \, \rho_{j-1}\end{aligned}$$ for $1 \leq i < j-1 \leq n-1$. We have decomposition $VB_n = VP_n \rtimes S_n$ and $S_n$ acts on $VP_n$ by the rules **Lemma 1** ([@B]). Let $a$ be an element of $\langle \rho_1, \rho_2, \ldots, \rho_{n-1} \rangle$ and $\bar{a}$ is its image in $S_n$ under the isomorphism $\rho_i \mapsto (i,i+1)$, $i = 1, 2, \ldots, n-1$, then for any generator $\lambda_{ij}$ of $VP_n$ the following holds $$a^{-1} \lambda_{ij} a = \lambda_{(i)\bar{a}, (j)\bar{a}},$$ where $(k)\bar{a}$ is the image of $k$ under the action of the permutation $\bar{a}$. It is possible to define another epimorphism $\mu \colon VB_n \to S_n$ as follows: $$\mu(\sigma_i)=e, \; \mu(\rho_i)=\rho_i, \;i=1,2,\dots, n-1\, ,$$ where $S_n$ is generated by $\rho_i$ for $i=1,2,\dots, n-1$. Let us denote by $VH_n$ the normal closure of $B_n$ in $VB_n$. It is evident that $\ker \mu$ coincides with $VH_n$. Let us define elements: $$x_{i,i+1}=\sigma_i,~~x_{i+1,i}=\rho_i \sigma_i \rho_i =\rho_i x_{i,i+1} \rho_i,$$ for $i= 1, 2, \ldots, n-1$, and $$x_{i,j}=\rho_{j-1} \cdots \rho_{i+1} \sigma_i \rho_{i+1} \cdots \rho_{j-1},$$ $$x_{j,i}=\rho_{j-1} \cdots \rho_{i+1} \rho_i \sigma_i \rho_i \rho_{i+1} \cdots \rho_{j-1},$$ for $1 \le i < j-1 \le n-1$. The group $VH_n$ admits a presentation with the generators $x_{k,\, l},$ $1 \leq k \neq l \leq n$, and the defining relations: $$\label{eq40} x_{i,j} \, x_{k,\, l} = x_{k,\, l} \, x_{i,j},$$ $$\label{eq41} x_{i,k} \, x_{k,j} \, x_{i,k} = x_{k,j} \, x_{i,k} \, x_{k,j},$$ where distinct letters stand for distinct indices. This presentation was found in [@R] (see also [@BB]). We have decomposition $VB_n = VH_n \rtimes S_n$, where $S_n=\langle \rho_1, \dots, \rho_{n-1} \rangle$ acts on the generators of $VH_n$ by permutation of indices and we have the next analogous of Lemma [Lemma 1](#form){reference-type="ref" reference="form"}. **Lemma 2** ([@BB]). Let $a$ be an element of $\langle \rho_1, \rho_2, \ldots, \rho_{n-1} \rangle$ and $\bar{a}$ is its image in the symmetric group $S_n$ under the isomorphism $\rho_i \mapsto (i,i+1)$, $i = 1, 2, \ldots, n-1$, then for any generator $x_{ij}$ of $VH_n$ the following holds $$a^{-1} x_{ij} a = x_{(i)\bar{a}, (j)\bar{a}},$$ where $(k)\bar{a}$ is the image of $k$ under the action of the permutation $\bar{a}$. ## The twisted virtual braid group {#tvbg} This group $TVB_n$ was defined in [@NPK]. We get this group if add to the presentation of $VB_n$ the set of new generators, $$\gamma_1, \gamma_2, \ldots, \gamma_n,$$ which satisfy the relations $$\label{rel-inverse-b} \gamma_i ^2 = e~~ \text{for}~~i=1,\ldots, n,$$ $$\label{rel-height-bb} \gamma_i \gamma_j = \gamma_j \gamma_i~~\textrm{for}~~ i, j \in \{1,2,\ldots,n\}.$$ Also, we add four types of mixed relations, $$\label{rel-height-bv} \gamma_j \rho_i = \rho_i \gamma_j~~\textrm{for}~~ j \not= i, i+1,$$ $$\label{rel-height-sb} \gamma_j \sigma_i = \sigma_i \gamma_j~~\textrm{for}~~ j \not= i, i+1,$$ $$\label{rel-bv} \rho_i \gamma_i = \gamma_{i+1} \rho_i,~~\textrm{for}~~ i \in \{1,2,\ldots,n-1\},$$ $$\label{rel-twist-III} ~~\rho_i \sigma_i \rho_i = \gamma_{i+1} \gamma_{i} \, \sigma_i \, \gamma_{i} \, \gamma_{i+1} ~~\textrm{for}~~ i \in \{1,2,\ldots,n-1\}.$$ From this presentation we see that the subgroup $A_n = \langle \gamma_1, \gamma_2, \ldots, \gamma_n \rangle$ of $TVB_n$ is the quotient of the abelian group $\mathbb{Z}_2^n$. Later (see Corollary [Corollary 9](#isom){reference-type="ref" reference="isom"}), we show that in fact, $A_n \cong \mathbb{Z}_2^n$. {#gen width="12cm" height="7cm"} # Some subgroups of $TVB_n$ {#pure} One can define some epimorphism of $TVB_n$ onto $S_n$. The first one is $$\varphi_P \colon TVB_n \to S_n,~~\sigma_i \mapsto \rho_i, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto e,~~j=1, 2, \ldots, n.$$ Its kernel $\ker(\varphi_P)$ is the twisted virtual pure braid group $TVP_n$. The second epimorphism is $$\varphi_H \colon TVB_n \to S_n,~~\sigma_i \mapsto e, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto e,~~j=1, 2, \ldots, n.$$ Its kernel $\ker(\varphi_H)$ is denoted by $TVH_n$. It is clear that $TVP_n$ and $TVH_n$ are a normal subgroup of index $n!$ of $TVB_n$ and we have short exact sequences, $$1 \to TVP_n \to TVB_n \to S_n \to 1,$$ $$1 \to TVH_n \to TVB_n \to S_n \to 1.$$ ## Structure of twisted virtual pure braid group $TVP_n$. Consider the following elements: $$\lambda_{i,i+1}=\rho_i\sigma_i^{-1},\text{ } \lambda_{i+1,i}=\rho_i\lambda_{i,i+1}\rho_i, \text{ } i=1, 2, \ldots, n-1,$$ $$\lambda_{i,j}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\lambda_{i,i+1}\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1},$$ $$\lambda_{j,i}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\lambda_{i+1,i}\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1},\text{ } 1 \leq i \leq j-1 \leq n-1,$$ which generate $VP_n$. Before proving presentation of $TVP_n$, we require some results. The next lemma is analogous to Lemma [Lemma 1](#form){reference-type="ref" reference="form"} and Lemma [Lemma 2](#form1){reference-type="ref" reference="form1"}. **Lemma 3**. The group $S_n$ acts by conjugation on the set $\{\gamma_i~ |~ 1\leq i\leq n\}$. This action is transitive. Let $a$ be an element of $\langle \rho_1, \rho_2, \ldots, \rho_{n-1} \rangle$ and $\bar{a}$ is its image in $S_n$ under the isomorphism $\rho_i \mapsto (i,i+1)$, $i = 1, 2, \ldots, n-1$, then for any generator $\gamma_{i}$ of $TVP_n$ the following holds $$a^{-1} \gamma_{i} a = \gamma_{(i)\bar{a}},$$ where $(k)\bar{a}$ is the image of $k$ under the action of the permutation $\bar{a}$. *Proof.* Using relations ([\[rel-inverse-b\]](#rel-inverse-b){reference-type="ref" reference="rel-inverse-b"}), ([\[rel-height-bv\]](#rel-height-bv){reference-type="ref" reference="rel-height-bv"}), and ([\[rel-bv\]](#rel-bv){reference-type="ref" reference="rel-bv"}), it is easy to see that the following conjugation rules hold in $TVB_n$: - $\rho_i\gamma_i\rho_i=\gamma_{i+1} \Leftrightarrow \rho_i\gamma_{i+1}\rho_i=\gamma_{i}$, for $1\leq i < n$; - $\rho_k\gamma_i\rho_k=\gamma_{i}$ for $1\leq i\neq k\leq n$. The lemma follows from these rules. ◻ **Remark 4**. To simplify the notation we will write $a^{-1} \gamma_i a =\gamma_{a(i)}$ instead $a^{-1} \gamma_{i} a = \gamma_{(i)\bar{a}}$ for $a \in S_n$. To find generators and defining relations for $TVP_n$, we use the Reidemeister-Schreier method (see, for example [@MKS Chapter 2.3]). As a Schreier set of coset representation of $TVP_n$ in $TVB_n$ we take the same set $\Lambda_n$, which is used in $VB_n$, $$\Lambda_n = \left\{ \prod\limits_{k=2}^n m_{k,j_k}~ |~ 1 \leq j_k \leq k \right\}$$ where $m_{kl}=\rho_{k-1}\rho_{k-2}\cdots \rho_l$ for $l<k$ and $m_{kl}=1$ in the other cases. The first main result of the present section is **Theorem 5**. *The group $TVP_n$ admits a presentation with the generators\ $\lambda_{kl},~1 \leq k\neq l \leq n$, and $\gamma_j$, $1\leq j\leq n$. The defining relations are as follows: $$\begin{aligned} \lambda_{ij}\lambda_{kl} &=\lambda_{kl}\lambda_{ij},\label{comm-clas}\\ \lambda_{ki}(\lambda_{kj}\lambda_{ij}) &=(\lambda_{ij}\lambda_{kj})\lambda_{ki},\label{classical}\\ \gamma_i^2 &=1,\label{g1}\\ \gamma_i\gamma_j &=\gamma_j\gamma_i,\label{g2}\\ \lambda_{ij}\gamma_k & =\gamma_k\lambda_{ij},\label{g3}\\ \lambda_{ij} &=\gamma_i\gamma_j\lambda_{ji}\gamma_j\gamma_i,\label{g4}\end{aligned}$$ where distinct letters stand for distinct indices.* *Proof.* [\[main\]]{#main label="main"} Define the map $\Bar{}: TVB_n \to \Lambda_n$ which takes an element $w \in TVB_n$ to its representative $\overline{w}$ from $\Lambda_n$. In this case the element $w\overline{w}^{-1}$ belongs to $TVP_n$. By Theorem 2.7 of [@MKS] the group $TVP_n$ is generated by $$s_{\lambda,a}=\lambda a \cdot (\overline{\lambda a})^{-1},~~~ \lambda \in \Lambda_n, ~~a \in \{\rho_1, \ldots, \rho_{n-1}, \sigma_1, \ldots, \sigma_{n-1}, \gamma_1, \ldots, \gamma_ n\}.$$ Then all $s_{\lambda, \rho_i}=e$ and $s_{\lambda, \sigma_i}=\lambda (s_{e,\sigma_i} )\lambda^{-1}=\lambda (\sigma_i\rho_i)\lambda^{-1}=\lambda (\lambda_{i,i+1}^{-1})\lambda^{-1}$, which is equal to some $\lambda_{kl}$, by Lemma [Lemma 1](#form){reference-type="ref" reference="form"}. These calculations are done in Theorem 1 [@B]. Now, consider the generators $$s_{\lambda,\gamma_i}=\lambda(s_{e,\gamma_i})\lambda^{-1}.$$ Since $s_{e,\gamma_i}=\gamma_i$. $s_{\lambda,\gamma_i}=\lambda(\gamma_i)\lambda^{-1}$, which is equal to some $\gamma_j$ by Lemma [Lemma 3](#gamma){reference-type="ref" reference="gamma"}. Therefore, generators of the group $TVP_n$ are $\lambda_{kl}$, $1 \leq k\neq l \leq n$ and, $\gamma_j$, $1\leq j\leq n$. To find the defining relations of $TVP_n$, we define a rewriting process $\tau$. It helps to rewrite a word $u$ to $\tau(u)$, where $u$ is written in the generators of $TVB_n$ but represents an element of $TVP_n$ and $\tau(u)$ is a word written in the generators of $TVP_n$. Let us associate to reduce word $$u=a_1^{\epsilon_1}a_2^{\epsilon_2}\cdots a_v^{\epsilon_v}, ~~\epsilon_l=\pm 1, ~~a_l \in \{\sigma_1,\sigma_2, \ldots, \sigma_{n-1}, \rho_1,\rho_2, \ldots, \rho_{n-1}, \gamma_1,\gamma_2, \ldots, \gamma_n\},$$ the word $$\tau(u)=s_{k_1,a_1}^{\epsilon_1}s_{k_2,a_2}^{\epsilon_2}\cdots s_{k_v,a_v}^{\epsilon_v},$$ in the generators of $TVP_n$, where $k_j$ is the $(j-1)$-th initial segment of the word $u$ if $\epsilon_j=1$, and a representative of the $j$-th initial segment of $u$ if $\epsilon_j=-1$. By Theorem 2.9 in [@MKS], the group $TVP_n$ is defined by the relations $$r_{\mu, \lambda}=\tau(\lambda r_\mu \lambda^{-1})=\lambda\tau( r_\mu) \lambda^{-1}, ~~~\lambda \in \Lambda_n,$$ where $r_\mu$ is a defining relation of $TVB_n$. Common relations of $VB_n$ and $TVB_n$ give rise to relations ([\[comm-clas\]](#comm-clas){reference-type="ref" reference="comm-clas"}), and ([\[classical\]](#classical){reference-type="ref" reference="classical"}) proved in Theorem 1 [@B]. Let us consider added relations stated in subsection [2.2](#tvbg){reference-type="ref" reference="tvbg"}. Denote by $r_1=\gamma_i^2$ the first relation of $TVB_n$ which is not a relation of $VB_n$. Then $$\begin{aligned} r_{1,e} = \tau(r_1)& = s_{e,\gamma_i}s_{\bar{\gamma_i}, \gamma_i}\\ & = s_{e,\gamma_i}s_{e, \gamma_i}\\ & = (e\cdot\gamma_i(\overline{e\cdot\gamma_i})^{-1})^2\\ & = \gamma_i^2. \end{aligned}$$ The remaining $r_{1,\lambda}$, $\lambda \in \Lambda_n$, can be obtained from this relation using conjugation by $\lambda^{-1}$ and it gives the same relation, by Lemma [Lemma 3](#gamma){reference-type="ref" reference="gamma"}. We have obtained ([\[g1\]](#g1){reference-type="ref" reference="g1"}). Now, consider the next relation $r_2=\gamma_i\gamma_j\gamma_i\gamma_{j}.$ We have, $$\begin{aligned} r_{2,e} = \tau(r_2)& = s_{e,\gamma_i}s_{\bar{\gamma_i},\gamma_j}s_{\overline{\gamma_i\gamma_j},\gamma_i}s_{\overline{\gamma_i\gamma_j\gamma_i}, \gamma_j}\\ & = s_{e,\gamma_i}s_{e,\gamma_j}s_{e, \gamma_i}s_{e,\gamma_j}\\ & = \gamma_i \gamma_j \gamma_i \gamma_j. \end{aligned}$$ The remaining $r_{2,\lambda}$, $\lambda \in \Lambda_n$, will give the same relations. We have obtained ([\[g2\]](#g2){reference-type="ref" reference="g2"}). Consider the next relation $r_3= \sigma_i\gamma_k\sigma_i^{-1}\gamma_k$, $k\neq i,i+1$, $$\begin{aligned} r_{3,e} = \tau(r_3)& = s_{e,\sigma_i}s_{\bar{\sigma_i},\gamma_k}s^{-1}_{\overline{\sigma_i\gamma_k\sigma_i^{-1}},\sigma_i}s_{\overline{ \sigma_i\gamma_k\sigma_i^{-1}}, \gamma_k} \\ & = s_{e,\sigma_i}s_{\rho_i,\gamma_k}s^{-1}_{e, \sigma_i}s_{e,\gamma_k}\\ & = \lambda^{-1}_{i,i+1} \rho_i \gamma_k \rho_i\lambda_{i,i+1}\gamma_k\\ & = \lambda^{-1}_{i,i+1}\gamma_k\lambda_{i,i+1}\gamma_k. \end{aligned}$$ The remaining relations $r_{3,\lambda}, \lambda \in \Lambda_n$, can be obtained from this relation using conjugation by $\lambda^{-1}$. By Lemma [Lemma 3](#gamma){reference-type="ref" reference="gamma"}, and Lemma [Lemma 1](#form){reference-type="ref" reference="form"}, we obtain ([\[g3\]](#g3){reference-type="ref" reference="g3"}). Let us consider the last relation $r_4=\rho_i\sigma_i\rho_i\gamma_{i+1}\gamma_i\sigma^{-1}_i\gamma_i\gamma_{i+1}$, $$\begin{aligned} r_{4,e} = \tau(r_4)= & s_{e,\rho_i}s_{\bar{\rho_i},\sigma_i}s_{\overline{\rho_i\sigma_i},\rho_i},s_{\overline{\rho_i\sigma_i\rho_i},\gamma_{i+1}}s_{\overline{\rho_i\sigma_i\rho_i\gamma_{i+1}},\gamma_i}s^{-1}_{\overline{\rho_i\sigma_i\rho_i\gamma_{i+1}\gamma_i\sigma^{-1}_i},\sigma_i}\\ & s_{\overline{\rho_i\sigma_i\rho_i\gamma_{i+1}\gamma_i\sigma^{-1}_i},\gamma_i}s_{\overline{\rho_i\sigma_i\rho_i\gamma_{i+1}\gamma_i\sigma^{-1}_i\gamma_i},\gamma_{i+1}}\\ = & s_{e,\rho_i}s_{\rho_i,\sigma_i}s_{e,\rho_i}s_{\rho_i,\gamma_{i+1}}s_{\rho_i,\gamma_{i}}s^{-1}_{e, \sigma_i}s_{e,\gamma_i}s_{e,\gamma_{i+1}}\\ = & \lambda^{-1}_{i+1,i} \gamma_{i}\gamma_{i+1}\lambda_{i,i+1}\gamma_i\gamma_{i+1}. \end{aligned}$$ Conjugating this relation by all representatives from $\Lambda_n$, we obtain ([\[g4\]](#g4){reference-type="ref" reference="g4"}). Therefore, the group $TVP_n$ is defined by the relations ([\[comm-clas\]](#comm-clas){reference-type="ref" reference="comm-clas"})-([\[g4\]](#g4){reference-type="ref" reference="g4"}). ◻ We can establish an alternative representation of the group $TVP_n$ by applying the relations ([\[g4\]](#g4){reference-type="ref" reference="g4"}) as outlined in Theorem [Theorem 5](#sTVP_n){reference-type="ref" reference="sTVP_n"}. **Corollary 6**. The group $TVP_n$ admits a presentation with the generators\ $\lambda_{kl},~1 \leq k< l \leq n$, and $\gamma_j$, $1\leq j\leq n$. The defining relations are as follows: $$\begin{aligned} \lambda_{ij}\lambda_{kl} &=\lambda_{kl}\lambda_{ij},\label{1comm-clas}\\ \lambda_{ki}(\lambda_{kj}\lambda_{ij}) &=(\lambda_{ij}\lambda_{kj})\lambda_{ki},\label{classical1}\\ \lambda_{ki}^{(ki)}(\lambda_{kj}^{(kj)}\lambda_{ij}^{(ij)}) &=(\lambda_{ij}^{(ij)}\lambda_{kj}^{(kj)})\lambda_{ki}^{(ki)},\label{classical2}\\ \lambda_{ki}^{(ki)}(\lambda_{ij}\lambda_{kj}) &=(\lambda_{kj}\lambda_{ij})\lambda_{ki}^{(ki)},\label{classical3}\\ \lambda_{ki}(\lambda_{ij}^{(ij)}\lambda_{kj}^{(kj)}) &=(\lambda_{kj}^{(kj)}\lambda_{ij}^{(ij)})\lambda_{ki},\label{classical4}\\ \lambda_{kj}(\lambda_{ki}\lambda_{ij}^{(ij)}) &=(\lambda_{ij}^{(ij)}\lambda_{ki})\lambda_{kj},\label{classical5}\\ \lambda_{kj}^{(kj)}(\lambda_{ki}^{(ki)}\lambda_{ij}) &=(\lambda_{ij}\lambda_{ki}^{(ki)})\lambda_{kj}^{(kj)},\label{classical6}\\ \gamma_i^2 &=1,\label{1g1}\\ \gamma_i\gamma_j &=\gamma_j\gamma_i,\label{1g2}\\ \lambda_{ij}\gamma_k & =\gamma_k\lambda_{ij}\label{1g3},\end{aligned}$$ where we denoted $\lambda_{ij}^{(ij)}=\lambda_{ij}^{\gamma_i\gamma_j}$ and, as usual, distinct letters stand for distinct indices. **Example 7**. Structure of $TVP_1$ and $TVP_2$ are shown as below: - $TVP_1=\langle \gamma_1~|~ \gamma_1^2 = e \rangle \cong \mathbb{Z}_2$. - $TVP_2= \langle \lambda_{12}, \lambda_{21}, \gamma_1, \gamma_2~ |~ \lambda_{12}=\gamma_1 \gamma_2 \lambda_{21} \gamma_2 \gamma_1, ~~\gamma_1^2=\gamma_2^2=e, ~~\gamma_1\gamma_2=\gamma_2\gamma_1 \rangle.$ We can remove the generator $\lambda_{21}$ and the first relation. Then, $$TVP_2= \langle \lambda_{12}, \gamma_1, \gamma_2~ |~ \gamma_1^2=\gamma_2^2=e, ~~\gamma_1\gamma_2=\gamma_2\gamma_1, \rangle \cong \mathbb{Z} * (\mathbb{Z}_2 \times \mathbb{Z}_2).$$ From the definition of $TVP_n$, Lemma [Lemma 3](#gamma){reference-type="ref" reference="gamma"}, and Lemma [Lemma 1](#form){reference-type="ref" reference="form"} it follows **Corollary 8**. $TVB_n=TVP_n \rtimes S_n.$ ## Decomposition of $TVP_n$ There exists an epimorphism $$\psi_P \colon TVP_n \to A_n,~~\lambda_{kl} \mapsto e,~~\gamma_j \mapsto \gamma_j,$$ where $$A_n = \langle \gamma_1, \ldots, \gamma_n \rangle.$$ Analysing the presentation of $TVP_n$ and the image of $\psi_P$, we get **Corollary 9**. The subgroup $A_n$ of $TVP_n$ has a presentation $$A_n = \langle \gamma_1, \ldots, \gamma_n ~|~\gamma_1^2 = \ldots = \gamma_n^2 = e, ~~\gamma_i \gamma_j = \gamma_j \gamma_i.~1 \leq i < j \leq n \rangle,$$ i.e. is isomorphic to $\mathbb{Z}_2^n$. Denote by $PL_n = \ker(\psi_P)$. Then $TVP_n = PL_n \rtimes A_n$. It is interesting to find a structure of $PL_n$. Using the relations ([\[g4\]](#g4){reference-type="ref" reference="g4"}) in Theorem [Theorem 5](#sTVP_n){reference-type="ref" reference="sTVP_n"}, $$\lambda_{ji}=\gamma_i\gamma_j\lambda_{ij}\gamma_j\gamma_i,~~~1 \leq i < j \leq n,$$ we can remove all the generators $\lambda_{ji}$, $1 \leq i < j \leq n,$ from the generating set of $TVP_n$. In particular, in the case $n = 3$, using the relations $$\lambda_{21} = \lambda_{12}^{ \gamma_1 \gamma_2},~~~\lambda_{31} = \lambda_{13}^{ \gamma_1 \gamma_3},~~~ \lambda_{32} = \lambda_{23}^{ \gamma_2 \gamma_3},$$ we get $$TVP_3 = \langle \lambda_{12}, \lambda_{13}, \lambda_{23}, \gamma_1, \gamma_2, \gamma_3~|~$$ $$\lambda_{12}\lambda_{13}\lambda_{23}=\lambda_{23}\lambda_{13}\lambda_{12}\text{, \hspace{0.3cm}}\lambda_{12}^{\gamma_1 \gamma_2} \lambda_{23}\lambda_{13}=\lambda_{13}\lambda_{23}\lambda_{12}^{\gamma_1 \gamma_2}\text{, \hspace{0.3cm}}\lambda_{13}\lambda_{12}\lambda_{23}^{\gamma_2 \gamma_3}=\lambda_{23}^{\gamma_2 \gamma_3} \lambda_{12}\lambda_{13},$$ $$\lambda_{13}^{\gamma_1 \gamma_3} \lambda_{23}^{\gamma_2 \gamma_3} \lambda_{12}=\lambda_{12} \lambda_{23}^{\gamma_2 \gamma_3} \lambda_{13}^{\gamma_1 \gamma_3} \text{, \hspace{0.3cm}}\lambda_{23} \lambda_{12}^{\gamma_1 \gamma_2} \lambda_{13}^{\gamma_1 \gamma_3}= \lambda_{13}^{\gamma_1 \gamma_3} \lambda_{12}^{\gamma_1 \gamma_2}\lambda_{23} \text{, \hspace{0.3cm}} \lambda_{23}^{\gamma_2 \gamma_3} \lambda_{13}^{\gamma_1 \gamma_3} \lambda_{12}^{\gamma_1 \gamma_2}=\lambda_{12}^{\gamma_1 \gamma_2}\lambda_{13}^{\gamma_1 \gamma_3} \lambda_{23}^{\gamma_2 \gamma_3},$$ $$\text{\hspace{0.3cm}} \gamma_1\lambda_{23}=\lambda_{23} \gamma_1 \text{, \hspace{0.3cm}} \gamma_1 \lambda_{23}^{\gamma_2 \gamma_3}=\lambda_{23}^{\gamma_2 \gamma_3}\gamma_1 \text{, \hspace{0.3cm}} \gamma_2\lambda_{13}^{\gamma_1 \gamma_3} = \lambda_{13}^{\gamma_1 \gamma_3} \gamma_2,$$ $$\text{\hspace{0.3cm}} \gamma_2\lambda_{13}=\lambda_{13} \gamma_2 \text{, \hspace{0.3cm}} \gamma_3 \lambda_{12} = \lambda_{12} \gamma_3 \text{, \hspace{0.3cm}} \gamma_3\lambda_{12}^{\gamma_1 \gamma_2}=\lambda_{12}^{\gamma_1 \gamma_2}\gamma_3,~~rel(A_3) \rangle,$$ where $rel(A_3)$ means the set of defining relations in $A_3$. The relations $$\gamma_1 \lambda_{23}^{\gamma_2 \gamma_3}=\lambda_{23}^{\gamma_2 \gamma_3}\gamma_1 \text{, \hspace{0.3cm}} \gamma_2\lambda_{13}^{\gamma_1 \gamma_3} = \lambda_{13}^{\gamma_1 \gamma_3} \gamma_2, \text{\hspace{0.3cm}} \gamma_3\lambda_{12}^{\gamma_1 \gamma_2}=\lambda_{12}^{\gamma_1 \gamma_2}\gamma_3$$ follows from other relations and we can remove them. We get $$TVP_3 = \langle \lambda_{12}, \lambda_{13}, \lambda_{23}, \gamma_1, \gamma_2, \gamma_3~|~$$ $$\lambda_{12}\lambda_{13}\lambda_{23}=\lambda_{23}\lambda_{13}\lambda_{12}\text{, \hspace{0.3cm}} \lambda_{12}^{\gamma_1 \gamma_2}\lambda_{13}^{\gamma_1 \gamma_3} \lambda_{23}^{\gamma_2 \gamma_3}=\lambda_{23}^{\gamma_2 \gamma_3} \lambda_{13}^{\gamma_1 \gamma_3} \lambda_{12}^{\gamma_1 \gamma_2},$$ $$\lambda_{12}^{\gamma_1 \gamma_2} \lambda_{23}\lambda_{13}=\lambda_{13}\lambda_{23}\lambda_{12}^{\gamma_1 \gamma_2}\text{, \hspace{0.3cm}} \lambda_{12} \lambda_{23}^{\gamma_2 \gamma_3} \lambda_{13}^{\gamma_1 \gamma_3} =\lambda_{13}^{\gamma_1 \gamma_3} \lambda_{23}^{\gamma_2 \gamma_3} \lambda_{12},$$ $$\lambda_{13}\lambda_{12}\lambda_{23}^{\gamma_2 \gamma_3}=\lambda_{23}^{\gamma_2 \gamma_3} \lambda_{12}\lambda_{13}\text{, \hspace{0.3cm}} \lambda_{13}^{\gamma_1 \gamma_3} \lambda_{12}^{\gamma_1 \gamma_2}\lambda_{23}=\lambda_{23} \lambda_{12}^{\gamma_1 \gamma_2} \lambda_{13}^{\gamma_1 \gamma_3},$$ $$\gamma_1\lambda_{23}=\lambda_{23} \gamma_1 \text{, \hspace{0.3cm}} \gamma_2\lambda_{13}=\lambda_{13} \gamma_2 \text{, \hspace{0.3cm}} \gamma_3 \lambda_{12} = \lambda_{12} \gamma_3 \text{, \hspace{0.3cm}}~~rel(A_3) \rangle.$$ Let us introduce elements $$\lambda_{12}^{(0)} = \lambda_{12},~~ \lambda_{12}^{(1)} = \lambda_{12}^{\gamma_1},~~\lambda_{12}^{(2)} = \lambda_{12}^{\gamma_2}, ~~\lambda_{12}^{(12)} = \lambda_{12}^{\gamma_1\gamma_2},$$ $$\lambda_{13}^{(0)} = \lambda_{13},~~\lambda_{13}^{(1)} = \lambda_{13}^{\gamma_1},~~\lambda_{13}^{(3)} = \lambda_{13}^{\gamma_3}, ~~\lambda_{13}^{(13)} = \lambda_{13}^{\gamma_1\gamma_3},$$ $$\lambda_{23}^{(0)} = \lambda_{23},~~\lambda_{23}^{(2)} = \lambda_{23}^{\gamma_2},~~\lambda_{23}^{(3)} = \lambda_{23}^{\gamma_3}, ~~\lambda_{23}^{(23)} = \lambda_{23}^{\gamma_2\gamma_3}.$$ Using the Reidemeister-Schreier method it is not difficult to prove **Proposition 10**. The group $PL_3$ is generated by elements $$\lambda_{ij}^{(0)}, ~\lambda_{ij}^{(i)},~ \lambda_{ij}^{(j)},~ \lambda_{ij}^{(ij)}, ~~1\leq i<j \leq 3,$$ and is defined by the relations which can be found from the relations $$\lambda_{12}^{(0)} \lambda_{13}^{(0)} \lambda_{23}^{(0)} =\lambda_{23} ^{(0)} \lambda_{13} ^{(0)} \lambda_{12}^{(0)} \text{, \hspace{0.3cm}} \lambda_{12}^{(12)}\lambda_{13}^{(13)} \lambda_{23}^{(23)}=\lambda_{23}^{(23)} \lambda_{13}^{(13)}\lambda_{12}^{(12)},$$ $$\lambda_{12}^{(12)} \lambda_{23}^{(0)} \lambda_{13}^{(0)} =\lambda_{13}^{(0)} \lambda_{23}^{(0)} \lambda_{12}^{(12)}\text{, \hspace{0.3cm}} \lambda_{12}^{(0)} \lambda_{23}^{(23)} \lambda_{13}^{(13)} =\lambda_{13}^{(13)} \lambda_{23}^{(23)} \lambda_{12}^{(0)},$$ $$\lambda_{13}^{(0)} \lambda_{12}^{(0)} \lambda_{23}^{(23)}=\lambda_{23}^{(23)} \lambda_{12}^{(0)} \lambda_{13}^{(0)} \text{, \hspace{0.3cm}} \lambda_{13}^{(13)} \lambda_{12}^{(12)}\lambda_{23}^{(0)} = \lambda_{23}^{(0)} \lambda_{12}^{(12)} \lambda_{13}^{(13)},$$ using the conjugations by elements $$\gamma_1^{\varepsilon_1} \gamma_2^{\varepsilon_2} \gamma_3^{\varepsilon_3},~~~\varepsilon_1, \varepsilon_2, \varepsilon_3 \in \{ 0, 1 \}.$$ The general case can be established using the Reidemeister-Schreier method **Theorem 11**. *The group $PL_n$, $n \geq 2$ is generated by elements* *$$\lambda_{ij}^{(0)} = \lambda_{ij}, ~\lambda_{ij}^{(i)} = \lambda_{ij}^{\gamma_i},~ \lambda_{ij}^{(j)} = \lambda_{ij}^{\gamma_j},~ \lambda_{ij}^{(ij)} = \lambda_{ij}^{\gamma_i \gamma_j}, ~~1\leq i<j \leq n,$$* *and relations are defined as follows,* *$$\lambda_{ij}^{(0)}\lambda_{kl}^{(0)} =\lambda^{(0)}_{kl}\lambda^{(0)}_{ij},~~ \{i,j\}\cap \{k,l\}=\phi$$ $$\lambda^{(i)}_{ij}\lambda^{(0)}_{kl} =\lambda^{(0)}_{kl}\lambda^{(i)}_{ij}, ~~\lambda^{(j)}_{ij}\lambda^{(0)}_{kl} =\lambda^{(0)}_{kl}\lambda^{(j)}_{ij},~~ \lambda^{(0)}_{ij}\lambda^{(k)}_{kl} =\lambda^{(k)}_{kl}\lambda^{(0)}_{ij}$$ $$\lambda^{(0)}_{ij}\lambda^{(l)}_{kl} =\lambda_{kl}^{(l)}\lambda^{(0)}_{ij}, ~~\lambda^{(ij)}_{ij}\lambda^{(0)}_{kl} =\lambda^{(0)}_{kl}\lambda_{ij}^{(ij)},~~ \lambda^{(0)}_{ij}\lambda^{(kl)}_{kl} =\lambda_{kl}^{(kl)}\lambda^{(0)}_{ij}$$ $$\lambda^{(i)}_{ij}\lambda^{(k)}_{kl} =\lambda^{(k)}_{kl}\lambda^{(i)}_{ij}, ~~\lambda^{(i)}_{ij}\lambda^{(l)}_{kl} =\lambda^{(l)}_{kl}\lambda^{(i)}_{ij},~~ \lambda^{(j)}_{ij}\lambda^{(k)}_{kl} =\lambda^{(k)}_{kl}\lambda^{(j)}_{ij}$$ $$\lambda^{(j)}_{ij}\lambda^{(l)}_{kl} =\lambda^{(l)}_{kl}\lambda^{(j)}_{ij}, ~~\lambda^{(ij)}_{ij}\lambda^{(k)}_{kl} =\lambda^{(k)}_{kl}\lambda^{(ij)}_{ij},~~ \lambda^{(ij)}_{ij}\lambda^{(l)}_{kl} =\lambda^{(l)}_{kl}\lambda^{(ij)}_{ij}$$ $$\lambda^{(i)}_{ij}\lambda^{(kl)}_{kl} =\lambda^{(kl)}_{kl}\lambda^{(i)}_{ij}, ~~\lambda^{(j)}_{ij}\lambda^{(kl)}_{kl} =\lambda^{(kl)}_{kl}\lambda^{(j)}_{ij},~~ \lambda^{(ij)}_{ij}\lambda^{(kl)}_{kl} =\lambda^{(kl)}_{kl}\lambda^{(ij)}_{ij}$$* *$$\lambda_{ij}^{(0)} \lambda_{ik}^{(0)} \lambda_{jk}^{(0)} =\lambda_{jk}^{(0)} \lambda_{ik}^{(0)} \lambda_{ij}^{(0)} \text{, \hspace{0.3cm}} \lambda_{ij}^{(i)} \lambda_{ik}^{(i)} \lambda_{jk}^{(0)} =\lambda_{jk} ^{(0)} \lambda_{ik} ^{(i)} \lambda_{ij}^{(i)}$$ $$\lambda_{ij}^{(j)} \lambda_{ik}^{(0)} \lambda_{jk}^{(j)} =\lambda_{jk}^{(j)} \lambda_{ik}^{(0)} \lambda_{ij}^{(j)} \text{, \hspace{0.3cm}} \lambda_{ij}^{(0)} \lambda_{ik}^{(k)} \lambda_{jk}^{(k)} =\lambda_{jk} ^{(k)} \lambda_{ik} ^{(k)} \lambda_{ij}^{(0)}$$ $$\lambda_{ij}^{(ij)} \lambda_{ik}^{(i)} \lambda_{jk}^{(j)} =\lambda_{jk} ^{(j)} \lambda_{ik} ^{(i)} \lambda_{ij}^{(ij)} \text{, \hspace{0.3cm}} \lambda_{ij}^{(j)} \lambda_{ik}^{(k)} \lambda_{jk}^{(jk)} =\lambda_{jk} ^{(jk)} \lambda_{ik} ^{(k)} \lambda_{ij}^{(j)}$$ $$\lambda_{ij}^{(i)} \lambda_{ik}^{(ik)} \lambda_{jk}^{(k)} =\lambda_{jk} ^{(k)} \lambda_{ik} ^{(ik)} \lambda_{ij}^{(i)} \text{, \hspace{0.3cm}} \lambda_{ij}^{(ij)}\lambda_{ik}^{(ik)} \lambda_{jk}^{(jk)}=\lambda_{jk}^{(jk)} \lambda_{ik}^{(ik)}\lambda_{ij}^{(ij)}$$* *$$\lambda_{ij}^{(ij)} \lambda_{jk}^{(0)} \lambda_{ik}^{(0)} =\lambda_{ik}^{(0)} \lambda_{jk}^{(0)} \lambda_{ij}^{(ij)}\text{, \hspace{0.3cm}} \lambda_{ij}^{(j)} \lambda_{jk}^{(0)} \lambda_{ik}^{(i)} =\lambda_{ik}^{(i)} \lambda_{jk}^{(0)} \lambda_{ij}^{(j)}$$ $$\lambda_{ij}^{(i)} \lambda_{jk}^{(j)}\lambda_{ik}^{(0)} =\lambda_{ik}^{(0)} \lambda_{jk}^{(j)} \lambda_{ij}^{(i)}\text{, \hspace{0.3cm}} \lambda_{ij}^{(0)} \lambda_{jk}^{(j)}\lambda_{ik}^{(i)} =\lambda_{ik}^{(i)} \lambda_{jk}^{(j)} \lambda_{ij}^{(0)}$$ $$\lambda_{ij}^{(ij)} \lambda_{jk}^{(k)} \lambda_{ik}^{(k)} =\lambda_{ik}^{(k)} \lambda_{jk}^{(k)} \lambda_{ij}^{(ij)}\text{, \hspace{0.3cm}} \lambda_{ij}^{(i)} \lambda_{jk}^{(jk)} \lambda_{ik}^{(k)} =\lambda_{ik}^{(k)} \lambda_{jk}^{(jk)} \lambda_{ij}^{(i)}$$ $$\lambda_{ij}^{(j)} \lambda_{jk}^{(k)}\lambda_{ik}^{(ik)} =\lambda_{ik}^{(ik)} \lambda_{jk}^{(k)} \lambda_{ij}^{(j)}\text{, \hspace{0.3cm}} \lambda_{ij}^{(0)} \lambda_{jk}^{(jk)}\lambda_{ik}^{(ik)} =\lambda_{ik}^{(ik)} \lambda_{jk}^{(jk)} \lambda_{ij}^{(0)}$$* *$$\lambda_{ik}^{(0)} \lambda_{ij}^{(0)} \lambda_{jk}^{(jk)}=\lambda_{jk}^{(jk)} \lambda_{ij}^{(0)} \lambda_{ik}^{(0)} \text{, \hspace{0.3cm}} \lambda_{ik}^{(i)} \lambda_{ij}^{(i)} \lambda_{jk}^{(jk)}=\lambda_{jk}^{(jk)} \lambda_{ij}^{(i)} \lambda_{ik}^{(i)}$$ $$\lambda_{ik}^{(0)} \lambda_{ij}^{(j)} \lambda_{jk}^{(k)}=\lambda_{jk}^{(k)} \lambda_{ij}^{(j)} \lambda_{ik}^{(0)} \text{, \hspace{0.3cm}} \lambda_{ik}^{(k)} \lambda_{ij}^{(0)} \lambda_{jk}^{(j)}=\lambda_{jk}^{(j)} \lambda_{ij}^{(0)} \lambda_{ik}^{(k)}$$ $$\lambda_{ik}^{(i)} \lambda_{ij}^{(ij)} \lambda_{jk}^{(k)}=\lambda_{jk}^{(k)} \lambda_{ij}^{(ij)} \lambda_{ik}^{(i)} \text{, \hspace{0.3cm}} \lambda_{ik}^{(k)} \lambda_{ij}^{(j)} \lambda_{jk}^{(jk)}=\lambda_{jk}^{(jk)} \lambda_{ij}^{(j)} \lambda_{ik}^{(j)}$$ $$\lambda_{ik}^{(ik)} \lambda_{ij}^{(i)}\lambda_{jk}^{(j)} = \lambda_{jk}^{(j)} \lambda_{ij}^{(i)} \lambda_{ik}^{(ik)}\text{, \hspace{0.3cm}} \lambda_{ik}^{(ik)} \lambda_{ij}^{(ij)}\lambda_{jk}^{(0)} = \lambda_{jk}^{(0)} \lambda_{ij}^{(ij)} \lambda_{ik}^{(ik)}$$* **Theorem 12**. *The map $$\lambda^{(0)}_{ij} \to \lambda_{ij},~~\lambda^{(j)}_{ij} \to e,~~\lambda^{(i)}_{ij} \to e,~~ \lambda^{(ij)}_{ij} \to \lambda_{ji},~~~1 \leq i < j \leq n,$$ defines an endomorphism of $PL_n$ onto $VP_n$. Consequently, it follows $VP_n \leq PL_n$.* *Proof.* If we observe, the defined endomorphism's image contains the following non-trivial relations: $$\lambda_{ij}\lambda_{kl} =\lambda_{kl}\lambda_{ij},~~\lambda_{ji}\lambda_{kl} =\lambda_{kl}\lambda_{ji}$$ $$\lambda_{ij}\lambda_{lk} =\lambda_{lk}\lambda_{ij},~~\lambda_{ji}\lambda_{lk} =\lambda_{lk}\lambda_{ji}$$ $$\lambda_{ij}\lambda_{ik} \lambda_{jk} =\lambda_{jk}\lambda_{ik} \lambda_{ij} \text{, \hspace{0.3cm}} \lambda_{ji}\lambda_{ki} \lambda_{kj}=\lambda_{kj} \lambda_{ki}\lambda_{ji},$$ $$\lambda_{ji} \lambda_{jk} \lambda_{ik}=\lambda_{ik} \lambda_{jk} \lambda_{ji}\text{, \hspace{0.3cm}} \lambda_{ij} \lambda_{kj}\lambda_{ki} =\lambda_{ki} \lambda_{kj}\lambda_{ij}$$ $$\lambda_{ik} \lambda_{ij} \lambda_{kj}=\lambda_{kj}\lambda_{ij}\lambda_{ik}\text{, \hspace{0.3cm}} \lambda_{ki}\lambda_{ji}\lambda_{jk} = \lambda_{jk} \lambda_{ji}\lambda_{ki}$$ All these relations satisfied in $VP_n$. Hence $VP_n \leq PL_n$. ◻ As corollary we get **Corollary 13**. The virtual pure braid group $VP_n$ is a subgroup of $TVP_n$. ## Presentation of $TVH_n$ {#tvhn} Consider the following elements of $TVB_n$: $$x_{i,i+1}=\sigma_i,\text{ } x_{i+1,i}=\rho_i\sigma_i\rho_i, \text{ } i=1,\ldots, n-1,$$ $$x_{i,j}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\sigma_{i,i+1}\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1},$$ $$x_{j,i}=\rho_{j-1}\rho_{j-2}\cdots\rho_{i+1}\rho_i\sigma_i\rho_i\rho_{i+1}\cdots\rho_{j-2}\rho_{j-1},\text{ } 1 \leq i \leq j-1 \leq n-1.$$ **Theorem 14**. *The group $TVH_n$ admits a presentation with the generators\ $x_{kl}$, $1 \leq k\neq l \leq n$, and $\gamma_j$, $1\leq j\leq n$. The defining relations are as follows: $$\begin{aligned} x_{ij}x_{kl} &=x_{kl}x_{ij}\label{comm-clas2}\\ x_{ik}x_{kj}x_{ik} &=x_{kj}x_{ik}x_{kj}\label{2classical2}\\ \gamma_i^2 &=1\label{g5}\\ \gamma_i\gamma_j &=\gamma_j\gamma_i\label{g6}\\ x_{ij}\gamma_k & =\gamma_kx_{ij}\label{g7}\\ x_{ij} &=\gamma_i\gamma_jx_{ji}\gamma_j\gamma_i\label{g8},\end{aligned}$$ where distinct letters stand for distinct indices.* *Proof.* It is similar to the proof of Theorem [Theorem 5](#sTVP_n){reference-type="ref" reference="sTVP_n"}. ◻ We can establish an alternative representation of the group $TVH_n$ by applying the relations ([\[g8\]](#g8){reference-type="ref" reference="g8"}) as outlined in Theorem [Theorem 14](#main2){reference-type="ref" reference="main2"}. **Corollary 15**. The group $TVH_n$ admits a presentation with the generators\ $x_{kl},~1 \leq k< l \leq n$, and $\gamma_j$, $1\leq j\leq n$. The defining relations are as follows: $$\begin{aligned} x_{ij} \, x_{kl} &=x_{kl} \, x_{ij},\label{2comm-clas}\\ x_{ik} \, x_{kj} \, x_{ik} &=x_{kj}\, x_{ik} \, x_{kj},\label{1classical}\\ x_{ij} \, x_{kj}^{(kj)} \, x_{ij} &=x_{kj}^{(kj)} \, x_{ij} \, x_{kj}^{(kj)},\label{2classical}\\ x_{ij}^{(ij)} \, x_{ik} \, x_{ij}^{(ij)} &=x_{ik} \, x_{ij}^{(ij)} \, x_{ik},\label{3classical}\\ x_{ik}^{(ik)} \, x_{kj}^{(kj)} \, x_{ik}^{(ik)} &=x_{kj}^{(kj)} \, x_{ik}^{(ik)} \, x_{kj}^{(kj)},\label{4classical}\\ \gamma_i^2 &=1,\label{2g1}\\ \gamma_i \, \gamma_j &=\gamma_j \, \gamma_i,\label{2g2}\\ x_{ij} \, \gamma_k & =\gamma_k \, \lambda_{ij}\label{2g3},\end{aligned}$$ where we denoted $x^{(ij)}_{ij}=x^{\gamma_i\gamma_j}$ and, as usual, distinct letters stand for distinct indices. It is easy to see that the map $$\psi_H \colon TVH_n \to A_n = \langle \gamma_1, \gamma_2, \ldots, \gamma_n \rangle,$$ which is defined on the generators $x_{kl} \mapsto e,~1 \leq k\neq l \leq n$, $\gamma_j \mapsto \gamma_j$, $1\leq j\leq n$, can be extended to an endomorphism of $TVH_n$. Similar to the methodology used previously for $\ker(\psi_P)$ it is ease to prove **Theorem 16**. *The group $HL_n = \ker(\psi_H)$ is generated by elements $$x_{ij}^{(0)} = x_{ij}, ~x_{ij}^{(i)} = x_{ij}^{\gamma_i},~ x_{ij}^{(j)} = x_{ij}^{\gamma_j},~ x_{ij}^{(ij)} = x_{ij}^{\gamma_i \gamma_j}, ~~1\leq i<j \leq n,$$ and relations are generated by conjugating the relations ([\[2comm-clas\]](#2comm-clas){reference-type="ref" reference="2comm-clas"}) to ([\[4classical\]](#4classical){reference-type="ref" reference="4classical"}) with elements from $A_n$.* As corollary we get **Corollary 17**. $TVH_n = HL_n \rtimes A_n$. The natural question that arises is whether $TVP_n$ is isomorphic to $TVH_n$. We will establish this in the subsequent part. It is not difficult to find, $$TVH_2= \langle x_{12},x_{21},\gamma_1,\gamma_2 | x_{12}=\gamma_1\gamma_2x_{21}\gamma_2\gamma_1, \gamma_1\gamma_2=\gamma_2\gamma_1, \gamma_1^2=\gamma_2^2=1\rangle.$$ Hence, $TVH_2 \cong TVP_2$. But in general case we have **Proposition 18**. The group $TVH_n$ and $TVP_n$ are not isomorphic for $n\geq 3$. *Proof.* It is sufficient to prove that the abelianisation of $TVP_n$ and $TVH_n$ are different. From the relation ([\[g4\]](#g4){reference-type="ref" reference="g4"}) follows that in the abelianisation of $TVP_n$, $\lambda_{ij}=\lambda_{ji}$. So, abelianisation of $TVP_n$ is isomorphic to $\mathbb{Z}^{\frac{n(n-1)}{2}}\bigoplus \mathbb{Z}_2^n$. By [@BB Proposition 19] and relation ([\[g8\]](#g8){reference-type="ref" reference="g8"}) in the abelianisation of $TVH_n$, all $x_{ij}$ are identified. Hence, abelianisation of $TVH_n$ is $\mathbb{Z} \bigoplus \mathbb{Z}_2^n$. ◻ # Maps $\varphi_{PT}$ and $\varphi_{HT}$ {#PT} Let us defined two endomorphism of $TVB_n$ by the actions on the generators, $$\varphi_{PT} \colon TVB_n \to TS_n,~~\sigma_i \mapsto \rho_i, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto \gamma_j,~~j=1, 2, \ldots, n,$$ $$\varphi_{HT} \colon TVB_n \to TS_n,~~\sigma_i \mapsto e, ~~\rho_i \mapsto \rho_i,~~i=1, 2, \ldots,n-1,~~\gamma_j \mapsto \gamma_j,~~j=1, 2, \ldots, n,$$ where $TS_n$ is the subgroup of $TVB_n$ that is generated by $$\rho_1, \rho_2, \ldots, \rho_{n-1}, \gamma_1, \gamma_2, \ldots, \gamma_n.$$ Considering the defining relations of $TVB_n$, we get **Proposition 19**. $$Im(\varphi_{PT}) = Im(\varphi_{HT}) = TS_n = \langle \rho_1, \rho_2, \ldots, \rho_{n-1}, \gamma_1, \gamma_2, \ldots, \gamma_n \rangle$$ is isomorphic to the extended symmetric group and hence, has a presentation $TS_n = A_n \rtimes S_n$. *Proof.* It is need to show that under these endomorphism any relation of $TVB_n$ goes to a relation which holds in extended symmetric group. For $\varphi_{PT}$ it is evident for all relations, except the relation ([\[rel-twist-III\]](#rel-twist-III){reference-type="ref" reference="rel-twist-III"}), $$\rho_i \sigma_i \rho_i = \gamma_{i+1} \gamma_{i} \, \sigma_i \, \gamma_{i} \, \gamma_{i+1},$$ which goes to the relation $$\rho_i = \gamma_{i+1} \gamma_{i} \, \rho_i\, \gamma_{i} \, \gamma_{i+1} ~\Leftrightarrow~e = (\rho_i \gamma_{i+1} \gamma_{i} \, \rho_i ) \, \gamma_{i} \, \gamma_{i+1},$$ and using ([\[rel-bv\]](#rel-bv){reference-type="ref" reference="rel-bv"}), $$\rho_i \gamma_i = \gamma_{i+1} \rho_i,$$ we get $$e = \gamma_{i} \gamma_{i+1} \, \gamma_{i} \, \gamma_{i+1}.$$ Since $\gamma_{i}$ and $\gamma_{i+1}$ are commute, we get the trivial relation. For the endomorphism $\varphi_{HT}$ the claim is evident. ◻ Further we will find sets of generators and defining relations for $\ker(\varphi_{PT})$ and $\ker(\varphi_{HT})$. **Lemma 20**. In $TVP_n$ the following conjugation rules hold $$\lambda_{ij}^{\gamma_k} = \left\{ \begin{array}{ll} \vspace{0.1 cm} \lambda_{ij} & \text{for }~ i<k<j \text{ or } k<i \text{ or } k>j, \\ \vspace{0.1 cm} \lambda_{ji}^{\gamma_j} & \text{ for } k=i, \\ \vspace{0.1 cm} \lambda_{ji}^{\gamma_i} & \text{ for } k=j. \end{array} \right.$$ Using this lemma and the presentation of $TVP_n$ it is easy to prove the following. **Proposition 21**. The kernel $PT_n = \ker(\varphi_{PT})$ is the group that is generated by the elements $$\lambda_{ij},~~\lambda^{\gamma_i}_{ij},~~\lambda^{\gamma_j}_{ij},~~1 \leq i\neq j \leq n,$$ with defining relations ([\[comm-clas\]](#comm-clas){reference-type="ref" reference="comm-clas"}), ([\[classical\]](#classical){reference-type="ref" reference="classical"}), relations from Lemma [Lemma 20](#conj1){reference-type="ref" reference="conj1"}, and the following relations: $$\lambda_{ij}^{m}\lambda_{kl}^{n}=\lambda_{kl}^{n}\lambda_{ij}^{m}, \text{ where } m\in \{e,\gamma_i,\gamma_j\}, \text{ and } n\in \{e, \gamma_k,\gamma_l\},$$ $$\lambda_{ki}\lambda_{kj}^{\gamma_j}\lambda_{ij}^{\gamma_j} =\lambda_{ij}^{\gamma_j}\lambda_{kj}^{\gamma_j}\lambda_{ki}\text{, \hspace{0.3cm}} \lambda_{ki}^{\gamma_k}\lambda_{kj}^{\gamma_k}\lambda_{ij}=\lambda_{ij}\lambda_{kj}^{\gamma_k}\lambda_{ki}^{\gamma_k},$$ $$\lambda_{ki}^{\gamma_i}\lambda_{kj}\lambda_{ij}^{\gamma_i} =\lambda_{ij}^{\gamma_i}\lambda_{kj}\lambda_{ki}^{\gamma_i}\text{, \hspace{0.3cm}} \lambda_{ik}\lambda_{kj}^{\gamma_k}\lambda_{ij}^{\gamma_i} =\lambda_{ij}^{\gamma_i}\lambda_{kj}^{\gamma_k}\lambda_{ik},$$ $$\lambda_{ki}^{\gamma_k}\lambda_{jk}\lambda_{ij}^{\gamma_j} =\lambda_{ij}^{\gamma_j}\lambda_{jk}\lambda_{ki}^{\gamma_k} \text{, \hspace{0.3cm}} \lambda_{ki}^{\gamma_i}\lambda_{kj}^{\gamma_j}\lambda_{ji}=\lambda_{ji}\lambda_{kj}^{\gamma_j}\lambda_{ki}^{\gamma_i},$$ $$\lambda_{ik}\lambda_{jk}\lambda_{ji} =\lambda_{ji}\lambda_{jk}\lambda_{ik}.$$ As corollary we get **Corollary 22**. $TVB_n = PT_n \rtimes TS_n$. Comparing the presentation of of $PT_n$ with the presentation of $PL_n$, we can see that these groups are isomorphic. **Proposition 23**. $PT_n \cong PL_n$ for all $n \geq 2$. *Proof.* From Lemma 4.2, we can notice, for $i>j$; $\lambda_{ij}^{\gamma_i}=\lambda_{ji}^{\gamma_j}$ and $\lambda_{ij}^{\gamma_j}=\lambda_{ji}^{\gamma_i}$. Also, $\lambda_{ij}^{\gamma_i}=\lambda_{ji}^{\gamma_j}$ implies $\lambda_{ij}=\lambda_{ji}^{\gamma_j\gamma_i}$. So, we can remove elements $\lambda^{\gamma_k}_{ij}$ for $i > j$ and replace $\lambda_{ij}$ by $\lambda_{ji}^{\gamma_j\gamma_i}$ for $i > j$ . Therefore, proved that the generators of the groups $PL_n$ and $PT_n$ are identical, it follows that the relations will also be the same. ◻ We want to find the kernel $\ker(\varphi_{HT})$. **Example 24**. Let us consider the case $n=3$. In this case, $\ker(\varphi_{HT})$ is the normal closure of $B_3$ in $TVB_3$. As we know, the normal closure of $B_3$ in $VB_3$ is the subgroup $VH_3$ which is generated by elements $$x_{12} = \sigma_1,~~x_{13} = \rho_2 \sigma_1 \rho_2,~~x_{23} = \sigma_2,$$ $$x_{21} = \rho_1 \sigma_1 \rho_1,~~x_{31} = \rho_2 \rho_1 \sigma_1 \rho_1 \rho_2,~~x_{32} = \rho_2 \sigma_2 \rho_2.$$ In $TVB_3$ we have relations $$\gamma_1 \sigma_2 = \sigma_2 \gamma_1,~~\gamma_3 \sigma_1 = \sigma_1 \gamma_3,$$ $$\rho_1 \sigma_1 \rho_1 = \gamma_2 \gamma_1 \sigma_1 \gamma_1 \gamma_2,~~\rho_2 \sigma_2 \rho_2 = \gamma_3 \gamma_2 \sigma_2 \gamma_2 \gamma_3.$$ We can rewrite these relations in the form $$x_{23}^{\gamma_1} = x_{23},~~~x_{12}^{\gamma_3} = x_{12},$$ $$x_{21}^{ \gamma_2} = x_{12}^{\gamma_1},~~~x_{32}^{ \gamma_3} = x_{23}^{\gamma_2}.$$ Conjugating these relations by $\rho_1$, we get $$x_{13}^{\gamma_2} = x_{13},~~~x_{21}^{\gamma_3} = x_{21},$$ $$x_{12}^{ \gamma_1} = x_{21}^{\gamma_2},~~~x_{31}^{ \gamma_3} = x_{13}^{\gamma_1}.$$ Conjugating these relations by $\rho_2$, we get $$x_{32}^{\gamma_1} = x_{32},~~~x_{13}^{\gamma_2} = x_{13},$$ $$x_{31}^{ \gamma_3} = x_{13}^{\gamma_1},~~~x_{23}^{ \gamma_2} = x_{32}^{\gamma_3}.$$ Analysing the conjugation rules from this example, one can prove the next lemma. **Lemma 25**. In $TVH_n$ the following conjugation rules hold $$x_{ij}^{\gamma_k} = \left\{ \begin{array}{ll} \vspace{0.1 cm} x_{ij} & \text{for }~ i<k<j \text{ or } k<i \text{ or } k>j, \\ \vspace{0.1 cm} x_{ji}^{\gamma_j} & \text{ for } k=i, \\ \vspace{0.1 cm} x_{ji}^{\gamma_i} & \text{ for } k=j. \end{array} \right.$$ Using this lemma and the presentation of $TVH_n$ it is easy to prove **Proposition 26**. The kernel $HT_n=\ker(\varphi_{HT})$ is the group that is generated by the elements $$x_{ij},~~x^{\gamma_i}_{ij},~~ x^{\gamma_j}_{ij},~~1 \leq i\neq j \leq n,$$ with defining relations ([\[comm-clas2\]](#comm-clas2){reference-type="ref" reference="comm-clas2"}), ([\[2classical2\]](#2classical2){reference-type="ref" reference="2classical2"}), relations from Lemma [Lemma 25](#conj){reference-type="ref" reference="conj"} and the following relations: $$x_{ij}^{\gamma_m}x_{kl}^{\gamma_n} =x_{kl}^{\gamma_n}x_{ij}^{\gamma_m}, \text{ where } m\in \{i,j\}, \text{ and } n\in \{k,l\},$$ $$x_{ik}^{\gamma_i}x_{kj}x_{ik}^{\gamma_i} =x_{kj}x_{ik}^{\gamma_i}x_{kj}\text{ , }x_{ik}^{\gamma_k}x_{kj}^{\gamma_k}x_{ik}^{\gamma_k} =x_{kj}^{\gamma_k}x_{ik}^{\gamma_k}x_{kj}^{\gamma_k},$$ $$x_{ik}x_{kj}^{\gamma_j}x_{ik} =x_{kj}^{\gamma_j}x_{ik}x_{kj}^{\gamma_j} \text{ , }x_{ki}x_{kj}^{\gamma_k}x_{ki} =x_{kj}^{\gamma_k}x_{ki}x_{kj}^{\gamma_k},$$ $$x_{ik}^{\gamma_i}x_{kj}^{\gamma_j}x_{ik}^{\gamma_i} =x_{kj}^{\gamma_j}x_{ik}^{\gamma_i}x_{kj}^{\gamma_j} \text{ , }x_{ik}^{\gamma_k}x_{jk}x_{ik}^{\gamma_k} =x_{jk}x_{ik}^{\gamma_k}x_{jk},$$ $$x_{ki}x_{jk}x_{ki}=x_{jk}x_{ki}x_{jk}.$$ **Corollary 27**. $TVB_n = HT_n \rtimes TS_n$. Likewise Proposition [Proposition 23](#isom1){reference-type="ref" reference="isom1"}, it can be proved the next proposition. **Proposition 28**. $HT_n \cong HL_n$ for all $n \geq 2$. From results of this paper we get the follows proposition **Proposition 29**. 1) In the next diagram rows and columns are short exact sequences: $$\begin{matrix} & & 1 & & 1 & & 1 & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 1 & \longrightarrow & PL_n = \ker(\psi_P)& \overset{\cong}{\longrightarrow} & PT_n = \ker(\varphi_{PT}) & \longrightarrow & 1 & \longrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 1 & \longrightarrow & TVP_n = \ker(\varphi_{P}) & \longrightarrow & TVB_n & \overset{\varphi_{P}}{\longrightarrow} & S_n & \longrightarrow & 1 \\ & & \downarrow {\text{\tiny $\psi_P$} } & & \downarrow {\text{\tiny $\varphi_{PT}$} } & & \| & & \\ 1 & \longrightarrow & A_n & \longrightarrow & TS_n & \longrightarrow & S_n & \longrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 1 & & 1 & & 1 & \\ \end{matrix}$$ 2\) In the next diagram rows and columns are short exact sequences: $$\begin{matrix} & & 1 & & 1 & & 1 & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 1 & \longrightarrow &HL_n = \ker(\psi_T) & \overset{\cong}{\longrightarrow} & HT_n = \ker(\varphi_{HT}) & \longrightarrow & 1 & \longrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 1 & \longrightarrow & TVH_n = \ker(\varphi_{T}) & \longrightarrow & TVB_n & \overset{\varphi_{T}}{\longrightarrow} & S_n & \longrightarrow & 1 \\ & & \downarrow {\text{\tiny $\psi_T$} } & & \downarrow {\text{\tiny $\varphi_{HT}$} } & & \| & & \\ 1 & \longrightarrow & A_n & \longrightarrow & TS_n & \longrightarrow & S_n & \longrightarrow & 1 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 1 & & 1 & & 1 & \\ \end{matrix}$$ # Concluding remarks {#concluding-remarks .unnumbered} In this paper, we study twisted virtual braid group and some of its subgroups. As a future work, it will be interesting to find an answers on the next questions. 1\) There exists some representations of $VB_n$ by automorphisms of some groups [@BN; @BMN; @BKV]. Is it possible to extend these representations to representations of $TVB_n$? 2\) M. O. Bourgoin [@Bour] introduced twisted knot theory as a generalization of the classical knot theory. Is it possible to use representations of $TVB_n$ construct group invariants for twisted virtual knots? For virtual knots, this has been done in the work [@BMN-1]. 3\) The concept of a subgroup of camomile-type were introduced in [@BK-1] (see also [@BW]). Is it true that $TVP_n$, $n\geq 4$ is a subgroup of camomile type in $TVB_n$ with the highlighted petal $TVP_4$? HD E. Artin, *Theory of Braids*, Annals of Math., 1946, 48, 101--126. M. O. Bourgoin, *Twisted link theory*, Algebr. Geom. Topol., 8, no. 3 (2008), 1249--1279. V. G. Bardakov, *The virtual and universal braids*, Fund. Math., 184, no. 1 (2004), 1--18. V. G. Bardakov, P. Bellingeri, *Combinatorial properties of virtual braids*, Topology Appl., 156, no. 6 (2009), 1071--1082. V. G. Bardakov, M. V. Neshchadim, *On a representation of virtual braids by automorphisms.* (Russian) Algebra Logika, 56, no. 5 (2017), 539--547; translation in Algebra Logic, 56, no. 5 (2017), 355--361. V. G. Bardakov, Yu. A. Mikhalchishina, and M. V. Neshchadim, *Representations of virtual braids by automorphisms and virtual knot group*, J. Knot Theory Ramif., 26, no. 1 (2017) 1750003 (17 pages). V. G. Bardakov, Yu. A. Mikhalchishina, and M. V. Neshchadim, *Virtual link groups*, Siberian Mathematical Journal, 58, no. 5 (2017), 765--777. V. G. Bardakov, P. Bellingeri, *On Representations of Braids as Automorphisms of Free Groups and Corresponding Linear Representations*, Contemporary Mathematics, 670, 2016, 285--298. V. G. Bardakov and T. A. Kozlovskaya, *On 3-strand singular pure braid group*, J. Knot Theory Ramif., 29, no. 10, (2020) 2042001(20 pages). V. G. Bardakov and T. A. Kozlovskaya, *Singular braids, singular links and subgroups of camomile type*, ArXiv:2212.08267. V. G. Bardakov, R. Mikhailov, V. V. Vershinin and J. Wu, *On the pure virtual braid group $PV_3$*, Commun. in Algebra, 44, no. 3 (2016), 1350--1378. V. Bardakov, I. Emel'yanenkov, M. Ivanov, T. Kozlovskaya, T. Nasybullov, A. Vesnin, *Virtual and universal braid groups, their quotients and representations*, J. Group Theory, 25, no 4 (2022), 679-712. DOI: 10.1515/jgth-2021-0114. V. G. Bardakov, J. Wu, *Lifting theorem for the virtual pure braid groups*, arXiv:2002.08686. J. S. Birman, *New points of view in knot theory*, Bull. Amer. Math. Soc. 28 (1993), 253--287. J. S. Birman, *Braids, links and mapping class group*, Annals of Mathematics Studies, no. 82. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. ix+228 pp. K. Gongopadhyay, T. Kozlovskaya, O. Mamonov, *On some decompositions of the 3-strand singular braid group*, Topology Appl. 283, 107398, 2020. L. Kauffman, *Virtual knot theory*, Eur. J. Comb., 20, no. 7 (1999), 663--690. L. Kauffman, S. Lambropoulou, *Virtual braids*, Fund. Math., 184, no. 1 (2004), 159--186. T. A. Kozlovskaya, *Structure of 4-strand singular pure braid group*, Siberian Electronic Mathematical Reports, 19, no. 1 (2022), 18--33, DOI: 10.33048/semi.2022.19.002 W. Magnus, A. Karrass, D. Solitar, *Combinatorial group theory*, Interscience Publishers, New York, 1996. A. A. Markoff, *Foundations of the algebraic theory of braids*, Trudy Mat. Inst. Steklova, no. 16 (1945), 1--54. K. Negi, M. Prabhakar, and S. Kamada, *Twisted virtual braids and twisted links*, arXiv:2302.13244. L. Rabenda, *Memoire de DEA (Master thesis)*, Universite de Bourgogne, 2003. N. Kamada, *Polynomial invariants and quandles of twisted links*, Topology Appl. 159, no. 4 (2012), 999--1006. N. Kamada and S. 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arxiv_math

--- abstract: | This short note provides explicit solutions to the linearized Boussinesq equations around the stably stratified Couette flow posed on ${\mathbb T}\times{\mathbb R}$. We consider the long-time behavior of such solutions and prove inviscid damping of the perturbed density and velocity field for any positive Richardson number, with optimal rates. The explicit solution is obtained through the limiting absorption principle whereas the inviscid damping is proved using oscillatory integral methods. address: - Department of Mathematics, Imperial College London, London, SW7 2AZ, UK - Department of Mathematics, Imperial College London, London, SW7 2AZ, UK author: - Michele Coti Zelati - Marc Nualart bibliography: - CZN-InvBoussStripBiblio.bib title: Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip --- # Introduction {#inviscid} The Euler equations under the Boussinesq approximation $$\label{eq:EBintro} \begin{aligned} (\partial_t+\tilde\boldsymbol{u}\cdot\nabla)\tilde{\omega}&= - \mathfrak{g}\,\partial_x\tilde\rho,\\ (\partial_t+\tilde\boldsymbol{u}\cdot\nabla)\tilde\rho&=0, \end{aligned}$$ models the evolution of an incompressible, non-homogeneous ideal fluid whose velocity field is $\tilde\boldsymbol{u}=\nabla^\perp\Delta^{-1}\tilde{\omega}$, with associated vorticity $\tilde{\omega}=\nabla^\perp \cdot \tilde\boldsymbol{u}$ and where the density of the fluid is given by $\tilde\rho$. Here, $\mathfrak{g}$ is the gravity constant. The physical domain in which we consider the Euler-Boussinesq system [\[eq:EBintro\]](#eq:EBintro){reference-type="eqref" reference="eq:EBintro"} is the periodic strip ${\mathbb T}\times{\mathbb R}$, where $$\label{eq:StratCouette} \bar{\boldsymbol{u}}=(y,0), \quad \bar{\rho}(y)=1-\vartheta y, \quad \partial_y p=-\mathfrak{g}\bar{\rho}(y),$$ constitutes a steady solution for the equations of motion and represents a stably stratified Couette flow whose density slope is $\vartheta>0$. Our interest lies in describing the linearized long-time dynamics of solutions to [\[eq:EBintro\]](#eq:EBintro){reference-type="eqref" reference="eq:EBintro"} that are near the stationary configuration [\[eq:StratCouette\]](#eq:StratCouette){reference-type="eqref" reference="eq:StratCouette"}. As such, we consider the perturbed velocity $\tilde{\boldsymbol{u}}=\bar{\boldsymbol{u}}+\boldsymbol{u}$ and density profile $\tilde{\rho}=\bar{\rho}+\vartheta\rho$, and define the corresponding vorticity perturbation ${\omega}=\nabla^\perp\cdot \boldsymbol{u}$. The linearized Euler-Boussinesq system [\[eq:EBintro\]](#eq:EBintro){reference-type="eqref" reference="eq:EBintro"} nearby the stably stratified Couette flow [\[eq:StratCouette\]](#eq:StratCouette){reference-type="eqref" reference="eq:StratCouette"} then takes the form $$\label{eq:linEulerBouss} \begin{cases} \partial_t{\omega}+ y\partial_x{\omega}=-\beta^2\partial_x\rho \\ \partial_t\rho + y\partial_x\rho =\partial_x\psi,\\ {\Delta}\psi={\omega}, \end{cases}$$ with $\psi$ being the stream-function of the velocity field $\boldsymbol{u}$ and $\beta=\sqrt{\vartheta \mathfrak{g}} >0$. In the periodic setting $x\in{\mathbb T}$, it is advantageous to write $$\label{xFourier} {\omega}(t,x,y)=\sum_{m\in{\mathbb Z}}{\omega}_m(t,y){\rm e}^{imx},\quad \rho(t,x,y)=\sum_{m\in{\mathbb Z}}\rho_m(t,y){\rm e}^{imx}, \quad \psi(t,x,y)=\sum_{m\in{\mathbb Z}}\psi_{m}(t,y){\rm e}^{imx}.$$ Thus, [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} now reads $$\begin{aligned} (\partial_t +imy) {\omega}_m &= -im\beta^2\rho_m \label{eq:lin om}\\ (\partial_t + imy) \rho_m &= im\psi_m, \label{eq: lin rhom}\end{aligned}$$ where further $$\label{eq:defDm} \begin{cases} {\Delta}_m\psi_m = {\omega}_m, \\ \lim_{|y|\to \infty}\psi_m=0, \end{cases} \qquad {\Delta}_m:= \partial_y^2 - m^2,$$ for all $m\in{\mathbb Z}$. Our first result shows that [\[eq:lin om\]](#eq:lin om){reference-type="eqref" reference="eq:lin om"}-[\[eq: lin rhom\]](#eq: lin rhom){reference-type="eqref" reference="eq: lin rhom"}, and thus [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} through [\[xFourier\]](#xFourier){reference-type="eqref" reference="xFourier"} can be solved explicitly in the physical space $y\in{\mathbb R}$ as truncated convolutions of oscillating Whittaker functions against suitable combinations of the initial data. The Whittaker functions $W_{0,\gamma}$ with $\gamma\in{\mathbb C}$ satisfy (see [@NIST; @Whittaker03]) $$\label{eq:W0gamma} \partial_\zeta^2W_{0,\gamma} + \left( -\frac14 + \frac{1/4 - \gamma^2}{\zeta^2}\right)W_{0,\gamma}=0, \qquad W_{0,\gamma}(\zeta)\sim {\rm e}^{-\frac12\zeta} \quad\text{ as }\quad \zeta\rightarrow\infty,$$ and constitute the main ingredient in the construction of the explicit solutions. **Theorem 1**. *Let $\beta> 0$. Given initial conditions $({\omega}^0, \rho^0)$ such that $$\label{eq:zeroxave} \int_{\mathbb T}{\omega}^0(x,y)\mathrm{d}x = \int_{\mathbb T}\rho^0(x,y)\mathrm{d}x =0,$$ the unique solution to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} is given through [\[xFourier\]](#xFourier){reference-type="eqref" reference="xFourier"} and [\[eq:defDm\]](#eq:defDm){reference-type="eqref" reference="eq:defDm"} by $\psi_0=\rho_0\equiv 0$, $$\label{eq:psim} \begin{aligned} \psi_m(t,y)&=\frac{{\rm e}^{-imyt}}{2|m|\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}W(\eta) \int_0^\infty W(\xi)G_{m}(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}W(\eta)\int_0^\infty W(\xi)G_{m}(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right) \end{aligned}$$ and $$\label{eq:rhom} \begin{aligned} \rho_m(t,y)&=\frac{{\rm e}^{-imyt}}{2|m|\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}\frac{W(\eta)}{\eta} \int_0^\infty W(\xi)G_{m}(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}\frac{W(\eta)}{\eta}\int_0^\infty W(\xi)G_{m}(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right), \end{aligned}$$ for all $m\neq 0$. Here, for $\gamma:=\sqrt{1/4-\beta^2}$ we denote $W(\cdot)=W_{0,\gamma}(2|m|\cdot)$, where $W_{0,\gamma}$ satisfies [\[eq:W0gamma\]](#eq:W0gamma){reference-type="eqref" reference="eq:W0gamma"}. Further, $$G_m(\eta,\xi,y)={\Delta}_m\left(\rho_m^0(\xi+y-\eta)-\frac{1}{\beta^2}\xi {\omega}_m^0(\xi+y-\eta)\right).$$* The zero modes in $x\in{\mathbb T}$ of initial data evolving according to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} are constants of motion, so that [\[eq:zeroxave\]](#eq:zeroxave){reference-type="eqref" reference="eq:zeroxave"} does not actually constitute a restriction on the initial data. The expressions [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} give rise to real-valued solutions $\psi$, ${\omega}$ and $\rho$ via [\[xFourier\]](#xFourier){reference-type="eqref" reference="xFourier"} due to the fact that $\psi_m = \overline{\psi_{-m}}$ and $\rho_m$ = $\overline{\rho_{-m}}$, which are straightforward consequences of ${\omega}^0$ and $\rho^0$ being real-valued. In particular, we shall assume without loss of generality throughout the article that $m\geq 1$. Our second result consists in the derivation of sharp decay estimates, which quantify the phenomenon of inviscid damping. **Theorem 2**. *Let $\beta>0$ and assume that the initial data $({\omega}^0,\rho^0)$ to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} satisfies [\[eq:zeroxave\]](#eq:zeroxave){reference-type="eqref" reference="eq:zeroxave"}. Let $\boldsymbol{u}=(u^x,u^y)=\nabla^\perp\psi=(-\partial_y\psi,\partial_x\psi)$ be the corresponding velocity field. We have the following estimates.* - *If $\beta^2\neq1/4$, let $\mu={\rm Re}\sqrt{1/4-\beta^2}$ and $\nu={\rm Im}\sqrt{1/4-\beta^2}$. Then, $$\begin{aligned} \Vert u^x(t) \Vert_{L^2}&\lesssim\frac{1}{ t^{\frac12-\mu}}\left( \Vert \rho^0 \Vert_{L^2_xH^3_y} + \Vert {\omega}^0 \Vert_{L^2_xH^3_y}\right), \label{eq:decayvx} \\ \Vert u^y(t) \Vert_{L^2}&\lesssim \frac{1}{ t^{\frac32-\mu}}\left( \Vert \rho^0 \Vert_{L^2_xH^4_y} + \Vert {\omega}^0 \Vert_{L^2_xH^4_y}\right), \label{eq:decayvy}\\ \Vert \rho(t) \Vert_{L^2}&\lesssim \frac{1}{t^{\frac12-\mu}}\left( \Vert \rho^0 \Vert_{L^2_xH^3_y} + \Vert {\omega}^0 \Vert_{L^2_xH^3_y}\right), \label{eq:decayrho} \end{aligned}$$ for every $t\geq 1$.* - *If $\beta^2=1/4$, then $$\begin{aligned} \Vert u^x(t) \Vert_{L^2}&\lesssim \frac{1+\log(t)}{ t^\frac12}\left( \Vert \rho^0 \Vert_{L^2_xH^3_y} + \Vert {\omega}^0 \Vert_{L^2_xH^3_y}\right), \label{eq:decayvxlog}\\ \Vert u^y(t) \Vert_{L^2}&\lesssim \frac{1+\log(t)}{ t^\frac32}\left( \Vert \rho^0 \Vert_{L^2_xH^4_y} + \Vert {\omega}^0 \Vert_{L^2_xH^4_y}\right), \label{eq:decayvylog}\\ \Vert \rho(t) \Vert_{L^2}&\lesssim\frac{1+\log(t)}{t^\frac12}\left( \Vert \rho^0 \Vert_{L^2_xH^3_y} + \Vert {\omega}^0 \Vert_{L^2_xH^3_y}\right), \label{eq:decayrholog} \end{aligned}$$ for every $t\geq 1$.* The inviscid damping estimates [\[eq:decayvx\]](#eq:decayvx){reference-type="eqref" reference="eq:decayvx"}-[\[eq:decayrholog\]](#eq:decayrholog){reference-type="eqref" reference="eq:decayrholog"} describe the long-time dynamics of solutions to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} and show the linear asymptotic stability of the stratified Couette configuration [\[eq:StratCouette\]](#eq:StratCouette){reference-type="eqref" reference="eq:StratCouette"} for the Euler-Boussinesq system [\[eq:EBintro\]](#eq:EBintro){reference-type="eqref" reference="eq:EBintro"}. The decay is produced by two phenomena. Firstly, there is *mixing* due to the background Couette flow and secondly there is *stratification* due to the background density. The effect of mixing has been thoroughly studied in the homogeneous Euler equations both at the linear [@WZZ18; @WZZ19; @Zillinger16; @JiaGev20] and non-linear level [@BM15; @IJnon20; @MZ20]. Estimates analogous to those of Theorem [Theorem 2](#thm:mainstrip){reference-type="ref" reference="thm:mainstrip"} have been already obtained in [@YL18] using an explicit formula for solutions on the Fourier side (inspired by an early work of Hartman [@Hartman] in 1975), and in [@BCZD22] via an energy method. Our approach is rather based on a stationary-phase type argument, exploiting the explicit solutions of Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"} in physical space and obtaining decay rates related to the regularity (and more precisely on the asymptotic expansion) of the Whittaker functions about the origin. While these formulae do not produce a new result in the periodic strip ${\mathbb T}\times{\mathbb R}$, our method allows to treat the physically relevant case of the periodic channel ${\mathbb T}\times [0,1]$, see [@CZN23chan], and it is therefore more robust in this sense. In [@CZN23chan] explicit solutions are not available, however one similarly can write solutions to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} through oscillatory integrals now involving a limiting absorption principle in which the regularity of the limiting functions (and thus the gained time-decay via stationary-phase arguments) is related to that of the Whittaker functions. Notation and assumptions Throughout the article, we assume that $\beta>0$ and $m\geq 1$. To quantify the regularity of the initial data, for $j\geq 0$ we introduce $$Q_{j,m}=\Vert \rho_m^0 \Vert_{H^{2+j}_y({\mathbb R})} + \Vert {\omega}_m^0 \Vert_{H^{2+j}_y({\mathbb R})}.$$ As usual, we say that $A\lesssim B$ when there exists $C>0$ such that $A\leq CB$. Plan of the article In Section [2](#sec: proof thm sol){reference-type="ref" reference="sec: proof thm sol"} we prove Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"} and in Section [3](#sec:heuristics){reference-type="ref" reference="sec:heuristics"} we provide an heuristic explanation for the form of the solutions [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"}, [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"}. Section [4](#sec: time decay estimates){reference-type="ref" reference="sec: time decay estimates"} is devoted to the proof of Theorem [Theorem 2](#thm:mainstrip){reference-type="ref" reference="thm:mainstrip"}. In the Appendix [5](#app: Whittaker){reference-type="ref" reference="app: Whittaker"} we provide the main asymptotic expansions for the Whittaker functions that are used to establish Theorem [Theorem 2](#thm:mainstrip){reference-type="ref" reference="thm:mainstrip"}. # Proof of Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"} {#sec: proof thm sol} The proof consists on showing that $\psi_m$, $\rho_m$ and ${\omega}_m$ given by [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"}, [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} and [\[eq:defDm\]](#eq:defDm){reference-type="eqref" reference="eq:defDm"} respectively, satisfy the linearized Euler-Boussinesq equations [\[eq:lin om\]](#eq:lin om){reference-type="eqref" reference="eq:lin om"}-[\[eq: lin rhom\]](#eq: lin rhom){reference-type="eqref" reference="eq: lin rhom"}. According to [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"}, [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} and [\[eq:defDm\]](#eq:defDm){reference-type="eqref" reference="eq:defDm"}, we write $$\label{eq:defPsiRho} \psi_m(t,y)={\rm e}^{-imyt}\Psi_m(t,y), \qquad \rho_m(t,y)={\rm e}^{-imty}{\mathrm{P}}_m(t,y), \qquad {\omega}_m(t,y)={\rm e}^{-imyt}{\Omega}_m(t,y),$$ where $$\label{eq:defOm} {\Omega}_m :=-m^2t^2\Psi_m -2imt\partial_y\Psi_m +{\Delta}_m\Psi_m .$$ Now, with this formulation we must check that ${\Omega}_m,\Psi_m$ satisfy $$\begin{aligned} \partial_t{\Omega}_m&=-im\beta^2{\mathrm{P}}_m \label{eq:Om}\\ \partial_t{\mathrm{P}}_m&=im\Psi_m. \label{eq:Rhom}\end{aligned}$$ Clearly, [\[eq:Rhom\]](#eq:Rhom){reference-type="eqref" reference="eq:Rhom"} follows directly from [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"}, [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} and [\[eq:defPsiRho\]](#eq:defPsiRho){reference-type="eqref" reference="eq:defPsiRho"}. To show [\[eq:Om\]](#eq:Om){reference-type="eqref" reference="eq:Om"}, we first notice that $$\partial_t{\Omega}_m=-2m^2t\Psi_m -m^2t^2\partial_t\Psi_m -2im\partial_y\Psi_m -2imt\partial_t\partial_y\Psi_{m}+ {\Delta}_m\partial_t\Psi_m,$$ where, from [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:defPsiRho\]](#eq:defPsiRho){reference-type="eqref" reference="eq:defPsiRho"}, we have that $$\begin{aligned} -2m^2t\Psi_m&=-2m^2t\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty \frac{1}{imt}\partial_\eta \left({\rm e}^{im\eta t}\right)W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty \frac{1}{imt}\partial_\eta \left( {\rm e}^{-im\eta t}\right)W(\eta )\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right) \\ &=-2mi\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t}\partial_\eta \left(W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}\partial_\eta \left(W(\eta )\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right), \end{aligned}$$ while $$\begin{aligned} -m^2t^2\partial_t\Psi_m&=-m^2t^2\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t}im\eta W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}im\eta W(\eta )\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right) \\ &=im\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t} \partial_\eta ^2\left( \eta W(\eta) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}\partial_\eta ^2\left(\eta W(\eta)\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right) \end{aligned}$$ and $$\begin{aligned} -2im\partial_y\Psi_m&=-2im\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t}W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}W(\eta )\int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right), \end{aligned}$$ with also $$\begin{aligned} -2imt\partial_t\partial_y\Psi_m&=-2imt\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t}im\eta W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}im\eta W(\eta )\int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right) \\ &=2im\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t} \partial_\eta \left( \eta W(\eta) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}\partial_\eta \left(\eta W(\eta)\int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi\right)\mathrm{d}\eta \right) \end{aligned}$$ and finally $$\begin{aligned} {\Delta}_m\partial_t\Psi_m&=im\frac{\cos(\gamma\pi)}{2m\pi} \left(\int_0^\infty {\rm e}^{im\eta t}\eta W(\eta ) \int_0^\infty W(\xi){\Delta}_mG_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}\eta W(\eta)\int_0^\infty W(\xi){\Delta}_mG_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right). \end{aligned}$$ Therefore, it is straightforward to see that $$\begin{aligned} \partial_t{\Omega}_m=im\frac{\cos(\gamma\pi)}{2m\pi}\left( \int_0^\infty {\rm e}^{im\eta t}{\Omega}_m^{(+)}(\eta )\mathrm{d}\eta + \int_0^\infty {\rm e}^{im\eta t}{\Omega}_m^{(-)}(\eta )\mathrm{d}\eta \right), \end{aligned}$$ where $$\begin{aligned} {\Omega}_m^{(+)}(\eta )&=-2\partial_\eta \left( W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi \right) + \partial_\eta ^2 \left( \eta W(\eta) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi \right) \\ &\quad- 2 W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi + 2\partial_\eta \left( \eta W(\eta) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi \right) \\ &\quad+ \eta W(\eta) \int_0^\infty W(\xi){\Delta}_mG_m(\eta,\xi,y)\mathrm{d}\xi \end{aligned}$$ and similarly $$\begin{aligned} {\Omega}_m^{(-)}(\eta )&=-2\partial_\eta \left( W(\eta ) \int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi \right) + \partial_\eta ^2 \left( \eta W(\eta) \int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi \right) \\ &\quad+ 2 W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi - 2\partial_\eta \left( \eta W(\eta) \int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi \right) \\ &\quad+ \eta W(\eta) \int_0^\infty W(\xi){\Delta}_mG_m(-\eta,-\xi,y)\mathrm{d}\xi. \end{aligned}$$ Now, note that $$\begin{aligned} {\Omega}_m^{(+)}(\eta )&=\eta \partial_\eta ^2 \left( W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi \right) + 2\eta\partial_\eta \left( W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi \right) \\ &\quad+ \eta W(\eta) \int_0^\infty W(\xi){\Delta}_mG_m(\eta,\xi,y)\mathrm{d}\xi \\ &=\eta W''(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi + 2 \eta W'(\eta ) \int_0^\infty W(\xi)\partial_\eta G_m(\eta,\xi,y)\mathrm{d}\xi \\ &\quad+ \eta W(\eta ) \int_0^\infty W(\xi)\partial_\eta ^2 G_m(\eta,\xi,y)\mathrm{d}\xi + 2\eta W'(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi \\ &\quad +2\eta W(\eta ) \int_0^\infty W(\xi)\partial_\eta \partial_yG_m(\eta,\xi,y)\mathrm{d}\xi + \eta W(\eta) \int_0^\infty W(\xi){\Delta}_mG_m(\eta,\xi,y)\mathrm{d}\xi \end{aligned}$$ and further observe that $\left(\partial_y+\partial_\eta \right) G_m(\eta ,\xi,y)=0$, from which we deduce that $$\left(\partial_\eta ^2 +2\partial_\eta \partial_y+\partial_y^2\right) G_m(\eta,\xi,y)= (\partial_\eta +\partial_y)^2{\mathcal{G}}_m(\eta ,\xi,y)=0$$ and $${\Omega}_m^{(+)}(\eta )=\eta {\Delta}_mW(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi$$ Now, using [\[eq:W0gamma\]](#eq:W0gamma){reference-type="eqref" reference="eq:W0gamma"}, we see that $${\Delta}_m W(\zeta) +\beta^2\frac{W(\zeta)}{\zeta^2}=0,$$ and thus we can write $${\Omega}_m^{(+)}(\eta )= -\beta^2\frac{W(\eta )}{\eta } \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi$$ and similarly for ${\Omega}_m^{(-)}(\eta )$. We finish by assembling and recognising ${\mathrm{P}}_m$, $$\begin{aligned} \partial_t{\Omega}_m&=im\frac{\cos(\gamma\pi)}{2m\pi}\left( -\int_0^{\infty} {\rm e}^{im\eta t}\beta^2\frac{W(\eta )}{\eta } \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi \mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^{\infty} {\rm e}^{-im\eta t}\beta^2\frac{W(\eta )}{\eta } \int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi \mathrm{d}\eta \right) \\ &=-im\beta^2 {\mathrm{P}}_m \end{aligned}$$ With this, the proof is concluded. # Heuristics for the explicit solutions {#sec:heuristics} The presence of the Whittaker functions in [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} is key in the proof of Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"}, since they relate ${\Omega}_m^{(\pm)}$ to ${\mathrm{P}}_m$ due to [\[eq:W0gamma\]](#eq:W0gamma){reference-type="eqref" reference="eq:W0gamma"}. In fact, this is essentially the main reason why [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} provide solutions to [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"}. However, the proof of Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"} does not explain why these Whittaker functions arise in [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"}. This is precisely the purpose of this section, which sets the framework for obtaining [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} via the method of the limiting absorption principle. Generalized stream-functions and densities Writing [\[eq:lin om\]](#eq:lin om){reference-type="eqref" reference="eq:lin om"}-[\[eq: lin rhom\]](#eq: lin rhom){reference-type="eqref" reference="eq: lin rhom"} in the compact stream-function formulation $$\partial_t \begin{pmatrix} \psi_m \\ \rho_m\end{pmatrix}+imL_m\begin{pmatrix} \psi_m \\ \rho_m \end{pmatrix}=0,$$ we directly obtain its solution as $$\begin{pmatrix} \psi_m \\ \rho_m\end{pmatrix}={\rm e}^{-imL_mt}\begin{pmatrix} \psi_m^0 \\ \rho_m^0 \end{pmatrix},$$ where $L_m$ is the linear operator defined by $$\label{eq:linOP} L_m =\begin{pmatrix} {\Delta}_m^{-1}(y{\Delta}_m) & \beta^2{\Delta}_m^{-1} \\ -1 & y \end{pmatrix}.$$ Using Dunford's formula [@Engel-Nagel; @Taylor-11], we have that $$\label{eq:Dunford} \begin{pmatrix} \psi_m(t,y) \\ \rho_m(t,y) \end{pmatrix} = \frac{1}{2\pi i} \int_{\partial\mathrm{D}}{\rm e}^{-imct} (c-L_m)^{-1}\begin{pmatrix} \psi_m^0(y) \\ \rho_m^0(y) \end{pmatrix} \,\mathrm{d}c,$$ where here $\mathrm{D}$ is any domain containing the spectrum $\sigma(L_m)$. On the periodic strip, the spectrum $\sigma(L_m)$ is continuous and consists on the real line ${\mathbb R}$. Hence, we can reduce the contour of integration to $$\begin{aligned} \label{eq:LAPpsirho} \begin{pmatrix} \psi_m(t,y) \\ \rho_m(t,y) \end{pmatrix} %&= \frac{1}{2\pi i}\lim_{\ep\rightarrow 0} \int_{-\infty}^{\infty} \e^{-imct}\l[ (c-i\ep-L_m)^{-1}-(c+i\ep-L_m)^{-1}\r]\begin{pmatrix} %\psi_m^0 \\ \rho_m^0 %\end{pmatrix} \,\d c \\ &=\frac{1}{2\pi i }\lim_{\varepsilon\rightarrow 0}\int_{-\infty}^{+\infty} {\rm e}^{-imy_0t}\left[(-y_0-i\varepsilon+L_m)^{-1}-(-y_0+i\varepsilon+L_m)^{-1}\right] \begin{pmatrix} \psi_m^0 \\ \rho_m^0 \end{pmatrix}\, \mathrm{d}y_0.\end{aligned}$$ For $\varepsilon>0$, we denote $$\begin{pmatrix} \psi^{\pm}_{m,\varepsilon}(y,y_0) \\ \rho^\pm_{m,\varepsilon}(y,y_0) \end{pmatrix}:=\left( -y_0\pm i\varepsilon+L_m\right)^{-1}\begin{pmatrix} \psi_m^0(y) \\ \rho_m^0(y) \end{pmatrix}$$ and obtain the coupled system of equations for the generalized stream-functions $\psi_{m,\varepsilon}^\pm$ and generalized densities $\rho_{m,\varepsilon}^\pm$ $$\begin{aligned} {\omega}_m^0(y)&=(y-y_0\pm i\varepsilon){\Delta}_m\psi^\pm_{m,\varepsilon}(y,y_0)+\beta^2 \rho^\pm_{m,\varepsilon}(y,y_0), \\ \rho_m^0(y)&=(y-y_0\pm i\varepsilon)\rho^\pm_{m,\varepsilon}(y,y_0) -\psi^\pm_{m,\varepsilon}(y,y_0). \end{aligned}$$ We first solve for the generalized densities $$\label{eq rho m ep} \rho^\pm_{m,\varepsilon}(y,y_0)=\frac{1}{y-y_0\pm i\varepsilon}\left( \rho_m^0(y)+\psi^\pm_{m,\varepsilon}(y,y_0)\right)$$ and from there we obtain the following inhomogeneous *Taylor-Goldstein equation* for the generalized stream-functions $\psi^\pm_{m,\varepsilon}$, $$\tag{TG}\label{eq:TG} {\Delta}_m\psi^{\pm}_{m,\varepsilon}+\beta^2\frac{\psi^\pm_{m,\varepsilon}}{(y-y_0\pm i\varepsilon)^2}=\frac{{\omega}_m^0}{y-y_0\pm i\varepsilon}-\beta^2\frac{\rho_m^0}{(y-y_0\pm i\varepsilon)^2},$$ along with the vanishing of $\psi_{m,\varepsilon}^\pm$ at infinity. Explicit solutions for the generalized stream-functions and densities The Taylor-Goldstein equation [\[eq:TG\]](#eq:TG){reference-type="eqref" reference="eq:TG"} admits a fairly explicit Green's function given by $$\label{eq: Greens} {\mathcal{G}}_{m,\varepsilon}^\pm(y,y_0,z) =-\frac{1}{2m}\begin{cases} W(y_0-z\mp i\varepsilon)W(y-y_0\pm i\varepsilon), & y\geq z, \\ W(z-y_0\pm i\varepsilon)W(y_0-y\mp i\varepsilon), & y\leq z, \end{cases}$$ where we recall $W(\zeta)=W_{0,\gamma}(2m\zeta)$ for $\gamma^2=\frac14 - \beta^2$ and it is such that $$\label{eq:Whittakereqn} \partial^2_{\zeta} W+\left(-{\frac {1}{4}} +{\frac {1/4- (1/4-\beta^2)}{4m^2\zeta^{2}}}\right)W=0, \quad W(\zeta)\sim {\rm e}^{-m\zeta}, \text{ as } \zeta\rightarrow \infty,$$ since the Whittaker function $W_{0,\gamma}$ satisfies [\[eq:W0gamma\]](#eq:W0gamma){reference-type="eqref" reference="eq:W0gamma"}. To obtain suitable formulas for the generalized stream-functions and densities, define $$\label{eq:Fdata} H_{m,\varepsilon}^\pm(z,y_0):={\Delta}_m\rho_m^0(z)-\frac{1}{\beta^2}{\Delta}_m\big((z-y_0\pm i\varepsilon){\omega}_m^0(z)\big).$$ and assume that the initial data vanish at infinity. Then, the solution $\psi^\pm_{m,\varepsilon}(y,y_0)$ to [\[eq:TG\]](#eq:TG){reference-type="eqref" reference="eq:TG"} is $$\label{def psi 2} \begin{aligned} \psi^\pm_{m,\varepsilon}(y,y_0)&= \frac{1}{\beta^2}(y-y_0\pm i\varepsilon){\omega}_m^0(y) -\rho_m^0(y) + \int_{-\infty}^{+\infty} {\mathcal{G}}^\pm_{m,\varepsilon}(y,y_0,z) H_{m,\varepsilon}^\pm(z,y_0) \mathrm{d}z. \end{aligned}$$ and the generalized density is given by $$\label{def rho 2} \rho_{m,\varepsilon}^\pm(y,y_0)=\frac{1}{\beta^2}{\omega}_m^0(y) +\frac{1}{y-y_0\pm i\varepsilon}\int_{-\infty}^{+\infty} {{\mathcal{G}}}_{m,\varepsilon}^\pm(y,y_0,z) H_{m,\varepsilon}^\pm(z,y_0)\mathrm{d}z.$$ The limiting absorption principle With [\[def psi 2\]](#def psi 2){reference-type="eqref" reference="def psi 2"} and [\[def rho 2\]](#def rho 2){reference-type="eqref" reference="def rho 2"} at hand, one may precisely compute the limiting absorption principle, that is, we may precisely compute [\[eq:LAPpsirho\]](#eq:LAPpsirho){reference-type="eqref" reference="eq:LAPpsirho"} For instance, to obtain $\psi_m(t,y)$ one may compute $$\lim_{\varepsilon\rightarrow0}\int_{\mathbb R}{\rm e}^{-imy_0t}\int_{\mathbb R}\left( {\mathcal{G}}_{m,\varepsilon}^-(y,y_0,z)-{\mathcal{G}}_{m,\varepsilon}^+(y,y_0,z)\right) H_{m,\varepsilon}^-(z,y_0) \mathrm{d}z \mathrm{d}y_0$$ and note that with the change of variables $\xi=z-y_0$ and $\eta =y-y_0$, $$\begin{aligned} \int_{\mathbb R}{\mathcal{G}}_m^\pm(y,z)H_{m,\varepsilon}^-(z,y_0)\mathrm{d}z&=-\frac{1}{2m}W(y-y_0\pm i\varepsilon)\int_{-\infty}^y W(y_0-z\mp i\varepsilon)H_{m,\varepsilon}^-(z,y_0)\mathrm{d}z \\ &\quad-\frac{1}{2m}W(y_0-y\mp i\varepsilon)\int_y^\infty W(z-y_0\pm i\varepsilon)H_{m,\varepsilon}^-(z,y_0)\mathrm{d}z \\ &=-\frac{1}{2m}W(\eta \pm i\varepsilon))\int_{-\infty}^\eta W(-\xi\mp i\varepsilon))H_{m,\varepsilon}^-(\xi+y-\eta ,y-\eta )\mathrm{d}\xi \\ &\quad-\frac{1}{2m}W(-\eta \mp i\varepsilon)\int_\eta ^\infty W(\xi\pm i\varepsilon)H_{m,\varepsilon}^-(\xi+y-\eta ,y-\eta )\mathrm{d}\xi.\end{aligned}$$ Setting $G_{m,\varepsilon}^\pm(\eta ,\xi,y):=H_{m,\varepsilon}^\pm(\xi+y-\eta ,y-\eta )$ and $G_m:=\lim_{\varepsilon\rightarrow 0} G_{m,\varepsilon}^\pm=G_{m,0}^\pm$, we find that $$\begin{aligned} \int_{\mathbb R}\left({\mathcal{G}}_m^-(y,z)-{\mathcal{G}}_m^+(y,z)\right)&H_{m,\varepsilon}^-(z,y_0)\mathrm{d}z \\ &=\frac{1}{2m}\Big( W(\eta +i\varepsilon)-W(\eta -i\varepsilon)\Big)\int_{-\infty}^\eta W(-\xi-i\varepsilon)G_{m,\varepsilon}^-(\eta ,\xi,y)\mathrm{d}\xi \\ &\quad+\frac{1}{2m} W(\eta -i\varepsilon)\int_{-\infty}^\eta \Big( W(-\xi-i\varepsilon)-W(-\xi+i\varepsilon)\Big)G_{m,\varepsilon}^-(\eta ,\xi,y)\mathrm{d}\xi \\ &\quad+ \frac{1}{2m}\Big( W(-\eta -i\varepsilon)-W(-\eta +i\varepsilon)\Big)\int_\eta ^\infty W(\xi+i\varepsilon)G_{m,\varepsilon}^-(\eta ,\xi,y)\mathrm{d}\xi \\ &\quad+\frac{1}{2m}W(-\eta +i\varepsilon)\int_\eta ^\infty \Big( W(\xi+i\varepsilon)-W(\xi-i\varepsilon)\Big) G_{m,\varepsilon}^-(\eta ,\xi,y)\mathrm{d}\xi.\end{aligned}$$ Taking the limit as $\varepsilon$ vanishes is not trivial. Indeed, $W$ has a branch cut in the negative real axis, see Appendix [5](#app: Whittaker){reference-type="ref" reference="app: Whittaker"}, and is thus not continuous there. For this reason, we need the analytic continuation of $W$, recorded in the following lemma, whose proof is postponed to Appendix [5](#app: Whittaker){reference-type="ref" reference="app: Whittaker"}. **Lemma 1** (Analytic continuation). *Let $\eta \geq0$ and $0<\varepsilon<1$. Then, $$\lim_{\varepsilon\rightarrow0}W(-\eta +i\varepsilon)-W(-\eta -i\varepsilon)=2i\cos(\gamma\pi)W(\eta).$$* The whole limiting procedure can be carried out rigorously and produces the explicit formulas exhibited in Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"}. However, for the sake of brevity, we opted for showing the validity of the explicit formulas by checking they satisfy the linearized system of equations. When the equations [\[eq:linEulerBouss\]](#eq:linEulerBouss){reference-type="eqref" reference="eq:linEulerBouss"} are posed in ${\mathbb T}\times[0,1]$, the limiting procedure becomes much more complicated. Nevertheless, it is still possible to obtain asymptotic expansions on the resulting stream-function and density near the critical layer that capture the same nature of the explicit formulas [\[eq:psim\]](#eq:psim){reference-type="eqref" reference="eq:psim"} and [\[eq:rhom\]](#eq:rhom){reference-type="eqref" reference="eq:rhom"} for the spatial setting ${\mathbb T}\times{\mathbb R}$, we refer the reader to [@CZN23chan]. # Proof of Theorem [Theorem 2](#thm:mainstrip){reference-type="ref" reference="thm:mainstrip"} {#sec: time decay estimates} In this section we obtain the point-wise decay rates in time for the stream function $\psi_m(t,y)$ and the density $\rho_m(t,y)$. These will be obtained as direct consequence of the following lemma, which concerns the time decay of general oscillatory integrals. Before stating it, we introduce the following spaces of functions. **Definition 2**. For $\delta_0>0$ we define $$X:= \left\lbrace f:[0,\infty)\times{\mathbb R}\rightarrow {\mathbb R}\text{ such that } \Vert f \Vert_{X}:= \Vert f \Vert_{L^\infty_\eta \left( 0,\delta_0;L^2_y({\mathbb R})\right)} < \infty \right\rbrace$$ and also $$Y:= \left\lbrace f:[0,\infty)\times{\mathbb R}\rightarrow {\mathbb R}\text{ such that } \Vert f \Vert_{Y}:= \Vert f \Vert_{L^1_\eta \left(\delta_0,\infty;L^2_y({\mathbb R})\right)} < \infty \right\rbrace.$$ **Lemma 3**. *Let $0<\alpha<1$, $\delta_0=\frac{1}{2m}$ and $t\geq1$. Let $F=F(\eta ,y):(0,\infty)\times{\mathbb R}\rightarrow {\mathbb R}$ be such that $\Vert F(\eta ,\cdot) \Vert_{L^2_y({\mathbb R})}$ vanishes as $\eta \rightarrow+\infty$ and $\partial_\eta F\in Y$. We have the following.* 1. *[\[item:i\]]{#item:i label="item:i"}Assume $F$ admits the decomposition $F(\eta ,y)=\eta ^{-\alpha}E_1(\eta ,y)$, for some $E_1\in X$ and $\partial_\eta F(\eta ,y)=\eta ^{-\alpha-1}E_2(\eta ,y)$ for some $E_2\in X$. Then, $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim \frac{1}{(mt)^{1-\alpha}} \left( \Vert E_1 \Vert_X + \Vert E_2 \Vert_X\right) + \frac{1}{mt}\Vert \partial_\eta F \Vert_Y.$$* 2. *[\[item:ii\]]{#item:ii label="item:ii"} Assume $F$ admits the decomposition $F(\eta ,y)=\eta ^{-\alpha}\left( E_{1,1}(\eta ,y) + \log(\eta )E_{1,2}(\eta ,y)\right)$ and\ $\partial_\eta F(\eta ,y)=\eta ^{-\alpha-1}(E_{2,1}(\eta ,y)+\log(\eta )E_{2,2}(\eta ,y))$ for some $E_{i,j}\in X$, with $i,j\in \left\lbrace 1,2 \right\rbrace$. Then, $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim \frac{1+\log(mt)}{(mt)^{1-\alpha}} \sum_{i,j\in \left\lbrace 1,2\right\rbrace} \Vert E_{i,j} \Vert_X + \frac{1}{mt}\Vert \partial_\eta F\Vert_Y.$$* *Proof.* Let $\delta\in (0,\delta_0)$ and set $$\begin{aligned} {\mathcal I}(y)=\int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta &= \left( \int_0^\delta + \int_\delta^\infty \right) {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta= {\mathcal I}_1(y) + {\mathcal I}_2(y). \end{aligned}$$ $\bullet$ We begin by estimating ${\mathcal I}_1(y)$. Since we integrate in $(0,\delta)$ and $\delta\in (0,\delta_0)$, we can write $F(\eta ,y)=\eta ^{-\alpha}E_1(\eta ,y)$ and directly estimate using Minkowsky inequality $$\left\Vert {\mathcal I}_1 \right\Vert_{L^2_y({\mathbb R})} \leq \Vert E_1 \Vert_X\int_0^\delta \eta ^{-\alpha}\mathrm{d}\eta = \frac{\Vert E_1 \Vert_X}{1-\alpha}\delta^{1-\alpha}.$$ On the other hand, since $F$ vanishes at infinity, integrating by parts we can write $$\begin{aligned} {\mathcal I}_2 &= \frac{1}{imt}\int_\delta^\infty \partial_\eta \left( {\rm e}^{im\eta t}\right) F(\eta ,y)\mathrm{d}\eta \\ &= -\frac{1}{imt} {\rm e}^{im\delta t}F(\delta,y) - \frac{1}{imt}\int_\delta^{\delta_0} {\rm e}^{im\eta t}\partial_\eta F(\eta ,y) \mathrm{d}\eta - \frac{1}{imt}\int_{\delta_0}^\infty {\rm e}^{im\eta t}\partial_\eta F(\eta ,y) \mathrm{d}\eta .\end{aligned}$$ and we estimate $$\left\Vert {\mathcal I}_2\right\Vert_{L^2_y({\mathbb R})} \lesssim \frac{1}{mt}\delta^{-\alpha} \left( \Vert E_1 \Vert_X + \Vert E_2 \Vert_X\right) +\frac{1}{mt}\Vert \partial_\eta F \Vert_Y.$$ Therefore, we conclude that $$\Vert {\mathcal I}\Vert_{L^2_y({\mathbb R})} \lesssim \left(\delta^{1-\alpha} + \frac{1}{mt}\delta^{-\alpha}\right) \left( \Vert E_1 \Vert_X + \Vert E_2 \Vert_X\right) + \frac{1}{mt}\Vert \partial_\eta F \Vert_Y.$$ For $\delta=\frac{1}{4mt}<{\delta_0}$ we obtain the desired decay estimate $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim \frac{1}{(mt)^{1-\alpha}} \left( \Vert E_1 \Vert_X + \Vert E_2 \Vert_X\right) + \frac{1}{mt}\Vert \partial_\eta F \Vert_Y.$$ $\bullet$ For ${\mathcal I}_1$, since we have the expansion $F(\eta ,y)=\eta ^{-\alpha}(E_{1,1}(\eta ,y)+\log(\eta )E_{1,2}(\eta ,y))$ for $\eta \in(0,{\delta_0})$, we have that since $\delta<1$, $$\begin{aligned} \Vert {\mathcal I}_1 \Vert_{L^2_y({\mathbb R})} &\lesssim \int_0^\delta \eta ^{-\alpha}\left(1 + |\log(\eta )|\right)\left(\Vert E_{1,1}\Vert_X + \Vert E_{1,2}\Vert_X\right) \mathrm{d}\eta \\ &\lesssim \delta^{1-\alpha}\left( 1 + \big| \log\left(\delta\right) \big|\right) \left(\Vert E_{1,1}\Vert_X + \Vert E_{1,2}\Vert_X\right) . \end{aligned}$$ As for ${\mathcal I}_2$, integrating by parts, since $F$ vanishes at infinity and using the asymptotic expansion $\partial_\eta F(\eta ,y)=\eta ^{-\alpha-1}(E_{2,1}(\eta ,y) + \log(\eta )E_{2,2}(\eta ,y))$, one can estimate $$\begin{aligned} \Vert {\mathcal I}_2 \Vert_{L^2_y({\mathbb R})} &\leq \frac{1}{mt}\left( \Vert F(\delta,\cdot)\Vert_{L^2_y({\mathbb R})}+\int_\delta^{\delta_0} \Vert \partial_\eta F(\eta ,\cdot)\Vert_{L^2_y({\mathbb R})} \mathrm{d}\eta + \int_{\delta_0}^\infty \Vert \partial_\eta F(\eta ,\cdot)\Vert_{L^2_y({\mathbb R})}\mathrm{d}\eta \right) \\ &\lesssim \frac{1}{mt} \delta^{-\alpha} \left( 1 + \big| \log \left(\delta\right) \big| \right) \sum_{i,j\in \left\lbrace 1,2\right\rbrace} \Vert E_{i,j} \Vert_X + \frac{1}{mt}\Vert \partial_\eta F\Vert_Y. \end{aligned}$$ Choosing once again $\delta=\frac{1}{4mt}$ yields the estimate $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim \frac{1}{(mt)^{1-\alpha}} (1+\log(mt)) \sum_{i,j\in \left\lbrace 1,2\right\rbrace} \Vert E_{i,j} \Vert_X + \frac{1}{mt}\Vert \partial_\eta F\Vert_Y,$$ which concludes the proof. ◻ We now obtain the decay estimates for the stream-function $\psi_m$. **Proposition 4**. *The following holds for all $t\geq 1$.* - *If $\beta^2\neq 1/4$, then $$\Vert \psi_m(t)\Vert_{L^2_y({\mathbb R})} \lesssim m^{-3}t^{-\frac32+\mu } Q_{2,m} .$$* - *If $\beta^2 = 1/4$, then $$\Vert \psi_m(t)\Vert_{L^2_y({\mathbb R})} \lesssim m^{-3}t^{-\frac32}(1+\log \left(mt\right)) Q_{2,m}.$$* *Proof.* We have from Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"} that $$\begin{aligned} \psi_m(t,y)&=\frac{{\rm e}^{-imyt}}{2m\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}W(\eta ) \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}W(\eta )\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right). \end{aligned}$$ We show the decay estimates for $$\label{eq:Tpsi} \mathcal{T}^{+}(y):= \int_0^\infty {\rm e}^{im\eta t}W(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi\mathrm{d}\eta$$ since one can directly replicate the arguments to obtain the same estimates for $$\mathcal{T}^{-}(y):= \int_0^\infty {\rm e}^{-im\eta t}W(\eta )\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta.$$ The time decay is achieved appealing to Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"}. Integrating [\[eq:Tpsi\]](#eq:Tpsi){reference-type="eqref" reference="eq:Tpsi"} by parts in $\eta$ provides $$\begin{aligned} \mathcal{T}^+(y)&=\int_0^\infty \frac{1}{imt}\partial_\eta ({\rm e}^{im\eta t})W(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi\mathrm{d}\eta \\ &=-\frac{1}{imt}\int_0^\infty {\rm e}^{im\eta t}\left( W'(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi + W(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \right)\mathrm{d}\eta . \\ \end{aligned}$$ and further define $$\label{eq: def F psi} F(\eta ,y):=W'(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi + W(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi.$$ Clearly, $$\begin{aligned} \partial_\eta F(\eta ,y)&=W''(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi + 2W'(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad+W(\eta )\int_0^\infty W(\xi)\partial_\eta ^2G_m(\eta ,\xi, y)\mathrm{d}\xi. \end{aligned}$$ We begin by checking that $\partial_\eta F\in Y$. For this, [\[eq:Whittakereqn\]](#eq:Whittakereqn){reference-type="eqref" reference="eq:Whittakereqn"} yields $$\int_{\delta_0}^\infty |W''(\eta )|\mathrm{d}\eta = 4m^2 \int_{\delta_0}^\infty \left|-\frac{1}{4}+\beta^2\frac{1}{(2m\eta)^2}\right| \left| W_{0,\gamma}(2m\eta)\right| \mathrm{d}\eta \lesssim m\Vert W_{0,\gamma} \Vert_{L^1}.$$ Similarly we easily estimate $$\int_{\delta_0}^\infty |W'(\eta )|\mathrm{d}\eta \leq \Vert W' \Vert_{L^1}= \Vert W'_{0,\gamma}\Vert_{L^1}$$ and $$\int_{\delta_0}^\infty |W(\eta )|\mathrm{d}\eta \leq \frac{1}{2m}\Vert W_{0,\gamma}\Vert_{L^1(0,\infty)}.$$ Moreover we have that, $$\label{bound int WdjG} \begin{aligned} \left\Vert \int_0^\infty W(\xi)\partial_\eta ^jG_m(\eta ,\xi, y)\mathrm{d}\xi\right\Vert_{L^2_y({\mathbb R})} &\leq \Vert W \Vert_{L^1} Q_{j,m} \leq \frac{1}{m}\Vert W_{0,\gamma} \Vert_{L^2} Q_{j,m}, \end{aligned}$$ for all $j\geq0$. With this, we infer that $\partial_\eta F\in Y$ and $$\Vert \partial_\eta F \Vert_Y \lesssim Q_{2,m}.$$ Next, we check the asymptotic expansions of $F(\eta ,y)$ and $\partial_\eta F(\eta ,y)$ for $\eta \in [0,{\delta_0}]$. For this, we will distinguish the two cases. $\bullet$ We can use Lemma [Lemma 7](#asymptotic expansion W){reference-type="ref" reference="asymptotic expansion W"} to write $$W(\eta )=\eta ^{\frac12-\mu} \mathcal{E}_{m,2}(\eta ),\qquad W'(\eta )=\eta ^{-\frac12-\mu} \mathcal{E}_{m,1}(\eta ), \qquad W''(\eta )=\eta ^{-\frac32-\mu} \mathcal{E}_{m,2}(\eta ),$$ which yields $$\begin{aligned} F(\eta ,y)&=\eta ^{-\frac12-\mu}\mathcal{E}_{m,1}(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi +\eta ^{\frac12+\mu}\mathcal{E}_{m,0}(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &=\eta ^{-\frac12 -\mu}E_1(\eta,y ), \end{aligned}$$ with $\Vert E_1 \Vert_X\lesssim m^{-\frac12-\mu} Q_{1,m}$ and $$\begin{aligned} \partial_\eta F(\eta ,y)&=\eta ^{-\frac32-\mu}\mathcal{E}_{m,2}(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi + 2u^{-\frac12-\mu}\mathcal{E}_{m,1}(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\qquad\qquad+ \eta ^{\frac12-\mu}\mathcal{E}_{m,0}(\eta )\int_0^\infty W(\xi)\partial_\eta ^2G_m(\eta ,\xi,y )\mathrm{d}\xi \\ &=\eta ^{-\frac32-\mu}E_2(\eta, y), \end{aligned}$$ with $\Vert E_2 \Vert_X \lesssim m^{-\frac12-\mu} Q_{2,m}$. With this, for $\alpha=\frac{1}{2}+\mu$, we show that $F(\eta ,y)$ defined above satisfies the conditions of Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"} and we conclude that $$\begin{aligned} \left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y) \mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} &\lesssim \frac{1}{mt^{1-\alpha}} Q_{2,m}, \end{aligned}$$ which yields the claimed bound for $\Vert \psi_m(t)\Vert_{L^2_y({\mathbb R})}$. $\bullet$ We shall now use the asymptotic expansions of Lemma [Lemma 8](#asymptotic expansion special W){reference-type="ref" reference="asymptotic expansion special W"}. We will use these expansions to check the validity of the hypothesis required to apply Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"}. In this direction, for $\eta \in (0,{\delta_0})$ we can write $$\begin{aligned} F(\eta ,y)&=\eta ^{-\frac12}\left[\mathcal{E}_{m,1,1}(\eta )+\log(\eta )\mathcal{E}_{m,1,2}(\eta ) \right]\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad+\eta ^{\frac12}\left[\mathcal{E}_{m,0,1}(\eta )+\log(\eta )\mathcal{E}_{m,0,2}(\eta )\right]\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &=\eta ^{-\frac12}(E_{1,1}(\eta ,y)+\log(\eta )E_{1,2}(\eta ,y)), \end{aligned}$$ with the uniform bounds $$\begin{aligned} \Vert E_{1,1}\Vert_X\lesssim m^{-\frac12}\left( 1 + \log \left(m\right) \right) Q_{1,m}, \quad \Vert E_{1,2}\Vert_X \lesssim m^{-\frac12} Q_{1,m}.\end{aligned}$$ Similarly, for $\partial_\eta F(\eta ,y)$ we can write $$\begin{aligned} \partial_\eta F(\eta ,y)&=\eta ^{-\frac32}\left[\mathcal{E}_{m,2,1}(\eta ) +\log(\eta ) \mathcal{E}_{m,2,2}(\eta )\right]\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad+ 2u^{-\frac12}\left[\mathcal{E}_{m,1,1}(\eta )+\log(\eta )\mathcal{E}_{m,1,2}(\eta ) \right]\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad+ \eta ^{\frac12}\left[\mathcal{E}_{m,0,1}(\eta )+\log(\eta )\mathcal{E}_{m,0,2}(\eta )\right]\int_0^\infty W(\xi)\partial_\eta ^2G_m(\eta ,v,\eta )\mathrm{d}\xi \\ &=\eta ^{-\frac32}\big(E_{2,1}(\eta ,y)+\log(\eta )E_{2,2}(\eta ,y)\big), \end{aligned}$$ with the bounds $$\begin{aligned} \Vert E_{2,1}\Vert_X &\lesssim m^{-\frac12}\left( 1 + \log \left(m\right) \right) Q_{2,m}, \quad \Vert E_{2,2}\Vert_X \lesssim m^{-\frac12} Q_{2,m}.\end{aligned}$$ Hence, we apply Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"} for $\alpha=\frac12$ and $\delta=\frac{1}{4mt}$ to obtain $$\begin{aligned} \left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} &\lesssim \frac{1}{(mt)^{\frac12}}(1+\log \left(mt\right) ) \sum_{i,j\in \left\lbrace 1,2\right\rbrace} \Vert E_{i,j} \Vert_X + \frac{1}{mt}\Vert \partial_\eta F\Vert_Y\\ &\lesssim \frac{1}{mt^{\frac12}}(1+\log \left(mt\right) ) Q_{2,m}. \end{aligned}$$ From here, the stated bound for $\Vert \psi_m(t)\Vert_{L^2_y({\mathbb R})}$ follows easily. ◻ From the explicit expression of $\psi_m(t,y)$ and replicating the proof of Proposition [Proposition 4](#prop: invdamp psi){reference-type="ref" reference="prop: invdamp psi"}, one obtains the following result. **Corollary 5**. *The following holds for all $t\geq 1$.* - *If $\beta^2\neq 1/4$, then $$\Vert \partial_y\psi_m(t)\Vert_{L^2_y({\mathbb R})}\lesssim m^{-2}t^{-\frac12+\mu} Q_{1,m}.$$* - *If $\beta^2 = 1/4$, then $$\Vert \partial_y \psi_m(t)\Vert \lesssim m^{-2}t^{-\frac12}\left( 1 + \log \left(mt\right) \right) Q_{1,m}.$$* *Proof.* Note that $$\begin{aligned} \partial_y\psi_m(t,y) &= -imt\psi_m(t,y) + \frac{{\rm e}^{-imyt}}{2m\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}W(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}W(\eta )\int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right). \end{aligned}$$ In particular, we observe that $$\begin{aligned} -imt &\frac{{\rm e}^{-imyt}}{2m\pi}\cos (\mu\pi)\int_0^\infty {\rm e}^{im\eta t}W(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y) \mathrm{d}\xi \mathrm{d}\eta \\ &= -\frac{{\rm e}^{-imyt}}{2m\pi}\cos (\mu\pi)\int_0^\infty \partial_\eta \left({\rm e}^{im\eta t}\right)W(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y) \mathrm{d}\xi \mathrm{d}\eta \\ &= \frac{{\rm e}^{-imyt}}{2m\pi}\cos (\mu\pi)\int_0^\infty {\rm e}^{im\eta t}W'(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y) \mathrm{d}\xi \mathrm{d}\eta \\ &\quad + \frac{{\rm e}^{-imyt}}{2m\pi}\cos (\mu\pi)\int_0^\infty {\rm e}^{im\eta t}W(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y) \mathrm{d}\xi \mathrm{d}\eta .\end{aligned}$$ Under the observation that $(\partial_\eta + \partial_y){\mathcal{G}}_m(\eta ,\xi, y)=0$, we conclude that $$\begin{aligned} \partial_y\psi_m(t,y) &=\frac{{\rm e}^{-imyt}}{2m\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}W'(\eta ) \int_0^\infty W(\xi)\partial_yG_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. -\int_0^\infty {\rm e}^{-im\eta t}W'(\eta )\int_0^\infty W(\xi)\partial_yG_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right). \end{aligned}$$ and the corollary follows applying Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"}, we omit the details. ◻ We now obtain the decay in time of the perturbed density. **Proposition 6**. *The following holds for all $t\geq 1$.* - *If $\beta^2\neq 1/4$, then $$\Vert \rho_m(t)\Vert_{L^2_y({\mathbb R})}\lesssim m^{-1}t^{-\frac12+\mu} Q_{1,m}.$$* - *If $\beta^2 = 1/4$, then $$\Vert \rho_m(t)\Vert_{L^2_y({\mathbb R})}\lesssim m^{-2}t^{-\frac12}\left( 1 + \log \left(mt\right) \right) Q_{1,m}.$$* *Proof.* From Theorem [Theorem 1](#thm: main solution formula){reference-type="ref" reference="thm: main solution formula"}, $$\begin{aligned} \rho_m(t,y)&=\frac{{\rm e}^{-imyt}}{2m\pi}\cos(\gamma\pi) \left(\int_0^\infty {\rm e}^{im\eta t}\frac{W(\eta )}{\eta } \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right. \\ &\qquad\qquad\qquad\qquad\qquad \left. +\int_0^\infty {\rm e}^{-im\eta t}\frac{W(\eta )}{\eta }\int_0^\infty W(\xi)G_m(-\eta,-\xi,y)\mathrm{d}\xi\mathrm{d}\eta \right). \end{aligned}$$ As before, we only show the decay estimate for $$\mathcal{T}:=\int_0^\infty {\rm e}^{im\eta t}\frac{W(\eta )}{\eta } \int_0^\infty W(\xi)G_m(\eta,\xi,y)\mathrm{d}\xi\mathrm{d}\eta.$$ Denoting $$F(\eta ,y)=\frac{W(\eta )}{\eta }\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \mathrm{d}\eta ,$$ we shall apply Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"}. We compute $$\partial_\eta F(\eta ,y)=\left(\frac{W'(\eta )}{\eta }-\frac{W(\eta )}{\eta ^2}\right)\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi + \frac{W(\eta )}{\eta }\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y) \mathrm{d}\xi$$ and we observe the following bounds: $$\begin{aligned} \int_{\delta_0}^\infty \left|\frac{W'(\eta )}{\eta }\right|\mathrm{d}\eta &\leq {\delta_0}^{-1}\Vert W' \Vert_{L^1({\delta_0},\infty)}={\delta_0}^{-1}\Vert W_{0,\gamma}' \Vert_{L^1(1,\infty)} \\ \int_{\delta_0}^\infty \left|\frac{W(\eta )}{\eta ^2}\right|\mathrm{d}\eta &\leq {\delta_0}^{-2}\Vert W \Vert_{L^1({\delta_0},\infty)}={\delta_0}^{-1}\Vert W_{0,\gamma} \Vert_{L^1(1,\infty)} \\ \int_{\delta_0}^\infty \left|\frac{W(\eta )}{\eta }\right|\mathrm{d}\eta &\leq {\delta_0}^{-1}\Vert W \Vert_{L^1({\delta_0},\infty)}=\Vert W_{0,\gamma} \Vert_{L^1(1,\infty)} \end{aligned}$$ since ${\delta_0}^{-1}=2m$. Together with [\[bound int WdjG\]](#bound int WdjG){reference-type="eqref" reference="bound int WdjG"} we deduce that $\partial_\eta F\in Y$ and we can estimate $$\Vert \partial_\eta F \Vert_Y \lesssim Q_{1,m}.$$ We next treat each $\beta^2$ case separately to obtain the correct asymptotic expansions. $\bullet$ Following the asymptotic expansions of Lemma [Lemma 7](#asymptotic expansion W){reference-type="ref" reference="asymptotic expansion W"}, we can write $$F(\eta ,y)=\eta ^{-\frac12-\mu}\mathcal{E}_{m,0}(\eta )\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi=\eta ^{-\frac12-\mu}E_{1}(\eta ,y),$$ with $\Vert E_1 \Vert_X \lesssim m^{-\frac12-\mu} Q_{0,m}$. Similarly, we have that $$\begin{aligned} \partial_\eta F(\eta ,y)&=\left( \eta ^{-\frac32-\mu}\mathcal{E}_{m,1}(\eta )-\eta ^{-\frac32-\mu}\mathcal{E}_{m,0}(\eta )\right)\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\qquad+ \eta ^{-\frac12}\mathcal{E}_{m,0}(\eta )\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &= \eta ^{-\frac32-\mu}E_2(\eta ,y), \end{aligned}$$ where $\Vert E_2 \Vert_X \lesssim m^{-\frac12-\mu} Q_{1,m}$. Hence, taking $\alpha=\frac12+\mu$ we apply Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"} swiftly and obtain the decay estimate $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y)\mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim m^{-1}t^{-\frac12+\mu} Q_{1,m},$$ from which the proof follows. $\bullet$ Thanks to the asymptotic expansions of Lemma [Lemma 7](#asymptotic expansion W){reference-type="ref" reference="asymptotic expansion W"}, we have $$\begin{aligned} F(\eta ,y)&=\eta ^{-\frac12}\big(\mathcal{E}_{m,0,1}(\eta )+\log(\eta )\mathcal{E}_{m,0,2}(\eta )\big)\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &=\eta ^{-\frac12}\big( E_{1,1}(\eta ,y)+\log(\eta )E_{1,2}(\eta ,y)\big), \end{aligned}$$ with the bounds $$\begin{aligned} \Vert E_{1,1}\Vert_X &\lesssim m^{-\frac12}\left( 1 + \log \left(m\right) \right) Q_{0,m}, \quad \Vert E_{1,2}\Vert_X \lesssim m^{-\frac12} Q_{0,m}.\end{aligned}$$ As for $\partial_\eta F$, we have that $$\begin{aligned} \partial_\eta F(\eta ,y)&= \eta ^{-\frac32}\big(\mathcal{E}_{m,1,1}(\eta ) + \log(\eta )\mathcal{E}_{m,1,2}(\eta )\big)\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad- \eta ^{-\frac32}\big(\mathcal{E}_{m,0,1}(\eta )+\log(\eta )\mathcal{E}_{m,0,2}(\eta )\big)\int_0^\infty W(\xi)G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &\quad+ \eta ^{-\frac12}\big(\mathcal{E}_{m,0,1}(\eta )+\log(\eta )\mathcal{E}_{m,0,2}(\eta )\big)\int_0^\infty W(\xi)\partial_\eta G_m(\eta ,\xi, y)\mathrm{d}\xi \\ &=\eta ^{-\frac32}\big( E_{2,1}(\eta ,y) + \log(\eta )E_{2,2}(\eta ,y)\big), \end{aligned}$$ where we can bound $$\begin{aligned} \Vert E_{2,1}\Vert_X &\lesssim m^{-\frac12} \left( 1 + \log \left(m\right) \right) Q_{1,m}, \quad \Vert E_{2,2}\Vert_X \lesssim m^{-\frac12} Q_{1,m}.\end{aligned}$$ Now, for $\alpha=1/2$, we have that $$\left\Vert \int_0^\infty {\rm e}^{im\eta t}F(\eta ,y) \mathrm{d}\eta \right\Vert_{L^2_y({\mathbb R})} \lesssim m^{-1}t^{-\frac12}\left( 1 + \log \left(mt\right) \right) Q_{1,m}$$ due to Lemma [Lemma 3](#lemma: generic decay-by-regularity){reference-type="ref" reference="lemma: generic decay-by-regularity"}. With this, the proof is complete. ◻ # The Whittaker functions {#app: Whittaker} Here we give a description of the Whittaker function $W_{0,\gamma}$ and its asymptotic expansions. For $\mu={\rm Re}\left( \sqrt{1/4 -\beta^2}\right)$ and $\nu={\rm Im}\left( \sqrt{1/4 - \beta^2} \right)$ we set $\gamma=\mu + i\nu$. For $\gamma\neq 0$ and $\zeta\in{\mathbb C}$, the Whittaker function $W_{0,\gamma}(\zeta)$ is given by $$\label{eq:WtoM} \begin{split} W_{0,\gamma}(\zeta)=\frac{\Gamma(-2\gamma)}{\Gamma(\frac{1}{2}-\gamma)}M_{0,\gamma}(\zeta)+\frac{\Gamma(2\gamma)}{\Gamma(\frac{1}{2}+\gamma)}M_{0,-\gamma}(\zeta) \end{split}$$ Here, $\Gamma(\cdot)$ stands for the Gamma function and the Whittaker functions $M_{0,\gamma}$ and $M_{0,-\gamma}$ are given by $$\label{eq: def Mgamma} M_{0,\pm\gamma}(\zeta) = {\rm e}^{-\frac12\zeta}\zeta^{\frac12\pm \gamma}M\left( \tfrac12\pm \gamma, 1\pm 2\gamma,\zeta\right), \quad M(a,b,\zeta)=\sum_{s=0}^\infty\frac{(a)_s}{(b)_s s!}\zeta^s,$$ where $(a)_s=a(a+1)(a+2)\dots (a+s-1)$. See [@CZN23chan; @NIST] for more details. The asymptotic estimates for $W_{0,\gamma}$ are deduced from the asymptotic estimates for $M_{0,\pm\gamma}$, recorded in Lemma A.3 from [@CZN23chan], due to the relation [\[eq:WtoM\]](#eq:WtoM){reference-type="eqref" reference="eq:WtoM"}. **Lemma 7**. *Let $\zeta\in{\mathbb C}$. Let $B_R\subset{\mathbb C}$ denote the closed unit ball of radius $R>0$ centered in the origin. Then, $$\begin{aligned} W_{0,\gamma}(\zeta)&=\zeta^{\frac12 -\gamma}\mathcal{E}_{0,\gamma}(\zeta),\quad W_{0,\gamma}'(\zeta)=\zeta^{-\frac12 -\gamma}\mathcal{E}_{1,\gamma}(\zeta), \end{aligned}$$ where $\mathcal{E}_{j,\gamma}\in L^\infty(B_R)$ and $\Vert \mathcal{E}_{j,\gamma}\Vert_{L^\infty(B_R)} \lesssim_{\gamma,R} 1$, for $j=0,1$.* For $\beta^2=1/4$, we have $\gamma=0$ and [\[eq:WtoM\]](#eq:WtoM){reference-type="eqref" reference="eq:WtoM"} is no longer valid. Then, $W_{0,0}$ is given by $$\label{eq:WtoK} W_{0,0}(\zeta)=\sqrt{\frac{\zeta}{\pi}}K_0\left(\frac{\zeta}{2}\right),$$ where $K_0$ is the modified Bessel function of second kind of order 0. See [@NIST] for more details on $K_0$. We next state the asymptotic expansions for $W_{0,0}$, which follow from [\[eq:WtoK\]](#eq:WtoK){reference-type="eqref" reference="eq:WtoK"} and are shown in [@CZN23chan]. **Lemma 8** ([@CZN23chan], Lemma A.4). *Let $\beta^2=1/4$ and $\zeta\in{\mathbb C}$. Let $B_R\subset{\mathbb C}$ denote the closed ball of radius $R>0$ centered at the origin. Then, $$W_{0,0}(\zeta) = \zeta^\frac12 \big( {\mathcal E}_{0,1}(\zeta) - \log (\zeta) {\mathcal E}_{0,2}(\zeta)\big), \quad W_{0,0}'(\zeta) = \zeta^{-\frac12} \big( {\mathcal E}_{1,1}(\zeta) - \log (\zeta) {\mathcal E}_{1,2}(\zeta)\big),$$ where ${\mathcal E}_{j,k}(\zeta)$ are entire functions in ${\mathbb C}$ and $\Vert {\mathcal E}_{j,k}\Vert_{L^\infty(B_R)}\lesssim 1$, for $j=0,1$ and $k=1,2$.* We finish our discussion with the proof of Lemma [Lemma 1](#analytic continuation lemma){reference-type="ref" reference="analytic continuation lemma"} when $\beta^2\neq 1/4$. *Proof of Lemma [Lemma 1](#analytic continuation lemma){reference-type="ref" reference="analytic continuation lemma"}.* From [\[eq:WtoM\]](#eq:WtoM){reference-type="eqref" reference="eq:WtoM"} we write $$\begin{split} W(\zeta)=A(\gamma)M_{0,\gamma}(2m\zeta) + B(\gamma)M_{0,-\gamma}(2m\zeta). \end{split}$$ The analytic continuation property of $M_{0,\gamma}(\zeta)$, see [@NIST], states that $$M_{0,\gamma}(\zeta {\rm e}^{\pm \pi i}) = \pm i {\rm e}^{\pm \gamma \pi i}M_{0,\gamma}(\zeta).$$ Therefore, we can write $$M_{0,\gamma}(2m(-\eta +i\varepsilon))=M_{0,\gamma}(2m(\eta -i\varepsilon){\rm e}^{i\pi})=i{\rm e}^{\gamma\pi i}M_{0,\gamma}(2m(\eta -i\varepsilon))$$ and $$M_{0,\gamma}(2m(-\eta -i\varepsilon))=M_{0,\gamma}(2m(\eta +i\varepsilon){\rm e}^{-i\pi})=-i{\rm e}^{-\gamma\pi i}M_{0,\gamma}(2m(\eta +i\varepsilon)).$$ Similarly, we have $$M_{0,-\gamma}(2m(-\eta +i\varepsilon))=i{\rm e}^{-\gamma\pi i}M_{0,-\gamma}(2m(\eta -i\varepsilon))$$ and $$M_{0,-\gamma}(2m(-\eta -i\varepsilon))=-i{\rm e}^{\gamma\pi i}M_{0,-\gamma}(2m(\eta +i\varepsilon)).$$ Now, we have that $$W(-\eta +i\varepsilon)=i{\rm e}^{\gamma\pi i}A(\gamma)M_{0,\gamma}(2m(\eta -i\varepsilon))+i{\rm e}^{-\gamma\pi i}B(\gamma)M_{0,-\gamma}(2m(\eta -i\varepsilon))$$ and $$W(-\eta -i\varepsilon)=-i{\rm e}^{-\gamma\pi i}A(\gamma)M_{0,\gamma}(2m(\eta +i\varepsilon))-i{\rm e}^{\gamma\pi i}B(\gamma)M_{0,-\gamma}(2m(\eta +i\varepsilon)).$$ Since both $M_{0,\gamma}(\zeta)$ and $M_{0,-\gamma}(\zeta)$ are continuous functions in the complex subset $\left\lbrace \zeta\in{\mathbb C}: {\rm Re}(\zeta)\geq0\right\rbrace,$ it is easily seen that $$\begin{split} \lim_{\varepsilon\rightarrow0}W(-\zeta +i\varepsilon)-W(-\zeta -i\varepsilon)&= i\left( {\rm e}^{\gamma\pi i}+ {\rm e}^{-\gamma\pi i}\right) \left(A(\gamma)M_{0,\gamma}(\zeta)+B(\gamma)M_{0,-\gamma}(\zeta)\right) \\ &=2i\cos(\gamma\pi)W(\zeta ). \end{split}$$ ◻ # Acknowledgments {#acknowledgments .unnumbered} The research of MCZ was partially supported by the Royal Society URF\\R1\\191492 and EPSRC Horizon Europe Guarantee EP/X020886/1.

arxiv_math

--- author: - - - bibliography: - multimatroid-tens.bib title: Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial --- # Introduction Let $G$ and $H$ be graphs, $G$ loopless and bridgeless, and $H$ with a distinguished edge $e$ that is neither a loop nor a bridge. The tensor product $G\otimes_e H$ is formed by, for each edge of $G$, identifying that edge with $e$ in a copy of $H$ then deleting the edge formed by the identification. So each edge of $G$ is replaced with a copy of $H \mathop{\mathrm{\backslash}}e$. *Brylawski's tensor product formula* [@Brylawski_2010] expresses the Tutte polynomial of $G\otimes_e H$ in terms of Tutte polynomaials coming from $G$ and $H$: $$T(G \otimes_e H;x,y) = \alpha^{n(G)} \beta^{r(G)} T\left(G;T(H\mathop{\mathrm{\backslash}}e ;x,y)/\beta,T(H\mathop{\mathrm{/}}e;x,y)/\alpha \right),$$ where $\alpha$ and $\beta$ are the unique solutions to $$\begin{aligned} (x-1)\alpha + \beta &= T(H\mathop{\mathrm{\backslash}}e;x,y), \\ \alpha + (y-1)\beta &= T(H\mathop{\mathrm{/}}e;x,y).\end{aligned}$$ Brylawski's formula has been extended to graphs embedded in surfaces in two different ways: in [@Huggett_2011] where a tensor product formula for the Bollobás--Riordan polynomial was given, and in [@ELLIS_MONAGHAN_2014] which gave a tensor product formula for the topological transition polynomial. These expressions were further developed in [@maya2]. In the classical case, Brylawski's tensor product formula for the Tutte polynomial is properly a result about matroids, rather than graphs. All of the structure required for the result for graphs resides in the cycle matroids of the graphs. Additionally, the formula holds for all matroids, not just graphic ones. Here we address the problem of finding the correct framework for the tensor product formulas for graphs embedded in surfaces. A reader familiar with topological analogues of the Tutte polynomial might expect this framework to be given by delta-matroids. (Background on delta-matroids and topological graph polynomials can be found in [@CMNR1; @CMNR2]). However, we shall see that the natural setting is in fact Bouchet's tight multimatroids. In this paper we introduce tensor products of tight multimatroids. This operation is compatible with the graph, matroid and embedded graph tensor products. Our main result is a generalisation of Brylawski's formula to the transition polynomial of multimatroids, given here as Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}. From this we deduce tensor product formulas for delta-matroids, including the recovery of Brylawski's result for matroids. We also deduce an extension of Ellis-Monaghan and Moffatt's tensor product formula [@ELLIS_MONAGHAN_2014] for the topological transition polynomial by removing the restriction on the type of edges that the tensor product acts on. This paper is structured as follows. Section [2](#sec:multim){reference-type="ref" reference="sec:multim"} provides an overview of multimatroids. Section [3](#sec:tensors){reference-type="ref" reference="sec:tensors"} discusses the 2-sum operation for multimatroids and introduces the tensor product. Section [4](#sec:results){reference-type="ref" reference="sec:results"} contains our main result, Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}, giving a version of Brylawski's formula for the multimatroid transition polynomial. In Section [5](#sec:delta){reference-type="ref" reference="sec:delta"} we apply our main result to deduce tensor product formulas for delta-matroid and matroid polynomials. Finally, in Section [6](#sec:rg){reference-type="ref" reference="sec:rg"} we review a connection between graphs embedded in surfaces and tight 3-matroids and deduce a tensor product formula for the topological transition polynomial. To aid the digestion of our results, we provide plenty of examples and we do not assume any familiarity with multimatroids. # Multimatroids {#sec:multim} Multimatroids were introduced by Bouchet in [@MM1zbMATH01116184]. They are defined on what he called a *carrier*, a pair $(U,\Omega)$ where $U$ is a finite set and $\Omega$ is a partition of $U$ into (non-empty) blocks called *skew classes*. If each skew class has size $q$, then $(U,\Omega)$ is a $q$-carrier. Informally, it is the set of skew classes rather than the underlying set $U$ which plays the role of the set of elements of a multimatroid, with the members of a skew class representing different states or roles that the corresponding element of the multimatroid might play. As an illustration, in Section [6](#sec:rg){reference-type="ref" reference="sec:rg"} we consider multimatroids arising from ribbon graphs (or, equivalently, graphs in surfaces). There each edge of the ribbon graph corresponds to a skew class, and each skew class consists of three elements that can be thought of corresponding to an edge being either present, absent, or present but given a "half-twist". We shall adopt notation that reflects this understanding. **Notation 1**. Skew classes will be named or indexed by some set $E$. For each element $e$ of $E$, the members in its skew class are named by indexing or decorating the symbol $e$. For example, for a skew class $e$ we might write $e=\{e_1, e_2,\ldots ,e_q\}$ or $e=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e}\}$. (See Example [Example 3](#example1){reference-type="ref" reference="example1"} for this convention in practice.) For a carrier $(U,\Omega)$, a pair of distinct elements belonging to the same skew class is called a *skew pair*. A *subtransversal* of $\Omega$ is a subset of $U$ meeting each skew class at most once. The set of subtransversals of $\Omega$ is denoted by $\mathcal S(\Omega)$. A *transversal* $T$ of $\Omega$ is a subtransversal of $\Omega$ with $|T|=|\Omega|$, that is, a set containing precisely one element from each skew class. The set of transversals of $\Omega$ is denoted by $\mathcal T(\Omega)$. We say that a subtransversal $S$ is a *near-transversal* if $|S|=|\Omega|-1$, that is, it meets all but one skew class. The set of near transversals of $\Omega$ is denoted by $\mathcal N(\Omega)$. A *multimatroid* is a triple $Z=(U,\Omega,r_Z)$ so that $(U,\Omega)$ is a carrier and $r_Z$ is a *rank function* $r_Z:\mathcal S(\Omega)\rightarrow \mathbb Z^+$ satisfying the two axioms below. 1. [\[r1\]]{#r1 label="r1"} For each transversal $T$ of $\Omega$, the pair comprising $T$ and the restriction of $r_Z$ to subsets of $T$ forms a matroid. 2. [\[r2\]]{#r2 label="r2"} For each subtransversal $S$ of $\Omega$ and skew pair $\{e_i,e_j\}$ from a skew class disjoint from $S$, we have $$r_Z(S\cup \{e_i\}) + r_Z(S\cup \{e_j\}) - 2r_Z(S) \geq 1.$$ We generally omit the subscript $Z$ from $r_Z$ whenever the context is clear. Axiom (R1) implies the following: $r(\emptyset)=0$; for $A\in\mathcal S(\Omega)$ and $e_i\in U$ such that $A\cup\{e_i\}\in\mathcal S(\Omega)$, $r(A)\leq r(A\cup\{e_i\})\leq r(A)+1$; and for $A$, $B\in\mathcal S(\Omega)$ such that $A\cup B\in\mathcal S(\Omega)$, $r(A\cup B)+r(A\cap B)\leq r(A)+r(B)$. It is convenient to extend the concepts we have defined for carriers to multimatroids. For example, by a transversal of a multimatroid we mean a transversal of its underlying carrier, and we use $\mathcal{T}(Z)$ do denote the set of transversals of a multimatroid $Z$, and $\mathcal{N}(Z)$ the set of near transversals. We say that a multimatroid is a *$q$-matroid* if its carrier is a $q$-carrier (i.e., each skew class has exactly $q$ members). (The special case of $2$-matroids was introduced earlier than general multimatroids in [@MR904585] under the name *symmetric matroids*.) An element $e_i$ of a multimatroid $(U,\Omega,r)$ is *singular* if $r(\{e_i\})=0$. A skew class is *singular* if it contains a singular element. A multimatroid is *non-degenerate* if every skew class has size at least two. We say a skew class is *ordered* if there is a total order on its elements. We shall write $e=\{e_1<e_2<\cdots< e_q\}$ to denote the skew class $e=\{e_1,e_2,\ldots, e_q\}$ with order $e_1<e_2<\cdots< e_q$. Isomorphism for multimatroids is defined in the obvious way. If $Z_1$ and $Z_2$ are isomorphic, we write $Z_1\cong Z_2$. If $S$ is a subtransversal of $Z$, then we define its *nullity* $n_Z(S)$ by $n_Z(S)=|S|-r_Z(S)$, again omitting the subscript when the context is clear. Let $Z=(U,\Omega,r)$, let $e_i$ be in $U$, and suppose that $e_i$ is in the skew class $e$. Then, following [@MM2zbMATH01119073], the *elementary minor* of $Z$ by $e_i$, denoted by $Z |e_i$, is the triple $(U',\Omega',r')$ with $\Omega'= \Omega- e$, $U'=\bigcup_{f\in \Omega'}f$ and for all $S$ in $\mathcal S(\Omega')$, $r'(S)=r(S\cup\{e_i\})-r(\{e_i\})$. It is straightforward to show that $Z|e_i$ is a multimatroid. An *independent set* of $Z$ is a subtransversal $S$ with $n(S)=0$. Subtransversals that are not independent are *dependent*. A *basis* is a maximal independent subtransversal, and we let $\mathcal{B}(Z)$ denote the set of all bases of $Z$. For any subtransversal $S$, we have $r(S)=\max_{B\in\mathcal B(Z)} |B\cap S|$, and consequently multimatroids can be defined by their carrier and collection of bases. By [@MM1zbMATH01116184 Proposition 5.5], the bases of a non-degenerate multimatroid $Z$ are transversals of $Z$. We are particularly interested in tight multimatroids. These were introduced by Bouchet in [@MR1845490] and they include the 3-matroids of ribbon graphs (as discussed in Section [6](#sec:rg){reference-type="ref" reference="sec:rg"}). **Definition 2**. A multimatroid $Z$ is *tight* if it is nondegenerate and for every near-transversal $S$, there is a unique element $\theta_{{Z}}({S})$ in the skew class disjoint from $S$ such that $r(S \cup \{\theta_{{Z}}({S})\}) = r(S)$. We shall call the element $\theta_{{Z}}({S})$ the *slack element* associated with $S$. We omit the subscript $Z$, when the context is clear. Note that the uniqueness condition is not included in Bouchet's definition of tight multimatroids, however uniqueness follows easily from [\[r2\]](#r2){reference-type="ref" reference="r2"}. Moreover, as shown in [@MR1845490 Proposition 4.1], if a multimatroid is tight then so are all its elementary minors. **Example 3**. Let $(U,\{a,b,c\})$ be the 3-carrier with ordered skew classes $a=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}< \overline{a}< \widehat{a}\}$, $b=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}< \overline{b}< \widehat{b}\}$ and $c=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}< \overline{c}< \widehat{c}\}$. An example of a 3-matroid $Z$ over this carrier has the following sixteen bases: ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------- $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \overline{b}, \overline{c}\}$, $\{ \widehat{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \overline{c}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \widehat{b}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$, $\{ \widehat{a}, \widehat{b}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$, $\{ \overline{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \overline{c}\}$, $\{ \widehat{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \widehat{c}\}$, $\{ \widehat{a}, \widehat{b}, \widehat{c}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \widehat{b}, \overline{c}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \overline{b}, \widehat{c}\}$, $\{ \widehat{a}, \overline{b}, \widehat{c}\}$, $\{ \widehat{a}, \overline{b}, \overline{c}\}$, $\{ \overline{a}, \widehat{b}, \overline{c}\}$, $\{ \overline{a}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}, \widehat{c}\}$, $\{ \overline{a}, \widehat{b}, \widehat{c}\}$. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------- Of the remaining eleven transversals, all of which are dependent, $\{ \overline{a}, \overline{b}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$, has rank one and the rest have rank two. This multimatroid can be shown to be tight, and, for example, $\theta({\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}, \overline{b}\}})=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}$. For use later, we give a second, similar example. **Example 4**. Let $(U,\{e,f,g\})$ be the 3-carrier with an ordered skew class $e=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}< \overline{e}< \widehat{e}\}$, and unordered skew classes $f=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \overline{f}, \widehat{f}\}$ and $g=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}, \overline{g}, \widehat{g}\}$. An example of a 3-matroid $Z$ over this carrier has the following sixteen bases: ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \widehat{g}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{f}, \overline{g}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{f}, \widehat{g}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \widehat{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \widehat{f}, \overline{g}\}$, $\{ \overline{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \overline{g}\}$, $\{ \overline{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \widehat{g}\}$, $\{ \overline{e}, \overline{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \overline{e}, \overline{f}, \widehat{g}\}$, $\{ \overline{e}, \widehat{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \overline{e}, \widehat{f}, \overline{g}\}$, $\{ \widehat{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \widehat{e}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \overline{g}\}$, $\{ \widehat{e}, \overline{f}, \accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}$, $\{ \widehat{e}, \overline{f}, \overline{g}\}$. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- # Tensor products and 2-sums {#sec:tensors} The 2-sum of multimatroids was introduced in [@2sumpaper]. The idea behind it is as follows. Suppose $Z_1$ and $Z_2$ are two tight multimatroids that share a common non-singular skew class $e$ but are otherwise disjoint. Suppose also that $I_1$ and $I_2$ are independent near transversals of $Z_1$ and $Z_2$, respectively, both avoiding $e$. Then $I_1 \cup I_2$ is a basis of the 2-sum $Z_1\oplus_{e} Z_2$ if and only if $\theta_{{Z_1}}({I_1}) \neq \theta_{{Z_2}}({I_2})$. The idea of constructing the tensor product $Z_1\otimes_{e} Z_2$ is then to 2-sum a copy of $Z_2$ along each skew class in $Z_1$. Although the ideas behind the 2-sum and tensor product, as just described, are straightforward, we shall need to carefully track the names of all of the elements $Z_1\otimes_{e} Z_2$ and which copies of $Z_2$ they arise from. Doing this means makes the formal definitions of the 2-sum and tensor product a little more involved than might seem to be needed. First we adapt the above idea of 2-sums to the situation where $Z_1$ and $Z_2$ do not share a common skew class. **Definition 5**. Let $Z_1=(U_1, \Omega_1, r_1)$ and $Z_2=(U_2, \Omega_2, r_2)$ be tight multimatroids. Suppose that $Z_1$ has a non-singular ordered skew class $f= \{f_1<f_2<\cdots <f_q\}$, and $Z_2$ has a non-singular ordered skew class $e= \{e_1<e_2<\cdots < e_q\}$ (in particular, $|f|=|e|=q$). Then the multimatroid $Z= (U, \Omega, r)$ is the *2-sum* along $f$ and $e$, denoted by $Z_1 \oplus_{f=e}Z_2$, if $U=(U_1\cup U_2)-(f\cup e)$ and $\Omega = (\Omega_1\cup \Omega_2)-(\{f,e\})$ and $\mathcal{B}(Z)$ is the set $$\begin{split} \{ B\in \mathcal{T}(\Omega) : B\cap U_1 \text{ and } B\cap U_2 \text{ are independent near transversals such that } \\ \theta_{{Z_1}}({B\cap U_1}) = f_i \text{ and } \theta_{{Z_2}}({B\cap U_2}) = e_j \text{ for some } i,j \text{ with } i\neq j \}. \end{split}$$ **Example 6**. Suppose that $Z_1$ is the multimatroid in Example [Example 3](#example1){reference-type="ref" reference="example1"}, and $Z_2$ that in Example [Example 4](#example1b){reference-type="ref" reference="example1b"}. We consider $Z_1 \oplus_{a=e}Z_2$. The set of independent near transversals $T_1$ of $Z_1$ with $\theta_{{Z_1}}({T_1})=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}$ is $X_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b},\overline{c}\}, \{\widehat{b}, \widehat{c}\}\}$; where $\theta_{{Z_1}}({T_1})=\overline{a}$ is $X_{\overline{a}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}, \{\overline{b},\overline{c}\}, \{\overline{b},\widehat{c}\} \{\widehat{b},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\},\}$; and where $\theta_{{Z_1}}({T_1})=\widehat{a}$ is $X_{\widehat{a}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b},\widehat{c}\}, \{\widehat{b},\overline{c}\}\}$ The set of independent near transversals $T_2$ of $Z_2$ with $\theta_{{Z_2}}({T_2})=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}$ is $Y_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f},\overline{g}\}, \{\overline{f},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}\}$; where $\theta_{{Z_2}}({T_2})=\overline{e}$ is $Y_{\overline{e}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}, \{\overline{f},\overline{g}\}\}$; and where $\theta_{{Z_2}}({T_2})=\widehat{e}$ is $Y_{\widehat{e}} = \{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f},\widehat{g}\}, \{\overline{f},\widehat{g}\}, \{\widehat{f},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}\}, \{\widehat{f},\overline{g}\}\}$. Then $Z_1 \oplus_{a=e}Z_2$ has skew classes $b=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}< \overline{b}< \widehat{b}\}$, $c=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}< \overline{c}< \widehat{c}\}$, $f=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{f}, \overline{f}, \widehat{f}\}$, and $g=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{g}, \overline{g}, \widehat{g}\}$, Its bases consist of all sets of the form $x\cup y$ where $x\in X_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}}$ and $y\in Y_{\overline{e}} \cup Y_{\widehat{e}}$, or $x\in X_{\overline{a}}$ and $y\in Y_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}} \cup Y_{\widehat{e}}$, or $x\in X_{\widehat{a}}$ and $y\in Y_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}} \cup Y_{\overline{e}}$. The following three results are from [@2sumpaper]. **Proposition 7**. *Let $Z_1$ and $Z_2$ be as in Definition [Definition 5](#def:2sum){reference-type="ref" reference="def:2sum"}. Suppose that $T_1$ is a near transversal of $Z_1$ avoiding $f$, and $T_2$ is a near transversal of $Z_2$ avoiding $e$. Then $$r_{Z_1 \oplus_{f=e}Z_2}(T_1\cup T_2) = \begin{cases} r_{Z_1}(T_1) +r_{Z_2}(T_2) &\text{if } \theta_{{Z_1}}({T_1})=f_i \text{ and } \theta_{{Z_2}}({T_2})=e_j \\ & \qquad \text{ with } i\neq j, \\ r_{Z_1}(T_1) +r_{Z_2}(T_2) -1 &\text{if } \theta_{{Z_1}}({T_1})=f_i \text{ and } \theta_{{Z_2}}({T_2})=e_i. \end{cases}$$* **Proposition 8**. *Let $Z_1$ and $Z_2$ be as in Definition [Definition 5](#def:2sum){reference-type="ref" reference="def:2sum"}. If $Z_1$ and $Z_2$ are both tight, then so is $Z_1 \oplus_{f= e} Z_2$.* **Proposition 9**. *Let $Z_1, Z_2, Z_3$ be tight multimatroids with non-singular skew classes $g$ and $h$ of $Z_1$, non-singular skew class $e$ of $Z_2$, and non-singular skew class $f$ of $Z_3$. Suppose further that these four skew classes are all ordered, $|e|=|h|$ and $|f|=|g|$. Then $$\big( Z_1 \oplus_{h= e} Z_2\big) \oplus_{g= f} Z_3 = \big( Z_1 \oplus_{g= f} Z_3\big) \oplus_{h= e } Z_2.$$* As a consequence of the previous proposition we can write expressions such as $Z_1 \oplus_{h= e} Z_2\oplus_{g= f} Z_3$ without fear of ambiguity. The following will be useful in the proof of the main result. **Lemma 10**. *Let $Z_0$, $Z_1$, ..., $Z_k$ be tight multimatroids so that $Z_0$ has non-singular ordered skew classes $x_1$, ..., $x_k$ and for each $i=1,\ldots, k$, $Z_i$ has a non-singular skew class $y_i$ with $|x_i|=|y_i|$. Let $q_i=|y_i|$ and suppose that for each $i$ we have $$x_i= \{x_{i1}< \cdots < x_{iq_i}\} \quad \text{and} \quad y_i = \{y_{i1}< \cdots < y_{iq_i}\}.$$ Let $$Z= Z_0\oplus_{x_1=y_1}Z_1 \oplus_{x_2=y_2} \cdots \oplus_{x_k=y_k} Z_k.$$ Now suppose that $T$ is a transversal of $Z$ and write $T = T_0 \sqcup T_1 \sqcup \cdots \sqcup T_k$, so that $T_0$ is a sub-transversal of $Z_0$ missing each $x_i$ and for each $i=1,\ldots,k$, we have that $T_i$ is a near transversal of $Z_i$ missing $y_i$. Finally let $$T' = T_0 \cup \{x_{ij} : y_{ij}:=\theta_{{Z_i}}({T_i}),\ 1\leq i \leq k\}.$$ (Note that $T'$ is a transversal of $Z_0$.) Then $$n_Z(T) = n_{Z_0} (T') + \sum_{i=1}^k n_{Z_i}(T_i).$$* *Proof.* We prove the result by induction on $k$. It is clearly true if $k=0$, so suppose that $k>0$ and let $T$ be a transversal of $Z$. Let $Z'=Z_0\oplus_{x_1=y_1}Z_1 \oplus_{x_2=y_2} \cdots \oplus_{x_{k-1}=y_{k-1}} Z_{k-1}$. Then $Z=Z'\oplus_{x_k=y_k} Z_k$. Let $T''=T_0 \sqcup T_1 \cdots \sqcup T_{k-1}$. Then $T''$ is a near transversal of $Z'$ and $T=T''\sqcup T_k$. Suppose that $\theta_{{Z_k}}({T_k})=y_{kj}$. We consider two cases. Suppose first that $\theta_{{Z'}}({T''})\ne x_{kj}$. Then $n_{Z'}(T'' \cup \{x_{kj}\}) = n_{Z'}(T'')$ and by Proposition [Proposition 7](#prop:twosumrank){reference-type="ref" reference="prop:twosumrank"} and the inductive hypothesis, we have $$n_Z(T) = n_{Z'}(T''_1 \cup \{x_{kj}\}) + n_{Z_k} (T_k) = \Big(n_{Z_0} (T') + \sum_{i=1}^{k-1} n_{Z_i}(T_i)\Big) +n_{Z_k}(T_k),$$ as required. Next suppose that $\theta_{{Z'}}({T''})= x_{kj}$. Then $n_{Z'}(T'' \cup \{x_{kj}\}) = n_{Z'}(T'')+1$ and by Proposition [Proposition 7](#prop:twosumrank){reference-type="ref" reference="prop:twosumrank"} and the inductive hypothesis, we have $$\begin{aligned} n_Z(T) &= n_{Z'}(T'') + n_{Z_k} (T_k) + 1 = n_{Z'}(T''\cup\{x_{kj}\}) + n_{Z_k} (T_k) \\ &= \Big(n_{Z_0} (T') + \sum_{i=1}^{k-1} n_{Z_i}(T_i)\Big) +n_{Z_k}(T_k),\end{aligned}$$ as required. The result follows by induction. ◻ In light of Proposition [Proposition 9](#prop:sums){reference-type="ref" reference="prop:sums"} we can introduce the tensor product $Z_1\otimes_{e} Z_2$ as the result of 2-summing a copy of $Z_2$ along each skew class in $Z_1$. **Definition 11**. For $q\geq 2$, let $Z_1$ be a tight $q$-matroid in which every every skew class is non-singular and ordered. Let $Z_2$ be a tight multimatroid with a non-singular ordered skew class $e$ of size $q$. The *tensor product* $Z_1\otimes_{e} Z_2$ is the multimatroid formed as follows. Take an isomorphic copy of $Z_2$ for each skew class in $Z_1$. For notational convenience we assume that in each such copy the skew class $e$ remains, but the copies are otherwise on disjoint sets of elements. The multimatroid $Z_1\otimes_{e} Z_2$ is then constructed by forming the 2-sum along $\alpha$ and $e$ for each skew class $\alpha$ of $Z_1$ and its corresponding copy of $Z_2$. Note that by Proposition [Proposition 8](#lem:slack){reference-type="ref" reference="lem:slack"}, the tensor product of tight multimatroids is tight. We now give a small example of the tensor product of two 2-matroids. In this example we adopt a notation that very carefully keeps track of the elements of the 2-matroids throughout the construction, and one that we shall extend in the next section. **Example 12**. Let $Z_1$ be the $2$-matroid whose ordered skew classes are $\alpha_1=\{ \alpha_{11}<\alpha_{12}\}$ and $\alpha_2=\{ \alpha_{21}<\alpha_{22}\}$, and whose bases are $\{\alpha_{11},\alpha_{22}\}$ and $\{\alpha_{12},\alpha_{21}\}$. Let $Z_2$ be the $2$-matroid whose skew classes are $e=\{e_1<e_2\}$, $\beta_1=\{\beta_{11},\beta_{12}\}$, and $\beta_2=\{\beta_{21},\beta_{22}\}$ and whose bases are $\{e_1,\beta_{12},\beta_{22}\} , \{e_2,\beta_{11},\beta_{22}\} ,\{e_2,\beta_{12},\beta_{21}\}$. To form the tensor product $Z_1\otimes_{e} Z_2$ we need a copy of $Z_2$ for each skew class $\alpha_i$ of $Z_1$. Denote these copies by $Z_2^{(\alpha_i)}$, for $i=1,2$, where the skew classes are $e=\{e_1<e_2\}$, $\beta_1^{(\alpha_i)}=\{\beta_{11}^{(\alpha_i)},\beta_{12}^{(\alpha_i)}\}$, and $\beta_2^{(\alpha_i)}=\{\beta_{21}^{(\alpha_i)},\beta_{22}^{(\alpha_i)}\}$ and whose bases are $\{e_1^{(\alpha_i)},\beta_{12}^{(\alpha_i)},\beta_{22}^{(\alpha_i)}\} , \{e_2^{(\alpha_i)},\beta_{11}^{(\alpha_i)},\beta_{22}^{(\alpha_i)}\} ,\{e_2^{(\alpha_i)},\beta_{12}^{(\alpha_i)},\beta_{21}^{(\alpha_i)}\}$. Then $Z_1\otimes_{e} Z_2$ is the 2-matroid with the four skew classes $\beta_1^{(\alpha_1)}$, $\beta_1^{(\alpha_2)}$, $\beta_2^{(\alpha_1)}$, and $\beta_2^{(\alpha_2)}$ and whose bases are $\{ \beta_{11}^{(\alpha_1)} ,\beta_{22}^{(\alpha_1)} ,\beta_{12}^{(\alpha_2)} , \beta_{22}^{(\alpha_2)} \}$, $\{ \beta_{12}^{(\alpha_1)} ,\beta_{21}^{(\alpha_1)} ,\beta_{12}^{(\alpha_2)} , \beta_{22}^{(\alpha_2)} \}$, $\{ \beta_{12}^{(\alpha_1)} ,\beta_{22}^{(\alpha_1)} ,\beta_{11}^{(\alpha_2)} , \beta_{22}^{(\alpha_2)} \}$, $\{ \beta_{12}^{(\alpha_1)} ,\beta_{22}^{(\alpha_1)} ,\beta_{12}^{(\alpha_2)} , \beta_{21}^{(\alpha_2)} \}$. # The transition polynomial {#sec:results} This section contains our main result, an extension of Brylawski's tensor product formula to multimatroids. The *weighted transition polynomial* of a multimatroid $Z=(U,\Omega,r)$, introduced in [@MR3191496], is $$Q(Z;\mathbf{x}, t):= \sum_{T\in \mathcal T(Z)} t^{n(T)}\mathbf{x}_T ,$$ where $\mathbf{x}$ is a set indexed by $U$ with an entry $x_u$ for each $u$ in $U$ and $\mathbf{x}_T=\prod_{u\in T} x_u$. Notice that if $U=\emptyset$, then $\mathcal T(\Omega)=\{\emptyset\}$, so $Q(Z;\mathbf{x},t)=1$. Here we treat the $x_u$ as formal variables, giving a polynomial in $\mathbb{Z}[t, \{x_u\}_{u\in U}]$. However they can, and often are, treated as parameters. **Example 13**. Let $Z$ be the multimatroid in Example [Example 3](#example1){reference-type="ref" reference="example1"}. Then the transition polynomial $Q(Z;\mathbf{x}, t)$ has 27 summands. Sixteen of these come from the bases and give the $t^0$ terms, the transversal $\{\overline{a},\overline{b},\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}\}$ gives the only $t^2$ term, and the remaining eleven transitions give the $t^1$ terms. So the transition polynomial takes the form $$(x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}}x_{\overline{b}}x_{\overline{c}} +x_{\overline{a}}x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}}x_{\overline{c}} +\cdots + x_{\overline{a}}x_{\widehat{b}}x_{\widehat{c}}) + t( x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}}x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}}x_{\overline{c}}+ \cdots + x_{\overline{a}}x_{\overline{b}}x_{\widehat{c}} ) + t^2(x_{\overline{a}}x_{\overline{b}}x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{c}}) .$$ In practice, it is common to consider some specialisation of the variables $x_T$. For example, taking, for each $e$, $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}}=u$, $x_{\overline{e}}=v$, $x_{\widehat{e}}=0$ gives a polynomial $$Q(Z;(u,v,0), t) = (u^3+2uv^2)+t(3u^2v+v^3) +t^2uv^2$$ that is, as described in Section [5](#sec:delta){reference-type="ref" reference="sec:delta"}, related to the ribbon graph polynomial of $Z$. As the variables of the transition polynomial are indexed by the elements of a multimatroid, it is convenient to use a naming convention for the elements and skew classes in a tensor product. The convention extends that used in Example [Example 12](#ex:smtens){reference-type="ref" reference="ex:smtens"}, and that example illustrates it. **Notation 14**. For $q\geq 2$, let $Z_1$ be a tight $q$-matroid in which every skew class is non-singular and ordered. Let $Z_2$ be a tight multimatroid with a non-singular ordered skew class $e$ of size $q$. Our notational convention is that: - the skew classes of $Z_1$ are $\alpha_1,\alpha_2, \ldots, \alpha_n$; - for each $i$, we have $\alpha_i = \{ \alpha_{i1}< \alpha_{i2}< \cdots <\alpha_{iq} \}$; and - the skew classes of $Z_2$, are $e,\beta_1,\beta_2, \ldots, \beta_m$; - where $e= \{e_1<e_2<\cdots < e_q\}$; - and $\beta_j = \{ \beta_{j1}, \beta_{j2}, \ldots , \beta_{js_j} \}$, for each $j$. The copy of $Z_2$ corresponding to the skew class $\alpha_i$ is denoted by $Z_2^{(\alpha_i)}$ and has: - skew classes $e,\beta_1^{(\alpha_i)},\beta_2^{(\alpha_i)}, \ldots, \beta_m^{(\alpha_i)}$; - with $e= \{e_1<e_2<\cdots < e_q\}$; - and $\beta_j^{(\alpha_i)} = \{ \beta_{j1}^{(\alpha_i)}, \beta_{j2}^{(\alpha_i)}, \ldots , \beta_{js_j}^{(\alpha_i)} \}$, for each $j$. Thus $Z_1\otimes_{e} Z_2$ has - skew classes $\beta_j^{(\alpha_i)}$ for $1\leq i \leq n$ and $1\leq j \leq m$; - elements $\beta_{jk}^{(\alpha_i)}$ where $1\leq i \leq n$, $1\leq j \leq m$, and for each value of $j$, we have $1\leq k\leq s_j$. Additionally we use the following notation related to subtransversals in $Z_1$, $Z_2$ and $Z_1\otimes_{e} Z_2$. - Every transversal $T$ of $Z_1\otimes_{e} Z_2$ can be written as $T= T^{(\alpha_1)} \sqcup \cdots \sqcup T^{(\alpha_n)}$ where each $T^{(\alpha_i)}$ is a near transversal of $Z_2^{(\alpha_i)}$ missing $e$. - For each $i$, the orderings of the skew classes $\alpha_i$ of $Z_1$ and $e$ of $Z_2^{(\alpha_i)}$ give a natural bijection $\psi^{(\alpha_i)}$ from $e$ to $\alpha_i$ whose action is $$\psi^{(\alpha_i)}(e_j) = \alpha_{ij}.$$ - A transversal $\psi(T)$ of $Z_1$ can be naturally associated with each transversal $T$ of $Z_1\otimes_{e} Z_2$ by setting $$\psi(T) := \{ \psi^{(\alpha_1)}(\theta({T^{(\alpha_1)}})), \ldots , \psi^{(\alpha_n)}(\theta({T^{(\alpha_n)}})) \},$$ where we recall $\theta({T^{(\alpha_i)}})$ denotes the slack element of $T^{(\alpha_i)}$. We now state our main result, a tensor product formula for the transition polynomial. As the variables of the transition polynomial are indexed by the elements of the multimatroid, we make use of a substitution function $\phi$ in the theorem statement. Although $\phi$ is awkward to formalise, its action is the obvious natural one. We illustrate it in an example just after the theorem statement. **Theorem 15**. *For $q\geq 2$, let $Z_1$ be a tight $q$-matroid in which every every skew class is non-singular and ordered. Let $Z_2$ be a tight multimatroid with a non-singular ordered skew class $e$ of size $q$. Suppose further that $Z_1$, $Z_2$ and $Z_1\otimes_{e} Z_2$ are given following Notation [Notation 14](#not:tens){reference-type="ref" reference="not:tens"}. Then $$Q(Z_1\otimes_{e} Z_2 ; \mathbf{x}{}, t) = \phi( Q(Z_1; \mathbf{x}, t )),$$ where $\phi$ is the map induced by the substitution $x_{\alpha_{ij}} \mapsto \phi^{(\alpha_i)} (y_j)$, for $\phi^{(\alpha_i)}$ the map induced by the action $x_{ \beta_{jk} } \mapsto x_{ \beta_{jk}^{(\alpha_i)}}$, and where $y_1,\dots,y_q$ are the unique solution to the system of linear equations $$\begin{aligned} Q(Z_2|e_1;\mathbf{x},t)&=ty_1+y_2+\dots +y_q, \\ Q(Z_2|e_2;\mathbf{x},t)&=y_1+ty_2+\dots +y_q, \\ & \vdots \\ Q(Z_2|e_q;\mathbf{x},t)&=y_1+y_2+\dots +ty_q. \end{aligned}$$* An example of an application of the theorem can be found at the end of this section. But before proving this theorem we comment on the map $\phi$. **Example 16**. In this illustration, we choose the indexing for clarity, and the particular terms may not correspond to any particular example. A summand of $Q(Z_1; \mathbf{x}, t )$ typically looks like $t^2x_{\alpha_{17}}x_{\alpha_{24}}$. (In this case $Z_1$ would have two skew classes.) And two components $y_7$ and $y_4$ of the unique solution to the linear equations might have the form $y_7=tx_{\beta_{13}}x_{\beta_{22}}x_{\beta_{31}}+t^2x_{\beta_{11}}x_{\beta_{21}}x_{\beta_{36}}$, and $y_4=x_{\beta_{14}}x_{\beta_{21}}x_{\beta_{34}}$. (In this case $Z_2$ would have four skew classes.) There are two maps $\phi^{(\alpha_1)}$ and $\phi^{(\alpha_2)}$, one for each skew class of $Z_1$. We have $\phi^{(\alpha_1)}(y_7) = tx_{\beta_{13}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{31}^{(\alpha_1)}}+t^2x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{36}^{(\alpha_1)}}$, and $\phi^{(\alpha_2)}(y_4) = x_{\beta_{14}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}}x_{\beta_{34}^{(\alpha_2)}}$. With this we have $$\begin{aligned} \phi(t^2x_{\alpha_{17}}x_{\alpha_{24}}) &=t^2 \,(\phi^{(\alpha_1)}(y_7))\,( \phi^{(\alpha_2)}(y_4) ) \\ &= t^2 (tx_{\beta_{13}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{31}^{(\alpha_1)}}+t^2x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{36}^{(\alpha_1)}}) (x_{\beta_{14}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}}x_{\beta_{34}^{(\alpha_2)}}).\end{aligned}$$ *Proof of Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}.* We follow the conventions given in Notation [Notation 14](#not:tens){reference-type="ref" reference="not:tens"}. In particular we recall that each transversal $T$ of $Z=Z_1\otimes_{e}Z_2$ can be written as $T=T^{(\alpha_1)} \sqcup \cdots \sqcup T^{(\alpha_n)}$ where each $T^{(\alpha_i)}$ is a near transversal of $Z_2^{(\alpha_i)}$ that avoids $e$. Also, by considering the slack element of these near transversals we write $\psi(T) = \{ \psi^{(\alpha_1)}(\theta({T^{(\alpha_1)}})), \ldots , \psi^{(\alpha_n)}(\theta({T^{(\alpha_n)}})) \}$, and then $\psi(T)$ is a transversal of $Z_1$. It follows from Lemma [Lemma 10](#lem:claim1){reference-type="ref" reference="lem:claim1"} that $$\label{eq:claim1} n_Z(T) = n_{Z_1}(\psi(T)) +\sum_{i=1}^n n_{Z_2^{(\alpha_i)}} (T^{(\alpha_i)}) .$$ For clarity, we denote the nullity $n_{Z_2^{(\alpha_i)}}$ by $n_i$ in the following equations. We have $$\begin{aligned} Q(Z;\mathbf{x}, t) &= \sum_{T\in \mathcal{T}(Z)} t^{n_Z(T)}\mathbf{x}_T \nonumber \\&= \sum_{A\in \mathcal{T}(Z_1)} \sum_{\substack{T\in \mathcal{T}(Z) \\ \psi(T)=A}} t^{n_Z(T)}\mathbf{x}_T \nonumber \\ &= \sum_{A\in \mathcal{T}(Z_1)} \sum_{\substack{T\in \mathcal{T}(Z) \\ \psi(T)=A}} t^{n_{Z_1}(A) + \sum_{i=1}^n n_{i}(T^{(\alpha_i)})}\mathbf{x}_T \nonumber \\ &= \sum_{A\in \mathcal{T}(Z_1)} t^{n_{Z_1}(A)} \sum_{\substack{T\in \mathcal{T}(Z) \\ \psi(T)=A}} \Big(\prod_{i=1}^n t^{n_i(T^{(\alpha_i)}) }\mathbf{x}_{T^{(\alpha_i)}}\Big) \nonumber \\ &= \sum_{A \in \mathcal{T}(Z_1) } t^{n_{Z_1}(A)} \prod_{i=1}^n \Big( \sum_{\substack{ T \in \mathcal{N}(Z_2^{(\alpha_i)}) \\ \{\psi^{(\alpha_i)}( \theta({T}))\}= A\cap \alpha_i}} t^{n_i(T) }\mathbf{x}_{T} \Big). \label{al:5}\end{aligned}$$ Now consider some $A \in \mathcal{T}(Z_1)$. Its contribution to the transition polynomial $Q(Z_1;\mathbf{x}, t)$ is $$t^{n_{Z_1}(A)} x_{A} = t^{n_{Z_1}(A)} \prod_{i=1}^n x_{\alpha_{ij_i}},$$ for some values of $j_i$ for $i=1,\ldots, n$. Thus we can obtain the expression [\[al:5\]](#al:5){reference-type="eqref" reference="al:5"} from $Q(Z_1;\mathbf{x}, t)$ by the substitutions $$\begin{aligned} x_{\alpha_{ij}} & \mapsto \sum_{\substack{ T \in \mathcal{N}(Z_2^{(\alpha_i)}) \\ \psi^{(\alpha_i)}( \theta({T}))= \alpha_{ij}}} t^{n_i(T) }\mathbf{x}_{T} \\ &= \sum_{\substack{ T \in \mathcal{N}(Z_2^{(\alpha_i)}) \\ \theta({T})= e_j}} t^{n_i(T) }\mathbf{x}_{T} \\ &= \phi^{(\alpha_i)} \Big( \sum_{\substack{ T \in \mathcal{N}(Z_2) \\ \theta({T})= e_j}} t^{n_{Z_2}(T) }\mathbf{x}_{T} \Big) \\ &= \phi^{(\alpha_i)}(y_j),\end{aligned}$$ where $y_j := \sum\limits_{\substack{T\in \mathcal{N}(Z_2)\\ \theta({T})= e_j }} t^{n_{Z_2}(T)} \mathbf{x}_T$ and $\phi^{(\alpha_i)}$ is the map induced by the action $x_{ \beta_{jk} } \mapsto x_{ \beta_{jk}^{(\alpha_i)}}$ that renames the variables. To determine the $y_j$ observe that as $e$ is a non-singular skew class and $Z_2$ is tight, when $T$ is a near transversal avoiding $e$ we have $$n_{Z_2|e_i}(T)= \begin{cases} n_{Z_2}(T) &\text{if }\theta({T})\neq e_i,\\ n_{Z_2}(T)+1 &\text{if }\theta({T})= e_i. \end{cases}$$ Thus $$\begin{aligned} Q(Z_2|e_i) &= \sum_{T\in \mathcal{T} (Z_2|e_i)} t^{n_{Z_2|e_i}(T)} \mathbf{x}_T \\&= \sum_{\substack{T\in \mathcal{N}(Z_2) \\ T\cap \,e=\emptyset}} t^{n_{Z_2|e_i}(T)} \mathbf{x}_T \\&= \sum_{\substack{T\in \mathcal{N}(Z_2) \\ \theta({T})=e_i }} t^{n_{Z_2}(T)+1} \mathbf{x}_T +\sum_{\substack{T\in \mathcal{N}(Z_2) \\ T\cap \,e=\emptyset \\ \theta({T})\neq e_i }} t^{n_{Z_2}(T)} \mathbf{x}_T \\&= \sum_{\substack{T\in \mathcal{N}(Z_2) \\ \theta({T})=e_i }} t^{n_{Z_2}(T)+1} \mathbf{x}_T + \sum_{\substack{j=1,\\ j\neq i}}^n \sum_{\substack{T\in \mathcal{N}(Z_2) \\ \theta({T})= e_j }} t^{n_{Z_2}(T)} \mathbf{x}_T \\ &= ty_i + \sum_{\substack{j=1\\ j\neq i}}^n y_j.\end{aligned}$$ It is routine to check that the $y_i$ can be obtained as the unique solution to the system of equations given in the theorem statement. ◻ **Example 17**. Let $Z_1$ and $Z_2$ be as in Example [Example 12](#ex:smtens){reference-type="ref" reference="ex:smtens"}. Then $$\begin{aligned} Q(Z_1;\mathbf{x},t) &= tx_{\alpha_{11}}x_{\alpha_{21}}+x_{\alpha_{11}}x_{\alpha_{22}}+x_{\alpha_{12}}x_{\alpha_{21}}+tx_{\alpha_{12}}x_{\alpha_{22}}, % \\ Q(Z_2|e_1;\mathbf{x},t) &= t^2x_{\beta_{11}}x_{\beta_{21}}+tx_{\beta_{11}}x_{\beta_{22}}+tx_{\beta_{12}}x_{\beta_{21}}+x_{\beta_{12}}x_{\beta_{22}}, % \\ Q(Z_2|e_2;\mathbf{x},t) &= tx_{\beta_{11}}x_{\beta_{21}}+x_{\beta_{11}}x_{\beta_{22}}+x_{\beta_{12}}x_{\beta_{21}}+tx_{\beta_{12}}x_{\beta_{22}}.\end{aligned}$$ Thus $$\begin{aligned} y_1&=tx_{\beta_{11}}x_{\beta_{21}}+x_{\beta_{11}}x_{\beta_{22}}+x_{\beta_{12}}x_{\beta_{21}}, % \\ y_2&=x_{\beta_{12}}x_{\beta_{22}}.\end{aligned}$$ The substitution function $\phi$ is given by $$\begin{aligned} x_{\alpha_{11}}&\mapsto \phi^{(\alpha_1)} (y_1 )= tx_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}+x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}+x_{\beta_{12}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}} \\ x_{\alpha_{12}}&\mapsto \phi^{(\alpha_1)} (y_2) = x_{\beta_{12}^{(\alpha_1)} }x_{\beta_{22}^{(\alpha_1)} } \\ x_{\alpha_{21}}&\mapsto \phi^{(\alpha_2)} (y_1) = tx_{\beta_{11}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}}+x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}}+x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} \\ x_{\alpha_{22}}&\mapsto \phi^{(\alpha_2)} (y_2) = x_{\beta_{12}^{(\alpha_2)} }x_{\beta_{22}^{(\alpha_2)} }.\end{aligned}$$ Thus $$\begin{aligned} Q(Z_1\otimes_{e} Z_2 ; \mathbf{x}, t) &= t^3x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + t^2x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} \\ &\phantom{=} {}+ t^2x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + t^2x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} \\ &\phantom{=} {}+ t^2x_{\beta_{12}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + tx_{\beta_{11}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} \\ &\phantom{=} {}+ tx_{\beta_{11}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} + tx_{\beta_{12}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} \\ &\phantom{=} {}+ tx_{\beta_{11}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + tx_{\beta_{12}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} \\ &\phantom{=} {}+ tx_{\beta_{12}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + x_{\beta_{11}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} \\ &\phantom{=} {}+ x_{\beta_{12}^{(\alpha_1)}}x_{\beta_{21}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} + x_{\beta_{12}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{11}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}} \\ &\phantom{=} {}+ x_{\beta_{12}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{21}^{(\alpha_2)}} + tx_{\beta_{12}^{(\alpha_1)}}x_{\beta_{22}^{(\alpha_1)}}x_{\beta_{12}^{(\alpha_2)}}x_{\beta_{22}^{(\alpha_2)}}.\end{aligned}$$ # Delta-matroids {#sec:delta} In this section we exploit connections between multimatroids and delta-matroids to obtain Brylawski-type formulas for the tensor product of delta-matroids. Theorem [Theorem 20](#thm:rgpdm){reference-type="ref" reference="thm:rgpdm"} gives the tensor product formula for the weighted transition polynomial for vf-safe delta-matroids; Theorem [Theorem 21](#thm:dmeven){reference-type="ref" reference="thm:dmeven"} gives the corresponding formula for even delta-matroids. We conclude the section by rewriting the latter formula in terms of Tutte polynomials of delta-matroids, deriving Brylawski's original formula for matroids as a consequence. A *delta-matroid* is a pair $(E,\mathcal{F})$ where $E$ is a finite set called the *ground set* and $\mathcal{F}$ is a non-empty collection of subsets of $E$ satisfying the *symmetric exchange axiom*: for all triples $(X,Y,u)$ with $X$ and $Y$ in $\mathcal{F}$ and $u\in X\bigtriangleup Y$ (where $\bigtriangleup$ denotes symmetric difference of sets), there is an element $v\in X\bigtriangleup Y$ (perhaps $u$ itself) such that $X\bigtriangleup \{u,v\}$ is in $\mathcal{F}$. The elements of $\mathcal{F}$ are called *feasible sets*. A *coloop* of a delta-matroid is an element belonging to every feasible set; a *loop* is an element belonging to no feasible set. Let $D$ be a delta-matroid and $e$ an element of $D$. Suppose first that $e$ is not a coloop of $D$. Then we define $D\mathop{\mathrm{\backslash}}e$, the *deletion* of $e$, to be the pair $$(E-\{e\},\{ F \in \mathcal {F} : e\notin F\}).$$ Now suppose that $e$ is not a loop of $D$. Then we define $D/ e$, the *contraction* of $e$, to be the pair $$(E-\{e\},\{ F-\{e\} : F \in \mathcal {F} \text{ and } e \in F\}).$$ If $e$ is either a coloop or a loop of $D$, then one of $D\mathop{\mathrm{\backslash}}e$ and $D/e$ is defined. In this case, we define whichever of $D\mathop{\mathrm{\backslash}}e$ and $D/e$ is so far undefined by setting $D\mathop{\mathrm{\backslash}}e = D/e$. It is easy to check that both $D\mathop{\mathrm{\backslash}}e$ and $D/e$ are delta-matroids. Let $D=(E,\mathcal F)$ be a delta-matroid. For a subset $A$ of $E$, the *twist* of $D$ by $A$, denoted by $D*A$, is given by $(E,\{X\bigtriangleup A: X\in\mathcal{F}\})$. It is straightforward to show that $D*A$ is indeed a delta-matroid. Given an element $e$ in $E$, following [@zbMATH05982480], we define the *loop complement* of $D$ by $e$, which we denote by $D+e$, to be the pair $(E,\mathcal{F}')$, where $$\mathcal{F}'=\mathcal{F}\bigtriangleup\{X\cup \{e\} : X\in\mathcal{F}, e\notin X\}.$$ Thus a set $A$ not containing $e$ is in $\mathcal F'$ if and only it is in $\mathcal F$, and a set $A$ containing $e$ is in $\mathcal F'$ if and only if precisely one of $A$ and $A-\{e\}$ is in $\mathcal F$. If $e_1, e_2\in E$, then $(D+e_1)+e_2=(D+e_2)+e_1$, and so for a subset $A$ of $E$ we can unambiguously define the loop complement $D+A$ of $D$ by $A$, by forming the loop complement with respect to each element of $A$ in turn. The set of delta-matroids is not closed under loop complementation. A delta-matroid $D=(E,\mathcal{F})$ is said to be *vf-safe* if the application of every sequence of twists and loop complements results in a delta-matroid. It was been shown in [@zbMATH05982480] that the operations of twisting $\ast e$ and loop complementation $+e$ with respect to an element $e$ are involutions and can be used to define an action of the symmetric group $S_3$ on the set of vf-safe delta-matroids. The relevance of this for us is that there is a third involution, $+e\ast e+e=\ast e+e\ast e$, which we denote by $\overline{\ast}e$. For a vf-safe delta-matroid $D$ with element $e$, we use $D\overline{\ast} e$ to denote the delta-matroid $D+e\ast e+e$. Throughout this section we shall work exclusively with 2-matroids and 3-matroids indexed by a set $E$. A 2-matroid is *indexed* by $E$ if its collection of skew classes has the form $\{\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}\}:e\in E\}$; a 3-matroid is *indexed* by $E$ if its collection of skew classes has the form $\{\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e}\}:e\in E\}$. We denote the skew class $\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}\}$ or $\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e}\}$ by its indexing element $e$. For a set $E$, let $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E}=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}: e\in E\}$, $\overline{E}=\{\overline e:e \in E\}$, and $\widehat{E}=\{\widehat{e}: e\in E\}$. For a subset $X$ of $E$, let $$\omega(X):=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}:e \in X\} \cup\{\overline {e}:e\notin X\}.$$ In [@MR904585], Bouchet established the relationship between delta-matroids and 2-matroids. Given a delta-matroid $D=(E,\mathcal{F})$, let $U=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E}\cup\overline{E}$ and $\Omega=\{\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e},\overline{e}\}:e\in E\}$. We use $Z_{[2]}(D)$ to denote the associated (indexed) 2-matroid $(U,\Omega,r)$ with collection of bases $\{\omega(X):X\in \mathcal F\}$. Up to isomorphism, every 2-matroid arises in this way. On the other hand, suppose $Z=(U,\Omega,r)$ is a $2$-matroid indexed by $E$. Now let $\mathcal{F}=\{X \subseteq E: \omega(X)\in \mathcal{B}(Z)\}$. Then $(E,\mathcal{F})$ is a delta-matroid and we use $D(Z)$ to denote it. We note that by naming the elements of each skew class in such a specific way, we implicitly partition the elements of $Z$ into two transversals. The resulting delta-matroid $D(Z)$ is dependent on the choice of partition. Any two delta-matroids that arise from different ways of naming the skew classes are related by twisting. For a delta-matroid $D$ with element $e$, $$\begin{aligned} Z_{[2]}(D\mathop{\mathrm{/}}e) = Z_{[2]}(D)|\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \quad\text{and} \quad Z_{[2]}(D\mathop{\mathrm{\backslash}}e) = Z_{[2]}(D)|\overline{e}.\end{aligned}$$ Although this was surely known to Bouchet, we cannot find an explicit statement in his work. However, details may be found in [@MR3191496 Lemma 19]. If the cardinalities of the feasible sets in a delta-matroid all have the same parity, then the delta-matroid is *even*. In [@MR1845490] it was shown that $Z$ is a tight 2-matroid if and only if $D(Z)$ is an even delta-matroid. More recently, in [@MR3191496] Brijder and Hoogeboom established the relationship between tight 3-matroids and vf-safe delta-matroids. Following Bouchet [@MM1zbMATH01116184], given a multimatroid $Z=(U,\Omega,r)$ and a subset $A$ of its elements, we define $Z[A]$ to be the multimatroid $(A,\Omega',r')$ so that $$\Omega' = \{ e \cap A : e \in \Omega, e \cap A \ne \emptyset\}$$ and $r'$ is the restriction of $r$ to transversals of $(A,\Omega')$. Given a vf-safe delta-matroid $D=(E,\mathcal{F})$, we obtain its corresponding (indexed) tight 3-matroid $Z_{[3]}(D)$ as follows. Let $U=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E}\cup\overline{E}\cup\widehat{E}$ and $\Omega=\{\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e},\overline{e},\widehat{e}\}:e\in E\}$. Brijder and Hoogeboom showed in [@MR3191496] that the tightness condition implies that for every delta-matroid $D$ there is at most one tight multimatroid $Z$ with carrier $(U,\Omega)$ such that $Z[\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E} \cup \overline{E}]=Z_{[2]}(D)$. They also showed in [@MR3191496], by giving an explicit construction, that $Z$ exists if and only if $D$ is vf-safe. When $D$ is vf-safe, we take $Z_{[3]}(D)$ to be this $Z$. Beyond the fact that a transversal $T$ of $Z_{[2]}(D)$ is a basis of $Z_{[3]}(D)$ if and only if it is a basis of $Z_{[2]}(D)$, the precise form of $\mathcal B(Z_{[3]}(D))$ will not concern us. Up to isomorphism, every tight 3-matroid arises in this way. Conversely, let $Z$ be a tight 3-matroid indexed by $E$. Then $Z[\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E}\cup\overline{E}]$ is a 2-matroid and we let $D(Z):=D(Z[\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{E}\cup\overline{E}])$. By [@MR3191496], $D(Z)$ is a vf-safe delta-matroid and every vf-safe delta-matroid arises in this way. Now the choice of naming implicitly yields a partition of the elements of $Z$ into three transversals and the delta-matroid $D(Z)$ depends on this choice. Any two delta-matroids that arise from different ways of naming the elements are related by a sequence of twistings and loop complementations. It has been shown by Brijder and Hoogeboom in [@MR3191496 Lemma 19] that for a vf-safe delta-matroid $D=(E,\mathcal{F})$ with element $e$, $$\label{eq:z3dmmin} Z_{[3]}(D\mathop{\mathrm{/}}e) = Z_{[3]}(D)|\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \quad Z_{[3]}(D\mathop{\mathrm{\backslash}}e) = Z_{[3]}(D)|\overline{e}, \quad Z_{[3]}(D+ e\mathop{\mathrm{/}}e) = Z_{[3]}(D)|\widehat{e}.$$ To define the two-sum of two delta-matroids, we need the concept of a singular element in a delta-matroid. An element $e$ of a delta-matroid $D$ is *singular* if $e$ is a coloop, $e$ is a loop or $e$ has the property that for every feasible set $F$ of $D$, $F\bigtriangleup e$ is also feasible. Notice that a singular element of an even delta-matroid is either a coloop or a loop. Let $D_1$ and $D_2$ be two even delta-matroids with disjoint ground sets, and let $f$ and $e$ be non-singular elements of $D_1$ and $D_2$, respectively. The *2-sum* along $f$ and $e$, $D_1\oplus_{f=e}D_2$, is defined as $$D_1\oplus_{f=e}D_2:=D(Z_{[2]}(D_1)\oplus_{f=e}Z_{[2]}(D_2)).$$ Similarly, let $D_1$ and $D_2$ be two vf-safe delta-matroids with disjoint ground sets, and let $f$ and $e$ be non-singular elements of $D_1$ and $D_2$, respectively. The *2-sum* along $f$ and $e$, $D_1\oplus_{f=e}D_2$, is defined as $$D_1\oplus_{f=e}D_2:=D(Z_{[3]}(D_1)\oplus_{f=e}Z_{[3]}(D_2)).$$ Notice that if $D_1$ and $D_2$ are both even and vf-safe, then we have apparently given two different definitions of the two-sum. Nevertheless it is shown in [@2sumpaper] that the two definitions coincide in this case. We can use 2-sums to define tensor products. Let $D_1=(E_1,\mathcal{F}_1)$ and $D_2=(E_2,\mathcal{F}_2)$ be delta-matroids that are both even or both vf-safe and such that $D_1$ has no non-singular elements. Let $e$ be a non-singular element in $D_2$. Take an isomorphic copy of $D_2$ for every element $f$ of $D_1$. For notational convenience we assume that in each copy the element $e$ remains, but the copies are otherwise on disjoint sets of elements. The *tensor product* $D_1\otimes_e D_2$ is the delta-matroid constructed by forming the 2-sum along $f$ and $e$ for each element $f$ of $E_1$ and its corresponding copy of $D_2$. We adopt a similar naming convention for the elements of $D_1\otimes_e D_2$ as that used in Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}. In $D_1\otimes_e D_2$, for each $a\in E_1$ and $b\in E_2$, we shall use $b^{(a)}$ to denote the copy of element $b$ that appears in the copy of $D_2$ arising from element $a$ of $D_1$. We have the following consequences of the definition of the two-sum. **Proposition 18**. *Let $D_1=(E_1,\mathcal{F}_1)$ be an even delta-matroid with no non-singular elements, and $D_2=(E_2,\mathcal{F}_2)$ be an even delta-matroid with a non-singular element $e$. Then, $$Z_{[2]}(D_1\otimes_{e}D_2)\cong Z_{[2]}(D_1)\otimes_{e} Z_{[2]}(D_2),$$ where the isomorphism identifies $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b^{(a)}}$ with $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}^{(a)}$, and $\overline{b^{(a)}}$ with $\overline{b}^{(a)}$.* **Proposition 19**. *Let $D_1=(E_1,\mathcal{F}_1)$ be a vf-safe delta-matroid with no non-singular elements, and $D_2=(E_2,\mathcal{F}_2)$ be a vf-safe delta-matroid with a non-singular element $e$. Then, $$Z_{[3]}(D_1\otimes_{e}D_2)\cong Z_{[3]}(D_1)\otimes_{e}Z_{[3]}(D_2),$$ where the isomorphism identifies $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b^{(a)}}$ with $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}^{(a)}$, $\overline{b^{(a)}}$ with $\overline{b}^{(a)}$, and $\widehat{b^{(a)}}$ with $\widehat{b}^{(a)}$.* For a delta-matroid $D$ and subset $X$ of its elements, define $$d_D(X):=\min_{F\in \mathcal{F}}|F\bigtriangleup X|.$$ When $D$ is unambiguous, we omit the subscript. Let $D=(E,\mathcal{F})$ be a vf-safe delta-matroid, $\boldsymbol{u}=\{u\}_{e\in E}$, $\boldsymbol{v}=\{v_e\}_{e\in E}$, $\boldsymbol{w}=\{w_e\}_{e\in E}$ be indexed families of formal variables, and $t$ be another formal variable. Then, from [@MR3191496], the *multivariate topological transition polynomial* $Q(D;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$ can be defined as $$Q(D;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t):=\sum_{(X,Y,Z) \in \mathcal{P}_3(E)} \Big( \prod_{e\in X}u_e\Big) \Big( \prod_{e\in Y} v_e\Big) \Big( \prod_{e\in Z} w_e\Big) t^{d_{D\bar{\ast}Z}(X)},$$ where $\mathcal{P}_3(E)$ denotes the set of ordered partitions of $E$ into three blocks. (The blocks may be empty.) Using [@MR3191496 Lemma 18] it may be shown, for a vf-safe delta-matroid $D$ and partition $(X,Y,Z)\in \mathcal{P}_3$, that $d_{D\overline{\ast}Z}(X)=n_{Z_{[3]}(D)}(\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{X}\cup\overline{Y}\cup\widehat{Z})$. So we can obtain the multivariate topological transition polynomial $Q(D;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$ as an instance of the weighted transition polynomial $Q(Z;\boldsymbol{x},t)$ as follows: $$\label{eq:dmttp}Q(D;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)=Q(Z_{[3]}(D);\boldsymbol{x},t),$$ when ${u}_e={x}_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}}$, ${v}_e={x}_{\overline{e}}$, and ${w}_e={x}_{\widehat{e}}$ for each $e\in E$. **Theorem 20**. *Let $D_1$ be a vf-safe delta-matroid having no non-singular elements and let $D_2$ be a vf-safe delta-matroid with non-singular element $e$. Then $$Q(D_1\otimes_e D_2;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)=\psi(Q(D_1;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)),$$ where $\psi$ is the map induced by the substitution $u_a\mapsto \psi^{(a)}(p)$, $v_a\mapsto\psi^{(a)}(q)$, and $w_a\mapsto\psi^{(a)}(r)$; and $\psi^{(a)}$ sends $u_b\mapsto u_{b^{(a)}}$, $v_b\mapsto v_{b^{(a)}}$, and $w_b\mapsto w_{b^{(a)}}$; and where $p$, $q$ and $r$ are the unique solutions to the system of linear equations $$\begin{aligned} Q(D_2\mathop{\mathrm{/}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)=&tp+q+r,\\ Q(D_2\mathop{\mathrm{\backslash}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)=&p+tq+r,\\ Q(D_2+e\mathop{\mathrm{/}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)=&p+q+tr. \end{aligned}$$* *Proof.* By Equation [\[eq: dmqtp\]](#eq: dmqtp){reference-type="eqref" reference="eq: dmqtp"}, and the definition of the transition polynomial, $$Q(D_1\otimes_e D_2;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t) = \sum_{T\in \mathcal{T}(Z_{[3]}(D_1\otimes_e D_2))} t^{n(T)} \prod_{s\in T} x_s,$$ when ${x}_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b^{(a)}}}={u}_{b^{(a)}}$, ${x}_{\overline{{b^{(a)}}}}={v}_{b^{(a)}}$, and ${x}_{\widehat{{b^{(a)}}}}={w}_{b^{(a)}}$. Then by Proposition [Proposition 19](#cor:vfdmtp){reference-type="ref" reference="cor:vfdmtp"} this is equal to $$\sum_{T\in \mathcal{T}(Z_{[3]}(D_1)\otimes_e Z_{[3]}(D_2))} t^{n(T)} \prod_{s\in T} x_s =Q(Z_{[3]}(D_1)\otimes_e Z_{[3]}(D_2)); \boldsymbol{x},t) ,$$ when ${x}_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}^{(a)}}={u}_{b^{(a)}}$, ${x}_{\overline{b}^{(a)}}={v}_{b^{(a)}}$, and ${x}_{\widehat{b}^{(a)}}={w}_{b^{(a)}}$. An application of Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"} gives that this equals $$\phi(Q(Z_{[3]}(D_1); \boldsymbol{x},t))$$ where $\phi$ is given by the action $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}}\mapsto \phi^{(a)}(\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y})$, $x_{\overline{a}}\mapsto \phi^{(a)}(\overline{y})$, $x_{\widehat{a}}\mapsto \phi^{(a)}(\widehat{y})$; $\phi^{(a)}$ is given by the action $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}}\mapsto u_{b^{(a)}}$, $x_{\widehat{b}}\mapsto v_{b^{(a)}}$, $x_{\overline{b}}\mapsto w_{b^{(a)}}$; and, making use of Equation [\[eq: dmqtp\]](#eq: dmqtp){reference-type="eqref" reference="eq: dmqtp"}, $$\begin{aligned} Q(Z_{[3]}(D_2\mathop{\mathrm{/}}e);\boldsymbol{x},t) =Q(Z_{[3]}(D_2)|\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e};\boldsymbol{x},t)&= t\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+\overline{y}+\widehat{y},\\ %%%%%%% Q(Z_{[3]}(D_2\mathop{\mathrm{\backslash}}e);\boldsymbol{x},t) =Q(Z_{[3]}(D_2)|\overline{e};\boldsymbol{x},t) &=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+t\overline{y}+\widehat{y},\\ %%%%%%%%% Q(Z_{[3]}(D_2+e\mathop{\mathrm{/}}e);\boldsymbol{x},t) =Q(Z_{[3]}(D_2)|\widehat{e};\boldsymbol{x},t)&=\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+\overline{y}+t\widehat{y}. \end{aligned}$$ When $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{a}}=u_a$, $x_{\overline{a}}=v_a$ and $x_{\widehat{a}}=w_a$ for each element $a$ of $D_1$, we have $\phi(Q(Z_{[3]}(D_1);\boldsymbol{x},t)=\phi(Q(D_1;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$. Similarly, when $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{b}}=u_b$, $x_{\overline{b}}=v_b$ and $x_{\widehat{b}}=w_b$ for each element $b$ of $D_2$ except $e$, we have $\phi(Q(Z_{[3]}(D_2\mathop{\mathrm{/}}e);\boldsymbol{x},t)=\phi(Q(D_2\mathop{\mathrm{/}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$, $\phi(Q(Z_{[3]}(D_2\mathop{\mathrm{\backslash}}e);\boldsymbol{x},t)=\phi(Q(D_2\mathop{\mathrm{\backslash}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$, and $\phi(Q(Z_{[3]}(D_2+e\mathop{\mathrm{/}}e);\boldsymbol{x},t)=\phi(Q(D_2\mathop{\mathrm{/}}e;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$. Applying the same substitution to $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}$, $\overline{y}$ and $\widehat{y}$ yields $p$, $q$ and $r$ respectively. Thus $\phi(Q(Z_{[3]}(D_1);\boldsymbol{x},t) = \psi(D_1;(\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}),t)$ and the theorem follows. ◻ A similar analysis gives a tensor product formula for the transition polynomial of Brijder and Hoogeboom from [@MR3191496]. Let $D=(E,\mathcal{F})$ be a delta-matroid, and $w,x,t$ be formal variables. Then the *transition polynomial* $Q(D;w,x,t)$ for delta-matroids is $$Q(D;w,x,t):= \sum_{A \subseteq E} w^{|E-A|}x^{|A|} t^{d(A)}.$$ (In [@MR3191496] this polynomial was denoted $Q_{w,x,0}(D;t)$.) It can be shown that for every delta-matroid $D$, we have $d_D(X)=n_{Z_{[2]}(D)}(\omega(X))$. (See Lemma 18 of [@MR3191496] and the comments below it.) Consequently it follows that $$\label{eq: dmqtp} Q(D;w,x,t)=Q(Z_{[2]}(D);\boldsymbol{x},t),$$ when $x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}}=x$ and $x_{\overline{e}}=w$, for each $e\in E$. **Theorem 21**. *Let $D_1$ be an even delta-matroid having no non-singular elements and let $D_2$ be an even delta-matroid with non-singular element $e$. Then $$Q(D_1\otimes_e D_2;w,x,t)=Q(D_1;p,q,t),$$ where $p,q$ are the unique solutions to the system of linear equations $$\begin{aligned} Q(D_2\mathop{\mathrm{/}}e;w,x,t)&=tp+q,\\ Q(D_2\mathop{\mathrm{\backslash}}e;w,x,t)&=p+tq. \end{aligned}$$* A proof of this result can be obtained following the method used for the proof of Theorem [Theorem 20](#thm:rgpdm){reference-type="ref" reference="thm:rgpdm"}. As such we omit its proof. The key points are that Equation [\[eq: dmqtp\]](#eq: dmqtp){reference-type="eqref" reference="eq: dmqtp"} and Proposition [Proposition 18](#cor:dmtp){reference-type="ref" reference="cor:dmtp"} are used in place of Equation [\[eq:dmttp\]](#eq:dmttp){reference-type="eqref" reference="eq:dmttp"} and Proposition [Proposition 19](#cor:vfdmtp){reference-type="ref" reference="cor:vfdmtp"}. The substitution functions that appear in Theorem [Theorem 20](#thm:rgpdm){reference-type="ref" reference="thm:rgpdm"} are not needed in the final theorem statement because the variables of $Q(D;w,x,t)$ are not indexed by its elements. Matroids are precisely delta-matroids in which all feasible sets have the same carnality. In this situation the feasible sets are exactly the bases of the matroid. We use $r(M)$ to denote the rank of a matorid $M$. (It will always be clear whether $r$ is referring to the rank of a matroid or of a multimatorid, so the double use of $r$ should cause no confusion.) For an arbitrary delta-matroid $D=(E,\mathcal{F})$, let $\mathcal{F}_{\max}$ and $\mathcal{F}_{\min}$ denote the set of feasible sets in $\mathcal{F}$ with maximum and minimum cardinality, respectively. As in [@zbMATH04185622], $D_{\max}:=(E,\mathcal{F}_{\max})$ and $D_{\min}:=(E,\mathcal{F}_{\min})$ are both matroids. By considering the ranks of these matroids we can define a function $\sigma$ on delta-matroids by setting $$\label{eq:defsigma} \sigma(D):=\tfrac{1}{2}(r(D_{\min})+r(D_{\max})).$$ For a subset $A\subseteq E$, we set $\sigma_D(A):=\sigma(D\backslash A^c)$, where $A^c:=E-A$. We omit the subscript when $D$ is clear from context. Using [@CMNR1 Proposition 5.39] we can deduce that $$\sigma_D(A)=\tfrac{1}{2}(|A|-d_D(A)+r(D_{\min})).$$ (Note that the parameter $\rho_D(A)$ appearing in [@CMNR1] equals $|E|-d_D(A)$ in our notation.) The *Tutte polynomial*, $R(D;x,y)$, of a delta-matroid $D=(E,\mathcal{F})$, introduced in [@CMNR1], is $$R(D;x,y)=\sum_{A\subseteq E} (x-1)^{\sigma(E)-\sigma(A)}(y-1)^{|A|-\sigma(A)}.$$ By using Equation [\[eq:defsigma\]](#eq:defsigma){reference-type="eqref" reference="eq:defsigma"} to expand the exponents, and the definition of $d_D(E)$ for the prefactor, it follows that $$\label{eq:dmrgp} R(D;x+1,y+1)=\sqrt{x}^{\, r(D_{\max})-|E|}\sqrt{y}^{\,-r(D_{\min})}Q(D;\sqrt{x},\sqrt{y},\sqrt{xy}).$$ **Theorem 22**. *Let $D_1=(E_1,\mathcal{F}_1)$ be an even delta-matroid having no non-singular element and let $D_2=(E_2,\mathcal{F}_2)$ be an even delta-matroid with non-singular element $e$. Denote their tensor product $D_1\otimes_e D_2$ by $\widetilde{D}=(E,\mathcal{F})$. Then $$\begin{aligned} R(\widetilde{D};x+1,y+1)=q^{|E_1|}(\tfrac{p}{q\sqrt{xy}})^{\sigma(D_1)}\sqrt{x}^{2\sigma(\widetilde{D})-|E|} \sqrt{xy}^{\,r({(D_1)}_{\min})-r(\widetilde{D}_{\min})}\\ \cdot R(D_1;\tfrac{q\sqrt{xy}}{p}+1,\tfrac{p\sqrt{xy}}{q}+1), \end{aligned}$$ where $p$ and $q$ are the unique solutions to the following set of equations $$\begin{aligned} R(D_2\mathop{\mathrm{/}}e;x+1,y+1)& = \sqrt{x}^{\,r((D_2\mathop{\mathrm{/}}e)_{\max})-|E_2|+1}\sqrt{y}^{\,-r((D_2\mathop{\mathrm{/}}e)_{\min})}(p\sqrt{xy}+q),\\ R(D_2\mathop{\mathrm{\backslash}}e;x+1,y+1)& = \sqrt{x}^{\,r((D_2\mathop{\mathrm{\backslash}}e)_{\max})-|E_2|+1}\sqrt{y}^{\,-r((D_2\mathop{\mathrm{\backslash}}e)_{\min})}(p + q\sqrt{xy} ) . \end{aligned}$$* *Proof.* Using Equation [\[eq:dmrgp\]](#eq:dmrgp){reference-type="eqref" reference="eq:dmrgp"} to express $R(\widetilde{D};x+1,y+1)$ in terms of $Q(\widetilde{D};\sqrt{x},\sqrt{y},\sqrt{xy})$, then applying Theorem [Theorem 21](#thm:dmeven){reference-type="ref" reference="thm:dmeven"} gives $$\begin{aligned} R(\widetilde{D};x+1,y+1)=\sqrt{x}^{\, r(\widetilde{D}_{\max})-|E|}\sqrt{y}^{\,-r(\widetilde{D}_{\min})}Q(D_1;q,p,\sqrt{xy}), \end{aligned}$$ where $p$ and $q$ are the unique solutions to the system of linear equations $$\begin{aligned} Q(D_2\mathop{\mathrm{/}}e;\sqrt{x},\sqrt{y},\sqrt{xy})&=p\sqrt{xy}+q,\\ Q(D_2\mathop{\mathrm{\backslash}}e;\sqrt{x},\sqrt{y},\sqrt{xy})&=p+q\sqrt{xy}. \end{aligned}$$ We apply Equation [\[eq:dmrgp\]](#eq:dmrgp){reference-type="eqref" reference="eq:dmrgp"} to this system of linear equations to obtain those in the theorem. Using $\sigma_D(A)=\tfrac{1}{2}(|A|-d_D(A)+r(D_{\min}))$ we can rewrite $Q(D_1;q,p,\sqrt{xy})$ as follows $$\begin{aligned} Q(D_1;q,p,\sqrt{xy})&=\sum_{A\subseteq E_1}q^{|E_1|-|A|}p^{|A|}\sqrt{xy}^{\, d_{D_1}(A)}\\ &=q^{|E_1|}\sum_{A\subseteq E_1}\tfrac{p}{q}^{|A|}\sqrt{xy}^{\, |A|+r({(D_1)}_{\min})-2\sigma_{D_1}(A)}\\ &=q^{|E_1|}\sqrt{xy}^{\, r({(D_1)}_{\min})}\sum_{A\subseteq E_1}\tfrac{p\sqrt{xy}}{q}^{|A|-\sigma_{D_1}(A)}\tfrac{q\sqrt{xy}}{p}^{\, -\sigma_{D_1}(A)}\\ &=q^{|E_1|}(\tfrac{p}{q\sqrt{xy}})^{\sigma(D_1)}\sqrt{xy}^{\, r({(D_1)}_{\min})}R(D_1;\tfrac{q\sqrt{xy}}{p}+1,\tfrac{p\sqrt{xy}}{q}+1). \end{aligned}$$ Hence, the result follows. ◻ Brylawski's original result for the tensor product of the Tutte polynomial of matroids may be obtained as a corollary of Theorem [Theorem 22](#thm:dmrg){reference-type="ref" reference="thm:dmrg"}. The *Tutte polynomial*, $T(M;x,y)$, of a matroid $M=(E,\mathcal{F})$ is defined as $$T(M;x,y):=\sum_{A\subseteq E}(x-1)^{r(E)-r(M\mathop{\mathrm{\backslash}}A^c)}(y-1)^{|A|-r(M\mathop{\mathrm{\backslash}}A^c)}.$$ If $M=(E,\mathcal{F})$ is a matroid and $A\subseteq E$, then $\sigma(A)=r(M\mathop{\mathrm{\backslash}}A^c)$ and so $T(M;x,y)=R(M;x,y)$. Hence, we get the following corollary of Theorem [Theorem 22](#thm:dmrg){reference-type="ref" reference="thm:dmrg"}. **Corollary 23**. *Let $M_1=(E_1,\mathcal{F}_1)$ be a matroid having no loops or coloops and let $M_2=(E_2,\mathcal{F}_2)$ be a matroid with element $e$ which is neither a loop nor a coloop. Denote their tensor product $M_1\otimes_e M_2$ by $\widetilde{M}=(E,\mathcal{F})$. Then $$T(\widetilde{M};x,y)=\alpha^{n(M_1)}\beta^{r(M_1)}T(M_1;T(M_2\mathop{\mathrm{\backslash}}e;x,y)/\beta,T(M_2\mathop{\mathrm{/}}e;x,y)/\alpha),$$ where $\alpha$ and $\beta$ are the unique solutions to $$\begin{aligned} T(M_2\mathop{\mathrm{\backslash}}e;x,y)&=(x-1)\alpha+\beta,\\ T(M_2\mathop{\mathrm{/}}e;x,y)&=\alpha+(y-1)\beta. \end{aligned}$$* *Proof.* This theorem follows from applying Theorem [Theorem 22](#thm:dmrg){reference-type="ref" reference="thm:dmrg"} to the matroid $M_1\otimes_e M_2$. We use the observation that $r({M}_{\max})=r({M}_{\min})=r(M)$ and $\sigma(A)=r(M\mathop{\mathrm{\backslash}}A^c)$, and that when $e$ is non-singular, $r(M_2\mathop{\mathrm{\backslash}}e)=r(M_2)$ and $r(M_2\mathop{\mathrm{/}}e)=r(M_2)-1$, to obtain $$T(\widetilde{M};x+1,y+1)=q^{|E_1|}(\tfrac{p}{q})^{r(M_1)}\sqrt{x}^{\, -|E|}\sqrt{\tfrac{x}{y}}^{\,r(\widetilde{M})}T(M_1;\tfrac{q\sqrt{xy}}{p}+1,\tfrac{p\sqrt{xy}}{q}+1),$$ where $p$ and $q$ are the unique solutions to $$\begin{aligned} T(M_2\mathop{\mathrm{/}}e;x+1,y+1)& = \sqrt{x}^{1-|E_2|}\sqrt{\tfrac{x}{y}}^{\, r(M_2)-1}(p\sqrt{xy}+q),\\ T(M_2\mathop{\mathrm{\backslash}}e;x+1,y+1)& = \sqrt{x}^{1-|E_2|}\sqrt{\tfrac{x}{y}}^{\, r(M_2)}(p + q\sqrt{xy} ). \end{aligned}$$ As a matroid is an even delta-matroid, it is equivalent to a tight 2-matroid and we can deduce from Lemma [Lemma 10](#lem:claim1){reference-type="ref" reference="lem:claim1"} that $r(\widetilde{M})=r(M_1)-|E_1|+|E_1|r(M_2)$. So by letting $\alpha=q \sqrt{x}^{\, 1-|E_2|}\sqrt{\tfrac{x}{y}}^{\, r(M_2)-1}$ and $\beta=p \sqrt{x}^{\, 1-|E_2|}\sqrt{\tfrac{x}{y}}^{\, r(M_2)}$ we obtain the result. ◻ # Ribbon graphs {#sec:rg} at 75 72 at 75 26 at 11 72 {#fig:rg1} at 34 57 at 108 55 at 50 19 at 111 17 at 18 35 at 56 46 {#fig:rg4} {#fig:rg2} {#fig:rg3} The *topological transition polynomial*, introduced in [@MR2869185], is a multivariate polynomial of ribbon graphs (or equivalently, of graphs embedded in surfaces). It contains both the 2-variable version of Bollobás and Riordan's ribbon graph polynomial [@MR1851080; @MR1906909] and the Penrose polynomial [@MR1428870; @MR2994409] as specialisations, and is intimately related to Jaeger's transition polynomial [@MR1096990] and the generalised transition polynomial of [@MR1980048]. A *ribbon graph* $\mathbb{G}=\left(V,E\right)$ is a surface with boundary, represented as the union of two sets of discs --- a set $V$ of *vertices* and a set $E$ of *edges* --- such that: (1) the vertices and edges intersect in disjoint line segments; (2) each such line segment lies on the boundary of precisely one vertex and precisely one edge; and (3) every edge contains exactly two such line segments. As every ribbon graph $\mathbb{G}$ can be regarded as a surface with boundary, it has some number of boundary components, which we shall denote by $b(\mathbb{G})$. The boundary components will be particularly important in what follows. We let $k(\mathbb{G})$ denote the number of connected components of $\mathbb{G}$, and $e(\mathbb{G})$ its number of edges. Some examples of ribbon graphs can be found in Figure [\[fig:rg\]](#fig:rg){reference-type="ref" reference="fig:rg"}. The ribbon graphs in Figures [1](#fig:rg1){reference-type="ref" reference="fig:rg1"} and [3](#fig:rg2){reference-type="ref" reference="fig:rg2"} each have one boundary component, while that in Figure [4](#fig:rg3){reference-type="ref" reference="fig:rg3"} has two. Additional background on ribbon graphs can be found in [@MR3086663; @zbMATH07553843]. Let $\mathbb{G}=(V,E)$ be a ribbon graph and $e\in E$. Then $\mathbb{G}\mathop{\mathrm{\backslash}}e$ denotes the ribbon graph obtained from $\mathbb{G}$ by *deleting* the edge $e$. For $A\subseteq E$, $\mathbb{G}\mathop{\mathrm{\backslash}}A$ is the result of deleting each edge in $A$ (in any order). Edges can also be contracted. If $u$ and $v$ are vertices incident to $e$ (it is possible that $u=v$ here), consider the boundary component(s) of $e\cup\{u,v\}$ as curves on $\mathbb{G}$. For each resulting curve, attach a disc (which will form a vertex of $\mathbb{G}\mathop{\mathrm{/}}e$) by identifying its boundary component with the curve. Delete $e$, $u$ and $v$ from the resulting complex, to get the ribbon graph $\mathbb{G}\mathop{\mathrm{/}}e$. The *partial Petrial* with respect to an edge of a ribbon graph $e$, introduced in [@MR2869185] and denoted here by $\mathbb{G}^{\tau(e)}$, informally is the result of detaching one end of the edge $e$ from a vertex, giving the edge a half-twist, and reattaching it in the same place. Formally, it is obtained by detaching an end of $e$ from its incident vertex $v$ creating arcs $[a,b]$ on $v$, and $[a',b']$ on $e$ (so that $\mathbb{G}$ is recovered by identifying $[a,b]$ with $[a',b']$), then reattaching the end by identifying the arcs antipodally (so that $[a,b]$ is identified with $[b',a']$). For $A\subseteq E(G)$, the partial Petrial $\mathbb{G}^{\tau(A)}$ is the result of forming the partial Petrial with respect to every element in $A$ (in any order). There is a natural correspondence between the edges of $\mathbb{G}$ and $\mathbb{G}^{\tau(A)}$, and so we can and will assume that they have the same edge set. Again, examples can be found in Figure [\[fig:rg\]](#fig:rg){reference-type="ref" reference="fig:rg"}. Let $\mathbb{G}=(V,E)$ be a ribbon graph, $\boldsymbol{u}=\{u_e\}_{e\in E}$, $\boldsymbol{v}=\{v_e\}_{e\in E}$, $\boldsymbol{w}=\{w_e\}_{e\in E}$ be indexed families of indeterminates, and $t$ be another indeterminate. Then the *topological transition polynomial*, $Q(\mathbb{G}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t)$, can be defined as $$Q(\mathbb{G}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) :=\sum_{(X,Y,Z) \in \mathcal{P}_3(E)} \Big( \prod_{e\in X}u_e\Big) \Big( \prod_{e\in Y} v_e\Big) \Big( \prod_{e\in Z} w_e\Big) t^{b( \mathbb{G}^{\tau(Z)}\mathop{\mathrm{\backslash}}Y)},$$ where $\mathcal{P}_3(E)$ denotes the set of ordered partitions of $E$ into three blocks (blocks may be empty), and where $b(\mathbb{G}^{\tau(Z)}\mathop{\mathrm{\backslash}}Y)$ denotes the number of boundary components of $\mathbb{G}^{\tau(Z)}\mathop{\mathrm{\backslash}}Y$. Additional background on the topological transition polynomial can be found in, for example, [@MR3086663]. **Example 24**. If $\mathbb{G}$ is a plane 2-cycle (i.e., the ribbon graph that is a 2-cycle of genus 0) with edges $a$ and $b$ then $$\begin{gathered} Q(\mathbb{G}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) \\= u_a u_b t^2+ u_a v_b t+ u_a w_b t+ v_a u_b t+ v_a v_b t^2+ v_a w_b t+ w_a u_b t+ w_a v_b t+ w_a w_b t^2.\end{gathered}$$ Following [@mmact Section 5], a tight 3-matroid $Z(\mathbb{G})=(U(\mathbb{G}), E, r)$ can be obtained from a ribbon graph $\mathbb{G}=(V,E)$. For this let $U:=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e}: e\in E\}$, the skew classes be $\Omega=\{\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e} \}:e \in E\}$, and the set of bases of $Z(\mathbb{G})$ be $$\{ \{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{x}: x\in X\} \cup \{\overline{y}: y\in Y\}\cup \{\widehat{z}: z\in Z\} : (X,Y,Z)\in\mathcal{P}_3(E) \text{ and } b(\mathbb{G}^{\tau(Z)} \mathop{\mathrm{\backslash}}Y) = k(\mathbb{G}) \}.$$ Following our notational conventions, we denote the skew class $\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}, \overline{e}, \widehat{e}\}$ by $e$. We order each skew class according to $\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e} < \overline{e}< \widehat{e}\}$ and assume this order without further comment. **Example 25**. Consider the ribbon graphs $\mathbb{G}$ and $\mathbb{H}$ shown in Figure [7](#fig:rgs1){reference-type="ref" reference="fig:rgs1"}. Then the 3-matroid of $\mathbb{G}$ is given in Example [Example 3](#example1){reference-type="ref" reference="example1"}, and the 3-matroid of $\mathbb{H}$ is given in Example [Example 4](#example1b){reference-type="ref" reference="example1b"}. We need to recognise which edges of $\mathbb{G}$ give rise to singular skew classes of $Z(\mathbb{G})$. For this we need some additional ribbon graph terminology. An edge $e$ of a ribbon graph $\mathbb{G}$ is a *bridge* if $k(\mathbb{G}\mathop{\mathrm{\backslash}}e)>k(\mathbb{G})$. It is a *loop* if it is incident to exactly one vertex. A loop $e$ is *orientable* if $e$ together with its incident vertex forms an annulus, and is *nonorientable* if it forms a Möbius band. A loop $e$ is said to be *interlaced* with a cycle $C$ if when travelling around the boundary of the vertex incident to $e$ we see edges in the cyclic order $e \, c_1\, e \,c_2$ where $c_1$ and $c_2$ are edges of $C$. A loop is *trivial* if it is not interlaced with any cycle. Observe that a bridge necessarily intersects one boundary component of $\mathbb{G}$, and a trivial orientable loop must meet two boundary components. It was shown in [@mmact] that for a ribbon graph $\mathbb{G}$ with an edge $e$, a skew class $e=\{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e},\overline{e},\widehat{e}\}$ of $Z(\mathbb{G})$ is singular if and only if the edge $e$ is either a bridge or trivial loop in $\mathbb{G}$. It was also shown there that for each edge $e$ of $\mathbb{G}$, $$\label{eq:rgzdc} Z(\mathbb{G}\mathop{\mathrm{/}}e) = Z(\mathbb{G})|_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}}, \quad Z(\mathbb{G}\mathop{\mathrm{\backslash}}e) = Z(\mathbb{G})|_{\overline{e}}, \quad Z(\mathbb{G}^{\tau(e)} \mathop{\mathrm{/}}e ) = Z(\mathbb{G})|_{\widehat{e}}.$$ As shown in [@mmact], transition polynomials for multimatroids and ribbon graphs agree. We have $$\label{eq:rztra} Q(\mathbb{G}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) = t^{k(\mathbb{G})}\cdot Q(Z(\mathbb{G});\mathbf{x}, t) ,$$ when $u_e=x_{\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{e}}$, $v_e= x_{\overline{e}}$, and $w_e=x_{\widehat{e}}$ for each edge $e$ of $\mathbb{G}$. In order to efficiently describe 2-sums of ribbon graphs we consider arrow presentations. An *arrow presentation* $\mathbb{G}$ consists of a set of circles (i.e., closed 1-manifolds) and a set of labels. For each label there are exactly two arrows lying along one or more of the circles. All arrows are disjoint. The set of labels is called the *edge set* and its elements are *edges*. Arrow presentations describe ribbon graphs as follows: if $\mathbb{G}$ is a ribbon graph, for each edge $e$, arbitrarily orient the boundary of that edge, place an $e$-labelled arrow on the arcs where $e$ intersects a vertex, pointing in the direction given by the edge's boundary orientation. Taking the boundaries of the vertices together with the labelled arrows gives an arrow presentation. On the other hand, given an arrow presentation, obtain a ribbon graph by identifying each circle with the boundary of a vertex disk. Then, for each label $e$, take a disc with an orientation of its boundary and identify an arc on its boundary with an $e$-labelled arrow such that the direction of each arrow agrees with the orientation. Thus, in an arrow presentation, each circle corresponds to a vertex of a ribbon graph, and each pair of $e$-labelled arrows corresponds to an edge. Let $\mathbb{G}$ and $\mathbb{H}$ be arrow presentations, let $f$ be an edge of $\mathbb{G}$, and $e$ be an edge of $\mathbb{H}$. Furthermore, suppose one of the $f$-labelled arrows of $\mathbb{G}$ is distinguished, and one of the $e$-labelled arrow of $\mathbb{G}$ is distinguished. Then the *2-sum* $\mathbb{G}\oplus_{f=e}\mathbb{H}$ is the arrow presentation constructed by splicing the arrow presentations as indicate in Figure [\[fig:ap\]](#fig:ap){reference-type="ref" reference="fig:ap"}. For this, add arcs from: the tail of the distinguished $f$-labelled arrow to the head of the distinguished $e$-labelled arrow; the head of the distinguished $f$-labelled arrow to the tail of the distinguished $e$-labelled arrow; the tail of the non-distinguished $f$-labelled arrow to the head of the non-distinguished $e$-labelled arrow; and the head of the non-distinguished $f$-labelled arrow to the tail of the non-distinguished $e$-labelled arrow. Then delete all four arrows as well as the arcs of the circles of the arrow presentation that they lie on (excluding the end points of these arcs). at 27 78 at 30 34 at 37 68 at 36 21 at 8 4 at 57 4 {#fig:ap1} at 31 4 {#fig:ap2} To define the 2-sums of ribbon graphs we suppose that $\mathbb{G}$ and $\mathbb{H}$ are ribbon graphs, $f$ is an edge of $\mathbb{G}$, and $e$ is an edge of $\mathbb{H}$. Further we suppose that both $e$ and $f$ have a local orientation (i.e., a notion of clockwise for that edge) and that they are both directed (i.e., both edges have a head and a tail). Convert $\mathbb{G}$ and $\mathbb{H}$ to arrow presentations reading off the directions of the $e$-labelled and $f$-labelled arrows so that that they agree with the local edge orientations. Distinguish the two arrows that arose from the heads of $e$ and $f$ in the ribbon graphs. Form the 2-sum of the arrow presentations, then convert back to ribbon graphs. at 58 66 at 30 66 at 6 128 at 102 66 at 155 90 at 155 35 at 46 2 at 115 2 {#fig:rgs1} at 36 40 at 2 26 at 26 85 at 70 31 at 89 47 at 112 30 at 46 106 at 2 98 at 10 125 at 73 92 at 112 108 at 95 85 at 21 2 at 90 2 {#fig:rgs2} at 2 26 at 26 85 at 89 47 at 112 30 at 2 98 at 10 125 at 112 108 at 95 85 {#fig:rgs3} at 32 66 at 10 128 at 80 113 at 81 33 {#fig:rgs5} at 36 40 at 2 26 at 89 47 at 112 30 at 46 106 at 2 98 at 112 108 at 95 85 {#fig:rgs4} at 32 60 at 58 60 at 5 122 at 62 130 {#fig:rgs6} at 80 118 at 85 35 at 7 40 at 45 60 at 2 122 at 65 136 {#fig:rgs7} As shown in [@serpar] the 2-sum operation for ribbon graphs and for multimatroids are compatible operations. **Theorem 26**. *Let $\mathbb{G}$ and $\mathbb{H}$ be ribbon graphs. Suppose that $\mathbb{G}$ has a locally oriented and directed edge $f$, and that $\mathbb{H}$ has a locally oriented and directed edge $e$. Further suppose that $e$ and $f$ are not bridges or trivial loops. Then $$Z(\mathbb{G}\oplus_{f=e}\mathbb{H}) = Z(\mathbb{G})\oplus_{f=e}Z(\mathbb{H}) .$$* We can use 2-sums to define tensor products. Let $\mathbb{G}$ be a ribbon graph in which every edge has a local orientation and is directed. Let $\mathbb{H}$ be a ribbon graph with a distinguished edge $e$ that has a local orientation and is directed. Then the *tensor product* $\mathbb{G}\otimes_{e} \mathbb{H}$ is the ribbon graph formed as follows. Take an isomorphic copy of $\mathbb{H}$ for each edge of $\mathbb{G}$. For convenience assume in each copy the edge $e$ remains, but the copies are otherwise on disjoint sets of edges. The ribbon graph $\mathbb{G}\otimes_{e} \mathbb{H}$ is then constructed by forming the 2-sum along $\alpha$ and $e$ for each edge $\alpha$ of $\mathbb{G}$ and its corresponding copy of $\mathbb{H}$. As a corollary of Theorem [Theorem 26](#thm:rgtp){reference-type="ref" reference="thm:rgtp"} we have the following. **Corollary 27**. *Let $\mathbb{G}$ be a ribbon graph in which every edge has a local orientation and is directed. Let $\mathbb{H}$ be a ribbon graph with a distinguished edge $e$ that has a local orientation and is directed. Further suppose that no edge of $\mathbb{G}$ is a bridge or trivial loop, nor is the edge $e$ of $\mathbb{H}$. Then $$Z(\mathbb{G}\otimes_{e}\mathbb{H}) = Z(\mathbb{G})\otimes_{e}Z(\mathbb{H}) .$$* As with Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"} it is convenient to have a naming convention for the elements of $\mathbb{G}\otimes_e \mathbb{H}$. For each edge $g$ of $\mathbb{G}$ and $h$ of $\mathbb{H}$, we shall use $h^{(g)}$ to denote the copy of the edge $h$ that appears in the the copy of $\mathbb{H}$ arising from the edge $g$ of $\mathbb{G}$. **Theorem 28**. *Let $\mathbb{G}$ be a ribbon graph in which every edge has a local orientation and is directed. Let $\mathbb{H}$ be a ribbon graph with a distinguished edge $e$ that has a local orientation and is directed. Further suppose that no edge of $\mathbb{G}$ is a bridge or trivial loop, nor is the edge $e$ of $\mathbb{H}$, and using the above naming convention for the edges of $\mathbb{G}\otimes \mathbb{H}$. Then $$Q(\mathbb{G}\otimes_e \mathbb{H}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) = %t^{k(\bG\otimes_e \bH) - k(\bG)} t^{e(\mathbb{G})(k(\mathbb{H})-1)} \phi( Q(\mathbb{G}; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t),$$ where $\phi$ is the map induced by the substitution $u_g \mapsto \phi^{(g)}(\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y})$, $v_g \mapsto \phi^{(g)}(\overline{y})$, and $w_g \mapsto \phi^{(g)}(\widehat{y})$; and where $\phi^{(g)}$ sends $u_h\mapsto u_{h^{(g)}}$, $v_h\mapsto v_{h^{(g)}}$, and $w_h\mapsto w_{h^{(g)}}$; and where $\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}$, $\overline{y}$ and $\widehat{y}$ are the are the unique solutions to the system of linear equations $$\begin{aligned} Q(\mathbb{H}\mathop{\mathrm{/}}e; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) &=t^{k(\mathbb{H})} (t\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+\overline{y}+\widehat{y}), \\ Q(\mathbb{H}\mathop{\mathrm{\backslash}}e; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) &=t^{k(\mathbb{H})} (\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+t\overline{y}+\widehat{y}), \\ Q(\mathbb{H}^{\tau(e)} \mathop{\mathrm{/}}e; (\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) , t) &=t^{k(\mathbb{H})} (\accentset{\mathbin{\vcenter{\hbox{\scalebox{0.4}{$\bullet$}}}}}{y}+\overline{y}+t\widehat{y}). \end{aligned}$$* *Proof.* The theorem follows by using Equation [\[eq:rztra\]](#eq:rztra){reference-type="eqref" reference="eq:rztra"} to express $Q(\mathbb{G}\otimes_e \mathbb{H})$ in terms of $Q(Z(\mathbb{G}\otimes_e \mathbb{H}))$, applying Corollary [Corollary 27](#cor:rgtp){reference-type="ref" reference="cor:rgtp"} to express this is in terms of $Q(Z(\mathbb{G})\otimes_e Z(\mathbb{H}))$, applying Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}, then Equation [\[eq:rztra\]](#eq:rztra){reference-type="eqref" reference="eq:rztra"} once again to return to ribbon graphs. The system of linear equations arises from those of Theorem [Theorem 15](#thm:main){reference-type="ref" reference="thm:main"}, by again using Equation [\[eq:rztra\]](#eq:rztra){reference-type="eqref" reference="eq:rztra"} to express them in terms of ribbon graphs, then applying Equations [\[eq:rgzdc\]](#eq:rgzdc){reference-type="eqref" reference="eq:rgzdc"}. The rewriting also uses the fact that $k(\mathbb{H}) = k(\mathbb{H}\mathop{\mathrm{/}}e) = k(\mathbb{H}\mathop{\mathrm{\backslash}}e)=k(\mathbb{H}^{\tau(e)} \mathop{\mathrm{/}}e)$, which follows because $e$ is neither a bridge nor a trivial loop. Finally, for the factors of $t$, we use that $k(\mathbb{G}\otimes_e \mathbb{H}) =e(\mathbb{G})(k(\mathbb{H})-1)+k(\mathbb{G})$. ◻ Note that by taking $\mathbb{G}$ and $\mathbb{H}$ to be connected, $e$ to be a non-loop edge, and each variable in $\boldsymbol{u}$, $\boldsymbol{v}$, and $\boldsymbol{w}$ to be identical we recover exactly Theorem 6.1 of [@ELLIS_MONAGHAN_2014]. Our proof here shows that the requirement for $e$ to be a non-loop edge in that theorem can be weakened. As in [@ELLIS_MONAGHAN_2014 Section 6], Theorem [Theorem 28](#thm:rgtpf){reference-type="ref" reference="thm:rgtpf"} can be specialised to give a tensor product formula for Bollobás and Riordan's ribbon graph polynomial. Alternatively, this tensor product formula can be deduced from Theorem [Theorem 20](#thm:rgpdm){reference-type="ref" reference="thm:rgpdm"} by considering the delta-matroid $D(\mathbb{G})$ of $\mathbb{G}$, following the construction in [@CMNR1 Section 4] and making use of [@CMNR1 Theorems 6.4 and 6.6] to relate the ribbon graph and delta-matroid polynomials. **Funding Information** Iain Moffatt and Steven Noble were supported the Engineering and Physical Sciences Research Council \[grant number EP/W033038/1\]. **Declarations** **Conflict of interest** There are no conflicts of interest. **Data** No underlying data is associated with this article. **Open Access** For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

arxiv_math

--- abstract: | We establish robust exponential convergence for $r p$--Finite Element Methods (FEMs) applied to fourth order singularly perturbed boundary value problems, in a *balanced norm* which is stronger than the usual energy norm associated with the problem. As a corollary, we get robust exponential convergence in the maximum norm. $r p$ FEMs are simply $p$ FEMs with possible repositioning of the (fixed number of) nodes. This is done for a $C^1$ Galerkin FEM in 1-D, and a $C^0$ mixed FEM in 2-D over domains with smooth boundary. In both cases we utilize the *Spectral Boundary Layer* mesh from [@MXO]. *Keywords:* fourth order singularly perturbed problem, boundary layers, $r p$-finite element method, uniform, exponential convergence, balanced norm *AMS subject classification (2000):* 65N30 author: - "Torsten Linß[^1]" - "Christos Xenophontos[^2]" title: Balanced norm estimates for $rp$-Finite Element Methods applied to singularly perturbed fourth order boundary value problems --- # Introduction {#intro} The quest for *balanced norm* estimates for singularly perturbed problems (SPPs) has been going on for a while now (see [@FR]--[@LS] and the references therein). When using the Finite Element Method (FEM), the error estimates are usually given in the *energy norm* defined by the bilinear form in the variational formulation of the problem. In certain cases, this norm is too weak and cannot see the layers present in the solution (see, e.g., [@LS]). While it is straight forward to define a balanced norm, the bilinear form lacks coercivity with respect to it, hence the usual methods of proof cannot be applied. Various approaches have been proposed [@FR]--[@LS], dealing mainly with reaction-diffusion type problems. The majority of these works pertains to fixed order FEMs; notable exceptions are [@CVX; @MX]. In this article we consider fourth order SPPs and the approximation of their solution by the $r p$ version of the FEM on the *Spectral Boundary Layer* mesh from [@MXO]. In this version the position (but not the number) of the nodes could possibly change, in addition to the polynomial degree $p$ of the approximating polynomials, being increased. In the literature, one finds results on robust exponential convergence in the energy norm [@PC; @CFLX; @PZMX]. We prove robust exponential convergence in a balanced norm and as a corollary we get robust exponential convergence in the maximum norm. In one-dimension this is achieved for a $C^1$ Galerkin FEM, used in [@PC; @PZMX], and in two-dimensions for the $C^0$ mixed formulation from [@CFLX]. The novelty in our approach lies with the choice of a certain projection of the solution onto the FE space. Specifically, we use an interpolant in the layer region and an appropriate projection in the rest of the domain. While this general approach has been used before (e.g., [@FR3]), our contribution differs twofold: first we consider the $rp$ version (hence exponential rates of convergence are obtained), and second, the projection is used *only* for the smooth parts of the solution, as opposed to the entire solution as was done in [@FR3]. Finally, our contribution proves the conjectures made in [@PC] and [@CFLX] based on numerical evidence. The rest of the paper is organized as follows: in Section [2](#sec:1D){reference-type="ref" reference="sec:1D"} we consider a one-dimensional problem and its discretization by a $C^1$ Galerkin FEM. We present the available estimates from the literature and in Section [2.2](#sec:balanced){reference-type="ref" reference="sec:balanced"} we improve them by establishing robust exponential convergence in a balanced, as well as the maximum norm. Section [3](#sec:2D){reference-type="ref" reference="sec:2D"} contains the two-dimensional case, over domains with smooth boundary, and the $C^0$ mixed method from [@CFLX]. We, again, describe the estimates from the literature and then in Section [3.2](#estimates){reference-type="ref" reference="estimates"} we improve them, just like in one-dimension. #### Notation With $D \subset \mathbb{R}^d$, $d=1,2$, a domain with boundary $\partial D$ and measure $\left|D\right|$, we will denote by $C^{k}(D)$ the space of continuous functions on $D$ with continuous derivatives up to order $k$. We will use the usual Sobolev spaces $W^{k,q}(D)$, $q\in[0,\infty]$, $k\in\mathbb{N}_0$ of functions on $D$ generalized derivatives of order $0,1,\dots,k$ in $L^{q}(D)$, equipped with the norm and seminorm $\left\|\cdot\right\|_{k,q,D}$ and $\left|\cdot\right|_{k,m,D}$, respectively. When $q=2$, we will write $H^k(D)$ instead of $W^{k,2}(D)$, and for the norm and seminorm, we will write $\left\|\cdot\right\|_{k,D}$ and $\left|\cdot\right|_{k,D}$, respectively. The usual $L^{2}(D)$ inner product will be denoted by $\left\langle \cdot , \cdot \right\rangle_{D}$, with the subscript omitted when $D=\Omega$ and there is no confusion. We will also use the space $$\begin{gathered} H_0^2\left(D\right) = \left\{ u\in H^2(D) \colon \left.u\right\vert _{\partial D} = \left.\frac{\partial u}{\partial n}\right\vert _{\partial D}=0\right\},\end{gathered}$$ where $\frac{\partial}{\partial n}$ denotes the normal derivative. The norm of the space $L^\infty(D) = W^{0,\infty}(D)$ of essentially bounded functions on $D$ is denoted by $\left\|\cdot\right\|_{L^\infty(D)} = \left\|\cdot\right\|_{0,\infty,D} = \left|\cdot\right|_{0,\infty,D}$. Finally, the notation "$a\lesssim b$" means "$a\leq Cb$" with $C$ being a generic positive constant, independent of any discretization or singular perturbation parameters. # One-dimensional problems {#sec:1D} Since the boundary layer effect is one-dimensional (in the direction normal to the boundary), studying one-dimensional problems is a necessary first step towards two (and higher) dimensions. In fact, most of the ideas are present in the one-dimensional case, and their extention to two-dimensions is (in most cases) straight forward. We consider the following problem (cf. [@PC; @PZMX]): find $u$ such that [\[de-bc\]]{#de-bc label="de-bc"} $$\begin{aligned} {2} \varepsilon^{2}u^{(4)}-\left( bu'\right)'+cu & = f &\quad& \text{in} \ \ \Omega \coloneqq \left(0,1\right), \\ %\label{de} \\ u(0)=u(1)=u'(0)=u'(1) & = 0, \label{bc} \end{aligned}$$ where $0<\varepsilon\leq 1$ is a given parameter that can approach zero and the functions $b$, $c$ and $f$ are given and sufficiently smooth. In particular, we assume that they are (real) *analytic* functions satisfying, for some positive constant $\gamma_d$, independent of $\varepsilon$, $$\begin{gathered} \label{analytic} \left|f\right|_{n,\infty,\Omega} + \left|c\right|_{n,\infty,\Omega} + \left|b\right|_{n,\infty,\Omega}\lesssim n!\gamma_d^n \quad \forall \; n\in \mathbb{N}_0.\end{gathered}$$ In addition, we assume that there exist positive constants $b_{\min}$ and $c_{\min}$, independent of $\varepsilon$, such that $$\begin{gathered} \label{data} b\ge b_{\min}^2, \quad c\geq c_{\min}^2 \quad \text{on} \ \ \bar{\Omega}, \end{gathered}$$ It is well known (see, e.g. [@OMalley]) that the solution $u$ to [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"} can be decomposed into a smooth part, two boundary layer parts and a remainder. We have the following result from [@PZMX]. **Proposition 1**. *Let $u$ be the solution to [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"}, and assume [\[analytic\]](#analytic){reference-type="eqref" reference="analytic"} holds. Then, there exist a positive constants $K$, $\gamma$ and $\delta$ such that for all $n\in \mathbb{N}_0$, $$\begin{gathered} \label{A1} \left|u\right|_{n,\infty,\Omega} \lesssim K^{n}\max \left\{ n^{n}, \varepsilon^{1-n}\right\}\,. \end{gathered}$$ Moreover, $u$ may be decomposed as* *[\[AA\]]{#AA label="AA"} $$\begin{gathered} % \label{A2} u=u_S+\tilde{u} + \bar{u} + u_R, \end{gathered}$$ with $$\begin{aligned} {2} % \label{A3}% \left|u_S\right|_{n,\infty,\Omega} & \lesssim K^nn^n, \\ \label{A4}% \left|\tilde{u}^{(n)}(x)\right| + \left|\bar{u}^{(n)}(1-x)\right| & \lesssim K^{n} \varepsilon^{1-n} \mathrm{e}^{-\gamma x/\varepsilon}\ \ \forall \ x\in\bar\Omega \\ %\intertext{and} \label{A6} \left\|u_R\right\|_{2,\Omega} & \lesssim \mathrm{e}^{-\delta /\varepsilon}. \end{aligned}$$* *In this decomposition, $u_S$ denotes the smooth part, $\tilde{u}$ the boundary layer at the left endpoint, $\bar{u}$ denotes the boundary layer at the right endpoint, and $u_R$ is the remainder.* We mention that [\[A1\]](#A1){reference-type="eqref" reference="A1"} corresponds to classical differentiability, while [\[AA\]](#AA){reference-type="eqref" reference="AA"} correspond to regularity through asymptotic expansions, see [@PC; @melenk]. ## Discretization by a $C^{1}$ $rp$-FEM The variational formulation of [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"} reads: Find $u\in H_{0}^{2}(\Omega)$ such that $$\begin{gathered} \label{BuvFv} \mathcal{B}_\varepsilon\left(u,v\right) = \mathcal{F}\left(v\right) \;\;\forall \;v\in H_{0}^{2}(\Omega), \intertext{where} \label{Buv} \mathcal{B}_\varepsilon\left(w,v\right) \coloneqq \varepsilon^2 \left\langle w'' , v'' \right\rangle_{\Omega} + \left\langle bw' , v' \right\rangle_{\Omega} + \left\langle cw , v \right\rangle_{\Omega}, \quad \mathcal{F}(v) \coloneqq \left\langle f , v \right\rangle_{\Omega}.\end{gathered}$$ Because of [\[data\]](#data){reference-type="eqref" reference="data"}, the bilinear form $\mathcal{B}_{\varepsilon}\left(\cdot,\cdot\right)$ induces a norm $$\begin{gathered} \label{energy} \left\|w\right\|_{E} \coloneqq \mathcal{B}_{\varepsilon}\left(w,w\right) ,\end{gathered}$$ the so called *energy norm*. Clearly, $\mathcal{B}_{\varepsilon}\left(\cdot,\cdot\right)$ is coercive (with constant $1$) with respect to this norm. We shall seek an approximation to $u$ in a finite dimensional subspace of $H_0^2(\Omega)$. To this end, let $\mathcal{P}_p$, $p\in\mathbb{N}_0$, denote the space of polynomials of maximum degree $p$, and let $\Delta \coloneqq \left\{x_i\right\} _{i=0}^{N}$ be an arbitrary subdivision of $\Omega$ with mesh intervals $\Omega_j\coloneqq(x_{i-1},x_i)$ and mesh sizes $h_i\coloneqq x_i-x_{i-1}>0$, $i=1,\dots,N$. We define the finite dimensional spaces $$\begin{gathered} \mathcal{S}^p_\Delta \coloneqq \Bigl\{w\in C^1(\bar\Omega) \colon w\vert_{\Omega_i}\in\mathcal{P}_p, \ i=1,\dots,N\Bigr\}\,, \quad \mathcal{S}_{0,\Delta}^p \coloneqq \mathcal{S}^p_\Delta \cap H_0^2(\Omega)\,.\end{gathered}$$ Our discretization of [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"} reads: Find $u_{p,\Delta}\in \mathcal{S}_{0,\Delta}^p$ such that $$\begin{gathered} \mathcal{B}_\varepsilon\left(u_{p,\Delta},v\right) = \mathcal{F}(v) \;\;\forall \;v\in \mathcal{S}_{0,\Delta}^p.\end{gathered}$$ This discretization possesses the Galerkin orthogonality property $$\begin{gathered} \mathcal{B}_{\varepsilon} \left( u-u_{p,\Delta},v\right) = 0\;\;\forall \;v\in \mathcal{S}_{0,\Delta}^p\end{gathered}$$ as well as best approximation property $$\begin{gathered} \label{bestapprox} \left\|u-u_{p,\Delta}\right\|_{E} = \inf_{v\in \mathcal{S}_{0,\Delta}^p} \left\|u-v\right\|_{E}\,.\end{gathered}$$ Given $w\in H^2(\Omega)$, an interpolant $I^p_\Delta w \in \mathcal{S}^p_\Delta$, $p\ge 3$, is uniquely defined by $$\begin{aligned} \left(I^p_\Delta w - w\right)^{(k)} (x_i) & = 0, \quad i=0,\dots,N, \ \ k=0,1 \\ \intertext{and} \int_{\Omega_j}\left(I^p_\Delta w - w\right)'' (x) q(x) \;\mathrm{d}x & = 0, \ \forall q \in \mathcal{P}_{p-2}, \ \ i=0,\dots,N.\end{aligned}$$ **Lemma 2**. *Let $w\in H^\ell(\Omega_j)$, $\ell\ge 2$. Then for all $s$, $0\le s \le \min(p-1,\ell-2)$, the interpolation error satisfies the local bounds $$\begin{gathered} (p-1)^{2-k}\left|w-I_\Delta^p w\right|_{k,\Omega_j}^2 \le \left(\frac{\left|\Omega_j\right|}{2}\right)^{2(s+2-k)} \frac{\left(p-1-s\right)!}{\left(p+1+s\right)!} \left|w\right|_{s+2,\Omega_j}^2\,, \ \ k=0,1,2. \end{gathered}$$* *Proof.* See, [@schwab] and [@BdVBRS]. ◻ We next give the definition of the *Spectral Boundary Layer Mesh* we will utilize throughout the article. **Definition 3**. *For $\kappa>0$, $p\in \mathbb{N}$ and $0 < \varepsilon\le 1$, define the Spectral Boundary Layer mesh $\Delta _{BL}$ as $$\begin{gathered} \Delta _{BL} \coloneqq \bigl\{0,\tau,1-\tau,1\bigr\}\,,\quad \tau \coloneqq \min\left\{\kappa p\varepsilon, 1/3\right\}\,, \end{gathered}$$ with the layer and coarse-mesh regions* *$$\begin{gathered} \Omega_\ell \coloneqq (0, \tau) \cup (1-\tau,1) \, , \ \ \Omega_c \coloneqq (\tau,1-\tau). \end{gathered}$$* We shall consider the sequence $u_p\in\mathcal{S}^p_{0,\Delta_{BL}}$, $p=3,4,\dots$, of approximations defined by $$\begin{gathered} \label{discrete} \mathcal{B}_\varepsilon\left(u_p,v\right) = \mathcal{F}(v) \;\;\forall \;v\in \mathcal{S}_{0,\Delta_{BL}}^p.\end{gathered}$$ Given $w\in H^2(\Omega)$, we define interpolants $I_p w \coloneqq I^p_{\Delta_{BL}} w$, $p=3,4,\dots$ The best approximation property [\[bestapprox\]](#bestapprox){reference-type="eqref" reference="bestapprox"} implies $$\begin{gathered} \left\|u-u_p\right\|_{E} \le \left\|u-I_p u\right\|_{E}\,, \ \ p=3,4,\dots\end{gathered}$$ **Proposition 4**. *Let $u$ be the solution to [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"} and assume that [\[analytic\]](#analytic){reference-type="eqref" reference="analytic"} holds. Then there exist positive constants $\kappa_0$, $\kappa _1$ and $\beta>0$ independent of $\varepsilon$ and $p$, such that the following is true: For every $p\in\mathbb{N}$ and every $\kappa \in (0,\kappa _{0}]$ with $\kappa p \geq \kappa_1$, the interpolation error satisfies $$\label{interp_est} \left\|u-I_{p}u\right\| _{1,\infty,\Omega} + \varepsilon^{1/2} \left|u-I_p u\right|_{2,\Omega} \lesssim \mathrm{e}^{-\beta p}.$$ Moreover, $$\label{interp_uS} \left\|u_S-I_{p}u_S\right\| _{2,\Omega} \lesssim \mathrm{e}^{-\beta p},$$ and $$\label{interp_lay} \left|u- I_p u\right|_{k,{\Omega_{BL}}} \lesssim \varepsilon^{3/2-k} \mathrm{e}^{-\beta p}.$$* *Proof.* Inequality [\[interp_est\]](#interp_est){reference-type="eqref" reference="interp_est"} was established in [@PZMX] (see also [@PC]), and inequalities [\[interp_uS\]](#interp_uS){reference-type="eqref" reference="interp_uS"}, [\[interp_lay\]](#interp_lay){reference-type="eqref" reference="interp_lay"} are shown in the Appendix. ◻ Using [\[bestapprox\]](#bestapprox){reference-type="eqref" reference="bestapprox"} and [\[interp_est\]](#interp_est){reference-type="eqref" reference="interp_est"} the following may be established, cf. [@PC]. **Proposition 5**. *Let $u$ solve [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"} and let $u_p$, $p=3,4,\dots$, be its finite element approximations obtained by [\[discrete\]](#discrete){reference-type="eqref" reference="discrete"}. Then there exists a constant $\sigma >0$, independent of $\varepsilon$ but depending on the data, such that $$\begin{gathered} \left\|u-u_p\right\|_{E} \lesssim \mathrm{e}^{-\sigma p}. \end{gathered}$$* ## Balanced norm estimates {#sec:balanced} The energy norm used in Proposition [Proposition 5](#thm:main){reference-type="ref" reference="thm:main"} is not the appropriate norm for this problem, since with $$\begin{gathered} u_{BL} \coloneqq \tilde{u}+ \bar{u},\end{gathered}$$ we calculate $$\begin{gathered} \left\|u_{BL}\right\|_{E} = \mathcal{O}\left(\varepsilon^{1/2}\right) \; \mbox{ while } \; \left\|u_{S}\right\|_{E} = \mathcal{O}\left(1\right)\,, \ \ \varepsilon\to 0.\end{gathered}$$ Thus, the energy norm does not see the layers present in the solution of [\[de-bc\]](#de-bc){reference-type="eqref" reference="de-bc"}, as $\varepsilon\to 0$. The correct weight for $\left|u\right|_{2,\Omega}$ is $\varepsilon^{1/2}$ (as opposed to $\varepsilon$, cf. [\[energy\]](#energy){reference-type="eqref" reference="energy"}). In which case, the resulting norm would be *balanced*, like the one below: $$\label{balanced-1D} \left\|u\right\|^2_{B} \coloneqq \varepsilon\left|u\right|_{2,\Omega}^2 + \left\|u\right\|_{1,\Omega}^2.$$ In this norm we have $$\begin{gathered} \left\|u_{BL}\right\|_{B} = \mathcal{O}\left(1\right) = \left\|u_{S}\right\|_{B}\,, \ \ \varepsilon\to 0.\end{gathered}$$ Unfortunately, the bilinear form $\mathcal{B}_{\varepsilon}$ given by [\[Buv\]](#Buv){reference-type="eqref" reference="Buv"}, is not coercive with respect to this norm. Nevertheless, numerical experiments reported in [@PC], suggest the following: $$\begin{gathered} \label{C1}% \label{C3} \left\|u - u_p\right\|_{1,\infty,\Omega} + \left\|u-u_p\right\|_B \lesssim \mathrm{e}^{-\sigma p}\end{gathered}$$ Our goal in this section is to establish [\[C1\]](#C1){reference-type="eqref" reference="C1"}. We begin by studying the auxiliary problem of finding $u_{S,p} \in \mathcal{P}_p$, $p\ge3$, such that $$\begin{aligned} \label{pi_p} \mathcal{B}_{0}\left(u_{S,p}-u_{S},v\right) = 0 \ \ \forall \; v \in \mathcal{P}_p\cap H_0^1(\Omega), \ \ u_{S,p}-u_{S} = 0 \ \text{on} \ \partial\Omega\end{aligned}$$ where $$\begin{gathered} \mathcal{B}_{0}\left(w,v\right) \coloneqq \left\langle b w' , v' \right\rangle + \left\langle c w , v \right\rangle.\end{gathered}$$ Note that $\mathcal{B}_{0}$ induces a norm $\left|\left|\left|w\right|\right|\right| \coloneqq \mathcal{B}_{0}(w,w)^{1/2}$ that is equivalent to the standard $H^1({\Omega_{REG}})$ norm because of [\[data\]](#data){reference-type="eqref" reference="data"}. Consequently, $u_{S,p}$ is uniquely defined and we have the following. **Lemma 6**. *Let $u_{S,p} \in \mathcal{P}_p$, $p\ge3$, be defined by [\[pi_p\]](#pi_p){reference-type="eqref" reference="pi_p"}. Then, $$\begin{gathered} \left|u_{S,p} - u_S\right|_{k} \lesssim \mathrm{e}^{-\beta p}, \quad k=0,1,2, \\ \left\|u_{S,p} - u_S\right\|_{C^1(\bar\Omega)} \lesssim \mathrm{e}^{-\beta p} \intertext{and} \left|\left(u_{S,p} - u_S\right)(x)\right| \lesssim \min(x,1-x) \mathrm{e}^{-\beta p}\,, \ \ x\in \bar\Omega. \end{gathered}$$* *Proof.* Note, that the bilinear form in [\[pi_p\]](#pi_p){reference-type="eqref" reference="pi_p"} is coercive and bounded. So standard techniques for spectral methods apply to give the bounds of the first inequality for $k=0$ and $k=1$. For $k=2$ we proceed as follows. We have $$\begin{gathered} \left|\left|\left|u_{S,p} - u_S\right|\right|\right| \le \left|\left|\left|v - u_S\right|\right|\right| \ \ \forall \ v\in\mathcal{P}_p \ \ \text{with} \ \ v= u_S \ \ \text{on} \ \partial\Omega, \end{gathered}$$ which implies $$\begin{gathered} \left|\left|\left|u_{S,p} - I_p u_S\right|\right|\right| \le 2 \left|\left|\left|I_p u_S - u_S\right|\right|\right|, \end{gathered}$$ by the triangle inequality. Next $$\begin{aligned} \left|u_{S,p} - u_S\right|_{2} & \le \left|u_{S,p} - I_p u_S\right|_{2} + \left|I_p u_S - u\right|_{2} \le (p-1)^2 \left|u_{S,p} - I_p u_S\right|_{1} + \left|I_p u_S - u\right|_{2} \\ & \le 2 b_0^{-1} (p-1)^2 \left|\left|\left|I_p u_S-u_S\right|\right|\right|_{1} + \left|I_p u_S - u\right|_{2}\,, \end{aligned}$$ where we have used an inverse inequality, see Lemma [Lemma 19](#lem:inverse-1D){reference-type="ref" reference="lem:inverse-1D"}. Now the desired result follows from [\[interp_uS\]](#interp_uS){reference-type="eqref" reference="interp_uS"}. A Sobolev embedding gives the bound in the $C^1$ norm, while the final bound follows upon noting that $\left(u_{S,p}-u_S\right)(x)=0$ for $x\in\{0,1\}$ and the maximum-norm bound for the derivative. ◻ Next, we construct a special FE-function $\hat{u}_p\in \mathcal{S}^p_{0,\Delta_{BL}}$ that is particularly close to $u$. To this end, define auxiliary functions $\chi_0,\chi_1\in \mathcal{P}_3$ by $$\begin{gathered} \chi_0(0)=\chi_0'(0)=\chi_0(\tau)=0, \quad \chi_0'(\tau)=1, \\ \intertext{and} \chi_1(0)=\chi_1'(0)=\chi_1'(\tau)=0, \quad \chi_1(\tau)=1.\end{gathered}$$ A direct calculation establishes the following bounds on the $\chi_i$: $$\begin{gathered} \label{bound-chi} \left|\chi_i\right|_{k,(0,\tau)} \lesssim \tau^{3/2-k-i}, \; k \in \{0, 1, 2\}, \; i \in \{0,1\}.\end{gathered}$$ Now, set $$\begin{gathered} \label{uhat} \hat{u}_p(x) \coloneqq \begin{cases} \left(I_p u\right)(x) + \chi_0(x)\left(u_{S,p} - u\right)'(\tau) % \\ %\qquad\qquad\quad + \chi_1(x)\left(u_{S,p} - u\right)(\tau), & x \in [0, \tau], \\ u_{S,p}(x), & x \in [\tau, 1-\tau], \\ (I_p u)(x)-\chi_0(1-x)\left(u_{S,p} - u\right)'(1-\tau), % \\ % \qquad\qquad\quad + \chi_1(1-x)\left(u_{S,p}-u\right)(1-\tau), & x \in[1-\tau,1]. \end{cases}\end{gathered}$$ We re-iterate that in the layer region, $\hat{u}_p$ is given as the interpolant $I_pu$ of $u$ plus corrections that ensure $\hat{u}_p\in C^1(\bar\Omega)$. In the regular region, we *only* apply the projection on $u_S$. We have the following. **Lemma 7**. *Assume that [\[AA\]](#AA){reference-type="eqref" reference="AA"} holds and that $\hat{u}_p$ is defined by [\[uhat\]](#uhat){reference-type="eqref" reference="uhat"}. Then the following bounds hold true:* *$$\begin{aligned} {2} \label{u-ut:bl} \left|u - \hat{u}_p\right|_{k, {\Omega_{BL}}} & \lesssim \varepsilon^{3/2-k} \mathrm{e}^{-\beta p}, & \ & k \in \{0,1,2\}, \\ \label{u-ut:reg} \left|u - \hat{u}_p\right|_{k, {\Omega_{REG}}} & \lesssim \max\left(1,\varepsilon^{3/2-k}\right) \mathrm{e}^{-\beta p}, & \ & k\in \{0,1,2\}. \end{aligned}$$* *Proof.* We shall study the two regions ${\Omega_{BL}}$ and ${\Omega_{REG}}$ separately. #### ${\Omega_{BL}}$: We present the argument for $(0,\tau)$, because identical bounds hold for the interval $(1-\tau,1)$. We have $$\begin{gathered} \left|u - \hat{u}_p\right|_{k,(0,\tau)} \le \left|u - I_p u\right|_{k,(0,\tau)} + \left|\chi_0\right|_{k,(0,\tau)} \cdot \left|\left(u_{S,p} - u\right)'(\tau)\right| + \left|\chi_1\right|_{k,(0,\tau)} \cdot \left|\left(u_{S,p} - u\right)(\tau)\right|, \ \ k\in\{0,1,2\}. \end{gathered}$$ Proposition [Proposition 4](#prop:interp){reference-type="ref" reference="prop:interp"} takes care of the first term above. For the other two terms, a triangle inequality, Lemma [Lemma 6](#lem:pi_p){reference-type="ref" reference="lem:pi_p"}, [\[A4\]](#A4){reference-type="eqref" reference="A4"} and [\[A6\]](#A6){reference-type="eqref" reference="A6"} yield $$\begin{gathered} \left|\left(u_{S,p} - u\right)^{(\ell)}(\tau)\right| = \left|\left(u_{S,p} - u_S - u_{BL} - u_R\right)^{(\ell)}(\tau)\right| \lesssim \tau^{1-\ell} \mathrm{e}^{-\beta p}\,, \ \ \ell\in{0,1}. \end{gathered}$$ Recalling [\[bound-chi\]](#bound-chi){reference-type="eqref" reference="bound-chi"}, we get $$\begin{gathered} \left|\chi_{1-\ell}^{}\right|_{k,(0,\tau)} \left|\left(u_{S,p} - u\right)^{(\ell)}(\tau)\right| \lesssim \tau^{3/2-k} \mathrm{e}^{-\beta p}\,, \ \ \ell\in{0,1}, \end{gathered}$$ and [\[u-ut:bl\]](#u-ut:bl){reference-type="eqref" reference="u-ut:bl"} follows. We mention here that we will routinely "hide" polynomial powers into exponentials, e.g. $p^s \mathrm{e}^{-\beta p} \lesssim \mathrm{e}^{-\beta p} \; \; \forall \;\; s \in \mathbb{R}$. #### ${\Omega_{REG}}$: We have from the definition of $\hat{u}_p$, $$\begin{gathered} \left|u -\hat{u}_p\right|_{k,{\Omega_{REG}}} \le \left|u_{S}-u_{S,p}\right|_{k,{\Omega_{REG}}} + \left|u_{BL}\right|_{k,{\Omega_{REG}}} + \left|u_R\right|_{k,{\Omega_{REG}}}, \ k=0,1,2. \end{gathered}$$ The first term has been bounded in Lemma [Lemma 6](#lem:pi_p){reference-type="ref" reference="lem:pi_p"}. For $u_{BL}$ we have by [\[A4\]](#A4){reference-type="eqref" reference="A4"}, and because $x\in[\tau,1-\tau]$, $$\begin{gathered} \left|u_{BL}\right|_{k,{\Omega_{REG}}} \lesssim \varepsilon^{3/2-k} \mathrm{e}^{-\gamma \kappa p}, \ k=0,1,2, \end{gathered}$$ while for $u_R$, ineq. [\[A6\]](#A6){reference-type="eqref" reference="A6"} gives $$\begin{gathered} \left|u_R\right|_{k,{\Omega_{REG}}} \lesssim \mathrm{e}^{-3\delta \kappa p}, \ k=0,1,2. \end{gathered}$$ Combining the last two inequalities with Lemma [Lemma 6](#lem:pi_p){reference-type="ref" reference="lem:pi_p"}, we obtain [\[u-ut:reg\]](#u-ut:reg){reference-type="eqref" reference="u-ut:reg"}. ◻ We now have the necessary tools for proving the following, main result of this section. **Theorem 8**. *Let $u$ be the solution of [\[BuvFv\]](#BuvFv){reference-type="eqref" reference="BuvFv"} and let $u_p$ be the approximation obtained by [\[discrete\]](#discrete){reference-type="eqref" reference="discrete"}. Then, there exists a positive constant $\beta$, independent of $\varepsilon$, such that $$\begin{gathered} \left\|u - u_p\right\|_{B} \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$* *Proof.* By [\[balanced-1D\]](#balanced-1D){reference-type="eqref" reference="balanced-1D"} $$\begin{gathered} \left\|u-u_p\right\|^2_{B} = \varepsilon\left|u - u_p\right|^2_{2,\Omega} + \left\|u-u_p\right\|^2_{1,\Omega}. \end{gathered}$$ The only "troublesome" term above is $\varepsilon\left|u-u_p\right|_{2,\Omega}$. (The other one is handled by Proposition [Proposition 5](#thm:main){reference-type="ref" reference="thm:main"}). To deal with it, we use the triangle inequality (with $\hat{u}_p$ defined by [\[uhat\]](#uhat){reference-type="eqref" reference="uhat"}) to get $$\begin{gathered} \varepsilon^{1/2} \left|u - u_p\right|_{2,\Omega} \le \varepsilon^{1/2} \left|u-\hat{u}_p\right|_{2,\Omega} + \varepsilon^{1/2} \left|\hat{u}_p-u_p\right|_{2,\Omega}. \end{gathered}$$ Only the term $\varepsilon^{1/2} \left|\hat{u}_p-u_p\right|_{2,\Omega}$ needs to be considered, since the rest can be bounded by Lemma [Lemma 7](#lem:u-ut){reference-type="ref" reference="lem:u-ut"}. Let $\eta \coloneqq \hat{u}_p-u_p \in \mathcal{S}^p_{0,\Delta_{BL}}$. Galerkin orthogonality gives $$\begin{gathered} \left\|\eta\right\|_E^2 = \varepsilon^2 \left\langle \left(\hat{u}_p-u\right)'' , \eta'' \right\rangle + \left\langle b(\hat{u}_p-u)' , \eta' \right\rangle + \left\langle c(\hat{u}_p-u) , \eta \right\rangle . \end{gathered}$$ On the coarse mesh region $\Omega_{REG}$, we have $$\begin{aligned} & \left\langle b\left(\hat{u}_p-u\right)' , \eta' \right\rangle_{{\Omega_{REG}}} + \left\langle c\left(\hat{u}_p-u\right) , \eta \right\rangle_{{\Omega_{REG}}} \\ & \qquad = \left\langle b\left(u_{S,p}-u_S\right)' , \eta' \right\rangle_{{\Omega_{REG}}} + \left\langle c\left(u_{S,p}-u_S\right) , \eta \right\rangle_{{\Omega_{REG}}} \\ & \qquad\qquad\qquad - \left\langle b\left(u_{BL}+u_R\right)' , \eta' \right\rangle_{{\Omega_{REG}}} - \left\langle c\left(u_{BL}+u_R\right) , \eta \right\rangle_{{\Omega_{REG}}} \\ & \qquad = - \left\langle b\left(u_{S,p}-u_S\right)' , \eta' \right\rangle_{{\Omega_{BL}}} - \left\langle c\left(u_{S,p}-u_S\right) , \eta \right\rangle_{{\Omega_{BL}}} \\ & \qquad\qquad\qquad - \left\langle b\left(u_{BL}+u_R\right)' , \eta' \right\rangle_{{\Omega_{REG}}} - \left\langle c\left(u_{BL}+u_R\right) , \eta \right\rangle_{{\Omega_{REG}}}, \end{aligned}$$ where the definition of $\hat{u}_p$ was used. Now, the $C^1$-norm bounds of Lemma [Lemma 6](#lem:pi_p){reference-type="ref" reference="lem:pi_p"}, [\[A4\]](#A4){reference-type="eqref" reference="A4"} and [\[A6\]](#A6){reference-type="eqref" reference="A6"} yield $$\begin{aligned} & \left|\left\langle b\left(\hat{u}_p-u\right)' , \eta' \right\rangle_{{\Omega_{REG}}} + \left\langle c\left(\hat{u}_p-u\right) , \eta \right\rangle_{{\Omega_{REG}}}\right| \\ & \qquad \lesssim \tau^{1/2} \left\|u_{S,p}-u_S\right\|_{C^1(\bar{\Omega}_{BL})} \left\|\eta\right\|_{1,{\Omega_{BL}}} + \left\|u_{BL}+u_R\right\|_{1,{\Omega_{REG}}} \left\|\eta\right\|_{1,{\Omega_{REG}}} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p} \left\|\eta\right\|_{1,\Omega}. \end{aligned}$$ Similarly, using the $H^k$-norm bounds of Lemma [Lemma 6](#lem:pi_p){reference-type="ref" reference="lem:pi_p"}, [\[A4\]](#A4){reference-type="eqref" reference="A4"} and [\[A6\]](#A6){reference-type="eqref" reference="A6"}, we get $$\begin{aligned} & \varepsilon^2\left|\left\langle (\hat{u}_p-u)'' , \eta'' \right\rangle_{{\Omega_{REG}}}\right| \\ & \qquad \le \varepsilon^2 \left|\left\langle (u_{S,p}-u_S)'' , \eta'' \right\rangle_{{\Omega_{REG}}}\right| + \varepsilon^2\left|\left\langle \left(u_{BL}+u_R\right)'' , \eta'' \right\rangle_{{\Omega_{REG}}}\right| \lesssim \varepsilon^{3/2} \mathrm{e}^{-\beta p} \left|\eta\right|_{2,{\Omega_{REG}}}. \end{aligned}$$ In the layer region ${\Omega_{BL}}$, we use the Cauchy-Schwarz inequality and [\[u-ut:bl\]](#u-ut:bl){reference-type="eqref" reference="u-ut:bl"} to obtain $$\begin{gathered} \left|\left\langle b\left(\hat{u}_p-u\right)' , \eta' \right\rangle_{{\Omega_{BL}}} + \left\langle c\left(\hat{u}_p-u\right) , \eta \right\rangle_{{\Omega_{BL}}}\right| \lesssim \varepsilon^{1/2} \mathrm{e}^{\beta p} \left\|\eta\right\|_{1,{\Omega_{BL}}} \intertext{and} \varepsilon^2\left|\left\langle (\hat{u}_p-u)'' , \eta'' \right\rangle_{{\Omega_{BL}}}\right| \lesssim \varepsilon^{1/2} \mathrm{e}^{\beta p} \varepsilon\left|\eta\right|_{2,{\Omega_{BL}}}. \end{gathered}$$ Combining the above we have $$\begin{gathered} \left\|\eta\right\|_E^2 \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p} \left\|\eta\right\|_E. \end{gathered}$$ Dividing by $\left\|\eta\right\|_E$ yields $$\begin{gathered} \varepsilon^{1/2} \left|u_p-\hat{u}_p\right|_{2,\Omega} \lesssim \left\|u_p-\hat{u}_p\right\|_E \lesssim \mathrm{e}^{-\beta p}, \end{gathered}$$ and from the triangle inequality we have $$\begin{gathered} \varepsilon^{1/2} \left|u_p-u\right|_{2,\Omega} \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$ This completes the proof. ◻ Once we have the balanced norm estimate, we may obtain a maximum norm estimate as follows. **Corollary 9**. *Let $u$ be the solution of [\[BuvFv\]](#BuvFv){reference-type="eqref" reference="BuvFv"} and let $u_p$ be the solution of [\[discrete\]](#discrete){reference-type="eqref" reference="discrete"}, based on the Spectral Boundary Layer mesh of Definition [Definition 3](#SBL){reference-type="ref" reference="SBL"}. Then, there exists a positive constant $\beta$, independent of $\varepsilon$, such that $$\begin{gathered} \left\|u - u_p\right\|_{C^1(\bar\Omega)} \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$* *Proof.* Let $x \in [0,\tau]$ be arbitrary. Then, using the boundary conditions [\[bc\]](#bc){reference-type="eqref" reference="bc"}, we have $$\begin{gathered} (u-u_p)^{(k)}(x) = \int_0^{x} \left(u-u_p\right)^{(k+1)}(t) \;\mathrm{d}t , \ k \in \{0, 1\}, \end{gathered}$$ and by the Cauchy-Schwarz inequality $$\begin{gathered} \left|(u-u_p)^{(k)}(x)\right| \lesssim \tau^{1/2} \left|u-u_p\right|_{k+1,{\Omega_{BL}}} \lesssim (p\varepsilon)^{1/2} \varepsilon^{-1/2} \mathrm{e}^{-\beta p} \lesssim \mathrm{e}^{-\beta p}, \end{gathered}$$ where Theorem [Theorem 8](#thm:C1balanced){reference-type="ref" reference="thm:C1balanced"} was used. The same technique works for the other layer region $[1-\tau, 1]$, so let us consider the coarse region $\Omega_{\mathrm{REG}}$. We use the triangle inequality, with $I_p$ the interpolant of Proposition [Proposition 4](#prop:interp){reference-type="ref" reference="prop:interp"}, to get $$\begin{gathered} \left|u-u_p\right|_{k,\infty,{\Omega_{REG}}} \leq \left|u-I_p u\right|_{k,\infty,{\Omega_{REG}}} + \left|I_p u-u_p\right|_{k,\infty,{\Omega_{REG}}}. \end{gathered}$$ The term $\left|u-I_p u\right|_{k,\infty,{\Omega_{REG}}}$ is handled by Proposition [Proposition 4](#prop:interp){reference-type="ref" reference="prop:interp"}, while for the second term an inverse inequality gives $$\begin{gathered} \left|I_p u-u_p\right|_{k,\infty,{\Omega_{REG}}} \lesssim \left|I_p u-u_p\right|_{k,2,{\Omega_{REG}}} \le \left|I_p u-u\right|_{k,2,{\Omega_{REG}}} + \left|u-u_p\right|_{k,2,{\Omega_{REG}}}. \end{gathered}$$ Application of Propositions [Proposition 4](#prop:interp){reference-type="ref" reference="prop:interp"} and [Proposition 5](#thm:main){reference-type="ref" reference="thm:main"} completes the proof. ◻ **Remark 10**. *Theorem [Theorem 8](#thm:C1balanced){reference-type="ref" reference="thm:C1balanced"} and Corollary [Corollary 9](#max_norm_C1){reference-type="ref" reference="max_norm_C1"} establish the conjecture [\[C1\]](#C1){reference-type="eqref" reference="C1"} made in [@PC].* # Two-dimensional problems {#sec:2D} In two dimensional smooth domains $\Omega \subset \mathbb{R}^2$, we would have great difficulty constructing a $C^1$ conforming FEM; instead, a $C^0$ mixed formulation is usually preferred. Before we present the problem under consideration along with existing results, we comment on the differences between the $C^1$ method and the $C^0$ mixed method, beyond the obvious ones. Specifically, we want to point out that in order to use the previous methodology, we should define two special representatives, since in the mixed formulation we will be seeking two unknown functions -- the second one will be auxiliary. Moreover, for the extra function we only need to use the $L^2$ projection, while for $u$ we (again) utilize the $H^1$ projection. We consider the following problem[^3] from [@CFLX]: Find $u$ such that [\[bvp2\]]{#bvp2 label="bvp2"} $$\begin{aligned} {2} \varepsilon^{2}\Delta ^{2}u-b\Delta u+cu & = f & \ \ & \text{in }\Omega \subset \mathbb{R}^{2}, \\ % \label{de2} \\ u=\frac{\partial u}{\partial n} & = 0 && \text{on} \ \partial \Omega , % \label{bc2} \end{aligned}$$ where $0<\varepsilon\leq 1$ is a given parameter, $\Delta$ denotes the Laplacian (and $\Delta^{2}$ the biharmonic) operator, $b,c>0$ are given constants, $\Omega$ is an open bounded domain with analytic boundary $\partial \Omega$ and $f$ is a given analytic function, which satisfies $$\left\|\nabla ^{n}f\right\|_{\infty ,\Omega} \lesssim n!\gamma_{f}^{n}\ \ \text{for} \ n=0,1,2,\dots, \label{analytic2}$$ for some positive constant $\gamma _{f}$ independent of $\varepsilon$. Here we have used the shorthand notation $$\left|\nabla ^{n}f\right|^{2} \coloneqq \sum_{\left|\alpha\right|=n} \frac{\left|\alpha\right|!}{\alpha !}\left|D^{\alpha} f\right|^{2}, % = \sum_{\beta _{1},...,\beta_{n}=1}^{2}\left\vert D^{\beta_{1} \cdots \beta _{n}}f\right\vert ^{2},$$ with $D^{\alpha}f$ denoting the generalized derivative of $f$ with respect to the multiindex $\alpha\in\mathbb{N}_0^2$. The regularity of the solution to [\[bvp2\]](#bvp2){reference-type="eqref" reference="bvp2"} was studied in [@PC; @MX], and will be described in Proposition [Proposition 1](#prop:reg){reference-type="ref" reference="prop:reg"} below. To this end, define *boundary fitted coordinates* $(\rho,\theta)$ in a neighborhood of the boundary as follows: Let $\bigl(X(\theta),Y(\theta)\bigr)$, $\theta \in [0,L]$ be a parametrization of $\partial \Omega$ by arclength and let $\Omega _{0}$ be a tubular neighborhood of $\partial \Omega$ in $\Omega$. For each point $z=(x,y)\in\Omega_{0}$ there is a unique nearest point $z_{0}\in \partial \Omega$, so with $\theta$ the arclength parameter (with counterclockwise orientation), we set $\rho=\left|z-z_{0}\right|$ which measures the distance from the point $z$ to $\partial \Omega$. Explicitely, let $\rho _{0}>0$ be less than the minimum radius of curvature of $\partial \Omega$ and set $$\begin{gathered} \Omega_{0} \coloneqq \Bigl\{ z-\rho \overrightarrow{n}_{z} \colon z\in \partial \Omega, \ \rho\in (0,\rho_{0}) \Bigr\}, \label{Omega0}\end{gathered}$$ where $\overrightarrow{n}_{z}$ is the outward unit normal at $z\in \partial \Omega$, and $$x=X(\theta )-\rho Y'(\theta ),\ y=Y(\theta )+\rho X'(\theta),$$ with $\rho \in (0,\rho _{0})$, $\theta \in (0,L)$; see Figure [1](#fig0){reference-type="ref" reference="fig0"}. {#fig0 width="0.6 \\textwidth"} **Proposition 11**. *[\[AAA\]]{#AAA label="AAA"} The solution $u$ of [\[bvp2\]](#bvp2){reference-type="eqref" reference="bvp2"} can be decomposed into a smooth part $u^{S}$, a boundary layer part $u^{BL}$ and a remainder $u^R$, viz. $$u = u^{S}+\chi u^{BL}+u^R,%\label{AA1}$$ where $\chi$ is a smooth cut-off function, satisfying $$\chi = \begin{cases} 1 & \text{for} \ 0<\rho <\rho _{0}/3, \\ 0 & \text{for} \ \rho >2\rho _{0}/3. \end{cases}$$ Moreover, there exist constants $K_{1},K_{2},\omega ,\delta >0$, independent of $\varepsilon$, such that $$\begin{gathered} \label{AA2} \left\|D^{n}u^{S}\right\|_{0,\Omega } \lesssim \left|n\right|! \, K_{1}^{\left|n\right|}\ \ \forall \ n\in \mathbb{N}_{0}^{2}, \\ \label{AA3} \left|\frac{\partial ^{m+n}u^{BL}(\rho ,\theta)}{\partial \rho^{m} \partial \theta ^{n}}\right| \lesssim n!\ K_{2}^{m+n} \varepsilon^{1-m} \mathrm{e}^{-\omega \rho /\varepsilon}\ \ \forall \ m,n\in \mathbb{N}, \ (\rho,\theta) \in \overline{\Omega}_0, \intertext{and} % \label{AA4} \varepsilon\left|u^R\right|_{2,\Omega} + \left\|u^R\right\|_{1,\Omega}\lesssim \mathrm{e}^{-\delta/\varepsilon}. \end{gathered}$$ Finally, there exist constants $C,K>0$, depending only on the data, such that $$% \label{AA6} \left\|D^{n}u\right\|_{0,\Omega } \leq CK^{\left|n\right|} \max \left\{ \left|n\right|^{\left|n\right|}, \varepsilon^{1-\left|n\right|}\right\} \ \ \forall \ n\in \mathbb{N}_{0}^{2}.$$* *Proof.* All estimates were shown in [@PC], except for [\[AA2\]](#AA2){reference-type="eqref" reference="AA2"} which appears in [@MX2]. ◻ Letting[^4] $w\coloneqq\varepsilon\Delta u\in H^{2}(\Omega)$, we shall study the *mixed* formulation of finding $(u,w)\in H_{0}^{1}(\Omega )\times H^{1}(\Omega)$ such that [\[mixed\]]{#mixed label="mixed"} $$\begin{aligned} {2} \varepsilon\left\langle \nabla u , \nabla \phi \right\rangle + \left\langle w , \phi \right\rangle & = 0 & \ \ & \forall \ \phi \in H^{1}(\Omega), \\ b\left\langle u , \nabla \right\rangle + c\left\langle u , \psi \right\rangle - \varepsilon\left\langle \nabla w , \nabla \right\rangle & = \left\langle f , \psi \right\rangle && \forall \ \psi \in H_{0}^{1}(\Omega ). \end{aligned}$$ We define the bilinear form $$\begin{gathered} \label{Buv2} \mathcal{A}_{\varepsilon} \bigl((u,w),(\psi ,\phi )\bigr) \coloneqq \varepsilon\left\langle \nabla u , \nabla \phi \right\rangle + \left\langle w , \phi \right\rangle + b \left\langle \nabla u , \nabla \psi \right\rangle + c\left\langle u , \psi \right\rangle - \varepsilon\left\langle \nabla w , \nabla \psi \right\rangle.\end{gathered}$$ Then the *energy norm* of $u$ is given by $$\begin{gathered} % \label{norm-coercive} \left\|u\right\|_{E}^{2} = \left|\left|\left|(u,\varepsilon\Delta u)\right|\right|\right|^{2} = \varepsilon^{2} \left\|\Delta u\right\|_{0,\Omega }^{2} + b\left\|\nabla u\right\|_{0,\Omega }^{2} + c\left\|u\right\|_{0,\Omega}^{2} = \left|\left|\left|(u,w)\right|\right|\right|^{2}.\end{gathered}$$ A regularity result for $w$, analogous to Proposition [Proposition 11](#assumption){reference-type="ref" reference="assumption"}, is given below. **Corollary 12**. *[\[B0\]]{#B0 label="B0"} Let $w = \varepsilon\Delta u$ and assume Proposition [Proposition 11](#assumption){reference-type="ref" reference="assumption"} holds. Then $$w=w^{S}+\chi w^{BL}+w^R, \label{B1}$$ where $\chi$ is the same cut-off function as in Proposition [Proposition 11](#assumption){reference-type="ref" reference="assumption"}. Moreover, there exist constants $K_{3},K_{4},\beta ,\gamma >0$, independent of $\varepsilon$, such that $$\begin{gathered} % \label{B2} \left\|D^{n}w^{S}\right\|_{0,\Omega } \lesssim \left|n\right|!\ K_{3}^{\left|n\right|} \ \ \forall \ n\in \mathbb{N}_{0}^{2}, \\ \label{B3} \left|\frac{\partial^{m+n}w^{BL}(\rho,\theta )}{\partial\rho^{m} \partial\theta ^{n}}\right| \lesssim n!K_{4}^{m+n}\varepsilon^{-m} \mathrm{e}^{-\beta\rho/\varepsilon}\ \ \forall \ m,n\in\mathbb{N}, \ (\rho,\theta) \in \overline{\Omega}_0, \intertext{and} % \label{B4} \left\|w^R\right\|_{1,\Omega} \lesssim \mathrm{e}^{-\gamma/\varepsilon}. \end{gathered}$$ Finally, there exists a constant $K>0$, depending only on the data, such that $$\begin{gathered} % \label{B5} \left\|D^{n}w\right\|_{0,\Omega }\lesssim K^{\left|n\right|} \max \left\{ \left|n\right|^{\left|n\right|}, \varepsilon^{-\left|n\right|}\right\} \ \ \forall \ n\in \mathbb{N}_{0}^{2}. \end{gathered}$$* *Proof.* Follows from Proposition [Proposition 11](#assumption){reference-type="ref" reference="assumption"} and the definition of $w=\varepsilon\Delta u$. ◻ We close this section with the following result which will be used in the sequel. **Lemma 13**. *Let $u^{BL}$ and $w^{BL}$ satisfy [\[AA3\]](#AA3){reference-type="eqref" reference="AA3"} and [\[B3\]](#B3){reference-type="eqref" reference="B3"}, respectively. With $\Omega_0$ given by [\[Omega0\]](#Omega0){reference-type="eqref" reference="Omega0"}, we have $$\begin{gathered} \varepsilon^{-1} \left\|u^{BL}\right\|_{0,\Omega \setminus \Omega_0} + \left\|w^{BL}\right\|_{0,\Omega \setminus \Omega_0} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\sigma/ \varepsilon}, \end{gathered}$$ for some positive constant $\sigma$, independent of $\varepsilon$.* *Proof.* This follows from direct calculations. ◻ ## Discretization by a mixed $rp$-FEM In order to define our finite dimensional discretization spaces $V_{1}^{N}\subset H_0^1(\Omega)$ and $V_{2}^{N}\subset H^1(\Omega)$, we let $\Delta \coloneqq \left\{\Omega _{i}\right\} _{i=1}^{N}$ be a mesh consisting of curvilinear quadrilaterals, subject to standard conditions (see, e.g. [@melenk]) and associate with each $\Omega _{i}$ a bijective mapping $M_{i}\colon S_{ST} \rightarrow \overline{\Omega }_{i}$, where $S_{ST}=[0,1]^{2}$ denotes the reference square. With $\mathcal{Q}_{p}(S_{ST})$ the space of polynomials of degree $p$ (in each variable) on $S_{ST}$, we define $$\begin{aligned} \mathcal{S}^{p}(\Delta ) & \coloneqq \Bigl\{ u\in H^{1}\left( \Omega \right) \colon u\vert_{\Omega_{i}}\circ M_{i}\in \mathcal{Q}_{p}(S_{ST}), \ \ i=1,\dots,N\Bigr\}, \\ \mathcal{S}_{0}^{p}(\Delta) & \coloneqq \mathcal{S}^{p}(\Delta )\cap H_{0}^{1}(\Omega ).\end{aligned}$$ The mesh $\Delta$ is chosen following the construction in [@CFLX; @MX]. We denote by $\Delta _{A}$ a *fixed* (asymptotic) mesh consisting of curvilinear quadrilateral elements $\Omega _{i}$, $i=1,\dots,N_{1}$. These elements $\Omega _{i}$ are the images of the reference square $S_{ST}$ under the element mappings $M_{A,i}$, $i=1,\dots,N_{1}\in \mathbb{N}$. Moreover, the element mappings $M_{A,i}$ are assumed to be analytic (with analytic inverses). We also assume that the elements do not have a single vertex on the boundary $\partial\Omega$, but only complete, single edges. For convenience, we number the elements along the boundary first, i.e., $\Omega_{i}$, $i=1,\dots,N_{2}<N_{1}$ for some $N_{2}\in \mathbb{N}$. We now give the definition of the layer-adapted mesh from [@MX], almost verbatim. **Definition 14**. *Given parameters $\kappa>0$, $p\in\mathbb{N}$, $\varepsilon\in (0,1]$ and the (asymptotic) mesh $\Delta _{A}$, the *Spectral Boundary Layer Mesh* $\Delta _{BL}$ is defined as follows:* 1. *If $\kappa p\varepsilon \geq 1/2$ then we are in the asymptotic range of p and we use the mesh $\Delta _{A}$.* 2. *If $\kappa p\varepsilon <1/2$, we need to define so-called needle elements. We do so by splitting the elements $\Omega_{i}$, $i=1,\dots,N_{2}$ into two elements $\Omega _{i}^{need}$ and $\Omega_{i}^{reg}$. To this end, split the reference square $S_{ST}$ into two rectangles $$S^{need} = \left[0,\kappa p\varepsilon\right] \times [0,1],\quad S^{reg} = \left[\kappa p\varepsilon, 1\right] \times [0,1],$$ and define the elements $\Omega _{i}^{need}$ and $\Omega _{i}^{reg}$ as the images of these two rectangles under the element map $M_{A,i}$ and the corresponding element maps as the concatination of the affine maps $$\begin{aligned} {5} & A^{need} && \colon S_{ST} \rightarrow S^{need}, && \quad (\xi ,\eta )\rightarrow (\kappa p\varepsilon \xi ,\eta ), \\ & A^{reg} && \colon S_{ST} \rightarrow S^{reg}, && \quad (\xi ,\eta )\rightarrow (\kappa p\varepsilon + (1-\kappa p\varepsilon )\xi ,\eta ) \end{aligned}$$ with the element map $M_{A,i}$, i.e., $M_{i}^{need}=M_{A,i}\circ A^{need}$ and $M_{i}^{reg}=M_{A,i}\circ A^{reg}$.* In total, the mesh $\Delta_{BL}$ consists of $N=N_{1}+N_{2}$ elements if $\kappa p\varepsilon <1/2$. By construction, the resulting mesh $$\begin{gathered} \Delta _{BL} = \Bigl\{\Omega_{1}^{need},\dots, \Omega_{N_{1}}^{need},\Omega _{1}^{reg},\dots, \Omega _{N_{1}}^{reg}, \Omega_{N_{1}+1},\dots,\Omega _{N}\Bigr\},\end{gathered}$$ is a regular admissible mesh. In Figure [2](#fig1){reference-type="ref" reference="fig1"}, we show an example of such a mesh from [@CFLX]. {#fig1 width="0.3 \\textwidth"} Next, we take $V_{1}^{N}=\mathcal{S}_{0}^{p}(\Delta_{BL})$ and $V_{2}^{N}=\mathcal{S}^{p}(\Delta_{BL})$, and seek an approximation $\left(u_{p,\Delta},w_{p,\Delta}\right)\in V_1^N\times V_2^N$ of [\[mixed\]](#mixed){reference-type="eqref" reference="mixed"} such that $$\begin{gathered} \label{discrete2} \mathcal{A}_{\varepsilon} \left(\left(u_{p,\Delta},w_{p,\Delta}\right)),(\psi ,\phi )\right) = \left\langle f , \psi \right\rangle \ \ \forall \ (\phi,\psi)\in V_1^N\times V_2^N.\end{gathered}$$ Subtracting [\[discrete2\]](#discrete2){reference-type="eqref" reference="discrete2"} from [\[mixed\]](#mixed){reference-type="eqref" reference="mixed"}, we get $$\begin{gathered} % \label{orthogonal} \mathcal{A}_{\varepsilon} \left( (u-u_{p,\Delta},w-w_{p,\Delta}),(\psi ,\phi )\right) =0 \ \ \forall \ (\psi ,\phi)\in V_{1}^{N}\times V_{2}^{N}. \end{gathered}$$ We have the following interpolation estimates, which were based on the Gauss-Lobbato interpolant from [@melenk]. **Lemma 15** ([@CFLX]). *Let $\left(u,w\right)$ be the solution to [\[mixed\]](#mixed){reference-type="eqref" reference="mixed"} and assume that [\[analytic2\]](#analytic2){reference-type="eqref" reference="analytic2"} holds. Then there exist constants $\kappa_{0},\kappa _{1}, \beta >0$ independent of both $\varepsilon\in (0,1]$ and of $p\in \mathbb{N}$, such that the following is true: For every $p$ and every $\kappa \in (0,\kappa _{0}]$ with $\kappa p\geq \kappa _{1}$, there exist $I _{p}u\in \mathcal{S}_{0}^{{p}}(\Delta _{BL})$ and $J_{p}w\in \mathcal{S}^{{p}}(\Delta _{BL})$ such that $$\begin{gathered} \left\|u-I_{p}u\right\|_{L^{\infty}(\Omega)} + \left\|\nabla (u-I_{p}u)\right\|_{L^{\infty}(\Omega)} + \left\|w-J_{p}w\right\|_{L^{\infty}(\Omega)} + \varepsilon^{1/2} \left\|\nabla(w-J_{p}w)\right\|_{0 ,\Omega } \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$* With $$\begin{gathered} \Omega_{BL} = \bigcup_i \Omega_i^{need} \, , \quad {\Omega_{REG}}= \bigcup_i \Omega_i^{reg}\end{gathered}$$ and $\left|\Omega_{BL}\right| = \mathcal{O}\left(\kappa p \varepsilon\right)$, $\left|{\Omega_{REG}}\right| = \mathcal{O}\left(1\right)$, we see that additionally there holds $$\label{BLregion_bound} \left\|\nabla (u-I_{p} u)\right\|_{0 ,\Omega_{BL}} + %\lesssim \varepsilon^{1/2} e^{-\beta p } \; , \; \left\|w-J_{p} w\right\|_{0 ,\Omega_{BL}} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p }$$ The following was the main result of [@CFLX]. **Theorem 16** ([@CFLX]). *Let $(u,w)$ and $\left( u_{p,\Delta},w_{p,\Delta}\right)$ be the solutions to [\[mixed\]](#mixed){reference-type="eqref" reference="mixed"} and [\[discrete2\]](#discrete2){reference-type="eqref" reference="discrete2"}, respectively. Assume Proposition [Proposition 11](#assumption){reference-type="ref" reference="assumption"} holds. Then there exists a positive constant $\beta$, independent of $\varepsilon$, such that $$\begin{gathered} \left|\left|\left|\left( u-u_{p,\Delta},w-w_{p,\Delta}\right)\right|\right|\right| \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$* Note that the norm used in Theorem [Theorem 16](#theorem_main){reference-type="ref" reference="theorem_main"} is not correctly balanced, because $$\left|\left|\left|(u^{BL},\varepsilon\Delta u^{BL})\right|\right|\right| = \mathcal{O}\left(\varepsilon^{1/2}\right)\,, \ \ \text{but} \ \ \left|\left|\left|(u^{S},\varepsilon\Delta u^{S})\right|\right|\right| = \mathcal{O}\left(1\right)\,, \ \ (\varepsilon\to 0).$$ Just like in the one-dimensional case, as $\varepsilon\rightarrow 0$, "the energy norm does not see the layers" [@FR; @FR3]. This is due to the fact that the weight on $\left\|w\right\|_{0,\Omega}$ is not the appropriate one. A more suitable, correctly balanced norm is $$\begin{gathered} \label{balanced} \left|\left|\left|(u,w)\right|\right|\right|_{B}^{2} \coloneqq \varepsilon^{-1} \left\|w\right\|_{0,\Omega}^{2} + b \left\|\nabla u\right\|_{0,\Omega }^{2} + c \left\|u\right\|_{0,\Omega }^{2},\end{gathered}$$ since then $$\left|\left|\left|\left(u^{BL},\varepsilon\Delta u^{BL}\right)\right|\right|\right|_B = \mathcal{O}\left(1\right) = \left|\left|\left|\left(u^{S},\varepsilon\Delta u^{S}\right)\right|\right|\right|_B, \ \ \ (\varepsilon\to0).$$ Numerical experiments using the above norm have been reported in [@CFLX], suggesting that indeed the method yields uniform exponential convergence in the balanced norm $\left|\left|\left|\cdot\right|\right|\right|_B$. Unfortunately, the bilinear form $\mathcal{A}_{\varepsilon}$ given by [\[Buv2\]](#Buv2){reference-type="eqref" reference="Buv2"}, is not uniformly coercive with respect to the norm [\[balanced\]](#balanced){reference-type="eqref" reference="balanced"}. To circumvent this obstacle, we will use the methodology derived in Section [2](#sec:1D){reference-type="ref" reference="sec:1D"}. ## Balanced norm estimates {#estimates} Recall the decomposition [\[B0\]](#B0){reference-type="eqref" reference="B0"}, set $w^{SR}\coloneqq w^S+w^R$ and define the $L^2$ projection of $w^{SR}$ on the regular region ${\Omega_{REG}}$, denoted by $\pi^1_p w^{SR} \in V^N_2|_{{\Omega_{REG}}}$, via $$\begin{gathered} % \label{pi_p2D} \left\langle \pi^1_p w^{SR} , v \right\rangle_{{\Omega_{REG}}}=0 \quad \forall \; v \in V^N_2|_{{\Omega_{REG}}}.\end{gathered}$$ The projection is uniquely defined and consequently, we have the following: $$\begin{gathered} \label{pi_p2} \left\|\pi^1_p w^{SR}-w^{S}\right\|_{0, \Omega_{REG}} + \left\|\pi^1_p w^{SR}-w^{SR}\right\|_{L^{\infty}(\Omega_{REG})} \lesssim \mathrm{e}^{-\kappa p}\end{gathered}$$ where the definition of the projection $\pi^1_p$ and Lemma [Lemma 13](#BLsize){reference-type="ref" reference="BLsize"} were used, along with $\kappa p \varepsilon< 1/2$, which is when we use the SBL mesh. Let analogously $u^{SR}\coloneqq u^S+u^R$ and define $\pi_p^2 u^{SR} \in V^N_2|_{\Omega_{REG}}$ as the weighted $H^1$ projection of $u^{SR}$ on $\Omega_{REG}$: $$\begin{aligned} %\label{pi_p2_2D} b \left\langle \nabla ( \pi^2_p u^{SR}-u^{SR}) , v \right\rangle_{\Omega_{REG}} + c \left\langle \pi^2_p u^{SR}-u^{SR} , v \right\rangle_{\Omega_{REG}}=0 \quad \forall \; v \in V^N_2|_{\Omega_{REG}},\end{aligned}$$ and we have $$\begin{gathered} % \label{lem:pi_p3} \left\|\pi^2_p u^{SR}-u^{SR}\right\|_{1, \Omega_{REG}} + \left\|\pi^2_p \nabla (u^{SR}-u^{SR})\right\|_{L^{\infty}(\Omega_{REG})} \lesssim \mathrm{e}^{-\beta p}. %\label{lem:pi_p4}\end{gathered}$$ We next define an auxiliary function $\chi_2 \in \mathcal{P}_1$ by the conditions $$\begin{gathered} \chi_2(0)=0, \quad \chi_2(\kappa p \varepsilon)=1.\end{gathered}$$ A direct calculation establishes the following bounds on $\chi_2$: $$\begin{gathered} \label{bound-chi2D} \left\|\chi_2\right\|_{0,\Omega_{BL}} \lesssim (\kappa p \varepsilon)^{1/2}, \quad \left\|\chi_2'\right\|_{0,\Omega_{BL}} \lesssim (\kappa p \varepsilon)^{-1/2}. \quad\end{gathered}$$ We are now in a position to define *special representatives* $\tilde{w}\in \mathcal{S}^p$ of $w$ and $\tilde{u}\in \mathcal{S}_0^p$ of $u$. Recalling the decomposition [\[B1\]](#B1){reference-type="eqref" reference="B1"}, define $$\begin{aligned} \label{wt} \tilde{w}(x) & \coloneqq \begin{cases}J_p w- \chi_2(\rho) \left(w^{SR}+w^{BL}-\pi^1_p w^{SR}\right)|_{\partial \Omega_{REG}} & \text{ in } \Omega_{BL}, \\ \pi_p^1 w^{SR} & \text{ in } \Omega_{REG}. \end{cases} \\ \label{ut} \tilde{u}(x) & \coloneqq \begin{cases}I_p u- \chi_2(\rho) \left(u^{SR}+u^{BL}-\pi^2_p u^{SR}\right)|_{\partial \Omega_{REG}} & \text{ in } \Omega_{BL}, \\ \pi_p^2 u^{SR} & \text{ in } \Omega_{REG}. \end{cases}\end{aligned}$$ We have the following. **Lemma 17**. *There exist a positive contant $\beta$ such that the following bounds holds true:* *$$\begin{aligned} \label{w-wt2D:bl} \left\|w - \tilde{w}\right\|_{0,\Omega_{BL}} & \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p}, \\ \label{w-wt2D:reg} \left\|w - \tilde{w}\right\|_{0,\Omega_{REG}} & \lesssim \mathrm{e}^{-\beta p}, \\ \label{u-ut2D:bl} \left\|u - \tilde{u}\right\|_{0,\Omega_{BL}} & \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p}, \\ \label{u-ut2D:reg} \left\|u - \tilde{u}\right\|_{0,\Omega_{REG}} & \lesssim \mathrm{e}^{-\beta p}. \end{aligned}$$* *Proof.* We study the two regions ${\Omega_{BL}}$ and ${\Omega_{REG}}$ separately. #### ${\Omega_{BL}}$: We have for $w-\tilde{w}$, $$\begin{gathered} \left\|w - \tilde{w}\right\|_{0,\Omega_{BL}} \le \left\|w- J_p w\right\|_{0,\Omega_{BL}} + \left\|\chi_2\right\|_{0,\Omega_{BL}} \left\{ \left\| w^{BL}\right\|_{L^{\infty}(\partial {\Omega_{REG}})} + \left\|\pi^1_p w^{SR} - w^{SR}\right\|_{L^{\infty}(\Omega_{BL})}\right\}\,. \end{gathered}$$ Equation [\[BLregion_bound\]](#BLregion_bound){reference-type="eqref" reference="BLregion_bound"} takes care of the first term above, and equations [\[AA3\]](#AA3){reference-type="eqref" reference="AA3"}, [\[pi_p2\]](#pi_p2){reference-type="eqref" reference="pi_p2"} and [\[bound-chi2D\]](#bound-chi2D){reference-type="eqref" reference="bound-chi2D"}, allow us to bound the remaining terms, as desired. This shows [\[w-wt2D:bl\]](#w-wt2D:bl){reference-type="eqref" reference="w-wt2D:bl"}. Similarly, for $u-\tilde{u}$ we have $$\begin{gathered} \left\|u - \tilde{u}\right\|_{0,\Omega_{BL}} \le \left\|u- I_p u\right\|_{0,\Omega_{BL}} + \left\|\chi_2\right\|_{0,\Omega_{BL}} \left\{ \Vert{ u^{BL} \Vert_{L^{\infty}(\partial {\Omega_{REG}})} } + \left\|\pi^2_p u^{SR} - u^{SR}\right\|_{L^{\infty}(\Omega_{BL})}\right\}. \end{gathered}$$ Eq. [\[u-ut2D:bl\]](#u-ut2D:bl){reference-type="eqref" reference="u-ut2D:bl"} follows from the same equations. #### ${\Omega_{REG}}$: We have $w=w^{BL} + w^{SR}$. Therefore, $$\begin{gathered} \left\|w - \tilde{w}\right\|_{0,{\Omega_{REG}}} \le % \norm{u_\mathrm{bl}}_{\Oreg} + \norm{u_\mathrm{reg} - \tilde{u}}_{\Oreg} = \left\|w^{BL}\right\|_{0,{\Omega_{REG}}} + \left\|w^{SR} - \pi^1_p w^{SR} \right\|_{0,{\Omega_{REG}}} . \end{gathered}$$ Lemma [Lemma 13](#BLsize){reference-type="ref" reference="BLsize"} and [\[pi_p2\]](#pi_p2){reference-type="eqref" reference="pi_p2"} then give [\[w-wt2D:reg\]](#w-wt2D:reg){reference-type="eqref" reference="w-wt2D:reg"}. Eq. [\[u-ut2D:reg\]](#u-ut2D:reg){reference-type="eqref" reference="u-ut2D:reg"} follows in the same fashion. ◻ We now have the necessary tools for proving the following. **Theorem 18**. *Let $(u,w)$ be the solution of [\[mixed\]](#mixed){reference-type="eqref" reference="mixed"} and let $(u_{p,\Delta},w_{p,\Delta})$ be the solution of [\[discrete2\]](#discrete2){reference-type="eqref" reference="discrete2"}, based on the Spectral Boundary Layer mesh of Definition [Definition 14](#SBL-2D){reference-type="ref" reference="SBL-2D"}. Then, there exists a positive constant $\beta$, independent of $\varepsilon$, such that $$\begin{gathered} \left|\left|\left|(u - u_{p,\Delta}, w - w_{p,\Delta})\right|\right|\right|_{B} \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$* *Proof.* Let $\tilde{w}, \tilde{u}$ be defined by [\[wt\]](#wt){reference-type="eqref" reference="wt"} and [\[ut\]](#ut){reference-type="eqref" reference="ut"}, respectively. We have from [\[balanced\]](#balanced){reference-type="eqref" reference="balanced"} and the triangle inequality, $$\begin{aligned} \left|\left|\left|(u-u_{p,\Delta},w-w_{p,\Delta})\right|\right|\right| _{B} & \leq \left|\left|\left|(u-\tilde{u},w-\tilde{w})\right|\right|\right| _{B} + \left|\left|\left|(\tilde{u}-u_{p,\Delta},\tilde{w}-w_{p,\Delta})\right|\right|\right| _{B} \\ & \lesssim \varepsilon^{-1/2} \left\|w - \tilde{w}\right\|_0 + \left\|u - \tilde{u}\right\|_1 + \varepsilon^{-1/2} \left\|\tilde{w}-w_{p,\Delta}\right\|_0 + \left\|\tilde{u} - u_{p,\Delta}\right\|_1 . \end{aligned}$$ We only need to treat the terms $\varepsilon^{-1/2} \left\|\tilde{w}-w_{p,\Delta}\right\|_0$ and $\left\|\tilde{u}-u_{p,\Delta}\right\|_1$ since the rest can be handled by Theorem [Theorem 16](#theorem_main){reference-type="ref" reference="theorem_main"} and Lemma [Lemma 17](#lem:u-ut2D){reference-type="ref" reference="lem:u-ut2D"}. Let $\eta \coloneqq w_{p,\Delta} - \tilde{w}$ and $\xi \coloneqq u_{p,\Delta} - \tilde{u}$. We have by Galerkin orthogonality, $$\begin{gathered} \label{GO} \left|\left|\left|(\eta, \xi)\right|\right|\right| = \left\langle w - \tilde{w} , \eta \right\rangle + b \left\langle \nabla (u - \tilde{u}) , \nabla \xi \right\rangle + c \left\langle u - \tilde{u} , \xi \right\rangle. \end{gathered}$$ We will consider the two regions separately. #### $\Omega_{BL}$: We first note that $$\begin{gathered} % \left\|\chi_2\right\|_{0,\Omega_{\mathrm{BL}}} \left\{\left\|w^{BL}\right\|_{L_\infty(\partial\Omega_{REG})} + \left\|\pi^1_p w^{SR}-w^{SR}\right\|_{L_\infty(\partial\Omega_{REG})} \right\} \lesssim \varepsilon^{1/ 2} \mathrm{e}^{-\kappa p}. \end{gathered}$$ Then, we combine the above with [\[BLregion_bound\]](#BLregion_bound){reference-type="eqref" reference="BLregion_bound"}, and use the Cauchy-Schwarz inequality, along with the defintion of $\hat{w}$, to obtain $$\begin{gathered} \left\langle w-\hat{w} , \eta \right\rangle_{\Omega_{BL}} \lesssim \varepsilon^{1/ 2} \mathrm{e}^{-\beta p} \left\|\eta\right\|_{\Omega_{BL}}. \end{gathered}$$ Similarly, we first note that $$\begin{gathered} \left|\chi_2'\right|_{0,\Omega_{\mathrm{BL}}} \left\{\left\|u^{BL}\right\|_{L_\infty(\partial\Omega_{REG}} + \left\|\pi^2_p u^{SR}-u^{SR}\right\|_{L_\infty(\partial\Omega_{REG}} \right\} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\kappa p}, \end{gathered}$$ hence, as above, we have $$\begin{gathered} b\left\langle \nabla (u - \tilde{u}) , \nabla \xi \right\rangle_{\Omega_{BL}} +c\left\langle u - \tilde{u} , \xi \right\rangle_{\Omega_{BL}} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p} \left\|\xi\right\|_{1,\Omega_{BL}}. \end{gathered}$$ #### $\Omega_{REG}$: We have by the definition of $\hat{w}$, $$\begin{aligned} \left\langle w-\hat{w} , \eta \right\rangle_{\Omega_{REG}} = \left\langle w_{BL} + w^{SR} - \pi_p w^{SR} , \eta \right\rangle_{{\Omega_{REG}}} = \left\langle w^{BL} , \eta \right\rangle_{\Omega_{REG}} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\kappa p} \left\|\eta\right\|_{0, {\Omega_{REG}}}. \end{aligned}$$ Similarly, $$\begin{aligned} & b \left\langle \nabla (u - \tilde{u}) , \nabla \xi \right\rangle_{{\Omega_{REG}}} + c \left\langle u - \tilde{u} , \xi \right\rangle_{{\Omega_{REG}}} \\ & \qquad = b \left\langle \nabla u^{BL} , \nabla \xi \right\rangle_{{\Omega_{REG}}} + c \left\langle u^{BL} , \xi \right\rangle_{{\Omega_{REG}}} \lesssim \left\|u^{BL}\right\|_{1,{\Omega_{BL}}} \left\|\xi\right\|_{1,{\Omega_{BL}}} \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p} \left\|\xi\right\|_{1,{\Omega_{BL}}}\,. \end{aligned}$$ Combining the above with [\[GO\]](#GO){reference-type="eqref" reference="GO"}, we see that $$\begin{gathered} \left|\left|\left|(\eta, \xi)\right|\right|\right|^2 \lesssim \varepsilon^{1/2} \mathrm{e}^{-\beta p} \left|\left|\left|(\eta, \xi)\right|\right|\right|\,, \end{gathered}$$ from which we have $$\begin{gathered} \varepsilon^{-1/2} \left\|w_{p,\Delta} - \tilde{w}\right\|_{0,\Omega} + \left\|u_{p,\Delta} - \tilde{u}\right\|_{1,\Omega} \lesssim \mathrm{e}^{-\beta p}. \end{gathered}$$ This completes the proof. ◻ # Conclusions {#concl} We presented balanced norm estimates for singularly perturbed fourth order problems in one-dimension as well as in smooth (analytic) domains in $\mathbb{R}^2$. As a corollary, we also got maximum norm estimates in the one-dimensional case. The methodology is not restricted to conforming methods, and in fact we are in the process of studying balanced norm estimates for non-conforming $rp$-version methods. Finally, we mention that this contribution proves the conjectures made in [@PC], [@CFLX] based on numerical evidence. 99 L. Beirão da Veiga, A. Buffa, J. Rivas and G. Sangalli, *Some estimates for h-p-k-refinement in isogeometric analysis*, Numer. Math. **118** (2011), 271--305. P. Constantinou, *The hp Finite Element Method for fourth order singularly perturbed problems*, Doctoral Dissertation, Department of Mathematics and Statistics, University of Cyprus, 2019. P. Constantinou, S. Franz, L. Ludwig and C. Xenophontos, *A mixed hp finite element method for a 4th order singularly perturbed problem*, Num. Meth. PDEs, **35** (2019) 114--127. P. Constantinou, C. Varnava and C. Xenophontos, *An hp finite element method for fourth order singularly perturbed problems*, Numer. Alg., **73** (2016) 567--590. P. Constantinou and C. Xenophontos, *An hp finite element method for a 4$^{th}$ order singularly perturbed boundary value problem in two dimensions*, Comp. Math. Appl., **72** (2017) ,1565--1575. D. A. Di Pietro and A. Ern, *Mathematical Aspects of Discontinuous Galerkin Methods*, Mathematique et Applications 69, Springer-Verlag, 2012. S. Franz and H.-G. Roos, *Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems*, Comp. Math. Appl., **72** (2016) 233--247. S. Franz and H.-G. Roos, *On robust error estimation for singularly perturbed fourth-order problems*, In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, Lecture Notes in Computational Science and Engineering **120** (2017), 77--85, Springer, Berlin Heidelberg, S. Franz and H.-G. Roos, *Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems*, Model. Anal. Inform. Sist., **23** (2016), 364--369. R. Lin and M. Stynes, *A balanced finite element method for singularly perturbed reaction--diffusion problems*, SIAM J. Numer. Anal., **50** (2012) 2729--2743. T. Linß, *Layer-adapted meshes for reaction-convection-diffusion problems*, Lecture Notes in Mathematics 1985, Springer-Verlag, 2010. J. M. Melenk, *hp-Finite Element Methods for Singular Perturbations*, Vol. 1796 of Springer Lecture Notes in Mathematics, Springer Verlag, 2002. J. M. Melenk and C. Xenophontos, *Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion equations*, Calcolo, **53** (2016) 105--132 (2016). J. M. Melenk and C. Xenophontos, *Analytic regularity results for a fourth singularly perturbed problem in smooth domains*, in preparation (2023). J. M. Melenk, C. Xenophontos and L. Oberbroeckling, *Robust exponential convergence of hp-FEM for singularly perturbed systems of reaction-diffusion equations with multiple scales*, IMA J. Num. Anal., **33** (2013) 609--628. R. E. O'Malley, *Introduction to Singular Perturbations*, Academic Press, New York, 1974. P. Panaseti, A. Zouvani, N. Madden and C. Xenophontos, *A $C^{1}$--conforming $hp$ finite element method for fourth order singularly perturbed boundary value problems*, Appl. Num. Math., **104** (2016) 81--97. C. Schwab, *p- and hp-Finite Element Methods*, Oxford University Press, 1998. # Auxiliary results **Lemma 19** (Inverse inequality). *For any polynomial $q\in\mathcal{P}_p$, $p\in\mathbb{N}$, and any domain $D$ there holds $$\begin{gathered} \left|q\right|_{k,D} \le \left(\frac{p!}{(p-k)!}\right)^2 \left|D\right|^{-k} \left\|q\right\|_{0,D}\,, k=0,1,\dots, p. \end{gathered}$$* *Proof.* Theorem 4.76 in [@schwab] gives the cases $k=0$ and $1$. Iterating on that result, we obtain the desired inequality. ◻ # Proof of [\[interp_uS\]](#interp_uS){reference-type="eqref" reference="interp_uS"} and [\[interp_lay\]](#interp_lay){reference-type="eqref" reference="interp_lay"} We begin with the following results: **Lemma 20**. *Let $v \in C^{\infty}(I_{ST})$, $I_{ST}=(-1,1)$ satisfy $$\begin{gathered} \left\|v^{(n)}\right\|_{{\infty},I_{ST}} \le C_v (\gamma_v h)^n K^{n-1}, \ \forall \; n \in \mathbb{N},\end{gathered}$$ for some constants $C_v, \gamma_v>0, K\ge 1$, and for $h \in (0,1]$. Then there exists an approximation $\mathcal{I}_p v \in \mathcal{P}_p$ with $\mathcal{I}^{(k)}_p v(\pm 1) = v^{(k)}(\pm 1)$, $k=0,1$ such that under the condition $$\begin{gathered} \frac{h K}{p} \leq \eta,\end{gathered}$$ for some $\eta > 0$, there holds $$\begin{gathered} \left\|v - \mathcal{I}_p v\right\|_{2,I_{ST}} \lesssim C_v K^{1/2} h \mathrm{e}^{-\beta p}.\end{gathered}$$* *Proof.* There exists (see, e.g. [@PZMX]) $\mathcal{I}_p v \in \mathcal{P}_p$ with $\left(\mathcal{I}_p v\right)^{(k)} (\pm 1) = v^{(k)}(\pm 1)$, $k=0,1$ and $$\begin{gathered} \left\|v - \mathcal{I}_p v\right\|^2_{2,I_{ST}} \leq \frac{(p-s)!}{(p+s)!} \left\|v^{(s+1)}\right\|^2_{0,I_{ST}}, \ \ \forall \ s \in (0,p). \end{gathered}$$ Choose $s=\lambda p$, with $\lambda \in (0,1]$ to be selected shortly. Then we calculate $$\begin{gathered} \frac{(p-\lambda p)!}{(p+\lambda p)!} \left\|v^{(\lambda p+1)}\right\|^2_{0,I_{ST}} \le \left[ \frac{(1-\lambda)^{1-\lambda}}{ (1+\lambda)^{1+\lambda}} \right]^p p^{-2\lambda p} \mathrm{e}^{2\lambda p + 1} C^2_v (h \gamma_v)^{2 (\lambda p+1)} K^{2\lambda p+1} \le \mathrm{e}C_v^2 h^2 \gamma_v^2 K \left( \frac{\mathrm{e}\gamma_v h K}{p} \right)^{2\lambda p} q^p , \end{gathered}$$ with $q = (1-\lambda)^{1-\lambda} / (1+\lambda)^{1+\lambda} \in (0,1)$. Using the hypothesis $hK/p \leq \eta$ and selecting $\lambda \in (0,1]$, $\eta>0$ such that $\left( \gamma_v \mathrm{e}\eta \right)^{2\lambda p}\leq 1$, gives $$\begin{gathered} \frac{(p-\lambda p)!}{(p+\lambda p)!} \left\|v^{(\lambda p+1)}\right\|^2_{0,I_{ST}} \lesssim C_v^2 h^2 K \gamma_v q^p, \end{gathered}$$ from which the desired result follows. ◻ By means of affine transformations the interpolation operator $\mathcal{I}_p$ of Lemma [Lemma 20](#lem_aux1){reference-type="ref" reference="lem_aux1"} can be applied to all subintervals of our mesh $\Delta$ to give a global piecewise polynomial interpolant. To this end introduce the affine transformations $$\begin{gathered} Q_j \colon I_{ST} \to I_j \colon t \mapsto Q_{j}(t) \coloneqq\frac{1-t}{2} x_{j-1}+\frac{1+t}{2} x_j, \ \ j=1,\dots,N,\end{gathered}$$ and define $$\begin{gathered} \label{interp:pw-poly} \left(\mathcal{I}_{p,\Delta}v\right)\big|_{I_j} \coloneqq \mathcal{I}_p\left(v\circ Q_j^{-1}\right), \ \ j=1,\dots,N,\end{gathered}$$ As there is no confusion, we drop the $\Delta$ from the notation. **Corollary 21**. *Let $j\in\{1,\dots,N\}$ and assume $v \in C^{\infty}(\bar{I}_j)$ satisfies $$\begin{gathered} \left\|v^{(n)}\right\|_{{\infty},I} \le C_v (h_j \gamma_v)^n K^{n-1}, \ \ \forall \ n \in \mathbb{N}, \end{gathered}$$ for some constants $C_v, \gamma_v>0$, $K\ge 1$. Then there exist constants $\eta, \beta > 0$ depending only on $\gamma_v$ such that under the condition $$\begin{gathered} \frac{h_j K}{p} \leq \eta, \end{gathered}$$ the polynomial approximation $\mathcal{I}_p v$ defined in [\[interp:pw-poly\]](#interp:pw-poly){reference-type="eqref" reference="interp:pw-poly"} satisfies $$\begin{gathered} \left\|\left(v-\mathcal{I}_p v \right)^{(k)}\right\|_{0,I_j} \lesssim C_v K^{1/2} h_j^{3/2-k} \mathrm{e}^{-\beta p}, \ \ k=0,1,2. \end{gathered}$$* *Proof.* Set $\hat{v} = v \circ Q_j$. Then $$\begin{gathered} \left\|\hat{v}^{(n)}\right\|_{0,I_{ST}} \le C_v (h_j/2)^n \gamma^n_v K^{n-1}, \end{gathered}$$ and by Lemma [Lemma 20](#lem_aux1){reference-type="ref" reference="lem_aux1"} we have $$\begin{gathered} \left\|\hat{v} - \mathcal{I}_p \hat{v}\right\|_{2,I_{ST}} \lesssim C_v K^{1/2} h_j \mathrm{e}^{-\beta p}. \end{gathered}$$ Transforming back to $I_j$ gives the result. ◻ For our spectral boundary layer mesh, the previous results give $$\begin{gathered} \left\|\left( u - \mathcal{I}_p u \right)^{(k)}\right\|_{0,\Omega_{BL}} \lesssim C_u \varepsilon^{-1/2} h_j^{3/2-k} \mathrm{e}^{-\beta p}, \ \ k=0,1,2,\end{gathered}$$ or equivalently, since $h_j = \kappa p \varepsilon$, $$\begin{gathered} \label{res1p} (\kappa p \varepsilon)^{-1} \left\|u-\mathcal{I}_p u\right\|_{0,\Omega_{BL}} + \left|u-\mathcal{I}_p u\right|_{1,\Omega_{BL}} + \kappa p \varepsilon\left|u-\mathcal{I}_p u\right|_{2,\Omega_{BL}} \lesssim C_u \left(\kappa p \right)^{1/2} \mathrm{e}^{-\beta p}. \end{gathered}$$ Now, since $\left(u- \mathcal{I}_p u \right)^{(k)}(0) = 0$, $k=0,1$, we may write $$\begin{gathered} \left|\left(u- \mathcal{I}_p u\right)^{(k)}(x)\right| = \left|\int_{0}^{x}\left(u- \mathcal{I}_p u\right)^{(k+1)}(t)\;\mathrm{d}t\right|, \ \ x \in \Omega_{BL}.\end{gathered}$$ Then, by the Cauchy-Schwarz inequality and [\[res1p\]](#res1p){reference-type="eqref" reference="res1p"} we get for $k=0,1$, $$\begin{gathered} \left|\left(u- \mathcal{I}_p u\right)^{(k)}(x)\right| \le (\kappa p \varepsilon)^{1/2} \left|\left(u- \mathcal{I}_p u\right)^{(k)}\right|_{1,\Omega_{BL}} \lesssim \begin{cases} C_u \varepsilon^{-1/2} (\kappa p \varepsilon)^{2} \mathrm{e}^{-\beta p}\;, & k=0, \\ C_u \varepsilon^{-1/2} \kappa p \varepsilon\mathrm{e}^{-\beta p} \;, & k=1 . \end{cases}\end{gathered}$$ Thus $$\begin{gathered} (\kappa p \varepsilon)^{-3/2} \left\|u-\mathcal{I}_p u\right\|_{\infty,\Omega_{BL}} + (\kappa p \varepsilon)^{-1/2} \left\|\left(u- \mathcal{I}_p u \right)'\right\|_{\infty,\Omega_{BL}} \lesssim C_u (\kappa p )^{1/2} \mathrm{e}^{-\beta p}. \end{gathered}$$ [^1]: FernUniversität in Hagen, Fakultät für Mathematik und Informatik, Universitätsstraße 1, 58084, Hagen, email: `torsten.linss@fernuni-hagen.de` [^2]: University of Cyprus, Department of Mathematics & Statistics, PO BOX 20537, 1678, Nicosia, email: `xenophontos.christos@ucy.ac.cy` [^3]: Henceforth, the symbols that appeared in the previous section will represent two-dimensional analogs. [^4]: The fact that $w \in H^2$ follows from the smoothness of $\partial\Omega$.

arxiv_math

--- abstract: | We derive stochastic compressible Euler Equation from a Hamiltonian microscopic dynamics. We consider systems of interacting particles with Hölder noise and potential whose range is large in comparison with the typical distance between neighbouring particles. It is shown that the empirical measures associated to the position and velocity of the system converge to the solutions of compressible Euler equations driven by additive Hölder path(noise), in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. Furthermore, explicit rates for the convergence are obtained in Besov and Triebel-Lizorkin spaces. Our proof is based on the Itô-Wentzell-Kunita formula for Young integral. author: - | Christian Olivera $^{1,}$ Juan D. Londoño $^{2}$\ Jesus M. Correa $^{3}$\ $^{1}$ Departamento de Matemática, Universidade Estadual de Campinas,\ 13.081-970-Campinas-SP-Brazil.\ colivera\@ime.unicamp.br\ $^{2}$ j209372\@dac.unicamp.br $^{3}$ j209398\@dac.unicamp.br date: - - title: " **From Hamiltonian Systems to Compressible Euler Equation driven by aditive Holder noise**" --- ***Key words and phrases:** Young Differential Equation, Compressible Euler Equation, Particle systems, Besov and Triebel-Lizorkin spaces, Itô-Kunita-Wentzell formula.* 60H15, 35R60, 60H30, 35Q31 . Data sharing not applicable to this article as no datasets were generated or analysed during the current study # Introduction This paper is aimed to derive macroscopic continuous models characterizing the asymptotic behavior of interacting particle systems as the number of particles goes to infinity. More precisely, we analyse the evolution of an indistinguishable $N-$ point particle system given by $$\begin{aligned} \label{xk} d^{2}X_{t}^{k,N}=-\frac{1}{N}\sum_{l=1}^{N}\nabla\phi_{N}\left(X_{t}^{k,N}-X_{t}^{l,N}\right)dt + \sigma\left(t,X_{t}^{k,N} \right) dY_{t} \end{aligned}$$ here $X^{i,N}$ and $V^{i,N}=\frac{d X^{i,N}}{dt}$ denote the position and velocity of $i$-particle at time t,respectively, $\{(Y_{t}^{q})_{t\in[0,T]}, \; q=1,\cdots,d\}$ is a $\mathbb{R}^d$-valued $\alpha$-Hölder path with $\alpha > \frac{1}{2}$. Follows [@Oes] the interaction potential $\phi_{N}$ is obtained from a function $\phi_{1}$ by the scaling $$\label{molli} \phi_{N}(x)=N^{\beta}\phi_{1}(N^{\beta/d}x),\ \beta\in(0,1),$$ where $\phi_{1}$ is symmetric and sufficiently smooth. The parameter $\beta\in (0,1)$ describes how $\phi_{1}$ is rescaled for the total number $N$ of individuals and expresses the so-called moderate interaction among the individuals; see Oelschläger [@Oes2]. On the other hand, $\beta = 0$ expresses an interaction of mean field type, whereas $\beta = 1$ generates the so-called nearest-neighbour interaction. We analyze the asymptotic behavior among many particles moderately interacting with each other. Our aim is the study of the asymptotic as $N\rightarrow\infty$ of the time evolution of the whole system of all particles. Therefore, we investigate the empirical processes $$\begin{aligned} \label{empp} S_{t}^{N}:=\frac{1}{N}\sum_{k=1}^{N}\delta_{X_{t}^{k,N}},\end{aligned}$$ $$\begin{aligned} \label{empp} V_{t}^{N}:=\frac{1}{N}\sum_{k=1}^{N}V_{t}^{k,N}\delta_{X_{t}^{k,N}}\qquad k=1,\cdots, N\end{aligned}$$ where $\delta_{a}$, denotes the Dirac measure at $a$. The measures $S_{t}^{N}$ and $V_{t}^{N}$ determine the distribution of the positions and the velocities in the $Nth$ system. We shall show that $S_{t}^{N}$ and $V_{t}^{N}$ converge as $N\rightarrow\infty$ to solutions of the continuity equation and the Euler equation, respectively driven by $\alpha$-Hölder path. More precisely we show the convergence to the system $$\begin{aligned} \label{eq1} \left\{ \begin{array}{lc} d\varrho =-\operatorname{div}_{x}(\varrho\upsilon)dt \\[0.3cm] d(\varrho\upsilon_{q})=-\left(\nabla^{q}p+\operatorname{div}_{x}(\varrho\upsilon_{q}\upsilon)\right)dt + \varrho \sigma_{q} dY_{t}^{q}\qquad q=1,\cdots, d \end{array} \right.\end{aligned}$$ or equivalently $$\begin{aligned} \label{eq2} \left\{ \begin{array}{lc} d\varrho =-\operatorname{div}_{x}(\varrho\upsilon)dt \\[0.3cm] d(\upsilon_{q})=-\left(\frac{1}{\varrho} \nabla^{q}p+\upsilon \nabla \upsilon_{q} \right) dt+ \sigma_{q} dY_{t}^{q}\qquad q=1,\cdots, d \end{array} \right.\end{aligned}$$ where the pressure is $p=\frac{1}{2}\varrho^{2}.$ A source of much research in mathematical physics is the problem of rigorously deriving the compressible Euler equation in dimensions $d\geq 2$. One of the main open problems in mechanics fluids is the derivation of the stochastic equations of fluids, the so-called stochastic Euler and Navier-Stokes equations, from the microscopic Hamiltonian dynamics. The main purpose of this work is to make the above formal derivation completely rigorous. More precisely, we will provide quantitative estimation between solutions and the empirical measures. The proof is based on Itô-Wentzell-Kunita formula for Young integral, see [@CatuognoCastrequini], and on the paper [@Oes] where the same equation without noise is considered. The noise in equation ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) has a very special form, compared to general abstract models of stochastic partial differential equations (SPDEs). Our aim is not an abstract generality We point out that in ([\[xk\]](#xk){reference-type="ref" reference="xk"}) we consider $\alpha$-Hölder path but we believe which can be easily extended for infinite noises. Concerning the problem of well-posedness of ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"})-([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) remains open, see [@Fei], [@Chae], [@Chae2] [@Lions] and [@Majda] on well-posedness results in the deterministic setting. #### Related Works. In [@Correa] we consider the case that the Hamiltonian system is perturbed by the Brownian motion, for related works in the deterministics setting see [@Carillo], [@Dil] and [@Franz]. For particle approximations for stochastic incompressible Euler-Navier stokes equations see [@Maurelli] and [@Kotelenez]. We also mention that [@Crisan] derive a class of rough geophysical fluid dynamics models as critical points of rough action functionals using the theory of controlled rough paths. Finally an excellent recent monograph [@Jabin] where some additional information can be found. #### Example of Hölder noise. Let $(B_{t}) _{t\in [0, T]}$ be a fractional Brownian motion with Hurst index $H\in (1/2, 1)$. Recall that $B$ is a centered Gaussian process with covariance $$\mathbf{E} B_{t} B_{s}= \frac{1}{2} \left( t ^{2H}+ s^{2H}-\vert t-s\vert ^{2H} \right) \mbox{ for every }s,t\in [0,T].$$ This process is self-similar of index $H$, and their paths are Hölder continuous of order $\alpha \in (0,H)$. # Definitions, preliminaries and hypothesis. ## Young Integral. Fix a time interval $[0,T]$. Let $U$, $V$ and $W$ be Banach spaces. Given a path $\phi:[0,T]\to V$ and $s,t\in[0,T]$ we write $\phi_{st}=\phi_{t}-\phi_{s}$. **Definition 1**. *Let $0<\alpha\leq 1$. $\mathcal{C}^{\alpha}([0,T];V)$ is the space of functions on $[0,T]$ taking values in $V$ such that the following $\alpha$-Hölder seminorm $$\|\phi\|_{\alpha}:=\sup_{0\leq s<t\leq T}\frac{|\phi_{st}|}{|t-s|^{\alpha}}.$$ is finite.* The space $\mathcal{C}^{\alpha}([0,T];V)$ is a Banach space with the norm $$|\phi|_{\alpha}=\|\phi\|_{\alpha}+\|\phi\|_{\infty},$$ where, as usual, $\|\phi\|_{\infty}=\sup_{t\in[0,T]}|\phi(t)|$.\ We will denote by $L(U,V)$ the space of continuous linear mappings from $U$ to $V$, and $$\Delta_{T}:=\left\{(s,t)\in[0,T]^{2}:0\leq s\leq t\leq T\right\}.$$ In this section we define the Young's integral $\int XdY$ when $Y\in C^{\alpha}([0,T];V)$ and $X\in C^{\beta}([0,T];L(V,W))$ with $\alpha+\beta>1$. We refer the reader to the paper [@Young] by L.C. Young.\ The map $$(X,Y)\mapsto\left( t\mapsto\int_{0}^{t}X_{s}dY_{s}\right)$$ is bilinear and continuous from $C^{\beta}([0,T];L(V,W))\times C^{\alpha}([0,T];V)$ to $C^{\alpha}([0,T];W)$. Finally, the cornerstone of this theory is the following Young-Loeve estimative, see Proposition 3 in [@Gubinelli]. **Theorem 2**. *Let $Y\in C^{\alpha}([0,T];V)$ and $X\in C^{\beta}([0,T];L(V,W))$ for some $\alpha,\beta\in(0,1]$ with $\alpha+\beta>1$. Then the limit $$\int_{0}^{t}X_{r}dY_{r}:=\lim_{|\pi|\to 0}\sum_{[r,s]\in\pi}X_{r}Y_{rs}$$ exists for every $t\in[0,T]$, where the limit is taken over any $\pi\in\mathcal{P}([0,t])$, and $\mathcal{P}([0,t])$ the set of all partitions $\pi$ of the interval $[0,t]$. This limit is called the Young integral of $X$ against $Y$, which moreover holds the following estimative $$\label{Loeve} \left|\int_{s}^{t}X_{r}dY_{r}-X_{s}Y_{st}\right|_{W}\leq C_{\alpha+\beta}\| Y\|_{\alpha}\| X\|_{\beta}|t-s|^{\alpha+\beta}$$ for all $(s,t)\in\Delta_{T}.$* We recall the independence of the Young integral with respect to the choice of the partitioning points. Let $r\in[s,\theta]$ denote an arbitrary point in the interval $[s,\theta]$. The Young integral of $X$ against $Y$ is equal to the limit $$\int_{0}^{t}X_{s}dY_{s}:=\lim_{|\pi|\to 0}\sum_{[s,\theta]\in\pi}X_{r}Y_{s\theta}$$ for any $t\in[0,T]$.\ As the next lemma shows, Young integration satisfies the classical integration by parts formula. **Lemma 3**. *Let $X\in C^{\alpha}$ and $Y\in C^{\beta}$ for some $\alpha, \beta\in(0,1]$ with $\alpha+\beta>1$. Then $$\label{produ} X_{T}Y_{T}=X_{0}Y_{0}+\int_{0}^{T}X_{u}dY_{u}+\int_{0}^{T}Y_{u}dX_{u}.$$* **Lemma 4**. *Let $X\in C^{\alpha}([0,T],V)$ and $f\in C^{1+\gamma}(V,W)$ for some $\alpha, \gamma\in(0,1]$, such that $\alpha(1+\gamma)>1$. Then $\int_{0}^{T}Df(X_{r})dX_{r}$ is well-defined Young integral, and $$\label{Ito} f(X_{T})=f(X_{0})+\int_{0}^{T}Df(X_{r})dX_{r}.$$* We finish this section with the generalized Itô-Wentzell formula for the Young integral, see Theorem 3.1 of [@CatuognoCastrequini]. **Theorem 5**. *Let $\alpha\in(\frac{1}{2},1]$, $Y\in C^{\alpha}([0,T];V)$ and $h:[0,T]\times U\to L(V,W)$ continuous and differentiable in $U$ such that* 1. *$(t,x)\to D_{x}h_{t}(x)$ is continuous,* 2. *$h\in C(U,C^{\beta}([0,T];L(V,W)))$ for some $\beta\in(\frac{1}{2},1]$.* *Let $$g_{t}(x)=g_{0}(x)+\int_{0}^{t}h_{s}(x)dY_{s}.$$ We assume that $g:[0,T]\times U\to W$ is twice differentiable in $U$and the functions $(t,x)\mapsto D_{x}^{2}g_{t}(x)$ are continuous. Then for any $X\in C^{\alpha}([0,T],U)$, $$\label{Ito-W} g_{t}(X_{t})=g_{0}(X_{0})+\int_{0}^{t}h_{s}(X_{s})dY_{s}+\int_{0}^{t}D_{x}g_{s}(X_{s})dX_{s},$$ where the integral $\int_{0}^{t}D_{x}g_{s}(X_{s})dX_{s}$ is understood in the Riemann-Stieltjes sense.\ If $D_{x}g\in C(U,C^{\gamma}([0,T],L(V,W)))$ for some $\gamma\in(\frac{1}{2},1]$ we have that the integral $\int_{0}^{t}D_{x}g_{s}(X_{s})dX_{s}$ is a Young integral, and $t\mapsto g_{t}(X_{t})\in C^{\alpha}([0,T],W)$.* ## Space of functions Let us first define a dyadic partition of unity, as follows: we consider two $C_{0}^{\infty}(\mathbb{R}^{d})$-functions $\chi$ and $\varphi$ which take values in $[0,1]$ and satisfy the following: there exists $\lambda \in (1,\sqrt 2)$ such that $$\mathrm{Supp}\; \chi=\left\{ |\xi |\leq\lambda \right\} \qquad \text{and} \qquad \mathrm{Supp} \ \varphi=\left\{ \frac{1}{\lambda} \leq|\xi|\leq\lambda \right\}.$$ Moreover, with the following notations, $$\varphi_{-1}(\xi):=\chi(\xi), \qquad \varphi_{i}(\xi):=\varphi(2^{-i}\xi), \ \text{for any } i\geq 0,$$ the sequence $\{\varphi_i\}$ satisfies $$\begin{aligned} & \mathrm{Supp} \ \varphi_{i}\cap \mathrm{Supp} \ \varphi_{j}=\emptyset \quad \text{ if } \ |i-j|>1, \\ & \sum_{i\geq-1} \varphi_{i}(\xi)=1, \quad \text{ for any } x \in\mathbb{R}^{d}.\end{aligned}$$ Take a fixed dyadic partition of unity $\{ \varphi_{i} \}$ with its inverse Fourier transforms $\{ \check{\varphi}_{i} \}$. For $u\in \mathcal{S}'(\mathbb{R}^{d})$ , the nonhomogeneous dyadic blocks are defined as $$\Delta_{i}\equiv 0, \quad \text{ if } i< -1, \qquad \text{and} \qquad \Delta_{i}u=\check{\varphi}_{i}\ast u, \quad \text{ if } i\geq -1.$$ The partial sum of dyadic blocks is defined as a nonhomogeneous low frequency cut-off operator: $$S_{j}:=\sum_{i\leq j-1} \Delta_{i}.$$ Having the theses smooth resolution of unity we are able to introduce the Triebel-Lizorkin and Besov Space. **Definition 6**. *Let $s\in \mathbb{R}$ and $1< p<\infty$.* 1. *If $1< q< \infty$, then* *$$F_{p,q}^{s}= \left\{ f\in \mathcal{S}^{\prime}(\mathbb{R}^{d}) : \ \big\| \| (2^{js} \Delta_{j}f)_{j \in \mathbb{Z} } \|_{l^q(\mathbb{Z})} \big\|_{\mathit{L}^{p}(\mathbb{R}^{d})}< \infty \right\}.$$* 2. *If $0< q\leq \infty$, then* *$$B_{p,q}^{s}= \left\{ f\in \mathcal{S}^{\prime}(\mathbb{R}^{d}) : \ \big\| (2^{js}\| \Delta_{j}u \|_{L^p} )_{j \in \mathbb{Z} } \big\|_{\mathit{l}^{q}(\mathbb{Z})}< \infty \right\}.$$* The spaces $F_{p,q}^{s}$ and $B_{p,q}^{s}$are independent of the dyadic partition chosen, see [@Triebel]. We denoted $E_{p,q}^{s}=F_{p,q}^{s}$ or $E_{p,q}^{s}=B_{p,q}^{s}$ with $q\geq 2$. We recall the Sobolev embedding $$\label{Sobolev} E_{p,q}^{s}\subset C_{b}^{s-\frac{d}{p}}$$ if $s> \frac{d}{p}$, see p. 203 of [@Triebel]. We denoted $C_{b}(\mathbb{R}^{d}; \mathbb{R}^{d})$ is the space of bounded continuous $\mathbb{R}^{d}$-valued functions on $\mathbb{R}^{d}$. Here we used the brackets as an abbreviation for an integral over $\mathbb{R}^d$. ## Definition of solution We say that $X_{t}^{i,N}$ solve the system ([\[xk\]](#xk){reference-type="ref" reference="xk"}) if it is solution of the system $$\begin{aligned} \label{VN} \left\{ \begin{array}{lc} d X_{t}^{k,N}=V_{t}^{k,N}dt \\[0.3cm] d V_{t}^{k,N}=-\nabla\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt+ \sigma (t, X_{t}^{k,N}) d Y_{t}\quad k=1,\cdots, N \end{array} \right.\end{aligned}$$ where the last integral is taken in the Young sense. **Definition 7**. *Let $\varrho_{0}\in E_{p,q}^{s}\cap L^{1}(\mathbb{R}^{d})$ with $\varrho_{0}>0$, and $\upsilon_{0}\in E_{p,q}^{s}$ with $s>\frac{d}{p}+2$. Then $(\varrho, \upsilon)$ is called a solution of ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}) if the following conditions are satisfied* 1. *$\left(\varrho, \upsilon\right)\in C([0,T], E_{p,q}^{s} ) \times C^{\alpha}([0,T], E_{p,q}^{s} )$* 2. *For all $x\in \mathbb{R}^{d}$, $\upsilon(x,.) \in C^{\alpha}([0,T])$.* 3. *holds* *$$\begin{aligned} &\varrho\left(t \right)=\varrho_{0}-\int_{0}^{t} \operatorname{div}_{x}(\varrho\upsilon)\mathrm{d} s, \\ &\upsilon_{q}\left(t\right)=\upsilon_{q,0}-\int_{0}^{t} \left(\frac{1}{\varrho} \nabla^{q}p+\upsilon \nabla \upsilon_{q} \right)\mathrm{d} s+\int_{0}^{t} \sigma_{q} dY_{t}^{q}\quad q=1,\cdots, d \end{aligned}$$ for all $t \in[0, T]$.* ## Technical hypothesis Next, we suppose that $\phi_{1}$ can be written as a convolution product $$\phi_{1}=\phi_{1}^{r}\ast\phi_{1}^{r}$$ where $\phi_{1}^{r}\in C_{b}^{2}(\mathbb{R}^{d})$ and it is symmetric probability density. **Definition 8**. *For all $q\in\{1,\cdots,d\}$ we define the function $$\begin{aligned} U_{1;\alpha}^{q}:\mathbb{R}^{d}&\longrightarrow&\mathbb{R}\\ x&\longmapsto&(-1)^{1+|\alpha|} \dfrac{x^{\alpha}}{\alpha!} \nabla^{q}\phi_{1}^{r}(x)\end{aligned}$$ where $0\leq|\alpha|\leq L+1$ with $L:=\left[\frac{d+2}{2}\right]$.* We assume that functions $\phi_{1}^{r}$ and $U_{1;\alpha}^{q}$ satisfy $$\begin{aligned} \label{c1} |\phi_{1}^{r}(x)|\leq\frac{C}{1+|x|^{d+2}}\qquad |x|\geq1,\end{aligned}$$ $$\begin{aligned} \label{cotawildeu} |\widetilde{U^{q}_{1;\alpha}}(\lambda)|\leq C|\widetilde{\phi_{1}^{r}}(\lambda)|,\qquad\text{para } 1\leq|\alpha|\leq L ,\end{aligned}$$ and $$\begin{aligned} \label{cotauj} |U^{q}_{1;\alpha}(x)|\leq C \left(\frac{1}{1+|x|^{d+1}}\right)^{1/2},\qquad\text{para } |\alpha|=L+1\ .\end{aligned}$$ We also assume that $\sigma\in C^{\alpha}([0,T], C_{b}^{1}( \mathbb{R}^{d}, \mathbb{R}^{d}))$. # Result First we present a simple lemma. **Lemma 9**. *We assume [\[c1\]](#c1){reference-type="eqref" reference="c1"}, then there exist $C_{0}>0$ such that $$\begin{aligned} \label{cotafNbeta} \Vert f-(f\ast\phi_{N}^{r})\Vert_{\infty}&\leq&C_{0}N^{-\beta/d}\Vert\nabla f\Vert_{\infty}\qquad\text{for all}\,\,\,\,\, f\in C_{b}^{1}(\mathbb{R}^{d})\end{aligned}$$* *Proof.* See [@Correa] ◻ We define the energy $$\label{QN} Q_{t}^{N}:=\frac{1}{N}\sum_{k=1}^{N}|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)|^{2} +\Vert S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t)\Vert_{L^{2}(\mathbb{R}^{d})}^{2}.$$ **Theorem 10**. *Let $s\geq\frac{d}{p}+3$, $1< p,q< \infty$ $\alpha> \frac{1}{2}$, $\eta > \frac{d}{2} +1$. We assume [\[molli\]](#molli){reference-type="eqref" reference="molli"}, [\[c1\]](#c1){reference-type="eqref" reference="c1"}-[\[cotauj\]](#cotauj){reference-type="eqref" reference="cotauj"}. Then there exist $C_{T}>0$ such that* *$$Q_{t}^{N}\leq C_{T}( Q_{0}^{N}+N^{-\beta/d})\quad\text{for all } N\in\mathbb{N},$$* *$$\|S_{t}^{N}- \varrho_{t}\|_{E_{2,\hat{q}}^{-\eta}}^{2} \leq C_{T}( Q_{0}^{N}+N^{-\beta/d}) \quad\text{for all } N\in\mathbb{N},$$* *and* *$$\|V_{t}^{N}- (\varrho \upsilon)_{t}\|_{E_{2,\hat{q}}^{-\lambda}} \leq C_{T} ( Q_{0}^{N} +N^{-\beta/d}) \quad\text{for all } N\in\mathbb{N},$$* *with $\frac{1}{q} +\frac{1}{\hat{q}}=1$.* *Proof.* We recall that by Sobolev embedding $\varrho, \upsilon\in C([0,T], C_{b}^{3}(\mathbb{R}^{d}))$, we use this regularity during the proof. We first observe that $$\begin{aligned} Q_{t}^{N}&=&\dfrac{1}{N}\sum_{k=1}^{N}\vert V_{t}^{k,N}\vert^{2}-\dfrac{2}{N}\sum_{k=1}^{N} V_{t}^{k,N}\cdot\upsilon(X_{t}^{k,N},t)\\ &&+\dfrac{1}{N}\sum_{k=1}^{N}\vert\upsilon(X_{t}^{k,N},t)\vert^{2}+\Vert S_{t}^{N}\ast\phi_{N}^{r}\Vert_{L^{2}(\mathbb{R}^{d})}^{2}-2\langle S_{t}^{N}\ast\phi_{N}^{r},\varrho(.,t)\rangle\\ &&+\Vert\varrho(.,t)\Vert_{0}^{2}\\ &=&\dfrac{1}{N}\sum_{k=1}^{N}\vert V_{t}^{k,N}\vert^{2}-\dfrac{2}{N}\sum_{k=1}^{N} V_{t}^{k,N}\cdot\upsilon(X_{t}^{k,N},t)\\ &&+\dfrac{1}{N}\sum_{k=1}^{N}\vert\upsilon(X_{t}^{k,N},t)\vert^{2}+\dfrac{1}{N^{2}}\sum_{k,l=1}^{N}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})+\Vert\varrho(.,t)\Vert_{L^{2}(\mathbb{R}^{d})}^{2}.\end{aligned}$$ We will calculate the differential of each term in the sum of $Q_{t}^{N}$ via chain rules for the Young integral. By formula ([\[Ito\]](#Ito){reference-type="ref" reference="Ito"}) we deduce $$\begin{aligned} d\left(\dfrac{1}{N}\sum_{k=1}^{N}\vert V_{t}^{k,N}\vert^{2}\right)&=& \dfrac{1}{N}\sum_{k=1}^{N}d\left(\vert V_{t}^{k,N}\vert^{2}\right)=\dfrac{1}{N}\sum_{k=1}^{N}2V_{t,q}^{k,N} d(V_{t,q}^{k,N})\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla^{q}\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N} \sigma_{q}(t,X_{t}^{k,N}) d Y_{t}^{q}.\end{aligned}$$ By formulas ([\[produ\]](#produ){reference-type="ref" reference="produ"}) and ([\[Ito-W\]](#Ito-W){reference-type="ref" reference="Ito-W"}) we have $$\begin{aligned} &&d\left(-\dfrac{2}{N}\sum_{k=1}^{N} V_{t}^{k,N}\cdot\upsilon(X_{t}^{k,N},t)\right) =-\dfrac{2}{N}\sum_{k=1}^{N}d[ V_{t,q}^{k,N}\upsilon_{q}(X_{t}^{k,N},t)]\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t) d(V_{t,q}^{k,N})-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N} d(\upsilon_{q}(X_{t}^{k,N},t))\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla^{q}\varrho(X_{t}^{k,N},t)dt-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla\upsilon_{q}(X_{t}^{k,N},t)\cdot V_{t}^{k,N}dt\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t) \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}\\ &&- \dfrac{2}{N}\sum_{k=1}^{N}\ V_{t,q}^{k,N} \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}.\\\end{aligned}$$ Again for formulas ([\[Ito\]](#Ito){reference-type="ref" reference="Ito"}) and ([\[Ito-W\]](#Ito-W){reference-type="ref" reference="Ito-W"}) we obtain $$\begin{aligned} &&d\left(\dfrac{1}{N}\sum_{k=1}^{N}\vert\upsilon(X_{t}^{k,N},t)\vert^{2}\right)=\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\ d(\upsilon_{q}(X_{t}^{k,N},t))\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\nabla^{q}\varrho(X_{t}^{k,N},t)dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t) \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\nabla \upsilon_{q}(X_{t}^{k,N},t)\cdot V_{N}^{k}(t)dt.\end{aligned}$$ By Leibniz rule simple manipulations we can deduce $$\begin{aligned} d\left(\dfrac{1}{N^{2}}\sum_{k,l=1}^{N}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right)&=&\dfrac{1}{N^{2}}\sum_{k,l=1}^{N}d\left(\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right)\\ &=&\dfrac{1}{N^{2}}\sum_{k,l=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N}) d \left(X_{t,q}^{k,N}-X_{t,q}^{l,N} \right)\\ &=&\dfrac{1}{N^{2}}\sum_{k,l=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N}) \left(V_{t,q}^{k,N}-V_{t,q}^{l,N} \right)dt\\ &=&\dfrac{1}{N}\sum_{k=1}^{N}\left(\dfrac{1}{N}\sum_{l=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right)\ V_{t,q}^{k,N}dt\\ & &-\dfrac{1}{N}\sum_{l=1}^{N}\left(\dfrac{1}{N}\sum_{k=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right) V_{t,q}^{l,N}dt \\ &=&\dfrac{1}{N}\sum_{k=1}^{N}\left(\dfrac{1}{N}\sum_{l=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,K})\right) V_{t,q}^{k,N}dt\\ & &-\dfrac{1}{N}\sum_{l=1}^{N}\left(-\dfrac{1}{N}\sum_{k=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{l,N}-X_{t}^{k,N})\right) V_{t,q}^{l,N}dt \\ &=&\dfrac{2}{N}\sum_{k=1}^{N}\left(\dfrac{1}{N}\sum_{l=1}^{N}\nabla^{q}\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right) V_{t,q}^{k,N}dt\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}\left(\dfrac{1}{N}\sum_{l=1}^{N}\nabla\phi_{N}(X_{t}^{k,N}-X_{t}^{l,N})\right)\cdot V_{t}^{k,N}dt\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}V_{N}^{k}\cdot\nabla\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt\\\end{aligned}$$ By Leibniz rule we get $$\begin{aligned} d\left(-\dfrac{2}{N}\sum_{k=1}^{N}\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right) &=&-\dfrac{2}{N}\sum_{k=1}^{N}d\left(\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right)\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}d\left(\int\varrho(x,t)\phi_{N}^{r}(x-X_{t}^{k,N})dx\right)\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}\left( d\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt\\ &=&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt.\end{aligned}$$ and $$\begin{aligned} d\left(\Vert\varrho(.,t)\Vert_{L^{2}(\mathbb{R}^{d})}^{2}\right)&=&-2\langle\varrho(\cdot,t),\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\rangle dt.\end{aligned}$$ Adding all terms we have $$\begin{aligned} d(Q_{t}^{N})&=&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla^{q}\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N} \sigma_{q}(t,X_{t}^{k,N}) d Y_{t}^{q}\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\left(\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)+\nabla^{q}\varrho(X_{t}^{k,N},t)\right)dt\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla\upsilon_{q}(X_{t}^{k,N},t)\cdot V_{t}^{k,N}\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t) \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}\\ &&- \dfrac{2}{N}\sum_{k=1}^{N} V_{t,q}^{k,N} \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\left(\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)+\nabla^{q}\varrho(X_{t}^{k,N},t)\right)dt\\ &&- \dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t) \sigma_{q}(t,X_{t}^{k,N}) dY_{t}^{q}\\ && +\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\nabla\upsilon_{q}(X_{t}^{k,N},t)\cdot V_{t}^{k,N}dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla^{q}\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt-2\langle\varrho(\cdot,t),\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\rangle dt\end{aligned}$$ Doing obvious cancellations we have $$\begin{aligned} &&d(Q_{t}^{N})\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\left(S_{t}^{N}\ast\phi_{N}\right)(X_{t}^{k,N})dt+\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\varrho(X_{t}^{k,N},t)dt-\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\nabla\upsilon_{q}(X_{t}^{k,N},t)\cdot V_{t}^{k,N} dt\\ &&-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)dt-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\varrho(X_{t}^{k,N},t)dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\nabla\upsilon_{q}(X_{t}^{k,N},t)\cdot V_{t}^{k,N}dt-\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\ast\phi_{N}^{r}\right)(X_{t}^{k,N})dt-2\langle\varrho(\cdot,t),\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(\cdot,t))\rangle dt\\ &&=\sum_{j=1}^{10} D_{N}(j,t).\end{aligned}$$ We observe that all term involving Young differential (integral) have been canceled. In the rest of the proof we follow line by line [@Correa] and [@Oes]. From the definition of the empirical measure and convolution, and integration by part we obtain $$\begin{aligned} &&D_{N}(9,t)=2\left\langle S_{t}^{N},\left(\left(\operatorname{div}_{x}\left(\varrho(.,t)\upsilon(.,t)\right)\right)\ast\phi_{N}^{r}\right)(.)\right\rangle\\ &=&2\left\langle (S_{t}^{N}\ast\phi_{N}^{r})(.),\operatorname{div}_{x}\left(\varrho(.,t)\upsilon(.,t)\right)\right\rangle=2\int_{\mathbb{R}^{d}}(S_{t}^{N}\ast\phi_{N}^{r})(x)\operatorname{div}_{x}\left(\varrho(x,t)\upsilon(x,t)\right)\thinspace dx\\ &=&2\int_{\mathbb{R}^{d}}\operatorname{div}_{x}\left((S_{t}^{N}\ast\phi_{N}^{r})(x)\varrho(x,t)\upsilon(x,t)\right)\thinspace dx-2\int_{\mathbb{R}^{d}}\nabla(S_{t}^{N}\ast\phi_{N}^{r})(x)\cdot\left(\varrho(x,t)\upsilon(x,t)\right)\thinspace dx\\ &=&-2\int_{\mathbb{R}^{d}}\nabla(S_{t}^{N}\ast\phi_{N}^{r})(x)\cdot\left(\varrho(x,t)\upsilon(x,t)\right)\thinspace dx\\ &=&-2\left\langle\nabla(S_{t}^{N}\ast\phi_{N}^{r})(.),\varrho(.,t)\upsilon(.,t)\right\rangle.\end{aligned}$$ From the definition of the empirical measure and integration by parts we deduce $$\begin{aligned} A_{N}(1,t)&:=D_{N}(1,t)+D_{N}(6,t)+D_{N}(9,t)+D_{N}(10,t)\nonumber\\ &\,\,=\dfrac{2}{N}\sum_{k=1}^{N} \upsilon(X_{t}^{k,N},t)\cdot\nabla(S_{t}^{N}\ast\phi_{N})(X_{t}^{k,N})-\dfrac{2}{N}\sum_{k=1}^{N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\varrho(,t)\nonumber\\ &\,\,\,\,\,\,\,\,\,-2\left\langle\nabla(S_{t}^{N}\ast\phi_{N}^{r})(.),\varrho(.,t)\upsilon(.,t)\right\rangle-2\left\langle\varrho(.,t),\operatorname{div}_{x}(\varrho(.,t)\upsilon(.,t))\right\rangle\nonumber\\ &\,\,=2\left\langle S_{t}^{N},\nabla(S_{t}^{N}\ast\phi_{N})(.)\cdot\upsilon(,t)\right\rangle-2\left\langle S_{t}^{N},\nabla\varrho(,t)\cdot\upsilon(.,t)\right\rangle\nonumber\\ &\,\,\,\,\,\,\,\,\,-2\left\langle\varrho(.,t),\nabla(S_{t}^{N}\ast\phi_{N}^{r})(.)\cdot\upsilon(.,t)\right\rangle+2\left\langle\varrho(.,t),\nabla\varrho(.,t)\cdot\upsilon(.,t)\right\rangle.\label{AN1}\end{aligned}$$ We observe $$\begin{aligned} \label{AN2} A_{N}(2,t)&:=&D_{N}(2,t)+D_{N}(4,t)+D_{N}(7,t)+D_{N}(5,t)\nonumber\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}V_{t,q}^{k,N}\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t)-\dfrac{2}{N} \sum_{k=1}^{N}V_{t,q}^{k,N}\nabla\upsilon_{q}(X_{t}^{k,N},t) \cdot V_{t}^{k,N}\nonumber\\ &&+\dfrac{2}{N}\sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\nabla\upsilon_{q}(X_{t}^{k,N},t) \cdot V_{t}^{k,N}\nonumber\\ &&-\dfrac{2}{N} \sum_{k=1}^{N}\upsilon_{q}(X_{t}^{k,N},t)\upsilon(X_{t}^{k,N},t)\cdot\nabla\upsilon_{q}(X_{t}^{k,N},t). %&=&2\left\langle S_{t}^{N},V_{N,q}(.,t)\nabla\upsilon_{q}(.,t)\cdot V_{N}(.,t)\right\rangle+2\left\langle S_{t}^{N},V_{N,q}(.,t)\nabla\upsilon_{q}(.,t)\cdot\upsilon(.,t)\right\rangle\nonumber\\ %&+&2\left\langle S_{t}^{N},\upsilon_{q}(.,t)\nabla\upsilon_{q}(.,t)\cdot V_{N}(.,t)\right\rangle-2\left\langle S_{t}^{N},\upsilon_{q}(.,t)\nabla\upsilon_{q}(.,t)\cdot\upsilon(.,t)\right\rangle\end{aligned}$$ Let $C$ be a constant that changes from line to line. Conveniently grouping and applying Young inequality we can deduce $$\begin{aligned} \label{cotaAN2} |A_{N}(2,t)|&=&\dfrac{2}{N}\bigg|\sum_{k=1}^{N}V_{t,q}^{k,N}\upsilon_{q^{\prime}}(X_{t}^{k,N},t)\nabla^{q^{\prime}}\upsilon_{q}(X_{t}^{k,N},t)-V_{t,q}^{k,N}\nabla^{q^{\prime}}\upsilon_{q}(X_{t}^{k,N},t) V_{t,q^{\prime}}^{k,N}\nonumber\\ &+&\upsilon_{q}(X_{t}^{k,N},t)\nabla^{q^{\prime}}\upsilon_{q}(X_{t}^{k,N},t) V_{t,q^{\prime}}^{k,N}-\upsilon_{q}(X_{t}^{k,N},t)\upsilon_{q^{\prime}}(X_{t}^{k,N},t)\nabla^{q^{\prime}}\upsilon_{q} (X_{t}^{k,N},t)\bigg|\nonumber\\ &=&\frac{2}{N}\bigg|\sum_{k=1}^{N}-\nabla^{q^{\prime}}\upsilon_{q}(X_{t}^{k,N},t)\left(V_{t,q}^{k,N}-\upsilon_{q}(X_{t}^{k,N},t)\right)\left(V_{t,q^{\prime}}^{k,N}-\upsilon_{q^{\prime}}(X_{t}^{k,N},t)\right)\bigg|\nonumber\\ &\leq& C\frac{1}{N}\sum_{q,q^{\prime}=1}^{d}\sum_{k=1}^{N}\Big|V_{t,q}^{k,N}-\upsilon_{q}(X_{t}^{k,N},t)\Big|\thinspace\Big|V_{t,q^{\prime}}^{k,N}-\upsilon_{q^{\prime}}(X_{t}^{k,N},t)\Big|\nonumber\\ &\leq& C\frac{1}{N}\sum_{k=1}^{N}\Big|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\Big|^{2}.\end{aligned}$$ Now, we observe $$\begin{aligned} \label{AN3} A_{N}(3,t)&:=& D_{N}(3,t)+D_{N}(8,t)\nonumber\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\varrho(X_{t}^{k,N},t)-\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\nonumber\\ &=&\dfrac{2}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot\left(\nabla\varrho(X_{t}^{k,N},t)-\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right).\end{aligned}$$ By Hölder inequality we obtain $$\begin{aligned} \label{cotaAN3} |A_{N}(3,t)|&=&\frac{2}{N}\left|\sum_{k=1}^{N}V_{t}^{k,N}\cdot\left(\nabla\varrho(X_{t}^{k,N},t)-\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right)\right|\nonumber\\ &\leq&\frac{2}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\cdot\left(\nabla\varrho(X_{t}^{k,N},t)-\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right)\right|\nonumber\\ &\leq&\frac{2}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\right|\left|\left(\nabla\varrho(X_{t}^{k,N},t)-\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}\right)(X_{t}^{k,N})\right)\right|\nonumber\\ &\leq&\frac{2}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\right|\Vert\nabla\varrho(.,t)-\nabla\varrho(.,t)\ast\phi_{N}^{r}(.)\Vert_{\infty}\nonumber\\ &=&2\Vert\nabla\varrho(.,t)-\nabla\varrho(.,t)\ast\phi_{N}^{r}(.)\Vert_{\infty}\sum_{k=1}^{N}\frac{\left|V_{t}^{k,N}\right|}{N}\nonumber\\ &\leq&2\Vert\nabla\varrho(.,t)-\nabla\varrho(.,t)\ast\phi_{N}^{r}(.)\Vert_{\infty}\left(\sum_{k=1}^{N}\frac{\left|V_{t}^{k,N}\right|^{2}}{N}\right)^{1/2}\left(\sum_{k=1}^{N}\frac{1}{N}\right)^{1/2}\nonumber\\ &=&2\Vert\nabla\varrho(.,t)-\nabla\varrho(.,t)\ast\phi_{N}^{r}(.)\Vert_{\infty}\left(\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\right|^{2}\right)^{1/2}\nonumber\\ &\leq&2\left(\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\right|^{2}\right)^{1/2}\sum_{i=1}^{d}\Vert\nabla^{i}\varrho(.,t)-\nabla^{i}\varrho(.,t)\ast\phi_{N}^{r}(.)\Vert_{\infty}\nonumber\\ &\leq&2\left(\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}\right|^{2}\right)^{1/2}N^{-\beta/d}\sum_{i=1}^{d}C_{0}\Vert\nabla(\nabla^{i}\varrho(.,t))\Vert_{\infty}\nonumber\\ &\leq&N^{-\beta/d}C\left(\frac{1}{N}\sum_{k=1}^{N}\left|\upsilon(X_{t}^{k,N},t)\right|^{2}+\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|^{2}\right)^{1/2}\nonumber\\ &\leq&N^{-\beta/d}C\left(\Vert\upsilon(.,t)\Vert_{\infty}^{2}+\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|^{2}\right)^{1/2}\nonumber\\ &\leq&N^{-\beta/d}C\max\{\Vert\upsilon(.,t)\Vert_{\infty},1\}\left(1+\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|^{2}\right)^{1/2}\nonumber\\ &\leq&N^{-\beta/d}C\max\{\Vert\upsilon(.,t)\Vert_{\infty},1\}\left(1+\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|^{2}\right)\nonumber\\ &\leq&C\left(N^{-\beta/d}+\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|^{2}\right)\end{aligned}$$ We have that $$\label{ANRN} A_{N}(1,t)=2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\left(S_{t}^{N}-\varrho(.,t)\right)\ast\phi_{N}\right)(.)\cdot\upsilon(.,t)\rangle+R_{N}(t)$$ where $$R_{N}(t)=2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t) \rangle$$ $$+2\langle \varrho(.,t),\nabla\left(S_{t}^{N}\ast\phi_{N}-S_{t}^{N}\ast\phi_{N}^{r}\right)(.)\cdot\upsilon(.,t)\rangle.$$ By integration by parts and simple calculations we obtain $$\begin{aligned} \label{RN} R_{N}(t)&=& 2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t)\rangle\nonumber\\ & &+2\langle \varrho(.,t),\nabla\left(S_{t}^{N}\ast\left(\phi_{N}-\phi_{N}^{r}\right)\right)(.)\cdot\upsilon(.,t)\rangle\nonumber\\ &=&2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t)\rangle\nonumber\\ & &-2\langle \operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t)),\left(S_{t}^{N}\ast\left(\phi_{N}-\phi_{N}^{r}\right)\right)(\cdot)\rangle\nonumber\\ &=&2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t)\rangle\nonumber\\ & &-2\langle S_{t}^{N},\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\ast\left(\phi_{N}-\phi_{N}^{r}\right)\right)(\cdot)\rangle\nonumber\\ &=&2\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t)\rangle\nonumber\\ & &+2\langle S_{t}^{N},\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\ast\left(\phi_{N}^{r}-\phi_{N}\right)\right)(\cdot)\rangle.\end{aligned}$$ Since $S_{t}^{N},\varrho$ are probability densities and [\[cotafNbeta\]](#cotafNbeta){reference-type="eqref" reference="cotafNbeta"} we obtian $$\begin{aligned} \label{cotaRN} |R_{N}(t)|&\leq&2|\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\varrho(.,t)\ast\phi_{N}^{r}-\varrho(.,t)\right)(.)\cdot\upsilon(.,t)\rangle|\nonumber\\ &&\qquad+2\langle S_{t}^{N},|\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\ast\left(\phi_{N}^{r}-\phi_{N}\right)\right)(\cdot)|\rangle\nonumber\\ &\leq&2\langle S_{t}^{N}+\varrho(.,t),1\rangle N^{-\beta/d}\sum_{i=1}^{d}C_{0}\Vert\nabla(\nabla^{i}\varrho(.,t))\Vert_{\infty}\Vert\upsilon(.,t)\Vert_{\infty}\nonumber\\ &&\qquad+2N^{-\beta/d}C_{0}\Vert\nabla\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\ast\phi_{N}^{r}\right)\Vert_{\infty}\nonumber\\ &=&2C N^{-\beta/d}\Vert\upsilon(.,t)\Vert_{\infty}+2N^{-\beta/d}C_{0}\Vert\left[\nabla\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\right)\right]\ast\phi_{N}^{r}\Vert_{\infty}\nonumber\\ &\leq&2C N^{-\beta/d}\Vert\upsilon(.,t)\Vert_{\infty}+2N^{-\beta/d}C_{0}d\Vert\nabla\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\right)\Vert_{\infty}\Vert\phi_{N}^{r}\Vert_{1}\nonumber\\ &=&2C N^{-\beta/d}\Vert\upsilon(.,t)\Vert_{\infty}+2N^{-\beta/d}C_{0}d\Vert\nabla\left(\operatorname{div}_{x}(\varrho(\cdot,t)\upsilon(.,t))\right)\Vert_{\infty}\Vert\phi_{1}^{r}\Vert_{1}\nonumber\\ &\leq&CN^{-\beta/d}.\end{aligned}$$ We follow [@Oes] and [@Correa] the estimation $$\begin{aligned} |A_{N}(1,t)-R_{N}(t)|&=2|\langle S_{t}^{N}-\varrho(.,t),\nabla\left(\left(S_{t}^{N}-\varrho(.,t)\right)\ast\phi_{N}\right)(.)\cdot\upsilon(.,t)\rangle|\nonumber\\ &\leq C\left(\Vert S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t)\Vert_{L^{2}(\mathbb{R}^{d})}^{2}+N^{-\beta/d}\right).\label{restoA1}\end{aligned}$$ From [\[cotaAN2\]](#cotaAN2){reference-type="eqref" reference="cotaAN2"}, [\[cotaAN3\]](#cotaAN3){reference-type="eqref" reference="cotaAN3"}, [\[cotaRN\]](#cotaRN){reference-type="eqref" reference="cotaRN"}, [\[restoA1\]](#restoA1){reference-type="eqref" reference="restoA1"} we conclude that there exist $C>0$ such that $$\begin{aligned} Q_{t}^{N} &\leq& Q_{0}^{N} + C\int_{0}^{t} \left( Q_{s}^{N}+N^{-\beta/d}\right)ds\qquad0\leq t\leq T\end{aligned}$$ Then by Gronwall lemma we conclude $$\begin{aligned} Q_{t}^{N}\leq C( Q_{0}^{N} +N^{-\beta/d})\qquad\text{for all}\,\,\,\,\, N\in\mathbb{N}. \end{aligned}$$ Finally we will show the convergence of $S_{t}^{N}$ e $V_{t}^{N}$ to $\upsilon$ e $\varrho \upsilon$ respectively. Let $f\in E_{2,q}^{s}$, we have $$\begin{aligned} |\langle S_{t}^{N},f\rangle &-\langle\varrho(.,t),f \rangle|\nonumber\\ &=|\langle S_{t}^{N},f-f\ast\phi_{N}^{r}+f\ast\phi_{N}^{r}\rangle-\langle\varrho(.,t),f \rangle|\nonumber\\ &=|\langle S_{t}^{N},f-f\ast\phi_{N}^{r}\rangle+\langle S_{t}^{N}\ast\phi_{N}^{r},f\rangle-\langle\varrho(.,t),f \rangle|\nonumber\\ &=|\langle S_{t}^{N},f-f\ast\phi_{N}^{r}\rangle+\langle S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t),f \rangle|\nonumber\\ &\leq|\langle S_{t}^{N},f-f\ast\phi_{N}^{r}\rangle|+|\langle S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t),f \rangle|\nonumber\\ &\leq|\langle S_{t}^{N},1\rangle|\left\Vert f-f\ast\phi_{N}^{r}\right\Vert_{\infty}+|\langle S_{t}^{N}\ast\phi_{N}^{r}-\varrho(,t),f \rangle|\nonumber\\ &\leq\left\Vert f-f\ast\phi_{N}^{r}\right\Vert_{\infty}+\left\Vert f\right\Vert_{L^{2}(\mathbb{R}^{d})}\left\Vert S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t)\right\Vert_{L^{2}(\mathbb{R}^{d})}\nonumber\\ &\leq CN^{-\beta/d}\left\Vert\nabla f\right\Vert_{\infty}+\left\Vert f\right\Vert_{L^{2}(\mathbb{R}^{d})}\left\Vert S_{t}^{N}\ast\phi_{N}^{r}-\varrho(.,t)\right\Vert_{L^{2}(\mathbb{R}^{d})}.\label{limXNvarr}\end{aligned}$$ From ([\[limXNvarr\]](#limXNvarr){reference-type="ref" reference="limXNvarr"}) and ([\[Sobolev\]](#Sobolev){reference-type="ref" reference="Sobolev"}) we deduce $$\|S_{t}^{N}- \varrho_{t} \|_{E_{2,\hat{q}}^{-s}}^{2} \leq C_{t} (N^{-\beta/ d} + Q_{0}^{N}).$$ Let $g\in E_{2,q}^{s}$, we obtain $$\begin{aligned} &\left|\left\langle V_{t}^{N}-\varrho(.,t)\upsilon(.,t),g\right\rangle\right|\nonumber\\ &\quad=\left|\frac{1}{N}\sum_{k=1}^{N}V_{t}^{k,N}\cdot g(X_{t}^{k,N})-\left\langle\varrho(.,t),\upsilon(.,t)\cdot g\right\rangle\right|\nonumber\\ &\quad=\left|\frac{1}{N}\sum_{k=1}^{N}(V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\cdot g(X_{t}^{k,N})+\left\langle S_{t}^{N}-\varrho(.,t),\upsilon(.,t)\cdot g\right\rangle\right|\nonumber\\ &\quad\leq\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|\left| g(X_{t}^{k,N})\right|\nonumber+\left|\left\langle S_{t}^{N}-\varrho(.,t),\upsilon(.,t)\cdot g\right\rangle\right|\nonumber\\ &\quad\leq\Vert g\Vert_{\infty}\frac{1}{N}\sum_{k=1}^{N}\left|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)\right|+|\langle S_{t}^{N}-\varrho(.,t),\upsilon(.,t)\cdot g\rangle|\nonumber\\ &\quad\leq C\Vert g\Vert_{\infty}\left(\frac{1}{N}\sum_{k=1}^{N}|V_{t}^{k,N}-\upsilon(X_{t}^{k,N},t)|^{2} \right)^{1/2}+|\langle S_{t}^{N}-\varrho(.,t),\upsilon(.,t)\cdot g\rangle |.\label{convV}\end{aligned}$$ From ([\[convV\]](#convV){reference-type="ref" reference="convV"}) and ([\[Sobolev\]](#Sobolev){reference-type="ref" reference="Sobolev"}) we obtain $$\|V_{t}^{N}- (\varrho \upsilon)_{t}\|_{E_{2, \hat{q}}^{-s}}^{2} \leq C_{T} (N^{-\beta/d} + Q_{0}^{N}).$$ ◻ # Acknowledgements {#acknowledgements .unnumbered} Author Jesus Correa has received research grants from CNPq through the grant 141464/2020-8. Author Juan D. Londoño has received grants from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001 Author Christian Olivera is partially supported by FAPESP by the grant 2020/04426-6 and by FAPESP-ANR by the grant Stochastic and Deterministic Analysis for Irregular Models-2022/03379-0. 99 D. Crisan, D. D. Holm , J. Leahy , T. Nilssen, *Variational principles for fluid dynamics on rough paths*, Advances in Mathematics, 404, 2022. J. A. Carrillo, *Mean-Field Limits: From Particle Descriptions to Macroscopic Equations*, Archive for Rational Mechanics and Analysis, 241,2021. D. Chae, *On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces*, Commun. Pure Appl. Math, 55, 2002. D. Chae, *Local existence and blow-up criterion for the Euler equations in the Besov spaces*, Asymptot. Anal, 38, 2004 J. Correa, C. Olivera , *From Stochastic Hamiltonian Systems to Stochastic Compressible Euler Equation*, arXiv preprint arXiv:2301.08101, 2022. Castrequini, Rafael A. and Catuogno, Pedro J. *A generalized change of variable formula for the Young integral*, Chaos, Solitons and Fractals, 158, 2022. Feireisl, E. *Maximal Dissipation and Well-posedness for the Compressible Euler System*, J. Math. Fluid Mech. 16, 447-461, 2014. P.E. Jabin, Z. Wang *Wang, Mean Field Limit for Stochastic Particle Systems*, In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Cham, 2017. M. Coghi, M. Maurelli , *Regularized vortex approximation for 2D Euler equations with transport noise*, Stochastics and Dynamics, 2020. R. Di Lisio *A particle method for a self-gravitating fluid: A convergence result*, Mathematical Methods in the Applied Sciences, 18, 1995. T Franz, H Wendland, *Convergence of the smoothed particle hydrodynamics method for a specific barotropic fluid flow: constructive kernel theory*, SIAM Journal on Mathematical Analysis, 50, 2018. Gubinelli, M. *Controlling rough paths*, Journal of Functional Analysis, 216, pages 86-140, 2004. P. Kotelenez, A Stochastic *Navier-Stokes Equation for the Vorticity of a Two-Dimensional Fluid*, The Annals of Applied Probability, 5, 1995. P.L.Lions, *Mathematical topis in fluid mechanics*,Vol 2,Compressible Models.New York:Oxford University Press, 1998. A.Majda *Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables*, Applied Mathematical Sciences, Springer-Verlag, NewYork, 1984 K. Oelschläger, *A law of large numbers for moderately interacting diffusion processes*, Zeitschrift fur Wahrscheinlichkeitstheorie verwandte Gebiete 69, 1985. K. Oelschläger *On the connection between Hamiltonian many-particle systems and the hydrodynamical equations*, 115, Archive for Rational Mechanics and Analysis, 1991. H. Triebel, *Interpolation Theory, Function Spaces, Differential Operators*, North-Holland Publishing Co., Amsterdam-New York, 1978. Young, L. C., *An inequality of the Hölder type connected with Stieltjes integration*, C. R. Congr. internat. Math., Oslo 1936, 2, 119 (1937).

arxiv_math

--- abstract: | In this paper, we study precise deviations including precise large deviations and moderate deviations for discrete Hawkes processes for large time asymptotics by using mod-$\phi$ convergence theory. address: - School of Mathematics, SHUFE - School of Mathematics, SHUFE author: - Ying-Li Wang - Ping He$^*$ bibliography: - bibtex.bib title: Precise deviations for a discrete Hawkes process --- [^1] # Introduction ## Continuous-time Hawkes processes and their limit theorems Hawkes process is a continuous-time stochastic model that captures temporal stochastic self-exciting phenomena which is first introduced by Hawkes[@hawkes1971spectra]. In particular, the linear Hawkes process has been well studied and widely used in practice because of its mathematical tractability especially the immigration-birth representation. There are applications in neuroscience, e.g. Johnson[@johnson1996point], DNA modeling, e.g. Gusto and Schbath[@gusto2005fado], finance, and many other fields. Applications of the Hawkes process in finance include market order modeling, e.g. Bauwens and Hautsch[@bauwens2009modelling], Bowsher[@bowsher2007modelling] and Large[@large2007measuring], value-at-risk, e.g. Chavez-Demoulin et al.[@chavez2005estimating], and credit risk, e.g. Errais et al.[@errais2010affine]. Let us introduce the Hawkes processes. Let $N$ be a simple point process on $\mathbb R$, and let $\mathcal F_t^{-\infty}:=\sigma(N(C),C\in \mathcal B(\mathbb R), C\subset (-\infty,t])$ be an increasing family of $\sigma$-algebras. Any nonnegative $\mathcal F_t^{-\infty}$-progressively measurable $\lambda_t$ with $$\mathbb E\left[N(a,b]|\mathcal F_a^{-\infty}\right]=\mathbb E\left[ \int_a^b\lambda_sds|\mathcal F_a^{-\infty} \right]$$ a.s. for all intevals $(a,b]$ is called an $\mathcal F_t^{-\infty}$-intensity of $N$. We use the notation $N_t:=N(0,t]$ to denote the number of points in the inteval $(0,t]$. A general Hawkes process is a simple point process $N$ admitting an $\mathcal F_t^{-\infty}$ intensity $$\lambda_t:=\lambda\left( \int_{-\infty}^th(t-s)N(ds) \right),$$ where $\lambda(\cdot):\mathbb R^+\rightarrow\mathbb R^+$ is locally integrable and left continuous, $h(\cdot):\mathbb R^+\rightarrow\mathbb R^+$, and we always assume that $\left\lVert h\right\rVert_{L^1}=\int_0^\infty h(t)dt<\infty$. We always assume that $N(-\infty,0]=0$, i.e. the Hawkes process has empty history. In the literature, $h(\cdot)$ and $\lambda(\cdot)$ are usually referred to as the exciting function and the rate function, respectively. The Hawkes process is linear if $\lambda(\cdot)$ is linear and it is nonlinear otherwise, in the linear case, the stochastic intensity can be written as $$\lambda_t=\nu+\int_0^{t-}h(t-s)N(ds).$$ Because the lack of immigration-birth representation and computational tractability, nonlinear Hawkes processes are much less studied. A nonlinear Hawkes processes are first introduced by Bremaud et al.[@bremaud1996stability]. Let us review the limit theorems for linear Hawkes processes in the literature. It is well known that we have the law of large numbers $\frac{N_t}{t}\rightarrow\frac{\nu}{1-\left\lVert h\right\rVert_{L^1}}$ as $t\rightarrow\infty$. Bacry et al.[@BACRY20132475] obtain a functional central limit theorem for multivariate Hawkes process and as a special case of their result, $$\label{clthawkes} \frac{N_t-\frac{\nu}{1-\left\lVert h\right\rVert_{L^1}}}{\sqrt t}\rightarrow N\left( 0,\frac{\nu}{(1-\left\lVert h\right\rVert_{L^1})^3} \right),$$ in distribution as $t\rightarrow\infty$ under the assumption that $\int_0^\infty t^{1/2}h(t)dt<\infty$. Bordenave and Torrisi[@bordenave2007large] prove that $\mathbb P(\frac{N_t}{t}\in \cdot)$ satisfies a large deviation principle with the rate function: $$\label{ldpcontinuoushawkes} I(x)=x\log\left( \frac{x}{\nu+x\left\lVert h\right\rVert_{L^1}} \right)-x+x\left\lVert h\right\rVert_{L^1}+\nu,$$ if $x\ge0$ and $I(x)=+\infty$ otherwise. The rate function is written in Legendre transform expression in Bordenave et al.[@bordenave2007large], and [\[ldpcontinuoushawkes\]](#ldpcontinuoushawkes){reference-type="eqref" reference="ldpcontinuoushawkes"} is first mentioned in Karabash and Zhu[@karabash2015limit]. Moderate deviations for linear Hawkes processes are studied in Zhu[@ZHU2013885]. For nonlinear Hawkes processes, Zhu[@zhu2013nonlinear] is a complete introduction. Zhu[@zhu2013central] studies the central limit theorem, and [@zhu2014process] obtains a level-3 large deviation principle, and hence has the scalar large deviations as a by-product. When the system is Markovian, Zhu[@zhu2015large] obtains an expression for the rate function. Zhu[@zhu2014limit] also studies limit theorems for a CIR process with Hawkes jumps. The large deviations and moderate deviations for linear Hawkes processes are of the Donsker-Varadhan type, which only gives the leading order term. In many occasions, more accurate estimates are desired, i.e. the precise deviations. Gao and Zhu[@gao2021precise] use the recent mod-$\phi$ convergence theory[@F_ray_2016] to compute the precise deviations including precise large deviations and moderate deviations for continuous-time linear Hawkes processes. The mod-$\phi$ convergence theory shows that if we can characterize the convergence speed of the moment generating function and verify the limit corresponds to an infinitely divisible distribution, we can obtain the mod-$\phi$ convergence which can derive the precise deviations. ## Discrete Hawkes processes In practical applications, data are always from discrete time observations. As a result, there has been some literatures to study discrete Hawkes models. Discrete Hawkes processes are first introduced in Seol[@seol2015limit], the literature studies the limit theorems for discrete time Hawkes-type model with 0-1 arrivals including law of large number, central limit theorem and the invariance principle. Wang[@wang2022limit] studies the one with Poisson arrivals and marked situation whose large and moderate deviations are stated in [@wang2023large]. In this paper, we also use mod-$\phi$ convergence theory to study precise deviations including precise large deviations and moderate deviations for discrete Hawkes processes which is stated in [@wang2022limit]. Our results extends the result in [@wang2023large] to some extent. Two important proof techniques when characterizing the mod-$\phi$ convergence are an another type of Abel's lemma and a discrete type generalized Gronwall's inequality. We will show the discrete case is very similar to the continuous case which is stated in [@gao2021precise]. # Main Results Before we introduce the discrete model and precise deviation results, let us first recall the definition of mod-$\phi$ convergence in [@F_ray_2016]. ## Mod-$\phi$ convergence Let $(X_n)_{n\ge1}$ be a sequence of real-valued random variables and $\mathbb E[e^{zX_n}]$ exist in a strip $\mathcal S_{(c,d)}:=\{z\in \mathbb C:c<\mathcal R(z)<d\}$, with $c<0<d$ extended real numbers, i.e. we allow $c=-\infty$ and $d=+\infty$ and $\mathcal R(z)$ denotes the real part of $z\in\mathbb C$ throughout this paper. We assume that there exists a non-constant infinitely divisible distribution $\phi$ with $\int_\mathbb Re^{zx}\phi(dx)=e^{\eta(z)}$, which is well defined on $\mathcal S_{(c,d)}$, and an analytic function $\psi(z)$ that does not vanish on the real part of $\mathcal S_{(c,d)}$ such that locally uniformly in $z\in\mathcal S_{(c,d)}$, $$e^{-t_n\eta(z)}\mathbb E[e^{zX_n}]\rightarrow \psi(z),$$ where $t_n\rightarrow\infty$ as $n\rightarrow\infty$. Then we say that $X_n$ converges mod-$\phi$ on $\mathcal S_{(c,d)}$ with parameters $(t_n)_{n\ge1}$ and limiting function $\psi$. Assume that $\phi$ is a lattice distribution i.e., a distribution with support included in $\gamma+\lambda\mathbb Z$ for some constants $\gamma,\lambda>0$. Also assume that the sequence of random variables $(X_n)_{n\ge1}$ converges mod-$\phi$ at speed $O(t_n^{-v})$, that is $$\sup_{z\in K}\left| e^{-t_n\eta(z)}\mathbb E[e^{zX_n}]-\psi(z) \right|\le C_Kt_n^{-v},$$ where $C_K>0$ is some constant, for any compact set $K\subset \mathcal S_{(c,d)}$. Then the Theorem 3.2.2 in [@F_ray_2016] states that for any $x\in\mathbb R$ in the inteval $(\eta'(c),\eta'(d))$ such that $t_nx\in\mathbb N$, we have $$\mathbb P(X_n=t_nx)=\frac{e^{-t_nF(x)}}{\sqrt{2\pi t_n\eta''(\theta^*)}}\left( \psi(\theta^*)+\frac{a_1}{t_n}+\frac{a_2}{t_n^2}+\cdots+\frac{a_{v-1}}{t_n^{v-1}}+O\left( \frac1{t_n^v} \right) \right),$$ as $n\rightarrow\infty$, where $\theta^*$ is defined via $\eta'(\theta^*)=x$, and $F(x):=\sup_{\theta\in\mathbb R}\{\theta x-\eta(\theta)\}$ is the Legendre transform of $\eta(\cdot)$, and if $x\in\mathbb R$ and $x\in(\eta'(0),\eta'(d))$, then, as $n\rightarrow\infty$, $$\mathbb P(X_n\ge t_nx)=\frac{e^{-t_nF(x)}}{\sqrt{2\pi t_n\eta''(\theta^*)}}\frac1{1-e^{-\theta^*}}\left( \psi(\theta^*)+\frac{b_1}{t_n}+\frac{b_2}{t_n^2}+\cdots+\frac{b_{v-1}}{t_n^{v-1}}+O\left( \frac1{t_n^v} \right) \right),$$ where $(a_k)_{k\ge1}$, $(b_k)_{k\ge1}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^*$. ## The discrete model For $t\in\mathbb N$, let $\alpha_t:=\alpha(t):\mathbb N\rightarrow\mathbb R_+$ be a positive function on $\mathbb N$. The process has an empty history and $X_0=N_0=0$. It is worth to mention that $\alpha(\cdot)$ is an exponential function in [@xu2022self], and the model proposed in [@wang2022limit] is in fact the extension of the model in [@xu2022self]. Define $\left\lVert\alpha\right\rVert_1:=\sum_{t=0}^\infty \alpha_t$ (for convenience, set $\alpha_0=0$) as the $l_1$ norm of $\alpha$. Conditional on $X_{t-1},X_{t-2},...,X_1$, we define $X_t$ as a Poisson random variable with mean $$\lambda_t:=\nu+\sum_{s=1}^{t-1}\alpha_sX_{t-s}.$$ Finally, we define $N_t:=\sum_{s=1}^tX_s$. And the moment generating function can be directly obtained from [@wang2022limit]. $$\begin{aligned} \label{nt} \mathbb E[e^{z N_t}] =&\exp\left(\nu\left(-t+\sum_{i=0}^{t-1}e^{f_i(z)}\right)\right),\end{aligned}$$ where $$f_0(z)=z,f_1(z)=z+(e^z-1)\alpha_1,\ f_s(z)=z+\sum_{i=1}^s\alpha_i(e^{f_{s-i}(z)}-1).$$ Wang[@wang2023large] proves that pointwisely we have $$f_\infty(z)=z+(e^{f_\infty(z)}-1)||\alpha||_{1}.$$ Set $x(z):=e^{f_\infty(z)}$, $x(z)$ satisfies the algebraic equation $$x(z)=e^{z+\left\lVert\alpha\right\rVert_1(x(z)-1)}.$$ We notice that in [@wang2023large], the rate function is written in Legendre transform expression since random marks are included. After removing marks, the rate function in [@wang2023large] can be expressed explicitly as follows, $$\label{ratefunctiondiscrete} I(x)=x\log\left( \frac{x}{\nu+x\left\lVert\alpha\right\rVert_1} \right)-x+x\left\lVert\alpha\right\rVert_1+\nu,$$ if $x\ge0$ and $I(x)=+\infty$ otherwise, we refer the readers to [@bordenave2007large] and [@karabash2015limit] for this result. Choose $\eta(z)=\nu\left( e^{z+(e^{f_\infty(z)}-1)||\alpha||_1}-1 \right)=\nu \left( x(z)-1 \right)$, and we have the following lemma. **Lemma 1**. *Assume there is a random variable $Y$ such that $\mathbb E[e^{zY}]=e^{\eta(z)}=e^{\nu \left( x(z)-1 \right)}$, then $Y$ has an infinitely divisible distrbution.* *Proof.* After replacing the $L^1$ norm of the exciting function $h$ in [@gao2021precise] by the discrete norm $l^1$ of $\alpha$. It is a direct result from Lemma 6 in [@gao2021precise], which shows $\eta(\cdot)$ is a moment generating function of an infinitely divisible random variable $Y$, i.e. $$\mathbb E[e^{zY}]=e^{\eta(z)}=e^{\nu \left( x(z)-1 \right)}.$$ ◻ We will show $\eta(\cdot)$ is the exactly the function we need in characterizing mod-$\phi$ convergence, i.e. $$e^{-t\eta(z)}\mathbb E[e^{zN_t}]\longrightarrow \psi(z):=e^{\nu \varphi(z)},$$ as $t\rightarrow\infty$, locally uniformly in $z$ for $\mathcal R(z)\le \theta_c:=\left\lVert\alpha\right\rVert_1-1-\log \left\lVert\alpha\right\rVert_1$ , where the limit function $$\psi(z)=\exp\left(\nu\sum_{i=0}^\infty \left(e^{f_i(z)}-e^{f_\infty(z)}\right)\right).$$ **Proposition 2**. *For any $\theta\in\mathbb R$, and $\theta\le \theta_c$, where $\theta_c:=\left\lVert\alpha\right\rVert_1-1-\log \left\lVert\alpha\right\rVert_1$, we have (i) $x(\theta)\left\lVert\alpha\right\rVert_1\le 1.$ (ii)$x'(\theta)\rightarrow\infty$ as $\theta\uparrow \theta_c$.* *Proof.* The proof is exactly the same as [@gao2021precise] after replacing $\left\lVert h\right\rVert_{L^1}$ with $\left\lVert\alpha\right\rVert_1$. ◻ **Lemma 3** (An another type of the Abel's lemma). *Assume $(b_i)_{i\ge1}\in l^1$, denote $B_k=\sum_{i=k+1}^\infty b_i,k\ge0$, then $$\sum_{k=1}^pa_kb_k=a_1B_0+\sum_{k=1}^{p-1}(a_{k+1}-a_k)B_k-a_pB_p,\ p\ge2.$$* *Proof.* The proof is similar to the proof of the classical Abel's lemma, $$\begin{aligned} \sum_{k=1}^pa_kb_k =&\sum_{k=1}^pa_k(B_{k-1}-B_k)\\ =&\sum_{k=1}^pa_kB_{k-1}-\sum_{k=1}^pa_kB_k\\ =&a_1B_0+\sum_{k=1}^{p-1}a_{k+1}B_k-\sum_{k=1}^{p-1}a_kB_k-a_pB_p\\ =&a_1B_0+\sum_{k=1}^{p-1}(a_{k+1}-a_k)B_k-a_pB_p. \end{aligned}$$ ◻ **Lemma 4** (Discrete generalized Gronwall's inequality). *Let $\left(p(n)\right)_{n\ge1}$ and $\left(q(n)\right)_{n\ge1}$ be two $l^1$ nonnegative sequences. If $p(i)\le g(i)$ and $$p(i)\le \sum_{j=1}^{i-1}q(i-j)p(j)+g(i),\ i\ge2,$$ then $$\begin{aligned} p(i) \le &\sum_{j=1}^{i-1}Q(i-j)g(j)+g(i),\ i\ge2, \end{aligned}$$ where $$Q(i)=\sum_{j=1}^\infty q^{*j}(i),\ q^{*j}(i)=\sum_{j=1}^{i-1}q^{*j-1}(j)q(i-j),\ q^{*0}(i)=q(i).$$* *Proof.* We can prove it like the method in [@chu1967gronwall], $$\begin{aligned} p(i) \le &\sum_{j=1}^{i-1}q(i-j)\left( \sum_{m=1}^{j-1}q(j-m)p(m)+g(j) \right)+g(i)\\ =&\sum_{j=1}^{i-1}q(i-j)\sum_{m=1}^{j-1}q(j-m)p(m)+\sum_{j=1}^{i-1}q(i-j)g(j)+g(i)\\ =&\sum_{j=1}^{i-1}p(j)q^{*2}(i-j)+\sum_{j=1}^{i-1}q^{*1}(i-j)g(j)+g(i). \end{aligned}$$ By iterating, we have $$\begin{aligned} p(i) \le &\sum_{j=1}^{i-1}Q(i-j)g(j)+g(i),\ i\ge2. \end{aligned}$$ ◻ **Lemma 5**. *For any $\mathcal R(z)\le \theta_c$, where $\theta_c:=\left\lVert\alpha\right\rVert_1-1-\log \left\lVert\alpha\right\rVert_1$, $$\varphi(z)=\sum_{i=0}^\infty \left(e^{f_i(z)}-x(z)\right)$$ is well-defined and analytic, and as $t\rightarrow\infty$, $$e^{-t\eta(z)}\mathbb E[e^{zN_t}]\longrightarrow \psi(z):=e^{\nu \varphi(z)},$$ locally uniformly in $z$. In addition, if $\sum_{i=0}^\infty i^{v+1}\alpha_i<\infty$, then for any compact set $K$, there exists some $C_K>0$ such that $\sup_{z\in K}|e^{-t\eta(z)}\mathbb E[e^{zN_t}]-e^{\nu \varphi(z)}|\le C_Kt^{-v}$.* *Proof.* First, it is obvious that $x(z)$ is analytic in $\mathcal S_{(-\infty,\theta_c)}$, and for any positive integer $t$, $\sum_{i=0}^t\left( e^{f_i(z)}-x(z) \right)$ are analytic in $\mathcal S_{(-\infty,\theta_c)}$. To show $\varphi(z)$ is well-defined and analytic in $\mathcal S_{(-\infty,\theta_c)}$, we need to prove $$\sum_{i=0}^t\left( e^{f_i(z)}-x(z) \right)\rightarrow \varphi(z)$$ as $t\rightarrow\infty$, locally uniformly in $z$ for $\mathcal R(z)<\theta_c$. In other words, we need to prove that for any compact set $K\subset \{z\in\mathbb C;\mathcal R(z)<\left\lVert\alpha\right\rVert_1-1-\log \left\lVert\alpha\right\rVert_1\}$, $$\sum_{i=0}^\infty\sup_{z\in K}|e^{f_i(z)}-x(z)|<\infty.$$ In fact, from page 11-12 in [@wang2023large], we know that $e^{f_i(z)}\rightarrow x(z)$ pointwisely as $i\rightarrow\infty$. Since $$\begin{aligned} e^{f_i(z)}-e^{f_\infty(z)} =&e^{f_\infty(z)}\cdot \left(e^{\sum_{j=1}^i\alpha_j(e^{f_{i-j}(z)}-e^{f_\infty(z)})-\sum_{j=i+1}^\infty\alpha_j(e^{f_\infty(z)}-1)}-1\right), \end{aligned}$$ which yields that $$\sum_{j=1}^i\alpha_j(e^{f_{i-j}(z)}-e^{f_\infty(z)})\rightarrow 0,\ \text{as}\ i\rightarrow\infty.$$ Furthermore, for any fixed $\delta>0$ such that $(1+\delta)|x(z)|\left\lVert\alpha\right\rVert_1<1$, there exists $M>0$, so that for any $i\ge M$ and $z\in K$, we have $$\begin{aligned} \label{fangsuo} \left| e^{f_i(z)}-x(z) \right| \le& (1+\delta)|x(z)|\left( \sum_{j=1}^i\alpha_j|e^{f_{i-j}(z)}-x(z)|+|x(z)-1|\sum_{j=i+1}^\infty\alpha_j \right). \end{aligned}$$ Therefore, we get that for any $T>M$, $$\begin{aligned} &\sum_{i=M}^T\sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|\\ \le &(1+\delta)\sup_{z\in K}|x(z)|\sum_{i=M}^T\sum_{j=1}^i\sup_{z\in K}|e^{f_{i-j}(z)}-x(z)|\alpha_j+(1+\delta)\sup_{z\in K}|x(z)||x(z)-1|\sum_{i=M}^T\sum_{j=i+1}^\infty\alpha_j\\ \le &(1+\delta)\sup_{z\in K}|x(z)|\sum_{i=1}^T\sum_{j=1}^i\sup_{z\in K}|e^{f_{i-j}(z)}-x(z)|\alpha_j+(1+\delta)\sup_{z\in K}|x(z)||x(z)-1|\sum_{i=1}^\infty\sum_{j=i+1}^\infty\alpha_j\\ =&(1+\delta)\sup_{z\in K}|x(z)|\sum_{j=1}^T\alpha_j\sum_{i=0}^{T-j}\sup_{z\in K}|e^{f_i(z)}-x(z)|+(1+\delta)\sup_{z\in K}|x(z)||x(z)-1|\sum_{j=1}^\infty j\alpha_j, \end{aligned}$$ which implies that $$\begin{aligned} &\sum_{i=0}^T\sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|\\ \le&\sum_{i=0}^M\sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|+(1+\delta)\sup_{z\in K}|x(z)|\sum_{j=1}^T\alpha_j\sum_{i=0}^{T-j}\sup_{z\in K}|e^{f_i(z)}-x(z)|\\ &+(1+\delta)\sup_{z\in K}|x(z)||x(z)-1|\sum_{j=1}^\infty j\alpha_j\\ \le&\sum_{i=0}^M\sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|+(1+\delta)\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1\sum_{i=0}^{T}\sup_{z\in K}|e^{f_i(z)}-x(z)|\\ &+(1+\delta)\sup_{z\in K}|x(z)|\left(\sup_{x\in K}|x(z)|+1\right)\sum_{j=1}^\infty j\alpha_j. \end{aligned}$$ Let $T\rightarrow\infty$, we have $$\begin{aligned} \label{fixbound} &\sum_{i=0}^\infty \sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|\\ \le &\frac{\sum_{i=0}^M\sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|+(1+\delta)\sup_{z\in K}|x(z)|\left(\sup_{x\in K}|x(z)|+1\right)\sum_{j=1}^\infty j\alpha_j}{1-(1+\delta)\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1}. \end{aligned}$$ Hence, we conclude that $\sum_{i=t}^\infty \sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|\rightarrow 0$ as $t\rightarrow\infty$, and so $$\label{locallyuniformly} \sum_{i=0}^t\left( e^{f_i(z)}-x(z) \right)\rightarrow \sum_{i=0}^\infty\left( e^{f_i(z)}-x(z) \right)=\varphi(z),$$ as $t\rightarrow\infty$, locally uniformly in $z$ for $\mathcal R(z)<\theta_c$. Hence, $\varphi(z)$ is well-defined and is analytic in $\mathcal S_{(-\infty,\theta_c)}$. By equations [\[nt\]](#nt){reference-type="eqref" reference="nt"}, [\[locallyuniformly\]](#locallyuniformly){reference-type="eqref" reference="locallyuniformly"} and the definitions of $\psi(z)$ and $\varphi(z)$, we have proved that locally uniformly in $z$ for $\mathcal R(z)<\theta_c$, $$e^{-t(\nu(x(z)-1))}\mathbb E[e^{zN_t}]=\exp\left(\nu\sum_{i=0}^{t-1}\left( e^{f_i(z)}-x(z) \right)\right)\longrightarrow \psi(z):=e^{\nu \varphi(z)},\ \text{as}\ t\rightarrow\infty.$$ To show the mod-$\phi$ convergence at speed $O(t^{-v})$, that is, for any compact set $K$, there exists some $C_K>0$ such that $\sup_{z\in K}|e^{-t(\nu(x(z)-1))}\mathbb E[e^{zN_t}]|\le C_K t^{-v}$, it suffices to show that for any compact set $K\subset \{z\in\mathbb C;\mathcal R(z)<\left\lVert\alpha\right\rVert_1-1-\log\left\lVert\alpha\right\rVert_1\}$, $$\label{2.5} \sum_{i=0}^\infty i^v \sup_{z\in K} \left| e^{f_i(z)}-x(z) \right|<\infty.$$ To see this, notice that $$\begin{aligned} \sup_{z\in K}\left| e^{-t(\nu(x(z)-1))}\mathbb E[e^{zN_t}]-e^{\nu\varphi(z)}\right| =&\sup_{z\in K}\left|e^{\nu\varphi(z)}\left( e^{\nu\sum_{i=t}^\infty\left( e^{f_i(z)}-x(z) \right)}-1\right)\right|\\ \le &\sup_{z\in K}e^{\nu|\varphi(z)|}\left( e^{\nu\sum_{i=t}^\infty\sup_{z\in K}\left| e^{f_i(z)}-x(z) \right|}-1 \right). \end{aligned}$$ Thus it suffices to show that $$\begin{aligned} \label{25} \sum_{i=t}^\infty\sup_{z\in K}\left| e^{f_i(z)}-x(z) \right|\le c_Kt^{-v}, \end{aligned}$$ for some $c_K>0$. By Lemma [Lemma 3](#Abel){reference-type="ref" reference="Abel"}, $$\begin{aligned} \label{23} &\sum_{i=0}^Ti^v\sup_{z\in K}|e^{f_i(z)}-x(z)|\\ =&\sum_{i=0}^\infty\sup_{z\in K}|e^{f_i(z)}-x(z)|+\sum_{i=0}^{T-1}((i+1)^v-i^v)\sum_{j=i+1}^\infty \sup_{z\in K}|e^{f_j(z)}-x(z)|\\ &-T^v\sum_{i=T+1}^\infty\sup_{z\in K}|e^{f_i(z)}-x(z)|. \end{aligned}$$ On the other hand, by Tonelli's theorem, $$\begin{aligned} \label{24} \sum_{i=0}^{T-1}((i+1)^v-i^v)\sum_{j=i+1}^\infty \sup_{z\in K}|e^{f_j(z)}-x(z)|=\sum_{j=0}^\infty j^v\sup_{z\in K}\left| e^{f_j(z)}-x(z) \right|<\infty. \end{aligned}$$ Hence, we get $\lim_{T\rightarrow\infty} T^v\sum_{i=T+1}^\infty \sup_{z\in K}|e^{f_i(z)}-x(z)|=0$ by letting $T\rightarrow\infty$ in [\[23\]](#23){reference-type="eqref" reference="23"} and applying [\[24\]](#24){reference-type="eqref" reference="24"}. This implies [\[25\]](#25){reference-type="eqref" reference="25"}. Next, let us prove [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"}. From [\[fangsuo\]](#fangsuo){reference-type="eqref" reference="fangsuo"}, we obtain that there exists $M>0$ so that for any $i\ge M$, $$\begin{aligned} &\sup_{z\in K}|e^{f_i(z)}-x(z)|\\ \le& (1+\delta)\sup_{z\in K}|x(z)|\sum_{j=1}^i\alpha_j\sup_{z\in K}|e^{f_{i-j}(z)}-x(z)|+(1+\delta)\sup_{z\in K}|x(z)|\sup_{z\in K}|x(z)-1|\sum_{j=i+1}^\infty \alpha_j. \end{aligned}$$ Therefore, for every $i\ge2$, $$\begin{aligned} \label{fangsuo1} &\sup_{z\in K}|e^{f_i(z)}-x(z)|\\ \le &(1+\delta)\sup_{z\in K}|x(z)|\sum_{j=1}^i\alpha_j\sup_{z\in K}|e^{f_{i-j}(z)}-x(z)|+g(i), \end{aligned}$$ where $$\begin{aligned} g(i):=C_1\sum_{j=i+1}^\infty \alpha_j+C_21_{\{i\le M\}}, \end{aligned}$$ where $$C_1:=(1+\delta)\sup_{z\in K}|x(z)|\sup_{z\in K}|x(z)-1|,\ C_2=\sup_{0\le i\le M}\sup_{z\in K}|e^{f_i(z)}-x(z)|.$$ Let $p(i)=\sup_{z\in K}|e^{f_i(z)}-x(z)|$ and $q(i):=(1+\delta)\sup_{z\in K}|x(z)|\alpha_i$ for every $i\ge2$. Then [\[fangsuo1\]](#fangsuo1){reference-type="eqref" reference="fangsuo1"} can be re-written as $$\begin{aligned} \label{iter} p(i)\le \sum_{j=1}^{i-1}q(i-j)p(j)+g(i). \end{aligned}$$ By Lemma [Lemma 4](#discretegronwall){reference-type="ref" reference="discretegronwall"}, we conclude that $$\begin{aligned} p(i) \le &\sum_{j=1}^{i-1}Q(i-j)g(j)+g(i), \end{aligned}$$ where $$Q(i)=\sum_{j=1}^\infty q^{*j}(i)=\sum_{j=1}^\infty ((1+\delta)\sup_{z\in K}|x(z)|)^j\alpha_i^{*j}.$$ It is equivalent to $$\sup_{z\in K}|e^{f_i(z)}-x(z)|\le \sum_{j=1}^{i-1}Q(i-j)g(j)+g(i),\ i\ge2.$$ Hence, it remains to show that $$\begin{aligned} \label{fequ1} \sum_{i=1}^\infty i^vg(i)<\infty, \end{aligned}$$ $$\begin{aligned} \label{fequ2} \sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)g(j)<\infty. \end{aligned}$$ Let us first prove [\[fequ1\]](#fequ1){reference-type="eqref" reference="fequ1"}. Note that $$\begin{aligned} \sum_{i=1}^\infty i^vg(i) =&\sum_{i=1}^\infty i^v\left( C_1\sum_{j=i+1}^\infty \alpha_j+C_21_{\{i\le M\}} \right)\\ =&C_1\sum_{i=1}^\infty i^v\sum_{j=i+1}^\infty \alpha_j+C_2\sum_{i=1}^M i^v, \end{aligned}$$ and by our assumption $\sum_{j=1}^\infty j^{v+1}\alpha_j<\infty$, we obtain $$\begin{aligned} \sum_{i=1}^\infty i^v\sum_{j=i+1}^\infty \alpha_j =&\sum_{j=2}^\infty \alpha_j\sum_{i=1}^{j-1}i^v\le \frac1{v+1}\sum_{j=1}^\infty j^{v+1}\alpha_j<\infty, \end{aligned}$$ thus [\[fequ1\]](#fequ1){reference-type="eqref" reference="fequ1"} follows. Next, let us prove [\[fequ2\]](#fequ2){reference-type="eqref" reference="fequ2"}. Note that $$\begin{aligned} &\sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)g(j)\\ =&C_1\sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)\sum_{m=j+1}^\infty \alpha_m+C_2\sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)1_{\{j\le M\}}, \end{aligned}$$ and it is easy to check that $$\begin{aligned} \sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)1_{\{j\le M\}} =&\sum_{j=1}^\infty 1_{\{j\le M\}}\sum_{i=j+1}^\infty Q(i-j)\cdot i^v\\ =&\sum_{j=1}^M\sum_{i=2}^\infty Q(i)(i+j)^v\\ \le &2^{v-1}(M+1)\sum_{i=2}^\infty Q(i)(i^v+j^v)\\ \le &2^{v-1}(M+1)\sum_{i=2}^\infty Q(i)(i^v+M^v). \end{aligned}$$ where we use the inequality $(a+b)^v\le 2^{v-1}(a^v+b^v)$ for any $a,b>0$ and $v\ge1$. We can compute that $$\begin{aligned} \sum_{i=1}^\infty Q(i) =&\sum_{i=1}^\infty\sum_{j=1}^\infty \left((1+\delta)\sup_{z\in K}|x(z)|\right)^j\alpha_i^{*j}\\ =&\sum_{j=1}^\infty \left((1+\delta)\sup_{z\in K}|x(z)|\right)^j\left\lVert\alpha\right\rVert_1^j, \end{aligned}$$ which is finite since $(1+\delta)\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1<1$. Next, let us show that $$\sum_{i=1}^\infty i^vQ(i)<\infty.$$ Notice that $$\begin{aligned} \sum_{i=1}^\infty i^vQ(i) =&\sum_{i=1}^\infty i^v\sum_{j=1}^\infty q^{*j}(i)\\ =&\sum_{i=1}^\infty\sum_{j=1}^\infty \left( (1+\delta)\sup_{z\in K}|x(z)| \right)^j\alpha_i^{*j}\\ =&\sum_{j=1}^\infty\left( (1+\delta)\sup_{z\in K}|x(z)| \right)^j\sum_{i=1}^\infty i^v\alpha_i^{*j}\\ =&\sum_{j=1}^\infty\left( (1+\delta)\sup_{z\in K}|x(z)| \right)^j\sum_{i=1}^\infty i^v\sum_{m=0}^i\alpha_m^{*j-1}\alpha_{i-m}\\ =&\sum_{j=1}^\infty\left( (1+\delta)\sup_{z\in K}|x(z)| \right)^j\sum_{m=0}^\infty \alpha_m^{*j-1}\sum_{i=0}^\infty(i+m)^v\alpha_i. \end{aligned}$$ Note that for any $\delta'>0$, there exists some $C(\delta')>0$ such that for any $s,u\ge0$, $$(s+u)^v\le C(\delta')s^v+(1+\delta')u^v.$$ Therefore, $$\begin{aligned} \sum_{m=0}^\infty m^v\alpha_m^{*k} \le &C(\delta')\sum_{m=0}^\infty \alpha_m^{*k-1}\sum_{i=0}^\infty i^v\alpha_i+(1+\delta')\sum_{m=0}^\infty m^v\alpha_m^{*k-1}\left\lVert\alpha\right\rVert_1. \end{aligned}$$ Let us define $A_k:=\sum_{m=0}^\infty m^v\alpha_m^{*k}$. Then, we have: $$A_k\le C(\delta')\left\lVert\alpha\right\rVert_1^{k-1}A_1+(1+\delta')A_{k-1}\left\lVert\alpha\right\rVert_1.$$ It follows that $$\begin{aligned} A_k \le &C(\delta')\left\lVert\alpha\right\rVert_1^{k-1}\left[ 1+(1+\delta')\left\lVert\alpha\right\rVert_1+((1+\delta')\left\lVert\alpha\right\rVert_1)^2+\cdots+((1+\delta')\left\lVert\alpha\right\rVert_1)^{k-2} \right]A_1\\ &+((1+\delta')\left\lVert\alpha\right\rVert_1)^{k-1}A_1. \end{aligned}$$ Choose $\delta'>0$ to be sufficiently small so that $(1+\delta')\left\lVert\alpha\right\rVert_1<1$. Then, we have $$\begin{aligned} A_k \le &\frac{C(\delta')}{1-(1+\delta')\left\lVert\alpha\right\rVert_1}\left\lVert\alpha\right\rVert_1^{k-1}A_1+(1+\delta')^{k-1}\left\lVert\alpha\right\rVert_1^{k-1}A_1\\ \le &\left( \frac{C(\delta')}{1-(1+\delta')\left\lVert\alpha\right\rVert_1}+1 \right)(1+\delta')^{k-1}\left\lVert\alpha\right\rVert_1^{k-1}A_1. \end{aligned}$$ Choose $\delta'>0$ to be sufficiently small so that $(1+\delta)(1+\delta')\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1<1$. Hence, we conclude that $$\begin{aligned} &\sum_{i=0}^\infty i^vQ(i)\\ =&\sum_{j=1}^\infty\left( (1+\delta)\sup_{z\in K}|x(z)| \right)^jA_j\\ \le&(1+\delta)\sup_{z\in K}|x(z)|A_1\left( \frac{C(\delta')}{1-(1+\delta')\left\lVert\alpha\right\rVert_1}+1 \right)\sum_{j=1}^\infty\left( (1+\delta)(1+\delta')\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1 \right)^{j-1}\\ =&\frac{(1+\delta)\sup_{z\in K}|x(z)|A_1\left( \frac{C(\delta')}{1-(1+\delta')\left\lVert\alpha\right\rVert_1}+1 \right)}{1-(1+\delta)(1+\delta')\sup_{z\in K}|x(z)|\left\lVert\alpha\right\rVert_1}<\infty. \end{aligned}$$ Finally, we can compute that $$\begin{aligned} &\sum_{i=2}^\infty i^v\sum_{j=1}^{i-1}Q(i-j)\sum_{m=j+1}^\infty \alpha_m\\ =&\sum_{j=1}^\infty\sum_{i=j+1}^\infty i^vQ(i-j)\sum_{m=j+1}^\infty\alpha_m\\ =&\sum_{j=1}^\infty\sum_{i=1}^\infty(i+j)^vQ(i)\sum_{m=j+1}^\infty\alpha_m\\ \le &\sum_{j=1}^\infty\sum_{i=1}^\infty 2^{v-1}(i^v+j^v)Q(i)\sum_{m=j+1}^\infty\alpha_m\\ =&\sum_{j=1}^\infty\sum_{m=j+1}^\infty\alpha_m\left( \sum_{i=1}^\infty 2^{v-1}i^vQ(i) \right)+\sum_{j=1}^\infty j^v\sum_{m=j+1}^\infty\alpha_m\left( \sum_{i=1}^\infty 2^{v-1}Q(i) \right)\\ =&\sum_{m=2}^\infty m\alpha_m\left( \sum_{i=1}^\infty 2^{v-1}i^vQ(i) \right)+\sum_{m=2}^\infty\sum_{j=0}^{m-1}\alpha_m j^v\left( \sum_{i=1}^\infty 2^{v-1}Q(i) \right)\\ \le&2^{v-1}\sum_{m=1}^\infty m\alpha_m\left( \sum_{i=1}^\infty i^vQ(i) \right)+\frac{2^{v-1}}{v+1}\sum_{m=1}^\infty m^{v+1}\alpha_m\left( \sum_{i=1}^\infty Q(i) \right)<\infty. \end{aligned}$$ This completes the proof. ◻ ## Precise Large Deviations **Theorem 6**. *Given $v\in\mathbb N$. Assume $\left\lVert\alpha\right\rVert_1<\infty$ and the following condition holds: $$\begin{aligned} \sum_{i=1}^\infty i^{v+1}\alpha_i<\infty. \end{aligned}$$* 1. *For any $x>0$, and $tx\in\mathbb N$, as $t\rightarrow\infty$, $$\mathbb P(N_t=tx)=e^{-tI(x)}\sqrt{\frac{I''(x)}{2\pi t}}\left( \psi(\theta^*)+\frac{a_1}{t}+\frac{a_2}{t^2}+\cdots+\frac{a_{v-1}}{t^{v-1}}+O\left(\frac{1}{t^v}\right) \right),$$ where for any $\mathcal R(z)\le \left\lVert\alpha\right\rVert_1-1-\log\left\lVert\alpha\right\rVert_1$, $$\psi(z):=e^{\nu\varphi(z)},\ \text{and}\ \varphi(z)=\sum_{i=0}^\infty \left(e^{f_i(z)}-x(z)\right),$$ which is analytic in $\mathcal S_{(-\infty,\left\lVert\alpha\right\rVert_1-1-\log\left\lVert\alpha\right\rVert_1)}$, where $$f_0(z)=z,f_1(z)=z+(e^z-1)\alpha_1,\ f_s(z)=z+\sum_{i=1}^s\alpha_i(e^{f_{s-i}(z)}-1),$$ and $x(z)=e^{f_\infty(z)}$ exists and it satisfies the equation $$x(z)=e^{z+\left\lVert\alpha\right\rVert_1(x(z)-1)},$$ and it is analytic in $\mathcal S_{(-\infty,\left\lVert\alpha\right\rVert_1-1-\log \left\lVert\alpha\right\rVert_1)}$. And $I(x)$ is defined in [\[ratefunctiondiscrete\]](#ratefunctiondiscrete){reference-type="eqref" reference="ratefunctiondiscrete"}, $I''(x)=\frac{\nu^2}{x(\nu+\left\lVert\alpha\right\rVert_1x)^2}$, and $$\theta^*=\log\left( \frac{x}{\nu+\left\lVert\alpha\right\rVert_1x} \right)-\frac{\left\lVert\alpha\right\rVert_1x}{\nu+\left\lVert\alpha\right\rVert_1x}+\left\lVert\alpha\right\rVert_1,$$ where $(a_k)_{k\ge1}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^*$ whose formulas are given in the Proposition 1(i) from [@gao2021precise], which is $$\begin{aligned} a_k =&\sum_{l=0}^{2k}\frac{\psi^{(2k-l)}(\theta^*)}{(2k-l)!}\sum_{\mathcal S_l}\frac{(-1)^{m_1+\cdots+m_l}}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_l!l!^{m_l}}\\ &\cdot \prod_{j=1}^l\left( \frac1{\eta''(\theta^*)}\frac{\eta^{(j+2)(\theta^*)}}{(j+2)(j+1)} \right)^{m_j}\frac{(-1)^k(2(k+m_1+\cdots+m_l)-1)!!}{(\eta''(\theta^*))^k},\ k\ge1, \end{aligned}$$ where $\eta(z):=\nu(x(z)-1)$.* 2. *For any $x>\frac{\nu}{1-\left\lVert\alpha\right\rVert_1}$ and $tx\in\mathbb N$, as $t\rightarrow\infty$, $$\mathbb P(N_t\ge tx)=e^{-tI(x)}\sqrt{\frac{I''(x)}{2\pi t}}\frac1{1-e^{-\theta^*}}\left( \psi(\theta^*)+\frac{b_1}{t}+\frac{b_2}{t^2}+\cdots+\frac{b_{v-1}}{t^{v-1}}+O\left( \frac1{t^v} \right) \right),$$ where $(b_k)_{k\ge1}$ are rational fractions in the derivatives of $\eta$ and $\psi$ at $\theta^*$ whose formulas are given in the Proposition 1(ii) from [@gao2021precise], which is $$\begin{aligned} b_k =&\sum_{n=0}^{2k}\sum_{\mathcal S_n}\frac{e^{-(m_1+\cdots+m_n)\theta^*}(m_1+\cdots+m_n)!(1-e^{-\theta^*})^{-(m_1+\cdots+m_n)-1}}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_n!n!^{m_n}}\cdot \prod_{j=1}^n (-1)^{j\cdot m_j}\\ &\cdot \sum_{l=0}^{2k-n}\frac{\psi^{(2k-n-l)}(\theta^*)}{(2k-n-l)!}\sum_{\mathcal S_l}\frac{(-1)^{m_1+\cdots+m_l}}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_l!l!^{m_l}}\\ \ &\cdot\prod_{j=1}^l\left( \frac1{\eta''(\theta^*)}\frac{\eta^{(j+2)}(\theta^*)}{(j+2)(j+1)} \right)^{m_j}\frac{(-1)^k(2(k+m_1+\cdots+m_l)-1)!!}{(\eta''(\theta^*))^k}. \end{aligned}$$* *Proof.* By Lemma [Lemma 1](#infinitelydivisible){reference-type="ref" reference="infinitelydivisible"}, and Lemma [Lemma 5](#modphidiscrete){reference-type="ref" reference="modphidiscrete"}, we have established the mod-$\phi$ convergence. The proof is exactly the same as [@gao2021precise] after replacing $\left\lVert h\right\rVert_{L^1}$ with $\left\lVert\alpha\right\rVert_1$. ◻ ## Precise Moderate Deviations **Theorem 7**. *$I(\cdot)$ is defined in [\[ratefunctiondiscrete\]](#ratefunctiondiscrete){reference-type="eqref" reference="ratefunctiondiscrete"} and for any $i\ge2$, $$I^{(i)}(x)=(i-2)!(-1)^{i-2}x^{1-i}\left( (i-1)\left( \frac{\left\lVert\alpha\right\rVert_1x}{\nu+\left\lVert\alpha\right\rVert_1x} \right)^i-i\left( \frac{\left\lVert\alpha\right\rVert_1x}{\nu+\left\lVert\alpha\right\rVert_1x} \right)^{i-1}+1 \right).$$ Assume $\left\lVert\alpha\right\rVert_1<\infty$ holds, if $y=o(t^{1/2-1/m})$, where $m\ge3$, then as $t\rightarrow\infty$, $$\mathbb P\left( N_t\ge \frac{\nu}{1-\left\lVert\alpha\right\rVert_1}t+\sqrt t\frac{\sqrt \nu}{(1-\left\lVert\alpha\right\rVert_1)^{3/2}}y \right)=\frac{(1+o(1))}{y\sqrt {2\pi}}e^{-\sum_{i=2}^{m-1}\frac{I^{(i)}(\eta'(0))}{i!}\frac{(\eta''(0))^{i/2}y^i}{t^{(i-2)/2}}},$$ where $\eta'(0)=\frac{\nu}{1-\left\lVert\alpha\right\rVert_1}$, and $\eta''(0)=\frac{\nu}{(1-\left\lVert\alpha\right\rVert_1)^3}$.* *Proof.* The proof is exactly the same as [@gao2021precise] after replacing $\left\lVert h\right\rVert_{L^1}$ with $\left\lVert\alpha\right\rVert_1$. ◻ **Proposition 8**. 1. *$\eta(\theta^*)=\nu (x(\theta^*)-1)$, and for $k\ge1$, $\eta^{(k)}(\theta^*)=\nu x^{(k)}(\theta^*)$. For $k\ge1$, $x^{(k)}(\theta^*)$ can be computed recursively as: $$\begin{aligned} x^{(k)}(\theta^*) =&\frac{x(\theta^*)}{1-\left\lVert\alpha\right\rVert_1x(\theta^*)}\sum_{\mathcal T_k}\frac{k!\cdot\left\lVert\alpha\right\rVert_1^{m_1+\cdots+m_{k-1}}}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_{k-1}!(k-1)!^{m_{k-1}}}\cdots\prod_{j=1}^{k-1}(x^{(j)}(\theta^*))^{m_j}\\ &+\frac{x(\theta^*)}{1-\left\lVert\alpha\right\rVert_1x(\theta^*)}\sum_{l=0}^{k-1}\binom{k}{l}\sum_{\mathcal S_l}\frac{l!\cdot \left\lVert\alpha\right\rVert_1^{m_1+\cdots+m_l}}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_l!l!^{m_l}}\cdot \prod_{j=1}^l(x^{(j)}(\theta^*))^{m_j}, \end{aligned}$$ where $\mathcal T_k$ denotes the set of $(k-1)$-tuples of nonnegative integers $(m_1,\cdots,m_{k-1})$ satisfying the constraint $1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+(k-1)\cdot m_{k-1}=k-1$.* 2. *For every $k\ge1$, $$\begin{aligned} \psi^{(k)}(\theta^*)=\sum_{\mathcal S_k}\frac{k!\cdot \nu^{m_1+\cdots+m_k}\cdot \psi(\theta^*)}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_k!k!^{m_k}}\cdot \prod_{j=1}^k\left( \sum_{i=0}^\infty \left(\left( e^{f_i(z)} \right)^{(j)}-x^{(j)}(z)\right) \right)^{m_j}, \end{aligned}$$ where $$\left( e^{f_i(z)} \right)^{(j)}=\sum_{\mathcal S_j}\frac{j!e^{f_i(z)}}{q_1!1!^{q_1}q_2!2!^{q_2}\cdots q_j!j!^{q_j}}\prod_{r=1}^j\left( f_i^{(r)}(z) \right)^{q_r}.$$* *Proof.* The expression of $x^{(k)}(\theta^*)$ is a direct result after replacing $\left\lVert h\right\rVert_{L^1}$ with $\left\lVert\alpha\right\rVert_1$. Next, let us compute the expression of $\psi^{(k)}(\theta^*)$, recall that $$\begin{aligned} \psi(z)=e^{\nu\sum_{i=0}^\infty \left(e^{f_i(z)}-x(z)\right)},\end{aligned}$$ where $$f_0(z)=z,f_1(z)=z+(e^z-1)\alpha_1,\ f_s(z)=z+\sum_{i=1}^s\alpha_i(e^{f_{s-i}(z)}-1).$$ Let $\left( e^{f_i(z)} \right)^{(k)}$ denote the $k$-th derivative of $e^{f_i(z)}$. By Faà di Bruno's formula, we have $$\begin{aligned} \psi^{(k)}(\theta^*)=\sum_{\mathcal S_k}\frac{k!\cdot \nu^{m_1+\cdots+m_k}\cdot \psi(\theta^*)}{m_1!1!^{m_1}m_2!2!^{m_2}\cdots m_k!k!^{m_k}}\cdot \prod_{j=1}^k\left( \sum_{i=0}^\infty \left(\left( e^{f_i(z)} \right)^{(j)}-x^{(j)}(z)\right) \right)^{m_j},\end{aligned}$$ where $\mathcal S_k$ consists of all the $k$-tuples of nonnegative integers $(m_1,...,m_k)$ satisfying the constraint $1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+k\cdot m_k=k$, $\left( e^{f_i(z)} \right)^{(j)}$ can be computed again by Faà di Bruno's formula, $$\left( e^{f_i(z)} \right)^{(j)}=\sum_{\mathcal S_j}\frac{j!e^{f_i(z)}}{q_1!1!^{q_1}q_2!2!^{q_2}\cdots q_j!j!^{q_j}}\prod_{r=1}^j\left( f_i^{(r)}(z) \right)^{q_r},$$ where $\mathcal S_j$ consists of all the $j$-tuples of nonnegative integers $(q_1,...,q_j)$ satisfying the constraint $1\cdot q_1+2\cdot q_2+3\cdot q_3+\cdots+j\cdot q_j=j$. ◻ # Acknowledgement The first author Yingli Wang would like to thank Professor Lingjiong Zhu for bringing up this topic and Assistant Professor Qinghua Wang for helpful discussions. [^1]: $^*$ Corresponding author

arxiv_math

--- abstract: | **Abstract:** We study implications and consequences of well-posed solutions of Cauchy problems of a Novikov equation describing pseudospherical surfaces. We show that if the co-frame of dual one-forms satisfies certain conditions for a given periodic initial datum, then there exists exactly two families of periodic one-forms satisfying the structural equations for a surface. Each pair then defines a metric of constant Gaussian curvature and a corresponding Levi-Civita connection form. We prove the existence of universal connection forms giving rise to second fundamental forms compatible with the metric. The main tool to prove our geometrical results is the Kato's semi-group approach, which is used to establish well-posedness of solutions of the Cauchy problem involved and ensure $C^1$ regularity for the first fundamental form and the Levi-Civita connection form. **2020 AMS Mathematics Classification numbers**: 35B10, 53A05, 58J60, 35A30. **Keywords:** Equations describing pseudospherical surfaces; First fundamental form; Second fundamental form; Cauchy problems; Kato's approach author: - Nilay Duruk Mutlubas - Igor Leite Freire title: Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems --- # Introduction {#sec1} In [@sasaki] Sasaki made a remarkable observation, showing that solutions of integrable equations solved by the AKNS $2\times2$ method [@akns] give rise to metrics of pseudospherical surfaces with Gaussian curvature ${\cal K}=-1$, see [@sasaki section 2] and [@chern section 1] for further details. Later on, Chern and Tenenblat, in their seminal paper [@chern], introduced the notion of equations describing pseudospherical surfaces (PSS equation) and gave a systematic way for finding them. The works by Sasaki [@sasaki] and Chern and Tenenblat [@chern] showed a deep connection between integrability and differential geometry of pseudospherical surfaces, unsurprisingly, leading to a new notion of integrability, see [@reyes2000 Definition 2] and [@reyes2006-sel page 245]. Notwithstanding its relevance in terms of integrability, the work by Chern and Tenenblat made an in-depth investigation on certain very peculiar equations having the following property: with some exceptions (that we will discuss later), their solutions give rise to metrics with constant Gaussian curvature. This fact *per se* has been known for a long time for the Sine-Gordon equation, see [@rogers Section 1], but its systematic study, implications and applications to other equations, potential links with integrable systems and construction of conserved quantities made [@chern] a paramount work. The metric and the Gaussian curvature are intrinsic properties of a surface, but alas insufficient to completely describe it. To this end, we need further information provided by its second fundamental form. While the first fundamental form (metric) can be though as the *way a two-dimensional being walks* on the surface (an intrinsic aspect), the second fundamental form tells us *how the surface behaves from, or looks like to, an observer located outside it*. Given the importance of the second fundamental form and the relevance of the observations and ideas introduced in [@sasaki] and [@chern], respectively, it is somewhat surprising that it was taken nearly three decades from [@chern] until the first works [@kah-book; @kah-cag; @kah] concerning second fundamental forms of the surfaces defined by the solutions of PSS equations. Although the work by Chern and Tenenblat was born in the context of integrability of differential equations, and most of the follow-up works, not to say all, were concerned with these connections, see [@cat; @reyes2000; @reyes2002; @reyes2006-sel; @reyes2006-jde; @reyes2011], along time the integrability aspects were put aside and the research carried out has been more focused on geometric aspects and classification of equations describing PSS, see [@tarcisio; @ding; @kah-book; @kah-cag; @kah; @keti2015; @tito] and references therein. A considerable number of relevant PSS equations can be seen as dynamical systems in certain Banach spaces, and from the point of view of analysis of PDEs, qualitative aspects of their solutions are obtained from Cauchy problems, meaning that not only the equation is relevant, but also a condition satisfied by a given solution of the equation at a given time, very often $t=0$ (initial condition or datum). Usually, the regularity of the initial datum determines that of the corresponding solution of the equation. From a geometric perspective, solutions emanating from Cauchy problems involving an equation for an unknown $u=u(x,t)$ can be seen as follows: given a certain curve $x\mapsto(x,0,u_0(x))$, can we find a solution $u$ for the equation such that the given curve belongs to the graph of $u$? Moreover, what does the regularity of the curve say about the graph of $u$? Is $u$ the only solution of the equation whose graph contains the given curve? Despite being a topic mostly concerned with analysis of PDEs, the paragraph above shows that the problem of existence and uniqueness of solutions (well-posedness) of PDEs makes sense in the context of PSS equations. Surprisingly, it seems this topic has been out of the agenda of the literature of PSS equations. The purpose of our paper is to shed light on it. The main motivation for us to undertake the research reported in the present work are recent results reported in [@nazime] concerned with the equation $$\label{1.0.1} u_t-u_{txx}=\partial_x(2-\partial_x)(1+\partial_x)u^2,$$ which was discovered in [@nov] and recently has been proved to be geometrically integrable [@freire-tito-sam], meaning that its solutions describe a non-trivial family of pseudospherical surfaces. Equation [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} was studied in [@guo-bvp; @li-na; @li-jmaa; @liu-jde] from the point of view of qualitative analysis, such as existence and uniqueness of solutions. More recently, in [@nazime Theorem 5.1] results from [@li-na; @liu-jde] were combined with [@freire-tito-sam] to prove the existence of $C^\omega$ (metrics for) pseudospherical surfaces arising from the solutions of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}. The aforementioned result proved in [@nazime], despite being established for $C^\omega$ solutions, strongly indicates the possibility of relating Cauchy problems and pseudospherical surfaces. Actually, it made such a connection, but by considering solutions emanating from an initial datum with $C^\omega$ regularity. The question is: Could we consider the same problem replacing a $C^\omega$ initial datum by one with lower regularity? Can we consider a periodic initial datum? In line with the comments above, the vast majority of works in the field of PSS equations considers explicit or implicitly $C^\infty$ solutions of the PSS equations, which technically avoid problems regarding regularity (that is, how much smooth the object is) and lead to $C^\infty$ metrics. A simple question then arises: What may happen if we consider solutions with regularity other than $C^\infty$? The answer to the questions above is given in our first result. **Theorem 1**. *Let $u_0\in H^4(\mathbb{S})$ be a non-trivial and non-constant initial datum, with $u-u_0''>0$, and consider the Cauchy problem $$\label{1.0.2} \left\{ \begin{array}{l} u_t-u_{txx}=\partial_x(2-\partial_x)(1+\partial_x)u^2,~~~~~~x\in \mathbb{R},~~~~~t>0,\\ \\ u(x,0)=u_0(x),~~~~~~x\in \mathbb{R},\\ \\ u(x,t)=u(x+1,t),~~~~~~x\in \mathbb{R},~~~~~t>0. \end{array}\right.$$* *Then there exists triads of $C^1$ one-forms $\omega_1,\,\omega_2,\,\omega_3$, with $$\label{1.0.3} \omega_i=f_{i1}dx+f_{i2}dt,\quad 1\leq i\leq 3,$$ $$\label{1.0.4} f_{p1}=\mu_pf_{11}+\eta_p,\quad 1\leq p\leq 2,$$ where $\mu_p,\,\eta_p\in\mathbb{R}$, such that the forms [\[1.0.3\]](#1.0.3){reference-type="eqref" reference="1.0.3"} are defined on $U=\mathbb{R}\times(0,\infty)$, periodic with respect to $x$, and define a PSS whenever $\nabla u\neq(0,0)$, with $\omega_3$ being the Levi-Civita connection of the metric determined by $\omega_1$ and $\omega_2$.* *Moreover, fixed a pair $\{\omega_1,\,\omega_2\}$ and $p\in U$, there exists connection forms $\omega_{13}=a\omega_1+b\omega_2$, $\omega_{23}=b\omega_1+c\omega_2$, where $a,\,b,\,c$ are $C^\infty$ functions defined on an open neighborhood $V\subseteq U$ of $p$, such that $\{\omega_1,\omega_2,\omega_{13},\,\omega_{23}\}$ defines a PSS of Gaussian curvature ${\cal K}=-1$.* In section [2](#sec2){reference-type="ref" reference="sec2"} we shall present all pertinent definitions and notions, but for now it suffices to say that a function belonging to $H^4(\mathbb{S})$ is a real valued periodic function, with period $1$, of class $C^3$. Our theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} can be seen as an existence and uniqueness result for PSS surfaces. In fact, it says that from solutions of equation [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} whose graphs contain the regular curve $x\mapsto(x,0,u_0(x))$, with $u_0\in H^4(\mathbb{S})$, we can obtain an open set $V\subseteq\mathbb{R}^2$ in which we have two possible choices to define a first fundamental form for a PSS surface with Gaussian curvature ${\cal K}=-1$. Moreover, we can locally define connection forms on each point of $V$. This fact, jointly with Bonnet theorem, tells us that we can locally define a PSS surface embedded in $\mathbb{R}^3$. A key point to understand and prove theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is determining whether the problem [\[1.0.2\]](#1.0.2){reference-type="eqref" reference="1.0.2"} is well-posed. To this end, recognising the presence of the Helmholtz operator $\Lambda^{2}=1-\partial_x^2$ in [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}, we can rewrite the problem [\[1.0.2\]](#1.0.2){reference-type="eqref" reference="1.0.2"} in an alternative form, given by $$\label{1.0.5} \left\{ \begin{array}{l} u_t-2uu_x=\partial_x\Lambda^{-2}(u^2+(u^2)_x),~~~~~~x\in \mathbb{R},~~~~~t>0,\\ \\ u(x,0)=u_0(x),~~~~~~x\in \mathbb{R},\\ \\ u(x,t)=u(x+1,t),~~~~~~x\in \mathbb{R},~~~~~t>0. \end{array}\right.$$ **Theorem 2**. *Let $u_0\in H^{s}(\mathbb{S})$, $s>3/2$ be a given initial datum. Then there exists a maximal time of existence $T>0$, depending on $u_0$, such that there is a unique solution $u$ to [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} satisfying $u\in C^0(H^{s}(\mathbb{S}),[0,T))\cap C^1(H^{s-1}(\mathbb{S}),[0,T))$. Moreover, the map $u_0\in H^{s}(\mathbb{S}) \rightarrow u$, is continuous from $H^{s}(\mathbb{S})$ to $C^0(H^{s}(\mathbb{S}),[0,T))\cap C^1(H^{s-1}(\mathbb{S}),[0,T))$ and $T$ is independent of $s$.* The inverse of the Helmholtz operator, denoted by $\Lambda^{-2}$ and acting on a function $f$, is defined by the convolution $g\ast f$, where $$\label{1.0.6} g(x)=\frac{\cosh{(x-\lfloor x\rfloor-1/2)}}{2\sinh{(1/2)}}$$ and $\lfloor \cdot\rfloor$ denotes the greatest integer function. Some previous results in the literature had already proved well-posedness results concerning periodic solutions of the problem [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"}, see [@liu-jde], but it was either shown that $u\in C^0([0,T),H^{s}(\mathbb{S}))$ or $u$ is $C^\omega$ in both variables on a certain domain, see [@liu-jde Theorem 1.1] and [@liu-jde Theorem 1.4], respectively. Although these results show the existence and uniqueness of solutions for a very large class of functions, they are unsuitable for our purposes because we need solutions with $C^1$ regularity with respect to $t$. The importance of our theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} comes from just the fact that it ensures we have $C^1$ solutions in both variables. Actually, this is a consequence of the Sobolev Lemma. In particular, it tells us that the solutions granted by theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} are strong solutions for the (non-local) first order PDE in [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"}. A natural question then arises: is a strong solution of the equation in [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} also a strong solution of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}? In general the answer is no! However, requiring enough regularity of the initial datum we can find solutions for the Cauchy problem [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} that are also strong solutions for [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}, and therefore, simultaneously strong solutions for both formulations of the equation. In fact, whenever we consider an initial datum in $H^4(\mathbb{S})$, the corresponding solution provided by theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} not only is a $C^1$ solution (in both variables), but the Sobolev Lemma also implies that $x\mapsto u(x,t)$, $t$ fixed, is $C^3$, meaning that the solution emanating from [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} is a strong, or classical, solution for [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}, which makes sense to be considered in the study of PSS and differential equations. Although the regularity of the initial datum is enough to make the corresponding solution of [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} a strong solution of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}, it is insufficient to guarantee that the solution is global, in the sense that it is defined for every $t>0$. We can have global solutions requiring little more from the initial datum. **Theorem 3**. *If $u_0\in H^4(\mathbb{S})$ is a non-trivial initial datum, and $u_0(x)-u_0''(x)>0$, $x\in\mathbb{R}$, then the solution of the problem [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} exists for any $t>0$. Moreover, $u\in C^1(\mathbb{R}\times(0,\infty))$ and $x\mapsto u(x,t)$ is a $C^3$ periodic function, for each fixed $t>0$.* It is important to note that in view of the Sobolev Lemma, $H^4(\mathbb{S})$ is continuously and densely embedded in $H^s(\mathbb{S})$, for $s\in(3/2,4)$. Therefore, both theorems [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} and [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"} tell us that the problem [\[1.0.2\]](#1.0.2){reference-type="eqref" reference="1.0.2"} has only one solution $u$. ## Novelty and challenges of the manuscript We study the problem of PSS determined by the solutions of a given equation from the perspective of well-posedness of Cauchy problems, which as far as we know, has not been considered yet. Let us highlight the relevance of our results by discussing the following problem: suppose we know a curve from the graph of a (unknown) solution of a PSS equation. Can we precisely describe the corresponding PSS? Let us exemplify by considering the function $u_c(x,t):=e^{x-ct}$. Any member of the family ${\cal U}=\{u_c,\,c\in\mathbb{R}\}$ is a solution of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} defined for $(x,t)\in\mathbb{R}\times(0,\infty)$, see [@nazime page 5]. For any $u\in{\cal U}$, let $\text{Gr}(u)=\{(x,t,u(x,t)),\,\,x\in\mathbb{R},\,\,t>0\}.$ Consider the curve $\Gamma$, given by $x\mapsto(x,0,e^x)$ and let $\partial\text{Gr}(u)$ denote the boundary of a set $\text{Gr}(u)$. Then it is easy to see that $\Gamma\subseteq \partial\text{Gr}(u)$, for any $u\in{\cal U}$. In particular, we have $$\Gamma\subseteq \bigcap_{u\in{\cal U}} \partial\text{Gr}(u).$$ If we allow $t=0$ in the domain of $u_c$, then the curve $\Gamma$ belongs to the corresponding graph, but for our purposes it is enough to consider it lying in the boundary, the latter being disjoint from the graph. On the other hand, [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} is a PSS equation (see [@freire-tito-sam Theorem 1]), and for any member $u_c$ of ${\cal U}$ we can construct a PSS $U_c$ in an intrinsic way (note that these solutions satisfy all required conditions for the existence of a PSS, see the comments after Theorem 1 in [@freire-tito-sam]). Due to the fact that the curve $\Gamma$ belongs to the boundary of any graph of the solutions $u_c$, we cannot determine any specific PSS surface only knowing $\Gamma$. Our theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} gives a rather different answer to the same question. In geometric terms, it says that given a curve $\Gamma$ of the form $x\mapsto(x,0,u_0(x))$, as long as $u_0\in H^4(\mathbb{S})$, we can precisely and intrinsically describe a PSS among all infinite surfaces emanating from all solutions of the PDE [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}. To address this problem we make use of techniques of existence and uniqueness of solutions for PDEs that can be seen as dynamical systems in certain Banach spaces. In view of this approach, we deal with solutions of the equation that are less regular than those usually considered in the literature of the PSS equations. One of the difficulties to be overcome is concerned with the regularity of the one-forms $\omega_1$ and $\omega_2$ defining the metric of the corresponding PSS. Most of the books in differential geometry require $C^\infty$ forms, although some of them require at least $C^2$ regularity. As we will better discuss in the next section, for the classical theory of curves and surfaces, we can have PSS surfaces from $C^1$ forms satisfying the structure equations for a surface. Last but not least, one of the challenges of this paper is that its main result is geometric, but the tools for proving it comes from modern approaches to prove qualitative aspects of solutions of Cauchy problems. For this reason, we tried our best to make clear and explain the technical aspects of each area, so that the readers can have a better reading and appreciation of our work. ## Outline of the manuscript Since this is a paper focusing on Analysis and Geometry, in the next section we provide an overview about PSS and functional analysis. We also fix the notation, present essential concepts and revisit Kato's semi-group approach, which is the main tool for proving theorems [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} and [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"}, whose demonstrations are given in sections [3](#sec3){reference-type="ref" reference="sec3"} and [4](#sec4){reference-type="ref" reference="sec4"}, respectively. Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is proved in section [5](#sec5){reference-type="ref" reference="sec5"}, whereas our conclusions are given in section [6](#sec6){reference-type="ref" reference="sec6"}. # Notation, notions and preliminaries {#sec2} In this section we introduce and fix the notation used throughout the manuscript. Given its plural and diverse aspects, we also present basic facts and concepts from differential geometry of surfaces and functional analysis, which are the main pillars of the work. Most of the geometric content can be better explored in [@ilka Chapter 5], [@cle Chapter 4] and [@keti-book Chapter 2], whereas our main references for functional analysis are [@iorio Chapter 3] and [@taylor Chapter 4]. ## Notation Given a function $u=u(x,t)$, by $u(x,\cdot)$ we mean the function $t\mapsto u(x,t)$, for fixed $x$, whereas $u(\cdot,t)$ denotes the function $x\mapsto u(x,t)$, for fixed $t$. Let $I,J$ two open, non-empty subsets of $\mathbb{R}$. We say that $u\in C^0(I\times J)$ if $u=u(x,t)$ is continuous with respect to both variables $(x,t)\in I\times J$. Partial derivative of $u$ with respect to its first argument will be denoted by $u_x$ or $\partial_x u$, whereas $u_t$ or $\partial_t u$ will denote partial derivative with respect to the second argument. Higher order derivatives can be considered using the standard conventions. For a positive integer $k$, we say that $u\in C^k(I\times J)$ if all partial derivatives of $u$ up to order $k$ (including the mixed ones) are continuous. Given a positive integer $n$, we denote by $u_{(n)}$ the set of ordered $n-th$ derivatives of $u$. Also, we say that $u$ is $C^k$ whenever all of its partial derivatives up to order $k$ are continuous on the domain of $u$. By $C^{3,1}(I\times J)$ we mean the set of function $u:I\times J\rightarrow\mathbb{R}$ such that $u$, $u_x$, $u_t$, $u_{xx}$, $u_{xt}$, $u_{xxx}$ and $u_{xxt}$ belong to $C^0(I\times J)$. Let $X$ be a Banach space of real valued functions and $I\subseteq\mathbb{R}$. The set $C^0(I,X)$ denotes collection of continuous functions such that $u(t,\cdot)\in X$. More generally, given a positive integer $n$, we say that $u\in C^n(I,X)$ if $\partial_t^ku(t,\cdot) \in C^0(I,X)$, $0\leq k\leq n$. ## Structure equations and pseudospherical surfaces Let $\langle\cdot,\cdot\rangle$ be the usual inner product in $\mathbb{R}^3$ and denote the pair $(\mathbb{R}^3,\langle\cdot,\cdot\rangle)$ by $\mathbb{E}^3$, that is, the Euclidean space. We recall that a surface is a two dimensional manifold in $\mathbb{E}^3$, which we generally denote by $\cal M$. Given a point $p\in\cal M$, the tangent and the co-tangent spaces to $\cal M$ at $p$ are denoted by $T_p\cal M$ and $T^\ast_p\cal M$, respectively. Let $\{e_1,e_2\}$ be vector (sufficiently differentiable) valued functions on $\cal M$, such that at each point $p\in\cal M$, we have: $\{e_1,e_2\}$ is orthonormal with respect to inner product $\langle\cdot,\cdot\rangle$; $\text{Span}\{e_1,e_2\}=T_p\cal M$; $\{\omega_1,\omega_2\}$ is the dual bases of $\{e_1,e_2\}$. In particular, $\text{Span}\{\omega_1,\omega_2\}=T_p^\ast\cal M$. Let $\{\omega_1,\omega_2\}$ be the corresponding dual of the basis $\{e_1,e_2\}$. Since $\langle e_i,e_j\rangle$ is either $0$ or $1$, depending on whether $i=j$ or not, we have $$\label{2.2.1} \langle d e_i,e_j\rangle+\langle e_i,de_j\rangle=0,$$ where $d(\cdot)$ denotes the usual differential, and we can then define one-forms $$\label{2.2.2} \omega_{ij}=\langle de_i,e_j\rangle,$$ called *connection forms*, and from [\[2.2.1\]](#2.2.1){reference-type="eqref" reference="2.2.1"} we see that $\omega_{ij}=-\omega_{ji}$. A one-form $\omega$ can be written as $\omega=f(x,t)dx+g(x,t)dt$, where $f$ and $g$ are certain functions, called coefficients of the form $\omega$. We say that $\omega$ is of class $C^k$ if and only if both $f$ and $g$ are $C^k$ functions. Let $\otimes$ and $\wedge$ be the tensor and wedge products (for further details, see [@cle page 39], respectively. The dual forms $\omega_1$ and $\omega_2$, jointly with the connection forms, satisfy the following relations: $$\label{2.2.3} d\omega_1=\omega_2\wedge\omega_{21},\quad d\omega_2=\omega_1\wedge\omega_{12},$$ $$\label{2.2.4} \omega_1\wedge\omega_{13}+\omega_2\wedge\omega_{23}=0,$$ and $$\label{2.2.5} d\omega_{12}=\omega_{13}\wedge\omega_{32},\quad d\omega_{13}=\omega_{12}\wedge\omega_{23},\quad d\omega_{23}=\omega_{21}\wedge\omega_{13}.$$ It is important to note that the connection form $\omega_{12}$ is completely determined by the forms $\omega_1$ and $\omega_2$, and is known as the Levi-Civita (connection form). For this reason, it is common write $\omega_3:=\omega_{12}$. Moreover, we can define the Gaussian curvature as being the function ${\cal K}$ satisfying the relation $$\label{2.2.6} d\omega_3=-{\cal K}\,\omega_1\wedge\omega_2.$$ Equation [\[2.2.6\]](#2.2.6){reference-type="eqref" reference="2.2.6"} is called the *Gauss equation*, and it reflects the fact that the Gaussian curvature is intrinsically determined by the surface, whereas we can rewrite equations [\[2.2.3\]](#2.2.3){reference-type="eqref" reference="2.2.3"} in terms of form $\omega_3$, which reads $$\label{2.2.7} d\omega_1=\omega_3\wedge\omega_{2},\quad d\omega_2=\omega_1\wedge\omega_{3}.$$ Equations [\[2.2.6\]](#2.2.6){reference-type="eqref" reference="2.2.6"}-[\[2.2.7\]](#2.2.7){reference-type="eqref" reference="2.2.7"} are called *structure equations* of the surface $\cal M$. **Definition 1**. *Let $\omega_1$, $\omega_2$, $\omega_{13}$, and $\omega_{23}$ be given one-forms on a surface $\cal M$ in $\mathbb{E}^3$, such that $\{\omega_1,\omega_2\}$ is LI, and $p\in\cal M$. The first and second fundamental forms of $\cal M$ are defined, on each $T_p\cal M$, by $I(v)=\omega_1(v)^2+\omega_2(v)^2$ and $II(v)=\omega_{13}(v)\omega_1(v)+\omega_{23}(v)\omega_2(v)$, for each $v\in T_p\cal M$.* Commonly one writes the first and the second fundamental forms as $I=\omega_1^2+\omega_2^2$ and $II=\omega_{13}\omega_1+\omega_{23}\omega_2$, with the convection $\alpha\beta=\alpha\otimes\beta$ and $\alpha^2=\alpha\alpha$, for any (one-)forms $\alpha$ and $\beta$. We observe that everything done so far refers to a given surface $\cal M$ in the euclidean space $\mathbb{E}^3$. A quite useful result for our purposes is **Lemma 1**. *Let $\omega_1$, $\omega_2$, $\omega_{12}$, $\omega_{13}$, and $\omega_{23}$ be $C^1$ one-forms. Then they determine a local surface up to a euclidean motion if and only if $\omega_1\wedge\omega_2\neq0$ and equations [\[2.2.3\]](#2.2.3){reference-type="eqref" reference="2.2.3"}--[\[2.2.5\]](#2.2.5){reference-type="eqref" reference="2.2.5"} are satisfied.* Lemma [Lemma 1](#lemma2.1){reference-type="ref" reference="lemma2.1"} (see [@gug Theorem 10-19, page 232] and also [@gug Theorem 10-18, page 232] for its proof) is a *sine qua non* result from classical differential geometry of surfaces that enabled us to consider solutions $u\in C^{3,1}(\mathbb{R}\times(0,\infty))$ and use them to prove our theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. **Remark 1**. *The relevance of lemma [Lemma 1](#lemma2.1){reference-type="ref" reference="lemma2.1"} for us is the following: it states that if a set of given one-forms in $\mathbb{R}^3$ satisfies its conditions, then they define, at least locally, a surface $\cal M$ in the euclidean space. Such a result is sometimes fundamental theorem of surface theory, see [@ilka theorem 11, page 143], or also called Bonnet theorem, see [@cle theorem 4.39, page 127] or [@ku theorem 4.24, page 153].* Surfaces for which their Gaussian curvatures are constant and negative are called *pseudospherical* surfaces [@cle page 9]. ## Equations describing pseudospherical surfaces If we take ${\cal K}=-1$ in the structure equations [\[2.2.6\]](#2.2.6){reference-type="eqref" reference="2.2.6"}--[\[2.2.7\]](#2.2.7){reference-type="eqref" reference="2.2.7"}, we then have $$\label{2.3.1} d\omega_1=\omega_3\wedge\omega_2,\quad d\omega_2=\omega_1\wedge\omega_3,\quad d\omega_3=\omega_1\wedge\omega_2.$$ Sasaki's observation [@sasaki] can be summed up as follows: if we denote $$\label{2.3.2}\omega_i=f_{i1}dx+f_{i2}dt,\quad 1\leq i\leq 3,$$ from the AKNS method [@akns] we can determine functions $f_{ij}$ for which the corresponding triad of one-forms satisfies [\[2.2.7\]](#2.2.7){reference-type="eqref" reference="2.2.7"} and, as a consequence, they determine a PSS in an intrinsic way. Let $(x,t)$ be independent variables. A differential equation for a real valued function $u=u(x,t)$ of order $n$ is generically denoted by $$\label{2.3.3} {\cal E}(x,t,u,u_{(1)},\cdots,u_{({n)}})=0.$$ **Definition 2**. *A differential equation [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"} is said to describe a pseudospherical surface, or it is said to be of pseudospherical type, if it is a necessary and sufficient condition for the existence of differentiable functions $f_{ij}$, $1\leq i,j\leq 3$, such that the forms [\[2.3.2\]](#2.3.2){reference-type="eqref" reference="2.3.2"} satisfy the structure equations of a pseudospherical surface [\[2.3.1\]](#2.3.1){reference-type="eqref" reference="2.3.1"}.* In practical terms, given a triad of one-forms $\omega_1$, $\omega_2$ and $\omega_3$, and an equation [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"}, we can check if they describe a PSS surface in the following way: let us define a matrix of one-forms $\Omega$ by $$\label{2.3.4}\Omega=\frac{1}{2}\begin{pmatrix} \omega_2 & \omega_1-\omega_3 \\ \omega_1-\omega_3 & -\omega_2 \end{pmatrix}=:(\Omega_{ij}),\quad (\Omega\wedge\Omega)_{ij}:=(\sum_{k=1}^2\Omega_{ik}\wedge\Omega_{kj}),$$ $$\label{2.3.5} \Sigma:=d\Omega-\Omega\wedge\Omega,\quad d\Omega:=(d\Omega_{ij}).$$ If, when restricted to the manifold determined by the solutions of [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"}, the matrix $\Sigma$ vanishes, we say that [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"} is a PSS equation, and the triad $\{\omega_1,\, \omega_2,\, \omega_3\}$ satisfies the structure equations of a PSS equation with Gaussian curvature ${\cal K}=-1$. **Example 1**. *Let $m_1\in\{-2,1\}$, $\mu\in\mathbb{R}$ and consider the triad of one-forms $$\label{2.3.6} \begin{array}{lcl} \omega_1&=&\Big(u-u_{xx}\Big)dx+\Big(2u(u-u_{xx})+\psi\Big)dt,\\ \\ \omega_2&=&\Big(\mu (u-u_{xx})\pm m_{1}\sqrt{1+\mu^{2}}\Big)dx+\displaystyle{\mu\big(2u(u-u_{xx})+\psi\big)}dt,\\ \\ \omega_3&=&\Big(\pm\sqrt{1+\mu^{2}}(u-u_{xx})+m_{1}\mu \Big)dx\\ \\ &&\pm\Big(\displaystyle{\sqrt{1+\mu^{2}}\big(2u(u-u_{xx})+\psi\big)}\Big)dt, \end{array}$$ where $$\label{2.3.7} \psi:=\frac{4}{m_{1}}uu_x-2u_x^{2}-2u^{2},$$ and $$\label{2.3.8} {\cal E}=u_{t}-u_{txx}-4uu_{x}-2u_{x}^{2}-2uu_{xx}+6u_{x}u_{xx}+2uu_{xxx}.$$* *Note that ${\cal E}=0$ is nothing but [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}. It is straightforward, but lengthy, to confirm that (see [@tito Theorem 4.5]) $$\label{2.3.9} \begin{array}{lcl} d\omega_1-\omega_3\wedge\omega_2&=&{\cal E}dx\wedge dt,\quad d\omega_2-\omega_1\wedge\omega_3={\cal E}dx\wedge dt,\\ \\ d\omega_3-\omega_1\wedge\omega_2&=&\pm\sqrt{1+\mu^2}{\cal E}dx\wedge dt. \end{array}$$* *Substituting [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"}--[\[2.3.7\]](#2.3.7){reference-type="eqref" reference="2.3.7"} into [\[2.3.4\]](#2.3.4){reference-type="eqref" reference="2.3.4"}, after reckoning we conclude that [\[2.3.5\]](#2.3.5){reference-type="eqref" reference="2.3.5"} is given by $$\label{2.3.10} \Sigma=\frac{{\cal E}}{2}\begin{pmatrix} 1 & 1\pm\sqrt{1+\mu^2} \\ 1\pm\sqrt{1+\mu^2} & -1 \end{pmatrix}.$$ Therefore, $\Sigma=0$ if and only if ${\cal E}=0$, but ${\cal E}=0$ if and only if the forms [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"} satisfies the structure equations [\[2.3.1\]](#2.3.1){reference-type="eqref" reference="2.3.1"} for a PSS with ${\cal K}=-1$ in view of [\[2.3.9\]](#2.3.9){reference-type="eqref" reference="2.3.9"}. In particular, $u$ must be a solution of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}.* It is important to highlight that a *sine qua non* condition for the existence of a PSS defined on an open set $\Omega$ contained in the domain of a solution $u$ is that $\omega_1\wedge\omega_2\neq0$ whenever $(x,t)\in\Omega$, otherwise the Gaussian curvature cannot be inferred from the Gauss equation [\[2.2.6\]](#2.2.6){reference-type="eqref" reference="2.2.6"}. **Definition 3**. *Suppose that [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"} is a PSS equation with corresponding one forms satisfying [\[2.3.1\]](#2.3.1){reference-type="eqref" reference="2.3.1"}. A solution $u$ of [\[2.3.3\]](#2.3.3){reference-type="eqref" reference="2.3.3"} for which $\omega_1\wedge\omega_2\neq0$ is called *generic*, whereas those satisfying $\omega_1\wedge\omega_2=0$ are said to be non-generic.* **Example 2**. *From the one-forms $\omega_1$ and $\omega_2$ in [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"}, we obtain $$\label{2.3.11} \omega_1\wedge\omega_2=\pm\sqrt{1+\mu^2}\Big(2m_1uu_{xx}-4uu_x+2m_1u_x^2\Big)dx\wedge dt.$$* *The condition $\omega_1\wedge\omega_2=0$ on an open set $\Omega$ contained in the domain of $u$ is satisfied in the following circumstances:* - *For $m_1=-2$, then $\phi(x,t)=\pm\sqrt{ae^{-x}+b}$;* - *For $m_1=1$, then $\phi(x,t)=\pm\sqrt{ae^{2x}+b}$ or $\phi(x,t)=f(t)e^{x}$.* *Above $a$ and $b$ are real constants, whereas $f\in C^1(\mathbb{R})$.* *Let $u$ be a solution of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"}, satisfying the condition $u(x+1,t)=u(x,t)$. Then it is non-generic on an open set $\Omega$ if and only if it is constant.* ## Sobolev spaces and a few of functional analysis Let ${\cal P}[0,1]$ be the collection of all periodic functions $f:\mathbb{R}\rightarrow\mathbb{C}$ with period $1$. Given a positive integer $k$, we denote by $f^{(n)}$ its $n-$th order derivative, while the set of functions $f$ for which $f^{(n)}\in C^0(\mathbb{R})$, $0\leq n\leq k$, is denoted by $C^k(\mathbb{R})$. If $k$ is a non-negative integer, we define $C_{\text{per}}^k[0,1]=C^k(\mathbb{R})\cap{\cal P}[0,1]$. For the very particular case $k=\infty$, we write ${\cal P}$ instead of $C_{\text{per}}^\infty[0,1]$, with topological dual denoted by ${\cal P}'$. Recall that a member of ${\cal P}'$ is a continuous linear functional $f:{\cal P}\rightarrow\mathbb{C}$. The Fourier transform of $f\in{\cal P}'$ is defined by $$\hat{f}(k)=\frac{1}{2\pi}\int_{0}^1 f(x)e^{-ikx}dx.$$ Let $\ell^2(\mathbb{Z})$ be the collection of sequences $\alpha=(\alpha_n)_{n\in\mathbb{Z}}$ such that $$\sum_{n\in\mathbb{Z}}|\alpha_n|^2<\infty.$$ Given $s\in\mathbb{R}$, we denote by $\ell^2_s(\mathbb{Z})$ as the collection of $\alpha\in\ell^2(\mathbb{Z})$ such that $$\sum_{k=-\infty}^\infty (1+|k|^2)^s|\alpha_k|^2<\infty,$$ which has a structure of a Banach space when endowed with norm $$\|\alpha\|_{\ell^2_s}=\sqrt{\sum_{k=-\infty}^\infty (1+|k|^2)|\alpha_k|^2}.$$ The periodic Sobolev space of order $s$ is $H^s_{\text{per}}=\{f\in{\cal P}';\,\,(\hat{f}(k))_{k\in\mathbb{Z}}\in\ell_s^2(\mathbb{Z})\}$, and the sesquilinear form $$\big(f\big|g\big)_s=\sum_{k=-\infty}^\infty(1+|k|^2)^2\hat{f}(k)\overline{\hat{g}(k)}$$ turns it into a Hilbert space. In particular, note that $H^0_{\text{per}}=L^2[0,1]$. Let us now define the following equivalence relation: given $a,b\in\mathbb{R}$, we say that $a\sim b$ if $b=a+k$, for some integer $k$. The quotient space $\mathbb{R}/\sim$ can be identified with the set $[0,1)$, which we shall denote by $\mathbb{S}$. For this reason, henceforth we define $H^s(\mathbb{S}):=H_{\text{per}}[0,1]$. The norm of $H^s(\mathbb{S})$ will be denoted by $\|\cdot\|_s$, whereas $\|\cdot\|_\infty$ is reserved for the norm in $L^\infty$. Given two Banach spaces $X$ and $Y$, we write $X\hookrightarrow Y$ to mean that $X$ is continuously and densely embedded in $Y$. **Lemma 2**. *`(Theorem 3.193, page 201)`[\[lemma2.2\]]{#lemma2.2 label="lemma2.2"} Let $s,r\in\mathbb{R}$, with $s\geq r$. Then $H^s(\mathbb{S})\hookrightarrow H^r(\mathbb{S})$ and $\|f\|_r\leq\|f\|_s,$ for any $f\in H^s(\mathbb{S})$. In particular, $H^s(\mathbb{S})\hookrightarrow L^2[0,1]$ for any $s\geq0$.* The next result is known as Sobolev Lemma, or also as Sobolev Embedding Theorem. **Lemma 3**. *`(Theorem 3.195, page 204, Proposition 3.3, page 329)`[\[lemma2.3\]]{#lemma2.3 label="lemma2.3"} If $s>1/2$, then $H^s(\mathbb{S})\hookrightarrow C_{\text{per}}^0[0,1]$ and $\|f\|_\infty\leq c\|f\|_s$, for some constant $c$ depending only on $s$, where $f\in H^s(\mathbb{S})$. More generally, if $s>1/2+m$, where $m$ is a positive integer, then $H^s(\mathbb{S})\hookrightarrow C_{\text{per}}^m[0,1]$.* We conclude our revision on Sobolev spaces by recalling the algebra property. **Lemma 4**. *`(Theorem 3.200, page 207)`[\[lemma2.4\]]{#lemma2.4 label="lemma2.4"} If $s>1/2$, for any $f,g\in H^s(\mathbb{S})$, we have $fg\in H^s(\mathbb{S})$ and their norm satisfies the estimate $\|fg\|_s\leq c\|f\|_s\|g\|_s,$ for some constant $c>0$ depending only on $s$.* ## Semigroup Approach {#sec2.4} Let us revisit basic aspects of Kato's theory [@KatoI; @KatoII; @Pazy], also known as semigroup approach, which is our main tool for proving well-posedness of solutions with the regularity we need to make them consistent with the geometric nature of our problem. Let $X$ be a Hilbert space, and let $u(\cdot,t)\in X$ such that $$\label{qlee} u_t +A(u)u = f(u), ~~~~t\geq 0,~~~~~u(0)=u_0.$$ Let $Y\hookrightarrow X$ and $S:Y\rightarrow X$ be a topological isomorphism. Assume that - For any given $r>0$ it holds that for all $u \in \mathrm B_r(0) \subseteq Y$ (the ball around the origin in $Y$ with radius $r$), the linear operator $A(u)\colon X \to X$ generates a strongly continuous semigroup $T_u(t)$ in $X$ which satisfies $\| T_u(t) \|_{\mathcal L(X)} \leq \mathrm e^{\omega_r t}$, for all $t\in [0,\infty)$, for a uniform constant $\omega_r > 0$; - $A$ maps $Y$ into $\mathcal L(Y,X)$, more precisely the domain $D(A(u))$ contains $Y$ and the restriction $A(u)|_Y$ belongs to $\mathcal L(Y,X)$ for any $u\in Y$. Furthermore $A$ is Lipschitz continuous in the sense that for all $r>0$ there exists a constant $C_1$ which only depends on $r$ such that $\| A(u) - A(v) \|_{\mathcal L(Y,X)} \leq C_1 \, \|u-v\|_X$ for all $u,~v\in\mathrm B_r(0) \subseteq Y$. - For any $u\in Y$ there exists a bounded linear operator $B(u) \in \mathcal L(X)$ satisfying $B(u) = S A(u) S^{-1} - A(u)$ and $B \colon Y \to \mathcal L(X)$ is uniformly bounded on bounded sets in $Y$. Furthermore for all $r>0$ there exists a constant $C_2$ which depends only on $r$ such that $\| B(u) - B(v)\|_{\mathcal L(X)} \leq C_2 \, \|u-v\|_Y,$ for all $u,~v \in \mathrm B_r(0)\subseteq Y$; - For all $t\in[0,\infty)$, $f$ is uniformly bounded on bounded sets in $Y$. Moreover, the map $f\colon Y \to Y$ is locally $X$-Lipschitz continuous in the sense that for every $r>0$ there exists a constant $C_3>0$, depending only on $r$, such that $\| f(u) - f(v)\|_{X} \leq C_3 \, \|u-v\|_X$, for all $u,~v \in \mathrm B_r(0) \subseteq Y$, and locally $Y$-Lipschitz continuous in the sense that for every $r>0$ there exists a constant $C_4>0$, depending only on $r$, such that $\| f(u) - f(v)\|_{Y} \leq C_4$ $\|u-v\|_Y$, for all $u,~v \in \mathrm B_r(0) \subseteq Y$. **Lemma 5**. *[@KatoI] [\[kato\]]{#kato label="kato"} Assume that (A1)-(A4) hold. Then for given $u_0\in Y$, there is a maximal time of existence $T>0$, depending on $u_0$, and a unique solution $u$ to ([\[qlee\]](#qlee){reference-type="ref" reference="qlee"}) in $X$ such that $u=u(u_0,.)\in C^0(Y,[0,T))\cap C^1(X,[0,T)).$ Moreover, the solution depends continuously on the initial data, i.e. the map $u_0\rightarrow u(u_0,.)$ is continuous from $Y$ to $C^0(Y,[0,T))\cap C^1(X,[0,T))$.* # Proof of theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} {#sec3} We begin by noticing that the equation in ([\[1.0.5\]](#1.0.5){reference-type="ref" reference="1.0.5"}) is in the quasi-linear equation form ([\[qlee\]](#qlee){reference-type="ref" reference="qlee"}), where $$\label{ql} A(u) = -2u\partial_x$$ and $$\label{nl} f(u) = \Lambda^{-2} \partial_x \big( u^2 + (u^2)_x\big) .$$ Let $$(\Lambda^sf)(k):=\sum_{k\in\mathbb{Z}}(1+n^2)^{s/2}\hat{f}(n)e^{ink}.$$ For any $s,s'\in\mathbb{R}$, $\Lambda^s:H^{s'}(\mathbb{S})\rightarrow H^{s'-s}(\mathbb{S})$ is an isomorphism [@taylor page 330]. Then, it is natural to choose as Hilbert spaces $X\coloneqq (H^{s-1}(\mathbb{S}), \|\cdot \|_{s-1})$ and $Y\coloneqq (H^{s}(\mathbb{S}),\|\cdot \|_s)$ with $s>\frac{3}{2}$, and $S=\Lambda$ as well, and work with them using Kato's approach. We aim at proving lemmas ensuring the validity of the assumptions (A1)-(A4). For convenience, in the remaining part of this section we simply write $H^s$ in place of $H^s(\mathbb{S})$. Moreover, for a given function $g\in H^r$ with $r>1/2$ let us denote by $M_g$ the corresponding multiplication operator on $H^r$, i.e. $M_g \colon H^r \to H^r, w\mapsto g w$. Since $H^r$, $r>1/2$, is closed under multiplication, $M_g$ is continuous. Now, we verify the assumptions needed for Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"}. We start with assumption (A1): **Lemma 6**. *Let $s>3/2$. For any given $r>0$, it holds that for all $u \in \mathrm B_r(0) \subseteq H^{s}$, the linear operator $A(u)\colon H^{s-1} \to H^{s-1}$, with domain $D(A(u))\coloneqq \{ w \in H^{s-1}\colon A(u) w \in H^{s-1} \}$, generates a strongly continuous semigroup $T_u(t)$ in $X$ which satisfies $\| T_u(t) \|_{\mathcal L(X)} \leq \mathrm e^{\omega_r t}$ for all $t\in [0,\infty)$, for a uniform constant $\omega_r > 0$. In particular, the operator $A(u)$ given in [\[ql\]](#ql){reference-type="eqref" reference="ql"}, with domain $\mathcal{D}(A)=\{\omega\in H^{s-1}:A(u)\omega\in H^{s-1}\}\subset H^{s-1}$ is quasi-m-accreative in $H^{s-1}$if $u\in H^{s}$, $s>\frac{3}{2}$.* For convenience, it would be good to mention that the coefficient in [\[ql\]](#ql){reference-type="eqref" reference="ql"} does not affect the analysis. It just plays a role in constant estimation which is not of our interest. Therefore, we will neglect it and keep the operator form as $u\partial_x$. We prove this lemma in two steps. First step is given as follows: **Lemma 7**. *The operator $A(u)= u\partial_x$ in $L^2$, with $u\in H^s$, $s>\frac{3}{2}$, is\ quasi-m-accreative.* *Proof.* A linear operator $A=A(u)$ in $X$ is quasi-m-accretive if and only if [@KatoII]: 1. There is a real number $\beta$ such that $(A\omega,\omega)_X\geq -\beta\|\omega\|_X^2$ for all $\omega\in D(A)$; 2. The range of $A(u)+\lambda I$ is all of $X$ for some (or equivalently, all) $\lambda>\beta$. Note that if the above property (a) holds, then $A+\lambda I$ is dissipative for all $\lambda>\beta$. Moreover, if $A$ is a closed operator, then $A+\lambda I$ has closed range in $X$ for all $\lambda>\beta$. Hence, in order to prove (b) in such a case, it is enough to show that $A+\lambda I$ has dense range in $X$ for all $\lambda>\beta$. First we show that $A$ is a closed operator in $L^2$. Let $(v_n)_{n\in\mathbb{N}}$ be a sequence in $D(A)$ with $v_n\to v$ in $L^2$ and $A v_n \to w$ in $L^2$. Then $u v_n \in H^{1}$ for all $n\in \mathbb{N}$ by definition of $D(A)$ since an alternative way of writing the domain is $D(A)=\{\omega\in L^{2}:u\omega\in H^{1}\}$ and $v_n\in D(A)$. Moreover, both $u v_n \to u v$ and $u_x v_n \to u_x v$ in $L^2$ by the continuity of the multiplication $H^r \times L^2 \to L^2$ for $r>1/2$. Therefore,$(u v_n)_x \to w+u_x v$ in $L^2$. Having sequences $(uv_n)_{n\in\mathbb{N}}$ and $((u v_n)_x)_{n\in\mathbb{N}}$ convergent in $L^2$ implies that $(uv_n)_{n\in\mathbb{N}}$ converges in $H^1$ with the limit $uv$, thus $v\in D(A)$. Moreover the continuity of $\partial_x\colon H^1 \to L^2$ implies that $\lim_{n\to\infty} (u v_n)_x = (u v)_x$, therefore $w=(uv)_x-u_x v = A v$. Now, we take the following $L^2$ inner product $$\begin{aligned} (A(u)\omega,\omega)_{0} &=(u\partial_x\omega ,\omega)_{0} \nonumber %&=(\Lambda^{s-1}u\partial_x\omega ,\Lambda^{s-1}\omega)_{0} \nonumber \\ %&=([\Lambda^{s-1},u]\partial_x\omega ,\Lambda^{s-1}\omega)_{0}+(u\partial_x\Lambda^{s-1}\omega ,\Lambda^{s-1}\omega)_{0} \label{est}.\end{aligned}$$ We refer to Lemma [Lemma 8](#L-core){reference-type="ref" reference="L-core"} to be stated below and use integration by parts to get: $$\begin{aligned} |(u\partial_x\omega ,\omega)_{0}| =|-\frac{1}{2}(u_x, \omega^2)_0|\leq C \|u_x\|_{L^\infty}\|w\|_{0}^2\leq \tilde{C}\|\omega\|_{0}^2.\end{aligned}$$ Having $\|u\|_s$ bounded allows us to choose $\beta=\tilde{C}(\|u\|_{H^{s}})$ and to show that the operator satisfies the inequality in (a). Thus, $A(u)+\lambda I$ is dissipative for all $\lambda>\beta$. Moreover, recall that $A(u)$ is a closed operator. Therefore, we now show that $A(u)+\lambda I$ has dense range in $L^2$ for all $\lambda>\beta$. It is known that if the adjoint of an operator has trivial kernel, then the operator has dense range [@RS]. For $A(u)=u\partial_x$, the adjoint operator can be expressed $A^*(u)=-u_x-u\partial_x$. Observe that $$A^*(u)\omega=-u_x\omega-u\omega_x=-(u\omega)_x.$$ Since $u_x\in L^{\infty}$ and $\omega\in L^2$, we have $u_x\omega\in L^2$. Having also $A(u)\omega=u\omega_x\in L^2$ for $\omega\in D(A)$ reveals that $\mathcal{D}(A^*)=\{\omega\in L^2:A^{*}(u)\omega\in L^2\}$. Assume that $A(u)+\lambda I$ does not have a dense range in $L^2$. Then, there exists $0\neq z\in L^2$ such that $((A(u)+\lambda I)\omega,z)_0=0$ for all $\omega\in \mathcal{D}(A)$. Since $H^1\subset \mathcal{D}(A)$, $\mathcal{D}(A)=\mathcal{D}(A^*)$ is dense in $L^2$. It means that there exists a sequence $z_k\in \mathcal{D}(A^*)$ such that it converges to an element $z\in L^2$. Recall that $D(A^*)$ is closed. So, $z\in \mathcal{D}(A^*)$. Moreover, $$((A(u)+\lambda I)\omega,z)_{0}=(\omega,(A(u)+\lambda I)^*z)_{0}=0$$ reveals that $(A^*(u)+\lambda I) z=0$ in $L^2$. Multiplying by $z$ and integrating by parts, we get $$0=((A^*(u)+\lambda I)z,z)_{0}=(\lambda z,z)_{0}+(z,A(u)z)_{0}\geq (\lambda-\beta)\|z\|_{0}^2~~~\forall \lambda>\beta$$ and thus, $z=0$, which contradicts our assumption. It completes the proof of (b). Therefore, the operator $A(u)$ is quasi-m-accreative. ◻ In the proof of Lemma [Lemma 6](#opA){reference-type="ref" reference="opA"} we use the fact that $C^{\infty}(\mathbb{S})$ is a *core* for $A$ in $H^{s-1}$, i.e. $A(u)v$ can be approximated by smooth functions in $H^{s-1}$ ([@CE2]): **Lemma 8**. *Given $v\in D(A)$ there exists a sequence $(v_n)_{n\in \mathbb{N}}$ in $\mathcal C^{\infty}$ such that both $v_n \to v$ and $A v_n \to Av$ in $H^{s-1}$.* *Proof.* Let $v\in D(A)$ and fix $\rho\in C_c^\infty$, where $C_c^\infty$ denotes the set of $C^\infty$ functions with compact support, with $\rho \geq 0$ and $\int_{\mathbb{R}} \rho =1$. Given $n\geq 1$, let $\rho_n= n\rho(nx)$. If we set $v_n\coloneqq \rho_n * v$, then $v_n \in C_c^\infty$ for $n\geq 1$ and $v_n \to v$ in $H_p^{s-1}$. We have to prove that $(uv_n)_x \to (uv)_x$ in $H^{s-1}$. Since $v\in D(A)$, we have that $uv_x\in H^{s-1}$ and hence $\rho_n *(uv_x)\rightarrow uv_x$. Moreover, since $uv_n\in H^s$ it follows that $(uv_n)_x = u_x v_n + u(v_n)_x \in H^{s-1}$, hence we have that $u_xv_n\rightarrow u_x v$ in $H^{s-1}$. Therefore $$\begin{aligned} (uv_n)_x-(uv)_x&=u_xv_n-u_xv +\rho_n *(uv_x)-uv_x + u(v_n)_x - \rho_n*(uv_x)\end{aligned}$$ holds true and it suffices to show that $u(v_n)_x-\rho_n*(uv_x) \to 0$ in $H^{s-1}$. To this end, denote $$P_n v \coloneqq u(v_n)_x-\rho_n*(uv_x), \quad n\geq 1.$$ We will show that there exists $K>0$ independent of $v$ such that $$\label{eq-Pn} \|P_n v \|_{s-1} \leq K\|v\|_{s-1}, \quad n\geq 1.$$ That will enable us to conclude that $P_n$ is uniformly bounded in $H^{s-1}$ by the uniform boundedness principle. When we approximate $v$ in $H^{s-1}$ by smooth functions, and use this conclusion, we will be able to prove the assertion $P_n\to 0$ for $v\in C_c^\infty$. Since the set of smooth functions is dense in $H^{s-1}$ and $P_n$ are uniformly bounded, the proof will be completed. We first notice that $$\begin{aligned} P_n v(x) &= \int_{\mathbb{R}}(\rho_n)_y(y) (u(x) - u(x-y))v(x-y) dy +(\rho_n*(u_x v))(x)\\ &= n^2\int_{\mathbb{R}} \rho_y(ny) (u(x)-u(x-y)) v(x-y)dy+(\rho_n*(u_x v))(x)\\ &= n\int_{-1}^1 \rho_y(y) (u(x)-u(x-\frac{y}{n})) v(x-\frac{y}{n})dy+(\rho_n*(u_x v))(x),\end{aligned}$$ where $\rm supp(\rho)\subset [-1,1]$. Moreover, using the mean value theorem, we obtain the estimate $$\begin{aligned} &\Big|n^2\int_{\mathbb{R}} \rho_y(ny) (u(x)-u(x-y)) v(x-y)dy\Big|= \Big|n^2\int_{\mathbb{R}} \rho_y(ny) u_x(x_0) y v(x-y)dy\Big|\\ &= \Big|\int_{-1}^1\rho_y(y) u_x(x_0) y v(x-y)dy\Big|\leq \|u_x\|_{L^\infty}\int_{-1}^1 |\rho_y(y)|\,|y|\,|v(x-\frac{y}{n})|dy,\end{aligned}$$ for some $x_0\in (x,x-y)$. Let now $C\coloneqq \sup_{x\in\mathbb{R}}\|u_x\|_{L^\infty}^2\int_{-1}^1 |\rho_y(y)y|^2dy$. Then the Cauchy-Schwarz inequality, Fubini's theorem and the fact that the operator $\Lambda^{s-1}$ commutes with integration yield that $$\begin{aligned} &\|n^2\int_{\mathbb{R}} \rho_y(ny) (u(x)-u(x-y)) v(x-y)dy\|^2_{s-1}\\ &=\|\Lambda^{s-1}\int_{-1}^1 \rho_y(y) u_x(x_0)y (v(x-\frac{y}{n}))dy\|^2_{2}\\ &=\int_{\mathbb{R}} \left|\int_{-1}^1 \rho_y(y) u_x(x_0)y \Lambda^{s-1}v(x-\frac{y}{n})dy\right|^2 dx\\ &\leq C \int_{-1}^1\int |\Lambda^{s-1}v(x-\frac{y}{n})|^2 dx dy\leq 2C\|v\|_{s-1}.\end{aligned}$$ Moreover, we obtain by Plancherel's theorem that $$\begin{aligned} \|\rho_n*(u_x v)\|_{s-1} &= \|\Lambda^{s-1}(\rho_n*(u_x v))\|_{2} = \|\rho_n*\Lambda^{s-1}(u_x v))\|_{2} &\leq \|\Lambda^{s-1}(u_x v)\|_2\\ &\leq \|u_x\|_{L^\infty}\|v\|_{s-1}.\end{aligned}$$ Therefore we conclude that $$\label{bound} \|P_n v\|_{s-1}\leq (\sqrt{2C}+\|u_x\|_{L^\infty})\,\|v\|_{s-1},~~n\geq 1.$$ For $K=\sqrt{2C}+\|u_x\|_{L^\infty}$ in ([\[eq-Pn\]](#eq-Pn){reference-type="ref" reference="eq-Pn"}), proof is completed by the estimate ([\[bound\]](#bound){reference-type="ref" reference="bound"}). ◻ Second step to prove Lemma [Lemma 6](#opA){reference-type="ref" reference="opA"} is making use of the following lemma proved in [@Pazy]: **Lemma 9**. *Let $X$ and $Y$ be two Banach spaces such that $Y$ is continuously and densely embedded in $X$. Let $-A$ be the infinitesimal generator of the $C_0$-semigroup $T(t)$ on $X$ and let $Q$ be an isomorphism from $Y$ onto $X$. Then $Y$ is $-A$-admissible (i.e. $T(t)Y\subset Y$ for all $t\geq 0$, and the restriction of $T(t)$ to $Y$ is a $C_0$-semigroup on $Y$) if and only if $-A_1=-QAQ^{-1}$ is the infinitesimal generator of the $C_0$-semigroup $T_1(t)=QT(t)Q^{-1}$ on $X$. Moreover, if $Y$ is $-A$-admissible, then the part of $-A$ in $Y$ is the infinitesimal generator of the restriction $T(t)$ to $Y$.* Before we proceed with the proof of Lemma [Lemma 6](#opA){reference-type="ref" reference="opA"}, we give the commutator estimate which will be used: **Lemma 10** ([@Taylor]). *Let $m> 0$, $s\geq 0$ and $3/2<s+m\leq \sigma$. Then for all $f\in H^{\sigma}$ and $g\in H^{s+m-1}$ one has $\|[\Lambda^{m},f]g\|_{s} \leq C\|f\|_{\sigma} \, \|g\|_{s+m-1},$ where $C$ is a constant which is independent of $f$ and $g$.* **Proof of Lemma [Lemma 6](#opA){reference-type="ref" reference="opA"}**: Following the arguments in the proof of Lemma [Lemma 7](#pre){reference-type="ref" reference="pre"}, we first take the following $H^{s-1}$ inner product $$\begin{aligned} (A(u)\omega,\omega)_{s-1} &=(u\partial_x\omega ,\omega)_{s-1}=(\Lambda^{s-1}u\partial_x\omega ,\Lambda^{s-1}\omega)_{0} \nonumber \\ &=([\Lambda^{s-1},u]\partial_x\omega ,\Lambda^{s-1}\omega)_{0}+(u\partial_x\Lambda^{s-1}\omega ,\Lambda^{s-1}\omega)_{0} \label{est}.\end{aligned}$$ Using Cauchy-Schwartz's inequality and Lemma [Lemma 10](#L-comm){reference-type="ref" reference="L-comm"} with $m=s-1$, $\sigma=s$, we get the following estimate for the first term of ([\[est\]](#est){reference-type="ref" reference="est"}): $$\begin{aligned} |([\Lambda^{s-1},u]\partial_x\omega ,\Lambda^{s-1}\omega)_{0}|&\leq C\|u\|_s\|\partial_x \omega\|_{s-2}\|\omega\|_{s-1}\leq \tilde{C}\|\omega\|_{s-1}^2,\end{aligned}$$ for some constant $\tilde{C}$ depending on $\|u\|_s$. For the second term of ([\[est\]](#est){reference-type="ref" reference="est"}), we again refer to Lemma [Lemma 8](#L-core){reference-type="ref" reference="L-core"} and use integration by parts to get: $$\begin{aligned} |(u\partial_x\Lambda^{s-1}\omega ,\Lambda^{s-1}\omega)_{0}| =|-\frac{1}{2}(u_x, (\Lambda^{s-1}\omega)^2)_0|\leq C \|u_x\|_{L^\infty}\|w\|_{s-1}^2\leq \tilde{C}\|\omega\|_{s-1}^2.\end{aligned}$$ Choosing $\beta=\tilde{C}(\|u\|_{H^{s}})$, the operator satisfies the required inequality. Moreover, let $Q:=\Lambda^{s-1}$ and notice that $Q$ is an isomorphism of $H^{s-1}$ to $L^2$ and $H^{s-1}$ is continuously and densely imbedded into $L^2$ as $s>\frac{3}{2}$.\ Define $$\begin{aligned} A_1(u)=QA(u)Q^{-1}=\Lambda^{s-1}A(u)\Lambda^{1-s}=\Lambda^{s-1}u\partial_x\Lambda^{1-s}=\Lambda^{s-1}u\Lambda^{1-s}\partial_x,\end{aligned}$$ and let $\omega\in L^2$ and $u\in H^s$, $s>\frac{5}{2}$. Then write $B_1(u)=A_1(u)-A(u)$ and consider the following estimate: $$\begin{aligned} \|B_1(u)\omega\|_0&=\|[\Lambda^{s-1}, A(u)]\Lambda^{1-s}\omega\|_0=\|[\Lambda^{s-1}, u]\Lambda^{1-s}\partial_x\omega\|_0\\ &\leq C \|u\|_s\|\Lambda^{1-s}\partial_x\omega\|_{s-2}\leq C\|u\|_s\|\omega\|_0,\end{aligned}$$ where we applied Lemma [Lemma 10](#L-comm){reference-type="ref" reference="L-comm"} with $m=s-1$ and $\sigma=s$. Hence, we obtain $B_1(u)\in \mathcal{L}(L^2).$ Recall from Lemma [Lemma 7](#pre){reference-type="ref" reference="pre"} that $A(u)$ is quasi-m-accretive in $L^2$, i.e. $-A(u)$ is the infinitesimal generator of a $C_0$-semigroup on $L^2$. Thus, $A_1(u)=A(u)+B_1(u)$ is also the infinitesimal generator of a $C_0$-semigroup in $L^2$ by means of a perturbation theorem for semigroups (see [@Pazy]). Lemma [Lemma 9](#adm){reference-type="ref" reference="adm"} reveals that for $Y=H^{s-1}$, $X=L^2$ and $Q=\Lambda^{s-1}$, $H^{s-1}$ is $A$-admissible. Hence, $-A(u)$ is the infinitesimal generator of a $C_0$-semigroup on $H^{s-1}$. 0◻ We continue with the proof of assumption (A2): **Lemma 11**. *$A$ maps $H^s$ into $\mathcal L(H^s,H^{s-1})$, more precisely the domain $D(A(u))$ contains $H^s$ and the restriction $A(u)|_{H^s}$ belongs to $\mathcal L(H^s,H^{s-1})$ for any $u\in H^{s}$. Furthermore $A$ is Lipschitz continuous in the sense that for all $r>0$ there exists a constant $C_1$ which only depends on $r$ such that $$\label{A_Lipschitz} \| A(u) - A(v) \|_{\mathcal L(H^{s},H^{s-1})} \leq C_1 \, \|u-v\|_{s-1}$$ for all $u,~v$ inside $\mathrm B_r(0) \subseteq H^{s}$.* *Proof.* The operator $A(u)|_{H^s}$ belongs to $\mathcal L(H^s,H^{s-1})$ for any $u\in H^s$, since $\partial_x \in \mathcal L(H^s,H^{s-1})$ and $M_{u}\in \mathcal L(H^{s-1})$. To see that the required estimate is satisfied, let $u,v, w \in H^s$ be arbitrary. Then, $$\begin{aligned} \|(A(u)-A(v))w\|_{s-1} &=\|(u - v ) \partial_x w\|_{s-1} \leq C\|u-v\|_{s-1} \, \|\partial_x w\|_{s-1} \\ &\leq C \|u-v\|_{s-1} \, \| w\|_s , \end{aligned}$$ where $C$ denotes a generic constant. This shows that for arbitrary $r>0$ one can always find a constant $C_1$ such that [\[A_Lipschitz\]](#A_Lipschitz){reference-type="eqref" reference="A_Lipschitz"} holds uniformly for all $u,v \in \mathrm B_r(0) \subseteq H^s$. ◻ The last two assumptions (A3)-(A4) are proved by the help of the following commutator and product estimates stated in [@Lannes Proposition B.10.(2)] and [@KatoI Lemma A1], respectively: **Lemma 12**. *Let $r>1/2$.* 1. *[\[com_est_2\]]{#com_est_2 label="com_est_2"} If $-1/2 < t \leq r +1$, there exists a constant $C_{r,t}>0$ such that $$\big\|[\Lambda^t,M_g]h\big\|_0 \leq C_{r,t} \, \|g\|_{r+1} \, \|h\|_{t-1}$$* *for all $g\in H^{r +1}$ and $h\in H^{t-1}$.* 2. *[\[com_est_3\]]{#com_est_3 label="com_est_3"} If $-r < t\leq r$, there exists a constant $C_{r,t}>0$ such that $\|fg\|_t\leq C_{r,t} \, \|f\|_r \, \|g\|_t$, for all $f\in H^{r}$ and $g\in H^{t}$.* Even though Lemma [Lemma 12](#L_comm){reference-type="ref" reference="L_comm"} is stated on the real line (in general $\mathbb{R}^m$), it holds on periodic domain as well. Now, we define a bounded linear operator and prove assumption (A3): **Lemma 13**. *For any $u\in H^{s}$ there exists a bounded linear operator $B(u) \in \mathcal L(H^{s-1})$ satisfying $B(u) = \Lambda A(u) \Lambda^{-1} - A(u)$ and $B \colon H^{s} \to \mathcal L(H^{s-1})$ is uniformly bounded on bounded sets in $H^{s}$. Furthermore for all $r>0$ there exists a constant $C_2$ which depends only on $r$ such that $$\label{lem_B_estimate} \| B(u) - B(v)\|_{\mathcal L(H^{s-1})} \leq C_2 \, \|u-v\|_{s}$$ for all $u,v \in \mathrm B_r(0)\subseteq H^{s-1}$. Here, $A(u)$ is the operator given by [\[ql\]](#ql){reference-type="eqref" reference="ql"}.* *Proof.* Let $u\in H^s$. Since $\partial_x$ commutes with $\Lambda$ and $\Lambda^{-1}$ we obtain that $$B(u)=\Lambda u\partial_x\Lambda^{-1}-u\partial_x =[\Lambda,u\partial_x]\Lambda^{-1} =[\Lambda,u]\Lambda^{-1}\partial_x.$$ Hence we can write $\Lambda^{s-1} B(u)$ as $$\begin{aligned} \Lambda^{s-1}[\Lambda,u]\Lambda^{-1}\partial_x&=\Lambda^{s}u\Lambda^{-1}\partial_x-\Lambda^{s-1}u\partial_x=[\Lambda^{s},u]\Lambda^{-1}\partial_x+u\Lambda^{s-1}\partial_x-\Lambda^{s-1}u\partial_x\\ &=[\Lambda^{s},u]\Lambda^{-1}\partial_x +[u,\Lambda^{s-1}]\partial_x.\end{aligned}$$ Let now $\omega\in H^{s-1}$ and $u,v \in H^s$ be arbitrary. In view of the above identity and Lemma [Lemma 12](#L_comm){reference-type="ref" reference="L_comm"}, we obtain the following estimate $$\begin{aligned} &\|(B(u)-B(v))\omega\|_{s-1} = \|\Lambda^{s-1}(B(u)-B(v))\omega\|_0 \\ &\leq\|[\Lambda^{s},u-v]\Lambda^{-1}\partial_x\omega\|_0 +\|[u-v,\Lambda^{s-1}]\partial_x\omega\|_0\\ &\leq C \|u-v\|_s(\|\Lambda^{-1}\partial_x\omega\|_{s-1}+\|\partial_x\omega\|_{s-2})\leq C \|u-v\|_s\|\omega\|_{s-1},\end{aligned}$$ where $C$ is a generic constant independent of $u,w$ and $w$. In particular, this shows that $B(u)$ extends to a bounded linear operator on $H^{s-1}$ for every $u\in H^s$ such that $B\colon H^s \to \mathcal L(H^{s-1})$ is uniformly bounded on bounded sets in $H^s$. Furthermore, this estimation proves that there exists a constant $C_2$ depending only on the radius of the ball $\mathrm B_r(0) \subseteq H^s$ such that [\[lem_B\_estimate\]](#lem_B_estimate){reference-type="eqref" reference="lem_B_estimate"} is satisfied for all $u,v\in B_r(0)$. ◻ The last assumption (A4) is proved in Lemma [Lemma 14](#A4){reference-type="ref" reference="A4"}: **Lemma 14**. *For all $t\in[0,\infty)$, $f$ is uniformly bounded on bounded sets in $H^s$. Moreover, the map $f\colon H^{s}\to H^{s}$ is locally $H^{s-1}$-Lipschitz continuous in the sense that for every $r>0$ there exists a constant $C_3>0$, depending only on $r$, such that $\| f(u) - f(v)\|_{s-1} \leq C_3 \, \|u - v\|_{s-1}$ for all $u,v \in \mathrm B_r(0) \subseteq H^{s}$ and locally $H^{s}$-Lipschitz continuous in the sense that for every $r>0$ there exists a constant $C_4>0$, depending only on $r$, such that $\| f(u) - f(v)\|_{s} \leq C_4 \, \|u - v\|_{s}$ for all $u,v \in \mathrm B_r(0) \subseteq H^{s}$.* *Proof.* Recall that $f(u) = \Lambda^{-2} \partial_x \big( u^2 + (u^2)_x\big).$ Therefore, with the help of Lemma [Lemma 12](#L_comm){reference-type="ref" reference="L_comm"} $$\begin{aligned} \|f(u)-f(v)\|_{s-1} &\leq C \|(u^2-v^2)+(u^2-v^2)_x\|_{s-2}\\ &\leq C \|(u+v)(u-v)+((u+v)(u-v))_x\|_{s-2}\\ &\leq C (\|(u+v)\|_{s-2}\|(u-v)\|_{s-1}+\|u+v\|_{s-1}\|u-v\|_{s-1})\\ &\leq C_3 \|u-v\|_{s-1}\end{aligned}$$ where $C_3$ is a constant depending on $\|u\|_{H^{s}}$ and $\|v\|_{H^{s}}$. This proves $H^{s-1}$-Lipschitz continuity. Similar arguments will show that we have the following estimates: $$\begin{aligned} \label{lips} \|f(u)-f(v)\|_{s}\leq& C_4 \|u-v\|_{s}\end{aligned}$$ where $C_4$ is also a constant depending on $\|u\|_{H^{s}}$ and $\|v\|_{H^{s}}$. Since we choose $u_0\in H^{s}$, this estimate actually corresponds to the proof of continuous dependence on the initial data. Note that boundedness of $f(u)$ on bounded subsets $\{u\in H^s: \|u\|_s\leq M\}$ of $H^s$ (for all $M$) can be obtained from ([\[lips\]](#lips){reference-type="ref" reference="lips"}) by choosing $v=0$. Hence, we get the estimates for (A4). ◻ # Proof of theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"} {#sec4} In order to prove theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"} we need some estimates for $u,\,u_x$ and $u_{xx}$. To this end, we need the next result. **Theorem 4**. *Assume that $u_0\in H^3(\mathbb{S})$, $m_0(x):=u_0(x)-u''_{0}(x)$, and let $u$ be the corresponding solution of [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"}. If $m_0(x)\geq0$, $x\in\mathbb{S}$, then $m(x,t):=u(x,t)-u_{xx}(x,t)$ is non-negative for any $t$ as long as the solution exists, and any $x\in\mathbb{S}$. Moreover, $u$ is also non-negative. In particular, if $m_0>0$, then $u>0$.* An analogous result for non-periodic problems was proved in [@li-na Lemma 5.5], and following the same steps we get the demonstration for Theorem [Theorem 4](#thm4.1){reference-type="ref" reference="thm4.1"}. For this reason it is omitted. There is one more fact regarding the $L^\infty$ norm of $u_x$. We begin by observing that $$\int_{\mathbb{S}}u_{xx}dx=0.$$ This fact is enough to guarantee the existence of a point $\xi_t-1\in(0,1)$ such that $u_x(t,\xi_t-1)=0$, for each $t\in(0,T)$,. **Lemma 15**. *If $u_0\in H^3(\mathbb{S})\cap L^1(\mathbb{S})$, is such that $m_0\geq 0$, then there exists a constant $K>0$ such that the solution of [\[1.0.5\]](#1.0.5){reference-type="eqref" reference="1.0.5"} satisfies $\|u_x\|_{L^\infty(\mathbb{S})}\leq K$.* *Proof.* Let us first assume that $\|m(\cdot,t)\|_{L^1}(\mathbb{S})$ is constant for any $t$ as long as the solution exists. Assume $m_0$ does not change sign and $m_0\geq 0$. Then, $$\begin{aligned} K_1&=&\|m_0\|_{L^1(\mathbb{S})}=\int_\mathbb{S}m_0(r)dr=\int_\mathbb{S}m(r,t)dr=\int_{\xi_{t}-1}^{\xi_t}m(r,t)dr \\ &\geq&\int_{\xi_{t}-1}^x (u-u_{xx})(r,t)dr=\int_{\xi_{t}-1}^x u(r,t)dr-u_x(x,t)\geq -u_x(x,t)\end{aligned}$$ holds for every $x\in[\xi_{t}-1, \xi_t]$. Here, we use Theorem [Theorem 4](#thm4.1){reference-type="ref" reference="thm4.1"}, which guarantee that $u$ does not change sign, under the assumption that $m_0$ does not change sign. Taking into account the final result, we observe that $u_x$ is bounded from below. Moreover, $$\begin{aligned} K_1=\int_{\xi_{t}-1}^{\xi_t}m(r,t)dr \geq\int_x^{\xi_{t}} m(r,t)dr=\int_x^{\xi_{t}} u(r,t)dr+u_x(x,t)\geq u_x(x,t).\end{aligned}$$ Hence, $u_x$ is bounded also from above. Therefore, we can conclude that $\|u_x\|_{\infty}$ norm is bounded provided that $m$ does not change sign , i.e. $\|u_x\|_{\infty}\leq K$. The case $m_0\leq0$ is proved in a similar way and, therefore, is omitted. We now complete the demonstration proving that $\|m(\cdot,t)\|_{L^1(\mathbb{S})}$ is constant. We begin by noticing that [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} is itself a conservation law, in the sense that $$\partial_t(u-u_{xx})=\partial_x\Big((2-\partial_x)(1+\partial_x)u^2\Big)=\partial_x\Big((1-\partial_x^2)u^2+u^2)\Big).$$ Integrating the relation above with respect to $x$ on $\mathbb{S}$, we obtain $$\frac{d}{dt}\int_\mathbb{S}(u-u_{xx})dx=\Big((1-\partial_x^2)u^2+u^2)\Big)\big|_\mathbb{S}=0,$$ meaning that the $\|m(\cdot,t)\|_{L^1(\mathbb{S})}=const.$. Since $m_0\in L^1(\mathbb{S})$, we conclude that $\|m(\cdot,t)\|_{L^1(\mathbb{S})}=\|m_0\|_{L^1(\mathbb{S})}.$ ◻ Now we start proving Theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"}: First, we rewrite the equation ([\[1.0.5\]](#1.0.5){reference-type="ref" reference="1.0.5"}) in the following form $$u_t-2uu_x+u^2=\Lambda^{-2}(u^2+(u^2)_x)$$ by using $\Lambda^{-2}(f(u))_{xx}=\Lambda^{-2}f(u)-f(u)$. Calling $f(u)=u^2+(u^2)_x$ and observing that $2uu_x=(u^2)_x$, we get $$\label{4.0.1} u_t+(1-\partial_x)u^2=\Lambda^{-2}f(u).$$ Now, we will differentiate ([\[4.0.1\]](#4.0.1){reference-type="ref" reference="4.0.1"}) with respect to $x$, simplify and write $$\label{4.0.2} u_{tx}+(\partial_x-\partial_x^2)u^2=\Lambda^{-2}f(u)-u^2.$$ We continue this process and get the following equations: $$\label{4.0.3} u_{txx}+(\partial_x^2-\partial_x^3)u^2=\Lambda^{-2}f(u)-f(u),$$ $$\label{4.0.4} u_{txxx}+(\partial_x^3-\partial_x^4)u^2=\partial_x\Lambda^{-2}f(u)-\partial_x f(u),$$ $$\label{4.0.5} u_{txxxx}+(\partial_x^4-\partial_x^5)u^2=\Lambda^{-2}f(u)-f(u)-\partial_x^2 f(u).$$ Moreover, we will multiply ([\[4.0.1\]](#4.0.1){reference-type="ref" reference="4.0.1"}) by $u$, ([\[4.0.2\]](#4.0.2){reference-type="ref" reference="4.0.2"}) by $u_x$, ([\[4.0.3\]](#4.0.3){reference-type="ref" reference="4.0.3"}) by $u_{xx}$, ([\[4.0.4\]](#4.0.4){reference-type="ref" reference="4.0.4"}) by $u_{xxx}$, ([\[4.0.5\]](#4.0.5){reference-type="ref" reference="4.0.5"}) by $u_{xxxx}$ and integrate all over $\mathbb{S}$. Let $I(u)=\int_{\mathbb{S}}(u^2+u_x^2+u_{xx}^2+u_{xxx}^2+u_{xxxx}^2) dx$. Therefore, summing up equations ([\[4.0.1\]](#4.0.1){reference-type="ref" reference="4.0.1"})-([\[4.0.5\]](#4.0.5){reference-type="ref" reference="4.0.5"}) we obtain $$\begin{aligned} &&\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u(1-\partial_x)u^2+u_x (\partial_x-\partial_x^2)u^2+u_{xx}(\partial_x^2-\partial_x^3)u^2\\ &&+u_{xxx}(\partial_x^3-\partial_x^4)u^2+u_{xxxx}(\partial_x^4-\partial_x^5)u^2)dx\\ &&=\int_{\mathbb{S}} (u\Lambda^{-2}f(u)+u_x\Lambda^{-2}f(u)-u^2u_x+u_{xx}(\Lambda^{-2}f(u)-f(u))\\ &&+u_{xxx}(\partial_x\Lambda^{-2}f(u)-\partial_x f(u))+u_{xxxx}(\Lambda^{-2}f(u)-f(u)-\partial_x^2f(u))dx.\end{aligned}$$ Our main aim is to obtain $I(u)$, which is equivalent to $H^4$ norm of $u$, within the equation so that Gronwall's inequality is applicable and we get an upper bound valid for all time. That bound will imply the global existence of solution. After integration by parts, we can rewrite the equality in the following form: $$\begin{aligned} &&\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^3+2(u^2)_x u_{xx}+(u^2)_{xx} u_{xxx}+2(u^2)_{xxx}u_{xxxx}\\ &&+(u^2)_{xxxx} u_{xxxx}+(u^2)_{xxxx} u_{xxxxx}+u^2u_{xxxx})dx=\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx.\end{aligned}$$ Since $$\begin{aligned} &&(u^2)_x=2uu_x,\quad (u^2)_{xx}= 2u_x^2+2uu_{xx},\quad (u^2)_{xxx}=6u_xu_{xx}+2uu_{xxx},\\ &&(u^2)_{xxxx}=6u_{xx}^2+8u_xu_{xxx}+2uu_{xxxx},\\ &&(u^2)_{xxxxx}=20u_{xx}u_{xxx}+10u_{x}u_{xxxx}+2uu_{xxxxx},\end{aligned}$$ and integrating by parts once more, the integral becomes $$\begin{aligned} &&\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^3-2(u^2)_{xx} u_{x}-(u^2)_{xxx} u_{xx}-2(u^2)_{xxxx}u_{xxx}\nonumber\\ &&-(u^2)_{xxxxx} u_{xxx}-(u^2)_{xxxxx} u_{xxxx}+(u^2)_{xx}u_{xx})dx\nonumber\\ &=&\frac{1}{2}\frac{d}{dt}I(u)+\int_{\mathbb{S}}(u^3-4u_x^3-4uu_xu_{xx}-4uu_{xx}^2-2uu_{xx}u_{xxx}\nonumber\\ &&-16u_xu_{xxx}^2-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^2\nonumber\\ &&-10u_xu_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_xu_{xxxx}^2\nonumber\\ &&-2uu_{xxxx}u_{xxxxx})dx=\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx\label{4.0.6}.\end{aligned}$$ By Lemma [Lemma 15](#lem4.1){reference-type="ref" reference="lem4.1"}, we have that $\|u_x\|_{\infty}$ is bounded. Moreover, we can rewrite ([\[1.0.1\]](#1.0.1){reference-type="ref" reference="1.0.1"}) as $$u_t-u_{txx}=4uu_x+2u_x^2+2uu_{xx}-6u_xu_{xx}-2uu_{xxx}.$$ Multiplying equation above by $u$, noting that $$\begin{aligned} 2uu_xu_{xx}&=&\partial_x(uu_x^2)-u_x^3, \quad u^2u_{xxx}=\partial_x(u^2u_{xx}-uu_x^2)+u_x^3,\\ u^2u_{xx}&=& \partial_x(u^2u_{x})-2uu_x^2,\end{aligned}$$ integrating over $\mathbb{S}$ and using the identities above, we obtain $$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int_\mathbb{S}(u^2+u_x^2)dx=\int_\mathbb{S}(4u_x^3-2uu_x^2)dx\leq 5\|u_x\|_{\infty} \int_\mathbb{S}(u^2+u_x^2)dx,\end{aligned}$$ which implies $$\|u\|_1^2\leq \|u_0\|_{1}^2 e^{10\int_0^t\|u_x\|_{\infty}d\tau} \leq \|u_0\|_{1}^2 e^{At}=C_0^2,$$ for some optimal constant $A>0$ since $\|u_x\|_{\infty}$ norm is bounded. This estimate is valid at any finite time, therefore is a global bound for $H^1$ norm of $u$. The reason we provide this inequality is to verify that $\|u\|_{\infty}$ is also bounded, since $\|u\|_{\infty} \leq \|u\|_{1} \leq C_0$ by Sobolev embedding theorem. Moreover, we can show that $H^2$ norm of $u$ will be bounded in finite time: Let $J(u)=\int_{\mathbb{S}}(u^2+u_x^2+u_{xx}^2)dx$. Following the arguments done above for ([\[4.0.1\]](#4.0.1){reference-type="ref" reference="4.0.1"})-([\[4.0.3\]](#4.0.3){reference-type="ref" reference="4.0.3"}) and using integration by parts, $$\begin{aligned} \frac{1}{2}\frac{d}{dt}J(u)&+&\int_{\mathbb{S}}(2u^3+(u^2)_{x} u_{x}+2(u^2)_{x} u_{xx}+(u^2)_{xx}u_{xx}+(u^2)_{xx}u_{xxx})dx\\ &\leq& \int_{\mathbb{S}}2u(\Lambda^{-2}f(u))dx.\end{aligned}$$ Since $\|u\|_\infty$ and $\|u_x\|_\infty$ are bounded, $$\begin{aligned} \frac{1}{2}\frac{d}{dt}J(u)&\leq&2\max{(\|u\|_{\infty}, \|u_x\|_{\infty})}J(u)+2\max{(\|u\|_\infty)}\int_{\mathbb{S}}(\Lambda^{-2}f(u))dx.\end{aligned}$$ As it was given in ([\[1.0.6\]](#1.0.6){reference-type="ref" reference="1.0.6"}), $\Lambda^{-2}f=g*f$. Now, we need to estimate the $L^\infty-$norm of the integrand of the second term in order to obtain a differential inequality and apply Gronwall's inequality. For this purpose, we first provide the following two estimates: $$\begin{aligned} \|g\|_{2}\leq \frac{1}{2}(\frac{e^2+2e-1}{e^2-2e+1})^{1/2}:=n_2, \quad \|g\|_{\infty}\leq \frac{1}{2}(\frac{e+1}{e-1}):=n_{\infty}.\end{aligned}$$ Hence, $$\begin{aligned} &&\|g*u^2\|_{\infty}\leq \|g\|_{\infty} \|u^2\|_{1}\leq \|g\|_{\infty}\|u\|_{2}^2\leq n_{\infty} C_0,\\ &&\|g*(u^2)_x\|_{\infty}\leq\|g\|_{2}\|(u^2)_x\|_{2}\leq \|g\|_{2} \|u^2\|_{1}\leq n_2 C_0^2.\end{aligned}$$ These estimates, together with Gronwall's inequality, provide the boundedness of $J(u)$ which is equivalent to $H^2$ norm. Therefore, we will be able to give an upper bound for $\|u_{x}\|_\infty$ norm as well since $\|u_x\|_{\infty}\leq \|u_x\|_1<\infty.$ Proving that $H^3$ norm is bounded in finite time will be the last issue to conclude the proof of Theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"}:\ Let $K(u)=\int_{\mathbb{S}}(u^2+u_x^2+u_{xx}^2++u_{xxx}^2)dx$. Like we did for $J(u)$, we can evaluate the following inequality: $$\begin{aligned} \frac{1}{2}\frac{d}{dt}K(u)&+&\int_{\mathbb{S}}(2u^3+(u^2)_{x} u_{x}+2(u^2)_{x} u_{xx}+(u^2)_{xx}u_{xx}\\ &+&(u^2)_{xx}u_{xxx}+(u^2)_{xxx}u_{xxx}+(u^2)_{xxx}u_{xxxx})dx\\ &\leq& \int_{\mathbb{S}}(2u(\Lambda^{-2}f(u))+u_x(\Lambda^{-2}f(u))+u_{xxx}\partial_x(\Lambda^{-2}f(u)))dx,\end{aligned}$$ and $$\begin{aligned} \frac{1}{2}\frac{d}{dt}K(u)&\leq& 2\max{(\|u\|_{\infty}, \|u_x\|_{\infty}})K(u)+\int_{\mathbb{S}}(2u(\Lambda^{-2}(u^2))+2u_x(\Lambda^{-2}(u^2)_x))dx\\ &\leq& 2\max{(\|u\|_{\infty}, \|u_x\|_{\infty}})K(u)+2\max{(\|u\|_{\infty}, \|u_{x}\|_\infty})\int_{\mathbb{S}}(\Lambda^{-2}f(u))dx.\end{aligned}$$ Similar arguments reveal that $K(u)$ is bounded in finite time and hence, $\|u_{xx}\|_\infty\leq \|u_{xx}\|_1<\infty.$ Recalling the equality ([\[4.0.6\]](#4.0.6){reference-type="ref" reference="4.0.6"}), and Theorem [Theorem 4](#thm4.1){reference-type="ref" reference="thm4.1"} which guarantees that $u>0$, we evaluate $$\begin{aligned} \frac{1}{2}\frac{d}{dt}I(u)&=&-(\int_{\mathbb{S}}(u^3-4u_x^3-4uu_xu_{xx}-4uu_{xx}^2-2uu_{xx}u_{xxx}\nonumber\\ &&-16u_xu_{xxx}^2-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^2\nonumber\\ &&-10u_xu_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_xu_{xxxx}^2\nonumber\\ &&-2uu_{xxxx}u_{xxxxx})dx)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx\\ &\leq&-(\int_{\mathbb{S}}(-u^3-4u_x^3-4uu_xu_{xx}-4uu_{xx}^2-2uu_{xx}u_{xxx}\nonumber\\ &&-16u_xu_{xxx}^2-4uu_{xxx}u_{xxxx}-20u_{xx}u_{xxx}^2\nonumber\\ &&-10u_xu_{xxx}u_{xxxx}-2uu_{xxx}u_{xxxxx}-20u_{xx}u_{xxx}u_{xxxx}-10u_xu_{xxxx}^2\nonumber\\ &&-2uu_{xxxx}u_{xxxxx})dx)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx\\ &\leq&\max{(\|u\|_{\infty}, \|u_x\|_{\infty}, \|u_{xx}\|_{\infty})}I(u)+\int_{\mathbb{S}}u(\Lambda^{-2}f(u))dx\\ &\leq& \max{(\|u\|_{\infty}, \|u_x\|_{\infty}, \|u_{xx}\|_{\infty})}I(u)+\max{(\|u\|_{\infty})\int_{\mathbb{S}}}(\Lambda^{-2}f(u))dx.\end{aligned}$$ Therefore, $$\begin{aligned} \frac{1}{2}\frac{d}{dt}I(u)&\leq& K_2I(u)+K_3\end{aligned}$$ for some optimal constants $K_2$, $K_3$. Gronwall's inequality implies $I(u)\leq [I(0)+K_3t]e^{K_2t}$, which is valid for any finite time $0<t\leq T$. Since we find an upper bound for $\|u\|_4$, this completes the proof of Theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"}. # Proof of theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} {#sec5} The proof of theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"} is divided in three major parts, namely, - Existence of $C^1$ periodic one-forms $\omega_1,\omega_2$ and $\omega_3$ satisfying [\[1.0.4\]](#1.0.4){reference-type="eqref" reference="1.0.4"}; - Existence of a domain $V$, depending on the initial datum, containing open sets endowed with a PSS structure; - Existence of local connection forms $\omega_{13},\omega_{23}$. **$\bullet$ Existence of $C^1$ periodic one-forms $\omega_1$, $\omega_2$ and $\omega_3$.** Example [Example 1](#example2.1){reference-type="ref" reference="example2.1"} exhibits two triads of one forms [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"} satisfying the condition [\[1.0.4\]](#1.0.4){reference-type="eqref" reference="1.0.4"}. For solutions $u$ emanating from an initial datum $u_0\in H^4(\mathbb{S})$, with $u_0-u_0''>0$, theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} implies that $u\in C(H^{4}(\mathbb{S}),[0,T))\cap C^1(H^{3}(\mathbb{S}),[0,T))$, whereas Theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"} informs us that $u$ is defined on $U=\mathbb{R}\times(0,\infty)$. Moreover, $u_t(\cdot,t)\in H^3(\mathbb{S})\subseteq C^2_{\text{per}}(\mathbb{R})$ and $u(\cdot,t)\in H^4(\mathbb{S})\subseteq C^3_{\text{per}}(\mathbb{R})$ in view of the Sobolev Lemma (see lemma [\[lemma2.3\]](#lemma2.3){reference-type="ref" reference="lemma2.3"}). Therefore, $u\in C^{3,1}(\mathbb{R})$ and then $f_{ij}\in C^1(\mathbb{R})$ and is periodic in the variable $x$, where $f_{ij}$ are the coefficients of the forms given in [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"}. =**$\bullet$ Existence of a domain $V$, depending on the initial datum, containing open sets endowed with a PSS structure**; From example [Example 2](#example2.2){reference-type="ref" reference="example2.2"}, a non-generic solution $u$ of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} can only be periodic if it is constant. Since the initial datum satisfies the condition $u_0-u_0''>0$, by theorem [Theorem 4](#thm4.1){reference-type="ref" reference="thm4.1"} we know that $u>0$ and it cannot be constant on $U$. Let us then suppose the existence of an open set $\Omega\subseteq U$ for which $u\big|_\Omega=k$, where $k>0$ is a real number. Without loss of generality, we may assume that $\Omega\subseteq(0,1)\times(0,\infty)=:U_1$ in view of the periodicity of $u$ with respect to $x$. For some $p\in U_1\setminus\Omega$ and $\epsilon>0$, we have $\nabla u(p)=(u_x(p),u_t(p))\neq(0,0)$ and $u\big|_{B_\epsilon(p)}$ is non-constant, where $B_\epsilon(p)$ denotes the disc of centre $p$ and radius $\epsilon$. As a result, $U_1$ has at least one connected component $V$ (and $U$ as well), with $B_\epsilon(p)\subseteq V$, endowed with a PSS structure determined by the forms $\omega_1$ and $\omega_2$. **$\bullet$ Existence of connection forms defined everywhere $\omega_1\wedge\omega_2\neq0$.** Henceforth we assume that the open sets under consideration are those that $\nabla u\neq(0,0)$ everywhere. Let us denote these sets generically by $V$. Our proof is based on, and follows, that made by Castro Silva and Kamran [@tarcisio Proposition 3.7]. We have two possible choices for the form $\omega_2$. For this reason, fix one of them and consider the frame $\{\omega_1,\omega_2\}$. Let $a$, $b$ and $c$ functions such that $$\label{5.0.4} \omega_{13}=a\omega_1+b\omega_2,\quad \omega_{23}=b\omega_1+c\omega_2.$$ Our task is to find functions locally defined on any open set of $U$ for which [\[5.0.4\]](#5.0.4){reference-type="eqref" reference="5.0.4"} and the Levi-Civita connection form $\omega_3$ given in [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"} satisfy [\[2.3.1\]](#2.3.1){reference-type="eqref" reference="2.3.1"}. In [@tarcisio Proposition 3.7] it was shown that the connection forms [\[5.0.4\]](#5.0.4){reference-type="eqref" reference="5.0.4"} for an equation of the type $$\label{5.0.5} u_t-u_{txx}=\lambda uu_{xxx}+G(u,u_x,u_{xx}),$$ satisfy a certain set of differential equations, see [@tarcisio Theorem 2.4] and also [@keti2015 Theorem 3.4]. The function $G$ has somewhat a specific dependence on its arguments, and also some parameters $\mu$, $m_1$ and $m_2$. In [@freire-tito-sam Theorem 1] it was shown that [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} is a PSS equation, and one of the steps for that demonstration was just to show that it falls in the class considered in [@keti2015 Theorem 3.4]. In particular, the mentioned parameters are $\mu\in\mathbb{R}$, $m_1\in\{-2,1\}$ and $m_2=0$. Therefore, in view of [@freire-tito-sam Equation (7)] and [@keti2015 Theorem 3.4], we fall either into [@tarcisio Proposition 3.7, case ii.)] or [@tarcisio Proposition 3.7, case iii.)], depending on whether $\mu=0$ or $\mu\neq0$. According to [@tarcisio Equation (181)], the functions $a$, $b$ and $c$ in [\[5.0.4\]](#5.0.4){reference-type="eqref" reference="5.0.4"} take the form (recall that $m_2=0$) $a=\phi_1(z)$, $b=\phi_2(z)$, and $c=\phi_3(z)$, $z=m_1x$, for some real valued and smooth functions $\phi_1$, $\phi_2$ and $\phi_3$ to be determined, satisfying the condition $$\label{5.0.6} \phi_1\phi_3\neq0.$$ The Codazzi-Mainardi equations give (see [@tarcisio Equations (182)--(183)]) $$\label{5.0.7} \phi_1'+\mu\phi_2'-\phi_1-2\mu\phi_2+\phi_3=0$$ and $$\label{5.0.8} \phi_2'+\mu\phi_3'+\mu\phi_1-2\phi_2-\mu\phi_3=0,$$ whereas the Gauss equation reads $$\label{5.0.9} \phi_1\phi_3-\phi_2^2=-1.$$ Equation [\[5.0.9\]](#5.0.9){reference-type="eqref" reference="5.0.9"}, jointly with condition [\[5.0.6\]](#5.0.6){reference-type="eqref" reference="5.0.6"}, imply that $b\neq0$ everywhere. From [\[5.0.7\]](#5.0.7){reference-type="eqref" reference="5.0.7"} we obtain $\phi_3$ in terms of $\phi_1$, $\phi_2$ and their first derivatives. Substituting the result into [\[5.0.8\]](#5.0.8){reference-type="eqref" reference="5.0.8"} and integrating once, we obtain (see [@tarcisio Equation (184)]) $$\label{5.0.10} \mu\phi_1'=(1+\mu^2)\phi_2-\mu^2\phi_2'-\beta e^{2z},$$ where $\beta\in\mathbb{R}$ is a constant. We now divide our proof in two different cases. **Case $\mu=0$.** From [\[5.0.10\]](#5.0.10){reference-type="eqref" reference="5.0.10"} we conclude that $\phi_2=\beta e^{2z},$ whereas [\[5.0.7\]](#5.0.7){reference-type="eqref" reference="5.0.7"} gives $$\label{5.0.11} \phi_3=\phi_1-\phi_1'.$$ Substituting $\phi_2$ and $\phi_3$ into the Gauss equation [\[5.0.9\]](#5.0.9){reference-type="eqref" reference="5.0.9"} we obtain the following Abel differential equation of the second kind $$\phi_1\phi_1'=\phi_1^2-\beta^2 e^{4z}+1.$$ Under the change $\phi_1=e^zw$, where $w$ is another function of $z$, we obtain the following simpler ODE $$ww'=e^{-2z}-\beta^2 e^ {2z},$$ that, after solving, substituting back the result for $\phi_1$, and proceeding some manipulation, gives $$\label{5.0.12} \phi_1(z)=\pm e^z\sqrt{\gamma-\beta^2 e^{2z}-e^{-2z}}=\pm\sqrt{\gamma e^{2z}-1 -\beta^2 e^{4z}}.$$ Substituting [\[5.0.12\]](#5.0.12){reference-type="eqref" reference="5.0.12"} into [\[5.0.11\]](#5.0.11){reference-type="eqref" reference="5.0.11"} and going back to the original functions $a$, $b$ and $c$, we obtain $$\label{5.0.13} \begin{array}{lcl} a(x,t)&=&\displaystyle{\pm \sqrt{\gamma e^{2m_1x}-\beta^2e^{4m_1x}-1}},\quad b(x,t)=\displaystyle{\beta e^{2m_1x}},\\ \\ c(x,t)&=&\pm\displaystyle{\frac{\beta^2 e^{2m_1x}-1}{ \sqrt{\gamma e^{2m_1x}-\beta^2 e^{4m_1x}-1}}}. \end{array}$$ \]**Case $\mu\neq0$.** From [\[5.0.10\]](#5.0.10){reference-type="eqref" reference="5.0.10"} we can write $\phi_1'$ in terms of $\phi_2$ and $\phi_2'$. Substituting it into [\[5.0.9\]](#5.0.9){reference-type="eqref" reference="5.0.9"} we obtain $$\label{5.0.14} \phi_3=\phi_1+\phi,\quad \phi=\displaystyle{\frac{\mu^2-1}{\mu}\phi_2-\frac{\beta}{\mu}e^{2z}.}$$ Substituting [\[5.0.14\]](#5.0.14){reference-type="eqref" reference="5.0.14"} into the Gauss equation [\[5.0.9\]](#5.0.9){reference-type="eqref" reference="5.0.9"} we obtain $$\phi_1^2+\phi\phi_1-\phi_2^2=-1,$$ which, after solved for $\phi_1$, yields $$\label{5.0.15} \phi_1=\frac{-\phi\pm\sqrt{\Delta}}{2},\quad \Delta=\phi^2-4(1-\phi_2^2),$$ which are well defined as long as $\Delta\geq0$. Substituting [\[5.0.15\]](#5.0.15){reference-type="eqref" reference="5.0.15"} into [\[5.0.14\]](#5.0.14){reference-type="eqref" reference="5.0.14"} and the result into [\[5.0.10\]](#5.0.10){reference-type="eqref" reference="5.0.10"} we obtain the following ODE for $\phi_2$ $$\label{5.0.16} [(1+\mu^2)\sqrt{\Delta}\pm(\mu^2-1)\phi\pm4\mu \phi_2]\phi_2' -2(1+\mu^2)\sqrt{\Delta}\phi_2\mp 2\beta e^{2z}\phi=0.$$ It was shown in [@tarcisio page 36] that the coefficient of $\phi_2'$ cannot vanish, otherwise we would conclude that $\phi_1=\phi_3$ and $\phi_2=0$, which contradicts [\[5.0.9\]](#5.0.9){reference-type="eqref" reference="5.0.9"}. By continuity, such a coefficient does not change its sign. Without loss of generality, we may assume it to be positive and the ODE above takes the form $b'=g(z,b)$. Given a point $p=(x_0,t_0)\in V$, arguing exactly as [@tarcisio page 37] we conclude that the ODE to $b$ subject to $b(x_0/m_1)=t_0$ has a unique (local) solution, that guarantee the local existence of $b$ in a neighborhood of each point of $V$.$\square$ **Remark 2**. *The uniqueness of the forms [\[2.3.6\]](#2.3.6){reference-type="eqref" reference="2.3.6"} follows from [@keti2015 Theorem 3.4], see also [@freire-tito-sam page 760], and the fact that the solution $u$ of [\[1.0.1\]](#1.0.1){reference-type="eqref" reference="1.0.1"} is unique in view of theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"}.* # Concluding remarks {#sec6} In this paper we studied an equation whose solutions define metrics for a PSS from the point of view of geometric analysis. More precisely, we used tools of semi-group theory to establish well-posedness of solutions and then study the corresponding surface qualitatively. From the point of view of analysis, our Theorem [Theorem 2](#thm1.2){reference-type="ref" reference="thm1.2"} ensures well-posedness of solutions, whereas our Theorem [Theorem 3](#thm1.3){reference-type="ref" reference="thm1.3"} ensures enough regularity of the solution in order to ensure geometric relevance in the one-forms given in Theorem [Theorem 1](#thm1.1){reference-type="ref" reference="thm1.1"}. In particular, this last theorem can be seen as a sort of existence and uniqueness theorem for periodic surfaces emanating from a given initial datum, which can be associated to a certain curve in the three-dimensional space. # Acknowledgements {#acknowledgements .unnumbered} N. D. Mutlubaş is supported by the Turkish Academy of Sciences within the framework of the Outstanding Young Scientists Awards Program (TÜBA-GEBIP-2022). I. L. Freire is thankful to Enrique Reyes, Marcio Fabiano da Silva and Stefano Nardulli for stimulating discussions and support. He is also grateful to CNPq (grant nº 310074/2021-5) and FAPESP (grants nº 2020/02055-0 and 2022/00163-6) for financial support. He is also grateful to the Department of Mathematical Sciences for warm hospitality and support during the development of this work. 10 M. J. Ablowitz, D. J. Kaup, A. C.Newell, H. Segur, and R. M. Miura, Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., vol. 31, 125-127, (1973). I. Agricola and T. Friedrich, Global Analysis, Graduate Studies in Mathematics, voll 52, AMS, (2002). T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Diff. Equ., vol. 259, 4897--4923, (2015). T. Castro Silva and N. Kamran, Third-order differential equations and local isometric immersions of pseudospherical surfaces, Communications in Contemporary Mathematics, vol. 18, paper 1650021, (2016). J. A. Cavalcante and K. Tenenblat, Conservation laws for nonlinear evolution equations, J. Math. 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Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). M. Reed, B. Simon: Methods of Modern Mathematical Physics, Volume I:Functional Analysis. Academic Press (1972). E. G. Reyes, Some geometric aspects of integrability of differential equations in two independent variables, Acta Appl. Math., vol. 64, 75--109, (2000). E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., vol. 59, 117--131, (2002). E. G. Reyes, Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations, Sel. Math., New Ser., vol. 12, 241--270, (2006). E. G. Reyes, Correspondence theorems for hierarchies of equations of pseudospherical type, J. Diff. Equ., vol 225, 26--56, (2006). E. G. Reyes, Equations of pseudospherical type (After S. S. Chern and K. Tenenblat), Results. Math., vol 60, 53--101, (2011). C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge University Press, (2014). N. Sales Filho and I. L. Freire, Structural and qualitative properties of a geometrically integrable equation, Commun. Nonlin. Sci. Num. Simul., vol. 114, paper 106668, (2022). R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Physics B, vol. 154, 343--357, (1979). M. Taylor, Commutator Estimates, Proc. of the Amer. Math. Soc. 131, 1501-1507 (2002). M. E. Taylor, Partial Differential Equations I, 2nd edition, Springer, (2011). K. Tenenblat, Transformations of Manifolds and Applications to Differential Equations, Addison--Wesley/Longman, London, UK, (1998). R. S. Tito, Equações descrevendo superfícies pseudo-esféricas, MSc dissertation, Universidade Federal do ABC (2022). (in Portuguese)

arxiv_math

--- abstract: | In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We prove that if the conjectured relations hold, then they must form a complete set of relations. Our main result is a proof of the conjectured presentation in the case of the incidence varieties. We also show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety. address: - Max-Planck-Institute für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany - Department of Mathematics, 225 Stanger Street, McBryde Hall, Virginia Tech University, Blacksburg, VA 24061 USA - Department of Physics MC 0435, 850 West Campus Drive, Virginia Tech University, Blacksburg VA 24061 USA - Department of Mathematics, 225 Stanger Street, McBryde Hall, Virginia Tech University, Blacksburg, VA 24061 USA - Department of Physics MC 0435, 850 West Campus Drive, Virginia Tech University, Blacksburg VA 24061 USA - Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China - Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China author: - Wei Gu - Leonardo C. Mihalcea - Eric Sharpe - Weihong Xu - Hao Zhang - Hao Zou bibliography: - biblio.bib title: Quantum K Whitney relations for partial flag varieties --- [^1] # Introduction {#sec:intro} In [@Gu:2020zpg], conjectural presentations by generators and relations for the quantum $\mathop{\mathrm{K}}$-theory rings of the ordinary Grassmanians were stated. These presentations come in two flavors: a "Coulomb branch presentation\" which arises in physics as the critical locus of a certain one-loop twisted superpotential associated to a gauged linear sigma model (GLSM) (cf. §[5.4](#sec:physics){reference-type="ref" reference="sec:physics"} below), and a "quantum K Whitney presentation\" which arises in mathematics from a quantization of the classical Whitney relations. A mathematical proof of these presentations was given in [@gu2022quantum], in the more general equivariant setting. Continuing this work, we recently conjectured in [@gu2023quantum] an extension of these presentations from Grassmannians to any partial flag variety, and we analyzed these conjectures from the physics point of view. Our goal in this paper is to investigate these presentations mathematically, and provide proofs. Notably, we prove that our conjectures hold in the case of incidence varieties $\mathop{\mathrm{Fl}}(1,n-1;n)$ which parametrize flags of vector spaces $F_1 \subset F_2 \subset {\mathbb C}^n$ with $\dim F_1 = 1$ and $\dim F_2=n-1$. The conjectural presentations also hold for the complete flag varieties, conditional on the validity of an unpublished conjecture by Buch and Mihalcea about a divisor axiom in quantum $\mathop{\mathrm{K}}$-theory; cf.  below. Furthermore, we show that if the relations conjectured in [@gu2023quantum] hold, then they generate the full ideal of relations. In other words, no other relations besides those already stated will be necessary. A presentation of the equivariant quantum K ring of the complete flag varieties $\mathop{\mathrm{Fl}}(n)$ was recently proved by Maeno, Naito and Sagaki [@maeno.naito.sagaki:QKideal], and it is related to the Toda lattice presentation from quantum cohomology [@givental.kim; @kim.toda]. Relations similar to those from *loc.cit.* appear in [@givental.lee:quantum], in relation to the finite-difference Toda lattice; in [@koroteev], in relation to the study of the quasimap quantum $\mathop{\mathrm{K}}$-theory of the cotangent bundle of $\mathop{\mathrm{Fl}}(n)$ and the Bethe ansatz; and in [@ikeda.iwao.maeno], in relation to the Peterson isomorphism in quantum $\mathop{\mathrm{K}}$-theory. Our presentation is generally different from those mentioned above, and, as we explain in [@gu2023quantum], is most closely related to a presentation obtained by Gu and Kalashnikov [@GuKa] of the quantum cohomology ring of quiver flag varieties. ## Statement of results We provide next a more precise account of our results. Let $X=\mathop{\mathrm{Fl}}(r_1,\dots,r_k;n)$ be a partial flag variety, equipped with the flag of tautological vector bundles $$0={\mathcal S}_{0}\subset{\mathcal S}_{1}\subset\ldots \subset {\mathcal S}_{k}\subset{\mathcal S}_{{k+1}}={\mathbb C}^n,$$ where ${\mathcal S}_{j}$ has rank $r_j$. The variety $X$ is a homogeneous space for the left action by $G:=\mathop{\mathrm{GL}}_n({\mathbb C})$, and we denote by $T \subset G$ the maximal torus consisting of diagonal matrices. The $T$-equivariant quantum K ring $\mathop{\mathrm{QK}}_T(X)$, defined by Givental and Lee [@givental:onwdvv; @lee:QK], is a deformation of the Grothendieck ring $\mathop{\mathrm{K}}_T(X)$ of $T$-equivariant vector bundles on $X$. Additively, $\mathop{\mathrm{QK}}_T(X) = \mathop{\mathrm{K}}_T(X) \otimes_{\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})} \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$ where $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}}) = \mathop{\mathrm{Rep}}(T)$ is the representation ring of $T$ and $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]:=\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1, \ldots , q_k]\!]$ is a power series ring in the sequence of quantum parameters $(q_i)$ indexed by a basis of $\mathop{\mathrm{H}}_2(X)$. This $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-module is equipped with a quantum product $\star$ which gives $\mathop{\mathrm{QK}}_T(X)$ the structure of a commutative, associative, $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-algebra. For $E \to X$ an equivariant vector bundle of rank $\mathrm{rk}~E$ we denote by $$\lambda_y(E) := 1 + y [E] + \ldots + y^{\mathrm{rk}\,E} [\wedge^{\mathrm{rk}\,E} E] \quad \in \mathop{\mathrm{K}}_T(X)[y]$$ the Hirzerbruch $\lambda_y$ class of $E$. This class is multiplicative for short exact sequences. In an abuse of notation, we sometimes write $E$ for the class $[E]$ in $\mathop{\mathrm{K}}_T(X)$. The following is our main conjecture, also stated in the companion paper [@gu2023quantum]. **Conjecture 1**. *For $j=1,\dots, k$, the following relations hold in $\mathop{\mathrm{QK}}_T(X)$: $$\label{eqn:lambda_y rel} \lambda_y({\mathcal S}_{j})\star\lambda_y({\mathcal S}_{{j+1}}/{\mathcal S}_{j})=\lambda_y({\mathcal S}_{{j+1}})-y^{r_{j+1}-r_j}\frac{q_j}{1-q_j}\det({\mathcal S}_{{j+1}}/{\mathcal S}_{j})\star(\lambda_y({\mathcal S}_{j})-\lambda_y({\mathcal S}_{{j-1}})).$$ Here, $\lambda_y({\mathcal S}_{k+1})=\lambda_y({\mathbb C}^n) = \prod_{i=1}^n (1+yT_i)$, where $T_i \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ are given by the decomposition of ${\mathbb C}^n$ into one dimensional $T$-modules.* If $k=1$, i.e., if $X=\mathop{\mathrm{Gr}}(r,n)$, these relations were conjectured in [@Gu:2020zpg], and proved in [@gu2022quantum]. Specializing all $q_j$ to $0$ recovers the usual Whitney relations in $\mathop{\mathrm{K}}_T(X)$ obtained from the short exact sequences $$0 \to {\mathcal S}_{j} \to {\mathcal S}_{{j+1}} \to {\mathcal S}_{{j+1}}/{\mathcal S}_{j} \to 0.$$ These relations may be formalized by introducing abstract variables $X^{(j)}=(X^{(j)}_1,\dots,X^{(j)}_{r_j})$ and $Y^{(j)}=(Y^{(j)}_1,\dots,Y^{(j)}_{s_j})$ for the exponentials of the Chern roots of ${\mathcal S}_j$ and ${\mathcal S}_{j+1}/{\mathcal S}_j$, respectively; see [\[defn:Iq\]](#defn:Iq){reference-type="eqref" reference="defn:Iq"} below. Here $s_j=r_{j+1}-r_j$. Set $$S\coloneqq \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[e_1(X^{(j)}),\dots, e_{r_j}(X^{(j)}),e_1(Y^{(j)}),\dots,e_{s_j}(Y^{(j)}), j=1,\dots, k],$$ and let $I_q\subset S[\![q]\!]$ be the ideal generated by the coefficients of $y$ in [([\[eqn:lambda_y rel\]](#eqn:lambda_y rel){reference-type="ref" reference="eqn:lambda_y rel"})](#eqn:lambda_y rel). Our first result is the following, cf. . **Theorem 2**. *Assume holds. Then the relations [([\[eqn:lambda_y rel\]](#eqn:lambda_y rel){reference-type="ref" reference="eqn:lambda_y rel"})](#eqn:lambda_y rel) form a complete set of relations, i.e., there is an isomorphism $$\Psi: {S[\![q]\!]}/I_q\to \mathop{\mathrm{QK}}_T(X)$$ of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-algebras sending $e_\ell(X^{(j)})$ to $\wedge^\ell({\mathcal S}_{j})$ and $e_\ell(Y^{(j)})$ to $\wedge^\ell({\mathcal S}_{{j+1}}/{\mathcal S}_{j})$.* The proof of this theorem follows a strategy developed in [@gu2022quantum], generalizing a classical result by Siebert and Tian [@siebert.tian] (see also [@fulton.pandh:notes]) about the quantum cohomology ring. Roughly speaking, in order to find the ideal of quantum relations, it suffices to find a presentation of the classical ring, then quantize each relation in this presentation. While the quantum cohomology version follows easily from a graded version of Nakayama's lemma, the version needed for quantum $\mathop{\mathrm{K}}$-theory, is more subtle. The precise statement is given in below. A key hypothesis needed in this proposition is that the claimed presentation is finitely generated as a $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-module. We prove this in Appendix [7](#app:fg){reference-type="ref" reference="app:fg"} (see ) in a rather general context about modules over formal power series rings; it is closely related to results from [@eisenbud:CAbook]. This method might be of use for proving presentations for other (flag) varieties. The main result of this paper is a proof of for the incidence varieties $\mathop{\mathrm{Fl}}(1,n-1;n)$, cf. and . We denote by $q_1, q_2$ the quantum parameters corresponding to the Schubert curves indexed by the simple reflections $s_1=(1,2), s_{n-1}=(n-1, n)$, respectively. **Theorem 3**. *holds for $\mathop{\mathrm{Fl}}(1,n-1;n)$. More explicitly, the following relations hold in $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(1,n-1;n))$, and they form a complete set of relations: $$\lambda_y({\mathcal S}_{1})\star\lambda_y({{\mathcal S}_{2}}/{\mathcal S}_{1})=\lambda_y({{\mathcal S}_{2}})-y^{n-2}\frac{q_1}{1-q_1}\det({{\mathcal S}_{2}}/{\mathcal S}_{1})\star(\lambda_y({\mathcal S}_{1})-1) \/;$$ $$\lambda_y({{\mathcal S}_{2}})\star\lambda_y({\mathbb C}^n/{{\mathcal S}_{2}})=\lambda_y({\mathbb C}^n)-y\frac{q_2}{1-q_2}\det({\mathbb C}^n/{{\mathcal S}_2})\star(\lambda_y({{\mathcal S}_{2}})-\lambda_y({\mathcal S}_{1})) \/.$$* *Here, $\lambda_y({\mathbb C}^n) = \prod_{i=1}^n (1+yT_i)$, where $T_i \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ are given by the decomposition of ${\mathbb C}^n$ into one dimensional $T$-modules.* The proof of relies on a recent result of Xu [@xu2021quantum], which proves a conjecture by Buch and Mihalcea in the case of incidence varieties $\mathop{\mathrm{Fl}}(1,n-1;n)$; cf.  below. This conjecture, which may be thought of as a divisor axiom in quantum $\mathop{\mathrm{K}}$-theory, gives a formula for computing $3$-pointed (equivariant) $\mathop{\mathrm{K}}$-theoretic Gromov-Witten invariants in ordinary (equivariant) $\mathop{\mathrm{K}}$-theory, when one of the insertions is a line bundle class such as $\det({\mathcal S}_j)$. Our strategy is to reduce the relations in $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(1,n-1;n))$ into a form involving no other quantum multiplications except for those of the form $\det({\mathcal S}_{j})\star a$, turn these relations into identities involving $3$-pointed (equivariant) $\mathop{\mathrm{K}}$-theoretic Gromov-Witten invariants, apply the aforementioned formula, and do some computations in ordinary (equivariant) $\mathop{\mathrm{K}}$-theory. In addition, Xu proved in [@xu2021quantum] that the Schubert classes in $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(1,n-1;n))$ are generated by ${\mathcal S}_1$ and $\det ({\mathcal S}_2)$ over $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q]$. As a consequence, one may replace the power series ring $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$ by the localized ring $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q]_{1 + \langle q \rangle}$, and conjecturally, by the polynomial ring $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q]$; see and below. Aside from the incidence varieties, the quantum K divisor axiom () is known for cominuscule Grassmannians [@buch.m:qk; @chaput.perrin:rationality]. A natural question is whether more cases of may be proved assuming the validity of . This leads to our next result, cf. . **Theorem 4**. *Assume holds for the complete flag variety $\mathop{\mathrm{Fl}}(n)$. Then the relations [([\[eqn:lambda_y rel\]](#eqn:lambda_y rel){reference-type="ref" reference="eqn:lambda_y rel"})](#eqn:lambda_y rel) hold in $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(n))$. Equivalently, implies for $\mathop{\mathrm{Fl}}(n)$.* The proof of this statement employs the technique of curve neighborhoods developed in [@buch.m:nbhds], see also [@BCMP:qkfin]. For an effective degree $d \in \mathop{\mathrm{H}}_2(X)$, we utilize an interpretation of the curve neighborhood $a[d]$ of an element $a \in \mathop{\mathrm{K}}_T(X)$ as $\partial_{z_d}(a)$, the result of applying an iterated Demazure operator, where $z_d$ is a permutation defined in [@buch.m:nbhds]. This interpretation, also utilized in [@li.mihalcea], allows us to compute the curve neighborhoods of the exterior powers of tautological bundles $\wedge^\ell {\mathcal S}_i$; see . The key calculation is , which establishes an equality between curve neighborhoods of exterior powers of adjacent tautological bundles, when the degrees differ by a simple (co)root. *Acknowledgments.* We thank Anders Buch, Linda Chen, Elana Kalashnikov, Peter Koroteev, Y.P. Lee, and Henry Liu for helpful discussions. Special thanks are due to Prof. Satoshi Naito for many inspiring discussions, and for pointing out some references utilized in the Appendix. Finally, we thank the Simons Center for Geometry and Physics for supporting the excellent workshop GLSM\@30, which made possible many stimulating discussions. # Equivariant K-theory {#sec:preliminaries} ## Preliminaries In this section we recall some basic facts about the equivariant K-theory of a variety with a group action. For an introduction to equivariant K theory, and more details, see [@chriss2009representation]. Let $X$ be a smooth projective variety with an action of a linear algebraic group $G$. The equivariant K theory ring $\mathop{\mathrm{K}}_G(X)$ is the Grothendieck ring generated by symbols $[E]$, where $E \to X$ is a $G$-equivariant vector bundle, modulo the relations $[E]=[E']+[E'']$ for any short exact sequence $0 \to E' \to E \to E'' \to 0$ of equivariant vector bundles. The additive ring structure is given by direct sum, and the multiplication is given by tensor products of vector bundles. Since $X$ is smooth, any $G$-linearized coherent sheaf has a finite resolution by (equivariant) vector bundles, and the ring $\mathop{\mathrm{K}}_G(X)$ coincides with the Grothendieck group of $G$-linearized coherent sheaves on $X$. In particular, any $G$-linearized coherent sheaf $\mathcal{F}$ on $X$ determines a class $[\mathcal{F}] \in \mathop{\mathrm{K}}_G(X)$. As an important special case, if $\Omega \subset X$ is a $G$-stable subscheme, then its structure sheaf determines a class $[{\mathcal O}_\Omega] \in \mathop{\mathrm{K}}_G(X)$. We shall abuse notation and sometimes write ${\mathcal F}$ or ${\mathcal O}_\Omega$ for the corresponding classes $[{\mathcal F}]$ and $[{\mathcal O}_\Omega]$ in $\mathop{\mathrm{K}}_G(X)$. The ring $\mathop{\mathrm{K}}_G(X)$ is an algebra over $\mathop{\mathrm{K}}_G(\mathop{\mathrm{pt}}) = \mathop{\mathrm{Rep}}(G)$, the representation ring of $G$. If $G=T$ is a complex torus, then this is the Laurent polynomial ring $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}}) = {\mathbb Z}[T_1^{\pm 1}, \ldots, T_n^{\pm 1}]$ where $T_i:={\mathbb C}_{t_i}$ are characters corresponding to a basis of the Lie algebra of $T$. Let $E \to X$ be an equivariant vector bundle of rank $\mathrm{rk}\,E$. The (Hirzebruch) $\lambda_y$ class is defined as $$\lambda_y(E) := 1 + y E + \ldots + y^{\mathrm{rk}\,E} \wedge^{\mathrm{rk}\,E} E \quad \in \mathop{\mathrm{K}}_T(X)[y].$$ This class was introduced by Hirzebruch [@hirzebruch:topological] in relation to the Grothendieck--Riemann--Roch theorem. The $\lambda_y$ class is multiplicative with respect to short exact sequences, i.e., if $0 \to E' \to E \to E'' \to 0$ is such a sequence of vector bundles, then $$\lambda_y(E) = \lambda_y(E') \cdot \lambda_y(E'').$$ This is part of the $\lambda$-ring structure of $\mathop{\mathrm{K}}_T(X)$, see e.g. [@nielsen:diag], referring to [@SGA6]. A particular case of this construction is when $V$ is a (complex) vector space with an action of a complex torus $T$. The $\lambda_y$ class of $V$ is the element $\lambda_y(V) =\sum_{i \ge 0} y^i \wedge^i V \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[y]$. Since $V$ decomposes into $1$-dimensional $T$-representations: $V = \bigoplus_i {\mathbb C}_{\mu_i}$, it follows from the multiplicative property of $\lambda_y$ classes that $\lambda_y(V) = \prod_i (1+y\,{\mathbb C}_{\mu_i})$. Since $X$ is proper, we can push the class of a sheaf forward to the point. This is given by the sheaf Euler characteristic, or, equivalently, the virtual representation $$\chi_X^T(\mathcal{F}) := \sum_i (-1)^i \mathop{\mathrm{H}}^i(X, \mathcal{F})\quad\in \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}}) \/.$$ ## (Equivariant) $\mathop{\mathrm{K}}$-theory of flag varieties {#sec:kflag} The partial flag variety $$X:=\mathop{\mathrm{Fl}}(r_1, \ldots, r_k;n)$$ parametrizes flags of vector spaces $F_{1} \subset F_{2} \subset \ldots \subset F_{k} \subset {\mathbb C}^n$ with $\dim F_{i} = r_i$ for $1 \le i \le k$. It is a projective manifold and admits a transitive action of $G=\mathop{\mathrm{GL}}_n({\mathbb C})$. We denote by $T$ the maximal torus in $G$ consisting of diagonal matrices. Let $S_n$ be the symmetric group in $n$ letters, and let $S_{r_1, \ldots, r_k} \le S_n$ be the subgroup generated by simple reflections $s_i = (i,i+1)$ where $i \notin \{ r_1, \ldots, r_k\}$. Denote by $\ell:S_n \to \mathbb{N}$ the length function, and by $S^{r_1, \ldots, r_k}$ the set of minimal length representatives of $S_n/S_{r_1, \ldots, r_k}$. This consists of permutations $w \in S_n$ which have descents at most at positions $r_1, \ldots, r_k$, i.e., $w(r_j+1)< \ldots <w(r_{j+1})$, for $j=0, \ldots, k$, with the convention that $r_0=0$ and $r_{k+1} =n$. The $T$-fixed points $e_w \in X$ are indexed by the permutations $w \in S_n^{r_1, \ldots, r_k}$. Let $B,\ B^-\subset G$ be the Borel subgroups of upper and lower triangular matrices, respectively. For each $T$-fixed point, the $B$-stable Schubert variety $X_w=\overline{B.e_w}$ and $B^-$-stable Schubert variety $X^w=\overline{B^-.e_w}$ are closures of the $B$ and $B^-$ orbits in $X$, respectively. We have $\dim X_w=\mathop{\mathrm{codim}}X^w=\ell(w)$. Let ${\mathcal O}_w=[{\mathcal O}_{X_w}]$ and ${\mathcal O}^w=[{\mathcal O}_{X^w}]$ be the classes in $\mathop{\mathrm{K}}_T(X)$ determined by the structure sheaves of $X_w$ and $X^w$, respectively. The ring $\mathop{\mathrm{K}}_T(X)$ is a free module over $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ with a basis given by these Schubert classes: $$\mathop{\mathrm{K}}_T(X) = \bigoplus_{w \in S_n^{r_1,\ldots, r_k}} \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}}) {\mathcal O}_w = \bigoplus_{w \in S_n^{r_1,\ldots, r_k}} \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}}) {\mathcal O}^w \/.$$ Denote by $0={\mathcal S}_{0}\subset{\mathcal S}_{1}\subset\dots \subset {\mathcal S}_{k}\subset{\mathcal S}_{{k+1}}={\mathbb C}^n$ the flag of tautological vector bundles on $X$, where ${\mathcal S}_{j}$ has rank $r_j$. Since we could not find a precise reference, we will take the opportunity to outline a proof for a (folklore) presentation by generators and relations of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(r_1, \ldots, r_k;n))$. The relations $$\lambda_y({\mathcal S}_{j})\cdot\lambda_y({\mathcal S}_{{j+1}}/{\mathcal S}_{j})=\lambda_y({\mathcal S}_{{j+1}}),\quad j=1,\dots,k$$ arise from the Whitney relations applied to the exact sequences $$0 \to {\mathcal S}_{{j}} \to {\mathcal S}_{j+1} \to {\mathcal S}_{j+1}/{\mathcal S}_{{j}} \to 0, \quad j=1,\dots,k,$$ and are specializations with $q_j =0, j=1,\dots,k$ of the relations from . This presentation is related to well-known presentations such as that in [@lascoux:anneau]\*§7. More precisely, let $${X}^{(j)}=(X^{(j)}_1,\dots,X^{(j)}_{r_j})\text{ and }{Y}^{(j)}=(Y^{(j)}_1,\dots,Y^{(j)}_{s_j})$$ denote formal variables for $j=1,\dots,k$, where $s_j \coloneqq r_{j+1}-r_j$. Let $X^{(k+1)}\coloneqq (T_1,\dots,T_n)$ be the equivariant parameters in $\mathop{\mathrm{K}}^T(\mathop{\mathrm{pt}})$. Geometrically, the variables $X_i^{(j)}$ and $Y_s^{(j)}$ arise from the splitting principle: $$\lambda_y({\mathcal S}_j) = \prod_i (1+y X_i^{(j)}) \/, \quad \lambda_y({\mathcal S}_{j+1}/{\mathcal S}_{j}) = \prod_s (1+y Y_s^{(j)}),$$ i.e., they are the K-theoretic Chern roots of ${\mathcal S}_{j}$ and ${\mathcal S}_{j+1}/{\mathcal S}_{{j}}$, respectively. Let $e_\ell({X}^{(j)})$ and $e_\ell({Y}^{(j)})$ be the $\ell$-th elementary symmetric polynomials in ${X}^{(j)}$ and ${Y}^{(j)}$, respectively. Denote by $S$ the (Laurent) polynomial ring $$S\coloneqq\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[e_i(X^{(j)}), e_s(Y^{(j)}); 1 \le j \le k, 1 \le i \le r_j, 1 \le s \le s_{j}]$$ and define the ideal $I \subset S$ generated by $$\label{eqn:rel_classical} \sum_{i+s=\ell} e_i(X^{(j)}) e_s(Y^{(j)}) - e_\ell(X^{(j+1)})\/, \quad 1 \le j \le k \/.$$ **Proposition 5**. *There is an isomorphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$-algebras: $$\Psi: S/I \to \mathop{\mathrm{K}}_T(X)$$ sending $e_i(X^{(j)}) \mapsto \wedge^i {\mathcal S}_{j}$ and $e_i(Y^{(j)}) \mapsto \wedge^i {({\mathcal S}_{{j+1}}/{\mathcal S}_{{j}})}$.* *Proof.* Denote the conjectured presentation ring by $A$. The K theoretic Whitney relations imply that $\lambda_y({\mathcal S}_j) \cdot \lambda_y({\mathcal S}_{j+1}/{\mathcal S}_j) = \lambda_y({\mathcal S}_{j+1})$. Then the geometric interpretation of the variables $X^{(j)}, Y^{(j)}$ in terms of the splitting principle before the theorem implies that $\Psi$ is a well-defined $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$-algebra homomorphism. To prove surjectivity of $\Psi$, we first consider the case when $X=\mathop{\mathrm{Fl}}(n)$ is the full flag variety, and we utilize the theory of double Grothendieck polynomials [@fulton.lascoux; @buch:grothendieck]. It was proved in [@buch:grothendieck]\*Thm. 2.1 that each Schubert class in $\mathop{\mathrm{K}}_T(X)$ may be written as a (double Grothendieck) polynomial in $$\label{E:vars} 1-({\mathbb C}^n/{\mathcal S}_{n-1})^{-1}, 1- ({\mathcal S}_{n-1}/{\mathcal S}_{n-2})^{-1}, \ldots , 1 - ({\mathcal S}_2/{\mathcal S}_1)^{-1}, 1-({\mathcal S}_1)^{-1}$$ with coefficients in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$. Note that in $\mathop{\mathrm{K}}_T(X)$, $$\begin{aligned} ({\mathcal S}_{i}/{\mathcal S}_{i-1})^{-1} =&\det({\mathbb C}^n)^{-1}\cdot{\mathbb C}^n/{\mathcal S}_{n-1}\cdot\ldots\cdot{{\mathcal S}_{i+1}/{\mathcal S}_{i}}\cdot{\mathcal S}_{i-1}/{\mathcal S}_{i-2}\cdot\ldots\cdot{\mathcal S}_2/{\mathcal S}_1\cdot{\mathcal S}_1 \end{aligned}$$ for $i=1,\dots,n$. Therefore, each Schubert class may be written as a polynomial in variables ${\mathcal S}_{i}/{\mathcal S}_{i-1}$ for $i=1,\dots,n$ with coefficients in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$. This proves the surjectivity in this case. For partial flag varieties, consider the injective ring homomorphism given by pulling back via the natural projection $p: \mathop{\mathrm{Fl}}(n) \to \mathop{\mathrm{Fl}}(r_1, \ldots, r_k;n)$. The pullbacks of Schubert classes and of the tautological bundles are $$p^*{\mathcal O}^w = {\mathcal O}^w \/, \quad p^*(\wedge^\ell {\mathcal S}_i) = \wedge^\ell {\mathcal S}_{r_i} \/, \quad p^*(\wedge^\ell ({\mathcal S}_i/{\mathcal S}_{i-1})) = \wedge^\ell ({\mathcal S}_{r_i}/{\mathcal S}_{r_{i-1}}) \quad \/,$$ for any $w \in S_n^{r_1, \ldots, r_k}$, any $1 \le i \le k$, and any $\ell$. On the other hand, since $w \in S_n^{r_1, \ldots, r_k}$, the Schubert classes $p^*{\mathcal O}^w$ may be written as (double Grothendieck) polynomials symmetric in each block of variables $1-({\mathcal S}_{r_i+1}/{\mathcal S}_{r_i})^{-1}, \ldots, 1-({\mathcal S}_{r_{i+1}}/{\mathcal S}_{r_{i+1}-1})^{-1}$, for $0 \le i \le k$, i.e., in the elementary symmetric functions $e_{\ell}((1-({\mathcal S}_{r_i+1}/{\mathcal S}_{r_i})^{-1}, \ldots, 1-({\mathcal S}_{r_{i+1}}/{\mathcal S}_{r_{i+1}-1})^{-1}))$ in these sets of variables. Each such elementary symmetric function may be further expanded as a ${\mathbb Z}$-linear combination of terms of the form $$\label{E:fractions} \frac{e_{s}({\mathcal S}_{r_i+1}/{\mathcal S}_{r_i}, \ldots, {\mathcal S}_{r_{i+1}}/{\mathcal S}_{r_{i+1}-1})}{{\mathcal S}_{r_i+1}/{\mathcal S}_{r_i}\cdot \ldots \cdot {\mathcal S}_{r_{i+1}}/{\mathcal S}_{r_{i+1}-1}} = \frac{\wedge^s({\mathcal S}_{r_{i+1}}/{\mathcal S}_{r_i})}{\det ({\mathcal S}_{r_{i+1}}) / \det ({\mathcal S}_{r_i})} \/.$$ Observe that $\det {\mathbb C}^n = \prod_{i=1}^{k+1} {\det} ({\mathcal S}_{r_i}/{\mathcal S}_{r_{i-1}}) {= \det({\mathcal S}_{r_{j}}) \prod_{i=j+1}^{k+1}\det({\mathcal S}_{r_i}/{\mathcal S}_{r_{i-1}})}$. Therefore, $$\det( {\mathcal S}_{r_{j}})^{-1} = (\det {\mathbb C}^n)^{-1} {\prod_{i=j+1}^{k+1}\det({\mathcal S}_{r_i}/{\mathcal S}_{r_{i-1}})}, \quad j=1,\dots, k.$$ We have shown that the pullbacks of Schubert classes $p^*({\mathcal O}^w)$ are polynomials in the pullbacks of the tautological bundles and their quotients, and we deduce that $\Psi$ is surjective for partial flag manifolds. Injectivity holds because $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ is an integral domain and both $A$ and $\mathop{\mathrm{K}}_T(X)$ have the same rank as free modules over $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$. To see the latter, consider the ring $$A':= {\mathbb Z}[T_1, \ldots, T_n][e_i(X^j), e_s(Y^j): 1 \le j \le k, 1 \le i \le r_j, 1 \le s \le s_{j}]/I',$$ where $I'\subseteq A'$ is the ideal generated by [([\[eqn:rel_classical\]](#eqn:rel_classical){reference-type="ref" reference="eqn:rel_classical"})](#eqn:rel_classical). It is classically known that ${\mathbb Z}[T_1, \ldots, T_n]$-algebra $A'$ is isomorphic to the equivariant cohomology algebra $\mathop{\mathrm{H}}^*_T(X)$, with $e_i(X^{(j)})$ being sent to the equivariant Chern class $c_i^T({\mathcal S}_j)$ and $e_i(Y^{(j)})$ to the equivariant Chern class $c_i^T({\mathcal S}_{j+1}/{\mathcal S}_j)$. (This follows from example by realizing the partial flag variety as a tower of Grassmann bundles, then using a description of the cohomology of the latter as in [@fulton:IT]\*Example 14.6.6.) In particular, $A'$ is a free ${\mathbb Z}[T_1, \ldots, T_n]$-algebra of rank equal to the number of Schubert classes in $X$. Then $A= A' \otimes_{{\mathbb Z}[T_1, \ldots, T_n]} \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ is a free $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$-module of the same rank. ◻ ## Push-forward formulae of Schur bundles Next, we recall some results about cohomology of Schur bundles on Grassmannians, which we will need later. Our main reference is Kapranov's paper [@kapranov:Gr]. A reference for basic definitions of Schur bundles is Weyman's book [@weyman]. Recall that if $X$ is a $T$-variety, $\pi:E \to X$ is any $T$-equivariant vector bundle of rank $e$, and $\lambda=(\lambda_1, \ldots, \lambda_k)$ is a partition with at most $e$ parts, the *Schur bundle* $\mathfrak{S}_\lambda(E)$ is a $T$-equivariant vector bundle over $X$. It has the property that if $x \in X$ is a $T$-fixed point, the fibre $({\mathfrak S}_\lambda(E))_x$ is the $T$-module with character the Schur function $s_\lambda$. For example, if $\lambda = (1^k)$, then ${\mathfrak S}_{(1^k)}(E) = \wedge^k E$, and if $\lambda = (k)$ then ${\mathfrak S}_{(k)}(E)=\mathrm{Sym}^k(E)$. Consider a $T$-variety $X$ equipped with a $T$-equivariant vector bundle $\mathcal{V}$ of rank $n$. Denote by $\pi: \mathbb{G}(k,\mathcal{V}) \to X$ the Grassmann bundle over $X$. It is equipped with a tautological sequence $0 \to \underline{{\mathcal S}} \to \pi^* \mathcal{V} \to \underline{{\mathcal Q}} \to 0$ over $\mathbb{G}(k,\mathcal{V})$. The next result follows from [@kapranov:Gr Prop. 2.2], see also [@gu2022quantum Prop. 3.2 and Cor. 3.3]. (Kapranov proved this when $X=\mathop{\mathrm{pt}}$; the relative version follows immediately using that $\pi$ is a $T$-equivariant locally trivial fibration). **Proposition 6** (Kapranov). *For any nonempty partition $\lambda= (\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_k \geq 0)$ such that $\lambda_1 \le n-k$, there are the following isomorphisms of $T$-equivariant vector bundles:* 1. *For all $i \ge 0$, the higher direct images, $R^i \pi_* {\mathfrak S}_\lambda(\underline{{\mathcal S}}) = 0$;* 2. *$$R^i \pi_*{\mathfrak S}_\lambda(\underline{{\mathcal S}}^*)) = \begin{cases} {\mathfrak S}_\lambda(\mathcal{V}^*) & i=0 \\ 0 & i>0 \/; \end{cases}$$* 3. *$$R^i \pi_*{\mathfrak S}_\lambda(\underline{{\mathcal Q}}) = \begin{cases} {\mathfrak S}_\lambda(\mathcal{V}) & i=0 \\ 0 & i>0 \/. \end{cases}$$* # (Equivariant) quantum K-theory of flag varieties In this section, we first recall the definition of the equivariant quantum K ring of a partial flag variety. We then proceed by proving a presentation of the quantum K ring by generators and relations of this ring, generalizing the one from ; cf. . At this time, the statement holds under the assumption that certain generalized (quantum K) Whitney relations do hold in the quantum K ring; cf. . In [Section [5](#sec:incidence){reference-type="ref" reference="sec:incidence"}](#sec:incidence), we will show that this assumption is satisfied for incidence varieties, and in [Section [6](#sec:fullflag){reference-type="ref" reference="sec:fullflag"}](#sec:fullflag) for the complete flag varieties, the latter under an extra assumption. Throughout this section, we continue with the notation $G= \mathrm{GL}_n$ and $X=\mathop{\mathrm{Fl}}(r_1,\dots,r_k;n)$. ## Preliminaries An effective degree is a $k$-tuple of nonnegative integers $d=(d_1,\dots,d_k)$, which is identified with $\sum_{i=1}^{k}d_i[X_{s_{r_i}}]\in\mathop{\mathrm{H}}_2(X,{\mathbb Z})$. We write $q^d$ for $q_1^{d_1}\dots q_k^{d_k}$, where $q=(q_1, \ldots, q_k)$ is a sequence of quantum parameters. We recall the definition of the $T$-equivariant (small) quantum K theory ring $\mathop{\mathrm{QK}}_T(X)$, following [@givental:onwdvv; @lee:QK]: $$\label{eqn:QKTfree} \mathop{\mathrm{QK}}_T(X) = \mathop{\mathrm{K}}_T(X) \otimes_{\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})} \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$$ is a free $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-module with a $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-basis given by Schubert classes. It is equipped with a commutative, associative product, denoted by $\star$, and determined by the condition $$\label{eqn:def} (\!(\sigma_1\star\sigma_2,\sigma_3)\!)=\sum_d q^d \angles{ \sigma_1,\sigma_2,\sigma_3 }^T_{d}$$ for all $\sigma_1,\sigma_2,\sigma_3\in \mathop{\mathrm{K}}_T(X)$, where $$(\!(\sigma_1,\sigma_2)\!)\coloneqq \sum_{d}q^d \angles{ \sigma_1,\sigma_2 }^T_{d}$$ is the quantum $\mathop{\mathrm{K}}$-metric and $\angles{ \sigma_1,\dots,\sigma_n }^T_{d}$ are $T$-equivariant $\mathop{\mathrm{K}}$-theoretic Gromov--Witten invariants. We define these invariants next. Let $d\in \mathop{\mathrm{H}}_2(X,{\mathbb Z})_+$ be an effective degree and $\overline{\mathcal M}_{0,n}(X,d)$ be the Kontsevich moduli space parametrizing $n$-pointed, genus $0$, degree $d$ stable maps to $X$. Let $$\mathop{\mathrm{ev}}_1, \mathop{\mathrm{ev}}_2, \dots, \mathop{\mathrm{ev}}_n: \overline{\mathcal M}_{0,n}(X,d)\to X$$ be evaluations at the $n$ marked points. Given $\sigma_1, \sigma_2,\dots, \sigma_n\in \mathop{\mathrm{K}}_T(X)$, we define the $T$-equivariant $\mathop{\mathrm{K}}$-theoretic Gromov--Witten invariant $$\angles{ \sigma_1,\sigma_2,\dots,\sigma_n }^T_{d}\coloneqq\chi_{\overline{M}_{0,n}(X,d)}^T(\mathop{\mathrm{ev}}_1^*\sigma_1\cdot\mathop{\mathrm{ev}}_2^*\sigma_2\cdots\mathop{\mathrm{ev}}_n^*\sigma_n),$$ where $\chi_Y^T: \mathop{\mathrm{K}}_T(Y)\to \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ is the pushforward to a point. We adopt the convention that when $d$ is not effective, the invariant $\angles{ \sigma_1,\sigma_2,\dots,\sigma_n }^T_{d}=0$. **Remark 7**. For $1 \le j \le k$, declare $\deg q_{j} = \deg ([X_{s_{r_j}}] \cap c_1(T_X)) = r_{j+1}- r_{j-1}$. For a multidegree $d = (d_{r_1}, \ldots, d_{r_k})$, set $\deg({\mathcal O}^w \otimes q^d) = \ell(w) + \sum \deg(q_i) \cdot d_{r_i}$. Together with the topological filtration on $\mathop{\mathrm{K}}_T(X)$, this equips $\mathop{\mathrm{QK}}_T(X)$ with a structure of a filtered ring; see [@buch.m:qk §5.1]. The associated graded of this ring is $\mathop{\mathrm{QH}}^*_T(X)$, the (small) $T$-equivariant quantum cohomology of $X$, a free $\mathop{\mathrm{H}}^*_T(\mathop{\mathrm{pt}})[q]$-algebra of the same rank as $\mathop{\mathrm{QK}}_T(X)$. ## A conjectural Whitney presentation for $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(r_1,\dots,r_k,n))$ As before, $X$ denotes $\mathop{\mathrm{Fl}}(r_1,\dots,r_k,n)$, equipped with the tautological vector bundles $$0={\mathcal S}_{0}\subset{\mathcal S}_{1}\subset\dots \subset {\mathcal S}_{k}\subset{\mathcal S}_{{k+1}}={\mathbb C}^n,$$ where ${\mathcal S}_j$ has rank $r_j$. Recall from [Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}](#sec:intro) the following conjecture from [@gu2023quantum]. **Conjecture 8**. *For $j=1,\dots, k$, the following relations hold in $\mathop{\mathrm{QK}}_T(X)$: $$\label{eqn:lambda_y_rel} \lambda_y({\mathcal S}_{j})\star\lambda_y({\mathcal S}_{{j+1}}/{\mathcal S}_{j})=\lambda_y({\mathcal S}_{{j+1}})-y^{r_{j+1}-r_j}\frac{q_j}{1-q_j}\det({\mathcal S}_{{j+1}}/{\mathcal S}_{j})\star(\lambda_y({\mathcal S}_{j})-\lambda_y({\mathcal S}_{{j-1}})).$$* *Assuming that this conjecture holds* we will state and prove a presentation by generators and relations of the ring $\mathop{\mathrm{QK}}_T(X)$. This conjecture was proved for Grassmannians [@gu2022quantum], and we will verify it later for the incidence varieties $X=\mathop{\mathrm{Fl}}(1,n-1;n)$ and the complete flag varieties $X=\mathop{\mathrm{Fl}}(n)\coloneqq\mathop{\mathrm{Fl}}(1,\dots,n-1;n)$, the latter under an additional assumption. We start by transforming [([\[eqn:lambda_y rel\]](#eqn:lambda_y rel){reference-type="ref" reference="eqn:lambda_y rel"})](#eqn:lambda_y rel) into an abstract presentation. As in [Section [2.2](#sec:kflag){reference-type="ref" reference="sec:kflag"}](#sec:kflag), let $${X}^{(j)}=(X^{(j)}_1,\dots,X^{(j)}_{r_j})\text{ and }{Y}^{(j)}=(Y^{(j)}_1,\dots,Y^{(j)}_{s_j})$$ denote formal variables for $j=1,\dots,k$, where $s_j \coloneqq r_{j+1}-r_j$. Let $X^{(k+1)}\coloneqq (T_1,\dots,T_n)$ be the equivariant parameters in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$. Let $e_\ell({X}^{(j)})$ and $e_\ell({Y}^{(j)})$ be the $\ell$-th elementary symmetric polynomials in ${X}^{(j)}$ and ${Y}^{(j)}$, respectively. **Definition 9**. As before, $$S=\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[e_1(X^{(j)}),\dots, e_{r_j}(X^{(j)}),e_1(Y^{(j)}),\dots,e_{s_j}(Y^{(j)}), j=1,\dots, k].$$ Let $I_q\subset S[\![q]\!]=S[\![q_1,\dots,q_k]\!]$ be the ideal generated by the coefficients of $y$ in $$\begin{gathered} \label{eqn:qkrel} \prod_{\ell=1}^{r_j}(1+y X^{(j)}_\ell)\prod_{\ell=1}^{s_j}(1+y Y^{(j)}_\ell)-\prod_{\ell=1}^{r_{j+1}}(1+y X^{(j+1)}_\ell)\\+y^{{s_j}}\frac{q_j}{1-q_j}\prod_{\ell=1}^{s_j}Y^{(j)}_\ell\left(\prod_{\ell=1}^{r_j}(1+yX^{(j)}_\ell)-\prod_{\ell=1}^{r_{j-1}}(1+yX^{(j-1)}_\ell)\right),\ j=1,\dots, k.\end{gathered}$$ **Theorem 10**. *Assume holds. Then there is an isomorphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-algebras $$\Psi: {S[\![q]\!]}/I_q\to \mathop{\mathrm{QK}}_T(X)$$ sending $e_\ell(X^{(j)})$ to $\wedge^\ell({\mathcal S}_{j})$ and $e_\ell(Y^{(j)})$ to $\wedge^\ell({\mathcal S}_{{j+1}}/{\mathcal S}_{j})$.* Different presentations have recently been obtained in [@maeno.naito.sagaki:QKideal] for the complete flag varieties, utilizing different methods. Note that the algebra homomorphism $\Psi$ is well-defined by . The proof of follows a method developed in [@gu2022quantum], where was proved for Grassmannians. For notions about filtered modules and completions we refer to [@gu2022quantum Appendix A] and [@AM:intro Ch. 10] for further details. The key fact we utilize is the following result proved in [@gu2022quantum Prop. A.3]. **Proposition 11** (Gu--Mihalcea--Sharpe--Zou). *Let $A$ be a Noetherian integral domain, and let $\mathfrak{a} \subset A$ be an ideal. Assume that $A$ is complete in the $\mathfrak{a}$-adic topology. Let $M,N$ be finitely generated $A$-modules.* *Assume that the $A$-module $N$, and the $A/\mathfrak{a}$-module $N/\mathfrak{a}N$, are both free modules of the same rank $p< \infty$, and that we are given an $A$-module homomorphism $f: M \to N$ such that the induced $A/\mathfrak{a}$-module map $\overline{f}: M/\mathfrak{a} M \to N / \mathfrak{a}N$ is an isomorphism of $A/\mathfrak{a}$-modules.* *Then $f$ is an isomorphism.* A key hypothesis needed in this proposition is that the claimed presentation is finitely generated as a $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q]\!]$-module. This is proved in Appendix [7](#app:fg){reference-type="ref" reference="app:fg"} (see ) in a rather general context about modules over formal power series rings. *Proof of .* We use , with $M$ is the conjectured presentation on the left-hand side of , $$A= \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1, \ldots, q_k]\!], \quad \mathfrak{a}= \langle q_1, \ldots, q_k \rangle\subset A \/, \quad N= \mathop{\mathrm{QK}}_T(X)\/, \quad f= \Psi \/.$$ Since $N$ and $N/\mathfrak{a}N = \mathop{\mathrm{K}}_T(X)$ are both free modules of rank equal to the number of Schubert classes (over $A/\mathfrak{a}$, and over $A$, respectively), the hypotheses are satisfied for $N$ and $N/\mathfrak{a}N$.  implies that the induced $A/\mathfrak{a}$-module map map $\overline{f}: M/\mathfrak{a} M \to N / \mathfrak{a}N$ is an isomorphism of $A/\mathfrak{a}$-modules. Since $R\coloneqq\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$ and $S$ are Noetherian, it then follows from that $M$ is a finitely generated $A$-module. Then the claim follows from . ◻ # Curve neighborhoods and some relations in $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(r_1,\dots,r_k;n))$ Given that depends on the validity , our goal from now on will be to prove this conjecture in some special cases, notably for the incidence varieties $\mathop{\mathrm{Fl}}(1,n-1;n)$ and -- under an extra assumption -- for the complete flag varieties $\mathop{\mathrm{Fl}}(n)\coloneqq\mathop{\mathrm{Fl}}(1,\dots,n-1;n)$. To this aim, we recall some geometric facts about *curve neighborhoods*, a notion introduced in [@buch.m:nbhds] (see also [@BCMP:qkfin]) to help study spaces of rational curves incident to Schubert varieties. For an effective degree $d$, we define the degree $d$ curve neighborhood of a (Schubert) variety $\Omega\subseteq X$ to be $$\Gamma_d(\Omega)\coloneqq\mathop{\mathrm{ev}}_2\left(\mathop{\mathrm{ev}}_1^{-1}(\Omega)\right) \subset X \/.$$ The degree $d$ curve neighborhood of a class $\sigma \in \mathop{\mathrm{K}}_T(X)$ is defined by $$\sigma[d]\coloneqq {\mathop{\mathrm{ev}}_2}_*\mathop{\mathrm{ev}}_1^*(\sigma) \in \mathop{\mathrm{K}}_T(X) \/.$$ It was proved in [@BCMP:qkfin]\*Proposition 3.2 that if $\Omega$ is a Schubert variety, then $\mathop{\mathrm{ev}}_2: \mathop{\mathrm{ev}}_1^{-1}(\Omega)\to\Gamma_d(\Omega)$ is a locally trivial fibration with unirational fibers. It follows that $${\mathcal O}_\Omega[d]={\mathcal O}_{\Gamma_d(\Omega)}.$$ Moreover, it was proved in [@BCMP:qkfin; @buch.m:nbhds] that $\Gamma_d(\Omega)$ is again a Schubert variety. The following proposition follows from [@BCMP:qkfin §2] and [@buch.m:nbhds §7]. **Proposition 12** (Buch--Chaput--Mihalcea--Perrin). *For any effective degree $d$, a Schubert variety $\Omega\subseteq X$, and $j=1,\dots, k$, we have $$\label{eqn:q=c1} \angles{ {\mathcal O}_\Omega }^T_{d}=\chi^T_X({\mathcal O}_{\Gamma_d(\Omega)})=1,$$ $$\label{eqn:q=c2} \angles{ \sigma, {\mathcal O}_\Omega }^T_{d}=\chi^T_X(\sigma\cdot{\mathcal O}_{\Gamma_d(\Omega)}),$$ and if $\Omega$ is $B$-stable, $$\label{eqn:q=c2divisor} \angles{ {\mathcal O}^{s_j}, {\mathcal O}_\Omega }^T_{d}= \begin{cases} 1 & d_j>0\\ \chi^T_X({\mathcal O}^{s_j}\cdot{\mathcal O}_\Omega) & d_j=0 .\end{cases}$$* For $i=1,\dots,k$, let $p_j: X\to Y_j\coloneqq \mathop{\mathrm{Gr}}(r_j,n)$ be the equivariant projection. Abusing notation, we denote by ${\mathcal S}_j$ the tautological bundle on $Y_j$. Note that on $Y_j$ we have the $T$-equivariant short exact sequence $$0\to \det({\mathcal S}_j)\otimes{\mathbb C}_{-t_1-\dots-t_j}\to {\mathcal O}_{Y_j}\to {\mathcal O}^{s_j}\to 0.$$ Pulling back along $p_j$ it gives the identity $$\label{eqn:div-bun} \det({\mathcal S}_j)={\mathbb C}_{t_1+\dots+t_j}(1-{\mathcal O}^{s_j})$$ in $\mathop{\mathrm{K}}_T(X)$. **Corollary 13**. *For any effective degree $d$, a $B$-stable Schubert variety $\Omega \subseteq X$, and $j=1,\dots, k$, we have $$\label{eqn:2point-det} \angles{ \det({\mathcal S}_j), {\mathcal O}_\Omega }^T_{d}= \begin{cases} 0 & d_i>0\\ \chi^T_X(\det({\mathcal S}_j)\cdot{\mathcal O}_\Omega) & d_j=0 .\end{cases}$$* *Proof.* By [([\[eqn:div-bun\]](#eqn:div-bun){reference-type="ref" reference="eqn:div-bun"})](#eqn:div-bun), we have $$\begin{aligned} \angles{ \det({\mathcal S}_j), {\mathcal O}_\Omega }^T_{d}={\mathbb C}_{t_1+\dots+t_j}\left(\angles{ {\mathcal O}_\Omega }^T_{d}-\angles{ {\mathcal O}^{s_j},{\mathcal O}_\Omega }^T_{d}\right), \end{aligned}$$ and [([\[eqn:2point-det\]](#eqn:2point-det){reference-type="ref" reference="eqn:2point-det"})](#eqn:2point-det) follows from [([\[eqn:q=c1\]](#eqn:q=c1){reference-type="ref" reference="eqn:q=c1"})](#eqn:q=c1) and [([\[eqn:q=c2divisor\]](#eqn:q=c2divisor){reference-type="ref" reference="eqn:q=c2divisor"})](#eqn:q=c2divisor). ◻ Buch and Mihalcea have the following unpublished conjecture about 3-pointed (equivariant) Chevalley-type $\mathop{\mathrm{K}}$-theoretic Gromov--Witten invariants. **Conjecture 14** (Buch--Mihalcea). *For any effective degree $d$ and $\sigma, \tau\in\mathop{\mathrm{K}}_T(X)$, we have $$\label{eqn:conj} \angles{ \det({\mathcal S}_j),\sigma,\tau }^T_{d}= \begin{cases} 0 & d_j>0\\\chi_X^T(\det({\mathcal S}_j)\cdot\sigma\cdot\tau[d]) & d_j=0. \end{cases}$$* The conjecture has been proved in several special situations. For (cominuscule) Grassmannians, it follows from the "quantum = classical" results in [@buch.m:qk; @chaput.perrin:rationality], and for incidence varieties $\mathop{\mathrm{Fl}}(1,n-1;n)$ it was recently proved in [@xu2021quantum]; see below. In the next proposition, we give a short proof that implies part of for all partial flag varieties. In [Section [6](#sec:fullflag){reference-type="ref" reference="sec:fullflag"}](#sec:fullflag), we prove that implies when $X$ is the complete flag variety $\mathop{\mathrm{Fl}}(n)$. **Proposition 15**. *Assuming that holds, the following relations hold in $\mathop{\mathrm{QK}}_T(X)$: $$\det({\mathcal S}_i)\star\det({\mathcal S}_{i+1}/{\mathcal S}_i)=(1-q_i)\det({\mathcal S}_{i+1}),\quad i=1,\dots,k-1.$$* *Proof.* Let $d=(d_1,\dots,d_k)$ be any degree and with $i$ fixed, define the degree $d'$ by $${d_j'}= \begin{cases} d_j & j\neq i\\d_i-1 & j=i. \end{cases}$$ We need to show that for any $\sigma \in \mathop{\mathrm{K}}_T(X)$, $$\label{eqn:toprove} \angles{ \det({\mathcal S}_i), \det({\mathcal S}_{i+1}/{\mathcal S}_i), \sigma }^T_{d} = \angles{ \det({\mathcal S}_{i+1}), \sigma }^T_{d} - \angles{ \det({\mathcal S}_{i+1}), \sigma }^T_{d'} \/,$$ with the convention that the term involving $d'$ is omitted if $d_i=0$. If $d_i\neq0$, then by , the left-hand side of [([\[eqn:toprove\]](#eqn:toprove){reference-type="ref" reference="eqn:toprove"})](#eqn:toprove) is equal to $0$. By [([\[eqn:2point-det\]](#eqn:2point-det){reference-type="ref" reference="eqn:2point-det"})](#eqn:2point-det), $$\begin{split} \angles{ \det({\mathcal S}_{i+1}), \sigma }^T_{d} &= \begin{cases} 0 & d_{i+1}>0 \\ \chi_X^T(\det({\mathcal S}_{i+1})\cdot\sigma) & d_{i+1}=0 \end{cases}\\ &=\angles{ \det({\mathcal S}_{i+1}), \sigma }^T_{d'}\ , \end{split}$$ which implies that the right-hand side of [([\[eqn:toprove\]](#eqn:toprove){reference-type="ref" reference="eqn:toprove"})](#eqn:toprove) is also equal to $0$. If $d_i=0$, then by and [([\[eqn:2point-det\]](#eqn:2point-det){reference-type="ref" reference="eqn:2point-det"})](#eqn:2point-det), the left-hand side of [([\[eqn:toprove\]](#eqn:toprove){reference-type="ref" reference="eqn:toprove"})](#eqn:toprove) equals $$\chi_X^T\left(\det({\mathcal S}_i)\cdot\det({\mathcal S}_{i+1}/{\mathcal S}_i)\cdot\sigma[d]\right)=\chi_X^T(\det({\mathcal S}_{i+1})\cdot\sigma[d])=\angles{ \det({\mathcal S}_{i+1}),\sigma }^T_{d},$$ which equals the right-hand side of [([\[eqn:toprove\]](#eqn:toprove){reference-type="ref" reference="eqn:toprove"})](#eqn:toprove). ◻ # The Whitney presentation for $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(1,n-1;n))$ {#sec:incidence} In this section, we prove the Whitney presentation for $\mathop{\mathrm{QK}}_T(X)$, where $X=\mathop{\mathrm{Fl}}(1,n-1;n)$ for $n \ge 3$ is the incidence variety. A key fact is that has been proved in this case [@xu2021quantum]. We start with a section introducing some preliminary results needed in the proof. ## Preliminaries Recall that $X$ is a projective manifold of dimension $2n-3$, and that $0\subset{\mathcal S}_1\subset{\mathcal S}_2\subset{\mathcal S}_3={\mathbb C}^n$ is the flag of tautological bundles over $X$, where the ${\mathcal S}_1, {\mathcal S}_2$ are of ranks $1, n-1$ respectively. It was proved in [@xu2021quantum]\*Corollary 4.6 that holds for incidence varieties. **Theorem 16** (Xu). *Let $d$ be an effective degree. For any Schubert variety $\Omega$, and $k=1,\ 2$, $$\label{eqn:q=c3} \angles{ {\mathcal O}^{s_k},\sigma,{\mathcal O}_\Omega }^T_{d}=\begin{cases} \chi^T_X(\sigma\cdot{\mathcal O}_{\Gamma_d(\Omega)}) & d_k>0\\ \chi^T_X({\mathcal O}^{s_k}\cdot\sigma\cdot{\mathcal O}_{\Gamma_d(\Omega)}) & d_k=0 .\end{cases}$$* **Corollary 17**. *For any effective degree $d$, and any Schubert variety $\Omega$, $$\label{eqn:q=c31-O1} \angles{ {\mathcal S}_1, \sigma, {\mathcal O}_\Omega }^T_{d}= \begin{cases} 0 & d_1>0\\\chi^T_X({\mathcal S}_1\cdot\sigma\cdot{\mathcal O}_{\Gamma_d(\Omega)}) & d_1=0 ,\end{cases}$$ $$\label{eqn:q=c31-O2} \angles{ \det({\mathcal S}_2), \sigma, {\mathcal O}_\Omega }^T_{d}=\begin{cases} 0 & d_2>0\\\chi^T_X(\det({\mathcal S}_2)\cdot\sigma\cdot{\mathcal O}_{\Gamma_d(\Omega)}) & d_2=0 .\end{cases}$$* *Proof.* Since ${\mathcal S}_1={\mathbb C}_{t_1}(1-{\mathcal O}^{s_1})$ by [([\[eqn:div-bun\]](#eqn:div-bun){reference-type="ref" reference="eqn:div-bun"})](#eqn:div-bun), we have $$\begin{aligned} \angles{ {\mathcal S}_1, \sigma, {\mathcal O}_\Omega }^T_{d}={\mathbb C}_{t_1}\left(\angles{ \sigma,{\mathcal O}_\Omega }^T_{d}-\angles{ {\mathcal O}^{s_1},\sigma,{\mathcal O}_\Omega }^T_{d}\right), \end{aligned}$$ and then [([\[eqn:q=c31-O1\]](#eqn:q=c31-O1){reference-type="ref" reference="eqn:q=c31-O1"})](#eqn:q=c31-O1) follows from [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2) and [([\[eqn:q=c3\]](#eqn:q=c3){reference-type="ref" reference="eqn:q=c3"})](#eqn:q=c3). The proof of [([\[eqn:q=c31-O2\]](#eqn:q=c31-O2){reference-type="ref" reference="eqn:q=c31-O2"})](#eqn:q=c31-O2) is similar. ◻ **Remark 18**. Let $\mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)\subseteq\mathop{\mathrm{QK}}_T(X)$ be the subring generated by ${\mathcal O}^{s_1}$ and ${\mathcal O}^{s_2}$ over the ground ring $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q_1,q_2]$. [@xu2021quantum]\*Algorithm 4.16 gives an algorithm that recursively expresses any Schubert class as a polynomial in ${\mathcal O}^{s_1},\ {\mathcal O}^{s_2}$ with coefficients in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q_1,q_2]$. Combined with [@xu2021quantum]\*Theorem 4.5, this means that when expressing the product of two Schubert classes as a linear combination of Schubert classes in $\mathop{\mathrm{QK}}_T(X)$, the coefficients are always in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q_1,q_2]$. Therefore, $\mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)$ can be identified with $\mathop{\mathrm{K}}_T(X)\otimes{\mathbb Z}[q_1,q_2]$ as a module. Because of [([\[eqn:div-bun\]](#eqn:div-bun){reference-type="ref" reference="eqn:div-bun"})](#eqn:div-bun), $\mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)$ is also generated by ${\mathcal S}_1,\ \det({\mathcal S}_2)$ over $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q_1,q_2]$. For convenience, we restate the following curve neighborhood computations from [@xu2021quantum §2.2.2]. **Lemma 19**. *The curve neighborhoods of a Schubert variety $\Omega \subseteq X$ are: $$\label{eqn:curvenbhd} \Gamma_d(\Omega)=\begin{cases} p_1^{-1}\left(p_1(\Omega)\right)& d_1=0,\ d_2>0\\ p_2^{-1}\left(p_2(\Omega)\right)&d_1>0,\ d_2=0\\ X& d_1>0,\ d_2>0. \end{cases}$$* ## Quantum $\mathop{\mathrm{K}}$ Whitney relations for incidence varieties We prove in of this section two equivalent quantized versions of the Whitney relations for the quantum K ring of incidence varieties. We shall use the classical Whitney relations in $\mathop{\mathrm{K}}_T(X)$: $$\label{eqn:kincidence} \lambda_y({\mathcal S}_{j})\cdot\lambda_y({\mathcal S}_{{j+1}}/{\mathcal S}_{j})=\lambda_y({\mathcal S}_{{j+1}}),\quad j=1,2.$$ Equivalently, $$\label{eqn:w1} \wedge^{\ell}({\mathcal S}_2/{\mathcal S}_1)+{\mathcal S}_1\cdot\wedge^{\ell-1}({\mathcal S}_2/{\mathcal S}_1)=\wedge^\ell({\mathcal S}_2),\quad \ell=1,\dots,n-1$$ and $$\label{eqn:w2} \wedge^\ell{\mathcal S}_2+\wedge^{\ell-1}{\mathcal S}_2\cdot{\mathbb C}^n/{\mathcal S}_2=\wedge^\ell{\mathbb C}^n, \quad\ell=1,\dots,n.$$ We also regard the incidence variety $X$ as a Grassmann bundle in two ways, via the equivariant projections $$p_1: X = \mathbb{G}(n-2, {\mathbb C}^n/{\mathcal S}_1) \to Y_1=\mathop{\mathrm{Gr}}(1,n)$$ and $$p_2: X = \mathbb{G}(1, {\mathcal S}_2)\to Y_2=\mathop{\mathrm{Gr}}(n-1,n).$$ The next proposition will turn out to be a restatement of the second relation in . **Proposition 20**. *For any $1 \le \ell \le n$, the following relation holds in $\mathop{\mathrm{QK}}_T(X)$: $$\label{eqn:3cases} \det({\mathcal S}_2)\star\left(\wedge^\ell{\mathbb C}^n-\wedge^\ell {\mathcal S}_2\right)=\wedge^n{\mathbb C}^n \left(\wedge^{\ell -1}{\mathcal S}_2-q_2\wedge^{\ell -1}{\mathcal S}_1\right).$$* *Proof.* To prove [([\[eqn:3cases\]](#eqn:3cases){reference-type="ref" reference="eqn:3cases"})](#eqn:3cases), it suffices to prove that $$\label{eqn:GW} \angles{ \det({\mathcal S}_2),(\wedge^\ell {\mathbb C}^n-\wedge^\ell {\mathcal S}_2),{\mathcal O}_\Omega }^T_{d}=\wedge^n{\mathbb C}^n\left(\angles{ \wedge^{\ell -1}{\mathcal S}_2,{\mathcal O}_\Omega }^T_{d}-\angles{ \wedge^{\ell -1}{\mathcal S}_1,{\mathcal O}_\Omega }^T_{d''}\right)$$ for any Schubert variety $\Omega\subseteq X$ and effective degree $d=(d_1,d_2)$, where $d''\coloneqq(d_1,d_2-1)$. This follows from [([\[eqn:def\]](#eqn:def){reference-type="ref" reference="eqn:def"})](#eqn:def) and the fact that Schubert classes form a basis for $\mathop{\mathrm{K}}_T(X)$ over $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$. First assume $d_2=0$. Then by [([\[eqn:q=c31-O2\]](#eqn:q=c31-O2){reference-type="ref" reference="eqn:q=c31-O2"})](#eqn:q=c31-O2), the left-hand side of [([\[eqn:GW\]](#eqn:GW){reference-type="ref" reference="eqn:GW"})](#eqn:GW) is equal to $$\begin{aligned} \chi^T_X\left(\det({\mathcal S}_2)\cdot(\wedge^\ell{\mathbb C}^n-\wedge^\ell {\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)&=\chi^T_X\left(\det({\mathcal S}_2)\cdot({\mathbb C}^n/{\mathcal S}_2)\cdot\wedge^{\ell-1}{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)\\ &=\chi_X^T\left(\wedge^n{\mathbb C}^n\cdot\wedge^{\ell-1}{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)\\ &=\wedge^n{\mathbb C}^n\cdot \chi^T_X\left(\wedge^{\ell-1}{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right), \end{aligned}$$ where the first equality utilizes [([\[eqn:w2\]](#eqn:w2){reference-type="ref" reference="eqn:w2"})](#eqn:w2), and the second follows again from [([\[eqn:w2\]](#eqn:w2){reference-type="ref" reference="eqn:w2"})](#eqn:w2) in the special case $\ell =n$. By [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2), this equals the right-hand side of [([\[eqn:GW\]](#eqn:GW){reference-type="ref" reference="eqn:GW"})](#eqn:GW). Now assume $d_2>0$. By [([\[eqn:q=c31-O2\]](#eqn:q=c31-O2){reference-type="ref" reference="eqn:q=c31-O2"})](#eqn:q=c31-O2), the left-hand side of [([\[eqn:GW\]](#eqn:GW){reference-type="ref" reference="eqn:GW"})](#eqn:GW) is equal to $0$. It suffices to show that for any $0 \le k \le n-1$, $$\label{eqn:right-hand side} \angles{ \wedge^k{\mathcal S}_2,{\mathcal O}_\Omega }^T_{d}=\angles{ \wedge^k{\mathcal S}_1,{\mathcal O}_\Omega }^T_{{d''}}.$$ By [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2), equation [([\[eqn:right-hand side\]](#eqn:right-hand side){reference-type="ref" reference="eqn:right-hand side"})](#eqn:right-hand side) is equivalent to $$\chi_X^T\left(\wedge^k{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi_X^T\left(\wedge^k{\mathcal S}_1\cdot{\mathcal O}_{\Gamma_{d''}(\Omega)}\right).$$ When $k=0$ both sides are equal to $1$, and we assume from now on that $k>0$. By [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2) and the projection formula, $$\begin{aligned} \chi^T_X\left(\wedge^k{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi^T_{Y_2}\left(\wedge^k{\mathcal S}_2\cdot{p_2}_*{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi^T_{Y_2}\left(\wedge^k{\mathcal S}_2\cdot{\mathcal O}_{p_2\left(\Gamma_d(\Omega)\right)}\right). \end{aligned}$$ When $d_1>0$, we have $\Gamma_d(\Omega)=X$ by [([\[eqn:curvenbhd\]](#eqn:curvenbhd){reference-type="ref" reference="eqn:curvenbhd"})](#eqn:curvenbhd) and $p_2\left(\Gamma_d(\Omega)\right)=Y_2$. Therefore, $$\chi^T_{Y_2}\left(\wedge^k{\mathcal S}_2\cdot{\mathcal O}_{p_2\left(\Gamma_d(\Omega)\right)}\right)=\chi^T_{Y_2}\left(\wedge^k{\mathcal S}_2\right)=0$$ by . By [([\[eqn:2point-det\]](#eqn:2point-det){reference-type="ref" reference="eqn:2point-det"})](#eqn:2point-det), we have $\chi_X^T\left(\wedge^k{\mathcal S}_1\cdot{\mathcal O}_{\Gamma_{d''}(\Omega)}\right)=0,$ proving [([\[eqn:right-hand side\]](#eqn:right-hand side){reference-type="ref" reference="eqn:right-hand side"})](#eqn:right-hand side) in this case. When $d_1=0$, by [([\[eqn:curvenbhd\]](#eqn:curvenbhd){reference-type="ref" reference="eqn:curvenbhd"})](#eqn:curvenbhd), $\Gamma_d(\Omega)=p_1^{-1}\left(p_1(\Omega)\right)$, and therefore $$p_1(\Omega)\supseteq p_1\left(\Gamma_d(\Omega)\right)\supseteq p_1\left(\Gamma_{d''}(\Omega)\right)\supseteq p_1(\Omega),$$ forcing all of them to be equal. By the projection formula and [([\[eqn:w1\]](#eqn:w1){reference-type="ref" reference="eqn:w1"})](#eqn:w1), $$\begin{aligned} \chi^T_X\left(\wedge^k{\mathcal S}_2\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=&\chi^T_X\left(\wedge^k{\mathcal S}_2\cdot p_1^*{\mathcal O}_{p_1(\Omega)}\right)=\chi_{Y_1}^T\left({p_1}_*(\wedge^k{\mathcal S}_2)\cdot{\mathcal O}_{p_1(\Omega)}\right)\\=&\chi_{Y_1}^T\left({p_1}_*\left({\mathcal S}_1\cdot\wedge^{k-1}({\mathcal S}_2/{\mathcal S}_1)+\wedge^k({\mathcal S}_2/{\mathcal S}_1)\right)\cdot{\mathcal O}_{p_1(\Omega)}\right)\\=&\chi_{Y_1}^T\left(\left({\mathcal S}_1\cdot{p_1}_*\left(\wedge^{k-1}({\mathcal S}_2/{\mathcal S}_1)\right)+{p_1}_*\left(\wedge^k({\mathcal S}_2/{\mathcal S}_1)\right)\right)\cdot{\mathcal O}_{p_1(\Omega)}\right)\\=& \begin{cases} \chi_{Y_1}^T\left({\mathcal S}_1\cdot{\mathcal O}_{p_1(\Omega)}\right) & k=1\\ 0 & k>1, \end{cases} \end{aligned}$$ where the last equality follows from because ${\mathcal S}_2/{\mathcal S}_1$ is the tautological subbundle of the Grassmann bundle $X=\mathbb{G}(n-2, {\mathbb C}^n/{\mathcal S}_1) \to Y_1$. The claim follows from combining this and $$\begin{split} \chi_X^T\left(\wedge^k{\mathcal S}_1\cdot{\mathcal O}_{\Gamma_{d''}(\Omega)}\right)& =\chi_{Y_1}^T\left(\wedge^k{\mathcal S}_1\cdot{\mathcal O}_{p_1(\Gamma_{d''}(\Omega))}\right)\\ & =\chi_{Y_1}^T\left(\wedge^k{\mathcal S}_1\cdot{\mathcal O}_{p_1(\Omega)}\right)\\ & =\begin{cases} \chi_{Y_1}^T\left({\mathcal S}_1\cdot{\mathcal O}_{p_1(\Omega)}\right) & k=1\\ 0 & k>1. \end{cases} \end{split}$$ ◻ The following theorem "quantizes" the classical $\mathop{\mathrm{K}}$-theoretic Whitney relations [([\[eqn:kincidence\]](#eqn:kincidence){reference-type="ref" reference="eqn:kincidence"})](#eqn:kincidence). **Theorem 21**. *The following relations hold in $\mathop{\mathrm{QK}}_T(X)$: $$\label{eqn:rel1} \lambda_y({\mathcal S}_1)\star\lambda_y({\mathcal S}_2/{\mathcal S}_1)=\lambda_y({\mathcal S}_2)-q_1 y^{n-1}\det({\mathcal S}_2),$$ $$\label{eqn:rel2} {\lambda}_y({\mathcal S}_2)\star{\lambda}_y({\mathbb C}^n/{\mathcal S}_2)={\lambda}_y({\mathbb C}^n)-q_2[{\lambda}_y({\mathbb C}^n)-{\lambda}_y({\mathcal S}_2)-({\lambda}_y({\mathbb C}^n/{\mathcal S}_2)-1)\star {\lambda}_y({\mathcal S}_1)].$$ As a consequence, holds for incidence varieties.* *Proof.* The proof of [([\[eqn:rel1\]](#eqn:rel1){reference-type="ref" reference="eqn:rel1"})](#eqn:rel1) is similar to that of , but we provide the details for completeness. To prove [([\[eqn:rel1\]](#eqn:rel1){reference-type="ref" reference="eqn:rel1"})](#eqn:rel1), it suffices to prove that $$\label{eqn:GW1} \angles{ {\lambda}_y({\mathcal S}_1),{\lambda}_y({\mathcal S}_2/{\mathcal S}_1),{\mathcal O}_\Omega }^T_{d}=\angles{ {\lambda}_y({\mathcal S}_2),{\mathcal O}_\Omega }^T_{d}-y^{n-1}\angles{ \det({\mathcal S}_2),{\mathcal O}_\Omega }^T_{d'}$$ for any Schubert variety $\Omega\subseteq X$ and effective degree $d=(d_1,d_2)$, where $d'\coloneqq (d_1-1,d_2)$. When $d_1=0$, equation [([\[eqn:GW1\]](#eqn:GW1){reference-type="ref" reference="eqn:GW1"})](#eqn:GW1) follows from [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2) and [([\[eqn:q=c31-O1\]](#eqn:q=c31-O1){reference-type="ref" reference="eqn:q=c31-O1"})](#eqn:q=c31-O1), because $$\chi_X^T\left({\lambda}_y({\mathcal S}_1)\cdot{\lambda}_y({\mathcal S}_2/{\mathcal S}_1)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi_X^T\left({\lambda}_y({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)\/$$ by [([\[eqn:kincidence\]](#eqn:kincidence){reference-type="ref" reference="eqn:kincidence"})](#eqn:kincidence). Now assume that $d_1>0$. By [([\[eqn:q=c2\]](#eqn:q=c2){reference-type="ref" reference="eqn:q=c2"})](#eqn:q=c2) and [([\[eqn:q=c31-O1\]](#eqn:q=c31-O1){reference-type="ref" reference="eqn:q=c31-O1"})](#eqn:q=c31-O1), the left-hand side of [([\[eqn:GW1\]](#eqn:GW1){reference-type="ref" reference="eqn:GW1"})](#eqn:GW1) is equal to $$\chi_X^T\left({\lambda}_y({\mathcal S}_2/{\mathcal S}_1)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right),$$ and the right-hand side of [([\[eqn:GW1\]](#eqn:GW1){reference-type="ref" reference="eqn:GW1"})](#eqn:GW1) is equal to $$\chi_X^T\left({\lambda}_y({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)-y^{n-1}\chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_{d'}(\Omega)}\right).$$ Note that $\Gamma_d(\Omega)=p_2^{-1}(p_2\left(\Gamma_d(\Omega)\right))$ by [([\[eqn:curvenbhd\]](#eqn:curvenbhd){reference-type="ref" reference="eqn:curvenbhd"})](#eqn:curvenbhd). By the projection formula, $$\begin{aligned} \chi_X^T\left({\lambda}_y({\mathcal S}_2/{\mathcal S}_1)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)&=\chi_X^T\left({\lambda}_y({\mathcal S}_2/{\mathcal S}_1)\cdot p_2^*{\mathcal O}_{p_2\left(\Gamma_d(\Omega)\right)}\right)\\ &=\chi_{Y_2}^T\left({p_2}_*\left({\lambda}_y({\mathcal S}_2/{\mathcal S}_1)\right)\cdot{\mathcal O}_{p_2\left(\Gamma_d(\Omega)\right)}\right)\\ &=\chi_{Y_2}^T\left({\lambda}_y({\mathcal S}_2)_{\leq n-2}\cdot{\mathcal O}_{p_2\left(\Gamma_d(\Omega)\right)}\right)\\ &=\chi_{Y_2}^T\left({\lambda}_y({\mathcal S}_2)_{\leq n-2}\cdot{p_2}_*{\mathcal O}_{\Gamma_d(\Omega)}\right)\\ &=\chi_X^T\left({\lambda}_y({\mathcal S}_2)_{\leq n-2}\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right), \end{aligned}$$ where $${\lambda}_y({\mathcal S}_2)_{\leq n-2}=1+y{\mathcal S}_2+\dots+y^{n-2}\wedge^{n-2}{\mathcal S}_2$$ and the third equality follows from . Therefore, it suffices to show that $$\label{eqn:d1>0} \chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_{d'}(\Omega)}\right).$$ When $d_2=0$, by [([\[eqn:curvenbhd\]](#eqn:curvenbhd){reference-type="ref" reference="eqn:curvenbhd"})](#eqn:curvenbhd), we have $\Gamma_d(\Omega)=p_2^{-1}(p_2(\Omega))$, and as in the proof of , we have $$p_2(\Omega)\supseteq p_2\left(\Gamma_d(\Omega)\right)\supseteq p_2\left(\Gamma_{d'}(\Omega)\right)\supseteq p_2(\Omega),$$ forcing all of them to be equal. By the projection formula, $$\begin{aligned} \chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=&\chi_{Y_2}^T\left(\det({\mathcal S}_2)\cdot{p_2}_*{\mathcal O}_{\Gamma_d(\Omega)}\right)\\ =&\chi_{Y_2}^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{{p_2}(\Gamma_d(\Omega))}\right). \end{aligned}$$ Similarly, $$\chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_{d'}(\Omega)}\right)=\chi_{Y_2}^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{{p_2}(\Gamma_{d'}(\Omega))}\right).$$ Equation [([\[eqn:d1\>0\]](#eqn:d1>0){reference-type="ref" reference="eqn:d1>0"})](#eqn:d1>0) follows because ${p_2}(\Gamma_d(\Omega))={p_2}(\Gamma_{d'}(\Omega))$. When $d_2>0$, by [([\[eqn:curvenbhd\]](#eqn:curvenbhd){reference-type="ref" reference="eqn:curvenbhd"})](#eqn:curvenbhd), $\Gamma_d(\Omega)=X$ and $\Gamma_{d'}(\Omega)=p_1^{-1}(p_1(\Gamma_{d'}(\Omega)))$. Therefore, by the projection formula and , $$\chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_d(\Omega)}\right)=\chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_X\right)=\chi_X^T\left(\det({\mathcal S}_2)\right)=\chi_{Y_2}^T\left(\det({\mathcal S}_2)\right)=0,$$ and $$\begin{aligned} \chi_X^T\left(\det({\mathcal S}_2)\cdot{\mathcal O}_{\Gamma_{d'}(\Omega)}\right)=&\chi_X^T\left(\wedge^{n-2}({\mathcal S}_2/{\mathcal S}_1)\cdot {\mathcal S}_1\cdot{\mathcal O}_{\Gamma_{d'}(\Omega)}\right)\\ =&\chi_X^T\left(\wedge^{n-2}({\mathcal S}_2/{\mathcal S}_1)\cdot p_1^*\left({\mathcal S}_1\cdot{\mathcal O}_{p_1(\Gamma_{d'}(\Omega))}\right)\right)\\ =&\chi_{Y_1}^T\left({p_1}_*\left(\wedge^{n-2}({\mathcal S}_2/{\mathcal S}_1)\right)\cdot {\mathcal S}_1\cdot{\mathcal O}_{p_1(\Gamma_{d'}(\Omega))}\right)=0, \end{aligned}$$ from which [([\[eqn:d1\>0\]](#eqn:d1>0){reference-type="ref" reference="eqn:d1>0"})](#eqn:d1>0) follows. We derive [([\[eqn:rel2\]](#eqn:rel2){reference-type="ref" reference="eqn:rel2"})](#eqn:rel2) from [([\[eqn:3cases\]](#eqn:3cases){reference-type="ref" reference="eqn:3cases"})](#eqn:3cases). First, note that $$\label{eqn:sum} {\lambda}_y({\mathcal S}_2)\star{\lambda}_y({\mathbb C}^n/{\mathcal S}_2)=\sum_{\ell=0}^n y^\ell(\wedge^\ell {\mathcal S}_2+\wedge^{\ell-1}{\mathcal S}_2\star({\mathbb C}^n/{\mathcal S}_2)).$$ Equation [([\[eqn:3cases\]](#eqn:3cases){reference-type="ref" reference="eqn:3cases"})](#eqn:3cases) applied to $\ell=1$ gives that $({\mathbb C}^n/{\mathcal S}_2)\star\det({\mathcal S}_2)={\wedge^n {\mathbb C}^n}(1-q_2)$. Then by associativity $$\begin{aligned} \frac{({\mathbb C}^n/{\mathcal S}_2)\star\det({\mathcal S}_2)\star(\wedge^\ell{\mathbb C}^n-\wedge^\ell {\mathcal S}_2)}{\wedge^n{\mathbb C}^n}&=(1-q_2)(\wedge^\ell{\mathbb C}^n-\wedge^\ell {\mathcal S}_2)\\ &=({\mathbb C}^n/{\mathcal S}_2)\star(\wedge^{\ell-1}{\mathcal S}_2-q_2\wedge^{\ell-1}{\mathcal S}_1). \end{aligned}$$ After rearranging the terms in the last equality, we obtain $$\label{eqn:subs} \begin{aligned} \wedge^\ell {\mathcal S}_2+\wedge^{\ell-1}{\mathcal S}_2\star({\mathbb C}^n/{\mathcal S}_2)=\wedge^\ell{\mathbb C}^n-q_2\left(\wedge^\ell{\mathbb C}^n-\wedge^\ell {\mathcal S}_2-({\mathbb C}^n/{\mathcal S}_2)\star\wedge^{\ell-1}{\mathcal S}_1\right). \end{aligned}$$ Plugging [([\[eqn:subs\]](#eqn:subs){reference-type="ref" reference="eqn:subs"})](#eqn:subs) into [([\[eqn:sum\]](#eqn:sum){reference-type="ref" reference="eqn:sum"})](#eqn:sum), we obtain [([\[eqn:rel2\]](#eqn:rel2){reference-type="ref" reference="eqn:rel2"})](#eqn:rel2): $${\lambda}_y({\mathcal S}_2)\star{\lambda}_y({\mathbb C}^n/{\mathcal S}_2)={\lambda}_y({\mathbb C}^n)-q_2[{\lambda}_y({\mathbb C}^n)-{\lambda}_y({\mathcal S}_2)-({\lambda}_y({\mathbb C}^n/{\mathcal S}_2)-1)\star {\lambda}_y({\mathcal S}_1)].$$ Finally, the proof ends by observing that the relations just proved are equivalent to the relations from . Indeed, the equivalence of [([\[eqn:rel1\]](#eqn:rel1){reference-type="ref" reference="eqn:rel1"})](#eqn:rel1) and the first relation ($j=1$) from uses that $$\det {\mathcal S}_1 \star \det {\mathcal S}_2/{\mathcal S}_1 = (1-q_1) \det {\mathcal S}_2 \/$$ is a special case of both [([\[eqn:rel1\]](#eqn:rel1){reference-type="ref" reference="eqn:rel1"})](#eqn:rel1) and . The equivalence of [([\[eqn:rel2\]](#eqn:rel2){reference-type="ref" reference="eqn:rel2"})](#eqn:rel2) and the second relation ($j=2$) from follows by multiplying by $1-q_2$ and rearranging terms. This finishes the proof. ◻ ## The QK Whitney presentation {#sec:presentation} The goal of this section is to prove, in and respectively, the Whitney presentation for $\mathop{\mathrm{QK}}_T(X)$ and a "localized" version, which holds over a subring of the power series ring. In we conjecture a "polynomial" version of these presentations. We start by recalling the Whitney relations in the case of incidence varieties. Recall that $$S = \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[X^{(1)}_1,e_1(X^{(2)}),\dots,e_{{n-1}}(X^{(2)}),e_1(Y^{(1)}),\dots,e_{n-2}(Y^{(1)}),Y^{(2)}_1] \/,$$ and let $I_q \subset S[\![q]\!]$ be the ideal generated by the coefficients of $y$ in $$\label{eqn:r-1} (1+yX^{(1)}_1)\prod_{\ell=1}^{n-2}(1+yY^{(1)}_\ell)-\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)+y^{n-1}\frac{q_1}{1-q_1}(\prod_{\ell=1}^{n-2}Y^{(1)}_\ell)X^{(1)}_1$$ and $$\label{eqn:r-2} \left(\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)\right)(1+yY^{(2)}_1)-\prod_{\ell=1}^{n}(1+yT_\ell)+y\frac{q_2}{1-q_2}Y^{(2)}_1\left(\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)-(1+yX^{(1)}_1)\right).$$ **Theorem 22**. *There is an isomorphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1,q_2]\!]$-algebras $$\Psi:S[\![q]\!]/I_q\to \mathop{\mathrm{QK}}_T(X)$$ sending $X^{(1)}_1$ to ${\mathcal S}_1$, $e_\ell(Y^{(1)})$ to $\wedge^\ell({\mathcal S}_2/{\mathcal S}_1)$, $e_\ell(X^{(2)})$ to $\wedge^\ell({\mathcal S}_2)$, and $Y^{(2)}_1$ to ${\mathbb C}^n/{\mathcal S}_2$.* *Proof.* This follows from and . ◻ Define the following polynomial and localized versions of submodules of $\mathop{\mathrm{QK}}_T(X)$: $$\mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)\coloneqq\mathop{\mathrm{K}}_T(X) \otimes{\mathbb Z}[q_1,q_2] \subset \mathop{\mathrm{QK}}_T^{\rm{loc}}(X)\coloneqq \mathop{\mathrm{K}}_T(X) \otimes{\mathbb Z}[q_1,q_2]_{1+\langle q_1,q_2\rangle} \subset \mathop{\mathrm{QK}}_T(X) \/.$$ Since the product of two Schubert classes in $\mathop{\mathrm{QK}}_T(X)$ involves only coefficients in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[q_1, q_2]$ (cf. , see also [@anderson2022finite] and [@kato:loop]\*Cor. 4.16), it follows that both $\mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)$ and $\mathop{\mathrm{QK}}_T^{\rm{loc}}(X)$ are in fact subalgebras of $\mathop{\mathrm{QK}}_T(X)$ over the appropriate ground rings. Consider the polynomial ring $S[q]$ and let $I_q^{{\mathrm{poly}}}\subset S[q]$ be the ideal generated by the coefficients of $y$ in $$\label{eqn:r1} (1+y X^{(1)}_1)\prod_{\ell=1}^{n-2}(1+yY^{(1)}_\ell)=\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)-q_1y^{n-1}\prod_{\ell=1}^{n-1}X^{(2)}_\ell$$ and $$\begin{gathered} \label{eqn:r2} \left(\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)\right)(1+yY^{(2)}_1)=\prod_{\ell=1}^{n}(1+yT_\ell)\\-q_2\left(\prod_{\ell=1}^{n}(1+yT_\ell)-\prod_{\ell=1}^{n-1}(1+yX^{(2)}_\ell)-yY^{(2)}_1(1+yX^{(1)}_1)\right).\end{gathered}$$ By [@AM:intro Prop. 10.13], the completion of the $S[q]$-module $S[q]/I_q^{\mathrm{poly}}$ along the ideal $\langle q\rangle\coloneqq\langle q_1, q_2 \rangle$ is $$S[q]/I_q^{\mathrm{poly}}\otimes_{S[q]} S[\![q]\!] = S[\![q]\!]/I_q \/,$$ where we have used that $I_q^{\mathrm{poly}}S[\![q]\!] = I_q$, by the proof of . Set $S[q]_{\mathrm{loc}}:=S[q]_{1+\langle q\rangle}$, and define the ideal $I_q^{\mathrm{loc}}\coloneqq I_q^{\mathrm{poly}}S[q]_{\mathrm{loc}}$ generated by [([\[eqn:r-1\]](#eqn:r-1){reference-type="ref" reference="eqn:r-1"})](#eqn:r-1) and [([\[eqn:r-2\]](#eqn:r-2){reference-type="ref" reference="eqn:r-2"})](#eqn:r-2) in the localized ring $S[q]_{\mathrm{loc}}$. Since $$S[q]_{\mathrm{loc}}/I_q^{\mathrm{loc}}= (S[q]/I_q^{\mathrm{poly}})_{1+\langle q\rangle} \/,$$ it follows from [@AM:intro p. 110] that $S[q]_{\mathrm{loc}}/I_q^{\mathrm{loc}}$ is s subring of $S[\![q]\!]/I_q$. **Corollary 23**. *The isomorphism $\Psi$ restricts to an isomorphism of $S[q]_{\mathrm{loc}}$-algebras $$\Psi_{\rm{loc}}: S[q]_{\mathrm{loc}}/I_q^{\mathrm{loc}}\to \mathop{\mathrm{QK}}_T^{\rm{loc}}(X).$$* *Proof.* The injectivity follows from the injetivity of $\Psi$. The surjectivity follows from , which implies that $\mathop{\mathrm{QK}}_T^{\rm{loc}}(X)$ is generated over $S[q]_{\mathrm{loc}}$ by the (line) bundles ${\mathcal S}_1$ and $\det {\mathcal S}_2$. ◻ **Remark 24**. The attentive reader may have noticed that, by , there is also a natural ring homomorphism $$\Psi_{\mathrm{poly}}: S[q]/I_q^{\mathrm{poly}}\to \mathop{\mathrm{QK}}_T^{\mathrm{poly}}(X)$$ defined by the same formula as $\Psi$. Furthermore, implies that $\Psi_{\mathrm{poly}}$ is surjective. We expect that $\Psi_{\mathrm{poly}}$ is also an isomorphism. This would follow if the natural map $S[q]/I_q^{\mathrm{poly}}\to S[\![q]\!]/I_q$ were injective. By [@AM:intro p.110], the kernel of this map coincides with the kernel of the localization map $S[q]/I_q^{\mathrm{poly}}\to S[q]_{\mathrm{loc}}/I_q^{\mathrm{loc}}$. We plan to study this elsewhere. ## A physics derivation of the QK Whitney presentation {#sec:physics} Continuing the circle of ideas first used for Grassmannians in [@Gu:2020zpg; @gu2022quantum], we derived from physics in [@gu2023quantum] a "Coulomb branch presentation" of the quantum K ring of any partial flag variety. This presentation was obtained as the critical locus of a certain *one-loop twisted superpotential* $\mathcal{W}$ which arises in the study of $3$d gauged linear sigma models (GLSM). In this section, we recall the derivation of the Coulomb branch presentation for incidence varieties and show that it is essentially the same as the Whitney presentation. The same ideas apply to any partial flag variety. As before, consider the variables $X^{(1)}_1, X^{(2)}_1, \ldots, X^{(2)}_{n-1}$. The twisted superpotential $\mathcal{W}$ is defined as $$\begin{aligned} \mathcal{W} & = & \frac{1}{2} (n-2) \sum_{a=1}^{n-1} \left( \ln X^{(2)}_a \right)^2 \: - \: \sum_{1 \le a < b \le n-1} \left(\ln X^{(2)}_{a}\right) \left(\ln X^{(2)}_{b}\right) \nonumber \\ & & \: + \: \left( \ln q_1 \right) \left( \ln X^{(1)}_1 \right) \: + \: \left( \ln \left( (-1)^{n-2} q_2 \right) \right) \sum_{a=1}^{n-1} \left( \ln X^{(2)}_a \right) \nonumber \\ & & \: + \: \sum_{a=1}^{n-1} {\rm Li}_2 \left( X^{(1)}_1 / X^{(2)}_a \right) \: + \: \sum_{a=1}^{n-1} \sum_{i=1}^n {\rm Li}_2 \left( X^{(2)}_a / T_i \right)\end{aligned}$$ on the subset of the torus $X^{(1)}_1 \cdot X^{(2)}_1 \cdot \ldots \cdot X^{(2)}_{n-1} \neq 0$ where $X^{(2)}_i \neq X^{(2)}_j$ for $i \neq j$. Here, $\mathrm{Li}_2$ is the dilogarithm function, satisfying $x \frac{\partial}{\partial x} \mathrm{Li}_2(x) = - \ln (1-x)$. As a side remark, the more general superpotential associated to any GLSM also depends on certain Chern-Simons levels. Here we have already chosen the levels giving the quantum K theory ring. The (unsymmetrized) Coulomb branch relations are given by $$\mathrm{exp} \left(\frac{\partial \mathcal{W}}{\partial \ln (X^{(1)}_1)}\right) = 1 \/; \quad \mathrm{exp} \left(\frac{\partial \mathcal{W}}{\partial \ln (X^{(2)}_i)}\right) = 1\/, \quad 1 \le i \le n-1 \/.$$ Calculating derivatives one obtains $$\label{eqn:deriv1} q_1 \: = \: \prod_{i=1}^{n-1} \left( 1 - \frac{ X_1^{(1)} }{ X_i^{(2)} } \right),$$ $$\label{eqn:deriv2} (-1)^{n-2} q_2 \left( 1- \frac{X_1^{(1)}}{X_k^{(2)}} \right) \: = \: \left( \prod_{i=1}^{n-1} \frac{X_i^{(2)}}{X_k^{(2)}} \right) \prod_{j = 1}^n \left( 1 - \frac{ X_k^{(2)} }{ T_{j} } \right),\ k= 1, \dots, n-1.$$ Equations [([\[eqn:deriv1\]](#eqn:deriv1){reference-type="ref" reference="eqn:deriv1"})](#eqn:deriv1) and [([\[eqn:deriv2\]](#eqn:deriv2){reference-type="ref" reference="eqn:deriv2"})](#eqn:deriv2) are a special case of the Bethe Ansatz equations [@koroteev]\*Equation (27), suggesting connections with integrable systems. Investigating these connections is of interest, but beyond the scope of the current paper. We observe that $\xi=X_1^{(1)}$ is a solution to the equation $$X_1^{(1)}\sum_{l=0}^{n-1}(-1)^\ell e_\ell(X^{(2)})\xi^{n-1-\ell}+(-1)^{n-2}q_1 e_{n-1}(X^{(2)})\xi=0,$$ and we denote the remaining $n-2$ solutions by $\overline{X}^{(1)}=(\overline{X}_1^{(1)},\dots,\overline{X}_{n-2}^{(1)})$; similarly, we have that $\xi=X^{(2)}_1,\dots,X^{(2)}_{n-1}$ are solutions to $$e_{n-1}(X^{(2)})\sum_{\ell=0}^{n}(-1)^\ell e_\ell(T)\xi^{n-\ell}+q_2 e_n(T)\left(-\xi^{n-1}+X_1^{(1)}\xi^{n-2}\right)=0,$$ and we denote the remaining solution by $\overline{X}^{(2)}_1$. Then, using Vieta's formulae we deduce equations $$e_\ell(\overline{X}^{(1)})+X_1^{(1)} e_{\ell-1}(\overline{X}^{(1)})-e_{\ell}( X^{(2)} ) =\begin{cases} 0 & \ell = 1, \dots, n-3, n-1 \\ q_1 e_{n-1}( X^{(2)} )/X^{(1)}_1 & \ell = n-2 \/, \end{cases}$$ and $$e_\ell(X^{(2)})+e_{\ell-1}(X^{(2)})\overline{X}^{(2)}_1- e_{\ell}(T) = \begin{cases} q_2 e_{n}(T)/e_{n-1}(X^{(2)}) & \ell=1\\ q_2 e_n(T)X^{(1)}_1/e_{n-1}(X^{(2)}) & \ell=2\\ 0 & \ell=3,\dots,n \/, \end{cases}$$ which simplify to the equations $$\label{eqn:inc:final:1} e_\ell(\overline{X}^{(1)})+X_1^{(1)} e_{\ell-1}(\overline{X}^{(1)})-e_{\ell}( X^{(2)} ) = \: \left\{ \begin{array}{cl} 0 & \ell = 1, \dots, n-3, n-1\\q_1 e_{n-2}( \overline{X}^{(1)} ) & \ell = n-2 \end{array} \right.$$ and $$\label{eqn:inc:final:2} e_\ell(X^{(2)})+e_{\ell-1}(X^{(2)})\overline{X}^{(2)}_1- e_{\ell}(T) = \begin{cases} q_2 \overline{X}_1^{(2)} & \ell=1\\ q_2 X^{(1)}_1 \overline{X}_1^{(2)} & \ell=2\\ 0 & \ell=3,\dots,n \/. \end{cases}$$ Let $$\overline{\mathop{\mathrm{QK}}}_T(X)\coloneqq \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1,q_2]\!][X^{(1)}_1, e_i(X^{(2)}), e_j(\overline{X}^{(1)}),\overline{X}^{(2)}_1; 1 \le i \le n-1,\ 1 \le j \le n-2]/J_q$$ be the "Coulomb branch" ring, where $J_q$ is the ideal generated by the relations given by [([\[eqn:inc:final:1\]](#eqn:inc:final:1){reference-type="ref" reference="eqn:inc:final:1"})](#eqn:inc:final:1) and [([\[eqn:inc:final:2\]](#eqn:inc:final:2){reference-type="ref" reference="eqn:inc:final:2"})](#eqn:inc:final:2). **Proposition 25**. *There is an isomorphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1,q_2]\!]$-algebras $\Phi: \overline{\mathop{\mathrm{QK}}}_T(X) \to \mathop{\mathrm{QK}}_T(X)$ given by $$\label{eqn:Phi1} X^{(1)}_1\mapsto{\mathcal S}_1;\quad e_k(X^{(2)})\mapsto\wedge^k({\mathcal S}_2),\ 1\leq k\leq n-1;$$ and $$\label{eqn:Phi2} e_\ell(\overline{X}^{(1)})\mapsto \begin{cases} \wedge^{\ell}( {\mathcal S}_2/{\mathcal S}_1 ) & 1\leq\ell < n-2\\ \det({\mathcal S}_2/{\mathcal S}_1)/(1-q_1) & \ell=n-2 \end{cases}; \quad \overline{X}^{(2)}_1\mapsto({\mathbb C}^n/{\mathcal S}_2)/(1-q_2).$$* *Proof.* Let the morphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1,q_2]\!]$-algebras $$\widetilde{\Phi}: \mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})[\![q_1,q_2]\!][X^{(1)}_1, e_i(X^{(2)}), e_j(\overline{X}^{(1)}),\overline{X}^{(2)}_1; 1 \le i \le n-1,\ 1 \le j \le n-2]\to\mathop{\mathrm{QK}}_T(X)$$ be defined by [([\[eqn:Phi1\]](#eqn:Phi1){reference-type="ref" reference="eqn:Phi1"})](#eqn:Phi1) and [([\[eqn:Phi2\]](#eqn:Phi2){reference-type="ref" reference="eqn:Phi2"})](#eqn:Phi2). Note that $\widetilde{\Phi}$ is surjective. The morphism $\Phi$ sends equations [([\[eqn:inc:final:1\]](#eqn:inc:final:1){reference-type="ref" reference="eqn:inc:final:1"})](#eqn:inc:final:1) and [([\[eqn:inc:final:2\]](#eqn:inc:final:2){reference-type="ref" reference="eqn:inc:final:2"})](#eqn:inc:final:2) to equations $$\label{eqn:increl1} \wedge^\ell({\mathcal S}_2/{\mathcal S}_1)+{\mathcal S}_1\star\wedge^{l-1}({\mathcal S}_2/{\mathcal S}_1)= \begin{cases} \wedge^\ell({\mathcal S}_2) & \ell=1,\dots,n-2\\ (1-q_1)\wedge^\ell({\mathcal S}_2) & \ell=n-1 \end{cases}$$ and $$\begin{gathered} \label{eqn:increl2} \wedge^\ell({\mathcal S}_2)+\wedge^{l-1}({\mathcal S}_2)\star({\mathbb C}^n/{\mathcal S}_2)\\=\wedge^\ell({\mathbb C}^n)-\frac{q_2}{1-q_2}\wedge^{\ell-1}({\mathbb C}^n/{\mathcal S}_2)\star(\wedge^{\ell-1}({\mathcal S}_2)-\wedge^{\ell-1}({\mathcal S}_1))\quad\ell=1,\dots,n. \end{gathered}$$ Note that [([\[eqn:increl1\]](#eqn:increl1){reference-type="ref" reference="eqn:increl1"})](#eqn:increl1) and [([\[eqn:increl2\]](#eqn:increl2){reference-type="ref" reference="eqn:increl2"})](#eqn:increl2) are equivalent to the relations in . Therefore, $\widetilde{\Phi}$ induces the desired isomorphism $\Phi:\overline{\mathop{\mathrm{QK}}}_T(X) \to \mathop{\mathrm{QK}}_T(X)$. ◻ # The Whitney presentation for $\mathop{\mathrm{QK}}_T(\mathop{\mathrm{Fl}}(n))$ {#sec:fullflag} In this section, we prove in the case when $X=\mathop{\mathrm{Fl}}(n)=\mathop{\mathrm{Fl}}(1,\dots,n-1;n)$ is the complete flag variety, *under the assumption that holds*. The main idea is to rewrite the relations in so that only relations involving multiplication by $\wedge^i {\mathcal S}_i$ appear (see ), then apply . This reduction is not available for arbitrary partial flag varietys. Another key ingredient in our proof is to realize curve neighborhoods by certain iterated Demazure operators, a technique possibly of independent interest. ## A first reduction In this subsection we rewrite the relations in in a way that highlights the role of $\wedge^i {\mathcal S}_i$. The results in this subsection are logically independent of . We start by recalling the relations [([\[eqn:lambda_y\_rel\]](#eqn:lambda_y_rel){reference-type="ref" reference="eqn:lambda_y_rel"})](#eqn:lambda_y_rel): $$\label{eqn2:lambda_y_full} \lambda_y(S_i)\star\lambda_y({\mathcal S}_{i+1}/S_i)=\lambda_y({\mathcal S}_{i+1})-y\frac{q_i}{1-q_i}{\mathcal S}_{i+1}/S_i\star(\lambda_y(S_i)-\lambda_y({\mathcal S}_{i-1})).$$ After multiplying both sides by $1-q_i$ and expanding, we can write the relations as $$\label{eqn:quot_full} (1-q_i)(\wedge^\ell {\mathcal S}_{i+1}-\wedge^\ell S_i)={\mathcal S}_{i+1}/S_i\star(\wedge^{\ell-1}S_i-q_i\wedge^{\ell-1}{\mathcal S}_{i-1})$$ for $\ell=1,\dots, i+1; \ i=1,\dots,n-1$. **Lemma 26**. *Relations [([\[eqn:quot_full\]](#eqn:quot_full){reference-type="ref" reference="eqn:quot_full"})](#eqn:quot_full) are equivalent to the relations $$\label{eqn:det_full} \wedge^i S_i\star(\wedge^\ell {\mathcal S}_{i+1}-\wedge^\ell S_i)=\wedge^{i+1}{\mathcal S}_{i+1}\star(\wedge^{\ell-1}S_i-q_i\wedge^{\ell-1}{\mathcal S}_{i-1})$$ for $\ell=1,\dots, i+1; \ i=1,\dots,n-1$.* *Proof.* Note that $$\label{eqn:special_rel} \wedge^i {\mathcal S}_i\star {\mathcal S}_{i+1}/{\mathcal S}_i=(1-q_i)\wedge^{i+1}{\mathcal S}_{i+1}$$ is the $\ell=1$ case of [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full) and the $\ell=i+1$ case of [([\[eqn:quot_full\]](#eqn:quot_full){reference-type="ref" reference="eqn:quot_full"})](#eqn:quot_full). It follows that $$\det({\mathcal S}_i)\star{\mathcal S}_{i+1}/{\mathcal S}_i\star\dots\star{\mathcal S}_{n-1}/{\mathcal S}_{n-2}\star{\mathbb C}^n/{\mathcal S}_{n-1}=(1-q_i)\cdots(1-q_{n-1})\wedge^n{\mathbb C}^n$$ and, in particular, $\det({\mathcal S}_i)$ and ${\mathcal S}_{i+1}/{\mathcal S}_i$ are units for $i=1,\dots, n-1$. Multiplying both sides of [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full) by ${\mathcal S}_{i+1}/{\mathcal S}_i$ and using [([\[eqn:special_rel\]](#eqn:special_rel){reference-type="ref" reference="eqn:special_rel"})](#eqn:special_rel), we have $$\begin{gathered} (1-q_i)\wedge^{i+1}{\mathcal S}_{i+1}\star(\wedge^{\ell}{\mathcal S}_{i+1}-\wedge^{\ell}{\mathcal S}_{i}) = {\mathcal S}_{i+1}/{\mathcal S}_i\star\det({\mathcal S}_i)\star(\wedge^\ell {\mathcal S}_{i+1}-\wedge^\ell S_i)\\={\mathcal S}_{i+1}/{\mathcal S}_i\star\wedge^{i+1}{\mathcal S}_{i+1}\star(\wedge^{\ell-1}S_i-q_i\wedge^{\ell-1}{\mathcal S}_{i-1})\quad\text{for }\ell=1,\dots, i+1, \end{gathered}$$ showing that [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full) implies [([\[eqn:quot_full\]](#eqn:quot_full){reference-type="ref" reference="eqn:quot_full"})](#eqn:quot_full). Multiplying both sides of [([\[eqn:quot_full\]](#eqn:quot_full){reference-type="ref" reference="eqn:quot_full"})](#eqn:quot_full) by $\det({\mathcal S}_i)$ and using [([\[eqn:special_rel\]](#eqn:special_rel){reference-type="ref" reference="eqn:special_rel"})](#eqn:special_rel), we have $$\begin{gathered} \label{eqn:rel_2} (1-q_i)(\wedge^\ell {\mathcal S}_{i+1}-\wedge^\ell S_i)\star\det({\mathcal S}_i)=\det({\mathcal S}_i)\star{\mathcal S}_{i+1}/{\mathcal S}_i\star(\wedge^{\ell-1}S_i-q_i\wedge^{\ell-1}{\mathcal S}_{i-1})\\ =(1-q_i)\det({\mathcal S}_{i+1})\star(\wedge^{\ell-1}{\mathcal S}_i-q_i\wedge^{\ell-1}{\mathcal S}_{i-1})\quad\text{for }\ell=1,\dots, i+1, \end{gathered}$$ showing that [([\[eqn:quot_full\]](#eqn:quot_full){reference-type="ref" reference="eqn:quot_full"})](#eqn:quot_full) implies [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full). ◻ ## Curve neighborhoods of tautological bundles The main result in this subsection is , which establishes an equality between curve neighborhoods of exterior powers of adjacent tautological bundles, when the degrees differ by a simple (co)root. This is the key calculation needed to prove the QK relations in the next subsection. In fact, some of these results work for generalized flag varieties $G/B$, and we try to utilize suggestive notation so that it will be easy for the cognizant reader to rewrite the arguments in that generality. This section is still logically independent of the validity of . Denote by $\alpha_i= (0, \ldots, 1, \ldots, 0)$ (with $1$ at position $i$) the degree of the (Schubert) curve $X_{s_i}$. The degree $\alpha_i$ may also be identified with the simple (co)root $\varepsilon_i - \varepsilon_{i+1}$. More generally, denote by $R^+ = \{ \varepsilon_i - \varepsilon_j: 1 \le i < j \le n \}$ the set of positive roots of type $A_{n-1}$, equipped with the partial order given by $\alpha \le \beta$ if $\beta - \alpha$ is a non-negative combination of simple roots. The Weyl group $W$ is equipped with an associative monoid structure given by the *Demazure product*. For $u \in W$ and $s_i=(i,i+1)$ a simple reflection, $$u \cdot s_i = \begin{cases} u s_i & us_i >u \/; \\ u & \textrm{ otherwise } \/. \end{cases}$$ More generally, if $v = s_{i_1} s_{i_2} \ldots s_{i_p}$ is a reduced expression, then $u \cdot v = (((u \cdot s_{i_1}) \cdot s_{i_2}) \ldots \cdot s_{i_p})$. For $\beta \in R^+$, the support $\mathop{\mathrm{supp}}(\beta)$ is the set of simple roots $\alpha_i$ such that $\beta - \alpha_i \ge 0$. We will also implicitly utilize the fact that if $\alpha, \beta \in R^+$, then $$\label{eqn:demcom} s_\alpha \cdot s_\beta = s_\beta \cdot s_\alpha \textrm{ if } \mathop{\mathrm{supp}}(\alpha) \cap \mathop{\mathrm{supp}}(\beta) = \emptyset \textrm{ or } \mathop{\mathrm{supp}}(\alpha) \subset \mathop{\mathrm{supp}}(\beta) \textrm{ or } \mathop{\mathrm{supp}}(\beta) \subset \mathop{\mathrm{supp}}(\alpha) \/.$$ It is easy to check this directly, and it also follows from [@buch.m:nbhds Prop. 4.8]. For an effective degree $d$, define the Weyl group element $z_d \in W$ by the requirement that $\Gamma_d(X_{s_0}) = X_{z_d}$, where $s_0$ denotes the identity element in $W$. The element $z_d$ may be calculated recursively utilizing a formula from [@buch.m:nbhds]: if $\beta$ is any maximal root such that $\beta\le d$, then $$\label{eqn:reczd} z_d = z_{d - \beta} \cdot s_\beta \/.$$ The support $\mathop{\mathrm{supp}}(d)$ of $d=\sum d_i \alpha_i$ is defined similarly as the set of simple roots $\alpha_i$ such that $d_i \neq 0$. If $\alpha_i \in \mathop{\mathrm{supp}}(d)$, denote by $d' = d - \alpha_{i}$. In other words, $$d_j'= \begin{cases} d_j & j\neq i\\d_i-1 & j=i. \end{cases}$$ (The index $i$ will be understood from the context.) The proof of the following lemma is a direct application of the recursive expression from [([\[eqn:reczd\]](#eqn:reczd){reference-type="ref" reference="eqn:reczd"})](#eqn:reczd) and is left to the reader. **Lemma 27**. *Let $d$ be a degree such that $d_i \neq 0$ and $d_{i+1}=0$. Assume that $z_d = s_{\beta_1}\cdot \ldots \cdot s_{\beta_k }\cdot \ldots \cdot s_{\beta_p}$ for some positive roots $\beta_1, \ldots, \beta_p$ such that $\alpha_i \in \mathop{\mathrm{supp}}(\beta_k) \subset \ldots \subset \mathop{\mathrm{supp}}(\beta_p)$ and this is the longest chain with this property. Then $\alpha_{i+1}$ is not in the support of any of the roots $\beta_j$, and, furthermore, $$z_{d'} = s_{\beta_1}\cdot \ldots \cdot s_{\beta_k - \alpha_i} \cdot \ldots \cdot s_{\beta_p} \/.$$* Let $p_i: \mathop{\mathrm{Fl}}(n) \to \mathop{\mathrm{Fl}}(1,\dots,\hat{i},\dots,n-1;n)$ be the natural projection. The *Demazure operator* is defined by $\partial_i = p_i^* {p_i}_*$, and it is a $\mathop{\mathrm{K}}_T(\mathop{\mathrm{pt}})$-linear endomorphism of $\mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(n))$. These operators satisfy $\partial_i^2=\partial_i$, and the usual commutation and braid relations. In particular, if $u= s_{i_1} \ldots s_{i_p}$ is a reduced decomposition of $u$, then there is a well-defined operator $\partial_u = \partial_{i_1} \ldots \partial_{i_p}$. The Demazure operators satisfy $\partial_i {\mathcal O}_u = {\mathcal O}_{u \cdot s_i}$, therefore $\partial_{v^{-1}} {\mathcal O}_u = {\mathcal O}_{u \cdot v}$. We will utilize this to obtain curve neighborhoods by iterated Demazure operators. Recall that the degree $d$ curve neighborhood of a class $\sigma \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(n))$ is $\sigma[d]={\mathop{\mathrm{ev}}_2}_*(\mathop{\mathrm{ev}}_1^*(a))$. Using that $z_d = z_d^{-1}$ (cf. [@buch.m:nbhds Cor.4.9]), we obtain that $${\mathcal O}_u[d]={\mathop{\mathrm{ev}}_2}_*(\mathop{\mathrm{ev}}_1^*({\mathcal O}_u))={\mathcal O}_{\Gamma_d(X_u)}={\mathcal O}_{u \cdot z_d} = {\mathcal O}_{u \cdot z_d^{-1}} = \partial_{z_d} ({\mathcal O}_u) \/.$$ Since the Schubert classes form a basis, it follows that for any $\sigma \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(n))$, $$\label{eqn:partial_zd} \sigma[d]={\mathop{\mathrm{ev}}_2}_*(\mathop{\mathrm{ev}}_1^*(\sigma)) = \partial_{z_d}(\sigma) \/.$$ (This equation easily generalizes to any $G/B$.) Our next goal is to calculate the curve neighborhoods of the bundles $\wedge^\ell {\mathcal S}_i$ in $\mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(n))$. To do this we need the following lemma. **Lemma 28**. *For $\ell=1,\dots,k$, the following hold:* 1. *[\[eqn:partial_wedge\]]{#eqn:partial_wedge label="eqn:partial_wedge"} $$\partial_i (\wedge^\ell {\mathcal S}_k) = \begin{cases} \wedge^\ell {\mathcal S}_k & \textrm{ if } i \neq k \\\wedge^{\ell} {\mathcal S}_{k-1} & \textrm{ if } i = k \/. \end{cases}$$* 2. *[\[eqn:partial_product\]]{#eqn:partial_product label="eqn:partial_product"} For $i \neq k$ and any $\sigma \in \mathop{\mathrm{K}}_T(\mathop{\mathrm{Fl}}(n))$, $$\partial_i(\wedge^\ell {\mathcal S}_k \cdot \sigma) = \wedge^\ell {\mathcal S}_k \cdot \partial_i(\sigma) \/.$$* *Proof.* If $i \neq k$, then ${\mathcal S}_k$ is also a bundle on $\mathop{\mathrm{Fl}}(1,\dots,\hat{k},\dots,n-1;n)$, therefore, by the projection formula ${p_i}_* (\wedge^\ell {\mathcal S}_k) = \wedge^\ell {\mathcal S}_k$. This implies part [([\[eqn:partial_product\]](#eqn:partial_product){reference-type="ref" reference="eqn:partial_product"})](#eqn:partial_product) and the first branch of part [([\[eqn:partial_wedge\]](#eqn:partial_wedge){reference-type="ref" reference="eqn:partial_wedge"})](#eqn:partial_wedge). If $i=k$, from the short exact sequence $0 \to {\mathcal S}_{k-1} \to {\mathcal S}_k \to {\mathcal S}_k/{\mathcal S}_{k-1} \to 0$ it follows that $$\label{eqn:wedge-eqs}\wedge^\ell {\mathcal S}_k = \wedge^\ell {\mathcal S}_{k-1} + \wedge^{\ell-1} {\mathcal S}_{k-1}\cdot {\mathcal S}_k/{\mathcal S}_{k-1} \/.$$ Then $$\begin{split} {p_k}_* ( \wedge^\ell {\mathcal S}_k) & = {p_k}_*(\wedge^\ell {\mathcal S}_{k-1} + \wedge^{\ell-1} {\mathcal S}_{k-1}\cdot {\mathcal S}_k/{\mathcal S}_{k-1})\\ & = {p_k}_*(\wedge^\ell {\mathcal S}_{k-1}) + \wedge^{\ell-1} {\mathcal S}_{k-1}\cdot {p_k}_*({\mathcal S}_k/{\mathcal S}_{k-1}) \\ & = \wedge^\ell {\mathcal S}_{k-1} \/. \end{split}$$ Here the last equality follows because ${p_k}_*({\mathcal S}_k/{\mathcal S}_{k-1})=0$ by . The second branch of part [([\[eqn:partial_wedge\]](#eqn:partial_wedge){reference-type="ref" reference="eqn:partial_wedge"})](#eqn:partial_wedge) follows from applying $p_k^*$ to this. ◻ **Corollary 29**. *For $\varepsilon_a-\varepsilon_{i+1}\in R^+$ and $\ell=1,\dots,k$, the following hold: $$\partial_{\varepsilon_a-\varepsilon_{i+1}}(\wedge^\ell{\mathcal S}_k)=\begin{cases} \wedge^\ell{\mathcal S}_{a-1} & a\leq k\leq i\\\wedge^\ell{\mathcal S}_k & \text{otherwise.}\\ \end{cases}$$* *Proof.* The reflection $s_{\varepsilon_a-\varepsilon_{i+1}}$ has reduced decomposition $s_{i}s_{i-1}\ldots s_a\ldots s_{i-1}s_i$. The claim follows from repeated application of . ◻ **Corollary 30**. *Let $d$ be a degree such that $d_i \neq 0, d_{i+1}=0$. Then $$\partial_{z_d}(\wedge^{\ell} {\mathcal S}_{i}) = \partial_{z_{d'}}(\wedge^{\ell} {\mathcal S}_{i-1})\text{ for }\ell=1,\dots, i.$$* *Proof.* Let $d$ be an effective degree such that $d_i \neq 0$ and $d_{i+1}=0$. From the recursive expression [([\[eqn:reczd\]](#eqn:reczd){reference-type="ref" reference="eqn:reczd"})](#eqn:reczd) we can write $$z_d = s_{\beta_1}\cdot \ldots \cdot s_{\beta_k }\cdot \ldots \cdot s_{\beta_p}$$ for some positive roots $\beta_1, \ldots, \beta_p$ such that $\alpha_i\in \mathop{\mathrm{supp}}(\beta_k)\subseteq\dots\subseteq \mathop{\mathrm{supp}}(\beta_p)$, $\alpha_i\not\in \mathop{\mathrm{supp}}(\beta_j)$ for $1\leq j\leq k-1$, and $\alpha_{i+1}\not\in \mathop{\mathrm{supp}}(\beta_j)$ for $1\leq j\leq p$. In practice, this means that for any $k \le j \le p$, we have $\beta_j = \varepsilon_{a_j} - \varepsilon_{i+1}$, where $i \ge a_k \ge a_{k+1} \ge \ldots \ge a_p$. By , we may write $$z_{d'} = s_{\beta_1}\cdot \ldots \cdot s_{\beta_k - \alpha_i}\cdot \ldots \cdot s_{\beta_p} \/.$$ Define $z'\coloneqq s_{\beta_1}\cdot \ldots \cdot s_{\beta_{k-1}}$ and $z''=s_{\beta_k-\alpha_i}\cdot \ldots \cdot s_{\beta_p}$. After writing $s_{\beta_k} = s_i \cdot s_{\beta_k - \alpha_i} \cdot s_i$, and observing that for any $j \ge k+1$, $s_i \cdot s_{\beta_{j}} = s_{\beta_{j}}\cdot s_i = s_{\beta_j}$ (by [([\[eqn:demcom\]](#eqn:demcom){reference-type="ref" reference="eqn:demcom"})](#eqn:demcom) above), it follows that $$z_d = z' \cdot s_i \cdot z'' \cdot s_i \textrm{ and } z_{d'} = z' \cdot z'' \/.$$ Since the indices $a_j > a_p -1$ for any $j \ge k$, it follows from repeated application of that $$\partial_{z'' \cdot s_i} (\wedge^{\ell} {\mathcal S}_{i}) = \partial_{z''} (\wedge^\ell {\mathcal S}_{i-1}) = \wedge^{\ell} {\mathcal S}_{a_p-1}\/,$$ and since $a_p-1<i$, we have $$\partial_{s_i \cdot z'' \cdot s_i} (\wedge^{\ell} {\mathcal S}_{i}) = \partial_i \partial_{z''} \partial_i (\wedge^{\ell} {\mathcal S}_{i}) = \partial_i (\wedge^{\ell} {\mathcal S}_{a_p-1}) = \wedge^{\ell} {\mathcal S}_{a_p-1} \/.$$ In particular, this shows that $\partial_{s_i \cdot z'' \cdot s_i} (\wedge^{\ell} {\mathcal S}_{i}) = \partial_{z''} (\wedge^{\ell} {\mathcal S}_{i-1})$. Then the claim follows from the calculation: $$\partial_{z_d} (\wedge^{\ell} {\mathcal S}_{i}) = \partial_{z'} \partial_{s_i \cdot z'' \cdot s_i} (\wedge^{\ell} {\mathcal S}_{i}) = \partial_{z'} \partial_{z''} (\wedge^{\ell} {\mathcal S}_{i-1}) = \partial_{z_{d'}} (\wedge^{\ell} {\mathcal S}_{i-1}) \/.$$ ◻ ## The quantum K Whitney relations In this subsection, we prove relations [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full) for the complete flag variety $X=\mathop{\mathrm{Fl}}(n)$ assuming . As a consequence, we get that implies the Whitney presentation for $\mathop{\mathrm{QK}}_T(X)$. Let $d$ be an effective degree and $\sigma, \tau\in\mathop{\mathrm{K}}_T(X)$. In light of [([\[eqn:partial_zd\]](#eqn:partial_zd){reference-type="ref" reference="eqn:partial_zd"})](#eqn:partial_zd), may be restated as $$\label{eqn:KGW_full} \angles{ \det({\mathcal S}_i),\sigma,\tau }^T_{d}=\begin{cases}0 & d_i\neq0\\\chi_X^T\left(\det({\mathcal S}_i)\cdot\sigma\cdot\partial_{z_d}\tau\right) & d_i =0 \/\end{cases}$$ for $i=1,\dots,n-1$. **Theorem 31**. *For any $\sigma \in \mathop{\mathrm{K}}_T(X)$, $\ell=1,\dots, i+1$, and $i=1,\dots,n-1$, implies $$\label{eqn:kgw_identity} \angles{ \det({\mathcal S}_i), \wedge^\ell{\mathcal S}_{i+1}-\wedge^\ell{\mathcal S}_i,\sigma }^T_{d}=\angles{ \det({\mathcal S}_{i+1}),\wedge^{\ell-1}{\mathcal S}_i,\sigma }^T_{d}-\angles{ \det({\mathcal S}_{i+1}),\wedge^{\ell-1}{\mathcal S}_{i-1},\sigma }^T_{d-\alpha_i} \/,$$ with the convention that $\angles{ \det({\mathcal S}_{i+1}),\wedge^{\ell-1}{\mathcal S}_{i-1},\sigma }^T_{d-\alpha_i} =0$ unless $\alpha_i \in \mathop{\mathrm{supp}}(d)$.* *In particular, the equalities [([\[eqn:det_full\]](#eqn:det_full){reference-type="ref" reference="eqn:det_full"})](#eqn:det_full) hold.* *Proof.* We distinguish three cases. - **$d_i \cdot d_{i+1} \neq 0$**. In this case, both sides of [([\[eqn:kgw_identity\]](#eqn:kgw_identity){reference-type="ref" reference="eqn:kgw_identity"})](#eqn:kgw_identity) are equal to $0$ by [([\[eqn:KGW_full\]](#eqn:KGW_full){reference-type="ref" reference="eqn:KGW_full"})](#eqn:KGW_full). - $d_i \neq 0,\ d_{i+1}=0$. By [([\[eqn:KGW_full\]](#eqn:KGW_full){reference-type="ref" reference="eqn:KGW_full"})](#eqn:KGW_full), the left-hand side of [([\[eqn:kgw_identity\]](#eqn:kgw_identity){reference-type="ref" reference="eqn:kgw_identity"})](#eqn:kgw_identity) is equal to $0$, and the right-hand side is equal to $$\chi_X^T\left(\det({\mathcal S}_{i+1}) \cdot \left(\partial_{z_d} (\wedge^{\ell-1} {\mathcal S}_i) - \partial_{z_{d-\alpha_i}} (\wedge^{\ell-1}{\mathcal S}_{i-1})\right)\cdot\sigma\right).$$ Finally, note that by , $\partial_{z_d} (\wedge^{\ell-1} {\mathcal S}_i) - \partial_{z_{d-\alpha_i}} (\wedge^{\ell-1}{\mathcal S}_{i-1})=0.$ - $d_i=0$. By [([\[eqn:KGW_full\]](#eqn:KGW_full){reference-type="ref" reference="eqn:KGW_full"})](#eqn:KGW_full) and [([\[eqn:wedge-eqs\]](#eqn:wedge-eqs){reference-type="ref" reference="eqn:wedge-eqs"})](#eqn:wedge-eqs), the left-hand side of [([\[eqn:kgw_identity\]](#eqn:kgw_identity){reference-type="ref" reference="eqn:kgw_identity"})](#eqn:kgw_identity) equals $$\begin{aligned} \chi_X^T\left(\det({\mathcal S}_i)\cdot(\wedge^\ell{\mathcal S}_{i+1}-\wedge^\ell{\mathcal S}_i)\cdot\partial_{z_d}\sigma\right)&=\chi_X^T\left(\det({\mathcal S}_i)\cdot({\mathcal S}_{i+1}/{\mathcal S}_i)\cdot\wedge^{\ell-1}{\mathcal S}_i\cdot\partial_{z_d}\sigma\right)\\ &=\chi_X^T\left(\det({\mathcal S}_{i+1})\cdot\wedge^{\ell-1}{\mathcal S}_i\cdot\partial_{z_d}\sigma\right),\end{aligned}$$ which is equal to the right-hand side of [([\[eqn:kgw_identity\]](#eqn:kgw_identity){reference-type="ref" reference="eqn:kgw_identity"})](#eqn:kgw_identity).  ◻ By and we have: **Corollary 32**. *implies for $\mathop{\mathrm{Fl}}(n)$.* # Finite generation over formal power series {#app:fg} The main result of this Appendix is , which is the key result needed in the proof of the presentation of the quantum K ring in . It gives mild conditions under which an algebra over a formal power series ring is finitely generated as a module, allowing one to apply , or, more generally, Nakayama-type results. One may also deduce directly from [@eisenbud:CAbook Exercise 7.8]. We thank Prof. S. Naito for providing us with this reference. For the convenience of the reader, we include a proof. We start with the following general result proved in [@matsumura:commutative Thm. 8.4], see also [@StacksProj [Tag 031D](https://stacks.math.columbia.edu/tag/031D)]. **Lemma 33**. *Let $A$ be a commutative ring and let $\mathfrak{a}\subset A$ be an ideal. Let $M$ be an $A$-module. Assume that $A$ is $\mathfrak{a}$-adically complete, $\bigcap_{n \ge 1} \mathfrak{a}^n M = (0)$, and that $M/\mathfrak{a}M$ is a finitely generated $A/\mathfrak{a}$-module. Then $M$ is a finitely generated $A$-module.* For a commutative ring $S$ with $1$ we denote by $\mathop{\mathrm{Jac}}(S)$ its Jacobson radical, i.e., the intersection of all its maximal ideals. It is proved in [@AM:intro Prop. 1.9] that $x \in \mathop{\mathrm{Jac}}(S)$ if and only if $1-xy$ is a unit in $S$ for all $y \in S$. **Lemma 34**. *Let $R,S$ be commutative rings with $1$ and $\pi: R \to S$ be a surjective ring homomorphism with $\pi(1)=1$. Then:* 1. *$\pi(\mathop{\mathrm{Jac}}(R)) \subseteq \mathop{\mathrm{Jac}}(S)$;* 2. *[\[item:ideal\]]{#item:ideal label="item:ideal"} If $J$ is an ideal in $R$, then $\pi(J)$ is an ideal in $S$.* *Proof.* Let $x \in \mathop{\mathrm{Jac}}(R)$. Then $1-xr$ is a unit in $R$ for all $r\in R$. This implies $f(1-xr)=1-f(x)f(r)$ is a unit in $S$. Since $f$ is surjective, this means $f(x)\in \mathop{\mathrm{Jac}}(S)$. Part [\[item:ideal\]](#item:ideal){reference-type="eqref" reference="item:ideal"} is immediate from the definitions. ◻ From now on, $S$ is a commutative Noetherian ring, and $I$ is an ideal of the formal power series ring $S[\![q_1, \ldots, q_k]\!]$. Let $$\pi: S[\![q_1, \ldots, q_k]\!] \to M\coloneqq S[\![q_1, \ldots, q_k]\!]/I$$ be the projection. Let $$J\coloneqq \langle q_1, \ldots, q_k \rangle \subset S[\![q_1, \ldots, q_k]\!].$$ **Lemma 35**. *The ideal $\pi(J)$ is contained in the Jacobson radical of $M$.* *Proof.* By [@AM:intro Prop. 10.15] $J$ is contained in the Jacobson radical of $S[\![q_1, \ldots, q_k]\!]$. Then the claim follows from . ◻ **Corollary 36**. *We have that $\bigcap_{n \ge 1} \pi(J)^n = (0)$.* *Proof.* Note that $S[\![q_1, \ldots, q_k]\!]$ is Noetherian from [@AM:intro Cor. 10.27]. Then its quotient $M$ is also Noetherian, and by we have that $\pi(J) \subset \mathop{\mathrm{Jac}}(M)$. The claim follows from a corollary a Krull's theorem, [@AM:intro Cor. 10.19], applied to $M$ as a module over $S[\![q_1, \ldots, q_k]\!]$ and the ideal $\pi(J)$. ◻ Let us assume further that $S$ is an $R$-algebra for a Noetherian ring $R$. Let $$A\coloneqq R[\![q_1, \ldots, q_k]\!] \subset S[\![q_1, \ldots, q_k]\!]$$ with ideal $\mathfrak{a}=\langle q_1, \ldots, q_k\rangle \subset A$. The goal of the Appendix is to prove the following Proposition, see also [@eisenbud:CAbook Exercise 7.8]. **Proposition 37**. *If $M/\mathfrak{a}M$ is a finitely generated $A/\mathfrak{a}$-module, then* 1. *[\[item:fg\]]{#item:fg label="item:fg"} $M$ is a finitely generated $A$-module;* 2. *[\[item:complete\]]{#item:complete label="item:complete"} $M$ is $\mathfrak{a}$-adically complete.* *Proof.* Note that $A$ is $\mathfrak{a}$-adically complete [@eisenbud:CAbook §7.1], and that $$\bigcap_{n \ge 1} \mathfrak{a}^nM = \bigcap_{n \ge 1} J^n M= \bigcap_{n \ge 1} \pi(J)^n =(0)$$ by . Then part [\[item:fg\]](#item:fg){reference-type="eqref" reference="item:fg"} follows from . Since $A$ is $\mathfrak{a}$-dically complete, it follows from [@AM:intro Prop. 10.13] that the $\mathfrak{a}$-adic completion of $M$ is $\widehat{M} = M \otimes_A \widehat{A}=M$, proving part [\[item:complete\]](#item:complete){reference-type="eqref" reference="item:complete"}. ◻ [^1]: WG was partially supported by NSF grant PHY-1720321. LM was partially supported by NSF grant DMS-2152294 and a Simons Collaboration Grant. ES was partially supported by NSF grant PHY-2310588. HZ was partially supported by the China Postdoctoral Science Foundation with grant No. 2022M720509

arxiv_math

--- abstract: | Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer $k$ there are finitely many isomorphism classes of groups that occur as the defect group of a block with $k$ irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with $3$ irreducible characters is necessarily the cyclic group of order $3$. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture. address: - Departament d'Àlgebra - Universitat de València, 46100 Burjassot, València, Spain - Departament d'Àlgebra - Universitat de València, 46100 Burjassot, València, Spain - Dept. Mathematics - University of Denver, Denver, CO 80210, USA; and Dept. Mathematics and Statistics - MSU Denver, Denver, CO 80217, USA author: - Alexander Moretó - Noelia Rizo - A. A. Schaeffer Fry title: Brauer's Problem 21 for Principal Blocks --- [^1] # Introduction In Problem 21 of his famous list of open problems in representation theory, R. Brauer asks whether for any positive integer $k$ there are finitely many isomorphism classes of groups that occur as the defect group of a block with $k$ irreducible characters ([@bra]). This is equivalent to the question of whether the order of a defect group can be bounded from above in terms of the number of irreducible characters in the block. This conjecture was proved for solvable groups by B. Külshammer [@kul1] in 1989 and then for $p$-solvable groups [@kul2] in 1990. On the other hand, using E. Zelmanov's solution of the restricted Burnside problem, it was proved by Külshammer and G. R. Robinson that the Alperin--McKay conjecture implies Brauer's Problem 21 [@Kulshammer-Robinson]. Hence, L. Ruhstorfer's recent solution of Alperin--McKay for $p=2$ [@ruh] implies that Brauer's Problem 21 holds for this prime. In the main result of this paper, we prove Brauer's Problem 21 for principal blocks. **Theorem 1**. *Brauer's Problem 21 has an affirmative answer for principal blocks for every prime.* Recall that Landau's theorem asserts that the order of a finite group is bounded from above in terms of the number of conjugacy classes. As pointed out by Brauer [@bra], Landau's argument provides the bound $|G|\leq 2^{2^{k(G)}}$. Brauer's Problem 3 asks for substantially better bounds. This problem has also generated a large amount of research. L. Pyber [@pyb] found an asymptotically substantially better bound, although it is still not known whether there exists a bound of the form $|G|\leq c^{k(G)}$ for some constant $c$. We refer the reader to [@bmt] for the best known bound as of the writing of this article. Note that Brauer's Problem 21 asks for a blockwise version of Landau's theorem. As Brauer did with Landau's theorem, it also seems interesting to ask for asymptotically good bounds for the order of a defect group in terms of the number of characters in the block. Our proof of Theorem [Theorem 1](#thm:BP21principal){reference-type="ref" reference="thm:BP21principal"} provides an explicit bound that surely will be far from best possible. For almost simple groups, we obtain a better bound in Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}. Given a Brauer $p$-block $B$ of a finite group $G$ with defect group $D$, we will write $k(B)$ to denote the number of irreducible complex characters in $B$. R. Brauer himself proved that if $k(B)=1$ then $D$ is the trivial group ([@nbook Theorem 3.18]). More than 40 years later, J. Brandt proved that if $k(B)=2$ then $D$ is the cyclic group of order $2$. However, despite a large amount of work in the area in recent years, the conjecture remains open when $k(B)\geq 3$. It has been speculated since Brandt's [@Br82] 1982 paper that if $k(B)=3$ then the defect group is cyclic of order $3$. It seems that it was known to Külshammer that this follows from the Alperin-McKay conjecture since 1990 [@kul2]. A proof of this fact appeared in [@KNST14], where Külshammer, G. Navarro, B. Sambale and P. H. Tiep formally state Brandt's speculation as a conjecture. We present a condition on quasisimple groups that would imply the Külshammer-Navarro-Sambale-Tiep conjecture (that is, that $k(B)=3$ implies that the defect group is of size $3$). **Condition 2**. *Let $p$ be an odd prime and let $S$ be a non-abelian simple group of order divisible by $p$. We say that Condition B holds for $(S,p)$ if the following holds: let $K$ be a quasisimple group of order divisible by $p$ with center $Z$, a cyclic $p'$-group, and $K/Z=S$. Let $B$ be a non-principal faithful $p$-block of $K$ with $|{\rm cd}(B)|>1$ and let $D$ be a defect group of $B$, not cyclic and elementary abelian. Then there are at least 4 irreducible characters in $B$ not ${\rm Aut}(K)$-conjugate.* **Theorem 3**. *Let $p$ be a prime. If $p$ is odd, suppose that Condition B holds for $(S,p)$ for all non-abelian composition factors $S$ of $G$. Then the Külshammer-Navarro-Sambale-Tiep conjecture holds for $G$.* We remark that this reduction and Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} have played an influential role in the recent solution of Brauer's height zero conjecture [@MNST]. In fact, the fundamental Theorem B of [@MNST] is a slightly weaker version of Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"}: it shows there always exist $3$ irreducible characters in $B$ not ${\rm Aut}(K)$-conjugate. In fact, we will see in Remark [Remark 30](#rem:counterexample){reference-type="ref" reference="rem:counterexample"} that this is tight. Although Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} seems to hold in many situations, we will see an example of a family of simple groups for which Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} does not hold, for $p=5$. In Section [2](#sec:sectionsimplesBP21){reference-type="ref" reference="sec:sectionsimplesBP21"}, we prove Brauer Problem 21 for the principal blocks of almost simple groups, which is used in Section [3](#sec:BP21proof){reference-type="ref" reference="sec:BP21proof"} to prove Theorem [Theorem 1](#thm:BP21principal){reference-type="ref" reference="thm:BP21principal"}. In Section [4](#sec:kb3reduction){reference-type="ref" reference="sec:kb3reduction"}, we prove Theorem [Theorem 3](#thm:kb3reduction){reference-type="ref" reference="thm:kb3reduction"}. We conclude the paper by discussing Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} in Section [5](#sec:orbits){reference-type="ref" reference="sec:orbits"}. # BP21 for almost simple groups {#sec:sectionsimplesBP21} The following is the main result of this section. **Theorem 1**. *Let $p$ be a prime. Let $S\leq A\leq \operatorname{Aut}(S)$, where $S$ is a finite nonabelian simple group, and $p\mid |S|$. Let $P\in {\operatorname{Syl}}_p(A)$ and let $k:=k(B_0(A))$ be the number of irreducible complex characters in the principal block of $A$. Then we have:* 1. *[\[thm:BP21simplemain\]]{#thm:BP21simplemain label="thm:BP21simplemain"} $|P|\leq k^{2(k^2+2k)}$.* 2. *[\[thm:BP21simple\]]{#thm:BP21simple label="thm:BP21simple"} $|P\cap S|\leq k^{2k^2}$.* Note that in the context of Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}, any character in ${\rm Irr}(B_0(S))$ lies below some character in ${\rm Irr}(B_0(A))$ by [@nbook Theorem (9.4)], so that $k(B_0(A))\geq k_{\operatorname{Aut}(S)}(B_0(S))$, where we write $k_{\operatorname{Aut}(S)}(B_0(S))$ for the number of distinct $\operatorname{Aut}(S)$-orbits intersecting ${\rm Irr}(B_0(S))$. *Remark 2*. We further remark that, by the results of [@KS21; @RSV21; @HSV23], we may assume for Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"} that $k(B_0(A))\geq 7$. The following is the main result of [@HSF21], from which we obtain a bound for $p$ in terms of the number of irreducible characters in a given principal block. **Lemma 3** (Hung--Schaeffer Fry). *Let $p$ be a prime and let $G$ be a finite group with $p\mid |G|$. Let $B_0$ denote the principal $p$-block of $G$. Then $$k(B_0)^2\geq 4(p-1).$$ In particular, $p\leq \frac{1}{4}k(B_0)^2+1\leq \frac{1}{2}k(B_0)^2$, with the last inequality strict for $k(B_0)>2$.* Next, we consider the case of cyclic Sylow subgroups. **Lemma 4**. *Let $p$ be a prime and let $G$ be a finite group with $p\mid |G|$. Assume that a Sylow $p$-subgroup $P\in{\operatorname{Syl}}_p(G)$ is cyclic, and let $B_0$ denote the principal $p$-block of $G$. Then $$|P|<k(B_0)^2.$$* *Proof.* In this case, by Dade's theory of blocks with cyclic defect group [@cravenbook Theorem 5.1.2], we have $k(B_0)=e+\frac{|P|-1}{e}$, where $e=l(B_0)$ is the number of irreducible $p$-Brauer characters in $B_0$. Since $1\leq l(B_0)<k(B_0)$ (see [@isbook2 Theorem 15.29]), this yields $$|P|=k(B_0)e-e^2+1\leq k(B_0)e<k(B_0)^2,$$ as claimed. ◻ ## Notation and Additional Preliminaries Let $q$ be a power of a prime. By a group of Lie type, we will mean a finite group obtained as the group ${\mathbf{G}}^F$ of fixed points of a connected reductive algebraic group ${\mathbf{G}}$ over $\bar{{\mathbb{F}}}_q$ under a Steinberg morphism $F\colon {\mathbf{G}}\rightarrow{\mathbf{G}}$ endowing ${\mathbf{G}}$ with an ${\mathbb{F}}_q$-structure. In our situation of finite simple groups, we will often take ${\mathbf{G}}$ to further be simple and simply connected, so that ${\mathbf{G}}^F$ is, with some exceptions dealt with separately, the full Schur covering group of a simple group $S={\mathbf{G}}^F/{\bf Z}({\mathbf{G}}^F)$. Writing $G={\mathbf{G}}^F$, we let $G^\ast$ denote the group $({\mathbf{G}}^\ast)^F$, where the pair $({\mathbf{G}}^\ast, F)$ is dual to $({\mathbf{G}}, F)$, with respect to some maximally split torus ${\mathbf{T}}$ of ${\mathbf{G}}$. Given a semisimple element $s\in G^\ast$ (that is, an element of order relatively prime to $q$), we obtain a rational Lusztig series $\mathcal{E}(G,s)$ of irreducible characters of $G$ associated to the $G^\ast$-conjugacy class of $s$. When $s=1$, the set $\mathcal{E}(G,1)$ is comprised of the so-called unipotent characters. Each series $\mathcal{E}(G,s)$ contains so-called semisimple characters, and if ${\bf C}_{{\mathbf{G}}^\ast}(s)$ is connected, there is a unique semisimple character, which we will denote by $\chi_s$. The following lemma will help us obtain many semisimple characters in the principal block. Here, we write ${\bf Z}(G)={\bf Z}( G)_p\times{\bf Z}(G)_{p'}$, where ${\bf Z}(G)_p\in{\operatorname{Syl}}_p({\bf Z}(G))$. **Lemma 5**. *Let $p$ be a prime and let $G:=\mathbf{G}^F$ be a group of Lie type defined over ${\mathbb{F}}_q$ with $p\nmid q$ and such that ${\bf Z}(\mathbf{G})$ is connected or such that $p$ is good for $\mathbf{G}$ and ${\bf C}_{\mathbf{G}^\ast}(s)$ is connected. Let $s\in G^\ast$ be a semisimple element with order a power of $p$. Then the corresponding semisimple character $\chi_s\in{\operatorname{Irr}}(G)$ lies in the principal $p$-block $B_0(G)$ of $G$ and is trivial on ${\bf Z}(G)_{p'}$.* *Proof.* The first statement, also noted in [@HSF21 Theorem 5.1], is due to Hiss [@hiss90 Corollary 3.4], and the second follows from [@bon06 11.1(d)]. ◻ Throughout, for $q$ an integer and $p$ a prime not dividing $q$, we let $d_p(q)$ denote the order of $q$ modulo $p$ if $p$ is odd, and $d_2(q)$ is the order of $q$ modulo $4$. For the remainder of Section [2](#sec:sectionsimplesBP21){reference-type="ref" reference="sec:sectionsimplesBP21"}, we will let $G={\mathbf{G}}^F$ for ${\mathbf{G}}$ a simple, simply connected reductive group and $F\colon {\mathbf{G}}\rightarrow{\mathbf{G}}$ a Steinberg endomorphism such that $G/{\bf Z}(G)$ is a simple group of Lie type. Further, we will address the case that $S$ is a simple group with an exceptional Schur multiplier (see [@GLS Table 6.1.3] for the list of such $S$), sporadic, or alternating separately in the proof of Theorem [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"} below, and hence until then, we assume further that ${\bf Z}(G)$ is a nonexceptional Schur multiplier for the simple group of Lie type $S:=G/{\bf Z}(G)$. Let $\widetilde{S}$ denote the group of inner-diagonal automorphisms of $S$. ## Exceptional Groups We first consider the exceptional groups, by which we mean the groups $S=\operatorname{G}_2(q)$, ${}^2\!\operatorname{B}_2(q^2)$, ${}^2\!\operatorname{G}_2(q^2)$, $\operatorname{F}_4(q)$, ${}^2\!\operatorname{F}_4(q^2)$, ${}^3\!\operatorname{D}_4(q)$, $\operatorname{E}_6(q)$, ${}^2\!\operatorname{E}_6(q)$, $\operatorname{E}_7(q)$, and $\operatorname{E}_8(q)$, when $p$ is a prime not dividing $q$. Let $P\in{\operatorname{Syl}}_p(S)$. Then either $P$ may be identified with a Sylow $p$-subgroup of $G$ or $(p,{\mathbf{G}})\in\{(3,\operatorname{E}_6), (2, \operatorname{E}_7)\}$ and $|P|=|\hat{P}|/p$ with $\hat{P}$ a Sylow $p$-subgroup of $G$. If $G$ is not of Suzuki or Ree type (i.e. $G$ is not one of ${}^2\!\operatorname{B}_2(q^2)$, ${}^2\!\operatorname{G}_2(q^2)$, or ${}^2\!\operatorname{F}_4(q^2)$), let $e:=d_p(q)$ and let $\Phi_e:=\Phi_e(q)$ denote the $e$th cyclotomic polynomial in $q$. If $G$ is a Suzuki or Ree group, instead let $\Phi_e:=\Phi^{(p)}$ as in [@malle07 Section 8]. In either case, let $p^b$ be the the highest power of $p$ dividing $\Phi_e$ and let $m_e$ denote the largest positive integer such that ${\Phi_e}^{m_e}$ divides the order polynomial of $({\mathbf{G}}, F)$. From [@GLS Theorem 4.10.2], we see that $\hat P$ contains a normal abelian subgroup $P_T\lhd \hat P$ such that $\hat P/P_T$ is isomorphic to a subgroup of the Weyl group $W={\bf N}_{{\mathbf{G}}}({\mathbf{T}})/{\mathbf{T}}$. We also have $\hat P=P_T$ if and only if $P_T$ is abelian (see [@malle14 Proposition 2.2]). Similarly, a Sylow $p$-subgroup of the dual group $G^\ast$ contains a group isomorphic to $P_T$. **Proposition 6**. *Let $p=2$ and let $S$ be an exceptional group of Lie type as above with $2\nmid q$. Let $P\in{\operatorname{Syl}}_2(S)$ and write $k_0:=k_{\operatorname{Aut}(S)}(B_0(S))$. Then $|P|\leq 2^{14+8k_0}$. (In particular, Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}[\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"} holds in this case.)* *Proof.* First consider the case $S={}^2\!\operatorname{G}_2(q^2)$ with $q^2=3^{2n+1}>3$. Then we have $|P|=8$, so the statement is clear. Hence we assume that $S$ is not of Suzuki or Ree type. Let $H=G^\ast=({\mathbf{G}}^\ast)^F$ and notice that $S=[H, H]$ and ${\bf Z}({\mathbf{G}}^\ast)$ is connected. Then notice that the semisimple characters $\chi_s\in{\operatorname{Irr}}(H)$ of $H$ for $s\in H^\ast=G$ of $2$-power order lie in $B_0(H)$ by Lemma [Lemma 5](#lem:ss){reference-type="ref" reference="lem:ss"}. Let $2^{b+1}\mid\mid(q^2-1)$ and note that $G$ contains an element of order $2^{b}$. For $1\leq i\leq b$, let $s_i\in G$ be of order $2^{i}$, so that the semisimple characters $\chi_{s_i}$ of $H$ for $1\leq i\leq b$ lie in $B_0(H)$. Further, since the $|s_i|$ are distinct, these lie in distinct $\operatorname{Aut}(S)$-conjugacy classes, using [@NTT08 Corollary 2.5]. Then choosing an irreducible constituent $\chi_{s_i}'$ on $S$ for each $i$, we obtain $b$ characters in $B_0(S)$ in distinct $\operatorname{Aut}(S)$-classes. Considering in addition the trivial character, we obtain $k_0\geq b+1$. On the other hand, letting $r$ be the rank of ${\mathbf{G}}$, we have $r\leq 8$ and $|P_T|\leq (2^{b+1})^r\leq (2^{b+1})^8$ by the description of $P_T$ in [@GLS Theorem 4.10.2]. Further, $|\hat P/P_T|\leq |W|_2\leq 2^{14}$. Hence $|P|\leq |\hat P|\leq 2^{14}\cdot 2^{8k_0}$, as stated. Recalling that $k\geq 7$ (see Remark [Remark 2](#rem:k7){reference-type="ref" reference="rem:k7"}), in the situation of Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}[\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"} we have $|P|\leq 7^5 \cdot 7^{3k}\leq k^{5+3k}$. ◻ Now, when $p$ is odd, a similar argument can be used. However, we aim for a better bound. In this case, [@GLS Theorem 4.10.2] further tells us that $P_T$ has a complement $P_W$ in $\hat P$ and we have $P_T\cong C_{p^b}^{m_e}$ unless $(p, G)= (3,{}^3\!\operatorname{D}_4(q))$, in which case $P_T\cong C_{3^a}\times C_{3^{a+1}}$. In the following, let $W(\operatorname{E}_8)$ denote the Weyl group $W$ obtained in the case that ${\mathbf{G}}=\operatorname{E}_8$. **Proposition 7**. *Let $S$ be an exceptional group of Lie type as above, and let $P\in {\operatorname{Syl}}_p(S)$ with $p$ an odd prime not dividing $q$. Let $k_0:=k_{\operatorname{Aut}(S)}(B_0(S))$. Then if $P$ is cyclic, we have $|P|< p^{k_0}$. Otherwise, we have $$|P|\leq C_{ex}\cdot k_0^2$$ for some constant $C_{ex}\leq 36|W(\operatorname{E}_8)|^2$. In particular, when $S\leq A\leq \operatorname{Aut}(S)$ with $k(B_0(A))\geq 5$, this yields $|P|\leq k(B_0(A))^{k(B_0(A))^2}$ in either case.* It should be noted, however, that in the last statement, $P\in{\operatorname{Syl}}_p(S)$, rather than ${\operatorname{Syl}}_p(A)$. *Proof.* Keep the notation above. Suppose first that a Sylow $p$-subgroup of $G$ is abelian. If $P$ is cyclic, then $P=\hat{P}=P_T=C_{p^b}$. Here we may argue similarly to Proposition [Proposition 6](#prop:BP21except2){reference-type="ref" reference="prop:BP21except2"} to obtain $b<k_0$, and hence $|P|<p^{k_0}$. Lemma [Lemma 4](#lem:BP21cyclic){reference-type="ref" reference="lem:BP21cyclic"} further yields the last statement in this case. Hence, we may assume that $P$ is not cyclic, so that $m_e\geq 2$. By the discussion preceeding [@HSF21 Theorem 5.4], we have $$\label{eq:HSFbnd} k_0\geq \frac{p^{b m_e}}{gd p^b|W_e|},$$ where $g$ is the size of the subgroup of ${\operatorname{Out}}(S)$ of graph automorphisms, $d:=[\widetilde{S}:S]$ is the size of the group of diagonal automorphisms, and $|W_e|$ is the so-called relative Weyl group for a Sylow $\Phi_e$-torus of $G$. Since $d\leq 3$, $g\leq 2$, and $|W_e|$ is bounded by the size of the largest Weyl group for the types under consideration, $|W(\operatorname{E}_8)|$, we have $$k_0\geq \frac{p^{bm_e}}{6|W(E_8)| p^b}.$$ Notice that $p^{b(m_e-1)}\geq p^{bm_e/2}$ for $m_e\geq 2$. Then we have $\sqrt{|P|}\leq \sqrt{|\hat P|}\leq 6|W(E_8)|k _0$, and hence the statement holds. We now assume that $\hat P$ is nonabelian. By considering only the semisimple characters of $G$ corresponding to elements of $G^\ast$ found in a copy of $P_T$, the exact same arguments as in [@HSF21 Section 5] yield that the bound [\[eq:HSFbnd\]](#eq:HSFbnd){reference-type="eqref" reference="eq:HSFbnd"} still holds in this case. By considering the degree polynomials, we see that in each case, we have $\sqrt{|P|}\leq p^{b(m_e-1)}$ again, except possibly if $G=\operatorname{G}_2(q)$, $p=3$, $m_e=2$, and $|P|=p^{2b+1}$. Then $\sqrt{|P|}=\sqrt{3}\cdot 3^b= \sqrt{3}\cdot p^{b(m_e-1)}\leq \sqrt{3}|W_e|k_0$, where the last inequality is because $d=1=g$ in this case. In all cases, then, we see that the statement holds. ◻ ## Classical Groups {#sec:classical} We now turn to the case of classical groups. In this section, let $G={\mathbf{G}}^F$ be a group of Lie type defined over ${\mathbb{F}}_q$, where $q$ is a power of a prime $q_0$ and ${\mathbf{G}}$ is a simple, simply connected reductive group of type $\operatorname{A}_{n-1}$ with $n\geq 2$, $\operatorname{C}_n$ with $n\geq 2$, $\operatorname{B}_n$ with $n\geq 3$, or type $\operatorname{D}_n$ with $n\geq 4$ but $G\neq {}^3\!\operatorname{D}_4(q)$, and such that $G$ is a nonexceptional Schur covering group for the simple group $S:=G/{\bf Z}(G)$. That is, $G={\operatorname{SL}}_n^\epsilon(q)$, ${\operatorname{Sp}}_{2n}(q)$, $\operatorname{Spin}_{2n+1}(q)$, or $\operatorname{Spin}_{2n}^\pm(q)$, and $S={\operatorname{PSL}}_n^\epsilon(q)$, ${\operatorname{PSp}}_{2n}(q)$, ${\operatorname{P\Omega}}_{2n+1}(q)$, or ${\operatorname{P\Omega}}_{2n}^\pm(q)$, respectively, for the corresponding values of $n$. Let $H$ be the related groups $H:=\operatorname{GL}_n^\epsilon(q), {\operatorname{Sp}}_{2n}(q), {\operatorname{SO}}_{2n+1}(q)$, respectively ${\operatorname{SO}}_{2n}^\pm(q)$. We remark that, taking $\Omega:={\operatorname{O}}^{q_0'}(H)$, we have $\Omega$ is perfect and $S=\Omega/{\bf Z}(\Omega)=G/{\bf Z}(G)$. We also have ${\bf Z}(\Omega)\leq {\bf Z}(H)$ and further $H/\Omega$ and ${\bf Z}(H)$ are both $2$-groups if $H\neq \operatorname{GL}_n^\epsilon(q)$. Note that the dual group of $H$ is $H^\ast = \operatorname{GL}_n^\epsilon(q)$, ${\operatorname{SO}}_{2n+1}(q)$, ${\operatorname{Sp}}_{2n}(q)$, and ${\operatorname{SO}}_{2n}^\pm(q)$, respectively. Let $p\neq q_0$ be a prime and write $\widetilde{P}$ for a Sylow $p$-subgroup of $H^\ast$. We remark that if $X\in\{G, S\}$, then $|P|\leq |\widetilde{P}|$ for $P\in{\operatorname{Syl}}_p(X)$. ### Sylow $p$-Subgroups of Symmetric Groups Since the Sylow $p$-subgroups of classical groups are closely related to those of symmetric groups, we begin with a discussion of the latter. Let $w$ be a positive integer with $p$-adic expansion $$\label{eq:wpadic} w=a_0+a_1p+a_2p^2+\cdots+a_tp^t,$$ where $0\leq a_i<p$ for $0\leq i\leq t-1$ and $0<a_t<p$. Let $Q\in{\operatorname{Syl}}_p(\mathfrak{S}_w)$. We have $Q=\prod_{i=0}^{t}Q_i^{a_i}$, , where $Q_i$ is a Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_{p^i}$. Moreover $|Q_i|=p^{p^{i-1}+p^{i-2}+\cdots+p+1}\leq p^{p^i}$ for each $1\leq i\leq t$. Then with this, we see $$\label{eq:sizeSylowSw} |Q|=(w!)_p\leq p^w.$$ ### Unipotent Characters Recall that $\mathcal{E}(G,1)$ is the set of unipotent characters of $G$. Since unipotent characters are trivial on ${\bf Z}(G)$, we may say that $\chi\in{\operatorname{Irr}}(S)$ is a unipotent character of $S$ if it is the deflation of some unipotent character of $G$. The following observation will be useful in the cases of defining characteristic and when $p=2$. **Lemma 8**. *Let $S$ be one of the groups $S={\operatorname{PSL}}_n^\epsilon(q)$ with $n\geq 2$, ${\operatorname{PSp}}_{2n}(q)$ with $n\geq 2$, ${\operatorname{P\Omega}}_{2n+1}(q)$ with $n\geq 3$, ${\operatorname{P\Omega}}_{2n}^+(q)$ with $n\geq 4$, or ${\operatorname{P\Omega}}_{2n}^-(q)$ with $n\geq 4$. Then there are at least $n$ non-$\operatorname{Aut}(S)$-conjugate unipotent characters of $S$.* *Proof.* The unipotent characters of $G$ are described in [@carter Section 13.8]. From this, we have the number of unipotent characters in the case ${\operatorname{PSL}}_n^\epsilon(q)$ is the number of partitions $\pi(n)$ of $n$. In the remaining cases, the unipotent characters of $G$ lying in the principal series are in bijection with the characters of the Weyl group $W(\operatorname{C}_n)$, $W(\operatorname{B}_n)$, $W(\operatorname{D}_n)$, or $W(\operatorname{B}_{n-1})$, respectively, each of which contains a quotient group isomorphic to a symmetric group $\mathfrak{S}_n$ (resp. $\mathfrak{S}_{n-1}$ in the case of $W(B_{n-1})$). In each of these cases, there are also non-principal series unipotent characters. Then the number of unipotent characters is more than $\pi(n)$ (resp. $\pi(n-1)$) in these cases. Note that $\pi(n)\geq n$, with strict inequality for $n\geq 4$, and that further $\pi(n)\geq 2n$ for $n\geq 7$. With the exception of ${\operatorname{PSp}}_{4}(q)$ with $q$ even and ${\operatorname{P\Omega}}_{2n}^+(q)$, all unipotent characters of the groups under consideration are $\operatorname{Aut}(S)$-invariant (see [@Malle08 Theorem 2.5]), and we see that there are at least $n$ such characters in each case. For ${\operatorname{PSp}}_{4}(q)$ with $q$ even, there are six unipotent characters, with two of them interchanged by the exceptional graph automorphism. For ${\operatorname{P\Omega}}_{2n}^+(q)$ with $n\geq 5$, we see there are at least $2n$ unipotent characters (explicitly for $n=5,6$, and since $\pi(n)\geq 2n$ for $n\geq 7$), and the $\operatorname{Aut}(S)$-orbits have size at most 2. The group ${\operatorname{P\Omega}}_8^+(q)$ has 14 unipotent characters and the $\operatorname{Aut}(S)$-orbits have size at most 3. In all cases, then, we see that there are at least $n$ $\operatorname{Aut}(S)$-orbits of unipotent characters. ◻ ### Bounds in the Case of Classical Groups for Defining Characteristic or $p=2$ **Corollary 9**. *Let $S$ be one of the groups as in Lemma [Lemma 8](#lem:classicalunips){reference-type="ref" reference="lem:classicalunips"}. Assume that $p\mid q$ or that $p=2$ and $q$ is odd. Then $k_{\operatorname{Aut}(S)}(B_0(S))\geq n$.* *Proof.* In defining characteristic, we have ${\rm Irr}(B_0(S))={\rm Irr}(S)\setminus\{\mathrm{St}_S\}$, where $\mathrm{St}_S$ is the Steinberg character (see [@CE04 Theorem 6.18]). In the case $p=2$ and $q$ is odd, we have $B_0(S)$ is the unique block containing unipotent characters, by [@CE04 Theorem 21.14]. Then the statement follows from Lemma [Lemma 8](#lem:classicalunips){reference-type="ref" reference="lem:classicalunips"} and the fact that ${\rm Irr}(S)$ contains non-unipotent characters. ◻ **Lemma 10**. *Let $S$ be as in Lemma [Lemma 8](#lem:classicalunips){reference-type="ref" reference="lem:classicalunips"} with $q$ odd, and let $2^{b+1}$ be the largest power of $2$ dividing $q^2-1$. Let $B_0(S)$ be the principal $2$-block of $S$. Then $b+1\leq k_{\operatorname{Aut}(S)}(B_0(S))$.* *Proof.* As before, let $S=G/{\bf Z}(G)$ with $G={\mathbf{G}}^F$ of simply connected type. Recall that $B_0(G)$ is the unique unipotent block of $G$ by [@CE04 Theorem 21.14], and hence $B_0(G)$ is exactly the union of rational Lusztig series $\mathcal{E}(G,s)$ with $s\in G^\ast$ having order a power of $2$. From the structure of the Sylow $2$-subgroup of $G^\ast$ described in [@GLS Theorem 4.10.2] (see also [@carterfong]), we can see that the group ${\mathbf O}^{q_0'}(G^\ast)$ contains an element of order $2^{b-1}$. For $1\leq i\leq b-1$, let $s_i\in {\mathbf O}^{q_0'}(G^\ast)$ be of order $2^{i}$. Then the semisimple characters $\chi_{s_i}$ for $1\leq i\leq b-1$ lie in $B_0(G)$ and are trivial on ${\bf Z}(G)$ by the dual version of [@DM20 Proposition 11.4.12 and Remark 11.4.14]. Further, these lie in distinct $\operatorname{Aut}(S)$-conjugacy classes, using [@NTT08 Corollary 2.5]. Combining with Lemma [Lemma 8](#lem:classicalunips){reference-type="ref" reference="lem:classicalunips"}, we see $k_{\operatorname{Aut}(S)}(B_0(S))\geq b-1+n\geq b+1$. ◻ ### Bounds in the Case of Classical Groups for Nondefining Characteristic with $p$ Odd {#sec:classicalBP21} Now let $p$ be an odd prime not dividing $q$ and let $d:=d_p(q)$. If $G={\operatorname{SL}}_n^\epsilon(q)$, let $e:=d_p(\epsilon q)$, and otherwise let $e:=d_p(q^2)$. Further, let $b\geq 1$ be the largest integer such that $p^b$ divides $(\epsilon q)^e-1$, respectively $q^{2e}-1$. We begin by discussing the Sylow $p$-subgroups $\widetilde{P}$ of $H^\ast$, which have been described by Weir [@weir]. First, consider the case $H=\operatorname{GL}_n^\epsilon(q)$. Let $n=ew+r$, where $r,w$ are positive integers with $0\leq r<e$ and $w$ is written with $p$-adic expansion as in [\[eq:wpadic\]](#eq:wpadic){reference-type="eqref" reference="eq:wpadic"}. A Sylow $p$-subgroup of $H=\operatorname{GL}_n^\epsilon(q)$ is then of the form $\widetilde{P}=\prod_{i=0}^t P_i^{a_i}$, where $P_i\in {\operatorname{Syl}}_p(\operatorname{GL}_{ep^i}^\epsilon(q))$ is of the form $C_{p^b}\wr Q_i$. Hence $\widetilde{P}$ contains a subgroup of the form $\bar P=C_{p^b}^w$. Further, $\widetilde{P}\cap {\operatorname{SL}}_n^\epsilon(q)$ is a Sylow $p$-subgroup of $G={\operatorname{SL}}_n^\epsilon(q)$. Now consider $H={\operatorname{Sp}}_{2n}(q)$, ${\operatorname{SO}}_{2n+1}(q)$, and ${\operatorname{SO}}_{2n}^\epsilon(q)$. The structure of $\widetilde{P}$ in these case builds off of the case of linear groups above. If $H^\ast\in\{{\operatorname{SO}}_{2n+1}(q), {\operatorname{Sp}}_{2n}(q)\}$, we have $\widetilde{P}$ is already a Sylow $p$-subgroup of $\operatorname{GL}_{2n+1}(q)$ (and hence of $\operatorname{GL}_{2n}(q)$) when $d$ is even, and are Sylow subgroups of the naturally-embedded $\operatorname{GL}_n(q)$ if $d$ is odd. In particular, writing $n=ew+r$ with $r, w$ as before, we have $\widetilde{P}$ is again of the form $\widetilde{P}\cong P_0^{a_0}\times\cdots\times P_t^{a_t}$, where each $P_i$ is a Sylow $p$-subgroup of $\operatorname{GL}_{dp^{i}}(q)$ (and hence again $P_i\cong C_{p^b}\wr Q_i$). If $H^\ast={\operatorname{SO}}_{2n}^\pm(q)$, then we have embeddings ${\operatorname{SO}}_{2n-1}(q)\leq H^\ast\leq {\operatorname{SO}}_{2n+1}(q)$, and $\widetilde{P}$ is a Sylow subgroup of either ${\operatorname{SO}}_{2n-1}(q)$ or ${\operatorname{SO}}_{2n+1}(q)$. In this case, letting $m\in\{n, n-1\}$ so that $\widetilde{P}$ is a Sylow subgroup of ${\operatorname{SO}}_{2m+1}(q)$ and now writing $m=ew+r$ with $w$ again written as in [\[eq:wpadic\]](#eq:wpadic){reference-type="eqref" reference="eq:wpadic"}, $\widetilde{P}$ can again be written $\widetilde{P}\cong P_0^{a_0}\times\cdots\times P_t^{a_t}$ with each $P_i$ a Sylow subgroup of $\operatorname{GL}_{dp^{i}}(q)$. In all cases, we remark that $p^t\leq w\leq p^{t+1}$ and that $t=0$ corresponds to the case that a Sylow $p$-subgroup of $H$ is abelian. Further, $\widetilde{P}$ contains a subgroup of the form $\bar P\cong C_{p^b}^w$. **Lemma 11**. *With the notation above, we have $b+1\leq k_{\operatorname{Aut}(S)}(B_0(S))$.* *Proof.* We will show that there are at least $b$ characters in ${\rm Irr}(B_0(S))\setminus\{1_S\}$ lying in distinct $\operatorname{Aut}(S)$-orbits. First, let $G={\operatorname{SL}}_n^\epsilon(q)$ and let $\widetilde{G}:=H=\operatorname{GL}_n^\epsilon(q)$, and note $\widetilde{G}^\ast\cong \widetilde{G}$. We have $\operatorname{Aut}(S)=\widetilde{S}\rtimes \mathcal{D}$, where $\mathcal{D}$ is an appropriate group of graph and field automorphisms and $\widetilde{S}:=\widetilde{G}/{\bf Z}(\widetilde{G})$. Recall that a Sylow $p$-subgroup $\widetilde{P}$ of $\widetilde{G}$ contains a subgroup of the form $C_{p^b}^w$. Assume for the moment that $e>1$, so that $p\nmid|{\bf Z}(\widetilde{G})|$ and $p\nmid [\widetilde{G}:G]$. Hence, for $1\leq j\leq b$, we may let $s_j\in \widetilde{G}^\ast\cong \widetilde{G}$ be an element of order $p^j$. The corresponding semisimple character $\chi_{s_j}$ of $\widetilde{G}$ is trivial on ${\bf Z}(\widetilde{G})$ and lies in $B_0(\widetilde{G})$, using Lemma [Lemma 5](#lem:ss){reference-type="ref" reference="lem:ss"}. Hence, each $\chi_{s_j}$ can be viewed as a character of $B_0(\widetilde{S})$. Further, note that since $p\nmid |{\bf Z}(\widetilde{G})|$, $s_i$ and $s_j^\alpha z$ cannot be $\widetilde{G}$-conjugate for any $i\neq j$ and any $\alpha\in \mathcal{D}$ and $z\in{\bf Z}(\widetilde{G})$. If instead $e=1$, we have $n=w\geq 2$. For $1\leq j\leq b$, let $\lambda_j\in C_{p^b}\leq {\mathbb{F}}_{q^2}^\times$ with $|\lambda_j|=p^j$, and let $s_j$ be an element of $C_{p^b}^n\leq \widetilde{P}$ of the form $\mathrm{diag}(\lambda_j, \lambda_j^{-1}, 1, \ldots 1)$, where $1$ appears as an eigenvalue with multiplicity $n-2$. Then again $\chi_{s_j}\in{\rm Irr}(B_0(\widetilde{G}))$ by Lemma [Lemma 5](#lem:ss){reference-type="ref" reference="lem:ss"} and is trivial on ${\bf Z}(\widetilde{G})$ by the dual version of [@DM20 Proposition 11.4.12, and Remark 11.4.14], since $s_j\in [\widetilde{G},\widetilde{G}]=G$. Further, we again see that $s_j$ is not $\widetilde{G}$-conjugate to $s_i^\alpha z$ for any $i\neq j$, $\alpha\in \mathcal{D}$, and $z\in {\bf Z}(\widetilde{G})$, by considering the eigenvalues. In either case, we let $\chi_i$ for $1\leq i\leq b$ be a constituent of $\chi_{s_i}$ restricted to $S$. Then $\chi_i$ cannot be $\operatorname{Aut}(S)$-conjugate to $\chi_j$ for $i\neq j$, using [@NTT08 Corollary 2.5] along with [@DM20 Proposition 11.4.12 and Remark 11.4.14]. Hence, we see at least $b$ distinct $\operatorname{Aut}(S)$-orbits represented in ${\rm Irr}(B_0(S))\setminus\{1_S\}$. Now let $G$ be one of the remaining groups as in the beginning of the section. Then $|{\bf Z}(G)|$ is a power of $2$. In each case, a Sylow $p$-subgroup of $G$ (or, equivalently, of $S$) and of $G^\ast$ contains a subgroup of the form $C_{p^b}$. Here, we may again, for each $1\leq j\leq b$, let $s_j\in G^\ast$ be a semisimple element of order $p^j$. Then since each $(|s_j|, |{\bf Z}(G)|)=1$, we have by [@MT11 Exercise 20.16] that ${\bf C}_{{\mathbf{G}}^\ast}(s_j)$ is connected since $p\geq 3$ is good for ${\mathbf{G}}$, and hence the corresponding semisimple character $\chi_{s_j}$ of $G$ lies in $B_0(G)$ and is trivial on ${\bf Z}(G)$ by Lemma [Lemma 5](#lem:ss){reference-type="ref" reference="lem:ss"}. That is, we may again view $\chi_{s_j}$ as a character in ${\rm Irr}(B_0(S))\setminus\{1_G\}$. Since $s_i$ cannot be $\operatorname{Aut}(G^\ast)$-conjugate to $s_j$ for any $i\neq j$, we see $\chi_{s_i}$ and $\chi_{s_j}$ cannot be $\operatorname{Aut}(S)$-conjugate as before and we again have $k_{\operatorname{Aut}(S)}(B_0(S))\geq b+1$. ◻ **Lemma 12**. *With the above notation, we have at least $w$ unipotent characters in $B_0(S)$ that are not $\operatorname{Aut}(S)$-conjugate, and hence $w\leq k_{\operatorname{Aut}(S)}(B_0(S))$. If $t\geq 1$, this yields $p\leq p^t\leq k_{\operatorname{Aut}(S)}(B_0(S))$.* *Proof.* We remark first that the unipotent characters of $H$ are irreducible on restriction to $\Omega$ and are trivial on ${\bf Z}(H)$. (See, e.g. [@GM20 Proposition 2.3.15].) In the case $H=\operatorname{GL}_n^\epsilon(q)$, we have the number of unipotent characters in $B_0(H)$ is $k(e,w)$, by [@MO83 Proposition (2.3)], where $k(e,w)$ can be computed as in [@olsson84 Lemma 1]. This yields at least $k(e,w) \geq w$ unipotent characters in $B_0(S)$, which are all $\operatorname{Aut}(S)$-invariant (see [@Malle08 Theorem 2.5]). If $H={\operatorname{Sp}}_{2n}(q)$ or ${\operatorname{SO}}_{2n+1}(q)$, we see from [@malle17 Section 5.2] that the number of unipotent characters in $B_0(H)$ is $k(2e, w)>2w$, which again are $\operatorname{Aut}(S)$-invariant by [@Malle08 Theorem 2.5] unless $H={\operatorname{Sp}}_{4}(q)$ with $q$ even. In the latter case, the unipotent characters are at worst permuted in pairs by $\operatorname{Aut}(S)$, and hence again there are at least $w$ non-$\operatorname{Aut}(S)$-conjugate such characters. If $H={\operatorname{SO}}_{2n}^\pm(q)$, then $B_0(H)$ contains either at least $k(2e,w)$ unipotent characters or at least $(k(2e,w)+3k(e,w/2))/2$ when $w$ is even, using [@malle17 Section 5.3 and Lemma 5.6]. One can see that these numbers are again at least $2w$, and by [@Malle08 Theorem 2.5], again the unipotent characters are at worst permuted in pairs by $\operatorname{Aut}(S)$ unless $H={\operatorname{SO}}_8^+(q)$. In the latter case, [@RSV21 Lemma 3.10] gives the claim. ◻ ## The Proof of Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"} {#the-proof-of-theorem-thmbp21almostsimple} The following will be useful in the proof of Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}, as well as in the proof of Theorem [Theorem 1](#thm:BP21principal){reference-type="ref" reference="thm:BP21principal"} below. Here for a group $G$, we write $k_p(G)$ to denote the number of conjugacy classes of $p$-elements of $G$. **Lemma 13**. *Let $G$ be a finite group. Then $k_p(G)\leq k(B_0(G))$. In particular, the number of chief factors of $G$ of order divisible by $p$ is at most $k(B_0(G))$.* *Proof.* Let $\{x_1,\dots,x_t\}$ be a set of representatives of the non-central conjugacy classes of $p$-elements of $G$. By [@nbook Theorem 4.14], $B_0({\mathbf{C}}_G(x_i))^G$ is defined for every $i=1,\dots,t$. By [@nbook Theorem 5.12]) and Brauer's third main theorem ([@nbook Theorem 6.7]), we have that $$k(B_0(G))=l(B_0(G))|{\mathbf{Z}}(G)|_p+\sum_{i=1}^t l(B_0({\mathbf{C}}_G(x_i)))\geq|{\mathbf{Z}}(G)|_p+t=k_p(G),$$ as wanted. ◻ Finally, we can prove Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}. *Proof of Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}.* Recall from Remark [Remark 2](#rem:k7){reference-type="ref" reference="rem:k7"} that we may assume that $k:=k(B_0(A))\geq 7$. If $S$ is a sporadic group, Tits group, group of Lie type with exceptional Schur multiplier, or alternating group $\mathfrak{A}_n$ with $n\leq 7$, then the result is readily checked using GAP and its Character Table Library [@GAP]. We therefore assume that $S$ is not one of these groups. Throughout, let $P\in{\operatorname{Syl}}_p(A)$ and $P_0\in{\operatorname{Syl}}_p(S)$ such that $P_0=P\cap S$. \(I\) If $S$ is an alternating group with $n\geq 8$, then $A\in\{\mathfrak{A}_n, \mathfrak{S}_n\}$. Note that $2k\geq k(B_0(\mathfrak{S}_n))$. Let $n=pw+r$ with $0\leq r<p$. Then we have $|P|\leq (2,p)\cdot|P_0|=(n!)_p=((pw)!)_p\leq p^{pw}$ by [\[eq:sizeSylowSw\]](#eq:sizeSylowSw){reference-type="eqref" reference="eq:sizeSylowSw"}. Further, by [@MO83 Theorem 1.10] and [@olsson84 Lemma 1 and p. 44], we have $$k(B_0(\mathfrak{S}_{n}))=k(B_0(\mathfrak{S}_{pw}))=k(p,w) >\pi(w)p\geq wp,$$ where $\pi(w)$ denotes the number of partitions of $w$. Then $$|P_0|\leq |P|\leq p^{pw}<p^{2k}\leq \left(\frac{k^4}{4}\right)^{k}$$ when combined with Lemma [Lemma 3](#lem:HSF){reference-type="ref" reference="lem:HSF"}, yielding a bound stronger than (a)-(c) in this case. From now on, we assume $S$ is a simple group of Lie type. Let $S=G/{\bf Z}(G)$, where $G={\mathbf{G}}^F$ for a simple, simply connected reductive group ${\mathbf{G}}$ and a Steinberg endomorphism $F\colon {\mathbf{G}}\rightarrow{\mathbf{G}}$, where ${\bf Z}(G)$ is the full, nonexceptional Schur covering group of $S$. Write $k_0:=k_{\operatorname{Aut}(S)}(B_0(S))$ so that $k_0\leq k$. \(II\) We will first show [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"}. First, assume $S$ is defined in characteristic $p$, so that $|P_0|=q^{|\Phi^+|}$, where $\Phi^+$ is the set of positive roots of ${\mathbf{G}}$ (see [@MT11 Proposition 24.3]). We have ${\rm Irr}(B_0(S))={\rm Irr}(S)\setminus\{\mathrm{St}_S\}$, where $\mathrm{St}_S$ is the Steinberg character (see [@CE04 Theorem 6.18]), and $k_0\geq \frac{q^r}{|{\bf Z}(G)|\cdot|\mathrm{Out}(S)|}$, as in [@HSF21 Section 2D]. Let $f$ be the integer (or half-integer, in the case of Suzuki and Ree groups ${}^2\!\operatorname{G}_2(q^2)$, ${}^2\!\operatorname{F}_4(q^2)$, ${}^2\!\operatorname{B}_2(q^2)$) such that $q=p^f$, and note that $\sqrt{q^r}=p^{rf/2}\leq p^{rf}/f=q^r/f$, unless $q=8$ and $r=1$. In the latter case, $S={\operatorname{PSL}}_2(8)$ and $|P_0|=8$, so the statement holds. So, we assume $(q,r)\neq (8,1)$. Here, we include the full argument for the groups ${\operatorname{PSL}}_n^\epsilon(q)$ ($n\geq 2$), which correspond to ${\mathbf{G}}$ of type $\operatorname{A}_{n-1}$. Table [1](#tab:defchar){reference-type="ref" reference="tab:defchar"} gives relevant values for various groups of Lie type, and from this information, the arguments in the other cases are similar. So, let $S={\operatorname{PSL}}_n^\epsilon(q)$. Then $|P_0|=q^{n(n-1)/2}$; $r=n-1$; $|{\operatorname{Out}}(S)|\leq 2f\cdot (n, q-\epsilon)\leq 2fn$; and $|{\bf Z}(G)|=(n, q-\epsilon)\leq n$. By Corollary [Corollary 9](#cor:nunips){reference-type="ref" reference="cor:nunips"}, we have $k_0\geq n$. Together, this gives $$k_0\geq \frac{q^{n-1}}{2n^2f}\geq \frac{q^{(n-1)/2}}{2n^2}\geq \frac{q^{(n-1)/2}}{2k_0^2}.$$ Then $q^{(n-1)/2}\leq 2k_0^3\leq k_0^4$, so $|P_0|=q^{n(n-1)/2}\leq k_0^{4n}\leq k_0^{4k_0}\leq k_0^{2k_0^2}$. Finally, we may assume $S$ is a group of Lie type defined in characteristic different than $p$. If $S$ is of exceptional type, then Propositions [Proposition 6](#prop:BP21except2){reference-type="ref" reference="prop:BP21except2"} and [Proposition 7](#prop:BP21except){reference-type="ref" reference="prop:BP21except"} yield [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"}. Hence, we may assume $S$ is of classical type, and we let $H, \widetilde{P}$, and $\bar{P}$ be as in Section [2.3](#sec:classical){reference-type="ref" reference="sec:classical"}. Recall that we have $|P_0|\leq |\widetilde{P}|$. If $p=2$, we further have $|\widetilde{P}|\leq |\operatorname{GL}_n(q^2)|_2\leq 2^{(b+1)n}(n!)_2\leq 2^{(b+2)n}$, where $2^{b+1}$ is the largest power of $2$ dividing $q^2-1$ and the last inequality is from [\[eq:sizeSylowSw\]](#eq:sizeSylowSw){reference-type="eqref" reference="eq:sizeSylowSw"}. In particular using Lemma [Lemma 10](#lem:boundclassicalb2){reference-type="ref" reference="lem:boundclassicalb2"} and Corollary [Corollary 9](#cor:nunips){reference-type="ref" reference="cor:nunips"}, in this case $|P_0|\leq 2^{k_0^2+k_0}<k_0^{2k_0^2}.$ Now we assume $p$ is odd. If $\widetilde{P}$ is abelian, note that $\widetilde{P}=\bar{P}$ in the notation before. Then $|P_0|\leq p^{bw}<k^{2k_0^2}$, from Lemmas [Lemma 3](#lem:HSF){reference-type="ref" reference="lem:HSF"}, [Lemma 11](#lem:boundclassicalb){reference-type="ref" reference="lem:boundclassicalb"}, and [Lemma 12](#lem:boundclassicalt){reference-type="ref" reference="lem:boundclassicalt"}, and the statement holds. We are left with the case that $S$ is classical and $t\geq 1$. Then by Lemmas [Lemma 11](#lem:boundclassicalb){reference-type="ref" reference="lem:boundclassicalb"} and [Lemma 12](#lem:boundclassicalt){reference-type="ref" reference="lem:boundclassicalt"}, along with [\[eq:sizeSylowSw\]](#eq:sizeSylowSw){reference-type="eqref" reference="eq:sizeSylowSw"}, we see that $$|P_0|\leq p^{bw}\cdot (w!)_p\leq p^{bw}\cdot p^{w}=p^{(b+1)w}\leq k_0^{k_0^2},$$ which completes the proof of [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"}. \(III\) We now complete the proof of [\[thm:BP21simplemain\]](#thm:BP21simplemain){reference-type="ref" reference="thm:BP21simplemain"}. Let $G$ be defined over $\mathbb{F}_q$, where $q=q_0^f$ for some prime $q_0$ and integer $f$. (In the case of Suzuki and Ree groups, we instead let $q^2:=q_0^f$ with $f$ an odd integer.) Further, write $f:=p^{f'}\cdot m$ with $(m,p)=1$. From part [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"}, recall that $|P_0|\leq k^{2k^2}$. Note that $|P/P_0|=|A/S|_p$ and this number is at most $p^{f'+1}$ unless $S=\operatorname{D}_n(q)$ or ${}^2\!\operatorname{D}_n(q)$ with $p=2$ and $|A/S|_2\leq 2^{f'+3}$ or $S={\operatorname{PSL}}_n^\epsilon(q)$ with $n\geq 3$ and $p\mid(n,q-\epsilon)$, in which case $|A/S|_p$ divides $2p^{b+f'}$ with $p^b\mid\mid (q-\epsilon)$. Recall that $\operatorname{Aut}(S)=\widetilde{S}\rtimes \mathcal{D}$ with $\mathcal{D}$ a group of field and graph automorphisms as before. A Sylow $p$-subgroup of $\widetilde{S}A\cap \mathcal{D}$ contains a cyclic group of size $p^{f''}$, where $f''\leq f'$ and $|\widetilde{S}A\cap\mathcal{D}|_p\leq p^{f''+1}$. Then $A$ must also contain an element of order $p^{f''}$, and hence elements of orders $p^i$ for $1\leq i\leq f''$. Then $k_p(A)\geq f''$, and hence $k\geq f''$ by Lemma [Lemma 13](#kp){reference-type="ref" reference="kp"}. Now, if $S$ is not one of the exceptions mentioned above, we have $|A|_p\leq |\widetilde{S}A|_p=|\widetilde{S}|_p\cdot |\widetilde{S}A\cap\mathcal{D}|_p\leq |P_0|\cdot p^{f''+1}$. If $S=\operatorname{D}_n(q)$ or ${}^2\!\operatorname{D}_n(q)$ with $p=2$, we have $|\widetilde{S}A|_2\leq |P_0|\cdot 2^{f''+3}$. If $S={\operatorname{PSL}}_n^\epsilon(q)$ with $n\geq 3$ and $p\mid(n,q-\epsilon)$, we have $|\widetilde{S}A|_p\leq |P_0|\cdot p^{b+f''+1}$, where $p^b\mid\mid(q-\epsilon)$. Then using [\[thm:BP21simple\]](#thm:BP21simple){reference-type="ref" reference="thm:BP21simple"} and Lemmas [Lemma 10](#lem:boundclassicalb2){reference-type="ref" reference="lem:boundclassicalb2"}, and [Lemma 11](#lem:boundclassicalb){reference-type="ref" reference="lem:boundclassicalb"}, we have in each case that $$|P|=|A|_p\leq k^{2k^2}\cdot p^{f''+k}.$$ Combining the above with Lemma [Lemma 3](#lem:HSF){reference-type="ref" reference="lem:HSF"}, we obtain $|P|\leq k^{2k^2}\cdot p^{2k}< k^{2k^2}\cdot k^{4k}$, completing the proof. ◻ [\[tab:defchar\]]{#tab:defchar label="tab:defchar"} Type of ${\mathbf{G}}$ Size of $\Phi^+$ Rank $r$ upper bound for $|{\bf Z}(G)|$ upper bound for $|{\operatorname{Out}}(S)|$ --------------------------------------------------------- ------------------ ---------- -------------------------------- --------------------------------------------- $\operatorname{A}_{n-1}$, $n\geq 2$ $n(n-1)/2$ $n-1$ $n$ $2fn$ $\operatorname{B}_n$ or $\operatorname{C}_n$, $n\geq 3$ $n^2$ $n$ 2 $2f$ $\operatorname{B}_2$ $4$ $2$ $2$ $2f$ $\operatorname{D}_n$, $n\geq 5$ $n(n-1)$ $n$ 4 $8f$ $\operatorname{D}_4$ $12$ $4$ 4 $24f$ $\operatorname{F}_4$ $24$ $4$ 1 $2f$ $\operatorname{G}_2$ $6$ $2$ 1 $2f$ $\operatorname{E}_6$ $36$ $6$ 3 $6f$ $\operatorname{E}_7$ $63$ $7$ 2 $2f$ $\operatorname{E}_8$ $120$ $8$ 1 $f$ : Relevant Data for Bounding $|P_0|$ in Defining Characteristic # Proof of Theorem A {#sec:BP21proof} In this section we complete the proof of Theorem [Theorem 1](#thm:BP21principal){reference-type="ref" reference="thm:BP21principal"}. We begin with some additional general observations that will be useful in the proof. **Lemma 14**. *Let $G$ be a finite group and let $N\trianglelefteq G$. If $b\in{\operatorname{Bl}}(N)$ is covered by $B\in{\operatorname{Bl}}(G)$ then $k(b)\leq|G:N|k(B)$.* *Proof.* This is a direct consequence of [@nbook Theorem 9.4] ◻ **Lemma 15**. *Let $G$ be a finite group and suppose that $N=S_1\times\cdots\times S_n$ is a normal subgroup, where where $S_i$ is simple nonabelian and $p$ divides $|S_i|$ for all $i$. Then $n\leq k(B_0(G))$.* *Proof.* Let $1\neq x_i\in S_i$ be a $p$-element for every $i$. Note that $G$ acts on $\{S_1,\dots, S_n\}$ by conjugation. Therefore, the elements $$(x_1,1,\dots,1), (x_1,x_2,1,\dots,1),\dots,(x_1,\dots,x_n)$$ are representatives of $n$ different conjugacy classes of $p$-elements of $G$. By Lemma [Lemma 13](#kp){reference-type="ref" reference="kp"}, $n\leq k(B_0(G))$. ◻ **Lemma 16**. *Suppose that $S_1,\dots, S_n$ are nonabelian simple groups of order divisible by a prime number $p$ and let $S_1\times\cdots\times S_n\leq G\leq\operatorname{Aut}(S_1)\times\cdots\times\operatorname{Aut}(S_n)$. Let $k=k(B_0(G))$, where $B_0(G)$ is the principal $p$-block of $G$. Then $$|G|_p\leq k^{4k^3}.$$* *Proof.* Write $A_i=\operatorname{Aut}(S_i)$ and $A=A_1\times\cdots\times A_n$. Let $\pi_i$ be the restriction to $G$ of the projection from $A$ onto $A_i$ for every $i$. Set $K_i=\operatorname{Ker}\pi_i$. Notice that $G/K_i$ is isomorphic to an almost simple group $G_i$ with socle $S_i$. Furthermore, the intersection of the $K_i's$ is trivial, so $G$ embeds into the direct product of the groups $G/K_i$. Furthermore, $B_0(G/K_i)\subseteq B_0(G)$ for every $i$. By Theorem [Theorem 1](#thm:BP21almostsimple){reference-type="ref" reference="thm:BP21almostsimple"}, we have that $$|G/K_i|_p\leq k^{4k^2}$$ for every $i$. Since $S_i$ is normal in $G$ for all $i$, by Lemma [Lemma 15](#lem:pelements){reference-type="ref" reference="lem:pelements"} we have that $n\leq k$, and hence $$|G|_p\leq\prod_{i=1}^n |G/K_i|_p\leq k^{4k^3},$$ as desired. ◻ We define the function $$f(k)=k^{k}\cdot (k!k)^{4(k!k)^3}.$$ **Theorem 17**. *Let $G$ be a finite group and let $R$ be the $p$-solvable radical of $G$. Then $|G:R|_p\leq f(k)$, where $k=k(B_0(G))$.* *Proof.* Without loss of generality, we may assume that $R=1$. Let $F={\mathbf F}^*(G)$ be the generalized Fitting subgroup, which in this case is a direct product of non-abelian simple groups of order divisible by $p$. Write $F=S_1\times\cdots\times S_n$. By Lemma [Lemma 15](#lem:pelements){reference-type="ref" reference="lem:pelements"}, we obtain that $n\leq k$. Since ${\mathbf{C}}_G(F)\leq{\mathbf{Z}}(F)=1$, it follows that $G$ embeds into $\Gamma=\operatorname{Aut}(F)$. Note that $A=\operatorname{Aut}(S_1)\times\cdots\times\operatorname{Aut}(S_n)$ is a normal subgroup of $\Gamma$ and $\Gamma/A$ is isomorphic to a subgroup of $\mathsf{S}_n$. In particular, $$|\Gamma/A|\leq n!\leq k!.$$ Put $N=G\cap A$ and note that $|G:N|\leq k!$. By the well-known Legendre's inequality, we have that $(k!)_p\leq p^k$, so $(k!)_p\leq k^k$. Write $k'=k(B_0(N))$. It follows from Lemma [Lemma 16](#proj){reference-type="ref" reference="proj"} that $$|G|_p=|G:N|_p|N|_p\leq k^{k}\cdot k'^{4k'^3}.$$ Now, Lemma [Lemma 14](#lem: boundingcovering){reference-type="ref" reference="lem: boundingcovering"} implies that $$|G|_p\leq k^{k}\cdot (k!k)^{4(k!k)^3},$$ as wanted. ◻ Recall that the socle ${\operatorname{Soc}}(G)$ of a finite group $G$ is the product of the minimal normal subgroups of $G$. We can write ${\operatorname{Soc}}(G)=A(G)\times T(G)$, where $A(G)$ is the product of the abelian minimal normal subgroups of $G$ and $T(G)$ is the product of the non-abelian minimal normal subgroups of $G$. Note that $T(G)$ is a direct product of non-abelian simple groups. Finally, we set $g(k)=2^{2^k}f(k)^k$. The following completes the proof of Theorem [Theorem 1](#thm:BP21principal){reference-type="ref" reference="thm:BP21principal"}. **Theorem 18**. *Let $G$ be a finite group. Let $k=k(B_0(G))$. Then $|G|_p\leq g(k)$.* *Proof.* Let $O_1={\mathbf O}_{p'}(G)$. Set $E_1/O_1=T(G/O_1)$, where $T(G/O_1)$ is the non-abelian part of the socle of $G/O_1$. For $i>1$, we define $O_i/E_{i-1}={\mathbf O}_{p'}(G/E_{i-1})$ and $E_i/O_i=T(G/O_i)$, so that we have a normal series $1\leq O_1\leq E_1\leq O_2\leq E_2\leq\cdots$. Note that if $O_i<E_i$ then $E_i/O_i$ is a direct product of simple groups of order divisible by $p$. By Lemma [Lemma 13](#kp){reference-type="ref" reference="kp"} we conclude that $O_{k+1}=E_{k+1}=O_{k+2}=\cdots$. Set $O=O_{k+1}$. Note that $F/O={\mathbf F}^*(G/O)$ is a $p$-group. Since ${\mathbf{C}}_{G/O}(F/O)\leq F/O$, [@fei Corollary V.3.11] implies that $B_0(G/O)$ is the unique $p$-block of $G/O$. Since $B_0(G/O)\subseteq B_0(G)$, Landau's theorem implies that $$|G/O|\leq2^{2^k}.$$ Now, for $i\leq k$, let $C_i/O_i={\mathbf{C}}_{G/O_i}(E_i/O_i)$, so that $G/C_i$ is isomorphic to a subgroup of $\operatorname{Aut}(E_i/O_i)$ that contains $E_iC_i/C_i\cong E_i/O_i$. Notice that the $p$-solvable radical of $G/C_i$ is trivial, so by Theorem [Theorem 17](#rad){reference-type="ref" reference="rad"} applied to $G/C_i$, we have that $$|E_i/O_i|_p\leq|G/C_i|_p\leq f(k).$$ It follows that $$|G|_p=|G:O|_p\prod_{i=1}^k|E_i/O_i|_p\leq 2^{2^k}f(k)^k,$$ as wanted. ◻ **Remark 19**. * Arguing in a similar way, we can see that if a finite group $G$ does not have simple groups of Lie type in characteristic different from $p$ as composition factors, then $|G:{\mathbf O}_{p'}(G)|$ can be bounded above in terms of $k(B_0(G))$. We sketch the proof. First, we know that if $p$ is a prime and $S$ is a simple group of Lie type in characteristic $p$ or an alternating group, then $|S|$ is bounded from above in terms of $|S|_p$. Therefore, the same happens for almost simple groups with socle of Lie type in characteristic $p$ or alternating. Now, let $R$ be the $p$-solvable radical of a finite group $G$. We can argue as in the proof of Theorem [Theorem 17](#rad){reference-type="ref" reference="rad"} to see that $|G:R|$ is bounded from above in terms of $k(B_0(G))$. Using Lemma [Lemma 14](#lem: boundingcovering){reference-type="ref" reference="lem: boundingcovering"}, we see that $k(B_0(R))$ is bounded from above in terms of $k(B_0(G))$. Since $R$ is $p$-solvable, ${\operatorname{Irr}}(B_0(R))={\operatorname{Irr}}(R/{\mathbf O}_{p'}(R))$. Using that ${\mathbf O}_{p'}(R)={\mathbf O}_{p'}(G)$ and Landau's theorem, we deduce that $|R:{\mathbf O}_{p'}(G)|$ is bounded from above in terms of $k(B_0(R))$. The result follows.* **Remark 20**. * We have already mentioned that the case $p=2$ of Brauer's Problem 21 was already known by [@Kulshammer-Robinson] and [@ruh]. However, this relies on Zelmanov solution of the restricted Burnside problem. As discussed in [@VZ99] the bounds that are attainable in this problem are of a magnitude that is incomprehensibly large. The bound that we have obtained for principal blocks, although surely far from best possible, is much better than any bound that relies on the restricted Burnside problem.* Recently, there has been a large interest in studying relations among (principal) blocks for different primes. For instance, what can we say about the set of irreducible characters that belong to some principal block? The groups with the property that all irreducible characters belong to some principal block were determined in [@BZ11]. As a consequence of Brauer's Problem 21 for principal blocks, we see that for any integer $k$ there are finitely many groups with at most $k$ irreducible characters in some principal block. Note that this is a strong form of Landau's theorem. In this corollary, given a prime $p$, we write $B_p(G)$ to denote the principal $p$-block of $G$. **Corollary 21**. *The order of a finite group is bounded from above in terms of $|\bigcup_p{\operatorname{Irr}}(B_p(G))|$.* *Proof.* By Theorem A, we know that for any prime $p$, $|G|_p$ is bounded from above in terms of $k(B_p(G))$. It follows that $|G|_p$ is bounded from above in terms of $|\bigcup_p{\operatorname{Irr}}(B_p(G))|$. In particular, if $p$ is a prime divisor of $|G|$, then $p$ is bounded from above in terms of $|\bigcup_p{\operatorname{Irr}}(B_p(G))|$. The result follows. ◻ # Blocks with three irreducible characters {#sec:kb3reduction} In this section, we prove Theorem C. As usual, if $B$ is a $p$-block of a finite group $G$, $l(B)$ is the number of irreducible $p$-Brauer characters in $B$. By [@K84], we know that if $k(B)=3$ and $l(B)=1$, then the defect group is cyclic of order $3$. So we are left with the case $l(B)=2$. **Lemma 22**. *Let $N\lhd G$ and let $B$ be a $p$-block of $G$ with defect group $D$. Suppose that $B$ covers a $G$-invariant block $b$ of $N$ such that $D$ is a defect group of $b$. If $b$ is nilpotent then $k(B)=k(B')$, where $B'\in{\rm Bl}({\bf N}_{G}(D))$ is the Brauer first main correspondent of $B$.* *Proof.* Since $D$ is a defect group of $b$, we have that the Harris-Knörr correspondent of $B$ (see [@nbook Theorem 9.28]) with respect to $b$ is $B'$, the Brauer first main correspondent of $B$. By the work in [@KP90] (see the explanation at the beginning of [@Rob02 Section 3], for instance) we have that $B$ and $B'$ are Morita equivalent, and hence, they have the same number of irreducible characters. ◻ We write ${\rm cd}(B)$ to denote the set of degrees of the irreducible (ordinary) characters in $B$. We write $k_0(B)$ to denote the number of irreducible (ordinary) characters of height zero in $B$. The following is Theorem [Theorem 3](#thm:kb3reduction){reference-type="ref" reference="thm:kb3reduction"}. **Theorem 23**. *Let $G$ be a finite group and let $B$ be a $p$-block of $G$. Suppose that Condition B holds for $(S,p)$ for all simple non-abelian composition factors $S$ of $G$. Let $D$ be a defect group of $B$. If $k(B)=3$, then $|D|=3$.* *Proof.* We proceed by induction on $|G|$. Notice that we may assume $D>1$ is elementary abelian by Brauer theorem (see [@nbook Theorem 3.18], for instance) and [@hkks Corollary 7.2] and that $l(B)=2$ by [@K84]. *Step 0. We may assume that $p$ is odd.* Suppose that $p=2$. By [@Lan81 Corollary 1.3(i)], if $|D|>2$ we have that $4$ divides $k_0(B)\leq k(B)=3$, which is absurd. Hence we have that $|D|=2$. But in this case we know that $k(B)=2$, by [@Br82]. This is a contradiction, so $p$ is odd. *Step 1. We may assume ${\mathbf O}_{p}(G)=1$.* Let $M={\rm \textbf{O}}_{p}(G)$, and let $\bar{B}\in {\rm Bl}(G/M)$ dominated by $B$ with defect group $D/M$ ([@nbook Theorem 9.9(b)]). Since $M$ is a $p$-group, it has just one $p$-block, the principal one, so $B$ covers $B_0(M)$. By [@nbook Theorem 9.4] if $1_M\neq \theta\in{\rm Irr}(M)$, there is $\chi\in{\rm Irr}(B)$ over $\theta$, hence $\chi$ does not lie in $\bar{B}$. Then $k(\bar{B})\leq 2$. If $k(\bar{B})=1$, then $D=M$ and $D$ is normal in $G$. In this case $|D|=3$ by [@KNST14 Theorem 4.1]. If $k(\bar{B})=2$, then by [@Br82], we have $p=2$. This contradicts Step 0. *Step 2. If $N$ is a normal subgroup of $G$, and $b$ is a $p$-block of $N$ covered by $B$, we may assume that $b$ is $G$-invariant.* Let $G_b$ be the stabilizer of $b$ in $G$. By the Fong-Reynolds correspondence ([@nbook Theorem 9.14]), if $c$ is the block of $G_b$ covering $b$ such that $c^G=B$, we have that $k(c)=k(B)=3$ and if $E$ is a defect group of $c$, then $E$ is a defect group of $B$. If $G_b<G$, by induction we are done. *Step 3. We may assume that if $N$ is a normal subgroup of $G$ and $b$ is a $p$-block of defect zero of $N$ covered by $B$, then $N$ is central and cyclic. In particular, we may assume that ${\bf Z}(G)={\bf O}_{p'}(G)$ is cyclic.* Write $b=\{\theta\}$. Since $\theta$ is of defect zero, we have that $(G,N,\theta)$ is an ordinary-modular character triple and there exists $(G^*,N^*,\theta^*)$ an isomorphic ordinary-modular character triple with $N^*$ a $p'$-group central in $G^*$ and cyclic (see [@nbook Problems 8.10 and 8.13]). Notice also that since $G^*/N^*\cong G/N$, the set of non-abelian composition factors of $G^*$ is contained in the set of non-abelian composition factors of $G$, so Condition B holds for all non-abelian composition factors of $G^*$. If $$*:{\rm Irr}(G|\theta)\rightarrow {\rm Irr}(G^*|\theta^*)$$ is the bijection given by the isomorphism of character triples and $B=\{\chi_1,\chi_2,\chi_3\}$, we have that $B^*=\{\chi_1^*,\chi_2^*,\chi_3^*\}$ is a $p$-block of $G^*$. Now, if $D^*$ is a defect group of $B^*$ and $|D^*|=3$, we claim that $|D|=3$ (notice that in this case $p=3$, so we just need to prove that $|D|=p$). Indeed, let $\chi\in{\rm Irr}(B)$ of height zero. Since isomorphism of character triples preserves ratios of character degrees and all the characters in $B^*$ are of height zero (because $D^*$ has prime order), we have $$\frac{|G:D|_p}{|N|_p}=\left(\frac{\chi(1)}{\theta(1)}\right)_p=\chi^*(1)_p=|G^*:D^*|_p=\frac{|G^*:N^*|_p}{p}=\frac{|G:N|_p}{p}.$$ Since $b$ is $G$-invariant, we have that $D\cap N$ is a defect group of $b$ by [@nbook Theorem 9.26], so $D\cap N=1$ because $b$ has defect zero. Now, we have $$|D|=\frac{|G|_p}{|G:D|_p}=\frac{|G:N|_p|N|_p}{|G:D|_p}=p,$$ as claimed. Hence we may assume that $N$ is central and cyclic. In particular, by Step 1 we have that ${\bf Z}(G)={\bf O}_{p'}(G)$ is cyclic. *Step 4. There is a unique $G$-conjugacy class of non-trivial elements in $D$.* Let $b$ be the $p$-block of $N$ covered by $B$. Since $b$ is $G$-invariant, we have that $D\cap N$ is a defect group of $b$ by [@nbook Theorem 9.26]. By a theorem of Brauer ([@nbook Theorem 5.12]) we have that $$k(B)=l(B)|{\bf Z}(G)|_p+\sum_{i=1}^k\sum_{\substack{b\in{\rm Bl}({\bf C}_{G}(x_i))\\ b^G=B}} l(b),$$ where $\{x_1,x_2,\ldots,x_k\}$ are the representatives of the non-central $G$-conjugacy classes of $p$-elements of $G$. Since $|{\bf Z}(G)|_p=1$ by Step 1, we have $$k(B)=l(B)+\sum_{i=1}^k\sum_{\substack{b\in{\rm Bl}({\bf C}_{G}(x_i))\\ b^G=B}} l(b).$$ By [@nbook Theorem 4.14], if $x_i\in D$, then there is $b\in{\rm Bl}({\bf C}_{G}(x_i))$ such that $b^G=B$. Since $l(B)=2$ and $k(B)=3$, we have that there is just one $G$-conjugacy class of non-trivial elements in $D$. *Step 5. If $N$ is a non-central normal subgroup of $G$, then $D\leq N$. In particular, if $b$ is the only block of $N$ covered by $B$, then $D$ is a defect group of $b$.* Let $b$ be the $p$-block of $N$ covered by $B$. Again, since $b$ is $G$-invariant, we have that $D\cap N$ is a defect group of $b$ (by [@nbook Theorem 9.26]). Since $D\cap N>1$ (otherwise $b$ is of defect zero and $N$ is central by Step 3), we have that there is an element $1\neq x\in D\cap N$. If $1\neq y\in D$, $y$ is $G$-conjugate to $x$ by Step 4 and thus $y\in N$, as wanted. *Step 6. If $N$ is a normal subgroup of $G$, $b$ is the block of $N$ covered by $B$ and all the irreducible characters in $b$ have the same degree, then $N$ is central.* Suppose that $N$ is not central. By Step 5 we have that $D$ is a defect group of $b$. By [@OT83 Proposition 1 and Theorem 3] we have that $D$ is abelian and has inertial index 1. By [@BP80 1.ex.3], we know that $b$ is nilpotent. Hence by Lemma [Lemma 22](#lemamoritanilpotent){reference-type="ref" reference="lemamoritanilpotent"} we have that $k(B')=k(B)=3$, where $B'$ is the Brauer first main correspondent of $B$ in ${\bf N}_{G}(D)$. If ${\bf N}_{G}(D)< G$, by induction we are done. Hence we may assume that $D\lhd G$, but this is a contradiction with Step 1. Therefore $N$ is central. *Step 7. If $N$ is a normal subgroup of $G$, and $b$ is the unique block of $N$ covered by $B$, then ${\rm Irr}(b)$ has at most three $G/{\bf C}_{G}(N)$-orbits*. Suppose that there are more than three $G/{\bf C}_{G}(N)$-orbits in ${\rm Irr}(b)$, and let $\theta_i\in{\rm Irr}(b)$ be a representative for these orbits (so there are at least four of them). By [@nbook Theorem 9.4] we can take $\chi_i\in{\rm Irr}(B)$ lying over $\theta_i$. By Clifford's theorem, the $\chi_i$ are all different. But this is a contradiction since $k(B)=3$. *Step 8. We may assume that $D$ is not cyclic*. Otherwise, by Dade's theory of blocks with cyclic defect [@Dad66], we have that $k(B)=k_0(B)=k_0(B')=k(B')$ where $B'\in{\rm Bl}({\bf N}_{G}(D)|D)$ is the Brauer correspondent of $B$, and hence we may assume that $D$ is normal in $G$. In this case we are done by Step 1. *Step 9. Write $Z={\bf Z}(G)$ and $\overline{G}=G/Z$. Then $\overline{G}$ has a unique minimal normal subgroup $\overline{K}=K/Z$, which is simple.* Let $K/Z$ be a minimal normal subgroup of $G/Z$. Since $Z={\rm \textbf{O}}_{p'}(G)$, we have that $K/Z$ is not a $p'$-group. Since ${\rm \textbf{O}}_p(G)=1$, $K/Z$ is not a $p$-group. Hence $K/Z$ is semisimple. Notice that $K/Z$ is the unique minimal normal subgroup of $G/Z$. Indeed, if $K_1/Z , K_2/Z$ are minimal normal subgroups of $G/Z$, then by Step 5, $D\subseteq K_1\cap K_2=Z ={\bf O}_{p'}(G)$ and hence $D=1$, a contradiction. Write $\overline{K}=K/Z$. Then $\overline{K}=\overline{S_1}\times\cdots\times\overline{S_t}$, where $\overline{S_i}$ is non-abelian simple and $\overline{S_i}=\overline{S_1}^{g_i}$ for some $g_i\in G$. Write $\overline{S_i}=S_i/Z$ and notice that $S_i=S_1^{g_i}$. Notice that since $S_i/{\bf Z}(S_i)=S_i/Z$ is simple, we have that $S_i'$ is a component of $G$ and hence $[S_i',S_j']=1$ whenever $i\neq j$ ([@isaacs Theorem 9.4]). Furthermore, $S_i=S_i'Z$, so $[S_i,S_j]=1$ whenever $i\neq j$. We want to show that $t=1$. By Step 5 we have that $D$ is a defect group of $b$, the only block in $K$ covered by $B$. If $D\cap S_i=1$ for all $i=1,\ldots,t$, then $D=1$, a contradiction. Hence there is $i$ such that $D\cap S_i>1$. Without loss of generality we may assume that $D\cap S_1>1$. Let $b_1$ be the only block of $S_1$ covered by $b$ and notice that, since $b_1$ is $K$-invariant, $D\cap S_1$ is a defect group of $b_1$. We claim that $D\not\subseteq S_1$. Suppose otherwise. Notice that $D^{g_i}$ is a defect group of $b^{g_i}=b$ and hence $D^{g_i}=D^k$ for some $k\in K$. Now, $D^{g_i}=D^k\subseteq S_1^{g_i}\cap S_1^k=S_i\cap S_1=Z$, which is a $p'$-group. This is a contradiction, so $D\not\subseteq S_1$. Let $1\neq x\in D\cap S_1$. If $D\cap S_i=1$ for all $i\neq 1$, we have that $D=D\cap S_1$ which is a contradiction by the previous paragraph. Hence there is $i\neq 1$ such that $D\cap S_i\neq 1$. Let $1\neq x_i\in D\cap S_i$. Now $xx_i,x\in D$ and by Step 4 we have that $x$ and $xx_i$ are $G$-conjugate, which is not possible. Hence $t=1$, as wanted. *Step 10. Final step.* Now $K'$ is a quasi simple group with center a cyclic $p'$-group. If $b$ is the unique block of $K'$ covered by $B$, we have that $D$ is a defect group of $b$ by Step 5 and hence is not cyclic elementary abelian by Step 8. We claim that $b$ is faithful. Let $X={\rm ker}(b)$. By Theorem [@nbook Theorem 6.10], we have that $X\leq Z\cap K'$. Now, let $\psi\in{\rm Irr}(b)$, then $\psi$ lies over $1_X$ and hence, there is $\chi\in{\rm Irr}(B)$ lying over $1_X$. Now, by [@nbook Theorem 9.9 (c)] we have that $k(\bar{B})=k(B)=3$, where $\bar{B}$ is the block of $G/X$ containing $\chi$. If $X>1$, by induction we obtain that $|D|=|DX/X|=3$, and we are done. Hence we may assume that $X=1$. By Condition B, there are at least four ${\rm Aut}(K')$-conjugacy classes of irreducible characters in $b$, which is a contradiction by Step 7. ◻ # On Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} {#sec:orbits} We end the paper with a discussion on Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"}. In [@MNST Theorem B], a statement similar to Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} but requiring only 3 distinct orbits is proven. Unfortunately, for groups of Lie type in non-defining characteristic, the strategy used there is not quite sufficient to obtain 4 orbits. In fact, we will see that this is not always attainable. However, here we address several situations in which we do obtain Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"}. **Proposition 24**. *Let $p\geq 3$ be prime. Let $K$ be a quasisimple group with ${\bf Z}(K)$ a cyclic $p'$-group and socle $K/{\bf Z}(K)$ a simple sporadic group, the Tits group ${}^2\!\operatorname{F}_4(2)'$, $\operatorname{G}_2(2)'$, ${}^2\!\operatorname{G}_2(3)'=\operatorname{A}_1(8)$, a simple group of Lie type with exceptional Schur multiplier, or an alternating group $\mathfrak{A}_n$ with $5\leq n\leq 13$. Let $B$ be a $p$-block for $K$ with noncyclic, positive defect. Then $|\mathrm{cd}(B)|\geq 4$, with the following exceptions when $p=3$:* - *$K=2.\mathfrak{A}_7$; $B$ is Block 3 in GAP; $|\mathrm{cd}(B)|= 3$; and $k_{\operatorname{Aut}(K)}(B)= 4$* - *$K=2.\mathfrak{A}_8$; $B$ is Block 5 in GAP; $|\mathrm{cd}(B)|= 3$; and $k_{\operatorname{Aut}(K)}(B)= 4$* - *$K=2.\mathfrak{A}_{11}$; $B$ is Block 5 in GAP; $|\mathrm{cd}(B)|= 3$; and $k_{\operatorname{Aut}(K)}(B)= 4$* - *$K=2.\mathfrak{A}_{13}$; $B$ is Block 5 in GAP; $|\mathrm{cd}(B)|= 3$; and $k_{\operatorname{Aut}(K)}(B)= 4$* - *$K={}^2\!\operatorname{G}_2(3)'=\operatorname{A}_1(8)$; $B=B_0(K)$; $|\mathrm{cd}(B)|= 3$, and $k_{\operatorname{Aut}(K)}(B)= 4$.* *In particular, Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} is true for $K$.* *Proof.* This can be seen using the GAP Character Table Library. We note that the groups with exceptional Schur multipliers are listed in [@GLS Table 6.1.3]. ◻ **Theorem 25**. *Let $p\geq 3$ be prime. Let $K$ be a quasisimple group with $K/{\bf Z}(K)\cong \mathfrak{A}_n$, an alternating group with $n>11$. Let $B$ be a $p$-block for $K$ with noncyclic, positive defect. Then $k_{\operatorname{Aut}(K)}(B)\geq 4$. In particular, Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} is true if $K$ is a covering group for $S\cong \mathfrak{A}_n$ for $n> 11$.* *Proof.* The proof here is essentially the same as that of [@MNST Proposition 3.4]. Let $\hat\mathfrak{A}_n$ and $\hat\mathfrak{S}_n$ denote the double covers, respectively, of $\mathfrak{A}_n$ and $\mathfrak{S}_n$. Recall that $\operatorname{Aut}(S)=\mathfrak{S}_n$ and $\operatorname{Aut}(\hat\mathfrak{A}_n)=\hat\mathfrak{S}_n$. Following [@Ol93], a $p$-block of $\mathfrak{S}_n$ has $k(p,w)$ ordinary irreducible characters, and a $p$-block of $\hat\mathfrak{S}_n$ lying over the nontrivial character of ${\bf Z}(\hat\mathfrak{S}_n)$ (a "spin block\") has $k^\pm(\bar p, w)$ ordinary irreducible characters, where $w$ is the so-called "weight\" of the block. We remark that our assumption that a defect group is noncyclic forces $w\geq 2$. From [@Ol93 (3.11) and Section 13], we see that these numbers are larger than 6 (and hence there are strictly more than 3 $\operatorname{Aut}(K)$-orbits represented in a given block $B$ of $K$) if $p\geq 3$ and $w\geq 2$, except for the case $(p,w)=(3,2)$ and $B$ is a spin block, in which case $k^\pm(\bar{3},2)=6$. In this case, [@Ol93 Proposition 13.19] forces at least one of the characters in the block of $\hat\mathfrak{S}_n$ to restrict to the sum of two characters of $\hat\mathfrak{A}_n$, and hence our block again contains characters from strictly more than 3 $\operatorname{Aut}(K)$-orbits. ◻ **Proposition 26**. *Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} holds for $K$ a quasisimple group with $S:=K/{\bf Z}(K)$ of Lie type defined in characteristic $p$ with a non-exceptional Schur multiplier.* *Proof.* We may assume that $K$ is not an exceptional cover of $S:=K/{\bf Z}(K)$, as the latter have been discussed in Proposition [Proposition 24](#prop:sporadic){reference-type="ref" reference="prop:sporadic"}. Now, every $p$-block of $K$ is either maximal defect or defect zero, by [@Hum Theorem.]. Hence the defect groups of $B$ are Sylow $p$-subgroups of $K$. Now, the condition that a Sylow $p$-subgroup is abelian and non-cyclic forces $S={\operatorname{PSL}}_2(p^a)$ for some integer $a\ge2$, so we may assume that $K={\operatorname{SL}}_2(p^a)$ is the Schur covering group of $S$. In this situation, the blocks of maximal defect are in bijection with the characters of ${\bf Z}(K)$. Namely, we have $B_0(K)$, which contains all members of ${\operatorname{Irr}}(K|1_{{\bf Z}(K)})\setminus \{\mathrm{St}\}$ and a second block of maximal defect containing all characters of $K$ that are nontrivial on ${\bf Z}(K)$. (See [@Hum Section 5].) By inspection (see [@GM20 Tab. 2.6]) there are four degrees for characters in $B_0(K)$, and three in the second block of maximal defect. Hence, it suffices to show that there are two semisimple characters $\chi_s$ of the same degree $q\pm1$ that are not $\operatorname{Aut}(K)$-conjugate and nontrivial on ${\bf Z}(K)$. (The latter is equivalent to $s\not\in [K^\ast, K^\ast]$ using [@DM20 Proposition 11.4.12 and Remark 11.4.14]). Since $a\geq 2$, $p^a-1$ must have at least two distinct divisors, so we consider $x_1, x_2\in C_{p^a-1}$ with these orders. Let $s_i:=\mathrm{diag}(x_i,1)\in\widetilde{K}^\ast:=\operatorname{GL}_2(p^a)$ for $i=1,2$. Note that $s_i\not\in[\widetilde{K}, \widetilde{K}]={\operatorname{SL}}_2(p^a)$ . Further, $s_1^\alpha$ cannot be conjugate to $s_2z$ for any $z\in{\bf Z}(\widetilde{K})$ and $\alpha\in\operatorname{Aut}(K)$. Hence the two semisimple characters $\chi_{s_i}$ of $\widetilde{K}$ for $i=1,2$ cannot be $\operatorname{Aut}(K)$-conjugate and restrict to distinct characters of $K$. Hence constituents of these restrictions are not $\operatorname{Aut}(K)$-conjugate. ◻ This leaves us to consider groups $S$ of Lie type in non-defining characteristic. Recall that by Proposition [Proposition 24](#prop:sporadic){reference-type="ref" reference="prop:sporadic"}, we may assume that $S$ does not have an exceptional Schur multiplier. Hence the Schur covering group of $S$ is of the form $G={\mathbf{G}}^F$, where ${\mathbf{G}}$ is a simple, simply connected algebraic group and $F\colon {\mathbf{G}}\rightarrow{\mathbf{G}}$ is a Frobenius endomorphism endowing ${\mathbf{G}}$ with an ${\mathbb{F}}_q$-rational structure, where $p\nmid q$. Given a semisimple $s\in G^\ast$ of $p'$-order, a fundamental result of Broué--Michel shows that the set $\mathcal{E}_p(G,s)$ is a union of $p$-blocks of $G$, where $\mathcal{E}_p(G,s)$ is obtained as the union of series $\mathcal{E}(G, st)$ as $t$ runs over elements of $p$-power order in ${\bf C}_{G^\ast}(s)$. (See [@CE04 Theorem 9.12].) We first dispense of the Suzuki and Ree groups. **Proposition 27**. *Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} holds when $S=K/Z$ a Suzuki, Ree, or triality group ${}^2\!\operatorname{B}_2(q)$, ${}^2\!\operatorname{G}_2(q)$, ${}^2\!\operatorname{F}_4(q)$, or ${}^3\!\operatorname{D}_4(q)$ with $p\geq 3$ a prime dividing $|S|$ and not dividing $q$.* *Proof.* Note that the Schur multiplier for $S$ is trivial or $S$ was considered already in Proposition [Proposition 24](#prop:sporadic){reference-type="ref" reference="prop:sporadic"}. Hence, we let $K=S$. Further, $\operatorname{Aut}(S)/S$ is cyclic, generated by field automorphisms. For $p\geq 3$ a prime not dividing $q^2$, the Sylow $p$-subgroups of $S={}^2\!\operatorname{B}_2(q^2)$ and $S={}^2\!\operatorname{G}_2(q^2)$ are cyclic. So, first let $K={}^2\!\operatorname{F}_4(q^2)$ with $q^2=2^{2n+1}$. Note that $K^\ast=K$ is self-dual. In this case, the semisimple classes, centralizers, and maximal tori are given in [@shinoda], and the blocks are studied in [@malle91]. First, suppose that $p\mid (q^2-1)$. Then $K$ has a unique unipotent block (namely, $B_0(K)$) with noncyclic defect group, which contains more than 3 characters of distinct degree. Similarly, there is a unique noncyclic block of positive defect in each series $\mathcal{E}(K, s)$ for $s\in\{t_1, t_2, t_3\}$, with $t_i$ as in [@shinoda], using [@malle91 Bem. 1]. The remaining blocks of positive defect are cyclic. If $s$ is one of the classes of the form $t_1$ or $t_2$, then this noncyclic block contains two characters from $\mathcal{E}(K, s)$ with distinct degrees. The centralizers ${\bf C}_{K}(s)$ contain the maximal torus ${\mathbb{Z}}_{q^2-1}^2$, from which we may obtain $t, t'\in{\bf C}_{K}(s)_p$ that are not $\operatorname{Aut}(K)$-conjugate (taking, for example, $p$-elements from classes $t_1$ and $t_2$). This yields four characters in the block that are not $\operatorname{Aut}(K)$-conjugate, as desired. For $s$ of the form $t_3$, we have ${\bf C}_{K}(s)$ is the full maximal torus ${\mathbb{Z}}_{q^2-1}^2$, and for any $t\in{\bf C}_{K}(s)_p$, we have ${\bf C}_{K}(st)={\bf C}_{K}(s)$. Hence we see that every irreducible character in this block has the same degree. When $p\nmid (q^2-1)$, each $\mathcal{E}_p(K, s)$ contains at most one block of positive defect (see [@malle91 Bem. 1]). First, assume $p\mid (q^2+1)$. Here, the noncyclic blocks correspond to $s\in\{t_4, t_5, t_{14}\}$. The set $\mathcal{E}(K, s)$ contains 3, 2, 1 distinct character degrees, respectively, in these cases, and each ${\bf C}_{K}(s)$ contains the maximal torus ${\mathbb{Z}}_{q^2+1}^2$. As before, there is only one character degree in the block in the latter case. In the other cases, we argue analogously to the previous paragraph to obtain four characters in the block that are not $\operatorname{Aut}(K)$-conjugate. If instead $p\mid (q^4+1)$, then there are three distinct character degrees in $\mathcal{E}(K, s)$ with $s\in\{t_7, t_9\}$. Then considering any character in $\mathcal{E}(K, st)$ with $t\in{\bf C}_{K}(s)_p$, we obtain a fourth character in the block that is not $\operatorname{Aut}(K)$-conjugate to these three. If instead $s\in\{t_{12}, t_{13}\}$, we obtain as before that every character in the block has the same degree. The remaining blocks in this case have cyclic defect groups. Now, let $K={}^3\!\operatorname{D}_4(q)$. In this case, the blocks have been studied in [@deriziotismichler]. Using the results there, we may argue analogously to the situation above. ◻ In the remaining cases, we would hope to appeal to the strategy employed in [@MNST Section 3]. Namely, with the above results, the results of loc. cit. largely reduce the problem of proving Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} to the following: **Condition 28**. *Let ${\mathbf{H}}$ be a simple, simply connected reductive group and $F\colon {\mathbf{H}}\rightarrow{\mathbf{H}}$ be a Frobenious morphism and $H={\mathbf{H}}^F$ the corresponding finite group of Lie type. Let $B$ be a quasi-isolated $p$-block of $H$ with an elementary abelian defect group $D$. Then $$k_{\operatorname{Aut}(H)}(B)\geq \left\{\begin{array}{cc} 4 & \hbox{ if $D$ is not cyclic}\\ 3 & \hbox{ if $D$ is cyclic}\\ \end{array}\right.$$* Indeed, from our above results, we may assume that $S$ is a group of Lie type defined in characteristic distinct from $p$ and that the Schur covering group for $S$ is $G={\mathbf{G}}^F$ where ${\mathbf{G}}$ is a simple, simply connected group with $F$ a Frobenius endomorphism. Note then that $K$ is a quotient of $G$ by some central subgroup and that, from our assumption that $p\nmid Z$ in Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"}, [@nbook Theorem 9.9(c)] tells us that it suffices to prove Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} when $K=G/{\bf Z}(G)_{p}$, where ${\bf Z}(G)_p$ is the Sylow $p$-subgroup of ${\bf Z}(G)$. Our assumption that $p\geq 3$ then means that we may assume that $K=G$ unless $S={\operatorname{PSL}}_n^\epsilon(q)$ with $p\mid (q-\epsilon)$ or $S=\operatorname{E}^\epsilon_6(q)$ with $p=3\mid (q-\epsilon)$. Then indeed, when $K=G$, Condition [Condition 28](#cond:MNSTorbitfurtherreduction){reference-type="ref" reference="cond:MNSTorbitfurtherreduction"} implies Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} by [@MNST Proposition 3.9 and Lemma 3.12] (see also Remarks 3.10, 3.11 of Loc. Cit.) and the fact that Bonnafe--Rouquier correspondence preserves isomorphism types of defect groups when the defect is abelian, by [@KM13 Theorem 7.16]. Of course, in the cases of $S={\operatorname{PSL}}_n^\epsilon(q)$ with $p\mid (q-\epsilon)$ or $S=\operatorname{E}^\epsilon_6(q)$ with $p=3\mid (q-\epsilon)$, some additional work is needed, as was the case in [@MNST Prop. 3.16(b) and Theorem 3.21]. However, this method is not quite sufficient for completing the proof of Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"}. Unfortunately, although we see by following the proofs in [@MNST Sections 3.4-3.6] that Condition [Condition 28](#cond:MNSTorbitfurtherreduction){reference-type="ref" reference="cond:MNSTorbitfurtherreduction"} holds in many situations, it turns out that there are indeed cyclic quasi-isolated blocks with $k_{\operatorname{Aut}(G)}(B)=2$. (This is already pointed out in [@MNST] after the statement of Theorem 3.1.) We list some additional situations in the following: *Example 29*. Let $G={\operatorname{SL}}_n^\epsilon(q)$ and let $\widetilde{G}=\operatorname{GL}_n^\epsilon(q)$. Let $p$ be an odd prime not dividing $q$ and let $B$ be a $p$-block of $G$ with positive defect. The following situations lead to exceptions to Condition [Condition 28](#cond:MNSTorbitfurtherreduction){reference-type="ref" reference="cond:MNSTorbitfurtherreduction"}: 1. If $B$ is noncyclic: - $n=3$, $p=5\mid\mid(q-\epsilon)$, and $B$ lies under a block $\widetilde{B}$ of $\widetilde{G}=\operatorname{GL}_3^\epsilon(q)$ indexed by a semisimple $5'$-element $\widetilde{s}\in \widetilde{G}$ with ${\bf C}_{\widetilde{G}}(\widetilde{s})\cong C_{q-\epsilon}^3$. In this case, $k_{\operatorname{Aut}(G)}(B)\geq 3$ (and equality can occur) and $|\mathrm{cd}({\widetilde{B}})|=1$. Note: this includes the quasi-isolated block of ${\operatorname{SL}}_3(q)$ under the block indexed by the semisimple element $\mathrm{diag}(1,\zeta_3, \zeta_3^{-1})$, where $|\zeta_3|=3$. (See also Remark [Remark 30](#rem:counterexample){reference-type="ref" reference="rem:counterexample"}.) - $n=3$, $p=3\mid\mid(q-\epsilon)$, and $B$ lies under a block $\widetilde{B}$ of $\widetilde{G}=\operatorname{GL}_3^\epsilon(q)$ indexed by a semisimple $3'$-element $\widetilde{s}\in \widetilde{G}$ with ${\bf C}_{\widetilde{G}}(\widetilde{s})\cong C_{q-\epsilon}^3$. In this case, $k_{\operatorname{Aut}(G)}(B)\geq 3$ (and equality can occur), $|{\bf Z}(G)|=3$, and the block of $S$ contained in $B$ is cyclic. - $p=3\mid\mid(q+\epsilon)$ and $B$ lies under a block $\widetilde{B}$ of $\widetilde{G}=\operatorname{GL}_n^\epsilon(q)$ with defect group $C_{3}^2 \leq C_{q+\epsilon}^2$. 2. If $B$ is cyclic: - $n=2$, $p\mid\mid(q-\epsilon)$, and $B$ lies under a block $\widetilde{B}$ of $\widetilde{G}=\operatorname{GL}_2^\epsilon(q)$ indexed by a semisimple $p'$-element $\widetilde{s}\in \widetilde{G}$ with ${\bf C}_{\widetilde{G}}(\widetilde{s})\cong C_{q-\epsilon}^2$. Here $k_{\operatorname{Aut}(G)}(B)\geq 2$ (and equality can occur). Note: this includes the quasi-isolated block of ${\operatorname{SL}}_2(q)$ under the block indexed by the semisimple element $\mathrm{diag}(-1,1)$. - $B$ lies under a block $\widetilde{B}$ of $\widetilde{G}$ indexed by a semisimple $p'$-element with ${\bf C}_{\widetilde{G}}(\widetilde{s})\cong C_{q^\delta-\eta}$ where $\delta=n$ and $p\mid\mid(q^\delta-\eta)$. Here $k_{\operatorname{Aut}(G)}(B)\geq 2$ (and equality can occur). *Remark 30*. We remark that some of the exceptions given in Example [Example 29](#ex:typeAexceptions){reference-type="ref" reference="ex:typeAexceptions"} mean that Condition [Condition 28](#cond:MNSTorbitfurtherreduction){reference-type="ref" reference="cond:MNSTorbitfurtherreduction"} is not always feasible. Further, the first exception of (I) yields examples of $5$-blocks with $k_{\operatorname{Aut}(S)}(B)=3$, meaning that Condition [Condition 2](#quasisimples2){reference-type="ref" reference="quasisimples2"} will also not always hold. For example, let $K={\operatorname{SL}}_3(q)$ where $q=q_0^4$, $q_0\equiv 2\pmod 3$, $q_0\equiv 3\pmod 5$, and let $p=5\mid\mid(q-1)$. Let $\zeta\in{\mathbb{F}}_q^\times$ with $|\zeta|=3$. Then a block $B$ of $K$ lying below the (unique) block $\widetilde{B}$ of $\operatorname{GL}_3(q)$ in $\mathcal{E}_5(\operatorname{GL}_3(q), s)$ with $s=\mathrm{diag}(\zeta, \zeta^{-1}, 1)$ satisfies $k_{\operatorname{Aut}(S)}(B)=3$ and $|D|=C_5\times C_5$. 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[^1]: The authors thank the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge and the organizers of the Summer 2022 INI program Groups, Representations, and Applications: New Perspectives, supported by EPSRC grant EP/R014604/1, where part of this work was completed. The second and third-named authors also thank the National Science Foundation Grant No. DMS-1928930, which supported them while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Summer of 2023. The first and second-named authors are supported by Ministerio de Ciencia e Innovación (Grants PID2019-103854GB-I00 and PID2022-137612NB-I00 funded by MCIN/AEI/10.13039/501100011033 and "ERDF A way of making Europe\" ). The first-named author also acknowledges support by Generalitat Valenciana CIAICO/2021/163. The second-named author is supported by a CDEIGENT grant CIDEIG/2022/29 funded by Generalitat Valenciana. The third-named author also gratefully acknowledges support from the National Science Foundation, Award No. DMS-2100912, and her former institution, Metropolitan State University of Denver, which holds the award and allows her to serve as PI. She also thanks the first and second authors and the CARGRUPS research team at U. Valencia for a productive stay in March 2023. The authors thank A. Maróti and G. Navarro for many useful conversations on Theorem C

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arxiv_math

--- author: - Oliver Hinder bibliography: - master.bib title: Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs --- # Introduction Consider the following linear program: [\[primal-LP\]]{#primal-LP label="primal-LP"} \_x \^m_2 c\^x\ Ax = b\ x and its dual: \_y \^m_1 b\^y\ A\^y c where $m_1$ and $m_2$ are positive integers, $x$ and $y$ are the primal and dual variables, and $A \in \mathbb{R}^{m_1\times m_2}$, $c \in \mathbb{R}^{m_2}$, $b \in \mathbb{R}^{m_1}$ are the problem parameters. Traditional methods for solving linear programs such as simplex and interior point methods require linear system factorizations, which have high memory overhead and are difficult to parallelize. Recently, there has been interest in using first-order methods for solving linear programs that use matrix-vector multiplication as their key primitive [@lin2021admm; @applegate2022faster; @o2016conic]. The advantage of matrix-vector multiplication is that it can be efficiently parallelized across multiple cores or machines. Moreover, the methods have a low memory footprint, using minimal additional memory beyond storing the problem data. These properties make these first-order methods suitable for tackling extreme-scale problems. These methods reformulate finding a primal and dual optimal solution to the linear program [\[primal-LP\]](#primal-LP){reference-type="eqref" reference="primal-LP"} as solving a minimax problem: $$\label{eq:poi-primal-dual} \min_{x \ge 0}\max_{y \in \mathbb{R}^{m_1}} \mathcal{L}(x,y) = c^\top x + b^\top y - y^\top A x$$ and then apply methods designed for solving minimax problems such as primal-dual hybrid gradient (PDHG) [@chambolle2011first] or the alternating direction method of multipliers (ADMM) [@douglas1956numerical]. See @applegate2021practical and @applegate2022faster for a more exhaustive introduction and list of references on this topic. Empirically, a promising first-order method for solving linear programs is PDLP [@applegate2021practical]. PDLP is based on restarted PDHG, combined with several other heuristics, for example, for preconditioning and adaptively choosing the primal and dual step sizes. Restarted PDHG (and other primal-dual algorithms) were analyzed by @applegate2022faster on linear programs, but their convergence bounds depend on the Hoffman constant of the KKT system. This bound is difficult to interpret[^1] and is not easily computable. In this paper, we extend the analysis of @applegate2022faster to provide an explicit complexity bound when this method is applied to totally unimodular linear programs [@hoffmann1956integral]. Totally, unimodular linear programs are an important subclass of linear programs which, for any integer right hand side and objective coefficients, all extreme points are integer. This subclass is of particular interest to the integer programming community [@conforti2014integer]. It also encapsulates the minimum cost flow problem, an important subclass of linear programs, for which almost linear-time algorithms exist [@chen2022maximum]. This work analyzes a general-purpose linear programming method on a specialized problem. Many papers perform this style of analysis for the simplex method. For example, even though the simplex method has worst-case exponential runtime on general linear programs, improved guarantees for the simplex method exist for subclasses such as Markov decision processes [@ye2011simplex; @post2015simplex], minimum cost flow [@orlin1997polynomial; @dantzig1951application] and totally unimodular linear programs [@kitahara2013bound; @mizuno2016simplex]. In particular, @kitahara2013bound shows that the number of iterations of simplex method to find an exact optimal solution on a totally unimodular linear program with nondegenerate primal is $m_2\lceil m_1\| b \|_1 \log( m_2\| b \|_1) \rceil$. Better complexities for this problem can be achieved by interior point methods [@lee2014path], although at the cost of potentially much higher memory usage. #### Notation Let $\mathbb{R}$ be the set of real numbers and $\mathbb{N}$ be the set of natural numbers starting from one. Denote $\{ 1, \dots, m \}$ by $[ m ]$. Let $\textbf{nnz}(A)$ be the number of nonzeros in $A$. Assume $m_2\ge m_1$ and that $\abs{b_i} \le H$ for all $i \in [m_1]$ and $\abs{c_j} \le H$ for all $j \in [m_2]$. Let $\| \cdot \|_2$ be the Euclidean norm for vector and spectral norm for a matrix. Let $\sigma_{\min}(M) := \min_{\| v \|_2 = 1} \| M v \|_2$ be the minimum singular value of a matrix $M$, $Z = \{ x \in \mathbb{R}^{m_2} : x \ge 0 \} \times \mathbb{R}^{m_1}$, $W_r(z) := \{ \hat{z} \in Z : \| z - \hat{z} \|_2 \le r \}$, and $\mathop{\bf dist{}}(z, Z) := \min_{\hat{z} \in Z} \| z - \hat{z} \|_2$. Let $X^\star$ be the set of optimal primal solutions to [\[primal-LP\]](#primal-LP){reference-type="eqref" reference="primal-LP"} and $Y^\star$ be the set of optimal dual solutions. Define $Z^{\star}= X^\star\times Y^\star$. Let $e_i$ be a vector containing a one in the $i$th entry and zero elsewhere. Let $\mathbf{1}$ be a matrix of ones and $\mathbf{0}$ be a matrix of zeros. #### Paper outline provides background on restarted PDHG, provides a new Hoffman bound that we will find useful and proves the main result. # Background on restarted PDHG {#sec:restarted-PDHG} This section introduces concepts from @applegate2022faster that will be useful for our analysis. For ease of exposition we specialized PDHG to linear programming (Algorithm [\[alg:one-step-PDHG\]](#alg:one-step-PDHG){reference-type="ref" reference="alg:one-step-PDHG"}). See @chambolle2011first for the general PDHG formula. A key concept for restarted PDHG is the *normalized duality gap* [@applegate2022faster] defined for $r > 0$ as $$\rho_r(z) := \frac{\max_{\hat{z} \in W_r(z)} \mathcal{L}(x,\hat{y}) - \mathcal{L}(\hat{x}, y)}{r}$$ where for conciseness we use the notation $(\hat{x},\hat{y}) =\hat{z}$ (this notation is used through the paper, i.e., we also have $(x,y) = z$), and for completeness define $\rho_0(z) := \limsup_{r \rightarrow 0^{+}} \rho_r(z)$. The normalized duality gap is preferable over the standard duality gap, $\max_{z \in Z} \mathcal{L}(x,\hat{y}) - \mathcal{L}(\hat{x}, y)$, because when $Z$ is unbounded the standard duality gap can be infinite even when we are arbitarily close to an optimal solution in both the primal and dual. Next, we introduce the definition of a primal-dual problem being sharp (Definition [Definition 1](#def:sharp-pd-problem){reference-type="ref" reference="def:sharp-pd-problem"}) and PDHG with adaptive restarts (Algorithm [\[alg:restarted-pdhg\]](#alg:restarted-pdhg){reference-type="ref" reference="alg:restarted-pdhg"}) along with three results from @applegate2022faster that we will use in this paper. **Definition 1** (Definition 1 of @applegate2022faster). *We say a primal-dual problem [\[eq:poi-primal-dual\]](#eq:poi-primal-dual){reference-type="eqref" reference="eq:poi-primal-dual"} is $\alpha$-sharp on the set $S \subseteq Z$ if for all $r \in (0, \mathop{\bf diam{}}(S)]$ and $z \in S$ that $\alpha \mathop{\bf dist{}}(z, Z^{\star}) \le \rho_r(z)$.* **Input:** $z^{0,0}, \tau^{0}$\ **Lemma 1**. *Algorithm [\[alg:restarted-pdhg\]](#alg:restarted-pdhg){reference-type="ref" reference="alg:restarted-pdhg"} for $z^{0,0} \in Z$ satisfies $z^{n,0} \in W_{\theta \mathop{\bf dist{}}(z^{0,0}, Z^{\star})}(z^{0,0})$ for $\theta = 2 \sqrt{\frac{1 + \eta \| A \|_2}{1 - \eta \| A \|_2}}$.* *Proof.* Proposition 9 of @applegate2022faster (which uses the norm $\| z \|_{\eta A} := \| x \|_2^2 - 2 \eta x^\top A y + \| y \|_2^2$) states that $\| z^{n,0} - z^{0,0} \|_{\eta A} \le 2 \| z^{0,0} - z^\star \|_{\eta A}$ for any starting point $z^{0,0} \in Z$ and optimal solution $z^\star \in Z^{\star}$. Proposition 7 of @applegate2022faster states that $(1 - \eta \| A \|_2 ) \| z \|_2^2 \le \| z \|_{\eta A} \le (1 + \eta \| A \|_2 ) \| z \|_2^2$ for all $z \in Z$. Combining these two statements yields $$(1 - \eta \| A \|_2) \| z^{n,0} - z^{0,0} \|_2^2 \le \| z^{n,0} - z^{0,0} \|_{\eta A}^2 \le 2^2 \| z^{0,0} - z^\star \|_{\eta A}^2 \le 2^2 (1 + \eta \| A \|_2) \| z^{0,0} - z^\star \|_2^2$$ for all $z^\star \in Z^{\star}$. Rearranging gives the result. ◻ **Theorem 1**. *Consider the sequence $\{z^{n,0}\}_{n=0}^{\infty}$, $\{\tau^{n} \}_{n=1}^{\infty}$ generated by Algorithm [\[alg:restarted-pdhg\]](#alg:restarted-pdhg){reference-type="ref" reference="alg:restarted-pdhg"} with $\eta \in (0, 1/ \| A \|_2)$ and $\beta \in (0,1)$. Suppose that there exists a set $S \subseteq Z$ such that $z^{n,0} \in S$ for any $n \ge 0$ and the primal-dual problem [\[eq:poi-primal-dual\]](#eq:poi-primal-dual){reference-type="eqref" reference="eq:poi-primal-dual"} is $\alpha$-sharp on the set $S$. Then, it holds for each outer iteration $n \in \mathbb{N}$ that* - *The restart length, $\tau^{n}$, is upper bounded by $t^{\star}$:* *\^n t\^ :=* *with $C := \frac{2}{\eta (1 - \eta \| A \|_2)}$ and $q := 4 \frac{1 + \eta \| A \|_2}{1 - \eta \| A \|_2}$.* - *The distance to the primal-dual optimal solution set decays linearly:* *(z\^n,0,Z\^) \^n (z\^0,0, Z\^).* *Proof.* See Theorem 2 and Corollary 2 of @applegate2022faster. ◻ **Lemma 2**. *For all $R \in (0,\infty)$, $z \in Z$, $r \in (0,R]$ with $\| z \|_2 \le R$ we have $$\frac{1}{2} \left\| \begin{pmatrix} \frac{1}{R} (c^\top x - b^\top y )^{+} \\ A x - b \\ (A^\top y - c)^{+} \end{pmatrix} \right\|_2 \le \rho_r(z).$$* *Proof.* This is a variant of Lemma 4 of @applegate2022faster. To prove the result it suffices to combine Equation (22) and (25) of @applegate2022faster. ◻ # A Hoffman bound that explicitly takes into account nonnegativity and inequality constraints {#sec:hoffman-bound} Hoffman bounds guarrantee how much the distance to feasibility decreases as the constraint violation decreases. Typical Hoffman bounds consider a linear inequality system of the form: $K z \le k$ where $K$ is a matrix and $k$ is a vector. For example, [@guler1995approximations Theorem 4.2.] states that for any matrix $K$ and vector $k$ we have: $$\mathop{\bf dist{}}(z, \{ z : K z \le k \}) \le \alpha \| (K z - k)^{+} \|_2$$ where $\left( \cdot \right)^{+} = \max\{ \cdot, \mathbf 0\}$ where the max operator is applied element-wise and $\alpha > 0$ is the minimum singular value across all nonsingular submatricies of $K$. In @applegate2022faster the authors employ Hoffman bounds show that $$\left\| \begin{pmatrix} (c^\top x - b^\top y )^{+} \\ A x - b \\ (A^\top y - c)^{+} \end{pmatrix} \right\|_2$$ is bounded below by a constant times the distance to optimality. Using , this establishes that $\rho_r$ is sharp. is a Hoffman bound that explicitly handles both inequality and nonnegativity constraints. Similar types of Hoffman bounds exist in the literature (e.g., [@pena2021new]) but we were unable to find a bound that could be readily adapted to our purpose. The proof of is a blackbox reduction to standard Hoffman bounds [@guler1995approximations Theorem 4.2.]. In contrast, @applegate2022faster treats the nonnegativity constraints as generic inequality constraints. However, in this paper we take advantage of the fact that the nonnegativity constraints on $x$ are never violated (due to PDHG performing a projection). Explicitly handling these nonnegativity constraints improves the quality of the Hoffman constant. In particular, the nonsingular submatricies considered in the calculation are smaller (because they do not contain the identity block that [@applegate2022faster Equation (20)] introduces). If we did employ the strategy of @applegate2022faster the remainder of the paper would remain essentially unchanged but the iteration bound in Theorem [Corollary 2](#thm:main-result){reference-type="ref" reference="thm:main-result"} would contain $m_2+ m_1$ instead of just $m_2$ because the nonsingular submatrix $G$ could be much bigger. This alternative worst-case bound is inferior for $m_2\gg m_1$. **Corollary 1**. *For any nonnegative integers $m, n$ and $q$ consider matricies $D\in \mathbb{R}^{m \times n}$, $F\in \mathbb{R}^{p \times n}$ and vectors $d \in \mathbb{R}^{m}$, $f \in \mathbb{R}^{p}$. Let $U:= \{ u\in \mathbb{R}^{n} : u_i \ge 0, \forall i \in S \}$ for some $S \subseteq [ n ]$. Define the polytope, $P:= \{ u\in U: Du\le d, Fu= f\}$. Let $\mathcal{G}$ be the set of all nonsingular submatricies of the matrix $\begin{pmatrix} D\\ F\end{pmatrix}$ and let $\alpha= \frac{1}{\max_{G \in \mathcal{G}} \| G^{-1} \|_2}$. Then, for all $u\in U$ we have $$\alpha\mathop{\bf dist{}}( u, P) \le \left\| \begin{pmatrix} (Du- d)^{+} \\ Fu- d \end{pmatrix} \right\|_2.$$* *Proof.* Consider the system $$\left[ K_{\lambda} := \begin{pmatrix} D\\ F\\ -F\\ -\lambda E \end{pmatrix} \right] u\le \begin{pmatrix} d\\ f\\ -f\\ \mathbf 0\end{pmatrix} =: k$$ for $\lambda > 0$, where $-\lambda E u\le \mathbf 0$ corresponds to the constraint $u\in U$ with each row of $E$ containing exactly one nonzero entry (either $-1$ or $1$). Let $S$ be a nonsingular submatrix of $K_{\lambda}$ decomposed into submatricies of $D$, $F$, $-F$ and $E$ which we call $S_{D}, S_{F}, S_{-F}, S_{E}$ respectively such that $$S = \begin{pmatrix} S_{D} \\ S_{F} \\ S_{-F} \\ -\lambda S_{E} \end{pmatrix}.$$ As each row of $E$ contains exactly one nonzero entry, each row of $S_E$ must contain at most one nonzero entry. Since $S$ is nonsingular it follows that $S_{E}^\top v \neq \mathbf 0$ for all $v \neq \mathbf 0$. By choosing $v = e_i$ for each row $i$, we deduce each row of $S_{E}$ must contain exactly one nonzero entry. Therefore there exists matricies $M_{11}, M_{12}$ and $M_{22}$ such that $M_{11}$ and $M_{12}$ are submatricies of $\begin{pmatrix} D\\ F\\ -F \end{pmatrix}$, $M_{22}$ is a nonsingular square submatrix of $S_{E}$, and $$\Pi S = \begin{pmatrix} M_{11} & M_{12} \\ \mathbf 0& \lambda M_{22} \end{pmatrix}$$ where $\Pi$ is some perturbation matrix. Let $\mathcal{M}$ be the set of all matricies $M$ such that $M_{11}$ and $M_{12}$ are submatricies of $\begin{pmatrix} D\\ F\\ -F \end{pmatrix}$, and $M_{22}$ is a square nonsingular submatrix of $E$. Note that since $S$ is square and $M_{22}$ is square we must have $M_{11}$ is square. Applying [@guler1995approximations Theorem 4.2.] gives $$\alpha_{\lambda} \mathop{\bf dist{}}( u, P) \le \left\| (K x - k)^{+} \right\|_2 \quad \forall u\in U$$ for $$\alpha_{\lambda} := \max_{M \in \mathcal{M}_{\lambda}} \left\| \begin{pmatrix} M_{11} & M_{12} \\ \mathbf 0& \lambda M_{22} \end{pmatrix}^{-1} \right\|_2.$$ It follows by Lemma [Lemma 6](#lem:linear-algebra-schur-fact){reference-type="ref" reference="lem:linear-algebra-schur-fact"} (see ) that $$\lim_{\lambda \rightarrow \infty} \alpha_{\lambda} = \lim_{\lambda \rightarrow \infty} \max_{M \in \mathcal{M}} \left\| \begin{pmatrix} M_{11} & M_{12} \\ \mathbf 0& \lambda M_{22} \end{pmatrix}^{-1} \right\|_2 = \max_{M \in \mathcal{M}} \lim_{\lambda \rightarrow \infty} \left\| \begin{pmatrix} M_{11} & M_{12} \\ \mathbf 0& \lambda M_{22} \end{pmatrix}^{-1} \right\|_2 = \max_{M \in \mathcal{M}} \| M_{11}^{-1} \|_2. %$$ Recall that $M_{22}$ is square, and also that $S$ and therefore $\Pi S$ is nonsingular. Therefore by Lemma [Lemma 6](#lem:linear-algebra-schur-fact){reference-type="ref" reference="lem:linear-algebra-schur-fact"} we deduce $M_{11}$ is nonsingular and consequently either the row $-F_{i,\cdot}$ or $F_{i,\cdot}$ appears in $M_{11}$. Moreover, negating rows of $M_{11}$ does not effect its maximum singular value. Therefore, there exists $G \in \mathcal{G}$ such that $\| G^{-1} \|_2 = \| M_{11}^{-1} \|_2$. ◻ # Analysis of restarted PDHG on totally unimodular linear programs {#sec:analysis-restarted-PDHG} This section analyzes the worst-case convergence rate of restarted PDHG on totally unimodular linear programs. For completeness we first define what it means for a linear program to be totally unimodular. **Definition 2** ([@hoffmann1956integral]). *A matrix $A$ is totally unimodular if every square non-singular submatrix is unimodular (has determinant $0$, $1$ or $-1$). The linear program [\[eq:poi-primal-dual\]](#eq:poi-primal-dual){reference-type="eqref" reference="eq:poi-primal-dual"} is totally unimodular if $A$ is totally unimodular and the entries of $c$ and $b$ are all integers.* The following standard result will be useful. **Lemma 3**. *If $A \in \mathbb{R}^{m_1\times m_2}$ is a totally unimodular matrix then (i) $[A ~ e_i]$ for any $i \in [m_1]$, (ii) $A^\top$ and (iii) $\begin{pmatrix} A & 0 \\ 0 & A^\top \end{pmatrix}$ are totally unimodular matricies. Moroever, (iv) if $A$ is a nonsingular then $(A^{-1})_{ij}\in \{-1, 0, 1 \}$ for all $i \in [m_1]$, $j \in [m_2]$ and $\| A^{-1} \|_2 \le m_1$.* *Proof.* The proof of (i)-(iii) appears in @seymour1986decomposition. To see (iv) note by Cramer's rule and Lemma [Lemma 3](#lem:total-unimodularity-preserved){reference-type="ref" reference="lem:total-unimodularity-preserved"}.i the inverse of a square nonsingular totally unimodular matrix has all entries equal to either $-1$, $0$ or $1$, and therefore $\| A^{-1} \|_2 \le \| A^{-1} \|_F \le m_1$. ◻ The key insight of this paper is Lemma [Lemma 4](#lem:spectral-norm-inverse){reference-type="ref" reference="lem:spectral-norm-inverse"} which allows us to reason about matricies that are both nonsingular and after removing one row, totally unimodular. The proof uses Lemma [Lemma 3](#lem:total-unimodularity-preserved){reference-type="ref" reference="lem:total-unimodularity-preserved"} to decompose the matrix into the sum of a totally unimodular matrix and a rank one component. The Sherman-Morrison formula is used to analyze the inverse and therefore spectral norm of the inverse. **Lemma 4**. *Suppose the matrix $\begin{pmatrix} v^\top\\ V \end{pmatrix}$ is nonsingular where $V$ is a totally unimodular matrix with $n$ rows, and $v$ is a vector of length $n+1$. Assume there exists a positive integer $M$ such that for each $i \in [n+1]$ there exists a integer $k_i$ such that $v_i = k_i / M$. Then $$\left\| \begin{pmatrix} v^\top\\ V \end{pmatrix}^{-1} \right\|_2 \le n+ 1 + M( (n+ 1)^{1.5} \| v\|_2 + n+ 1 ).$$* *Proof.* First observe that $\begin{pmatrix} v^\top\\ V \end{pmatrix} = \begin{pmatrix} e_j^\top\\ V \end{pmatrix} + e_1 (v- e_j)^\top$ where $j$ is choosen such that $W:= \begin{pmatrix} e_j^\top\\ V \end{pmatrix}$ is nonsingular[^2]. Note that by Lemma [Lemma 3](#lem:total-unimodularity-preserved){reference-type="ref" reference="lem:total-unimodularity-preserved"}, $W$ is totally unimodular. By the Sherman-Morrison formula [@sherman1950adjustment]: [\[eq:apply-sherman-morrison\]]{#eq:apply-sherman-morrison label="eq:apply-sherman-morrison"} v\^\ V \^-1 = W\^-1 - . As $W^{-1} e_1$ is an integer vector, and $v_i = k_i / M$ where $k_i$ is an integer and $M$ is a nonnegative integer then there exists some integer $z$ such that $1 + (v- e_j)^\top W^{-1} e_1 = z / M$. Since $1 + (v- e_j)^\top W^{-1} e_1 \neq 0$ it follows that [\[eq:M-inv-bound\]]{#eq:M-inv-bound label="eq:M-inv-bound"} . Using .iv we have $\| W^{-1} \|_2 \le n+ 1$, $\| W^{-1} e_1 vW^{-1} \|_2 \le \| W^{-1} e_1 \|_2 \| vW^{-1} \|_2 \le (n+ 1)^{0.5} \| W^{-1} \|_2 \| v\|_2 \le (n+ 1)^{0.5} (n+ 1) \| v\|_2$ and $\| W^{-1} e_1 e_j^\top W^{-1} \|_2 \le \| W^{-1} e_1 \|_2 \| W^{-1} e_j \|_2 \le n+ 1$. Therefore, by [\[eq:apply-sherman-morrison\]](#eq:apply-sherman-morrison){reference-type="eqref" reference="eq:apply-sherman-morrison"} and [\[eq:M-inv-bound\]](#eq:M-inv-bound){reference-type="eqref" reference="eq:M-inv-bound"} we get $$\left\| \begin{pmatrix} v^\top\\ V \end{pmatrix}^{-1} \right\|_2 \le n+ 1 + M((n+ 1)^{1.5} \| v\|_2 + n+ 1).$$ ◻ characterizes the sharpness constant of the normalized duality gap. The proof uses and . **Lemma 5**. *Let $R := \lceil 8 m_1^{1.5} H\rceil$. If [\[primal-LP\]](#primal-LP){reference-type="eqref" reference="primal-LP"} is a totally unimodular linear program with an optimal solution then there exists $\alpha > 0$ such that $$\alpha= \Omega\left( \frac{1}{m_1^{2.5} H} \right)$$ and $\alpha\mathop{\bf dist{}}(z, Z^{\star}) \le \rho_r(z)$ for all $z \in W_R(\mathbf 0)$ and $r \in (0, R]$. Moreover, there exists $z^\star \in Z^{\star}$ such that $\| z^\star \|_2 \le R/4$.* *Proof.* First, we get a bound on the norm of an optimal solution. As there is an optimal solution to the linear program there exists an optimal basic feasible solution [@bertsimas1997introduction Chapter 2]. Let $B$ be an optimal basis with corresponding optimal solutions $x^\star$ and $y^\star$, it follows that $\| x^{\star} \|_2 = \| x^\star_B \|_2 = \| A_B^{-1} b \|_2 \le \| A_B^{-1} \|_2 \| b \|_2 \le m_1\| b \|_2 \le m_1^{1.5} H$ and $\| y^\star \|_2 = \| (A_B^{-1})^\top c_B \|_2 \le \| (A_B^{-1})^\top\|_2 \| c_B \|_2 \le m_1\| c \|_2 \le m_1^{1.5} H$ where $\| A_{B}^{-1} \|_2 \le m_1$ by .iv. With $z^\star = (x^\star, y^\star)$ we conclude $\| z^\star \|_2 \le \| x^{\star} \|_2 + \| y^{\star} \|_2 \le 2 m_1^{1.5} H\le R/4$. Consider a square nonsingular submatrix $\begin{pmatrix} v^\top\\ V \end{pmatrix}$ of the matrix $$\begin{pmatrix} \frac{1}{R} c^\top& - \frac{1}{R} b^\top\\ A & 0 \\ 0 & A^\top \end{pmatrix}$$ where $v$ is a subvector of $\begin{pmatrix} \frac{1}{R} c^\top& - \frac{1}{R} b^\top\end{pmatrix}$ and $V$ is a submatrix of $\begin{pmatrix} A & 0 \\ 0 & A^\top \end{pmatrix}$. Note that $V$ is totally unimodular by Lemma [Lemma 3](#lem:total-unimodularity-preserved){reference-type="ref" reference="lem:total-unimodularity-preserved"}. We now prove $V$ contains at most $2 m_1$ rows. Note $A$ contains at most $m_1$ rows by definition and the submatrix of $V$ corresponding to $A^{\top}$ contains at most $m_1$ columns. Consequently, the submatrix of $V$ corresponding to $A^{\top}$ contains at most $m_1$ rows otherwise it would be row rank-defficient rendering $\begin{pmatrix} v^\top\\ V \end{pmatrix}$ row rank-deficient and contridicting our assumption that $\begin{pmatrix} v^\top\\ V \end{pmatrix}$ is nonsingular. By with $M= R = O(H m_1^{1.5})$, $n= 2 m_1$ and using that $\| v \|_2 \le \frac{1}{R} (\| c \|_2 + \| b \|_2) \le 2 Hm_1^{0.5} / R$ we get This implies by that Combining [\[eq:lower-bound-primal-system\]](#eq:lower-bound-primal-system){reference-type="eqref" reference="eq:lower-bound-primal-system"} with Lemma [Lemma 2](#lem:normalized-duality-gap-LP-basic-result){reference-type="ref" reference="lem:normalized-duality-gap-LP-basic-result"} shows $\alpha\mathop{\bf dist{}}(z, Z^{\star}) \le \rho_r(z)$. ◻ We are now ready to prove the main result, . Note that $\| A \|_2$ can be readily estimated by power iteration with high probability in $\tilde{O}(1)$ matrix-vector multiplications [@kuczynski1992estimating] and therefore it is possible to select a step size that meets the requirements of the Theorem. **Corollary 2**. *If [\[primal-LP\]](#primal-LP){reference-type="eqref" reference="primal-LP"} is a totally unimodular linear program with an optimal solution then Algorithm [\[alg:restarted-pdhg\]](#alg:restarted-pdhg){reference-type="ref" reference="alg:restarted-pdhg"} starting from $z^{0,0} = \mathbf 0$ with $\frac{1}{4 \| A \|_2} \le \eta \le \frac{1}{2 \| A \|_2}$ and $\tau^{0} = 1$ requires at most $$O\left( Hm_1^{2.5} \| A \|_2 \log\left( \frac{m_2H}{\epsilon} \right) \right) \le O\left( Hm_1^{2.5} \sqrt{\textbf{nnz}(A)} \log\left( \frac{m_2H}{\epsilon} \right) \right)$$ matrix-vector multiplications to obtain a point satisfying $\mathop{\bf dist{}}(Z^{\star}, z^{n,0}) \le \epsilon$.* *Proof.* By we have $z^{n,0} \in W_{\theta \mathop{\bf dist{}}(z^{0,0}, Z^{\star})}(\mathbf 0)$ for $\theta = 2 \sqrt{\frac{1 + \eta \| A \|_2}{1 - \eta \| A \|_2}} \le 2 \sqrt{2} \le 4$. By , $\rho_r(z)$ is $\Omega\left(m_1^{-2.5} H^{-1} \right)$-sharp for all $\| z \|_2 \le 4 \mathop{\bf dist{}}(\mathbf 0, Z^{\star})$ and $r \le 4 \mathop{\bf dist{}}(\mathbf 0, Z^{\star})$. Combining this with (using $\tau^0 = 1$) gives $$t^{\star} = \Bigg\lceil \frac{2 C (q + 2)}{\alpha \beta} \Bigg\rceil = O( m_1^{2.5} H\| A \|_2) \quad \text{ and } \quad \mathop{\bf dist{}}(z^{n,0},Z^{\star}) \le \beta^{n} t^{\star} \mathop{\bf dist{}}(z^{0,0}, Z^{\star}).$$ Thus for $n \ge \log_{1/\beta}(t^\star / \epsilon)$ we have $\mathop{\bf dist{}}(z^{n,0},Z^{\star}) \le \epsilon$ and the total number of iterations is $O\left( Hm_1^{2.5} \| A \|_2 \log\left( \frac{m_1H\| A \|_2}{\epsilon} \right) \right)$. Finally, $\log\left( \frac{m_1H\| A \|_2}{\epsilon} \right) \le \log\left( \frac{m_2^2 H}{\epsilon} \right) \le 2 \log\left( \frac{m_2H}{\epsilon} \right)$ because $\| A \|_2 \le \| A \|_F \le \sqrt{\textbf{nnz}(A)} \le \sqrt{m_1m_2} \le m_2$ where the last inequality uses the assumption that $m_1\le m_2$. ◻ # Acknowledgments {#acknowledgments .unnumbered} The author thanks Yinyu Ye and Javier Peña for useful feedback. The author was supported by a Google Research Scholar award, AFOSR grant \#FA9550-23-1-0242 and by the NSF-BSF program, under NSF grant \#2239527. # Proof of {#sec:lem:linear-algebra-schur-fact} **Lemma 6**. *Define $$M_{\lambda} := \begin{pmatrix} M_{11} & M_{12} \\ \mathbf 0& \lambda M_{22} \end{pmatrix}$$ where $M_{22}$ is a square matrix, and assume $M_{\lambda}$ is nonsingular for some $\lambda \in (0,\infty)$. Then $M_{11}$ and $M_{22}$ are nonsingular and $$\lim_{\lambda \rightarrow \infty} M_{\lambda}^{-1} = \begin{pmatrix} M_{11}^{-1} & \mathbf 0\\ \mathbf 0& \mathbf 0 \end{pmatrix} \ .$$* *Proof.* Since $M_{\lambda}$ is nonsingular and $M_{22}$ is square we deduce that $M_{11}$ is also square. Also, we have for all $u \neq \mathbf 0$ that $\mathbf 0\neq M_{\lambda} \begin{pmatrix} u \\ \mathbf 0\end{pmatrix} = M_{11} u$ which implies that $M_{11}$ is nonsingular. Similarly, for all $v \neq \mathbf 0$ we have $\mathbf 0\neq M_{\lambda}^\top\begin{pmatrix} v \\ \mathbf 0\end{pmatrix} = \lambda M_{22}^\top v$ which implies that $M_{22}$ is nonsingular. Next, the Schur complement of $M_{\lambda}$ with respect to the $\lambda M_{22}$ block is $S := M_{11} - \mathbf 0\frac{1}{\lambda} M_{11}^{-1} M_{12} = M_{11}$. Therefore using the standard formula for the matrix inverse using the Schur complement [@haynsworth1968schur] we have $$M_{\lambda}^{-1} = \begin{pmatrix} M_{11}^{-1} & \frac{1}{\lambda} M_{11}^{-1} M_{12} M_{22}^{-1} \\ \mathbf 0& \frac{1}{\lambda} M_{22}^{-1} \end{pmatrix},$$ which taking $\lambda \rightarrow \infty$ yields the desired result. ◻ [^1]: Recent work by @lu2023geometry develops a different, more interpretable linear convergence bound for PDHG. However, it is unclear if these new bounds would be useful for analyzing totally unimodular linear programs. [^2]: Such a $j$ exists because the rows of $V$ are linearly independent and $V$ has $n- 1$ rows. Therefore, there must exist some $e_j$ that is not in the span of the rows of $V$ and makes the dimension of the subspace spanned by the rows of $W$ to $n$ which implies it is nonsingular.

arxiv_math

--- abstract: | We consider a superposition operator of the form $$\int_{[0, 1]} (-\Delta)^s u\, d\mu(s),$$ for a signed measure $\mu$ on the interval of fractional exponents $[0,1]$, joined to a nonlinearity whose term of homogeneity equal to one is "jumping", i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of "critical" type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure $\mu$. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators "with the wrong sign" is also a complete novelty in this setting. address: - "Serena Dipierro: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" - "Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901-6975, USA" - "Caterina Sportelli: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" - "Enrico Valdinoci: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia" author: - Serena Dipierro - Kanishka Perera - Caterina Sportelli - Enrico Valdinoci title: | An existence theory\ for superposition operators of mixed order\ subject to jumping nonlinearities --- # Introduction The aim of this paper is to address the study of critical problems involving a nonlocal operator obtained through the linear superposition of fractional operators of different orders. Specifically, we consider two nonnegative finite (Borel) measures $\mu^+$ and $\mu^-$ in $[{{ 0 }}, 1]$, as well as the corresponding signed measure $\mu:=\mu^+ -\mu^-$. The main operator of interest for us takes the form $$\label{AMMU} A_\mu u:=\int_{[{{ 0 }}, 1]} (-\Delta)^s u\, d\mu(s) .$$ As customary, the notation $(-\Delta)^s$ is reserved to the fractional Laplacian, defined, for all $s\in(0,1)$ as $$\label{AMMU0} (- \Delta)^s\, u(x) = c_{N,s}\int_{\mathbb{R}^N} \frac{2u(x) - u(x+y)-u(x-y)}{|y|^{N+2s}}\, dy.$$ The positive normalizing constant $c_{N,s}$ is chosen in such a way that, for $u$ smooth and rapidly decaying, the Fourier transform of $(-\Delta)^s u$ returns $(2\pi|\xi|)^{2s}$ times the Fourier transform of $u$ and provides consistent limits as $s\nearrow1$ and as $s\searrow0$, namely $$\lim_{s\nearrow1}(-\Delta)^su=(-\Delta)^1u=-\Delta u \qquad{\mbox{and}}\qquad \lim_{s\searrow0}(-\Delta)^s u=(-\Delta)^0u=u.$$ Particular cases for the operator in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"} are (minus) the Laplacian (corresponding to the choice of $\mu$ being the Dirac measure concentrated at $1$), the fractional Laplacian $(-\Delta)^{s_\star}$ (corresponding to the choice of $\mu$ being the Dirac measure concentrated at some fractional power $s_\star$), the "mixed order operator" $-\Delta+(-\Delta)^{s_\star}$ (when $\mu$ is the sum of two Dirac measures), etc. The "continuous" superposition of operators of different fractional orders has also been recently considered in the literature, see e.g. [@MR3485125]. A list of interesting cases for this operator will be discussed in detail in Section [5](#EXA:A){reference-type="ref" reference="EXA:A"}. For the moment, let us recall that operators arising from the superpositions of local and nonlocal operators are a topic intensively studied in the contemporary research, under different perspectives, including regularity theory (see [@MR2911421; @MR4381148; @MR4387204; @MR4469224; @MR4530314; @MIN]), existence and nonexistence results (see [@MR4275496]), viscosity solution theory (see [@MR2129093; @MR2653895]), symmetry results (see [@MR4313576]), geometric and variational inequalities (see [@MR4391102; @FABE]), etc. Moreover, operators of this type naturally arise in concrete applications: for instance they model the dispersal of a biological population whose individuals are subject to different kinds of diffusive strategies (such as Gaussian and Lévy flights), see [@MR4249816; @EDOARDO]. Interestingly, the signed measure $\mu$ allows each single fractional Laplacian to take part in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"} with possibly different signs. However, in our setting, we will assume that the interval $[{{ 0 }}, 1]$ can be divided into two subintervals, the one on the right of a specific sign and suitably dominating the one on the left. More precisely, we assume that there exist $\overline s\in ({{ 0 }}, 1]$ and $\gamma\geqslant 0$ such that $$\label{mu00} {\mu^+}([\overline s, 1])>0,$$ $$\label{mu3} {\mu^-}_{\big|_{[\overline s, 1]}}=0$$ and $$\label{mu2} \mu^-\big([{{ 0 }}, \overline s]\big)\leqslant\gamma \mu^+\big([\overline s, 1]\big).$$ Roughly speaking, conditions [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"} and [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} state that the component of the signed measure $\mu$ supported on higher fractional exponents is positive, and condition [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} prescribes that the negative components of the signed measure $\mu$ (if any) must be conveniently "reabsorbed" into the positive ones. Our main assumption will thus be that $\gamma$ in [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} is sufficiently small. It is worth pointing out that, by assumption [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, there exists $s_\sharp\in [\overline s, 1]$ such that $$\label{scritico} \mu^+ ([s_\sharp, 1])> 0.$$ We will see below that the exponent $s_\sharp$ plays the role of a critical exponent (therefore, roughly speaking, while some arbitrariness is allowed in the choice of $s_\sharp$ here above, the results obtained will be stronger if one picks $s_\sharp$ "as large as possible" but still fulfilling [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}). In this paper, we investigate the existence of nontrivial solutions of the critical growth elliptic problem with a jumping nonlinearity $$\label{mainab} \left\{\begin{aligned} \int_{[{{ 0 }}, 1]} (-\Delta)^s u\, d\mu(s) & = bu^+ - au^- + |u|^{2_{s_\sharp}^\ast - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, with $N\geqslant 3$. Moreover, $$2_{s_\sharp}^\ast = \frac{2N}{N - 2s_\sharp}$$ is the fractional critical Sobolev exponent corresponding to the parameter $s_\sharp$, $a$ and $b$ are positive real numbers and $u^\pm := \max \left\{\pm u,0\right\}$ are the positive and negative parts of $u$, respectively. When $a = b$, the right-hand side of the equation in [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} reduces to a Brézis-Nirenberg nonlinearity. When $a\ne b$, an additional difficulty arises in the analysis of [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"}, due to the lack of regularity of the source term. As a matter of fact, the study of non-differentiable nonlinearities (sometimes called "jump" nonlinearities) has a very consolidated tradition, see e.g. [@MR499709; @MR1181350; @MR1215262; @MR1725568]. Interestingly, this type of nonlinearity frequently occurs in concrete problems, including singular perturbations of classical nonlinearities, plasma problems, mathematical biology, etc., see [@MR1303035]. Our goal in this paper is to find pairs $(a,b)$ which guarantee the existence of nontrivial solutions for [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"}. To this end, we need to identify suitable regions of the space of parameters (corresponding to $\mathbb{R}^2$) that are conveniently related to the spectral properties of the operator $A_\mu$ in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"}. To this end, we briefly recall the construction of the minimal and maximal curves of the Dancer-Fučı́k spectrum (see [@MR3012848 Chapter 4] for a general setting). One looks at the operator $A_\mu$ in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"} (the setting can be actually generalized to include the case of monotone, self-adjoint operators with compact inverse, coupled to potentials). The classical spectrum of $A_\mu$ consists of isolated eigenvalues $\lambda_l$, with $l \geqslant 1$, with finite multiplicity, satisfying $0 < \lambda_1 < \cdots < \lambda_l < \cdots$. Instead, the Dancer-Fučı́k spectrum of $A_\mu$ consists of the couples $(a,b) \in \mathbb{R}^2$ for which the equation $$\label{2.3} A_\mu u = bu^+ - au^-$$ has a nontrivial solution. The Dancer-Fučı́k spectrum is a closed subset of $\mathbb{R}^2$ (see [@MR3012848 Proposition 4.4.3]). We also point out that equation [\[2.3\]](#2.3){reference-type="eqref" reference="2.3"} reduces to $A_\mu u = \lambda u$ when $a = b = \lambda$, and therefore the Dancer-Fučı́k spectrum of $A_\mu$ contains points of the form $(\lambda_l,\lambda_l)$. The Dancer-Fučı́k spectrum presents an interesting geometry, see [@MR3012848 Theorem 4.7.9]. Namely, there exist two continuous and strictly decreasing functions $\nu_{l-1}$ and $\mu_l$, such that: - for all $a\in(\lambda_{l-1},\lambda_{l+1})$, we have that $\nu_{l-1}(a)\leqslant\mu_l(a)$, - $\nu_{l-1}(\lambda_l) = \lambda_l=\mu_l(\lambda_l)$, - for all $a\in(\lambda_{l-1},\lambda_{l+1})$, we have that both $(a,\nu_{l-1}(a))$ and $(a,\mu_{l}(a))$ belong to the Dancer-Fučı́k spectrum, - if $a\in(\lambda_{l-1},\lambda_{l+1})$ and $b\in (\lambda_{l-1},\lambda_{l+1})$, with either $b < \nu_{l-1}(a)$ or $b > \mu_l(a)$, then $(a,b)$ does not belong to the Dancer-Fučı́k spectrum. In particular, setting, for any $l \geqslant 2$, $$\label{QELL} Q_l := (\lambda_{l-1},\lambda_{l+1}) \times (\lambda_{l-1},\lambda_{l+1}),$$ we have that the graphs of $\nu_{l-1}$ and $\mu_l$ are strictly decreasing curves in $Q_l$ that belong to the Dancer-Fučı́k spectrum. Also, both these curves pass through the point $(\lambda_l,\lambda_l)$, while the region $\left\{(a,b) \in Q_l : b < \nu_{l-1}(a)\right\}$ below the lower curve and the region $\left\{(a,b) \in Q_l : b > \mu_l(a)\right\}$ above the upper curve lie outside the Dancer-Fučı́k spectrum. Points in the region $\left\{(a,b) \in Q_l : \nu_{l-1}(a) < b < \mu_l(a)\right\}$ between these two graphs (when such region is nonempty) may or may not belong to the Dancer-Fučı́k spectrum. The geometry related to the Dancer-Fučı́k spectrum is sketched in Figure [1](#FI11){reference-type="ref" reference="FI11"}. {#FI11} For our purposes, for all $l \geqslant 2$, the region in $Q_l$ below the lower curve of the Dancer-Fučı́k spectrum is of particular importance, since a portion of this region contains the pairs $(a,b)$ allowing for nontrivial solutions of [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"}. To describe this portion of the plane, we define $$\label{best_rho}\begin{split}{\mathcal{S}}:=&\inf\Bigg\{ \mu^+(0)\,\|u\|^2_{L^2(\Omega)} +\mu^+(1)\,\|\nabla u\|^2_{L^2(\Omega)} \\&\qquad\qquad+ \int_{({{ 0 }},1)}\left[ {c_{N,s}} \iint_{\mathbb{R}^{2N}} \frac{| u(x)-u(y)|^2}{|x-y|^{N+2s}}\, dx\,dy\right]\,d\mu^+(s) \Bigg\}.\end{split}$$ The infimum above[^1] is taken over all the functions $u\in C^\infty_0(\Omega)$ satisfying $\|u\|_{L^{2^*_{s_\sharp}}(\mathbb{R}^N)}=1$, with $s_\sharp$ as in [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}. Roughly speaking, one can consider ${\mathcal{S}}$ as the analogue of the Sobolev constant for the operator $A_\mu$ in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"}. For our purpose this generalized Sobolev constant is useful to identify the pairs $(a,b)$ allowing for a nontrivial solution of [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"}. Specifically, these pairs are precisely the ones lying in $Q_l$ below the lower curve of the Dancer-Fučı́k spectrum and satisfying $$\label{sotto} \min \left\{a,b\right\} > \lambda_l - \frac{\mathcal S}{|\Omega|^{(2 s_\sharp)/N}}.$$ Here above and in the rest of this paper, $|\Omega|$ stands for the Lebesgue measure of $\Omega$. The corresponding region of interest is sketched in Figure [2](#FI12){reference-type="ref" reference="FI12"}. The result that we obtain is thus as follows: **Theorem 1**. *Let $\mu=\mu^+-\mu^-$ with $\mu^+$ and $\mu^-$ satisfying [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}.* *Let $(a,b) \in Q_l$. Assume that $b < \nu_{l-1}(a)$ and that [\[sotto\]](#sotto){reference-type="eqref" reference="sotto"} is satisfied.* *Then, there exists $\gamma_0>0$, depending only on $N$, $\Omega$, $s_\sharp$, $a$ and $b$, such that if $\gamma\in[0,\gamma_0]$ then problem [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} admits a nontrivial solution.* {reference-type="ref" reference="main1"}.](spectrum.pdf){#FI12} We stress that Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"} is not only new in its wide generality, but it also possesses many specific cases which are also new. In particular: - If $\mu:=\delta_1$, i.e. if $A_\mu$ reduces to the classical Laplacian, then problem [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} has been recently studied in [@MR4535437]. In particular, [@MR4535437 Theorem 1.3] provided the existence of a nontrivial solution when $N\geqslant 4$. Our result gives an existence result in a small region below the lower curve and holds for $N\geqslant 3$ (hence improving the known condition on the dimension). A detailed discussion of this type of results will be given in Corollary [Corollary 10](#ILS){reference-type="ref" reference="ILS"}. - If $\mu:=\delta_s$, i.e. if the operator is the fractional Laplacian $(-\Delta)^s$ for some $s\in(0,1)$, our results are still new, to the best of our knowledge (in fact, they seem to be new even in the case $a=b$ in which no jumping nonlinearity is present, but see [@MR3060890] for related results). We treat this case in detail in Corollary [Corollary 11](#ILFRA){reference-type="ref" reference="ILFRA"}. - If $\mu:=\delta_1+\delta_s$, i.e. if $A_\mu=-\Delta+(-\Delta)^s$ for some $s\in(0,1)$, then Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"} is new. In this setting, the particular case $a=b=\lambda$ has been recently studied in [@arXiv.2209.07502 Theorem 1.4], where a nontrivial solution was found for a suitable range of $\lambda$. The application of Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"} for this choice of $\mu$ will be discussed in Corollary [Corollary 12](#mix_corollary){reference-type="ref" reference="mix_corollary"}. - The case of the superposition of two nonlocal operators with different orders, corresponding to the choice of the measure $\mu:=\delta_{s_1}+\delta_{s_2}$ and to an operator of the type $(-\Delta)^{s_1}+(-\Delta)^{s_2}$ for some $s_1$, $s_2\in(0,1)$, is also new to our best knowledge. - The case in which the measure $\mu$ changes sign is also new to our best knowledge. This seems to be new even in the case $\mu:= \delta_1 -\alpha\delta_s$, corresponding to an operator of the form $-\Delta-\alpha(-\Delta)^s$, where $s\in(0,1)$ and $\alpha$ is a small positive constant (notice the "wrong" sign in the second term of this operator). A simple example in this setting is provided in Corollary [Corollary 13](#mix_signed){reference-type="ref" reference="mix_signed"}. Actually, we think that our strategy on how to deal with "wrong" sign contributions may be of general interest and lead to the study of a rather general class of operators with competing diffusive trends. - The case of a convergent series $$\sum_{k=0}^{+\infty} c_k (-\Delta)^{s_k} u,\qquad{\mbox{where }}\, \sum_{k=0}^{+\infty} c_k\in(0,+\infty),$$ with - either $c_k\geqslant 0$ for all $k\in\mathbb{N}$, - or $$\begin{aligned} & & c_k>0\ \text{ for all } k\in\{1,\dots, \overline k\} \text{ and } \sum_{k=\overline k +1}^{+\infty} c_k \leqslant\gamma \sum_{k=0}^{\overline k} c_k,\\ &&{\mbox{for some~$\overline k\in\mathbb{N}$ and~$\gamma\geqslant 0$,}}\end{aligned}$$ are also new (see Corollaries  [Corollary 14](#serie1){reference-type="ref" reference="serie1"} and  [Corollary 15](#serie2){reference-type="ref" reference="serie2"}). - The continuous superposition of fractional operators of the form $$\int_0^1 (-\Delta)^s u \,f(s)\,ds,$$ where $f$ is a measurable and non identically zero function, is also new (see Corollary [Corollary 16](#function){reference-type="ref" reference="function"}). In the forthcoming paper [@CATERINA-PP], we will also consider the case of nonlinear fractional operators of mixed order of $p$--Laplacian type. The rest of this paper is organized as follows. Section [2](#op_section){reference-type="ref" reference="op_section"} gathers several estimates of Sobolev type which will constitute the functional analytic core of our study. Then, we present in Section [3](#KAMqw){reference-type="ref" reference="KAMqw"} the variational framework in which we work and we complete the proof of the main result in Section [4](#sec_main1){reference-type="ref" reference="sec_main1"}. In Section [5](#EXA:A){reference-type="ref" reference="EXA:A"} we apply the general result in Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"} to several specific cases of interest, which are also new in the literature. An interesting technical aspect of the proofs presented is that our arguments are (for the first time in the literature, to our best knowledge[^2]) capable also of dealing with operators "with the wrong sign", i.e. the ones coming from the measure $\mu^-$ and which have to be "reabsorbed" through quantitative estimates into $\mu^+$. We think that it is particularly remarkable that no extra assumption on the equation is needed for this. Given its interest also in practical situations (in which competing operators could participate to a complex model with opposite[^3] diffusion and concentration features) we believe that this novelty can open a new direction of research and apply to other problems as well. # Sobolev-type estimates {#op_section} In this section, we consider a bounded open set $\Omega\subset\mathbb{R}^N$ and we develop suitable energy estimates to deal with the operator in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"}. For this, for $s\in(0,1)$, we let $$\label{semisp} [u]_{s}: = \left({c_{N,s}}\iint_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right)^{\frac12}$$ be the Gagliardo seminorm of a measurable function $u : \mathbb{R}^N \to \mathbb{R}$, see e.g. [@MR2944369]. The consistent choice of the normalizing constant $c_{N,s}$ is such that $$\lim_{s\nearrow1}[u]_{s}=[u]_1:=\|\nabla u\|_{L^2(\mathbb{R}^N)} \qquad{\mbox{and}}\qquad \lim_{s\searrow0}[u]_s=[u]_0:=\|u\|_{L^2(\mathbb{R}^N)}.$$ We now observe that higher exponents in fractional norms control lower exponents, with uniform constants, according to the next observation: **Lemma 2**. *Let $0\leqslant s_1 \leqslant s_2 \leqslant 1$.* *Then, for any measurable function $u:\mathbb{R}^N\to\mathbb{R}$ with $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega$ we have that $$\label{spp} [u]_{s_1}\leqslant c \, [u]_{s_2},$$ for a suitable positive constant $c=c(N,\Omega)$.* *Proof.* First, we suppose that $u\in C^\infty_0(\Omega)$. A rather delicate issue is that we can take the constant $c(N,\Omega)$ independent of $s_1$ and $s_2$. To check this, it is convenient to write the Gagliardo seminorm in terms of the Fourier transform (see e.g. [@MR2944369]) as $$[u]_s=\left(\;\int_{\mathbb{R}^{N}} |2\pi \xi|^{2s}\,|\widehat u(\xi)|^2\,d\xi\right)^{\frac12}.$$ We stress that, by Plancherel Theorem, this is also valid when $s=0$ and $s=1$. One can combine this with the fractional Sobolev constant, which can be explicitly computed in the Hilbert setting, see [@MR3179693 formula (2)], according to which $$[u]_s^2\geqslant C^\star(N,s)\,\|u\|_{L^{2^*_s}(\mathbb{R}^N)}^2,$$ with $$C^\star(N,s):= 2^{4s} \pi^{3s} \, \frac{\Gamma((N+2s)/2)}{\Gamma((N-2s)/2)} \, \left(\frac{\Gamma(N/2)}{\Gamma(N)}\right)^{2s/N},$$ where we used the standard notation for the Gamma Function. We recall that the Gamma Function on the real line has a minimum at $r_\star:=1.46...$, with $\Gamma(r_\star)>0.88$, rising to either side of this minimum. Thus, $$C^\star(N,s)\geqslant\frac{0.88}{\Gamma(N+10)} \, \left(\frac{0.88}{\Gamma(N)}\right)^{2s/N}\geqslant\frac{0.88}{\Gamma(N+10)} \, \left(\frac{0.88}{\Gamma(N)}\right)^{2/N}=:C^\star(N).$$ Consequently, using the Sobolev and Hölder inequalities, $$\begin{aligned} _{s_1}^2&\leqslant& \int_{B_{1/(2\pi)}} |2\pi \xi|^{2s_1}\,|\widehat u(\xi)|^2\,d\xi+\int_{\mathbb{R}^N\setminus B_{1/(2\pi)}} |2\pi \xi|^{2s_2}\,|\widehat u(\xi)|^2\,d\xi\\ &\leqslant& \int_{\mathbb{R}^N} |\widehat u(\xi)|^2\,d\xi+\int_{\mathbb{R}^N} |2\pi \xi|^{2s_2}\,|\widehat u(\xi)|^2\,d\xi \\&=& \|u\|^2_{L^2(\Omega)}+[u]^2_{s_2} \\&\leqslant& |\Omega|^{\frac{2s_2}{N}}\|u\|^2_{L^{2^*_{s_2}}(\Omega)}+[u]^2_{s_2} \\&\leqslant&\frac{(1+|\Omega|)^{\frac{2s_2}{N}}}{C^\star(N,s_2)}[u]^2_{s_2}+[u]^2_{s_2} \\&\leqslant&\frac{(1+|\Omega|)^{\frac{2}{N}}}{C^\star(N)}[u]^2_{s_2}+[u]^2_{s_2}\end{aligned}$$ and this proves [\[spp\]](#spp){reference-type="eqref" reference="spp"} when $u\in C^\infty_0(\Omega)$. Now we perform a density argument to establish [\[spp\]](#spp){reference-type="eqref" reference="spp"} in the general case. To this end, let $u:\mathbb{R}^N\to\mathbb{R}$ be a measurable function with $u=0$ a.e. in $\mathbb{R}^N\setminus\Omega$. We can assume that $[u]_{s_2}<+\infty$, otherwise there is nothing to prove. Then, by the density of the smooth functions in the (possibly fractional) Sobolev spaces, we find a sequence of functions $u_k\in C^\infty_0(\Omega)$ such that $[u_k-u]_{s_2}\to0$ as $k\to+\infty$. Thus, by (possibly fractional) Sobolev embeddings, up to a subsequence we can assume that $u_k\to u$ in $L^2(\Omega)$ and a.e. in $\mathbb{R}^N$. Accordingly, we can use the already proved version of [\[spp\]](#spp){reference-type="eqref" reference="spp"} to infer that $[u_k]_{s_1}\leqslant c \, [u_k]_{s_2}$ and, as a consequence, $$\begin{aligned} \liminf_{k\to+\infty}[u_k]_{s_1}\leqslant c \liminf_{k\to+\infty} [u_k]_{s_2}\leqslant c\left([u]_{s_2}+ \liminf_{k\to+\infty}[u_k-u]_{s_2}\right)=c\,[u]_{s_2}.\end{aligned}$$ Hence, the desired result in [\[spp\]](#spp){reference-type="eqref" reference="spp"} follows from Fatou's Lemma. ◻ We define the space $\mathcal{X}(\Omega)$ as the set of measurable functions $u:\mathbb{R}^N\to\mathbb{R}$ such that $u=0$ in $\mathbb{R}^N\setminus \Omega$ and $$\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s) <+\infty.$$ **Lemma 3**. *Suppose that $\mu^+$ satisfies [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}. Then, $\mathcal{X}(\Omega)$ is a Hilbert space.* *Proof.* Let $u_n$ be a Cauchy sequence and take $\epsilon>0$. Thus, there exists $\overline{n}\in\mathbb{N}$ such that if $m$, $k\geqslant\overline{n}$ then $$\label{jdiewohuggbrdbv} \int_{[{{ 0 }}, 1]} [u_m-u_k]^2_{s}\, d\mu^+ (s)\leqslant\epsilon.$$ Now, we distinguish two cases, either $\mu^+(1)\neq0$ or $\mu^+(1)=0$. If $\mu^+(1)\neq0$, then we obtain from [\[jdiewohuggbrdbv\]](#jdiewohuggbrdbv){reference-type="eqref" reference="jdiewohuggbrdbv"} that $$\epsilon\geqslant\mu^+(1)\int_{\Omega}|\nabla u_m-\nabla u_k|^2\,dx,$$ and therefore $u_n$ is a Cauchy sequence in $H^1_0(\Omega)$. Accordingly, there exists $u\in H^1_0(\Omega)$ such that $u_n\to u$ in $H^1_0(\Omega)$ as $n\to+\infty$. Also, exploiting Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"}, we see that, for all $s\in[0,1)$, $$[u-u_n]_{s}\leqslant c(N,\Omega) \, [u-u_n]_{1}.$$ As a consequence, $$\int_{[0,1]}[u-u_n]_s^2\,d\mu^+(s) \leqslant c^2(N,\Omega) \mu^+([0,1]) [u-u_n]_{1}^2,$$ which gives that $$\lim_{n\to+\infty} \int_{[0,1]}[u-u_n]_s^2\,d\mu^+(s)=0,$$ as desired. If instead $\mu^+(1)=0$, then we deduce from [\[jdiewohuggbrdbv\]](#jdiewohuggbrdbv){reference-type="eqref" reference="jdiewohuggbrdbv"} and Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"} that $$\begin{aligned} &&\epsilon\geqslant \int_{[{{ 0 }}, 1)} [u_m-u_k]^2_{s}\, d\mu^+ (s)\geqslant \int_{[\overline{s}, 1)} [u_m-u_k]^2_{s}\, d\mu^+ (s)\\&&\qquad \geqslant\frac{1}{c^2(N,\Omega)} \int_{[\overline{s}, 1)} [u_m-u_k]^2_{\overline{s}}\, d\mu^+ (s) =\frac{\mu^+([\overline{s}, 1))}{c^2(N,\Omega)}\, [u_m-u_k]^2_{\overline{s}} .\end{aligned}$$ We point out that $\mu^+([\overline{s}, 1))>0$ in light of [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"} and the fact that $\mu^+(1)=0$. Accordingly, $u_n$ is a Cauchy sequence in $H^{\overline{s}}_0(\Omega)$ and therefore it converges to some $u$ in $H^{\overline{s}}_0(\Omega)$. Hence, we can extract a subsequence $u_{n_j}$ converging to $u$ in $L^2(\Omega)$ and a.e. in $\mathbb{R}^N$. Then, if $m\geqslant\overline{n}$, we have that $$\begin{aligned} \epsilon&\geqslant& \lim_{j\to+\infty} \int_{[{{ 0 }}, 1)} [u_m-u_{n_j}]^2_{s}\, d\mu^+ (s) \\&\geqslant& \lim_{j\to+\infty} \left(\mu^+(0)\|u_m- u_{n_j}\|^2_{L^2(\Omega)}+ \int_{({{ 0 }}, 1)} [u_m-u_{n_j}]^2_{s}\, d\mu^+ (s)\right)\\ &=&\mu^+(0)\|u_m- u\|^2_{L^2(\Omega)}\\&&\qquad +\lim_{j\to+\infty} \int_{({{ 0 }}, 1)} \left({c_{N,s}}\iint_{\mathbb{R}^{2N}} \frac{|(u_m-u_{n_j})(x) - (u_m-u_{n_j})(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+ (s).\end{aligned}$$ As a result, by Fatou's Lemma, $$\begin{aligned} \epsilon&\geqslant& \mu^+(0)\|u_m- u\|^2_{L^2(\Omega)}\\&&\qquad + \int_{({{ 0 }}, 1)} \left({c_{N,s}}\iint_{\mathbb{R}^{2N}} \liminf_{j\to+\infty} \frac{|(u_m-u_{n_j})(x) - (u_m-u_{n_j})(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+ (s)\\&=& \mu^+(0)\|u_m- u\|^2_{L^2(\Omega)}\\&&\qquad + \int_{({{ 0 }}, 1)} \left({c_{N,s}}\iint_{\mathbb{R}^{2N}} \frac{|(u_m-u)(x) - (u_m-u)(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+ (s)\\ &=& \int_{[{{ 0 }}, 1]} [u_m-u]^2_{s}\, d\mu^+ (s),\end{aligned}$$ which says that the sequence $u_n$ converges to $u$ in ${\mathcal{X}}(\Omega)$, as desired. ◻ In this setting, we can "reabsorb" the negative part of the signed measure $\mu$, according to the following result: **Proposition 4**. *Assume [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}.* *Then, there exists $c_0=c_0(N,\Omega)>0$ such that, for any $u\in\mathcal{X}(\Omega)$, we have $$\int_{[{{ 0 }}, \overline s]} [u]_{s}^2 \, d\mu^- (s) \leqslant c_0\,\gamma \int_{[\overline s, 1]} [u]^2_{s} \, d\mu(s).$$* *Proof.* We notice that if $\mu^+([\overline s,1])=0$, then condition [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} would give that $\mu^-([0,\overline s])=0$, and therefore Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"} would be trivially satisfied. Thus, from now on we suppose that $\mu^+([\overline s,1])>0$. By applying Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"} with $s_1: = \overline s$ and $s_2: = s$ we infer that, for all $s\in[\overline s,1]$, $$\begin{aligned} ^2_{\overline s}\leqslant c^2(N,\Omega) \, [u]^2_{s}.\end{aligned}$$ Similarly, applying Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"} with $s_1: = s$ and $s_2: = \overline s$, for all $s\in[0,\overline s]$ we have that $$\begin{aligned} ^2_{s}\leqslant c^2(N,\Omega) \, [u]^2_{\overline s}.\end{aligned}$$ Consequently, recalling [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}, $$\begin{aligned} && \int_{[{{ 0 }}, \overline s]} [u]_{s}^2 \, d\mu^- (s)\leqslant c^2(N,\Omega) \int_{[{{ 0 }}, \overline s]} [u]^2_{\overline s}\, d\mu^- (s)=c^2(N,\Omega) \,[u]^2_{\overline s}\,\mu^-\big({[{{ 0 }}, \overline s]}\big)\\&&\qquad\leqslant c^2(N,\Omega) \,\gamma\,[u]^2_{\overline s}\, \mu^+\big([\overline s, 1]\big)= c^2(N,\Omega) \,\gamma \int_{[\overline s, 1]}[u]^2_{\overline s}\,d\mu^+(s) \\&&\qquad\leqslant c^4(N,\Omega) \,\gamma \int_{[\overline s, 1]}[u]^2_{s}\,d\mu^+(s) .\end{aligned}$$ This is the desired result, with $c_0:=c^4(N,\Omega)$. ◻ **Proposition 5**. *Assume [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}. Let $s_\sharp\in [\overline s, 1]$ be as in [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"}.* *Then, there exists a positive constant $\bar c =\bar c (N,\Omega, s_\sharp)$ such that, for any $u\in \mathcal{X}(\Omega)$, $$\label{pqwodfl134rt} [u]_{s_\sharp}\leqslant\bar{c}\left(\, \int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s)\right)^{\frac12}.$$ In particular, the space $\mathcal{X}(\Omega)$ is continuously embedded in $L^r(\Omega)$ for any $r\in [1, 2^*_{s_\sharp}]$ and compactly embedded in $L^r(\Omega)$ for any $r\in [1, 2^*_{s_\sharp})$.* *Proof.* By Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"}, used here with $s_1: = s_\sharp$ and $s_2 := s$, for all $s\in[s_\sharp,1]$ we have that $[u]_{s_\sharp}\leqslant c(N,\Omega) \, [u]_{s}$. As a result, $$\mu^+ \big([s_\sharp, 1]\big)\,[u]_{s_\sharp}^2\leqslant c^2(N,\Omega) \, \int_{[s_\sharp, 1]}[u]_{s}^2\,d\mu^+(s)\leqslant c^2(N,\Omega)\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s).$$ This and [\[scritico\]](#scritico){reference-type="eqref" reference="scritico"} yield the desired result. ◻ # Variational setting {#KAMqw} In this section, we cast problem [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} into a suitable variational setting. To start, we state the following definition. **Definition 6**. A weak solution of problem [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} is a function $u \in \mathcal{X}(\Omega)$ such that, for all $v \in \mathcal{X}(\Omega)$, $$\begin{split} &\int_{[{{ 0 }}, 1]}\left(c_{N,s}\iint_{\mathbb{R}^{2N}} \frac{(u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+(s)\\ &\qquad- \int_{[{{ 0 }}, \overline s]}\left(c_{N,s}\iint_{\mathbb{R}^{2N}} \frac{(u(x) - u(y))\, (v(x) - v(y))}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^-(s)\\ &\qquad\qquad= \int_{\Omega} (bu^+ - a u^-) v\, dx + \int_\Omega |u|^{2_{s_\sharp}^\ast - 2}\, uv\, dx. \end{split}$$ The variational functional $E: \mathcal{X}(\Omega)\to \mathbb{R}$ associated with problem [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} is defined by $$\label{fun} E(u) = \frac{1}{2}\,\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s) -\frac{1}{2}\int_{[{{ 0 }}, \overline s]} [u]^2_s\, d\mu^-(s) - \frac{1}{2}\int_\Omega \left[a\, (u^-)^2 + b\, (u^+)^2\right]\, dx - \frac{1}{2_{s_\sharp}^*} \int_\Omega |u|^{2_{s_\sharp}^*}\, dx.$$ **Remark 7**. Note that in the functional [\[fun\]](#fun){reference-type="eqref" reference="fun"} the term arising from the negative part of the measure $\mu$ can be absorbed in the norm. In fact, by Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"} we have that $$\int_{[{{ 0 }}, \overline s]} [u]_s^2 \, d\mu^- (s) \leqslant c_0(N,\Omega) \,\gamma\int_{[\overline s, 1]} [u]^2_s \, d\mu(s) \leqslant c_0(N,\Omega) \,\gamma\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s).$$ In particular, if $\gamma$ is sufficiently small (possibly depending on $N$ and $\Omega$) it follows that $$E(u) \geqslant\frac{1}{4}\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s) - \frac{1}{2}\int_\Omega \left[a\, (u^-)^2 + b\, (u^+)^2\right] \,dx - \frac{1}{2_{s_\sharp}^*} \int_\Omega |u|^{2_{s_\sharp}^*}\, dx.$$ We now state a weak convergence result (to be used below in the analysis of Palais-Smale sequences in the forthcoming Proposition [Proposition 9](#P51){reference-type="ref" reference="P51"}): **Lemma 8**. *Let $u_n$ be a bounded sequence in $\mathcal X(\Omega)$.* *Then, there exists $u:\mathbb{R}^N\to\mathbb{R}$ such that, up to a subsequence, for any $v\in\mathcal X(\Omega)$, $$\label{Vconv} \begin{split} &\lim_{n\to+\infty}\int_{[{{ 0 }}, \overline s]} \left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s}(u_n(x)-u_n(y)) (v(x)-v(y))}{|x-y|^{N+2s}} \, dx\, dy\right)\, d\mu^-(s) \\ &\qquad =\int_{[{{ 0 }}, \overline s]} \left(\;\iint_{\mathbb{R}^{2N}} \frac{c_{N,s}(u(x)-u(y)) (v(x)-v(y))}{|x-y|^{N+2s}} \, dx\, dy\right)\, d\mu^-(s). \end{split}$$ Also, $$\label{lunoconv} {\mbox{$u_n$ converges to~$u$ in~$L^1(\Omega)$ as~$n\to+\infty$.}}$$* *Proof.* By [\[semisp\]](#semisp){reference-type="eqref" reference="semisp"} and Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"}, $$\begin{aligned} && \int_{[{{ 0 }}, \overline s]}\left( {c_{N,s}} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right)\,d\mu^-(s) =\int_{[{{ 0 }}, \overline s]} [u_n]_{s}^2 \, d\mu^- (s)\\&&\qquad\qquad \leqslant c_0\,\gamma \int_{[\overline s, 1]} [u_n]^2_{s} \, d\mu(s)\leqslant c_0\,\gamma \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s),\end{aligned}$$ which is bounded uniformly in $n$. The desired result in [\[Vconv\]](#Vconv){reference-type="eqref" reference="Vconv"} now follows from the Banach-Alaoglu Theorem. It remains to establish [\[lunoconv\]](#lunoconv){reference-type="eqref" reference="lunoconv"}. To this aim, we consider three cases: $\mu^-$ is the zero measure, $\mu^-$ is a Dirac measure at $0$ and, as a last possibility, $\mu^-((0,\overline{s}])>0$. In the first two cases, we could have used directly Proposition [Proposition 5](#emb){reference-type="ref" reference="emb"}, obtaining that, up to a subsequence, $u_n$ converges to some $u$ in $L^2(\mathbb{R}^N)$. This $u$ would have satisfied both [\[Vconv\]](#Vconv){reference-type="eqref" reference="Vconv"} and [\[lunoconv\]](#lunoconv){reference-type="eqref" reference="lunoconv"} (because in the first case [\[Vconv\]](#Vconv){reference-type="eqref" reference="Vconv"} is void and in the second case, recalling footnote [\[footpahetehfh\]](#footpahetehfh){reference-type="ref" reference="footpahetehfh"} on page , it has to be interpreted as a weak convergence in $L^2(\mathbb{R}^N)$). So, we can focus on the third case, namely we suppose that $\mu^-((0,\overline{s}])>0$. Hence, by the Dominated Convergence Theorem, $$\lim_{\epsilon\searrow0} \mu^-([\epsilon,\overline{s}]) =\mu^-((0,\overline{s}])>0.$$ Accordingly, there exists $\epsilon_0\in(0,\overline{s}]$ such that $\mu^-([\epsilon_0,\overline{s}])>0$. From this and Lemma [Lemma 2](#nons){reference-type="ref" reference="nons"}, we have that $$\mu^-([\epsilon_0,\overline{s}])[u_n]^2_{\epsilon_0}\leqslant c^2(N,\Omega) \, \int_{[\epsilon_0,\overline{s}]}[u_n]_{s}^2\,d\mu^-(s),$$ which is bounded uniformly in $n$, thanks to Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"}. Thus, by the compactness result for fractional Sobolev spaces (see e.g. [@MR2944369]), we obtain [\[lunoconv\]](#lunoconv){reference-type="eqref" reference="lunoconv"}, as desired. ◻ Now we address the convergence of the Palais-Smale sequences. For this, we first point out that, in view of [\[sotto\]](#sotto){reference-type="eqref" reference="sotto"}, there exists $\epsilon_0=\epsilon_0(N,\Omega, s_\sharp, a,b)\in\left(0,\frac{{\mathcal{S}}}{|\Omega|^{(2 s_\sharp)/N}}\right)$ such that $$\min \left\{a,b\right\} > \lambda_l - \frac{\mathcal S}{|\Omega|^{(2 s_\sharp)/N}}+\epsilon_0.$$ Hence, we define $$\theta_0=\theta_0(N,\Omega, s_\sharp, a,b):=\frac{|\Omega|^{(2 s_\sharp)/N}}{\mathcal S}\,\epsilon_0\in(0,1)$$ and we see that $$\label{sottoBIS} \min \left\{a,b\right\} > \lambda_l - \frac{\mathcal S}{|\Omega|^{(2 s_\sharp)/N}}(1-\theta_0).$$ With this notation, we have: **Proposition 9**. *Let $\mathcal S$ be as in [\[best_rho\]](#best_rho){reference-type="eqref" reference="best_rho"} and $$\label{CASTA} c^\ast:= \frac{s_\sharp}{N}\, \big((1-\theta_0)\mathcal S \big)^{\frac{N}{2 s_\sharp}}.$$* *Then, there exists $\gamma_0>0$, depending on $N$, $\Omega$, $s_\sharp$, $a$ and $b$, such that if $\gamma\in[0,\gamma_0]$ and $c \in (0,c^\ast)$, then every $(\text{PS})_{c}$ sequence of the functional [\[fun\]](#fun){reference-type="eqref" reference="fun"} has a subsequence that converges weakly to a nontrivial critical point of [\[fun\]](#fun){reference-type="eqref" reference="fun"}.* *Proof.* Let $u_n$ be a $(\text{PS})_{c}$ sequence of the functional $E$, i.e. $$\label{c1} \begin{split}& \lim_{n\to+\infty} E(u_n) \\ = \;&\lim_{n\to+\infty}\frac{1}{2}\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\frac{1}{2}\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s) \\&\qquad - \frac{1}{2}\int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx - \frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx\\= \;&c \end{split}$$ and $dE (u_n)$ converges to $0$ in the dual of $\mathcal{X}(\Omega)$, namely $$\label{c2} \lim_{n\to+\infty}\sup_{ v \in \mathcal{X}(\Omega)}\big|\langle dE (u_n), v\rangle\big| =0.$$ Since, for all $v \in \mathcal{X}(\Omega)$, $$\begin{split} \langle dE (u_n), v\rangle = \;& \int_{[{{ 0 }}, 1]}\left(c_{N,s} \iint_{\mathbb{R}^{2N}} \frac{(u_n(x) - u_n(y))\, (v(x) - v(y))}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+(s)\\ &\qquad- \int_{[{{ 0 }}, \overline s]}\left(c_{N,s} \iint_{\mathbb{R}^{2N}} \frac{(u_n(x) - u_n(y))\, (v(x) - v(y))}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^-(s)\\ &\qquad -\int_\Omega \big(b\, u_n^+ -a\, u_n^-\big)v \, dx - \int_\Omega |u_n|^{2_{s_\sharp}^\ast - 2}\, u_n v\, dx, \end{split}$$ choosing $v:=u_n$ in [\[c2\]](#c2){reference-type="eqref" reference="c2"}, we obtain that $$\label{hufewghwutgui0987654} \begin{split}0=\;& \lim_{n\to+\infty} \langle dE (u_n), u_n\rangle\\= \;& \lim_{n\to+\infty}\int_{[{{ 0 }}, 1]}\left(c_{N,s} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^+(s)\\ &\qquad- \int_{[{{ 0 }}, \overline s]}\left(c_{N,s} \iint_{\mathbb{R}^{2N}} \frac{|u_n(x) - u_n(y)|^2}{|x - y|^{N+2s}}\, dx\, dy\right) d\mu^-(s)\\ &\qquad -\int_\Omega \big[b\, (u_n^+)^2 +a\, (u_n^-)^2\big]\, dx - \int_\Omega |u_n|^{2_{s_\sharp}^\ast}\, dx\\=\;& \lim_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s) - \int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx - \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx \\=\;& \lim_{n\to+\infty} 2E(u_n)+\left(\frac2{2^*_{s_\sharp}}-1\right)\int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx. \end{split}$$ Combining this and [\[c1\]](#c1){reference-type="eqref" reference="c1"}, we infer that $$\label{jdewfguewogewuootr6u768i0} 0=2c+ \left(\frac2{2^*_{s_\sharp}}-1\right)\lim_{n\to+\infty}\int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx,$$ yielding that, for large $n$, $$\label{prima} \left(\frac{1}{2}- \frac{1}{2_{s_\sharp}^*}\right) \|u_n\|^{2^*_{s_\sharp}}_{L^{2^*_{s_\sharp}}(\Omega)} \leqslant c+1.$$ Moreover, from this and the Hölder inequality, it follows that $$\label{prima22}\begin{split}& \int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right]\, dx \leqslant\max\{ a, b\}\|u_n\|^2_{L^2(\Omega)} \leqslant\max\{ a, b\} |\Omega|^{(2 s_\sharp)/N} \|u_n\|^2_{L^{2^*_{s_\sharp}}(\Omega)}\\ &\qquad\leqslant\left(\frac{(c+1)N}{s_\sharp}\right)^{\frac2{2^*_{s_\sharp}}} \max\{ a, b\} |\Omega|^{(2 s_\sharp)/N}. \end{split}$$ Now, by [\[c1\]](#c1){reference-type="eqref" reference="c1"} and Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"}, we have that, as soon as $n$ is big enough, $$\begin{aligned} &&\frac12\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\\ &\leqslant&\frac{1}{2}\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s) +\frac{1}{2}\int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx +\frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx+c+1\\&\leqslant& \frac{c_0(N,\Omega)\gamma}2\int_{[ \overline s,1]} [u_n]^2_s \, d\mu(s) +\frac{1}{2}\int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx +\frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx+c+1,\end{aligned}$$ and therefore, if $\gamma$ is sufficiently small (possibly depending on $N$ and $\Omega$), $$\frac14 \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\leqslant \frac{1}{2}\int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx +\frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx+c+1.$$ From this, [\[prima\]](#prima){reference-type="eqref" reference="prima"} and [\[prima22\]](#prima22){reference-type="eqref" reference="prima22"}, we obtain that $$\frac14 \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\leqslant \frac{1}{2}\left(\frac{(c+1)N}{s_\sharp}\right)^{\frac2{2^*_{s_\sharp}}} \max\{ a, b\} |\Omega|^{(2 s_\sharp)/N} +\frac{N(c+1)}{2s_\sharp},$$ which says that $\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)$ is uniformly bounded in $n$. Hence, in view of Lemma [Lemma 3](#hilbert){reference-type="ref" reference="hilbert"} and Proposition [Proposition 5](#emb){reference-type="ref" reference="emb"}, there exists $u\in \mathcal{X}(\Omega)$ such that, up to subsequences, $$\label{weakun}\begin{split} u_n\rightharpoonup u &\text{ in } \mathcal{X}(\Omega),\\ u_n\to u &\text{ in } L^{r}(\Omega) \text{ for every } r\in [1, 2_{s_\sharp}^*),\\ u_n\to u &\text{ a.e. in } \Omega. \end{split}$$ Furthermore, we observe that $u$ is a weak solution of [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"}, according to Definition [Definition 6](#wsol2){reference-type="ref" reference="wsol2"}, thanks to the convergence statements in [\[weakun\]](#weakun){reference-type="eqref" reference="weakun"} and Lemma [Lemma 8](#Vitali){reference-type="ref" reference="Vitali"}. It remains to prove that $$\label{uzerosper00} u\not\equiv 0.$$ To this end, suppose by contradiction that $u\equiv 0$. We recall from [\[hufewghwutgui0987654\]](#hufewghwutgui0987654){reference-type="eqref" reference="hufewghwutgui0987654"} that $$\begin{aligned} 0&=& \lim_{n\to+\infty}\langle dE(u_n), u_n\rangle \\ &=&\lim_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s) - \int_\Omega \left[a\, (u_n^-)^2 + b\, (u_n^+)^2\right] \,dx - \int_\Omega |u_n|^{2_{s_\sharp}^\ast}\, dx \\&=& \lim_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s) - \int_\Omega |u_n|^{2_{s_\sharp}^\ast}\, dx.\end{aligned}$$ Thus, exploiting Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"}, we have that $$\begin{aligned} 0&\geqslant& \lim_{n\to+\infty} \big(1-c_0(N,\Omega)\gamma\big)\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) - \int_\Omega |u_n|^{2_{s_\sharp}^\ast}\, dx.\end{aligned}$$ Accordingly, recalling the definition of ${\mathcal{S}}$ in [\[best_rho\]](#best_rho){reference-type="eqref" reference="best_rho"}, we infer that $$\begin{aligned} 0 &\geqslant& \lim_{n\to+\infty} \big(1-c_0(N,\Omega)\gamma\big)\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\mathcal{S}^{-\frac{2_{s_\sharp}^*}2} \left(\, \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\right)^{\frac{2_{s_\sharp}^*}2}\\ &=& \big(1-c_0(N,\Omega)\gamma\big)\\&&\qquad\times\lim_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) \left(1 - \frac{\mathcal{S}^{-\frac{2_{s_\sharp}^*}2}}{\big(1-c_0(N,\Omega)\gamma\big)} \left(\, \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\right)^{\frac{2_{s_\sharp}^*}2-1 }\right).\end{aligned}$$ Now, choosing $\gamma$ sufficiently small (possibly in dependence of $N$ and $\Omega$) so that $1-c_0(N,\Omega)\gamma>0$, we conclude that $$\label{jdiweogbfgsdkgjk00} 0\geqslant\lim_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) \left(1 - \frac{\mathcal{S}^{-\frac{2_{s_\sharp}^*}2}}{\big(1-c_0(N,\Omega)\gamma\big)} \left(\, \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\right)^{\frac{2_{s_\sharp}^*}2-1 }\right).$$ We observe that $$\label{jdiweogbfgsdkgjk} \liminf_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)>0.$$ Indeed, suppose by contradiction that $$\liminf_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)=0.$$ Then, by Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"} we would also have that $$\liminf_{n\to+\infty} \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^- (s)=0,$$ and therefore, by [\[c1\]](#c1){reference-type="eqref" reference="c1"}, $$0<c=\liminf_{n\to+\infty}\left(-\frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx\right)\leqslant 0,$$ which is a contradiction and thus establishes [\[jdiweogbfgsdkgjk\]](#jdiweogbfgsdkgjk){reference-type="eqref" reference="jdiweogbfgsdkgjk"}. Thanks to [\[jdiweogbfgsdkgjk00\]](#jdiweogbfgsdkgjk00){reference-type="eqref" reference="jdiweogbfgsdkgjk00"} and [\[jdiweogbfgsdkgjk\]](#jdiweogbfgsdkgjk){reference-type="eqref" reference="jdiweogbfgsdkgjk"}, we conclude that $$0\geqslant\limsup_{n\to+\infty} \left(1 - \frac{\mathcal{S}^{-\frac{2_{s_\sharp}^*}2}}{\big(1-c_0(N,\Omega)\gamma\big)} \left(\, \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\right)^{\frac{2_{s_\sharp}^*}2-1 }\right),$$ which in turn gives that $$\label{dhuwegi4utg4u3gtu43} \liminf_{n\to+\infty}\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s)\geqslant\big(1-c_0(N,\Omega)\gamma\big)^{\frac2{2_{s_\sharp}^*-2} } \mathcal{S}^{\frac{2_{s_\sharp}^*}{2_{s_\sharp}^*-2}} =\big(1-c_0(N,\Omega)\gamma\big)^{\frac{N-2s_\sharp}{2s_\sharp} } \mathcal{S}^{\frac{N}{2_{s_\sharp}}}.$$ Additionally, using again [\[c1\]](#c1){reference-type="eqref" reference="c1"}, and recalling the strong convergence statement in [\[weakun\]](#weakun){reference-type="eqref" reference="weakun"}, $$c= \lim_{n\to+\infty}\frac{1}{2}\int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) -\frac{1}{2}\int_{[{{ 0 }}, \overline s]} [u_n]^2_s \, d\mu^-(s)- \frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx.$$ Hence, exploiting Proposition [Proposition 4](#crucial){reference-type="ref" reference="crucial"}, this gives that $$c\geqslant \lim_{n\to+\infty}\frac{1-c_0(N,\Omega)\gamma}2 \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) - \frac{1}{2_{s_\sharp}^*} \int_\Omega |u_n|^{2_{s_\sharp}^*}\, dx.$$ From this and [\[jdewfguewogewuootr6u768i0\]](#jdewfguewogewuootr6u768i0){reference-type="eqref" reference="jdewfguewogewuootr6u768i0"} it follows that $$c\geqslant \lim_{n\to+\infty}\frac{1-c_0(N,\Omega)\gamma}2 \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s) - \frac{(N-2s_\sharp)c}{2{s_\sharp}},$$ and therefore $$\frac{Nc}{2s_\sharp}\geqslant \lim_{n\to+\infty}\frac{1-c_0(N,\Omega)\gamma}2 \int_{[{{ 0 }}, 1]} [u_n]^2_{s}\, d\mu^+ (s).$$ This and [\[dhuwegi4utg4u3gtu43\]](#dhuwegi4utg4u3gtu43){reference-type="eqref" reference="dhuwegi4utg4u3gtu43"} give that $$\frac{Nc}{s_\sharp}\geqslant \big(1-c_0(N,\Omega)\gamma\big)^{\frac{N}{2s_\sharp}} \mathcal{S}^{\frac{N}{2_{s_\sharp}}},$$ and thus, recalling the definition of $c^\ast$ in [\[CASTA\]](#CASTA){reference-type="eqref" reference="CASTA"}, $$c^\ast>c\geqslant \big(1-c_0(N,\Omega)\gamma\big)^{\frac{N}{2s_\sharp}}\frac{s_\sharp}{N} \mathcal{S}^{\frac{N}{2_{s_\sharp}}} =\left(\frac{1-c_0(N,\Omega)\gamma}{1-\theta_0}\right)^{\frac{N}{2s_\sharp}} c^\ast.$$ We point out that this implies that $$c_0(N,\Omega)\gamma\geqslant\theta_0,$$ hence, choosing $\gamma$ sufficiently small, possibly in dependence of $N$, $\Omega$, $s_\sharp$, $a$ and $b$, we obtain the desired contradiction. Then, the claim in [\[uzerosper00\]](#uzerosper00){reference-type="eqref" reference="uzerosper00"} is established, completing the proof of Proposition [Proposition 9](#P51){reference-type="ref" reference="P51"}. ◻ # Existence theory and proof of Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"} {#sec_main1} With the preliminary work carried out so far, we are now in position of proving the existence result in Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. *Proof of Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}.* The aim is to exploit [@MR4535437 Theorem 4.1]. To this end, we set $E_l$ to be the eigenspace associated with the eigenvalue $\lambda_l$ and we remark that, for all $u\in E_j$ with $j\in\{1,\dots,l\}$, $$\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu (s) =\lambda_j \|u\|^2_{L^2(\Omega)}\leqslant\lambda_l \|u\|^2_{L^2(\Omega)}.$$ Now, we observe that, by the Hölder inequality, $$\|u\|^{2^*_{s_\sharp}}_{L^2(\Omega)}\leqslant|\Omega|^{\frac{2 s_\sharp}{N-2 s_\sharp}} \|u\|_{L^{2_{s_\sharp}^*}(\Omega)}^{2^*_{s_\sharp}},$$ and thus $$\int_\Omega |u|^{2_{s_\sharp}^*}\, dx = \|u\|_{L^{2_{s_\sharp}^*}(\Omega)}^{2^*_{s_\sharp}}\geqslant|\Omega|^{-\frac{2 s_\sharp}{N-2s_\sharp}} \|u\|^{2^*_{s_\sharp}}_{L^2(\Omega)}.$$ Consequently, recalling the definition of the functional in [\[fun\]](#fun){reference-type="eqref" reference="fun"}, we have that, for all $u\in E_j$ with $j\in\{1,\dots,l\}$, $$\label{diywtr843yt9876543}\begin{split} &E(u)\\ =\;& \frac{1}{2}\,\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu^+ (s) -\frac{1}{2}\int_{[{{ 0 }}, \overline s]} [u]^2_s\, d\mu^-(s) - \frac{1}{2}\int_\Omega \left[a\, (u^-)^2 + b\, (u^+)^2\right]\, dx - \frac{1}{2_{s_\sharp}^*} \int_\Omega |u|^{2_{s_\sharp}^*}\, dx \\ \leqslant\;&\frac{1}{2}\,\int_{[{{ 0 }}, 1]} [u]^2_{s}\, d\mu (s) - \frac{ \min \left\{a,b\right\}}2\|u\|_{L^2(\Omega)}^2 - \frac{1}{2_{s_\sharp}^*} |\Omega|^{-\frac{2 s_\sharp}{N-2s_\sharp}} \|u\|^{2^*_{s_\sharp}}_{L^2(\Omega)}\\ \leqslant\;&\frac{\lambda_l}2 \|u\|^2_{L^2(\Omega)} - \frac{ \min \left\{a,b\right\}}2\|u\|_{L^2(\Omega)}^2 - \frac{1}{2_{s_\sharp}^*} |\Omega|^{-\frac{2 s_\sharp}{N-2s_\sharp}} \|u\|^{2^*_{s_\sharp}}_{L^2(\Omega)}. \end{split}$$ Now we consider the function $$h(t):=\frac12\big( \lambda_l - \min \left\{a,b\right\}\big)t^2 - \frac{1}{2_{s_\sharp}^*} |\Omega|^{-\frac{2 s_\sharp}{N-2s_\sharp}}t^{2^*_{s_\sharp}}$$ and we observe that $\lambda_l- \min \left\{a,b\right\}>0$. Accordingly, we obtain that $$\max_{t\geqslant 0} h(t) =\left(\frac12-\frac1{2_{s_\sharp}^*}\right) |\Omega| \big(\lambda_l - \min \left\{a,b\right\}\big)^{\frac{N}{2s_\sharp}}.$$ Plugging this information into [\[diywtr843yt9876543\]](#diywtr843yt9876543){reference-type="eqref" reference="diywtr843yt9876543"}, we conclude that $$E(u)\leqslant\left(\frac12-\frac1{2_{s_\sharp}^*}\right) |\Omega| \big(\lambda_l - \min \left\{a,b\right\}\big)^{\frac{N}{2s_\sharp}}.$$ Therefore, exploiting the assumption in [\[sottoBIS\]](#sottoBIS){reference-type="eqref" reference="sottoBIS"} and recalling the definition of $c^\ast$ in [\[CASTA\]](#CASTA){reference-type="eqref" reference="CASTA"}, we obtain that, for all $u\in E_j$ with $j\in\{1,\dots,l\}$, $$E(u)< \left(\frac12-\frac1{2_{s_\sharp}^*}\right) |\Omega| \left(\frac{(1-\theta_0)\mathcal S}{|\Omega|^{(2 s_\sharp)/N}}\right)^{\frac{N}{2s_\sharp}} =\left(\frac12-\frac1{2_{s_\sharp}^*}\right) \big( (1-\theta_0){\mathcal S}\big)^{\frac{N}{2s_\sharp}}=c^\ast.$$ Thus, Propositions [Proposition 9](#P51){reference-type="ref" reference="P51"} ensures the convergence of Palais-Smale sequences below the threshold $c^\ast$, provided that $\gamma$ is sufficiently small. This allows us to use [@MR4535437 Theorem 4.1], from which the desired result follows. ◻ # Examples and applications {#EXA:A} Note that the operator introduced in [\[mainab\]](#mainab){reference-type="eqref" reference="mainab"} is very general and we can employ it to produce a wide number of new interesting existence results for critical problems, depending on the particular choice of the measure $\mu$. We showcase some of these cases here below. We start proving that, by choosing $\mu$ in a proper way, our result can be compared to [@MR4535437 Theorem 1.3] and [@MPSS Theorem 1.2]. **Corollary 10**. *Let $\lambda_l$ be the sequence of Dirichlet eigenvalues of $-\Delta$ and $2^* = 2N/(N-2)$ be the classical Sobolev exponent.* *If $(a, b)\in Q_l$, $b <\nu_{l-1}(a)$ and $$\min \left\{a,b\right\} > \lambda_l -\frac{S}{|\Omega|^{2/N}}$$ where $S$ as denotes the classical best Sobolev constant, then problem $$\left\{\begin{aligned} -\Delta \, u & = bu^+ - au^- + |u|^{2^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \ \partial\Omega, \end{aligned}\right.$$ possesses a nontrivial solution.* *Proof.* Let $\mu:= \delta_1$ be the Dirac measure centred at the point $1$. In this case $\mu^-=0$ as well and $\mu$ satisfies [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"}. Furthermore, we can take $\overline s:=1$ and $s_\sharp:=1$, so that ${\mathcal{S}}$ in [\[best_rho\]](#best_rho){reference-type="eqref" reference="best_rho"} reduces to the classical Sobolev constant. The desired result now follows from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ To our best knowledge, our main result is new even for the case of the fractional Laplacian, and even for the case $a=b$ in which the jumping nonlinearity is not present. For the reader convenience, we state this result here below: **Corollary 11**. *Let $s\in [{{ 0 }}, 1)$ and $2^*_s = (2N)/(N-2s)$ be the critical fractional Sobolev exponent.* *Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of $(-\Delta)^s$ and by $S(s)$ the fractional Sobolev constant corresponding to $(-\Delta)^s$.* *If $(a, b)\in Q_l$, $b <\nu_{l-1}(a)$ and $$\min \left\{a,b\right\} > \lambda_l - \frac{S(s)}{|\Omega|^{2s/N}},$$ then problem $$\left\{\begin{aligned} (- \Delta)^s\, u & = bu^+ - au^- + |u|^{2_{s}^\ast - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* Here, one takes $\mu:= \delta_s$, $\overline s:=s$ and $s_\sharp:=s$, and the desired result is a consequence of Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ Now we show how to relate our new results to [@arXiv.2209.07502 Theorem 1.4]: **Corollary 12**. *Let $s\in [{{ 0 }}, 1)$. Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of the mixed operator $-\Delta +(-\Delta)^s$, by $S$ the classical Sobolev constant and by $2^* = 2N/(N-2)$ the classical critical Sobolev exponent.* *Then, if $(a, b)\in Q_l$, $b <\nu_{l-1}(a)$ and $$\min \left\{a,b\right\} > \lambda_l -\frac{S}{|\Omega|^{2/N}},$$ then problem $$\label{mixedab} \left\{\begin{aligned} -\Delta \, u + (-\Delta)^s \, u & = bu^+ - au^- + |u|^{2^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* We set $\mu:= \delta_1 +\delta_s$, where $\delta_1$ and $\delta_s$ denote the Dirac measures centered at the points $1$ and $s$ respectively. As done in the proof of Corollary [Corollary 10](#ILS){reference-type="ref" reference="ILS"}, we can take $\overline s:=1$ and $s_\sharp:=1$ and deduce the desired result from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ Interestingly, our setting is general enough to include also operators containing small terms with the "wrong" sign. As a paradigmatic example, we showcase the following result: **Corollary 13**. *Let $s\in [{{ 0 }}, 1)$ and $\alpha\in\mathbb{R}$. Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of the mixed operator $-\Delta - \alpha(-\Delta)^s$, by $2^* = 2N/(N-2)$ the classical critical Sobolev exponent, and by $S$ the classical Sobolev constant.* *Let $(a, b)\in Q_l$ and $b <\nu_{l-1}(a)$ and suppose that $$\min \left\{a,b\right\} > \lambda_l -\frac{S}{|\Omega|^{2/N}},$$* *Then, there exists $\alpha_0>0$, depending only on $N$, $\Omega$, $a$ and $b$, such that if $\alpha\leqslant\alpha_0$, then problem $$\left\{\begin{aligned} -\Delta \, u - \alpha(-\Delta)^s \, u & = bu^+ - au^- + |u|^{2^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* We define $\mu:= \delta_1 -\alpha\delta_s$, $\overline s:=1$ and $s_\sharp:=1$. Once again, the desired result follows from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ One more interesting application arises taking $\mu$ as a convergent series of Dirac measures. On this matter, we provide the next two results: **Corollary 14**. *Let $1\geqslant s_0 > s_1> s_2 >\dots \geqslant 0$. Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of the operator $$\sum_{k=0}^{+\infty} c_k (-\Delta)^{s_k} \qquad{\mbox{with $\,c_k\geqslant 0\,$ and }} \,\sum_{k=0}^{+\infty} c_k \in (0, +\infty),$$ by $S_0$ the best Sobolev constant corresponding to the exponent $s_0$ and by $2_{s_0}^* = 2N/(N-2s_0)$ the critical Sobolev exponent.* *Then, if $(a, b)\in Q_l$, $b <\nu_{l-1}(a)$ and $$\min \left\{a,b\right\} > \lambda_l -\frac{S_0}{|\Omega|^{2s_0/N}},$$ then problem $$\left\{\begin{aligned} \sum_{k=0}^{+\infty} c_k (-\Delta)^{s_k} u & = bu^+ - au^- + |u|^{2_{s_0}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* We set $$\mu:=\sum_{k=0}^{+\infty} c_k \,\delta_{s_k} ,$$ where $\delta_{s_k}$ denote the Dirac measures centered at each $s_k$. In this case, we can take $\overline s:=0$ and $s_\sharp:=s_0$ and deduce the desired result from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ **Corollary 15**. *Let $1\geqslant s_0 > s_1> s_2 >\dots \geqslant 0$ and $c_k\in\mathbb{R}$ for all $k\in\mathbb{N}$ be such that $$\sum_{k=0}^{+\infty} c_k \in (0, +\infty).$$ Assume that there exists $\gamma\geqslant 0$ and $\overline k\in\mathbb{N}$ such that $$\label{serft87604} c_k>0\ \text{ for all } k\in\{1,\dots, \overline k\}\quad \text{ and }\quad \sum_{k=\overline k +1}^{+\infty} c_k \leqslant\gamma \sum_{k=0}^{\overline k} c_k.$$ Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of the operator $$\sum_{k=0}^{+\infty} c_k (-\Delta)^{s_k}$$ by $S_0$ the best Sobolev constant corresponding to the exponent $s_0$ and by $2_{s_0}^* = 2N/(N-2s_0)$ the critical Sobolev exponent.* *Let $(a, b)\in Q_l$ and $b <\nu_{l-1}(a)$ and suppose that $$\min \left\{a,b\right\} > \lambda_l -\frac{S_0}{|\Omega|^{2s_0/N}},$$* *Then, there exists $\gamma_0>0$, depending only on $N$, $\Omega$, $s_0$, $a$ and $b$, such that if $\gamma\in[0,\gamma_0]$ then problem $$\left\{\begin{aligned} \sum_{k=0}^{+\infty} c_k (-\Delta)^{s_k} u & = bu^+ - au^- + |u|^{2_{s_0}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* We set $$\mu:=\sum_{k=0}^{+\infty} c_k \,\delta_{s_k}$$ where $\delta_{s_k}$ denote the Dirac measures centered at each $s_k$. Notice that [\[serft87604\]](#serft87604){reference-type="eqref" reference="serft87604"} guarantees that the assumptions on $\mu$ in [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} are satisfied. Thus, we can take $\overline s:=s_{\overline k}$ and $s_\sharp:=s_0$ and infer the desired result from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ It is worth nothing that the wide generality of our setting enables us to address also the case of the continuous superposition of fractional operators. To be more precise, the following result holds true. **Corollary 16**. *Let $s_\sharp\in [{{ 0 }}, 1)$, $\gamma\geqslant 0$ and $f$ be a measurable and non identically zero function such that $$\label{fun-int}\begin{split} & {\mbox{$f\geqslant 0$ in~$(s_\sharp,1)$,}}\\& \int_{s_\sharp}^1 f(s) \,ds >0 \\ {\mbox{and }}\qquad& \int_0^{s_\sharp} \max\{0,-f(s)\} \,ds \leqslant\gamma \int_{s_\sharp}^1 f(s) \,ds . \end{split}$$* *Denote by $\lambda_l$ the sequence of Dirichlet eigenvalues of the operator $$\label{djweiohtgierogh597865432} \int_0^1 f(s) (-\Delta)^s \, u \, ds ,$$ by $S_\sharp$ the best Sobolev constant corresponding to the exponent $s_\sharp$ and by $2_{s_\sharp}^* = 2N/(N-2s_\sharp)$ the fractional critical Sobolev exponent.* *Let $(a, b)\in Q_l$ and $b <\nu_{l-1}(a)$ and suppose that $$\min \left\{a,b\right\} > \lambda_l -\frac{S_\sharp}{|\Omega|^{2s_\sharp/N}},$$* *Then, there exists $\gamma_0>0$, depending only on $N$, $\Omega$, $s_\sharp$, $a$ and $b$, such that if $\gamma\in[0,\gamma_0]$ then problem $$\left\{\begin{aligned} \int_0^1 f(s) (-\Delta)^s \, u \, ds & = bu^+ - au^- + |u|^{2_{s_\sharp}^* - 2}\, u && \text{in } \Omega,\\ u & = 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned}\right.$$ admits a nontrivial solution.* *Proof.* We observe that the operator in [\[djweiohtgierogh597865432\]](#djweiohtgierogh597865432){reference-type="eqref" reference="djweiohtgierogh597865432"} is a particular case of $A_\mu$ as defined in [\[AMMU\]](#AMMU){reference-type="eqref" reference="AMMU"}, where $d\mu(s)$ boils down to $f(s)\,ds$. Additionally, [\[fun-int\]](#fun-int){reference-type="eqref" reference="fun-int"} guarantees that the assumptions in [\[mu00\]](#mu00){reference-type="eqref" reference="mu00"}, [\[mu3\]](#mu3){reference-type="eqref" reference="mu3"} and [\[mu2\]](#mu2){reference-type="eqref" reference="mu2"} are satisfied. Thus, we can take $\overline s:=s_\sharp$, which plays the role of the fractional critical exponent. The desired result then follows from Theorem [Theorem 1](#main1){reference-type="ref" reference="main1"}. ◻ # Acknowledgements {#acknowledgements .unnumbered} SD and EV are members of the Australian Mathematical Society (AustMS). EV is supported by the Australian Laureate Fellowship FL190100081 "Minimal surfaces, free boundaries and partial differential equations". CS is member of INdAM-GNAMPA. This work was partially completed while KP was visiting the Department of Mathematics and Statistics at the University of Western Australia, and he is grateful for the hospitality of the host department. His visit to the UWA was supported by the Simons Foundation Award 962241 "Local and nonlocal variational problems with lack of compactness". 100 [^1]: From now on, [\[footpahetehfh\]]{#footpahetehfh label="footpahetehfh"} with a slight abuse of notation, the quantity in brackets in formula [\[best_rho\]](#best_rho){reference-type="eqref" reference="best_rho"} (and similar quantities) will be abbreviated into $$\int_{[{{ 0 }},1]}\left[ {c_{N,s}} \iint_{\mathbb{R}^{2N}} \frac{| u(x)-u(y)|^2}{|x-y|^{N+2s}}\, dx\,dy\right]\,d\mu^+(s),$$ with the understanding that the measure evaluation at $s=0$ and $s=1$ (if nonvoid) returns the classical expressions. [^2]: For instance, the measure on fractional exponents studied in [@MR3485125] was supposed to be positive and supported away from $s=0$, while we do not need either of these assumptions here. [^3]: For example, an interesting model of mixed operator with opposite sign could be that of a biological population in which different individuals have different dispersal behaviors. Indeed, on the one hand, in view of the Lévy flight foraging hypothesis, mixed diffusive operators are an interesting tool to describe searching patters of predators; on the other hand, it would be tempting to have a simplified, but effective, description of social behaviors via a reversed fractional heat equation (while diffusion tends to spread mass around, inverse diffusion favors concentration and it may therefore be a helpful tool to include animals' tendency to cluster into groups of conspecifics). In this sense, mixed operators with different signs can turn out to be handy in mathematical biology, population dynamics, social sciences, etc.

arxiv_math

--- abstract: | We study pattern avoidance for permutations of $[n]$ with a fixed leading term and find exact formulas for the number of permutations avoiding single and pairs of patterns of length three. Our counting argument is then adapted to enumerating permutations avoiding both 3412 and 3421 by leading terms. In so doing, we obtain a new combinatorial proof of a recurrence relation for the large Schröder numbers. We also define $r$-Wilf equivalence for permutations with leading term $r$ and classify the $r$-Wilf-equivalence classes for both classical and vincular patterns of length three. address: - Department of Computer Science, University of California, Santa Barbara, CA 93106 - Department of Mathematics, University of Denver, Denver, CO 80208 - Department of Mathematics, University of Denver, Denver, CO 80208 author: - Ömer Eğecİoğlu - Collier Gaiser - Mei Yin title: | Enumerating pattern-avoiding permutations\ by leading terms --- # Introduction Let $A\subseteq\mathbb{N}:=\{1,2,\ldots\}$ be a finite set. We use $|A|$ to denote the number of elements in $A$. A permutation $\sigma$ on $A$ is a sequence $(\sigma(1),\ldots,\sigma(|A|))$ of length $|A|$ consisting of distinct numbers in $A$. When $A\subseteq\{1,\ldots,9\}$ or when there is no confusion, we simply write a permutation/sequence without commas or parentheses in single-line notation. The first term of a sequence is called the leading term of the sequence. We use $S_A$ to denote the set of permutations on $A$. When $A=[n]:=\{1,2,\ldots,n\}$ for some $n\in\mathbb{N}$, we write $S_n$ for $S_{[n]}$. For any $\tau\in S_n$ and $\sigma\in S_k$, if there exist $1\leq i_1<i_2<\cdots<i_k\leq n$ such that for all $1\leq a<b\leq k$, $\tau(i_a)<\tau(i_b)$ if and only if $\sigma(a)<\sigma(b)$, then we say that $\tau$ contains $\sigma$ as a pattern and that $(\tau(i_1),\ldots,\tau(i_k))$ is a $\sigma$ pattern. The permutation $\tau$ is said to avoid $\sigma$ if $\tau$ does not contain $\sigma$ as a pattern. For example, the permutation $\tau=12453\in S_5$ contains the pattern $132$ because $(\tau(1),\tau(3),\tau(5))=143$ is a $132$ pattern; however, $\tau$ avoids the pattern $321$. For any $m,n,k\in\mathbb{N}$ and $\sigma_1,\ldots,\sigma_m\in S_k$, we use $S_n(\sigma_1,\ldots,\sigma_m)$ to denote the set of permutations on $[n]$ which avoid all of the patterns $\sigma_1,\ldots,\sigma_m$. The interest in the study of pattern avoidance can be traced back to stack-sortable permutations in computer science [@Kitaev2011 Section 2.1]. One of the earliest results is the enumeration of permutations avoiding $\sigma\in S_3$, i.e., patterns of length three. D. Knuth proved that the number of permutations in $S_n$ avoiding any given pattern of length three is counted by the Catalan numbers $C_n$ (see also [@Bona2022 Theorem 4.7]). **Theorem 1**. *[@Knuth1973 p. 238] [\[Theorem:ClassicalSingle3\]]{#Theorem:ClassicalSingle3 label="Theorem:ClassicalSingle3"} For all $n\geq1$ and $\sigma\in S_3$, we have $$|S_n(\sigma)|=C_n=\frac{1}{n+1} \binom{2n}{n}.$$* Two permutation patterns $\sigma$ and $\sigma'$ are said to be *Wilf equivalent*, denoted $\sigma\sim\sigma'$, if $|S_n(\sigma)|=|S_n(\sigma')|$ for all $n\in\mathbb{N}$. By [\[Theorem:ClassicalSingle3\]](#Theorem:ClassicalSingle3){reference-type="ref" reference="Theorem:ClassicalSingle3"}, all permutation patterns of length three are Wilf equivalent: $123\sim132\sim213\sim231\sim312\sim321$. In other words, there is only one Wilf-equivalence class for permutation patterns of length three. Recently, pattern avoidance in rooted labeled forests and trees has also been studied [@AndersArcher2019; @GargPeng2023; @Panholzer2021; @Ren2020]. Many interesting results emerged in this setting. For example, Anders and Archer [@AndersArcher2019 Section 3] defined forest-Wilf equivalence, an analogue of Wilf equivalence for forests, and showed that there are two forest-Wilf-equivalence classes for patterns of length three. In this setting, permutations can be viewed as rooted paths where the leading terms are the roots. To our knowledge, pattern-avoiding permutations with a fixed leading term have not been extensively explored among the pattern-avoidance problems studied in the literature. Our aim is to fill this gap in this paper. For $r,n\in\mathbb{N}$ with $r\leq n$, let $S_{n,r}$ denote the set of permutations $\tau\in S_n$ with $\tau(1)=r$. We choose letter $r$ because of the representation of the leading term as the root of a path as discussed in the previous paragraph. It is clear that $|S_{n,r}|=(n-1)!$ for all $r\in[n]$. For any $r,n,m,k\in\mathbb{N}$ and $\sigma_1,\ldots,\sigma_m\in S_k$, we use $S_{n,r}(\sigma_1,\ldots,\sigma_m)$ to denote the set of permutations in $S_{n,r}$ which avoid the patterns $\sigma_1,\ldots,\sigma_m$ simultaneously. For a fixed $r\in\mathbb{N}$, two patterns $\sigma$ and $\sigma'$ are called *$r$-Wilf equivalent* if $|S_{n,r}(\sigma)|=|S_{n,r}(\sigma')|$ for all $n\geq r$. We write $\sigma\stackrel{r}{\sim}\sigma'$ if $\sigma$ and $\sigma'$ are $r$-Wilf equivalent. As an example, two patterns $\sigma$ and $\sigma'$ are $2$-Wilf equivalent, denoted $\sigma\stackrel{2}{\sim}\sigma'$, if for all $n\geq2$, $|S_{n,2}(\sigma)|=|S_{n,2}(\sigma')|$. For all $1\leq r\leq n$ and $\sigma\in S_3$, we obtain formulas for $|S_{n,r}(\sigma)|$ using elementary counting arguments. The key ingredient in our proofs is the shuffle of two permutations of smaller length. A shuffle of two shorter permutations is a longer permutation in which the relative order of each shorter permutation is preserved. We also show that there are two $1$-Wilf-equivalence classes for patterns of length three, $123\stackrel{1}{\sim} 132$ and $321\stackrel{1}{\sim}312\stackrel{1}{\sim}213\stackrel{1}{\sim}231$; and, for all $r\geq2$, there are three $r$-Wilf-equivalence classes for patterns of length three, $213\stackrel{r}{\sim}231$, $123\stackrel{r}{\sim}132$, and $321\stackrel{r}{\sim}312$. The combinatorial argument we use for enumerating $|S_{n,r}(213)|$ and $|S_{n,r}(231)|$ can be adapted to enumerating $|S_{n,r}(3412,3421)|$, which is then used to obtain a new combinatorial proof of a recurrence relation for large Schröder numbers $\mathbb{S}_n$ [@Bona2022 p. 446]. Following the classical paper of Simion and Schmidt [@SimionSchmidt1985], we also enumerate $|S_{n,r}(\sigma_1,\sigma_2)|$ where $\sigma_1$ and $\sigma_2$ are two different patterns of length three.\ This paper is organized as follows. In [2](#Section:Preliminaries){reference-type="ref" reference="Section:Preliminaries"}, we define basic concepts and state our preliminary results. We then enumerate permutations avoiding patterns of length three and classify $r$-Wilf-equivalence classes for these patterns in [3](#Section:Single3){reference-type="ref" reference="Section:Single3"}. In [4](#Section:Pair3){reference-type="ref" reference="Section:Pair3"}, permutations avoiding pairs of patterns of length three are enumerated. Permutations avoiding both 3412 and 3421 are then studied in [5](#Section:Pair4){reference-type="ref" reference="Section:Pair4"}. Finally, in [6](#Section:Concluding){reference-type="ref" reference="Section:Concluding"}, we classify $r$-Wilf-equivalence classes for vincular patterns of length three as studied in [@BabsonSteingrimsson2000; @Claesson2001] and discuss further research. # Preliminaries {#Section:Preliminaries} **Definition 2**. For a permutation $\tau\in S_n$, the *complement* $\tau^c$ of $\tau$ is a permutation in $S_n$ defined by setting $\tau^c(i)=n+1-\tau(i)$. The following result which relates permutations avoiding certain patterns with permutations avoiding the complement of these patterns is elementary but will be crucial in our later derivations. **Lemma 3**. *Let $r$, $n$, $m$, and $k$ be positive integers with $r\leq n$, and let $\sigma_1,\dots,\sigma_m\in S_k$ be permutation patterns. Then $|S_{n,r}(\sigma_1,\dots,\sigma_m)|=|S_{n,n+1-r}(\sigma_1^c,\dots,\sigma_m^c)|$.* *Proof.* It is clear that, by [Definition 2](#Definition:Complements){reference-type="ref" reference="Definition:Complements"}, if $\tau_1,\tau_2\in S_{n,r}$ with $\tau_1\neq \tau_2$, then $\tau_1^c\neq \tau_2^c$. Let $\tau\in S_{n,r}(\sigma_1,\ldots,\sigma_m)$. Then $\tau^c(1)=n+1-\tau(1)=n+1-r$. So $\tau^c\in S_{n,n+1-r}$. If $\tau^c$ contains the pattern $\sigma_i^c$ for some $i\in[m]$, then $\tau$ would contain the pattern $\sigma_i$ by [Definition 2](#Definition:Complements){reference-type="ref" reference="Definition:Complements"}. So $\tau^c$ must avoid all the patterns $\sigma_1^c,\ldots,\sigma_m^c$. Hence $\tau^c\in S_{n,n+1-r}(\sigma_1^c,\ldots,\sigma_m^c)$. So we have $|S_{n,n+1-r}(\sigma_1^c,\ldots,\sigma_m^c)|\geq|S_{n,r}(\sigma_1,\ldots,\sigma_m)|$. Now let $\tau\in S_{n,n+1-r}(\sigma_1^c,\dots,\sigma_m^c)$. Then, by the result in the previous paragraph, we have $\tau^c\in S_{n,n+1-(n+1-r)}((\sigma_1^c)^c,\dots,(\sigma_m^c)^c)=S_{n,r}(\sigma_1,\dots,\sigma_m)$. Hence, we have $|S_{n,n+1-r}(\sigma_1^c,\ldots,\sigma_m^c)|\leq|S_{n,r}(\sigma_1,\ldots,\sigma_m)|$. ◻ **Definition 4**. Let $A$ and $B$ be two finite subsets of $\mathbb{N}$ with $A\subseteq B$, $\sigma\in S_A$, and $\tau\in S_B$. We say that $\sigma$ is a *subpermutation* of $\tau$ on $A$ if there exist indices $1 \leq i_1<i_2<\cdots<i_{|A|} \leq |B|$ such that $$(\tau(i_1),\tau(i_2),\ldots,\tau(i_{|A|}))=(\sigma(1),\sigma(2),\ldots,\sigma(|A|)).$$ For example if $\tau=543621\in S_6$, then $\sigma=462\in S_{\{2,4,6\}}$ is a subpermutation of $\tau$ on $\{2,4,6\}$. **Definition 5**. Suppose $\sigma$ is a permutation on a set $A$ and $\tau$ is a permutation on a set $B$ with $|A\cap B|=\emptyset$. A *shuffle* of $\sigma$ and $\tau$ is a permutation $\alpha$ on $A\cup B$ such that $\sigma$ is a subpermutation of $\alpha$ on $A$ and $\tau$ is a subpermutation of $\alpha$ on $B$. For example, let $A= \{4,5,7\}$, $B=\{1,3,6\}$, $\sigma=457\in S_A$, and $\tau=631\in S_B$. Then $\alpha=643571\in S_{A\cup B}$ and $\alpha'=456317\in S_{A\cup B}$ are shuffles of $\sigma$ and $\tau$. The following result is elementary but again will be crucial in our later derivations. We state it without proof. **Lemma 6**. *Suppose $A,B\subseteq \mathbb{N}$ with $|A\cap B|=\emptyset$, $|A|=k$, and $|B|=\ell$. If $\sigma\in S_A$ and $\tau\in S_B$, then the number of shuffles of $\sigma$ and $\tau$ is $\binom{k+\ell}{k}$.* We will use the following terminology. Let $A\subseteq\mathbb{N}$ be a finite set and $\tau\in S_{A}$. If $\tau \in S_A$ and $\tau(i)=a$, then we use $\mathcal{A}_\tau(a)=\{\tau(1),\ldots,\tau(i-1)\}$ to denote the set of *ancestors* of $a$ in $\tau$ and $\mathcal{D}_\tau(a)=\{\tau(i+1),\ldots,\tau(|A|)\}$ to denote the set of *descendants* of $a$ in $\tau$. For example, if $\tau=2785$ then $\mathcal{A}_\tau(8)=\{2,7\}$ and $\mathcal{D}_\tau(7)=\{5,8\}$. The first few Catalan numbers $C_n$, Bell numbers $B_n$, and large Schröder numbers $\mathbb{S}_n$ are listed in Table [1](#first_table){reference-type="ref" reference="first_table"} for later reference. $n$ 0 1 2 3 4 5 6 7 8 9 10 OEIS [@OEIS] ---------------- --- --- --- ---- ---- ----- ------ ------ ------- -------- --------- -------------- $C_n$ 1 1 2 5 14 42 132 429 1430 4862 16796 A000108 $B_n$ 1 1 2 5 15 52 203 877 4140 21147 115975 A000110 $\mathbb{S}_n$ 1 2 6 22 90 394 1806 8558 41586 206098 1037718 A006318 : $C_n$, $B_n$, and $\mathbb{S}_n$ for $n\leq 10$. We also need the following elementary results on the Catalan and the Bell numbers. **Lemma 7**. *For all $n\geq4$, we have $C_n<B_n$.* *Proof.* It is well-known that $C_n$ counts the number of noncrossing partitions of $[n]$ and $B_n$ counts the total number of partitions of $[n]$. For these facts and the definitions of partitions and noncrossing partitions, see for example [@Mezo2020 Section 1.1] and [@Simion2000]. For $n\geq4$, there is at least one crossing partition of $[n]$ and therefore $C_n<B_n$. ◻ **Lemma 8**. *For all $n\geq3$, we have $B_n>2B_{n-1}$.* *Proof.* The Bell numbers $B_n$ satisfy the following recurrence relation [@EgeciogluGarsia2021 p. 49]: $$B_n=\sum_{k=0}^{n-1}\binom{n-1}{k}B_k.$$ Let $n\geq3$. Then we have $$\begin{split} B_n=&~B_{n-1}+\binom{n-1}{n-2}B_{n-2}+\binom{n-1}{n-3}B_{n-3}+\cdots+\binom{n-1}{0}B_0\\> &~B_{n-1}+\binom{n-2}{n-2}B_{n-2}+\binom{n-2}{n-3}B_{n-3}+\cdots+\binom{n-2}{0}B_0\\=&~B_{n-1}+\sum_{k=0}^{n-2}\binom{n-2}{k}B_k=2B_{n-1}. \end{split}$$ ◻ # Permutations Avoiding a Single Pattern of Length Three {#Section:Single3} We start with some elementary results summarized in the table below: $r$ $|S_{n,r}(123)|$ $|S_{n,r}(321)|$ $|S_{n,r}(132)|$ $|S_{n,r}(312)|$ $|S_{n,r}(213)|$ $|S_{n,r}(231)|$ ------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ $n$ $C_{n-1}$ $1$ $C_{n-1}$ $1$ $C_{n-1}$ $C_{n-1}$ $n-1$ $C_{n-1}$ $n-1$ $C_{n-1}$ $n-1$ $C_{n-2}$ $C_{n-2}$ $2$ $n-1$ $C_{n-1}$ $n-1$ $C_{n-1}$ $C_{n-2}$ $C_{n-2}$ $1$ $1$ $C_{n-1}$ $1$ $C_{n-1}$ $C_{n-1}$ $C_{n-1}$ : Single Patterns of Length $3$ for $n\geq2$. It is easy to check the correctness of the expressions in [2](#Table:SingleClassical){reference-type="ref" reference="Table:SingleClassical"}. As an example, we sketch the proof of the fact that $|S_{n,n-1}(123)|=C_{n-1}$ for all $n\geq2$. For any $i,j\in\{2,\ldots,n\}$ with $i<j$, either $\tau(i)<n-1$ or $\tau(j)<n-1$. It follows that $(n-1,\tau(i),\tau(j))$ will never form a $123$ pattern for any $i,j\in\{2,\ldots,n\}$ with $i<j$. Hence, $\tau\in S_{n,n-1}$ avoids $123$ if and only if $(\tau(2),\ldots,\tau(n))$ avoids $123$. Therefore, by [\[Theorem:ClassicalSingle3\]](#Theorem:ClassicalSingle3){reference-type="ref" reference="Theorem:ClassicalSingle3"}, we have $|S_{n,n-1}(123)|=|S_{n-1}(123)|=C_{n-1}$. We next enumerate the number of permutations avoiding a single pattern of length three by their leading term $r$. The easiest ones are the patterns $213$ and $231$. **Theorem 9**. *For all $1\leq r\leq n$, we have $$|S_{n,r}(213)|=|S_{n,r}(231)|=C_{r-1}C_{n-r}.$$* *Proof.* Let $1\leq r\leq n$. We first show that $|S_{n,r}(231)|=C_{r-1}C_{n-r}$. Let $\mathcal{P}$ be a subset of $S_{n,r}$ such that every $\tau\in\mathcal{P}$ has the following properties: - $\{\tau(2),\ldots,\tau(r)\}=\{1,\ldots,r-1\}$; - $\{\tau(r+1),\ldots,\tau(n)\}=\{r+1,\ldots,n\}$; - $(\tau(2),\ldots,\tau(r))$ avoids $231$; - and $(\tau(r+1),\ldots,\tau(n))$ avoids $231$. By [\[Theorem:ClassicalSingle3\]](#Theorem:ClassicalSingle3){reference-type="ref" reference="Theorem:ClassicalSingle3"}, there are $C_{r-1}$ ways for a permutation on $\{1,\ldots,r-1\}$ to avoid $231$ and $C_{n-r}$ ways for a permutation on $\{r+1,\ldots,n\}$ to avoid $231$. Hence we have $|\mathcal{P}|=C_{r-1}C_{n-r}$. We will show that $S_{n,r}(231)=\mathcal{P}$. Let $\tau\in S_{n,r}(231)$. Then for all $a\in\{r+1,\ldots,n\}$, $\mathcal{D}_\tau(a)\cap\{1,\ldots,r-1\}=\emptyset$, since otherwise for any $b\in\mathcal{D}_\tau(a)\cap\{1,\ldots,r-1\}$, $rab$ is a 231 pattern which is a contradiction. Hence we have $\{\tau(2),\ldots,\tau(r)\}=\{1,\ldots,r-1\}$ and $\{\tau(r+1),\ldots,\tau(n)\}=\{r+1,\ldots,n\}$. Since $\tau\in S_{n,r}(231)$, $(\tau(2),\ldots,\tau(r))$ and $(\tau(r+1),\ldots,\tau(n))$ both avoid $231$. Hence we have $S_{n,r}(231)\subseteq\mathcal{P}$. Now let $\tau\in\mathcal{P}$. To show that $\tau\in S_{n,r}(231)$, it suffices to show that any subpermutation $abc$ of $\tau$ is not a $231$ pattern. Suppose, by way of contradiction, that a subpermutation $abc$ of $\tau$ is a 231 pattern. Then we must have $c<a<b$ and $c\in D_\tau(b)$. We split the argument into three cases. Case 1: $a=r$. Then we have $b>a=r$ and $c<a=r$. Since $\{\tau(r+1),\ldots,\tau(n)\}=\{r+1,\ldots,n\}$, we have $b=\tau(i)$ for some $i\geq r+1$. Since $c\in D_\tau(b)$, we must have $c>r$. This is a contradiction. Case 2: $a<r$. Then $c<a<r$. Since the subpermutation of $\tau$ on $\{1,\ldots,r-1\}$ avoids $231$, we must have $b>r$. Since $\{\tau(r+1),\ldots,\tau(n)\}=\{r+1,\ldots,n\}$, we have $b=\tau(i)$ for some $i\geq r+1$. Since $c\in D_\tau(b)$, we must have $c>r$. This is again a contradiction. Case 3: $a>r$. Then $b>a>r$. Since $\{\tau(r+1),\ldots,\tau(n)\}=\{r+1,\ldots,n\}$, we have $b=\tau(i)$ for some $i\geq r+1$. Since $c\in D_\tau(b)$, we must have $c>r$. So $abc$ is a subpermutation of $(\tau(r+1),\ldots,\tau(n))$ which is a contradiction because $(\tau(r+1),\ldots,\tau(n))$ avoids $231$. This completes the proof that $\mathcal{P}\subseteq S_{n,r}(231)$. Now we have $S_{n,r}(231)=\mathcal{P}$. Therefore, we have $$|S_{n,r}(231)|=|\mathcal{P}|=C_{r-1}C_{n-r}.$$ Now by [Lemma 3](#lemma:complements){reference-type="ref" reference="lemma:complements"}, we have $$|S_{n,r}(213)|=|S_{n,n+1-r}((213)^c)|=|S_{n,n+1-r}(231)|= C_{(n+1-r)-1}C_{n-(n+1-r)}= |S_{n,r}(231)|.$$ ◻ **Remark 10**. For all $n\geq1$, by [\[Theorem:ClassicalSingle3,Theorem:213;231\]](#Theorem:ClassicalSingle3,Theorem:213;231){reference-type="ref" reference="Theorem:ClassicalSingle3,Theorem:213;231"}, we have $$C_n=|S_n(231)|=\sum_{r=1}^n|S_{n,r}(231)|=\sum_{r=1}^nC_{n-r}C_{r-1}.$$ This offers an alternative interpretation for the well-known recurrence relation for the Catalan numbers, see [@EgeciogluGarsia2021 Section 3.2] and [@Stanley2015 Section 1.2]. Next, we consider the patterns $123$ and $132$. We need a definition first. **Definition 11**. For all $n\in\mathbb{N}$, define $a_{n}(i):[n]\to[n]$ as follows: - for all $i<n$, let $a_{n}(i)$ be the number of permutations $\tau\in S_n(123)$ such that $i$ is the smallest index with $\tau(i)<\tau(i+1)$; - and set $a_{n}(n)=1$. That is, if $i<n$, then $a_n(i)$ is the number of permutations $\tau\in S_n$ avoiding $123$ such that $\tau(1)>\tau(2)>\cdots>\tau(i-1)>\tau(i)<\tau(i+1)$. If $i=n$, then $a_n(n)=1$ because there is exactly one decreasing sequence $(n,n-1,\ldots,2,1)\in S_n$. Simion and Schmidt [@SimionSchmidt1985 Lemma 2] proved the following result for $a_n(i)$: **Lemma 12**. *For all $1\leq i\leq n$, $$a_n(i)=\binom{2n-i-1}{n-1}-\binom{2n-i-1}{n}.$$* Now we are ready to state and prove our result for the enumeration of $|S_{n,r}(123)|$ and $|S_{n,r}(132)|$. **Theorem 13**. *For all $n\geq 1$, we have $$|S_{n,1}(123)|=|S_{n,1}(132)|=1;$$ and for all $2 \leq r \leq n$, we have $$|S_{n,r}(123)|=|S_{n,r}(132)|=\sum_{i=1}^{r-1}\binom{i+n-r}{i}\left[\binom{2r-i-3}{r-2}-\binom{2r-i-3}{r-1}\right].$$* *Proof.* Let $1\leq r\leq n$. The well-known bijection between $S_n(123)$ and $S_n(132)$ by Simion and Schmidt [@SimionSchmidt1985] preserves the leading terms. We refer to [@Bona2022 Lemma 4.4] for a proof of this fact. Because of this, we have $|S_{n,r}(123)|=|S_{n,r}(132)|$ and we can focus on enumerating $|S_{n,r}(123)|$. If $\tau\in S_{n,1}(123)$, then $(\tau(2),\ldots,\tau(n))$ must be a decreasing sequence and hence $\tau=(1,n,n-1,\ldots,2)$. Therefore, we have $|S_{n,1}(123)|=1$. Now suppose $2\leq r\leq n$. Let $\mathcal{Q}$ be a subset of $S_{n,r}$ such that every $\tau\in\mathcal{Q}$ has the following properties: - the subpermutation $\tau'$ of $\tau$ on $\{1,\ldots,r-1\}$ avoids $123$; - the subpermutation $\tau''$ of $\tau$ on $\{r+1,\ldots,n\}$ is $(n,n-1,\ldots,r+1)$; - if $r>2$ and $i<r-1$ is the smallest index with $\tau'(1)>\tau'(2)>\cdots>\tau'(i)<\tau'(i+1)$, then $\{r+1,\ldots,n\}\subseteq\mathcal{A}_\tau(\tau'(i+1))$. In Property iii., we do not impose any extra condition on the location of $\{r+1,\ldots,n\}$ when $\tau'=(r-1,\ldots,1)$. We first show that $$|\mathcal{Q}|=\sum_{i=1}^{r-1}\binom{i+n-r}{i}a_{r-1}(i).$$ Let $\tau\in\mathcal{Q}$. If $r>2$, then for all $i\in\{1,\ldots,r-2\}$ and a fixed $\tau'$ with $\tau'(1)>\tau'(2)>\cdots>\tau'(i)<\tau'(i+1)$, the number of shuffles of $(\tau'(1),\ldots,\tau'(i))$ and $\tau''=(n,n-1,\ldots,r+1)$ is $\binom{i+n-r}{i}$. If $\tau'=(r-1,\ldots,1)$, then the number of shuffles of $\tau'$ and $\tau''$ is $\binom{r-1+n-r}{r-1}$. As the number of such $\tau'$ is $a_{r-1}(i)$ for all $i\in\{1,\ldots,r-1\}$, we have $|\mathcal{Q}|=\sum_{i=1}^{r-1}\binom{i+n-r}{i}a_{r-1}(i)$. It remains to show that $S_{n,r}(123)=\mathcal{Q}$. Let $\tau\in S_{n,r}(123)$, and let $\tau'$ be the subpermutation of $\tau$ on $\{1,2,\ldots,r-1\}$ and $\tau''$ be the subpermutation of $\tau$ on $\{r+1,\ldots,n\}$. Since $\tau$ avoids $123$, $\tau'$ avoids $123$ as well. We now show that $\tau''$ avoids $12$. If this is not the case then there exist $a<b\leq n-r$ such that $\tau''(a)<\tau''(b)$. Since $\tau''(a)>r$, $(r,\tau''(a),\tau''(b))$ is a 123 pattern, and this is a contradiction. Therefore $\tau''=(n,n-1,\ldots,r+1)$. Now suppose $r>2$ and let $i<r-1$ be the smallest index such that $\tau'(1)>\tau'(2)>\cdots>\tau'(i)<\tau'(i+1)$. We still need to show that $\{r+1,\ldots,n\}\subseteq\mathcal{A}_\tau(\tau'(i+1))$. Suppose, by way of contradiction, $\{r+1,\ldots,n\}\not\subseteq\mathcal{A}_\tau(\tau'(i+1))$. Then there exists $a\in\{r+1,\ldots,n\}\cap\mathcal{D}_\tau(\tau'(i+1))$. Since $(\tau'(i),\tau'(i+1),a)$ is a 123 pattern, this would be a contradiction. Hence, we have $S_{n,r}(123)\subseteq\mathcal{Q}$. Now let $\tau\in\mathcal{Q}$. We need to show that $\tau\in S_{n,r}(123)$. Let $\tau'$ be the subpermutation of $\tau$ on $\{1,2,\ldots,r-1\}$ and $\tau''$ be the subpermutation of $\tau$ on $\{r+1,\ldots,n\}$. If $r>2$, then let $i<r-1$ be the smallest index such that $\tau'(1)>\tau'(2)>\cdots>\tau'(i)<\tau'(i+1)$; if $\tau'=(r-1,\ldots,1)$, then we set $i=r-1$. Since $\tau\in\mathcal{Q}$, $\tau=(r,\alpha(1),\ldots,\alpha(n-r+i),\tau'(i+1),\ldots,\tau'(r-1))$ where $\alpha=(\alpha(1),\ldots,\alpha(n-r+i))$ is a shuffle of $(\tau'(1),\ldots,\tau'(i))$ and $\tau''=(n,n-1,\ldots,r+1)$. We need to show that any subpermutation $abc$ of $\tau$ is not a $123$ pattern. Suppose, by way of contradiction, there exists a subpermutation $abc$ of $\tau$ which is a $123$ pattern. Then $a<b<c$. We split the argument into two cases. Case 1: $a\geq r$. Then $c>b>r$. It follows that $bc$ is an increasing subpermutation of $\tau''$. But this contradicts the fact that $\tau''=(n,\ldots,r+1)$. Case 2: $a<r$. Since $\tau'$ avoids $123$, either $b>r$ or $c>r$. If $b>r$, then $c>b>r$. Using a similar argument as in Case 1, we have a contradiction. So we suppose $b<r$ and $c>r$. Here $b\neq r$ because $b\in D_\tau(a)$. Since $ab$ is an increasing subpermutation of $\tau'$, we have $b=\tau'(j)$ for some $j\geq i+1$. Hence $c\in \mathcal{A}_\tau(\tau'(i+1)) \subseteq \mathcal{A}_\tau(b)$ which is again a contradiction. This proves that $\mathcal{Q}\subseteq S_{n,r}(123)$, and consequently $S_{n,r}(123)=\mathcal{Q}$. Therefore, by [Lemma 12](#Lemma:AscendIn123Avoiding){reference-type="ref" reference="Lemma:AscendIn123Avoiding"}, $$|S_{n, r}(123)|=|\mathcal{Q}|=\sum_{i=1}^{r-1}\binom{i+n-r}{i}a_{r-1}(i)=\sum_{i=1}^{r-1}\binom{i+n-r}{i}\left[\binom{2r-i-3}{r-2}-\binom{2r-i-3}{r-1}\right].$$ ◻ **Example 14**. Consider $S_{n,r}(123)$ with $n=6$. From [\[first_table,Table:SingleClassical\]](#first_table,Table:SingleClassical){reference-type="ref" reference="first_table,Table:SingleClassical"}, we have $|S_{6,1}(123)|=1$, $|S_{6,2}(123)|=6-1=5$, $|S_{6,5}(123)|=C_5=42$, and $|S_{6,6}(123)|=C_5=42$. It remains to calculate $|S_{6,3}(123)|$ and $|S_{6,4}(123)|$. If $\tau\in S_{6,3}(123)$, then the subpermutation $\tau''$ of $\tau$ on $\{4,5,6\}$ is $654$ and the subpermutation $\tau'$ of $\tau$ on $\{1,2\}$ could be either $12$ or $21$. If $\tau'=21$, then any shuffle of $\tau'$ and $\tau''$ will guarantee that $\pi$ avoids 123. In this case, we have $\binom{3+2}{2}$ choices. If $\tau'=12$, then $4,5,6$ must be ancestors of $2$. The number of shuffles of $654$ and $1$ is $\binom{3+1}{1}$. So we have $$|S_{6,3}(123)|=\binom{3+1}{1}+\binom{3+2}{2}=4+10=14.$$ If $\tau\in S_{6,4}(123)$, then the subpermutation $\tau''$ of $\tau$ on $\{5,6\}$ is $65$. Since the subpermutation $\tau'$ of $\tau$ on $\{1,2,3\}$ avoids $123$, there are 5 choices for $\tau'$. They are $132$, $231$, $312$, $213$, and $321$. Out of these 5 choices, two are of the form $\tau'(1)<\tau'(2)>\tau'(3)$, two are of the form $\tau'(1)>\tau'(2)<\tau'(3)$, and one is of the form $\tau'(1)>\tau'(2)>\tau'(3)$. If $\tau'(1)<\tau'(2)>\tau'(3)$, then we must have $\{5,6\}\in\mathcal{A}_\tau(\tau'(2))$ and hence we have $\binom{2+1}{2}$ shuffles of $65$ and $\tau'(1)$ for any given $\tau'$. If $\tau'(1)>\tau'(2)<\tau'(3)$, then we must have $\{5,6\}\in \mathcal{A}_\tau(\tau'(3))$ and hence we have $\binom{2+2}{2}$ shuffles of $65$ and $\tau'(1)\tau'(2)$ for any given $\tau'$. If $\tau'(1)>\tau'(2)>\tau'(3)$, then we have $\binom{2+3}{3}$ shuffles of $65$ and $\tau'(1)\tau'(2)\tau'(3)=321$. So we have $$|S_{6,4}(123)|=2\binom{2+1}{1}+2\binom{2+2}{2}+\binom{2+3}{3}=6+12+10=28.$$ Adding up the contributions, we have $$\sum_{r=1}^6|S_{n,r}(123)|=1+5+14+28+42+42=132=C_6.$$  \ The formulas for $|S_{n,r}(321)|$ and $|S_{n,r}(312)|$ follow directly from [Lemma 3](#lemma:complements){reference-type="ref" reference="lemma:complements"} and [Theorem 13](#Theorem:123;132){reference-type="ref" reference="Theorem:123;132"}. **Corollary 15**. *For all $n\geq 1$, $$|S_{n,n}(321)|=|S_{n,n}(312)|=1;$$ and for all $n\geq 2$ and $1 \leq r\leq n-1$, we have $$|S_{n,r}(321)|=|S_{n,r}(312)|=\sum_{i=1}^{n-r}\binom{i+r-1}{i}\left[\binom{2n-2r-i-1}{n-r-1}-\binom{2n-2r-i-1}{n-r}\right].$$* Next we classify the $r$-Wilf-equivalence classes for patterns of length three for all $r\in\mathbb{N}$. **Theorem 16**. *There are two $1$-Wilf-equivalence classes for patterns of length three: $123\stackrel{1}{\sim} 132$ and $321\stackrel{1}{\sim}312\stackrel{1}{\sim}213\stackrel{1}{\sim}231$. For $r\geq2$, there are three $r$-Wilf-equivalence classes for patterns of length three: $213\stackrel{r}{\sim}231$, $123\stackrel{r}{\sim}132$, and $321\stackrel{r}{\sim}312$.* *Proof.* The fact that there are two $1$-Wilf-equivalence classes for patterns of length three follows from the last row of [2](#Table:SingleClassical){reference-type="ref" reference="Table:SingleClassical"}. Let $r\geq2$. By [\[Theorem:213;231,Theorem:123;132\]](#Theorem:213;231,Theorem:123;132){reference-type="ref" reference="Theorem:213;231,Theorem:123;132"}, and [Corollary 15](#Corollary:321;312){reference-type="ref" reference="Corollary:321;312"}, we have $213\stackrel{r}{\sim}231$, $123\stackrel{r}{\sim}132$, and $321\stackrel{r}{\sim}312$. We need to show that there exist $n_1,n_2,n_3\geq r$ such that $|S_{n_1,r}(213)|\neq|S_{n_1,r}(123)|$, $|S_{n_2,r}(213)|\neq|S_{n_2,r}(321)|$, and $|S_{n_3,r}(123)|\neq|S_{n_3,r}(321)|$. There are three cases to consider. Case 1: $r=2$. Set $n_1=n_2=n_3=4$. By [\[first_table,Table:SingleClassical\]](#first_table,Table:SingleClassical){reference-type="ref" reference="first_table,Table:SingleClassical"}, we have $|S_{4,2}(123)|=4-1=3$, $|S_{4,2}(321)|=C_3=5$, and $|S_{4,2}(213)|=C_2=2$. Hence we have the desired result. Case 2: $r=3$. Set $n_1=n_2=n_3=4$. By [\[first_table,Table:SingleClassical\]](#first_table,Table:SingleClassical){reference-type="ref" reference="first_table,Table:SingleClassical"}, we have $|S_{4,3}(123)|=C_4=5$, $|S_{4,3}(321)|=4-1=3$, and $|S_{4,3}(213)|=C_2=2$. Hence we have the desired result. Case 3: $r\geq3$. Set $n_1=n_2=n_3=r+1$. By [2](#Table:SingleClassical){reference-type="ref" reference="Table:SingleClassical"}, we have $|S_{r+1,r}(123)|=C_r$, $|S_{r+1,r}(321)|=r$, and $|S_{r+1,r}(213)|=C_{r-1}$. By [1](#first_table){reference-type="ref" reference="first_table"}, we have $C_r>C_{r-1}>r$ and the theorem follows. ◻ **Remark 17**. Anders and Archer [@AndersArcher2019 p. 7] showed that, for rooted labeled forests, there are three forest-Wilf-equivalence classes for consecutive patterns[^1]. However, the equivalence classes in [@AndersArcher2019] are different from the ones in [Theorem 16](#Theorem:rWilf3){reference-type="ref" reference="Theorem:rWilf3"}. # Permutations Avoiding Pairs of Patterns of Length Three {#Section:Pair3} In this section we enumerate $|S_{n,r}(\sigma_1,\sigma_2)|$, where $\sigma_1$ and $\sigma_2$ are two distinct patterns of length three. We need the following results by Simion and Schmidt [@SimionSchmidt1985 Section 3]: **Theorem 18**. *For all $n\geq1$, $$\begin{split} |S_n(123,132)|=&|S_n(321,312)|=|S_n(123,213)|=|S_n(321,231)|=|S_n(132,213)|=|S_n(312,231)|\\=&|S_n(132,231)|=|S_n(312,213)|=|S_n(132,312)|=|S_n(213,231)|=2^{n-1}, \end{split}$$ $$|S_n(123,312)|=|S_n(321,132)|=|S_n(123,231)|=|S_n(321,213)|=\binom{n}{2}+1,$$ and $$|S_n(123,321)|=\left\{\begin{array}{ll} 0 & \text{ if }~~n\geq5, \\ n & \text{ if }~~n=1\text{ or }n=2,\\ 4 & \text{ if }~~n=3\text{ or }n=4. \end{array}\right.$$* Out of the 15 pairs of patterns of length 3, there are three self-complementary pairs: $\{123,321\}$, $\{132,312\}$, and $\{213,231\}$. That is, $\{123^c,321^c\}=\{123,321\}$, $\{132^c,312^c\}=\{132,312\}$, and $\{213^c,231^c\}=\{213,231\}$. We start with these patterns. **Proposition 19**. *For all $1\leq r\leq n$, $$|S_{n,r}(123,321)|=\left\{\begin{array}{ll} 0 &\text{ if }~~n\geq 5,\text{ or } n=4 \text{ and } r=1,\text{ or }n=4 \text{ and } r=4, \\ 1 &\text{ if }~~n=1, \text{ or }n=2, \text{ or }n=3 \text{ and }r=1,\text{ or }n=3\text{ and }r=3,\\ 2 &\text{ if }~~n=3 \text{ and }r=2,\text{ or }n=4 \text{ and }r=2,\text{ or }n=4 \text{ and }r=3. \end{array}\right.$$* *Proof.* By the Erdős-Szekeres theorem [@ErdosSzekeres1935 p. 467], for $n\geq5$, every $\pi\in S_n$ has either an increasing or a decreasing subpermutation/subseqeunce of length three. Hence $|S_{n,r}(123,321)|=0$ if $n\geq5$. The rest follows from routine calculation. ◻ **Proposition 20**. *For all $1\leq r\leq n$, $$|S_{n,r}(132,312)|=\binom{n-1}{r-1}.$$* *Proof.* Let $\tau\in S_{n,r}(132,312)$. Since $\tau$ avoids $132$, the subpermutation of $\tau$ on $\{r+1,\ldots,n\}$ is $(r+1,\ldots,n)$. Since $\tau$ avoids $312$, the subpermutation of $\tau$ on $\{1,\ldots,r-1\}$ is $(r-1,\ldots,1)$. The number of shuffles of $(r-1,\ldots,1)$ and $(r+1,\ldots,n)$ is $\binom{n-1}{r-1}$. So we have $$|S_{n,r}(132,312)|\leq\binom{n-1}{r-1}.$$ By [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"}, we have $$2^{n-1}=|S_n(132,312)|=\sum_{r=1}^n|S_{n,r}(132,312)|\leq\sum_{r=1}^n\binom{n-1}{r-1}=2^{n-1}.$$ Therefore $$|S_{n,r}(132,312)|=\binom{n-1}{r-1}.$$ ◻ **Proposition 21**. *For all $1\leq r\leq n$, $$|S_{n,r}(213,231)|=\left\{\begin{array}{ll} 0 & \text{ if }~~r\neq1,n,\\ 2^{n-2} &\text{ if }~~r=1\text{ or }n. \end{array}\right.$$* *Proof.* First suppose $r=1$ or $n$. For any $\tau\in S_{n,r}$ and $2\leq i<j\leq n$, $(r,\tau(i),\tau(j))$ is not a 213 or 231 pattern because $\tau(i)$ and $\tau(j)$ are either both greater than $r$ or both smaller than $r$. It follows that $\tau\in S_{n,r}(213,231)$ if and only if $(\tau(2),\ldots,\tau(n))$ avoids both $213$ and $231$. So by [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"}, we have $$|S_{n,1}(213,231)|=|S_{n,n}(213,231)|=|S_{n-1}(213,231)|=2^{n-2}.$$ By [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"} again, for all $r\neq1,n$, we have $$|S_{n,r}(213,231)|\leq|S_n(213,231)|-|S_{n,1}(213,231)|-|S_{n,n}(213,231)|=2^{n-1}-2^{n-2}-2^{n-2}=0.$$ Therefore for all $r\neq 1,n$, we have $|S_{n,r}(213,231)|=0$. ◻ We have 12 pairs left to consider. By [Lemma 3](#lemma:complements){reference-type="ref" reference="lemma:complements"}, it suffices to look at $\{123,132\}$, $\{123,213\}$, $\{132,213\}$, $\{132,231\}$, $\{123,312\}$, and $\{123,231\}$. **Proposition 22**. *For all $1\leq r\leq n$, $$|S_{n,r}(123,132)|=\left\{\begin{array}{ll} 0 &\text{ if }~~r<n-1, \\ 2^{n-2} &\text{ if }~~r=n-1\text{ or }r=n. \end{array} \right.$$* *Proof.* Let $1\leq r\leq n$ and let $\tau\in S_{n,r}$. If $r<n-1$, then $\tau$ contains either $(r,n-1,n)$ or $(r,n,n-1)$ as a subpermutation and hence $\tau$ contains either the pattern $123$ or the pattern $132$. Therefore we have $|S_{n,r}(123,132)|=0$ if $r<n-1$. Now suppose $r=n-1$ or $n$. Then for any $2\leq i<j\leq n$, $(r,\tau(i),\tau(j))$ avoids both $123$ and $132$. So $\tau\in S_{n,r}(123,132)$ if and only if $(\tau(2),\ldots,\tau(n))$ avoids both $123$ and $132$. Consequently by [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"} $$|S_{n,n-1}(123,132)|=|S_{n,n}(123,132)|=|S_{n-1}(123,132)|=2^{n-2}.$$ ◻ **Proposition 23**. *For all $1\leq r\leq n$, $$|S_{n,r}(123,213)|=|S_{n,r}(132,213)|=|S_{n,r}(132,231)|=\left\{\begin{array}{ll} 1 &\text{ if }~~r=1, \\ 2^{r-2} &\text{ if }~~r\geq2. \end{array}\right.$$* *Proof.* We will prove that $|S_{n,1}(123,213)|=1$ and, for all $r\geq2$, $|S_{n,r}(123,213)|=2^{r-2}$. The proofs for the other two pairs are similar and will be omitted. Let $1\leq r\leq n$ and $\tau\in S_{n,1}(123,213)$. Since $\tau$ avoids $123$, the subpermutation of $\tau$ on $\{2,\ldots,n\}$ is $(n,n-1,\ldots,2)$. This proves that $|S_{n,1}(123,213)|=1$. Now suppose $r\geq2$. Let $\tau\in S_{n,r}(123,213)$. Since $\tau$ avoids $123$, the subpermutation of $\tau$ on $\{r+1,\ldots,n\}$ is $(n,n-1,\ldots,r+1)$. Since $\tau$ avoids $213$, the numbers $r+1$, ..., $n$ are ancestors of all numbers in $\{1,\ldots,r-1\}$. At the same time, the subpermutation on $\{1,\ldots,r-1\}$ avoids both $123$ and $213$. So by [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"}, we have $$|S_{n,r}(123,213)|\leq |S_{r-1}(123,213)|=2^{r-2}.$$ It is easy to check that for all $\tau'\in S_{r-1}(123,213)$, we have $(r,n,n-1,\ldots,r+1,\tau'(1),\ldots,\tau'(r-1))\in S_{n,r}(123,213)$. Therefore $$|S_{n,r}(123,213)|=|S_{r-1}(123,213)|=2^{r-2}.$$ ◻ **Proposition 24**. *For all $1\leq r\leq n$, $$|S_{n,r}(123,312)|=\left\{\begin{array}{ll} 1 &\text{ if }~~r=n, \\ r &\text{ if }~~r<n. \end{array}\right.$$* *Proof.* Let $n\geq1$ and $\tau\in S_{n,n}(123,312)$. Since $\tau$ avoids $312$, $\tau=(n,n-1,\ldots,1)$ and therefore $|S_{n,n}(123,312)|=1$. Now let $1\leq r<n$ and $\tau\in S_{n,r}(123,312)$. Let $\tau'$ be the subpermutation on $\{1,\ldots,r-1\}$ and let $\tau''$ be the subpermutation on $\{r+1,\ldots,n\}$. Since $\tau$ avoids $123$, $\tau''=(n,n-1,\ldots,r+1)$, and since $\tau$ avoids $312$, $\tau'=(r-1,\ldots,1)$. Moreover, if $\tau(i)\in\{1,..,r-1\}$ and $\tau(j),\tau(k)\in\{r+1,\ldots,n\}$ with $j<k$, either $\tau(j),\tau(k)\in\mathcal{A}_\tau(\tau(i))$ or $\tau(j),\tau(k)\in\mathcal{D}_\tau(\tau(i))$. Otherwise, we would have a $312$ pattern from $(\tau(j), \tau(i), \tau(k))$. Therefore the number of shuffles of $\tau'$ and $\tau''$ that do not create a $312$ pattern is simply $\binom{r-1+1}{1}=r$. It is easy to check that none of these shuffles creates a $123$ pattern. Hence $|S_{n,r}(123,312)|=r$. ◻ **Proposition 25**. *For all $1\leq r\leq n$, $$|S_{n,r}(123,231)|=\left\{\begin{array}{ll} 1 &\text{ if }~~r<n,\\ \binom{n-1}{2}+1 &\text{ if }~~r=n, \end{array}\right.$$ where we use the convention that $\binom{0}{2}=\binom{1}{2}=0$.* *Proof.* Let $n\geq1$. First note that $\tau\in S_{n,n}(123,231)$ if and only if $(\tau(2),\ldots,\tau(n))\in S_{n-1}(123,231)$. So by [Theorem 18](#Theorem:Pairs3){reference-type="ref" reference="Theorem:Pairs3"}, $$|S_{n,n}(123,231)|=|S_{n-1}(123,231)|=\binom{n-1}{2}+1.$$ Now suppose $n\geq2$ and $1\leq r<n$. Let $\tau\in S_{n,r}(123,231)$. Since $\tau$ avoids $123$, the subpermutation $\tau'$ of $\tau$ on $\{r+1,\ldots,n\}$ is $(n,n-1,\ldots,r+1)$. Since $\tau$ avoids $231$, $\{r+1,\ldots,n\}\subseteq\mathcal{D}_\tau(i)$ for all $i<r$. Therefore $(\tau(2),\ldots,\tau(r))$ is a subpermutation of $\tau$ on $\{1,\ldots,r-1\}$. If $\tau(i)<\tau(j)$ for some $i,j\in\{2,\ldots,r\}$, then $(\tau(i),\tau(j),\tau(r+1))$ would be a $123$ pattern. Hence $(\tau(2),\ldots,\tau(r))=(r,r-1,\ldots,2)$ and $|S_{n,r}(123,231)|=1$. ◻ # Permutations Avoiding 3412 and 3421 {#Section:Pair4} The goal of this section is to show that the counting argument used in the proof of [Theorem 9](#Theorem:213;231){reference-type="ref" reference="Theorem:213;231"} can be generalized to permutations avoiding both 3412 and 3421. More specifically, we obtain exact formulas for $|S_{n,r}(3412,3421)|$, which can then be used to obtain a new proof of a recurrence relation satisfied by the large Schröder numbers $\mathbb{S}_n$. We need the following result proved by Kremer [@Kremer2000 Corollary 9]: **Theorem 26**. *For all $n\geq1$, $$|S_n(3412,3421)|=\mathbb{S}_{n-1},$$ where $\mathbb{S}_{n-1}$ is the $(n-1)$st large (big) Schröder number.* We can use [Theorem 26](#Theorem:34123421){reference-type="ref" reference="Theorem:34123421"} to enumerate $|S_{n,r}(3412,3421)|$. **Theorem 27**. *For all $n\geq2$ and $r\in\{1,2,n\}$, we have $$|S_{n,r}(3412,3421)|=\mathbb{S}_{n-2};$$ and for all $n\geq4$ and $2<r<n$, we have $$|S_{n,r}(3412,3421)|=\mathbb{S}_{r-2}\mathbb{S}_{n-r}.$$* *Proof.* Fist, suppose $n\geq 1$ and $r\in\{1,n\}$. Let $\tau\in S_{n,r}$. If $r=1$, then $r<a$ for all $a\in\mathcal{D}_\tau(r)$. If $r=n$, then $r>a$ for all $a\in\mathcal{D}_\tau(r)$. If $r=2$, then there is exactly one $a\in \mathcal{D}_\tau(r)$. In any case, $rabc$ is not a 3412 pattern or a 3421 pattern for any $a,b,c\in\mathcal{D}_\tau(r)$. Hence $\tau\in S_{n,r}(3412,3421)$ if and only if $(\tau(2),\ldots,\tau(n))$ avoids both $3412$ and $3421$. Therefore, by [Theorem 26](#Theorem:34123421){reference-type="ref" reference="Theorem:34123421"}, we have $|S_{n,r}(3412,3421)|=|S_{n-1}(3412,3421)|=\mathbb{S}_{n-2}$. Now suppose $n\geq4$ and $2<r<n$. Let $\mathcal{R}$ be a subset of $S_{n,r}$ such that every $\tau\in\mathcal{R}$ has the following properties: - $\{\tau(2),\ldots,\tau(r-1)\}\subseteq\{1,\ldots,r-1\}$; - the subpermutation $\tau'$ of $\tau$ on $\{1,\ldots,r-1\}$ avoids both $3412$ and $3421$; - $\tau''=(\tau(r),\ldots,\tau(n))$ avoids both $3412$ and $3421$. Let $\tau\in \mathcal{R}$. By [Theorem 26](#Theorem:34123421){reference-type="ref" reference="Theorem:34123421"}, there are $\mathbb{S}_{r-2}$ ways for $\tau'$ to avoid both $3412$ and $3421$, and for each fixed $\tau'$, there are $\mathbb{S}_{n-r}$ ways for $\tau''$ to avoid both $3412$ and $3421$. Hence, we have $|\mathcal{R}|=\mathbb{S}_{r-2}\mathbb{S}_{n-r}$. Now we show that $S_{n,r}(3412,3421)=\mathcal{R}$. Let $\tau\in S_{n,r}(3412,3421)$, $\tau'$ the subpermutation of $\tau$ on $\{1,\ldots,r-1\}$, and $\tau''=(\tau(r),\ldots,\tau(n))$. Since $\tau$ avoids both $3412$ and $3421$, $\tau'$ avoids both $3412$ and $3421$ as well. Similarly, $\tau''$ avoids both $3412$ and $3421$. We now show that $\tau(i)\in\{1,\ldots,r-1\}$ for all $i=2,...,r-1$. Suppose, by way of contradiction, that $\tau(i)>r$ for some $i\in\{2,\ldots,r-1\}$. Then, since $\tau(1)=r$, at most $r-3$ numbers in $\{\tau(1),\ldots,\tau(r-1)\}$ are less than $r$. So there exist $k>j>r-1$ such that $\tau(j),\tau(k)<r$. Now $(r,\tau(i),\tau(j),\tau(k))$ is either a 3412 pattern or a 3421 pattern. This is a contradiction. Hence, we have $S_{n,r}(3412,3421)\subseteq\mathcal{R}$. On the other hand, suppose $\tau\in \mathcal{R}$. We will show that $\tau\in S_{n,r}(3412,3421)$. Suppose, by way of contradiction, that $xyzw$ is a subpermutation of $\tau$ which is a $3412$ pattern or a $3421$ pattern. Then we have $z, w<x<y$ and $z,w\in D_\tau(y)$. We split the argument into three cases. Case 1: $x=r$. Then $y>r$. Since $\{\tau(2),\ldots,\tau(r-1)\} \subseteq \{1,\ldots r-1\}$, we must have $y=\tau(i)$ for some $i>r-1$ and at most one $j>i$ with $\tau(j)<r$. So either $z>r=x$ or $w>r=x$ which is a contradiction. Case 2: $x<r$. Since the subpermutation on $\{1,\ldots,r-1\}$ avoids both $3412$ and $3421$, we must have $y>r$. The rest of the argument is then the same as Case 1. Case 3: $x>r$. Since $\{\tau(2),\ldots,\tau(r-1)\}\subseteq\{1,\ldots,r-1\}$, $xyzw$ is a subpermutation of $(\tau(r),\ldots,\tau(n))$. Since $(\tau(r),\ldots,\tau(n))$ avoids both $3412$ and $3421$, $xyzw$ is not a $3412$ or $3421$ pattern. This is a contradiction. This completes the proof that $\mathcal{R}\subseteq S_{n,r}(3412,3421)$. Hence we have $S_{n,r}(3412,3421)=\mathcal{R}$, and therefore $$|S_{n,r}(3412,3421)|=|\mathcal{R}|=\mathbb{S}_{r-2}\mathbb{S}_{n-r}.$$ ◻ By [Lemma 3](#lemma:complements){reference-type="ref" reference="lemma:complements"}, we also have the following: **Corollary 28**. *For all $n\geq2$ and $r\in\{1,n-1,n\}$, we have $$|S_{n,r}(2143,2134)|=\mathbb{S}_{n-2};$$ and for all $n\geq4$ and $1<r<n-1$, $$|S_{n,r}(2143,2134)|=\mathbb{S}_{r-2}\mathbb{S}_{n-r}.$$* Summing over $r$ in [Theorem 27](#Theorem:34123421leading){reference-type="ref" reference="Theorem:34123421leading"}, we have the following recurrence relation for $\mathbb{S}_n$: **Corollary 29**. *For all $n\geq1$, $$\mathbb{S}_{n+1}=\mathbb{S}_n+\sum_{r=0}^{n}\mathbb{S}_{r}\mathbb{S}_{n-r}.$$* *Proof.* Let $n\geq1$. Note that by [1](#first_table){reference-type="ref" reference="first_table"}, we have $\mathbb{S}_0=1$. So by [Theorem 27](#Theorem:34123421leading){reference-type="ref" reference="Theorem:34123421leading"}, we have $|S_{n+2,2}(3412,3421)|=|S_{n+2,n+2}(3412,3421)|=\mathbb{S}_n=\mathbb{S}_n\mathbb{S}_0$. Now, by [\[Theorem:34123421,Theorem:34123421leading\]](#Theorem:34123421,Theorem:34123421leading){reference-type="ref" reference="Theorem:34123421,Theorem:34123421leading"}, we have $$\begin{split} \mathbb{S}_{n+1}=|S_{n+2}(3412,3421)|=\sum_{r=1}^{n+2}|S_{n+2,r}(3412,3421)|=\mathbb{S}_{n}+\sum_{r=2}^{n+2}\mathbb{S}_{r-2}\mathbb{S}_{n+2-r}=\mathbb{S}_n+\sum_{r=0}^{n}\mathbb{S}_{r}\mathbb{S}_{n-r}. \end{split}$$ ◻ **Remark 30**. Qi and Guo [@QiGuo2017 Theorem 5] proved [Corollary 29](#Corollary:RecurrenceSchroder){reference-type="ref" reference="Corollary:RecurrenceSchroder"} using generating functions. In [@Bona2022 p. 446], it is also noted that [Corollary 29](#Corollary:RecurrenceSchroder){reference-type="ref" reference="Corollary:RecurrenceSchroder"} can also be derived from the recurrence $\mathbb{S}_n=\sum_{i=0}^n\binom{2n-i}{i}C_{n-i}$ which was proved by West [@West1996 p. 255]. Our proof of this identity does not use the Catalan numbers and is purely combinatorial. # Extensions and Concluding Remarks {#Section:Concluding} We have only scratched the surface of enumerating pattern-avoiding permutations by leading terms, mostly concentrating on patterns of length three. It would be interesting to study permutations avoiding other patterns by leading terms. All single patterns of length greater than three are open. Another direction would be generalized patterns. For example, many papers studied consecutive patterns [@ElizaldeNoy2003], bivincular patterns [@BCDK2010], and mesh patterns [@BrandenClaesson2011; @HJSVU2015]. Here we briefly describe, for all $r\geq5$, the $r$-Wilf equivalence classes for vincular patterns of length three studied by Babson and Steingrímsson [@BabsonSteingrimsson2000] and later, Claesson [@Claesson2001]. In vincular patterns [@HJSVU2015 Section 2], some consecutive elements in a permutation pattern are required to be adjacent. We use overlines to indicate that the elements under the overlines are required to be adjacent. There are twelve vincular patterns of length three where one requires exactly two numbers to be adjacent. For example, a permutation $\tau\in S_n$ contains the pattern $1\overline{32}$ if there exist indices $i<j$ such that $(\tau(i),\tau(j),\tau(j+1))$ is a $132$ pattern. Other vincular patterns are defined similarly. **Example 31**. In the permutation $\tau=13542\in S_5$, $(\tau(2),\tau(3),\tau(5))=352$ is a $\overline{23}1$ pattern and $(\tau(1),\tau(4),\tau(5))=142$ is a $1\overline{32}$ pattern, but $\tau$ avoids the pattern $\overline{21}3$. Claesson [@Claesson2001 Propositions 1-3 and Lemma 2] proved that there are two Wilf-equivalence classes for the twelve vincular patterns. They are counted either by the Catalan numbers or by the Bell numbers: **Theorem 32**. *For all $n\geq1$, $$\begin{split} |S_n(1\overline{23})|=&|S_n(3\overline{21})|=|S_n(\overline{12}3)|=|S_n(\overline{32}1)|=|S_n(1\overline{32})|\\=&|S_n(3\overline{12})|=|S_n(\overline{21}3)|=|S_n(\overline{23}1)|=B_n, \end{split}$$ and $$|S_n(2\overline{13})|=|S_n(2\overline{31})|=|S_n(\overline{13}2)|=|S_n(\overline{31}2)|=C_n,$$ where $B_n$ is the $n$th Bell number and $C_n$ is the $n$th Catalan number.* We first adapt some results in Claesson [@Claesson2001] to show $r$-Wilf equivalence for several vincular patterns. **Proposition 33**. *For all $r\in\mathbb{N}$, $2\overline{13}\stackrel{r}{\sim}2\overline{31}$, $1\overline{23}\stackrel{r}{\sim}1\overline{32}$, and $3\overline{21}\stackrel{r}{\sim}3\overline{12}$.* *Proof.* Let $1\leq r\leq n$. Using a short combinatorial argument, Claesson [@Claesson2001 Lemma 2] showed that for all $n\geq1$, $\tau\in S_n$ avoids $2\overline{13}$ if and only if it avoids $213$. Taking complements, for all $n\geq1$, $\tau\in S_n$ avoids $2\overline{31}$ if and only if it avoids $231$. Hence for all $\tau\in S_{n,r}$, $\tau$ avoids $2\overline{13}$ if and only if it avoids $213$ and $\tau$ avoids $2\overline{31}$ if and only if it avoids $231$. Then we have $|S_{n,r}(2\overline{13})|=|S_{n,r}(213)|$ and $|S_{n,r}(2\overline{31})|=|S_{n,r}(231)|$. Now by [Theorem 9](#Theorem:213;231){reference-type="ref" reference="Theorem:213;231"}, we have $|S_{n,r}(2\overline{13})|=|S_{n,r}(213)|=|S_{n,r}(231)|=|S_{n,r}(2\overline{31})|$. Therefore, $2\overline{13}\stackrel{r}{\sim}2\overline{31}$. Claesson [@Claesson2001 Propositions 2 and 4] constructed bijections between $S_n(1\overline{23})$ and the partitions of $[n]$, and then between $S_n(1\overline{32})$ and the partitions of $[n]$. These bijections preserve the leading terms of permutations. So for all $1\leq r\leq n$, we have $|S_{n,r}(1\overline{23})|=|S_{n,r}(1\overline{32})|$. Taking the complements, we also have $|S_{n,r}(3\overline{21})|=|S_{n,r}(3\overline{12})|$. Therefore, we have $1\overline{23}\stackrel{r}{\sim}1\overline{32}$ and $3\overline{21}\stackrel{r}{\sim}3\overline{12}$. ◻ By [Proposition 33](#Proposition:rWilfClassesWithMoreThan1){reference-type="ref" reference="Proposition:rWilfClassesWithMoreThan1"}, there are at most nine $r$-Wilf-equivalence classes for vincular patterns. The following table lists the results we need to classify $r$-Wilf equivalence classes for all twelve vincular patterns. $n=r$ $n=r+1$ $n=r+2$ ------------------------------------------------------- ----------- --------------- ------------------- -- -- $|S_{n,r}(2\overline{13})|=|S_{n,r}(2\overline{31})|$ $C_{r-1}$ $C_{r-1}$ $|S_{n,r}(\overline{13}2)|$ $C_{r-1}$ $C_r$ $|S_{n,r}(3\overline{21})|=|S_{n,r}(3\overline{12})|$ $1$ $2^{r-1}$ $|S_{n,r}(\overline{31}2)|$ $1$ $r$ $|S_{n,r}(1\overline{23})|=|S_{n,r}(1\overline{32})|$ $B_{r-1}$ $B_r$ $B_{r+1}-B_{r-1}$ $|S_{n,r}(\overline{12}3)|$ $B_{r-1}$ $B_r$ $B_{r+1}-B_r$ $|S_{n,r}(\overline{21}3)|$ $B_{r-1}$ $B_{r-1}$ $|S_{n,r}(\overline{23}1)|$ $B_{r-1}$ $B_r-B_{r-1}$ $|S_{n,r}(\overline{32}1)|$ $B_{r-2}$ : Avoiding Vincular Patterns by Leading Terms for $r\geq3$. (We leave some entries in the table blank and only include results that are needed to classify $r$-Wilf equivalence classes for the twelve vincular patterns.) Most of the expressions in [3](#Table:GeneralizedPatterns){reference-type="ref" reference="Table:GeneralizedPatterns"} can be obtained by straightforward calculation using [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"} and [2](#Table:SingleClassical){reference-type="ref" reference="Table:SingleClassical"}. We will only sketch the proofs of a few of them. **Lemma 34**. *For all $r\geq3$, $$|S_{r,r}(\overline{32}1)|=B_{r-2}.$$* *Proof.* Let $\tau\in S_{r,r}(\overline{32}1)$. If $\tau(2)\neq 1$, then $(\tau(1),\tau(2),1)$ is a $\overline{32}1$ pattern. So we must have $\tau(2)=1$. At the same time, $(\tau(3),\ldots,\tau(n))$ avoids the pattern $\overline{32}1$. So by [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"}, we have $|S_{r,r}(\overline{32}1)|\leq|S_{r-2}(\overline{32}1)|=B_{r-2}$. Now let $\tau\in S_{r,r}$ with $\tau(2)=1$ and $(\tau(3),\ldots,\tau(n))$ avoiding the pattern $\overline{32}1$. Since $r1x$ and $1xy$ are never $\overline{32}1$ patterns, we must have $\tau\in S_{r,r}(\overline{32}1)$. Hence we have $B_{r-2}=|S_{r-2}(\overline{32}1)|\leq |S_{r,r}(\overline{32}1)|$. This completes the proof of the lemma. ◻ **Lemma 35**. *For all $r\geq1$, $$|S_{r+2,r}(1\overline{23})|=B_{r+1}-B_{r-1} \text{ and } |S_{r+2,r}(\overline{12}3)|=B_{r+1}-B_{r}.$$* *Proof.* We first prove that $|S_{r+2,r}(1\overline{23})|=B_{r+1}-B_{r-1}$. Let $\tau\in S_{r+2,r}(1\overline{23})$. Then $(\tau(2),\ldots,\tau(r+2))$ avoids $1\overline{23}$. By [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"}, there are $|S_{r+1}(1\overline{23})|=B_{r+1}$ ways for $(\tau(2),\ldots,\tau(r+2))$ to avoid $1\overline{23}$. For these permutations on $\{1,\ldots,r-1,r+1,r+2\}$, the only way that $r+1$ and $r+2$ are adjacent and the subpermutation on $\{r+1,r+2\}$ is $(r+1,r+2)$ is when $\tau(2)=r+1$ and $\tau(3)=r+2$ because otherwise $(\tau(2),\ldots,\tau(r+2))$ would contain a $1\overline{23}$ pattern. This is the only case that $(r,r+1,r+2)$ is a $1\overline{23}$ pattern. Since $(\tau(4),\ldots,\tau(r+2))$ also need to avoid $1\overline{23}$, by [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"}, the number of permutations $(\tau(2),\ldots,\tau(r+2))$ avoiding $1\overline{23}$, with $\tau(2)=r+1$ and $\tau(3)=r+2$, is $|S_{r-1}(1\overline{23})|=B_{r-1}$. Here it is easy to check that if $\tau(2)=r+1$, $\tau(3)=r+2$, and $(\tau(4),\ldots,\tau(r+2))$ avoids $1\overline{23}$, then $(\tau(2),\ldots,\tau(r+2))$ avoids $1\overline{23}$ as well. Therefore $$|S_{r+2,r}(1\overline{23})|=|S_{r+1}(1\overline{23})|-|S_{r-1}(1\overline{23})|=B_{r+1}-B_{r-1}.$$ Next, we prove that $|S_{r+2,r}(\overline{12}3)|=B_{r+1}-B_r$. Let $\tau\in S_{r+2,r}(\overline{12}3)$. Then $(\tau(2),\ldots,\tau(r+2))$ avoids $\overline{12}3$. By [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"}, there are $|S_{r+1}(\overline{12}3)|=B_{r+1}$ ways for $(\tau(2),\ldots,\tau(n))$ to avoid $\overline{12}3$. For these permutations, the only way that we have a $\overline{12}3$ pattern starting with $r$ is when $\tau(2)=r+1$, then $(\tau(1),\tau(2),r+2)$ is a $\overline{12}3$ pattern. Here, it is easy to see that if $\tau(2)=r+1$, then, for all $2<i<j\leq r+2$, $(\tau(2),\tau(i),\tau(j))$ is never a $\overline{12}3$ pattern. Hence, by [Theorem 32](#Theorem:GeneralizedSingle3){reference-type="ref" reference="Theorem:GeneralizedSingle3"}, the number of permutations $(\tau(2),\ldots,\tau(r+2))$, with $\tau(2)=r+1$, avoiding $\overline{12}3$ is $|S_{r}(\overline{12}3)|=B_r$. Using subtraction, we have $$|S_{r+2,r}(\overline{12}3)|=|S_{r+1}(\overline{12}3)|-|S_{r}(\overline{12}3)|=B_{r+1}-B_{r}.$$ ◻ **Lemma 36**. *For all $r\geq1$, $$|S_{r+1,r}(3\overline{21})|=2^{r-1}.$$* *Proof.* Let $\tau\in S_{r+1,r}(3\overline{21})$ and let $i>1$ be such that $\tau(i)=r+1$. Then for all $j\in\{2,\ldots,i-2\}$, we must have $\tau(j)<\tau(j+1)$. To see this, suppose that $\tau(j)>\tau(j+1)$ for some $j\in\{2,\ldots,i-2\}$. Then $(r,\tau(j),\tau(j+1))$ is a $3\overline{21}$ pattern which is a contradiction. Similarly, for all $j\in\{i+1,\ldots,n-1\}$, we must have $\tau(j)<\tau(j+1)$. Hence, we have $\tau(2)<\tau(3)<\cdots<\tau(i-1)$ and $\tau(i+1)<\tau(i+2)<\cdots<\tau(r+1)$. On the other hand, it is easy to check that for all $\tau\in S_{r+1,r}$, if $\tau(i)=r+1$, $\tau(2)<\tau(3)<\cdots<\tau(i-1)$ , and $\tau(i+1)<\tau(i+2)<\cdots<\tau(r+1)$ for some $i>1$, then $\tau$ avoids $3\overline{21}$. So $|S_{r+1,r}(3\overline{21})|$ is equal to the number of permutations $\tau\in S_{r+1,r}$ such that for some $i\in\{2,\ldots,r+1\}$, we have $\tau(i)=r+1$, $\tau(2)<\tau(3)<\cdots<\tau(i-1)$, and $\tau(i+1)<\tau(i+2)<\cdots<\tau(r+1)$. Let $\tau$ be such a permutation and $i\in\{2,\ldots,r+1\}$. Then there are $\binom{r-1}{i-2}$ ways to choose $i-2$ numbers from $\{1,\ldots,r-1\}$ and assign them to $\tau(2),\ldots,\tau(i-1)$ so that $\tau(2)<\tau(3)<\cdots<\tau(i-1)$; once $\tau(2),\ldots,\tau(i-1)$ are determined, $\tau(i+1),\ldots,\tau(r+1)$ are uniquely determined as well. Hence we have $$|S_{r+1,r}(3\overline{21})|=\sum_{i=2}^{r+1}\binom{r-1}{i-2}=\sum_{i=0}^{r-1}\binom{r-1}{i}=2^{r-1}.$$ ◻ If $r\geq5$, by [\[first_table,Table:GeneralizedPatterns\]](#first_table,Table:GeneralizedPatterns){reference-type="ref" reference="first_table,Table:GeneralizedPatterns"} and [\[Lemma:CatalanBell,Lemma:Bell\]](#Lemma:CatalanBell,Lemma:Bell){reference-type="ref" reference="Lemma:CatalanBell,Lemma:Bell"}, there are nine $r$-Wilf equivalence classes. To see this, it suffices to note that for each $r\geq5$ and for any two distinct generalized patterns $\sigma$ and $\sigma'$ in different rows, either $|S_{r,r}(\sigma)|\neq|S_{r,r}(\sigma')|$, or $|S_{r+1,r}(\sigma)|\neq|S_{r+1,r}(\sigma')|$, or $|S_{r+2,r}(\sigma)|\neq|S_{r+2,r}(\sigma')|$. We briefly describe several of them as examples. **Example 37**. By [3](#Table:GeneralizedPatterns){reference-type="ref" reference="Table:GeneralizedPatterns"} and [Lemma 7](#Lemma:CatalanBell){reference-type="ref" reference="Lemma:CatalanBell"}, for all $r\geq5$, $|S_{r,r}(\overline{13}2)|=C_{r-1}<B_{r-1}=|S_{r,r}(\overline{12}3)|$. Hence, for all $r\geq5$, $\overline{13}2$ and $\overline{12}3$ are not $r$-Wilf equivalent. **Example 38**. By [3](#Table:GeneralizedPatterns){reference-type="ref" reference="Table:GeneralizedPatterns"} and [Lemma 8](#Lemma:Bell){reference-type="ref" reference="Lemma:Bell"}, for all $r\geq5$, $|S_{r,r}(\overline{21}3)|=B_{r-1}=|S_{r,r}(\overline{23}1)|$, but $|S_{r+1,r}(\overline{21}3)|=B_{r-1}<B_r-B_{r-1}=|S_{r+1,r}(\overline{23}1)|$. Hence, for all $r\geq5$, $\overline{21}3$ and $\overline{23}1$ are not $r$-Wilf equivalent. **Example 39**. By [3](#Table:GeneralizedPatterns){reference-type="ref" reference="Table:GeneralizedPatterns"}, we have $|S_{r,r}(1\overline{23})|=B_{r-1}=|S_{r,r}(\overline{12}3)|$ and $|S_{r+1,r}(1\overline{23})|=B_{r}=|S_{r+1,r}(\overline{12}3)|$, but $|S_{r+2,r}(1\overline{23})|=B_{r+1}-B_{r-1}>B_{r+1}-B_r=|S_{r+2,r}(\overline{12}3)|$ for all $r\geq5$. Hence $1\overline{23}$ and $\overline{12}3$ belong to two different equivalence classes when $r\geq5$. **Example 40**. By [3](#Table:GeneralizedPatterns){reference-type="ref" reference="Table:GeneralizedPatterns"}, we have $|S_{r,r}(2\overline{13})|=C_{r-1}$ and $|S_{r,r}(\overline{32}1)|=B_{r-2}$ for all $r\geq5$. By [1](#first_table){reference-type="ref" reference="first_table"} and the generating functions of the Catalan and Bell numbers [@EgeciogluGarsia2021 Sections 3.2 and 6.1], we have $B_{r-1}\neq C_r$ for all $r\geq5$. Hence $2\overline{13}$ and $\overline{32}1$ belong to two different equivalence classes when $r\geq5$. The following theorem completely classifies, for all $r\geq5$, the $r$-Wilf-equivalence classes for the twelve vincular patterns of length three. **Theorem 41**. *For all $r\geq5$, there are nine $r$-Wilf-equivalence classes for vincular patterns of length three: $2\overline{13}\stackrel{r}{\sim}2\overline{31}$, $1\overline{23}\stackrel{r}{\sim}1\overline{32}$, $3\overline{21}\stackrel{r}{\sim}3\overline{12}$, and the other six classes each contains a single vincular pattern.* # Acknowledgements {#acknowledgements .unnumbered} We would like to thank Lanqing Zhao for discussions during early stages of the project, and Miklós Bóna for pointing out the source of [Lemma 12](#Lemma:AscendIn123Avoiding){reference-type="ref" reference="Lemma:AscendIn123Avoiding"}. Ö. Eğecioğlu would like to acknowledge his sabbatical time at Reykjavik University in 2019 during which he had a chance to learn about the combinatorics of pattern avoidance. C. Gaiser and M. Yin were supported by the University of Denver's Professional Research Opportunities for Faculty Fund 80369-145601. 10 K. Anders and K. Archer, Rooted forests that avoid sets of permutations, *European J. Combin.* **77** (2019), 1-16. E. Babson and E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, *Sém. Lothar. Combin.* **44** (2000), B44b, 18 pp. M. Bóna, *Combinatorics of Permutations*, Third Edition, CRC Press, Boca Raton, 2022. M. Bousquet-Mélou, A. Claesson, M. Dukes, and S. Kitaev, $(2+2)$-free posets, ascent sequences and pattern avoiding permutations, *J. Combin. Theory Ser. A* **117** (2010), 884-909. P. Brädén and A. Claesson, Mesh patterns and the expansion of permutation statistics as sums of permutation patterns, *Electron. J. Combin.* **18** (2011), no. 2, \#P5. A. Claesson, Generalized pattern avoidance, *European J. Combin.* **22** (2001), 961-971. Ö. Eğecioğlu and A. Garsia, *Lessons in Enumerative Combinatorics*, Springer, Graduate Texts in Mathematics, 2021. S. Elizalde and M. Noy, Consecutive patterns in permutations, *Adv. in Appl. Math.* **30** (2003), 110-125. P. Erdős and G. Szekeres, A combinatorial problem in geometry, *Compos. Math.* **2** (1935), 463-470. S. Garg and A. Peng, Classical and consecutive pattern avoidance in rooted forests, *J. Combin. Theory Ser. A* **194** (2023), 105699. Í. Hilmarsson, I. Jónsdóttir, S. Sigurðardóttir, L. Viðarsdóttir and H. Ulfarsson, Wilf-classification of mesh patterns of short length, *Electron. J. Combin.* **22** (2015), no. 4, \#P4.13. S. Kitaev, *Patterns in Permutations and Words*, Monographs in Theoretical Computer Science. An EATCS Series, Springer, 2011. D.E. Knuth, *The Art of Computer Programming*, vol. 3, Addison-Wesley, Reading, MA, 1973. D. Kremer, Permutations with forbidden subsequences and a generalized Schröder number, *Discrete Math.* **218** (2000), 121-130. I. Mező, *Combinatorics and Number Theory of Counting Sequences*, CRC Press, Boca Raton, 2020. OEIS Foundation Inc., *The On-Line Encyclopedia of Integer Sequences*, Published electronically at <https://oeis.org>, 2023. A. Panholzer, Consecutive permutation patterns in trees and mappings, *J. Combin.* **12** (2021), no. 1, 17-54. F. Qi and B.N. Guo, Some explicit and recursive formulas of the large and little Schröder numbers, *Arab J. Math. Sci.* **23** (2017), 141-147. M. Ren, Wilf equivalences and Stanley-Wilf limits for patterns in rooted labeled forests, (2020), <https://arxiv.org/abs/2007.12690>. R. Simion, Noncrossing partitions, *Discrete Math.* **217** (2000), 367-409. R. Simion and F. Schmidt, Restricted permutations, *European J. Combin.* **6** (1985), 383-406. R.P. Stanley, *Catalan Numbers*, Cambridge University Press, Cambridge, 2015. J. West, Generating trees and forbidden subsequences, *Discrete Math.* **157** (1996), 363-374. [^1]: For the definition of consecutive patterns, see for example [@ElizaldeNoy2003].

arxiv_math

--- author: - Elías Mochán bibliography: - Bibliografia.bib title: Polytopality of 2-orbit maniplexes --- # Introduction {#s:Intro} Abstract polytopes are posets with some properties that generalize those of the face lattice of convex and skeletal polytopes. Since their introduction by Schulte in [@EgonPHD], the main approach to study abstract polytopes has been by their symmetries. It is known that the group of symmetries (or automorphism group) of an abstract polytope acts freely on its *flags*: chains consisting of one element of each rank (dimension). Therefore, we may measure how symmetric a polytope is by counting how many flag orbits it has under the action of its automorphism group. A *$k$-orbit polytope* is one that has $k$ flag orbits under this action. The smaller the number of flag orbits, the more symmetric the polytope. We call 1-orbit polytopes *regular*, and they have been by far the most studied. The book [@ARP] is the standard reference and it is dedicated solely to abstract regular polytopes. It is quite clear that 2-orbit polytopes are the second most symmetric kind of polytopes after the regular ones, so, naturally, they have been the second most studied family. Nevertheless, the general theory of 2-orbit polytopes has been much more challenging than that of regular ones. One can classify 2-orbit polytopes of rank $n$ in $2^n-1$ different *symmetry types*. One of those symmetry types consists of the so called *chiral* polytopes: those with all possible "rotational" symmetry but no "reflection" symmetry at all. Historically, chiral polytopes have been the most studied type of 2-orbit polytopes. The main theory of (abstract) chiral polytopes was developed in 1991 by Schulte and Weiss [@QuiralesEgonAsia] but the existence of chiral polytopes in any rank was one of the main questions to consider. Rank 3 chiral polytopes had been studied in the context of maps on surfaces, and in the 1970's Coxeter gave examples of rank 4 chiral polytopes arising as quotients of hyperbolic tessellations [@coxeter_twisted]. However, it is until 2008 that algebraic methods are developed to find (finite) chiral polytopes of rank 5 [@QuiralesConstruct]. In 2010, almost 20 years after the publication of [@QuiralesEgonAsia], Pellicer proves that there exist abstract chiral polytopes in any rank $n\geq 3$ [@Quirales]. Also in 2010, Hubard [@2OrbPol1] describes a way to construct 2-orbit polyhedra of any type from groups and examples of 2-orbit polyhedra are well known for all 7 symmetry types. In 2016, Pellicer [@Cuboctaedro] finds geometric examples of 2-orbit polytopes of any rank with a fixed symmetry type. However, the challenge of finding 2-orbit polytopes of any possible type has not had that much of an advancement. *Maniplexes* are a generalization of polytopes where some conditions on connectivity are relaxed. In 2019, Pellicer, Potočnik and Toledo [@2OrbMani] find a way to construct 2-orbit maniplexes of any type. To do this, they use a *voltage assignment* on the symmetry type graph. However, they do not try to answer the question of whether or not their examples are flag graphs of polytopes. This is due to the fact that there was no method to find the intersection properties that the so called voltage group should satisfy for the derived maniplex to be polytopal. However, in [@IntPropYo], Hubard and the author of this paper describe a method to find the intersection properties for every symmetry type for any number of orbits (see Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"}). Using this method for 2 orbits, we have a way to try and determine the polytopality of these 2-orbit maniplexes. The 2-orbit $n$-maniplexes constructed in [@2OrbMani] depend on making some choices: a regular $(n-1)$-maniplex and a *monodromy* (to be defined in [2](#s:basic){reference-type="ref" reference="s:basic"}) of that maniplex, both satisfying some conditions. They show that given a 2-vertex symmetry type, there exists a maniplex $\mathcal{M}$ and a monodromy $\eta$ of $\mathcal{M}$ satisfying such conditions and thus, the $n$-maniplex obtained has the desired symmetry type graph. In this paper we shall show that some of the maniplexes constructed in [@2OrbMani] are in fact polytopal. More specifically, we shall see that there exists a maniplex $\mathcal{M}$ and some particular monodromy $\eta$ of $\mathcal{M}$ not only satisfying the conditions required in [@2OrbMani], but also that for the case when the symmetry type forbids exactly two kinds of reflection, the constructed maniplex is the flag graph of a polytope. The first sections of this paper will introduce all the concepts that we will use. In Section [2](#s:basic){reference-type="ref" reference="s:basic"} we define the basic concepts such as *abstract polytopes, flags, maniplexes, premaniplexes, automorphisms, symmetry type graphs, monodromies* and *coverings*. In Section [3](#s:volts){reference-type="ref" reference="s:volts"} we define the concepts of *fundamental groupoid* and *fundamental group* of a graph, *voltage graphs, derived graphs*, and we enunciate Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"}, which will be our main tool. In Section [4](#s:2alaM){reference-type="ref" reference="s:2alaM"} we describe the construction $\hat{2}^\mathcal{M}$, which gives rise to a family of regular polytopes of all ranks that we will be using for all our examples. In Section [5](#s:2OrbMani){reference-type="ref" reference="s:2OrbMani"} we describe the construction given in [@2OrbMani] to get 2-orbit maniplexes, and we also add some obserbations that will be helpful with the proof of the main result. Finally, in Section [6](#s:Poli2){reference-type="ref" reference="s:Poli2"} we use this construction and Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} to prove that if $X$ is a 2-vertex premaniplex with exactly 2 links, then there exists a polytope $\mathcal{P}$ with $X$ as its symmetry type graph. # Basic concepts {#s:basic} An *abstract polytope* is defined as a poset $(\mathcal{P},\leq)$ that is flagged, strongly flag-connected and satisfies the diamond property. We clarify each of these properties now: A poset is *flagged* if it has a least element $F_{-1}$, a greatest element $F_n$, and all maximal chains, called *flags* have the same finite size. If the size of the flags is $n+2$, we say that the flagged poset has *rank $n$*. Moreover, given an element $F$ of a flagged poset, we can define its *rank* ${\bf rank}(F)$, as two less than the cardinality of a maximal chain having $F$ as its biggest element. For the purposes of this paper, an element of rank $i$ will be called an *$i$-face*, and in general, the elements of a flagged poset will be called *faces*. The faces of rank one less than the poset are called *facets*. If $\Phi$ is a flag, we denote its face of rank $i$ by $(\Phi)_i$. We say that a flagged poset has the *diamond condition* if given two faces $F,G$ such that $F<G$ and ${\bf rank}(F)<{\bf rank}(G)$, there exist exactly two faces $H_1,H_2$ such that $F<H_i<G$ for $i\in\{1,2\}$. We say that two flags $\Phi$ and $\Psi$ in a flagged poset are *$i$-adjacent* if they have the same faces with ranks different than $i$ but they have different $i$-faces. If the flagged poset satisfies the diamond condition, every flag $\Phi$ has exactly one $i$-adjacent flag for $i\in\{0,1,\ldots,n-1\}$. We denote the $i$-adjacent flag to $\Phi$ by $\Phi^i$. We say that a flagged poset is *strongly flag-connected* if given two flags $\Phi, \Psi$, there is a sequence $\Phi_0=\Phi, \Phi_1, \Phi_2, \ldots, \Phi_k=\Psi$, such that $\Phi_{j-1}$ and $\Phi_j$ are $i_j$-adjacent for some $i_j$ satisfying that $\Phi$ and $\Psi$ have different $i_j$-faces. From now on, we will refer to abstract polytopes just as *polytopes*. We will also abuse notation and call $\mathcal{P}$ an abstract polytope assuming that the symbol $\leq$ will be used for the order relation (and $<$ for the strict order relation). A polytope of rank $n$ is called an *$n$-polytope*. Given a polytope $\mathcal{P}$, the *flag graph of $\mathcal{P}$*, denoted $\mathcal{G}(\mathcal{P})$, is the edge-colored graph whose vertices are the flags of $\mathcal{P}$ and $i$-adjacent flags are joined by an edge of color $i$. The flag graph of an $n$-polytope is an example of what is called an *$n$-maniplex* or a *maniplex of rank $n$*, that is, a connected simple graph with edges colored with the numbers $\{0,1,\ldots,n-1\}$, in such a way that: - every vertex is incident to an edge of each color, and - given two colors $i$ and $j$ such that $|i-j|>1$, the paths of length 4 that consist of edges alternating between these two colors are closed. The vertices of a maniplex are usually referred to as *flags*. At this point, we point out that, although infinite maniplexes and polytopes can be very interesting, for the purposes of this paper, all maniplexes and polytopes are assumed to be finite. Some of the results and definitions may also apply for the infinite case, but not all of them (particularly, the main definition of Section [4](#s:2alaM){reference-type="ref" reference="s:2alaM"} needs some fine-tuning for the infinite case). Given a set of colors $I\subset\{0,1,\ldots,n-1\}$, and an $n$-maniplex $\mathcal{M}$, we use the notation $\mathcal{M}_I$ for the subgraph of $\mathcal{M}$ induced by the edges with colors in $I$. We will denote the complement of $I$ in $\{0,1,\ldots,n-1\}$ by $\overline{I}$, and if $I=\{i\}$, we denote its complement just by $\overline{i}$. The connected components of $M_{\overline{i}}$ are the *$i$-faces of $\mathcal{M}$*. We say that an $i$-face $F$ and a $j$-face $G$ of a maniplex are *incident* if they have a non-empty intersection. If in addition $i\leq j$, we write $F\leq G$. In this way we have defined a poset $\mathcal{P}(\mathcal{M})$. If $\mathcal{P}$ is a polytope, then $\mathcal{P}(\mathcal{G}(\mathcal{P}))$ is isomorphic to $\mathcal{P}$. On the other hand, in [@PolyMani] it is proven that if $\mathcal{M}$ satisfies certain conditions called *intersection properties*, then $\mathcal{P}(\mathcal{M})$ is a polytope and $\mathcal{G}(\mathcal{P}(\mathcal{M}))$ is isomorphic to $\mathcal{M}$, showing that such maniplexes are exactly the flag graphs of polytopes, therefore being called *polytopal*. An *isomorphism* of $n$-maniplexes is just a graph isomorphism that preserves the colors of the edges. Naturally, an automorphism of a maniplex is just an isomorphism into itself. The automorphism group of a maniplex $\mathcal{M}$ is denoted by $\Gamma(\mathcal{M})$. It is known that the automorphism group of a maniplex acts freely on its flags [@ARP Proposition 2A4]. Moreover, the automorphisms of the flag graph of a polytope correspond to the automorphisms of the polytope itself (as a poset) (see [@ARP Lemma 2A3] for one inclusion. The other inclusion can be proved using the construction in [@PolyMani]). The *symmetry type graph (STG)* of a maniplex $\mathcal{M}$, denoted by $\mathcal{T}(\mathcal{M})$, is simply its quotient by its automorphism group. That is, the vertices of $\mathcal{T}(\mathcal{M})$ are the flag orbits of $\mathcal{M}$ under its automorphism group, and two orbits are connected by an edge of color $i$ if there is a pair of $i$-adjacent flags one in each of those orbits. If two $i$-adjacent flags are on the same orbit, we draw a semi-edge on the vertex corresponding to that orbit. Notice that $\mathcal{T}(\mathcal{M})$ is not necessarily simple, but it satisfies the other conditions that define an $n$-maniplex. We call such a graph an *$n$-premaniplex*, or just a *premaniplex* if the rank $n$ is implicit. The *symmetry type graph (STG)* of a polytope $\mathcal{P}$ is just the STG of its flag graph, and it is denoted by $\mathcal{T}(\mathcal{P})$. A $k$-orbit polytope (or maniplex) is one with exactly $k$ flag orbits under the action of its automorphism group, in other words, one whose STG has exactly $k$ vertices. A 1-orbit polytope (or maniplex) is called *regular*. Given a maniplex $\mathcal{M}$, we define $r_i$ as the flag permutation that maps each flag to its $i$-adjacent flag. It is easy to see that $r_i$ is an involution and that, if $|i-j|>1$, then $r_ir_j$ is also an involution. In particular $r_i$ and $r_j$ commute whenever $i$ and $j$ are not consecutive. The group $Mon(\mathcal{M})=\langle r_i:0\leq i \leq n-1 \rangle$ is called the *monodromy group of $\mathcal{M}$*, and we will call each of its elements a *monodromy*. It is worth mentioning that some authors prefer the term *connection group* and the notation $Con(\mathcal{M})$. An alternative way of defining the automorphism group of a maniplex $\mathcal{M}$ is as those flag permutations that commute with the elements of the monodromy group. For this paper, both the automorphism and the monodromy group will act on the right. If $\mathcal{M}$ is a regular maniplex (sometimes called *reflexible*), for a fixed flag $\Phi$ we can define $\rho_i$ as the (unique) automorphism that maps $\Phi$ to $\Phi^i$. It is known (see [@ARP Theorem 2B8] for the polytopal case) that $\{\rho_i:i\in\{0,1,\ldots,n-1\}\}$ is a generating set for the automorphism group of $\mathcal{M}$. Moreover, the function mapping $\rho_i$ to $r_i$ can be extended to a group anti-isomorphism between $\Gamma(\mathcal{M})$ and $Mon(\mathcal{M})$ (see, for example, [@MixAndMon Theorem 3.9] or [@Maniplexes Section 7]). In particular, the monodromy group of a regular maniplex acts regularly on its flags. A *covering projection*, or just *covering* for short, from a premaniplex $\mathcal{M}$ to another premaniplex $X$ of the same rank, is just a function $p$ from the vertices of $\mathcal{M}$ to the vertices of $X$ that preserves $i$-adjacencies. It can be proved that all coverings are surjective. If $d$ is a dart (directed edge) of $M$ starting at a vertex $x$, we define $p(d)$ as the dart starting at $p(x)$ that has the same color as $d$. If there is a covering from $\mathcal{M}$ to $X$, we say that *$\mathcal{M}$ covers $X$*. Note that if $\mathcal{M}$ is a maniplex and $X$ is its symmetry type graph, the natural projection from $\mathcal{M}$ to $X$ is always a covering. # Voltage graphs and intersection properties {#s:volts} A *dart* in a graph is just a directed edge. The *inverse* of a dart $d$, denoted by $d^{-1}$ is the dart that corresponds to the same edge with different orientation. We think of semi-edges as having only one orientation, having thus only one dart which is inverse to itself. An edge with two different endpoints (and therefore, with two different darts) will be called a *link*. Every dart $d$ has an *initial vertex*, or *star-point* $I(d)$, and a *terminal vertex* or *endpoint* $T(d)$, having the property that its underlying edge is incident to both $I(d)$ and $T(d)$, and that $I(d^{-1}) = T(d)$ and $T(d^{-1}) = I(d)$. A *path* is a sequence of darts $W=d_1d_2\ldots d_k$, such that $T(d_i) = I(d_{i+1})$ for every $i<k$. The *start-point of $W$* is $I(W)=I(d_1)$, and the *endpoint of $W$* is $T(W)=T(d_k)$. We say that *$W$ goes from $u$ to $v$* if the start-point of $W$ is $u$ and its endpoint is $v$. In this case we write $W:u\to v$. If $W:x\to x$ for some vertex $x$ we say that it is *closed* and *based at $x$*, and if a path is not closed we say that it is *open*. We also consider a formal empty path for each vertex that goes from that vertex to itself. If $W$ is the empty path from $v$ to $v$, we abuse notation and denote it by $v$ as well. If $T(W)=I(V)$ we may concatenate the paths $W$ and $V$ and get a new path denoted by $WV$. Two paths $W$ and $W'$ with the same start-point and endpoint are said to be *homotopic* if one can transform $W$ into $W'$ by a finite sequence of the following operations: - Inserting two consecutive inverse darts at any point, that is $$d_1d_2\ldots d_i d_{i+1}\ldots d_k\mapsto d_1\ldots d_i d d^{-1} d_{i+1}\ldots d_k,$$ where $I(d)=T(d_i)$; - Deleting two consecutive inverse darts at any point, that is $$d_1\ldots d_i d d^{-1} d_{i+1}\ldots d_k \mapsto d_1d_2\ldots d_i d_{i+1}\ldots d_k;$$ In this case we write $W\sim W'$. It is known that homotopy is an equivalence relation and that it is preserved by concatenation of paths. We often abuse notation and use $W$ as the name for a path or for its homotopy class. Furthermore, even if we refer to $W$ as a path, we will always be thinking of it up to homotopy. The set of all homotopy classes of paths in a graph $X$, together with the concatenation operation forms a groupoid called *the fundamental groupoid of $X$*, which we will denote by $\Pi(X)$. The subset of closed paths based at a vertex $x$ forms a group denoted by $\Pi^x(X)$ and called *the fundamental group of $X$ based at $x$*. It is easy to show that if $X$ is connected, all its fundamental groups are isomorphic (moreover, they are conjugates by elements of the fundamental groupoid). Given a spanning tree $T$ and a vertex of a graph $X$ we can find a distinguished set of generators for the fundamental group $\Pi^x(X)$. For each dart $d$ in $X$ but not in $T$, take the path $C_d$ that goes from $x$ to the starting point of $d$ through $T$, then takes $d$, and then goes back to $x$ from the endpoint of $d$ trough $T$. It is easy to see that $\Pi^x(X)$ is generated by $\{C_d\}$, where $d$ runs among the darts in $X$ not in $T$. Given a graph $X$ and a group $\Gamma$, a *voltage assignment (with voltage group $\Gamma$)* is a groupoid anti-morphism $\xi:\Pi(X)\to \Gamma$. The pair $(X,\xi)$ is called a *voltage graph*. To construct a voltage assignment, it is enough to define it for the darts of $X$ (in such a way that inverse darts have inverse voltages). Then, the voltage of the path $W=d_1d_2,\ldots d_k$ is simply $\xi(d_k)\ldots \xi(d_2)\xi(d_1)$. Note that $\xi(\Pi^x(X))=\langle \{\xi(C_d)\} \rangle$, where $d$ runs among the darts of the graph $X$ not in a given spanning tree $T$. This will be important later, since very often we want to know the voltages of some sets of closed paths. Knowing the voltages of a fundamental group will also help us to calculate the set of voltages of open paths with fixed starting point and endpoint as a coset of the voltages of a fundamental group. Given a voltage graph $(X,\xi)$ with voltage group $\Gamma$, we can construct the *derived graph* $X^\xi$ as follows: - The vertex set is $V\times \Gamma$ where $V$ is the vertex set of $X$. - The dart set is $D\times \Gamma$ where $D$ is the dart set of $X$. - If the dart $d$ goes from $x$ to $y$, the dart $(d,\gamma)$ goes from the vertex $(x,\gamma)$ to $(y,\xi(d)\gamma)$. - (Optional) If a dart (or vertex) $d$ has a color $c$, then the dart (or vertex) $(d,\gamma)$ also has color $c$. Note that the inverse of the dart $(d,\gamma)$ is the dart $(d^{-1},\xi(d)\gamma)$. The voltage group $\Gamma$ always acts by automorphisms on the derived graph $X^\xi$, with the action given by $(x,\gamma)\sigma = (x,\gamma\sigma)$, where $x$ is a dart or vertex of $X$ and $\gamma,\sigma\in \Gamma$. It should also be clear that if $X$ has some coloring of its vertices or darts, then $\Gamma$ preserves the induced coloring when acting on $X^\xi$. In [@IntPropYo], Hubard and the author of this paper prove the following results: **Lemma 1**. *[@IntPropYo Lemma 3.1] Let $X$ be a premaniplex and $\eta:\Pi(X)\to \Gamma$ a voltage assignment. Then $X^\xi$ is a maniplex if and only if the following conditions hold:* - *The set $\xi(D)$ generates $\Gamma$ where $D$ is the set of darts of $X$.* - *If $d$ is the dart of a semi-edge, then $\xi(d)$ has order exactly two.* - *If $d$ and $d'$ share both their start-point and their endpoint, then $\xi(d)\neq \xi(d')$.* - *If $|i-j|>1$ every (closed) path $W$ of length 4 whose darts alternate between these two colors has trivial voltage.* Before giving the next result we need to introduce some notation. Given two integers $k$ and $m$ we define $[k,m]:=\{k,k+1,\ldots, m\}$. If $k>m$, then we define $[k,m]$ as the empty set. If $X$ is an $n$-premaniplex, $a,b$ are two vertices in $X$, and $I\subset [0,n-1]$, then we define $\Pi^{a,b}_I(X)$ as the subset of $\Pi(X)$ consisting of all the paths from $a$ to $b$ that only use darts with colors in $I$. In particular, $\Pi^{a,b}_{[k,m]}(X)$ is the set of paths from $a$ to $b$ that only use colors between $k$ and $m$. **Theorem 2**. *[@IntPropYo Theorem 4.2] Let $X$ be a premaniplex of rank $n$ and $\eta:\Pi(X)\to \Gamma$ a voltage assignment. Then $X^\xi$ is polytopal if and only if it is a maniplex and the following equation holds for all $k,m\in\{0,1,\ldots,n-1\}$ and all pairs of vertices $a,b\in X$: $$\label{eq:IntProp} \xi(\Pi^{a,b}_{[0,m]}(X)) \cap \xi(\Pi^{a,b}_{[k,n-1]}(X)) = \xi(\Pi^{a,b}_{[k,m]}(X)).$$* In this paper we will use Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} to show that we can construct 2-orbit polytopes with some prescribed symmetry type graphs. # The construction $\hat{2}^\mathcal{M}$ {#s:2alaM} The main theorem in [@2OrbMani] states that given a regular maniplex $\mathcal{M}$ of rank $n$ with some conditions, and a premaniplex $X$ of rank $n+1$ with 2 vertices, there is a voltage assignment $\xi$ on $X$ that gives a 2-orbit maniplex $\widetilde{\mathcal{M}}$ with symmetry type graph $X$. The voltage group is a group acting on the set $\mathcal{M}_W\times \mathbb{Z}_{2k}$, where $\mathcal{M}_W$ consists of a specific half of the flags of $\mathcal{M}$ and $k$ is a very large number. To be more precise, the conditions on $\mathcal{M}$ are (equivalent to) the following: 1. $\mathcal{M}$ has to cover $X_{\overline{n}}$ (the premaniplex of rank $n$ obtained by deleting the darts of color $n$ from $X$). 2. There is an involutory monodromy $\eta$ in $\mathcal{M}$ that maps all the flags of any given facet to different facets. A concrete family of maniplexes satisfying these conditions is given and its elements are called $\mathcal{M}_n$ where $n$ denotes the rank. This family is constructed recursively and it does not depend on the choice of $X$. More concretely, $\mathcal{M}_2$ is the flag graph of the square, and $\mathcal{M}_{n+1}$ is constructed from $\mathcal{M}_n$ as $\hat{2}^{\mathcal{M}_n}$ (which we will define shortly). In [@2ala] it is proved that if $\mathcal{P}$ is a regular polytope then $\hat{2}^{\mathcal{G}(\mathcal{P})}$ is the flag graph of a regular polytope, and we will see shortly a proof of the fact that if $\mathcal{M}$ is regular $\hat{2}^\mathcal{M}$ is regular too. This implies that the family $\{\mathcal{M}_n\}_{n\geq 2}$ consists of flag graphs of regular polytopes. Since there are known examples of 2-orbit polyhedra (rank 3) with any given symmetry type, we are only concerned about the family $\{\mathcal{M}_n\}_{n\geq 3}$, which are the maniplexes used to construct 2-orbit maniplexes of ranks 4 and higher. The construction $\hat{2}^\mathcal{M}$ works as follows: Given a maniplex $\mathcal{M}$, the flags of $\hat{2}^\mathcal{M}$ are $\mathcal{F}(\mathcal{M})\times \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$, where $\mathcal{F}(\mathcal{M})$ is the set of flags of $\mathcal{M}$ and ${\rm Fac}(\mathcal{M})$ is the set of facets. We will think of $\mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$ as the set of functions from ${\rm Fac}(\mathcal{M})$ to the cyclic group $\mathbb{Z}_2$. Then the adjacencies in $\hat{2}^\mathcal{M}$ are defined by: $$\begin{aligned} (\Phi,x)^i&:=&(\Phi^i,x) \quad {\rm if}\ i<n,\\ (\Phi,x)^n&:=&(\Phi,x+\chi_{{\rm Fac}(\Phi)}),\end{aligned}$$ where ${\rm Fac}(\Phi)$ denotes the facet of $\Phi$ and given a facet $F\in {\rm Fac}(\mathcal{M})$ the vector $\chi_F$ is the one associated with the characteristic function of $F$, that is, the vector with 1 in the coordinate corresponding to $F$ and 0 in every other one. If we remove the edges of color $n$ from $\hat{2}^\mathcal{M}$ we get one connected component for each vector $x\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$. Each component consists of all flags of type $(\Phi,x)$ where $\Phi$ is a flag of $\mathcal{M}$. In particular, every facet of $\hat{2}^\mathcal{M}$ is isomorphic to $\mathcal{M}$. Given a polytope $\mathcal{P}$ we can give a construction $\hat{2}^\mathcal{P}$ such that the flag graph of $\hat{2}^\mathcal{P}$ is the maniplex $\hat{2}^{\mathcal{G}(\mathcal{P})}$, where $\mathcal{G}(\mathcal{P})$ denotes the flag graph of $\mathcal{P}$. This construction is the following: For $-1\leq i \leq n$, denote by $F_i$ the set of $i$-faces of $\mathcal{P}$. If $f\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$ the *support of $f$*, denoted by ${\rm \bf supp}(f)$, is defined as the set of facets $F\in {\rm Fac}(\mathcal{M})$ such that $f(F) = 1$. If we denote by $\hat{F}_i$ the set of $i$-faces of $\hat{2}^\mathcal{P}$, then for $-1\leq i \leq n$ let $\hat{F}_i:=F_i\times \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}/\sim$ where $(F,x)\sim (F',x')$ if and only if $F=F'$ and for every facet $G\in {\rm \bf supp}(x+x')$ we have that $F\leq G$. We then add a formal greatest face $F_{n+1}$. Finally, the incidence relation on $\hat{2}^\mathcal{P}$ is given by $(A,x)<(B,y)$ if and only if $A<B$ and $(A,x)\sim (A,y)$. We should remark that the construction $\hat{2}^\mathcal{P}$ is actually $(2^{\mathcal{P}^*})^*$, where the superscript $*$ denotes duality and $2^\mathcal{P}$ is a better-known construction (see [@2alaDanzer]). There is a natural bijection between the flags of $\hat{2}^{\mathcal{G}(\mathcal{P})}$ and the flags of $\mathcal{G}(\hat{2}^\mathcal{P})$, and it is given by $(\Phi,x)\mapsto \{((\Phi)_j,x)|j\leq n\}$. This bijection maps $(\Phi,x)^i = (\Phi^i,x)$ to $\{((\Phi^i)_j,x)|j\leq n\} = \{((\Phi)_j,x)|j\leq n\}^i$ for $i<n$ and it maps $(\Phi,x)^n = (\Phi,x+\chi_{{\rm Fac}(\Phi)})$ to $\{((\Phi)_j,x+\chi_{{\rm Fac}(\Phi)})|j\leq n\}$, and since $(\Phi)_j\leq {\rm Fac}(\Phi)$ for $j<n$ this is $\{((\Phi)_j,x)|j\leq n\}^n$, so this bijection is an isomorphism between $\hat{2}^{\mathcal{G}(\mathcal{P})}$ and $\mathcal{G}(\hat{2}^\mathcal{P})$. We can prove that if $\mathcal{M}$ is a regular maniplex then $\hat{2}^\mathcal{M}$ is a regular maniplex too. It is not difficult to see that the automorphisms of $\mathcal{M}$ have a natural action on $\hat{2}^\mathcal{M}$. First we define a left action of $\Gamma(\mathcal{M})$ on $\mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$ as follows: given an automorphism $\sigma$ of $\mathcal{M}$, a vector $x$ in $\mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$ and a facet $F$ of $\mathcal{M}$ we define that $\sigma x:f\mapsto x(f\sigma)$. Now, if $\sigma$ is an automorphism of $\mathcal{M}$, we can extended it to an automorphism of $\hat{2}^\mathcal{M}$ by defining $(\Phi,x)\sigma := (\Phi\sigma, \sigma^{-1} x)$. Note that ${\rm \bf supp}(\sigma^{-1}x) = {\rm \bf supp}(x)\sigma$, so we are actually using the natural action of an automorphism on a set of facets. Let us show that this is in fact a maniplex automorphism: If $i<n$ then $$\begin{aligned} ((\Phi,x)^i)\sigma &=& (\Phi^i,x)\sigma\\ &=& ((\Phi^i)\sigma, \sigma^{-1} x)\\ &=& ((\Phi\sigma)^i,\sigma^{-1} x)\\ &=& ((\Phi,x)\sigma)^i.\end{aligned}$$ And for $i=n$ we get $$\begin{aligned} ((\Phi,x)^n)\sigma &=& (\Phi,x+\chi_{{\rm Fac}(\Phi)})\sigma\\ &=& (\Phi\sigma, \sigma^{-1}(x+\chi_{{\rm Fac}(\Phi)}))\\ &=& (\Phi\sigma, \sigma^{-1} x + \sigma^{-1}\chi_{{\rm Fac}(\Phi)})\\ &=& (\Phi\sigma, \sigma^{-1} x + \chi_{{\rm Fac}(\Phi\sigma)})\\ &=& ((\Phi,x)\sigma)^n.\end{aligned}$$ This implies that the automorphism group of $\mathcal{M}$ acts transitively on the flags of the facet $(F_n,0)$ where $F_n$ is the greatest face of $\mathcal{M}$. But for any $y\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}$ we have that the function $T_y$ given by $(\Phi,x)T_y = (\Phi,x+y)$ is also an automorphism. In fact for $i<n$ we get $$\begin{aligned} ((\Phi,x)^i)T_y &=& (\Phi^i,x)T_y\\ &=& (\Phi^i,x+y)\\ &=& (\Phi,x+y)^i\\ &=& ((\Phi,x)T_y)^i.\end{aligned}$$ And for $i=n$ we get: $$\begin{aligned} ((\Phi,x)^n)T_y &=& (\Phi,x+\chi_{{\rm Fac}(\Phi)})T_y\\ &=&(\Phi,x+y+\chi_{{\rm Fac}(\Phi)})\\ &=&(\Phi,x+y)^n\\ &=&((\Phi,x)T_y)^n.\end{aligned}$$ This implies that $\hat{2}^\mathcal{M}$ is facet-transitive and in conclusion regular. Following this same logic, in [@2OrbMani Proposition 11] it is also proved that if an $n$-maniplex $\mathcal{M}$ has symmetry type graph $X$ then the symmetry type graph of $\hat{2}^\mathcal{M}$ is obtained by adding semi-edges of color $n$ to each vertex of $X$. In fact, in [@twist] it is proved that the automorphism group of $\hat{2}^\mathcal{M}$ is a semi-direct product $T\rtimes \Gamma(\mathcal{M})$, where $T$ is the group $\{T_y:y\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M})}\}$. When we let $\mathcal{M}_2$ be the flag graph of a square and define $\mathcal{M}_{i+1}:=\hat{2}^{\mathcal{M}_i}$ for $i\geq 2$, $\mathcal{M}_3$ happens to be the map on the torus called $\{4,4\}_{(4,0)}$ which can be thought of as a $4\times 4$ chess board in which we identify opposite sides (without twisting). The 3-faces of every subsequent $\mathcal{M}_n$ will be of this type and we will make use of this in our proofs. A *lattice* is a poset in which every pair of elements $\{A,B\}$ has a lowest upper bound $A\vee B$ (called the *join* of $A$ and $B$) and a greatest lower bound $A\wedge B$ (called the *meet* of $A$ and $B$). It is known that if $\mathcal{P}$ is a lattice then $\hat{2}^\mathcal{P}$ is also a lattice. One can verify that if $A$ and $B$ are faces of $\mathcal{P}$, $x,y \in \mathbb{Z}_2^{{\rm Fac}(\mathcal{P})}$ and there is at least one facet of $\hat{2}^\mathcal{P}$ containing both $(A,x)$ and $(B,y)$ then, $$(A,x) \vee (B,y) = (A\vee B,x)\sim (A\vee B, y),$$ and $$(A,x)\wedge (B,y) = (C,x) \sim (C,y)$$ where $C$ is the greatest lower bound of the set ${\rm \bf supp}(x+y) \cup \{A,B\}$ . For a full proof of a more general result see [@Egon2alaLattice Theorem 5]. **Corollary 3**. *For $n\geq 2$, the poset associated with $\mathcal{M}_n$ is a lattice.* Notice that the face lattice of the square, which is associated to $\mathcal{M}_2$, is a lattice. The result follows by induction on $n$. In [@2OrbMani Proposition 14, Lemma 16] it is proved that the family $\{\mathcal{M}_n\}_{n\geq 3}$ satisfies both conditions mentioned at the beginning of this section. To prove that it satisfies the second condition, the authors prove and use the following lemma: **Lemma 4**. *[@2OrbMani Lemma 15] If $\mathcal{M}$ has a set of facets $S$ which is not invariant under any non-trivial automorphism, then there is an involutory monodromy $\eta$ of $\hat{2}^\mathcal{M}$ that maps all the flags of any given facet to different facets.* Such a set $S$ is constructed for $\mathcal{M}_3$, and recursively constructed for $\mathcal{M}_{n+1}$ in terms of one constructed for $\mathcal{M}_n$. The construction is as follows: If $S\subset {\rm Fac}(\mathcal{M})$ is a set of facets that is not invariant under any non-trivial automorphism, then $\hat{S}:=\{(F_n,\chi_F)|F\in S\}\cup\{(F_n,0)\}$ is a set of facets of $\hat{2}^\mathcal{M}$ and it satisfies the same condition. In [@2OrbMani] the authors only care that $S$ is not invariant under any non-trivial automorphism, however we shall choose an $S$ that satisfies some extra conditions, which will prove useful when dealing with the polytopality of the constructed 2-orbit maniplexes. From now on, we call the shaded set in Figure [1](#f:S3){reference-type="ref" reference="f:S3"} $S_3$, and we call $S_n\subset{\rm Fac}(\mathcal{M}_n)$ the set constructed recursively as $S_n:=\hat{S}_{n-1}$ for $n\geq 4$. Note that $S_3$ is not invariant under non-trivial automorphisms of $\mathcal{M}_3$, and therefore $S_n$ is not invariant under non-trivial automorphisms of $\mathcal{M}_n$. In [@2OrbMani], the authors actually use the compliment of the set we have chosen as $S_3$. The reason why we have chosen the compliment will be apparent when we prove Lemma [Lemma 6](#l:DosCaras){reference-type="ref" reference="l:DosCaras"}. ![[\[f:S3\]]{#f:S3 label="f:S3"} The set $S_3$ consisting of the shaded 2-faces is not invariant under any non-trivial symmetry of $\mathcal{M}_3$ and it is not contained in the closure of two 0-faces.](S3.pdf){#f:S3 width="5cm"} **Definition 5**. Given a polytope $\mathcal{P}$ and a face $f\in \mathcal{P}$ let us define the *closure of $f$*, denoted by $\overline{f}$, as the set of all the facets of $\mathcal{P}$ which are incident to $f$. Given a pre-ordered set $(\mathcal{P},\leq)$, one can give it a topology by defining that a subset $C$ is closed if and only if whenever $x\in C$ and $x\leq y$ we get that $y\in C$. In fact every topology on a finite set can be obtained from a pre-order in this way (see, for example, [@ETF2]). In such context, the closure of a set $S$ is just $\overline{S}=\{y\in \mathcal{P}: \exists x\in S, x\leq y\}$. If we remove the least and greatest faces of $\mathcal{P}$ and equip it with this topology, then what we are calling the closure of $f$ is actually just the set of facets contained in the topological closure of $\{f\}$. **Lemma 6**. *Given any two proper faces $u,v$ of $\mathcal{M}_n$ with $n\geq 3$, the set $S_n$ is not contained in $\overline{u} \cup \overline{v}$.* We proceed by induction on $n$. For $n=3$ this is a simple observation obtained from Figure [1](#f:S3){reference-type="ref" reference="f:S3"}. Note that the closure of a face is contained in the closure of any incident face of smaller rank, so we only need to prove the lemma for 0-faces. Suppose the lemma is true for $\mathcal{M}_n$. Let $u,v$ be 0-faces of $\mathcal{M}_{n+1}$, so that $u=(u',x)$ and $v=(v',y)$ for some 0-faces $u',v'$ of $\mathcal{M}_n$ and $x,y \in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_n)}$. We will prove that $\overline{u} \cup \overline{v}$ does not cover even $S_{n+1}\setminus \{(F_n,0)\}$, so we may assume that both $u$ and $v$ are each incident to at least one element of $S_{n+1}$ other than $(F_n,0)$, say for example that $u=(u',x)<(F_n,\chi_G)$ for some facet $G$ of $\mathcal{M}_n$. By our definition of the order $<$, this implies that $u'<F_n$ (which is tautological) and $(u',x)\sim(u',\chi_G)$, so we may assume without loss of generality that $x = \chi_G$. Analogously we may assume that there exists a facet $F$ in $\mathcal{M}_n$ such that $y =\chi_F$. A facet $(F_n,z)$ is in $\overline{u} \cup \overline{v}$ if and only if ${\rm \bf supp}(z+\chi_G)\subset \overline{u'}$ or ${\rm \bf supp}(z+\chi_F)\subset \overline{v'}$. By induction hypothesis, $S_n$ is not contained in $\overline{u'} \cup \overline{v'}$, so there exists a facet $D \in S_n \setminus (\overline{u'} \cup \overline{v'})$. Then $(F_n,\chi_D)$ is a facet in $S_{n+1}$ not in $\overline{u} \cup \overline{v}$. 0◻ In topological terms, Lemma [Lemma 6](#l:DosCaras){reference-type="ref" reference="l:DosCaras"} tells us that no set of two elements is *dense* in $S_n$. Now we turn our attention to the monodromy $\eta$. In the proof of Lemma [Lemma 4](#l:EtaExiste){reference-type="ref" reference="l:EtaExiste"} found in [@2OrbMani], $\eta$ is constructed as follows: Let $\mathcal{M}$ be a regular maniplex and let $S$ be a set of facets not invariant under non-trivial automorphisms. For every facet $F \in S$ let $\Phi_F$ be a fixed base flag in that facet. Let $F_0$ be a base facet in $S$ and let $\Phi = \Phi_{F_0}$. For every $F\in S$, let $\omega_F$ be a monodromy of $\mathcal{M}$ that maps $\Phi$ to $\Phi_F$. Note that since $\mathcal{M}$ is regular, its monodromy group acts regularly on its flags, so $\omega_F$ is actually unique. Then for every flag $(\Psi,x)$ in $\hat{2}^\mathcal{M}$ define $(\Psi,x)\eta:=(\Psi,x+\sum_{F\in S}\chi_{{\rm Fac}(\Psi\omega_F)})$. The action of $\eta$ will be more clear with the following lemma. **Lemma 7**. *Let $(\Psi,x) \in \hat{2}^\mathcal{M}$. Let $\gamma$ be the automorphism of $\mathcal{M}$ mapping the base flag $\Phi$ of $F_0$ to $\Psi$. Then, the vector corresponding to the facet of $(\Psi,x)\eta$ differs from $x$ only in the coordinates corresponding to $S\gamma$, that is, if $(\Psi,x)\eta = (\Psi,y)$ then ${\rm \bf supp}(x+y) = S\gamma$.* For every $F \in S$ we have that $$\Psi\omega_F = (\Phi\gamma)\omega_F = (\Phi\omega_F)\gamma,$$ so $${\rm Fac}(\Psi\omega_F) = {\rm Fac}((\Phi\omega_F)\gamma) = ({\rm Fac}(\Phi\omega_F))\gamma = ({\rm Fac}(\Phi_F))\gamma = F\gamma.$$ This implies that $(\Psi,x)\eta = (\Psi,x+\sum_{F\in S} \chi_{{\rm Fac}(\Psi\omega_F)}) = (\Psi,x+\sum_{F\in S} \chi_{F\gamma})$ so $y = x+\sum_{F\in S}\chi_{F\gamma}$. By doing the change of variable $G=F\gamma$ we get $y=x+\sum_{G\in S\gamma}\chi_G$ and ${\rm \bf supp}(x+y) = S\gamma$. 0◻ Given two sets $S$ and $S'$ of facets of a polytope $\mathcal{P}$, we say that $S'$ is *a copy of $S$* if there is an automorphism $\gamma$ of $\mathcal{P}$ such that $S'=S\gamma$. Lemma [Lemma 7](#l:AccionEta){reference-type="ref" reference="l:AccionEta"} tells us that if $(\Psi,x)\eta = (\Psi,y)$ then ${\rm \bf supp}(x+y)$ is a copy of $S$. In [@2OrbMani Lemma 15] it is proved that $\eta$ is in fact a monodromy of $\hat{2}^\mathcal{M}$ and that if $S$ is not invariant under non-trivial automorphisms of $\mathcal{M}$ then $\eta$ maps all the flags of a facet of $\hat{2}^\mathcal{M}$ to different facets. Note that using a set of facets of $\mathcal{M}_n$ not invariant under non-trivial automorphisms we constructed the monodromy $\eta$ for $\mathcal{M}_{n+1}$. We have found such sets of facets for $\mathcal{M}_n$ with $n\geq 3$, which means that we have found the monodromy $\eta$ for $n\geq 4$. We have not found the monodromy $\eta$ for $\mathcal{M}_3$, since every set of facets (edges) of $\mathcal{M}_2$ (the square) is invariant under some non-trivial automorphism. Actually $\mathcal{M}_3$ does not have such a monodromy. However, the map on the torus $\{4,4\}_{(8,0)}$ (a chess board of regular size where each border is identified with its opposite) does admit such a monodromy: simply take $\eta=r_2r_1r_0r_1r_2r_1r_2r_1$. This monodromy is indeed involutory (use the fact that $r_1r_2$ has order 4) and it maps the 8 flags of a facet to the 8 squares where a knight could legally move in a game of chess. This example also covers all 2-vertex premaniplexes in rank 3, so, to construct 2-orbit polytopes of rank 4, one may use this example instead of $\mathcal{M}_3$. # 2-orbit maniplexes {#s:2OrbMani} The symmetry type of a 2-orbit maniplex is denoted by $2_I^n$ where $n$ denotes the rank and $I$ is a proper subset of $\{0,1,\ldots, n-1\}$. Some authors assume the rank is implicit and just write $2_I$, but we shall not do that in this paper as we will be working in several different ranks at once. In this notation, $I$ is the set of colors $i$ such that any flag is in the same orbit as its $i$-adjacent flag. In terms of the symmetry type graph, $I$ is the set of colors of the semi-edges. Informally, we may think of $I$ as the set of "allowed kinds of reflections". In particular, chiral $n$-polytopes are those of type $2_\emptyset^n$, i.e. those in which no reflections are allowed. In order to find 2-orbit polytopes (resp. maniplexes) with every possible 2-vertex premaniplex as its symmetry type graph, we only need to find those where $0,n-1 \notin I$. To prove this, one should observe that, given an $n$-polytope $\mathcal{P}$, there are several constructions (for example $\hat{2}^\mathcal{P}$) that give an $(n+1)$-polytope with the same symmetry type graph but with an extra semi-edge of color $n$ at each vertex. Conjugating with duality, one can construct an $(n+1)$-polytope which has the same symmetry type as $\mathcal{P}$ but with colors shifted by 1, and an extra semi-edge of color 0 at each vertex. Suppose that we want to find a polytope (resp. maniplex) with symmetry type $2^k_J$. Let $r$ be the least color not in $J$ and let $s$ be the greatest color not in $J$. Then it is enough to find a polytope (resp. maniplex) with symmetry type $2^n_I$ where $I:=J-r=\{j-r|j\in J\} \cap [0,n-1]$ and $n=s-r+1$. Note that $I$ satisfies that $0, n-1 \notin I$ since $0=r-r$ and $n-1=s-r$ and by construction $r,s\notin J$. Therefore, using a polytope (resp. maniplex) with symmetry type $2^n_I$ as well as the operations $\hat{2}^\mathcal{M}$ and duality, one can obtain the desired polytope (resp. maniplex) of symmetry type $2^k_J$. Given a 2-vertex premaniplex $X = 2_I^{n+1}$ with $0,n\notin I$, the authors of [@2OrbMani] give a voltage assignment to its darts such that the derived graph is a 2-orbit maniplex with $X$ as its symmetry type graph. We discuss this voltage assignment next. Let us color the vertices of $X$ one white and one black. Let $\{r_0,r_1,\ldots,r_{n-1}\}$ be the distinguished generators of the monodromy group of $\mathcal{M}_n$. If $d$ is a semi-edge incident on the white vertex we assign to it the voltage $r_i$ where $i$ is its color. If $d$ is a semi-edge of color $i$ incident to the black vertex, we assign to it the voltage $r_0r_ir_0$, or in other words, $r_i$ if $i>1$ and $r_0r_1r_0$ if $i=1$. Finally, if $d$ is a dart of color $i<n$ from the white vertex to the black vertex we assign to it the voltage $r_0r_i$, in particular, the voltage of the edge of color 0 is trivial. We shall give the voltage of the edge of color $n$ shortly (which will be an involution, so orientation is irrelevant). From now on, we call this voltage assignment $\xi$. ![[\[f:Volts2orb\]]{#f:Volts2orb label="f:Volts2orb"} The voltage $\xi$ we use on $X=2^{n+1}_I$.](Volts2orb.pdf){#f:Volts2orb width="5cm"} Let $p:\mathcal{M}_n\to X_{\overline{n}}$ be a covering. We color the vertices of $\mathcal{M}_n$ white or black according to the color of their image under $p$. Since $X_{\overline{n}}$ has exactly 2 vertices and $\mathcal{M}_n$ covers $X$, then every monodromy of $\mathcal{M}_n$ either preserves the color of every flag or it changes the color of every flag. Note that the voltages of every dart preserve the color of the flags when thought of as monodromies. ## The voltage of the darts of color $n$ {#s:y_n} Now we turn our attention to the voltage of the darts of color $n$. In [@2OrbMani] the authors make the voltage group act on $(\mathcal{M}_n)_w \times \mathbb{Z}_{2k}$ for some large $k$, where $(\mathcal{M}_n)_w$ is the set of white flags of $\mathcal{M}_n$ (recall that all voltages preserve the color of the flags). The action is simply defined by $(\Psi,x)\omega = (\Psi\omega, x)$ for all $(\Psi,x) \in (\mathcal{M}_n)_w \times \mathbb{Z}_{2k}$ and every monodromy $\omega$. In other words, each monodromy acts as usual in the $(\mathcal{M}_n)_w$-coordinate and as the identity on the $\mathbb{Z}_{2k}$-coordinate. The voltage $\widetilde{y}_n$ of the darts of color $n$ is defined as the composition of three commuting involutions $\widetilde{\rho}_0$, $r_0$ and $s$. Both $r_0$ and $\widetilde{\rho}_0$ act only on the $(\mathcal{M}_n)_w$ coordinate and are independent of the $\mathbb{Z}_{2k}$ one, while $s$ acts only on the $\mathbb{Z}_{2k}$ coordinate. To avoid future confusion, the reader must recall that $\mathcal{F}(\mathcal{M}_n) = \mathcal{F}(\mathcal{M}_{n-1}) \times \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$, so $\mathcal{F}(\mathcal{M}_n)\times \mathbb{Z}_{2k}$ may be regarded as $\mathcal{F}(\mathcal{M}_{n-1}) \times \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})} \times \mathbb{Z}_{2k}$. The importance of the $\mathbb{Z}_{2k}$ coordinate lies in that it is used to prove that the derived maniplex is in fact a 2-orbit maniplex instead of a regular one, but it has no importance when proving that the derived graph is a maniplex. For our purposes, we will ignore $s$ for now, consider as if all actions were on $\mathcal{F}(\mathcal{M}_n) = \mathcal{F}(\mathcal{M}_{n-1}) \times \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$, and assign the voltage $y_n := \widetilde{\rho}_0r_0$ to the edge of color $n$. In order to define $\widetilde{\rho}_0$ we first need to choose a *base flag* $\Phi_F$ for each facet $F$ of $\mathcal{M}_n$. For the proof in [@2OrbMani] to work, we first choose a *base facet* $F_0=(f_{n-1},0)$ of $\mathcal{M}_n$ (where $f_{n-1}$ is the greatest face of $\mathcal{M}_{n-1}$) and then for every *white* flag $\Psi$ in $F_0$ we set $\Psi r_0 \eta r_0$ to be the base flag of its facet. Recall that $\eta$ maps all the flags of $F_0$ to different facets, so there is no ambiguity. In [@2OrbMani] all the other base flags are chosen arbitrarily, however we will later have a preferred choice for them too. Then we define $\widetilde{\rho}_0$ as the flag-permutation that acts on each facet $F$ as the reflection (facet automorphism) that fixes all the faces of the base flag $\Phi_F$ but its 0-face. Note that if we replace a base flag with any flag sharing the same edge (1-face) and the same facet, we get the same permutation $\widetilde{\rho}_0$. So we may define a *base edge* of a facet $F$ as the 1-face of the corresponding base flag and forget about the base flag. Now the previous choice of base edges would be equivalent to the following: For every *black* flag $\Psi$ in $F_0$ we set $(\Psi\eta)_1$ to be the base edge of its facet. Let $\Psi$ be a black flag in $F_0$. Since $\mathcal{M}_n = \hat{2}^{\mathcal{M}_{n-1}}$, and by the choice of $F_0$, the flag $\Psi$ can be written as $(\psi,0)$ for some flag $\psi$ in $\mathcal{M}_{n-1}$. Lemma [Lemma 7](#l:AccionEta){reference-type="ref" reference="l:AccionEta"} tells us that $\Psi\eta = (\psi,x)$ for some $x\in\mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$ satisfying that ${\rm \bf supp}(x)$ is a copy of $S_{n-1}$. So for facets corresponding to vectors whose support is a copy of $S_{n-1}$ we are forced to choose a specific base edge, but for any other facet we may choose the base edge as we want. Let $(e,0)$ be the base edge of the base facet $F_0 = (f_{n-1},0)$. Then for every $x$ whose support is not a copy of $S_{n-1}$ we choose $(e,x)$ as the base edge of the facet $(f_{n-1},x)$. We summarize this in the following definition: **Definition 8**. *Base edges:* Let $(e,0)$ be the base edge of the base facet of $\mathcal{M}_n$, let $x\in\mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_n)}$ and let $f_{n-1}$ be the greatest face of $\mathcal{M}_{n-1}$. If ${\rm \bf supp}(x)$ is not a copy of $S_{n-1}$, then we define the base edge of the facet $(f_{n-1},x)$ of $\mathcal{M}_n$ to be $(e,x)$. If ${\rm \bf supp}(x)=S_{n-1}\gamma$ for some $\gamma\in \Gamma(\mathcal{M}_{n-1})$, then we define the base edge of $(f_{n-1},x)$ to be $(e\gamma,x)$. **Corollary 9**. *If $x\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_n)}$ is such that ${\rm \bf supp}(x)\subset \overline{u} \cup \overline{v}$ for some faces $u,v$ in $\mathcal{M}_{n-1}$, then the base edge of the facet $(f_{n-1},x)$ is $(e,x)$.* If ${\rm \bf supp}(x)\subset \overline{u} \cup \overline{v}$, then, because of Lemma [Lemma 6](#l:DosCaras){reference-type="ref" reference="l:DosCaras"}, ${\rm \bf supp}(x)$ cannot be a copy of $S_{n-1}$. Then, Definition [Definition 8](#d:AristaBase){reference-type="ref" reference="d:AristaBase"} gives us the desired result. **Corollary 10**. *If $(e,0)$ is the base edge of the base facet $(f_{n-1},0)$ of $\mathcal{M}_n$, then it is also the base edge of any other facet containing it.* If $(f_{n-1},x)$ is a facet containing $(e,0)$ then ${\rm \bf supp}(x) \subset \overline{e}$. Then, Corollary [Corollary 9](#c:AristaBase){reference-type="ref" reference="c:AristaBase"} tells us that base edge of $(f_{n-1},x)$ is $(e,x) \sim (e,0)$.0◻ Finally we let $y_n:=\widetilde{\rho}_0r_0$ and extend the voltage assignment $\xi$ by assigning $y_n$ as the voltage of the edge of color $n$ of $X$. Note that since $\widetilde{\rho}_0$ acts as an automorphism in each facet, it commutes with all the monodromies that do not use the generator $r_n$. In particular $\widetilde{\rho}_0$ commutes with $r_0$, implying that $y_n$ is an involution. As previously discussed, in [@2OrbMani] the authors consider the voltage group as acting on $(\mathcal{M}_n)_w\times \mathbb{Z}_{2k}$ where $(\mathcal{M}_n)_w$ is the set of white flags of $\mathcal{M}_n$ and $k$ is some large integer. If $(\Psi,a) \in (\mathcal{M}_n)_w\times \mathbb{Z}_{2k}$ and $\omega$ is a color preserving monodromy of $\mathcal{M}_n$ then $(\Psi,a)\omega$ is simply defined as $(\Psi\omega,a)$. Then they use $\widetilde{y}_n:=\widetilde{\rho}_0r_0s$ as the voltage of the edge of color $n$, where $s$ is an involution acting only on the $\mathbb{Z}_{2k}$ coordinate. We will call the voltage assignment used in [@2OrbMani] $\xi'$. In [@2OrbMani] the authors prove that $X^{\xi'}$ is a maniplex with STG $X$. The proof of the fact that $X^{\xi'}$ is a maniplex also applies to $X^\xi$, but the proof of the fact $X^{\xi'}$ is not regular relies heavily on $k$ being large. On the other hand $X^\xi$ is either regular or has STG $X$. However our proofs to show polytopality of some derived maniplexes will be much clearer if we work with the voltage assignment $\xi$ and consider that $\xi=\pi_{\mathcal{M}_n}\xi'$ where $\pi_{\mathcal{M}_n}$ is the projection to the $\mathcal{M}_n$-coordinate. That is, the $\xi$-voltage of a path is just the first coordinate of the $\xi'$-voltage of the same path. Note that two paths may have the same $\xi$-voltage while having different $\xi'$-voltages, but not the other way around. # Polytopality {#s:Poli2} In this section we prove that if $I=\{1,2,\ldots,n-1\}$, then the derived graph from the voltage graph $X = 2^{n+1}_I$ in Figure [2](#f:Volts2orb){reference-type="ref" reference="f:Volts2orb"} is polytopal. We will also prove that most of the intersection properties are also satisfied for arbitrary $I$, implying that there are only a few intersection properties that would be needed to check to prove that there are 2-orbit polytopes of all symmetry types in rank $n\geq 3$. In order to deal with the intersection properties in Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} for this particular voltage assignment, we should first understand better what are the voltages of paths in the voltage graph in Figure [2](#f:Volts2orb){reference-type="ref" reference="f:Volts2orb"} with respect to the colors they use. To do this we prove Lemmas [Lemma 12](#l:VoltsCerr){reference-type="ref" reference="l:VoltsCerr"} and [Lemma 13](#l:VoltsAbs){reference-type="ref" reference="l:VoltsAbs"}, which describe the voltages of (most) closed and open paths respectively. Let us first prove the following: **Claim 11**. Let $n\geq 3$. Let $W=d_1d_2\ldots d_m$ be a path in the voltage graph of Figure [2](#f:Volts2orb){reference-type="ref" reference="f:Volts2orb"}. Let $c_i$ be the color of the dart $d_i$. If $c_i\neq 1,n$ for all $i\in\{1,2,\ldots,m\}$ then $\xi(W)=r_0^\varepsilon r_{c_m}r_{c_{m-1}}\ldots r_{c_1}$ where $\varepsilon$ is 0 or 1 depending on whether the path $W$ is closed or open, respectively. Note that $W$ is closed if it uses an even number of links, and it is open if it uses an odd number of links. If $d_m$ is a semi-edge based on the white vertex, then $$\xi(d_1d_2 \ldots d_m) = r_{c_m}\xi(d_1d_1\ldots d_{m-1}).$$ If $d_m$ is a dart from the white vertex to the black one, then $$\xi(d_1d_2 \ldots d_m) = r_0r_{c_m}\xi(d_1d_1\ldots d_{m-1}).$$ If $d_m$ is a dart from the black vertex to the white one, then $$\xi(d_1d_2 \ldots d_m) = r_{c_m}r_0\xi(d_1d_1\ldots d_{m-1})$$ but since $c_m\neq 1$ we know that $r_{c_m}r_0=r_0r_{c_m}$ and we get $$\xi(d_1d_2 \ldots d_m) = r_0r_{c_m}\xi(d_1d_1\ldots d_{m-1}).$$ And if $d_m$ is a semi-edge on the black vertex then $$\xi(d_1d_2 \ldots d_m) = r_0r_{c_m}r_0\xi(d_1d_1\ldots d_{m-1})$$ but since $c_m\neq 1$ this means $\xi(d_1d_2 \ldots d_m) = r_{c_m}\xi(d_1d_1\ldots d_{m-1})$. If we repeat this argument for $d_{m-1}$, then $d_{m-2}$ and so on, and note that since $c_i\neq 1$ then $r_{c_i}$ commutes with $r_0$ for all $i$, we get the desired result. 0◻ If $\Phi$ is a flag and $K\subset\{0,1,\ldots,n-1\}$, we will denote by $(\Phi)_K$ the set of faces in $\Phi$ whose rank is in $K$. The following lemma characterizes the voltages of *closed* paths that do not use edges of color $1$ or the edge of color $n$ and voltage $y_n$. **Lemma 12**. *Let $\omega$ be a monodromy of $\mathcal{M}_n$ that preserves the color of its flags and $K\subset \{0,1,\ldots,n-1\}$. Suppose $1\in K$. Then for every white flag $\Phi$ of $\mathcal{M}_n$ we have that $(\Phi)_K = (\Phi\omega)_K$ if and only if $\omega$ is the voltage of a *closed* path based on the white vertex of $X$ that does not use colors in $K\cup \{n\}$.* If $(\Phi)_K = (\Phi\omega)_K$, by strong connectedness of $\mathcal{M}_n$ there is a path $\widetilde{W}$ from $\Phi$ to $\Phi\omega$ not using colors in $K$. Let $c_1c_2\ldots c_k$ be the sequence of colors of $\widetilde{W}$. Then $\Phi\omega = \Phi r_{c_1}r_{c_2}\ldots r_{c_k}$, but since $\mathcal{M}_n$ is regular, the action of the monodromy group on the flags is regular, so we conclude that $\omega = r_{c_1}r_{c_2}\ldots r_{c_k}$. Let $W$ be the path on $X$ that starts on the white vertex and follows the sequence of colors $c_1c_2\ldots c_k$, that is $W = p(\widetilde{W})$. We know that $\Phi\omega$ and $\Phi$ are both white, so $W$ must be a closed path. Recall that the voltage of $W$ is the product of the voltages of its darts but in reverse order. Let $a_1a_2\ldots a_k$ be the dart sequence of $W$. Let us consider $W^{-1} = a_ka_{k-1}\ldots a_1$. Since $W$ does not use colors in $K$ and $1\in K$ Claim [Claim 11](#a:Volts2){reference-type="ref" reference="a:Volts2"} tells us that $\xi(W^{-1}) = r_0^\varepsilon\omega$ and since $W$ is closed, $W^{-1}$ is closed too, so $\varepsilon = 0$ and $\xi(W^{-1})=\omega$. Thus we have found a closed path that does not use colors in $K\cup\{n\}$ and has voltage $\omega$. For the converse, let $W=a_1a_2\ldots a_k$ be a closed path based on the white vertex of $X$ and suppose that $W$ does not use colors in $K\cup \{n\}$. Since $W$ does not use the color 1 (because $1\in K$) Claim [Claim 11](#a:Volts2){reference-type="ref" reference="a:Volts2"} tells us that $\xi(W)=r_{c_k}r_{c_{k-1}}\ldots r_{c_1}=:\omega$ where $c_i$ is the color of $a_i$. Since $c_i\notin K\cup \{n\}$ we know that $(\Phi\omega)_K = (\Phi)_K$. 0◻ The following lemma characterizes the voltages of *open* paths that do not use edges of color $1$ or the edge of color $n$ and voltage $y_n$. **Lemma 13**. *Let $\omega$ be a monodromy of $\mathcal{M}_n$ that preserves the color of the flags and $K\subset \{0,1,\ldots,n-1\}$. Then $(\Phi)_K = (\Phi^0\omega)_K$ if and only if $\omega$ is the voltage of an *open* path in $X$ that does not use colors in $K \cup \{n\}$.* If $(\Phi)_K = (\Phi^0\omega)_K$, by strong connectedness of $\mathcal{M}_n$ there is a path $\widetilde{W}$ from $\Phi$ to $\Phi^0\omega$ not using colors in $K$. Let $c_1c_2\ldots c_k$ be the sequence of colors of $\widetilde{W}$. Then $\Phi r_0\omega = \Phi r_{c_1}r_{c_2}\ldots r_{c_k}$, but since $\mathcal{M}_n$ is regular, the action of the monodromy group on the flags is regular, so we conclude that $r_0\omega = r_{c_1}r_{c_2}\ldots r_{c_k}$, which means $\omega = r_0r_{c_1}r_{c_2}\ldots r_{c_k}$. Once again let $W^{-1}$ be the path on $X$ that starts on the white vertex and follows the sequence of colors $c_kc_{k-1}\ldots c_1$, that is $W=p(\widetilde{W})$. Since $\omega$ preserves the color of the flags $\Phi^0\omega$ has a different color from $\Phi$, so $W$ is an open path. The rest of the proof is analogous to that of Lemma [Lemma 12](#l:VoltsCerr){reference-type="ref" reference="l:VoltsCerr"}. 0◻ We want to show that if $X$ is a 2-vertex premaniplex with only a 0-link and an $n$-link, and $\xi$ is the voltage assignment defined in Section [5](#s:2OrbMani){reference-type="ref" reference="s:2OrbMani"}, then the condition of Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} is satisfied, that is, we want to prove that if $X = \hat{2}^{n+1}_{[1,n-1]}$, we have that $$\xi(\Pi^{a,b}_{[0,m]}(X)) \cap \xi(\Pi^{a,b}_{[k,n-1]}(X)) = \xi(\Pi^{a,b}_{[k,m]}(X)),$$ for all $k,m \in [0,n]$ and all pairs of vertices $(a,b)$. In [@2OrbMani] the authors only use the sets $S_n$ to find the monodromy $\eta$ of $\mathcal{M}_n$ sending each flag of a facet to a different facet. To ensure that $\eta$ acts this way they only need to use the fact that $S_n$ is not invariant under non-trivial automorphisms, and they do not consider any other properties. However, for our purposes this condition is not enough. We need $\eta$ to also send the flags of the base facet "very far away". This is to ensure that "close facets" have the same base edge as the base facet. This is why we have proved Corollary [Corollary 9](#c:AristaBase){reference-type="ref" reference="c:AristaBase"}. First we will prove that the intersection condition is satisfied for $k>1$ for every 2-vertex premaniplex $2^{n+1}_I$ where $0,n\notin I$. **Theorem 14**. *Let $X$ be an $(n+1)$-premaniplex with two vertices and with links of color 0 and $n$, that is $X=2^{n+1}_I$ with $0,n\notin I$. Let $\xi$ be the voltage assignment defined in Section [5](#s:2OrbMani){reference-type="ref" reference="s:2OrbMani"}. Then, for $k>1$ and for all $m\in [0,n]$ we have that $$\xi(\Pi^{a,b}_{[0,m]}(X)) \cap \xi(\Pi^{a,b}_{[k,n]}(X)) = \xi(\Pi^{a,b}_{[k,m]}(X)),$$ for all pairs of vertices $(a,b)$ in $X$.* Before proceeding with the proof, let us introduce a few new concepts. Let $\mathcal{M}$ be a maniplex of rank $n$ and let $\mu$ be a flag-permutation of $\mathcal{M}$. Let $i \in \{0,1,\ldots,n-1\}$ and let $F$ and $G$ be $i$-faces of $\mathcal{M}$. If $\mu$ maps every flag with $i$-face $F$ to a flag with $i$-face $G$ we will say that *$\mu$ maps $F$ to $G$*. In the case where $F=G$ we will say that *$\mu$ fixes $F$*. If $\mu$ fixes $F$ for every $i$-face $F$ we will say that *$\mu$ fixes $i$-faces*. To save some notation, in what follows we will write $\Pi^{a,b}_I$ instead of $\Pi^{a,b}_I(X)$, taking for granted that we are speaking about paths on $X$. Recall that the interval $[k,m]$ is considered to be the empty set when $k>m$ and that $\xi(\Pi^{a,b}_\emptyset)$ is the trivial group 1 if $a=b$ and it is the empty set if $a\neq b$. If $m=n$ there is nothing to prove. If $m<n$ then $\xi(\Pi^{a,b}_{[0,m]})$ is generated by monodromies of $\mathcal{M}_n$ mapping any white flag $\Phi$ in $\mathcal{M}_n$ to a white flag with the same $i$-faces for $i>m$. Now let $W \in \Pi^{a,b}_{[k,n]}$. If $\xi(W) \in \xi(\Pi^{a,b}_{[0,m]})$, then it must be a monodromy of $\mathcal{M}_n$ that preserves $i$-faces for $i>m$. Let $\Phi$ be the base flag of the base facet of $\mathcal{M}_n$. Note that, since $k>1$, the elements of $\xi(\Pi^{a,b}_{[k,n]})$ are products of flag permutations that do not change the 1-face of $\Phi$, that is, for every $\omega \in \xi(\Pi^{a,b}_{[k,n]})$ we have that $(\Phi\omega)_1 = (\Phi)_1$. This implies that $\Phi\omega y_n = \Phi\omega$, as all facets containing $(\Phi)_1$ must have it as their base edge because of Corollary [Corollary 10](#c:AristaBaseBase){reference-type="ref" reference="c:AristaBaseBase"}. Note also that if $1<i<n$, then the voltage of all darts of color $i$ is the same. This means that, if we write $\xi(W) \in \xi(\Pi^{a,b}_{[k,n]})$ as the product of the voltages of the darts of $W$, it acts on $\Phi$ the same way as the voltage of the path $W'$ that follows the same colors as $W$ but ignoring each occurrence of a dart of color $n$. If $W$ uses an even number of darts of color $n$, then $W'$ has the same end-points as $W$ and its voltage $\xi(W')$ acts in the same way as $\kappa = \xi(W)$ on $\Phi$. Since $\mathcal{M}_n$ is regular, if $\kappa$ is a monodromy of $\mathcal{M}_n$ it must coincide with $\xi(W')$, since it is also a monodromy acting the same way on some flag. But $W'\in \Pi^{a,b}_{[k,n-1]}$, so then we have that $\kappa \in \xi(\Pi^{a,b}_{[0,m]}) \cap \xi(\Pi^{a,b}_{[k,n-1]})$. Since $\kappa\in \xi(\Pi^{a,b}_{[k,n-1]})$, it must be a monodromy that preserves $i$-faces for $i<k$, and since $\kappa \in \xi(\Pi^{a,b}_{[0,m]})$, it also preserves $i$-faces for $i>m$. Then, by Lemmas [Lemma 12](#l:VoltsCerr){reference-type="ref" reference="l:VoltsCerr"} and [Lemma 13](#l:VoltsAbs){reference-type="ref" reference="l:VoltsAbs"}, $\kappa \in \xi(\Pi^{a,b}_{[k,m]})$. Now we want to prove that if $\xi(W)\in \xi(\Pi^{a,b}_{[0,m]})$ then $W$ cannot use an odd number of darts of color $n$, thus proving that $\xi(\Pi^{a,b}_{[0,m]}) \cap \xi(\Pi^{a,b}_{[k,n]}) = \xi(\Pi^{a,b}_{[k,m]})$. Let $(e,0)$ be the 1-face of $\Phi$, $(e_0,0)$ be the 1-face of $\Phi^1$ and $(e_1,0)$ be the 1-face of $\Phi^{010}$. Here we are thinking of $e,e_0$ and $e_1$ as 1-faces of the polytope $\mathcal{M}_{n-1}$, which is naturally isomorphic to any facet of $\mathcal{M}_n$ (see Figure [3](#f:Aristas){reference-type="ref" reference="f:Aristas"}). ![[\[f:Aristas\]]{#f:Aristas label="f:Aristas"} The edges $e,e_0$ and $e_1$ illustrated on a 3-face of $\mathcal{M}_{n-1}$.](Aristas.pdf){#f:Aristas width="5cm"} If $F=(f_{n-1},x)$ is a facet of $M_n$ which has $(e,x)$ as its base edge, then $y_n$ interchanges flags with 1-face $(e_0,x)$ with flags with 1-face $(e_1,x)$, while it fixes flags with 1-face $(e,x)$ (see Figure [4](#f:ynAristas){reference-type="ref" reference="f:ynAristas"}). ![[\[f:ynAristas\]]{#f:ynAristas label="f:ynAristas"} If $(e,x)$ is the base edge of a facet, then $y_n$ interchanges the edges $(e_0,x)$ and $(e_1,x)$ while it fixes the edge $(e,x)$.](ynAristas.pdf){#f:ynAristas width="5cm"} Let $\omega \in \xi(\Pi^{a,b}_{[k,n-1]})$, let $F$ be a facet with base edge $(e,x)$, and let $\Psi$ be a flag with facet $F$ and 1-face $(e_j,x)$ for $j\in\{0,1\}$. Let $(\psi,y) = \Psi\omega$. If we write $\omega$ as a product of the voltages of darts, every time we change the facet of $\mathcal{M}_n$ (that is, every occurrence of $r_{n-1}$ or $r_0r_{n-1}$), we must change to a new facet with the same edge. This means that the edge of $\Psi\omega$ must be the same as the edge of $\Psi$, or in other words, that $(e_j,y)\sim (e_j,x)$ where $\sim$ is the equivalence relation we used when defining $\hat{2}^\mathcal{P}$ for a polytope $\mathcal{P}$. Then ${\rm \bf supp}(x+y)$ is contained in $\overline{e_j}$. Now let $\kappa \in \xi(\Pi^{a,b}_{[k,n]})$ and consider $\Phi^1\kappa$ and $\Phi^{10}\kappa$. Since $0\notin I$, then $\{\Phi^1\kappa,\Phi^{10}\kappa\}$ has exactly one white flag and one black flag. Set $(\psi,x)$ to be the white flag in $\{\Phi^1\kappa,\Phi^{10}\kappa\}$. Using again the fact that the voltage of a dart with color greater than 1 does not depend on its start-point, we may write $\kappa$ as $\omega_1 y_n \omega_2 y_n \ldots \omega_{s-1} y_n \omega_s$ where $\omega_i \in \xi(\Pi^{a,b}_{[k,n-1]})$ . Each $\omega_i$ may change the facet to one where the support of the corresponding vector differs in coordinates corresponding to a set contained in $\overline{e_0} \cup \overline{e_1}$. Let $(\psi_i,x_i) = \Phi^1\omega_1 y_n \omega_2 y_n \ldots \omega_i$ if $(\psi,x)=\Phi^1\kappa$ or $\Phi^{10}\omega_1 y_n \omega_2 y_n \ldots \omega_i$ if $(\psi,x)=\Phi^{10}\kappa$. We claim that ${\rm \bf supp}(x_i) \subset \overline{e_0} \cup \overline{e_1}$. This is proved by a simple induction on $i$. For $i=0$ we have that $x_0=0$, which has support ${\rm \bf supp}(0)=\emptyset \subset \overline{e_0} \cup \overline{e_1}$. If the claim is true for $i$, since ${\rm \bf supp}(x_i) \subset \overline{e_0} \cup \overline{e_1}$, Corollary [Corollary 9](#c:AristaBase){reference-type="ref" reference="c:AristaBase"} tells us that the facet of $\mathcal{M}_n$ corresponding to $x_i$ has $(e,x_i)$ as its base edge. This implies that, if $(\psi,x_i)$ has $(e_j,x_i)$ as its 1-face, then $(\psi_{i+1},x_{i+1}) = (\psi,x_i)y_n\omega_{i+1}$ has $e_{1-j}$ as its 1-face. This implies that ${\rm \bf supp}(x_i+x_{i+1}) \subset \overline{e_0} \cup \overline{e_1}$. Then $${\rm \bf supp}(x_{i+1}) = {\rm \bf supp}((x_i+x_{i+1})+x_i) \subset {\rm \bf supp}(x_i+x_{i+1}) \cup {\rm \bf supp}(x_i) \subset \overline{e_0} \cup \overline{e_1}.$$ Thus we have proved our claim. Note that $x=x_s$, so our claim and Corollary [Corollary 9](#c:AristaBase){reference-type="ref" reference="c:AristaBase"} tell us that the facet $(f_{n-1},x)$ of $\mathcal{M}_n$ (where $f_{n-1}$ is the greatest face of $\mathcal{M}_{n-1}$) has base edge $(e,x)$. Since we assumed that $W$ uses an odd number of darts of color $n$, we know by our claim that its voltage $\kappa = \xi(W)$ maps $\Phi^1$ or $\Phi^{10}$, with edge $(e_0,0)$, to a flag with edge $(e_1,x)$ (for some $x$). But we know also that it maps $\Phi$ to a flag with the same edge $(e,0)$. Since $M_n$ is regular, $\kappa$ cannot be a monodromy of $\mathcal{M}_n$ (if a monodromy of a regular polytope fixes one edge, it must fix all edges), so it cannot be in $\xi(\Pi^{a,b}_{[0,m]})$. 0◻ **Corollary 15**. *Let $X$ be an $(n+1)$-premaniplex with two vertices and with links of color 0 and $n$, that is $X=2^{n+1}_I$ with $0,n\notin I$. Let $\xi'$ be the voltage assignment defined as in Section [5](#s:2OrbMani){reference-type="ref" reference="s:2OrbMani"} . Then, for $k>1$ and for all $m\in [0,n]$ we have that $$\xi'(\Pi^{a,b}_{[0,m]}) \cap \xi'(\Pi^{a,b}_{[k,n]}) = \xi'(\Pi^{a,b}_{[k,m]}),$$ for all pairs of vertices $(a,b)$ in $X$.* Since for $m=n$ there is nothing to prove, let us assume that $m<n$. Let $\widetilde{\omega}\in \xi'(\Pi^{a,b}_{[0,m]}) \cap \xi'(\Pi^{a,b}_{[k,n]})$. Let $\pi_{\mathcal{M}_n}\omega=\widetilde{\omega}$, that is, $\omega$ is the same function as $\widetilde{\omega}$ but considering only its action on the $(\mathcal{M}_n)_w$-coordinate. By Theorem [Theorem 14](#t:k>1){reference-type="ref" reference="t:k>1"} $\omega \in \xi(\Pi^{a,b}_{[k,m]})$. We also know that since $m<n$ and $\widetilde{\omega}\in \xi'(\Pi^{a,b}_{[0,m]})$ then $\widetilde{\omega}$ is a monodromy of $\mathcal{M}_n$ and does not change the $\mathbb{Z}_{2k}$-coordinate of the elements of $(\mathcal{M}_n)_w\times \mathbb{Z}_{2k}$. Let $W\in\Pi^{a,b}_{[k,m]}$ be a path such that $\xi(W)=\omega$. Then $\pi_{\mathcal{M}_n}(\xi'(W)) = \xi(W) = \omega = \pi_{\mathcal{M}_n}\widetilde{\omega}$, but also $\pi_{\mathbb{Z}_{2k}}(\xi'(W)) = Id_{\mathbb{Z}_{2k}} = \pi_{\mathbb{Z}_{2k}}\widetilde{\omega}$. Therefore $\xi'(W) = \widetilde{\omega} \in \xi'(\Pi^{a,b}_{[k,m]})$. 0◻ **Theorem 16**. *If $X=2^{n+1}_{[1,n-1]}$, then $X$ is the symmetry type graph of a polytope.* We will prove that the voltage assignment $\xi$ defined in Sections [5](#s:2OrbMani){reference-type="ref" reference="s:2OrbMani"} and [5.1](#s:y_n){reference-type="ref" reference="s:y_n"} satisfies that $$\label{IntPropk1} \xi(\Pi^{a,b}_{[0,m]}) \cap \xi(\Pi^{a,b}_{[k,n]}) = \xi(\Pi^{a,b}_{[k,m]})$$ for all $k,m\in [0,n]$ and all pairs of vertices $(a,b)$. This will imply that $\xi'$ also satisfies this intersection property (exactly as in the proof of Corollary [Corollary 15](#c:k>1){reference-type="ref" reference="c:k>1"}). Then, Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} will imply that both $X^\xi$ and $X^{\xi'}$ are polytopal, and since we already know that $X^{\xi'}$ has STG $X$ the theorem follows. Since Theorem [Theorem 14](#t:k>1){reference-type="ref" reference="t:k>1"} tells us that ([\[IntPropk1\]](#IntPropk1){reference-type="ref" reference="IntPropk1"}) holds for $k>1$ and it trivially holds for $k=0$, we only need to prove the case when $k=1$. Let us first find the distinguished generators of $\xi(\Pi^a_{[1,n]})$, where $a$ is the white vertex in Figure [2](#f:Volts2orb){reference-type="ref" reference="f:Volts2orb"}. When we remove the link of color 0, we are left with only one spanning tree, consisting only of the link of color $n$ and voltage $y_n = r_0\widetilde{\rho}_0$ (see Figure [5](#f:Gen2links){reference-type="ref" reference="f:Gen2links"}). Then, the generator corresponding to a semi-edge of color $i\in [1,n-1]$ on the white vertex is its voltage $r_i$. The generator corresponding to a semi-edge $e$ on the black vertex is the voltage of the path $a_nea_n^{-1}$, where $a_n$ is the dart from the white vertex to the black one with color $n$. This voltage is $$\xi(a_nea_n^{-1}) = \xi(a_n)^{-1}\xi(e)\xi(a_n) = (y_n)(r_0r_ir_0)(y_n) = \widetilde{\rho}_0r_i\widetilde{\rho}_0.$$ Since $\widetilde{\rho}_0$ acts as an automorphism in each facet, for $i<n-1$ we get $\widetilde{\rho}_0r_i\widetilde{\rho}_0 = r_i$. So in conclusion, $\xi(\Pi^a_{[1,n]})$ is equal to $\langle \{r_i\}_{i=1}^{n-2}\cup \{\widetilde{\rho}_0 r_{n-1} \widetilde{\rho}_0\} \rangle$. Note that each $r_i$ fixes the vertex of any flag. ![[\[f:Gen2links\]]{#f:Gen2links label="f:Gen2links"} The premaniplex $(2^{n+1}_{[1,n-1]})_{\overline{0}}$ with its voltage assignment. ](Gen2links.pdf){#f:Gen2links width="5cm"} Now let us turn our attention to the generator $\widetilde{\rho}_0 r_{n-1} \widetilde{\rho}_0$. Let $(u,0) = (\Phi)_0$ be the base vertex, and let $(v,0)$ be the other vertex incident to the base edge $(e,0)$ (see Figure [6](#f:Vertices){reference-type="ref" reference="f:Vertices"}). Let $x\in \mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$ have support contained in $\overline{u} \cup \overline{v}$. We know by Corollary [Corollary 9](#c:AristaBase){reference-type="ref" reference="c:AristaBase"} that the facet $(f_{n-1},x)$ of $\mathcal{M}_n$ has base edge $(e,x)$. Let $\Psi = (\psi,x)$ be a flag with vertex $(u,x)$. Let $(\psi',x) := \Psi\widetilde{\rho}_0$. We must have that $(\psi')_0 = v$. Then $\Psi\widetilde{\rho}_0 r_{n-1} = (\psi', x+\chi_F)$ for some $F\in \overline{v}$. Since ${\rm \bf supp}(x+\chi_F) \subset \overline{u}\cup \overline{v}$ we know that the facet $(f_{n-1}, x+\chi_F)$ has base edge $(e, x+\chi_{F})$. This implies that $\Psi\widetilde{\rho}_0 r_{n-1}\widetilde{\rho}_0 = (\psi', x+\chi_F) \widetilde{\rho}_0 = (\psi, x+\chi_F)$. In conclusion we have proved the following lemma: **Lemma 17**. *If ${\rm \bf supp}(x) \subset \overline{u} \cup \overline{v}$ and $\psi$ is a flag in $\mathcal{M}_{n-1}$ with vertex $u$, then $(\psi,x)\widetilde{\rho}_0 r_{n-1} \widetilde{\rho}_0 = (\psi, y)$ for a vector $y$ satisfying ${\rm \bf supp}(y) \subset \overline{u} \cup \overline{v}$.* *Moreover, if $\omega \in \xi(\Pi^a_{[1,n]})$ and $(\psi,x)\omega = (\psi',y)$ then $\psi'_0 = u$ and ${\rm \bf supp}(y) \subset \overline{u} \cup \overline{v}$.* Now we are ready to prove the case for closed paths, that is, we may prove that $$\xi(\Pi^a_{[1,n]}) \cap \xi(\Pi^a_{[0,m]}) = \xi(\Pi^a_{[1,m]}).$$ Once again, for $m=n$ there is nothing to prove, so let $m<n$. We prove first the case when $m<n-1$. Let $\omega \in \xi(\Pi^a_{[1,n]}) \cap \xi(\Pi^a_{[0,m]})$. Since $\omega \in \xi(\Pi^a_{[0,m]}) = \langle r_i|i\leq m \rangle$, it must be a monodromy of $\mathcal{M}_n$ and since $m<n-1$, it must fix facets. If $\Phi$ is the base flag of the base facet of $\mathcal{M}_n$, Lemma  [Lemma 17](#l:FijaVertice){reference-type="ref" reference="l:FijaVertice"} implies that $\Phi\omega = (\psi,x)$ for some flag $\psi$ in $\mathcal{M}_{n-1}$ and some vector $x\in\mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$ satisfying $(\psi)_0 = u$ and ${\rm \bf supp}(x) \subset \overline{u} \cup \overline{v}$. But since $\omega$ must fix facets, $x$ is actually 0. This means that $(\Phi\omega)_0 = (\Phi)_0$, and since $\mathcal{M}_n$ is regular, this means that $\omega$ fixes all 0-faces. Also because of the regularity of $\mathcal{M}_n$, the monodromies in $\xi(\Pi^a_{[0,m]}) = \langle r_i|i\leq m \rangle$ that fix 0-faces are $\langle r_i|1\leq i \leq m \rangle$, but this is precisely $\xi(\Pi^a_{[1,m]})$. Now, if $m=n-1$ we have a little more work to do. We proceed as in the previous case, but we cannot ensure that $x=0$. However, we have that $\Phi\omega = (\psi,x)$ and the inclusion ${\rm \bf supp}(x) \subset \overline{u} \cup \overline{v}$ still holds. By writing the 2-face of $\Phi$ as $(\Phi)_2=(Q,0)$, we know that $Q$ is a square. Let $w$ be the opposite vertex of $u$ in $Q$ and $q$ be the opposite vertex to $v$, so that the vertices of $Q$ are $uvwq$ in cyclical order (see Figure [6](#f:Vertices){reference-type="ref" reference="f:Vertices"}). ![[\[f:Vertices\]]{#f:Vertices label="f:Vertices"} The 2-face $Q$ and its vertices $u,v,w$ and $q$.](Vertices.pdf){#f:Vertices width="5cm"} Since $\omega$ is a monodromy of $\mathcal{M}_n$, it must commute with the automorphism $\rho_1$ of $\mathcal{M}_n$, which maps $\Phi$ to $\Phi^1$. This implies that $$\Phi^1\omega = (\Phi\omega)\rho_1 = (\psi, x)\rho_1 = (\psi\rho_1, \rho_1 x).$$ Here we have used the way $\Gamma(\mathcal{M}_{n-1})$ acts on $\mathcal{M}_n$ discussed on Section [4](#s:2alaM){reference-type="ref" reference="s:2alaM"}. Note that we have used the same symbol $\rho_1$ to denote an automorphism of $\mathcal{M}_n$ and also an automorphism of $\mathcal{M}_{n-1}$, but because of the way $\Gamma(\mathcal{M}_{n-1})$ acts on $\mathcal{M}_n$ this is actually not ambiguous. Notice that ${\rm \bf supp}(\rho_1 x) = ({\rm \bf supp}(x)) \rho_1$ must be contained in $(\overline{u} \cup \overline{v})\rho_1 = \overline{u} \cup \overline{q}$. On the other hand, the vector corresponding to $\Phi^1$ is 0, so Lemma [Lemma 17](#l:FijaVertice){reference-type="ref" reference="l:FijaVertice"} tells us that ${\rm \bf supp}(\rho_1 x) \subset \overline{u} \cup \overline{v}$. Every facet incident to both $v$ and $q$ must be incident to $Q=v\vee q$ (see Corollary [Corollary 3](#c:lattice){reference-type="ref" reference="c:lattice"}), and hence also to $u$. Then, the intersection $(\overline{u} \cup \overline{q}) \cap (\overline{u} \cup \overline{v})$ is just $\overline{u}$ which means that $\omega$ fixes the vertex of $\Phi^1$, and therefore it must fix the vertex of every flag (again, $\mathcal{M}_n$ is regular). Hence, we conclude that $\xi(\Pi^a_{[1,n]}) \cap \xi(\Pi^a_{[0,m]}) = \xi(\Pi^a_{[1,m]})$. Now let us solve the case for open paths, that is, let us prove that $\xi(\Pi^{a,b}_{[0,m]}) \cap \xi(\Pi^{a,b}_{[1,n]}) = \xi(\Pi^{a,b}_{[1,m]})$ when $a$ is the white vertex and $b$ the black vertex. First notice that since we only have links of colors 0 and $n$, we know that $\xi(\Pi^{a,b}_{[1,m]})$ is in fact empty. So what we really want to prove is that there are no monodromies in $\xi(\Pi^{a,b}_{[1,n]})$. Notice that $\Pi^{a,b}_{[1,n]} = \Pi^a_{[1,n]}a_n$, where once again, $a_n$ is the dart of color $n$ from $a$ to $b$. Then $$\xi(\Pi^{a,b}_{[1,n]}) = \xi(\Pi^a_{[1,n]}a_n) = y_n\xi(\Pi^a_{[1,n]}).$$ Let $\omega \in \xi(\Pi^a_{[1,n]})$. We want to prove that $y_n\omega$ is not a monodromy of $\mathcal{M}_n$. If it was, it would also act as a monodromy on $\mathcal{M}_{n-1}$ (just ignore the $\mathbb{Z}_2^{{\rm Fac}(\mathcal{M}_{n-1})}$-coordinate). Take the base flag $\phi$ of $\mathcal{M}_{n-1}$. Then $\phi y_n\omega = \phi\omega$, and as we have noted before (Lemma [Lemma 17](#l:FijaVertice){reference-type="ref" reference="l:FijaVertice"}), this must be a flag with vertex $u$ (the same vertex as $\phi$). Now, if we consider $\phi^1$ (with vertex $u$) we get that $\phi^1y_n\omega = \phi^1\widetilde{\rho}_0 r_0 \omega = \phi^{010}\omega$ is a flag with vertex $w$. Then $y_n\omega$ cannot act as a monodromy on $\mathcal{M}_{n-1}$, since it fixes the vertex of some flags but it changes the vertex of others and $\mathcal{M}_{n-1}$ is regular. Therefore, $y_n\omega$ is not a monodromy of $\mathcal{M}_n$. We have proved that $\xi(\Pi^{a,b}_{[0,m]}) \cap \xi(\Pi^{a,b}_{[1,n]}) = \emptyset$. Thus we have proved that ([\[IntPropk1\]](#IntPropk1){reference-type="ref" reference="IntPropk1"}) holds. By doing a proof analogous to that of Corollary [Corollary 15](#c:k>1){reference-type="ref" reference="c:k>1"} we get that $\xi'(\Pi^{a,b}_{[0,m]}) \cap \xi'(\Pi^{a,b}_{[k,n]}) = \xi'(\Pi^{a,b}_{[k,m]})$ for all $k,m\in [0,n]$. Theorem [Theorem 2](#t:IntProp){reference-type="ref" reference="t:IntProp"} then implies that $X^{\xi'}$ is polytopal. Since we already knew that $\mathcal{T}(X^{\xi'}) = X$, we have found a polytope whose symmetry type graph is $X$. 0◻ Recall once again that examples of polyhedra (rank 3) of all possible 2-orbit symmetry types exist, in particular those with 1 or 2 links. Applying the constructions $2^\mathcal{P}$ or $\hat{2}^\mathcal{P}$ repeatedly to the examples of 2 links of Theorem [Theorem 16](#t:k=1){reference-type="ref" reference="t:k=1"} or the examples of rank 3 one gets that all symmetry types with 1 or 2 links exist in any rank higher than two. In addition to this, as previously discussed, in [@Quirales] Pellicer proves that chiral polytopes (i.e. those with symmetry type $2^n_\emptyset$) exist in rank 3 and higher, and if we use the constructions $2^\mathcal{P}$ or $\hat{2}^\mathcal{P}$ to those, we get all symmetry types where all the links have consecutive colors. In conclusion, we have the following theorem: **Theorem 18**. *Let $n\geq 3$ and let $X=2^n_I$ be a 2-vertex premaniplex of rank $n$. Let $\overline{I}:=\{0,1,\ldots,n-1\}\setminus I$ be the set of the colors of the links of $X$. Then, in any of the following cases, $X$ is the symmetry type graph of a polytope.* - *$\overline{I}$ has exactly 1 or 2 elements.* - *$\overline{I}$ is an interval $[k,\ell] = \{k,k+1,\ldots,\ell\}$.* Theorem [Theorem 18](#t:TiposExisten){reference-type="ref" reference="t:TiposExisten"} ensures that of the total of $2^n-1$ premaniplexes of rank $n$ with 2 vertices, at least $n^2-n+1$ are the symmetry type of a polytope ($n$ with 1 link, $\frac{n(n-1)}{2}$ with 2 links and $\frac{n(n-1)}{2}-n+1$ with an interval of links of size at least 3). It appears that there is still a long way to go, nevertheless, Theorem [Theorem 14](#t:k>1){reference-type="ref" reference="t:k>1"} and Corollary [Corollary 15](#c:k>1){reference-type="ref" reference="c:k>1"} ensure that to prove that there are polytopes with symmetry type $2^n_I$, one should only check that  ([\[IntPropk1\]](#IntPropk1){reference-type="ref" reference="IntPropk1"}) is satisfied for $k=1$ for the voltage $\xi$ (and this would imply that it is also satisfied for $\xi'$). Sadly, the proof of Theorem [Theorem 16](#t:k=1){reference-type="ref" reference="t:k=1"} cannot be easily generalized to arbitrary $I$. This is because the voltage of a link of color $1\leq i <n$ would be $r_0r_i$ and this affects most arguments used, mainly because there would be paths whose voltage change 0-faces but they do not use the color 0. The author of this thesis conjectures that if this challenge is solved for one example $2^n_I$ with 3 or more links of non-consecutive colors, the same solution must work for all others, and that would prove that there exist 2-orbit polytopes of any possible symmetry type (and rank $n\geq 3$). # Acknowledgments {#acknowledgments .unnumbered} The author would like to thank Gabe Cunningham, Antonio Montero, Daniel Pellicer, for helpful comments on early versions of this paper. The author also gives special thanks to Isabel Hubard, for her guidance during the writing of the thesis that gave birth to this paper (and beyond). The research for this paper was done with the financial support of CONACyT grant A1-S-21678, and PAPIIT- DGAPA grant IN109023. This paper was completed while the author had a postdoctoral position at the Department of Mathematics of Northeastern University.

arxiv_math

--- abstract: | We will develop some elements in stochastic analysis in the Wasserstein space $\mathbb{P}_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic Itô formulae, stochastic regular curves and parallel translations along them. We will establish the existence of parallel translations along regular curves, or stochastic regular curves in case of $\mathbb{P}_2(\mathbb{T})$. Surprisingly enough, in this last case, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE. author: - | Hao DING$^{1}$[^1] Shizan FANG$^2$[^2] Xiang-dong LI$^{3,4}$[^3]\ $^1$National Center for Mathematics and Interdisciplinary Sciences, \ Chinese Academy of Sciences, 55, Zhongguancun East Road, Beijing, 100190, China \ $^2$Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, \ Université de Bourgogne Franche-Comté, F-21000 Dijon, France\ $^3$Academy of Mathematics and Systems Science,\ Chinese Academy of Sciences, 55, Zhongguancun East Road, Beijing, 100190, China \ $^4$School of Mathematical Sciences, \ University of Chinese Academy of Sciences, Beijing, 100049, China title: | **Stochastic differential equations and\ stochastic parallel translations\ in the Wasserstein space** --- **MSC 2010**: 58B20, 60J45 **Keywords**: constant vector fields, stochastic regular curves, Itô['s]{style="color: black"} formula, stochastic parallel translations, internal energy functional. # Introduction {#sect1} During recent decades, the study of the optimal transportation problem has obtained great successes. In particular, in a seminar paper [@Brenier], Y. Brenier proved that the optimal transport map for the long standing Monge-Kantorovich problem with the quadratic distance cost function on Euclidean spaces is given by the gradient of a convex function, which satisfies a Monge-Ampère equation. In [@McCann], R. McCann extended Brenier's result to compact Riemannian manifolds. In [@Caffarelli], L. A. Caffarelli et al. obtained the $C^{2,\alpha}$ regularity estimates for the solutions of Brenier Monge-Ampère equation. Here we will not enter the details of this topic, but only refer to two monographs by C. Villani [@Villani1; @Villani2]. For infinite dimensional case, see [@FeyelUstunel; @FangShao]. On the other hand, R. Jordan, D. Kinderlehrer and F. Otto [@JKO] gave a steepest descent formulation of the Fokker-Planck equation on Euclidean space. In [@Otto], F. Otto introduced an infinite dimensional Riemannian structure on the Wasserstein space of probabilities measures with finite second moments, and interpreted the Fokker-Planck equation and the porous medium equation as the gradient flows associated to the Boltzmann or Rényi type entropy functionals on the Wasserstein space $\mathbb{P}_{2,\infty}(M)$, which is the space of probability measures having positive smooth density. To obtain the long time behavior of above equations, he developed heuristically the Hessian calculus on the Wasserstein space $\mathbb{P}_{2,\infty}(M)$. A striking result is that the Hessian of the Boltzmann entropy functional is given by the Bakry-Emery Ricci curvature [@BakryEmery]. See Otto [@Otto], Otto-Villani [@OV], Renesse-Sturm [@RS2] and Lott [@Lott1]. In [@Sturm; @LottVillani; @RS2], J. Lott, C. Villani and K.T. Sturm developed a synthetic geometric analysis on metric measured spaces. In [@AGS], L. Ambrosio, N. Gigli and G. Savaré developed the theory of generalized gradient flows on the Wasserstein space over metric measured spaces. In [@LiLi; @LiLi2], S. Li and the third named author of this paper introduced the Langevin deformation between the Wasserstein geodesic flow and the gradient flows on $\mathbb{P}_{2,\infty}(M)$ and proved some $W$-entropy-information formulas. The definition of Levi-Civita covariant derivatives on $\mathbb{P}_{2,\infty}(M)$ was introduced by J. Lott in [@Lott1]. He also derived the linear partial differential equation for parallel translations and identified the equation of the associated Riemannian geodesics with those introduced by Benamou and Brenier [@BB] for the $L^2$-(Monge-Kantorovich-Rubinstein-)Wasserstein distance from a hydrodynamic point of view. However, the issue of existence and uniqueness of parallel transports has not been studied in [@Lott1]. In [@AG], L. Ambrosio and N. Gigli established the existence of parallel translations along regular curves induced by flows of diffeomorphisms, but in a weak sense; the question whether the strong solutions for parallel translations exist remains open. Concerning parallel translations along geodesics in $\mathbb{P}_{2,\infty}(M)$, the situation seems much more complicated, see [@Lott2]. [In [@DingFang], the first two authors]{style="color: black"} revisited these works by proving that vector fields along a regular curve are restrictions on this curve of some vector fields defined on the whole space; this allowed us to introduce parallel translations as in differential geometry. The main purpose of this work is to introduce parallel translations along stochastic regular curves, we will establish their existence and uniqueness in the case of $\mathbb{P}_{2,\infty}(\mathbb{T})$. Finally we remark that M.K Von Renesse and K.T. Sturm rigorously constructed in [@RS1] a quasi-invariant measure on $\mathbb{P}_2(\mathbb{T})$. # Framework {#sect2} In this work, we consider a connected compact Riemannian manifold $M$, of the Riemannian distance $d_M$, together with the normalized Riemannian measure $dx$: $\int_M dx=1$. As usual, we denote by $\mathbb{P}_2(M)$ the space of probability measures on $M$, endowed with the Wasserstein distance $W_2$ defined by $$W_2^2(\mu_1, \mu_2) =\inf\Bigl\{ \int_{M\times M} d_M^2(x,y)\,\pi(dx,dy),\quad \pi\in \mathcal{C}(\mu_1, \mu_2)\Bigr\},$$ where $\mathcal{C}(\mu_1, \mu_2)$ is the set of probability measures $\pi$ on $M\times M$, having $\mu_1, \mu_2$ as two marginal laws. It is well known that $\mathbb{P}_2(M)$ endowed with $W_2$ is a compact space. For tangent spaces $\bar{\mathbf{T}}_\mu$ of $\mathbb{P}_2(M)$ at $\mu$, we adopt the definition given in [@AGS], that is, $$\label{eq1.1} \bar{\mathbf{T}}_\mu = \overline{\bigl\{ \nabla\psi,\ \psi\in C^\infty(M) \bigr\}}^{L^2(\mu)},$$ the closure of gradients of smooth functions in the space $L^2(\mu)$ of vector fields on $M$. A curve $\{c(t);\ t\in [0,1]\}$ in $\mathbb{P}_2(M)$ is said to be absolutely continuous if there exists $k\in L^2([0,1])$ such that $$W_2\bigl( c(t_1), c(t_2)\bigr)\leq \int_{t_1}^{t_2}k(s)\, ds,\quad t_1<t_2.$$ For such a curve, Ambrosio, Gigli and Savaré [@AGS] proved that there exists a Borel vector field $Z_t$ on $M$ in $\displaystyle L^2([0,1]\times M)$, that is, $$\int_0^1 \Bigl[\int_M |Z_t(x)|_{T_xM}^2\, c_t(dx)\Bigr]\ dt<+\infty,$$ which satisfies the continuity equation $$\label{eq1.2} \frac{dc_t}{dt} +\nabla\cdot (Z_tc_t)=0.$$ The uniqueness of solutions to [\[eq1.2\]](#eq1.2){reference-type="eqref" reference="eq1.2"} holds if $Z_t\in \bar{\mathbf{T}}_{c_t}$ for almost all $t\in [0,1]$. We say that $Z_t$ is the intrinsic derivative of $\{c_t\}$ and denote it by $$\label{eq1.3} \frac{d^I c_t}{dt}.$$ For reader's convenience, we introduce now two classes of absolutely continuous curves in $\mathbb{P}_2(M)$. The first example is those $\{c_t\}$ generated by flows of maps. More precisely, $c_t=(U_t)_\#c_0$ with $$\label{eq1.3.1} \frac{dU_t}{dt}=\nabla\psi(U_t), \quad U_0(x)=x,$$ for regular function $\psi$ on $M$. In this case, $\displaystyle\frac{d^Ic_t}{dt}=\nabla\psi$. In what follows, we use the notation $V_\psi$ when $\nabla\psi$ is seen as a vector field on $\mathbb{P}_2(M)$, just as J. Lott did in [@Lott1] in order to clarify different roles played by $\nabla\psi$. For a functional $F$ on $\mathbb{P}_2(M)$, we say that $F$ is derivable along $V_\psi$ if $$\label{eq1.4} (\bar{D}_{V_\psi}F)(\mu)=\Bigl\{\frac{d}{dt}F((U_t)_{\#\mu})\Bigr\}_{ t=0}\quad\hbox{\rm exists}.$$ We say that the gradient $\bar{\nabla}F(\mu)\in \bar{\mathbf{T}}_\mu$ exists if for each $\psi\in C^\infty(M)$, $\bar{D}_{V_\psi}F$ exists and $(\bar{D}_{V_\psi}F)(\mu)=\langle\bar{\nabla}F,V_\psi\rangle_{\bar{\mathbf{T}}_\mu}$. The second class of examples of such $\{c_t\}$ is the class of geodesics in $\mathbb{P}_2(M)$. More precisely, for two probability measures $c_0, c_1$ on $M$ having density, according to R. McCann [@McCann], there is a function $\phi$ in the Sobolev space ${\mathbb D}_1^2(M)$ such that the optimal transport map ${\cal T}$ pushing $c_0$ to $c_1$, admits the expression $$\label{eq1.5} {\cal T}(x)=\exp_x(\nabla\phi(x)).$$ Then $\{c_t\}$ is defined by $\displaystyle c_t=({\cal T}_t)_\#c_0$ where ${\cal T}_t(x)=\exp_x(t\nabla\phi(x))$, and $\displaystyle\frac{d^Ic_t}{dt}=\nabla\phi({\cal T}_t^{-1}(x))$. In order to intrinsically formulate stochastic differential equations (SDE) on $\mathbb{P}_2(M)$, we need to introduce suitable functionals. Here are usual functionals considered in literature (see for example [@AGS]). 1\) Potential energy functional. For any $\varphi\in C^2(M)$, we set $F_\varphi(\mu)=\int_M \varphi\ \mu(dx)$. 2\) Internal energy functional. Let $\chi: [0,+\infty[\rightarrow]-\infty, +\infty]$ be a proper, continuous convex function satisfying $\chi(0)=0, \ \liminf_{s\rightarrow 0} \frac{\chi(s)}{s^\alpha}>-\infty$ for some $\alpha>\frac{d}{d+2}$, where $d$ is the dimension of $M$. The internal energy $\mathcal{F}$ is defined as follows $$\mathcal{F}(\mu)=\int_M \chi(\rho(x))\, dx,\quad\hbox{\rm if }\ d\mu=\rho\, dx,$$ and $\mathcal{F}(\mu)=+\infty$ otherwise. Two important examples are $\chi(s)=s\log(s)$ and $\displaystyle\chi(s)=\frac{s^m}{m-1}$ for $m> 1$. 3\) Interaction energy functional. Let $W: M^2 \rightarrow]-\infty, +\infty]$ be a l.s.c function, we define $${\mathcal W}(\mu)=\int_{M\times M} W(x,y)\mu(dx)\mu(dy).$$ The results below are not new, but for the sake of self-contained of our paper, we give a complete proof in what follows. We collect first and second order derivatives of these functionals in the following **Proposition 1**. *Let $V_\psi$ be a constant vector field on $\mathbb{P}_2(M)$, we have* *(i)$\displaystyle(\bar{D}_{V_\psi}F_\varphi)(\mu)=\langle V_\psi, \bar{\nabla}F_\varphi\rangle_{\bar{\mathbf{T}}_\mu}$ and $$\label{eq1.6} (\bar{D}_{V_\psi}\bar{D}_{V_\psi}F_\varphi)(\mu)=\int_M \mathcal L_{\nabla\psi}^2\varphi\ \mu(dx).$$* *(ii) For $\chi\in C^2(\mathbb{R}^*)$ such that $|\chi(s)| + s|\chi'(s)|+s^2|\chi''(s)|$ is bounded over $[0,1]$, we have $\displaystyle(\bar{D}_{V_\psi}\mathcal{F})(\mu)=-\int_M\bigl(\chi'(\rho)\rho-\chi(\rho)\bigr)\, \Delta\psi\, dx$ and $$\label{eq1.7} (\bar{D}_{V_\psi}\bar{D}_{V_\psi}\mathcal{F})(\mu)=\int_M \textcolor{black}{p}'(\rho)(\Delta\psi)^2\rho^2\, dx -\int_M \textcolor{black}{p}(\rho)\langle\nabla\psi, \nabla\Delta\psi>\, \rho\, dx,$$ where $\displaystyle\textcolor{black}{p}(s)=\chi'(s)-\frac{\chi(s)}{s}$. For $\chi(s)=s\log s$, $\displaystyle(\bar{D}_{V_\psi}\bar{D}_{V_\psi}\mathcal{F})(\mu)=-\int_M \langle\nabla\psi, \nabla\Delta\psi\rangle\, \rho\, dx$.* *(iii)Set $$\Phi(x,\mu)=\int_M \bigl( W(x,y)+W(y,x)\bigr)\, \mu(dy).$$ Then $\displaystyle(\bar{D}_{V_\psi}{\mathcal W})(\mu)=\int_M \langle\nabla\Phi(x,;\mu), \nabla\psi(x)\rangle\, \mu(dx)$, and $$(\bar{D}_{V_\psi}\bar{D}_{V_\psi}{\cal W})(\mu)= \int_M \bigl((\bar{D}_{V_\psi}\mathcal L_{\nabla\psi}\Phi)(x,\mu)+(\mathcal L^2_{\nabla\psi}\Phi)(x,\mu)\bigr)\, \mu(dx).$$ Where $\mathcal L_{\nabla\psi}$ denotes the Lie derivative with respect to $\nabla\psi$ on $M$.* *Proof.* Item (i) is well-known. For proving (ii), we consider the flow $(U_t)$ associated to $\nabla\psi$ (see [\[eq1.3.1\]](#eq1.3.1){reference-type="eqref" reference="eq1.3.1"}). Let $\mu_t=(U_t)_\#(\rho dx)=\rho_t\, dx$. It is known (see [@ABC]) that $$\rho_t=\rho(U_{-t})\,e^{-\int_0^t \Delta\psi(U_{-s})ds} \quad \hbox{\rm or}\quad \rho_t(U_t)=\rho\,e^{-\int_0^t \Delta\psi(U_{t-s})ds}.$$ It follows that $$\label{eq1.9} \Bigl\{\frac{d}{dt}\rho_t(U_t)\Bigr\}_{t=0}=- (\Delta\psi)\, \rho.$$ We have $\displaystyle\mathcal{F}(\mu_t)=\int_M \chi(\rho_t)\,dx=\int_M \frac{\chi(\rho_t)}{\rho_t}\, \rho_tdx=\int_M \hat\chi(\rho_t(U_t))\, \rho dx$, where $\hat\chi(s)=\chi(s)/s$. Therefore by Relation [\[eq1.9\]](#eq1.9){reference-type="eqref" reference="eq1.9"}, we get $$\Bigl\{\frac{d}{dt}\mathcal{F}(\mu_t)\Bigr\}_{t=0}=-\int_M \hat\chi'(\rho) (\Delta\psi\, \rho)\, \rho dx =-\int_M (\chi'(\rho)\rho-\chi(\rho))\Delta\psi\, dx.$$ For the sake of simplicity, we denote for a moment by $\tilde\mathcal{F}$ the right hand of above relation. Then $$\tilde\mathcal{F}(\mu_t)=-\int_M \bigl(\chi'(\rho_t)\rho_t-\chi(\rho_t)\bigr)\Delta\psi\, dx =-\int_M \textcolor{black}{p}(\rho_t)\, \Delta\psi\, \rho_t\, dx$$ which is equal to $$-\int_M \textcolor{black}{p}(\rho_t(U_t))\, \Delta\psi(U_t)\, \rho dx.$$ Again using Relation [\[eq1.9\]](#eq1.9){reference-type="eqref" reference="eq1.9"}, we get $$\Bigl\{\frac{d}{dt}\tilde\mathcal{F}(\mu_t)\Bigr\}_{t=0}=\int_M \textcolor{black}{p}'(\rho) (\Delta\psi\, \rho)\, \Delta\psi\rho dx -\int_M \textcolor{black}{p}(\rho)\,\langle\nabla\Delta\psi, \nabla\psi\rangle\,\rho dx,$$ which is nothing but the second formula in (ii). Item (iii) comes from direct calculation. ◻ # Stochastic differential equations on $\mathbb{P}_2(M)$ {#sect3} Let's first say a few words on SDE on a Riemannian manifold $M$. Given a family of vector fields $\{A_0(t,\cdot), A_1(t,\cdot), \ldots, A_N(t,\cdot)\}$ on $M$, and $B_t^0=t$ and a standard Brownian motion $t\rightarrow(B_t^1, \ldots, B_t^N)$ on $\mathbb{R}^N$, how to understand the following SDE on $M$ $$\label{eq2.1} dX_{t,s}=\sum_{i=0}^N A_i(t, X_{t,s})\circ dB_t^i,\quad X_{s,s}(x)=x \ ?$$ The equality [\[eq2.1\]](#eq2.1){reference-type="eqref" reference="eq2.1"} formally holds in the tangent space $T_{X_{t,s}}M$. Rigorously, a stochastic process $\{X_{t,s},\ t\geq s\}$ is a solution to SDE [\[eq2.1\]](#eq2.1){reference-type="eqref" reference="eq2.1"} if for any test function $f\in C^2(M)$, it holds $$\label{eq2.2} \begin{split} f(X_{t,s})=&f(x)+\sum_{i=1}^N \int_s^t (\mathcal L_{A_i(u)}f)(X_{u,s})\,dB_u^i\\ &+ \int_s^t \Bigl(\bigl( \mathcal L_{A_0(u)}f+\frac{1}{2}\sum_{i=1}^N \mathcal L^2_{A_i(u)}f\bigr)(X_{u,s})\, \Bigr)\, du, \end{split}$$ where $\mathcal L_{A_i(u)}$ denotes the Lie derivative on $M$ with respect to $x\rightarrow A_i(u,x)$, see [@Elworthy; @IW; @Malliavin]. In what follows, we develop this concept and use three types of functionals in the preceding section to introducing SDE on $\mathbb{P}_2(M)$. Let $\{\phi_0, \phi_1, \ldots, \phi_N\}$ be a family of functions on $[0,1]\times M$, continuous in $t\in [0,1]$ and smooth in $x\in M$. In this work, $\nabla$ always denotes the gradient operator on $M$. First we consider the following Stratanovich SDE on $M$: $$\label{eq2.3} dX_{t,s}=\sum_{i=0}^N \nabla\phi_i(t, X_{t,s})\circ dB_t^i,\quad t\geq s, \quad X_{s,s}(x)=x.$$ Let $d\mu=\rho\, dx$ be a probability measure on $M$, we set $\displaystyle\mu_t(\omega)=\bigl(X_{t,0}(\omega)\bigr)_\#\mu$. Let $\varphi\in C^2(M)$. After first using Itô['s]{style="color: black"} formula to $\varphi(X_{t,0})$, then integrating the two hand sides respect to $d\mu$, we get $$\circ d_t\,F_\varphi(\mu_t)=\sum_{i=0}^N \Bigl( \int_M \langle\nabla\varphi, \nabla\phi_i(t, \cdot)\rangle\ \mu_t(dx)\Bigr) \circ dB_t^i =\sum_{i=0}^N \langle V_\varphi, V_{\phi_i(t,\cdot)} \rangle_{\bar{\mathbf{T}}_{\mu_t}}\circ dB_t^i.$$ **Definition 2**. *We say that the intrinsic Itô stochastic differential of $\mu_t$, denoted by $\circ d_t^I\mu_t$, admits the following expression $$\label{eq2.4} \circ d_t^I\mu_t=\sum_{i=0}^N V_{\phi_i(t,\cdot)}\ \circ dB_t^i.$$* Recall that $\bar{\nabla}F_\varphi=V_\varphi$; using notation [\[eq2.4\]](#eq2.4){reference-type="eqref" reference="eq2.4"}, $\circ d_tF_\varphi(\mu_t)$ can be written in the form $$\circ d_t\, F_\varphi(\mu_t)=\langle \bar{\nabla}F_\varphi,\ \circ d_t^I \mu_t\rangle_{\bar{\mathbf{T}}_{\mu_t}},$$ symbolically read in the inner product of $\bar{\mathbf{T}}_{\mu_t}$, in the same way as we did on Riemannian manifolds. By Itô['s]{style="color: black"} formula [\[eq2.2\]](#eq2.2){reference-type="eqref" reference="eq2.2"}, $$d_t\varphi(X_{t,0})= \sum_{i=0}^N (\mathcal L_{\nabla \phi_i(t)} \varphi)(X_{t,0})\, dB_t^i +\frac{1}{2}\ \sum_{i=1}^N \bigl(\mathcal L_{\nabla\phi_i(t)}^2\varphi\bigr)(X_{t,0})\, dt.$$ According to Proposition [Proposition 1](#prop1.1){reference-type="ref" reference="prop1.1"}, above relation yields $$d_t F_\varphi (\mu_t)=\sum_{i=0}^N \langle\bar{\nabla}F_\varphi, V_{\phi_i(t)}\rangle_{\bar{\mathbf{T}}_{\mu_t}}\, dB_t^i +\frac{1}{2} \sum_{i=1}^n (\bar{D}_{V_{\phi_i(t)}}^2F_\varphi)(\mu_t)\, dt.$$ **Proposition 3**. *For any polynomial $F$ on $\mathbb{P}_2(M)$, we have $$\label{eq2.5} d_t F(\mu_t) = \sum_{i=0}^N \langle \bar{\nabla}F, V_{\phi_i(t,\cdot)}\rangle_{\bar{\mathbf{T}}_{\mu_t}}\ dB_t^i +\frac{1}{2}\sum_{i=1}^N (\bar D_{V_{\phi_i(t,\cdot)}}^2F)(\mu_t)\ dt.$$* *Proof.* For two functionals $F_1$ and $F_2$ satisfying Formula [\[eq2.5\]](#eq2.5){reference-type="eqref" reference="eq2.5"}, by Itô formula, $$d_t(F_1F_2)(\mu_t)=d_tF_1(\mu_t)\, F_2(\mu_t)+F_1(\mu_t)\, d_tF_2(\mu_t)+ d_tF_1(\mu_t)\cdot d_tF_2(\mu_t).$$ Note that $$\bar D_{V_{\phi_i(t,\cdot)}}^2 (F_1F_2) =F_2\bar D_{V_{\phi_i(t,\cdot)}}^2F_1 +F_1 \bar D_{V_{\phi_i(t,\cdot)}}^2F_2 +2 \langle \bar{\nabla}F_1, V_{\phi_i(t,\cdot)}\rangle\cdot \langle \bar{\nabla}F_2, V_{\phi_i(t,\cdot)}\rangle,$$ and $\displaystyle d_tF_1(\mu_t)\cdot d_tF_2(\mu_t) =\sum_{i=1}^N \langle \bar{\nabla}F_1, V_{\phi_i(t,\cdot)}\rangle\cdot \langle \bar{\nabla}F_2, V_{\phi_i(t,\cdot)}\rangle\, dt$; so Formula [\[eq2.5\]](#eq2.5){reference-type="eqref" reference="eq2.5"} holds for $F_1F_2$. A polynomial $F$ on $\mathbb{P}_2(M)$ is a finite sum of $F_{\varphi_1}\cdots F_{\varphi_k}$, Formula [\[eq2.5\]](#eq2.5){reference-type="eqref" reference="eq2.5"} remains true for $F$. We complete the proof. ◻ Now are going to see what happens with internal energy functional $\mathcal{F}$, which is not continuous. Note that if $\displaystyle\rho=\frac{d\mu}{dx}>0$, then for almost all $\omega$, $\mu_t(\omega)$ has a positive density $\rho_t$ with respect to $dx$. **Proposition 4**. *The stochastic process $\{\rho_t, t\geq 0\}$ satisfies the following SPDE $$\label{eq2.6} d\rho_t=-\sum_{i=0}^N \textup{div}\bigl(\rho_t\,\nabla\phi_i(t)\bigr)\, dB_t^i +\frac{1}{2} \sum_{i=1}^N \textup{div}\bigl(\textup{div}(\rho_t\nabla\phi_i(t))\nabla\phi_i(t)\bigr)\, dt.$$* *Proof.* We have $$\begin{split} \int_M \langle\nabla\varphi, \nabla\phi_i(t)\rangle\, \mu_t(dx) &=\int_M \langle\nabla\varphi, \rho_t\nabla\phi_i(t)\rangle\, dx\\ &=-\int_M \varphi\, \textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)\, dx. \end{split}$$ In the same way, $$\int_M \mathcal L_{\nabla\phi_i(t)}^2\varphi\, \mu_t(dx)=\int_M \varphi\, \textup{div}\bigl(\textup{div}(\rho_t\nabla\phi_i(t))\nabla\phi_i(t)\bigr)\ dx.$$ Formula [\[eq2.6\]](#eq2.6){reference-type="eqref" reference="eq2.6"} follows by considering $F_\varphi(\mu_t)=\int_M \varphi\, \rho_t\, dx$ for any $\varphi\in C^2(M)$, together with Formula [\[eq2.5\]](#eq2.5){reference-type="eqref" reference="eq2.5"}. ◻ Let $\chi$ be a $C^2$ function defined on $]0,+\infty[$. According to [\[eq2.6\]](#eq2.6){reference-type="eqref" reference="eq2.6"}, we have $$\begin{split} d\chi(\rho_t)=-\sum_{i=0}^N \chi'(\rho_t)\, \textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)\, dB_t^i &+\frac{1}{2}\sum_{i=1}^N \chi'(\rho_t)\,\textup{div}\bigl(\textup{div}(\rho_t\nabla\phi_i(t))\nabla\phi_i(t) \bigr)\,dt\\ &+\frac{1}{2}\sum_{i=1}^N \chi''(\rho_t)\bigl(\textup{div}(\rho_t\nabla\phi_i(t)) \bigr)^2\, dt. \end{split}$$ Remarking that $\displaystyle\textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)=\rho_t\Delta\phi_i(t)+\langle\nabla\rho_t, \nabla\phi_i(t)\rangle$, that we will use several times in the sequel, we have $$\int_M \chi'(\rho_t)\, \textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)\, dx=\int_M \chi'(\rho_t)\, \rho_t\Delta\phi_i(t)\, dx +\int_M \chi'(\rho_t)\langle\nabla\rho_t, \nabla\phi_i(t)\rangle\, dx,$$ this last term is equal to $$\int_M \langle\nabla\bigl(\chi(\rho_t)\bigr), \nabla\phi_i(t)\rangle\, dx =-\int_M \chi(\rho_t)\, \Delta\phi_i(t)\, dx.$$ Therefore $$\int_M \chi'(\rho_t)\, \textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)\, dx=\int_M\bigr( \chi'(\rho_t)\rho_t - \chi(\rho_t)\bigr)\,\Delta\phi_i(t)\, dx,$$ which is $-(\bar{D}_{V_{\phi_i(t)}} \mathcal{F})(\mu_t)$ by Proposition [Proposition 1](#prop1.1){reference-type="ref" reference="prop1.1"}. Now we are going to deal with drift terms. First of all, we compute $$I_1=\int_M \chi'(\rho_t)\,\textup{div}\bigl(\textup{div}(\rho_t\nabla\phi_i(t))\nabla\phi_i(t) \bigr)\,dx =-\int_M \chi''(\rho_t)\,\langle\nabla\rho_t, \nabla\phi_i(t)\rangle\, \textup{div}(\rho_t\nabla\phi_i(t))\, dx.$$ Again using $\displaystyle\textup{div}\bigl(\rho_t\nabla\phi_i(t)\bigr)=\rho_t\Delta\phi_i(t)+\langle\nabla\rho_t, \nabla\phi_i(t)\rangle$, we write down $I_1$ as $$I_1=-\int_M \chi''(\rho_t) \rho_t\, \langle\nabla\rho_t, \nabla\phi_i(t)\rangle\Delta\phi_i(t)\,dx -\int_M \chi''(\rho_t)\,\langle\nabla\rho_t, \nabla\phi_i(t)\rangle^2\, dx.$$ Put $\displaystyle I_2= \int_M \chi''(\rho_t)\, \bigl( \textup{div}(\rho_t\nabla\phi_i(t))\bigr)^2\, dx$. In the same way, $$\begin{split} I_2=\int_M \chi''(\rho_t) \rho_t^2(\Delta\phi_i(t))^2\, dx &+ 2\int_M \chi''(\rho_t)\rho_t\langle\nabla\rho_t, \nabla\phi_i(t)\rangle\,\Delta\phi_i(t)\, dx\\ &+\int_M \chi''(\rho_t) \langle\nabla\rho_t, \nabla\phi_i(t)\rangle^2\, dx. \end{split}$$ Combining above terms gives $$\label{eq2.7} I_1+I_2=\int_M \chi''(\rho_t) \rho_t^2(\Delta\phi_i(t))^2\, dx + \int_M \chi''(\rho_t)\rho_t\,\langle\nabla\rho_t, \nabla\phi_i(t)\rangle\,\Delta\phi_i(t)\, dx.$$ On the other hand, according to the expression for $\bar{D}_{V_{\phi_i(t)}}^2\mathcal{F}$ in Proposition [Proposition 1](#prop1.1){reference-type="ref" reference="prop1.1"}, we set $$I_3=\int_M \textcolor{black}{p}'(\rho_t)\,(\Delta\phi_i(t))^2\, \rho_t^2\,dx.$$ Replacing $\tilde\chi$ by its expression, $I_3$ becomes $$\begin{split} I_3=\int_M \chi''(\rho_t)\,(\Delta\phi_i(t))^2\, \rho_t^2\,dx&-\int_M \chi'(\rho_t)(\Delta\phi_i(t))^2\, \rho_t\,dx\\ &+\int_M \chi(\rho_t)\,(\Delta\phi_i(t))^2\, dx. \end{split}$$ We put $$I_4 =\int_M \chi'(\rho_t)\, \langle\nabla\phi_i(t), \nabla\Delta\phi_i(t)\rangle\, \rho_t\, dx -\int_M \chi(\rho_t)\, \langle\nabla\phi_i(t), \nabla\Delta\phi_i(t)\rangle\, dx.$$ In order to more clearly see the relation between $I_1+I_2$ and $I_3-I_4$, we now deal with the second term in $I_1+I_2$, that is, $$J=\int_M \chi''(\rho_t)\rho_t\,\langle\nabla\rho_t, \nabla\phi_i(t)\rangle\,\Delta\phi_i(t)\, dx =\int_M \langle\nabla\bigl(\chi'(\rho_t)),\nabla\phi_i(t)\rangle\, \rho_t\Delta\phi_i(t)\, dx$$ Via divergence operator $\textup{div}$, we get $$J=-\int_M \chi'(\rho_t)\, \textup{div}\bigl(\rho_t\Delta\phi_i(t)\,\nabla\phi_i(t)\bigr)\ dx.$$ Note that $\displaystyle\textup{div}\bigl(\rho_t\Delta\phi_i(t)\,\nabla\phi_i(t)\bigr)=\rho_t\, (\Delta\phi_i(t))^2+ \rho_t\, \langle\nabla\Delta\phi_i(t),\nabla\phi_i(t)\rangle+ \langle\nabla\rho_t,\nabla\phi_i(t)\rangle\Delta\phi_i(t)$, and $$\begin{split} -&\int_M \chi'(\rho_t)\langle\nabla\rho_t,\nabla\phi_i(t)\rangle\Delta\phi_i(t)\, dx =\int_M \chi(\rho_t)\, \textup{div}\bigl(\Delta\phi_i(t)\nabla\phi_i(t)\bigr)\,dx\\ &=\int_M \chi(\rho_t)\, \bigl((\Delta\phi_i(t))^2+\langle\nabla\phi_i(t), \nabla\Delta\phi_i(t)\rangle\bigr)\, dx. \end{split}$$ Combining all of these terms, finally we prove that $I_1+I_2=I_3-I_4$. In other words, we get the following result. **Proposition 5**. *For any internal energy functional $\mathcal{F}$ with $\chi\in C^2(]0,+\infty[)$, we have $$d_t\mathcal{F}(\mu_t)=\sum_{i=0}^N \langle\bar{\nabla}\mathcal{F}, V_{\phi_i(t)}\rangle_{\bar{\mathbf{T}}_{\mu_t}}\, dB_t^i +\frac{1}{2}\sum_{i=1}^N \bigl(\bar{D}_{V_{\phi_i(t)}}^2\mathcal{F}\bigr)(\mu_t)\, dt.$$* **Proposition 6**. *Itô['s]{style="color: black"} formula [\[eq2.5\]](#eq2.5){reference-type="eqref" reference="eq2.5"} also holds for interaction energy functional ${\mathcal W}$.* *Proof.* Applying Itô formula to $\displaystyle W\bigl(X_{t,0}(x), X_{t,0}(y)\bigr)$ and proceeding as above yields the result. ◻ Now having these results in hand, we say now that the stochastic process $\{\mu_t;\ t\geq 0\}$ solves the following SDE on $\mathbb{P}_{2,\infty}(M)$, $$\label{eq2.8} \circ d_t^I\mu_t=\sum_{i=0}^N V_{\phi_i(t)}(\mu_t)\circ dB_t^i,\quad \mu_0=\rho\, dx.$$ In what follows, we introduce quite general vector fields on $\mathbb{P}_2(M)$. **Definition 7**. *We say that $Z$ is a vector field on $\mathbb{P}_2(M)$ if there exists a Borel map $\Phi: M\times\mathbb{P}_2(M)\rightarrow\mathbb{R}$ such that, for any $\mu\in\mathbb{P}_2(M)$, $x\rightarrow\Phi(x,\mu)$ is $C^1$ and $\displaystyle Z(\mu)=V_{\Phi(\cdot,\mu)}$.* Recall that for two probability measures $\mu, \nu\in \mathbb{P}_{2,\infty}(M)$, there is a unique optimal transport map $T_{\mu,\nu}: M\rightarrow M$, which pushes $\mu$ to $\nu$ and has the expression (see [@McCann; @GMcCann; @Villani1]): $$T_{\mu,\nu}(x)=\exp_x\bigl(\nabla\xi(x)\bigr).$$ Let $\displaystyle\xi_x^{\mu,\nu}(t)=\exp_x\bigl(t\nabla\xi(x)\bigr)$ and let $//_t^{\xi_x^{\mu,\nu}}$ be the parallel translation along $\{\xi_x^{\mu,\nu}(t);\ t\in [0,1]\}$. **Definition 8**. *We say that $Z$ is Lipschitzian if there exists a constant $\kappa>0$ such that $$\label{eq2.9} \int_M \Bigl|//_1^{\xi_x^{\mu,\nu}}\nabla\Phi(x,\mu)-\nabla\Phi\bigl(T_{\mu,\nu}(x),\nu\bigr)\Bigr|^2\, \mu(dx) \leq \kappa^2\, W_2^2(\mu,\nu),$$ for any couple of probability measures $(\mu,\nu)\in \mathbb{P}_{2,\infty}(M)\times \mathbb{P}_{2,\infty}(M)$.* It was proved in [@DingFang] that, under above Lipschitzian condition, the ODE on $\mathbb{P}_{2,\infty}(M)$ $$\label{eq2.10} \frac{d^I \nu_t}{dt}=Z(\nu_t);\quad \nu_0=\rho\, dx \ \hbox{\rm given}$$ admits a unique solution. Condition [\[eq2.9\]](#eq2.9){reference-type="eqref" reference="eq2.9"} is satisfied [@DingFang] if (i) $x\rightarrow\nabla^2\Phi(x,\mu)$ exists such that $$\label{eq2.11} \Lambda_1=\sup_{\mu\in \mathbb{P}_2(M)}||\nabla^2\Phi(\cdot, \mu)||_\infty <+\infty,$$ and (ii) there exists a constant $\Lambda_2$ such that $$\label{eq2.12} |\nabla\Phi(x,\mu)-\textcolor{black}{\nabla} \Phi(x,\nu)|\leq \Lambda_2\, W_2(\mu, \nu),\quad \hbox{\rm for any }\ x\in M.$$ **Remark 9**. *The function $\Phi$ involved in the interaction energy functional ${\mathcal W}$ satisfies Condition [\[eq2.11\]](#eq2.11){reference-type="eqref" reference="eq2.11"} and [\[eq2.12\]](#eq2.12){reference-type="eqref" reference="eq2.12"} if $W\in C^2(M\times M)$. This functional ${\mathcal W}$ plays an important role in the work [@Liming1]. A definition of absolutely continuous vector field using parallel translations was given in Chapter 3 in [@Gigli].* In the sequel, for simplicity, we suppose that $\phi_0, \ldots \phi_N$ are time-independent. The following result was established by F. Y. Wang in [@Wang] in a quite general setting. For the own interest, we give a construction of solutions. **Proposition 10**. *Suppose that $\Phi$ satisfies Conditions [\[eq2.11\]](#eq2.11){reference-type="eqref" reference="eq2.11"} and [\[eq2.12\]](#eq2.12){reference-type="eqref" reference="eq2.12"}, then there is a unique solution $(X_t, \mu_t)$ to the following Mckean-Vlasov SDE: $$\label{eq2.13} dX_t=\sum_{i=0}^N \nabla\phi_i(X_t)\circ dB_t^i + \nabla\Phi(X_t, \mu_t)\,dt,\ \mu_t=(X_t)_\#\mu_0,$$* *Proof.* We will construct a solution to [\[eq2.13\]](#eq2.13){reference-type="eqref" reference="eq2.13"}. Let $(U_t)_{t\geq 0}$ be the stochastic flow associated to the folllowing SDE $$dU_t = \sum_{i=0}^N \nabla\phi_i(U_t)\circ dB_t^i.$$ Define the stochastic measure-dependent vector fields $V_t(\omega, x, \mu)$ on $M$ by $$V_t(\omega, x, \mu)=\bigl(U_t^{-1}(\omega, \cdot) \bigr)_*\nabla\Phi(x, (U_t)_\#\mu) =(U_t^{-1})'(\omega, U_t(x))\nabla\Phi\bigl(U_t(x), (U_t)_\#\mu\bigr),$$ where the prime denotes the differential with respect to $x$. Since the manifold $M$ is compact, we have, for $\omega$ given, $$|V_t(\omega, x, \mu)-V_t(\omega, x, \nu)| \leq ||(U_t^{-1})'||_\infty\ |\nabla\Phi\bigl(U_t(x), (U_t)_\#\mu\bigr)-\nabla\Phi\bigl(U_t(x), (U_t)_\#\nu\bigr)|.$$ Note that $$W_2\bigl( (U_t)_\#\mu, (U_t)_\#\nu\bigr)\leq ||U_t'||_\infty\,, W_2(\mu, \nu);$$ then under Condition [\[eq2.12\]](#eq2.12){reference-type="eqref" reference="eq2.12"}, $\mu\rightarrow V_t(\omega, x, \mu)$ is Lipschitzian uniformly in $(t,x)\in [0,1]\times M$; moreover by Condition [\[eq2.11\]](#eq2.11){reference-type="eqref" reference="eq2.11"}, $x\rightarrow V_t(\omega, x, \mu)$ is Lipschitzian uniformly in $(t,\mu)\in [0,1]\times \mathbb{P}_2(M)$. So there is a unique solution $(Y_t, \nu_t)$ to the following Mckean-Vlasov ODE on $M$ $$\frac{d}{dt}Y_t=V_t(Y_t, \nu_t),\quad \nu_t=(Y_t)_\#\mu_0.$$ Let $\tilde X_t=U_t(Y_t)$. By Itô-Wentzell formula, $$d\tilde X_t=\sum_{i=0}^N \nabla\phi_i(U_t(Y_t))\circ dB_t^i + U_t'(Y_t)\, V_t(Y_t, \nu_t),$$ the last term in above equality is $\displaystyle\nabla\Phi\bigl( \tilde X_t, (U_t)_\#\nu_t\bigr)$. Note that $(\tilde X_t)_\#\mu_0=(U_t)_\#\,(Y_t)_\#\mu_0=(U_t)_\#\nu_t$; therefore $\big(\tilde X_t, (U_t)_\#\nu_t\bigr)$ is a solution to Mckean-Vlasov SDE [\[eq2.13\]](#eq2.13){reference-type="eqref" reference="eq2.13"} on $M$. ◻ **Proposition 11**. *The stochastic process $\{\mu_t;\ t\in [0,1]\}$ obtained in Proposition [Proposition 10](#prop2.8){reference-type="ref" reference="prop2.8"} solves the following SDE on $\mathbb{P}_2(M)$:* *$$\label{eq2.14} \circ d_t^I\mu_t=\sum_{i=0}^N V_{\phi_i}(\mu_t)\ \circ dB_t^i+ Z(\mu_t)\,dt,\quad \mu_{|_{t=0}}=\mu_0,$$ where $Z(\mu)=V_{\Phi(\cdot,\mu)}$, see Definition [Definition 7](#def2.2){reference-type="ref" reference="def2.2"}.* # Regular curves and parallel translations on $\mathbb{P}_{2,\infty}(M)$ {#sect3} In this section, we only consider the space $\mathbb{P}_{2,\infty}$. According to J. Lott [@Lott1], the Levi-Civita covariant derivative defined by usual formula $$\begin{split} 2\langle \bar{\nabla}_{V_{\psi_1}}V_{\psi_2}, V_{\psi_3}\rangle_{\bar{\mathbf{T}}_\mu} &= \bar{D}_{V_{\psi_1}}\langle V_{\psi_2}, V_{\psi_3}\rangle_{\bar{\mathbf{T}}_\mu} + \bar{D}_{V_{\psi_2}}\langle V_{\psi_3}, V_{\psi_1}\rangle_{\bar{\mathbf{T}}_\mu} - \bar{D}_{V_{\psi_3}}\langle V_{\psi_1}, V_{\psi_2}\rangle_{\bar{\mathbf{T}}_\mu} \\ &+ \langle V_{\psi_3}, [V_{\psi_1},V_{\psi_2}]\rangle_{\bar{\mathbf{T}}_\mu} - \langle V_{\psi_2}, [V_{\psi_1},V_{\psi_3}]\rangle_{\bar{\mathbf{T}}_\mu} - \langle V_{\psi_1}, [V_{\psi_2},V_{\psi_3}]\rangle_{\bar{\mathbf{T}}_\mu}, \end{split}$$ admits the following expression $$\label{eq3.1} \langle\bar{\nabla}_{V_{\psi_1}}V_{\psi_2}, V_{\psi_3}\rangle_{\bar{\mathbf{T}}_\mu}=\int_M\langle \nabla^2\psi_2, \nabla\psi_1\otimes\nabla\psi_3\rangle\, \mu(dx).$$ Also as usual, above bracket $[V_{\psi_1}, V_{\psi_2}]$ denotes the Lie bracket of two vector fields on $\mathbb{P}_{2,\infty}(M)$. Note that $[V_{\psi_1}, V_{\psi_2}]$ as well as $\bar{\nabla}_{V_{\psi_1}}V_{\psi_2}$ are not constant vector fields. Let $$\Pi_\mu: L^2(M,TM;\mu)\rightarrow\bar{\mathbf{T}}_\mu$$ be the orthogonal projection; then $$\label{eq3.2} \bar{\nabla}_{V_{\psi_1}}V_{\psi_2}(\mu)=\Pi_\mu\bigl(\nabla_{\nabla\psi_1}\nabla\psi_2\bigr).$$ For $\mu\in \mathbb{P}_{2,\infty(M)}$ with $d\mu=\rho\, dx,\ \rho>0$, we denote by $\Delta_\mu$ the Witten Laplacian: $$\Delta_\mu=\Delta +\langle\nabla\log\rho,\ \nabla\cdot\rangle,$$ and $\textup{div}_\mu$ the divergence operator defined by $\int_M \langle\nabla\varphi, Z\rangle\,\mu(dx)=-\int_M\varphi\, \textup{div}_\mu(Z)\, \mu(dx)$ for any $\varphi\in C^\infty(M)$. We have the relation $\textup{div}_\mu(Z)=\textup{div}(Z)+\langle\nabla\log\rho, Z\rangle$. By Hodge decomposition [@Li], there is a function $f$ and a vector field $Y$ of $\textup{div}_\mu(Y)=0$ such that $Z=\nabla f + Y$. Therefore $\textup{div}_\mu(Z)=\Delta_\mu f$ and $f=\Delta_\mu^{-1}\bigl(\textup{div}_\mu(Z)\bigr)$, or $$\label{eq3.3} \Pi_\mu(Z)=\nabla \Delta_\mu^{-1}\bigl(\textup{div}_\mu(Z)\bigr).$$ Formula [\[eq3.3\]](#eq3.3){reference-type="eqref" reference="eq3.3"} first holds for smooth $Z$, then extends to the space $L^2(M,TM; \mu)$. Now let $\{c_t\}$ be a curve in $\mathbb{P}_{2,\infty}(M)$ defined by a flow of diffeomorphisms $X_{t,s}$ associate to ODE: $$\label{ODE} dX_{t,s}=\nabla\phi_t(X_{t,s})\, dt, \quad t\geq s, \quad X_s(x)=x,$$ with $c_t=(X_{t,0})_\#(\rho\,dx)$. The following derivative formula of $\Pi_{c_{t}}$ was obtained in chapter 5 of [@Gigli]. For reader's convenience, we include a proof here. **Theorem 12**. *For a smooth vector $Z$ on $M$, $t\rightarrow\mathbb{P}_{c_t}(Z)$ is absolutely continuous and $$\label{eq3.4} \frac{d}{dt}\Pi_{c_t}(Z)=-\Pi_{c_t}\Bigl(\Delta_{c_t}\phi_t\, \Pi_{c_t}^\perp(Z) \Bigr),$$ where $\Pi_\mu^\perp=I-\Pi_\mu$.* *Proof.* Denote for a moment $\hat\rho_t$ the density of $c_t$ with respect to $c_0$: $\displaystyle\hat\rho_t=\frac{dc_t}{dc_0}$, then $\hat\rho_t$ satisfies the following equation $$\label{eq3.5} \frac{d\hat\rho_t}{dt}=-\textup{div}_{c_t}(\nabla\phi_t)\, \hat\rho_t=-(\Delta_{c_t}\phi_t)\, \hat\rho_t.$$ Let $f\in C^\infty(M)$, we have the relation $$\int_M\langle\nabla f, Z\rangle\, c_t(dx)=\int_M \langle\nabla f, \Pi_{c_t}(Z)\rangle\, c_t(dx).$$ Using the density $\hat\rho_t$ in the two hand sides, above equality becomes $$\int_M\langle\nabla f, Z\rangle\, \hat\rho_t\ c_0(dx)=\int_M \langle\nabla f, \Pi_{c_t}(Z)\rangle\, \hat\rho_t\ c_0(dx).$$ Taking the derivative on the two hand sides with respect to $t$, and using [\[eq3.5\]](#eq3.5){reference-type="eqref" reference="eq3.5"}, we get $$\begin{split} -\int_M \langle\nabla f, Z\rangle(\Delta_{c_t}\phi_t)\, \hat\rho_t\ c_0(dx) =&\int_M \langle\nabla f, \frac{d}{dt}\Pi_{c_t}(Z)\rangle\, \hat\rho_t\ c_0(dx)\\ &-\int_M \langle\nabla f, \Pi_{c_t}(Z)\rangle\, \Delta_{c_t}\phi_t\, \hat\rho_t\ c_0(dx), \end{split}$$ which implies the equality $$\int_M \langle\nabla f, \frac{d}{dt}\Pi_{c_t}(Z)\rangle\, \hat\rho_t\ c_0(dx) =-\int_M \langle\nabla f, \Pi_{c_t}^\perp(Z)\rangle\, \Delta_{c_t}\phi_t\, \hat\rho_t\ c_0(dx).$$ By [\[eq3.3\]](#eq3.3){reference-type="eqref" reference="eq3.3"}, $\displaystyle\frac{d}{dt}\Pi_{c_t}(Z)\in \bar{\mathbf{T}}_{c_t}$. Since $f$ is arbitrary, we get the result [\[eq3.4\]](#eq3.4){reference-type="eqref" reference="eq3.4"} from above equality. ◻ Now let's briefly describe results obtained in the literature for parallel translations on $\mathbb{P}_2(M)$. Let $\{Y_t;\ t\in [0,1]\}$ be a family of vector fields along $\{c_t;\ t\in [0,1]\}$, that is, $\displaystyle Y_t\in \bar{\mathbf{T}}_{c_t}$. Suppose there are smooth functions $(t,x)\rightarrow\Phi_t(x)$ and $(t,x)\rightarrow\Psi_t(x)$ such that $$\frac{d^Ic_t}{dt}=V_{\Phi_t},\quad Y_t=V_{\Psi_t}.$$ J. Lott obtained in [@Lott1] that if $\{Y_t;\ t\in [0,1]\}$ is parallel along $\{c_t;\ t\in [0,1]\}$, then $\{\nabla\Psi_t;\ t\in [0,1]\}$ is a solution to the following linear PDE $$\label{eq3.6} \frac{d}{dt}\nabla\Psi_t+\Pi_{c_t}\Bigl(\nabla_{\nabla\Phi_t}\nabla\Psi_t\Bigr)=0.$$ Existence [of weak solutions]{style="color: black"} to [\[eq3.6\]](#eq3.6){reference-type="eqref" reference="eq3.6"} were established by L. Ambrosio and N. Gigli in [@AG]. By mimicing the section 5 of the paper [@AG], [the first two authors]{style="color: black"} obtained in [@DingFang] the followng result **Theorem 13**. *For any $\nabla\Psi_0\in L^2(c_0)$, there is a unique weak solution $\{\nabla\Psi_t, t\in [0,1]\}$ in the sense that $V_{\Psi_t}\in \bar{\mathbf{T}}_{c_t}$ and $$\label{eq3.7} \Pi_{c_t} \Bigl( \lim_{\varepsilon\downarrow 0}\frac{\tau_{\varepsilon}^{-1}\,\nabla\Psi_{t+\varepsilon}(X_{t+\varepsilon,t})-\nabla\Psi_t}{\varepsilon} \Bigr)=0$$ holds in $L^2(c_t)$ for almost all $t\in [0,1]$, where $\tau_\varepsilon$ is the parallel translation along $\{s\rightarrow X_{t+s,t}, s\in [0,\varepsilon]\}$, that is equivalent to say that $t\rightarrow\nabla\Psi_t$ is absolutely continuous and $$\label{eq3.8} \frac{d}{dt}\int_M \langle \nabla f, \nabla\Psi_t\rangle\, c_t(dx)=\int_M \langle \nabla^2f, \nabla\phi_t\otimes \nabla\Psi_t\rangle\,c_t(dx), \quad f\in C^\infty(M).$$* **Proposition 14**. *Let $\{\nabla\Psi_t;\ t\in [0,1]\}$ be the parallel translation along $\{c_t;\ t\in [0,1]\}$ in Theorem [Theorem 13](#th3.2){reference-type="ref" reference="th3.2"}, then for any smooth vector field $Z$ on $M$, $$\label{eq3.9} \begin{split} \frac{d}{dt}\int_M \langle Z,\nabla\Psi_t\rangle\, c_t(dx) =&-\int_M\langle(\Delta_{c_t}\phi_t)\, \Pi_{c_t}^\perp(Z),\ \nabla\Psi_t\rangle\ c_t(dx)\\ &+\int_M \langle\nabla_{\nabla\phi_t}\bigl(\Pi_{c_t}(Z) \bigr),\ \nabla\Psi_t\rangle\ c_t(dx). \end{split}$$* *Proof.* Let $\displaystyle I_t=\int_M \langle\Pi_{c_t}(Z),\nabla\Psi_t\rangle\, c_t(dx)$. For any $\varepsilon>0$, we have $$I_{t+\varepsilon}=\int_M \langle\Pi_{c_{t+\varepsilon}}(Z),\nabla\Psi_{t+\varepsilon}\rangle\, c_{t+\varepsilon}(dx) =\int_M \langle\tau_\varepsilon^{-1}\Pi_{c_{t+\varepsilon}}(Z),\tau_\varepsilon^{-1}\nabla\Psi_{t+\varepsilon}\rangle(X_{t+\varepsilon,t})\, c_t(dx).$$ Then $$\begin{split} I_{t+\varepsilon}-I_t&=\int_M \langle\tau_\varepsilon^{-1}\Pi_{c_{t+\varepsilon}}(Z)(X_{t+\varepsilon,t})-\Pi_{c_t}(Z)(x), \ \tau_\varepsilon^{-1}\nabla\Psi_{t+\varepsilon}(X_{t+\varepsilon,t})\rangle\ c_t(dx)\\ &+\int_M \langle \Pi_{c_t}(Z), \tau_\varepsilon^{-1}\nabla\Psi_{t+\varepsilon}(X_{t+\varepsilon,t})-\nabla\Psi_t(x)\rangle\ c_t(dx) =J_\varepsilon^1+J_\varepsilon^2\quad \hbox{\rm respectively}. \end{split}$$ As $\varepsilon\rightarrow 0$, $\displaystyle\tau_\varepsilon^{-1}\nabla\Psi_{t+\varepsilon}(X_{t+\varepsilon,t})$ converges to $\nabla\Psi_t(x)$ and $J_\varepsilon^2/\varepsilon$ converges to $0$ by [\[eq3.7\]](#eq3.7){reference-type="eqref" reference="eq3.7"}. For $J_\varepsilon^1$, note that $$\begin{split} \frac{1}{\varepsilon}\Bigl(\tau_\varepsilon^{-1}\Pi_{c_{t+\varepsilon}}(Z)(X_{t+\varepsilon,t})-\Pi_{c_t}(Z)(x) \Bigr) =&\frac{1}{\varepsilon} \Bigl(\tau_\varepsilon^{-1}\Pi_{c_{t+\varepsilon}}(Z)(X_{t+\varepsilon,t}) -\tau_\varepsilon^{-1}\Pi_{c_{t}}(Z)(X_{t+\varepsilon,t})\Bigr)\\ &+\frac{1}{\varepsilon}\Bigl(\tau_\varepsilon^{-1}\Pi_{c_{t}}(Z)(X_{t+\varepsilon,t})-\Pi_{c_t}(Z)(x)\Bigr) \end{split}$$ As $\varepsilon\rightarrow 0$, the last term in above equality converges to $\displaystyle\nabla_{\nabla\phi_t}\Pi_{c_t}(Z)$, while the first term on the right hand gives $\displaystyle\frac{d}{dt}\Pi_{c_t}(Z)$. Finally using [\[eq3.4\]](#eq3.4){reference-type="eqref" reference="eq3.4"}, we get the result [\[eq3.9\]](#eq3.9){reference-type="eqref" reference="eq3.9"}. ◻ Now we are going to establish the existence of strong solution to Equation [\[eq3.6\]](#eq3.6){reference-type="eqref" reference="eq3.6"} in the case of $\mathbb{P}_{2,\infty}(\mathbb{T})$, the base manifold $M$ being the torus $\mathbb{T}$. In this case, we can explicit the orthogonal projection $\Pi_{c_t}$. A function $v$ on $\mathbb{T}$ is the derivative of a function $\phi$ if and only if $\int_\mathbb{T}v(x)\, dx=0$. In order to make explicit dependence on $x$, the derivative of $\phi$ on $\mathbb{T}$ is denoted by $\partial_x\phi$. Let $\mu\in \mathbb{P}_{2,\infty}(\mathbb{T})$ with $\displaystyle\rho=\frac{d\mu}{dx}>0$. Let $\partial_x\phi=\Pi_\mu(v)$; then for any function $f$, $$\int_\mathbb{T}\partial_xf\, v(x)\rho(x)\, dx=\int_\mathbb{T}\partial_xf\, \partial_x\phi\, \rho(x)\, dx.$$ This implies that $\partial_x(v\rho)=\partial_x(\partial_x\phi\ \rho)$, so that for a constant $K$, $$v\rho=\partial_x\phi\, \rho+K\quad\hbox{\rm or}\quad v=\partial_x\phi+\frac{K}{\rho}.$$ Integrating the two hand sides over $\mathbb{T}$ yields $\displaystyle K=\frac{\int_\mathbb{T}v(x)dx}{\int_\mathbb{T}\frac{dx}{\rho}}$. Therefore $$\Pi_\mu(v)=v-\frac{\int_\mathbb{T}v(x)dx}{\int_\mathbb{T}\frac{dx}{\rho}}\cdot \frac{1}{\rho}.$$ For simplifying notations, we put $$\label{eq3.10} \hat \rho=\frac{1}{\Bigl(\int_\mathbb{T}\frac{dx}{\rho}\Bigr)\, \rho}.$$ It is obvious that $\int_\mathbb{T}\hat\rho\, dx=1$. In the sequel, we use notation $\Pi_\rho$ instead of $\Pi_\mu$. Then $$\label{eq3.11} \Pi_\rho(v)=v-\Bigl(\int_\mathbb{T}v(x)dx\Bigr)\ \hat\rho.$$ Let $\phi_t\in C^\infty(\mathbb{T})$ and $(X_t)$ be the flow associated to $$\frac{dX_t}{dt}=\partial_x\phi_t(X_t).$$ Let $c_t=(X_t)_\#\mu$ with $d\mu=\rho\, dx$. Set $\displaystyle\rho_t =\frac{dc_t}{dx}$ the density of $c_t$ with respect to $dx$. Let $g_t\in C^2(\mathbb{T})$ such that $\int_\mathbb{T}g_t(x)\,dx=0$. Then $\{g_t;\ t\in [0,1]\}$ is a solution to [\[eq3.6\]](#eq3.6){reference-type="eqref" reference="eq3.6"} if $$\frac{dg_t}{dt}+\Pi_{\rho_t}\Bigl(\partial_xg_t\, \partial_x\phi_t\Bigr)=0.$$ According to [\[eq3.11\]](#eq3.11){reference-type="eqref" reference="eq3.11"}, we have $$\label{eq3.12} \frac{dg_t}{dt}=-\partial_xg_t\, \partial_x\phi_t+\Bigl(\int_\mathbb{T}\partial_xg_t\,\partial_x\phi_t\,dx\Bigr)\, \hat\rho_t.$$ From above equation, it is easy to see that $\displaystyle\frac{d}{dt}\int_\mathbb{T}g_t(x)\,dx=0$ since $\int_\mathbb{T}\hat\rho_t(x)dx=1$. It follows that $\displaystyle\int_\mathbb{T}g_t(x)\, dx=\int_\mathbb{T}g_0(x)\, dx=0$. In other words, $g_t$ is derivative of a function on $\mathbb{T}$ if the initial condition $g_0$ does. Put $\displaystyle f_t=g_t(X_t)$. Then $$\frac{df_t}{dt}=\Bigl(\int_\mathbb{T}\partial_x g_t\,\partial_x\phi_t\,dx\Bigr)\, \hat\rho_t(X_t).$$ Remark that $$\int_\mathbb{T}\partial_x g_t\,\partial_x\phi_t\,dx=-\int_\mathbb{T}g_t\, \partial_x^2\phi_t\, dx=-\int_\mathbb{T}\frac{g_t\,\partial_x^2\phi_t}{\rho_t}\,\rho_t\, dx =\int_\mathbb{T}g_t(X_t)\, \Bigr(\frac{\partial_x^2\phi_t}{\rho_t}\Bigr)(X_t)\, \rho\,dx.$$ Then $f_t$ satisfies the following equation $$\label{eq3.13} \frac{df_t}{dt}=-\Bigl(\int_\mathbb{T}f_t\ \frac{\partial_x^2\phi_t}{\rho_t}(X_t)\, \rho\,dx\Bigr)\, \hat\rho_t(X_t).$$ Define $\displaystyle\Lambda(t,f) = -\Bigl(\int_\mathbb{T}f \ \frac{\partial_x^2\phi_t}{\rho_t}(X_t)\, \rho\,dx\Bigr)\, \hat\rho_t(X_t)$. Then above equation can be written in the form $$\frac{df_t}{dt}=\Lambda(t, f_t).$$ **Lemma 15**. *We have $$\label{eq3.14} ||\Lambda(t, f)-\Lambda(t, g)||_{L^2(\rho\, dx)}\leq \Bigl(\sup_{t\in [0,1]}||\partial_x^2\phi_t||_\infty\Bigr)\, ||f-g||_{L^2(\rho\, dx)},\quad t\in [0,1].$$* *Proof.* Note that $$\int_\mathbb{T}\Bigl(\frac{\partial_x^2\phi_t}{\rho_t}\Bigr)^2(X_t)\, \rho\,dx=\int_\mathbb{T}\frac{(\partial_x^2\phi_t)^2}{\rho_t}\, dx \leq ||\partial_x^2\phi_t||_\infty^2\, \int_\mathbb{T}\frac{dx}{\rho_t},$$ and $\displaystyle\int_\mathbb{T}\hat\rho_t(X_t)^2\, \rho\,dx=(\int_\mathbb{T}\frac{dx}{\rho_t})^{-1}$; it follows that $$\int_\mathbb{T}\Bigl|\int_\mathbb{T}f\ \frac{\partial_x^2\phi_t}{\rho_t}(X_t)\, \rho\,dx\Bigr|^2\, \hat\rho_t(X_t)^2\, \rho\,dx \leq ||\partial_x^2\phi_t||_\infty^2\, ||f||_{L^2(\rho\,dx)}^2$$ and global Lipschitz condition [\[eq3.14\]](#eq3.14){reference-type="eqref" reference="eq3.14"} holds. ◻ By classical theory of ODE on Banach spaces, for $f_0\in L^2(\rho\,dx)$ given, there is a unique solution $f_t$ to Equation [\[eq3.13\]](#eq3.13){reference-type="eqref" reference="eq3.13"}. Now having this solution $(f_t)_{t\in [0,1]}$ in hand, we set $$g_t=f_t(X_t^{-1}).$$ We have $$\int_\mathbb{T}|g_t|^2\rho_t\,dx=\int_\mathbb{T}|f_t|^2\,\frac{\rho_t}{\rho}(X_t)\, \tilde\rho_t\, dx,$$ where $\displaystyle\tilde\rho_t =\frac{d(X_t^{-1})_\#(\rho\,dx)}{dx}$. It is known that $\displaystyle\frac{\rho_t}{\rho}(X_t)\, \tilde\rho_t=\rho$. Hence $$\label{eq3.15} \int_\mathbb{T}|g_t|^2\rho_t\,dx=\int_\mathbb{T}|f_t|^2\,\rho\, dx.$$ Now by a quite standard computation, we prove that $$\frac{dg_t}{dt}=-\Bigl(\int_\mathbb{T}g_t\partial_x^2\phi_t\, dx\Bigr)\hat\rho_t-\partial_xg_t\, \partial_x\phi_t,$$ $g_t$ is a solution to Equation [\[eq3.12\]](#eq3.12){reference-type="eqref" reference="eq3.12"}, therefore $\int_\mathbb{T}g_t\, dx=0$. **Proposition 16**. *The solution $\{g_t;\ t\in [0,1\}]$ to Equation [\[eq3.12\]](#eq3.12){reference-type="eqref" reference="eq3.12"} preserves norms, that is $$\label{eq3.16} \int_\mathbb{T}|g_t|^2\, \rho_t\, dx=\int_\mathbb{T}|g_0|^2\, \rho\,dx,\quad t\in [0,1].$$* *Proof.* By [\[eq3.15\]](#eq3.15){reference-type="eqref" reference="eq3.15"}, it is sufficient to check $\displaystyle\int_\mathbb{T}f_t\, \frac{df_t}{dt}\, \rho\,dx=0$. But by [\[eq3.13\]](#eq3.13){reference-type="eqref" reference="eq3.13"}, we compute $$\int_\mathbb{T}\hat\rho_t(X_t)\, f_t\, \rho\,dx=\int_\mathbb{T}\hat\rho_t(X_t)g_t(X_t)\,\rho\,dx =\int_\mathbb{T}\hat\rho_t g_t\rho_t\, dx=\frac{\int_\mathbb{T}g_t\, dx}{\int_\mathbb{T}\frac{dx}{\rho_t}},$$ this last term is equal to $0$. ◻ Finally we get the main result of this section. **Theorem 17**. *For any $g_0\in \bar{\mathbf{T}}_{\rho dx}$ given, there is a unique solution $g_t\in \bar{\mathbf{T}}_{\rho_t dx}$ to parallel translation equation [\[eq3.12\]](#eq3.12){reference-type="eqref" reference="eq3.12"} such that $\int_\mathbb{T}|g_t|^2\, \rho_t\, dx=\int_\mathbb{T}|g_0|^2\rho\,dx$ for any $t\in [0,1]$.* # Stochastic parallel translations {#sect4} Let $\{\phi_0, \phi_1, \ldots, \phi_N\}$ be a finite family of smooth function on $M$ and $\{\mu_t;\ t\in [0,1]\}$ a solution to SDE [\[eq2.8\]](#eq2.8){reference-type="eqref" reference="eq2.8"}, which comes from SDE on $M$ $$dX_t=\sum_{i=0}^N\nabla\phi_i(X_t)\circ dB_t^i,\quad\hbox{\rm with }\ B_t^0=t.$$ According to [\[eq3.6\]](#eq3.6){reference-type="eqref" reference="eq3.6"}, if a stochastic process $\{\nabla\Psi_t;\ t\in [0,1]\}$ is parallel along $\{\mu_t;\ t\in [0,1]\}$, Equation [\[eq3.6\]](#eq3.6){reference-type="eqref" reference="eq3.6"} would be replaced by the following formal Stratanovich SDE $$\label{eq4.1} \circ d_t(\nabla\Psi_t)=-\sum_{i=0}^N \Pi_{\mu_t}\Bigl( \nabla_{\nabla\phi_i}\nabla\Psi_t\Bigr)\circ dB_t^i.$$ Using notation [\[eq3.2\]](#eq3.2){reference-type="eqref" reference="eq3.2"}, the above equation becomes $$\label{eq4.2} \circ d_t V_{\Psi_t}=-\sum_{i=0}^N \bigl(\bar{\nabla}_{V_{\phi_i}}V_{\Psi_t}\bigr)\circ dB_t^i.$$ [If [\[eq4.2\]](#eq4.2){reference-type="eqref" reference="eq4.2"} has a smooth solution, it preserves norm. More precisely, we have the following proposition.]{style="color: black"} **Proposition 18**. *Suppose that $\{\nabla\Psi_t;\ t\in [0,1]\}$ is a solution to Equation [\[eq4.2\]](#eq4.2){reference-type="eqref" reference="eq4.2"}, then for any $t\in [0,1]$, $$\label{eq4.3} \int_M |\nabla\Psi_t(x)|^2\mu_t(dx)=\int_M |\nabla\Psi_0(x)|^2\, \mu(dx).$$* *Proof.* We give a heuristic proof of [\[eq4.3\]](#eq4.3){reference-type="eqref" reference="eq4.3"}. Let $\displaystyle\rho_t=\frac{d\mu_t}{d\mu}$ be the density with respect to the initial probability measure $\mu$, then $\{\rho_t;\ t\in [0,1]\}$ satisfies the following SPDE: $$\circ d_t\rho_t =-\sum_{i=0}^N \Bigl(\textup{div}_{\mu_t}(\nabla\phi_i)\rho_t \Bigr)\, \circ dB_t^i.$$ Now $$\circ d_t\int_M |\nabla\Psi_t|^2\, \rho_t\, \mu(dx)=2\int_M \langle\nabla\Psi_t, \circ d_t\nabla\Psi_t\rangle\rho_t\, \mu(dx) +\int_M \langle\nabla\Psi_t, \nabla\Psi_t\rangle\, \circ d_t\rho_t\ \mu(dx).$$ Note that $$\int_M \langle\nabla\Psi_t, \nabla\Psi_t\rangle\, \textup{div}_{\mu_t}(\nabla\phi_i)\rho_t\,\mu(dx) =2\int_M \langle\nabla\Psi_t, \nabla_{\nabla\phi_i}\nabla\Psi_t\rangle\ \mu_t(dx).$$ Combining these equalities yields $$\circ d_t\int_M |\nabla\Psi_t|^2\mu_t(dx) =2\int_M\langle\nabla\Psi_t, \circ d_t\nabla\Psi_t +\sum_{i=0}^N \nabla_{\nabla\phi_i}\nabla\Psi_t\circ dB_t^i\rangle\ \mu_t(dx)=0.$$ We get [\[eq4.3\]](#eq4.3){reference-type="eqref" reference="eq4.3"}. ◻ The weak form of [\[eq4.2\]](#eq4.2){reference-type="eqref" reference="eq4.2"}, the stochastic counterpart of [\[eq3.8\]](#eq3.8){reference-type="eqref" reference="eq3.8"}, would be $$d_t\int_M \langle \nabla f, \nabla\Psi_t\rangle\, \mu_t(dx) =\sum_{i=0}^N \Bigl( \int_M\langle\nabla_{\nabla\phi_i}(\nabla f), \nabla\Psi_t\rangle\, \mu_t(dx)\Bigr)\circ dB_t^i .$$ Since $\displaystyle\nabla_{\nabla\phi_i}(\nabla f)$ is not a vector field of gradient type, the last term in above equality really is $$\sum_{i=0}^N \Bigl( \int_M\langle\Pi_{\mu_t}\bigl(\nabla_{\nabla\phi_i}(\nabla f)\bigr), \nabla\Psi_t\rangle\, \mu_t(dx)\Bigr)\circ dB_t^i .$$ **Proposition 19**. *For any $f\in C^3(M)$, set $$\label{eq4.4} R_t^f=\sum_{i=1}^N \Pi_{\mu_t}\Bigl(\nabla_{\nabla\phi_i}\Pi_{\mu_t} \bigl(\nabla_{\nabla_{\phi_i}}(\nabla f)\bigr) \Bigr),$$* *$$\label{eq4.5} S_t=\sum_{i=1}^N\Pi_{\mu_t}\Bigl((\Delta_{\mu_t}\phi_i)\,\Pi_{\mu_t}^\perp\bigl(\bigl(\nabla_{\nabla_{\phi_i}}(\nabla f)\bigr) \bigr) \Bigr).$$ Then the stochastic counterpart of [\[eq3.8\]](#eq3.8){reference-type="eqref" reference="eq3.8"} has the following form* *$$\begin{split} \int_M \langle\nabla f, \nabla\Psi_t\rangle\, \mu_t(dx)=& \int_M \langle\nabla f,\nabla\Psi_0\rangle\, \mu(dx) + \sum_{i=0}^N \int_0^t \Bigl(\int_M \langle \nabla_{\nabla \phi_i}(\nabla f),\ \nabla\Psi_s\rangle\, \mu_s(dx) \Bigr)\, dB_s^i\\ &+ \frac{1}{2}\, \int_0^t \Bigl(\int_M \langle R_s^f-S_s^f,\ \nabla\Psi_s\rangle \, \mu_s(dx)\Bigr)\,ds, \end{split}$$ or instrinsically* *$$\begin{split} \langle V_f, V_{\Psi_t}\rangle_{\bar{\mathbf{T}}_t} =&\langle V_f, V_{\Psi_0}\rangle_{\bar{\mathbf{T}}_\mu}+ \sum_{i=0}^N \int_0^t \langle \bar{\nabla}_{V_{\phi_i}}V_f,\ V_{\Psi_s}\rangle_{\bar{\mathbf{T}}_{\mu_s}}\, dB_s^i \\ &+ \frac{1}{2}\sum_{i=1}^N \int_0^t \langle\bar{\nabla}_{V_{\phi_i}}\bar{\nabla}_{V_{\phi_i}}V_f, \ V_{\Psi_s}\rangle_{\bar{\mathbf{T}}_{\mu_s}}\ ds -\frac{1}{2}\int_0^t \langle S_s^f, V_{\Psi_s}\rangle_{\bar{\mathbf{T}}_{\mu_s}}\ ds. \end{split}$$* *Proof.* We only remark that the right hand of [\[eq3.9\]](#eq3.9){reference-type="eqref" reference="eq3.9"} is the sum of two terms, the first term involving variations of orthogonal projection along the time $t$, while the second one provides the term $R_t^f$. ◻ Now we are going to see what happens in the case of $\mathbb{P}_{2,\infty}(\textcolor{black}{\mathbb{T}})$. For simplicity, we consider SDE on $\mathbb{T}$, $$\label{eq4.6} dX_t =\partial_x\phi_t(X_t)\circ dB_t,$$ where $B_t$ is a one-dimensional Brownian motion. Let $d\mu=\rho\, dx$ and $\mu_t=(X_t)_\#\mu$; set $\displaystyle\rho_t = \frac{d\mu_t}{dx}$ the density with respect to $dx$. Suppose that $\{\partial_x\Psi_t;\ t\in [0,1]\}$ is a solution to the equation of parallel translations: $$\label{eq4.7} d_t\partial_x\Psi_t=-\Pi_{\rho_t}\bigl( \partial_x^2\Psi_t\ \partial_x\phi_t)\, dB_t + \Big(\frac{1}{2}R_t^{\Psi_t}+\frac{1}{2}S_t^{\Psi_t}\Bigr)\, dt.$$ We first explicit $R_t$ and $S_t$ in this special case. Using the expression of $\Pi_\rho$ (see [\[eq3.11\]](#eq3.11){reference-type="eqref" reference="eq3.11"}), we have $$\begin{split} R_t^{\Psi_t}=&\partial_x\bigl(\partial_x^2\Psi_t\, \partial_x\phi_t\bigr) \partial_x\phi_t -\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\,\partial_x\phi_t\, dx\Bigr)\,\partial_x\hat\rho_t\, \partial_x\phi_t\\ &-\Bigl(\int_\mathbb{T}\partial_x\bigl(\partial_x^2\Psi_t\,\partial_x\phi_t \bigr)\, \partial_x\phi_t\, dx\Bigr)\textcolor{black}{\hat{\rho}_t} +\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\,dx \Bigr)\Bigl(\int_\mathbb{T}\partial_x\hat\rho_t\, \partial_x\phi_t\ dx \Bigr)\textcolor{black}{\hat{\rho}_t}\\ &=I_1+I_2+I_3+I_4 \end{split}$$ respectively, and $$\begin{split} S_t^{\Psi_t}=& \bigl(\partial_x^2\phi_t+\partial_x\log(\rho_t)\,\partial_x\phi_t\bigr) \Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t \ dx\Bigr)\, \hat\rho_t\\ &\textcolor{black}{-} \Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\, dx \Bigr) \Bigl(\int_\mathbb{T}\partial_x^2\phi_t\, \hat\rho_t\, dx \Bigr)\textcolor{black}{\hat{\rho}_t}\\ &\textcolor{black}{-\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\,dx \Bigr)} \Bigl(\int_\mathbb{T}\partial_x\log(\rho_t)\, \partial_x\phi_t\, \hat\rho_t\, dx\Bigr)\textcolor{black}{\hat{\rho}_t}\\ &=J_1+J_2+J_3+J_4 \end{split}$$ respectively. We have $$I_2+J_2 =-\textcolor{black}{2\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\, dx \Bigr)\,\partial_x\phi_t \partial_x\hat\rho_t},$$ and $$I_4+J_4 =\textcolor{black}{2\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\, dx \Bigr)\, \Bigl(\int_\mathbb{T}\partial_x\phi_t\, \partial_x\hat\rho_t dx\Bigr)\hat{\rho}_t}.$$ Therefore we get the following expression for $R_t^{\Psi_t}\textcolor{black}{+}S_t^{\Psi_t}$: $$\label{eq4.8} \begin{split} &R_t^{\Psi_t}\textcolor{black}{+}S_t^{\Psi_t}=\partial_x \bigl(\partial_x^2\Psi_t\, \partial_x\phi_t\bigr)\, \partial_x\phi_t -\Bigl(\int_\mathbb{T}\partial_x\bigl(\partial_x^2\Psi_t\,\partial_x\phi_t \bigr)\, \partial_x\phi_t\, dx\Bigr)\textcolor{black}{\hat{\rho}_t}\\ & \textcolor{black}{+}\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t \ dx\Bigr)\, \partial_x^2\phi_t\,\hat\rho_t \textcolor{red}{-}\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t \ dx\Bigr)\Bigl(\int_\mathbb{T}\partial_x^2\phi_t\, \hat\rho_t\, dx \Bigr)\textcolor{black}{\hat{\rho}_t}\\ &\textcolor{black}{-2\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\, dx \Bigr)\,\partial_x\phi_t \partial_x\hat\rho_t+2\Bigl(\int_\mathbb{T}\partial_x^2\Psi_t\, \partial_x\phi_t\, dx \Bigr)\, \Bigl(\int_\mathbb{T}\partial_x\phi_t\, \partial_x\hat\rho_t dx\Bigr)\hat{\rho}_t}. \end{split}$$ Let $\displaystyle f_t=\partial\Psi_t(X_t)$. Then by Kunita-Itô-Wentzell formula, we get $$\begin{split} d_tf_t=&-\Bigl(\int_\mathbb{T}\partial_x\Psi_t\, \partial_x^2\phi_t\, dx\Bigr)\, \hat\rho_t(X_t)\, dB_t -\frac{1}{2} \Bigl(\int_\mathbb{T}\partial_x\Psi_t\, \partial_x^2\phi_t\, dx\Bigr)\, (\partial_x^2\phi_t)(X_t) \hat\rho_t(X_t)\, dt\\ &-\frac{1}{2}\Bigl(\int_\mathbb{T}\partial_x\Psi_t\, \partial_x\bigl( \partial_x^2\phi_t\, \partial_x\phi_t\bigr)\,dx\Bigr)\hat\rho_t(X_t)\,dt +\frac{3}{2}\Bigl(\int_\mathbb{T}\partial_x\Psi_t\,\partial_x^2\phi_t\, dx \Bigr)\Bigl( \int_\mathbb{T}\partial_x^2\phi_t\, \hat\rho_t\, dx\Bigr)\hat\rho_t(X_t)dt. \end{split}$$ As in Section [4](#sect3){reference-type="ref" reference="sect3"}, we remark that $$\int_\mathbb{T}\partial_x\Psi_t\, \partial_x^2\phi_t\, dx=\int_\mathbb{T}f_t\times \frac{\partial_x^2\phi_t}{\rho_t}(X_t)\ \rho\,dx,$$ and aslo $$\int_\mathbb{T}\partial_x\Psi_t\, \partial_x\bigl(\partial_x^2\phi_t\,\partial_x\phi_t\bigr)\, dx =\int_\mathbb{T}f_t\times \frac{\partial_x (\partial_x^2\phi_t\, \partial_x\phi_t)}{\rho_t}(X_t)\ \rho\,dx.$$ We introduce two notations $$\label{eq4.9} a_t=\frac{\partial_x^2\phi_t}{\rho_t}(X_t),\quad b_t=\frac{\partial_x (\partial_x^2\phi_t\, \partial_x\phi_t)}{\rho_t}(X_t).$$ Then $\{f_t; t\in [0,1]\}$ satisfies the following equation $$\label{eq4.10} \begin{split} d_tf_t=&-\Bigl(\int_\mathbb{T}f_t a_t\, \rho dx\Bigr)\hat\rho_t(X_t)\, dB_t -\frac{1}{2}\Bigl(\int_\mathbb{T}f_t a_t\, \rho dx\Bigr)\bigl(\hat\rho_t\,\partial_x^2\phi_t\bigr)(X_t)\, dt\\ &-\frac{1}{2}\Bigl(\int_\mathbb{T}f_tb_t\, \rho dx\Bigr)\hat\rho_t(X_t)\, dt +\frac{3}{2}\Bigl(\int_\mathbb{T}f_ta_t \rho dx\Bigr)\Bigl(\int_\mathbb{T}\partial_x^2\phi_t\hat\rho_t\, dx\Bigr)\, \hat\rho_t(X_t)\, dt. \end{split}$$ Let $\displaystyle\Lambda(t,f)=-\Bigl(\int_\mathbb{T}f a_t\, \rho\, dx\Bigr)\, \hat\rho_t(X_t)$ and $$\begin{split} \Theta(t,f)=&-\frac{1}{2}\Bigl(\int_\mathbb{T}f_t a_t\, \rho dx\Bigr)\bigl(\hat\rho_t\,\partial_x^2\phi_t\bigr)(X_t) -\frac{1}{2}\Bigl(\int_\mathbb{T}f_tb_t\, \rho dx\Bigr)\hat\rho_t(X_t)\\ &+\frac{3}{2}\Bigl(\int_\mathbb{T}f_ta_t \rho dx\Bigr)\Bigl(\int_\mathbb{T}\partial_x^2\phi_t\hat\rho_t\, dx\Bigr)\, \hat\rho_t(X_t). \end{split}$$ We put $d_tf_t$ in the form $$d_tf_t=\Lambda(t, f_t)\, dB_t+\Theta(t, f_t)\, dt.$$ **Lemma 20**. *We have, for any $t\in [0,1]$ and $f\in L^2(\rho\, dx)$, $$\label{eq4.10-1} ||\Lambda(t, f)||_{L^2(\rho dx)} \leq \bigl(\sup_{t\in [0,1]}||\partial_x^2\phi_t||_\infty\bigr)\, ||f||_{L^2(\rho dx)},$$ $$\label{eq4.10-2} ||\Theta(t, f)||_{L^2(\rho dx)} \leq \Bigl(2\, \sup_{t\in [0,1]}||\partial_x^2\phi_t||_\infty^2\ + \sup_{t\in [0,1]}|| \partial_x (\partial_x^2\phi_t\, \partial_x\phi_t)||_\infty \Bigr)\, ||f||_{L^2(\rho dx)}.$$* *Proof.* We proceed in the same way as in the proof of Lemma [Lemma 15](#lemma3.4){reference-type="ref" reference="lemma3.4"}. ◻ By standard Picard iteration or by SDE on Hilbert spaces, finally we get the following result. **Theorem 21**. *There is a unique solution $\{f_t; t\in [0,1]\}$ to Equation [\[eq4.10\]](#eq4.10){reference-type="eqref" reference="eq4.10"}.* Define $\displaystyle g_t=f_t(X_t^{-1})$. Contrary to ODE, we have no SDE directly expressing $X_t^{-1}$. The following result will be used several times in the sequel. **Lemma 22**. *Let $d\mu=\rho\, dx$ be a probability measure on a compact Riemannian manifold $M$ such that $\rho>0$ and $\Phi: M\rightarrow M$ a diffeomorphism. Set $$\rho_\Phi =\frac{d\Phi_\#(\rho dx)}{dx}, \quad \tilde K=\frac{d (\Phi^{-1})_\#(dx)}{dx},$$ then $\quad\displaystyle\rho_\Phi(\Phi)\tilde K=\rho$.* *Proof.* Let $f\in C(M)$, we have $$\int_M f \rho dx =\int_M f(\Phi^{-1}(\Phi))\, \rho dx=\int_M f(\Phi^{-1})\rho_\Phi\, dx = \int_M f\ \rho_\Phi(\Phi)\, \tilde K\, dx,$$ the result follows. ◻ **Proposition 23**. *Suppose that $\displaystyle\int_\mathbb{T}g_0(x) dx=0$, then for any $t\in [0,1]$, $\displaystyle\int_\mathbb{T}g_t(x)\, dx=0$.* *Proof.* Let $\displaystyle\tilde K_t=\frac{d(X_t^{-1})_\#(dx)}{dx}$; then by Kunita [@Kunita] (see also [@FangLuoTh]), we have the following explicit formula: $$\tilde K_t=\exp\Bigl(\int_0^t (\partial_x^2\phi_s)(X_s)\circ dB_s\Bigr).$$ Using $\tilde K_t$, $\displaystyle\int_\mathbb{T}g_t(x)\, dx=\int_\mathbb{T}f_t\, \tilde K_t\, dx$. Remark that all of drift terms in $f_t$ came from Itô's stochastic contraction; therefore in Stratanovich form $$\label{eq4.11} \circ d_tf_t = -\Bigl(\int_\mathbb{T}f_t a_t\, \rho dx\Bigr)\hat\rho_t(X_t)\circ dB_t.$$ Now by Itô['s ]{style="color: black"} formula, $$\label{eq4.12} \circ d_t(f_t\tilde K_t)=- \Bigl(\int_\mathbb{T}f_t a_t\, \rho dx\Bigr)\hat\rho_t(X_t) \tilde K_t\circ dB_t +f_t\, \partial_x^2\phi_t(X_t)\, \tilde K_t\circ dB_t.$$ Note that $\displaystyle\int_\mathbb{T}\hat\rho_t(X_t)\tilde K_t\, dx=\int_\mathbb{T}\hat\rho_t(x)\, dx=1$. On the other hand, $$\int_\mathbb{T}f_t\, \partial_x^2\phi_t(X_t)\, \tilde K_t\, dx =\int_\mathbb{T}f_t\times \frac{\partial_x^2\phi_t}{\rho_t}(X_t)\, \rho_t(X_t)\,\tilde K_t\, dx =\int_\mathbb{T}f_t a_t\, \rho\, dx,$$ the last equality being due to $\displaystyle\rho_t(X_t)\,\tilde K_t=\rho$ by Lemma [Lemma 22](#LEMMA){reference-type="ref" reference="LEMMA"}. Now we get $\circ d_t\int_\mathbb{T}f_t\tilde K_t dx=0$ using [\[eq4.12\]](#eq4.12){reference-type="eqref" reference="eq4.12"}. Therefore $\displaystyle\int_\mathbb{T}g_t dx=\int_\mathbb{T}g_0 dx=0$. ◻ **Theorem 24**. *We have, for any $t\in [0,1]$, $$\label{eq4.13} \int_\mathbb{T}|g_t(x)|^2\rho_t(x)\, dx=\int_\mathbb{T}|g_0(x)|^2\, \rho dx.$$* *Proof.* By [\[eq4.11\]](#eq4.11){reference-type="eqref" reference="eq4.11"}, $$\begin{split} \circ d_t \int_\mathbb{T}f_t^2\, \rho dx=&- 2\Bigl[ \Bigl( \int_\mathbb{T}f_ta_t\rho dx\Bigr) \hat\rho_t(X_t)\, f_t\, \rho dx\Bigr]\circ dB_t\\ &=- 2\Bigl[ \Bigl( \int_\mathbb{T}f_ta_t\rho dx\Bigr) \Bigl( \int_\mathbb{T}\hat\rho_t(X_t) f_t(x)\, dx\Bigr)\Bigr]\, \circ dB_t, \end{split}$$ But we have seen that $$\int_\mathbb{T}\hat\rho_t(X_t) f_t(x)\, dx=\Bigl(\int_\mathbb{T}g_t(x)dx\Bigr)\Bigl(\int_\mathbb{T}\frac{dx}{\rho_t}\Bigr)^{-1}$$ which is equal to $0$ by Proposition [Proposition 23](#prop4.4){reference-type="ref" reference="prop4.4"}. ◻ Combining all above results, finally we get **Theorem 25**. *Let $\partial_x\Psi_t=g_t$. Then for $\mu=\rho\, dx$ and $\mu_t=(X_t)_\#(\rho dx)$, $\{\partial_x\Psi_t;\ t\in [0,1]\}$ is the parallel translation along the stochastic regular curve $\{\mu_t;\ t\in [0,1]\}$, that is, $\displaystyle\partial_x\Psi_t\in \bar{\mathbf{T}}_{\mu_t}$ and $$\int_{\mathbb{T}} |\partial_x\Psi_t|^2\, \mu_t(dx)=\int_{\mathbb{T}}|\partial_x\Psi_0|^2\, \rho dx,\quad t\in [0,1].$$* Actually, it is well-known that some quasi-invariant non-degenerated diffusion processes have been constructed in $\mathbb{P}_2(\mathbb{T})$, see for example [@RS1; @Wang]. It seems that these diffusion processes do not charge the subspace $\mathbb{P}_{2,\infty}(\mathbb{T})$. In what follows, we construct a non-degenerated diffusion process $\{\mu_t; t\in [0,1]\}$ on $\mathbb{P}_{2,\infty}(\mathbb{T})$ and parallel translations along it. For $k\in \mathbb{N}^*$, set $$\phi_{2k-1}(x)=\frac{\sin(kx)}{k},\quad \phi_{2k}(x)=-\frac{\cos(kx)}{k},\quad\hbox{\rm and }\quad a_k=k^q.$$ For an integer $N\geq 2$, we consider the SDE on $\mathbb{T}$, $$dX_t^N=\sum_{k=1}^N \frac{1}{\alpha_k} \Bigl( \partial_x\phi_{2k-1}(X_t^N)\circ dB_{2k-1}(t) + \partial_x\phi_{2k}(X_t^N)\circ dB_{2k}(t)\Bigr).$$ Note that $\displaystyle d_t \partial_x\phi_{2k-1}(X_t^N)\cdot dB_{2k-1}+ d_t \partial_x\phi_{2k}(X_t^N)\cdot dB_{2k}=0$; therefore the above Stratanovich SDE becomes the below Itô SDE: $$\label{eq4.14} dX_t^N=\sum_{k=1}^N \frac{1}{\alpha_k} \Bigl( \partial_x\phi_{2k-1}(X_t^N)\, dB_{2k-1}(t) + \partial_x\phi_{2k}(X_t^N)\, dB_{2k}(t)\Bigr).$$ It is well-known (see [@Elworthy; @IW; @Kunita; @Malliavin], especially in [@AR; @Fang]) that for $q>2$, almost surely, as $N\rightarrow+\infty$, $\displaystyle X_t^N$ converges in $\displaystyle C\bigl([0,1], \hbox{\rm Diff}(\mathbb{T})\bigr)$ to $X_t$, which solves the following SDE: $$\label{eq4.15} dX_t=\sum_{k=1}^\infty \frac{1}{\alpha_k} \Bigl( \partial_x\phi_{2k-1}(X_t)\, dB_{2k-1}(t) + \partial_x\phi_{2k}(X_t)\, dB_{2k}(t)\Bigr).$$ Let $d\mu=\rho\, dx\in \mathbb{P}_{2,\infty}(\mathbb{T})$ be given; for any $N>2$, we denote by $\rho_t^N$ the density of the measure $\displaystyle(X_t^N)_\#(\mu)$ with respect to $dx$. It is obvious that almost surely, $\rho_t^N$ converges to the density $\rho_t$ of $\displaystyle(X_t)_\#(\mu)$ uniformly in $(t,x)\in [0,1]\times\mathbb{T}$. It is quite automatic that results in Theorem [Theorem 25](#th4.6){reference-type="ref" reference="th4.6"} remain valid for SDE [\[eq4.14\]](#eq4.14){reference-type="eqref" reference="eq4.14"}. More precisely, for any $\partial_x\Psi_0$ given in $\displaystyle L^2(\rho\,dx)$, there exists the parallel translation $\{\partial_x\Psi_t^N;\ t\in [0,1]\}$ along $\{\mu_t^N;\ t\in [0,1]\}$, that is, $\partial_x\Psi_t\in \bar{\mathbf{T}}_{\mu_t^N}$ and $$\int_\mathbb{T}|\partial_x\Psi_t^N|^2\, \rho_t^N\,dx=\int_\mathbb{T}|\partial_x\Psi_0|^2\, \rho\,dx.$$ For simplicity, we again use the notation $\displaystyle g_t^N=\partial_x\Psi_t^N$ and $\displaystyle f_t^N=g_t^N(X_t^N)$. We introduce $$a_k^N(t)=\Bigl( \frac{\partial_x^2\phi_k}{\rho_t^N}\Bigr)(X_t^N), \quad b_k^N(t)=\Bigl( \frac{\partial_x\bigl(\partial_x^2\phi_k\,\partial_x\phi_k\bigr)}{\rho_t^N}\Bigr)(X_t^N),$$ and $$\Lambda_k^N(t,f)=-\Bigl(\int_\mathbb{T}f\, a_k^N(t)\, \rho\,dx\Bigr)\, \hat\rho_t^N(X_t^N),$$ $$\begin{split} \Theta_k^N(t,f)=-&\Bigl(\int_\mathbb{T}f\, a_k^N(t)\, \rho\,dx\Bigr)\, \bigl(\hat\rho_t^N\,\partial_x^2\phi_k\bigr)(X_t^N) -\Bigl(\int_\mathbb{T}f\, b_k^N(t)\, \rho\,dx\Bigr)\, \hat\rho_t^N(X_t^N)\\ +& 3\ \Bigl(\int_\mathbb{T}f\, a_k^N(t)\, \rho\,dx\Bigr)\, \Bigl(\int_\mathbb{T}\partial_x^2\phi_k\, \hat\rho_t^N\, dx\Bigr)\hat\rho_t^N(X_t^N). \end{split}$$ Then by [\[eq4.10\]](#eq4.10){reference-type="eqref" reference="eq4.10"}, $f_t^N$ satisfies the following SDE $$\label{eq4.16} d_tf_t^N=\sum_{k=1}^{2N} \frac{1}{\alpha_k}\, \Lambda_k^N(t, f_t^N)\, dB_t^k +\sum_{k=1}^{2N} \frac{1}{2\alpha_k^2}\,\Theta_k^N(t,f_t^N)\, dt,$$ with $\displaystyle\alpha_k = [\frac{k+1}{2}]^q$. In the sequel, we will use the notation: $\displaystyle\xi(s)=\sum_{k=1}^{+\infty} \frac{1}{k^s}$ for $s>1$. **Theorem 26**. *Let $\displaystyle q>\frac{5}{2}$; then almost surely, as $N\rightarrow+\infty$, $f_\cdot^N$ converges in $\displaystyle C\bigl([0,1], L^2(\mathbb{T}, \rho\,dx)\bigr)$. More precisely, there exists $f\in C\bigl([0,1], L^2(\mathbb{T}, \rho\,dx)\bigr)$ such that $$\label{eq4.17} \lim_{N\rightarrow+\infty} \sup_{t\in [0,1]}\int_\mathbb{T}|f_t^N-f_t|^2\rho\, dx =0.$$* *Proof.* We first remark that $$||\partial_x^2\phi_k||_\infty \leq k,\quad ||\partial_x(\partial_x^2\phi_k\, \partial_x\phi_k)||_\infty\leq k^2.$$ Then, by Lemma [Lemma 20](#lemma4.4){reference-type="ref" reference="lemma4.4"}, for any $N\geq 2$, $$\label{eq4.18} ||\Lambda_k^N(t,f)||_{L^2(\rho dx)}\leq k\, ||f||_{L^2(\rho dx)},\quad ||\Theta_k^N(t,f)||_{L^2(\rho dx)}\leq 3k^2\, ||f||_{L^2(\rho dx)}.$$ Using [\[eq4.18\]](#eq4.18){reference-type="eqref" reference="eq4.18"}, we have, $$\sum_{k=1}^{+\infty} \Bigl\| \frac{1}{\alpha_k}\, \Lambda_k^N(t,f)\Bigr\|^2_{L^2(\rho dx)} \leq \xi(2q-2)\, ||f||_{L^2(\rho dx)},$$ as well as $$\sum_{k=1}^{+\infty} \Bigl\| \frac{1}{\alpha_k^2}\Theta_k^N(t,f)\Bigr\|_{L^2(\rho dx)} \leq 3\xi(2q-2)\, ||f||_{L^2(\rho dx)}.$$ Combining these estimates, together with SDE [\[eq4.16\]](#eq4.16){reference-type="eqref" reference="eq4.16"}, we get that, for any $p\geq 1$, $$\label{eq4.21} D_p=\sup_N\sup_{t\in [0,1]} \mathbb{E}\Bigl(||f_t^N||_{L^2(\rho dx)}^p\Bigr) <+\infty.$$ Now let $N'>N\geq 2$, we have $$\begin{split} d_t\bigl(f_t^{N'}-f_t^N\bigr)=& \sum_{k=1}^{2N} \frac{1}{\alpha_k}\Bigl( \Lambda_k^{N'}(t,f_t^{N'})- \Lambda_k^{N}(t,f_t^{N})\Bigr)\, dB_t^k \\&+ \sum_{k=1}^{2N} \frac{1}{2\alpha_k^2} \Bigl( \Theta_k^{N'}(t,f_t^{N'})- \Theta_k^{N}(t,f_t^{N})\Bigr)\ dt\\ &\hskip -10mm + \sum_{k=2N+1}^{2N'} \frac{1}{\alpha_k} \Lambda_k^{N'}(t,f_t^{N'})\, dB_t^k + \sum_{k=2N+1}^{2N'} \frac{1}{2\alpha_k^2} \Theta_k^{N'}(t,f_t^{N'})\, dt\\ &=dI_1(t)+dI_2(t)+dI_3(t)+dI_4(t)\quad\hbox{\rm respectively}. \end{split}$$ By [the]{style="color: black"} Bürkhölder inequality, $$\mathbb{E}\Bigl(\sup_{s\in [0,t]}||I_3(s)||_{L^2(\rho dx)}^2\Bigr) \leq 4\ \sum_{k>2N} \mathbb{E}\Bigl( \int_0^t \frac{1}{\alpha_k^2}||\Lambda_k^{N'}(s, f_s^{N'})||_{L^2(\rho dx)}^2\, ds\Bigr),$$ which is dominated, using [\[eq4.18\]](#eq4.18){reference-type="eqref" reference="eq4.18"}, $$4 \sum_{k>2N} \frac{k^2}{k^{2q}}\, \mathbb{E}\Bigl(\int_0^t ||f_s^{N'}||_{L^2(\rho dx)}^2\, ds\Bigr) \leq \frac{4}{(2N)^{2q-3}}\, \mathbb{E}\Bigl(\int_0^1 ||f_s^{N'}||_{L^2(\rho dx)}^2\Bigr).$$ According to [\[eq4.21\]](#eq4.21){reference-type="eqref" reference="eq4.21"}, we get $$\label{eq4.22} \mathbb{E}\Bigl(\sup_{s\in [0,t]}||I_3(s)||_{L^2(\rho dx)}^2\Bigr) \leq \frac{4D_1}{(2N)^{2q-3}}.$$ Secondly, according to second estimate in [\[eq4.18\]](#eq4.18){reference-type="eqref" reference="eq4.18"}, we have $$\begin{split} &||I_4(t)||_{L^2(\rho dx)}\leq \sum_{k=2N+1}^{2N'} \frac{1}{2\alpha_k^2}\, \int_0^t ||\Theta_k^{N'}(s, f_s^{N'})||_{L^2(\rho dx)}\,ds\\ & \leq \sum_{k=2N+1}^{2N'} \frac{3k^2}{2 k^{2q}}\,\int_0^t ||f_s^{N'}||_{L^2(\rho dx)}\, ds \leq \frac{6}{(2N)^{2q-3}}\,\int_0^t ||f_s^{N'}||_{L^2(\rho dx)}\, ds. \end{split}$$ It follows that, for $N$ big enough, $$\label{eq4.23} \mathbb{E}\Bigl( \sup_{s\in [0,t]} ||I_4(s)||_{L^2(\rho dx)}^2\Bigr) \leq \Big( \frac{6}{(2N)^{2q-3}}\Bigr)^2\, D_1\leq \frac{4D_1}{(2N)^{2q-3}}.$$ Estimating $I_1(t)$ and $I_2(t)$ will make appear Gronwall type inequality for $$\mathbb{E}\Bigl( \sup_{s\in [0,t]}||f_s^{N'}-f_s^N||_{L^2(\rho dx)}^2\Bigr),$$ which will yield desired result. Again by Bürkhölder inequality, we have $$\label{eq4.24} \mathbb{E}\Bigl( \sup_{s\in [0,t]}||I_1(s)||_{L^2(\rho dx)}^2\Bigr)\leq \sum_{k=1}^{2N} \frac{4}{\alpha_k^2} \mathbb{E}\Bigl(\int_0^t ||\Lambda_k^{N'}(s, f_s^{N'})-\Lambda_k^{N}(s, f_s^{N})||_{L^2(\rho dx)}^2\, ds\Bigr).$$ By expression of $\Lambda_k$, we write down $$\label{eq4.24.1} \begin{split} \Lambda_k^{N'}(s, f_s^{N'})-\Lambda_k^{N}(s, f_s^{N}) =& \Bigl[ -\int_\mathbb{T}f_s^{N'}a_k^{N'}(s)\, \rho dx+ \int_\mathbb{T}f_s^{N}a_k^{N}(s)\, \rho dx\Bigr]\, \hat\rho_s^{N'}(X_s^{N'})\\ &+\Bigl( \int_\mathbb{T}f_s^{N}a_k^{N}(s)\, \rho dx\Bigr)\, \Bigl( \hat\rho_s^{N}(X_s^{N}) - \hat\rho_s^{N'}(X_s^{N'})\Bigr)\\ &=J_1(s)+J_2(s)\quad\hbox{respectively.} \end{split}$$ As for getting estimates in [\[eq4.18\]](#eq4.18){reference-type="eqref" reference="eq4.18"}, we have $$\label{eq4.25} \Bigl\|\Bigl( \int_\mathbb{T}(f_s^N-f_s^{N'})a_k^{N'}(s)\, \rho dx\Bigr)\, \hat\rho_s^{N'}(X_s^{N'})\Bigr\|_{L^2(\rho dx)} \leq k\, ||f_s^N-f_s^{N'}||_{L^2(\rho dx)}.$$ Hence $\displaystyle||\Lambda_k^{N'}(s, f_s^{N'})-\Lambda_k^{N}(s, f_s^{N})||_{L^2(\rho dx)}^2$ is dominated by the sum of two terms in the following way $$2k^2\, ||f_s^N-f_s^{N'}||_{L^2(\rho dx)}^2 + o_k(s, N, N'),$$ so that the right hand side of [\[eq4.24\]](#eq4.24){reference-type="eqref" reference="eq4.24"} has the following upper bound $$\sum_{k=1}^{2N} \frac{8k^2}{k^{2q}}\, \int_0^t \mathbb{E}\Bigl( ||f_s^N-f_s^{N'}||^2_{L^2(\rho dx)}\Bigr)\, ds +\sum_{k=1}^{2N} \frac{4}{k^{2q}}\, \int_0^t \mathbb{E}(o_k(s,N,N'))\, ds.$$ Therefore the right hand side of [\[eq4.24\]](#eq4.24){reference-type="eqref" reference="eq4.24"} is dominated by $$8\xi(2q-2)\, \int_0^t \mathbb{E}\Bigl( ||f_s^N-f_s^{N'}||^2_{L^2(\rho dx)}\Bigr)\, ds +o(N).$$ We have in fact the following inequality $$\label{eq4.26} \mathbb{E}\Bigl( \sup_{s\in [0,t]}||I_1(s)||_{L^2(\rho dx)}^2\Bigr)\leq 8\xi(2q-2)\, \int_0^t \mathbb{E}\Bigl( ||f_s^N-f_s^{N'}||^2_{L^2(\rho dx)}\Bigr)\, ds +\frac{C}{N^{q-\frac{1}{2}}}.$$ Since the detail of the proof of [\[eq4.26\]](#eq4.26){reference-type="eqref" reference="eq4.26"} is lengthy, we will do it in the following proposition. In the same way, there is a constant [$C_2>0$]{style="color: black"} such that $$\label{eq4.27} \mathbb{E}\Bigl( \sup_{s\in [0,t]}||I_2(s)||_{L^2(\rho dx)}^2\Bigr) \leq \textcolor{black}{C_2\xi(2q-2)}\, \int_0^t \mathbb{E}\Bigl( ||f_s^N-f_s^{N'}||^2_{L^2(\rho dx)}\Bigr)\, ds + \frac{C}{N^{q-\frac{1}{2}}}.$$ Also the proof of [\[eq4.27\]](#eq4.27){reference-type="eqref" reference="eq4.27"} is postponed in the following proposition. For $\displaystyle q>\frac{5}{2}$, $2q-3>q-\frac{1}{2}$, [combining]{style="color: black"} [\[eq4.22\]](#eq4.22){reference-type="eqref" reference="eq4.22"}, [\[eq4.23\]](#eq4.23){reference-type="eqref" reference="eq4.23"}, [\[eq4.26\]](#eq4.26){reference-type="eqref" reference="eq4.26"} and [\[eq4.27\]](#eq4.27){reference-type="eqref" reference="eq4.27"}, we finally get $$\mathbb{E}\Bigl( \sup_{s\in [0,t]}||f_s^N-f_s^{N'}||_{L^2(\rho dx)}^2\Bigr) \leq C_1\, \int_0^t \mathbb{E}\Bigl( ||f_s^N-f_s^{N'}||_{L^2(\rho dx)}^2\Bigr)\, ds + C_2 N^{-(q-\frac{1}{2})}.$$ By Gronwall['s]{style="color: black"} lemma, there is a constant $C>0$ such that, for any $N'\geq N$, $$\label{eq4.28} \mathbb{E}\Bigl( \sup_{s\in [0,t]}||f_s^N-f_s^{N'}||_{L^2(\rho dx)}^2\Bigr) \leq C\, N^{-(q-\frac{1}{2})} \textcolor{black}{e^{C_1t}}.$$ It follows that $\{f^N; N\geq 2\}$ is a Cauchy sequence in $\displaystyle L^2\Bigl(\Omega, C\bigl([0,1], L^2(\rho dx)\bigr)\Bigr)$; then there exists $f\in L^2\Bigl(\Omega, C\bigl([0,1], L^2(\rho dx)\bigr)\Bigr)$ such that $$\label{eq4.28} \mathbb{E}\Bigl( \sup_{s\in [0,1]}||f_s^N-f_s||_{L^2(\rho dx)}^2\Bigr) \leq C\, N^{-(q-\frac{1}{2})}.$$ This estimate is useful in order to obtain the almost surely convergence. Let $\beta \in (0, 1/2)$, we put $$\Omega_N=\Bigl\{\omega;\ \sup_{t\in [0,1]}||f_t^N-f_t||_{L^2(\rho dx)}\geq N^{-\beta}\Bigr\}.$$ By [\[eq4.28\]](#eq4.28){reference-type="eqref" reference="eq4.28"}, $\displaystyle{\mathbf P}(\Omega_N)\leq C\, N^{-(q-(1/2)-2\beta)}$. For $\displaystyle q>\frac{5}{2}$ and $\beta\in (0,1/2)$, the series $\displaystyle\sum_{N>1} N^{-(q-(1/2)-2\beta)}$ converges; therefore the Borel-Cantelli lemma yields the almost sure convergence with a convergence rate $N^{-\beta}$. ◻ **Proposition 27**. *Above inequality [\[eq4.26\]](#eq4.26){reference-type="eqref" reference="eq4.26"} as well as inequality [\[eq4.27\]](#eq4.27){reference-type="eqref" reference="eq4.27"} hold true.* *Proof.* Recall the expression of $J_1(s)$ in [\[eq4.24.1\]](#eq4.24.1){reference-type="eqref" reference="eq4.24.1"}: $$J_1(s)= \Bigl[ -\int_\mathbb{T}f_s^{N'}a_k^{N'}(s)\, \rho dx+ \int_\mathbb{T}f_s^{N}a_k^{N}(s)\, \rho dx\Bigr]\, \hat\rho_s^{N'}(X_s^{N'}).$$ Put $$J_{11}(s)= \Bigl[ \int_\mathbb{T}\bigl(f_s^N-f_s^{N'}\bigr)a_k^{N'}(s)\, \rho dx\Bigr]\,\hat\rho_s^{N'}(X_s^{N'}),$$ and $$J_{12}(s)= \Bigl[ \int_\mathbb{T}f_s^N\bigl( a_k^N(s)-a_k^{N'}(s)\bigr)\, \rho dx\Bigr]\,\hat\rho_s^{N'}(X_s^{N'}).$$ Then $\displaystyle J_1(s)=J_{11}(s)+J_{12}(s)$. Obviously $$\label{eq4.29.1} ||J_{11}(s)||_{L^2(\rho dx)}\leq k\, ||f_s^N-f_s^{N'}||_{L^2(\rho dx)}.$$ Remark that for any [strictly]{style="color: black"} positive probability density $\sigma$, $\displaystyle\int_\mathbb{T}\frac{dx}{\sigma}\geq 1$, since $$1=\int_\mathbb{T}\sqrt{\sigma}\cdot \frac{1}{\sqrt{\sigma}}\,dx \leq \Bigl( \int_\mathbb{T}\frac{dx}{\sigma}\Bigr)^{1/2}.$$ Therefore $$\int_\mathbb{T}\bigl(\hat\rho_s^N\bigr)^2(X_s^{N'})\, \rho dx=1/\int_\mathbb{T}\frac{dx}{\rho_s^{N'}}\leq 1.$$ So by expression of $J_{12}(s)$ and Cauchy-Schwarz inequality, we get $$\label{eq4.29.2} \int_\mathbb{T}|J_{12}(s)|^2\, \rho dx \leq ||f_s^N||_{L^2(\rho dx)}^2\, \int_\mathbb{T}\Bigl( a_k^N(s)-a_k^{N'}(s)\Bigr)^2\, \rho dx.$$ Now $$a_k^N(s)-a_k^{N'}(s) =\frac{\partial_x^2\phi_k(X_s^N)-\partial_x^2\phi_k(X_s^{N'})}{\rho_s^{N'}(X_s^{N'})} + \partial_x^2\phi_k(X_s^N)\Bigl(\frac{1}{\rho_s^{N}(X_s^{N})} -\frac{1}{\rho_s^{N'}(X_s^{N'})}\Bigr).$$ Remark that $$\frac{|\partial_x^2\phi_k(X_s^N)-\partial_x^2\phi_k(X_s^{N'})|}{\rho_s^{N'}(X_s^{N'})} \leq k^2\, \frac{ |X_s^N-X_s^{N'}|}{\rho_s^{N'}(X_s^{N'})}.$$ Put $\displaystyle V_k(s)= \int_\mathbb{T}\Bigl( a_k^N(s)-a_k^{N'}(s)\Bigr)^2\, \rho dx$, then, according to above two relations, $$V_k(s) \leq 2\,\Bigl[ k^4 \int_\mathbb{T}\frac{ |X_s^N-X_s^{N'}|^2}{(\rho_s^{N'}(X_s^{N'}))^2}\, \rho dx +k^2 \int_{\mathbb{T}} \Bigl(\frac{1}{\rho_s^{N}(X_s^{N})} -\frac{1}{\rho_s^{N'}(X_s^{N'})}\Bigr)^2\, \rho dx\Bigr].$$ Putting $$V_{k1}(s)=\int_\mathbb{T}\frac{ |X_s^N-X_s^{N'}|^2}{(\rho_s^{N'}(X_s^{N'}))^2}\, \rho dx,\quad V_{k2}(s)=\int_{\mathbb{T}} \Bigl(\frac{1}{\rho_s^{N}(X_s^{N})} -\frac{1}{\rho_s^{N'}(X_s^{N'})}\Bigr)^2\, \rho dx,$$ then $\displaystyle V_k(s) \leq 2 k^4 V_{k1}(s)+ 2 k^2 V_{k2}(s)$. Using [\[eq4.29.2\]](#eq4.29.2){reference-type="eqref" reference="eq4.29.2"}, we write down $$\label{eq4.29.3} \int_\mathbb{T}|J_{12}(s)|^2\, \rho dx\leq 2k^4 ||f_s^N||_{L^2(\rho dx)}^2\, V_{k1}(s)+ 2 k^2 ||f_s^N||_{L^2(\rho dx)}^2 V_{k2}(s).$$ Let $r>1$ and $\tilde r>1$ such that $\displaystyle\frac{1}{r}+ \frac{1}{\tilde r}=1$. By Hölder inequality, we have $$\int_0^t \mathbb{E}\Bigl( ||f_s^N||_{L^2}^2 V_{k1}(s)\Bigr)\,ds \leq \Bigl[ \int_0^t \mathbb{E}\Bigl( ||f_s^N||_{L^2}^{2\tilde r}\Bigr)\,ds\Bigr]^{1/\tilde r} \Bigl[\int_0^t \mathbb{E}\Bigl(\int_\mathbb{T}\frac{ |X_s^N-X_s^{N'}|^{2r}}{(\rho_s^{N'}(X_s^{N'}))^{2r}}\, \rho dx\Bigr)\,ds\Bigr]^{1/r}.$$ Denote by $\displaystyle o_N(t)^{1/r}$ the last term in the product in the right hand, again by Hölder inequality, we have $$\label{eq4.29.4} o_N(t)\leq \Bigl[\int_0^t\int_\mathbb{T}\mathbb{E}\bigl( |X_s^N-X_s^{N'}|^{2r^2}\bigr)\, \rho dx ds\Bigr]^{1/r} \Bigl[\int_0^t\int_\mathbb{T}\mathbb{E}\Bigl( \frac{1}{\bigl(\rho_s^{N'}(X_s^{N'})\bigr)^{2r\tilde r}}\Bigr)\, \rho dx ds\Bigr]^{1/\tilde r}.$$ Now using [\[eq5.3\]](#eq5.3){reference-type="eqref" reference="eq5.3"} in the next section and for $N'\geq N$, we have $$\Bigl( \mathbb{E}\Bigl[ |X_s^N-X_s^{N'}|^{2r^2}\Bigr]\Bigr)^{1/2r^2} \leq 2\, \Bigl( \frac{C_{r^2}}{N^{2q-1}} \Bigr)^{1/r^2};$$ therefore for some constant $C>0$, independent of $\omega, x, t$ such that $$\label{eq4.29.5} \Bigl[\int_0^t\int_\mathbb{T}\mathbb{E}\bigl( |X_s^N-X_s^{N'}|^{2r^2}\bigr)\, \rho dx ds\Bigr]^{1/r^2} \leq \frac{C}{N^{(2q-1)/r^2}}=\frac{C}{N^{q-\frac{1}{2}}},$$ if we take $r^2=2$. For estimating the second factor on the right hand of [\[eq4.29.4\]](#eq4.29.4){reference-type="eqref" reference="eq4.29.4"}, we introduce $\tilde K_s^N$ the density of $(X_s^N)^{-1}_\#(dx)$ relative to $dx$. By Lemma [Lemma 22](#LEMMA){reference-type="ref" reference="LEMMA"}, $\displaystyle\rho_s^{N'}(X_s^{N'})\, \tilde K_s^N=\rho$, which implies that $$\frac{1}{(\rho_s^{N'}(X_s^{N'}))^{2r\tilde r}}=\frac{(\tilde K_s^{N'})^{2r\tilde r}}{\rho^{2r\tilde r}}.$$ Since $0<\delta_1\leq \rho\leq \delta_2$ and using [\[eq5.6\]](#eq5.6){reference-type="eqref" reference="eq5.6"} in Appendice, we get $$\sup_N \sup_{s\in [0,1]} \mathbb{E}\Bigl(\int_\mathbb{T}\frac{(\tilde K_s^{N})^{2r\tilde r}}{\rho^{2r\tilde r}}\rho dx\Bigr) <+\infty.$$ Now combining this result with [\[eq4.29.4\]](#eq4.29.4){reference-type="eqref" reference="eq4.29.4"}, [\[eq4.29.5\]](#eq4.29.5){reference-type="eqref" reference="eq4.29.5"}, we get $$\label{eq4.29.6} \int_0^t \mathbb{E}\Bigl( ||f_s^N||_{L^2}^2 V_{k1}(s)\Bigr)\,ds\leq \frac{C k^4}{N^{q-\frac{1}{2}}}.$$ Again by Lemma [Lemma 22](#LEMMA){reference-type="ref" reference="LEMMA"}, we have $\displaystyle V_{k2}(s)=\int_\mathbb{T}(\tilde K_s^N-\tilde K_s^{N'})^2\, \rho^{-1}\, dx$, so that $$\int_0^t\mathbb{E}\Bigl(||f_s^N||_{L2}^2 V_{k2}(s)\Bigr)\, ds \leq \Bigl(\int_0^t \mathbb{E}\Big[||f_s^N||_{L^2}^{2\tilde r}\Bigr] ds\Bigr)^{1/\tilde r} \Bigl(\int_0^t\int_\mathbb{T}\mathbb{E}\Bigl[|\tilde K_s^N-\tilde K_s^{N'}|^{2r}\Bigr]\rho^{-r+1} dxds\Bigr)^{1/r}.$$ By Proposition [Proposition 33](#prop5.2){reference-type="ref" reference="prop5.2"} in Appendice, for some $\delta>0$ such that $\displaystyle q-\frac{1}{2}<q-\delta<2q-3$, $$\mathbb{E}\Bigl[|\tilde K_s^N-\tilde K_s^{N'}|^{2r}\Bigr] \leq \frac{C}{N^{q-\delta}}.$$ For some $r>1$ close to $1$, we have $\displaystyle\frac{1}{N^{(q-\delta)/r}}=\frac{1}{N^{q-\frac{1}{2}}}$. Finally we get $$\label{eq4.29.7} \int_0^t\mathbb{E}\Bigl(||f_s^N||_{L2}^2 V_{k2}(s)\Bigr)\, ds\leq \frac{C}{N^{q-\frac{1}{2}}}.$$ Finally combining [\[eq4.29.3\]](#eq4.29.3){reference-type="eqref" reference="eq4.29.3"}, [\[eq4.29.6\]](#eq4.29.6){reference-type="eqref" reference="eq4.29.6"} and [\[eq4.29.7\]](#eq4.29.7){reference-type="eqref" reference="eq4.29.7"}, we obtain $$\label{eq4.29.8} \int_0^t\mathbb{E}\Bigl[\int_{\mathbb{T}} J_{12}(s)^2\, \rho dx\Bigr]\,ds \leq \frac{Ck^4}{N^{q-\frac{1}{2}}}.$$ For estimating $J_2(s)$, we remark that $$\begin{split} \hat\rho_s^N(X_s^N)-\hat\rho_s^{N'}(X_s^{N'}) &= \Bigl( \frac{1}{\rho_s^N(X_s^N)}-\frac{1}{\rho_s^{N'}(X_s^{N'})}\Bigr) \frac{1}{\int_\mathbb{T}\frac{dx}{\rho_s^{N}}} +\frac{1}{\rho_s^{N'}(X_s^{N'})}\Bigl(\frac{1}{\int_\mathbb{T}\frac{dx}{\rho_s^{N}}}-\frac{1}{\int_\mathbb{T}\frac{dx}{\rho_s^{N'}}}\Bigr) \\ &= \frac{\tilde K_s^N-\tilde K_s^{N'}}{\rho \int_\mathbb{T}\frac{dx}{\rho_s^{N}}} +\frac{1}{\rho_s^{N'}(X_s^{N'})}\, \frac{\int_\mathbb{T}\frac{dx}{\rho_s^{N'}} -\int_\mathbb{T}\frac{dx}{\rho_s^{N}}}{\int_\mathbb{T}\frac{dx}{\rho_s^{N'}}\int_\mathbb{T}\frac{dx}{\rho_s^{N'}}}. \end{split}$$ Note that $$\int_\mathbb{T}\frac{dx}{\rho_s^{N}}=\int_\mathbb{T}\frac{\rho dx}{(\rho_s^N(X_s^N))^2} =\int_\mathbb{T}\frac{(\tilde K_s^N)^2}{\rho}\, dx;$$ then $$\int_\mathbb{T}\frac{dx}{\rho_s^{N'}}-\int_\mathbb{T}\frac{dx}{\rho_s^{N}} =\int_\mathbb{T}\frac{(\tilde K_s^{N'})^2- (\tilde K_s^N)^2}{\rho}\, dx.$$ On the other hand, $$\int_\mathbb{T}(a_k^N(s))^2\, \rho dx \leq k^2 \int_\mathbb{T}\frac{\rho dx}{(\rho_s^N(X_s^N))^2} =k^2 \int_\mathbb{T}\frac{dx}{\rho_s^{N}},$$ so that $$\Bigl|\int_\mathbb{T}f_s^N a_k^N(s)\, \rho dx\Bigr|\leq k ||f_s^N||_{L^2} \Bigl(\int_\mathbb{T}\frac{dx}{\rho_s^{N}} \Bigr)^{1/2}.$$ Recall that $$J_2(s)=\Bigl(\int_\mathbb{T}f_s^N a_k^N(s)\, \rho dx \Bigr) \Bigl(\hat\rho_s^N(X_s^N)-\hat\rho_s^{N'}(X_s^{N'}) \Bigr).$$ According to above calculation, we get $$|J_2(s)|\leq k ||f_s^N||_{L^2}\, \frac{|\tilde K_s^N-\tilde K_s^{N'}|}{\rho} +\frac{ k ||f_s^N||_{L^2}}{\rho_s^{N'}(X_s^{N'})}\, \Bigl|\int_\mathbb{T}\frac{(\tilde K_s^{N'})^2- (\tilde K_s^N)^2}{\rho}\, dx\Bigr|.$$ Proceeding as for estimating $J_1(s)$, we get finally $$\label{eq4.29.9} \int_0^t \mathbb{E}\Bigl[ |J_2(s)|^2\rho dx\Bigr]\, ds \leq \frac{C k^2}{N^{q-\frac{1}{2}}}.$$ Now using [\[eq4.29.1\]](#eq4.29.1){reference-type="eqref" reference="eq4.29.1"}, [\[eq4.29.8\]](#eq4.29.8){reference-type="eqref" reference="eq4.29.8"} and [\[eq4.29.9\]](#eq4.29.9){reference-type="eqref" reference="eq4.29.9"} and the fact that the series $\displaystyle\sum_{k\geq 1}\frac{k^4}{k^{2q}}$ converges for $\displaystyle q>\frac{5}{2}$, we get [\[eq4.26\]](#eq4.26){reference-type="eqref" reference="eq4.26"}. Inequality [\[eq4.27\]](#eq4.27){reference-type="eqref" reference="eq4.27"} can be proved in a similar way. ◻ **Definition 28**. *We define $\displaystyle g_t=f_t(X_t^{-1})$.* It is obvious that $\displaystyle\int_\mathbb{T}g_t^2\,\rho_t\, dx<+\infty$. In waht follows, we justify that $\{g_t; t\in [0,1]\}$ is the stochastic parallel translation along the curve $\{(X_t)_\#(\rho dx); \ t\in [0,1]\}$. We have no explicit SDE for $g_t$, but $f_t=g_t(X_t)$ satisfies a SDE. More precisely, $\{f_t; t\in [0,1]\}$ is a solution to the following SDE, with $\displaystyle q>\frac{5}{2}$, $$\label{eq4.29.10} d_tf_t=\sum_{k=1}^{+\infty} \frac{1}{k^q}\, \Lambda_k(t, f_t)\, dB_t^k +\sum_{k=1}^{+\infty} \frac{1}{2 k^{2q}}\,\Theta_k(t,f_t)\, dt,$$ where $$\Lambda_k(t,f)=-\Bigl(\int_\mathbb{T}f\, a_k(t)\, \rho\,dx\Bigr)\, \hat\rho_t(X_t),$$ $$\begin{split} \Theta_k(t,f)=-&\Bigl(\int_\mathbb{T}f\, a_k(t)\, \rho\,dx\Bigr)\, \bigl(\hat\rho_t\,\partial_x^2\phi_k\bigr)(X_t) -\Bigl(\int_\mathbb{T}f\, b_k(t)\, \rho\,dx\Bigr)\, \hat\rho_t(X_t)\\ +& 3\ \Bigl(\int_\mathbb{T}f\, a_k(t)\, \rho\,dx\Bigr)\, \Bigl(\int_\mathbb{T}\partial_x^2\phi_k\, \hat\rho_t\, dx\Bigr)\hat\rho_t(X_t), \end{split}$$ with $$a_k(t)=\Bigl( \frac{\partial_x^2\phi_k}{\rho_t}\Bigr)(X_t), \quad b_k(t)=\Bigl( \frac{\partial_x\bigl(\partial_x^2\phi_k\,\partial_x\phi_k\bigr)}{\rho_t}\Bigr)(X_t).$$ **Corollary 29**. *Under hypothesis of theorem [Theorem 26](#th4.8){reference-type="ref" reference="th4.8"}, almost surely, $$\label{eq4.29} \lim_{N\rightarrow+\infty} \int_\mathbb{T}(g_t^N)^2\, \rho_t^N\, dx=\int_\mathbb{T}g_t^2\, \rho_t\, dx.$$* *Proof.* We have $$\int_\mathbb{T}(g_t^N)\rho_t^n\, dx=\int_\mathbb{T}(f_t^N)^2\, \rho dx$$ which converges, by [\[eq4.17\]](#eq4.17){reference-type="eqref" reference="eq4.17"}, to $$\int_\mathbb{T}f_t^2\, \rho dx=\int_\mathbb{T}g_t^2\, \rho_t\, dx.$$ ◻ An immediate consequence of above result is **Corollary 30**. *For any $t\in [0,1]$, $$\label{eq4.30} \int_\mathbb{T}g_t^2\, \rho_t\, dx=\int_\mathbb{T}g_0^2\, \rho dx.$$* **Theorem 31**. *Almost surely, for $t\in [0,1]$, $$\label{eq4.31} \int_\mathbb{T}g_t(x)\, dx=0.$$* *Proof.* Let $\xi$ be a bounded Random variable, using the uniform (relative to $N$) estimate of the density of $(X_t^N)^{-1}_\#(\rho dx)$, it is straightforward (see [@FangLuoTh]) to prove that $$\lim_{N\rightarrow+\infty}\mathbb{E}\Bigl( \xi \int_\mathbb{T}f_t^N\bigl((X_t^N)^{-1}\bigr)\, dx\Bigr) =\mathbb{E}\Bigl( \xi \int_\mathbb{T}f_t\bigl((X_t)^{-1}\bigr)\, dx\Bigr).$$ But by Proposition [Proposition 23](#prop4.4){reference-type="ref" reference="prop4.4"}, $\displaystyle\int_{\mathbb{T}} f_t^N\bigl((X_t^N)^{-1}\bigr)\, dx=\int_\mathbb{T}g_t^N(x)\, dx=0$. Then for any bounded $\xi$, $$\mathbb{E}\Bigl( \xi \int_\mathbb{T}f_t\bigl((X_t)^{-1}\bigr)\, dx\Bigr)=0.$$ The result [\[eq4.31\]](#eq4.31){reference-type="eqref" reference="eq4.31"} follows. ◻ # Appendice {#sect5} First we recall the notations: $\displaystyle\phi_{2k-1}(x)=\frac{\sin(kx)}{k},\ \phi_{2k}(x)=-\frac{\cos(kx)}{k}$ and $\alpha_k=k^q$ for some $q>1$. Let $(X_t^N)$ and $(X_t)$ be solutions to SDE: $$\label{eq5.1} dX_t^N=\sum_{k=1}^N \frac{1}{\alpha_k} \Bigl( \partial_x\phi_{2k-1}(X_t^N)\, dB_{2k-1}(t) + \partial_x\phi_{2k}(X_t^N)\, dB_{2k}(t)\Bigr),$$ $$\label{eq5.2} dX_t=\sum_{k=1}^\infty \frac{1}{\alpha_k} \Bigl( \partial_x\phi_{2k-1}(X_t)\, dB_{2k-1}(t) + \partial_x\phi_{2k}(X_t)\, dB_{2k}(t)\Bigr)$$ respectively. **Proposition 32**. *Let $p\geq 1$ be an integer, there exists a constant $C_p>0$ independent of $x$ such that $$\label{eq5.3} \mathbb{E}\Bigl[ \Bigl( X_t^N(x)-X_t(x)\Bigr)^{2p}\Bigr] \leq \frac{C_p}{N^{2q-1}}.$$* *Proof.* Let $\eta_t= X_t(x)-X_t^N(x)$. Then $$\begin{split} d\eta_t=& \sum_{k=1}^N \frac{1}{\alpha_k} \Bigl[ \bigl( \cos(kX_t)-\cos(kX_t^N)\bigr)\,dB_{2k-1}(t)+ \bigl( \sin(kX_t)-\sin(kX_t^N)\bigr)\,dB_{2k}(t)\Bigr]\\ &+ \sum_{k>N} \frac{1}{\alpha_k} \Bigl[ \cos(kX_t)\,dB_{2k-1}(t)+ \sin(kX_t)\,dB_{2k}(t)\Bigr]. \end{split}$$ Then Itô stochastic contraction $d\eta_t\cdot d\eta_t$ admits the expression $$\sum_{k=1}^N \frac{1}{\alpha_k^2} \Bigl[ \bigl( \cos(kX_t)-\cos(kX_t^N)\bigr)^2 + \bigl( \sin(kX_t)-\sin(kX_t^N)\bigr)^2\Bigr]+ \sum_{k>N} \frac{1}{\alpha_k^2},$$ which is equal to $$\sum_{k=1}^N \frac{4}{k^{2q}}\, \sin^2\Bigl(k\, \frac{X_t-X_t^N}{2}\Bigr) + \sum_{k>N}\frac{1}{k^{2q}};$$ the first above sum is dominated by $$\sum_{k=1}^{+\infty} \frac{1}{k^{2q-2}}(X_t-X_t^N)^2,$$ while the second sum has the upper bound $\displaystyle\frac{1}{N^{2q-1}}$. Hence there is a constant $C>0$ independent of $x$ such that $$\label{eq5.4} d\eta_t\cdot d\eta_t \leq \Bigl(C\, \eta_t^2 + \frac{1}{N^{2q-1}}\bigr)\, dt.$$ By Itô['s]{style="color: black"} formula, $\displaystyle d\eta_t^{2p}=2p\eta_t^{2p-1}d\eta_t + 2p(2p-1)\eta_t^{2p-2} \, d\eta_t\cdot d\eta_t$, which is dominated, according to [\[eq5.4\]](#eq5.4){reference-type="eqref" reference="eq5.4"}, by $$2p\eta_t^{2p-1}d\eta_t + 2p(2p-1)C\, \eta_t^{2p}\, dt + 2p(2p-1)\frac{\eta_t^{2p-2}}{N^{2q-1}}dt.$$ Note that $\displaystyle 2p(2p-1)\eta_t^{2p-2}\leq 2p(2p-1)(2\pi)^{2p-2}$ that is denoted by $C_p'$. Therefore $$d\eta_t^{2p}\leq 2p\eta_t^{2p-1}\, d\eta_t + C_p'\, \eta_t^{2p} dt + \frac{C_p'}{N^{2q-1}} dt.$$ It follows that $$\mathbb{E}(\eta_t^{2p})\leq C_p'\, \int_0^T \mathbb{E}(\eta_s^{2p})\, ds + \frac{C_p' t}{N^{2q-1}}.$$ Then Gronwall['s]{style="color: black"} lemma yields $$\mathbb{E}(\eta_t^{2p})\leq \frac{C_p' t}{N^{2q-1}}\, e^{C_p' t},$$ that is nothing but [\[eq5.3\]](#eq5.3){reference-type="eqref" reference="eq5.3"}. ◻ Now we denote by $\displaystyle\tilde K_t^N$ the density of $(X_t^N)^{-1}_\#(dx)$ with respect to $dx$ and $\displaystyle\tilde K_t$ the density of $(X_t^{-1})_\#(dx)$ with respect to $dx$. By Kunita formula [@Kunita], we have $$\tilde K_t^N = \exp\Bigl( \sum_{k=1}^{2N}\frac{1}{\alpha_k}\int_0^t \partial_x^2\phi_k(X_s^N)\circ dB_s^k\Bigr),$$ and $$\tilde K_t = \exp\Bigl( \sum_{k=1}^{+\infty}\frac{1}{\alpha_k}\int_0^t \partial_x^2\phi_k(X_s)\circ dB_s^k\Bigr).$$ First we consider $$\hat K_t^N = \exp\Bigl( \sum_{k=1}^{2N}\frac{1}{\alpha_k}\int_0^t \partial_x^2\phi_k(X_s^N)\, dB_s^k\Bigr),$$ and $$\hat K_t = \exp\Bigl( \sum_{k=1}^{+\infty}\frac{1}{\alpha_k}\int_0^t \partial_x^2\phi_k(X_s)\, dB_s^k\Bigr).$$ Then they are linked by $$\label{L} \tilde K_t^N=\hat K_t^N\,exp\Bigl(-\sum_{k=1}^{2N} \frac{k^2}{k^{2q}}\Bigr).$$ To estimate $\hat K_t^N$ in $L^p$, we use exponential martingale. More precisely, for a continuous martingale $(M_t)$ with quadratic variation $<M>_t$, we write down, for $p\geq 1$, $$e^{pM_t}=e^{pM_t-p^2<M>_t}\, e^{p^2 <M>_t}.$$ By Cauchy-Schwarz inequality and the fact $\displaystyle\mathbb{E}\bigl( e^{2pM_t-2p^2<M>_t}\bigr)=1$, we get $$\label{eq5.5} \mathbb{E}(e^{pM_t})\leq \Bigl( E\bigl( e^{2p^2<M>_t}\bigr)\Bigr)^{1/2}.$$ For $\displaystyle M_t=\sum_{k=1}^{2N}\frac{1}{\alpha_k}\int_0^t \partial_x^2\phi_k(X_s^N)\, dB_s^k$, we have $$<M>_t =\sum_{k=1}^N \int_0^t \frac{1}{\alpha_k^2} \bigl( \partial_x^2\phi_k(X_s^N)\bigr)^2\, ds \leq \sum_{k=1}^{+\infty} \frac{t}{k^{2q-2}}=\xi(2q-2)\, t.$$ Then by [\[eq5.5\]](#eq5.5){reference-type="eqref" reference="eq5.5"}, we get $$\label{eq5.6} \mathbb{E}\Bigl( (\hat K_t^N)^p\Bigr) \leq e^{p^2\xi(2q-2)\, t}.$$ Obviously above estimate holds true for $\tilde K_t$, that is, $$\label{eq5.7} \mathbb{E}\bigl( \hat K_t^p\bigr) \leq e^{p^2\xi(2q-2)\, t}.$$ **Proposition 33**. *For $\displaystyle q>\frac{5}{2}$, choose $\delta>0$ such that $q-\frac{1}{2}<q-\delta<2q-3$; then there is a constant $C_p>0$ such that $$\label{eq5.8} \mathbb{E}\Bigl( (\tilde K_t^N-\tilde K_t)^{2p}\Bigr)\leq \frac{C}{N^{q-\delta}}.$$* *Proof.* Let $\displaystyle\zeta_t =\hat K_t^N-\hat K_t$. we have $$d\hat K_t^N =\hat K_t^N\, \sum_{k=1}^{2N}\frac{1}{\alpha_k}\partial_x^2\phi_k(X_t^N)\, dB_t^k,$$ and $$d\hat K_t=\hat K_t\, \sum_{k=1}^{+\infty}\frac{1}{\alpha_k}\partial_x^2\phi_k(X_t)\, dB_t^k.$$ So $$\begin{split} d\zeta_t=&\sum_{k=1}^{2N}\frac{1}{\alpha_k}\Bigl( \hat K_t^N\,\partial_x^2\phi_k(X_t^N) -\hat K_t\,\partial_x^2\phi_k(X_t)\Bigr)\, dB_t^k\\ &-\sum_{k>2N} \frac{1}{\alpha_k}\, \textcolor{black}{\hat K}_t\, \partial_x^2\phi_k(X_t)\, dB_t^k. \end{split}$$ Now the Itô stochastic contraction $d\zeta_t\cdot d\zeta_t$ admits the expression $$\begin{split} d\zeta_t\cdot \textcolor{black}{d}\zeta_t=&\sum_{k=1}^{2N}\frac{1}{\alpha_k^2}\Bigl( \hat K_t^N\,\partial_x^2\phi_k(X_t^N)\, -\hat K_t\,\partial_x^2\phi_k(X_t)\Bigr)^2\, dt\\ &+\sum_{k>2N} \frac{1}{\alpha_k^2}\, \bigl(\textcolor{black}{\hat K}_t\, \partial_x^2\phi_k(X_t)\bigr)^2\, dt. \end{split}$$ Remark that $$\hat K_t^N\,\partial_x^2\phi_k(X_t^N)\, -\hat K_t\,\partial_x^2\phi_k(X_t) = (\hat K_t^N-\hat K_t)\, \partial_x^2\phi_k(X_s^N) + \hat K_t\, \bigl(\partial_x^2\phi_k(X_s^N)-\partial_x^2\phi_k(X_s)\bigr),$$ then $$|\hat K_t^N\,\partial_x^2\phi_k(X_t^N)\, -\hat K_t\,\partial_x^2\phi_k(X_t)| \leq k\,|\hat K_t^N-\hat K_t| + k^2\,\hat K_t\, |X_s^N-X_s|.$$ Therefore we get the following upper bound $$\label{eq5.9} d\zeta_t\cdot d\zeta_t \leq 2\xi(2q-2)\, |\hat K_t^N-\hat K_t|^2+ 2\, \xi(2q-4)\, \hat K_t^2\, |X_t^N-X_t|^2 +\frac{\hat K_t^2}{N^{2q-3}}.$$ For $p\geq 1$ an integer, by Itô['s]{style="color: black"} formula and above estimate, there are three constants $C_1, C_2, C_3$ only dependent of $p$ such that $$\begin{split} d\zeta_t^{2p}&=2p\zeta_t^{2p-1}d\zeta_t + 2p(2p-1)\zeta_t^{2p-2}\, d\zeta_t\cdot \textcolor{black}{d}\zeta_t\\ &\leq 2p\zeta_t^{2p-1}d\zeta_t+ C_1(p) \zeta_t^{2p}\,dt + C_2(p) \zeta_t^{2p-2}\hat K_t^2 (X_t^N-X_t)^2\, dt +C_3(p) \frac{\zeta_t^{2p-2}\hat K_t^2}{N^{2q-3}}. \end{split}$$ It follows that $$\begin{split} \mathbb{E}(\zeta^{2p})&\leq C_1(p)\int_0^t \mathbb{E}(\zeta_s^{2p})\, ds + C_2(p)\int_0^t \mathbb{E}\Bigl( \zeta_s^{2p-2}\hat K_s^2 (X_s^N-X_s)^2\Bigr)ds\\ &+\frac{C_3(p)}{N^{2q-3}}\int_0^t \mathbb{E}\Bigl( \zeta_s^{2p-2}\hat K_s^2\Bigr)\, ds. \end{split}$$ Let $r>1$ and $\hat r>1$ such that $(1/r) + (1/\hat r)=1$. Now by [\[eq5.5\]](#eq5.5){reference-type="eqref" reference="eq5.5"} and [\[eq5.6\]](#eq5.6){reference-type="eqref" reference="eq5.6"}, we have $$C=\sup_{s\in [0,1]}\mathbb{E}\Bigl( \bigl(\zeta_s^{2p-2}\hat K_s^2\bigr)^{\hat r}\Bigr) <+\infty.$$ On the other hand, by [\[eq5.3\]](#eq5.3){reference-type="eqref" reference="eq5.3"}, $\displaystyle\mathbb{E}\bigl( (X_s^N-X_s)^{2r}\bigr)\leq \frac{C_r}{N^{2q-1}}$. Taking $\displaystyle r=\frac{2q-1}{q-\delta}$ yields $$\mathbb{E}(\textcolor{black}{\zeta_t}^{2p})\leq C_1(p)\int_0^t \mathbb{E}(\zeta_s^{2p})\, ds + \frac{C_2(p)}{N^{q-\delta}}\, t.$$ Gronwall lemma gives $$\mathbb{E}\Bigl( (\hat K_t^N-\hat K_t)^{2p}\Bigr)\leq \frac{C}{N^{q-\delta}}.$$ Combing this with Relation [\[L\]](#L){reference-type="eqref" reference="L"}, we obtain the desired estimate [\[eq5.8\]](#eq5.8){reference-type="eqref" reference="eq5.8"}. ◻ **Acknowledgement:** This work has been taken from a part of the PhD thesis [@Ding] of the first named author by a joint PhD program between the Academy of Mathematics and Systems Science, Chinese Academy of Sciences and the Institute of Mathematics of Burgundy, University of Burgundy. He is grateful to the hospitality of these two institutions. The financial support from China Scholarship Council and National Center for Mathematics and Interdisciplinary Sciences are particularly acknowledged. 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Villani, Ricci curvature for metric-measure spaces via optimal transport, *Ann of Math.*, 169 (2009), 903-991. P. Malliavin, Stochastic analysis, Grund. Math. Wissen., vol. 313, Springer, 1997. R. McCann, Polar factorization of maps on Riemannian manifolds, *Geo. Funct. Anal.*, 11 (2001), 589-608. F. Otto: The geometry of dissipative evolution equations: The porous medium equation, Comm. partial Diff. equations, **26** (2001), 101-174. F. Otto and Villani, *Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,* J. Funct. Anal. 173(2000), 361-400. M. K. Von Renesse, K. T. Sturm, Entropic measure and Wasserstein diffusion, *Ann. Prob.*, 37 (2009), 1114-1191. M.K. Von Renesse, K.T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, *Comm. Pures Appl. Math.*, 58 (2005), 923-940. K. T. Sturm, On the geometry of metric measure spaces, *Acta Math.*, 196 (2006), 65-131. C. Villani, *Optimal transport, Old and New*, vol. 338, Grund. Math. Wiss., Springer-Verlag, Berlin, 2009. C. Villani, *Topics in optimal transportation*, Graduate Studies in Mathematics, 58 (2003), AMS, Providence, Ehode Island. Feng-Yu Wang, Image-dependent conditional McKean-Vlasov SDEs for measure-valued diffusion processes, *J. Evol. Equ.*, 21 (2021), 2009-2045. [^1]: Email: dinghao16\@mails.ucas.ac.cn [^2]: Email:Shizan.Fang\@u-bourgogne.fr [^3]: Email: xdli\@amt.ac.cn

arxiv_math

--- abstract: | In this work, we consider the generalized Benjamin-Bona-Mahony equation $$\partial_t u+\partial_x u+\partial_x( |u|^pu)-\partial_t \partial_x^{2}u=0, \quad(t,x) \in \mathbb{R} \times \mathbb{R},$$ with $p>4$. This equation has the traveling wave solutions $\phi_{c}(x-ct),$ for any frequency $c>1.$ It has been proved by Souganidis and Strauss [@Strauss-1990] that, there exists a number $c_{0}(p)>1$, such that solitary waves $\phi_{c}(x-ct)$ with $1<c<c_{0}(p)$ is orbitally unstable, while for $c>c_{0}(p),$ $\phi_{c}(x-ct)$ is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein [@Pego-1991-eigenvalue]. In this paper, we prove the orbital instability in the critical case $c=c_{0}(p)$. address: - | (R. Jia) Center for Applied Mathematics\ Tianjin University\ Tianjin 300072, China - | (Y.Wu) Center for Applied Mathematics\ Tianjin University\ Tianjin 300072, China author: - Rui Jia - Yifei Wu title: Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation --- # Introduction It is well known that the KdV equation is a classical model used to describe the characteristics of water waves of long wave length in river channels. When studying nonlinear dispersive long wave unidirectional propagation, Benjamin, Bona, and Mahony [@T.; @B.benjamin-1972] considered a new model named the Benjamin-Bona-Mahony (BBM) equation, which can describe physical properties of long waves better. The BBM equation reads $$\begin{aligned} u_t+u_x+uu_x-u_{txx}=0.\end{aligned}$$ In this paper, we consider the following generalized Benjamin-Bona-Mohony (gBBM) equation $$\begin{aligned} \label{1.1} \partial_t u+\partial_x u+\partial_x(|u|^pu)-\partial_t \partial_x^{2}u=0 \quad(t,x) \in \mathbb{R} \times \mathbb{R}\end{aligned}$$ with $p>0.$ For $H^1$-solution, the momentum $Q$ and the energy $E$ are conserved under the flow, where $$\begin{aligned} \label{momentum Q} &Q(u)=\frac{1}{2}\int_{\mathbb{R}} u^{2}+u_{x}^{2} \,\mathrm{d}x;\\ \label{energy E} &E(u)=\frac{1}{2}\int_{\mathbb{R}} u^{2}\,\mathrm{d}x+\frac{1}{p+2}\int_{\mathbb{R}}|u|^{p+2} \,\mathrm{d}x. \end{aligned}$$ In particular, the equation [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} can be expressed in the following Hamiltonian form $$\begin{aligned} \label{equa-hami} \partial_t u= JE'(u),\qquad \mbox{ where } \qquad J=-(1-\partial_x^2)^{-1}\partial_x.\end{aligned}$$ In [@Strauss-1990], Souganidis and Strauss proved that if $u_{0}=u(0,x)\in H^1(\mathbb{R}),$ there exists a unique global solution $u$ of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} in $C(\mathbb{R}; H^1(\mathbb{R}))$. The equation [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} has the solitary waves solution given by $u(x, t)=\phi_{c}(x-ct)$ for any $c>1$, where $\phi_{c}$ is the ground state solution of the following elliptic equation $$\begin{aligned} \label{1.3} -c\partial_{xx}\phi_{c}+(c-1)\phi_{c}-\phi_{c}^{p+1}=0.\end{aligned}$$ The ground state solution $\phi_{c}$ is a smooth, even, and positive function, which decays exponentially as $|x|\rightarrow \infty$, in the sense that $|\phi_{c}|\leq C_{1}e^{-C_{2}|x|}, |\partial_x\phi_{c}|\leq C_{1}e^{-C_{2}|x|}$ for some $C_{1}, C_{2}>0.$ Then a natural problem is the stability theory of the solitary waves solution $\phi_{c}(x-ct)$, which is defined as follows. For $\varepsilon>0,$ we denote the set $U_{\varepsilon}(\phi_{c})$ as: $$\begin{aligned} \label{1.9} U_{\varepsilon}(\phi_{c})&= \{ u \in C(\mathbb{R}; H^1(\mathbb{R})): \sup_{t \in \mathbb{R}}\inf_{y\in\mathbb{R}}\big\|u-\phi_{c}(\cdot-y)\big\|_{H^1(\mathbb{R})}<\varepsilon \}. \end{aligned}$$ Then we define the orbital stability/instability of the solitary waves as **Definition 1**. *We say that the solitary waves solution $\phi_{c}(x-ct)$ of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} is orbitally stable if for any $\varepsilon>0,$ there exists $\delta>0$ such that if $\big\|u_{0}-\phi_{c}\big\|_{H^1(\mathbb{R})}<\delta,$ then the solution $u$ of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} with $u(0, x)=u_{0}(x)$ satisfies $u \in U_{\varepsilon}(\phi_{c}).$ Otherwise, $\phi_{c}(x-ct)$ is said to be orbitally unstable.* Regarding the stability theory of these solitary waves, Souganidis and Strauss [@Strauss-1990] proved that when $0<p\leq 4$, the solitary waves solution $\phi_{c}(x-ct)$ is orbitally stable for all $c>1$, while when $p>4$, the solitary waves solution $\phi_{c}(x-ct)$ is orbitally unstable in $H^1(\mathbb{R})$ for $1<c<c_{0}(p)$ and orbitally stable in $H^1(\mathbb{R})$ for $c>c_{0}(p)$. Here $$c_{0}(p):=\frac{p}{4+2p}\left(1+\sqrt{2+\frac{1}{2}p}\right).$$ Denote $d(c):=E(\phi_{c})-cQ(\phi_{c})$, the critical parameter $c_0(p)$ is determined by $$d''(c)\Big|_{c=c_0(p)}=0.$$ Since the operator $J$ is not onto, the framework of Grillakis, Shatah, and Strauss [@GrShStr-87; @GrShStr-90] cannot be directly applied to study the stability of the solitary waves $\phi_{c}(x-ct)$. Therefore, the work [@Strauss-1990] is not a direct application of the theories established in [@GrShStr-87; @GrShStr-90]. For further discussion on these cases, readers are referred to the more recent paper [@LZ-2022-MAMS] by Lin and Zeng. In [@Pego-1991-eigenvalue], Pego and Weinstein established criteria for the linear exponential instability of solitary waves solution of the gBBM equation. They further proved the linear exponential instability of $\phi_{c}(x-ct)$ for each $p>4$ when $1<c<c_{0}(p)$. So far, the stability of the solitary waves $\phi_{c}(x-ct)$ has been nearly established, except for the critical case $c=c_0(p)$, which corresponds to the degenerate case where $d''(c)=0$. In this paper, our aim is to fully establish the stability of the solitary waves $\phi_{c}(x-ct)$ by studying the degenerate case of $c=c_0(p)$. Before presenting our theorem, let us clarify some definitions that will be used. We define the functional $S_c$ as $$\begin{aligned} \label{S_c} S_c(u):=E(u)-cQ(u).\end{aligned}$$ Then the equation [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"} is equivalent to $S_c'(\phi_{c})=0$. For convenience, we denote that $$\omega=c^{-\frac12},$$ and $$\begin{aligned} \label{1.4} \psi_{\omega}(x)=c^{-\frac{1}{p}}\phi_{c}(x).\end{aligned}$$ Then by [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}, we find that $\psi_{\omega}$ satisfies the following equation: $$\begin{aligned} \label{1.5} -\partial_{xx}\psi_{\omega}+(1-\omega^{2})\psi_{\omega}-\psi_{\omega}^{p+1}=0.\end{aligned}$$ As mentioned earlier, our objective is to demonstrate the instability of the solitary waves in the critical frequency case: $c=c_0(p)$. Our argument relies on the assumption of the negativity of a specific direction of the Hessian operator $S_c''(\phi_{c})$, which is confirmed numerically. More precisely, let $$\begin{aligned} \label{Psi_c} \Psi_{c}&:=-\frac{1}{2}\omega^{1-\frac{2}{p}}\partial_{\omega} \psi_{\omega},\\ \label{Gamma_c} \Gamma_c &:=B(c)\left[c^2\Psi_{c}+\frac{c}{2}x\partial_{x}\phi_{c}+c\phi_{c}\right]+D(c)(3x^2\phi_{c}+x^3\partial_{x}\phi_c), \end{aligned}$$ where $$\begin{aligned} &D(c)=-\frac{4pc+4c-3p}{2(p+4)}\big\|\phi_{c}\big\|^2_{L^2}, \quad B(c)=\frac{3}{2}\big\|x\phi_{c}\big\|^2_{L^2}+\frac{9}{2}\big\|x\partial_{x}\phi_{c}\big\|^2_{L^2}-3\big\|\phi_{c}\big\|^2_{L^2}. \end{aligned}$$ We assume that $$\begin{aligned} \label{second-orth-condition} \langle S''_c(\phi_{c})\Gamma_c, \Gamma_c \rangle <0 \quad \mbox{ holds for } c=c_0(p) \mbox{ with }p>4, \end{aligned}$$ which is checked numerically [^1] in Appendix [6.4](#numerical result){reference-type="ref" reference="numerical result"}. The main result in the present paper is the following. **Theorem 2**. *Let $p>4, c=c_0(p)$ and $\phi_{c}$ be the ground state of [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}. Assume [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}, then the solitary waves solution $\phi_{c}(x-ct)$ is orbitally unstable.* *Remark 3*. (1) In the previous works [@Pego-1991-eigenvalue] and [@Strauss-1990], the stability and instability results of solitary waves solution $\phi_{c}(x-ct)$ for the non-degenerate case $d''(c)\neq 0$ in the gBBM equation have already been established. Under [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}, Theorem [Theorem 2](#main:thm){reference-type="ref" reference="main:thm"} closes the only remaining gap for the degenerate case $d''(c)= 0$ and thereby completes the entire stability theory of solitary waves solution for the gBBM equation. We also give an element numerical computation to check [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}. \(2\) The instability of the solitary waves $\phi_{c}(x-ct)$ with $1<c<c_{0}(p)$ has been demonstrated using the Lyapunov stability argument based on the monotonicity of the Lyapunov functional in the non-degenerate cases $d''(c)\neq 0.$ However, this argument does not apply to the degenerate cases, as the Lyapunov functional loses monotonicity when $d''(c)=0.$ Therefore, we need to construct a new monotonic functional. The outline of the proof will be provided in the following subsection. ## Sketch of the proof The main approach is to construct a monotonic quantity based on virial quantities and the modulation argument, drawing inspiration from [@wu-KG], which established the instability of standing wave solutions of the Klein-Gordon equation in the degenerate case. The methodology involves analyzing the orthogonality conditions and the dynamics of the modulated parameters. However, due to the intricate structure of the gBBM equation, constructing the monotonic functional in this paper is much more complex than the Klein-Gordon equation case. In particular, the non-onto property of the skew symmetry operator $J$ poses significant obstacles. The key ingredients of the proof can be summarized as follows. *Step1: Modulation.* First of all, we assume the solitary wave is stable. The modulation argument allows us to find two parameters, $y(t)$ and $\lambda(t),$ and a perturbation function $\xi$, such that the solution $u$ can be expressed as $$\begin{aligned} \label{perturbation-fun} u(t,x)=(\phi_{\lambda}+\xi)\big(x-y(t)\big), \end{aligned}$$ where $\lambda(t)$ is a scaling parameter suitably defined, and $y(t)$ is a spatial translation parameter. We also need to find two different orthogonality conditions, namely: $$\begin{aligned} \label{intro-orth-vague} \langle \xi, \psi_1 \rangle=\langle \xi, \psi_2 \rangle=0. \end{aligned}$$ To find suitable $\psi_1$ and $\psi_2,$ it is natural to consider the spectrum of the Hessian of the action $S_{c}''(\phi_{c})$. The study of [@weinstein-1985-modulational] indicates that $\ker S''_c(\phi_{c})=\{\alpha\partial_{x}\phi_{c}, \alpha\in \mathbb{R}\}$ and $S''_c(\phi_{c})$ has a unique negative eigenvalue. The properties of spectrum of $S''_c(\phi_{c})$ helpfully identify the origin of $\psi_1$ and $\psi_2,$ specifically: $$\begin{aligned} \label{orth-orign} \psi_1 \in \ker S''_c(\phi_{c}), \quad \langle S''_c(\phi_{c})\psi_2, \psi_2\rangle <0. \end{aligned}$$ It is worth noting that $\psi_2$ is not unique and it is not necessary to choose the negative eigenfunction. Indeed, the choice of $\psi_2$ is crucial as its concrete expression has a significant impact on the construction of monotonicity, which will be addressed in Step 6 below. *Step2: Coercivity.* Having determined the properties of $\psi_1$ and $\psi_2$ (i.e.,[\[orth-orign\]](#orth-orign){reference-type="eqref" reference="orth-orign"}), we shall prove the the coercivity of the Hessian $S_{\lambda}''(\phi_{\lambda})$ as shown in Proposition [Proposition 20](#prop3.4){reference-type="ref" reference="prop3.4"} for a general criterion by means of spectral decomposition argument, which can be expressed as follows: $$\begin{aligned} \langle S_{\lambda}''(\phi_{\lambda})\xi, \xi\rangle\gtrsim \|\xi\|^2_{H^1}. \end{aligned}$$ In addition, in the degenerate case $c= c_0(p)$, we have $$\begin{aligned} \label{flatness} \langle S_{c}''(\phi_{c})\partial_{c}\phi_{c}, \partial_{c}\phi_{c}\rangle= 0. \end{aligned}$$ This flatness equality [\[flatness\]](#flatness){reference-type="eqref" reference="flatness"}, combined with the coercivity of $S_{\lambda}''(\phi_{\lambda})$, implies an important estimate: $$\begin{aligned} \label{control-xi} \|\xi\|_{H^1}\ll |\lambda-c|. \end{aligned}$$ This means that the perturbation of the solution $\xi,$ can be controlled by the scaling increment $\lambda-c$. We emphasize that the Step 1 and Step 2 here are similar to the paper in [@wu-KG], however, the following steps are of much problem-dependence and much more complicated for BBM mainly due to its poor Hamiltonian structure. *Step 3: Dynamic of the modulation parameters.* Directly following the Implicit Function Theorem, the (translation) modulation parameter $y$ has a trivial bound given by $$\begin{aligned} \label{rough-y-dot} \dot y-\lambda=O(\|\xi\|_{H^1}). \end{aligned}$$ In simpler terms, $\dot y-\lambda$ is actually the first-order of $\xi.$ However, the rough estimate is not enough to support the later analysis. To obtain a more accurate estimate, we apply [\[perturbation-fun\]](#perturbation-fun){reference-type="eqref" reference="perturbation-fun"} to [\[equa-hami\]](#equa-hami){reference-type="eqref" reference="equa-hami"}, which yields $$\begin{aligned} \label{intro-dynamic} \dot \xi+\partial_\lambda \phi_\lambda \cdot \dot \lambda-\partial_x \phi_\lambda\cdot (\dot y-\lambda) =JS_\lambda''(\phi_\lambda)\xi+\mbox{``high-order term''}. \end{aligned}$$ This gives us the key expression of the dynamic of $\dot y-\lambda$: $$\begin{aligned} \label{dot-y-Roughdynamic} \dot y-\lambda= c_1(\lambda)\left\langle \xi, S_\lambda''(\phi_\lambda)\big(Jf_{\lambda}\big)\right\rangle+c_1(\lambda)\partial_t \big\langle \xi, f_{\lambda}\big\rangle+\mbox{``high-order term''}, \end{aligned}$$ for any $f_{\lambda}$ satisfying $$\begin{aligned} \label{condition-f_lambda} \langle f_{\lambda},\partial_\lambda \phi_\lambda\rangle=0,\quad \langle f_{\lambda},\partial_x \phi_\lambda\rangle\ne 0. \end{aligned}$$ Obviously, the function $f_{\lambda}$ satisfying condition [\[condition-f_lambda\]](#condition-f_lambda){reference-type="eqref" reference="condition-f_lambda"} is not unique. The estimate [\[dot-y-Roughdynamic\]](#dot-y-Roughdynamic){reference-type="eqref" reference="dot-y-Roughdynamic"} is therefore relatively flexible, depending on the choice of $f_\lambda$. Indeed, the latter almost determines the expression of the first-order of $\xi$ which appears in $I'(t)$ defined later. This constitutes the first key ingredient in our proof. *Step 4: Design of the virial identity.* We are now in the position to consider the construction of virial identity $I(t).$ Our goal is to show that it exhibits monotonic behavior, i.e., $I'(t)>0$ or $I'(t)<0.$ The virial identity typically arises from conservation laws such as [\[momentum Q\]](#momentum Q){reference-type="eqref" reference="momentum Q"}-[\[energy E\]](#energy E){reference-type="eqref" reference="energy E"} and the dynamic of the modulated function as in [\[intro-dynamic\]](#intro-dynamic){reference-type="eqref" reference="intro-dynamic"}. The ideal form of $I'(t)$ is as follows: $$\begin{aligned} \label{idealform} I'(t)= \beta(u_{0})+\gamma(\lambda)+\mbox{``high-order term"},\end{aligned}$$ where the high-order term is in fact $\|\xi\|^{2}_{H^1}$ that has been estimated in Step 2. If $\beta(u_0)$ is a positive quantity, and $\gamma(\lambda)$ is also a positive quantity satisfying $\gamma(\lambda)\gtrsim(\lambda-c)^2$ which requires that $$\begin{aligned} \label{positive-gamma(lambda)} \gamma(c)=\gamma'(c)=0 \quad \mbox{and}\quad \gamma''(c)>0,\end{aligned}$$ then the monotonicity of virial identity is guaranteed. This constitutes the second key ingredient in our proof. *Step 5: Construction of the monotonicity.* Unlike the Lyapunov functional, the main monotonic functional here comes from the localized virial identity. Specifically, we first define $$I(t)=\int_{\mathbb{R}} \chi(x-y(t))(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}) \,\mathrm{d}x.$$ where $\chi$ is a suitable smooth cutoff function. By the expansion [\[perturbation-fun\]](#perturbation-fun){reference-type="eqref" reference="perturbation-fun"}, we observe that $I'(t)$ has the following structure: $$I'(t)=\beta(u_{0})+\gamma(\lambda)+c_2(\lambda)\cdot(\dot{y}-\lambda)+c_3(\lambda)\langle \xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle+c_4(\lambda)\langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\rangle +O\big(\|\xi\|^2_{H^1}\big).$$ Since the precise estimate of $\dot{y}-\lambda$ is already known in Step 3, we obtain the structure of $I'(t)$ as follows: $$\begin{aligned} I'(t) &= \beta(u_{0})+\gamma(\lambda) +\langle \xi, S''_{\lambda}(\phi_{\lambda})\big[c_1(\lambda)c_2(\lambda)Jf_{\lambda}+c_4(\lambda)\phi_{\lambda}\big] \rangle \\ &\quad +c_1(\lambda)c_2(\lambda)\partial_t \big\langle \xi, f_{\lambda}\big\rangle +c_3(\lambda)\langle \xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle +O\big(\|\xi\|^2_{H^1}\big).\end{aligned}$$ Move the term $c_1(\lambda)c_2(\lambda)\partial_t \big\langle \xi, f_{\lambda}\big\rangle$ to the left-hand side, and we further obtain that $$\begin{aligned} \label{stru-I'(t)} \frac{d}{dt}\Big(I(t)-c_1(\lambda)c_2(\lambda)\big\langle \xi, f_{\lambda}\big\rangle\Big) &= \beta(u_{0})+\gamma(\lambda) +\langle \xi, S''_{\lambda}(\phi_{\lambda})\big[c_1(\lambda)c_2(\lambda)Jf_{\lambda}+c_4(\lambda)\phi_{\lambda}\big] \rangle \notag\\ &\quad +c_3(\lambda)\langle \xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle +O\big(\|\xi\|^2_{H^1}\big).\end{aligned}$$ This inspires us to make a bold assumption: After suitably choosing $f_{\lambda}$, if $$\begin{aligned} \label{intro-orth} \left\langle \xi, S''_{\lambda}(\phi_{\lambda})\big[c_1(\lambda)c_2(\lambda)Jf_{\lambda}+c_4(\lambda)\phi_{\lambda}\big] +c_3(\lambda)\big(\phi_{\lambda}+\partial_{xx}\phi_{\lambda}\big)\right\rangle=0, \end{aligned}$$ then $I'(t)$ will become the ideal form [\[idealform\]](#idealform){reference-type="eqref" reference="idealform"}. In order to match the form in [\[orth-orign\]](#orth-orign){reference-type="eqref" reference="orth-orign"}, we need the existence and the explicit expressions of the pre-images of $\phi_{\lambda}$ and $\partial_{xx}\phi_{\lambda}$. As a matter of fact, we need to find $\Psi_1$ and $\Psi_2,$ such that $$\begin{aligned} \phi_{\lambda}=S_{\lambda}''(\phi_{\lambda})\Psi_1, \quad \partial_{xx}\phi_{\lambda}=S_{\lambda}''(\phi_{\lambda})\Psi_2.\end{aligned}$$ It is not an easy task, but we accomplished it. In fact, we observe that $$\begin{aligned} \phi_{\lambda}&=S_{\lambda}''(\phi_{\lambda})\big(-\frac{1}{2}\omega^{1-\frac{2}{p}}\partial_{\omega}\psi_{\omega}\big), \\ \partial_{xx}\phi_{\lambda}&=S_{\lambda}''(\phi_{\lambda})\big(\frac{1}{2}x\partial_x\phi_{\lambda}\big),\end{aligned}$$ where $\omega=\lambda^{-\frac{1}{2}},$ $\psi_{\omega}$ satisfying [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"}. This constitutes the third key ingredient in our proof. *Step 6: Verification of the negative direction.* Inspired by [\[intro-orth\]](#intro-orth){reference-type="eqref" reference="intro-orth"}, we denote $$\Upsilon_{\lambda} =c_1(\lambda)c_2(\lambda)Jf_{\lambda}+c_3(\lambda)\big(-\frac{1}{2}\omega^{1-\frac{2}{p}}\partial_{\omega}\psi_{\omega}+\frac{1}{2}x\partial_x\phi_{\lambda}\big)+c_4(\lambda)\phi_{\lambda}.$$ It is time to verify the negativity of $S''_{\lambda}(\phi_{\lambda})$ on $\Upsilon_{\lambda}$. More precisely, the problem finally reduces to the following claim: **Claim 4**. *There exists a function $f_{\lambda}$ verifying [\[condition-f_lambda\]](#condition-f_lambda){reference-type="eqref" reference="condition-f_lambda"}, such that $$\begin{aligned} \label{negative direction} \langle S''_{\lambda}(\phi_{\lambda})\Upsilon_{c},\Upsilon_{c}\rangle<0.\end{aligned}$$* This claim is established by choosing $f_\lambda=(1-\partial_x^2)(x^3\phi_{\lambda})$ as presented in the assumption [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}. Due to the complexity of the expression $\Upsilon_{\lambda}$, we decide to check it by numerical experiments. Suppose the claim is true, then we choose $\psi_2=S''_{\lambda}(\phi_{\lambda})\Upsilon_{\lambda}$ and thus obtain the form of $I'(t)$ in [\[idealform\]](#idealform){reference-type="eqref" reference="idealform"} as we expected. This constitutes the fourth key ingredient in our proof. *Step 7: Contradiction.* Based on the works before, the structure of monotonicity becomes clear. Indeed, $$I'(t)= \beta(u_{0})+\gamma(\lambda)+O(\|\xi\|^2_{H^1}).$$ Then using [\[control-xi\]](#control-xi){reference-type="eqref" reference="control-xi"}, we can infer that $$I'(t)\ge \beta(u_{0})+ \frac12 C(\lambda-c)^{2}.$$ The positivity of $\beta(u_{0})$ can be verified by suitably choosing initial data $u_0.$ Thus we establish the monotonicity of $I(t).$ The contradiction between uniformly boundedness and monotonicity of $I(t)$ proves the instability in the end. ## Organization of the paper The remainder of the paper is organized as follows. In Section [2](#2){reference-type="ref" reference="2"}, we provide some preliminaries. In Section [3](#3){reference-type="ref" reference="3"}, we establish the coercivity of the Hessian $S_\lambda''(\phi_{\lambda})$ and control the modulation parameters. In Section [4](#4){reference-type="ref" reference="4"}, we demonstrate the localized virial identities and define the monotonicity functional. In Section [5](#5){reference-type="ref" reference="5"}, we establish the monotonicity of the functional obtained in Section [4](#4){reference-type="ref" reference="4"} and prove the main theorem. Finally, in Appendix [6](#appendix){reference-type="ref" reference="appendix"}, we present a general coercivity property of the Hessian of the action $S_c''(\phi_{c})$ and the numerical result of the negative eigenfunction of $S_c''(\phi_{c})$. # Notations {#2} ## Notations {#notations} For $f, g\in L^2(\mathbb{R}),$ we define $$\langle f,g\rangle =\int_{\mathbb{R}}f(x)g(x)\,\mathrm{d}x$$ and regard $L^2(\mathbb{R})$ as a real Hilbert Space. For a function $f(x)$, its $L^{q}$-norm $\|f\|_{q} =\left(\int_{\mathbb{R}}|f(x)|^{q} \,\mathrm{d}x\right)^{\frac{1}{q}}$ and its $H^{1}$-norm $\|f\|_{H^{1}}=(\|f\|^{2}_{L^{2}}+\|\partial_x f\|^{2}_{L^{2}})^\frac{1}{2}.$ Further, we write $X\lesssim Y$ or $Y\lesssim X$ to indicate $X\leqslant CY$ for some constant $C>0.$ We use the notation $X\sim Y$ to denote $X\lesssim Y\lesssim X.$ We also use $O(Y)$ to denote any quantity $X$ such that $|X|\lesssim Y$ and use $o(Y)$ to denote any quantity $X$ such that $X/Y\rightarrow 0$ if $Y\rightarrow 0.$ Throughout the whole paper, the letter $C$ will denote various positive constants which are of no importance in our analysis. ## Some basic definitions and properties In the rest of this paper, we consider the case of $p>4,$ and $c=c_{0}(p). %=\frac{p}{4+2p}\left(1+\sqrt{2+\frac{1}{2}p}\right).$ Recall the expression of conserved equality and the functional $S_{c}$, we have $$\begin{aligned} E(u) &=\frac{1}{2}\int_{\mathbb{R}} u^{2}\,\mathrm{d}x+\frac{1}{p+2}\int_{\mathbb{R}}|u|^{p+2} \,\mathrm{d}x, \notag\\ Q(u) &=\frac{1}{2}\int_{\mathbb{R}}\big( u^{2}+u_{x}^{2}\big) \,\mathrm{d}x; \notag \\ \label{2.5} S_{c}(u)&=E(u)-cQ(u). \end{aligned}$$ Taking derivative, then we have $$\begin{aligned} \label{2.6} E'(u) &=u+|u|^{p}u,\\ \label{2.7} Q'(u) &=u-\partial_{xx}u,\\ S_{c}'(u) &=E'(u)-cQ'(u)=c\partial_{xx} u +(1-c)u+|u|^{p}u.\notag\end{aligned}$$ Note that $S_{c}'(\phi_{c})=0.$ Moreover, for the real-valued function $f,$ a direct computation shows $$\begin{aligned} \label{2.8} S_{c}''(\phi_{c})f =c\partial_{xx} f+(1-c)f+(p+1)\phi_{c}^{p}f.\end{aligned}$$ Taking the derivative of $S_{c}'\big(\phi_{c}(\cdot-x)\big)=0$ with respect to $x$ gives $$\begin{aligned} \label{2.9} S_{c}''(\phi_{c})(\partial_x \phi_{c})=0.\end{aligned}$$ For any function $\xi, \eta,$ we have $$\begin{aligned} \label{2.10} \langle S_{c}''(\phi_{c})\xi, \eta\rangle =\langle S_{c}''(\phi_{c})\eta, \xi\rangle.\end{aligned}$$ Moreover, taking the derivative of $S_{c}'(\phi_{c})=0$ with respect to $c$ gives $$\begin{aligned} \label{2.11} S_{c}''(\phi_{c})\partial_{c}\phi_{c} =Q'(\phi_{c}).\end{aligned}$$ Next, we give some basic properties on the momentum, energy and the functional $S_c$. **Lemma 5**. *Let $c=c_{0}(p);$ then the following equality holds: $$\begin{aligned} \partial_{c}Q\left(\phi_{c}\right)\Big|_{c=c_{0}(p)}=0. \end{aligned}$$* *Proof.* Note that $$\begin{aligned} Q(u) =\frac{1}{2}\int_{\mathbb{R}} u^{2}+u_{x}^{2} \,\mathrm{d}x. \end{aligned}$$ Taking inner product of [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"} and $\phi_{c}, x\partial_{x}\phi_{c}$ respectively, by integration-by-parts, we can get $$\begin{aligned} &c\big\|\partial_{x}\phi_{c}\big\|^{2}_{L^{2}}+(c-1)\big\|\phi_{c}\big\|^{2}_{L^{2}}-\big\|\phi_{c}\big\|^{p+2}_{L^{p+2}}=0,\\ &c\big\|\partial_{x}\phi_{c}\big\|^{2}_{L^{2}}-(c-1)\big\|\phi_{c}\big\|^{2}_{L^{2}}+\frac{2}{p+2}\big\|\phi_{c}\big\|^{p+2}_{L^{p+2}}=0.\end{aligned}$$ This gives that $$\begin{aligned} \label{2.1} \big\|\phi_{c}\big\|^{p+2}_{L^{p+2}} =\frac{2(p+2)(c-1)}{p+4}\big\|\phi_{c}\big\|^{2}_{L^{2}}; \quad %\label{2.2} \big\|\partial_{x}\phi_{c}\big\|^{2}_{L^{2}} =\frac{p(c-1)}{(p+4)c}\big\|\phi_{c}\big\|^{2}_{L^{2}}.\end{aligned}$$ This further yields that $$Q(\phi_{c}) =\frac{1}{2} \left[1+\frac{p(c-1)}{(p+4)c} \right]\big\| \phi_{c}\big\| ^{2}_{L^{2}}.$$ By scaling, we find $$\begin{aligned} \label{2.3} \big\| \phi_{c}\big\| ^{2}_{L^{2}} &=c^{\frac{1}{2}}(c-1)^{\frac{2}{p}-\frac{1}{2}}\big\|\psi_{0}\big\|^{2}_{L^{2}},\end{aligned}$$ where $\psi_0$ is the solution of $$\begin{aligned} -\partial_{xx}\psi_0+\psi_0-\psi_0^{p+1}=0.\end{aligned}$$ Hence, $$\begin{aligned} \label{2.4} \partial_{c}\big\| \phi_{c}\big\| ^{2}_{L^{2}} =\frac{4c-p}{2pc(c-1)}\big\| \phi_{c}\big\| ^{2}_{L^{2}}.\end{aligned}$$ By a straightforward computation, we have $$\begin{aligned} \label{partial_c_Q(phi_c)} \partial_{c} Q(\phi_{c}) =\frac{8(p+2)c^{2}-8pc-p^{2}}{4p(p+4)c^2(c-1)} \big\|\phi_{c}\big\|^{2}_{L^{2}}. \end{aligned}$$ Finally, we substitute $c=c_{0}(p)$ into the equality above, and thus we complete the proof. ◻ Then a consequence of Lemma [Lemma 5](#lem2.1){reference-type="ref" reference="lem2.1"} is **Corollary 6**. *Let $\lambda>1, c=c_{0}(p)$, then $$\begin{aligned} S_{\lambda}(\phi_{\lambda})-S_{\lambda}(\phi_{c}) =o((\lambda-c)^{2}).\end{aligned}$$* *Proof.* From the definition of $S_{c}$ in [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"}, we have $$\begin{aligned} \label{2.14} S_{\lambda}(\phi_{\lambda})-S_{\lambda}(\phi_{c}) =S_{c}(\phi_{\lambda})-S_{c}(\phi_{c}) -(\lambda-c)\big[Q(\phi_{\lambda})-Q(\phi_{c})\big]. \end{aligned}$$ Recall that $S_{c}'(\phi_{c})=0,$ then we use Taylor's expansion to calculate $$\begin{aligned} \label{2.15} S_{\lambda}(\phi_{\lambda})-S_{\lambda}(\phi_{c}) =\frac{1}{2}\langle S_{c}''(\phi_{c})(\phi_{\lambda}-\phi_{c}), (\phi_{\lambda}-\phi_{c})\rangle -(\lambda-c)\big[Q(\phi_{\lambda})-Q(\phi_{c})\big] +o((\lambda-c)^{2}). \end{aligned}$$ Note that $$\begin{aligned} \phi_{\lambda}-\phi_{c} =(\lambda-c)\partial_{c} \phi_{c}+o(\lambda-c), \end{aligned}$$ then we find $$\begin{aligned} \langle S_{c}''(\phi_{c})(\phi_{\lambda}-\phi_{c}), (\phi_{\lambda}-\phi_{c})\rangle &=(\lambda-c)^{2}\langle S_{c}''(\phi_{c})\partial_{c} \phi_{c}, \partial_{c}\phi_{c}\rangle +o((\lambda-c)^{2})\\ &=(\lambda-c)^{2} \langle Q'(\phi_{c}), \partial_{c}\phi_{c}\rangle +o((\lambda-c)^{2})\\ &=(\lambda-c)^{2}\cdot\partial_{c} Q(\phi_{c})\big|_{c=c_{0}(p)} +o\big((\lambda-c)^{2}\big),\end{aligned}$$ where we used [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"} in the second step. Using Lemma [Lemma 5](#lem2.1){reference-type="ref" reference="lem2.1"}, we have $$\begin{aligned} \partial_{c}Q(\phi_{c})|_{c=c_{0}(p)}=0. \end{aligned}$$ Hence, $$\begin{aligned} Q(\phi_{\lambda})-Q(\phi_{c})=o(\lambda-c), \end{aligned}$$ and $$\begin{aligned} \langle S_{c}''(\phi_{c})(\phi_{\lambda}-\phi_{c}), (\phi_{\lambda}-\phi_{c})\rangle =o\left((\lambda-c)^{2}\right). \end{aligned}$$ Taking these two results into [\[2.15\]](#2.15){reference-type="eqref" reference="2.15"}, we obtain the desired estimate. ◻ The next lemma gives pairs of pre-image and image of $S_{c}''(\phi_{c}).$ **Lemma 7**. *It holds that $$\begin{aligned} \label{3.5} S_{c}''(\phi_{c})(x\partial_{x}\phi_{c}) &=2c\partial_{xx}\phi_{c}. \end{aligned}$$ Moreover, let $\psi_{\omega}$ be defined in [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"}, and denote that $$\Psi_{c}=-\frac{1}{2}\omega^{1-\frac{2}{p}}\partial_{\omega} \psi_{\omega}.$$ Then $$\begin{aligned} \label{3.6} S_{c}''(\phi_{c})\Psi_{c} &=\phi_{c}. \end{aligned}$$* *Proof.* First, by the expression of $S_{c}''(\phi_{c})$ in [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"}, we have $$\begin{aligned} S_{c}''(\phi_{c})(x\partial_{x}\phi_{c}) &=c\partial_{xx}(x\partial_{x}\phi_{c})+(1-c)(x\partial_{x}\phi_{c})+(p+1)\phi_{c}^{p}(x\partial_{x}\phi_{c})\\ &=2c\partial_{xx}\phi_{c}+x\partial_{x}(c\partial_{xx}\phi_{c}+(1-c)\phi_{c}+\phi_{c}^{p+1})\\ &=2c\partial_{xx}\phi_{c}.\end{aligned}$$ So we obtain [\[3.5\]](#3.5){reference-type="eqref" reference="3.5"}. Second, taking the derivative of [\[1.5\]](#1.5){reference-type="eqref" reference="1.5"} with respect to $\omega,$ we have $$\begin{aligned} \label{3.7} -\partial_{xx}(\partial_{\omega} \psi_{\omega}) +(1-\omega^{2})\partial_{\omega}\psi_{\omega} -(p+1)\psi_{\omega}^{p}\partial_{\omega}\psi_{\omega} =2\omega \psi_{\omega}.\end{aligned}$$ Moreover, denoting that $$\begin{aligned} \label{2.12} L_{\omega}f =-\partial_{xx} f+(1-\omega^{2})f-(p+1)\psi_{\omega}^{p}f, \end{aligned}$$ then we have that $$\begin{aligned} \label{2.13} S_{c}''(\phi_{c})f=-cL_{\omega}f.\end{aligned}$$ Using the expression of $S_{c}''(\phi_{c})$ in [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"}, we have $$\begin{aligned} S_{c}''(\phi_{c})\Psi_{c} &=-cL_{\omega}\Psi_{c}\\ &=\frac{1}{2}\omega^{-1-\frac{2}{p}} \big[-\partial_{xx}(\partial_{\omega} \psi_{\omega}) +(1-\omega^{2})\partial_{\omega}\psi_{\omega} -(p+1)\psi_{\omega}^{p}\partial_{\omega}\psi_{\omega}\big]. \end{aligned}$$ This combined with [\[3.7\]](#3.7){reference-type="eqref" reference="3.7"} and [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} gives $$\begin{aligned} S_{c}''(\phi_{c})\Psi_{c} =\omega^{-\frac{2}{p}}\psi_{\omega} =\phi_{c}. \end{aligned}$$ Thus we obtain the desired results. ◻ # Modulation and dynamic of the parameter {#3} Under the assumption [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}, in order to obtain a contradiction, we assume that the solitary waves solution is stable, that is: for any $\varepsilon>0$, there exists $\delta>0$ such that when $$\begin{aligned} \big\|u_{0}-\phi_{c}\big\|_{H^1(\mathbb{R})}<\delta, \end{aligned}$$ we have $$\begin{aligned} \label{4.1} u \in U_{\varepsilon}(\phi_{c}). \end{aligned}$$ **Proposition 8**. *Let $c=c_{0}(p).$ Suppose that $u(t) \in U_{\varepsilon}(\phi_{c})$ for any $t\in \mathbb{R}$. Then there exist $C^1$ functions $$\begin{aligned} y: \mathbb{R}\rightarrow \mathbb{R}, \quad \lambda: \mathbb{R}\rightarrow \mathbb{R}^{+} \end{aligned}$$ such that for $$\begin{aligned} \label{xi} \xi(t)=u(t, \cdot +y(t))-\phi_{\lambda(t)}, \end{aligned}$$ the following orthogonality conditions hold: $$\begin{aligned} \label{orth} \langle \xi, \partial_{x}\phi_{\lambda(t)}\rangle =\langle \xi, \kappa_{\lambda(t)}\rangle =0, \end{aligned}$$ where $$\begin{aligned} \label{kappa_c} \kappa_{\lambda}=S_\lambda''(\phi_{\lambda})\Gamma_{\lambda}, \end{aligned}$$ and $\xi$ lies in the positive direction of $S_\lambda''(\phi_{\lambda}),$ that is, $$\begin{aligned} \label{posi} \langle S_\lambda''(\phi_{\lambda})\xi, \xi\rangle \gtrsim \|\xi\|^{2}_{H^{1}}. \end{aligned}$$ Furthermore, the following estimate holds: $$\begin{aligned} \label{vare} \|\xi\|_{H^1(\mathbb{R})}+|\lambda-c| \lesssim \varepsilon. \end{aligned}$$* *Proof.* From Proposition [Proposition 22](#prop4.1){reference-type="ref" reference="prop4.1"}, we first verify $\kappa_{c}$ satisfying [\[Assume-Modulation\]](#Assume-Modulation){reference-type="eqref" reference="Assume-Modulation"}. By [\[2.11\]](#2.11){reference-type="eqref" reference="2.11"} and Lemma [Lemma 5](#lem2.1){reference-type="ref" reference="lem2.1"}, we have $$\begin{aligned} \langle S''_c(\phi_{c})\partial_c\phi_{c}, \partial_c\phi_{c}\rangle=\frac{d}{dc}Q(\phi_{c})\Big|_{c=c_0(p)}=0.\end{aligned}$$ Then by Corollary [Corollary 21](#cor:a.4){reference-type="ref" reference="cor:a.4"}, we obtain $$\begin{aligned} \langle\partial_{c}\phi_{c}, \kappa_{c}\rangle \neq 0.\end{aligned}$$ Therefore, there exists $\varepsilon_{0}>0$ such that for $\varepsilon \in (0, \varepsilon_{0}), u\in U_{\varepsilon}(\phi_{c}),$ there exists unique $C^{1}$-functions $$\begin{aligned} y: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}, \quad \lambda: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}^{+}, \end{aligned}$$ such that $$\begin{aligned} \langle \xi, \partial_{x}\phi_{\lambda}\rangle =\langle \xi, \kappa_{\lambda}\rangle =0.\end{aligned}$$ By [\[second-orth-condition\]](#second-orth-condition){reference-type="eqref" reference="second-orth-condition"}, we have that $\Gamma_{\lambda}$ satisfying [\[Assume-Coer\]](#Assume-Coer){reference-type="eqref" reference="Assume-Coer"}. From Proposition [Proposition 20](#prop3.4){reference-type="ref" reference="prop3.4"}, we obtain [\[posi\]](#posi){reference-type="eqref" reference="posi"}. Furthermore, $$\begin{aligned} \left( \begin{array}{cccc} \partial_{u}\lambda & \partial_{v}\lambda \\ \partial_{u}y & \partial_{v}y \end{array} \right) =J^{-1} \left( \begin{array}{cccc} \partial_{u}F_{1} & \partial_{v}F_{1} \\ \partial_{u}F_{2} & \partial_{v}F_{2} \end{array} \right). \end{aligned}$$ This implies that $$\begin{aligned} |\lambda-c| \lesssim \big\|u-\phi_{c}\big\|_{H^1(\mathbb{R})}<\varepsilon.\end{aligned}$$ This finishes the proof of the proposition. ◻ Some consequences of Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"} are the follows. The first one is the rough estimate on $\dot y$ and $\dot \lambda.$ **Corollary 9**. *Let $u$ be the solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"} with $u\in U_{\varepsilon}(\phi_{c}),$ where $\varepsilon$ is obtained in Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"}. Let $y, \lambda, \xi$ be the parameters and function obtained in Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"}, then $$\begin{aligned} \dot{y} -\lambda =O\left(\|\xi\|_{H^1(\mathbb{R})}\right) \end{aligned}$$ and $$\begin{aligned} \dot{\lambda}=O\left(\|\xi\|_{H^1(\mathbb{R})}\right). \end{aligned}$$* *Proof.* Recall the definition $\xi(t)=u(t, \cdot +y(t))-\phi_{\lambda(t)}$ in [\[xi\]](#xi){reference-type="eqref" reference="xi"}, that is $$\begin{aligned} \label{5.3} u(t, x)=\phi_{\lambda}(x-y(t))+\xi(t,x-y(t)). \end{aligned}$$ Taking the derivative of [\[5.3\]](#5.3){reference-type="eqref" reference="5.3"} with respect to $t,$ we have $$\begin{aligned} \label{ut1} u_t=\dot{\xi}+\dot{\lambda}\partial_{\lambda}\phi_{\lambda}-\dot{y}\partial_{x}(\phi_{\lambda}+\xi). \end{aligned}$$ Inserting [\[ut1\]](#ut1){reference-type="eqref" reference="ut1"} into equation [\[equa-hami\]](#equa-hami){reference-type="eqref" reference="equa-hami"}, we get $$\begin{aligned} \label{ut2} \dot{\xi}+\dot{\lambda}\partial_{\lambda}\phi_{\lambda}-\dot{y}\partial_{x}(\phi_{\lambda}+\xi)=JE'(\phi_{\lambda}+\xi), \end{aligned}$$ where we note that $J=-(1-\partial_{x}^2)^{-1}\partial_{x}.$ Adding $\lambda\partial_{x}(\phi_{\lambda}+\xi)$ to both sides of [\[ut2\]](#ut2){reference-type="eqref" reference="ut2"}, we have $$\begin{aligned} \dot{\xi}+\dot{\lambda}\partial_{\lambda}\phi_{\lambda}-(\dot{y}-\lambda)\partial_{x}(\phi_{\lambda}+\xi) &=JE'(\phi_{\lambda}+\xi)+\lambda\partial_{x}(\phi_{\lambda}+\xi) \notag\\ &=J\left[E'(\phi_{\lambda}+\xi)-\lambda(1-\partial_{x}^2)(\phi_{\lambda}+\xi)\right].\end{aligned}$$ We note that $(1-\partial_{x}^2)(\phi_{\lambda}+\xi)=Q'(\phi_{\lambda}+\xi),$ then the above equality can be rewritten as follows: $$\begin{aligned} \label{dynamic-1} \dot{\xi}+\dot{\lambda}\partial_{\lambda}\phi_{\lambda}-(\dot{y}-\lambda)\partial_{x}(\phi_{\lambda}+\xi) &=J\left[E'(\phi_{\lambda}+\xi)-\lambda Q'(\phi_{\lambda}+\xi)\right] \notag\\ &=JS'_{\lambda}(\phi_{\lambda}+\xi). \end{aligned}$$ Using Taylor's type expansion, we have $$\begin{aligned} \label{taylor-S'(phi-la+xi)} S'_{\lambda}(\phi_{\lambda}+\xi) &=S'_{\lambda}(\phi_{\lambda})+S''_{\lambda}(\phi_{\lambda})\xi+O\big(\xi^2\big) \notag\\ &=S''_{\lambda}(\phi_{\lambda})\xi+O\big(\xi^2\big),\end{aligned}$$ where we used $S'_{\lambda}(\phi_{\lambda})=0.$ Inserting [\[taylor-S\'(phi-la+xi)\]](#taylor-S'(phi-la+xi)){reference-type="eqref" reference="taylor-S'(phi-la+xi)"} into [\[dynamic-1\]](#dynamic-1){reference-type="eqref" reference="dynamic-1"}, we have $$\begin{aligned} \label{m5.3} \dot{\xi}+\dot{\lambda}\partial_{\lambda}\phi_{\lambda}-(\dot{y}-\lambda)\partial_{x}(\phi_{\lambda}+\xi)=JS''_{\lambda}(\phi_{\lambda})\xi+\mathcal{N}_1(\xi),\end{aligned}$$ where $\mathcal{N}_1(\xi)$ verifies $$\begin{aligned} \langle \mathcal{N}_1(\xi), f\rangle =O\left(\|\xi\|^{2}_{H^{1}}\|f\|_{H^1}\right) ,\quad \mbox{for any } f\in H^1. \end{aligned}$$ Taking inner product by [\[m5.3\]](#m5.3){reference-type="eqref" reference="m5.3"} and $\partial_{x}\phi_{\lambda}, \kappa_{\lambda}$ respectively, by integration-by-parts, we have $$\begin{aligned} \label{5.5} \langle \dot{\xi},\partial_{x}\phi_{\lambda} \rangle +\dot{\lambda} \langle \partial_{\lambda}\phi_{\lambda},\partial_{x}\phi_{\lambda} \rangle -(\dot{y}-\lambda) \langle \partial_{x}\left(\phi_{\lambda}+\xi\right),\partial_{x}\phi_{\lambda} \rangle &=\langle JS_{\lambda}''(\phi_{\lambda})\xi,\partial_{x}\phi_{\lambda} \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right), \\ \label{dynamic_phi_lambda} \langle \dot{\xi}, \kappa_{\lambda} \rangle +\dot{\lambda} \langle \partial_{\lambda}\phi_{\lambda}, \kappa_{\lambda} \rangle -(\dot{y}-\lambda) \langle \partial_{x}\left(\phi_{\lambda}+\xi\right), \kappa_{\lambda} \rangle &=\langle JS_{\lambda}''(\phi_{\lambda})\xi,\kappa_{\lambda} \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right). \end{aligned}$$ By the even property of $\phi_{\lambda},$ it is known that $\Gamma_{c}=B(c)\left(c^2\Psi_{c}+\frac{c}{2}x\partial_{x}\phi_{c}+c\phi_{c}\right)+D(c)(3x^2\phi_{c}+x^3\partial_{x}\phi_c)$ is an even function. Moreover, we note that $\kappa_{\lambda}$ is also an even function since $S_{\lambda}''(\phi_{\lambda})\Gamma_{\lambda}$ has the same parity as $\Gamma_{\lambda}$. Using orthogonality conditions in [\[orth\]](#orth){reference-type="eqref" reference="orth"}, we simplify [\[5.5\]](#5.5){reference-type="eqref" reference="5.5"} and [\[dynamic_phi_lambda\]](#dynamic_phi_lambda){reference-type="eqref" reference="dynamic_phi_lambda"} as $$\begin{aligned} \label{sim_parameter1} -\dot{\lambda}\langle \xi, \partial_{\lambda}\partial_{x}\phi_{\lambda}\rangle -(\dot{y}-\lambda)\Big(\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}-\langle\xi, \partial_{xx}\phi_{\lambda}\rangle\Big) & =-\langle \xi, S_{\lambda}''(\phi_{\lambda})\big(J\partial_{x}\phi_{\lambda}\big) \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right), \\ \label{sim_parameter2} \dot{\lambda} \left[ -\langle \xi, \partial_{\lambda}\kappa_{\lambda} \rangle +\langle\partial_{\lambda}\phi_{\lambda}, \kappa_{\lambda}\rangle \right] +(\dot{y}-\lambda)\langle \xi, \partial_{x}\kappa_{\lambda}\rangle & =-\langle \xi,S_{\lambda}''(\phi_{\lambda})\big(J\kappa_{\lambda}\big) \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right), \end{aligned}$$ where $\langle\partial_{\lambda}\phi_{\lambda}, \kappa_{\lambda}\rangle$ is a constant denoted by $C(\lambda)$ which only depends on $\lambda.$ The term $\dot{\lambda} \langle \partial_{\lambda}\phi_{\lambda},\partial_{x}\phi_{\lambda} \rangle$ vanishes as $\phi_{\lambda}$ is an even function. By [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"} we can simplify the third term on the left side of equality [\[5.5\]](#5.5){reference-type="eqref" reference="5.5"} as $$\begin{aligned} -(\dot{y}-\lambda) \langle \partial_{x}(\phi_{\lambda}+\xi),\partial_{x}\phi_{\lambda} \rangle &=-(\dot{y}-\lambda) \big(\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} -\langle \xi,\partial_{xx}\phi_{\lambda}\rangle \big) \\ &=-(\dot{y}-\lambda) \left[ \frac{p(\lambda-1)}{(p+4)\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} -\langle \xi,\partial_{xx}\phi_{\lambda}\rangle \right]. \end{aligned}$$ Using the property of $S''_{\lambda}(\phi_{\lambda})$ in [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"}, we have $$\begin{aligned} -\langle \partial_{x}S_{\lambda}''(\phi_{\lambda})\xi,(1-\partial_{x}^{2})^{-1}\partial_{x}\phi_{\lambda} \rangle =\langle \xi, S_{\lambda}''(\phi_{\lambda})(1-\partial_{x}^{2})^{-1}\partial_{xx}\phi_{\lambda}\rangle. \end{aligned}$$ Thus we obtain $$\begin{aligned} \label{5.6} -\dot{\lambda}\langle \xi, \partial_{\lambda}\partial_{x}\phi_{\lambda} \rangle &-(\dot{y}-\lambda) \left[ \frac{p(\lambda-1)}{(p+4)\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} -\langle \xi,\partial_{xx}\phi_{\lambda}\rangle \right]\notag \\ &=\langle \xi, S_{\lambda}''(\phi_{\lambda})(1-\partial_{x}^{2})^{-1}\partial_{xx}\phi_{\lambda}\rangle +O(\|\xi\|^{2}_{H^{1}}). \end{aligned}$$ Using the similar method, taking inner product by [\[m5.3\]](#m5.3){reference-type="eqref" reference="m5.3"} and $\phi_{\lambda},$ by integration-by-parts and using the property of $S''_{\lambda}(\phi_{\lambda})$ in [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"}, we can get $$\begin{aligned} \label{5.7} \dot{\lambda} \left[ -\langle \xi, \partial_{\lambda}\phi_{\lambda} \rangle +\frac{4\lambda-p}{4p\lambda(\lambda-1)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} \right] =\langle \xi, S_{\lambda}''(\phi_{\lambda})(1-\partial_{x}^{2})^{-1}\partial_{x}\phi_{\lambda}\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right). \end{aligned}$$ We denote $$\begin{aligned} A= \begin{pmatrix} &-\langle \xi, \partial_{\lambda}\partial_{x}\phi_{\lambda} \rangle &-\big\|\partial_{x}\phi_{\lambda}\big\|^2_{L^2} +\langle \xi,\partial_{xx}\phi_{\lambda}\rangle \\ &-\langle \xi, \partial_{\lambda}\kappa_{\lambda} \rangle +C(\lambda) &\langle \xi, \partial_{x}\kappa_{\lambda}\rangle \end{pmatrix}. \end{aligned}$$ Combining [\[sim_parameter1\]](#sim_parameter1){reference-type="eqref" reference="sim_parameter1"} and [\[sim_parameter2\]](#sim_parameter2){reference-type="eqref" reference="sim_parameter2"}, by a direct computation, we have $$\begin{aligned} \label{5.8} \begin{pmatrix} \dot{\lambda} \\ \dot{y}-\lambda \end{pmatrix} &=A^{-1} \begin{pmatrix} &-\langle \xi, S_{\lambda}''(\phi_{\lambda})\big(J\partial_{x}\phi_{\lambda}\big)\rangle \\ &-\langle \xi, S_{\lambda}''(\phi_{\lambda})\big(J\kappa_{\lambda}\big)\rangle \end{pmatrix} +\begin{pmatrix} O\left(\|\xi\|^2_{H^{1}}\right) \\ O\left(\|\xi\|^2_{H^{1}}\right) \end{pmatrix} \notag\\ &=\begin{pmatrix} O\left(\|\xi\|_{H^{1}}\right) \\ O\left(\|\xi\|_{H^{1}}\right) \end{pmatrix}. \end{aligned}$$ Thus we obtain the desired results. ◻ The second is a precise estimate on the spatial transform parameter $y(t)$. **Corollary 10**. *Under the same assumption as in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"}; let $f_\lambda=x^3\phi_{\lambda},$ then $$\begin{aligned} \label{estimate_doty-lambda} \dot{y}-\lambda &=\frac{1}{B(\lambda)}\langle \xi, S''_{\lambda}(\phi_{\lambda})\partial_{x}f_\lambda\rangle -\frac{1}{B(\lambda)}\partial_{t}\langle \xi, (1-\partial_{x}^2)f_\lambda\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right),\end{aligned}$$ where $$\begin{aligned} B(\lambda) =\frac{3}{2}\big\|x\phi_{\lambda}\big\|^{2}_{L^{2}}+\frac{9}{2}\big\|x\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}-3\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$* *Proof.* Taking inner product by [\[m5.3\]](#m5.3){reference-type="eqref" reference="m5.3"} and $(1-\partial_{x}^2)f_\lambda,$ by integration-by-parts, we have $$\begin{aligned} \label{dynamic_doty} \langle (1-\partial_{x}^{2})\dot{\xi}, f_\lambda\rangle +\dot{\lambda}\langle (1-\partial_{x}^{2})\partial_{\lambda}\phi_{\lambda}, f_\lambda\rangle -(\dot{y}-\lambda)&\langle (1-\partial_{x}^{2})\partial_{x}\phi_{\lambda}, f_\lambda\rangle\notag\\ &=-\langle \partial_{x}\left(S_{\lambda}''(\phi_{\lambda})\xi\right), f_\lambda \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right). \end{aligned}$$ It's worth noting that $f_\lambda=x^3\phi_{\lambda}\in L^2(\mathbb{R})$ since $\phi_{\lambda}$ is exponential decaying. Now we consider terms in [\[dynamic_doty\]](#dynamic_doty){reference-type="eqref" reference="dynamic_doty"} one by one. First, from the rough estimate $\dot{\lambda}=O\left(\|\xi\|_{H^{1}}\right)$ in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"}, we have $$\begin{aligned} \label{first-oder-term} \langle (1-\partial_{x}^{2})\dot{\xi}, f_\lambda\rangle &=\partial_{t}\langle \xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle -\dot{\lambda}\langle \xi, (1-\partial_{x}^{2})\left(x^3\partial_{\lambda}\phi_{\lambda}\right)\rangle\notag\\ &=\partial_{t}\langle \xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right). \end{aligned}$$ The term $\dot{\lambda}\langle (1-\partial_{x}^{2})\partial_{\lambda}\phi_{\lambda}, x^3\phi_{\lambda}\rangle$ vanishes since $\phi_{\lambda}$ is an even function. Then, direct calculation gives that $$\begin{aligned} \label{cofficient_doty-lam} -(\dot{y}-\lambda)\langle (1-\partial_{x}^{2})\partial_{x}\phi_{\lambda}, x^3\phi_{\lambda}\rangle &=(\dot{y}-\lambda)\langle (1-\partial_{x}^{2})\phi_{\lambda}, 3x^2\phi_{\lambda}+x^3\partial_{x}\phi_{\lambda}\rangle\notag\\ &=(\dot{y}-\lambda)\left[\frac{3}{2}\big\|x\phi_{\lambda}\big\|^{2}_{L^{2}}+\frac{9}{2}\big\|x\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}-3\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}\right].\end{aligned}$$ Using the property of $S_\lambda''(\phi_{\lambda})$ in [\[2.10\]](#2.10){reference-type="eqref" reference="2.10"}, we have $$\begin{aligned} \label{S''partial_xf} -\langle \partial_x\left(S''_{\lambda}(\phi_{\lambda})\xi\right), f_\lambda\rangle &=\langle S''_{\lambda}(\phi_{\lambda})\xi, \partial_xf_\lambda\rangle \notag\\ &=\langle \xi, S''_{\lambda}(\phi_{\lambda})\left(3x^2\phi_{\lambda}+x^3\partial_{x}\phi_{\lambda}\right) \rangle.\end{aligned}$$ Combining [\[first-oder-term\]](#first-oder-term){reference-type="eqref" reference="first-oder-term"}--[\[S\'\'partial_xf\]](#S''partial_xf){reference-type="eqref" reference="S''partial_xf"}, and thus we complete the proof. ◻ # The localized virial identity {#4} The following lemma is the localized virial identity. Let $y, \lambda, \xi$ be the parameters and function obtained in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"}, $f_\lambda, B(\lambda)$ are the same as Corollary [Corollary 10](#cor:5.2){reference-type="ref" reference="cor:5.2"}. Denote $$\begin{aligned} \label{defi-H(u)} H(u)=-(1-\partial_{x}^2)^{-1}(u+|u|^{p}u). \end{aligned}$$ From the equation [\[equa-hami\]](#equa-hami){reference-type="eqref" reference="equa-hami"}, we obtain that $\partial_{x}H(u)=u_{t}.$ Inserting the expression of $u$ in [\[5.3\]](#5.3){reference-type="eqref" reference="5.3"} into [\[defi-H(u)\]](#defi-H(u)){reference-type="eqref" reference="defi-H(u)"}, we have $$\begin{aligned} H(u)=-(1-\partial_{x}^2)^{-1}\big(\phi_{\lambda}+\xi+|\phi_{\lambda}+\xi|^{p}(\phi_{\lambda}+\xi)\big).\end{aligned}$$ Noting that $\phi_{\lambda}$ satisfies $$\begin{aligned} \label{equa-lambda} -\lambda\partial_{xx}\phi_{\lambda}+(\lambda-1)\phi_{\lambda}-\phi_{\lambda}^{p+1}=0,\end{aligned}$$ and $$\begin{aligned} \label{S''la} S''_{\lambda}(\phi_{\lambda})f=\lambda\partial_{xx}f+(1-\lambda)f+(p+1)\phi_{\lambda}^{p}f.\end{aligned}$$ From [\[equa-lambda\]](#equa-lambda){reference-type="eqref" reference="equa-lambda"} and [\[S\'\'la\]](#S''la){reference-type="eqref" reference="S''la"}, we obtain that $$\begin{aligned} \label{H(u)} H(u) =-\lambda(\phi_{\lambda}+\xi)-(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)+(1-\partial_{x}^2)^{-1}\mathcal{N}_2(\xi),\end{aligned}$$ where $\mathcal{N}_2(\xi)$ has the same property as $\mathcal{N}_1(\xi)$ in [\[m5.3\]](#m5.3){reference-type="eqref" reference="m5.3"} which verifies that $$\begin{aligned} \langle \mathcal{N}_2(\xi), f\rangle =O\left(\|\xi\|^{2}_{H^{1}}\|f\|_{H^1}\right),\quad \mbox{for any } f\in H^1(\mathbb{R}).\end{aligned}$$ We also denote that $$\begin{aligned} &I_{1}(t)=\int_{\mathbb{R}} \varphi(x-y(t))(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}) \,\mathrm{d}x,\\ &I_{2}(t)=\frac{D(\lambda)}{B(\lambda)}\langle\xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle, \end{aligned}$$ where $$\begin{aligned} D(\lambda)&=-\frac{4p\lambda+4\lambda-3p}{2(p+4)}\big\|\phi_{\lambda}\big\|^2_{L^2}, \\ B(\lambda)&=\frac{3}{2}\big\|x\phi_{\lambda}\big\|^2_{L^{2}}+\frac{9}{2}\big\|x\partial_x\phi_{\lambda}\big\|^{2}_{L^2}-3\big\|\phi_{\lambda}\big\|^2_{L^{2}}.\end{aligned}$$ **Lemma 11**. *Let $\varphi \in C^{3}(\mathbb{R}), u\in H^1(\mathbb{R})$ be the solution of [\[1.1\]](#1.1){reference-type="eqref" reference="1.1"}, then $$\begin{aligned} I_{1}'(t) &= -\dot{y}\int_{\mathbb{R}} \varphi'(x-y(t))\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right)\,\mathrm{d}x +\frac{1}{2}\int_{\mathbb{R}}\varphi'(x-y(t)) \left[\big(H(u)\big)^{2}-u_{t}^{2}\right] \,\mathrm{d}x, \\ I_{2}'(t) &=\frac{D(\lambda)}{B(\lambda)}\partial_{t}\langle\xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle +O\big(\|\xi\|_{H^1}^2\big).\end{aligned}$$* *Proof.* First, a direct computation gives that $$\begin{aligned} I_{1}'(t)= -\dot{y}\int_{\mathbb{R}} \varphi'(x-y(t))\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right)\,\mathrm{d}x +\int_{\mathbb{R}} \varphi(x-y(t))\partial_{t}\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right)\,\mathrm{d}x.\end{aligned}$$ Multiplying [\[equa-hami\]](#equa-hami){reference-type="eqref" reference="equa-hami"} by $u+|u|^{p}u$ gives: $$\begin{aligned} \partial_{t}\big(\frac{1}{2}u^2+\frac{1}{p+2}|u|^{p+2}\big) =-(1-\partial_{x}^{2})^{-1}\partial_{x}\big(u+|u|^{p}u \big)\cdot\big(u+|u|^{p}u \big).\end{aligned}$$ Further, noting that $$u+|u|^{p}u=-(1-\partial_x^2)H(u)=-H(u)+\partial_{t}\partial_{x}u,$$ we get that $$\begin{aligned} \partial_{t}\big(\frac{1}{2}u^2+\frac{1}{p+2}|u|^{p+2}\big) &=\partial_{x}H(u)\cdot\big(-H(u)+\partial_{t}\partial_{x}u\big) \notag\\ &=-\frac{1}{2}\partial_{x}\big[\big(H(u)\big)^{2}-u_t^{2}\big],\end{aligned}$$ where we used $\partial_{x}H(u)=u_t$ in the last step. Then by integration-by-parts, we obtain that $$\begin{aligned} I_{1}'(t)= &-\dot{y}\int_{\mathbb{R}} \varphi'(x-y(t))\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right)\,\mathrm{d}x +\frac{1}{2}\int_{\mathbb{R}}\varphi'(x-y(t)) \left[\big(H(u)\big)^{2}-u_{t}^{2}\right]\,\mathrm{d}x.\end{aligned}$$ Second, a direct computation shows that $$\begin{aligned} I_{2}'(t)= &\dot{\lambda}\partial_{\lambda}\Big[\frac{D(\lambda)}{B(\lambda)}\Big] \langle \xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle +\frac{D(\lambda)}{B(\lambda)}\partial_{t}\langle\xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle +O\big(\|\xi\|_{H^1}^2\big) \\ =&\frac{D(\lambda)}{B(\lambda)}\partial_{t}\langle\xi, (1-\partial_{x}^{2})(x^3\phi_{\lambda})\rangle +O\big(\|\xi\|_{H^1}^2\big),\end{aligned}$$ here we used the estimate of $\dot{\lambda}=O\big(\|\xi\|_{H^1}\big)$ in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"} in the last step. This completes the proof. ◻ # The monotonic functional {#5} This section is devoted to prove our main theorem. ## Virial identities Let $\varphi(x)$ be a smooth cutoff function, where $$\begin{aligned} \label{7.1} \varphi(x)=\left\{ \begin{aligned} &x, \qquad |x|\leqslant R, \\ &0, \qquad |x|\geqslant 2R, \end{aligned} \right.\end{aligned}$$ $0 \leqslant \varphi' \leqslant 1$ for any $x\in\mathbb{R},$ and $R$ is a large constant decided later. Moreover, we denote $$\begin{aligned} I(t)=I_{1}(t)+I_{2}(t).\end{aligned}$$ Then we have the following lemma. **Lemma 12**. *Let $R>0, y, \lambda,\xi$ be the parameters and function obtained in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"}. Then* *$$\begin{aligned} %\label{7.2} I_1'(t) =&-\lambda E(u_{0}) +\frac{1}{2}\lambda^{2} \left( \big\|\phi_{\lambda}\big\|^{2}_{L^{2}}-\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} \right) \label{I1-1} \\ &+\lambda^{2}\langle\xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle +\lambda \langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda} \rangle -(\dot{y}-\lambda)\Big[E(u_0)+2\lambda \big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}\Big]\\ % -\frac{1}{B(\lambda)}\partial_{t}\Big(\big\langle \xi, (1-\partial_{x}^2)f\big\rangle\Big) \label{I1-2}\\ &+O\left(\frac{1}{R}+\|\xi\|^{2}_{H^{1}}\right).\notag\end{aligned}$$* *Proof.* From [\[lem6.1\]](#lem6.1){reference-type="eqref" reference="lem6.1"} and the conversation law of energy, we change the form of $I_{1}'(t)$ as $$\begin{aligned} I_{1}'(t) = &-\dot{y}\int_{\mathbb{R}} \varphi'(x-y(t))\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right)\,\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}} \varphi'(x-y(t)) \left[\big(H(u)\big)^{2}-u_{t}^{2}\right]\,\mathrm{d}x \notag\\ = &-\dot{y}E(u_{0}) +\frac{1}{2}\int_{\mathbb{R}} \left[\big(H(u)\big)^{2}-u_{t}^{2}\right]\,\mathrm{d}x+R(u),\end{aligned}$$ where $$\begin{aligned} \label{7.3} R(u) &=\int_{\mathbb{R}}\left[1-\varphi'(x-y(t))\right] \left[ \dot{y}\left(\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right) -\frac{1}{2}\left(H(u)\right)^{2} +\frac{1}{2}u_{t}^{2} \right]\,\mathrm{d}x.\end{aligned}$$ Then we need to consider the terms $\frac{1}{2}\int_{\mathbb{R}} \left[\big(H(u)\big)^{2}-u_{t}^{2}\right]\,\mathrm{d}x$ and $R(u).$ $\bullet$ *Estimate on $\frac{1}{2}\int_{\mathbb{R}} \left[(H(u))^{2}-u_{t}^{2}\right]\,\mathrm{d}x.$* Now we consider terms $\int_{\mathbb{R}} (H(u))^{2}\,\mathrm{d}x$ and $\int_{\mathbb{R}}u_{t}^{2}\,\mathrm{d}x$ respectively. We recall the expression of $H(u)$ in [\[H(u)\]](#H(u)){reference-type="eqref" reference="H(u)"}, we have $$\begin{aligned} %\label{-1u} \int_{\mathbb{R}} (H(u))^{2}\,\mathrm{d}x &=\int_{\mathbb{R}}\left[-\lambda(\phi_{\lambda}+\xi)-(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)+(1-\partial_{x}^2)^{-1}\mathcal{N}_2(\xi)\right]^2\,\mathrm{d}x \notag\\ &=\lambda^2\big\|\phi_{\lambda}+\xi\big\|^{2}_{L^{2}} +2\lambda\left\langle \phi_{\lambda}, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle \\ &\quad +2\lambda\left\langle \xi, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle +\big\|(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\big\|^{2}_{L^{2}} +O\big(\|\xi\|_{H^{1}}^2\big).\end{aligned}$$ Now we estimate the terms above one by one. \(i\) The term $\big\|\phi_{\lambda}+\xi\big\|^{2}_{L^{2}}$. $$\begin{aligned} \label{term1} \big\|\phi_{\lambda}+\xi\big\|^{2}_{L^{2}} =\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}+2\langle \xi, \phi_{\lambda}\rangle+O\big(\|\xi\|_{H^{1}}^2\big).\end{aligned}$$ \(ii\) The term $\left\langle \phi_{\lambda}, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle$. We recall [\[sim_parameter1\]](#sim_parameter1){reference-type="eqref" reference="sim_parameter1"} and insert the rough estimates of $\dot{\lambda}$ and $\dot{y}-\lambda$ obtained in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"} into [\[sim_parameter1\]](#sim_parameter1){reference-type="eqref" reference="sim_parameter1"}, we have that $$\begin{aligned} -(\dot{y}-\lambda)\big\|\partial_x\phi_{\lambda}\big\|^{2}_{L^{2}} =\langle \xi, S''_{\lambda}(\phi_{\lambda})\left[(1-\partial_{x}^2)^{-1}\partial_{xx}\phi_{\lambda}\right] \rangle +O\big(\|\xi\|_{H^{1}}^2\big).\end{aligned}$$ We note that $(1-\partial_{x}^2)^{-1}\partial_{xx}\phi_{\lambda}=-\phi_{\lambda}+(1-\partial_{x}^2)^{-1}\phi_{\lambda},$ then we have $$\begin{aligned} \label{1-partial_xx^{-1}} \left\langle \phi_{\lambda}, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle &=\left\langle \xi, S''_{\lambda}(\phi_{\lambda})\big((1-\partial_{x}^2)^{-1}\phi_{\lambda}\big)\right\rangle \notag\\ &=\left\langle \xi, S''_{\lambda}(\phi_{\lambda})\big((1-\partial_{x}^2)^{-1}\partial_{xx}\phi_{\lambda}\big)\right\rangle +\left\langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\right\rangle \notag\\ &= -(\dot{y}-\lambda)\big\|\partial_x\phi_{\lambda}\big\|^{2}_{L^{2}} +\langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\rangle +O\big(\|\xi\|_{H^{1}}^2\big).\end{aligned}$$ \(iii\) The term $\big\|(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\big\|^{2}_{L^{2}}$. Using the expression of $S''_{\lambda}(\phi_{\lambda})$ in [\[S\'\'la\]](#S''la){reference-type="eqref" reference="S''la"}, we have $$\begin{aligned} (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big) =-\lambda\xi+(1-\partial_{x}^2)^{-1}\left[\xi+(p+1)\phi_{\lambda}^{p}\xi\right]\end{aligned}$$ Thus by Young's inequality, we have $$\begin{aligned} \label{1-partial_xx-S''xi} \big\|(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\big\|^{2}_{L^{2}} &\lesssim \big\|\xi\big\|^{2}_{L^{2}} +\big\|(1-\partial_{x}^2)^{-1}\left[\xi+(p+1)\phi_{\lambda}^{p}\xi\right]\big\|^{2}_{L^{2}} \notag\\ &\lesssim \big\|\xi\big\|^{2}_{L^{2}}+\big\|\xi+(p+1)\phi_{\lambda}^{p}\xi\big\|^{2}_{L^{2}} \notag\\ &\lesssim \big\|\xi\big\|^{2}_{L^{2}},\end{aligned}$$ where we used $\phi_{\lambda}\in L^{\infty}(\mathbb{R})$ in the last step. \(iv\) The term $\left\langle \xi, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle$. By Hölder's inequality and [\[1-partial_xx-S\'\'xi\]](#1-partial_xx-S''xi){reference-type="eqref" reference="1-partial_xx-S''xi"}, we obtain that $$\begin{aligned} \label{xi-1-partial_xx-S''xi} \left\langle \xi, (1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\right\rangle &\lesssim \big\|\xi\big\|_{L^{2}} \cdot \big\|(1-\partial_{x}^2)^{-1}\big(S''_{\lambda}(\phi_{\lambda})\xi\big)\big\|_{L^{2}} \notag\\ &\lesssim \big\|\xi\big\|^2_{L^{2}}.\end{aligned}$$ Collecting all the estimates above, we obtain that $$\begin{aligned} \label{-1ut} \int_{\mathbb{R}}(H(u))^{2}\,\mathrm{d}x &=\lambda^{2}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} +2\lambda^{2}\langle \xi, \phi_{\lambda}\rangle +2\lambda \langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\rangle -2\lambda(\dot{y}-\lambda)\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} +O\left(\|\xi\|^{2}_{H^{1}}\right). \end{aligned}$$ Arguing similarly, taking the derivative of [\[H(u)\]](#H(u)){reference-type="eqref" reference="H(u)"} with respect to $x$, we have that $$\begin{aligned} \label{u_t} u_t &=-(1-\partial_{x}^2)^{-1}\partial_{x}(u+|u|^{p}u) \notag\\ &=-\lambda\partial_x(\phi_{\lambda}+\xi)-(1-\partial_{x}^2)^{-1}\partial_{x}\left[ S''_{\lambda}(\phi_{\lambda})\xi\right] +(1-\partial_{x}^2)^{-1}\partial_{x}\mathcal{N}_2(\xi).\end{aligned}$$ Repeating the process above, we obtain that $$\begin{aligned} \label{int_u_t^2} \int_{\mathbb{R}}u_{t}^{2}\,\mathrm{d}x =\lambda^{2}\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} -2\lambda^2\langle \xi, \partial_{xx}\phi_{\lambda} \rangle +2\lambda(\dot{y}-\lambda)\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} +O\left(\|\xi\|_{H^{1}}^2\right). \end{aligned}$$ From [\[-1ut\]](#-1ut){reference-type="eqref" reference="-1ut"} and [\[int_u\_t\^2\]](#int_u_t^2){reference-type="eqref" reference="int_u_t^2"}, we have $$\begin{aligned} \frac{1}{2}\int_{\mathbb{R}} \left[(H(u))^{2}-u_{t}^{2}\right]\,\mathrm{d}x &=\frac{1}{2}\lambda^{2} \left( \big\|\phi_{\lambda}\big\|^{2}_{L^{2}} -\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} \right) +\lambda^{2}\langle\xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle \notag\\ &\quad +\lambda \langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda} \rangle -2\lambda(\dot{y}-\lambda)\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} +O\left(\|\xi\|^{2}_{H^{1}}\right). \label{est:main-part1}\end{aligned}$$ $\bullet$*Estimate on $R(u)$.*\ Using the definition of the cutoff function $\varphi$ in [\[7.1\]](#7.1){reference-type="eqref" reference="7.1"}, we have $$\begin{aligned} \big|R(u)\big| &\leqslant \frac{1}{2}\Bigg|\int_{|x-y(t)|>R} \left[1-\varphi'(x-y(t))\right] \left[\big(H(u)\big)^{2}-u_{t}^{2}\right] \,\mathrm{d}x\Bigg|\\ &\quad +|\dot{y}| \Bigg|\int_{|x-y(t)|>R} \left[1-\varphi'(x-y(t))\right] \left[\frac{1}{2}u^{2}+\frac{1}{p+2}|u|^{p+2}\right]\,\mathrm{d}x \\ &\lesssim \int_{|x|>R} \left[u^{2}+|u|^{p+2}+\big|H(u)\big|^{2}+u_{t}^{2} \right]\,\mathrm{d}x.\end{aligned}$$ By Hölder's inequality, Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"}, [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}, [\[S\'\'la\]](#S''la){reference-type="eqref" reference="S''la"}, [\[-1ut\]](#-1ut){reference-type="eqref" reference="-1ut"} and [\[int_u\_t\^2\]](#int_u_t^2){reference-type="eqref" reference="int_u_t^2"}, we have $$\begin{aligned} \big|R(u)\big| &\lesssim \int_{|x|>R} \left[ (\phi_{\lambda}+\xi)^{2}+|\phi_{\lambda}+\xi|^{p+2} +|\phi_{\lambda}|^{2}+|\partial_{x}\phi_{\lambda}|^{2}] \right]\,\mathrm{d}x\\ &\qquad +\Big|\int_{|x|>R} \xi \cdot (\phi_{\lambda}+\partial_{xx}\phi_{\lambda}) \,\mathrm{d}x\Big| +O(\|\xi\|^{2}_{H^{1}}).\end{aligned}$$ Further, using the property of exponential decay of $\phi_{\lambda}, \partial_{xx}\phi_{\lambda}$ we have $$\begin{aligned} \int_{|x|>R} |\phi_{\lambda}|^{2}+|\partial_{x}\phi_{\lambda}|^{2} \,\mathrm{d}x \leqslant C\int_{|x|>R}e^{-C|x|}\,\mathrm{d}x \leqslant \frac{C}{R}. \end{aligned}$$ Then Young's inequality gives that $$\begin{aligned} \label{7.8} \int_{|x|>R}\xi \cdot (\phi_{\lambda}+\partial_{xx}\phi_{\lambda})\,\mathrm{d}x \lesssim \frac{1}{R}+\|\xi\|^{2}_{L^{2}},\\ \label{7.9} \int_{|x|>R} (\phi_{\lambda}+\xi)^{2} \,\mathrm{d}x \lesssim \frac{1}{R}+\|\xi\|^{2}_{L^{2}},\\ \label{7.10} \int_{|x|>R} (\phi_{\lambda}+\xi)^{p+2} \,\mathrm{d}x \lesssim \frac{1}{R}+\|\xi\|^{2}_{H^{1}}.\end{aligned}$$ Therefore, we combine [\[7.8\]](#7.8){reference-type="eqref" reference="7.8"}--[\[7.10\]](#7.10){reference-type="eqref" reference="7.10"} to obtain $$\begin{aligned} \label{est:remainder} \big|R(u)\big| \leqslant C\left(\frac{1}{R}+\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ Now inserting the estimates in [\[est:main-part1\]](#est:main-part1){reference-type="eqref" reference="est:main-part1"} and [\[est:remainder\]](#est:remainder){reference-type="eqref" reference="est:remainder"} into [\[7.3\]](#7.3){reference-type="eqref" reference="7.3"}, we give the desired estimate and thus complete the proof of the lemma. ◻ ## Structure of $I'(t)$ Denote $$\begin{aligned} \label{7.11} \beta(u_{0}) &=-\lambda\Big[E(u_{0})-E(\phi_{c})\Big], \\ \label{7.12} \gamma(\lambda) &=-\lambda E(\phi_{c}) +\frac{1}{2}\lambda^{2} \left(\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}-\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}\right), \\ \label{7.13} \tilde{R}(u) &=R(u)+\lambda^{2}\langle\xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle +\lambda \langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\rangle -(\dot{y}-\lambda)\left[E(u_0)+2\lambda\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}\right] \notag\\ &\quad +\frac{D(\lambda)}{B(\lambda)}\partial_{t}\langle \xi, (1-\partial_{x}^2)(x^3\phi_{\lambda})\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right),\end{aligned}$$ where $$\begin{aligned} D(\lambda)=-\frac{4p\lambda+4\lambda-3p}{2(p+4)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}, \quad B(\lambda)=\frac{3}{2}\big\|x\phi_{\lambda}\big\|^2_{L^{2}}+\frac{9}{2}\big\|x\partial_x\phi_{\lambda}\big\|^{2}_{L^2}-3\big\|\phi_{\lambda}\big\|^2_{L^{2}}.\end{aligned}$$ **Lemma 14**. *It holds that $$\begin{aligned} I'(t)=\beta(u_{0})+\gamma(\lambda)+\tilde{R}(u).\end{aligned}$$* **Proof.* It follows from Lemma [Lemma 11](#lem6.1){reference-type="ref" reference="lem6.1"} and Lemma [Lemma 12](#lem7.1){reference-type="ref" reference="lem7.1"} directly. ◻* ***Lemma 13**. *We estimate $\tilde{R}(u)$ as follows: $$\begin{aligned} \label{7.14} \tilde{R}(u) =O(\frac{1}{R}+\|\xi\|^{2}_{H^{1}}). \end{aligned}$$** **Proof.* Recall the definition of $\tilde{R}(u)$ in [\[7.13\]](#7.13){reference-type="eqref" reference="7.13"}: $$\begin{aligned} \tilde{R}(u) &= R(u) +\lambda^{2}\langle\xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle +\lambda \langle \xi, S''_{\lambda}(\phi_{\lambda})\phi_{\lambda}\rangle -(\dot{y}-\lambda)\left[ E(u_{0}) +2\lambda\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} \right] \\ &\quad +\frac{D(\lambda)}{B(\lambda)}\partial_{t}\langle \xi, (1-\partial_{x}^2)(x^3\phi_{\lambda})\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ First, from Lemma [Lemma 7](#lem3.3){reference-type="ref" reference="lem3.3"}, we have already known that $\phi_{\lambda}$ and $\partial_{xx}\phi_{\lambda}$ both have pre-image with respect to $S''_{\lambda}(\phi_{\lambda}),$ then we have $$\begin{aligned} \label{tidel-R-part1} \langle\xi, \phi_{\lambda}+\partial_{xx}\phi_{\lambda}\rangle=\langle\xi, S''_{\lambda}(\phi_{\lambda})\big(\Psi_{\lambda}+\frac{1}{2\lambda}x\partial_{x}\phi_{\lambda}\big)\rangle.\end{aligned}$$ By [\[5.3\]](#5.3){reference-type="eqref" reference="5.3"} and Taylor's type expansion, we have $$\begin{aligned} E(u_{0})=E(u) &=E(\phi_{\lambda})+\langle E'(\phi_{\lambda}),\xi\rangle +O(\|\xi\|^{2}_{H^{1}})\\ &=E(\phi_{\lambda})+\langle \xi, \phi_{\lambda}+\phi_{\lambda}^{p+1} \rangle +O\left(\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ From Lemma [Lemma 5](#lem2.1){reference-type="ref" reference="lem2.1"} and [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"}, we have $$\begin{aligned} E(\phi_{\lambda})&=\frac{4\lambda+p}{2(p+4)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}; \\ E(\phi_{\lambda})+2\lambda\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} &=\frac{4p\lambda+4\lambda-3p}{2(p+4)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ Combining the rough estimate of $\dot{y}-\lambda=O\left(\|\xi\|_{H^1}\right)$ in Corollary [Corollary 9](#cor:5.1){reference-type="ref" reference="cor:5.1"} and the precise estimate of $\dot{y}-\lambda$ in [\[estimate_doty-lambda\]](#estimate_doty-lambda){reference-type="eqref" reference="estimate_doty-lambda"}, we have $$\begin{aligned} \label{7.16} &-(\dot{y}-\lambda)\left[ E(u_{0}) +2\lambda\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} \right] \notag\\ =& -(\dot{y}-\lambda)\left[ E(\phi_{\lambda}) +2\lambda\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} +\langle \xi, \phi_{\lambda}+\phi_{\lambda}^{p+1} \rangle+O\left(\|\xi\|^{2}_{H^{1}}\right) \right]\notag\\ =&-\frac{4p\lambda+4\lambda-3p}{2(p+4)B(\lambda)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} \cdot \langle\xi, S''_{\lambda}(\phi_{\lambda})(3x^2\phi_{\lambda}+x^3\partial_{x}\phi_{\lambda})\rangle \notag\\ &\quad +\frac{4p\lambda+4\lambda-3p}{2(p+4)B(\lambda)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} \cdot \partial_{t}\langle \xi, (1-\partial_{x}^{2})\big(x^3\phi_{\lambda}\big)\rangle +O\left(\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ Inserting [\[tidel-R-part1\]](#tidel-R-part1){reference-type="eqref" reference="tidel-R-part1"} and [\[7.16\]](#7.16){reference-type="eqref" reference="7.16"} into the expression of $\tilde{R}(u)$, we have $$\begin{aligned} \tilde{R}(u) =R(u)+ \langle\xi, S''_{\lambda}(\phi_{\lambda}) \big[\lambda^2\Psi_{\lambda}+\frac{\lambda}{2}x\partial_{x}\phi_{\lambda}+\lambda\phi_{\lambda}+\frac{D(\lambda)}{B(\lambda)}(3x^2\phi_{\lambda}+x^3\partial_{x}\phi_{\lambda})\big] \rangle.\end{aligned}$$ We note that $S''_{\lambda}(\phi_{\lambda}) \big[B(\lambda)\big(\lambda^2\Psi_{\lambda}+\frac{\lambda}{2}x\partial_{x}\phi_{\lambda}+\lambda\phi_{\lambda}\big)+D(\lambda)(3x^2\phi_{\lambda}+x^3\partial_{x}\phi_{\lambda})\big]=\kappa_{\lambda}$. By the second orthogonality condition [\[orth\]](#orth){reference-type="eqref" reference="orth"} in Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"}, we complete the proof. ◻* ### Lower bound of $\beta(u_0)$ **Lemma 15**. *Let $u_{0}=(1-a)\phi_{c}$ for some small positive constant $a$. Then there exist a constant $C_{1}>0$, such that $$\beta(u_{0})\geq C_{1}a.$$* *Proof.* Recall the definition of $\beta(u_{0})$ in [\[7.11\]](#7.11){reference-type="eqref" reference="7.11"}: $$\begin{aligned} \beta(u_{0}) =-\lambda\Big[E(u_{0})-E(\phi_{c})\Big]. \end{aligned}$$ Using the expression in [\[2.6\]](#2.6){reference-type="eqref" reference="2.6"} and Taylor's type expansion, we have $$\begin{aligned} \label{7.17} E(u_{0})-E(\phi_{c}) &=\langle E'(\phi_{c}), u_{0}-\phi_{c}\rangle +O(\big\|u_{0}-\phi_{c}\big\|^{2}_{H^{1}}) \notag\\ &=-a\int_{\mathbb{R}} \left(\phi_{c}+\phi_{c}^{p+1}\right) \cdot \phi_{c}\,\mathrm{d}x +O(a^{2})\notag\\ &=-a\left[ \frac{2(p+2)c}{p+4}-\frac{p}{p+4}\right]\big\|\phi_{c}\big\|^{2}_{L^{2}} +O(a^{2}). \end{aligned}$$ Then we put [\[7.17\]](#7.17){reference-type="eqref" reference="7.17"} into the expression of $\beta(u_{0}),$ $$\begin{aligned} \beta(u_{0}) &=-\lambda\Big[E(u_{0})-E(\phi_{c})\Big] \notag\\ &=a\lambda\left[ \frac{2(p+2)c}{p+4}-\frac{p}{p+4}\right] \big\|\phi_{c}\big\|^{2}_{L^{2}}+O(a^{2})\notag\\ &=ac\left[ \frac{2(p+2)c}{p+4}-\frac{p}{p+4}\right] \big\|\phi_{c}\big\|^{2}_{L^{2}}+O(a|\lambda-c|)+O(a^{2}). \end{aligned}$$ Note that $c>1,$ and choosing $a$ and $\varepsilon_{0}$ small enough, where $\varepsilon_{0}$ is the constant in Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"}, and by [\[vare\]](#vare){reference-type="eqref" reference="vare"}, we obtain the conclusion of this lemma. ◻ ### Lower bound of $\gamma(\lambda)$ **Lemma 16**. *There exists a positive constant $C_2$ such that $$\gamma(\lambda)\geq C_{2}(\lambda-c)^{2}+o\left((\lambda-c)^{2}\right).$$* *Proof.* Recall the definition of $\gamma(\lambda)$ from [\[7.12\]](#7.12){reference-type="eqref" reference="7.12"}: $$\begin{aligned} \gamma(\lambda) &=-\lambda E(\phi_{c}) +\frac{1}{2}\lambda^{2} \left(\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}-\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}}\right).\end{aligned}$$ We claim that $$\begin{aligned} \label{7.18} \gamma(c)=0, \quad \gamma'(c)=0, \quad \gamma''(c)>0.\end{aligned}$$ We prove the claim by the following three steps. *Step 1. $\gamma(c)=0.$* From [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"}, we have $$\begin{aligned} E(\phi_{c}) =\left(\frac{1}{2}+\frac{2(c-1)}{p+4}\right)\big\|\phi_{c}\big\|^{2}_{L^{2}}; \\ \big\|\phi_{\lambda}\big\|^{2}_{L^{2}}-\big\|\partial_{x}\phi_{\lambda}\big\|^{2}_{L^{2}} =\frac{4\lambda+p}{(p+4)\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ So, we have $$\begin{aligned} \label{7.19} \gamma(\lambda) =-\lambda\left( \frac{1}{2}+\frac{2(c-1)}{p+4}\right) \big\|\phi_{c}\big\|^{2}_{L^{2}} +\frac{4\lambda^{2}+p\lambda}{2(p+4)} \big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ A direct computation gives $$\begin{aligned} \gamma(c) &=-c\left( \frac{1}{2}+\frac{2(c-1)}{p+4}\right) \big\|\phi_{c}\big\|^{2}_{L^{2}} +\frac{4c^{2}+pc}{2(p+4)} \big\|\phi_{c}\big\|^{2}_{L^{2}} =0.\end{aligned}$$ *Step 2. $\gamma'(c)=0.$* Using the expression of $\gamma(\lambda)$ in [\[7.19\]](#7.19){reference-type="eqref" reference="7.19"}, we have $$\begin{aligned} \label{7.20} \gamma'(\lambda) &=-\left( \frac{1}{2}+\frac{2(c-1)}{p+4}\right) \big\|\phi_{c}\big\|^{2}_{L^{2}} +\frac{8\lambda+p}{2(p+4)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}\ +\frac{4\lambda^{2}+p\lambda}{2(p+4)}\partial_{\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ By [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"}, we have $$\begin{aligned} \label{dla} \partial_{\lambda}\big(\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}\big) =\frac{4\lambda-p}{2p\lambda(\lambda-1)}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}, \end{aligned}$$ so we have $$\begin{aligned} \gamma'(c) &=\left[ -\left(\frac{1}{2}+\frac{2(c-1)}{p+4}\right) +\frac{8c+p}{2(p+4)} +\frac{(4c+p)(4c-p)}{4p(p+4)(c-1)} \right] \big\|\phi_{c}\big\|^{2}_{L^{2}} =0.\end{aligned}$$ *Step 3. $\gamma''(c)>0.$* From the expression of $\gamma(\lambda)$ in [\[7.19\]](#7.19){reference-type="eqref" reference="7.19"}, we have $$\begin{aligned} \gamma''(\lambda) =\frac{8\lambda+p}{p+4}\partial_{\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} +\frac{4}{p+4}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} +\frac{4\lambda^{2}+p\lambda}{2(p+4)}\partial_{\lambda\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ From [\[dla\]](#dla){reference-type="eqref" reference="dla"}, we have $$\begin{aligned} \partial_{\lambda\lambda}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}} &=\frac{-4\lambda^2+2p\lambda-p}{2p\lambda^2(\lambda-1)^2}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}+\left[\frac{4\lambda-p}{2p\lambda(\lambda-1)}\right]^2\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}\\ &=\frac{8(2-p)\lambda^{2}+4p(p-2)\lambda-p^{2}}{4p^{2}\lambda^{2}(\lambda-1)^{2}}\big\|\phi_{\lambda}\big\|^{2}_{L^{2}}.\end{aligned}$$ So we get $$\begin{aligned} \gamma''(c) =\frac{p-4c}{2p(c-1)^{2}}\big\|\phi_{c}\big\|^{2}_{L^{2}},\end{aligned}$$ noting that $p-4c=\frac{p}{p+2}\left[p-2\sqrt{2+\frac{1}{2}p}\right]>0$ when $p>4,$ then $\gamma''(c)>0.$ This proves the claim [\[7.18\]](#7.18){reference-type="eqref" reference="7.18"}. Using [\[7.18\]](#7.18){reference-type="eqref" reference="7.18"} and Taylor's type expansion, we get $$\begin{aligned} \gamma(\lambda) &=\gamma(c)+\gamma'(c)(\lambda-c)+\frac{1}{2}\gamma''(c)(\lambda-c)^{2}+o\left((\lambda-c)^{2}\right)\\ &\geq C_{2}(\lambda-c)^{2}+o\left((\lambda-c)^{2}\right),\end{aligned}$$ where $C_{2}=\frac{1}{2}\gamma''(c)>0.$ Thus we obtain the conclusion of this lemma. ◻ Hence, combining Lemmas [Lemma 14](#lem7.3){reference-type="ref" reference="lem7.3"}--[Lemma 15](#lem7.5){reference-type="ref" reference="lem7.5"}, and [\[7.14\]](#7.14){reference-type="eqref" reference="7.14"}, we have $$\begin{aligned} \label{7.21} I'(t) \geq C_{1}a+C_{2}(\lambda-c)^{2} +O\left(\frac{1}{R}+\|\xi\|^{2}_{H^{1}}\right) +o\left((\lambda-c)^{2}\right).\end{aligned}$$ ### Upper bound of $\|\xi\|_{H^{1}}$ **Lemma 17**. *Let $\xi$ be defined in [\[xi\]](#xi){reference-type="eqref" reference="xi"}, then for any $t \in \mathbb{R},$ $$\begin{aligned} \|\xi\|^{2}_{H^{1}} \lesssim O\left(a|\lambda-c|+a^{2}\right) +o\left((\lambda-c)^{2}\right).\end{aligned}$$* *Proof.* First, since $u=\left(\phi_{\lambda}+\xi\right)(x-y)$ in [\[5.3\]](#5.3){reference-type="eqref" reference="5.3"}, by Taylor's type extension and $S'_{c}(\phi_{c})=0$, we have $$\begin{aligned} S_{\lambda}(u)-S_{\lambda}(\phi_{\lambda}) &=\langle S'_{\lambda}(\phi_{\lambda}), \xi\rangle +\frac{1}{2}\langle S''_{\lambda}(\phi_{\lambda})\xi, \xi\rangle +o\left(\|\xi\|^{2}_{H^{1}}\right) \\ &=\frac{1}{2}\langle S''_{\lambda}(\phi_{\lambda})\xi, \xi\rangle +o\left(\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ Using $S'_{c}(\phi_{c})=0$ and Taylor's type extension, we have $$\begin{aligned} S_{\lambda}(u)-S_{\lambda}(\phi_{\lambda}) =\frac{1}{2}\langle S''_{\lambda}(\phi_{\lambda})\xi, \xi\rangle +o\left(\|\xi\|^{2}_{H^{1}}\right).\end{aligned}$$ Then by Proposition [Proposition 8](#prop3.1){reference-type="ref" reference="prop3.1"}, we get $$\begin{aligned} S_{\lambda}(u)-S_{\lambda}(\phi_{\lambda}) \gtrsim \|\xi\|^{2}_{H^{1}}.\end{aligned}$$ Second, note that $$\begin{aligned} S_{\lambda}(u)-S_{\lambda}(\phi_{\lambda}) =S_{\lambda}(u_{0})-S_{\lambda}(\phi_{c})+S_{\lambda}(\phi_{c})-S_{\lambda}(\phi_{\lambda}),\end{aligned}$$ and the expression of $S_c$ in [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"} gives that $$\begin{aligned} S_{\lambda}(u_{0})-S_{\lambda}(\phi_{c}) =S_{c}(u_{0})-S_{c}(\phi_{c}) -(\lambda-c)\big[Q(u_{0})-Q(\phi_{c})].\end{aligned}$$ Using the Taylor's type expansion, by $S'_{c}(\phi_{c})=0,$ [\[2.7\]](#2.7){reference-type="eqref" reference="2.7"} and [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"}, we have $$\begin{aligned} S_{c}(u_{0})-S_{c}(\phi_{c}) &=\langle S'_{c}(\phi_{c}),u_{0}-\phi_{c}\rangle +O\left(\big\|u_{0}-\phi_{c}\big\|^{2}_{H^{1}}\right)\\ &=O(a^{2}); \\ Q(u_{0})-Q(\phi_{c}) &=\langle Q'(\phi_{c}),u_{0}-\phi_{c}\rangle +O\left(\big\|u_{0}-\phi_{c}\big\|^{2}_{H^{1}}\right)\\ &=-a\left[1+\frac{p(c-1)}{c(p+4)}\right]\big\|\phi_{c}\big\|^{2}_{L^{2}}+O(a^{2}).\end{aligned}$$ So we obtain $$\begin{aligned} S_{\lambda}(u_{0})-S_{\lambda}(\phi_{c}) &=(\lambda-c)a\left[1+\frac{p(c-1)}{c(p+4)}\right]\big\|\phi_{c}\big\|^{2}_{L^{2}}+O(a^{2})\\ &=O\left(a^{2}+a|\lambda-c|\right).\end{aligned}$$ Moreover, by Corollary [Corollary 6](#cor2.2){reference-type="ref" reference="cor2.2"}, we have $$\begin{aligned} S_{\lambda}(\phi_{c})-S_{\lambda}(\phi_{\lambda}) =o\left((\lambda-c)^{2}\right).\end{aligned}$$ Finally, we get the desired result $$\begin{aligned} \|\xi\|^{2}_{H^{1}} &\lesssim S_{\lambda}(u)-S_{\lambda}(\phi_{\lambda}) =S_{\lambda}(u_{0})-S_{\lambda}(\phi_{c})+S_{\lambda}(\phi_{c})-S_{\lambda}(\phi_{\lambda})\\ &=O\left(a^{2}+a|\lambda-c|\right) +o\left((\lambda-c)^{2}\right).\end{aligned}$$ This completes the proof. ◻ ## Proof of Theorem 1.2 As in the discussion above, we assume that $u\in U_{\varepsilon}(\phi_{c}),$ and thus $|\lambda-c|\lesssim \varepsilon.$ We note that from the definition of $I(t)$ and [\[2.1\]](#2.1){reference-type="eqref" reference="2.1"} we have the uniform boundedness of $I(t):$ $$\begin{aligned} \label{7.22} \sup_{t \in \mathbb{R}} I(t) \lesssim R(\big\|\phi_{c}\big\|^{2}_{L^{2}}+1). \end{aligned}$$ Now we estimate $I'(t).$ From [\[7.21\]](#7.21){reference-type="eqref" reference="7.21"} and Lemma [Lemma 17](#lem7.6){reference-type="ref" reference="lem7.6"}, we have $$\begin{aligned} I'(t) &\geq C_{1}a+C_{2}(\lambda-c)^{2} +O\left(\frac{1}{R}+\|\xi\|^{2}_{H^{1}}\right) +o\left((\lambda-c)^{2}\right)\\ &\geq C_{1}a+C_{2}(\lambda-c)^{2} +O\left(a^{2}+a|\lambda-c|\right) +o\left((\lambda-c)^{2}\right) +O\left(\frac{1}{R}\right).\end{aligned}$$ By [\[vare\]](#vare){reference-type="eqref" reference="vare"}, choosing $R$ satisfying $\frac{1}{R}\leqslant a^{2}$ and $\varepsilon, a_0$ small enough, we obtain that for any $a\in (0, a_{0}),$ $$\begin{aligned} I'(t) &\geq C_{1}a+C_{2}(\lambda-c)^{2} +O\left(a^{2}+a|\lambda-c|\right) +o\left((\lambda-c)^{2}\right)\\ &\geq \frac{1}{2}C_{1}a+\frac{1}{2}C_{2}(\lambda-c)^{2}.\end{aligned}$$ This implies $I(t)\rightarrow +\infty$ when $t\rightarrow +\infty,$ which is contradicted with [\[7.22\]](#7.22){reference-type="eqref" reference="7.22"}. Hence we prove the instability of solitary wave solution $\phi_{c}(x-ct)$ and thus give the proof of Theorem [Theorem 2](#main:thm){reference-type="ref" reference="main:thm"}. # {#appendix} ## Spectrum of $S_{c}''(\phi_{c})$ First, we study the kernel of $S_{c}''(\phi_{c})$ in the following lemma. The proof is standard, and it is a consequence of the result from [@weinstein-1985-modulational]. **Lemma 18**. *The kernel of $S_{c}''(\phi_{c})$ satisfies that $$\begin{aligned} \ker S_{c}''(\phi_{c}) =\{\alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}\}. \end{aligned}$$* *Proof.* First, we need to show the relationship $``\supset".$ For any $f \in \{{\alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}}\},$ using [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}, we have $$\begin{aligned} \label{3.1} S_{c}''(\phi_{c})f &=S_{c}''(\phi_{c})(\alpha \partial_x \phi_{c})\notag\\ &=\alpha \partial_x (c\partial_{xx} \phi_{c}+(1-c)\partial_{c} \phi_{c}+\phi_{c}^{p+1})\notag \\ &=0\end{aligned}$$ Then [\[3.1\]](#3.1){reference-type="eqref" reference="3.1"} implies that $f$ is in the kernel of $S_{c}''(\phi_{c}),$ and we have the conclusion $$\begin{aligned} \ker S_{c}''(\phi_{c}) \supset \{ \alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}\}.\end{aligned}$$ Second, we prove the reverse relationship $``\subset".$ By the expression of $S_{c}''(\phi_{c})$ in [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"}, we have $$\begin{aligned} S_{c}''(\phi_{c})f=0 \Leftrightarrow L_{\omega}f =0, \end{aligned}$$ for any $f \in \ker (S_{c}''(\phi_{c})),$ that is $$\begin{aligned} \label{3.2} -\partial_{xx}f+(1-\omega^{2})f+(p+1)\psi_{\omega}^{p}f=0. \end{aligned}$$ By the work of Weinstein [@weinstein-1985-modulational], the only solution to [\[3.2\]](#3.2){reference-type="eqref" reference="3.2"} are $$\begin{aligned} f=\alpha \partial_x \psi_{\omega}, \quad \alpha \in \mathbb{R}. \end{aligned}$$ Note that $$\begin{aligned} \partial_x \psi_{\omega} =c^{-\frac{1}{p}} \partial_x \phi_{c}. \end{aligned}$$ This implies that $f \in \{ \alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}\},$ and we have $$\begin{aligned} \ker (S_{c}''(\phi_{c})) \subset \{ \alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}\}.\end{aligned}$$ Finally, combining the two relationship gives us $$\begin{aligned} \ker (S_{c}''(\phi_{c})) = \{ \alpha \partial_x \phi_{c}:\alpha \in \mathbb{R}\}.\end{aligned}$$ This gives the proof of the lemma. ◻ The second lemma is the uniqueness of the negative eigenvalue of $S_{c}''(\phi_{c}).$ **Lemma 19**. *$S_{c}''(\phi_{c})$ exists only one negative eigenvalue.* *Proof.* It is known that the operator $-\partial_{xx} +(1-\omega^{2})+(p+1)\psi_{\omega}^{p}$ has only one negative eigenvalue(see [@weinstein-1985-modulational]), and we denote it by $\lambda_{-1}.$ Then there exists a unique associated eigenfunction $\theta \in H^1(\mathbb{R})$ such that $$\begin{aligned} \label{3.3} -\partial_{xx} \theta +(1-\omega^{2})\theta-(p+1)\psi_{\omega}^{p}\theta =\lambda_{-1} \theta.\end{aligned}$$ Using the expression of $S_{c}''(\phi_{c})$ in [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"}, we have $$\begin{aligned} \langle S_{c}''(\phi_{c})\partial_{\omega}\psi_{\omega}, \partial_{\omega}\psi_{\omega}\rangle &=-c \langle L_{\omega}\partial_{\omega}\psi_{\omega}, \partial_{\omega}\psi_{\omega}\rangle \\ &=-c \int_{\mathbb{R}} (-\partial_{xx} \partial_{\omega}\psi_{\omega} +(1-\omega^{2})\partial_{\omega}\psi_{\omega}-(p+1)\psi_{\omega}^{p}\partial_{\omega}\psi_{\omega}) \cdot \partial_{\omega}\psi_{\omega} \,\mathrm{d}x\\ &=-c \int_{\mathbb{R}} 2\omega\psi_{\omega} \cdot \partial_{\omega}\psi_{\omega} \,\mathrm{d}x\\ &=-c \omega \frac{d}{d\omega} \Big(\big\|\psi_{\omega} \big\|^{2}_{L^{2}}\Big)\\ &=2(\frac{2}{p}-\frac{1}{2}) (1-\omega^{2})^{\frac{2}{p}-\frac{3}{2}} \big\|\psi_{0} \big\|^{2}_{L^{2}}. \end{aligned}$$ Note that $p>4,$ then we have $$\begin{aligned} \langle S_{c}''(\phi_{c})\partial_{\omega}\psi_{\omega}, \partial_{\omega}\psi_{\omega}\rangle < 0. \end{aligned}$$ This implies that $S_{c}''(\phi_{c})$ has at least one negative eigenvalue $\mu_{0}.$ Assume its associated eigenfunction $\xi_{0},$ that is, $$\begin{aligned} S_{c}''(\phi_{c})\xi_{0} =\mu_{0}\xi_{0}. \end{aligned}$$ Using the expression of $S_{c}''(\phi_{c})$ in [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"} again, the last equality yields $$\begin{aligned} -c\big[-\partial_{xx} \xi_{0} +(1-\omega^{2})\xi_{0} -(p+1)\psi_{\omega}^{p} \xi_{0}\big] =\mu_{0} \xi_{0}.\end{aligned}$$ Then we have $\xi_{0}=-\frac{1}{c} \theta.$ Hence, by [\[3.3\]](#3.3){reference-type="eqref" reference="3.3"}, $(\mu_{0}, \xi_{0})$ is exactly the pair satisfying $$\begin{aligned} \mu_{0}=\lambda_{-1}, \quad \xi_{0}=-\frac{1}{c} \theta.\end{aligned}$$ This implies that $S_{c}''(\phi_{c})$ has exactly one simple negative eigenvalue. This completes the proof of Lemma [\[3.2\]](#3.2){reference-type="eqref" reference="3.2"}. ◻ ## Coercivity In this subsection, we give a general coercivity property on the Hessian of the action $S_{c}''(\phi_{c}).$ **Proposition 20**. *Let $\tau_c, \Psi$ be any functions satisfying that $$\begin{aligned} \label{Assume-Coer} & \tau_{c}=S_{c}''(\phi_{c})\Psi,\quad \mbox{and}\quad \langle S_{c}''(\phi_{c})\Psi, \Psi\rangle <0.\end{aligned}$$ Suppose that $\xi\in H^1(\mathbb{R})$ satisfies $$\begin{aligned} \label{3.10} \langle \xi, \partial_x\phi_c\rangle =\langle \xi, \tau_{c}\rangle =0. \end{aligned}$$ Then $$\begin{aligned} \langle S_{c}''(\phi_{c})\xi, \xi\rangle \gtrsim \|\xi\|^{2}_{H^1(\mathbb{R})}. \end{aligned}$$* *Proof.* From the expression of $S_{c}''(\phi_{c})$ in [\[2.13\]](#2.13){reference-type="eqref" reference="2.13"}, we can write $S_{c}''(\phi_{c})$ as $$\begin{aligned} S_{c}''(\phi_{c})=-c(L+V), \end{aligned}$$ where $L=-\partial_{xx}+(1-\omega^{2}),$ and $V=-(p+1)\psi_{\omega}^{p}.$ Hence $V$ is a compact perturbation of the self-adjoint operator $L.$\ *Step 1. Analyze the spectrum of $S_{c}''(\phi_{c}).$* We first compute the essential spectrum of $L.$ Note that for any $g\in H^1(\mathbb{R}),$ $$\begin{aligned} \label{3.11} \langle Lg, g\rangle &=\int_{\mathbb{R}} \big(-\partial_{xx}g+(1-\omega^{2})g\big) \cdot g \,\mathrm{d}x\notag \\ &=\big\|\partial_x g\big\|^{2}_{L^2} +(1-\omega^{2})\big\|g\big\|^2_{L^{2}}. \end{aligned}$$ Since $c=\omega^{-2}, c>1,$ we can get $|\omega|<1,$ and thus $$\begin{aligned} \langle Lg, g \rangle \gtrsim \big\|g\big\|^{2}_{H^1(\mathbb{R})}.\end{aligned}$$ This means that there exists $\delta>0$ such that the essential spectrum of $L$ is $[\delta, +\infty).$ By Weyl Theorem, $S_{c}''(\phi_{c})$ and $L$ share the same essential spectrum. So we obtain the essential spectrum of $S_{c}''(\phi_{c}).$ Recall that we have obtained the only one negative eigenvalue $\mu_{0}$ of $S_{c}''(\phi_{c})$ in Lemma [Lemma 19](#lem3.2){reference-type="ref" reference="lem3.2"} and the kernel of $S_{c}''(\phi_{c})$ in Lemma [Lemma 18](#lem3.1){reference-type="ref" reference="lem3.1"}. So the discrete spectrum of $S_{c}''(\phi_{c})$ is $\mu_{0}, 0,$ and the essential spectrum is $[\delta, +\infty).$\ *Step 2. Positivity.* By Lemma [Lemma 19](#lem3.2){reference-type="ref" reference="lem3.2"}, we have the unique negative eigenvalue $\mu_{0}$ and the eigenfunction $\xi_{0}$ of $S_{c}''(\phi_{c}).$ For convenience, we normalize the eigenfunction $\xi_{0}$ such that $\big\|\xi_{0}\big\|_{L^{2}}=1.$ Hence, for $\xi_{0}\in H^1(\mathbb{R}),$ by the spectral decomposition theorem we can write the decomposition of $\xi$ along the spectrum of $S_{c}''(\phi_{c}),$ $$\begin{aligned} \xi=a_{\xi}\xi_{0} +b_{\xi}\partial_x\phi_{c} +g_{\xi}, \end{aligned}$$ where $a_{\xi}, b_{\xi}\in \mathbb{R},$ and $g_{\xi}$ lies in the positive eigenspace of $S_{c}''(\phi_{c}),$ that is, $g_{\xi}$ satisfies $$\begin{aligned} \langle g_{\xi}, \xi_{0}\rangle =\langle g_{\xi}, \partial_x\phi_{c}\rangle =0, \end{aligned}$$ and there exists an absolute constant $\sigma>0$ such that $$\begin{aligned} \label{3.12} \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle \geqslant \sigma \big\|g_{\xi}\big\|^{2}_{L^{2}}.\end{aligned}$$ Since $\xi$ satisfies the orthogonality condition $\langle\xi, \partial_x\phi_{c}\rangle$ in [\[3.10\]](#3.10){reference-type="eqref" reference="3.10"} and $\langle\xi_{0}, \partial_x\phi_{c} \rangle =0,$ we have $b_{\xi}=0,$ and thus $$\begin{aligned} \label{3.13} \xi=a_{\xi}\xi_{0}+g_{\xi}. \end{aligned}$$ Substituting [\[3.13\]](#3.13){reference-type="eqref" reference="3.13"} into $\langle S_{c}''(\phi_{c})\xi, \xi\rangle ,$ we get $$\begin{aligned} \langle S_{c}''(\phi_{c})\xi, \xi\rangle &=\langle S_{c}''(\phi_{c})(a_{\xi}\xi_{0}+g_{\xi}), a_{\xi}\xi_{0}+g_{\xi}\rangle \notag\\ &=a_{\xi}^{2}\langle S_{c}''(\phi_{c})\xi_{0}, \xi_{0}\rangle +2\mu_{0}a_{\xi}\langle \xi_{0},g_{\xi}\rangle +\langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle. \end{aligned}$$ Due to the orthogonality property of $\langle \xi_{0}, g_{\xi}\rangle =0,$ we have $$\begin{aligned} \label{3.14} \langle S_{c}''(\phi_{c})\xi, \xi\rangle =\mu_{0}a_{\xi}^{2} +\langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle.\end{aligned}$$ To $\Psi,$ by spectral decomposition theorem again, we may write $$\begin{aligned} \Psi=a\xi_{0}+b\partial_x\phi_{c}+g, \end{aligned}$$ where $a, b\in\mathbb{R},$ and $g$ lies in the positive eigenspace of $S_{c}''(\phi_{c}).$ We note that $$\begin{aligned} \label{S_c-psi} S''_c(\phi_{c})\Psi=S''_c(\phi_{c})(a\xi_{0}+b\partial_x\phi_{c}+g)=S''_c(\phi_{c})(a\xi_{0}+g).\end{aligned}$$ Therefore, a similar computation as above shows that $$\begin{aligned} \langle S_{c}''(\phi_{c})\Psi, \Psi\rangle &=\langle S_{c}''(\phi_{c})(a\xi_{0}+g), (a\xi_{0}+g)\rangle \\ &=\mu_{0}a^{2}+\langle S_{c}''(\phi_{c})g, g\rangle.\end{aligned}$$ For convenience, let $-\delta_{0}=\langle S_{c}''(\phi_{c})\Psi, \Psi\rangle.$ Then by [\[Assume-Coer\]](#Assume-Coer){reference-type="eqref" reference="Assume-Coer"}, we know that $\delta_{0}>0.$ Moreover, we have $$\begin{aligned} \label{3.15} -\delta_{0} =\mu_{0}a^{2}+\langle S_{c}''(\phi_{c})g, g\rangle.\end{aligned}$$ By [\[3.13\]](#3.13){reference-type="eqref" reference="3.13"} and [\[S_c-psi\]](#S_c-psi){reference-type="eqref" reference="S_c-psi"}, using the orthogonality assumption $\langle \xi, \tau_c \rangle =0$ in [\[3.10\]](#3.10){reference-type="eqref" reference="3.10"} we have $$\begin{aligned} 0=\langle \xi, \tau_c\rangle &=\langle a_{\xi}\xi_{0}+g_{\xi}, S_{c}''(\phi_{c})\Psi\rangle \\ &=\langle a_{\xi}\xi_{0}+g_{\xi}, S_{c}''(\phi_{c})(a\xi_{0}+g)\rangle \\ &=\langle a_{\xi}\xi_{0}, S_{c}''(\phi_{c})a\xi_{0}\rangle +\langle S_{c}''(\phi_{c})g, g_{\xi}\rangle \\ &=\mu_{0}aa_{\xi} \langle \xi_{0},\xi_{0}\rangle +\langle S_{c}''(\phi_{c})g, g_{\xi}\rangle \\ &=\mu_{0}aa_{\xi} +\langle S_{c}''(\phi_{c})g, g_{\xi}\rangle. \end{aligned}$$ So we get the equality $$\begin{aligned} 0=\mu_{0}aa_{\xi} +\langle S_{c}''(\phi_{c})g, g_{\xi}\rangle. \end{aligned}$$ By the Cauchy-Schwartz inequality, we have $$\begin{aligned} (\mu_{0}aa_{\xi})^{2} &=\langle S_{c}''(\phi_{c})g, g_{\xi}\rangle ^{2}\\ &\leqslant \langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle.\end{aligned}$$ This gives $$\begin{aligned} \label{3.16} (-\mu_{0}a^{2})(-\mu_{0}a_{\xi}^{2}) \leqslant \langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle.\end{aligned}$$ The last inequality combining with [\[3.15\]](#3.15){reference-type="eqref" reference="3.15"} implies that $$\begin{aligned} -\mu_{0}a_{\xi}^{2} &\leqslant \frac{\langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle } {-\mu_{0}a^{2}}\\ &=\frac{\langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle } {\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle },\end{aligned}$$ that is $$\begin{aligned} \label{3.17} \mu_{0}a_{\xi}^{2} \geqslant -\frac{\langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle } {\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle }. \end{aligned}$$ Inserting [\[3.17\]](#3.17){reference-type="eqref" reference="3.17"} into [\[3.14\]](#3.14){reference-type="eqref" reference="3.14"}, we obtain $$\begin{aligned} \langle S_{c}''(\phi_{c})\xi, \xi\rangle &\geqslant -\frac{\langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle } {\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle } +\langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle \\ &=\langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle \left(1- \frac{\langle S_{c}''(\phi_{c})g, g\rangle } {\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle }\right)\\ &=\langle S_{c}''(\phi_{c})g_{\xi}, g_{\xi}\rangle \frac{\delta_{0}}{\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle }. \end{aligned}$$ Recalling that $g_{\xi}$ satisfies [\[3.12\]](#3.12){reference-type="eqref" reference="3.12"}, we have $$\begin{aligned} \label{3.18} \langle S_{c}''(\phi_{c})\xi, \xi\rangle \geqslant \frac{\delta_{0} \sigma} {\delta_{0}+\langle S_{c}''(\phi_{c})g, g\rangle } \big\|g_{\xi}\big\|^{2}_{L^{2}}, \quad \sigma>0. \end{aligned}$$ From the expression of $\xi$ in [\[3.13\]](#3.13){reference-type="eqref" reference="3.13"} and the inequality [\[3.14\]](#3.14){reference-type="eqref" reference="3.14"}, we have $$\begin{aligned} \|\xi\|^{2}_{L^{2}} &=\big\|a_{\xi}\xi_{0}+g_{\xi}\big\|^{2}_{L^{2}} =a_{\xi}^{2}+\big\|g_{\xi}\big\|^{2}_{L^{2}}\\ &\leqslant -\frac{\langle S_{c}''(\phi_{c})g, g\rangle \langle S_{c}''(\phi_{c})\xi, \xi\rangle } {\mu_{0}\delta_{0}} +\big\|g_{\xi}\big\|^{2}_{L^{2}}\\ &\lesssim \langle S_{c}''(\phi_{c})\xi, \xi\rangle \end{aligned}$$ Therefore, this gives $$\begin{aligned} \label{3.19} \langle S_{c}''(\phi_{c})\xi, \xi\rangle \gtrsim \|\xi\|^{2}_{L^{2}}. \end{aligned}$$ To obtain the final conclusion, we still need to estimate $$\begin{aligned} \langle S_{c}''(\phi_{c})\xi, \xi\rangle \gtrsim \|\xi\|^{2}_{H^1(\mathbb{R})}. \end{aligned}$$ Using the expression of $S_{c}''(\phi_{c})$ in [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"}, we have $$\begin{aligned} \langle S_{c}''(\phi_{c})\xi, \xi\rangle &=\int_{\mathbb{R}} (c\partial_{xx} \xi +(1-c)\xi+(p+1)\phi_{c}^{p}\xi) \cdot \xi \,\mathrm{d}x\\ &=-c\big\|\partial_x\xi\big\|^{2}_{L^{2}}+(1-c)\|\xi\|^{2}_{L^{2}} +(p+1)\int_{\mathbb{R}} |\phi_{c}|^{p}\xi^{2} \,\mathrm{d}x\end{aligned}$$ Thus by [\[3.19\]](#3.19){reference-type="eqref" reference="3.19"}, we get $$\begin{aligned} \label{3.20} \big\|\partial_x\xi\big\|^{2}_{L^{2}} &=-\frac{1}{c} \left[\langle S_{c}''(\phi_{c})\xi, \xi\rangle -(1-c)\|\xi\|^{2}_{L^{2}} -(p+1)\int_{\mathbb{R}}|\phi_{c}|^{p}\xi^{2} \,\mathrm{d}x\right]\notag\\ &\leqslant -\frac{1}{c}\langle S_{c}''(\phi_{c})\xi, \xi\rangle +(\frac{1}{c}-1)\|\xi\|^{2}_{L^{2}} +\frac{p+1}{c}\big\|\phi_{c}\big\|^{p}_{L^{\infty}} \|\xi\|^{2}_{L^{2}} \notag\\ &\leqslant -\frac{1}{c}\langle S_{c}''(\phi_{c})\xi, \xi\rangle +\left(\frac{1}{c}-1+\frac{p+1}{c}\big\|\phi_{c}\big\|^{p}_{L^{\infty}}\right) \|\xi\|^{2}_{L^{2}} \notag\\ &\lesssim \langle S_{c}''(\phi_{c})\xi, \xi\rangle + \|\xi\|^{2}_{L^{2}} \notag\\ &\lesssim \langle S_{c}''(\phi_{c})\xi, \xi\rangle. \end{aligned}$$ Therefore, together [\[3.19\]](#3.19){reference-type="eqref" reference="3.19"} and [\[3.20\]](#3.20){reference-type="eqref" reference="3.20"}, we obtain $$\begin{aligned} \|\xi\|^{2}_{H^1(\mathbb{R})} &= \|\xi\|^{2}_{L^{2}}+\big\|\partial_x\xi\big\|^{2}_{L^{2}}\\ &\lesssim \langle S_{c}''(\phi_{c})\xi, \xi\rangle.\end{aligned}$$ Thus we obtain the desired result. ◻ **Corollary 21**. *Assume $$\begin{aligned} \label{condition_1} \langle S''_c(\phi_{c})\eta, \eta \rangle \leqslant 0, \quad \eta \notin \ker S''_c(\phi_{c}), \quad \eta\in H^1(\mathbb{R}), \end{aligned}$$ then for any $\zeta \in H^1(\mathbb{R}),$ s.t. $$\begin{aligned} \label{orth-condition 2} \langle S_c''(\phi_{c})\zeta, \eta\rangle =0,\end{aligned}$$ we have $$\begin{aligned} \langle S_c''(\phi_{c})\zeta, \zeta\rangle \geqslant 0.\end{aligned}$$* *Proof.* Using the similar spectral decomposition argument as in Proposition [Proposition 20](#prop3.4){reference-type="ref" reference="prop3.4"}, we apply the notation from Proposition [Proposition 20](#prop3.4){reference-type="ref" reference="prop3.4"}, that is: the unique negative eigenvalue $\mu_{0}$ and its corresponding normalized eigenfunction $\xi_{0}$ of $S_c''(\phi_{c}).$ So for $\eta \in H^1(\mathbb{R}),$ we can write the decomposition of $\eta$ as $$\begin{aligned} \label{decomposition_eta} \eta=a_{\eta}\xi_{0}+b_{\eta}\partial_{x}\phi_{c}+g_{\eta}, \end{aligned}$$ where $a_{\eta}, b_{\eta} \in \mathbb{R},$ and $g_{\eta}$ lies in the positive eigenspace of $S''_c(\phi_{c}),$ that is $g_{\eta}$ satisfies $$\begin{aligned} \label{orth-property} \langle g_{\eta}, \xi_{0}\rangle=\langle g_{\eta}, \partial_{x}\phi_{c}\rangle=0,\end{aligned}$$ and there exists an absolute constant $\sigma_1>0$ such that $$\begin{aligned} \label{positive-eigen1} \langle S''_c(\phi_{c})g_{\eta}, g_{\eta} \rangle\geqslant \sigma_1\|g_{\eta}\|^{2}_{L^{2}}.\end{aligned}$$ Since $\eta$ satisfies $\eqref{condition_1},$ there exists an absolute constant $\delta_1\geqslant0,$ such that $$\begin{aligned} \langle S''_c(\phi_{c})\eta, \eta \rangle=-\delta_1.\end{aligned}$$ By Lemma [Lemma 18](#lem3.1){reference-type="ref" reference="lem3.1"}, and combining [\[decomposition_eta\]](#decomposition_eta){reference-type="eqref" reference="decomposition_eta"} and [\[orth-property\]](#orth-property){reference-type="eqref" reference="orth-property"}, we have $$\begin{aligned} S''_c(\phi_{c})\eta =a_{\eta}S''_c(\phi_{c})\xi_{0}+S''_c(\phi_{c})g_{\eta}.\end{aligned}$$ So we obtain that $$\begin{aligned} \label{negative property of eta} \langle S''_c(\phi_{c})\eta, \eta \rangle &=\langle a_{\eta}S''_c(\phi_{c})\xi_{0}+S''_c(\phi_{c})g_{\eta}, a_{\eta}\xi_{0}+b_{\eta}\partial_{x}\phi_{c}+g_{\eta}\rangle \notag\\ &=\mu_{0}a_{\eta}^2+\langle S''_c(\phi_{c})g_{\eta}, g_{\eta} \rangle =-\delta_1.\end{aligned}$$ Similarly, we write $\zeta$ as $\zeta=a_1\xi_{0}+b_1\partial_{x}\phi_{c}+g_1,$ where $a_1, b_1\in \mathbb{R},$ and $g_1$ lies in the positive eigenspace of $S''_c(\phi_{c}).$ We note that $$\begin{aligned} \label{S_c(phi_c)zeta} \langle S''_c(\phi_{c})\zeta,\zeta \rangle=\mu_{0}a_{1}^2+\langle S''_c(\phi_{c})g_{1}, g_1\rangle.\end{aligned}$$ From condition [\[orth-condition 2\]](#orth-condition 2){reference-type="eqref" reference="orth-condition 2"}, we have $$\begin{aligned} \langle S''_c(\phi_{c})\zeta, \eta \rangle &=\langle a_{1}S''_c(\phi_{c})\xi_{0}+S''_c(\phi_{c})g_{1}, a_{\eta}\xi_{0}+b_{\eta}\partial_{x}\phi_{c}+g_{ \eta}\rangle \\ &=\mu_{0}a_{1}a_{\eta}+\langle S''_c(\phi_{c})g_{1}, g_{\eta}\rangle=0.\end{aligned}$$ So we get $$\begin{aligned} \big(\mu_{0}a_{1}a_{\eta}\big)^2 &=\big(\mu_{0}a_{1}^2\big)\big(\mu_{0}a_{\eta}^2\big) \\ &=\langle S''_c(\phi_{c})g_{1}, g_{\eta}\rangle^2 \\ &\leqslant \langle S''_c(\phi_{c})g_{1}, g_{1}\rangle \langle S''_c(\phi_{c})g_{\eta}, g_{\eta}\rangle,\end{aligned}$$ where we used Cauchy-Schwartz inequality in the last step. Combining [\[negative property of eta\]](#negative property of eta){reference-type="eqref" reference="negative property of eta"}, the last inequality implies that $$\begin{aligned} \label{lemma a.4-1} \mu_{0}a_{1}^2 &\geqslant \frac{\langle S''_c(\phi_{c})g_{1}, g_{1}\rangle \langle S''_c(\phi_{c})g_{\eta}, g_{\eta}\rangle}{\mu_{0}a_{\eta}^2} \notag\\ &= -\frac{\langle S''_c(\phi_{c})g_{1}, g_{1}\rangle \langle S''_c(\phi_{c})g_{\eta}, g_{\eta}\rangle} {\langle S''_c(\phi_{c})g_{\eta}, g_{\eta} \rangle+\delta_1}.\end{aligned}$$ Inserting [\[lemma a.4-1\]](#lemma a.4-1){reference-type="eqref" reference="lemma a.4-1"} into [\[S_c(phi_c)zeta\]](#S_c(phi_c)zeta){reference-type="eqref" reference="S_c(phi_c)zeta"}, we have $$\begin{aligned} \langle S''_c(\phi_{c})\zeta,\zeta \rangle &\geqslant -\frac{\langle S''_c(\phi_{c})g_{1}, g_{1}\rangle \langle S''_c(\phi_{c})g_{\eta}, g_{\eta}\rangle} {\langle S''_c(\phi_{c})g_{\eta}, g_{\eta} \rangle+\delta_1} +\langle S''_c(\phi_{c})g_{1}, g_{1}\rangle \\ &=\frac{\delta_1\langle S''_c(\phi_{c})g_{1}, g_{1}\rangle}{\langle S''_c(\phi_{c})g_{\eta}, g_{\eta} \rangle+\delta_1} \geqslant0.\end{aligned}$$ Thus we complete the proof. ◻ ## Modulation The modulation theory shows that by choosing suitable parameters, some orthogonality conditions as in [\[prop3.4\]](#prop3.4){reference-type="eqref" reference="prop3.4"} can be verified. **Proposition 22**. *Assume that $\tau_c$ be the function satisfying $$\begin{aligned} \label{Assume-Modulation} \langle \partial_{c}\phi_{c}, \tau_c\rangle \neq 0. \end{aligned}$$ Moreover, suppose that there exists $\varepsilon_{0}>0$ such that for any $\varepsilon \in (0, \varepsilon_{0})$, and any $u \in U_{\varepsilon}(\phi_{c}),$ then the following properties are verified. There exist $C^{1}$-functions $$\begin{aligned} y: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}, \quad \lambda: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}^{+}\end{aligned}$$ such that if we define $\xi$ by $$\begin{aligned} \label{4.2} \xi=u(\cdot +y)-\phi_{\lambda}, \end{aligned}$$ then $\xi$ satisfies the following orthogonality conditions: $$\begin{aligned} \label{4.3} \langle \xi, \partial_x\phi_{\lambda}\rangle = \langle \xi, \tau_{\lambda}\rangle =0. \end{aligned}$$* *Proof.* We use the Implicit Function Theorem to prove this proposition. Here we only give the important steps of the proof and refer the readers to [@weinstein-1985-modulational; @weinstein-1986-lyapunov] for the similar argument. Define $$\begin{aligned} \vec p=(u; \lambda, y), \qquad \vec p_{0}=(\phi_{c}; c, 0).\end{aligned}$$ Let $\varepsilon$ be the parameter decided later, and define the functional pair $(F_{1}, F_{2}): U_{\varepsilon}(\phi_{c})\times \mathbb{R}^{+} \times \mathbb{R}\rightarrow \mathbb{R}^{2}$ as $$\begin{aligned} F_{1}(\vec p)=\langle \xi, \partial_x\phi_{\lambda}\rangle , \quad F_{2}(\vec p)=\langle \xi, \tau_{\lambda}\rangle.\end{aligned}$$ We claim that there exists $\varepsilon_{0}>0,$ such that for any $\varepsilon \in (0, \varepsilon_{0}),$ there exists a unique $C^{1}$ map: $U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}^{+} \times \mathbb{R}$ such that $(F_{1}(\vec p), F_{2}(\vec p))=0.$ Indeed, we have $$\begin{aligned} F_{1}(\vec p_{0})= F_{2}(\vec p_{0})=0. \end{aligned}$$ Second, we prove that $$\begin{aligned} |J|= \left| \begin{array}{cccc} \partial_{\lambda}F_{1} & \partial_{y}F_{1}\\ \partial_{\lambda}F_{2} & \partial_{y}F_{2} \end{array} \right|_{\vec p=\vec p_{0}} \neq 0.\end{aligned}$$ Indeed, a direct computation gives that $$\begin{aligned} \partial_{\lambda}F_{1}(\vec p) =\partial_{\lambda}\langle \xi, \partial_x\phi_{\lambda}\rangle &=\partial_{\lambda}\langle u(t, x+y(t))-\phi_{\lambda}, \partial_x\phi_{\lambda}\rangle \\ &=\langle -\partial_{\lambda}\phi_{\lambda}, \partial_x\phi_{\lambda}\rangle +\langle u(t,x+y(t))-\phi_{\lambda(t)}, \partial_{\lambda}\partial_x\phi_{\lambda}\rangle.\end{aligned}$$ When $\vec p=\vec p_{0},$ we observe that $u(t,x+y(t))-\phi_{\lambda(t)}=0,$ and the second term vanishes. So we get $$\begin{aligned} \partial_{\lambda}F_{1}(\vec p)\big|_{\vec p=\vec p_{0}} =-\langle \partial_{c}\phi_{c}, \partial_x\phi_{c}\rangle=0 \end{aligned}$$ as $\phi_{c}$ is an even function. A similar computation shows that $$\begin{aligned} \partial_{y}F_{1}(\vec p)\big|_{\vec p=\vec p_{0}} &=\langle \partial_x u(t,x+y(t)),\partial_x\phi_{\lambda} \rangle \big|_{\vec p=\vec p_{0}} =\big\| \partial_x \phi_{c}\big\|_{L^2}^2; \\ \partial_{\lambda}F_{2}(\vec p)\big|_{\vec p=\vec p_{0}} &=\partial_{\lambda}\langle u(t,x+y(t))-\phi_{\lambda},\tau_{\lambda} \rangle \big|_{\vec p=\vec p_{0}} =-\langle \partial_{c}\phi_{c}, \tau_c \rangle; \\ \partial_{y}F_{2}(\vec p)\big|_{\vec p=\vec p_{0}} &=\langle \partial_x u(t,x+y(t)), \tau_{\lambda} \rangle \big|_{\vec p=\vec p_{0}} =\langle \partial_{x}\phi_{c}, \tau_c\rangle. \end{aligned}$$ By [\[Assume-Modulation\]](#Assume-Modulation){reference-type="eqref" reference="Assume-Modulation"}, we find that $$\begin{aligned} \left| \begin{array}{cccc} \partial_{\lambda}F_{1} & \partial_{y}F_{1}\\ \partial_{\lambda}F_{2} & \partial_{y}F_{2} \end{array} \right|_{\vec p=\vec p_{0}} &= \left| \begin{array}{cccc} 0 & \big\| \partial_x \phi_{c}\big\|_{L^2}^2 \\ -\langle \partial_{c}\phi_{c}, \tau_c \rangle & \langle \partial_{x}\phi_{c}, \tau_c\rangle \end{array} \right| \\ &=\big\| \partial_x \phi_{c}\big\|_{L^2}^2 \langle \partial_{c}\phi_{c}, \tau_c\rangle \neq 0.\end{aligned}$$ Therefore, the Implicit Function Theorem implies that there exists $\varepsilon_{0}>0$ such that for $\varepsilon \in (0, \varepsilon_{0}), u\in U_{\varepsilon}(\phi_{c}),$ there exist unique $C^{1}$-functions $$\begin{aligned} y: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}, \quad \lambda: U_{\varepsilon}(\phi_{c}) \rightarrow \mathbb{R}^{+}, \end{aligned}$$ such that $$\begin{aligned} \label{4.5} \langle \xi, \partial_x\phi_{\lambda}\rangle =\langle \xi, \tau_{\lambda}\rangle =0.\end{aligned}$$ This proves the Proposition. ◻ ## The negativity of $\langle S''_c(\phi_{c})\Gamma_c, \Gamma_c\rangle$ (numerically checked). {#numerical result} We recall the expression of $\Gamma_c$ and $\kappa_c$ which introduced in [\[Gamma_c\]](#Gamma_c){reference-type="eqref" reference="Gamma_c"} and [\[kappa_c\]](#kappa_c){reference-type="eqref" reference="kappa_c"}: $$\begin{aligned} &\Gamma_c =B(c)\big(c^2\Psi_{c}+\frac{c}{2}x\partial_{x}\phi_{c}+c\phi_{c}\big)+D(c)(3x^2\phi_{c}+x^3\partial_{x}\phi_c), \\ &S''_c(\phi_{c})\Gamma_c=\kappa_c.\end{aligned}$$ $\bullet$*The expression of $\kappa_c.$* From Lemma [Lemma 7](#lem3.3){reference-type="ref" reference="lem3.3"}, we have already known that $$\begin{aligned} \label{f_1} S''_c(\phi_{c})\Psi_{c}=\phi_{c},\\ \label{f_2} S''_c(\phi_{c})(\frac{1}{2c}x\partial_{x}\phi_{c})=\partial_{xx}\phi_{c}.\end{aligned}$$ By the expression of $S''_c(\phi_{c})$ in [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"}, we have $$\begin{aligned} S''_c(\phi_{c})\phi_{c}=c\partial_{xx}\phi_{c}+(1-c)\phi_{c}+(p+1)\phi_{c}^{p+1}.\end{aligned}$$ From equation [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}, we have $$\begin{aligned} \label{phi_c^p+1} \phi_{c}^{p+1}=-c\partial_{xx}\phi_{c}+(c-1)\phi_{c}.\end{aligned}$$ Thus we obtain $$\begin{aligned} \label{S''phi_c} S''_c(\phi_{c})\phi_{c}=-pc\partial_{xx}\phi_{c}+p(c-1)\phi_{c}.\end{aligned}$$ Using the expression of $S''_c(\phi_{c})$ in [\[2.8\]](#2.8){reference-type="eqref" reference="2.8"} again, we obtain $$\begin{aligned} \label{S''f_4-step1} &\quad S''_c(\phi_{c})(3x^2\phi_{c}+x^3\partial_{x}\phi_c)\notag\\ &=c\partial_{xx}(3x^2\phi_{c}+x^3\partial_{x}\phi_c)+(1-c)(3x^2\phi_{c}+x^3\partial_{x}\phi_c)+(p+1)\phi_{c}^p(3x^2\phi_{c}+x^3\partial_{x}\phi_c)\notag\\ &=c(6\phi_{c}+18x\partial_{x}\phi_{c}+9x^2\partial_{xx}\phi_{c}+x^3\partial^3_{x}\phi_{c})+(1-c)(3x^2\phi_{c}+x^3\partial_{x}\phi_c)+(p+1)\phi_{c}^p(3x^2\phi_{c}+x^3\partial_{x}\phi_c)\notag\\ &=6c\phi_{c}+18cx\partial_{x}\phi_{c}+6cx^2\partial_{xx}\phi_{c}+3x^2\big[c\partial_{xx}\phi_{c}+(1-c)\phi_{c}+(p+1)\phi_{c}^{p+1}\big] \notag\\ &\quad +x^3\partial_{x}(c\partial_{xx}\phi_{c}+(1-c)\phi_{c}+\phi_{c}^{p+1}).\end{aligned}$$ Inserting [\[phi_c\^p+1\]](#phi_c^p+1){reference-type="eqref" reference="phi_c^p+1"} into [\[S\'\'f_4-step1\]](#S''f_4-step1){reference-type="eqref" reference="S''f_4-step1"}, and by [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"}, we have $$\begin{aligned} \label{S''f_4} S''_c(\phi_{c})(3x^2\phi_{c}+x^3\partial_{x}\phi_c) =6c\phi_{c}+18cx\partial_{x}\phi_{c}+(6c-3pc)x^2\partial_{xx}\phi_{c}+3p(c-1)x^2\phi_{c}.\end{aligned}$$ Combining [\[f_1\]](#f_1){reference-type="eqref" reference="f_1"}, [\[f_2\]](#f_2){reference-type="eqref" reference="f_2"}, [\[S\'\'phi_c\]](#S''phi_c){reference-type="eqref" reference="S''phi_c"} and [\[S\'\'f_4\]](#S''f_4){reference-type="eqref" reference="S''f_4"}, we finally obtain the concrete expression of $\kappa_c,$ that is $$\begin{aligned} \label{expression-kappa_c} \kappa_c&=S''_c(\phi_{c})\Gamma_c\notag\\ &=\big[B(c)(p+1)c^2-B(c)pc+6cD(c)\big]\phi_{c}+B(c)(1-p)c^2\partial_{xx}\phi_{c}+18cD(c)x\partial_{x}\phi_{c} \notag\\ &\quad +(6c-3pc)D(c)x^2\partial_{xx}\phi_{c}+3p(c-1)D(c)x^2\phi_{c}.\end{aligned}$$ $\bullet$*The numerical result of $\langle \kappa_{c}, \Gamma_{c} \rangle.$* According to [@Pego-1991-eigenvalue], the solution of elliptic equation [\[1.3\]](#1.3){reference-type="eqref" reference="1.3"} $\phi_{c}$ is explicitly given by $$\begin{aligned} \label{phi_c} \phi_{c}(x)=\left[\frac{1}{2}(c-1)(p+2)\right]^{\frac{1}{p}}\,\mathrm{sech}^{\frac{2}{p}}\big(\frac{1}{2}xp\sqrt{\frac{c-1}{c}}\big).\end{aligned}$$ By [\[1.4\]](#1.4){reference-type="eqref" reference="1.4"} and the expression of $\Psi_{c}$ in [\[Psi_c\]](#Psi_c){reference-type="eqref" reference="Psi_c"}, we have $$\begin{aligned} \Psi_{c} &=-\frac{1}{2}\omega^{1-\frac{2}{p}}\partial_{\omega}\psi_{\omega} =-\frac{1}{2}\omega^{1-\frac{2}{p}}\frac{d}{dc}(c^{-\frac{1}{p}}\phi_{c})\cdot\frac{dc}{d\omega} \\ &=c^{1+\frac{1}{p}}\frac{d}{dc}(c^{-\frac{1}{p}}\phi_{c}).\end{aligned}$$ Using [\[phi_c\]](#phi_c){reference-type="eqref" reference="phi_c"}, and by direct computation we obtain that $$\begin{aligned} \label{num-Psi_c} \Psi_{c} &=\phi_{c}\cdot \Big[\frac{1}{p(c-1)}-\frac{x}{2\sqrt{c(c-1)}}\tanh\big(\frac{1}{2}xp\sqrt{\frac{c-1}{c}}\big)\Big], \\ \label{partial_xphi_{c}} \partial_x\phi_{c} &=-\sqrt{\frac{c-1}{c}}\phi_{c}\cdot \tanh\big(\frac{1}{2}xp\sqrt{\frac{c-1}{c}}\big), \\ \label{partial_xxphi_c} \partial_{xx}\phi_c &=\frac{c-1}{c}\phi_{c}\cdot \Big[\tanh^2\big(\frac{1}{2}xp\sqrt{\frac{c-1}{c}}\big)-\frac{1}{2}p \,\mathrm{sech}^{2}\big(\frac{1}{2}xp\sqrt{\frac{c-1}{c}}\big)\Big].\end{aligned}$$ We check $\langle S''_c(\phi_{c})\Gamma_{c},\Gamma_{c}\rangle <0$ in the following case:\ *The limit of integration is $[-50\pi, 50\pi].$* \(1\) for $p=4.1, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-1024.83,$ \(2\) for $p=4.5, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-362.82,$ \(3\) for $p=5, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-292.10,$ \(4\) for $p=6, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-274.60,$ \(5\) for $p=6.5, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-276.36,$ \(6\) for $p=10, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-303.22,$ \(7\) for $p=30, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-445.07,$ \(8\) for $p=50, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-609.47,$ \(9\) for $p=70, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-790.46,$ \(10\) for $p=100, \langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle=-1083.61.$\ The graph of $\langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle$ as a function of p is shown as below: {width="0.8\\linewidth"} # Acknowledgment {#acknowledgment .unnumbered} R. Jia and Y. Wu are partially supported by NSFC 12171356. The authors would like to thank Professor Zeng Chongchun for many valuable discussions and suggestions. # Data Availability {#data-availability .unnumbered} There is no additional data associated to this article. # Declarations {#declarations .unnumbered} ## Conflict of interest {#conflict-of-interest .unnumbered} On behalf of all authors, the corresponding author states that there is no conflict of interest. Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. 99 [T. B.  Benjamin, J. L.  Bona, and J. J.  Mahony,]{.smallcaps} Model equations for long waves in nonlinear dispersive equations, *Philos. Trans. Roy. Soc. London Ser. A*, **272** (1972), 47-78. M. Grillakis, J. Shatah, and W. Strauss, *Stability theory of solitary waves in the presence of symmetry, I*, J. Funct. Anal., 74 (1987), pp. 160--197. M. Grillakis, J. Shatah, and W. Strauss, *Stability theory of solitary waves in the presence of symmetry, II*, J. Funct. Anal., 94 (1990), pp. 308--348. [Z. Lin; C. Zeng,]{.smallcaps} Instability, index theorem, and exponential trichotomy for linear Hamiltonian PDEs. *Mem. Amer. Math. Soc.,* **275** (2022), no. 1347, v+136 pp. [R. Pego and M. Weinstein,]{.smallcaps} Eigenvalues, and instabilities of solitary waves. *Philos. Trans. Roy. Soc. London Ser. A*, **340** (1992), 47--94. [P. E.  Souganidis, and W. A.  Strauss,]{.smallcaps} Instability of a class of dispersive solitary waves, *Proc. R. Soc. Edinb*, **114** (1990), 195-212. [M. Weinstein,]{.smallcaps} Modulational stability of ground states of nonlinear Schrödinger equations, *SIAM J. Math. Anal.*, **16** (1985), 472-491. [M. Weinstein,]{.smallcaps} Lyapunov stability of ground states of nonlinear Schrödinger equations, *Comm. Pure Appl. Math*, **39** (1986), 51-67. [Y. Wu,]{.smallcaps} Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension, *Trans. Amer. Math.Soc.*,**376** (2023), no. 6, 4085--4103. [^1]: According to Appendix [6.4](#numerical result){reference-type="ref" reference="numerical result"}, we use Matlab to compute $\langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle <0.$ It suffices in practice to run the computations until $p=100$ to check $\langle S''_c(\phi_{c})\Gamma_{c}, \Gamma_{c}\rangle <0$ as the inner product is decreasing fastly as a power function when $p$ is bigger than $10.$ We refer to "Appendix [6.4](#numerical result){reference-type="ref" reference="numerical result"}\" for more details.

arxiv_math

--- abstract: | Under investigation in this paper is the dynamics and probability of heads in the toss of a coin with symmetric inhomogeneous density. Such coins are assumed to have diagonal inertia matrix. The rotational motion of the coin is determined by the initial angular momentum and initial position of the coin. We described the dynamic behavior of the unit normal vector and calculated the limiting probability of heads as time goes to infinity with respect to the fixed initial parameters. Our probability formula extends the formula for homogeneous coins by Keller and Diaconis et al. author: - "Shilun Li[^1]" bibliography: - mybibfile.bib title: | Dynamics and Probability in the Toss of a Coin\ with Symmetric Inhomogeneous Density --- coin toss, rigid body, limiting probability, dynamic equations # Introduction The motion of a coin toss can be modeled with a dynamical system governed by mechanics laws, determined entirely on the initial configuration. The outcomes can be random due to the variations in the initial parameters. Several physical mechanisms for randomness in coin toss have been reported, see [@mahadevan2011probability]. Keller considered a specific uniform coin with initial velocity and angular velocity imparted at the instant of tossing [@keller1986probability]. The uniform coin has inertia matrix given by $\text{diag}(I_{x}, I_{y}, I_{z})$ with $I_{x}=I_{y}<I_{z}$, spins without air resistance and lands without bouncing. Assuming that the coin rotates about a horizontal axis that lies along a diameter of the coin, Keller proved that the limiting probability of heads is $50\%$. Building upon Keller's work, Diaconis et al found dynamical bias in the toss of a uniform coin which depends on the angle $\theta$ between initial the angular momentum $\bm{L}$ and normal of heads $\bm{n}$ [@diaconis2007dynamical]. The probability of heads if 50% if and only if $\theta=\frac{\pi}{2}$. If a coin starts out heads, it ends up heads more often. Diaconis et al also measured empirical distributions of $\theta$ from real coin flip experiments and estimated that the probability of heads is $50.83\%$ given the coin starts out heads. While Keller and Diaconis et al neglects air resistance and bouncing of the coin, Vulović and Prange[@vulovic1986randomness] analysed the effect of bouncing on the probability. They found that bouncing adds randomness to the toss which results in an increase in fairness. Yue and Zhang[@zeng1985sensitive] takes into account both bouncing and air resistance. The non-linearity of air resistance and bouncing causes acute sensitivity to initial conditions, adding randomness to the coin toss. On the other hands, Lindley[@lindley1981coin] followed by Gelman et al[@gelman2002you] considered non-uniform coins with mass inhomogeneously distributed. They gave informal arguments without rigorous proofs suggesting that the inhomogeneity of the coin will not affect the probability if the coin is caught in hand. In this paper, we will investigate the dynamical bias of coins with symmetric inhomogeneous density, which is also referred to as non-uniform coins. Non-uniform coins are coins with inertia matrix given by $\text{diag}(I_{x}, I_{y}, I_{z})$ where $I_{x}<I_{y}<I_{z}$. We will neglect the influence of air resistance and bouncing, assuming that the coin rotates freely in the air. # Preliminaries {#sec::2} We will first introduce three coordinate systems centered at the centroid of the coin with orthonormal basis: - Reference frame $\{\bm{i}, \bm{j}, \bm{k}\}$ where $-\bm{k}$ is the direction of gravity and $\bm{i},\bm{j}$ independent of time. - Body fixed frame $\{{\bm{e_1}}, {\bm{e_2}}, \bm{n}\}$ where $\bm{n}$ is the normal to the heads of the coin. - Intermediate frame $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$ where ${\bm{\varepsilon_1}}=\frac{\bm{k}-<\bm{k}, \bm{l}>\bm{l}}{||\bm{k}-<\bm{k}, \bm{l}>\bm{l}||}$, ${\bm{\varepsilon_2}}=\bm{n}\times{\bm{\varepsilon_1}}$, as shown in Figure [2](#coordinate){reference-type="ref" reference="coordinate"}. {#coordinate width="5cm"} {#coordinate width="5cm"} We use superscript r refer to the coordinates of vectors in the reference frame, b for the body fixed frame, and no superscript for basis independent situations. The intermediate frame is only introduced for calculating the rotational matrix between the body fixed frame and the reference frame and its existence will be suppressed in section [3](#sec::normal vec){reference-type="ref" reference="sec::normal vec"}. Angular momentum theorem applies in the reference frame [@murray1993probability; @strzalko2008dynamics]. It tells us the angular momentum $\bm{L^r}$ is conserved in the reference frame since the coin is torque free if we ignore air resistance. Then the coordinates of $\bm{L^r}$ and $\bm{l^r}=\frac{\bm{L^r}}{||\bm{L^r}||}$ are time independent in the reference frame. But the coordinates in body frame are time dependent. Using spherical coordinates, we can write $$\bm{l^r}= (\cos\alpha\sin\beta, \sin\alpha\sin\beta, \cos\beta), \label{lr}$$ and $$\bm{l^b}=(\cos{\varphi_t}\sin{\theta_t}, \sin{\varphi_t}\sin{\theta_t}, \cos{\theta_t}), \label{lb}$$ where $\alpha$, $\beta$ and ${\theta_t}$ are shown in Figure [2](#coordinate){reference-type="ref" reference="coordinate"}. Any orthonormal basis can be rotated to another orthonormal basis by a sequence of three Euler angles [@goldstein2011classical], precession pr, nutation nu, and rotation rt, as shown in the right panel of Figure [2](#coordinate){reference-type="ref" reference="coordinate"}. The rotation matrix (acting by left multiplication) in terms of Euler angles is $$A=\begin{pmatrix} C_{pr}C_{rt}-S_{pr}C_{nu}S_{rt} & -C_{pr}S_{rt}-S_{pr}C_{nu}C_{rt} & S_{pr} S_{nu}\\ S_{pr}C_{rt}+C_{pr}C_{nu}S_{rt} & -S_{pr}S_{rt}+C_{pr}C_{nu}C_{rt} & -C_{pr}S_{nu}\\ S_{nu}S_{rt} & S_{nu}C_{rt} & C_{nu}\\ \end{pmatrix}, \label{Amatrix}$$ where $S$ and $C$ denote the trigonometric functions $\sin$ and $\cos$, e.g. $S_{nu},\ C_{nu}$ denote $\sin(nu),\ \cos(nu)$, respectively. Intermediate frame acts as a bridge between the body frame and the reference frame. Let $A_1$ be the rotation matrix described by Euler angles $\{pr_1, nu_1, rt_1\}$ from $\{\bm{i},\bm{j},\bm{k}\}$ to $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$, $A_2$ be the rotation matrix described by Euler angles $\{pr_2, nu_2, rt_2\}$ from $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$ to $\{{\bm{e_1}},{\bm{e_2}},\bm{n}\}$. We have, $$(\bm{i},\bm{j},\bm{k})\xrightarrow[]{(pr_1, nu_1, rt_1)} ({\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}) \xrightarrow[]{(pr_2, nu_2, rt_2)} ({\bm{e_1}},{\bm{e_2}},\bm{n})$$ From the definition of the basis $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$, we have $\bm{k}= (\sin\beta){\bm{\varepsilon_1}}+ (\cos\beta)\bm{l}$. So the coordinates of $\bm{k}$ in the system $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$ and $\{\bm{i},\bm{j},\bm{k}\}$ are $\big(\sin\beta,0,\cos\beta\big)$ and $(0,0,1)$, respectively. In addition, the coordinates of $\bm{l}$ in the system $\{\bm{i},\bm{j},\bm{k}\}$ and $\{{\bm{\varepsilon_1}},{\bm{\varepsilon_2}},\bm{l}\}$ are $\big(\cos\alpha\sin\beta,\sin\alpha\sin\beta,\cos\beta\big)$ and $(0,0,1)$, respectively. Then the rotation matrix $A_1$ satisfies $$\begin{cases} \left(\sin\beta,0,\cos\beta\right)^T = A_1^T (0,0,1)^T,\\ \left(\cos\alpha\sin\beta,\sin\alpha\sin\beta,\cos\beta\right)^T = A_1 (0,0,1)^T. \end{cases}$$ The components of $A_1$ have form ([\[Amatrix\]](#Amatrix){reference-type="ref" reference="Amatrix"}). Those equations imply $(pr_1,\ nu_1, \ rt_1)=(\alpha + \frac{\pi}{2},\beta, \frac{\pi}{2})$ or $(\alpha - \frac{\pi}{2},-\beta, -\frac{\pi}{2})$. Then we obtain $$\label{A1 matrix} A_1 = \begin{pmatrix} -C_\alpha C_\beta & S_\alpha & C_\alpha S_\beta\\ -S_\alpha C_\beta & -C_\alpha & S_\alpha S_\beta\\ S_\beta & 0 & C_\beta\\ \end{pmatrix}$$ Let ${\psi_t}$ be the dihedral angle $\bm{k}-\bm{l}-\bm{n}$, the dynamic angle of the plane spanned by $\bm{n}(t)$ and $\bm{l}$ rotating around the plane spanned by $\bm{k}$ and $\bm{l}$, as shown in Figure [2](#coordinate){reference-type="ref" reference="coordinate"}. Then ${\psi_t}\text{ mod } 2\pi$ is also the longitude of $\bm{n}$ in the intermediate frame. Similarly, we have $(pr_2,\ nu_2, \ rt_2)= ( {\psi_t}+ \frac{\pi}{2},{\theta_t}, \frac{\pi}{2} - {\varphi_t})$ or $( {\psi_t}- \frac{\pi}{2},-{\theta_t}, -\frac{\pi}{2} - {\varphi_t})$. Therefore $$\label{A2 matrix} A_2 = \begin{pmatrix} -S_{\psi_t}S_{\varphi_t}-C_{\psi_t}C_{\theta_t}C_{\varphi_t}& S_{\psi_t}C_{\varphi_t}- C_{\psi_t}C_{\theta_t}S_{\varphi_t}& C_{\psi_t}S_{\theta_t} \\ C_{\psi_t}S_{\varphi_t}- S_{\psi_t}C_{\theta_t}C_{\varphi_t}& -C_{\psi_t}C_{\varphi_t}- S_{\psi_t}C_{\theta_t}S_{\varphi_t}& S_{\psi_t}S_{\theta_t} \\ S_{\theta_t}C_{\varphi_t}& S_{\theta_t}S_{\varphi_t}& C_{\theta_t} \end{pmatrix}$$ # The evolution of normal vector $\bm{n^r}$ {#sec::normal vec} ## Dynamic equations of angular momentum $\bm{l^b}$ The coin rotates freely not subject to any net forces or torques around the fixed centroid. This is a classical Euler-Poinsot problem. The dynamic equations are given in Landau[@landau1969mechanics] by $$\begin{cases} \frac{d}{dt}{\bm{L^b}_x}= (I_{z}^{-1}- I_{y}^{-1}){\bm{L^b}_x}{\bm{L^b}_z},\\ \frac{d}{dt}{\bm{L^b}_y}= (I_{x}^{-1}- I_{z}^{-1}){\bm{L^b}_z}{\bm{L^b}_x},\\ \frac{d}{dt}{\bm{L^b}_z}= (I_{y}^{-1}- I_{x}^{-1}){\bm{L^b}_x}{\bm{L^b}_y}. \end{cases} \label{dynamic_equation_L}$$ Or in terms of Euler angles, $$\begin{cases} \frac{d{\psi_t}}{dt}= ||\bm{L}||\left(\frac{\cos^2{\varphi_t}}{I_{x}}+\frac{\sin^2{\varphi_t}}{I_{y}}\right),\\ \frac{d{\varphi_t}}{dt}=||\bm{L}||\cos {\theta_t}\left(\frac{\cos^2{\varphi_t}}{I_{x}}+\frac{\sin^2{\varphi_t}}{I_{y}}-\frac{1}{I_{z}}\right),\\ \frac{d{\theta_t}}{dt}=\frac{||\bm{L}||}{2}\left(I_{x}^{-1}-I_{y}^{-1}\right)\sin{\theta_t}\sin(2{\varphi_t}).\\ \end{cases} \label{dynamic_euqation_angles}$$ Note that for non-uniform coins, there is no explicit analytical solution for $\bm{L^b}(t)$. The rotational kinetic energy of the coin is given by $$E =\frac{1}{2}\bigg(\frac{({\bm{L^b}_x})^2}{I_{x}}+\frac{({\bm{L^b}_y})^2}{I_{y}}+\frac{({\bm{L^b}_z})^2}{I_{z}}\bigg) = \frac{||\bm{L}||^2}{2}\bigg(\frac{({\bm{l^b}_x})^2}{I_{x}}+\frac{({\bm{l^b}_y})^2}{I_{y}}+\frac{({\bm{l^b}_z})^2}{I_{z}}\bigg) \label{energy_sphere}$$ which is constant with respect to $t$. Therefore, $\bm{l^b}$ must lie on the fixed ellipsoid $\frac{x^2}{I_{x}}+\frac{y^2}{I_{y}}+\frac{z^2}{I_{z}}=\frac{2E}{||\bm{L}||}$ and the sphere $x^2+y^2+z^2=1$ in the body fixed frame for all $t$. The intersection is a closed curve as shown in Figure [4](#angular momentum traj){reference-type="ref" reference="angular momentum traj"}. So $\bm{l^b}$ is periodic. In the special case of uniform coins, the angular velocity or angular momentum rotates and traces out a circle in the body-fixed frame. {#angular momentum traj width="5cm"} {#angular momentum traj width="5cm"} ## The normal $\bm{n^r}$ Based on the evolution of $\bm{L^b}$ in the body frame, and the motion of $\bm{L^b}$ relative to normal vector $\bm{n}$, we can further derive the evolution of $\bm{n^r}$ in reference frame. **Theorem 1**. *Given an initial angular momentum $$\bm{L^r}= ||\bm{L}||\left(\cos\alpha\sin\beta, \sin\alpha\sin\beta, \cos\beta\right).$$ Then at time $t$, the unit normal vector $$\begin{aligned} \bm{n^r}=& \begin{pmatrix} -C_{\alpha}C_\beta C_{\psi_t}S_{\theta_t}+S_{\alpha} S_{\psi_t}S_{\theta_t}+ C_{\alpha}S_\beta C_{\theta_t}\\ -S_{\alpha}C_\beta C_{\psi_t}S_{\theta_t} -C_{\alpha} S_{\psi_t}S_{\theta_t} +S_{\alpha} S_{\beta} C_{\theta_t}\\ S_{\beta} C_{\psi_t}S_{\theta_t} +C_{\beta} C_{\theta_t}\\ \end{pmatrix},\end{aligned}$$ where $({\varphi_t}, {\theta_t}, {\psi_t})$ are determined by equations ([\[dynamic_euqation_angles\]](#dynamic_euqation_angles){reference-type="ref" reference="dynamic_euqation_angles"}).* *Proof.* The theorem directly follows from equations ([\[A1 matrix\]](#A1 matrix){reference-type="ref" reference="A1 matrix"}), ([\[A2 matrix\]](#A2 matrix){reference-type="ref" reference="A2 matrix"}) and $\bm{n^r}= A_1A_2(0,0,1)^T$. ◻ In Figure [6](#normal traj){reference-type="ref" reference="normal traj"}, the coin is heads up when $\bm{n^r}$ is at the north hemisphere and tails is up otherwise. The figures show that $\bm{n}$ precesses around the angular momentum $\bm{l}$. For uniform coins, $\bm{n}$ spin around $\bm{l}$ in a circle, and the angle ${\theta_t}$ between $\bm{l}$ and $\bm{n}$ stay constant. For non-uniform coins, $\bm{n}$ spin with nutation around $\bm{l}$ in a ring between 2 parallel circles. {#normal traj width="5cm"} {#normal traj width="5cm"} From Theorem [Theorem 1](#normal evolution){reference-type="ref" reference="normal evolution"}, we obtain the criterion for the coin landing heads up: **Corollary 1**. *$\bm{n^r}(t)$ satisfies $$\bm{n^r}(t)\cdot\bm{k}= \cos\beta\cos{\theta_t}+ \sin\beta\sin{\theta_t}\cos{\psi_t}, \label{criteria_eqation}$$ and the coin is head up at time $t$ if and only if $$\sin\beta\sin{\theta_t}\cos{\psi_t}>-\cos\beta\cos{\theta_t}.$$ [\[col::criteria\]]{#col::criteria label="col::criteria"}* Equation ([\[criteria_eqation\]](#criteria_eqation){reference-type="ref" reference="criteria_eqation"}) is just the law of cosines for the spherical triangle, the shaded part in the left panel in Figure [2](#coordinate){reference-type="ref" reference="coordinate"}, which is formed by the endpoints of unit vectors $\bm{n},\bm{k}$ and $\bm{l}$, **Remark 1**. ***(property on precession ${\psi_t}$)** [\[remark::precession\]]{#remark::precession label="remark::precession"} For the uniform coins with $I_{x}=I_{y}$, $\bm{n}$ precesses around $\bm{l}$ at a constant speed $||\bm{L}||I_{x}^{-1}$. However, for non-uniform coins, the $\bm{n}$ precesses around $\bm{l}$ at speed varying from $||\bm{L}||I_{y}^{-1}$ to $||\bm{L}||I_{x}^{-1}$.* **Remark 2**. ***(property on nutation ${\theta_t}$)** [\[remark::nutation\]]{#remark::nutation label="remark::nutation"} For uniform coins, ${\theta_t}$ is constant. For non-uniform coins ${\theta_t}$ varies periodically from $\theta_m$ to $\theta_M$, which is given by: $$\begin{aligned} \theta_m&=\arccos(\sqrt{c_2}),\ \theta_M=\pi-\arccos(\sqrt{c_2}), &\text{ if } c_1<0,\\ \theta_m&=\arccos(\sqrt{c_2}),\ \theta_M=\arccos(\sqrt{c_1}), &\text{ if } c_2\geq 0 \text{ and } {\theta_0}\in[0,\frac{\pi}{2}],\\ \theta_m&=\pi-\arccos(\sqrt{c_1}),\ \theta_M=\pi-\arccos(\sqrt{c_2}), &\text{ if }c_1\geq 0\text{ and } {\theta_0}\in(\frac{\pi}{2},\pi].\end{aligned}$$ with $c_1,c_2$ given by ([\[c1\]](#c1){reference-type="ref" reference="c1"}), ([\[c2\]](#c2){reference-type="ref" reference="c2"}).* *Proof.* Since $||\bm{l^b}||=1$ and $\bm{l^b}$ satisfies equation ([\[energy_sphere\]](#energy_sphere){reference-type="ref" reference="energy_sphere"}). Adopting Lagrange method, the extremums of ${\bm{l^b}_z}$, locating on ${\bm{l^b}_x}=0$ or ${\bm{l^b}_y}=0$, are given by $$c_1=\cos^2{\theta_0}-\frac{I_{x}^{-1}-I_{y}^{-1}}{I_{y}^{-1}-I_{z}^{-1}}\cos^2{{\varphi_0}}\sin^2{{\theta_0}} \label{c1}$$ $$c_2=\cos^2{{\theta_0}}+\frac{I_{x}^{-1}-I_{y}^{-1}}{I_{x}^{-1}-I_{z}^{-1}}\sin^2{{\varphi_0}}\sin^2{{\theta_0}} \label{c2}$$ Since ${\bm{l^b}_z}=\cos{\theta_t}$, For the uniform case, $c_1=c_2$ so ${\theta_t}$ is constant. For the non-uniform case, we have the desired result by considering the three situations: $c_1<0$, $c_2\geq 0$ and ${\theta_0}\in[0,\frac{\pi}{2}]$, $c_1\geq 0$ and ${\theta_0}\in(\frac{\pi}{2},\pi]$. ◻ $\theta_m$ and $\theta_M$ are are independent of $||\bm{L}||$. They are determined by $I_{x}, I_{y}$ and $I_{z}$. For non-uniform coins, $\theta_m$ and $\theta_M$ are either supplementary, both acute, or both obtuse. When the initial ${\theta_0}$ is close enough to $\frac{\pi}{2}$, contained in $$S_F=\{({\varphi_0}, {\theta_0})|\cot^2{\theta_0} <\frac{I_{x}^{-1}-I_{y}^{-1}}{I_{y}^{-1}-I_{z}^{-1}}\cos^2{{\varphi_0}}\}, \label{SF}$$ then $\theta_m$ and $\theta_M$ are supplementary. Under this condition, the two boundary circles perpendicular to $\bm{l}$ corresponding to $\theta_m$ and $\theta_M$ in Figure [6](#normal traj){reference-type="ref" reference="normal traj"} are centered symmetrically around the spherical center. Let us denote the fair region as the set of initial parameters $({\varphi_0}, {\theta_0})$ such that the proportions of \"heads\" zone and \"tails\" zone of $\bm{n^r}$ are equal (to 50%). $S_F$ in ([\[SF\]](#SF){reference-type="ref" reference="SF"}) is the fair region for non-uniform coins. On the other hand, for uniform coins, the proportion of \"heads\" zone is 50% if and only if ${\theta_0}=\frac{\pi}{2}$. The fair region is shown in Figure [7](#fair region){reference-type="ref" reference="fair region"}. {#fair region width="7cm"} The probability of heads (which we will formulate in section [4](#sec::probability){reference-type="ref" reference="sec::probability"}) is approximately the proportion of the \"heads\" zone of $\bm{n^r}$. So we can assume the coin is fair when the initial parameters $({\varphi_0},{\theta_0})$ is inside the fair region. # Probability of Heads {#sec::probability} As the orientation of the coin is determined by $||\bm{L}||$, $({\varphi_0},{\theta_0})$ and $t$, we will define the probability of heads $p$ as the limiting probability of $\bm{n^r}_z>0$ as $t\rightarrow\infty$ given $({\varphi_0},{\theta_0})$ and a distribution on $||\bm{L}||$. When $({\varphi_0}, {\theta_0})$ is in the fair region, referred to as the fair case, we assume $p$ is 50%. So let us now consider the situation where $({\varphi_0}, {\theta_0})$ is outside the fair region. Notice that in this situation, either ${\theta_t}<\frac{\pi}{2}$ for all $t$ as show in the left of Figure [4](#angular momentum traj){reference-type="ref" reference="angular momentum traj"}, which we refer to as the acute case, or ${\theta_t}>\frac{\pi}{2}$ for all $t$, which we refer to as the obtuse case. The key to obtaining $p$ is to obtain the limiting joint distribution of $({\psi_t},{\varphi_t})$. However, it is suffice to obtain the limiting distribution of $({\psi_t}\text{ mod } 2\pi,{\varphi_t}\text{ mod } 2\pi)$. This will rely on the following lemma about limit distributions. **Lemma 2**. *If $X$ is a random variable with characteristic function vanishing at infinity. Let $g_1,...,g_n:\mathbb{R}\rightarrow\mathbb{R}$ be real valued continuous function such that $\lim_{t\rightarrow \infty}|\sum_{i=1}^n m_i g_i(t)|=\infty$ for any $(m_1,..,m_n)\in \mathbb{Z}^n\setminus \{0\}$. Then $(g_1(t)X\mod 1,...,g_n(t)X\mod 1)$ converges in distribution to $\mathcal{U}[0,1]^n$ as $t\rightarrow\infty$.* *Proof.* Let us consider the case where $n=2$, cases where $n>2$ follows similarly. Let $Y_1=(g_1(t)X\text{ mod } 1)$ and $Y_2=(g_2(t)X\text{ mod } 1)$. The cumulative distribution function and characteristic function of $X$ are denoted by $F(x)$ and $\Phi_X$, respectively. The characteristic function $\Phi_Y$ of $(Y_1,Y_2)$ is determined by $\Phi_X$. Consider $\Phi_Y(2\pi m_1,2\pi m_2)$ for any $(m_1,m_2)\in \mathbb{Z}^2$, which are called Fourier coefficients in Engel [@engel1992road], we have $$\begin{aligned} &\Phi_Y(2\pi m_1,2\pi m_2)\\ =&\mathbb{E}\left[\exp\{i(2\pi m_1Y_1+2\pi m_2Y_2\}\right]\\ =&\int_{-\infty}^\infty \exp\{2\pi i( m_1Y_1+ m_2Y_2)\} dF(x)\\ =&\sum_{k_1,k_2\in\mathbb{Z}} \int_{I_{k_1,k_2}} \exp\{2\pi i( m_1Y_1+ m_2Y_2)\} dF(x) \\ =&\sum_{k_1,k_2\in\mathbb{Z}} \int_{I_{k_1,k_2}} \exp\{2\pi i[ m_1(g_1(t)X-k_1)+ m_2(g_2(t)X-k_2)]\} dF(x) \\ =&\sum_{k_1,k_2\in\mathbb{Z}} \int_{I_{k_1,k_2}} \exp\{2\pi i(m_1g_1(t)X+ m_2g_2(t)X)\} dF(x) \\ =&\int_{-\infty}^\infty \exp\{2\pi i( m_1g_1(t)X+ m_2g_2(t)X)\} dF(x)\\ =&\Phi_X[m_1g_1(t)+m_2g_2(t)]\end{aligned}$$ where $I_{k_1,k_2}=[k_1,k_1+1)\times [k_2,k_2+1)$. When $t\rightarrow\infty$, $m_1g_1(t)+m_2g_2(t)\rightarrow\infty$ and $\Phi_X[m_1g_1(t)+m_2g_2(t)]\rightarrow 0$. So $$\lim_{t\rightarrow\infty} \Phi_Y(2\pi m_1,2\pi m_2) = 0$$ for all $(m_1,m_2)\in\mathbb{Z}^2\setminus\{(0,0)\}$. Since $Y$ is supported by $[0,1]\times[0,1]$, and the Fourier coefficients of $\mathcal{U}[0,1]^2$ are zero, according to page 361 of Billingsley[@billingsley1968probability], $$Y \xrightarrow[]{\mathcal{D}} \mathcal{U}[0,1]^2$$ which completes the proof. ◻ **Lemma 3**. *Suppose ${\theta_t}<\frac{\pi}{2}$ for all $t$ or ${\theta_t}>\frac{\pi}{2}$ for all $t$. For all Schwartz densities of $||\bm{L}||$, all initial parameters $({\varphi_0},{\theta_0},I_{x},I_{y},I_{z})$ excluding a measure 0 set, when $t\rightarrow \infty$, we have $({\psi_t}\text{ mod } 2\pi,{\varphi_t}\text{ mod } 2\pi)\xrightarrow[]{\mathcal{D}} \mathcal{U}[0,2\pi]^2$ in distribution. [\[joint uniform distribution\]]{#joint uniform distribution label="joint uniform distribution"}* *Proof.* Recall that $({\bm{l^b}_x},{\bm{l^b}_y},{\bm{l^b}_z})$ lies on a closed curve and is periodic with some period $T$ and $({\varphi_t}, {\psi_t})$ is its spherical coordinates. So $({\varphi_t}, {\psi_t})$ has period $T$. Let us denote $c_1$ as the maximum of ${\bm{l^b}_z}$ on the curve and $c_2$ as the minimum of ${\bm{l^b}_z}$ on the curve. Define $$\label{h} h(t):=\int_{0}^t \cos\theta_\tau \left(\frac{\cos^2\varphi_\tau}{I_{x}}+\frac{\sin^2\varphi_\tau}{I_{y}}-\frac{1}{I_{z}}\right)d\tau,$$ and $$\label{g} g(t):=\int_0^t \left(\frac{\cos^2\varphi_\tau}{I_{x}}+\frac{\sin^2\varphi_\tau}{I_{y}}\right)d\tau.$$ From ([\[dynamic_euqation_angles\]](#dynamic_euqation_angles){reference-type="ref" reference="dynamic_euqation_angles"}), we have $${\varphi_t}= {\varphi_0}+ ||\bm{L}||h(t),\quad {\psi_t}= {\psi_0}+ ||\bm{L}||g(t)$$ Now since $||\bm{L}||$ is Schwartz, the characteristic function vanishes at infinity. By Lemma [Lemma 2](#lem::limit lemma){reference-type="ref" reference="lem::limit lemma"}, it is suffice for us to show $$\lim_{t\rightarrow\infty}|m_1 h(t)+m_2 g(t)|=\infty,\quad \forall m_1,m_2\in\frac{1}{2\pi}\mathbb{Z}^2\setminus\{0\}$$ which is equivalent to the condition $$m_1 h(T)+m_2 g(T)\neq 0,\quad \forall m_1,m_2\in\frac{1}{2\pi}\mathbb{Z}^2\setminus\{0\} \label{condition not 0}$$ since $h'(t)$ and $g'(t)$ is periodic with period $T$. By ([\[energy_sphere\]](#energy_sphere){reference-type="ref" reference="energy_sphere"}), we get the equality: $$%\begin{align*} \frac{\sin^2{\theta_t}}{\sin^2{\theta_0}} = \frac{\frac{2E}{||\bm{L}||^2}-I_{z}^{-1}}{I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_t}}\\ = \frac{I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_0}}{I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_t}}. \label{sin theta_t} % \end{align*}$$ Using equation ([\[sin theta_t\]](#sin theta_t){reference-type="ref" reference="sin theta_t"}), we can rewrite $m_1h'(t)+m_2g'(t)$ as $$m_1h'(t)+m_2g'(t) = \frac{a_0 (m_1\cos{\theta_t}+m_2)}{\sin^2{\theta_t}}+\frac{m_2}{I_{z}}=\frac{a_0 (m_1{\bm{l^b}_z}+m_2)}{1-({\bm{l^b}_z})^2}+\frac{m_2}{I_{z}}$$ where $a_0=\sin^2{\theta_0}\left[I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_0}\right]$. To make a change of variable, we need $\frac{dz(t)}{dt}$. From ([\[dynamic_euqation_angles\]](#dynamic_euqation_angles){reference-type="ref" reference="dynamic_euqation_angles"}) and the relation $$({\bm{l^b}_x})^2=\frac{a_0-I_{y}^{-1}+(I_{y}^{-1}-I_{z}^{-1})({\bm{l^b}_z})^2}{I_{x}^{-1}-I_{y}^{-1}},\quad ({\bm{l^b}_y})^2=\frac{-a_0+I_{x}^{-1}-(I_{x}^{-1}-I_{z}^{-1})({\bm{l^b}_z})^2}{I_{x}^{-1}-I_{y}^{-1}}$$ given by ([\[energy_sphere\]](#energy_sphere){reference-type="ref" reference="energy_sphere"}), we have $$\frac{d{\bm{l^b}_z}}{dt}=\pm||\bm{L}||\sqrt{A({\bm{l^b}_z})^4+B({\bm{l^b}_z})^2+C}$$ for $$\begin{aligned} A=& -(I_{x}^{-1}-I_{z}^{-1})(I_{y}^{-1}-I_{z}^{-1}),\\ B=& \frac{2I_{z}-I_{x}-I_{y}}{I_{x}I_{y}I_{z}}-a_0(I_{x}^{-1}+I_{y}^{-1}-2I_{z}^{-1}),\\ C=& -(a_0-I_{x}^{-1})(a_0-I_{y}^{-1}),\end{aligned}$$ where $+$ is taken when ${\bm{l^b}_z}$ is going from $c_1$ to $c_2$ and $-$ is taken when ${\bm{l^b}_z}$ is going from $c_2$ to $c_1$. So with a change of variables from $t$ to ${\bm{l^b}_z}$, we can finally express the condition in ([\[condition not 0\]](#condition not 0){reference-type="ref" reference="condition not 0"}) as $$m_1h(T)+m_2g(T)=\frac{2}{||\bm{L}||}\int_{c_1}^{c_2}\frac{\frac{a_0 (m_1z+m_2)}{1-z^2}+\frac{m_2}{I_{z}}}{\sqrt{Az^4+Bz^2+C}}dz\neq 0,\quad\forall (m_1,m_2)\in\frac{1}{2\pi}\mathbb{Z}^2\setminus\{0\}$$ It is very rare that one of the countable the integral equations $$\int_{c_1}^{c_2}\frac{\frac{a_0 (m_1z+m_2)}{1-z^2}+\frac{m_2}{I_{z}}}{\sqrt{Az^4+Bz^2+C}}dz = 0 \label{integral equation}$$ has a solution. We will assume that the set of $({\varphi_0},{\theta_0},I_{x},I_{y},I_{z})$ such that ([\[integral equation\]](#integral equation){reference-type="ref" reference="integral equation"}) has a solution for some $(m_1,m_2)\in\frac{1}{2\pi}\mathbb{Z}^2\setminus\{0\}$ has Lebesgue measure 0 in $\mathbb{R}^5$. The proof will be left as an open problem. So for all $({\varphi_0},{\theta_0},I_{x},I_{y},I_{z})$ excluding a measure 0 set, the condition to Lemma [Lemma 2](#lem::limit lemma){reference-type="ref" reference="lem::limit lemma"} is satisfied. According to the lemma, $$({\psi_t}\text{ mod } 2\pi,{\varphi_t}\text{ mod } 2\pi)\xrightarrow[]{\mathcal{D}} \mathcal{U}[0,2\pi]^2$$ as $t\rightarrow\infty$ which completes the proof. ◻ Let us assume from now on that the parameters $({\varphi_0},{\theta_0},I_{x},I_{y},I_{z})$ are not in the measure 0 set of Lemma [Lemma 4](#lem::limit distribution theta){reference-type="ref" reference="lem::limit distribution theta"}. Now it remains for us to find the distribution of ${\theta_t}$ before we can calculate the probability of heads. The following lemma obtains the distribution of ${\theta_t}$ via its relation with ${\varphi_t}$: **Lemma 4**. *Suppose ${\theta_t}<\frac{\pi}{2}$ for all $t$ or ${\theta_t}>\frac{\pi}{2}$ for all $t$. ${\theta_t}$ is a function of $\cos (2{\varphi_t})$ independent of $({\psi_t}\text{ mod } 2\pi)$. When $t\rightarrow\infty$, we have $\csc^2{\theta_t}\xrightarrow[]{\mathcal{D}}\text{Arcsine}(a, b)$ for some $a,b$ where $\text{Arcsine}(a, b)$ denotes the Arcsine distribution on $(a,b)$. Moreover, the limiting pdf of ${\theta_t}$ as $t\rightarrow\infty$ is given by $$f_\theta(y)=\frac{2|\cot y|}{\pi\sqrt{|(1-\csc^2\theta_m\sin^2 y)(\csc^2\theta_M\sin^2 y-1)|}}.$$* *Proof.* By ([\[sin theta_t\]](#sin theta_t){reference-type="ref" reference="sin theta_t"}) we get $$\sin^2{\theta_t}=\frac{\sin^2{\theta_0}\left[I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_0}\right]}{I_{x}^{-1}-I_{z}^{-1}-(I_{x}^{-1}-I_{y}^{-1})\sin^2{\varphi_t}} =\frac{1}{k_1\cos(2{\varphi_t})+k_2}$$ for some $k_1,k_2$ determined by $I_{x}, I_{y}, I_{z}$,${\varphi_0}$ and ${\theta_0}$. By Lemma [Lemma 3](#lem::limit distribution phi psi){reference-type="ref" reference="lem::limit distribution phi psi"}, we know that ${\varphi_t}\xrightarrow[]{\mathcal{D}} \mathcal{U}[0,2\pi]$ as $t\rightarrow\infty$. So we have $$\csc^2{\theta_t}\xrightarrow[]{\mathcal{D}} \text{Arcsin}(k_2-k_1, k_2+k_1)$$ as $t\rightarrow\infty$. Comparing the expression of $k_1,k_2$ with the formula of $\theta_m, \theta_M$, we conclude that in the acute case: $$\csc^2{\theta_t}\xrightarrow[]{\mathcal{D}}\text{Arcsine}(\csc^2\theta_M,\csc^2\theta_m),$$ and in the obtuse case: $$\csc^2{\theta_t}\xrightarrow[]{\mathcal{D}}\text{Arcsine}(\csc^2\theta_m,\csc^2\theta_M),$$ as $t\rightarrow\infty$. So in both acute and obtuse cases, the limiting pdf of $\csc^2{\theta_t}$ is given by: $$f(x)=\frac{1}{\pi\sqrt{|(x-\csc^2\theta_m)(\csc^2\theta_M-x)|}}.$$ So the limiting pdf of ${\theta_t}$ is given by: $$\label{pdf of thetat} f_\theta(y)=\frac{2|\cot y|}{\pi\sqrt{|(1-\csc^2\theta_M\sin^2 y)(\csc^2\theta_m\sin^2 y-1)|}}.$$ which completes the proof. ◻ Now with the result of Lemma [Lemma 3](#lem::limit distribution phi psi){reference-type="ref" reference="lem::limit distribution phi psi"} and Lemma [Lemma 4](#lem::limit distribution theta){reference-type="ref" reference="lem::limit distribution theta"}, we can start calculating the probability of heads. **Theorem 5**. *For all Schwartz densities of $||\bm{L}||$, the limiting probability of heads as $t\rightarrow\infty$ with $({\varphi_0}, {\theta_0})$ fixed, is given by $$\label{probability formula} p(\beta,{\varphi_0},{\theta_0})=\frac{1}{2}+\frac{1}{\pi^2}\int_{\theta_m({\varphi_0},{\theta_0})}^{\theta_M({\varphi_0},{\theta_0})}\arcsin(\min\{1, \cot\beta\cot y\})f_\theta(y) dy,$$ with $f_\theta(y)$ is given in Lemma [Lemma 4](#lem::limit distribution theta){reference-type="ref" reference="lem::limit distribution theta"}. In the special case when $I_{x}=I_{y}<I_{z}$(uniform coins), $$\label{probability formula uniform} p(\beta,{\varphi_0},{\theta_0})=p(\beta,{\theta_0})=\frac{1}{2}+\frac{1}{\pi}\arcsin(\min\{1, \cot\beta\cot {\theta_0}\}).$$* *Proof.* Let us first consider the case where ${\theta_t}<\frac{\pi}{2}$ for all $t$ or ${\theta_t}>\frac{\pi}{2}$ for all $t$. By Lemma [Lemma 4](#lem::limit distribution theta){reference-type="ref" reference="lem::limit distribution theta"}, the limiting pdf of ${\theta_t}$ is $f_\theta(y)$, given in [\[pdf of thetat\]](#pdf of thetat){reference-type="ref" reference="pdf of thetat"}.By Lemma [Lemma 3](#lem::limit distribution phi psi){reference-type="ref" reference="lem::limit distribution phi psi"}, the limiting pdf of ${\psi_t}\text{ mod } 2\pi$ is $f_\psi(x)=\frac{1}{2\pi}$. Let us define $$\begin{aligned} D=\{x\in(0,2\pi),y\in(\theta_m,\theta_M)|\cos(x)>-\cot(\beta)\cot(y)\},\\ D_1=\{x\in(0,\pi),y\in(\theta_m,\theta_M)|\cos(x)>-\cot(\beta)\cot(y)\}.\end{aligned}$$ By Corollary [\[col::criteria\]](#col::criteria){reference-type="ref" reference="col::criteria"}, $D$ is the region where the coin is heads up. Thus the limiting probability of heads is given by $$\begin{aligned} p(\beta,{\varphi_0},{\theta_0}) =&\iint_D f_\psi(x)f_\theta(y) dx dy\\ =& 2\iint_{D_1} f_\psi(x)f_\theta(y) dx dy\\ =&\frac{1}{\pi}\iint_{D_1} f_\theta(y) dx dy\\ =&\frac{1}{\pi}\int_{\theta_m}^{\theta_M} \arccos(-\min\{1, \cot\beta\cot y\})f_\theta(y)dy\\ =&\frac{1}{2}+\frac{1}{\pi}\int_{\theta_m}^{\theta_M}\arcsin(\min\{1, \cot\beta\cot y\})f_\theta(y)dy\end{aligned}$$ as desired. When $({\varphi_0},{\theta_0})$ lies in the fair region, $\theta_m$ and $\theta_M$ are complementary and $p=\frac{1}{2}$. Notice that ([\[probability formula\]](#probability formula){reference-type="ref" reference="probability formula"}) still holds as $$\int_{\theta_m}^{\theta_M}\arcsin(\min\{1, \cot\beta\cot y\})f_\theta(y)dy =0 \label{integral is 0}$$ since $$\arcsin(\min\{1, \cot\beta\cot y\})f_\theta(y)=-\arcsin(\min\{1, \cot\beta\cot (\pi-y)\})f_\theta(\pi-y). \label{odd function equation}$$ So ([\[probability formula\]](#probability formula){reference-type="ref" reference="probability formula"}) holds for all situations of ${\theta_t}$. In the special case when $I_{x}=I_{y}<I_{z}$, $\theta_m=\theta_M$ and the integral is integrated at a direct delta distribution at ${\theta_0}$, which gives us $$p(\beta,{\varphi_0},{\theta_0})=p(\beta,{\theta_0})=\frac{1}{2}+\frac{1}{\pi}\arcsin(\min\{1, \cot\beta\cot {\theta_0}\})$$ as desired. ◻ We can also see from Equation ([\[odd function equation\]](#odd function equation){reference-type="ref" reference="odd function equation"}) that it is natural to assume the probability is 50% in the fair case. Note that $\bm{l^b}$ traces out a curve symmetric along the x-y plane and as shown on the right of Figure [4](#angular momentum traj){reference-type="ref" reference="angular momentum traj"}. If the limiting distribution of $\bm{l^b}_z$ as $t\rightarrow\infty$ exists, then since $\bm{l^b}_z(t\mod T)=-\bm{l^b}_z(-t\mod T)$, the limiting distribution of $\bm{l^b}_z$ will be symmetric along the x-y plane. Thus by Equation ([\[odd function equation\]](#odd function equation){reference-type="ref" reference="odd function equation"}) and ${\theta_t}=\arccos(\bm{l^b}_z)$, the integral in ([\[integral is 0\]](#integral is 0){reference-type="ref" reference="integral is 0"}) is 0, giving us a probability of heads of 50%. Usually, coin flips tend to start with $\beta={\theta_0}$, i.e. face of the coin facing straight up. An immediate corollary to Theorem [Theorem 5](#thm::probability heads){reference-type="ref" reference="thm::probability heads"} is **Corollary 2**. *With the assumptions of Theorem [Theorem 5](#thm::probability heads){reference-type="ref" reference="thm::probability heads"} and further assuming that heads is facing straight up at the initial position, the limiting probability of heads as $t\rightarrow\infty$ with $({\varphi_0},{\theta_0})$ fixed, is given by $$\label{probability formula straight} p({\varphi_0},{\theta_0})=\frac{1}{2}+\frac{1}{\pi^2}\int_{\theta_m({\varphi_0},{\theta_0})}^{\theta_M({\varphi_0},{\theta_0})}\arcsin(\min\{1, \cot{\theta_0}\cot y\})f_\theta(y) dy,$$ In the special case when $I_{x}=I_{y}<I_{z}$(uniform coins), $$\label{probability formula uniform straight} p({\varphi_0},{\theta_0})=p({\theta_0})=\frac{1}{2}+\frac{1}{\pi}\arcsin(\min\{1, \cot^2{\theta_0}\}).$$* Formula ([\[probability formula uniform\]](#probability formula uniform){reference-type="ref" reference="probability formula uniform"}) for uniform coins is also shown in Theorem 2 by Diaconis et al.[@diaconis2007dynamical] Furthermore, if the flip is Keller flip (${\theta_0}=\frac{\pi}{2}$), then $p$ is just $\frac{1}{2}$. We can assume that a normal coin toss starts with heads facing straight up. So it remains for us to find the distribution of the initial parameters $({\varphi_0},{\theta_0})$. We use the ${\theta_0}$ values from the 27 real flip experiments by Diaconis et al[@diaconis2007dynamical] to be the empirical distribution of ${\theta_0}$. And we will assume that ${\varphi_0}$ is uniform distributed in $[0,2\pi)$. Then using the probability formula in Corollary [Corollary 2](#col::probability straight){reference-type="ref" reference="col::probability straight"}, we can calculate the probability of heads of a normal non-uniform coin toss with the assistance of the computer. Like Diaconis et al[@diaconis2007dynamical], we use an American half dollar which has $I_{x}=6.68g\cdot cm^2$, $I_{z}=13.24g\cdot cm^2$ and assume that $I_{y}=7.35g\cdot cm^2$. As a result, we obtain the probability of heads of a non-uniform coin $P=50.45\%$. This is closer to $50\%$ compared to the probability of the uniform coin calculated by Diaconis et al [@diaconis2007dynamical] which is $50.83\%$. This shows that non-uniform coins are fairer than uniform coins. # Conclusions While Coin-tossing is often used to make a decision between two options, the tossed coins are usually not absolutely uniform in our daily life. In this work, we investigated the dynamic behavior of non-uniform coins whose inertia matrix is given by $\text{diag}(I_{x}, I_{y}, I_{z})$ where $I_{x}<I_{y}<I_{z}$. These coins include homogeneous coins with axis symmetrical convex parts, such as ellipse, rectangular, oblong shapes, on the surface and symmetrical inhomogeneous coins. We expressed the status, heads or tails, in terms of the initial direction of the angular momentum, the precession and nutation of the normal vector. We provided calculation of the limiting probability of heads as $t\rightarrow\infty$, with fixed initial direction of the angular momentum and distribution on magnitude of the angular momentum. The results from Keller[@keller1986probability] and Diaconis[@diaconis2007dynamical] are special cases of our study. In Figure [7](#fair region){reference-type="ref" reference="fair region"}, the fair region of initial parameters $(\varphi_0,\theta_0)$ of non-uniform coin has positive area while the fair region of the uniform coins is only a line in $\mathbb{R}^2$. The area of the fair region for non-uniform coin depends on $I_{x},I_{y},I_{z}$. So there are much more situations of initial conditions where the non-uniform coin is fair and the uniform in not fair. {#compare_traj width="5cm"} {#compare_traj width="5cm"} In addition, Equation ([\[probability formula uniform\]](#probability formula uniform){reference-type="ref" reference="probability formula uniform"}) implies the probability of heads for uniform coin is 100% if $\theta_0 \leq \frac{\pi}{4}$. Figure [9](#compare_traj){reference-type="ref" reference="compare_traj"} shows a situation ($\beta=\theta_0=\pi/4$) where the non-uniform coin is clearly fairer than the uniform coin. The two figures are possible regions of the normal vector with the same angular momentum for the uniform coin, non-uniform coin, respectively. Note that in left panel the possible region of unit vector of the uniform coin is inside the northern hemisphere. The coin never turns over and therefore the probability of heads is 100%. But note that in the right panel, there is a small region inside the southern hemisphere due to nutation. Intuitively, we see there should be a small probability of the coin landing in tails. Corollary [Corollary 2](#col::probability straight){reference-type="ref" reference="col::probability straight"} proves that in this situation, the probability of heads is strictly less than 100% for non-uniform coins.\ \ # Acknowledgements {#acknowledgements .unnumbered} I would like to express my deepest thanks to Prof. Persi Diaconis from Stanford University for his guidance and mentorship on this research. His lectures on Mathematics and Statistics of Gambling greatly inspired me and gave me valuable insights on this topic. [^1]:

arxiv_math

--- abstract: | Finite dimensional (FD) models, i.e., deterministic functions of time/space and finite sets of random variables, are constructed for target vector-valued random processes/fields. They are required to have two properties. First, standard Monte Carlo algorithms can be used to generate their samples, referred to as FD samples. Second, under some conditions specified by several theorems, FD samples can be used to estimate distributions of extremes and other functionals of target random functions. Numerical illustrations involving two-dimensional random processes and apparent properties of random microstructures are presented to illustrate the implementation of FD models for these stochastic problems and show that they are accurate if the conditions of our theorems are satisfied. author: - | Hui Xu$^{1}$    Mircea D. Grigoriu$^{1,2}$\ [hx223\@cornell.edu](hx223@cornell.edu)   [mdg12\@cornell.edu](mdg12@cornell.edu)\ *1. Center for Applied Mathematics, Cornell University, Ithaca, NY, USA*\ *2. Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY, USA* title: "**Extremes of vector-valued processes by finite dimensional models**" --- **Keywords:** extremes; finite dimensional model; Karhunen-Loève (KL) representation; Monte Carlo algorithms **MSC 2010 Subject Classification:** 60,62 # Introduction Extremes of random functions are used extensively in science and engineering to, e.g., predict the intensity of natural hazards and their consequences, design physical systems, and characterize material microstructures. Analytical expressions of the distributions of extremes of random functions are available in special cases of limited practical use. Numerical methods need to be employed to find these distributions. Their implementation requires to represent the random functions under consideration, referred to as target random functions, by finite dimensional (FD) models, i.e., deterministic functions of time/space and finite sets of random variables. In contrast to target functions which, generally, have infinite stochastic dimensions as uncountable families of random variables indexed by time/space arguments, FD models have finite stochastic dimensions equal to the numbers of random variables in their definitions. Our objective is to construct FD models for target random functions which have the following two properties. First, large sets of samples of FD models, referred to as FD samples, can be generated efficiently by standard Monte Carlo algorithms. Second, distributions of extremes of target random functions can be estimated from FD samples to any desired accuracy. We establish conditions under which the FD models under consideration have these two properties. Three examples are presented to illustrate the construction of FD models and assess the accuracy of the estimates of extremes obtained from FD samples. The first example is a two-dimensional random process whose components are linear forms of two independent non-Gaussian processes (Example [Example 1](#exam1){reference-type="ref" reference="exam1"}). The process of the second example is defined by the joint solution of two linear oscillators subjected to scaled versions of the same non-Gaussian input process (Example [Example 2](#exam2){reference-type="ref" reference="exam2"}). The third example estimates the apparent conductivity of a two-dimensional material specimen with random properties (Example [Example 3](#exam3){reference-type="ref" reference="exam3"}). The paper is organized as follows. The FD models for vector-valued processes considered in this study are discussed in the subsequent section (Sect. 2). Our main result is in Sect. 3. Conditions are established under which extremes of FD models converge weakly to those of target processes as their stochastic dimensions increase indefinitely. Numerical illustrations are in Sect. 4. Final comments can be found in Sect. 5. # Finite dimension (FD) models Throughout this paper, we denote the $L_2$ and $L_{\infty}$ norms by $||\cdot||_2$ and $||\cdot||$. For $n$-dimensional vectors $x\in\mathbb{R}^n$, they have the expressions $||x||_2=(\sum_{i=1}^{l}x_i^2)^{1/2}$ and $||x||=\max_{1\le i \le l} |x_i|$. These norms are equivalent since we deal with matrices and vectors in finite dimensional linear spaces [@R1987 Chap. 3]. Let $X(t)=(X_1(t),\cdots,X_n(t))^T, t\in[0,\tau]$, $0<\tau<\infty$, be an $\mathbb{R}^n$-valued process defined on a probability space $\big(\Omega,{\cal F},P\big)$ with means $\mu_i(t)=E[X_i(t)]=0$ and continuous correlation functions $c_{ij}(s,t)=E[X_i(s)X_j(t)]$, $i,j=1,\cdots,n$, $t,s\in[0,\tau]$. The assumption $\mu_i(t)=0$ is not restrictive since, if $\mu_i(t)\not=0$, the deterministic function $\mu(t)=(\mu_1(t),\cdots,\mu_n(t))^T$ can be added to the samples of $X(t)$. Denote by $\{\lambda_{i,k}\}$ and $\{\varphi_{i,k}\}$ the eigenvalues and the eigenfunctions of the correlation functions $\{c_{ii}(s,t)=E[X_i(s)\,X_i(t)\}$ of the components of $X(t)$, i.e., the solution of the integral equations $\int_{0}^{\tau}c_{ii}(s,t)\varphi_{i,k}(t)dt=\lambda_{i,k}\varphi_{i,k}(s)$, $s\in[0,\tau]$ for $k=1,2,\cdots$ and $i=1,\cdots,n$. Since the correlation functions of $X(t)$ are continuous, the series $c_{ii}(s,t)=\sum_{k=1}^{\infty}\lambda_{i,k}\varphi_{i,k}(s)\varphi_{i,k}(t)$ converge absolutely and uniformly $L_2([0,\tau]^2)$ for all $i=1,\cdots,n$ by Mercer's Theorem [@J1909]. Consider the family of $\mathbb{R}^n$-valued FD models $\{X_d(t)=(X_{1,d_1}(t),\cdots,X_{1,d_n}(t))\}$ of $X(t)$ defined by $$\label{201b} X_{i,d_i}(t)=\sum_{k=1}^{d_i} Z_{i,k}\,\varphi_{i,k}(t),\quad d_i=1,2,\ldots, \quad i=1,\cdots,n, \quad t\in[0,\tau],$$ where $d_i\geq 1$ are integers, $d=(d_1,\cdots,d_n)$, $\varphi_{i,k}(t)$, $k=1,\cdots,d_i$, are the top $d_i$ eigenfunctions of $c_{ii}(s,t)$, i.e., the eigenfunctions corresponding to the largest $d_i$ eigenvalues of $c_{ii}(s,t)$, and $\{Z_{i,k}\}$ are random coefficients whose samples are obtained by projecting samples $X_i(t,\omega)$ of $X_i(t)$ on the basis functions $\{\varphi_{i,k}(t)\}$, i.e., $$\begin{aligned} \label{202} Z_{i,k}(\omega)=\int_{0}^{\tau}X_i(t,\omega)\varphi_{i,k}(t)dt, \quad k=1,2,\cdots, \quad i=1,\cdots,n, \quad \omega\in\Omega.\end{aligned}$$ The first two moments of the random variables $\{Z_{i,k}\}$ are $$\begin{aligned} E[Z_{i,k}]=E\bigg[\int_0^\tau X_i(t)\,\varphi_{i,k}(t)\,dt\bigg]=\int_0^\tau E[X_i(t)] \,\varphi_{i,k}(t)\,dt=0\end{aligned}$$ and $$\begin{aligned} E[Z_{i,k}Z_{i,l}]&=&E\bigg[\int_{[0,\tau]^2} X_i(s)X_i(t)\varphi_{i,k}(s)\varphi_{i,l}(t)dsdt\bigg] =\int_{[0,\tau]^2} E[X_i(s)X_i(t)]\varphi_{i,k}(s)\varphi_{i,l}(t)dsdt\nonumber\\ &=&\int_0^\tau \varphi_{i,k}(s)\bigg[\int_0^\tau c_{ii}(s,t)\varphi_{i,l}(t)dt\bigg]ds =\lambda_{i,l}\int_0^\tau \varphi_{i,k}(s)\varphi_{i,l}(s)ds=\lambda_{i,l}\delta_{kl},\end{aligned}$$ where the change of order of integration holds by Fubini's theorem. The latter equality holds by the orthonormality of the eigenfunctions, i.e., $\langle \varphi_{i,k},\varphi_{i,l}\rangle=\int_0^\tau \varphi_{i,k}(t)\,\varphi_{i,l}(t)\,dt=\delta_{kl}$. The FD model $X_d(t)$ of ([\[201b\]](#201b){reference-type="ref" reference="201b"}) has two notable properties. First, the random vector $X_d(t)$ converges in m.s. to $X(t)$ for any $t$, i.e., $E[||X_d(t)-X(t)||_2^2]$ as $\min_{1\le i \le n}d_i\to\infty$, where $||\cdot||_2$ denotes the $L_2$ norm. This results from the equalities $$\begin{aligned} \label{mean-error} E[ \ ||X_d(t)-X(t)||_2^2 \ ]&=&\sum_{i=1}^{n}E[ \ (X_{i,d_i}(t)-X_i(t))^2 \ ] =\sum_{i=1}^{n}\sum_{k=d_i+1}^{\infty}E[Z_{i,k}^2]\varphi_{i,k}(t)^2\nonumber\\ &=&\sum_{i=1}^{n}\sum_{k=d_i+1}^{\infty}\lambda_{i,k}\varphi_{i,k}(t)^2\end{aligned}$$ and the convergence $\sum_{k=d_i+1}^{\infty}\lambda_{i,k}\varphi_{i,k}(t)^2\to 0$ as $d_i\to 0$, $i=1,\ldots,n$, implied by the Mercer Theorem [@J1909]. The second property is that the finite dimensional distributions of $X_d(t)$ converge to those of $X(t)$ as $\min_{1\le i \le n}d_i\to\infty$. Let $l\geq 1$ be an arbitrary integer and $(t_1,\ldots,t_l)$ be $l$ arbitrary times in $[0,\tau]$. Denote by $F_{\mathcal{X}}$ and $F_{\mathcal{X}_d}$ the distributions of the $nl$-dimensional vectors $\mathcal{X}=(X(t_1)^T,\ldots,X(t_l)^T)^T$ and $\mathcal{X}_d=(X_d(t_1)^T,\ldots,X_d(t_l)^T)^T$, i.e., the finite dimensional distributions of $X(t)$ and $X_d(t)$. Since $X_{i,d_i}(t_j)$ converges in m.s. to $X_i(t_j)$ as $d_i\to\infty$ for any $i=1,\cdots,n$ and $j=1,\cdots,l$, then $\mathcal{X}_d$ converges in m.s. to $\mathcal{X}$ in $L_{2}$ norm as $\min_{1\le i \le n}d_i\to\infty$. Then, $\mathcal{X}_d$ converges to $\mathcal{X}$ in probability since for any $\varepsilon>0$, $P(||\mathcal{X}_d-\mathcal{X}||_2>\varepsilon)\leq E[ \ ||\mathcal{X}_d-\mathcal{X}||_2 \ ]/\varepsilon$ by Chebyshev's inequality [@F1982]. This implies the convergence $F_{\mathcal{X}_d}\to F_{\mathcal{X}}$ as $\min_{1\le i \le n}d_i\to\infty$, i.e., the convergence of the finite dimensional distributions of $X_d(t)$ to those of $X(t)$, see Theorem 18.10 in [@v1998]. # Main results Let $X(t)$ and $X_d(t)$, $0\leq t\leq \tau$, be $\mathbb{R}^n$-valued zero-mean processes defined on the same probability space $\big(\Omega,{\cal F},P\big)$. It is assumed that $X(t)$ has continuous samples and continuous correlation functions $c_{ij}(s,t)$ on $[0,\tau]^2$, so that the samples of $X(t)$ and $X_d(t)$ are elements of the space of $n$-dimensional continuous function $C[0,\tau]$. Note also that the samples of these two processes are paired by construction since the samples of $\{Z_{i,k}\}$ are obtained from samples of $X_i(t)$ by projection, see ([\[202\]](#202){reference-type="ref" reference="202"}). Our objective is to show that $X_d(t)$ converges weakly to $X(t)$ in the metric of $C[0,\tau]$, a convergence which is denoted by $X_d \Rightarrow X$. According to Theorem 8.1 in [@billingsley68], the family of processes $\{X_d(t)\}$ converges weakly to $X(t)$ in $C[0,\tau]$ if (1) the finite dimensional distributions $F_{\mathcal{X}_d}$ of $X_d(t)$ converge to $F_{\mathcal{X}}$, a convergence already established, and (2) the family of processes $\{X_d(t)\}$ is tight in $C[0,\tau]$. We use the tightness criterion of Theorem 8.2 in [@billingsley68]. The following theorem is our main result. **Theorem 1**. *If $X(t)$ has continuous samples, the correlation functions of the components of $X(t)$ are continuous, $F_{\mathcal{X}_d}\to F_{\mathcal{X}}$ as $\min_{1\le i \le n}d_i\to\infty$ and $\sum_{k=1}^{\infty}\lambda_{i,k}C_{i,k}<\infty$ for any $i=1,\cdots,n$, then $$\begin{aligned} \label{thm2-2} \sup_{t\in[0,\tau]}||X_d(t)||\overset{w}{\to}\sup_{t\in[0,\tau]}||X(t)||, \ \min_{1\le i \le n} d_i\to\infty,\end{aligned}$$ and $$\begin{aligned} \label{thm2-1} \sup_{t\in[0,\tau]}|X_{i,d_i}(t)|\overset{w}{\to}\sup_{t\in[0,\tau]}|X_i(t)|, \ d_i\to\infty, \ i=1,\cdots,n\end{aligned}$$ where $w$ denotes weak convergence, $X_d(t)$ is given by $(\ref{201b})$, $C_{i,k}=\sup_{t\in[0,\tau]}\varphi_{i,k}(t)^2$ and $||\cdot||$ denotes the $L_{\infty}$ norm.* *Proof.* We first show that the family of random vectors $\{X_d(0)\}$ is tight, i.e., for arbitrary $\varepsilon>0$, there exists $L>0$ such that $P(||X_d(0)||>L)\leq\varepsilon$, see Theorem 8.2 in [@billingsley68]. Note that $$\begin{aligned} E[X_{i,d_i}(0)^2]=\sum_{k=1}^{d}\lambda_{i,k}\varphi_{i,k}(0)^2\leq\sum_{k=1}^{\infty}\lambda_{i,k}\varphi_{i,k}(0)^2=E[X_i(0)^2]<\infty, \ \forall \ d\ge1, \ i=1,\cdots,n,\end{aligned}$$ since $X_i(t)$ has finite variance, so that $$\begin{aligned} P(||X_d(0)||>L)\leq \frac{1}{L}E[ \ ||X_d(0)|| \ ]\leq\frac{1}{L}\sum_{i=1}^{n}E[ \ |X_{i,d_i}(0)| \ ]\leq \frac{1}{L}\sum_{i=1}^{n}\Big(E[X_{i,d_i}(0)^2]\Big)^{1/2}, \forall d\ge1 %\leq\varepsilon, \\end{aligned}$$ by Chebyshev's inequality. Since $\sum_{i=1}^{n}\Big(E[X_{i,d_i}(0)^2]\Big)^{1/2}$ is finite, there exits $L>0$ such that $P(||X_d(0)||>L)\leq\varepsilon$ for any $\varepsilon>0$, so that $\{X_d(0)\}$ is tight. We consider now the second condition of Theorem 8.2, which requires to show that, for given $\varepsilon,\eta>0$, there exists $\delta>0$ such that $P\big(W(\delta)\geq \varepsilon\big)\leq \eta$, where $W(\delta)=\sup_{|s-t|\leq\delta}||X_d(t)-X_d(s)||$ denotes the modulus of continuity of $X_d(t)$ and $||\cdot||$ is the $L_{\infty}$ norm. Note that $$\begin{aligned} P\big(W(\delta)\geq \varepsilon\big)&=&P\bigg(\sup_{|s-t|\leq\delta}||X_d(t)-X_d(s)||\geq\varepsilon\bigg)\nonumber\\ &\leq&P\bigg(\bigcup_{i=1}^{n}\sup_{|s-t|\leq\delta}|X_{i,d_i}(t)-X_{i,d_i}(s)|\geq\varepsilon\bigg)\nonumber\\ &\leq&\sum_{i=1}^{n}P\bigg(\sup_{|s-t|\leq\delta}|X_{i,d_i}(t)-X_{i,d_i}(s)|\geq\varepsilon\bigg)\nonumber\\ &\leq&\frac{1}{\varepsilon^2}\sum_{i=1}^{n}E\bigg[\bigg(\sup_{|s-t|\leq\delta}|X_{i,d_i}(t)-X_{i,d_i}(s)|\bigg)^2\bigg]\end{aligned}$$ by Chebyshev's inequality. Since $\sup_{s,t\in[0,\tau]}f(s,t)^2=(\sup_{s,t\in[0,\tau]}|f(s,t)|)^2$ and $\sup_{s,t\in[0,\tau]}(f(s,t)+g(s,t))\leq\sup_{s,t\in[0,\tau]}f(s,t)+\sup_{s,t\in[0,\tau]}g(s,t)$, then we have $$\begin{aligned} &&E\bigg[\bigg(\sup_{|s-t|\leq\delta}|X_{i,d_i}(t)-X_{i,d_i}(s)| \bigg)^2\bigg] =E\bigg[\sup_{|s-t|\leq\delta}\Big(X_{i,d_i}(t)-X_{i,d_i}(s)\Big)^2\bigg]\nonumber\\ &=&E\bigg[\sup_{|s-t|\leq\delta}\Big(\sum_{k=1}^{d_i}Z_{i,k}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)\Big)^2\bigg]\nonumber\\ &=&E\bigg[\sup_{|s-t|\leq\delta}\Big(\sum_{k=1}^{d_i}Z_{i,k}^2\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2 +\sum_{1\le k\not=l\le d_i}Z_{i,k}Z_{i,l}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)\big(\varphi_{i,l}(t)-\varphi_{i,l}(s)\big)\Big)\bigg]\nonumber\\ &\leq&E\bigg[\sum_{k=1}^{d_i}Z_{i,k}^2\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2\nonumber\\ & &+\sum_{1\le k\not=l\le d_i}Z_{i,k}Z_{i,l}\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)\big(\varphi_{i,l}(t)-\varphi_{i,l}(s)\big)\bigg]\nonumber\\ &=&\sum_{k=1}^{d_i}E[Z_{i,k}^2]\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2 \leq\sum_{k=1}^{\infty}\lambda_{i,k}\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2.\end{aligned}$$ Let $L_{i,k}(\delta)=\lambda_{i,k}\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2$, $\delta\in[0,\tau]$, and note that $$\begin{aligned} \sum_{k=1}^{\infty}L_{i,k}(\delta)\leq2\sum_{k=1}^{\infty}\lambda_{i,k}\sup_{s,t\in[0,\tau]}\big(\varphi_{i,k}(t)^2+\varphi_{i,k}(s)^2\big) \leq4\sum_{k=1}^{\infty}\lambda_{i,k}C_{i,k}<\infty.\end{aligned}$$ Since $L_{i,k}(\delta)\leq 4\lambda_{i,k}C_{i,k}$, for $k=1,2,\ldots$ and $i=1,\ldots,n$, and the series $\sum_{k=1}^{\infty}\lambda_{i,k}C_{i,k}$ is convergent, $\sum_{k=1}^\infty L_{i,k}(\delta)$ converges uniformly on $[0,\tau]$ by Theorem 7.10 in [@R1976]. Since $L_{i,k}(\delta)\to 0$ as $\delta\to0$ by the continuity of the eigenfunctions $\{\varphi_{i,k}\}$, we have $$\begin{aligned} \lim_{\delta\to0}\sum_{k=1}^{\infty}L_{i,k}(\delta)=\sum_{k=1}^{\infty}\lim_{\delta\to0}L_{i,k}(\delta)=0, \quad i=1,\cdots,n\end{aligned}$$ by Theorem 7.11 in [@R1976]. Therefore, for given $\varepsilon,\eta>0$, there exists $\delta$ such that $$\begin{aligned} P\big(W(\delta)\geq \varepsilon\big)\leq \frac{1}{\varepsilon^2}\sum_{i=1}^{n}\sum_{k=1}^{\infty}\lambda_{i,k}\sup_{|s-t|\leq\delta}\big(\varphi_{i,k}(t)-\varphi_{i,k}(s)\big)^2<\eta,\end{aligned}$$ which means that $X_d(t)$ is tight in $C[0,\tau]$. If $X(t)$ is a real-valued stochastic process, i.e., $n=1$, the conditions of the above theorem reduce to $\sum_{k=1}^{\infty}\lambda_{k}C_{k}<\infty$, where $C_{k}=\sup_{t\in[0,\tau]}\varphi_{k}(t)^2$, $\lambda_k$ and $\varphi_k(t)$ are the eigenvalue and eigenfunction of the correlation function of $X(t)$, and ([\[thm2-2\]](#thm2-2){reference-type="ref" reference="thm2-2"}) holds under these conditions. This implies that ([\[thm2-1\]](#thm2-1){reference-type="ref" reference="thm2-1"}) holds for each real-valued processes $X_{i}(t)$. ◻ The practical implication of the weak convergence in ([\[thm2-2\]](#thm2-2){reference-type="ref" reference="thm2-2"}) and ([\[thm2-1\]](#thm2-1){reference-type="ref" reference="thm2-1"}) is that the distributions of $\sup_{t\in[0,\tau]}||X(t)||$ and $\sup_{t\in[0,\tau]}|X_i(t)|$ can be estimated from FD samples for sufficiently large truncation levels $\{d_i\}$. Generally, the conditions of the above theorem are difficult to check, since eigenvalues and eigenfunctions are not available analytically. An example in which these conditions can be checked is that of a real-valued, zero-mean weakly stationary process $X(t)$, $t\in[-\tau,\tau]$, with correlation function $c(s,t)=(1+e^{-2\gamma|s-t|})/4$, where $\gamma>1/(2\tau)$ and $s,t\in[-\tau,\tau]$. We define $X_d(t)$ as ([\[201b\]](#201b){reference-type="ref" reference="201b"}). The eigenfunction of $c(s,t)$ are (see Example 6-4.1 in [@dr1987]) $$\begin{aligned} \varphi_{2k-1}(t)=\frac{\cos(2\gamma a_k t)}{\sqrt{\tau+\sin(4\gamma\tau a_k)/(4\gamma a_k)}} \ \ \text{and} \ \ \varphi_{2k}(t)=\frac{\sin(2\gamma b_k t)}{\sqrt{\tau-\sin(4\gamma\tau b_k)/(4\gamma b_k)}}, \ \ k=1,2,\cdots,\end{aligned}$$ where $a_k$ and $b_k$ are solutions of $a_k\tan(2\gamma \tau a_k)=1$ and $b_k\cot(2\gamma \tau b_k)=1$. Note that $$\begin{aligned} |\varphi_{2k-1}(t)|=\frac{|\cos(2\gamma a_k t)|}{\sqrt{\tau+\sin^2(2\gamma\tau a_k)/(2\gamma)}}\leq \frac{1}{\sqrt{\tau}}, \ \ k=1,2,\cdots\end{aligned}$$ and $$\begin{aligned} |\varphi_{2k}(t)|=\frac{|\sin(2\gamma b_k t)|}{\sqrt{\tau-\cos^2(2\gamma\tau b_k)/(2\gamma)}}\leq \frac{1}{\sqrt{\tau-1/(2\gamma)}}, \ \ k=1,2,\cdots,\end{aligned}$$ which implies $|\varphi_k(t)|\leq 1/\sqrt{\tau-1/(2\gamma)}$ for all $k\ge1$. Then $$\begin{aligned} \sum_{k=1}^{\infty}\lambda_{k}C_{k}\leq\frac{1}{\sqrt{\tau-1/(2\gamma)}}\sum_{k=1}^{\infty}\lambda_{k}<\infty,\end{aligned}$$ since $\sum_{k=1}^{\infty}\lambda_{k}<\infty$ [@GG1980 Chap. 4]. Therefore, from Theorem $\ref{thm2}$, we have $\sup_{t\in[0,\tau]}|X_d(t)|\overset{w}{\to}\sup_{t\in[0,\tau]}|X(t)|$ as $d\to\infty$. The following two theorems give alternative conditions for the weak convergence of FD models to target processes. These theorems are useful in applications since their conditions are simpler to check and are satisfied by a broad range of processes. **Theorem 2**. *If the second derivative of the correlation functions $\{c_{ii}(s,t)\}$ of the components of $X(t)$ exist and are continuous in $[0,\tau]^2$, then $$\begin{aligned} \sup_{t\in[0,\tau]}||X_d(t)-X(t)||\overset{p}{\to}0, \ \min_{1\le i \le n} d_i\to\infty,\end{aligned}$$ where $p$ denotes convergence in probability.* *Proof.* For any $\varepsilon>0$, $$\begin{aligned} &&P\bigg(\sup_{t\in[0,\tau]}||X_d(t)-X(t)||>\varepsilon\bigg)\leq\frac{1}{\varepsilon}E\bigg[\sup_{t\in[0,\tau]}||X_d(t)-X(t)||\bigg]\nonumber\\ &\leq&\frac{1}{\varepsilon}E\bigg[\sup_{t\in[0,\tau]}\sum_{i=1}^{n}|X_{i,d_i}(t)-X_i(t)|\bigg] \leq\frac{1}{\varepsilon}\sum_{i=1}^{n}E\bigg[\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\bigg].\end{aligned}$$ by the relationship $\sup_{t\in[0,\tau]}(f(t)+g(t))\leq\sup_{t\in[0,\tau]}f(t)+\sup_{t\in[0,\tau]}g(t)$ and Chebyshev's inequality. Since $\dot{X}(t)$ exists in the mean square sense, we have $$\begin{aligned} &&E\bigg[\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\bigg]\nonumber\\ &=&E\bigg[\sup_{t\in[0,\tau]}\bigg|\int_{0}^{t}\dot{X}_{i,d_i}(s)ds-\int_{0}^{t}\dot{X}_i(s)ds+X_{i,d_i}(0)-X_i(0)\bigg|\bigg]\nonumber\\ &\leq&E\bigg[\sup_{t\in[0,\tau]}\int_{0}^{t}|\dot{X}_{i,d_i}(s)-\dot{X}_i(s)|ds\bigg]+E[ \ |X_{i,d_i}(0)-X_i(0)| \ ]\nonumber\\ &=&E\bigg[\int_{0}^{\tau}|\dot{X}_{i,d_i}(s)-\dot{X}_i(s)|ds\bigg]+E[ \ |X_{i,d_i}(0)-X_i(0)| \ ]\nonumber\\ &=&\int_{0}^{\tau}E[ \ |\dot{X}_{i,d_i}(s)-\dot{X}_i(s)| \ ]ds+E[ \ |X_{i,d_i}(0)-X_i(0)| \ ]\nonumber\\ &\leq&\int_{0}^{\tau}\Big(E[ \ (\dot{X}_{i,d_i}(s)-\dot{X}_i(s))^2 \ ]\Big)^{1/2}ds+\Big(E[ \ (X_{i,d_i}(0)-X_i(0))^2 \ ]\Big)^{1/2}, \ i=1,\cdots,n,\end{aligned}$$ where the change of order of integration holds by Fubini's theorem. Note that $$\begin{aligned} X_{i,d_i}(s)=\sum_{k=1}^{d_i}\lambda_{i,k}\varphi_{i,k}(s)^2 \ \ \text{and} \ \ \dot{X}_{i,d_i}(s)=\sum_{k=1}^{d_i}\lambda_{i,k}\dot{\varphi}_{i,k}(s)^2, \ i=1,\cdots,n, \ s\in[0,\tau].\end{aligned}$$ Since the second derivative of $c_{ii}(s,t)$ is continuous for each $i=1,\cdots,N$, the eigenfunctions $\{\varphi_{i,k}(s)\}$ and $\{\dot{\varphi}_{i,k}(s)\}$ are continuous [@K1967] so that calculations as in ([\[mean-error\]](#mean-error){reference-type="ref" reference="mean-error"}) give $$\begin{aligned} E[ \ (X_{i,d_i}(s)-X_i(s))^2 \ ]=\sum_{k=d_i+1}^{\infty}\lambda_{i,k}\varphi_{i,k}(s)^2\to0, \ d_i\to\infty\end{aligned}$$ and $$\begin{aligned} E[ \ (\dot{X}_{i,d_i}(s)-\dot{X}_i(s))^2 \ ]=\sum_{k=d_i+1}^{\infty}\lambda_{i,k}\dot{\varphi}_{i,k}(s)^2\to0, \ d_i\to\infty,\end{aligned}$$ where the convergence is uniform in $s\in[0,\tau]$ for each $i=1,\cdots,n$ by Mercer's theorem. This implies $$\begin{aligned} \label{thm2-101} E\bigg[\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\bigg]\to0, \ d_i\to\infty, \ i=1,\cdots,n.\end{aligned}$$ so that for any $\varepsilon>0$, $$\begin{aligned} P\bigg(\sup_{t\in[0,\tau]}||X_d(t)-X(t)||>\varepsilon\bigg)&=&P\bigg(\bigcup_{i=1}^{n}\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|>\varepsilon\bigg)\nonumber\\ &\leq&\sum_{i=1}^{n}P\bigg(\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|>\varepsilon\bigg)\nonumber\\ &\leq&\frac{1}{\varepsilon}\sum_{i=1}^{n}E\bigg[\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\bigg]\to0, \ \min_{1\le i\le n}d_i\to\infty\end{aligned}$$ by Chebyshev's inequality and ([\[thm2-101\]](#thm2-101){reference-type="ref" reference="thm2-101"}). Hence, $$\begin{aligned} \sup_{t\in[0,\tau]}||X_d(t)-X(t)||\overset{p}{\to}0, \ \min_{1\le i \le n} d_i\to\infty.\end{aligned}$$ ◻ **Theorem 3**. *Let $G(t)=(G_1(t),\cdots,G_n(t))^T$ be a zero-mean Gaussian process with continuous samples and continuous correlation function and let $\{G_d(t)\}$ be the FD models of $G(t)$ defined by ([\[201b\]](#201b){reference-type="ref" reference="201b"}). Then $$\begin{aligned} \label{thm4-eq-1} \sup_{t\in[0,\tau]}||G_d(t)-G(t)||\overset{p}{\to}0, \ \min_{1\le i \le n}d_i\to\infty,\end{aligned}$$ where $p$ denotes convergence in probability.* *Proof.* Note that $G_{i,d_i}(t)-G_i(t)$ is a zero-mean Gaussian process defined on the closed interval $[0,\tau]$ for any $i=1,2,\cdots,n$, so that there exists constant $K_i$ such that $$\begin{aligned} P\bigg(\sup_{t\in[0,\tau]}\Big(G_{i,d_i}(t)-G_i(t)\Big)>\varepsilon\bigg)\leq K_ie^{-\varepsilon^2/4\sigma_{i,d_i}^2},\quad \varepsilon>0\end{aligned}$$ by [@ms1971] lemma 3.1 and [@gs1991], where $$\begin{aligned} \sigma_{i,d_i}^2&=&\sup_{t\in[0,\tau]}{\rm Var}[G_{i,d_i}(t)-G_i(t)]=\sup_{t\in[0,\tau]}E\Big[\Big(G_{i,d_i}(t)-G_i(t)\Big)^2\Big]\nonumber\\ &=&\sup_{t\in[0,\tau]}\sum_{k=d_i+1}^{\infty}\lambda_{i,k}\varphi_{i,k}(t)^2\to0, \ d_i\to\infty\end{aligned}$$ by Mercer's theorem. Then, for any $\varepsilon>0$, we have $$\begin{aligned} &&P\bigg(\sup_{t\in[0,\tau]}|G_{i,d_i}(t)-G_i(t)|>\varepsilon\bigg)\nonumber\\ &=&P\bigg(\bigg\{\sup_{t\in[0,\tau]}\Big(G_{i,d_i}(t)-G_i(t)\Big)>\varepsilon\bigg\}\bigcup \bigg\{\sup_{t\in[0,\tau]}\Big(G_i(t)-G_{i,d_i}(t)\Big)>\varepsilon\bigg\}\bigg)\nonumber\\ &\leq&P\bigg(\sup_{t\in[0,\tau]}\Big(G_{i,d_i}(t)-G_i(t)\Big)>\varepsilon\bigg)+ P\bigg(\sup_{t\in[0,\tau]}\Big(G_i(t)-G_{i,d_i}(t)\Big)>\varepsilon\bigg)\nonumber\\ &\leq&2K_ie^{-\varepsilon^2/4\sigma_{i,d_i}^2}\to0, \ d_i\to\infty, \ i=1,\cdots,n,\end{aligned}$$ which means that $\sup_{t\in[0,\tau]}|G_{i,d_i}(t)-G_i(t)|$ converges in probability to zero, as $d_i\to\infty$, $i=1,\cdots,n$. Therefore $$\begin{aligned} \sup_{t\in[0,\tau]}||G_d(t)-G(t)||\leq \sum_{i=1}^{n}\sup_{t\in[0,\tau]}|G_{i,d_i}(t)-G_i(t)|\overset{p}{\to}0, \ \min_{1\le i \le n}d_i\to\infty.\end{aligned}$$ ◻ The latter result extends directly to the class of non-Gaussian translation processes $X(t)$. The components of these processes are defined by $$\begin{aligned} \label{t-1} X_i(t)=F_i^{-1}\circ\Phi(G_i(t)),\quad i=1,\ldots,n,\end{aligned}$$ where $\big(G_1(t),\ldots,G_n(t)\big)$ is a stationary vector-valued Gaussian process whose components have zero means and unit variances, $\Phi$ denotes the distribution of the standard normal variable and $F_i$ is the marginal distribution of $X_i(t)$. The FD models $\{X_d(t)\}$ of $X(t)$ have the form $$\begin{aligned} \label{t-2} X_{i,d_i}(t)=F_i^{-1}\circ \Phi(G_{i,d_i}(t)),\quad i=1,\ldots,n,\end{aligned}$$ where $G_{i,d_i}(t)$ is a finite dimensional model of a Gaussian process $G_i(t)$, see ([\[201b\]](#201b){reference-type="ref" reference="201b"}). **Corollary 1**. *Let $X(t)$, $X_d(t)$ be defined in $(\ref{t-1})$ and $(\ref{t-2})$. If $G_i(t)$ in $(\ref{t-1})$ satisfies the conditions of Theorem [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"} and $F_i$ is continuous and strictly monotonically increasing for each $i=1,\cdots,n$, then* *$$\begin{aligned} \sup_{t\in[0,\tau]}||X_d(t)-X(t)||\overset{p}{\to}0, \ \min_{1\le i \le n}d_i\to\infty.\end{aligned}$$* *Proof.* Let $V_d(t)=(V_{1,d_1}(t),\cdots,V_{n,d_n}(t))^T$ and $V(t)=(V_1(t),\cdots,V_n(t))^T$, where $V_{i,d_i}(t)=\Phi(G_{i,d_i}(t))$ and $V_{i}(t)=\Phi(G_{i}(t))$, $i=1,\cdots,n$. Then according to Theorem [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"} and mean value theorem, $$\begin{aligned} \sup_{t\in[0,\tau]}|V_{i,d_i}(t)-V_i(t)|\leq\frac{1}{\sqrt{2\pi}}\sup_{t\in[0,\tau]}|G_{i,d_i}(t)-G_i(t)|\overset{p}{\to}0, \ d_i\to\infty, \ i=1,\cdots,n.\end{aligned}$$ Since $F_i^{-1}$ is continuous, then $F_i^{-1}$ is uniformly continuous on $[0,1]$ for each $i=1,\cdots,n$, which leads to $$\begin{aligned} \sup_{t\in[0,\tau]}||X_d(t)-X(t)||&\leq&\sum_{i=1}^{n}\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X(t)|\nonumber\\ &=&\sum_{i=1}^{n}\sup_{t\in[0,\tau]}|F_i^{-1}(V_{i,d_i}(t))-F_i^{-1}(V_i(t))|\overset{p}{\to}0, \ \min_{1\le i\le n}d_i\to\infty.\end{aligned}$$ ◻ We also consider the FD model $X^{(N)}(t)=(X^{(N)}_1(t),\cdots,X^{(N)}_n(t))^T$ whose samples interpolate linearly between values of $X(t)$ at the times $(t_0,t_1,\ldots,t_N)$, where $t_i=i\Delta t$, $0\le i\le N$ and $\Delta t=\tau/N$. The following theorem shows that, under some proper conditions, the discrepancy between the samples of $X^{(N)}(t)$ and $X(t)$ measured by the metric of $C[0,\tau]$ can be made as small as desired by increasing $N$. **Theorem 4**. *Let $X(t)$ be a $n$-dimensional vector valued process on $[0,\tau]$. If there exists $\kappa_1,\kappa_2,C>0$ such that $$\begin{aligned} \label{thm5-5} E[ \ ||X(t+\delta)-X(t)||^{\kappa_1} \ ]\leq C\delta^{1+\kappa_2}, \quad t,t+\delta\in[0,\tau],\end{aligned}$$ then $$\begin{aligned} \label{thm5-3} \sup_{t\in[0,\tau]}||X^{(N)}(t)-X(t)||\overset{a.s.}{\to}0, \ N\to\infty,\end{aligned}$$ where a.s. denotes almost sure convergence.* *Proof.* For any $t\in[t_{i-1},t_i]$, $i=1,\cdots,N$, let $t=t_{i-1}+\xi$, $\xi\in[0,\Delta t]$, we have $$\begin{aligned} X^{(N)}(t)-X(t) &=&X(t_{i-1})+\frac{1}{\Delta t} \Big(X(t_i)-X(t_{i-1})\Big)(t-t_{i-1})-X(t)\nonumber\\ &=&\Big(1-\frac{\xi}{\Delta t}\Big)\Big(X(t_{i-1})-X(t_{i-1}+\xi)\Big)+\frac{\xi}{\Delta t} \Big(X(t_i)-X(t_{i-1}+\xi)\Big) \end{aligned}$$ for almost all samples so that $$\begin{aligned} &&\sup_{t\in[0,\tau]}||X^{(N)}(t)-X(t)|| =\max_{1\le i\le N}\sup_{t\in[t_{i-1},t_i]}||X^{(N)}(t)-X(t)||\nonumber\\ &=&\max_{1\le i\le N}\sup_{\xi\in[0,\Delta t]}\bigg|\bigg|\Big(1-\frac{\xi}{\Delta t}\Big)\Big(X(t_{i-1})-X(t_{i-1}+\xi)\Big)+\frac{\xi}{\Delta t} \Big(X(t_i)-X(t_{i-1}+\xi)\Big)\bigg|\bigg|\nonumber\\ &\leq&\max_{1\le i\le N}\sup_{\xi\in[0,\Delta t]}\Big(1-\frac{\xi}{\Delta t}\Big)\bigg|\bigg|X(t_{i-1})-X(t_{i-1}+\xi)\bigg|\bigg| +\max_{1\le i\le N}\sup_{\xi\in[0,\Delta t]}\frac{\xi}{\Delta t} \bigg|\bigg|X(t_i)-X(t_{i-1}+\xi)\bigg|\bigg|\nonumber\\ &\leq&2\sup_{|s-t|\leq\Delta t}||X(s)-X(t)||=2\sup_{|s-t|\leq\tau/N}||X(s)-X(t)||.%, \ a.s.\end{aligned}$$ Under the condition ([\[thm5-5\]](#thm5-5){reference-type="ref" reference="thm5-5"}), which constitutes the Kolmogorov continuity criterion ([@WH1985], Proposition 4.2), we have $\sup_{|s-t|\leq\tau/N}||X(s)-X(t)||\to0$ almost surely, as $N\to\infty$ so that $\sup_{t\in[0,\tau]}||X^{(N)}(t)-X(t)||\to0$ almost surely, as $N\to\infty$. ◻ # Numerical illustrations The previous section provides conditions under which the distribution of extremes of FD models converge to that of extremes of target processes (Theorem [Theorem 1](#thm2){reference-type="ref" reference="thm2"}), the probability of the discrepancy between FD and target processes can be made as small as desired (Theorems [Theorem 2](#thm3){reference-type="ref" reference="thm3"} and [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"}) and the discrepancy between FD and target samples can be eliminated (Theorem [Theorem 4](#thm5-2){reference-type="ref" reference="thm5-2"}). The conditions of Theorem [Theorem 1](#thm2){reference-type="ref" reference="thm2"} are difficult to validate and, if satisfied, they guarantee that the distribution of extremes of target functions can be approximated by that of FD models for a sufficiently large stochastic dimension. If the conditions of Theorems [Theorem 2](#thm3){reference-type="ref" reference="thm3"} and [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"} hold, then the subset $$\Omega_d(\varepsilon)=\{\omega: \sup_{t\in[0,\tau]}||X_d(t,\omega)-X(t,\omega)||>\varepsilon\}$$ in which the discrepancy between FD and target samples in the metric of $C[0,\tau]$ exceeds an arbitrary $\varepsilon>0$ can be made as small as desired by increasing $\min_{1\le i \le n}d_i$ since $P\big(\Omega_d(\varepsilon)\big)\to 0$ as $\min_{1\leq i\leq n}d_i\to 0$. Accordingly, most of FD samples provide accurate representations of the corresponding target samples. **Example 1**. *Let $X_1(t)=Y_1(t)$ and $X_2(t)=Y_1(t)+Y_2(t)$, $t\in[0,\tau]$, where $Y_i(t)=F_i^{-1}\circ\Phi(G_i(t))$, follows the Gumbel distribution $F_i(x)=\exp\{-\exp\{(x-\mu_i)/\gamma_i\}\}$ with parameters $\mu_i\in\mathbb{R}$, $\gamma_i>0$ and $\{G_i(t)\}$ are zero-mean independent Gaussian processes with correlation functions $c_i(s,t)=(1+\nu_i|s-t|)e^{-\nu_i|s-t|}$, $\nu_i>0$, $i=1,2$. The following numerical results are for $\mu_1=0$, $\mu_2=1$, $\gamma_1=1$, $\gamma_2=2$, $\nu_1=0.1$, $\nu_2=0.2$ and $\tau=50$. All reported statistics are based on 5000 samples. The reference samples are those of $X^{(N)}(t)$ in Theorem 3.4 with $\Delta t=\tau/N=0.01$ and $N=5000$. * We construct two types of FD models for $X(t)=(X_1(t),X_2(t))^T$. The first FD model has the form $$\begin{aligned} X_{1,d_1}(t)=Y_{1,d_1}(t) \ \text{and} \ X_{2,d_2}(t)=Y_{1,d_2}(t)+Y_{2,d_2}(t),\end{aligned}$$ where $Y_{1,d_1}(t)$, $Y_{1,d_2}(t)$ and $Y_{2,d_2}(t)$ are FD models of $Y_1(t)$ and $Y_2(t)$ defined by ([\[201b\]](#201b){reference-type="ref" reference="201b"}) with $n=1$. Since the distributions $F^{-1}$ and $\Phi$ and the correlation functions of $Y_1(t)$ and $Y_2(t)$ are differentiable, $\sup_{t\in[0,\tau]}|Y_{i,d_i}(t)-Y_i(t)|$ converges weakly to $0$ for $i=1,2$ as $\min\{d_1,d_2\}\to\infty$ by Theorem [Theorem 2](#thm3){reference-type="ref" reference="thm3"}, which implies the weak convergence $\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\to 0$, $i=1,2$. This means that most of the FD samples will approximate accurately the corresponding target samples for a sufficiently large stochastic dimension. The left, middle and right panels of Figs. [3](#ex1-fig1){reference-type="ref" reference="ex1-fig1"} and [6](#ex1-fig6){reference-type="ref" reference="ex1-fig6"} show with solid and dotted lines five samples of $X_i(t)$ and $X_{i,d_i}(t)$, $i=1,2$ for $d_1=5,10,15$ and $d_2=15,20,25$. Scatter plots of $\big(\sup_{0\leq t\leq \tau}|X_i(t)|,\sup_{0\leq t\leq \tau}|X_{i,d_i}(t)|\big)$ are in Figs. [9](#ex1-fig4){reference-type="ref" reference="ex1-fig4"} and [12](#ex1-fig5){reference-type="ref" reference="ex1-fig5"} for the same values of $d_i,i=1,2$ (left, middle and right panels). The thin solid lines of the left and right panels of Fig. [14](#ex1-fig7){reference-type="ref" reference="ex1-fig7"} are estimates of $P(\sup_{t\in[0,\tau]}|X_i(t)|$\ $>x)$ for $i=1$ and $i=2$ which are obtained directly from data. These probabilities are viewed as reference. The other lines of the figure are calculated from FD models with $d_1=5$ and $d_2=15$ (heavy solid lines), $d_1=10$ and $d_2=20$ (dotted lines) and $d_1=15$ and $d_2=25$ (dashed lines) for the first and second components (left and right panels). The dashed lines are the closest to the reference. These plots show, in agreement with our theoretical results, that the discrepancy between samples and extremes of $X_i(t)$ and $X_{i,d_i}(t)$ can be made as small as desired by increasing the stochastic dimension $d_i$, $i=1,2$. The FD models of $X_1(t)$ require a lower truncation level since the samples of $X_1(t)$ are smoother than those of $X_2(t)$, see Figs. [3](#ex1-fig1){reference-type="ref" reference="ex1-fig1"} and [6](#ex1-fig6){reference-type="ref" reference="ex1-fig6"} {#ex1-fig1} {#ex1-fig1} {#ex1-fig1} {#ex1-fig6} {#ex1-fig6} {#ex1-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex1-x1-scatter-5.png "fig:"){#ex1-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex1-x1-scatter-10.png "fig:"){#ex1-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex1-x1-scatter-15.png "fig:"){#ex1-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex1-x2-scatter-15.png "fig:"){#ex1-fig5} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex1-x2-scatter-20.png "fig:"){#ex1-fig5} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex1-x2-scatter-25.png "fig:"){#ex1-fig5} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex1-x1-log.png "fig:"){#ex1-fig7} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex1-x2-log.png "fig:"){#ex1-fig7} The second FD model of $X(t)$ has the fom $$\begin{aligned} X_{1,d_1}(t)=F_1^{-1}\circ\Phi(G_{1,d_1}(t)) \ \text{and} \ X_{2,d_2}(t)=F_1^{-1}\circ\Phi(G_{1,d_1}(t))+F_2^{-1}\circ\Phi(G_{2,d_2}(t)),\end{aligned}$$ where $G_{1,d_1}(t)$ and $G_{2,d_2}(t)$ are FD models of $G_1(t)$ and $G_2(t)$ given by ([\[201b\]](#201b){reference-type="ref" reference="201b"}) with $n=1$. According to Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}, $\sup_{t\in[0,\tau]}|F_i^{-1}\circ\Phi(G_{i,d_i}(t))-F_i^{-1}\circ\Phi(G_{i}(t))|\overset{w}{\to}0$, as $\min\{d_1,d_2\}\to\infty$, which implies $\sup_{t\in[0,\tau]}|X_{i,d}(t)-X_i(t)|\overset{w}{\to}0$, as $\min\{d_1,d_2\}\to\infty$. This means that most of the FD samples will approximate accurately the corresponding target samples for a sufficiently large stochastic dimension. The left, middle and right panels of Figs. [17](#ex2-fig1){reference-type="ref" reference="ex2-fig1"} and [20](#ex2-fig6){reference-type="ref" reference="ex2-fig6"} show with solid and dotted lines five samples of $X_i(t)$ and $X_{i,d_i}(t)$, $i=1,2$ for $d_1=5,10,15$ and $d_2=15,20,25$. Scatter plots of $\big(\sup_{0\leq t\leq \tau}|X_i(t)|,\sup_{0\leq t\leq \tau}|X_{i,d_i}(t)|\big)$ are in Figs. [23](#ex2-fig4){reference-type="ref" reference="ex2-fig4"} and [26](#ex2-fig5){reference-type="ref" reference="ex2-fig5"} for the same values of $d_i$, $i=1,2$ (left, middle and right panels). The thin solid lines of the left and right panels of Fig. [28](#ex2-fig7){reference-type="ref" reference="ex2-fig7"} are estimates of $P(\sup_{t\in[0,\tau]}|X_i(t)|$\ $>x)$ for $i=1$ and $i=2$ which are obtained directly from data. These probabilities are viewed as reference. The other lines of the figure are calculated from FD models for $d_1=5$ and $d_2=15$ (heavy solid lines), $d_1=10$ and $d_2=20$ (dotted lines) and $d_1=15$ and $d_2=25$ (dashed lines) for the first and second components (left and right panels). The dashed lines are the closest to the reference. These plots show, in agreement with our theoretical results, that the discrepancy between samples and extremes of $X_i(t)$ and $X_{i,d_i}(t)$ can be made as small as desired by increasing the stochastic dimension $d_i$, $i=1,2$. The FD models of $X_1(t)$ require a lower truncation level since the samples of $X_1(t)$ are smoother than those of $X_2(t)$, see Figs. [17](#ex2-fig1){reference-type="ref" reference="ex2-fig1"} and [20](#ex2-fig6){reference-type="ref" reference="ex2-fig6"}. {#ex2-fig1} {#ex2-fig1} {#ex2-fig1} {#ex2-fig6} {#ex2-fig6} {#ex2-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex2-x1-scatter-5.png "fig:"){#ex2-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex2-x1-scatter-10.png "fig:"){#ex2-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,10,15$ (left, middle and right panels).](ex2-x1-scatter-15.png "fig:"){#ex2-fig4} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex2-x2-scatter-15.png "fig:"){#ex2-fig5} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex2-x2-scatter-20.png "fig:"){#ex2-fig5} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=15,20,25$ (left, middle and right panels).](ex2-x2-scatter-25.png "fig:"){#ex2-fig5} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex2-x1-log.png "fig:"){#ex2-fig7} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex2-x2-log.png "fig:"){#ex2-fig7} We conclude this example with the observation that, according to the results of Figs. [9](#ex1-fig4){reference-type="ref" reference="ex1-fig4"} to [14](#ex1-fig7){reference-type="ref" reference="ex1-fig7"} and Figs. [23](#ex2-fig4){reference-type="ref" reference="ex2-fig4"} to [28](#ex2-fig7){reference-type="ref" reference="ex2-fig7"}, the performance of the two FD models of $X(t)$ is similar. **Example 2**. *Let $X(t)=(X_1(t),X_2(t))^T$, $0\leq t\leq \tau$, be an $\mathbb{R}^2$-valued stochastic process defined by the differential equations $$\begin{aligned} \label{100} \ddot{X}_1(t)+\alpha_1\dot{X}_1(t)+\beta_1X_1(t)&=&\gamma_1Y(t)^2, \nonumber\\ \ddot{X}_2(t)+\alpha_2\dot{X}_2(t)+\beta_2X_2(t)&=&\gamma_2Y(t)^2,\end{aligned}$$ with the initial conditions $X_i(0)=0$ and $\dot{X}_i(0)=0$ for $i=1,2$, where $\alpha_i,\beta_i,\gamma_i>0$, $i=1,2$, are constants, $\beta_i-\alpha_i^2/4>0$, $Y(t)$ is the stationary solution of $dY(t)=-\rho \,Y(t)\,dt+\sqrt{2\,\rho} \,dB(t)$, $\rho>0$, and $B$ denotes the standard Brownian motion. The following numerical results are for $\alpha_1=0.5$, $\alpha_2=0.2$, $\beta_1=10$, $\beta_2=5$, $\gamma_1=1$, $\gamma_2=2$, $\rho=1$, $\tau=10$ and the time step $\Delta t=0.01$. The reported statistics are based on $5000$ samples of $X(t)$.* The processes $X_i(t)$, $\dot{X}_i(t)$ and $Y^2(t)$ have continuous samples as they are obtained from samples of $B(t)$ by integration and $B(t)$ has continuous samples. We construct two types of FD models for $X(t)=(X_1(t),X_2(t))^T$. The first FD model is for the solution $X(t)$ of the differential equation ([\[100\]](#100){reference-type="ref" reference="100"}) by using ([\[201b\]](#201b){reference-type="ref" reference="201b"}) with $n=2$. Since the second derivatives of the correlation functions of $X_i(t)$ exist and are continuous in $[0,\tau]^2$, $\sup_{t\in[0,\tau]}|X_{i,d_i}(t)-X_i(t)|\overset{w}{\to}0$, as $d_i\to\infty$, $i=1,2$, see Theorem [Theorem 2](#thm3){reference-type="ref" reference="thm3"}. This suggests that most of the samples of $X(t)$ can be approximated accurately by FD samples and that the subset of samples of $X(t)$ which do not have this property decreases with the stochastic dimension of $X_d(t)$ . The solid and dotted lines of Fig. [31](#ex4-fig1){reference-type="ref" reference="ex4-fig1"}  show a sample of $X(t)$ and $X_d(t)$ starting at the origin for $d_1=d_2=5$, $15$ and $25$ (left, middle and right panels). Scatter plots of $\big(\sup_{0\leq t\leq \tau}|X_i(t)|,\sup_{0\leq t\leq \tau}|X_{i,d_i}(t)|\big)$ are in Figs. [34](#ex4-fig6){reference-type="ref" reference="ex4-fig6"} and [37](#ex4-fig7){reference-type="ref" reference="ex4-fig7"} for the same values of $d_i$, $i=1,2$ (left, middle and right panels). The thin solid lines of the left and right panels of Fig. [39](#ex4-fig9){reference-type="ref" reference="ex4-fig9"} are estimates of $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ for $i=1$ and $i=2$ which are obtained directly from data. These probabilities are viewed as reference. The other lines of the figure are calculated from FD models for $d_1=d_2=5$ (heavy solid lines), $d_1=d_2=10$ (dotted lines) and $d_1=d_2=15$ (dashed lines) for the first and second components (left and right panels). The dashed lines are the closest to the reference. These plots show, in agreement with our theoretical results, that the discrepancy between samples and extremes of $X_i(t)$ and $X_{i,d_i}(t)$ can be made as small as desired by increasing the stochastic dimension $d_i$, $i=1,2$. {#ex4-fig1} {#ex4-fig1} {#ex4-fig1} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,15,25$ (left, middle and right panels).](ex4-x1-scatter-5.png "fig:"){#ex4-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,15,25$ (left, middle and right panels).](ex4-x1-scatter-15.png "fig:"){#ex4-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d_1}(t)|$ for $d_1=5,15,25$ (left, middle and right panels).](ex4-x1-scatter-25.png "fig:"){#ex4-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=5,15,25$ (left, middle and right panels).](ex4-x2-scatter-5.png "fig:"){#ex4-fig7} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=5,15,25$ (left, middle and right panels).](ex4-x2-scatter-15.png "fig:"){#ex4-fig7} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d_2}(t)|$ for $d_2=5,15,25$ (left, middle and right panels).](ex4-x2-scatter-25.png "fig:"){#ex4-fig7} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex4-x1-log.png "fig:"){#ex4-fig9} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex4-x2-log.png "fig:"){#ex4-fig9} The second FD model of $X(t)$ is constructed from FD models of the input $Y(t)$. We have $$\begin{aligned} \label{solution1} X_{i}(t)=\int_{0}^{t}\frac{\gamma_i}{\psi_i}e^{-\alpha_i(t-u)/2}\sin(\psi_i(t-u))Y(u)^2du, \ i=1,2, \quad 0\leq t\leq\tau,\end{aligned}$$ where $\psi_i=(\beta_i-\alpha_i^2/4)^{1/2}$, $i=1,2$, [@mg-ld Chap. 2]. The FD models $Y_d(t)$ of $Y(t)$ are given by ([\[201b\]](#201b){reference-type="ref" reference="201b"}) with $n=1$, then the FD models of $X_d(t)=(X_{1,d}(t),X_{2,d}(t))^T$ result from ([\[solution1\]](#solution1){reference-type="ref" reference="solution1"}) with $Y_d(t)$ in place of $Y(t)$, i.e., $$\begin{aligned} \label{solution2} X_{i,d}(t)=\int_{0}^{t}\frac{\gamma_i}{\psi_i}e^{-\alpha_i(t-u)/2}\sin(\psi_i(t-u))Y_d(u)^2du, \ i=1,2, \quad 0\leq t\leq\tau.\end{aligned}$$ Since $Y(t)$ is a Gaussian process with continuous samples and continuous correlation function $c(s,t)=e^{-\rho|s-t|}$, we have $\sup_{t\in[0,\tau]}|Y_d(t)-Y(t)|\overset{w}{\to}0$ by Theorem [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"}. Therefore, $$\begin{aligned} &&\sup_{t\in[0,\tau]}|X_{i,d}(t)-X_i(t)|\nonumber\\ &=&\sup_{t\in[0,\tau]}\bigg|\int_{0}^{t}\frac{\gamma_i}{\psi_i}e^{-\alpha_i(t-u)/2}\sin(\psi_i(t-u))\Big(Y_d(u)^2-Y(u)^2\Big)du\bigg|\nonumber\\ &\leq&\int_{0}^{\tau}\frac{\gamma_i}{\psi_i}|Y_d(u)^2-Y(u)^2|du\leq\frac{\gamma_i\tau}{\psi_i}\sup_{t\in[0,\tau]}|Y_d(u)^2-Y(u)^2|\overset{w}{\to}0.\end{aligned}$$ This suggest that most of the samples of $X(t)$ can be approximated accurately by FD samples and that the subset of samples of $X(t)$ which do not have this property decreases with the stochastic dimension of $X_d(t)$ . The solid and dotted lines of Fig. [42](#ex5-fig1){reference-type="ref" reference="ex5-fig1"}  show a sample of $X(t)$ and $X_d(t)$ starting at the origin for $d_1=d_2=5$, $15$ and $25$ (left, middle and right panels). Scatter plots of $\big(\sup_{0\leq t\leq \tau}|X_i(t)|,\sup_{0\leq t\leq \tau}|X_{i,d_i}(t)|\big)$ are in Figs. [45](#ex5-fig6){reference-type="ref" reference="ex5-fig6"} and [48](#ex5-fig7){reference-type="ref" reference="ex5-fig7"} for the same values of $d_i$, $i=1,2$ (left, middle and right panels). The thin solid lines of the left and right panels of Fig. [50](#ex5-fig9){reference-type="ref" reference="ex5-fig9"} are estimates of $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ for $i=1$ and $i=2$ which are obtained directly from data. These probabilities are viewed as reference. The other lines of the figure are calculated from FD models for $d_1=d_2=5$ (heavy solid lines), $d_1=d_2=10$ (dotted lines) and $d_1=d_2=15$ (dashed lines) for the first and second components (left and right panels). The dashed lines are the closest to the reference. These plots show, in agreement with our theoretical results, that the discrepancy between samples and extremes of $X_i(t)$ and $X_{i,d_i}(t)$ can be made as small as desired by increasing the stochastic dimension $d_i$, $i=1,2$. {#ex5-fig1} {#ex5-fig1} {#ex5-fig1} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x1-scatter-5.png "fig:"){#ex5-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x1-scatter-15.png "fig:"){#ex5-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_1(t)|$ and $\sup_{t\in[0,\tau]}|X_{1,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x1-scatter-25.png "fig:"){#ex5-fig6} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x2-scatter-5.png "fig:"){#ex5-fig7} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x2-scatter-15.png "fig:"){#ex5-fig7} ![Scatter plots of $\sup_{t\in[0,\tau]}|X_2(t)|$ and $\sup_{t\in[0,\tau]}|X_{2,d}(t)|$ for $d=5,15,25$ (left, middle and right panels).](ex5-x2-scatter-25.png "fig:"){#ex5-fig7} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex5-x1-log.png "fig:"){#ex5-fig9} ![Estimates of the target probability $P(\sup_{t\in[0,\tau]}|X_i(t)|>x)$ (thin solid line), estimates based on FD model for different values of $d_i$ (heavy solid line, dotted line and dashed line) in logarithmic scale for $X_1(t)$ and $X_2(t)$ (left and right panels).](ex5-x2-log.png "fig:"){#ex5-fig9} Figures [34](#ex4-fig6){reference-type="ref" reference="ex4-fig6"} to [39](#ex4-fig9){reference-type="ref" reference="ex4-fig9"} and Figs. [45](#ex5-fig6){reference-type="ref" reference="ex5-fig6"} to [50](#ex5-fig9){reference-type="ref" reference="ex5-fig9"} suggest that the first FD models are superior in the sense that they approximate more accurately the extremes of the target processes. Note that the first FD model is constructed on the differentiable process $X(t)$, in contrast to the second FD model, which is constructed from the non-differentiable process $Y(t)$. The differentiability creates smoothness of the sample paths which may explain the superior performance of the first FD model. **Example 3**. *Let $U(t),t\in D=[0,\tau_1]\times[0,\tau_2]$, be the solution of the stochastic boundary value problem $$\begin{aligned} \label{pde1} \nabla\cdot(X(t,\omega)\nabla U(t,\omega))=0,\ t\in D\end{aligned}$$ with the boundary conditions $U(0,t_2)=0$, $U(\tau_1,t_2)=1$, $\partial U(t_1,t_2)/\partial t_2=0$ for $t_2=0$ and $t_2=\tau_2$, a.s. Note that $t=(t_1,t_2)\in D$ is a space coordinate in this example. The random field $X(t)$ is defined by $$\begin{aligned} \label{beta1} X(t)=\alpha+(\beta-\alpha)F^{-1}_{{\rm Beta}(p,q)}\circ\Phi(G(t)), \quad t\in D,\end{aligned}$$ where $0<\alpha<\beta<\infty$, $F_{{\rm Beta}(p,q)}$ denotes the distribution of a standard Beta random variable with shape parameters $(p,q)$ and $G(t),t\in\mathbb{R}^2$, is a homogeneous Gaussian field with zero-mean, unit variance, spectral density $$\begin{aligned} s(\nu)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg\{-\frac{\nu_1^2-2\rho \nu_1\nu_2+\nu_2^2}{2(1-\rho^2)}\bigg\},\ \rho\in(-1,1), \ \nu=(\nu_1,\nu_2)\in\mathbb{R}^2\end{aligned}$$ and correlation function $$\begin{aligned} \label{corr} c(s,t)=\exp\bigg\{-\frac{(t_1-s_1)^2+2\rho(t_1-s_1)(t_2-s_2)+(t_2-s_2)^2}{2}\bigg\}, \ t, s\in\mathbb{R}^2.\end{aligned}$$* Our objective is to estimate the distribution of the apparent conductivity [@G2005; @H2007; @M2007 Chap.7] $$\begin{aligned} \label{app1} X_{\rm{app}}=\frac{1}{\tau_2}\int_{0}^{\tau_2}\int_{0}^{\tau_1}X(t)\frac{\partial U(t)}{\partial t_1}dt_1dt_2.\end{aligned}$$ This objective cannot be achieved directly since the distribution of $X_{\rm{app}}$ is not available analytically and it is not possible to generate samples of this random variable as the integrand in its definition has infinite stochastic dimension. To characterize $X_{\rm{app}}$, we construct a family of surrogates $\{X_{\rm{app},d}\}$, $d=1,2,\cdots$, which have the following two properties. First, samples of $X_{\rm{app},d}$ can be generated by standard Monte Carlo algorithms. Second, the distribution of $X_{\rm{app},d}$ converges to that of $X_{\rm{app}}$ as $d\to\infty$. The surrogates $\{X_{\rm{app},d}\}$ are defined by $$\begin{aligned} \label{app2} X_{\rm{app},d}=\frac{1}{\tau_2}\int_{0}^{\tau_2}\int_{0}^{\tau_1}X_d(t)\frac{\partial U_d(t)}{\partial t_1}dt_1dt_2,\end{aligned}$$ where $U_d(t)$ is the solution of the stochastic boundary value problem $$\begin{aligned} \label{pde2} \nabla\cdot(X_d(t,\omega)\nabla U_d(t,\omega))=0,\ t\in D\end{aligned}$$ with the boundary conditions of $(\ref{pde1})$, $X_d(t)$ is defined by $$\begin{aligned} \label{beta2} X_d(t)=\alpha+(\beta-\alpha)F^{-1}_{{\rm Beta}(p,q)}\circ\Phi(G_d(t)), \quad t\in D\end{aligned}$$ and $G_d(t)$ is the FD model of $G(t)$. The random field $G_d(t)$ has the form $$\label{ex3-1} G_{d}(t)=\sum_{k=1}^{d} Z_{k}\,\varphi_{k}(t), \quad d=1,2,\ldots, \quad t\in D,$$ where $\varphi_{k}(t)$, $k=1,\cdots,d$ are the top $d$ eigenfunctions of $c(s,t)$, i.e., the eigenfunctions corresponding to the largest $d$ eigenvalues, and the random coefficients $\{Z_{k}\}$ are defined sample-by-sample from samples of $G(t)$ by projection, i.e., $$\begin{aligned} \label{ex3-2} Z_{k}(\omega)=\int_{D}G(t,\omega)\varphi_{k}(t)dt, \quad k=1,2,\cdots, \quad \omega\in\Omega.\end{aligned}$$ Note that the random elements $X_d(t)$, $U_d(t)$ and $X_{\rm{app},d}$ depend on the finite set of random variables $(Z_1,\cdots,Z_d)$ so that samples of $X_{\rm{app},d}$ can be generated by standard Monte Carlo algorithms. We now show that $X_{\rm{app},d}$ also has the second property, i.e., its samples can be used to estimate the distribution of $X_{\rm app}$ provided that $d$ is sufficiently large. From ([\[pde1\]](#pde1){reference-type="ref" reference="pde1"}) and ([\[pde2\]](#pde2){reference-type="ref" reference="pde2"}), we have $$\begin{aligned} \nabla\cdot(X(t)\nabla U(t))=\nabla\cdot(X_d(t)\nabla U_d(t)),\end{aligned}$$ which implies that $U_d(t)-U(t)$ satisfies the stochastic equation $$\begin{aligned} \label{pde3} \nabla\cdot\Big(X(t)\nabla\big(U_d(t)-U(t)\big)\Big)=\xi_d(t),\end{aligned}$$ with the boundary conditions $U(0,t_2)-U_d(0,t_2)=0$, $U(\tau_1,t_2)-U_d(\tau_1,t_2)=0$, $\partial (U(t_1,t_2)-U_d(t_1,t_2))/\partial t_2=0$ for $t_2=0$ and $t_2=\tau_2$, a.s., where $\xi_d(t)=-\big(\nabla\big(X_d(t)-X(t)\big)\big)\cdot\nabla U_d(t)-\big(X_d(t)-X(t)\big)\Delta U_d(t)$. The solution of ([\[pde3\]](#pde3){reference-type="ref" reference="pde3"}) is $$\begin{aligned} U_d(t,\omega)-U(t,\omega)=\int_{D}g(t,s,\omega)\xi_d(s,\omega)ds, \quad \text{\rm a.s.},\end{aligned}$$ where $g(t,s,\omega)$ is the Green function of the operator $\nabla\cdot(X(t,\omega)\nabla(\cdot))$ [@CH1953] Section V.14. The Green function $g(t,s,\omega)$ is bounded in $D$ for a fixed $\omega\in\Omega$ since it is continuous and $D$ is bounded. We assume that (1) there is a single finite bound for the entire family of Green functions, i.e., $\sup_{t,s\in D}|g(t,s)|$ is almost surely bounded, (2) $\sup_{t\in D}|U(t)|$, $\sup_{t\in D}|\partial U_d(t)/\partial t_i|$, $i=1,2$ and $\sup_{t\in D}|\Delta U_d(t)|$ are also almost surely bounded, and (3) the random fields $G(t)$ and $\partial G(t)/\partial t_i$, $i=1,2$ have continuous samples. Under these conditions, $$\begin{aligned} \label{convergence1} \sup_{t\in D}|U_d(t)-U(t)|&=&\bigg|\int_{D}g(t,s)\xi_d(s)ds\bigg|\nonumber\\ &\leq&\tau_1\tau_2\sup_{t,s\in D\times D}|g(t,s)|\sup_{s\in D}|\xi_d(s)|\overset{w}{\to}0, \ d\to\infty,\end{aligned}$$ provided that $\sup_{s\in D}|\xi_d(s)|\overset{w}{\to}0$, as $d\to\infty$. Since $$\begin{aligned} \sup_{t\in D}|\xi_d(t)|\leq \sum_{i=1}^{2}\sup_{t\in D}\bigg|\frac{\partial}{\partial t_i}X_d(t)-\frac{\partial}{\partial t_i}X(t)\bigg|\sup_{t\in D}\bigg| \frac{\partial}{\partial t_i}U_d(t)\bigg|+\sup_{s\in D}|X_d(t)-X(t)|\sup_{t\in D}|\Delta U_d(t)|\end{aligned}$$ and $\sup_{t\in D}|X_{d}(t)-X(t)|\overset{w}{\to}0$ as $d\to\infty$ by Theorem [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"} and Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"} which also hold for random fields, it remains to show that $$\begin{aligned} \label{convergence} \sup_{t\in D}\bigg|\frac{\partial}{\partial t_i}X_d(t)-\frac{\partial}{\partial t_i}X(t)\bigg|\overset{w}{\to}0, \ i=1,2, \ d\to\infty.\end{aligned}$$ Let $h(x)=\alpha+(\beta-\alpha)F^{-1}_{{\rm Beta}(p,q)}\circ\Phi(x)$, $x\in\mathbb{R}$, then $$\begin{aligned} \sup_{t\in D}\bigg|\frac{\partial}{\partial t_i}X_d(t)-\frac{\partial}{\partial t_i}X(t)\bigg| &=&\sup_{t\in D}\bigg|h^{\prime}(G_d(t))\frac{\partial}{\partial t_i}G_d(t)-h^{\prime}(G(t))\frac{\partial}{\partial t_i}G(t)\bigg|\nonumber\\ &\leq&\bigg(\sup_{t\in D}\bigg|\frac{\partial}{\partial t_i}G_d(t)\bigg|\bigg)\sup_{t\in D}|h^{\prime}(G_d(t))-h^{\prime}(G(t))|\nonumber\\ & &+\bigg(\sup_{t\in D}|h^{\prime}(G(t))|\bigg)\sup_{t\in D}\bigg|\frac{\partial}{\partial t_i}G_d(t)-\frac{\partial}{\partial t_i}G(t)\bigg|.\end{aligned}$$ It can be shown that $h^{\prime}(\cdot)$ is bounded and Lipschitz continuous by using L'H$\rm \hat{o}$pital's Rule. Since the Gaussian random fields $G(t)$ and $\partial G(t)/\partial t_i$, $i=1,2$ have continuous samples, and the correlation function $c(s,t)$ in ([\[corr\]](#corr){reference-type="ref" reference="corr"}) and its second derivative are continuous, Theorem [Theorem 3](#thm4-1){reference-type="ref" reference="thm4-1"} applies to $G(t)$ and $\partial G(t)/\partial t_i$ so that $\sup_{t\in D}|G_d(t)-G(t)|\overset{w}{\to}0$ and $\sup_{t\in D}|(\partial G_d(t)/\partial t_i)-(\partial G(t)/\partial t_i)|\overset{w}{\to}0$, $i=1,2$, as $d\to\infty$. The Lipschitz continuity of $h^{\prime}(\cdot)$ and $\sup_{t\in D}|G_d(t)-G(t)|\overset{w}{\to}0$ as $d\to\infty$ imply the convergence $\sup_{t\in D}|h^{\prime}(G_d(t))-h^{\prime}(G(t))|\overset{w}{\to}0$, as $d\to\infty$. Since $\sup_{t\in D}{\rm Var}[\partial G_d(t)/\partial t_i]<\infty$ for all $d\geq 1$, then $P(\sup_{t\in D}|\partial G_d(t)/\partial t_i|>x)\to0$, as $x\to\infty$ [@gs1991], which implies $\sup_{t\in D}|\partial G_d(t)/\partial t_i|$, $i=1,2$, is almost surely bounded, $\sup_{t\in D}|h^{\prime}(G(t))|$ is also almost surely bounded since $h^{\prime}(\cdot)$ is bounded. Therefore, $\sup_{t\in D}|U_d(t)-U(t)|\overset{w}{\to}0$ as $d\to\infty$. The discrepancy between the target and FD apparent conductivities can be bounded by $$\begin{aligned} &&|X_{\rm{app},d}-X_{\rm{app}}|\nonumber\\ &=&\frac{1}{\tau_2}\bigg|\int_{0}^{\tau_2}\int_{0}^{\tau_1}\bigg(X_d(t)\frac{\partial U_d(t)}{\partial t_1}- X(t)\frac{\partial U(t)}{\partial t_1}\bigg)dt_1dt_2\bigg|\nonumber\\ &=&\frac{1}{\tau_2}\bigg|\int_{0}^{\tau_2}\int_{0}^{\tau_1}\bigg(U_d(t)\frac{\partial X_d(t)}{\partial t_1}- U(t)\frac{\partial X(t)}{\partial t_1}\bigg)dt_1dt_2+\int_{0}^{\tau_2}\Big(X_d(\tau_1,t_2)-X(\tau_1,t_2)\Big)dt_2\bigg|\nonumber\\ &\leq&\frac{1}{\tau_2}\int_{0}^{\tau_2}\int_{0}^{\tau_1}\bigg|\Big(U_d(t)-U(t)\Big) \frac{\partial X_d(t)}{\partial t_1}\bigg|dt_1dt_2+\frac{1}{\tau_2}\int_{0}^{\tau_2}\int_{0}^{\tau_1}\bigg|U(t)\bigg(\frac{\partial X_d(t)}{\partial t_1}- \frac{\partial X(t)}{\partial t_1}\bigg)\bigg|dt_1dt_2\nonumber\\ & &+\sup_{t_2\in[0,\tau_2]}|X_d(\tau_1,t_2)-X(\tau_1,t_2)|\nonumber\\ &\leq&\tau_1\sup_{t\in D}|U_d(t)-U(t)|\sup_{t\in D}\bigg|\frac{\partial X_d(t)}{\partial t_1}\bigg|+\tau_1\sup_{t\in D}|U(t)|\sup_{t\in D}\bigg|\frac{\partial X_d(t)}{\partial t_1}- \frac{\partial X(t)}{\partial t_1}\bigg|\nonumber\\ & &+\sup_{t\in D}|X_d(t)-X(t)|\end{aligned}$$ by using boundary conditions, i.e., $U(0,t_2)=U_d(0,t_2)=0$ and $U(\tau_1,t_2)=U_d(\tau_1,t_2)=1$, and integration by parts. The above inequality implies the convergence $|X_{\rm{app},d}-X_{\rm{app}}|\overset{w}{\to}0$ as $d\to\infty$ since $(i)$ $\sup_{t\in D}|U(t)|<\infty$ by assumption, $(ii)$ $\sup_{t\in D}|U_d(t)-U(t)|\overset{w}{\to}0$, $\sup_{t\in D}|X_d(t)-X(t)|\overset{w}{\to}0$ and $\sup_{t\in D}|(\partial X_d(t)/\partial t_1)-(\partial X(t)/\partial t_1)|\overset{w}{\to}0$ as $d\to\infty$, a property already shown, and $(iii)$  $\sup_{t\in D}|\partial X_d(t)/\partial t_1|\leq \sup_{t\in D}|h^{\prime}(G_d(t))|\sup_{t\in D}|\partial G_d(t)/\partial t_1|$, where $\sup_{t\in D}|\partial G_d(t)/\partial t_1|<\infty$ almost surely is already shown, $\sup_{t\in D}|h^{\prime}(G_d(t))|<\infty$ almost surely holds by the boundedness of $h^{\prime}(\cdot)$. The following numerical results are for $\alpha=1$, $\beta=20$, $p=0.5$, $q=1.5$, $\rho=0.7$, $\tau_1=20$ and $\tau_2=15$. The reference conductivity samples, i.e., the samples of $X_{\rm app}$ have been calculated from the corresponding samples of a discrete version of $G(t)$ corresponding to a mesh with sides $\Delta t_1=0.4$ and $\Delta t_2=0.3$. The reported statistics are based on $1000$ samples of $X_{\rm{app}}$. Figure [53](#ex6-fig1){reference-type="ref" reference="ex6-fig1"}  shows a sample of $G(t)$ and $G_d(t)$ for $d=50,150$ (left, middle and right panels). Figure [56](#ex6-fig2){reference-type="ref" reference="ex6-fig2"}  shows the corresponding samples of $X(t)$ and $X_d(t)$ for $d=50,150$ (left, middle and right panels) obtained from ([\[beta1\]](#beta1){reference-type="ref" reference="beta1"}) and ([\[beta2\]](#beta2){reference-type="ref" reference="beta2"}). Figure [59](#ex6-fig3){reference-type="ref" reference="ex6-fig3"}  shows the corresponding samples of $U(t)$ and $U_d(t)$ for $d=50,150$ (left, middle and right panels). The left and middle panels of Fig. [62](#ex6-fig5){reference-type="ref" reference="ex6-fig5"} show the scatter plots of $X_{\rm{app}}$ and $X_{\rm{app},d}$ for $d=50,150$. The thin solid line in the right panel of Fig. [62](#ex6-fig5){reference-type="ref" reference="ex6-fig5"} is an estimate of $P(X_{\rm{app}}>x)$ which is obtained from ([\[app1\]](#app1){reference-type="ref" reference="app1"}) by using the reference samples of $X(t)$ and $G(t)$. It is viewed as reference. The other lines of the figure are calculated from FD models for $d=50$ (heavy solid line) and $d=150$ (dotted line). The dotted line is the closest to the reference. These plots show, in agreement with our theoretical results, that the discrepancy between samples of $G(t)$ and $G_{d}(t)$, $X(t)$ and $X_{d}(t)$, $U(t)$ and $U_{d}(t)$, and distributions of $X_{\rm{app}}$ and $X_{\rm{app},d}$ can be made as small as desired by increasing the stochastic dimension $d$. {#ex6-fig1} {#ex6-fig1} {#ex6-fig1} {#ex6-fig2} {#ex6-fig2} {#ex6-fig2} {#ex6-fig3} {#ex6-fig3} {#ex6-fig3} {#ex6-fig5} {#ex6-fig5} {#ex6-fig5} # Conclusions Finite dimensional (FD) models, i.e., deterministic functions of time/space and finite sets of random variables, have been constructed for target vector-valued random processes and real-valued random fields. Samples of these models, referred to as FD samples, can be generated by standard Monte Carlo algorithms since, in contrast to target random functions which are uncountable families of random variables, as they depend on finite sets of random variables. Conditions have been established under which FD samples can be used as surrogates for target samples and can be employed to estimate distributions of functionals of target random functions, e.g., distributions of extremes of these functions. Numerical examples are presented to illustrate the implementation of FD models and show that the distributions of extremes of target random functions can be estimated from FD samples provided that they satisfy the conditions of our theorems. The numerical examples include estimates of the distributions of extremes of two vector-valued non-Gaussian processes and of the distribution of the apparent property of a two dimensional material specimen with random properties. It is shown that FD estimates of extremes and other functionals of target random functions are accurate for FD models satisfying the conditions of our theorems.\ # Declarations The work reported in this paper has been partially supported by the National Science Foundation under the grant CMMI-2013697. This support is gratefully acknowledged.\ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\ The datasets generated during the current study are not publicly available, since they have been generated for particular applications, but are available from the corresponding author on reasonable request.\ Hui Xu and Mircea D. Grigoriu confirm sole responsibility for the following: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation 99 P. Billingsley. . Wiley, New York, 1968. R. Courant and D. Hilbert. Methods of Mathematical Physics, volume 1. , Inc., New York, 1953. Wilbur B. Davenport and William L. Root. . Wiley, New York, 1987. K. Ferentios. On Tcebycheff's type inequalities. , 33, 125-132, 1982. I. Gohberg and S. Goldberg. . Birkhäuser, 2013. M. Grigoriu and K. D. Papoulia. Effective conductivity by a probability-based local method. , Vol.98, No.3, 033706, 2005. M. Grigoriu. Linear Dynamical Systems 1st Edition. , 2021. Y. Hakobyan, K. D. Papoulia and M. Grigoriu. Physical and geometrical percolation of effective coductivity on a lattice. , Vol.76, No.14, 144205, 2007. T. T. Kadota. Term-by-Term Differentiability of Mercer's Expansion. , Vol.18, No.1, pp.69-72, 1967. M. B. Marcus and L. A. Shepp. Sample behavior of Gaussian processes. pp. 423-442, 1971. J. Mercer. Functions of positive and negative type and their connection with the theory of integral equations. 209 (441-458): 415-446, doi:10.1098/rsta.1909.0016, 1909. M. Ostoja-Starzewski. . Chapman and Hall/CRC, 2007. W. Rudin. . McGraw-Hill, Inc., New York, 1976. W. Rudin. . McGraw-Hill, Inc., New York, 1987. G. Samorodnitsky. Probability tails of Gaussian extrema. , 38:55-84, 1991. A. W. van der Vaart. . New York: Cambridge University Press. ISBN 978-0-521-49603-2, 1998. E. Wong and B. Hajek. . Springer-Verlag, New York, 1985.

arxiv_math

--- abstract: | This paper studies a class of strongly monotone games involving non-cooperative agents that optimize their own time-varying cost functions. We assume that the agents can observe other agents' historical actions and choose actions that best respond to other agents' previous actions; we call this a best response scheme. We start by analyzing the convergence rate of this best response scheme for standard time-invariant games. Specifically, we provide a sufficient condition on the strong monotonicity parameter of the time-invariant games under which the proposed best response algorithm achieves exponential convergence to the static Nash equilibrium. We further illustrate that this best response algorithm may oscillate when the proposed sufficient condition fails to hold, which indicates that this condition is tight. Next, we analyze this best response algorithm for time-varying games where the cost functions of each agent change over time. Under similar conditions as for time-invariant games, we show that the proposed best response algorithm stays asymptotically close to the evolving equilibrium. We do so by analyzing both the equilibrium tracking error and the dynamic regret. Numerical experiments on economic market problems are presented to validate our analysis. author: - "Zifan Wang, Yi Shen, Michael M. Zavlanos, and Karl H. Johansson [^1] [^2] [^3]" bibliography: - ref.bib title: | **Convergence Analysis of the Best Response Algorithm\ for Time-Varying Games** --- # Introduction Online convex games study the interplay between game theory and online learning, and find many applications ranging from traffic routing [@sessa2019no] to economic market optimization [@wang2022risk; @lin2020finite]. In these games, agents simultaneously take actions to minimize their loss functions, which depend on the other agents' actions. Generally, every agent in an online convex game adapts its actions to the actions of other agents in a dynamic manner with the objective to minimize its regret, defined as the cumulative difference in performance between the agent's online actions and the best single action in hindsight. An algorithm is said to achieve no-regret learning if every agent's regret generated by this algorithm is sub-linear in the total number of episodes. If the agents in an online game reach a stationary point from which no agent has an incentive to deviate, then we say the game has reached a Nash equilibrium. There is a growing literature [@tatarenko2018learning; @bravo2018bandit; @drusvyatskiy2021improved; @mertikopoulos2019learning; @wang2022zeroth] that analyzes the Nash equilibrium convergence in strongly monotone games which admit a unique Nash equilibrium as shown in [@rosen1965existence]. In non-cooperative games, a common strategy used by competitive agents that selfishly minimize their own cost functions is the best response algorithm since it produces the most favorable outcome given the other agents plays. The best response algorithm has been shown to converge under a spectral condition associated with the best-response map [@shanbhag2016inexact; @facchinei201012]. In general, best response algorithms have been studied for several classes of games, including supermodular games [@milgrom1990rationalizability], potential games [@pass2019course; @swenson2018best; @lei2017randomized] and zero-sum games [@leslie2020best]. For example, [@swenson2018best] shows that in almost every potential game with finite actions, the best response dynamics converges to the unique Nash equilibrium with linear rate. Similarly, [@leslie2020best] shows the convergence of several best response dynamics in two-player zero-sum games. In this paper, we study the regret and equilibrium tracking error of the best response algorithm for time-varying games. Specifically, we consider a class of strongly monotone games [@rosen1965existence; @bravo2018bandit], which guarantee the uniqueness of the well-defined Nash equilibrium. To the best of our knowledge, the best response algorithm has not been explored in the literature for time-varying games. Instead, time-varying games have been analyzed using gradient-based algorithms for, e.g., strongly monotone games [@duvocelle2022multiagent] and zero-sum games [@zhang2022no]. Specifically, [@duvocelle2022multiagent] analyzes the Nash equilibrium convergence and the equilibrium tracking properties of the mirror descent algorithm for games that converge and diverge, respectively. In [@zhang2022no], a gradient-type algorithm is proposed that achieves performance guarantees under three different measures. As gradient-based algorithms are fundamentally different compared to the best response method, the techniques developed in these works cannot be applied here to analyze the best response algorithm. To address this challenge, we first start with time-invariant games. Specifically, we assume games that satisfy the so-called strong monotonicity condition with parameter $m>0$, which guarantees the uniqueness of the Nash equilibrium [@rosen1965existence]. We provide a sufficient condition $m>L\sqrt{N-1}$ under which the best response algorithm achieves linear convergence to the static Nash equilibrium, where $L$ is the Lipschitz constant related to the gradient of the individual loss functions and $N$ is the number of agents. Moreover, we show numerically that when this condition fails to hold, the best response algorithm may oscillate. Compared to [@facchinei201012], here we characterize the convergence in terms of the strong monotonicity parameter. For simple problems, we can show that our proposed condition is equivalent to the spectral condition proposed in [@facchinei201012]. Then, we analyze the best response algorithm for time-varying games where the Nash equilibrium evolves over time. Specifically, under similar conditions as for time-invariant games, we show that the average distance from the evolving equilibrium is bounded by the equilibrium variation. We also show that the dynamic regret is bounded by the cumulative variations of the loss functions. The rest of the paper is organized as follows. In Section [2](#sec:problem){reference-type="ref" reference="sec:problem"}, we provide some preliminaries and formally define the problem. In Section [3](#sec:BR){reference-type="ref" reference="sec:BR"}, we present the regret and equilibrium convergence of the best response algorithm for time-invariant games. In Section [4](#sec:TV_BR){reference-type="ref" reference="sec:TV_BR"}, we extend our result to time-varying games and analyze the equilibrium tracking error and the dynamic regret. In Section [5](#sec:simulation){reference-type="ref" reference="sec:simulation"}, numerical experiments on a Cournot game are presented to verify our method. Finally, in Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}, we conclude the paper. # Preliminaries and Problem Definition {#sec:problem} ## Online Convex Games Consider an online convex game $\mathcal{G}$ with $N$ agents, whose goal is to learn their best individual actions that minimize their local loss functions. For each agent $i\in \mathcal{N}=\{1,\ldots,N\}$, denote by $\mathcal{C}_i(x_i,x_{-i}) : \mathcal{X} \rightarrow \mathbb{R}$ its individual loss function, where $x_i \in \mathcal{X}_i$ is the action of agent $i$, $x_{-i}$ are the actions of all agents excluding agent $i$, and we define by $\mathcal{X} =\Pi_{i=1}^N\mathcal{X}_i$ the joint action space since each agent takes actions independently. For ease of notation, we collect all agents' actions in a vector $x:=(x_1,\ldots,x_N)$. We assume that $\mathcal{C}_i(x)$ is convex in $x_i$ for all $x_{-i} \in \mathcal{X}_{-i}$, where $\mathcal{X}_{-i}$ is the joint action space excluding agent $i$. The goal of every agent $i$ is to determine the action $x_i$ that minimizes its individual cost function, i.e., $$\begin{aligned} \label{eq:def:game} \mathop{{\rm{min}}}_{x_i \in \mathcal{X}_i} \mathcal{C}_{i}(x_i,x_{-i}).\end{aligned}$$ As shown in [@rosen1965existence], convex games always have at least one Nash equilibrium. In what follows, we denote by $x^{*}$ a Nash equilibrium of the game [\[eq:def:game\]](#eq:def:game){reference-type="eqref" reference="eq:def:game"}. Then, for each agent $i$, we have $\mathcal{C}_i(x^{*})\leq \mathcal{C}_i(x_i,x_{-i}^{*}),$ $\forall x_i \in \mathcal{X}_i$, $i\in\mathcal{N}$. At this Nash equilibrium point, agents are strategically stable in the sense that each agent lacks incentive to change its action. Since the agents' loss functions are convex, the Nash equilibrium can also be characterized by the first-order optimality condition, i.e., $\langle \nabla_{x_i} \mathcal{C}_i(x^{*}), x_i - x_i^{*} \rangle \geq 0, \; \forall x_i \in \mathcal{X}_i, i\in\mathcal{N},$ where $\nabla_{x_i} \mathcal{C}_i(x)$ is the partial derivative of the loss function with respect to each agent's action. We write $\nabla_{i} \mathcal{C}_i(x)$ instead of $\nabla_{x_i} \mathcal{C}_i(x)$ whenever it is clear from the context. In general, it is not easy to show convergence to a Nash equilibrium for games with multiple Nash Equilibria. For this reason, recent studies often focus on games that are so-called strongly monotone and are well-known to have a unique Nash equilibrium [@rosen1965existence]. The game [\[eq:def:game\]](#eq:def:game){reference-type="eqref" reference="eq:def:game"} is said to be $m$-strongly monotone if for $\forall x,x'\in \mathcal{X}$ we have that $$\begin{aligned} \label{eq:strong_monotone} \sum_{i=1}^N \langle \nabla_i \mathcal{C}_i(x) -\nabla_i \mathcal{C}_i(x'),x_i-x_i' \rangle \geq m \left\|x -x' \right\|^2.\end{aligned}$$ The ability of the agents to efficiently learn their optimal actions can be quantified using the notion of (static) regret that captures the cumulative loss of the learned online actions compared to the best actions in hindsight, and can be formally defined as $$\begin{aligned} \label{eq:def:regret:game} {\rm{SR}}_i(T)= \sum_{t=1}^T \mathcal{C}_i(x_t) - \mathop{\rm{min}}_{x_i} \sum_{t=1}^T\mathcal{C}_i(x_i,x_{-i,t}),\end{aligned}$$ for sequences of actions $\{x_{i,t} \}_{t=1}^T, i=1,\ldots,N$. An algorithm is said to be no-regret if the regret of each agent is sub-linear in the total number of episodes $T$, i.e., ${\rm{SR}}_i(T)=\mathcal{O}(T^a), a\in[0,1)$, $\forall i \in \mathcal{N}$. ## Problem Definition In this work, we consider the time-varying game $\mathcal{G}_t$ where at episode $t$ each agent aims to minimize its time-varying cost function, i.e., $$\begin{aligned} \label{eq:def:TV:game} \mathop{{\rm{min}}}_{x_i \in \mathcal{X}_i} \mathcal{C}_{i,t}(x_i,x_{-i}).\end{aligned}$$ Then, we can define the best response algorithm for time-varying games as $$\begin{aligned} \label{eq:TVBR:update} x_{i,t+1} = \mathop{\rm{arg min}}_{x_i \in \mathcal{X}_i} \mathcal{C}_{i,t} (x_i, x_{-i,t}).\end{aligned}$$ To attain the best response action $x_{i,t+1}$, for each agent $i$, we assume the cost function $\mathcal{C}_{i,t}$ is known and all other agents' previous actions are provided. This is not a very strong assumption. For example, in supply chain problems [@cachon2006game], $\mathcal{C}_{i,t}$ can represent an agent's local revenue model that depends on all competitors' actions and unknown market demands. At the beginning of episode $t+1$, the agents may not be able to observe the other agents' actions and precisely predict the market demands. However, previous actions and demands can be obtained from public revenue reports. Thus, it is reasonable to implement a strategy where the agents take actions that best respond to the other agents' actions from the previous episode. In addition, we assume that at every episode $t$, the time-varying game with the cost function $\mathcal{C}_{i,t}$ is strongly monotone and thus has a unique Nash equilibrium, which we denote by $x_t^{*}$. To analyze the performance of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} for time-varying games, we define the equilibrium tracking error $$\begin{aligned} \label{eq:BRTV:trackingerror} {\rm{Err}}(T):=\sum_{t=1}^T\left\| x_t - x_t^{*}\right\|^2,\end{aligned}$$ and the dynamic regret $$\begin{aligned} \label{eq:BRTV:dynamic:regret} {\rm{DR}}_i(T) := \sum_{t=1}^T \Big( \mathcal{C}_{i,t}(x_t) - \mathop{\rm{min}}_{y_i} \mathcal{C}_{i,t}(y_i,x_{-i,t})\Big),\end{aligned}$$ where $T$ is the total number of episodes. If the game $\mathcal{G}_t$ changes significantly over time, it is reasonable to expect that it may become impossible to track the evolving equilibrium. The time-varying problem becomes meaningful only when the variation of the game $\mathcal{G}_t$ is reasonably small. To capture the effect of the variation of the game $\mathcal{G}_t$ on the performance of the best response algorithm, we first define the equilibrium variation $$\begin{aligned} \label{eq:def:VT} V_T:=\sum_{t=1}^T\left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2,\end{aligned}$$ which tracks the changes of Nash equilibria. It is possible that the cost function $\mathcal{C}_{i,t}$ changes over time but the equilibrium stays constant, i.e., $V_T=0$. To further capture the variations of the cost functions, we define the function variation $$\begin{aligned} \label{eq:def:WT} W_{i,T} = \sum_{t=1}^T \sup_{x\in\mathcal{X}}|C_{i,t}(x) - C_{i,t+1}(x)|.\end{aligned}$$ Our goal in this paper is to analyze the equilibrium tracking error and the dynamic regret of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} for time-varying games. To do so, we start with the analysis of time-invariant games and then extend our results to the time-varying case. # Time-Invariant Games {#sec:BR} In this section, we provide sufficient conditions for Nash equilibrium convergence of the best response algorithm for time-invariant games. The best response algorithm in this case becomes $$\begin{aligned} \label{eq:BR:update} x_{i,t+1} = \mathop{\rm{arg min}}_{x_i \in \mathcal{X}_i} \mathcal{C}_{i} (x_i, x_{-i,t}).\end{aligned}$$ **Proposition 1**. *Suppose that the game $\mathcal{G}$ is $m$-strongly monotone, and $\nabla_i \mathcal{C}_i(x_i,x_{-i})$ is $L$-Lipschitz continuous in $x_{-i}$ for every $x_i \in \mathcal{X}_i$, with parameter $m>L \sqrt{N-1}$. Then, the best response algorithm [\[eq:BR:update\]](#eq:BR:update){reference-type="eqref" reference="eq:BR:update"} satisfies that $$\begin{aligned} \label{eq:BR:convergence} \left\| x_T - x^{*}\right\| \leq \rho^{T-1} \left\| x_1 - x^{*}\right\|,\end{aligned}$$ where $\rho:= \frac{L\sqrt{N-1}}{m}$.* Applying the first order optimality condition to the cost function $\mathcal{C}_i$ at the optimal point $x_{i,t+1}$ and using the update rule [\[eq:BR:update\]](#eq:BR:update){reference-type="eqref" reference="eq:BR:update"}, we have that $$\begin{aligned} \label{eq:BR_temp1} \langle \nabla_i \mathcal{C}_i (x_{i,t+1},x_{-i,t}),x_i - x_{i,t+1} \rangle \geq 0, \; \; \forall x_i \in \mathcal{X}_i.\end{aligned}$$ Since the game is strongly monotone, we have that for all $x_i \in \mathcal{X}_i$, $$\begin{aligned} \label{eq:BR_temp2} \langle \nabla_i \mathcal{C}_i (x_{i},x_{-i,t})- \nabla_i \mathcal{C}_i (x_{i,t+1},x_{-i,t}), x_i - x_{i,t+1} \rangle \nonumber \\ \geq m \left\| x_i -x_{i,t+1} \right\|^2,\end{aligned}$$ which follows from the definition [\[eq:strong_monotone\]](#eq:strong_monotone){reference-type="eqref" reference="eq:strong_monotone"} by setting $x=(x_{i},x_{-i,t})$ and $x'=(x_{i,t+1},x_{-i,t})$. Combining [\[eq:BR_temp2\]](#eq:BR_temp2){reference-type="eqref" reference="eq:BR_temp2"} with [\[eq:BR_temp1\]](#eq:BR_temp1){reference-type="eqref" reference="eq:BR_temp1"} and replacing $x_i$ with $x_i^{*}$, we get $$\begin{aligned} \label{eq:BR_temp3} &m \left\| x_i^{*} -x_{i,t+1} \right\|^2 % \leq & \langle \nabla_i \mathcal{C}_i (x_{i}^{*},x_{-i,t})- \nabla_i \mathcal{C}_i (x_{i,t+1},x_{-i,t}), x_i^{*} - x_{i,t+1} \rangle \nonumber \\ \leq \langle \nabla_i \mathcal{C}_i (x_{i}^{*},x_{-i,t}) , x_i^{*} - x_{i,t+1} \rangle .\end{aligned}$$ Summing the both sides of inequality [\[eq:BR_temp3\]](#eq:BR_temp3){reference-type="eqref" reference="eq:BR_temp3"} over $i=1,\ldots,N$, we have that $$\begin{aligned} \label{eq:BR_temp4} & \left\| x_{t+1} - x^{*} \right\|^2 \leq \frac{1}{m} \sum_i \langle \nabla_i \mathcal{C}_i (x_{i}^{*},x_{-i,t}) , x_i^{*} - x_{i,t+1} \rangle \nonumber \\ & \leq \frac{1}{m} \sum_i \langle \nabla_i \mathcal{C}_i (x_{i}^{*},x_{-i,t}) - \nabla_i \mathcal{C}_i (x^{*}) , x_i^{*} - x_{i,t+1} \rangle \nonumber \\ & \leq \frac{1}{m} \sum_i L \left\| x_{-i,t} - x_{-i}^{*}\right\| \left\| x_{i}^{*} - x_{i,t+1}\right\| \nonumber \\ & \leq \frac{L\sqrt{N-1}}{m} \left\| x_{t} - x^{*}\right\| \left\| x^{*} - x_{t+1}\right\|,\end{aligned}$$ where the second inequality follows from the Nash equilibrium condition $\langle \nabla_i \mathcal{C}_i (x^{*}),x_i - x_{i}^{*} \rangle \geq 0$, $\forall x_i \in \mathcal{X}_i$ and the third inequality is due to the Lipschitz continuous property of the function $\mathcal{C}_i$ in $x_{-i}$. The last inequality follows from the Cauchy-Schwarz inequality. Dividing the inequality [\[eq:BR_temp4\]](#eq:BR_temp4){reference-type="eqref" reference="eq:BR_temp4"} by $\left\| x_{t+1} - x^{*} \right\|$ yields $$\begin{aligned} \label{eq:BR_temp5} \left\| x_{t+1} - x^{*} \right\| \leq \frac{L \sqrt{N-1}}{m}\left\| x_{t} - x^{*} \right\|.\end{aligned}$$ Note, if $\left\| x_{t+1} - x^{*} \right\| = 0$, then [\[eq:BR_temp5\]](#eq:BR_temp5){reference-type="eqref" reference="eq:BR_temp5"} holds trivially. Applying inequality [\[eq:BR_temp5\]](#eq:BR_temp5){reference-type="eqref" reference="eq:BR_temp5"} iteratively over $t=1,\ldots,T-1$ completes the proof. In what follows, we provide some intuition and explain the condition $m>L \sqrt{N-1}$. First, suppose that $L_1$ is the Lipschitz constant of the function $\nabla_i \mathcal{C}_i(x)$ with respect to $x$. From its definitions we conclude that $L\leq L_1$. Therefore, the Lipschitz constant $L_1$ provides an upper bound on the variation of the gradients and is always greater than the strongly monotone parameter $m$ which provides a lower bound, i.e., $m\leq L_1$. However, it is still possible to have $m>L \sqrt{N-1}$. For example, if $\mathcal{C}_i$ only depends on $x_i$, we have that $L = 0$ and thus the condition naturally holds as long as $m>0$. On the other hand, consider the condition $m>L\sqrt{N-1}$ and rearrange the terms to get $L<\frac{m}{ \sqrt{N-1}}$. Recall that $L$ is the Lipschitz constant of the function $\nabla_i \mathcal{C}_i(x_i,x_{-i})$ with respect to $x_{-i}$, which can be interpreted as the maximum influence of the other agents' actions on agent $i$. The condition $L<\frac{m}{ \sqrt{N-1}}$ requires that this influence is small enough for the game to converge. The presence of multiple agents ($N$ is large) reduces the upper bound on the influence of other agents' actions which , effectively, increases the difficulty of the game. Note that [@facchinei201012] also provides a sufficient condition for convergence of the best response algorithm, that involves the spectral norm of a matrix composed of parameters related to the second-order partial derivative of the cost function. In this work, we analyze the best response algorithm from a different perspective that relies on strong monotonicity to characterize convergence. In simple cases such as two-player potential games, it is easy to show that our condition is equivalent to the condition in [@facchinei201012]. However, in general, strong monotonicity provides a more intuitive condition for convergence. Finally, we experimentally show that when the condition $m>L\sqrt{N-1}$ does not hold, the best-response algorithm may lead to cycles. This result further validates the utility of the proposed condition. Proposition [Proposition 1](#prop:BR){reference-type="ref" reference="prop:BR"} shows that the best response algorithm converges to the Nash equilibrium at an exponential rate. Indeed, it is a no-regret learning algorithm for each agent as well, as shown in the following proposition. **Proposition 2**. *Suppose that the game $\mathcal{G}$ is $m$-strongly monotone with parameter $m>L \sqrt{N-1}$, the cost $C_i(x_i,x_{-i})$ is $L_0$-Lipschitz continuous in $x_{-i}$ for every $x_i \in \mathcal{X}_i$, and the diameter of the convex set $\mathcal{X}_i$ is bounded by $D$, for all $i=1,\ldots,N$ Then, the static regret of the best response algorithm satisfies $$\begin{aligned} {\rm{SR}}_i(T) \leq \sum_{t=1}^T \mathcal{C}_i(x_t) - \sum_{t=1}^T \min_{x_i}\mathcal{C}_i(x_i,x_{-i,t}) = \mathcal{O}(1).\end{aligned}$$* The first inequality holds due to the fact that $\sum_{t=1}^T \min_{x_i}\mathcal{C}_i(x_i,x_{-i,t})\leq \min_{x_i}\sum_{t=1}^T \mathcal{C}_i(x_i,x_{-i,t})$. Observe that $\mathcal{C}_i(x_{i,t+1},x_{-i,t})=\min_{x_i}\mathcal{C}_i(x_i,x_{-i,t})$ since $x_{i,t+1}=\mathop{\rm{arg min}}_{x_i \in \mathcal{X}_i} \mathcal{C}_{i} (x_i, x_{-i,t})$. Then, it follows that $$\begin{aligned} \label{eq:BR_no_temp1} &{\rm{SR}}_i(T) \leq \sum_{t=1}^T \mathcal{C}_i(x_t) - \sum_{t=1}^T \min_{x_i}\mathcal{C}_i(x_i,x_{-i,t}) \nonumber\\ & = \sum_{t=1}^T \Big( \mathcal{C}_i(x_t)- \mathcal{C}_i(x_{t+1})+ \mathcal{C}_i(x_{t+1}) - \mathcal{C}_i(x_{i,t+1},x_{-i,t}) \Big) \nonumber \\ & \leq \mathcal{C}_i(x_1) + \sum_{t=1}^T \Big(\mathcal{C}_i(x_{t+1}) - \mathcal{C}_i(x_{i,t+1},x_{-i,t}) \Big) \nonumber \\ & \leq \mathcal{C}_i(x_1) + L_0 \sum_{t=1}^T \left\|x_{-i,t+1} - x_{-i,t}\right\| \nonumber \\ & \leq \mathcal{C}_i(x_1) + L_0 \sum_{t=1}^T \left\|x_{t+1} - x_{t}\right\|,\end{aligned}$$ where the second to the last inequality follows from the Lipschitz continuous property of the function $\mathcal{C}_i$ in $x$. By virtue of [\[eq:BR:convergence\]](#eq:BR:convergence){reference-type="eqref" reference="eq:BR:convergence"} in Proposition 1, we have $$\begin{aligned} \label{eq:BR_no_temp2} &\left\|x_{t+1} - x_{t}\right\|^2 = \left\|x_{t+1} -x^{*}+x^{*} -x_{t}\right\|^2 \nonumber \\ &\leq 2 \left\|x_{t+1} -x^{*}\right\|^2 + 2 \left\|x^{*} -x_{t}\right\|^2 \leq 2(\rho^2+1)\left\|x_{t}-x^{*} \right\|^2.\end{aligned}$$ Substituting the inequality [\[eq:BR_no_temp2\]](#eq:BR_no_temp2){reference-type="eqref" reference="eq:BR_no_temp2"} into [\[eq:BR_no_temp1\]](#eq:BR_no_temp1){reference-type="eqref" reference="eq:BR_no_temp1"}, we have $$\begin{aligned} \label{eq:BR_no_temp3} {\rm{SR}}_i(T) \leq & \mathcal{C}_i(x_1) + L_0 \sum_{t=1}^T \sqrt{2(\rho^2+1)} \left\|x_{t} - x^{*}\right\| \nonumber \\ \leq & \mathcal{C}_i(x_1) + L_0 \sqrt{2(\rho^2+1)} \sum_{t=1}^T \rho^t D \nonumber \\ \leq & \mathcal{C}_i(x_1)+\frac{ DL_0 \sqrt{2(\rho^2+1)}} {1-\rho},\end{aligned}$$ which completes the proof. Proposition [Proposition 2](#prop:BR:no_regret){reference-type="ref" reference="prop:BR:no_regret"} indeed provides a stronger bound than the static regret defined in [\[eq:def:regret:game\]](#eq:def:regret:game){reference-type="eqref" reference="eq:def:regret:game"}. Instead of comparing to a single best action in hindsight, it compares with a sequence of episode-wise best actions, which is equivalent to the dynamic regret with time-invariant cost functions. This strong result own itself to the best response algorithm. # Time-varying games {#sec:TV_BR} In this section, we analyze time-varying games $\mathcal{G}_t$ where the cost functions of the agents change over time. Since the equilibrium of these games also varies, in what follows we analyze the ability of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} to generate actions that track the evolving equilibrium. If the game $\mathcal{G}_t$ changes significantly, it is reasonable to expect that it will be hard to track the evolving equilibrium. Therefore, as in related literature [@duvocelle2022multiagent; @zhang2022no], we assume that both the equilibrium variation $V_T$ in [\[eq:def:VT\]](#eq:def:VT){reference-type="eqref" reference="eq:def:VT"} and the function variation $W_{i,T}$ in [\[eq:def:WT\]](#eq:def:WT){reference-type="eqref" reference="eq:def:WT"} are sub-linear in $T$, for $i=1,\ldots,N$. In what follows, we analyze the equilibrium tracking error of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} in terms of the equilibrium variation. **Theorem 1**. *Suppose that the time-varying game $\mathcal{G}_t$ is $m_t$-strongly monotone and $\nabla_i \mathcal{C}_{i,t}(x_i,x_{-i})$ is $L_t$-Lipschitz continuous in $x_{-i}$ for every $x_i \in \mathcal{X}_i$ with parameter $m_t>L_t \sqrt{N-1}$, for $\forall t$. Then, the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} satisfies that $$\begin{aligned} \label{eq:BRTV:convergence} {\rm{Err}}(T) \leq \frac{\left\| x_{1}- x_{1}^{*}\right\|^2}{1-\rho_m} + \frac{V_T}{(1-\rho_m)^2}=\mathcal{O}\left( 1+ V_T\right),\end{aligned}$$ where $\rho_m:= \mathop{\rm{max}}_t \left\{ \frac{L_t\sqrt{N-1}}{m_t} \right\}$.* Applying the same arguments as in Proposition [Proposition 1](#prop:BR){reference-type="ref" reference="prop:BR"} to the cost function $\mathcal{C}_{i,t}$, we can obtain an inequality similar to [\[eq:BR_temp5\]](#eq:BR_temp5){reference-type="eqref" reference="eq:BR_temp5"} as $$\begin{aligned} \label{eq:BRTV_temp4} \left\| x_{t+1}-x_{t}^{*} \right\| \leq \rho_t \left\| x_{t} - x_{t}^{*}\right\|,\end{aligned}$$ where $\rho_t:= \frac{L_t\sqrt{N-1}}{m_t}$. Observe that $$\begin{aligned} &\left\| x_{t+1}-x_{t+1}^{*} \right\|^2 = \left\| x_{t+1}- x_{t}^{*} + x_{t}^{*}- x_{t+1}^{*} \right\|^2 \nonumber \\ \leq &(1+\lambda) \left\| x_{t+1}- x_{t}^{*}\right\|^2 + (1+\frac{1}{\lambda}) \left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2,\end{aligned}$$ for $\forall \lambda >0$. Setting $\lambda = \frac{1}{\rho_t}-1>0$ yields $$\begin{aligned} \label{eq:BRTV_temp5} &\left\| x_{t+1}-x_{t+1}^{*} \right\|^2 \leq \frac{1}{\rho_t } \left\| x_{t+1}- x_{t}^{*}\right\|^2 + \frac{1}{1-\rho_t } \left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2 \nonumber \\ & \leq \rho_t \left\| x_{t}- x_{t}^{*}\right\|^2 + \frac{1}{1-\rho_t } \left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2 \nonumber \\ & \leq \rho_m \left\| x_{t}- x_{t}^{*}\right\|^2 + \frac{1}{1-\rho_m } \left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2,\end{aligned}$$ where the second inequality follows from [\[eq:BRTV_temp4\]](#eq:BRTV_temp4){reference-type="eqref" reference="eq:BRTV_temp4"} and the last inequality is due to the fact that $\rho_t \leq \rho_m <1$. Rearranging and summing [\[eq:BRTV_temp5\]](#eq:BRTV_temp5){reference-type="eqref" reference="eq:BRTV_temp5"} over $t=1,\ldots,T$, we have that $$\begin{aligned} &(1-\rho_m) \sum_{t=1}^T \left\| x_{t}- x_{t}^{*}\right\|^2 \nonumber \\ \leq & \sum_{t=1}^T \left(\left\| x_{t}- x_{t}^{*}\right\|^2 - \left\| x_{t+1}-x_{t+1}^{*} \right\|^2 + \frac{\left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2}{1-\rho_m } \right) \nonumber \\ \leq & \left\| x_{1}- x_{1}^{*}\right\|^2 +\frac{1}{1-\rho_m } \sum_{t=1}^T\left\|x_{t}^{*}- x_{t+1}^{*} \right\|^2 \nonumber \\ \leq & \left\| x_{1}- x_{1}^{*}\right\|^2 + \frac{1}{1-\rho_m } V_T.\end{aligned}$$ Dividing both sides of the above inequality by $(1-\rho_m)$ completes the proof. Theorem [Theorem 1](#theorem:BRTV){reference-type="ref" reference="theorem:BRTV"} shows that $V_T$ dominates the equilibrium tracking error. If $V_T$ is sub-linear in $T$, so is the equilibrium tracking error. In what follows, we analyze the dynamic regret of each agent in terms of the equilibrium variation and the function variation. **Theorem 2**. *Suppose that the time-varying game $\mathcal{G}_t$ is $m_t$-strongly monotone, $\nabla_i \mathcal{C}_{i,t}(x_i,x_{-i})$ is $L_t$-Lipschitz continuous in $x_{-i}$ for every $x_i \in \mathcal{X}_i$ with parameter $m_t>L_t \sqrt{N-1}$, and the cost $\mathcal{C}_{i,t}(x)$ is $L_0$-Lipschitz continuous in $x_{-i}$ for every $x_i \in \mathcal{X}_i$ for $\forall t$. Then, the dynamic regret of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"} satisfies $$\begin{aligned} {\rm{DR}}_i(T) = \mathcal{O}\left( W_{i,t} +\sqrt{TV_T} \right), \; i=1,\ldots,N.\end{aligned}$$* Using the update rule of the best response algorithm [\[eq:TVBR:update\]](#eq:TVBR:update){reference-type="eqref" reference="eq:TVBR:update"}, we have $$\begin{aligned} &{\rm{DR}}_i(T) = \sum_{t=1}^T \Big( \mathcal{C}_{i,t}(x_t) -\mathcal{C}_{i,t}(x_{i,t+1},x_{-i,t})\Big) \nonumber \\ =& \sum_{t=1}^T \Big( \mathcal{C}_{i,t}(x_t) - \mathcal{C}_{i,t+1}(x_{t+1}) + \mathcal{C}_{i,t+1}(x_{t+1}) \nonumber \\ &- \mathcal{C}_{i,t}(x_{t+1}) + \mathcal{C}_{i,t}(x_{t+1}) - \mathcal{C}_{i,t}(x_{i,t+1},x_{-i,t}) \Big) \nonumber \\ \leq & \mathcal{C}_{i,1}(x_1) + W_{i,T} + \sum_{t=1}^T \Big(\mathcal{C}_{i,t}(x_{t+1}) - \mathcal{C}_{i,t}(x_{i,t+1},x_{-i,t}) \Big) \nonumber \\ \leq & \mathcal{C}_{i,1}(x_1)+ W_{i,T} + L_0 \sum_{t=1}^T\left\| x_{-i,t+1}-x_{-i,t}\right\| \nonumber \\ \leq & \mathcal{C}_{i,1}(x_1)+ W_{i,T} + L_0 \sum_{t=1}^T \left\| x_{t+1}-x_{t}\right\|.\end{aligned}$$ Using the inequality [\[eq:BRTV_temp4\]](#eq:BRTV_temp4){reference-type="eqref" reference="eq:BRTV_temp4"} and the fact that $\rho_t \leq \rho_m <1$, we have $$\begin{aligned} &\sum_{t=1}^T\left\| x_{t+1}-x_{t}\right\|^2 =\sum_{t=1}^T\left\| x_{t+1}- x_t^{*} +x_t^{*} - x_{t}\right\|^2 \nonumber \\ & \leq \sum_{t=1}^T \big( (1+\frac{1}{\rho_m}) \left\| x_{t+1}- x_t^{*}\right\|^2 + (1+\rho_m) \left\| x_t^{*} - x_{t}\right\|^2 \big) \nonumber \\ & \leq (\rho_m+1)^2 \sum_{t=1}^T \left\| x_{t} - x_t^{*} \right\|^2,\end{aligned}$$ which further yields $$\begin{aligned} &{\rm{DR}}_i(T) \nonumber \\ \leq & \mathcal{C}_{i,1}(x_1)+ W_{i,T} + L_0 \sqrt{T} \sqrt{\sum_{t=1}^T \left\| x_{t+1}-x_{t}\right\|^2} \nonumber \\ \leq & \mathcal{C}_{i,1}(x_1)+ W_{i,T} + L_0 \sqrt{T} \sqrt{(\rho_m+1)^2 \sum_{t=1}^T \left\| x_{t} - x_t^{*} \right\|^2} \nonumber \\ =& \mathcal{O}\left( W_{i,t} +\sqrt{TV_T} \right),\end{aligned}$$ where in the last inequality we use the results from Theorem [Theorem 1](#theorem:BRTV){reference-type="ref" reference="theorem:BRTV"}. The proof is complete. Theorem [Theorem 2](#theorem:BRTV:no_regret){reference-type="ref" reference="theorem:BRTV:no_regret"} shows that the dynamic regret is sublinear in $T$ if the variation of the game satisfies $W_{i,T}=\mathcal{O}(T^a)$ and $V_{T}=\mathcal{O}(T^b)$ with $a,b\in[0,1)$. **Remark 1**. *(Connection between dynamic regret and equilibrium tracking error). In the single agent case, equilibrium tracking error is equivalent to the dynamic regret. However, this is not true for games involving multiple agents. This is due to the fact that the function $\mathcal{C}_{i,t}(\cdot,x_{-i,t})$ is time-varying due to changes in the function $\mathcal{C}_{i,t}$ itself and changes in other agents' actions $x_{-i,t}$. To see this, consider the class of time-varying games with time-varying cost functions but constant equilibrium, i.e., $V_T=0$, $W_{i,T}=\mathcal{O}(T^a)$ for some $a>0$. In this case, we have ${\rm{Err}}(T) = \mathcal{O}(1)$ but ${\rm{DR}}_i(T) = \mathcal{O}(T^a)$.* # Numerical Experiments {#sec:simulation} In this section, we validate our analysis on a Cournot game for both time-invariant and time-varying losses. ## Time-invariant game We first focus on the time-invariant case. We consider a Cournot game with two agents whose goal is to minimize their local losses by appropriately setting the production quantity $x_i$, $i=1,2$. The loss function of each agent is given by $\mathcal{C}_i(x) = x_i(\frac{a_i x_i}{2} + b_i x_{-i} - e_i)+ 1$, where $a_i>0$ , $b_i$, $e_i$ are constant parameters, and $x_{-i}$ denotes the production quantity of the opponent of agent $i$. It is easy to show that $\nabla_i \mathcal{C}_i(x) = a_i x_i + b_i x_{-i} -e_i$. Recalling that $L$ is the Lipschitz constant of the function $\nabla_i \mathcal{C}_i(x)$ with respect to $x_{-i}$, we have $L={\rm{max}}\{ |b_1|,|b_2|\}$. Define $g(x) = (\nabla_1 \mathcal{C}_1(x), \nabla_2 \mathcal{C}_2(x))$ and let $G(x)$ denote the Jacobian of $g(x)$, i.e., $G(x)=[a_1,b_1;b_2,a_2]$. According to [@rosen1965existence], the strong monotonicity parameter $m$ coincides with the smallest eigenvalue of the matrix $\frac{G(x)+G'(x)}{2}$. We validate our methods for three different selections of parameters $\theta^k:=(a_1^k,a_2^k,b_1^k,b_2^k,e_1^k,e_2^k)$ for $k=1,2,3$. Specifically, We select $\theta^1 = (1,1,0.6,-0.5,1.2,0.8)$, $\theta^2 = (1,1,1,-1,1.2,0.8)$ and $\theta^3 = (1,1,2,-1,1.2,0.8)$. It is easy to verify that $\theta^1$, $\theta^2$ and $\theta^3$ correspond to the cases $m>L\sqrt{N-1}$, $m=L\sqrt{N-1}$, and $m<L\sqrt{N-1}$, respectively. The convergence results are shown in Figure [1](#fig_TI){reference-type="ref" reference="fig_TI"}. We observe that when $m>L\sqrt{N-1}$, the best response converges with exponential rate. When $m\leq L\sqrt{N-1}$, the best response algorithm fails to converge, which indicates the tightness of our theoretical results. {#fig_TI width="0.9\\columnwidth"} ## Time-varying games {#time-varying-games} For the time-varying case, the loss function of agent $i$ is defined as $\mathcal{C}_{i,t}(x) = x_i(\frac{a_i x_i}{2} + b_{i,t} x_{-i} - e_{i,t})+ 1$, where $a_i=2$, $i=1,2$, and $b_{i,t}$, $e_{i,t}$ are time-varying parameters. The time-varying parameters are selected as $$\begin{aligned} b_{i,t}=\left\{ \begin{array}{cc} 0.3+ 0.1\times (-1)^t & t\in[1,T^{0.6}] \\ 0.3 & t\in(T^{0.6},T] \end{array},\right.\end{aligned}$$ $$\begin{aligned} e_{i,t}=\left\{ \begin{array}{cc} 0.4 & t\in[1,T^{0.6}] \\ 0.4+ 0.1\times (-1)^t t^{-1/4} & t\in(T^{0.6},T] \end{array}.\right.\end{aligned}$$ We select $T=1000$ and thus $T^{0.6}\approx63$. It can be verified that the selection of parameters yields $m_t\geq L_t \sqrt{N-1}$ for $\forall t$, and $V_T = \mathcal{O}(T^{3/4})$, $W_{i,T} = \mathcal{O}(T^{3/4})$, $i=1,2$. Figures [2](#fig_tracking){reference-type="ref" reference="fig_tracking"}--[3](#fig_DR){reference-type="ref" reference="fig_DR"} illustrate the equilibrium tracking error and the dynamic regret of the best response algorithm, respectively. We observe that, when $t\in[1,T^{0.6}]$, both the equilibrium tracking error and the dynamic regret grow rapidly due to the oscillations of $b_{i,t}$; when $t\in(T^{0.6},T]$, they grow slowly since $b_{i,t}$ is a constant and the variation of $e_{i,t}$ is decreasing over time. Moreover, both the equilibrium tracking error and the dynamic regret are sub-linear in the total number of episodes, which supports our theoretical results. {#fig_tracking width="0.9\\columnwidth"} {#fig_DR width="0.9\\columnwidth"} # Conclusion {#sec:conclusion} In this work, we analyzed the best response algorithm for the class of strongly monotone games. We first considered standard time-invariant games and obtained a sufficient condition under which the best response algorithm converges at an exponential rate. We provided numerical experiments that showed the best response algorithm can diverge if this condition fails to hold, which indicates that the condition is tight. Subsequently, we analyzed the best response algorithm for time-varying games with evolving equilibria. We showed that the equilibrium tracking error and the dynamic regret can be bounded in terms of the variations of evolving equilibria and loss functions. Moreover, we provided additional numerical simulations to verify our results. [^1]: \* This work was supported in part by Swedish Research Council Distinguished Professor Grant 2017-01078, Knut and Alice Wallenberg Foundation, Wallenberg Scholar Grant, the Swedish Strategic Research Foundation CLAS Grant RIT17-0046, AFOSR under award \#FA9550-19-1-0169, and NSF under award CNS-1932011. [^2]: Zifan Wang and Karl H. Johansson are with Division of Decision and Control Systems, School of Electrical Enginnering and Computer Science, KTH Royal Institute of Technology, and also with Digital Futures, SE-10044 Stockholm, Sweden. Email: {zifanw,kallej}\@kth.se. [^3]: Yi Shen and Michael M. Zavlanos are with the Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA. Email: {yi.shen478, michael.zavlanos}\@duke.edu

arxiv_math

--- abstract: | We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if $\mathbb{A}$ is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include $-\mathrm{id}$ (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement. author: - Alexandru Chirvasitu title: Extensible endomorphisms of compact groups --- # Introduction {#introduction .unnumbered} The automorphisms of a group ${\mathbb G}$ that extend to automorphisms of every larger group ${\mathbb H}\ge {\mathbb G}$ are (as one might have expected), precisely the inner ones: [@schupp_inner Theorem]. The analogous statements hold [@pettet_inner-fin Corollary] for other classes of groups (i.e. with ${\mathbb G}$ and ${\mathbb H}$ both restricted to the relevant class), such as finite, or finite solvable, or finite nilpotent, perhaps further constrained to orders divisible only by pre-specified sets of primes. The relevant notion is as follows: **Definition 1**. *A endomorphism $\varphi\in \mathop{\mathrm{\mathrm{End}}}_{{\mathcal C}}(c)$ of an object $c$ in a category ${\mathcal C}$ *extends (or is extensible) along* a morphism $c\xrightarrow{\theta}c'$ if it fits into a commutative diagram $$\begin{tikzpicture}[auto,baseline=(current bounding box.center)] \path[anchor=base] (0,0) node (l) {$c$} +(2,.5) node (u) {$c$} +(2,-.5) node (d) {$c'$} +(4,0) node (r) {$c'$} ; \draw[->] (l) to[bend left=6] node[pos=.5,auto] {$\scriptstyle \varphi$} (u); \draw[->] (u) to[bend left=6] node[pos=.5,auto] {$\scriptstyle \theta$} (r); \draw[->] (l) to[bend right=6] node[pos=.5,auto,swap] {$\scriptstyle \theta$} (d); \draw[->] (d) to[bend right=6] node[pos=.5,auto,swap] {$\scriptstyle \varphi'$} (r); \end{tikzpicture}$$ We also refer to a map $\varphi'$ as in the preceding diagram as an *extension of $\varphi$ along $\theta$*.* An adjacent theme is taken up in [@bg], which centers around extending endomorphisms *functorially* in the above diagram's $\theta$ (regarded as an object of the *comma category* [@ahs Exercise 3K] $c\downarrow{\mathcal C}$). It is that paper that motivated and provided the scaffolding for [@chi_inner_xv1_toappear], with [@chi_inner_xv1_toappear Section 2] focused on functorial extensions in the category of compact groups. The two setting tend to not *quite* be comparable, though they frequently almost are: the functorial versions of the results (e.g. [@bg Corollary 3]) have stronger hypotheses *and* somewhat stronger conclusions than the non-functorial counterparts (such as the already-cited [@schupp_inner Theorem]). In the present note, retaining the compact-group setting, we revert back to "one-off" (non-functorial) extensibility in the sense of . The connected-compact-group version of [@schupp_inner Theorem] (or rather of a hypothetical analogue thereof concerning *endo*morphisms) reads (): **Theorem 1**. *Let ${\mathbb G}$ be a compact connected group and $\varphi$ an endomorphism thereof. The following conditions are equivalent.* (a) *$\varphi$ is either trivial or an inner automorphism.* (b) *$\varphi$ is extensible along every morphism to a compact group.* (c) *$\varphi$ is extensible along every embedding into a compact group.* (d) *$\varphi$ is extensible along any morphism to a compact Lie group.* (e) *$\varphi$ is extensible along any morphism to a compact Lie group of the form $U(n)\rtimes\Gamma$ for finite $\Gamma$ acting on $U(n)$. $\blacksquare$* The careful listing of the various mutually equivalent conditions is not (always, or only) pedantry: they genuinely make a difference. Were ${\mathbb G}$ above abelian for instance, the result would not hold as such for endomorphisms extensible only along maps to *connected* compact groups: per , there is also the perhaps surprising additional option of $-\mathop{\mathrm{id}}$: **Theorem 2**. *The endomorphisms of a compact connected group extensible along all morphisms to (or embeddings into) compact connected (Lie) groups are precisely the trivial one and $\pm\mathop{\mathrm{id}}$. $\blacksquare$* Note incidentally that connectedness on *both* sides of the morphisms is essential in a result such as the one just stated: the discussions in [Examples [Examples 7](#exs:adisc){reference-type="ref" reference="exs:adisc"}](#exs:adisc) and each drop one type of connectedness. As for morphisms whose domain and codomain are *both* abelian, the surprises (if any) are not major, some care is needed when handling the profinite case. Compressing for brevity: **Theorem 3**. (1) *The endomorphisms extensible in the category of compact abelian groups are precisely the scaling operators by integers.* (2) *For non-profinite compact abelian groups these are also the morphisms extensible along maps into finite-dimensional tori.* (3) *While for profinite groups the latter are the scaling operators by profinite integers, i.e. elements of the profinite completion $\widehat{{\mathbb Z}}$ of $({\mathbb Z},+)$. $\blacksquare$* ## Acknowledgements {#acknowledgements .unnumbered} I am grateful for helpful comments from R. Kanda, I. Penkov, D. Sage, A. Sikora and S. P. Smith. This work is partially supported by NSF grant DMS-2001128. # Extending compact-group endomorphisms {#se:cpct} For (connected) Lie groups, simplicity is understood as that of the Lie algebra [@hm4 following Proposition 6.5]: the non-trivial normal subgroups are discrete but possibly non-trivial. We refer to the stronger property that the group have no proper non-trivial normal subgroups as *algebraic* simplicity. For compact groups, this is also equivalent [@hm4 Theorem 9.90] to asking that *closed* proper normal subgroups be trivial. One of the main results, announced above in the Introduction, is **Theorem 2**. *For a compact connected group ${\mathbb G}$ the following conditions on $\varphi\in \mathop{\mathrm{\mathrm{End}}}({\mathbb G})$ are equivalent.* (a) *[\[item:th:conninner-a\]]{#item:th:conninner-a label="item:th:conninner-a"} $\varphi$ is either trivial or an inner automorphism.* (b) *[\[item:th:conninner-b\]]{#item:th:conninner-b label="item:th:conninner-b"} $\varphi$ is extensible along every morphism to a compact group.* (c) *[\[item:th:conninner-c\]]{#item:th:conninner-c label="item:th:conninner-c"} $\varphi$ is extensible along every embedding into a compact group.* (d) *[\[item:th:conninner-d\]]{#item:th:conninner-d label="item:th:conninner-d"} $\varphi$ is extensible along any morphism to a compact Lie group.* (e) *[\[item:th:conninner-e\]]{#item:th:conninner-e label="item:th:conninner-e"} $\varphi$ is extensible along any morphism to a compact Lie group of the form $U(n)\rtimes\Gamma$ for finite $\Gamma$ acting on $U(n)$.* Some small diversions are needed. The following result extends part of [@chi_inner_xv1_toappear Proposition 2.3] (which in turn proves slightly more than it states). Its aim is to rid us of the mild annoyance of endomorphisms that fail to be bijective. **Proposition 3**. *For an endomorphism $\varphi\in\mathop{\mathrm{\mathrm{End}}}({\mathbb G})$ of a compact group, consider the following conditions:* (a) *[\[item:embinj\]]{#item:embinj label="item:embinj"} $\varphi$ is extensible along every embedding into a compact connected group.* (b) *[\[item:morbij\]]{#item:morbij label="item:morbij"} $\varphi$ is extensible along any morphism to a compact connected (algebraically simple) Lie group.* (c) *[\[item:isbij\]]{#item:isbij label="item:isbij"} $\varphi$ is either trivial or an automorphism.* *We then have $$\text{ \Cref{item:embinj} $\xRightarrow{\quad}$ \Cref{item:morbij} $\xRightarrow{\quad}$ \Cref{item:isbij} }.$$ Similarly, dropping connectedness in both and , the former implies the other.* *Proof.* Regarding and , we focus on the connected version of the result; the other admits a virtually identical proof. **$\xRightarrow{\quad}$ :** The claim is that follows from the *weakest* version of . To see this, consider a morphism ${\mathbb G}\xrightarrow{\theta}{\mathbb H}$ to a compact connected Lie group. On the one hand ${\mathbb G}$ embeds into a product of compact connected Lie groups [@hm4 Corollary 2.36], while on the other, every compact Lie group embeds [@chi_inner_xv1_toappear Lemma 2.4] into a compact, connected, algebraically simple Lie group. This means that there is an embedding ${\mathbb G}\lhook\joinrel\xrightarrow{\iota}{\mathbb K}$ into a product of compact, connected, algebraically simple Lie groups, all of dimension larger than $\dim {\mathbb H}$. The hypothesis ensures the extensibility of $\varphi$ along the *embedding* $$\label{eq:thetapi} {\mathbb G}\xrightarrow{\quad (\theta,\iota)\quad}{\mathbb H}\times {\mathbb K}.$$ Because the individual simple factors of ${\mathbb K}$ have dimension larger than $\dim {\mathbb H}$, they only map to the latter trivially. This means that every endomorphism of ${\mathbb H}\times {\mathbb K}$ leaves the factor ${\mathbb K}$ invariant, so that *every* endomorphism of ${\mathbb H}\times {\mathbb K}$ extends along the first projection ${\mathbb H}\times {\mathbb K}\xrightarrow{\pi_1} {\mathbb H}$. Composing the two extensions of $\varphi$ (first along and then along $\pi_1$), we obtained the desired extensibility along $\theta$. **$\xRightarrow{\quad}$ :** We prove the strictest version of the claim. Assuming $\varphi$ is not trivial, its injectivity follows from [@chi_inner_xv1_toappear Proposition 2.3, part (1) of the proof]. The argument for surjectivity, on the other hand, is a slight modification of [@chi_inner_xv1_toappear Proposition 2.3, part (2) of the proof]. Assume that $\varphi$ is *not* onto (and also non-trivial, i.e. ${\mathbb G}\ne \{1\}$). We can then find a morphism ${\mathbb G}\xrightarrow{\iota}{\mathbb H}$ into a compact connected algebraically simple Lie group with $$\{1\} \lneq \iota(\varphi({\mathbb G})) \lneq \iota({\mathbb G})\le {\mathbb H}.$$ In the commutative diagram $$\begin{tikzpicture}[auto,baseline=(current bounding box.center)] \path[anchor=base] (0,0) node (l) {${\mathbb G}$} +(2,.5) node (u) {${\mathbb G}$} +(2,-.5) node (d) {${\mathbb H}$} +(4,0) node (r) {${\mathbb H}$} ; \draw[->] (l) to[bend left=6] node[pos=.5,auto] {$\scriptstyle \varphi$} (u); \draw[->] (u) to[bend left=6] node[pos=.5,auto] {$\scriptstyle \iota$} (r); \draw[->] (l) to[bend right=6] node[pos=.5,auto,swap] {$\scriptstyle \iota$} (d); \draw[->] (d) to[bend right=6] node[pos=.5,auto,swap] {$\scriptstyle \varphi'$} (r); \end{tikzpicture}$$ the top path is non-trivial and hence so is $\varphi'$. But then the latter must be an *auto*morphism of ${\mathbb H}$ (injective by algebraic simplicity and onto for dimension reasons), so it cannot shrink $\iota({\mathbb G})$ properly into $$\iota(\varphi({\mathbb G})) = \varphi'(\iota({\mathbb G})).$$ This gives the desired contradiction, finishing the proof.  ◻ **Remark 4**. *[@chi_inner_xv1_toappear Proposition 2.3, part (2) of the proof] at one point relies on the *functorial* extensibility assumed there, hence the need for the slight alteration in the argument above.* As part of a series of remarks that will eventually plug back into the proof of , note that requiring extensibility along morphisms into possibly *dis*connected compact groups is crucial. provides a first inkling of this. Recall [@hm4 Definitions 9.30] that a *pro-torus* is a compact, connected, abelian group. **Lemma 5**. *Given a pro-torus ${\mathbb A}\le {\mathbb G}$ of a compact, connected group, there is an automorphism of ${\mathbb G}$ that restricts to $-\mathop{\mathrm{id}}$ on ${\mathbb A}$.* *Proof.* Per standard structure theory [@hm4 Theorem 9.24], ${\mathbb G}$ is a quotient of a compact product $${\mathbb T}\times \prod_{i\in I}{\mathbb S}_i ,\quad {\mathbb T}\text{ a pro-torus} ,\quad {\mathbb S}_i\text{ simple, connected, simply-connected Lie}$$ by a 0-dimensional central subgroup. The image of ${\mathbb A}$ in each of the factors ${\mathbb S}_i$ is a torus (being compact, connected, abelian and Lie). Because maximal tori in ${\mathbb S}_i$ (being their own centralizers [@hm4 Theorem 9.32 or Corollary 6.33]) respectively contain the centers of ${\mathbb S}_i$, it is enough to argue that for a compact, connected, simple Lie group ${\mathbb S}$ there is an automorphism restricting to $-\mathop{\mathrm{id}}$ on any given maximal torus ${\mathbb T}\subset {\mathbb S}$. We may as well assume without loss of generality that ${\mathbb S}$ is simply-connected, whereupon the claim is a consequence of its complex-Lie-algebra analogue [@hmph_liealg Proposition 14.3]: a complex, simple Lie algebra has an automorphism acting as $-\mathop{\mathrm{id}}$ on any given Cartan subalgebra. ◻ **Remarks 6**. (1) *As already hinted at, it is natural to wonder to what extent connectedness is crucial to . Since the only automorphisms of a finite group extensible along embeddings into finite groups are inner [@pettet_inner-fin Corollary], the two connectedness constraints can certainly not *both* be dropped. In fact, we will see that in fact neither condition can be dropped in isolation (while retaining the full generality of otherwise): [Examples [Examples 7](#exs:adisc){reference-type="ref" reference="exs:adisc"}](#exs:adisc) address the (dis)connected of ${\mathbb A}$, while is concerned with that of ${\mathbb G}$.* (2) *Once the setup of , the requisite automorphism of ${\mathbb G}$ can even be chosen inner, so long as $-1$ belongs to the Weyl group. This happens [@hmph_liealg §13.4, Exercise 5] in types $A_1$, $B_{\ell}$, $C_{\ell}$, $D_{2k}$, $E_7$, $E_8$, $F_4$ and $G_2$.* The subgroup ${\mathbb A}\le {\mathbb G}$ of cannot in general be assumed disconnected, even if ${\mathbb G}$ is connected. **Examples 7**. *There compact, simply-connected Lie groups containing (necessarily disconnected) maximal abelian subgroups whose inversion automorphism $x\xmapsto{}-x$ does not lift to the ambient group.* (1) ***: Type $F_4$.** If ${\mathbb G}$ is the simple compact Lie group of type $F_4$, there is a maximal abelian subgroup [@yu_maxab equation (75)] $$({\mathbb Z}/3)^3\cong {\mathbb A}\subset {\mathbb G}$$ whose Weyl group (i.e. quotient of the normalizer) consists precisely of the *unimodular* $3\times 3$ matrices over the field ${\mathbb F}_3$ [@yu_maxab Proposition 3.14] (see also [@ek_liegradings Theorem 5.22]). In particular, the inversion automorphism of ${\mathbb A}$ does not extend to an inner automorphism of ${\mathbb G}$; but *all* automorphisms are inner [@afg_max p.953].* (2) *[\[item:exs:adisc-e6\]]{#item:exs:adisc-e6 label="item:exs:adisc-e6"} **: Type $E_6$.** Let ${\mathbb G}$ be compact, simply-connected of type $E_6$. Its *adjoint* quotient ${\mathbb G}_{ad}:={\mathbb G}/Z({\mathbb G})$ (by the order-3 center: see the list immediately preceding [@hmph_liealg §13.2]) has an abelian subgroup $$({\mathbb Z}/3)^3\cong \overline{{\mathbb A}}\subset {\mathbb G}_{ad}$$ which, when lifted to ${\mathbb G}$, generates a maximal abelian (finite) subgroup ${\mathbb A}$ together with the center $Z({\mathbb G})$: $${\mathbb A}:=\text{preimage of $\overline{{\mathbb A}}$ through }{\mathbb G}% \xrightarrow[]{\quad}\mathrel{\mkern-14mu}\rightarrow {\mathbb G}/Z({\mathbb G}).$$ The group $\overline{{\mathbb A}}$ is the one recorded in [@yu_maxab $F_2$, Table 4, second row], and its Weyl group is again $SL_3({\mathbb F}_3)$ [@yu_maxab Proposition 4.4]. It thus follows that no inner automorphism of ${\mathbb G}_{ad}$ can operate on $\overline{{\mathbb A}}$ as $-\mathop{\mathrm{id}}$. Since three generators of $\overline{{\mathbb A}}$ lift to a *rank-0 commuting triple* [@bfm_almostcomm §1.9] in ${\mathbb G}$ and the latter's automorphisms operate on such triples as conjugations [@bfm_almostcomm Proposition 5.3.1], it follows that no automorphism of ${\mathbb G}$ can invert ${\mathbb A}$.* (3) ***: Type $E_8$.** The only compact Lie group of type $E_8$ is simply-connected (and center-less, e.g. by the selfsame [@hmph_liealg discussion preceding §13.2]), so no issues of lifting arise.* *The group listed in [@yu_maxab $F_1$, Table 7, row 1] is maximal abelian and isomorphic to $({\mathbb Z}/5)^3$, with Weyl group $SL_3({\mathbb F}_5)$ [@yu_maxab Proposition 7.2] (hence excluding $-\mathop{\mathrm{id}}$). Because the automorphisms of $E_8$ are all inner (e.g. by [@hmph_liealg §16.5]; the type-$E_8$ Dynkin diagram has no non-trivial automorphisms), the conclusion follows.* **Remark 8**. *[@ek_liegradings Propositions 1.28 and 1.32] explain how (possibly disconnected) maximal abelian subgroups of adjoint simple compact Lie groups relate to the *fine gradings* [@ek_liegradings §1.3] of the corresponding simple Lie algebras (the gradings are by the discrete abelian *Pontryagin duals* [@de §3.5] of the respective maximal groups).* *The literature on fine gradings is vast (as the above monograph's reference list will confirm). So far as type $E_6$ goes, the full list of such gradings appears, for instance, in [@ek_liegradings Figure 6.2, p.268, first column] or [@ev_finegre6 Theorem 1, table]. Note, though, that those lists do *not* contain the group denoted by $\overline{{\mathbb A}}$ in : the reason is that the class of subgroups of a compact simple Lie group classified in [@yu_maxab] (those satisfying [@yu_maxab Introduction, condition (\*)]) includes [@yu_maxab §2.5] that of maximal subgroups *strictly*: $\overline{{\mathbb A}}$ is not maximal in the adjoint compact form of $E_6$, but its lift to the simply-connected form is maximal therein.* We next need what must be a straightforward observation, added here for completeness and because I have been unable to find it in the literature (not in the desired form, ready-citable). Following standard practice (e.g. [@rot_halg Definition preceding Theorem 9.70]), we refer to a group morphism $$\label{eq:projrep} \Gamma \xrightarrow{\quad\rho\quad} {\mathbb G}/Z({\mathbb G})$$ as a *projective representation* of $\Gamma$ in ${\mathbb G}$ ($\Gamma$ will typically be discrete). $\rho$ gives rise ([@rot_halg proof of Theorem 9.70] or [@ser_locf §VII.3]) to a *2-cocycle* $$\label{eq:correspcocyc} z=z_{\rho}\in Z^2(\Gamma,\ Z({\mathbb G})) ,\quad z(\gamma,\gamma'):=\widetilde{\rho}(\gamma)\widetilde{\rho}(\gamma')\widetilde{\rho}(\gamma\gamma')^{-1}$$ for a lift $\Gamma\xrightarrow{\widetilde{\rho}}{\mathbb G}$ of $\rho$ (the $\rho\xleftrightarrow{\quad}\widetilde{\rho}$ notation will recur). In parsing the following statement, note that a projective representation induces an action of $\Gamma$ on ${\mathbb G}$ by conjugation: $${\mathbb G}\ni x\xmapsto{\quad \gamma\quad} \tensor[^{\gamma}]{x}{} := \widetilde{\rho}(\gamma)x\widetilde{\gamma}(\gamma)^{-1}.$$ It thus makes sense to speak of the resulting semidirect product ${\mathbb G}\rtimes \Gamma$. **Lemma 9**. *Let be a projective representation with corresponding cocycle .* *An automorphism $\alpha$ of ${\mathbb G}$ lifts to ${\mathbb G}\rtimes \Gamma$ if and only if the cohomology class of $\alpha(z)$ is in the orbit of $\mathop{\mathrm{\mathrm{Aut}}}(\Gamma)$ acting on $H^2(\Gamma,\ Z({\mathbb G}))$.* *Proof.* An extension of $\alpha$ to ${\mathbb G}\rtimes \Gamma$, restricted to $\Gamma$, will be of the form $$\Gamma\ni \gamma \xmapsto{\quad} a_{\gamma}\theta(\gamma) \in {\mathbb G}\rtimes \Gamma$$ for - an automorphism $\theta\in \mathop{\mathrm{\mathrm{Aut}}}(\Gamma)$; - and a (non-abelian) 1-cocycle $$(a_{\gamma})_{\gamma}\in Z^1(\Gamma,\ {\mathbb G}),$$ where the action of $\Gamma$ on ${\mathbb G}$ pertinent to the cocycle space is the original one, twisted by $\theta$: $${\mathbb G}\ni x \xmapsto{\quad\gamma\quad} \tensor[^{\theta\gamma}]{x}{} = \widetilde{\rho}(\theta\gamma)x\widetilde{\rho}(\theta\gamma)^{-1}.$$ This is a version of the correspondence [@ser_galcoh §I.5.1, Exercise 1] between 1-cocycles and lifts of the surjection ${\mathbb G}\rtimes\Gamma\to \Gamma$; the requisite cocycle condition is $$\label{eq:aiscocyc} a_{\gamma\gamma'} = a_{\gamma}\cdot \tensor[^{\theta\gamma}]{a}{_{\gamma'}} ,\quad \forall \gamma,\ \gamma'\in \Gamma.$$ The compatibility condition between $\alpha$ and the desired extension (after some processing) reads $$a_{\gamma}c_{\gamma} = \alpha\left(\widetilde{\rho}\gamma\right)\cdot \widetilde{\rho}\left(\theta \gamma\right)^{-1} ,\quad \forall \gamma\in \Gamma ,\quad \text{where }c_{\gamma}\in Z({\mathbb G}).$$ The cocycle constraint now simply means that the 2-cocycle $$\left(\Gamma^2\ni (\gamma,\gamma') \xmapsto{\quad} \alpha(z(\gamma,\gamma'))\cdot z(\theta\gamma,\theta\gamma')^{-1}\right) \in Z^2(\Gamma,\ Z({\mathbb G}))$$ is the coboundary of the 1-cochain $(c_{\gamma})_{\gamma}$. In short, finding such $(c_{\gamma})_{\gamma}$ and $\theta$ is equivalent to the class in $H^2(\Gamma,\ Z({\mathbb G}))$ of $\alpha(z)$ coinciding with that of $z$ twisted by $\theta\in \mathop{\mathrm{\mathrm{Aut}}}(\Gamma)$. ◻ **Corollary 10**. *Let $\Gamma\xrightarrow{\rho}PU(n)$ be a projective representation, $z\in Z^2(\Gamma,\ Z(U(n)))$ the corresponding 2-cocycle , and $\overline{z}\in H^2(\Gamma,\ Z(U(n)))$ its cohomology class.* *The following conditions are equivalent:* (a) *[\[item:allautos\]]{#item:allautos label="item:allautos"} Every automorphism of $U(n)$ lifts to an automorphism of $U(n)\rtimes\Gamma$.* (b) *[\[item:allouterautos\]]{#item:allouterautos label="item:allouterautos"} All outer automorphisms of $U(n)$ lift to automorphisms of $U(n)\rtimes\Gamma$.* (c) *[\[item:1outerauto\]]{#item:1outerauto label="item:1outerauto"} Any one specific outer automorphism of $U(n)$ (e.g. complex conjugation) lifts to $U(n)\rtimes\Gamma$.* (d) *[\[item:z-1\]]{#item:z-1 label="item:z-1"} $\overline{z}^{-1}$ is in the orbit of $\overline{z}$ under $\mathop{\mathrm{\mathrm{Aut}}}(\Gamma)$.* *Proof.* The equivalence $\xLeftrightarrow{\quad}$ of course holds in complete generality, since *inner* automorphisms of $U(n)$ certainly extend to any overgroup. That this is all further equivalent to follows from the fact that $$\mathop{\mathrm{\mathrm{Aut}}}(U(n))\cong PU(n)\rtimes {\mathbb Z}/2 ,\quad {\mathbb Z}/2\text{ acting by complex conjugation}$$ (analogous to the classification of automorphisms for the type-$A$ simple Lie algebras, e.g. [@jc §IX.5, Theorem 5]). Finally, $\xLeftrightarrow{\quad}$ follows from and the fact that complex conjugation operates on $Z(U(n))$ as inversion. ◻ Returning, now, to the failure of for disconnected ${\mathbb G}$: **Example 11**. *We consider the setting of , with finite $\Gamma$. Every cohomology class in $$H^2(\Gamma,\ {\mathbb C}^{\times})\cong H^2(\Gamma,\ {\mathbb S}^1)\cong H^2(\Gamma,\ {\mathbb Q}/{\mathbb Z})$$ is realizable via a projective representation into some $PU(n)$ (consider a central extension of $\Gamma$ by some finite subgroup of ${\mathbb Q}/{\mathbb Z}$ corresponding [@gruen_coh §5.3, Theorem 1] to said cohomology class, represent that extension in some $U(n)$, etc.). Because inner automorphisms act trivially on cohomology [@ser_locf §VII.5, Proposition 3], it will be enough to produce a group $\Gamma$ with* - *only inner automorphisms;* - *and a cohomology class $\overline{z}\in H^2(\Gamma,\ {\mathbb C}^{\times})$ of order $>2$, so that its inverse is not in the (singleton!) orbit of $\overline{z}$ under $\mathop{\mathrm{\mathrm{Aut}}}(\Gamma)$.* *This is not difficult to do, with $\Gamma$ *metabelian*, for instance (i.e. an extension of an abelian group by another). The metabelian groups of order $>2$ with only inner automorphisms are [@gr_metabinner Theorem 1] precisely those of the form $$\prod_{i}\left({\mathbb Z}/p_i^{n_i},+\right)\rtimes \left(({\mathbb Z}/p_i^{n_i})^{\times},\cdot\right),$$ where the $p_i^{n_i}$ are finitely many distinct odd prime powers and $$\left(({\mathbb Z}/p_i^{n_i})^{\times},\cdot\right)\cong \mathop{\mathrm{\mathrm{Aut}}}\left({\mathbb Z}/p_i^{n_i}, +\right).$$ Consider, say, a product of two such factors: $$\Gamma = ({\mathbb Z}/9,+)\rtimes ({\mathbb Z}/6,+) \times ({\mathbb Z}/27,+)\rtimes ({\mathbb Z}/18,+).$$ The quotient ${\mathbb Z}/6\times {\mathbb Z}/18$ of $\Gamma$ is split, so $$\label{eq:lhs} H^2({\mathbb Z}/6\times {\mathbb Z}/18,\ {\mathbb C}^{\times})\cong {\mathbb Z}/6$$ is a summand in $H^2(\Gamma,\ {\mathbb C}^{\times})$ ( is a simple computation via Lyndon-Hochschild-Serre [@rot_halg Theorem 10.52] say, given the well-known cohomology of finite cyclic groups [@rot_halg Theorem 9.27]). In particular, the latter group has non-involutive elements.* It is perhaps worth noting explicitly what types of uses one can put to. **Lemma 12**. *For every positive integer $d$, there are compact Lie groups of the form $U(n)\rtimes\Gamma$ with $n$ divisible by $d$ to which no outer automorphism of $U(n)$ lifts.* *Proof.* Once we have a projective representation affording an appropriate cocycle as in , one can simply replace it with a sum of $d$ copies thereof. ◻ **Proof of 1**. *The downstream implications $$\text{ \Cref{item:th:conninner-a} $\xRightarrow{\ }$ \Cref{item:th:conninner-b} $\xRightarrow{\ }$ \Cref{item:th:conninner-c} $\xRightarrow{\ }$ \Cref{item:th:conninner-d} $\xRightarrow{\ }$ \Cref{item:th:conninner-e} }$$ are self-evident, so addressing $\xRightarrow{\ }$ will suffice. We assume $\varphi$ non-trivial so that ${\mathbb G}$ is non-trivial and () $\varphi$ is an automorphism. Consider unitary representations $$\begin{aligned} {\mathbb G}&\xrightarrow{\quad \rho'\quad}U(d')\text{ non-trivial, irreducible, arbitrary and}\\ {\mathbb G}&\xrightarrow{\quad \rho\quad}U(d)\text{ defined by} \begin{cases} \rho=\rho'&\text{if }d'>1\\ \rho=\left(\rho'\right)^{\oplus d}&\text{otherwise}, \end{cases} \end{aligned}$$ where in the latter case $d$ is chosen so that the image $\varphi(\rho')$ of $\rho'$ under the action of $\varphi\in\mathop{\mathrm{\mathrm{Aut}}}({\mathbb G})$ on irreducible representations is *not* a power of $(\rho')^{\otimes d}$ (not a typo: that is a tensor power of a 1-dimensional representation, hence another such).* *Next, consider the morphism* *$$\begin{tikzpicture}[auto,baseline=(current bounding box.center)] \path[anchor=base] (0,0) node (l) {${\mathbb G}$} +(3,.5) node (u) {$U(n)$} +(5,0) node (r) {$U(n)\rtimes\Gamma$} ; \draw[->] (l) to[bend left=6] node[pos=.5,auto] {$\scriptstyle \rho^{\oplus (n/d)}$} (u); \draw[right hook->] (u) to[bend left=6] node[pos=.5,auto] {$\scriptstyle $} (r); \draw[->] (l) to[bend right=6] node[pos=.5,auto,swap] {$\scriptstyle \theta$} (r); \end{tikzpicture}$$ for $d|n$ and $\Gamma$ as in . The choices made above ensure that an extension of $\varphi$ along $\theta$, when restricted to $U(n)$, would have to be an automorphism. That automorphism cannot be outer because of the choice of semidirect product via . This means that (the isomorphism class of) $\rho^{\oplus(n/d)}$ and hence also $\rho'$ is fixed by $\varphi$; since $\rho'$ was an arbitrary irreducible representation of the *connected* compact group ${\mathbb G}$, $\varphi$ is inner by [@mcm-dual Corollary 2].* The discussion preceding alludes to possible "misbehavior" in when imposing extensibility only along morphisms into connected groups. itself suggests that these phenomena might obtain when the base group is abelian, and the following result spells this out in full. **Theorem 13**. *The following conditions on an endomorphism $\varphi\in\mathop{\mathrm{\mathrm{End}}}({\mathbb A})$ of a pro-torus are equivalent:* (a) *[\[item:ab-trivpm1\]]{#item:ab-trivpm1 label="item:ab-trivpm1"} $\varphi$ is trivial or $\pm\mathop{\mathrm{id}}$.* (b) *[\[item:ab-extmor\]]{#item:ab-extmor label="item:ab-extmor"} $\varphi$ extends along any morphism ${\mathbb A}\xrightarrow{}{\mathbb H}$ to a compact connected group.* (c) *[\[item:ab-extemb\]]{#item:ab-extemb label="item:ab-extemb"} $\varphi$ extends along any embedding ${\mathbb A}\lhook\joinrel\xrightarrow{}{\mathbb H}$ into a compact connected group.* (d) *[\[item:ab-extmor-lie\]]{#item:ab-extmor-lie label="item:ab-extmor-lie"} $\varphi$ extends along any morphism ${\mathbb A}\xrightarrow{}{\mathbb H}$ to a compact connected (algebraically simple) Lie group.* *Proof.* **$\xRightarrow{\quad}$ :** Only the case $\varphi=-\mathop{\mathrm{id}}$ is interesting, and it is delivered by . **$\xRightarrow{\quad}$** formally. **$\xRightarrow{\quad}$ :** This is part of . **$\xRightarrow{\quad}$ :** Indeed, $\varphi$ is either trivial or an automorphism by , as well as scaling by an integer by . The only options are to scale by $0$ or $\pm 1$. $\blacksquare$  ◻ # Extensibility in the category of compact abelian groups {#se:cpctab} The ensuing discussion assumes some background on *profinite* groups [@rz_prof §2.1], i.e. *cofiltered (or inverse, or projective) limits* ([@rz_prof §1.1] or [@stacks-project [Tag 04AY](https://stacks.math.columbia.edu/tag/04AY)]) of finite quotient groups. These are also precisely [@hm4 Theorem 1.34] the totally disconnected compact groups, and, when abelian, those whose Pontryagin duals are torsion [@hm4 Corollary 8.5]. Torsion abelian groups being modules in an obvious fashion over the *profinite completion* [@rz_prof Example 2.1.6(2)] $$\widehat{{\mathbb Z}}:=\varprojlim_n {\mathbb Z}/n,\quad n\in {\mathbb Z}_{>0}\text{ ordered by divisibility},$$ their Pontryagin duals (i.e. the profinite abelian groups) are similarly modules over $\widehat{{\mathbb Z}}$ [@rz_prof discussion preceding §5.2]. In other words: the usual integer-scaling endomorphisms $$\label{eq:scale} {\mathbb A}\ni a\xmapsto{\quad}na\in {\mathbb A},\quad n\in {\mathbb Z},$$ of abelian groups ${\mathbb A}$, extend to actions of the profinite ring $\widehat{{\mathbb Z}}\supset {\mathbb Z}$ when ${\mathbb A}$ is profinite. What is more, we have an isomorphism [@rz_prof Example 2.3.11] $$\widehat{{\mathbb Z}}\xrightarrow[\cong]{\qquad}\prod_{\text{primes }p}{\mathbb Z}_p$$ with $${\mathbb Z}_p:=\varprojlim_n {\mathbb Z}/p^n=\text{the ring of {\it $p$-adic integers} \cite[Example 2.1.6(2)]{rz_prof}}$$ (e.g. as a particular instance of the decomposition [@rz_prof Proposition 2.3.8] of any profinite pro-nilpotent group as the product of its *$p$-Sylow subgroups*). $\widehat{{\mathbb Z}}$ acts on *$p$-primary* torsion groups (i.e. [@kap_infab preceding Theorem 1] those whose elements have $p$-power orders) via its ${\mathbb Z}_p$ quotient. **Theorem 14**. *Let $({\mathbb A},+)$ be a compact abelian group and $\varphi\in\mathop{\mathrm{\mathrm{End}}}({\mathbb A})$.* (1) *[\[item:abendo-all\]]{#item:abendo-all label="item:abendo-all"} $\varphi$ is extensible along morphisms into arbitrary compact abelian groups if and only if it is an integer-scaling map .* (2) *[\[item:abendo-tor\]]{#item:abendo-tor label="item:abendo-tor"} If the identity component ${\mathbb A}_0$ is non-trivial (i.e. ${\mathbb A}$ is not profinite), $\varphi$ is extensible along morphisms into (finite-dimensional) tori if and only if it is an integer-scaling map .* (3) *[\[item:abendo-prof\]]{#item:abendo-prof label="item:abendo-prof"} If ${\mathbb A}$ is profinite, $\varphi$ extends along morphisms to (finite-dimensional) tori if and only it is scaling by some $n\in\widehat{{\mathbb Z}}$.* (4) *[\[item:abendo-protor\]]{#item:abendo-protor label="item:abendo-protor"} The conclusion of also holds for profinite ${\mathbb A}$ if it either* - *has a non-trivial torsion-free quotient;* - *or infinitely many Sylow subgroups.* It will be convenient to spell out some auxiliary partial results; first a simple observation: **Lemma 15**. *Consider the following conditions on an endomorphism $\varphi\in\mathop{\mathrm{\mathrm{End}}}({\mathbb A})$ of a compact abelian group.* (a) *[\[item:extor\]]{#item:extor label="item:extor"} $\varphi$ extends along any morphism into a finite-dimensional torus.* (b) *[\[item:excirc\]]{#item:excirc label="item:excirc"} $\varphi$ extends along any morphism into ${\mathbb S}^1$.* (c) *[\[item:allin\]]{#item:allin label="item:allin"} $\varphi$ leaves invariant every closed subgroup of ${\mathbb A}$, and hence induces an endomorphism on every quotient of ${\mathbb A}$.* (d) *[\[item:fininv\]]{#item:fininv label="item:fininv"} $\varphi$ leaves invariant every finite-codimensional closed subgroup of ${\mathbb A}$, and hence induces an endomorphism on every Lie quotient of ${\mathbb A}$.* *We have $$\text{ \Cref{item:extor} $\xRightarrow{\quad}$ \Cref{item:excirc} $\xLeftrightarrow{\quad}$ \Cref{item:allin} $\xLeftrightarrow{\quad}$ \Cref{item:fininv} }.$$* *Proof.* That implies is obvious. As for the rest, it will become transparent upon (Pontryagin-)dualizing. Extensibility of $\varphi$ along ${\mathbb A}\xrightarrow{\theta}{\mathbb B}$, for instance, can also be phrased as extensibility of the Pontryagin dual endomorphism $\widehat{\varphi}\in\mathop{\mathrm{\mathrm{End}}}(\widehat{{\mathbb A}})$ along the morphism $$\widehat{{\mathbb B}}\xrightarrow{\quad\widehat{\theta}\quad}\widehat{{\mathbb A}}$$ of *discrete* abelian groups in the obvious sense, dual to that of . With that in place, says that $\widehat{\varphi}$ extends along every morphism ${\mathbb Z}\to\widehat{{\mathbb A}}$. Since $\mathop{\mathrm{\mathrm{End}}}({\mathbb Z})={\mathbb Z}$, this means exactly that every element of $\widehat{{\mathbb A}}$ is mapped into an integer multiple of itself, or that all (cyclic, or finitely-generated, or unrestricted) subgroups of $\widehat{{\mathbb A}}$ are $\widehat{\varphi}$-invariant. ◻ **Lemma 16**. *holds for pro-tori (i.e. when ${\mathbb A}={\mathbb A}_0$).* *Proof.* One direction is obvious (scaling maps, of course, extend along morphisms of abelian groups), so we will be focusing on the converse claims. The claim is immediate for a *torus* ${\mathbb A}={\mathbb T}^d=({\mathbb S}^1)^d$: those endomorphisms of its Pontryagin dual [@de Examples 3.5.1] $\widehat{{\mathbb T}^d}\cong {\mathbb Z}^d$ are elements of $$\mathop{\mathrm{\mathrm{End}}}({\mathbb Z}^d)\cong M_d({\mathbb Z}):=\text{$d\times d$ matrices with integer entries}$$ preserving *every* subgroup of ${\mathbb Z}^d$. That such a morphism is scalar (i.e. multiplication by some $n$) then follows immediately. Now write $${\mathbb A}\cong \varprojlim_{i\in I}{\mathbb T}_i\quad\left(\text{cofiltered limit of tori by \cite[Corollary 2.36]{hm4}}\right),$$ apply the preceding observation to each individual torus ${\mathbb T}_i$, and note that $n\in {\mathbb Z}$ is uniquely determined by its scaling action on any torus. This means that there must be *one* $n$ by which $\varphi$ operates on each of the quotients ${\mathbb A}% \xrightarrow[]{}\mathrel{\mkern-14mu}\rightarrow {\mathbb T}_i$. ◻ **Remark 17**. *Which morphisms one seeks to extend along (e.g. embedding or arbitrary, or morphisms into specific classes of groups) can make a big difference to the classification, as should already be clear from comparing and , say. Other examples of such drastic differences (to the preceding results):* (a) *Tori ${\mathbb T}^I:=({\mathbb S}^1)^I$ (for possibly infinite index sets $I$) being *injective objects* [@hm4 Theorem 8.78] in the category of compact abelian groups, their embeddings therein split. It follows immediately that *all* endomorphisms of a torus extend along its embeddings into compact abelian groups.* (b) *By the same token (injectivity of tori) every endomorphism of every compact abelian group extends along every embedding into a torus.* Because centers around distinguishing between scaling by profinite integers and honest integers, it will be helpful to isolate the relevant instances of that problem, focusing on particularly pleasant classes of groups. **Proposition 18**. *Let ${\mathbb A}$ be a profinite abelian group and $n\in\widehat{{\mathbb Z}}$, inducing a scaling endomorphism $\varphi=\varphi_n$ on ${\mathbb A}$.* *$\varphi$ is in fact induced by an integer if and only if this is so for the corestrictions of $\varphi$ to countable quotients $${\mathbb A}% \xrightarrow[]{\quad}\mathrel{\mkern-14mu}\rightarrow \prod_{i\in\aleph_0}{\mathbb A}_i,$$ where ${\mathbb A}_i$ are either finite cyclic or copies of the $p$-adic groups $({\mathbb Z}_p,+)$ [@rz_prof Example 2.1.6(2)] for various primes $p$.* *Proof.* There is no need to address one of the implications, so assume $\varphi$ is *not* an integer scaling; or, for convenience, we say that $\varphi$ scales by a *proper* profinite integer. Switching perspective to the Pontryagin dual $\widehat{{\mathbb A}}$, the claim is one regarding subgroups thereof expressible as countable direct *sums* (rather than products): of finite cyclic groups and *divisible* [@kap_infab §5] indecomposable summands of the form $${\mathbb Z}/p^{\infty}:=\bigcup_{n\in {\mathbb Z}_{>0}}{\mathbb Z}/p^n\cong \left\{\text{$p$-power-order roots of unity}\right\}\subset {\mathbb S}^1$$ (the $Z(p^{\infty})$ of [@kap_infab §2 (g)]), respectively dual to ${\mathbb Z}_p$. We have [@kap_infab Theorems 3 and 4] $$\label{eq:divdec} \widehat{{\mathbb A}}\cong {\mathbb D}\oplus {\mathbb E},\quad {\mathbb D}=\bigoplus{\mathbb Z}/p^{\infty},\quad {\mathbb E}\text{ {\it reduced}},$$ meaning ([@kap_infab Definition preceding Theorem 4], [@fuchs_abgp following Lemma 4.1.3]) that ${\mathbb E}$ has no divisible summands. The hypothesis is that $$\sup_{\text{finite }{\mathbb F}\le \widehat{{\mathbb A}}} \min\left\{|m|\quad : \quad m\in {\mathbb Z},\ \widehat{\varphi}|_{{\mathbb F}}=\text{scaling by $m$}\right\}=\infty$$ That supremum will also be infinite when ${\mathbb F}$ ranges over a *countable* set of finite subgroups, so there is no loss in assuming $\widehat{{\mathbb A}}$ countable to begin with (as we henceforth will). ${\mathbb E}$ and the set of summands in , then, will be countable. Because primary cyclic groups and the ${\mathbb Z}/p^{\infty}$ are [@kap_infab Theorem 10] the only indecomposable torsion abelian groups, the claim is that if $$\widehat{\varphi} \in \mathop{\mathrm{\mathrm{End}}}(\text{countable abelian group }\widehat{{\mathbb A}})$$ scales by a proper profinite integer, then it already does so on a direct sum of indecomposable subgroups of $\widehat{{\mathbb A}}$. The summands ${\mathbb D}$ and ${\mathbb E}$ also decompose [@kap_infab Theorem 1] as the direct sums of their respective *primary components* ${\mathbb D}_p$ and ${\mathbb E}_p$, i.e. their maximal primary subgroups: $$\begin{aligned} {\mathbb D}&=\bigoplus_{\text{primes p}}{\mathbb D}_p ,\quad {\mathbb D}_p=({\mathbb Z}/p^{\infty})^{\oplus \alpha_p},\quad 0\le \alpha_p\le \aleph_0\\ {\mathbb E}&=\bigoplus_{\text{primes p}}{\mathbb E}_p ,\quad {\mathbb E}_p\text{ reduced, countable, $p$-primary}. \end{aligned}$$ Recall [@kap_infab §9] the notion of *height* $$h(x) = h_{{\mathbb E}_p}(x):=\sup\left\{d\in {\mathbb Z}_{>0}\ |\ \exists y\in {\mathbb E}_p\text{ with }p^d y=x\right\}$$ for an element $x$ of a primary group (such as ${\mathbb E}_p$, in this case). We will now construct a subgroup $$\label{eq:dirsumine} {\mathbb E}_p':=\bigoplus_{i}{\mathbb Z}/p^i\le {\mathbb E}_p$$ branching over two cases: (I) [\[item:bddh\]]{#item:bddh label="item:bddh"} the heights of $x\in {\mathbb E}_p$ with $px=0$ are bounded; (II) [\[item:unbddh\]]{#item:unbddh label="item:unbddh"} or not. Either way, we construct recursively, choosing one summand at a time as in the proof of [@kap_infab Theorem 9]. In case we can choose countably many summands of unbounded orders and stop. In case , we continue the recursion transfinitely, taking unions for limit ordinals. Every step will produce *pure* [@kap_infab §7] subgroups of ${\mathbb E}_p$, which, being by assumption of bounded order, are again summands of ${\mathbb E}_p$ [@kap_infab Theorem 7]. But this means that the recursion can proceed until we have exhausted ${\mathbb E}_p$, so that *it* will be ${\mathbb E}'_p$. The subgroup $$\bigoplus_{\text{primes }p}\left({\mathbb D}_p\oplus {\mathbb E}'_p\right)\le \widehat{{\mathbb A}},$$ by construction a direct sum of indecomposable groups, is now sufficient to distinguish between proper profinite integers and plain integers. ◻ **Proof of 1**. *Once more, the extensibility of *integer*-scaling maps is not at issue. Nor is the corresponding direction in significantly more difficult: morphisms from a profinite group into a Lie group factor through a finite quotient, on which scaling by $n\in\widehat{{\mathbb Z}}$ induces scaling by an integer; that then extends, etc. The rest of the proof is devoted, then, to the interesting implication(s).* *Assume for the moment that $\varphi$ extends along morphisms into finite-dimensional tori, a hypothesis common to all items. then implies that $\varphi$ induces an endomorphism $\varphi_{\pi}$ on every quotient $${\mathbb A}% \xrightarrow[]{\quad\pi\quad}\mathrel{\mkern-14mu}\rightarrow {\mathbb B}:={\mathbb T}^d\times{\mathbb F},\quad {\mathbb F}\text{ finite},$$ while further implies that $\varphi_{\pi}$ scales ${\mathbb T}^d$ by some $n\in {\mathbb Z}$. At this point, the proofs of the various items diverge (with most conveniently saved for last).* 1. ***:** I claim first that if $d>0$, then $\varphi_{\pi}$ is scaling by $n$ (globally, so on ${\mathbb F}$ as well as ${\mathbb T}^d$).* *Indeed, we have just concluded that all closed subgroups of ${\mathbb B}={\mathbb T}^d\times {\mathbb F}$ are $\varphi_{\pi}$-invariant, which already implies that $\varphi_{\pi}$ restricted to ${\mathbb F}\subset {\mathbb B}$ scaling by some $n'\in {\mathbb Z}$. If $$y\in {\mathbb F},\quad x\in {\mathbb T}^d,\quad\mathrm{ord}(y)\ |\ \mathrm{ord}(x)$$ then the finite subgroup of ${\mathbb B}$ generated by $(x,y)$ has no non-trivial intersection with the right-hand factor ${\mathbb F}$. But then $$n(x,y)=(nx,ny)\quad\text{and}\quad(nx,n'y) = \varphi_{\pi}(x,y)$$ cannot both belong to the group generated by $(x,y)$ unless $n'y=ny$.* 2. ***:** We have already established that $\varphi$ induces an endomorphism $\varphi_{\mathbb F}$ on every finite quotient ${\mathbb A}% \xrightarrow[]{\pi_{{\mathbb F}}}\mathrel{\mkern-14mu}\rightarrow {\mathbb F}$, and part then ensures that that endomorphism is multiplication by some $n_{{\mathbb F}}\in {\mathbb Z}$. Now, $(n_{{\mathbb F}})_{{\mathbb F}}\subset {\mathbb Z}\subset \widehat{{\mathbb Z}}$ constitutes a *net* [@will_top Definition 11.1] if the finite quotients are ordered by $$\pi_{{\mathbb F}}\le \pi_{{\mathbb F}'} \iff \text{the latter factors through the former},$$ which, $\widehat{{\mathbb Z}}$ being compact, has a cluster point [@will_top Theorem 17.4] $n\in\widehat{{\mathbb Z}}$; that $n$ will do.* 3. ***:** Since part supplies the conclusion even under the weaker hypothesis of extensibility along morphisms into *tori* unless ${\mathbb A}$ is profinite, only that case is left. Furthermore, we may then assume by that the map scales by some $n\in\widehat{{\mathbb Z}}$. The hypothesis can also be phrased as the extensibility of $\widehat{\varphi}\in\mathop{\mathrm{\mathrm{End}}}(\widehat{{\mathbb A}})$ along every morphism ${\mathbb D}\to \widehat{{\mathbb A}}$ from a discrete torsion-free group ${\mathbb D}$ (proof of ). In that context, reduces the problem to the case when $\widehat{{\mathbb A}}$ is a countable sum of indecomposable torsion abelian groups.* *Dualizing, the two properties listed in translate [@hm4 Corollary 8.5] to the following.* (I) ***: $\widehat{{\mathbb A}}$ is not reduced.** That is, it has at least one summand ${\mathbb Z}/p^{\infty}$. Note that the action of ${\mathbb Z}_p$ (and *a fortiori* also that of ${\mathbb Z}\subset {\mathbb Z}_p$) is faithful on ${\mathbb Z}/p^{\infty}$, so an integer scaling a subgroup of the form $$\label{eq:pinfq} {\mathbb Z}/p^{\infty}\oplus {\mathbb Z}/q^{\ell}\le \widehat{{\mathbb A}} ,\quad q\text{ prime}$$ is uniquely determined. It follows that it is enough to work with only such groups.* *If $q\ne p$ then is the quotient of $${\mathbb D}:= \varinjlim\left( {\mathbb Z} \lhook\joinrel\xrightarrow{\ q^{\ell}p\ } {\mathbb Z} \lhook\joinrel\xrightarrow{\ p\ } {\mathbb Z} \lhook\joinrel\xrightarrow{\ p\ } \cdots \right),$$ by the leftmost copy of ${\mathbb Z}$. The endomorphism group of ${\mathbb D}$ is the *localization* [@am_comm Chapter 3] of ${\mathbb Z}$ with respect to the multiplicative set $\{p^n\ |\ n\in {\mathbb Z}_{\ge 0}\}$; or: the rationals with $p$-power denominators. Since the only such rationals that descend to an endomorphisms of ${\mathbb Z}_p$ are integers, we are done.* *If, on the other hand, $q=p$, then the argument can proceed as in the proof of part above, with the two factors ${\mathbb Z}/p^{\infty}$ and ${\mathbb Z}/p^{\ell}$ in place of ${\mathbb T}^d$ and ${\mathbb F}$ respectively (for we can then find $y\in {\mathbb Z}/p^{\ell}$ and $x\in {\mathbb Z}/p^{\infty}$ with the former's order dividing the latter's, etc.).* *We henceforth assume $\widehat{{\mathbb A}}$ reduced.* (II) ***: $\widehat{{\mathbb A}}$ has infinitely many primary components.** The proof is very similar in spirit to the preceding argument. An $$\label{eq:infprod} \text{infinite product of the form }\prod_{\text{infinitely many distinct }p}{\mathbb Z}/p^{\ell_p}\le\widehat{{\mathbb A}}$$ is a quotient (by the leftmost copy of ${\mathbb Z}$) of $$\varinjlim\left( {\mathbb Z} \lhook\joinrel\xrightarrow{\ p^{\ell_p}\ } {\mathbb Z} \lhook\joinrel\xrightarrow{\ (p')^{\ell'}\ } {\mathbb Z} \lhook\joinrel\xrightarrow{\ (p'')^{\ell''}\ } \cdots \right),$$ whose endomorphism ring is ${\mathbb Z}$. It follows that $\widehat{\varphi}$ scales every by an integer; since ${\mathbb Z}$ acts faithfully thereon, said integers all coincide with, say, some $n\in {\mathbb Z}$. But then $\widehat{\varphi}$ scales *every* sum of indecomposable abelian groups by $n$, including those with infinitely many $p$-primary summands for single primes $p$. The conclusion follows from .* 4. ***:** The preceding cases (and ) have reduced the issue to endomorphisms of subgroups $${\mathbb D}\cong\bigoplus_{\text{prime powers }q}{\mathbb Z}/q\le \widehat{{\mathbb A}},$$ extensible along morphisms ${\mathbb E}\to {\mathbb D}$ in the sense dual to (as in the proof of ). To settle that case, consider the extension $$\label{eq:zed} 0\to {\mathbb Z}\xrightarrow{\quad}{\mathbb E}\xrightarrow{\quad}{\mathbb D}\to 0$$ corresponding [@fuchs_abgp §9.1] the an element of $$\mathop{\mathrm{\mathrm{Ext}}}({\mathbb D},{\mathbb Z})\cong \mathop{\mathrm{\mathrm{Hom}}}({\mathbb D},{\mathbb Q}/{\mathbb Z}) \quad\text{\cite[Corollary 9.3.6]{fuchs_abgp}}$$ that identifies every summand ${\mathbb Z}/q$ of ${\mathbb D}$ isomorphically (in any fashion whatever) with the unique order-$q$ cyclic subgroup of ${\mathbb Q}/{\mathbb Z}$. Because the endomorphism ring of the ${\mathbb Z}$ kernel in is ${\mathbb Z}$, any endomorphism of ${\mathbb E}$ extending a $\widehat{{\mathbb Z}}$-scaling of ${\mathbb D}$ must in fact be a ${\mathbb Z}$-scaling. $\blacksquare$* **Remark 19**. *The group ${\mathbb E}$ of will generally *not* be torsion-free, so the above argument does indeed use the full force of the hypothesis of : the compact abelian group $\widehat{{\mathbb E}}$ will not, typically, be connected.* 10 Jiřı́ Adámek, Horst Herrlich, and George E. Strecker. . Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, 1990. The joy of cats, A Wiley-Interscience Publication. Fernando Antoneli, Michael Forger, and Paola Gaviria. Maximal subgroups of compact Lie groups. , 22(4):949--1024, 2012. Michael F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Reading, Mass.-Menlo Park, Calif.-London-Don Mills, Ont.: Addison-Wesley Publishing Company (1969)., 1969. The Stacks Project Authors. Stacks project. George M. Bergman. An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange. , 56(1):91--126, 2012. Armand Borel, Robert Friedman, and John W. Morgan. , volume 747 of *Mem. Am. Math. Soc.* Providence, RI: American Mathematical Society (AMS), 2002. Alexandru Chirvasitu. Naturality and innerness for morphisms of compact groups and (restricted) lie algebras, 2022. to appear. Anton Deitmar and Siegfried Echterhoff. . Universitext. Springer, Cham, second edition, 2014. Cristina Draper and Antonio Viruel. Fine gradings on $\mathfrak e_6$. , 60(1):113--170, 2016. Alberto Elduque and Mikhail Kochetov. , volume 189 of *Math. Surv. Monogr.* Providence, RI: American Mathematical Society (AMS); Halifax, NS: Atlantic Association for Research in the Mathematical Sciences, 2013. László Fuchs. . Springer Monogr. Math. Cham: Springer, 2015. Terence M. Gagen and Derek J. S. Robinson. Finite metabelian groups with no outer automorphisms. , 32:417--423, 1979. K. W. Gruenberg. , volume 143 of *Lect. Notes Math.* Springer, Cham, 1970. Karl H. Hofmann and Sidney A. Morris. , volume 25 of *De Gruyter Studies in Mathematics*. De Gruyter, Berlin, \[2020\] ©2020. Fourth edition \[of 1646190\]. James E. Humphreys. , volume 9 of *Grad. Texts Math.* Springer, Cham, 1980. Nathan Jacobson. . Dover Publications, Inc., New York, 1979. Republication of the 1962 original. Irving Kaplansky. Infinite abelian groups. Ann Arbor: The University of Michigan Press. VII, 95 p. (1969)., 1969. John R. McMullen. On the dual object of a compact connected group. , 185(4):539--552, 1984. Martin R. Pettet. On inner automorphisms of finite groups. , 106(1):87--90, 1989. Luis Ribes and Pavel Zalesskii. , volume 40 of *Ergeb. Math. Grenzgeb., 3. Folge*. Berlin: Springer, 2nd ed. edition, 2010. Joseph J. Rotman. . Universitext. Berlin: Springer, 2nd ed. edition, 2009. Paul E. Schupp. A characterization of inner automorphisms. , 101(2):226--228, 1987. Jean-Pierre Serre. , volume 67 of *Grad. Texts Math.* Springer, Cham, 1979. Jean-Pierre Serre. . Berlin: Springer, 1997. Stephen Willard. General topology. Addison-Wesley Series in Mathematics. Reading, Mass. etc.: Addison-Wesley Publishing Company. XII, 369 p. (1970)., 1970. Jun Yu. Maximal abelian subgroups of compact simple Lie groups of type E. , 185:205--269, 2016. [Department of Mathematics, University at Buffalo]{.smallcaps} [Buffalo, NY 14260-2900, USA]{.smallcaps} *E-mail address*: `achirvas@buffalo.edu`

arxiv_math

--- abstract: | We answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show that for any interval $I\subset(0,1]$, there exist $q,p\in I$, $q<p$, and a measure $\mu$ for which the $q$-log-Sobolev inequality holds, while the $p$-log-Sobolev inequality is violated. As a tool we develop certain necessary and closely related sufficient conditions characterizing those inequalities in the case of birth-death processes on $\mathbb{N}$. author: - Bartłomiej Polaczyk bibliography: - p-logSobolev.bib title: P-log-Sobolev inequalities on $\mathbb{N}$ --- # Preliminaries ## Background Functional inequalities are one of the central objects of modern probability theory. They arise naturally when studying mixing times of Markov chains and are an important tool in proving concentration and hypercontractive estimates. Arguably, the most prominent examples are the Poincaré and log-Sobolev inequalities since most of the other functional inequalities known in the literature arise either as a modification of one of them (e.g., various modified log-Sobolev inequalities or inequalities with defects) or as a product of some sort of procedure that interpolates between the above two. An important example of the latter type is a family of Beckner inequalities introduced by Beckner [@MR954373] who showed they hold true in the case of the Gaussian measure. They were further studied, e.g., by Latała and Oleszkiewicz [@MR1796718] who used them to obtain intermediate concentration estimates between subexponential and subgaussian. Another particular example is a family of $p$-log-Sobolev inequalities introduced by Gross [@MR420249] for $p>1$. His definition was then extended by Bakry [@MR1307413] to any real $p$. Mossel et al. [@MosselOleszkiewiczSen] studied $p$-log-Sobolev inequalities in the context of reverse hypercontraction. A vibrant area of research is the study of the relations between various inequalities. E.g., it is by now classical that the log-Sobolev inequality implies the Poincaré inequality. Such sort of results can often have far-reaching consequences as demonstrated, e.g., in [@MosselOleszkiewiczSen], where the authors prove that for $0\le q<p\le 2$, the $p$-log-Sobolev inequality is stronger than the $q$-log-Sobolev inequality (cf. Theorem [Theorem 6](#T:MOS){reference-type="ref" reference="T:MOS"} below). They exploit this relation further to prove the reverse hypercontraction for measures satisfying the modified log-Sobolev inequality, which found various applications, cf., e.g., [@DBLP:journals/tit/Raginsky16; @MR3876877]. A similar situation occurred in [@MR4372142], where the authors proved that the modified log-Sobolev inequality implies a particular family of Beckner inequalities, which allows deriving strong concentration and moment bounds from the modified log-Sobolev inequality. Both of the mentioned results are examples of positive results (i.e., claiming that one inequality implies the other) and follow from abstract arguments that involve direct comparison of Dirichlet forms. Similar results can be found in, e.g., the works of Diaconis--Saloff-Coste [@MR1410112] or Bobkov--Tetali [@MR2283379]. Another approach needs to be taken in the case of negative results, i.e., when showing that one inequality does not imply another. E.g., it is classical that the Poincaré inequality does not imply the log-Sobolev inequality. In that case, one counterexample is the exponential measure which satisfies the Poincaré inequality and does not satisfy the log-Sobolev inequality as the latter implies subgaussian concentration. To prove such statements, it is often useful to derive some characterization of the functional inequality in question. These characterizations are often of independent interest, as the conditions they are expressed in are usually much more accessible than a direct proof of the functional inequalities. Some of the results that provide such characterizations include: the work by Bobkov and Götze [@MR1682772] who, by viewing the log-Sobolev inequality in the general framework of Orlicz spaces, characterize the log-Sobolev inequality on $\mathbb{R}$; the work by Miclo [@miclo] who uses Hardy inequalities on $\mathbb{Z}$ to characterize the Poincaré inequality on trees or the work by Barthe--Roberto [@bartheroberto] who treat the case of the modified log-Sobolev inequalities on $\mathbb{R}$. The aim of this note is to answer an open question from [@MosselOleszkiewiczSen] on the relation between $p$-log-Sobolev inequalities for $p\in(0,1]$, cf. Problem [Problem 7](#Pr:MOS){reference-type="ref" reference="Pr:MOS"} below. Our result falls into the category of negative examples described above. As a byproduct, we develop a sufficient condition and complement it with a closely related necessary condition for $p$-log-Sobolev inequalities, which are of independent interest. Below we describe our setting and results in more detail. ## General setup {#sec:general_setup} Let $(\Omega, \mathcal{B}, \mu)$ denote some discrete probability space. We assume that $\mu$ is fully supported on $\Omega$. Let $P\colon [0,\infty)\times \Omega \times\mathcal{B} \to [0,1]$ be a homogeneous Markov transition function for which $\mu$ is an invariant measure. We assume that $P$ is reversible with respect to $\mu$ and that it induces a strongly continuous semigroup $(P_t)_{t\ge 0}$ of operators on $L_2(\Omega, \mu)$, defined as $$P_tf(x) = \int_\Omega f(y) P(t, x, dy).$$ It can be then shown that for each $f\in L_2(\Omega,\mu)$, the mapping $$(0,\infty) \ni t\mapsto \frac{1}{2t} \int_\Omega\int_\Omega (f(x)-f(y))^2 P(t,x,dy)\mu(dx)$$ is non-increasing. Denote $$\label{eq:calh_def} \mathop{\mathrm{\mathcal{H}}} = \Bigl\{\, f\in L_2(\Omega,\mu) \,\colon\, \sup_{t\ge 0} \frac{1}{2t} \int_\Omega\int_\Omega (f(x)-f(y))^2 P(t,x,dy)\mu(dx) < \infty \,\Bigr\}$$ to be a domain of the Dirichlet form $\mathop{\mathrm{\mathcal{E}}}$ associated with this semigroup, given by the formula $$\label{eq:cale_def} \mathop{\mathrm{\mathcal{E}}}(f,g) = \lim_{t\to 0^+} \frac{1}{2t} \int_\Omega\int_\Omega (f(x)-f(y)) (g(x)-g(y)) P(t,x,dy)\mu(dx)$$ for $f,g\in\mathop{\mathrm{\mathcal{H}}}$. If $L$ is the infinitesimal generator of the semigroup $(P_t)_{t\ge 0}$, defined via $$Lf = \lim_{t\to 0^+} \frac{P_t f - f}{h}$$ with the convergence in the $L_2$ sense, then for $f,g$ belonging to the domain $\mathop{\mathrm{\mathcal{H}}}_L$ of $L$, $$\mathop{\mathrm{\mathcal{E}}}(f,g) = - \int fLg\,d\mu.$$ We refer the Reader to [@MR2091955; @MR2778606; @MR2574430; @MR3155209; @MR1258492; @MR4372142] for a detailed treatment of Markov processes, generators, Dirichlet forms and associated domains. Set $\mathop{\mathrm{\mathcal{H}}}_+= \mathop{\mathrm{\mathcal{H}}}\cap \mathbb{R}_+^{\Omega}$, where $\mathbb{R}_+ = (0,\infty)$. Below we gather some notation and results from [@MosselOleszkiewiczSen]. Note that while the results in [@MosselOleszkiewiczSen] are stated for finite spaces, the extension to general discrete spaces is straightforward as discussed in Section 12.3 therein. **Definition 1**. For $p\in\mathbb{R}\setminus\{0,1\}$, the $p$-log-Sobolev inequality is satisfied with constant $C>0$ if $$\label{eq:p-LS} \mathop{\mathrm{Ent}}_\mu(f^p) \le \frac{Cp^2}{4(p-1)}\mathop{\mathrm{\mathcal{E}}}(f^{p-1},f),$$ for all $f\in\mathop{\mathrm{\mathcal{H}}}_+$, such that $f^{p-1}\in\mathop{\mathrm{\mathcal{H}}}_+$. The 1-log-Sobolev inequality is satisfied with constant $C>0$ if $$\label{eq:1-LS} \mathop{\mathrm{Ent}}_\mu(f) \le \frac{C}{4}\mathop{\mathrm{\mathcal{E}}}(f,\log f)$$ for all $f\in\mathop{\mathrm{\mathcal{H}}}_+$, such that $\log f\in\mathop{\mathrm{\mathcal{H}}}$. The 0-log-Sobolev inequality is satisfied with constant $C>0$ if $$\label{eq:0-LS} \mathop{\mathrm{Var}}_\mu(\log f) \le \frac{C}{2}\mathop{\mathrm{\mathcal{E}}}(f,-1/f)$$ for all $f\in\mathop{\mathrm{\mathcal{H}}}_+$, such that $1/f\in\mathop{\mathrm{\mathcal{H}}}_+$. The Poincaré inequality is satisfied with constant $C>0$ if $$\label{eq:Poincare} \mathop{\mathrm{Var}}_\mu(f) \le \frac{C}{2}\mathop{\mathrm{\mathcal{E}}}(f,f)$$ for all $f\in\mathop{\mathrm{\mathcal{H}}}$. We write $p$-LS($C$) for short to denote the $p$-log-Sobolev inequality with constant $C$. We say that a pair $(\mu, \mathop{\mathrm{\mathcal{E}}})$ satisfies the $p$-log-Sobolev inequality if $p$-LS($C$) is satisfied with some finite $C>0$. If the underlying Dirichlet form $\mathop{\mathrm{\mathcal{E}}}$ is clear from context, we omit it and simply say the $\mu$ satisfies the $p$-log-Sobolev inequality. Note that the $1$-log-Sobolev inequality is often referred to as the *modified* log-Sobolev (or entropic) inequality in the literature and that the $0$-log-Sobolev and $1$-log-Sobolev inequalities are limiting cases for $p$-log-Sobolev inequalities for $p\in\mathbb{R}\setminus\{0,1\}$. Results below provide some basic relations between the inequalities introduced in Definition [Definition 1](#def:ineqs){reference-type="ref" reference="def:ineqs"}. **Proposition 2** ([@MosselOleszkiewiczSen Lemma 3.1]). *The Poincaré inequality [\[eq:Poincare\]](#eq:Poincare){reference-type="eqref" reference="eq:Poincare"} with constant $C>0$ is equivalent to the 0-log-Sobolev inequality with the same constant $C$.* We set $p'=p/(p-1)$ for the Hölder conjugate of $p\in\mathbb{R}\setminus\{1\}$. **Proposition 3** ([@MosselOleszkiewiczSen Lemma 3.2]). *For $p\in\mathbb{R}\setminus\{1\}$, $p$-LS(C) is equivalent to $p'$-LS(C).* In particular, Proposition [Proposition 3](#P:holder-equivalence){reference-type="ref" reference="P:holder-equivalence"} implies that the study of relations between various $p$-log-Sobolev inequalities can be reduced to the case $p\in[0,2]$. Moreover, since in the continuous setting (in the presence of the chain rule) all $p$-log-Sobolev inequalities for $p\neq 0$ are equivalent to the usual log-Sobolev inequality[^1] and since by Proposition [Proposition 2](#P:MOS-Poincare-equivalence){reference-type="ref" reference="P:MOS-Poincare-equivalence"}, 0-log-Sobolev inequality is equivalent to the Poincaré inequality which is strictly weaker than the log-Sobolev inequality, the only interesting range of $p$ is in fact $p\in(0,2]$. Denote $$\label{eq:calep_def} \mathop{\mathrm{\mathcal{E}}}_p(f) = \begin{cases} pp'\mathop{\mathrm{\mathcal{E}}}(f^{1/p}, f^{1/p'}) & \text{if } p\in (0,2] \setminus \{1\},\\ \mathop{\mathrm{\mathcal{E}}}(f,\log f) & \text{if } p =1, \end{cases}$$ so that $p$-LS($C$) for $p\in (0,2]$ is equivalent to $$\mathop{\mathrm{Ent}}_\mu(f)\le \frac{C}{4}\mathop{\mathrm{\mathcal{E}}}_p(f).$$ **Proposition 4** ([@MosselOleszkiewiczSen Theorem 2.1]). *For any positive $f$, the mapping $(0,2] \ni p\mapsto \mathop{\mathrm{\mathcal{E}}}_p(f)$ is non-increasing.* *Remark 5*. Note that [@MosselOleszkiewiczSen Theorem 2.1] is stated for $p\in(0,2]\setminus\{1\}$ but a natural extension to the case $p=1$ is straightforward as discussed in Remark 2.2 therein. Proposition [Proposition 4](#P:MOS_main){reference-type="ref" reference="P:MOS_main"} serves as a tool for obtaining the following main result of [@MosselOleszkiewiczSen]. **Theorem 6** ([@MosselOleszkiewiczSen Theorem 1.7]). *For any $0\le q\le p\le 2$, $p$-LS($C$) implies $q$-LS($C$). Moreover, for any $1<q\le p \le 2$, $q$-LS($C$) implies $p$-LS($\tilde{C}$) with $\tilde{C}=Cqq'/pp'$.* The main goal of this paper is to provide an answer to the following question posed in [@MosselOleszkiewiczSen Section 12]. **Problem 7**. Is there any subset $I\subset (0,1]$ with non-empty interior, such that for any $p,q\in I$, $p$-LS$(C)$ implies $q$-LS$(c(I)C)$, where $c(I)>0$ depends on $I$ only? # Main result {#sec:main-result} Let $\mu$ be a measure on $\mathbb{N}$ with full support. We use the convention that $\mu_k=\mu(\{k\})$ and $\mu[k,\infty) = \mu([k,\infty))$. For any $f\colon\mathbb{N}\to\mathbb{R}$ and $k\in\mathbb{N}$, denote $Df(k) = f(k+1) - f(k)$. Consider the following Dirichlet form $$\begin{aligned} \label{eq:calE_birth-death} \mathop{\mathrm{\mathcal{E}}}(f,g) = \sum_{k\ge 0} Df(k)Dg(k)\mu_k\end{aligned}$$ defined for $f,g\in \{\, h\in L_2(\Omega,\mu)\colon \sum_{k\ge 0} (Dh(k))^2\mu_k < \infty \,\} =: \mathop{\mathrm{\mathcal{H}}}$. The corresponding birth-death dynamics generator is given by the formula $$\label{eq:logSobolev-generator} Lf(k) = Df(k) - {\bf 1}_{\{ k>0 \} } \frac{\mu_{k-1}}{\mu_k} Df(k-1),$$ so that $\mathop{\mathrm{\mathcal{E}}}(f,g)=-\int fLg\,d\mu$ for $f,g\in\mathop{\mathrm{\mathcal{H}}}_L\subset\mathop{\mathrm{\mathcal{H}}}$. A Markov process generated by $L$ given by [\[eq:logSobolev-generator\]](#eq:logSobolev-generator){reference-type="eqref" reference="eq:logSobolev-generator"} and Dirichlet form given by [\[eq:calE_birth-death\]](#eq:calE_birth-death){reference-type="eqref" reference="eq:calE_birth-death"} with stationary measure $\mu$ being a geometric measure is investigated in [@MR1944012] as an example of a pair $(\mu,\mathop{\mathrm{\mathcal{E}}})$ satisfying the Poincaré inequality [\[eq:Poincare\]](#eq:Poincare){reference-type="eqref" reference="eq:Poincare"} and violating the modified log-Sobolev inequality [\[eq:1-LS\]](#eq:1-LS){reference-type="eqref" reference="eq:1-LS"}. *Remark 8*. It follows from the general theory of birth and death processes (see, e.g., [@MR2091955; @MR2574430]), that $\mathcal{E}$ is indeed a Dirichlet form corresponding to a Markov process on $\mathbb{N}$. As a consequence, when proving a functional inequality in question it actually suffices to consider simple (i.e., having finitely many jumps) functions. Therefore, henceforth we will sometimes restrict from specifying particular domains and simply assume that the expressions we introduce are considered for functions for which they are well-defined as one can always restrict the attention to the class of simple functions. For $x>0$, set $$\label{eq:H_p_def} H_p(x) = \begin{cases} pp'(x^{1/p}-1)(x^{1/p'}-1) & \text{if } p\in(0,1),\\ (x-1)\log(x) & \text{if } p=1 \end{cases}$$ so that $\mathop{\mathrm{\mathcal{E}}}_p$ defined in [\[eq:calep_def\]](#eq:calep_def){reference-type="eqref" reference="eq:calep_def"} is given by $$\label{eq:calep_birth_death} \mathop{\mathrm{\mathcal{E}}}_p(f) = \sum_{k\ge 0} f(k) H_p\bigl(\frac{f(k+1)}{f(k)}\bigr)\mu_k$$ and $p$-LS($C$) for $p\in (0,1]$ is equivalent to $$\label{eq:pLS_birth_death} \mathop{\mathrm{Ent}}_\mu(f) \le \frac{C}{4} \sum_{k\ge 0} f(k) H_p\bigl(\frac{f(k+1)}{f(k)}\bigr)\mu_k,$$ while $0$-LS($C$) (i.e., the Poincaré inequality) is equivalent to $$\label{eq:poincare_N} \mathop{\mathrm{Var}}_\mu(f) \le \frac{C}{2}\sum_{l=0}^\infty (Df(l))^2\mu_l.$$ Theorem below is our main result. **Theorem 9**. *For any $p\in(0,1)$, there exists a measure $\mu$ on $\mathbb{N}$ that does not satisfy the $p$-log-Sobolev inequality but satisfies the $q$-log-Sobolev inequality for all $q\in (0, p)$. In particular, there are no intervals $I$ that meet the conditions posed in Problem [Problem 7](#Pr:MOS){reference-type="ref" reference="Pr:MOS"}.* We construct the required counterexample and verify that it satisfies the appropriate $p$-log-Sobolev inequalities with the use of the theorem below, which is of independent interest. **Theorem 10**. *Choose any $p\in(0,1]$ and a measure $\mu$ on $\mathbb{N}$ with full support and with associated Dirichlet form $\mathop{\mathrm{\mathcal{E}}}$, given by the birth-death process generator [\[eq:calE_birth-death\]](#eq:calE_birth-death){reference-type="eqref" reference="eq:calE_birth-death"}.* *If $\mu$ satisfies the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} with some finite constant $C_P>0$ and $$\label{eq:main_condition} \hat{C} := \sup_{n\ge 1} \Big\{ \Bigl[ H_p\bigl( \frac{\mu[n-1,\infty)}{\mu[n,\infty)} \bigr) \Bigr]^{-1} \cdot \log\bigl( \frac{2}{\mu[n,\infty)} \bigr) \Big\} < \infty,$$ then $\mu$ satisfies $p$-LS($C$) [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} with some finite $C>0$.* *Contrarily, if there exists an increasing sequence $\tau_0<\tau_1<\ldots$ such that $$\label{eq:necessary_cond} \lim_{n\to\infty} \Big\{ \Bigl[ H_p\bigl( \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_n,\infty)} \bigr) \Bigr]^{-1} \cdot \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_{n}-1,\infty)} \cdot \log\bigl( \frac{2}{\mu[\tau_n,\infty)} \bigr) \Big\} = \infty,$$ then $\mu$ does not satisfy $p$-LS($C$) [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} with any finite $C>0$.* *Remark 11*. The Poincaré inequality is implied by the $p$-log-Sobolev inequality for any $p\in (0,2]$, cf. Theorem [Theorem 6](#T:MOS){reference-type="ref" reference="T:MOS"} and Proposition [Proposition 2](#P:MOS-Poincare-equivalence){reference-type="ref" reference="P:MOS-Poincare-equivalence"}, therefore making it a part of the sufficient condition of Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"} (alongside [\[eq:main_condition\]](#eq:main_condition){reference-type="eqref" reference="eq:main_condition"}) is non-restrictive. *Remark 12*. The negation of [\[eq:main_condition\]](#eq:main_condition){reference-type="eqref" reference="eq:main_condition"} is equivalent to the existence of an increasing sequence $\tau_0<\tau_1<\ldots$ such that $$\lim_{n\to\infty} \Big\{ \Big[ H_p\bigl( \frac{\mu[\tau_{n}-1,\infty)}{\mu[\tau_n,\infty)} \bigr) \Big]^{-1} \cdot \log\bigl( \frac{2}{\mu[\tau_n,\infty)} \bigr) \Big\} = \infty.$$ If $\mu$ verifies the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} with constant $C_P$, then by Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"} (cf. also Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}) and by Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"}, [\[L:H_p\_c_increasing\]](#L:H_p_c_increasing){reference-type="ref" reference="L:H_p_c_increasing"} below, for any $\tau_{n-1}<\tau_n$ and some $c>0$, $$\Bigl[ H_p\bigl( \frac{\mu[\tau_{n}-1,\infty)}{\mu[\tau_n,\infty)} \bigr) \Bigr]^{-1} \ge \frac{1}{1+c} \Bigl[ H_p\bigl( \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_n,\infty)} \bigr) \Bigr]^{-1} \cdot \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_{n}-1,\infty)}.$$ Whence, condition [\[eq:necessary_cond\]](#eq:necessary_cond){reference-type="eqref" reference="eq:necessary_cond"} implies that $\hat{C}=\infty$ and thus [\[eq:main_condition\]](#eq:main_condition){reference-type="eqref" reference="eq:main_condition"} does not hold, but the reverse needs not to be true. Thus, providing a full characterization of $p$-log-Sobolev inequalities on $\mathbb{N}$ for $p\in(0,1]$ remains open. # Auxiliary results In this section we gather some lemmas and known results required for the proof of Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"}. ## Hardy inequality In the sequel we put $0\cdot\infty = 0$ and $c/0=\infty$ for any $c>0$. **Definition 13**. We say that a probability measure $\mu$ on $\mathbb{N}$ satisfies the Hardy inequality with constant $C$, if $$\label{eq:Hardy} \sum_{l=0}^\infty (f(l)-f(0))^2\mu_l \le C \sum_{l=0}^\infty (Df(l))^2\mu_l = C \mathop{\mathrm{\mathcal{E}}}(f,f)$$ for all $f\in\mathop{\mathrm{\mathcal{H}}}$. Define $$\label{eq:C_mu} C_\mu = \sup_{k\ge 1} \Bigl\{ \mu[k,\infty) \cdot \sum_{l=0}^{k-1} \frac{1}{\mu_l} \Bigr\}.$$ The following result states that $C_\mu$ characterizes the Hardy inequality [\[eq:Hardy\]](#eq:Hardy){reference-type="eqref" reference="eq:Hardy"}. **Theorem 14** (Miclo [@miclo]). *The best constant $\hat{C}_H$ in the Hardy inequality [\[eq:Hardy\]](#eq:Hardy){reference-type="eqref" reference="eq:Hardy"}, $$\label{eq:C_H_best} \hat{C}_H = \sup_{f\in\mathop{\mathrm{\mathcal{H}}}} \Bigl\{ \frac{\sum_{l=0}^\infty (f(l)-f(0))^2\mu_l}{\mathop{\mathrm{\mathcal{E}}}(f,f)} \colon \mathop{\mathrm{\mathcal{E}}}(f,f)>0 \Bigr\},$$ verifies $$C_\mu \le \hat{C}_H \le 4C_\mu.$$* *Remark 15*. It is easy to see that for fully supported measures, the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} is satisfied if and only if the Hardy inequality [\[eq:Hardy\]](#eq:Hardy){reference-type="eqref" reference="eq:Hardy"} is satisfied. More precisely, the best constant $\hat{C}_P$ in the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"}, $$\label{eq:C_P_best} \hat{C}_P = 2 \sup_{f\in\mathop{\mathrm{\mathcal{H}}}} \Bigl\{ \frac{\mathop{\mathrm{Var}}_\mu(f)}{\mathop{\mathrm{\mathcal{E}}}(f,f)} \colon \mathop{\mathrm{\mathcal{E}}}(f,f)>0 \Bigr\},$$ satisfies $$2\mu_0 \hat{C}_H \le \hat{C}_P \le 2\hat{C}_H.$$ Indeed, $\hat{C}_P \le 2\hat{C}_H$ follows from the estimate $\mathop{\mathrm{Var}}_\mu(f) \le \int (f-f(0))^2\,d\mu$. To see that $2\mu_0 \hat{C}_H \le \hat{C}_P$, let $f_\varepsilon\in\mathop{\mathrm{\mathcal{H}}}$ for any $\varepsilon > 0$ be such that $f_\varepsilon(0)=0$, $\mathop{\mathrm{\mathcal{E}}}(f_\varepsilon,f_\varepsilon)>0$ and $\int f_\varepsilon^2\,d\mu \ge (\hat{C}_H-\varepsilon)\mathop{\mathrm{\mathcal{E}}}(f_\varepsilon, f_\varepsilon)$. Then, by the Cauchy--Schwarz inequality, $(\int f_\varepsilon\,d\mu)^2 \le (1-\mu_0) \int f_\varepsilon^2\,d\mu$, whence $$\begin{aligned} \frac{\hat{C}_P}{2}\mathop{\mathrm{\mathcal{E}}}(f_\varepsilon,f_\varepsilon) \ge \mathop{\mathrm{Var}}_\mu(f_\varepsilon) \ge \mu_0 \int f_\varepsilon^2\,d\mu \ge \mu_0 (\hat{C}_H-\varepsilon)\mathop{\mathrm{\mathcal{E}}}(f_\varepsilon, f_\varepsilon), \end{aligned}$$ and we conclude by taking $\varepsilon\to 0^+$. The quantity $C_\mu$ is useful for controlling the tail behavior of $\mu$ as demonstrated in the lemmas below. **Lemma 16**. *If $\mu$ is fully supported, then $$\label{eq:quotient_upper_bound} \sup_{k\ge 0} \frac{\mu[k,\infty)}{\mu_k} \le 1+C_\mu$$ and $$\label{eq:rho_lower_bound} \inf_{k\ge 1} \Bigl\{ \frac{\mu[k-1,\infty)}{\mu[k,\infty)} \Bigr\} \ge 1+\frac{1}{C_\mu}.$$* *Proof.* The estimate [\[eq:quotient_upper_bound\]](#eq:quotient_upper_bound){reference-type="eqref" reference="eq:quotient_upper_bound"} follows from the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$. Using [\[eq:quotient_upper_bound\]](#eq:quotient_upper_bound){reference-type="eqref" reference="eq:quotient_upper_bound"}, we obtain for any $k\ge 1$, $$\begin{aligned} \mu[k,\infty) &= \mu[k-1,\infty) - \mu_{k-1} \\&\le \mu[k-1,\infty) - \frac{\mu[k-1,\infty)}{1+C_\mu} = \bigl( 1+\frac{1}{C_\mu} \bigr)^{-1} \mu[k-1,\infty) \end{aligned}$$ and [\[eq:rho_lower_bound\]](#eq:rho_lower_bound){reference-type="eqref" reference="eq:rho_lower_bound"} follows. ◻ **Lemma 17**. *If $\mu$ is fully supported, then for $X\sim\mu$, $\mathop{\mathrm{\mathbb{E}}}X \le 1 + C_\mu$.* *Proof.* By Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"}, $\mathop{\mathrm{\mathbb{E}}}X \le \sum_{k\in\mathbb{N}} \mu[k,\infty) \le (1+C_\mu) \sum_{k\in\mathbb{N}} \mu_k = 1+C_\mu$. ◻ ## Tail estimates Proposition below states that the $p$-log-Sobolev inequality implies Poisson-type tail behavior for $p\in(0,1]$. It is deduced by using a variant of Herbst's argument. **Proposition 18**. *If $\mu$ is fully supported and satisfies $p$-LS($C$) [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} for some $p\in(0,1]$ and some $C<\infty$, then there exists $\varepsilon_p\colon (0,\infty) \to [0,1]$, such that $\varepsilon_p(t)\to 0$ as $t\to\infty$ and $$\label{eq:poisson_tails} \log\bigl( \mu[t,\infty) \bigr) \le -(1-\varepsilon_p(t)) tp\log (t+1)$$ for any $t\ge 0$.* For the proof of Proposition [Proposition 18](#P:poisson_tails){reference-type="ref" reference="P:poisson_tails"}, we need the following lemma. **Lemma 19**. *For any $\lambda > 1$ and $p\in(0,1)$, $$\label{eq:integral_est} \int_1^\lambda \frac{e^{s/p}-1}{s^2}\,ds \le \frac{p}{1-p} \cdot \frac{e^{\lambda/p}}{\lambda} = -p' \cdot \frac{e^{\lambda/p}}{\lambda}.$$* *Proof.* For any $s>1$, $$\frac{1-p}{p}\cdot \frac{e^{s/p}-1}{s^2} \le \frac{1-p}{p} \cdot \frac{e^{s / p}}{s^2} \le \frac{s-p}{p} \cdot \frac{e^{s / p}}{s^2} = \frac{d}{ds}\Bigl[ \frac{e^{s/p}}{s} \Bigr],$$ whence $$\int_1^\lambda \frac{e^{s/p}-1}{s^2} \,ds \le \frac{p}{1-p} \Bigl( \frac{e^{\lambda/p}}{\lambda} - e^{1/p} \Bigr) \le \frac{p}{1-p} \frac{e^{\lambda/p}}{\lambda}$$ as desired. ◻ *Proof of Proposition [Proposition 18](#P:poisson_tails){reference-type="ref" reference="P:poisson_tails"}.* Consider first the case $p\in(0,1)$. For $s, h>0$, $x\in\mathbb{R}$, denote $\phi_h(x) = \min(x,h)$, $f_{s,h}(x)=\exp(s \phi_h(x)/p)$ and $g_{s, h}(x) = \exp(s \phi_h(x)/p')$. Let $X\sim\mu$ and for $h>0$, set $X_h = \min(X,h)$. For any $s,h>0$, $h\in\mathbb{N}$, applying $p$-LS($C$) to $f_{s,h}$ and using the Dirichlet form formula [\[eq:calE_birth-death\]](#eq:calE_birth-death){reference-type="eqref" reference="eq:calE_birth-death"}, we obtain $$\begin{aligned} \label{eq:X_exp_integrability_derivative} \begin{split} \frac{d}{ds} \Bigl[ \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{s X_h}}{s} \Bigr] &= \frac{\mathop{\mathrm{Ent}}(e^{s X_h})}{s^2 \mathop{\mathrm{\mathbb{E}}}e^{s X_h}} \\&\le \frac{Cpp'}{4s^2 \mathop{\mathrm{\mathbb{E}}}e^{s X_h}} \mathop{\mathrm{\mathcal{E}}}(f_{s, h}, g_{s, h}) \\&= \frac{Cpp'}{4s^2 \mathop{\mathrm{\mathbb{E}}}e^{s X_h}} \sum_{k=0}^{h-1} \bigl( e^{s(k+1)/p} - e^{sk/p} \bigr) \bigl( e^{s(k+1)/p'} - e^{sk/p'} \bigr)\mu_k \\&= \frac{Cpp'}{4s^2} \bigl( e^{s/p} - 1 \bigr) \bigl( e^{s/p'} - 1 \bigr) \frac{\sum_{k=0}^{h-1} e^{sk/p + sk/p'}\mu_k}{\mathop{\mathrm{\mathbb{E}}}e^{s X_h}} \\&\le \frac{Cpp'}{4s^2} \bigl( e^{s/p} - 1 \bigr) \bigl( e^{s/p'} - 1 \bigr), \end{split}\end{aligned}$$ where in the last inequality we have also used that $p'(e^{s/p'} - 1) > 0$. By Theorem [Theorem 6](#T:MOS){reference-type="ref" reference="T:MOS"}, $\mu$ satisfies the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"}, whence by Lemma [Lemma 17](#L:poincare_implies_integrability){reference-type="ref" reference="L:poincare_implies_integrability"}, $\mathop{\mathrm{\mathbb{E}}}X < 1+C_\mu < \infty$, cf. Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}. Consequently, for any $\lambda,h>0$, $$\begin{aligned} \begin{split}\label{eq:X_exp_integrability} \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{\lambda X_h}}{\lambda} &= \mathop{\mathrm{\mathbb{E}}}X_h + \int_0^\lambda \frac{d}{ds}\bigl[ \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{sX_h}}{s} \bigr]\,ds \\&\le \mathop{\mathrm{\mathbb{E}}}X_h + \frac{Cpp'}{4} \int_0^\lambda \frac{ \bigl( e^{s/p} - 1 \bigr) \bigl( e^{s/p'} - 1 \bigr) }{s^2} \,ds \\&\le \mathop{\mathrm{\mathbb{E}}}X + \frac{Cpp'}{4} \int_0^\lambda \frac{ \bigl( e^{s/p} - 1 \bigr) \bigl( e^{s/p'} - 1 \bigr) }{s^2} \,ds < \infty. \end{split}\end{aligned}$$ Taking $h\to +\infty$ and using monotone convergence theorem in [\[eq:X_exp_integrability\]](#eq:X_exp_integrability){reference-type="eqref" reference="eq:X_exp_integrability"}, we obtain that for any $\lambda > 0$, $\mathop{\mathrm{\mathbb{E}}}e^{\lambda X}<\infty$. Therefore, we can repeat reasoning from [\[eq:X_exp_integrability_derivative\]](#eq:X_exp_integrability_derivative){reference-type="eqref" reference="eq:X_exp_integrability_derivative"} to get that for any $s>0$, $$\label{eq:herbst} \frac{d}{ds} \Bigl[ \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{s X}}{s} \Bigr] \le \frac{Cpp'}{4s^2} \bigl( e^{s/p} - 1 \bigr) \bigl( e^{s/p'} - 1 \bigr).$$ For any $\lambda>1$, by [\[eq:herbst\]](#eq:herbst){reference-type="eqref" reference="eq:herbst"}, estimating $1-e^{s/p'}\le 1$ for $s>1$ and using Lemma [Lemma 19](#L:integral_est){reference-type="ref" reference="L:integral_est"}, $$\begin{aligned} \int_1^\lambda \frac{d}{ds} \bigl[ \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{s X}}{s} \bigr]\,ds &\le \frac{C pp'}{4} \int_1^\lambda \frac{(e^{s/p}-1)(e^{s/p'}-1)}{s^2} \,ds \\&\le \frac{C p(-p')}{4} \int_1^\lambda \frac{e^{s/p}-1}{s^2} \,ds \le \frac{C pp'^2}{4\lambda} e^{\lambda/p},\end{aligned}$$ whence, for $t>e^{1/p}-1$, by the Chernoff bound and setting $\lambda = p\log (t+1)>1$, $$\begin{aligned} \log\bigl( \mathop{\mathrm{\mathbb{P}}}(X \ge t) \bigr) &\le \log \mathop{\mathrm{\mathbb{E}}} \exp(-\lambda t + \lambda X) \\ &= -\lambda t + \lambda \log \mathop{\mathrm{\mathbb{E}}}e^X + \lambda \int_1^\lambda \frac{d}{ds} \bigl[ \frac{\log \mathop{\mathrm{\mathbb{E}}}e^{s X}}{s} \bigr]\,ds \\ &\le -\lambda t + \lambda \log \mathop{\mathrm{\mathbb{E}}}e^X + \frac{C pp'^2}{4} e^{\lambda/p} \\ &= - pt\log (t+1) + p (\log \mathop{\mathrm{\mathbb{E}}}e^X)\log (t+1) + \frac{C pp'^2}{4}(t+1) \\ &= \notag -\Bigl( 1 - \frac{\log\mathop{\mathrm{\mathbb{E}}}e^X}{t} - \frac{Cp'^2(t+1)}{4t\log(t+1)} \Bigr)tp \log (t+1).\end{aligned}$$ Therefore, for any $p\in(0,1)$ and $$\label{eq:varepsilon_p_def} \varepsilon_p(t) = \begin{cases} 1 &\text{if}\quad t\le e^{1/p}-1,\\ \min\bigl( 1, \frac{\log\mathop{\mathrm{\mathbb{E}}}e^X}{t} + \frac{C}{4} \cdot \frac{1}{(1/p-1)^2} \cdot \frac{t+1}{t\log(t+1)} \bigr) &\text{if}\quad t> e^{1/p}-1, \end{cases}$$ the estimate [\[eq:poisson_tails\]](#eq:poisson_tails){reference-type="eqref" reference="eq:poisson_tails"} holds for any $t\ge 0$ as desired. We turn to the case $p=1$. Recall that by Theorem [Theorem 6](#T:MOS){reference-type="ref" reference="T:MOS"}, for any $p\in(0,1)$ the 1-log-Sobolev inequality implies the $p$-log-Sobolev inequality with the same constant. Let $\varepsilon_p$ be given by [\[eq:varepsilon_p\_def\]](#eq:varepsilon_p_def){reference-type="eqref" reference="eq:varepsilon_p_def"} and let $\varphi\colon (0,\infty)\to (0,1)$ be such that $\varphi(t)\to 1$ and $\varepsilon_{\varphi(t)}(t)\to 0$ as $t\to\infty$ (one can take, e.g., $\varphi(t)=0.5$ for $t\le e^e$ and $\varphi(t) = (1+1/\log\log t)^{-1}$ for $t>e^e$). For $t\in(0,\infty)$, set $$\varepsilon_1(t) = 1 - \varphi(t) \bigl( 1 - \varepsilon_{\varphi(t)}(t) \bigr).$$ Then $\varepsilon_1(t)\to 0$ as $t\to\infty$ and for any $t> 0$, we apply [\[eq:poisson_tails\]](#eq:poisson_tails){reference-type="eqref" reference="eq:poisson_tails"} with $p=\varphi(t)$ to obtain the conclusion. ◻ ## Reduction to increasing functions The proposition below is an adaptation of [@bartheroberto Proposition 3] to the discrete case. The proof goes along similar lines -- we present it for completeness. **Proposition 20**. *Recall the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$ and assume $C_\mu<\infty$. For any $f\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps and for any $\rho>1$, $$\mathop{\mathrm{Ent}}_\mu(f) \le 16C_\mu(1+\sqrt{\rho})^2 \mathop{\mathrm{\mathcal{E}}}(\sqrt{f}, \sqrt{f}) + \sum_{k\colon g(k)\ge \rho \mathop{\mathrm{\mathbb{E}}}_\mu g} g(k)\log\bigl(\frac{g(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu g}\bigr)\mu_k,$$ where $g(n) = f(0) + \sum_{k=0}^{n-1}(Df(k))_+.$* *Proof.* We use the assumption that $f$ has a finite number of jumps to make sure that every quantity below is well-defined. Note that $g\ge f$ since $(Df(k))_+\ge Df(k)$ for all $k\in\mathbb{N}$. For $x,t>0$, denote $\psi(t,x)=x\log\frac{x}{t} - (x-t)$, $\phi(x)=x\log x$ and set $\Theta = \{ n\in \mathbb{N}\colon f(n) \ge \rho \mathop{\mathrm{\mathbb{E}}}g \}$. Convexity of $\phi$ implies that $\phi(\mathop{\mathrm{\mathbb{E}}}_\mu f) \ge \phi(\mathop{\mathrm{\mathbb{E}}}_\mu g) + \phi'(\mathop{\mathrm{\mathbb{E}}}_\mu g)(\mathop{\mathrm{\mathbb{E}}}_\mu f - \mathop{\mathrm{\mathbb{E}}}_\mu g)$, so that $$\begin{aligned} \label{eq:BR_ent_split} \begin{split} \mathop{\mathrm{Ent}}(f) &= \mathop{\mathrm{\mathbb{E}}}_\mu \bigl[ \phi(f) - \phi(\mathop{\mathrm{\mathbb{E}}}_\mu f) \bigr] \\ &\le \mathop{\mathrm{\mathbb{E}}}_\mu\bigl[ \phi(f)-\phi(\mathop{\mathrm{\mathbb{E}}}_\mu g) + \phi'(\mathop{\mathrm{\mathbb{E}}}_\mu g)\cdot (\mathop{\mathrm{\mathbb{E}}}_\mu g - \mathop{\mathrm{\mathbb{E}}}_\mu f) \bigr] = \mathop{\mathrm{\mathbb{E}}}_\mu \psi(\mathop{\mathrm{\mathbb{E}}}_\mu g, f). \end{split}\end{aligned}$$ Set $U_\rho=(1+\sqrt{\rho})^2$. Since for $x\in [0,\rho t]$, $$\begin{aligned} \psi(t,x) \le x\Bigl(\frac{x}{t}-1\Bigr) - (x-t) &= t\Bigl( \frac{x}{t}-1 \Bigr)^2 \\&= \Bigl( 1+\sqrt{\frac{x}{t}} \Bigr)^2 (\sqrt{x}-\sqrt{t})^2 \le U_\rho (\sqrt{x}-\sqrt{t})^2,\end{aligned}$$ we obtain $$\begin{aligned} \label{eq:entropy_bound_small_f} \begin{split} \mathop{\mathrm{\mathbb{E}}}_\mu \psi(\mathop{\mathrm{\mathbb{E}}}_\mu g, f){\bf 1}_{\Theta^c} &\le U_\rho \mathop{\mathrm{\mathbb{E}}}_\mu (\sqrt{f}-\sqrt{\mathop{\mathrm{\mathbb{E}}}_\mu g})^2 \\&\le 2 U_\rho \bigl( \mathop{\mathrm{\mathbb{E}}}_\mu (\sqrt{f}-\sqrt{g})^2 + \mathop{\mathrm{\mathbb{E}}}_\mu (\sqrt{g}-\sqrt{\mathop{\mathrm{\mathbb{E}}}_\mu g})^2 \bigr) \\&\le 2 U_\rho \bigl( \mathop{\mathrm{\mathbb{E}}}_\mu (\sqrt{f}-\sqrt{g})^2 + 2\mathop{\mathrm{Var}}_\mu(\sqrt{g}) \bigr). \end{split}\end{aligned}$$ Let $\hat{C}_H$ and $\hat{C}_P$ be the best constants in the Hardy inequality [\[eq:Hardy\]](#eq:Hardy){reference-type="eqref" reference="eq:Hardy"} and the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} respectively, cf. the definitions [\[eq:C_H\_best\]](#eq:C_H_best){reference-type="eqref" reference="eq:C_H_best"} and [\[eq:C_P\_best\]](#eq:C_P_best){reference-type="eqref" reference="eq:C_P_best"}. By Theorem [Theorem 14](#T:Miclo){reference-type="ref" reference="T:Miclo"} and as indicated in Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}, $\hat{C}_P \le 2\hat{C}_H\le 8C_\mu$. Recall that $\sqrt{f(0)}=\sqrt{g(0)}$, whence the RHS of [\[eq:entropy_bound_small_f\]](#eq:entropy_bound_small_f){reference-type="eqref" reference="eq:entropy_bound_small_f"} can be estimated from above by $$2U_\rho \bigl( 4C_\mu \sum_{k=0}^\infty (D\sqrt{f}-D\sqrt{g})^2(k)\mu_k + 8C_\mu \sum_{k=0}^\infty (D\sqrt{g})^2(k)\mu_k \bigr) \le 16C_\mu U_\rho \mathop{\mathrm{\mathcal{E}}}(\sqrt{f}, \sqrt{f}),$$ where we have used the fact that $4(D\sqrt{f}-D\sqrt{g})^2+8(D\sqrt{g})^2\le 8(D\sqrt{f})^2$, which follows from the pointwise estimate $0\le D\sqrt{g}\le (D\sqrt{f})_+$ and the fact that for a fixed $y$, the convex mapping $x\mapsto 4(y-x)^2 + 8x^2$ on a closed interval admits a maximum at an end of this interval. Turning to the remaining part of the RHS of [\[eq:BR_ent_split\]](#eq:BR_ent_split){reference-type="eqref" reference="eq:BR_ent_split"}, we get $$\begin{aligned} \mathop{\mathrm{\mathbb{E}}}_\mu \psi(\mathop{\mathrm{\mathbb{E}}}g, f){\bf 1}_\Theta &= \mathop{\mathrm{\mathbb{E}}}_\mu \bigl[f\log \frac{f}{\mathop{\mathrm{\mathbb{E}}}_\mu g} - (f-\mathop{\mathrm{\mathbb{E}}}_\mu g) \bigr]{\bf 1}_\Theta \\ &\le \mathop{\mathrm{\mathbb{E}}}_\mu \bigl[g \log\frac{g}{\mathop{\mathrm{\mathbb{E}}}_\mu g} \bigr]{\bf 1}_\Theta \le \sum_{k\colon g(k)\ge \rho \mathop{\mathrm{\mathbb{E}}}_\mu g} g(k)\log\frac{g(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu g}\mu_k,\end{aligned}$$ where in both inequalities we have used the definition of $\Theta$ and the facts that $f\le g$ and $\rho>1$. Combining all the above estimates yields the conclusion. ◻ When proving that some $p$-log-Sobolev inequality is satisfied, Proposition [Proposition 20](#P:Barthe-Roberto){reference-type="ref" reference="P:Barthe-Roberto"} allows us to restrict our attention to a special subclass of functions from $\mathop{\mathrm{\mathcal{H}}}_+$. This idea is formalized in the corollary below. **Corollary 21**. *If $C_\mu<\infty$ and $$\label{eq:Barthe-Roberto_cond} \sum_{k\colon g(k)\ge \rho \mathop{\mathrm{\mathbb{E}}}_\mu g} g(k)\log\bigl(\frac{g(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu g}\bigr)\mu_k \le C_\rho \mathop{\mathrm{\mathcal{E}}}_p(g)$$ for some $\rho>1$, $p\in(0,2]$, some constant $C_\rho>0$ and any non-decreasing function $g\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps, then $\mu$ satisfies the $p$-log-Sobolev inequality [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} with constant $$C=64C_\mu(1+\sqrt{\rho})^2+4C_\rho.$$* *Proof.* By applying Proposition [Proposition 20](#P:Barthe-Roberto){reference-type="ref" reference="P:Barthe-Roberto"} together with condition [\[eq:Barthe-Roberto_cond\]](#eq:Barthe-Roberto_cond){reference-type="eqref" reference="eq:Barthe-Roberto_cond"}, we get that for any $f\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps $$\label{eq:f-g-replacement} \mathop{\mathrm{Ent}}_\mu(f) \le 16C_\mu(1+\sqrt{\rho})^2 \mathop{\mathrm{\mathcal{E}}}(\sqrt{f}, \sqrt{f}) + C_\rho \mathop{\mathrm{\mathcal{E}}}_p(g),$$ where $g(0)=f(0)$ and $g(n) = f(0) + \sum_{k=0}^{n-1} (Df(k))_+$ for $n\ge 1$. Note that $f\le g$ by definition and that for any $x, \Delta > 0$, the mapping $x\mapsto x H_p(1+\Delta/x)$ is non-increasing, which follows from the convexity of $H_p$ on $[1,\infty)$, cf. Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"}, [\[L:H_p\_monotone\]](#L:H_p_monotone){reference-type="ref" reference="L:H_p_monotone"} below, and the fact that $H_p(1)=0$. Using that $H_p\ge 0$ and monotonicity of $x\mapsto x H_p(1+\Delta/x)$, we obtain $$\begin{aligned} \label{eq:calef-caleg-comp} \begin{split} \mathop{\mathrm{\mathcal{E}}}_p(f) &= \sum_{k=0}^\infty f(k) H_p\Bigl( 1 + \frac{Df(k)}{f(k)} \Bigr) \mu_k \\&\ge \sum_{k=0}^\infty f(k) H_p\Bigl( 1 + \frac{Df(k)}{f(k)} \Bigr) \mu_k {\bf 1}_{\{ Df(k) > 0 \}} \\&\ge \sum_{k=0}^\infty g(k) H_p\Bigl( 1 + \frac{Df(k)}{g(k)} \Bigr) \mu_k {\bf 1}_{\{ Df(k) > 0 \}} \\&= \sum_{k=0}^\infty g(k) H_p\Bigl( 1 + \frac{Dg(k)}{g(k)} \Bigr) \mu_k = \mathop{\mathrm{\mathcal{E}}}_p(g). \end{split} \end{aligned}$$ By Proposition [Proposition 4](#P:MOS_main){reference-type="ref" reference="P:MOS_main"}, $\mathop{\mathrm{\mathcal{E}}}(\sqrt{f}, \sqrt{f}) \le \mathop{\mathrm{\mathcal{E}}}_2(f) \le \mathop{\mathrm{\mathcal{E}}}_p(f)$ for any $p\in (0,2]$. Therefore, combining [\[eq:f-g-replacement\]](#eq:f-g-replacement){reference-type="eqref" reference="eq:f-g-replacement"} and [\[eq:calef-caleg-comp\]](#eq:calef-caleg-comp){reference-type="eqref" reference="eq:calef-caleg-comp"}, we obtain that for any $f\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps, $$\label{eq:Barthe-Roberto-final-step} \mathop{\mathrm{Ent}}_\mu(f) \le \bigl( 16C_\mu(1+\sqrt{\rho})^2 + C_\rho \bigr) \mathop{\mathrm{\mathcal{E}}}_p(f).$$ We conclude the result for arbitrary $f$ by the monotone convergence theorem (applied to the RHS of [\[eq:Barthe-Roberto-final-step\]](#eq:Barthe-Roberto-final-step){reference-type="eqref" reference="eq:Barthe-Roberto-final-step"}) and Fatou's lemma (applied to the LHS of [\[eq:Barthe-Roberto-final-step\]](#eq:Barthe-Roberto-final-step){reference-type="eqref" reference="eq:Barthe-Roberto-final-step"}). ◻ ## Technical lemmas Recall the definition [\[eq:H_p\_def\]](#eq:H_p_def){reference-type="eqref" reference="eq:H_p_def"} of $H_p$ and note that for $p\in(0,1)$ and $x>0$, $H_p(x)=pp'(x-x^{1/p}-x^{1/p'}+1)$. **Lemma 22**. *For any $p\in(0,1]$, the following properties are true:* 1. *[\[L:H_p\_monotone\]]{#L:H_p_monotone label="L:H_p_monotone"} $H_p$ is increasing and convex on $[1,\infty)$;* 2. *[\[L:H_p\_lower_bound\]]{#L:H_p_lower_bound label="L:H_p_lower_bound"} $H_p(x)\ge (\log x)^2$ for $x\ge 1$.* *If additionally $x\ge \lambda$ for some $\lambda>1$, then* 1. *$H_p(x) \ge (x \log x) \cdot \min\bigl\{ \frac{H_p(\lambda)}{\lambda\log\lambda}, \frac{H'_p(\lambda)}{1+\log\lambda}, 1 \bigr\}$; [\[L:H_p\_xlogx\]]{#L:H_p_xlogx label="L:H_p_xlogx"}* 2. *$H_p(x) \ge H_p(xc) \cdot \min\bigl\{ \frac{H_p(\lambda)}{H_p(\lambda c)}, \frac{H_p'(\lambda)}{cH_p'(\lambda c)}, c^{-1/p} \bigr\}$ for any $c>1$; [\[L:H_p\_cx\]]{#L:H_p_cx label="L:H_p_cx"}* 3. *$\frac{H_p(y)}{y} \le \frac{\lambda}{\lambda-1} \frac{H_p(x)}{x}$ for any $\lambda \le y \le x$. [\[L:H_p\_c_increasing\]]{#L:H_p_c_increasing label="L:H_p_c_increasing"}* *Proof.* We start with [\[L:H_p\_monotone\]](#L:H_p_monotone){reference-type="ref" reference="L:H_p_monotone"} and [\[L:H_p\_lower_bound\]](#L:H_p_lower_bound){reference-type="ref" reference="L:H_p_lower_bound"}. For $p=1$, $H_1(x)=(x-1)\log x$, $H_1(1) = H_1'(1)=0$ and $H_1''(x) = \frac{1}{x^2} + \frac{1}{x} > 0$, yielding [\[L:H_p\_monotone\]](#L:H_p_monotone){reference-type="ref" reference="L:H_p_monotone"}, while [\[L:H_p\_lower_bound\]](#L:H_p_lower_bound){reference-type="ref" reference="L:H_p_lower_bound"} follows immediately as $\log x < x-1$. For $p\in(0,1)$, denote $h(x)=(\log x)^2$. Then, $H_p(1)= H_p'(1) = h(1) = h'(1) = 0$ and, using AM-GM inequality, $$\label{eq:H_p_2nd_der} H''_p(x) = \frac{x^{1/p} + x^{1/p'}}{x^2} \ge \frac{2 x^{1/2p+1/2p'}}{x^2} \ge \frac{2}{x^2} \ge 2\frac{1-\log x}{x^2} =h''(x),$$ whence [\[L:H_p\_lower_bound\]](#L:H_p_lower_bound){reference-type="ref" reference="L:H_p_lower_bound"} follows. Moreover, [\[eq:H_p\_2nd_der\]](#eq:H_p_2nd_der){reference-type="eqref" reference="eq:H_p_2nd_der"} implies that $H_p''(x)\ge 0$, yielding also [\[L:H_p\_monotone\]](#L:H_p_monotone){reference-type="ref" reference="L:H_p_monotone"}. To see [\[L:H_p\_xlogx\]](#L:H_p_xlogx){reference-type="ref" reference="L:H_p_xlogx"}, denote the function on its RHS by $\tilde{h}(x)$ and note that by the definition of $\tilde{h}$, $H_p(\lambda) \ge \tilde{h}(\lambda)$ and $H_p'(\lambda) \ge \tilde{h}'(\lambda)$. Since for any $x\ge 1$, $$H''_p(x) = \frac{x^{1/p} + x^{1/p'}}{x^2} \ge x^{1/p - 2} \ge \frac{1}{x} = (x\log x)'',$$ then also $H''_p(x) \ge \tilde{h}''(x)$ for any $x\ge \lambda$ and [\[L:H_p\_xlogx\]](#L:H_p_xlogx){reference-type="ref" reference="L:H_p_xlogx"} follows (note that the calculation above also covers the case $p=1$, as then $1/p'=0$). Similarly, if $\tilde{h}(x)$ is the RHS of [\[L:H_p\_cx\]](#L:H_p_cx){reference-type="ref" reference="L:H_p_cx"}, then $H_p(\lambda) \ge \tilde{h}(\lambda)$ and $H_p'(\lambda) \ge \tilde{h}'(\lambda)$ by the definition of $\tilde{h}$. As $$H''_p(x) = \frac{x^{1/p} + x^{1/p'}}{x^2} \ge c^{-1/p} \cdot c^2 \frac{(cx)^{1/p} + (cx)^{1/p'}}{(cx)^2} = \frac{d^2}{dx^2} \bigl( c^{-1/p} H_p(cx) \bigr),$$ it follows that $H_p''(x)\ge \tilde{h}''(x)$ for any $x\ge \lambda$, yielding [\[L:H_p\_cx\]](#L:H_p_cx){reference-type="ref" reference="L:H_p_cx"}. Finally, since $H_p(1)=0$ and $H_p$ is convex by [\[L:H_p\_monotone\]](#L:H_p_monotone){reference-type="ref" reference="L:H_p_monotone"}, we have for any $\lambda \le y\le x$, $$\frac{H_p(y)}{y} \le \frac{H_p(y)}{y-1} \le \frac{H_p(x)}{x-1} \le \frac{\lambda}{\lambda-1} \frac{H_p(x)}{x}$$ yielding [\[L:H_p\_c_increasing\]](#L:H_p_c_increasing){reference-type="ref" reference="L:H_p_c_increasing"}. ◻ For $1<\rho\le x$ and $k\ge 1$, denote $$\label{eq:alpha_def} \alpha_{x,\rho}(k) = \inf\Bigl\{\, \sum_{s=0}^{k-1} H_p\bigl( \frac{g_{s+1}}{g_s} \bigr) \mu_s \; :\; g_0=1\le\ldots\le g_{k-1} < \rho \le x \le g_k \,\Bigr\}.$$ This quantity plays a crucial role in providing sufficient condition for the $p$-log-Sobolev inequalities in Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"}. Its definition is partially inspired by an analogous quantity defined in [@bartheroberto] in the continuous setting. **Lemma 23**. *For any $1<\rho\le x$ and $k\ge 1$, $$\label{eq:alpha_1_estimate} \alpha_{x,\rho}(k) \ge \Bigl[ {\sum_{s=0}^{k-1} \mu_s^{-1}} \Bigr]^{-1} \cdot {(\log x)^2}$$ and $$\label{eq:alpha_2_estimate} \alpha_{x,\rho}(k) \ge H_p(x\rho^{-1})\mu_{k-1}.$$* *Proof.* We start with [\[eq:alpha_1\_estimate\]](#eq:alpha_1_estimate){reference-type="eqref" reference="eq:alpha_1_estimate"}. By Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"}, [\[L:H_p\_lower_bound\]](#L:H_p_lower_bound){reference-type="ref" reference="L:H_p_lower_bound"}, $$\begin{aligned} \alpha_{x,\rho}(k) &\ge \inf\Bigl\{ \sum_{s=0}^{k-1} \Bigl( \log \frac{g_{s+1}}{g_s} \Bigr)^2 \mu_s \; :\; g_0=1\le\ldots\le g_{k-1} < \rho \le x \le g_k \Bigr\} \\&\ge \inf\Bigl\{ \sum_{s=0}^{k-1} \lambda_s^2 \mu_s \; :\; \sum_{s=0}^{k-1}\lambda_s \ge \log x \Bigr\} \\&\ge \inf\Bigl\{ \Bigl[\sum_{s=0}^{k-1} \lambda_s \Bigr]^2 \Bigl[ {\sum_{s=0}^{k-1} \mu_s^{-1}} \Bigr]^{-1} \; :\; \sum_{s=0}^{k-1}\lambda_s \ge \log x \Bigr\} = \Bigl[ {\sum_{s=0}^{k-1} \mu_s^{-1}} \Bigr]^{-1} \cdot (\log x)^2, \end{aligned}$$ where in the last estimate we used the Cauchy-Schwarz inequality. We turn to [\[eq:alpha_2\_estimate\]](#eq:alpha_2_estimate){reference-type="eqref" reference="eq:alpha_2_estimate"}. By Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"}, $H_p$ is non-negative and increasing on $[1,\infty)$, whence $$\begin{aligned} \alpha_{x.\rho}(k) &\ge \inf\Bigl\{\, H_p\bigl( \frac{g_{k}}{g_{k-1}} \bigr) \mu_{k-1} \; :\; 1\le g_{k-1} < \rho \le x \le g_k \,\Bigr\} = H_p(x\rho^{-1})\mu_{k-1} \end{aligned}$$ as desired. ◻ # Proof of Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"} {#proof-of-theorem-tmain} ## Sufficient condition Fix $p\in(0,1]$ and assume that $\mu$ satisfies the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} with constant $C_P<\infty$ and that $\hat{C}<\infty$, i.e., the condition [\[eq:main_condition\]](#eq:main_condition){reference-type="eqref" reference="eq:main_condition"} holds. Recall the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$ and that for fully supported measures, $C_\mu \le C_P/2\mu_0 < \infty$, cf. Theorem [Theorem 14](#T:Miclo){reference-type="ref" reference="T:Miclo"} and Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}. We show that $\mu$ satisfies the $p$-log-Sobolev inequality [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} with constant $C$ bounded from above by a quantity depending on $C_\mu$ and $\hat{C}$ only. Define $\rho$ as $$\label{eq:rho_def} \rho = \min\Bigl(\bigl( \frac{1+C_{\mu}}{C_\mu} \bigr)^{1/3}, 2\Bigr).$$ By Corollary [Corollary 21](#C:Barthe-Roberto){reference-type="ref" reference="C:Barthe-Roberto"}, it suffices to show that for any non-decreasing function $f\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps, $$\label{eq:increasing_fn_cond} \sum_{k\colon f(k)\ge \rho\mathop{\mathrm{\mathbb{E}}}f} f(k)\log\big(\frac{f(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu_k \le C_{\rho}\mathop{\mathrm{\mathcal{E}}}_p(f) = C_{\rho} \sum_{k=0}^\infty f(k) H_p\bigl( \frac{f(k+1)}{f(k)} \bigr) \mu_k$$ for some constant $C_{\rho}>0$ independent of $f$. By the homogeneity of [\[eq:increasing_fn_cond\]](#eq:increasing_fn_cond){reference-type="eqref" reference="eq:increasing_fn_cond"}, we can assume that $f(0)=1$. Let us consider such $f$ and denote $$\begin{aligned} \tau_{0}=0, \quad \tau_k = \inf\{\, l > \tau_{k-1} \colon f(l) \ge \rho f(\tau_{k-1}) \,\} \end{aligned}$$ for $k\ge 1$. Since $f$ has a finite number of jumps, then there exists $M\in\mathbb{N}\setminus\{0\}$ such that $\tau_{M-1} < \tau_{M} =\infty$. If $M=1$, then the LHS of [\[eq:increasing_fn_cond\]](#eq:increasing_fn_cond){reference-type="eqref" reference="eq:increasing_fn_cond"} equals zero as $\mathop{\mathrm{\mathbb{E}}}f \ge f(0)=1$ and [\[eq:increasing_fn_cond\]](#eq:increasing_fn_cond){reference-type="eqref" reference="eq:increasing_fn_cond"} holds with any $C_{\rho}>0$. Assume therefore from now on that $M > 1$. For $k\in \{1,\ldots, M-1\}$, denote $$\gamma_k = \rho f(\tau_k) \log_+\big(\frac{\rho f(\tau_k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu[\tau_k,\infty),$$ where $\log_+(x) := \max(\log x, 0)$ and let $$\delta_k = f(\tau_{k-1}) \sum_{l=\tau_{k-1}}^{\tau_k-1} H_p\bigl( \frac{f(l+1)}{f(l)} \bigr) \mu_l.$$ Since $\mathop{\mathrm{\mathbb{E}}}f \ge f(0) = 1$ and as for $k\ge 1$ and $l\in [\tau_k,\tau_{k+1})$, $f(\tau_k)\le f(l) < \rho f(\tau_k)$, we have $$\begin{aligned} \sum_{k\colon f(k)\ge \rho\mathop{\mathrm{\mathbb{E}}}_\mu f} f(k)\log\big(\frac{f(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu_k &\le \sum_{k\colon f(k)\ge \rho} f(k)\log_+\big(\frac{f(k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu_k \\&= \sum_{k=1}^{M-1} \sum_{l=\tau_k}^{\tau_{k+1}-1} f(l)\log_+\big(\frac{f(l)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu_l \\&< \sum_{k=1}^{M-1} \rho f(\tau_k) \log_+\big(\frac{\rho f(\tau_k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\big) \mu[\tau_k,\tau_{k+1}) \le \sum_{k=1}^{M-1} \gamma_k \end{aligned}$$ and as $H_p\ge 0$ and by the monotonicity of $f$, $$\begin{aligned} \sum_{k=0}^\infty f(k) H_p\bigl( \frac{f(k+1)}{f(k)} \bigr) \mu_k % \ge % \sum_{k=1}^{M} \delta_k \ge \sum_{k=1}^{M-1} \delta_k. \end{aligned}$$ Therefore, to prove [\[eq:increasing_fn_cond\]](#eq:increasing_fn_cond){reference-type="eqref" reference="eq:increasing_fn_cond"}, it suffices to show that $$\label{eq:necessary-cond-reduction-gamma-delta} \sum_{k=1}^{M-1} \gamma_k \le C_{\rho}\sum_{k=1}^{M-1}\delta_k.$$ Recall that $H_p(1)=0$, $H_p\ge 0$ and recall the definition [\[eq:alpha_def\]](#eq:alpha_def){reference-type="eqref" reference="eq:alpha_def"} of $\alpha_{x,\rho}$. Consider $g_l=1$ for $l=0,1,\ldots,\tau_{k-1}$ and $g_l=f(l)/f(\tau_{k-1})$ for $l=\tau_{k-1}+1,\ldots,\tau_k$. Then, by the definition of $\delta_k$, we have for any $k\in \{1,\ldots, M-1\}$, $$\begin{aligned} \label{eq:alpha_delta_relation} \begin{split} \alpha_{f(\tau_k)/f(\tau_{k-1}), \rho}(\tau_k) \le \delta_k / f(\tau_{k-1}). \end{split} \end{aligned}$$ For $k=1$, using the monotonicity of $f$, estimate [\[eq:alpha_delta_relation\]](#eq:alpha_delta_relation){reference-type="eqref" reference="eq:alpha_delta_relation"} and the fact that $f(\tau_0)=1 < \rho \le f(\tau_1)$, we get $$\begin{aligned} \label{eq:C1_def} \begin{split} \gamma_1 &\le \rho f(\tau_1) \log_+\big(\frac{\rho f(\tau_1)}{\mu[0,\tau_1) + f(\tau_1)\mu[\tau_1,\infty)}\big) \mu[\tau_1,\infty) \cdot \frac{\delta_1}{\delta_1} \\&\le \rho f(\tau_1) \log_+\big(\frac{\rho f(\tau_1)}{\mu[0,\tau_1) + f(\tau_1)\mu[\tau_1,\infty)}\big) \mu[\tau_1,\infty) \bigl[ {\alpha_{f(\tau_1),\rho}(\tau_1)} \bigr]^{-1} \cdot \delta_1 \\&\le \sup \Bigl\{ {\rho x\log_+\bigl(\frac{\rho x}{\mu[0,l)+x\mu[l,\infty)}\bigr) \mu[l,\infty)} \bigl[ {\alpha_{x,\rho}(l)} \bigr]^{-1} \; : \; x \ge \rho,\; l \ge 1 \Bigr\} \cdot \delta_1 \\&=: C_1\delta_1. \end{split} \end{aligned}$$ For $2\le k \le M-1$ (if such exist), we estimate each $\gamma_k$ based on two cases. To that end, choose $\varepsilon = \rho^{-1}$ and note that by the definition [\[eq:rho_def\]](#eq:rho_def){reference-type="eqref" reference="eq:rho_def"} of $\rho$ and by Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"}, for any $l\ge 1$, $$\label{eq:rho_eps_relation} \varepsilon < 1 < \rho < \rho^2 \le \varepsilon \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) }.$$ Consider the case $f(\tau_k)/f(\tau_{k-1}) > \varepsilon \mu[\tau_{k}-1,\infty)/\mu[\tau_k,\infty)$. Using monotonicity of $f$, Markov's inequality implies that $\frac{\rho f(\tau_k)}{\mathop{\mathrm{\mathbb{E}}}_\mu f}\le \frac{\rho}{\mu[\tau_k,\infty)}$. Using this estimate together with [\[eq:alpha_delta_relation\]](#eq:alpha_delta_relation){reference-type="eqref" reference="eq:alpha_delta_relation"}, we get that $$\begin{aligned} \begin{split} \gamma_k &\le % {\rho f(\tau_k) \log \big(\frac{\rho }{\mu[\tau_k,\infty)}\big) \mu[\tau_k,\infty)} % \\&= {\rho f(\tau_k) \log \big(\frac{\rho }{\mu[\tau_k,\infty)}\big) \mu[\tau_k,\infty)} \cdot \frac{\delta_k}{\delta_k} \\&\le \frac{\rho f(\tau_k)}{f(\tau_{k-1})} { \log \bigl( \frac{\rho}{\mu[\tau_k,\infty)} \bigr) \mu[\tau_k,\infty) } \bigl[ {\alpha}_{f(\tau_k)/f(\tau_{k-1}),\rho}(\tau_k) \bigr]^{-1} \cdot \delta_k \\&\le \sup\Bigl\{ \frac{ \rho x \log\bigl( \frac{\rho}{\mu[l,\infty)} \bigr) \mu[l,\infty) }{ {\alpha}_{x,\rho}(l) } \,:\, l \ge 1,\, x > \varepsilon \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \Bigr\} \cdot \delta_k \\&=: C_2 \delta_k. \end{split} \label{eq:C2_def} \end{aligned}$$ If $f(\tau_k)/f(\tau_{k-1}) \le \varepsilon \mu[\tau_{k}-1,\infty)/\mu[\tau_k,\infty)$, then using the estimate $\log_+(xy)\le \log_+(x)+\log_+(y)$, we split $$\gamma_k \le \underbrace{ \rho f(\tau_k) \log_+\big(\frac{\rho f(\tau_{k-1})}{\mathop{\mathrm{\mathbb{E}}}f}\big) \mu[\tau_k,\infty) }_{=: A} + \underbrace{ \rho f(\tau_k) \log_+\big(\frac{f(\tau_k)}{f(\tau_{k-1})}\big) \mu[\tau_k,\infty) }_{=: B},$$ and we estimate $A$ and $B$ separately using the assumption as follows: $$\begin{aligned} \begin{split}\label{eq:gamma_recursion} A &\le \varepsilon \rho f(\tau_{k-1}) \log_+\big(\frac{\rho f(\tau_{k-1})}{\mathop{\mathrm{\mathbb{E}}}f}\big) \mu[\tau_{k}-1,\infty) \\&\le \varepsilon \rho f(\tau_{k-1}) \log_+\big(\frac{\rho f(\tau_{k-1})}{\mathop{\mathrm{\mathbb{E}}}f}\big) \mu[\tau_{k-1},\infty) =\varepsilon \gamma_{k-1} \end{split} \end{aligned}$$ and, using monotonicity of $f$ and [\[eq:alpha_delta_relation\]](#eq:alpha_delta_relation){reference-type="eqref" reference="eq:alpha_delta_relation"}, $$\begin{aligned} \begin{split} B &= \rho f(\tau_k) \log\big(\frac{f(\tau_k) }{f(\tau_{k-1})}\big) \mu[\tau_k,\infty) \cdot \frac{\delta_k}{\delta_k} \\&\le \frac{\rho f(\tau_k)}{f(\tau_{k-1})} \log\big(\frac{f(\tau_k) }{f(\tau_{k-1})}\big) \mu[\tau_k,\infty) \bigl[ {\alpha}_{f(\tau_k)/f(\tau_{k-1}),\rho}(\tau_k) \bigr]^{-1} \cdot \delta_k \\&\le \sup \Bigl\{ \frac{\rho x \log(x)}{\alpha_{x,\rho}(l)}\mu[l,\infty) \; : \; l \ge 1,\, \rho \le x \le \varepsilon\frac{\mu[{l-1},\infty)}{\mu[l,\infty)} \Bigr\} \cdot \delta_k \\&=: C_3\delta_k. \end{split} \label{eq:C3_def} \end{aligned}$$ Combine estimates [\[eq:C2_def\]](#eq:C2_def){reference-type="eqref" reference="eq:C2_def"}, [\[eq:gamma_recursion\]](#eq:gamma_recursion){reference-type="eqref" reference="eq:gamma_recursion"} and [\[eq:C3_def\]](#eq:C3_def){reference-type="eqref" reference="eq:C3_def"} to get that for $k > 1$, $$\gamma_k \le \varepsilon \gamma_{k-1} + (C_2+C_3) \delta_k.$$ By [\[eq:C1_def\]](#eq:C1_def){reference-type="eqref" reference="eq:C1_def"}, $\gamma_1\le C_1\delta_1$, whence $$\label{eq:gamma-delta-relation} (1-\varepsilon) \sum_{k=1}^{M-1}\gamma_k \le (C_1+C_2+C_3) \sum_{k=1}^{M-1}\delta_k.$$ It suffices therefore to estimate the terms $C_1$, $C_2$ and $C_3$. ## Estimating $C_3$ {#estimating-c_3 .unnumbered} Recall the definition [\[eq:C3_def\]](#eq:C3_def){reference-type="eqref" reference="eq:C3_def"} of $C_3$ and the relation [\[eq:rho_eps_relation\]](#eq:rho_eps_relation){reference-type="eqref" reference="eq:rho_eps_relation"} -- we split the supremum based on whether $x\le\rho^2$ or not. By Lemma [Lemma 23](#L:alpha_estimate){reference-type="ref" reference="L:alpha_estimate"}, eq. [\[eq:alpha_1\_estimate\]](#eq:alpha_1_estimate){reference-type="eqref" reference="eq:alpha_1_estimate"}, and by the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$, $$\begin{gathered} \label{eq:C3_1st_estimate} \sup \Bigl\{ \frac{\rho x \log(x)}{\alpha_{x,\rho}(l)}\mu[l,\infty) \; : \; l \ge 1,\, \rho \le x \le \rho^2 \Bigr\} \\ \le \sup\Bigl\{ \frac{\rho x }{\log x} \bigl[ \sum_{s=0}^{l-1}\mu_s^{-1} \bigr] \mu[l,\infty) \; : \; l \ge 1,\, \rho \le x \le \rho^2 \Bigr\} \le C_\mu\frac{\rho^3}{\log \rho}. \end{gathered}$$ Similarly, by Lemma [Lemma 23](#L:alpha_estimate){reference-type="ref" reference="L:alpha_estimate"}, eq. [\[eq:alpha_2\_estimate\]](#eq:alpha_2_estimate){reference-type="eqref" reference="eq:alpha_2_estimate"}, and by the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$, $$\begin{gathered} \label{eq:C3_2nd_estimate} \sup \Bigl\{ \frac{\rho x \log(x)}{\alpha_{x,\rho}(l)}\mu[l,\infty) \; : \; l \ge 1,\, \rho^2 \le x \le \varepsilon \frac{\mu[{l-1},\infty)}{\mu[l,\infty)} \Bigr\} \\ \le \sup \Bigl\{ \frac{\rho x \log(x)}{H_p(x\rho^{-1})} \frac{\mu[l,\infty)}{\mu_{l-1}} \; : \; l \ge 1,\, \rho^2 \le x \le \varepsilon \frac{\mu[{l-1},\infty)}{\mu[l,\infty)} \Bigr\} \le C_\mu \sup_{x\ge \rho^2} \frac{\rho x \log(x)}{H_p(x\rho^{-1})}. \end{gathered}$$ Combining [\[eq:C3_1st_estimate\]](#eq:C3_1st_estimate){reference-type="eqref" reference="eq:C3_1st_estimate"} and [\[eq:C3_2nd_estimate\]](#eq:C3_2nd_estimate){reference-type="eqref" reference="eq:C3_2nd_estimate"} and using [\[L:H_p\_xlogx\]](#L:H_p_xlogx){reference-type="ref" reference="L:H_p_xlogx"} of Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"} with $\lambda=\rho$ and $x\rho^{-1} \ge \rho$ in place of $x$, we obtain $$\begin{aligned} \label{eq:C3_bound} \begin{split} C_3 &\le C_\mu\Bigl( \frac{\rho^3}{\log \rho} + \max\bigl\{ 1, \frac{\rho\log\rho}{H_p(\rho)}, \frac{1+\log \rho}{H_p'(\rho)} \bigr\} \cdot \sup_{x\ge \rho^2} \frac{\rho^2 \log x}{\log (x\rho^{-1})} \Bigr) \\&= C_\mu\Bigl( \frac{\rho^3}{\log \rho} + 2 \max\bigl\{ 1, \frac{\rho\log\rho}{H_p(\rho)}, \frac{1+\log \rho}{H_p'(\rho)} \bigr\} \cdot \rho^2 \Bigr). \end{split} \end{aligned}$$ ## Estimating $C_2$ {#estimating-c_2 .unnumbered} Recall the definition [\[eq:C2_def\]](#eq:C2_def){reference-type="eqref" reference="eq:C2_def"} of $C_2$ and the relation [\[eq:rho_eps_relation\]](#eq:rho_eps_relation){reference-type="eqref" reference="eq:rho_eps_relation"}. By Lemma [Lemma 23](#L:alpha_estimate){reference-type="ref" reference="L:alpha_estimate"}, eq. [\[eq:alpha_2\_estimate\]](#eq:alpha_2_estimate){reference-type="eqref" reference="eq:alpha_2_estimate"} and using that $\rho\le 2$ by definition, we get $$\begin{aligned} \label{eq:C2_step1} \begin{split} C_2 &\le \sup\Bigl\{ \frac{ \rho x }{ H_p(x\rho^{-1}) } \cdot \log\bigl( \frac{\rho}{\mu[l,\infty)} \bigr) \cdot \frac{ \mu[l,\infty) }{ \mu_{l-1} } \,:\, l \ge 1,\, x > \varepsilon \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \Bigr\} \\&\le \sup\Bigl\{ \frac{ \rho x }{ H_p(x\rho^{-1}) } \cdot \log\bigl( \frac{2}{\mu[l,\infty)} \bigr) \cdot \frac{ \mu[l,\infty) }{ \mu_{l-1} } \,:\, l \ge 1,\, x > \varepsilon \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \Bigr\}. \end{split} \end{aligned}$$ For any $l\ge 1$ and $x > \varepsilon \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) }$, applying [\[L:H_p\_c_increasing\]](#L:H_p_c_increasing){reference-type="ref" reference="L:H_p_c_increasing"} of Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"} with $\rho^{-1}x\ge \rho$ in place of $x$, $\frac{\varepsilon}{\rho} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \ge \rho$ in place of $y$ and $\lambda=\rho$ (recall [\[eq:rho_eps_relation\]](#eq:rho_eps_relation){reference-type="eqref" reference="eq:rho_eps_relation"}) we get that $$\frac{\rho}{\varepsilon} \frac{ \mu[l,\infty) }{ \mu[l-1,\infty) } H_p\bigl( \frac{\varepsilon}{\rho} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \le \frac{\rho^2}{\rho-1} \frac{ H_p(x\rho^{-1}) }{x},$$ which after rearrangement (recall that $\varepsilon=\rho^{-1}$) is equivalent to $$\frac{\rho x}{H_p(x\rho^{-1})} \cdot {\mu[l,\infty)} \le \frac{\rho}{\rho -1} \Big[ H_p\bigl( \frac{1}{\rho^2} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \Big]^{-1} \mu[l-1,\infty),$$ which combined with Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"} allows estimating further [\[eq:C2_step1\]](#eq:C2_step1){reference-type="eqref" reference="eq:C2_step1"} as follows: $$\begin{aligned} \label{eq:C2_step2} \begin{split} C_2 &\le \frac{\rho}{\rho-1} \sup_{l\ge 1} \Bigl\{ \Bigl[ H_p\bigl( \frac{1}{\rho^2} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \Bigr]^{-1} \cdot \log\bigl( \frac{2}{\mu[l,\infty)} \bigr) \cdot \frac{\mu[l-1,\infty)}{\mu_{l-1}} \Bigr\} \\&\le (1+C_\mu) \frac{\rho}{\rho-1} \sup_{l\ge 1} \Bigl\{ \Bigl[ H_p\bigl( \frac{1}{\rho^2} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \Bigr]^{-1} \cdot \log\bigl( \frac{2}{\mu[l,\infty)} \bigr) \Bigr\}. \end{split} \end{aligned}$$ For any $l\ge 1$, applying [\[L:H_p\_cx\]](#L:H_p_cx){reference-type="ref" reference="L:H_p_cx"} of Lemma [Lemma 22](#L:H_p_properties){reference-type="ref" reference="L:H_p_properties"} with $c=\rho^2$, $\lambda=\rho$ and $\frac{1}{\rho^2} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) }\ge \rho$ in place of $x$ (recall [\[eq:rho_eps_relation\]](#eq:rho_eps_relation){reference-type="eqref" reference="eq:rho_eps_relation"}) gives $$H_p\bigl( \frac{1}{\rho^2} \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \ge H_p\bigl( \frac{ \mu[l-1,\infty) }{ \mu[l,\infty) } \bigr) \min\bigl\{ \frac{H_p(\rho)}{H_p(\rho^3)}, \frac{H_p'(\rho)}{\rho^2 H_p'(\rho^3)}, \rho^{-2/p} \bigr\},$$ which combined with [\[eq:C2_step2\]](#eq:C2_step2){reference-type="eqref" reference="eq:C2_step2"} and assumption [\[eq:main_condition\]](#eq:main_condition){reference-type="eqref" reference="eq:main_condition"} results in $$\begin{aligned} \label{eq:C2_bound} C_2 &\le (1+C_\mu) \frac{\rho}{\rho-1} \max\bigl\{ \frac{H_p(\rho^3)}{H_p(\rho)}, \frac{\rho^2H_p'(\rho^3)}{H_p'(\rho)}, \rho^{2/p} \bigr\} \cdot \hat{C} <\infty. \end{aligned}$$ ## Estimating $C_1$. {#estimating-c_1. .unnumbered} Recall the definition [\[eq:C1_def\]](#eq:C1_def){reference-type="eqref" reference="eq:C1_def"} of $C_1$. We use the same ideas as above by considering two cases. For any $l\ge 1$, if $\rho \le x\le \varepsilon \mu[l-1,\infty)/\mu[l,\infty)$, then $$\begin{aligned} \label{eq:C1-C3-estimate} \begin{split} \rho x\log\bigl(\frac{\rho x}{\mu[0,l)+x\mu[l,\infty)}\bigr) \mu[l,\infty) &\le \rho x\log(\rho x) \mu[l,\infty) \\&\le 2\rho x\log( x) \mu[l,\infty) \le 2C_3 \cdot \alpha_{x,\rho}(l), \end{split} \end{aligned}$$ where in the fist and second step we used that $x\ge \rho > 1$ and in the last step we used the definition [\[eq:C3_def\]](#eq:C3_def){reference-type="eqref" reference="eq:C3_def"} of $C_3$. If $x>\varepsilon \mu[l-1,\infty)/\mu[l,\infty)$, then by the definition [\[eq:C2_def\]](#eq:C2_def){reference-type="eqref" reference="eq:C2_def"} of $C_2$, $$\label{eq:C1-C2-estimate} \rho x\log\bigl(\frac{\rho x}{\mu[0,l)+x\mu[l,\infty)}\bigr) \mu[l,\infty) \le \rho x\log\bigl(\frac{\rho }{\mu[l,\infty)}\bigr) \mu[l,\infty) \le C_2 \cdot \alpha_{x,\rho}(l)$$ and thus combining [\[eq:C1-C3-estimate\]](#eq:C1-C3-estimate){reference-type="eqref" reference="eq:C1-C3-estimate"} and [\[eq:C1-C2-estimate\]](#eq:C1-C2-estimate){reference-type="eqref" reference="eq:C1-C2-estimate"}, we arrive at $$\label{eq:C1-C2-C3-estimate} C_1\le C_2+2C_3.$$ ## Final estimate {#final-estimate .unnumbered} Combining [\[eq:C3_bound\]](#eq:C3_bound){reference-type="eqref" reference="eq:C3_bound"} and [\[eq:C2_bound\]](#eq:C2_bound){reference-type="eqref" reference="eq:C2_bound"} together with bounds [\[eq:gamma-delta-relation\]](#eq:gamma-delta-relation){reference-type="eqref" reference="eq:gamma-delta-relation"} and [\[eq:C1-C2-C3-estimate\]](#eq:C1-C2-C3-estimate){reference-type="eqref" reference="eq:C1-C2-C3-estimate"} yields [\[eq:necessary-cond-reduction-gamma-delta\]](#eq:necessary-cond-reduction-gamma-delta){reference-type="eqref" reference="eq:necessary-cond-reduction-gamma-delta"} with $C_\rho=\frac{2C_2+3C_3}{1-\varepsilon}$, which is bounded from above by $$\begin{gathered} \frac{\rho}{\rho-1} \Bigl[ 2 (1+C_\mu) \hat{C} \frac{\rho}{\rho-1} \max\bigl\{ \frac{H_p(\rho^3)}{H_p(\rho)}, \frac{\rho^2H_p'(\rho^3)}{H_p'(\rho)}, \rho^{2/p} \bigr\} \\ + 3 C_\mu\Bigl( \frac{\rho^3}{\log\rho} + 2\rho^2 \max\bigl\{ 1, \frac{\rho\log\rho}{H_p(\rho)}, \frac{1+\log \rho}{H_p'(\rho)} \bigr\} \Bigr) \Bigr] < \infty. \end{gathered}$$ Therefore, we obtain [\[eq:increasing_fn_cond\]](#eq:increasing_fn_cond){reference-type="eqref" reference="eq:increasing_fn_cond"} for any non-decreasing $f\colon\mathbb{N}\to\mathbb{R}_+$ with a finite number of jumps and $C_\rho$ as above. Thus, Corollary [Corollary 21](#C:Barthe-Roberto){reference-type="ref" reference="C:Barthe-Roberto"} implies that $$\mathop{\mathrm{Ent}}_\mu(f) \le [64C_\mu(1+\sqrt{\rho})^2 + 4C_\rho] \mathop{\mathrm{\mathcal{E}}}_p(f)$$ for any $f\colon\mathbb{N}\to\mathbb{R}_+$, as desired. ## Necessary condition For the sake of contradiction, assume that there exists a sequence $0=\tau_0<\tau_1<\ldots$ such that [\[eq:necessary_cond\]](#eq:necessary_cond){reference-type="eqref" reference="eq:necessary_cond"} holds, i.e., $$\beta_k := \Bigl[ H_p\bigl( \frac{\mu[\tau_{k-1},\infty)}{\mu[\tau_k,\infty)} \bigr) \Bigr]^{-1} \cdot \frac{\mu[\tau_{k-1},\infty)}{\mu[\tau_{k}-1,\infty)} \cdot \log\bigl( \frac{2}{\mu[\tau_k,\infty)} \bigr) \to \infty$$ as $k\to\infty$, and that $\mu$ verifies $p$-LS($C$) with some finite constant $C>0$. By Theorem [Theorem 6](#T:MOS){reference-type="ref" reference="T:MOS"}, $\mu$ satisfies the Poincaré inequality [\[eq:Poincare\]](#eq:Poincare){reference-type="eqref" reference="eq:Poincare"} with the same constant $C$ and therefore $C_\mu<\infty$ (recall the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$ and Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}). For $M\ge 1$, set $$f_M = \Bigl[ \sum_{k=0}^{M-1} \frac{ {\bf 1}_{[\tau_k, \tau_{k+1})} }{ \mu[\tau_k,\infty) } \Bigr] + \frac{ {\bf 1}_{[\tau_M, \infty)} }{ \mu[\tau_M,\infty) }.$$ Then $\mathop{\mathrm{\mathbb{E}}}_\mu f_M \le M+1$. Moreover, by Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"}, $\frac{\mu[\tau_k,\tau_{k+1})}{\mu[\tau_k,\infty)}\ge\frac{1}{1+C_\mu}$ for any $k\in\mathbb{N}$, whence $$\mathop{\mathrm{Ent}}_\mu(f_M) \ge -(M+1)\log(M+1) + \frac{1}{1+C_\mu} \sum_{k=1}^{M}\log\bigl( \frac{1}{\mu[\tau_k,\infty)} \bigr).$$ Similarly (recall that $H_p(1)=0$), $$\begin{aligned} \mathop{\mathrm{\mathcal{E}}}_p(f_M) &= \sum_{k=1}^M \mu_{\tau_k-1}f(\tau_k-1)H_p\bigl( \frac{f(\tau_k)}{f(\tau_k-1)} \bigr) \\&\le \sum_{k=1}^M \frac{\mu[\tau_k-1,\infty)}{\mu[\tau_{k-1},\infty)} H_p\bigl( \frac{f(\tau_k)}{f(\tau_k-1)} \bigr) = \sum_{k=1}^M \beta_k^{-1} \log\bigl( \frac{2}{\mu[\tau_k,\infty)} \bigr)\end{aligned}$$ and consequently, since $\mathop{\mathrm{Ent}}_\mu(f_M)\le \frac{C}{4}\mathop{\mathrm{\mathcal{E}}}_p(f_M)$ by assumption, $$\begin{gathered} \label{eq:beta-M-estimate} \sum_{k=1}^M \Bigl[ \log\bigl( \frac{1}{\mu[\tau_k,\infty)} \bigr) \cdot \bigl( 1 - \frac{C(1+C_\mu)}{4\beta_k} \bigr) \Bigr] \le (1+C_\mu)\Bigl[ \frac{C}{4}M(\log 2)\sup_{k}\beta_k^{-1} + (M+1)\log(M+1) \Bigr]\end{gathered}$$ (recall that $\beta_k\to\infty$, whence $\sup_{k}\beta_k^{-1}<\infty$). Let $k_0$ be such that $2\beta_k\ge C(1+C_\mu)$ for every $k\ge k_0$. By Proposition [Proposition 18](#P:poisson_tails){reference-type="ref" reference="P:poisson_tails"}, $\log\bigl( \frac{1}{\mu[\tau_k,\infty)} \bigr)\ge cp\tau_k\log\tau_k$ for some constant $c>0$, whence by [\[eq:beta-M-estimate\]](#eq:beta-M-estimate){reference-type="eqref" reference="eq:beta-M-estimate"} $$\begin{aligned} \sum_{k=k_0}^{M}k\log k &\le \sum_{k=k_0}^{M}\tau_k\log \tau_k \\&\le \frac{1}{cp} \sum_{k=k_0}^M \log\bigl( \frac{1}{\mu[\tau_k,\infty)} \bigr) \le \tilde{c} \cdot (M+1)\log(M+1)\end{aligned}$$ for $M$ big enough and some constant $\tilde{c}>0$ independent of $M$. We arrive at the desired contradiction by taking a limit as $M\to\infty$ and by noting that $\int_{k_0}^M x\log x\,dx \ge \hat{c}M^2\log M$ for $M$ big enough and some constant $\hat{c}>0$ independent of $M$. # Proof of Theorem [Theorem 9](#C:main){reference-type="ref" reference="C:main"} {#proof-of-theorem-cmain} As a counterexample we will take the Conway--Maxwell--Poisson distribution with parameter $\nu>0$, defined as $\mu_\nu(k) = \frac{1}{Z_\nu}(k!)^{-\nu}$ for $k \in \mathbb{N}$, where $Z_\nu = \sum_{k\ge 0} (k!)^{-\nu}$ is the normalizing constant, cf. [@conway1962queuing]. For any $n\in\mathbb{N}$, using the estimate $$\label{eq:factorial-estimate} b! \ge a!(b-a)!$$ valid for $b \ge a \ge 0$ we obtain $$\begin{aligned} \label{eq:CMP_est} \begin{split} \mu_\nu(\{n\}) \le \mu_\nu[n,\infty) &= \frac{1}{Z_\nu} \sum_{k\ge n} \frac{1}{(k!)^\nu} \\&\le \frac{1}{Z_\nu (n!)^\nu} \sum_{k \ge 0} \frac{1}{(k!)^\nu} = \frac{1}{(n!)^\nu} = Z_\nu \mu_\nu(\{n\}). \end{split}\end{aligned}$$ Recall the definition [\[eq:C_mu\]](#eq:C_mu){reference-type="eqref" reference="eq:C_mu"} of $C_\mu$. Using [\[eq:CMP_est\]](#eq:CMP_est){reference-type="eqref" reference="eq:CMP_est"} and [\[eq:factorial-estimate\]](#eq:factorial-estimate){reference-type="eqref" reference="eq:factorial-estimate"}, we obtain for $n\in\mathbb{N}\setminus\{0\}$, $$\begin{aligned} \mu_\nu[n,\infty) \sum_{k=0}^{n-1} \frac{1}{\mu_\nu(\{k\})} \le Z_\nu \mu_\nu(\{n\}) \sum_{k=0}^{n-1} \frac{1}{\mu_\nu(\{k\})} = Z_\nu \frac{\sum_{k=0}^{n-1}(k!)^{\nu}}{(n!)^{\nu}} \le Z_\nu^2,\end{aligned}$$ whence $C_{\mu_\nu}<\infty$ and thus $\mu_\nu$ satisfies the Poincaré inequality [\[eq:poincare_N\]](#eq:poincare_N){reference-type="eqref" reference="eq:poincare_N"} with constant $C_P=8Z_\nu^2$, cf. Theorem [Theorem 14](#T:Miclo){reference-type="ref" reference="T:Miclo"} and Remark [Remark 15](#rk:hardy-poincare){reference-type="ref" reference="rk:hardy-poincare"}. We first show that for any $\nu\in(0,1]$, $\mu_\nu$ verifies the $p$-log-Sobolev inequality [\[eq:pLS_birth_death\]](#eq:pLS_birth_death){reference-type="eqref" reference="eq:pLS_birth_death"} for any $p<\nu$. Fix some $0<p<\nu\le 1$. By the first part of Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"}, it suffices to show that $$\label{eq:CMP_cond} \sup_{n\ge 1} \Bigl\{ \Bigl[ H_p\bigl( \frac{\mu_\nu[n-1,\infty)}{\mu_\nu[n,\infty)} \bigr) \Bigr]^{-1} \cdot \log\bigl( \frac{2}{\mu_\nu[n,\infty)} \bigr) \Bigr\} <\infty .$$ By [\[eq:CMP_est\]](#eq:CMP_est){reference-type="eqref" reference="eq:CMP_est"}, for any $n\ge 1$ $$\begin{aligned} \label{eq:CMP_tails} \begin{split} \frac{n^\nu}{Z_\nu} = \frac{\mu_\nu(\{n-1\})}{\mu_\nu(\{n\})} \frac{1}{Z_\nu} &\le \frac{\mu_\nu(\{n-1\})}{\mu_\nu[n,\infty)} \\&\le \frac{\mu_\nu[n-1,\infty)}{\mu_\nu[n,\infty)} \\&\le Z_\nu \frac{\mu_\nu(\{n-1\})}{\mu_\nu[n,\infty)} \le Z_\nu \frac{\mu_\nu(\{n-1\})}{\mu_\nu(\{n\})} = Z_\nu n^\nu. \end{split}\end{aligned}$$ This, together with Lemma [Lemma 16](#L:rho_lower_bound){reference-type="ref" reference="L:rho_lower_bound"} and the definition [\[eq:H_p\_def\]](#eq:H_p_def){reference-type="eqref" reference="eq:H_p_def"} of $H_p$ implies that $$\label{eq:CMP_Hp1} \Bigl[ H_p\bigl( \frac{\mu_\nu[n-1,\infty)}{\mu_\nu[n,\infty)} \bigr) \Bigr]^{-1} \le \Bigl[ H_p\bigl( \max\bigl\{ \frac{1+C_{\mu_\nu}}{C_{\mu_\nu}}, \frac{n^\nu}{Z_\nu} \bigr\} \bigr) \Bigr]^{-1} \le \frac{C'}{n^{\nu/p}}$$ for any $n\ge 1$ and some big enough constant $C'>0$ (independent on $n$ but dependent on $\nu$ and $p$). By [\[eq:CMP_est\]](#eq:CMP_est){reference-type="eqref" reference="eq:CMP_est"} and Stirling's formula $$\label{eq:stirling} \lim_{n\to\infty} \frac{ \log \bigl(\frac{2}{\mu_\nu[n,\infty)}\bigr) }{ n\log n } = \nu.$$ Combining [\[eq:CMP_Hp1\]](#eq:CMP_Hp1){reference-type="eqref" reference="eq:CMP_Hp1"} with [\[eq:stirling\]](#eq:stirling){reference-type="eqref" reference="eq:stirling"} and recalling that $p<\nu$, we obtain [\[eq:CMP_cond\]](#eq:CMP_cond){reference-type="eqref" reference="eq:CMP_cond"}. Finally, we show that for any $\nu\in(0,1)$, $\mu_\nu$ does not satisfy the $\nu$-log-Sobolev inequality. By Theorem [Theorem 10](#T:main){reference-type="ref" reference="T:main"}, it suffices to show that there exists an increasing sequence $\tau_0<\tau_1<\ldots$ such that $$\label{eq:CMP_cond2} \lim_{n\to\infty} \Big\{ \Bigl[ H_\nu\bigl( \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_n,\infty)} \bigr) \Bigr]^{-1} \cdot \frac{\mu[\tau_{n-1},\infty)}{\mu[\tau_{n}-1,\infty)} \cdot \log\bigl( \frac{2}{\mu[\tau_n,\infty)} \bigr) \Big\} = \infty.$$ We choose $\tau_n = n$ and [\[eq:CMP_cond2\]](#eq:CMP_cond2){reference-type="eqref" reference="eq:CMP_cond2"} becomes $$\label{eq:CMP_cond3} \lim_{n\to\infty} \Big\{ \Bigl[ H_\nu\bigl( \frac{\mu[{n-1},\infty)}{\mu[n,\infty)} \bigr) \Bigr]^{-1} \cdot \log\bigl( \frac{2}{\mu[n,\infty)} \bigr) \Big\} = \infty.$$ Analogously as in [\[eq:CMP_Hp1\]](#eq:CMP_Hp1){reference-type="eqref" reference="eq:CMP_Hp1"}, using [\[eq:CMP_tails\]](#eq:CMP_tails){reference-type="eqref" reference="eq:CMP_tails"} and the definition [\[eq:H_p\_def\]](#eq:H_p_def){reference-type="eqref" reference="eq:H_p_def"} of $H_p$, we get that $$\label{eq:CMP_Hp2} \Bigl[ H_\nu\bigl( \frac{\mu_\nu[n-1,\infty)}{\mu_\nu[n,\infty)} \bigr) \Bigr]^{-1} \ge \Bigl[ H_\nu\bigl( Z_\nu n^\nu \bigr) \Bigr]^{-1} \ge \frac{C''}{n}$$ for any $n\ge 1$ and some small enough constant $C''>0$ (independent on $n$ but dependent on $\nu$). We conclude [\[eq:CMP_cond3\]](#eq:CMP_cond3){reference-type="eqref" reference="eq:CMP_cond3"} by combining [\[eq:stirling\]](#eq:stirling){reference-type="eqref" reference="eq:stirling"} with [\[eq:CMP_Hp2\]](#eq:CMP_Hp2){reference-type="eqref" reference="eq:CMP_Hp2"}. # Acknowledgements {#acknowledgements .unnumbered} I would like to thank Michał Strzelecki for telling me about Problem [Problem 7](#Pr:MOS){reference-type="ref" reference="Pr:MOS"}. I am also very grateful to Radosław Adamczak, Krzysztof Oleszkiewicz and Michał Strzelecki for their valuable remarks on the preliminary versions of this manuscript. [^1]: This can be seen by substituting $f^2 \leftarrow f^p$ in the definition of the $p$-log-Sobolev inequality for $p\in\mathbb{R}\setminus\{0\}$.

arxiv_math

--- abstract: | Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two. author: - Jonas Bergström and Fabien Cléry title: Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two --- # Introduction In this paper, we refine the previously known dimension formulas for spaces $M_{k,j}(\Gamma[2])$ of vector-valued Siegel modular forms of degree 2 and level 2, by determining their isotypical decomposition under the action of ${\mathrm{Sp}}(4,\mathbb F_2)\cong \mathfrak{S}_6$. This extends previous work, see for instance [@Igusa64; @Arakawa; @Tsushima1982; @Tsushima; @Ibukiyama84; @Ibukiyama07; @Wakatsuki; @CvdGG]. In particular, Tsushima gave in [@Tsushima Theorems 2, 3] a formula for the dimension of the space $S_{k,j}(\Gamma[N])$ for any $N$ under the conditions $j\geqslant 1$ and $k\geqslant 5$ or $j=0$ and $k\geqslant 4$. The ranges for $j$ and $k$ in Tsushima's dimension formula for $N=2$ have been slightly extended in [@CvdGG Theorem 12.1]. There is an overview of dimension formulas such as these on the webpage [@Schmidt]. In [@BFvdG], there is a conjectural description of the motivic Euler characteristic, with its isotypical decomposition under the action of $\mathfrak{S}_6$, of any symplectic local system on $\mathcal A_2[2]$ the moduli space of abelian surfaces with a full level two structure. These conjectures were later proven in [@Roesner]. In particular, this gives us the integer-valued Euler characteristic of an isotypical component under $\mathfrak{S}_6$ of any local system on $\mathcal A_2[2]$ as a sum of a well-known value (in terms of dimensions of spaces of elliptic modular cusp forms) plus four times the dimension of the (isotypical component of the) vector space of Siegel modular cusp forms of degree 2 and level 2, see Theorem [Theorem 7](#thm-main){reference-type="ref" reference="thm-main"}. In Section [4](#sec-euler){reference-type="ref" reference="sec-euler"} we then find an effective formula to compute these integer-valued Euler characteristics. This is achieved by stratifying the moduli space $\mathcal A_2[2]$ in terms of the automorphism groups of principally polarized abelian surfaces, which are Jacobians of smooth projective curves of genus $2$, or products of elliptic curves. By computing the action of these automorphism groups and of $\mathfrak{S}_6$, on the first cohomology group of the corresponding abelian surfaces, we can find a formula for the integer-valued Euler characteristic, see Equation [\[eq-numeric\]](#eq-numeric){reference-type="eqref" reference="eq-numeric"}. This is a method previously used for instance in [@Getzler]. In Section [2](#sec-smf){reference-type="ref" reference="sec-smf"} we give an overview of the Siegel modular forms we are interested in, together with a short description of the Arthur packets for $\mathrm{GSp}(4)$. Then, in Section [3](#sec-sv){reference-type="ref" reference="sec-sv"} and Section [5](#sec-vv){reference-type="ref" reference="sec-vv"} we include isotypical decompositions of the spaces of Siegel modular forms of degree $2$ and level $2$ to give a comprehensive reference for these results. Computer programs, written in Sage, which compute all results of this paper, are provided on a GitHub repository [@BC]. Tables with some of the results of this paper can also be found on the webpage [@BCFvdGweb]. # Siegel modular forms {#sec-smf} The level 2 congruence subgroups we are concerned with are the following ones $$\Gamma[2] = \left \{ \gamma \in \Gamma : \gamma \equiv 1_4 \bmod 2 \right \}, \quad \Gamma_1[2] = \left \{ \gamma \in \Gamma: \gamma \equiv \left(\begin{smallmatrix}1_2 & * \\ 0 & 1_2\end{smallmatrix}\right) \bmod 2 \right \}, \quad \Gamma_0[2] = \left \{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \Gamma: c \equiv 0 \bmod 2 \right \},$$ where $\Gamma={\mathrm{Sp}}(4,\mathbb{Z})=\{\gamma \in {\mathrm{GL}}(4,\mathbb{Z}): \gamma^t J \gamma =J\}$ with $J=\left(\begin{smallmatrix}0 & 1_2 \\ -1_2 & 0\end{smallmatrix}\right)$ and $1_n$ the identity matrix of size $n$. We clearly have the following inclusions $\Gamma[2] < \Gamma_1[2] < \Gamma_0[2] < \Gamma$ and the successive quotients can be identified as follows $$\Gamma_1[2]/\Gamma[2] \cong (\mathbb{Z}/2\mathbb{Z})^3, \quad \Gamma_0[2]/\Gamma[2] \cong \mathbb{Z}/2\mathbb{Z}\times \mathfrak{S}_4, \quad \Gamma_0[2]/\Gamma_1[2] \cong \mathfrak{S}_3, \quad \Gamma/\Gamma[2] \cong \mathfrak{S}_6,$$ with $\mathfrak{S}_n$ the symmetric group on $n$ letters. As usual, theses groups act on the Siegel upper half space $\mathfrak{H}_2$ of degree 2: $\mathfrak{H}_2=\{\tau \in \text{Mat}(2\times 2,\mathbb{C}) : \tau^t=\tau, \text{Im}(\tau)>0 \}$ via $\tau \mapsto \gamma\,\tau=(a\tau+b)(c\tau+d)^{-1}$. More details on the orbifolds of the action of the previous groups can be found in [@CvdGG Section 2]. We let $G$ be one of the groups $\Gamma[2],\Gamma_1[2], \Gamma_0[2]$ or $\Gamma$. For any integer $k$ and non-negative integer $j$, the space of modular forms of weight $(k,j)$ on $G$ is denoted by $M_{k,j}(G)$ and is defined by $$\begin{aligned} &M_{k,j}(G)= \{ f \colon \mathfrak{H}_2 \to \mathbb{C}^{j+1}\, \text{holomorphic}\, | \, \\ &f((a\tau+b)(c\tau +d)^{-1}) = \text{Sym}^j(c\tau +d)\otimes \det(c\tau +d)^{k}f(\tau)\, \text{ for all }\, \gamma=\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)\in G \}.\end{aligned}$$ The subspace of cusp forms of $M_{k,j}(G)$ will be denoted by $S_{k,j}(G)$, this is the kernel of the (global) Siegel $\Phi$-operator. Let us make a couple of easily verified remarks. Firstly, since $-1_4$ belongs to the group $\Gamma[2]$, we have that $M_{k,j}(\Gamma[2])=\{0\}$ if $j$ is odd. This can be directly read off from the functional equation satisfied by any element of $M_{k,j}(\Gamma[2])$. Therefore, from now on, we assume that *$j$ is even*. Secondly, if $k$ is odd then $M_{k,j}(\Gamma[2])=S_{k,j}(\Gamma[2])$: let $\Gamma(2)$ be the principal congruence subgroup of level $2$ of ${\mathrm{SL}}(2,\mathbb{Z})$, the (global) Siegel $\Phi$-operator maps $M_{k,j}(\Gamma[2])$ to $M_{j+k}(\Gamma(2))^{\oplus 15}$ (note $15$ is the number of $1$-dimensional cusps of the group $\Gamma[2]$) and since $-1_2$ belongs to $\Gamma(2)$, the space $M_{j+k}(\Gamma(2))$ reduces to ${0}$ when $k$ is odd ($j$ is even). These two facts also hold for the groups $\Gamma_1[2], \Gamma_0[2]$ and $\Gamma$. A less easy fact is the generalisation of the Koecher principle to vector-valued Siegel modular forms, see [@Freitag79 Satz 1], which implies that $$M_{k,j}(\Gamma[2])=\left\{0\right\} \text{ for any } k<0 \text{ for any } j.$$ The Petersson inner product provides an orthogonal decomposition $$\label{OrthoDecom} M_{k,j}(G)=E_{k,j}(G)\oplus S_{k,j}(G).$$ We call $E_{k,j}(G)$ the space of Eisenstein series. The decomposition ([\[OrthoDecom\]](#OrthoDecom){reference-type="ref" reference="OrthoDecom"}) can be refined according to the classification of automorphic representations of G${\mathrm{Sp}}(4)$: Arthur packets. There are six different types of Arthur packets (see [@Arthur], or [@Schmidt18 pp. 3088--3089]): - $\textbf{(G)}$: general type. They can only appear in $S_{k,j}(G)$. - $\textbf{(Y)}$: Yoshida type. They can only appear in $S_{k,j}(G)$. Modular forms of this type are also called **Yoshida lifts**. - $\textbf{(Q)}$: Soudry type (Klingen parabolic). They can only appear in $E_{k,j}(G)$. Modular forms of this type are also called **Klingen-Eisenstein series**. - $\textbf{(P)}$: Saito-Kurokawa type (Siegel parabolic). They can only appear in $S_{k,0}(G)$, i.e. they are scalar-valued cusp forms. Modular forms of this type are also called **Saito-Kurokawa lifts**. - $\textbf{(B)}$: Howe--Piatetski-Shapiro type (Borel parabolic). They can only appear in the space of cusp forms, but they do not appear in $S_{k,j}(G)$. - $\textbf{(F)}$: Finite type. They can only appear in $E_{k,j}(G)$. Modular forms of this type are also called **Siegel-Eisenstein series**. So we have $$\label{DecompPackets} E_{k,j}(G)=E^{\textbf{(F)}}_{k,j}(G)\oplus E^{\textbf{(Q)}}_{k,j}(G) \quad \text{and} \quad S_{k,j}(G)= S^{\textbf{(G)}}_{k,j}(G)\oplus S^{\textbf{(P)}}_{k,j}(G)\oplus S^{\textbf{(Y)}}_{k,j}(G).$$ Since the group $\Gamma[2]$ is a normal subgroup of $\Gamma$ (it is the kernel of the reduction modulo $2$), we get an action of $\Gamma$ on the space $M_{k,j}(\Gamma[2])$ $$\begin{smallmatrix} {\mathrm{Sp}}(4,\mathbb{Z}) \times M_{k,j}(\Gamma[2]) & \to & M_{k,j}(\Gamma[2])\\ (\gamma,f) & \mapsto & f\vert_{k,j} \gamma^{-1} \end{smallmatrix}$$ From this action, we deduce a group homomorphism $$\begin{smallmatrix} \Gamma & \to & {\mathrm{GL}}(M_{k,j}(\Gamma[2]))\\ \gamma & \mapsto & \left( \begin{smallmatrix} M_{k,j}(\Gamma[2]) & \to & M_{k,j}(\Gamma[2])\\ f & \mapsto & f\vert_{k,j} \gamma^{-1} \end{smallmatrix} \right) \end{smallmatrix}$$ whose kernel obviously contains the group $\Gamma[2]$. So the previous homomorphism factors through the group $\Gamma[2]$ and we obtain a group homomorphism $$\Gamma/\Gamma[2] \cong {\mathrm{Sp}}(4,\mathbb F_2) \cong \mathfrak{S}_6 \to {\mathrm{GL}}(M_{k,j}(\Gamma[2]))$$ i.e. a representation of the group $\mathfrak{S}_6$ on the space $M_{k,j}(\Gamma[2])$. Note that the second isomorphism is ambiguous due to the outer automorphism of $\mathfrak{S}_6$ so we need to fix this isomorphism. We fix this isomorphism as follows: $\mathfrak{S}_6=\langle(12),(123456)\rangle$ and as in [@CvdGG Equation (3.2)] (see also [@Igusa64 pp. 398-399]), we set $$(12) \mapsto \left( \begin{smallmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right) \bmod 2 \quad \text{and} \quad (123456) \mapsto \left( \begin{smallmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ -1 & 1 & 0 & 1 \end{smallmatrix} \right) \bmod 2.$$ The irreducible representations of $\mathfrak{S}_n$ correspond bijectively with the partitions of $n$. The representation of the symmetric group $\mathfrak{S}_n$ corresponding with the partition $\varpi$ will be denoted by $s[\varpi]$, with $s[n]$ the trivial one and $s[1^n]$ the alternating one. [\[Irrep_S\_6\]]{#Irrep_S_6 label="Irrep_S_6"} $[\varpi]$ $[6]$ $[5,1]$ $[4,2]$ $[4,1^2]$ $[3^2]$ $[3,2,1]$ $[3,1^3]$ $[2^3]$ $[2^2,1^2]$ $[2,1^4]$ $[1^6]$ ------------------ ------- --------- --------- ----------- --------- ----------- ----------- --------- ------------- ----------- --------- $\dim s[\varpi]$ 1 5 9 10 5 16 10 5 9 5 1 [\[table:IrrepList\]]{#table:IrrepList label="table:IrrepList"} For a representation $V$ of the symmetric group $\mathfrak{S}_n$ we write $$\label{dim_Sn} \dim_{\mathfrak{S}_n} V=\sum_{\varpi} m_{s[\varpi]}(V) \cdot s[\varpi] \in \mathbb{Z}[\mathfrak{S}_n],$$ where $\mathbb{Z}[\mathfrak{S}_n]$ is the representation ring and $m_{\varpi}(V)$ is the multiplicity of the representation $s[\varpi]$ appearing in $V$. We call the right hand side of ([\[dim_Sn\]](#dim_Sn){reference-type="ref" reference="dim_Sn"}) the isotypical decomposition of $V$. Knowing the isotypical decomposition of a space $M_{k,j}(\Gamma[2])$ gives us all the information we want about the spaces $M_{k,j}(\Gamma_1[2])$, $M_{k,j}(\Gamma_0[2])$ and $M_{k,j}(\Gamma)$ by representation theory: to an isotypical decomposition $\dim_{\mathfrak{S_6}}M_{k,j}(\Gamma[2])= m_{s[6]}\,s[6]+m_{s[5,1]}\,s[5,1]+\cdots+m_{s[1^6]}\,s[1^6]$ contributes $$\begin{aligned} &\dim_{\mathfrak{S_3}}M_{k,j}(\Gamma_1[2]) = (m_{s[6]}+m_{s[4,2]}+m_{s[2^3]})s[3]+ (m_{s[5,1]}+m_{s[4,2]}+m_{s[3,2,1]})s[2,1]+ (m_{s[4,1^2]}+m_{s[3^2]})s[1^3], \\ &\dim M_{k,j}(\Gamma_0[2])=m_{s[6]}+m_{s[4,2]}+m_{s[2^3]},\\ &\dim M_{k,j}(\Gamma)=\dim M_{k,j}(\Gamma[2])^{s[6]}=m_{s[6]},\\ &\dim M_{k,j}(\Gamma,\varepsilon)=\dim M_{k,j}(\Gamma[2])^{s[1^6]}=m_{s[1^6]}.\end{aligned}$$ Here, $\epsilon$ denotes the unique non-trivial character of $\Gamma$, see [@CFvdG19 Section 12] for a brief description of this character. Therefore we focus on the spaces $M_{k,j}(\Gamma[2])$ in the sequel. In the case of scalar-valued modular forms, the previous decompositions allow us to recover the results given in [@Roy_Schmidt_Yi Appendix A] for the groups $\Gamma, \Gamma_0[2]$ and $\Gamma[2]$. In the case of scalar-valued modular forms we write $M_{k}(\Gamma[2])=M_{k,0}(\Gamma[2])$ and $S_{k}(\Gamma[2])=S_{k,0}(\Gamma[2])$. # Isotypical decompositions in the scalar-valued case {#sec-sv} By the Koecher principle, we know that $M_{k}(\Gamma[2])=\{0\}$ for $k<0$. The following theorem is due to Igusa, see [@Igusa64 p.398]. **Theorem 1** (Igusa). *We have $$\dim M_{k}(\Gamma[2])= \begin{cases} \frac{(k+1)(k^2+2k+12)}{12} \hspace{20pt} \text{if}\,\,\, k\geqslant 0 \,\,\, even\\ \dim M_{k-5}(\Gamma[2]) \hspace{13pt} \text{if} \,\,\,\, k\geqslant 1\,\,\, \text{odd}. \end{cases}$$* For $k$ odd, the last equality comes from $M_{k}(\Gamma[2])=S_{k}(\Gamma[2])=\chi_5 \cdot M_{k-5}(\Gamma[2])$ where $\chi_5$ denotes the unique cusp form, up to a multiplicative constant, generating the space $S_{5}(\Gamma[2])$. In fact, Igusa did more than computing the dimension of the spaces $M_{k}(\Gamma[2])$. He also computed the characters of $\mathfrak{S}_6$ on the space $M_{k}(\Gamma[2])$ (see [@Igusa64 Theorem 2]) and he showed that as $\mathfrak{S}_6$-representation for $k$ even we have $$\dim_{\mathfrak{S}_6} M_{k}(\Gamma[2])={\text{Sym}}^{k/2}(s[2^3])- \begin{cases} 0 \hspace{2.75cm}\text{ if } k \in \{0,2,4,6\}\\ {\text{Sym}}^{k/2-4}(s[2^3]) \,\,\,\,\,\,\,\,\,\,\,\, \text{ if } k \geqslant 8, \end{cases}$$ where we put ${\mathrm{Sym}}^0(s[\varpi])=s[n]$ for any irreducible representation $s[\varpi]$ of $\mathfrak{S}_n$. Note that the relation appearing in weight $8$ defines the Igusa quartic. From the results of Igusa, we deduce the generating series for the multiplicity of the irreducible representations of $\mathfrak{S}_6$ in $M_{k}(\Gamma[2])$: $s[\varpi]$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(M_k(\Gamma[2]))\,t^k$\ $s[6]$ & $\frac{1+t^{35}}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ $s[5,1]$ & $\frac{t^{11}(1+t)}{((1-t^4)(1-t^6))^2}$\ $s[4,2]$ & $\frac{t^4(1+t^{15})}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[4,1^2]$ & $\frac{t^{11}(1+t^4)}{(1-t)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3^2]$ & $\frac{t^7(1+t^{13})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3,2,1]$ & $\frac{t^8(1-t^8)}{(1-t^2)^2(1-t^5)(1-t^6)^2}$\ $s[3,1^3]$ & $\frac{t^6(1+t^4+t^{11}+t^{15})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^3]$ & $\frac{t^2(1+t^{23})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^2,1^2]$ & $\frac{t^9}{(1-t^2)(1-t^4)^2(1-t^5)}$\ $s[2,1^4]$ & $\frac{t^6(1+t^{11})}{((1-t^4)(1-t^6))^2}$\ $s[1^6]$ & $\frac{t^5(1+t^{25})}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ **Remark 2**. A couple of sanity checks. Firstly, $$\sum_{k \geqslant 0} \sum_{s[\varpi]}\dim (s[\varpi]) m_{{s[\varpi]}}(M_k(\Gamma[2]))\,t^k=\sum_{k \geqslant 0}\dim M_{k}(\Gamma[2])\,t^k=\frac{(1+t^2)(1+t^4)(1+t^5)}{(1-t^2)^4},$$ which is in agreement with Theorem [Theorem 1](#Igusa){reference-type="ref" reference="Igusa"}. Secondly, the generating series for the dimension of spaces of modular forms on $\Gamma_{0}[2]$ is given by (see [@Roy_Schmidt_Yi Appendix A.1] and the references therein) $$\sum_{k \geqslant 0}\dim M_{k}(\Gamma_0[2])\,t^k =\frac{1+t^{19}}{(1-t^2)(1-t^4)^2(1-t^6)}.$$ We checked that by adding the generating series for the multiplicities of the irreducible representations $s[6], s[4,2]$ and $s[2^3]$ we recover this formula. Let us give the first few isotypical decompositions of $M_k(\Gamma[2])$, we put $d=\dim M_k(\Gamma[2])$ $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} s[\varpi] & s[6] & s[5,1] & s[4,2] & s[4,1^2] & s[3^2] & s[3,2,1] & s[3,1^3] & s[2^3] & s[2^2,1^2] & s[2,1^4] & s[1^6] & \\ \dim s[\varpi]& 1& 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1&\\ \hline \hline k & & & & & & & & & & & & d \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 5\\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 15\\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 6 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 35\\ 7 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ 8 & 1 & 0 & 3 & 0 & 0 & 1 & 1 & 3 & 0 & 0 & 0 & 69\\ 9 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 15\\ 10 & 2 & 0 & 3 & 0 & 0 & 2 & 3 & 4 & 0 & 2 & 0 & 121\\ 11 & 0 & 1 & 0 & 1 & 2 & 0 & 0 & 0 & 1 & 0 & 1 & 35 \end{array}$$ Next, we give the dimension of the various pieces of the space $M_{k}(\Gamma[2])$ as in ([\[OrthoDecom\]](#OrthoDecom){reference-type="ref" reference="OrthoDecom"}) and ([\[DecompPackets\]](#DecompPackets){reference-type="ref" reference="DecompPackets"}) and also their isotypical decomposition. We distinguish two cases according to the parity of $k$. ## Isotypical decomposition of $M_{2k+1}(\Gamma[2])$ We already have seen that $M_{2k+1}(\Gamma[2])=S_{2k+1}(\Gamma[2])$ and from Theorem [Theorem 1](#Igusa){reference-type="ref" reference="Igusa"} we deduce $\dim S_{1}(\Gamma[2])=\dim S_{3}(\Gamma[2])=0$ and $$\dim S_{2k+1}(\Gamma[2])= (2k^3-9k^2+19k-15)/3 \quad \text{for} \quad k\geqslant 2.$$ Therefore the generating series for $\dim S_{2k+1}(\Gamma[2])$ is given by $$\sum_{k\geqslant 0} \dim S_{2k+1}(\Gamma[2])\, t^{2k+1}= \frac{t^5(1+t^2+t^4+t^{6})}{(1-t^2)^4}.$$ We have seen that for $k\geqslant 0$ we have $M_{2k+1}(\Gamma[2])=S_{2k+1}(\Gamma[2])=\chi_5\cdot M_{2k-4}(\Gamma[2])$ and since the cusp form $\chi_5$ is $\mathfrak{S}_6$-anti-invariant (i.e. it occurs in the alternating representation $s[1^6]$) we get $$\dim_{\mathfrak{S}_6}M_{2k+1}(\Gamma[2])=\dim_{\mathfrak{S}_6}S_{2k+1}(\Gamma[2])= s[1^6] \otimes \dim_{\mathfrak{S}_6} M_{2k-4}(\Gamma[2]).$$ From the generating series of the multiplicities of the irreducible representations of $\mathfrak{S}_6$ previously given, we deduce $s[\varpi]$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))\,t^{2k+1}$\ $s[6]$ & $\frac{t^{35}}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ $s[5,1]$ & $\frac{t^{11}}{((1-t^4)(1-t^6))^2}$\ $s[4,2]$ & $\frac{t^{19}}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[4,1^2]$ & $\frac{t^{11}(1+t^4)}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3^2]$ & $\frac{t^7}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3,2,1]$ & $\frac{t^{13}(1+t^2+t^4+t^6)}{(1-t^2)(1-t^6)^2(1-t^{10})}$\ $s[3,1^3]$ & $\frac{t^{17}(1+t^4)}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^3]$ & $\frac{t^{25}}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^2,1^2]$ & $\frac{t^9}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[2,1^4]$ & $\frac{t^{17}}{((1-t^4)(1-t^6))^2}$\ $s[1^6]$ & $\frac{t^5}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ Let us give the first few isotypical decompositions of $S_{2k+1}(\Gamma[2])$, we put $d=\dim S_{2k+1}(\Gamma[2])$. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} s[\varpi] & s[6] & s[5,1] & s[4,2] & s[4,1^2] & s[3^2] & s[3,2,1] & s[3,1^3] & s[2^3] & s[2^2,1^2] & s[2,1^4] & s[1^6] & \\ \dim s[\varpi]& 1& 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1&\\ \hline \hline 2k+1 & & & & & & & & & & & & d \\ \hline 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 7 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ 9 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 15\\ 11 & 0 & 1 & 0 & 1 & 2 & 0 & 0 & 0 & 1 & 0 & 1 & 35\\ 13 & 0 & 0 & 0 & 1 & 3 & 1 & 0 & 0 & 3 & 0 & 1 & 69\\ 15 & 0 & 2 & 0 & 3 & 4 & 2 & 0 & 0 & 3 & 0 & 2 & 121 \end{array}$$ We further refine these formulas according to the decomposition ([\[DecompPackets\]](#DecompPackets){reference-type="ref" reference="DecompPackets"}). Conjecture 6.6 of [@BFvdG], now proved by Rösner, see [@Roesner Section 5], tells us that only the Arthur packets $\textbf{(P)}$ and $\textbf{(G)}$ can occur in $S_{2k+1}(\Gamma[2])$. Hence, $$S_{2k+1}(\Gamma[2])= S^{\textbf{(G)}}_{2k+1}(\Gamma[2])\oplus S^{\textbf{(P)}}_{2k+1}(\Gamma[2]).$$ Moreover this conjecture gives the isotypical decomposition of $S^{\textbf{(P)}}_{2k+1}(\Gamma[2])$: $$\dim_{\mathfrak{S}_6} S^{\textbf{(P)}}_{2k+1}(\Gamma[2])= d^{-}_{2,4k} s[5,1] + d_{4,4k} s[3^2] + d^{+}_{2,4k} s[1^6]$$ where $$\label{def_d_k} d_{N,k}= \dim S_{k}(\Gamma_0(N))^{\text{new}} \quad \text{and} \quad d^{\pm}_{N,k}=\dim S^{\pm}_{k}(\Gamma_0(N))^{\text{new}}.$$ Here $\Gamma_0(N)$ denotes the Hecke congruence subgroup of level $N$ of ${\mathrm{SL}}(2,\mathbb{Z})$ and $S^{\pm}_{k}(\Gamma_0(N))^{\text{new}}$ denotes the space of new cusp forms of weight $k$ on $\Gamma_0(N)$ with eigenvalues $\pm 1$ for the Fricke involution. Note that for $N=1$, we have $d_{1,k}= \dim S_{k}(\Gamma_0(1))^{\text{new}}=\dim S_k({\mathrm{SL}}(2,\mathbb{Z}))$. The dimension of the space $S_{k}(\Gamma_0(N))^{\text{new}}$ is classical, see [@Stein Chapter 6] and for example we have: $$\label{dimnew2new4} \dim S_{4k}(\Gamma_0(2))^{\text{new}}=k-1-2\,\lfloor k/3 \rfloor \quad \text{and} \quad \dim S_{4k}(\Gamma_0(4))^{\text{new}}=\lfloor k/3 \rfloor \quad \text{for} \quad k\geqslant 1.$$ For $k>2$, the dimensions of $S^{\pm}_{k}(\Gamma_0(2))^{\text{new}}$ are given by (see [@MartinK Theorem 2.2]) $$d^{\pm}_{2,k}= \left\{ \begin{tabular}{ccc} $(d_{2,k}\pm 1)/2$ & \text{if} & $k\equiv 0 \mod 8$\\ $(d_{2,k}\mp 1)/2$ & \text{if} & $k\equiv 2 \mod 8$\\ $d_{2,k}/2$ & \text{if} & $k\equiv 4, 6 \mod 8$. \end{tabular} \right.$$ So the generating series for the multiplicities of the irreducible representations $s[5,1]$, $s[3^2]$ and $s[1^6]$ in $S^{\textbf{(P)}}_{2k+1}(\Gamma[2])$ for $k\geqslant 0$ are given by $s[\varpi]$ & $s[5,1]$ & $s[3^2]$ & $s[1^6]$\ $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k+1}(\Gamma[2]))\,t^{2k+1}$ & $\frac{t^{11}}{(1-t^4)(1-t^6)}$ & $\frac{t^7}{(1-t^2)(1-t^6)}$ & $\frac{t^5}{(1-t^4)(1-t^6)}$ Keeping in mind that $\dim s[5,1]=\dim s[3^2]=5$ and $\dim s[1^6]=1$, we get $$\sum_{k\geqslant 0} \dim S^{\textbf{(P)}}_{2k+1}(\Gamma[2]) t^{2k+1}= \frac{t^5+5t^7+5t^9+5t^{11}}{(1-t^4)(1-t^6)}.$$ From this we deduce the generating series for $\dim S^{\textbf{(G)}}_{2k+1}(\Gamma[2])$ $$\begin{aligned} \sum_{k\geqslant 0} \dim S^{\textbf{(G)}}_{2k+1}(\Gamma[2])\,t^{2k+1}=& \sum_{k\geqslant 0} \bigl(\dim S_{2k+1}(\Gamma[2])-\dim S^{\textbf{(P)}}_{2k+1}(\Gamma[2])\bigr)\,t^{2k+1} = \frac{t^9(t^8-2t^6+10t^4+6t^2+9)}{(1-t^2)^3(1-t^6)(1+t^2)}.\end{aligned}$$ We also deduce the generating series for the multiplicities of the irreducible representations in $S^{\textbf{(G)}}_{2k+1}(\Gamma[2])$ for $k\geqslant 0$ - for $s[\varpi]\in \left\{ s[6], s[4,2], s[4,1^2], s[3,2,1], s[3,1^3], s[2^3], s[2^2,1^2], s[2,1^4] \right\}$ we have $$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))$$ - for $s[\varpi] \in \left\{ s[5,1], s[3^2], s[1^6] \right\}$ we have $$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))= m_{{s[\varpi]}}(S_{2k+1}(\Gamma[2]))- m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k+1}(\Gamma[2]))$$ so $s[\varpi]$ & $s[5,1]$ & $s[3^2]$ & $s[1^6]$\ $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k+1}(\Gamma[2]))\,t^{2k+1}$ & $\frac{t^{15}(1+t^2-t^6)}{((1-t^4)(1-t^6))^2}$ & $\frac{t^{11}(1+t^{8}-t^{12})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$ & $\frac{t^{15}(1+t^{2}-t^{12})}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$ The next table gives the first few isotypical decompositions of $S_{2k+1}(\Gamma[2])$, we indicate the multiplicities of Saito-Kurakawa lifts in blue and those of the general type in black, we put ${\color{blue} d_P}=\dim S^{\textbf{(P)}}_{k}(\Gamma[2])$ and $d_G=\dim S^{\textbf{(G)}}_{k}(\Gamma[2])$ $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} s[\varpi] & s[6] & s[5,1] & s[4,2] & s[4,1^2] & s[3^2] & s[3,2,1] & s[3,1^3] & s[2^3] & s[2^2,1^2] & s[2,1^4] & s[1^6] & \\ \dim s[\varpi]& 1& 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1&\\ \hline \hline 2k+1 & & & & & & & & & & & & d_G+{\color{blue} d_P} \\ \hline 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0+{\color{blue} 0}\\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0+{\color{blue} 0}\\ 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\color{blue} 1} & 0+{\color{blue} 1} \\ 7 & 0 & 0 & 0 & 0 & {\color{blue} 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0+ {\color{blue} 5}\\ 9 & 0 & 0 & 0 & 0 & {\color{blue} 1} & 0 & 0 & 0 & 1 & 0 & {\color{blue} 1} & 9+ {\color{blue} 6}\\ 11 & 0 & {\color{blue} 1} & 0 & 1 & 1+ {\color{blue} 1} & 0 & 0 & 0 & 1 & 0 & {\color{blue} 1} & 24+ {\color{blue} 11}\\ 13 & 0 & 0 & 0 & 1 & 1+ {\color{blue} 2} & 1 & 0 & 0 & 3 & 0 & {\color{blue} 1} & 58+ {\color{blue} 11}\\ 15 & 0 & 1+ {\color{blue} 1} & 0 & 3 & 2+ {\color{blue} 2} & 2 & 0 & 0 & 3 & 0 & 1+ {\color{blue} 1} & 105+ {\color{blue} 16} \end{array}$$ ## Isotypical decomposition of $M_{2k}(\Gamma[2])$ We start by giving the generating series for the multiplicities of the irreducible representations of $\mathfrak{S}_6$ in $M_{2k}(\Gamma[2])$. $s[\varpi]$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))\,t^{2k}$\ $s[6]$ & $\frac{1}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ $s[5,1]$ & $\frac{t^{12}}{((1-t^4)(1-t^6))^2}$\ $s[4,2]$ & $\frac{t^4}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[4,1^2]$ & $\frac{t^{12}(1+t^4)}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3^2]$ & $\frac{t^{20}}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3,2,1]$ & $\frac{t^8(1+t^2+t^4+t^8)}{(1-t^2)(1-t^6)^2(1-t^{10})}$\ $s[3,1^3]$ & $\frac{t^6(1+t^4)}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^3]$ & $\frac{t^2}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^2,1^2]$ & $\frac{t^{14}}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[2,1^4]$ & $\frac{t^6}{((1-t^4)(1-t^6))^2}$\ $s[1^6]$ & $\frac{t^{30}}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ **Remark 3**. As a sanity check, we verify $$\sum_{k \geqslant 0} \sum_{s[\varpi]}\dim (s[\varpi]) m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))\,t^{2k}=\frac{(1+t^2)(1+t^4)}{(1-t^2)^4} =\sum_{k \geqslant 0}\dim M_{2k}(\Gamma[2])\,t^{2k}$$ in agreement with Theorem [Theorem 1](#Igusa){reference-type="ref" reference="Igusa"}. Let us give the first few isotypical decompositions of $M_{2k}(\Gamma[2])$, we put $d=\dim M_{2k}(\Gamma[2])$ $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} s[\varpi] & s[6] & s[5,1] & s[4,2] & s[4,1^2] & s[3^2] & s[3,2,1] & s[3,1^3] & s[2^3] & s[2^2,1^2] & s[2,1^4] & s[1^6] & \\ \dim s[\varpi]& 1& 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1&\\ \hline \hline k & & & & & & & & & & & & d \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 5\\ 4 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 15\\ 6 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 35\\ 8 & 1 & 0 & 3 & 0 & 0 & 1 & 1 & 3 & 0 & 0 & 0 & 69\\ 10 & 2 & 0 & 3 & 0 & 0 & 2 & 3 & 4 & 0 & 2 & 0 & 121\\ 12 & 3 & 1 & 6 & 1 & 0 & 3 & 4 & 5 & 0 & 2 & 0 & 195\\ 14 & 2 & 0 & 7 & 1 & 0 & 6 & 6 & 8 & 1 & 3 & 0 & 295\\ 16 & 4 & 2 & 11 & 3 & 0 & 8 & 8 & 9 & 1 & 4 & 0 & 425 \end{array}$$ From ([\[OrthoDecom\]](#OrthoDecom){reference-type="ref" reference="OrthoDecom"}) we have the following decomposition of the space $M_{2k}(\Gamma[2])$: $$M_{2k}(\Gamma[2])=E_{2k}(\Gamma[2]) \oplus S_{2k}(\Gamma[2]) \quad \text{for} \quad k\geqslant 0$$ so we are going to give the isotypical decomposition of the Eisenstein and the cuspidal parts according to ([\[DecompPackets\]](#DecompPackets){reference-type="ref" reference="DecompPackets"}). ### **Isotypical decomposition of $E_{2k}(\Gamma[2])$** {#EevenQ} From ([\[DecompPackets\]](#DecompPackets){reference-type="ref" reference="DecompPackets"}), we have $$E_{2k}(\Gamma[2])=E^{\textbf{(F)}}_{2k}(\Gamma[2])\oplus E^{\textbf{(Q)}}_{2k}(\Gamma[2]) \quad \text{for} \quad k\geqslant 0.$$ Recall that $E^{\textbf{(F)}}_{2k}(\Gamma[2])$ corresponds to Siegel-Eisenstein series while $E^{\textbf{(Q)}}_{2k}(\Gamma[2])$ to Klingen-Eisenstein series. From [@CvdGG Section 13], we deduce that $$\dim_{\mathfrak{S}_6}E^{\textbf{(F)}}_{0}(\Gamma[2])=s[6], \; \dim_{\mathfrak{S}_6}E^{\textbf{(F)}}_{2}(\Gamma[2])=s[2^3] \text{ and } \dim_{\mathfrak{S}_6}E^{\textbf{(F)}}_{2k}(\Gamma[2])=s[6]+ s[4,2] + s[2^3] \text{\, for\, } k\geqslant 2.$$ By construction, Klingen-Eisenstein series of weight $2k$ for $k>2$ (to ensure convergence) on $\Gamma[2]$ come from cusp forms of weight $2k$ on $\Gamma(2)$. Since ${\mathrm{SL}}(2,\mathbb{Z})/\Gamma(2)\cong \mathfrak{S}_3$, the space $S_{2k}(\Gamma(2))$ can be decomposed into irreducible representations of $\mathfrak{S}_3$ and this is given by $$\label{DecomCuspLevel2} \dim_{\mathfrak{S}_3}S_{2k}(\Gamma(2))=d_{1,2k}s[3]+(d_{1,2k}+d_{2,2k})s[2,1]+d_{4,2k}s[1^3],$$ where the integers $d_{N,k}$ are defined as in ([\[def_d\_k\]](#def_d_k){reference-type="ref" reference="def_d_k"}). Note that the previous formula corrects [@Petersen Proposition 6.1]. Recall that the dimension of the space $S_{2k}(\Gamma(2)) \cong S_{2k}(\Gamma_0(4))$ is $k-2$ for $k\geqslant 3$ and $0$ otherwise. From the previous table, Theorem [Theorem 1](#Igusa){reference-type="ref" reference="Igusa"} and the isotypical decomposition of $E^{\textbf{(F)}}_{2k}(\Gamma[2])$, we deduce that $\dim_{\mathfrak{S}_6}E^{\textbf{(Q)}}_{0}(\Gamma[2]) =\dim_{\mathfrak{S}_6}E^{\textbf{(Q)}}_{2}(\Gamma[2]) =0.$ For $k\geqslant 2$, Proposition 13.1 of [@CvdGG] gives $$\dim_{\mathfrak{S}_6}E^{\textbf{(Q)}}_{2k}(\Gamma[2])= \text{Ind}_{H}^{\mathfrak{S}_6}\left(\dim_{\mathfrak{S}_3}S_{2k}(\Gamma(2))\right)$$ where $H$ denotes the stabiliser in $\mathfrak{S}_6$ of one of the $1$-dimensional boundary components of the Satake compactification of $\Gamma[2] \backslash\mathfrak{H}_2$. Note that $H$ is of order 48 and recall (for more details see [@CvdGG Section 2]) that we have $s[\varpi]$ $s[3]$ $s[2,1]$ $s[1^3]$ --------------------------------------------------------- ---------------------------------- -------------------------------------- --------------------------- ${\rm Ind}_{H}^{\mathfrak{S}_6}\left(s[\varpi] \right)$ $s[6]\oplus s[5,1]\oplus s[4,2]$ $s[4,2]\oplus s[3,2,1]\oplus s[2^3]$ $s[3,1^3]\oplus s[2,1^4]$ Therefore, for $k\geqslant 2$, the isotypical decomposition of the space $E^{\textbf{(Q)}}_{2k}(\Gamma[2])$ is as follows $$\begin{aligned} \dim_{\mathfrak{S}_6}E^{\textbf{(Q)}}_{2k}(\Gamma[2])=& d_{1,2k}(s[6] + s[5,1])+ (2d_{1,2k}+d_{2,2k})s[4,2] + (d_{1,2k}+d_{2,2k})(s[3,2,1]+s[2^3])\\ &+ d_{4,2k}(s[3,1^3]+s[2,1^4]).\end{aligned}$$ Putting this together we get the generating series for the multiplicities of the irreducible representations of $\mathfrak{S}_6$ in $E^{\textbf{(F)}}_{2k}(\Gamma[2])$ and $E^{\textbf{(Q)}}_{2k}(\Gamma[2])$ $s[\varpi]$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(E^{\textbf{(F)}}_{2k}(\Gamma[2]))\,t^{2k}$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2]))\,t^{2k}$\ $s[6]$ & $\frac{1-t^2+t^4}{1-t^2}$ & $\frac{t^{12}}{(1-t^4)(1-t^6)}$\ $s[5,1]$ & $0$ & $\frac{t^{12}}{(1-t^4)(1-t^6)}$\ $s[4,2]$ & $\frac{t^4}{1-t^2}$ & $\frac{t^{8}}{(1-t^2)(1-t^4)}$\ $s[3,2,1]$ & $0$ & $\frac{t^{8}}{(1-t^2)(1-t^6)}$\ $s[3,1^3]$ & $0$ & $\frac{t^{6}}{(1-t^4)(1-t^6)}$\ $s[2^3]$ & $\frac{t^2}{1-t^2}$ & $\frac{t^{8}}{(1-t^2)(1-t^6)}$\ $s[2,1^4]$ & $0$ & $\frac{t^{6}}{(1-t^4)(1-t^6)}$\ and $0$ for $s[4,1^2], s[3^2], s[2^2,1^2]$ and $s[1^6]$. **Remark 4**. The isotypical decomposition of the space $S_{2k}(\Gamma(2))$ in [@CvdGG Proposition 13.1] was written as $$\dim_{\mathfrak{S}_3}S_{2k}(\Gamma(2))=\text{{Sym}}^k(s[2,1])- \left\{ \begin{tabular}{ccc} $s[2,1]$ & \text{if} & $k=1$\\ $s[3]+s[2,1]$ & \text{if} & $k\geqslant 2$. \end{tabular} \right.$$ This directly gives $\dim S_{4k}(\Gamma(2))=2(k-1)$ for $k\geqslant 1$, this can also be checked by using ([\[dimnew2new4\]](#dimnew2new4){reference-type="ref" reference="dimnew2new4"}), ([\[DecomCuspLevel2\]](#DecomCuspLevel2){reference-type="ref" reference="DecomCuspLevel2"}) and $\dim S_{4k}({\mathrm{SL}}(2,\mathbb{Z}))=\lfloor k/3 \rfloor$. As a sanity check, we verify $$\begin{aligned} \sum_{k \geqslant 0} \sum_{s[\varpi]}\dim (s[\varpi]) m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2]))t^{2k} &= 15\frac{t^6}{(1-t^2)^2}=15 \sum_{k \geqslant 0} \dim S_{2k}(\Gamma(2))t^{2k}\end{aligned}$$ in agreement with $E^{\textbf{(Q)}}_{2k}(\Gamma[2]) \cong S_{2k}(\Gamma(2))^{\oplus 15}$. The next table gives the first few isotypical decompositions of $E_{2k}(\Gamma[2])$ where we indicate the multiplicities of the Siegel-Eisenstein part in blue and those of the Klingen-Eisenstein part in black, we put ${\color{blue} d_F}=\dim E^{\textbf{(F)}}_{k}(\Gamma[2])$ and $d_Q=\dim E^{\textbf{(Q)}}_{k}(\Gamma[2])$ $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c} s[\varpi] & s[6] & s[5,1] & s[4,2] & s[4,1^2] & s[3^2] & s[3,2,1] & s[3,1^3] & s[2^3] & s[2^2,1^2] & s[2,1^4] & s[1^6] & \\ \dim s[\varpi]& 1& 5 & 9 & 10 & 5 & 16 & 10 & 5 & 9 & 5 & 1&\\ \hline \hline k & & & & & & & & & & & & d={\color{blue} d_F}+d_Q \\ \hline 0 & {\color{blue} 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\color{blue} 1}+0\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\color{blue} 1} & 0 & 0 & 0 & {\color{blue} 5}+0\\ 4 & {\color{blue} 1} & 0 & {\color{blue} 1} & 0 & 0 & 0 & 0 & {\color{blue} 1} & 0 & 0 & 0 & {\color{blue} 15}+0\\ 6 & {\color{blue} 1} & 0 & {\color{blue} 1} & 0 & 0 & 0 & 1 & {\color{blue} 1} & 0 & 1 & 0 & {\color{blue} 15}+ 15\\ 8 & {\color{blue} 1} & 0 & {\color{blue} 1}+1 & 0 & 0 & 1 & 0 & {\color{blue} 1}+1 & 0 & 0 & 0 & {\color{blue} 15}+ 30\\ 10 & {\color{blue} 1} & 0 & {\color{blue} 1}+1 & 0 & 0 & 1 & 1 & {\color{blue} 1}+1 & 0 & 1 & 0 & {\color{blue} 15}+ 45\\ 12 & {\color{blue} 1}+1 & 1 & {\color{blue} 1}+2 & 0 & 0 & 1 & 1 & {\color{blue} 1}+1 & 0 & 1 & 0 & {\color{blue} 15}+ 60\\ 14 & {\color{blue} 1} & 0 & {\color{blue} 1}+2 & 0 & 0 & 2 & 1 & {\color{blue} 1}+2 & 0 & 1 & 0 & {\color{blue} 15} + 75\\ %16 & {\color{blue} 1}+1 & 1 & {\color{blue} 1}+3 & 0 & 0 & 2 & 1 & {\color{blue} 1}+2 & 0 & 1 & 0 & {\color{blue} 15} + 90 \end{array}$$ ### **Isotypical decomposition of $S_{2k}(\Gamma[2])$** For $k\geqslant 2$, we know, see [@Tsushima1982 pp. 882-883], that $$\label{Dim_S2k} \dim S_{2k}(\Gamma[2])=\dim M_{2k}(\Gamma[2]) -15(k-2)-15=(k-2)(2k^2+7k-24)/3.$$ We also know $S_{0}(\Gamma[2])=S_{2}(\Gamma[2])=\{0\}$. The generating series for the dimension of the spaces $S_{2k}(\Gamma[2])$ is therefore given by $$\sum_{k \geqslant 0}\dim S_{2k}(\Gamma[2])\,t^{2k}= \frac{t^6(5+4t^2-5t^4)}{(1-t^2)^4}.$$ By definition of cusp forms, for $k \geqslant 0$ we have $$m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))= m_{{s[\varpi]}}(M_{2k}(\Gamma[2]))- \Big( m_{{s[\varpi]}}(E^{\textbf{(F)}}_{2k}(\Gamma[2]))+ m_{{s[\varpi]}}(E^{\textbf{(Q)}}_{2k}(\Gamma[2])) \Big).$$ So the generating series for the multiplicities of the irreducible representations of $\mathfrak{S}_6$ in $S_{2k}(\Gamma[2])$ are given by $s[\varpi]$ & $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))\,t^{2k}$\ $s[6]$ & $\frac{t^{10}(1+t^2-t^{12})}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ $s[5,1]$ & $\frac{t^{16}(1+t^2-t^6)}{((1-t^4)(1-t^6))^2}$\ $s[4,2]$ & $\frac{t^8(1+t^6-t^{10})}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[4,1^2]$ & $\frac{t^{12}(1+t^4)}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3^2]$ & $\frac{t^{20}}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[3,2,1]$ & $\frac{t^{10}(1+t^2+2t^4+t^8-t^{14})}{(1-t^2)(1-t^6)^2(1-t^{10})}$\ $s[3,1^3]$ & $\frac{t^8(1+t^2+t^{10}-t^{12})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^3]$ & $\frac{t^6(1+t^8-t^{12})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$\ $s[2^2,1^2]$ & $\frac{t^{14}}{(1-t^2)(1-t^4)^2(1-t^{10})}$\ $s[2,1^4]$ & $\frac{t^{10}(1+t^2-t^6)}{((1-t^4)(1-t^6))^2}$\ $s[1^6]$ & $\frac{t^{30}}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$\ Again, Conjecture 6.6 of [@BFvdG] now proved by Rösner (see [@Roesner Section 5]) tells us that only the Arthur packets $\textbf{(P)}$ and $\textbf{(G)}$ can occur in $S_{2k}(\Gamma[2])$ and so $$S_{2k}(\Gamma[2])= S^{\textbf{(G)}}_{2k}(\Gamma[2])\oplus S^{\textbf{(P)}}_{2k}(\Gamma[2]).$$ Moreover Conjecture 6.6 of [@BFvdG] gives the isotypical decomposition of $S^{\textbf{(P)}}_{2k}(\Gamma[2])$: $$\begin{aligned} \dim_{\mathfrak{S}_6}S^{\textbf{(P)}}_{2k}(\Gamma[2])=& d_{1,4k-2}\,s[6] + (d_{1,4k-2}+d^+_{2,4k-2}) \, s[4,2] + (d_{2,4k-2}+d^{-}_{2,4k-2}) \, s[2^3] \end{aligned}$$ where the integers $d_{N,k}$ and $d^{\pm}_{N,k}$ are defined as in ([\[def_d\_k\]](#def_d_k){reference-type="ref" reference="def_d_k"}). The generating series for the multiplicity of the irreducible representations $s[6]$, $s[4,2]$ and $s[2^3]$ in $S^{\textbf{(P)}}_{2k}(\Gamma[2])$ for $k\geqslant 0$ are therefore given by $s[\varpi]$ & $s[6]$ & $s[4,2]$ & $s[2^3]$\ $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k}(\Gamma[2]))\,t^{2k}$ & $\frac{t^{10}}{(1-t^2)(1-t^{6})}$ & $\frac{t^{8}}{(1-t^2)(1-t^4)}$ & $\frac{t^{6}}{(1-t^2)(1-t^4)}$ Keeping in mind that $\dim s[6]=1$, $\dim s[4,2]=9$ and $\dim s[2^3]=5$, we deduce $$\sum_{k\geqslant 0} \dim S^{\textbf{(P)}}_{2k}(\Gamma[2]) t^{2k}= \frac{t^6(5+14t^2+15t^4+10t^6)}{(1-t^4)(1-t^{6})}.$$ From this we get the generating series for $\dim S^{\textbf{(G)}}_{2k}(\Gamma[2])$: $$\sum_{k\geqslant 0} \dim S^{\textbf{(G)}}_{2k}(\Gamma[2]) t^{2k}= \sum_{k\geqslant 0} \bigl(\dim S_{2k}(\Gamma[2])-\dim S^{\textbf{(P)}}_{2k}(\Gamma[2])\bigr) t^{2k}\\ = \frac{t^8(10+21t^2+9t^4-t^6-15t^8)}{(1-t^2)^2(1-t^4)(1-t^{6})}$$ and also the generating series for the multiplicity of the irreducible representations in $S^{\textbf{(G)}}_{2k}(\Gamma[2])$ for $k\geqslant 0$ - for $s[\varpi]\in \left\{s[5,1], s[4,1^2], s[3^2], s[3,2,1], s[3,1^3], s[2^2,1^2], s[2,1^4], s[1^6]\right\}$ we have $$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))=m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))$$ - for $s[\varpi] \in \left\{s[6], s[4,2], s[2^3]\right\}$ we have $$m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))= m_{{s[\varpi]}}(S_{2k}(\Gamma[2]))- m_{{s[\varpi]}}(S^{\textbf{(P)}}_{2k}(\Gamma[2]))$$ so $s[\varpi]$ & $s[6]$ & $s[4,2]$ & $s[2^3]$\ $\sum_{k \geqslant 0}m_{{s[\varpi]}}(S^{\textbf{(G)}}_{2k}(\Gamma[2]))\,t^{2k}$ & $\frac{t^{20}(1+t^2+t^4-t^{12}-t^{14})}{(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})}$ & $\frac{t^{12}(1+t^2-t^{10})}{(1-t^2)(1-t^4)^2(1-t^{10})}$ & $\frac{t^{12}(1+t^2-t^{12})}{(1-t^2)(1-t^4)(1-t^6)(1-t^{12})}$ # Euler characteristics of local systems {#sec-euler} Let $\mathcal{A}_2[2]$ be the moduli space of principally polarized abelian surfaces equipped with a full level two structure. This is a smooth Deligne-Mumford stack defined over ${\rm Spec}(\mathbb{Z}[1/2])$. The space $\mathcal{A}_2[2]$ comes equipped with a natural action of the symmetric group $\mathfrak{S}_6 \cong {\rm GSp}(4,\mathbb{Z}/2)$. Let $\pi: \mathcal X \to \mathcal{A}_2[2]$ denote the universal object and define the local system $\mathbb{V}=R^1\pi_* \mathbb{C}$ on $(\mathcal{A}_2[2])_{\mathbb{C}}$. To each pair of integers $(l,m)$, with $l \geq m \geq 0$, we get a local system $\mathbb{V}_{l,m}$ from the corresponding irreducible representation of $\mathrm{GSp}(4)$. The moduli space $\mathcal{A}_2[2]$ can be identified with the union of the moduli space of tuples $(C,r_1,\ldots,r_6)$ where $C$ is either a genus $2$ curve or an unordered pair of elliptic curves (note that curves are assumed to be projective, irreducible and smooth) intersecting in the point at infinity, and where $(r_1,\ldots,r_6)$ is a $6$-tuple of marked Weierstraß  points (distinct from infinity, in the elliptic curve case). Denote the strata respectively by $\mathcal M_2[2]$ and $\mathcal{A}_{1,1}[2]$. With this identification, the action of $\mathfrak{S}_6$ is by permutation of the marked Weierstraß  points. For more details about the above cf. [@BFvdG]. Let $\psi: (\mathcal{A}_2[2])_{\mathbb{C}} \to A_2[2]$ denote the coarse moduli space, respectively $M_2[2]$ and $A_{1,1}[2]$, and put $V_{l,m}=\psi_*\mathbb{V}_{l,m}$. There is an induced action of $\mathfrak{S}_6$ on the compactly supported Betti cohomology groups $H^i_c$ of these spaces with coefficients in $V_{l,m}$. For any partition $\varpi=[6^{\varpi_6},\ldots,1^{\varpi_1}]$ of $6$, put $h^i_{c,\varpi}=m_{s[\varpi]}(H^i_c)$. We will now identify the representation ring $\mathbb{Z}[\mathfrak{S}_6]$ of $\mathfrak{S}_6$ with the ring of symmetric polynomials. With this interpretation, for a partition $\varpi$ of $6$, $s[\varpi]$ equals the corresponding Schur polynomial. Let also $p_i$ denote the $i$th power sum polynomial and put $p_{\varpi}=p_1^{\varpi_1}\cdots p_6^{\varpi_6}$. Moreover, for any $\lambda=(l,m)$, let $s_{<\lambda>}$ denote the symplectic Schur polynomial in four variables associated to $\lambda$, see [@FH Appendix A]. ## Formulas for the Euler characteristics The aim of this section is to give a formula, for any $\lambda=(l,m)$, of the $\mathfrak{S}_6$-equivariant Euler characteristic, $$E_c(A_2[2],V_{\lambda})= \sum_{\varpi \vdash 6} E_{c,\varpi}(A_2[2],V_{\lambda}) s[\varpi] \in \mathbb{Z}[\mathfrak{S}_6],$$ where $$E_{c,\varpi}(A_2[2],V_{\lambda})=\sum^4_{i=0} (-1)^i h^i_{c,\varpi}(A_2[2], V_{\lambda}) \in \mathbb{Z}.$$ Stratify the spaces $X_1=M_2[2]$ and $X_2=A_{1,1}[2]$ (or equivalently $M_2$ and $A_{1,1}$, the corresponding coarse moduli spaces without a level two structure), into strata $\Sigma_i(G)$, for $G$ a finite group, consisting of the curves corresponding to points of $X_i$ whose automorphism group equals $G$. Let $E_c(\Sigma_i(G))$ denote the Euler characteristic of $\Sigma_i(G)$. Say that $g \in G$ has eigenvalues $\xi_1(g)$, $\xi_2(g)$, $\xi_3(g)$ and $\xi_4(g)$ when acting on $H^1(C,\mathbb{C})$ of a curve $C \in \Sigma_i(G)$. Say furthermore that the induced action of $g \in G$ on the six Weierstraß  points of a curve $C \in \Sigma_i(G)$ has $\mu_j$ cycles of length $j$ for $j=1,\ldots,6$, giving a partition $\mu(g,G,i)$. Note that this data will be constant on the strata, i.e. independent of the choice of $C \in \Sigma_i(G)$. On a strata $\Sigma_i(G)$ the Euler characteristic $E_c(\Sigma_i(G),V_{\lambda})=E_c(\Sigma_i(G)) \cdot \dim V_{\lambda}^{G}$ and hence $$E_c(A_2,V_{\lambda})=\sum_{i=1}^2 \sum_{G} \frac{E_c \bigl(\Sigma_i(G) \bigr)}{|G|} \sum_{g \in G} s_{<\lambda>} \bigl(\xi_1(g), \xi_2(g),\xi_3(g),\xi_4(g) \bigr) \in \mathbb{Z}.$$ This method was used in [@Getzler] to find a formula for $E_c(M_2,V_{\lambda})$ for any $\lambda$. Adding the level two structure we need to take the action of $\mathfrak{S}_6$ on the Weierstraß  points into account and one finds that, $$\label{eq-numeric} E_c(A_2[2],V_{\lambda})=\sum_{i=1}^2 \sum_{G} \frac{E_c \bigl(\Sigma_i(G) \bigr)}{|G|} \sum_{g \in G} s_{<\lambda>} \bigl(\xi_1(g), \xi_2(g),\xi_3(g),\xi_4(g) \bigr) p_{\mu(g,G,i)} \in \mathbb{Z}[\mathfrak{S}_6].$$ This formula can be compared to the one in [@BvdG Section 9]. In the two following sections, we will describe how to find the necessary information to compute [\[eq-numeric\]](#eq-numeric){reference-type="eqref" reference="eq-numeric"} for any $\lambda$. ## Smooth curves of genus two The stratification by automorphism group $G$ for $M_2$ was found by Bolza [@Bolza], see below. We follow the description in [@Getzler Section 4]. Curves $C$ of genus $2$ are described by equations $C_f:y^2-f(x)=0$, where $f$ square-free polynomial of degree $5$ or $6$. The automorphism group $G_f$ of a curve $C_f$ is equal to the subgroup of $\mathrm{SL}(2,\mathbb{C}) \times \mathbb{C}^{\times}$ consisting of elements $$(\gamma,u)=\Bigl(\Bigl( \begin{array}{cc} a & b \\ c & d \end{array} \Bigr),u\Bigr) \in \mathrm{SL}(2,\mathbb{C}) \times \mathbb{C}^{\times} \quad \text{such that} \quad f(x)=(\gamma,u) \cdot f(x)=\frac{(cx+d)^6}{u^2} \, f\Bigl(\frac{ax+b}{cx+d}\Bigr)$$ quotiented by the subgroup generated by the element $(-\mathrm{id},-1) \in \mathrm{SL}(2,\mathbb{C}) \times \mathbb{C}^{\times}$. These groups will be given as pairs $(\Gamma_f,\rho_f)$, where $\Gamma_f$ is a subgroup of $\mathrm{SL}(2,\mathbb{C})$ that preserves the set of roots of $f$ and $\rho_f$ is a character of $\Gamma_f$ such that $G_f \cong \Gamma_f(\rho_f)/<(-\mathrm{id},-1)>$ where $$\Gamma_f(\rho_f)=\{(\gamma,u) \in \mathrm{SL}(2,\mathbb{C}) \times \mathbb{C}^{\times}:u^2=\rho_f(\gamma) \}.$$ There is an isomorphism $H^1(C_f,\mathbb{C}) \cong H^0(C_f,\Omega) \oplus H^0(C_f,\Omega)^{\vee}$ and $H^0(C_f,\Omega)$ has a basis consisting of the differentials $\omega_0=dx/y$, $\omega_1=xdx/y$. The action of $(\gamma,u) \in \Gamma_f(\rho_f)$ on the basis $(\omega_0,\omega_1)$ equals $$(\gamma,u)(\omega_0,\omega_1)=(u^{-1}(c\omega_1+d\omega_0),u^{-1}(a\omega_1+b\omega_0)),$$ see [@Getzler Proposition 2]. This tells us that if $\lambda_{\gamma}$ is an eigenvalues of $\gamma \in \Gamma_f$ then $\xi_1=\lambda_{\gamma}u^{-1}$, $\xi_2=\lambda_{\gamma}^{-1}u^{-1}$, $\xi_3=\lambda_{\gamma}^{-1}u$ and $\xi_4=\lambda_{\gamma}u$, are the eigenvalues of $(\gamma,u) \in G_f$ acting on $H^1(C_f,\mathbb{C})$. Finally, we need to determine the action of every $\gamma \in \Gamma_f$ on the roots of $f$, together with the point at infinity in the case that the degree of $f$ equals five. We will choose an ordering of the roots of $f$ (and possibly infinity) and denote the induced permutation by $\sigma_{\gamma}(f)$. There are seven strata for $M_2[2]$ corresponding to the different automorphism groups $(\Gamma,\rho)$: $(C_2,\mathrm{id})$, $(C_4,\chi^2)$, $(Q_8,\chi_0)$, $(Q_{12},\chi_0)$, $(O,\chi)$, $(Q_{24},\chi_+)$ and $(C_{10},\chi^6)$. Here $C_n$ denotes the cyclic group with $n$ elements, $Q_{4n}$ the quaternionic group with $4n$ elements and $O$ is the binary octahedral group with $48$ elements. The characters are defined as in [@Getzler pp. 124--125]. The groups $\Gamma \subset \mathrm{SL}(2,\mathbb{C})$ can be generated by one element $S\in \mathrm{SL}(2,\mathbb{C})$ in the abelian case, and two elements $S$ and $U=\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)$ in the non-abelian case. Put $\epsilon_n=e^{2\pi i/n}$. For further descriptions of these groups and characters, together with the computation of the Euler characteristics of the different strata, we refer to [@Getzler Section 2]. The information in the following table can be gotten from straightforward computations. The table above provides sufficient information to compute the contribution of $\Sigma_1(G)$ to [\[eq-numeric\]](#eq-numeric){reference-type="eqref" reference="eq-numeric"}, for all abelian groups $G$. For the non-abelian groups $G$, the information that is missing is an eigenvalue $\lambda_{\gamma}$ for all $\gamma \in G$. This problem is solved for the quaternionic groups $Q_{4n}$ by noting that it consists of the matrices $\pm S^j$ and $\pm US^j$ for $j=1,\ldots,n$, and the latter all have eigenvalues $\epsilon_4,-\epsilon_4$. Eigenvalues for the elements of the binary octahedral group can be gotten from straightforward computation. ## Pairs of elliptic curves The stratification by automorphism group for $A_1$, the moduli space of elliptic curves, is given by the three groups $C_2$, $C_4$ and $C_6$. The corresponding strata have Euler characteristics $-1$, $1$ and $1$ respectively. The two latter strata are points which can be represented by the curves $y^2=x(x^2-1)$ and $y^2=x^3-1$ respectively. The automorphism group is generated by the element $y \mapsto -y$ for $C_2$, by $y \mapsto \epsilon_4 y, x\mapsto -x$ for $C_4$ and $y \mapsto -y, x\mapsto \epsilon_3 x$ for $C_6$. For all elliptic curves of the form $y^2=f(x)$, $H^0(C_f,\Omega)$ has a basis consisting of the differential $\omega_0=dx/y$. The eigenvalues of the induced action on $H^1(C_f,\mathbb{C})$ of the generators of the automorphism groups given above then equals $-1,-1$ for $C_2$, $\epsilon_4,-\epsilon_4$ for $C_4$ and $\epsilon_6,-\epsilon_6$ for $C_6$. The induced action of the generators on the Weierstraß  points (after choosing an ordering), which correspond to the roots of $f(x)$ (together with infinity) equals $\mathrm{id}$, $(12)$ and $(123)$ respectively. Consider now $A_{1,1} \cong (A_1 \times A_1)/\mathfrak{S}_2$, which is the moduli space of unordered pairs of elliptic curves. This has the consequence that a pair of equal (or isomorphic) elliptic curves $E \times E$ will have an extra automorphism that sends $(p_1,p_2) \in E \times E$ to $(p_2,p_1) \in E \times E$. There will therefore be seven possible automorphism groups for $A_{1,1}$, namely $C_2 \times C_2$, $C_2 \wr \mathfrak{S}_2$, $C_2 \times C_4$, $C_2 \times C_6$, $C_4 \wr \mathfrak{S}_2$, $C_4 \times C_6$ and $C_6 \wr \mathfrak{S}_2$, where $\wr$ denotes the wreath product. The Euler characteristics of the corresponding strata are directly found to be $1,-1,-1,-1,1,1$ and $1$, respectively. Take any two elliptic curves $E_1$ and $E_2$ with automorphism groups $G_1$ and $G_2$. Since $H^1(E_1 \times E_2,\mathbb{C}) \cong H^1(E_1,\mathbb{C}) \oplus H^1(E_2,\mathbb{C})$ it is straightforward, using the information for $A_1$ above, to compute the action of $G_1 \times G_2$ if $E_1$ and $E_2$ are not isomorphic, and of $G_1 \wr \mathfrak{S}_2$ if $E_1$ and $E_2$ are isomorphic. The action of $G_1 \times G_2$ (and of $G_1 \wr \mathfrak{S}_2$) on the six Weierstraß  point that are distinct from infinity on both elliptic curves is also straightforward. # Isotypical decomposition in the vector-valued case {#sec-vv} In this section we assume that $j>0$, so we are dealing with vector-valued Siegel modular forms. As a consequence of [@BGZ Proposition 1], we have $$M_{0,j}(\Gamma[2])=\left\{0\right\} \text{ for any } j>0.$$ Theorem A.5 by G. Chenevier in [@CvdG] tells us that $$M_{1,j}(\Gamma[2])=S_{1,j}(\Gamma[2])=\left\{0\right\} \text{ for any } j>0.$$ For $k=2$, there is no dimension formula for the space $M_{2,j}(\Gamma[2])$ in general. A conjectural description of the isotypical decomposition of the space $S_{2,j}(\Gamma[2])$ is given in [@CvdG Conjecture 1.2]. As this conjecture has only been verified for $j<12$ we decided to not implement the isotypical decomposition of the space $M_{2,j}(\Gamma[2])$ in our code. For $k=3$, the situation is also still conjectural but with more evidence. In fact only the isotypical decomposition of $E_{c,\mathrm{Eis}}(A_2[2],V_{j,0})$ (see Theorem [Theorem 7](#thm-main){reference-type="ref" reference="thm-main"} to understand how this part contributes to the isotypical decomposition of the space $S_{3,j}(\Gamma[2])$, and then further in Remark [Remark 8](#rem-k3){reference-type="ref" reference="rem-k3"}) is still conjectural so we decided to implement the isotypical decomposition of the space $S_{3,j}(\Gamma[2])$ in our code. Evidence towards this conjecture are given for example by the results of Petersen in [@Petersen2017] or in Section 6.6 of [@BFvdG2014]. We start by recalling the dimension formula for the space $M_{k,j}(\Gamma[2])$ which can also be used to check its conjectural isotypical decomposition for $k=3$. **Theorem 5** ([@Tsushima Theorems 2 and 3] and [@CvdGG Theorem 12.1]). *For $k\geqslant 3$ odd and $j\geqslant 2$ even we have $$\begin{aligned} \dim M_{k,j}(\Gamma[2])=\dim S_{k,j}(\Gamma[2])= \frac{1}{24} \big(& 2(j+ 1)k^3 + 3(j^2-2j-8)k^2+(j^3-9j^2-42j+118)k\\ & -2j^3-9j^2+152j-216 \big).\end{aligned}$$ For $k\geqslant 4$ even and $j\geqslant 2$ even we have $$\begin{aligned} \dim M_{k,j}(\Gamma[2])= \frac{1}{24} \big(& 2(j+ 1)k^3 + 3(j^2-2j+2)k^2+(j^3-9j^2-12j+28)k -2j^3-9j^2+182j-336 \big).\end{aligned}$$* **Remark 6**. For $k=3$, this formula is rather pretty $$\dim S_{3,j}(\Gamma[2])=(j-2)(j-3)(j-4)/24.$$ ## Isotypical decomposition of $M_{k,j}(\Gamma[2])$ For $k\geqslant 0$, the results of Rösner (see [@Roesner Section 5]) and those of [@CvdGG Section 13] tell us $$M_{k,j}(\Gamma[2])=E_{k,j}(\Gamma[2]) \oplus S_{k,j}(\Gamma[2])= E^{\textbf{(Q)}}_{k,j}(\Gamma[2]) \oplus S^{\textbf{(Y)}}_{k,j}(\Gamma[2]) \oplus S^{\textbf{(G)}}_{k,j}(\Gamma[2])$$ Theorem 5.13 and Remark 5.14 in [@Roesner] which prove Conjecture 6.4 of [@BFvdG] give us the isotypical decomposition of the space $S^{\textbf{(Y)}}_{k,j}(\Gamma[2])$ for $k\geqslant 3$ and $j>0$ $$\begin{aligned} \dim_{\mathfrak{S}_6} S^{\textbf{(Y)}}_{k,j}(\Gamma[2]) = \mu_1\, s[2^3]\, + \, \mu_2\,s[2,1^4] + \, \mu_3\,s[1^6] \quad \text{with} \quad \mu_1 & = d^+_{2,j+2k-2}d^+_{2,j+2}+d^-_{2,j+2k-2}d^-_{2,j+2}\\ \mu_2 & = d_{4,j+2k-2}d_{4,j+2}\\ \mu_3 & = d^+_{2,j+2k-2}d^-_{2,j+2}+d^-_{2,j+2k-2}d^+_{2,j+2}\end{aligned}$$ where the integers $d_{N,k}$ and $d^{\pm}_{N,k}$ are defined as in ([\[def_d\_k\]](#def_d_k){reference-type="ref" reference="def_d_k"}). Proposition 13.1 in [@CvdGG] gives us $E^{\textbf{(Q)}}_{k,j}(\Gamma[2])={0}$ for $k$ odd. For $k\geqslant 2$, this proposition (note that there is a typo in [@CvdGG]) tells us that $$\begin{aligned} \dim_{\mathfrak{S}_6} E^{\textbf{(Q)}}_{2k,j}(\Gamma[2])=& \text{Ind}_{H}^{\mathfrak{S}_6}\left( \text{{Sym}}^{j/2+k}\bigl(s[2,1]\bigr)-s[3]-s[2,1]\right)= \text{Ind}_{H}^{\mathfrak{S}_6}\left(S_{2k+j}(\Gamma(2))\right)\\ =&d_{1,2k+j}(s[6] + s[5,1])+ (2d_{1,2k+j}+d_{2,2k+j})s[4,2] + (d_{1,2k+j}+d_{2,2k+j})(s[3,2,1]+s[2^3])\\ &+ d_{4,2k+j}(s[3,1^3]+s[2,1^4])\end{aligned}$$ where the last identity followed from Section [3.2.1](#EevenQ){reference-type="ref" reference="EevenQ"}. So to get the isotypical decomposition of the space $M_{k,j}(\Gamma[2])$ it remains to determine it for the space $S^{\textbf{(G)}}_{k,j}(\Gamma[2])$. This is done in the next section. An isotypical dimension formula for $S^{\textbf{(G)}}_{k,j}(\Gamma[2])$ First we introduce some notation from [@BFvdG]. Let $$\begin{array}{ll} A=s[6]\oplus s[5,1]+s[4,2], &\qquad A'=s[6]\oplus s[4,2]\oplus s[2^3], \\ B=s[4,2]\oplus s[3,2,1]+s[2^3], &\qquad B'=s[5,1]\oplus s[4,2]\oplus s[3,2,1], \\ C=s[3,1^3]\oplus s[2,1^4], &\qquad C'=s[4,1^2]\oplus s[3^2]. \end{array}$$ For any $l,m$, with $l > m > 0$, put $n=l+m+4$, $n'=l-m+2$ and define $$\begin{aligned} \label{EisCohom} \nonumber E_{c,\mathrm{Eis}}(A_2[2],V_{l,m})=& (d_{1,n'}-d_{1,n})\,(A'+B')+\,(d_{2,n'}-d_{2,n})\,B'+(d_{4,n'}-d_{4,n})\, C' +\frac{1}{2}\bigl(1+(-1)^m\bigr)\, (A+B)\\ &+2\bigl((d_{1,m+2}-d_{1,l+3})\,(A+B) +(d_{2,m+2}-d_{2,l+3})\,B+ (d_{4,m+2}-d_{4,l+3})\,C\bigr)\end{aligned}$$ and $$\begin{aligned} E_{c,\mathrm{endo}}(A_2[2],V_{l,m})= & -2\Bigl( d_{4,n'} \, \bigl(d_{4,n} \, s[3,1^3]+d_{1,n} \, s[3^2]+(d_ {1,n}+d_{2,n}) \, s[4,1^2] \bigr)\\ & + d_{2,n'} \, \bigl((d_{1,n}+d_{2,n}) \, s[3,2,1]+d_{4,n} \, s[4,1^2] + d_{1,n} \, s[4,2]+d_{1,n} \, s[5,1] \bigr) \\ & + d^+_{2,n'}\, \bigl(d^+_{2,n} \, s[4,2]+d^-_{2,n}\, s[5,1]\bigr) +d^-_{2,n'}\, \bigl(d^-_{2,n} \, s[4,2]+d^+_{2,n} \, s[5,1]\bigr) \\ & + d_{1,n'} \, \bigl(d_{1,n}\,(A'+B') +d_{2,n}\,B'+ d_{4,n}\,C'\bigr) \Bigr)\end{aligned}$$ as elements of the representation ring $\mathbb{Z}[\mathfrak{S}_6]$. **Theorem 7**. *For any $k\geq 4$ and $j >0$, put $l=j+k-3$ and $m=k-3$. Then $$\mathrm{dim}_{\mathfrak{S}_6} S^{\textbf{(G)}}_{k,j}(\Gamma[2])=-\frac{1}{4}\Bigl(E_c(A_2[2],V_{l,m})-E_{c,\mathrm{Eis}}(A_2[2],V_{l,m})-E_{c,\mathrm{endo}}(A_2[2],V_{l,m})+2 \, \mathrm{dim}_{\mathfrak{S}_6} S^{\textbf{(Y)}}_{k,j}(\Gamma[2])\Bigr).$$* *Proof.* In [@BFvdG], the compactly supported $\ell$-adic Euler characteristic of local systems $\mathbb{V}_{l,m}$ on $\mathcal A_2[2]$ taking values in the Grothendieck group of (absolute) Galois representations is decomposed into the following pieces, $$e_c(\mathcal A_2[2],\mathbb{V}_{l,m})=e_{c,\mathrm{Eis}}(\mathcal A_2[2],\mathbb{V}_{l,m})+e_{c,\mathrm{endo}}(\mathcal A_2[2],\mathbb{V}_{l,m})- S[l-m,m+3,\Gamma[2]].$$ The formula for $E_{c,\mathrm{Eis}}(A_2[2],V_{l,m})$ (respectively $E_{c,\mathrm{endo}}(A_2[2],V_{l,m})$) is found by taking dimensions in the formula for $e_{c,\mathrm{Eis}}(\mathcal A_2[2],\mathbb{V}_{l,m})$ (respectively $e_{c,\mathrm{endo}}(\mathcal A_2[2],\mathbb{V}_{l,m})$) in [@BFvdG Theorem 4.4] (respectively [@BFvdG Conjecture 7.1]). The representation $S[l-m,m+3,\Gamma[2]]$ should conjecturally consist of $2$-dimensional pieces for each Hecke eigenvector in $S^{\textbf{(Y)}}_{k,j}(\Gamma[2])$, with isotypic decomposition given in [@BFvdG Conjecture 6.4], and $4$-dimensional pieces for each Hecke eigenvector in $S^{\textbf{(G)}}_{k,j}(\Gamma[2])$. The conjectural description in [@BFvdG] described above, has been proven in [@Roesner]. Conjectures 7.1 and 6.4 of [@BFvdG] are proved by Theorem 5.13 of [@Roesner], see Remark 5.14 of [@Roesner]. The result then follows from [@Roesner Corollary 5.20]. ◻ **Remark 8**. In [@BFvdG], directly after Theorem 4.4, it is conjectured that $E_{c,\mathrm{Eis}}(A_2[2],V_{l,0})$ for any $l>0$ is given by ([\[EisCohom\]](#EisCohom){reference-type="ref" reference="EisCohom"}), with the difference that one needs to put $d_{1,2}=-1$. If we assume this conjecture to be true, and we define $E_{c,\mathrm{endo}}(A_2[2],V_{l,0})$ for any $l>0$ using the formula above, then Theorem [Theorem 7](#thm-main){reference-type="ref" reference="thm-main"} also holds for $k=3$ and $j>0$ using the same proof (and the same results of [@Roesner]). # Acknowledgement {#acknowledgement .unnumbered} The second author was supported by the Simons Foundation Award 546235 at the Institute for Computational and Experimental Research in Mathematics at Brown University. We thank Eran Assaf and Gerard van der Geer for useful discussions. 9999 T. Arakawa: *Vector valued Siegel modular forms of degree two and the associated Andrianov L-functions.* Manuscripta Math. 44(1983), 155--185. J. Arthur: *Automorphic representations of GSp(4).* Contributions to automorphic forms, geometry, and number theory, 65--81, Johns Hopkins Univ. Press, Baltimore, MD, 2004. J. Bergström, F. 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Petersen: *Cohomology of local systems on the moduli of principally polarized abelian surfaces.* Pacific J. Math. **275** (2015), no.1, 39--61. M. Rösner: *Parahoric Restriction for GSp(4) and the Inner Cohomology of Siegel Modular Threefolds.* Thesis-Ruprecht-Karls-Universität Heidelberg (Germany), DOI: 10.11588/heidok.00021401 <https://archiv.ub.uni-heidelberg.de/volltextserver/21401/1/Dissertation_Roesner.pdf> M. Roy, R. Schmidt, S. Yi: *Dimension formulas for Siegel modular forms of level 4.* Mathematika **69** (2023), no. 3, 795--840. R. Schmidt: *Packet structure and paramodular forms.* Trans. Amer. Math. Soc. 370 (2018), no. 5, 3085--3112. R. Schmidt: *Dimension formulas for spaces of Siegel modular forms of degree 2.* <https://math.ou.edu/~rschmidt/dimension_formulas/> W. Stein: *Modular forms, a computational approach.* With an appendix by Paul E. Gunnells Grad. Stud. Math., **79** American Mathematical Society, Providence, RI, 2007. xvi+268 pp. R. 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arxiv_math

--- abstract: | In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used. address: - Department of Mathematics, California State University, Bakersfield, CA 93311 - Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 - Department of Mathematics, The Ohio State University, Columbus, OH 43210 author: - Qiwei Sheng - Cory Hauck - Yulong Xing bibliography: - AP&unif_Conv_SH_DG_RTE.bib title: Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering --- [^1] # Introduction The radiative transfer equation (RTE) is a type of kinetic equation that models the scattering and absorption of radiation propagating through a material medium. In this paper, we consider the scaled, steady-state, linear RTE with isotropic scattering and periodic boundary conditions for the angular flux $u$ that depends on $\mathbf{x}\in X\subset\mathbb{R}^d$ and $\bm{\omega}\in \mathbb{S}$, the unit sphere in $\mathbb{R}^d$. For simplicity we assume $X=(0,1)^d$, and for physical problems, $d=3$. However, if $u$ possesses special symmetries, reduced equations in one or two-dimensional spatial domains can be derived [@LM1984]. We denote by $\partial X$ the boundary of $X$ and by $\mathbf{k}(\mathbf{x})$ the unit outward normal to $X$ at $\mathbf{x}\in \partial X$. We then set $\Gamma^-=\{(\mathbf{x},\bm{\omega})\in \partial X \times \mathbb{S} \colon \bm{\omega}\cdot \mathbf{k}(\mathbf{x}) < 0\}$. With this notation, the RTE with periodic boundary conditions takes the form [\[eq:rte_scale_all\]]{#eq:rte_scale_all label="eq:rte_scale_all"} $$\begin{aligned} {2} \bm{\omega}\cdot\nabla u(\mathbf{x},\bm{\omega})+\frac{\sigma_{\mathrm{t}}}{\varepsilon}u(\mathbf{x},\bm{\omega}) &=\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\overline{u}(\mathbf{x})+\varepsilon f, &&\quad (\mathbf{x},\bm{\omega}) \in X \times \mathbb{S},\label{eq:rte_scale}\\ u(\mathbf{x},\bm{\omega})&=u(\mathbf{x}- \mathbf{k}(\mathbf{x}),\bm{\omega}),&& \quad (\mathbf{x},\bm{\omega}) \in \Gamma^-, \label{eq:rte_scale_pbc} \end{aligned}$$ where the scalar flux $\overline{u}=\frac{1}{\int_{\mathbb{S}}\,\mathrm{d}\bm{\omega}}\int_\mathbb{S} u \,\mathrm{d}{\bm{\omega}}$ is the average of $u$ over $\mathbb{S}$, the functions $\sigma_{\mathrm{a}} = \sigma_{\mathrm{a}}(\mathbf{x})$ and $\sigma_{\mathrm{t}}=\sigma_{\mathrm{t}}(\mathbf{x})$ are (known) absorption and total cross sections, respectively, and $f=f(\mathbf{x},\bm{\omega})$ is a (known) source. The parameter $\varepsilon>0$ is a scaling parameter that measures the relative strength of scattering. We assume throughout the paper that $0 < \varepsilon< 1$. Although $u$ depends on $\varepsilon$, we suppress this dependence in the notation for ease of presentation. We assume that [\[as:rte_scale\]]{#as:rte_scale label="as:rte_scale"} $$\begin{aligned} &\sigma_{\mathrm{t}},\sigma_{\mathrm{a}}\in L^\infty(X),\quad \sigma_{\mathrm{t}}(\mathbf{x})-\varepsilon^2\sigma_{\mathrm{a}}(\mathbf{x}) > 0 ~\text{a.e.}~ \mathbf{x} \in X,\label{as:rte_scale_1}\\ &\sigma_{\mathrm{t}}>\sigma_{\mathrm{a}} \ge \sigma_{\mathrm{a}}^{\mathrm{min}} \text{ in } X \text{ for some constant } \sigma_{\mathrm{a}}^{\mathrm{min}} >0,\label{as:rte_scale_2}\\ &f(\mathbf{x},\bm{\omega})\in L^2(X\times\mathbb{S}).\label{as:rte_scale_3} \end{aligned}$$ The quantity $\sigma_{\mathrm{s},\varepsilon} = \sigma_{\mathrm{t}}(\mathbf{x})-\varepsilon^2\sigma_{\mathrm{a}}(\mathbf{x})$ is a (non-dimensional) scattering cross-section. While the condition [\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} is not strictly necessary, it is frequently used in a priori estimates. Under the assumptions in [\[as:rte_scale\]](#as:rte_scale){reference-type="eqref" reference="as:rte_scale"}, the system [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} has a unique solution [@agoshkov1998boundary] in $\mathring{H}^1_2(X\times\mathbb{S})$---the subspace of $H^1_2(X\times\mathbb{S}):=\{v \in L^2(X\times \mathbb{S})\colon {\bm{\omega}}\cdot \nabla v \in L^2(X\times \mathbb{S})\}$ containing all functions that are periodic in each spatial component $x_i$, $i \in \{1,\cdots,d\}$, with norm $$\|v\|_{\mathring{H}^1_2(X\times\mathbb{S})}=\|v\|_{H^1_2(X\times\mathbb{S})}=(\|v\|^2_{L^2(X\times\mathbb{S})}+\|\bm{\omega}\cdot \nabla v\|^2_{L^2(X\times\mathbb{S})})^{1/2}.$$ If $\varepsilon \ll 1$ and $\sigma_{\mathrm{t}} / \varepsilon \gg 1$ uniformly across the spatial domain $X$, then solution $u$ of the RTE can be approximated accurately by the solution of a diffusion equation that is much cheaper to compute [@habetler1975uniform; @LarsenKeller; @bensoussan1979boundary]. However, for true multiscale applications, the size of $\sigma_{\mathrm{t}}$ varies significantly as a function of $\mathbf{x}$, so that both diffusive and non-diffusive regions may coexist. In such situations, it is necessary to utilize kinetic models like the RTE for accurate simulations. Unfortunately, traditional numerical approaches for solving the RTE may be highly inefficient efficient in diffusive regions. In particular, local truncation errors may scale like $h^p / \varepsilon$, where $h>0$ is the spatial mesh size and $p>0$ is an integer corresponding to the order of the method [@jin2010asymptotic; @lowrie2002methods]. In such cases, accuracy may degrade dramatically as the parameter $\varepsilon\to 0$. For this reason, asymptotic preserving (AP) schemes [@Jin1999; @jin2010asymptotic] have been developed. Such schemes transition to consistent and stable numerical schemes for the macroscopic model as $\varepsilon \to 0$. Numerical discretization of the RTE is often carried out separately for the angular and spatial variables. In this paper, we consider angular discretizations using spherical harmonics expansions. The spherical harmonics ($P_N$) method [@CZ1967; @davison1957neutron] is one of the most widely-employed angular discretization methods for the RTE. It is also used to discretize kinetic semiconductor equations [@rupp2016review], which share many features with the RTE. The $P_N$ method is a class of spectral methods that approximates the solution to the RTE by a linear combination of spherical harmonics basis functions in angle. In the steady-state case, the coefficients of this expansion are functions of space that satisfy a symmetrizable, linear hyperbolic system of equations. Due to the properties of the spherical harmonics, the matrices in this system are very sparse and thereby cheap to assemble and apply. The $P_N$ method enjoys several beneficial properties. Among them: (i) it preserves the rotational symmetry of the transport equation [@garrett2016eigenstructure]; (ii) for steady-state problems, it is equivalent to the diffusion limit when $N=1$; and (iii) for sufficiently smooth solutions, the convergence of the method is spectral [@FHK2016]. Moreover, under certain conditions, the $P_N$ solution converges to the solution of the RTE at a rate $\mathcal{O}(\varepsilon^N)$ as $\varepsilon \to 0$, and its angular moments (which correspond to physically meaningful quantities) may converge even faster [@CH2019]. While the current paper focuses on periodic conditions, inflow boundary conditions that prescribe known data on $\Gamma^-$ are more physically relevant. The treatment of such boundary conditions is possible with direct approaches [@bunger2020stable] and variational frameworks [@ES2012; @MRS1999]. For vacuum (zero inflow) conditions, an extended computational domain can be employed to maintain sparsity of the $P_N$ system [@SW2021; @egger2019perfectly] or to handle complicated geometries [@powell2019pseudospectral] in a periodic setting. Spatial discretizations strategies for the $P_N$ equations include finite differences [@seibold2014starmap], least squares [@brown2003moment; @varin2005spherical] and mixed finite elements [@egger2019perfectly], discontinuous Galerkin methods [@SW2021], and psuedo-spectral methods [@powell2019pseudospectral]. Methods for capturing the diffusion limit for the time-dependent $P_N$ equations [@hauck2010methods] have been explored in both finite-volume [@hauck2009temporal] and discontinuous Galerkin (DG) [@mcclarren2008effects; @mcclarren2008semi] contexts; and a rigorous analysis of the steady-state problem, based on a least-squares finite element formulation, can be found in [@manteuffel1999boundary]. In the current paper, we focus on DG discretizations of the $P_N$ equations ($P_N$-DG). DG methods were first introduced in [@reed1973triangular] to simulate transport equations like [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"}, and a rigorous analysis was carried out in [@LR1974] for a simplified equation without scattering. A complete space-angle convergence analysis of discrete ordinate DG methods was conducted in [@HHE2010] for problems with scattering, and analogous results for the spherical harmonic DG method were recently proved in [@SW2021]. While the analysis studied in [@HHE2010; @SW2021] is valid for any fixed $\varepsilon= \mathcal{O}(1)$, it does not address the behavior of the DG method when $\varepsilon\to 0$. In the context of discrete ordinates angular discretization, it is known that DG methods can capture the diffusion limit whenever the underlying approximation space supports globally continuous linear polynomials [@LM1989; @Adams2001; @GK2010]. This requirement translates to local $\mathbb{P}_1$ and $\mathbb{Q}_1$ approximations for triangular and rectangular elements, respectively. In [@jin2010asymptotic], a general framework was provided to construct uniform error estimates from conventional error bounds and asymptotic analysis, although these estimates are typically not sharp. Sharper estimates for discrete-ordinate DG discretizations that are uniform in $\varepsilon$ were recently established in [@SH2021]. In this paper, we investigate the convergence of $P_N$-DG methods for the scaled RTE with isotropic scattering and periodic boundary conditions. The analysis builds upon ideas from [@SW2021] for establishing angular error estimates and from [@SH2021] for establishing spatial error estimates. The main contribution is to derive spectral estimates for the $P_N$ discretization that are uniform in $\varepsilon$ as well as $\mathcal{O}(h^k)$ error estimates for the DG scheme that are uniform in $\varepsilon$ whenever the DG space contains $\mathbb{Q}_1$ for Cartesian cells and $\mathbb{P}_1$ for triangles. Under this assumption on $k$, the DG estimate improves upon the general strategy in [@jin2010asymptotic] (which leads to an error bound of $\mathcal{O}(h^{k/2 + 1/4})$) and it implies that the $P_N$-DG scheme is necessarily accurate in the diffusion limit. Another contribution of this paper is that, to the best of the authors' knowledge, we prove for the first time a uniform optimal error estimate $\mathcal{O}(h^{k+1})$ for the spatial discretization on Cartesian meshes with tensor product elements. The analysis presented here does have some limitations. For example, the uniform error estimates may come at a price: the error obtained in [@SW2021] is $\mathcal{O}(h^{k+1/2})$ for a fixed $\varepsilon$, while here we can obtain a uniform $\mathcal{O}(h^{k})$ bound across all $\varepsilon\in (0,1)$. In addition, the assumptions in [\[as:rte_scale\]](#as:rte_scale){reference-type="eqref" reference="as:rte_scale"} imply that voids (i.e. $\sigma_{\rm{t}}=0$) are not allowed and, moreover, that $\sigma_{\rm{t}} / \varepsilon\gg 1$ whenever $\varepsilon\ll 1$. Hence as $\varepsilon\to 0$, the problem in [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} becomes uniformly diffusive. In other words, regions of very thick and thin materials do not coexist at the same time, as is often the case in realistic problems. Therefore the analysis here should be considered as a first step in analyzing more realistic problems. The rest of this article is organized as follows. In , the scaled RTE with isotropic scattering and its spherical harmonic discretization are introduced. We also present the notations to be used in the remainder of this paper, and the a priori estimates regarding the solutions of the spherical harmonic equation. In the last part of this section, we define the DG scheme for the spherical harmonic equation and prove its stability and well-posedness. In , uniform convergence and error estimates of the spherical harmonic DG method for the scaled RTE are established. In , we prove an optimal error estimate for the upwind DG scheme in one-dimensional slab geometries and on multidimensional Cartesian mesh with tensor product elements. Conclusions are given in . # Preliminaries and problem setting {#sec:settings} We start with presenting some notations. Throughout the paper, the symbol $a\lesssim b$ abbreviates $a\le Cb$ for any two real quantities $a$ and $b$, with $C>0$ being a nonessential constant independent of the finite element mesh size, which may take different values at different appearances. The conventional notation $H^r(D)$ was adopted to indicate Sobolev spaces on (possibly lower-dimensional) subdomain $D\subset X$ with the norm $\|\cdot\|_{r,D}$. Clearly, we have $H^0(D)=L^2(D)$ whose norm is denoted by $\|\cdot\|_D$. ## Spherical harmonic method for angular discretization {#sec:SH_PN} In this section, the spherical harmonic functions and the $P_N$ method will be briefly reviewed, we refer the interested readers to, e.g., [@AH2012] and [@Claus1966]. The angle bracket is introduced as a short-hand notation for the angular integration over $\mathbb{S}$: $\langle \cdot\rangle = \int_\mathbb{S} (\cdot) \,\mathrm{d}\bm{\omega}$. For any $\bm{\omega}\in \mathbb{S}$, let $\bm{\omega}=\begin{bmatrix} \omega_1 & \omega_2 &\omega_3 \end{bmatrix}^{\mathsf{T}} = \begin{bmatrix} \sqrt{1-\mu^2}\cos(\varphi) & \sqrt{1-\mu^2}\sin(\varphi) & \mu \end{bmatrix}^{\mathsf{T}}$, where $\mu:=\omega_3\in [-1,1]$ and $\varphi\in [0,2\pi)$ is the angle between the $x_1$-axis and the projection of $\bm{\omega}$ onto the $x_1$-$x_2$ plane, respectively. Given integers $\ell\ge 0$ and $\kappa\in [-\ell,\ell]$, the normalized, real-valued spherical harmonic of degree $\ell$ and order $\kappa$ is expressed in terms of $\mu$ and $\varphi$ as $m^\kappa_\ell(\bm{\omega})=\alpha_\ell^\kappa P^\kappa_\ell(\mu)\, T^\kappa(\varphi)$, where $P^\kappa_\ell$ is an associated Legendre function, $T^\kappa$ is a sinusoidal function [@FHK2016], and $\alpha_\ell^\kappa = \sqrt{\frac{(2\ell+1)}{4 \pi} \frac{(\ell - |\kappa|)!}{(\ell+|\kappa|)!}}$ is a constant. We collect the $n_\ell := 2\ell+1$ real-valued normalized harmonics of degree $\ell$ together into a vector-valued function $\bm{m}_\ell=\begin{bmatrix} m^{-\ell}_\ell & m^{-\ell+1}_\ell & \cdots & m^0_\ell & \cdots & m^{\ell-1}_\ell & m^\ell_\ell \end{bmatrix}^{\mathsf{T}}$ and for any given $N$, we set $\bm{m} = \begin{bmatrix}\bm{m}_0^{\mathsf{T}} & \bm{m}_1^{\mathsf{T}} & \cdots & \bm{m}_N^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}}$. Note that the normality of $\bm{m}$ is not necessary but adopting it would bring great convenience in the later deductions. The vector $\bm{m}$ has $L := \sum^N_{\ell=0} n_\ell= (N+1)^2$ components which form an orthogonal basis for the space $\mathbb{P}_N=\left\{\sum^N_{\ell=0} \sum^\ell_{\kappa=-\ell} c^\kappa_\ell m^\kappa_\ell : c^\kappa_\ell\in \mathbb{R} \text{ or } L^2(X), \text{ and } 0\le\ell\le N, |\kappa|\le\ell\right\}$. Furthermore, the spherical harmonics fulfill a recursion relation of the form [@AH2012] $$\label{eq:SH_rec} \omega_i \bm{m}_\ell = \bm{A}^{(i)}_{\ell,\ell+1}\bm{m}_{\ell+1} + \bm{A}^{(i)}_{\ell,\ell-1} \bm{m}_{\ell-1},$$ where $\bm{A}^{(i)}_{\ell,\ell'} = \langle\omega_i\bm{m}_\ell\bm{m}^{\mathsf{T}}_{\ell'}\rangle$ and $\big(\bm{A}^{(i)}_{\ell,\ell'}\big)^{\mathsf{T}}=\bm{A}^{(i)}_{\ell',\ell}$. The $P_N$ equations for the RTE [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} can be derived by firstly approximating $u(\mathbf{x},\bm{\omega})$ with a function $u_{P_N} \in \mathbb{P}_N$ in the form of $$\label{eq:u_appx} u_{P_N}(\mathbf{x},\bm{\omega}) := \bm{m}^{\mathsf{T}}(\bm{\omega})\bm{u}(\mathbf{x}),$$ such that, for any $v\in \mathbb{P}_N$, $$\label{eq:rte_scale_pre_Pn} \left\langle v\left(\bm{\omega}\cdot\nabla u_{P_N}(\mathbf{x},\bm{\omega})+\frac{\sigma_{\mathrm{t}}(\mathbf{x})}{\varepsilon} u_{P_N}(\mathbf{x},\bm{\omega}) -\left(\frac{\sigma_{\mathrm{t}}(\mathbf{x})}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}(\mathbf{x})\right)\bar{u}_{P_N}\right) \right\rangle = \varepsilon\langle v f \rangle.$$ We denote $u_\ell^k := \int_{\mathbb{S}}m_\ell^k u_{P_N} \,\mathrm{d}\bm{\omega}, \quad \ell=0,\cdots,N, \; |k|\le\ell$, and $\bm{u}_\ell = \begin{bmatrix}u_\ell^{-\ell}&\cdots&u_\ell^\ell\end{bmatrix}^{\mathsf{T}}$. Then $\bm{u}= \begin{bmatrix}\bm{u}_0^{\mathsf{T}} & \bm{u}_1^{\mathsf{T}} & \cdots & \bm{u}_N^{\mathsf{T}} \end{bmatrix}^{\mathsf{T}}$. We also use a single index to denote the components in $\bm{m}$ and $\bm{u}$, e.g., $\bm{u}=\begin{bmatrix}u_1 & u_2 & \cdots & u_L\end{bmatrix}^{\mathsf{T}}$. The function $u_{P_N}$ can be expressed as $$u_{P_N} = \bm{u}^{\mathsf{T}}\bm{m} = \sum_{\ell=0}^N\bm{u}^{\mathsf{T}}_\ell\bm{m}_\ell = \sum_{\ell=0}^N\sum_{|k|\leq \ell} {u}^k_\ell{m}_\ell^k=\sum_{i=1}^{L}m_iu_i.$$ Setting $v=\bm{m}$ in [\[eq:rte_scale_pre_Pn\]](#eq:rte_scale_pre_Pn){reference-type="eqref" reference="eq:rte_scale_pre_Pn"} and using [\[eq:u_appx\]](#eq:u_appx){reference-type="eqref" reference="eq:u_appx"} and the fact that $\langle \bm{m}\bm{m}^{\mathsf{T}}\rangle=\bm{I}$, we can reformulate [\[eq:rte_scale_pre_Pn\]](#eq:rte_scale_pre_Pn){reference-type="eqref" reference="eq:rte_scale_pre_Pn"} into a system of hyperbolic equations ($P_N$ equations): [\[eq:rte_scale_Pn_all\]]{#eq:rte_scale_Pn_all label="eq:rte_scale_Pn_all"} $$\begin{aligned} \bm{A}\cdot \nabla \bm{u}(\mathbf{x}) + \varepsilon\sigma_{\mathrm{a}}\bm{u}(\mathbf{x}) + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \bm{R}\bm{u}(\mathbf{x}) &= \varepsilon\bm{f}(\mathbf{x}) \qquad \text{ in } X, \label{eq:rte_scale_Pn}\\ \bm{u}(\mathbf{x})&=\bm{u}(\mathbf{x}+\mathbf{k}) \quad \text{ on } \partial X, \label{eq:rte_scale_Pn_bc} \end{aligned}$$ for arbitrary $\mathbf{x}$ and certain $\mathbf{k}$ such that if $\mathbf{x}\in \partial X$, then $\mathbf{x}+\mathbf{k}\in \partial X$. Here, $\bm{u}(\mathbf{x})\in \mathbb{R}^L$, $\bm{f}=\langle\bm{m}f\rangle$, $\bm{R}=\bm{I}-\mathop{\mathrm{diag}}(1,0,\cdots,0)=\mathop{\mathrm{diag}}(0,1,\cdots,1)$ is diagonal and positive semi-definite, and the dot product between $\bm{A}:=\begin{bmatrix} \left(\bm{A}^{(1)}\right)^\mathsf{T}& \left(\bm{A}^{(2)}\right)^\mathsf{T}& \left(\bm{A}^{(3)}\right)^\mathsf{T}\end{bmatrix}^\mathsf{T}$ and the gradient is understood as $\bm{A}\cdot \nabla := \sum^3_{i=1}\bm{A}^{(i)}\partial_i$ with $\bm{A}^{(i)}=\int_{\mathbb{S}}\omega_i\bm{m}\bm{m}^{\mathsf{T}}\,\mathrm{d}\bm{\omega}$, $i=1,2,3$. Note that $\bm{A}^{(i)},i=1,2,3$ is symmetric and sparse. Indeed, the recursion relation [\[eq:SH_rec\]](#eq:SH_rec){reference-type="eqref" reference="eq:SH_rec"} and orthogonality conditions for the associated Legendre functions [@AH2012; @Claus1966] imply that $\bm{A}^{(i)}_{\ell,\ell'}$ is nonzero only if $\ell' = \ell \pm 1$. Therefore $$\label{eq:A_struc} \bm{A}^{(i)} = \begin{bmatrix} 0 & \bm{A}^{(i)}_{0,1} & 0 & 0 & \dots & 0 \\ \bm{A}^{(i)}_{1,0} & 0 & \bm{A}^{(i)}_{1,2} & 0 & \dots & 0\\ 0 & \bm{A}^{(i)}_{2,1} & 0 & \bm{A}^{(i)}_{2,3} & \dots & 0\\ 0 & 0 & \ddots & \ddots & \ddots & 0\\ \vdots & \vdots & \ddots &\bm{A}^{(i)}_{N-1,N-2}& 0 &\bm{A}^{(i)}_{N-1,N} \\ 0 & 0 & \dots &0 &\bm{A}^{(i)}_{N,N-1}& 0 \\ \end{bmatrix}.$$ Moreover, since $\bm{A}^{(i)}$, $i=1,2,3$ are symmetric, they can be diagonalized as: $$\label{eq:A_sym} \bm{A}^{(i)}=\mathcal{Q}_i\varLambda^{(i)} \mathcal{Q}_{i}^{\mathsf{T}},$$ where $\mathcal{Q}_i$ is a real orthogonal matrix and $\varLambda^{(i)}=\mathop{\mathrm{diag}}(\lambda^{(i)}_1,\lambda^{(i)}_2,\cdots,\lambda^{(i)}_L)$ is real. Let $|\varLambda^{(i)}|:=\mathop{\mathrm{diag}}(|\lambda^{(i)}_1|,|\lambda^{(i)}_2|,\cdots,|\lambda^{(i)}_L|)$, then we define $|\bm{A}^{(i)}|=\mathcal{Q}_i |\varLambda^{(i)}| \mathcal{Q}_i^{\mathsf{T}}$, which will be used for the definition of numerical flux later. Let $D\subseteq X$ be a (possibly lower-dimensional) subdomain of $X$. Given $\bm{u}, \bm{v}\in \left[L^2(D)\right]^L$, define the inner product $$\label{eq:vector_inner} (\bm{u},\bm{v})_D=\int_D \bm{u}^{\mathsf{T}} \bm{v}\,\mathrm{d}\mathbf{x}= \sum_{\ell=0}^N\sum_{|k|\leq \ell} \int_D u^k_\ell\,v^k_\ell \,\mathrm{d}\mathbf{x},$$ as well as the norms $$\label{eq:norm_def} \|\bm{u}\|_{r,D}=\left(\sum_{\ell=0}^N\sum_{|k|\leq \ell}\|u^k_\ell\|_{r,D}^2\right)^{1/2} \quad\text{ for } \bm{u}\in \left[H^r(D)\right]^L.$$ When $r=0$ and $D=X$, we omit the subscripts $0$ and $X$, i.e., $(\bm{u},\bm{v})=(\bm{u},\bm{v})_X$ and $\|\bm{u}\|:=\|\bm{u}\|_{0,X}$. Since $\langle \bm{m}\bm{m}^{\mathsf{T}}\rangle=\bm{I}$, it follows that $$\|u_{P_N}\|_{L^2(X\times\mathbb{S})} =\left(\int_X\langle (\bm{m}^{\mathsf{T}}\bm{u})^2\rangle \,\mathrm{d}\mathbf{x}\right)^{1/2}=\left(\int_X \bm{u}^{\mathsf{T}}\langle\bm{m}\bm{m}^{\mathsf{T}}\rangle\bm{u} \,\mathrm{d}\mathbf{x}\right)^{1/2}=\|\bm{u}\|.$$ Define the space $\bm{V}:=\left\{\bm{u}\in [L^2(X)]^{L}: \bm{A}\cdot\nabla\bm{u}\in [L^2(X)]^{L}\right\}$ with the associated norm $\|\bm{u}\|_{\bm{V}} := \|\bm{u}\| + \|\bm{A}\cdot\nabla\bm{u}\|$, and the space $\bm{W}:=\left\{\bm{u}\in\bm{V}:\bm{u}\right.$ is $1$-periodic in each spatial argument $x_i$, $\left.i=1,2,3\right\}$. We have the following result [@SW2021 Theorem 6]. **Theorem 1**. *Assume that [\[as:rte_scale\]](#as:rte_scale){reference-type="eqref" reference="as:rte_scale"} holds. Then, for any fixed $\varepsilon>0$, the system of $P_N$ equations [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"} has a unique solution $\bm{u}\in\bm{W}$.* The space $\bm{V}$ has a well-defined trace, and the following integration by parts result [@EG2004 Corollary B.57] holds: $\forall \bm{u},\bm{v}\in \bm{V}$, $$\label{eq:int_by_part} (\bm{A}\cdot\nabla\bm{u}, \bm{v})_{D} = (\mathbf{n}\cdot\bm{A}\bm{u}, \bm{v})_{\partial D} - \sum_{i=1}^3(\bm{A}^{(i)}\bm{u},\partial_i \bm{v})_D,$$ for any Lipschitz domain $D\subseteq X$, where $\mathbf{n}\cdot\bm{A} = \sum_{i=1}^3 n_i\bm{A}^{(i)}$. Letting $\bm{v}=\mathbf{e}_i$, $i=1,\cdots,L$ in [\[eq:int_by_part\]](#eq:int_by_part){reference-type="eqref" reference="eq:int_by_part"}, where $\mathbf{e}_i$ is the $i$th column of the identity matrix, yields the divergence formula: $$\label{eq:div} \int_D \bm{A}\cdot\nabla\bm{u} \,\mathrm{d}\mathbf{x}= \int_{\partial D}\mathbf{n}\cdot\bm{A}\bm{u}\,\mathrm{d}\mathbf{x}.$$ The following lemma describes the continuity of the functions in $\bm{V}$. **Lemma 2** ([@SW2021 Lemma 2]). *Assume $\bm{u}\in\bm{V}$. Then for any Lipschitz surface $E\subset X$, there holds $$\label{eq:cont} \int_{E} \mathbf{n}\cdot\bm{A}\big(\bm{u}|_{D_1}-\bm{u}|_{D_2}\big)\,\mathrm{d}\mathbf{x}=\mathbf{0},$$ where $D_1$ and $D_2$ are subdomains of a Lipschitz domain $D\subseteq X$ such that $D=D_1\cup D_2$ with $D_1\cap D_2=\emptyset$ and $\overline{D}_1\cap\overline{D}_2=E$, i.e., $E$ is a shared surface of $D_1$ and $D_2$.* Since $E\subset X$ is arbitrary, by , $\mathbf{n}\cdot\bm{A}\big(\bm{u}|_{D_1}-\bm{u}|_{D_2}\big)=\mathbf{0}$ almost everywhere on any surface in $X$. ## Variational Formulation Multiplying [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"} by an arbitrary function $\bm{v}\in \left[H^1_2(X)\right]^L$, integrating over $X$, using integration-by-parts, and employing the periodic boundary condition [\[eq:rte_scale_Pn_bc\]](#eq:rte_scale_Pn_bc){reference-type="eqref" reference="eq:rte_scale_Pn_bc"}, we get a variational formulation of [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"}: $$\label{eq:rte_var_form} \mathfrak{a}(\bm{u},\bm{v})=\ell(\bm{v}),\quad \forall \bm{v}\in \left[H^1_2(X)\right]^L,$$ where $$\mathfrak{a}(\bm{u},\bm{v})= (\bm{A}\cdot \nabla\bm{u},\bm{v}) + (\bm{Q}\bm{u},\bm{v}) \;\text{ and }\; \ell(\bm{v})= \varepsilon (\bm{f},\bm{v}).$$ Here $$\label{eq:def_Q} \bm{Q} = \varepsilon\sigma_{\mathrm{a}}\bm{I} + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \bm{R} = \begin{bmatrix}\varepsilon\sigma_{\mathrm{a}} & \\ & \frac{\sigma_{\mathrm{t}}}{\varepsilon}\bm{I}_{L-1} \end{bmatrix},$$ where $I$ is the identity matrix. We also define $\sqrt{\bm{Q}} = \mathop{\mathrm{diag}}(\sqrt{\varepsilon\sigma_{\mathrm{a}}}, \; \sqrt{\frac{\sigma_{\mathrm{t}}}{\varepsilon}}\bm{I}_{L-1})$, and denote by $$\label{eq:Q_norm} (\bm{u},\bm{v})_{\bm{Q}}=(\bm{Q}\bm{u},\bm{v}) \quad\text{ and } \quad \|\bm{u}\|_{\bm{Q}}=(\bm{u},\bm{u})_{\bm{Q}}^{1/2}=\|\sqrt{\bm{Q}}\bm{u}\|.$$ By re-scaling $\bm{u}_0$ and $\bm{u}_\ell$, $\ell=1,\cdots, N$ with $\sqrt{\varepsilon}$ and $1/\sqrt{\varepsilon}$, respectively, we have the following lemma on the relation between estimates in terms of $\|\cdot\|$ and $\|\cdot\|_{\bm{Q}}$. **Lemma 3**. *For $\|\cdot\|_{\bm{Q}}$ defined in [\[eq:Q_norm\]](#eq:Q_norm){reference-type="eqref" reference="eq:Q_norm"} with $\bm{Q}$ defined in [\[eq:def_Q\]](#eq:def_Q){reference-type="eqref" reference="eq:def_Q"}, we have $$\|\bm{u}\|\lesssim \left(\sqrt{\varepsilon}+\frac{1}{\sqrt{\varepsilon}}\right)\|\bm{u}\|_{\bm{Q}}.$$* Since $\bm{Q}$ is symmetric and strictly positive definite, $(\cdot, \cdot)_{\bm{Q}}$ is an inner product. Due to the periodic boundary condition, a direct calculation shows that $$\label{eq:a_stab} \mathfrak{a}(\bm{u},\bm{u}) = (\bm{Q}\bm{u},\bm{u}).$$ ## A priori Estimates The purpose of this subsection is to derive some a priori estimates for the spherical harmonic equation [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"}. **Lemma 4**. *Assume that [\[as:rte_scale\]](#as:rte_scale){reference-type="eqref" reference="as:rte_scale"} holds. Let $\bm{u}=\begin{bmatrix}u_1 & u_2 & \cdots & u_{L}\end{bmatrix}^{\mathsf{T}}$ be the solution of [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"}. Then the following estimates hold: $$\label{lem:priori_est_1} \|u_1\| = \mathcal{O}(1) \text{ and } \|u_i\| = \mathcal{O}(\varepsilon), \; i=2,3,\cdots,L.$$ Furthermore, we have $$\|\bm{u}\|=\mathcal{O}(1+\varepsilon).$$* *Proof.* Taking $\bm{v}=\bm{u}$ in [\[eq:rte_var_form\]](#eq:rte_var_form){reference-type="eqref" reference="eq:rte_var_form"} and noting [\[eq:a_stab\]](#eq:a_stab){reference-type="eqref" reference="eq:a_stab"}, we have $$\label{eq:est_iden_1} (\bm{Q}\bm{u},\bm{u}) = \varepsilon (\bm{f},\bm{u}).$$ Applying the Cauchy--Schwarz inequality to [\[eq:est_iden_1\]](#eq:est_iden_1){reference-type="eqref" reference="eq:est_iden_1"} yields $$\begin{gathered} \varepsilon(\sigma_{\mathrm{a}}u_1,u_1)+\frac{1}{\varepsilon}\sum_{i=2}^{L}(\sigma_{\mathrm{t}}u_i,u_i) =\varepsilon\sum_{i=1}^{L}(f_i,u_i)\\ \le \varepsilon(\|f_1/\sqrt{\sigma_{\mathrm{a}}}\|^2/2+\|\sqrt{\sigma_{\mathrm{a}}} u_1\|^2/2) + \varepsilon\sum_{i=2}^{L}\left(\varepsilon^2\|f_i/\sqrt{\sigma_{\mathrm{t}}}\|^2/2+\frac{\|\sqrt{\sigma_{\mathrm{t}}}u_i\|^2}{2\varepsilon^2}\right), \end{gathered}$$ from which we obtain $$\frac{\varepsilon}{2}(\sigma_{\mathrm{a}}u_1,u_1)+\frac{1}{2\varepsilon}\sum_{i=2}^{L}(\sigma_{\mathrm{t}}u_i,u_i) \le \frac{\varepsilon \|f_1/\sqrt{\sigma_{\mathrm{a}}}\|^2}{2} + \varepsilon^3\sum_{i=2}^{L}\|f_i/\sqrt{\sigma_{\mathrm{t}}}\|^2/2,$$ i.e., $$\left\{\begin{aligned} \varepsilon (\sigma_{\mathrm{a}}u_1,u_1)&\le \varepsilon \|f_1/\sqrt{\sigma_{\mathrm{a}}}\|^2 + \varepsilon^3\sum_{i=2}^{L}\|f_i/\sqrt{\sigma_{\mathrm{t}}}\|^2,\\ \frac{1}{\varepsilon}(\sigma_{\mathrm{t}}u_i,u_i)&\le \varepsilon \|f_1/\sqrt{\sigma_{\mathrm{a}}}\|^2 + \varepsilon^3\sum_{i=2}^{L}\|f_i/\sqrt{\sigma_{\mathrm{t}}}\|^2,\quad i=2,3,\cdots,L. \end{aligned}\right.$$ We deduce that $$\label{eq:priori_est_1_entry} \|u_1\| \lesssim \|f_1\|+\varepsilon\sum_{i=2}^{L}\|f_i\|,\quad \|u_i\| \lesssim \varepsilon\|f_1\|+\varepsilon^2\sum_{i=2}^{L}\|f_i\|, \; i=2,3,\cdots,L,$$ which proves the lemma. ◻ ## Discontinuous Galerkin method for spatial discretization {#sec:DG} Equation [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"} is a first-order symmetric hyperbolic system in space, which will be discretized by the DG method in this section. Let $\mathcal{T}_h$ be a regular family of partition of the domain $X=(0,1)^3$ with elements $K$. To avoid unnecessary technicalities, we assume the meshes on the boundary to be periodic, i.e., the surface meshes on two opposite parallel faces of $\partial X$ are identical. Let $\mathcal{E}^{\mathrm{int}}_h$ be the set of all interior faces of $\mathcal{T}_h$ which includes all inner surfaces in each $K$. Define $h_K:= \textrm{diam}(K)$ and $h:=\max_{K\in \mathcal{T}_h}h_K$. Denote by $\mathbf{n}_K=\begin{bmatrix}n_1^K & n_2^K & n_3^K\end{bmatrix}^{\mathsf{T}}$ the unit outward normal to $\partial K$ with respect to the element $K$, and $e$ the generic interface of a mesh element $K\in \mathcal{T}_h$. For an interface $e\subset \partial K$, we use $\mathbf{n}_e$ to denote the unit outward normal to $e$ with respect to $K$. Let $V_{h}^k$ be a discontinuous Galerkin finite element space whose elements are polynomials of degree no more than $k$ when restricted to any element $K$, i.e., $$\label{eq:poly_space} V_h^k=\begin{cases} \{v\in L^2(\mathbb{S}); \;v|_K\in \mathbb{P}_k(K)\;\forall K\in \mathcal{T}_h\}, &\text{ if } \mathcal{T}_h \text{ is triangular/tetrahedral}, \\ \{v\in L^2(\mathbb{S}); \;v|_K\in \mathbb{Q}_k(K)\;\forall K\in \mathcal{T}_h\}, &\text{ if } \mathcal{T}_h \text{ is rectangular/cuboidal}, \end{cases}$$ where $k$ is a nonnegative integer, $\mathbb{P}_k(K)$ denotes the set of all polynomials on $K$ of a total degree no more than $k$, and $\mathbb{Q}_k(K) = \{\sum_j c_j p_j(x)q_j(y)r_j(z): p_j,q_j,r_j$ polynomials of degree $\le k\}$. Define $\bm{V}_{h}^\ell=[V_{h}^k]^{2\ell+1}$, $\ell=0,1,\cdots,N$, and the Cartesian product $\bm{V}_h= \bm{V}_{h}^0\times \bm{V}_{h}^1\times \cdots \times \bm{V}_{h}^N = [V_{h}^k]^{L}$. Denote $E+F:=\{u+v:\, u\in E, \, v\in F\}$, for any two function spaces $E$ and $F$. For any given element $K\in\mathcal{T}_h\subset \mathbb{R}^3$, and any function $\bm{f}\in \bm{V}_h + [C(\overline{X})]^L$, we define the inside and outside values of $\bm{f}$ (with respect to $K$) on the interface $\partial K$ as, respectively, $$\begin{aligned} \bm{f}^- (\mathbf{x}) &=\lim\limits_{\epsilon \to 0^+}\bm{f}(\mathbf{x}- \epsilon\,\mathbf{n}_K(\mathbf{x})), \quad \;\;\;\forall \mathbf{x}\in \partial K,\\ \bm{f}^+ (\mathbf{x}) &=\begin{cases} \displaystyle\lim_{\epsilon \to 0^+}\bm{f}(\mathbf{x}+ \epsilon\,\mathbf{n}_K(\mathbf{x})), & \forall \mathbf{x}\in \partial K\backslash\partial X,\\ \bm{f}^- (\tilde{\mathbf{x}}), & \forall \mathbf{x}\in \partial K\cap\partial X, \end{cases} \end{aligned}$$ where $\tilde{\mathbf{x}}$ is a point on $\partial X$ corresponding to $\mathbf{x}$ associated with the periodic boundary condition. Note that the extensions of the outside values on $\partial X$ reflect the periodic boundary conditions. The jump in $\bm{f}\in \bm{V}_h$ with respect to $K$ is $[\![\bm{f}]\!]=\bm{f}^+-\bm{f}^-$, and the average of $\bm{f}$ is denoted by $\{\!\!\{\bm{f}\}\!\!\}=(\bm{f}^+ + \bm{f}^-)/2$. For any $\bm{u}_h,\bm{v}_h\in \bm{V}_h + [C(\overline{X})]^L$, we define $$\begin{aligned} \label{eq:a} \mathfrak{a}_h(\bm{u}_h,\bm{v}_h) &= \sum_{K\in\mathcal{T}_h}\bigg\{ \Big(\mathbf{n}_K\cdot\overrightarrow{\bm{A}\bm{u}_h}, \, \bm{v}_h^-\Big)_{\partial K} - \sum_{i=1}^3(\bm{A}^{(i)}\bm{u}_h, \partial_i \bm{v}_h)_K + (\bm{Q}\bm{u}_h,\bm{v}_h)_K \bigg\}, \\ \mathfrak{f}(\bm{v}_h) &= \varepsilon\sum_{K\in\mathcal{T}_h}(\bm{f},\bm{v}_h)_K. \end{aligned}$$ Here, following [@HLL1983], we define the numerical flux $\mathbf{n}_K\cdot\overrightarrow{\bm{A}\bm{u}_h}$ as $$\label{eq:flux_Pn} \mathbf{n}_K\cdot\overrightarrow{\bm{A}\bm{u}_h} = \mathbf{n}_K\cdot\bm{A}\{\!\!\{\bm{u}_h\}\!\!\} -\frac{1}{2}|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{u}_h]\!],$$ where $|\mathbf{n}_K|\cdot\bm{D} = \sum_{i=1}^3 |n_i^K|\bm{D}^{(i)}$ and, assuming the upwind flux, $\bm{D}^{(i)}=|\bm{A}^{(i)}|$. Applying integration by parts to [\[eq:a\]](#eq:a){reference-type="eqref" reference="eq:a"} leads to an equivalent form of the bilinear form $\mathfrak{a}_h$, which will be useful later: $$\label{eq:def_a_2} \mathfrak{a}_h(\bm{u}_h,\bm{v}_h) = \sum_{K\in\mathcal{T}_h}\bigg\{(\bm{A}\cdot\nabla\bm{u}_h, \bm{v}_h)_K - (\bm{u}_h^-, \mathbf{n}_K\cdot\overleftarrow{\bm{A}\bm{v}_h})_{\partial K} + (\bm{Q}\bm{u}_h,\bm{v}_h)_K\bigg\},$$ where the downwind trace $\mathbf{n}_K\cdot\overleftarrow{\bm{A}\bm{v}_h}$ is defined by $$\mathbf{n}_K\cdot\overleftarrow{\bm{A}\bm{v}_h} = \mathbf{n}_K\cdot\bm{A}\{\!\!\{\bm{v}_h\}\!\!\} +\frac{1}{2}|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{v}_h]\!].$$ We can present the fully discrete spherical harmonic DG scheme for [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} as follows. **Problem 1**. *Find $\bm{u}_h\in \bm{V}_h$ such that $$\label{eq:bilinear} \mathfrak{a}_h(\bm{u}_h,\bm{v}_h) = \mathfrak{f}(\bm{v}_h) \quad \forall \bm{v}_h\in \bm{V}_h.$$* **Remark 1**. *Note that the periodic boundary condition [\[eq:rte_scale_Pn_bc\]](#eq:rte_scale_Pn_bc){reference-type="eqref" reference="eq:rte_scale_Pn_bc"} is not imposed into the definition of the DG finite element space. Instead, we use it implicitly in the definition of the numerical flux [\[eq:flux_Pn\]](#eq:flux_Pn){reference-type="eqref" reference="eq:flux_Pn"} during the construction of the spherical harmonics DG method.* **Remark 2**. *Following , it is easy to validate that the discrete problem [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} is consistent, i.e., given the exact solution $\bm{u}$ of the $P_N$ model [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"}, we have $\mathfrak{a}_h(\bm{u},\bm{v}_h) = \mathfrak{f}(\bm{v}_h)$ for any $\bm{v}_h\in \bm{V}_h$, and therefore, the following Galerkin orthogonality holds $$\label{eq:go} \mathfrak{a}_h(\bm{u}-\bm{u}_h,\bm{v}_h) = 0 \quad \forall \bm{v}_h\in \bm{V}_h.$$* To study the stability and well-posedness of , we define the following norm: $$\label{def:norm} |||\bm{v}_h|||_h = \left(\frac{1}{4}\sum_{K\in\mathcal{T}_h} \Big(|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{v}_h]\!], [\![\bm{v}_h]\!]\Big)_{\partial K} + (\bm{Q}\bm{v}_h,\bm{v}_h)\right)^\frac{1}{2}, \quad \forall \bm{v}_h\in \bm{V}_{h}.$$ Note that, for $i=1,2,3$, $0\le |n_i^K|\le 1$ and $\|\mathbf{n}_K\|=1$. Since $\bm{D}^{(i)}$ are positive semi-definite and $\bm{Q}$ is positive definite for any $\varepsilon>0$, we verify that $|||\cdot|||_h$ is a norm. **Lemma 5** (Stability). *Under conditions [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"} and [\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"}, we have $$\label{eq:stab} |||\bm{v}_h|||^2_h = \mathfrak{a}_h(\bm{v}_h,\bm{v}_h), \quad \forall \bm{v}_h\in \bm{V}_h.$$* *Proof.* For any $\bm{v}_h\in \bm{V}_h$, we have, by the definition [\[eq:a\]](#eq:a){reference-type="eqref" reference="eq:a"}, that $\mathfrak{a}_h(\bm{v}_h,\bm{v}_h) = \mathrm{I} +\Pi + \mathrm{III}$, where $$\begin{aligned} %\label{eq:a_decomp_1} \mathrm{I} &:= \sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 \Big(n_i^K\bm{A}^{(i)}\{\!\!\{\bm{v}_h\}\!\!\} -\frac{1}{2}|n_i^K|\bm{D}^{(i)}[\![\bm{v}_h]\!], \bm{v}^{-}_h\Big)_{\partial K}, \\ \mathrm{II} &:= -\sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 (\bm{A}^{(i)}\bm{v}_h,\partial_i \bm{v}_h)_K, \quad \mathrm{III} := \sum_{K\in\mathcal{T}_h} (\bm{Q}\bm{v}_h,\bm{v}_h)_K. \end{aligned}$$ For the first term, since $A$ is symmetric, we can simplify it as $$%\label{eq:rte_test_1} \mathrm{I} = \sum_{K\in\mathcal{T}_h} \frac{1}{2}\Big(\mathbf{n}_K\cdot\bm{A} \bm{v}^{-}_h, \bm{v}^{-}_h\Big)_{\partial K} + \sum_{K\in\mathcal{T}_h} \frac{1}{4} \Big(|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{v}_h]\!], [\![\bm{v}_h]\!]\Big)_{\partial K}.$$ Since $(\bm{A}^{(i)}\bm{v}_h,\partial_i \bm{v}_h)_K=(\varLambda^{(i)} \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h,\partial_i \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h)_K$, the second term can be handled as: $$%\label{eq:rte_test_2} \mathrm{II} = -\sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 \frac{1}{2} (n_i^K\varLambda^{(i)} \mathcal{Q}_i^{\mathsf{T}}\bm{v}^{-}_h, \mathcal{Q}_i^{\mathsf{T}}\bm{v}^{-}_h)_{\partial K} = -\sum_{K\in\mathcal{T}_h} \frac{1}{2} (\mathbf{n}_K\cdot\bm{A}\bm{v}^{-}_h, \bm{v}^{-}_h)_{\partial K}.$$ Therefore, we have $\mathrm{I} + \mathrm{II} = \sum_{K\in\mathcal{T}_h} \frac{1}{4} \Big(|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{v}_h]\!], [\![\bm{v}_h]\!]\Big)_{\partial K}$, and [\[eq:stab\]](#eq:stab){reference-type="eqref" reference="eq:stab"} follows. ◻ All norms in the finite-dimensional space $\bm{V}_h$ should be equivalent, which ensures the continuity of the bilinear form. The following corollary is a direct consequence of the above result, due to the well-known Lax--Milgram lemma. **Corollary 6**. *Under the assumptions [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"} and [\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"}, the spherical harmonic DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} has a unique solution.* # Error analysis {#sec:err_ana} In this section, we provide the error analysis of the SH-DG approximation for solving the RTE [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} with periodic boundary conditions. Let $u$ and $\bm{u}_h$ be the solutions of [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} and [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"}, respectively. The total numerical error $u-\bm{m}^{\mathsf{T}}\bm{u}_h$ can be divided into two parts: $$u-\bm{m}^{\mathsf{T}}\bm{u}_h = (u- \bm{m}^{\mathsf{T}}\bm{u}) + (\bm{m}^{\mathsf{T}}\bm{u}-\bm{m}^{\mathsf{T}}\bm{u}_h).$$ Since $\bm{m}^{\mathsf{T}}\bm{u}=u_{P_N}$ in [\[eq:u_appx\]](#eq:u_appx){reference-type="eqref" reference="eq:u_appx"}, the first part stands for the error due to the angular discretization, while the second part accounts for the error of the spatial discretization. ## Error analysis for the angular discretization We start with unraveling the angular discretization error $\tau(\mathbf{x},\bm{\omega}) := u(\mathbf{x},\bm{\omega}) - u_{P_N}(\mathbf{x},\bm{\omega})$ arising from the semi-discretization [\[eq:rte_scale_pre_Pn\]](#eq:rte_scale_pre_Pn){reference-type="eqref" reference="eq:rte_scale_pre_Pn"}. Define $\mathcal{P}_N u = \bm{m}^{\mathsf{T}}\langle \bm{m}u\rangle$, then $\mathcal{P}_N$ is the $L^2$-orthogonal projection of a generic function $u$ on $\mathbb{S}$ onto $\mathbb{P}_N$. The following two lemmas will be useful in the proof of the main result. **Lemma 7**. *[@STW2011 Theorem 2.1] Assume $v\in H^q(\mathbb{S})$ for some $q>0$, then $$\|v-\mathcal{P}_N v\|_{L^2(\mathbb{S})} \lesssim N^{-q}\|v\|_{H^q(\mathbb{S})}.$$* **Lemma 8**. *[@FHK2016] The matrix $\bm{A}^{(i)}_{\ell,\ell+1}$, $i=1,2,3$ defined in [\[eq:SH_rec\]](#eq:SH_rec){reference-type="eqref" reference="eq:SH_rec"} is uniformly bounded in the induced $2$-norm.* The following bound for $\bm{A}^{(i)}$, $i=1,2,3$ follows from the recursion relation [\[eq:SH_rec\]](#eq:SH_rec){reference-type="eqref" reference="eq:SH_rec"}. **Corollary 9**. *The matrix $\bm{A}^{(i)}$, $i=1,2,3$ is uniformly bounded in the induced $2$-norm.* The main result is the following theorem on the error of angular discretization. The proof follows [@SW2021] with additional attention given to the parameter $\varepsilon$. **Theorem 10**. *Let $u_{P_N}=\bm{m}^{\mathsf{T}}\bm{u}$ be the solution to [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"}. Assume the solution $u$ to [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} satisfies the additional regularity assumptions $$\label{as:reg_Omega} \text{for some } q>0, \, u\in L^2(X; H^q(\mathbb{S})), \; \partial_i u\in L^2(X; H^q(\mathbb{S})),\; i=1,2,3.$$ Then $$\| u- u_{P_N}\|_{L^2(X\times\mathbb{S})}\lesssim N^{-q} \left( \|u\|_{L^2(X; H^q(\mathbb{S}))} + \sum_{i=1}^{3}\|\partial_i u\|_{L^2(X; H^q(\mathbb{S}))} \right).$$* *Proof.* We split the error $\tau = u - u_{P_N}$ into the projection error $\eta=u-\mathcal{P}_N u$ and a remainder $\xi=\mathcal{P}_N u - u_{P_N} = \bm{m}^{\mathsf{T}}(\langle \bm{m}u\rangle - \bm{u})$ which is an element of $\mathbb{P}_N$, that is, $\tau = \eta + \xi$. Therefore, $$\label{eq:ang_err_split} \|\tau\|_{L^2(X\times\mathbb{S})} \le \|\eta\|_{L^2(X\times\mathbb{S})} + \|\xi\|_{L^2(X\times\mathbb{S})}.$$ By the assumption [\[as:reg_Omega\]](#as:reg_Omega){reference-type="eqref" reference="as:reg_Omega"} and , we have $$\label{eq:ang_intp} \|\eta\|_{L^2(X\times\mathbb{S})}=\| u-\mathcal{P}_N u \|_{L^2(X\times\mathbb{S})} \lesssim N^{-q}\|u\|_{L^2(X; H^q(\mathbb{S}))}.$$ The next step is to estimate $\xi$. First, it follows from [\[eq:rte_scale\]](#eq:rte_scale){reference-type="eqref" reference="eq:rte_scale"} and [\[eq:rte_scale_pre_Pn\]](#eq:rte_scale_pre_Pn){reference-type="eqref" reference="eq:rte_scale_pre_Pn"} that, for all $v\in \mathbb{P}_N$, $$\label{eq:error_total} \left\langle\left(\bm{\omega}\cdot \nabla \tau +\frac{\sigma_{\mathrm{t}}}{\varepsilon}\tau\right)v\right\rangle = \left\langle\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)v\bar{\tau}\right\rangle.$$ Denote $\bm{\xi}=\langle \bm{m}u\rangle - \bm{u}$, then $\xi = \bm{m}^{\mathsf{T}}\bm{\xi}$. Taking $v=\xi$ in [\[eq:error_total\]](#eq:error_total){reference-type="eqref" reference="eq:error_total"} gives $$\begin{gathered} \label{eq:error_eq} \langle \xi(\bm{\omega}\cdot \nabla \eta)\rangle + \frac{1}{2}\langle \bm{\omega}\cdot \nabla (\xi)^2\rangle + \frac{\sigma_{\mathrm{t}}}{\varepsilon}\langle\xi^2\rangle + \frac{\sigma_{\mathrm{t}}}{\varepsilon}\langle\eta\xi\rangle \\ = \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\langle \bm{\xi}^{\mathsf{T}}\bm{m}\, \bar{\eta}\rangle + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\langle \bm{\xi}^{\mathsf{T}}\bm{m}\, \overline{\bm{m}^{\mathsf{T}}\bm{\xi}}\rangle. \end{gathered}$$ By the recursion relation [\[eq:SH_rec\]](#eq:SH_rec){reference-type="eqref" reference="eq:SH_rec"} of the spherical harmonics, $$\begin{gathered} \langle \xi(\bm{\omega}\cdot \nabla \eta)\rangle = \langle \bm{\xi}^{\mathsf{T}}\bm{m}\,(\bm{\omega}\cdot \nabla \eta)\rangle = \bm{\xi}^{\mathsf{T}} \nabla\cdot\langle\bm{\omega}\, \bm{m}\eta\rangle = \bm{\xi}^{\mathsf{T}} \nabla\cdot\langle \bm{A}\, \bm{m}\eta\rangle \\ = \bm{\xi}^{\mathsf{T}} \nabla\cdot \langle \bm{A}\, \bm{m} (u-\mathcal{P}_N u)\rangle = \bm{\xi}^{\mathsf{T}}_N \bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle. \end{gathered}$$ Since $\mathcal{P}_N$ is an orthogonal projection on $\mathbb{S}$, $\sigma_{\mathrm{t}}\langle\eta\xi\rangle=0$, and $$\langle \bm{\xi}^{\mathsf{T}}\bm{m}\, \bar{\eta}\rangle = \bm{\xi}^{\mathsf{T}}\langle\bm{m}\, \bar{\eta}\rangle = \bm{\xi}^{\mathsf{T}}(\bm{I}-\bm{R})\langle \bm{m}(u-\mathcal{P}_N u)\rangle =0.$$ The last term on the right-hand side of [\[eq:error_eq\]](#eq:error_eq){reference-type="eqref" reference="eq:error_eq"} can be rewritten as $$\langle \bm{\xi}^{\mathsf{T}}\bm{m}\, \overline{\bm{m}^{\mathsf{T}}\bm{\xi}}\rangle = \bm{\xi}^{\mathsf{T}}\langle \bm{m}\, \overline{\bm{m}^{\mathsf{T}}}\rangle\bm{\xi} = \bm{\xi}^{\mathsf{T}} \langle \bm{m}\, (\bm{I}-\bm{R})\bm{m}^{\mathsf{T}}\rangle\bm{\xi} = \bm{\xi}^{\mathsf{T}} (\bm{I}-\bm{R})\bm{\xi}.$$ Therefore, [\[eq:error_eq\]](#eq:error_eq){reference-type="eqref" reference="eq:error_eq"} can be reformulated into $$\bm{\xi}^{\mathsf{T}}_N \bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle + \frac{1}{2}\langle \bm{\omega}\cdot \nabla (\bm{\xi}^{\mathsf{T}}\bm{\xi})\rangle + \varepsilon\sigma_{\mathrm{a}}\bm{\xi}^{\mathsf{T}}\bm{\xi} + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\bm{\xi}^{\mathsf{T}} \bm{R}\bm{\xi} = 0.$$ Integrating the above equation over $X$ and noting that $\xi$ is periodic on $\partial X$, and therefore $\int_{\partial X}\bm{\omega}\cdot\mathbf{n}\, (\bm{\xi}^{\mathsf{T}}\bm{\xi})\,\mathrm{d}\mathbf{x}\,\mathrm{d}\bm{\omega}=0$, we have $$(\bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle, \bm{\xi}_N) + (\varepsilon\sigma_{\mathrm{a}}\bm{\xi},\bm{\xi}) + \left(\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\bm{R}\bm{\xi}, \bm{\xi}\right) = 0.$$ Since $\bm{R}$ is positive semi-definite, applying the Cauchy--Schwarz inequality yields $$(\bm{Q}\bm{\xi},\bm{\xi})\le \|\bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle\|\cdot \|\bm{\xi}_N\|\le \frac{\varepsilon}{2\sigma_{\mathrm{t}}}\|\bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle\|^2 + \frac{\sigma_{\mathrm{t}}}{2\varepsilon}\|\bm{\xi}_N\|^2.$$ Therefore, by the definition [\[eq:def_Q\]](#eq:def_Q){reference-type="eqref" reference="eq:def_Q"} of $\bm{Q}$, $$\label{eq:err_cls} \|\sqrt{\bm{Q}}\bm{\xi}\| \lesssim \varepsilon\|\bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle\|.$$ The term on the right-hand side of the above inequality can be considered as the closure error. Using Lemma [Lemma 7](#lem:interp_ang){reference-type="ref" reference="lem:interp_ang"} and Lemma [Lemma 8](#lem:uni_bnd_A){reference-type="ref" reference="lem:uni_bnd_A"}, the closure error in [\[eq:err_cls\]](#eq:err_cls){reference-type="eqref" reference="eq:err_cls"} can be further estimated as $$\begin{aligned} \|\bm{A}_{N,N+1}\cdot \nabla \langle \bm{m}_{N+1} u\rangle\| &\lesssim \sum_{i=1}^{3}\|\langle \bm{m}_{N+1} \partial_i u\rangle\| = \sum_{i=1}^{3}\|\langle (\mathcal{P}_{N+1}- \mathcal{P}_N) \partial_i u\rangle\|\\ & \le \sum_{i=1}^{3}\left(\|\langle (\mathcal{I} - \mathcal{P}_{N+1} )\partial_i u\rangle\| + \|\langle (\mathcal{I} - \mathcal{P}_{N} )\partial_i u\rangle\| \right)\\ & \le 2\sum_{i=1}^{3}\|\langle (\mathcal{I}- \mathcal{P}_N) \partial_i u\rangle\| \lesssim N^{-q}\sum_{i=1}^{3}\|\partial_i u\|_{L^2(X; H^q(\mathbb{S}))}. \end{aligned}$$ This, together with [\[eq:ang_err_split\]](#eq:ang_err_split){reference-type="eqref" reference="eq:ang_err_split"}, [\[eq:ang_intp\]](#eq:ang_intp){reference-type="eqref" reference="eq:ang_intp"} and [\[eq:err_cls\]](#eq:err_cls){reference-type="eqref" reference="eq:err_cls"}, completes the proof. ◻ ## Error Analysis for the numerical solution of the spherical harmonic equation {#sec:asym_error_analysis} For the purpose of error analysis, we make the following assumptions on the regularity of $\bm{u}$. **Assumption 1**. *For some $r>1$, $\bm{u}\in \left[H^r(X)\right]^L$, and furthermore, $$\label{eq:reg_assump} \|u_1\|_{r,X} = \mathcal{O}(1) \text{ and } \|u_i\|_{r,X} = \mathcal{O}(\varepsilon), \; i=2,3,\cdots,L.$$ In other words, we assume $$\label{eq:reg_assump_u} \|\bm{u}\|_{r,X}=\mathcal{O}(1+\varepsilon).$$* **Remark 3**. *From , we know that the above assumptions are true when $r=0$. Assume $\sigma_{\mathrm{t}}(\mathbf{x})$, $\sigma_{\mathrm{s}}(\mathbf{x})$, and $f(\mathbf{x})$ are smooth enough and $\partial_j\bm{u}(\mathbf{x}) = \partial_j\bm{u}(\mathbf{x}+\mathbf{k})$ on $\partial X$. Taking any partial derivative $\partial_j$ ($j=1,2,3$) of [\[eq:rte_scale_Pn\]](#eq:rte_scale_Pn){reference-type="eqref" reference="eq:rte_scale_Pn"} with respect to $x$, $y$ or $z$, respectively, we have $$\bm{A}\cdot \nabla (\partial_j\bm{u}) + \bm{Q}(\partial_j\bm{u}) = \varepsilon\partial_j\bm{f} -\varepsilon(\partial_j\sigma_{\mathrm{a}})\bm{u} - \partial_j\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \bm{R}\bm{u} \quad \text{ in } X.$$ By the estimate [\[eq:priori_est_1\_entry\]](#eq:priori_est_1_entry){reference-type="eqref" reference="eq:priori_est_1_entry"}, we have $$\|\partial_j u_1\|\lesssim \|\partial_j f_1\|+\|u_1\|= \mathcal{O}(1),$$ and $$\|\partial_j u_i\|\lesssim \varepsilon (\|f_1\|+\|u_1\|) + \varepsilon^2\sum_{i=2}^{L}\left(\|f_i\|+\|u_i\|+\|u_i\|/\varepsilon^2\right)=\mathcal{O}(\varepsilon), \quad i=2,3,\cdots,L.$$ Hence, remains true for $\partial_i\bm{u}$, i.e., holds for $r=1$. Results for $r\ge2$ can also be derived analogously.* We begin with two lemmas which will be used later. The first lemma gives the convergence rate of the interpolation. For any $K\in\mathcal{T}_h$, define $\mathcal{I}_h$ to be an interpolation operator from $L^2(K)$ onto $\mathbb{P}_k(K)$ which satisfies the following requirements: (i) the restriction of $\mathcal{I}_h$ to any face $e\subset K$ is uniquely determined by the interpolation points on $e$; and (ii) two adjacent elements have the same interpolation points on their shared edge/surface. For example, the commonly employed nodal finite elements satisfy such requirements (see, e.g., [@BS2008 Chapter 3] or [@C2002 Theorem 2.2.1]). Then by using the trace theorem and the scaling argument, we can easily obtain the following result (cf. [@AH2009; @BS2008; @C2002]). **Lemma 11** (Polynomial interpolation error estimate). *For all $v\in H^r(K)$ with $r>0$ and $K\in \mathcal{T}_h$, $$\label{eq:interp_est} \|(I-\mathcal{I}_h) v\|_{q,K}\lesssim h_K^{\min \{r,k+1\}-q}\|v\|_{r,K}, \quad \|(I-\mathcal{I}_h) v\|_{0,\partial K}\lesssim h_K^{\min \{r,k+1\}-1/2}\|v\|_{r,K},$$ where $k$ is the degree of the polynomial used in [\[eq:poly_space\]](#eq:poly_space){reference-type="eqref" reference="eq:poly_space"} for the definition of $V^k_h$.* The second lemma is similar to the second Strang lemma in the error analysis of nonconforming finite element methods but in the format of vector arguments (cf. [@HHE2010]). **Lemma 12**. *Let $\{\bm{Z}_h\}_{h>0}:=\{[Z_h]^L\}_{h>0}$ be a family of finite-dimensional product spaces equipped with norms $\{\|\cdot\|_h\}_{h>0}$. Let $\mathfrak{b}_h(\cdot,\cdot)$ be a uniformly coercive bilinear form over $\bm{Z}_h\times \bm{Z}_h$, i.e., there exists a positive constant $\gamma$ independent of $h$ such that $$\label{eq:stab_lem} \gamma\|\bm{z}_h\|^2_h\le \mathfrak{b}_h(\bm{z}_h,\bm{z}_h), \quad \forall \bm{z}_h\in \bm{Z}_h.$$ Let $\bm{Z}=[Z]^L$ be the product space of an (infinite dimensional) function space $Z$ and assume $\bm{v}\in \bm{Z}$ is a vector of functions and $\mathfrak{b}_h$ can be extended to $(\bm{Z}+\bm{Z}_h)\times \bm{Z}_h$ such that $|\mathfrak{b}_h(\bm{v},\bm{z}_h)|<C$ for all $\bm{z}_h\in \bm{Z}_h$, and $\bm{v}_h$ is an element in $\bm{Z}_h$ satisfying $$\label{eq:rte_scale_lem_1_asspt2} \mathfrak{b}_h(\bm{v}-\bm{v}_h,\bm{z}_h)=0, \quad \forall \bm{z}_h\in \bm{Z}_h.$$ Furthermore, norms $\|\cdot\|_h$ can also be defined on $\bm{Z}+\bm{Z}_h$. Then $$\label{eq:error_inq1} \|\bm{v}-\bm{v}_h\|_h\le \inf_{\bm{z}_h\in \bm{Z}_h}\left\{\|\bm{v}-\bm{z}_h\|_h + \frac{1}{\gamma}\sup_{\hat{\bm{z}}_h\in \bm{Z}_h}\frac{\mathfrak{b}_h(\bm{v}-\bm{z}_h,\hat{\bm{z}}_h)}{\|\hat{\bm{z}}_h\|_h}\right\}.$$* ### Error analysis for non-constant elements We now establish the following error bound between the DG approximation $\bm{u}_h$ and the solution $\bm{u}$ to the $P_N$ system [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"} when $k\ge 1$. **Theorem 13**. *Assume that [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"}--[\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} and hold. Consider only $k\ge 1$. The spherical harmonic DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} admits the following error estimate: $$\label{eq:est_total_general} \|\bm{u}-\bm{u}_h\|_{\bm{Q}} \lesssim \sqrt{\varepsilon}\Big(h^{\min\{r,k+1\}-1} \big(1+\mathcal{O}(\varepsilon)\big) + h^{\min\{r,k+1\}} \Big).$$* *Proof.* Since $\mathfrak{a}_h(\bm{v}_h,\bm{v}_h) = |||\bm{v}_h|||^2_h\ge \|\bm{v}_h\|^2_{\bm{Q}}$ by , the $\bm{Q}$ norm can be employed in [\[eq:error_inq1\]](#eq:error_inq1){reference-type="eqref" reference="eq:error_inq1"}. Then it follows from the Galerkin orthogonality [\[eq:go\]](#eq:go){reference-type="eqref" reference="eq:go"} and that $$\label{eq:error_inq2} \|\bm{u}-\bm{u}_h\|_{\bm{Q}} \lesssim \|\bm{u} - \mathcal{I}_h\bm{u}\|_{\bm{Q}} + \sup_{\bm{v}_h\in \bm{V}_h}\frac{\mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h)}{\|\bm{v}_h \|_{\bm{Q}}},$$ where $\mathcal{I}_h$ denotes the interpolation operator onto $\bm{V}_h$ in an element-wise way, i.e., for $\bm{v}\in L^2(X)$, $\mathcal{I}_h\bm{v}|_K = \mathcal{I}_K(\bm{v})$. By the definition of $\|\cdot\|_{\bm{Q}}$, , and , we have $$\label{eq:error_inq_part_one} \|\bm{u} - \mathcal{I}_h\bm{u}\|_{\bm{Q}} \lesssim h^{\min\{r,k+1\}} \|\sqrt{\bm{Q}}\bm{u}\|_{r,X} \lesssim \sqrt{\varepsilon} h^{\min\{r,k+1\}}.$$ On the other hand, by the definition of $\mathfrak{a}_h(\cdot,\cdot)$, we have $$\label{eq:error_inq_bi_parts} \mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h) = \mathbb{I}_1 +\mathbb{I}_2 + \mathbb{I}_3,$$ where $$\begin{aligned} \label{eq:a_decomp_2} \mathbb{I}_1 &:= -\sum_{K\in\mathcal{T}_h}\sum_{i=1}^3 ((\bm{u}-\mathcal{I}_h\bm{u})^-, n_i^K\overleftarrow{\bm{A}^{(i)}\bm{v}_h})_{\partial K}, \\ \mathbb{I}_2 &:= \sum_{K\in\mathcal{T}_h} (\bm{A}\cdot \nabla (\bm{u}-\mathcal{I}_h\bm{u}),\bm{v}_h)_K, \quad \mathbb{I}_3 := \sum_{K\in\mathcal{T}_h} \left(\bm{Q}(\bm{u}-\mathcal{I}_h\bm{u}),\bm{v}_h\right)_K. \end{aligned}$$ Owing to , $\mathbf{n}\cdot\bm{A}\,[\![\bm{u}]\!]=\mathbf{0}$ almost everywhere. Furthermore, due to the requirement of the interpolation operator $\mathcal{I}_h$, $\mathcal{I}_h\bm{u}$ is also continuous a.e. when $k\ge 1$, which implies $[\![\mathcal{I}_h\bm{u}]\!]=0$ a.e. when $k\ge 1$. Therefore, for the first term, $$\label{eq:error_inq_bi_1} \mathbb{I}_1 = \frac{1}{2}\sum_{K\in\mathcal{T}_h} \Big( [\![\bm{u}-\mathcal{I}_h\bm{u}]\!], \mathbf{n}_K\cdot\overleftarrow{\bm{A}\bm{v}_h}\Big)_{\partial K} = \frac{1}{2}\sum_{K\in\mathcal{T}_h} \Big( [\![\bm{u}]\!]-[\![\mathcal{I}_h\bm{u}]\!], \mathbf{n}_K\cdot\overleftarrow{\bm{A}\bm{v}_h}\Big)_{\partial K} = 0.$$ To bound $\mathbb{I}_2$, recalling that $\bm{A}^{(i)}$ has the fine structure [\[eq:A_struc\]](#eq:A_struc){reference-type="eqref" reference="eq:A_struc"}, we have $$\begin{aligned} \mathbb{I}_2 &= \sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 \left(\bm{A}^{(i)}\partial_i (\bm{u}-\mathcal{I}_h\bm{u}),\bm{v}_h\right)_K \nonumber\\ &= \sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 \left(\sqrt{\bm{Q}}^{-1}\begin{bmatrix}\bm{A}^{(i)}_{0,1}\partial_i (\bm{u}_1-\mathcal{I}_h\bm{u}_1)\\ \bm{A}^{(i)}_{1,0}\partial_i (\bm{u}_0-\mathcal{I}_h\bm{u}_0) + \bm{A}^{(i)}_{1,2}\partial_i (\bm{u}_2-\mathcal{I}_h\bm{u}_2) \\ \vdots \\ \bm{A}^{(i)}_{N,N-1}\partial_i (\bm{u}_{N-1}-\mathcal{I}_h\bm{u}_{N-1}) \end{bmatrix},\sqrt{\bm{Q}}\begin{bmatrix}\bm{v}_{0,h}\\\bm{v}_{1,h}\\\vdots\\\bm{v}_{N,h}\end{bmatrix}\right)_K. \nonumber \end{aligned}$$ We use an inverse inequality and the definition of $\bm{Q}$ in [\[eq:def_Q\]](#eq:def_Q){reference-type="eqref" reference="eq:def_Q"}, and proceed as follows: $$\begin{aligned} \label{eq:error_inq_bi_2} |\mathbb{I}_2| &\lesssim \sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 h^{-1}\left\|\begin{bmatrix}\varepsilon^{-1/2}\bm{A}^{(i)}_{0,1} (\bm{u}_1-\mathcal{I}_h\bm{u}_1)\\ \sqrt{\varepsilon}\bm{A}^{(i)}_{1,0} (\bm{u}_0-\mathcal{I}_h\bm{u}_0) + \sqrt{\varepsilon}\bm{A}^{(i)}_{1,2} (\bm{u}_2-\mathcal{I}_h\bm{u}_2) \\ \vdots \\ \sqrt{\varepsilon}\bm{A}^{(i)}_{N,N-1} (\bm{u}_{N-1}-\mathcal{I}_h\bm{u}_{N-1}) \end{bmatrix}\right\|_K \left\|\sqrt{\bm{Q}}\bm{v}_h\right\|_K \nonumber\\ &\lesssim h^{-1}\sum_{K\in\mathcal{T}_h} h^{\min\{r,k+1\}} \Big(\varepsilon\|\bm{u}_0\|^2_{r,K} + (\varepsilon+\varepsilon^{-1})\|\bm{u}_1\|^2_{r,K} \nonumber\\ &\phantom{=}\qquad + 2\varepsilon\|\bm{u}_2\|^2_{r,K} + \cdots + \varepsilon\|\bm{u}_N\|^2_{r,K}\Big)^{1/2} \|\bm{v}_h\|_{\bm{Q}} \nonumber\\ &\lesssim \sqrt{\varepsilon}h^{\min\{r,k+1\}-1} \big(1+\mathcal{O}(\varepsilon)\big)\|\bm{v}_h\|_{\bm{Q}}. \end{aligned}$$ To bound $\mathbb{I}_3$, using the Cauchy-Schwarz inequality, $$\begin{aligned} \label{eq:error_inq_bi_3} |\mathbb{I}_3| &\lesssim \sum_{K\in\mathcal{T}_h}\left\|\sqrt{\bm{Q}}(\bm{u}-\mathcal{I}_h\bm{u})\right\|_K\left\|\sqrt{\bm{Q}}\bm{v}_h\right\|_K \lesssim \sum_{K\in\mathcal{T}_h}\left\|\sqrt{\bm{Q}}(\bm{u}-\mathcal{I}_h\bm{u})\right\|_K\|\bm{v}_h\|_{\bm{Q}} \nonumber\\ &\lesssim h^{\min\{r,k+1\}} \left\|\sqrt{\bm{Q}}\bm{u}\right\|_{r,X}\|\bm{v}_h\|_{\bm{Q}} \lesssim \sqrt{\varepsilon}h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}. \end{aligned}$$ Therefore, by [\[eq:error_inq_bi_parts\]](#eq:error_inq_bi_parts){reference-type="eqref" reference="eq:error_inq_bi_parts"}, [\[eq:error_inq_bi_1\]](#eq:error_inq_bi_1){reference-type="eqref" reference="eq:error_inq_bi_1"}, [\[eq:error_inq_bi_2\]](#eq:error_inq_bi_2){reference-type="eqref" reference="eq:error_inq_bi_2"}, and [\[eq:error_inq_bi_3\]](#eq:error_inq_bi_3){reference-type="eqref" reference="eq:error_inq_bi_3"}, we have $$\label{eq:error_inq_bi} \left|\mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h)\right|\lesssim \sqrt{\varepsilon}\Big(h^{\min\{r,k+1\}-1} \big(1+\mathcal{O}(\varepsilon)\big) + h^{\min\{r,k+1\}} \Big) \|\bm{v}_h\|_{\bm{Q}}.$$ Combining [\[eq:error_inq2\]](#eq:error_inq2){reference-type="eqref" reference="eq:error_inq2"}, [\[eq:error_inq_part_one\]](#eq:error_inq_part_one){reference-type="eqref" reference="eq:error_inq_part_one"} and [\[eq:error_inq_bi\]](#eq:error_inq_bi){reference-type="eqref" reference="eq:error_inq_bi"} leads to the stated error estimate. ◻ Re-scaling the $\|\cdot\|_{\bm{Q}}$ in [\[eq:est_total_general\]](#eq:est_total_general){reference-type="eqref" reference="eq:est_total_general"} by , we have the following theorem. **Theorem 14** (Uniform error estimate). *Assume that [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"}--[\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} and hold. Consider only $k\ge 1$. The solution $\bm{u}_h$ of the DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} for the spherical harmonic radiative transfer equation [\[eq:rte_scale_Pn_all\]](#eq:rte_scale_Pn_all){reference-type="eqref" reference="eq:rte_scale_Pn_all"} converges to $\bm{u}$ uniformly in $\varepsilon$ as $h\rightarrow 0$, and admits the following error estimate: $$\left\|\bm{u}-\bm{u}_h\right\| \lesssim h^{\min\{r,k+1\}-1} (1+\varepsilon+\varepsilon^2) + h^{\min\{r,k+1\}}(1+\varepsilon), \quad k\ge 1.$$* combined with and noting that $$\|u-\bm{m}^{\mathsf{T}}\bm{u}_h\|_{L^2(X\times\mathbb{S})} \le \|u - \bm{m}^{\mathsf{T}}\bm{u}\|_{L^2(X\times\mathbb{S})} + \|\bm{u}-\bm{u}_h\|$$ lead to the following error estimate between the solutions to the scaled RTE [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} and the SH-DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"}. **Corollary 15**. *Assume the conditions of Theorems [Theorem 10](#thm:ang_est){reference-type="ref" reference="thm:ang_est"} and [Theorem 13](#thm:X){reference-type="ref" reference="thm:X"} hold. Then, we have $$\begin{gathered} \|u-\bm{m}^{\mathsf{T}}\bm{u}_h\|_{L^2(X\times\mathbb{S})} \lesssim h^{\min\{r,k+1\}-1} (1+\varepsilon+\varepsilon^2) + h^{\min\{r,k+1\}}(1+\varepsilon) \\ + N^{-q} \left( \|u\|_{L^2(X; H^q(\mathbb{S}))}+\sum_{i=1}^{3}\|\partial_i u\|_{L^2(X; H^q(\mathbb{S}))} \right). \end{gathered}$$* ### Error analysis for constant elements Consider the case of $k=0$. **Theorem 16**. *Assume that [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"}--[\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} and hold. For $k= 0$, the spherical harmonic DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} admits the following error estimate: $$\label{eq:est_total_general_k=0} |||\bm{u}-\bm{u}_h|||_h \lesssim (1+\varepsilon) h^{\min\{r,1\}-1/2} + \sqrt{\varepsilon} h^{\min\{r,1\}},$$ and $$\label{eq:est_total_k=0} \left\|\bm{u}-\bm{u}_h\right\| \lesssim (\varepsilon^{-1/2} + \varepsilon^{1/2} + \varepsilon^{3/2}) h^{\min\{r,1\}-1/2} + (1 + \varepsilon) h^{\min\{r,1\}}, \quad k = 0.$$* *Proof.* It follows from the Galerkin orthogonality [\[eq:go\]](#eq:go){reference-type="eqref" reference="eq:go"}, the definition of $|||\cdot|||_h$ norm in [\[def:norm\]](#def:norm){reference-type="eqref" reference="def:norm"}, and Lemma [Lemma 12](#lem:stab_0){reference-type="ref" reference="lem:stab_0"} that $$\label{eq:error_inq3} |||\bm{u}-\bm{u}_h|||_h \lesssim |||\bm{u} - \mathcal{I}_h\bm{u}|||_h + \sup_{\bm{v}_h\in \bm{V}_h}\frac{\mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h)}{|||\bm{v}_h |||_h}.$$ Note that for $k=0$, $[\![\mathcal{I}_h\bm{u}]\!]\big|_{\partial K}\neq 0$ since the interpolant cannot be generated by edge points. Therefore, $[\![\bm{u} - \mathcal{I}_h\bm{u}]\!]\big|_{\partial K}\neq 0$. Instead, we have $$\begin{aligned} |||\bm{u} - \mathcal{I}_h\bm{u}|||^2_h &=\frac{1}{4}\sum_{K\in\mathcal{T}_h} \Big(|\mathbf{n}_K|\cdot\bm{D}\,[\![\bm{u} - \mathcal{I}_h\bm{u}]\!], [\![\bm{u} - \mathcal{I}_h\bm{u}]\!]\Big)_{\partial K} \\ &\phantom{=}\quad + \left(\bm{Q}(\bm{u} - \mathcal{I}_h\bm{u}),\bm{u} - \mathcal{I}_h\bm{u}\right) \nonumber\\ &\lesssim h^{2\min\{r,1\}-1}\|\bm{u}\|^2_{r,X} + h^{2\min\{r,1\}} \left\|\sqrt{\bm{Q}}\bm{u}\right\|^2_{r,X}, \\ &\lesssim (1+\varepsilon)^2 h^{2\min\{r,1\}-1} + \varepsilon h^{2\min\{r,1\}}, \end{aligned}$$ from , i.e., $$\label{eq:error_inq3_k=0} |||\bm{u} - \mathcal{I}_h\bm{u}|||_h \lesssim (1+\varepsilon) h^{\min\{r,1\}-1/2} + \sqrt{\varepsilon} h^{\min\{r,k+1\}}.$$ For the term $\mathfrak{a}_h(\cdot,\cdot)$, we have $\mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h) = \mathbb{I}_1 +\mathbb{I}_2 + \mathbb{I}_3$, where $$\begin{aligned} \mathbb{I}_1 &:= \sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 \frac{1}{2}\Big(n_i^K\overrightarrow{\bm{A}^{(i)}(\bm{u}-\mathcal{I}_h\bm{u})}, [\![\bm{v}_h]\!]\Big)_{\partial K} \\ \mathbb{I}_2 &:= -\sum_{K\in\mathcal{T}_h} \sum_{i=1}^3 (\bm{A}^{(i)}(\bm{u}-\mathcal{I}_h\bm{u}),\partial_i \bm{v}_h)_K, \quad \mathbb{I}_3 := \sum_{K\in\mathcal{T}_h} \left(\bm{Q}(\bm{u}-\mathcal{I}_h\bm{u}),\bm{v}_h\right)_K.\end{aligned}$$ For the first term, since $\mathcal{Q}_i$, $i=1,2,3$ are orthogonal, we have $$\begin{aligned} \label{eq:error_inq_bi_1_k=0} |\mathbb{I}_1| &\le \frac{1}{2}\left(\sum_{K\in\mathcal{T}_h} \sum_{i=1}^3\Big(|n_i^K|\,|\varLambda|^{(i)}\overrightarrow{\mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\mathcal{I}_h \bm{u})},\overrightarrow{\mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\mathcal{I}_h\bm{u})}\Big)_{\partial K}\right)^{1/2} \nonumber\\ &\phantom{\qquad\qquad\qquad\qquad}\cdot\left(\sum_{K\in\mathcal{T}_h} \sum_{i=1}^3\Big(\big|n_i^K\big|\,\bm{D}^{(i)} [\![\bm{v}_h]\!],[\![\bm{v}_h]\!]\Big)_{\partial K}\right)^{1/2} \nonumber\\ &\lesssim \sum_{i=1}^3 h^{\min\{r,1\}-1/2}\left\|\mathcal{Q}_i^{\mathsf{T}}\bm{u}\right\|_{r,X} |||\bm{v}_h|||_h = 3h^{\min\{r,1\}-1/2}\|\bm{u}\|_{r,X} |||\bm{v}_h|||_h \nonumber \\ &\lesssim (1+\varepsilon) h^{\min\{r,1\}-1/2} |||\bm{v}_h|||_h,\end{aligned}$$ where the last inequality is due to the assumption [\[eq:reg_assump_u\]](#eq:reg_assump_u){reference-type="eqref" reference="eq:reg_assump_u"}. Since $k=0$, we have $\partial_i \bm{v}_h=0$ for any $\bm{v}_h\in \bm{V}_h$. Therefore, $$\label{eq:error_inq_bi_2_k=0} \mathbb{I}_2=0.$$ The third term can be handled similarly to [\[eq:error_inq_bi_3\]](#eq:error_inq_bi_3){reference-type="eqref" reference="eq:error_inq_bi_3"}: $$\label{eq:error_inq_bi_3_k=0} |\mathbb{I}_3| \lesssim \sqrt{\varepsilon}h^{\min\{r,1\}} |||\bm{v}_h|||_h.$$ Combining [\[eq:error_inq_bi_1\_k=0\]](#eq:error_inq_bi_1_k=0){reference-type="eqref" reference="eq:error_inq_bi_1_k=0"}, [\[eq:error_inq_bi_2\_k=0\]](#eq:error_inq_bi_2_k=0){reference-type="eqref" reference="eq:error_inq_bi_2_k=0"}, and [\[eq:error_inq_bi_3\_k=0\]](#eq:error_inq_bi_3_k=0){reference-type="eqref" reference="eq:error_inq_bi_3_k=0"} gives $$\label{eq:error_inq_bi_k=0} \left|\mathfrak{a}_h(\bm{u}-\mathcal{I}_h\bm{u},\bm{v}_h)\right|\lesssim \left((1+\varepsilon) h^{\min\{r,1\}-1/2} + \sqrt{\varepsilon}h^{\min\{r,1\}} \right)|||\bm{v}_h|||_h.$$ The estimate [\[eq:est_total_general_k=0\]](#eq:est_total_general_k=0){reference-type="eqref" reference="eq:est_total_general_k=0"} follows by gathering and inserting the estimates for [\[eq:error_inq3_k=0\]](#eq:error_inq3_k=0){reference-type="eqref" reference="eq:error_inq3_k=0"} and [\[eq:error_inq_bi_k=0\]](#eq:error_inq_bi_k=0){reference-type="eqref" reference="eq:error_inq_bi_k=0"} obtained above into [\[eq:error_inq3\]](#eq:error_inq3){reference-type="eqref" reference="eq:error_inq3"}. By the definition of $|||\cdot|||_h$, [\[eq:est_total_k=0\]](#eq:est_total_k=0){reference-type="eqref" reference="eq:est_total_k=0"} can be directly deduced from [\[eq:est_total_general_k=0\]](#eq:est_total_general_k=0){reference-type="eqref" reference="eq:est_total_general_k=0"}. ◻ **Remark 4**. *It is well known that the DG method with piecewise constant approximations does not achieve the diffusion limit [@LMM1987]. This fact is reflected by the term $\varepsilon^{-1/2}$ in [\[eq:est_total_k=0\]](#eq:est_total_k=0){reference-type="eqref" reference="eq:est_total_k=0"} that tends to infinity as $\varepsilon\to 0$.* # Error analysis for tensor product elements on Cartesian mesh {#sec:err_ana_optimal} The framework for the error analysis developed in is applicable to fairly general settings. We obtain a uniform $\mathcal{O}(h^k)$ bound across all $\varepsilon\in(0,1)$ when local polynomials of degree $k\ge 1$ are employed. However, for tensor product elements on Cartesian meshes of dimensions one, two, or three, optimal convergence results can be derived. Here, we consider the one-dimensional slab geometry model (since its error analysis has a unique approach different from the multidimensional cases) and the two-dimensional plane-parallel model [@agoshkov1998boundary; @Modest] to simplify the analysis, which are briefly introduced below. Note that these reduced models correspond to a three-dimensional problem with certain symmetries. Since it is already known that DG methods do not perform well when $k = 0$, we focus only on the case $k \ge 1$. **One-dimensional slab geometry problems.** In slab geometry, the RTE [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} can reduce to the following form [\[eq:1d\]]{#eq:1d label="eq:1d"} $$\begin{aligned} \mu\frac{\partial u}{\partial z}+\frac{\sigma_{\mathrm{t}}}{\varepsilon} u &= \frac{1}{2}\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \int^{1}_{-1} u(z,\mu')\,\mathrm{d}\mu' + f,\quad \forall z\in X=I:=(0,1),\\ u(0,\mu)&=u(1,\mu) \quad \forall \mu\in [-1,1], \end{aligned}$$ where $\mu\in[-1,1]$ is the $z$-coordinate of $\bm{\omega}$, $u=u(z,\mu)$, $f=f(z,\mu)$, and $\,\mathrm{d}\mu'$ is the Lebesgue measure on $(-1,1)$. We set $\bm{A}^{(1)}=\bm{A}^{(2)}=\bm{0}$ and therefore $\bm{A}\cdot\nabla=\bm{A}^{(3)}\partial_z$, and all the notation and formulas in Sections [2.1](#sec:SH_PN){reference-type="ref" reference="sec:SH_PN"} and [2.4](#sec:DG){reference-type="ref" reference="sec:DG"} can be kept. Especially, [\[eq:rte_scale_Pn\]](#eq:rte_scale_Pn){reference-type="eqref" reference="eq:rte_scale_Pn"} takes the following form: $$\label{eq:rte_scale_Pn_1D} \bm{A}^{(3)}\partial_3\bm{u} + \varepsilon\sigma_{\mathrm{a}}\bm{u} + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \bm{R}\bm{u} = \varepsilon \bm{f}.$$ **2-D plane-parallel model.** We consider the 2-dimensional case $X=(0,1)^2$. The angular variable $\bm{\omega}$ is determined by polar angle $\theta\in[0,\pi]$ and azimuth $\varphi\in[0,2\pi)$ by use of the standard spherical coordinates, i.e. $\bm{\omega}=\begin{bmatrix} \sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \end{bmatrix}^{\mathsf{T}}$. Set $\mu=\cos\theta$, and the RTE [\[eq:rte_scale_all\]](#eq:rte_scale_all){reference-type="eqref" reference="eq:rte_scale_all"} takes the form [\[eq:2d\]]{#eq:2d label="eq:2d"} $$\begin{aligned} \sqrt{1-\mu^2}\cos\varphi\frac{\partial u}{\partial x} + \sqrt{1-\mu^2}\sin\varphi\frac{\partial u}{\partial y} + \frac{\sigma_{\mathrm{t}}}{\varepsilon}u &=\left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right)\bar{u}+\varepsilon f,\\ u(0,y,\mu,\varphi)&=u(1,y,\mu,\varphi), y\in(0,1), \\ u(x,0,\mu,\varphi)&=u(x,1,\mu,\varphi), x\in(0,1), \end{aligned}$$ where $\mu=\cos\theta\in[-1,1]$, $\varphi\in [0,2\pi)$, $u=u(x,y,\mu,\varphi)$, and $f=f(x,y)$. Then [\[eq:rte_scale_Pn\]](#eq:rte_scale_Pn){reference-type="eqref" reference="eq:rte_scale_Pn"} can be rewritten as $$\label{eq:rte_scale_Pn_2D} \bm{A}^{(1)}\partial_1\bm{u} + \bm{A}^{(2)}\partial_2\bm{u} + \varepsilon\sigma_{\mathrm{a}}\bm{u} + \left(\frac{\sigma_{\mathrm{t}}}{\varepsilon}-\varepsilon\sigma_{\mathrm{a}}\right) \bm{R}\bm{u} = \varepsilon \bm{f}.$$ ## Error analysis in spatial discretization with upwind flux for one-dimensional reduced radiative transfer equations Let us first introduce a special Radau projection $\mathcal{R}$, defined on an interval $J$, which will be very useful in the optimal error analysis. The Radau projection is defined as follows: Given a value of $\mu$, the simplex $J$ has a unique outflow point $x^{\mathrm{out}}_J$ based on the sign of $\mu$, and, if $k\ge 1$, [\[eq:Radau_proj\]]{#eq:Radau_proj label="eq:Radau_proj"} $$\begin{aligned} (u-\mathcal{R} u,v)_J &=0 \quad \forall v\in \mathbb{P}_{k-1}(J), \label{eq:Radau_proj_a}\\ u(x^{\mathrm{out}}_{J})-\mathcal{R} u(x^{\mathrm{out}}_{J}) &=0. \label{eq:Radau_proj_b} \end{aligned}$$ In fact, if $\mu>0$, we define $x^{\mathrm{out}}_{J}$ to be the right boundary point of $J$; If $\mu<0$, $x^{\mathrm{out}}_{J}$ is then the left boundary point of $J$; If $\mu=0$, $x^{\mathrm{out}}_{J}$ could be either the right or the left boundary point of $J$, or $\mathcal{R}$ can be defined as the usual $L_2$-orthogonal projection in this case, since the advection terms vanish. The following lemma is known in [@CDG2008_SINUM Lemma 2.1]. **Lemma 17**. *For all $u\in H^{r}(J)$ and $J\in\mathcal{T}_h$, we have $$\|u-\mathcal{R} u\|_{J}\le C h^{\min \{r,k+1\}}\|u\|_{r,J},$$ where $C$ depends only on $k$.* We have the following optimal spatial error estimate for the DG approximation in the one-dimensional slab geometry setting. **Theorem 18**. *If [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"}--[\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} hold and $\bm{u}$ satisfies , the spherical harmonic DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} with the upwind flux for the one-dimensional slab geometry problem [\[eq:1d\]](#eq:1d){reference-type="eqref" reference="eq:1d"} admits the following error estimate for $k\geq 1$: $$\|\bm{u}-\bm{u}_h\|_I \lesssim (1+\varepsilon)h^{\min\{r,k+1\}}.$$* *Proof.* To derive the error estimate, we first observe that since $\mathcal{R}\bm{u}\in \bm{V}_h$, the inequality [\[eq:error_inq1\]](#eq:error_inq1){reference-type="eqref" reference="eq:error_inq1"} holds in terms of the norm $\|\cdot\|_{\bm{Q}}$: $$\label{eq:error_inq2_1d} \|\bm{u}-\bm{u}_h\|_{\bm{Q}} \lesssim \|\bm{u} - \mathcal{R}\bm{u}\|_{\bm{Q}} + \sup_{\bm{v}_h\in \bm{V}_h}\frac{\mathfrak{a}_h(\bm{u}-\mathcal{R}\bm{u},\bm{v}_h)}{\|\bm{v}_h \|_{\bm{Q}}}.$$ We estimate each term in the right-hand side of the above inequality. For the first term, owning to the and , we have $$\label{eq:error_inq3_1d} \|\bm{u}-\mathcal{R}\bm{u}\|_{\bm{Q}} \lesssim \sqrt{\varepsilon} h^{\min\{r,k+1\}}.$$ For the second term, similar to the proof of , we have $$\label{eq:error_inq_bi_parts_1d} \mathfrak{a}_h(\bm{u}-\mathcal{R}\bm{u},\bm{v}_h) = \mathbb{I}_1 +\mathbb{I}_2 + \mathbb{I}_3,$$ where $$\begin{aligned} {2}%\label{eq:a_decomp_2_1d} \mathbb{I}_1 &:= \sum_{J\in\mathcal{T}_h} \Big(n_3^J\overrightarrow{\bm{A}^{(3)}(\bm{u}-\mathcal{R}\bm{u})}, \bm{v}_h^{-}\Big)_{\partial J}, & &(n_3=\pm 1),\\ \mathbb{I}_2 &:= -\sum_{J\in\mathcal{T}_h} (\bm{A}^{(3)}(\bm{u}-\mathcal{R}\bm{u}),\partial_z \bm{v}_h)_J, &\quad\mathbb{I}_3 &:= \sum_{J\in\mathcal{T}_h} \left(\bm{Q}(\bm{u}-\mathcal{R}\bm{u}),\bm{v}_h\right)_J. \end{aligned}$$ Since the upwind flux is assumed, for the term $\mathbb{I}_1$, due to [\[eq:A_sym\]](#eq:A_sym){reference-type="eqref" reference="eq:A_sym"}, we have $$\begin{aligned} \Big(n_3^J\overrightarrow{\bm{A}^{(3)}(\bm{u}-\mathcal{R}\bm{u})}, \bm{v}_h^{-}\Big)_{\partial J} &= \Big(n_3^J\mathcal{Q}_3 \varLambda^{(3)} \overrightarrow{\mathcal{Q}_3^{\mathsf{T}}(\bm{u}-\mathcal{R}\bm{u})}, \bm{v}_h^{-}\Big)_{\partial J}\\ &= \Big(n_3^J \varLambda^{(3)} \overrightarrow{\mathcal{Q}_3^{\mathsf{T}}(\bm{u}-\mathcal{R}\bm{u})}, \mathcal{Q}_3^{\mathsf{T}}\bm{v}_h^{-}\Big)_{\partial J}\\ [\text{Set }\bm{w}=\mathcal{Q}_3^{\mathsf{T}}\bm{u}] &= \Big(n_3^J \varLambda^{(3)} \overrightarrow{\bm{w} - \mathcal{R}\bm{w}}, \,\mathcal{Q}_3^{\mathsf{T}}\bm{v}_h^-\Big)_{\partial J}. \end{aligned}$$ Since $\varLambda^{(3)}$ is a diagonal matrix, by use of the property [\[eq:Radau_proj_b\]](#eq:Radau_proj_b){reference-type="eqref" reference="eq:Radau_proj_b"}, we infer $\mathbb{I}_1=0$. Noting $\partial_z \bm{v}_h\in \mathbb{P}_{k-1}(J)$ and the property [\[eq:Radau_proj_a\]](#eq:Radau_proj_a){reference-type="eqref" reference="eq:Radau_proj_a"}, we get $\mathbb{I}_2=\sum_{J\in\mathcal{T}_h}\left(\bm{u}-\mathcal{R}\bm{u},\bm{A}^{(3)}\partial_z \bm{v}_h\right)_J=0$. By Lemma [Lemma 17](#lem:Radau_interp){reference-type="ref" reference="lem:Radau_interp"}, we have $|\mathbb{I}_3|\lesssim \sqrt{\varepsilon} h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}$. Combining all the bounds that have been derived gives $$\label{eq:error_inq_bi_1d} \left|\mathfrak{a}_h(\bm{u}-\mathcal{R}\bm{u},\bm{v}_h)\right|\lesssim \sqrt{\varepsilon} h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}.$$ By inserting [\[eq:error_inq3_1d\]](#eq:error_inq3_1d){reference-type="eqref" reference="eq:error_inq3_1d"} and [\[eq:error_inq_bi_1d\]](#eq:error_inq_bi_1d){reference-type="eqref" reference="eq:error_inq_bi_1d"} into [\[eq:error_inq2_1d\]](#eq:error_inq2_1d){reference-type="eqref" reference="eq:error_inq2_1d"} and noting , we infer $$\|\bm{u}-\bm{u}_h\|_I \lesssim \left(\sqrt{\varepsilon}+\frac{1}{\sqrt{\varepsilon}}\right)\|\bm{u}-\bm{u}_h\|_{\bm{Q}} \lesssim (1+\varepsilon) h^{\min\{r,k+1\}},$$ which completes the proof. ◻ ## Error analysis in spatial discretization with upwind flux for tensor product polynomials on rectangular elements of Cartesian mesh The optimal estimate also holds in the two- and three-dimensional cases when the tensor product polynomials on rectangular elements of Cartesian mesh are employed. We focus here only on the two-dimensional case ($d=2$). The extension of our analysis to the case $d=3$ is straightforward. On a rectangle $K=J_1\times J_2$, for $w\in C^0(\overline{K})$, we define $$\Pi w:=\mathcal{R}_1\otimes\mathcal{R}_2 w$$ with the subscripts of $\mathcal{R}$ indicating the application of the one-dimensional operators $\mathcal{R}$ in [\[eq:Radau_proj\]](#eq:Radau_proj){reference-type="eqref" reference="eq:Radau_proj"} with respect to the corresponding variable. The following approximation result is known (see, e.g., [@CKPS2001 Lemma 3.2]). **Lemma 19**. *For all $w\in H^{r}(J)$ and $J\in\mathcal{T}_h$, we have, for $k\ge 1$, $$\|w-\Pi w\|_K\le C h^{\min \{r,k+1\}}\|w\|_{r,K},$$ where $C$ depends only on $k$.* Consider a cuboid $K\in \mathcal{T}_h$ with four edges $\gamma_1^-$, $\gamma_1^+$, $\gamma_2^-$, and $\gamma_2^+$ as in . Given a direction vector $\bm{\omega}$, the boundary of $K$ is partitioned into two parts: $\partial K=\gamma^-\cup\gamma^+$, where $\gamma^-=\gamma_1^-\cup\gamma_2^-$ is the incoming boundary and $\gamma^+=\gamma_1^+\cup\gamma_2^+$ is the outgoing one. Note that the sets $\gamma^-$ and $\gamma^+$ depend on the choice of $\bm{\omega}$. Define [\[eq:Z_K\]]{#eq:Z_K label="eq:Z_K"} $$\begin{aligned} Z_{K,1}(u,v_h)&= (u-\mathcal{R}_2 u, v_h^-)_{\gamma_2^+} - (u-\mathcal{R}_2 u, v_h^-)_{\gamma_2^-} - (u-\Pi u,\partial_1 v_h)_K, \\ Z_{K,2}(u,v_h)&=(u-\mathcal{R}_1 u, v_h^-)_{\gamma_1^+} - (u-\mathcal{R}_1 u, v_h^-)_{\gamma_1^-} - (u-\Pi u,\partial_2 v_h)_K. \end{aligned}$$ Note that $v_h^-$ indicates the inside value of $v_h$ with respect to $K$. When vector arguments are employed in $Z_{K,i}$, $i=1,2$, the right-hand sides of [\[eq:Z_K\]](#eq:Z_K){reference-type="eqref" reference="eq:Z_K"} are understood in terms of [\[eq:vector_inner\]](#eq:vector_inner){reference-type="eqref" reference="eq:vector_inner"}. The following superconvergence result [@CKPS2001], essentially due to LeSaint and Raviart [@LR1974], plays an essential role in obtaining the optimal estimate. **Lemma 20**. *Let $Z_{K,1}$ and $Z_{K,2}$ be defined by [\[eq:Z_K\]](#eq:Z_K){reference-type="eqref" reference="eq:Z_K"}. Assume $u\in H^r(K)$ and $v_h\in \mathbb{Q}_k(K)$. Then we have for $r>0$, $$|Z_{K,i}(u,v_h)|\le C h^{\min\{r,k+1\}} \|u\|_{r,K}\|v _h\|_K, \quad i=1,2,$$ where the constant $C$ depends only on $r$ and $k$.* **Theorem 21**. *Assume all the elements in $\mathcal{T}_h$ are rectangles and tensor product polynomials of degree at most $k$ are used. If [\[as:rte_scale_1\]](#as:rte_scale_1){reference-type="eqref" reference="as:rte_scale_1"}--[\[as:rte_scale_2\]](#as:rte_scale_2){reference-type="eqref" reference="as:rte_scale_2"} hold and $\bm{u}$ satisfies , the spherical harmonic DG method [\[eq:bilinear\]](#eq:bilinear){reference-type="eqref" reference="eq:bilinear"} with the upwind flux for the two-dimensional plane-parallel problem [\[eq:2d\]](#eq:2d){reference-type="eqref" reference="eq:2d"} admits the following error estimate for $k\ge 1$: $$\|\bm{u}-\bm{u}_h\| \lesssim \left(1+\mathcal{O}(\varepsilon)\right)h^{\min\{r,k+1\}}.$$* *Proof.* To derive the error estimate, we first observe that since $\Pi\bm{u}\in \bm{V}_h$ and $\mathfrak{a}_h(\bm{v}_h,\bm{v}_h) = |||\bm{v}_h|||^2_h\ge \|\bm{v}_h\|_{\bm{Q}}^2$, the inequality [\[eq:error_inq1\]](#eq:error_inq1){reference-type="eqref" reference="eq:error_inq1"} holds in term of the norm $\|\cdot\|_{\bm{Q}}$: $$\label{eq:error_inq2_2d} \|\bm{u}-\bm{u}_h\|_{\bm{Q}} \lesssim \|\bm{u} - \Pi \bm{u}\|_{\bm{Q}} + \sup_{\bm{v}_h\in \bm{V}_h}\frac{\mathfrak{a}_h(\bm{u}-\Pi \bm{u},\bm{v}_h)}{\|\bm{v}_h\|_{\bm{Q}}}.$$ We estimate each term on the right-hand side of the above inequality. For the first term, owning to and , we have $$\label{eq:error_inq3_2d} \|\bm{u}-\Pi \bm{u}\|_{\bm{Q}} \lesssim \sqrt{\varepsilon} h^{\min\{r,k+1\}}.$$ For the second term, we have, similar to the proof of , $$\label{eq:error_inq_bi_parts_2d} \mathfrak{a}_h(\bm{u}-\Pi \bm{u},\bm{v}_h) = \sum_{K\in\mathcal{T}_h}\mathbf{Z}_K(\bm{u},\bm{v}_h) + \mathbb{I}_3,$$ where $\mathbb{I}_3 := \sum_{K\in\mathcal{T}_h} \left(\bm{Q}(\bm{u}-\Pi \bm{u}),\bm{v}_h\right)_K$, and $$\mathbf{Z}_K(\bm{u},\bm{v}_h):= -\sum_{i=1}^2 \left(\bm{A}^{(i)}(\bm{u}-\Pi \bm{u}),\partial_i \bm{v}_h\right)_K + \sum_{i=1}^2 \left(n_i^K\overrightarrow{\bm{A}^{(i)}(\bm{u}-\Pi \bm{u})}, \bm{v}_h^-\right)_{\partial K}.\label{eq:ZZ_K}$$ By and , we infer $$\label{eq:est_I_3} |\mathbb{I}_3|\lesssim \sqrt{\varepsilon}h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}.$$ Next, we estimate $\mathbf{Z}_K(\bm{u},\bm{v}_h)$ and proceed as follows: $$\begin{aligned} \mathbf{Z}_K(\bm{u},\bm{v}_h)&= \sum_{i=1}^2\left( \big(n_i^K\overrightarrow{\bm{A}^{(i)}(\bm{u}-\Pi \bm{u})}, \bm{v}_h^-\big)_{\partial K} - \big(\bm{A}^{(i)}(\bm{u}-\Pi \bm{u}),\partial_i \bm{v}_h\big)_K\right) \\ &= \sum_{i=1}^2 \left(\big(n_i^K\mathcal{Q}_i \varLambda^{(i)} \overrightarrow{\mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\Pi \bm{u})}, \bm{v}_h^-\big)_{\partial K} - \big(\mathcal{Q}_i \varLambda^{(i)} \mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\Pi \bm{u}),\partial_i \bm{v}_h\big)_K \right) \\ &= \sum_{i=1}^2 \left(\big(n_i^K\varLambda^{(i)} \overrightarrow{\mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\Pi \bm{u})}, \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h^-\big)_{\partial K} - \big(\varLambda^{(i)} \mathcal{Q}_i^{\mathsf{T}}(\bm{u}-\Pi \bm{u}),\partial_i \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h\big)_K \right)\\ \big[&\text{Set }\bm{w}_i=\mathcal{Q}_i^{\mathsf{T}}\bm{u}\big]\\ &= \sum_{i=1}^2 \left(n_i^K \overrightarrow{\varLambda^{(i)}(\bm{w}_i-\Pi \bm{w}_i)}, \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h^-\right)_{\partial K} -\sum_{i=1}^2 \left( \varLambda^{(i)} (\bm{w}_i-\Pi \bm{w}_i),\partial_i \mathcal{Q}_i^{\mathsf{T}}\bm{v}_h\right)_K\\ &= \sum_{i=1}^2 \varLambda^{(i)}Z_{K,i}(\bm{w}_i,\mathcal{Q}_i^{\mathsf{T}}\bm{v}_h). \end{aligned}$$ The last equality is due to the fact that $\overrightarrow{\varLambda^{(i)}\bm{w}_i}|_{\gamma_i^\pm}=\varLambda^{(i)}\bm{w}_i|_{\gamma_i^\pm}$ and $\overrightarrow{\varLambda^{(i)}\Pi\bm{w}_i}|_{\gamma_i^-} =\varLambda^{(i)}\mathcal{R}_i(\bm{w}_i|_{\gamma_i^-})$ since the edges of the rectangle $K\in\mathcal{T}_h$ are parallel to the $x/y$-axes and $\varLambda_i$ is diagonal. By and $\left\|\mathcal{Q}_i^{\mathsf{T}}\bm{v}_h\right\|_K=\|\bm{v}_h\|_K$ since $\mathcal{Q}_i$, $i=1,2$ are orthogonal matrices, we infer that $$\label{eq:ZZ_est} \left|\mathbf{Z}_K(\bm{u},\bm{v}_h)\right|\le C h^{\min\{r,k+1\}} \|\bm{w}\|_{r,K}\left\|\mathcal{Q}_i^{\mathsf{T}}\bm{v}_h\right\|_K\le C h^{\min\{r,k+1\}} \|\bm{u}\|_{r,K}\|\bm{v}_h\|_K.$$ Define $\bm{Q}'= \mathop{\mathrm{diag}}\left(\sqrt{\varepsilon}/\sqrt{\sigma_{\mathrm{t}}},\sqrt{\varepsilon}/\sqrt{\sigma_{\mathrm{t}}}+(\sqrt{\varepsilon\sigma_{\mathrm{a}}})^{-1}, \sqrt{\varepsilon}/\sqrt{\sigma_{\mathrm{t}}}, \cdots, \sqrt{\varepsilon}/\sqrt{\sigma_{\mathrm{t}}}\right)$. A direct calculation shows that for $i=1,2$, $$\sqrt{\bm{Q}}^{-1}\bm{A}^{(i)}=\bm{A}^{(i)}\bm{Q}'-\begin{bmatrix} 0 & \frac{\sqrt{\varepsilon}}{\sqrt{\sigma_{\mathrm{t}}}}\bm{A}^{(i)}_{0,1} & 0 &\dots & 0 \\ 0 & 0 & 0 & \dots & 0\\ 0 & \sqrt{\varepsilon\sigma_{\mathrm{a}}}^{-1}\bm{A}^{(i)}_{2,1} & 0 & \dots & 0 \\ 0 & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix},$$ and therefore $$\begin{gathered} \sqrt{\bm{Q}}^{-1}\bm{A}^{(i)}\bm{u}=\bm{A}^{(i)}\bm{Q}'\bm{u}\\ -\mathop{\mathrm{diag}}(\sqrt{\varepsilon}/\sqrt{\sigma_{\mathrm{t}}},0,\sqrt{\varepsilon\sigma_{\mathrm{a}}}^{-1},0,\cdots,0)\bm{A}^{(i)}\begin{bmatrix} \mathbf{0} & \bm{u}_1^\mathsf{T}& \mathbf{0} &\cdots & \mathbf{0} \end{bmatrix}^{\mathsf{T}}. \end{gathered}$$ We deduce $$\begin{aligned} \sum_{K\in\mathcal{T}_h}&\big|\mathbf{Z}_K(\bm{u},\bm{v}_h)\big| =\sum_{K\in\mathcal{T}_h}\big|\sqrt{\bm{Q}}^{-1}\mathbf{Z}_K(\bm{u},\sqrt{\bm{Q}}\bm{v}_h)\big|\\ &= \sum_{K\in\mathcal{T}_h} \bigg|\mathbf{Z}_K(\bm{Q}'\bm{u},\sqrt{\bm{Q}}\bm{v}_h)\\ &\qquad\qquad -\mathbf{Z}_K(\begin{bmatrix} \mathbf{0} & \bm{u}_1^\mathsf{T}& \mathbf{0} &\cdots & \mathbf{0} \end{bmatrix}^{\mathsf{T}},\mathop{\mathrm{diag}}(\sqrt{\varepsilon},0,\sqrt{\varepsilon}^{-1},0,\cdots,0)\sqrt{\bm{Q}}\bm{v}_h)\bigg| \\ &\le \sum_{K\in\mathcal{T}_h} \big|\mathbf{Z}_K(\bm{Q}'\bm{u},\sqrt{\bm{Q}}\bm{v}_h)\big| \\ &\quad\quad +\sum_{K\in\mathcal{T}_h}\big|\mathbf{Z}_K(\begin{bmatrix} \mathbf{0} & \bm{u}_1^\mathsf{T}& \mathbf{0} &\cdots & \mathbf{0} \end{bmatrix}^{\mathsf{T}},\mathop{\mathrm{diag}}(\sqrt{\varepsilon},0,\sqrt{\varepsilon}^{-1},0,\cdots,0)\sqrt{\bm{Q}}\bm{v}_h)\big|. \end{aligned}$$ Note that the estimate [\[eq:ZZ_est\]](#eq:ZZ_est){reference-type="eqref" reference="eq:ZZ_est"} holds for any $\bm{u}\in [H^r(X)]^L$ and $\bm{v}_h\in V_h$. Therefore, $$\begin{aligned} \sum_{K\in\mathcal{T}_h} |\mathbf{Z}_K(\bm{Q}'\bm{u},\sqrt{\bm{Q}}\bm{v}_h)|&\le \sum_{K\in\mathcal{T}_h} h^{\min\{r,k+1\}} \|\bm{Q}'\bm{u}\|_{r,K}\|\sqrt{\bm{Q}}\bm{v}_h\|_K \\ &\lesssim \sqrt{\varepsilon}h^{\min\{r,k+1\}} \big(1+\mathcal{O}(\varepsilon)\big)\|\bm{v}_h\|_{\bm{Q}}. \end{aligned}$$ Finally, $$\begin{aligned} \sum_{K\in\mathcal{T}_h}\bigg|\mathbf{Z}_K&\left(\begin{bmatrix} 0 & \bm{u}_1^\mathsf{T}& \mathbf{0} &\cdots & \mathbf{0} \end{bmatrix}^{\mathsf{T}},\mathop{\mathrm{diag}}(\sqrt{\varepsilon},0,\sqrt{\varepsilon}^{-1},0,\cdots,0)\sqrt{\bm{Q}}\bm{v}_h\right)\bigg| \\ &\le (\sqrt{\varepsilon}+\sqrt{\varepsilon}^{-1})\sum_{K\in\mathcal{T}_h}h^{\min\{r,k+1\}}\|\bm{u}_1\|_{r,K} \|\sqrt{\bm{Q}}\bm{v}_h\|_K \\ &\lesssim \sqrt{\varepsilon}(1+\varepsilon)h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}. \end{aligned}$$ Hence, $$\label{eq:est_Z_K} \sum_{K\in\mathcal{T}_h}\big|\mathbf{Z}_K(\bm{u},\bm{v}_h)\big|\lesssim \sqrt{\varepsilon}(1+\varepsilon)h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}.$$ With the bounds [\[eq:est_I\_3\]](#eq:est_I_3){reference-type="eqref" reference="eq:est_I_3"} and [\[eq:est_Z\_K\]](#eq:est_Z_K){reference-type="eqref" reference="eq:est_Z_K"}, we conclude from [\[eq:error_inq_bi_parts_2d\]](#eq:error_inq_bi_parts_2d){reference-type="eqref" reference="eq:error_inq_bi_parts_2d"} that $$\label{eq:error_inq_bi_2d} \left|\mathfrak{a}_h(\bm{u}-\mathcal{R}\bm{u},\bm{v}_h)\right|\lesssim \sqrt{\varepsilon}(1+\varepsilon)h^{\min\{r,k+1\}} \|\bm{v}_h\|_{\bm{Q}}.$$ The result follows by inserting [\[eq:error_inq3_2d\]](#eq:error_inq3_2d){reference-type="eqref" reference="eq:error_inq3_2d"} and [\[eq:error_inq_bi_2d\]](#eq:error_inq_bi_2d){reference-type="eqref" reference="eq:error_inq_bi_2d"} into [\[eq:error_inq2_2d\]](#eq:error_inq2_2d){reference-type="eqref" reference="eq:error_inq2_2d"} and re-scaled by $\sqrt{\bm{Q}}$ with . ◻ # Conclusions {#sec:conclusion} In this paper, we analyze the convergence of a spherical harmonic DG scheme for scaled radiative transfer equations with isotropic scattering. We first prove that the spherical harmonic approximations for the angular variable converge uniformly with respect to $\varepsilon$. For sufficiently rich approximation spaces, we prove uniform convergence rates with respect to $\varepsilon$ for the DG scheme in the spacial variable. However, this convergence rate is in general not optimal. By employing the Radau projection and previous results for the DG method [@LR1974] for linear hyperbolic problems, we are able to further obtain the optimal and uniform convergence rate on Cartesian grids with tensor product polynomials of degree at least one. In future work, we hope to leverage the current analysis for more physically realistic scenarios that do not rely on the assumptions in [\[as:rte_scale\]](#as:rte_scale){reference-type="eqref" reference="as:rte_scale"}, but rather allow for arbitrarily thin and thick materials in the same problem. In addition, problems with more realistic boundary conditions and less optimistic regularity assumptions will be considered. [^1]: This work of the second author is supported by the DOE Office of Advanced Scientific Computing Research and by the National Science Foundation under Grant No. 1217170. ORNL is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The work of the third author is partially supported by the NSF grant DMS-1753581.

arxiv_math

--- abstract: | We consider probability measures arising from the Cauchy summation identity for the LLT (Lascoux--Leclerc--Thibon) symmetric polynomials of rank $n \geqslant 1$. We study the asymptotic behaviour of these measures as one of the two sets of polynomials in the Cauchy identity stays fixed, while the other one grows to infinity. At $n=1$, this corresponds to an analogous limit of the Schur process, which is known to be given by the Gaussian Unitary Ensemble (GUE) corners process. Our main result states that, for $n>1$, our measures asymptotically split into two parts: a continuous one and a discrete one. The continuous part is a product of $n$ GUE corners processes; the discrete part is an explicit finite distribution on interlacing $n$-colourings of $n$ interlacing triangles, which has weights that are rational functions in the LLT parameter $q$. The latter distribution has a number of interesting (partly conjectural) combinatorial properties, such as $q$-nonnegativity and enumerative phenomena underlying its support. Our main tools are two different representations of the LLT polynomials, one as partition functions of a fermionic lattice model of rank $n$, and the other as finite-dimensional contour integrals, which were recently obtained in `arXiv:2012.02376`, `arXiv:2101.01605`. author: - Amol Aggarwal, Alexei Borodin and Michael Wheeler bibliography: - references.bib title: Coloured corner processes from asymptotics of LLT polynomials --- # Introduction ## Preface The Gaussian Unitary Ensemble (or GUE, for short) is one of the cornerstones of Random Matrix Theory that goes back to Wigner [@Wigner65]. It consists of Hermitian matrices $H$ distributed according to the Gaussian measure $P(dH) \sim \exp(-Tr(H^2)) dH$, which is the essentially unique[^1] distribution on this set that satisfies two natural conditions: (a) It is invariant under any unitary conjugation; and (b) Linearly independent real and imaginary parts of matrix elements are statistically independent, see [@Mehta Section 2.5]. One important feature of the GUE is that it can be viewed as a universal limiting object for discrete probabilistic systems related to representation theory. The first limiting relation of this kind goes back to Kerov [@Kerov88] who studied the distribution of symmetry types of tensors in high tensor powers of a finite-dimensional vector space. In this case the limit is described by the distribution of spectra of traceless GUE matrices, and the condition of vanishing trace can be naturally removed by randomizing the number of tensor factors (also known as Poissonization). The result was later rediscovered by Tracy--Widom [@TracyWidom01] in the first wave of works related to the asymptotics of longest increasing sequences. A somewhat more conceptual way to view this result is that of the quasi-classical limit in representation theory, see *e.g.* Heckman [@Heckman82]. GUEs of different sizes can be coupled by viewing them as upper-left corners of the same infinite Hermitian matrices. Such measures on infinite Hermitian matrices naturally appear in Asymptotic Representation Theory, see Olshanski--Vershik [@OlshanskiVershik]. In the framework of tiling models, thus coupled GUEs were first obtained by Johansson--Nordenstam [@JohanssonNordenstam] and Okounkov--Reshetikhin [@OkounkovReshetikhin], and their universality in such contexts was recently shown by Aggarwal--Gorin [@AggarwalGorin]. The terms "GUE minors process\" and "GUE corners process\" have both been introduced for the resulting ensemble; we will use the latter one. If one translates the problem of analyzing tensor symmetry types to the language of symmetric functions (which in this case represent the characters of both general linear and symmetric groups), then one is looking at probability measures on partitions obtained from summands in the Cauchy summation identity for the Schur symmetric polynomials. There are two sets of Schur polynomials in the game; one of them remains fixed, in correspondence with the fixed dimension of the vector space that is being tensored, while the specialization of the other one is growing in the way corresponding to the growing tensor power[^2]. The measures on partitions arising from specializations of this Cauchy identity have been known as *Schur measures* since the work of Okounkov [@Okounkov01]. The goal of the present work is to perform asymptotic analysis in a similar setup, but with the role of the Schur polynomials played instead by *LLT symmetric polynomials*. The LLT polynomials were introduced by Lascoux--Leclerc--Thibon in [@LLT]; an insightful and easy-to-read account of their first 25 years by Thibon can be found at [@Thibon]. Cauchy-type summation identities for the LLT polynomials were later obtained by Lam [@Lam], and the probability measures that we study have weights proportional to the summands of such an identity. While the origins of the LLT polynomials were representation theoretic, *cf.* [@CarreLeclerc], their most transparent definition is combinatorial --- they are generating functions of ribbon Young tableaux, where monomials in the variables of the polynomials are used to track the weight of the tableaux, and powers of a new parameter $q$ track the so-called *spin* statistics introduced in [@LLT]. When $q=1$, the LLT polynomials reduce to products of Schur polynomials, the number of which (also equal to the size of the ribbons) will be called the *rank*; we will denote it by $n$ throughout the paper. Thus, one can think of the LLT polynomials as a higher rank $q$-analogue of (products of) Schur polynomials. Neither the combinatorial nor the representation theoretic definitions of the LLT polynomials seem suitable for the asymptotic problem in question. On the other hand, we recently found an integral representation for these polynomials in [@ABW21 Chapter 11]. It is the steepest descent analysis of those integral representations that allowed us to reach our main result. The limit that we obtained carried a couple of surprises, the main one being that it splits into a continuous and a discrete part. The continuous part is a direct product of $n$ GUE corners processes. The discrete part is a probability distribution on the (finitely many) ways to colour $n$ interlacing triangular arrays[^3] by $n$ colors so that each color interlaces (an exact definition is below). The latter distribution has a few interesting properties. First, its weights can be represented as certain partition functions of a *fermionic lattice model of rank $n$*. The connection is in no way immediate, and it is related to the vertex model representations for the LLT polynomials obtained in [@ABW21], see also Corteel--Gitlin--Keating--Meza [@CGKM22]. This vertex model interpretation of the limiting distribution ends up being crucial for our proof. Second, these weights, which are *a priori* rational functions of the deformation parameter $q$, appear to be given by polynomials in $q$ with positive integer coefficients divided by a power of the $q$-factorial of $n$. We conjecture that this is always the case, even though we were only able to observe this phenomenon on the few examples we tested on a computer. The combinatorial meaning of the coefficients of the resulting polynomials also remains unclear. See Figure [\[fig:n=2\]](#fig:n=2){reference-type="ref" reference="fig:n=2"} in Appendix [8](#sec:app){reference-type="ref" reference="sec:app"} below for a quick example in rank $2$. Third, the size of the support of the distributions, *i.e.*, the number of interlacing $n$-colourings of $n$ triangular arrays appears to be combinatorially interesting. It is easy to compute for $n=1$ and $2$, when it is equal to $1$ and to a simple power of $2$, respectively. However, for $n=3$ it turns out to be equal to the number of 4-colourings of a triangle in the triangular lattice. We originally conjectured this coincidence on the basis of numerics, and it was later proved via an elegant bijective construction by Gaetz--Gao [@GaetzGao]. For $n=4$ the numerics suggest a similar relationship with $5$-colourings of squares in the "king graph", see Conjecture [Conjecture 71](#conj:a5){reference-type="ref" reference="conj:a5"}, although no proof is currently available. Finally, for $n\geqslant 5$ we were not able to find similar matchings. Recalling the appearance of the GUE corners process in random tilings, it is natural to ask if the limiting object we observed has a meaning in the world of tiling models. We believe it is indeed so, and in particular, the limiting behaviour of the random $n$-tilings of Aztec diamonds introduced by Corteel--Gitlin--Keating [@CGK] should have the same limit, as the size of the Aztec diamond tends to infinity, near the tangency points of the "arctic curve\" that bounds the frozen regions. The reason is that these $n$-tilings can be described via a closely related *dual* Cauchy identity for the LLT polynomials. The focus on a tangency point of the arctic curve results in one set of the LLT polynomials within the identity staying fixed, while the specialization of the other one is growing with the size of the domain, much like in the limit that we investigated. We will, however, leave this connection to future studies. Let us now describe our results in more detail. ## Fermionic vertex models, coloured compositions and partition functions The vertex models that we consider in this work assign weights to collections of paths drawn on a square grid. Each vertex that is traversed by at least one path produces a weight that depends on the configuration of all the paths that go through it. The total weight for a collection of paths is the product of weights of the vertices that the paths traverse (we assume the normalization in which the weight of an empty vertex is equal to unity). Each path carries a colour that is a number between $1$ and $n$, where $n\geqslant 1$ is the rank of the model. Let us first assume that each horizontal edge of the underlying square grid can carry no more than one path, while vertical edges can be occupied by multiple paths of *distinct colours*. Thus, the states of the horizontal edges can be encoded by an integer between $0$ and $n$, with $0$ denoting an edge that is not occupied by a path, while the states of the vertical edges can be encoded by $n$-dimensional binary strings which specify whether each colour $\{1,\dots,n\}$ appears (or not) at that edge. Our paths will always travel upward in the vertical direction, and in the horizontal direction a path can travel rightward or leftward, depending on the specific type of vertices that are used; this choice will always be explicitly stated. Let us now specify our vertex weights more precisely. In regions of rightward horizontal travel, our vertex weights take the following form: $$\begin{aligned} \label{generic-L-intro} \tilde{L}^{(s)}_{x,q}(\bm{A},b;\bm{C},d) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[left] at (-1.5,0) {$x \rightarrow$}; \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $d$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; \end{tikzpicture} \quad\quad b,d \in \{0,1,\dots,n\}, \quad \bm{A},\bm{C}\in \{0,1\}^n,\end{aligned}$$ where $\tilde{L}^{(s)}_{x,q}$ is a rational function of three parameters $x,q,s$. Here $x$ is the *spectral parameter* associated to a row of the lattice (a different parameter may be used for each row), $q$ is the *quantum deformation parameter* (a global parameter that is common to all vertices), and $s$ is the *spin parameter*, which arises due to the fact that the vertical line of the vertex is a higher-spin module for the underlying quantized affine Lie algebra $U_q(\widehat{\mathfrak{sl}}(1|n))$. For the explicit form of these weights, see equation [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}. In regions of leftward horizontal travel, our vertex weights are given by $$\begin{aligned} \label{generic-M-intro} \tilde{M}^{(s)}_{x,q}(\bm{A},b;\bm{C},d) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[right] at (1.5,0) {$\leftarrow x$}; \draw[lgray,line width=1.5pt,<-] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $d$};\node[right] at (1,0) {\tiny $b$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; \end{tikzpicture} \quad\quad b,d \in \{0,1,\dots,n\}, \quad \bm{A},\bm{C}\in \{0,1\}^n,\end{aligned}$$ where $\tilde{M}^{(s)}_{x,q}$ is again a rational function of the three parameters $x,q,s$ defined above. The weights [\[generic-L-intro\]](#generic-L-intro){reference-type="eqref" reference="generic-L-intro"} and [\[generic-M-intro\]](#generic-M-intro){reference-type="eqref" reference="generic-M-intro"} are related via the simple identity $$\begin{aligned} \label{LM-sym-intro} \tilde{M}^{(s)}_{x,q}(\bm{A},b;\bm{C},d) = \tilde{L}^{(1/s)}_{1/x,1/q}(\bm{A},b;\bm{C},d),\end{aligned}$$ which holds for all $\bm{A},\bm{C}\in \{0,1\}^n$ and $b,d \in \{0,1,\dots,n\}$; see equation [\[LM-sym\]](#LM-sym){reference-type="eqref" reference="LM-sym"}. For full details about the weights [\[generic-L-intro\]](#generic-L-intro){reference-type="eqref" reference="generic-L-intro"} and [\[generic-M-intro\]](#generic-M-intro){reference-type="eqref" reference="generic-M-intro"}, including their Yang--Baxter equations, see Sections [2.2](#ssec:L){reference-type="ref" reference="ssec:L"}--[2.4](#ssec:YB){reference-type="ref" reference="ssec:YB"}. We note that the fermionic weights [\[generic-L-intro\]](#generic-L-intro){reference-type="eqref" reference="generic-L-intro"} and [\[generic-M-intro\]](#generic-M-intro){reference-type="eqref" reference="generic-M-intro"} appeared previously in [@ABW21], and bosonic counterparts of them date even further back to [@BorodinW]. The partition functions (and ultimately, probability measures) that we consider are all indexed by a set of objects called *coloured compositions*: **Definition 1** (Definition [Definition 14](#def:cc){reference-type="ref" reference="def:cc"} below). Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of length $n$. We introduce the set $\mathcal{S}_{\lambda}$ of $\lambda$-coloured compositions as follows: $$\begin{aligned} \label{lambda-col-intro} \mathcal{S}_{\lambda} = \Big\{ \mu = \Big( 0 \leqslant\mu^{(1)}_1 < \cdots < \mu^{(1)}_{\lambda_1} \Big| 0 \leqslant\mu^{(2)}_1 < \cdots < \mu^{(2)}_{\lambda_2} \Big| \cdots \Big| 0 \leqslant\mu^{(n)}_1 < \cdots < \mu^{(n)}_{\lambda_n}\Big) \Big\}.\end{aligned}$$ One may think of the elements of $\mathcal{S}_{\lambda}$ as $n$-tuples $\left(\mu^{(1)},\dots,\mu^{(n)}\right)$ of strict compositions. For each $1 \leqslant i \leqslant n$, the superscript of $\mu^{(i)}$ is its *colour*, and its length is $\lambda_i$. Our first partition function of interest is denoted $f_{\mu}(\lambda;x_1,\dots,x_m;s)$. This is a (nonsymmetric) rational function in an alphabet $(x_1,\dots,x_m)$, indexed by a composition $\lambda = (\lambda_1,\dots,\lambda_n)$ satisfying $\sum_{i=1}^{n} \lambda_i = m$, as well as a coloured composition $\mu \in \mathcal{S}_{\lambda}$. Up to an overall multiplicative factor, $f_{\mu}(\lambda;x_1,\dots,x_m;s)$ is defined as a partition function using the vertex weights [\[generic-L-intro\]](#generic-L-intro){reference-type="eqref" reference="generic-L-intro"}: $$\begin{aligned} \label{f-def-intro} (-s)^{|\mu|} \cdot f_{\mu}(\lambda;x_1,\dots,x_m;s) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \foreach\y in {0,...,5}{ \draw[lgray,line width=1.5pt,->] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,-1) -- (\x,6); } %spectral parameters \node[left] at (-0.5,0) {$x_1 \rightarrow$}; \node[left] at (-0.5,2) {$\vdots$}; \node[left] at (-0.5,3) {$\vdots$}; \node[left] at (-0.5,5) {$x_m \rightarrow$}; %bottom labels \node[below] at (7,-1) {$\cdots$}; \node[below] at (6,-1) {$\cdots$}; \node[below] at (5,-1) {$\cdots$}; \node[below] at (4,-1) {\footnotesize$\bm{e}_0$}; \node[below] at (3,-1) {\footnotesize$\bm{e}_0$}; \node[below] at (2,-1) {\footnotesize$\bm{e}_0$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{A}(2)$}; \node[above] at (3,6) {\footnotesize$\bm{A}(1)$}; \node[above] at (2,6) {\footnotesize$\bm{A}(0)$}; %right labels \node[right] at (8,0) {$0$}; \node[right] at (8,1) {$0$}; \node[right] at (8,2) {$0$}; \node[right] at (8,3) {$0$}; \node[right] at (8,4) {$0$}; \node[right] at (8,5) {$0$}; %left labels \node[left] at (1,0) {$1$}; \node[left] at (1.5,0.6) {$\vdots$}; \node[left] at (1,1) {$1$}; \node[left] at (1,2) {$\vdots$}; %\node[left] at (1.5,2.6) {$\vdots$}; \node[left] at (1,3) {$\vdots$}; %\node at (2.5,3.6) {$\vdots$}; %\node at (6.5,3.6) {$\vdots$}; \node[left] at (1,4) {$n$}; \node[left] at (1.5,4.6) {$\vdots$}; \node[left] at (1,5) {$n$}; \end{tikzpicture}\end{aligned}$$ where $\bm{e}_0$ denotes the $n$-dimensional zero vector and $\bm{A}(k) = \sum_{i=1}^{n} \bm{1}_{k \in \mu^{(i)}} \bm{e}_i$ is a binary string that encodes whether $k$ is present (or not) as a part in $\mu^{(i)}$, for all $1 \leqslant i \leqslant n$ and $k \geqslant 0$. A convenient visualization aid is that for each $1 \leqslant i \leqslant n$, a collection of $\lambda_i$ paths of colour $i$ enter the partition [\[f-def-intro\]](#f-def-intro){reference-type="eqref" reference="f-def-intro"} via its left boundary and travel through the lattice, ultimately exiting via the top of the columns $\mu^{(i)}_1 < \cdots < \mu^{(i)}_{\lambda_i}$. In a similar vein, one may define multivariate (nonsymmetric) rational functions as partition functions constructed from the weights [\[generic-M-intro\]](#generic-M-intro){reference-type="eqref" reference="generic-M-intro"}. We denote these by $g_{\mu}(\lambda;x_1,\dots,x_m;s)$, where the specification of $(x_1,\dots,x_m)$, $\lambda$ and $\mu \in \mathcal{S}_{\lambda}$ is exactly as above. Up to an overall multiplicative factor, $g_{\mu}(\lambda;x_1,\dots,x_m;s)$ is defined as follows: $$\begin{aligned} \label{g-def-intro} (-s)^{-|\mu|} \cdot g_{\mu}(\lambda;x_1,\dots,x_m;s) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \foreach\y in {0,...,5}{ \draw[lgray,line width=1.5pt,<-] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,-1) -- (\x,6); } %spectral parameters \node[right] at (8.5,0) {$\leftarrow x_1$}; \node[right] at (8.5,2) {$\vdots$}; \node[right] at (8.5,3) {$\vdots$}; \node[right] at (8.5,5) {$\leftarrow x_m$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{e}_0$}; \node[above] at (3,6) {\footnotesize$\bm{e}_0$}; \node[above] at (2,6) {\footnotesize$\bm{e}_0$}; %bottom labels \node[below] at (7,-1) {$\cdots$}; \node[below] at (6,-1) {$\cdots$}; \node[below] at (5,-1) {$\cdots$}; \node[below] at (4,-1) {\footnotesize$\bm{A}(2)$}; \node[below] at (3,-1) {\footnotesize$\bm{A}(1)$}; \node[below] at (2,-1) {\footnotesize$\bm{A}(0)$}; %right labels \node[right] at (8,0) {$0$}; \node[right] at (8,1) {$0$}; \node[right] at (8,2) {$0$}; \node[right] at (8,3) {$0$}; \node[right] at (8,4) {$0$}; \node[right] at (8,5) {$0$}; %left labels \node[left] at (1,0) {$1$}; \node[left] at (1.5,0.6) {$\vdots$}; \node[left] at (1,1) {$1$}; \node[left] at (1,2) {$\vdots$}; %\node[left] at (1.5,2.6) {$\vdots$}; \node[left] at (1,3) {$\vdots$}; %\node at (2.5,3.6) {$\vdots$}; %\node at (6.5,3.6) {$\vdots$}; \node[left] at (1,4) {$n$}; \node[left] at (1.5,4.6) {$\vdots$}; \node[left] at (1,5) {$n$}; \end{tikzpicture}\end{aligned}$$ where (as above) $\bm{A}(k) = \sum_{i=1}^{n} \bm{1}_{k \in \mu^{(i)}} \bm{e}_i$ for all $1 \leqslant i \leqslant n$ and $k \geqslant 0$. The functions $f_{\mu}(\lambda;x_1,\dots,x_m;s)$ and $g_{\mu}(\lambda;x_1,\dots,x_m;s)$ are also not new; they were introduced in [@ABW21]. They have a number of key properties, including exchange relations under the action of Hecke algebra (Section [3.5](#ssec:hecke){reference-type="ref" reference="ssec:hecke"}) and antisymmetrization identities (Section [3.6](#ssec:anti){reference-type="ref" reference="ssec:anti"}). They also have meaningful $s=0$ degenerations, when they both reduce to (certain antisymmetrizations of) nonsymmetric Hall--Littlewood polynomials. Moreover, the $s=0$ degenerations of $f_{\mu}(\lambda;x_1,\dots,x_m;s)$ and $g_{\mu}(\lambda;x_1,\dots,x_m;s)$ pair together to provide an integral formula for the LLT polynomials; it is the latter fact that shall be of most interest to us in the current text. ## Two formulas for LLT polynomials In this section we recall two formulas for the LLT polynomials. The first is as partition functions in a fermionic $U_q (\widehat{\mathfrak{sl}}(1|n))$ vertex model, following [@CGKM22; @ABW21]. The second is as a contour integral, following [@ABW21]. We begin with the partition function representation of the LLT polynomials. To state it, we extend our previous notion of vertex models to the situation where both horizontal and vertical edges may admit multiple paths of distinct colours; as such, every edge of the underlying square grid is now labelled by an $n$-dimensional binary string which specifies whether each colour $\{1,\dots,n\}$ appears (or not) at that edge. For arbitrary binary strings $\bm{A}= (A_1,\dots,A_n)$, $\bm{B}= (B_1,\dots,B_n)$, $\bm{C}= (C_1,\dots,C_n)$, $\bm{D}= (D_1,\dots,D_n)$ we then introduce the vertex weights $$\begin{aligned} \label{LLT-weights-intro} \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[left] at (-1.5,0) {$x \rightarrow $}; \draw[lgray,line width=4pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $\bm{B}$};\node[right] at (1,0) {\tiny $\bm{D}$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; \end{tikzpicture} = \bm{1}_{\bm{C}+\bm{D} \in \{0,1\}^n} \cdot x^{|\bm{D}|} q^{\varphi(\bm{D},\bm{C})+\varphi(\bm{D},\bm{D})} \qquad \bm{A},\bm{B},\bm{C},\bm{D}\in \{0,1\}^n,\end{aligned}$$ where $|\bm{D}| = \sum_{i=1}^{n} D_i$ and $\varphi(\bm{X},\bm{Y}) = \sum_{1 \leqslant i<j \leqslant n} X_i Y_j$ for any two vectors $\bm{X},\bm{Y} \in \mathbb{Z}^n$. Fix a composition $\lambda = (\lambda_1,\dots,\lambda_n)$ and two coloured compositions $\mu,\nu \in \mathcal{S}_{\lambda}$. The skew LLT (symmetric) polynomial $\mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p)$ is given by the following partition function in the model [\[LLT-weights-intro\]](#LLT-weights-intro){reference-type="eqref" reference="LLT-weights-intro"}: $$\begin{aligned} \label{G-pf-fused-intro} \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p) &= \begin{tikzpicture}[scale=0.75,baseline=(current bounding box.center),>=stealth] \foreach\y in {1,...,5}{ \draw[lgray,line width=4pt,->] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,0) -- (\x,6); } %spectral parameters \node[left] at (0.5,1) {$x_1 \rightarrow$}; \node[left] at (0.5,2) {$x_2 \rightarrow$}; \node[left] at (0.5,3) {$\vdots$}; \node[left] at (0.5,4) {$\vdots$}; \node[left] at (0.5,5) {$x_p \rightarrow$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{A}(2)$}; \node[above] at (3,6) {\footnotesize$\bm{A}(1)$}; \node[above] at (2,6) {\footnotesize$\bm{A}(0)$}; %bottom labels \node[below] at (7,0) {$\cdots$}; \node[below] at (6,0) {$\cdots$}; \node[below] at (5,0) {$\cdots$}; \node[below] at (4,0) {\footnotesize$\bm{B}(2)$}; \node[below] at (3,0) {\footnotesize$\bm{B}(1)$}; \node[below] at (2,0) {\footnotesize$\bm{B}(0)$}; %right labels \node[right] at (8,1) {$\bm{e}_0$}; \node[right] at (8,2) {$\bm{e}_0$}; \node[right] at (8,3) {$\vdots$}; \node[right] at (8,4) {$\vdots$}; \node[right] at (8,5) {$\bm{e}_0$}; %left labels \node[left] at (1,1) {$\bm{e}_0$}; \node[left] at (1,2) {$\bm{e}_0$}; \node[left] at (1,3) {$\vdots$}; \node[left] at (1,4) {$\vdots$}; \node[left] at (1,5) {$\bm{e}_0$}; \end{tikzpicture}\end{aligned}$$ where $\bm{A}(k) = \sum_{i=1}^{n} \bm{1}_{k \in \mu^{(i)}} \bm{e}_i$, $\bm{B}(k) = \sum_{i=1}^{n} \bm{1}_{k \in \nu^{(i)}} \bm{e}_i$ for all $1 \leqslant i \leqslant n$ and $k \geqslant 0$. As with our previous partition functions, there is a simple lattice path interpretation of [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"}: for each $1 \leqslant i \leqslant n$, a collection of $\lambda_i$ paths of colour $i$ enter the partition function [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} via the base of columns $\nu^{(i)}_1 < \cdots < \nu^{(i)}_{\lambda_i}$ and exit at the top of columns $\mu^{(i)}_1 < \cdots < \mu^{(i)}_{\lambda_i}$. As such, [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} provides a realization of the LLT polynomials in terms of $n$ overlapping ensembles of non-intersecting lattice paths. **Theorem 2** (Theorem [Theorem 45](#thm:LLT-int){reference-type="ref" reference="thm:LLT-int"} below). *Fix a composition $\lambda = (\lambda_1,\dots,\lambda_n)$ such that $\sum_{i=1}^{n} \lambda_i = m$, and choose two coloured compositions $\mu,\nu \in \mathcal{S}_{\lambda}$. The LLT polynomials [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} are given by the following integral expression: $$\begin{gathered} \label{LLTint-intro} \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p) = \frac{q^{m(m+1)/2}}{(q-1)^{m}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{m} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1};0) g_{\nu}(\lambda;y_1,\dots,y_m;0) \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1}{1-x_i y_j},\end{gathered}$$ where the contours $\{C_1,\dots,C_m\}$ are certain $q$-nested contours that all surround the origin; see the discussion at the start of Section [3.7](#ssec:orthog){reference-type="ref" reference="ssec:orthog"}. We have also used the notation $1^m = (1,\dots,1)$ (where $1$ appears with multiplicity $m$) and have defined $\breve\mu$ to be the unique element of $\mathcal{S}_{1^m}$ obtained by ordering the parts of $\mu$ in increasing order; see Definition [Definition 26](#def:rainbow-rec){reference-type="ref" reference="def:rainbow-rec"}.* Throughout most of the text, we consider LLT polynomials [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} in which $\lambda = N^n$ for some $N \geqslant 1$; that is, each colour is represented exactly $N$ times within the partition function [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"}. Whenever we make this choice, we write $$\mathbb{G}_{\mu/\nu}(N^n;x_1,\dots,x_p) = \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p).$$ We also assign a special notation to the coloured composition in $\mathcal{S}_{N^n}$ whose parts are as small as they can be, by writing $$\begin{aligned} \label{nothing-intro} \Delta = (0,1,\dots,N-1 | 0,1,\dots,N-1 | \cdots | 0,1,\dots,N-1) \in \mathcal{S}_{N^n}.\end{aligned}$$ ## LLT Cauchy identity and Markov kernels The Markov kernels that we study in this work are built from the (skew) Cauchy identity for the LLT polynomials [@Lam; @CGKM22; @ABW21]: **Theorem 3** (Theorem [Theorem 47](#thm:cauchy){reference-type="ref" reference="thm:cauchy"} below). *Fix two positive integers $p$ and $N$, and two alphabets $(x_1,\dots,x_p)$ and $(y_1,\dots,y_N)$. Let $\nu \in \mathcal{S}_{N^n}$ be a coloured composition. The LLT polynomials [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} satisfy the Cauchy summation identity $$\begin{aligned} \label{skew-cauchy-intro} \sum_{\mu \in \mathcal{S}_{N^n}} q^{-2\psi(\mu)} \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p) \mathbb{G}_{\mu}(y_1,\dots,y_N) = \prod_{i=1}^{p} \prod_{j=1}^{N} \frac{1}{(x_i y_j;q)_n} \cdot q^{-2\psi(\nu)} \mathbb{G}_{\nu}(y_1,\dots,y_N),\end{aligned}$$ where $(z;q)_n = \prod_{k=1}^{n} (1-q^{k-1}z)$ denotes the standard $q$-Pochhammer function, the exponents on the left and right hand side are defined as $$\begin{aligned} \psi(\mu) = \frac{1}{2} \sum_{1 \leqslant i<j \leqslant n}\ \sum_{a \in \mu^{(i)}}\ \sum_{b \in \mu^{(j)}} \bm{1}_{a>b},\end{aligned}$$ and $\mathbb{G}_{\mu}(y_1,\dots,y_N) \equiv \mathbb{G}_{\mu/\Delta}(y_1,\dots,y_N)$. This holds either as a formal power series, or as a numeric equality as long as $|q| <1$ and $|x_i y_j| <1$ for all $i,j$.* If one divides equation [\[skew-cauchy-intro\]](#skew-cauchy-intro){reference-type="eqref" reference="skew-cauchy-intro"} by the expression appearing on the right hand side, the resulting summands comprise a probability measure on coloured compositions $\mu \in \mathcal{S}_{N^n}$, assuming that they are nonnegative. One simple choice of the alphabets $(x_1,\dots,x_p)$ and $(y_1,\dots,y_N)$ which respects this nonnegativity requirement is to set the former all equal to $1$ and the latter to a Plancherel specialization ${\rm Pl}_t$, where $t \in \mathbb{R}_{>0}$ (see Section [5.4](#ssec:planch){reference-type="ref" reference="ssec:planch"}). This choice will be our sole focus in the current work; we denote the resulting Markov kernels as follows: $$\begin{aligned} \label{skew-cauchy3-intro} \mathbb{P}_{t,p}(\nu \rightarrow \mu) = q^{-2(\psi(\mu)-\psi(\nu))} \exp\left( -\frac{p(1-q^n)}{1-q}t \right) \mathbb{G}_{\mu/\nu}(1^p) \dfrac{\mathbb{G}_{\mu}({\rm Pl}_t)} {\mathbb{G}_{\nu}({\rm Pl}_t)},\end{aligned}$$ for any pair of coloured compositions $\mu,\nu \in \mathcal{S}_{N^n}$. In particular, will be interested in strings of random coloured compositions generated by the repeated action of $\mathbb{P}_{t,1}$ on the initial state [\[nothing-intro\]](#nothing-intro){reference-type="eqref" reference="nothing-intro"}: $$\begin{aligned} \label{chain-intro} \Delta \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[1]} \xrightarrow{\mathbb{P}_{t,1}} \cdots \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[m]} \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[m+1]} \xrightarrow{\mathbb{P}_{t,1}} \cdots \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[N]}.\end{aligned}$$ It is worth noting that any individual coloured composition within [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"} is distributed according to a non-skew version of [\[skew-cauchy3-intro\]](#skew-cauchy3-intro){reference-type="eqref" reference="skew-cauchy3-intro"}; see, in particular, Proposition [Proposition 49](#prop:non-skew){reference-type="ref" reference="prop:non-skew"} of the text. Our main result is a complete description of the asymptotic behaviour of the coloured compositions $\lambda^{[i]}$, $1 \leqslant i \leqslant N$, as $t \rightarrow \infty$ (with the number of steps in the chain [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"} remaining finite). ## Asymptotic analysis of Markov kernels {#intro:asymptotics} Before proceeding with the asymptotics, we introduce a convenient way to encode the coloured compositions appearing in the chain [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"}; we focus our attention on two neighbours in this sequence, namely $\lambda^{[m]}$ and $\lambda^{[m+1]}$. We shall begin with the assumption that these coloured compositions have pairwise distinct parts.[^4] Considering firstly $\lambda^{[m]} \in \mathcal{S}_{m^n}$, we see that it may be expressed uniquely in terms of its *coordinates* $\ell^{[m]} = \left\{\ell^{[m]}_1 < \cdots < \ell^{[m]}_{nm}\right\} \subset \mathbb{Z}_{\geqslant 0}$, which are simply the parts of $\lambda^{[m]}$ listed in increasing order, and its *colour sequence* $c^{[m]} = \left(c^{[m]}_1,\dots,c^{[m]}_{nm}\right) \in \{1,\dots,n\}^{nm}$, which is a vector that records the colour $c^{[m]}_i$ attributed to the part $\ell^{[m]}_i$ as it occurs within $\lambda^{[m]}$; the reader is referred to Definition [Definition 51](#def:coord){reference-type="ref" reference="def:coord"} for a precise formulation of these objects. After performing a similar identification for $\lambda^{[m+1]} \in \mathcal{S}_{(m+1)^n}$, we have the correspondences $$\begin{aligned} \label{identify-intro} \lambda^{[m]} \leftrightarrow \left( \ell^{[m]}_1,\dots,\ell^{[m]}_{nm} \Big| c^{[m]}_1,\dots,c^{[m]}_{nm} \right), \qquad \lambda^{[m+1]} \leftrightarrow \left( \ell^{[m+1]}_1,\dots,\ell^{[m+1]}_{n(m+1)} \Big| c^{[m+1]}_1,\dots,c^{[m+1]}_{n(m+1)} \right),\end{aligned}$$ and work directly with the right hand sides of these expressions in our calculations. Our next assumption concerning the coordinates $\ell^{[m]}$ and $\ell^{[m+1]}$ is that, in the limit $t \rightarrow \infty$, they arrange into $n$ *interlacing bundles* as follows: $$\begin{aligned} \label{interlace-disc-intro} \ell^{[m+1]}_{j(m+1)+i} < \ell^{[m]}_{jm+i} < \ell^{[m+1]}_{j(m+1)+i+1}, \qquad \forall\ i \in \{1,\dots,m\},\quad j \in \{0,\dots,n-1\}.\end{aligned}$$ A schematic illustration of such an arrangement, for $n=3$ and varying $m$, is provided below (see also Figure [\[fig:gue\]](#fig:gue){reference-type="ref" reference="fig:gue"} in the main body of the text): $$\begin{aligned} \label{interlace-picture} \begin{tikzpicture}[scale=0.75] %level 1 \draw[red,line width=0.7pt,->] (-0.05,1) -- (-0.05,2.05) -- (16,2.05) -- (16,3.05) -- (17.3,3.05) -- (17.3,4.05) -- (18.1,4.05) -- (18.1,5); \draw[green,line width=0.7pt,->] (0,1) -- (0,2) -- (4,2) -- (4,3) -- (15.1,3) -- (15.1,4) -- (16.2,4) -- (16.2,5); \draw[blue,line width=0.7pt,->] (0.05,1) -- (0.05,1.95) -- (8,1.95) -- (8,2.95) -- (8.9,2.95) -- (8.9,3.95) -- (14.6,3.95) -- (14.6,5); %%level 2 \draw[red,line width=0.7pt,->] (-0.25,1) -- (-0.25,3.05) -- (4.3,3.05) -- (4.3,4.05) -- (9.5,4.05) -- (9.5,5); \draw[green,line width=0.7pt,->] (-0.2,1) -- (-0.2,3) -- (3.4,3) -- (3.4,4) -- (4.8,4) -- (4.8,5); \draw[blue,line width=0.7pt,->] (-0.15,1) -- (-0.15,2.95) -- (7.3,2.95) -- (7.3,3.95) -- (8.1,3.95) -- (8.1,5); %%level 3 \draw[red,line width=0.7pt,->] (-0.45,1) -- (-0.45,4.05) -- (3.9,4.05) -- (3.9,5); \draw[green,line width=0.7pt,->] (-0.4,1) -- (-0.4,4) -- (3,4) -- (3,5); \draw[blue,line width=0.7pt,->] (-0.35,1) -- (-0.35,3.95) -- (7,3.95) -- (7,5); %nodes \node at (4,2) {$\bullet$}; \node[below] at (4,2) {$\ell_1^{[1]}$}; \node at (8,2) {$\bullet$}; \node[below] at (8,2) {$\ell_2^{[1]}$}; \node at (16,2) {$\bullet$}; \node[below] at (16,2) {$\ell_3^{[1]}$}; % \node at (3.4,3) {$\bullet$}; \node[below] at (3.4,3) {$\ell_1^{[2]}$}; \node at (4.3,3) {$\bullet$}; \node[below] at (4.3,3) {$\ell_2^{[2]}$}; \node at (7.3,3) {$\bullet$}; \node[below] at (7.3,3) {$\ell_3^{[2]}$}; \node at (8.9,3) {$\bullet$}; \node[below] at (8.9,3) {$\ell_4^{[2]}$}; \node at (15.1,3) {$\bullet$}; \node[below] at (15.1,3) {$\ell_5^{[2]}$}; \node at (17.3,3) {$\bullet$}; \node[below] at (17.3,3) {$\ell_6^{[2]}$}; % \node at (3,4) {$\bullet$}; \node[below] at (3,4) {$\ell_1^{[3]}$}; \node at (3.9,4) {$\bullet$}; \node[below] at (3.9,4) {$\ell_2^{[3]}$}; \node at (4.8,4) {$\bullet$}; \node[below] at (4.8,4) {$\ell_3^{[3]}$}; \node at (7,4) {$\bullet$}; \node[below] at (7,4) {$\ell_4^{[3]}$}; \node at (8.1,4) {$\bullet$}; \node[below] at (8.1,4) {$\ell_5^{[3]}$}; \node at (9.5,4) {$\bullet$}; \node[below] at (9.5,4) {$\ell_6^{[3]}$}; \node at (14.6,4) {$\bullet$}; \node[below] at (14.6,4) {$\ell_7^{[3]}$}; \node at (16.2,4) {$\bullet$}; \node[below] at (16.2,4) {$\ell_8^{[3]}$}; \node at (18.1,4) {$\bullet$}; \node[below] at (18.1,4) {$\ell_9^{[3]}$}; \end{tikzpicture}\end{aligned}$$ More precisely, we will assume that the coordinates $\ell^{[m]}$ and $\ell^{[m+1]}$ scale as $$\begin{aligned} \label{coord-scal-intro} \ell^{[k]}_i \mapsto q^{n-\lceil i/k \rceil} t + (q^{n-\lceil i/k \rceil} t)^\frac{1}{2} x^{[k]}_i, \qquad 1 \leqslant i \leqslant nk, \qquad k \in \{m,m+1\},\end{aligned}$$ as $t \rightarrow \infty$. Here $x^{[m]} = \left\{x^{[m]}_1 < \cdots x^{[m]}_{nm}\right\}$ and $x^{[m+1]} = \left\{x^{[m+1]}_1 < \cdots < x^{[m+1]}_{n(m+1)} \right\}$ are sets of reals that obey the relations [\[interlace-disc-intro\]](#interlace-disc-intro){reference-type="eqref" reference="interlace-disc-intro"} (with $\ell$ replaced by $x$), while $\lceil i/k \rceil$ denotes the ceiling function. Let $\theta_1^{[j]} \leqslant\cdots \leqslant\theta_j^{[j]}$ denote the eigenvalues of the top-left $j \times j$ corner of a random $N \times N$ matrix in the Gaussian Unitary Ensemble. The joint law of the eigenvalues $\theta_i^{[j]}$, $1 \leqslant i \leqslant j$, $j \in [1,N]$ is known as the *GUE corners process* of rank $N$. We let $$\begin{aligned} \rho_{\rm GUE}\left( x^{[1]}\prec \cdots \prec x^{[N]} \right) & := \rho\left(\theta_i^{[j]} = x_i^{[j]},1 \leqslant i \leqslant j \leqslant N\right)\end{aligned}$$ denote the associated joint probability density, and write $$\begin{aligned} \rho_{\rm GUE}\left( x^{[m]} \rightarrow x^{[m+1]} \right) & := \rho\left( \theta_i^{[m+1]} = x_i^{[m+1]}, 1 \leqslant i \leqslant m+1 \Big| \theta_i^{[m]} = x_i^{[m]}, 1 \leqslant i \leqslant m \right)\end{aligned}$$ for the conditional probability density for the eigenvalues of top-left $(m+1) \times (m+1)$ corner, given those of the $m \times m$ one. See [@gorin-notes] and Section [6.3](#ssec:main){reference-type="ref" reference="ssec:main"} of the current text for more information on these definitions. We are now able to state the main result of this paper. **Theorem 4** (Theorem [Theorem 54](#thm:main){reference-type="ref" reference="thm:main"} below). *In the asymptotic regime described by [\[coord-scal-intro\]](#coord-scal-intro){reference-type="eqref" reference="coord-scal-intro"}, the Markov kernel $\mathbb{P}_{t,1}$ weakly converges to a product of $n$ independent probability measures with densities in the GUE corners process, multiplied by a factor that depends only on the colour sequences [\[identify-intro\]](#identify-intro){reference-type="eqref" reference="identify-intro"}: $$\begin{gathered} \label{main-result-intro} \mathbb{P}_{t,1}\left(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}\right) \\ \rightarrow \prod_{i=1}^{n} \rho_{\rm GUE}\left( x_{(i-1)m+1}^{[m]},\dots,x_{im}^{[m]} \rightarrow x_{(i-1)(m+1)+1}^{[m+1]},\dots,x_{i(m+1)}^{[m+1]} \right) dx^{[m+1]} \cdot \mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right)\end{gathered}$$ as $t \rightarrow \infty$, where $dx^{[m+1]}$ denotes the $n(m+1)$-dimensional Lebesgue measure. The final multiplicative factor in [\[main-result-intro\]](#main-result-intro){reference-type="eqref" reference="main-result-intro"} is given explicitly by equation [\[col-markov-intro\]](#col-markov-intro){reference-type="eqref" reference="col-markov-intro"} below, and defines a discrete transition probability in a process on colour sequences: $$\begin{aligned} \label{discrete-sum-to1-intro} \sum_{c^{[m+1]}} \mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right) = 1,\end{aligned}$$ where the sum is taken over all $c^{[m+1]} \in \{1,\dots,n\}^{n(m+1)}$.* Our proof of Theorem [Theorem 4](#thm:main-intro){reference-type="ref" reference="thm:main-intro"} is by explicit analysis of [\[skew-cauchy3-intro\]](#skew-cauchy3-intro){reference-type="eqref" reference="skew-cauchy3-intro"} at $p=1$, employing the lattice model formula [\[G-pf-fused-intro\]](#G-pf-fused-intro){reference-type="eqref" reference="G-pf-fused-intro"} for the factor $\mathbb{G}_{\mu/\nu}(1)$ and (a Plancherel-specialized version of) the integral formula [\[LLTint-intro\]](#LLTint-intro){reference-type="eqref" reference="LLTint-intro"} for the functions $\mathbb{G}_{\mu}({\rm Pl}_t)$ and $\mathbb{G}_{\nu}({\rm Pl}_t)$. The study of the latter integrals proceeds by steepest descent analysis, combined with certain crucial algebraic properties of the functions [\[f-def-intro\]](#f-def-intro){reference-type="eqref" reference="f-def-intro"} and [\[g-def-intro\]](#g-def-intro){reference-type="eqref" reference="g-def-intro"} which appear in their integrands. As $t \rightarrow \infty$ one observes a remarkable factorization of these integrals into purely coordinate dependent and colour sequence dependent parts; the former can then be matched directly with transition densities for the GUE corners process. At the end of this procedure we have a leftover factor valued on colour sequences (see the second line of equation [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"}, below) and *a priori* it is by no means obvious that this quantity defines a valid discrete probability measure. Resolution of this particular issue is the topic of Section [7](#sec:discrete-dist){reference-type="ref" reference="sec:discrete-dist"} (see also Section [1.6](#ssec:intro-triangle){reference-type="ref" reference="ssec:intro-triangle"}, below). As a direct consequence of Theorem [Theorem 4](#thm:main-intro){reference-type="ref" reference="thm:main-intro"} we obtain the following corollary, completely describing the behaviour of the chain of coloured compositions [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"} as $t \rightarrow \infty$: **Corollary 5** (Corollary [Corollary 55](#cor:main){reference-type="ref" reference="cor:main"} below). *Let $\mathbb{P}_{t,N} (\Delta \rightarrow \lambda^{[1]} \rightarrow \cdots \rightarrow \lambda^{[N]})$ denote the joint distribution of coloured compositions $\lambda^{[1]},\dots, \lambda^{[N]}$ generated by $N$ applications of the kernel $\mathbb{P}_{t,1}$ to the trivial state $\Delta$. In the asymptotic regime described by [\[coord-scal-intro\]](#coord-scal-intro){reference-type="eqref" reference="coord-scal-intro"}, we have the following weak convergence of measures: $$\begin{gathered} \label{cor-joint} \mathbb{P}_{t,N} \left(\Delta \rightarrow \lambda^{[1]} \rightarrow \cdots \rightarrow \lambda^{[N]}\right) \\ \rightarrow \prod_{i=1}^{n} \rho_{\rm GUE} \left( (x^{[1]})_i \prec (x^{[2]})_i \prec \cdots \prec (x^{[N]})_i \right) dx^{[1,N]} \cdot \mathbb{P}_{\rm col}\left(c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]}\right)\end{gathered}$$ as $t \rightarrow \infty$, with $dx^{[1,N]} = \prod_{i=1}^{N} dx^{[i]}$ denoting the $nN(N+1)/2$-dimensional Lebesgue measure. Here we have introduced the shorthand $$\begin{aligned} \left(x^{[k]}\right)_i = \left(x^{[k]}_{(i-1)k+1},\dots,x^{[k]}_{ik}\right), \qquad \forall\ 1 \leqslant i \leqslant n,\ \ 1 \leqslant k \leqslant N,\end{aligned}$$ and $\mathbb{P}_{\rm col}(c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]})$ is a joint distribution on colour sequences given explicitly by [\[joint-distr-col-intro\]](#joint-distr-col-intro){reference-type="eqref" reference="joint-distr-col-intro"} below.* ## Distribution on interlacing triangles {#ssec:intro-triangle} While Theorem [Theorem 4](#thm:main-intro){reference-type="ref" reference="thm:main-intro"} and Corollary [Corollary 5](#cor:main-intro){reference-type="ref" reference="cor:main-intro"} provide a complete description of the asymptotic behaviour of coordinates of the coloured compositions [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"} as $t \rightarrow \infty$, we are left with the task of understanding the factors $\mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right)$ and $\mathbb{P}_{\rm col}\left(c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]}\right)$ that occur therein. These factors provide information about how colours distribute themselves within interlacing diagrams of the form [\[cor:main-intro\]](#cor:main-intro){reference-type="eqref" reference="cor:main-intro"}, as $t \rightarrow \infty$. Let $i^{[m]} \in \{1,\dots,n\}^{nm}$ and $j^{[m+1]} \in \{1,\dots,n\}^{n(m+1)}$ be two sequences such that each colour $\{1,\dots,n\}$ is represented exactly $m$ times in $i^{[m]}$ and $m+1$ times in $j^{[m+1]}$. We say that these colour sequences interlace, and write $i^{[m]} \prec j^{[m+1]}$, provided they can be stacked to form an *admissible diagram*: $$\begin{aligned} \label{admiss-diag} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,<-] (0.5,0) -- (12,0); \foreach\x in {1,2,3,5,6,7,9,10,11}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$j_1$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$j_{m+1}$}; % \node[above] at (5,0.5) {$\cdots$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \node[above] at (9,0.5) {$j_{(n-1)(m+1)+1}$}; \node[above] at (10.2,0.5) {$\cdots$}; \node[above] at (11,0.5) {$j_{n(m+1)}$}; % \foreach\x in {1.5,2.5,5.5,6.5,9.5,10.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$i_1$}; \node[below] at (2,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$i_m$}; % \node[below] at (5.5,-0.5) {$\cdots$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \node[below] at (9.2,-0.5) {$i_{(n-1)m+1}$}; \node[below] at (10.1,-0.5) {$\cdots$}; \node[below] at (10.7,-0.5) {$i_{nm}$}; % \node[left] at (0.5,0) {$\emptyset$}; \node[right] at (12,0) {$[1,n]$}; \end{tikzpicture}\end{aligned}$$ In the above diagram the incoming/outgoing vertical arrows are grouped into a total of $n$ bundles, each of width $m$ or $m+1$, respectively. The colours $i^{[m]}=(i_1,\dots,i_{nm})$ enter sequentially via the arrows at the base, while colours $j^{[m+1]}=(j_1,\dots,j_{n(m+1)})$ exit sequentially via the arrows at the top. A copy of all colours $[1,n] \equiv \{1,\dots,n\}$ enters via the right, and no colours exit via the left. The diagram is admissible provided that, after one draws the trajectories of all coloured paths, each colour $\{1,\dots,n\}$ never occurs more than once at any point along the thick horizontal line. Given two colour sequences $i^{[m]} \prec j^{[m+1]}$ we define a statistic $\xi\left( i^{[m]}; j^{[m+1]} \right)$ which enumerates the number of events of the form $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,<-] (0.5,0) -- (1.5,0); \draw[lgray,line width=1.5pt,->] (1,-0.5) -- (1,0); \node[below] at (1,-0.5) {$i$}; \node[left] at (0.5,0) {$c$}; \end{tikzpicture}\end{aligned}$$ in a path of colour $c$ passes over a colour $i$, with $c>i$, within the diagram [\[admiss-diag\]](#admiss-diag){reference-type="eqref" reference="admiss-diag"}. **Theorem 6**. *The factor $\mathbb{P}_{\rm col} \left( c^{[m]} \rightarrow c^{[m+1]} \right)$ appearing in [\[main-result-intro\]](#main-result-intro){reference-type="eqref" reference="main-result-intro"} is given by $$\begin{gathered} \label{col-markov-intro} \mathbb{P}_{\rm col} \left( c^{[m]} \rightarrow c^{[m+1]} \right) \\ = \bm{1}_{c^{[m]} \prec c^{[m+1]}} (-1)^n q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}-\xi\left( c^{[m]}; c^{[m+1]} \right)} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \frac{ g^{c^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right) } { g^{c^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) },\end{gathered}$$ where $g^{c^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right)$ denotes a partition function of the form [\[g-def-intro\]](#g-def-intro){reference-type="eqref" reference="g-def-intro"}, with $m \mapsto nm$, $\lambda = m^n$, $\mu=\Delta$, $s=0$, $$\begin{aligned} (x_1,\dots,x_{nm}) = \underbrace{(q^{n-1},\dots,q^{n-1})}_{m\ {\rm times}} \cup \cdots \cup \underbrace{(q,\dots,q)}_{m\ {\rm times}} \cup \underbrace{(1,\dots,1)}_{m\ {\rm times}} \equiv \vec{Q}^{[m]},\end{aligned}$$ and in which colour $c^{[m]}_i$ exits via the left edge of row $i$ within [\[g-def-intro\]](#g-def-intro){reference-type="eqref" reference="g-def-intro"} (rather than in totally ordered fashion). An analogous definition applies to $g^{c^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right)$. The expression [\[col-markov-intro\]](#col-markov-intro){reference-type="eqref" reference="col-markov-intro"} constitutes a valid discrete transition probability; namely, it satisfies the sum-to-unity property [\[discrete-sum-to1-intro\]](#discrete-sum-to1-intro){reference-type="eqref" reference="discrete-sum-to1-intro"}.* The sum-to-unity property in Theorem [Theorem 6](#thm:discr-distr){reference-type="ref" reference="thm:discr-distr"} is not immediate and plays a substantial role in the proof of Theorem [Theorem 4](#thm:main-intro){reference-type="ref" reference="thm:main-intro"} and Corollary [Corollary 5](#cor:main-intro){reference-type="ref" reference="cor:main-intro"}. In particular, as briefly mentioned in the beginning of Section [1.5](#intro:asymptotics){reference-type="ref" reference="intro:asymptotics"}, we only compute the asymptotics of the Markov kernel [\[skew-cauchy3-intro\]](#skew-cauchy3-intro){reference-type="eqref" reference="skew-cauchy3-intro"} under a certain ansatz for the behaviour of the coordinates $\ell^{[m]},\ell^{[m+1]}$ and colour sequences $c^{[m]},c^{[m+1]}$. To show that this ansatz asymptotically exhausts the full measure induced by the kernel [\[skew-cauchy3-intro\]](#skew-cauchy3-intro){reference-type="eqref" reference="skew-cauchy3-intro"} requires the above sum-to-unity property, whose proof hinges upon a rather unusual expansion property of the partition functions in question; see, in particular, Theorem [Theorem 58](#thm:expand){reference-type="ref" reference="thm:expand"}. The main tool behind this proof are commutation relations between the row operators used to build our partition functions, which in turn are a consequence of the underlying Yang--Baxter integrability. More generally, one may consider collections of colour sequences $\emptyset \prec c^{[1]} \prec \cdots \prec c^{[N]}$ such that for all $1 \leqslant k \leqslant N$, $c^{[k]} \in \{1,\dots,n\}^{nk}$ and each colour $\{1,\dots,n\}$ is represented exactly $k$ times in $c^{[k]}$. We refer to such a collection of positive integers as an *interlacing triangular array* of *rank* $n$ and *height* $N$, and let $\mathcal{T}_N(n)$ denote the set of all such objects; see Definition [Definition 60](#def:interlace){reference-type="ref" reference="def:interlace"} for a more precise formulation. As a direct consequence of Theorem [Theorem 6](#thm:discr-distr){reference-type="ref" reference="thm:discr-distr"} we obtain the following result: **Corollary 7** (Corollary [Corollary 64](#cor:joint-col){reference-type="ref" reference="cor:joint-col"} below). *Let $\emptyset \prec c^{[1]} \prec \cdots \prec c^{[N]}$ be an interlacing triangular array generated by $N$ successive applications of the Markov kernel [\[col-markov-intro\]](#col-markov-intro){reference-type="eqref" reference="col-markov-intro"} on the empty sequence $\emptyset$. This array has joint distribution $$\begin{aligned} \label{joint-distr-col-intro} \mathbb{P}_{\rm col}\left(c^{[1]} \prec \cdots \prec c^{[N]}\right) = \bm{1}_{c^{[1]} \prec \cdots \prec c^{[N]}} (-1)^{nN} q^{\binom{nN+1}{2}} \frac{(1-q)^{n \binom{N}{2}}}{(q;q)_{n}^{N^2}} g^{c^{[N]}}_{\Delta}\left(N^n;\vec{Q}^{[N]}\right) \prod_{i=1}^{N} q^{-\xi \left( c^{[i-1]} ; c^{[i]} \right)}.\end{aligned}$$* ## Positivity and enumeration conjectures A number of interesting observations arise concerning the measure [\[joint-distr-col-intro\]](#joint-distr-col-intro){reference-type="eqref" reference="joint-distr-col-intro"}, as well as the set $\mathcal{T}_N(n)$ of interlacing triangular arrays on which it is supported. The first is a positivity property that we noticed from explicit implementation of the Markov kernel [\[col-markov-intro\]](#col-markov-intro){reference-type="eqref" reference="col-markov-intro"} on a computer: **Conjecture 8** (Conjecture [Conjecture 66](#conj:pos){reference-type="ref" reference="conj:pos"} below). *Fix integers $m,n \geqslant 1$ and a colour sequence $c^{[m]} \in \{1,\dots,n\}^{nm}$. Let $\mathbb{P}_{\rm col}(c^{[m]})$ denote the probability of arriving at the colour sequence $c^{[m]}$ after $m$ applications of the Markov kernel [\[col-markov-intro\]](#col-markov-intro){reference-type="eqref" reference="col-markov-intro"} to the trivial sequence $c^{[0]}=\emptyset$. Then one has that $$\begin{aligned} \label{pos-conj-intro} \mathbb{P}_{\rm col}\left(c^{[m]}\right) = \mathcal{P}\left(c^{[m]}\right) \cdot \left( \prod_{i=1}^{n} \frac{1-q}{1-q^i} \right)^{m^2} \quad \text{where}\ \ \mathcal{P}\left(c^{[m]}\right) \in \mathbb{N}[q].\end{aligned}$$* In fact, one sees that [\[pos-conj-intro\]](#pos-conj-intro){reference-type="eqref" reference="pos-conj-intro"} expresses $\mathbb{P}_{\rm col}(c^{[m]})$ as a ratio of two positive polynomials in $q$; the denominator is nothing but the Poincaré polynomial associated to $\mathfrak{S}_n$ raised to the power $m^2$. An explicit illustration of this conjecture, for $n=2$, is given in Figure [\[fig:n=2\]](#fig:n=2){reference-type="ref" reference="fig:n=2"}. At this stage we do not know of any combinatorial interpretation of $\mathcal{P}\left(c^{[m]}\right)$, although it would be very interesting to find one. There is also the purely combinatorial problem of enumerating the number of elements in the set $\mathcal{T}_N(n)$. It a trivial fact that $|\mathcal{T}_N(1)| = 1$,[^5] and one can easily show that $|\mathcal{T}_N(2)|=2^N$; see Proposition [Proposition 67](#prop:n=2){reference-type="ref" reference="prop:n=2"}. While for $n \geqslant 3$ we have no direct enumeration of $|\mathcal{T}_N(n)|$, we do present two conjectures relating to $(n+1)$-colourings of certain graphs: **Conjecture 9** (Conjecture [Conjecture 69](#conj:triangle){reference-type="ref" reference="conj:triangle"} below). *Let $G^{\triangle}_N$ denote the triangular graph $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (1,0) {$\bullet$}; \node at (1.5,0) {$\bullet$}; % \node at (0.25,0.5) {$\bullet$}; \node at (0.75,0.5) {$\bullet$}; \node at (1.25,0.5) {$\bullet$}; % \node at (0.5,1) {$\bullet$}; \node at (1,1) {$\bullet$}; % \node at (0.75,1.5) {$\bullet$}; %%%%%%%%%% \draw (0,0) -- (1.5,0); \draw (0.25,0.5) -- (1.25,0.5); \draw (0.5,1) -- (1,1); % \draw (0.25,0.5) -- (0.5,0); \draw (0.5,1) -- (1,0); \draw (0.75,1.5) -- (1.5,0); % \draw (0,0) -- (0.75,1.5); \draw (0.5,0) -- (1,1); \draw (1,0) -- (1.25,0.5); \end{tikzpicture}\end{aligned}$$ where the number of vertices along one side of the triangle is equal to $N+1$. Let $\mathfrak{g}^{\triangle}_N(4)$ denote the number of $4$-colourings of $G^{\triangle}_N$ (adjacent vertices must have different colours). We conjecture that $$\begin{aligned} 4\cdot|\mathcal{T}_N(3)| = \mathfrak{g}^{\triangle}_N(4), \qquad \forall\ N \geqslant 1.\end{aligned}$$* **Conjecture 10** (Conjecture [Conjecture 71](#conj:a5){reference-type="ref" reference="conj:a5"} below). *Let $G^{\varhexstar}_N$ denote the graph $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (1,0) {$\bullet$}; \node at (1.5,0) {$\bullet$}; %%% \node at (0,0.5) {$\bullet$}; \node at (0.5,0.5) {$\bullet$}; \node at (1,0.5) {$\bullet$}; \node at (1.5,0.5) {$\bullet$}; %%% \node at (0,1) {$\bullet$}; \node at (0.5,1) {$\bullet$}; \node at (1,1) {$\bullet$}; \node at (1.5,1) {$\bullet$}; %%% \node at (0,1.5) {$\bullet$}; \node at (0.5,1.5) {$\bullet$}; \node at (1,1.5) {$\bullet$}; \node at (1.5,1.5) {$\bullet$}; %%%%%%% \draw (0.5,0) -- (0,0.5); \draw (1,0) -- (0,1); \draw (1.5,0) -- (0,1.5); \draw (1.5,0.5) -- (0.5,1.5); \draw (1.5,1) -- (1,1.5); % \draw (1,0) -- (1.5,0.5); \draw (0.5,0) -- (1.5,1); \draw (0,0) -- (1.5,1.5); \draw (0,0.5) -- (1,1.5); \draw (0,1) -- (0.5,1.5); % \draw (0,0) -- (0,1.5); \draw (0.5,0) -- (0.5,1.5); \draw (1,0) -- (1,1.5); \draw (1.5,0) -- (1.5,1.5); % \draw (0,0) -- (1.5,0); \draw (0,0.5) -- (1.5,0.5); \draw (0,1) -- (1.5,1); \draw (0,1.5) -- (1.5,1.5); \end{tikzpicture}\end{aligned}$$ where two vertices share an edge if they are connected via a king move on the chessboard (that is, they a connected via a unit horizontal, vertical, or diagonal step), and the number of vertices along one side of the square is equal to $N+1$. Let $\mathfrak{g}^{\varhexstar}_N(5)$ denote the number of $5$-colourings of $G^{\varhexstar}_N$ (adjacent vertices must have different colours). We conjecture that $$\begin{aligned} 5 \cdot |\mathcal{T}_N(4)| = \mathfrak{g}^{\varhexstar}_N(5), \qquad \forall\ N \geqslant 1.\end{aligned}$$* Conjecture [Conjecture 9](#conj:triangle-intro){reference-type="ref" reference="conj:triangle-intro"} has recently been proved bijectively [@GaetzGao], but Conjecture [Conjecture 10](#conj:a5-intro){reference-type="ref" reference="conj:a5-intro"} remains open. Based on these conjectures, it is tempting to speculate about the possibility of assigning meaningful probability measures to graph colourings, but this lies outside the scope of the current work. ## Acknowledgments {#acknowledgments .unnumbered} Amol Aggarwal was partially supported by a Clay Research Fellowship, a Packard Fellowship, and the IAS School of Mathematics. Alexei Borodin was partially supported by the NSF grants DMS-1664619, DMS-1853981, and the Simons Investigator program. Michael Wheeler was supported by an Australian Research Council Future Fellowship, grant FT200100981. # Fermionic vertex models In this section we review the basic vertex models that will be used throughout the text; these are *fermionic vertex models*, as introduced in [@ABW21]. We give the explicit form of our vertex weights in Sections [2.2](#ssec:L){reference-type="ref" reference="ssec:L"}--[2.3](#ssec:M){reference-type="ref" reference="ssec:M"}, as well as the Yang--Baxter equations that they satisfy, in Section [2.4](#ssec:YB){reference-type="ref" reference="ssec:YB"}. We conclude by introducing *row operators* and studying algebraic relations between them, in Sections [2.5](#ssec:row){reference-type="ref" reference="ssec:row"}--[2.6](#ssec:cr){reference-type="ref" reference="ssec:cr"}; these results will be needed in the subsequent material on partition functions in Section [3](#sec:pf){reference-type="ref" reference="sec:pf"}. ## Notation {#ssec:not} For all pairs of positive integers $i,j$ such that $i \leqslant j$ let $[i,j] \subset \mathbb{N}$ denote the interval $\{i,i+1,\dots,j\}$. Similarly, we define $(i,j] = [i+1,j]$ when $i<j$, and $(i,j] = \emptyset$ when $i=j$. For all $1\leqslant i \leqslant n$, let $\bm{e}_i \in \mathbb{R}^n$ denote the $i$-th Euclidean unit vector. Let $\bm{e}_0 \in \mathbb{R}^n$ denote the zero vector. Define $\bm{e}_{[i,j]} = \sum_{i \leqslant k \leqslant j} \bm{e}_k$; more generally, for any non-empty set $I \subset \mathbb{N}$ we write $\bm{e}_I = \sum_{i \in I} \bm{e}_i$. For any vector $\bm{A}= (A_1,\dots,A_n) \in (\mathbb{Z}_{\geqslant 0})^n$ and indices $i,j \in \{1,\dots,n\}$ we define $$\begin{aligned} \bm{A}^{+}_{i} = \bm{A}+ \bm{e}_i, \quad \bm{A}^{-}_{i} = \bm{A}- \bm{e}_i, \quad \bm{A}^{+-}_{ij} = \bm{A}+ \bm{e}_i - \bm{e}_j, \quad A_{[i,j]} = \sum_{k=i}^{j} A_k, \quad |\bm{A}| = A_{[1,n]} = \sum_{k=1}^{n} A_k,\end{aligned}$$ where in the second last case it is assumed that $i \leqslant j$. By agreement, we choose $A_{[i,j]} = 0$ for $i>j$. Let $\mathfrak{S}_m$ denote the symmetric group of degree $m$. For any set $I \subset \mathbb{N}$ we define $\mathfrak{S}_I$ to be the set of all permutations of the elements in $I$; in particular, we then have $\mathfrak{S}_{[1,m]}\equiv \mathfrak{S}_m$. ## $L$-weights {#ssec:L} Our partition functions will be expressed in terms of two families of vertex weights. The first of these were introduced in [@ABW21 Example 8.1.2 and Figure 8.2] and we call them *$L$-weights*; they are denoted by[^6] $$\begin{aligned} \label{generic-L} \tilde{L}^{(s)}_{z,q}(\bm{A},b;\bm{C},d) \equiv \tilde{L}_z(\bm{A},b;\bm{C},d) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[left] at (-1.5,0) {$z \rightarrow$}; \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $d$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; % \node[below] at (0,-1.4) {$(s)$}; \end{tikzpicture} \quad\quad b,d \in \{0,1,\dots,n\}, \quad \bm{A},\bm{C}\in \{0,1\}^n.\end{aligned}$$ Labels assigned to the left and right horizontal edges take values in $\{0,1,\dots,n\}$, while labels assigned to the bottom and top vertical edges are $n$-dimensional binary strings. We define $$\begin{aligned} \label{conserve-L} \tilde{L}^{(s)}_{z,q}(\bm{A},b;\bm{C},d) = 0, \qquad \text{unless} \qquad \bm{A}+ \bm{e}_b = \bm{C}+ \bm{e}_d.\end{aligned}$$ The property [\[conserve-L\]](#conserve-L){reference-type="eqref" reference="conserve-L"} expresses conservation of particles as one traverses through the vertex in the SW $\rightarrow$ NE direction. For the cases where the constraint $\bm{A}+ \bm{e}_b = \bm{C}+ \bm{e}_d$ is obeyed, we have the following table of weights: $$\begin{aligned} \label{fund-weights} \begin{tabular}{|c|c|c|} \hline \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $0$};\node[right] at (1,0) {\tiny $0$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $i$};\node[right] at (1,0) {\tiny $i$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $0$};\node[right] at (1,0) {\tiny $i$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}^{-}_i$}; \end{tikzpicture} \quad \\[1.3cm] \quad $\dfrac{1-q^{A_{[1,n]}}sz}{1-s z}$ \quad & \quad $\dfrac{(-1)^{A_i}(s-q^{A_i}z)q^{A_{(i,n]}}s}{1-s z}$ \quad & \quad $\dfrac{(q^{A_i}-1) q^{A_{(i,n]}}sz}{1-s z}$ \quad \\[0.7cm] \hline \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $i$};\node[right] at (1,0) {\tiny $0$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}^{+}_i$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $i$};\node[right] at (1,0) {\tiny $j$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}^{+-}_{ij}$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $j$};\node[right] at (1,0) {\tiny $i$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{A}^{+-}_{ji}$}; \end{tikzpicture} \quad \\[1.3cm] \quad $\dfrac{1-s^2 q^{A_{[1,n]}}}{1-s z}$ \quad & \quad $\dfrac{(q^{A_j}-1) q^{A_{(j,n]}}sz}{1-s z}$ \quad & \quad $\dfrac{(q^{A_i}-1) q^{A_{(i,n]}}s^2}{1-s z}$ \quad \\[0.7cm] \hline \end{tabular} \end{aligned}$$ where it is assumed that $1 \leqslant i < j \leqslant n$. The weights [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"} take a very similar form to the weights $\tilde{L}_z(\bm{A},b;\bm{C},d)$ defined in [@BorodinW Sections 2.2 and 2.5]; in fact, the two sets of weights differ only with respect to two details. The first is that the weights [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"} are defined only for $\bm{A},\bm{C}\in \{0,1\}^n$ (that is, for *fermionic* states), whereas in [@BorodinW Section 2.2] one has $\bm{A},\bm{C}\in (\mathbb{Z}_{\geqslant 0})^n$ (*bosonic* states). The second is that the specific weight $\tilde{L}_z(\bm{A},i;\bm{A},i)$ is different across the two works[^7], when $i \in [1,n]$ and $A_i > 0$. In certain partition functions that we subsequently define, the boundary conditions inject into the lattice exactly one particle of each colour $\{1,\dots,n\}$. In such partition functions, each colour $\{1,\dots,n\}$ flows at most once through a vertex of the lattice; in this setting, both of the differences between the weights [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"} and those of [@BorodinW Sections 2.2 and 2.5], pointed out above, are no longer apparent. This fact will allow us to deduce matchings between certain functions that we define in the present work and those of [@BorodinW], in spite of the fact that the model used in the current text is *a priori* different. ## $M$-weights {#ssec:M} The second family of vertex weights we call *$M$-weights*; they are denoted by $$\begin{aligned} \label{generic-M} \tilde{M}^{(s)}_{z,q}(\bm{A},b;\bm{C},d) \equiv \tilde{M}_z(\bm{A},b;\bm{C},d) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[right] at (1.5,0) {$\leftarrow z$}; \draw[lgray,line width=1.5pt,<-] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $d$};\node[right] at (1,0) {\tiny $b$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; % \node[below] at (0,-1.4) {$(s)$}; \end{tikzpicture} \quad\quad b,d \in \{0,1,\dots,n\}, \quad \bm{A},\bm{C}\in \{0,1\}^n.\end{aligned}$$ As in the case of $L$-weights, labels assigned to the left and right horizontal edges take values in $\{0,1,\dots,n\}$, while labels assigned to the bottom and top vertical edges are $n$-dimensional binary strings. In contrast to $L$-weights, particle conservation for $M$-type vertices happens in the SE $\rightarrow$ NW direction, namely: $$\begin{aligned} \label{conserve-M} \tilde{M}^{(s)}_{z,q}(\bm{A},b;\bm{C},d) = 0, \qquad \text{unless} \qquad \bm{A}+ \bm{e}_b = \bm{C}+ \bm{e}_d.\end{aligned}$$ For all $\bm{A},\bm{C}\in \{0,1\}^n$ and $b,d \in \{0,1,\dots,n\}$, we define $$\begin{aligned} \label{LM-sym} \tilde{M}^{(s)}_{z,q}(\bm{A},b;\bm{C},d) = \tilde{L}^{(1/s)}_{1/z,1/q}(\bm{A},b;\bm{C},d),\end{aligned}$$ expressing every $M$-weight in terms of a corresponding $L$-weight, under reflection about the thick vertical line of the vertex, and reciprocation of the parameters $z$, $q$, $s$. ## Yang--Baxter equations {#ssec:YB} We introduce one further set of vertex weights which arise from the fundamental $R$-matrix for the quantum affine superalgebra $U_q(\widehat{\mathfrak{sl}}(1|n))$ [@BazhanovShadrikov]; these we call *fundamental weights*. They are denoted by the crossing of two thin lines: $$\begin{aligned} \label{R-vert} R_{z,q}(a,b;c,d) \equiv R_z(a,b;c,d) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $d$}; \node[below] at (0,-1) {\tiny $a$};\node[above] at (0,1) {\tiny $c$}; \end{tikzpicture}, \quad a,b,c,d \in \{0,1,\dots,n\}.\end{aligned}$$ These vertices have the conservation property $$\begin{aligned} R_{z,q}(a,b;c,d) = 0, \qquad \text{unless} \qquad \bm{e}_a + \bm{e}_b = \bm{e}_c + \bm{e}_d.\end{aligned}$$ For the cases where the constraint $\bm{e}_a + \bm{e}_b = \bm{e}_c + \bm{e}_d$ is obeyed, we have the following table of weights: $$\begin{aligned} \label{fund-vert} \begin{tabular}{|c|c|c|} \hline \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $0$};\node[right] at (1,0) {\tiny $0$}; \node[below] at (0,-1) {\tiny $0$};\node[above] at (0,1) {\tiny $0$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $a$};\node[right] at (1,0) {\tiny $a$}; \node[below] at (0,-1) {\tiny $b$};\node[above] at (0,1) {\tiny $b$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $a$};\node[right] at (1,0) {\tiny $b$}; \node[below] at (0,-1) {\tiny $b$};\node[above] at (0,1) {\tiny $a$}; \end{tikzpicture} \quad \\[1.3cm] \quad $1$ \quad & \quad $\dfrac{q(1-z)}{1-qz}$ \quad & \quad $\dfrac{1-q}{1-qz}$ \quad \\[0.7cm] \hline \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $b$}; \node[below] at (0,-1) {\tiny $b$};\node[above] at (0,1) {\tiny $b$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $b$}; \node[below] at (0,-1) {\tiny $a$};\node[above] at (0,1) {\tiny $a$}; \end{tikzpicture} \quad & \quad \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); \draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $b$};\node[right] at (1,0) {\tiny $a$}; \node[below] at (0,-1) {\tiny $a$};\node[above] at (0,1) {\tiny $b$}; \end{tikzpicture} \quad \\[1.3cm] \quad $\dfrac{z-q}{1-qz}$ \quad & \quad $\dfrac{1-z}{1-qz}$ \quad & \quad $\dfrac{(1-q)z}{1-qz}$ \quad \\[0.7cm] \hline \end{tabular}\end{aligned}$$ where we assume that $0 \leqslant a < b \leqslant n$. The $L$-weights, $M$-weights and fundamental weights satisfy a collection of Yang--Baxter equations, that we record as a single theorem below. These Yang--Baxter equations underpin the algebraic relations between the row operators that we define in Section [2.5](#ssec:row){reference-type="ref" reference="ssec:row"}. **Theorem 11**. *For any fixed integers $a_1,a_2,a_3,b_1,b_2,b_3 \in \{0,1,\dots,n\}$ and vectors $\bm{A},\bm{B}\in \{0,1\}^n$, the vertex weights [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"}, [\[generic-M\]](#generic-M){reference-type="eqref" reference="generic-M"}, [\[R-vert\]](#R-vert){reference-type="eqref" reference="R-vert"} satisfy the relations $$\begin{gathered} \label{RLLa} \sum_{0 \leqslant c_1,c_2 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} R_{y/x}(a_2,a_1;c_2,c_1) \tilde{L}_x(\bm{A},c_1;\bm{C},b_1) \tilde{L}_y(\bm{C},c_2;\bm{B},b_2) \\ = \sum_{0 \leqslant c_1,c_2 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} \tilde{L}_y(\bm{A},a_2;\bm{C},c_2) \tilde{L}_x(\bm{C},a_1;\bm{B},c_1) R_{y/x}(c_2,c_1;b_2,b_1),\end{gathered}$$* *$$\begin{gathered} \label{RLLb} \sum_{0 \leqslant c_1,c_3 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} \tilde{L}_x(\bm{A},a_1;\bm{C},c_1) R_{1/(qxz)}(a_3,c_1;c_3,b_1) \tilde{M}_z(\bm{C},c_3;\bm{B},b_3) \\ = \sum_{0 \leqslant c_1,c_3 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} \tilde{M}_z(\bm{A},a_3;\bm{C},c_3) R_{1/(qxz)}(c_3,a_1;b_3,c_1) \tilde{L}_x(\bm{C},c_1;\bm{B},b_1),\end{gathered}$$* *$$\begin{gathered} \label{RLLc} \sum_{0 \leqslant c_2,c_3 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} \tilde{M}_y(\bm{A},a_2;\bm{C},c_2) \tilde{M}_z(\bm{C},a_3;\bm{B},c_3) R_{y/z}(c_3,c_2;b_3,b_2) \\ = \sum_{0 \leqslant c_2,c_3 \leqslant n} \ \sum_{\bm{C}\in \{0,1\}^n} R_{y/z}(a_3,a_2;c_3,c_2) \tilde{M}_z(\bm{A},c_3;\bm{C},b_3) \tilde{M}_y(\bm{C},c_2;\bm{B},b_2).\end{gathered}$$* *Proof.* All three equations may be recovered from the master Yang--Baxter equation [\[master\]](#master){reference-type="eqref" reference="master"}; we will comment briefly on this in Section [4.3](#ssec:master-yb){reference-type="ref" reference="ssec:master-yb"}. The equations [\[RLLa\]](#RLLa){reference-type="eqref" reference="RLLa"}--[\[RLLc\]](#RLLc){reference-type="eqref" reference="RLLc"} are the fermionic cousins of equations (2.3.1)--(2.3.3) in [@BorodinW Section 2.3]; the latter being valid for the bosonic counterparts of the models [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"} and [\[generic-M\]](#generic-M){reference-type="eqref" reference="generic-M"}. ◻ ## Row operators {#ssec:row} Let $V$ be the vector space obtained by taking the formal linear span of all $n$-dimensional binary strings: $$\begin{aligned} V= \bigoplus_{\bm{A}\in \{0,1\}^n} \mathbb{C} \left|\bm{A}\right\rangle,\end{aligned}$$ and for any $N \geqslant 0$ consider the $(N+1)$-fold tensor product of this space: $$\begin{aligned} \mathbb{V}(N) = \underbrace{V \otimes \cdots \otimes V}_{N+1\ {\rm times}}.\end{aligned}$$ For each $0 \leqslant i,j \leqslant n$ we introduce a linear operator $T^{\rightarrow}_{i,j}(x;N) \in {\rm End}(\mathbb{V}(N))$ with the action $$\begin{aligned} \label{C-row} T^{\rightarrow}_{i,j}(x;N) : \bigotimes_{k=0}^{N} \left|\bm{B}(k)\right\rangle \mapsto \sum_{\bm{A}(0),\ldots,\bm{A}(N) \in \{0,1\}^n} \left( \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (-0.2,0) {$x \rightarrow$}; \node[left] at (0.5,0) {\footnotesize$i$};\node[right] at (5.5,0) {\footnotesize$j$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture} \right) \bigotimes_{k=0}^{N} \left|\bm{A}(k)\right\rangle.\end{aligned}$$ The quantity $$\begin{aligned} \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (-0.2,0) {$x \rightarrow$}; \node[left] at (0.5,0) {\footnotesize$i$};\node[right] at (5.5,0) {\footnotesize$j$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture}\end{aligned}$$ is a one-row partition function in the model [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"}, and can be calculated by multiplying the weights of each vertex from left to right, noting that the integer values prescribed to all internal vertical edges are fixed by the local conservation property [\[conserve-L\]](#conserve-L){reference-type="eqref" reference="conserve-L"}. In a similar vein, for each $0 \leqslant i,j \leqslant n$ we introduce a linear operator $T^{\leftarrow}_{i,j}(x;N) \in {\rm End}(\mathbb{V}(N))$ with the action $$\begin{aligned} \label{B-row} T^{\leftarrow}_{i,j}(x;N) : \bigotimes_{k=0}^{N} \left|\bm{B}(k)\right\rangle \mapsto \sum_{\bm{A}(0),\ldots,\bm{A}(N) \in \{0,1\}^n} \left( \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,<-] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (6.2,0) {$\leftarrow x$}; \node[left] at (0.5,0) {\footnotesize$i$};\node[right] at (5.5,0) {\footnotesize$j$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture} \right) \bigotimes_{k=0}^{N} \left|\bm{A}(k)\right\rangle,\end{aligned}$$ where the quantity $$\begin{aligned} \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (6.2,0) {$\leftarrow x$}; \node[left] at (0.5,0) {\footnotesize$i$};\node[right] at (5.5,0) {\footnotesize$j$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture}\end{aligned}$$ is a one-row partition function in the model [\[generic-M\]](#generic-M){reference-type="eqref" reference="generic-M"}. ## Commutation relations {#ssec:cr} We introduce a lift of $\mathbb{V}(N)$ to an infinite tensor product: $$\begin{aligned} \mathbb{V}(\infty) = {\rm Span}_{\mathbb{C}} \left\{ \bigotimes_{k=0}^{\infty} \left|\bm{A}(k)\right\rangle \right\}\end{aligned}$$ where the binary strings $\bm{A}(k) \in \{0,1\}^n$, $k \geqslant 0$ have the stability property $$\begin{aligned} \exists\ M \in \mathbb{N} \ : \ \bm{A}(k) = \bm{e}_0,\ \forall\ k \geqslant M.\end{aligned}$$ Let $T^{\rightarrow}_{i,0}(x;\infty) = \mathcal{C}_i(x)$ and $T^{\leftarrow}_{i,0}(x;\infty) = \mathcal{B}_i(x)$ denote the corresponding lifts of the operators [\[C-row\]](#C-row){reference-type="eqref" reference="C-row"} and [\[B-row\]](#B-row){reference-type="eqref" reference="B-row"}, in the case where the right index $j$ is set to $0$. We shall only ever consider the case where $\mathcal{C}_i(x)$ and $\mathcal{B}_i(x)$ act on stable states in the infinite tensor product, *i.e.*, on the elements of $\mathbb{V}(\infty)$. **Theorem 12**. *Fix two nonnegative integers $i,j$ such that $1 \leqslant i<j \leqslant n$, and two arbitrary complex parameters $x,y$. The following exchange relations hold: $$\begin{aligned} \label{CC<} \frac{x-qy}{x-y} \mathcal{C}_i(y) \mathcal{C}_j(x) &= \frac{(1-q)y}{x-y} \mathcal{C}_i(x) \mathcal{C}_j(y) + \mathcal{C}_j(x) \mathcal{C}_i(y), \\ \label{BB<} \frac{y-qx}{q(y-x)} \mathcal{B}_j(y) \mathcal{B}_i(x) &= \frac{(1-q)x}{q(y-x)} \mathcal{B}_j(x) \mathcal{B}_i(y) + \mathcal{B}_i(x) \mathcal{B}_j(y).\end{aligned}$$* *Proof.* The proof of [\[CC\<\]](#CC<){reference-type="eqref" reference="CC<"} makes use of the first Yang--Baxter equation [\[RLLa\]](#RLLa){reference-type="eqref" reference="RLLa"}, applied successively to the two-row partition function that arises by joining the operators $\mathcal{C}_i(y)$ and $\mathcal{C}_j(x)$; the proof of [\[BB\<\]](#BB<){reference-type="eqref" reference="BB<"} employs the third Yang--Baxter equation [\[RLLc\]](#RLLc){reference-type="eqref" reference="RLLc"}, applied to the two-row partition function that arises by joining operators $\mathcal{B}_j(y)$ and $\mathcal{B}_i(x)$. For full details, we refer the reader to [@BorodinW Section 3.2, Theorems 3.2.1 and 3.2.5]. ◻ **Theorem 13**. *Fix two nonnegative integers $i,j$ such that $0 \leqslant i<j \leqslant n$, and complex parameters $x,y$ such that $$\begin{aligned} \label{weight-condition} \left| \frac{x-s}{1-sx} \cdot \frac{y-s}{1-sy} \right| < 1.\end{aligned}$$ The row operators $\mathcal{C}_i(x)$ and $\mathcal{B}_j(y)$ obey the following commutation relation: $$\begin{aligned} \label{CB} \mathcal{C}_i(x) \mathcal{B}_j(y) &= \frac{1-qxy}{1-xy} \mathcal{B}_j(y)\mathcal{C}_i(x).\end{aligned}$$* *Proof.* The proof makes use of the second Yang--Baxter equation [\[RLLb\]](#RLLb){reference-type="eqref" reference="RLLb"}, applied successively to the two-row partition function that arises by joining the operators $\mathcal{C}_i(x)$ and $\mathcal{B}_j(y)$. For the full details, we refer the reader to [@BorodinW Section 3.2, Theorem 3.2.3]. ◻ # Partition functions {#sec:pf} This section brings together a number of partition function definitions, as well as fundamental results related to them, for use throughout the remainder of the text. Most of the facts summarized here were first obtained in [@BorodinW Chapters 3--5 and Chapter 8], and where a theorem is directly transcribed from there, we refer the reader to that earlier text for a full proof. We begin by defining *coloured compositions* in Section [3.1](#ssec:cc){reference-type="ref" reference="ssec:cc"}; these are used to index many of the quantities that we subsequently define. Sections [3.2](#ssec:G){reference-type="ref" reference="ssec:G"}--[3.4](#ssec:perm){reference-type="ref" reference="ssec:perm"} introduce the partition functions required; we then state a number of properties of these partition functions in Sections [3.5](#ssec:hecke){reference-type="ref" reference="ssec:hecke"}--[3.9](#ssec:int){reference-type="ref" reference="ssec:int"}. ## Coloured compositions {#ssec:cc} **Definition 14**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of length $n$ such that $|\lambda| = \sum_{i=1}^{n} \lambda_i = m$; $m$ is called its *weight*. We introduce the set $\mathcal{S}_{\lambda}$ of (strict, nonnegative) $\lambda$-coloured compositions as follows: $$\begin{aligned} \label{lambda-col} \mathcal{S}_{\lambda} = \Big\{ \mu = \Big( 0 \leqslant\mu^{(1)}_1 < \cdots < \mu^{(1)}_{\lambda_1} \Big| 0 \leqslant\mu^{(2)}_1 < \cdots < \mu^{(2)}_{\lambda_2} \Big| \cdots \Big| 0 \leqslant\mu^{(n)}_1 < \cdots < \mu^{(n)}_{\lambda_n}\Big) \Big\}.\end{aligned}$$ The elements of $\mathcal{S}_{\lambda}$ are vectors of length $n$ whose $i$-th component $\mu^{(i)}$ is a strict[^8], nonnegative signature of length $\lambda_i$, for all $1 \leqslant i \leqslant n$. These components, or blocks, demarcate the colouring of $\mu$; the colour of each block is indicated by the superscript attached to it. We refer to $\lambda$ as the *colour profile* of $\mu$. **Definition 15**. With the same assumptions as in Definition [Definition 14](#def:cc){reference-type="ref" reference="def:cc"}, we also define the set $\mathcal{S}^{+}_{\lambda} \subset \mathcal{S}_{\lambda}$ as follows: $$\begin{aligned} \mathcal{S}^{+}_{\lambda} = \{\mu \in \mathcal{S}_{\lambda} : \mu_1^{(j)} \geqslant 1,\ \forall\ 1 \leqslant j \leqslant n \}.\end{aligned}$$ This is the restriction to coloured compositions that have positive parts only. For any coloured composition $\mu \in \mathcal{S}^{+}_{\lambda}$ we define its *padding* $0\cup\mu \in \mathcal{S}_{\lambda+1^n}$ by prepending a part of size $0$ in each of the $n$ blocks of $\mu$. Let $\mu \in \mathcal{S}_{\lambda}$ be a $\lambda$-coloured composition. We associate to $\mu$ a vector $\left|\mu\right\rangle_{\lambda} \in \mathbb{V}(\infty)$, defined as follows: $$\begin{aligned} \label{A(k)} \left|\mu\right\rangle_{\lambda} = \bigotimes_{k=0}^{\infty} \left|\bm{A}(k)\right\rangle, \qquad \bm{A}(k) = \sum_{j=1}^{n} A_j(k) \bm{e}_j, \qquad A_j(k) = \left\{ \begin{array}{ll} 1, & \quad k \in \mu^{(j)}, \\ \\ 0, & \quad {\rm otherwise}. \end{array} \right.\end{aligned}$$ In other words, the component $A_j(k)$ is equal to $1$ if the integer $k$ is present in the strict signature $\mu^{(j)}$, and equal to $0$ if not. We shall also make use of dual vectors $\left\langle \mu\right|_{\lambda} \in \mathbb{V}(\infty)^{*}$, defined to act linearly on elements of the form [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"} via the relation $\left\langle \mu\right|_{\lambda} \cdot \left|\nu\right\rangle_{\lambda} = \delta_{\mu,\nu}$ for all $\mu,\nu \in \mathcal{S}_{\lambda}$. **Definition 16** (Rainbow compositions). The elements of $\mathcal{S}_{1^n}$ are called *rainbow compositions*; we have $$\begin{aligned} \mathcal{S}_{1^n} = \Big\{ \mu = (\mu_1 | \mu_2 | \cdots | \mu_n) \Big\}.\end{aligned}$$ That is, a rainbow composition consists of $n$ blocks, each of unit length; no constraint is imposed on the relative ordering of the parts. ## Functions $G_{\mu/\nu}$ {#ssec:G} Fix a $\lambda$-coloured composition $\nu \in \mathcal{S}_{\lambda}$ with component signatures $\nu^{(i)}$, $1 \leqslant i \leqslant n$, and define, similarly to [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"}, a vector $\left|\nu\right\rangle_{\lambda} \in \mathbb{V}(\infty)$: $$\begin{aligned} \label{B(k)} \left|\nu\right\rangle_{\lambda} = \bigotimes_{k=0}^{\infty} \left|\bm{B}(k)\right\rangle, \qquad \bm{B}(k) = \sum_{j=1}^{n} B_j(k) \bm{e}_j, \qquad B_j(k) = \left\{ \begin{array}{ll} 1, & \quad k \in \nu^{(j)}, \\ \\ 0, & \quad {\rm otherwise}. \end{array} \right.\end{aligned}$$ **Definition 17**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition, and fix two $\lambda$-coloured compositions $\mu \in \mathcal{S}_{\lambda}$ and $\nu \in \mathcal{S}_{\lambda}$. Let the corresponding vectors in $\mathbb{V}(\infty)$, $\left|\mu\right\rangle_{\lambda}$ and $\left|\nu\right\rangle_{\lambda}$, be given by [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"} and [\[B(k)\]](#B(k)){reference-type="eqref" reference="B(k)"} respectively. For any integer $p \geqslant 1$ we define the following family of symmetric rational functions: $$\begin{aligned} \label{G-def} (-s)^{|\mu|-|\nu|} \cdot G_{\mu/\nu}(\lambda;x_1,\dots,x_p) = \left\langle \nu\right|_{\lambda} \prod_{i=1}^{p} \mathcal{C}_0(x_i) \left|\mu\right\rangle_{\lambda}.\end{aligned}$$ In the case $\lambda = (1,\dots,1) = 1^n$, we drop the notational dependence on $\lambda$, and write $$\begin{aligned} G_{\mu/\nu}(1^n;x_1,\dots,x_p) \equiv G_{\mu/\nu}(x_1,\dots,x_p).\end{aligned}$$ The symmetry in $(x_1,\dots,x_p)$ follows from the commutativity of the $\mathcal{C}_0(x_i)$ operators; for a proof of the latter fact, see [@BorodinW Theorem 3.2.1]. Translating the row operators in [\[G-def\]](#G-def){reference-type="eqref" reference="G-def"} into their graphical form, we obtain the following partition function representation of $G_{\mu/\nu}$: $$\begin{aligned} \label{G-pf} (-s)^{|\mu|-|\nu|} \cdot G_{\mu/\nu}(\lambda;x_1,\dots,x_p) &= \begin{tikzpicture}[scale=0.75,baseline=(current bounding box.center),>=stealth] \foreach\y in {1,...,5}{ \draw[lgray,line width=1.5pt,->] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,0) -- (\x,6); } %spectral parameters \node[left] at (0.5,1) {$x_1 \rightarrow$}; \node[left] at (0.5,2) {$x_2 \rightarrow$}; \node[left] at (0.5,3) {$\vdots$}; \node[left] at (0.5,4) {$\vdots$}; \node[left] at (0.5,5) {$x_p \rightarrow$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{A}(2)$}; \node[above] at (3,6) {\footnotesize$\bm{A}(1)$}; \node[above] at (2,6) {\footnotesize$\bm{A}(0)$}; %bottom labels \node[below] at (7,0) {$\cdots$}; \node[below] at (6,0) {$\cdots$}; \node[below] at (5,0) {$\cdots$}; \node[below] at (4,0) {\footnotesize$\bm{B}(2)$}; \node[below] at (3,0) {\footnotesize$\bm{B}(1)$}; \node[below] at (2,0) {\footnotesize$\bm{B}(0)$}; %right labels \node[right] at (8,1) {$0$}; \node[right] at (8,2) {$0$}; \node[right] at (8,3) {$\vdots$}; \node[right] at (8,4) {$\vdots$}; \node[right] at (8,5) {$0$}; %left labels \node[left] at (1,1) {$0$}; \node[left] at (1,2) {$0$}; \node[left] at (1,3) {$\vdots$}; \node[left] at (1,4) {$\vdots$}; \node[left] at (1,5) {$0$}; \end{tikzpicture}\end{aligned}$$ The factor of $(-s)^{|\mu|-|\nu|}$ incorporated into the definition [\[G-def\]](#G-def){reference-type="eqref" reference="G-def"} is due to the fact that, in the partition function representation [\[G-pf\]](#G-pf){reference-type="eqref" reference="G-pf"}, each right edge of the lattice that is occupied by a nonzero colour produces a residual factor of $-s$; this can be seen by analysing the final two columns of the table [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}. Since we ultimately intend to take the limit $s \rightarrow 0$, it is important to excise this overall factor from our partition function; we do this by multiplying the left hand side of [\[G-def\]](#G-def){reference-type="eqref" reference="G-def"} by $(-s)^{|\mu|-|\nu|}$, since the total number of occupied right edges in the lattice [\[G-pf\]](#G-pf){reference-type="eqref" reference="G-pf"} is equal to $|\mu|-|\nu|$. ## Functions $f_{\mu}$ and $g_{\mu}$ {#ssec:fg} **Definition 18**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of weight $m$, and fix a $\lambda$-coloured composition $\mu \in \mathcal{S}_{\lambda}$. Write $\ell_k = \sum_{i=1}^{k} \lambda_i$ for the $k$-th partial sum of $\lambda$. We define the following family of non-symmetric rational functions: $$\begin{aligned} \label{generic-f} (-s)^{|\mu|} \cdot f_{\mu}(\lambda; x_1,\dots,x_m) = \left\langle \emptyset\right| \prod_{j\in [1,\ell_1]} \mathcal{C}_1(x_j) \prod_{j\in (\ell_1,\ell_2]} \mathcal{C}_2(x_j) \cdots \prod_{j\in(\ell_{n-1},\ell_n]} \mathcal{C}_n(x_j) \left|\mu\right\rangle_{\lambda},\end{aligned}$$ where $\left|\mu\right\rangle_{\lambda} \in \mathbb{V}(\infty)$ is given by [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"} and $\left\langle \emptyset\right| \in \mathbb{V}(\infty)^{*}$ denotes the (dual) vacuum state $$\begin{aligned} \label{dual-vac} \left\langle \emptyset\right| = \bigotimes_{k=0}^{\infty} \left\langle \bm{e}_0\right|,\end{aligned}$$ which is completely devoid of particles. Translating the row operators in [\[generic-f\]](#generic-f){reference-type="eqref" reference="generic-f"} into their graphical form, we obtain the following partition function representation of $f_{\mu}$: $$\begin{aligned} \label{f-def} (-s)^{|\mu|} \cdot f_{\mu}(\lambda;x_1,\dots,x_m) = \begin{tikzpicture}[scale=0.75,baseline=(current bounding box.center),>=stealth] \foreach\y in {0,...,5}{ \draw[lgray,line width=1.5pt,->] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,-1) -- (\x,6); } %spectral parameters \node[left] at (-0.5,0) {$x_1 \rightarrow$}; \node[left] at (-0.5,2) {$\vdots$}; \node[left] at (-0.5,3) {$\vdots$}; \node[left] at (-0.5,5) {$x_m \rightarrow$}; %bottom labels \node[below] at (7,-1) {$\cdots$}; \node[below] at (6,-1) {$\cdots$}; \node[below] at (5,-1) {$\cdots$}; \node[below] at (4,-1) {\footnotesize$\bm{e}_0$}; \node[below] at (3,-1) {\footnotesize$\bm{e}_0$}; \node[below] at (2,-1) {\footnotesize$\bm{e}_0$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{A}(2)$}; \node[above] at (3,6) {\footnotesize$\bm{A}(1)$}; \node[above] at (2,6) {\footnotesize$\bm{A}(0)$}; %right labels \node[right] at (8,0) {$0$}; \node[right] at (8,1) {$0$}; \node[right] at (8,2) {$0$}; \node[right] at (8,3) {$0$}; \node[right] at (8,4) {$0$}; \node[right] at (8,5) {$0$}; %left labels \node[left] at (1,0) {$1$}; \node[left] at (1.5,0.6) {$\vdots$}; \node[left] at (1,1) {$1$}; \node[left] at (1,2) {$\vdots$}; %\node[left] at (1.5,2.6) {$\vdots$}; \node[left] at (1,3) {$\vdots$}; %\node at (2.5,3.6) {$\vdots$}; %\node at (6.5,3.6) {$\vdots$}; \node[left] at (1,4) {$n$}; \node[left] at (1.5,4.6) {$\vdots$}; \node[left] at (1,5) {$n$}; \end{tikzpicture}\end{aligned}$$ The factor of $(-s)^{|\mu|}$ introduced into the definition [\[generic-f\]](#generic-f){reference-type="eqref" reference="generic-f"} has analogous origins to the factor $(-s)^{|\mu|-|\nu|}$ in [\[G-pf\]](#G-pf){reference-type="eqref" reference="G-pf"}; see the explanation in the paragraph immediately following [\[G-pf\]](#G-pf){reference-type="eqref" reference="G-pf"}. *Remark 19*. In the case $\lambda = (1,\dots,1) = 1^n$, we drop the notational dependence on $\lambda$, and write $$\begin{aligned} f_{\mu}(1^n; x_1,\dots,x_n) \equiv f_{\mu}(x_1,\dots,x_n).\end{aligned}$$ The function $f_{\mu}(x_1,\dots,x_n)$ then matches identically with the family of *non-symmetric spin Hall--Littlewood functions* defined in [@BorodinW Section 3.4]; see Definition 3.4.3 therein. The reason for the match is the fact that when $\lambda = 1^n$, each colour $\{1,\dots,n\}$ enters the partition function [\[f-def\]](#f-def){reference-type="eqref" reference="f-def"} exactly once, which is precisely the regime when the weights [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"} and those of [@BorodinW Sections 2.2 and 2.5] agree (see the discussion below equation [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}). *Remark 20*. In the case $\lambda = (1,\dots,1) = 1^n$, and assuming a *weakly increasing* rainbow composition $\mu = (\mu_1 \leqslant\cdots \leqslant\mu_n)$, one has the factorization $$\begin{aligned} \label{factorize} f_{\mu}(1^n;x_1,\dots,x_n) = \frac{\prod_{j \geqslant 0} (s^2;q)_{\#_j(\mu)}}{\prod_{i=1}^{n}(1-s x_i)} \prod_{i=1}^{n} \left( \frac{x_i-s}{1-sx_i} \right)^{\mu_i},\end{aligned}$$ where $\#_j(\mu)$ denotes the number of parts in $\mu$ which are equal to $j$. This is proved by a simple freezing argument applied to the partition function [\[f-def\]](#f-def){reference-type="eqref" reference="f-def"}; we refer the reader to [@BorodinW Section 5.1]. **Definition 21**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of weight $m$, and fix a $\lambda$-coloured composition $\mu \in \mathcal{S}_{\lambda}$. Write $\ell_k = \sum_{i=1}^{k} \lambda_i$ for the $k$-th partial sum of $\lambda$. Define a further family of non-symmetric rational functions: $$\begin{aligned} \label{generic-g} (-s)^{-|\mu|} \cdot g_{\mu}(\lambda; x_1,\dots,x_m) = \left\langle \mu\right|_{\lambda} \prod_{j\in [1,\ell_1]} \mathcal{B}_1(x_j) \prod_{j\in (\ell_1,\ell_2]} \mathcal{B}_2(x_j) \cdots \prod_{j\in(\ell_{n-1},\ell_n]} \mathcal{B}_n(x_j) \left|\emptyset\right\rangle,\end{aligned}$$ where $\left\langle \mu\right|_{\lambda} \in \mathbb{V}(\infty)^{*}$ is the dual of the vector [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"}, and $\left|\emptyset\right\rangle \in \mathbb{V}(\infty)$ denotes the vacuum state $$\begin{aligned} \label{vac} \left|\emptyset\right\rangle = \bigotimes_{k=0}^{\infty} \left|\bm{e}_0\right\rangle.\end{aligned}$$ Translating the row operators in [\[generic-g\]](#generic-g){reference-type="eqref" reference="generic-g"} into their graphical form, we obtain the following partition function representation of $g_{\mu}$: $$\begin{aligned} \label{g-def} (-s)^{-|\mu|} \cdot g_{\mu}(\lambda;x_1,\dots,x_m) = \begin{tikzpicture}[scale=0.75,baseline=(current bounding box.center),>=stealth] \foreach\y in {0,...,5}{ \draw[lgray,line width=1.5pt,<-] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,-1) -- (\x,6); } %spectral parameters \node[right] at (8.5,0) {$\leftarrow x_1$}; \node[right] at (8.5,2) {$\vdots$}; \node[right] at (8.5,3) {$\vdots$}; \node[right] at (8.5,5) {$\leftarrow x_m$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{e}_0$}; \node[above] at (3,6) {\footnotesize$\bm{e}_0$}; \node[above] at (2,6) {\footnotesize$\bm{e}_0$}; %bottom labels \node[below] at (7,-1) {$\cdots$}; \node[below] at (6,-1) {$\cdots$}; \node[below] at (5,-1) {$\cdots$}; \node[below] at (4,-1) {\footnotesize$\bm{A}(2)$}; \node[below] at (3,-1) {\footnotesize$\bm{A}(1)$}; \node[below] at (2,-1) {\footnotesize$\bm{A}(0)$}; %right labels \node[right] at (8,0) {$0$}; \node[right] at (8,1) {$0$}; \node[right] at (8,2) {$0$}; \node[right] at (8,3) {$0$}; \node[right] at (8,4) {$0$}; \node[right] at (8,5) {$0$}; %left labels \node[left] at (1,0) {$1$}; \node[left] at (1.5,0.6) {$\vdots$}; \node[left] at (1,1) {$1$}; \node[left] at (1,2) {$\vdots$}; %\node[left] at (1.5,2.6) {$\vdots$}; \node[left] at (1,3) {$\vdots$}; %\node at (2.5,3.6) {$\vdots$}; %\node at (6.5,3.6) {$\vdots$}; \node[left] at (1,4) {$n$}; \node[left] at (1.5,4.6) {$\vdots$}; \node[left] at (1,5) {$n$}; \end{tikzpicture}\end{aligned}$$ *Remark 22*. In the case $\lambda = (1,\dots,1) = 1^n$, we drop the notational dependence on $\lambda$, and write $$\begin{aligned} g_{\mu}(1^n; x_1,\dots,x_n) \equiv g_{\mu}(x_1,\dots,x_n).\end{aligned}$$ The function $g_{\mu}(x_1,\dots,x_n)$ matches identically with the family of *dual non-symmetric spin Hall--Littlewood functions* defined in [@BorodinW Section 3.4]; see Definition 3.4.6 therein. This match may be deduced by the same reasoning as in Remark [Remark 19](#rainbow-rmk){reference-type="ref" reference="rainbow-rmk"}, above. ## Permuted boundary conditions {#ssec:perm} **Definition 23**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of weight $m$, and fix a $\lambda$-coloured composition $\mu \in \mathcal{S}_{\lambda}$. Fix also a vector $\sigma = (\sigma_1,\dots,\sigma_m)$ such that $|\{k : \sigma_k = i\}| = \lambda_i$ for all $1 \leqslant i \leqslant n$. We define the following families of non-symmetric rational functions: $$\begin{aligned} \label{generic-f-sigma} (-s)^{|\mu|} \cdot f_{\mu}^{\sigma}(\lambda; x_1,\dots,x_m) &= \left\langle \emptyset\right| \prod_{j=1}^{m} \mathcal{C}_{\sigma_j}(x_j) \left|\mu\right\rangle_{\lambda}, \\ \label{generic-g-sigma} (-s)^{-|\mu|} \cdot g_{\mu}^{\sigma}(\lambda; x_1,\dots,x_m) &= \left\langle \mu\right|_{\lambda} \prod_{j=1}^{m} \mathcal{B}_{\sigma_j}(x_j) \left|\emptyset\right\rangle.\end{aligned}$$ The first family [\[generic-f-sigma\]](#generic-f-sigma){reference-type="eqref" reference="generic-f-sigma"} matches with that of Definition [Definition 18](#defn:gen-sector){reference-type="ref" reference="defn:gen-sector"}, and the second family [\[generic-g-sigma\]](#generic-g-sigma){reference-type="eqref" reference="generic-g-sigma"} matches with that of Definition [Definition 21](#defn:generic-sector-dual){reference-type="ref" reference="defn:generic-sector-dual"}, when $\sigma = (1^{\lambda_1},2^{\lambda_2},\dots,n^{\lambda_n})$. ## Hecke generators and recursion relations {#ssec:hecke} Recall the definition of the Hecke algebra of type $A_{n-1}$. It is the algebra generated by a family $T_1,\dots,T_{n-1}$, modulo the relations $$\begin{aligned} \label{hecke1} (T_i - q)(T_i + 1) = 0, \quad 1 \leqslant i \leqslant n-1, \qquad T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}, \quad 1 \leqslant i \leqslant n-2,\end{aligned}$$ as well as the commutativity property $$\begin{aligned} \label{hecke2} [T_i,T_j] = 0, \quad \forall\ i,j\ \ \text{such that}\ \ |i-j| > 1.\end{aligned}$$ Introduce the simple transpositions $\mathfrak{s}_i$, acting on arbitrary functions $h$ of the alphabet $(x_1,\dots,x_n)$: $$\begin{aligned} \mathfrak{s}_i \cdot h(x_1,\dots,x_n) = h(x_1,\dots,x_{i+1},x_i,\dots,x_n), \quad 1 \leqslant i \leqslant n-1.\end{aligned}$$ Making use of these, we define the *Demazure--Lusztig operators* $$\begin{aligned} \label{hecke-poly} T_i = q - \frac{x_i-q x_{i+1}}{x_i-x_{i+1}} (1-\mathfrak{s}_i), \quad 1 \leqslant i \leqslant n-1,\end{aligned}$$ which provide a faithful representation of the Hecke algebra on the field of rational functions $\mathbb{Q}(x_1,\dots,x_n)$.[^9] From the quadratic identity $(T_i-q)(T_i+1) = 0$, multiplied by $T_i^{-1}$, one gets an explicit formula for inverse Hecke generators: $$\begin{aligned} T_i^{-1} = q^{-1}(T_i-q+1) = q^{-1}\left(1 - \frac{x_i-q x_{i+1}}{x_i-x_{i+1}} (1-\mathfrak{s}_i) \right), \quad 1 \leqslant i \leqslant n-1.\end{aligned}$$ In what follows, we will need another version of the Demazure--Lusztig operators [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"} in which the variables $(x_i,x_{i+1})$ get reciprocated. We reserve a special notation for this: $$\begin{aligned} \label{hecke-tilde} \tilde{T}_i = q - \frac{x_{i+1}-q x_i}{x_{i+1}-x_i} (1-\mathfrak{s}_i), \quad \tilde{T}^{-1}_i = q^{-1} \left(1 - \frac{x_{i+1}-q x_i}{x_{i+1}-x_i} (1-\mathfrak{s}_i)\right), \quad 1 \leqslant i \leqslant n-1,\end{aligned}$$ Clearly, the generators $\tilde{T}_i$ also satisfy the basic relations [\[hecke1\]](#hecke1){reference-type="eqref" reference="hecke1"}--[\[hecke2\]](#hecke2){reference-type="eqref" reference="hecke2"} of the Hecke algebra. **Theorem 24**. *Fix an integer $1 \leqslant i \leqslant n$ and a composition $\mu = (\mu_1|\mu_2|\dots|\mu_n) \in \mathcal{S}_{1^n}$ such that $\mu_i < \mu_{i+1}$. The functions $f_{\mu}(1^n;x_1,\dots,x_n) \equiv f_{\mu}(x_1,\dots,x_n)$ and $g_{\mu}(1^n;x_1,\dots,x_n) \equiv g_{\mu}(x_1,\dots,x_n)$ transform under the action of [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"}--[\[hecke-tilde\]](#hecke-tilde){reference-type="eqref" reference="hecke-tilde"} in the following way: $$\begin{aligned} \label{T-f} T_i \cdot f_{\mu}(x_1,\dots,x_n) &= f_{\mathfrak{s}_i \cdot \mu}(x_1,\dots,x_n), \\ \label{T-g} \tilde{T}_i \cdot g_{\mu}(x_1,\dots,x_n) &= q \cdot g_{\mathfrak{s}_i \cdot \mu}(x_1,\dots,x_n),\end{aligned}$$ where $\mathfrak{s}_i \cdot \mu$ denotes the composition obtained by switching $\mu_i$ and $\mu_{i+1}$.* *Proof.* Both statements [\[T-f\]](#T-f){reference-type="eqref" reference="T-f"} and [\[T-g\]](#T-g){reference-type="eqref" reference="T-g"} are proved in [@BorodinW]; see equations (5.3.1) and (8.2.24) therein, respectively. ◻ **Theorem 25**. *Fix a coloured composition $\nu \in \mathcal{S}_{\lambda}$, where $\lambda = (\lambda_1,\dots,\lambda_n)$ is a composition such that $|\lambda| = m$, as well as a vector $\sigma = (\sigma_1,\dots,\sigma_m)$ such that $|\{k:\sigma_k = i\}| = \lambda_i$ for all $1 \leqslant i \leqslant n$. Assuming that $\sigma_j < \sigma_{j+1}$ for some $1 \leqslant j \leqslant m-1$, there holds $$\begin{aligned} \label{invT-f} T_j \cdot f_{\nu}^{\mathfrak{s}_j \cdot \sigma}(\lambda;x_1,\dots,x_m) &= q \cdot f_{\nu}^{\sigma}(\lambda;x_1,\dots,x_m), \\ \label{invT-g} \tilde{T}_j \cdot g_{\nu}^{\sigma}(\lambda;x_1,\dots,x_m) &= g_{\nu}^{\mathfrak{s}_j \cdot \sigma}(\lambda;x_1,\dots,x_m),\end{aligned}$$ where $\mathfrak{s}_j \cdot \sigma$ denotes the vector obtained by switching $\sigma_j$ and $\sigma_{j+1}$.* *Proof.* The proof of [\[invT-f\]](#invT-f){reference-type="eqref" reference="invT-f"} is by isolating the action of $T_j^{-1}$ on the pair of operators $\mathcal{C}_{\sigma_j}(x_j) \mathcal{C}_{\sigma_{j+1}}(x_{j+1})$, which is the only place that $f_{\nu}^{\sigma}(\lambda;x_1,\dots,x_m)$ depends on $(x_j,x_{j+1})$. Using the explicit form of $T_j^{-1}$, we have $$\begin{aligned} \label{invT-pf1} q\cdot T_j^{-1} \cdot \mathcal{C}_{\sigma_j}(x_j) \mathcal{C}_{\sigma_{j+1}}(x_{j+1}) = \frac{(q-1) x_{j+1}}{x_j-x_{j+1}} \mathcal{C}_{\sigma_j}(x_j) \mathcal{C}_{\sigma_{j+1}}(x_{j+1}) + \frac{x_j-qx_{j+1}}{x_j-x_{j+1}} \mathcal{C}_{\sigma_j}(x_{j+1}) \mathcal{C}_{\sigma_{j+1}}(x_j).\end{aligned}$$ In view of the fact that $\sigma_j < \sigma_{j+1}$, we may use the commutation relation [\[CC\<\]](#CC<){reference-type="eqref" reference="CC<"} to combine the right hand side of [\[invT-pf1\]](#invT-pf1){reference-type="eqref" reference="invT-pf1"} into a single term: $$\begin{aligned} q\cdot T_j^{-1} \cdot \mathcal{C}_{\sigma_j}(x_j) \mathcal{C}_{\sigma_{j+1}}(x_{j+1}) = \mathcal{C}_{\sigma_{j+1}}(x_j) \mathcal{C}_{\sigma_j}(x_{j+1}).\end{aligned}$$ Substitution of this identity into [\[generic-f-sigma\]](#generic-f-sigma){reference-type="eqref" reference="generic-f-sigma"} immediately proves [\[invT-f\]](#invT-f){reference-type="eqref" reference="invT-f"}. In a similar vein, one proves [\[invT-g\]](#invT-g){reference-type="eqref" reference="invT-g"} by isolating the action of $\tilde{T}_j^{-1}$ on the pair $\mathcal{B}_{\sigma_{j+1}}(x_j) \mathcal{B}_{\sigma_j}(x_{j+1})$, which is the only place that $g_{\nu}^{\mathfrak{s}_j \cdot \sigma}(\lambda;x_1,\dots,x_m)$ depends on $(x_j,x_{j+1})$. Using the explicit form of $\tilde{T}_j^{-1}$, we have $$\begin{aligned} \label{invT-pf2} \tilde{T}_j^{-1} \cdot \mathcal{B}_{\sigma_{j+1}}(x_j) \mathcal{B}_{\sigma_j}(x_{j+1}) = \frac{(q-1) x_j}{q(x_{j+1}-x_j)} \mathcal{B}_{\sigma_{j+1}}(x_j) \mathcal{B}_{\sigma_j}(x_{j+1}) + \frac{x_{j+1}-qx_j}{q(x_{j+1}-x_j)} \mathcal{B}_{\sigma_{j+1}}(x_{j+1}) \mathcal{B}_{\sigma_j}(x_j).\end{aligned}$$ Since $\sigma_j < \sigma_{j+1}$, we use the commutation relation [\[BB\<\]](#BB<){reference-type="eqref" reference="BB<"} to combine the right hand side of [\[invT-pf2\]](#invT-pf2){reference-type="eqref" reference="invT-pf2"} into a single term: $$\begin{aligned} \tilde{T}_j^{-1} \cdot \mathcal{B}_{\sigma_{j+1}}(x_j) \mathcal{B}_{\sigma_j}(x_{j+1}) = \mathcal{B}_{\sigma_j}(x_j) \mathcal{B}_{\sigma_{j+1}}(x_{j+1}).\end{aligned}$$ Substitution of this identity into [\[generic-g-sigma\]](#generic-g-sigma){reference-type="eqref" reference="generic-g-sigma"} proves [\[invT-g\]](#invT-g){reference-type="eqref" reference="invT-g"}. ◻ ## Antisymmetrization {#ssec:anti} A key property of the vertex models [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"} and [\[generic-M\]](#generic-M){reference-type="eqref" reference="generic-M"} is that of *colour-merging*; this is the combinatorial statement that partition functions in the models [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"} and [\[generic-M\]](#generic-M){reference-type="eqref" reference="generic-M"}, with $n$ colours, become equal to partition functions with $m < n$ colours under a certain antisymmetrization procedure applied to the boundary conditions. The most general colour-merging statement is given and proved as [@ABW21 Theorem 5.2.2]; here we will reproduce this statement only at the level that we need, namely, for two of the families of rational functions that we have defined. To state our antisymmetrization results, we require some definitions. **Definition 26** (Rainbow recolouring). Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition such that $|\lambda|=m$, and fix a coloured composition $\mu \in \mathcal{S}_{\lambda}$. Denoting $$\begin{aligned} \mu = \Big(\mu^{(1)}_1 < \cdots < \mu^{(1)}_{\lambda_1} \Big| \mu^{(2)}_1 < \cdots < \mu^{(2)}_{\lambda_2} \Big| \cdots \Big| \mu^{(n)}_1 < \cdots < \mu^{(n)}_{\lambda_n}\Big),\end{aligned}$$ we associate to this a rainbow composition $\breve\mu = \left( \breve{\mu}_1| \breve{\mu}_2 | \cdots | \breve{\mu}_m \right) \in \mathcal{S}_{1^m}$ such that for each $1 \leqslant i \leqslant m$ we have $$\begin{aligned} \breve{\mu}_i = \mu^{(k)}_j,\end{aligned}$$ where $1 \leqslant k \leqslant n$, $1 \leqslant j \leqslant\lambda_k$ are the unique integers such that $$\begin{aligned} i = j+\sum_{a=1}^{k-1} \lambda_a.\end{aligned}$$ In simpler terms, $\breve{\mu}$ is the composition obtained from recolouring the parts of $\mu$ sequentially from $1$ to $m$ into pairwise distinct colours, while keeping the magnitude of all parts fixed. **Definition 27**. Fix a positive integer $m$ and let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition such that $|\lambda| = m$, with partial sums $\ell_k = \sum_{i=1}^{k} \lambda_i$. We say that $\sigma \in \mathfrak{S}_{\lambda} \subset \mathfrak{S}_m$ provided that $\sigma$ fixes $(\ell_{k-1},\ell_k]$ for each integer $1 \leqslant k \leqslant n$, that is, $$\begin{aligned} \sigma \in \mathfrak{S}_{[1,\ell_1]} \times \mathfrak{S}_{(\ell_1,\ell_2]} \times \cdots \times \mathfrak{S}_{(\ell_{n-1},\ell_n]}.\end{aligned}$$ **Proposition 28**. *Fix a coloured composition $\nu \in \mathcal{S}_{\lambda}$ and let $\breve{\nu}$ denote its rainbow recolouring, as in Definition [Definition 26](#def:rainbow-rec){reference-type="ref" reference="def:rainbow-rec"}. We then have the following result, relating the functions [\[generic-g\]](#generic-g){reference-type="eqref" reference="generic-g"} for rainbow colour profiles with those of non-rainbow type: $$\begin{aligned} \sum_{\sigma \in \mathfrak{S}_{\lambda}} (-1)^{{\rm inv}(\sigma)} g_{\sigma(\breve{\nu})}(1^m;x_1,\dots,x_m) = g_{\nu}(\lambda;x_1,\dots,x_m),\end{aligned}$$ where the sum is taken over all elements in $\mathfrak{S}_{\lambda}$. Here we have defined ${\rm inv}(\sigma) = {\rm card}\{ (i,j) : i<j,\ \sigma_i > \sigma_j\}$ and $\sigma(\breve{\nu}) = \left(\breve{\nu}_{\sigma(1)} | \breve{\nu}_{\sigma(2)}| \cdots | \breve{\nu}_{\sigma(m)}\right)$.* **Proposition 29**. *Fix two coloured compositions $\mu,\nu \in \mathcal{S}_{\lambda}$ and let $\breve{\mu},\breve{\nu}$ denote their respective rainbow recolouring, as in Definition [Definition 26](#def:rainbow-rec){reference-type="ref" reference="def:rainbow-rec"}. The functions [\[G-def\]](#G-def){reference-type="eqref" reference="G-def"} have the following sum property: $$\begin{aligned} \sum_{\sigma \in \mathfrak{S}_{\lambda}} (-1)^{{\rm inv}(\sigma)} G_{\breve{\mu}/\sigma(\breve{\nu})}(1^m;x_1,\dots,x_p) = G_{\mu/\nu}(\lambda;x_1,\dots,x_p)\end{aligned}$$ where the sum is taken over all elements in $\mathfrak{S}_{\lambda}$.* A similar antisymmetrization result can be stated for the functions [\[generic-f\]](#generic-f){reference-type="eqref" reference="generic-f"}, but we omit it from this section since we shall not require it in what follows. *Remark 30*. Propositions [Proposition 28](#prop:f-ant){reference-type="ref" reference="prop:f-ant"} and [Proposition 29](#prop:G-ant){reference-type="ref" reference="prop:G-ant"} are both statements about partition functions constructed from $M$-weights, as defined in Section [2.3](#ssec:M){reference-type="ref" reference="ssec:M"}. In order to recover them as corollaries of [@ABW21 Theorem 5.2.2] one should first apply the symmetry [\[LM-sym\]](#LM-sym){reference-type="eqref" reference="LM-sym"}, which converts them to statements about partition functions built from $L$-weights, and the matching with [@ABW21] then goes through in a straightforward way. ## Orthogonality {#ssec:orthog} In this section we directly transcribe an orthogonality result for non-symmetric spin Hall--Littlewood functions, from [@BorodinW Chapter 8]. Throughout, we denote the imaginary unit by ${\tt i} = \sqrt{-1}$. Let $\{C_1,\dots,C_n\}$ be a collection of contours in the complex plane, and fix two complex parameters $q,s \in \mathbb{C}$. We say that the set $\{C_1,\dots,C_n\}$ is admissible with respect to $(q,s)$ if the following conditions are met: - The contours $\{C_1,\dots,C_n\}$ are closed, positively oriented and pairwise non-intersecting; - The contours $C_i$ and $q \cdot C_i$ are both contained within contour $C_{i+1}$ for all $1 \leqslant i \leqslant n-1$, where $q \cdot C_i$ denotes the image of $C_i$ under multiplication by $q$; - All contours surround the point $s$. **Theorem 31**. *Fix two rainbow compositions $\mu,\nu \in \mathcal{S}_{1^n}$, and let $\{C_1,\dots,C_n\}$ be contours admissible with respect to $(q,s)$. We then have $$\begin{aligned} \label{f-g-orthog} \left( \frac{1}{2\pi{\tt i}} \right)^n \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_n} \frac{dy_n}{y_n} \prod_{1 \leqslant i<j \leqslant n} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\mu}(y_1^{-1},\dots,y_n^{-1}) g_{\nu}(y_1,\dots,y_n) = \frac{{\bm 1}_{\mu = \nu}\cdot(q-1)^n}{q^{n(n+1)/2}}.\end{aligned}$$* *Proof.* This is Theorem 8.2.1 of [@BorodinW Chapter 8]. ◻ Closely related to the orthogonality statement [\[f-g-orthog\]](#f-g-orthog){reference-type="eqref" reference="f-g-orthog"}, and in fact instrumental in its proof, is the following property of the Hecke generators [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"}, [\[hecke-tilde\]](#hecke-tilde){reference-type="eqref" reference="hecke-tilde"} with respect to such integrals: **Proposition 32**. *Fix an integer $1 \leqslant k \leqslant n-1$, and three functions $a(y_1,\dots,y_n)$, $b(y_1,\dots,y_n)$ and $c(y_1,\dots,y_n)$, the last of which is symmetric in its alphabet $(y_1,\dots,y_n)$. We have the following equality of integrals: $$\begin{gathered} \label{adjoint} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_n} \frac{dy_n}{y_n} \prod_{1 \leqslant i<j \leqslant n} \left( \frac{y_j-y_i}{y_j-q y_i} \right) (T_k \cdot a)(y_1^{-1},\dots,y_n^{-1}) b(y_1,\dots,y_n) c(y_1,\dots,y_n) \\ = \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_n} \frac{dy_n}{y_n} \prod_{1 \leqslant i<j \leqslant n} \left( \frac{y_j-y_i}{y_j-q y_i} \right) a(y_1^{-1},\dots,y_n^{-1}) (\tilde{T}_k \cdot b)(y_1,\dots,y_n) c(y_1,\dots,y_n),\end{gathered}$$ with $T_k$, $\tilde{T}_k$ given by [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"}, [\[hecke-tilde\]](#hecke-tilde){reference-type="eqref" reference="hecke-tilde"}, respectively.* *Proof.* The proof of this result, for $c(y_1,\dots,y_n) \equiv 1$, is given in Proposition 8.1.3 in [@BorodinW Chapter 8]. The extension of the result to generic symmetric functions $c(y_1,\dots,y_n)$ follows immediately, in view of the fact that acting with Hecke generators $T_k$, $\tilde{T}_k$ commutes with multiplication by functions which are symmetric in $(y_k,y_{k+1})$. ◻ ## Cauchy identity {#ssec:cauchy} It is possible to derive a number of summation identities of Cauchy-type for the non-symmetric spin Hall--Littlewood functions; see [@BorodinW Chapter 4]. In this section we state a Cauchy identity that did not previously appear in that text, although it is similar in flavour to [@BorodinW Proposition 4.5.1], and proved in precisely the same fashion. **Theorem 33**. *Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition such that $|\lambda| = m$. Fix a coloured composition $\nu \in \mathcal{S}_{\lambda}$ and two alphabets $(x_1,\dots,x_p)$, $(y_1,\dots,y_m)$ of complex parameters satisfying the constraint $$\begin{aligned} \label{converge} \left| \frac{x_i-s}{1-s x_i} \cdot \frac{y_j-s}{1-s y_j} \right| < 1, \qquad \forall\ 1 \leqslant i \leqslant p,\ 1 \leqslant j \leqslant m.\end{aligned}$$ The following summation identity holds: $$\begin{aligned} \label{Gg-cauchy} \sum_{\kappa \in \mathcal{S}_{\lambda}} G_{\kappa/\nu}(\lambda;x_1,\dots,x_p) g_{\kappa}(\lambda;y_1,\dots,y_m) = \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1-q x_i y_j}{1-x_i y_j} \cdot g_{\nu}(\lambda;y_1,\dots,y_m).\end{aligned}$$* *Proof.* The left hand side of [\[Gg-cauchy\]](#Gg-cauchy){reference-type="eqref" reference="Gg-cauchy"} may be represented algebraically as $$\begin{gathered} \sum_{\kappa \in \mathcal{S}_{\lambda}} G_{\kappa/\nu}(\lambda;x_1,\dots,x_p) g_{\kappa}(\lambda;y_1,\dots,y_m) \\ = \left\langle \nu\right|_{\lambda} \prod_{i=1}^{p} \mathcal{C}_0(x_i) \prod_{j\in [1,\ell_1]} \mathcal{B}_1(x_j) \prod_{j\in (\ell_1,\ell_2]} \mathcal{B}_2(x_j) \cdots \prod_{j\in(\ell_{n-1},\ell_n]} \mathcal{B}_n(x_j) \left|\emptyset\right\rangle.\end{gathered}$$ We use the commutation relation [\[CB\]](#CB){reference-type="eqref" reference="CB"} in the case $i=0$, $j \geqslant 1$ to transfer all $\mathcal{B}$-operators to the left of the product; this results in the equation $$\begin{gathered} \label{gG-proof1} \sum_{\kappa \in \mathcal{S}_{\lambda}} G_{\kappa/\nu}(\lambda;x_1,\dots,x_p) g_{\kappa}(\lambda;y_1,\dots,y_m) \\ = \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1-q x_i y_j}{1-x_i y_j} \cdot \left\langle \nu\right|_{\lambda} \prod_{j\in [1,\ell_1]} \mathcal{B}_1(x_j) \prod_{j\in (\ell_1,\ell_2]} \mathcal{B}_2(x_j) \cdots \prod_{j\in(\ell_{n-1},\ell_n]} \mathcal{B}_n(x_j) \left|\emptyset\right\rangle,\end{gathered}$$ where we have used the fact that $$\begin{aligned} \prod_{i=1}^{p} \mathcal{C}_0(x_i) \left|\emptyset\right\rangle = \left|\emptyset\right\rangle.\end{aligned}$$ The expression obtained, [\[gG-proof1\]](#gG-proof1){reference-type="eqref" reference="gG-proof1"}, matches with the right hand side of [\[Gg-cauchy\]](#Gg-cauchy){reference-type="eqref" reference="Gg-cauchy"}. ◻ ## Integral formula for $G_{\mu/\nu}$ {#ssec:int} Combining the results of Sections [3.7](#ssec:orthog){reference-type="ref" reference="ssec:orthog"}--[3.8](#ssec:cauchy){reference-type="ref" reference="ssec:cauchy"}, we now obtain an integral formula[^10] for the rational symmetric functions [\[G-pf\]](#G-pf){reference-type="eqref" reference="G-pf"}: **Theorem 34**. *We have the following integral formula for the function $G_{\mu/\nu}(\lambda;x_1,\dots,x_p)$: $$\begin{gathered} \label{G-int} G_{\mu/\nu}(\lambda;x_1,\dots,x_p) = \frac{q^{m(m+1)/2}}{(q-1)^m} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^m \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(\lambda;y_1,\dots,y_m) \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1-q x_i y_j}{1-x_i y_j},\end{gathered}$$ where $\{C_1,\dots,C_m\}$ are contours admissible with respect to $(q,s)$.* *Proof.* This result is essentially given by [@ABW21 Corollary 11.3.2], though we reproduce its proof here for the reader's convenience. We begin by proving [\[G-int\]](#G-int){reference-type="eqref" reference="G-int"} in the case where $\mu,\nu$ are rainbow compositions. Start from the Cauchy identity [\[Gg-cauchy\]](#Gg-cauchy){reference-type="eqref" reference="Gg-cauchy"} with $\lambda = 1^m$, multiply it by $f_{\mu}(y_1^{-1},\dots,y_m^{-1})$, prior to integrating as in the left hand side of [\[f-g-orthog\]](#f-g-orthog){reference-type="eqref" reference="f-g-orthog"}.[^11] In view of the orthogonality property [\[f-g-orthog\]](#f-g-orthog){reference-type="eqref" reference="f-g-orthog"}, this filters the $\kappa = \mu$ term from the sum and we read off the identity $$\begin{gathered} \label{G-int-rainbow} G_{\mu/\nu}(1^m;x_1,\dots,x_p) = \frac{q^{m(m+1)/2}}{(q-1)^m} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^m \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(1^m;y_1,\dots,y_m) \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1-q x_i y_j}{1-x_i y_j}.\end{gathered}$$ This proves [\[G-int\]](#G-int){reference-type="eqref" reference="G-int"} in the case $\mu,\nu \in \mathcal{S}_{1^m}$. The general case [\[G-int\]](#G-int){reference-type="eqref" reference="G-int"} then follows by antisymmetrization of [\[G-int-rainbow\]](#G-int-rainbow){reference-type="eqref" reference="G-int-rainbow"}; the left hand side antisymmetrization is obtained using Proposition [Proposition 28](#prop:f-ant){reference-type="ref" reference="prop:f-ant"}, while that of the right hand side is carried out using Proposition [Proposition 29](#prop:G-ant){reference-type="ref" reference="prop:G-ant"}. ◻ # Fusion In this section we briefly recall some of the basics regarding the fusion procedure, when applied to the model [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"}. For full details, we refer the reader to [@ABW21 Chapter 3] and [@BorodinW Appendices B and C]. ## Definition of fused vertices To define fused vertices we require some additional notation; introduce *column vertices* by taking towers of height $N$ of the $L$-weights [\[generic-L\]](#generic-L){reference-type="eqref" reference="generic-L"}. In particular, for all $\bm{A}, \bm{C}\in \{0,1\}^n$ and $b_1,\dots,b_N,d_1,\dots,d_N \in [0,n]$ we define $$\begin{aligned} \label{tower} \tilde{L}^{(s)}_z \Big( \bm{A},(b_1,\dots,b_N);\bm{C},(d_1,\dots,d_N) \Big) = \begin{tikzpicture}[scale=0.9,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,5) -- (1,5); %\draw[lgray,line width=1.5pt,->] (-1,4) -- (1,4); \draw[lgray,line width=1.5pt,->] (-1,3) -- (1,3); %\draw[lgray,line width=1.5pt,->] (-1,2) -- (1,2); \draw[lgray,line width=1.5pt,->] (-1,1) -- (1,1); \draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0); % \draw[lgray,line width=4pt,->] (0,-1) -- (0,6); \node[left] at (-1,0) {\tiny $b_1$};\node[right] at (1,0) {\tiny $d_1$}; \node[left] at (-1,1) {\tiny $b_2$};\node[right] at (1,1) {\tiny $d_2$}; \node[left] at (-1,2.5) {$\vdots$};\node[right] at (1,2.5) {$\vdots$}; \node[left] at (-1,3.5) {$\vdots$};\node[right] at (1,3.5) {$\vdots$}; \node[left] at (-1,5) {\tiny $b_N$};\node[right] at (1,5) {\tiny $d_N$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,6) {\tiny $\bm{C}$}; \node[left] at (-2,0) {$z \rightarrow$}; \node[left] at (-2,1) {$q z \rightarrow$}; \node[left] at (-2,5) {$q^{N-1} z \rightarrow$}; \end{tikzpicture}\end{aligned}$$ where the spectral parameters associated to horizontal lines, read from bottom to top, form the geometric progression $(z,qz,\dots,q^{N-1}z)$. **Definition 35**. Fix four binary strings $\bm{A}= (A_1,\dots,A_n)$, $\bm{B}= (B_1,\dots,B_n)$, $\bm{C}= (C_1,\dots,C_n)$ and $\bm{D}= (D_1,\dots,D_n)$ in $\{0,1\}^n$. Choose an integer $N \geqslant 1$ and introduce the notation $r=q^{-N/2}$. We define *fused vertex weights* as follows: $$\begin{aligned} \label{fused-def} \tilde{L}^{(r,s)}_z(\bm{A},\bm{B}; \bm{C},\bm{D}) = \frac{1}{Z_q(N;\bm{B})} \sum_{\substack{ (b_1,\dots,b_N) \\ (d_1,\dots,d_N) }} q^{{\rm inv}(b_1,\dots,b_N)} \tilde{L}^{(s)}_z \Big( \bm{A},(b_1,\dots,b_N);\bm{C},(d_1,\dots,d_N) \Big),\end{aligned}$$ where the sum is taken over vectors $(b_1,\dots,b_N)$ and $(d_1,\dots,d_N)$ such that $\sum_{i=1}^{N} \bm{e}_{b_i} = \bm{B}$ and $\sum_{i=1}^{N} \bm{e}_{d_i} = \bm{D}$, we recall that ${\rm inv}(b_1,\dots,b_N) = {\rm card}\{ (i,j) : i<j,\ b_i > b_j\}$, and where the normalization takes the form $$\begin{aligned} Z_q(N;\bm{B}) = \frac{(q;q)_N}{(q;q)_{B_0} (q;q)_{B_1} \dots (q;q)_{B_n}}, \quad B_0 = N - \sum_{i=1}^{n} B_i.\end{aligned}$$ We represent the fused vertices [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} graphically as follows: $$\begin{aligned} \tilde{L}^{(r,s)}_z(\bm{A},\bm{B}; \bm{C},\bm{D}) = \begin{tikzpicture}[scale=0.7,baseline=(current bounding box.center),>=stealth] \node[left] at (-1.5,0) {$(z;r) \rightarrow $}; \draw[lgray,line width=4pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $\bm{B}$};\node[right] at (1,0) {\tiny $\bm{D}$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{C}$}; %\node[below] at (0,-1.4) {$\uparrow$}; \node[below] at (0,-1.4) {$(s)$}; \end{tikzpicture} \qquad \bm{A},\bm{B},\bm{C},\bm{D}\in \{0,1\}^n.\end{aligned}$$ **Proposition 36**. *For all integers $b,d \in [0,n]$ and binary strings $\bm{A},\bm{C}\in \{0,1\}^n$, one has $$\begin{aligned} \tilde{L}^{(r,s)}_z(\bm{A},\bm{e}_b; \bm{C},\bm{e}_d)\Big|_{r=q^{-1/2}} = \tilde{L}^{(s)}_z(\bm{A},b;\bm{C},d).\end{aligned}$$* *Proof.* Setting $r=q^{-1/2}$ is equivalent to taking $N=1$. For $N=1$ the sum on the right hand side of [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} trivializes and $Z_q(1;\bm{e}_b)=1$ for all $b \in [0,n]$; the claimed equality is then manifest. ◻ ## Fused vertex weights The fused vertex weights [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} were explicitly evaluated as [@ABW21 Theorem 4.3.2]; we recall this explicit formula here. For any pair of vectors $\bm{A},\bm{B}\in \mathbb{Z}^n$, define the function $$\begin{aligned} \label{phi-def} \varphi(\bm{A},\bm{B}) = \sum_{1 \leqslant i<j \leqslant n} A_i B_j.\end{aligned}$$ Fix binary strings $\bm{A}= (A_1,\dots,A_n)$, $\bm{B}= (B_1,\dots,B_n)$, $\bm{C}= (C_1,\dots,C_n)$, $\bm{D}= (D_1,\dots,D_n)$. Construct another vector $\bm{V}= (V_1,\dots,V_n)$, where $V_i = {\rm min}\{A_i,B_i,C_i,D_i\}$ for $i \in [1,n]$. The fused weights [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} are then given by[^12] $$\begin{gathered} \label{fused-weights} \tilde{L}^{(r,s)}_{sz}(\bm{A},\bm{B}; \bm{C},\bm{D}) = \bm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}} \\ \times (-1)^{|\bm{V}|} z^{|\bm{D}|-|\bm{B}|} r^{-2|\bm{A}|} s^{2|\bm{D}|} q^{-\varphi(\bm{A},\bm{V})-|\bm{V}|} \frac{(r^{-2}q^{-|\bm{V}|+1}z;q)_{|\bm{V}|}}{(s^2 r^{-2} q^{-|\bm{V}|}z;q)_{|\bm{V}|}} W^{(r,s)}_z(\bm{A},\bm{B}-\bm{V};\bm{C},\bm{D}-\bm{V})\end{gathered}$$ where we have defined $$\begin{gathered} \label{w-weight} W^{(r,s)}_z(\bm{A},\bm{B}-\bm{V};\bm{C},\bm{D}-\bm{V}) = \\ \sum_{\bm{P}} \Phi(\bm{C}-\bm{P},\bm{C}+\bm{D}-\bm{V}-\bm{P};s^2 r^{-2} q^{-|\bm{V}|} z,s^2 z) \Phi(\bm{P},\bm{B}-\bm{V};r^2q^{|\bm{V}|}z^{-1},r^2q^{|\bm{V}|}),\end{gathered}$$ with the sum over all $\bm{P}= (P_1,\dots,P_n)$ such that $P_i \leqslant{\rm min}\{C_i,B_i-V_i\}$ for all $i \in [1,n]$. The functions appearing in the summand of [\[w-weight\]](#w-weight){reference-type="eqref" reference="w-weight"} are defined for any two vectors $\bm{S} = (S_1,\dots,S_n)$, $\bm{T} = (T_1,\dots,T_n)$ such that $S_i \leqslant T_i$ for all $i \in [1,n]$: $$\begin{aligned} \Phi(\bm{S},\bm{T};u,v) = \frac{(u;q)_{|\bm{S}|} (v/u;q)_{|\bm{T}|-|\bm{S}|}}{(v;q)_{|\bm{T}|}} (v/u)^{|\bm{S}|} q^{\varphi(\bm{T}-\bm{S},\bm{S})} \prod_{i=1}^{n} \binom{T_i}{S_i}_q,\end{aligned}$$ where we have used the standard $q$-binomial coefficient $$\binom{b}{a}_q = \dfrac{(q;q)_b}{(q;q)_a(q;q)_{b-a}}, \quad a \leqslant b.$$ The weights [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"} provide an explicit evaluation of the right hand side of [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"}, under the identification $r=q^{-N/2}$; however, the formula [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"} makes sense for arbitrary values of $r \in \mathbb{C}$ (that is, as a rational function in $r$), and we tacitly assume this in what follows. ## Master Yang--Baxter equation {#ssec:master-yb} The fused vertex weights satisfy a master Yang--Baxter equation, that contains the previous three Yang--Baxter relations [\[RLLa\]](#RLLa){reference-type="eqref" reference="RLLa"}--[\[RLLc\]](#RLLc){reference-type="eqref" reference="RLLc"} as special cases. **Theorem 37**. *Fix a collection of binary strings $\bm{A}(1),\bm{A}(2),\bm{A}(3),\bm{B}(1),\bm{B}(2),\bm{B}(3) \in \{0,1\}^n$ and arbitrary parameters $x,y,r,s,t \in \mathbb{C}$. The weights [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"} satisfy the equation $$\begin{gathered} \label{master} \sum_{\bm{C}(1),\bm{C}(2),\bm{C}(3)} \tilde{L}^{(r,s)}_{sx/y}\Big(\bm{A}(2),\bm{A}(1);\bm{C}(2),\bm{C}(1)\Big) \tilde{L}^{(r,t)}_{x}\Big(\bm{A}(3),\bm{C}(1);\bm{C}(3),\bm{B}(1)\Big) \tilde{L}^{(s,t)}_{y}\Big(\bm{C}(3),\bm{C}(2);\bm{B}(3),\bm{B}(2)\Big) \\ = \sum_{\bm{C}(1),\bm{C}(2),\bm{C}(3)} \tilde{L}^{(s,t)}_{y}\Big(\bm{A}(3),\bm{A}(2);\bm{C}(3),\bm{C}(2)\Big) \tilde{L}^{(r,t)}_{x}\Big(\bm{C}(3),\bm{A}(1);\bm{B}(3),\bm{C}(1)\Big) \tilde{L}^{(r,s)}_{sx/y}\Big(\bm{C}(2),\bm{C}(1);\bm{B}(2),\bm{B}(1)\Big),\end{gathered}$$ where $\bm{C}(1),\bm{C}(2),\bm{C}(3)$ are summed over all binary strings in $\{0,1\}^n$.* *Proof.* See [@ABW21 Proposition 5.1.4] for full details. ◻ The master Yang--Baxter equation [\[master\]](#master){reference-type="eqref" reference="master"} reduces to the three given earlier, namely [\[RLLa\]](#RLLa){reference-type="eqref" reference="RLLa"}--[\[RLLc\]](#RLLc){reference-type="eqref" reference="RLLc"}, by choosing any two of $r,s,t$ to be equal to $q^{-1/2}$, keeping the remaining parameter arbitrary (and up to further relabelling of the spectral parameters $x$ and $y$). Details of these reductions, for the bosonic counterpart of the models discussed in the current text, may be found in [@BorodinW Appendix C]. In what follows, we will make use of yet another reduction: **Corollary 38**. *Fix two integers $a,b \in [0,n]$ and binary strings $\bm{A}(2),\bm{A}(3),\bm{B}(2),\bm{B}(3) \in \{0,1\}^n$. The weights [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"} and [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"} satisfy the equation $$\begin{gathered} \sum_{c,\bm{C}(2),\bm{C}(3)} \tilde{L}^{(r)}_{rx/y}\Big(\bm{A}(2),a;\bm{C}(2),c\Big) \tilde{L}^{(s)}_{x}\Big(\bm{A}(3),c;\bm{C}(3),b\Big) \tilde{L}^{(r,s)}_{y}\Big(\bm{C}(3),\bm{C}(2);\bm{B}(3),\bm{B}(2)\Big) \\ = \sum_{c,\bm{C}(2),\bm{C}(3)} \tilde{L}^{(r,s)}_{y}\Big(\bm{A}(3),\bm{A}(2);\bm{C}(3),\bm{C}(2)\Big) \tilde{L}^{(s)}_{x}\Big(\bm{C}(3),a;\bm{B}(3),c\Big) \tilde{L}^{(r)}_{rx/y}\Big(\bm{C}(2),c;\bm{B}(2),b\Big),\end{gathered}$$ where $c$ is summed over all integers in $[0,n]$ and $\bm{C}(2),\bm{C}(3)$ are summed over all binary strings in $\{0,1\}^n$. This equation has the following graphical version: $$\begin{aligned} \label{yb-reduce1} \sum_{c,\bm{C}(2),\bm{C}(3)} \begin{tikzpicture}[scale=0.9,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-2,1) node[above,scale=0.6] {\color{black} $a$} -- (-1,0) node[below,scale=0.6] {\color{black} $c$} -- (1,0) node[right,scale=0.6] {\color{black} $b$}; \draw[lgray,line width=4pt,->] (-2,0) node[below,scale=0.6] {\color{black} $\bm{A}(2)$} -- (-1,1) node[above,scale=0.6] {\color{black} $\bm{C}(2)$} -- (1,1) node[right,scale=0.6] {\color{black} $\bm{B}(2)$}; \draw[lgray,line width=4pt,->] (0,-1) node[below,scale=0.6] {\color{black} $\bm{A}(3)$} -- (0,0.5) node[scale=0.6] {\color{black} $\bm{C}(3)$} -- (0,2) node[above,scale=0.6] {\color{black} $\bm{B}(3)$}; % \node[left] at (-2.2,1) {$x \rightarrow$}; \node[left] at (-2.2,0) {$(y;r) \rightarrow$}; %\node[below] at (0,-1.4) {$\uparrow$}; \node[below] at (0,-1.4) {$(s)$}; \end{tikzpicture} \quad = \quad \sum_{c,\bm{C}(2),\bm{C}(3)} \begin{tikzpicture}[scale=0.9,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1.5pt,->] (-1,1) node[left,scale=0.6] {\color{black} $a$} -- (1,1) node[above,scale=0.6] {\color{black} $c$} -- (2,0) node[below,scale=0.6] {\color{black} $b$}; \draw[lgray,line width=4pt,->] (-1,0) node[left,scale=0.6] {\color{black} $\bm{A}(2)$} -- (1,0) node[below,scale=0.6] {\color{black} $\bm{C}(2)$} -- (2,1) node[above,scale=0.6] {\color{black} $\bm{B}(2)$}; \draw[lgray,line width=4pt,->] (0,-1) node[below,scale=0.6] {\color{black} $\bm{A}(3)$} -- (0,0.5) node[scale=0.6] {\color{black} $\bm{C}(3)$} -- (0,2) node[above,scale=0.6] {\color{black} $\bm{B}(3)$}; % \node[left] at (-1.5,1) {$x \rightarrow$}; \node[left] at (-1.5,0) {$(y;r) \rightarrow$}; %\node[below] at (0,-1.4) {$\uparrow$}; \node[below] at (0,-1.4) {$(s)$}; \end{tikzpicture}\end{aligned}$$* *Proof.* This is the reduction $r=q^{-1/2}$, $\bm{A}(1) = \bm{e}_a$, $\bm{B}(1) = \bm{e}_b$ of equation [\[master\]](#master){reference-type="eqref" reference="master"}, followed by the relabelling $s \mapsto r$, $t \mapsto s$. ◻ ## Fused row operators {#sec:fused-row} For any integer $N \geqslant 0$ and non-empty set $I \subset [0,n]$, define the following analogue of the row operators [\[C-row\]](#C-row){reference-type="eqref" reference="C-row"}: $$\begin{gathered} \label{Cfused-row} \mathcal{D}_{I}(x;r) : \bigotimes_{k=0}^{N} \left|\bm{B}(k)\right\rangle \\ \mapsto \sum_{\bm{A}(0),\ldots,\bm{A}(N) \in \{0,1\}^n} \left( \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (-0.5,0) {$(x;r) \rightarrow$}; \node[left] at (0.5,0) {\footnotesize$\bm{e}_0$};\node[right] at (5.5,0) {\footnotesize$\bm{e}_I$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture} \right) \bigotimes_{k=0}^{N} \left|\bm{A}(k)\right\rangle,\end{gathered}$$ where the quantity $$\begin{aligned} \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node at (-0.5,0) {$(x;r) \rightarrow$}; \node[left] at (0.5,0) {\footnotesize$\bm{e}_0$};\node[right] at (5.5,0) {\footnotesize$\bm{e}_I$}; \node[below] at (5,-0.5) {\footnotesize$\bm{A}(N)$};\node[above] at (5,0.5) {\footnotesize$\bm{B}(N)$}; \node[below] at (4,-0.5) {\footnotesize$\cdots$};\node[above] at (4,0.5) {\footnotesize$\cdots$}; \node[below] at (3,-0.5) {\footnotesize$\cdots$};\node[above] at (3,0.5) {\footnotesize$\cdots$}; \node[below] at (2,-0.5) {\footnotesize$\cdots$};\node[above] at (2,0.5) {\footnotesize$\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{A}(0)$};\node[above] at (1,0.5) {\footnotesize$\bm{B}(0)$}; \end{tikzpicture}\end{aligned}$$ is a one-row partition function using the vertex weights [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"}.[^13] When $r$ is specialized to $r = q^{-p/2}$, with $p \in [1,n]$, we refer to $\mathcal{D}_I(x;r)$ as a row operator of *width* $p$; this is in reference to the fact that the horizontal line of the row operator can now carry at most $p$ paths. The case $r=q^{-1/2}$ has a particular significance in what follows; in this case the capacity constraint of the horizontal line imposes that $|I| = 1$, and we have $$\begin{aligned} \label{unfused} \mathcal{D}_{\{i\}}(x;q^{-1/2}) \equiv \mathcal{D}_i(x) = T^{\rightarrow}_{0,i}(x;N),\end{aligned}$$ for all $i \in [0,n]$, where the operator on the right hand side is given by [\[C-row\]](#C-row){reference-type="eqref" reference="C-row"}. ## Commutation relations {#commutation-relations} This subsection documents several types of commutation relations between the fused row operators [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"} of varying widths. The majority of these results will not be needed until Section [7](#sec:discrete-dist){reference-type="ref" reference="sec:discrete-dist"} of the text, where they are used to compute a certain class of partition functions that play a role in our subsequent probability distributions. The reader may prefer to skip this subsection and return to it, as needed, in Section [7](#sec:discrete-dist){reference-type="ref" reference="sec:discrete-dist"}. **Proposition 39**. *Fix an integer $i \in [1,n]$ and a set $J \subset [1,n]$ such that $i \in J$. We have the following exchange relation between fused row operators [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"} and their unfused counterparts [\[unfused\]](#unfused){reference-type="eqref" reference="unfused"}: $$\begin{aligned} \label{Cfused-com} \mathcal{D}_i(x) \mathcal{D}_{J}(y;r) = \tilde{L}_{rx/y}^{(r)}(\bm{e}_J,i;\bm{e}_J,i) \cdot \mathcal{D}_{J}(y;r) \mathcal{D}_i(x),\end{aligned}$$ where the coefficient appearing on the right hand side is given by the top-middle entry of the table [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}.* *Proof.* We give the proof in the case of row operators of unit length, namely, for $N=0$; however, for generic $N$ the proof follows in exactly the same way. Starting from the relation [\[yb-reduce1\]](#yb-reduce1){reference-type="eqref" reference="yb-reduce1"}, we set $a=0$, $\bm{A}(2) = \bm{e}_0$, $b=i$, $\bm{B}(2) = \bm{e}_J$, keeping $\bm{A}(3)$ and $\bm{B}(3)$ arbitrary. The diagonally-oriented vertex on the left hand side freezes; it is given by $\tilde{L}^{(r)}_{rx/y}(\bm{e}_0,0;\bm{e}_0,0) = 1$. Due to the fact that $i \in J$, the diagonally-oriented vertex on the right hand side also freezes; colour $i$ is present in both of the outgoing edges of this vertex, meaning that it must be present in both of the incoming edges (since $\bm{C}(2)$ is a binary string). The weight of this frozen vertex is $\tilde{L}_{rx/y}^{(r)}(\bm{e}_J,i;\bm{e}_J,i)$, completing the proof. ◻ **Proposition 40**. *Fix an integer $i \in [1,n]$ and a set $J \subset [1,n]$ such that $i \not\in J$. We have the following exchange relation between fused row operators [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"} and their unfused counterparts [\[unfused\]](#unfused){reference-type="eqref" reference="unfused"}: $$\begin{aligned} \label{Cfused-com2} \mathcal{D}_i(x) \mathcal{D}_J(y;r) = \sum_{j \in \{0,i\} \cup J} \tilde{L}_{rx/y}^{(r)}(\bm{e}_J+\bm{e}_i-\bm{e}_j,j;\bm{e}_J,i) \cdot \mathcal{D}_{\{i\}\cup J\backslash \{j\}}(y;r) \mathcal{D}_j(x),\end{aligned}$$ where the coefficients appearing in the sum are given by the bottom-middle and bottom-right entries of the table [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}.* *Proof.* Similarly to the proof of Proposition [Proposition 40](#prop:row-com2){reference-type="ref" reference="prop:row-com2"}, one starts from the relation [\[yb-reduce1\]](#yb-reduce1){reference-type="eqref" reference="yb-reduce1"} and sets $a=0$, $\bm{A}(2) = \bm{e}_0$, $b=i$, $\bm{B}(2) = \bm{e}_J$, keeping $\bm{A}(3)$ and $\bm{B}(3)$ arbitrary. The diagonally-oriented vertex on the left hand side again freezes with weight $1$. This time, however, the diagonally-oriented vertex on the right hand side is not frozen; this is due to the fact that $i \not\in J$, meaning that colour $i$ is only present in one of the outgoing edges of this vertex. The weight of the diagonally-oriented vertex is seen to be $\tilde{L}_{rx/y}^{(r)}(\bm{e}_J+\bm{e}_i-\bm{e}_j,j;\bm{e}_J,i)$, and the result follows by summing over all possible values of $c=j$. ◻ Combining Propositions [Proposition 39](#prop:row-com1){reference-type="ref" reference="prop:row-com1"} and [Proposition 40](#prop:row-com2){reference-type="ref" reference="prop:row-com2"} we obtain the following important result: **Proposition 41**. *Fix two integers $p,i \in [1,n]$ and a set $J \subset [1,n]$ of cardinality $|J|=p$. We then have the commutation relation $$\begin{aligned} \label{important-relation} \mathcal{D}_i(x) \mathcal{D}_{J}(x;q^{-p/2}) = \frac{1-q}{1-q^p} \cdot \left\{ \begin{array}{rl} q^{\alpha_i(J)} \mathcal{D}_J(x;q^{-p/2}) \mathcal{D}_i(x), & i \in J, \\ \\ \displaystyle{\sum_{j \in J}} q^{\alpha_i\left(J^{+-}_{ij}\right)} \mathcal{D}_{J^{+-}_{ij}}(x;q^{-p/2}) \mathcal{D}_j(x), & i \not\in J, \end{array} \right.\end{aligned}$$ between row operators of width 1 and width $p$ respectively. Here we have defined $J^{+-}_{ij} = \{i\} \cup J \backslash \{j\}$ and $\alpha_i(K)$ denotes the number of elements in the set $K \subset \mathbb{N}$ which exceed $i$; namely, $\alpha_i(K) = |\{k \in K : k>i\}|$.* *Proof.* We analyse the cases $i \in J$ and $i \not\in J$ separately. For $i \in J$ we use [\[Cfused-com\]](#Cfused-com){reference-type="eqref" reference="Cfused-com"} with $x=y$, $r=q^{-p/2}$; under this choice of parameters the coefficient on the right hand side reads $$\begin{aligned} \tilde{L}_{rx/y}^{(r)}(\bm{e}_J,i;\bm{e}_J,i) \Big|_{x=y} \Big|_{r=q^{-p/2}} = \frac{r^2(qx/y-1)q^{\alpha_i(J)}}{1-r^2 x/y} \Big|_{x=y} \Big|_{r=q^{-p/2}} = \frac{1-q}{1-q^p} \cdot q^{\alpha_i(J)},\end{aligned}$$ and we recover the first line of [\[important-relation\]](#important-relation){reference-type="eqref" reference="important-relation"}. For $i \not\in J$ we use [\[Cfused-com2\]](#Cfused-com2){reference-type="eqref" reference="Cfused-com2"} with $x=y$, $r=q^{-p/2}$; this allows two of the terms on the right hand of [\[Cfused-com2\]](#Cfused-com2){reference-type="eqref" reference="Cfused-com2"} to be eliminated. First, we may eliminate the $j=0$ term from the summation; this follows from the fact that $\mathcal{D}_{\{i\}\cup J}(y;r) = 0$ at $r=q^{-p/2}$, since $|\bm{e}_J+\bm{e}_i| = p+1$. Second, we may eliminate the $j=i$ term from the summation, since for $i \not\in J$ one has $$\begin{aligned} \tilde{L}_{rx/y}^{(r)}(\bm{e}_J,i;\bm{e}_J,i) \Big|_{x=y} \Big|_{r=q^{-p/2}} = \frac{r^2(1-x/y)q^{\alpha_i(J)}}{1-r^2 x/y} \Big|_{x=y} \Big|_{r=q^{-p/2}} = 0.\end{aligned}$$ The remaining terms in the summation are those for which $j \in J$; for those we obtain $$\begin{aligned} \tilde{L}_{rx/y}^{(r)}(\bm{e}_J+\bm{e}_i-\bm{e}_j,j;\bm{e}_J,i) \Big|_{x=y} \Big|_{r=q^{-p/2}} = \frac{1-q}{1-q^p} \cdot q^{\alpha_i\left(J^{+-}_{ij}\right)},\end{aligned}$$ and the second line of [\[important-relation\]](#important-relation){reference-type="eqref" reference="important-relation"} holds. ◻ **Proposition 42**. *Fix an integer $p \in [1,n]$ and a set $I = \{i_1,\dots,i_p\} \subset [1,n]$ of cardinality $|I|=p$. We have the following "peeling" property between the fused row operators [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"} and their unfused counterparts [\[unfused\]](#unfused){reference-type="eqref" reference="unfused"}: $$\begin{aligned} \label{row-relation2} \mathcal{D}_I(x;q^{-p/2}) = \sum_{j \in I} \mathcal{D}_{I \backslash\{j\}}\left(qx;q^{-(p-1)/2}\right) \mathcal{D}_j(x),\end{aligned}$$ allowing us to extract a row operator of width 1 from a row operator of width $p$.* *Proof.* Extending the definition [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} of fused vertex weights to row operators (*cf.* [@BorodinW Appendix B]), one finds that $$\begin{aligned} \label{fused-to-sigma0} \mathcal{D}_I(x;q^{-p/2}) = \sum_{\sigma \in \mathfrak{S}_p} \mathcal{D}_{i_{\sigma(1)}}(x) \mathcal{D}_{i_{\sigma(2)}}(qx) \cdots \mathcal{D}_{i_{\sigma(p)}}(q^{p-1}x),\end{aligned}$$ where the objects appearing on the right hand side are unfused row operators [\[unfused\]](#unfused){reference-type="eqref" reference="unfused"}. In particular, for row operators of unit length (namely, for $N=0$), the relation [\[fused-to-sigma0\]](#fused-to-sigma0){reference-type="eqref" reference="fused-to-sigma0"} matches precisely with the definition [\[fused-def\]](#fused-def){reference-type="eqref" reference="fused-def"} for $r=q^{-p/2}$, $\bm{B}= \bm{e}_0$ and $\bm{D}=\bm{e}_I$. Converting the sum over $\mathfrak{S}_p$ into summation over $\mathfrak{S}_{p-1}$ subgroups, and re-fusing the final $p-1$ operators in the resulting summand, we may rewrite [\[fused-to-sigma0\]](#fused-to-sigma0){reference-type="eqref" reference="fused-to-sigma0"} as $$\begin{aligned} \label{fused-to-sigma} \mathcal{D}_I(x;q^{-p/2}) = \sum_{i \in I} \mathcal{D}_i(x) \mathcal{D}_{I \backslash\{i\}}\left(qx;q^{-(p-1)/2}\right).\end{aligned}$$ Now from [\[Cfused-com2\]](#Cfused-com2){reference-type="eqref" reference="Cfused-com2"} with $J = I \backslash \{i\}$ and $r=q^{-(p-1)/2}$, one has that $$\begin{aligned} \label{fused-to-sigma2} \mathcal{D}_i(x) \mathcal{D}_{I\backslash\{i\}}\left(qx;q^{-(p-1)/2}\right) = \sum_{j \in I} \left( \left. \tilde{L}_{rq^{-1}}^{(r)}(\bm{e}_I-\bm{e}_j,j;\bm{e}_I-\bm{e}_i,i) \right|_{r=q^{-(p-1)/2}} \right) \mathcal{D}_{I\backslash\{j\}}\left(qx;q^{-(p-1)/2}\right) \mathcal{D}_j(x);\end{aligned}$$ the $j=0$ term was dropped from the above sum because $\mathcal{D}_{I}(qx;q^{-(p-1)/2})=0$, in view of the fact that this row operator has width $p-1$ while $I$ has cardinality $p$. Summing both sides of [\[fused-to-sigma2\]](#fused-to-sigma2){reference-type="eqref" reference="fused-to-sigma2"} over $i \in I$ yields $$\begin{aligned} \label{reverse-order} \sum_{i \in I} \mathcal{D}_i(x) \mathcal{D}_{I\backslash\{i\}}\left(qx;q^{-(p-1)/2}\right) = \sum_{j \in I} \mathcal{D}_{I\backslash\{j\}}\left(qx;q^{-(p-1)/2}\right) \mathcal{D}_j(x),\end{aligned}$$ in view of the stochasticity property $\sum_{i \in I} \tilde{L}_{rq^{-1}}^{(r)}(\bm{e}_I-\bm{e}_j,j;\bm{e}_I-\bm{e}_i,i) = 1$ (see [@BorodinW Proposition 2.5.1]). Combining [\[fused-to-sigma\]](#fused-to-sigma){reference-type="eqref" reference="fused-to-sigma"} and [\[reverse-order\]](#reverse-order){reference-type="eqref" reference="reverse-order"}, we have completed the proof. ◻ # LLT measures and Plancherel specialization In this section we introduce the probability measures that will be central to this text; they are based on the Lascoux--Leclerc--Thibon polynomials [@LLT; @ABW21] and their associated Cauchy identity [@Lam; @ABW21], and accordingly we refer to them as *LLT measures*. In analogy with the Schur and Macdonald processes [@OR03; @BorodinC], one may introduce a class of Markov kernels that preserve the form of the LLT measure when they act upon it. Acting consecutively with these Markov kernels, we obtain $n$-tuples of random Gelfand--Tsetlin patterns; one Gelfand--Tsetlin pattern is produced for each of the $n$ colours in our partition functions. The main result of this paper is a complete description of the behaviour of these patterns under a certain asymptotic regime of the underlying measure; this is carried out in Section [6](#sec:asymp){reference-type="ref" reference="sec:asymp"}. The layout of this section is as follows. In Sections [5.1](#ssec:LLTdef){reference-type="ref" reference="ssec:LLTdef"}--[5.3](#ssec:intLLT){reference-type="ref" reference="ssec:intLLT"} we recall a partition function representation for the LLT polynomials, recently obtained in [@CGKM22; @ABW21], and use it to present an integral formula for the latter. In Section [5.4](#ssec:planch){reference-type="ref" reference="ssec:planch"} we apply the Plancherel specialization of the ring of symmetric functions to the integral obtained in Section [5.3](#ssec:intLLT){reference-type="ref" reference="ssec:intLLT"}, yielding an integral formula for the Plancherel-specialized LLT polynomials. In Sections [5.5](#ssec:cauchyLLT){reference-type="ref" reference="ssec:cauchyLLT"}--[5.6](#ssec:markov){reference-type="ref" reference="ssec:markov"} we recall the (skew) Cauchy identity for LLT polynomials, and use it to define our LLT measures and associated Markov kernels. ## Functions $\mathbb{G}_{\lambda/\mu}$ and reduction to LLT polynomials {#ssec:LLTdef} In Section [3.9](#ssec:int){reference-type="ref" reference="ssec:int"} we introduced the symmetric rational functions $G_{\lambda/\mu}$ as matrix elements of products of the row operators [\[C-row\]](#C-row){reference-type="eqref" reference="C-row"}; we now generalize these, by replacing the row operators in the algebraic construction with their fused analogues [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"}. **Definition 43**. Let $\lambda = (\lambda_1,\dots,\lambda_n)$ be a composition of weight $m$, and fix two $\lambda$-coloured compositions $\mu \in \mathcal{S}_{\lambda}$ and $\nu \in \mathcal{S}_{\lambda}$. Let the corresponding vectors in $\mathbb{V}(\infty)$, $\left|\mu\right\rangle_{\lambda}$ and $\left|\nu\right\rangle_{\lambda}$, be given by [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"} and [\[B(k)\]](#B(k)){reference-type="eqref" reference="B(k)"}, respectively. For any integer $p \geqslant 1$ we define the following family of symmetric rational functions: $$\begin{aligned} \label{G-def-fused} (-s)^{|\mu|-|\nu|} \cdot \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;r_1,\dots,r_p) = \left\langle \nu\right|_{\lambda} \prod_{i=1}^{p} \mathcal{D}_{\{0\}}(x_i;r_i) \left|\mu\right\rangle_{\lambda},\end{aligned}$$ where the operators $\mathcal{D}_{\{0\}}(x_i;r_i)$ are given by [\[Cfused-row\]](#Cfused-row){reference-type="eqref" reference="Cfused-row"}. In graphical notation, the definition [\[G-def-fused\]](#G-def-fused){reference-type="eqref" reference="G-def-fused"} reads $$\begin{aligned} \label{G-pf-fused} (-s)^{|\mu|-|\nu|} \cdot \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;r_1,\dots,r_p) &= \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \foreach\y in {1,...,5}{ \draw[lgray,line width=4pt,->] (1,\y) -- (8,\y); } \foreach\x in {2,...,7}{ \draw[lgray,line width=4pt,->] (\x,0) -- (\x,6); } %spectral parameters \node[left] at (0.5,1) {$(x_1;r_1) \rightarrow$}; \node[left] at (0.5,2) {$(x_2;r_2) \rightarrow$}; \node[left] at (0.5,3) {$\vdots$}; \node[left] at (0.5,4) {$\vdots$}; \node[left] at (0.5,5) {$(x_p;r_p) \rightarrow$}; %top labels \node[above] at (7,6) {$\cdots$}; \node[above] at (6,6) {$\cdots$}; \node[above] at (5,6) {$\cdots$}; \node[above] at (4,6) {\footnotesize$\bm{A}(2)$}; \node[above] at (3,6) {\footnotesize$\bm{A}(1)$}; \node[above] at (2,6) {\footnotesize$\bm{A}(0)$}; %bottom labels \node[below] at (7,0) {$\cdots$}; \node[below] at (6,0) {$\cdots$}; \node[below] at (5,0) {$\cdots$}; \node[below] at (4,0) {\footnotesize$\bm{B}(2)$}; \node[below] at (3,0) {\footnotesize$\bm{B}(1)$}; \node[below] at (2,0) {\footnotesize$\bm{B}(0)$}; %right labels \node[right] at (8,1) {$\bm{e}_0$}; \node[right] at (8,2) {$\bm{e}_0$}; \node[right] at (8,3) {$\vdots$}; \node[right] at (8,4) {$\vdots$}; \node[right] at (8,5) {$\bm{e}_0$}; %left labels \node[left] at (1,1) {$\bm{e}_0$}; \node[left] at (1,2) {$\bm{e}_0$}; \node[left] at (1,3) {$\vdots$}; \node[left] at (1,4) {$\vdots$}; \node[left] at (1,5) {$\bm{e}_0$}; \end{tikzpicture}\end{aligned}$$ where the vectors $\bm{A}(k), \bm{B}(k)$, $k \geqslant 0$ are given by [\[A(k)\]](#A(k)){reference-type="eqref" reference="A(k)"}--[\[B(k)\]](#B(k)){reference-type="eqref" reference="B(k)"}. Two reductions of [\[G-def-fused\]](#G-def-fused){reference-type="eqref" reference="G-def-fused"} are of interest. The first is obtained by setting $r_i = q^{-1/2}$ for all $1 \leqslant i \leqslant p$; in this case, each fused row operator reduces to its unfused analogue, as described in equation [\[unfused\]](#unfused){reference-type="eqref" reference="unfused"}, and we find that $$\begin{aligned} \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;q^{-1/2},\dots,q^{-1/2}) = G_{\mu/\nu}(\lambda;x_1,\dots,x_p).\end{aligned}$$ The second we record as a theorem, below. **Theorem 44**. *The function $\mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;r_1,\dots,r_p)$ has well-defined $s \rightarrow 0$ and $r_1,\dots,r_p \rightarrow \infty$ limits. Under these limits, it becomes a polynomial in $(x_1,\dots,x_p)$ with monomial coefficients living in $\mathbb{N}[q]$.* *Proof.* To compute the limit $s \rightarrow 0$, we divide both sides of [\[G-def-fused\]](#G-def-fused){reference-type="eqref" reference="G-def-fused"} by $(-s)^{|\mu|-|\nu|}$; on the right hand side, we may distribute the resulting $(-s)^{|\nu|-|\nu|}$ factor within the partition function by assigning a factor of $(-s)^{-1}$ to each horizontal unit step by a path. By [@ABW21 Corollary 8.3.6], $$\begin{aligned} \label{llt-weights} \lim_{r \rightarrow\infty} \lim_{s \rightarrow 0} (-s)^{-|\bm{D}|} \tilde{L}^{(r,s)}_x(\bm{A},\bm{B};\bm{C},\bm{D}) = \bm{1}_{\bm{C}+\bm{D} \in \{0,1\}^n} \cdot x^{|\bm{D}|} q^{\varphi(\bm{D},\bm{C})+\varphi(\bm{D},\bm{D})}, \qquad \forall\ \bm{A},\bm{B},\bm{C},\bm{D}\in \{0,1\}^n.\end{aligned}$$ Since this limit exists at the level of the individual vertices, it follows that the $s \rightarrow 0$ and $r_1,\dots,r_p \rightarrow \infty$ limits exist when applied to the whole partition function. The fact that the resulting function is a polynomial in $(x_1,\dots,x_p)$, with nonnegative polynomial coefficients in $q$, is manifest from the right hand side of [\[llt-weights\]](#llt-weights){reference-type="eqref" reference="llt-weights"}. ◻ Throughout the remainder of the text, we refer to $$\begin{aligned} \label{LLTdef} \mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;\infty,\dots,\infty)\Big|_{s \rightarrow 0} \equiv \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p)\end{aligned}$$ as a Lascoux--Leclerc--Thibon (LLT) polynomial[^14], and tacitly assume that the $s \rightarrow 0$ and $r_1,\dots,r_p \rightarrow \infty$ limits have been taken, unless it is specifically stated otherwise. ## Padding and shifting LLT polynomials To this point, LLT polynomials were indexed by coloured compositions, as given by Definiton [Definition 14](#def:cc){reference-type="ref" reference="def:cc"}. There is a natural way to extend their definition to allow indexing by *coloured signatures* (the extension of the set [\[lambda-col\]](#lambda-col){reference-type="eqref" reference="lambda-col"} that allows parts to take any integer values, including negative ones) which will be convenient when we come to stating the Cauchy identity for LLT polynomials (see Section [5.5](#ssec:cauchyLLT){reference-type="ref" reference="ssec:cauchyLLT"}). One may consider the effect of appending an extra column to the left of the partition function [\[G-pf-fused\]](#G-pf-fused){reference-type="eqref" reference="G-pf-fused"}, with the boundary conditions at the top and bottom of this column prescribed as $\bm{A}(-1) = 1^n$ and $\bm{B}(-1) = 1^n$, respectively. One sees that the appended column freezes with weight $1$ (assuming the limit where the weights [\[llt-weights\]](#llt-weights){reference-type="eqref" reference="llt-weights"} are used), and therefore does not contribute to the overall evaluation of the partition function. This invariance property may be expressed as $$\begin{aligned} \label{pad} \mathbb{G}_{-1\cup\mu/-1\cup\nu}(x_1,\dots,x_p) = \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p),\end{aligned}$$ for any $\mu,\nu \in \mathcal{S}_{\lambda}$, where $-1 \cup \mu$ and $-1 \cup \nu$ mean prepending a part of size $-1$ in each of the $n$ blocks of $\mu$ and $\nu$, respectively (similarly to Definition [Definition 15](#defn:padding){reference-type="ref" reference="defn:padding"}). The procedure [\[pad\]](#pad){reference-type="eqref" reference="pad"} may clearly be iterated, allowing us to prepend arbitrarily many negative parts to the coloured compositions in question. One also notes that, on the resulting coloured signatures, there holds $$\begin{aligned} \label{shift} \mathbb{G}_{(\mu-1)/(\nu-1)}(x_1,\dots,x_p) = \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p),\end{aligned}$$ where $(\mu-1)$ and $(\nu-1)$ mean subtracting $1$ from every part of $\mu$ and $\nu$, respectively. It is then easy to see that [\[pad\]](#pad){reference-type="eqref" reference="pad"} and [\[shift\]](#shift){reference-type="eqref" reference="shift"} completely determine the value of $\mathbb{G}_{\mu/\nu}(x_1,\dots,x_p)$ for any coloured signatures $\mu$ and $\nu$ (possibly containing infinitely many negative parts[^15]). ## Integral formula for LLT polynomials {#ssec:intLLT} Applying the fusion procedure to the integral formula obtained in Section [3.9](#ssec:int){reference-type="ref" reference="ssec:int"}, one may obtain an integral formula for the LLT polynomials; we reproduce that result here, in essentially the same form as it appeared in [@ABW21 Corollary 11.5.3]. **Theorem 45**. *Fix a composition $\lambda = (\lambda_1,\dots,\lambda_n)$ such that $|\lambda| = m$, and choose two coloured compositions $\mu,\nu \in \mathcal{S}_{\lambda}$. The LLT polynomials [\[LLTdef\]](#LLTdef){reference-type="eqref" reference="LLTdef"} are given by the following integral expression: $$\begin{gathered} \label{LLTint} \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p) = \frac{q^{m(m+1)/2}}{(q-1)^{m}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{m} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(\lambda;y_1,\dots,y_m) \prod_{i=1}^{p} \prod_{j=1}^{m} \frac{1}{1-x_i y_j},\end{gathered}$$ where the contours $\{C_1,\dots,C_m\}$ are admissible with respect to $(q,0)$. It is implicit that $s=0$ in the functions $f_{\breve\mu}$ and $g_{\nu}$.* *Proof.* We focus on the proof for $p=1$, as this captures the essence of the proof for generic $p$. Fix an integer $P \geqslant 1$. Using [\[G-def-fused\]](#G-def-fused){reference-type="eqref" reference="G-def-fused"}, the one-variable function $\mathbb{G}_{\mu/\nu}(\lambda; x; q^{-P/2})$ is given by $$\begin{aligned} (-s)^{|\mu|-|\nu|} \cdot \mathbb{G}_{\mu/\nu}(\lambda;x;q^{-P/2}) &= \left\langle \nu\right|_{\lambda} \mathcal{D}_{\{0\}}(x;q^{-P/2}) \left|\mu\right\rangle_{\lambda} \\ &= \left\langle \nu\right|_{\lambda} \mathcal{D}_0(x) \mathcal{D}_0(q x) \ldots \mathcal{D}_0(q^{P-1} x) \left|\mu\right\rangle_{\lambda} = (-s)^{|\mu|-|\nu|} G_{\mu/\nu}(\lambda;x,qx,\dots,q^{P-1}x)\end{aligned}$$ where we have replaced the fused row operator $\mathcal{D}_{\{0\}}(x;q^{-P/2})$ by the bundle of unfused row operators [\[C-row\]](#C-row){reference-type="eqref" reference="C-row"} which comprise it. From the integral formula [\[G-int\]](#G-int){reference-type="eqref" reference="G-int"}, we then have that $$\begin{gathered} \mathbb{G}_{\mu/\nu}(\lambda;x;q^{-P/2}) = G_{\mu/\nu}(\lambda;x,qx,\dots,q^{P-1}x) = \frac{q^{m(m+1)/2}}{(q-1)^m} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^m \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(\lambda;y_1,\dots,y_m) \prod_{j=1}^{m} \frac{1-q^P x y_j}{1-x y_j}.\end{gathered}$$ This yields an expression for $\mathbb{G}_{\mu/\nu}(\lambda;x;r)$ by performing the analytic continuation $q^P \mapsto r^{-2}$; the $p=1$ case of [\[LLTint\]](#LLTint){reference-type="eqref" reference="LLTint"} then follows by sending $r \rightarrow \infty$. The generic $p$ version of [\[LLTint\]](#LLTint){reference-type="eqref" reference="LLTint"} may be proved along similar lines; namely, we split each of the $p$ fused operators $\mathcal{D}_{\{0\}}(x_i;q^{-P/2})$ in [\[G-def-fused\]](#G-def-fused){reference-type="eqref" reference="G-def-fused"} into a bundle of $P$ unfused row operators, and carry out the analysis above on each of the bundles. ◻ ## Plancherel specialization {#ssec:planch} Let $\Lambda$ denote the ring of symmetric functions in the (infinite) alphabet $x := (x_1,x_2,\dots)$. As described in [@Macdonald Chapter I], the power sum basis of $\Lambda$ is the set of functions $$\begin{aligned} p_{\lambda}(x) := \prod_{i \geqslant 1} p_{\lambda_i}(x), \qquad p_k(x) := \sum_{i \geqslant 1} x_i^k, \ \ \forall\ k \geqslant 0,\end{aligned}$$ where $\lambda$ ranges over all partitions. Any function in $\Lambda$ is expressed as a unique linear combination of the functions $p_{\lambda}(x)$. Fix an indeterminate $t \in \mathbb{C}$. The Plancherel specialization of $\Lambda$ is the map ${\rm Pl}_t : \Lambda \rightarrow \mathbb{C}$ under which the power sums transform as follows: $$\begin{aligned} p_k(x) \mapsto \left\{ \begin{array}{ll} t, & \quad k=1, \\ \\ 0, & \quad k \geqslant 2. \end{array} \right.\end{aligned}$$ Following standard notational practice for specializations of the ring of symmetric functions, we denote the image of a function $f \in \Lambda$ under ${\rm Pl}_t$ by $f({\rm Pl}_t)$. The LLT polynomials [\[LLTdef\]](#LLTdef){reference-type="eqref" reference="LLTdef"} admit a natural lift to $\Lambda$, obtained by replacing the finite alphabet $(x_1,\dots,x_p)$ by the infinite one $x = (x_1,x_2,\dots)$. Making this replacement in [\[LLTint\]](#LLTint){reference-type="eqref" reference="LLTint"}, we have $$\begin{gathered} \label{LLTint-inf} \mathbb{G}_{\mu/\nu}(x_1,x_2,\dots) = \frac{q^{m(m+1)/2}}{(q-1)^{m}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{m} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(\lambda;y_1,\dots,y_m) \prod_{j=1}^{m} \exp\left(\sum_{k \geqslant 1} \frac{p_k(x) y_j^k}{k} \right)\end{gathered}$$ where we have used the fact that (as a formal power series) there holds $$\begin{aligned} \prod_{i \geqslant 1} \frac{1}{1-x_i y} = \exp\left(-\sum_{i \geqslant 1} \log(1-x_i y) \right) = \exp\left( \sum_{i \geqslant 1} \sum_{k=1}^{\infty} \frac{x_i^k y^k}{k} \right) = \exp\left(\sum_{k \geqslant 1} \frac{p_k(x) y^k}{k} \right).\end{aligned}$$ We then read off the Plancherel specialization of $\mathbb{G}_{\mu/\nu}$: $$\begin{gathered} \label{LLTplanch} \mathbb{G}_{\mu/\nu}({\rm Pl}_t) = \frac{q^{m(m+1)/2}}{(q-1)^{m}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{m} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_m} \frac{dy_m}{y_m} \\ \times \prod_{1 \leqslant i<j \leqslant m} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\mu}(1^m;y_1^{-1},\dots,y_m^{-1}) g_{\nu}(\lambda;y_1,\dots,y_m) \prod_{j=1}^{m} e^{t y_j}.\end{gathered}$$ In what follows, we shall further restrict to $t \in \mathbb{R}_{\geqslant 0}$, where $t$ could be viewed as playing the role of continuous time. ## Skew Cauchy identity for LLT polynomials {#ssec:cauchyLLT} Up until now we dealt with coloured compositions of arbitrary colour profile $\lambda = (\lambda_1,\dots,\lambda_n)$. Throughout the rest of the paper we shall restrict our attention to the case $\lambda_i = N$ for all $1 \leqslant i \leqslant n$, where $N$ is some given positive integer; this means that each colour within a coloured composition $\mu$ is represented exactly $N$ times. We denote the corresponding set of coloured compositions as follows: $$\begin{aligned} \mathcal{S}_{N^n} = \Big\{ \mu = \Big(\mu^{(1)}_1 < \cdots < \mu^{(1)}_{N} \Big| \mu^{(2)}_1 < \cdots < \mu^{(2)}_{N} \Big| \cdots \Big| \mu^{(n)}_1 < \cdots < \mu^{(n)}_{N}\Big) \Big\}.\end{aligned}$$ One element of $\mathcal{S}_{N^n}$ plays a special role; this is the element in which all parts of a coloured composition are as small as they can be. We assign this element the notation $\Delta$: $$\begin{aligned} \label{nothing} \Delta = (0,1,\dots,N-1 | 0,1,\dots,N-1 | \cdots | 0,1,\dots,N-1) \in \mathcal{S}_{N^n}.\end{aligned}$$ Whenever the lower coloured composition $\nu$ in an LLT polynomial $\mathbb{G}_{\mu/\nu}$ is set equal to $\Delta$, we employ the lighter notation $$\begin{aligned} \mathbb{G}_{\mu/\Delta}(x_1,\dots,x_p) \equiv \mathbb{G}_{\mu}(x_1,\dots,x_p).\end{aligned}$$ **Definition 46**. For any coloured composition $\mu = \Big( \mu^{(1)}_1 < \cdots < \mu^{(1)}_{N} \Big| \cdots \Big| \mu^{(n)}_1 < \cdots < \mu^{(n)}_{N} \Big) \in \mathcal{S}_{N^n}$ we define the statistic $$\begin{aligned} \psi(\mu) = \frac{1}{2} \sum_{1 \leqslant i<j \leqslant n}\ \sum_{a \in \mu^{(i)}}\ \sum_{b \in \mu^{(j)}} \bm{1}_{a>b}.\end{aligned}$$ **Theorem 47**. *Fix two positive integers $p$ and $N$, and two alphabets $(x_1,\dots,x_p)$ and $(y_1,\dots,y_N)$. Let $\nu \in \mathcal{S}_{N^n}$ be a coloured composition. The LLT polynomials [\[LLTdef\]](#LLTdef){reference-type="eqref" reference="LLTdef"} satisfy the Cauchy summation identity $$\begin{aligned} \label{skew-cauchy} \sum_{\mu \in \mathcal{S}_{N^n}} q^{-2\psi(\mu)} \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p) \mathbb{G}_{\mu}(y_1,\dots,y_N) = \prod_{i=1}^{p} \prod_{j=1}^{N} \frac{1}{(x_i y_j;q)_n} \cdot q^{-2\psi(\nu)} \mathbb{G}_{\nu}(y_1,\dots,y_N),\end{aligned}$$ where $\mathbb{G}_{\mu}(y_1,\dots,y_N) \equiv \mathbb{G}_{\mu/\Delta}(y_1,\dots,y_N)$ and $\mathbb{G}_{\nu}(y_1,\dots,y_N) \equiv \mathbb{G}_{\nu/\Delta}(y_1,\dots,y_N)$. This holds either as a formal power series, or as a numeric equality as long as $|q| <1$ and $|x_i y_j| <1$ for all $i,j$.* *Proof.* This was originally obtained in [@Lam Theorem 35]. For a formulation in terms of the vertex model setup of the current text, we refer to [@ABW21 Corollary 9.4.1] and [@CGKM22 Proposition 6.12]. ◻ We make a small but important adjustment to the Cauchy identity [\[skew-cauchy\]](#skew-cauchy){reference-type="eqref" reference="skew-cauchy"}. For any $N \geqslant 1$, introduce the set of coloured signatures $$\begin{aligned} \label{col-signat} \mathcal{S}(N) = \left\{ \mu = \left(\mu^{(1)} | \mu^{(2)} | \cdots | \mu^{(n)}\right) \right\},\end{aligned}$$ where for all $1 \leqslant i \leqslant n$ the components $$\begin{aligned} \label{inf-sig} \mu^{(i)} = \left(\cdots < \mu^{(i)}_{-1} < \mu^{(i)}_0 < \mu^{(i)}_1 < \cdots < \mu^{(i)}_N \right)\end{aligned}$$ are left-infinite strict signatures such that $\mu^{(i)}_k \not= k-1$ for only finitely many $k \in \mathbb{Z}$. We continue to denote by $\Delta$ the unique element in $\mathcal{S}(N)$ in which all signature parts are minimal. For any fixed $\nu \in \mathcal{S}(N)$, one then has that $$\begin{aligned} \label{skew-cauchy-pad} \sum_{\mu \in \mathcal{S}(N)} q^{-2\psi(\mu,\nu)} \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p) \mathbb{G}_{\mu}(y_1,y_2,\dots) = \prod_{i=1}^{p} \prod_{j\geqslant 1} \frac{1}{(x_i y_j;q)_n} \mathbb{G}_{\nu}(y_1,y_2,\dots),\end{aligned}$$ in which the size of the alphabet $(y_1,y_2,\dots)$ is now infinite.[^16] Here $\mathbb{G}_{\mu}(y_1,y_2,\dots) \equiv \mathbb{G}_{\mu/\Delta}(y_1,y_2,\dots)$ and $\mathbb{G}_{\nu}(y_1,y_2,\dots) \equiv \mathbb{G}_{\nu/\Delta}(y_1,y_2,\dots)$ as previously, and $$\begin{aligned} \label{new-psi} \psi(\mu,\nu) = \frac{1}{2} \sum_{1 \leqslant i<j \leqslant n}\ \sum_{a \in \mu^{(i)}}\ \sum_{b \in \mu^{(j)}} \bm{1}_{a>b>m} - \frac{1}{2} \sum_{1 \leqslant i<j \leqslant n}\ \sum_{a \in \nu^{(i)}}\ \sum_{b \in \nu^{(j)}} \bm{1}_{a>b>m},\end{aligned}$$ with $m$ chosen to be any integer such that $\mu^{(i)}_k =\nu^{(i)}_k = k-1$ for all $k \leqslant m$ and $1 \leqslant i \leqslant n$. Equation [\[skew-cauchy-pad\]](#skew-cauchy-pad){reference-type="eqref" reference="skew-cauchy-pad"} holds as an identity of formal power series, which converges if $|q| <1$ and $|x_i y_j| <1$ for all $i,j$. The claim [\[skew-cauchy-pad\]](#skew-cauchy-pad){reference-type="eqref" reference="skew-cauchy-pad"} is established by taking [\[skew-cauchy\]](#skew-cauchy){reference-type="eqref" reference="skew-cauchy"} with $N$ becoming arbitrarily large, and applying [\[pad\]](#pad){reference-type="eqref" reference="pad"} and [\[shift\]](#shift){reference-type="eqref" reference="shift"} appropriately to convert the indices of all functions to members of the set [\[col-signat\]](#col-signat){reference-type="eqref" reference="col-signat"}. It is easily verified that the quantity $\psi(\mu)-\psi(\nu)$ is invariant under such paddings and shifts, and may be written in the form [\[new-psi\]](#new-psi){reference-type="eqref" reference="new-psi"}. ## Markov kernels {#ssec:markov} We proceed to introduce probability measures from the skew Cauchy identity [\[skew-cauchy\]](#skew-cauchy){reference-type="eqref" reference="skew-cauchy"}. Normalizing so that the right hand side of [\[skew-cauchy\]](#skew-cauchy){reference-type="eqref" reference="skew-cauchy"} is equal to $1$, we have $$\begin{aligned} \label{skew-cauchy2} \sum_{\mu \in \mathcal{S}(N)} \prod_{i=1}^{p} \prod_{j \geqslant 1} (x_i y_j;q)_n \cdot q^{-2\psi(\mu,\nu)} \mathbb{G}_{\mu/\nu}(x_1,\dots,x_p) \frac{\mathbb{G}_{\mu}(y_1,y_2,\dots)}{\mathbb{G}_{\nu}(y_1,y_2,\dots)} = 1.\end{aligned}$$ In view of this sum-to-unity property, the summands in [\[skew-cauchy2\]](#skew-cauchy2){reference-type="eqref" reference="skew-cauchy2"} may be viewed as probabilities of transitioning from an initial coloured signature $\nu \in \mathcal{S}(N)$ to a final one $\mu \in \mathcal{S}(N)$. Many choices of the parameters $(x_1,\dots,x_p)$ and $(y_1,y_2,\dots)$ are possible, leading to a variety of interesting distributions, but in this work we focus on one particular choice; namely, we let $(x_1,\dots,x_p) = (1,\dots,1) \equiv 1^p$ and take the ${\rm Pl}_t$ specialization of the alphabet $(y_1,y_2,\dots)$. Under this choice, [\[skew-cauchy2\]](#skew-cauchy2){reference-type="eqref" reference="skew-cauchy2"} becomes $$\begin{aligned} \sum_{\mu \in \mathcal{S}(N)} \exp\left( -\frac{p(1-q^n)}{1-q} t \right) \mathbb{G}_{\mu/\nu}(1^p) q^{-2\psi(\mu,\nu)} \frac{\mathbb{G}_{\mu}({\rm Pl}_t)}{\mathbb{G}_{\nu}({\rm Pl}_t)} = 1.\end{aligned}$$ From this we introduce the Markov kernel $\mathbb{P}_{t,p} : V(\mathcal{S}(N)) \rightarrow V(\mathcal{S}(N))$ with matrix elements given by $$\begin{aligned} \label{skew-cauchy3} \mathbb{P}_{t,p}(\nu \rightarrow \mu) = q^{-2\psi(\mu,\nu)} \exp\left( -\frac{p(1-q^n)}{1-q}t \right) \mathbb{G}_{\mu/\nu}(1^p) \dfrac{\mathbb{G}_{\mu}({\rm Pl}_t)} {\mathbb{G}_{\nu}({\rm Pl}_t)},\end{aligned}$$ where $V(\mathcal{S}(N))$ denotes the complex linear span of the elements of $\mathcal{S}(N)$. Abusing notation slightly, whenever we write $\mathbb{P}_{t,p}(\nu)$ for some $\nu \in \mathcal{S}(N)$, this means a random coloured signature $\mu \in \mathcal{S}(N)$ sampled from the distribution [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"}. *Remark 48*. Throughout the rest of the text, we shall only be concerned with evaluating the kernel [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"} on coloured signatures $\mu,\nu$ such that $\mu^{(i)}_k = \nu^{(i)}_k = k-1$ for all $k \leqslant 0$ and $1 \leqslant i \leqslant n$. When coloured signatures have such a property, by slight abuse of notation we continue to write $\mu,\nu \in \mathcal{S}_{N^n} \subset \mathcal{S}(N)$ and shall still refer to these objects as coloured compositions. Below we collect some elementary facts about the Markov kernel [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"}. **Proposition 49**. *For any integer $p \geqslant 1$, real parameter $t \in \mathbb{R}_{\geqslant 0}$ and coloured composition $\mu \in \mathcal{S}_{N^n}$, we have $$\begin{aligned} \label{mu-meas} \mathbb{P}_{t,p}(\Delta \rightarrow \mu) = q^{-2\psi(\mu)+\binom{n}{2}\binom{N}{2}} \exp\left( -\frac{p(1-q^n)}{1-q} t \right) \mathbb{G}_{\mu}(1^p) \mathbb{G}_{\mu}({\rm Pl}_t).\end{aligned}$$* *Proof.* This is just the $\nu = \Delta$ case of [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"}, noting that $\mathbb{G}_{\Delta} = 1$ and $\psi(\Delta)=\frac{1}{2}\binom{n}{2}\binom{N}{2}$. ◻ **Proposition 50**. *For any two integers $p_1,p_2 \geqslant 1$ and real parameter $t \in \mathbb{R}_{\geqslant 0}$, the maps $\mathbb{P}_{t,p_1}$ and $\mathbb{P}_{t,p_2}$ compose according to the rule $$\begin{aligned} \label{compose} \mathbb{P}_{t,p_1} \circ \mathbb{P}_{t,p_2} = \mathbb{P}_{t,p_1+p_2}.\end{aligned}$$* *Proof.* For fixed $\lambda, \nu \in \mathcal{S}_{N^n}$ one computes $$\begin{aligned} & \sum_{\mu \in \mathcal{S}_{N^n}} \mathbb{P}_{t,p_1}(\nu \rightarrow \mu) \mathbb{P}_{t,p_2}(\mu \rightarrow \lambda) \\ \\ &= q^{-2(\psi(\lambda)-2\psi(\nu))} \exp\left( -\frac{(p_1+p_2)(1-q^n)}{1-q}t \right) \frac{\mathbb{G}_{\lambda}({\rm Pl}_t)}{\mathbb{G}_{\nu}({\rm Pl}_t)} \sum_{\mu \in \mathcal{S}_{N^n}} \mathbb{G}_{\mu/\nu}(1^{p_1}) \mathbb{G}_{\lambda/\mu}(1^{p_2}) \\ \\ &= q^{-2(\psi(\lambda)-2\psi(\nu))} \exp\left( -\frac{(p_1+p_2)(1-q^n)}{1-q}t \right) \frac{\mathbb{G}_{\lambda}({\rm Pl}_t)}{\mathbb{G}_{\nu}({\rm Pl}_t)} \mathbb{G}_{\lambda/\nu}(1^{p_1+p_2}) = \mathbb{P}_{t,p_1+p_2}(\nu \rightarrow \lambda),\end{aligned}$$ where we have used the branching rule for LLT polynomials (see [@ABW21 Remark 9.1.1]) to produce the second equality. ◻ In view of the property [\[compose\]](#compose){reference-type="eqref" reference="compose"}, we may view the Markov kernel $\mathbb{P}_{t,p}$ as the composition of $p$ kernels $\mathbb{P}_{t,1}$. Starting from the trivial state $\Delta \in \mathcal{S}_{N^n}$, we may either act directly with $\mathbb{P}_{t,p}$ to obtain a random coloured composition $\lambda^{[p]} = \mathbb{P}_{t,p}(\Delta)$, distributed according to [\[mu-meas\]](#mu-meas){reference-type="eqref" reference="mu-meas"}, or we may act $p$ times with $\mathbb{P}_{t,1}$, producing a chain of $p$ random coloured compositions $$\begin{aligned} \label{chain} \Delta \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[1]} \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[2]} \xrightarrow{\mathbb{P}_{t,1}} \cdots \xrightarrow{\mathbb{P}_{t,1}} \lambda^{[p]}.\end{aligned}$$ Our goal in the following section is to study the asymptotic behaviour of the distribution of the whole sequence $(\lambda^{[1]},\dots,\lambda^{[p]})$, as $t \rightarrow \infty$, with $p$ kept finite. # Asymptotics {#sec:asymp} In this section we carry out an asymptotic analysis of the Markov kernel [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"} with $p=1$, as $t \rightarrow \infty$; this analysis proceeds in several steps. We begin by rewriting coloured compositions in terms of a pair of vectors $\vec{\ell}$ and $\vec{c}$ in Section [6.1](#ssec:coord-not){reference-type="ref" reference="ssec:coord-not"}; $\vec{\ell}$ encodes the coordinates of the parts in a coloured composition, while $\vec{c}$ encodes the colour sequencing of its parts. In Section [6.2](#ssec:scaling){reference-type="ref" reference="ssec:scaling"}, we specify a particular time-dependent scaling of the coordinates associated to $\mu$ and $\nu$ within the function $\mathbb{P}_{t,1}$. We also impose certain interlacing constraints on the coordinates of $\mu$ and $\nu$; for finite $t$, these constraints prohibit certain coloured compositions on which the measure is non-zero, but it later transpires that as $t \rightarrow \infty$ these forbidden compositions naturally occur with vanishingly small probability, allowing us to omit them from our considerations. Having fixed our choice of scaling and our interlacing assumptions, we proceed to the analysis of the individual factors in the measure [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"}. Section [6.5](#ssec:analysis1){reference-type="ref" reference="ssec:analysis1"} deals with the factor $\mathbb{G}_{\mu/\nu}(1)$, whose analysis can be accessed by direct combinatorial means, while Section [6.6](#ssec:analysis2){reference-type="ref" reference="ssec:analysis2"} deals with the factors $\mathbb{G}_{\mu}({\rm Pl}_t)$ and $\mathbb{G}_{\nu}({\rm Pl_t})$, which are analysed by steepest descent method applied to the integral formula [\[LLTplanch\]](#LLTplanch){reference-type="eqref" reference="LLTplanch"}. Our final formula is presented in Section [6.7](#ssec:formula){reference-type="ref" reference="ssec:formula"}; we show that in the $t \rightarrow \infty$ limit being studied, the measure [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"} degenerates into the product of transition densities for $n$ independent GUE corners processes, multiplied by a discrete measure that is valued on colour sequences. ## Coordinate and colour sequence notation {#ssec:coord-not} **Definition 51**. To every coloured composition $\mu = \left(\mu^{(1)}_1 < \cdots < \mu^{(1)}_N | \cdots | \mu^{(n)}_1 < \cdots < \mu^{(n)}_N\right) \in \mathcal{S}_{N^n}$ we associate three vectors $\vec{\ell} = (\ell_1,\dots,\ell_{nN}) \in \mathbb{N}^{nN}$, $\vec{c} = (c_1,\dots,c_{nN}) \in [1,n]^{nN}$, $\vec{b} = (b_1,\dots,b_{nN}) \in [1,N]^{nN}$ satisfying the relation $$\begin{aligned} \ell_i = \mu^{(c_i)}_{b_i}, \qquad \forall\ 1 \leqslant i \leqslant nN,\end{aligned}$$ and satisfying the properties **(a)** $\ell_i \leqslant\ell_{i+1}$ for all $1 \leqslant i < nN$; **(b)** $c_i < c_{i+1}$ if $\ell_i = \ell_{i+1}$, for all $1 \leqslant i < nN$; **(c)** $b_i \not= b_j$ if $c_i = c_j$, for all $1 \leqslant i < j < nN$. More informally, $\vec{\ell}$ is the unique vector obtained by sorting the parts of $\mu$ in increasing order; we refer to it as the *coordinate vector* of $\mu$. The vector $\vec{c}$ records the colours of the parts of $\mu$ once it has been sorted in increasing order, with an increasing criterion imposed on these colours in the case of ties; we refer to it as the *colour sequence* of $\mu$. The vector $\vec{b}$ has been introduced only for the purpose of the making our definitions unambiguous, and plays no role in the rest of the paper. ## Starting assumptions and scaling {#ssec:scaling} Throughout the rest of this section we will be concerned with the analysis of the Markov kernel $\mathbb{P}_{t,1}\left(\nu \rightarrow \mu \right)$ as given by [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"}, with $\nu = 0\cup\lambda^{[m]}$ and $\mu = \lambda^{[m+1]}$, where we have chosen $\lambda^{[m]} \in \mathcal{S}^{+}_{m^n}$ and $\lambda^{[m+1]} \in \mathcal{S}^{+}_{(m+1)^n}$ (we remind the reader that the meaning of these notations is given by Definition [Definition 15](#defn:padding){reference-type="ref" reference="defn:padding"}). Under such choices, the kernel [\[skew-cauchy3\]](#skew-cauchy3){reference-type="eqref" reference="skew-cauchy3"} becomes $$\begin{aligned} \mathbb{P}_{t,1}\left(0 \cup \lambda^{[m]} \rightarrow \lambda^{[m+1]}\right) = q^{-2\left(\psi\left(\lambda^{[m+1]}\right)-\psi\left(0\cup\lambda^{[m]}\right)\right)} \exp\left( -\frac{1-q^n}{1-q}t \right) \mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1) \dfrac{\mathbb{G}_{\lambda^{[m+1]}}({\rm Pl}_t)} {\mathbb{G}_{0\cup\lambda^{[m]}}({\rm Pl}_t)}.\end{aligned}$$ Noting that $$\begin{aligned} \mathbb{G}_{0\cup\lambda^{[m]}}({\rm Pl}_t) \equiv \mathbb{G}_{0\cup\lambda^{[m]}/\Delta}({\rm Pl}_t) = \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t),\end{aligned}$$ where $\lambda^{[m]}-1$ means subtraction of $1$ from every part of $\lambda^{[m]}$, we have that $$\begin{aligned} \label{kernel} \mathbb{P}_{t,1}\left(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}\right) = q^{-2\left(\psi\left(\lambda^{[m+1]}\right)-\psi\left(0\cup\lambda^{[m]}\right)\right)} \exp\left( -\frac{1-q^n}{1-q}t \right) \mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1) \dfrac{\mathbb{G}_{\lambda^{[m+1]}}({\rm Pl}_t)} {\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)}.\end{aligned}$$ We shall make some assumptions concerning the coloured compositions $\lambda^{[m]} \in \mathcal{S}^{+}_{m^n}$ and $\lambda^{[m+1]} \in \mathcal{S}^{+}_{(m+1)^n}$ appearing within this formula. Following Definition [Definition 51](#def:coord){reference-type="ref" reference="def:coord"} we represent them in terms of their corresponding coordinate vectors and colour sequences: $$\begin{aligned} \label{identify} \lambda^{[m]} \leftrightarrow \left( \ell^{[m]}_1,\dots,\ell^{[m]}_{nm} \Big| c^{[m]}_1,\dots,c^{[m]}_{nm} \right), \qquad \lambda^{[m+1]} \leftrightarrow \left( \ell^{[m+1]}_1,\dots,\ell^{[m+1]}_{n(m+1)} \Big| c^{[m+1]}_1,\dots,c^{[m+1]}_{n(m+1)} \right),\end{aligned}$$ and we work directly with these vectors in what follows. Our first assumption is that the coordinates $\{\ell_i^{[m]}\}_{1 \leqslant i \leqslant nm}$ and $\{\ell_j^{[m+1]}\}_{1 \leqslant j \leqslant n(m+1)}$ are strictly increasing and obey the interlacing constraints $$\begin{aligned} \label{interlace-disc} \ell^{[m+1]}_{j(m+1)+i} < \ell^{[m]}_{jm+i} < \ell^{[m+1]}_{j(m+1)+i+1}, \qquad \forall\ i \in [1,m],\quad j \in [0,n-1].\end{aligned}$$ Informally, this means that the coordinates $\{\ell_i^{[m]}\}_{1 \leqslant i \leqslant nm}$ and $\{\ell_j^{[m+1]}\}_{1 \leqslant j \leqslant n(m+1)}$ are each grouped into $n$ bundles of equal size, and coordinates within those bundles interlace; see Figure [\[fig:gue\]](#fig:gue){reference-type="ref" reference="fig:gue"}. We will subsequently see that [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} depends on the coordinates $\{\ell_i^{[m]}\}_{1 \leqslant i \leqslant nm}$ and $\{\ell_j^{[m+1]}\}_{1 \leqslant j \leqslant n(m+1)}$ analytically. Our second assumption will be that these coordinates are analytically continued to real values, by setting $$\begin{aligned} \label{coord-scal} \ell^{[k]}_i \mapsto q^{n-\lceil i/k \rceil} t + (q^{n-\lceil i/k \rceil} t)^\frac{1}{2} x^{[k]}_i, \qquad 1 \leqslant i \leqslant nk, \qquad k \in \{m,m+1\},\end{aligned}$$ with $\lceil i/k \rceil$ denoting the ceiling function, and where $$\begin{aligned} \label{real-seq} \left( x^{[m]}_1 < \cdots < x^{[m]}_{nm} \right) \in \mathbb{R}^{nm}, \qquad \left( x^{[m+1]}_1 < \cdots < x^{[m+1]}_{n(m+1)} \right) \in \mathbb{R}^{n(m+1)}\end{aligned}$$ are sequences of reals that obey the interlacing constraints $$\begin{aligned} \label{interlace-real} x^{[m+1]}_{j(m+1)+i} < x^{[m]}_{jm+i} < x^{[m+1]}_{j(m+1)+i+1}, \qquad \forall\ i \in [1,m],\quad j \in [0,n-1].\end{aligned}$$ Note that [\[interlace-real\]](#interlace-real){reference-type="eqref" reference="interlace-real"} is simply the translation of the earlier interlacing constraint [\[interlace-disc\]](#interlace-disc){reference-type="eqref" reference="interlace-disc"} to the real variables that now parametrize our coordinates. We note that there exist choices of the coordinates $\{\ell_j^{[m+1]}\}_{1 \leqslant j \leqslant n(m+1)}$ which violate the constraints [\[interlace-disc\]](#interlace-disc){reference-type="eqref" reference="interlace-disc"} and yet have non-zero weight in the measure [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"}. We refer to such choices as *unfavourable coordinates*. Our main result will be to show that under the scaling [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}, unfavourable coordinates do not occur with probability converging to $1$ as $t \rightarrow \infty$. We do this by showing that as $t \rightarrow \infty$ the quantity [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} weakly converges to the product of a continuous transition density $\rho_{\rm GUE}\left(x^{[m]} \rightarrow x^{[m+1]}\right)$ valued on interlacing real sequences [\[real-seq\]](#real-seq){reference-type="eqref" reference="real-seq"} and a discrete transition probability $\mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right)$ valued on colour sequences [\[identify\]](#identify){reference-type="eqref" reference="identify"}. In demonstrating that the resulting quantity integrates to unity, we prove that [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"} captures the correct law of large numbers of the coordinates, with $\rho_{\rm GUE}\left(x^{[m]} \rightarrow x^{[m+1]}\right)$ providing the fluctuations. ## Main result {#ssec:main} **Definition 52** (GUE corners process). The Gaussian Unitary Ensemble (GUE) of rank $m$ is the collection of $m \times m$ Hermitian matrices $M = (M_{ij})_{i,j=1}^{m}$, where $M=(X+X^{*})/2$ and $X = (X_{ij})_{i,j=1}^{m}$ denotes an $m \times m$ matrix of i.i.d. complex Gaussian random variables $X_{ij} \sim \mathcal{N}(0,1) + {\tt i} \mathcal{N}(0,1)$. For all $1 \leqslant k \leqslant m$, write the eigenvalues of the $k \times k$ top-left sub-matrix of $M$ as $\theta_1^{[k]} \leqslant\cdots \leqslant\theta_k^{[k]}$. The joint law of the eigenvalues $\theta_i^{[j]}$, $1 \leqslant i \leqslant j$, $j \in [1,m]$ is called the *GUE corners process* of rank $m$. Following [@gorin-notes Theorem 20.1], one has the following explicit formula for the density of the GUE corners process: **Proposition 53**. *The array $\theta_i^{[j]}$, $1 \leqslant i \leqslant j$, $j \in [1,m]$ has joint density $$\begin{aligned} \rho\left(\theta_i^{[j]} = x_i^{[j]},1 \leqslant i \leqslant j \leqslant m\right) = \bm{1}_{x^{[1]} \prec \cdots \prec x^{[m]}} \left( \frac{1}{2\pi} \right)^{m/2} \prod_{1 \leqslant i<j \leqslant m} (x^{[m]}_j-x^{[m]}_i) \prod_{i=1}^{m} e^{-\frac{1}{2} \left( x^{[m]}_i \right)^2}\end{aligned}$$ with respect to the $m(m+1)/2$-dimensional Lebesgue measure.* Proposition [Proposition 53](#prop:gue-dens){reference-type="ref" reference="prop:gue-dens"} implies (see also [@gorin-notes Equation (20.2)]) the conditional probability density for the eigenvalues $\theta_i^{[m+1]}$, $1 \leqslant i \leqslant m+1$ of the $(m+1) \times (m+1)$ top-left sub-matrix, given those of the $m \times m$ one: $$\begin{gathered} \rho\left( \theta_i^{[m+1]} = x_i^{[m+1]}, 1 \leqslant i \leqslant m+1 \Big| \theta_i^{[m]} = x_i^{[m]}, 1 \leqslant i \leqslant m \right) \\ = \bm{1}_{x^{[m+1]} \succ x^{[m]}} \frac{1}{(2\pi)^{1/2}} \frac{\prod_{1 \leqslant i<j \leqslant m+1} \left(x^{[m+1]}_j-x^{[m+1]}_i\right) \cdot \prod_{i=1}^{m+1} e^{-\frac{1}{2} \left( x^{[m+1]}_i \right)^2}} {\prod_{1 \leqslant i<j \leqslant m} \left(x^{[m]}_j-x^{[m]}_i\right) \cdot \prod_{i=1}^{m} e^{-\frac{1}{2} \left( x^{[m]}_i \right)^2}},\end{gathered}$$ and for notational compactness, we shall write $$\begin{aligned} \rho_{\rm GUE}\left( x^{[1]}\prec \cdots \prec x^{[m]} \right) & := \rho\left(\theta_i^{[j]} = x_i^{[j]},1 \leqslant i \leqslant j \leqslant m\right), \\ \rho_{\rm GUE}\left( x^{[m]} \rightarrow x^{[m+1]} \right) & := \rho\left( \theta_i^{[m+1]} = x_i^{[m+1]}, 1 \leqslant i \leqslant m+1 \Big| \theta_i^{[m]} = x_i^{[m]}, 1 \leqslant i \leqslant m \right).\end{aligned}$$ **Theorem 54**. *In the asymptotic regime described by [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}, the Markov kernel [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} weakly converges to a product of $n$ independent probability measures with densities in the GUE corners process, multiplied by a factor that depends only on the colour sequences [\[identify\]](#identify){reference-type="eqref" reference="identify"}: $$\begin{gathered} \label{main-result} \mathbb{P}_{t,1}\left(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}\right) \\ \rightarrow \prod_{i=1}^{n} \rho_{\rm GUE}\left( x_{(i-1)m+1}^{[m]},\dots,x_{im}^{[m]} \rightarrow x_{(i-1)(m+1)+1}^{[m+1]},\dots,x_{i(m+1)}^{[m+1]} \right) dx^{[m+1]} \cdot \mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right)\end{gathered}$$ as $t \rightarrow \infty$, where $dx^{[m+1]}$ denotes the $n(m+1)$-dimensional Lebesgue measure. The final multiplicative factor in [\[main-result\]](#main-result){reference-type="eqref" reference="main-result"} is given explicitly by equation [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"} below, and defines a discrete transition probability in a process on colour sequences: $$\begin{aligned} \label{discrete-sum-to1} \sum_{c^{[m+1]}} \mathbb{P}_{\rm col}\left(c^{[m]} \rightarrow c^{[m+1]}\right) = 1,\end{aligned}$$ where the sum is taken over all $c^{[m+1]} = \left(c^{[m+1]}_1,\dots,c^{[m+1]}_{n(m+1)}\right) \in [1,n]^{n(m+1)}$.* **Corollary 55**. *Let $\mathbb{P}_{t,N} (\Delta \rightarrow \lambda^{[1]} \rightarrow \cdots \rightarrow \lambda^{[N]})$ denote the joint distribution of coloured compositions $\lambda^{[1]},\dots, \lambda^{[N]}$ generated by $N$ applications of the kernel [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} to the trivial state $\Delta$. In the asymptotic regime described by [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}, we have the following weak convergence of measures: $$\begin{gathered} \mathbb{P}_{t,N} \left(\Delta \rightarrow \lambda^{[1]} \rightarrow \cdots \rightarrow \lambda^{[N]}\right) \\ \rightarrow \prod_{i=1}^{n} \rho_{\rm GUE} \left( (x^{[1]})_i \prec (x^{[2]})_i \prec \cdots \prec (x^{[N]})_i \right) dx^{[1,N]} \cdot \mathbb{P}_{\rm col}\left(c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]}\right)\end{gathered}$$ as $t \rightarrow \infty$, with $dx^{[1,N]} = \prod_{i=1}^{N} dx^{[i]}$ denoting the $nN(N+1)/2$-dimensional Lebesgue measure. Here we have introduced the shorthand $$\begin{aligned} \left(x^{[k]}\right)_i = \left(x^{[k]}_{(i-1)k+1},\dots,x^{[k]}_{ik}\right), \qquad \forall\ 1 \leqslant i \leqslant n,\ \ 1 \leqslant k \leqslant N,\end{aligned}$$ and $\mathbb{P}_{\rm col}(c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]})$ is a joint distribution on colour sequences given explicitly by [\[joint-distr-col\]](#joint-distr-col){reference-type="eqref" reference="joint-distr-col"} below.* The remainder of the paper is devoted to the proof of this theorem. Throughout the rest of Section [6](#sec:asymp){reference-type="ref" reference="sec:asymp"}, we exhibit the splitting of the Markov kernel [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} as shown on the right hand side of [\[main-result\]](#main-result){reference-type="eqref" reference="main-result"}; the proof of the sum-to-unity property [\[discrete-sum-to1\]](#discrete-sum-to1){reference-type="eqref" reference="discrete-sum-to1"} is deferred to Section [7](#sec:discrete-dist){reference-type="ref" reference="sec:discrete-dist"}. ## Functions $\psi(\lambda^{[m+1]})$ and $\psi(0\cup\lambda^{[m]})$ We begin by studying the exponents $\psi(\lambda^{[m+1]})$ and $\psi(0\cup\lambda^{[m]})$ that appear within [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"}. Under the set of assumptions [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}--[\[real-seq\]](#real-seq){reference-type="eqref" reference="real-seq"}, the coordinates $\{\ell_i^{[m]}\}_{1 \leqslant i \leqslant nm}$ and $\{\ell_j^{[m+1]}\}_{1 \leqslant j \leqslant n(m+1)}$ are strictly increasing. This makes the computation of $\psi(\lambda^{[m+1]})$ and $\psi(0\cup\lambda^{[m]})$ particularly simple; one easily sees that $$\begin{aligned} \label{psi-inv} 2\psi\left(\lambda^{[m+1]}\right) = {\rm inv}\left(c^{[m+1]}\right), \qquad 2\psi\left(0\cup\lambda^{[m]}\right) = {\rm inv}\left((1,...,n)\cup c^{[m]}\right) = {\rm inv}\left(c^{[m]}\right)+m \binom{n}{2},\end{aligned}$$ where $(1,...,n)\cup c^{[m]}$ means concatenation of the two participating vectors. ## Factor $\mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1)$ {#ssec:analysis1} Next, we analyse the quantity $\mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1)$ within [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"}. It is given by the one-row partition function $$\begin{aligned} \label{G-one-row} \mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1) = \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,0) -- (5.5,0); \foreach\x in {1,...,5}{ \draw[lgray,line width=4pt,->] (\x,-0.5) -- (\x,0.5); } % \node[left] at (0.5,0) {\footnotesize$\bm{e}_0$};\node[right] at (5.5,0) {\footnotesize$\bm{e}_0$}; %\node[below] at (5,-0.5) {\fs $\cdots$};\node[above] at (5,0.5) {\fs $\cdots$}; %\node[below] at (4,-0.5) {\fs $\cdots$};\node[above] at (4,0.5) {\fs $\cdots$}; %\node[below] at (3,-0.5) {\fs $\cdots$};\node[above] at (3,0.5) {\fs $\cdots$}; %\node[below] at (2,-0.5) {\fs $\cdots$};\node[above] at (2,0.5) {\fs $\cdots$}; \node[below] at (1,-0.5) {\footnotesize$\bm{e}_{[1,n]}$};\node[above] at (1,0.5) {\footnotesize$\bm{e}_0$}; \node at (3.5, -0.7) {$\underbrace{\hspace{4cm}}$}; \node[below] at (3.5,-0.7) {$\lambda^{[m]}$}; \node at (3.5, 0.7) {$\overbrace{\hspace{4cm}}$}; \node[above] at (3.5,0.7) {$\lambda^{[m+1]}$}; \end{tikzpicture}\end{aligned}$$ where within the area marked $\lambda^{[k]}$, $k \in \{m,m+1\}$, the vector $\bm{e}_{c^{[k]}_i}$ is present at coordinate $\ell^{[k]}_i$ (in other words, a path of colour $c^{[k]}_i$ is present at position $\ell^{[k]}_i$), for all $1 \leqslant i \leqslant kn$. Following [\[llt-weights\]](#llt-weights){reference-type="eqref" reference="llt-weights"}, we have assumed the vertex weights $$\begin{aligned} \label{llt-wts} \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (-0.5,0) -- (0.5,0); \draw[lgray,line width=4pt,->] (0,-0.5) -- (0,0.5); % \node[left] at (-0.5,0) {\footnotesize$\bm{B}$}; \node[right] at (0.5,0) {\footnotesize$\bm{D}$}; \node[below] at (0,-0.5) {\footnotesize$\bm{A}$}; \node[above] at (0,0.5) {\footnotesize$\bm{C}$}; \end{tikzpicture} = \bm{1}_{\bm{C}+\bm{D} \in \{0,1\}^n} \cdot q^{\varphi(\bm{D},\bm{D})+\varphi(\bm{D},\bm{C})}, \qquad \bm{A},\bm{B},\bm{C},\bm{D}\in \{0,1\}^n,\end{aligned}$$ where the function $\varphi$ is as defined in [\[phi-def\]](#phi-def){reference-type="eqref" reference="phi-def"}. Let us study each of the factors appearing in [\[llt-wts\]](#llt-wts){reference-type="eqref" reference="llt-wts"} individually. First, we note that the indicator function $\bm{1}_{\bm{C}+\bm{D} \in \{0,1\}^n}$ prevents two paths of the same colour from traversing a vertex. Second, the factor $q^{\varphi(\bm{D},\bm{D})}$ assigns one power of $q$ for every pair of colours which pass through edge $\bm{D}$ of a vertex; there are $\binom{|\bm{D}|}{2}$ such pairs. Finally, the factor $q^{\varphi(\bm{D},\bm{C})}$ assigns one power of $q$ to each pair of colours $(i,j)$ passing through edges $(\bm{D},\bm{C})$, respectively, with $i<j$. Now we examine the contribution of each of the factors in the weights [\[llt-wts\]](#llt-wts){reference-type="eqref" reference="llt-wts"}, when they are multiplied together to form the one-row partition function [\[G-one-row\]](#G-one-row){reference-type="eqref" reference="G-one-row"}. Multiplying all indicator functions $\bm{1}_{\bm{C}+\bm{D} \in \{0,1\}^n}$ yields the the property that paths of the same colour do not intersect; at the level of the coloured compositions $\lambda^{[m]}$ and $\lambda^{[m+1]}$, this translates into the condition that $$\begin{aligned} \lambda^{[m+1](i)}_j < \lambda^{[m](i)}_j < \lambda^{[m+1](i)}_{j+1}, \qquad\forall\ 1 \leqslant i \leqslant n, \quad 1 \leqslant j \leqslant m,\end{aligned}$$ which we denote simply by writing $c^{[m]} \prec c^{[m+1]}$. Multiplying all factors $q^{\varphi(\bm{D},\bm{D})}$ requires us to compute the total number of paths $d_i$ going through the $i$-th horizontal edge of the partition function [\[G-one-row\]](#G-one-row){reference-type="eqref" reference="G-one-row"}, for all $i \geqslant 1$. The total contribution from these factors is then $$\begin{aligned} \prod_{i \geqslant 1} q^{\binom{d_i}{2}} = \prod_{j=1}^{n} q^{\binom{j}{2} p_j},\end{aligned}$$ where $p_j$ counts the number of horizontal edges in [\[G-one-row\]](#G-one-row){reference-type="eqref" reference="G-one-row"} that are occupied by $j$ paths. It is clear that the set $\{p_j\}_{1 \leqslant j \leqslant n}$ depends only on the values of the coordinates $\{\ell^{[m]}_i\}_{1 \leqslant i \leqslant nm}$ and $\{\ell^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$, and not on the colour sequences $\{c^{[m]}_i\}_{1 \leqslant i \leqslant nm}$ and $\{c^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$. In view of the interlacing [\[interlace-disc\]](#interlace-disc){reference-type="eqref" reference="interlace-disc"} of the coordinates $\{\ell^{[m]}_i\}_{1 \leqslant i \leqslant nm}$, $\{\ell^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$, we may routinely compute $p_j$ for all $1 \leqslant j \leqslant n$. We find that, *cf.* Figure [\[fig:gue\]](#fig:gue){reference-type="ref" reference="fig:gue"}, $$\begin{aligned} p_n &= \sum_{k=1}^{m+1} \ell^{[m+1]}_k - \sum_{k=1}^{m} \ell^{[m]}_k, \\ p_{n-j} &= \sum_{k=1}^{m+1} \left( \ell^{[m+1]}_{j(m+1)+k} - \ell^{[m+1]}_{(j-1)(m+1)+k} \right) - \sum_{k=1}^{m} \left( \ell^{[m]}_{jm+k} - \ell^{[m]}_{(j-1)m+k} \right), \qquad\forall\ j \in [1,n-1].\end{aligned}$$ We then have $$\begin{aligned} \sum_{j=1}^{n} \binom{j}{2} p_j &= \sum_{j=1}^{n} \left[ \binom{n-j+1}{2} - \binom{n-j}{2} \right] \sum_{k=1}^{m+1} \ell^{[m+1]}_{(j-1)(m+1)+k} - \sum_{j=1}^{n} \left[ \binom{n-j+1}{2} - \binom{n-j}{2} \right] \sum_{k=1}^{m} \ell^{[m]}_{(j-1)m+k} \\ &= \sum_{j=1}^{n} (n-j) \sum_{k=1}^{m+1} \ell^{[m+1]}_{(j-1)(m+1)+k} - \sum_{j=1}^{n} (n-j) \sum_{k=1}^{m} \ell^{[m]}_{(j-1)m+k}\end{aligned}$$ as the total exponent of $q$ coming from factors of the form $q^{\varphi(\bm{D},\bm{D})}$. Finally we need to examine the contribution from all factors $q^{\varphi(\bm{D},\bm{C})}$, when multiplying the weights of all vertices in the row [\[G-one-row\]](#G-one-row){reference-type="eqref" reference="G-one-row"}. In direct contrast to the factors $q^{\varphi(\bm{D},\bm{D})}$, this contribution only depends on the colour sequences $\{c^{[m]}_i\}_{1 \leqslant i \leqslant nm}$ and $\{c^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$, and not on the coordinates $\{\ell^{[m]}_i\}_{1 \leqslant i \leqslant nm}$ and $\{\ell^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$. Rather than attempting to write down an explicit formula for this contribution, we express it in terms of the following diagram[^17]: $$\begin{gathered} \label{ups} \Upsilon\left(c^{[m]};c^{[m+1]}\right) = \\ \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,0) -- (12,0); \foreach\x in {1,2,3,5,6,7,9,10,11}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \foreach\x in {1.5,2.5,5.5,6.5,9.5,10.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[left] at (0.5,0) {$[1,n]$}; \node[right] at (12,0) {$\emptyset$}; \node[below] at (1.5,-0.5) {$c^{[m]}_1$}; \node[above] at (1,0.5) {$c^{[m+1]}_1$}; \node[below] at (2.5,-0.5) {$c^{[m]}_m$}; \node[above] at (3,0.5) {$c^{[m+1]}_{m+1}$}; \node[below] at (2,-0.7) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; % \node[below] at (6,-0.7) {$\cdots$}; \node[above] at (6,0.5) {$\cdots$}; % \node[below] at (9.5,-0.5) {$c^{[m]}_{(n-1)m+1}$}; \node[above] at (9,0.5) {$c^{[m+1]}_{(n-1)(m+1)+1}$}; \node[below] at (10.5,-0.5) {$c^{[m]}_{nm}$}; \node[above] at (11,0.5) {$c^{[m+1]}_{n(m+1)}$}; \node[below] at (10,-0.6) {$\cdots$}; \node[above] at (10.2,0.5) {$\cdots$}; \end{tikzpicture}\end{gathered}$$ In this diagram, one of each of the colours $\{1,\dots,n\}$ enters via the leftmost horizontal edge; there is also a single colour $c^{[m]}_i$ entering via the $i$-th vertical edge along the base. No colours exit via the rightmost horizontal edge; a single colour $c^{[m+1]}_i$ exits via the $i$-th vertical edge along the top. Note that the diagram is also arranged to denote $n$ individual bundles of interlacing colours, with flow of colours possible along the horizontal line that connects bundles. The function $\Upsilon\left(c^{[m]};c^{[m+1]}\right)$ is a pure power of $q$. We compute it by tracing all colours in the diagram [\[ups\]](#ups){reference-type="eqref" reference="ups"} from their starting to finishing location, and assigning a power of $q$ to each of the following events: $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,0) -- (1.5,0); \draw[lgray,line width=1.5pt,->] (1,0) -- (1,0.5); \node[above] at (1,0.5) {$c$}; \node[right] at (1.5,0) {$i$}; \end{tikzpicture}\end{aligned}$$ which denotes a path of colour $i$ passing underneath a colour $c$, with $i<c$. Bringing all contributions together, we have $$\begin{aligned} \label{one-row-formula} \mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1) = \bm{1}_{c^{[m]} \prec c^{[m+1]}} \cdot \Upsilon\left(c^{[m]};c^{[m+1]}\right) \cdot \prod_{j=1}^{n} \left( \dfrac{\displaystyle{\prod_{k=1}^{m+1}} q^{\ell^{[m+1]}_{(j-1)(m+1)+k}}} {\displaystyle{\prod_{k=1}^{m}} q^{\ell^{[m]}_{(j-1)m+k}}} \right)^{n-j}\end{aligned}$$ We draw attention to the fact that dependence on the colour sequences $\{c^{[m]}_i\}_{1 \leqslant i \leqslant nm}$, $\{c^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$ and coordinates $\{\ell^{[m]}_i\}_{1 \leqslant i \leqslant nm}$, $\{\ell^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$ is completely separated in [\[one-row-formula\]](#one-row-formula){reference-type="eqref" reference="one-row-formula"}; further, $\mathbb{G}_{\lambda^{[m+1]}/0\cup\lambda^{[m]}}(1)$ is an analytic function of the coordinates. ## Factors $\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)$ and $\mathbb{G}_{\lambda^{[m+1]}}({\rm Pl}_t)$ {#ssec:analysis2} We now proceed to the analysis of the factors $\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)$ and $\mathbb{G}_{\lambda^{[m+1]}}({\rm Pl}_t)$, in the denominator and numerator of [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"}, under the replacements [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}. This computation is essentially the same for each of these factors, up to some straightforward relabelling of parameters, so we will focus on the analysis of $\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)$ first. ### Recasting the integrand Our starting point is [\[LLTplanch\]](#LLTplanch){reference-type="eqref" reference="LLTplanch"} with $\nu = \Delta \in \mathcal{S}_{m^n}$ and $\mu = \lambda^{[m]}-1 \in \mathcal{S}_{m^n}$: $$\begin{gathered} \label{LLTplanch2} \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t) = \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_{nm}} \frac{dy_{nm}}{y_{nm}} \\ \times \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\breve\lambda^{[m]}-1}(1^{nm};y_1^{-1},\dots,y_{nm}^{-1}) g_{\Delta}(m^n;y_1,\dots,y_{nm}) \prod_{j=1}^{nm} e^{t y_j}.\end{gathered}$$ In the first step, we use the action [\[T-f\]](#T-f){reference-type="eqref" reference="T-f"} of the Hecke generators to express the function $f_{\breve\lambda^{[m]}-1}$ in terms of $f_{\ell^{[m]}-1}$, noting that $\ell^{[m]}$ is just obtained by sorting the parts of $\breve\lambda^{[m]}$ in increasing order. We have $$\begin{gathered} \label{LLTplanch3} \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t) = \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_{nm}} \frac{dy_{nm}}{y_{nm}} \\ \times \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) (T_{\sigma} \cdot f_{\ell^{[m]}-1})(1^{nm};y_1^{-1},\dots,y_{nm}^{-1}) g_{\Delta}(m^n;y_1,\dots,y_{nm}) \prod_{j=1}^{nm} e^{t y_j},\end{gathered}$$ where we have denoted $T_{\sigma} = T_{a_1} \dots T_{a_p}$ with $T_a$ given by [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"}, and where $(a_1,\dots,a_p) \in [1,nm)^p$ is a minimal-length word such that $$\begin{aligned} \label{reorder} \mathfrak{s}_{a_1} \cdots \mathfrak{s}_{a_p} \cdot \ell^{[m]} = \breve\lambda^{[m]}.\end{aligned}$$ Using the property [\[adjoint\]](#adjoint){reference-type="eqref" reference="adjoint"} of Hecke generators, and the fact that the product $\prod_{j=1}^{nm} e^{t y_j}$ is symmetric with respect to $(y_1,\dots,y_{nm})$, we may recast this as $$\begin{gathered} \label{LLTplanch4} \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t) = \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_{nm}} \frac{dy_{nm}}{y_{nm}} \\ \times \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) f_{\ell^{[m]}-1}(1^{nm};y_1^{-1},\dots,y_{nm}^{-1}) (\tilde{T}_{\sigma} \cdot g_{\Delta})(m^n;y_1,\dots,y_{nm}) \prod_{j=1}^{nm} e^{t y_j},\end{gathered}$$ where we have denoted $\tilde{T}_{\sigma} = \tilde{T}_{a_p} \dots \tilde{T}_{a_1}$ with $\tilde{T}_a$ given by [\[hecke-tilde\]](#hecke-tilde){reference-type="eqref" reference="hecke-tilde"}, and where the word $(a_1,\dots,a_p)$ is specified as in [\[reorder\]](#reorder){reference-type="eqref" reference="reorder"}. Finally, we note that if the coordinates $\ell^{[m]}$ may be reordered to yield $\breve{\lambda}^{[m]}$ as in [\[reorder\]](#reorder){reference-type="eqref" reference="reorder"}, it also follows that the corresponding colour sequence $c^{[m]}$ reorders according to the rule $$\begin{aligned} \mathfrak{s}_{a_1} \cdots \mathfrak{s}_{a_p} \cdot c^{[m]} = (1^m,2^m,\dots,n^m),\end{aligned}$$ or equivalently, $\mathfrak{s}_{a_p} \cdots \mathfrak{s}_{a_1} \cdot (1^m,2^m,\dots,n^m) = c^{[m]}$. Using this relation in [\[LLTplanch4\]](#LLTplanch4){reference-type="eqref" reference="LLTplanch4"}, together with the action [\[invT-g\]](#invT-g){reference-type="eqref" reference="invT-g"} of Hecke generators on the function $g_{\Delta}(m^n;y_1,\dots,y_{nm})$, we recover the formula $$\begin{gathered} \label{LLTplanch5} \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t) = \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \oint_{C_1} \frac{dy_1}{y_1} \cdots \oint_{C_{nm}} \frac{dy_{nm}}{y_{nm}} \\ \times \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) \prod_{i=1}^{nm} y_i^{-\ell^{[m]}_i+1} g_{\Delta}^{c^{[m]}}(m^n;y_1,\dots,y_{nm}) \prod_{j=1}^{nm} e^{ty_j},\end{gathered}$$ where $g_{\Delta}^{c^{[m]}}(m^n;y_1,\dots,y_{nm})$ denotes a permuted-boundary function of the form [\[generic-g-sigma\]](#generic-g-sigma){reference-type="eqref" reference="generic-g-sigma"}, and where we have also used the fact that $f_{\ell^{[m]}-1}(1^{nm};y_1^{-1},\dots,y_{nm}^{-1})$ factorizes as in [\[factorize\]](#factorize){reference-type="eqref" reference="factorize"} with $s=0$, since $\ell^{[m]}$ is increasing. The formula [\[LLTplanch5\]](#LLTplanch5){reference-type="eqref" reference="LLTplanch5"} explicitly separates the coordinates $\ell^{[m]}$ and the colour sequence $c^{[m]}$; in view of the analytic dependence on the former, we now use it to carry out steepest descent asymptotics. ### $t \rightarrow \infty$ asymptotics via steepest descent In this section we compute the $t \rightarrow \infty$ asymptotics of the quantities $$\begin{aligned} \label{H-funct} \tilde{H}_{\lambda^{[m]}}(t) := \prod_{j=1}^{n} \prod_{k=1}^{m} q^{(n-j)\ell^{[m]}_{(j-1)m+k}} \cdot \mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t), \\ \label{H-funct2} H_{\lambda^{[m+1]}}(t) := \prod_{j=1}^{n} \prod_{k=1}^{m+1} q^{(n-j)\ell^{[m+1]}_{(j-1)(m+1)+k}} \cdot \mathbb{G}_{\lambda^{[m+1]}}({\rm Pl}_t),\end{aligned}$$ under the assumption that the coordinates $\{\ell^{[m]}_i\}_{1 \leqslant i \leqslant nm}$ and $\{\ell^{[m+1]}_i\}_{1 \leqslant i \leqslant n(m+1)}$ scale as [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"}. Note that, by virtue of [\[psi-inv\]](#psi-inv){reference-type="eqref" reference="psi-inv"} and the expression [\[one-row-formula\]](#one-row-formula){reference-type="eqref" reference="one-row-formula"}, the Markov kernel [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} may be expressed as $$\begin{gathered} \label{kernel-reexpress} \mathbb{P}_{t,1}\left(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}\right) \\ = \bm{1}_{c^{[m]} \prec c^{[m+1]}} \cdot q^{{\rm inv}(c^{[m]})-{\rm inv}(c^{[m+1]})+m \binom{n}{2}} \Upsilon\left( c^{[m]}; c^{[m+1]} \right) \dfrac{H_{\lambda^{[m+1]}}(t)}{\tilde{H}_{\lambda^{[m]}}(t)} \exp\left( -\frac{1-q^n}{1-q}t \right)\end{gathered}$$ where the colour sequences $c^{[m]}$, $c^{[m+1]}$ are independent of $t$; this means that the $t \rightarrow \infty$ asymptotics of our Markov kernel is indeed recovered by analysis of [\[H-funct\]](#H-funct){reference-type="eqref" reference="H-funct"} and [\[H-funct2\]](#H-funct2){reference-type="eqref" reference="H-funct2"}. We lighten our notation by writing $\ell^{[m]}_i \equiv \ell_i$ and $x^{[m]}_i \equiv x_i$ for all $1 \leqslant i \leqslant nm$. Distributing the $q$-dependent prefactor in [\[H-funct\]](#H-funct){reference-type="eqref" reference="H-funct"} within the integral [\[LLTplanch5\]](#LLTplanch5){reference-type="eqref" reference="LLTplanch5"}, we have $$\begin{gathered} \label{LLTplanch6} \tilde{H}_{\lambda^{[m]}}(t) = \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \oint_{C_1} dy_1 %\frac{dy_1}{y_1} \cdots \oint_{C_{nm}} dy_{nm} %\frac{dy_{nm}}{y_{nm}} \\ \times \prod_{i=1}^{nm} \left(\frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) g_{\Delta}^{c^{[m]}}(m^n;y_1,\dots,y_{nm}),\end{gathered}$$ where we have defined the vector $\vec{Q} \in \mathbb{C}^{nm}$ by $$\begin{aligned} \label{vecQ} \vec{Q} = (Q_1,\dots,Q_{nm}) = \underbrace{(q^{n-1},\dots,q^{n-1})}_{m\ {\rm times}} \cup \cdots \cup \underbrace{(q,\dots,q)}_{m\ {\rm times}} \cup \underbrace{(1,\dots,1)}_{m\ {\rm times}}.\end{aligned}$$ Using the formula [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"} and the notation [\[vecQ\]](#vecQ){reference-type="eqref" reference="vecQ"}, the coordinates $\ell_i$ are written as $\ell_i = Q_i t + (Q_i t)^{\frac{1}{2}} x_i$ for all $1 \leqslant i \leqslant nm$. Making use of this, the univariate factors in the integrand of [\[LLTplanch6\]](#LLTplanch6){reference-type="eqref" reference="LLTplanch6"} read $$\begin{aligned} \label{univar} \left( \frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} = \exp\left[t y_i-\ell_i \log y_i + \ell_i \log Q_i \right] = \exp\left[t(y_i-Q_i \log y_i + Q_i \log Q_i) + O(t^{1/2}) \right], \quad \text{as}\ \ t \rightarrow \infty.\end{aligned}$$ The $t \rightarrow \infty$ behaviour of [\[LLTplanch6\]](#LLTplanch6){reference-type="eqref" reference="LLTplanch6"} may now be recovered from steepest descent analysis applied to each of the $nm$ integrals. Neglecting for the moment the $O(t^{1/2})$ term above (which gives a sub-leading contribution to the $t \rightarrow \infty$ behaviour), we evaluate the critical point[^18] of the function $y_i-Q_i \log y_i + Q_i \log Q_i$, which is found to be $y_i = Q_i$. Computing the corresponding Taylor series about this point, we have that $$\begin{aligned} \left( \frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} = \exp\left[t \left( Q_i + \frac{(y_i-Q_i)^2}{2Q_i} + O(y_i-Q_i)^3 \right) + O(t^{1/2}) \right], \quad \text{as}\ \ t \rightarrow \infty,\end{aligned}$$ in a neighbourhood of the point $y_i=Q_i$. Following standard steepest descent analysis, the dominant contribution to the $t \rightarrow \infty$ asymptotics of the integral [\[LLTplanch6\]](#LLTplanch6){reference-type="eqref" reference="LLTplanch6"} is obtained by deforming each contour $C_i$ to pass through $Q_i$[^19], and reducing the resulting contour integrals to line integrals over small segments $D_i \subset C_i$ travelling through $Q_i$ and traversed in the direction where the function $\frac{(y_i-Q_i)^2}{2Q_i}$ has zero imaginary part. Accordingly[^20], we may write $$\begin{gathered} \label{LLTplanch7} \tilde{H}_{\lambda^{[m]}}(t) \sim \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \int_{Q_1-{\tt i}\epsilon}^{Q_1+{\tt i}\epsilon} dy_1 \cdots \int_{Q_{nm}-{\tt i}\epsilon}^{Q_{nm}+{\tt i}\epsilon} dy_{nm} \\ \prod_{i=1}^{nm} \left(\frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) g_{\Delta}^{c^{[m]}}(m^n;y_1,\dots,y_{nm}), \quad \text{as}\ \ t \rightarrow \infty,\end{gathered}$$ where $\epsilon$ is a small positive real number. Now switching to the local variables $y_i = Q_i+z_i t^{-\frac{1}{2}}$, the univariate factors in [\[LLTplanch7\]](#LLTplanch7){reference-type="eqref" reference="LLTplanch7"} become $$\begin{aligned} \left. \left( \frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} \right|_{y_i = Q_i+z_i t^{-\frac{1}{2}}} = \exp\left[ t(Q_i+z_i t^{-\frac{1}{2}}) - (Q_i t + (Q_i t)^{\frac{1}{2}} x_i) \log(Q_i+z_i t^{-\frac{1}{2}}) + (Q_i t + (Q_i t)^{\frac{1}{2}} x_i) \log Q_i \right],\end{aligned}$$ and using the fact that $$\begin{aligned} \log(Q+\epsilon) = \log(Q) + \frac{\epsilon}{Q} - \frac{1}{2}\frac{\epsilon^2}{Q^2} + O(\epsilon^3), \quad \text{as} \ \ \epsilon \rightarrow 0,\end{aligned}$$ we have $$\begin{aligned} & \left. \left( \frac{Q_i}{y_i} \right)^{\ell_i} e^{ty_i} \right|_{y_i = Q_i+z_i t^{-\frac{1}{2}}} \\ &= \exp\left[ t(Q_i+z_i t^{-\frac{1}{2}}) - (Q_i t + (Q_i t)^{\frac{1}{2}} x_i) \left(\log Q_i + \frac{z_i t^{-\frac{1}{2}}}{Q_i} - \frac{1}{2} \frac{z_i^2 t^{-1}}{Q_i^2} + O(t^{-\frac{3}{2}})\right) + (Q_i t + (Q_i t)^{\frac{1}{2}} x_i) \log Q_i \right] \\ &= \exp\left[ Q_i t - \frac{x_i z_i}{Q_i^{\frac{1}{2}}} + \frac{z_i^2}{2Q_i} + O(t^{-\frac{1}{2}}) \right], \quad \text{as} \ \ t \rightarrow \infty.\end{aligned}$$ Turning to other terms in the integral [\[LLTplanch7\]](#LLTplanch7){reference-type="eqref" reference="LLTplanch7"}, we have $$\begin{aligned} \label{q-vand} \left. \frac{y_j-y_i}{y_j-q y_i} \right|_{\vec{y} = \vec{Q}+\vec{z} t^{-\frac{1}{2}}} = \frac{Q_j-Q_i+z_j t^{-1/2}-z_i t^{-1/2}}{Q_j-qQ_i+z_j t^{-1/2} -qz_i t^{-1/2}}, \qquad 1 \leqslant i< j \leqslant nm.\end{aligned}$$ Using the fact that $Q_i = q^{n-\lceil i/m \rceil}$, we see that $Q_j - q Q_i$ in the denominator of [\[q-vand\]](#q-vand){reference-type="eqref" reference="q-vand"} is always nonzero; on the other hand, $Q_j - Q_i$ in the numerator vanishes whenever $\lceil i/m \rceil = \lceil j/m \rceil$ (and is nonzero otherwise). The $t \rightarrow \infty$ behaviour of [\[q-vand\]](#q-vand){reference-type="eqref" reference="q-vand"} then splits into two cases: $$\begin{aligned} \left. \frac{y_j-y_i}{y_j-q y_i} \right|_{\vec{y} = \vec{Q}+\vec{z} t^{-\frac{1}{2}}} \sim \frac{1}{q^{n-\lceil j/m \rceil} - q^{n-\lceil i/m \rceil +1}} \times \left\{ \begin{array}{ll} t^{-1/2} (z_j-z_i), \qquad & \lceil i/m \rceil = \lceil j/m \rceil, \\ \\ q^{n-\lceil j/m \rceil} - q^{n-\lceil i/m \rceil}, \qquad & \lceil i/m \rceil \not= \lceil j/m \rceil, \end{array} \right.\end{aligned}$$ and multiplying these factors over all indices $1 \leqslant i<j \leqslant nm$, we have $$\begin{gathered} \left. \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) \right|_{\vec{y} = \vec{Q}+\vec{z} t^{-\frac{1}{2}}} \sim t^{-\frac{n}{2} \binom{m}{2}} (1-q)^{-n \binom{m}{2}} \prod_{j=1}^{n} q^{(j-n) \binom{m}{2}} \\ \times \prod_{i=0}^{n-1} \prod_{1 \leqslant j<k \leqslant m} (z_{im+k}-z_{im+j}) \prod_{1 \leqslant i<j \leqslant n} \left( \frac{q^{n-j}-q^{n-i}}{q^{n-j}-q^{n-i+1}} \right)^{m^2}.\end{gathered}$$ After accounting for telescopic cancellations in the final product of this expression, it may be written as $$\begin{aligned} \prod_{1 \leqslant i<j \leqslant n} \left( \frac{q^{n-j}-q^{n-i}}{q^{n-j}-q^{n-i+1}} \right)^{m^2} = \left[ \frac{(1-q)^{n}}{(q;q)_n} \right]^{m^2}\end{aligned}$$ and accordingly we have that $$\begin{gathered} \left. \prod_{1 \leqslant i<j \leqslant nm} \left( \frac{y_j-y_i}{y_j-q y_i} \right) \right|_{\vec{y} = \vec{Q}+\vec{z} t^{-\frac{1}{2}}} \sim t^{-\frac{n}{2} \binom{m}{2}} (1-q)^{-n \binom{m}{2}} q^{-\binom{n}{2} \binom{m}{2}} \left[ \frac{(1-q)^{n}}{(q;q)_n} \right]^{m^2} \prod_{i=0}^{n-1} \prod_{1 \leqslant j<k \leqslant m} (z_{im+k}-z_{im+j}),\end{gathered}$$ as $t \rightarrow \infty$. Under the change to local variables, the remaining piece of the integral [\[LLTplanch7\]](#LLTplanch7){reference-type="eqref" reference="LLTplanch7"} (which is polynomial in the variables $y_1,\dots,y_{nm}$) becomes $$\begin{aligned} \left. g_{\Delta}^{c^{[m]}}(m^n;y_1,\dots,y_{nm}) \right|_{\vec{y} = \vec{Q}+\vec{z} t^{-\frac{1}{2}}} \sim g_{\Delta}^{c^{[m]}}(m^n;\vec{Q}), \qquad \text{as}\ \ t\rightarrow \infty.\end{aligned}$$ Combining everything (including a factor of $t^{-nm/2}$ for the change of integration variables), we read off the $t \rightarrow \infty$ asymptotic behaviour of $\tilde{H}_{\lambda^{[m]}}(t)$: $$\begin{gathered} \label{LLTplanch8} \tilde{H}_{\lambda^{[m]}}(t) \sim t^{-nm/2} \exp\left[ \frac{1-q^n}{1-q} mt \right] t^{-\frac{n}{2} \binom{m}{2}} (1-q)^{-n \binom{m}{2}} q^{-\binom{n}{2} \binom{m}{2}} \left[ \frac{(1-q)^{n}}{(q;q)_n} \right]^{m^2} g_{\Delta}^{c^{[m]}}(m^n;\vec{Q}) \\ \times \frac{q^{nm(nm+1)/2}}{(q-1)^{nm}} \cdot \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \int_{-{\tt i}\infty}^{{\tt i}\infty} dz_1 \cdots \int_{-{\tt i}\infty}^{{\tt i}\infty} dz_{nm} \prod_{i=1}^{nm} \exp\left[- \frac{x_i z_i}{Q_i^{1/2}} + \frac{z_i^2}{2 Q_i} \right] \prod_{i=0}^{n-1} \prod_{1 \leqslant j<k \leqslant m} (z_{im+k}-z_{im+j}).\end{gathered}$$ This simplifies to yield $$\begin{gathered} \label{LLTplanch9} \tilde{H}_{\lambda^{[m]}}(t) \sim (-1)^{nm} t^{-\frac{n}{2}\binom{m+1}{2}} q^{\binom{nm+1}{2}-\frac{1}{2}\binom{m+1}{2} \binom{n}{2}+m\binom{n}{2}} \exp\left[ \frac{1-q^n}{1-q} mt \right] \frac{(1-q)^{n\binom{m}{2}}}{(q;q)_n^{m^2}} g_{\Delta}^{c^{[m]}}(m^n;\vec{Q}) \\ \times \left( \frac{1}{2\pi{\tt i}} \right)^{nm} \int_{-{\tt i}\infty}^{{\tt i}\infty} dz_1 \cdots \int_{-{\tt i}\infty}^{{\tt i}\infty} dz_{nm} \prod_{i=1}^{nm} \exp\left[-x_i z_i + \frac{z_i^2}{2} \right] \prod_{i=0}^{n-1} \prod_{1 \leqslant j<k \leqslant m} (z_{im+k}-z_{im+j}),\end{gathered}$$ where we have rescaled the integration variables $z_i \mapsto Q_i^{1/2} z_i$ to obtain the final formula. Repeating these steps, one easily finds that $$\begin{aligned} \label{m-to-m+1} H_{\lambda^{[m+1]}}(t) = \left( q^{-m \binom{n}{2}} \tilde{H}_{\lambda^{[m]}}(t) \right)_{m \mapsto m+1},\end{aligned}$$ which is to be interpreted as taking the right hand side of [\[LLTplanch9\]](#LLTplanch9){reference-type="eqref" reference="LLTplanch9"}, modulo division by $q^{m \binom{n}{2}}$, and replacing all instances of $m$ by $m+1$ in the obvious way. The removal of the factor $q^{m \binom{n}{2}}$ requires the following justification. From [\[H-funct\]](#H-funct){reference-type="eqref" reference="H-funct"} and [\[H-funct2\]](#H-funct2){reference-type="eqref" reference="H-funct2"}, we see that $\tilde{H}_{\lambda^{[m]}}(t)$ depends on the function $\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)$ rather than $\mathbb{G}_{\lambda^{[m]}}({\rm Pl}_t)$, the latter being the desired quantity that leads to $H_{\lambda^{[m+1]}}(t)$ after the $m \mapsto m+1$ relabelling. Consulting the integral formula [\[LLTplanch5\]](#LLTplanch5){reference-type="eqref" reference="LLTplanch5"}, we see that the only difference between $\mathbb{G}_{\lambda^{[m]}-1}({\rm Pl}_t)$ and $\mathbb{G}_{\lambda^{[m]}}({\rm Pl}_t)$ is that the integrand used for the former contains an extra factor of $\prod_{i=1}^{nm} y_i$ compared with that of the latter. Carrying through steepest descent analysis of $\mathbb{G}_{\lambda^{[m]}}({\rm Pl}_t)$ therefore results in an overall factor $\prod_{i=1}^{nm} Q_i = q^{m \binom{n}{2}}$ less compared with the calculations above, which is the reason that we divide out this factor in [\[m-to-m+1\]](#m-to-m+1){reference-type="eqref" reference="m-to-m+1"}. ### Factorization into GUE corners Up to the multiplicative terms in the first line, equation [\[LLTplanch9\]](#LLTplanch9){reference-type="eqref" reference="LLTplanch9"} reveals the factorization of our starting integral [\[LLTplanch6\]](#LLTplanch6){reference-type="eqref" reference="LLTplanch6"} into $n$ identical $m$-dimensional integrals of the form $$\begin{aligned} \label{I-int} I(x_1,\dots,x_m) &= \left(\frac{1}{2\pi{\tt i}}\right)^{m} \int_{{\tt i} \cdot \mathbb{R}} dz_1 \cdots \int_{{\tt i} \cdot \mathbb{R}} dz_m \prod_{1 \leqslant i < j \leqslant m} (z_j-z_i) \prod_{i=1}^{m} e^{-x_i z_i + \frac{1}{2} z_i^2}, \\ \nonumber &= \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \left(\frac{1}{2\pi{\tt i}}\right)^{m} \int_{{\tt i} \cdot \mathbb{R}} dz_1 \cdots \int_{{\tt i} \cdot \mathbb{R}} dz_m \prod_{1 \leqslant i < j \leqslant m} (z_j-z_i) \prod_{i=1}^{m} e^{\frac{1}{2}(z_i-x_i)^2}.\end{aligned}$$ It is possible to explicitly evaluate the integral [\[I-int\]](#I-int){reference-type="eqref" reference="I-int"}, as we now show. Replacing the Vandermonde factor in [\[I-int\]](#I-int){reference-type="eqref" reference="I-int"} by its determinant form and using the multilinearity of the determinant, we recover $$\begin{aligned} \label{I-int2} I(x_1,\dots,x_m) &= \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \left(\frac{1}{2\pi {\tt i}}\right)^{m} \int_{{\tt i} \cdot \mathbb{R}} dz_1 \cdots \int_{{\tt i} \cdot \mathbb{R}} dz_m \det_{1 \leqslant i,j \leqslant m}(z_i^{j-1}) \prod_{i=1}^{m} e^{\frac{1}{2}(z_i-x_i)^2}, \\ \nonumber &= \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \det_{1 \leqslant i,j \leqslant m} \left( \frac{1}{2\pi {\tt i}} \int_{{\tt i} \cdot \mathbb{R}} z^{j-1} e^{\frac{1}{2}(z-x_i)^2} dz \right).\end{aligned}$$ Making the change of integration variables ${\tt i}u = z - x_i$ within the second line of [\[I-int2\]](#I-int2){reference-type="eqref" reference="I-int2"}, we have that $$\begin{aligned} I(x_1,\dots,x_m) &= \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \det_{1 \leqslant i,j \leqslant m} \left( \frac{1}{2\pi} \int_{-\infty+{\tt i}x_i}^{\infty+{\tt i}x_i} ({\tt i}u+x_i)^{j-1} e^{-\frac{1}{2}u^2} du \right).\end{aligned}$$ Expanding the factor $({\tt i}u+x_i)^{j-1}$ as a polynomial in $x_i$, this becomes $$\begin{aligned} I(x_1,\dots,x_m) &= \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \det_{1 \leqslant i,j \leqslant m} \left( \frac{x_i^{j-1}}{2\pi} \int_{-\infty+{\tt i}x_i}^{\infty+{\tt i}x_i} e^{-\frac{1}{2}u^2} du + O(x_i^{j-2}) \right),\end{aligned}$$ and the polynomial term of the form $O(x_i^{j-2})$ can be removed by elementary column transformations. The final result is thus $$\begin{aligned} I(x_1,\dots,x_m) &= \left(\frac{1}{2\pi}\right)^{\frac{m}{2}} \prod_{i=1}^{m} e^{-\frac{1}{2} x_i^2} \prod_{1 \leqslant i <j \leqslant m} (x_j-x_i).\end{aligned}$$ ## Final formula {#ssec:formula} We are now in a position to write the full asymptotic behaviour of the Markov kernel $\mathbb{P}_{t,1}(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]})$ as $t\rightarrow\infty$. Using [\[kernel-reexpress\]](#kernel-reexpress){reference-type="eqref" reference="kernel-reexpress"} with $H_{\lambda^{[m]}-1}(t)$ given by $$\begin{gathered} H_{\lambda^{[m]}-1}(t) \sim (-1)^{nm} t^{-\frac{n}{2}\binom{m+1}{2}} q^{\binom{nm+1}{2}-\frac{1}{2}\binom{m+1}{2} \binom{n}{2}+m\binom{n}{2}} \\ \times \exp\left[ \frac{1-q^n}{1-q} mt \right] \frac{(1-q)^{n\binom{m}{2}}}{(q;q)_n^{m^2}} g_{\Delta}^{c^{[m]}}(m^n;\vec{Q}^{[m]}) \prod_{i=0}^{n-1} I\left(x^{[m]}_{im+1},\dots,x^{[m]}_{i(m+1)}\right)\end{gathered}$$ and $H_{\lambda^{[m+1]}}(t)$ given by [\[m-to-m+1\]](#m-to-m+1){reference-type="eqref" reference="m-to-m+1"}, we obtain $$\begin{gathered} \label{final-formula} \mathbb{P}_{t,1}(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}) \sim \bm{1}_{c^{[m]} \prec c^{[m+1]}} \Upsilon\left(c^{[m]};c^{[m+1]}\right) q^{{\rm inv}(c^{[m]})-{\rm inv}(c^{[m+1]})} \cdot (-1)^{n} t^{-\frac{n}{2}(m+1)} \times \\ q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}-\frac{1}{2}(m+1)\binom{n}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \cdot \frac{g_{\Delta}^{c^{[m+1]}}\left((m+1)^n;\vec{Q}^{[m+1]}\right)} {g_{\Delta}^{c^{[m]}}\left(m^n;\vec{Q}^{[m]}\right)} \prod_{i=1}^{n} \frac{I\left(x^{[m+1]}_{(i-1)(m+1)+1},\dots,x^{[m+1]}_{i(m+1)}\right)} {I\left(x^{[m]}_{(i-1)m+1},\dots,x^{[m]}_{im}\right)},\end{gathered}$$ as $t \rightarrow \infty$. Recall from [\[coord-scal\]](#coord-scal){reference-type="eqref" reference="coord-scal"} that there is a factor of $(q^{n-\lceil i/(m+1) \rceil} t)^{1/2}$ present in the change of variables from $\ell^{[m+1]}_i \in \mathbb{Z}$ to $x^{[m+1]}_i \in \mathbb{R}$, for all $1 \leqslant i \leqslant n(m+1)$. In order to obtain transition densities valid on the scale of the $x^{[m+1]}_i$ variables, we must multiply the above formula by the product of all such factors; namely, by $$\begin{aligned} t^{\frac{n}{2} (m+1)} \prod_{i=1}^{n(m+1)} (q^{n-\lceil i/(m+1) \rceil})^{1/2} = t^{\frac{n}{2} (m+1)} q^{\frac{1}{2}(m+1)\binom{n}{2}}.\end{aligned}$$ We then read off the result $$\begin{gathered} \label{final-formula2} \mathbb{P}_{t,1}(0\cup\lambda^{[m]} \rightarrow \lambda^{[m+1]}) \rightarrow \prod_{i=1}^{n} \rho_{\rm GUE} \left(x^{[m]}_{(i-1)m+1},\dots,x^{[m]}_{im} \rightarrow x^{[m+1]}_{(i-1)(m+1)+1},\dots,x^{[m+1]}_{i(m+1)} \right) dx^{[m+1]} \\ \times \bm{1}_{c^{[m]} \prec c^{[m+1]}} (-1)^{n} \Upsilon\left(c^{[m]};c^{[m+1]}\right) q^{{\rm inv}(c^{[m]})-{\rm inv}(c^{[m+1]})} q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \frac{g_{\Delta}^{c^{[m+1]}}\left((m+1)^n;\vec{Q}^{[m+1]}\right)} {g_{\Delta}^{c^{[m]}}\left(m^n;\vec{Q}^{[m]}\right)}\end{gathered}$$ as $t \rightarrow \infty$. The convergence in [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"} is uniform provided that the $x^{[m]}$ and $x^{[m+1]}$ parameters are chosen to vary over compact subsets of $\mathbb{R}$. This completes the first part of the proof of Theorem [Theorem 54](#thm:main){reference-type="ref" reference="thm:main"}; it remains to show that the factors present in the second line of [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"} constitute a valid probability distribution on colour sequences. # Distribution on colour sequences {#sec:discrete-dist} In the previous section we showed (see [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"} above) that the Plancherel-specialized LLT Markov kernel [\[kernel\]](#kernel){reference-type="eqref" reference="kernel"} splits, under the $t \rightarrow \infty$ asymptotic regime studied, into a product of $n$ independent GUE corners processes multiplied by a further factor valued on colour sequences. Our aim in this section is to show that this extra factor constitutes a discrete probability measure on colour sequences; in showing this, we demonstrate that the right hand side of [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"} integrates to unity, validating the fact that the set of coloured compositions to which we have restricted our attention captures the full asymptotic behaviour as $t \rightarrow \infty$. Our primary task will be to better understand the ratio $g_{\Delta}^{c^{[m+1]}} \Big{/}g_{\Delta}^{c^{[m]}}$ appearing in [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"}. To that end, we begin by defining a family of partition functions that are related to the functions $g_{\Delta}^{c^{[m]}}$ via an explicit symmetry. ## Partition function $Z$ Fix two integers $n,m \geqslant 1$ and a vector $i^{[m]} = (i_1,\dots,i_{nm}) \in [1,n]^{nm}$ such that for all $1 \leqslant k \leqslant n$ we have $|\{a: i_a = k\}| = m$. We define the following partition function in the model [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"}: $$\begin{aligned} \label{Z-def} Z\left(x_1,\dots,x_{nm} ; i^{[m]}\right) = \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \foreach\y in {1,...,4}{ \draw[lgray,line width=1pt,->] (1.5,\y) -- (4.5,\y); } \foreach\x in {1,...,3}{ \draw[lgray,line width=4pt,->] (\x+1,0.5) -- (\x+1,4.5); } %spectral parameters \node[left] at (1,1) {$x_1 \rightarrow$}; \node[left] at (1,2) {$x_2 \rightarrow$}; \node[left] at (1,3) {$\vdots$}; \node[left] at (1,4) {$x_{nm} \rightarrow $}; %bottom labels \node[below] at (2,0.5) {$\bm{e}_{[1,n]}$}; \node[below] at (3,0.5) {$\cdots$}; \node[below] at (4,0.5) {$\bm{e}_{[1,n]}$}; %top labels \node[above] at (2,4.5) {$\bm{e}_0$}; \node[above] at (3,4.5) {$\cdots$}; \node[above] at (4,4.5) {$\bm{e}_0$}; %right labels \node[right] at (4.5,1) {$i_1$}; \node[right] at (4.5,2) {$i_2$}; \node[right] at (4.5,3) {$\vdots$}; \node[right] at (4.5,4) {$i_{nm}$}; %left labels \node[left] at (1.5,1) {$0$}; \node[left] at (1.5,2) {$0$}; \node[left] at (1.5,3) {$\vdots$}; \node[left] at (1.5,4) {$0$}; \end{tikzpicture}\end{aligned}$$ where each vertex in the $a$-th row of the lattice is assigned rapidity parameter $z = x_a$, for $1 \leqslant a \leqslant nm$. We may represent the partition function [\[Z-def\]](#Z-def){reference-type="eqref" reference="Z-def"} algebraically, as follows: $$\begin{aligned} Z\left(x_1,\dots,x_{nm} ; i^{[m]}\right) = \left\langle \bm{e}_{[1,n]}\right|^{\otimes m} \mathcal{D}_{i_1}(x_1) \dots \mathcal{D}_{i_{nm}}(x_{nm}) \left|\bm{e}_0\right\rangle^{\otimes m},\end{aligned}$$ where we recall the row operator definition $\mathcal{D}_i(x) = T_{0,i}^{\rightarrow}(x;m-1)$ from Section [4.4](#sec:fused-row){reference-type="ref" reference="sec:fused-row"}. The partition functions thus defined may be related to those of [\[generic-g-sigma\]](#generic-g-sigma){reference-type="eqref" reference="generic-g-sigma"}, via the following symmetry: **Proposition 56**. *Recall the definition [\[nothing\]](#nothing){reference-type="eqref" reference="nothing"} of the trivial element $\Delta \in \mathcal{S}_{m^n}$. For all vectors $i^{[m]} \in [1,n]^{nm}$ we have that $$\begin{aligned} \label{gZ-sym} g^{i^{[m]}}_{\Delta}(m^n;x_1,\dots,x_{nm};s) = (-s)^{n\binom{m}{2}} \cdot Z\left(x_1^{-1},\dots,x_{nm}^{-1} ; i^{[m]}\right) \Big|_{q \mapsto q^{-1},s \mapsto s^{-1}}\end{aligned}$$ where the variables $q$, $s$ are replaced by their reciprocals in the final partition function.* *Proof.* This is an immediate consequence of the symmetry [\[LM-sym\]](#LM-sym){reference-type="eqref" reference="LM-sym"} between the vertex weights used to define [\[generic-g-sigma\]](#generic-g-sigma){reference-type="eqref" reference="generic-g-sigma"} and [\[Z-def\]](#Z-def){reference-type="eqref" reference="Z-def"}. ◻ **Corollary 57**. *The $s=0$ and $$\begin{aligned} \label{x-spec} (x_1,\dots,x_{nm}) = \underbrace{(q^{n-1},\dots,q^{n-1})}_{m\ {\rm times}} \cup \cdots \cup \underbrace{(q,\dots,q)}_{m\ {\rm times}} \cup \underbrace{(1,\dots,1)}_{m\ {\rm times}} \equiv \vec{Q}^{[m]}\end{aligned}$$ specializations of [\[gZ-sym\]](#gZ-sym){reference-type="eqref" reference="gZ-sym"} are given by $$\begin{aligned} \label{gZ-sym2} g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) = \lim_{s \rightarrow \infty} (-s)^{-n\binom{m}{2}} \cdot Z\left(\vec{Q}^{[m]};i^{[m]}\right) \Big|_{q \mapsto q^{-1}}.\end{aligned}$$* ## Expansion formula **Theorem 58**. *Fix a vector $i^{[m]} = (i_1,\dots,i_{nm}) \in [1,n]^{nm}$ such that $|\{a: i_a = k\}| = m$ for all $1 \leqslant k \leqslant n$. Then there exist explicit rational functions in $q$, denoted $\Theta\left(i^{[m]};j^{[m+1]}\right)$, such that the following expansion formula holds: $$\begin{aligned} \label{Z-exp} g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) = \sum_{j^{[m+1]}} \Theta\left(i^{[m]};j^{[m+1]}\right) g^{j^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right),\end{aligned}$$ where the sum is over vectors $j^{[m+1]} = \left(j_1,\dots,j_{n(m+1)}\right) \in [1,n]^{n(m+1)}$ such that $|\{a : j_a = k\}| = m+1$ for all $1 \leqslant k \leqslant n$.* The proof of this theorem is split over the subsequent three subsections. In view of the relation [\[gZ-sym2\]](#gZ-sym2){reference-type="eqref" reference="gZ-sym2"}, all properties of the functions $g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right)$ and $g^{j^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right)$ may be deduced from those of $Z\left(\vec{Q}^{[m]};i^{[m]}\right) \equiv Z\left(i^{[m]}\right)$ and $Z\left(\vec{Q}^{[m+1]};j^{[m+1]}\right) \equiv Z\left(j^{[m+1]}\right)$; we adopt this approach in our proof of [\[Z-exp\]](#Z-exp){reference-type="eqref" reference="Z-exp"}. ## Partition function $\tilde{Z}$ Our strategy for proving [\[Z-exp\]](#Z-exp){reference-type="eqref" reference="Z-exp"} is to define another type of partition function, similar to [\[Z-def\]](#Z-def){reference-type="eqref" reference="Z-def"}, and calculate it in two different ways; the two different evaluations effectively yield the left and right hand sides of [\[Z-exp\]](#Z-exp){reference-type="eqref" reference="Z-exp"}. To that end, for all vectors $i^{[m]} = (i_1,\dots,i_{nm}) \in [1,n]^{nm}$ we introduce $$\begin{aligned} \label{tilde-Z} \tilde{Z}\left(u;x_1,\dots,x_{nm};i^{[m]}\right) = \begin{tikzpicture}[scale=1.2,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,5) -- (4.5,5); \foreach\y in {1,...,4}{ \draw[lgray,line width=1pt,->] (0.5,\y) -- (4.5,\y); } \foreach\x in {0,...,3}{ \draw[lgray,line width=4pt,->] (\x+1,0.5) -- (\x+1,5.5); } %spectral parameters \node[left] at (0,1) {$x_1 \rightarrow$}; \node[left] at (0,2) {$x_2 \rightarrow$}; \node[left] at (0,3) {$\vdots$}; \node[left] at (0,4) {$x_{nm} \rightarrow$}; \node[left] at (0,5) {$(su;r) \rightarrow$}; %bottom labels \node[below] at (1,0.5) {$\bm{e}_{[1,n]}$}; \node[below] at (2,0.5) {$\bm{e}_{[1,n]}$}; \node[below] at (3,0.5) {$\cdots$}; \node[below] at (4,0.5) {$\bm{e}_{[1,n]}$}; %top labels \node[above] at (1,5.5) {$\bm{e}_0$}; \node[above] at (2,5.5) {$\bm{e}_0$}; \node[above] at (3,5.5) {$\cdots$}; \node[above] at (4,5.5) {$\bm{e}_0$}; %right labels \node[right] at (4.5,1) {$i_1$}; \node[right] at (4.5,2) {$i_2$}; \node[right] at (4.5,3) {$\vdots$}; \node[right] at (4.5,4) {$i_{nm}$}; \node[right] at (4.5,5) {$\bm{e}_{[1,n]}$}; %left labels \node[left] at (0.5,1) {$0$}; \node[left] at (0.5,2) {$0$}; \node[left] at (0.5,3) {$\vdots$}; \node[left] at (0.5,4) {$0$}; \node[left] at (0.5,5) {$\bm{e}_0$}; \end{tikzpicture}\end{aligned}$$ which is effectively obtained by appending an extra fused row and column to the original partition function [\[Z-def\]](#Z-def){reference-type="eqref" reference="Z-def"}. We represent this partition function algebraically as $$\begin{aligned} \label{tilde-Z-alg} \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) = \left\langle \bm{e}_{[1,n]}\right|^{\otimes m+1} \mathcal{D}_{i_1}(x_1) \dots \mathcal{D}_{i_{nm}}(x_{nm}) \mathcal{D}_{[1,n]}(su;r) \left|\bm{e}_0\right\rangle^{\otimes m+1}.\end{aligned}$$ ## First evaluation of $\tilde{Z}$ Let us begin by analysing the dependence of the partition function [\[tilde-Z\]](#tilde-Z){reference-type="eqref" reference="tilde-Z"} on the parameter $u$. To do so, we need only study the vertices in the top row of [\[tilde-Z\]](#tilde-Z){reference-type="eqref" reference="tilde-Z"}. All of these vertices have weight given by [\[fused-weights\]](#fused-weights){reference-type="eqref" reference="fused-weights"}--[\[w-weight\]](#w-weight){reference-type="eqref" reference="w-weight"} in which $\bm{C}= \bm{V}= \bm{e}_0$; in that special case the weights simplify as follows: $$\begin{aligned} \tilde{L}^{(r,s)}_{su}(\bm{A},\bm{B}; \bm{e}_0,\bm{D}) = \bm{1}_{\bm{A}+\bm{B}=\bm{D}} \cdot u^{|\bm{D}|-|\bm{B}|} r^{-2|\bm{A}|} s^{2|\bm{D}|} W^{(r,s)}_u(\bm{A},\bm{B};\bm{e}_0,\bm{D}),\end{aligned}$$ with $$\begin{aligned} W^{(r,s)}_u(\bm{A},\bm{B};\bm{e}_0,\bm{D}) &= \Phi(\bm{e}_0,\bm{D};s^2 r^{-2} u,s^2 u) \Phi(\bm{e}_0,\bm{B};r^2 u^{-1},r^2) = \frac{(r^2;q)_{|\bm{D}|} (u;q)_{|\bm{B}|}}{(s^2 u;q)_{|\bm{D}|} (r^2;q)_{|\bm{B}|}}.\end{aligned}$$ From these expressions we see that $u^{|\bm{B}|-|\bm{D}|} \cdot (s^2 u;q)_n \cdot \tilde{L}^{(r,s)}_{su}(\bm{A},\bm{B}; \bm{e}_0,\bm{D})$ is a polynomial in $u$ of degree $n-|\bm{D}|+|\bm{B}|$. It follows that $u^{-n} \cdot (s^2 u;q)_n^{m+1} \cdot \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right)$ is a polynomial in $u$ of degree $nm$; this can be seen by telescoping the degrees of the individual vertices in the top row of the partition function. It turns out to be possible to determine all of the zeros of this polynomial explicitly, using the commutation relation [\[Cfused-com\]](#Cfused-com){reference-type="eqref" reference="Cfused-com"} with $J=[1,n]$ and $i=i_k$ for each $1 \leqslant k \leqslant nm$; indeed, using this relation in [\[tilde-Z-alg\]](#tilde-Z-alg){reference-type="eqref" reference="tilde-Z-alg"}, we find that $$\begin{gathered} \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) = r^{2nm} \prod_{k=1}^{nm} q^{n-i_k} \left(\frac{q x_k -su}{su-r^2 x_k}\right) \\ \times \left\langle \bm{e}_{[1,n]}\right|^{\otimes m+1} \mathcal{D}_{[1,n]}(su;r) \mathcal{D}_{i_1}(x_1) \dots \mathcal{D}_{i_{nm}}(x_{nm}) \left|\bm{e}_0\right\rangle^{\otimes m+1}\end{gathered}$$ which allows us to determine that $$\begin{aligned} \label{all-zeros} (s^2 u;q)_n^{m+1} \cdot \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) &= \alpha \cdot u^n \cdot \prod_{k=1}^{nm} (su-q x_k)\end{aligned}$$ where $\alpha$ is independent of $u$ but may depend on all other parameters. To determine $\alpha$, we seek an appropriate choice for the parameter $u$ in [\[all-zeros\]](#all-zeros){reference-type="eqref" reference="all-zeros"}. Our choice is motivated by studying the vertex in the top-left corner of the partition function [\[tilde-Z\]](#tilde-Z){reference-type="eqref" reference="tilde-Z"}; this vertex is of the form $$\begin{aligned} \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $\bm{e}_0$};\node[right] at (1,0) {\tiny $\bm{A}$}; \node[below] at (0,-1) {\tiny $\bm{A}$};\node[above] at (0,1) {\tiny $\bm{e}_0$}; \node[left] at (-1.5,0) {$(su;r) \rightarrow$}; \end{tikzpicture} &= \tilde{L}_{su}^{(r,s)}(\bm{A},\bm{e}_0;\bm{e}_0,\bm{A}) = u^{|\bm{A}|} r^{-2|\bm{A}|} s^{2|\bm{A}|} \Phi(\bm{e}_0,\bm{A};s^2 r^{-2} u,s^2 u) \Phi(\bm{e}_0,\bm{e}_0;r^2 u^{-1},r^2) \\ &= u^{|\bm{A}|} r^{-2|\bm{A}|} s^{2|\bm{A}|} \frac{(r^2;q)_{|\bm{A}|}}{(s^2 u;q)_{|\bm{A}|}}.\end{aligned}$$ From the explicit form of this vertex weight we find that $$\begin{aligned} \lim_{u \rightarrow s^{-2} q^{-n+1}} (s^2 u;q)_n \cdot \tilde{L}_{su}^{(r,s)}(\bm{A},\bm{e}_0;\bm{e}_0,\bm{A}) = \bm{1}_{\bm{A}= \bm{e}_{[1,n]}} \cdot q^{-(n-1)n} r^{-2n} (r^2;q)_n;\end{aligned}$$ it follows that if we set $u = s^{-2} q^{-n+1}$ in the left hand side of [\[all-zeros\]](#all-zeros){reference-type="eqref" reference="all-zeros"}, this produces a freezing of the top row and leftmost column in the partition function [\[tilde-Z\]](#tilde-Z){reference-type="eqref" reference="tilde-Z"}: $$\begin{gathered} \label{pf-split} \lim_{u \rightarrow s^{-2} q^{-n+1}} (s^2 u;q)_n^{m+1} \cdot \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) \\ = \lim_{u \rightarrow s^{-2} q^{-n+1}} \left[ (s^2 u;q)_n^{m+1} \times \begin{tikzpicture}[scale=1.1,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (0.5,5) -- (4.5,5); \foreach\y in {1,...,4}{ \draw[lgray,line width=1pt,->] (0.5,\y) -- (4.5,\y); } \foreach\x in {0,...,3}{ \draw[lgray,line width=4pt,->] (\x+1,0.5) -- (\x+1,5.5); } %spectral parameters \node[left] at (0,1) {$x_1 \rightarrow$}; \node[left] at (0,2) {$x_2 \rightarrow$}; \node[left] at (0,3) {$\vdots$}; \node[left] at (0,4) {$x_{nm} \rightarrow$}; \node[left] at (0,5) {$(su;r) \rightarrow$}; %bottom labels \node[below] at (1,0.5) {$\bm{e}_{[1,n]}$}; \node[below] at (2,0.5) {$\bm{e}_{[1,n]}$}; \node[below] at (3,0.5) {$\cdots$}; \node[below] at (4,0.5) {$\bm{e}_{[1,n]}$}; %top labels \node[above] at (1,5.5) {$\bm{e}_0$}; \node[above] at (2,5.5) {$\bm{e}_0$}; \node[above] at (3,5.5) {$\cdots$}; \node[above] at (4,5.5) {$\bm{e}_0$}; %right labels \node[right] at (4.5,1) {$i_1$}; \node[right] at (4.5,2) {$i_2$}; \node[right] at (4.5,3) {$\vdots$}; \node[right] at (4.5,4) {$i_{nm}$}; \node[right] at (4.5,5) {$\bm{e}_{[1,n]}$}; %left labels \node[left] at (0.5,1) {$0$}; \node[left] at (0.5,2) {$0$}; \node[left] at (0.5,3) {$\vdots$}; \node[left] at (0.5,4) {$0$}; \node[left] at (0.5,5) {$\bm{e}_0$}; %paths \draw[red,line width=0.5pt,->] (0.95,0.5) -- (0.95,5.05) -- (4.3,5.05); \draw[green,line width=0.5pt,->] (1,0.5) -- (1,5) -- (4.3,5); \draw[blue,line width=0.5pt,->] (1.05,0.5) -- (1.05,4.95) -- (4.3,4.95); \end{tikzpicture} \right]\end{gathered}$$ where coloured lines flowing through the leftmost column and top row represent the vector $\bm{e}_{[1,n]}$. The quantity [\[pf-split\]](#pf-split){reference-type="eqref" reference="pf-split"} splits into several pieces; apart from the partition function $Z(x_1,\dots,x_{nm};i^{[m]})$ which emerges in the bottom-right corner, there is a contribution from the vertex in the top-left corner, the remaining vertices in the top row, and the remaining vertices in the leftmost column. This leads us to the expression $$\begin{gathered} \lim_{u \rightarrow s^{-2} q^{-n+1}} (s^2 u;q)_n^{m+1} \cdot \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) = \lim_{u \rightarrow s^{-2} q^{-n+1}} \left[ (s^2 u;q)_n \times \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $\bm{e}_0$};\node[right] at (1,0) {\tiny $\bm{e}_{[1,n]}$}; \node[below] at (0,-1) {\tiny $\bm{e}_{[1,n]}$};\node[above] at (0,1) {\tiny $\bm{e}_0$}; \node[left] at (-1.5,0) {$(su;r) \rightarrow$}; %\node[below] at (0,-1.5) {$(y_0,s)$}; \end{tikzpicture} \right] \\ \times \lim_{u \rightarrow s^{-2} q^{-n+1}} \left[ (s^2 u;q)_n \times \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $\bm{e}_{[1,n]}$};\node[right] at (1,0) {\tiny $\bm{e}_{[1,n]}$}; \node[below] at (0,-1) {\tiny $\bm{e}_0$};\node[above] at (0,1) {\tiny $\bm{e}_0$}; \node[left] at (-2.1,0) {$(su;r) \rightarrow$}; %\node[below] at (0,-1.5) {$(y_j,s)$}; \end{tikzpicture} \right]^m \prod_{k=1}^{nm} \left[ \begin{tikzpicture}[scale=0.6,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=1pt,->] (-1,0) -- (1,0); \draw[lgray,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\tiny $0$};\node[right] at (1,0) {\tiny $0$}; \node[below] at (0,-1) {\tiny $\bm{e}_{[1,n]}$};\node[above] at (0,1) {\tiny $\bm{e}_{[1,n]}$}; \node[left] at (-1.5,0) {$x_k \rightarrow$}; \end{tikzpicture} \right] Z\left(x_1,\dots,x_{nm}; i^{[m]}\right).\end{gathered}$$ Explicitly computing the weight of each bracketed expression above, we arrive at the following evaluation of the left hand side of [\[all-zeros\]](#all-zeros){reference-type="eqref" reference="all-zeros"}: $$\begin{gathered} \lim_{u \rightarrow s^{-2} q^{-n+1}} (s^2 u;q)_n^{m+1} \cdot \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) \\ = q^{-(n-1)n} r^{-2n} (r^2;q)_n \Big[ s^{2n} (s^{-2}q^{-n+1};q)_n \Big]^m \prod_{k=1}^{nm} \frac{1-s q^n x_k}{1-s x_k} \cdot Z\left(x_1,\dots,x_{nm}; i^{[m]}\right).\end{gathered}$$ Equating this with the corresponding limit of the right hand side of [\[all-zeros\]](#all-zeros){reference-type="eqref" reference="all-zeros"}, we determine the constant $\alpha$ to be $$\begin{aligned} \alpha = s^{nm} \left( \frac{s}{r} \right)^{2n} (r^2;q)_n \frac{\prod_{k=1}^{n} (s^2 q^{n-1}-q^{k-1})^m}{\prod_{k=1}^{nm}(1-sx_k)} Z\left(x_1,\dots,x_{nm}; i^{[m]}\right).\end{aligned}$$ Substituting this into [\[all-zeros\]](#all-zeros){reference-type="eqref" reference="all-zeros"}, we obtain our first evaluation of the partition function: $$\begin{gathered} \label{first-eval} \tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right) \\ = u^n s^{nm} \left( \frac{s}{r} \right)^{2n} \frac{(r^2;q)_n}{(s^2 u;q)_n} \prod_{k=1}^{nm} \left( \frac{su-q x_k}{1-sx_k} \right) \prod_{k=1}^{n} \left( \frac{s^2 q^{n-1}-q^{k-1}}{1-s^2 q^{k-1} u} \right)^m Z\left(x_1,\dots,x_{nm}; i^{[m]}\right).\end{gathered}$$ Since our ultimate aim is to prove Theorem [Theorem 58](#thm:expand){reference-type="ref" reference="thm:expand"}, we shall be most interested in setting $u=s^{-1}$, $r=q^{-n/2}$ and choosing $(x_1,\dots,x_{nm})$ as in [\[x-spec\]](#x-spec){reference-type="eqref" reference="x-spec"}. Making these specializations in [\[first-eval\]](#first-eval){reference-type="eqref" reference="first-eval"}, we have $$\begin{aligned} \label{first-eval2} \left. \tilde{Z}\left(s^{-1}; \vec{Q}^{[m]}; i^{[m]} \right) \right|_{r = q^{-n/2}} = (-1)^{nm} s^{n(m+1)} q^{n^2+\binom{n}{2}m} \frac{ (q^{-n};q)_n (q;q)_n^m (s^2;q)_n^m } {(s;q)_n^{2m+1}} Z\left(\vec{Q}^{[m]}; i^{[m]}\right).\end{aligned}$$ ## Second evaluation of $\tilde{Z}$ For the second evaluation of $\tilde{Z}\left(u; x_1,\dots,x_{nm}; i^{[m]}\right)$, we take from the outset $u=s^{-1}$, $r=q^{-n/2}$ and specialize $(x_1,\dots,x_{nm})$ as in [\[x-spec\]](#x-spec){reference-type="eqref" reference="x-spec"}. We then compute [\[tilde-Z-alg\]](#tilde-Z-alg){reference-type="eqref" reference="tilde-Z-alg"} via a sequence of manipulations of the row operators; the relations that we need are [\[important-relation\]](#important-relation){reference-type="eqref" reference="important-relation"} and [\[row-relation2\]](#row-relation2){reference-type="eqref" reference="row-relation2"}. Commutation relation [\[important-relation\]](#important-relation){reference-type="eqref" reference="important-relation"} allows us to reverse the order of a pair of operators $\mathcal{D}_i(x) \mathcal{D}_J(x;q^{-p/2})$, where $J$ has cardinality $p$; [\[row-relation2\]](#row-relation2){reference-type="eqref" reference="row-relation2"} allows us to split off an unfused row operator from the row operator $\mathcal{D}_I(x;q^{-p/2})$ of width $p$, reducing it to a row operator of width $p-1$, and $q$-shifting its spectral parameter $x$. For any integer $k \in [1,n]$, let $J(k)$ be a subset of $[1,n]$ with $|J(k)| = k$; this means that, in particular, $J(n) = [1,n]$. Using [\[important-relation\]](#important-relation){reference-type="eqref" reference="important-relation"} repeatedly, we may start from a product of row operators of the form $$\begin{aligned} \mathcal{D}_{i_{(k-1)m+1}}(q^{n-k}) \cdots \mathcal{D}_{i_{km}}(q^{n-k}) \mathcal{D}_{J(k)}(q^{n-k};q^{-k/2})\end{aligned}$$ and drag the width $k$ operator $\mathcal{D}_{J(k)}(q^{n-k};q^{-k/2})$ towards the left of the product. After that, we apply [\[row-relation2\]](#row-relation2){reference-type="eqref" reference="row-relation2"} and reduce the leftward-emerging row operator to one of width $k-1$ (at the expense of splitting off a single unfused row operator, to its right); this changes its argument from $q^{n-k}$ to $q^{n-k+1}$, and we denote the resulting operator by $\mathcal{D}_{J(k-1)}(q^{n-k+1};q^{-(k-1)/2})$. Repeating this process for all $k \in [1,n]$, beginning with $k=n$ and reducing $k$ by $1$ at each step, it is straightforward to derive the following expansion: $$\begin{gathered} \label{psi-expand} \left. \tilde{Z}\left(s^{-1}; \vec{Q}^{[m]}; i^{[m]} \right) \right|_{r = q^{-n/2}} = \left[ \frac{(1-q)^{n}}{(q;q)_n} \right]^m \\ \times \sum_{j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \left\langle \bm{e}_{[1,n]}\right|^{\otimes m+1} \prod_{k=1}^{n} \Big[ \mathcal{D}_{j_{(k-1)(m+1)+1}}(q^{n-k}) \cdots \mathcal{D}_{j_{k(m+1)}}(q^{n-k}) \Big] \left|\bm{e}_0\right\rangle^{\otimes m+1}\end{gathered}$$ where the sum is taken over all vectors $j^{[m+1]} = (j_1,\dots,j_{n(m+1)}) \in [1,n]^{m+1}$ and with the coefficients $\Psi\left( i^{[m]}; j^{[m+1]} \right)$ given by the following one-row partition function: $$\begin{gathered} \Psi\left( i^{[m]}; j^{[m+1]} \right) = \\ \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,<-] (0.5,0) -- (12,0); \foreach\x in {1,2,3,5,6,7,9,10,11}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$j_1$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$j_{m+1}$}; % \node[above] at (5,0.5) {$\cdots$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \node[above] at (9,0.5) {$j_{(n-1)(m+1)+1}$}; \node[above] at (10.2,0.5) {$\cdots$}; \node[above] at (11,0.5) {$j_{n(m+1)}$}; % \foreach\x in {1.5,2.5,5.5,6.5,9.5,10.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$i_1$}; \node[below] at (2,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$i_m$}; % \node[below] at (5.5,-0.5) {$\cdots$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \node[below] at (9.2,-0.5) {$i_{(n-1)m+1}$}; \node[below] at (10.1,-0.5) {$\cdots$}; \node[below] at (10.7,-0.5) {$i_{nm}$}; % \node[left] at (0.5,0) {$\emptyset$}; \node[right] at (12,0) {$[1,n]$}; \end{tikzpicture}\end{gathered}$$ where colours are conserved in the direction of arrow flow, and with a weight of $q$ assigned to each of the events $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt,<-] (0.5,0) -- (1.5,0); \draw[lgray,line width=1.5pt,->] (1,-0.5) -- (1,0); \node[below] at (1,-0.5) {$i$}; \node[left] at (0.5,0) {$c$}; \end{tikzpicture}\end{aligned}$$ which denotes a path of colour $c$ passing above a colour $i$, with $c>i$. Recasting [\[psi-expand\]](#psi-expand){reference-type="eqref" reference="psi-expand"} in terms of the family of partition functions [\[Z-def\]](#Z-def){reference-type="eqref" reference="Z-def"}, we recover the expansion $$\begin{aligned} \label{second-eval} \left. \tilde{Z}\left(s^{-1}; \vec{Q}^{[m]}; i^{[m]} \right) \right|_{r = q^{-n/2}} = \left[ \frac{(1-q)^{n}}{(q;q)_n} \right]^m \times \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) Z\left( \vec{Q}^{[m+1]}; j^{[m+1]} \right).\end{aligned}$$ ## Comparing evaluations of $\tilde{Z}$ Comparing equations [\[first-eval2\]](#first-eval2){reference-type="eqref" reference="first-eval2"} and [\[second-eval\]](#second-eval){reference-type="eqref" reference="second-eval"}, we have shown that $$\begin{gathered} Z\left(\vec{Q}^{[m]}; i^{[m]}\right) = (-1)^{n(m+1)} s^{-n(m+1)} q^{-\binom{n}{2}(m+1)} \\ \times \left( \frac{(1-q)^{n}}{(s^2;q)_n} \right)^m \left( \frac{(s;q)_n}{(q;q)_n} \right)^{2m+1} \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) Z\left( \vec{Q}^{[m+1]}; j^{[m+1]} \right).\end{gathered}$$ Inverting $q$ in the previous equation and multiplying both sides by $(-s)^{-n \binom{m}{2}}$, we obtain $$\begin{gathered} (-s)^{-n \binom{m}{2}} Z\left(\vec{Q}^{[m]}; i^{[m]}\right) \Big|_{q \mapsto q^{-1}} = q^{\binom{n}{2}(m+1)} (-s)^{-n} \left( \frac{(1-q^{-1})^{n}}{(s^2;q^{-1})_n} \right)^m \left( \frac{(s;q^{-1})_n}{(q^{-1};q^{-1})_n} \right)^{2m+1} \\ \times \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}} (-s)^{-n \binom{m+1}{2}} Z\left( \vec{Q}^{[m+1]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}},\end{gathered}$$ and we now take the limit $s \rightarrow \infty$ using [\[gZ-sym2\]](#gZ-sym2){reference-type="eqref" reference="gZ-sym2"}: $$\begin{aligned} \label{4548} g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) = \frac{(q^{-1}-1)^{nm}}{(q^{-1};q^{-1})_n^{2m+1}} \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}} g^{j^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right).\end{aligned}$$ Finally, rearranging the factors in [\[4548\]](#4548){reference-type="eqref" reference="4548"}, one recovers $$\begin{gathered} g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) = (-1)^n q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \\ \times \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}} g^{j^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right).\end{gathered}$$ This completes the proof of Theorem [Theorem 58](#thm:expand){reference-type="ref" reference="thm:expand"}, with the coefficients in [\[Z-exp\]](#Z-exp){reference-type="eqref" reference="Z-exp"} identified as $$\begin{aligned} \label{4792} \Theta\left( i^{[m]}; j^{[m+1]} \right) = \bm{1}_{i^{[m]} \prec j^{[m+1]}} (-1)^n q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}}.\end{aligned}$$ ## Completing the proof of Theorem [Theorem 54](#thm:main){reference-type="ref" reference="thm:main"} {#completing-the-proof-of-theorem-thmmain} Rearrangement of [\[Z-exp\]](#Z-exp){reference-type="eqref" reference="Z-exp"}, using [\[4792\]](#4792){reference-type="eqref" reference="4792"}, yields the fact that $$\begin{aligned} \label{sum-to-1-seqs} \sum_{j^{[m+1]}} \bm{1}_{i^{[m]} \prec j^{[m+1]}} (-1)^n q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \Psi\left( i^{[m]}; j^{[m+1]} \right) \Big|_{q \mapsto q^{-1}} \frac{ g^{j^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right) } { g^{i^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) } =1.\end{aligned}$$ This is precisely the result that we need to complete the proof of Theorem [Theorem 54](#thm:main){reference-type="ref" reference="thm:main"}; all that remains is to match the summand of [\[sum-to-1-seqs\]](#sum-to-1-seqs){reference-type="eqref" reference="sum-to-1-seqs"} with the second line of equation [\[final-formula2\]](#final-formula2){reference-type="eqref" reference="final-formula2"}. We observe that all factors match perfectly, modulo the following proposition that takes care of the factors that are not yet manifestly equal: **Proposition 59**. *Fix two colour sequences $c^{[m]} \in [1,n]^{nm}$ and $c^{[m+1]} \in [1,n]^{n(m+1)}$ which satisfy the constraints $|\{a: c^{[m]}_a = k\}| = m$ and $|\{a: c^{[m+1]}_a = k\}| = m+1$ for all $1 \leqslant k \leqslant n$. Assuming also that $c^{[m]} \prec c^{[m+1]}$, the following relation holds: $$\begin{aligned} \label{statistic-match} \Psi\left( c^{[m]}; c^{[m+1]} \right) \Upsilon\left(c^{[m]};c^{[m+1]}\right) q^{{\rm inv}(c^{[m]})-{\rm inv}(c^{[m+1]})} = 1.\end{aligned}$$* *Proof.* This statement is equivalent to [@ABW21 Lemma 10.1.2], but rather than making a detailed match with that result, we give a standalone proof. As in [@ABW21], we shall proceed by induction on $n$. The $n=1$ case of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} is trivial; in that case we must have $c^{[m]} = 1^m$ and $c^{[m+1]} = 1^{m+1}$, which yields $$\begin{aligned} \Psi\left( c^{[m]}; c^{[m+1]} \right) = \Upsilon\left(c^{[m]};c^{[m+1]}\right) = q^{{\rm inv}(c^{[m]})-{\rm inv}(c^{[m+1]})} = 1.\end{aligned}$$ We shall take as our inductive assumption that [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} is valid for $n = p-1$, for given $p \geqslant 2$. Then for any $c^{[m]} \in [1,p-1]^{(p-1)m}$ and $c^{[m+1]} \in [1,p-1]^{(p-1)(m+1)}$, we also have $$\begin{aligned} \label{ind-step} \Psi\left( c^{[m]}\cup p^m; c^{[m+1]} \cup p^{m+1} \right) \Upsilon\left(c^{[m]}\cup p^m;c^{[m+1]} \cup p^{m+1}\right) q^{{\rm inv}(c^{[m]}\cup p^m)-{\rm inv}(c^{[m+1]} \cup p^{m+1})} = 1.\end{aligned}$$ Indeed, one can verify that appending a bundle of (maximal) colours $p$ to both $c^{[m]}$ and $c^{[m+1]}$ does not affect either of the partition functions $\Psi$ and $\Upsilon$, neither does it affect the value of the statistic ${\rm inv}$. Equation [\[ind-step\]](#ind-step){reference-type="eqref" reference="ind-step"} then constitutes a solution of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} at $n=p$; to prove that [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} holds generally at $n=p$, we seek "local moves" that can be applied to $c^{[m]}\cup p^m$ and $c^{[m+1]} \cup p^{m+1}$ to bring them to generic colour sequences in $[1,p]^{pm}$ and $[1,p]^{p(m+1)}$, respectively. These local moves will have the property that they preserve the interlacing property of the colour sequences, and applying them to a solution of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} will yield a new solution. The first two local moves that one requires are jumps across bundles: $$\begin{aligned} \label{move1} \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (7.5,0); \foreach\x in {1,2,3,5,6,7}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$a$}; % \node[above] at (5,0.5) {$p$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,5.5,6.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$\cdots$}; % \node[below] at (5.5,-0.5) {$\cdots$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.75,0) -- (5.25,0); \end{tikzpicture} \mapsto \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (7.5,0); \foreach\x in {1,2,3,5,6,7}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$p$}; % \node[above] at (5,0.5) {$a$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,5.5,6.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$\cdots$}; % \node[below] at (5.5,-0.5) {$\cdots$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.75,0) -- (5.25,0); \end{tikzpicture}\end{aligned}$$ $$\begin{aligned} \label{move2} \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (7.5,0); \foreach\x in {1,2,3,5,6,7}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$b$}; % \node[above] at (5,0.5) {$c$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,5.5,6.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$a$}; % \node[below] at (5.5,-0.5) {$p$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.25,0) -- (5.75,0); \end{tikzpicture} \mapsto \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (7.5,0); \foreach\x in {1,2,3,5,6,7}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$b$}; % \node[above] at (5,0.5) {$c$}; \node[above] at (6,0.5) {$\cdots$}; \node[above] at (7,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,5.5,6.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$p$}; % \node[below] at (5.5,-0.5) {$a$}; \node[below] at (6.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.25,0) -- (5.75,0); \end{tikzpicture}\end{aligned}$$ where arrows at the bottom of the diagram indicate colours in $c^{[m]}$, while those at the top indicate colours in $c^{[m+1]}$. All marked colours $a,b,c,p$ are assumed to be distinct (were they not distinct in the second case, the interlacing property of colours would be violated in at least one of the pictures shown), with $p$ being the largest. All colours remain fixed under these moves, apart from $a$ and $p$, which switch places. Let $\Psi_{{\sf L}/{\sf R}}$ and $\Upsilon_{{\sf L}/{\sf R}}$ denote the contributions to the functions $\Psi$ and $\Upsilon$ coming only from the indicated colours and marked regions of the diagrams, on the left/right hand side of both [\[move1\]](#move1){reference-type="eqref" reference="move1"} and [\[move2\]](#move2){reference-type="eqref" reference="move2"}. In the case of [\[move1\]](#move1){reference-type="eqref" reference="move1"}, one finds that $$\begin{aligned} \Psi_{\sf L}\cdot \Upsilon_{\sf L} = 1, \qquad \Psi_{\sf R}\cdot \Upsilon_{\sf R} = q,\end{aligned}$$ but since ${\rm inv}(c^{[m+1]})$ also increases by $1$ under the move [\[move1\]](#move1){reference-type="eqref" reference="move1"}, we find that [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} is preserved. In a similar vein, in the case of [\[move2\]](#move2){reference-type="eqref" reference="move2"} one obtains $$\begin{aligned} \Psi_{\sf L}\cdot \Upsilon_{\sf L} = q\cdot q^{\bm{1}_{b>c}+\bm{1}_{b>a}+\bm{1}_{c>a}}, \qquad \Psi_{\sf R}\cdot \Upsilon_{\sf R} = q^{\bm{1}_{b>a}+\bm{1}_{c>a}} \cdot q^{\bm{1}_{b>c}},\end{aligned}$$ and since ${\rm inv}(c^{[m]})$ also increases by $1$ under the move [\[move2\]](#move2){reference-type="eqref" reference="move2"}, we again find that [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} is preserved. The remaining two local moves needed are jumps within bundles: $$\begin{aligned} \label{move3} \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (4.5,0); \foreach\x in {1,2,3,4}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$a$}; \node[above] at (3,0.5) {$p$}; \node[above] at (4,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,3.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$b$}; \node[below] at (3.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (1.75,0) -- (3.25,0); \end{tikzpicture} \mapsto \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (4.5,0); \foreach\x in {1,2,3,4}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$p$}; \node[above] at (3,0.5) {$a$}; \node[above] at (4,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,3.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$b$}; \node[below] at (3.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (1.75,0) -- (3.25,0); \end{tikzpicture}\end{aligned}$$ $$\begin{aligned} \label{move4} \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (5.5,0); \foreach\x in {1,2,3,4,5}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$b$}; \node[above] at (4,0.5) {$\cdots$}; \node[above] at (5,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,3.5,4.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$a$}; \node[below] at (3.5,-0.5) {$p$}; \node[below] at (4.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.25,0) -- (3.75,0); \end{tikzpicture} \mapsto \begin{tikzpicture}[scale=0.8,baseline=(current bounding box.center),>=stealth] \draw[lgray,line width=4pt] (0.5,0) -- (5.5,0); \foreach\x in {1,2,3,4,5}{ \draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,0.5); } \node[above] at (1,0.5) {$\cdots$}; \node[above] at (2,0.5) {$\cdots$}; \node[above] at (3,0.5) {$b$}; \node[above] at (4,0.5) {$\cdots$}; \node[above] at (5,0.5) {$\cdots$}; % \foreach\x in {1.5,2.5,3.5,4.5}{ \draw[lgray,line width=1.5pt,->] (\x,-0.5) -- (\x,0); } % \node[below] at (1.5,-0.5) {$\cdots$}; \node[below] at (2.5,-0.5) {$p$}; \node[below] at (3.5,-0.5) {$a$}; \node[below] at (4.5,-0.5) {$\cdots$}; % \draw[line width=2pt] (2.25,0) -- (3.75,0); \end{tikzpicture}\end{aligned}$$ where all marked colours $a,b,p$ are assumed to be distinct, with $p$ being the largest. As previously, all colours remain fixed under these moves apart from $a$ and $p$, which exchange their places. We again let $\Psi_{{\sf L}/{\sf R}}$ and $\Upsilon_{{\sf L}/{\sf R}}$ denote the contributions to the functions $\Psi$ and $\Upsilon$ coming only from the indicated colours and marked regions of the diagrams, on the left/right hand side of both [\[move3\]](#move3){reference-type="eqref" reference="move3"} and [\[move4\]](#move4){reference-type="eqref" reference="move4"}. In the case of [\[move3\]](#move3){reference-type="eqref" reference="move3"}, we get $$\begin{aligned} \Psi_{\sf L}\cdot \Upsilon_{\sf L} = q^{\bm{1}_{a>b}} \cdot q, \qquad \Psi_{\sf R}\cdot \Upsilon_{\sf R} = q \cdot q^{1+\bm{1}_{a>b}},\end{aligned}$$ with the discrepancy between left and right hand sides cured by the fact that ${\rm inv}(c^{[m+1]})$ increases by $1$ under the move in question; hence [\[move3\]](#move3){reference-type="eqref" reference="move3"} preserves solutions of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"}. Finally, in the case of [\[move4\]](#move4){reference-type="eqref" reference="move4"} one finds that $$\begin{aligned} \Psi_{\sf L}\cdot \Upsilon_{\sf L} = q \cdot q^{\bm{1}_{a<b}}, \qquad \Psi_{\sf R}\cdot \Upsilon_{\sf R} = q^{\bm{1}_{a<b}} \cdot 1;\end{aligned}$$ left and right hand sides match after accounting for the fact that ${\rm inv}(c^{[m]})$ is increased by $1$ under this move. Therefore, [\[move4\]](#move4){reference-type="eqref" reference="move4"} also preserves solutions of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"}. This suffices to prove [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} generally at $n=p$, since we have already exhibited one solution [\[ind-step\]](#ind-step){reference-type="eqref" reference="ind-step"}, and it is clear that successive application of the four local moves generates all possible colour sequences. The proof of [\[statistic-match\]](#statistic-match){reference-type="eqref" reference="statistic-match"} is completed, and with it, the proof of Theorem [Theorem 54](#thm:main){reference-type="ref" reference="thm:main"}. ◻ ## Probability distribution on interlacing triangular arrays The results of this section allow us to conclude that the quantity $$\begin{gathered} \label{col-markov} \mathbb{P}_{\rm col} \left( c^{[m]} \rightarrow c^{[m+1]} \right) \\ := \bm{1}_{c^{[m]} \prec c^{[m+1]}} (-1)^n q^{\binom{nm+n+1}{2}-\binom{nm+1}{2}} \frac{(1-q)^{nm}}{(q;q)_n^{2m+1}} \Psi\left( c^{[m]}; c^{[m+1]} \right) \Big|_{q \mapsto q^{-1}} \frac{ g^{c^{[m+1]}}_{\Delta}\left((m+1)^n;\vec{Q}^{[m+1]}\right) } { g^{c^{[m]}}_{\Delta}\left(m^n;\vec{Q}^{[m]}\right) }\end{gathered}$$ defines a transition probability on colour sequences; this allows one to construct a discrete-time Markov process living on *interlacing triangular arrays*, as we define below. **Definition 60** (Interlacing triangular array). Fix integers $n \geqslant 1$, $N \geqslant 1$. For all $1 \leqslant i \leqslant n$, $1 \leqslant j \leqslant k \leqslant N$, fix positive integers $T^{(i)}_{j,k} \in [1,n]$ subject to two constraints: **(a)** For each $k \in [1,N]$, the collection $\{T^{(i)}_{j,k} \}_{1 \leqslant i \leqslant n,1 \leqslant j \leqslant k}$ is equal to $\{1^k\} \cup \{2^k\} \cup \cdots \cup \{n^k\}$, with the equality being at the level of sets; **(b)** Let the *horizontal coordinate* of the integer $T^{(i)}_{j,k}$ be defined as $c(i,j,k) = iN+j-(N+k)/2$. If $T^{(i)}_{j,k} = T^{(i')}_{j',k} = a \in [1,n]$, $c(i,j,k) < c(i',j',k)$ for some $i,j,i',j'$ and $1< k \leqslant N$, then we assume that there exists $i'',j''$ such that $T^{(i'')}_{j'',k-1} = a$ and $c(i,j,k) < c(i'',j'',k-1) < c(i',j',k)$; this is the *interlacing* property of our array. We refer to such a collection of positive integers as an *interlacing triangular array* of *rank* $n$ and *height* $N$. Let $\mathcal{T}_N(n)$ denote the set of all interlacing triangular arrays of rank $n$ and height $N$. *Remark 61*. Every interlacing triangular array in $\mathcal{T}_N(n)$ is in one-to-one correspondence with a string $c^{[1]} \prec c^{[2]} \prec \cdots \prec c^{[N]}$ of interlacing colour sequences $c^{[k]} \in [1,n]^{nk}$. The colour sequence $c^{[k]}$ is obtained simply by reading off the $k$-th row of the interlacing triangular array. **Example 62** ($n=2$, $N=3$). A permissible interlacing triangular array of rank $2$ and height $3$: $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,2) {$i=1$}; \node at (4,2) {$i=2$}; %%%%%%%%% \node at (0.5,-0.5) {\footnotesize$j=1$}; \node at (1.5,-0.5) {\footnotesize$j=2$}; \node at (2.5,-0.5) {\footnotesize$j=3$}; % \node at (4.5,-0.5) {\footnotesize$j=1$}; \node at (5.5,-0.5) {\footnotesize$j=2$}; \node at (6.5,-0.5) {\footnotesize$j=3$}; %%%%%%%%% \node at (-2.5,0) {\footnotesize$k=1$}; \node at (-2.5,0.5) {\footnotesize$k=2$}; \node at (-2.5,1) {\footnotesize$k=3$}; %%%%%%%%% \draw[densely dotted,->] (-2,0) -- (7,0); \draw[densely dotted,->] (-2,0.5) -- (7,0.5); \draw[densely dotted,->] (-2,1) -- (7,1); % \draw[densely dotted,->] (0.3,-0.3) -- (-1.5,1.5); \draw[densely dotted,->] (1.3,-0.3) -- (-0.5,1.5); \draw[densely dotted,->] (2.3,-0.3) -- (0.5,1.5); % \draw[densely dotted,->] (4.3,-0.3) -- (2.5,1.5); \draw[densely dotted,->] (5.3,-0.3) -- (3.5,1.5); \draw[densely dotted,->] (6.3,-0.3) -- (4.5,1.5); %%%%%%%%% \node at (0,0) {2}; % \node at (-0.5,0.5) {2}; \node at (0.5,0.5) {1}; % \node at (-1,1) {2}; \node at (0,1) {1}; \node at (1,1) {1}; %%%%%%%%% \node at (4,0) {1}; % \node at (3.5,0.5) {2}; \node at (4.5,0.5) {1}; % \node at (3,1) {2}; \node at (4,1) {2}; \node at (5,1) {1}; \end{tikzpicture}\end{aligned}$$ Recall that the integers in the array are written collectively in the form $T^{(i)}_{j,k}$. In this picture, the index $1 \leqslant i \leqslant 2$ increases from left to right and labels individual triangular arrays; the index $1 \leqslant k \leqslant 3$ labels the row in question, and the index $1 \leqslant j \leqslant k$ is used to label diagonals in each triangular array. Reading the numbers in the $k$-th row we recover the elements of the set $\{1^k\} \cup \{2^k\}$, which is constraint **(a)**; also, the interlacing constraint **(b)** holds separately both for the coordinates of the numbers $1$ and $2$. In this example one has $c^{[1]} = (2,1)$, $c^{[2]} = (2,1,2,1)$, $c^{[3]} = (2,1,1,2,2,1)$. **Example 63** ($n=3$, $N=3$). Under the translation red=1, green=2 and blue=3, Figure [\[fig:gue\]](#fig:gue){reference-type="ref" reference="fig:gue"} corresponds with the following interlacing triangular array: $$\begin{aligned} \begin{tikzpicture}[scale=1,baseline=(current bounding box.center),>=stealth] \node at (0,2) {$i=1$}; \node at (4,2) {$i=2$}; \node at (8,2) {$i=3$}; %%%%%%%%% \node at (0.5,-0.5) {\footnotesize$j=1$}; \node at (1.5,-0.5) {\footnotesize$j=2$}; \node at (2.5,-0.5) {\footnotesize$j=3$}; % \node at (4.5,-0.5) {\footnotesize$j=1$}; \node at (5.5,-0.5) {\footnotesize$j=2$}; \node at (6.5,-0.5) {\footnotesize$j=3$}; % \node at (8.5,-0.5) {\footnotesize$j=1$}; \node at (9.5,-0.5) {\footnotesize$j=2$}; \node at (10.5,-0.5) {\footnotesize$j=3$}; %%%%%%%%% \node at (-2.5,0) {\footnotesize$k=1$}; \node at (-2.5,0.5) {\footnotesize$k=2$}; \node at (-2.5,1) {\footnotesize$k=3$}; %%%%%%%%% \draw[densely dotted,->] (-2,0) -- (11,0); \draw[densely dotted,->] (-2,0.5) -- (11,0.5); \draw[densely dotted,->] (-2,1) -- (11,1); % \draw[densely dotted,->] (0.3,-0.3) -- (-1.5,1.5); \draw[densely dotted,->] (1.3,-0.3) -- (-0.5,1.5); \draw[densely dotted,->] (2.3,-0.3) -- (0.5,1.5); % \draw[densely dotted,->] (4.3,-0.3) -- (2.5,1.5); \draw[densely dotted,->] (5.3,-0.3) -- (3.5,1.5); \draw[densely dotted,->] (6.3,-0.3) -- (4.5,1.5); % \draw[densely dotted,->] (8.3,-0.3) -- (6.5,1.5); \draw[densely dotted,->] (9.3,-0.3) -- (7.5,1.5); \draw[densely dotted,->] (10.3,-0.3) -- (8.5,1.5); %%%%%%%%% \node at (0,0) {2}; % \node at (-0.5,0.5) {2}; \node at (0.5,0.5) {1}; % \node at (-1,1) {2}; \node at (0,1) {1}; \node at (1,1) {2}; %%%%%%%%% \node at (4,0) {3}; % \node at (3.5,0.5) {3}; \node at (4.5,0.5) {3}; % \node at (3,1) {3}; \node at (4,1) {3}; \node at (5,1) {1}; %%%%%%%%% \node at (8,0) {1}; % \node at (7.5,0.5) {2}; \node at (8.5,0.5) {1}; % \node at (7,1) {3}; \node at (8,1) {2}; \node at (9,1) {1}; \end{tikzpicture}\end{aligned}$$ In this example one has $c^{[1]} = (2,3,1)$, $c^{[2]} = (2,1,3,3,2,1)$, $c^{[3]} = (2,1,2,3,3,1,3,2,1)$. **Corollary 64**. *Let $T \in \mathcal{T}_n(N)$ be an interlacing triangular array generated by $N$ successive applications of the Markov kernel [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"}. Then the array $T^{(i)}_{j,k}$, $1 \leqslant i \leqslant n$, $1 \leqslant j \leqslant k \leqslant N$, has joint distribution $$\begin{gathered} \label{joint-distr-col} \mathbb{P}_{\rm col} \left( T^{(i)}_{j,k} = c^{[k]}_{(i-1)k+j} ; 1 \leqslant i \leqslant n,\ 1 \leqslant j \leqslant k \leqslant N \right) \\ = \bm{1}_{c^{[1]} \prec \cdots \prec c^{[N]}} (-1)^{nN} q^{\binom{nN+1}{2}} \frac{(1-q)^{n \binom{N}{2}}}{(q;q)_{n}^{N^2}} g^{c^{[N]}}_{\Delta}\left(N^n;\vec{Q}^{[N]}\right) \prod_{i=1}^{N} \Psi \left( c^{[i-1]} ; c^{[i]} \right) \Big|_{q \mapsto q^{-1}}.\end{gathered}$$* ## Explicit calculations In this subsection we document some explicit calculations in the case $n=2$, based on direct application of the formula [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"}. All factors appearing in [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"} are straightforwardly computed, with the exception of the functions $g^{c^{[m]}}_{\Delta}$ and $g^{c^{[m+1]}}_{\Delta}$. To evaluate the latter, we make use of the following factorization result, which follows from [@ABW21 Proposition 11.6.1]: **Proposition 65**. *Choosing $c^{[m]}$ to be the increasing colour sequence $1^m \cup \cdots \cup n^m \in [1,n]^{nm}$, one has the following formula: $$\begin{aligned} \label{factored-g} g^{1^m \cup \cdots \cup n^m}_{\Delta}(m^n;x_1,\dots,x_{nm}) = q^{-m^2 \binom{n}{2}} (1-q^{-1})^{nm} \prod_{k=0}^{n-1} \prod_{1 \leqslant i<j \leqslant m} (q^{-1} x_{mk+j}-x_{mk+i}).\end{aligned}$$* The initial condition [\[factored-g\]](#factored-g){reference-type="eqref" reference="factored-g"}, used in conjunction with the exchange relation [\[invT-g\]](#invT-g){reference-type="eqref" reference="invT-g"}, allows $g^{c^{[m]}}_{\Delta}$ and $g^{c^{[m+1]}}_{\Delta}$ to be efficiently implemented on a computer. We are then in a position to explicitly evaluate the transition probabilities [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"}; see Figure [\[fig:n=2\]](#fig:n=2){reference-type="ref" reference="fig:n=2"}. ## A positivity conjecture Studying the probabilities that appear in Figure [\[fig:n=2\]](#fig:n=2){reference-type="ref" reference="fig:n=2"}, one notices that they are always positive polynomials in $q$ over a common denominator that is easily predicted. Analysis of examples for $n \geqslant 3$ reveals that this structure appears to hold generally. This leads us to formulate the following positivity conjecture: **Conjecture 66**. *Fix integers $m,n \geqslant 1$ and a colour sequence $c^{[m]} \in [1,n]^{nm}$. Let $\mathbb{P}_{\rm col}(c^{[m]})$ denote the probability of arriving at the colour sequence $c^{[m]}$ after $m$ applications of the Markov kernel [\[col-markov\]](#col-markov){reference-type="eqref" reference="col-markov"} to the trivial sequence $c^{[0]}=\emptyset$. Then one has that $$\begin{aligned} \mathbb{P}_{\rm col}\left(c^{[m]}\right) = \mathcal{P}\left(c^{[m]}\right) \cdot \left( \prod_{i=1}^{n} \frac{1-q}{1-q^i} \right)^{m^2} \quad \text{where}\ \ \mathcal{P}\left(c^{[m]}\right) \in \mathbb{N}[q].\end{aligned}$$* # Interlacing triangles and graph colourings {#sec:app} In this appendix we turn to the problem of enumerating the number of elements in the set $\mathcal{T}_N(n)$ from Definition [Definition 60](#def:interlace){reference-type="ref" reference="def:interlace"} (that is, computing the size of the support of $\mathbb{P}_{\rm col}$). This turns out to be a triviality for $n=1$ and $n=2$; for $n=3$ and $n=4$ we are able to conjecture a relation between the cardinality of $\mathcal{T}_N(n)$ and certain graph colourings. An elegant bijective proof of the $n=3$ conjecture was already given in [@GaetzGao]. The $n=4$ case remains open. For $n=1$, an interlacing triangular array consists of a single triangle filled with the number $1$; as there is only one such arrangement, it follows that $|\mathcal{T}_N(1)| = 1$ for all $N \geqslant 1$. For $n=2$, we have the following elementary result: **Proposition 67**. *For all $N \geqslant 1$, $|\mathcal{T}_N(2)| = 2^N$.* *Proof.* The only triangular arrays in rank $2$ which respect the interlacing constraint **(b)** are those in which numbers remain constant along diagonals in the left triangle, and along anti-diagonals in the right; further, once we choose the numbers assigned to diagonals in the left triangle, this completely determines the right one, in view of constraint **(a)** (and the fact that numbers remain constant along its anti-diagonals). Hence there are exactly $2^N$ possibilities. ◻ For $n=3$ and $n=4$ we have no direct results around enumeration. We do, however, make two conjectures relating the cardinality of the set $\mathcal{T}_N(n)$ to colourings of certain families of graphs. **Definition 68**. Given a graph $G$ and an integer $m \geqslant 1$, an $m$-colouring of $G$ is an assignment of a label $l \in [1,m]$ to each vertex $v \in G$ such that $l \not= l'$ if $v$ and $v'$ are connected by an edge. **Conjecture 69**. *Let $G^{\triangle}_N$ denote the triangular graph $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (1,0) {$\bullet$}; \node at (1.5,0) {$\bullet$}; % \node at (0.25,0.5) {$\bullet$}; \node at (0.75,0.5) {$\bullet$}; \node at (1.25,0.5) {$\bullet$}; % \node at (0.5,1) {$\bullet$}; \node at (1,1) {$\bullet$}; % \node at (0.75,1.5) {$\bullet$}; %%%%%%%%%% \draw (0,0) -- (1.5,0); \draw (0.25,0.5) -- (1.25,0.5); \draw (0.5,1) -- (1,1); % \draw (0.25,0.5) -- (0.5,0); \draw (0.5,1) -- (1,0); \draw (0.75,1.5) -- (1.5,0); % \draw (0,0) -- (0.75,1.5); \draw (0.5,0) -- (1,1); \draw (1,0) -- (1.25,0.5); \end{tikzpicture}\end{aligned}$$ where the number of vertices along one side of the triangle is equal to $N+1$. Let $\mathfrak{g}^{\triangle}_N(4)$ denote the number of $4$-colourings of $G^{\triangle}_N$.[^21] We conjecture that $$\begin{aligned} 4\cdot|\mathcal{T}_N(3)| = \mathfrak{g}^{\triangle}_N(4), \qquad \forall\ N \geqslant 1.\end{aligned}$$* **Example 70** ($n=3$, $N=1$). For $n=3$ and $N=1$, triangular tuples just correspond with arrangements of $\{1,2,3\}$ along a line; hence $|\mathcal{T}_1(3)| = |\mathfrak{S}_3| = 6$. On the other hand, the possible $4$-colourings of the graph $G^{\triangle}_1 = \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \end{tikzpicture}$ (in which we fix the top vertex to have label $1$, which means an undercounting by an overall factor of $4$) are $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {2}; \node[below] at (0.5,0) {3}; \end{tikzpicture} % \quad \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {2}; \node[below] at (0.5,0) {4}; \end{tikzpicture} % \quad \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {3}; \node[below] at (0.5,0) {4}; \end{tikzpicture} % \quad \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {3}; \node[below] at (0.5,0) {2}; \end{tikzpicture} % \quad \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {4}; \node[below] at (0.5,0) {2}; \end{tikzpicture} % \quad \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (0.25,0.5) {$\bullet$}; \draw (0,0) -- (0.5,0) -- (0.25,0.5) -- (0,0); \node[above] at (0.25,0.5) {1}; \node[below] at (0,0) {4}; \node[below] at (0.5,0) {3}; \end{tikzpicture}\end{aligned}$$ and indeed $4\cdot|\mathcal{T}_1(3)| = \mathfrak{g}^{\triangle}_1(4)$. **Conjecture 71**. *Let $G^{\varhexstar}_N$ denote the graph $$\begin{aligned} \begin{tikzpicture}[scale=1.3,baseline=(current bounding box.center),>=stealth] \node at (0,0) {$\bullet$}; \node at (0.5,0) {$\bullet$}; \node at (1,0) {$\bullet$}; \node at (1.5,0) {$\bullet$}; %%% \node at (0,0.5) {$\bullet$}; \node at (0.5,0.5) {$\bullet$}; \node at (1,0.5) {$\bullet$}; \node at (1.5,0.5) {$\bullet$}; %%% \node at (0,1) {$\bullet$}; \node at (0.5,1) {$\bullet$}; \node at (1,1) {$\bullet$}; \node at (1.5,1) {$\bullet$}; %%% \node at (0,1.5) {$\bullet$}; \node at (0.5,1.5) {$\bullet$}; \node at (1,1.5) {$\bullet$}; \node at (1.5,1.5) {$\bullet$}; %%%%%%% \draw (0.5,0) -- (0,0.5); \draw (1,0) -- (0,1); \draw (1.5,0) -- (0,1.5); \draw (1.5,0.5) -- (0.5,1.5); \draw (1.5,1) -- (1,1.5); % \draw (1,0) -- (1.5,0.5); \draw (0.5,0) -- (1.5,1); \draw (0,0) -- (1.5,1.5); \draw (0,0.5) -- (1,1.5); \draw (0,1) -- (0.5,1.5); % \draw (0,0) -- (0,1.5); \draw (0.5,0) -- (0.5,1.5); \draw (1,0) -- (1,1.5); \draw (1.5,0) -- (1.5,1.5); % \draw (0,0) -- (1.5,0); \draw (0,0.5) -- (1.5,0.5); \draw (0,1) -- (1.5,1); \draw (0,1.5) -- (1.5,1.5); \end{tikzpicture}\end{aligned}$$ where two vertices share an edge if they are connected via a king move on the chessboard (that is, they a connected via a unit horizontal, vertical, or diagonal step), and the number of vertices along one side of the square is equal to $N+1$. Let $\mathfrak{g}^{\varhexstar}_N(5)$ denote the number of $5$-colourings of $G^{\varhexstar}_N$.[^22] We conjecture that $$\begin{aligned} 5 \cdot |\mathcal{T}_N(4)| = \mathfrak{g}^{\varhexstar}_N(5), \qquad \forall\ N \geqslant 1.\end{aligned}$$* [^1]: Up to shifting and scaling. [^2]: More exactly, the Poissonization parameter tends to infinity. [^3]: These arrays originate from $n$ Gelfand--Tsetlin patterns drawn next to each other. [^4]: This is the first of several assumptions that we make prior to performing our asymptotic analysis. The justification for these assumptions is an *a posteori* one: any sequence of coloured compositions [\[chain-intro\]](#chain-intro){reference-type="eqref" reference="chain-intro"} which violates our assumptions will be shown to take up a vanishingly small part of the measure, in the limit $t \rightarrow \infty$. [^5]: In the case $n=1$, LLT measures degenerate to their Schur counterparts. In that situation, the asymptotic analysis carried through in this text leads to a single GUE corners process, which has a trivial interlacing $1$-colouring. [^6]: We use a tilde when writing our weights for consistency with the work of [@BorodinW]. In that earlier work, which dealt with models based on $U_q(\widehat{\mathfrak{sl}}(n+1))$ rather than $U_q(\widehat{\mathfrak{sl}}(1|n))$ of the current text, the notation $\tilde{L}^{(s)}_{z,q}(\bm{A},b;\bm{C},d)$ was reserved for vertex weights in the *stochastic gauge* (that is, with a sum-to-unity property). While the weights [\[fund-weights\]](#fund-weights){reference-type="eqref" reference="fund-weights"} no longer satisfy a sum-to-unity property, it is easily seen that they have a completely analogous structure to their tilde analogues in [@BorodinW Chapter 2]. [^7]: Indeed, in [@BorodinW Sections 2.2 and 2.5], one has $$\tilde{L}_z(\bm{A},i;\bm{A},i)=\dfrac{(sq^{A_i}-z)q^{A_{(i,n]}}s}{1-s z}$$ [^8]: That is, with strict inequalities in [\[lambda-col\]](#lambda-col){reference-type="eqref" reference="lambda-col"}; this corresponds to the fermionicity of our model. [^9]: Normally, one takes the operators $T_i$ to act on polynomials in the alphabet $(x_1,\dots,x_n)$, since they preserve polynomiality. In this work our partition functions are *a priori* rational, which poses no problem, since the action [\[hecke-poly\]](#hecke-poly){reference-type="eqref" reference="hecke-poly"} is still faithful on $\mathbb{Q}(x_1,\dots,x_n)$. [^10]: A more general version of this integral formula appears in [@ABW21 Proposition 11.3.1]. [^11]: The summation convergence in [\[Gg-cauchy\]](#Gg-cauchy){reference-type="eqref" reference="Gg-cauchy"} is uniform on compact sets as long as the inequalities [\[converge\]](#converge){reference-type="eqref" reference="converge"} are satisfied. [^12]: We state this formula for spectral parameter $sz$, rather than $z$, in order to match with [@ABW21 Theorem 4.3.2]. [^13]: Note that unless $I = \{0\}$, it is now essential for $N$ to be finite, unlike in the definitions of $\mathcal{C}_i(x)$ and $\mathcal{B}_i(x)$ where $N$ is taken to $\infty$. [^14]: The LLT polynomials have two combinatorial definitions; either in terms of ribbon tilings of a Young diagram, or in terms of $n$-tuples of semi-standard Young tableaux. For both of these definitions, we refer to [@ABW21 Sections 9.1 and 9.2]; for the matching of $\mathbb{G}_{\mu/\nu}(\lambda;x_1,\dots,x_p;\infty,\dots,\infty)\Big|_{s \rightarrow 0}$ with the resulting polynomials we refer to [@ABW21 Theorem 9.3.2 (1)]. [^15]: In situations where infinitely many negative parts occur, only finitely many negative integers will be omitted from our coloured signatures; for more information, see equation [\[inf-sig\]](#inf-sig){reference-type="eqref" reference="inf-sig"} and the sentence that follows. [^16]: In particular, this will allow us to Plancherel-specialize this alphabet. [^17]: This diagram is not a partition function in the traditional sense, however it turns out to be quite expedient for our subsequent needs. [^18]: The critical point is the value where the first derivative with respect to $y_i$ vanishes. [^19]: We require that each integration contour $C_i$ may be freely deformed to a contour passing through the corresponding critical point $Q_i$, along which the real part of the exponent decreases as one travels away from $Q_i$. One choice that meets this requirement is to take the $C_i$ to be concentric circles of radii $Q_i$, for all $1 \leqslant i \leqslant nm$. [^20]: We will be slightly informal here, omitting the proofs of the tail estimates justifying the below approximations. The latter are fairly standard, and have already been applied numerous times in the literature. [^21]: *The sequence $\mathfrak{g}^{\triangle}_N(4)$ appears as A153467 in the Online Encyclopaedia of Integer Sequences; `https://oeis.org/A153467`.* [^22]: *The sequence $\mathfrak{g}^{\varhexstar}_N(5)$ appears as A068294 in the Online Encyclopaedia of Integer Sequences; `https://oeis.org/A068294`.*

arxiv_math

--- abstract: | A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone measures is proved. Without any presumptive condition, we obtain a necessary and sufficient condition for the ordered pair $(\mu, \nu)$ of finite monotone measures to have the so-called Radon-Nikodym property related to a nonnegative measurable function $f$. If $\nu$ is null-continuous and weakly null-additive, then $f$ is uniquely determined almost everywhere by $\nu$ and thus is called the Radon-Nikodym derivative of $\mu$ w.r.t. $\nu$. For $\sigma$-finite monotone measures, a Radon-Nikodym type theorem is also obtained under the assumption that the monotone measures are lower continuous and null-additive. *Keywords:*Radon-Nikodym theorem; Monotone measure; Choquet integral; lower continuous; null-additive author: - | Yao Ouyang[^1]\ *Faculty of Science, Huzhou Teacher's College, Huzhou, Zhejiang 313000,\ *People's Republic of China\ Jun Li[^2]\ *State Key Laboratory of Media Convergence and Communication,\ *Communication University of China, Beijing 100024, China**** title: "**A Radon-Nikodym theorem for monotone measures** " --- # Introduction Suppose that $\nu$ is a $\sigma$-additive measure and $f$ is a nonnegative integrable function. The measure $\mu$ defined by $$\mu(A)=\int_A fd\nu$$ for all measurable sets $A$ is said to be the indefinite integral of $f$ w.r.t. $\nu$. In this case, $\mu$ is absolutely continuous w.r.t. $\nu$. Under what conditions a measure can be expressed as the indefinite integral w.r.t. another measure is quite interesting. This pertains to the scope of the Radon-Nikodym theorem. Radon-Nikodym theorem, one of the most important theorems in measure theory, states that $\mu$ is the indefinite integral w.r.t. $\nu$ if and only if $\mu$ is absolutely continuous w.r.t. $\nu$, see Halmos [@Hal68] for example. We note that the Radon-Nikodym theorem has various proofs and all these proofs are highly dependent on the $\sigma$-additivity of measures. When one of the measures is only finitely additive, the Radon-Nikodym theorem does not hold in general. Since in this case, the Hahn decomposition does not hold in general and the implications "$\nu(A)=0\Rightarrow\mu(A)=0$" and "$\nu(A_n)\to 0\Rightarrow\mu(A_n)\to 0$" are not equivalent. Various conditions [@BelHag88; @CanMar92; @May79] have been derived in the literature for the validity of finitely additive measures-based Radon-Nikodym theorem. For example, in [@BelHag88] the Radon-Nikodym theorem was proved under absolute continuity and a property called Hahn separation (a variant of Hahn decomposition). Graf [@Gra80] proved a Radon-Nikodym theorem for the Choquet integral w.r.t. capacities (lower continuous subadditive monotone measure), while Nguyen et al. [@Ngu06; @NguNguWan97] investigated a Radon-Nikodym theorem for $\sigma$-subadditive monotone measures. Greco [@Gre81] (see also [@CanVol02]) obtained necessary and sufficient conditions of this theorem for null-additive monotone measures. Roughly speaking, these conditions include a variant of Hahn decomposition and some other conditions. We also note that Rébillé [@Reb13] discussed the superior Radon-Nikodym derivative of a set function w.r.t. a $\sigma$-additive measure. In this paper, a new version of Radon-Nikodym theorem for the Choquet integral is proved. It should be stressed that our result generalizes the corresponding ones in [@Gra80; @Gre81; @Ngu06]. Concretely, we introduce the concept of decomposition property of monotone measures in Section 3. This property concerns an ordered pair $(\mu, \nu)$ of monotone measures and a decreasing family $\{A_\alpha\}_{\alpha\in\mathbb{Q}_+}$ of measurable sets and is a natural generalization of the Hahn decomposition for $\sigma$-additive measures. The decomposition property together with $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$ is demonstrated to be the necessary and sufficient conditions for $(\mu, \nu)$ to have Radon-Nikodym property based on the Choquet integral (i.e., there is a nonnegative measurable function $f$ such that $\mu(E)=\int_E fd\nu$ for each measurable set $E$), where $\mu, \nu$ are finite monotone measures. This result is obtained without any presumptive condition other than the monotonicity of set functions, thus it is a generalization of the results of Greco [@Gre81] and Nguyen et al. [@NguNguWan97]. When $\nu$ is further weakly null-additive (which is weaker than subadditive) and null-continuous (which is implied by lower continuous), then the function $f$ is unique a.e.\[$\nu$\] and is called the Radon-Nikodym derivative of $\mu$ w.r.t. $\nu$. Thus, Graf's result is also generalized. The Radon-Nikodym theorem for $\sigma$-finite monotone measures are considered in Section 4. The existence and uniqueness of the Radon-Nikodym derivative is obtained when $\mu, \nu$ are lower continuous and $\nu$ is further null-additive. # Preliminaries {#Sec-Preliminaries} Let $(U, {\mathcal U})$ denote a measurable space, that is, a nonempty set $U$ equipped with a $\sigma$-algebra ${\mathcal U}$ of subsets of $U$. A subset $A$ of $U$ is called measurable (w.r.t. ${\mathcal U}$) if $A\in{\mathcal U}$. A nonnegative extended real-valued function $f\colon U\to \overline{\mathbb{R}}_{+}$ is called measurable if for each $\alpha\in [0, +\infty]$, $\{f\geq\alpha\}\in {\mathcal U}$ (here $\{f\geq\alpha\}$ is the abbreviation for $\{t\in U\, | \, f(t)\geq\alpha\}$). **Definition 1**. A set function $\mu: {\mathcal U}\rightarrow \overline{\mathbb{R}}_{+}$ is called a *monotone measure* if it satisfies the following two conditions: (i) $\mu(\emptyset)=0$; (vanishing at $\emptyset$) (ii) $\mu(A) \leq \mu(B)$ whenever $A \subset B$ and $A, B \in {\mathcal{U}}$. (monotonicity) The triple $(U, {\mathcal U}, \mu)$ is called a *monotone measure space*. A monotone measure $\mu$ on $(U, {\mathcal U})$ is said to be (i) *finite* if $\mu(U)<\infty$; (ii) *$\sigma$-finite* if there is $\{U_n\}_{n=1}^\infty\subset{\mathcal U}$ with $U_n\nearrow U$ (*i.e.*, $U_1\subset U_2\subset\cdots\subset U_n\subset\cdots$ and $\bigcup\limits_{n=1}^\infty U_n=U$) such that $\mu(U_n)<\infty$ for each $n$. Let $f$ be a nonnegative measurable function, $\nu$ be a monotone measure and $A$ be a measurable set. The Choquet integral of $f$ w.r.t. $\nu$ is defined as follows, see [@Ch53; @Den94; @Pap95]. **Definition 2**. The Choquet integral of $f$ w.r.t. $\nu$ on $A$ is given by $$\int_{A} fd\nu=\int_0^\infty \nu(\{f\geq\alpha\}\cap A)d\alpha,$$ where the integral on the right side is the improper Riemann integral. When $A=U$, we write $\int fd\nu$ instead of $\int_{U} fd\nu$. If $\int fd\nu<\infty$, then $f$ is called Choquet integrable w.r.t. $\nu$ on $U$. When $\nu$ is a $\sigma$-additive measure, the Choquet integral coincides with the Lebesgue integral. Throughout this paper, unless otherwise stated, all the considered integrals are assumed to be the Choquet integrals. The following are some basic properties of the Choquet integrals ([@Den94; @Pap95; @WanKli09][\[Prot-BasicPropofChoInt\]]{#Prot-BasicPropofChoInt label="Prot-BasicPropofChoInt"}): **Proposition 3**. *Let $(U, {\mathcal U}, \nu)$ be a monotone measure space and $f, g$ be nonnegative measurable functions. Then* *(i) $\int_A fd\nu=0$ whenever $\nu(A)=0$;* *(ii) $f\leq g$ implies $\int fd\nu\leq\int gd\nu$; (monotonicity)* *(iii) $\int cfd\nu=c\int fd\nu$ for any constant $c\geq 0$; (homogeneity)* *(iv) $\int_A\chi_A d\nu=\nu(A), \forall\, A\in{\mathcal U}$, where $\chi\sb{A}$ denotes the characteristic function of $A$;* *(v) $\int_A f d\nu=\int f\chi_A d\nu$;* *(vi) $\int_A f d\nu=\lim\limits_{n\to\infty}\int_A (f\wedge n)d\nu$.* We only give the proof of (vi). For any $A\in{\mathcal U}$, $$\begin{aligned} \int_A f d\nu &=& \int_0^\infty\nu(A\cap\{f\geq \alpha\})d\alpha=\lim_{n\to\infty}\int_0^n\nu(A\cap\{f\geq \alpha\})d\alpha\\ &=& \lim_{n\to\infty}\int_0^n\nu(A\cap\{f\wedge n\geq \alpha\})d\alpha\\ &=& \lim_{n\to\infty}\left(\int_0^n\nu(A\cap\{f\wedge n\geq \alpha\})d\alpha+\int_n^\infty\nu(A\cap\{f\wedge n\geq \alpha\})d\alpha\right)\\ &=&\lim_{n\to\infty}\int_0^\infty\nu(A\cap\{f\wedge n\geq \alpha\})d\alpha= \lim_{n\to\infty}\int_A (f\wedge n)d\nu.\end{aligned}$$   $\Box$ Two functions $f, g$ on $U$ are said to be *comonotone* if for any $t_1, t_2\in U$, $(f(t_1)- f(t_2))(g(t_1)- g(t_2))\geq 0$. The following proposition is known as *comonotonic additivity* of Choquet integral, which is a distinguishing feature of the Choquet integral, see [@Den94; @Sch86]. **Proposition 4**. *Let $(U, {\mathcal U}, \nu)$ be a monotone measure space and $f, g$ be nonnegative measurable functions. If $f$ and $g$ are comonotone, then $$\int(f+g)d\nu=\int fd\nu+\int gd\nu.$$* Note that two increasing (decreasing, resp.) functions are comonotone, and a constant function $c$ is comonotone with arbitrary functions. Moreover, for any function $f$ and any constant $c$, $(f-c)\vee 0$ and $f\wedge c$ are comonotone, where $(f\vee c)(t)=\max\{f(t), c\}$ and $(f\wedge c)(t)=\min\{f(t), c\}$. # Radon-Nikodym theorem for finite monotone measures In this section we present a new version of Radon-Nikodym theorem for finite monotone measures. To do this, we introduce the following concept of *decomposition property* relating to an ordered pair of monotone measures. **Definition 5**. Let $\mu, \nu$ be two monotone measures on $(U, {\mathcal U})$. The ordered pair $(\mu, \nu)$ is said to have *decomposition property* if there is a decreasing family $\{A_\alpha\}_{\alpha\in\mathbb{Q}^{+}}$ of measurable sets with $A_0=U$ such that $$\begin{aligned} \label{decompro-inequality-1} \alpha\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big) &\leq& \mu(A\cap A_\alpha)-\mu(A\cap A_\beta) \\ \label{decompro-inequality-2} &\leq& \beta\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big)\end{aligned}$$ holds for any $A\in{\mathcal U}$ with finite measures for $\mu$ and $\nu$, and any $\alpha, \beta\in \mathbb{Q}^{+}$ with $\alpha<\beta$, where $\mathbb{Q}^{+}$ is the set of all nonnegative rational numbers. **Example 6**. For any $\sigma$-additive finite measures $\mu, \nu$, the ordered pair $(\mu, \nu)$ has decomposition property w.r.t. $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$, where $(A_\alpha, A^{\rm c}_\alpha)$ is a Hahn decomposition of the signed measure $\mu-\alpha\nu$ (see Remark [Remark 12](#Nte-sigmaaddmeasure){reference-type="ref" reference="Nte-sigmaaddmeasure"} for detail). This is why we call the ordered pair $(\mu, \nu)$ having decomposition property if $\mu, \nu$ satisfy inequalities ([\[decompro-inequality-1\]](#decompro-inequality-1){reference-type="ref" reference="decompro-inequality-1"}) and ([\[decompro-inequality-2\]](#decompro-inequality-2){reference-type="ref" reference="decompro-inequality-2"}). **Lemma 7**. *Let $(\mu, \nu)$ have decomposition property w.r.t. $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$. If $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$, then $\lim\limits_{\alpha\to\infty}\alpha\nu(A_\alpha)=0$.* We can assume that $\mu(A_\alpha)\vee\nu(A_\alpha)<\infty$. From the first inequality in Definition [Definition 5](#Dfnt-decompro){reference-type="ref" reference="Dfnt-decompro"}, we get $$\alpha\Big(\nu(A_\alpha)-\nu(A_\beta)\Big)\leq \mu(A_\alpha)-\mu(A_\beta)$$ holds for any $\alpha<\beta$, and hence we have $\alpha\nu(A_\alpha)\leq\mu(A_\alpha)$ by letting $\beta\to\infty$ as $\lim\limits_{\beta\to\infty}\mu(A_\beta)\vee\nu(A_\beta)=0$. Thus we also have $\alpha\nu(A_\alpha)$ tends to $0$ whenever $\alpha$ tends to $\infty$.   $\Box$ Now we show our main result --- a version of Radon-Nikodym theorem for finite monotone measures. **Theorem 8**. *Let $\mu, \nu$ be two finite monotone measures on $(U, {\mathcal U})$. Then the following two assertions are equivalent:* *(i) The ordered pair $(\mu, \nu)$ has *Radon-Nikodym property*, i.e., there is a nonnegative measurable function $f\colon U\to \overline{\mathbb{R}}_{+}$ such that $$\label{Eq_R-N} \mu(A)=\int_A fd\nu, \ \ \forall\, A\in{\mathcal U}.$$* *(ii) The ordered pair $(\mu, \nu)$ has decomposition property w.r.t. a sets system $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$ and $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee \nu(A_\alpha)=0$.* (i)$\Longrightarrow$ (ii). Suppose there is a nonnegative measurable function $f$ such that Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}) holds, then $f$ is Choquet integrable w.r.t. $\nu$ on $U$, i.e., $\int fd\nu=\mu(U)<\infty$. First we show that $(\mu, \nu)$ has decomposition property. Put $A_\alpha=\{f\geq\alpha\}$, then $\{A_\alpha\}$ is decreasing and $A_0=U$. For any $A\in{\mathcal U}$ with finite measure (i.e., $\mu(A)\vee\nu(A)<\infty$), and any $\alpha<\beta$ it holds $$\begin{aligned} \mu(A\cap A_\alpha)-\mu(A\cap A_\beta)&=&\int_{A\cap A_\alpha}fd\nu-\int_{A\cap A_\beta}fd\nu\\ &=&\int_0^\infty \Big(\nu(A\cap A_\alpha\cap A_t)-\nu(A\cap A_\beta\cap A_t)\Big)dt\\ &\geq&\int_0^\alpha \Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big)dt\\ &=&\alpha\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big).\end{aligned}$$ On the other hand, we have $$\begin{aligned} \mu(A\cap A_\alpha)-\mu(A\cap A_\beta)&=&\int_0^\beta \Big(\nu(A\cap A_\alpha\cap A_t)-\nu(A\cap A_\beta)\Big)dt\\ &&+ \int_\beta^\infty \Big(\nu(A\cap A_t)-\nu(A\cap A_t)\Big)dt\\ &\leq&\int_0^\beta \Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big)dt\\ &=&\beta\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big).\end{aligned}$$ The assertion $\lim\limits_{\alpha\to\infty}\nu(A_\alpha)=0$ follows from $\alpha\nu(A_\alpha)\leq\int_{A_\alpha}f d\nu=\mu(A_\alpha)<\infty$. Since $f=((f-n)\vee 0)+ (f\wedge n)$ and $f=((f-n)\vee 0)$ and $(f\wedge n)$ are comonotone, then $$\int fd\nu=\int ((f-n)\vee 0)d\nu+\int (f\wedge n)d\nu$$ holds for each $n$. Therefore, from $\int fd\nu=\lim\limits_{n\to\infty}\int (f\wedge n)d\nu$ and noting that $\int fd\nu<\infty$, we conclude that $$\lim\limits_{n\to\infty}\int ((f-n)\vee 0)d\nu=0.$$ Also, $$\begin{aligned} \mu(A)=\int_A fd\nu &=& \int_A ((f-n)\vee 0)d\nu+\int_A (f\wedge n)d\nu \\ &\leq& \int ((f-n)\vee 0)d\nu+ n\nu(A)\end{aligned}$$ for each $A\in\mathcal{U}$. Specifically, $$\mu(A_n)\leq \int ((f-n)\vee 0)d\nu+ n\nu(A_n)$$ for each $n$, it follows that $\lim\limits_{n\to\infty}\mu(A_n)=0$. Therefore, $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)=0$ as $\{A_\alpha\}_{\alpha\geq 0}$ is a decreasing family of measurable sets. (ii)$\Longrightarrow$(i). Suppose that $(\mu, \nu)$ has decomposition property and $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$ is the corresponding sets system satisfying $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$. We show that there is a nonnegative measurable function $f\colon U\to \overline{\mathbb{R}}_{+}$ such that Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}) holds. Define $f\colon U\to [0, \infty]$ as $$\label{Eq-DecomFun} f(x)=\sup\{\alpha\,|\,x\in A_\alpha\}.$$ Then $f$ is measurable and $\mu(\{f=\infty\})=0$ as $\{f=\infty\}=\bigcap\limits_{\alpha\in\mathbb{N}}A_\alpha$. For each positive integer $n$, define $$f_n(x)=\left\{ \begin{array}{ll} \frac{k-1}{2^n}, &\mbox{if\,\,} x\in A_{\frac{k-1}{2^n}}\setminus A_{\frac{k}{2^n}}, k=1, 2, \cdots, n2^n, \\ n, &\mbox{if\,\,} x\in A_n. \end{array} \right.$$ Then $\{f_n\}_{n\in\mathbb{N}}$ is an increasing sequence and $f\wedge n-\frac{1}{2^n}\leq f_n\leq f$. Since $f_n$ can be rewritten as $$f_n=\frac{1}{2^n}\sum_{k=1}^{n\cdot 2^n}\chi_{A_\frac{k}{2^n}},$$ for any given $A\in{\mathcal U}$, we have $$\begin{aligned} & & \int_A f_nd\nu = \frac{1}{2^n}\sum_{k=1}^{n\cdot 2^n}\nu(A\cap A_\frac{k}{2^n})\\ &=& \sum_{k=1}^{n\cdot 2^n-1}\frac{k}{2^n}\Big(\nu(A\cap A_\frac{k}{2^n})-\nu(A\cap A_\frac{k+1}{2^n})\Big)+n\nu(A\cap A_n)\\ &\leq& \sum_{k=1}^{n\cdot 2^n-1}\Big(\mu(A\cap A_\frac{k}{2^n})-\mu(A\cap A_\frac{k+1}{2^n})\Big)+n\nu(A\cap A_n)\\ &=& \mu(A\cap A_\frac{1}{2^n})-\mu(A\cap A_n)+ n\nu(A\cap A_n).\end{aligned}$$ It follows from the assumption and Lemma [Lemma 7](#Lemm-lemma4-2){reference-type="ref" reference="Lemm-lemma4-2"} that $$\lim\limits_{n\to\infty}\int_A f_nd\nu\leq \lim\limits_{n\to\infty}\mu(A\cap A_\frac{1}{2^n})\leq\mu(A).$$ The inequality $f\wedge n\leq f_n+ \frac{1}{2^n}$ implies that $$\int_A (f\wedge n)d\nu\leq \int_A f_nd\nu+ \int_A \frac{1}{2^n}d\nu= \int_A f_nd\nu+ \frac{1}{2^n}\nu(A).$$ By virtue of Proposition [\[Prot-BasicPropofChoInt\]](#Prot-BasicPropofChoInt){reference-type="ref" reference="Prot-BasicPropofChoInt"}(vi) we get $$\int_A fd\nu=\lim\limits_{n\to\infty}\int_A (f\wedge n)d\nu\leq\lim\limits_{n\to\infty}\int_A f_nd\nu\leq\mu(A).$$ On the other hand, $$\begin{aligned} & & \int_A f_nd\nu = \frac{1}{2^n}\sum_{k=1}^{n\cdot 2^n}\nu(A\cap A_\frac{k}{2^n})\\ &=& \sum_{k=1}^{n\cdot 2^n-1}\frac{k+1}{2^n}\Big(\nu(A\cap A_\frac{k}{2^n})-\nu(A\cap A_\frac{k+1}{2^n})\Big) \\ & & \ \ \ \ \ \ \ \ \ + \ (n+\frac{1}{2^n})\nu(A\cap A_n)-\frac{1}{2^n}\nu(A\cap A_{\frac{1}{2^n}})\\ &\geq& \sum_{k=1}^{n\cdot 2^n-1}\Big(\mu(A\cap A_\frac{k}{2^n})-\mu(A\cap A_\frac{k+1}{2^n})\Big)\\ & & \ \ \ \ \ \ \ \ \ + \ (n+\frac{1}{2^n})\nu(A\cap A_n)-\frac{1}{2^n}\nu(A\cap A_{\frac{1}{2^n}})\\ &\geq& \mu(A\cap A_\frac{1}{2^n})-\mu(A\cap A_n)+(n+\frac{1}{2^n})\nu(A\cap A_n)-\frac{1}{2^n}\nu(A\cap A_{\frac{1}{2^n}}).\end{aligned}$$ By the decomposition property we have $$\begin{aligned} \mu(A)-\mu(A\cap A_\frac{1}{2^n}) &=&\mu(A\cap A_0)-\mu(A\cap A_\frac{1}{2^n})\\ &\leq& \frac{1}{2^n}(\nu(A\cap A_0)-\nu(A\cap A_\frac{1}{2^n}))\to 0\, (n\to\infty),\end{aligned}$$ *i.e.*, $\mu(A\cap A_\frac{1}{2^n})\to\mu(A)\, (n\to\infty)$. Since both $\mu(A_n)$ and $n\nu(A_n)$ tend to $0$ when $n\to\infty$, it then holds that $$\int_A fd\nu\geq\lim\limits_{n\to\infty}\int_A f_nd\nu\geq\mu(A)$$ as $f\geq f_n$ for each $n$. Thus we reach Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}), $$%\label{Eq_R-N} \mu(A)=\int_A fd\nu, \ \ \forall\, A\in{\mathcal U}.$$ The proof is complete.   $\Box$ The Radon-Nikodym theorem for classical measures concerns the absolute continuity of measures [@Hal68]. For monotone measures, there are various types of absolute continuity (see [@LMZ2010; @OuyLi04; @Wan96; @Wang1996a]). Let $\mu, \nu$ be two monotone measures on $(U, {\mathcal U})$. (1) If for any $A\in{\mathcal U}$, $\nu(A)=0$ implies $\mu(A)=0$, then we say that $\mu$ is absolutely continuous w.r.t. $\nu$ and denoted by $\mu\ll\nu$. (2) If for each $\epsilon>0$ there is a $\delta>0$ such that $\mu(A)<\epsilon$ for all sets $A\in{\mathcal U}$ satisfying $\nu<\delta$, then we say that $\mu$ is strongly absolutely continuous w.r.t. $\nu$ and denoted by $\mu\ll^{s}\nu$ ([@LMZ2010]). Obviously, $\mu\ll^{s}\nu$ implies $\mu\ll\nu$, but the converse is not true. Observe that Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}(ii) implies $\mu\ll\nu$ and $\mu\ll^{s}\nu$. In fact, assume $\nu(A)=0$. From the second inequality in Definition [Definition 5](#Dfnt-decompro){reference-type="ref" reference="Dfnt-decompro"}, we take $\alpha=0, \beta>0$, then $$\mu(A)- \mu(A\cap A_{\beta})\leq \beta\Big(\nu(A)- \nu(A\cap A_{\beta})\Big),$$ which implies $\mu(A)- \mu(A\cap A_{\beta})=0$ for any $\beta>0$. Therefore, $\mu(A)=0$ as $\lim_{\beta\to\infty}\mu(A_\beta)=0$. Similarly, $\mu\ll^{s}\nu$ is also true. Thus, we obtain necessary conditions that the Radon-Nikodym theorem in classical measure theory remains valid for the Choquet integral w.r.t. monotone measures (see also [@Wang1996a]). **Corollary 9**. *Let $\mu, \nu$ be two finite monotone measures on $(U, {\mathcal U})$. If there is a nonnegative measurable function $f\colon U\to \overline{\mathbb{R}}_{+}$ such that Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}) holds, i.e., $$\mu(A)=\int_A fd\nu, \ \ \forall\, A\in{\mathcal U},$$ then $\mu\ll\nu$ and $\mu\ll^{s}\nu$.* Note that the measurable function $f$ in Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"} is not unique in general. **Example 10**. Let $U$ be the set of all positive integers, $\mathcal U$ the power set of $U$ and $$A_\alpha=\left \{ \begin {array}{ll} U, &\quad \text{if}\ \alpha\in [0, 1]\cap\mathbb{Q}, \\[1mm] \{2, 4, 6, \cdots\}, &\quad \text{if}\ \alpha\in (1, 2]\cap\mathbb{Q}, \\[1mm] \emptyset, &\quad \text{if}\ \alpha\in (2, \infty)\cap\mathbb{Q}. \end {array} \right.$$ Define $$\mu(A)=\nu(A)=\left \{ \begin {array}{ll} 1, &\quad \text{if\,} A=U, \\[1mm] 0, &\quad \text{otherwise.} \end {array} \right.$$ It is routine to verify that $(\mu, \nu)$ has decomposition property w.r.t. $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$. The condition $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$ is obviously satisfied. By Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"} there exists a nonnegative function $f$ such that $\mu(A)=\int_A fd\nu$ for each $A$. In fact $$f_1(x)=\sup\{\alpha \ | \ x\in A_\alpha\}=\left \{ \begin {array}{ll} 1, &\quad \text{if\, $x$ is odd} \\[1mm] 2, &\quad \text{if\, $x$ is even} \end {array} \right.$$ is such a function. Note that $f_2$ defined by $$f_2(x)=\left \{ \begin {array}{ll} 1, &\quad \text{if\, $x$ is even} \\[1mm] 2, &\quad \text{if\, $x$ is odd} \end {array} \right.$$ also satisfies $\mu(A)=\int_A f_2d\nu$ (there are in fact infinitely many such functions). Interestingly, $f_1$ and $f_2$ are different at every point and thus $\nu(\{f_1\neq f_2\})=\nu(U)=1$. To ensure the uniqueness of $f$, we have to impose some additional conditions. Recall that a monotone measure $\mu$ is said to be (i)  *weakly null-additive* [@WanKli09], if $\mu(A_{1}\cup A_{2})=0$ for any $A_{1}, A_{2}\in{\mathcal U}$ with $\mu(A_{1})=\mu(A_{2})=0$; (ii)  *null-continuous* [@AsaMuro2006], if $\mu(\bigcup_{n=1}^{\infty}A_n) = 0$ for every increasing sequence $\{A_n\}_{n\in N} \subset {\mathcal{A}}$ such that $\mu(A_n) = 0, n=1,2,\cdots.$ The monotone measure $\mu$ is both weakly null-additive and null-continuous if and only if $\mu(\bigcup_{n=1}^{\infty}A_n) = 0$ whenever $\{A_n\}_{n\in \mathbb{N}} \subset {\mathcal{A}}$ and $\mu(A_n) = 0, n=1,2,\cdots$, see [@Li2000]. Such a monotone measure $\mu$ is called to have *property* ($\sigma$), i.e., the set of all $\mu$-null sets is a $\sigma$-ideal, see [@CanVol02]. **Proposition 11**. *Let $\mu, \nu$ be two monotone measures and $\nu$ is weakly null-additive and null-continuous. If measurable functions $f, g\colon U\to [0, \infty]$ satisfy Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}), i.e., $$\mu(A)=\int_A fd\nu=\int_A gd\nu, \ \ \forall\,A\in{\mathcal U},$$ then $f=g \ a.e.[\nu]$ (i.e., $\nu(\{f\neq g\})=0$).* Let $A=\{f>g\}$ and $A_n=\{f>g+\frac{1}{n}\}$, then $A_n\nearrow A$. We conclude that $\nu(A)= 0$, otherwise there is some $n$ such that $\nu(A_n)>0$ as $\nu$ is null-continuous. Then $$\begin{aligned} \int_{A_n}fd\nu\geq\int_{A_n}(g+\frac{1}{n})d\nu &=& \int_{A_n}gd\nu+\int_{A_n}\frac{1}{n}d\nu\\ &=& \int_{A_n}gd\nu+\frac{1}{n}\nu(A_n)>\int_{A_n}gd\nu,\end{aligned}$$ a contradiction. It holds similarly that $\nu(B)=0$, where $B=\{g>f\}$. Since $\nu$ is weakly null-additive, we have $\nu(A\cup B)=0$. As a consequence $\nu(\{f\neq g\}) = \nu(A\cup B)=0.$   $\Box$ Note: We also obtain $f=g \ a.e.[\mu]$ (i.e., $\mu(\{f\neq g\})=0$). Now by using Proposition [Proposition 11](#Prot-uniqueness){reference-type="ref" reference="Prot-uniqueness"} we can propose the concept of Radon-Nikodym derivative for monotone measures. In Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}, we consider that $\nu$ is weakly null-additive and null-continuous (i.e., $\nu$ has *property* ($\sigma$)), then the measurable function $f$ on $U$ for which Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}) holds is called a Radon-Nikodym derivative (or Radon-Nikodym density) of $\mu$ w.r.t. $\nu$, and denoted by $\frac{d\mu}{d\nu}$ and Eq. ([\[Eq_R-N\]](#Eq_R-N){reference-type="ref" reference="Eq_R-N"}) will be written as $f=\frac{d\mu}{d\nu}$ or $d\mu =fd\nu$. Thus, the preceding Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"} asserts that if the ordered pair $(\mu, \nu)$ satisfies the condition (ii) and $\nu$ has *property* ($\sigma$), then any two Radon-Nikodym derivatives of $\mu$ w.r.t. $\nu$ are equal $a.e. \ [\nu]$, and so the notation $\frac{d\mu}{d\nu}$ is only ambiguous up to a $\nu$-null set. We can discuss some properties of Radon-Nikodym derivative. For example, suppose that $\mu, \lambda$ and $\nu$ are finite monotone measures on $(U, {\mathcal U})$ and $\nu$ has *property* ($\sigma$), and $(\mu,\nu)$ and $(\lambda,\nu)$ satisfy the condition (ii) in Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}, respectively, then $\frac{d\mu}{d\nu}$ and $\frac{d\lambda}{d\nu}$ exist. We write $\frac{d\mu}{d\nu}=f$ and $\frac{d\lambda}{d\nu}=g$, if $f$ and $g$ are comonotone, then we have $$\frac{d(\mu+\lambda)}{d\nu}= \frac{d\mu}{d\nu} + \frac{d\lambda}{d\nu} \ \ a.e.[\nu].$$ **Remark 12**. Let $\mu, \nu$ be $\sigma$-additive finite measures. For each nonnegative rational number $\tau$ the signed measure $\mu-\tau\nu$ has a Hahn decomposition, *i.e.*, there is a measurable set $A_\tau$ such that $A_\tau$ is a positive set of $\mu-\tau\nu$ and $A^{\rm c}_\tau$ is a negative set of $\mu-\tau\nu$. Since $A_\tau$ is also a positive set of $\mu-\gamma\nu$ for any $\gamma\leq\tau$, without loss of generality we can suppose that $\{A_\tau\}_{\tau\in \mathbb{Q}^{+}}$ is decreasing. If $\tau=0$, then $U$ itself is a positive set of $\mu-\tau\nu=\mu$ and so $A_0=U$. Let $\alpha<\beta$ be given. Since $A_\alpha$ is a positive set of $\mu-\alpha\nu$, for any $A\in{\mathcal U}$, $(\mu-\alpha\nu)(A\cap (A_\alpha\setminus A_\beta))\geq 0$, *i.e.*, $$\alpha\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big)\leq \mu(A\cap A_\alpha)-\mu(A\cap A_\beta).$$ Similarly, $A^{\rm c}_\beta$ is a negative set of $\mu-\beta\nu$ and hence is a positive set of $\beta\nu-\mu$. For any $A\in{\mathcal U}$, $(\beta\nu-\mu)(A\cap (A_\alpha\setminus A_\beta))\geq 0$, *i.e.*, $$\mu(A\cap A_\alpha)-\mu(A\cap A_\beta)\leq\beta\Big(\nu(A\cap A_\alpha)-\nu(A\cap A_\beta)\Big).$$ Thus $(\mu, \nu)$ has decomposition property and $(A_\alpha, A^{\rm c}_\alpha)$ is a Hahn decomposition of the signed measure $\mu-\alpha\nu$. If $\mu, \nu$ further satisfy $\mu\ll\nu$, then we have $\lim\limits_{\alpha\to\infty}\nu(A_\alpha)=0$ and hence $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)=0$. In fact, if $\lim\limits_{\alpha\to\infty}\nu(A_\alpha)>0$, then $(\mu-\alpha\nu)(A_\alpha)\to-\infty$ when $\alpha\to\infty$, contradicting with the fact that $A_\alpha$ is a positive set of $\mu-\alpha\nu$. On the other hand, if $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$, then $\mu\ll\nu$. To see this, let $\nu(A)=0$ be given. For any $\beta>0$, $$\mu(A\cap A_0)-\mu(A\cap A_\beta)\leq\beta(\nu(A\cap A_0)-\nu(A\cap A_\beta))=0,$$ which implies $\mu(A)=0$ as $A_0=U$ and $\mu(A_\beta)\to 0\, (\beta\to\infty)$. In conclusion, for two $\sigma$-additive finite measures $\mu, \nu$, the pair $(\mu,\nu)$ has decomposition property, and $\mu\ll\nu$ if and only if $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)\vee\nu(A_\alpha)=0$. As a special case of Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}, we thus obtain a classical Radon-Nikodym theorem. **Corollary 13**. *Let $\mu, \nu$ be $\sigma$-additive finite measures on $(U, {\mathcal U})$. There exists a nonnegative measurable function $f\colon U\to\overline{\mathbb{R}}_{+}$ such that $$\mu(A)=\int_A fd\nu, \ \ \forall\, A\in{\mathcal U}$$ if and only if $\mu\ll\nu$.* Let $\mu, \nu$ be bounded finitely additive measures such that for every $\epsilon>0$ there exists a finite decomposition of $U$, $\{A_1,\cdots, A_n\}\subset{\mathcal U}$, satisfying $\mu(A_i)\vee\nu(A_i)<\epsilon$. For such measures, Candeloro and Martellotti proved in [@CanMar92] that if $\mu\ll^s\nu$ and the set $\{(\mu(A), \nu(A))| A\in{\mathcal U}\}$ is closed, then $(\mu, \nu)$ satisfies all requirements in Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}(ii) and thus $(\mu, \nu)$ has Radon-Nikodym property. **Remark 14**. There are several papers dealing with Radon-Nikodym theorem for monotone measures (see, for example, Graf [@Gra80], Greco [@Gre81] and Nguyen et al. [@Ngu06; @NguNguWan97]). Graf obtained his result under the assumption that the monotone measures are subadditive and lower continuous, while Nguyen et al. demanded the monotone measures being $\sigma$-subadditive. Greco [@Gre81] (see also Theorem 1.2 in Candeloro, Volčič [@CanVol02]) posed an additional requirement that $$(*) \ \ \ \ \ \mu(S)=\nu(S)=0\Rightarrow \nu(A\cup S)=\nu(A),\, \ \forall A\in{\mathcal U}.$$ Specifically, under the condition $(*)$, Greco proved that there is a nonnegative function $f$ such that $$\mu(A)=\int_A fd\nu,\, \ \forall A\in{\mathcal U}$$ if and only if $(\mu, \nu)$ satisfies a strong decomposition property (S.D.P. for short, see [@Gra80]) w.r.t. a sets system $\{A_\alpha\}$ and $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)=0$. It is not difficult to see that S.D.P. together with $\{A_\alpha\}$ and $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)=0$ implies that $\mu\ll\nu$, hence the condition $(*)$ says in fact that $\nu$ is null-additive. In contrast to these results, our result have no additional requirements for monotone measures other than a set of sufficient and necessary conditions. Interestingly, Example [Example 10](#Exa-withoutWeaklyNull-add.){reference-type="ref" reference="Exa-withoutWeaklyNull-add."} shows that the Radon-Nikodym theorem can hold even for monotone measures without weakly null-additivity. # Radon-Nikodym theorem for $\sigma$-finite monotone measures Before presenting a Radon-Nikodym theorem for $\sigma$-finite monotone measures, we need some further properties of the Choquet integral. Note that if $(\mu, \nu)$ has decomposition property w.r.t. a sets system $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$, then for any nonempty set $V\in{\mathcal U}$ the ordered pair $(\mu|_V, \nu|_V)$ also has decomposition property w.r.t. the system $\{A_\alpha\cap V\}_{\alpha\in \mathbb{Q}^{+}}$. A monotone measure $\mu$ is said to be (i) *lower continuous* (or *continuous from below*) if for any $\{A_n\}\subset{\mathcal U}$ with $A_n\nearrow A$, it holds that $\mu(A)=\lim\limits_{n\to\infty}\mu(A_n)$; (ii) *null-additive* if $\mu(A\cup N)=\mu(A)$ for any $A, N\in{\mathcal U}$ with $\mu(N)=0$. Obviously, lower continuity implies null-continuity and null-additivity implies weak null-continuity, but not vice versa (see [@Li2000]). **Proposition 15**. *[@Den94; @SonLi05][\[Prot-furtherpropofCho\]]{#Prot-furtherpropofCho label="Prot-furtherpropofCho"} Let $(U, {\mathcal U}, \nu)$ be a monotone measure space and $f, g, f_n \ (n=1, 2,\cdots)$ be nonnegative measurable functions.* *(i) If $f=g \ a.e.[\nu]$ and $\nu$ is null-additive, then $\int fd\nu=\int gd\nu$.* *(ii) If $\nu$ is lower continuous and $f_n\nearrow f$, then $\lim\limits_{n\to\infty}\int f_nd\nu=\int fd\nu$.* In the following we suppose that the monotone measures $\mu, \nu$ are $\sigma$-finite. Without loss of generality, we can assume that there is $\{U_n\}_{n=1}^\infty\subset{\mathcal U}$ with $U_n\nearrow U$ such that for every $n$, $\mu(U_n)<\infty$ and $\nu(U_n)<\infty$ hold simultaneously. **Theorem 16**. *Let $\mu, \nu$ be $\sigma$-finite and lower continuous and $\nu$ be null-additive. If $(\mu, \nu)$ has decomposition property w.r.t. the system $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$ and $\lim\limits_{\alpha\to\infty}\mu(A_\alpha\cap U_n)\vee\nu(A_\alpha\cap U_n)=0$ for each $n$, then there is a nonnegative and finite a.e.\[$\nu$\] measurable function $f$ such that $$\mu(A)=\int_A fd\nu, \ \ \forall\, A\in{\mathcal U}.$$ In this case, $f$ is unique $a.e.[\nu]$.* For each $n$, $(\mu|_{U_n}, \nu|_{U_n})$ also has decomposition property w.r.t. the system $\{A_\alpha\cap U_n\}_{\alpha\in \mathbb{Q}^{+}}$. Since $\lim\limits_{\alpha\to\infty}\mu(A_\alpha\cap U_n)\vee\nu(A_\alpha\cap U_n)=0$ also holds, according to Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}, there is a nonnegative measurable function $f_n$ on $U_n$ such that $$\mu|_{U_n}(E)=\int_E f_nd\nu|_{U_n}$$ for each measurable subset $E$ of $U_n$. Equivalently, for each $A\in{\mathcal U}$ we have $$\mu|_{U_n}(A_n)=\int_{A_n} f_nd\nu|_{U_n},$$ where $A_n=A\cap U_n$. By Proposition [Proposition 11](#Prot-uniqueness){reference-type="ref" reference="Prot-uniqueness"}, for $n>m$ we have $f_n|_{U_m}=f_m \ a.e.[\nu]$. Without loss of generality, by Proposition [\[Prot-furtherpropofCho\]](#Prot-furtherpropofCho){reference-type="ref" reference="Prot-furtherpropofCho"}(i) we can assume that $f_n|_{U_m}=f_m$ as $\nu$ is null-additive. Let $\tilde{f}_n(u)=f_n(u)$ for $u\in U_n$ and $\tilde{f}_n(u)=0$ for $u\in U\setminus U_n$. Then $$\mu(A_n)=\int_{A_n} \tilde{f}_nd\nu$$ holds for each $n$. Note that the sequence $\{\tilde{f}_n\}$ is nondecreasing and thus it is convergent everywhere. Denote $f=\lim\limits_{n\to\infty} \tilde{f}_n$, then $f|_{U_n}=\tilde{f}_n$ for each $n$. Thus $$\{f=\infty\}=\bigcup_{n=1}^\infty\Big(U_n\cap\{f=\infty\}\Big)=\bigcup_{n=1}^\infty\{\tilde{f}_n=\infty\},$$ which implies that $f$ is finite a.e.\[$\nu$\] as $\nu$ is null-additive and lower continuous. Moreover, $f|_{U_n}=\tilde{f}_n$ also implies that $$\mu(A_n)=\int_{A_n} fd\nu.$$ Since $A_n\nearrow A$, again by the lower continuity of $\mu, \nu$ we reach the final conclusion $$\mu(A)=\lim_{n\to\infty}\mu(A_n)=\lim_{n\to\infty}\int_{A_n} fd\nu=\int_{A} fd\nu.$$ The uniqueness of $f$ follows from (i) of Proposition [\[Prot-furtherpropofCho\]](#Prot-furtherpropofCho){reference-type="ref" reference="Prot-furtherpropofCho"}. This completes the proof.   $\Box$ **Example 17**. Let $U$ be the set of natural numbers and ${\mathcal U}$ be the power set of $U$. Let $\nu(A)=1$ for $A\neq\emptyset$, $\mu(A)=\max A$ if $A$ is finite and $\mu(A)=\infty$ if $A$ is infinite. Then (i) $\nu$ is lower continuous and null-additive; (ii)$\mu$ is lower continuous; (iii) $\nu$ is finite and $\mu$ is $\sigma$-finite. Let $A_\alpha=[\alpha, \infty)\cap U$ for each $\alpha\in \mathbb{Q}^{+}$. Then we can verify that $(\mu, \nu)$ has decomposition property w.r.t. the system $\{A_\alpha\}_{\alpha\in \mathbb{Q}^{+}}$. It is easy to see that $\mu(A)=\int_A fd\nu, \forall\, A\in{\mathcal U}$ for $f(x)=\sup\{\alpha|x\in A_\alpha\}=x$. Note that $\lim\limits_{\alpha\to\infty}\mu(A_\alpha)=\infty$ and $\lim\limits_{\alpha\to\infty}\nu(A_\alpha)=1$. But $\lim\limits_{\alpha\to\infty}\mu(A_\alpha\cap U_n)\vee \nu(A_\alpha\cap U_n)=0$ for each $n$, where $U_n=\{0, 1, 2, \cdots, n\}$. # Concluding remarks We have presented a version of the Radon-Nikodym theorem for finite monotone measures (Theorem [Theorem 8](#R-N-thrm-finitecase){reference-type="ref" reference="R-N-thrm-finitecase"}). As we have seen, we introduced the decomposition property of the ordered pair $(\mu,\nu)$ of monotone measures (Definition [Definition 5](#Dfnt-decompro){reference-type="ref" reference="Dfnt-decompro"}) and showed a necessary and sufficient condition that the Radon-Nikodym theorem holds for the Choquet integral w.r.t. finite monotone measures. We point out that our version has no additional conditions for finite monotone measures (such as, subadditivity, or $\sigma$-subadditivity, or continuity from below, etc.) other than monotonicity. The proof of this result is dependent on a distinguished feature, namely, the comonotonic additivity, of the Choquet integral. The uniqueness of Radon-Nikodym derivative and the case of $\sigma$-finite monotone measures have also been considered. Apart from the Choquet integral, there are other nonlinear integrals (the concave integral [@LehTep08] and pan-integral [@WanKli09] for example) in the literature that extend the Lebesgue integral. So, it would be an interesting topic to explore the Radon-Nikodym theorem for these integrals. As these integrals lack the comonotonic additivity, we need to seek new decomposition properties and techniques. The known relationships among these integrals [@LehTep08; @LiMesOuyWu23; @OuyLiMes15] may be useful. 9 S. Asahina, K. Uchino, T. Murofushi,  Relationship among continuity conditions and null-additivity conditions in non-additive measure theory, Fuzzy Sets Syst. 157 (2006) 691--698. W. C. Bell, J.W. Hagood, Separation properties and exact Radon-Nikodym derivatives for bounded finitely additive measures, Pacific J. Math. 131(2) (1988) 237-248. D. Candeloro, A. Martellotti, Geometric properties of the range of two-dimensional quasi-measures with respect to Radon-Nikodým property, Adv. Math. 93 (1992), 9-24. D. Candeloro, A. Volčič, Radon-Nikodým theorems, In Handbook of Measure Theory, edited by E. Pap, Chap. 6, 249-294. Amsterdam: ElsevierScience. G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295. D. Denneberg, Non-additive Measures and Integrals, Kluwer, Dordrecht, 1994. S. Graf, A Radon-Nikodym theorem for capacities, J. Reine Ang. Math. 320 (1980) 192-214. G. H. Greco, Un teorema di Radon-Nikodym per funzioni d'insieme subadditive, Ann. Univ. Ferrara 27 (1981) 13-19. P. R. Halmos, Measure Theory, Van Nostrand, New York, 1968. E. Lehrer, R. Teper, The concave integral over large spaces, Fuzzy Sets Syst. 159 (2008) 2130-2144. J. Li, On null-continuity of monotone measures, Mathematics 8(2)(2020) 205. J. Li, R. Mesiar, Q. Zhang, Absolute continuity of monotone measure and convergence in measure, Communications in Computer and Information Science (CCIS), vol. 80, pp. 500-504, Springer, 2010. J. Li, R. Mesiar, Y. Ouyang, L. Wu, On the coincidence of the pan-integral and the Choquet integral, Fuzzy Sets Syst. 467 (2023) 108577. H. B. Maynard, A Radon-Nikodym theorem for finitely additive bounded measures, Pacific J. Math. 83(2) (1979) 401-403. H. T. Nguyen, An Introduction to Radom Sets, CRC Press, Boca Raton (2007). H. T. Nguyen, N.T. Nguyen, T. Wang, On capacity functions in interval probabilities, Int. J. Uncertain. Fuzz. 5(3) (1997) 359-377. Y. Ouyang, J. Li, A note on the monotone set functions defined by Choquet integral, Fuzzy Sets Syst. 146 (2004) 147-151. Y. Ouyang, J. Li, R. Mesiar, Relationship between the concave integrals and the pan-integrals on finite spaces, J. Math. Anal. Appl. 424 (2015) 975-987. E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995. Y. Rébillé, A super Radon-Nikodym derivative for almost subadditive set functions, Int. J. Uncertain. Fuzz. 21 (3) (2013) 347-365. D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986) 255-261. J. Song, J. Li, Lebesgue theorems in non-additive measure theory, Fuzzy Sets Syst. 149 (2005) 543-548 Z. Wang, G. J. Klir, Fuzzy measures defined by fuzzy integral and their absolute continuity, J. Math. Anal. Appl. 203 (1996) 150-165. Z. Wang, G. J. Klir, W. Wang, Monotone set functions defined by Choquet integra, Fuzzy Sets Syst. 81 (1996) 241-250. Z. Wang, G. J. Klir, Generalized Measure Theory, Springer, New York, 2009. [^1]: E-mail:oyy\@zjhu.edu.cn(Y. Ouyang) [^2]: Corresponding author. E-mail:lijun\@cuc.edu.cn(J. Li)

arxiv_math

--- abstract: | We provide a criterion for certain algebraic objects over Jacobson schemes to be forms of each other based on their behaviour at closed fibres. This criterion permits to answer a question that I. Burban had asked the authors. Reductive group scheme, torsor, Lie algebras.\ 14L30, 17B67, 11E72, 14E20. author: - P. Gille$^{\rm 1}$ and A. Pianzola$^{\rm 2,3}$ title: Fiberwise Criteria for Twisted Forms of Algebraic Structures --- *In memoriam Professor Georgia Benkart* $^{\rm 1}$*UMR 5208 du CNRS - Institut Camille Jordan - Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex - France.* $^{\rm 2}$*Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada.* $^{\rm 3}$*Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, (1084) Buenos Aires, Argentina.* # Introduction Let $\text{\rm \bf G}$ be a group scheme over a base scheme $\text{\rm{S}}.$ The concept of $\text{\rm \bf G}$--torsors over $\text{\rm{S}}$ is to be found in many areas of mathematics and mathematical physics; they are useful tools to frame problems in a language that is partial to powerful methods from algebraic geometry. $\text{\rm \bf G}$ acts on itself by right multiplication. A $\text{\rm \bf G}$--torsor is a scheme $\text{\rm \bf X}$ with a right action of $\text{\rm \bf G}$ that "locally\" looks like $\text{\rm \bf G}$ with this action. By locally we mean that there exists a faithfully flat and locally finitely presented scheme morphism $\text{\rm{T}}\to \text{\rm{S}}$ such that $\text{\rm \bf X}_\text{\rm{T}}$ and $\text{\rm \bf G}_\text{\rm{T}}$ are isomorphic as $\text{\rm{T}}$--schemes with the corresponding induced $\text{\rm \bf G}_\text{\rm{T}}$--action. The treatise [@SGA3] shows how a deep understanding of $\text{\rm \bf G}$ can be had through the study of its fibres $\text{\rm \bf G}_s := \text{\rm \bf G}\times_\text{\rm{S}}\mathop{\rm Spec}\nolimits\big(\kappa(s)\big)$ where $\kappa(s)$ is the residue field of $s \in \text{\rm{S}}.$ It thus seems natural, given a right action of $\text{\rm \bf G}$ on an $\text{\rm{S}}$--scheme $\text{\rm \bf X},$ to study situations under which the fibres $\text{\rm \bf X}_s$ being $\text{\rm \bf G}_s$--torsors yield information about $\text{\rm \bf X}$ itself being a $\text{\rm \bf G}$--torsor. In practice the fibres of closed points are more tractable. The existence of "enough\" closed points brings up the natural working hypothesis that $\text{\rm{S}}$ be a Jacobson scheme. It is under this assumption that we are able to create a fiberwise criterion for torsors under certain reductive group schemes. This work grew up out of the desire to answer a question that I. Burban posed to the authors a couple of years ago [@Bu]. His question and the (positive) answer that can be given using our main result is given in the last section. The second author wishes to sincerely thank CNRS for facilitating an invited researcher visit to the Camille Jordan Institute. # Preliminaries on group schemes and smoothness Throughout this paper $\text{\rm{S}}$ is a (base) scheme with structure sheaf ${\cal O}_\text{\rm{S}}$. By an $\text{\rm{S}}$*---functor* (resp. *monoid, group*) we mean a contravariant functor $\text{\rm \bf F}$ from the category of schemes over $\text{\rm{S}}$ to the category of sets (resp. monoids, groups). When $\text{\rm \bf F}$ is representable, i.e. a scheme, we say that $\text{\rm \bf F}$ is an $\text{\rm{S}}$--*scheme* (resp. *monoid scheme, group scheme*). The use of bold face characters in the text is used to emphasize the functorial nature of the object under consideration. In particular, if ${\rm X}$ is an $\text{\rm{S}}$--scheme then its functor of points $$\text{\rm{T}}\mapsto \mathop{\rm Hom}\nolimits_{\text{\rm{S}}-{\rm Sch}}(\text{\rm{T}},{\rm X})$$ will be denoted by $\text{\rm \bf X}.$ If $\text{\rm \bf F}$ is an $\text{\rm{S}}$--functor and $\text{\rm{T}}$ a scheme over $\text{\rm{S}}$, we denote by $\text{\rm \bf F}_\text{\rm{T}}$ the $\text{\rm{T}}$--functor obtained by base change. If $\text{\rm{T}}= \mathop{\rm Spec}\nolimits(R)$ we denote $\text{\rm \bf F}(\text{\rm{T}})$ and $\text{\rm \bf F}_\text{\rm{T}}$ by $\text{\rm \bf F}(R)$ and $\text{\rm \bf F}_R$ respectively. If $\text{\rm \bf F}$ is an $\text{\rm{S}}$--scheme, then $\text{\rm \bf F}_\text{\rm{T}}= \text{\rm \bf F}\times_\text{\rm{S}}\text{\rm{T}}$ and, as it is customary, we denote $\text{\rm \bf F}\times_{\mathop{\rm Spec}\nolimits(R)} \text{\rm{T}}$ by $\text{\rm \bf F}\times_R \text{\rm{T}}$ and $\text{\rm \bf F}\times_\text{\rm{S}}\mathop{\rm Spec}\nolimits(R)$ by $\text{\rm \bf F}\times_\text{\rm{S}}R$ We denote as usual by $\text{\rm \bf O}_\text{\rm{S}}$ the affine $\text{\rm{S}}$--ring scheme $\text{\rm{S}}[t] = \mathop{\rm Spec}\nolimits(\Bbb Z[t])\times_\Bbb Z\text{\rm{S}}.$ Thus $\text{\rm \bf O}_\text{\rm{S}}(\text{\rm{T}}) = {\cal O}_\text{\rm{T}}(\text{\rm{T}}).$ Recall that an $\text{\rm \bf O}_\text{\rm{S}}$-*module* is an abelian $\text{\rm{S}}$--group ${\bf M}$ together with an $\text{\rm \bf O}_\text{\rm{S}}(\text{\rm{T}})$--module structure on ${\bf M}(\text{\rm{T}})$ that is functorial on $\text{\rm{T}}.$ The concept of an $\text{\rm \bf O}_\text{\rm{S}}$--algebra is defined similarly. Base change is defined and denoted as it is for functors. We mainly use the terminology and notation of Grothendieck--Dieudonné [@EGA-neu], which for the most part agrees with that of Demazure--Grothendieck used in [@SGA3 Exp. I and II]. Below we briefly review those concepts and results that are relevant to this paper. ## Groups attached to quasi--coherent modules Let ${\cal E}$ be a quasi--coherent module over $\text{\rm{S}}$. We denote its dual by ${\cal E}^{^{\vee}}$. For each morphism $f: \text{\rm{T}}\to \text{\rm{S}}$ we let ${\cal E}_{\text{\rm{T}}}=f^*({\cal E})$ be the inverse image of ${\cal E}$ under the morphism $f$, and define an abelian $\text{\rm{S}}$--group $\text{\rm \bf V}({\cal E})$ by $\text{\rm \bf V}({\cal E})(\text{\rm{T}}) = \mathop{\rm Hom}\nolimits_{{\cal O}_\text{\rm{T}}}({\cal E}_\text{\rm{T}}, {\cal O}_\text{\rm{T}}) = \Gamma\big(\text{\rm{T}}, ({\cal E}_\text{\rm{T}})^{^{\vee}}\big)$. $\text{\rm \bf V}$ is actually an $\text{\rm{S}}$--scheme; it is represented by the affine scheme $\mathop{\rm Spec}\nolimits\bigl( \mathrm{\bf Sym}({\cal E})\bigr)$ where $\mathrm{\bf Sym}({\cal E})$ is the symmetric ${\cal O}_\text{\rm{S}}$--algebra of ${\cal E}$ [@EGA-neu 9.4.9]. If ${\cal E}$ is of finite type (resp. of finite presentation), then $\mathbf V({\cal E})$ is an $\text{\rm{S}}$--scheme of finite type (resp. of finite presentation), ibid, 9.4.11. The abelian $\text{\rm{S}}$--group $\text{\rm \bf W}({\cal E})$ is defined by $\text{\rm \bf W}({\cal E})(\text{\rm{T}}) = \Gamma(\text{\rm{T}}, {\cal E}_{\text{\rm{T}}}).$ Recall that if ${\cal E}$ is locally free of finite type then $\text{\rm \bf W}({\cal E}) \simeq \text{\rm \bf V}({\cal E}^{^{\vee}}).$ In particular $\text{\rm \bf W}({\cal E})$ is in this case an affine $\text{\rm{S}}$--scheme. Note that the abelian $\text{\rm{S}}$--groups $\text{\rm \bf V}({\cal E})$ and $\text{\rm \bf W}({\cal E})$ have natural $\text{\rm \bf O}_\text{\rm{S}}$--module structures. **Example 1**. Assume $\text{\rm{S}}= \mathop{\rm Spec}\nolimits(R).$ Let ${\cal L}$ be a quasi--coherent ${\cal O}_\text{\rm{S}}$--module and denote by ${\rm L}$ the corresponding $R$--module. We denote $\text{\rm \bf W}({\cal L})$ by $\text{\rm \bf W}({\rm L}).$ For all $\text{\rm{S}}$--scheme $\text{\rm{T}}$ by definition $\text{\rm \bf W}({\rm L})(\text{\rm{T}})$ is the ${\cal O}_\text{\rm{T}}(\text{\rm{T}})$--module ${\rm L}\otimes_R {\cal O}_\text{\rm{T}}(\text{\rm{T}}).$ Similarly if ${\cal L}$ is an ${\cal O}_\text{\rm{S}}$--algebra. **Remark 2**. We can view $\text{\rm \bf V}$ (resp. $\text{\rm \bf W}$) as a contravariant (resp. covariant) functor from the category of quasi-coherent ${\cal O}_\text{\rm{S}}$-modules to the category of $\text{\rm \bf O}_\text{\rm{S}}$-modules. These functors are full and faithful [@SGA3 I Prop. 4.6.2] To ${\cal E}$ we attach the $\text{\rm{S}}$-functor $\text{\bf{End}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal E})$ whose functor of points is given by $\text{\rm{T}}\mapsto \mathop{\rm End}\nolimits_{{{\cal O}_\text{\rm{T}}}-{\rm mod}}({\cal E}\otimes_{{\cal O}_\text{\rm{S}}} {\cal O}_\text{\rm{T}}).$ This can be viewed as an abelian $\text{\rm{S}}$-group or an $\text{\rm \bf O}_\text{\rm{S}}$--module. We also have the $\text{\rm{S}}$--group $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal E})$ given by $\text{\rm{T}}\mapsto \text{\rm{Aut}}_{{{\cal O}_\text{\rm{T}}}-{\rm mod}}({\cal E}\otimes_{{\cal O}_\text{\rm{S}}} {\cal O}_\text{\rm{T}}).$ If in addition ${\cal E}$ is an ${\cal O}_\text{\rm{S}}$--algebra, one defines the $\text{\rm{S}}$--group $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm alg}}({\cal E})$ in the obvious way. It is an $\text{\rm{S}}$--subgroup of $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal E})$. Similarly to an $\text{\rm \bf O}_\text{\rm{S}}$-module $\text{\rm \bf L}$ we attach an abelian $\text{\rm{S}}$--group (in fact an $\text{\rm \bf O}_\text{\rm{S}}$--module) $\text{\bf{End}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm mod}}(\text{\rm \bf L})$ via $\text{\rm{T}}\mapsto \mathop{\rm End}\nolimits_{\text{\rm \bf O}_\text{\rm{T}}-{\rm mod}}(\text{\rm \bf L}_\text{\rm{T}}).$ Finally if $\text{\rm \bf L}$ is an $\text{\rm \bf O}_\text{\rm{S}}$--algebra, the $\text{\rm{S}}$--group $\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L})$ is given by the functor of points $\text{\rm{T}}\mapsto \text{\rm{Aut}}_{\text{\rm \bf O}_\text{\rm{T}}-{\rm alg}}(\text{\rm \bf L}_\text{\rm{T}}).$ **Lemma 3**. *Let ${\cal L}$ be an ${\cal O}_\text{\rm{S}}$--algebra. Assume that as an ${\cal O}_\text{\rm{S}}$--module ${\cal L}$ is locally free of finite rank. Then.* *(1) The natural maps $$\text{\bf{End}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L}) \to \text{\bf{End}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm mod}}\big(\text{\rm \bf W}({\cal L})\big)$$ and $$\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm alg}}({\cal L}) \to \text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}\big(\text{\rm \bf W}({\cal L})\big)$$ are $\text{\rm{S}}$-functor isomorphisms.* *(2) $\text{\bf{End}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L})$ and $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm alg}}({\cal L})$ are representable by affine $\text{\rm{S}}$--schemes of finite presentation.* *Proof.* (1) From their definition we see that the maps under consideration are functorial . That they are bijective on points follows from Remark [Remark 2](#FuFa){reference-type="ref" reference="FuFa"}. \(2\) Assume that $\text{\rm{S}}= \mathop{\rm Spec}\nolimits(R).$ Then ${\cal L}$ corresponds to an $R$-module ${\rm L}$ which is projective of finite rank. The ${\cal O}_\text{\rm{S}}$--module $\mathcal End _{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L})$ corresponds to the $R$-module $\mathop{\rm End}\nolimits_{R-{\rm mod}}({\rm L}) \simeq {\rm L}^* \otimes_R {\rm L}.$ Because the $R$-module map $\mathop{\rm End}\nolimits_{R-{\rm mod}}({\rm L}) \otimes_R R' \to \mathop{\rm End}\nolimits_{R'-{\rm mod}}({\rm L}\otimes_R R')$ is an isomorphism for all $R'/R,$ it follows that $\text{\bf{End}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}\big({\cal L})$ is represented by the affine $R$-scheme of finite presentation $\text{\rm \bf W}({\rm L}^*\otimes_R {\rm L}).$ It is clear that $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm alg}}({\cal L})$ is a closed subscheme of $\text{\bf{End}}_{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L}),$ hence also affine, which is of finite presentation since $L$ is locally free of finite rank. From the foregoing considerations it follows that our two functors are affine $\text{\rm{S}}$-schemes which are locally of finite presentation. Since their structure morphisms are affine, they are quasi-compact and separated, hence of finite presentation. ◻ **Remark 4**. Let ${\cal L}$ be a quasi-coherent ${\cal O}_\text{\rm{S}}$--module, and consider the corresponding ${\cal O}_\text{\rm{S}}$--module $\mathcal End _{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L}).$ There is a natural $\text{\rm \bf O}_\text{\rm{S}}$--module morphism $$\text{\rm \bf W}\big(\mathcal End _{{\cal O}_\text{\rm{S}}-{\rm mod}}({\cal L})\big) \to \text{\bf{End}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm mod}}\big(\text{\rm \bf W}({\cal L})\big).$$ This morphism *need not* be an isomorphism. It is if ${\cal L}$ is locally free of finite type. ## Group schemes and Lie algebras Throughout $\text{\rm \bf G}$ will denote an $\text{\rm{S}}$-group scheme and we denote by $e \in \text{\rm \bf G}(\text{\rm{S}})$ its unit section. We refer to [@SGA3 I and II], [@DG] and [@LLR §1] for details in what follows. If $\text{\rm{T}}$ is a scheme we denote by $\text{\rm{T}}[\epsilon]$ the corresponding scheme of dual number [@SGA3 II.2]. The functor $\text{\rm{T}}\to \mathop{\bf Lie}\nolimits(\text{\rm \bf G})(\text{\rm{T}}) := \ker\big( \text{\rm \bf G}(\text{\rm{T}}[\epsilon]) \to \text{\rm \bf G}(\text{\rm{T}})\big)$ is an $\text{\rm \bf O}_\text{\rm{S}}$--module isomorphic to $\text{\rm \bf V}(\omega^1_{\text{\rm \bf G}/\text{\rm{S}}})$ where $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}= e^*( \Omega^1_{\text{\rm \bf G}/\text{\rm{S}}})$ ibid. Prop. 3.3 and 3.6. Furthermore, the $\text{\rm \bf O}_\text{\rm{S}}(\text{\rm{T}})$--module $\mathop{\bf Lie}\nolimits(\text{\rm \bf G})(\text{\rm{T}})$ has a natural Lie algebra structure. It is thus an $\text{\rm \bf O}_\text{\rm{S}}$--Lie algebra. Recall that for all scheme morphisms $\text{\rm{T}}\to \text{\rm{S}}$ we have a natural $\text{\rm \bf O}_\text{\rm{T}}$--Lie algebra isomorphism $$\label{Lialg} \mathop{\bf Lie}\nolimits(\text{\rm \bf G}) \times_\text{\rm{S}}\text{\rm{T}}\simeq \mathop{\bf Lie}\nolimits(\text{\rm \bf G}_\text{\rm{T}}).$$ The ${\cal O}_\text{\rm{S}}(\text{\rm{S}})$--Lie algebra $\mathop{\bf Lie}\nolimits(\text{\rm \bf G})(\text{\rm{S}})$ is denoted by $\mathop{\rm Lie}\nolimits(\text{\rm \bf G}).$ From the above isomorphism we have $$\label{Lialg1} \mathop{\bf Lie}\nolimits(\text{\rm \bf G})( \text{\rm{T}}) = \mathop{\rm Lie}\nolimits(\text{\rm \bf G}_\text{\rm{T}}).$$ We denote by $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$ the vector group sheaf (fibration vectorielle) of sections of the affine scheme $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}) \to \text{\rm{S}}.$ In other words, $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$ is the ${\cal O}_\text{\rm{S}}$-module $({\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}})^{\vee} = \mathop{\rm Hom}\nolimits_{{\cal O}_\text{\rm{S}}}(\omega^1_{\text{\rm \bf G}/S}, {\cal O}_\text{\rm{S}}).$ Note that $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$ is naturally an ${\cal O}_\text{\rm{S}}$--Lie algebra. **Remark 5**. In general $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$ does not determine $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}),$ but it does if $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}$ is locally free of finite type, in particular if $\text{\rm \bf G}$ is smooth. In this case $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}) = \text{\rm \bf W}\big(\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})\big).$ See [@SGA3 II Lemma 4.11.7]. If $\text{\rm{S}}= \mathop{\rm Spec}\nolimits(R)$ the $R$--Lie algebra $\text{\rm Lie}(\text{\rm \bf G}) = \mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})(\text{\rm{S}})$ is a locally free $R$--module of finite type. For any ring extension $R'/R$ we have $\text{\rm Lie}(\text{\rm \bf G}_{R'}) = \mathop{\bf Lie}\nolimits(\text{\rm \bf G})(R') = \text{\rm Lie}(\text{\rm \bf G}) \otimes_R R'$. **Lemma 6**. *Let ${\cal L}$ be an ${\cal O}_\text{\rm{S}}$-algebra which is locally free of finite type (as an ${\cal O}_\text{\rm{S}}$--module). Let $\text{\rm \bf L}= \text{\rm \bf W}({\cal L}).$ There is a natural $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebra isomorphism $\text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L}) \simeq \mathop{\bf Lie}\nolimits\big(\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L})\big).$* *Proof.* Since $\text{\rm \bf L}$ is a good $\text{\rm \bf O}_\text{\rm{S}}$-module [@SGA3 II Def. 4.4 and Ex. 4.4.2], we have an $\text{\rm \bf O}_\text{\rm{S}}$-module isomorphisms $\eta: \text{\bf{End}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm mod}}(\text{\rm \bf L}) \simeq \mathop{\bf Lie}\nolimits\big(\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm mod}}(\text{\rm \bf L})\big)$ [@SGA3 II Prop. 4.5]. Since ${\cal L}$ is locally free of finite type we can appeal to Lemma [Lemma 3](#repres){reference-type="ref" reference="repres"}(1) to conclude that this is in fact an isomorphism of $\text{\rm{S}}$-schemes. We claim that the restriction of $\eta$ to $\text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L})$ is our isomorphism. The proof reduces to the case when $\text{\rm{S}}$ is affine, which can be found in [@DG II §4 2.3]. ◻ ## Connected component of the identity, forms and type of a reductive group scheme. {#CCFT} Let $\text{\rm \bf G}$ be an $\text{\rm{S}}$-group scheme which is locally of finite presentation. We consider the $\text{\rm{S}}$--subgroup (functor) $\text{\rm \bf G}^\circ$ of $\text{\rm \bf G}$ [@SGA3 VI$_B$.3.1] defined by $$\text{\rm \bf G}^\circ(\text{\rm{T}}) = \Bigl\{ u \in \text{\rm \bf G}(\text{\rm{T}}) \, \mid \, \forall s \in \text{\rm{S}}, u_s(\text{\rm{T}}_s) \subset \text{\rm \bf G}_s^\circ \Bigr\}.$$ where $\text{\rm \bf G}_s^\circ$ is the connected component of the identity of the $\kappa(s)$-algebraic group $\text{\rm \bf G}_s.$ If $\text{\rm \bf G}$ is smooth along the unit section, $\text{\rm \bf G}^\circ$ is representable by a smooth $\text{\rm{S}}$--group scheme called *the connected component of the identity of $\text{\rm \bf G}$* [@SGA3 VI$_{\rm B}$.4.1]. Furthermore, the fibre $(\text{\rm \bf G}^{\circ})_s$ is naturally isomorphic to $\text{\rm \bf G}_s^\circ.$ Let $\text{\rm \bf G}$ be a reductive $\text{\rm{S}}$--group scheme, that is a smooth and affine $\text{\rm{S}}$--group whose geometric fibres are *connected* reductive algebraic groups. An $\text{\rm{S}}$--group scheme $\text{\rm \bf G}'$ is called a *(twisted) form of* $\text{\rm \bf G},$ if there exist a faithfully flat and localy presented extension $\text{\rm{T}}\to \text{\rm{S}}$ such that $\text{\rm \bf G}_\text{\rm{T}}$ and $\text{\rm \bf G}'_\text{\rm{T}}$ are isomorphic $\text{\rm{T}}$--group schemes. In a similar fashion one defines forms of $\text{\rm \bf O}_\text{\rm{S}}$--algebras. Let $\text{\rm \bf G}$ be a reductive $K$-group scheme, where $K$ is an algebraically closed field. If $\text{\rm \bf T}$ is a maximal torus of $\text{\rm \bf G}$, then $\text{\rm \bf T}$ is split and defines a root datum which is up to isomorphism independent of the choice of $\text{\rm \bf T}.$ It is called the *type of* $\text{\rm \bf G}$ and is denoted by $\mathcal R(\text{\rm \bf G}).$ Let $\text{\rm \bf G}$ be a reductive $\text{\rm{S}}$--group scheme. If $s \in \text{\rm{S}},$ the *type of $\text{\rm \bf G}$ at $s$* is the type of the reductive $\overline{\kappa(s)}$--group $\text{\rm \bf G}_{\overline{s}} := \text{\rm \bf G}_s \times_{\kappa(s)} \overline{\kappa(s)}.$ Because $\text{\rm \bf G}$ posseses maximal tori locally for the étale topology, one knows that the type function $s \mapsto \text{ type of} \,\, \text{\rm \bf G}_{\overline{s}}$ is locally constant [@SGA3 XXII 2.8]. We say that $\text{\rm \bf G}$ is of *constant type* if the type function is constant. This is the case, for example, if $\text{\rm{S}}$ is connected. ## Specific properties in characteristic zero {#subsec_specific} [\[2.4\]]{#2.4 label="2.4"} We assume in this section that $\text{\rm{S}}$ is a $\Bbb Q$--scheme. Let $\text{\rm \bf G}$ be a semisimple adjoint $\text{\rm{S}}$--group scheme of constant type. We will denote the $\text{\rm \bf O}_\text{\rm{S}}$--Lie algebra $\mathop{\bf Lie}\nolimits(\text{\rm \bf G})$ by $\text{\rm \bf L},$ and the ${\cal O}_\text{\rm{S}}$--Lie algebra $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$ by ${\cal L}.$ Recall (see Remark [Remark 5](#Liesmooth){reference-type="ref" reference="Liesmooth"}) that ${\cal L}$ is locally free of finite type and that $\text{\rm \bf W}({\cal L}) = \text{\rm \bf L}.$ Since $\text{\rm \bf G}$ and its simply connected cover $\text{\rm \bf G}^{\rm sc}$ have the same Lie algebra, we have isomorphisms of affine $\text{\rm{S}}$--group schemes [@SGA3 XXIV 3.6 and 7.3.1] $$\label{AutG=AutL} \text{\bf{Aut}}(\text{\rm \bf G}) \simeq \text{\bf{Aut}}(\text{\rm \bf G}^{\rm sc}) \simeq \text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}(\text{\rm \bf L}).\footnote{\, La ``définition évidente" of the $\text{\rm{S}}$-group scheme $\underline{\mathcal{A}\mathcal{u}\mathcal{t}}_{{\cal O}_\text{\rm{S}}-alg-de-Lie}({\cal L})$ of 7.3.1(iii) is precisely our $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm Lie}}({\cal L}),$ so that the asertion is that the natural map $\text{\bf{Aut}}(\text{\rm \bf G}^{\rm sc}) \to \text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm Lie}}({\cal L})$ is an isomorphism. Finally $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}-{\rm Lie}}({\cal L}) \simeq \text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm Lie}}(\text{\rm \bf L})$ by Lemma \ref{repres}.}$$ In particular, the $\text{\rm{S}}$-scheme $\text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}(\text{\rm \bf L})$ is smooth and affine. By ([\[AutG=AutL\]](#AutG=AutL){reference-type="ref" reference="AutG=AutL"}) and [@SGA3 XXIV 1.3 and 1.8] we have an isomorphism $$\label{G} \text{\rm \bf G}\simeq \text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}(\text{\rm \bf L})^{\circ}.$$ This yields an $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebra isomorphism $$\label{identif} \text{\rm \bf L}\simeq \mathop{\bf Lie}\nolimits\big(\text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}(\text{\rm \bf L})\big) = \mathop{\bf Lie}\nolimits\big(\text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}(\text{\rm \bf L})^\circ\big).$$ Recall [@SGA3 II Theo. 4.7 and Prop. 4.8] the adjoint representation $\mathop{\mathrm{Ad}}: \text{\rm \bf G}\to \text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}} - {\rm mod}}(\text{\rm \bf L})$ and the induced a $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebra homomorphism $\text{\rm ad}: \text{\rm \bf L}\to \text{\bf{End}}_{{\text{\rm \bf O}_\text{\rm{S}}}- {\rm mod}}(\text{\rm \bf L})$. **Lemma 7**. *Under the identification $\text{\rm \bf L}\simeq \mathop{\bf Lie}\nolimits\big( \text{\bf{Aut}}_{{\text{\rm \bf O}_\text{\rm{S}}}-\text{\rm Lie}}( \text{\rm \bf L})\big)$ of ([\[identif\]](#identif){reference-type="ref" reference="identif"}) the map $\text{\rm ad}$ induces an $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebra isomorphism between $\text{\rm \bf L}$ and $\text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}- {\rm Lie}}(\text{\rm \bf L})$.* *Proof.* All the morphisms under consideration are $\text{\rm{S}}$--scheme morphisms, so the question is local on $\text{\rm{S}}$ and we may assume that $\text{\rm{S}}= \mathop{\rm Spec}\nolimits(R)$ for some $\Bbb{Q}$-ring $R.$ Consider the $R$-Lie algebra $L =\text{\rm \bf L}(R).$ Since $\text{\rm \bf L}= \text{\rm \bf W}({\rm L})$ where $\text{\rm \bf W}({\rm L})(\text{\rm{T}}) = {\rm L}\otimes_R {\cal O}_\text{\rm{T}}(\text{\rm{T}}),$ it will suffice to show that for all ring extensions $R'/R$ the $R'$-Lie algebra homomorphism $d =\text{\rm ad}_R' : {\rm L}\otimes_R R'\to \text{\rm Der}_{R'-\text{\rm Lie}}({\rm L}\otimes_R R')$ is an isomorphism. By replacing $\text{\rm \bf G}$ by $\text{\rm \bf G}_R'$ we may assume that $R= R'.$ By [@SGA3 II 4.7.2] the map $d$ is nothing but the adjoint representation $d(x) = \text{\rm ad}_{\rm L}(x).$ Since $\text{\rm \bf G}$ is of constant type there exists a unique Chevalley group $\text{\rm \bf G}_0$ such that $\text{\rm \bf G}$ is a twisted form of $\text{\rm \bf G}_0 \times _\Bbb{Z} R.$ Let $\mathfrak g= \mathop{\bf Lie}\nolimits(\text{\rm \bf G}_0)(\Bbb{Z}) \otimes _\Bbb{Z} \Bbb{Q}.$ Then $\mathfrak g$ is a finite dimensional split semisimple Lie algebra over $\Bbb{Q}$ and ${\rm L}$ is a twisted form of $\mathfrak g\otimes_\Bbb{Q} R,$ that is, there exists an fppf extension (in fact étale cover) $R'$ of $R$ such that the $R'$-Lie algebras ${\rm L}\otimes_R R'$ and $(\mathfrak g\otimes_\Bbb{Q} R)\otimes_R R' \simeq \mathfrak g\otimes_\Bbb{Q} R'$ are isomorphic. Fix one such isomorphism $$\label{iso} \psi : {\rm L}\otimes_R R' \to \mathfrak g\otimes_\Bbb{Q} R'.$$ If $d(x) = 0,$ then $x$ belongs to the centre of ${\rm L}$ and therefore $\psi(x \otimes 1)$ belongs to the centre of $\mathfrak g\otimes_\Bbb{Q} R',$ which is trivial. Since $R'/R$ is faithfully flat $x = 0$ so that $d$ is injective and we henceforth identify ${\rm L}$ with an $R$-submodule of $\mathop{\rm Der}\nolimits({\rm L}).$ To show that $d$ is surjective, that is that every derivation of ${\rm L}$ is inner, we reason as follows. If ${\rm L}= \mathfrak g\otimes_\Bbb QR$ every derivation is inner by Whitehead's Lemma (see [@P1 Example 4.9]).[^1] For ${\rm L}$ arbitrary, we appeal to the isomorphism ([\[iso\]](#iso){reference-type="ref" reference="iso"}). By the split case we have $(\mathop{\rm Der}\nolimits({\rm L})/{\rm L})\otimes_R R' = 0.$ Thus ${\rm L}= \mathop{\rm Der}\nolimits({\rm L})$ as desired. ◻ ## Semicontinuity property for Lie algebras For convenience we recall the following well-known fact. **Lemma 8**. *Let ${\cal F}$ be a quasi-coherent ${\cal O}_\text{\rm{S}}$--module of finite presentation. Then the function $s \mapsto \mathop{\rm dim}\nolimits_{\kappa(s)}\big( {\cal F}\otimes_{{\cal O}_\text{\rm{S}}} \kappa(s)\big)$ is upper semi-continuous.* *Proof.* The statement is local and therefore reduces then to the case of a ring $R$ and of an $R$--module $M$ of finite presentation [@Stacks Tag 01PC]. If $R$ is noetherian then $M$ is coherent and the result can be found in [@Ha 12.7.2]. For lack of a reference we provide a proof in the general case. We consider an exact sequence $R^m \xrightarrow{A} R^n \to M \to 0$ where $A$ is a matrix of size $(m,n)$ with entries in $R.$ Let $\mathfrak q\in \mathop{\rm Spec}\nolimits(R)$ and $r=\mathop{\rm dim}\nolimits_{\kappa(\mathfrak q)} (M \otimes_R \kappa(\mathfrak q) ).$ Since the sequence ${\kappa(\mathfrak q)}^m \xrightarrow{A_{\kappa(\mathfrak q)}} {\kappa(\mathfrak q)}^n \to M_{\kappa(\mathfrak q)} \to 0$ is exact, there exists a minor $B$ of $A$ of size $(n-r,n-r)$ such that $\det(B_{\kappa(\mathfrak{q})} )\neq 0 \in \kappa(\mathfrak{q}).$ Consider the basic open set $\text{\rm U}$ of $\mathop{\rm Spec}\nolimits(R)$ consisting of prime ideals of $R$ that do not contain $f = \det(B) \in R.$ From $\det B_{\kappa(\mathfrak{q})} \neq 0$ it follows that $\mathfrak{q}\in \text{\rm U}.$ It is clear that for all $\mathfrak p\in \text{\rm U}$ the image of $A_{\kappa(\mathfrak p)}$ is a $\kappa(\mathfrak p)$-space of dimension at least $n-r.$ Thus $$\mathop{\rm dim}\nolimits_{\kappa(\mathfrak p)} \big(M \otimes_R \kappa(\mathfrak p) \big) \leq r = \mathop{\rm dim}\nolimits_{\kappa(\mathfrak q)} \big(M \otimes_R \kappa(\mathfrak q) \big)$$ for all $\mathfrak p\in \text{\rm U},$ so that our function is thus upper semi-continuous as desired. ◻ **Lemma 9**. *Let $\text{\rm \bf G}$ be an $\text{\rm{S}}$--group scheme that is locally of finite presentation. Then $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}$ is an $\text{\rm \bf O}_\text{\rm{S}}$--module of finite presentation and the map $s \mapsto \mathop{\rm dim}\nolimits_{\kappa(s)}\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_s)$ is upper semi-continuous.* *Proof.* That $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}$ is of finite presentation follows from [@Stacks Tag 01V2, 01V3]. By ([\[Lialg1\]](#Lialg1){reference-type="ref" reference="Lialg1"}) we have a $\kappa(s)-$Lie algebra isomorphism $\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_s) \simeq \mathop{\bf Lie}\nolimits(\text{\rm \bf G})\big(\kappa(s)\big).$ From the isomorphism $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}) \simeq\text{\rm \bf V}( \omega^1_{\text{\rm \bf G}/\text{\rm{S}}})$, we see that the dimension of $\mathop{\bf Lie}\nolimits(\text{\rm \bf G})\big(\kappa(s)\big)$ at a point $s \in \text{\rm{S}}$ is the dimension of the $\kappa(s)$--space $\mathop{\rm Hom}\nolimits_{\kappa(s)}\big( \omega^1_{\text{\rm \bf G}/\text{\rm{S}}} \otimes_{{\cal O}_S} \kappa(s), \kappa(s)\big),$ which is the dimension of $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}} \otimes_{{\cal O}_S} \kappa(s)$ since $\omega^1_{\text{\rm \bf G}/\text{\rm{S}}}$ is an ${\cal O}_\text{\rm{S}}$--module of finite type. Furthermore, since $\omega^1_{G/S}$ is in fact of finite presentation, Lemma [Lemma 8](#lem_upper){reference-type="ref" reference="lem_upper"} implies that the map $s \mapsto \mathop{\rm dim}\nolimits_{\kappa(s)}\bigl( \mathop{\bf Lie}\nolimits(\text{\rm \bf G})(\kappa(s)) \bigr)$ is upper semi-continuous. ◻ # Group schemes over Jacobson schemes ## Jacobson schemes and smoothness We refer the reader to [@EGAIV §10] for general results about Jacobson schemes. Let $\text{\rm X}$ be a topological space. We recall that a subset $\text{\rm X}_0$ of $\text{\rm X}$ is called *very dense* if for every closed subset $\text{\rm Z}\subset \text{\rm X}$ we have $\text{\rm Z}= \overline{\text{\rm Z}\cap \text{\rm X}_0}.$ Let $\text{\rm{S}}$ be a scheme. View $\text{\rm{S}}$ as a topological space and let $\text{\rm{S}}_0 \subset \text{\rm{S}}$ the set of closed points. The scheme $\text{\rm{S}}$ is said to be *Jacobson* if $\text{\rm{S}}_0$ is a very dense subset of $\text{\rm{S}}.$ Let us recall for the sake of completeness that: \(i\) Very dense subsets of a topological space are dense. \(ii\) If $\text{\rm{S}}= \mathop{\rm Spec}\nolimits(R)$, then $\text{\rm{S}}$ is Jacobson if and only if $R$ is a Jacobson ring, that is every ideal of $R$ is an intersection of maximal ideals. \(iii\) If $\text{\rm{S}}= \bigcup_{i \in I} \mathop{\rm Spec}\nolimits(R_i)$, the $\text{\rm{S}}$ is Jacobson if and only if every $R_i$ is a Jacobson ring. \(iv\) Every radical ideal $I$ is the intersection of the maximal ideals containing it [@Stacks Tag 00G4].[^2] **Proposition 10**. *Let $f : \text{\rm X}\to \text{\rm{S}}$ be a scheme morphisms which is locally of finite presentation. Assume that $x \in \text{\rm X}$ is such that:* *(i) $f$ is flat at $x$.* *(ii) $f_{\kappa(x)}$ is smooth.* *Then $f$ is smooth on an open neighborhood of $x.$* *Proof.* By assumption there exists affine open subschemes $\mathop{\rm Spec}\nolimits(A) = \text{\rm U}\subset \text{\rm X}$ and $\mathop{\rm Spec}\nolimits(B) = \text{\rm V}\subset \text{\rm{S}}$ such that $x \in \text{\rm U},$ $f(\text{\rm U}) \subset \text{\rm V}$ and $A$ is a finitely presented $B$-algebra. Let $\mathfrak{q}\in \mathop{\rm Spec}\nolimits(A)$ and $\mathfrak{p}\in \mathop{\rm Spec}\nolimits(B)$ be the points corresponding to $x$ and $f(x)$ respectively. By (i) $A_\mathfrak{q}$ is flat over $B_\mathfrak{p}$, while by (ii) $A \otimes_B \kappa(\mathfrak{p})$ is smooth over $\kappa(\mathfrak{p}).$ It follows from [@Stacks Tag 00TF] that the restriction of $f$ to $\text{\rm U}$ is smooth at $\mathfrak{q}$, hence (by definition of smoothness) smooth also in a neighbourhood of $\mathfrak{q}.$ ◻ **Corollary 11**. *Let $f : \text{\rm X}\to \text{\rm{S}}$ be as above. Assume that (i) and (ii) hold for all closed points of $\text{\rm X}$. If $\text{\rm{S}}$ is Jacobson, then $f$ is smooth.* *Proof.* By [@EGAIV Cor. 10.4.7] $\text{\rm X}$ is Jacobson. Let $\text{\rm X}_0$ be the set of closed points of $\text{\rm X}$ viewed as a topological space. By the Proposition there exists an open $\text{\rm U}\subset \text{\rm X}$ containing $\text{\rm X}_0$ in which $f$ is smooth. Let $\text{\rm Z}= \text{\rm X}\setminus \text{\rm U}.$ If $\text{\rm Z}\neq \emptyset$, then $\text{\rm Z}\cap \text{\rm X}_0 \neq \emptyset$ because $X_0$ is very dense. This contradicts $\text{\rm X}_0 \subset \text{\rm U}.$ Thus $\text{\rm Z}= \emptyset$ so that $\text{\rm U}= \text{\rm X}$ as desired. ◻ **Proposition 12**. *Let $\text{\rm{S}}$ be an integral Jacobson scheme, and $\text{\rm \bf G}$ be an $\text{\rm{S}}$-group scheme of finite presentation. Assume that for all closed point $b \in \text{\rm{S}}$ the fibres $\text{\rm \bf G}_b$ are smooth and of the same dimension $d$.[^3] Then.* *(1) For all $s \in \text{\rm{S}}$ the algebraic $\kappa(s)$--group $\text{\rm \bf G}_s$ is smooth of dimension $d.$* *(2) The $\text{\rm{S}}$-functor $\text{\rm \bf G}^\circ$ is representable by a smooth $\text{\rm{S}}$--group scheme of relative dimension $d$ which is open in $\text{\rm \bf G}$.* *(3) If for each closed point $b$ of $\text{\rm{S}}$ the fibre $\text{\rm \bf G}_b$ is connected, then $\text{\rm \bf G}$ is smooth and has connected geometric fibres (i.e., $\text{\rm \bf G}_{\overline{b}}$ is connected).* *Proof.* (1) We denote by $\eta: \mathop{\rm Spec}\nolimits(F) \to \text{\rm{S}}$ the generic point of $\text{\rm{S}}$. Chevalley's generic flatness theorem [@GW 10.85] shows that there exists an open dense subset $\text{\rm U}$ of $\text{\rm{S}}$ such that $\text{\rm \bf G}\times_\text{\rm{S}}\text{\rm U}$ is flat over $\text{\rm U}$. We know that $\text{\rm U}$ is also Jacobson [@EGAIV Prop. 10.3.3]. Corollary [Corollary 11](#cor_smooth_jacobson){reference-type="ref" reference="cor_smooth_jacobson"} shows that $\text{\rm \bf G}\times_\text{\rm{S}}\text{\rm U}$ is smooth over $\text{\rm U}$. By [@Stacks Tag 05F7], up to shrinking $\text{\rm U}$ if necessary, the dimension function of the fibres of $\text{\rm \bf G}\times_\text{\rm{S}}\text{\rm U}\to \text{\rm U}$ is constant. Since by assumption this dimension has value $d$ on the closed points of $\text{\rm U}$, we get that $\text{\rm \bf G}\times_\text{\rm{S}}\text{\rm U}$ is smooth of relative dimension $d$.[^4] In particular the algebraic $F$-group $\text{\rm \bf G}_F$ is smooth of dimension $d$. Let $s \in \text{\rm{S}}$ and let $c = \mathop{\rm dim}\nolimits(\text{\rm \bf G}_s).$ According to Chevalley's semicontinuity theorem [@SGA3 VI$_B$.4.1] $$F_c = \{x \in \text{\rm{S}}: \mathop{\rm dim}\nolimits(\text{\rm \bf G}_x) \geq c\} \subset \text{\rm{S}}$$ is a closed set. Since $F_c \cap \overline{ \{s\}}$ contains $s$ and is closed $$\label{closedpoint} \overline{ \{s\}} \subset F_c.$$ Since $\text{\rm{S}}$ is Jacobson, it follows that there exists a closed point $b$ in $\overline{ \{s\}}.$ We have $$\label{inneq1} \mathop{\rm dim}\nolimits(\text{\rm \bf G}_s) \leq \mathop{\rm dim}\nolimits(\text{\rm \bf G}_b)=d.$$ On the other hand $s$ belongs to $\overline{\{\eta\}}.$ The same semicontinuity reasoning used above shows that $$\label{inneq2} \mathop{\rm dim}\nolimits(\text{\rm \bf G}_F) \leq \mathop{\rm dim}\nolimits(\text{\rm \bf G}_s).$$ Since $\text{\rm \bf G}_F$ is of dimension $d$ it follows from ([\[inneq1\]](#inneq1){reference-type="ref" reference="inneq1"}) and ([\[inneq2\]](#inneq2){reference-type="ref" reference="inneq2"}) that $\mathop{\rm dim}\nolimits(\text{\rm \bf G}_s)=d$. For establishing smoothness we use a variation of the previous argument by applying semicontinuity considerations to the Lie algebras. Lemma [Lemma 9](#lem_upper_Lie){reference-type="ref" reference="lem_upper_Lie"} yields the inequalities $$\mathop{\rm dim}\nolimits_F \mathop{\rm Lie}\nolimits(\text{\rm \bf G}_F) \leq \mathop{\rm dim}\nolimits_{\kappa(s)}\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_s)\leq \mathop{\rm dim}\nolimits_{\kappa(b)}\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_b).$$ Since $\text{\rm \bf G}$ is smooth of dimension $d$ at $\eta$ and at $b$ we obtain that $d=\mathop{\rm dim}\nolimits_{\kappa(s)}\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_s).$ Thus $\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_s)$ and $\text{\rm \bf G}_s$ have the same dimension (namely $d$). By the smoothness criterion [@DG II.5.2.1] $\text{\rm \bf G}_s$ is smooth. This completes the proof of (1). \(2\) By (1) we see that assumption (ii) of [@SGA3 VI$_B$ Cor. 4.4] holds. It follows that the $\text{\rm{S}}$-functor $\text{\rm \bf G}^\circ$ is representable by a smooth $\text{\rm{S}}$--group scheme which is open in $\text{\rm \bf G}$. \(3\) According to [@Stacks Tag 055I], the level set $$E_n= \bigl\{ s \in \text{\rm{S}}\mid \text{\rm \bf G}_s \enskip \hbox{has $n$ geometrically connected components} \bigr\}$$ is a locally constructible subset of $\text{\rm{S}}$ for each $n \geq 1$. Since $\text{\rm{S}}$ in a Jacobson scheme, the set $\text{\rm{S}}_0$ of closed points of the underlying topological space of $\text{\rm{S}}$ is very dense in $\text{\rm{S}}.$ Thus $\text{\rm{S}}_0 \cap E_n$ is dense in $E_n$ for each $n$ [@EGAIV 10.1.2, (b')]. Our assumption implies that $E_n = \emptyset$ for $n \geq 2$ so that $E_1=\text{\rm{S}}$. That $\text{\rm \bf G}$ is smooth now follows from (2). ◻ ## Forms **Proposition 13**. *Let $\text{\rm{S}}$ is an integral Jacobson scheme and $\text{\rm \bf G}$ an affine $\text{\rm{S}}$--group scheme of finite presentation. Assume that $\text{\rm \bf G}_b$ is reductive and of dimension $d$ for all closed point $b \in \text{\rm{S}}$. Then $\text{\rm \bf G}$ is a reductive $\text{\rm{S}}$--group scheme of constant type. In particular, there exists a unique Chevalley group $\text{\rm \bf G}_0$ such that $\text{\rm \bf G}$ is an $\text{\rm{S}}$--form of $\text{\rm \bf G}_0 \times_\Bbb Z\text{\rm{S}}$.* *Proof.* Proposition [Proposition 12](#prop_smooth){reference-type="ref" reference="prop_smooth"}.(3) shows that $\text{\rm \bf G}$ is smooth of relative dimension $d$ and has connected geometric fibres. According to [@Co1 Prop. 3.1.9.(1)] $\text{\rm \bf G}$ is reductive in a neighborhood of each closed point $b \in \text{\rm{S}}$. Since $\text{\rm{S}}$ is a Jacobson scheme, it follows that $\text{\rm \bf G}$ is reductive. Since the type function is locally constant, the connectedness of $\text{\rm{S}}$ implies that $\text{\rm \bf G}$ has constant type $t_0$ (see §[2.3](#CCFT){reference-type="ref" reference="CCFT"}). Let $\text{\rm \bf G}_0$ be the corresponding Chevalley group scheme. According to [@SGA3 XXIII 5.6] $\text{\rm \bf G}$ is an $\text{\rm{S}}$--form of $\text{\rm \bf G}_0 \times_\Bbb Z\text{\rm{S}}$. ◻ **Corollary 14**. *Let $\text{\rm \bf G}$ and $\text{\rm \bf G}'$ be group schemes over an integral Jacobson scheme $\text{\rm{S}}.$ Assume that $\text{\rm \bf G}$ is reductive and that $\text{\rm \bf G}'$ is affine and of finite presentation. If $\text{\rm \bf G}'_b$ is a form of $\text{\rm \bf G}_b$ for each closed point $b \in \text{\rm{S}}$, then $\text{\rm \bf G}'$ is an $\text{\rm{S}}$--form of $\text{\rm \bf G}$. In particular, $\text{\rm \bf G}'$ is a reductive $\text{\rm{S}}$--group scheme 0◻* ## Lie algebras If ${\cal L}$ is an ${\cal O}_\text{\rm{S}}$--Lie algebra and $s \in \text{\rm{S}},$ then the ${\cal O}_{\mathop{\rm Spec}\nolimits\kappa(s)}$--algebra ${\cal L}\otimes_{{\cal O}_\text{\rm{S}}} \kappa(s)$ obtained by base change is simply a $\kappa(s)$--algebra that we will denote by ${\cal L}_s.$[^5] Let $\text{\rm \bf L}=\text{\rm \bf W}({\cal L})$ and denote by a harmless abuse of notation the $\kappa(s)$--algebra $\text{\rm \bf L}\big(\kappa(s)\big)$ by $\text{\rm \bf L}_s.$ Note that ${\cal L}_s =\text{\rm \bf L}_s.$ **Proposition 15**. *Assume that $\text{\rm{S}}$ is an integral Jacobson $\Bbb Q$--scheme. Let ${\cal L}$ be an ${\cal O}_\text{\rm{S}}$--Lie algebra which is locally free of rank $d$, and let $\text{\rm \bf L}= \text{\rm \bf W}({\cal L})$ be its corresponding $\text{\rm \bf O}_\text{\rm{S}}$--Lie algebra. Assume that for each closed point $b \in \text{\rm{S}}$ the $\kappa(b)$--Lie algebra $\text{\rm \bf L}_b$ is semisimple.[^6] Then there exists a unique finite dimensional split semisimple Lie $\Bbb Q$-algebra $\rm{L}_0$ such that $\text{\rm \bf L}$ is a form of $\text{\rm \bf L}_0 := \rm{L}_0 \otimes_\Bbb Q\text{\rm \bf O}_\text{\rm{S}}$.[^7]* *Proof.* (1) By Lemma [Lemma 3](#repres){reference-type="ref" reference="repres"} $\text{\rm \bf G}:=\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}{\rm - Lie}}(\text{\rm \bf L})$ is an affine $\text{\rm{S}}$--group of finite presentation. For each closed point $b \in \text{\rm{S}}$, the algebraic group $\text{\rm \bf G}_b= \text{\bf{Aut}}_{\kappa(b)-{\rm Lie}}(\text{\rm \bf L}_b)$ is smooth of dimension $d$ and $\text{\rm \bf G}_b^\circ$ is semisimple adjoint. Proposition [Proposition 12](#prop_smooth){reference-type="ref" reference="prop_smooth"}.(2) shows that $\text{\rm \bf G}^\circ$ is representable by an open subgroup scheme of $\text{\rm \bf G}$ which is smooth of relative dimension $d.$ Appealing now to Proposition [Proposition 13](#prop_form){reference-type="ref" reference="prop_form"} yields that $\text{\rm \bf G}^\circ$ is adjoint semisimple and is a form of the $\text{\rm{S}}$-group corresponding to a (unique) semisimple adjoint Chevalley group scheme $\text{\rm \bf G}_0^\circ.$ We denote by $\rm{L}_0$ the $\Bbb Q$--Lie algebra of $\mathop{\rm Lie}\nolimits(\text{\rm \bf G}_0^\circ) \otimes_\Bbb Z\Bbb Q.$ This yields the $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebra that we denoted by $\text{\rm \bf L}_0.$ Observe that $\mathop{\bf Lie}\nolimits(\text{\rm \bf G})$ is a form of $\text{\rm \bf L}_0$ as an $\text{\rm \bf O}_\text{\rm{S}}$-Lie algebras. By Lemma [Lemma 7](#MZ){reference-type="ref" reference="MZ"} we have $\text{\rm \bf L}\simeq \text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-\text{\rm Lie}}(\text{\rm \bf L}).$ Finally since $\text{\rm \bf G}\simeq \text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-\mathop{\rm Lie}\nolimits}(\text{\rm \bf L})$ we can apply Lemma [Lemma 6](#derLie){reference-type="ref" reference="derLie"} to conclude that $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}) \simeq \text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-\text{\rm Lie}}(\text{\rm \bf L}).$ It follows that $\text{\rm \bf L}$ is a form of $\text{\rm \bf L}_0$ as desired. ◻ The following variant of the ideas presented heretofore will allow us to answer Burban's question (see §[\[Burban\]](#Burban){reference-type="ref" reference="Burban"} below). **Proposition 16**. *Assume that $\text{\rm{S}}$ is an integral Jacobson $\Bbb Q$--scheme. Let $\text{\rm \bf G}$ be a semisimple adjoint $\text{\rm{S}}$--group scheme and consider its ${\cal O}_\text{\rm{S}}$--Lie algebra ${\cal L}=\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})$.* *Let ${\cal L}'$ be a sheaf of ${\cal O}_\text{\rm{S}}-$Lie algebras. We assume that as an ${\cal O}_\text{\rm{S}}$--module ${\cal L}'$ is locally free of rank $d,$ and that for each closed point $b\in \text{\rm{S}}$, ${\cal L}'_b$ is a form of ${\cal L}_b$.* *(1) The $\text{\rm{S}}$--functor $\text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}- {\rm Lie}}({\cal L}')$ is a smooth affine $\text{\rm{S}}$--group scheme whose connected component of the identity $\text{\rm \bf G}'$ is an $\text{\rm{S}}$--form of $\text{\rm \bf G}$. In particular $\text{\rm \bf G}'$ is a semisimple adjoint $\text{\rm{S}}$--group scheme.* *(2) There is a natural isomorphism $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}') \simeq {\cal L}'$.* *(3) ${\cal L}'$ is an $\text{\rm{S}}$--form of ${\cal L}$.* *(3 bis) $\text{\rm \bf L}'$ is an $\text{\rm{S}}$--form of $\text{\rm \bf L},$ where $\text{\rm \bf L}' = \text{\rm \bf W}({\cal L}')$ and $\text{\rm \bf L}= \text{\rm \bf W}({\cal L})$.* *Proof.* (1) Let $\text{\rm \bf L}' = \text{\rm \bf W}({\cal L}').$ By Lemma [Lemma 3](#repres){reference-type="ref" reference="repres"} $\text{\rm \bf H}:= \text{\bf{Aut}}_{{\cal O}_\text{\rm{S}}- {\rm Lie}}({\cal L}') \simeq \text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}- {\rm Lie}}(\text{\rm \bf L}')$ is an affine $\text{\rm{S}}-$group scheme of finite presentation. Since $\text{\rm \bf H}_\text{\rm{T}}= {\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{T}}- {\rm Lie}}(\text{\rm \bf L}'_\text{\rm{T}})}$ for all $\text{\rm{S}}$--scheme $\text{\rm{T}},$ we have $\text{\rm \bf H}_s = \text{\bf{Aut}}(\text{\rm \bf L}'_s)$ for all $s \in \text{\rm{S}},$ where $\text{\rm \bf L}'_s := \text{\rm \bf L}' \otimes_{{\cal O}_\text{\rm{S}}} \kappa(s).$ These algebraic groups are smooth since $\kappa(s)$ is of characteristic $0.$ Let $b \in \text{\rm{S}}$ be a closed point. Since the $\kappa(b)$--Lie algebra ${\cal L}'_b$ is a twisted form of ${\cal L}_b$, the corresponding $\kappa(b)$-algebraic group $\text{\bf{Aut}}({\cal L}'_b)$ is a twisted form of $\text{\bf{Aut}}({\cal L}_b).$ Similarly for their connected component of the identity. As we have seen that $\text{\bf{Aut}}({\cal L}'_b) = \text{\rm \bf H}_b.$ On the other hand since $\text{\rm \bf G}$ is semisimple adjoint $\text{\bf{Aut}}({\cal L}_b)^\circ = \text{\rm \bf G}_b.$ This yields that $\text{\rm \bf H}_b^\circ$ is a twisted form of $\text{\rm \bf G}_b$. In particular all the $\text{\rm \bf H}_b$ are smooth groups of the same dimension. Proposition [Proposition 12](#prop_smooth){reference-type="ref" reference="prop_smooth"}.(2) then shows that the $\text{\rm{S}}$--subfunctor $\text{\rm \bf H}^\circ$ of $\text{\rm \bf H}$ is representable by a smooth affine $\text{\rm{S}}$--group scheme $\text{\rm \bf G}'$. According to [@Co1 prop. 3.1.9.(1)], $\text{\rm \bf G}'$ is reductive on a neighborhood of each closed point $b \in \text{\rm{S}}$. Since $\text{\rm{S}}$ is a Jacobson scheme, it follows that $\text{\rm \bf G}'$ is reductive. Since $\text{\rm{S}}$ is connected the type of $\text{\rm \bf G}'$ is constant. Since $\text{\rm \bf G}'$ and $\text{\rm \bf G}$ have the same type at closed points, their (constant) types coincide. Thus $\text{\rm \bf G}'$ is a twisted form of $\text{\rm \bf G}$ [@SGA3 XXIII 5.6]. In particular, $\text{\rm \bf G}'$ is semisimple adjoint. \(2\) By Lemma [Lemma 6](#derLie){reference-type="ref" reference="derLie"} $$\text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L}') \simeq \mathop{\bf Lie}\nolimits\big(\text{\bf{Aut}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L}')\big) = \mathop{\bf Lie}\nolimits(\text{\rm \bf H}^\circ) = \mathop{\bf Lie}\nolimits(\text{\rm \bf G}').$$ On the other hand we have an isomorphism of $\text{\rm \bf O}_\text{\rm{S}}$--Lie algebras $\text{\rm \bf L}' \simeq \text{\bf{Der}}_{\text{\rm \bf O}_\text{\rm{S}}-{\rm alg}}(\text{\rm \bf L}')$ (Lemma [Lemma 7](#MZ){reference-type="ref" reference="MZ"}), whence an isomorphism $\text{\rm \bf L}' \simeq \mathop{\bf Lie}\nolimits(\text{\rm \bf G}')$. It follows that $\text{\rm \bf W}({\cal L}') \simeq \text{\rm \bf W}(\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}'))$. Since $\text{\rm \bf W}$ is full and faithful ${\cal L}'$ and $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}')$ are isomorphic ${\cal O}_\text{\rm{S}}$--Lie algebras. \(3\) From (2) and Remark [Remark 5](#Liesmooth){reference-type="ref" reference="Liesmooth"} we get that $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}') = \text{\rm \bf W}\big(\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}')\big)$ is a form of $\mathop{\bf Lie}\nolimits(\text{\rm \bf G}) = \text{\rm \bf W}\big(\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G})\big).$ It follows that $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}')$ is a form of $\mathcal{L}\mathcal{i}\mathcal{e}(\text{\rm \bf G}).$ Thus (3) follows from (2). (3 bis.) Follows from (3) since $\text{\rm \bf W}$ commutes with base change. ◻ Burban's question[\[Burban\]]{#Burban label="Burban"} In order to formulate Burban's question we need to recall the concept of loop algebra of a simple Lie finite dimensional complex Lie algebra $\mathfrak g.$ Let $R= \mathbb C[t^{\pm 1}].$ Fix a positive integer $d,$ and set $R_d = \mathbb C[t^{\pm \frac{1}{d}}].$ The natural map $R\to R_d$ is faithfully flat and finite étale. Let $\xi \in \mathbb C$ be a primitive $d$-th root of unity. Then the elements of $\Gamma = \Bbb Z/d\Bbb Z$ act as automorphisms of $R_d$ over $R$ via $$^{\overline{e}} t^{\frac{1}{d}}= \xi ^e t^{\frac{1}{d}}.$$ for $e \in \Bbb Z.$ This action makes $R_d$ into a Galois extension of $R$ with Galois group $\Gamma.$ Let $\sigma$ be an automorphism of $\mathfrak g$ of order $d.$ For $i \in \Bbb Z$ consider the eigenspace $$\mathfrak g_i =\{x\in \mathfrak g:\sigma(x) = \xi ^ix\}$$ Then $\mathfrak g= \underset{0 \leq i <d}\oplus \mathfrak g_i$. Out of this data we define the corresponding *loop algebra* $$\label{defmulti} L(\mathfrak g,\sigma) = \underset{i \in \Bbb Z}\oplus\, \mathfrak g_i \otimes t^{\frac{i}{d}} \subset \mathfrak g\otimes _\mathbb CR_d.$$ The simple but crucial observation is that $L(\mathfrak g,\sigma)$ is stable under the scalar action of $R.$ Thus $L(\mathfrak g,\sigma)$ is not only an infinite dimensional complex Lie algebra, but also an $R$-Lie algebra. As we shall see, it is the algebra structure over this ring that allows non-abelian cohomological considerations to enter into the picture. It is an easy linear algebra exercise to verify that we have a natural $R_d$--algebra isomorphism $$\label{iso1} L(\mathfrak g, \sigma) \otimes_R R_d \simeq \mathfrak g\otimes_\mathbb CR_d \simeq \mathfrak g_R \otimes_R R_d.$$ where $\mathfrak g_R := \mathfrak g\otimes_\mathbb CR.$ This shows that the $R$-Lie algebra ${\rm L}:= L(\mathfrak g, \sigma)$ is a twisted form of $\mathfrak g\otimes_\mathbb CR.$ It therefore corresponds to an $\text{\bf{Aut}}(\mathfrak g_R)$--torsor $\text{\rm \bf X}$ over $\mathop{\rm Spec}\nolimits(R).$ More precisely $\text{\rm \bf X}= \text{\bf{Isom}}_{R-{\rm Lie}}(\mathfrak g_R, {\rm L}).$[^8] We can now formulate Burban's questions (essentially verbatim except for some notation changes): **Question 1:** Let ${\rm L}'$ be a Lie algebra over the ring $\mathbb C[t]$. Assume ${\rm L}'$ is free as a module and that for any complex number $b$ the quotient Lie algebra ${\rm L}'/(t-b){\rm L}'$ is isomorphic to $\mathfrak g.$ Does it follow that ${\rm L}'$ is isomorphic to $\mathfrak g\otimes_\mathbb C\mathbb C[t]$ (as a Lie algebra over $\mathbb C[t]$)? **Question 2:** Similarly, let ${\rm L}'$ be a Lie algebra over the ring $\mathbb C[t^{\pm 1}]$ which we assume is free as a module and such that for any $b \in \mathbb C^\times$ the quotient ${\rm L}'/(t-b){\rm L}'$ is isomorphic to $\mathfrak g.$ Does it follow that ${\rm L}'$ is isomorphic to a loop algebra (with respect to an automorphism of $\mathfrak g$ of finite order)? To answer these questions we take $\text{\rm{S}}=\mathop{\rm Spec}\nolimits(R)$ where $R =\mathbb C[t]$ or $\mathbb C[t^{\pm 1}]$, and where ${\cal L}'$ and ${\cal L}$ are the ${\cal O}_\text{\rm{S}}$--Lie algebra corresponding to ${\rm L}'$ and $\mathfrak g\otimes_\mathbb CR$ respectively. By Proposition [Proposition 16](#prop_lie_form2){reference-type="ref" reference="prop_lie_form2"} we see that ${\cal L}'$ corresponds to a torsor $\text{\rm \bf X}'$ over $\mathop{\rm Spec}\nolimits(R)$ whose class is an element of $H^1(R, \text{\bf{Aut}}(\mathfrak g_R)).$ Recall ([@SGA3 XXIV 1.3 and 7.3.1]. See also §[\[2.4\]](#2.4){reference-type="ref" reference="2.4"}) the split exact sequence of $R$--group schemes $$\label{splitexact} 1 \to \text{\rm \bf G}\to \text{\bf{Aut}}({\rm L}) \to \text{\bf{Out}}({\rm L}) \to 1$$ where $\text{\rm \bf G}$ is the split adjoint semisimple $R$-group scheme corresponding to $\mathfrak g$, and $\text{\bf{Out}}({\rm L})$ is the constant $R$-group scheme corresponding to the finite (abstract) group ${\rm Out}(\mathfrak g)$ of the symmetries of the Coxeter-Dynkin diagram of $\mathfrak g.$ This allow us to compute the relevant $H^1$ an thus determine the nature of $\text{\rm \bf X}'$, hence ${\rm L}'.$ According to [@CGP Cor. 3.3], we have a bijection $$H^1(R,\text{\bf{Aut}}({\rm L})) \buildrel\sim\over\longrightarrow H^1(R,\text{\bf{Out}}({\rm L})).$$ **Answer 1:** If $R =\mathbb C[t]$ then $H^1\big(R, \text{\bf{Out}}(L))=1$ since $R$ is simply connected. It follows that $H^1\big(R, \text{\bf{Aut}}({\rm L})\big) = 1$ and therefore that $L' \simeq \mathfrak g\otimes_k R$. **Answer 2:** If $R = \mathbb C[t^{\pm 1}]$ then furthermore $H^1(R, \text{\bf{Out}}({\rm L}))$ is the set of conjugacy classes of the (abstract) group ${\rm Out}(\mathfrak g)$ which, in terms of forms of ${\rm L}$, correspond to the loop algebras $L(\mathfrak g, \sigma)$ with $\sigma \in {\rm Out}(\mathfrak g)$ (see [@P2] for details, or more generally [@P3]). Thus ${\rm L}'$ is a loop algebra. 99 I. Burban, Open correspondence, 2020. V. Chernousov, P. Gille and A. Pianzola, *Three-point Lie algebras and Grothendieck's dessins d'enfants*, Mathematical Research Letters **23** (2016), 81-104. B. Conrad, *Reductive group schemes*, in *Autour des schémas en groupes, vol. I*, Panoramas et Synthèses **42-43**, Soc. Math. France 2014. M. Demazure et P. Gabriel, *Groupes algébriques*, Masson (1970). A. Grothendieck and J. Dieudonné, *Éléments de géométrie algébrique: I. Le langage des schémas*, Grundlehren der Mathematischen Wissenschaften **166** (2nd ed.). Berlin; New York: Springer-Verlag, 1971. A. Grothendieck (avec la collaboration de J. Dieudonné), *Eléments de Géométrie Algébrique IV: Étude locale des schémas et des morphismes de schémas*, Publications mathématiques de l'I.H.É.S. no 20, 24, 28 and 32 (1964 - 1967). U. Görtz and T. Wedhorn, *Algebraic Geometry I*, Viehweg+Teubner 2010. R. Hartshorne, *Algebraic Geometry*, Graduate Texts in Mathematics, Springer. Q. Liu, D. Lorenzini, M. Raynaud, *Néron models, Lie algebras, and reduction of curves of genus one*, Invent. Math. **157** (2004), 455-518. A. Pianzola, *Derivations of certain algebras defined by $\acute{e}$tale descent*. Math. Z. **264** (2010). A. Pianzola, *Affine Kac-Moody Lie algebras as torsors over the punctured line*, Indagationes Mathematicae N.S. **13**(2) (2002) 249-257. A. Pianzola, *Vanishing of $H^1$ for Dedekind rings and applications to loop algebras*, C. R. Acad. Sci. Paris, Ser. I **340** (2005), 633-638. *Séminaire de Géométrie algébrique de l'I.H.E.S., 1963-1964, schémas en groupes, dirigé par M. Demazure et A. Grothendieck*, Lecture Notes in Math. 151-153. Springer (1970). Stacks project, http://stacks.math.columbia.edu/ [^1]:   Note that our ${\rm L}$ is denoted by ${\cal L}$ in [@P1]. [^2]: AP. We will not use these properties in what follows. [^3]: *  Since $\text{\rm \bf G}$ is of finite presentation $\text{\rm \bf G}_s$ is an algebraic $\kappa(s)$-group. Thus $\text{\rm \bf G}_s$ has a finite number of irreducible components. They all have the same (finite) dimension and this is also the dimension of $\text{\rm \bf G}_s.$* [^4]:   That is, all the fibres $\text{\rm \bf G}_s, s \in \text{\rm U}$ are equidimensional and this dimension is $d.$ [^5]:   A ${\cal O}_{\mathop{\rm Spec}\nolimits\kappa(s)}$--algebra "is the same\" as a $\kappa(s)$--algebra. [^6]: *  Necessarily of finite dimension $d.$* [^7]: *  By definition $\text{\rm \bf L}_0(\text{\rm{T}})$ is the ${\cal O}_\text{\rm{T}}(\text{\rm{T}})$-Lie algebra ${\rm L}_0\otimes_\Bbb Q{\cal O}_\text{\rm{T}}(\text{\rm{T}}).$* [^8]:   See [@DG III §4] for the material on torsors used in this section.

arxiv_math

--- abstract: | We prove that two closed subsets of complex space $\mathbb{C}^n$ with corresponding complex homothetic sections (projections) are complex homothetic. The proof uses a new Helly-type theorem for cosets of closed subgroups of $\mathbb{S}^1$. address: UNAM, Mexico author: - Jorge Luis Arocha - Javier Bracho - Luis Montejano title: | Complex homothetic sections and projections\ through a Helly type Theorem for cosets of $\mathbb{S}^1$ --- # Introduction *Let $V$ be an $n$-dimensional vector space and let $K_1, K_2$ be two closed sets, $n\geq3.$ Suppose that for every 2-dimensional subspace (hyperplane) $\Lambda$ the corresponding sections $$\Lambda\cap K_1 \mbox{ and } \Lambda\cap K_2$$ are "equivalent". Is it true that $$K_1 \mbox{ is ``equivalent'' to } K_2?$$ Where the meaning of "equivalent" will be clarified in the sequel.* Suppose for example that $K_1, K_2$ are convex bodies containing the origin in euclidean $n$-space, $n\geq 3$, and the meaning of "equivalent" is congruence. That is, suppose that for every hyperplane $\Lambda$ through the origin there is a linear isometry $g\in O(\Lambda)$ and a vector $v\in \Lambda$ (depending on $\Lambda$) such that $$g(\Lambda\cap K_1) + v = \Lambda\cap K_2.$$ Is it true that $K_1$ is either a translated copy of $K_2$ or $-K_2$? V.P. Golubyatnikov [@G] and D. Ryagobin [@R0], [@R1], [@R3], [@R2], have studied this problem extensively. Perhaps the most representative result in this direction is Ryagobin's Theorem in [@R0]. Suppose $K_1, K_2$ are convex bodies in euclidean $n$-space such that for every plane $\Lambda$ through the origin there is a rotation $r$ of $\Lambda$ about the origin (depending on $\Lambda$) for which $r(\Lambda\cap K_1) = (\Lambda\cap K_2)$, then either $K_1=K_2$ or $K_1=-K_2$. Ryagobin's proof rests heavily on the fact that there are only a countable number of closed subgroups of $SO(2)$. The analogous is not true for $SO(k)$, $k>2$, and this is a reason why the result is unknown for $k$-subspaces in $\mathbb{R}^n$. In the charming paper [@R], Rogers proved the following. Suppose $K_1, K_2$ are convex bodies in euclidean $n$-space, $n\geq 3$, such that for every hyperplane $\Lambda$ through the origin $$\Lambda\cap K_1 \mbox{ is positively homothetic to } \Lambda\cap K_2,$$ then $K_1$ is positively homothetic to $K_2.$ Note that the center of each homothecy between $\Lambda\cap K_1$ and $\Lambda\cap K_2$ is not fixed and of course depends on the hyperplane $\Lambda$. This result was crucial in many characterizations of ellipsoids in the second half of the 20th century. Studying complex ellipsoids in [@ABMCE], it was necessary to prove a complex version of Rogers' Theorem for real homotheties. More precisely, suppose $K_1, K_2$ are convex bodies in complex $n$-space, $\mathbb{C}^n$, $n\geq 3$, such that for every complex hyperplane $\Lambda$ through the origin $$\Lambda\cap K_1 \mbox{ is positively homothetic to } \Lambda\cap K_2,$$ then $K_1$ is positively homothetic to $K_2.$ It would be interesting to know if the following complex version of Ryagobin's Theorem is true. Suppose two convex bodies $K_1, K_2$ in complex $n$-space $\mathbb{C}^n$ have the property that for every complex plane $\Lambda$, there is $g\in U(\Lambda)$ such that $$g(\Lambda\cap K_1)= \Lambda\cap K_2\,.$$ Is there $z\in \mathbb{S}^1$ with $|z|=1$ such that $zK_1=K_2$? It is precisely in this direction that we obtain the following theorem as a special case of the above. We note that our result is true not only for convex subsets but also for closed subsets of $\mathbb{C}^n$ (including of course finite sets). **Theorem 1**. *Let $K_1, K_2$ be two closed subsets of $\mathbb{C}^n$, $n\geq3$. Suppose that for every complex plane $\Lambda$ through the origin there is a complex number $z_\Lambda\in\mathbb{C}$ such that $$z_\Lambda(\Lambda\cap K_1)=\Lambda\cap K_2.$$ Then, there is $z\in \mathbb{C}$, such that $$zK_1 = K_2.$$* The analogue of Theorem [Theorem 1](#mainsec){reference-type="ref" reference="mainsec"} in the real setting is essentially Lemma 1 of [@R0]. The proof of Theorem [Theorem 1](#mainsec){reference-type="ref" reference="mainsec"} makes use of the following Helly-type theorem for cosets of closed subgroups of $\mathbb{S}^1$. **Theorem 2**. *Let $\mathcal{F}$ be a pairwise intersecting family of cosets of closed subgroups of $\mathbb{S}^1$. Then $\mathcal{F}$ has non-empty intersection.* It would be interesting to know which groups have this Helly property. That is, for which groups $G$ there exists an integer $m>2$ with the property that if $\mathcal{F}$ is a family of m+1 cosets of closed subgroups of $G$ and each $m$ members de $\mathcal{F}$ is intersecting, then $\mathcal{F}$ is intersecting. Indeed, the integers $\mathbb{Z}$ have this property. We will prove that every three pairwise intersecting arithmetic sequences is intersecting (see Theorem [Theorem 4](#Teo_Helly_integers){reference-type="ref" reference="Teo_Helly_integers"}). Similarly, using Theorem [Theorem 2](#Teo_Helly_orbital){reference-type="ref" reference="Teo_Helly_orbital"}, we prove the corresponding result for projections. Let $\Lambda$ be a complex subspace of $\mathbb{C}^n$, we denote by $\pi_\Lambda:\mathbb{C}^n\to \Lambda$, the orthogonal projection. **Theorem 3**. *Let $K_1, K_2\subset \mathbb{C}^n$ be either two closed convex sets or two finite sets, $n\geq3$. Suppose that for every complex plane $\Lambda$ through the origin there is a complex number $z_\Lambda\in\mathbb{C}$ such that $$z_\Lambda(\pi_\Lambda(K_1))=\pi_\Lambda(K_2).$$ Then, there is $z\in \mathbb{C}$, such that $$zK_1 = K_2.$$* Regarding projections of convex bodies in $\mathbb{R}^5$ whose projections are $SU(2)$-congruent, Ryagobin obtained interesting results in [@R3]. # The group of complex homotheties From now on, unless otherwise stated, all our lines, planes, hyperplanes pass through the origin. Let $V$ be a complex vector space and let $K\subset V$ be a closed subset. Define the *group of homotheties of* $K$ as: $$H(K)=\{\,z\in\mathbb{S}^1\,|\, zK=K\,\},$$ which is a closed subgroup of $\mathbb{S}^1$. For every subspace $\Delta\subset \mathbb{C}^n$, $$H(K)\mbox{ is a subgroup of } H(\Delta\cap K),$$ $$\label{Eq:Sub_proy} H(K)\mbox{ is a subgroup of } H(\pi_\Delta (K)).$$ Note that (1) is trivially true. On the other hand, if $\pi_\Delta:\mathbb{C}^n\to \Delta$ is the orthogonal projection, then $z\pi_\Delta(x)=\pi_\Delta(zx)$, for every $z\in \mathbb{C}$ and every $x\in \mathbb{C}^n$. Hence, $zK=K$ implies $z\pi_\Delta K=\pi_\Delta K$ and consequently (2) is true. Now, if $K_1, K_2$ are closed subsets of $V$, let us define $$H(K_1, K_2)=\{z\in\mathbb{S}^1|zK_1=K_2\}.$$ For every $z_0\in H(K_1,K_2)$, $$\label{Eq:Cosets} z_0H(K_1)=H(K_1,K_2)\,.$$ Therefore, if $H(K_1,K_2)$ is nonempty, it is a coset of the closed subgroup $H(K_1)$ of $\mathbb{S}^1$. Let $\mathcal{L}$ denote the set of complex lines through the origin in $\mathbb{C}^n$. Of course, it is the complex projective space of dimensión $n-1$, but we need to refer to its elements as complex lines. **Lemma 1**. *Let $K\subset \mathbb{C}^n$, $n\geq3$, be a closed subset. Then $$H(K)=\bigcap_{L\in\mathcal{L}}H(L\cap K).$$* By (1), $H(K)\subset \bigcap_{L\in\mathcal{L}}H(L\cap K)$. Suppose now, $z\in \bigcap_{L\in\mathcal{L}}H(L\cap K)$, that is, $z(L\cap K)=L\cap K$, for every line $L\in\mathcal{L}$. Consequently, $zK=K$, which implies that $z\in H(K)$ as we wished. 0◻ # Helly type results For the proof of our main Helly type theorem (Theorem [Theorem 2](#Teo_Helly_orbital){reference-type="ref" reference="Teo_Helly_orbital"}), we need two lemmas. **Lemma 2**. *Let $a, b \in \mathbb{Z}$, then $a \mathbb{Z}$, $b \mathbb{Z}$ and $a + (b-a)\mathbb{Z}$ have non empty intersection.* Let $d$ be the greatest common divisor of $a$ and $b$. There exist $a^\prime, b^\prime, x, y \in \mathbb{Z}$ such that $$a=d a^\prime \quad \text{,}\quad b=d b^\prime$$ and $$x a^\prime + y b^\prime=1\,.$$ Adding and subtracting $x b^\prime$ $$x (a^\prime-b^\prime) + (x+y) b^\prime=1\,.$$ From which $$(x+y) b^\prime=1+x (b^\prime-a^\prime)\,,$$ and multiplying by $d a^\prime=a$, we get $$(x+y) d a^\prime b^\prime=a+x a^\prime (b-a)\,.$$ The left hand side is in $a \mathbb{Z}\cap b \mathbb{Z}$; the right hand side is in $a + (b-a)\mathbb{Z}$. 0◻ **Lemma 3**. *Let $A, B, C$ be cosets of closed subgroups of $\mathbb{S}^1$ such that each pair is intersecting, then they intersect.* If one of the given cosets is $\mathbb{S}^1$, we are done, so we may assume that the three cosets are finite. Without loss of generality, we can suppose that $A$ and $B$ are subgroups of $\mathbb{S}^1$ (so that ${1\in A\cap B}$), for otherwise, take $z\in A\cap B$, and consider $z^{-1}A, z^{-1}B, z^{-1}C$. By hypothesis, there exist $\alpha\in A\cap C$ and $\beta\in B\cap C$. Observe that $\alpha\langle \beta\alpha^{-1} \rangle\subset C$, where we denote by $\langle z \rangle$ the subgroup of $\mathbb{S}^1$ generated by $z\in\mathbb{S}^1$. Let $p=order(\alpha)$ and $q=order(\beta)$. So that $\langle \alpha \rangle\subset A$ and $\langle \beta \rangle\subset B$ are the cyclic groups of order $p$ and $q$ respectively. Let $m$ be the minimum common multiple of $p$ and $q$. And let $$\begin{aligned} \varphi: \mathbb{Z}&\to\mathbb{S}^1 \\ \varphi(n)&=exp(n\frac{2\pi}{m})\end{aligned}$$ where $exp:\mathbb{R}\to\mathbb{S}^1$ is the standard exponential map. By construction, there exist $a, b\in\mathbb{Z}$ such that $\varphi(a)=\alpha$ and $\varphi(b)=\beta$. By Lemma [Lemma 2](#Helly-series){reference-type="ref" reference="Helly-series"}, there exists $c\in a\mathbb{Z}\cap b\mathbb{Z}\cap (a+(b-a)\mathbb{Z})$, so that $$\varphi(c)\in\langle \alpha \rangle\cap\langle \beta \rangle\cap\alpha\langle \beta\alpha^{-1} \rangle \subset A\cap B\cap C\,.$$ 0◻ . Let $\mathcal{F}$ be a pairwise intersecting family of cosets of closed subgroups of $\mathbb{S}^1$. First, we prove the case where $\mathcal{F}$ is finite by induction on the cardinality, $r$, of $\mathcal{F}$. For $r=3$ it is Lemma [Lemma 3](#Helly-3){reference-type="ref" reference="Helly-3"}. Suppose the Theorem true for families of size $r-1$. Consider $A\in\mathcal{F}$ and let $\mathcal{F}^\prime=\{\,A\cap X\,|\,X\in\mathcal{F}\setminus\{A\}\,\}$. By Lemma [Lemma 3](#Helly-3){reference-type="ref" reference="Helly-3"}, every pair of elements of $\mathcal{F}^\prime$ intersect and by induction they all have non-empty intersection which is also the intersection of all the elements of $\mathcal{F}$. Finally, consider the general case. Suppose that $\mathcal{F}$ is not intersecting. Then the family of its complements cover $\mathbb{S}^1$. They are open sets, so that the compactness of $\mathbb{S}^1$ gives us a finite open subcover. By the preceding paragraph, in the corresponding finite family of closed cosets of groups of $\mathbb{S}^1$ there must be a non intersecting pair, which is a contradiction. 0◻ Again using Lemma [Lemma 2](#Helly-series){reference-type="ref" reference="Helly-series"}, it is easy to prove that not only the group $\mathbb{S}^1$ has this Helly property but also the group $\mathbb{Z}$ for finite families. **Theorem 4**. *Let ${\mathcal{F}}$ be a finite pairwise intersecting family of cosets of subgroups of $\mathbb{Z}$. Then ${\mathcal{F}}$ has non empty intersection.* Following the proof of Theorem [Theorem 2](#Teo_Helly_orbital){reference-type="ref" reference="Teo_Helly_orbital"}, it is enough to consider the case when ${\mathcal{F}}=\{A,B,C\}$ consists of three cosets of subgroups of $\mathbb{Z}$. Therefore $A, B$ and $C$ are arithmetic sequences and without loss of generality $A=a\mathbb{Z}$, and $B=b\mathbb{Z}$. Assume $a'\in C\cap A$ and $b'\in C\cap B$, Consequently, $a'\mathbb{Z}\subset A$, $b'\mathbb{Z}\subset B$ and $a'+(b'-a')\mathbb{Z}\subset C$. By Lemma [Lemma 2](#Helly-series){reference-type="ref" reference="Helly-series"}, we have that ${\mathcal{F}}=\{A,B,C\}$ is intersecting. This concludes the proof of the theorem. 0◻ # Complex homothetic sections and projections Let $K_1, K_2$ be two non-homothetic complex ellipsoids centered at the origin in $\mathbb{C}^n$. Hence, for every line $L$, $L\cap K_1$ and $L\cap K_2$ are disks centered at the origin and consequently there is $z_L$ such that $$z_L(L\cap K_1)= L\cap K_2.$$ Nevertheless, $zK_1\not=K_2$, for $z\in \mathbb{C}$. This shows that Theorem 1 is false if we replace complex planes with complex lines. For the proof of Theorem [Theorem 1](#mainsec){reference-type="ref" reference="mainsec"}, we will start by noting that in its hypothesis we can assume, without loss of generality, that the module of all $z_\Lambda$ is 1. **Lemma 4**. *Let $K_1, K_2$ be closed subsets of $\mathbb{C}^n, n\geq3$ and suppose that for every complex plane $\Lambda$ there is $z_\Lambda\in \mathbb{C}$, such that $$z_\Lambda(\Lambda\cap K_1)=\Lambda\cap K_2.$$ Then, we may assume that $|z_\Lambda|$ is constant.* Let us call a subset of $\mathbb{C}^n$ *nontrivial* if it is neither empty nor the origin, and observe that for a nontrivial set $A\subset\mathbb{C}^n$, if $z_1A=z_2A$ then $|z_1|=|z_2|$. We may clearly assume $K_1$ is nontrivial, so that there exists a line $L$ such that $L\cap K_1$ is nontrivial. If $\Lambda_1$ and $\Lambda_2$ are two complex planes which contain the complex line $L$, then by hypothesis, $$z_{\Lambda_1}(L\cap K_1)=L\cap K_2=z_{\Lambda_2}(L\cap K_1)$$ and then, $|z_{\Lambda_1}|=|z_{\Lambda_2}|=r_L$. Now consider a plane $\Lambda$ that does not contain $L$. If $\Lambda\cap K_1$ is trivial, we may choose $z_{\Lambda}=r_L$. Otherwise, there is a line $L^\prime\subset\Lambda$ such that $L^\prime\cap K_1$ is nontrivial. Let $\Lambda^\prime$ be the plane generated by $L$ and $L^\prime$. Then, $$z_{\Lambda}(L^\prime\cap K_1)=L^\prime\cap K_2=z_{\Lambda^\prime}(L^\prime\cap K_1)\,,$$ and therefore, $|z_{\Lambda}|=|z_{\Lambda^\prime}|=r_L$. 0◻ . First, by Lemma [Lemma 4](#lemwlg){reference-type="ref" reference="lemwlg"} we may assume that $|z_\Lambda|=1$ for every complex plane $\Lambda$. Let $L$ be a complex line and $L \subset \Lambda$, where $\Lambda$ is a complex plane. Then $z_\Lambda(L\cap K_1)=(L\cap K_2).$ Therefore $z_\Lambda$ lies in the coset $H(L\cap K_1,L\cap K_2)\subset \mathbb{S}^1$ which is not empty. Let $$\mathcal{F}=\{\, H(L\cap K_1,L\cap K_2)\,\mid\, L\in\mathcal{L}\,\}\,.$$ We shall prove that $\mathcal{F}$ is a pairwise intersecting family of cosets of closed subgroups of $\mathbb{S}^1$. Let $L_1, L_2$ be two different complex lines and let $\Lambda$ be the complex plane generated by $L_1$ and $L_2$. Hence $$z_\Lambda\in H(L_1\cap K_1,L_1\cap K_2)\cap H(L_2\cap K_1,L_2\cap K_2)\not= \emptyset\,.$$ By Theorem [Theorem 2](#Teo_Helly_orbital){reference-type="ref" reference="Teo_Helly_orbital"} and Lemma [Lemma 1](#lemlines){reference-type="ref" reference="lemlines"}, there exists $$z\in \bigcap_{L\in\mathcal{L}}H(L\cap K_1,L\cap K_2)=H(K_1,K_2)\,.$$ Therefore, $zK_1=K_2$ as we wished. 0◻ For the proof of Theorem [Theorem 3](#mainproj){reference-type="ref" reference="mainproj"} we need two lemmas. **Lemma 5**. *If $K_1, K_2\subset \mathbb{C}^n$ are either two closed convex sets or two finite sets such that $\pi_L(K_1)= \pi_L(K_2)$ for every complex line $L$, then $K_1=K_2$.* Suppose first $K_1, K_2\subset \mathbb{C}^n$ are two closed convex sets. If $K_1\not=K_2$, then without loss of generality there is a point $x\in K_1$ and a real affine hyperplane $\Gamma$ such that $x$ is contained in the open semispace generated by $\Gamma$ and $K_2$ is contained in its complement. Let $\Delta$ be the unique complex affine hyperplane contained in $\Gamma$ and let $L$ be the complex line through the origin orthogonal to $\Delta$. Consider $\pi_L:\mathbb{C}^n\to L$ be the orthogonal projection. Then $\pi_L(x)\in \pi_L(K_1)$ is contained in the open half plane generated by the real line $\pi_L(\Gamma)$ of $L$ and $\pi_L(K_2)$ is contained in the complement, contradicting the assumption that $\pi_L(K_1)=\pi_L(K_2)$. Suppose now $K_1, K_2\subset \mathbb{C}^n$ are two finite sets. If they are different, then without loss of generality we may chose $x\in K_1\setminus K_2$. Since $K_1\cup K_2$ is finite there exists a complex line through the origin $L$ such that $\pi_L\mid_{K_1\cup K_2}$ is injective. Then $\pi_L(x)\notin K_2$, contradicting the assumption that $\pi_L(K_1)=\pi_L(K_2)$. 0◻ **Lemma 6**. *Let $K, K_1, K_2 \subset \mathbb{C}^n$, $n\geq3$, be either closed convex sets or finite sets. Then* 1. *$H(K)=\bigcap_{L\in\mathcal{L}}H(\pi_L(K))\,,$* 2. *$H(K_1, K_2)=\bigcap_{L\in\mathcal{L}}H(\pi_L(K_1), \pi_L(K_2) ).$* We know that $H(K)\subset \bigcap_{L\in\mathcal{L}}H(\pi_L(K))$ from [\[Eq:Sub_proy\]](#Eq:Sub_proy){reference-type="eqref" reference="Eq:Sub_proy"}. Suppose now, ${z\in \bigcap_{L\in\mathcal{L}}H(\pi_L(K))}$, that is, $z\pi_L(K)=\pi_L(K)$, for every line $L\in\mathcal{L}$. Consequently, $\pi_L(zK)=\pi_L(K)$, for every line $L$ and therefore, by Lemma [Lemma 5](#lemA){reference-type="ref" reference="lemA"}, $zK=K$, which implies that $z\in H(K)$ as we wished. Moreover, for $z_0\in H(K_1, K_2)$, by [\[Eq:Cosets\]](#Eq:Cosets){reference-type="eqref" reference="Eq:Cosets"} and item *(1)* of this lemma, we have: $$\begin{aligned} H(K_1, K_2)&=z_0H(K_1)=z_0\bigcap_{L\in\mathcal{L}}H(\pi_L(K_1))=\\ &=\bigcap_{L\in\mathcal{L}}z_0H(\pi_L(K_1))=\bigcap_{L\in\mathcal{L}}H(\pi_L(K_1), \pi_L(K_2) ).\end{aligned}$$ 0◻ . As in Lemma [Lemma 4](#lemwlg){reference-type="ref" reference="lemwlg"}, let us first prove that we may assume that $|z_\Lambda|$ is constant and does not depend on $\Lambda$. Remember from Section 2 that if $A$ is a closed subset of $\mathbb{C}^n$, $\Delta$ is a subspace of $\mathbb{C}^n$, $z\in \mathbb{C}$ and $zA=A$, then $z\pi_\Delta(A)=\pi_\Delta(A)$. Now, call a line $L$ *nontrivial* if $\pi_L(K_1)$ is nontrivial. If $L$ is a nontrivial complex line contained in the complex planes $\Lambda_1$ and $\Lambda_2$, then $z_{\Lambda_1}(\pi_L(K_1))=\pi_L(K_2)=z_{\Lambda_2}(\pi_L(K_1))$ and so, $|z_{\Lambda_1}|=|z_{\Lambda_2}|.$ If $\Lambda_1$ and $\Lambda_2$ are two complex planes with nontrivial $\pi_{\Lambda_i}(K_1)$, there is a third complex plane $\Lambda_3$ such that $\Lambda_i\cap\Lambda_3$ is a nontrivial complex line, $i=1,2$, so we obtain that $|z_{\Lambda_1}|= |z_{\Lambda_3}|=|z_{\Lambda_2}|$. Consequently, we may assume without loss of generality that $|z_\Lambda|=1$, for every complex plane. This time, let $$\mathcal{F}=\{ H(\pi_L(K_1), \pi_L (K_2))\mid L\in\mathcal{L}\}\,.$$ We shall prove that $\mathcal{F}$ is a pairwise intersecting family of cosets of closed subgroups of $\mathbb{S}^1$. Let $L_1, L_2$ be two different complex lines and let $\Lambda$ be the complex plane generated by $L_1$ and $L_2$. Hence, $$z_\Lambda\in H(\pi_{L_1}(K_1), \pi_{L_1} (K_2))\cap H(\pi_{L_2}(K_1), \pi_{L_2} (K_2))\,.$$ By Theorem [Theorem 2](#Teo_Helly_orbital){reference-type="ref" reference="Teo_Helly_orbital"}, there is $$z\in \bigcap_{L\in\mathcal{L}}H(\pi_L( K_1), \pi_L(K_2))\,.$$ Therefore, by Lemma [Lemma 6](#lemI){reference-type="ref" reference="lemI"}*(2), $z\in H(K_1,K_2)$. 0◻* Luis Montejano acknowledges support from CONACyT under project 166306 and from PAPIIT-UNAM under project IN112614. Javier Bracho acknowledges support from PAPIIT-UNAM under project IN109023. 11 M. Alfonseca, M. Cordier and D. Ryabogin, *On bodies with directly congruent projections and sections*, Israel J. Math., 215 (2016), no. 2, 765-799 J. L. Arocha, J. Bracho, and L. Montejano, *Complex ellipsoids and complex symmetry*, Aequationes mathematicae, 97, 295--306 (2023) V. P. Golubyatnikov, *Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography Type Projection* Data, Inverse and Posed Problems Series. Utrecht, Boston (2000). D. Ryabogin, *On the continual Rubik's cube* Advances in Mathematics 231 (2012) 3429--3444 D. Ryabogin, *On symmetries of projections and sections of convex bodies*, Discrete Geometry and Symmetry. In: Marston, D.E., Conder, A.D. and Weiss, A.I. (eds.) Honor of Károly Bezdek's and Egon Schulte's 60th Birthdays. Springer Proceedings in Mathematics and Statistics, 2017. D. Ryabogin, *On a functional equation related to convex bodies with SU(2)-congruent projections* To appear in Geometriae Dedicata. C. A. Rogers, *Sections and projections of convex bodies.* Port. Math. 24, 99--103 (1965)

arxiv_math

--- abstract: | We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups and in particular, on cotangent bundles of simple Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle $TG$ and the cotangent bundle $T^*G$ of a Lie group $G$, will always be endowed with their Lie group structures induced by the right trivialization. We show that $TG$ and $T^*G$ are isomorphic if $G$ itself possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to $T^*SU(2)$, the set of unit dual split quaternions is isomorphic to the cotangent bundle of the group of unit split quaternions. The group SE(3) of special rigid displacements of the Euclidean $3$-space is isomorphic to $T^*$SO(3). The group $SE(2,1)$ of special rigid displacements of the Minkowski $3$-space is isomorphic to $T^*$SO(2,1). So some results on SE($3$) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to $SE(2,1)$ and to $T^*G$, for any simple Lie group $G.$ author: - "André Diatta$^{( 1)}$[^1] ; Bakary Manga$^{( 2)}$ and Fatimata Sy$^{( 2)}$[^2]" title: On dual quaternions, dual split quaternions and Cartan-Schouten metrics on perfect Lie groups --- **Keywords:** Dual quaternion, dual split quaternion, rigid motion, screw motion, Minkowski space, cotangent bundle, Cartan-Schouten metric, Cartan-Schouten connection, geodesic, bi-invariant metric, covariant derivative. # Introduction One of the interests in the canonical Cartan-Schouten symmetric connection on Lie groups, introduced by E. Cartan and J. A. Schouten in [@Cartan-Schouten], is the inverse problem of Lagrangian dynamics for the system of second order differential equations corresponding to the induced geodesic spay ([@crampin-mestdag; @ghanam-thompson-miller; @bi-invariant-and-noninvariant-metrics-ghanam-hindeleh-thompson; @muzsnay-thompson; @muzsnay; @rawashdeh-thompson; @thompson2-3D; @thompson3D; @strugar-thompson]). A particular case is when there exists at least one Lagrangian which is quadratic in the velocities such that the corresponding Hessian $\mu$ is a metric on the Lie group $G.$ In that case, such a metric $\mu$ is covariantly constant with respect to the connection. In the present paper, (pseudo-) Riemannian metrics $\mu$ which are covariantly constant $\nabla\mu =0,$ where $\nabla$ is the canonical Cartan-Schouten connection, are referred to as Cartan-Schouten metrics (Definition [Definition 2](#def:Cartan-Schouten-Metrics){reference-type="ref" reference="def:Cartan-Schouten-Metrics"}). Lie groups of dimension $\leq 6,$ possessing a Cartan-Schouten metric have already been classified ([@bi-invariant-and-noninvariant-metrics-ghanam-hindeleh-thompson; @ghanam-thompson-miller; @muzsnay; @rawashdeh-thompson; @strugar-thompson; @thompson2-3D; @thompson3D]). In [@Zefran], in the framework of kinematic analysis and robot trajectory planning, the authors address the problem of finding Riemannian or pseudo-Riemannian metrics all of whose geodesics are screw motions in the special Euclidean group $SE(3,\mathbb R)$ of rigid motions. Cartan-Schouten metrics turn out to be the only solutions to such a problem. On the other hand, a more general statistical framework based on Riemannian Geometry has been introduced, and studied by many authors (see e.g. [@lauritzen87; @Lorentzi-Pennec; @Matsuzoe2007; @Matsuzoe2010; @pennec2]), which takes into account both Euclidean and curved spaces. It has been proposed that such a framework be used on Lie groups ([@gallier-quaintance; @miolane-pennec2015; @pennec1]). This entails that the underlying Riemannian or pseudo-Riemannian metric be compatible with the group structure. So far, the researchers and experts have linked such a compatibility with the condition that the metric should be biinvariant, that is, invariant under both left and right translations of the Lie group. However, as pointed out in [@miolane-pennec2015], some of the Lie groups which are commonly used in some applications do not carry such metrics. To overcome such an apparent limitation, we propose (see [@diatta-manga-sy-jmp; @FatimataSy]) to drop the invariance assumption on the metric. We only require that the metric abides by the Lie group features. Namely, we require that 1-parameter subgroups be geodesics, the Riemannian mean coincide with the Lie group biinvariant exponential barycenter as in [@pennec-biinvariant-means], etc. We thus propose the natural framework of Cartan-Schouten metrics. From the geometric viewpoint, Cartan-Schouten metrics are the natural generalization of biinvariant metrics as in [@bi-invariant-and-noninvariant-metrics-ghanam-hindeleh-thompson; @medina85; @medina-revoy-ENS; @pennec-biinvariant-means]. In the present paper, we prove the following. Endowed with their Lie group structures given by the right trivialization, the tangent bundle $TG$ and the cotangent bundle $T^*G$ of a Lie group $G$, are isomorphic if $G$ possesses a biinvariant Riemannian or pseudo-Riemannian metric (Theorem [Theorem 1](#thm:cotangentbundlerighttrivialization){reference-type="ref" reference="thm:cotangentbundlerighttrivialization"}). The converse will be discussed elsewhere. We prove that every Cartan-Schouten metric on a perfect Lie group is necessarily biinvariant (Theorem [Theorem 2](#Theorem:parallel-metric-perfect-Lie-groups){reference-type="ref" reference="Theorem:parallel-metric-perfect-Lie-groups"}). Importantly, we also prove that every Cartan-Schouten metric on the cotangent bundle of a simple Lie group is necessarily biinvariant. We further compute all such biinvariant metrics on $T^*G$ when G is a simple Lie group (Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"}). With their canonical Lie group structures, the set of unit dual quaternions (resp. unit dual split quaternions) is isomorphic to the cotangent bundle of set of unit quaternions (resp. unit split quaternions). The group SE(3) of special rigid displacements of the Euclidean $3$-space is isomorphic to the Lie group $T^*$SO(3). The group $SE(2,1)$ of rigid displacements of the Minkowski $3$-space is isomorphic to the Lie group $T^*$SO(2,1). See Theorems [Theorem 4](#theorem:commutingdiagramSO(3)){reference-type="ref" reference="theorem:commutingdiagramSO(3)"}, [Theorem 6](#theorem:cummutingdiagramSL(2)){reference-type="ref" reference="theorem:cummutingdiagramSL(2)"}. In the mathematical setting, this work also implies, in particular, that the sets of unit dual quaternions, unit dual split quaternions and of the rigid displacements $SO(3)\ltimes \mathbb R^3$, $SO(2,1)\ltimes \mathbb R^3$, are all symplectic manifolds and as Lie groups, they all carry biinvariant pseudo-Riemannian structures. The isomorphism between $T^*SO(3)$ and $SE(3)$ could be used to look at the result on Cartan-Schouten metrics on cotangent bundles of simple Lie groups, as a generalization of some results on SE($3$) by N. Miolane and Pennec ([@miolane-pennec2015], [@pennec1]), and M. Zefran, V. Kumar and C. Croke ([@Zefran]). The equivalences proved here also open more routes towards applications of dual quaternions and dual split quaternions in many areas such as thermodynamics, statistical mechanics, information geometry and machine learning. The paper is organized as follows. Section [2](#chap:Onstatistical-structures-on-groups){reference-type="ref" reference="chap:Onstatistical-structures-on-groups"} is devoted to a few definitions, examples, a characterization of Lie groups with a biinvariant metrics (Subsection [2.3](#chap:Liegroups-with-biinvariant-metric){reference-type="ref" reference="chap:Liegroups-with-biinvariant-metric"}) and Cartan-Schouten metrics on perfect Lie groups (Subsection [2.4](#chap:Cartan-Schouten-metrics-on-perfect-Liegroups){reference-type="ref" reference="chap:Cartan-Schouten-metrics-on-perfect-Liegroups"}). We discuss the Cartan-Schouten metrics on cotangent bundles of simple Lie groups in Section [3](#chap:Cartan-Schouten-of-cotangent-bundles-of-simple-Lie-groups){reference-type="ref" reference="chap:Cartan-Schouten-of-cotangent-bundles-of-simple-Lie-groups"}. Section [4.1](#chap:reminders-quaternions-plit-quaternions){reference-type="ref" reference="chap:reminders-quaternions-plit-quaternions"} is a reminder on quaternions, dual quaternions, split quaternions, dual split quaternions and some of their relationships with the groups of rigid motions. In Section [4.2](#chap:rigid-motions-dual){reference-type="ref" reference="chap:rigid-motions-dual"}, we discuss the isomorphisms between the cotangent bundles and the groups of rigid motions, as well as the problem of metrics all of whose geodesics are screw motions. Throughout this work, the word 'metric' refers to both Riemannian and pseudo-Riemannian metrics. We will let $E_{i,j}$ stand for the elementary $n\times n$ matrix with zero in all entries except the $(i,j)$ entry which is equal to $1$. If $V$ is a vector space, we denote its linear dual by $V^*$ and by $\mathbb I_{V}$ its identity map. We let $(e_1^*,\cdots,e_n^*)$ stand for the dual basis of a basis $(e_1,\cdots,e_n)$. All Lie groups and Lie algebras considered here are real. # On Cartan-Schouten metrics on Lie groups {#chap:Onstatistical-structures-on-groups} Let us remind that Cartan-Schouten connections on a Lie group $G,$ are the left invariant connections whose geodesics through the identity are $1$-parameter subgroups of $G$. The most prolific examples are the classical $+$, $-$ and $0$ Cartan-Schouten connections ([@Cartan-Schouten]) respectively given in the Lie algebra $\mathcal G$ of $G$ by $$\begin{aligned} \label{Cartan-connections} \nabla_x y:=\lambda [x,y], \end{aligned}$$ $\lambda=1,0,\frac{1}{2},$ $\forall \; x,y\in\mathcal G.$ **Definition 1**. *The $0$-connection given by $\nabla_xy=\frac{1}{2} [x,y],$ $\forall \; x,y\in\mathcal G,$ is also termed the Cartan-Schouten canonical connection. It is the unique symmetric (torsion free) Cartan-Schouten connection which is bi-invariant.* Consider a metric $\mu$ on $G$, which is covariantly constant (or equivalently, parallel) with respect to the Cartan-Schouten canonical connection $\nabla$. The latter property simply reads $\nabla \mu=0$ or, equivalently, $$\begin{aligned} \label{eq:parallel} x^+\cdot\mu(y^+,z^+)=\frac{1}{2}\Big(\mu([x^+,y^+],z^+)+\mu(y^+,[x^+,z^+])\Big) \end{aligned}$$ for any left invariant vector fields $x^+,y^+,z^+$ on $G.$ **Definition 2**. *If a metric on a Lie Group, is parallel with respect to the Cartan-Schouten canonical connection, we call it a Cartan-Schouten metric.* ## Example: The Heisenberg Lie group $\mathbb H_3$ of dimension 3 {#Heisenberg} Consider the Heisenberg group $\mathbb H_3:=\Big\{ \begin{pmatrix}1&x&z \\ 0&1&y\\ 0&0&1 \end{pmatrix}, \; x,\; y,\; z\in\mathbb R\Big\}$. Its Lie algebra $\mathcal{H}_3$ is spanned by the $3\times 3$ elementary matrices $e_1:=E_{1,2},$ $e_2:=E_{2,3},$ $e_3:=E_{1,3}.$ So the Lie bracket reads $[e_1,e_2]=e_3.$ We identify $\mathbb H_3$ with $\mathbb R^3$, with the multiplication $(x,y,z) (x',y',z')=(x+x',y+y',z+z'+xy').$ The left-invariant vector fields corresponding to $e_1$, $e_2,$ $e_3,$ are respectively $e_1^+=\frac{\partial}{\partial x}, \quad e_2^+=\frac{\partial}{\partial y}+x\frac{\partial}{\partial z}, \quad e_3^+=\frac{\partial}{\partial z}.$ The left-invariant $1$-forms associated to $e_1^*,$ $e_2^*,$ $e_3^*,$ are: $(e_1^*)^+=dx, \quad (e_2^*)^+=dy,\quad (e_3^*)^+=dz-xdy.$ The Heisenberg Lie group does not have any biinvariant metric, since $[\mathcal H_3, \mathcal H_3]=\mathbb R e_3$ is also the center of $\mathcal H_3.$ However, it does possess infinitely many Cartan-Schouten metrics. **Proposition 1**. *Any metric $\mu$ on $\mathbb H_3$ which is compatible with the Cartan-Schouten canonical connection is of the form $$\begin{aligned} \mu&=&\left(\frac{1}{4}ay^2-cy+m\right)dx^2 +\left(\frac{1}{4}ax^2-bx+e\right)dy^2 +a\;dz^2 \cr & & +(\frac{1}{4}axy-\frac{1}{2}cx-\frac{1}{2}by+d)\;dxdy % \cr %& & - (\frac{1}{2}ay-c)\;dxdz - (\frac{1}{2}ax-b) \;dydz\;,\end{aligned}$$ where $a,b,c,d,e,m$ are real constants such that $-ad^2+aem-b^2m+2bcd-c^2e\neq 0.$* **Remark 1**. *(A) If we set $a=e=m=1$ and $b=c=d=0$, we recover the metric given in [@bi-invariant-and-noninvariant-metrics-ghanam-hindeleh-thompson] : $\mu=dx^2 +dy^2 +\left(dz-\frac{y}{2}dx-\frac{x}{2}dy\right)^2,$ which is a Riemannian metric. (B) For $m=e=-a=1$ and $b=c=d=0$ we get $\mu=( 1-\frac{1}{4}y^2)dx^2 +( 1-\frac{1}{4}x^2)\;dy^2-dz^2 - \frac{1}{4}xy\;dxdy + \frac{1}{2}y\;dxdz+ \frac{1}{2}x\;dydz,$ which is a Lorentzian metric. Note that $\det(\mu)=-1$ and $\mu(\vec{v},\vec{v})=0$, where $\vec{v}=\frac{\partial}{\partial y}+(\frac{x}{2}+1)\frac{\partial}{\partial z}.$ Both metrics have the same Levi-Civita connection, although one is Riemannian and the other Lorentzian.* ## Example: Lorentz Lie group SO(3,1) and Lie algebra $\mathfrak{so}(3,1)$ {#LorentzGroup} Let $G$ be the group of Lorentz transformations (isometries of space-time) of Minkowski 4-space, with determinant $+1.$ A basis of its Lie algebra $\mathcal G$ is made of the $4\times 4$ matrices $S_1=E_{1,4}+E_{4,1},$ $S_2=E_{2,4}+E_{4,2},$ $S_3=E_{3,4}+E_{4,3},$ $S_4=E_{2,3}-E_{3,2},$ $S_5=E_{3,1}-E_{1,3},$ $S_6=E_{2,1}-E_{1,2}.$ Its Lie bracket reads $[S_1,S_2]= - S_6,$ $[S_1,S_3]= - S_5,$ $[S_1,S_5]= - S_3,$ $[S_1,S_6]= - S_2,$ $[S_2,S_3]= S_4,$ $[S_2,S_4]= S_3,$ $[S_2,S_6]=S_1,$ $[S_3,S_4]=-S_2,$ $[S_3,S_5]=S_1,$ $[S_4,S_5]=S_6,$ $[S_4,S_6]= - S_5,$ $[S_5,S_6]=S_4.$ Since $G$ is a simple Lie group, Theorem [Theorem 2](#Theorem:parallel-metric-perfect-Lie-groups){reference-type="ref" reference="Theorem:parallel-metric-perfect-Lie-groups"} ensures that every Cartan-Schouten metric $\mu$ on $G$ is biinvariant. A direct calculation shows that the matrix of $\mu,$ in the basis $(S_j),$ is $$\begin{aligned} (\mu(S_i,S_j))&=&k_1(E_{1,1}+E_{2,2}+E_{3,3}-E_{4,4}-E_{5,5}-E_{6,6})\nonumber\\ && + k_2(E_{1,4}+E_{4,1}+E_{2,5}+E_{5,2}-E_{3,6}-E_{6,3})\end{aligned}$$ $k_1,k_2\in\mathbb R.$ Note that $4(E_{1,1}+E_{2,2}+E_{3,3}-E_{4,4}-E_{5,5}-E_{6,6})$ is the matrix of the Killing form $K_0$ and $E_{1,4}+E_{4,1}+E_{2,5}+E_{5,2}-E_{3,6}-E_{6,3}$ is the matrix of $K_J:=K_0(J(\cdot),\cdot)$ where $J=E_{1,4}+E_{4,1}+E_{2,5}+E_{5,2}-E_{3,6}-E_{6,3}$ satisfies $J^2=-\mathbb I_{\mathcal G}$ and $J[x,y]=[Jx,y]$, for any $x,y\in\mathcal G.$ ## A characterization of Lie groups with a biinvariant metric {#chap:Liegroups-with-biinvariant-metric} Recall that the trivialization by right translations, or simply the right trivialization, of the cotangent bundle $T^*G$ of a Lie group $G$, is given by the isomorphism $\zeta$ of vector bundles, $\zeta: T^*G \to G\times \mathcal{G}^*$, $(\sigma,\nu_\sigma)\mapsto (\sigma,\nu_\sigma\circ T_\epsilon R_\sigma),$ where $R_\sigma$ is the right multiplication $R_\sigma(\tau):=\tau \sigma$ by $\sigma$ in $G$ and $T_\epsilon R_\sigma$ is the derivative of $R_\sigma$ at the unit $\epsilon$ of $G.$ Likewise, the right trivialization of the tangent bundle $TG$ is given by the isomorphism $\xi$ of vector bundles, $\xi: TG \to G\times \mathcal{G}$, $(\sigma,X_\sigma)\mapsto (\sigma, T_\sigma R_{\sigma^{-1}} X_\sigma).$ Let $G\ltimes_{Ad} \mathcal G$ and $G\ltimes_{Ad^*} \mathcal G^*$ (or simply $G\ltimes \mathcal G$ and $G\ltimes \mathcal G^*$, for short) stand for the manifolds $G\times \mathcal G$ and $G\times \mathcal G^*$, endowed with their Lie group structures respectively defined by the following products $$\begin{aligned} \label{Adjoint}(\sigma_1,x)(\sigma_2,y)&:=&\Big(\sigma_1\sigma_2, x+ Ad_{\sigma_1} y\Big)\;\text{ and } (\sigma_1,f)(\sigma_2,g):=\Big(\sigma_1\sigma_2,f+Ad_{\sigma_1}^* g\Big)\; ,\end{aligned}$$ for any $(\sigma_1,x),\;(\sigma_2,y)\in G\times \mathcal G$ and any $(\sigma_1,f),\;(\sigma_2,g)\in G\times \mathcal G^*.$ Here, $Ad:G\times \mathcal G\to \mathcal G$ and $Ad^*:G\times \mathcal G^*\to \mathcal G^*$ are respectively the adjoint and coadjoint action of $G$ on $\mathcal G$ and $\mathcal G^*$, given by $Ad_{\sigma}x=\frac{d}{dt}\vert_{t=0}\sigma \exp(tx) \sigma^{-1}$ and $Ad_{\sigma}^*f:=f\circ Ad_{\sigma^{-1}}$, for $\sigma\in G ,$ $x\in\mathcal G,$ $f\in\mathcal G^*$ and $\exp: \mathcal G \to G$ is the Lie group exponential. Thus, $T^*G$ inherits a Lie group structure obtained by pulling back the product ([\[Adjoint\]](#Adjoint){reference-type="ref" reference="Adjoint"}) in $G\ltimes \mathcal G^*$ via $\zeta$, so that $$\begin{aligned} \label{coadjoint2}(\sigma,\nu_\sigma)(\tau,\alpha_\tau)&:=&\zeta^{-1}\Big(\zeta(\sigma,\nu_\sigma)\zeta(\tau,\alpha)\Big) =\Big( \sigma\tau,\nu_\sigma\circ T_{\sigma\tau} R_{\tau^{-1}}\;+\;\; \alpha_\tau\circ T_{\sigma\tau} L_{\sigma^{-1}}\Big) \;.\end{aligned}$$ Likewise, $TG$ also inherits a Lie group structure, the pullback of ([\[Adjoint\]](#Adjoint){reference-type="ref" reference="Adjoint"}) using $\xi$, so that $$\begin{aligned} \label{adjoint1}(\sigma,X_\sigma)(\tau,Y_\tau)&:=&\xi^{-1}\Big(\xi(\sigma,X_\sigma)\xi(\tau,Y_\tau)\Big) = \Big( \sigma\tau, T_\sigma R_{\tau}X_\sigma + T_\tau L_{\sigma} Y_\tau\Big)\;.\end{aligned}$$ **Theorem 1**. *Let $G$ be a Lie group, $T^*G$ and $TG$ its cotangent and tangent bundles endowed with their respective Lie group structures ([\[coadjoint2\]](#coadjoint2){reference-type="ref" reference="coadjoint2"}) and ([\[adjoint1\]](#adjoint1){reference-type="ref" reference="adjoint1"}). If $G$ has a biinvariant metric, then the Lie groups $T^*G$ and $TG$ are isomorphic.* *Proof.* Suppose a Lie group $G$ is endowed with a biinvariant metric, say $\mu.$ By abuse of notation, we denote again by $\mu$ its value $\mu_\epsilon$ at the unit $\epsilon$ of $G$. Define the linear invertible map $\Theta:\mathcal G \to \mathcal G^*,$ $x\mapsto \Theta(x),$ where $\Theta(x):=\mu(x,\cdot)$ is the linear form that maps every $y\in\mathcal G$ to $\langle \Theta(x),y\rangle = \mu(x,y).$ The property $\mu(Ad_\sigma x, Ad_\sigma y) =\mu(x,y)$ is equivalent to $\Theta(Ad_\sigma x) = Ad_\sigma^*\Theta(x)$, for every $\sigma\in G$ and $x,y\in\mathcal G.$ Let $\Phi: TG\to T^*G$ be the isomorphism of vector bundles defined by $\Phi(\sigma,X_\sigma)= \Big(\sigma, \Theta(T_{\sigma}R_{\sigma^{-1}}X_{\sigma})\circ T_{\sigma}R_{\sigma^{-1}}\Big) = \Big(\sigma, \Theta(T_{\sigma}L_{\sigma^{-1}}X_{\sigma})\circ T_{\sigma}L_{\sigma^{-1}}\Big)$. From the definition of $\Phi$, the image $\Phi\Big((\sigma,X_\sigma)(\tau,Y_\tau)\Big)$ of the product ([\[adjoint1\]](#adjoint1){reference-type="ref" reference="adjoint1"}) comes to $$\begin{aligned} \Big( \sigma\tau, \Theta( T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \; T_\sigma R_{\tau}X_\sigma )\circ T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \;+\; \Theta( T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \; T_\tau L_{\sigma} Y_\tau)\circ T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}}\Big) .\end{aligned}$$ We further use the equalities $T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \; T_\sigma R_{\tau}=T_{\sigma} R_{\sigma^{-1}}$ and $$\begin{aligned} \Theta( T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \; T_\tau L_{\sigma} Y_\tau)\circ T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}}= \Theta( T_{\sigma\tau} L_{\tau^{-1} \sigma^{-1}} \; T_\tau L_{\sigma} Y_\tau)\circ T_{\sigma\tau} L_{\tau^{-1} \sigma^{-1}},\end{aligned}$$ the latter simplifying to $\Theta( T_{\tau} L_{\tau^{-1}} Y_\tau)\circ T_{\sigma\tau} L_{\tau^{-1} \sigma^{-1}}.$ So we get $$\begin{aligned} \Phi\Big((\sigma,X_\sigma)(\tau,Y_\tau)\Big)= \Big( \sigma\tau, \Theta( T_{\sigma} R_{\sigma^{-1}} X_\sigma )\circ T_{\sigma\tau} R_{\tau^{-1} \sigma^{-1}} \;+\; \Theta( T_{\tau} L_{\tau^{-1}} Y_\tau)\circ T_{\sigma\tau} L_{\tau^{-1} \sigma^{-1}}\Big) .\nonumber\end{aligned}$$ On the other hand, the product $\Phi(\sigma,X_\sigma)\Phi(\tau,Y_\tau)$ is equal to $$\begin{aligned} \Big( \sigma\tau, \Theta(T_{\sigma}R_{\sigma^{-1}}X_{\sigma})\circ T_{\sigma}R_{\sigma^{-1}}\circ T_{\sigma\tau} R_{\tau^{-1}}\;+\; \Theta(T_{\tau}L_{\tau^{-1}}Y_{\tau})\circ T_{\tau}L_{\tau^{-1}} \circ T_{\sigma\tau} L_{\sigma^{-1}} \Big)\nonumber\end{aligned}$$ and visibly, the latter coincides with $\Phi\Big((\sigma,X_\sigma)(\tau,Y_\tau)\Big).$ ◻ The Lie algebra $T^*\mathcal G$ of $T^*G$, is then isomorphic to the semi-direct product $\mathcal{G}\ltimes\mathcal{G}^*,$ the Lie bracket of two elements $(x,f)$ and $(y,g)$ of $T^*\mathcal G$ being $$\begin{aligned} \label{cotangent-bundle-Lie-bracket} [(x,f),(y,g)]:=([x,y],ad^*_xg-ad^*_yf).\end{aligned}$$ Similarly, the Lie algebra $T\mathcal G$ of $TG$, is the semi-direct sum $\mathcal{G}\ltimes\mathcal{G}$, the Lie bracket of two elements $(x_1,y_1)$ and $(x_2,y_2)$ being $[(x_1,y_1),(x_2,y_2)]=([x_1,x_2], [x_1,y_2]-[x_2,y_1]).$ ## Cartan-Schouten metrics on perfect Lie groups {#chap:Cartan-Schouten-metrics-on-perfect-Liegroups} In this section, we prove the following (without loss of generality, we can work with connected Lie groups). **Theorem 2**. *Let $G$ be a perfect Lie group, that is, its Lie algebra $\mathcal G$ satisfies $[\mathcal G,\mathcal G]=\mathcal G.$ If $G$ possesses a Cartan-Schouten metric $\mu$, then $\mu$ is necessarily a biinvariant metric.* *Proof.* From Ambrose-Singer holonomy theorem, the curvature tensor $R^\nabla$ of $\mu$ must be skew-symmetric with respect to $\mu.$ That is, $$\begin{aligned} \label{skew-symmetric-curvature} \mu\Big(R^\nabla (X_1,X_2)Y,Z\Big)+\mu\Big(Y,R^\nabla (X_1,X_2)Z \Big) =0\;,\end{aligned}$$ for any smooth vector fields $X_1,X_2,Y,Z$ on $G.$ In particular, if $X_1,X_2,Y,Z$ are all left invariant vector fields on $G,$ Equality ([\[skew-symmetric-curvature\]](#skew-symmetric-curvature){reference-type="ref" reference="skew-symmetric-curvature"}) becomes $$\begin{aligned} \label{skew-symmetric-curvature-invariant} \mu\Big([[X_1,X_2],Y],Z\Big)+\mu\Big(Y,[[X_1,X_2],Z] \Big) =0\;.\end{aligned}$$ Since $[\mathcal G,\mathcal G]=\mathcal G$, the equality ([\[skew-symmetric-curvature-invariant\]](#skew-symmetric-curvature-invariant){reference-type="ref" reference="skew-symmetric-curvature-invariant"}) linearly extends to $$\begin{aligned} \label{skew-symmetric-curvature-invariant2} \mu\Big([X,Y],Z\Big)+\mu\Big(Y,[X,Z] \Big) =0\;,\end{aligned}$$ for any left invariant vector fields $X,Y,Z$ on $G.$ According to ([\[eq:parallel\]](#eq:parallel){reference-type="ref" reference="eq:parallel"}), the left hand-side of ([\[skew-symmetric-curvature-invariant2\]](#skew-symmetric-curvature-invariant2){reference-type="ref" reference="skew-symmetric-curvature-invariant2"}) is precisely $2 X\cdot \mu(Y,Z)$. This implies that the differential of the function $\mu(Y,Z)$ satisfies $d\mu(Y,Z)=0$ for any left invariant vector fields $Y,Z$ on $G.$ Equivalently, $\mu$ is constant on left invariant vector fields and hence $\mu$ is itself left invariant on $G,$ in addition to satisfying ([\[skew-symmetric-curvature-invariant2\]](#skew-symmetric-curvature-invariant2){reference-type="ref" reference="skew-symmetric-curvature-invariant2"}). Consequently, $\mu$ is biinvariant. ◻ # Cartan-Schouten metrics on cotangent bundles of simple Lie groups {#chap:Cartan-Schouten-of-cotangent-bundles-of-simple-Lie-groups} In what follows, if $G$ is a Lie group, $\mathcal G$ its Lie algebra, the Killing form $K_0$ of $\mathcal G$ will also be looked at as a bilinear form on the Lie algebra $T^*\mathcal G$ of $T^*G$, with $K_0\Big((x,f),(y,g)\Big)=K_0(x,y)$ for any $(x,f),(y,g)\in T^*\mathcal G$. The following Lemma (see [@medina85], for part (B)) will be useful in the proof of Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"}. Example [2.2](#LorentzGroup){reference-type="ref" reference="LorentzGroup"} is a typical illustration of part (B)-(2). **Lemma 1**. *Let $\mathcal G$ be a simple real Lie algebra, $K_0$ its Killing form and $\dim \mathcal G=n$. Denote by $K(\mathcal G)$ the space of linear maps $L:\mathcal G\to \mathcal G$ satisfying $L[x,y]=[L(x),y]$, for any $x,y\in\mathcal G.$ (A) Every element of $K(\mathcal G)$ is symmetric with respect to $K_0.$* *(B) $\dim_{\mathbb R} K(\mathcal G)\leq 2$ and $K(\mathcal G)$ is a commutative subfield of the ring End($\mathcal G$) of endomorphims of the (real) vector space underlying $\mathcal G.$ (1) If $n$ is odd, then $K(\mathcal G)=\mathbb R\mathbb I_{\mathcal G}$. In particular, up to a constant factor, $K_0$ is the unique ad-invariant metric on $\mathcal G.$ (2) If $n$ is even, then $\dim K(\mathcal G)=2.$ More precisely, $K(\mathcal G)$ is isomorphic to $\mathbb C$, it is spanned by $\mathbb I_{\mathcal G}$ and some $J$ with $J^2=-\mathbb I_{\mathcal G}.$* *Proof.* (A) Write $L=L_1+L_2$, where $K_0( L_{1}(x),y) =\frac{1}{2} (K_0( L(x),y)+ K_0( x, L(y)))$ and $K_0( L_{2}(x),y) =\frac{1}{2} (K_0( L(x),y)- K_0( x, L(y)))$, for any $x,y\in\mathcal G.$ The following hold $L_{2}[x,y]= [ L_{2}(x),y]$ and $K_0( L_{2}([x,y]),z) = K_0( L_{2}(x),[y,z]) = - K_0( x,[ L_{2}(y),z]) = - K_0( [x, L_{2}(y)],z) = - K_0( L_{2}( [x,y]),z)$, for any $x,y,z\in\mathcal G.$ In other words $L_{2}( [x,y])=0$, for any $x,y\in\mathcal G.$ Hence, we have $L_{2}=0$ and $L=L_1$ is thus symmetric with respect to $K_0.$ Part (B) has been proved in [@medina85], Sect. 5, p.409-410. ◻ **Lemma 2**. *Let $G$ be a perfect Lie group, $\mathcal G$ its Lie algebra. Suppose $G$ has a Cartan-Schouten metric. Then $T^*G$ is a perfect Lie group. In particular, if $G$ is semisimple, then $T^*G$ is a perfect Lie group.* *Proof.* Let $\mu$ be a Cartan-Schouten metric on $G.$ From Theorem [Theorem 2](#Theorem:parallel-metric-perfect-Lie-groups){reference-type="ref" reference="Theorem:parallel-metric-perfect-Lie-groups"}, $\mu$ can be considered as an ad-invariant metric on $\mathcal G.$ The invertible linear map $\hat \Theta: \mathcal G\to\mathcal G^*,$ $x\mapsto\mu(x,\cdot)$, satisfies $\hat \Theta ([x_1,x_2])=ad_{x_1}^*\hat \Theta (x_2)$, and as $[\mathcal G,\mathcal G]=\mathcal G$, thus for any $\bar g\in\mathcal G^*$ we can suppose, without any loss of generality, that there exist $x_1,x_2\in \mathcal G$ such that $\bar g=\hat \Theta ([x_1,x_2])=ad_{x_1}^*\hat \Theta (x_2)=ad_{x}^* g=[x,g],$ where $x_1=: x$ and $\hat \Theta (x_2)=:g.$ So in particular, $[\mathcal G,\mathcal G^*]=\mathcal G^*$ and the Lie algebra $T^*\mathcal G$ of $T^*G$ satisfies $[T^*\mathcal G,T^*\mathcal G]=T^*\mathcal G.$ ◻ For an integer $p\ge 0$, let $\mathbb I_{p,n}$ stand for the diagonal $n\times n$ matrix with $-1$ in the first $p$ diagonal entries and $1$ in the last $n-p$ entries. As above, for a Lie algebra $\mathcal G$, we let $K_0$ stand for its Killing form and $\langle,\rangle$ the duality paring between $\mathcal G$ and $\mathcal G^*.$ For a linear map $J:\mathcal G\to \mathcal G$ as in (B)-(2) in Lemma [Lemma 1](#ad-invariant-endomorphisms){reference-type="ref" reference="ad-invariant-endomorphisms"}, we define the symmetric bilinear forms $K_J$ and $\langle,\rangle_J$ on $T^*\mathcal G$ as $K_J(x,y)=K_0(J x,y)$, $K_J(x,g)=K_J(f,g)=0$, $\langle x,y\rangle_J =\langle f,g\rangle_J =0$ and $\langle x,g\rangle_J =\langle J(x),g\rangle$, for $x,y\in \mathcal G,$ $f,g\in\mathcal G^*.$ **Theorem 3**. *Let $G$ be an $n$-dimensional simple Lie group, $\mathcal G$ its Lie algebra. Then every Cartan-Schouten metric $\mu$ on $T^*\mathcal G$ is biinvariant and has signature $(n,n)$.* *(A) If $n$ is odd, then $\mu$ is a linear combination $\mu=sK_0+t\langle,\rangle$ of $K_0$ and $\langle,\rangle$, with $s,t\in\mathbb R,$ $t\neq 0$. Let $(p,n-p)$ be the signature of $K_0$ and $(e_1,\dots,e_n)$ a basis of $\mathcal G$ in which the matrix of $K_0$ is $\mathbb I_{p,n}.$ Then $\mu$ has a matrix of the form $\begin{pmatrix} s\mathbb I_{p,n}&t \mathbb I_{n}\\ t \mathbb I_{n}&{\mathbf 0}_{n}\end{pmatrix}$ in the basis $(e_1,\dots,e_n, e_1^*,\dots,e_n^*)$ of $T^*\mathcal G$, where $s,t\in\mathbb R,$ $t\neq 0$ and ${\mathbf 0}_{n}$ is the zero $n\times n$ matrix.* *(B) If $n$ is even, then the space of Cartan-Schouten metrics on $T^*G$ is 4-dimensional. More precisely, each such metric is a linear combination $\mu=s_1K_0+s_2K_J+t_1\langle ,\rangle+t_2\langle,\rangle_J$, with $s_1,s_2,t_1,t_2\in\mathbb R,$ with $J$ as in Lemma [Lemma 1](#ad-invariant-endomorphisms){reference-type="ref" reference="ad-invariant-endomorphisms"}. That is, for any $x,y\in\mathcal G,$ $f,g\in\mathcal G^*,$ $$\begin{aligned} \mu\Big((x,f),(y,g)\Big)=s_1K_0(x,y)+s_2K_J(x,y) +t_1\langle (x,f),(y,g)\rangle+ t_2\langle (x,f),(y,g)\rangle_J \;.\end{aligned}$$* *Proof.* Let $\mu$ be a Cartan-Schouten metric on $T^* G.$ Lemma [Lemma 2](#lem:perfectT*G){reference-type="ref" reference="lem:perfectT*G"}, ensures that $T^*G$ is a perfect Lie group. Thus, according to Theorem [Theorem 2](#Theorem:parallel-metric-perfect-Lie-groups){reference-type="ref" reference="Theorem:parallel-metric-perfect-Lie-groups"}, $\mu$ must be biinvariant. So the restriction of $\mu$ to $T^*\mathcal G$ is ad-invariant. On $T^*\mathcal G=\mathcal G\ltimes \mathcal G^*$ with its Lie bracket ([\[cotangent-bundle-Lie-bracket\]](#cotangent-bundle-Lie-bracket){reference-type="ref" reference="cotangent-bundle-Lie-bracket"}), an ad-invariant metric $\mu$ is given by a linear invertible map $\phi: T^*\mathcal G\to T^*\mathcal G$ such that $\mu((x,f),(y,g))= \langle \phi(x,f),(y,g) \rangle=\langle(x,f), \phi(y,g) \rangle.$ We write $\phi(x,f)= (\phi_{1,1}(x)+\phi_{2,1}(f), \phi_{1,2}(x)+\phi_{2,2}(f))$ where $\phi_{1,1}:\mathcal G\to\mathcal G$, $\phi_{1,2}:\mathcal G\to\mathcal G^*$, $\phi_{2,1}:\mathcal G^*\to\mathcal G$ and $\phi_{2,2}:\mathcal G^*\to\mathcal G^*$, are linear maps. The ad-invariance of $\mu$ is equivalent to the relations $$\begin{aligned} \label{eq:ad-invariance} \phi([(x,f),(y,g)]) = [\phi(x,f),(y,g)]= [(x,f),\phi(y,g)],\end{aligned}$$ true for any $(x,f)$ and $(y,g)$ in $T^*\mathcal G.$ The latter equations, in the case $f=g=0,$ become $\phi[x,y] = (\phi_{1,1} [x,y], \phi_{1,2}([x,y]))= [\phi(x),y] = [\phi_{1,1}(x),y] - ad_{y}^* \phi_{1,2}(x) .$ A component-wise comparison leads to the equalities $\phi_{1,1} [x,y]= [\phi_{1,1}(x),y]$ and $\phi_{1,2}([x,y])=- ad_{y}^* \phi_{1,2}(x)$. The bilinear form $\mu_{\phi_{1,2}}$ on $\mathcal G$ given by $\mu_{\phi_{1,2}} (x,y):=\langle \phi_{1,2}(x),y \rangle$, is ad-invariant. Thus it is of the form $\mu_{\phi_{1,2}} (x,y)=K_0(\psi(x),y)$ for some linear map $\psi$ satisfying $\psi([x,y])=[\psi(x),y],$ for every $x,y\in\mathcal G.$ Now taking $y=0,f=0$ in ([\[eq:ad-invariance\]](#eq:ad-invariance){reference-type="ref" reference="eq:ad-invariance"}), leads to $\phi[x,g] = \phi_{2,1} (ad_x^*g) + \phi_{2,2}(ad_x^*g)=[\phi(x),g]= ad_{\phi_{1,1}(x)}^*g,$ or equivalently, $\phi_{2,1} (ad_x^*g) =0$ and $\phi_{2,2}(ad_x^*g)= ad_{\phi_{1,1}(x)}^*g$, for any $x\in\mathcal G$ and $g\in\mathcal G^*.$ Since every element of $\mathcal G^*$ can be taken in the form $ad_{x}^* g$, for some $x\in\mathcal G$ and $g\in\mathcal G^*$, this implies the equality $\phi_{2,1}=0.$ So $\mathcal G^*$ is totally isotropic for the metric $\mu.$ As a consequence, $\mu$ is of signature $(n,n).$ We also check that $\phi_{2,2}=\phi_{1,1}^T.$ Indeed $\langle \phi_{1,1}^Tg,x\rangle = \langle \phi_{1,1}(x), g\rangle = \langle \phi_{1,1}(x)+\phi_{1,2}(x), g\rangle = \langle \phi(x), g\rangle =\mu(x,g)= \langle x, \phi(g)\rangle= \langle x, \phi_{2,2}(g)\rangle,$ for any $x\in\mathcal G$, $g\in\mathcal G^*.$ Now we discuss the above in the two different scenari where $n$ is odd or even. \(A\) Suppose $n$ is odd. By Lemma [Lemma 1](#ad-invariant-endomorphisms){reference-type="ref" reference="ad-invariant-endomorphisms"}, we must have $\phi_{1,1} = t\mathbb I_{\mathcal G}$ and $\psi = s\mathbb I_{\mathcal G}$ for some $t,s\in\mathbb R,$ and thus $\mu_{\phi_{1,2}} = s K_0$, $s\in \mathbb R$ and $\phi_{2,2}(ad_x^*g)=t ad_{x}^* g$, for any $x\in \mathcal G$ and $g\in\mathcal G^*,$ or equivalently, $\phi_{2,2}=t\mathbb I_{\mathcal G^*}$, where $t\in\mathbb R$ and $\mathbb I_{\mathcal G},$ $\mathbb I_{\mathcal G^*}$ are the identity maps of $\mathcal G$ and $\mathcal G^*,$ respectively. Altogether, we have $\mu((x,f), (y,g))= \langle \phi_{1,2}(x),y\rangle+ \langle \phi_{1,1}(x),g\rangle +\langle \phi_{2,2}(f),y\rangle$ which also reads $\mu((x,f), (y,g))=\mu_{\phi_{1,2}}(x ,y) + t( \langle x,g\rangle +\langle f,y\rangle) = sK_0(x,y) + t \langle (x,f),(y,g)\rangle .$ Now choosing a basis $(e_1,\dots,e_n)$ of $\mathcal G$ in which the matrix $K_0(e_i,e_j)$ of $K_0$ is of the form $\mathbb I_{p,n}$, leads to the matrix $\Big(\mu_{\phi_{1,2}} (e_i,e_j) \Big)= s\mathbb I_{p,n}$ of $\mu_{\phi_{1,2}}.$ In the basis $(e_1,\dots,e_n, e_1^*,\dots,e_n^*)$ of $T^*\mathcal G$, where $(e_1^*,\dots,e_n^*)$ is the dual basis of $(e_1,\dots,e_n)$, the matrix $[\mu]$ of $\mu$ reads $$\begin{aligned} \label{eq:biinvariantmetricTsl(2,1)} [\mu]:=\begin{pmatrix}\mu(e_i,e_j) &\mu(e_i,e_j^*)\\ \mu(e_i,e_j^*) &\mu(e_i^*,e_j^*) \end{pmatrix}= \begin{pmatrix}s\mathbb I_{p,q} & t\mathbb I_{n} \\ t\mathbb I_{n} &{\mathbf 0}_n \end{pmatrix}\;.\end{aligned}$$ The characteristic polynomial of $[\mu]$ is $P(X) =(X+\lambda_1)^{p}(X+\lambda_2)^{p}(X-\lambda_1)^{n-p}(X-\lambda_2)^{n-p},$ with $\lambda_1:=\frac{1}{2}(s-\sqrt{s^2+4t^2})$ and $\lambda_2:=\frac{1}{2}(s+\sqrt{s^2+4t^2}).$ So $[\mu]$ has $n$ positive eigenvalues, $p$ of which being equal to $-\lambda_1$ and $n-p$ of which being equal to $\lambda_2,$ and $n$ negative eigenvalues, $p$ of which coinciding with $-\lambda_2$ and $n-p$ of which being equal to $\lambda_1.$ Hence confirms that $\mu$ is of signature $(n,n).$ \(B\) If $n$ is even, by Lemma [Lemma 1](#ad-invariant-endomorphisms){reference-type="ref" reference="ad-invariant-endomorphisms"}, $\psi$, $\phi_{1,1}$, $\phi_{2,2}$ are of the forms $\psi=s_1\mathbb I_{\mathcal G} +s_2 J,$ $\phi_{1,1}=t_1\mathbb I_{\mathcal G} +t_2 J$ and $\phi_{2,2}=t_1\mathbb I_{\mathcal G^*}+t_2J^T,$ for some fixed $J$ satisfying $J[x,y]=[Jx,y],$ for any $x,y\in\mathcal G$ and $s_1,s_2,t_1,t_2\in\mathbb R.$ Thus, we have for any $x,y\in\mathcal G,$ $f,g\in\mathcal G^*,$ $$\begin{aligned} \mu\Big((x,f),(y,g)\Big)&=&\mu(x,y)+\mu(x,g)+\mu(f,y)\nonumber\\ &=& s_1K_0(x,y)+s_2K_0(J(x),y) \nonumber\\ &&+t_1\langle x,g\rangle+ t_2\langle J(x),g\rangle +t_1\langle y,f\rangle+ t_2\langle J(y),f\rangle \nonumber\\ &=&s_1K_0(x,y)+s_2K_J(x,y) +t_1\langle (x,f),(y,g)\rangle+ t_2\langle (x,f),(y,g)\rangle_J . \nonumber\end{aligned}$$ ◻ # Dual quaternions, dual split quaternions, rigid motions, screws motions {#chap:rigid-motions-dual} ## A short tour on dual quaternions, dual split quaternions {#chap:reminders-quaternions-plit-quaternions} ### Quaternions, rigid motions, screw motions, dual quaternions Consider the following three vectors ${\mathbf i}$, ${\mathbf j}$, ${\mathbf k}$ called the pure imaginary units and satisfying the relation ${\mathbf i}^2={\mathbf j}^2={\mathbf k}^2={\mathbf i}{\mathbf j}{\mathbf k}=-1$. A quaternion is of the form $Q=q_0+q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k} =q_0+\vec{q}$, where $q_0,q_x,q_y,q_z\in\mathbb R$ and $\vec{q}:=q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k}$ is called the vector part or the pure imaginary part of $Q.$ One $\mathbb R$-linearly extends the above relation to define a product on the space $\mathbb H$ of quaternions, given for any $P=p_0+\vec{p}$, $Q=q_0+\vec{q}\in\mathbb H$, by $$\begin{aligned} \label{Eq:productquaternions} PQ =q_0p_0 - \vec{q}\cdot\vec{p}+ q_0\vec{p} + p_0\vec{q} + \vec{q}\times\vec{p},\end{aligned}$$ where $\vec{q}\cdot\vec{p}$ and $\vec{q}\times\vec{p}$ are respectively the ordinary dot and vector products of $\vec{q}$ and $\vec{p}$ seen as vectors in $\mathbb R^3.$ A vector $u$ with coordinates $(u_x,u_y,u_z)$ in the canonical basis of $\mathbb R^3,$ is identified with the pure quaternion $\vec{u}=u_x{\mathbf i}+u_y{\mathbf j}+u_y{\mathbf k}$. The product ([\[Eq:productquaternions\]](#Eq:productquaternions){reference-type="ref" reference="Eq:productquaternions"}) makes $\mathbb H$ an associative $\mathbb R$-algebra. The conjugate of $Q$ is the quaternion $Q^*=q_0-q_x{\mathbf i}-q_y{\mathbf j}-q_z{\mathbf k} =q_0-\vec{q}$. As one can see $QQ^*=q_0^2+q_x^2+q_y^2+q_z^2=\vert\vert Q\vert\vert^2,$ thus $Q$ is invertible if and only if $\vert\vert Q\vert\vert\neq 0$ and in this case, $Q^{-1} =\frac{1}{\vert\vert Q\vert\vert^2} Q^*.$ The set of invertible quaternions is an analytic 4-dimenional Lie group, of whom the subset $\mathbb H_1$ of those $Q$ with $\vert\vert Q\vert\vert=1$ (unit quaternions) is a closed subgroup, hence a Lie subgroup (according to Cartan's theorem). Consider the $8$-dimensional $\mathbb R$-algebra $\mathfrak{gl}(2,\mathbb C)$ of $2\times 2$ matrices with coefficients in $\mathbb C$ and its $4$-dimensional $\mathbb R$-subalgebra $\mathcal H$ spanned by the $2\times 2$ matrices $X_0:=E_{1,1}+E_{2,2},$ $X_1:= \sqrt{-1}\: (E_{2,2}-E_{1,1}),$ $X_2:=E_{2,1}-E_{1,2},$ $X_3:=\sqrt{-1}\;(E_{1,2}+E_{2,1}).$ The algebra $\mathbb H$ is isomorphic to $\mathcal H$, via the linear map $\psi$, defined by $$\begin{aligned} \label{isomorphimSU(2)-quaternions}\psi({\mathbf 1})=X_0, \; \psi({\mathbf i})=X_1, \; \psi({\mathbf j})=X_2, \; \psi({\mathbf k})=X_3.\end{aligned}$$ That is, $\det \psi\neq 0$ and $\psi(Q_1Q_2)=\psi(Q_1)\psi(Q_2)$, for any $Q_1,Q_2\in\mathbb H.$ Given a unit quaternion $Q = q_0+ \vec{q},$ one sets $\theta = 2\cos^{-1}(q_0),$ $0\leq \theta\le 2\pi,$ and $\vec{u} =\frac{\vec{q}}{\sin(\frac{\theta}{2})}$ if $\theta \neq 0,\;2\pi$ and $\vec{u} =0$ otherwise, so that $Q=\cos(\frac{\theta}{2})+\vec{u}\sin(\frac{\theta}{2})$, with $\vert\vert \vec{ u}\vert\vert=1$ when $\theta \neq 0,\;2\pi.$ One notes that $\mathcal H$ contains SU(2) and the restriction $\psi: \mathbb H_1\to SU(2)$ is an isomorphism between the Lie groups $\mathbb H_1$ and $SU(2).$ More precisely, let $Q=\cos(\frac{\theta}{2})+\sin(\frac{\theta}{2}) \vec{u}$$\in\mathbb H_1,$ $\vec{u} = u_x{\mathbf i} + u_y{\mathbf j}+ u_z{\mathbf k}$ with $u_x^2+u_y^2+u_z^2=1,$ then $$\begin{aligned} \psi(Q)=\sigma:= \begin{pmatrix}\cos(\frac{\theta}{2})-iu_x\sin(\frac{\theta}{2})&-u_y\sin(\frac{\theta}{2})+ iu_z\sin(\frac{\theta}{2})\\ u_y\sin(\frac{\theta}{2}) + iu_z\sin(\frac{\theta}{2})&\cos(\frac{\theta}{2})+iu_x\sin(\frac{\theta}{2})\end{pmatrix} =\begin{pmatrix}z&-\bar w\\ w&\bar z\end{pmatrix}\;, \end{aligned}$$ where $z=\cos(\frac{\theta}{2})-iu_x\sin(\frac{\theta}{2}),$ $w=u_y\sin(\frac{\theta}{2}) + iu_z\sin(\frac{\theta}{2})$ and $\vert z\vert^2 + \vert w\vert^2 =1.$ On the other hand, one constructs a Lie group double covering, more precisely, the well known Lie group homomorphism $\tilde \Pi: SU(2)\to SO(3)$ with kernel $\ker(\tilde \Pi)=\{1,-1\},$ using the adjoint representation Ad: $SU(2)\to GL(\mathfrak{su}(2))$, of $SU(2)$ with Ad$_\sigma X=\sigma X\sigma^{-1}$. Indeed, the metric defined by $K(M,N):=$Trace$(MN)$ is negative definite and Ad-invariant on the Lie algebra $\mathfrak{su}(2)$ of $SU(2),$ as $K(Ad_\sigma X,Ad_\sigma Y)=K(X,Y)$. Thus, $Ad_\sigma$ lies in $SO(3)$, for any $\sigma\in SU(2)$ and $\ker(Ad)=\{1,-1\}$. We set $Ad=:\tilde \Pi.$ Note that $(X_1,X_2,X_3)$ above is a basis of the Lie algebra $\mathfrak{su}(2)$ of $SU(2).$ For $Q=\cos(\frac{\theta}{2})+\vec{u}\sin(\frac{\theta}{2})\in\mathbb H_1$, the matrix of $\tilde \Pi\circ\psi (Q)$ in the basis $(X_1,X_2,X_3),$ is $$\begin{aligned} \label{MatrixAdSU(2)}\begin{pmatrix} \cos\theta+u_x^2(1-\cos\theta)&-u_z\sin\theta+u_xu_y(1-\cos\theta) &(1-\cos\theta)u_xu_z+ u_y\sin\theta\\ u_x u_y (1-\cos\theta)+u_z \sin\theta &u_y^2(1-\cos\theta)+ \cos\theta &u_yu_z(1-\cos \theta)-u_x\sin\theta \\ u_x u_z (1-\cos\theta)-u_y \sin\theta & u_y u_z (1- \cos\theta)+ u_x \sin\theta & \cos\theta+u_z^2(1-\cos\theta) \end{pmatrix}\end{aligned}$$ which is exactly the matrix of the rotation of angle $\theta$ around the axis defined by the unit vector $\vec{u},$ where the vector space underlying $\mathfrak{so}(3)$ has been identified with $\mathbb R^3.$ For a given unit $Q\in\mathbb H_1$, consider the linear map $$\begin{aligned} \label{definition-RQ} \mathcal R_Q:\mathbb R^3\to\mathbb R^3,\;\; \mathcal R_Q(\vec{u})=Q\vec{u}Q^{-1}. \end{aligned}$$ Each $\mathcal R_Q$ is a rotation of $\mathbb R^3$. More explicitly one has $Q \vec{u} = ( - \vec{q}\cdot\vec{u}, q_0\vec{u} + \vec{q}\times\vec{u} )$ and $$\begin{aligned} \label{rotation-unit-quaternion1} Q \vec{u}Q^{-1}& =&\ (q_0^2+F(\vec{q}))^2\vec{u} + (\vec{q}\cdot\vec{u})\vec{q} = (q_0^2 - \vert\vert\vec{q}\vert\vert^2)\vec{u} + 2q_0\vec{q}\times\vec{u} +2 (\vec{q}\cdot\vec{u})\vec{q} \;,\end{aligned}$$ where $F(x): \mathbb R^3\to \mathbb R^3,$ $y\mapsto F(x)y= x\times y,$ stands for the cross product by $x$. So $F: \mathbb R^3\to \mathfrak{so}(3)$, $x\mapsto F(x),$ is an isomorphism of vector spaces. If the tensor product $x\otimes y$ is seen as a linear map whose matrix has coefficients $(x\otimes y)_{i,j} = x_iy_j,$ then $$\begin{aligned} (\vec{q}\cdot\vec{u})\vec{q}&=&q_x(q_xu_x+q_yu_y+q_zu_z) e_1+q_y(q_xu_x+q_yu_y+q_zu_z) e_2 \nonumber\\ &&+q_z(q_xu_x+q_yu_y+q_zu_z) e_3=(\vec{q}\otimes \vec{q}) \vec{u}.\end{aligned}$$ So we get $Q \vec{u}Q^{-1} =\Big((q_0+F(\vec{q}))^2 + \vec{q}\otimes\vec{q} \Big)\vec{u}$. So the matrix of $\mathcal R_Q=(q_0+F(\vec{q}))^2 + \vec{q}\otimes\vec{q}$, in the canonical basis of $\mathbb R^3,$ is $\begin{pmatrix}q_0^2+q_x^2-q_y^2-q_z^2& -2q_0q_z+2q_xq_y& 2q_0q_y+2q_xq_z\\ 2q_0q_z+2q_xq_y& q_0^2-q_x^2+q_y^2-q_z^2& -2q_0q_x+2q_yq_z\\ -2q_0q_y+2q_xq_z& 2q_0q_x+2q_yq_z& q_0^2-q_x^2-q_y^2+q_z^2 \end{pmatrix},$ hence $\mathcal R_Q\mathcal R_Q^T=\vert\vert Q\vert\vert^2\; \mathbb I_{\mathbb R^3}=\mathbb I_{\mathbb R^3},$ which confirms that $\mathcal R_Q$ is a rotation in $\mathbb R^3.$ If one sets $Q=\cos(\frac{\theta}{2})+\vec{\omega} \sin(\frac{\theta}{2})$, where $\vec{\omega}=\omega_x{\mathbf i}+\omega_y{\mathbf j}+\omega_y{\mathbf k}$, $\omega_x^2+\omega_y^2+\omega_z^2=1$ and $0\leq\theta\leq 2\pi,$ then the matrix of $\mathcal R_Q$ now reads $$\begin{aligned} \label{Ad2} \begin{pmatrix}\cos\theta+\omega_x^2(1-\cos\theta) &\omega_x\omega_y(1-\cos\theta)-\omega_z\sin\theta&\omega_x\omega_z(1-\cos\theta)+\omega_y\sin\theta\\ \omega_x\omega_y(1-\cos\theta)+\omega_z\sin\theta & \cos\theta+\omega_y^2(1-\cos\theta) &\omega_y\omega_z(1-\cos\theta)-\omega_x\sin\theta\\ \omega_x\omega_z(1-\cos\theta)-\omega_y\sin\theta &\omega_y\omega_z(1-\cos\theta)+\omega_x\sin\theta&\cos\theta+\omega_z^2(1-\cos\theta)\end{pmatrix}.\end{aligned}$$ Clearly, this corresponds to the matrix of the rotation with angle $\theta$ around the axis defined by the unit vector $\omega$ with components $(\omega_x,\omega_y,\omega_z)$ in the canonical basis of $\mathbb R^3$. So $\mathcal R:\mathbb H_1\to SO(3), Q\mapsto \mathcal R_Q,$ is a Lie group homomorphism, with kernel $\{1,-1\}$. Indeed one has $\mathcal R_{Q_1Q_2}(\vec{u})=Q_1Q_2uQ_2^{-1} Q_1^{-1}=\mathcal R_{Q_1}\mathcal R_{Q_2}(\vec{u})$ for any $Q_1,Q_2\in \mathbb H_1$ and any $u\in \mathbb R^3.$ That is, $\mathcal R_{Q_1Q_2} = \mathcal R_{Q_1}\mathcal R_{Q_2}.$ On the other hand, suppose that $\mathcal R_{Q} =\mathbb I_{\mathbb R^3}$. Then choosing $\vec{u}$ perpendicular to $\vec{q}$ so that $\vec{u}$ and $\vec{q}\times\vec{u}$ are linearly independent and $\vec{q}\cdot\vec{u}=0$, implies that the expression $\vec{u}=Q \vec{u}Q^{-1} = (q_0^2 - \vert\vert\vec{q}\vert\vert^2)\vec{u} + 2q_0\vec{q}\times\vec{u} +2 (\vec{q}\cdot\vec{u})\vec{q}$ becomes $\vec{u}= (q_0^2 - \vert\vert\vec{q}\vert\vert^2)\vec{u} + 2q_0\vec{q}\times\vec{u} ,$ or equivalently, $\vec{q}=0$ and $q_0^2=1.$ This is, $Q=\pm 1\;.$ Conversely, to any given rotation $\hat{\mathcal R}$ in $3$D, of angle $\theta$ around an axis of rotation defined by a chosen unit vector, say $\omega\in\mathbb R^3$, one associates a unit quaternion $Q =\cos(\frac{\theta}{2}) +\omega\sin(\frac{\theta}{2})$. Further applying to the corresponding rotation $\mathcal R_Q$, the expression $Q \vec{u}Q^{-1} = (q_0^2 - \vert\vert\vec{q}\vert\vert^2)\vec{u} + 2q_0\vec{q}\times\vec{u} +2 (\vec{q}\cdot\vec{u})\vec{q}$ obtained in ([\[rotation-unit-quaternion1\]](#rotation-unit-quaternion1){reference-type="ref" reference="rotation-unit-quaternion1"}), leads to the following equalities, valid for any $\vec{u}:$ $$\begin{aligned} \mathcal R_Q(\vec{u}) &=& \Big( \cos\theta\; + \sin\theta\; F(\vec{\omega}) +(1-\cos\theta)\;\vec{\omega}\otimes \vec{\omega} \Big)\;\vec{u} =\hat{\mathcal R}(\vec{u}) \;.\end{aligned}$$ So $\hat{\mathcal R}=\mathcal R_Q,$ and hence $\mathcal R$ is surjective. Consider the coset $\mathbb H_1\slash \{+1,-1\}$ of elements $\bar Q=\{Q,-Q\}=Q\mod \pm1.$ From the above, the map $\bar {\mathcal R}:\mathbb H_1\slash \{+1,-1\}\to SO(3)$, $\bar Q\mapsto \mathcal R_Q,$ is a Lie group isomorphism. Thus $\mathbb H_1$ is the universal cover of SO($3$) and $\mathcal R$ can be considered as the corresponding universal covering. We note that, comparing ([\[MatrixAdSU(2)\]](#MatrixAdSU(2)){reference-type="ref" reference="MatrixAdSU(2)"}) and ([\[Ad2\]](#Ad2){reference-type="ref" reference="Ad2"}), one sees that $$\begin{aligned} \label{equality}\tilde \Pi\circ \psi = \mathcal R \;. \end{aligned}$$ We now extend $\tilde \Pi$ to the following map, also denote by $\tilde \Pi$ and defined as follows $$\begin{aligned} \label{isomorphism-T*SU(2)-T*SO(3)}\tilde \Pi: T^*SU(2)\to T^*SO(3)\;. \end{aligned}$$ Dual quaternions are defined as combinations of ordinary quaternions and dual numbers. More precisely, a dual quaternion is of the form $\hat Q={Q}_r+ \mathfrak{e}{Q}_d$ where ${Q}_r$ and ${Q}_d$ are quaternions and $\mathfrak{e}$, termed the dual operator, satisfies the properties $\mathfrak{e}\neq 0$ and $\mathfrak{e}^2=0.$ We have the following rules. The sum and product of two dual quaternions $\hat Q_1=Q_{r,1} + \mathfrak{e}Q_{d,1}$ and $\hat Q_2=Q_{r,2} + \mathfrak{e}Q_{d,2},$ are respectively $$\begin{aligned} \label{sum-product-dual-quaternions} \hat Q_1+\hat Q_2=Q_{r,1}+Q_{r,2} + \mathfrak{e}(Q_{d,1}+Q_{d,2}) \; ,\; \hat Q_1\hat Q_2=Q_{r,1}Q_{r,2} + \mathfrak{e}(Q_{r,1}Q_{d,2}+Q_{d,1}Q_{r,2}).\end{aligned}$$ Dual quaternions were introduced by Clifford in 1882 (see [@clifford]). They have since gained a lot interest due to their applications in many areas such as Robotics, Computer graphics and video games. They appear as a natural unified framework for both the translations and rotations, with computational advantages of being robust and presenting singularity free solutions. The dual conjugation (analogous to complex conjugation) of $\hat{Q}$ is $\hat{Q}^*={Q}_r^*+ \mathfrak{e}{Q}_d^*$ and its dual quaternionic conjugation (analogous to quaternionic conjugation) is $\bar{\hat{Q}}={Q}_r- \mathfrak{e}Q_d.$ One also defines the magnitude of a dual quaternion $\hat{Q}={Q}_r+ \mathfrak{e}{Q}_d$ using the expression $\vert\vert \hat{Q} \vert\vert^2 :=\hat{Q}\hat{Q}^*$$= {Q}_{r}{Q}_{r}^* + \mathfrak{e}({Q}_{r}{Q}_{d}^*+{Q}_{d}{Q}_{r}^*) .$ If we set ${Q}_r=s_r+x_r{\mathbf i} +y_r{\mathbf j}+z_r{\mathbf k}$, ${Q}_d=s_d+x_d{\mathbf i} +y_d{\mathbf j}+z_d{\mathbf k}$, we can see the following $$\begin{aligned} {Q}_r^*{Q}_d+{Q}_d^*{Q}_r&=& 2(s_rs_d+x_rx_d+y_ry_d+z_rz_d)=2\langle{Q}_r,{Q}_d\rangle\;.\end{aligned}$$ So we have $\vert\vert \hat{Q} \vert\vert^2 =\vert\vert {Q}_r \vert\vert^2 +2 \mathfrak{e}\langle{Q}_r,{Q}_d\rangle .$ We also further have $$\begin{aligned} \label{norm-dual-quaternion-multiplicative}\vert\vert\hat Q_1\hat{Q}_2\vert\vert^2&=& \vert\vert \hat{Q}_1 \vert\vert^2 \; \vert\vert \hat{Q}_2 \vert\vert^2. \end{aligned}$$ A dual quaternion $\hat Q:=Q_r+\mathfrak{e}Q_d$ has an inverse if and only if $Q_r\neq 0.$ In that case, $\hat Q^{-1}= Q_r^{-1}-\mathfrak{e} (Q_r^{-1}Q_dQ_r^{-1})$. With the sum and the product ([\[sum-product-dual-quaternions\]](#sum-product-dual-quaternions){reference-type="ref" reference="sum-product-dual-quaternions"}), the $8$-dimensional vector space underlying the space $\hat{\mathbb H}$ of dual quaternions is an associative algebra with a unit, namely $1$. Hence its (open) subset consisting of invertible dual quaternions is an analytic Lie group of dimension $8$, (see e.g. J. P. Serre [@serre], p. 103, for the general case of the set of invertible elements of an associative algebra with unit) of which the subset $\hat{\mathbb H}_1$ of unit dual quaternions is a closed subgroup and hence a closed Lie subgroup, according to Cartan's Theorem. Another useful way of representing unit dual quaternions is as follows $\hat Q=Q+\mathfrak{e}(\vec{u} Q)$, where $Q$ is a unit quaternion, $\vec{u}$ a pure quaternion. From e.g. [@sastry], 'a rigid motion of an object (in the Euclidean 3-space $\mathbb R^3$) is a continuous movement of the particles in the object such that the distance between any two particles remains fixed at all times. A rigid motion is one that preserves the distance between points and the angle between vectors. The net movement of a rigid body from one location to another via a rigid motion is called a rigid displacement. In general, a rigid displacement may consist of both translation and rotation of the object.' A rigid displacement is represented by an element of the special affine group SE($3$):=SO($3$)$\ltimes \mathbb R^3$, that is, by an affine map $x\mapsto L(x)+v$, with linear part $L\in SO(3)$ and fixed vector $v\in\mathbb R^3.$ A rigid motion is represented by a curve on SE($3$). It is well known that, together with the composition of maps, SE($3$) is a Lie group. A screw displacement is a rigid displacement consisting of a rotation at constant angular velocity around an axis (the screw or twist axis) followed by a translation with constant (translationnal) velocity along the same axis. From Chasles theorem, any rigid motion can be realized as a screw motion. One extends the map $\mathcal R$ discussed above, to the map $\Pi: \hat{\mathbb H}_1\to SE(3)$ as follows, $$\begin{aligned} \label{formula:unit-dual-quaternions2a}\Pi: \hat{\mathbb H}_1\to SE(3) \;,\; \hat Q=Q+\mathfrak{e}(\vec{u} Q) \;\mapsto \; (\mathcal R_Q, \vec{u}). \end{aligned}$$ If $\hat Q=Q+\mathfrak{e}\vec{u} Q$ and $\hat P=P+\mathfrak{e}\vec{v} P$ are unit dual quaternions, then the unit dual quaternion $\hat Q\hat P$ is represented as $\hat Q\hat P= QP+ \mathfrak{e}(Q\vec{v}P +\vec{u}QP ) = QP+ \mathfrak{e}(Q\vec{v}Q^{-1} +\vec{u})QP = QP+ \mathfrak{e}(\vec{X}QP)$ with $\vec{X} = \vec{u}+\mathcal R_Q\vec{v}.$ Thus, we have $\Pi(\hat Q\hat P)= (\mathcal R_{QP}, \vec{u}+\mathcal R_Q\vec{v})$, which we can rewrite as $\Pi(\hat Q\hat P)= (\mathcal R_Q, \vec{u})(\mathcal R_P, \vec{v})= \Pi(\hat Q)\Pi(\hat P)$. A unit dual quaternion $\hat Q:=Q+\mathfrak{e}(\vec{u} Q)$ satisfies $\Pi(\hat Q) =(\mathbb I_{\mathbb R^3},0)$ if and only if $\mathcal R_Q=\mathbb I_{\mathbb R^3}$ and $\vec{u}=0,$ or equivalently $\hat Q=\pm 1.$ Of course, for any rigid displacement $(\hat{\mathcal R},v)$, where $\hat {\mathcal R}$ is a rotation of angle $\theta$ around an axis defined by a unit vector $u$, the unit dual quaternion $\hat Q=Q+\mathfrak{e}\vec{v} Q$, where $Q=\cos(\frac{\theta}{2})+\sin(\frac{\theta}{2})\;\vec{u}$, satisfies $\Pi(\hat Q) =(\hat {\mathcal R},v).$ So $\Pi$ is a surjective homomorphism between the Lie groups $\hat{\mathbb H}_1$ and $SE(3),$ with kernel $\ker\Pi=\{1,-1\},$ hence a double cover. As $\hat{\mathbb H}_1$ is simply connected, $\Pi: \hat{\mathbb H}_1\to$ SE($3$) can be seen as the universal covering of SE($3$). This allows one to represent rigid motions by dual quaternions. Now our Theorem [Theorem 4](#theorem:commutingdiagramSO(3)){reference-type="ref" reference="theorem:commutingdiagramSO(3)"} asserts that, up to isomorphism, the Lie group $\hat{\mathbb H}_1$ is in fact the cotangent bundle $T^*SU(2)$ of the compact simple Lie group SU(2), endowed with its Lie group structure induced by right trivialization. Theorem [Theorem 4](#theorem:commutingdiagramSO(3)){reference-type="ref" reference="theorem:commutingdiagramSO(3)"} further shows that the group SE(3) of Rigid motions of the Euclidean space is the cotangent bundle of the compact simple Lie group $SO(3)$, up to isomorphism of Lie groups. ### Split quaternions, rigid motion in Minkowski space, dual split quaternions A split quaternion is of the form $Q=q_0+q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k} =q_0+\vec{q}$, where $q_0,q_x,q_y,q_z\in\mathbb R$ and $\vec{q}:=q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k}$ is called the vector part or the pure imaginary part of $Q.$ The vectors ${\mathbf i}$,${\mathbf j}$, ${\mathbf k}$ satisfy ${\mathbf i}^2=-1$, ${\mathbf j}^2={\mathbf k}^2=1$ and ${\mathbf i}{\mathbf j}{\mathbf k}=1.$ One $\mathbb R$-linearly extends the latter relations to define a product in the space of split quaternions, hereafter denoted by $\mathbb S,$ as follows. If $P=p_0+p_x{\mathbf i}+p_y{\mathbf j}+p_z{\mathbf k}=p_0+\vec{p}$ is another split quaternion, the product $Q P$ of $Q$ and $P,$ is $$\begin{aligned} \label{eq:product-split-quaternions1}QP &=& q_0p_0 - (p_xq_x - p_yq_y - p_zq_z) +q_0\vec{p}+p_0\vec{q}\nonumber\\ &&+(q_zp_y-q_yp_z){\mathbf i} +(q_zp_x-q_xp_z){\mathbf j}+(q_xp_y-q_yp_x){\mathbf k}.\end{aligned}$$ One identifies $\mathbb S$ with the vector space $\mathbb R^4=\mathbb R\oplus \mathbb R^3$, where a split quaternion $Q=q_0+q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k}$ is identified with the vector with coordinates $(q_0,q_x,q_y,q_z)$ in the canonical basis of $\mathbb R^4$. One also considers the metric $\langle , \rangle$ on $\mathbb R^4$ with neutral signature $(2,2)$ defined by $\langle Q, P\rangle =q_0p_0+q_xp_x - p_yq_y - p_zq_z,$ the restriction to the space $\mathbb R^3$ of which is the Lorentzian metric given by $\langle \vec{q}, \vec{p}\rangle:=p_xq_x - p_yq_y - p_zq_z.$ We will let $\mathbb R_1^3$ stand for $\mathbb R^3$ together with the Lorentzian metric $\langle,\rangle$, whereby the canonical basis of $\mathbb R^3$ is identified with $({\mathbf i, \; \mathbf j, \; \mathbf k}),$ and refer to it as the Minkowski $3$-space. Let us consider the cross product $\times_s$ in the Minkowski $3$-space, defined for $\vec{q}=q_x{\mathbf i}+q_y{\mathbf j}+q_z{\mathbf k}$ and $\vec{p}=p_x{\mathbf i}+p_y{\mathbf j}+p_z{\mathbf k}$ as, $$\begin{aligned} \label{cros-product-Minkowski}\vec{q}\times_s\vec{p}:=(q_zp_y-q_yp_z){\mathbf i} +(q_zp_x-q_xp_z){\mathbf j}+(q_xp_y-q_yp_x){\mathbf k}.\end{aligned}$$ The product ([\[eq:product-split-quaternions1\]](#eq:product-split-quaternions1){reference-type="ref" reference="eq:product-split-quaternions1"}) can then be rewritten as $$\begin{aligned} \label{eq:product-split-quaternions2}QP &=&q_0p_0-\langle \vec{q}, \vec{p}\rangle +q_0\vec{p}+p_0\vec{q}+\vec{q}\times_s\vec{p}\;.\end{aligned}$$ From ([\[eq:product-split-quaternions2\]](#eq:product-split-quaternions2){reference-type="ref" reference="eq:product-split-quaternions2"}), one notes that $QP-PQ=\vec{q} \vec{p} - \vec{p} \vec{q} = 2\vec{q}\times_s\vec{p}$, so one can also define the cross product $\times_s$ as $$\begin{aligned} \label{eq:cross-product-split-quaternions3}\vec{q}\times_s\vec{p}:=\frac{1}{2}(\vec{q} \vec{p} - \vec{p} \vec{q})\;,\end{aligned}$$ where $\vec{q} \vec{p} =- p_xq_x + p_yq_y + p_zq_z +(q_zp_y-q_yp_z){\mathbf i} +(q_zp_x-q_xp_z){\mathbf j}+(q_xp_y-q_yp_x){\mathbf k}$ is the product of $\vec{q}$ and $\vec{p}$. Like in the Euclidean case, one also has $\langle\vec{q},\vec{q}\times_s \vec{p}\rangle = \langle\vec{p},\vec{q}\times_s \vec{p}\rangle=0$ for any $\vec{q}, \vec{p}\in\mathbb R^3_1.$ The conjugate of a split quaternion $Q$ is defined as the split quaternion $Q^*=q_0-q_x{\mathbf i}-q_y{\mathbf j}-q_z{\mathbf k} =q_0-\vec{q}$. Considering the product $QQ^*= q_0^2+q_x^2 -q_y^2 - q_z^2 =\langle Q, Q\rangle$, one sees that a split quaternion $Q$ admits an inverse if and only if $\langle Q, Q\rangle\neq 0$ and in that case $Q^{-1} =\frac{1}{\langle Q, Q\rangle} Q^*.$ We say that $Q$ is a unit split quaternion, if $\langle Q,Q\rangle =1.$ A pure split quaternion $\vec{q}$ is said to be timelike if $\langle\vec{q},\vec{q}\rangle> 0,$ lightlike if $\langle\vec{q},\vec{q}\rangle =0$ and $\vec{q}\neq 0,$ and spacelike if $\langle\vec{q},\vec{q}\rangle < 0$ or $\vec{q}=0.$ From ([\[eq:product-split-quaternions2\]](#eq:product-split-quaternions2){reference-type="ref" reference="eq:product-split-quaternions2"}), we see that any pure split quaternion $\vec{q}$ satisfies $\vec{q} ^2= - \langle \vec{q}, \vec{q}\rangle$. Let $Q=q_0+\vec{q}$ be a unit split quaternion, then either $Q=\pm 1$ or $\langle \vec{q}, \vec{q}\rangle \neq 0$. In the case $\langle \vec{q}, \vec{q}\rangle \neq 0$, one sets $\vec{u}=\frac{\vec{q}}{\sqrt{\vert \langle\vec{q},\vec{q}\rangle\vert}}$. So if $\langle\vec{q},\vec{q}\rangle\geq 0,$ one sets $\theta=\cos^{-1} q_0$, with $0\leq \theta\leq \pi$, so that $q_0=\cos(\theta)$ and $\sin(\theta)=\sqrt{\langle\vec{q},\vec{q}\rangle}$. Furthermore, in the case $\langle\vec{q},\vec{q}\rangle> 0,$ the unit pure split quaternion $\vec{u}=\frac{\vec{q}}{\sqrt{\langle\vec{q},\vec{q}\rangle}}$ satisfies $\vec{u}^2= -1$ and $Q=q_0+\vec{q}=\cos\theta+\sin\theta\;\vec{u} =\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^{2k}}{(2k)!}+\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^{2k+1}}{(2k+1)!}=\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^k}{k!}=e^{\theta\vec{u}}.$ So a unit split quaternion $Q$ with a timelike vector part can always be written in the form $Q=\cos\theta+\sin\theta\;\vec{u}=e^{\theta\vec{u}},$ with $0\leq \theta\leq\pi$ and $\vec{u}$ is a unit (timelike) pure split quaternion. Now if $\langle\vec{q},\vec{q}\rangle < 0,$ one sets $\gamma=\cosh^{-1} q_0$, the pure split quaternion $\vec{u}=\frac{\vec{q}}{\sqrt{-\langle\vec{q},\vec{q}\rangle}}$ satisfies $\vec{u}^2= 1=-\langle \vec{u}, \vec{u}\rangle$ and $Q=q_0+\vec{q}=\cosh\theta+\sinh\theta\;\vec{u} =\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^{2k}}{(2k)!}+\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^{2k+1}}{(2k+1)!}=\displaystyle\sum_{k=0}^\infty\frac{(\theta\vec{u})^k}{k!}=e^{\theta\vec{u}}.$ Thus, a unit split quaternion $Q$ with a spacelike vector part can be written in the form $Q=\cosh\theta+\sinh\theta\;\vec{u}=e^{\theta\vec{u}},$ with $0\leq \theta\leq\pi$ and $\vec{u}$ is a unit (spacelike) pure split quaternion. This allows to define an exponential of a split quaternion and to write any unit split quaternion as the exponential of some pure split quaternion. For more details, see [@kula-yayli], [@kula-yayli-screw-motion], where rotations on the Minkowski space are also discussed using split quaternions. Similarly to dual quaternions, dual split quaternions are combinations of split quaternions and dual numbers. A dual split quaternion is of the form $\tilde Q={Q}_r+\mathfrak{e}{Q}_d$ where ${Q}_r$ and ${Q}_d$ are split quaternions and the dual operator $\mathfrak{e}$ satisfies $\mathfrak{e}\neq 0$ and $\mathfrak{e}^2=0,$ as above. The following rules apply. The sum and product of two split dual quaternions $\tilde Q_1=Q_{r,1} + \mathfrak{e} Q_{d,1}$ and $\tilde Q_2=Q_{r,2} + \mathfrak{e}Q_{d,2},$ are respectively defined as $$\begin{aligned} \label{sum-product-dual-split-quaternions} \tilde Q_1+\tilde Q_2=Q_{r,1}+Q_{r,2} + \mathfrak{e} (Q_{d,1}+Q_{d,2})\; ,\; \tilde Q_1\tilde Q_2=Q_{r,1}Q_{r,2} + \mathfrak{e} (Q_{r,1}Q_{d,2}+Q_{d,1}Q_{r,2}).\end{aligned}$$ Here too, the dual conjugation of $\tilde{Q}$ is $\tilde{Q}^*={Q}_r^*+ \mathfrak{e} {Q}_d^*$ and its dual split quaternionic conjugaison is $\bar{\tilde{Q}}={Q}_r- \mathfrak{e}{Q}_d.$ The magnitude of a dual split quaternion is defined using the expression $\vert\vert \tilde{Q} \vert\vert^2 :=\tilde{Q}\tilde{Q}^*$$= {Q}_{r}{Q}_{r}^* + \mathfrak{e} ({Q}_{r}{Q}_{d}^*+{Q}_{d}{Q}_{r}^*) .$ If we set ${Q}_r=s_r+x_r{\mathbf i} +y_r{\mathbf j}+z_r{\mathbf k}$, ${Q}_d=s_d+x_d{\mathbf i} +y_d{\mathbf j}+z_d{\mathbf k}$, we can see the following $$\begin{aligned} {Q}_r^*{Q}_d+{Q}_d^*{Q}_r &=&2(s_rs_d+x_rx_d+y_ry_d+z_rz_d)=2\langle{Q}_r,{Q}_d\rangle\;.\end{aligned}$$ So we also have $\vert\vert \tilde{Q} \vert\vert^2 =\vert\vert {Q}_r \vert\vert^2 +2 \mathfrak{e} \langle{Q}_r,{Q}_d\rangle.$ Exactly in the same way as in ([\[norm-dual-quaternion-multiplicative\]](#norm-dual-quaternion-multiplicative){reference-type="ref" reference="norm-dual-quaternion-multiplicative"}), we also have $$\begin{aligned} \label{norm-dual-split-quaternion-multiplicative} \vert\vert\tilde Q_1\tilde{Q}_2\vert\vert^2&=&({Q}_{r,1}{Q}_{r,2}) ({Q}_{r,1}{Q}_{r,2} )^*+ 2 \mathfrak{e} \langle{Q}_{r,1}{Q}_{r,2} \;,\;{Q}_{r,1}{Q}_{d,2}+{Q}_{d,1}{Q}_{r,2}\rangle \nonumber\\ &=&\vert\vert \tilde{Q}_1 \vert\vert^2 \; \vert\vert \tilde{Q}_2 \vert\vert^2. \end{aligned}$$ A dual split quaternion $\tilde Q:=Q_r+\mathfrak{e}Q_d$ has an inverse if and only if $\vert\vert Q_r\vert\vert \neq 0.$ In that case, $\tilde Q^{-1}= Q_r^{-1}-\mathfrak{e} (Q_r^{-1}Q_dQ_r^{-1})$. With the sum and the product ([\[sum-product-dual-split-quaternions\]](#sum-product-dual-split-quaternions){reference-type="ref" reference="sum-product-dual-split-quaternions"}), the space $\tilde{\mathbb S}$ is an associative algebra with a unit, namely $1$. Hence its subset consisting of invertible dual split quaternions is an analytic Lie group and the subset $\tilde{\mathbb S}_1$ of unit dual split quaternions is a closed Lie subgroup of the latter. As for dual quaternions, a unit dual split quaternion $\tilde Q,$ can be represented as follows $\tilde Q=Q+\mathfrak{e}(\vec{u} Q)$, where $Q$ is a unit split quaternion, $\vec{u}$ a pure split quaternion. The Lie algebra $\mathfrak{so}(2,1)$ of $SO(2,1)$ can be represented as the following set of $3\times 3$ matrices $$\begin{aligned} \Big\{N(n_1,n_2,n_3):=-n_3(E_{1,2}+E_{2,1})+n_2(E_{1,3}+E_{3,1})+n_1(E_{2,3}-E_{3,2}), \; n_1,n_2,n_3\in\mathbb R\Big\},\nonumber\end{aligned}$$ which is also the Lie algebra $\mathfrak{sl}(2,\mathbb R)$. Let us consider the linear map $$\begin{aligned} \label{defineH}H:\mathbb R^3\to\mathfrak{so}(2,1), \;\; \;x\mapsto H(x),\end{aligned}$$ where $H(x)$ is the cross-product by $x,$ that is, $$\begin{aligned} \label{defineH(x)} H(x): \mathbb R^3\to\mathbb R^3\;,\;\; y\mapsto H(x)y:=x\times_s y.\end{aligned}$$ One sees that, if $(n_1,n_2,n_3)$ are the coordinates of ${\mathbf n}\in\mathbb R^3$ in the canonical basis of $\mathbb R^3$, then $-n_3(E_{1,2}+E_{2,1})+n_2(E_{1,3}+E_{3,1})+n_1(E_{2,3}-E_{3,2})$ is the matrix of $H({\mathbf n})$ in the same basis. Clearly, $H$ is an isomorphism between $\mathbb R^3$ and the vector space underlying $\mathfrak{so}(2,1).$ We define $$\begin{aligned} \label{defineU}U:SO(2,1)\to SO(2,1)\;,\;\;U(\sigma):\mathbb R^3\to\mathbb R^3,\; U(\sigma)x:=H^{-1}(\sigma H(x)\sigma^{-1}).\end{aligned}$$ We also have $U(\sigma\tau)x:=H^{-1}(\sigma H(H^{-1}(\tau H(x)\tau^{-1}))\sigma^{-1})=U(\sigma)U(\tau)x$. So $U$ is a homomorphism of the Lie group $SO(2,1).$ In fact, $U$ is an isomorphism of the Lie group $SO(2,1).$ Recall that the universal cover of $SL(2,\mathbb R)$ and SO(2,1), is not a Lie group of matrices. This is a well-known common fact for all special linear groups SL($n,\mathbb R$) for any $n\ge 2.$ We will let $\widetilde{SL(n,\mathbb R)}$ stand for such a universal cover. We consider the Lie subgroup $SE(2,1):=SO(2,1)\ltimes \mathbb R^3$ of the Lie group Aff($3,\mathbb R$) $=GL(3,\mathbb R)\ltimes \mathbb R^3$, made of invertible affine displacements of $\mathbb R^3$ whose linear parts are (oriented and) preserve the Lorentz metric in $\mathbb R_1^3$. A natural way to represent the Lie group $SE(2,1)$ is as the following group of $4\times 4$ real matrices $$\begin{aligned} \label{defineSE(2,1)} SE(2,1)=\Big\{\begin{pmatrix}A&v\\ 0&1\end{pmatrix}, \;\; A\in SO(2,1),\; v\in\mathbb R^3\Big\}.\end{aligned}$$ By analogy with the group $SE(3):=SO(3)\ltimes \mathbb R^3$ of rigid displacement of the Euclidean $3$-space, we call $SE(2,1)$ the group of rigid displacements of the Minkowski $3$-space. By rigid motion in the Minkowski $3$-space, one means a curve in $SE(2,1).$ One also establishes a homomorphism $$\begin{aligned} \label{isomorphism-unitquaternions-SE(2,1)}\hat{\mathbb S}_1\to SE(2,1)\;,\;\;\hat Q:=Q+ \mathfrak{e}\vec{u}Q\to (\tilde {\mathcal R}_Q,\vec{u}),\end{aligned}$$ between the Lie group $\hat{\mathbb S}_1$ of unit dual split quaternions and the Lie group SE($2,1$). In a similar way as for dual quaternions, Theorem [Theorem 6](#theorem:cummutingdiagramSL(2)){reference-type="ref" reference="theorem:cummutingdiagramSL(2)"} says that, up to isomorphism, the Lie group $\hat{\mathbb S}_1$ is the cotangent bundle $T^* \widetilde{SL(2,\mathbb R)}$ of the simple Lie group $\widetilde{SL(2,\mathbb R)}$, endowed with its Lie group structure induced by right trivialization. ## Main results on dual quaternions, dual split quaternions {#chap:rigid-motions-dual} **Theorem 4**. *The Lie group $SE(3)$ of rigid motions of the Euclidean space $\mathbb R^3$, is isomorphic to both the cotangent bundle $T^*SO(3)$ and the tangent bundle $TSO(3)$ of $SO(3)$, endowed with their Lie group structure induced by the right trivialization. The Lie group $\hat{\mathbb H}_1$ of unit dual quaternions is isomorphic to both the cotangent bundle $T^*SU(2)$ and the tangent bundle $TSU(2)$ of $SU(2)$, endowed with their Lie group structure induced by the right trivialization.* *Proof.* Consider the linear map $F: \mathbb R^3\to \mathfrak{so}(3)$, $x\mapsto F(x),$ where $F(x)$ is the cross product by $x$, that is, $F(x)y:= x\times y$. So $F$ is an isomorphism between $\mathbb R^3$ and the vector space underlying $\mathfrak{so}(3)$. Let $S:SO(3)\to SO(3)$, $\sigma\mapsto S_\sigma,$ where $S(\sigma) x=F^{-1}(\sigma F(x)\sigma^{-1})$, $x\in\mathbb R^3$. Note that $S$ is a smooth map, for it only involves smooth maps, the linear map $F$ and the adjoint map. We further have $S(\sigma\tau)x=$$F^{-1}(\sigma\tau F(x)\tau^{-1}\sigma^{-1})=$ $F^{-1}(\sigma F(F^{-1}(\tau F(x)\tau^{-1}))\sigma^{-1})$ $= F^{-1}(\sigma F(S(\tau)x)\sigma^{-1})$ $= S(\sigma) S(\tau)x ,$ and $S$ is thus a homomorphism of the Lie group $SO(3).$ As $F$ is a bijection, and $SO(3)$ has a trivial center, $S$ is an isomorphism (automorphism) of the Lie group $SO(3).$ As in Section [2.3](#chap:Liegroups-with-biinvariant-metric){reference-type="ref" reference="chap:Liegroups-with-biinvariant-metric"}, denote by $\mu^+$ the (unique, up to a constant factor) biinvariant metric on $SO(3)$ with value $\mu_\epsilon^+:=\mu$ at $\epsilon$ and $\Theta:\mathfrak{so}(3) \to \mathfrak{so}^*(3),$ $x\mapsto \Theta(x),$ where $\Theta(x)=i_x\mu:=\mu(x,\cdot).$ We now consider the map $T: SO(3)\ltimes_{Ad^*}\mathfrak{so}^*(3)\to SE(3),$ $T(\sigma,f)=\Big(S(\sigma), F^{-1}( \Theta^{-1} (f))\Big)$. Clearly, $T$ is a bijection and is further a homomorphism. Indeed, on the one hand, we have $T(\sigma,f)\;T(\tau,g)$ $= \Big(S(\sigma)S(\tau), F^{-1}( \Theta^{-1} (f)) + S(\sigma) F^{-1}( \Theta^{-1} (g)) \Big)$. On the other hand, we have $T\Big((\sigma,f)(\tau,g)\Big)= \Big(S(\sigma\tau), F^{-1}( \Theta^{-1} (f)) + F^{-1}( \Theta^{-1} (Ad_\sigma^*g)) \Big)$. The equality $\Theta^{-1} (Ad_\sigma^*g))=Ad_\sigma\Theta^{-1} (g)$ implies $F^{-1}( \Theta^{-1} (Ad_\sigma^*g))= F^{-1}( \sigma F(F^{-1}(\Theta^{-1} (g)))\sigma^{-1}),$ which we rewrite as $F^{-1}( \Theta^{-1} (Ad_\sigma^*g))= S(\sigma)(F^{-1}(\Theta^{-1} (g)))$. Altogether, we have $T\Big((\sigma,f)(\tau,g)\Big)=\Big(S(\sigma\tau), F^{-1}( \Theta^{-1} (f)) + S(\sigma)(F^{-1}(\Theta^{-1} (g))) \Big) =T(\sigma,f)\;T(\tau,g)$. By construction, $T$ is smooth, as $S$ is smooth and $F^{-1}$, $\Theta^{-1}$ are linear maps. Hence $T\circ \zeta: T^*SO(3)\to SE(3)$ is an isomorphism between the Lie groups $T^*SO(3)$ and $SE(3),$ where $\zeta: T^*SO(3)\to SO(3)\ltimes_{Ad^*} \mathfrak{so}^*(3),$ is the right trivialization. We now consider $\bar \varphi:\hat{\mathbb H}_1\to SU(2)\ltimes \mathfrak{su}^*(2),$ $\bar\varphi (\hat Q) = \Big(\psi(Q), \Theta(\psi(\vec{u}))\Big)$ and show that it is an isomorphism between the Lie groups $\hat{\mathbb H}_1$ and $SU(2)\ltimes \mathfrak{su}^*(2).$ For any unit dual quaternions $\hat Q=Q+\mathfrak{e} \vec{u}Q$, $\hat P=P+\mathfrak{e} \vec{v}P$, we have $\bar\varphi(\hat Q)\bar\varphi(\hat P)= \Big(\psi(Q)\psi(P), \Theta(\psi(\vec{u})) + Ad_{\psi(Q)}^*\Theta(\psi(\vec{v}))\Big) .$ Applying the linearity of $\Theta$ and the identity $\Theta(Ad_\sigma x) = Ad_\sigma^*\Theta(x)$, the above come to $\bar \varphi(\hat Q)\bar\varphi(\hat P)=\Big(\psi(Q)\psi(P), \Theta(\psi(\vec{u}) + Ad_{\psi(Q)}\psi(\vec{v}))\Big)$. We further plug in the equality $\psi(\mathcal R_Q\vec{v})=\psi(Q\vec{v}Q^{-1})= \psi(Q)\psi(\vec{v}) (\psi(Q))^{-1} = Ad_{\psi(Q)}\psi(\vec{v})$, to finally get $\bar\varphi(\hat Q)\bar\varphi(\hat P) = \Big( \psi(QP), \Theta\psi(\vec{u}+\mathcal R_Q\vec{v})\Big)=\bar\varphi(\hat Q\hat P) .$ So $\bar\varphi$ is a Lie group homomorphism. By construction of $\Theta$ and $\psi$, we see that $\bar\varphi$ is in fact, an isomorphism between the Lie groups $\hat{\mathbb H}_1$ and $SU(2)\ltimes \mathfrak{su}^*(2)$. Again using the trivialization $\zeta_{SU}: T^*SU(2)\to SU(2)\ltimes_{Ad^*} \mathfrak{su}^*(3),$ we get the isomorphism $\varphi:=\zeta_{SU}^{-1}\circ \bar\varphi: \hat{\mathbb H}_1\to T^*SU(2)$ between the Lie groups $\hat{\mathbb H}_1$ and $T^*SU(2).$ More explicitly, for $\hat Q=Q+\mathfrak{e}(\vec{u} Q)$ in $\hat{\mathbb H}_1$, we have $\varphi(\hat Q)= \zeta_{SU}^{-1}\Big(\psi(Q), \Theta(\psi(\vec{u})\Big)= \Big(\psi(Q), \Theta(\psi(\vec{u}))\circ T_{\psi(Q)}R_{\psi(Q)^{-1}}\Big).$ From Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"}, the cotangent bundle $T^*SO(3)$ (resp. $T^*SU(2)$) and the tangent bundle $TSO(3)$ (resp. $TSU(2)$) of the compact Lie group $SO(3)$ (resp. $SU(2)$), are isomorphic. ◻ Applying Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"} to $T^*SO(3)$ and further identifying the latter with SE(3) according to Theorem [Theorem 4](#theorem:commutingdiagramSO(3)){reference-type="ref" reference="theorem:commutingdiagramSO(3)"}, we recover some of the results on SE$(3)$ obtained in [@pennec1], [@Zefran]. In particular, we recover Theorem 3.6 of [@Zefran]: **Theorem 5**. *[@Zefran] Let $\mu$ be a metric on SE($3$) such that every screw motion is a geodesic. Then $\mu$ is biinvariant and furthermore, its matrix is of the form ([\[eq:biinvariantmetricTsl(2,1)\]](#eq:biinvariantmetricTsl(2,1)){reference-type="ref" reference="eq:biinvariantmetricTsl(2,1)"}) in some basis of SE($3$). There is no Riemannian metric on SE($3$) for which every screw motion is a geodesic.* Consider the linear map $H:\mathbb R^3\to\mathfrak{so}(2,1),$ $x\mapsto H(x),$ where $H(x)$ is the cross-product by $x$ in the Minkowski $3$-space, $H(x)y:=x\times_s y,$ $x,y\in\mathbb R^3$. Clearly, $H$ is an isomorphism between $\mathbb R^3$ and the vector space underlying $\mathfrak{so}(2,1).$ We define $U:SO(2,1)\to SO(2,1)\;,\;\;U(\sigma):\mathbb R^3\to\mathbb R^3,\; U(\sigma)x:=H^{-1}(\sigma H(x)\sigma^{-1}).$ It is easy to see that $U$ is an isomorphism (automorphism) of the Lie group $SO(2,1).$ Let $\mu^+$ stand for the biinvariant Lorentzian metric on $SO(2,1)$, $\mu_\epsilon^+:=\mu$ and $\Theta:\mathfrak{so}(3) \to \mathfrak{so}^*(3),$ $x\mapsto \Theta(x):=\mu(x,\cdot).$ Further define $T': SO(2,1)\ltimes_{Ad^*}\mathfrak{so}^*(2,1)\to SE(2,1),$ $T'(\sigma,f)=\Big(U(\sigma), H^{-1}( \Theta^{-1} (f))\Big)$. Clearly, $T'$ is a bijection and is further a homomorphism. Indeed, the product $T'(\sigma,f)\;T'(\tau,g)$ reads $\Big(U(\sigma)U(\tau), H^{-1}( \Theta^{-1} (f)) + U(\sigma) H^{-1}( \Theta^{-1} (g)) \Big)$. On the other hand, $T'\Big((\sigma,f)(\tau,g)\Big)$ can be written as $\Big(U(\sigma\tau), H^{-1}( \Theta^{-1} (f)) + H^{-1}( \Theta^{-1} (Ad_\sigma^*g)) \Big)$. Applying the property $\Theta^{-1} (Ad_\sigma^*g)=Ad_\sigma\Theta^{-1} (g)= \sigma\Theta^{-1} (g)\sigma^{-1}$ to $T'\Big((\sigma,f)(\tau,g)\Big)$, the latter now becomes $\Big(U(\sigma\tau), H^{-1}( \Theta^{-1} (f)) + H^{-1}( \sigma H(H^{-1}(\Theta^{-1} (g)))\sigma^{-1}) \Big)$. We rewrite it as $T'\Big((\sigma,f)(\tau,g)\Big)=\Big(U(\sigma)U(\tau), H^{-1}( \Theta^{-1} (f)) + U(\sigma)(H^{-1}(\Theta^{-1} (g))) \Big)$, which is exactly $T'(\sigma,f)\;T'(\tau,g).$ Finally, the trivialization $\zeta: T^*SO(2,1)\to SO(2,1)\ltimes_{Ad^*} \mathfrak{so}^*(2,1),$ allows to get the needed isomorphism $T'\circ \zeta$ between the Lie groups $T^*SO(2,1)$ and $SE(2,1) .$ By Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"}, the Lie groups $T^*SO(2,1)$ and $TSO(2,1)$ are isomorphic. We have proved the following. **Theorem 6**. *The special affine Lie group $SE(2,1)=SO(2,1)\ltimes \mathbb R^3$ is isomorphic to both the tangent bundle $TSO(2,1)$ and the cotangent bundle $T^*SO(2,1)$ of $SO(2,1)$ endowed with their Lie group structure induced by the right trivialization. The group of unit dual split quaternions is isomorphic to the cotangent bundle T$^*\widetilde{SL(2,\mathbb R)}$ endowed with its Lie group structure induced by the right trivialization.* On $\mathfrak{sl}(2,\mathbb R)$, consider the Lorentzian metric $K_0(x,y) = trace(xy)$, which is also its Killing form. In the basis $e_1=\frac{\sqrt{2}}{2}(E_{1,2}-E_{2,1}),$ $e_2=\frac{\sqrt{2}}{2}(E_{1,1}-E_{2,2}),$ $e_3=\frac{\sqrt{2}}{2}(E_{1,2}+E_{2,1}),$ the matrix of $K_0$ is diag($-1,1,1$)$=\mathbb I_{1,2}.$ From Theorem [Theorem 3](#thm:biinvariant-metric-on-dual){reference-type="ref" reference="thm:biinvariant-metric-on-dual"}, every metric on $T^*SL(2,\mathbb R)$ which is parallel with respect to the Cartan-Schouten canonical connection, has the following nonzero coefficients, $\mu(e_2,e_2)=\mu(e_3,e_3)= -\mu(e_1,e_1)=s,$ $\mu(e_1,e_1^*)= \mu(e_2,e_2 ^*)=\mu(e_3,e_3^*)=t,$ for some $s,t\in\mathbb R$, with $t\neq 0,$ where $(e_1^*,e_2^*,e_3^*)$ is the dual basis of $(e_1,e_2,e_3)$. Applying the isomorphism between $T^*SL(2,\mathbb R)$ and SE(2,1) proved in Theorem [Theorem 6](#theorem:cummutingdiagramSL(2)){reference-type="ref" reference="theorem:cummutingdiagramSL(2)"}, we get the following. **Theorem 7**. *Let $\mu$ be a metric on SE($2,1$) such that every screw motion on $\mathbb R^3_1$ is a geodesic. Then $\mu$ is biinvariant and furthermore, its matrix is of the form ([\[eq:biinvariantmetricTsl(2,1)\]](#eq:biinvariantmetricTsl(2,1)){reference-type="ref" reference="eq:biinvariantmetricTsl(2,1)"}) in some basis of SE($2,1$). There is no Riemannian metric on SE($2,1$) for which every screw motion is a geodesic.* This could be seen as the SE($2,1)$ version of the results on SE$(3)$ in [@pennec1], [@Zefran]. **Remark 2**. *Note that Theorems [Theorem 4](#theorem:commutingdiagramSO(3)){reference-type="ref" reference="theorem:commutingdiagramSO(3)"} and [Theorem 6](#theorem:cummutingdiagramSL(2)){reference-type="ref" reference="theorem:cummutingdiagramSL(2)"} imply, in particular, that $SE(3)$, $SE(2,1)$, $\hat{\mathbb H}_1$ and $\tilde{\mathbb H}_1$ all have exact symplectic structures and as Lie groups, they possess biinvariant metrics.* 99 Amari, S.: Lecture Notes in Statistics 28 (Springer, New York, 1985). Cartan, E. and Schouten, J. A.: Proc. Akad. Wekensch, Amsterdam 29, 803-815 (1926). W. Clifford: Mathematical Papers. London: Macmillan, 1882. Crampin, M. and Mestdag, T.: J. Lie Theory 18, no. 2, 471-502 (2008). Diatta, A.; Manga B. and Sy, F.: Preprint (2023). Gallier J. and Quaintance, J.: Geometry and Computing, 12. Springer, Cham, 2020, 777pp. Ghanam, R.; Hindeleh, F. and Thompson, G.: J. Math. Phys. 48, no. 10, 102903 (2007). Ghanam, R.; Thompson, G. and Miller, E. J.: J. Lie Theory 14, no. 2, 395425 (2004). Kula, L. and Yayli, Y.: J. Korean Math. Soc. 44, no. 6, 1313-1327 (2007). Kula, L. and Yayli, Y.: Iran. J. Sci. Technol. Trans. A Sci. 30 (3), 245-258 (2006). Lauritzen, S. L.: In Differential Geometry in Statistical Inferences. IMS Lecture Notes Monogr. Ser., 10, Inst. Math. Statist., Hayward California, 1987, pp. 96-163. Lorenzi, M. and Pennec, X.: Int. J. Comput. Vis. 105, no. 2, 111-127 (2013). Manga, B.: Ph.D. Thesis . Université d'Abomey-Calavi, 2010. Matsuzoe, H.: Advanced Studies in Pure Mathematics 57, 2010 Probabilistic Approach to Geometry pp. 303-321. Matsuzoe, H.: In Proceedings of the 8th International Workshop on Complex Structures and Vector Fields. World Scientific, 2007, pp. 244-251 Medina, A.: Tôhoku Math. J. 37, 405-421 (1985). Medina, A. and Revoy, Ph.: Ann. Scient. Ec. Norm. Sup., 4e serie, **18**, no 3, 553-561 (1985) Miolane, N. and Pennec, X.: Entropy 17, no. 4, 1850-1881 (2015). Miolane, N. and Pennec X.: AIP Conference Proceedings 1641 (1) 59-66 (2015). Murray, R.M.; Li, Z. and Sastry, S.S.: CRC Press (1994). Muzsnay, Z. and Thompson, G.: Diff. Geom. Appl. **23**, 257-281 (2005). Muzsnay, Z.: J. Math. Phys. 46, no. 11, 112902, 11 pp (2005). Özdemir, M.: Appl. Math. Lett. 22, 258-263 (2009). Pennec, X.: Geometric Science of Information, 59-67, Lecture Notes in Comput. Sci, 8085, Springer, Heidelberg, 2013. Pennec, X.: J. Math. Imaging Vision 25 (1), 127-154 (2006). Rawashdeh, M. and Thompson, G.: J. Math. Phys. 47, no. 11, 112901, 29 pp (2006). Samereh, L.; Peyghan, E. and Mihai, I.: Entropy 24 (2022), no. 6, Paper No. 758, 10 pp. Serre, J.P.: Lectures notes in Math., 1500. Springer-Verlag, Berlin, 2006. Strugar, I. and Thompson, G.: Houston J. Math. 35, no. 2, 373-409 (2009). Sy, F.: Ph.D Thesis. Université C.A. Diop. In preparation. Thompson, G.: Differential Geom. Appl. 18, no. 3, 255-270 (2003). Thompson, G.: J. Geom. Phys. 19, 1-17 (1996). Zefran,M.; Kumar, V. and Croke, C.: The International Journal of Robotics Research, 18 (2) 242 (1999). [^1]: \(1\) Aix-Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, 13013 Marseille, France. Email: andre.diatta\@fresnel.fr; andrediatta\@gmail.com. [^2]: \(2\) Département de Mathématiques et Informatique, Université Cheikh Anta Diop de Dakar, BP 5005 Dakar-Fann, Dakar, Sénégal. Email: bakary.manga\@ucad.edu.sn; syfatima89\@gmail.com

arxiv_math

--- abstract: | We produce the first regular unimodular triangulation of an arbitrary matroid base polytope. We then extend our triangulation to integral generalized permutahedra. Prior to this work it was unknown whether each matroid base polytope admitted a unimodular cover. author: - Spencer Backman - Gaku Liu bibliography: - references.bib title: A Regular Unimodular Triangulation of the Matroid Base Polytope --- # Introduction Despite considerable interest, very little is known about triangulations of matroid base polytopes. There are a few motivations for wanting to have nice triangulations of matroid base polytopes. The first motivation comes from White's conjecture whose weakest version states that the toric ideal of a matroid base polytope is quadratically generated [@white1977basis][@lason2014toric]. Herzog and Hibi asked whether every matroid base polytope has a quadratic Gröbner basis [@herzog2002discrete]. A slight strengthening of this question is whether every matroid base polytope admits a quadratic triangulation, i.e. a regular unimodular flag triangulation---such a triangulation would give a square-free quadratic Gröbner basis for the toric ideal of a matroid base polytope. The strongest result known in this area is that the alcoved triangulation of a positroid produces a quadratic triangulation [@stanley1977eulerian; @sturmfels1996grobner; @lam2007alcoved; @lam2018alcoved]. The second motivation comes from Ehrhart theory. A formula for the volume of a matroid base polytope was calculated by Ardila--Doker--Benedetti [@ardila2010matroid], but no formula is currently known which is cancellation free, i.e. involves no subtraction. If a polytope $P$ admits a unimodular triangulation $\mathcal T$, then the volume of $P$ is equal to the number of maximal simplices in $\mathcal T$. The volume of a polytope occurs as the leading coefficient of the Ehrhart polynomial. Several researchers have investigated Ehrhart polynomials for matroid base polytopes [@castillo2018berline][@jochemko2022generalized] [@ferroni2022ehrhart] largely motivated by the conjecture of De Loera--Haws--Köppe [@de2009ehrhart] that matroid base polytopes are Ehrhart positive---this conjecture was recently disproven by Ferroni [@ferroni2022matroids], but various other questions about these polynomials remain open. The volume of a polytope $P$ is also given by the evaluation of the $h^*$-polynomial at 1. Another conjecture by De Loera--Haws--Köppe, which remains open, is that the $h^*$-vectors of matroid base polytopes are unimodal [@de2009ehrhart]. It has also been conjectured that if a polytope $P$ has the integer decomposition property (is IDP), then $P$ has a unimodal $h^\ast$-vector [@schepers2013unimodality], and it is known that every matroid base polytope is IDP [@herzog2002discrete]. We note that the property of admitting a unimodular triangulation is strictly stronger than the property of being IDP [@bruns1999normality]. It is known that the $h^*$-vector of a polytope is equal to the $h$-vector of any unimodular triangulation of the polytope [@stanley1980decompositions][@betke1985lattice], thus one might hope that such a triangulation could shed some light on this conjecture. A natural question which sits in between these various results and conjectures is whether each matroid base polytope admits a (not necessarily flag) regular unimodular triangulation. That the matroid base polytope admits a (not necessarily regular) unimodular triangulation was conjectured by Haws in their 2009 thesis [@haws2009matroid]. In this paper we give an affirmative answer to this question by providing a regular unimodular triangulation of an arbitrary matroid base polytope. We then apply this result to produce a regular unimodular triangulation of an arbitrary integral generalized permutahedron, and explain how this gives a regular unimodular triangulation of the matroid independence polytope. We emphasize that prior to this work it was unknown whether every matroid base polytope admitted a unimodular cover (this was also conjectured by Haws [@haws2009matroid]) let alone a unimodular triangulation. Our construction produces many different triangulations, but at the time of writing we do not know if any of them are flag. In the future we hope to investigate this question as well as applications of our triangulation to some of the topics listed above. # Preliminaries We recommend the following texts for an introduction to matroids [@Oxley2011matroid], polytope theory [@ziegler2012lectures], and triangulations [@de2010triangulations][@haase2021existence]. Let $[n]$ denote the set of integers $\{1,\dots,n\}$. Given $S \subseteq [n]$ we will employ the notation $x_S := \sum_{i \in S} x_i$. We identify $\{0,1\}^n$ with the collection of all subsets of $[n]$. We denote the standard basis vectors for $\mathbb{R}^n$ by $e_i$ for $1\leq i\leq n$. **Definition 1**. A *matroid* is a pair $M =(E,\mathcal{B})$ where $E$ is a finite set called the *ground set*, and $\mathcal{B}$ is a nonempty collection of subsets of $E$ called the *bases* which satisfy the following basis exchange condition: - For any $B_1,B_2 \in \mathcal{B}$, and $x \in B_1\setminus B_2$, there exists some $y \in B_2 \setminus B_1$ such that $B_1 \setminus \{x\} \cup \{y\} \in \mathcal{B}$. A set $I\subseteq E$ is *independent* if there exits some basis $B \in \mathcal B$ such that $I \subseteq B$. The collection of independent sets is denoted $\mathcal I$. The *rank* of a set $S \subseteq E$, written $r(S)$, is the maximum cardinality of an independent set contained in $S$. Matroid independence polytopes and the matroid base polytopes were introduced by Edmonds [@edmonds2003submodular]. **Definition 2**. Given a matroid $M$ on groundset $E = [n]$, the *matroid base polytope* $P_M$ is the convex hull of the indicator vectors for the bases of $M$, and the *matroid independence polytope* $P_{\mathcal I}$ is the convex hull of the indicator vectors of the independent sets. More explicitly, given $S\subseteq E$, we define the indicator vector $\chi_S \in \mathbb{R}^n$ with $i$th entry $$\chi_S(i)= \begin{cases} 1 & i \in S \\ 0 & i \notin S \end{cases}$$ Thus $P_M = \mathop{\mathrm{conv}}\{\chi_B: B \in \mathcal B\}$ and $P_{\mathcal I}= \mathop{\mathrm{conv}}\{\chi_I: I \in \mathcal I\}$. The matroid base polytope is the distinguished face of the matroid independence polytope where the sum of the coordinates is maximized. The matroid independence polytope will be discussed at the end of this article (see Corollary [Corollary 18](#indep){reference-type="ref" reference="indep"}). Gelfand--Goresky--MacPherson--Serganova uncovered a connection between matroid base polytopes and the geometry of the Grassmannian [@gelfand1987combinatorial]. They showed that torus orbit closure of a linear space $L$ in the Grassmannian is a normal toric variety whose weight polytope is the matroid base polytope $P_{M(L)}$ where $M(L)$ is the matroid determined by $L$. See Katz [@katz2016matroid] for an overview of this story. By standard toric theory, our regular unimodular triangulation of $P_M$ gives a projective Crepant resolution of the toric variety associated to the cone over a matroid base polytope. Matroid bases polytopes allow for a polytopal characterization of matroids. **Theorem 3**. *[@edmonds2003submodular][@gelfand1987combinatorial] A polytope $P$ is a matroid base polytope for some matroid $M$ if and only if $P$ is a 0-1 polytope whose edge directions are of the form $e_i-e_j$.* Polymatroids are a generalization of matroids described by monotonic submodular fuctions taking values in the nonnegative reals. Their base polytopes are equivalent by translation to the generalized permutahedra of Postnikov [@postnikov2009permutohedra]. See [@ardila2020coxeter] for a careful treatment of the following definition. **Definition 4**. A *generalized permutahedron* $P \subseteq \mathbb{R}^n$ is a polytope defined by any one of the following equivalent conditions: 1. [\[cond1\]]{#cond1 label="cond1"} The edge directions for $P$ are all of the form $e_i-e_j$, 2. [\[cond2\]]{#cond2 label="cond2"} The normal fan of $P$ is a coarsening of the braid arrangement, 3. [\[cond3\]]{#cond3 label="cond3"}$P$ is defined by inequalities $x_S \leq f(S)$ where $f:\{0,1\}^n\rightarrow \mathbb{R}$ is a submodular function, together with a single equation $x_{[n]} =f([n])$. An *integral generalized permutahedron* $P$ is a generalized permutahedron whose vertices have integer coordinates. The following is well-known, and follows from the unimodularity of the set of primitive ray generators of each chamber in the braid arrangement. **Lemma 5**. *Let $P$ be a generalized permutahedron determined by a submodular function $f$ as in condition ([\[cond3\]](#cond3){reference-type="ref" reference="cond3"}) of Definition [Definition 4](#genper){reference-type="ref" reference="genper"}. If $f$ is an integer-valued function then $P$ is an integral generalized permutahedron. Moreover, if $P$ is an integral generalized permutahedron then $f$ may be chosen to be integer-valued.* In our proof we will use condition ([\[cond2\]](#cond2){reference-type="ref" reference="cond2"}) from Definition [Definition 4](#genper){reference-type="ref" reference="genper"} as this allows us to describe the affine span of a face of a matroid base polytope. **Lemma 6**. *Let $P$ be an integral generalized permutahedron and $\mathop{\mathrm{aff}}(P)$ its affine span. Then* *$$\mathop{\mathrm{aff}}(P) = \bigcap_{i=1}^j \{ x_{S_i} = b_i \}$$ for some flag of subsets $\emptyset = S_0 \subsetneq S_1 \subsetneq \dots \subsetneq S_j = [n]$ and some $b_i \in \mathbb Z$.* We note that when $P$ is a matroid base polytope, the $b_i$ in the lemma above is equal to the rank of the set $S_i$ viewed as a subset of the ground set of the matroid. **Definition 7**. A *subdivision* of a polytope $P$ is a collection of polytopes $\mathcal{S} = \{P_1, \ldots, P_k\}$ such that 1. $\bigcup_{i=1}^k P_i =P$ 2. for each $P_i \in \mathcal{S}$ and $F$ a face of $P_i$, there exists some $j$ such that $F=P_j$ 3. for any $i$ and $j$ with $1\leq i,j \leq k$, the intersection $P_i\cap P_j$ is a face of both $P_i$ and $P_j.$ A maximal polytope in $\mathcal{S}$ is a *cell* of $\mathcal{S}$. **Definition 8**. A *triangulation* of a polytope $P$ is a subdivision $\mathcal{T} = \{T_1, \ldots, T_k\}$ of $P$ such that each polytope $T_i$ is a simplex. **Definition 9**. Let $P \subset \mathbb R^n$ be a polytope and $S$ a finite subset of $P$ containing the vertices of $P$. Given a function $f : S \to \mathbb R$, the subdivision *induced* by $f$ is the subdivision of $P$ formed by projecting the lower faces of the polytope $$\mathop{\mathrm{conv}}\{ (x,f(x)) : x \in S \} \subset \mathbb R^{n+1}.$$ A subdivision is *regular* if it is induced by some function $f$. We establish some notation. Given an affine subspace $A \subseteq \mathbb R^n$, let $\vec{A}$ denote the unique linear subspace such that $A = x + \vec{A}$ for some $x \in \mathbb R^n$. Similarly, given an affine sublattice $L \subseteq \mathbb Z^n$, let $\vec{L}$ denote the unique sublattice of $\mathbb Z^n$ such that $L = x + \vec{L}$ for some $x \in \mathbb Z^n$. Given $S\subset \mathbb{R}^n$ we let $\mathop{\mathrm{aff}}(S)$ denote the affine span of $S$. **Definition 10**. A lattice simplex $T$ is *unimodular* if it has normalized volume 1. Equivalently, if $T$ has vertices $v_0,\ldots, v_n \in \mathbb{Z}^n$, then $T$ is unimodular whenever a maximal linearly independent set of edge vectors $\{v_i-v_j\}$ form a lattice basis for $\vec{\mathop{\mathrm{aff}}}(T)\cap \mathbb{Z}^n$. **Definition 11**. The *resonance arrangement* $\mathcal{A}_n$ is the hyperplane arrangement in $\mathbb{R}^n$ consisting of all hyperplanes $H_S =\{x \in \mathbb{R}^n: x_S = 0\}$ where $\emptyset \subsetneq S\subseteq [n]$. For an introduction to the resonance arrangement (also called the *all subsets arrangement*) we refer the reader to [@kuhne2023universality]. **Definition 12**. We say that an affine functional $\ell:\mathbb{R}^n\rightarrow \mathbb{R}$ is *generic* if it is non constant on each positive dimensional flat of the resonance arrangement. We note that a generic point $p$ on the $n$-th moment curve $$C_n = \{(t, t^2, \ldots, t^{n}): t \in \mathbb{R}\}$$ produces a generic linear functional $x \mapsto \langle x,p \rangle$. # A deletion-contraction triangulation In this section we establish the main result of this paper. **Theorem 13**. *Every matroid base polytope has a regular unimodular triangulation.* Before providing a proof, we briefly give some context for our construction. Two fundamental operations on a matroid are the deletion and contraction of an element, and many important constructions in matroid theory proceed by an inductive appeal to these operations. If $e$ is a loop or coloop, then the matroid base polytope $P_M$ is translation equivalent to $P_{M/e}$ and $P_{M\setminus e}$. If $e$ is neither a loop nor a coloop then $P_M$ is the convex hull of $P_{M/e}$ and $P_{M\setminus e}$. In this way, our recursive construction fits into the paradigm of deletion-contraction. *Proof of Theorem [Theorem 13](#main){reference-type="ref" reference="main"}.* Let $M=(E,\mathcal{B})$ be a matroid with ground set $E=[n]$, and $P_M \subset \mathbb R^n$ its matroid base polytope. We will use $V(P_M)$ to denote the vertices of $P_M$. We show $P_M$ has a unimodular triangulation by induction on $n$. If $n = 1$, then $P_M$ is a point and we are done. Assume $n \ge 2$. Let $P_0$ and $P_1$ be the polytopes in $\mathbb R^{n-1}$ such that $P_0 \times \{0\} = P_M \cap \{x_1 = 0\}$ and $P_1 \times \{1\} = P_M \cap \{x_1 = 1\}$. Note that $P_0$ or $P_1$ may be empty, which occurs if 1 is a loop or coloop. If $P_0$ is nonempty then it is the matroid base polytope of $M \setminus 1$, and if $P_1$ is nonempty then it is the matroid base polytope of $M / 1$. By the inductive hypothesis, $P_0$ and $P_1$ have regular unimodular triangulations. (We assume an empty polytope has a regular unimodular triangulation, induced by a function with empty domain.) Let $f_0 : V(P_0) \to \mathbb R$ and $f_1 : V(P_1) \to \mathbb R$ be functions which induce these triangulations. Let $\ell_0, \ell_1 : \mathbb R^{n-1} \to \mathbb R$ be affine functionals such that $\ell_0 - \ell_1$ is generic. Let $\epsilon>0$ be sufficiently small, and define $f : V(P) \to \mathbb R$ to be the function $$f(x) = \begin{dcases*} \ell_0(x_2,\dots,x_n) + \epsilon f_0(x_2,\dots,x_n) & if $x_1 = 0$ \\ \ell_1(x_2,\dots,x_n) + \epsilon f_1(x_2,\dots,x_n) & if $x_1 = 1$. \end{dcases*}$$ We claim that $f$ induces a unimodular triangulation of $P_M$. We will need the following lemmas. We say that two affine subspaces $A, A'\subseteq \mathbb{R}^n$ are *independent* if $\vec{A} \cap \vec{A'} = \{0\}$. We say that two affine sublattices $L,L'\subseteq \mathbb{Z}^n$ are *complementary* if $\mathop{\mathrm{span}}{\vec L}$ and $\mathop{\mathrm{span}}{\vec{L'}}$ are independent and for some (any) lattice bases $B$ and $B'$ of $\vec{L}$ and $\vec{L'}$, respectively, $B \cup B'$ is a lattice basis for $(\mathop{\mathrm{span}}(\vec{L} \cup \vec{L'})) \cap \mathbb Z^n$. **Lemma 14**. *Let $V$ and $W$ be independent rational linear subspaces in $\mathbb{R}^n$ such that $V \cap \mathbb Z^n$ and $W \cap \mathbb Z^n$ are complementary. If $X$ and $Y$ are rational subspaces of $V$ and $W$, respectively, then $X \cap \mathbb Z^n$ and $Y \cap \mathbb Z^n$ are complementary.* *Proof.* Take lattice bases $B_X$ of $X \cap \mathbb Z^n$ and $B_Y$ of $Y \cap \mathbb Z^n$, and extend them to lattice bases $B_V$ of $V \cap \mathbb Z^n$ and $B_W$ of $W \cap \mathbb Z^n$, respectively. Because $V \cap \mathbb Z^n$ and $W \cap \mathbb Z^n$ are complementary, $B_V \cup B_W$ gives a basis for $(\mathop{\mathrm{span}}(\vec{W} \cup \vec{V})) \cap \mathbb Z^n$. For any $x \in (\mathop{\mathrm{span}}(\vec{X} \cup \vec{Y})) \cap \mathbb Z^n$ we know that $x$ is an integral combination of elements of $B_V \cup B_W$ and thus must be an integral combination of elements of $B_X \cup B_Y$. Because $V$ and $W$ are independent linear subspaces, so are $X$ and $Y$, hence the elements of $B_X \cup B_Y$ are linearly independent. By rationality of $X$ and $Y$, the elements of $B_X \cup B_Y$ are integrally independent and the result follows. ◻ The next lemma is essentially equivalent to Edmond's celebrated matroid intersection theorem [@edmonds2003submodular]. (We warn the reader that $P$ and $Q$ in the following lemma will not correspond to $P_0$ and $P_1$.) **Lemma 15**. *Let $P$, $Q$ be two matroid base polytopes such that $\mathop{\mathrm{aff}}(P)$ and $\mathop{\mathrm{aff}}(Q)$ are independent. Then $\mathop{\mathrm{aff}}(P) \cap \mathbb Z^n$ and $\mathop{\mathrm{aff}}(Q) \cap \mathbb Z^n$ are complementary.* *Proof.* Since $P$ is a generalized permutahedron, Lemma [Lemma 6](#flag){reference-type="ref" reference="flag"} states that $$\mathop{\mathrm{aff}}(P) = \bigcap_{i=1}^j \{ x_{S_i} = b_i \}$$ for some flag of subsets $\emptyset = S_0 \subsetneq S_1 \subsetneq \dots \subsetneq S_j = [n]$ and some $b_i \in \mathbb Z$. Letting $T_i = S_i \setminus S_{i-1}$ and $c_i = b_i - b_{i-1}$, we therefore have $$\mathop{\mathrm{aff}}(P) = \bigcap_{i=1}^j \{ x_{T_i} = c_i \}$$ Similarly, we have $$\mathop{\mathrm{aff}}(Q) = \bigcap_{i=1}^k \{ x_{U_i} = d_i \}$$ for some partition $(U_1,\dots,U_k)$ of $[n]$ and $d_i \in \mathbb Z$. Therefore, $$\begin{aligned} \vec \mathop{\mathrm{aff}}(P) &= \bigcap_{i=1}^j \{ x_{T_i} = 0 \} \\ \vec \mathop{\mathrm{aff}}(Q) &= \bigcap_{i=1}^k \{ x_{U_i} = 0 \}.\end{aligned}$$ Now, since $\mathop{\mathrm{aff}}(P)$ and $\mathop{\mathrm{aff}}(Q)$ are independent, $\{e_{T_i}\} \cup \{e_{U_i}\}$ spans $\mathbb R^n$. Consider the matrix $N$ whose rows are the elements of $\{e_{T_i}\} \cup \{e_{U_i}\}$. Since $(T_i)$ and $(U_i)$ partition $[n]$, this matrix $N$ is a 0-1 matrix where each column has exactly one 1 in the first $j$ rows and one 1 in the last $k$ rows. Hence, $N$ is the incidence matrix of a bipartite graph, and therefore totally unimodular. Let $(T_i')$ and $(U_i')$ be subsets of $(T_i)$ and $(U_i)$ such that $(e_{T_i'}) \cup (e_{U_i'})$ is a basis of $\mathbb R^n$. By the unimodularity of the matrix $N$, we have that $(e_{T_i'}) \cup (e_{U_i'})$ is a basis of $\mathbb Z^n$. Let $$\begin{aligned} V &= \bigcap \{ x_{T_i'} = 0 \} \\ W &= \bigcap\{ x_{U_i'} = 0 \}.\end{aligned}$$ We claim that $V \cap \mathbb Z^n$ and $W \cap \mathbb Z^n$ are complementary. Suppose that $A \in GL_n(\mathbb{Z})$ and let $U$ be a linear subspace of $\mathbb{R}^n$, then $A(U) \cap \mathbb Z^n = A(U \cap \mathbb Z^n)$. It is easy to check that $V \cap \mathbb Z^n$ and $W \cap \mathbb Z^n$ are complementary if and only if $A(V \cap \mathbb Z^n)$ and $A(W \cap \mathbb Z^n)$ are complementary. Let $A$ be the matrix whose rows are $(e_{T_i'}) \cup (e_{U_i'})$. We know that $A \in GL_n(\mathbb{Z})$, and we may observe that $A(V)$ and $A(W)$ are independent coordinate subspaces[^1], hence $A(V\cap \mathbb Z^n)$ and $A(W\cap \mathbb Z^n)$ are clearly complementary. Since $\vec \mathop{\mathrm{aff}}(P) \subset V$ and $\vec \mathop{\mathrm{aff}}(Q) \subset W$, we can apply Lemma [Lemma 14](#rational){reference-type="ref" reference="rational"} to conclude that $\mathop{\mathrm{aff}}(P) \cap \mathbb Z^n$ and $\mathop{\mathrm{aff}}(Q) \cap \mathbb Z^n$ are complementary as well. ◻ We now return to the main proof. If either $P_0$ or $P_1$ is empty, then $f$ induces a unimodular triangulation on $P_M$ by the definition of $f_0$ and $f_1$. Thus, assume $P_0$ and $P_1$ are not empty. Let $g : V(P) \to \mathbb R$ be the function $$g(x) = \begin{dcases*} \ell_0(x_2,\dots,x_n) & if $x_1 = 0$ \\ \ell_1(x_2,\dots,x_n) & if $x_1 = 1$. \end{dcases*}$$ Let $\mathcal S$ be the subdivision of $P_M$ induced by $g$. Since $g$ is affine on $P_0$ and $P_1$, it must be that $P_0, P_1 \in \mathcal S$, hence every cell of $\mathcal S$ is of the form $\mathop{\mathrm{conv}}( F_0 \times \{0\} \cup F_1 \times \{1\} )$, where $F_0$ and $F_1$ are faces of $P_0$ and $P_1$, respectively. We claim that because $\ell_0 - \ell_1$ is generic, $\mathop{\mathrm{aff}}(F_0)$ and $\mathop{\mathrm{aff}}(F_1)$ are independent. Note that the function $g$ and the function $$g'(x) = \begin{dcases*} 0 & if $x_1 = 0$ \\ \ell_1(x_2,\dots,x_n) - \ell_0(x_2,\dots,x_n) & if $x_1 = 1$. \end{dcases*}$$ differ by an affine function on $\mathbb R^n$, and therefore they induce the same subdivision on $P$. Moreover, $0 - (\ell_1 - \ell_0) = \ell_0 - \ell_1$. Therefore, by replacing $g$ with $g'$, we may assume $\ell_0 = 0$. Let $A = \mathop{\mathrm{aff}}(F_0)$ and $A' = \mathop{\mathrm{aff}}(F_1)$ and suppose that $A$ and $A'$ are not independent. Then $L = \vec{A} \cap \vec{A'}$ is a positive dimensional linear space which is a flat in the resonance arrangement. Furthermore, $x+L \subset A$ and $y+L \subset A'$ for some $x$, $y \in \mathbb Z^{n-1}$. Now, let $\tilde{g} : \mathbb R^n \to \mathbb R$ be an affine function which agrees with $g$ on $\mathop{\mathrm{conv}}( F_0 \times \{0\} \cup F_1 \times \{1\} )$. (This exists by the assumption that $\mathop{\mathrm{conv}}( F_0 \times \{0\} \cup F_1 \times \{1\} )$ is a cell in the subdivision induced by $g$.) Since $\ell_0 = 0$, we have that $\tilde{g}$ is 0 on $A \times \{0\}$, and hence it is 0 on $(x+L)\times\{0\} \subset A \times \{0\}$. Since $\tilde{g}$ is affine, it is therefore constant on $(y+L) \times \{1\}$. Since $(y+L) \times \{1\} \subset A' \times \{1\}$, it follows that $\ell_1$ is constant on $y+L$, and hence it is constant on $L$. However, this contradicts the assumption that $\ell_0-\ell_1$ is non-constant on positive dimensional flats of the resonance arrangement. Thus $\mathop{\mathrm{aff}}(F_0)$ and $\mathop{\mathrm{aff}}(F_1)$ are independent. Each face of a matroid base polytope is a matroid base polytope, hence by Lemma [Lemma 15](#complementary){reference-type="ref" reference="complementary"}, $\mathop{\mathrm{aff}}(F_0) \cap \mathbb Z^{n-1}$ and $\mathop{\mathrm{aff}}(F_1) \cap \mathbb Z^{n-1}$ are complementary. Now, let $\mathcal T$ be the subdivision of $P_M$ induced by $f$. Because $f$ is a perturbation of $g$, it follows that $\mathcal T$ refines $\mathcal S$. Furthermore, since $f$ restricted to $P_0 \times \{0\}$ is an affine function plus $\epsilon f_1(x_2,\dots,x_n)$, the restriction of $\mathcal T$ to $P_0 \times \{0\}$ is a unimodular triangulation. Similarly the restriction of $\mathcal T$ to $P_1 \times \{1\}$ is a unimodualar triangulation. Let $T$ be a cell of $\mathcal T$, and let $S = \mathop{\mathrm{conv}}( F_0 \times \{0\} \cup F_1 \times \{1\} )$ be a cell of $\mathcal S$ containing $T$. We have $T = \mathop{\mathrm{conv}}( T_0 \times \{0\} \cup T_1 \times \{1\} )$, where $T_i$ is a unimodular simplex contained in $F_i$. Because $\mathop{\mathrm{aff}}(F_0)$ and $\mathop{\mathrm{aff}}(F_1)$ are independent, $\mathop{\mathrm{aff}}(T_0)$ and $\mathop{\mathrm{aff}}(T_1)$ are independent, hence $T$ is a simplex. Moreover, since $\mathop{\mathrm{aff}}(F_0) \cap \mathbb Z^{n-1}$ and $\mathop{\mathrm{aff}}(F_1) \cap \mathbb Z^{n-1}$ are complementary, it follows by Lemma [Lemma 14](#rational){reference-type="ref" reference="rational"} that $\mathop{\mathrm{aff}}(T_0) \cap \mathbb Z^{n-1}$ and $\mathop{\mathrm{aff}}(T_1) \cap \mathbb Z^{n-1}$ are complementary. Thus we can form a basis $B$ of $(\vec{\mathop{\mathrm{aff}}}(T_0)+\vec{\mathop{\mathrm{aff}}}(T_1)) \cap \mathbb Z^{n-1}$ from maximal collections of linearly independent edge vectors of $T_0$ and $T_1$. Each edge vector of $T$ from a vertex in $T_0$ to a vertex in $T_1$ has first coordinate 1. We claim that we can add any such edge vector $v$ to $B$ to give a basis $B'$ of $\vec{\mathop{\mathrm{aff}}}(T) \cap \mathbb Z^n$. Given $x \in \vec{\mathop{\mathrm{aff}}}(T) \cap \mathbb Z^n$ we explain that $x$ is in the integral span of $B'$. We can subtract an integer multiple of $v$ from $x$ so that the first coordinate is zero. The resulting vector must be in the span of $\vec{\mathop{\mathrm{aff}}}(T_0)+\vec{\mathop{\mathrm{aff}}}(T_1)$ and is integral, hence it is in the integral span of $B$. We conclude that $T$ is a unimodular simplex and $\mathcal T$ is a unimodular triangulation. ◻ We note that the proof of [@haase2021existence]\[Lemma 4.15\] also involves showing that the convex hull of two unimodular simplices is unimodular. There, the unimodularity is provided by the properties of the $A_n$ root system, while here it is provided by Lemma [Lemma 15](#complementary){reference-type="ref" reference="complementary"}. The previous proof implies the following more explicit statement: **Theorem 16**. *Let $P \in \mathbb R^n$ be a matroid base polytope. For each string $s \in \bigsqcup_{k=1}^{n-1} \{0,1\}^k$, let $\ell_s : \mathbb R^{n-|s|} \to \mathbb R$ be an affine functional, where $|s|$ is the length of $s$. Assume that $\ell_{s'0} - \ell_{s'1}$ is generic for all strings $s'$. Then for $1 \gg \epsilon_1 \gg \epsilon_2 \gg \dots \gg \epsilon_{n-1} > 0$, the function $f : V(P) \to \mathbb R$ defined by $$f(x) = \sum_{k=1}^{n-1} \epsilon_k \ell_{x_1 \dots x_k}(x_{k+1},\dots,x_n)$$ induces a regular unimodular triangulation on $P_M$.* *Proof.* This is obtained by unwinding the induction in the proof of Theorem [Theorem 13](#main){reference-type="ref" reference="main"}. ◻ We now explain how to extend our triangulation to all integral generalized permutahedra. **Corollary 17**. *Every integral generalized permutahedron has a regular unimodular triangulation.* *Proof.* Let $P \in \mathbb R^n$ be an integral generalized permutahedron. By translating $P$ if necessary, we may assume without loss of generality that there is some positive integer $R$ such that $P \subset \{x : 0 \le x_k \le R \text{ for all } 1 \le k \le n \}$. It is known that dicing $P$ by the hyperplanes $\{x_k = c\}$ where $1 \le k \le n$ and $0 \le c \le R$ are integers gives a regular integral subdivision $\mathcal X$ of $P$, and every cell of the subdivision is a translation of a matroid base polytope[^2]. Let $g : P \cap \mathbb Z^n \to \mathbb R$ be a function which induces $\mathcal X$. For each $s \in \bigsqcup_{k=1}^{n-1} \{0,\dots,R\}^k$, choose an affine functional $\ell_s : \mathbb R^{n-|s|} \to \mathbb R$ so that $\ell_{s'i} - \ell_{s'(i+1)}$ is generic for all strings $s'$ and integers $i$. For $1 \gg \epsilon_1 \gg \epsilon_2 \gg \dots \gg \epsilon_{n-1} > 0$, define the function $f : P \cap \mathbb Z^n \to \mathbb R$ by $$f(x) = g(x) + \sum_{k=1}^{n-1} \epsilon_k \ell_{x_1 \dots x_k}(x_{k+1},\dots,x_n).$$ Then $f$ induces a subdivision of $P$ which refines $\mathcal X$. Moreover, by Theorem [Theorem 16](#functiondesc){reference-type="ref" reference="functiondesc"}, the restriction of $f$ to each cell of $\mathcal X$ induces a unimodular triangulation. ◻ **Corollary 18**. *Every matroid independence polytope has a regular unimodular triangulation.* *Proof.* Each matroid independence polytope $P_{\mathcal{I}}$ is unimodularily equivalent to an integral generalized permutahedron: given a point $v =(v_1, \ldots v_n) \in P_{\mathcal{I}}$, let $\psi(v) = (v_0, v_1, \dots v_n) \in \mathbb{R}^{n+1}$, where $v_0 = r(E)-\sum_{i=1}^n v_i$. The map $\psi$ is unimodular and its image is an integral generalized permutahedron[^3]. We can apply our triangulation to $\psi(P_{\mathcal{I}})$ and then map this triangulation back to $P_{\mathcal{I}}$ to obtain a regular unimodular triangulation of the latter. ◻ # Acknowledgements The authors would like to thank the University of Vermont, the University of Washington, and the Simons Center for Geometry and Physics Workshop "Combinatorics and Geometry of Convex Polyhedra\" for excellent working conditions where part of this work was completed. The authors also thank Mateusz Michalek for noting some typos in an earlier draft of this work. The first author was supported by a Simons Collaboration Gift \# 854037 and an NSF Grant (DMS-2246967). [^1]:  This follows from the basic fact that given an invertible linear transformation $A$ and vectors $u,v \in \mathbb{R}^n$, we have $\langle u, v\rangle=0$ if and only if $\langle (A^T)^{-1}(u), A(v)\rangle=0$. [^2]:  This can be verified by appealing to the submodularity description of generalized permutahedra, Lemma [Lemma 5](#permint){reference-type="ref" reference="permint"}, and Theorem [Theorem 3](#gelmat){reference-type="ref" reference="gelmat"}. [^3]:  It is implicit in [@ardila2013lifted] that the independence polytope is unimodularily equivalent to a generalized permutahedron.

arxiv_math

--- abstract: | We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates. address: - Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea - School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea - Department of Mathematics, Yeungnam University, Gyeongsan 38541, Republic of Korea author: - Sun-Sig Byun - Kyeong Song - Yeonghun Youn title: Singular elliptic measure data problems with irregular obstacles --- [^1] # Introduction In this paper, we study obstacle problems related to nonlinear elliptic equations of the type $$\label{model} -\mathrm{div}\,A(Du) = \mu \quad \text{in } \Omega.$$ Here $\Omega \subset \mathbb{R}^{n}$ ($n \ge 2$) is a bounded domain and $\mu$ belongs to $\mathcal{M}_{b}(\Omega)$, that is, the space of all signed Borel measures with finite total mass on $\Omega$. In the following, we extend $\mu$ to $\mathbb{R}^{n}$ by letting $|\mu|(\mathbb{R}^{n}\setminus\Omega) = 0$. The continuous vector field $A: \mathbb{R}^{n} \to \mathbb{R}^{n}$ is $C^{1}$-regular on $\mathbb{R}^{n}\setminus\{0\}$ and satisfies the following $p$-growth and ellipticity assumptions: $$\label{growth} \left\{ \begin{aligned} |A(z)| + |\partial A(z)|(|z|^{2}+s^{2})^{\frac{1}{2}} &\le L(|z|^{2} +s^{2})^{\frac{p-1}{2}},\\ \nu (|z|^{2} + s^{2})^{\frac{p-2}{2}} |\xi|^{2} &\le \partial A(z) \xi \cdot \xi \\ \end{aligned} \right.$$ for every $z , \xi \in \mathbb{R}^{n}$, where $0<\nu \le L <\infty$ and $s \ge 0$ are fixed constants. Throughout this paper, we assume $$\label{p.bound} 1 < p \le 2-\frac{1}{n}.$$ Roughly speaking, the obstacle problem we are going to consider is [\[model\]](#model){reference-type="eqref" reference="model"} coupled with a unilateral constraint of the form $u \ge \psi$ a.e. in $\Omega$, with $\psi \in W^{1,p}(\Omega)$ being a given obstacle. Note that if $\mu \in W^{-1,p'}(\Omega)$, then our obstacle problem is represented as the following variational inequality: $$\label{opmu} \int_{\Omega} A(Du) \cdot D(\phi - u)\,dx \ge \int_{\Omega}(\phi - u)\, d\mu$$ for every $\phi \in u + W^{1,p}_{0}(\Omega)$ with $\phi \ge \psi$ a.e. in $\Omega$. Moreover, the existence and uniqueness of a weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} are well known consequences of the monotone operator theory [@KS00]. However, when $\mu \notin W^{-1,p'}(\Omega)$, we cannot consider such a variational inequality. In this case, a different notion of solutions to the obstacle problem will be given in [Definition](#def.sol) [Definition 1](#def.sol){reference-type="ref" reference="def.sol"} below. ## Nonlinear potential estimates Pointwise estimates for solutions to nonlinear elliptic measure data problems like [\[model\]](#model){reference-type="eqref" reference="model"} originated from [@KM92; @KM94]. More precisely, these papers fundamentally considered $A$-superharmonic functions and corresponding elliptic problems involving nonnegative measures, by employing the maximum principle approach, to show the necessity part of the Wiener criterion. Subsequently, in [@TW02], an alternative approach was employed to prove analogous results for subelliptic problems. Later, in the papers [@DM11AJM; @KM12JFA], pointwise estimates were shown for the case of signed Radon measures with finite total mass using perturbation arguments. By combining the findings from the aforementioned papers, we can provide the following summary: if $u$ solves [\[model\]](#model){reference-type="eqref" reference="model"}, and either $p>2-1/n$ or $\mu \ge 0$, then there holds $$\label{KM.result} |u(x_{0})| \le c\mathbf{W}_{1,p}^{\mu}(x_{0},R) + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}(|u|+Rs)\,dx$$ whenever $B_{R}(x_{0}) \Subset \Omega$ is a ball and the right-hand side is finite, where $$\mathbf{W}_{\beta,p}^{\mu}(x_{0},R) \coloneqq \int_{0}^{R}\left[\frac{|\mu|(B_{\rho}(x_{0}))}{\rho^{n-\beta p}}\right]^{\frac{1}{p-1}}\frac{d\rho}{\rho}, \qquad \beta >0,$$ is the nonlinear Wolff potential of $\mu$. Moreover, when both $\mu$ and $u$ are nonnegative in $B_{R}(x_{0})$, we also have the lower bound $$\label{KM.result2} \mathbf{W}^{\mu}_{1,p}(x_{0},R) \le cu(x_{0}),$$ which shows that the estimate [\[KM.result\]](#KM.result){reference-type="eqref" reference="KM.result"} via $\mathbf{W}^{\mu}_{1,p}$ is sharp. We also refer to [@KM18JEMS] for the extension of [\[KM.result\]](#KM.result){reference-type="eqref" reference="KM.result"} to the $p$-Laplace system with measure data, $p>2-1/n$. However, as far as we are concerned, no vectorial analog of [\[KM.result2\]](#KM.result2){reference-type="eqref" reference="KM.result2"} is available due to the lack of maximum principle. Later, pointwise estimates were also obtained for the gradient of solutions to [\[model\]](#model){reference-type="eqref" reference="model"}. The first result was proved in [@Min11JEMS], which asserts that pointwise gradient bounds, like those available for the Poisson equation, hold for [\[model\]](#model){reference-type="eqref" reference="model"} in the case $p=2$: $$|Du(x_{0})| \le c\mathbf{I}^{\mu}_{1}(x_{0},R) + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}(|Du|+s)\,dx,$$ where $$\mathbf{I}_{1}^{\mu}(x_{0},R) \coloneqq \int_{0}^{R}\frac{|\mu|(B_{\rho}(x_{0}))}{\rho^{n-1}}\frac{d\rho}{\rho}$$ is the truncated 1-Riesz potential of $\mu$. For the superquadratic case $p>2$, in [@DM11AJM] the following Wolff potential estimate $$\label{DM.result} |Du(x_{0})| \le c\mathbf{W}^{\mu}_{\frac{1}{p},p}(x_{0},R) + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}(|Du|+s)\,dx$$ was proved. See also [@KM12JFA] for "universal" potential estimates that interpolate [\[KM.result\]](#KM.result){reference-type="eqref" reference="KM.result"} and [\[DM.result\]](#DM.result){reference-type="eqref" reference="DM.result"}. Surprisingly, in contrast with the zero-order estimate [\[KM.result\]](#KM.result){reference-type="eqref" reference="KM.result"}, it was proved in [@DM10JFA; @KM13ARMA] that pointwise gradient estimates via Riesz potentials hold for nonlinear, possibly degenerate equations like [\[model\]](#model){reference-type="eqref" reference="model"}. More precisely, we have the following: if $u$ solves [\[model\]](#model){reference-type="eqref" reference="model"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $$\label{p.sola} p > 2-\frac{1}{n},$$ then it holds that $$\label{riesz-intro} |Du(x_{0})| \le c[\mathbf{I}^{\mu}_{1}(x_{0},R)]^{\frac{1}{p-1}} + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}(|Du|+s)\,dx,$$ whenever $B_{R}(x_{0}) \Subset \Omega$ and the right-hand side is finite. Moreover, [\[riesz-intro\]](#riesz-intro){reference-type="eqref" reference="riesz-intro"} improves [\[DM.result\]](#DM.result){reference-type="eqref" reference="DM.result"} when $p>2$. Note that, in light of [\[growth\]](#growth){reference-type="eqref" reference="growth"}, estimate [\[riesz-intro\]](#riesz-intro){reference-type="eqref" reference="riesz-intro"} can be rephrased as $$|A(Du(x_{0}))| \le c\mathbf{I}^{\mu}_{1}(x_{0},R) + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}|A(Du)|\,dx.$$ We also remark that the results in [@DM10JFA; @DM11AJM; @KM12JFA; @KM13ARMA] are concerned with SOLA (Solutions Obtained as Limits of Approximations) introduced in [@BG89], for which the lower bound [\[p.sola\]](#p.sola){reference-type="eqref" reference="p.sola"} is indispensable; see also the discussions after [Definition](#def.sol) [Definition 1](#def.sol){reference-type="ref" reference="def.sol"} below. Estimate [\[riesz-intro\]](#riesz-intro){reference-type="eqref" reference="riesz-intro"}, known to be the sharp gradient potential estimate for $p$-Laplacian type equations, was further extended to elliptic equations with nonstandard growth [@Ba15; @BaHa14; @BY17; @BY18] and parabolic $p$-Laplacian type equations [@KM13Pisa; @KM14ARMA] with $p>2-1/(n+1)$. Later in [@KM18JEMS], estimate [\[riesz-intro\]](#riesz-intro){reference-type="eqref" reference="riesz-intro"} was also established for measure data systems involving the $p$-Laplacian, $p \ge 2$. Additionally, in the case when the data $\mu$ possesses sufficient regularity to guarantee the existence of weak solutions, it is possible to derive Riesz potential type estimates for elliptic systems without a quasi-diagonal structure in the context of partial regularity, see [@BY19; @D22; @DS23JFA; @KM16JEP]. In the recent papers [@DZ; @NP20JFA; @NP23ARMA], potential estimates for [\[model\]](#model){reference-type="eqref" reference="model"} were investigated for the range [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}, where different notions of solutions, such as renormalized solutions or approximable solutions, should be considered. We refer to the recent papers [@DMOP99; @CM17NA] for more details about each notion of solutions. The papers [@DZ; @NP20JFA; @NP23ARMA] proposed new methods in obtaining comparison estimates, which address the difficulties coming from the lack of integrability of $Du$ and the failure of Sobolev-Poincaré type inequalities. In these papers, such difficulties are overcome by initially establishing Marcinkiewicz type estimates and then proving new reverse Hölder type estimates. Furthermore, a modified excess functional in the form of [\[mod.exs\]](#mod.exs){reference-type="eqref" reference="mod.exs"} below was employed. ## Main results Here we describe the formulation of our obstacle problem, $OP(\psi;\mu)$, and the concept of solutions used in this paper. As mentioned above, since $\mu$ does not in general belong to $W^{-1,p'}(\Omega)$, the variational inequality [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} is not available for $OP(\psi;\mu)$. In this paper, we consider *limits of approximating solutions* introduced in [@Sch12JFA]. For other several notions of solutions, see [@Sch12JFA Section 1.1] and related references therein. For each $k>0$, we consider the truncation operator $T_{k}:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$\label{truncation.op} T_{k}(t) \coloneqq \min\{k,\max\{t,-k\}\}, \qquad t \in \mathbb{R}.$$ Given a boundary data $g \in W^{1,p}(\Omega)$, we set $$\mathcal{T}^{1,p}_{g}(\Omega) \coloneqq \left\{ u: \Omega \rightarrow \mathbb{R} \mid T_{k}(u-g) \in W^{1,p}_{0}(\Omega) \text{ for every } k>0 \right\}.$$ It is well known that for any $u \in \mathcal{T}_{g}^{1,p}(\Omega)$, there exists a unique measurable map $Z_u: \Omega \to \mathbb{R}^n$ satisfying $$D[T_k(u)] = \chi_{\{|u| < k\}} Z_u \qquad \text{a.e. in }\Omega$$ for every $k>0$, see [@BBGGPV1995 Lemma 2.1]. If $u \in \mathcal{T}_{g}^{1,p}(\Omega) \cap W^{1,1}(\Omega)$, then $Z_u$ coincides with the weak derivative $Du$ of $u$. In this paper, we denote $Z_u$ by $Du$ for any $u \in \mathcal{T}^{1,p}_{g}(\Omega)$. **Definition 1**. Suppose that an obstacle $\psi \in W^{1,p}(\Omega)$, measure data $\mu \in \mathcal{M}_{b}(\Omega)$ and boundary data $g \in W^{1,p}(\Omega)$ with $(\psi-g)_{+} \in W^{1,p}_{0}(\Omega)$ are given. We say that a function $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ with $u \ge \psi$ a.e. in $\Omega$ is a limit of approximating solutions to the obstacle problem $OP(\psi;\mu)$ under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$, if there exist a sequence of functions $\{\mu_{k}\} \subset W^{-1,p'}(\Omega)\cap L^{1}(\Omega)$ with $$\label{muk.conv} \left\{ \begin{aligned} &\mu_{k} \overset{\ast}{\rightharpoonup} \mu \;\; \textrm{in } \mathcal{M}_{b}(\Omega), \\ &\limsup_{k\rightarrow\infty} |\mu_{k}|(B) \le |\mu|(\bar{B}) \quad \textrm{for every ball }B \subset \mathbb{R}^{n} \end{aligned} \right.$$ and weak solutions $u_{k} \in g + W^{1,p}_{0} (\Omega)$ with $u_{k} \ge \psi$ a.e. in $\Omega$ to the variational inequalities $$\int_{\Omega} A(Du_{k})\cdot D(\phi - u_{k}) \,dx \ge \int_{\Omega}(\phi - u_{k})\, d\mu_{k}$$ for every $\phi \in u_{k} + W^{1,p}_{0}(\Omega)$ with $\phi \ge \psi$ a.e. in $\Omega$, such that $$\label{uk.conv} \left\{ \begin{aligned} & u_k \to u && \text{a.e. in }\Omega,\\ & \int_{\Omega} |u_k - u|^{\gamma} \,dx \to 0 && \text{for every } 0< \gamma < \frac{n(p-1)}{n-p}, \\ & \int_{\Omega} |Du_k - Du|^{q} \,dx \to 0 && \text{for every } 0< q < \frac{n(p-1)}{n-1}. \end{aligned} \right.$$ The existence of limits of approximating solutions to $OP(\psi;\mu)$ was proved in [@Sch12JFA] by extending the classical approach in [@BG89]; see also [@SY] for a uniqueness result in the case $\mu \in L^{1}(\Omega)$. Now it is easy to see the role of [\[p.sola\]](#p.sola){reference-type="eqref" reference="p.sola"}: $$p > 2-\frac{1}{n} \;\; \Longleftrightarrow \;\; \frac{n(p-1)}{n-1}>1.$$ We indeed have $u \in W^{1,1}(\Omega)$ if and only if [\[p.sola\]](#p.sola){reference-type="eqref" reference="p.sola"} is in force. Note that, while the convergence property [\[uk.conv\]](#uk.conv){reference-type="eqref" reference="uk.conv"} is very similar as in the case of SOLA, limits of approximating solutions can be defined for the range [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"} as well. This is because we do not require $u$ itself to satisfy a distributional formulation. ### Gradient potential estimates Gradient potential estimates for $OP(\psi;\mu)$ in the range [\[p.sola\]](#p.sola){reference-type="eqref" reference="p.sola"} were first obtained in [@Sch12JFA], under the assumption that $$\psi \in W^{1,p}(\Omega)\cap W^{2,1}(\Omega) \quad \text{satisfies} \quad \mathcal{D}\Psi \coloneqq \mathrm{div}\,A(D\psi) \in L^{1}(\Omega).$$ Such a higher regularity assumption allows one to apply the methods in [@DM10JFA; @DM11AJM] to $OP(\psi;\mu)$, treating the obstacle and the measure in the same way. Indeed, the main estimates in [@Sch12JFA] involve Wolff potentials (when $p>2$) and Riesz potentials (when $2-1/n < p \le 2$) of $\mu$ and $\mathcal{D}\Psi$. We also refer to [@BCP21; @BSY2] for integrability and differentiability results for elliptic double obstacle problems with measure data, under similar assumptions on the double obstacles. In the recent paper [@BSY], a new form of gradient potential estimates for $OP(\psi;\mu)$ was proved under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.sola\]](#p.sola){reference-type="eqref" reference="p.sola"}, without any higher regularity assumptions on the obstacle. Moreover, Wolff potentials of $\mu$ appearing in [@Sch12JFA Theorem 4.3] were replaced by Riesz potentials: $$\begin{aligned} |Du(x_{0})|^{p-1} & \le c\mathbf{I}^{\mu}_{1}(x_{0},R) + c\left[\int_{0}^{R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}}|)\,dx\right)^{\frac{1}{m}}\frac{d\rho}{\rho}\right]^{\frac{m}{p'}}\\ & \quad + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{R}(x_{0})}}(|Du|+s)^{p-1}\,dx,\end{aligned}$$ where $m \coloneqq \max\{p',2\}$, and the function $\varphi^{*}(\cdot)$ is defined in [\[phistar\]](#phistar){reference-type="eqref" reference="phistar"} below. The approach in [@BSY] is based on an intrinsic linearization technique motivated from those in [@AKM18; @BCDKS18] (see also [@BDW20; @DKS12]), which enables one to treat both measure data and irregular obstacles simultaneously. We also note that all the estimates were actually formulated in terms of the natural quantity $A(Du)$. In this paper, we extend the gradient potential estimate in [@BSY Theorem 1.2] to the range [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}, as mentioned in [@BSY]. To this aim, we first extend the approaches in [@NP23ME; @PS22] to the setting of obstacle problems, by employing new test functions, to establish comparison estimates for $Du$. We then apply an analog of the alternative scheme in [@BSY] to linearize such estimates, which gives an intrinsic form of estimates for $A(Du)$. Note that, while $Du$ need not be an $L^{1}$-function, we have $A(Du) \in L^{1}(\Omega)$ by [\[uk.conv\]](#uk.conv){reference-type="eqref" reference="uk.conv"}. Here we set the exponent $$\label{def.kappa} \kappa \coloneqq \frac{(p-1)^{2}}{2}.$$ **Theorem 2**. *Let $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ be a limit of approximating solutions to the problem $OP(\psi;\mu)$ under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then there exists a constant $c \equiv c(n,p,\nu,L)$ such that the pointwise estimate $$\begin{aligned} |A(Du)(x_{0})| & \le c\mathbf{I}^{\mu}_{1}(x_{0},2R) + c\int_{0}^{2R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(x_{0})}|)\,dx\right)^{\frac{1}{p'}}\frac{d\rho}{\rho} \\ & \quad + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}|A(Du)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\end{aligned}$$ holds whenever $B_{2R}(x_{0}) \subset \Omega$ and $x_{0}\in\Omega$ is a Lebesgue point of $A(Du)$.* The above theorem can be actually obtained as a corollary of a more general result, which we state as follows. See [\[mod.exs\]](#mod.exs){reference-type="eqref" reference="mod.exs"} below for the definition of $\mathcal{P}_{\kappa,B_{\rho}(x_{0})}(\cdot)$. **Theorem 3**. *Let $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ be a limit of approximating solutions to the problem $OP(\psi;\mu)$ under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}.* - *If $$\label{mainthm.1.asmp1} \lim_{\rho \to 0} \left[\frac{|\mu|(B_{\rho}(x_{0}))}{\rho^{n-1}} + \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(x_{0})}|) \,dx \right)^{\frac{1}{p'}} \right] = 0$$ holds for a point $x_{0}\in\Omega$, then $$\label{vmo.x0} \lim_{\rho \to 0} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_0)}}|A(Du) - \mathcal{P}_{\kappa,B_{\rho}(x_0)}(A(Du))|^{\kappa} \,dx= 0.$$* - *If $$\label{mainthm.1.asmp2} \mathbf{I}_{1}^{\mu}(x_0,2R) + \int_{0}^{2R} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(x_{0})}|)\,dx \right)^{\frac{1}{p'}}\frac{d\rho}{\rho} < \infty$$ holds for a ball $B_{2R}(x_{0})\subset \Omega$, then the limit $$\label{Lebesgue.pt} A_{0} \coloneqq \lim_{\rho\rightarrow0}\mathcal{P}_{\kappa,B_{\rho}(x_{0})}(A(Du))$$ exists. Moreover, the estimate $$\begin{aligned} \label{mainest.1} \lefteqn{ |A_{0} - \mathcal{P}_{\kappa,B_{2R}(x_0)}(A(Du))| } \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}|A(Du) - \mathcal{P}_{\kappa,B_{2R}(x_0)}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c \int_{0}^{2R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(x_{0})}|)\,dx \right)^{\frac{1}{p'}}\frac{d\rho}{\rho}\end{aligned}$$ holds for a constant $c \equiv c(n,p,\nu,L)$.* - *Finally, if $x_{0}$ is a Lebesgue point of $A(Du)$, then the limit $A_{0}$ defined in [\[Lebesgue.pt\]](#Lebesgue.pt){reference-type="eqref" reference="Lebesgue.pt"} is equal to $A(Du)(x_{0})$.* **Remark 4**. In the proof of [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"}, we can also obtain the following $C^{1}$-regularity criterion (see for instance [@DM10CV Theorem 1] and [@KM13ARMA Theorem 4]): if $\mu \in L(n,1)$ locally in $\Omega$ and $A(D\psi)$ has Dini mean oscillation, which means that $$\int_{0}[\omega(\rho)]^{\frac{1}{p'}}\frac{d\rho}{\rho} < \infty, \quad \text{where} \quad \omega(\rho) \coloneqq \sup_{y \in \Omega}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(y)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(y)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(y)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(y)}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(y)}|)\,dx,$$ then $Du$ is continuous in $\Omega$. We also refer to [@KM14CV] for a different proof that avoids potentials. ### Zero-order potential estimates We can also obtain potential estimates for $u$, which extend the results in [@Sch12PM] to the case [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. For simplicity, we only state an analog of [Theorem](#pointwise.est) [Theorem 2](#pointwise.est){reference-type="ref" reference="pointwise.est"}. **Theorem 5**. *Let $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ be a limit of approximating solutions to $OP(\psi;\mu)$, with the Carathéodory vector field $A:\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ satisfying $$\left\{ \begin{aligned} |A(x,z)| & \le L(|z|^{2}+s^{2})^{\frac{p-1}{2}} \\ \nu(|z_{1}|^{2}+|z_{2}|^{2}+s^{2})^{\frac{p-2}{2}}|z_{1}-z_{2}|^{2} & \le (A(x,z_{1})-A(x,z_{2}))\cdot(z_{1}-z_{2}) \end{aligned} \right.$$ for every $z,z_{1},z_{2} \in \mathbb{R}^{n}$ and a.e. $x\in\Omega$. Assume that $p$ satisfies [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then there exists a constant $c\equiv c(n,p,\nu,L)$ such that the pointwise estimate $$\begin{aligned} |u(x_{0})| & \le c\mathbf{W}^{\mu}_{1,p}(x_{0},2R) + c\int_{0}^{2R}\left[\rho^{p}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}(|D\psi|+s)^{p}\,dx\right]^{\frac{1}{p}}\frac{d\rho}{\rho} \\ & \quad + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_{0})}}(|u|+Rs)^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\end{aligned}$$ holds whenever $B_{2R}(x_{0})\subset\Omega$, for a.e. $x_{0}\in\Omega$.* **Remark 6**. Note that comparison estimates between homogeneous obstacle problems and obstacle-free problems in [@Sch12PM Section 3.2] are valid for every $p>1$, since they are concerned with weak solutions. Thus, once we have the comparison estimate given in [Lemma](#Du-Dw1) [Lemma 20](#Du-Dw1){reference-type="ref" reference="Du-Dw1"} below, the above theorem can be proved by the arguments in [@Sch12PM Section 4], see also [@CS18; @DM11AJM]. Moreover, the $C^{0}$-regularity criterion in [@Sch12PM Theorem 4.6] can be also extended to the range [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}: $$\mu \in L\left(\frac{n}{p},\frac{1}{p-1}\right), \, D\psi \in L(n,1) \text{ locally in }\Omega \;\; \Longrightarrow \;\; u \text{ is continuous in } \Omega.$$ The organization of this paper is as follows. In the next section, we introduce some notations and preliminary materials. [Section](#sec.reg.homo) [3](#sec.reg.homo){reference-type="ref" reference="sec.reg.homo"} is devoted to regularity results for homogeneous obstacle problems and homogeneous equations. In [Section](#sec.comparison.est) [4](#sec.comparison.est){reference-type="ref" reference="sec.comparison.est"} and [Section](#sec.lin.comp) [5](#sec.lin.comp){reference-type="ref" reference="sec.lin.comp"}, we establish several comparison estimates between [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and the corresponding reference problems. Finally, in [Section](#sec.pf.thm1) [6](#sec.pf.thm1){reference-type="ref" reference="sec.pf.thm1"} we prove [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"}. # Preliminaries ## Notation We denote by $c$ a general constant greater than or equal to one; special occurrences will be denoted by $c_{*},c_{0}$, etc. The value of $c$ may vary from line to line. Specific dependencies of constants are denoted by parentheses, and we use the abbreviation $$\mathop{\mathrm{\mathtt{data}}}\coloneqq (n, p,\nu,L).$$ Additionally, we write $a \approx b$ if there is a constant $c \geq 1$ depending only on $\mathop{\mathrm{\mathtt{data}}}$ such that $c^{-1} a \leq b \leq c a$. For any $q > 1$, we denote its Hölder conjugate exponent by $q' \coloneqq q/(q-1)$. As usual, with $x = (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}$, we denote by $$B_{r}(x) \coloneqq \left\{ y\in\mathbb{R}^{n} : |y-x| < r \right\} \quad \text{and} \quad Q_{r}(x) \coloneqq \left\{ y\in\mathbb{R}^{n} : \sup_{1\le i \le n}|y_{i}-x_{i}| < r \right\}$$ the open ball and cube, respectively, with center $x$ and "radius" $r>0$. If there is no confusion, we omit the centers and simply write $B_{r} \equiv B_{r}(x)$ and $Q_{r} \equiv Q_{r}(x)$. Also, given a ball $B$ and a cube $Q$, we denote by $\gamma B$ and $\gamma Q$ the concentric ball and cube, respectively, with radius magnified by a factor $\gamma>0$. Unless otherwise stated, different balls or cubes in the same context are concentric. Moreover, when considering cubes, we identify $\mathbb{R}^{n} \equiv \mathbb{R}^{n-1}\times \mathbb{R}$, denoting each element as $x = (x',x_{n})$. We accordingly denote $$Q_{r}'(x') \coloneqq \left\{ y' \in \mathbb{R}^{n-1}:\sup_{1\le i \le n-1}|y_{i} - x_{i}| < r \right\}$$ so that $Q_{r}(x) = Q_{r}'(x') \times (x_{n}-r,x_{n}+r)$. The ($n$-dimensional) Lebesgue measure of a measurable set $S \subset \mathbb{R}^n$ is denoted by $|S|$. For an integrable map $f:S \to \mathbb{R}^{k}$, with $k \ge 1$ and $0<|S|<\infty$, we write $$(f)_{S} \coloneqq \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}f\,dx \coloneqq \frac{1}{|S|} \int_{S}f\,dx$$ to mean the integral average of $f$ over $S$. The oscillation of $f$ on $S$ is defined by $$\mathop{\mathrm{osc}}_{S}f \coloneqq \sup_{x,y\in S}|f(x)-f(y)|.$$ We shall identify a function $\mu \in L^{1}(\Omega)$ with a signed measure, by denoting $$|\mu|(S) = \int_{S}|\mu|\,dx \quad \textrm{for each measurable subset } S \subseteq \Omega,$$ and thereby identify $L^{1}(\Omega)$ with a subset of $\mathcal{M}_{b}(\Omega)$. We use the following short notations for the admissible sets of the problem $OP(\psi;\mu)$: given an open set $\mathcal{O} \subseteq \Omega$ and a function $g \in W^{1,p}(\mathcal{O})$ with $g \ge \psi$ a.e. in $\mathcal{O}$, we denote $$\begin{aligned} \mathcal{A}_{\psi}(\mathcal{O}) &\coloneqq \left\{ \phi \in W^{1,p}(\mathcal{O}): \phi \ge \psi \textrm{ a.e. in } \mathcal{O} \right\}, \\ \mathcal{A}_{\psi}^{g}(\mathcal{O}) &\coloneqq \left\{ \phi \in g + W^{1,p}_{0}(\mathcal{O}): \phi \ge \psi \textrm{ a.e. in } \mathcal{O} \right\}.\end{aligned}$$ ## Basic properties of the vector fields $V(\cdot)$ and $A(\cdot)$ Recall that the ellipticity assumption in [\[growth\]](#growth){reference-type="eqref" reference="growth"} implies the following monotonicity property: $$(A(z_{1})-A(z_{2}))\cdot(z_{1}-z_{2}) \approx (|z_{1}|^{2} + |z_{2}|^{2} + s^{2})^\frac{p-2}{2} |z_{1}-z_{2}|^{2}$$ for any $z_{1},z_{2} \in \mathbb{R}^{n}$. We now consider the auxiliary vector field $V \equiv V_{s}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined by $$V(z) \equiv V_{s}(z) \coloneqq (|z|^{2}+s^{2})^{\frac{p-2}{4}}z,\qquad z \in \mathbb{R}^{n}.$$ It is well known that $$\label{V} |V(z_{1})-V(z_{2})| \approx (|z_{1}|^2 + |z_{2}|^2 + s^{2})^\frac{p-2}{4} |z_{1}-z_{2}|$$ holds for any $z_{1},z_{2} \in \mathbb{R}^n$, where the implicit constant depends only on $p$. Specifically, in view of [\[V\]](#V){reference-type="eqref" reference="V"}, the vector field $V(\cdot)$ is naturally linked to the monotonicity of $A(\cdot)$. Namely, for any $z_{1},z_{2}\in \mathbb{R}^n$ there holds $$\label{mono.V} (A(z_{1})-A(z_{2}))\cdot(z_{1}-z_{2}) \approx |V(z_{1})-V(z_{2})|^{2}.$$ We further recall some properties of the vector field $A(\cdot)$; see [@AKM18 Lemma 2.1]. **Lemma 7**. *The following inequalities hold for every choice of $z, z_{1}, z_{2} \in \mathbb{R}^{n}$: $$\begin{aligned} |A(z)| + s^{p-1} & \approx |z|^{p-1} + s^{p-1} \approx (|z|+s)^{p-1}, \\ |A(z_{1})-A(z_{2})| & \approx (|z_{1}|^{2}+|z_{2}|^{2}+s^{2})^{\frac{p-2}{2}}|z_{1}-z_{2}|. \label{a.diff} \end{aligned}$$ In particular, $A(\cdot)$ is a locally bi-Lipschitz bijection, and it holds that $$|A(z_{1})-A(z_{2})| \leq c |z_{1}-z_{2}|^{p-1} \quad \textrm{when } 1 < p \le 2,$$ for some $c = c(\mathop{\mathrm{\mathtt{data}}})$.* We also recall several properties of shifted power functions which are useful in dealing with divergence type data. For a comprehensive introduction, see [@BDW20; @BCDKS18; @DFTW; @DKS12] and references therein. For each $a \geq 0$, we define the function $\varphi_{a}(\cdot)$ by $$\varphi_{a}(t) \coloneqq (a+s+t)^{p-2}t^{2}, \qquad t \ge 0.$$ We simply denote $\varphi_{0} \equiv \varphi$. Then $\varphi_{a}(\cdot)$ is an $N$-function, i.e., it has a right continuous, non-decreasing derivative $\varphi_{a}'(\cdot)$ which satisfies $\varphi_{a}'(0) = 0$ and $\varphi_{a}'(t) > 0$ for $t>0$. Moreover, a direct calculation shows that $$\label{shifted.ftn.o} \min\{p-1,1\} \le \frac{t\varphi_{a}''(t)}{\varphi_{a}'(t)} \le \max\{p-1,1\} \quad \text{and} \quad \min\{p,2\} \le \frac{t\varphi_{a}'(t)}{\varphi_{a}(t)} \le \max\{p,2\}$$ hold for any $t \ge 0$. In particular, $\eqref{shifted.ftn.o}_{2}$ implies that the family $\{\varphi_{a}\}_{a\ge0}$ satisfies the $\Delta_{2}$ and $\nabla_{2}$ conditions uniformly in $a$, i.e., $\varphi_{a}(2t) \approx \varphi_{a}(t)$ uniformly in $a,t\ge0$. Accordingly, we can consider the complementary $N$-function of $\varphi_{a}(\cdot)$ which is defined by $$\label{phistar} (\varphi_{a})^{*}(t) \coloneqq \sup_{\tau \ge 0}\left(\tau t - \varphi_{a}(\tau)\right), \qquad t \ge 0.$$ We indeed have $$(\varphi_{a})^{*}(t) \approx ((a+s)^{p-1} + t)^{p'-2}t^{2}, \qquad t \ge 0.$$ Shifted $N$-functions are especially useful when describing the monotonicity property of $A(\cdot)$: $$\label{mono.shift} \begin{aligned} (A(z_{1})-A(z_{2}))\cdot(z_{1}-z_{2}) & \approx |V(z_{1})-V(z_{2})|^{2} \\ & \approx \varphi_{|z_{1}|}(|z_{1}-z_{2}|) \approx (\varphi_{|z_{1}|})^{*}(|A(z_{1})-A(z_{2})|). \end{aligned}$$ We also note the following "shift change formula" $$\label{shift.change} \begin{aligned} \varphi_{|z_{1}|}(t) & \le c\varepsilon^{1-\max\{p',2\}}\varphi_{|z_{2}|}(t) + \varepsilon|V(z_{1})-V(z_{2})|^{2}, \\ (\varphi_{|z_{1}|})^{*}(t) & \le c\varepsilon^{1-\max\{p,2\}}(\varphi_{|z_{2}|})^{*}(t) + \varepsilon|V(z_{1})-V(z_{2})|^{2}, \end{aligned}$$ valid for any $z_{1},z_{2} \in \mathbb{R}^{n}$, $\varepsilon \in (0,1]$ and $t \ge 0$. ## A modified excess functional We recall the following inequality: if $S \subset \mathbb{R}^{n}$ is a measurable set with $0<|S|<\infty$ and $f\in L^{q}(S;\mathbb{R}^{k})$ for some $q \in [1,\infty)$, then we have $$\label{mean.min} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-(f)_{S}|^{q}\,dx\right)^{\frac{1}{q}} \le 2\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-z_{0}|^{q}\,dx\right)^{\frac{1}{q}} \qquad \forall \; z_{0} \in \mathbb{R}^{k}.$$ The quantity on the left-hand side of [\[mean.min\]](#mean.min){reference-type="eqref" reference="mean.min"} is often called an excess functional. Such a quantity naturally appears in various subjects including Campanato's theory. In view of [\[mean.min\]](#mean.min){reference-type="eqref" reference="mean.min"}, we consider, this time for any $q \in (0,\infty)$, the following "modified excess functional" $$\inf_{z_{0} \in \mathbb{R}^{k}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-z_{0}|^{q}\,dx \right)^{\frac{1}{q}}.$$ Then there exists a vector $\mathcal{P}_{q,S}(f) \in \mathbb{R}^{k}$ such that $$\label{mod.exs} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-\mathcal{P}_{q,S}(f)|^{q}\,dx\right)^{\frac{1}{q}} = \inf_{z_{0}\in\mathbb{R}^{k}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-z_{0}|^{q}\,dx\right)^{\frac{1}{q}}.$$ It is well known that $\mathcal{P}_{2,S}(f) = (f)_{S}$. However, even if $f \in L^{1}(S)$, [\[mean.min\]](#mean.min){reference-type="eqref" reference="mean.min"} may fail for $q<1$, see [@DJL92 Section .A]. We also note that $\mathcal{P}_{q,S}(f)$ is not in general uniquely determined, for instance, when $q<1$. In this paper, when referring to $\mathcal{P}_{q,S}(f)$, we take any possible value of it. We note that $$\begin{aligned} \label{av.min} |\mathcal{P}_{q,S}(f) - z_{0}| &= \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|\mathcal{P}_{q,S}(f) - z_{0}|^{q}\,dx\right)^{\frac{1}{q}} \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|\mathcal{P}_{q,S}(f)-f|^{q}\,dx\right)^{\frac{1}{q}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-z_{0}|^{q}\,dx\right)^{\frac{1}{q}} \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emS}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{S}}|f-z_{0}|^{q}\,dx\right)^{\frac{1}{q}}\end{aligned}$$ holds for a constant $c\equiv c(q)$, whenever $z_{0} \in \mathbb{R}^{k}$. Moreover, the following analog of Lebesgue's differentiation theorem holds (see for instance [@DVS84 Lemma 4.1]): If $f \in L^{q}_{{\operatorname{loc}}}(\mathbb{R}^{n})$, then $$\lim_{\rho \rightarrow 0}\mathcal{P}_{q,Q_{\rho}(x_{0})}(f) = f(x_{0}) \qquad \text{for a.e. } x_{0} \in \mathbb{R}^{n}.$$ # Regularity for reference problems {#sec.reg.homo} We first note a reverse Hölder type inequality for the following homogeneous obstacle problem: $$\label{homogeneous.obstacle} \left\{ \begin{aligned} \int_{\Omega} A(Dw_{1})\cdot D(\phi - w_{1})\,dx &\ge 0 \qquad \forall\;\phi \in \mathcal{A}^{w_{1}}_{\psi}(\Omega)\\ w_{1} & \ge \psi \qquad \textrm{a.e. in } \Omega. \end{aligned} \right.$$ **Lemma 8**. *Let $w_{1} \in \mathcal{A}_{\psi}(\Omega)$ be a weak solution to [\[homogeneous.obstacle\]](#homogeneous.obstacle){reference-type="eqref" reference="homogeneous.obstacle"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then, with $\kappa$ defined in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"}, there exists a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$ such that $$\begin{aligned} \lefteqn{\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}|V(Dw_{1})-V(z_{0})|^{2}\,dx} \nonumber \\ & \le c(\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}|A(Dw_{1})-A(z_{0})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\end{aligned}$$ holds for every $z_{0},\xi_{0} \in \mathbb{R}^{n}$, whenever $2Q \Subset \Omega$.* *Proof.* By following the proof of [@BSY Lemma 3.3], with considering cubes instead of balls, we have $$\begin{aligned} \lefteqn{\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}|V(Dw_{1})-V(z_{0})|^{2}\,dx} \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}|V(Dw_{1})-V(z_{0})|^{2\sigma}\,dx\right)^{\frac{1}{\sigma}} + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx \\ \overset{\eqref{mono.shift}}&{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}[(\varphi_{|z_{0}|})^{*}(|A(Dw_{1})-A(z_{0})|)]^{\sigma}\,dx\right)^{\frac{1}{\sigma}} + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx,\end{aligned}$$ for any $\sigma \in (0,1)$, where $c\equiv c(\mathop{\mathrm{\mathtt{data}}},\sigma)$. We then observe that $$\label{exp.down} t \mapsto [((\varphi_{|z_{0}|})^{*})^{-1}(t^{1/\sigma})]^{\kappa} \text{ is convex for } \sigma>0 \text{ small enough.}$$ Hence, we apply Young's inequality to the first integral on the right-hand side, thereby getting the desired estimate. ◻ We next examine some various regularity estimates for the homogeneous equation $$\label{limiting.equation} -\mathrm{div}\,A(Dv)=0 \quad \textrm{in }\Omega.$$ The following reverse Hölder's inequality can be found in [@Min07 Lemma 3.2]. **Lemma 9**. *Let $v\in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution to [\[limiting.equation\]](#limiting.equation){reference-type="eqref" reference="limiting.equation"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then for any $\sigma \in (0,1)$ there exists a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}},\sigma)$ such that $$\label{revhol.v} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}|V(Dv)-V(z_{0})|^{2}\,dx \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4em2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{2Q}}|V(Dv)-V(z_{0})|^{2\sigma}\,dx\right)^{\frac{1}{\sigma}}$$ holds for every $z_{0} \in \mathbb{R}^{n}$, whenever $2Q \Subset \Omega$.* We then recall a gradient Hölder regularity result for [\[limiting.equation\]](#limiting.equation){reference-type="eqref" reference="limiting.equation"}. We state it as in [@AKM18 Theorem 3.3] with a slight modification. **Lemma 10**. *Let $v \in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution to [\[limiting.equation\]](#limiting.equation){reference-type="eqref" reference="limiting.equation"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then $v \in C^{1,\alpha}_{\mathrm{loc}}(\Omega)$ for some $\alpha \equiv \alpha(\mathop{\mathrm{\mathtt{data}}}) \in (0,1)$. Moreover, for every $t>0$, there exists a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}},t)$ such that $$\sup_{\varepsilon Q}(|Dv|+s) \le \frac{c}{(1-\varepsilon)^{n/t}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}(|Dv|+s)^{t}\,dx\right)^{\frac{1}{t}}$$ holds for every cube $Q \Subset \Omega$ and $\varepsilon \in (0,1)$. Finally, there exists a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$ such that $$|Dv(x_{1})-Dv(x_{2})| \le c \varepsilon^{\alpha}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q}}|Dv-(Dv)_{Q}|\,dx$$ holds for every cube $Q\Subset \Omega$ and $x_{1},x_{2} \in \varepsilon Q$ with $\varepsilon \in (0,1/2]$.* We recall [\[mod.exs\]](#mod.exs){reference-type="eqref" reference="mod.exs"} to further establish a decay estimate for a modified excess functional of $A(Dv)$. **Lemma 11**. *Let $v\in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution to [\[limiting.equation\]](#limiting.equation){reference-type="eqref" reference="limiting.equation"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then, with $\kappa$ given in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"}, there exists an exponent $\alpha_{A} \equiv \alpha_{A}(\mathop{\mathrm{\mathtt{data}}}) \in (0,1)$ such that $$\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-\mathcal{P}_{\kappa,Q_{\rho}}(A(Dv))|^{\kappa}\,dx \right)^{\frac{1}{\kappa}} \le c\left(\frac{\rho}{R}\right)^{\alpha_{A}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dv)-\mathcal{P}_{\kappa,Q_{R}}(A(Dv))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}$$ holds whenever $Q_{\rho} \subset Q_{R} \Subset \Omega$ are concentric cubes, where $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$.* *Proof.* We may assume $\rho \le R/2$ without loss of generality, and recall the following $L^{1}$-excess decay estimate that follows from [@BSY Theorem 4.4]: $$\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-(A(Dv))_{Q_{\rho}}|\,dx \le c\left(\frac{\rho}{R}\right)^{\alpha_{A}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}|A(Dv)-(A(Dv))_{Q_{R/2}}|\,dx.$$ Using this, we have $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-\mathcal{P}_{\kappa,Q_{\rho}}(A(Dv))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \le \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-(A(Dv))_{Q_{\rho}}|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} }\\ & \le \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-(A(Dv))_{Q_{\rho}}|\,dx \\ & \le c\left(\frac{\rho}{R}\right)^{\alpha_A}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}|A(Dv)-(A(Dv))_{Q_{R/2}}|\,dx \\ \overset{\eqref{mean.min}}&{\le} c\left(\frac{\rho}{R}\right)^{\alpha_{A}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}|A(Dv)-A(z_{0})|\,dx \\ & \le c\left(\frac{\rho}{R}\right)^{\alpha_{A}}((\varphi_{|z_{0}|})^{*})^{-1}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}(\varphi_{|z_{0}|})^{*}(|A(Dv)-A(z_{0})|)\,dx\right) \\ \overset{\eqref{mono.shift},\eqref{revhol.v}}&{\le} c\left(\frac{\rho}{R}\right)^{\alpha_{A}}((\varphi_{|z_{0}|})^{*})^{-1}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}[(\varphi_{|z_{0}|})^{*}(|A(Dv)-A(z_{0})|)]^{\sigma}\,dx\right)^{\frac{1}{\sigma}}\right] \end{aligned}$$ whenever $z_{0} \in \mathbb{R}^{n}$ and $\sigma \in (0,1)$, where $c\equiv c(\mathop{\mathrm{\mathtt{data}}},\sigma)$. Then, recalling [\[exp.down\]](#exp.down){reference-type="eqref" reference="exp.down"}, the desired estimate follows by applying Jensen's inequality and then taking infimum with respect to $z_{0}$. ◻ # Basic comparison estimates {#sec.comparison.est} In this section, we derive several comparison estimates under an additional assumption $$\label{regular} \mu \in W^{-1,p'}(\Omega) \cap L^{1}(\Omega), \qquad u \in \mathcal{A}^{g}_{\psi}(\Omega).$$ This assumption will be eventually removed in [Section](#sec.pf.thm1) [6](#sec.pf.thm1){reference-type="ref" reference="sec.pf.thm1"} below. Here we introduce the mixed norm $$\|f\|_{L^{s_{2}}_{x'}L^{s_{1}}_{x_{n}}(Q_{\rho}(x_{0}))} \coloneqq \left(\int_{Q_{\rho}'(x_{0}')}\left(\int_{x_{0,n}+(-\rho,\rho)}|f(x',x_{n})|^{s_{1}}\,dx_{n}\right)^\frac{s_{2}}{s_{1}}\,dx'\right)^{\frac{1}{s_{2}}}$$ and its averaged version $$\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.545em \|{f}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{f}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{f}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{f}\|}}_{L^{s_2}_{x'}L^{s_1}_{x_n}(Q_\rho(x_0))}=\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}'(x_{0}')}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}'(x_{0}')}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}'(x_{0}')}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}'(x_{0}')}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emx_{0,n}+(-\rho,\rho)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{x_{0,n}+(-\rho,\rho)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{x_{0,n}+(-\rho,\rho)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{x_{0,n}+(-\rho,\rho)}} |f(x',x_n)|^{s_1}\,dx_n\right)^{\frac{s_2}{s_1}} dx'\right)^{\frac{1}{s_2}}.$$ In [@BSY], the starting point of various comparison estimates and further linearization was the weighted type energy estimate given in [@BSY Lemma 5.1]. It is valid for [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"} as well, but the proof of subsequent comparison estimates in [@BSY Section 5] do not work in the case [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. We therefore develop a slightly different approach motivated from those in [@BSY; @NP23ARMA; @PS22], at some stage dividing the cases $$\label{divide.case} \frac{3n-2}{2n-1} < p \le 2-\frac{1}{n} \qquad \text{and} \qquad 1< p \le \frac{3n-2}{2n-1}.$$ ## Some technical results The following lemma is analogous to [@NP23ARMA Lemma 2.1], see also the proof of [@KM18JEMS Theorem 4.1]. Note that the estimate in [@NP23ARMA Lemma 2.1] is concerned with the case $k=0$ only, as $u-k$ also solves equation [\[model\]](#model){reference-type="eqref" reference="model"} for other values of $k$. Since this is not the case for obstacle problems, we have to consider general $k$ in the estimate. Also, due to the obstacle constraint, we need different choices of test functions. **Lemma 12**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then for any $\varepsilon>0$, $k\in\mathbb{R}$ and any nonnegative $\eta \in C^{\infty}_{0}(\Omega)$, we have $$\begin{aligned} \lefteqn{ \int_{\Omega}\left|D\left[(1+|u-k|)^{\frac{p-1-\varepsilon}{p}}\eta\right]\right|^{p}\,dx } \\ & \le \frac{c}{\varepsilon^{p}}\int_{\Omega}(1+|u-k|)^{(\varepsilon+1)(p-1)}|D\eta|^{p}\,dx + \frac{c}{\varepsilon}\int_{\Omega}\eta^{p}d|\mu| + c\int_{\Omega}s^{p}\eta^{p}\,dx\end{aligned}$$ for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$.* *Proof.* We first test [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} with $$\begin{aligned} \phi & = u + \frac{1}{\varepsilon}[1-(1+(u-k)_{-})^{-\varepsilon}]\eta^{p} \ge u \ge \psi, \end{aligned}$$ to have $$\begin{aligned} \int_{\{u \le k\}}\frac{-A(Du)\cdot Du}{(1+|u-k|)^{\varepsilon+1}}\eta^{p}\,dx & \ge -\frac{p}{\varepsilon}\int_{\Omega}A(Du)\cdot[1-(1+(u-k)_{-})^{-\varepsilon}]\eta^{p-1}D\eta\,dx \\ & \quad + \frac{1}{\varepsilon}\int_{\Omega}[1-(1+(u-k)_{-})^{-\varepsilon}]\eta^{p}\,d\mu\end{aligned}$$ and so $$\begin{aligned} \label{test.minus} \lefteqn{ \int_{\{u \le k\}}\frac{(|Du|+s)^{p}\eta^{p}}{(1+|u-k|)^{\varepsilon+1}}\,dx } \nonumber \\ & \le \frac{c}{\varepsilon}\int_{\Omega}(|Du|+s)^{p-1}\eta^{p-1}|D\eta|\,dx + \frac{c}{\varepsilon}\int_{\Omega}\eta^{p}\,d|\mu| + c\int_{\Omega}s^{p}\eta^{p}\,dx.\end{aligned}$$ We next test [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} with $$\begin{aligned} \phi & = u + \frac{1}{\varepsilon}(1+(u-k)_{+})^{-\varepsilon}\eta^{p} \ge u \ge \psi, \end{aligned}$$ and estimate in a similar way to obtain $$\begin{aligned} \label{test.plus} \lefteqn{ \int_{\{u \ge k\}}\frac{(|Du|+s)^{p}\eta^{p}}{(1+|u-k|)^{\varepsilon+1}}\,dx } \nonumber \\ & \le \frac{c}{\varepsilon}\int_{\Omega}(|Du|+s)^{p-1}\eta^{p-1}|D\eta|\,dx + \frac{c}{\varepsilon}\int_{\Omega}\eta^{p}\,d|\mu| + c\int_{\Omega}s^{p}\eta^{p}\,dx.\end{aligned}$$ Combining [\[test.minus\]](#test.minus){reference-type="eqref" reference="test.minus"} and[\[test.plus\]](#test.plus){reference-type="eqref" reference="test.plus"}, we arrive at $$\begin{aligned} \int_{\Omega}\frac{(|Du|+s)^{p}\eta^{p}}{(1+|u-k|)^{\varepsilon+1}}\,dx \le \frac{c}{\varepsilon}\int_{\Omega}(|Du|+s)^{p-1}\eta^{p-1}|D\eta|\,dx + \frac{c}{\varepsilon}\int_{\Omega}\eta^{p}\,d|\mu| + c\int_{\Omega}s^{p}\eta^{p}\,dx.\end{aligned}$$ Applying Young's inequality to the first term on the right-hand side, and then recalling the identity $$\begin{aligned} \lefteqn{ D\left((1+|u-k|)^{\frac{p-1-\varepsilon}{p}}\eta\right) }\\ & = \eta\frac{p-1-\varepsilon}{p}(1+|u-k|)^{-\frac{1+\varepsilon}{p}}\text{sign}(u-k)Du + (1+|u-k|)^{\frac{p-1-\varepsilon}{p}}D\eta,\end{aligned}$$ we have the desired estimate. ◻ [Lemma](#caccio.u) [Lemma 12](#caccio.u){reference-type="ref" reference="caccio.u"} gives a reverse Hölder type estimate for $u$ and a mixed norm estimate for $Du$; their proofs are exactly the same as in [@NP23ARMA Section 2]. They will play a crucial role in [Lemma](#Du-Dw1) [Lemma 20](#Du-Dw1){reference-type="ref" reference="Du-Dw1"} below. **Lemma 13**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $1<p<n$. Then for any $$0 < q_{1} < q < \frac{n(p-1)}{n-p},$$ $k \in \mathbb{R}$ and $\sigma \in (0,1)$, we have $$\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}(|u-k|+rs)^{q}\,dx\right)^{\frac{1}{q}} \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}(|u-k|+rs)^{q_{1}}\,dx\right)^{\frac{1}{q_{1}}} + c\left[\frac{|\mu|(Q_{r})}{r^{n-p}}\right]^{\frac{1}{p-1}}$$ for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}},q,q_{1},\sigma)$, whenever $Q_{\sigma r} \subset Q_{r} \subset \Omega$ are concentric cubes.* **Lemma 14**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $1<p<n$. Then for any exponents $q_{1}, s_{1}, s_{2}$ satisfying $$0<q_{1} < \frac{n(p-1)}{n-p}, \quad \frac{p-1}{n-1} < s_{1} < p, \quad 0 < s_{2} < \frac{s_{1}(n-1)(p-1)}{s_{1}(n-1)-p+1},$$ and any $k \in \mathbb{R}$, we have $$\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.545em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}_{L^{s_{2}}_{x'}L^{s_{1}}_{x_{n}}(Q_{\sigma r})} \le c\left[\frac{|\mu|(Q_{r})}{r^{n-1}}\right]^{\frac{1}{p-1}} + \frac{c}{r}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}(|u-k|+rs)^{q_{1}}\,dx\right)^{\frac{1}{q_{1}}}$$ for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}},q_{1},s_{1},s_{2},\sigma)$, whenever $Q_{\sigma r} \subset Q_{r} \subset \Omega$ are concentric cubes.* For a fixed cube $Q_{4R} \subset \Omega$, we first consider the homogeneous obstacle problem $$\label{homogeneous} \left\{ \begin{aligned} \int_{Q_{4R}} A(Dw_{1})\cdot D(\phi - w_{1})\,dx & \ge 0 \quad \forall \; \phi \in \mathcal{A}^{u}_{\psi}(Q_{4R}), \\ w_{1} & \ge \psi \quad \textrm{a.e. in } Q_{4R},\\ w_{1} & = u \quad \textrm{on } \partial Q_{4R}. \end{aligned} \right.$$ We obtain a preliminary comparison estimate between [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"}. ## Comparison with [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"} in the case $\eqref{divide.case}_{1}$ In this case, we extend the approaches in [@BSY; @PS22]. We first obtain the following lemma, which generalizes [@BSY Lemma 5.1]. **Lemma 15**. *Let $u\in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$, and let $w_{1} \in \mathcal{A}^{u}_{\psi}(Q_{4R})$ be as in [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"}. Then $$\label{weighted.comparison} \int_{Q_{4R}}\frac{|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\,dx \le c\frac{h^{(1-\gamma)(1-\xi)}}{(1-\gamma)(\xi-1)}|\mu|(Q_{4R})$$ holds for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$, whenever $h>0$, $\xi >1$ and $\gamma \in [0,1)$.* *Proof.* For any positive constants $\varepsilon$ and $\tilde{\varepsilon}$ satisfying $\varepsilon > \tilde{\varepsilon}^{1-\gamma}$, consider the function $$\zeta_{\pm} \coloneqq \min\left\{1,\max\left\{\frac{(u-w_{1})_{\pm}^{1-\gamma}-\tilde{\varepsilon}^{1-\gamma}}{\varepsilon-\tilde{\varepsilon}^{1-\gamma}},0\right\}\right\}.$$ We immediately see that ${\rm supp}\, \zeta_{\pm} = Q_{4R} \cap \{(u-w_{1})_{\pm} \ge \tilde{\varepsilon} \}$ and $$D\zeta_{\pm} = \frac{1-\gamma}{\varepsilon-\tilde{\varepsilon}^{1-\gamma}}\chi_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}(u-w_{1})_{\pm}^{-\gamma}D(u-w_{1})_{\pm},$$ $$\text{where}\;\; \mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon) \coloneqq Q_{4R} \cap \left\{\tilde{\varepsilon} < (u-w_{1})_{\pm} < \varepsilon^{\frac{1}{1-\gamma}} \right\}.$$ We also consider the function $$\eta_{\pm} \coloneqq \frac{1}{\xi-1}\left[1-\left(1+\frac{(u-w_{1})_{\pm}^{1-\gamma}}{h^{1-\gamma}}\right)^{1-\xi}\right].$$ The mean value theorem, applied to the function $t \mapsto t^{1-\xi}/(1-\xi)$, gives $$\eta_{\pm}(x) = \left(\frac{(u-w_{1})_{\pm}(x)}{h}\right)^{1-\gamma}(\tilde{\eta}_{\pm}(x))^{-\xi} \quad \text{for some } 1< \tilde{\eta}_{\pm}(x) < 1+\left(\frac{(u-w_{1})(x)}{h}\right)^{1-\gamma}.$$ Then, since $$\label{admissible} \tilde{\varepsilon}^{\gamma}(u-w_{1})_{\pm}^{1-\gamma} \le (u-w_{1})_{\pm} \;\; \text{in} \;\; {\rm supp}\,\zeta_{\pm},$$ we observe that $$\begin{aligned} u- \tilde{\varepsilon}^{\gamma}h^{1-\gamma}\eta_{+}\zeta_{+} & = u - \tilde{\varepsilon}^{\gamma}(u-w_{1})_{+}^{1-\gamma}\tilde{\eta}_{+}^{-\xi}\zeta_{+} \ge u - (u-w_{1})_{+} = \min\{u,w_{1}\}, \\ w_{1} - \tilde{\varepsilon}^{\gamma}h^{1-\gamma}\eta_{-}\zeta_{-} & = w_{1} - \tilde{\varepsilon}^{\gamma}(u-w_{1})_{-}^{1-\gamma}\tilde{\eta}_{-}^{-\xi}\zeta_{-} \ge w_{1} - (u-w_{1})_{-} = \min\{u,w_{1}\}\end{aligned}$$ a.e. in $Q_{4R}$. From this and [\[admissible\]](#admissible){reference-type="eqref" reference="admissible"}, we see that the functions $$u \pm \tilde{\varepsilon}^{\gamma}h^{1-\gamma}\eta_{\mp}\zeta_{\mp} \quad \text{and} \quad w_{1} \pm \tilde{\varepsilon}^{\gamma}h^{1-\gamma}\eta_{\pm}\zeta_{\pm}$$ belong to the admissible set $\mathcal{A}^{u}_{\psi}(Q_{4R})$. We now test [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} with $\phi \equiv u \pm \tilde{\varepsilon}^{\gamma}h^{1-\gamma}\eta_{\mp}\zeta_{\mp}$ to get $$\begin{aligned} \lefteqn{ \int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{|u-w_{1}|^{-\gamma}A(Du)\cdot (Du-Dw_{1})}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\zeta_{\pm}\,dx }\\ & \quad + \int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{h^{(1-\gamma)(1-\xi)}}{\varepsilon-\tilde{\varepsilon}^{1-\gamma}}\eta_{\pm}|u-w_{1}|^{-\gamma}A(Du)\cdot(Du-Dw_{1})\,dx \le \frac{h^{(1-\gamma)(1-\xi)}}{(1-\gamma)(\xi-1)}|\mu|(Q_{4R}) .\end{aligned}$$ In a similar way, testing [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"} with $\phi \equiv w_{1} \pm \tilde{\varepsilon}^{\gamma}h^{1-\gamma}h^{1-\gamma}\eta_{\pm}\zeta_{\pm}$, we have $$\begin{aligned} \lefteqn{ -\int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{|u-w_{1}|^{-\gamma}A(Dw_{1})\cdot (Du-Dw_{1})}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\zeta_{\pm}\,dx }\\ & \quad - \int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{h^{(1-\gamma)(1-\xi)}}{\varepsilon-\tilde{\varepsilon}^{1-\gamma}}\eta_{\pm}|u-w_{1}|^{-\gamma}A(Dw_{1})\cdot(Du-Dw_{1})\,dx \le 0.\end{aligned}$$ Combining the above two displays and using [\[mono.V\]](#mono.V){reference-type="eqref" reference="mono.V"}, we see that $$\begin{aligned} \lefteqn{ \int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\zeta_{\pm}\,dx }\\ & \quad + \int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{h^{(1-\gamma)(1-\xi)}}{\varepsilon-\tilde{\varepsilon}^{1-\gamma}}\eta_{\pm}|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}\,dx \le c\frac{h^{(1-\gamma)(1-\xi)}}{(1-\gamma)(\xi-1)}|\mu|(Q_{4R})\end{aligned}$$ holds for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$. In particular, since the second term on the left-hand side is nonnegative, we have $$\int_{\mathcal{A}_{\pm}(\tilde{\varepsilon},\varepsilon)}\frac{|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\zeta_{\pm}\,dx \le c\frac{h^{(1-\gamma)(1-\xi)}}{(1-\gamma)(\xi-1)}|\mu|(Q_{4R}).$$ As $\tilde{\varepsilon} \rightarrow 0$, recalling the definition of $\zeta_{\pm}$, we arrive at $$\int_{Q_{4R}}\frac{|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}}{(h^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi}}\min\left\{1,\frac{|u-w_{1}|^{1-\gamma}}{\varepsilon}\right\}\,dx \le c\frac{h^{(1-\gamma)(1-\xi)}}{(1-\gamma)(\xi-1)}|\mu|(Q_{4R})$$ with $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$. Thus, letting $\varepsilon\rightarrow0$ in the last display gives [\[weighted.comparison\]](#weighted.comparison){reference-type="eqref" reference="weighted.comparison"}. ◻ **Lemma 16**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $\eqref{divide.case}_{1}$, and let let $w_{1} \in \mathcal{A}^{u}_{\psi}(Q_{4R})$ be as in [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"}. Then for any $$\label{qrange} q \in \left(\frac{n}{2n-1}, \frac{n(p-1)}{n-1} \right),$$ the estimate $$\begin{aligned} \label{1st.comparison} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{q}\,dx\right)^{\frac{1}{q}} + \frac{1}{R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{q}\,dx\right)^{\frac{1}{q}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{1}{p-1}} + c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}\end{aligned}$$ holds for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}},q)$.* *Proof.* Given a constant $\varepsilon>0$, define $\mathcal{B}_{\varepsilon} \coloneqq Q_{4R} \cap \{|u-w_{1}| > \varepsilon \}$. We set the exponent $$\label{def.beta} \beta \coloneqq \frac{np(1-q)}{n-q} \;\; \Longleftrightarrow \;\; \frac{\beta q}{(1-q)(p-\beta)} = \frac{n}{n-1}$$ and define $$M_{\varepsilon} \coloneqq \frac{p}{p-\beta}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\left|D\left[(u-w_{1})^{\frac{p-\beta}{p}}\right]\right|\chi_{\mathcal{B}_{\varepsilon}}\,dx.$$ Note that $M_{\varepsilon} < \infty$ since $|u-w_{1}| > \varepsilon$ in $\mathcal{B}_{\varepsilon}$. We start by estimating $$\begin{aligned} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{q}\chi_{\mathcal{B}_{\varepsilon}}\,dx & = \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\left(|u-w_{1}|^{-\frac{\beta}{p}}|Du-Dw_{1}|\right)^{q}|u-w_{1}|^{\frac{\beta q}{p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx \\ & \le M_{\varepsilon}^{q}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{\beta q}{(1-q)p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{1-q}\end{aligned}$$ Here, recalling [\[def.beta\]](#def.beta){reference-type="eqref" reference="def.beta"}, we apply Sobolev-Poincaré inequality to have $$\begin{aligned} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{\beta q}{(1-q)p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\left(|u-w_{1}|^{\frac{p-\beta}{p}}-\varepsilon^{\frac{p-\beta}{p}}\right)_{+}^{\frac{\beta q}{(1-q)(p-\beta)}} + c\varepsilon^{\frac{\beta q}{(1-q)p}} \\ & \le c(RM_{\varepsilon})^{\frac{\beta q}{(1-q)(p-\beta)}} + c\varepsilon^{\frac{\beta q}{(1-q)p}}.\end{aligned}$$ Then, letting $$\label{def.he} h_{\varepsilon} \coloneqq (RM_{\varepsilon})^{\frac{p}{p-\beta}} + \varepsilon,$$ we arrive at $$\label{u-w.M} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{\beta q}{(1-q)p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx \le ch_{\varepsilon}^{\frac{\beta q}{(1-q)p}}$$ and $$\label{Du-Dw.M} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{q}\chi_{\mathcal{B}_{\varepsilon}}\,dx \le cM_{\varepsilon}^{q}h_{\varepsilon}^{\frac{\beta q}{p}}$$ for some $c \equiv c(\mathop{\mathrm{\mathtt{data}}},q).$ We now estimate $M_{\varepsilon}$. Recalling the inequality (see for instance [@Min11Milan (9.39)]) $$|Du-Dw_{1}| \le c|V(Du)-V(Dw_{1})|^{\frac{2}{p}} + c(|Du|+s)^{\frac{2-p}{2}}|V(Du)-V(Dw_{1})|,$$ we directly have $$\begin{aligned} \label{est.I1.I2} M_{\varepsilon} & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{-\frac{\beta}{p}}|V(Du)-V(Dw_{1})|^{\frac{2}{p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx \nonumber \\ & \quad + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{-\frac{\beta}{p}}(|Du|+s)^{\frac{2-p}{2}}|V(Du)-V(Dw_{1})|\chi_{\mathcal{B}_{\varepsilon}}\,dx \nonumber \\ & \eqqcolon cI_{1} + cI_{2}.\end{aligned}$$ Then, with $\xi_{1} > 1$ to be chosen, we use Hölder's inequality and [\[weighted.comparison\]](#weighted.comparison){reference-type="eqref" reference="weighted.comparison"} to obtain $$\begin{aligned} \label{I1.start} I_{1} & = \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\left(\frac{|u-w_{1}|^{-\beta}|V(Du)-V(Dw_{1})|^{2}}{(h_{\varepsilon}^{1-\beta}+|u-w_{1}|^{1-\beta})^{\xi_{1}}}\right)^{\frac{1}{p}}(h_{\varepsilon}^{1-\beta}+|u-w_{1}|^{1-\beta})^{\frac{\xi_{1}}{p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx \nonumber \\ & \le \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\frac{|u-w_{1}|^{-\beta}|V(Du)-V(Dw_{1})|^{2}}{(h_{\varepsilon}^{1-\beta}+|u-w_{1}|^{1-\beta})^{\xi_{1}}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{1}{p}} \nonumber \\ & \qquad \cdot \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(h_{\varepsilon}^{1-\beta}+|u-w_{1}|^{1-\beta})^{\frac{\xi_{1}}{p-1}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{p-1}{p}} \nonumber \\ & \le ch_{\varepsilon}^{\frac{(1-\beta)(1-\xi_{1})}{p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p}}\left\{ h_{\varepsilon}^{\frac{(1-\beta)\xi_{1}}{p}} + \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{(1-\beta)\xi_{1}}{p-1}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{p-1}{p}} \right\}.\end{aligned}$$ Since $$q < \frac{n(p-1)}{n-1} \;\; \Longleftrightarrow \;\; \frac{1-\beta}{p-1} < \frac{\beta q}{(1-q)p},$$ we can choose $\xi_{1} >1$, depending only on $\mathop{\mathrm{\mathtt{data}}}$ and $q$, such that $$\frac{(1-\beta)\xi_{1}}{p-1} < \frac{\beta q}{(1-q)p}.$$ Then, applying Hölder's inequality, we obtain $$\begin{aligned} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{(1-\beta)\xi_{1}}{p-1}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{p-1}{p}} & \le \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{\beta q}{(1-q)p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{(1-q)(1-\beta)\xi_{1}}{\beta q}} \nonumber \\ \overset{\eqref{u-w.M}}&{\le} ch_{\varepsilon}^{\frac{(1-\beta)\xi_{1}}{p}}\end{aligned}$$ for some $c \equiv c(\mathop{\mathrm{\mathtt{data}}},q)$. Plugging this into [\[I1.start\]](#I1.start){reference-type="eqref" reference="I1.start"} gives the following estimate of $I_{1}$: $$\label{I1.est} I_{1} \le ch_{\varepsilon}^{\frac{1-\beta}{p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p}}.$$ On the other hand, with $\gamma = 2\beta/p \in (0,1)$ and $\xi_{2} >1$ to be chosen, a similar calculation as in [\[I1.start\]](#I1.start){reference-type="eqref" reference="I1.start"} gives $$\begin{aligned} \label{I2.start} I_{2} & = \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\left(\frac{|u-w_{1}|^{-\gamma}|V(Du)-V(Dw_{1})|^{2}}{(h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi_{2}}}\,dx\right)^{\frac{1}{2}} \nonumber \\ & \qquad \qquad \cdot (h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\frac{\xi_{2}}{2}}(|Du|+s)^{\frac{2-p}{2}}\chi_{\mathcal{B}_{\varepsilon}}\,dx \nonumber \\ & \le ch_{\varepsilon}^{\frac{(1-\gamma)(1-\xi_{2})}{2}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{2}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi_{2}}(|Du|+s)^{2-p}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{1}{2}}.\end{aligned}$$ We then apply Hölder's inequality to the integral appearing on the right-hand side as follows: $$\begin{aligned} \label{I2.rhs} \lefteqn{ \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\xi_{2}}(|Du|+s)^{2-p}\chi_{\mathcal{B}_{\varepsilon}}\,dx } \nonumber \\ & \le \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\frac{\xi_{2} q}{q-2+p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{q-2+p}{q}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}\end{aligned}$$ and observe that $$q < \frac{n(p-1)}{n-1} \;\; \Longleftrightarrow \;\; \frac{(1-\gamma)q}{q-2+p} = \frac{(2n-1)q-n}{q-2+p}\frac{q}{n-q} < \frac{nq}{n-q}.$$ Thus, we can choose the constant $\xi_{2} > 1$, depending only on $\mathop{\mathrm{\mathtt{data}}}$ and $q$, such that $$\frac{(1-\gamma)\xi_{2} q}{q-2+p} < \frac{nq}{n-q}.$$ We note that [\[def.beta\]](#def.beta){reference-type="eqref" reference="def.beta"} implies $\beta q/[(1-q)p] = nq/(n-q)$. Then Hölder's inequality and [\[u-w.M\]](#u-w.M){reference-type="eqref" reference="u-w.M"} imply $$\begin{aligned} \label{I2.rhs2} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(h_{\varepsilon}^{1-\gamma}+|u-w_{1}|^{1-\gamma})^{\frac{\xi_{2} q}{q-2+p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{q-2+p}{q}} } \nonumber \\ & \le ch_{\varepsilon}^{(1-\gamma)\xi_{2}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{nq}{n-q}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{(1-\gamma)\xi_{2}(n-q)}{nq}} \le ch_{\varepsilon}^{(1-\gamma)\xi_{2}}.\end{aligned}$$ Connecting [\[I2.rhs\]](#I2.rhs){reference-type="eqref" reference="I2.rhs"} and [\[I2.rhs2\]](#I2.rhs2){reference-type="eqref" reference="I2.rhs2"} to [\[I2.start\]](#I2.start){reference-type="eqref" reference="I2.start"}, $I_{2}$ is estimated as $$\label{I2.est} I_{2} \le ch_{\varepsilon}^{\frac{p-2\beta}{2p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{2}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{2q}}.$$ We note that $$\lim_{\varepsilon\rightarrow0}M_{\varepsilon} = 0 \;\; \Longrightarrow \;\; Du=Dw_{1} \text{ a.e. in }Q_{4R},$$ and in this case there is nothing to prove. Hence, we may assume that $\inf_{\varepsilon}M_{\varepsilon} > 0$, which implies that there exists a constant $\varepsilon_{0} > 0$ such that $\varepsilon < (RM_{\varepsilon})^{p/(p-\beta)}$ whenever $\varepsilon \in (0,\varepsilon_{0})$. In turn, [\[def.he\]](#def.he){reference-type="eqref" reference="def.he"} gives $$\label{hepsilonest} h_{\varepsilon} < 2(RM_{\varepsilon})^{\frac{p}{p-\beta}} \qquad \forall \; \varepsilon \in (0,\varepsilon_{0}).$$ With such a value of $\varepsilon$, we connect [\[I1.est\]](#I1.est){reference-type="eqref" reference="I1.est"}, [\[I2.est\]](#I2.est){reference-type="eqref" reference="I2.est"}, and [\[hepsilonest\]](#hepsilonest){reference-type="eqref" reference="hepsilonest"} to [\[est.I1.I2\]](#est.I1.I2){reference-type="eqref" reference="est.I1.I2"}, and then apply Young's inequality to have $$\begin{aligned} \lefteqn{ M_{\varepsilon} \le ch_{\varepsilon}^{\frac{1-\beta}{p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p}} + ch_{\varepsilon}^{\frac{p-2\beta}{2 p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{2}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{2q}} } \\ & \le cM_{\varepsilon}^{\frac{1-\beta}{p-\beta}}R^{\frac{1-\beta}{p-\beta}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p}} + cM_{\varepsilon}^{\frac{p-2\beta}{2(p-\beta)}}R^{\frac{p-2\beta}{2(p-\beta)}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{2}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{2q}} \\ & \le \frac{1}{2}M_{\varepsilon} + cR^{\frac{1-\beta}{p-1}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{p-\beta}{p(p-1)}} + cR^{\frac{p-2\beta}{p}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{p-\beta}{p}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{(p-\beta)(2-p)}{pq}}\end{aligned}$$ and therefore $$M_{\varepsilon}^{\frac{p}{p-\beta}} \le cR^{\frac{p(1-\beta)}{(p-\beta)(p-1)}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p-1}} + cR^{\frac{p-2\beta}{p-\beta}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}.$$ This with [\[Du-Dw.M\]](#Du-Dw.M){reference-type="eqref" reference="Du-Dw.M"} implies $$\begin{aligned} \label{sdsdsd} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{q}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{1}{q}} \le cM_{\varepsilon}h_{\varepsilon}^{\frac{\beta}{p}} \overset{\eqref{hepsilonest}}{\le} cR^{\frac{\beta}{p-\beta}}M_{\varepsilon}^{\frac{p}{p-\beta}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{1}{p-1}} + c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}.\end{aligned}$$ In a similar way, this time using [\[u-w.M\]](#u-w.M){reference-type="eqref" reference="u-w.M"}, we also have $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\frac{\beta q}{(1-q)p}}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{(1-q)p}{\beta q}} \le ch_{\varepsilon} \overset{\eqref{hepsilonest}}{\le} c(RM_{\varepsilon})^{\frac{p}{p-\beta}} } \\ & \le cR^{\frac{p}{p-1}}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]^{\frac{1}{p-1}} + cR^{2}\left[\frac{|\mu|(Q_{4R})}{(4R)^{n}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}.\end{aligned}$$ Then Hölder's inequality and some elementary manipulations lead to $$\begin{aligned} \label{szszsz} \lefteqn{ \frac{1}{R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{q}\chi_{\mathcal{B}_{\varepsilon}}\,dx\right)^{\frac{1}{q}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{1}{p-1}} + c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{q}\,dx\right)^{\frac{2-p}{q}}.\end{aligned}$$ Combining [\[sdsdsd\]](#sdsdsd){reference-type="eqref" reference="sdsdsd"} and [\[szszsz\]](#szszsz){reference-type="eqref" reference="szszsz"}, and then letting $\varepsilon \rightarrow 0$, we conclude with the desired estimate. ◻ ## Comparison with [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"} in the case $\eqref{divide.case}_{2}$ In this case, the arguments in the proof of [@NP23ARMA Lemma 2.5] can be applied to $OP(\psi;\mu)$, which gives: **Lemma 17**. *Let $u\in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $\eqref{divide.case}_{2}$, and let $w_{1} \in \mathcal{A}^{u}_{\psi}(Q_{4R})$ be as in [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"}. Then $$\begin{aligned} \label{Du-Dw.mixed} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + \frac{1}{R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]^{\frac{1}{p-1}} + c\left[\frac{|\mu|(Q_{4R})}{(4R)^{n-1}}\right]\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.545em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}_{L^{\frac{(p-1)(2-p)}{3-p}}_{x'}L^{2-p}_{x_{n}}(Q_{4R})}\end{aligned}$$ holds for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}})$, where $\kappa$ is as in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"}.* *Proof.* By using a standard scaling argument, we may assume that $Q_{4R} \equiv Q_{1}(0) \equiv Q_{1}$ and $$^{\frac{1}{p-1}} + [|\mu|(Q_{1})]\||Du|+s\|_{L^{\frac{(p-1)(2-p)}{3-p}}_{x'}L^{2-p}_{x_{n}}(Q_{1})}^{2-p} \le 1.$$ For any $k>0$, we recall the truncation operator $T_{k}$ given in [\[truncation.op\]](#truncation.op){reference-type="eqref" reference="truncation.op"}. Testing [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"} with $\phi \equiv u + T_{2k}(w_{1}-u)$ and $\phi \equiv w_{1} - T_{2k}(w_{1}-u)$, respectively, we have $$\int_{Q_{1} \cap \{|u-w_{1}|<2k\}}|V(Du)-V(Dw_{1})|^{2}\,dx \le ck$$ for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. Then, by following the proof of [@NP23ARMA Lemma 2.5], we have the desired estimate. ◻ ## Reverse Hölder type inequalities for $OP(\psi;\mu)$ To proceed further, we need certain reverse Hölder type inequalities for $Du$. Once we have [Lemma](#reverse.holder.w1) [Lemma 8](#reverse.holder.w1){reference-type="ref" reference="reverse.holder.w1"}, [Lemma](#revhol.u) [Lemma 13](#revhol.u){reference-type="ref" reference="revhol.u"}, [Lemma](#ccp.mixed) [Lemma 14](#ccp.mixed){reference-type="ref" reference="ccp.mixed"} and the above two comparison estimates, we can obtain the following two lemmas, see [@NP23ME Lemma 2.1] and [@NP23ARMA Lemma 2.6 and Remark 2.7] for each case. We note that [Lemma](#reverse.holder.w1) [Lemma 8](#reverse.holder.w1){reference-type="ref" reference="reverse.holder.w1"}, [Lemma](#revhol.u) [Lemma 13](#revhol.u){reference-type="ref" reference="revhol.u"} and [Lemma](#ccp.mixed) [Lemma 14](#ccp.mixed){reference-type="ref" reference="ccp.mixed"} also hold in the case $p>2-1/n$, which along with [@BSY Lemma 5.2] give a new proof of [@BSY Lemma 5.3]. **Lemma 18**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $\eqref{divide.case}_{1}$. Then for any $q$ as in [\[qrange\]](#qrange){reference-type="eqref" reference="qrange"}, $\varepsilon \in (0,q]$ and $\sigma \in (0,1)$, we have $$\begin{aligned} \label{revhol.u.1} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma r}}}(|Du|+s)^{q}\,dx\right)^{\frac{1}{q}} & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}(|Du|+s)^{\varepsilon}\,dx\right)^{\frac{1}{\varepsilon}} \nonumber \\ & \quad + c\left[\frac{|\mu|(Q_{r})}{r^{n-1}}\right]^{\frac{1}{p-1}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}\varphi^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}},q,\varepsilon,\sigma)$, whenever $Q_{\sigma r} \subset Q_{r} \subset \Omega$ are concentric cubes and $\xi_{0} \in \mathbb{R}^{n}$.* **Lemma 19**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $\eqref{divide.case}_{2}$. With $\kappa$ given in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"}, let $$\theta \in \left(0,\frac{2\kappa(p-1)}{(2-p)(p-\kappa)} \right)$$ and define $s_{1}$ and $s_{2}$ by $$\frac{1}{2-p} = \frac{\theta}{\kappa} + \frac{1-\theta}{s_{1}}, \qquad \frac{3-p}{(p-1)(2-p)} = \frac{\theta}{\kappa} + \frac{1-\theta}{s_{2}}.$$ Then $$2-p < s_{1} < p, \quad s_{1} > s_{2} > \frac{(p-1)(2-p)}{3-p}, \quad s_{2} < \frac{s_{1}(n-1)(p-1)}{s_{1}(n-1) -p+1}.$$ Moreover, for any $\varepsilon \in (0,\kappa]$ and $\sigma \in (0,1)$, we have $$\begin{aligned} \label{revhol.mixed} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.545em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.495em \|{|Du|+s}\|}}_{L^{s_{2}}_{x'}L^{s_{1}}_{x_{n}}(Q_{\sigma r})} & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}(|Du|+s)^{\varepsilon}\,dx\right)^{\frac{1}{\varepsilon}} \nonumber \\ & \quad + c\left[\frac{|\mu|(Q_{r})}{r^{n-1}}\right]^{\frac{1}{p-1}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r}}}\varphi^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}},s_{1},s_{2},\sigma,\varepsilon)$, whenever $Q_{\sigma r} \subset Q_{r} \subset \Omega$ are concentric cubes and $\xi_{0} \in \mathbb{R}^{n}$.* From [\[1st.comparison\]](#1st.comparison){reference-type="eqref" reference="1st.comparison"}, [\[Du-Dw.mixed\]](#Du-Dw.mixed){reference-type="eqref" reference="Du-Dw.mixed"}, [\[revhol.u.1\]](#revhol.u.1){reference-type="eqref" reference="revhol.u.1"}, and [\[revhol.mixed\]](#revhol.mixed){reference-type="eqref" reference="revhol.mixed"}, we conclude with the following comparison estimate. **Lemma 20**. *Let $u$ and $w_{1}$ be the weak solutions to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"}, respectively, under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then for any $q,\varepsilon \in (0,\kappa]$ and $\xi_{0} \in \mathbb{R}^{n}$, we have $$\begin{aligned} \label{final.comparison} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{q}\,dx\right)^{\frac{1}{q}} + \frac{1}{R}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|u-w_{1}|^{q}\,dx\right)^{\frac{1}{q}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]^{\frac{1}{p-1}} + c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{\varepsilon}\,dx\right)^{\frac{2-p}{\varepsilon}} \nonumber \\ & \quad + c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}\varphi^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\right)^{\frac{2-p}{p}}\end{aligned}$$ for a constant $c\equiv c(\mathop{\mathrm{\mathtt{data}}},q,\varepsilon)$.* ## Comparison with obstacle-free problems Next, we consider the two Dirichlet problems: $$\left\{ \begin{aligned} -\mathrm{div}\, A(Dw_{2}) &= -\mathrm{div}\, A(D\psi) &\textrm{in } & Q_{2R}, \\ w_{2} & = w_{1} & \textrm{on } & \partial Q_{2R}, \end{aligned} \right.$$ and $$\label{homoeq} \left\{ \begin{aligned} -\mathrm{div} \, A(Dv) & =0 & \textrm{in } & Q_{R}, \\ v & = w_{2} & \textrm{on } & \partial Q_{R}. \end{aligned} \right.$$ The following comparison estimate can be proved in a completely similar way as in [@BSY Lemma 5.8], with the help of [\[exp.down\]](#exp.down){reference-type="eqref" reference="exp.down"}. **Lemma 21**. *Let $w_{1}$, $w_{2}$, and $v$ be defined as above, under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $p>1$. Then we have $$\begin{aligned} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|V(Dw_{1})-V(Dv)|^{2}\,dx & \le \varepsilon (\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})-A(z_{0})|^{\sigma}\,dx\right)^{\frac{1}{\sigma}}\right] \nonumber \\ & \quad + c\varepsilon^{1-\max\{p,2\}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\end{aligned}$$ for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$, whenever $z_{0}, \xi_{0} \in \mathbb{R}^{n}$ and $\varepsilon, \sigma \in (0,1]$.* We then establish a comparison estimate between $A(Dw_{1})$ and $A(Dv)$. **Lemma 22**. *Let $w_{1}$ and $v$ be as in [\[homogeneous\]](#homogeneous){reference-type="eqref" reference="homogeneous"} and [\[homoeq\]](#homoeq){reference-type="eqref" reference="homoeq"}, respectively, under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} with $1<p \le 2$. Then, with $\kappa$ given in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"}, we have $$\begin{aligned} \label{ADw1.ADv} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}} \nonumber \\ & \le \varepsilon\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})-A(z_{0})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c_{\varepsilon} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\varphi^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for any $\varepsilon \in (0,1)$ and $z_{0},\xi_{0}\in\mathbb{R}^{n}$, where $c_{\varepsilon} \equiv c_{\varepsilon}(\mathop{\mathrm{\mathtt{data}}},\varepsilon)$ is proportional to some negative power of $\varepsilon$.* *Proof.* We first estimate $$\begin{aligned} \lefteqn{ (\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] \le \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}(\varphi_{|z_{0}|})^{*}(|A(Dw_{1})-A(Dv)|)\,dx }\\ \overset{\eqref{shift.change}}&{\le} c\gamma_{1}^{-1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}(\varphi_{|Dw_{1}|})^{*}(|A(Dw_{1})-A(Dv)|)\,dx + \gamma_{1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|V(Dw_{1})-V(z_{0})|^{2}\,dx \\ \overset{\eqref{mono.shift}}&{\le} c\gamma_{1}^{-1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|V(Dw_{1})-V(Dv)|^{2}\,dx + \gamma_{1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|V(Dw_{1})-V(z_{0})|^{2}\,dx\end{aligned}$$ for any $\gamma_{1} \in (0,1)$. We then apply [Lemma](#w1.v.comparison) [Lemma 21](#w1.v.comparison){reference-type="ref" reference="w1.v.comparison"} and [Lemma](#reverse.holder.w1) [Lemma 8](#reverse.holder.w1){reference-type="ref" reference="reverse.holder.w1"} to estimate each term on the right-hand side, thereby obtaining $$\begin{aligned} \lefteqn{ (\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] }\\ & \le c\gamma_{1}^{-1}\gamma_{2}(\varphi_{|W_{R}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})-A(z_{0})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] \\ & \quad+ c\gamma_{1}^{-1}\gamma_{2}^{-1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx \\ & \quad + c\gamma_{1}(\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}|A(Dw_{1})-A(z_{0})|^{\kappa}\,dx \right)^{\frac{1}{\kappa}}\right] \\ & \quad + c\gamma_{1}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{2R}}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\end{aligned}$$ for any $\gamma_{2} \in (0,1)$. Choosing $\gamma_{2} = \gamma_{1}^{2}$, we arrive at $$\begin{aligned} (\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] & \le c\gamma_{1}(\varphi_{|z_{0}|})^{*}\left[\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})-A(z_{0})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}\right] \\ & \quad + c\gamma_{1}^{-3}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(\varphi_{|z_{0}|})^{*}(|A(D\psi)-A(\xi_{0})|)\,dx.\end{aligned}$$ Finally, in the proof of [@BSY Lemma 5.8], it is shown that $t \mapsto [((\varphi_{|z_{0}|})^{*})^{-1}(t)]^{p'}$ is quasi-convex. Therefore, with a suitable choice of $\gamma_{1}$, we apply Jensen's inequality to the last term and then use the fact that $t^{p'} \leq c \varphi^{*}(t)$ for any $1<p\le2$ and some $c=c(p)$, in order to conclude with [\[ADw1.ADv\]](#ADw1.ADv){reference-type="eqref" reference="ADw1.ADv"}. ◻ # Comparison estimates under alternatives {#sec.lin.comp} In this section, we linearize the comparison estimates between [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and [\[homoeq\]](#homoeq){reference-type="eqref" reference="homoeq"} established in the previous section. Throughout this section, we keep assuming [\[regular\]](#regular){reference-type="eqref" reference="regular"} to ensure the existence of weak solutions to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"}. We then fix a cube $$\label{ball.parameter} Q_{4MR} \equiv Q_{4MR}(x_{0}) \Subset \Omega \quad \textrm{with} \quad M\ge8 \quad \textrm{and} \quad R\le1,$$ where $M$ is a free parameter whose relevant value will be determined later in this section. ## The two-scales degenerate alternative We first consider the case when $$\label{deg} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \ge \theta\left[ |\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))| + s^{p-1} \right]$$ holds for another free parameter $\theta \in (0,1)$, where $\kappa$ and $M$ are as in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"} and [\[ball.parameter\]](#ball.parameter){reference-type="eqref" reference="ball.parameter"}, respectively. The values of $M$ and $\theta$ will be determined in the next section, and their specific values do not affect the results in this section. We observe that $$\begin{aligned} \label{jj} \lefteqn{\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} \overset{\eqref{a.diff}}{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|A(Du)|+s^{p-1})^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}|A(Du)-\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|+s^{p-1}\right] \nonumber \\ & \le cM^{\frac{2n}{\kappa}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|+s^{p-1}\right] \nonumber \\ \overset{\eqref{deg}}&{\le} cM^{\frac{2n}{\kappa}}\left(1+\frac{1}{\theta}\right)\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \end{aligned}$$ holds for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. Using this, we establish the following comparison estimate. **Lemma 23**. *Let $\theta \in (0,1)$ be such that [\[deg\]](#deg){reference-type="eqref" reference="deg"} holds with $M \ge 8$ as in [\[ball.parameter\]](#ball.parameter){reference-type="eqref" reference="ball.parameter"}. Then we have $$\begin{aligned} \label{lincomp.homo.deg} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Du)-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le \varepsilon M^{\frac{2n}{\kappa}}\left(1+\frac{1}{\theta}\right)\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c_{\varepsilon}\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right] + c_{\varepsilon}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}\varphi^{*}(|A(D\psi)-A(\xi_{0})|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for any $\xi_{0} \in \mathbb{R}^{n}$ and $\varepsilon \in (0,1]$, where $c_{\varepsilon} \equiv c_{\varepsilon}(\mathop{\mathrm{\mathtt{data}}},\varepsilon)$ is proportional to some negative power of $\varepsilon$.* *Proof.* We use [\[final.comparison\]](#final.comparison){reference-type="eqref" reference="final.comparison"} and Young's inequality to have $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Du)-A(Dw_{1})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \overset{\eqref{a.diff}}{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} }\\ & \le c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right] + c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]^{p-1}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{2-p}{\kappa}} \\ & \le \varepsilon \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\varepsilon^{\frac{p-2}{p-1}}\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]\end{aligned}$$ for any $\varepsilon \in (0,1]$. Combining this estimate with [\[ADw1.ADv\]](#ADw1.ADv){reference-type="eqref" reference="ADw1.ADv"} and using [\[jj\]](#jj){reference-type="eqref" reference="jj"}, we obtain [\[lincomp.homo.deg\]](#lincomp.homo.deg){reference-type="eqref" reference="lincomp.homo.deg"}. ◻ ## The two-scales non-degenerate alternative Here we consider the case when [\[deg\]](#deg){reference-type="eqref" reference="deg"} fails, namely $$\label{ndeg} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} < \theta\left[ |\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))| + s^{p-1} \right]$$ holds for a number $\theta \in (0,1)$. In the following, we denote $$\label{lambda.choice} \lambda \coloneqq \left|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))\right|^{\frac{1}{p-1}} + s.$$ Then we have the following: **Lemma 24**. *Let $\lambda$ be as in [\[lambda.choice\]](#lambda.choice){reference-type="eqref" reference="lambda.choice"}. For every $M \ge 8$ as in [\[ball.parameter\]](#ball.parameter){reference-type="eqref" reference="ball.parameter"}, there exists a number $\theta \equiv \theta(n,M)$ such that if [\[ndeg\]](#ndeg){reference-type="eqref" reference="ndeg"} is in force, then $$\label{mean.bound} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} \le c\lambda^{p-1},\quad \forall\,\,\sigma \in [1/M,4M]$$ holds for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$.* *Proof.* Using [\[ndeg\]](#ndeg){reference-type="eqref" reference="ndeg"}, we have $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} \overset{\eqref{a.diff}}{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}(|A(Du)|+s^{p-1})^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left|\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) -\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))\right| \\ & \quad + c\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))| + s^{p-1}\right] \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \\ & \quad + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|+s^{p-1}\right] \\ & \le c\left[\left(\frac{M}{\sigma}\right)^{n}+M^{2n}\right]^{\frac{1}{\kappa}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \\ & \quad + c\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|+s^{p-1}\right] \\ & \le c(1+M^{2n}\theta)^{\frac{1}{\kappa}}\left[|\mathcal{P}_{\kappa,Q_{R/M}}(A(Du))|+s^{p-1}\right].\end{aligned}$$ Then we choose the constant $\theta$ so small that $$\label{theta.M} M^{2n}\theta \le 1$$ in order to conclude with [\[mean.bound\]](#mean.bound){reference-type="eqref" reference="mean.bound"}. ◻ We now prove a counterpart of [Lemma](#a.comp.deg) [Lemma 23](#a.comp.deg){reference-type="ref" reference="a.comp.deg"} after fixing the values of $\theta$ and $M$. **Lemma 25**. *It is possible to determine $\theta$ and $M$ as functions of $\mathop{\mathrm{\mathtt{data}}}$ such that if [\[ndeg\]](#ndeg){reference-type="eqref" reference="ndeg"} is in force, then there holds $$\begin{aligned} \label{lincomp.homo.ndeg} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$.* In the proof of [Lemma](#a.comp.ndeg) [Lemma 25](#a.comp.ndeg){reference-type="ref" reference="a.comp.ndeg"}, we will distinguish two cases, making use of another free parameter $\sigma_{1} \in (0,1)$. The first one is when the following inequality holds: $$\label{measure.small} \left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi) - (A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}} \le \sigma_{1}\lambda^{p-1}.$$ The second one is when the above inequality fails; that is, $$\label{measure.large} \lambda^{p-1} < \frac{1}{\sigma_{1}}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + \frac{1}{\sigma_{1}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}.$$ The value of $\sigma_{1}$ will be determined in [Lemma](#grad.bound.lem) [Lemma 26](#grad.bound.lem){reference-type="ref" reference="grad.bound.lem"} below. ### Proof of [Lemma](#a.comp.ndeg) [Lemma 25](#a.comp.ndeg){reference-type="ref" reference="a.comp.ndeg"} in the first case [\[measure.small\]](#measure.small){reference-type="eqref" reference="measure.small"} and determination of $\sigma_{1}$ {#proof-of-lemma-a.comp.ndeg-in-the-first-case-measure.small-and-determination-of-sigma_1} **Lemma 26**. *There exists a choice of the parameters $$M \equiv M(\mathop{\mathrm{\mathtt{data}}}) \ge 8 \qquad \text{and} \qquad \sigma_{1} \equiv \sigma_{1}(\mathop{\mathrm{\mathtt{data}}},M) \in (0,1)$$ such that, if $\theta \equiv \theta(n,M)$ is the constant determined in [Lemma](#scale.change) [Lemma 24](#scale.change){reference-type="ref" reference="scale.change"} and [\[ndeg\]](#ndeg){reference-type="eqref" reference="ndeg"} is in force, then the following bounds hold: $$\label{v.bound} \frac{\lambda}{c} \le |Dv|+s \;\; \textrm{in } Q_{4R/M} \quad \textrm{and} \quad |Dv|+s \le c \lambda \;\; \textrm{in } Q_{R/2},$$ with constants $c$ depending only on $\mathop{\mathrm{\mathtt{data}}}$.* *Proof.* We first prove the upper bound. Using [Lemma](#grad.hol.v) [Lemma 10](#grad.hol.v){reference-type="ref" reference="grad.hol.v"} and [Lemma](#lin.comp.w1.v) [Lemma 22](#lin.comp.w1.v){reference-type="ref" reference="lin.comp.w1.v"}, we have $$\begin{aligned} \left[\sup_{Q_{R/2}}(|Dv|+s)\right]^{(p-1)\kappa} \overset{\eqref{a.diff}}&{\le} c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}(|A(Dv)|+s^{p-1})^{\kappa}\,dx \\ & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}(|A(Dw_{1})|+s^{p-1})^{\kappa}\,dx + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx \\ & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|A(Dw_{1})|+s^{p-1})^{\kappa}\,dx \\ & \quad + cM^{\frac{n\kappa}{p'}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{\kappa}{p'}}.\end{aligned}$$ We then apply [\[final.comparison\]](#final.comparison){reference-type="eqref" reference="final.comparison"}, [\[mean.bound\]](#mean.bound){reference-type="eqref" reference="mean.bound"}, and [\[measure.small\]](#measure.small){reference-type="eqref" reference="measure.small"} in order to estimate $$\begin{aligned} \label{ADw.mean} \lefteqn{ \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|A(Dw_{1})|+s^{p-1})^{\kappa}\,dx \overset{\eqref{a.diff}}{\le} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Dw_{1}|+s)^{(p-1)\kappa}\,dx } \nonumber \\ & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{(p-1)\kappa}\,dx + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{(p-1)\kappa}\,dx \nonumber \\ & \le c\lambda^{(p-1)\kappa} + cM^{(n-1)\kappa}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{\kappa} \nonumber \\ & \quad + cM^{(n-1)(p-1)\kappa}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{(p-1)\kappa}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{2-p}{\kappa}} \nonumber \\ & \quad + cM^{\frac{2n-p}{p'}\kappa}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{(p-1)\kappa}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{(2-p)\kappa}{p'}} \nonumber \\ & \le c\left[ 1 + M^{n-1}\sigma_{1} + \left(M^{n-1}\sigma_{1}\right)^{p-1} + M^{\frac{2n-p}{p'}}\sigma_{1} \right]^{\kappa}\lambda^{(p-1)\kappa}.\end{aligned}$$ Combining the above two estimates and using [\[measure.small\]](#measure.small){reference-type="eqref" reference="measure.small"}, we arrive at $$\begin{aligned} \left[\sup_{Q_{R/2}}(|Dv|+s)\right]^{(p-1)\kappa} & \le c\left[1+M^{n-1}\sigma_{1} + \left(M^{n-1}\sigma_{1}\right)^{p-1} + M^{\frac{2n-p}{p'}}\sigma_{1} + M^{\frac{n}{p'}}\sigma_{1}\right]^{\kappa}\lambda^{(p-1)\kappa}\end{aligned}$$ for a constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. By choosing $\sigma_{1} \equiv \sigma_{1}(\mathop{\mathrm{\mathtt{data}}},M)$ such that $$\label{sigma.cond2} M^{n-1}\sigma_{1} + M^{\frac{n}{p'}}\sigma_{1} + M^{\frac{2n-p}{p'}}\sigma_{1} \le 1,$$ we conclude that $$\label{Dv.upper} \sup_{Q_{R/2}}(|Dv|+s) \le c\lambda$$ holds with $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. To prove the lower bound, By using [\[mean.bound\]](#mean.bound){reference-type="eqref" reference="mean.bound"}, we fix a constant $c_{0}\equiv c_{0}(\mathop{\mathrm{\mathtt{data}}})>1$ satisfying $$\frac{\lambda^{(p-1)\kappa}}{c_{0}} \le (|A(Du)|^{\kappa})_{Q_{4R/M}} + s^{(p-1)\kappa} \le c_{0}\lambda^{(p-1)\kappa}$$ to find $$\begin{aligned} \label{ADv.mean.lower} (|A(Dv)|^{\kappa})_{Q_{4R/M}} + s^{(p-1)\kappa} & \ge (|A(Du)|^{\kappa})_{Q_{4R/M}} + s^{(p-1)\kappa} - (|A(Du)-A(Dv)|^{\kappa})_{Q_{4R/M}} \nonumber \\ & \ge \frac{\lambda^{(p-1)\kappa}}{c_{0}} - \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}|A(Du)-A(Dv)|^{\kappa}\,dx,\end{aligned}$$ where we have used the fact that $\kappa \in (0,1)$. In order to estimate the last integral, we split as follows: $$\begin{aligned} \label{i1i2} \lefteqn{ \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R/M}}}|A(Dv)-A(Du)|^{\kappa}\,dx } \nonumber \\ & \le cM^{n}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|A(Dv)-A(Dw_{1})|^{\kappa}\,dx + cM^{n}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})-A(Du)|^{\kappa}\,dx \nonumber \\ & \eqqcolon I_{1} + I_{2}.\end{aligned}$$ We estimate $I_{2}$ as $$\begin{aligned} \lefteqn{ I_{2} \overset{\eqref{a.diff}}{\le} cM^{n}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Dw_{1}-Du|^{(p-1)\kappa}\,dx } \\ \overset{\eqref{final.comparison}}&{\le} cM^{n+(n-1)\kappa}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{\kappa} \\ & \qquad + cM^{n+(n-1)(p-1)\kappa}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{(p-1)\kappa}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{2-p} \\ & \qquad + cM^{n+\frac{(2n-p)\kappa}{p'}}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right]^{(p-1)\kappa}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{(2-p)\kappa}{p'}} \\ \overset{\eqref{measure.small}}&{\le} c_{1}\left[M^{\frac{n}{\kappa}+n-1}\sigma_{1} + M^{\frac{n}{\kappa}+(n-1)(p-1)}\sigma_{1}^{p-1} + M^{\frac{n}{\kappa}+\frac{2n-p}{p'}}\sigma_{1}\right]^{\kappa}\lambda^{(p-1)\kappa}\end{aligned}$$ for a constant $c_{1}\equiv c_{1}(\mathop{\mathrm{\mathtt{data}}})$. Choosing $\sigma_{1} \equiv \sigma_{1}(\mathop{\mathrm{\mathtt{data}}},M)$ such that $$\label{sigma.cond4} c_{1}\left[M^{\frac{n}{\kappa}+n-1}\sigma_{1} + M^{\frac{n}{\kappa}+(n-1)(p-1)}\sigma_{1}^{p-1} + M^{\frac{n}{\kappa}+\frac{2n-p}{p'}}\sigma_{1}\right]^{\kappa} \le \frac{1}{4c_{0}},$$ we arrive at $$\label{itwo} I_{2} \le \frac{\lambda^{(p-1)\kappa}}{4c_{0}}.$$ As for $I_{1}$, we have $$\begin{aligned} I_{1} \overset{\eqref{ADw1.ADv}}&{\le} cM^{n}\varepsilon\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})|^{\kappa}\,dx + c_{\varepsilon} M^{n} \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{\kappa}{p'}} \\ \overset{\eqref{ADw.mean},\eqref{sigma.cond2}}&{\le} cM^{n}\varepsilon\lambda^{(p-1)\kappa} + c_{\varepsilon} M^{n\left(1+\frac{\kappa}{p'}\right)}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{\kappa}{p'}} \\ \overset{\eqref{measure.small}}&{\le} c_{2}\left[M^{n}\varepsilon + c_{\varepsilon} M^{n\left(1+\frac{\kappa}{p'}\right)}\sigma_{1}^{\kappa}\right]\lambda^{(p-1)\kappa}\end{aligned}$$ for some constants $c_{2}\equiv c_{2}(\mathop{\mathrm{\mathtt{data}}})$ and $c_{\varepsilon} \equiv c_{\varepsilon}(\mathop{\mathrm{\mathtt{data}}},\varepsilon)$, whenever $\varepsilon \in (0,1)$. Choosing $\varepsilon = 1/(8M^{n}c_{2}c_{0})$ and then $\sigma_{1} \equiv \sigma_{1}(\mathop{\mathrm{\mathtt{data}}},M)$ satisfying $$\label{sigma.cond5} c_{\varepsilon} c_{2}M^{n\left(1+\frac{\kappa}{p'}\right)} \sigma_{1}^{\kappa} \le \frac{1}{8c_{0}},$$ it follows that $$\label{ione} I_{1} \le \frac{\lambda^{(p-1)\kappa}}{4c_{0}}.$$ Connecting [\[i1i2\]](#i1i2){reference-type="eqref" reference="i1i2"}, [\[itwo\]](#itwo){reference-type="eqref" reference="itwo"} and [\[ione\]](#ione){reference-type="eqref" reference="ione"} to [\[ADv.mean.lower\]](#ADv.mean.lower){reference-type="eqref" reference="ADv.mean.lower"}, we have $$(|A(Dv)|^{\kappa})_{Q_{4R/M}} + s^{(p-1)\kappa} \ge \frac{\lambda^{(p-1)\kappa}}{2c_{0}}.$$ We now choose a point $x_{0} \in Q_{4R/M}$ satisfying $$\label{1pt.bound2} |A(Dv(x_{0}))|^{\kappa} + s^{(p-1)\kappa} \ge \frac{\lambda^{(p-1)\kappa}}{2c_{0}}.$$ Then, using the oscillation estimate [@BSY Corollary 4.5], [\[mean.min\]](#mean.min){reference-type="eqref" reference="mean.min"} and [\[Dv.upper\]](#Dv.upper){reference-type="eqref" reference="Dv.upper"}, we find that $$\mathop{\mathrm{osc}}_{Q_{4R/M}}A(Dv) \le \frac{c}{M^{\alpha_{A}}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/2}}}|A(Dv)|\,dx \le \frac{c_{3}}{M^{\alpha_{A}}}\lambda^{p-1}$$ holds for a constant $c_{3}\equiv c_{3}(\mathop{\mathrm{\mathtt{data}}})$. Choosing $M$ such that $$\label{M.cond2} \frac{c_{3}}{M^{\alpha_{A}}} \le \left(\frac{1}{4c_{0}}\right)^{\frac{1}{\kappa}}$$ and then combining the resulting inequality with [\[1pt.bound2\]](#1pt.bound2){reference-type="eqref" reference="1pt.bound2"}, we obtain the lower bound in [\[v.bound\]](#v.bound){reference-type="eqref" reference="v.bound"} with some constant $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. ◻ **Remark 27**. The process of fixing the constants $\theta$, $M$ and $\sigma_{1}$ can be summarized as follows. We first fix $M \equiv M(\mathop{\mathrm{\mathtt{data}}})$ as in [Lemma](#grad.bound.lem) [Lemma 26](#grad.bound.lem){reference-type="ref" reference="grad.bound.lem"} satisfying [\[M.cond2\]](#M.cond2){reference-type="eqref" reference="M.cond2"}. Then, by [Lemma](#scale.change) [Lemma 24](#scale.change){reference-type="ref" reference="scale.change"}, we choose $\theta \equiv \theta(\mathop{\mathrm{\mathtt{data}}})$ such that [\[theta.M\]](#theta.M){reference-type="eqref" reference="theta.M"} holds. In a similar way, we finally determine $\sigma_{1} \equiv \sigma_{1}(\mathop{\mathrm{\mathtt{data}}})$ as in [Lemma](#grad.bound.lem) [Lemma 26](#grad.bound.lem){reference-type="ref" reference="grad.bound.lem"}, by requiring that [\[sigma.cond2\]](#sigma.cond2){reference-type="eqref" reference="sigma.cond2"}, [\[sigma.cond4\]](#sigma.cond4){reference-type="eqref" reference="sigma.cond4"} and [\[sigma.cond5\]](#sigma.cond5){reference-type="eqref" reference="sigma.cond5"} are satisfied. Consequently, we have fixed all the parameters $\theta$, $M$ and $\sigma_{1}$ as universal constants depending only on $\mathop{\mathrm{\mathtt{data}}}$, for which the assertions of [Lemma](#scale.change) [Lemma 24](#scale.change){reference-type="ref" reference="scale.change"} and [Lemma](#grad.bound.lem) [Lemma 26](#grad.bound.lem){reference-type="ref" reference="grad.bound.lem"} hold simultaneously. These values of the parameters will be used in the rest of the paper. We now prove estimate [\[lincomp.homo.ndeg\]](#lincomp.homo.ndeg){reference-type="eqref" reference="lincomp.homo.ndeg"}. We have $$\begin{aligned} \label{ADu-ADw} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ \overset{\eqref{a.diff}}&{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}(|Du|+|Dv|+s)^{(p-2)\kappa}|Du-Dv|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ \overset{p<2}&{\le} c\left[\inf_{Q_{R/M}}(|Dv|+s)\right]^{p-2}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|Du-Dv|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ \overset{\eqref{v.bound}}&{\le} c\lambda^{p-2}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{\kappa}\,dx + \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|Dw_{1}-Dv|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}.\end{aligned}$$ We now estimate the two integrals in the right-hand side of [\[ADu-ADw\]](#ADu-ADw){reference-type="eqref" reference="ADu-ADw"}. We estimate the first one as $$\begin{aligned} \label{ADu.ADw.1st} \lefteqn{ \lambda^{p-2}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ \overset{\eqref{final.comparison}}&{\le} c\lambda^{p-2}\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]^{\frac{1}{p-1}} + c\lambda^{p-2}\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{2-p}{p-1}} \nonumber \\ & \qquad + c\lambda^{p-2}\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{8R}}|)\,dx\right)^{\frac{2-p}{p}} \nonumber \\ \overset{\eqref{measure.small}}&{\le} c\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right].\end{aligned}$$ The second one is estimated by applying Hölder's inequality and then [@BSY (6.34)-(6.36)]: $$\begin{aligned} \label{ADu.ADw.2nd} \lambda^{p-2}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|Dw_{1}-Dv|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} & \le \lambda^{p-2}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R}}}|Dw_{1}-Dv|\,dx \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ Combining [\[ADu-ADw\]](#ADu-ADw){reference-type="eqref" reference="ADu-ADw"}, [\[ADu.ADw.1st\]](#ADu.ADw.1st){reference-type="eqref" reference="ADu.ADw.1st"}, and [\[ADu.ADw.2nd\]](#ADu.ADw.2nd){reference-type="eqref" reference="ADu.ADw.2nd"}, we obtain the desired estimate [\[lincomp.homo.ndeg\]](#lincomp.homo.ndeg){reference-type="eqref" reference="lincomp.homo.ndeg"}. ### Proof of [Lemma](#a.comp.ndeg) [Lemma 25](#a.comp.ndeg){reference-type="ref" reference="a.comp.ndeg"} in the second case [\[measure.large\]](#measure.large){reference-type="eqref" reference="measure.large"} {#proof-of-lemma-a.comp.ndeg-in-the-second-case-measure.large} We observe that, from [\[mean.bound\]](#mean.bound){reference-type="eqref" reference="mean.bound"} and [\[measure.large\]](#measure.large){reference-type="eqref" reference="measure.large"}, $$\begin{aligned} \label{lambda.bound} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\sigma R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le c\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ holds whenever $\sigma \in [1/M,4M]$, where $c \equiv c(\mathop{\mathrm{\mathtt{data}}})$. Now we prove [\[lincomp.homo.ndeg\]](#lincomp.homo.ndeg){reference-type="eqref" reference="lincomp.homo.ndeg"}. We have $$\begin{aligned} \label{uw} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-A(Dw_{1})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \overset{\eqref{a.diff}}{\le} cM^{\frac{n}{\kappa}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|Du-Dw_{1}|^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ \overset{\eqref{final.comparison}}&{\le} c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right] + c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]^{p-1}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{2-p}{\kappa}} \nonumber \\ & \qquad + c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right]^{p-1}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{8R}}|)\,dx\right)^{\frac{2-p}{p'}} \nonumber \\ &\le c\left[\frac{|\mu|(Q_{8R})}{(8R)^{n-1}}\right] + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}(|Du|+s)^{(p-1)\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \qquad + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{8R}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{8R}}|)\,dx\right)^{\frac{1}{p'}} \nonumber \\ \overset{\eqref{lambda.bound}}&{\le} c\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ and $$\begin{aligned} \label{wv} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Dw_{1})-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ \overset{\eqref{ADw1.ADv}}&{\le} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Dw_{1})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}} \nonumber \\ & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4R}}}|A(Du)-A(Dw_{1})|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}} \nonumber \\ \overset{\eqref{lambda.bound},\eqref{uw}}&{\le} c\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}.\end{aligned}$$ Combining [\[uw\]](#uw){reference-type="eqref" reference="uw"} and [\[wv\]](#wv){reference-type="eqref" reference="wv"} gives [\[lincomp.homo.ndeg\]](#lincomp.homo.ndeg){reference-type="eqref" reference="lincomp.homo.ndeg"}, and the proof is complete. ## Combining the two alternatives Combining [Lemma](#a.comp.deg) [Lemma 23](#a.comp.deg){reference-type="ref" reference="a.comp.deg"} and [Lemma](#a.comp.ndeg) [Lemma 25](#a.comp.ndeg){reference-type="ref" reference="a.comp.ndeg"}, we conclude with the following comparison estimate. **Lemma 28**. *Let $u$ and $v$ be the weak solutions to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} and [\[homoeq\]](#homoeq){reference-type="eqref" reference="homoeq"}, respectively, under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then we have $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{R/M}}}|A(Du)-A(Dv)|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le \varepsilon\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}|A(Du)-\mathcal{P}_{\kappa,Q_{4MR}}(A(Du)) |^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c_{\varepsilon}\left[\frac{|\mu|(Q_{4MR})}{(4MR)^{n-1}}\right] + c_{\varepsilon}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{4MR}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{Q_{4MR}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ for any $\varepsilon \in (0,1)$, where $c_{\varepsilon} \equiv c_{\varepsilon} (\mathop{\mathrm{\mathtt{data}}},\varepsilon)$ is proportional to some negative power of $\varepsilon$.* # Proof of [Theorem](#pointwise.est) [Theorem 2](#pointwise.est){reference-type="ref" reference="pointwise.est"} and [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"} {#sec.pf.thm1} ## Excess decay estimates for $OP(\psi;\mu)$ In [Section](#sec.comparison.est) [4](#sec.comparison.est){reference-type="ref" reference="sec.comparison.est"} and [Section](#sec.lin.comp) [5](#sec.lin.comp){reference-type="ref" reference="sec.lin.comp"} above, we assumed [\[regular\]](#regular){reference-type="eqref" reference="regular"} and obtained comparison estimates for weak solutions to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"}. In this section, we first obtain an excess decay estimate for weak solutions to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"}. Note that we have chosen the constant $M$ depending only on $\mathop{\mathrm{\mathtt{data}}}$ in the previous section. **Lemma 29**. *Let $u \in \mathcal{A}^{g}_{\psi}(\Omega)$ be the weak solution to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then $$\begin{aligned} \label{lin.ed.est} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}|A(Du)-\mathcal{P}_{\kappa,B_{\rho}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ &\le c_{\mathrm{ex}}\left(\frac{\rho}{r}\right)^{\alpha_{A}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}|A(Du)-\mathcal{P}_{\kappa,B_{r}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c\left(\frac{r}{\rho}\right)^{n+\gamma}\left[\frac{|\mu|(B_{r})}{r^{n-1}}\right] + c\left(\frac{r}{\rho}\right)^{n+\gamma}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{r}}|)\,dx\right)^{\frac{1}{p'}}\end{aligned}$$ holds whenever $B_{\rho} \subset B_{r} \subset \Omega$ are concentric balls, where $c, c_{\mathrm{ex}}\ge 1$ and $\gamma \ge 0$ depend only on $\mathop{\mathrm{\mathtt{data}}}$, $\kappa$ is as in [\[def.kappa\]](#def.kappa){reference-type="eqref" reference="def.kappa"} and $\alpha_{A} \in (0,1)$ is the exponent determined in [Lemma](#ed.ADv) [Lemma 11](#ed.ADv){reference-type="ref" reference="ed.ADv"}.* *Proof.* Without loss of generality, we may assume that $\rho \le r/(4\sqrt{n}M^2)$. With the comparison map $v$ as in [\[homoeq\]](#homoeq){reference-type="eqref" reference="homoeq"} with $R = r/(4\sqrt{n}M)$, we apply [Lemma](#ed.ADv) [Lemma 11](#ed.ADv){reference-type="ref" reference="ed.ADv"} to find $$\begin{aligned} \lefteqn{ \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}|A(Du)-\mathcal{P}_{\kappa,B_{\rho}}(A(Du))|^{\kappa}\,dx \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Du)-\mathcal{P}_{\kappa,Q_{\rho}}(A(Dv))|^{\kappa}\,dx } \\ & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Dv)-\mathcal{P}_{\kappa,Q_{\rho}}(A(Dv))|^{\kappa}\,dx + c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{\rho}}}|A(Du)-A(Dv)|^{\kappa}\,dx \\ & \le c\left(\frac{\rho}{r}\right)^{\alpha_{A}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}\left|A(Dv)-\mathcal{P}_{\kappa,Q_{r/(4\sqrt{n}M^2)}}(A(Dv))\right|^{\kappa}\,dx \\ & \quad + c\left(\frac{r}{\rho}\right)^{n}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}|A(Du)-A(Dv)|^{\kappa}\,dx \\ & \le c\left(\frac{\rho}{r}\right)^{\alpha_{A}}\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}\left|A(Du)-\mathcal{P}_{\kappa,Q_{r/(4\sqrt{n}M^2)}}(A(Du))\right|^{\kappa}\,dx \\ & \quad + c\left(\frac{r}{\rho}\right)^{n} \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emQ_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{Q_{r/(4\sqrt{n}M^2)}}}|A(Du)-A(Dv)|^{\kappa}\,dx.\end{aligned}$$ Applying [Lemma](#lin.comp.u.v) [Lemma 28](#lin.comp.u.v){reference-type="ref" reference="lin.comp.u.v"} to the last integral with the choice $\varepsilon = (\rho/r)^{\alpha_{A}}$ and then making elementary manipulations, we get the desired estimate. ◻ To proceed further, we now consider any limit of approximating solutions $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ to $OP(\psi;\mu)$ with $\mu \in \mathcal{M}_{b}(\Omega)$. Then there exist a sequence of functions $\{\mu_k\} \subset W^{-1,p'}(\Omega) \cap L^{1}(\Omega)$ and corresponding sequence of weak solutions $\{u_k\} \subset \mathcal{A}^{g}_{\psi}(\Omega)$ to [\[opmu\]](#opmu){reference-type="eqref" reference="opmu"} described in [Definition](#def.sol) [Definition 1](#def.sol){reference-type="ref" reference="def.sol"}. Then the convergence properties [\[muk.conv\]](#muk.conv){reference-type="eqref" reference="muk.conv"} and [\[uk.conv\]](#uk.conv){reference-type="eqref" reference="uk.conv"} imply that [\[lin.ed.est\]](#lin.ed.est){reference-type="eqref" reference="lin.ed.est"} holds for $u$ as well. **Lemma 30**. *Let $u \in \mathcal{T}^{1,p}_{g}(\Omega)$ be a limit of approximating solutions to $OP(\psi;\mu)$ under assumptions [\[growth\]](#growth){reference-type="eqref" reference="growth"} and [\[p.bound\]](#p.bound){reference-type="eqref" reference="p.bound"}. Then [\[lin.ed.est\]](#lin.ed.est){reference-type="eqref" reference="lin.ed.est"} still holds whenever $B_{\rho} \subset B_{r} \subset \Omega$ are concentric balls.* We now prove our main results. It suffices to prove [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"}, which with [\[av.min\]](#av.min){reference-type="eqref" reference="av.min"} easily implies [Theorem](#pointwise.est) [Theorem 2](#pointwise.est){reference-type="ref" reference="pointwise.est"}. ## Proof of [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"} {#proof-of-theorem-mainthm.1} We start by fixing a ball $B_{2 R} \equiv B_{2 R}(x_{0}) \subset \Omega$ as in the statement. In the following, all the balls considered will be centered at $x_{0}$. We choose an integer $K \equiv K(\mathop{\mathrm{\mathtt{data}}}) \geq 4M$ such that $$\frac{c_{\mathrm{ex}}}{K^{\alpha_{A}}} \le \frac{1}{2}.$$ Applying [Lemma](#lin.ed.u.1-sola) [Lemma 30](#lin.ed.u.1-sola){reference-type="ref" reference="lin.ed.u.1-sola"} on arbitrary balls $B_{\rho} = B_{r/K} \subset B_{r} \Subset \Omega$, we have $$\begin{aligned} \label{EE} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r/K}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r/K}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r/K}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r/K}}}|A(Du)-\mathcal{P}_{\kappa,B_{r/K}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} } \nonumber \\ & \le \frac{1}{2}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}|A(Du)-\mathcal{P}_{\kappa,B_{r}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \nonumber \\ & \quad + c \left[\frac{|\mu|(B_{r})}{r^{n-1}}\right] + c \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{r}}|)\,dx\right)^{\frac{1}{p'}}.\end{aligned}$$ For $i = 0,1,2,\ldots$, we define $R_{i} \coloneqq R/K^{i}$, $B_{i} \coloneqq B_{R_{i}}(x_{0})$, $$\quad k_{i} \coloneqq \mathcal{P}_{\kappa,B_{i}}(A(Du)) \quad \text{and} \quad E_{i} \coloneqq \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}|A(Du)-\mathcal{P}_{\kappa,B_{i}}(A(Du))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}}.$$ *Step 1: Proof of [\[vmo.x0\]](#vmo.x0){reference-type="eqref" reference="vmo.x0"}.* Applying [\[EE\]](#EE){reference-type="eqref" reference="EE"} with $r \equiv R_{i-1}$ for any $i \ge 1$, we obtain $$\label{EE2} E_{i} \le \frac{1}{2}E_{i-1} + c\left[\frac{|\mu|(B_{i-1})}{R_{i-1}^{n-1}} + \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{i-1}}|)\,dx\right)^{\frac{1}{p'}}\right].$$ Iterating the above inequality, we have for any $k \geq 0$ $$\begin{aligned} E_{k} & \leq \frac{1}{2^{k}} E_{0} + c \sum_{i=1}^{k} \frac{1}{2^{k-i}} \left[\frac{|\mu|(B_{i-1})}{R_{i-1}^{n-1}} + \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i-1}}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{i-1}}|) \,dx \right)^{\frac{1}{p'}} \right] \\ & \leq \frac{1}{2^{k}} E_{0} + c \sup_{0 < \rho \leq R} \left[\frac{|\mu|(B_{\rho})}{\rho^{n-1}} + \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}}|) \,dx \right)^{\frac{1}{p'}} \right].\end{aligned}$$ From [\[mainthm.1.asmp1\]](#mainthm.1.asmp1){reference-type="eqref" reference="mainthm.1.asmp1"}, for any $\delta>0$, we temporarily fix the radius $R \equiv R(\delta)>0$ in this step to satisfy $$\sup_{0 < \rho \leq R} \left[\frac{|\mu|(B_{\rho})}{\rho^{n-1}} + \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}}|) \,dx \right)^{\frac{1}{p'}} \right] < \delta.$$ We then choose $k_0 \in \mathbb{N}$ so large that $$\frac{1}{2^{k_0}} E_0 \leq \delta.$$ Consequently, for any $0 < r \leq R_{k_0}$, we obtain $$\begin{aligned} \lefteqn{ \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{r}}} |A(Du) - \mathcal{P}_{\kappa,B_{r}}(A(Du))|^{\kappa} \, dx\right)^{\frac{1}{\kappa}} } \\ & \leq \frac{K^{n}}{2^{k_0 - 1}}E_0 + c \sup_{0 < \rho \leq R} \left[\frac{|\mu|(B_{\rho})}{\rho^{n-1}} + \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}}|) \,dx \right)^{\frac{1}{p'}}\right] \\ & \leq c \delta.\end{aligned}$$ Since $\delta>0$ was arbitrary, [\[vmo.x0\]](#vmo.x0){reference-type="eqref" reference="vmo.x0"} follows. *Step 2: Proof of [\[Lebesgue.pt\]](#Lebesgue.pt){reference-type="eqref" reference="Lebesgue.pt"} and [\[mainest.1\]](#mainest.1){reference-type="eqref" reference="mainest.1"}.* Let us first show [\[Lebesgue.pt\]](#Lebesgue.pt){reference-type="eqref" reference="Lebesgue.pt"}. Taking any $m_1<m_2 \in \mathbb{N}$ and then summing up [\[EE2\]](#EE2){reference-type="eqref" reference="EE2"} over $i \in \{m_{1}+1,\ldots,m_{2}\}$, we have $$\sum_{i=m_{1}+1}^{m_2}E_{i} \le \frac{1}{2}\sum_{i=m_{1}}^{m_2-1}E_{i} + c\sum_{i=m_{1}}^{m_2-1}\left[\frac{|\mu|(B_{i})}{R_{i}^{n-1}} + \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{i}}|)\,dx\right)^{\frac{1}{p'}} \right]$$ and hence $$\label{sumEi} \sum_{i=m_{1}}^{m_2}E_{i} \le 2 E_{m_{1}} + 2 c\sum_{i=m_{1}}^{m_2-1}\left[\frac{|\mu|(B_{i})}{R_{i}^{n-1}} + \left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{i}}|)\,dx\right)^{\frac{1}{p'}} \right].$$ We observe the following elementary inequalities (see for instance [@KM14BMS (115)]): $$\label{sum.int.mu} \sum_{i=m_{1}}^{m_{2}-1}\left[\frac{|\mu|(B_{i})}{R_{i}^{n-1}}\right] \le c(K)\mathbf{I}^{\mu}_{1}(x_{0},2 R_{m_{1}})$$ and $$\begin{aligned} \label{sum.int.psi} \lefteqn{ \sum_{i=m_{1}}^{m_{2}-1}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{i}}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{i}}|)\,dx\right)^{\frac{1}{p'}} } \nonumber \\ &\le c(K)\int_{0}^{2 R_{m_{1}}}\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}\varphi^{*}(|A(D\psi)-(A(D\psi))_{B_{\rho}(x_{0})}|)\,dx\right)^{\frac{1}{p'}}\frac{d\rho}{\rho}.\end{aligned}$$ Plugging [\[sum.int.mu\]](#sum.int.mu){reference-type="eqref" reference="sum.int.mu"} and [\[sum.int.psi\]](#sum.int.psi){reference-type="eqref" reference="sum.int.psi"} into [\[sumEi\]](#sumEi){reference-type="eqref" reference="sumEi"}, we have $$\begin{aligned} \label{sumEi.2} \lefteqn{ |k_{m_{1}} - k_{m_{2}}| \leq \sum_{i=m_{1}}^{m_{2}-1}|k_{i}-k_{i+1}| \leq cK^{\frac{n}{\kappa}} \sum_{i = m_{1}}^{m_{2}-1} E_i } \nonumber \\ & \leq c E_{m_{1}} + c \mathbf{I}_{1}^{\mu}(x_0, 2R_{m_{1}}) + c \int_{0}^{2R_{m_{1}}} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*} (|A(D\psi) - (A(D\psi))_{B_{\rho}}|) \,dx\right)^{\frac{1}{p'}} \frac{d\rho}{\rho}.\end{aligned}$$ Note that [\[mainthm.1.asmp2\]](#mainthm.1.asmp2){reference-type="eqref" reference="mainthm.1.asmp2"} implies [\[mainthm.1.asmp1\]](#mainthm.1.asmp1){reference-type="eqref" reference="mainthm.1.asmp1"} and $$\lim_{r \to 0} \left[ \mathbf{I}_{1}^{\mu}(x_0,r) + \int_{0}^{r} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*} (|A(D\psi) - (A(D\psi))_{B_{\rho}}|) \,dx\right)^{\frac{1}{p'}} \frac{d\rho}{\rho} \right] = 0.$$ In particular, as a consequence of Step 1, we have [\[vmo.x0\]](#vmo.x0){reference-type="eqref" reference="vmo.x0"}. Accordingly, for every $\varepsilon>0$, we can take $N \in \mathbb{N}$ such that $$E_{N} + \mathbf{I}_{1}^{\mu}(x_0, 2R_{N}) + \int_{0}^{2R_{N}} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}}} \varphi^{*} (|A(D\psi) - (A(D\psi))_{B_{\rho}}|) \,dx\right)^{\frac{1}{p'}} \frac{d\rho}{\rho} < \varepsilon.$$ From this and [\[sumEi.2\]](#sumEi.2){reference-type="eqref" reference="sumEi.2"}, we see that $$|k_{m_{1}} - k_{m_{2}}| < c \varepsilon \quad \text{whenever }N \leq m_1 < m_2,$$ which implies that $\{ k_{i} \}$ is a Cauchy sequence in $\mathbb{R}^n$. We therefore obtain [\[Lebesgue.pt\]](#Lebesgue.pt){reference-type="eqref" reference="Lebesgue.pt"}. Now, in order to show [\[mainest.1\]](#mainest.1){reference-type="eqref" reference="mainest.1"}, we again take an arbitrary small constant $\varepsilon>0$. In light of [\[Lebesgue.pt\]](#Lebesgue.pt){reference-type="eqref" reference="Lebesgue.pt"}, we can take $m \in \mathbb{N}$ large enough to satisfy $$|A_{0} - \mathcal{P}_{\kappa,B_m}(A(Du))| \leq \varepsilon.$$ It then follows from [\[sumEi.2\]](#sumEi.2){reference-type="eqref" reference="sumEi.2"} that $$\begin{aligned} \label{pt.bound} \lefteqn{ |A_{0} - \mathcal{P}_{\kappa,B_0}(A(Du))| \leq |A_{0} - \mathcal{P}_{\kappa,B_m}(A(Du))| + |\mathcal{P}_{\kappa,B_m}(A(Du)) - \mathcal{P}_{\kappa,B_0}(A(Du))| } \nonumber \\ & \leq \varepsilon + c E_{0} + c \mathbf{I}_{1}^{\mu}(x_0, 2R) + c \int_{0}^{2R} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}} \varphi^{*} (|A(D\psi) - (A(D\psi))_{B_{\rho}(x_{0})}|) \,dx\right)^{\frac{1}{p'}} \frac{d\rho}{\rho}.\end{aligned}$$ Recalling that $\varepsilon$ is arbitrary, we obtain [\[mainest.1\]](#mainest.1){reference-type="eqref" reference="mainest.1"} as follows: $$\begin{aligned} |A_{0} - \mathcal{P}_{\kappa,B_{2R}(x_0)}(A(Du))| & \leq |A_{0} - \mathcal{P}_{\kappa,B_0}(A(Du))| + |\mathcal{P}_{\kappa,B_0}(A(Du)) - \mathcal{P}_{\kappa,B_{2R}(x_0)}(A(Du))| \\ \overset{\eqref{pt.bound}}&{\leq} c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{2R}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_0)}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{2R}(x_0)}}|A(Du) - \mathcal{P}_{\kappa,B_{2R}(x_0)}(A(Du))|^{\kappa} \,dx\right)^{\frac{1}{\kappa}} + c \mathbf{I}_{1}^{\mu}(x_0, 2R) \\ & \quad + c \int_{0}^{2R} \left( \mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}} \varphi^{*} (|A(D\psi) - (A(D\psi))_{B_{\rho}(x_{0})}|) \,dx \right)^{\frac{1}{p'}} \frac{d\rho}{\rho}.\end{aligned}$$ Finally, if $x_{0}$ is a Lebesgue point of $A(Du)$, then [\[av.min\]](#av.min){reference-type="eqref" reference="av.min"} implies $$\begin{aligned} |A(Du(x_{0}))-\mathcal{P}_{\kappa,B_{\rho}(x_{0})}(A(Du))| & \le c\left(\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}|A(Du)-A(Du(x_{0}))|^{\kappa}\,dx\right)^{\frac{1}{\kappa}} \\ & \le c\mathchoice% {\mathop{\kern 0.2em\vrule width 0.6em height 0.69678ex depth -0.58065ex \kern -0.8em \intop}\nolimits_{\kern -0.4emB_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}% {\mathop{\kern 0.1em\vrule width 0.5em height 0.69678ex depth -0.60387ex \kern -0.6em \intop}\nolimits_{B_{\rho}(x_{0})}}|A(Du)-A(Du(x_{0}))|\,dx.\end{aligned}$$ Hence, letting $\rho\rightarrow0$, the last assertion in [Theorem](#mainthm.1) [Theorem 3](#mainthm.1){reference-type="ref" reference="mainthm.1"} follows.   0◻   **Conflict of interest.** The authors declare that they have no conflict of interest. **Data availability.** Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. 10 B. Avelin, T. Kuusi, and G. Mingione, *Nonlinear Calderón-Zygmund theory in the limiting case*, Arch. 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arxiv_math

--- abstract: | We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least $10$, we prove the absolute convergence (and hence positivity) under the assumption of Davenport's Geometric Condition, improving on a result of Heath-Brown. For the case of $9$ variables, we give a conditional treatment. We also provide a new short and elementary proof of Davenport's Shrinking Lemma which has been a crucial tool in previous literature on this and related problems. address: Mathematisches Institut, Bunsenstraße 3-5, 37073 Göttingen, Germany author: - Christian Bernert bibliography: - bibliography.bib title: The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma --- # Introduction Let $C(x_1,\dots,x_n) \in \mathbb{Z}[x_1,\dots,x_n]$ be a cubic form. We are interested in the existence of nontrivial integer solutions, i.e. nonzero vectors $\mathbf{x} \in \mathbb{Z}^n$ with $C(\mathbf{x})=0$. Davenport [@davenport63] proved that if $n \ge 16$, such nontrivial solutions always exist. This remained the state of the art for almost half a century until Heath-Brown [@heath2007cubic] could extend the admissible range to $n \ge 14$, this has not been improved to date. Given that $10$ variables suffice to guarantee local solubility [@lewis_p_adic_zeroes], it is generally expected that the same result should hold already when $n \ge 10$. The Hardy-Littlewood Circle Method aims to prove the existence of solutions by proving that there are indeed many. If it works, it provides us with an asymptotic formula of the shape $$\label{asymp} \#\{\mathbf{x} \in \mathbb{Z}^n, x \in P\mathcal{B}\} =(1+o(1)) \cdot \mathfrak{I} \cdot \mathfrak{S} \cdot P^{n-3}$$ as $P \to \infty$. Here $\mathcal{B} \subset \mathbb{R}^n$ is a suitably chosen box and $\mathfrak{I}$ and $\mathfrak{S}$ denote the usual *singular integral* and the *singular series* of the cubic form $C$, respectively, measuring the local solubility of $C$ over the fields $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. The singular integral is rather unimportant for this paper, so we refer the reader to [@davenport63] for its precise definition and only mention that it is known to be positive for a suitable choice of $\mathcal{B}$ as soon as $n \ge 4$. The singular series is the key object of the present paper and will be defined and discussed in more detail in the next section. For now, let us continue discussing the heuristic asymptotic formula [\[asymp\]](#asymp){reference-type="eqref" reference="asymp"} and let us note that it clearly fails in certain degenerate situations. Indeed, when $C$ is reducible, it is easy to see that the count on the left-hand side is already $\gg P^{n-1}$. More generally, if our cubic form is of the shape $C(\mathbf{x})=x_1Q_1(\mathbf{x})+x_2Q_2(\mathbf{x})$ for certain quadratic forms $Q_1$ and $Q_2$, we still have $\gg P^{n-2}$ solutions and hence too many for $\eqref{asymp}$ to possibly hold. The ingenious idea of Davenport to circumvent this fundamental problem was to establish a certain dichotomy: If the circle method fails to produce the asymptotic [\[asymp\]](#asymp){reference-type="eqref" reference="asymp"}, then this failure could be turned into an alternative proof of the existence of solutions, though not in such a precise quantitative manner. To describe Davenport's idea in more detail, we write $C(\mathbf{x})=\sum_{i,j,k} c_{ijk} x_ix_jx_k$ where we assume the $c_{ijk}$ to be symmetric and integers (as we may by multiplying $C$ by $6$ if necessary). We then define the bilinear forms $$B_i(\mathbf{x},\mathbf{y})=\sum_{j,k=1}^n c_{ijk} x_jy_k$$ and the matrix $M(\mathbf{x})$ with entries $$M(\mathbf{x})_{jk}=\sum_{i=1}^n c_{ijk} x_i$$ so that $M(\mathbf{x})\mathbf{y}$ is the vector with entries $B_i(\mathbf{x},\mathbf{y})$. For later use we let $D(\mathbf{x})=\det M(\mathbf{x})$ and $r(\mathbf{x})=\text{rk} M(\mathbf{x})$. For a prime $p$, we will also need to consider the $\mathbb{F}_p$-rank of $M(\mathbf{x})$ which we denote by $r_p(\mathbf{x})$. Let us now say that $C$ satisfies *Davenport's Geometric Condition* if $$\label{gc} \#\{\mathbf{x} \in \mathbb{Z}^n: \|x\|_{\infty} \le P, r(\mathbf{x})=r\} \ll P^{r+\varepsilon}$$ is satisfied for all integers $r$ with $0 \le r \le n$. We can then describe Davenport's result more concisely as follows: **Theorem 1**. *If $C$ does not satisfy Davenport's Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, then the equation $C(\mathbf{x})=0$ has a non-trivial integer solution.* **Theorem 2**. *If $C$ satisfies Davenport's Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, then the asymptotic formula $\eqref{asymp}$ holds with $\mathfrak{I}, \mathfrak{S}>0$ as soon as $n \ge 16$. In particular, there are non-trivial integer solutions to $C(\mathbf{x})=0$.* Note that Theorem [Theorem 1](#A){reference-type="ref" reference="A"} does not make any assumption on the number of variables $n$. This means that in trying to improve on the constraint on the number of variables, we are free to assume that the Geometric Condition is satisfied. Indeed, this is what Heath-Brown did, showing **Theorem 3**. *If $C$ satisfies Davenport's Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, then the asymptotic formula $\eqref{asymp}$ holds with $\mathfrak{I}, \mathfrak{S}>0$ as soon as $n \ge 14$. In particular, there are non-trivial integer solutions to $C(\mathbf{x})=0$.* In view of the above discussion, it is natural to conjecture that this should extend to $n \ge 10$: **Conjecture 1**. *If $C$ satisfies Davenport's Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, then the asymptotic formula $\eqref{asymp}$ holds with $\mathfrak{I}, \mathfrak{S}>0$ as soon as $n \ge 10$. In particular, there are non-trivial integer solutions to $C(\mathbf{x})=0$.* # Main results We now describe our main results. To this end, we need to return to the singular series $\mathfrak{S}$. It is defined in terms of the *Gauß sums* $$S(q,a)=\sum_{\mathbf{x} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod){q}} e\left(\frac{aC(\mathbf{x})}{q}\right)$$ via $$\mathfrak{S}=\sum_{q=1}^{\infty} \sum_{(a;q)=1} \frac{S(q,a)}{q^n}.$$ By standard multiplicativity properties of the Gauß sums, this can (at least formally) also be written as an Euler product $$\mathfrak{S}=\prod_p \chi_p$$ over all primes $p$ where $$\chi_p=\sum_{k=0}^{\infty} \sum_{(a;p^k)=1} \frac{S(p^k,a)}{p^{kn}}$$ is known as the $p$-adic density. By classical arguments it follows that $\chi_p>0$ if and only if $C(\mathbf{x})=0$ has a non-trivial solution over $\mathbb{Q}_p$. In particular, from [@lewis_p_adic_zeroes] we conclude that $\chi_p>0$ for all $p$ whenever $n \ge 10$. So far we have ignored all convergence issues. The rearrangement between the series and the product representation of $\mathfrak{S}$ is only valid when either of the two is known to be absolutely convergent. Proving absolute convergence of $\mathfrak{S}$ is therefore crucial for switching between the two representations and also to conclude its positivity from the positivity of all individual factors $\chi_p$. Only then, the formula [\[asymp\]](#asymp){reference-type="eqref" reference="asymp"} truly captures the expected Local-Global Principle. Previously, the absolute convergence for $\mathfrak{S}$ under the assumption of Davenport's Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"} was known for $n \ge 11$ by work of Heath-Brown [@heath2007cubic]. We begin by giving a new short and self-contained proof of this result. This new method then allows us to improve on previous work and establish the following. **Theorem 1**. *Assume that $n \ge 10$ and that $C$ satisfies Davenport's geometric condition. Then the singular series $\mathfrak{S}$ is absolutely convergent. In particular, $\mathfrak{S}>0$.* This can be seen as giving further evidence to Conjecture [Conjecture 1](#mainconj){reference-type="ref" reference="mainconj"}. Moreover, the Gauß sums featuring in the definition of the singular series are closely related to the Weyl sums that would appear in a circle method proof of $\eqref{asymp}$. It is therefore to be hoped that the study of the Gauß sums and hence of the singular series can serve as a good model problem for our understanding of the more difficult Circle Method Problem. We can also say something about the case $n=9$. We begin by proving that the only possible obstructions to absolute convergence are the Gauß sums with prime moduli. To deal with them, we then propose the following conjecture: **Conjecture 2**. *Assume that $C$ satisfied the Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}. Then for all $n$ and uniformly in $1 \le H \le R$, we have $$\#\{\mathbf{h} \le H, R<p \le 2R: r_p(\mathbf{h}) \le r\} \ll H^r \cdot R^{1+\varepsilon}.$$* We are able to prove the following: **Theorem 2**. *Under the assumption of Conjecture [Conjecture 2](#conj){reference-type="ref" reference="conj"}, the singular series is absolutely convergent for $n=9$.* In the last section, we return to the work of Davenport and Heath-Brown and give a short and elementary proof of *Davenport's Shrinking Lemma*, which is a crucial ingredient in the circle method approach to the cubic forms problem as pioneered by Davenport. The only previous proof of the Shrinking Lemma is due to Davenport and uses rather intricate tools from the geometry of numbers. ## Notation We use the usual notation $\mathcal{O}(\dots)$ and $\ll$ where the implicit constants are always allowed to depend on the cubic form $C(\mathbf{x})$. Moreover, whenever a bound involves $\varepsilon$, it means that the bound is true for all sufficiently small $\varepsilon>0$, but the implicit constant is allowed to depend on $\varepsilon$. Moreover, we use the notation $e(x)=e^{2\pi ix}$ and $\|x\|=\min_{n \in \mathbb{Z}} \vert x-n\vert$. Whenever we write something like $\sum_{\mathbf{h}}$, the sum is restricted to integer vectors $\mathbf{h}$ and the given restrictions on the summation are to be read component-wise. Finally, the condition $r \sim R$ denotes a restriction of $r$ to a dyadic interval $(R,2R]$. # Review of previous bounds for $S(q,a)$ The following simple lemma is good enough to recover all results previously obtained: **Lemma 1**. *Let $q$ and $n$ be positive integers and let $M$ be a $n \times n$ matrix with integer coefficients. Then the size of the kernel of $M$ viewed as a map from $(\mathbb{Z}/q\mathbb{Z})^n$ to itself divides $\det M$. In particular, if $q$ is a prime and $M$ has $\mathbb{F}_q$-rank at most $n-r$, then $p^r \mid \det M$.* **Proof.* Without loss of generality (that is, up to multiplication from both sides by invertible matrices), we may assume that $M$ is in Smith Normal Form with diagonal entries $a_1,\dots,a_n$. Then the kernel has size $\prod_{i=1}^n (a_i;q)$ which divides $\det M=\prod_{i=1}^n a_i$. ◻* We now recall the classical van der Corput differencing: **Lemma 2** (Initial van der Corput Bound). *Let $H \ge 1$ be arbitrary. Then, in the above notation, we have $$\label{vdc} \left(\frac{S(q,a)}{q^n}\right)^2 \ll \frac{1}{H^n} \sum_{1 \le \mathbf{h} \le H} \sqrt{\frac{1}{q^n}\#\{\mathbf{y} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}: q \mid B_i(\mathbf{y}, \mathbf{h})\}}.$$* *Proof.* We set out by applying Cauchy-Schwarz to the identity $$S(q,a)=\frac{1}{H^n}\sum_{\mathbf{x}\@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}} \sum_{1 \le \mathbf{h} \le H} e\left(\frac{aC(\mathbf{\mathbf{x}+\mathbf{h}})}{q}\right)$$ to obtain after some manipulations $$\vert S(q,a)\vert^2 \ll \frac{q^n}{H^n} \sum_{-H \le \mathbf{h} \le H} \left\vert\sum_{\mathbf{x} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}} e\left(\frac{a\left(C(\mathbf{x}+\mathbf{h})-C(\mathbf{x})\right)}{q}\right)\right\vert.$$ The lemma now follows by noting that the square of the absolute value of the inner sum is just $$\sum_{\mathbf{x},\mathbf{y}} e\left(\frac{a\left(C(\mathbf{x}+\mathbf{y}+\mathbf{h})-C(\mathbf{x}+\mathbf{y})-C(\mathbf{x}+\mathbf{h})+C(\mathbf{x}\right)}{q}\right)=\sum_{\mathbf{x},\mathbf{y}} e\left(\frac{a\sum_i x_iB_i(\mathbf{y},\mathbf{h})}{q}\right)$$ and using orthogonality. ◻ Next, from Lemma [Lemma 1](#lemma1){reference-type="ref" reference="lemma1"} we see that $q^{r(\mathbf{h})-n}\#\{\mathbf{y} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}: q \mid B_i(\mathbf{y}, \mathbf{h})\}$ divides a non-zero $r(\mathbf{h}) \times r(\mathbf{h})$ minor of $M$ so that in particular $$\frac{1}{q^n}\#\{\mathbf{y} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}: q \mid B_i(\mathbf{y}, \mathbf{h})\} \ll \left(\frac{H}{q}\right)^{r(\mathbf{h})}.$$ Inserting this into Lemma [Lemma 2](#lemma2){reference-type="ref" reference="lemma2"} and using the geometric condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, we find that $$\left(\frac{S(q,a)}{q^n}\right)^2 \ll \frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \left(\frac{H}{q}\right)^{r(\mathbf{h})/2} \ll \frac{q^{\varepsilon}}{H^n} \sum_{r=0}^n \left(\frac{H^3}{q}\right)^{r/2}\ll q^{\varepsilon}\left(\frac{1}{H^n}+\frac{H^{n/2}}{q^{n/2}}\right)$$ and putting $H=q^{1/3}$, we recover Heath-Brown's pointwise bound $S(q,a) \ll q^{5n/6+\varepsilon}$. Recalling the definition of the $p$-adic factor in the product expansion of $\mathfrak{S}$, we now find that $$\begin{aligned} \chi_p&=\sum_{k=0}^{\infty} \sum_{(a;p^k)=1} \frac{S(p^k,a)}{p^{nk}}\\ &=1+\sum_{(a;p)=1} \frac{S(p,a)}{p^n}+\mathcal{O}\left(\sum_{k=2}^{\infty} p^{k(1-n/6)+\varepsilon}\right)\\ &=1+\sum_{(a;p)=1} \frac{S(p,a)}{p^n}+\mathcal{O}\left(p^{2-n/3+\varepsilon}\right)\end{aligned}$$ so that the estimation of the terms with $k \ge 2$ is satisfactory for the question of absolute convergence of $\mathfrak{S}$ as soon as $n>9$. To establish Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}, it therefore remains to show that $$\sum_{p} \sum_{(a;p)=1} \frac{S(p,a)}{p^n}$$ converges absolutely for $n \ge 10$. It would therefore clearly suffice to show that $$\sum_{p \sim R} \max_{(a;p)=1} \left\vert \frac{S(p,a)}{p^n}\right\vert \ll R^{-1-\delta}$$ for which, by Cauchy-Schwarz, it suffices to establish $$\sum_{p \sim R} \max_{(a;p)=1} \left\vert \frac{S(p,a)}{p^n}\right\vert^2 \ll R^{-3-\delta}$$ for all choices of $R \ge 1$. Using our previous line of argument, the LHS is bounded by $$\frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \sum_{R \le p<2R} \sqrt{\frac{1}{p^n}\#\{\mathbf{y} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{p}: p \mid B_i(\mathbf{y}, \mathbf{h})\}}=\frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \sum_{R \le p<2R} p^{-r_p(\mathbf{h})/2}$$ where $r_p(\mathbf{h})$ is the $\mathbb{F}_p$-rank of $M(\mathbf{h})$. Heath-Brown's idea is now to distinguish two cases: Those pairs $(\mathbf{h},p)$ with $r_p(\mathbf{h})=r(\mathbf{h})$ give a contribution bounded by $$\label{fall1} \frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \sum_{R \le p<2R} p^{-r(\mathbf{h})/2} \ll \frac{1}{H^n}\sum_{r=0}^n H^{r+\varepsilon} R^{1-r/2} \ll H^{\varepsilon}\left(\frac{R}{H^n}+\frac{R}{R^{n/2}}\right)$$ by the Geometric Condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}. The last term is satisfactory for $n>8$. On the other hand, we need to estimate the contribution from those pairs $(\mathbf{h},p)$ with $r_p(\mathbf{h})<r(\mathbf{h})$. Here we use the implication from Lemma [Lemma 1](#lemma1){reference-type="ref" reference="lemma1"} that $p^{r(\mathbf{h})-r_p(\mathbf{h})}$ must divide a non-zero $r(\mathbf{h}) \times r(\mathbf{h})$-minor of $M(\mathbf{h})$ and is hence $\mathcal{O}(H^{r(\mathbf{h})})$ so that $$\label{fall2.1} p^{-r_p(\mathbf{h})} \ll \left(\frac{H}{p}\right)^{r(\mathbf{h})}.$$ Moreover, $p$ being a divisor of such a minor, there are at most $H^{\varepsilon}$ choices of such $p$ for any fixed $\mathbf{h}$. The total contribution of such pairs $(\mathbf{h},p)$ can therefore be bounded by $$\label{fall2} \frac{H^{\varepsilon}}{H^n} \sum_{-H \le \mathbf{h} \le H} \left(\frac{H}{R}\right)^{r(\mathbf{h})/2} \ll H^{\varepsilon-n} \sum_{r=0}^n \left(\frac{H}{R}\right)^{r/2} \ll H^{\varepsilon}\left(\frac{1}{H^n}+\left(\frac{H^3}{R}\right)^{n/2}\right)$$ again using the geometric condition. Comparing the contributions from [\[fall1\]](#fall1){reference-type="eqref" reference="fall1"} and [\[fall2\]](#fall2){reference-type="eqref" reference="fall2"} we find that the optimal choice is $H=R^{\frac{n+2}{3n}}$ leading to the bound $R^{-(n-1)/3+\varepsilon}$ which is satisfactory when $n>10$. # The case of ten variables When $n=10$, we observe that $H=R^{2/5+\delta}$ for sufficiently small $\delta>0$ leads to a satisfatory contribution from [\[fall1\]](#fall1){reference-type="eqref" reference="fall1"} and from all terms in [\[fall2\]](#fall2){reference-type="eqref" reference="fall2"} except when $r=n=10$. Moreover, even for this term it suffices to save another small power of $R$, which we do in [\[fall2.1\]](#fall2.1){reference-type="eqref" reference="fall2.1"} unless $r_p(\mathbf{h})=6$. It therefore suffices to show that $$\#\{\mathbf{h} \le H, p \sim R: r(\mathbf{h})=10, r_p(\mathbf{h})=6\} \ll H^{10-\delta'}$$ for some $\delta'>0$ whenever $H=R^{2/5+\delta}$ for sufficiently small $\delta>0$. To prove this, we use an argument inspired by a trick of Davenport [@davenport63] which he used to go from $17$ to $16$ variables. However, the presence of the extra averaging over $p$ requires a new idea. By Lemma [Lemma 1](#lemma1){reference-type="ref" reference="lemma1"}, we have $p^4 \mid D( \mathbf{h})$ for all vectors $\mathbf{h}$ in question. Moreover, there are $p^4$ vectors $\mathbf{y} \in \{0,1,2,\dots,p-1\}^n$ with $p \mid B_i(\mathbf{y},\mathbf{h})$ for all $i$. By the Pigeonhole principle, two of them differ by $\mathcal{O}(p^{3/5})$ in each component and by linearity of the $B_i$, this means that for each such $\mathbf{h}$ we get one solution $\mathbf{y}=\mathbf{y}(\mathbf{h}) \ne 0$ with $\|\mathbf{y}\|_{\infty} \ll p^{3/5}$ and $p \mid B_i(\mathbf{y},\mathbf{h})$. Writing $B_i(\mathbf{y},\mathbf{h})=pm_i$, we find that $m_i=m_i(\mathbf{h}) \ll R^{\delta}$. Moreover, not all $m_i$ are zero since we assumed $r(\mathbf{h})=10$. We can now count the number of pairs $(\mathbf{h},p)$ in question as follows: There are $\ll R^{10\delta}$ possible choices of the $m_i$. For a fixed choice of $(m_1,\dots,m_n)$, we then study the number of possible choices of $(\mathbf{h},p)$. The general solution of the system $B_i(\mathbf{y},\mathbf{h})=pm_i$ is given by $$y_j=p \cdot \frac{\sum_k m_k E_{jk}(\mathbf{h})}{D(\mathbf{h})}$$ where the $E_{jk}$ are certain $9 \times 9$ minors of $M(\mathbf{h})$, in particular homogeneous forms of degree $9$ in $\mathbf{h}$. Now certainly, for our given choice of the $m_i$, there is one $j$ such that the degree-$9$ form $E(\mathbf{h}):=\sum_k m_k E_{j,k}(\mathbf{h})$ is not identically zero. We conclude that $D(\mathbf{h}) \mid p \cdot E(\mathbf{h})$. Let $G$ be the greatest common divisor of $D$ and $E$ and write $D=GD'$ and $E=GE'$ so that $D'(\mathbf{h}) \mid p \cdot E'(\mathbf{h})$ and $D'$ is coprime to $E'$. We thus find by Bézout's Theorem a non-zero linear combination $F$ of $D'$ and $E'$ that depends only on $h_2,\dots,h_n$. Hence $D'(\mathbf{h}) \mid p \cdot F(h_2,\dots,h_n)$. Note that the coefficients of all the polynomials depend on the $m_i$, but are all polynomially bounded in terms of $R$ which is sufficient for our application. Now there are $\mathcal{O}(H^9)$ values of $\mathbf{h}$ where $F$ is zero and then $p$ as a divisor of $D(\mathbf{h})$ is determined up to $H^{\varepsilon}$ many choices, leading to a total bound of $H^{9+\varepsilon}$ for the number of pairs $(\mathbf{h},p)$ in this case. On the other hand, if $F(h_2,\dots,h_{10})$ is non-zero, we see that $p \mid F(h_2,\dots,h_{10})$ by the following ad-hoc bootstrapping argument: Since $p^4 \mid D(\mathbf{h})=G(\mathbf{h}) \cdot D'(\mathbf{h})$ and $\deg G \le 9$ we have $G(\mathbf{h}) \ll H^9<p^4$ if $\delta>0$ is sufficiently small. Hence $p \mid D'(\mathbf{h})$. But if $\delta$ is small, this forces $\deg D' \ge 3$ and hence $\deg G \le 7$ so that $G(\mathbf{h}) \ll H^7<p^3$, again if $\delta$ is small. Hence $p^2 \mid D'(\mathbf{h})$ and hence $p \mid F(h_2,\dots,h_{10})$ as desired. Finally, for any choice of $h_2,\dots,h_{10}$ with $F(\mathbf{h}) \ne 0$, this determines $p$ and $D'(\mathbf{h})$ up to $H^{\varepsilon}$ many choices and then also $h_1$ is determined up to finitely many choices, unless we are in a proper Zariski-closed subset of $h_2,\dots,h_{10}$. In any case, the total number of pairs $(\mathbf{h},p)$ can be bounded by $H^{9+\varepsilon}$. Summing up, we have thus shown that $$\#\{\mathbf{h} \le H, p \sim R: r(\mathbf{h})=10, r_p(\mathbf{h})=6\} \ll R^{10\delta} \cdot H^{9+\varepsilon}$$ which is satisfactory for $\delta>0$ sufficiently small. This finishes the proof of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}. # The case of nine variables We now set out to discuss the case $n=9$, aiming for a proof of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}. To begin with, we need to discuss the case of higher prime powers. The contribution to $\chi_p$ of $S(p^k,a)$ for $k \ge 3$ is seen to be satisfactory even for $n=9$. For the contribution of the terms with $k=2$, our pointwise bound $S(p^2,a) \ll p^{5n/3+\varepsilon}$ just fails to be good enough when $n=9$. However, we can use the averaging trick introduced in the previous section to also improve on this bound and therefore reduce the problem of absolute convergence of $\mathfrak{S}$ for $n=9$ to the study of $S(p,a)$ for primes $p$: **Lemma 3**. *For $n=9$, the sum $$\sum_p \sum_{(a;p^2)=1} \frac{S(p^2,a)}{p^{2n}}$$ is absolutely convergent. In particular, the singular series for $n=9$ converges absolutely if and only if $$\sum_p \sum_{(a;p)=1} \frac{S(p,a)}{p^n}$$ is absolutely convergent.* *Proof.* As before, a dyadic decomposition and an application of Cauchy-Schwarz reduce the problem to showing that $$\sum_{p \sim R} \max_{(a;p^2)=1} \left\vert \frac{S(p^2,a)}{p^{2n}}\right\vert^2 \ll R^{-5-\delta}.$$ From Lemma [Lemma 2](#lemma2){reference-type="ref" reference="lemma2"}, we see that the LHS is bounded by $$\frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \sum_{p \sim R} \sqrt{\frac{1}{p^{2n}} \#\{\mathbf{y} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{p^2}: p^2 \mid B_i(\mathbf{y},\mathbf{h})}\}.$$ We continue by separating the cases $r_p(\mathbf{h})=r(\mathbf{h})$ and $r_p(\mathbf{h})<r(\mathbf{h})$. In the first case, the expression under the root is $(p^2)^{-r(\mathbf{h})}$ and using the geometric condition [\[gc\]](#gc){reference-type="eqref" reference="gc"} we obtain a contribution bounded by $$\frac{RH^{\varepsilon}}{H^n} \sum_{r=0}^n \frac{H^r}{(R^2)^{r/2}} \ll R^{1+\varepsilon} \left(\frac{1}{H^n}+\frac{1}{R^n}\right)$$ (compare this with [\[fall1\]](#fall1){reference-type="eqref" reference="fall1"}). In the second case, for each $\mathbf{h}$, there are at most $R^{\varepsilon}$ choices of $p$ and for each such pair the expression under the root is bounded by $\left(\frac{H}{p^2}\right)^{r(\mathbf{h})/2}$ so that the contribution in this case can be bounded by $$\frac{R^{\varepsilon}}{H^n} \sum_{r=0}^n H^r \left(\frac{H}{R^2}\right)^{r/2} \ll R^{\varepsilon} \cdot \left(\frac{1}{H^n}+\left(\frac{H}{R^2}\right)^{n/2}\right)$$ (compare this with [\[fall2\]](#fall2){reference-type="eqref" reference="fall2"}) and choosing $H=(R^2)^{\frac{n+1}{3n}}$ we end up with the total contribution of $\ll R^{1-\frac{2(n+1)}{3}}$ from both cases together, which is satisfactory as soon as $n>8$. ◻ We are now ready to prove Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}: *Proof of Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"}.* By Lemma [Lemma 3](#lemma){reference-type="ref" reference="lemma"} and the arguments from the previous discussion, it suffices to prove that $$\sum_{p \sim R} \max_{(a;p)=1} \left\vert \frac{S(p,a)}{p^n}\right\vert^2 \ll R^{-1-\delta}$$ for all choices of $R \ge 1$. Using Lemma [Lemma 2](#lemma2){reference-type="ref" reference="lemma2"}, the LHS is bounded by $$\frac{1}{H^n} \sum_{-H \le \mathbf{h} \le H} \sum_{p \sim R} p^{-r_p(\mathbf{h})/2} \ll \frac{1}{H^n} \sum_{r=0}^n R^{-r/2} \#\{\mathbf{h} \le H, p \sim R: r_p(\mathbf{h})=r\}.$$ Assuming Conjecture [Conjecture 2](#conj){reference-type="ref" reference="conj"}, this can be further estimated as $$\ll \frac{1}{H^n} \sum_{r=0}^n R^{-r/2} H^rR^{1+\varepsilon} \ll \frac{R^{1+\varepsilon}}{H^n}+\frac{R^{1+\varepsilon}}{R^{n/2}}.$$ Choosing e.g. $H=R^{1/2}$ we see that this is satisfactory as soon as $n>8$. ◻ Indeed, as can be seen from the above proof, only something weaker than Conjecture [Conjecture 2](#conj){reference-type="ref" reference="conj"} is actually required. However, we do believe that this is the 'right' way to put the conjecture, as the proposed upper bound is exactly the contribution that we a priori get from the terms with $r(\mathbf{h})=r$ and $p$ arbitrary, using the geometric condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}. We close this section by a few more remarks regarding Conjecture [Conjecture 2](#conj){reference-type="ref" reference="conj"}. To start with, the cases $r=0$ and $r=n$ are easy to establish. Moreover, we can also prove the case $r=n-1$: Those $\mathbf{h}$ with $r(\mathbf{h})=n-1$ produce a satisfactory contribution by the geometric condition [\[gc\]](#gc){reference-type="eqref" reference="gc"}, as explained above. On the other hand, there can be only $\mathcal{O}(H^{n+\varepsilon})$ pairs $(\mathbf{h},p)$ with $r(\mathbf{h})=n$ and $r_p(\mathbf{h})=n-1$ as then $p \mid D(\mathbf{h})$ and so $p$ is determined by $\mathbf{h}$ up to $\mathcal{O}(H^{\varepsilon})$ many choices. # A new proof of Davenport's Shrinking Lemma In previous work on general cubic forms, a crucial tool for dealing with the bilinear counting problems as seen in [\[vdc\]](#vdc){reference-type="eqref" reference="vdc"} as well as more general versions for the Weyl sums was the following result of Davenport, also known as the Shrinking Lemma. **Lemma 4** (Davenport's Shrinking Lemma). *Let $L=(L_1,\dots,L_n) \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Let $P \ge 1$ and $0<Z<1$ be real numbers. Then $$\#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le P, \|L_i(\mathbf{x})\| <\frac{1}{2nP} \forall i\right\} \le \left(\frac{4}{Z}\right)^n \cdot \#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le ZP, \|L_i(\mathbf{x})\| <\frac{Z}{2nP}\forall i\right\}.$$ Here, $\|z\|$ denotes the distance of $z$ to the nearest integer.* In only dealing with the Gauß sums in the above discussion we were able to circumvent the use of the lemma, using Lemma [Lemma 1](#lemma1){reference-type="ref" reference="lemma1"} and the Pigeonhole Principle as a substitute, but for the Weyl sums it remains an essential ingredient. Since the only previous proof uses rather intricate tools from the geometry of numbers, it is therefore desirable to present a short and elementary proof which we do in this section. *Proof of Lemma [Lemma 4](#shrink){reference-type="ref" reference="shrink"}.* We begin by choosing a prime $q$ such that $\frac{2}{Z} \le q \le \frac{4}{Z}$ which is always possible. Then it will suffice to prove that $$\#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le P, \|L_i(\mathbf{x})\| <\frac{1}{2nP} \forall i\right\} \le q^n \cdot \#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le \frac{2P}{q}, \|L_i(\mathbf{x})\| <\frac{1}{nqP}\forall i\right\}.$$ Denote by $[z]$ the nearest integer to $z$. For each $(\mathbf{a},\mathbf{b}) \in (\mathbb{Z}/q\mathbb{Z})^2$ let $$N_{\mathbf{a},\mathbf{b}}=\#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le P, \|L_i(\mathbf{x})\| <\frac{1}{2nP} \forall i, \mathbf{x} \equiv \mathbf{a} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}, ([L_i(\mathbf{x})])_i \equiv \mathbf{b} \@ifstar\mkern 4mu({\operator@font mod}\mkern 6mu\@@pmod)*{q}\right\}.$$ Clearly the LHS of our inequality now decomposes as $\sum_{\mathbf{a},\mathbf{b}} N_{\mathbf{a},\mathbf{b}}$. Now observe that if $\mathbf{x}_1$ and $\mathbf{x}_2$ are counted by $N_{\mathbf{a},\mathbf{b}}$, then $\mathbf{x}:=\frac{\mathbf{x}_2-\mathbf{x}_1}{q}$ is counted by the RHS of our inequality. Hence it follows that $$N_{\mathbf{a},\mathbf{b}} \le \#\left\{\mathbf{x} \in \mathbb{Z}^n: \vert \mathbf{x}\vert \le \frac{2P}{q}, \|L_i(\mathbf{x})\| <\frac{1}{nqP}\forall i\right\}$$ which is already enough to deduce our claimed inequality with a factor of $q^{2n}$ instead of $q^n$, since there are $q^{2n}$ choices of $(\mathbf{a},\mathbf{b})$. To conclude the stronger claim, it will thus suffice to show that $N_{\mathbf{a},\mathbf{b}} \ne 0$ only for at most $q^n$ choices of $(\mathbf{a},\mathbf{b})$. Indeed, this will follow immediately if we can show that the $2n \times 2n$ matrix with columns $\left(\mathbf{x},[L_i(\mathbf{x})]\right)$ for $\mathbf{x}$ counted by the LHS of our inequality has rank at most $n$. However, note that by our estimate on $\|L_i(\mathbf{x})\|$ and the symmetry of $L$ we have $$\mathbf{y} \cdot ([L_i(\mathbf{x})])_i=\mathbf{x} \cdot ([L_i(\mathbf{y})])_i$$ for all $\mathbf{x}$ and $\mathbf{y}$ counted, since both sides are integers and differ by less than $2n \cdot P \cdot \frac{1}{2nP}=1$. Hence, if we add to our matrix the columns $\left(-[L_i(\mathbf{x})],\mathbf{x}\right)$ each column of the new part will be orthogonal to each column of the old part, and since they both have the same rank, both parts can have rank at most $n$, as desired. ◻ # Acknowledgments {#acknowledgments .unnumbered} This work was carried out while the author was a Ph.D. student at the University of Göttingen, supported by the DFG Research Training Group 2491 'Fourier Analysis and Spectral Theory'. I would like to thank my supervisor Jörg Brüdern for introducing me to the topic and for encouraging me to work on it.

arxiv_math

--- abstract: | Classifying the nonflat hypersurfaces in Euclidean space $f\colon M^n\to\mathbb{R}^{n+1}$ that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten [@Sc] in 1928. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler. author: - M. Dajczer and M. I. Jimenez title: | The infinitesimal deformations of hypersurfaces\ that preserve the Gauss map --- The local study of the smooth deformations of the submanifolds of the Euclidean space comprises several types, among them isometric, conformal and affine variations, as well as their somewhat less demanding infinitesimal versions. The case of surfaces was already a hot topic during the $19^{th}$ century, whereas work for higher dimensions can be traced back to the final decade of that century. A rather large amount of papers on these and even more general types of deformations have since been published. Related to this paper, we recall that a necessary condition for a hypersurface $M^n$ in $\mathbb{R}^{n+1}$, $n\geq 3$, without flat points to admit a non-trivial infinitesimal variation is to have at any point precisely two nonzero principal curvatures. This rigidity result is already in the book by Cesàro [@Ce] published in 1896, but it seems that it was first obtained by Killing in 1885. A parametric local classification of the Euclidean hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 3$, that admit smooth isometric deformations was obtained by Sbrana [@Sb1] in 1909. His achievement was the conclusion of a long quest for such a classification. Among several partial results there is one due to Bianchi [@Bi] for dimension $n=3$. As an aside, it seems that Bianchi had been Sbrana's adviser. Around the same time Sbrana [@Sb] obtained the local classification of the nonflat hypersurfaces that admit infinitesimal isometric deformations. These deformations are the ones that preserve lengths just "up to the first order\". At the very end of his paper, Sbrana showed that these hypersurfaces form a *much larger* class than the one of the isometrically deformable hypersurfaces. Still in the isometric case, in 1916 Cartan [@Ca1] obtained a classification equivalent to Sbrana's by means of his own methods. Then, the following year he gave in [@Ca2] a parametric classification for the more general conformal case. For modern versions of all these classical results for hypersurfaces as well as further developments we refer to [@DFT], [@DJ1], [@DJ2], [@DJV], [@DT] and [@DV]. The local basic question: *To what extent an Euclidean submanifold in arbitrary codimension $f\colon M^n\to\mathbb{R}^{n+p}$, $n\geq 2$, is determined by its Gauss map into the Grassmannian $G_{np}$* was answered in [@DG1], with the later complement given in [@DR]. Of course, for $\mathbb{R}^3$ there was the very classical result that a nonflat surface has to be minimal and that any other isometric immersion with the same Gauss map belongs to the associated one-parameter family given by the Weierstrass parametrization. For instance, see page 383 in Darboux [@Dar]. In the case of hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 3$, we have from Corollary $4.12$ in [@DG1], besides the minimality of the immersion, that the manifold has to be Kaehler up to a cylindrical Euclidean factor; see also the more general result given in [@DG2]. The class of minimal Kaehler hypersurfaces and their associated one-parameter families in a Weierstrass type parametrization is discussed with details in Section [2](#Kaehler){reference-type="ref" reference="Kaehler"}. This paper is devoted to the study of the hypersurfaces $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 2$, that admit infinitesimal isometric deformations that infinitesimally preserve the Gauss map. Although we were not able to find in the literature any previous attempt to give a classification in the isometric case other than the one aforementioned, it was for us quite unexpected to discover that there is one for the infinitesimal case. In fact, in terms of a rather difficult notation Schouten [@Sc] gave in 1928 a necessary and sufficient condition. His main achievement is that we must be in the presence of a minimal hypersurface. Then, there is a second condition somehow involving the second fundamental form of the submanifold. Of course, Schouten's result does not fully describe the hypersurfaces involved. The following result gives a complete local parametric classification of the Euclidean hypersurfaces that admit infinitesimal isometric deformations which infinitesimally preserve the Gauss map. Although, as aforementioned, while the class of hypersurface in the infinitesimally deformable class is much richer than in the isometrically deformable class, it turns out that that under the Gauss map condition we are reduced to the same family of hypersurfaces, namely, the minimal ones that, up to a possible Euclidean cylindrical factor, are Kaehler. Since the presence of the former does not really significantly change the situation, that possibility is not considered in the following statement. **Theorem 1**. *Let $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 2$, be an isometric immersion that admits a nowhere trivial infinitesimal variation that infinitesimally preserves the Gauss map. Assume that $M^n$ is free of flat points and that $f$ is not part of an Euclidean cylinder when restricted to any open subset of $M^n$. Then $f$ is a minimal immersion of a Kaehler manifold. In addition, if $M^n$ is simply connected, then the variational vector field associated to the infinitesimal variation is, up to constants, the conjugate to $f$ in its associate family.* The assumptions that a hypersurface has no local Euclidean factor, as well as that the infinitesimal variation is nowhere trivial, are explained in the next section. Two local parametric descriptions of the minimal Kaehler hypersurfaces are provided in Section 2, among them a Weierstrass type representation that extends the classical one for minimal surfaces. # Infinitesimal isometric variations On this section, we recall from [@DJ1] and [@DT] several concepts as well as some of the basic facts on infinitesimal deformations of Euclidean hypersurfaces. Let $F\colon I\times M^n\to\mathbb{R}^{n+1}$ denote a smooth variation of a given isometric immersion $f\colon M^n\to\mathbb{R}^{n+1}$ of a Riemannian manifold $M^n$, $n\geq 2$, such that $0\in I\subset\mathbb{R}$ is an interval and $F(0,x)=f(x)$. We assume that $f_t=F(t,\cdot)\colon M^n\to\mathbb{R}^m$ is an immersion for any $t\in I$. The variational vector field of the variation $F$ of $f$ is the section ${\cal T}\in\Gamma(f^*T\mathbb{R}^m)$ defined as $${\cal T}=F_*\partial/\partial t|_{t=0}=\tilde\nabla_{\partial/\partial t}f_t|_{t=0},$$ where $\tilde\nabla$ denotes the Euclidean connection. We assume that $F\colon I\times M^n\to\mathbb{R}^{n+1}$ is an *infinitesimal variation*. This means that the metrics $g_t$, $t\in I$, induced by $f_t$ satisfy $\partial/\partial t|_{t=0}g_t=0$, that is, $$\label{varcond2} \frac{\partial}{\partial t}|_{t=0}{\langle}f_{t*}X,f_{t*}Y{\rangle}=0\;\; \mbox{for any}\;\; X,Y\in\mathfrak{X}(M).$$ That is what it is meant by saying that the variation preserves lengths "up to the first order\". Equivalent to [\[varcond2\]](#varcond2){reference-type="eqref" reference="varcond2"} is that the variational vector field ${\cal T}\in\Gamma(f^*T\mathbb{R}^{n+1})$ of $F$ satisfies the condition $$\label{infben} {\langle}{\cal T}_*X,f_*Y{\rangle}+{\langle}f_*X,{\cal T}_*Y{\rangle}=0\;\; \mbox{for any}\;\; X,Y\in\mathfrak{X}(M).$$ A vector field ${\cal T}\in \Gamma(f^*T\mathbb{R}^{n+1})$ satisfying [\[infben\]](#infben){reference-type="eqref" reference="infben"} is called an *infinitesimal bending* of $f$. It is known already from classical differential geometry that the convenient approach to study infinitesimal variations is to focus on the infinitesimal bending determined by the variation. Associated to a given infinitesimal bending ${\cal T}$ of $f\colon M^n\to\mathbb{R}^{n+1}$ we have the infinitesimal variation $F\colon\mathbb{R}\times M^n\to\mathbb{R}^{n+1}$ given by $$\label{unique0} F(t,x)=f(x)+t{\cal T}(x)$$ which has ${\cal T}$ as variational vector field. By no means this infinitesimal variation is unique with this property, although it may be seen as the simplest one. In fact, new infinitesimal variations with variational vector field ${\cal T}$ are obtained by adding to [\[unique0\]](#unique0){reference-type="eqref" reference="unique0"} terms of the type $t^k\delta$, $k\geq 2$, where $\delta\in\Gamma(f^*T\mathbb{R}^{n+1})$. An infinitesimal bending is called a *trivial* infinitesimal bending if it is the restriction to the submanifold of a Killing vector field of the ambient space. That is, it is the variational vector field of a variation by Euclidean motions. Hence, there is a skew-symmetric linear endomorphism ${\cal D}$ of $\mathbb{R}^{n+1}$ and a vector $w\in\mathbb{R}^{n+1}$ such that ${\cal T}={\cal D}f+w$. Conversely, given a trivial infinitesimal bending we have the trivial isometric variation of $f$ given by $F(t,x)=e^{t{\cal D}}f(x)+tw$. Let $A(t)$, $t\in I$, with $A(0)=A$ denote the shape operators associated to the smooth family of Gauss maps $N_t$ of $f_t$ where $N_0=N$. Then let $B\in\Gamma(\mbox{End}(TM))$ be the symmetric Codazzi tensor defined by $B=\partial/\partial t\vert_{t=0}A(t)$. The Gauss formula for each $f_t$ states $$\label{Gaussform} \tilde\nabla_Xf_{t*}Y=f_{t*}\nabla^t_XY+{\langle}f_{t*}A_tX,f_{t*}Y{\rangle}N_t \;\;\mbox{for any}\;\;X,Y\in\mathfrak{X}(M),$$ where $\nabla^t$ is the Levi-Civita connection of the metric induced by $f_t$. Notice that $$\partial/\partial t|_{t=0}\tilde\nabla_Xf_{t*}Y =\partial/\partial t|_{t=0}\tilde\nabla_X\tilde\nabla_Yf_t=\tilde\nabla_X{\cal T}_*Y.$$ Since $F$ is an infinitesimal variation then $\partial/\partial t|_{t=0}\nabla^t_XY=0$. Thus the derivative with respect to $t$ at $t=0$ of the right hand side of [\[Gaussform\]](#Gaussform){reference-type="eqref" reference="Gaussform"} gives $$\partial/\partial t|_{t=0}(f_{t*}\nabla^t_XY +{\langle}f_{t*}A_tX,f_{t*}Y{\rangle}N_t) ={\cal T}_*\nabla_XY+{\langle}BX,Y{\rangle}N+{\langle}AX,Y{\rangle}\partial/\partial t|_{t=0}N_t.$$ Hence, $$\label{B} (\tilde\nabla_X{\cal T}_*)Y={\langle}AX,Y{\rangle}\partial/\partial t|_{t=0}N_t+{\langle}BX,Y{\rangle}N \;\;\mbox{for any}\;\;X,Y\in\mathfrak{X}(M).$$ In addition, we have from equation $(2.21)$ in [@DJ1] that the tensors $A$ and $B$ satisfy $$\label{infGauss2} BX\wedge AY-BY\wedge AX=0\;\;\mbox{for any}\;\;X,Y\in\mathfrak{X}(M).$$ This equation together with the Codazzi equation for $B$ form what is called the fundamental equations of an infinitesimal bending for a hypersurface. **Proposition 2**. *The infinitesimal bending ${\cal T}$ is trivial if and only if $B=0$.* This is Proposition $2.10$ in [@DJ1]. pt  pt width4 pt height6 pt depth1.5 pt  Let $N_t$ be the Gauss maps of $f_t$, $t\in I$. That the non-trivial infinitesimal variation $F$ of $f$ is *infinitesimal with respect to the Gauss map* means that $\partial/\partial t|_{t=0}N_t=0$. In other words, $N_t$ coincides with the Gauss map $N$ of $f$ up to the first order. In this situation, the derivative with respect to $t$ of ${\langle}N_t,f_{t*}X{\rangle}=0$ computed at $t=0$ yields $$\label{normal} {\langle}N,{\cal T}_*X{\rangle}=0\;\;\text{for any}\;\;X\in\mathfrak{X}(M).$$ Hence ${\cal T}_*X\in\Gamma(f_*TM)$ for any $X\in\mathfrak{X}(M)$, and thus determines a skew symmetric endomorphism of the tangent space also denoted by ${\cal T}_*\in\Gamma(\mbox{End}(TM))$ for simplicity. Let $M^n$ be a Riemannian product $N^k\times V$ where $N^k$ is a simply connected manifold free of flat points and $V\subset\mathbb{R}^{n-k}$ is an open subset. Let $g\colon N^k\to\mathbb{R}^{k+1}$ be an isometric immersion and $f\colon M^n\to\mathbb{R}^{k+1}\oplus\mathbb{R}^{n-k}$ the cylinder given by $f(y,z)=(g(y),z)$. Let $G$ be an infinitesimal variation of $g$ and let $F$ be the variation of $f$ defined by $F(t,y,z)=(G(t,y),\Psi_t(z))$, where $\Psi_t$ is a smooth one-parameter family of isometries of $\mathbb{R}^{n-k}$ with $\Psi_0=I$. Then $F$ is an infinitesimal variation of $f$ by cylinders. The corresponding infinitesimal bending is of the form $$\label{bending} {\cal T}=({\cal T}_1,{\cal T}_2)$$ where ${\cal T}_1$ corresponds to $G$ and ${\cal T}_2$ is a Killing vector field of $V$. The next result shows that this trivial situation always holds in the presence of a cylinder and explains why the existence of a local Euclidean cylindrical has been excluded in the statement of Theorem [Theorem 1](#main){reference-type="ref" reference="main"}. **Proposition 3**. *Any non-trivial infinitesimal bending of $f$ as above is as in [\[bending\]](#bending){reference-type="eqref" reference="bending"} up to a trivial infinitesimal bending.* Let ${\cal T}$ be a nontrivial infinitesimal bending of $f$ with associated tensor $B$. From Proposition [Proposition 2](#trivial){reference-type="ref" reference="trivial"} we know that $B\neq 0$. Since $M^n$ is free of flat points, it follows from [\[infGauss2\]](#infGauss2){reference-type="eqref" reference="infGauss2"} that $\Delta\subset\ker B$ at any point of $M^n$. In particular, we have that $BS=0$ for any $S\in\Gamma(TV)$. Then the Codazzi equation $(\nabla_SB)X=(\nabla_XB)S$ for $X\in\Gamma(\Delta^\perp)$ gives that $B$ is a parallel tensor along $V$. Recall that the second fundamental form of $f$ is given by the second fundamental form of $g$ on $\Delta^\perp$. Thus $B$ is a Codazzi tensor defined on $N^k$ which, together with the second fundamental form of $g$, satisfies [\[infGauss2\]](#infGauss2){reference-type="eqref" reference="infGauss2"}. Hence, it follows from the Fundamental Theorem of infinitesimal bendings, namely, Theorem $2.11$ in [@DJ1], that $B$ is the associated tensor of an infinitesimal bending ${\cal T}_1$ of $g$. Finally, it follows from Proposition [Proposition 2](#trivial){reference-type="ref" reference="trivial"} that ${\cal T}-({\cal T}_1,0)$ is a trivial infinitesimal bending of $f$. pt  pt width4 pt height6 pt depth1.5 pt  # Euclidean Kaehler hypersurfaces {#Kaehler} In this section, we first discuss a local parametrization of the nonflat Euclidean Kaehler hypersurfaces in terms of the so called Gauss parametrization, in particular, when the submanifold is minimal. Then, we give a Weierstrass type representation for the hypersurfaces in the latter situation. Let $f\colon M^n\to\mathbb{R}^{n+1}$, $n\geq 3$, be an Euclidean hypersurface of rank two, that is, with precisely two nonzero principal curvatures at each point. Such a hypersurface can be locally parametrized by means of the Gauss parametrization, first used by Sbrana in his two aforementioned papers, that is briefly discussed next. For details about this parametrization we refer to Theorem $7.18$ in [@DT]. Let $g\colon L^2\to\mathbb{S}^n$ be an isometric immersion of a surface into the unit sphere with normal bundle $\Lambda$ and let $\gamma\in C^\infty(L^2)$ be an arbitrary function. Set $h=i\circ g$ where $i\colon\mathbb{S}^n\to\mathbb{R}^{n+1}$ denotes the inclusion map. Then the smooth map $\Psi\colon\Lambda\to\mathbb{R}^{n+1}$ given by $$\label{Gausspar} \Psi(x,w)=\gamma(x)h(x)+h_*\mbox{grad\,}\gamma(x)+i_*w$$ is along the open subset of $\Lambda$ of regular points a parametrization of a hypersurface of rank two. Conversely, any hypersurface of rank two admits such a local parametrization. According to Theorem $2.5$ in [@DG1] or Theorem $15.14$ in [@DT] the hypersurface is Kaehler if and only if the surface $g$ is pseudoholomorphic in the sense of Calabi [@Ca]. The latter means that there exists an orthogonal tensor $T$ in $\Lambda$ that is parallel with respect to the normal connection and satisfies $A_{T\xi}=J_0\circ A_\xi$ for all $\xi\in\Lambda$, where $J_0$ is the almost complex structure on $L^2$. In particular, the surface $g$ is minimal. The submanifold given by [\[Gausspar\]](#Gausspar){reference-type="eqref" reference="Gausspar"} is minimal if and only if $\Delta\gamma+2\gamma=0$ is satisfied. Notice that in this case, the hypersurface is real analytic. Moreover, the Gauss map of the hypersurface is $N(x,w)=g(x)$ and $\gamma={\langle}f,N{\rangle}$ is its support function. From Chapter $15$ in [@DT] we have that any minimal simply connected Kaehler hypersurface without flat points $f\colon M^n\to\mathbb{R}^{n+1}$ admits a one-parameter associated family of noncongruent isometric minimal immersions $f_\theta\colon M^n\to\mathbb{R}^{n+1}$, $\theta\in[0,\pi)$ with $f_0=f$. In fact, if $J_\theta=\cos\theta I + \sin\theta J$ then the *associated family* is given by the line integral $$\label{assfam} f_\theta(x)=\int_{x_0}^xf_*\circ J_\theta$$ where $x_0$ is any fixed point of $M^n$. Since [\[assfam\]](#assfam){reference-type="eqref" reference="assfam"} yields that ${f_\theta}_*=f_*\circ J_\theta$ then all the hypersurfaces in the family have the same Gauss map. Moreover, the hypersurface can be realized as the real part of its *holomorphic representative* $F\colon M^n\to\mathbb{C}^{n+1}$ given by $\sqrt{2}F(x)=(f(x),\bar{f}(x)) \in\mathbb{R}^{n+1}\oplus\mathbb{R}^{n+1}\cong\mathbb{C}^{n+1}$, where $\bar{f} =f_{\pi/2}$ is called the *conjugate* hypersurface to $f$. Finally, we have that $A(\theta)=A\circ J_\theta$ is the shape operator of $f_\theta$. In the unpublished work of Hennes [@He] it was observed that the arguments used in [@DG3] can be adopted in order to produce the local Weierstrass type representation of all nonflat minimal Kaehler hypersurfaces discussed next. Let $U\subset\mathbb{C}$ be a simply connected domain. Start with a nonzero holomorphic function $\alpha_0\colon U\to\mathbb{C}$ and let $\phi_0(z)=\int\alpha_0(z)dz$. Assuming that the functions $\alpha_r, \phi_r\colon U\to\mathbb{C}^{2r+1}$ have been defined for some $0\leq r\leq n-1$ choose any nonzero holomorphic function $\mu_{r+1}\colon U\to\mathbb{C}$ and set $$\alpha_{r+1}=\mu_{r+1} \begin{pmatrix} \frac{1-\phi_r^2}{2}\\ i\frac{1+\phi_r^2}{2}\\ \phi_r\end{pmatrix} \;\;\mbox{and}\;\;\phi_{r+1}=\int\alpha_{r+1}(z)dz,$$ where $\phi_r^2=\phi_r.\phi_r$ with respect to the standard symmetric inner product in $\mathbb{C}^{2r+1}$. Let $W$ be an open subset in $\mathbb{C}^{n-1}$ containing the origin. Then let $F\colon U\times W\to\mathbb{C}^{n+1}$ be given by $$F(z,w_1,\ldots,w_{n-1}) =\sum_{j=0}^{n-1}\int b_j(z)\delta^{(j)}(z)dz +\sum_{j=1}^{n-1}w_j\delta^{(j-1)}(z)$$ where $\delta=\alpha_n$, $\delta'=d\delta/d z$ and $b_0,b_1,\ldots,b_{n-1}\colon U\to\mathbb{C}$ are holomorphic functions such that $b_{n-1}$ is never zero. Set $w_j=u_j+iv_j$, $u=(u_1,\ldots,u_{n-1})$ and $v=(v_1,\ldots,v_{n-1})$. Let $f(z,w_1,\ldots,w_{n-1})=\sqrt{2}Re(F(z,w_1,\ldots,w_{n-1}))$, that is, $$\label{param} \frac{1}{\sqrt{2}}f(z,u,v) =Re\sum_{j=0}^{n-1}\int b_j(z)\delta^{(j)}(z)dz +\sum_{j=1}^{n-1} \left(u_j Re\,\delta^{(j-1)}(z)-v_jIm\,\delta^{(j-1)}(z)\right).$$ Then $M^{2n}=(U\times W, f^*{\langle}\,,\,{\rangle})$ is a Kaehler manifold and $f\colon M^{2n}\to\mathbb{R}^{2n+1}$ is minimal. The conjugate minimal hypersurface of $f$ is $\bar{f}(z,w_1,\ldots,w_{n-1})=\sqrt{2}Im(F)$. Hence $F$ is the holomorphic representative of $f$. Moreover, $f_\theta=\cos\theta f+\sin\theta \bar{f}$ is the full associated family. Since $d f_\theta/d\theta|_{\theta=0}=\bar{f}$ thus the conjugate hypersurface $\bar{f}=f_{\pi/2}$ in the associated family to $f$ is the variational vector field of the variation $f_\theta$. **Remark 4**. **Two examples on case $M^4$ in $\mathbb{R}^5$ of [\[param\]](#param){reference-type="eqref" reference="param"} are computed explicitly in [@He].* * # The proof of Theorem [Theorem 1](#main){reference-type="ref" reference="main"} {#the-proof-of-theorem-main} We have that the non-trivial infinitesimal variation $F$ of $f$ is infinitesimal with respect to the Gauss map. Then [\[B\]](#B){reference-type="eqref" reference="B"} yields $$(\tilde\nabla_X{\cal T}_*)Y={\langle}BX,Y{\rangle}N\;\;\text{for any}\;\;X,Y\in\mathfrak{X}(M).$$ We have seen that ${\cal T}_*\in\Gamma(\mbox{End}(TM))$, therefore ${\cal T}_*$ is a parallel tensor on $M^n$ and it holds that ${\langle}AX,{\cal T}_*Y{\rangle}={\langle}BX,Y{\rangle}$ for any $X,Y\in\mathfrak{X}(M)$. In other terms, we have $$\label{Tpar} (\nabla_X{\cal T}_*)Y=0\;\;\mbox{for any}\;\;X,Y\in\mathfrak{X}(M)$$ and $$\label{BAT} B=A\circ{\cal T}_*=-{\cal T}_*\circ A.$$ From our assumptions and the classical rigidity theorem the hypersurface possesses constant rank $2$. By [\[infGauss2\]](#infGauss2){reference-type="eqref" reference="infGauss2"} the relative nullity vector subspace $\Delta=\ker A$ satisfies $\Delta\subset\ker B$ at any point. It follows from [\[BAT\]](#BAT){reference-type="eqref" reference="BAT"} that ${\cal T}_*$ leaves $\Delta$ invariant and being ${\cal T}_*$ skew-symmetric we have that also $\Delta^\perp$ is left invariant. Since $\Delta^\perp$ is two-dimensional, then we necessarily have that ${\cal T}_*|_{\Delta^\perp}$ is a multiple of a rotation of angle $\pi/2$, say $$\label{rot} {\cal T}_*|_{\Delta^\perp}=cR_{\pi/2},$$ where $c\in\mathbb{R}$ from [\[Tpar\]](#Tpar){reference-type="eqref" reference="Tpar"}. Proposition [Proposition 2](#trivial){reference-type="ref" reference="trivial"} gives that $B\neq0$ and hence $c\neq 0$. Expressing $A$ and ${\cal T}_*|_{\Delta^\perp}$ in an orthonormal basis of principal directions, we have from [\[BAT\]](#BAT){reference-type="eqref" reference="BAT"} that $$B|_{\Delta^\perp}=A|_{\Delta^\perp}{\cal T}_*|_{\Delta^\perp} =\begin{bmatrix} \lambda_1 & 0\\ 0 & \lambda_2 \end{bmatrix} \begin{bmatrix} 0 & -c\\ c & 0 \end{bmatrix}= c\begin{bmatrix} 0 & -\lambda_1\\ \lambda_2 & 0 \end{bmatrix}.$$ Then the symmetry of $B$ yields that $\lambda_1+\lambda_2=0$, that is, $f$ is a minimal hypersurface and, in particular, it is real analytic. Notice that this concludes the proof for the surface case. We have from [\[infben\]](#infben){reference-type="eqref" reference="infben"} that the vector subspaces $\ker {\cal T}_*$ and $(\ker{\cal T}_*)^\perp$ are ${\cal T}_*$-invariant. Then, on any open subset of $M^n$ where their dimensions are constant it follows from [\[Tpar\]](#Tpar){reference-type="eqref" reference="Tpar"} that they form totally geodesic distributions. Since ${\cal T}_*|_{\Delta^\perp}\neq 0$ and leaves $\Delta^\perp$ invariant then $\ker{\cal T}_*\subset\Delta$. Hence, if $\dim\ker{\cal T}_*=\ell>0$ is constant, we necessarily have that $f$ is locally a cylinder over an isometric immersion $g\colon N^{2k}\to\mathbb{R}^{2k+1}$ of an even dimensional Riemannian manifold $N^{2k}$. But this possibility has been ruled out by assumption. Let $f_t\colon M^n\to\mathbb{R}^{n+1}$ be the variation of $f$ defined by $f_t=f+t{\cal T}$ for $t\in\mathbb{R}$. Using [\[infben\]](#infben){reference-type="eqref" reference="infben"} we have $$\|f_{t*}X\|^2=\|f_*X\|+t^2\|{\cal T}_*X\|^2.$$ Thus $f_t$ and $f_{-t}$ for each $t$ induce the same Riemannian metric $g_t$ on $M^n$. Moreover, being ${\cal T}$ non-trivial it follows from either Proposition $14.3$ in [@DT] or Proposition $2.12$ in [@DJ1] that these two immersions are not congruent. In our situation, we have from [\[normal\]](#normal){reference-type="eqref" reference="normal"} that the unit normal vector field $N$ is in fact normal to $f_t$ for all $t$, i.e., the $f_t$, $t\in\mathbb{R}$, share the same Gauss map. Then, from Corollary $4.12$ in [@DG1] we have that the Riemannian manifold $(M^n,g_t)$ admits an almost complex structure $J_t$ which is a parallel orthogonal tensor and that the immersions $f_t$ and $f_{-t}$ are both minimal. Thus $f_t$ it is a minimal Kaehler hypersurface of the Euclidean space for each $t$. Then, from an aforementioned result it follows that the Gauss map $N_t=N$ of $f_t$ determines a pseudoholomorphic surface of $\mathbb{S}^n$. Since $f$ shares the same Gauss map with all $f_t$, thus $(M^n,g_0)$ is a Kaehler manifold of dimension $n=2k$. Let $J$ be the almost complex structure of $M^{2k}$. Since the immersion $f$ is minimal we have from either Theorem $1.2$ in [@DR] or Theorem $15.7$ in [@DT] that $A\circ J=-J\circ A$. Therefore $\Delta$ and $\Delta^\perp$ are $J$-invariant and thus $J|_{\Delta^\perp}=R_{\pi/2}$. Hence, it follows from [\[assfam\]](#assfam){reference-type="eqref" reference="assfam"} that the conjugate minimal immersion $\bar{f}=f_{\pi/2}$ satisfies $\bar{f}_*X=f_*JX$ for any $X\in\mathfrak{X}(M)$. In particular, $${\langle}f_*X,\bar{f}_*X{\rangle}=0\;\;\mbox{for any}\;\;X\in\mathfrak{X}(M)$$ and thus $\bar{f}$ is an infinitesimal bending of $f$. Moreover, recall from Section [2](#Kaehler){reference-type="ref" reference="Kaehler"} that the associated family to $f$ has $\bar{f}$ as its variational vector field and that all the immersions share the same Gauss map. Thus, in particular, [\[normal\]](#normal){reference-type="eqref" reference="normal"} holds for $\bar{f}$. It follows from [\[B\]](#B){reference-type="eqref" reference="B"} that its associated tensor $\bar{B}$ is the second fundamental form of $\bar{f}$, that is, $\bar{B}=A\circ J$. On the other hand, from [\[BAT\]](#BAT){reference-type="eqref" reference="BAT"} and [\[rot\]](#rot){reference-type="eqref" reference="rot"} the tensor associated to ${\cal T}$ satisfies $B|_{\Delta^\perp}=cA|_{\Delta^\perp}\circ R_{\pi/2}$. Then Proposition [Proposition 2](#trivial){reference-type="ref" reference="trivial"} yields that the infinitesimal bending ${\cal T}-c\bar{f}$ is trivial. Therefore, we have ${\cal T}-c\bar{f}=\mathcal{D}f+w$ where $\mathcal{D}\in\mbox{End}(\mathbb{R}^{2k+1})$ is skew-symmetric and $w\in\mathbb{R}^{2k+1}$. Since ${\cal T}_*$ and $c\bar{f}_*$ coincide on $\Delta^\perp$ then $\Delta^\perp\subset\ker\mathcal{D}$. Moreover, we have from [\[normal\]](#normal){reference-type="eqref" reference="normal"} for ${\cal T}$ and $\bar{f}$ that ${\langle}\mathcal{D}X,N{\rangle}=0$ for all $X\in\mathfrak{X}(M)$ and hence $N\in\ker\mathcal{D}$. Therefore $\ker\mathcal{D}^\perp$, seen in $T_{f(x)}\mathbb{R}^{2k+1}$, lies in $f_*\Delta(x)$ at any point $x\in M^{2k}$. Being $\mathcal{D}$ constant we have that $f$ would necessarily be a cylinder with Euclidean factor given by $\ker\mathcal{D}^\perp$. Thus $\mathcal{D}$ vanishes and therefore ${\cal T}$ coincides with $c\bar{f}+w$ for some $c\in\mathbb{R}$ and $w\in\mathbb{R}^{2k+1}$. pt  pt width4 pt height6 pt depth1.5 pt  Funding statements: Marcos Dajczer is partially supported by the grant PID2021-124157NB-I00 funded by MCIN/AEI/10.13039/501100011033/ 'ERDF A way of making Europe', Spain, and is also supported by Comunidad Autónoma de la Región de Murcia, Spain, within the framework of the Regional Programme in Promotion of the Scientific and Technical Research (Action Plan 2022), by Fundación Séneca, Regional Agency of Science and Technology, REF, 21899/PI/22. Miguel I. Jimenez is supported by FAPESP with the grant 2022/05321-9. lll Bianchi, L., *Sulle varietà a tre dimensioni deformabili entro lo spazio euclideo a quattro dimensioni*, Memorie di Matematica e di Fisica della Società Italiana delle Scienze III, XIII (1905), 261--323. Calabi, E., *Isometric embeddings of complex manifolds*, Ann. of Math. **58** (1953), 1--23. Cartan, E., *La déformation des hypersurfaces dans l'espace euclidien réel a $n$ dimensions*, Bull. Soc. Math. France **44** (1916), 65--99. Cartan, E., *La déformation des hypersurfaces dans l'espace conforme réel a $n\ge 5$ dimensions*, Bull. Soc. Math. France **45** (1917), 57--121. Cesàro, E., "Lezioni di Geometria intrinseca". Napoli, 1896. Dajczer, M., Florit, L. and Tojeiro, R., *On deformable hypersurfaces in space forms*, Ann. Mat. Pura Appl. **174** (1998), 361--390. Dajczer, M. and Gromoll, D., *Real Kaehler submanifolds and uniqueness of the Gauss map*, J. Differ. Geom. **22** (1985), 13--28. Dajczer, M. and Gromoll, D., *Euclidean hypersurfaces with isometric Gauss maps*, Math. Z. **191** (1986), 201--205. Dajczer, M. and Gromoll, D., *The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two*, Invent. Math. **119** (1995), 235--242. Dajczer, M. and Jimenez, M.I., *Infinitesimal variations of submanifolds*, Ensaios Matemáticos, **35** (2021), 1--156.\ Free access at <https://doi.org/10.21711/217504322021/em35> Dajczer, M. and Jimenez, M.I., *A construction of Sbrana-Cartan hypersurfaces in the discrete class*, Proc. Amer. Math. Soc. **151** (2023) 3069--3081. Dajczer M., Jimenez M. I. and Vlachos, Th., *Conformal infinitesimal variations of Euclidean hypersurfaces*. Ann. Mat. Pura Appl. **201** (2022), 743--768. Dajczer, M. and Rodríguez, L., *Rigidity of real Kaehler submanifolds*. Duke Math.J. **53** (1986), 211--220. Dajczer, M. and Tojeiro, R., "Submanifold theory beyond an introduction\". Universitext. Springer, New York, 2019. Dajczer, M. and Vlachos, Th., Infinitesimally bendable Euclidean hypersurfaces, Ann. Mat. Pura Appl. **196** (2017), 1961--1979 and Ann. Mat. Pura Appl. **196** (2017), 1981--1982. Darboux, G., "Leçons sur la théorie générale des surfaces I, II\". Gauthier-Villars, Paris 1941. Hennes, P., *Weierstrass representations of minimal real Kähler submanifolds*, Thesis Ph.D., State University of New York at Stony Brook. 2001. Sbrana, U., *Sulla deformazione infinitesima delle ipersuperficie*, Ann. Mat. Pura Appl. **15** (1908), 329--348. Sbrana, U., *Sulle varietà ad $n-1$ dimensioni deformabili nello spazio euclideo ad $n$ dimensioni*, Rend. Circ. Mat. Palermo **27** (1909), 1--45. Schouten, J. A. *On infinitesimal deformations of $V^m$ in $V^n$*. Proceedings Amsterdam **31** (1928), 208--218. Marcos Dajczer\ Departamento de Matemáticas\ Universidad de Murcia, Campus de Espinardo\ E-30100 Espinardo, Murcia, Spain\ e-mail: marcos\@impa.br Miguel Ibieta Jimenez\ Instituto de Ciências Matemáticas e de Computação\ Universidade de São Paulo\ São Carlos\ SP 13566-590-- Brazil\ e-mail: mibieta\@icmc.usp.br

arxiv_math

--- abstract: | The performance of sparse matrix computation highly depends on the matching of the matrix format with the underlying structure of the data being computed on. Different sparse matrix formats are suitable for different structures of data. Therefore, the first challenge is identifying the matrix structure before the computation to match it with an appropriate data format. The second challenge is to avoid reading the entire dataset before classifying it. This can be done by identifying the matrix structure through samples and their features. Yet, it is possible that global features cannot be determined from a sampling set and must instead be inferred from local features. To address these challenges, we develop a framework that generates sparse matrix structure classifiers using graph convolutional networks. The framework can also be extended to other matrix structures using user-provided generators. The approach achieves 97% classification accuracy on a set of representative sparse matrix shapes. author: - Khaled Abdelaal - Richard Veras bibliography: - refs.bib title: "Observe Locally, Classify Globally: Using GNNs to Identify Sparse Matrix Structure" --- # Introduction {#fig:diag-orig width="\\linewidth"} {#fig:diag-scrm width="\\linewidth"} Sparse matrices represent a fundamental building block used throughout the field of scientific computing in applications, such as graph analytics, machine learning, fluid mechanics, and finite element analysis [@snapnets; @10.1145/2049662.2049663]. Such matrices appear as operands in numerous fundamental computational kernels such as sparse matrix-vector multiplication (SpMV), Cholesky factorization, LU factorization, sparse matrix -dense matrix multiplication, and matricized tensor times Khatri-Rao product (MTTKRP) among others. Building efficient algorithms for this class of kernels mainly depends on the storage format used for the sparse matrix as observed in different studies [@10.1145/1693453.1693471; @6375570]. A variety of such formats are proposed in literature [@10.1145/3017994; @7036061]. Hence, it is crucial to identify the structure of the matrix to choose the ideal sparse format, and eventually tailor the algorithm to that format to optimize the workload performance. However, identifying the structure of the matrix is not always trivial. Figure [1](#fig:diag-orig){reference-type="ref" reference="fig:diag-orig"} shows a spy plot of an off-diagonal sparse matrix, and Figure [2](#fig:diag-scrm){reference-type="ref" reference="fig:diag-scrm"} shows the same matrix, with some of the original row indices and column indices re-labelled. It is less obvious for the latter figure to provide an insight of the original structure of the non-zeros within the matrix. Additionally, in case of huge sparse matrices, we might only have access to samples of the matrix. This could be because of computational or storage restrictions, or missing data. In these two cases (re-labelling and sub-sampling), we need efficient techniques to recognize the shape of the input matrix. To tackle mentioned issues, we propose a framework to identify sparse matrices structures, using graph neural networks. Figure [\[fig:conf-mat\]](#fig:conf-mat){reference-type="ref" reference="fig:conf-mat"} shows the confusion matrix for the proposed framework using four sample classes on re-labelled variants. The framework design is modular, allowing users to easily augment it with new structures generators or feature sets. The main contributions of this paper are as follows: - Proposing a novel, modular Graph Neural Network framework to accurately predict the shapes of sparse matrices, including partial samples, and re-labelled variants of original matrices. - Presenting a new balanced synthetic dataset for structured sparse matrices. - Providing a performance analysis of graph-level classification on sparse matrices, using different feature sets. - Introducing two new compact and efficient feature sets for matrices as graphs, namely: Linear and Exponential Binned One-Hot Degree Encoding. The rest of this paper is organized as follows: Section [2](#sec:bg){reference-type="ref" reference="sec:bg"} introduces the necessary background, Section [3](#sec:methods){reference-type="ref" reference="sec:methods"} details the design of the proposed framework, Section [4](#sec:analysis){reference-type="ref" reference="sec:analysis"} discusses the evaluation and results of the framework, while Section [5](#sec:related){reference-type="ref" reference="sec:related"} describes related work. Finally, Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} summarizes the findings of the paper. # Background {#sec:bg} ## Graph Neural Networks Graph neural networks (GNNs) [@4700287] are a class of deep learning models that operate on graphs or networks. Unlike traditional neural networks that operate on structured data such as images or sequences, GNNs can handle arbitrary graph structures with varying node and edge attributes, enabling them to learn powerful representations of graph-structured data. The key idea behind GNNs is to iteratively update node embeddings by aggregating information from the embeddings of their neighbors through the \"graph convolution\" operation. By stacking multiple layers of graph convolution and non-linear activation functions, GNNs can learn hierarchical representations of the graph that capture both local and global information. ## Structured Matrices Several common structures are observed in sparse matrices, such as: **Diagonal** all non-zeros are located on the main or a secondary diagonal. This structure represents a 1D mesh and commonly appears in various scientific and engineering applications. **Random** the non-zero elements are randomly distributed across the matrix, with variable density. Such matrices have no specific identifiable structure. **Kronecker Graphs** [@JMLR:v11:leskovec10a] are a class of synthetic graphs that have been widely used to model real-world networks, and are generated by recursively applying the Kronecker product of a small base graph with itself. Let $A$ and $B$ be two matrices. Then, their Kronecker product $A \otimes B$ is given by $$A \otimes B = \begin{pmatrix} a_{11}B & \cdots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \cdots & a_{mn}B \end{pmatrix}$$ where $a_{ij}$ are the entries of $A$. We use these three classes of structures, and a combination of them, as a representative set that can be combined to form more complex relationships [@1386651; @van1992computational]. Our framework is not limited to only these structures, and they serve as an example to evaluate its performance. {#fig:diag-deg width=".8\\textwidth"} {#fig:rand-deg width=".8\\textwidth"} {#fig:rand-diag-deg width=".8\\textwidth"} {#fig:rand-deg width=".8\\textwidth"} **Degree as a representative node feature** Figure [\[fig:deg-dist\]](#fig:deg-dist){reference-type="ref" reference="fig:deg-dist"} illustrates that one can accurately distinguish between the different classes based on the degree distribution of the representative graph. For example, for Diagonal matrices (Figure [3](#fig:diag-deg){reference-type="ref" reference="fig:diag-deg"}), the degree for all nodes is low, and is either constant or linear across all nodes. Kronecker graphs follow a power-law degree distribution, with only a few nodes having many connections (high degree) and most of the nodes having relatively few connections (low degree). However, only the local per-node degree view may be immediately available, and not the global graph view. An example of such a case is only having a sample of the graph and not the entire graph due to storage or computational limitations. The power of GNNs can be leveraged to carry out the required task: the prediction of the sample matrix structure. # Framework Description {#sec:methods} {width="\\textwidth"} The goal of the proposed framework is to predict the structure of the input sparse matrix through its classification into one of the configured target classes. We use diagonal, random, diagonal+random, and Kronecker graph as examples of these classes to evaluate the performance of the framework. New structures can be seamlessly integrated. Figure [\[fig:high-level\]](#fig:high-level){reference-type="ref" reference="fig:high-level"} shows a high-level overview of the proposed framework. It consists of three main stages: Dataset generation, Model Training, and Inference. A synthetic dataset is generated using different generators for different shapes of matrices, which are then represented as graphs. In the training phase, we use GNN with 5-fold cross validation to evaluate the model performance. Finally, the trained model instance is used for later inference. ## Dataset Generation {#sec:gen} We generate a balanced dataset of 40K synthetic sparse matrices, covering the four sample classes through individual generators. Each of the generators returns a Coordinate (COO) representation for the matrix, excluding the actual non-zero values. The COO representation is then used as the adjacency list to build the graph representation. ## Feature Set Selection {#sec:feat} A per-node feature vector is necessary for the graph neural network to classify matrices. Node degree can be calculated for rows/columns in input matrices from their graph representation. **One-Hot Degree Encoding** uses a number of features equal to the maximum degree + 1. A limitation of this encoding is that it requires the knowledge of the maximum degree in the entire dataset before training. Also, the required storage is proportional to the maximum degree recorded in the dataset, which increases memory requirements and reduces maximum possible batch size during training. Moreover, it poses complications during inference if the input matrix has a degree greater than the maximum degree in the training set. In our dataset, the maximum training set degree is 7710, so the length of one-hot encoding feature vector per node is 7711, limiting the maximum batch size on GPU to only 1 graph. **Local Degree Profile** (LDP) [@cai2018simple] captures the local structural information of nodes in their immediate neighborhood. LDP is calculated for each node as a five-feature vector: the node degree, the minimum degree of its neighbors, the maximum degree of its neighbors, the mean degree for its neighbors, and the standard deviation of the degrees of each neighbors. LDP features are easy to compute for any given graph. Additionally, the number of features per node is fixed, regardless of the used training data. LDP incurs low storage overhead. **Linear Binned One-Hot Degree Encoding** [\[sec:linear-onehot\]]{#sec:linear-onehot label="sec:linear-onehot"} (LBOH) We implement a modified version of one-hot encoding, to address its limitations. LBOH works by having a fixed number of buckets for representing one-hot degrees. Buckets ranges are designed as follows: a set individual sequential buckets from 0 t $\alpha$ (inclusive) where $\alpha$ is a small integer (less than 10). Then, we add a set of buckets with fixed step $\beta$ from $\alpha$: $(\alpha + \beta)$, $(\alpha + 2\beta)$, \..., $(\alpha + k\beta)$ where $(\alpha + k\beta)$ is the maximum degree threshold. Any degree greater than $(\alpha + k\beta)$ is mapped to the final bucket. {#fig:linear-onehot} Figure [7](#fig:linear-onehot){reference-type="ref" reference="fig:linear-onehot"} shows an example of LBOH encoding. As opposed to One-Hot Encoding, LBOH provides a fixed number of features regardless of the maximum degree in the training dataset. At the inference stage, only the values of $\alpha$, $\beta$, and $k$ are needed. **Exponential Binned One-Hot Degree Encoding (EBOH)** The main difference between EBOH and LBOH is the kind of step between buckets ranges. Instead of a linear step in LBOH, EBOH uses an exponential step to cover more degree values with a small number of features. First, the value of $\alpha$ is chosen such that $1 \leq \alpha \leq 3$. Then for the buckets, a sequential one-to-one mapping is performed for values 0 through $2^\alpha$. For the following buckets, the upper bound (inclusive) is $2^{\alpha + i}$ where $i \in [1,k]$ and $k \in \mathbb{N}^*$. {#fig:expo-onehot} Figure [8](#fig:expo-onehot){reference-type="ref" reference="fig:expo-onehot"} shows an example of EBOH encoding. EBOH encoding still provides the benefit of having the number of features independent of the maximum degree in the training set. ## The Graph Neural Network Architecture To identify the structure of the input matrix, the matrix is viewed as the adjacency list of a graph, enabling the use of machine learning methods designed for graph data. GNNs provide additional benefits such as allowing the use of matrices (graphs) of arbitrary sizes as input, Also, GNNs are agnostic to node ordering. This powerful property enables re-labelling or permuting nodes in a graph representing a sparse matrix, while maintaining accurate predictions. The machine learning task of interest is graph-level prediction since a single label (class) is needed for an entire graph (matrix). The GNN architecture is shown in Figure [\[fig:GCN\]](#fig:GCN){reference-type="ref" reference="fig:GCN"}. The hidden layers are three graph convolution layers and one linear (output) layer. Graph convolution is an operation where node embeddings are iteratively generated as the aggregations from the node neighborhoods. This operation is used to capture complex features of the graph. The first convolution layer aggregates information from the local neighborhood of each node. This operation is repeated in subsequent convolution layers in order to propagate information to increasingly larger neighborhoods. By the end of three convolution layers, the model has learned a hierarchical representation of the graph, where the features at each layer capture increasingly complex structural patterns. The learned representation so far is \"node embeddings\". Then, learned node embeddings are reduced into a single graph embedding using a global mean pooling operation (called readout layer). Samples are randomly dropped out to reduce overfitting. Finally, a linear classifier is applied to the graph embedding. ``` {.python language="Python"} def generateDiagRandom(size, threshold=2): """ A function to generate a Diag+Random square matrix """ tuples = [(x,y) for x in range(size) for y in range(size) if (random.randint(0,10) <= threshold or x == y)] # seperate tuples into two lists: the row array and the column array coo_rep = list(map(list, zip(*tuples))) return coo_rep[0], coo_rep[1], [size, size] ``` ``` {.python language="Python"} def process(self): catMap = [...., { # Number of instances to generate for this class 'num_iter':10000, # Name of the generator function 'generator':generateDiagRandom, # A string list of required generator function param 'gen_params':['random.randint(MIN_DIM_SIZE,MAX_DIM_SIZE)'] }] ``` **Modularity** New shapes of matrices can be easily integrated in our framework. To achieve this, two steps are needed as shown in Figure [\[fig:addClass\]](#fig:addClass){reference-type="ref" reference="fig:addClass"}: (1) write a generator for that new shape, and (2) add an entry to the categories (shapes) map in the dataset class for this shape, containing the number of dataset instances to generate, the name of the generator function, and the different required parameters. The generator is required to return the COO representation excluding values, and the matrix dimensions. After generating the new dataset instances for this class, one does not need to re-train the entire model again. Transfer learning [@weiss2016survey] can be used to replace the last layer of the trained model with a new layer that has the appropriate number of outputs, after introducing the new shape(s). Then, the weights of all previous layers are frozen and only the new layer is trained. Another aspect of modularity in our framework is the ability to seamlessly attach different feature sets. Feature sets are only computed when the graph is queried. To implement a new feature set, a modification to the `get` method of the dataset is needed. This method first reads in the graph file from disk, calculates the new feature set, and attaches it to the graph. # Analysis {#sec:analysis} We run a set of experiments to evaluate the accuracy of our approach in detecting structures, using the different feature representation discussed in Section [3.2](#sec:feat){reference-type="ref" reference="sec:feat"}. ## Evaluation **Experimental Setup** Table [1](#tab:experimental-setup){reference-type="ref" reference="tab:experimental-setup"} shows the experimental setup and learning parameters used in the experiments. We use PyTorch Geometric [@Fey/Lenssen/2019] for the GNN. ------------------ ------------------- ------------------------ --------------- **Component** **Specification** **Parameter** **Value** (lr)1-2 GPU NVIDIA RTX A6000 Optimizer Adam GPU Memory 48 GB GDDR6 Learning Rate 0.01 CUDA Version 11.8 Error Criterion Cross Entropy Main Memory 64 GB DDR4 Batch Size 256\* Operating System Ubuntu 22.04 Cross Validation Folds 5 ------------------ ------------------- ------------------------ --------------- : Experimental setup and Training parameters used in the experiments. \* Batch size used for traditional one-hot encoding is 1. [\[tab:experimental-setup\]]{#tab:experimental-setup label="tab:experimental-setup"} **Evaluation Metrics** [\[sec:eval-metrics\]]{#sec:eval-metrics label="sec:eval-metrics"} To evaluate the prediction accuracy of the framework, four derived metrics are used: accuracy, precision, recall, and F1-score. We report per-class and overall accuracy and F1 score numbers, since the latter is the harmonic mean of precision and recall. Using both accuracy and F1-score helps provide a more comprehensive evaluation of the framework's performance. Accuracy gives an overall view of how well the classifier is performing, while the F1-score provides insights into its ability to correctly classify positive instances. ## Results {#sec:results} **Classification Performance** Table [\[tab:performance\]](#tab:performance){reference-type="ref" reference="tab:performance"} shows the accuracy and F1 score for the classifier using the different degree representations discussed in Section [3.2](#sec:feat){reference-type="ref" reference="sec:feat"}. Performance results show that both LBOH, and EBOH provide high prediction accuracy of around 97% and a F1 score of around 98%. On the other hand, traditional one-hot encoding exhibits a lower accuracy of around 90%. One-Hot Encoding requires a significantly large number of features per node (7711), limiting the training batch size on the A6000 GPU to only one graph. This forces the optimizer to adjust the neural network weights very frequently, hence hurting the overall accuracy. Using LDP as a feature set exhibits variant model performance across folds depending on the validation set being used. In some folds, LDP provides high accuracy of around 97% to 98% similar to EBOH. In other folds, LDP fails to converge to an acceptable loss value, and ends up with an accuracy of around 74% on the last few epochs. This performance variance across folds deems LDP unfit for the purposes of our application. It significantly fails in two classes: Random and Kronecker. It predicts Random matrices as Random+Diagonal for more than 32.5% of the instances. This is likely due to the prevalence of the local degree neighbor summary features (the last four LDP features) instead of focusing on the node degree. This eventually results in failing to discover the global hierarchical structures in the matrix. LDP still shows perfect accuracy in case of diagonal matrices since almost all nodes in the matrix's graph have the same degree. LDP prediction quality for Kronecker graphs is also lower than other evaluated feature sets (around 81% in some folds) for the same reasons. {#fig:loss-folds width="\\textwidth"} {#fig:loss-ldp width="\\textwidth"} Figure [9](#fig:loss-folds){reference-type="ref" reference="fig:loss-folds"} demonstrates the validation loss across the 5 different folds for EBOH. It shows almost no variance in the loss across the different folds, indicating the stability of the model's performance across folds. On the other hand, Figure [10](#fig:loss-ldp){reference-type="ref" reference="fig:loss-ldp"} shows the validation loss for LDP and illustrates that the loss does not converge in 2 out of 5 folds. **Classifying Sub-samples and Re-labelled Subgraphs** To test the efficacy of GNN on both aspects, we generate 200 new matrices: 50 for each of the four classes. For each of them, we generate 10 subgraphs and 10 re-labelled variants. To generate the subgraphs, we use uniform random node sampling (URNS) [@10.1145/1150402.1150479]: nodes are randomly selected with uniform probability, as well as the edges connecting the selected nodes. Re-labelling of a graph $G$ simply renames the nodes $V$ of the graph, and produces a new graph $G'$ with the same size and degree distribution of the original graph $G$. Figure [\[fig:subgraph-example\]](#fig:subgraph-example){reference-type="ref" reference="fig:subgraph-example"} shows an example of both URNS and random re-labelling. **Class** **Node Sampling** **Node Re-labelling** **Original Graphs** ----------------- ------------------- ----------------------- --------------------- Diagonal Random Random+Diagonal Kronecker **Overall** **0.92** **0.97** **0.97** : Accuracy comparison for node sampling, node re-labelling, and original graphs using EBOH feature set. [\[tab:matrix_accuracy\]]{#tab:matrix_accuracy label="tab:matrix_accuracy"} Table [2](#tab:matrix_accuracy){reference-type="ref" reference="tab:matrix_accuracy"} shows the model's performance on subgraphs and re-labelled variants as compared to original full graphs. The table shows that re-labelling node has no impact on the classification accuracy; it shows the same overall accuracy of around 97% which is observed for the original graphs. This is expected because the arrangement of nodes in a graph is irrelevant, since the graph has the same degree distribution. For subgraphs generated using URNS of larger graphs, the overall accuracy drops to around 92%. The reason being that random node sampling can alter the degree distribution of the graph. The random choice of nodes can result in either isolated nodes (no edges) or much lower degree nodes as compared to the original graph. This affects the accuracy specially for complex shapes such as random and Kronecker graphs. One way to reduce the accuracy loss for samples is to use a more sophisticated graph sampling technique rather than randomly selecting nodes or edges. # Related Work {#sec:related} **Prediction on Sparse Matrices** Several studies investigated the use of machine learning to predict the optimal sparse format for SpMV on CPU and GPU [@10.1145/3178487.3178495; @10.1145/3218823; @10.1145/2304576.2304624; @6748055; @10.1145/2499370.2462181; @7573853]. Our framework does not directly predict the best sparse format, instead, we only predict the structure of the input matrix. This allows de-coupling the sparsity pattern from the sparse format, following the argument adopted by AlphaSparse [@10.5555/3571885.3571972] since our framework also allows the seamless integration of new classes. Existing techniques collect a set of features from each matrix such as: the number of diagonals, the ratio of true diagonals to total diagonals, the (maximum) number of non-zeros per row, the variation of the number of nonzeros per row, the ratio of nonzeros in DIA and ELL data structures, and a factor or power-law distribution. We only need to calculate one feature per node: its degree. Also, [@10.1145/3178487.3178495] uses a CNN approach to treat matrices as images, and in order to fix the size of the matrix, they normalize input matrices into fixed size blocks, losing partial matrix information in the process. In contrast, our approach handles arbitrary sizes of matrices, without losing precision, leveraging the power of Graph Neural Networks. We can optionally sample large matrices and maintain high prediction accuracy. An additional benefit to our framework is that it is order in-variant, since matrices are represented as graphs. **Graph Representation for Learning** Representing non-attribute graphs is an open problem [@10.1145/3511808.3557661]. Common approaches employ graph properties such as node degree, more specifically a one-hot encoding of the degree [@xu2018how]. One-hot encoding suffers from numerous limitations (Section [3.2](#sec:feat){reference-type="ref" reference="sec:feat"}). LDP [@cai2018simple] provides a compact representation for graph using five features per node. Although the computation of such feature vector is efficient, using LDP results in unreliable model performance for our task (Section [4.2](#sec:results){reference-type="ref" reference="sec:results"}). Both our representations (LBOH and EBOH) outperform one-hot encoding and LDP while addressing their shortcomings. # Summary {#sec:conclusion} In this paper, we proposed a GNN based framework to classify structured sparse matrices. We introduced two novel non-attribute graph representations based on node degrees: LBOH, and EBOH. We evaluated the efficacy of our framework on a synthetic, balanced dataset of matrices that we generated containing random matrices from four sample classes: diagonal, random, random+diagonal, and Kronecker graphs. Performance results demonstrate a high classification accuracy of 97% for the framework when using our feature sets: LBOH and EBOH. They also show high accuracy of 92% and 97% on random node subsamples and re-labelled variants respectively. Our framework is modular, allowing the inclusion of additional classes with minimal user effort. Future endeavors target the automatic generation of the optimal sparse data format and algorithm for sparse matrix kernels, using the obtained prediction results from the current framework.

arxiv_math

--- abstract: | We generalize a Furstenberg-type result of Orponen-Shmerkin to higher dimensions, leading to an $\varepsilon$-improvement in Kaufman's projection theorem for hyperplanes and an unconditional discretized radial projection theorem in the spirit of Orponen-Shmerkin-Wang. Our proof relies on a new incidence estimate for $\delta$-tubes and a quasi-product set of $\delta$-balls in $\mathbb{R}^d$. author: - Kevin Ren bibliography: - main.bib date: June 2022 title: Discretized Radial Projections in $\R^d$ --- # Introduction Let $X$ be a set in $\mathbb{R}^n$, and define the radial projection $\pi_x (y) := \frac{y-x}{|y-x|} \in S^{n-1}$. We wish to study the size of radial projections $\pi_x (Y)$ of $Y$, where $x$ is taken in some set $X$. Recently, Orponen, Shmerkin, and Wang [@orponen2022kaufman] proved a strong radial projection theorem in two dimensions, but they prove a conditional result in higher dimensions. In this paper, we shall remove the condition $\dim_H (X) \ge k - \frac{1}{k} + \eta(k)$ in higher dimensions, which answers Conjecture 1.5 of [@shmerkin2021distance] and improves Theorem 1.9 of [@orponen2022kaufman]. We also improve upon the previously known result $\frac{d-1}{d} \min(\dim_H (X), \dim_H (Y)) + \eta(d, \dim_H (X), \dim_H (Y))$ of [@shmerkin2022non Theorem 6.15]. **Theorem 1**. *Let $X, Y \subset \mathbb{R}^d$ be Borel sets with $\dim_H (X), \dim_H (Y) \le k$. If $X$ is not contained in a $k$-plane, then $$\sup_{x \in X} \dim_H (\pi_x (Y \setminus \{ x \})) \ge \min(\dim_H (X), \dim_H (Y)).$$* In fact, we can prove the following slicing result, which improves Proposition 6.8 of [@orponen2022kaufman] and makes progress towards answering Conjecture 1.10 of [@orponen2022kaufman]. **Corollary 2**. *Let $s \in (d-2, d]$, then there exists $\varepsilon(s, d) > 0$ such that the following holds. Let $\mu, \nu$ be Borel probability measures on $\mathbb{R}^d$ with disjoint supports that satisfy $\mathcal{E}_s (\mu), \mathcal{E}_s (\nu) < \infty$ and $\dim_H (\mathrm{spt}(\nu)) < s + \varepsilon(s, d)$. Further, assume that $\mu, \nu$ don't simultaneously give full measure to any affine $(d-1)$-plane $H \subset \mathbb{R}^d$. Then there exist restrictions of $\mu, \nu$ to subsets of positive measure (which we keep denoting $\mu, \nu$) such that the following holds. For almost every affine 2-plane $W \subset \mathbb{R}^d$ (with respect to the natural measure on the affine Grassmanian), if the sliced measures $\mu_W$, $\nu_W$ on $W$ is non-trivial, then they don't simultaneously give full measure to any line. In other words, $$(\gamma_{d,2} \times \mu) \{ (V, x) : \mu_{V,x} (\ell) \nu_{V,x} (\ell) = |\mu_{V,x}| |\nu_{V,x}| > 0 \text{ for some } \ell \in \mathbb{A}(V + x, 1) \} = 0,$$ where we parametrize affine 2-planes as $V + x$, for $x \in \mathbb{R}^d$ and $V$ in the Grassmannian $\mathrm{Gr}(d, 2)$ with the rotationally invariant Haar measure $\gamma_{d,2}$.* We also deduce an $\varepsilon$-improvement in Kaufman's projection theorem for hyperplanes. The proof is a standard higher-dimensional generalization of the details in [@orponen2021hausdorff Section 3.2] and we will omit it. For $\sigma \in S^{n-1}$, let $\pi_\sigma$ be projection in the direction orthogonal to $\sigma$. **Theorem 3**. *For every $k < s < t \le d$, there exists $\varepsilon(s, t)$ such that the following holds. Let $E$ be an analytic set in $\mathbb{R}^d$ with $\dim_H (E) = t$. Then $$\dim_H \{ \sigma \in S^{d-1} : \dim_H (\pi_\sigma (E)) \le s \} \le s - \varepsilon.$$* **Remark 4**. *Kaufman's theorem is sharp when $s = k$ and $t \in (k, k+1]$ because $E$ can be contained within a $(k+1)$-plane.* We also derive a higher-dimensional version of Beck's theorem (unlike in the discrete setting, the higher-dimensional version cannot proved by projection onto a generic 2D plane). The proof again follows similarly to the 2D version presented in [@orponen2022kaufman Corollary 1.4]. **Corollary 5**. *Let $X \subset \mathbb{R}^d$ be a Borel set such that $\dim_H (X \setminus H) = \dim_H X$ for all $k$-planes $H$. Then, the line set $\mathcal{L}(X)$ spanned by pairs of distinct points in $X$ satisfies $$\dim_H (\mathcal{L}(X)) \ge \min\{ 2 \dim_H X, 2k \}.$$* ## Connections and related work Radial projections have also been used to study the Falconer distance set problem, which asks for lower bounds on the Hausdorff dimension of the distance set $\Delta(X) := \{ |x - y| : x, y \in X \}$ given the value of $\dim_H (X)$ for some $X \in \mathbb{R}^d$. In two dimensions, Wolff [@wolff1999decay] used Fourier analysis to show that if $\dim_H (X) \ge \frac{4}{3}$, then $\Delta(X)$ has positive Lebesgue measure. Using Orponen's radial projection theorem [@orponen2018radial], Guth-Iosevich-Ou-Wang [@guth2020falconer] used a good-bad tube decomposition and decoupling to improve the threshold to $\dim_H (X) \ge \frac{5}{4}$. See also works of Keleti-Shmerkin [@keleti2019new] [@keleti2019new], Shmerkin [@shmerkin2021improved], Liu [@liu2020hausdorff], and Stull [@stull2022pinned] which provide better lower bounds for $\dim_H (\Delta(X))$ given that $\dim_H (X) \in (1, \frac{5}{4})$. In higher dimensions, the works of Du-Iosevich-Ou-Wang-Zhang [@du2021improved] and Wang-Zheng [@wang2022improvement] used a good-bad tube decomposition using Orponen's radial projection theorem and decoupling techniques [@orponen2018radial] to provide state-of-the-art results when the dimension $d$ is even; when $d$ is odd, a more classical approach purely based on decoupling gave the best estimates [@du2019sharp], [@harris2021low]. More recently, Shmerkin and Wang [@shmerkin2022dimensions] prove a radial projection theorem in the spirit of this paper to provide an improved lower bound when $\dim_H (X) = \frac{d}{2}$, $d = 2, 3$; using their framework combined with updated results of [@orponen2022kaufman], one can show for example that $\dim_H (\Delta(X)) \ge \frac{5}{8}$ when $X \subset \mathbb{R}^3$ satisfies $\dim_H (X) = \frac{3}{2}$. In fact, all of these works prove lower bounds on the size of the pinned distance set, $\Delta_x (X) := \{ |x - y| : y \in X \}$. In the forthcoming companion papers [@DOKZFalconer], [@DOKZFalconerDec], we use Theorem [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"} to improve the lower bounds for the Falconer distance set problem in all dimensions $d$. Very recently, radial projections in dimension $2$ have been used to prove the ABC sum-product conjecture and Furstenberg set conjecture, and yield progress on the discretized sum-product problem [@orponen2023projections], [@ren2023furstenberg]. It is natural to wonder whether the exciting progress in 2 dimensions will generalize to higher dimensions. The starting point of the breakthrough work of [@orponen2023projections] (which was also used in [@ren2023furstenberg]) is a sharp radial projection theorem in 2 dimensions, [@orponen2022kaufman Theorem 1.1]. We hope to use our higher dimensional radial projection theorem to prove analogous results to [@orponen2023projections], [@ren2023furstenberg] in all dimensions. ## Discretized results We deduce Theorem [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"} from $\delta$-discretized versions. The following notation will be used throughout this paper. **Definition 6**. *Let $P \subset \mathbb{R}^d$ be a bounded nonempty set, $d \ge 1$. Let $\delta > 0$ be a dyadic number, and let $0 \le s \le d$ and $C > 0$. We say that $P$ is a $(\delta, s, C, k)$-set if for every $(r, k)$-plate $H$ with $r \in [\delta, 1]$, we have $$|P \cap H|_\delta \le C \cdot |P|_\delta \cdot r^s.$$ If $k$ is not specified, we default to $k = 0$ (which becomes a standard definition from [@orponen2021hausdorff] because $(r, 0)$-plates are $r$-balls).* **Definition 7**. *Let $\mathcal{T}\subset \mathbb{R}^d$ be a bounded nonempty set of dyadic $\delta$-tubes, $d \ge 2$. Let $\delta > 0$ be a dyadic number, and let $0 \le s \le d$, $0 \le k \le d-2$, and $C > 0$. We say that $\mathcal{T}$ is a $(\delta, s, C, k)$-set of tubes if for every $(r, k+1)$-plate $H$ and $\delta \le r \le 1$, we have $$\label{eqn:tube non-concentration} |\mathcal{T}\cap H| \le C \cdot |\mathcal{T}| \cdot r^s.$$ If $k$ is not specified, we default to $k = 0$. We also say $\mathcal{T}$ is a $(\delta, s, C, k)$-set of tubes from scale $r_1$ to $r_2$ if the non-concentration condition [\[eqn:tube non-concentration\]](#eqn:tube non-concentration){reference-type="eqref" reference="eqn:tube non-concentration"} holds for $r_2 \le r \le r_1$.* A $(\delta, s, C, k)$-set of balls cannot be concentrated in a small neighborhood of a $k$-plane, while a $(\delta, s, C, k)$-set of tubes cannot be concentrated in a small neighborhood of a $(k+1)$-plane. The main ingredient in the proof of Theorem [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"} is an $\varepsilon$-improvement to the (dual) Furstenberg set problem that generalizes Theorem 1.3 in [@orponen2021hausdorff] to higher dimensions. **Theorem 8**. *For any $0 \le k < d-1$, $0 \le s < k+1$, $s < t \le d$, $\kappa > 0$, there exists $\varepsilon(s, t, \kappa, k, d) > 0$ such that the following holds for all small enough $\delta \in 2^{-\mathbb{N}}$, depending only on $s, t, \kappa, k, d$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a $(\delta, t, \delta^{-\varepsilon})$-set with $\cup \mathcal{P}\subset [0, 1)^d$, and let $\mathcal{T}\subset \mathcal{T}^\delta$ be a family of $\delta$-tubes. Assume that for every $p \in \mathcal{P}$, there exists a $(\delta, s, \delta^{-\varepsilon}, 0)$ and $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set $\mathcal{T}(p) \subset \mathcal{T}$ such that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$. Then $|\mathcal{T}| \ge \delta^{-2s-\varepsilon}$.* **Remark 9**. *The condition of $\mathcal{T}(p)$ being a $(\delta, \kappa, \delta^{-\varepsilon}, k+1)$-set is to prevent the counterexample in (say) $\mathbb{R}^3$ when $s = 1, t \in (1, 2]$, and $\mathcal{T}$ is a maximal set of $\delta^{-2}$ many essentially distinct tubes in $[0, 1]^2$. This condition is automatically taken care of when $s > k$: any $(\delta, s, \delta^{-\varepsilon}, 1)$-set is a $(\delta, \kappa, \delta^{-\varepsilon}, k+1)$-set with $\kappa = s-k$.* **Remark 10**. *We can make this decay around $k$-plane assumption assuming that (1) $P$ is a $(\delta, \kappa, \delta^{-\varepsilon}, k+1)$-set and (2) for $p \in P$, $|\mathcal{T}(p) \cap P| \ge \delta^\varepsilon|P|$. This will be useful for radial projection estimates, since we can guarantee (1) by Theorem B.1 of [@shmerkin2022non] and (2) because we can get rid of $\lesssim\delta^{\varepsilon} |P|$ many pairs $(p, q)$ for a fixed $p$.* In fact, we can prove the following refined version of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"}. **Theorem 11**. *For any $0 \le k < d-1$, $0 \le s < k+1$, $\max(s, k) < t \le d$, $\kappa > 0$, $r_0 \le 1$, there exists $\varepsilon(s, t, \kappa, k, d) > 0$ such that the following holds for all small enough $\delta/r_0 \in 2^{-\mathbb{N}} \cap (0, \delta_0)$, with $\delta_0$ depending only on $s, t, \kappa, k, d$. Let $H$ be a $(r_0, k+1)$-plate, $\mathcal{P}\subset \mathcal{D}_\delta \cap H$ be a $(\delta, t, (\delta/r_0)^{-\varepsilon})$-set with $\cup \mathcal{P}\subset [0, 1)^d$, and let $\mathcal{T}\subset \mathcal{T}^\delta \cap H$ be a family of $\delta$-tubes. Assume that for every $p \in \mathcal{P}$, there exists a set $\mathcal{T}(p) \subset \mathcal{T}$ such that:* - *$T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$;* - *$\mathcal{T}(p)$ is a $(\delta, s, (\delta/r_0)^{-\varepsilon} r_0^{k-s}, 0)$-set down from scale $r$;* - *$\mathcal{T}(p)$ is a $(\delta, \kappa, (\delta/r_0)^{-\varepsilon} r_0^{-\kappa}, k)$-set.* *Then $|\mathcal{T}| \ge (\frac{\delta}{r_0})^{-\varepsilon} \delta^{-2s} r_0^{2(s-k)}$.* **Remark 12**. *(a) Given fixed $k, \kappa$, the value of $\varepsilon$ can be chosen uniformly in a compact subset of $\{ (s, t) : 0 \le s < k+1, \max(s, k) < t \le d \}$. Indeed, if $\varepsilon> 0$ works for $(s, t)$, then $\frac{\varepsilon}{2}$ works in the $\frac{\varepsilon}{2}$-neighborhood of $(s, t)$.* *(b) Conjecture: can we replace the condition of being in $H$ by $\mathcal{T}(p)$ being a $(\delta, k, (\delta/r_0)^{-\varepsilon}, 0)$-set from scales $1$ to $r_0$?* Using Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}, a bootstrap argument based on [@orponen2022kaufman] gives the following. **Theorem 13**. *Let $k \in \{ 1, 2, \cdots, d-1 \}$, $k-1 < \sigma < s \le k$, and $\varepsilon> 0$. There exist $N, K_0$ depending on $\sigma, s, k$, and $\eta(\varepsilon) > 0$ (with $\eta(1) = 1$) such that the following holds. Fix $r_0 \le 1$, and $K \ge K_0$. Let $\mu, \nu$ be $\sim 1$-separated $s$-dimensional measures with constant $C_\mu, C_\nu$ supported on $E_1, E_2$, which lie in $B(0, 1)$. Assume that $|\mu|, |\nu| \le 1$. Let $A$ be the pairs of $(x, y) \in E_1 \times E_2$ that lie in some $K^{-1}$-concentrated $(r_0, k)$-plate. Then there exists a set $B \subset E_1 \times E_2$ with $\mu \times \nu (B) \lesssim K^{-\eta}$ such that for every $x \in E_1$ and $r$-tube $T$ through $x$, we have $$\nu(T \setminus (A|_x \cup B|_x)) \lesssim\frac{r^\sigma}{r_0^{\sigma-(k-1)+N\varepsilon}} K^N.$$ The implicit constant may depend on $C_\mu, C_\nu, \sigma, s, k$.* **Remark 14**. *(a) It is not assumed that $\mu, \nu$ are a probability measures, just that $\mu(B(0,1)), \nu(B(0,1)) \le 1$.* *(b) If $\alpha>d-1$, then the numerology of Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"} doesn't apply. Instead, Orponen's radial projection theorem [@orponen2018radial] in dimension $d$ applies. The result (stated in [@guth2020falconer Lemma 3.6] for $d = 2$, but can be generalized to all dimensions $d$) is that for $\gamma = \varepsilon/C$, there exists a set $B \subset E_1 \times E_2$ with $\mu_1 \times \mu_2 (B) \le r^{\gamma}$ such that for every $x \in E_1$ and $\delta$-tube $T$ through $x$, we have $$\mu_2 (T \setminus B|_x) \lesssim r^{d-1-\varepsilon}.$$ Note that the set $A$ of "concentrated pairs" is not needed here.* *(c) If $r \sim r_0$, we can obtain a slightly better result by projecting to a generic $k$-dimensional subspace and following the argument in [@du2021improved Section 3.2]. The result is that for $\gamma = \varepsilon/C$, there exists a set $B \subset E_1 \times E_2$ with $\mu_1 \times \mu_2 (B) \le \delta^{\gamma}$ such that for every $x \in E_1$ and $r$-tube $T$ through $x$, we have $$\mu_2 (T \setminus B|_x) \lesssim r^{k-1-\varepsilon}.$$ The set $A$ is again not needed in this case. The main novelty of Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"} comes when $r < r_0$.* ## Proof ideas The main proof ideas for Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} are as follows: 1. Perform a standard multiscale decomposition argument due to [@orponen2021hausdorff] to reduce the original problem to two building blocks: the case when $\mathcal{P}$ is a $(\delta, s)$-set and when $\mathcal{P}$ is a $t$-regular set. The first case doesn't happen all the time and has no loss by an elementary incidence argument, so we focus on gaining an $\varepsilon$-improvement in the second case. A $t$-regular set $\mathcal{P}$ has the special property that $|\mathcal{P}\cap Q|$ is still a $(\Delta, t)$-set for $Q \in \mathcal{D}_{\Delta} (\mathcal{P})$, $\Delta = \delta^{1/2}$. 2. If $\mathcal{P}$ is $t$-regular with $\Delta = \delta^{1/2}$, we may find a $\Delta$-tube $\mathbf{T}$ such that upon dilation of $\mathbf{T}$ to $[0, 1]^d$, we obtain a new Furstenberg problem with the ball set having a quasi-product structure. See Appendix A of [@orponen2021hausdorff]. 3. Finally, we will use discretized sum-product type arguments to conclude an $\varepsilon$-improvement to the dual Furstenberg problem assuming $\mathcal{P}= X \times Y \subset \mathbb{R}^{d-1} \times \mathbb{R}$ has a quasi-product structure. In very rough terms, we shall lift $Y$ to have dimension close to $1$, and apply multi-linear Kakeya. This idea of lifting the dimension was found in He's work on a higher-rank discretized sum-product theorem [@he2016discretized] in a slightly different context. To prove Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}, we use a similar multiscale decomposition argument as in (1) to reduce to two building blocks: a smaller version of the setting of Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} and a smaller version of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"}. The smaller version of Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} has no loss by an elementary incidence argument, and the smaller version of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} admits a gain. For Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"}, we first prove the case when $\mu, \nu$ are supported in a $r_0 K$ plate (where $K$ is a small power of $r_0^{-1}$). This uses a similar argument as in [@orponen2022kaufman Lemma 2.8]. The general case follows from applying this special case many times. ## Structure of the paper In Section [2](#sec:prelims){reference-type="ref" reference="sec:prelims"}, we introduce some key concepts that will be used throughout the paper. In Sections [3](#sec:quasi-product sets){reference-type="ref" reference="sec:quasi-product sets"} through [5](#sec:main thm general){reference-type="ref" reference="sec:main thm general"}, we prove Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} first for quasi-product sets following ideas of [@he2020orthogonal], and then for regular sets and finally for general sets following [@orponen2021hausdorff]. In Section [6](#sec:refined from main){reference-type="ref" reference="sec:refined from main"}, we prove Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} from Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"}. In Section [7](#sec:power decay){reference-type="ref" reference="sec:power decay"}, we generalize a radial projection theorem of Shmerkin [@shmerkin2022non Theorem 6.3] that enables us to assume our sets have power decay around $k$-planes. In Section [8](#sec:threshold){reference-type="ref" reference="sec:threshold"}, we prove Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"} following ideas from [@orponen2022kaufman]. Finally, in Section [9](#sec:corollaries){reference-type="ref" reference="sec:corollaries"}, we prove Theorem [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"} and [Corollary 2](#cor:shm_conj_2){reference-type="ref" reference="cor:shm_conj_2"} from the discretized results. **Acknowledgments.** The author is supported by a NSF GRFP fellowship. The author would like to thank Xiumin Du, Tuomas Orponen, Yumeng Ou, Pablo Shmerkin, Hong Wang, and Ruixiang Zhang for helpful discussions. We thank Paige Bright and Yuqiu Fu for suggesting to include a higher-dimensional version of Beck's theorem in this paper. # Preliminaries {#sec:prelims} This section will summarize the argument of [@orponen2021hausdorff], and in lieu of proofs (with the exception of Proposition [Proposition 23](#prop:easy_est){reference-type="ref" reference="prop:easy_est"}), we either refer the reader to [@orponen2021hausdorff] or defer the proof to a later section. ## Definitions We use $A \lesssim B$ to denote $A \le CB$ for some constant $C$. We use $A \lesssim_N B$ to indicate the constant $C$ can depend on $N$. We will also use $A \lessapprox B$ in future proofs; its exact meaning will always be clarified when used. For a finite set $A$, let $|A|$ denote the cardinality of $A$. If $A$ is infinite, let $|A|$ denote the Lebesgue measure of $A$. For a set $A$, let $A^c = \mathbb{R}^d \setminus A$. For a tube $T$, let $\ell(T)$ denote the central line segment of $T$. For a set $E$, let $E^{(\delta)}$ be the $\delta$-neighborhood of $E$. For $A \subset X \times Y$ and $x \in X$, define the slice $A|_x = \{ y \in Y : (x, y) \in A \}$ and $A|^y = \{ x \in X : (x, y) \in A \}$. For a measure $\mu$ and a set $G$, define the restricted measure $\mu|_G$ by $\mu|_G (A) = \mu(G \cap A)$. The renormalized restricted measure is $\mu_G = \frac{1}{\mu(G)} \mu|_G$. For vectors $v_1, \cdots, v_i \in \mathbb{R}^d$, $1 \le i \le d$, the quantity $|v_1 \wedge \cdots \wedge v_i|$ is the non-negative volume of the parallelepiped spanned by $v_1$ through $v_i$. $B(x, r)$ is the ball in $\mathbb{R}^d$ of radius $r$ centered at $x$. We also use the notation $B_r$ for an arbitrary $r$-ball in $\mathbb{R}^d$. For sets $A, B$ and $P \subset A \times B$, let $A \overset{P}{+}B := \{ a + b : (a, b) \in P \}$. **Definition 15**. *We say $\mu$ supported in $\mathbb{R}^d$ is an $\alpha$-dimensional measure with constant $C_\mu$ if $\mu(B_r) \le C_\mu r^\alpha$ for all $r \le 1$ and balls $B_r$ of radius $r$.* ## Plates {#subsec:r-net} We work in $\mathbb{R}^d$. An $(r, k)$-plate is the $r$-neighborhood of a $k$-dimensional hyperplane in $\mathbb{R}^d$. We construct a set $\mathcal{E}_{r,k}$ of $(r, k)$-plates with the following properties: - Each $(\frac{r}{2}, k)$-plate intersecting $B(0, 1)$ lies in at least one plate of $\mathcal{E}_{r,k}$; - For $s \ge r$, every $(s, k)$-plate contains $\lesssim\left( \frac{s}{r} \right)^{(k+1)(d-k)}$ many $(r, k)$-plates of $\mathcal{E}_{r,k}$. For example, when $k = 1$ and $d = 2$, we can simply pick $\sim r^{-1}$ many $r$-tubes in each of an $r$-net of directions. This generalizes to higher $k$ and $d$ via a standard $r$-net argument, but we haven't seen it in the literature, so we provide a precise construction. An $r$-net of a metric space is a subset $S$ such that $B(x, r) \cap B(y, r) = \emptyset$ for $x \neq y, x, y \in S$. The affine Grassmanian manifold $\mathbb{A}(k, d)$ is the set of all $k$-planes in $\mathbb{R}^d$. By counting degrees of freedom, we see that $\dim \mathbb{A}(k, d) = (k+1)(d-k)$. Any such plane is uniquely $V = V_0 + a$ for some $k$-dimensional subspace $V_0$ and $a \in V_0^\perp$. For $V=V_0 + a$ and $W=W_0 + b$, define their distance $d_\mathbb{A}$ to be (following Section 3.16 of [@mattila1999geometry]): $$d_\mathbb{A}(V, W) = \| \pi_{V_0} - \pi_{W_0} \|_{op} + |a - b|,$$ where $\pi_{V_0}:\mathbb{R}^d\to V_0$ and $\pi_{W_0}:\mathbb{R}^d\to W_0$ are orthogonal projections, and $\|\cdot\|_{op}$ is the usual operator norm for linear maps. Let $\mathbb{A}_0 (k, d)$ be the submanifold of $k$-planes $V_0 + a$ with $a \in B(0, 10)$. Since the manifold $(\mathbb{A}_0 (k, d), d_\mathbb{A})$ is compact and smooth, it can be covered by finitely many charts that are $\sim 1$-bilipschitz to a subset of $\mathbb{R}^{(k+1)(d-k)}$. From a maximal $cr$-net $\mathcal{N}$ of the set of affine planes of $\mathbb{A}_0 (k, d)$ with $c > 0$ a sufficiently small constant, we can construct a set $\mathcal{E}_{r,k}$ of $(r, k)$-plates whose central planes are the elements of $\mathcal{N}$. We now check the two properties for $\mathcal{E}_{r,k}$. To prove the first property, let $H$ be a $(\frac{r}{2}, k)$-plate intersecting $B(0, 1)$. Then the central plane $P = P_H$ must lie at distance $\le 2cr$ from some element $Q$ of $\mathcal{N}$ (otherwise, we can add it to the net). Let $P = P_0 + a$ and $Q = Q_0 + b$. Hence, $\| \pi_{P_0} - \pi_{Q_0} \|_{op} \le 2cr$ and $|a - b| \le 2cr$, so for $x \in P \cap B(0, 10)$ (so $x-a \in P_0$), $$|\pi_{Q_0} (x-a) - (x-a)| \le 2cr |x-a| \le 2cr(|x| + |a|) \le 40cr.$$ Now, note that $\pi_{Q_0} (x-a) + b \in Q$. It is close to $x$ if $c < \frac{1}{100}$: $$|\pi_{Q_0} (x-a) + b - x| \le 40cr + |a-b| \le 50cr < \frac{r}{2}.$$ We have proved $P \cap B(0, 10) \subset Q^{(r/2)}$ and thus $P^{(r/2)} \cap B(0, 10) \subset Q^{(r)}$. Hence, $H$ is contained in the $(r, k)$-plate with central plane $Q$. To prove the second property, we note that the set of $k$-planes in $\mathbb{A}(k, d)$ whose intersection with $B(0, 10)$ is contained in a given $(s, k)$-plate is contained in an $O(s)$-ball $B$ of $\mathbb{A}(k, d)$. First suppose $B$ is contained within some coordinate chart; we would like to prove that $|\mathcal{N}\cap B| \lesssim\left( \frac{s}{r} \right)^{(k+1)(d-k)}$. To show this, note that $\{ B(x, r) : x \in \mathcal{N}\cap B \}$ is a packing of $B^{(r)}$ with finitely overlapping $r$-balls. Now map the chart to $\mathbb{R}^{(k+1)(d-k)}$. Since the map only distorts distances by a constant factor, we can pack $|\mathcal{N}\cap B|$ many finitely overlapping $c_1 r$-balls into a ball of radius $O(s)$. Thus by a volume argument, we have $|\mathcal{N}\cap B| \lesssim\left( \frac{s}{r} \right)^{(k+1)(d-k)}$. Since there are finitely many charts, we can apply the argument to $B$ intersecting each chart, which proves the second property. We specialize our discussion to tubes. For each scale $\delta$, let $\mathcal{T}^\delta$ be a cover of $[0, 1]^d$ with $\delta$-tubes such that every $\frac{\delta}{2}$-tube (and in particular every $r$-tube with $r < \frac{\delta}{2}$) is contained in at least $1$ and at most $C_d$ many tubes of $\mathcal{T}^\delta$. Slightly abusing notation (á la [@orponen2021hausdorff]), we will also use $\mathcal{T}, \mathcal{T}_\delta, \mathcal{T}_\Delta$ to represent sets of tubes, where the subscript $\delta$ helpfully indicates a set of $\delta$-tubes. In Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"}, we pay attention to certain plates with disproportionately much mass. **Definition 16**. *We say that a $(r, k)$-plate $H$ is $c$-concentrated on $\mu$ if $\mu(H) \ge c$.* Other notation is following [@orponen2021hausdorff]. Unlike [@orponen2021hausdorff], we work with ordinary rather than dyadic tubes. The advantage of dyadic tubes is that every $2^{-n}$-tube is in a unique $2^{-m}$-tube if $n > m$; thus, dyadic tubes will avoid the $C_d$ loss incurred by the finitely overlapping cover $\mathcal{T}^\delta$. However, dyadic tubes have the disadvantage that they don't behave well under rotations or dilations, and it would be more cumbersome to define $(\delta, s, C, k)$-sets of dyadic tubes (whereas the definition for ordinary tubes is more geometric). Thus, in principle it is possible to work with dyadic tubes and save on the $C_d$ loss, but it doesn't affect our numerology in the end (since our losses will depend badly on $d$ anyway), so we chose to work with ordinary tubes throughout. **Definition 17**. *[@orponen2021hausdorff] Let $P \subset \mathbb{R}^d$ be a bounded nonempty set, $d \ge 1$. Let $\delta > 0$ be a dyadic number, and let $0 \le s \le d$ and $C > 0$. We say that $P$ is a $(\delta, s, C)$-set if $$|P \cap Q|_\delta \le C \cdot |P|_\delta \cdot r^s, \qquad Q \in \mathcal{D}_r (\mathbb{R}^d), r \in [\delta, 1].$$* **Definition 18**. *Let $\mathcal{T}\subset \mathbb{R}^d$ be a bounded nonempty set of dyadic $\delta$-tubes, $d \ge 2$. Let $\delta > 0$ be a dyadic number, and let $0 \le s \le d$, $0 \le k \le d-2$, and $C > 0$. We say that $\mathcal{T}$ is a $(\delta, s, C, k)$-set of tubes if for every $(r, k+1)$-plate $H$ and $\delta \le r \le 1$, we have $$|\mathcal{T}\cap H| \le C \cdot |\mathcal{T}| \cdot r^s.$$ If $k$ is not specified, we default to $k = 0$.* The following is a simpler interpretation of $(\delta, s, C, k)$-set if the tubes all pass through the same point. **Definition 19**. *Let $\sigma(t) \in S^{d-1}$ be the slope of the central axis of $t$.* **Lemma 20**. *Let $\mathcal{T}$ be a set of $\delta$-tubes intersecting $p$. Then if $\mathcal{T}$ is a $(\delta, s, C, k)$-set, then $\sigma(\mathcal{T})$ is a $(\delta, s, O(C), k)$-set. Conversely, if $\sigma(\mathcal{T})$ is a $(\delta, s, C, k)$-set, then $\mathcal{T}$ is a $(\delta, s, O(C), k)$-set.* *Proof.* Let $\pi_p : \mathbb{R}^d \to S^{d-1}$ denote spherical projection through $p$. Then $\pi_p (t \setminus B(p, 1/2))$ is well-defined and equals $\sigma(t)$, up to an additive loss of $C\delta$. Fix a $(r, k)$-plate $H \in S^{d-1}$. Then the set of tubes with slope in $H$ and passing through $p$ must lie in a $(r + C\delta, k+1)$-plate $p^{(C\delta)} + \pi_p^{-1} (H)$. Conversely, for any $(r, k+1)$-plate $W$ containing $p$, the set of possible slopes of tubes through $p$ contained in $W$ is contained in a $(r + C\delta, k)$-plate $(\pi_p (W - p))^{C\delta}$. ◻ We will need the following lemma from [@orponen2021hausdorff]. **Lemma 21** ([@orponen2021hausdorff], Lemma 2.7). *Let $P \subset [-2, 2]^d$ be a $(\delta, s, C)$-set. Then $P$ contains a $\delta$-separated $(\delta, s, O_d (C))$-subset $P'$ with $|P'| \le \delta^{-s}$.* First, since $(\delta, \kappa, \delta^{-\varepsilon}, k)$-sets are $(\delta, \kappa, \delta^{-\varepsilon}, k')$-sets for any $k' < k$, we can assume that $k \le s < k+1$. Next, since $(\delta, t, \delta^{-\varepsilon})$-sets are $(\delta, t', \delta^{-\varepsilon})$-sets for $t' < t$, we may assume $t \le k+1$. In particular, we get $t - s \le 1$, a useful assumption. We record a useful geometric fact about $(r, k)$-plates. **Lemma 22**. *Fix $C_\mathrm{sep}\ge 1$, then there exists $r_0$ depending on $C_\mathrm{sep}$ such that the following is true for $r < r_0$. If $(x, y)$ lie in an $(r, k)$-plate $H$ and $|x-y| = C_\mathrm{sep}^{-1}$, then any $r$-tube $T$ through $x, y$ will lie in $H^{(CC_\mathrm{sep}r)}$, which is a $(CC_\mathrm{sep}r, k)$-plate.* *Proof.* For $C$ sufficiently large: If $T$ does not lie in $H^{(CC_\mathrm{sep}r)}$, then $H \cap T$ will be contained in a $(2C_\mathrm{sep})^{-1}$-tube segment of $T$. ◻ ## An Elementary Estimate We prove a classical estimate which can be viewed as Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} with $\varepsilon= 0$. We won't need the fact that $\mathcal{T}(p)$ is a $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set. The $d = 2$ case is proven as Proposition 2.13 and Corollary 2.14 of [@orponen2021hausdorff]. For higher dimensions, the proof is similar and we sketch the details. Let $A \lessapprox_\delta B$ denote the inequality $$A \le C \cdot \log (\frac{1}{\delta})^C B.$$ **Proposition 23**. *Let $0 \le s \le t \le d-1$, and let $C_P, C_T \ge 1$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a $(\delta, t, C_P)$-set. Assume that for every $p \in \mathcal{P}$ there exists a $(\delta, s, C_T)$-family $\mathcal{T}(p) \subset \mathcal{T}^\delta$ of dyadic $\delta$-tubes with the property that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$, and $|\mathcal{T}(p)| = M$ for some $M \ge 1$.* *Let $\mathcal{T}\subset \mathcal{T}^\delta$ be arbitrary, and define $I(\mathcal{P}, \mathcal{T}) = \{ (p, T) \in \mathcal{P}\times \mathcal{T}: T \in \mathcal{T}(p) \}$. Then $$|I(\mathcal{P}, \mathcal{T})| \lessapprox_\delta \sqrt{C_P C_T} \cdot (M\delta^s)^{\theta/2} \cdot |\mathcal{T}|^{1/2} |\mathcal{P}|,$$ where $\theta = \theta(s, t) = \frac{d-1-t}{d-1-s} \in [0,1]$. (If $s=t=d-1$, then $\theta(s,t) = 0$.)* The following corollary of Proposition [Proposition 23](#prop:easy_est){reference-type="ref" reference="prop:easy_est"} is the form we will use. **Corollary 24**. *Let $0 \le s \le t \le d-1$, and let $C_P, C_T \ge 1$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a $(\delta, t, C_P)$-set. Assume that for every $p \in \mathcal{P}$ there exists a $(\delta, s, C_T)$-family $\mathcal{T}(p) \subset \mathcal{T}^\delta$ of dyadic $\delta$-tubes with the property that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$, and $|\mathcal{T}(p)| = M$ for some $M \ge 1$. If $\mathcal{T}= \cup_{p \in \mathcal{P}} \mathcal{T}(p)$, then $$|\mathcal{T}| \gtrapprox_\delta (C_P C_T)^{-1} \cdot M\delta^{-s} \cdot (M\delta^s)^{\frac{t-s}{d-1-s}}.$$ (If $s = t = d-1$, then $\frac{t-s}{d-1-s} = 0$.)* **Remark 25**. *To use Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}, we need $t \le d-1$. Fortunately, this is a harmless assumption because $s < d-1$, and changing $t$ to $\min(t, d-1)$ makes the hypothesis of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} weaker.* *Proof.* We begin with an application of Cauchy-Schwarz. $$\begin{aligned} |I(\mathcal{P},\mathcal{T})| &= \sum_{T \in \mathcal{T}} |\{p \in \mathcal{P} : T \in \mathcal{T}(p)\}| \\ &\leq |\mathcal{T}|^{1/2} \left| \{(T,P,P'): T\in\mathcal{T}(p)\cap \mathcal{T}(p') \}\right|^{1/2}.\end{aligned}$$ Note that we have the following bounds: $$\label{form94} |\mathcal{T}(p) \cap \mathcal{T}(p')| \lesssim \min \left\{C_{T} \cdot M \cdot \left(\tfrac{\delta}{d(p,p')+\delta} \right)^{s}, \left( \tfrac{1}{d(p,p')+\delta} \right)^{d-1} \right\},$$ where $d(p,p')$ stands for the distance of the midpoints of $p$ and $p'$. To prove [\[form94\]](#form94){reference-type="eqref" reference="form94"}, observe that if $T \in \mathcal{T}(p) \cap \mathcal{T}(p')$, then $T$ lies in a $\frac{\delta}{d(p,p')+\delta}$-tube with central line being the line between $p$ and $p'$. Thus, the first bound in [\[form94\]](#form94){reference-type="eqref" reference="form94"} follows from $\mathcal{T}(p)$ being a $(\delta, s, C_T)$-set with $|\mathcal{T}(p)| = M$, and the second bound is the maximum number of essentially distinct $\delta$-tubes that can fit inside a $\frac{\delta}{d(p,p')+\delta}$-tube. Write $\theta := \theta(s,t) := \frac{(d-1) - t}{(d-1) - s} \in [0,1]$. (If $s = t = d-1$, we set $\theta := 0$.) The parameter $\theta$ is chosen so that $t = s\theta+(d-1)(1-\theta)$. Then [\[form94\]](#form94){reference-type="eqref" reference="form94"} and the inequality $\min\{a,b\} \leq a^{\theta}b^{1 - \theta}$ imply that $$|\mathcal{T}(p) \cap \mathcal{T}(p')| \lesssim (C_{T}M\delta^{s})^{\theta} \cdot d(p,p')^{-t}.$$ Since $\mathcal{P}$ is a $(\delta,t,C_{P})$-set, for fixed $p\in\mathcal{P}$ we have $$\sum_{p'} (d(p,p')+\delta)^{-t} \lesssim \sum_{ \sqrt{2}\cdot\delta \le 2^{-j} \le \sqrt{2} } 2^{t j}|\{p'\in \mathcal{P}: d(p,p') \le 2^{-j}\}| \lessapprox_{\delta} C_{P}\cdot |\mathcal{P}|.$$ We deduce that $$\sum_{p, p'} |\mathcal{T}(p) \cap \mathcal{T}(p')| \lesssim (C_{T}M\delta^{s})^{\theta} \sum_{p, p'} (d(p,p')+\delta)^{-t} \lessapprox_{\delta} C_{P}(C_{T}M\delta^{s})^{\theta} \cdot |\mathcal{P}|^{2},$$ so $$|I(\mathcal{P},\mathcal{T})| \lessapprox_{\delta} C_{P}^{1/2}(C_{T}M\delta^{s})^{\theta/2} \cdot |\mathcal{T}|^{1/2}|\mathcal{P}| \leq \sqrt{C_{P}C_{T}} \cdot (M\delta^{s})^{\theta/2} \cdot |\mathcal{T}|^{1/2}|\mathcal{P}|.$$ This proves Proposition [Proposition 23](#prop:easy_est){reference-type="ref" reference="prop:easy_est"}, and Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"} follows by observing $|I(\mathcal{P}, \mathcal{T})| \ge M |\mathcal{P}|$. ◻ ## Multiscale analysis Following Section 4 of [@orponen2021hausdorff], we would like to change scale from $\delta$ to $\Delta > \delta$, while preserving the properties of $\mathcal{T}(p)$. We say $A \lessapprox_\delta B$ if there exists an absolute constant $C \ge 1$ such that $A \le C \cdot [\log (1/\delta)]^C$. We start by naming the objects in Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"}. **Definition 26**. *Fix $\delta \in 2^{-\mathbb{N}}, s \in [0, d-1], C > 0, M \in \mathbb{N}$. We say that a pair $(\mathcal{P}_0, \mathcal{T}_0) \subset \mathcal{D}_\delta \times \mathcal{T}^\delta$ is a $(\delta, s, C_1, \kappa, C_2, M)$-nice configuration if for every $p \in \mathcal{P}_0$, there exists a $(\delta, s, C_1, 0)$ and $(\delta, \kappa, C_2, k)$-set $\mathcal{T}(p) \subset \mathcal{T}_0$ with $|\mathcal{T}(p)| = M$ and such that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$.* Using the method of induction on scales, we would like to relate nice configurations at scale $\delta$ to nice configurations at scales $\Delta, \frac{\delta}{\Delta}$, where $\delta < \Delta \le 1$. The following proposition, which combines Propositions 4.1 and 5.2 of [@orponen2021hausdorff], gives a way of doing so with only polylog losses. Our proof relies on the same ideas as [@orponen2021hausdorff], with some technical simplifications. We defer the proof to Section [6](#sec:refined from main){reference-type="ref" reference="sec:refined from main"}, where we prove a slightly more general version. **Proposition 27**. *Fix dyadic numbers $0 < \delta < \Delta \le 1$. Let $(\mathcal{P}_0, \mathcal{T}_0)$ be a $(\delta, s, C_1, \kappa, C_2, M)$-nice configuration. Then there exist sets $\mathcal{P}\subset \mathcal{P}_0$, $\mathcal{T}(p) \subset \mathcal{T}_0 (p), p \in \mathcal{P}$, and $\mathcal{T}_\Delta \subset \mathcal{T}^\Delta$ such that denoting $\mathcal{T}= \cup_{p \in \mathcal{P}} \mathcal{T}(p)$ the following hold:* (i) *[\[item1\]]{#item1 label="item1"} $|\mathcal{D}_\Delta (\mathcal{P})| \approx_\delta |\mathcal{D}_\Delta (\mathcal{P}_0)|$ and $|\mathcal{P}\cap Q| \approx_\delta |\mathcal{P}_0 \cap Q|$ for all $Q \in \mathcal{D}_\Delta (\mathcal{P})$.* (ii) *[\[item21\]]{#item21 label="item21"} There exists $\mathbf{N}$ such that $|\mathcal{T}\cap \mathbf{T}| \sim \mathbf{N}$ for all $\mathbf{T}\in \mathcal{T}_\Delta$.* (iii) *[\[item3\]]{#item3 label="item3"} $(\mathcal{D}_\Delta (\mathcal{P}), \mathcal{T}_\Delta)$ is $(\Delta, s, C^1_\Delta, \kappa, C^2_\Delta, M_\Delta)$-nice for some $C^1_\Delta \approx_\delta C_1$, $C^2_\Delta \approx_\delta C_2$, and $M_\Delta \ge 1$.* (iv) *[\[item4\]]{#item4 label="item4"} For each $Q \in \mathcal{D}_\Delta (\mathcal{P})$, let $\mathcal{T}_\Delta (Q)$ be the tubes in $\mathcal{T}_\Delta$ through $Q$. Then for all $\mathbf{T}\in \mathcal{T}_\Delta (Q)$, we have $$|\{ (p, T) \in (\mathcal{P}\cap Q) \times \mathcal{T}: T \in \mathcal{T}(p) \text{ and } T \subset \mathbf{T}\} | \gtrapprox_\delta \frac{M \cdot |\mathcal{P}\cap Q|}{|\mathcal{T}_\Delta (Q)|}.$$* (v) *[\[item5\]]{#item5 label="item5"} For each $Q \in \mathcal{D}_\Delta (\mathcal{P})$, there exist $C^1_Q \approx_\delta C_1$, $C^2_Q \approx_\delta C_2$, $M_Q \ge 1$, a subset $\mathcal{P}_Q \subset \mathcal{P}\cap Q$ with $|\mathcal{P}_Q| \gtrapprox_\Delta |\mathcal{P}\cap Q|$, and a family of tubes $\mathcal{T}_Q \subset \mathcal{T}^{\delta/\Delta}$ such that $(S_Q (\mathcal{P}_Q), \mathcal{T}_Q)$ is $(\delta/\Delta, s, C^1_Q, \kappa, C^2_Q, M_Q)$-nice.* *Furthermore, the families $\mathcal{T}_Q$ can be chosen so that $$\label{eqn:item6} \frac{|\mathcal{T}_0|}{M} \gtrapprox_\delta \frac{|\mathcal{T}_\Delta|}{M_\Delta} \cdot \left( \max_{Q \in \mathcal{D}_\Delta (\mathcal{P})} \frac{|\mathcal{T}_Q|}{M_Q} \right).$$* Iterate this proposition to get (for details, see [@shmerkin2022dimensions Corollary 4.1]) **Corollary 28**. *Fix $N \ge 2$ and a sequence $\{ \Delta_j \}_{j=0}^n \subset 2^{-\mathbb{N}}$ with $$0 < \delta = \Delta_N < \Delta_{N-1} < \cdots < \Delta_1 < \Delta_0 = 1.$$ Let $(\mathcal{P}_0, \mathcal{T}_0) \subset \mathcal{D}_\delta \times \mathcal{T}^\delta$ be a $(\delta, s, C_1, \kappa, C_2, M)$-nice configuration. Then there exists a set $\mathcal{P}\subset \mathcal{P}_0$ such that:* 1. *$|\mathcal{D}_{\Delta_j} (\mathcal{P})| \approx_\delta |\mathcal{D}_{\Delta_j} (\mathcal{P}_0)|$ and $|\mathcal{P}\cap \textbf{p}| \approx_\delta |\mathcal{P}_0 \cap \textbf{p}|$, $1 \leq j \leq N$, $\textbf{p} \in \mathcal{D}_{\Delta_j} (\mathcal{P})$.* 2. *For every $0 \leq j \leq N-1$ and $\textbf{p} \in \mathcal{D}_{\Delta_j}$, there exist numbers $C_{\textbf{p}}^1 \approx_\delta C^1$, $C_{\textbf{p}}^2 \approx_\delta C^2$, and $M_{\textbf{p}} \geq 1$, and a family of tubes $\mathcal{T}_{\textbf{p}} \subset \mathcal{T}^{\Delta_{j+1}/\Delta_j}$ with the property that $(S_{\textbf{p}} (\mathcal{P}\cap \textbf{p}), \mathcal{T}_{\textbf{p}})$ is a $(\Delta_{j+1}/\Delta_j, s, C_{\textbf{p}}^1, \kappa, C_{\textbf{p}}^2, M_{\textbf{p}})$-nice configuration.* *Furthermore, the families $\mathcal{T}_{\textbf{p}}$ can be chosen such that if $\textbf{p}_j \in \mathcal{D}_{\Delta_j} (\mathcal{P})$ for $0 \le j \le N-1$, then $$\frac{|\mathcal{T}_0|}{M} \gtrapprox_\delta \prod_{j=0}^{N-1} \frac{|\mathcal{T}_{\textbf{p}_j}|}{M_{\textbf{p}_j}}.$$ Here, $\gtrapprox_\delta$ means $\gtrsim_N \log(1/\delta)^C$, and likewise for $\lessapprox_\delta, \approx_\delta$.* ## Uniform sets and branching numbers The following exposition borrows heavily from [@orponen2023projections Section 2.3]. **Definition 29**. *Let $n \ge 1$ and $$\delta = \Delta_n < \Delta_{n-1} < \cdots < \Delta_1 \le \Delta_0 = 1$$ be a sequence of dyadic scales. We say that a set $P \subset [0, 1)^d$ is $\{ \Delta_j \}_{j=1}^n$-uniform if there is a sequence $\{ N_j \}_{j=1}^n$ such that $N_j \in 2^{\mathbb{N}}$ and $|P \cap Q|_{\Delta_j} = N_j$ for all $j \in \{ 1, 2, \cdots, n \}$ and $Q \in \mathcal{D}_{\Delta_{j-1}} (P)$.* **Remark 30**. *By uniformity, we have $|P|_{\Delta_m} = |P \cap Q|_{\Delta_m} |P|_{\Delta_\ell}$ for $0 \le \ell < m \le n$ and $Q \in \mathcal{D}_{\Delta_\ell} (P)$.* As a result, we can always refine a set $P$ to be uniform: **Lemma 31**. *Let $P \subset [0, 1)^d$, $m, T \in \mathbb{N}$, and $\delta = 2^{-mT}$. Let $\Delta_j := 2^{-jT}$ for $0 \le j \le m$, so in particular $\delta = \Delta_m$. Then there is a $\{ \Delta_j \}_{j=1}^m$-uniform set $P' \subset P$ such that $$|P'|_\delta \ge (2T)^{-m} |P|_\delta.$$ In particular, if $\varepsilon> 0$ and $T^{-1} \log (2T) \le \varepsilon$, then $|P'| \ge \delta^\varepsilon|P|$.* Uniform sets can be encoded by a branching function. **Definition 32**. *Let $T\in \mathbb{N}$, and let $\mathcal{P}\subset [0,1)^d$ be a $\{\Delta_j\}_{j=1}^n$-uniform set, with $\Delta_j: = 2^{-jT}$, and with associated sequence $\{N_j\}_{j=1}^n\subset \{ 1, \dots, 2^{dT}\}^n$. We define the *branching function* $f: [0, n]\rightarrow [0, dn]$ by setting $f(0)=0$, and $$f(j):=\frac{\log |\mathcal{P}|_{2^{-jT}}}{T} =\frac{1}{T}\sum_{i=1}^j \log N_i, \quad i \in \{1, \dots n\},$$ and then interpolating linearly between integers.* **Definition 33**. *Let $s_f (a, b) = \frac{f(b) - f(a)}{b - a}$ denote the slope of a line segment between $(a, b)$ and $(f(a), f(b))$. We say that a function $f: [0,n]\rightarrow \mathbb{R}$ is $\varepsilon$-superlinear on $[a,b]\subset [0,n]$, or that $(f, a, b)$ is $\varepsilon$-superlinear, if $$f(x)\geq f(a) + s_f (a, b) (x-a) - \varepsilon(b - a), x\in [a, b].$$ We say that $(f, a, b)$ is $\varepsilon$-linear if $$|f(x) - f(a) - s_f (a, b) (x-a)| \le \varepsilon(b - a), x\in [a, b].$$* The following lemma converts between branching functions and the uniform structure of $P$. It is [@orponen2021hausdorff Lemma 8.3] (or an immediate consequence of the definitions) **Lemma 34**. *Let $P$ be a $(\Delta^i)_{i=1}^m$-uniform set in $[0, 1)^d$ with associated branching function $f$, and let $\delta = \Delta^m$.* (i) *If $f$ is $\varepsilon$-superlinear on $[0, m]$, then $P$ is a $(\delta, s_f (0, m), O_\Delta (1) \delta^{-\varepsilon})$-set.* (ii) *If $f$ is $\varepsilon$-linear on $[0, m]$, then $P$ is a $(\delta, s_f (0, m), O_\Delta (1) \delta^{-\varepsilon}, O_\Delta (1) \delta^{-\varepsilon})$-regular set between scales $\delta$ and $1$.* The crucial branching lemma is [@orponen2021hausdorff Lemma 8.5] applied to the function $\frac{2}{d} \cdot f$: **Lemma 35**. *Fix $s \in (0, 1)$ and $t \in (s, d]$. For every $\varepsilon> 0$ there is $\tau = \tau(\varepsilon, s, t) > 0$ such that the following holds: for every piecewise affine $d$-Lipschitz function $f : [0, m] \to \mathbb{R}$ such that $$f(x) \ge tx - \varepsilon m \text{ for all } x \in [0, m],$$ there exists a family of non-overlapping intervals $\{ [c_j, d_j] \}_{j=1}^n$ contained in $[0, m]$ such that:* 1. *For each $j$, at least one of the following alternatives holds:* 1. *$(f, c_j, d_j)$ is $\varepsilon$-linear with $s_f (c_j, d_j) \ge s$;* 2. *$(f, c_j, d_j)$ is $\varepsilon$-superlinear with $s_f (c_j, d_j) = s$.* 2. *$d_j - c_j \ge \tau m$ for all $j$;* 3. *$|[0, m] \setminus \cup_j [c_j, d_j]| \lesssim_{s,t} \varepsilon m$.* ## Combinatorial and probabilistic preliminaries In this section, we collect a few of the results from additive combinatorics and probability that will be used in the following sections. First, we make the following observation (Lemma 19 of [@he2020orthogonal]) about intersections of high-probability events. (That lemma was stated for Lebesgue measure but the same proof works for general measures $\nu$.) **Lemma 36**. *Let $A \subset \mathbb{R}^d$ equipped with a measure $\nu$ and $\Theta$ be an index set equipped with a probability measure $\mu$. Suppose there is $K \ge 1$ and for each $\theta \in \Theta$, a Borel subset $A_\theta$ with $\nu(A_\theta) \ge \nu(A)/K$. Then $$\mu^{\otimes q} (\{ (\theta_1, \theta_2, \cdots, \theta_q) : \nu(A_{\theta_1} \cap A_{\theta_2} \cap \cdots \cap A_{\theta_q}) \ge \frac{\nu(A)}{2K^q} \})\ge \frac{1}{2K^q}.$$* Next, we state Rusza's triangular inequality [@he2020orthogonal Lemma 21] (see also [@ruzsa1978cardinality]): **Lemma 37**. *For any sets $A, B, C \subset \mathbb{R}^d$, we have $$|B|_\delta |A-C|_\delta \lesssim_d |A-B|_\delta |B-C|_\delta.$$* We also would like the Plünnecke-Rusza inequality, in the form stated by [@he2020orthogonal Lemma 22]: **Lemma 38**. *Let $A, B$ be bounded subsets of $\mathbb{R}^d$. For all $K \ge 1$, $\delta > 0$, if $|A + B|_\delta \le K |B|_\delta$, then for all $k, \ell \ge 1$, we have $$|kA - \ell A|_\delta \lesssim_d K^{k+\ell} |B|_\delta.$$ Here, $kA = \underbrace{A + \cdots + A}_{k \text{ times}}$.* In a similar spirit, the set of $w$ such that $X + wX$ is small compared to $|X|$ forms a ring. The following is a restatement of [@he2016discretized Lemma 30(i,ii)] for $\mathbb{R}$. Note that $\mathrm{End}(\mathbb{R}) \simeq \mathbb{R}$ with identity $1$. **Lemma 39**. *Define $S_\delta (X; K) = \{ w \in [-K, K] : |X + wX|_\delta \le K |X|_\delta \}$.* (i) *If $a \in S_\delta (X; \delta^{-\varepsilon})$ and $b \in \mathbb{R}$ such that $|a-b| \le \delta^{1-\varepsilon}$, then $b \in S_\delta (X; \delta^{-O(\varepsilon)})$.* (ii) *If $1, a, b \in S_\delta (X; \delta^{-\varepsilon})$, then $a+b, a-b, ab$ all belong to $S_\delta (X; \delta^{-O(\varepsilon)})$.* The following theorem (a special case of Theorem 5 of [@he2016discretized]) is a quantitative statement that $\frac{1}{2}$-dimensional subrings of $\mathbb{R}$ don't exist. In fact, by repeated sum-product operations, we can get all of $\mathbb{R}$. **Theorem 40**. *We work in $\mathbb{R}^1$. Given $\kappa, \varepsilon_0 > 0$, there exist $\varepsilon> 0$ and an integer $s \ge 1$ such that for $\delta < \delta_0 (\kappa, \varepsilon_0)$, the following holds. For every $(\kappa, \delta^{-\varepsilon})$-set $A \subset B(0, \delta^{-\varepsilon})$, we have $$B(0, \delta^{\varepsilon_0}) \subset \langle A \rangle_s + B(0, \delta),$$ where $\langle A \rangle_1 := A \cup (-A)$ and for any integer $s \ge 1$, define $\langle A \rangle_{s+1} := \langle A \rangle_s \cup (\langle A \rangle_s + \langle A \rangle_1) \cup (\langle A \rangle_s \cdot \langle A \rangle_1)$.* Finally, we shall need a discretized variant of the Balog-Szemerédi-Gowers theorem. Our version is closest to [@orponen2020improved Theorem 4.38], which is taken from [@bourgain2010discretized p. 196], which in turn refers to Exercise 6.4.10 in [@tao2006additive]. But the exercise is only sketched in [@tao2006additive], so for completeness, we provide a proof in Appendix [10](#appendix:proof of bsg){reference-type="ref" reference="appendix:proof of bsg"}. **Theorem 41**. *Let $K \ge 1$ and $\delta > 0$ be parameters. Let $A, B$ be bounded subsets of $\mathbb{R}^d$, and let $P \subset A \times B$ satisfy $$|P|_\delta \ge K^{-1} |A|_\delta |B|_\delta \text{ and } |\{ a + b : (a, b) \in P \}|_\delta \le K (|A|_\delta |B|_\delta)^{1/2}.$$ Then one can find subsets $A' \subset A, B' \subset B$ satisfying* - *$|A'|_\delta \gtrsim_d K^{-2} |A|_\delta, |B'|_\delta \gtrsim_d K^{-2} |B|_\delta$,* - *$|A' + B'|_\delta \lesssim_d K^8 (|A|_\delta |B|_\delta)^{1/2}$,* - *$|P \cap (A' \times B')| \gtrsim_d \frac{|A|_\delta |B|_\delta}{K^2}$.* *(Implicit constants depend on $d$ but not on $\delta, K$.)* We also need the following version of multi-linear Kakeya. **Theorem 42** (Theorem 1 in [@carbery2013endpoint]). *Let $2 \le k \le d$ and $\mathcal{T}_1, \mathcal{T}_2, \cdots, \mathcal{T}_k$ be families of $1$-tubes in $\mathbb{R}^d$. Then $$\int_{\mathbb{R}^d} \left( \sum_{T_1 \in \mathcal{T}_1} \cdots \sum_{T_k \in \mathcal{T}_k} |e(T_1) \wedge \cdots \wedge e(T_k)| \chi_{T_1 \cap \cdots \cap T_k} (x) \right)^{1/(k-1)} \, dx \lesssim_{k,d} \left( \prod_{i=1}^k |\mathcal{T}_i| \right)^{1/(k-1)}.$$ Here, $e(T_i)$ is the unit vector in the direction of tube $T_i$.* ## Energy **Definition 43**. *The $(s, k)$-Riesz energy of a finite Borel measure $\mu$ on $\mathbb{R}^d$ is $$I_{s,k}^\delta (\mu) = \int (|(x_0 - x_1) \wedge \cdots \wedge (x_0 - x_k)| + \delta)^{-s} \, d\mu(x_0) \cdots d\mu(x_k).$$ If $k = 1$ and $\delta = 0$, we recover the usual $s$-dimensional Riesz energy.* **Lemma 44**. (a) *Fix $0 < s < t$ and a measure $\mu$ with total mass $C$. If $\mu(H_r) \le Cr^t$ for every $(r, k-1)$-plate $H_r$ and $r > 0$, then $I_{s,k}^0 (\mu) \lesssim_{s,t} C^k$.* (b) *Fix $0 < \delta < \frac{1}{2}$. If $I_{s_i,k_i}^\delta (\mu) \le C$ for $1 \le i \le m$, then $\mathrm{spt}(\mu)$ contains a set which is simultaneously a $(\delta, \frac{s_i}{k_i}, O(1) \cdot (Cm)^{1/k_i} \log \delta^{-1}, k_i-1)$-set for each $i$.* **Remark 45**. *If $k_1 = m = 1$ in part (b), then we can drop the log factor (c.f. proof of Lemma A.6 in [@orponen2021hausdorff]). We don't know if we can drop the log factor for $k > 1$ or $m > 1$.* *Proof.* (a) Let $\rho_i$ be the distance between $x_i$ and the plane spanned by $x_0, \cdots, x_{i-1}$; notice that $|(x_0 - x_1) \wedge \cdots \wedge (x_0 - x_k)| = \prod_{i=1}^k \rho_i$. Thus, we can rewrite $I_{s,k} (\mu)$ as an iterated integral $$\int d\mu(x_0) \int \rho_1^{-s} d\mu(x_1) \int \rho_2^{-s} \, d\mu(x_2) \cdots \int \rho_k^{-s} \, d\mu(x_k).$$ We will be done if we show for all $1 \le i \le k$ and choices of $x_0, \cdots, x_{i-1}$, that $\int \rho_i^{-s} \, d\mu(x_i) \lesssim C$. Let $H$ be the span of $x_0$ through $x_{i-1}$, and observe that by definition, $\{ x_i : \rho_i \ge r \} \subset H^{(r)}$, which is contained in a $(r, k-1)$-plate. Thus, $$\int \rho_i^{-s} \, d\mu(x_i) \lesssim C + \sum_{\rho = 2^{-n}, n \ge 1} C\rho^{t-s} \lesssim_{s,t} C.$$ \(b\) Let $P_i = \{ x_0 \in \mathrm{spt}(\mu) : \int (|(x_0 - x_1) \wedge \cdots \wedge (x_0 - x_{k_i})| + \delta)^{-s_i} \, d\mu(x_1) \cdots d\mu(x_{k_i}) < 2mC \}$. By Markov's inequality, $\mu(P_i) > 1 - \frac{1}{2m}$, so by the union bound, $P = \cap_{i=1}^m P_i$ satisfies $\mu(P) > \frac{1}{2}$. We claim that $\mu(P \cap H_r) \le C r^{s_i/k_i}$ for all $(r, k_i-1)$-plates $H_r$ and $\delta \le r \le 1$, $1 \le i \le m$. Indeed, if $P \cap H_r = \emptyset$, then we are done. Otherwise, pick $x_0 \in P \cap H_r$ and observe that if $x_1, x_2, \cdots, x_{k_i} \in H_r$, then $|(x_0 - x_1) \wedge \cdots \wedge (x_0 - x_{k_i})| + \delta \lesssim r$. Thus, we get $\mu(H_r)^{k_i} \cdot r^{-s} \le 2C$, so $\mu(P \cap H_r) \le (2C r^s)^{1/k_i}$. Finally, let $P'_c \subset \mathcal{D}_\delta (P)$ be those dyadic $\delta$-cubes $p$ such that $\mu(p) \sim c$. We know $\sum_{c = 2^{-n} \in [\delta^d, 1]} \mu(P'_c) \ge \frac{1}{4}$, so by dyadic pigeonholing, some $\mu(P'_c) \gtrsim(\log \delta^{-1})^{-1}$. Then $P'_c$ will be a $(\delta, \frac{s_i}{k_i}, O(1) \cdot (Cm)^{1/k_i} \log \delta^{-1}, k_i-1)$-set for all $1 \le i \le m$. ◻ # Improved incidence estimates for quasi-product sets {#sec:quasi-product sets} The main novelty of this paper is the following Proposition, which is a higher-dimensional refinement of [@orponen2020improved Proposition 4.36] (see also [@orponen2021hausdorff Proposition A.7]). It can be viewed as a variant of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} for quasi-product sets. **Proposition 46**. *Given $0 \le k < d-1$, $0 \le s < k+1$, $\tau, \kappa > 0$, there exist $\eta(s, k, \kappa, \tau, d) > 0$ and $\delta_0 (s, k, \kappa, \tau, d) > 0$ such that the following holds for all $\delta \in (0, \delta_0]$.* *Let $\mathbf{Y}\subset (\delta \cdot \mathbb{Z}) \cap [0,1)$ be a $(\delta, \tau, \delta^{-\eta})$-set, and for each $\mathbf{y}\in \mathbf{Y}$, assume that $\mathbf{X}_{\mathbf{y}} \subset (\delta \cdot \mathbb{Z})^{d-1} \cap [0,1)^{d-1}$ is a $(\delta, \kappa, \delta^{-\eta}, k)$-set with cardinality $\ge \delta^{-s+\eta}$. Let $$\mathbf{Z}= \bigcup_{\mathbf{y}\in \mathbf{Y}} \mathbf{X}_{\mathbf{y}} \times \{ \mathbf{y}\}.$$ For every $\mathbf{z}\in \mathbf{Z}$, assume that $\mathcal{T}(\mathbf{z})$ is a set of $\delta$-tubes each making an angle $\ge \frac{1}{100}$ with the plane $y = 0$ with $|\mathcal{T}(\mathbf{z})| \ge \delta^{-s+\eta}$ such that $\mathbf{z}\in T$ for all $T \in \mathcal{T}(\mathbf{z})$. Then $|\mathcal{T}| \ge \delta^{-2s-\eta}$, where $\mathcal{T}= \cup_{\mathbf{z}\in \mathbf{Z}} \mathcal{T}(\mathbf{z})$.* **Remark 47**. *In contrast to Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} and [@orponen2020improved Proposition 4.36], we (perhaps surprisingly) don't need any non-concentration assumptions on the tube sets $\mathcal{T}(\mathbf{z})$ (even when $d = 2$). Instead, it suffices to have weak non-concentration assumptions on $\mathbf{Y}$ and $\mathbf{X}_\mathbf{y}$ for each $\mathbf{y}\in \mathbf{Y}$. The non-concentration assumption on $\mathbf{X}_\mathbf{y}$ is necessary: otherwise, we can take $s = k$, and let $\mathbf{Z}$ to be the $\delta$-balls contained in some $(\delta, k+1)$-plate $H$, and $\mathcal{T}$ to be the $\delta$-tubes contained in $H$.* ## An improved slicing estimate {#subsec:slicing} We will eventually deduce Proposition [Proposition 46](#prop:improved_incidence_weaker){reference-type="ref" reference="prop:improved_incidence_weaker"} from the following slicing estimate. **Theorem 48**. *For $0 \le k \le d-2$, $0 \le s < k+1$, and $0 < \kappa \le 1$, there exists $\varepsilon> 0$ such that the following holds for sufficiently small $\delta < \delta_0 (s, k, d, \varepsilon)$. Let $\mathcal{T}$ be a $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set of $\delta$-tubes each making angle $\ge \frac{1}{100}$ with the plane $y = 0$ with $|\mathcal{T}| \ge \delta^{-2s+\varepsilon}$. Let $\mu$ be a probability measure on $\mathbb{R}$ such that for all $\delta \le r \le 1$, we have $\mu(B_r) \le \delta^{-\varepsilon} r^\kappa$. Then there is a set $\mathcal{D}\subset \mathbb{R}$ with $\mu(\mathcal{D}) \ge 1 - \delta^\varepsilon$ such that the slice of $\cup \mathcal{T}'$ at $z = z_0$ has $\delta$-covering number $\ge \delta^{-s-\varepsilon}$, for every subset $\mathcal{T}' \subset \mathcal{T}$ with $|\mathcal{T}'| \ge \delta^\varepsilon|\mathcal{T}|$ and $x \in \mathcal{D}$.* **Remark 49**. *One should compare Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} to [@he2020orthogonal Theorem 1]. Indeed, if $k = 0$ and $d = 2$, Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} is a direct corollary of [@he2020orthogonal Theorem 1]. We can see this by using ball-tube duality, which turns $\mathcal{T}$ into a subset of $\mathbb{R}^2$. Under this duality, the slice of $\cup \mathcal{T}'$ at $z = z_0$ becomes the orthogonal projection $\pi_{\tilde{z}_0}$ to a line in the dual space, for some $\tilde{z}_0 \in S^1$. The map $z_0 \to \tilde{z}_0$ induces a pushforward measure $\tilde{\mu}$ of $\mu$ which still satisfies the non-concentration condition $\tilde{\mu}(B_r) \lesssim\delta^{-\varepsilon} r^\kappa$, so we can apply [@he2020orthogonal Theorem 1]. (For more details, see the proof of Proposition A.7 in [@orponen2021hausdorff].)* *In higher dimensions, we can still use duality to turn $\mathcal{T}$ into a subset of $\mathbb{A}(d, 1) \sim \mathbb{R}^{2(d-1)}$, and then slices of $\cup \mathcal{T}'$ become orthogonal projections to $(d-1)$-planes. Unfortunately, [@he2020orthogonal Theorem 1] does not apply because the pushforward measure $\tilde{\mu}$ is still supported on a line in $S^{d-1}$. This approach is bound to fail because [@he2020orthogonal Theorem 1] does not use the strong assumption that $\mathcal{T}$ is non-concentrated around $(k+1)$-planes. Using this assumption is the key novelty of this proof.* Nonetheless, Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} will borrow many ideas from the proof of [@he2020orthogonal Theorem 1] and He's previous work [@he2016discretized]. Roughly, the strategy is as follows. - As in [@he2020orthogonal], reduce to the following slightly weaker statement: given $\mathcal{T}$ and $\mu$, we can find a subset $\mathcal{T}' \subset \mathcal{T}$ such that the conclusion of Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} holds for $\mathcal{T}'$ in place of $\mathcal{T}$. This relies on a formal exhaustion argument. - Then, as in [@he2020orthogonal], reduce this slightly weaker to the following even weaker statement: there exists $z_0 \in E := \mathrm{spt}\mu$ such that the slice of $\cup \mathcal{T}$ at $z = z_0$ has $\delta$-covering number $\ge \delta^{-s-\varepsilon}$. This relies on additive combinatorics (e.g. the Balog-Szemerédi-Gowers theorem) and some probability. - Assume this is false: that for all $z_0 \in E$, the slice of $\cup \mathcal{T}$ at $z = z_0$ has $\delta$-covering number $\lessapprox\delta^{-s}$. Using additive combinatorics as in [@he2016discretized], the same conclusion is true for all $z_0 \in E'$, which is the set of sums or differences of $m$ many terms, each of which is a product of $m$ elements of $E$. (Here, $m$ will be a fixed large integer.) - Finally, if $m$ is sufficiently large in terms of $\kappa, \varepsilon$, then $E'$ contains a large interval $[0, \delta^{\varepsilon}]$ (c.f. [@he2016discretized Theorem 5]). Essentially, we have a set of $\gtrapprox\delta^{-2s}$ many tubes $\mathcal{T}$, each containing $\gtrapprox\delta^{-1}$ many $\delta$-balls, such that the union of the $\delta$-balls has cardinality $\lessapprox\delta^{-(s+1)}$. Without further restrictions, this Furstenberg-type problem doesn't lead to a contradiction: take $s = k$ and $\mathcal{T}$ to be the set of $\delta$-tubes in a $(\delta, k+1)$-plate. Luckily, our set of tubes $\mathcal{T}$ is still a $(\delta, \kappa, \delta^{-O(\varepsilon)}, k)$-set, which rules out this counterexample. Indeed, we may finish using multi-linear Kakeya. The reader be warned: we shall execute this strategy in reverse order. This is mainly because the main innovation of the paper is the fourth bullet point. ## An improved Furstenberg estimate The following estimate complements work of Zahl [@zahl2022unions]: we prove an $\varepsilon$-improvement on the union of tubes under a mild $(k+1)$-plane non-concentration for the set of tubes. As in Zahl [@zahl2022unions], the key technique is multilinear Kakeya. **Theorem 50**. *For any $0 \le k < d-1$, $0 \le s < k+1$, $0 < \kappa \le 1$, there exists $\varepsilon> 0$ such that the following holds for sufficiently small $\delta > 0$. Let $\mathcal{T}$ be a $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set of $\delta$-tubes with $|\mathcal{T}| \ge \delta^{-2s+\varepsilon}$, and for each $t \in \mathcal{T}$, let $P_t$ be a set of $\delta$-balls intersecting $t$ such that $|P_t| \ge \delta^{-1+\varepsilon}$. Then $|\cup P_t| \gtrsim\delta^{-(s+1)-\varepsilon}$.* *Proof.* The proof below is lossy and can possibly be improved (say by induction on scale). Also, the $\varepsilon$ can be determined explicitly in terms of the parameters but we choose not to do so here. We use $\gtrapprox$ notation to hide $\delta^{-C \varepsilon}$ terms, where $C$ can depend on the other parameters. Let $P = \cup P_t$, and suppose $|P| \lessapprox\delta^{-(s+1)}$. Let $\mathcal{T}(p)$ be the set of tubes in $\mathcal{T}$ through $p$. Use a bush argument to upper bound $|\mathcal{T}(p)|$: $$\delta^{-(s+1)} \gtrapprox|P| \ge |\cup_{t \ni p} (P_t \setminus B(p, \delta^{2\varepsilon}))| \ge \delta^{2d\varepsilon} \sum_{t \ni p} |P_t \setminus B(p, \delta^{2\varepsilon})| \ge \delta^{-1+(2d+1) \varepsilon} |\mathcal{T}(p)|.$$ Thus, $|\mathcal{T}(p)| \lessapprox\delta^{-s}$ for all $p \in P$. We get the following inequality chain $$\delta^{-2s-1} \lessapprox\delta^{-1} |\mathcal{T}| \lessapprox I(P, \mathcal{T}) \lessapprox\delta^{-s} |P| \lessapprox\delta^{-2s-1.}$$ This means $I(P, \mathcal{T}) \approx\delta^{-2s-1}$, $|P| \approx\delta^{-s-1}$, and $|\mathcal{T}| \approx\delta^{-2s}$. Now perform a dyadic pigeonholing to extract a subset $P' \subset P$ such that $|\mathcal{T}(p)| \in [M, 2M]$ for all $p \in P'$ and $I(P', \mathcal{T}) \approx\delta^{-2s-1}$. We know from before that $M \le \delta^{-s}$, and $\delta^{-2s-1} \lessapprox I(P', \mathcal{T}) \approx M |P'| \lessapprox M |P| \lessapprox\delta^{-2s-1}$, so $M \approx\delta^{-s}$ and $|P'| \approx\delta^{-s}$. (This type of dyadic pigeonholing will also be used later. We also remark that dyadic pigeonholing was not necessary to achieve this step; simply let $P'$ be the set of $p \in P$ satisfying $|\mathcal{T}(p)| \ge \delta^{-s+C\eta}$ for some large $C$, and use the bound on $I(P, \mathcal{T})$ to get a lower bound for $|P'|$.) Now, we claim that $P'$ is a $(\delta, \kappa, \delta^{-O(\varepsilon)}, k+1)$-set. Fix $\delta < r < 1$ and let $H_r$ be a $(r, k+1)$-plate. We first bound $I(P' \cap H_r, \mathcal{T})$. Letting $H_{r'}$ be the $(r', k+1)$-plate that is a dilate of $H_r$ with the same center, we have $$\begin{aligned} I(P' \cap H_r, \mathcal{T}) &\le \sum_{r' = 2^{-\mathbb{N}} \cap [r, 1]} I(P' \cap H_r, H_{r'} \setminus H_{r'/2}) \\ &\le \sum_{r'} \frac{r}{r' \delta} \cdot |\mathcal{T}\cap H_{r'}| \\ &\le \sum_{r'} \frac{r}{r' \delta} \cdot |\mathcal{T}| \delta^{-\varepsilon} (r')^\kappa \\ &\lesssim\delta^{-1} |\mathcal{T}| \delta^{-\varepsilon} r^\kappa.\end{aligned}$$ Thus, since $I(P' \cap H_r, \mathcal{T}) \gtrapprox\delta^{-s} |P' \cap H_r|$, we have $|P' \cap H_r| \lessapprox\delta^{s-1} |\mathcal{T}| r^\kappa \approx\delta^{-(s+1)} r^\kappa \approx|P'| r^\kappa$. Finally, since $I(P', \mathcal{T}) \gtrapprox|P'| \delta^{-s} \gtrapprox\delta^{-2s-1}$ and $|\mathcal{T}| \lessapprox\delta^{-2s}$, by dyadic pigeonholing there exists a subset $\mathcal{T}' \subset \mathcal{T}$ with $|\mathcal{T}'| \approx|\mathcal{T}|$ such that each $t \in \mathcal{T}'$ contains $\approx\delta^{-1}$ many $\delta$-balls in $P'$. Now since $I(P', \mathcal{T}') \gtrapprox\delta^{-1} |\mathcal{T}'| \gtrapprox\delta^{-2s-1}$, $|P'| \lessapprox\delta^{-s-1}$, and $|\mathcal{T}(p)| \lessapprox\delta^{-s}$ for all $p \in P'$, by dyadic pigeonholing we can find $\tilde{P}\subset P'$ with $|\tilde{P}| \gtrapprox|P'|$ such that each $p \in \tilde{P}$ lies in $\approx\delta^{-s}$ many tubes in $\mathcal{T}'$. Now, we are in good shape to apply multilinear Kakeya. For $p \in \tilde{P}$, let $\mathcal{T}(p)$ be the tubes in $\mathcal{T}'$ through $p$. By a bush argument, $\cup \mathcal{T}(p)$ contains $\gtrapprox\delta^{-(s+1)}$ many $\delta$-balls in $P$. Since $P'$ is a $(\delta, \kappa, \delta^{-O(\varepsilon)}, k+1)$-set, there are $\gtrapprox\delta^{-(s+1)(k+3)}$ many $(k+3)$-tuples of points $(p_0, p_1, \cdots, p_{k+2})$ such that $p_0$ and $p_i$ lie on some tube $t_i \in \mathcal{T}_i$ and $|e(t_1) \wedge \cdots \wedge e(t_{k+2})| \gtrapprox 1$ (where $e(t)$ is the unit vector in the direction of tube $t$). Thus, there is a choice of $p_1, \cdots, p_{k+2}$ such that there are $\gtrapprox\delta^{-(s+1)}$ many valid choices for $p_0$. But this leads to a contradiction by the following argument. Let $\mathcal{T}_i$ be the tubes of $\mathcal{T}$ through $p_i$, $1 \le i \le k+2$; then by a rescaled version of Multilinear Kakeya (Theorem [Theorem 42](#thm:multlinear kakeya){reference-type="ref" reference="thm:multlinear kakeya"}), the number of valid choices for $p_0$ is $\lessapprox\left( \prod_{i=1}^{k+2} |\mathcal{T}_i| \right)^{1/(k+1)} \lessapprox\delta^{-s(k+2)/(k+1)}$, which (using $s < k+1$) is much smaller than $\delta^{-(s+1)}$ provided that $\varepsilon, \delta$ are sufficiently small in terms of the parameters. This contradiction completes the proof. ◻ ## From Furstenberg to weak slicing *This subsection contains ideas from [@he2016discretized] and [@he2020orthogonal].* **Theorem 51**. *For $0 \le k \le d-2$, $0 \le s < k+1$, and $0 < \kappa \le 1$, there exists $\varepsilon> 0$ such that the following holds for sufficiently small $\delta < \delta_0 (s, k, d, \varepsilon)$. Let $\mathcal{T}$ be a $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set of $\delta$-tubes each making angle $\ge \frac{1}{100}$ with the plane $y = 0$ with $|\mathcal{T}| \ge \delta^{-2s+\varepsilon}$. Let $\mu$ be a probability measure on $\mathbb{R}$ such that for all $\delta \le r \le 1$, we have $\mu(B_r) \le \delta^{-\varepsilon} r^\kappa$. Then there exists $z_0 \in \mathrm{spt}\mu$ such that the slice of $\cup \mathcal{T}$ at $z = z_0$ has $\delta$-covering number $\ge \delta^{-s-\varepsilon}$.* *Proof.* We use $\lessapprox$ to denote $\le C \delta^{-C\varepsilon}$, where $C$ may depend on $\kappa, s$. Let $E := \mathrm{spt}\mu$; without loss of generality, assume $E$ is closed. Let $z_1 = \inf E$ and $z_2 = \sup E$; then $d(z_1, z_2) \ge \delta^{2\varepsilon/\kappa}$ since $\mu(B(z_1, \delta^{2\varepsilon/\kappa})) \le \delta^{-\varepsilon} \cdot \delta^{2\varepsilon} \le \delta^{\varepsilon} < 1$. Let $X = \cup \mathcal{T}(z = z_1)$ and $Y = \cup \mathcal{T}(z = z_2)$; we are given that $|X|_\delta, |Y|_\delta \lessapprox\delta^{-s}$. On the other hand, since each $T \in \mathcal{T}$ passes through $O(1)$ many elements in $X$ and $O(1)$ many elements in $Y$, we get that $$\delta^{-2s} \gtrapprox|X|_\delta |Y|_\delta \gtrsim|\mathcal{T}| \gtrapprox\delta^{-2s},$$ so in fact, $|X|, |Y| \approx\delta^{-s}$ and $|\mathcal{T}| \approx\delta^{-2s}$. Let $E' := [z_1, z_2] \setminus (B(z_1, \delta^{2\varepsilon/\kappa}) \cup B(z_2, \delta^{2\varepsilon/\kappa}))$; then $\mu(E') \ge 1 - 2 \delta^\varepsilon\ge \frac{1}{2}$ for $\delta$ small enough. Let $f(z) = \frac{z-z_1}{z_2-z}$; note that on $E'$, we have that $f$ is $\approx 1$-bilipschitz, and $f(z) \approx 1$ for all $z \in E'$. The problem condition literally states $|(z_2 - z) X + (z - z_1) Y|_\delta \lessapprox\delta^{-s}$ for $z \in E'$; since $(z_2 - z) \approx 1$, we can divide through by $(z_2 - z)$ to get $$|X + f(z) Y|_\delta \lessapprox\delta^{-s} \text{ for } z \in E'.$$ Now pick an arbitrary $z' \in E'$. In particular, we get $|X + f(z') Y|_\delta \lessapprox\delta^{-s}$, so by Lemma [Lemma 37](#lem:rusza triangle){reference-type="ref" reference="lem:rusza triangle"}, we have for all $z \in E'$, $$|X - \frac{f(z)}{f(z')} X|_\delta \le \frac{|X + f(z) Y|_\delta |f(z) Y + \frac{f(z)}{f(z')} X|_\delta}{|f(z) Y|_\delta} \lessapprox\delta^{-s}.$$ In addition, since $|X + f(z') Y|_\delta \lessapprox\delta^{-s} \lessapprox|X|_\delta$, the Plünnecke-Rusza inequality (Lemma [Lemma 38](#lem:plunnecke-rusza){reference-type="ref" reference="lem:plunnecke-rusza"}) gives $|X + X|_\delta \lessapprox|Y|_\delta \lessapprox\delta^{-s}$. Define $\tilde{\mu}= g_* (\mu)$, the pushforward of $g(z) = \frac{f(z)}{f(z')}$; then $g$ (like $f$) is $\gtrapprox 1$-bilipschitz on $E'$, so $\tilde{\mu}$ also satisfies a non-concentration condition $\tilde{\mu}(B_r) \lessapprox r^\kappa$ for $\delta \le r \le 1$. Now pick $\varepsilon_0 > 0$, and assume $\varepsilon$ is chosen sufficiently small in terms of $\varepsilon_0$. By the iterated sum-product Theorem [Theorem 40](#thm:iterated sum-product){reference-type="ref" reference="thm:iterated sum-product"}, we can find an integer $m \ge 1$ such that for $\delta < \delta_0 (\kappa, \varepsilon, \varepsilon_0)$, $$B(0, \delta^{\varepsilon_0}) \subset \langle A \rangle_m + B(0, \delta),$$ where $\langle A \rangle_1 := A \cup (-A)$ and for any integer $m \ge 1$, define $\langle A \rangle_{m+1} := \langle A \rangle_m \cup (\langle A \rangle_m + \langle A \rangle_1) \cup (\langle A \rangle_m \cdot \langle A \rangle_1)$. By applying the ring structure Lemma [Lemma 39](#lem:ring_structure){reference-type="ref" reference="lem:ring_structure"} many times, we see that $B(0, \delta^{\varepsilon_0}) \subset S_\delta (X; \delta^{-O_m (\varepsilon)})$ and since $1 \in S_\delta (X; \delta^{-O(\varepsilon)})$, that $B(1, \delta^{\varepsilon_0}) \subset S_\delta (X; \delta^{-O_m (\varepsilon)})$. By definition of $S_\delta$ and Lemma [Lemma 37](#lem:rusza triangle){reference-type="ref" reference="lem:rusza triangle"}, we get for $w \in B(1, \delta^{\varepsilon_0})$, $$|X + w f(z') Y|_\delta \le \frac{|X - w X|_\delta |w X + w f(z') Y|_\delta}{|w X|_\delta} \lessapprox\delta^{-s}.$$ In other words, for all $z_0 \in I := f^{-1} ([(1-\delta^{\varepsilon_0}) f(z'), (1+\delta^{\varepsilon_0}) f(z')])$, the slice $\cup \mathcal{T}(z = z_0)$ has $\delta$-covering number $\lessapprox\delta^{-s}$. Since $f$ is $\approx 1$-bilipschitz and $f(z') \approx 1$, we have $|I| \approx\delta^{\varepsilon_0}$. Now, we seek a contradiction to Theorem [Theorem 50](#thm:mult_kakeya){reference-type="ref" reference="thm:mult_kakeya"}. For every $t \in \mathcal{T}$, we let $P_t$ be the $\delta$-balls on $t$ with $z$-coordinate in $I$. We observe the following: - Recall our assumption that $\mathcal{T}$ is a $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set of $\delta$-tubes with $|\mathcal{T}| \ge \delta^{-2s+\varepsilon}$. - $|P_t| \gtrsim\delta^{-1} |I| \gtrapprox\delta^{-1+\varepsilon_0}$. - $|\cup P_t| \le |\cup_{z \in I} (X_* + f(z) Y_*)| \lessapprox\delta^{-(s+1)}$. Thus, if $\varepsilon, \varepsilon_0$ are sufficiently small in terms of $s, k, \kappa$, then we contradict Theorem [Theorem 50](#thm:mult_kakeya){reference-type="ref" reference="thm:mult_kakeya"}. ◻ ## An intermediate slicing result *This subsection contains ideas from [@he2020orthogonal].* Let $\mathcal{E}(\mathcal{T}, \varepsilon)$ be the set of exceptional slices, $$\mathcal{E}(\mathcal{T}, \varepsilon) = \{ z_0 \in \mathbb{R}: \exists \mathcal{T}' \subset \mathcal{T}, |\mathcal{T}'| \ge \delta^\varepsilon|\mathcal{T}|, |\cup \mathcal{T}'(z=z_0)| < \delta^{-s-\varepsilon} \}.$$ Just like in [@he2020orthogonal Proposition 25], we will prove a weaker version of Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"}; the stronger version follows from a formal exhaustion argument which we present in the next subsection. **Theorem 52**. *With assumptions of Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"}, there exists $\mathcal{T}' \subset \mathcal{T}$ such that $\mu(\mathcal{E}(\mathcal{T}')) \le \delta^\varepsilon$.* *Proof.* We use $\lessapprox$ to denote $\le C \delta^{-C\varepsilon}$, where $C$ may depend on $\kappa, s$. Let $\pi : \mathbb{R}^d \to \mathbb{R}^{d-1}$ be the projection onto the plane orthogonal to the $z$-axis. For a tube $t$, let $t(z') = \pi(t \cap \{ z = z' \})$, and for a set of tubes $\mathcal{T}$, let $\mathcal{T}(z')$ denote the slice $\pi(\mathcal{T}\cap \{ z = z' \})$. We follow the argument in [@he2020orthogonal Proof of Proposition 7]. Suppose Theorem [Theorem 52](#thm:projection_reduced){reference-type="ref" reference="thm:projection_reduced"} is false. We can find $z_1$ and a subset $\mathcal{T}''' \subset \mathcal{T}$ with $|\mathcal{T}'''| \ge \delta^\varepsilon|\mathcal{T}|$ such that $|\cup \mathcal{T}'''(z_1)| < \delta^{-s-\varepsilon}$. For this $\mathcal{T}'''$ we have $\mu(\mathcal{E}(\mathcal{T}''')) \ge \delta^\varepsilon$, hence $\mu(\mathcal{E}(\mathcal{T}''') \setminus B(z_1, \delta^{3\varepsilon/\kappa})) \ge \delta^\varepsilon- \delta^{2\varepsilon} > 0$ by the non-concentration property of $\mu$. Thus, we can find $z_2$ with $|z_1 - z_2| \gtrapprox 1$ and $\mathcal{T}'' \subset \mathcal{T}'''$ with $|\mathcal{T}''| \ge \delta^{2\varepsilon} |\mathcal{T}|$ such that $X := |\cup \mathcal{T}''(z_1)| < \delta^{-s-\varepsilon}$ and $Y := |\cup \mathcal{T}''(z_2)| < \delta^{-s-\varepsilon}$. Since every $t \in \mathcal{T}''$ passes through a point in $X$ and a point in $Y$, and since there are $\lessapprox 1$ many tubes through given points $x \in X$ and $y \in Y$, we can find $\mathcal{T}' \subset \mathcal{T}''$ with $|\mathcal{T}'| \gtrapprox|\mathcal{T}| \gtrapprox\delta^{-2s}$ such that for every $x \in X, y \in Y$, there is at most one tube in $\mathcal{T}'$ through $x, y$. In particular, $$\delta^{-2s} \lessapprox|\mathcal{T}'| \le |X|_\delta |Y|_\delta \lessapprox\delta^{-2s},$$ and so $|X|_\delta, |Y|_\delta \gtrapprox\delta^{-s}$, $|\mathcal{T}| \lessapprox|\mathcal{T}'| \lessapprox\delta^{-2s}$. For this $\mathcal{T}'$ we have $\mu(\mathcal{E}(\mathcal{T}')) \ge \delta^\varepsilon$, so defining $\mathcal{D}= \mathcal{E}(\mathcal{T}') \setminus (B(z_1, \delta^{3\varepsilon/\kappa}) \cup B(z_2, \delta^{3\varepsilon/\kappa}))$, we have $\mu(\mathcal{D}) \ge \delta^\varepsilon- 2\delta^{2\varepsilon} > \delta^{2\varepsilon}$. **Claim 1.** For $z = az_1 + (1-a)z_2 \in \mathcal{D}$, we have $a, 1-a \gtrapprox 1$. Furthermore, there exists $X_z \subset X$, $Y_z \subset X$, and $\mathcal{T}_z \subset \mathcal{T}'$ with $|X_z|_\delta, |Y_z|_\delta \gtrapprox\delta^{-s}, |\mathcal{T}_z| \gtrapprox\delta^{-2s}$ such that $|X_z + \frac{1-a}{a} Y_z| \lessapprox\delta^{-s}$ and for each $t \in \mathcal{T}_z$, we have $t(z_1) \in X_z^{(\delta)}$ and $t(z_2) \in Y_z^{(\delta)}$. *Proof.* The first claim is evident by definition of $\mathcal{D}$. For the second claim, since $z \in \mathcal{E}(\mathcal{T}')$, there exists $\mathcal{T}'_z \subset \mathcal{T}'$ such that $|\mathcal{T}'_z| \gtrapprox\delta^{-2s}$ and $|\mathcal{T}'_z (z)|_\delta \lessapprox\delta^{-s}$. Now notice that for each $x \in X, y \in Y$ there is at most one tube $t \in \mathcal{T}'$ passing through $x, y$. Let $P$ be the set of $(x, y) \in X \times Y$ with exactly one tube $t_{x,y} \in \mathcal{T}'_z$ passing through $x, y$. So $|P| \ge |\mathcal{T}'_z| \gtrapprox\delta^{-2s}$. We also observe that $|ax + (1-a)y - t_{x,y} (z)| \le \delta$, and so $|aX \overset{P}{+}(1-a)Y|_\delta \le |\mathcal{T}'_z (z)|_{2\delta} \lessapprox\delta^{-2s}$. Thus, by the Balog-Szemerédi-Gowers theorem [Theorem 41](#thm:bsg){reference-type="ref" reference="thm:bsg"}, we can find $X_z \subset X$, $Y_z \subset Y$, and $\mathcal{T}_z \subset \mathcal{T}'_z$ such that $|aX_z|_\delta, |(1-a)Y_z|_\delta \gtrapprox\delta^{-s}, |\mathcal{T}_z| \gtrapprox\delta^{-2s}$, $|a X_z + (1-a) Y_z| \lessapprox\delta^{-s}$, and for each $t \in \mathcal{T}_z$, we have $t(z_1) \in X_z^{(\delta)}$ and $t(z_2) \in Y_z^{(\delta)}$. Then $|X_z|_\delta, |Y_z|_\delta \gtrapprox\delta^{-s}$ and $|X_z + \frac{(1-a)}{a} Y_z| \lessapprox\delta^{-s}$, proving the Claim. 0◻ Now, we apply Lemma [Lemma 36](#lem:intersections_of_events){reference-type="ref" reference="lem:intersections_of_events"} to the sets $X_z^{(\delta)} \times Y_z^{(\delta)}$, the measure $\frac{1}{\mu(\mathcal{D})} \mu|_\mathcal{D}$, and $K = \delta^{-C\varepsilon}$ for a sufficiently large $C$. The result, after applying Fubini's theorem, is that we can find $z_*$, $X_* := X_{z_*} \subset X$, $Y_* := Y_{z_*} \subset Y$, and a subset $\mathcal{D}' \subset \mathcal{D}$ with $\mu(\mathcal{D}') \gtrapprox\mu(\mathcal{D}) \gtrapprox 1$ and $z_* \in \mathcal{D}'$ such that for all $z \in \mathcal{D}'$, we have $$|X_*^{(\delta)} \cap X_z^{(\delta)}| |Y_*^{(\delta)} \cap Y_z^{(\delta)}| \gtrapprox\delta^{2(n-1)} \delta^{-2s}.$$ Since $|X_*^{(\delta)} \cap X_z^{(\delta)}| \lesssim|X|_\delta \lessapprox\delta^{-s}$ and $|Y_*^{(\delta)} \cap Y_z^{(\delta)}| \lesssim|Y|_\delta \lessapprox\delta^{-s}$, we have in fact $|X_*^{(\delta)} \cap X_z^{(\delta)}|, |Y_*^{(\delta)} \cap Y_z^{(\delta)}| \approx\delta^{-s}$. In particular, $|X_z^{(\delta)}|, |Y_z^{(\delta)}| \approx\delta^{-s}$ for all $z \in \mathcal{D}'$. The next leg of the proof is to show: **Claim 2.** For all $z \in \mathcal{D}'$, if we write $z = az_1 + (1-a)z_2$, then $|X_* + \frac{1-a}{a} Y_*|_\delta \lessapprox\delta^{-s}$. *Proof.* Note that Claim 1 tells us $|X_z + \frac{1-a}{a} Y_z|_\delta \lessapprox\delta^{-s}$. Combining this with the Rusza triangle inequality (Lemma [Lemma 37](#lem:rusza triangle){reference-type="ref" reference="lem:rusza triangle"}), $X_*^{(\delta)} \cap X_z^{(\delta)} \subset X_z^{(\delta)}$, and $|A^{(\delta)}|_\delta \sim_d |A|_\delta$ for any subset $A$ of the doubling metric space $\mathbb{R}^d$, we have $$|X_z - X_*^{(\delta)} \cap X_z^{(\delta)}|_\delta \lesssim|X_z^{(\delta)} - X_z^{(\delta)}|_\delta \lesssim|X_z - X_z|_\delta \lesssim\frac{|X_z + \frac{1-a}{a} Y_z|_\delta^2}{|\frac{1-a}{a} Y_z|} \lessapprox\delta^{-s}.$$ The same argument shows (where $z_* = a_* z_1 + (1-a_*) z_2$): $$|X_* - X_*^{(\delta)} \cap X_z^{(\delta)}|_\delta \lesssim|X_* - X_*|_\delta \lesssim\frac{|X_* + \frac{1-a_*}{a_*} Y_*|_\delta^2}{|\frac{1-a_*}{a_*} Y_*|} \lessapprox\delta^{-s}.$$ Thus, by Lemma [Lemma 37](#lem:rusza triangle){reference-type="ref" reference="lem:rusza triangle"} again, we have $$|X_* - X_z|_\delta \lesssim\frac{|X_z - X_*^{(\delta)} \cap X_z^{(\delta)}|_\delta |X_* - X_*^{(\delta)} \cap X_z^{(\delta)}|_\delta}{|X_*^{(\delta} \cap X_z^{(\delta)}|_\delta} \lessapprox\delta^{-s}.$$ Similarly, we have $|Y_* - Y_z|_\delta \lessapprox\delta^{-s}$. A final application of Lemma [Lemma 37](#lem:rusza triangle){reference-type="ref" reference="lem:rusza triangle"} gives $$\begin{aligned} |X_* + \frac{1-a}{a} Y_*|_\delta &\lesssim\frac{|X_z + \frac{1-a}{a} Y_*|_\delta |X_z - X_*|_\delta}{|X_z|_\delta} \\ &\lesssim\frac{|X_z + \frac{1-a}{a} Y_z|_\delta |X_z - X_*|_\delta |\frac{1-a}{a} (Y_z - Y_*)|_\delta}{|X_z|_\delta |\frac{1-a}{a} Y_z|_\delta} \\ &\lessapprox\frac{\delta^{-s} \delta^{-s} \delta^{-s}}{\delta^{-s} \delta^{-s}} \le \delta^{-s}. \end{aligned}$$ This proves Claim 2. 0◻ Finally, we seek a contradiction by applying Theorem [Theorem 51](#thm:projection_weaker){reference-type="ref" reference="thm:projection_weaker"} to $\mathcal{T}_{z_*}$ and $\mu|_{\mathcal{D}'}$. We satisfy the condition (if $\varepsilon$ is sufficiently small) because Claim 1 and $|\mathcal{T}| \lessapprox\delta^{-2s}$ tell us that $\mathcal{T}_{z_*}$ is a $(\delta, \kappa, \delta^{-O(\varepsilon)}, k)$-set with $|\mathcal{T}_{z_*}| \gtrapprox\delta^{-2s}$. But we violate the conclusion (if $\varepsilon$ is sufficiently small) because Claim 2 tells us that $|aX_* + (1-a) Y_*|_\delta \lessapprox\delta^{-s}$. This contradiction finishes the proof of Theorem [Theorem 52](#thm:projection_reduced){reference-type="ref" reference="thm:projection_reduced"}. ◻ ## Formal exhaustion argument Using Theorem [Theorem 52](#thm:projection_reduced){reference-type="ref" reference="thm:projection_reduced"}, we prove the following proposition, which implies Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} with a different value for $\varepsilon$. **Proposition 53**. *For $0 \le k < d-1$, $0 \le s < k+1$, and $0 < \kappa \le 1$, there exists $\varepsilon> 0$ such that the following holds for sufficiently small $\delta < \delta_0 (s, k, d, \varepsilon)$. Let $\mathcal{T}$ be a $(\delta, \kappa, \delta^{-\varepsilon/2}, k)$-set of $\delta$-tubes each making angle $\ge \frac{1}{100}$ with the plane $y = 0$ with $|\mathcal{T}| \ge \delta^{-2s+\varepsilon/2}$. Let $\mu$ be a probability measure on $\mathbb{R}$ such that for all $\delta \le r \le 1$, we have $\mu(B_r) \le \delta^{-\varepsilon} r^\kappa$. Then $\mu(\mathcal{E}(\mathcal{T}, \frac{\varepsilon}{3})) \le \delta^{\varepsilon/2}$.* The idea is the following. A first application of Theorem [Theorem 52](#thm:projection_reduced){reference-type="ref" reference="thm:projection_reduced"} gives a subset $\mathcal{T}' \subset \mathcal{T}$ with $\mu(\mathcal{E}(\mathcal{T}',\epsilon)) \leq \delta^\epsilon$. Either $\mathcal{T}'$ is large enough in which case we are done or we can cut $\mathcal{T}'$ out of $\mathcal{T}$ and apply Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} again. This will give us another subset $\mathcal{T}'$. Then we iterate until the union of these sets $\mathcal{T}'$ is large enough. *Proof.* Let $N\geq 0$ be an integer. Suppose we have already constructed pairwise disjoint sets $\mathcal{T}_1,\cdots,\mathcal{T}_N$ such that $\mu(\mathcal{E}(\mathcal{T}_i,\epsilon)) \leq \delta^\epsilon$ for every $i = 1,\cdots,N$. Either we have $$\label{eq:AprimeBIG} \left| \mathcal{T}\setminus \bigcup_{i=1}^N \mathcal{T}_i \right| \leq \delta^{\frac{\epsilon}{2}} |\mathcal{T}|,$$ in which case we stop, or the set $\mathcal{T}\setminus \bigcup_{i=1}^N \mathcal{T}_i$ satisfies the conditions of Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"}. In the latter case Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} gives us $\mathcal{T}_{N+1} \subset \mathcal{T}\setminus \bigcup_{i=1}^N \mathcal{T}_i$ with $\mu(\mathcal{E}(A_{N+1},\epsilon)) \leq \delta^\epsilon$. By construction, $\mathcal{T}_{N+1}$ is disjoint with any of the $\mathcal{T}_i$, $i = 1,\cdots,N$. When this procedure ends write $\mathcal{T}_0 = \bigcup_{i=1}^N \mathcal{T}_i$. Then [\[eq:AprimeBIG\]](#eq:AprimeBIG){reference-type="eqref" reference="eq:AprimeBIG"} says $|\mathcal{T}\setminus \mathcal{T}_0| \leq \delta^\frac{\varepsilon}{2} |\mathcal{T}|$. Moreover, since the $\mathcal{T}_i$'s are disjoint, $|\mathcal{T}_0| = \sum_{i=1}^N |\mathcal{T}_i|$. Set $a_i = \frac{|\mathcal{T}_i|}{|\mathcal{T}_0|}$. We claim that $$\mathcal{E}(\mathcal{T},\frac{\varepsilon}{3}) \subset \bigcup_I \bigcap_{i \in I} \mathcal{E}(\mathcal{T}_i,\varepsilon),$$ where the index set $I$ runs over subsets of $\{1, 2, \cdots, n \}$ with $\sum_{i \in I}a_i \geq \delta^\frac{\varepsilon}{2}$. Since $\mu(\mathcal{E}(\mathcal{T}_i))) \le \delta^\varepsilon$ for all $i$, the desired upper bound $\mu(\mathcal{E}(\mathcal{T}, \frac{\varepsilon}{3})) \le \delta^{\varepsilon/2}$ then follows immediately from Markov's inequality applied to the event $\sum a_i \mathbbm{1}_{\mathcal{T}_i}$ (or [@he2020orthogonal Lemma 20]). We will now show the claim. Let $z_0 \in \mathcal{E}(A,\frac{\varepsilon}{3})$, so there exists $\mathcal{T}' \subset \mathcal{T}$ with $|\mathcal{T}'| \geq \delta^\frac{\varepsilon}{3} |\mathcal{T}|$ and $|\pi_{z_0} (\mathcal{T}')|_\delta \leq \delta^{-s - \frac{\varepsilon}{3}}$. Consider the index set $I$ defined as $$I = \{1 \le i \le n \mid |\mathcal{T}' \cap \mathcal{T}_i| \geq \delta^\varepsilon|\mathcal{T}_i|\}.$$ We have $$\begin{aligned} \delta^{\varepsilon/2} |\mathcal{T}| &\le |\mathcal{T}'| - |\mathcal{T}\setminus \mathcal{T}_0| \\ &\leq \sum_{i=1}^n |\mathcal{T}' \cap \mathcal{T}_i|\\ &\lesssim\sum_{i \in I} |\mathcal{T}_i| + \sum_{i \notin I} \delta^\varepsilon|\mathcal{T}_i| \\ &\lesssim\sum_{i\in I} a_i |\mathcal{T}| + \delta^\varepsilon|\mathcal{T}|\end{aligned}$$ Hence $\sum_{i \in I}a_i \geq \delta^\frac{\varepsilon}{2}$. On the other hand, for all $i \in I$, since $$|\pi_{z_0}(\mathcal{T}' \cap \mathcal{T}_i))|_\delta \le |\pi_{z_0} (\mathcal{T}')|_\delta \le \delta^{-s-\frac{\varepsilon}{3}},$$ we have $z_0 \in \mathcal{E}(\mathcal{T}_i,\varepsilon)$ for all $i \in I$. This finishes the proof of the claim. ◻ ## Proof of Proposition [Proposition 46](#prop:improved_incidence_weaker){reference-type="ref" reference="prop:improved_incidence_weaker"} {#proof-of-proposition-propimproved_incidence_weaker} *This subsection is based on Section A.7 of [@orponen2021hausdorff].* We restate Proposition [Proposition 46](#prop:improved_incidence_weaker){reference-type="ref" reference="prop:improved_incidence_weaker"}. **Proposition 54**. *Given $0 \le k < d-1$, $0 \le s < k+1$, $\tau, \kappa > 0$, there exist $\eta(s, k, \kappa, \tau, d) > 0$ and $\delta_0 (s, k, \kappa, \tau, d) > 0$ such that the following holds for all $\delta \in (0, \delta_0]$.* *Let $\mathbf{Y}\subset (\delta \cdot \mathbb{Z}) \cap [0,1)$ be a $(\delta, \tau, \delta^{-\eta})$-set, and for each $\mathbf{y}\in \mathbf{Y}$, assume that $\mathbf{X}_{\mathbf{y}} \subset (\delta \cdot \mathbb{Z})^{d-1} \cap [0,1)^{d-1}$ is a $(\delta, \kappa, \delta^{-\eta}, k)$-set with cardinality $\ge \delta^{-s+\eta}$. Let $$\mathbf{Z}= \bigcup_{\mathbf{y}\in \mathbf{Y}} \mathbf{X}_{\mathbf{y}} \times \{ \mathbf{y}\}.$$ For every $\mathbf{z}\in \mathbf{Z}$, assume that $\mathcal{T}(\mathbf{z})$ is a set of $\delta$-tubes each making an angle $\ge \frac{1}{100}$ with the plane $y = 0$ with $|\mathcal{T}(\mathbf{z})| \ge \delta^{-s+\eta}$ such that $\mathbf{z}\in T$ for all $T \in \mathcal{T}(\mathbf{z})$. Then $|\mathcal{T}| \ge \delta^{-2s-\eta}$, where $\mathcal{T}= \cup_{\mathbf{z}\in \mathbf{Z}} \mathcal{T}(\mathbf{z})$.* *Proof.* Let $A \lessapprox B$ denote $A \le C \delta^{-C \eta} B$ for some absolute constant $C \ge 1$. A $(\delta, u, m)$-set stands for a $(\delta, u, C\delta^{-C\eta}, m)$-set. First, without loss of generality, assume $|\mathcal{T}(\mathbf{z})| = \delta^{-s+\eta}$ for each $\mathbf{z}\in \mathbf{Z}$. Suppose $|\mathcal{T}| \le \delta^{-2s-\eta}$. Let $$\mathcal{T}(\mathbf{y}) = \bigcup_{\mathbf{x}\in \mathbf{X}_{\mathbf{y}}} \mathcal{T}(\mathbf{x},\mathbf{y}).$$ Since each tube in $\mathcal{T}(\mathbf{y})$ has angle $\ge \frac{1}{100}$ with the plane $y = 0$, it only intersects $O(1)$ many $\delta$-balls $(\mathbf{x}, \mathbf{y})$ for a given $\mathbf{y}$. Since $|\mathcal{T}(\mathbf{x}, \mathbf{y})| \gtrapprox\delta^{-s}$ for each $\mathbf{x}\in \mathbf{X}_\mathbf{y}$, we get $|\mathcal{T}(\mathbf{y})| \gtrapprox\delta^{-s} |\mathbf{X}_{\mathbf{y}}|$. With the counter-assumption $|\mathcal{T}| \lessapprox\delta^{-2s}$, this forces $|\mathbf{X}_{\mathbf{y}}| \lessapprox\delta^{-s}$ for each $\mathbf{y}\in \mathbf{Y}$. On the other hand, $|\mathbf{X}_{\mathbf{y}}| \gtrapprox\delta^{-s}$ and so $|\mathcal{T}| \approx\delta^{-2s}$. Now, we check that $\mathcal{T}(\mathbf{y})$ is a $(\delta, \kappa, \delta^{-O(\eta)}, k)$-set. Pick a $(r, k+1)$-plane $H$. We claim that either $\mathcal{T}(\mathbf{y}) \cap H = \emptyset$ or $H(y = \mathbf{y})$ is contained in a $(O(r), k)$-plate. Indeed, if $H(y = \mathbf{y})$ is not contained within a $(Cr, k)$-plate, then $H$ is contained within the $O(C^{-1})$-neighborhood of the plane $y = \mathbf{y}$, which means that $H$ cannot contain any tubes of $\mathcal{T}(\mathbf{y})$ if $C$ is large enough (since the tubes of $\mathcal{T}(\mathbf{y})$ have angle $\ge \frac{1}{100}$ with that plane). Thus, we may assume $H(y = \mathbf{y})$ is contained within a $(Cr, k)$-plate, which means $$\begin{gathered} |\mathcal{T}(\mathbf{y}) \cap H| = |\bigcup_{\mathbf{x}\in \mathbf{X}_\mathbf{y}\cap H} \mathcal{T}(\mathbf{x}, \mathbf{y}) \cap H| \\ \le |\mathbf{X}_\mathbf{y}\cap H| \cdot \delta^{-s+\eta} \lessapprox|\mathbf{X}_\mathbf{y}| r^\kappa \cdot \delta^{-s+\eta} \lessapprox r^\kappa |\mathcal{T}(\mathbf{y})|. \end{gathered}$$ Since $|\mathcal{T}(\mathbf{y})| \approx|\mathcal{T}|$ for each $\mathbf{y}\in \mathbf{Y}$, there is a subset $\overline{\mathcal{T}}\subset \mathcal{T}$ such that $|\mathcal{T}| \approx|\overline{\mathcal{T}}|$ and each $T \in \overline{\mathcal{T}}$ belongs to $\approx|\mathbf{Y}|$ of the sets $\mathcal{T}(\mathbf{y})$. We show $\overline{\mathcal{T}}$ is a $(\delta, \kappa, \delta^{-O(\eta)}, k)$-set. Indeed, given a $(r, k+1)$-plate $H$, we have $$\begin{gathered} |\overline{\mathcal{T}}\cap H| \approx\sum_{T \in \overline{\mathcal{T}}\cap H} \frac{1}{|\mathbf{Y}|} \sum_{\mathbf{y}\in \mathbf{Y}} \mathbbm{1}_{\mathcal{T}(y)} (T)\\ \lessapprox\frac{1}{|\mathbf{Y}|} \sum_{\mathbf{y}\in \mathbf{Y}} |\overline{\mathcal{T}}(\mathbf{y}) \cap H| \lessapprox\frac{1}{|\mathbf{Y}|} \sum_{\mathbf{y}\in \mathbf{Y}} r^\kappa |\mathcal{T}(y)| \le r^\kappa |\overline{\mathcal{T}}|. \end{gathered}$$ Finally, we refine $\mathbf{Y}$ further: since $$\sum_{\mathbf{y}\in \mathbf{Y}} |\overline{\mathcal{T}}\cap \mathcal{T}(y)| = \sum_{T \in \overline{\mathcal{T}}} |\{ \mathbf{y}\in \mathbf{Y}: T \in \mathcal{T}(\mathbf{Y}) \}| \approx|\overline{\mathcal{T}}| |\mathbf{Y}|,$$ we can find a subset $\overline{\mathbf{Y}}\subset \mathbf{Y}$ with the property that $|\overline{\mathcal{T}}(y)| := |\overline{\mathcal{T}}\cap \mathcal{T}(y)| \approx|\overline{\mathcal{T}}|$ for each $y \in \overline{\mathbf{Y}}$. Also, $\overline{\mathbf{Y}}$ is still a $(\delta, \tau, \delta^{-O(\eta)})$-set. Now for each $\mathbf{y}\in \overline{\mathbf{Y}}$, the large subset $\overline{\mathcal{T}}(y) \subset \overline{\mathcal{T}}$ has small covering number $|\mathbf{X}_{\mathbf{y}}| \approx\delta^{-s}$. On the other hand, $|\overline{\mathcal{T}}| \approx\delta^{-2s}$. This contradicts Theorem [Theorem 48](#thm:projection){reference-type="ref" reference="thm:projection"} if $\eta$ is chosen sufficiently small in terms of the $\varepsilon$ of the theorem. ◻ # Improved incidence estimates for regular sets {#sec:regular sets} In this section, we prove a version of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} for regular sets. **Definition 55**. *Let $\delta \in 2^{-2\mathbb{N}}$ be a dyadic number. Let $C, K > 0$, and let $0 \le s \le d$. A non-empty set $\mathcal{P}\subset \mathcal{D}_\delta$ is called $(\delta, s, C, K)$-regular if $\mathcal{P}$ is a $(\delta, s, C, 0)$-set, and $$|\mathcal{P}|_{\delta^{1/2}} \le K \cdot \delta^{-s/2}.$$* **Theorem 56**. *For any $0 \le s, k < d-1$, $\max(s, k) < t \le d$, $\kappa > 0$, there exists $\varepsilon(s, t, \kappa, k, d) > 0$ such that the following holds for all small enough $\delta \in 2^{-\mathbb{N}}$, depending only on $s, t, \kappa, k, d$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a $(\delta, t, \delta^{-\varepsilon}, \delta^{-\varepsilon})$-regular set. Assume that for every $p \in \mathcal{P}$, there exists a $(\delta, s, \delta^{-\varepsilon}, 0)$ and $(\delta, \kappa, \delta^{-\varepsilon}, k)$-set $\mathcal{T}(p) \subset \mathcal{T}$ with $|\mathcal{T}(p)| = M$ such that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$. Then $|\mathcal{T}| \ge M \delta^{-s-\varepsilon}$.* ## Initial reductions *This subsection is based on Sections 6 and A.1-A.3 of [@orponen2021hausdorff].* In this section, let $A \lessapprox B$ denote $A \le C \delta^{-C \varepsilon} B$ for some constant $C \ge 1$ depending only on $s, t, \kappa, k, d$. Also, let $\mathcal{P}\cap Q := \{ p \in \mathcal{P}: p \subset Q \}$. The proof will be based on contradiction, so assume $|\mathcal{T}| \le M\delta^{-s-\varepsilon}$. Let's rename $\mathcal{P}$ to $\mathcal{P}_0$ and $\mathcal{T}$ to $\mathcal{T}_0$, reserving $\mathcal{P}, \mathcal{T}$ for the use of Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}. By Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}, we have $1 \gtrapprox(M\delta^s)^{\frac{t-s}{d-1-s}}$, so $M \lessapprox\delta^{-s}$ and $|\mathcal{T}| \lessapprox\delta^{-2s}$. But $\mathcal{T}(p)$ is a $(\delta, s, \delta^{-\varepsilon})$-set, so $M \approx\delta^{-s}$. Finally, by Lemma [Lemma 21](#lem:small sets){reference-type="ref" reference="lem:small sets"}, we may assume $|\mathcal{P}_0| \approx\delta^{-t}$ (passing to subsets will preserve the $(\delta, t, \delta^{-\varepsilon}, \delta^{-\varepsilon})$-regularity of $\mathcal{P}_0$). The next reduction will make the value $|\mathcal{P}_0 \cap Q|$ uniform for different $Q \in \mathcal{D}_\Delta (\mathcal{P}_0)$. Let $\mathcal{Q}_0 = \mathcal{D}_\Delta (\mathcal{P}_0)$. By $(\delta, t, \delta^{-\varepsilon}, \delta^{-\varepsilon})$-regularity of $\mathcal{P}_0$, we have $|\mathcal{Q}_0| \lessapprox\Delta^{-t}$. On the other hand, since $\mathcal{P}_0$ is a $(\delta, t)$-set, we have that for all $Q \in \mathcal{Q}_0$, $$\label{eqn:light squares} |\mathcal{P}_0 \cap Q| \lessapprox\Delta^{-t}$$ This means $|\mathcal{Q}_0| \gtrapprox\Delta^{-t}$. Hence, $|\mathcal{Q}_0| \approx\Delta^{-t}$. Now using [\[eqn:light squares\]](#eqn:light squares){reference-type="eqref" reference="eqn:light squares"} again and $|\mathcal{P}_0| \approx\Delta^{-2t}$, there exists $\mathcal{Q}_0' \subset \mathcal{Q}_0$ with $|\mathcal{Q}_0'| \gtrapprox|\mathcal{Q}_0|$ such that for each $Q \in \mathcal{Q}_0'$, $$\label{eqn:good squares} |\mathcal{P}_0 \cap Q| \approx\Delta^{-t}$$ Using [\[eqn:good squares\]](#eqn:good squares){reference-type="eqref" reference="eqn:good squares"}, we quickly check that $\mathcal{Q}_0'$ is a $(\Delta, t)$-set. Indeed, for $r \in (\Delta, 1)$ and $Q_r \in \mathcal{D}_r$, we have $$\label{eqn:cQ0 dim check} |\mathcal{Q}_0' \cap Q_r| \overset{\eqref{eqn:good squares}}{\approx} \Delta^t \cdot |\mathcal{P}_0 \cap Q_r| \lessapprox\Delta^t \cdot |\mathcal{P}_0| \cdot r^t \approx|\mathcal{Q}_0'| \cdot r^t.$$ (The second inequality uses that $\mathcal{P}_0$ is a $(\Delta, t)$-set.) Let $\mathcal{P}_0' = \bigcup_{Q \in \mathcal{Q}_0'} \mathcal{P}\cap Q$; then $|\mathcal{P}_0'| \approx|\mathcal{P}|$ and $|\mathcal{P}_0' \cap Q| = |\mathcal{P}_0 \cap Q| \approx\Delta^{-t}$ for $Q \in \mathcal{P}_0'$. Apply Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"} to find $\mathcal{P}\subset \mathcal{P}_0', \mathcal{T}(p) \subset \mathcal{T}_0 (p)$, $\mathcal{T}_\Delta$, $\mathcal{T}$, and $\mathcal{T}_Q$. Let $\mathcal{Q}= \mathcal{D}_\Delta (\mathcal{P})$. **Claim.** $M_\Delta \approx\Delta^{-s}$ and $|\mathcal{T}_\Delta| \lessapprox\delta^{-s}$. *Proof.* By Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}[\[item3\]](#item3){reference-type="ref" reference="item3"}, we know that $(\mathcal{D}_\Delta (\mathcal{P}), \mathcal{T}_\Delta)$ is $(\Delta, s, C_\Delta^1, \kappa, C_\Delta^2, M_\Delta)$-nice, so $M_\Delta \gtrapprox\Delta^{-s}$. Also, by Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}, we have that $$\label{eqn:lower T_Delta} |\mathcal{T}_\Delta| \gtrapprox M_\Delta \delta^{-s/2} \cdot (M_\Delta \delta^{s/2})^{\frac{t-s}{d-1-s}}.$$ Next, for any $Q \in \mathcal{Q}$, we know that $(S_Q (\mathcal{P}\cap Q), \mathcal{T}_Q)$ is $(\Delta, s, C_Q^1, \kappa, C_Q^2, M_Q)$-nice. Recall that $$S_Q (\mathcal{P}\cap Q) = \{ S_Q (p) : p \in \mathcal{P}, p \subset Q \} \subset \mathcal{D}_\Delta.$$ We also know $|\mathcal{P}\cap Q| \approx|\mathcal{P}_0' \cap Q| \approx\delta^{-t/2}$ and $\mathcal{P}$ is a $(\delta, t)$-set, so by a similar check to [\[eqn:cQ0 dim check\]](#eqn:cQ0 dim check){reference-type="eqref" reference="eqn:cQ0 dim check"}, we get that $S_Q (\mathcal{P}\cap Q)$ is a $(\Delta, t)$-set. Thus by Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}, we have $$|\mathcal{T}_Q| \gtrapprox M_Q \cdot \delta^{-s/2}$$ But by our counterassumption $|\mathcal{T}_0| \lessapprox\delta^{-2s}$, we get from [\[eqn:item6\]](#eqn:item6){reference-type="eqref" reference="eqn:item6"} in Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"} and $M \gtrapprox\delta^{-s}$, $$\delta^{-2s} \gtrapprox\frac{|\mathcal{T}_\Delta|}{M_\Delta} \cdot \frac{|\mathcal{T}_Q|}{M_Q} \cdot M \gtrapprox\frac{|\mathcal{T}_\Delta|}{M_\Delta} \cdot \delta^{-3s/2}.$$ Thus, $|\mathcal{T}_\Delta| \lessapprox M_\Delta \delta^{-s/2}$. Substitute into [\[eqn:lower T_Delta\]](#eqn:lower T_Delta){reference-type="eqref" reference="eqn:lower T_Delta"} to get $$\delta^{-s/2} \gtrapprox\frac{|T_\Delta|}{M_\Delta} \gtrapprox\delta^{-s/2} \cdot (M_\Delta \delta^{s/2})^{\frac{t-s}{d-1-s}}.$$ Thus, $M_\Delta \delta^{s/2} \lessapprox 1$, so $M_\Delta \lessapprox\Delta^{-s}$ and $|\mathcal{T}_\Delta| \lessapprox\delta^{-s}$, proving the Claim. 0◻ Thus, we get the higher-dimensional analogues of properties (H1-2), (G1-4) of [@orponen2021hausdorff] except we only know $|\mathcal{T}| \lessapprox\delta^{-2s}$ and not $|\mathcal{T}| \gtrapprox\delta^{-2s}$. But this is not a limitation. We repeat and relabel these properties here: 1. [\[g1\]]{#g1 label="g1"} $|\mathcal{Q}| \approx\Delta^{-t}$ and $|\mathcal{P}\cap Q| \approx\Delta^{-t}$ for all $Q \in \mathcal{Q}$. 2. [\[g2\]]{#g2 label="g2"} Every tube $\mathbf{T}\in \mathcal{T}_\Delta$ satisfies $|\mathcal{T}\cap \mathbf{T}| \lessapprox\delta^{-s}$. 3. [\[g3\]]{#g3 label="g3"} For every square $Q \in \mathcal{Q}$, there corresponds a $(\Delta, s, 0)$-set and $(\Delta, \kappa, k)$-set $\mathcal{T}_\Delta (Q) \subset \mathcal{T}_\Delta$ of cardinality $\approx M_\Delta \approx\Delta^{-s}$ such that $\mathbf{T}\cap Q \neq \emptyset$ for all $\mathbf{T}\in \mathcal{T}_\Delta (Q)$. 4. [\[g4\]]{#g4 label="g4"} $|\mathcal{T}| \lessapprox\delta^{-2s}$ and $|\mathcal{T}_\Delta| \approx\Delta^{-2s}$. 5. [\[g5\]]{#g5 label="g5"} For $\mathbf{T}\in \mathcal{T}_\Delta (Q)$, we have $$|\{ (p, T) \in (\mathcal{P}\cap Q) \times \mathcal{T}: T \in \mathcal{T}(p) \cap \mathbf{T}\}| \gtrapprox\Delta^{-s-t}.$$ Item [\[g1\]](#g1){reference-type="ref" reference="g1"} follows from Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}[\[item1\]](#item1){reference-type="ref" reference="item1"}. Item [\[g3\]](#g3){reference-type="ref" reference="g3"} follows from Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}[\[item3\]](#item3){reference-type="ref" reference="item3"} and Claim. Item [\[g4\]](#g4){reference-type="ref" reference="g4"} follows from $|\mathcal{T}_0| \lessapprox\delta^{-2s}$ and Claim. Item [\[g5\]](#g5){reference-type="ref" reference="g5"} follows from Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}[\[item4\]](#item4){reference-type="ref" reference="item4"} and the estimation $M \cdot |\mathcal{P}\cap Q| / |\mathcal{T}_\Delta (Q)| \approx\Delta^{-s-t}$, which uses item [\[g1\]](#g1){reference-type="ref" reference="g1"}, item [\[g3\]](#g3){reference-type="ref" reference="g3"}, and the fact $M \approx\delta^{-s}$ we proved at the beginning of the argument. Item [\[g2\]](#g2){reference-type="ref" reference="g2"} follows from Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}[\[item21\]](#item21){reference-type="ref" reference="item21"}, the fact that a given $\delta$-tube lies in $\lesssim 1$ many of the $\mathbf{T}$'s in $\mathcal{T}_\Delta$, and item [\[g4\]](#g4){reference-type="ref" reference="g4"}: $$\delta^{-2s} \gtrapprox|\mathcal{T}| \gtrsim\sum_{\mathbf{T}\in \mathcal{T}_\Delta} |\mathcal{T}\cap \mathbf{T}| \sim |\mathcal{T}_\Delta| \cdot \mathbf{N}\approx\Delta^{-2s} \cdot \mathbf{N}.$$ ## Transferring angular non-concentration to ball non-concentration *This subsection is based on Section A.4 of [@orponen2021hausdorff].* We first recall some notation. For a unit vector $\sigma \in \mathbb{R}^d$, define $\pi_\sigma (\vec{v}) := \vec{v} - (\vec{v} \cdot \sigma) \sigma$ to be the orthogonal projection to the orthogonal complement of $\sigma$. For a $\delta$-tube $T$, let $\sigma(T) \in S^{d-1}$ denote the direction of $T$. In this subsection, we fix a $Q \in \mathcal{D}_\Delta (\mathcal{P})$. Our goal is to show that for many $\mathbf{T}\in \mathcal{T}_\Delta (Q)$, the $\Delta^{-1}$-rescaled version of $\pi_{\sigma(\mathbf{T})} (\cup (\mathcal{P}\cap Q))$ contains a $(\Delta, s, 0)$ and $(\Delta, \kappa', k)$-set for some $\kappa' > 0$. This is the content of the next Proposition [Proposition 57](#prop:projections){reference-type="ref" reference="prop:projections"}, which is a higher-dimensional extension of Lemma A.6 of [@orponen2021hausdorff]. The proposition encodes the following principle: If we have a set of orthogonal projections in $\mathrm{Gr}(d, d-1)$ (which we view as $S^{d-1}$) that don't concentrate around $k$-planes, and we have a $t$-dimensional set $X$ with $t > k$, then many projections of $X$ will not concentrate around $k$-planes. **Proposition 57**. *Let $0 \le \max(s, k) < t \le d$, $\kappa > 0$, and $\mathbf{A}, \mathbf{B}> 0$. Let $\mathcal{P}$ be a $(\Delta, t, \Delta^{-\mathbf{A}\varepsilon})$-set in $[0,1)^d$, and let $\Gamma \subset S^{d-1}$ be a $(\Delta, s, \Delta^{-\mathbf{A}\varepsilon}, 0)$-set and $(\Delta, \kappa, \Delta^{-\mathbf{A}\varepsilon}, k)$-set. There exists a subset $\Sigma \subset \Gamma$ with $|\Sigma| \ge \frac{1}{2} |\Gamma|$ such that the following holds for all $\sigma \in \Sigma$: if $\mathcal{P}' \subset \mathcal{P}$ is an arbitrary subset of cardinality $|\mathcal{P}'| \ge \Delta^{\mathbf{B}\varepsilon} |\mathcal{P}|$, then $\pi_\sigma(\mathcal{P}')$ contains a $(\Delta, \frac{1}{k+1} \min(\frac{t-k}{2}, \kappa), \Delta^{-\mathbf{C}(\mathbf{A}+ \mathbf{B}) \varepsilon}, k)$ and $(\Delta, s, \Delta^{-\mathbf{C}(\mathbf{A}+ \mathbf{B})}, 0)$-set, where $\mathbf{C}\ge 1$ is absolute depending on $k$.* *Proof.* We will use a variation of the energy argument due to Kaufman [@kaufman1968hausdorff] in the form used to prove [@orponen2021hausdorff Lemma A.6]. An alternate proof can follow [@he2020orthogonal Lemma 27], but this approach would give weaker bounds. Let $\mu$ be the $\Delta$-discretized probability measure corresponding to $\mathcal{P}$, $$\mu := \frac{1}{|\mathcal{P}|} \sum_{q \in \mathcal{P}} \frac{\mathcal{L}^d|_q}{\Delta^d},$$ where $\mathcal{L}^d$ is $d$-dimensional Lebesgue measure. Since $\mathcal{P}$ is a $(\Delta, t, \Delta^{-\mathbf{A}\varepsilon})$-set, we have $\mu(B(x, r)) \lessapprox r^t$ for all $r > \delta$, and it's also true for $r < \delta$ since $\mu$ behaves like Lebesgue measure at small scales. We will choose a uniformly random $\sigma \in \Gamma$ and consider what happens to the energy of $\mu$ under projection by $\sigma$. By linearity of expectation and definition of energy, $$E_{s,1} := \mathbb{E}_\sigma [I_{s,1}^\Delta (\pi_\sigma \mu)] = \int \mathbb{E}_\sigma [(|\pi_\sigma (x_0 - x_1)| + \delta)^{-s}] \, d\mu(x_0) d\mu(x_1).$$ Since $\Gamma$ is a $(\Delta, s)$-set, we have $\mathbb{E}_\sigma [(|\pi_\sigma (x_0 - x_1)| + \Delta)^{-s}] \lesssim(\log \Delta^{-1}) \cdot \Delta^{-\mathbf{A}\varepsilon} |x_0 - x_1|^{-s}$ (c.f. [@kaufman1968hausdorff]), and so $E_{s,1} \lessapprox I_{s,1}^0 (\mu) \lessapprox 1$ by Lemma [Lemma 44](#lem:energy){reference-type="ref" reference="lem:energy"}(a) and $s < t$. Analogously, we have (let $\beta = \min(\kappa, \frac{t-k}{2})$): $$E_{\beta,k+1} := \mathbb{E}_\sigma [I_{\beta,k+1}^\Delta (\pi_\sigma \mu)] = \int \mathbb{E}_\sigma \left[\left(\left|\bigwedge_{i=1}^{k+1} \pi_\sigma (x_0 - x_i) \right| + \Delta \right)^{-\beta} \right] \, d\mu(x_0) \cdots d\mu(x_{k+1}).$$ Observe that $$\left|\bigwedge_{i=1}^{k+1} \pi_\sigma (x_0 - x_i)\right| = \left|\sigma \wedge \bigwedge_{i=1}^{k+1} \pi_\sigma (x_0 - x_i)\right| = \left|\sigma \wedge \bigwedge_{i=1}^{k+1} (x_0 - x_i)\right| = \left|\bigwedge_{i=1}^{k+1} (x_0 - x_i)\right| \cdot \rho,$$ where $\rho$ is the distance from $\sigma$ to the plane spanned by $x_1 - x_0$ through $x_{k+1} - x_0$. (The first equality follows since $\sigma$ is orthogonal to each $\pi_\sigma (x_0 - x_i)$. The second equality follows since $\wedge$ is multilinear and $\sigma \wedge \sigma = 0$. The third equality follows by the geometric definition of wedge product as a volume of a parallelepiped.) Thus, since $\Gamma$ is a $(\Delta, \kappa, k)$-set and $\beta \le \kappa$, we have $$\begin{aligned} \mathbb{E}_\sigma \left[\left(\left|\bigwedge_{i=1}^{k+1} \pi_\sigma (x_0 - x_i) \right| + \Delta \right)^{-\beta} \right] &\lesssim\sum_{\rho = 2^{-n} \in (\Delta, 1)} \Delta^{-\mathbf{A}\varepsilon} \cdot \rho^{\kappa-\beta} \left|\bigwedge_{i=1}^{k+1} (x_0 - x_i)\right|^{-\beta} \\ &\lesssim(\log \Delta^{-1}) \cdot \Delta^{-\mathbf{A}\varepsilon} \left|\bigwedge_{i=1}^{k+1} (x_0 - x_i)\right|^{-\beta}, \end{aligned}$$ and so $E_{\beta, k+1} \lessapprox I_{\beta, k+1}^0 (\mu) \lessapprox 1$ by Lemma [Lemma 44](#lem:energy){reference-type="ref" reference="lem:energy"}(a) and $\beta < t-k$. Consequently, by Markov's inequality we can find $\Sigma \subset \Gamma$ with $|\Sigma| \ge \frac{1}{2} |\Gamma|$ such that for each $\sigma \in \Sigma$, we have $I_{s,1}^\Delta (\pi_\sigma \mu) \le \Delta^{-2C_1 \mathbf{A}\varepsilon}$ and $I_{\beta,k+1}^\Delta (\pi_\sigma \mu) \le \Delta^{-2C_1 \mathbf{A}\varepsilon}$. For any $\mathcal{P}' \subset \mathcal{P}$ with $|\mathcal{P}'| \ge \Delta^{\mathbf{B}\varepsilon} |\mathcal{P}|$, we have $I_{s,1}^\Delta (\pi_\sigma \mu_{\mathcal{P}'}) \le \Delta^{-(2C_1 \mathbf{A}+ 2\mathbf{B}) \varepsilon}$ and $I_{\beta,k+1}^\Delta (\pi_\sigma \mu_{\mathcal{P}'}) \le \Delta^{-(2C_1 \mathbf{A}+ (k+2) \mathbf{B}) \varepsilon}$, where $\mu_{\mathcal{P}'} = \frac{1}{\mu(\mathcal{P}')} \mu|_{\mathcal{P}'}$ is the renormalized restriction of $\mu$ to $\mathcal{P}'$. Then Lemma [Lemma 44](#lem:energy){reference-type="ref" reference="lem:energy"}(b) gives the desired conclusion. ◻ ## Finding a special $\Delta$-tube *This subsection is based on Section A.4 of [@orponen2021hausdorff].* Apply Proposition [Proposition 57](#prop:projections){reference-type="ref" reference="prop:projections"} to $S_Q (\mathcal{P}\cap Q)$, which is a $(\Delta, t)$-set using [\[g1\]](#g1){reference-type="ref" reference="g1"} and the fact that $\mathcal{P}$ is a $(\Delta, t)$-set. Define $$\mathcal{T}_\Delta^\pi (Q) = \{ \mathbf{T}\in \mathcal{T}_\Delta (Q) : \sigma(\mathbf{T}) \in \Sigma(Q) \}, \qquad Q \in \mathcal{Q},$$ where $\Sigma(Q)$ is the set of good directions of cardinality $|\Sigma(Q)| \ge \frac{1}{4} |\sigma(Q)| \sim |\mathcal{T}_\Delta (Q)|$ (since for a given direction, there are $\sim 1$ many $\Delta$-tubes in that direction that intersect $Q$). Then $\mathcal{T}_\Delta^\pi (Q), Q \in \mathcal{Q}$ remain $(\Delta, s)$-sets of cardinality $\approx\Delta^{-s}$, and so the properties [\[g1\]](#g1){reference-type="ref" reference="g1"}-[\[g5\]](#g5){reference-type="ref" reference="g5"} remain valid upon replacing $\mathcal{T}_\Delta (Q)$ with $\mathcal{T}_\Delta (Q)$. (We leave $\mathcal{T}_\Delta$ unchanged, so only [\[g3\]](#g3){reference-type="ref" reference="g3"} and [\[g5\]](#g5){reference-type="ref" reference="g5"} are affected.) Thus, $\mathcal{P}\cap Q$ for $Q \in \mathcal{Q}$ and their large subsets have nice projections in the sense of Proposition [Proposition 57](#prop:projections){reference-type="ref" reference="prop:projections"} in every direction orthogonal to the tubes $\mathbf{T}\in \mathcal{T}_\Delta^\pi (Q)$. We keep the symbol "$\pi$" as a reminder of this fact. The next goal is to find a tube $\mathbf{T}_0$ with the following properties: 1. [\[P1\]]{#P1 label="P1"} The set $\{ Q \in \mathcal{Q}: \mathbf{T}_0 \in \mathcal{T}_\Delta^\pi (Q) \}$ contains a $(\Delta, t-s)$-subset, which we denote $\mathbf{T}_0 (\mathcal{Q})$. 2. [\[P2\]]{#P2 label="P2"} $|\mathcal{T}\cap \mathbf{T}_0| \lessapprox\Delta^{-2s}$. 3. [\[P3\]]{#P3 label="P3"} For each $Q \in \mathbf{T}_0 (\mathcal{Q})$, there exists a subset $\mathcal{P}_Q \subset \mathcal{P}\cap Q$ such that $$|\mathcal{P}_Q| \approx\Delta^{-t} \text{ and } |\mathcal{T}(p) \cap \mathbf{T}_0| \approx\Delta^{-s} \text{ for all } p \in \mathcal{P}_Q.$$ 4. [\[P4\]]{#P4 label="P4"} Let $\sigma$ be the direction of $\mathbf{T}$. Then $\pi_\sigma (S_Q (\mathcal{P}_Q))$ contains a $(\Delta, \kappa', k)$-set with cardinality $\gtrapprox\Delta^{-s}$, where $\kappa' := \frac{1}{k+1} \min(\frac{t-k}{2}, \kappa)$. To get [\[P1\]](#P1){reference-type="ref" reference="P1"}- [\[P3\]](#P3){reference-type="ref" reference="P3"}, we will mostly follow Section A.4 of [@orponen2021hausdorff]. (We have used the fact that $\mathcal{T}$ is a $(\Delta, \kappa, \Delta^{-\mathbf{A}\varepsilon}, k)$-set, by converting it into ball concentration near $(k+1)$-planes in Proposition [Proposition 57](#prop:projections){reference-type="ref" reference="prop:projections"}; the rest of the argument will only use the fact that $\mathcal{T}$ is a $(\Delta, s, \Delta^{-\mathbf{A}\varepsilon}, 0)$-set.) First, we refine the sets $\mathcal{Q}$ and $\mathcal{T}_\Delta^\pi (Q)$ further to ensure that the family $\{ Q \in \mathcal{Q}: \mathbf{T}\in \mathcal{T}_\Delta^\pi (Q) \}$ will be $(\Delta, t-s)$-sets for $\mathbf{T}\in \mathcal{T}_\Delta$. Indeed, we have $$\begin{aligned} \sum_{\mathbf{T}\in \mathcal{T}_\Delta} \sum_{\substack{Q, Q' \in \mathcal{Q}\\Q \neq Q'}} \frac{\mathbbm{1}_{\mathcal{T}_\Delta^\pi (Q) \cap \mathcal{T}_\Delta^\pi (Q')} (\mathbf{T})}{d(Q, Q')^{t-s}} &= \sum_{Q, Q' \in \mathcal{Q}, Q \neq Q'} \frac{|\mathcal{T}_\Delta^\pi (Q) \cap \mathcal{T}_\Delta^\pi (Q')|}{d(Q, Q')^{t-s}} \\ &\lessapprox\sum_{Q, Q' \in \mathcal{Q}, Q \neq Q'} \frac{1}{d(Q, Q')^t} \lessapprox\Delta^{-2t}.\end{aligned}$$ The first $\lessapprox$ inequality uses the fact that $\mathcal{T}_\Delta^\pi (Q)$ is a $(\Delta, s)$-set of tubes with $|\mathcal{T}_\Delta^\pi (Q)| \approx\Delta^{-s}$, and the second $\lessapprox$ inequality uses the fact that $\mathcal{Q}$ is a $(\Delta, t)$-set with $|\mathcal{Q}| \approx\Delta^{-t}$. Thus, by Markov's inequality, for a fixed absolute large constant $C \ge 1$, we have $$\label{eqn:exception} \sum_{\substack{Q, Q' \in \mathcal{Q}\\Q \neq Q'} }\frac{\mathbbm{1}_{\mathcal{T}_\Delta^\pi (Q) \cap \mathcal{T}_\Delta^\pi (Q')} (\mathbf{T})}{d(Q, Q')^{t-s}} \ge \Delta^{-C\varepsilon+ 2(s-t)}$$ can only hold for $\lessapprox\Delta^{C\varepsilon-2s}$ many tubes $\mathbf{T}\in \mathcal{T}_\Delta$. **Claim 2.** If $C \ge 1$ is sufficiently large, then there exists a subset $\overline{\mathcal{Q}}\subset \mathcal{Q}$ with $|\overline{\mathcal{Q}}| \ge \frac{1}{2} |\mathcal{Q}|$ such that for all $Q_0 \in \overline{\mathcal{Q}}$, at most half of the tubes $\mathbf{T}\in \mathcal{T}_\Delta^\pi (Q_0)$ satisfy [\[eqn:exception\]](#eqn:exception){reference-type="eqref" reference="eqn:exception"}. *Proof.* Suppose this is not true: there exists a set $\mathcal{Q}_{\text{bad}}$ such that for $Q_0 \in \mathcal{Q}$, at least $\frac{1}{2} |\mathcal{T}_\Delta^\pi (Q_0)|$ many tubes $\mathbf{T}\in \mathcal{T}_\Delta^\pi (Q_0)$ satisfy [\[eqn:exception\]](#eqn:exception){reference-type="eqref" reference="eqn:exception"}. Then apply Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"} to $\mathcal{Q}_{\text{bad}}$ and the bad parts of $\mathcal{T}_\Delta^\pi (Q_0)$, which are still $(\Delta, s$)-sets. By Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}, we have $\gtrapprox\Delta^{-2s}$ many $\Delta$-tubes in $\mathcal{T}_\Delta$ that satisfy [\[eqn:exception\]](#eqn:exception){reference-type="eqref" reference="eqn:exception"}. But we observed before that [\[eqn:exception\]](#eqn:exception){reference-type="eqref" reference="eqn:exception"} only holds for $\lessapprox\Delta^{C\varepsilon-2s}$ many tubes $\mathbf{T}\in \mathcal{T}_\Delta$. By choosing $C$ large enough (and $\delta$ small enough), we obtain a contradiction. 0◻ In what follows, the $C$ in Claim 2 will be absorbed into the $\lessapprox$ notation. Replace $\mathcal{Q}$ by $\overline{\mathcal{Q}}$ and $\mathcal{T}_\Delta^\pi (Q)$ by their good subsets without changing notation. All of the properties [\[g1\]](#g1){reference-type="ref" reference="g1"}-[\[g5\]](#g5){reference-type="ref" reference="g5"} remain valid, and $$\label{eqn:good energy} \sum_{\substack{Q, Q' \in \mathcal{Q}\\Q \neq Q'}} \frac{\mathbbm{1}_{\mathcal{T}_\Delta^\pi (Q) \cap \mathcal{T}_\Delta^\pi (Q')} (\mathbf{T})}{d(Q, Q')^{t-s}} \lessapprox\Delta^{2(s-t)}, \qquad \mathbf{T}\in \mathcal{T}_\Delta^\pi (Q_0), Q_0 \in \mathcal{Q}.$$ Now, we will find $\mathbf{T}_0 \in \mathcal{T}_\Delta$ satisfying $$\label{eqn:card s-t} |\mathbf{T}_0 (\mathcal{Q})| := |\{ Q \in \mathcal{Q}: \mathbf{T}_0 \in \mathcal{T}_\Delta^\pi (Q) \}| \gtrapprox\Delta^{s-t}.$$ Indeed, the average tube works, because of the following: since $|\mathcal{T}_\Delta| \approx\Delta^{-2s}, |\mathcal{Q}| \approx\Delta^{-t}$, and $|\mathcal{T}_\Delta^\pi (Q)| \approx\Delta^{-s}$ (by [\[g4\]](#g4){reference-type="ref" reference="g4"}, [\[g1\]](#g1){reference-type="ref" reference="g1"}, [\[g3\]](#g3){reference-type="ref" reference="g3"} respectively), we have $$\frac{1}{|\mathcal{T}_\Delta|} \sum_{\mathbf{T}\in \mathcal{T}_\Delta} |\{ Q \in \mathcal{Q}: \mathbf{T}_0 \in \mathcal{T}_\Delta^\pi (Q) \}| = \frac{1}{|\mathcal{T}_\Delta|} \sum_{Q \in \mathcal{Q}} |\mathcal{T}_\Delta^\pi (Q)| \approx\frac{|\mathcal{Q}| \cdot \Delta^{-s}}{\Delta^{-2s}} \approx\Delta^{s-t}.$$ Now, we show that using [\[eqn:good energy\]](#eqn:good energy){reference-type="eqref" reference="eqn:good energy"} and [\[eqn:card s-t\]](#eqn:card s-t){reference-type="eqref" reference="eqn:card s-t"}, the family $\mathbf{T}_0 (\mathcal{Q}) \subset \{ Q \in \mathcal{Q}: Q \cap \mathbf{T}_0 \neq \emptyset \}$ contains a $(\Delta, t-s)$-set, which proves item [\[P1\]](#P1){reference-type="ref" reference="P1"}. Indeed, rewrite [\[eqn:good energy\]](#eqn:good energy){reference-type="eqref" reference="eqn:good energy"} as $$\label{eqn:good energy'} \sum_{\substack{Q, Q' \in \mathbf{T}_0 (\mathcal{Q})\\Q \neq Q'}} \frac{1}{d(Q, Q')^{t-s}} \lessapprox\Delta^{2(s-t)}.$$ Let $$\label{eqn:bt0'} \mathbf{T}_0' (\mathcal{Q}) := \{ Q \in \mathbf{T}_0 (\mathcal{Q}) : \sum_{Q' \in \mathbf{T}_0 (\mathcal{Q}) \setminus \{ Q \}} d(Q, Q')^{s-t} \le \Delta^{s-t-C\varepsilon} \}.$$ By Markov's inequality on [\[eqn:good energy\'\]](#eqn:good energy'){reference-type="eqref" reference="eqn:good energy'"}, we have $|\mathbf{T}_0 (\mathcal{Q}) \setminus \mathbf{T}_0' (\mathcal{Q})| \lessapprox\Delta^{s-t+C\varepsilon}$. Hence, if $C$ is chosen large enough, we have by [\[eqn:card s-t\]](#eqn:card s-t){reference-type="eqref" reference="eqn:card s-t"}, $|\mathbf{T}_0' (\mathcal{Q})| \ge \frac{1}{2} |\mathbf{T}_0 (\mathcal{Q})| \gtrapprox\Delta^{s-t}$. By Markov's inequality on [\[eqn:bt0\'\]](#eqn:bt0'){reference-type="eqref" reference="eqn:bt0'"}, we have that for all $Q \in \mathbf{T}_0' (\mathcal{Q})$ and $r \in (\delta, 1)$, $$|\{ Q' \in \mathbf{T}_0 (\mathcal{Q}) : d(Q, Q') \le r \}| \le \Delta^{s-t-C\varepsilon} r^{t-s}.$$ Thus, $\mathbf{T}_0' (\mathcal{Q})$ is a $(\Delta, t-s)$-set, which proves [\[P1\]](#P1){reference-type="ref" reference="P1"}. To get [\[P2\]](#P2){reference-type="ref" reference="P2"}, we use [\[g2\]](#g2){reference-type="ref" reference="g2"}. $$|\mathcal{T}\cap \mathbf{T}_0| \lessapprox\delta^{-s} = \Delta^{-2s}.$$ By [\[g5\]](#g5){reference-type="ref" reference="g5"}, we have $$\label{eqn:incidences} |\{ (p, T) \in (\mathcal{P}\cap Q) \times \mathcal{T}: T \in \mathcal{T}(p) \cap \mathbf{T}_0 \}| \gtrapprox\Delta^{-s-t}.$$ Fix $Q \in \mathbf{T}_0 (\mathcal{Q})$. Since $|\mathcal{P}\cap Q| \approx\Delta^{-t}$ by [\[g1\]](#g1){reference-type="ref" reference="g1"} and $|\mathcal{T}(p) \cap \mathbf{T}_0| \lessapprox\Delta^{-s}$ since $\mathcal{T}(p)$ is a $(\delta, s)$-set, we use [\[eqn:incidences\]](#eqn:incidences){reference-type="eqref" reference="eqn:incidences"} to find a subset $\mathcal{P}_Q \subset \mathcal{P}\cap Q$ with $$|\mathcal{P}_Q| \approx|\mathcal{P}\cap Q| \approx\Delta^{-t} \text{ and } |\mathcal{T}(p) \cap \mathbf{T}_0| \approx\Delta^{-s} \text{ for all } p \in \mathcal{P}_Q.$$ This verifies [\[P3\]](#P3){reference-type="ref" reference="P3"}. Finally, we get [\[P4\]](#P4){reference-type="ref" reference="P4"} by $|\mathcal{P}_Q| \ge \Delta^{\mathbf{B}\varepsilon} |\mathcal{P}\cap Q|$ for some constant $\mathbf{B}\ge 1$ and Proposition [Proposition 57](#prop:projections){reference-type="ref" reference="prop:projections"}. ## Product-like structure *This subsection is based on Section A.6 of [@orponen2021hausdorff].* Our goal is to find a product-type structure and apply Proposition [Proposition 46](#prop:improved_incidence_weaker){reference-type="ref" reference="prop:improved_incidence_weaker"}. Choose coordinates such that the $y$-axis is in the direction of $\mathbf{T}_0$, and let $\pi(\mathbf{x}, y) := \mathbf{x}\in \mathbb{R}^{d-1}$ denote the orthogonal projection to the orthogonal complement of the $y$-axis. Define the function $\Delta^{-1} (\mathbf{x}, y) = (\Delta^{-1} \mathbf{x}, y)$. If $T \in \mathbf{T}_0$, then $\Delta^{-1} T$ is roughly a $\Delta$-tube: it is contained in some $C\Delta$-tube and contains a $c\Delta$-tube for some universal constants $c, C > 0$. This technicality will not cause issues in what follows. For each $Q \in \mathbf{T}_0 (\mathcal{Q})$, let $\mathbf{y}_Q \in \Delta \cdot \mathbb{Z}\cap [0, 1)$ be a point such that the plane $y = \mathbf{y}_Q$ intersects $Q$. By [\[P1\]](#P1){reference-type="ref" reference="P1"}, we know that $\mathbf{Y}= \{ \mathbf{y}_Q : Q \in \mathbf{T}_0 (\mathcal{Q}) \}$ is a $(\Delta, t-s)$-set. By [\[P4\]](#P4){reference-type="ref" reference="P4"}, we know that for each $\mathbf{y}\in \mathbf{Y}$ that $\pi(\Delta^{-1} (\mathcal{P}\cap Q))$ contains a $(\Delta, \kappa', k)$-set $\mathbf{X}_\mathbf{y}'$ with cardinality $\gtrapprox\Delta^{-s}$. Let $\mathbf{X}_\mathbf{y}\subset (\Delta \cdot \mathbb{Z})^d \cap [0, 1]^d$ that is $\mathbf{X}_\mathbf{y}'$ rounded to the nearest multiple of $\Delta$. Now, let $L = (\Delta \cdot \mathbb{Z}) \cap B(0, \Delta (\sqrt{d}+1))$ and $\mathcal{T}(\mathbf{Z}) = \{ \sigma(T) + x : T \in \mathcal{T}\cap \mathbf{T}_0, x \in L \}$. Clearly, $|\mathcal{T}(\mathbf{Z})| \lesssim_d |\mathcal{T}\cap \mathbf{T}_0| \lessapprox\Delta^{-2s}$ by [\[P2\]](#P2){reference-type="ref" reference="P2"}. On the other hand, we show that $|\mathcal{T}(\mathbf{z})| := |\{ \mathbf{T}\in \mathcal{T}(\mathbf{Z}) : \mathbf{z}\in \mathbf{T}\}| \gtrapprox\Delta^{-s}$ for any $(\mathbf{x}, \mathbf{y}) \in \mathbf{Z}$. This follows since $\mathbf{z}= (\mathbf{x}, \mathbf{y}_Q)$ for some $Q$ and $\mathbf{x}\in \mathbf{X}_\mathbf{y}$. Let $p \in \mathcal{P}_Q$ such that $d(\pi(\Delta^{-1} p), \mathbf{x}) \le \Delta$. We know $d((\pi(\Delta^{-1} p), \mathbf{y}_Q), \Delta^{-1} p) \le \Delta$ since $Q$ has diameter $\Delta$, so by triangle inequality, we have $d(\Delta^{-1} p, \mathbf{z}) \le (\sqrt{d}+1)\Delta$. Thus, $\mathcal{T}(\mathbf{z})$ contains $\{ \sigma(T) + x : T \in \mathcal{T}(p) \cap \mathbf{T}_0 \}$ for some suitable $x \in L$. By [\[P3\]](#P3){reference-type="ref" reference="P3"}, we get the desired cardinality estimate $|\mathcal{T}(\mathbf{z})| \approx\Delta^{-s}$. Finally, we apply Proposition [Proposition 46](#prop:improved_incidence_weaker){reference-type="ref" reference="prop:improved_incidence_weaker"} to the sets $\mathbf{Z}$ and $\mathcal{T}(\mathbf{Z})$ to obtain a contradiction if $\varepsilon> 0$ is sufficiently small. This proves Theorem [Theorem 56](#thm:improved_incidence){reference-type="ref" reference="thm:improved_incidence"}. # Improved incidence estimates for general sets {#sec:main thm general} In this section, we will prove the following refinement of Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"}, following Sections 7-9 of [@orponen2021hausdorff]. **Theorem 58**. *For any $0 \le k < d-1$, $0 \le s < k+1$, $s < t \le d$, $\kappa > 0$, there exist $\varepsilon(s, t, \kappa, k, d) > 0$ and $\eta(s, t, \kappa, k, d) > 0$ such that the following holds for all small enough $\delta \in 2^{-\mathbb{N}}$, depending only on $s, t, \kappa, k, d$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a $(\delta, t, \delta^{-\varepsilon})$-set with $\cup \mathcal{P}\subset [0, 1)^d$, and let $\mathcal{T}\subset \mathcal{T}^\delta$ be a family of $\delta$-tubes. Assume that for every $p \in \mathcal{P}$, there exists a $(\delta, s, \delta^{-\lambda}, 0)$ and $(\delta, \kappa, \delta^{-\lambda}, k)$-set $\mathcal{T}(p) \subset \mathcal{T}$ with $|\mathcal{T}(p)| = M$ such that $T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$. Then $|\mathcal{T}| \ge M\delta^{-s-\varepsilon}$.* The original theorem follows from taking $\varepsilon= \eta$ and pigeonholing, since $M \in (\delta^{-s+\varepsilon}, \delta^{-d})$. *Proof.* Before anything else, we state the dependencies of the parameters: $\varepsilon_0 (s, t, \kappa, k, d)$, $\varepsilon(\varepsilon_0, s, t, \kappa, k, d), T(\varepsilon), \tau(s, t, \varepsilon), \eta(\varepsilon_0, \tau)$. First, choose $T=T(\varepsilon)$ such that $\frac{2\log T}{T}\leq \varepsilon$. By Lemma [Lemma 31](#lem:uniform){reference-type="ref" reference="lem:uniform"} we may find a subset $\mathcal{P}' \subset \mathcal{P}$ with $|\mathcal{P}'| \ge \delta^\varepsilon|\mathcal{P}|$ that is $\{2^{-jT}\}_{j=1}^m$-uniform for $2^{-mT}=\delta$ with associated sequence $\{N_j\}_{j=1}^m$. Thus, $\mathcal{P}'$ is a $(\delta, t, \delta^{-2\varepsilon})$-set. Replacing $\mathcal{P}$ with $\mathcal{P}'$ and $\varepsilon$ with $\frac{\varepsilon}{2}$, we may assume from the start that $\mathcal{P}$ is $\{2^{-jT}\}_{j=1}^m$-uniform. Let $f$ be the corresponding branching function. Since $\mathcal{P}$ is a $(\delta, t, \delta^{-\varepsilon})$-set, we have $f(x) \ge tx - \varepsilon m$ for all $x \in [0, m]$. Let $\{ [c_j, d_j] \}_{j=1}^n$ be the intervals from Proposition [Corollary 28](#cor:multiscale){reference-type="ref" reference="cor:multiscale"} applied with parameters $s, t, \varepsilon$, corresponding to a sequence $0 < \delta = \Delta_n < \Delta_{n-1} < \cdots < \Delta_1 < \Delta_0 = 1$. We can partition $\{ 0, 1, \cdots, n-1 \} = \mathcal{S}\cup \mathcal{B}$, "structured" and "bad" scales such that: - $\frac{\Delta_{j}}{\Delta_{j+1}} \ge \delta^{-\tau}$ for all $j \in \mathcal{S}$, and $\prod_{j \in \mathcal{B}} (\Delta_{j}/\Delta_{j+1}) \le \delta^{-\varepsilon}$; - For each $j \in \mathcal{S}$ and $\mathbf{p}\in \mathcal{D}_{\Delta_j} (\mathcal{P})$, the set $\mathcal{P}_j := S_{\mathbf{p}} (\mathcal{P}\cap \mathbf{p})$ is either 1. an $(t_j , \Delta_{j+1}/\Delta_j, (\Delta_j/ \Delta_{j+1})^{\varepsilon}, (\Delta_j/ \Delta_{j+1})^{\varepsilon})$-regular set, where $t_j \in (s, 2)$; 2. a $(s , \Delta_{j+1}/\Delta_j, (\Delta_j/ \Delta_{j+1})^{\varepsilon})$-set. - $\prod_{j \in S} (\Delta_j/\Delta_{j+1})^{t_j} \ge |\mathcal{P}| \cdot \prod_{j \in \mathcal{B}} (\Delta_{j+1}/\Delta_{j})^d \ge |\mathcal{P}| \delta^{O_{s,t,d} (\varepsilon)}$. Apply Proposition [Corollary 28](#cor:multiscale){reference-type="ref" reference="cor:multiscale"} and $\frac{\Delta_{j}}{\Delta_{j+1}} \ge \delta^{-\tau}$ to get a family of tubes $\mathcal{T}_{\textbf{p}} \subset \mathcal{T}^{\Delta_{j+1}/\Delta_j}$ with the property that $(S_{\textbf{p}} (\mathcal{P}\cap \textbf{p}), \mathcal{T}_{\textbf{p}})$ is a $(\Delta_{j+1}/\Delta_j, s, C_j^1, \kappa, C_j^2 M_{\textbf{p}})$-nice configuration for some $C_j^1, C_j^2 \lessapprox_\delta (\Delta_{j+1}/\Delta_j)^{-\tau^{-1} \eta}$ and $$\frac{|\mathcal{T}_0|}{M} \gtrapprox_\delta \prod_{j=0}^{N-1} \frac{|\mathcal{T}_{\textbf{p}_j}|}{M_{\textbf{p}_j}}.$$ Let $\mathcal{S}_1 = \{ j \in S : t_j \ge \frac{s+t}{2} \}$ and $\mathcal{S}_2 = \mathcal{S}\setminus \mathcal{S}_1$. Then $$\prod_{j \in S_1} (\Delta_j/\Delta_{j+1})^{t_j} \ge |\mathcal{P}| \delta^{O_{s,t,d} (\varepsilon)} \prod_{j \in S_2} (\Delta_j/\Delta_{j+1})^{-\frac{s+t}{2}} \ge \delta^{\frac{t-s}{2} + O_{s,t,d} (\varepsilon)}.$$ For $j \in \mathcal{S}_1$ we apply Theorem [Theorem 56](#thm:improved_incidence){reference-type="ref" reference="thm:improved_incidence"} with parameters $s, \frac{s+t}{2}$, and for $j \in \mathcal{S}_2$ we apply Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}. If $\varepsilon_0 (s, t, \kappa, k, d)$ is the $\eta$ from Theorem [Theorem 56](#thm:improved_incidence){reference-type="ref" reference="thm:improved_incidence"}, then for $\tau^{-1} \eta < \varepsilon_0$, we get $$\frac{|\mathcal{T}_0|}{M} \gtrapprox_\delta \prod_{j \in \mathcal{S}_1} \left( \frac{\Delta_j}{\Delta_{j+1}} \right)^{-s-\varepsilon_0} \cdot \prod_{j \in \mathcal{S}_2} \left( \frac{\Delta_j}{\Delta_{j+1}} \right)^{-s+O(\varepsilon)} \ge \delta^{-s(1-\varepsilon) - (\frac{t-s}{2} + O_{s,t,d} (\varepsilon)) \varepsilon_0 + O(\varepsilon)} \ge \delta^{-s-\varepsilon}$$ as long as $\varepsilon$ is taken small enough in terms of $\varepsilon_0, s, t, d$. ◻ # Sets contained in an $(r_0, k)$-plate {#sec:refined from main} We restate Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}. **Theorem 59**. *For any $0 \le k < d-1$, $0 \le s < k+1$, $\max(s, k) < t \le d$, $\kappa > 0$, $r_0 \le 1$, there exists $\varepsilon(s, t, \kappa, k, d) > 0$ such that the following holds for all small enough $\delta/r_0 \in 2^{-\mathbb{N}} \cap (0, \delta_0)$, with $\delta_0$ depending only on $s, t, \kappa, k, d$. Let $H$ be a $(r_0, k+1)$-plate, $\mathcal{P}\subset \mathcal{D}_\delta \cap H$ be a $(\delta, t, (\delta/r_0)^{-\varepsilon})$-set with $\cup \mathcal{P}\subset [0, 1)^d$, and let $\mathcal{T}\subset \mathcal{T}^\delta \cap H$ be a family of $\delta$-tubes. Assume that for every $p \in \mathcal{P}$, there exists a set $\mathcal{T}(p) \subset \mathcal{T}$ such that:* - *$T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$;* - *$\mathcal{T}(p)$ is a $(\delta, s, (\delta/r_0)^{-\varepsilon} r_0^{k-s}, 0)$-set down from scale $r$;* - *$\mathcal{T}(p)$ is a $(\delta, \kappa, (\delta/r_0)^{-\varepsilon} r_0^{-\kappa}, k)$-set.* *Then $|\mathcal{T}| \ge (\frac{\delta}{r_0})^{-\varepsilon} \delta^{-2s} r_0^{2(s-k)}$.* ## Multiscale analysis We will use Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} to prove Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}. Let $S_H$ be the dilation sending $H$ to $[0,1]^d$. Then $\mathcal{P}, \mathcal{T}(p)$, and $\mathcal{T}$ become deformed under $S_H$, but they satisfy the following statistics assumptions for $r \in [\frac{\delta}{r_0}, 1]$: $$\begin{aligned} |\mathcal{P}\cap S_H^{-1} (Q)| \le \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \cdot |\mathcal{P}| \cdot r^t, & \qquad Q \in \mathcal{D}_r (\mathbb{R}^d), \label{eqn:non_conc1} \\ |\mathcal{T}(p) \cap S_H^{-1} (\mathbf{T})| \le \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \cdot |\mathcal{T}(p)| \cdot r^s, & \qquad \mathbf{T}\quad r \text{-tube}, \label{eqn:non_conc2} \\ |\mathcal{T}(p) \cap S_H^{-1} (W)| \le \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \cdot |\mathcal{T}(p)| \cdot r^\kappa, & \qquad W \quad (r, k+1) \text{-plate}. \label{eqn:non_conc3}\end{aligned}$$ To prove [\[eqn:non_conc1\]](#eqn:non_conc1){reference-type="eqref" reference="eqn:non_conc1"}, observe that $S_H^{-1} (Q)$ is contained in an $r$-ball, and then we use that $\mathcal{P}$ is a $(\delta, t, (\delta/r_0)^{-\varepsilon}, 0)$-set. To prove [\[eqn:non_conc2\]](#eqn:non_conc2){reference-type="eqref" reference="eqn:non_conc2"}, observe that $S_H^{-1} (\mathbf{T})$ is contained in a box with $k$ sides of length $r$ and $d-k$ sides of length $rr_0$. This box can be covered by $\sim r_0^{-k}$ many $rr_0$-balls. Finally, use that $\mathcal{T}(p)$ is a $(\delta, s, (\delta/r_0)^{-\varepsilon} r_0^{k-s}, 0)$-set. To prove [\[eqn:non_conc3\]](#eqn:non_conc3){reference-type="eqref" reference="eqn:non_conc3"}, observe that $S_H^{-1} (W)$ is contained in a $(rr_0, k)$-plate. Using these observations, we obtain the following refinement of Proposition [Proposition 27](#prop:nice_tubes){reference-type="ref" reference="prop:nice_tubes"}. We use $(\delta, s, C_1 r_0^{k-s}, \kappa, C_2, M)$-nice configuration down from scale $r_0$ to indicate that $\mathcal{T}(p)$ is a $(\delta, s, C_1 r_0^{k-s}, 0)$-set down from scale $r_0$. **Proposition 60**. *Fix dyadic numbers $0 < \delta' = \frac{\delta}{r_0} < \Delta \le 1$. Let $(\mathcal{P}_0, \mathcal{T}_0)$ be a $(\delta, s, C_1 r_0^{k-s}, \kappa, C_2, M)$-nice configuration down from scale $r_0$, and assume $\mathcal{P}_0 \subset H$ for some $(r_0, k)$-plate $H$. Then there exist refinements $\mathcal{P}\subset \mathcal{P}_0$, $\mathcal{T}(p) \subset \mathcal{T}_0 (p), p \in \mathcal{P}$, and $\mathcal{T}_\Delta (Q) \subset \mathcal{T}^\Delta$ such that denoting $\mathcal{T}_\Delta = \cup_{Q \in \mathcal{D}_{\Delta} (S_H (\mathcal{P}))} \mathcal{T}_\Delta (Q)$ and $\mathcal{T}= \cup_{p \in \mathcal{P}} \mathcal{T}(p)$ the following hold:* (i) *$|\mathcal{D}_\Delta (S_H(\mathcal{P}))| \approx_{\Delta} |\mathcal{D}_\Delta (S_H(\mathcal{P}_0))|$ and $|S_H(\mathcal{P}) \cap Q| \approx_{\Delta} |S_H(\mathcal{P}_0) \cap Q|$ for all $Q \in \mathcal{D}_\Delta (\mathcal{P})$.* (ii) *We have $|\mathcal{T}\cap \mathbf{T}| \lessapprox\frac{|\mathcal{T}_0|}{|\mathcal{T}_\Delta|}$ for all $\mathbf{T}\in \mathcal{T}_\Delta$.* (iii) *$(\mathcal{D}_\Delta (S_H(\mathcal{P})), \mathcal{T}_\Delta)$ is $(\Delta, s, C^1_\Delta, \kappa, C^2_\Delta, M_\Delta)$-nice for some $C^1_\Delta \approx_{\Delta} C_1$, $C^2_\Delta \approx_{\Delta} C_2$, and $M_\Delta \ge 1$.* (iv) *For all $\mathbf{T}\in \mathcal{T}_\Delta (Q)$, we have $$|\{ (p, T) \in \mathcal{P}\times \mathcal{T}: T \in \mathcal{T}(p) \text{ and } T \subset S_H^{-1} (\mathbf{T}) \} | \gtrapprox_\Delta \frac{M \cdot |S_H (\mathcal{P}) \cap Q|}{|\mathcal{T}_\Delta (Q)|}.$$* (v) *For each $Q \in \mathcal{D}_\Delta (S_H(\mathcal{P}_2))$, there exist $C^1_Q \approx_{\Delta} C_1$, $C^2_Q \approx_{\Delta} C_2$, $M_Q \ge 1$, a subset $\mathcal{P}_Q \subset \mathcal{P}\cap Q$ with $|\mathcal{P}_Q| \gtrapprox_\Delta |\mathcal{P}\cap Q|$ and a family of tubes $\mathcal{T}_Q \subset \mathcal{T}^{\delta/\Delta}$ such that $(S_H^{-1} \circ S_Q (S_H(\mathcal{P}_2 ) \cap Q), \mathcal{T}_Q)$ is $(\delta/\Delta, s, C^1_Q r_0^{k-s}, \kappa, C^2_Q, M_Q)$-nice down from scale $r_0$.* *Furthermore, the families $\mathcal{T}_Q$ can be chosen so that $$\label{eqn:item6'} \frac{|\mathcal{T}_0|}{M} \gtrapprox_{\Delta} \frac{|\mathcal{T}^\Delta (\mathcal{T})|}{M_\Delta} \cdot \left( \max_{Q \in \mathcal{D}_\Delta (\mathcal{P}_2)} \frac{|\mathcal{T}_Q|}{M_Q} \right).$$* *Proof.* The proof will involve many dyadic pigeonholing steps. **Step 1: construct $\mathcal{T}_\Delta (Q)$.** For a given $Q \in \mathcal{D}_\Delta (S_H (\mathcal{P}_0)) := \mathcal{Q}_0$, we claim that we can find a subset $\mathcal{P}_Q \subset \mathcal{P}_0 \cap S_H^{-1} (Q)$ with $|\mathcal{P}_Q| \approx_\Delta |\mathcal{P}_0 \cap S_H^{-1} (Q)|$ and a family of dyadic $\Delta$-tubes $\overline{\mathcal{T}}_\Delta (Q)$ intersecting $Q$ such that the following holds: 1. [\[T1\]]{#T1 label="T1"} $\overline{\mathcal{T}}_\Delta (Q)$ is a $(\Delta, s, C_\Delta^1, 0)$-set and $(\Delta, \kappa, C_\Delta^2, k)$-set for some $C_\Delta^1, C_\Delta^2 \approx_\Delta C_1$. 2. [\[T2\]]{#T2 label="T2"} there exists a constant $H_Q \approx_\Delta M \cdot |\mathcal{P}_Q| / |\overline{\mathcal{T}}_\Delta (Q)|$ such that $$|\{ (p, T) \in \mathcal{P}_Q \times \mathcal{T}_0 : T \in \mathcal{T}_0 (p) \text{ and } T \subset S_H^{-1} (\mathbf{T}) \} | \gtrsim H_Q, \qquad \mathbf{T}\in \overline{\mathcal{T}}_\Delta (Q).$$ This claim generalizes [@orponen2021hausdorff Proposition 4.1] and relies on the same dyadic pigeonholing steps; for brevity, we only state these steps and refer the reader to [@orponen2021hausdorff] for the detailed proof. (We essentially follow the same proof for [\[T2\]](#T2){reference-type="ref" reference="T2"}, and we introduce a nice shortcut to derive [\[T1\]](#T1){reference-type="ref" reference="T1"} from [\[T2\]](#T2){reference-type="ref" reference="T2"}.) Let $\mathcal{T}_\Delta (Q) \subset \mathcal{T}^\Delta$ be a minimal finitely overlapping cover of $S_H (\mathcal{T}_Q) := \cup_{p \in \mathcal{P}_0 \cap Q} S_H (\mathcal{T}_0(p))$ by $\Delta$-tubes. For $p \in \mathcal{P}_0 \cap Q$, define $$\mathcal{T}_{\Delta, j} (p) = \{ \mathbf{T}\in \mathcal{T}_\Delta (Q) : 2^{j-1} < |\{ T \in \mathcal{T}(p) : T \subset S_H^{-1} (\mathbf{T}) \}| \le 2^j \}.$$ Since $|\mathcal{T}_\Delta (Q)| \lesssim 100\Delta^{-2(d-1)}$ and $M \lesssim\sum_j 2^j \cdot |\mathcal{T}_{\Delta,j} (p)|$, we in fact have $$M \lesssim\sum_{M\Delta^{2(d-1)}/200 \le 2^j \le M} 2^j \cdot |\mathcal{T}_{\Delta,j} (p)|$$ Thus, by dyadic pigeonholing, there exists $j = j(p)$ such that $2^j \cdot |\mathcal{T}_{\Delta,j} (p)| \approx_\Delta M$. Another dyadic pigeonholing allows us to find $\mathcal{P}_Q \subset \mathcal{P}_0 \cap Q$ such that $j(p)$ is constant for $p \in \mathcal{P}_Q$. This is the desired refinement $\mathcal{P}_Q$ of $\mathcal{P}_0 \cap Q$. Finally, let $$\mathcal{T}_{\Delta,i} (Q) := \{ \mathbf{T}\in \mathcal{T}_\Delta (Q) : 2^{i-1} < |\{ p \in \mathcal{P}_Q : \mathbf{T}\in \mathcal{T}_\Delta (p) \}| \le 2^i \}.$$ Then by a similar dyadic pigeonholing (for calculations, see [@orponen2021hausdorff Proposition 4.1]), there is $i$ such that $$\label{eqn:TDeltaj} \frac{1}{200} |\mathcal{P}_Q| \Delta^{d-1} \le 2^i \le |\mathcal{P}_Q| \text{ and } 2^{i+j} \cdot |\mathcal{T}_{\Delta,i} (Q)| \approx_\Delta M \cdot |\mathcal{P}_Q|.$$ Finally, we define $\overline{\mathcal{T}}_\Delta (Q) := \mathcal{T}_{\Delta, i} (Q)$, which is the desired refinement of $\mathcal{T}_\Delta (Q)$. We check [\[T2\]](#T2){reference-type="ref" reference="T2"} holds with $H_Q = 2^{i+j}$, which satisfies $H_Q \approx_\Delta M \cdot |\mathcal{P}\cap Q|/|\overline{\mathcal{T}}_\Delta|$ by [\[eqn:TDeltaj\]](#eqn:TDeltaj){reference-type="eqref" reference="eqn:TDeltaj"} and $|\mathcal{P}_Q| \approx_\Delta |\mathcal{P}\cap Q|$. With this choice of $H_Q$, fix $\mathbf{T}\in \overline{\mathcal{T}}_\Delta$ and note that $$\begin{gathered} |\{ (p, T) \in \mathcal{P}_Q \times \mathcal{T}_0 : T \in \mathcal{T}_0 (p), T \subset S_H^{-1} (\mathbf{T}) \}| = \sum_{p \in \mathcal{P}_Q} |\{ T \in \mathcal{T}(p) : T \subset S_H^{-1} (\mathbf{T}) \}| \\ \ge 2^j |\{ p \in \mathcal{P}_Q : \mathbf{T}\in \mathcal{T}_\Delta (p) \}| \ge 2^{i+j} = H. \end{gathered}$$ To check [\[T1\]](#T1){reference-type="ref" reference="T1"}, we first pick a $r$-tube $\mathbf{T}_r$ with $r \ge \Delta$. Then by [\[T2\]](#T2){reference-type="ref" reference="T2"} and [\[eqn:non_conc2\]](#eqn:non_conc2){reference-type="eqref" reference="eqn:non_conc2"}, $$\begin{gathered} |\{ \mathbf{T}\in \overline{\mathcal{T}}_\Delta : \mathbf{T}\subset \mathbf{T}_r \}| \lesssim\frac{1}{H} |\{ (p, T) \in \mathcal{P}_Q \times \mathcal{T}_0 : T \in \mathcal{T}_0 (p), T \subset S_H^{-1} (\mathbf{T}_r) \}| \\ \lesssim\frac{1}{H} |\mathcal{P}_Q| \cdot C_1 M r^s \lessapprox C_1 |\overline{\mathcal{T}}_\Delta| r^s. \end{gathered}$$ Thus, $\overline{\mathcal{T}}_\Delta (Q)$ is a $(\Delta, s, C_\Delta^1, 0)$-set with $C_\Delta^1 \approx_\Delta C_1$. Doing the same calculation with an $(r, k+1)$-plank instead of an $r$-tube, we get that $\overline{\mathcal{T}}_\Delta (Q)$ is a $(\Delta, \kappa, C_\Delta^2, k)$-set with $C_\Delta^2 \approx_\Delta C_1$. This proves [\[T1\]](#T1){reference-type="ref" reference="T1"} and thus the claim. **Step 2: uniformity of $|\mathcal{T}_0 \cap \mathbf{T}|$.** By the pigeonhole principle, we can find $\overline{M}_\Delta \ge 1$ and a subset $\mathcal{Q}\subset \mathcal{D}_\Delta (\mathcal{P})$ with $|\mathcal{Q}| \approx_\Delta |\mathcal{Q}_0|$ such that $|\overline{\mathcal{T}}_\Delta (Q)| \sim \overline{M}_\Delta$ for all $Q \in \mathcal{Q}$. Write $$\overline{\mathcal{T}}_\Delta = \bigcup_{Q \in \mathcal{Q}} \overline{\mathcal{T}}_\Delta (Q).$$ Next, by another dyadic pigeonholing, we can find a subset $\overline{\mathcal{T}}_\Delta' \subset \overline{\mathcal{T}}_\Delta$ such that $I(\mathcal{Q}, \overline{\mathcal{T}}_\Delta') \gtrapprox I(\mathcal{Q}, \overline{\mathcal{T}}_\Delta)$ and $|\mathcal{T}_0 \cap \mathbf{T}| \sim N_\Delta$ for all $\mathbf{T}\in \overline{\mathcal{T}}_\Delta'$. Also, $|\overline{\mathcal{T}}_\Delta (Q)| \lesssim\overline{M}_\Delta$ for all $Q \in \mathcal{Q}$. Thus, we can find $\mathcal{Q}' \subset \mathcal{Q}$ with $|\mathcal{Q}'| \approx_\Delta |\mathcal{Q}|$, and for each $Q \in \mathcal{Q}'$ a subset $\mathcal{T}_\Delta (Q)$ of cardinality $\approx\overline{M}_\Delta$, such that $\mathcal{T}_\Delta (Q) \subset \overline{\mathcal{T}}_\Delta'$. In other words, $$|\mathcal{T}_0 \cap \mathbf{T}| \sim N_\Delta \text{ for } \mathbf{T}\in \mathcal{T}_\Delta (Q).$$ Thus, we obtain item [\[item21\]](#item21){reference-type="ref" reference="item21"}. $$\label{eqn: step 2} |\mathcal{T}_0| \ge |\mathcal{T}_\Delta| \cdot \min_{\mathbf{T}\in \mathcal{T}_\Delta} |\mathcal{T}_0 \cap \mathbf{T}| \sim |\mathcal{T}_\Delta| \cdot N_\Delta.$$ Reduce the families $\mathcal{T}_\Delta (Q)$ such that their cardinality is $M_\Delta := \min(|\mathcal{T}_\Delta (Q)| : Q \in \mathcal{Q}\}) \approx_\delta \overline{M}_\Delta$. By [\[T1\]](#T1){reference-type="ref" reference="T1"}, $\mathcal{T}_\Delta (Q)$ remains a $(\Delta, s, C_\Delta^1, 0)$ and $(\Delta, \kappa, C_\Delta^2, k)$-set with $C_\Delta^1, C_\Delta^2 \approx_\delta C_1$. Finally, define $$\mathcal{P}= \bigcup_{Q \in \mathcal{Q}} \mathcal{P}_Q,$$ where $\mathcal{Q}$ is the latest refinement of $\mathcal{Q}_0$. Since $|\mathcal{P}_Q| \approx_\Delta |\mathcal{P}_0 \cap S_H^{-1} (Q)|$, we get that item [\[item1\]](#item1){reference-type="ref" reference="item1"} holds. For $p \in \mathcal{P}_Q = \mathcal{P}\cap Q$, $Q \in \mathcal{Q}$, define $$\mathcal{T}(p) = \bigcup_{\mathbf{T}\in \mathcal{T}_\Delta (Q)} (\mathcal{T}_0 (p) \cap \mathbf{T}), \mathcal{T}= \bigcup_{p \in \mathcal{P}} \mathcal{T}(p), \mathcal{T}_\Delta = \bigcup_{Q \in \mathcal{Q}} \mathcal{T}_\Delta (Q).$$ Thus, $(\mathcal{D}_\Delta (\mathcal{P}), \mathcal{T}_\Delta) = (\mathcal{Q}, \mathcal{T}_\Delta)$ is a $(\Delta, s, C_\Delta^1, \kappa, C_\Delta^2, M_\Delta)$-nice configuration, establishing item [\[item3\]](#item3){reference-type="ref" reference="item3"}. To summarize, in this step, we refined $\mathcal{Q}$ and $\mathcal{T}_\Delta (Q)$ for $Q \in \mathcal{Q}$, so [\[T2\]](#T2){reference-type="ref" reference="T2"}/[\[item4\]](#item4){reference-type="ref" reference="item4"} still holds (with same $H_Q$ and a weaker implied constant). **Step 3: uniformity of $\mathcal{T}(p)$ and construct $\mathcal{T}_Q$.** This step will be devoted to verifying [\[item5\]](#item5){reference-type="ref" reference="item5"} and [\[eqn:item6\'\]](#eqn:item6'){reference-type="eqref" reference="eqn:item6'"}. We will not change $\mathcal{P}, \mathcal{T}$, or $\mathcal{T}_\Delta$. Fix $Q \in \mathcal{Q}$, and let $\mathcal{P}_Q = \mathcal{P}\cap S_H^{-1} (Q)$. Define $$\mathcal{T}(Q) = \bigcup_{p \in \mathcal{P}_Q} \mathcal{T}(p).$$ By dyadic pigeonholing and [\[T2\]](#T2){reference-type="ref" reference="T2"}, we can find a $\approx_\Delta$-comparable subset of $\mathcal{P}_Q$ (which we keep denoting $\mathcal{P}_Q$) such that $$|\mathcal{T}(p)| \approx_\Delta M, \qquad p \in \mathcal{P}_Q.$$ Next, $$\label{eqn: step 3} |\mathcal{T}(Q)| \le \sum_{\mathbf{T}\in \mathcal{T}_\Delta (Q)} |\mathcal{T}\cap \mathbf{T}| \lessapprox_\Delta M_\Delta \cdot N_\Delta.$$ For a given $p \in Q$, we consider the tube packet $\mathbb{U}(p) := \mathcal{T}(p) \cap S_H^{-1} (Q)$ (discarding duplicate tubelets). Each tubelet $u \in \mathbb{U}(p)$ lies in at most $\Delta^{-2(d-1)}$ many tubes of $\mathcal{T}(p)$, so by dyadic pigeonholing, we can refine $\mathcal{T}(p)$ by a $\log \Delta^{-1}$ factor to ensure that each tubelet $u \in \mathbb{U}(p)$ lies in $\sim m(p)$ many tubes of $\mathcal{T}(p)$, and there are $M(p) \approx_\Delta \frac{M}{m(p)}$ many distinct tubelets through $p$. By refining $\mathcal{P}_Q$ by a $(\log \Delta^{-1})$-factor, we may assume $m(p) \approx m_Q$ for each $p \in \mathcal{P}_Q$. Now, define $$\mathcal{P}^Q := S_H^{-1} \circ S_Q \circ S_H (\mathcal{P}_Q) \text{ and } \mathcal{T}_Q := \bigcup_{p \in \mathcal{P}_Q} S_H^{-1} \circ S_Q \circ S_H (\mathbb{U}(p)).$$ Since tubelets are essentially distinct and each tubelet in any $\mathbb{U}(p)$ corresponds to $\approx m_Q$ many tubes in $\mathcal{T}(Q)$, we obtain: $$\label{eqn:step4} |\mathcal{T}(Q)| \gtrapprox_\Delta \left|\bigcup_{p \in \mathcal{P}_Q} \mathbb{U}(p) \right| \cdot m_Q \gtrsim|\mathcal{T}_Q| \cdot \frac{M}{M_Q}.$$ Then [\[eqn:item6\'\]](#eqn:item6'){reference-type="eqref" reference="eqn:item6'"} will follow by combining [\[eqn: step 2\]](#eqn: step 2){reference-type="eqref" reference="eqn: step 2"}, [\[eqn: step 3\]](#eqn: step 3){reference-type="eqref" reference="eqn: step 3"}, and [\[eqn:step4\]](#eqn:step4){reference-type="eqref" reference="eqn:step4"}. We finally check $(\mathcal{P}^Q, \mathcal{T}_Q)$ is a $(\delta/\Delta, s, C_Q^1 r_0^{k-s}, \kappa, C_Q^2, M_Q)$-nice configuration down from scale $r_0$. First, for any $\overline{\delta}< r < r_0$, we have for any $(r, 0)$-plank $H$ in $S^{d-1}$, $$|\sigma(\mathcal{T}_Q) \cap H| \sim_\Delta \frac{1}{m_Q} |\sigma(\mathcal{T}(p)) \cap H| \lessapprox_\Delta \frac{1}{m_Q} \cdot C \cdot M \cdot r^s = C \cdot M_Q \cdot r^s.$$ Thus, $\sigma(\mathcal{T}_Q)$ is a $(\overline{\delta}, s, C_Q^1, 0)$-set down from scale $r_0$ with $C_Q^1 \approx_\Delta C_1$. Similarly, $\sigma(\mathcal{T}_Q)$ is a $(\overline{\delta}, \kappa, C_Q^2, k)$-set with $C_Q^2 \approx_\Delta C_2$. This shows item [\[item5\]](#item5){reference-type="ref" reference="item5"} and thus the proof of the Proposition. ◻ ## Good multiscale decomposition The idea is to apply Proposition [Proposition 60](#prop:nice_tubes_2){reference-type="ref" reference="prop:nice_tubes_2"}, then apply Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} to bound $|\mathcal{T}_\Delta|$ and Corollary [Corollary 63](#cor:easy_est_2){reference-type="ref" reference="cor:easy_est_2"} to bound $|\mathcal{T}_Q|$. Unfortunately, while we use pigeonholing to ensure that $\mathcal{D}_\Delta (S_H(\mathcal{P}))$ is a $(\Delta, t)$-set, we don't know that $S_H^{-1} \circ S_Q (S_H(\mathcal{P}) \cap Q)$ is a $(\frac{\delta}{\Delta}, t)$-set. In fact, we won't show this statement, but rather a slightly weaker statement that is good enough. For this, a good choice of $\Delta$ based on the branching structure of $\mathcal{P}$ is needed. First, we explain the pigeonholing preliminaries. **Lemma 61**. *Given $P \subset H_r$, a $(r_0, k)$-plane, there is a subset $P' \subset P$ with $|P'|_\delta \ge (\log (\frac{r_0}{\delta}))^{-1} |P|_\delta$ such that $|Q \cap S_H(P)|$ is constant for all $Q \in \mathcal{D}_{\delta/r_0} (S_H(P))$.* *Proof.* Let $f(N) = \sum \{ |P \cap S_H^{-1} (Q)|_\delta : Q \in \mathcal{D}_{\delta/r_0} ([0, 1]^d), |P \cap S_H^{-1} (Q)|_\delta \in [N, 2N] \}$. Then $\sum_{N \text{ dyadic}} f(N) = |P|_\delta$. For each $N$, either $f(N) = 0$ or $N \le f(N) \le (r_0/\delta)^d \cdot N$. Hence, if $N_0$ is the largest $N$ for which $f(N) > 0$, we get $f(N_0) \ge N_0 > \sum_{M < N_0 (\delta/r_0)^d/100 \text{ dyadic}} f(M)$. Thus, we have $$\sum_{N_0 (\delta/r_0)^d/100 < M < N_0 \text{ dyadic}} f(M) > \frac{1}{2} |P|_\delta.$$ Thus, by dyadic pigeonholing, there exists $M \in (N_0 (\delta/r_0)^d/100, N_0)$ such that $f(M) \ge \frac{1}{20d} (\log (\frac{r_0}{\delta}))^{-1} |P|_\delta$. ◻ The next step is to make $S_H^{-1} \circ S_Q (S_H(\mathcal{P}) \cap Q)$ satisfy a $t'$-dimensional spacing condition with $t'$ just slightly less than $t$, for all $Q \in \mathcal{D}_\Delta (S_H (\mathcal{P}))$ at a certain scale $\Delta$. To do so, we need the following lemma. **Lemma 62**. *Fix $C, \varepsilon> 0$, and let $\frac{\delta}{r_0} = \Delta^m$ and $P \subset H_r$ be a $(\delta, t, C, 0)$-set in $H_r$, a $(r_0, k)$-plane. Let $L = \log(\frac{r_0}{\delta}) \cdot (\log(1/\Delta))^m$. If $t' < \frac{t - d\varepsilon}{1-\varepsilon}$, then there exists $m\varepsilon\le k \le m$ and a subset $P' \subset P$ with $|P'| \ge L^{-1} |P|$ such that for any $k \le j \le m$, $Q \in \mathcal{D}_{\Delta^k} (P')$, and $R \in \mathcal{D}_{\Delta^j} (P') \cap Q$, we have $$\label{eqn:extract1} |P' \cap S_H^{-1} (R)| \le |P' \cap S_H^{-1} (Q)| \cdot \Delta^{(j-k)t'},$$ and for $\delta \le r \le \frac{\delta}{r_0}$ and a ball $B_r$, we have $$\label{eqn:extract2} |P' \cap B_r| \le C \cdot L \cdot |P' \cap S_H^{-1} (Q)| \left( \frac{r}{\Delta^k} \right)^{t'}.$$* *Proof.* Throughout this proof we will not distinguish between $m\varepsilon$ and $\lceil m\varepsilon\rceil$. First, we will make $S_H(P)$ uniform at scales $1, \Delta, \Delta^2, \cdots, \Delta^m = \frac{\delta}{r_0}$. By Lemmas [Lemma 61](#lem:uniform1){reference-type="ref" reference="lem:uniform1"} and [Lemma 31](#lem:uniform){reference-type="ref" reference="lem:uniform"}, we can find $|P'| \ge L^{-1} |P|$ such that there is a sequence $(N_j)_{j=1}^n$ with $|S_H(P') \cap Q|_{\Delta^k} = N_k$ for all $1 \le k \le n$ and $Q \in \mathcal{D}_{\Delta^k} (P')$. Let $m\varepsilon\le k \le m$ be the largest index such that $N_k \ge |P'| \Delta^{m\varepsilon\cdot d+(k-m\varepsilon)t'}$ for one (equivalently all) $Q \in \mathcal{D}_{\Delta^k} (P')$. Certainly $k = m\varepsilon$ is a valid index since $|\mathcal{D}_{\Delta^{m\varepsilon}}| = \Delta^{-dm\varepsilon}$. Now, we will check the given conditions. By maximality of $k$, we have for $k \le j \le m$, $$N_j \le |P'| \Delta^{dm\varepsilon+(j-m\varepsilon)t'} \le N_k \Delta^{(j-k)t'}.$$ Noticing that $|P' \cap S_H^{-1} (Q)| = |S_H(P') \cap Q|$ and likewise for $R \in \mathcal{D}_{\Delta^j} (P') \cap Q$, this proves [\[eqn:extract1\]](#eqn:extract1){reference-type="eqref" reference="eqn:extract1"}. To check [\[eqn:extract2\]](#eqn:extract2){reference-type="eqref" reference="eqn:extract2"}, we recall that $L N_k \ge L \cdot |P'| \Delta^{dm\varepsilon+(k-m\varepsilon)t'} \ge |P| \Delta^{dm\varepsilon+(k-m\varepsilon)t'}$. Using $r \le \frac{\delta}{r_0} = \Delta^m$, $t' \le \frac{t - d\varepsilon}{1-\varepsilon}$, and that $P$ is a $(\delta, t, C)$-set, we have $$|P' \cap B_r| \le |P \cap B_r| \le C |P| r^t \le C \cdot N_k L \left( \frac{r}{\Delta^k} \right)^{t'}.$$ ◻ Finally, we will need the following variant of Corollary [Corollary 24](#cor:easy_est){reference-type="ref" reference="cor:easy_est"}. **Corollary 63**. *Let $0 \le \max(s, k) < t \le d-1$, $\delta \le r \le 1$, and let $C_P \ge 1, C_T \ge 0$. Let $\mathcal{P}\subset \mathcal{D}_\delta$ be a set contained in an $(r_0, k+1)$-plate $H$ satisfying the following conditions:* - *For all $\frac{\delta}{r_0} \le r \le 1$ and balls $B_r$, we have $$\label{eqn:p_condition1} |\mathcal{P}\cap S_H^{-1} (B_r)| \le C_P \cdot |\mathcal{P}| \cdot r^t.$$* - *For all $\delta \le r \le \frac{\delta}{r_0}$ and balls $B_r$, we have $$\label{eqn:p_condition2} |\mathcal{P}\cap B_r| \le C_P \cdot |\mathcal{P}| \cdot r^t.$$* *Assume that for every $p \in \mathcal{P}$ there exists a family $\mathcal{T}(p) \subset \mathcal{T}^\delta$ of dyadic $\delta$-tubes satisfying the following conditions:* - *$T \cap p \neq \emptyset$ for all $T \in \mathcal{T}(p)$;* - *$|\mathcal{T}(p) \cap \mathbf{T}| \le C_T \cdot |\mathcal{T}(p)| \cdot r_0^{k-s} x^s$ for all $x$-tubes $\mathbf{T}$ with $\delta \le x \le r_0$.* *Further assume that $|\mathcal{T}(p)| = M$ for some $M \ge 1$. If $\mathcal{T}= \cup_{p \in \mathcal{P}} \mathcal{T}(p)$, then $$|\mathcal{T}| \gtrsim(C_P C_T)^{-1} \cdot Mr_0^{s-k} \delta^{-s}.$$* *Proof.* Let $$j_P (\mathcal{P}, \mathcal{T}) = \{ (q, t) \in \mathcal{P}\times \mathcal{T}(p) : t \in \mathcal{T}(q) \}$$ We have the following: **Lemma 64**. *For all $p \in \mathcal{P}$, we have $j_p (\mathcal{P}, \mathcal{T}) \lesssim_{s,t,k} C_P C_T |\mathcal{P}| \cdot Mr_0^{k-s} \delta^s$.* *Proof.* We count $j_p (\mathcal{P}, \mathcal{T})$ by first choosing a dyadic $\delta < r < 1$, then counting the number of $q \in \mathcal{P}$ with $|p - q| \sim r$, then finally counting the number of $t \in \mathcal{T}$ that pass through $p, q$. If $r > \frac{\delta}{r_0}$, we claim that if $|x - y| \in [r, 2r]$ and some tube through $x, y$ lies in $H$, then $|S_H(x) - S_H(y)| \le 100r$, so $y \in S_H^{-1} (B_{100r} (S_H(x)))$. To prove this, we may assume $r_0 \le \frac{1}{50}$, as otherwise we can use the simple fact $|S_H(x) - S_H(y)| \le r_0^{-1} |x-y| \le 100r$. Now choose a coordinate system such that the first $k+1$ axes correspond to the long sides of $H$, and the remaining axes correspond to the short sides of $H$. Let $x - y = (\vec{a}, \vec{b}) \in \mathbb{R}^{k+1} \times \mathbb{R}^{d-k-1}$. Then $|\vec{a}| \le |x-y| \le r$. Furthermore, we have $|\vec{b}| \le 50r_0 |\vec{a}|$, otherwise any tube through $x, y$ would be roughly orthogonal to $H$ and intersect $H$ in a subtube with length $2r_0 \le 1$, contradiction. Thus, we have $|S_H(x) - S_H(y)| \le |\vec{a}| + r_0^{-1} |\vec{b}| \le 100r$. Using the claim and condition [\[eqn:p_condition1\]](#eqn:p_condition1){reference-type="eqref" reference="eqn:p_condition1"}, we see that there are $\lesssim C_P |\mathcal{P}| \cdot r^t$ many choices for $q$. For each $q$, the set of tubes $t \in \mathcal{T}(p)$ passing through $q$ lies in a $\frac{\delta}{r}$-tube, so by the tube non-concentration condition (and noting that $\frac{\delta}{r} < r_0$), we have $C_T \cdot Mr_0^{k-s} \left( \frac{\delta}{r} \right)^s$ choices for $t$. Thus, the contribution to $j_p (\mathcal{P}, \mathcal{T})$ for a given dyadic $r > \frac{\delta}{r_0}$ is $C_P C_T |\mathcal{P}| \cdot Mr_0^{k-s} \delta^s \cdot r^{s-t}$, and summing over dyadic $r$ gives $C_P C_T |\mathcal{P}| \cdot Mr_0^{k-s} \delta^s \cdot O_{s-t} (1)$. If $r < \frac{\delta}{r_0}$, then by condition [\[eqn:p_condition2\]](#eqn:p_condition2){reference-type="ref" reference="eqn:p_condition2"} we see that there are $\lesssim C_P |\mathcal{P}| \cdot r^t$ many choices for $q$. For each $q$, the set of tubes $t \in \mathcal{T}(p)$ passing through $q$ lies in a $\frac{\delta}{r}$-tube $\mathbf{T}_{\delta/r}$. We note that $\frac{\delta}{r} > r_0$, so the tube non-concentration doesn't apply directly, but luckily we note that $\mathbf{T}_{\delta/r} \cap H$ can be covered by $(\frac{\delta}{rr_0})^k$ many $r_0$-tubes. Thus, by using tube non-concentration at scale $r_0$, we have $C_T \cdot Mr_0^k \cdot (\frac{\delta}{rr_0})^k$ choices for $t$. Thus, the contribution to $j_p (\mathcal{P}, \mathcal{T})$ for a given dyadic $r < \frac{\delta}{r_0}$ is (after some manipulation) $$C_P C_T |P| Mr_0^k \left( \frac{\delta}{r_0} \right)^t \cdot \left( \frac{rr_0}{\delta} \right)^{t-k}.$$ Since $t > \max(k, s)$, the sum is $\lesssim C_P C_T |P| Mr_0^k \left( \frac{\delta}{r_0} \right)^s$. Adding up both $r > \frac{\delta}{r_0}$ and $r < \frac{\delta}{r_0}$ contributions, we prove the Lemma. ◻ For $t \in \mathcal{T}$, let $\mathcal{P}(t) = \{ p \in \mathcal{P}: t \in \mathcal{T}(p) \}$. By Cauchy-Schwarz, we have $$(M|\mathcal{P}|)^2 = \left( \sum_{t \in \mathcal{T}} |\mathcal{P}(t)| \right)^2 \le |\mathcal{T}| \sum_{t \in \mathcal{T}} |\mathcal{P}(t)|^2 = |\mathcal{T}| \sum_{p \in \mathcal{P}} j_p (\mathcal{P}, \mathcal{T}).$$ By Lemma [Lemma 64](#lem:jp_est){reference-type="ref" reference="lem:jp_est"}, we get $$|\mathcal{T}| \ge \frac{M^2 |\mathcal{P}|^2}{C_P C_T |\mathcal{P}|^2 Mr_0^{k-s} \delta^s} = (C_P C_T)^{-1} M r_0^{s-k} \delta^{-s}.$$ ◻ *Proof of Theorem [Theorem 59](#thm:main_refined'){reference-type="ref" reference="thm:main_refined'"}.* A small reduction: we would like to assume $|\mathcal{T}(p)| \sim M$ for all $p \in \mathcal{P}$. To assume this, we first observe that $|\mathcal{T}(p)| \ge M_0 = (\delta/r_0)^{-\varepsilon} r_0^{k-s} \delta^{-s}$ for all $p \in \mathcal{P}$. On the other hand, if for at least half of the $p \in \mathcal{P}$ (call them $\mathcal{P}'$) we have $|\mathcal{T}(p)| \ge M_0 (\delta/r_0)^{-1}$, then we are immediately done by Corollary [Corollary 63](#cor:easy_est_2){reference-type="ref" reference="cor:easy_est_2"} applied to $\mathcal{P}'$ and $\mathcal{T}(p)$. Thus, by reducing $\mathcal{P}$ if necessary, we may assume $|\mathcal{T}(p)| \in (M_0, M_0 (\delta/r_0)^{-1})$. Then by reducing $\mathcal{P}$ further by a $\lesssim\log (\delta/r_0)^{-1}$ factor, we may assume $|\mathcal{T}(p)| \in (M, 2M)$ for some $M \in (M_0, M_0 (\delta/r_0)^{-1})$. Finally, we may remove some tubes from each $\mathcal{T}(p)$ to make $|\mathcal{T}(p)| = M$. Then $(\mathcal{P}_0, \mathcal{T}_0)$ is a $(\delta, s, C_1 r_0^{k-s}, \kappa, C_2, M)$-nice configuration. Pick $\beta(s, t, k) > 0$ such that $\frac{t-d\beta}{1-\beta} > \max(s, k)$, and let $t' = \frac{1}{2} (\frac{t-d\beta}{1-\beta} + \max(s, k))$. Pick $\Delta > 0$ such that $\log(1/\Delta) < \Delta^{-\varepsilon}$. Find $\Delta' = \Delta^k \in (\delta/r_0, (\delta/r_0)^\beta)$ such that the conclusion of Lemma [Lemma 62](#lem:extract_t'_set){reference-type="ref" reference="lem:extract_t'_set"} holds. Now by Proposition [Proposition 60](#prop:nice_tubes_2){reference-type="ref" reference="prop:nice_tubes_2"}, we have $$\frac{|\mathcal{T}|}{M} \ge \frac{|\mathcal{T}_Q|}{M_Q} \cdot \frac{|\mathcal{T}^{\Delta'} (\mathcal{T})|}{M_{\Delta'}}.$$ If $\varepsilon< \beta \eta^2$, where $\eta(s, t, \kappa, k, d)$ is the parameter in Theorem [Theorem 58](#thm:main'){reference-type="ref" reference="thm:main'"}, we have $\frac{|\mathcal{T}^{\Delta'} (\mathcal{T})|}{M_{\Delta'}} \ge (\Delta')^{-s-\sqrt{\varepsilon}}$. Pick $Q$. Then $S_H^{-1} \circ S_Q (S_H(\mathcal{P}) \cap Q)$ satisfies the conditions of Corollary [Corollary 63](#cor:easy_est_2){reference-type="ref" reference="cor:easy_est_2"} with $C_P = \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \cdot L \cdot \Delta^{-d}$. Thus, we have $\frac{|\mathcal{T}_Q|}{M_Q} \ge \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \Delta^d L^{-1} \cdot r_0^{s-k} \left( \frac{\delta}{\Delta'} \right)^{-s}$. Using these two bounds and $M \ge (\delta/r_0)^\varepsilon r_0^{s-k} \delta^{-s}$, we get $$|\mathcal{T}| \ge \left( \frac{\delta}{r_0} \right)^{2\varepsilon-\sqrt{\varepsilon}\beta} \Delta^{-d} L^{-1} r_0^{2(s-k)} \cdot \delta^{-2s}.$$ It remains to choose $\varepsilon< \beta^2/100$ and also for $\frac{\delta}{r_0}$ small enough, we have $\Delta^{-d} < \left( \frac{\delta}{r_0} \right)^{-\varepsilon}$ and $L \le \left( \frac{\delta}{r_0} \right)^{-\varepsilon} \Delta^{-\varepsilon m} \le \left( \frac{\delta}{r_0} \right)^{-2\varepsilon}$. Thus, $|\mathcal{T}| \ge \left( \frac{\delta}{r_0} \right)^{-\varepsilon} r_0^{2(s-k)} \cdot \delta^{-2s}$ and we are done. ◻ # Power decay around $k$-planes {#sec:power decay} In this section, we will roughly deal with the following situation: - $\mu, \nu$ are $s$-Frostman measures with $k-1 < s \le k$; - $\nu$ gives mass $\le \varepsilon$ to any $(r_0, k)$-plate. In other words, $\nu$ does not concentrate around $(r_0, k)$-plates. We would like to understand the $\nu$-mass of $(r, k)$-plates for $r$ much smaller than $r_0$. A result of Shmerkin [@shmerkin2022non Proposition B.1] says that there exist $r_1 (r_0, s, k), \kappa(s, k) > 0$, a subset $X \subset \mathrm{spt}\mu$ with $\mu(X) > 1 - O(\varepsilon)$, and for each $x \in X$, a subset $Y_x \subset \mathrm{spt}\nu$ with $\mu(Y_x) > 1 - O(\varepsilon)$ such that $\nu(H \cap Y_x) \le r^\eta$ for all $r \le r_1$ and $(r, k)$-plates $H$ through $x$. Thus, we do obtain a power decay for sufficiently small $r$. But what is the optimal starting point of the power decay? Can we hope for a power decay $\nu(H \cap Y_x) \lesssim K (\frac{r}{r_0})^\eta$ for all $(r, k)$-plates through $x$? The answer is yes, and indeed we shall prove it by making small but meaningful tweaks to Shmerkin's argument. But before stating our result, we shall introduce some convenient notation. We define thin $k$-plates, a generalization of thin tubes, as follows. **Definition 65**. *Let $K, t \ge 0$, $1 \le k \le d-1$, and $c \in (0, 1]$. Let $\mu, \nu \in \mathbb{P}(\mathbb{R}^d)$ supported on $X, Y$. Fix $G \subset X \times Y$. We say $(\mu, \nu)$ has $(t, K, c)$-thin $k$-plates on $G$ down from scale $r_0$ if $$\label{eqn:thin tubes} \nu(H \cap G|_x) \le K \cdot r^t \quad \text{ for all } r \in (0, r_0) \text{ and all } (r, k)\text{-plates } H \text{ containing } x.$$* **Remark 66**. *In this paper, we will choose $G = (A \cup B)^c$ where $\mu \times \nu(B)$ is small. (The complement is taken with respect to $\mathbb{R}^d \times \mathbb{R}^d$.) In this case, the equation [\[eqn:thin tubes\]](#eqn:thin tubes){reference-type="eqref" reference="eqn:thin tubes"} becomes $$\nu(H \setminus (A|_x \cup B|_x)) \le K \cdot r^t \quad \text{ for all } r \in (0, r_0) \text{ and all } (r, k)\text{-plates } H \text{ containing } x.$$* Now, we can state the main proposition, which generalizes and extends Proposition B.1 of [@shmerkin2022non]. It may be of independent interest. **Proposition 67**. *Let $1 \le k \le d-1$ and $k-1 < s \le k$. There exist $\eta(\kappa, k, d) > 0$ and $K_0 (\kappa, k, d) > 0$ with the following property. Fix $r_0 \le 1$ and $K \ge K_0$. Suppose that $\mu, \nu$ are positive measures with $|\mu|, |\nu| \ge 1$ and for any $(r, k-1)$-plate $H$, we have $$\begin{gathered} \mu(H) \le C_\mu r^\kappa, \\ \nu(H) \le C_\nu r^\kappa. \end{gathered}$$ Let $A \subset X \times Y$ be the pairs of points that lie in some $K^{-1}$-concentrated $(r_0, k)$-plate. Then there exists $B$ with $\mu \times \nu(B) \le K_0 K^{-1}$ such that $(\mu, \nu)$ have $(\eta, K r_0^{-\eta})$-thin $k$-plates on $(A \cup B)^c$. (The complement is taken with respect to $\mathbb{R}^d \times \mathbb{R}^d$.)* **Remark 68**. *(a) We can apply Proposition [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"} in case $\mu, \nu$ are $s$-dimensional with $s > k-1$.* *(b) In Proposition B.1 of [@shmerkin2022non], the exponents for $\mu, \nu$ are allowed to differ. The proof of Proposition [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"} is easily modified to include this detail.* To prove Proposition [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"}, we need the following two lemmas. Fix $r \le r_0$. The first says that there are few dense $(r, k)$-plates, and the second says that for most $x \in X$, the dense $(r, k)$-plates through $x$ lie in some $(r_0, k)$-plate. **Lemma 69**. *There is $N=N(\kappa, k, d)$ such that the following holds: let $\nu$ be a measure with mass $\le 1$ such that $\nu(W) \le C_\nu \rho^\kappa$ for all $(\rho, k-1)$-plates $W$, $1 > \rho > r$. Let $\mathcal{E}_{r,k}$ be a set of $(r, k)$-plates such that every $(s, k)$-plate contains $\lesssim\left( \frac{s}{r} \right)^{(k+1)(d-k)}$ many $r$-plates of $\mathcal{E}_{r,k}$ (as in Section [2.2](#subsec:r-net){reference-type="ref" reference="subsec:r-net"}). Let $\mathcal{H}= \{ H \in \mathcal{E}_{r,k} : \nu(H) \ge a \}$. Then $|\mathcal{H}| \lesssim(\frac{C_\nu}{a})^N$.* Lemma [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"} follows from the condition on $\mathcal{E}_{r,k}$ and the following generalization of [@shmerkin2022non Lemma B.3]. In the case $a = \delta^\eta$, the resulting bound is stronger but the assumption is also stronger. **Lemma 70**. *Suppose $\nu(W) \le C_\nu \rho^\kappa$ for all $(\rho, k-1)$-plates $W$, $1 > \rho > r$. Then there exists a family of $\lesssim a^{-1}$ many $(r(C_\nu/a^2)^{1/\kappa}, k)$-plates $\{ T_j \}$ such that every $(r, k)$-plate $H$ with $\nu(H) \ge a$ is contained in some plate $T_j$.* *Proof.* Choose a maximal set of $(r, k)$-plates $\{ Y_j \}_{j=1}^m$ such that 1. $\nu(Y_i) \ge a$, 2. $\nu(Y_i \cap Y_j) \le a^2/2$ for $1 \le i < j \le m$. We claim $m \le 2a^{-1}$. Indeed, if $S = \sum_{i=1}^m \nu(Y_i)$ and $f = \sum_{i=1}^m \mathbbm{1}_{Y_i}$, then, then $$\label{eqn:L2} S^2 = \left( \int f \, d\nu \right)^2 \le \int f^2 \, d\nu = S + \sum_{1 \le i < j \le m} \nu(Y_i \cap Y_j) \le S + m^2 a^2/2.$$ Now, $S \ge ma > 2$, so $S^2 - S > \frac{S^2}{2}$. Combining with [\[eqn:L2\]](#eqn:L2){reference-type="eqref" reference="eqn:L2"} gives $S^2 < m^2 a^2$, a contradiction. Let $\{ T_j \}_{j=1}^m$ be the $(r(C_\nu/a^2)^{1/\kappa}, k)$-plates with same central $k$-plane as $Y_j$. We show the problem condition. Given an $(r, k)$-plate $H$ with $\nu(H) \ge a$, by maximality there exists $Y_j$ such that $\nu(H \cap Y_j) \ge a^2/2$. Thus, if $\angle(H, Y_j)$ is the largest principal angle between the central planes of $H$ and $Y_j$, then $H \cap Y_j$ is contained in a box of dimensions $$\underbrace{1 \times \cdots \times 1}_{(k-1) \text{ times}} \times r/\angle(H, Y_j) \times \underbrace{r \times \cdots \times r}_{d-k \text{ times}}.$$ Thus, $H \cap Y_j$ is contained in a $(r/\angle(H, Y_j), k-1)$-plate, so $\nu(H \cap Y_j) \le C_\nu (r/\angle(H, Y_j))^\kappa$. Thus, $\angle(H, Y_j) \lesssim r(C_\nu/a^2)^{1/\kappa}$, so $H$ is contained in $T_j$. ◻ **Remark 71**. *We would like to present an alternative proof of Lemma [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"}, which was the original one found by the author. It gives slightly worse bounds but we believe it is slightly more motivated.* *If $a > 1$ then $\mathcal{H}= \emptyset$, so assume $a \le 1$. Let $\xi = \left(\frac{a}{2C_\nu}\right)^{1/\kappa} \le 1$. By induction, for each $0 \le i \le k$, there exist $x_0, \cdots, x_i$ such that $|x_0 \wedge x_1 \wedge \cdots \wedge x_i| \ge \xi^i$ and that lie in at least $|\mathcal{H}| (\frac{a}{2})^{i+1}$ many elements of $\mathcal{H}$.* *The base case $i = 1$ is trivial. For the inductive step, suppose $x_0, \cdots, x_i$ are found. Let $\Omega$ be the $\tilde{r}$-neighborhood of the span of $x_1, \cdots, x_i$. Then since $\mu(\Omega) \le C_\nu \xi^\kappa < \frac{1}{2} a$ for every $H \in \mathcal{H}$, we have $\mu(H \setminus \Omega) \ge \frac{1}{2} a$. Thus, there is $x_{i+1} \in \mathbb{R}^d \setminus \Omega$ such that $x_0, \cdots, x_{i+1}$ lie in at least $|\mathcal{H}| (\frac{a}{2})^{(i+2)}$ many elements of $\mathcal{H}$, and by construction, $|x_0 \wedge x_1 \wedge \cdots \wedge x_{i+1}| \ge \xi^{i+1}$. This completes the inductive step and thus the proof of the claim.* *Finally, the set of $(r,k)$-plates through $x_1, \cdots, x_k$ must lie in a $(r \xi^{-k}, k)$-plate, so at most $\xi^{-k(k+1)(d-k)}$ many $(r,k)$-plates of $\mathcal{E}_{r,k}$ can lie in it. Thus, $|\mathcal{H}| \le (\frac{a}{2})^{-(k+1)} \xi^{-k(k+1)(d-k)} \lesssim(\frac{C_\nu}{a})^N$.* The following lemma is in the same spirit as [@shmerkin2022non Proposition B.2]. **Lemma 72**. *Let $\mathcal{H}$ be a collection of $(r,k)$-plates, and suppose $\mu(W) \le C_\mu \rho^\kappa$ for all $(\rho, k-1)$-plates $W$, $1 > \rho > r$. Then for all $x \in X$ except a set of $\mu$-measure $\le C_\mu \left( \frac{r}{r_0} \right)^\kappa |\mathcal{H}|^2$, there exists an $(r_0,k)$-plate that contains every $(r,k)$-plate in $\mathcal{H}$ that passes through $x$.* *Proof.* The exceptional set is contained in the set of $x \in X$ that lies in two plates of $\mathcal{H}$ with "angle" $\ge \frac{1}{r_0}$. The intersection of two such plates is contained in a box with dimensions $\underbrace{r \times \cdots \times r}_{d-k \text{ times}} \times \frac{r}{r_0} \times \underbrace{1 \times \cdots \times 1}_{k-1 \text{ times}}$, which in turn is contained in a $(\frac{r}{r_0}, k-1)$-plate (since $r_0 \le 1$). Thus, by assumption on $\mu$, this box has mass $\lesssim C_\mu \left( \frac{r}{r_0} \right)^{s-(k-1)}$. Finally, there are $|\mathcal{H}|^2$ pairs of plates in $\mathcal{H}$. ◻ *Proof of Proposition [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"}.* Fix $r \le r_0$, and let $\eta = \frac{\kappa}{4N}$, where $N$ is the constant in Lemma [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"}. We may assume $N \ge 2$. By Lemmas [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"} and [Lemma 72](#lem:concentrate_in_r0){reference-type="ref" reference="lem:concentrate_in_r0"}, we can find a set $E_r$ with $\mu(E_r) \le K^{-2} \left( \frac{r}{r_0} \right)^{\eta}$ and, for each $x \notin E_r$, a set $P_r (x) \subset Y$ that is either empty or a $(r_0^{1/2} r^{1/2}, k)$-plate through $x$ such that $\nu(W) \le K \left( \frac{r}{r_0} \right)^\eta$ for every $W$ intersecting $Y \setminus P_r (x)$. Now, let $E = \cup_{n \ge 0} E_{r_0 K^{-2^n}}$ and $P(x) = \cup_{n \ge 0} P_{r_0 K^{-2^n}} (x)$. We claim that $\mu(E) \le K^{-1}$ and if $x \notin E$, then $\nu(P(x) \setminus A|_x) \lesssim K^{-1}$. Then if $r \ge r_0 K^{-1}$, then $\nu(W) \le 1 \le K \left( \frac{r}{r_0} \right)^\eta$ for $\eta < 1$; for any $r_0 K^{-2^n} \le r < r_0 K^{-2^{n-1}}$, we have for any $(r, k)$-plate $W$, $$\nu(W \setminus P_{r_0 K^{-2^n}} (x)) \le K \left( \frac{r_0 K^{-2^{n-1}}}{r_0} \right)^\eta \le K \left( \frac{r}{r_0} \right)^{\eta/2}$$ Then $(\mu, \nu)$ have $(\eta/2, K^2 r_0^{-\eta/2}, 1-K^{-1})$-thin $k$-plates relative to $A$. To prove the first claim, we observe that $\mu(E) \le K^{-2} \sum_{n=0}^\infty K^{-\eta 2^n} \le K^{-1}$ if $K_0$ is sufficiently large in terms of $\eta$. Next, by definition of $P_{r_0 K^{-2^{n-1}}}$, we have $\nu(P_{r_0 K^{-2^n}} (x) \setminus P_{r_0 K^{-2^{n-1}}} (x)) \le K \left(\frac{r_0 K^{-2^{n-1}}}{r_0} \right)^{\eta/2} \le K^{1-2^{n-2} \eta}$. We also have the bound $\nu(P_{r_0 2^{-n}} (x) \setminus A|_x) \le K^{-1}$ from the given condition (note that $P_{r_0 2^{-n}} (x)$ is a $(r_0 2^{-(n-1)}, k)$-plate). Thus, $$\begin{aligned} \nu(P(x) \setminus A|_x) &\le \sum_{n=0}^{\log \eta^{-1}} \nu(P_{r_0 K^{-2^n}} (x) \setminus A|_x) + \sum_{n=\log \eta^{-1}}^\infty \nu(P_{r_0 K^{-2^n}} (x) \setminus P_{r_0 K^{-2^{n-1}}} (x)) \\ &\le \log \eta^{-1} \cdot K^{-1} + \sum_{n=\log \eta^{-1}}^\infty K^{1-2^{n-2} \eta} \\ &\lesssim K^{-1}, \end{aligned}$$ if $K_0$ is chosen large enough. ◻ # Radial projection estimates {#sec:threshold} In this section, we will first prove a key special case, and then the general case of Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"}. ## Maximal plate concentration case *This subsection is based on ideas from [@orponen2022kaufman].* **Theorem 73**. *Let $k \in \{ 1, 2, \cdots, d-1 \}$, $k-1 < \sigma < s \le k$, and fix $K \ge 1$. There exists $N \in \mathbb{N}$ and $K_0$ depending on $\sigma, s, k$ such that the following holds. Fix $r_0 \le 1$ and $K_1, K_2 \ge K_0$. Let $\mu, \nu$ be $\sim 1$-separated $s$-dimensional measures with constant $C_\mu, C_\nu$ supported on $E_1, E_2$, which lie in an $(r_0, k)$-plate $H_r$. Assume that $|\mu|, |\nu| \le 1$. Let $A$ be the pairs of $(x, y) \in E_1 \times E_2$ that lie in some $K_1^{-1}$-concentrated $(\frac{r_0}{K_2}, k)$-plate. Then there exists a set $B \subset E_1 \times E_2$ with $\mu \times \nu (B) \lesssim K_1^{-1}$ such that for every $x \in E_1$ and $r$-tube $T$ through $x$, we have $$\mu(T \setminus (A|_x \cup B|_x)) \lesssim\frac{r^\sigma}{r_0^{\sigma-(k-1)}} (K_1 K_2)^N.$$ The implicit constant may depend on $s, k$.* Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"} is the special case of Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"} where $(\mu, \nu)$ are concentrated in a $(r_0 K, k)$-plate for some small $K \ll r_0$ (we call this the maximal plate concentration case). For this, we closely follow the bootstrapping approach of [@orponen2022kaufman]. There are three ingredients. - The next Proposition [Proposition 74](#prop:k-thin){reference-type="ref" reference="prop:k-thin"} will be the base case for the bootstrapping argument ($\sigma = 0$). - Proposition [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"} will ensure power decay for $\mu, \nu$ around $k$-planes. - Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} will be used in the bootstrapping step to upgrade $\sigma$ to $\sigma + \eta$. **Proposition 74**. *Let $1 \le k \le d-1$ and $k-1 < s \le k$, then there exists $N = N(s, k)$ such that the following holds. Fix $K \ge K_0$. Then for any $s$-dimensional measures $\mu, \nu$ with constant $\sim 1$ contained in the $r_0$-neighborhood of a $k$-plane and $d(\mu, \nu) \gtrsim 1$, there exists $B \subset X \times Y$ with $\mu \times \nu(B) \le K^{-1}$ such that $(\mu, \nu)$ has $(0, K^N r_0^{k-1})$-thin tubes on $B^c$ down from scale $r_0$.* *Proof.* Let $\tilde{\mu}, \tilde{\nu}$ be the projected measures on the $k$-plane. Then $\tilde{\mu}, \tilde{\nu}$ satisfy $s$-dimensional Frostman conditions for $r_0 \le r \le 1$. Let $$B = \{ (x, y) : x, y \in T \text{ for some } r_0\text{-tube } T \text{ with } \nu(T) \ge K^N r_0^{k-1}. \}$$ The rest is a standard argument following [@guth2019incidence Proof of Lemma 3.6]. Define the radial projection $P_y (x) = \frac{x-y}{|x-y|}$. Orponen's radial projection theorem [@orponen2018radial Equation (3.5)] can be written in the form (where $p = p(s, k) > 1$): $$\label{eqn:orponen radial} \int \| P_x \tilde{\mu} \|_{L^p}^p \, d\tilde{\mu}(x) \lesssim 1.$$ To effectively use [\[eqn:orponen radial\]](#eqn:orponen radial){reference-type="eqref" reference="eqn:orponen radial"}, we will show that $|P_x (B|_x)|$ is small for $x \in X$. Indeed, let $\mathcal{T}_x$ be a minimal set of finitely overlapping $2r_0$-tubes through $x$ such that any $r_0$-tube through $x$ with $\nu(T) \ge K^N r_0^{k-1}$ lies in a $2r_0$-tube in $\mathcal{T}_x$. Then each $2r_0$-tube in $\mathcal{T}_x$ has $\nu$-measure $\ge K^N r_0^{k-1}$. Since $d(x, \nu) \gtrsim 1$, we conclude that $|\mathcal{T}_x| \lesssim K^{-N} r_0^{1-k}$. Therefore, since the Lebesgue measure $|P_x (T)| \lesssim r_0^{k-1}$ for a $2r_0$-tube $T$ through $x$, we obtain $|P_x (B|_x)| \lesssim K^{-N}$. Finally, we can use Holder's inequality and [\[eqn:orponen radial\]](#eqn:orponen radial){reference-type="eqref" reference="eqn:orponen radial"} to upper bound $\mu \times \nu(B)$: $$\begin{aligned} \mu \times \nu(B) &= \int \nu(B|_x) d\mu(x) \\ &= %\text{\color{red} this is equality right?} \int \left( \int_{P_x (B|_x)} P_x (\nu) \right) d\mu(x) \\ &\le \sup_x |P_x (B|_x)|^{1-1/p} \int \| P_x \nu \|_{L^p} d\mu(x) \\ &\lesssim K^{-N(1-1/p)}. \end{aligned}$$ Choose $N = 1 + (1 - 1/p)^{-1}$ to finish (the implicit constant is dominated by $K \ge K_0$ if $K_0$ is large enough). ◻ The bootstrapping step is as follows: **Proposition 75**. *Let $k \in \{ 1, \cdots, d-1 \}$, $0 \le \sigma \le k$, $\max(\sigma, k-1) < s \le k$, $\kappa > 0$. There exist $\eta(\sigma, s, \kappa, k, d)$ and $K_0(\eta, k) > 0$ such that the following holds. Fix $r_0 \le 1$ and $K \ge K_0$. Let $\mu, \nu$ be $\sim K^{-1}$-separated $s$-dimensional measures with constant $K$ supported on $X, Y$, which lie in an $(r_0, k)$-plate $H$. Let $G \subset X \times Y$. Suppose that $(\mu, \nu)$ and $(\nu, \mu)$ have $(\sigma, K r_0^{-(\sigma-(k-1))})$-thin tubes and $(\kappa, K r_0^{-\kappa})$-thin $k$-plates on $G$ down from scale $r_0$. Then there exists a set $B \subset X \times Y$ with $\mu \times \nu(B) \le K^{-1}$ such that $(\mu, \nu)$ and $(\nu, \mu)$ have $(\sigma+\eta, K^{d+1} r_0^{-(\sigma+\eta-(k-1))})$-thin tubes on $G \setminus B$ down from scale $r_0$. Furthermore, $\eta(\sigma, s, \kappa, k, d)$ is bounded away from zero on any compact subset of $\{ (\sigma, s, \kappa, k) : \max(\sigma, k-1) < s \le k \le d-1 \}$.* **Remark 76**. *The reader is advised to set $r_0 = 1$ in the following argument, in which case it is a straightforward modification of [@orponen2022kaufman Lemma 2.8], with one small technical exception in the proof of the concentrated case, where we improve upon the dyadic pigeonholing step. Also if $r_0 = 1$, then the simpler Theorem [Theorem 8](#thm:main){reference-type="ref" reference="thm:main"} can be used instead of Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} in the proof.* *Proof.* We are given that for all $r \in (0, r_0]$, $$\begin{gathered} \nu(T \cap G|_x) \le K \cdot \frac{r^{\sigma}}{r_0^{\sigma-(k-1)}} \text { for all } r\text{-tubes } T \text{ containing } x \in X, \label{gather1}\\ \nu(W \cap G|_x) \le K \cdot \frac{r^{\sigma}}{r_0^{\sigma-(k-1)}} \text { for all } (r, k)\text{-plates } W \text{ containing } x \in X, \label{gather2}\\ \mu(T \cap G|^y) \le K \cdot \frac{r^{\sigma}}{r_0^{\sigma-(k-1)}} \text { for all } r\text{-tubes } T \text{ containing } y \in Y, \label{gather3}\\ \mu(W \cap G|^y) \le K \cdot \frac{r^{\sigma}}{r_0^{\sigma-(k-1)}} \text { for all } (r, k)\text{-plates } W \text{ containing } y \in Y. \label{gather4} \end{gathered}$$ For $x \in X$ and $r \le r_0$, let $\mathcal{T}''_{x,r}$ denote the $r$-tubes through $x$ such that $$\label{eqn:T''} \nu(T \cap G|_x) \ge K^{d+1} \cdot \frac{r^{\sigma+\eta}}{r_0^{\sigma+\eta-(k-1)}}.$$ Now, let $\mathcal{T}'_{x,r}$ denote a covering of $\mathcal{T}''_{x,r}$ by essentially distinct $2r$-tubes. Then for $x \in X$, since $d(x, Y) \ge K^{-1}$, we have that the tubes in $\mathcal{T}'_{x,r}$ have $\lesssim K^{d-1}$-overlap on $\nu$, so $|\mathcal{T}'_{x,r}| \lesssim\frac{r^{-(\sigma+\eta)}}{r_0^{-(\sigma+\eta-(k-1))}}$. For a dyadic $r \in (0, r_0]$, let $H_r = \{ (x, y) \in G : y \in \cup \mathcal{T}'_{x,r} \}$, where $\cup \mathcal{T}'_{x,r}$ denotes the union of the tubes in $\mathcal{T}'_{x,r}$. **Claim.** There are $\eta(\sigma, s, \kappa, k, d) > 0$ and $K_0 (\eta) > 0$ such that the following holds for $K \ge K_0$. If $\frac{r}{r_0} < K^{-1/\eta}$, then $\mu \times \nu(H_r) \le 2\left(\frac{r}{r_0}\right)^\eta$. Furthermore, $\eta(\sigma, s, \kappa, k, d)$ is bounded away from zero on any compact subset of $\{ (\sigma, s, \kappa, k, d) : \max(\sigma, k-1) < s \le k \le d-1 \}$. We will be done if we show the claim. Indeed, let $B_1 = \cup_{r \le r_0 \text{ dyadic }} H_r$; then for any dyadic $r \le r_0$ and any $r$-tube $T$ through some $x \in X$, we either have $T \in \mathcal{T}'_{x,r}$, which means $T \cap G|_x \setminus B_1|_x = \emptyset$, or the negation of [\[eqn:T\'\'\]](#eqn:T''){reference-type="eqref" reference="eqn:T''"} holds. In either case, we get $$\label{eqn:T'' neg} \nu(T \cap G|_x \setminus B_1|_x) \le K^{d+1} \cdot \frac{r^{\sigma+\eta}}{r_0^{\sigma+\eta-(k-1)}}.$$ We have [\[eqn:T\'\' neg\]](#eqn:T'' neg){reference-type="eqref" reference="eqn:T'' neg"} for dyadic $r \le r_0$, but it also holds for all $r \le r_0$ at the cost of introducing a multiplicative factor of $2^{\sigma+\eta} \le 2^{k+1}$ on the RHS of [\[eqn:T\'\' neg\]](#eqn:T'' neg){reference-type="eqref" reference="eqn:T'' neg"}. Thus, $(\mu, \nu)$ have $(\sigma+\eta, 2^{k+1} \cdot K^d r_0^{-(\sigma+\eta-(k-1))})$-thin tubes on $G \setminus B_1$ down from scale $r_0$. Now we move to upper-bounding $\mu \times \nu(B_1)$. By [\[gather1\]](#gather1){reference-type="eqref" reference="gather1"} and [\[eqn:T\'\'\]](#eqn:T''){reference-type="eqref" reference="eqn:T''"}, we have $H_r \neq \emptyset$ for all $r > r_0 K^{-d/\eta}$, and so if $K \ge K_0$ from Claim, then $$\mu \times \nu(B_1) \le \sum_{r \le r_0 K^{-d/\eta} \text{ dyadic }} \mu \times \nu(H_r) \le\sum_{r \le r_0 K^{-d/\eta} \text{ dyadic }} 2\left( \frac{r}{r_0} \right)^{\eta} \le C_\eta K^{-d}.$$ Let $K_0$ be the maximum of the value of $K_0$ from Claim, $2C_\eta$, and $2^{k+1}$. Since $d \ge 2$, we get $\mu \times \nu(B_1) \le \frac{1}{2} K^{-1}$ and $(\mu, \nu)$ have $(\sigma+\eta, K^{d+1} r_0^{-(\sigma+\eta-(k-1))})$-thin tubes on $G \setminus B_1$ down from scale $r_0$. We can analogously find $B_2 \subset X \times Y$ with $\mu \times \nu(B_2) \le \frac{1}{2} K^{-1}$ such that $(\nu, \mu)$ have $(\sigma+\eta, K^{d+1} r_0^{-(\sigma+\eta-(k-1))})$-thin tubes on $G \setminus B_2$ down from scale $r_0$, and so $B = B_1 \cup B_2$ would be a good choice. Now we turn to proving the Claim. *Proof of Claim.* We will choose $\eta = \min\{ \frac{1}{2} (6 + \frac{15(d-1)}{s - \max(\sigma, k-1)} )^2, \frac{1}{5} \varepsilon^2\}$, where $\varepsilon$ is obtained from Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}. From Remark [Remark 12](#rmk:uniform eps){reference-type="ref" reference="rmk:uniform eps"} and the continuity of the function $(s, \sigma, k) \mapsto (s - \max(\sigma, k-1))^{-1}$, we see that $\eta(\sigma, s, \kappa, k, d)$ is bounded away from zero on any compact subset of $\{ (\sigma, s, \kappa, k, d) : \max(\sigma, k-1) < s \le k \le d-1 \}$. Suppose that Claim is false. Let $\mathbf{X}= \{ x \in X : \nu(H_r) \ge \left( \frac{r}{r_0} \right)^\eta \}$. Then $\mu(\mathbf{X}) \ge \left( \frac{r}{r_0} \right)^\eta$. Recall that for $x \in X$, the fiber $H_r|_x$ is covered by $\mathcal{T}'_{x,r}$, which is a set of cardinality $\lesssim\frac{r^{-(\sigma+\eta)}}{r_0^{-(\sigma+\eta-(k-1))}}$. Let $$\mathcal{T}_x = \{ T \in \mathcal{T}'_{x,r} : \nu(T \cap H_r|_x) \ge \frac{r^{\sigma+3\eta}}{r_0^{\sigma+3\eta-(k-1)}} \}, \qquad Y_x = (H_r|_x) \cap \bigcup \mathcal{T}_x.$$ Then $\nu(Y_x) \ge \left( \frac{r}{r_0} \right)^\eta - \left( \frac{r}{r_0} \right)^{2\eta} \ge \left( \frac{r}{r_0} \right)^{2\eta}$ for all $x \in \mathbf{X}$. Furthermore, for every $T \in \mathcal{T}_x$, we have $$\label{eqn:T cap Hr} \frac{r^{\sigma+3\eta}}{r_0^{\sigma+3\eta-(k-1)}} \le \nu(T \cap Y_x) \le \frac{r^{\sigma-\eta}}{r_0^{\sigma-\eta-(k-1)}}.$$ The upper bound follows from $Y_x \subset H_r|_x \subset G|_x$, [\[gather1\]](#gather1){reference-type="eqref" reference="gather1"}, and $K \le \left( \frac{r}{r_0} \right)^{-\eta}$. In fact, we have in general, $$\nu(T^{(\rho)} \cap Y_x) \le \left( \frac{r}{r_0} \right)^{-\eta} \rho^\eta, \qquad \rho \in [r, 1], T \in \mathcal{T}_x.$$ We also take the time to state the thin plates assumption: $$\nu(W^{(\rho)} \cap Y_x) \le \left( \frac{r}{r_0} \right)^{\kappa-\eta} \qquad \rho \in [r, 1], W \text{ is } (\rho, k)\text{-plate}.$$ Since $\cup \mathcal{T}_x$ covers $Y_x$, we get by the upper bound in [\[eqn:T cap Hr\]](#eqn:T cap Hr){reference-type="eqref" reference="eqn:T cap Hr"}, $|\mathcal{T}_x| \gtrsim\frac{r^{-\sigma+\eta}}{r_0^{-\sigma+\eta+(k-1)}} \nu(Y_x) \ge \frac{r^{-\sigma+3\eta}}{r_0^{-\sigma+3\eta+(k-1)}}$. Hence, $\mathcal{T}_x$ is a $(r, \sigma, r_0^{-(\sigma-(k-1))} \left(\frac{r}{r_0} \right)^{-5\eta})$-set and $(r, \kappa, r_0^{-\kappa} \left(\frac{r}{r_0} \right)^{-5\eta}, k-1)$-set for each $x \in \mathbf{X}$. Let $\gamma = \frac{15\eta}{s-\max(\sigma,k-1)}$. Call a tube $T \in \mathcal{T}_x$ concentrated if there is a ball $B_T$ with radius $\left(\frac{r}{r_0}\right)^\gamma$ such that $$\label{eqn:non-conc} \nu(T \cap B_T \cap Y_x) \ge \frac{1}{3} \cdot \nu(T \cap Y_x).$$ Suppose that there is $\mathbf{X}' \subset \mathbf{X}$ with $\mu(\mathbf{X}') \ge \mu(\mathbf{X})/2$ such that for each $x \in \mathbf{X}'$, at least half the tubes of $\mathcal{T}_x$ are non-concentrated. Since $\mu(\mathbf{X}') \ge \frac{1}{2} \mu(\mathbf{X})/2 \ge \frac{1}{2} \left( \frac{r}{r_0} \right)^{2\eta}$ and $\mu$ is Frostman with constant $K \le \left( \frac{r}{r_0} \right)^{-\eta}$, we can find a $(r, \sigma, \left( \frac{r}{r_0} \right)^{-3\eta})$-set $P \subset \mathbf{X}'$. For each $x \in \mathbf{X}'$, the set of non-concentrated tubes $\mathcal{T}'_x \subset \mathcal{T}_x$ is a $(r, \sigma, 2r_0^{-(\sigma-(k-1))} \left(\frac{r}{r_0} \right)^{-5\eta})$-set and $(r, \kappa, 2r_0^{-\kappa} \left(\frac{r}{r_0} \right)^{-5\eta}, k-1)$-set. Let $\mathcal{T}= \cup_{x \in P} \mathcal{T}'_x$. By Lemma [Lemma 22](#lem:concentrated_points){reference-type="ref" reference="lem:concentrated_points"}, since $d(X, Y) \ge K^{-1}$, we have that $\mathcal{T}$ is contained in the $O(K) \cdot r_0$-neighborhood of $H$. Now, we apply Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"} with $\overline{r}_0 := \min(O(K) \cdot r_0, 1)$. Since $K \le \left( \frac{r}{r_0} \right)^{-\eta}$ and $\sigma \le k$, we still have that for each $x \in \mathbf{X}'$, the set of non-concentrated tubes $\mathcal{T}_x'$ is a $(r, \sigma, 2\overline{r}_0^{-(\sigma-(k-1))} \left(\frac{r}{\overline{r}_0} \right)^{-7\eta})$-set and $(r, \kappa, 2\overline{r}_0^{-\kappa} \left(\frac{r}{\overline{r}_0} \right)^{-7\eta}, k-1)$-set. At this point, let us remark that implicit constants are dominated by $\left( \frac{r}{\overline{r}_0} \right)^{-\eta} \ge K^\eta$ if $K \ge K_0 (\eta)$ is chosen large enough. Then if $\eta \le \varepsilon^2/4$, where $\varepsilon$ is obtained from Theorem [Theorem 11](#thm:main_refined){reference-type="ref" reference="thm:main_refined"}, then $$|\mathcal{T}| \ge \frac{r^{-2\sigma - 2\sqrt{\eta}}}{\overline{r}_0^{-2(\sigma-(k-1))- 2\sqrt{\eta}}} \ge \frac{r^{-2\sigma - \sqrt{\eta}}}{r_0^{-2(\sigma-(k-1))-\sqrt{\eta}}}.$$ In other words, we get a gain of $\left( \frac{r}{r_0} \right)^{-\sqrt{\eta}}$, which means a two-ends argument gives an immediate contradiction. Specifically, by [\[eqn:T cap Hr\]](#eqn:T cap Hr){reference-type="eqref" reference="eqn:T cap Hr"} and [\[eqn:non-conc\]](#eqn:non-conc){reference-type="eqref" reference="eqn:non-conc"}, we have for each non-concentrated $T \in \mathcal{T}$, $\nu \times \nu(\{(x, y) : x, y \in T, d(x, y) \ge \left( \frac{r}{r_0} \right)^\gamma \}) \ge \frac{2}{3} \nu(T \cap Y_x)^2 \ge \frac{r^{2\sigma+6\eta}}{r_0^{2\sigma+6\eta-2(k-1)}}$. Thus, by Fubini, there exists a pair $(x, y)$ with $d(x, y) \ge \left( \frac{r}{r_0} \right)^\gamma$ such that $x, y \in T$ for $\gtrsim\frac{r^{2\sigma+6\eta}}{r_0^{2\sigma+6\eta-2(k-1)}} |\mathcal{T}| \ge \left( \frac{r}{r_0} \right)^{-\sqrt{\eta}+6\eta}$ many tubes $T \in \mathcal{T}$. However, since $d(x, y) \ge \left( \frac{r}{r_0} \right)^\gamma$, we have that $x, y$ can only lie in $\lesssim\left( \frac{r}{r_0} \right)^{-(d-1)\gamma}$ many essentially distinct $2r$-tubes. Since $\sqrt{\eta} - 6\eta \ge (d-1)\gamma$, we get a contradiction. Now we focus on the concentrated case: assume there is a subset $\mathbf{X}' \subset \mathbf{X}$ with $\mu(\mathbf{X}') \ge \mu(\mathbf{X})/2$ such that at least half of the tubes in $\mathcal{T}_x$ are concentrated for all $x \in \mathbf{X}'$. This case is where we use the fact that $\nu$ is a $s$-dimensional measure. Let $\mathcal{T}_x'$ denote the concentrated tubes and $\{ B_T : T \in \mathcal{T}_x' \}$ denote the corresponding heavy $\left( \frac{r}{r_0} \right)^\gamma$-balls. Because the family $\mathcal{T}_x$ has $K$-overlap on $\mathrm{spt}(\nu)$, the set $$H' = \{ (x, y) : x \in \mathbf{X}', y \in T \cap B_T \cap Y_x \text{ for some } T \in \mathcal{T}_x' \}$$ has measure $$\begin{gathered} (\mu \times \nu)(H') \gtrsim K^{-1} \cdot \mu(\mathbf{X}') \cdot \inf_{x \in \mathbf{X}'} |\mathcal{T}_x'| \cdot \inf_{x \in \mathbf{X}', T \in \mathcal{T}_x'} \nu(T \cap B_T \cap Y_x) \\ \gtrsim\left( \frac{r}{r_0} \right)^{2\eta} \cdot \frac{r^{-\sigma+3\eta}}{r_0^{-(\sigma-3\eta-(k-1))}} \cdot \frac{r^{\sigma + 3\eta}}{r_0^{\sigma + 3\eta-(k-1)}} = \left( \frac{r}{r_0} \right)^{8\eta}. \end{gathered}$$ Notice that if $(x, y) \in H'$, then there is a tube $T(x, y) \in \mathcal{T}^r$ containing $x, y$ such that $$\nu(B(y, 2(r/r_0)^\gamma) \cap T(x, y)) \gtrsim\frac{r^{\sigma + 3\eta}}{r_0^{\sigma + 3\eta-(k-1)}}.$$ Thus, $\nu$ can't be too concentrated near $y$: $$\nu(B(y, r)) \le K \cdot r^s \le \frac{1}{2} \nu(B(y, 2\left( \frac{r}{r_0} \right)^\gamma) \cap T(x, y)),$$ assuming $4\eta < s - \sigma$ and $k-1 < s$. (The relevant inequalities are $K \le \left( \frac{r}{r_0} \right)^{-\eta}$ and $r^{s-\sigma-3\eta} \le r_0^{s-\sigma-3\eta} \le r_0^{k-1-\sigma-3\eta}$.) Therefore, for each $(x, y) \in H'$, we can choose a dyadic number $r \le \xi(x, y) \le (r/r_0)^\gamma$ such that $$\nu(A(y, \xi(x, y), 2\xi(x, y)) \cap T(x,y)) \ge \left( \frac{r}{r_0} \right)^{\sigma+4\eta} \left( \frac{\xi(x,y)}{(r/r_0)^\gamma} \right)^\eta r_0^k,$$ where the annulus $A(y, \xi, 2\xi) := B(y, 2\xi) \setminus B(y, \xi)$. (One remark: [@orponen2022kaufman] used dyadic pigeonholing at this step, but we can't do this because then we would introduce a $\log r_0^{-1}$ factor. Fortunately, we are allowed to introduce the decaying tail $\left( \frac{\xi(x,y)}{(r/r_0)^\gamma} \right)^\eta$, which is summable in $\xi(x,y)$.) Then, recalling that $(\mu \times \nu)(H') \gtrsim\left( \frac{r}{r_0} \right)^{7\eta}$, we can further find $r \le \xi \le \left( \frac{r}{r_0} \right)^\gamma$ such that $$(\mu \times \nu)(H'') \ge \left( \frac{r}{r_0} \right)^{8\eta} \left( \frac{\xi(x,y)}{(r/r_0)^\gamma} \right)^\eta, \text{ where } H'' = \{ (x, y) \in H' : \xi(x,y) = \xi \} \subset G.$$ By Fubini, we can find $y \in Y$ such that $\mu(H''|^y) \ge \left( \frac{r}{r_0} \right)^{8\eta} \left( \frac{\xi(x,y)}{(r/r_0)^\gamma} \right)^\eta$. Then by construction, $H''|^y$ can be covered by a collection of tubes $\mathcal{T}_y \subset \mathcal{T}^r$ containing $y$ that satisfy $$\nu(A(y, \xi, 2\xi) \cap T) \ge \nu(A(y, \xi(x, y), 2\xi(x, y)) \cap T(x,y)) \ge \left( \frac{r}{r_0} \right)^{\sigma+4\eta} \left( \frac{\xi(x,y)}{(r/r_0)^\gamma} \right)^\eta r_0^k.$$ Finally, we claim that $\mathcal{T}_y$ contains a subset $\mathcal{T}_y'$ whose directions are separated by $\ge (r/\xi)$, such that $|\mathcal{T}_y'| \gtrsim\mu(H''|^y) \cdot r^\eta \cdot \left( \frac{\xi r_0}{r} \right)^\sigma r_0^{-k}$ if $\xi > \frac{r}{r_0}$ and $|\mathcal{T}_y'| \gtrsim\mu(H''|^y) \cdot r^\eta \cdot \left( \frac{\xi r_0}{r} \right)^\sigma r_0^{-k}$ if $\xi > \frac{r}{r_0}$ if $r < \xi < \frac{r}{r_0}$. Indeed, if $\xi > \frac{r}{r_0}$, then any $r/\xi$-tube $\mathbf{T}$ containing $y$ has $$\mu(\mathbf{T}\cap H''|^y) \le \mu(\mathbf{T}\cap A|^y) \le K \cdot \left( \frac{r}{\xi r_0} \right)^\sigma r_0^k \le \left( \frac{r}{r_0} \right)^\eta \cdot \left( \frac{r}{\xi r_0} \right)^\sigma r_0^k.$$ If $\xi < \frac{r}{r_0}$, then any $r/\xi$-tube $\mathbf{T}$ containing $y$ lies in the union of $(\frac{r}{\xi r_0})^k$ many $r_0$-tubes, and so $$\mu(\mathbf{T}\cap H''|^y) \le \mu(\mathbf{T}\cap A|^y) \le K \cdot \left( \frac{r}{\xi r_0} \right)^{-k} r_0^k \le \left( \frac{r}{r_0} \right)^\eta \cdot \left( \frac{r}{\xi} \right)^k.$$ Thus, if $\xi > \frac{r}{r_0}$, then it takes $\gtrsim\mu(H''|^y) \cdot r^\eta \cdot \left( \frac{\xi r_0}{r} \right)^\sigma r_0^{-k}$ many $(r/\xi)$-tubes to cover $H''|^y$, and perhaps even more to cover $\cup \mathcal{T}_y$. We may now choose $\mathcal{T}_y' \subset \mathcal{T}_y$ to be a maximal subset with $(r/\xi)$-separated directions to prove the claim for $\xi > \frac{r}{r_0}$. A similar argument holds for $\xi < \frac{r}{r_0}$. Finally, let's first assume $\xi > \frac{r}{r_0}$. Since $\mathcal{T}^y$ has bounded overlap in $\mathbb{R}^d \setminus B(y, \xi)$, we obtain $$\begin{gathered} \left( \frac{r}{r_0} \right)^{\sigma+13\eta} \left( \frac{\xi}{(r/r_0)^\gamma} \right)^{2\eta} \cdot \left( \frac{\xi r_0}{r} \right)^\sigma r_0^{-k} \\ \lesssim\inf_{T \in \mathcal{T}_y'} \nu(A(y, \xi, 2\xi) \cap T) \cdot |\mathcal{T}_y'| \lesssim\nu(B(y, 2\xi)) \le C \cdot (2\xi)^s. \end{gathered}$$ We will obtain a contradiction if we show the opposite inequality holds, for $\gamma = \frac{15\eta}{s - \max(k, \sigma)}$. Since $2\eta + \sigma < s$ and $\xi \le (\frac{r}{r_0})^\gamma$, it suffices to check $\xi = (\frac{r}{r_0})^\gamma$. If $\xi < \frac{r}{r_0}$, then we obtain $$\begin{gathered} \left( \frac{r}{r_0} \right)^{\sigma+13\eta} \left( \frac{\xi}{(r/r_0)^\gamma} \right)^{2\eta} \cdot \left( \frac{\xi r_0}{r} \right)^k r_0^{-k} \\ \lesssim\inf_{T \in \mathcal{T}_y'} \nu(A(y, \xi, 2\xi) \cap T) \cdot |\mathcal{T}_y'| \lesssim\nu(B(y, 2\xi)) \le C \cdot (2\xi)^s. \end{gathered}$$ Again, since $2\eta + k < s$, it suffices to check $\xi = \frac{r}{r_0}$. This proves the result. ◻ *Proof of Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"}.* By Propositions [Proposition 67](#prop:b1){reference-type="ref" reference="prop:b1"} (with $\frac{r_0}{K_2}$ for $r_0$) and [Proposition 74](#prop:k-thin){reference-type="ref" reference="prop:k-thin"}, there exists a set $B_0 \subset X \times Y$ with $\mu \times \nu(B_0) \lesssim K_1^{-1}$ such that $(\mu, \nu)$ and $(\nu, \mu)$ have $(0, K_1^N r_0^{k-1})$-thin tubes on $B_0^c$ down from scale $r_0$, and $(\mu, \nu)$ and $(\nu, \mu)$ have $(\kappa, K_1 \left( \frac{r_0}{K_2} \right)^{-\kappa})$-thin $k$-plates on $(A \cup B_0)^c$. Then iterate Proposition [Proposition 75](#prop:bootstrap){reference-type="ref" reference="prop:bootstrap"} applied to a uniform $\eta(\sigma, s, \kappa, k, d)$. So initially we have $K = \max(K_1^N K_2^\kappa, K_0 (\eta, k))$, and after each iteration, $K$ becomes $K^{d+1}$. After iterating $\lesssim\eta^{-1}$ many times and letting $B_1 \subset X \times Y$ be the union of the $B$'s outputted from the Proposition (so $\mu \times \nu(B_1) \lesssim K^{-1} \le K_1^{-1}$), we find that $(\mu, \nu)$ and $(\nu, \mu)$ have $(\sigma, K^{(d+1)\eta^{-1}} r_0^{k-1})$-thin tubes on $(A \cup B_0 \cup B_1)^c$. Then we can take $B := B_0 \cup B_1$ to be our desired set. ◻ ## Proof of Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"}, general case {#proof-of-theorem-thmintermediate-general-case} We will prove Theorem [Theorem 13](#thm:intermediate){reference-type="ref" reference="thm:intermediate"}, which we restate here. **Theorem 77**. *Let $k \in \{ 1, 2, \cdots, d-1 \}$, $k-1 < \sigma < s \le k$, and $\varepsilon> 0$. There exist $N, K_0$ depending on $\sigma, s, k$, and $\eta(\varepsilon) > 0$ (with $\eta(1) = 1$) such that the following holds. Fix $r_0 \le 1$, and $K \ge K_0$. Let $\mu, \nu$ be $\sim 1$-separated $s$-dimensional measures with constant $C_\mu, C_\nu$ supported on $E_1, E_2$, which lie in $B(0, 1)$. Assume that $|\mu|, |\nu| \le 1$. Let $A$ be the pairs of $(x, y) \in E_1 \times E_2$ that lie in some $K^{-1}$-concentrated $(r_0, k)$-plate. Then there exists a set $B \subset E_1 \times E_2$ with $\mu \times \nu (B) \lesssim K^{-\eta}$ such that for every $x \in E_1$ and $r$-tube $T$ through $x$, we have $$\nu(T \setminus (A|_x \cup B|_x)) \lesssim\frac{r^\sigma}{r_0^{\sigma-(k-1)+N\varepsilon}} K^N.$$ The implicit constant may depend on $C_\mu, C_\nu, \sigma, s, k$.* **Remark 78**. *Note that in Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"}, we demand the stronger conclusion $\mu \times \nu(B) \lesssim K^{-1}$.* The idea is to apply Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"} at different scales. As a start, if $\varepsilon= 1$, then we can directly apply Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"} with $K_1 = K_2 = K$ (and thus we may take $\eta(1) = 1$). We may assume $\varepsilon= \frac{1}{M}$ for some $M$. Let $N$ be the large constant in Lemma [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"}, and let $\eta_n = (N+2)^{n-M}$. For $1 \le n \le M$, let $A_n$ be the pairs of $(x, y) \in E_1 \times E_2$ that lie in some $K^{-\eta_n}$-concentrated $(r_0^{n\varepsilon}, k)$-plate. We remark that $A_M = A$. **Lemma 79**. *Fix $n \ge 1$. There exists a set $B_n \subset A_n$ with $\mu \times \nu(B_n) \lesssim K^{-\eta_n}$ such that for every $x \in E_1$ and $r$-tube through $x$ that intersects $A_n|_x$, we have $$\nu(T \setminus (A_{n+1}|_x \cup B|_x)) \lesssim\frac{r^\sigma}{r_0^{n\varepsilon(\sigma-(k-1)) + N\varepsilon}} K^N.$$* *Proof.* By Lemma [Lemma 22](#lem:concentrated_points){reference-type="ref" reference="lem:concentrated_points"}, there exists an absolute constant $C$ such that every $r$-tube through some $(x, y) \in A_n$ lies in some $K^{-\eta_n}$-concentrated $(CK^{-n}, k)$-plate. We can find a collection $\mathcal{H}$ of essentially distinct $K^{-\eta_n}$-concentrated $(2CK^{-n}, k)$-plates such that each $K^{-\eta_n}$-concentrated $(CK^{-n}, k)$-plate is contained within some element of $\mathcal{H}$. By Lemma [Lemma 69](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"}, $|\mathcal{H}| \lesssim K^{-N\eta_n}$. By construction, every $r$-tube through some $(x, y) \in A_n$ is contained in some member of $\mathcal{H}$. Apply Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"} to each $H \in \mathcal{H}$ with measures $\mu|_H, \nu|_H$ and $K_1 \rightarrow K^{-\eta_{n+1}}$, $K_2 \rightarrow 2Cr_0^{-\varepsilon}$, and $r_0 \rightarrow r_0^{n\varepsilon}$ to obtain a set $B_H$ with $\mu \times \nu(B_H) \lesssim K^{-\eta_{n+1}}$. Let $B_n = \cup_{H \in \mathcal{H}} B_H$, and then $\mu \times \nu(B_n) \le K^{N\eta_n} \cdot K^{-\eta_{n+1}} < K^{-\eta_n}$ since $(N+1)\eta_n < \eta_{n+1}$. ◻ *Proof of Theorem [Theorem 77](#thm:intermediate_refined){reference-type="ref" reference="thm:intermediate_refined"}.* Let $B = \cup_{n=1}^M B_n$; then $\mu \times \nu(B) \le K^{-\eta_0}$. Fix an $r$-tube $T$ and $x \in E_1$. Let $n \le M-1$ be the largest number such that $T$ passes through points in $A_n|_x$. Then by Lemma [Lemma 79](#lem:intermediate_r){reference-type="ref" reference="lem:intermediate_r"}, we have $\nu(T \setminus (A_{n+1}|_x \cup B|_x)) \lesssim\frac{r^\sigma}{K^{-n (\sigma-(k-1)}} K^N$. If $m < M-1$, then $T \cap A_{n+1}|_x = \emptyset$. In any case, we have $\nu(T \setminus (A|_x \cup B|_x)) \lesssim\frac{r^\sigma}{r_0^{\sigma-(k-1)+N\varepsilon}} K^N$, completing the proof of Theorem [Theorem 77](#thm:intermediate_refined){reference-type="ref" reference="thm:intermediate_refined"}. ◻ # Corollaries of Radial Projection Estimates {#sec:corollaries} We prove a variant of Corollary [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"}. **Proposition 80**. *Fix $s \in (k-1, k]$ and $\eta > 0$. Let $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$ be measures with $\mathcal{E}_s (\mu), \mathcal{E}_s (\nu) < \infty$ and $\sim 1$-separated supports. Suppose that $\mu(H) = \nu(H) = 0$ for each $k$-plane $H \in \mathbb{A}(\mathbb{R}^d, k)$. Then for $\mu$-almost all $x$, for all sets $Y$ of positive $\nu$-measure, $$\dim_H (\pi_x Y) \ge s - \eta.$$* *Proof.* The proof is standard and follows [@shmerkin2022dimensions Proof of Proposition 6.9]. By Lemma [Lemma 44](#lem:energy){reference-type="ref" reference="lem:energy"}, by passing to subsets of nearly full measure and replacing $s$ by an arbitrary $s' < s$, we may assume that $\mu(B_r), \nu(B_r) \lesssim r^s$ for all $r \in (0, 1]$. Fix $\varepsilon> 0$. By a compactness argument, there exists $r_0 > 0$ such that $\mu(H), \nu(H) < \varepsilon$ for all $(r_0, k)$-plates $H$. In Theorem [Theorem 73](#thm:threshold_refined){reference-type="ref" reference="thm:threshold_refined"}, we know that for $\varepsilon> 0$ sufficiently small, the set $A = \emptyset$. Thus, there exists $B \subset X \times Y$ with $\mu \times \nu(B) \lesssim\varepsilon$ such that for every $x \in X$ and $r$-tube through $x$, we have $$\nu(T \setminus B|_x) \lesssim_{\eta,\varepsilon,s} r^{s - \eta}.$$ Thus, there is a set $X$ with $\mu(X) > 1-O(\varepsilon)$ such that if $x \in X$, then $$\dim_H (\pi_x Y) \ge s - \eta \text{ for all } Y \text{ with } \nu(Y) \ge O(\varepsilon).$$ Taking $\varepsilon\to 0$ completes the proof. ◻ Using this, we prove Corollary [Corollary 2](#cor:shm_conj_2){reference-type="ref" reference="cor:shm_conj_2"}. **Corollary 81**. *Let $s \in (d-2, d]$, then there exists $\varepsilon(s, d) > 0$ such that the following holds. Let $\mu, \nu$ be Borel probability measures on $\mathbb{R}^d$ with disjoint supports that satisfy $\mathcal{E}_s (\mu), \mathcal{E}_s (\nu) < \infty$ and $\dim_H (\mathrm{spt}(\nu)) < s + \varepsilon(s, d)$. Further, assume that $\mu, \nu$ don't simultaneously give full measure to any affine $(d-1)$-plane $H \subset \mathbb{R}^d$. Then there exist restrictions of $\mu, \nu$ to subsets of positive measure (which we keep denoting $\mu, \nu$) such that the following holds. For almost every affine 2-plane $W \subset \mathbb{R}^d$ (with respect to the natural measure on the affine Grassmanian), if the sliced measures $\mu_W$, $\nu_W$ on $W$ is non-trivial, then they don't simultaneously give full measure to any line. In other words, $$(\gamma_{d,2} \times \mu) \{ (V, x) : \mu_{V,x} (\ell) \nu_{V,x} (\ell) = |\mu_{V,x}| |\nu_{V,x}| > 0 \text{ for some } \ell \in \mathbb{A}(V + x, 1) \} = 0$$ where we parametrize affine 2-planes as $V + x$, for $x \in \mathbb{R}^d$ and $V$ in the Grassmannian $\mathrm{Gr}(d, 2)$ with the rotationally invariant Haar measure $\gamma_{d,2}$.* *Proof.* First, if $\mu(H) > 0$ for some affine $(d-1)$-plane $H$, then $\nu(H^c) > 0$ where $H^c$ denotes the complement of $H$ in $\mathbb{R}^d$. By restricting $\mu$ to $H$ and $\nu$ to $H^c$ (and calling the results $\mu, \nu$), we see that the sliced measures $\mu_W$ and $\nu_W$ can't give full mass to any line $\ell$ for any affine $(d-1)$-plane $W$, for the simple reason that $\mu_W (\ell) > 0$ forces $\ell \subset H$, and $\nu_W (\ell) > 0$ forces $\ell \subset H^c$. Likewise, we are done if $\nu(H) > 0$ for some affine $(d-1)$-plane $H \subset \mathbb{R}^d$. Thus, assume $\mu(H) = \nu(H) = 0$ for all affine $(d-1)$-planes $H$. With this assumption, the remainder of the proof is nearly identical to the proof of Proposition 6.8 in [@shmerkin2022dimensions], except using Proposition [Proposition 80](#cor:shm_conj'){reference-type="ref" reference="cor:shm_conj'"} instead of [@shmerkin2022dimensions Proposition 6.9]. One can take $\varepsilon(s, d)$ to be arbitrarily close to $s - (d-2)$. ◻ Finally, we can deduce Theorem [Theorem 1](#cor:shm_conj){reference-type="ref" reference="cor:shm_conj"} from either Proposition [Proposition 80](#cor:shm_conj'){reference-type="ref" reference="cor:shm_conj'"} or Proposition [Corollary 81](#cor:shm_conj_2'){reference-type="ref" reference="cor:shm_conj_2'"}, see [@orponen2022kaufman Section 4] for details. The only case not yet considered in this paper is when either $\mu, \nu$ gives positive mass to a $k$-plane. But this special case was considered in [@orponen2022kaufman Section 4] (briefly, if $X$ gives positive mass to some $k$-plane, then radial projections become orthogonal projections and then we apply Kaufman's projection theorem; if $Y$ gives positive mass to some $k$-plane $H$, then for $x \notin H$, we have $\dim_H (\pi_x (Y)) = \dim_H (Y)$.) # Proof of Balog-Szemerédi-Gowers {#appendix:proof of bsg} By a standard covering argument (e.g. see Section 3 of [@katz2001some]), Theorem [Theorem 41](#thm:bsg){reference-type="ref" reference="thm:bsg"} follows from the case $\delta = 0$, which we prove below. **Theorem 82** (refined Theorem 4.1 of [@sudakov2005question]). *Let $K \ge 1$ be a parameter. Let $A, B$ be finite subsets of $\mathbb{R}^n$, and let $P \subset A \times B$ satisfy $|P| \ge K^{-1} |A| |B|$. Suppose that $|A \overset{P}{+}B| \le K (|A| |B|)^{1/2}$, where $A \overset{P}{+}B = \{ a + b : (a, b) \in P \}$. Then one can find subsets $A' \subset A, B' \subset B$ with $|A'| \ge \frac{1}{16K^2} |A|, |B'| \ge \frac{1}{16K^2} |B|$ such that $|A' + B'| \le 2^{12} K^8 (|A| |B|)^{1/2}$ and $|P \cap (A' \times B')| \ge \frac{|A| |B|}{16K^2}$.* *Proof.* We follow the exposition in [@sheffer2016balog]. **Claim.** There exist subsets $A' \subset A, B' \subset B$ with $|P \cap (A' \times B')| \ge \frac{|A| |B|}{16K^2}$, such that for each $a \in A', b \in B'$, there are $\ge \frac{|A||B|}{2^{12} K^5}$ many pairs $(a', b') \in A \times B$ such that $(a, b')$, $(a', b')$, and $(a', b) \in P$. Assuming the claim, we will see how the theorem follows. First, we get $|A'| |B'| \ge |P \cap (A' \times B')| \ge \frac{|A| |B|}{16K^2}$. Since $|A'| \le |A|$ and $|B'| \le |B|$, we get $|A| \ge \frac{|A|}{16K^2}$ and $|B| \ge \frac{|B|}{16K^2}$. Next, for $a \in A', b \in B'$, we have $$a + b = (a + b') - (a' + b') + (a' + b).$$ Thus, there are $\ge |A| |B| 2^{-12} K^{-5}$ many solutions to $a + b = x - y + z$ with $x, y, z \in A \overset{P}{+}B$. Since $|A \overset{P}{+}B| \le K(|A| |B|)^{1/2}$, we get $|A' + B'| \lesssim\frac{K^3 (|A| |B|)^{3/2}}{|A| |B| 2^{-12} K^{-5}} = 2^{12} K^8 |A|^{1/2} |B|^{1/2}$. Now, we prove the claim. For convenience, we can prune $P$ to satisfy $|P| = K^{-1} |A| |B|$ (this is not necessary but will make the proof look nicer). Treat $(A \cup B, P)$ as a bipartite graph with an edge between $a \in A$ and $b \in B$ if $(a, b) \in P$. Then we want to find $A', B'$ such that there are many paths of length $3$ between any $a \in A', b \in B'$. The average degree of a vertex in $A$ is $K^{-1} |B|$. Thus, if we delete the vertices in $A$ with degree $\le \frac{1}{2} K^{-1} |B|$, then at least $\frac{1}{2K} |A| |B|$ many edges remain. Let $E$ be the set of edges. For $v \in A \cup B$, let $N(v)$ be the set of neighbors of $v$. Now pick a vertex $b \in B$. On average, it has $\frac{|E|}{|B|} \ge \frac{1}{2K} |A|$ many neighbors. Now, we say $(a, a') \in A^2$ is bad if $|N(a) \cap N(a')| < \frac{1}{128K^3} |B|$. For $v \in B$, let $\mathrm{Bad}_v$ be the set of bad pairs in $N(v)^2$. There are $\binom{|A|}{2}$ many pairs in $A$, so (expectation is taken over uniformly chosen $v \in B$) $$\mathbb{E}[|\mathrm{Bad}_v|] < \binom{|A|}{2} \cdot \frac{1}{128K^3} < \frac{|A|^2}{256K^3}.$$ If $A_{bad,v}$ is the set of vertices of $A$ that lie in at least $\frac{|A|}{32K^2}$ many pairs of $B_v$, then $$\mathbb{E}[|A_{bad,v}|] \le \frac{2\mathbb{E}[|B_v|]}{|A|/(32K^2)} < \frac{|A|}{4K}.$$ Finally, let $A_v = N(v) \setminus A_{bad,v}$. Then by linearity of expectation, $$\mathbb{E}[|A_v|] = \mathbb{E}[|N(v)|] - \mathbb{E}[|A_{bad,v}|] > \frac{|A|}{2K} - \frac{|A|}{4K} = \frac{|A|}{4K}.$$ Thus, there exists $v \in B$ such that $|A_v| > \frac{|A|}{4K}$. Then, let $A' = A_v$ and $$B' = \{ w \in B : |N(w) \cap A'| \ge \frac{|A|}{16K^2}.$$ Let $E(X, Y)$ be the number of edges between $X$ and $Y$. We first check that $E(A', B') \ge \frac{|A||B|}{16K^2}$. Indeed, since every vertex of $A$ has degree $\ge \frac{|B|}{2K}$, we have $$|E(A', B)| \ge \frac{|A'| |B|}{2K} \ge \frac{|A||B|^2}{8K^2}.$$ On the other hand, every vertex in $B \setminus B'$ corresponds to fewer than $\frac{|A|}{16K^2}$ many edges of $A'$, so $|E(A', B \setminus B')| \le \frac{|A||B|^2}{16K^2}$. Hence, $|E(A', B')| \ge \frac{|A||B|^2}{16K^2}$. Finally, for any $v \in A'$, $w \in B'$, we know that $w$ has at least $\frac{|A|}{16K^2}$ many neighbors in $A'$, and fewer than $\frac{|A|}{32K^2}$ of those form a bad pair with $w$. For the remaining $\ge \frac{|A|}{32K^2}$ vertices $v'$ that do not form a bad pair with $w$, there are $\ge \frac{|B|}{128K^3}$ many vertices $w' \in B$ that are common neighbors of $v, v'$. Thus, we get at least $\frac{|A|}{32K} \cdot \frac{|B|}{128K^3} = \frac{|A||B|}{2^{12} K^5}$ many paths $(v, w', v', w)$ between $v$ and $w$. ◻

arxiv_math

--- abstract: | We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the rôle of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators. **MSC: 58A10, 58A20, 83D05** **Keywords:** Jet bundle, Poincaré--Cartan form, Lepage equivalent of a Lagrangian, variational bicomplex author: - D.J. Saunders title: Lepage equivalents and the Variational Bicomplex --- # Dedication {#dedication .unnumbered} In the Notes to Chapter 5 of [@Olv86], Peter Olver wrote about the variational complex and the variational bicomplex 'It is hoped that these methods will inspire further research in the geometric theory of the calculus of variations'. A few years later [@Olv93] he wrote 'In the geometric theory of the calculus of variations in mechanics, the Cartan form, which first arose as the integrand in Hilbert's invariant integral, plays a ubiquitous role'. Lepage equivalents are generalizations of Cartan forms, and I hope that this paper will be a small contribution to Peter's project. # Introduction {#Sintro} In recent years there has been a revival of interest in the 'fundamental Lepage equivalent' of a Lagrangian, a differential form on a jet bundle which (as with any such Lepage equivalent) provides a geometrical construction leading to the Euler--Lagrange equations of the corresponding variational problem, but which has the additional property that it is closed precisely when the Lagrangian is null [@Pal22; @Urb22]. The original formulation of the fundamental Lepage equivalent was given for first order Lagrangians (in [@Kru77], and then independently in [@Bet84]). Although expressed in local coordinates, the form is in fact invariant under changes of coordinates, and so is a global geometric object. There had, however, been been no similar construction for higher order Lagrangians. A construction for Lagrangians of arbitrary order has now been proposed in [@Voi22], giving a Lepage form of order no greater than $4k-2$ for a Lagrangian of order $k$. The construction is again given in local coordinates, but now there is no guarantee that it will be defined globally. In addition, if the original Lagrangian happens to be first order, the new Lepage form will in general be of second order and will differ from the original, first order, fundamental Lepage equivalent. In the first part of this paper, after giving some background on the different types of Lepage equivalent, we propose a new method of constructing a fundamental Lepage equivalent for a Lagrangian of arbitrary order by using homotopy operators for the horizontal differential in the variational bicomplex on the infinite jet manifold. This has the disadvantage that any bound on the order of the resulting Lepage form, although necessarily finite, will depend on the number of independent variables. On the other hand, the choice of a symmetric linear connection will allow the construction of a global form, and in the case of a first order Lagrangian the result will be independent of any connection and we recover the classic fundamental Lepage equivalent. In the second part of the paper we explore the potential for clarifying this construction by using 'vertical endomorphisms' on jet bundles, tensorial objects depending on a closed differential form, which are related to the canonical isomorphism between the tangent space at any point of an affine space, and the vector space on which the affine space is modelled [@Sau87]. We recall how local homotopy operators for the horizontal differential can be constructed from these vertical endomorphisms, and then we show how a symmetric linear connection can be used to remove the dependence on the differential form to produce a globally-defined, fully tensorial object. (A related but technically different approach has been described in [@Bet93].) Such a 'vertical tensor' can be used to give an infinitesimal rigidity to nonholonomic jet bundles, allowing the construction of a global Lepage equivalent for a Lagrangian of arbitrary order. Finally we offer a conjecture regarding how these vertical tensors, together with a covariant version of the horizontal differential, might be used to construct a global homotopy operator for the ordinary horizontal differential. # Notation {#Snot} We adopt a modified version of the notation used in [@Sau89]. We let $\pi : E \to M$ be a fibred manifold with $\dim M = m$ and $\dim E = m + n$. The $k$-jet manifold of $\pi$ will be denoted $J^k\pi$ with projections $\pi_k: J^k\pi\to M$, $\pi_{k,0}: J^k\pi\to E$ and $\pi_{k,l}: J^k\pi\to J^l\pi$ where $l < k$. A typical element of $J^k\pi$ will be denoted $j^k_p\phi$. We use similar notation for jets of the cotangent bundle $\tau : T^* M \to M$. We let $\mathfrak{X}(J^k\pi)$ denote the module of vector fields on $J^k\pi$, and $\Omega^r(J^k\pi)$ the module of $r$-forms. Regarding the jet bundle $\pi_{k-1}: J^{k-1}\pi\to M$ as the starting bundle, we shall let $(\pi_{k-1})_1: J^1\pi_{k-1}\to M$ denote its first jet bundle, and we let $\mathrm{i}_{1,k-1}: J^k\pi\to J^1\pi_{k-1}$ be the canonical inclusion. There is also an intermediate submanifold $\widehat{J}^k\pi\subset J^1\pi_{k-1}$, the semiholonomic manifold [@Lib97], with a canonical symmetrization projection $\mathrm{p}_k : \widehat{J}^k\pi\to J^k\pi$. We also use the infinite jet bundle $\pi_\infty: J^\infty \pi\to M$ where $J^\infty \pi$, although infinite dimensional, is a Fréchet manifold and so is reasonably well behaved. We let $\Omega^r$ (without specifying a manifold) denote the module of $r$-forms on $J^\infty \pi$ of globally finite order, so that each such form is projectable to some $J^k\pi$. Any differential form $\omega\in \Omega^r$ can be decomposed uniquely into its contact components $$\omega= \omega^{(0)} + \omega^{(1)} + \cdots + \omega^{(p)} + \cdots + \omega^{(r)}$$ where if $r > m$ then $\omega^{(p)} = 0$ for $p < r-m$. We let $\Omega^{p,q} \subset \Omega^r$ with $p+q=r$ denote the submodule of $p$-contact $r$-forms. In a similar way, a differential form $\omega\in \Omega^r(J^k\pi)$ on a finite order jet manifold may be decomposed into contact components, but these will normally be defined on $J^{k+1}\pi$ rather than on $J^k\pi$. When using coordinates, we take fibred coordinates $(x^i,u^\aG)$ on $E$ over base coordinates $(x^i)$ on $M$. Jet coordinates will be denoted $(u^\aG_i)$ on $J^1\pi$ and $(u^\aG_i, u^\aG_{(ij)})$ on $J^2\pi$ with parentheses denoting symmetrization because $u^\aG_{(ji)}$ is the same coordinate as $u^\aG_{(ij)}$. For this reason we use the symbol $\#(ij)$ to equal $1$ when $i=j$ and to equal $2$ when $i \ne j$, in order to avoid double counting during summation. (We use the standard summation convention for repeated indices.) On higher order jet manifolds this notation becomes unwieldy and we write $(u^\aG_I)$ instead, where $I \in \mathbb{N}^m$ is a multi-index with $I(i)$ giving the number of copies of the index $i$, so that this notation automatically takes care of symmetrization. We let $1_i$ denote the multi-index with a single $1$ in the $i$-th position; $\lvert I \rvert = \sum_{i=1}^n I(i)$ is the length of $I$, and $I! = I(1)! I(2)! \cdots I(m)!$ is its factorial. Any summation involving multi-indices will be indicated explicitly, including the zero multi-index where appropriate. Sometimes we need to use a mixed notation, and converting to or from multi-index notation requires coefficients to be adjusted. If $F(J)$ is some object depending on the multi-index $J$ then $$\sum_{\lvert J \rvert = r+1} \frac{\lvert J \rvert!}{J!} F(J) = \sum_{i=1}^m \sum_{\lvert I \rvert = r} \frac{\lvert I \rvert!}{I!} F(I+1_i)$$ where the quotient $\lvert J \rvert! / J!$ is the 'weight' of the multi-index $J$. We use notation $$\theta^\aG = du^\aG - u^\aG_j dx^j \, , \qquad \theta^\aG_i = du^\aG_i - u^\aG_{(ij)} dx^j \, , \qquad \theta^\aG_I = du^\aG_I - u^\aG_{I+1_j} dx^j$$ for local contact $1$-forms, and $$\omega_0 = dx^1 \wedge \cdots \wedge dx^m \, , \qquad \omega_i = i_{\partial/ \partial x^i} \, \omega_0 = (-1)^{i-1} dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^m$$ (where the circumflex indicates an omitted factor) for local forms horizontal over $M$. Local total derivatives, dual to the local contact forms, will be denoted $d_i$ and are given explicitly as $$\frac{\partial}{\partial x^i} + u^\aG_i \frac{\partial}{\partial u^\aG} \, , \qquad \frac{\partial}{\partial x^i} + u^\aG_i \frac{\partial}{\partial u^\aG_i} + u^\aG_{(ij)} \frac{\partial}{\partial u^\aG_j} \, , \qquad \frac{\partial}{\partial x^i} + \sum_I u^\aG_{I+1_i} \frac{\partial}{\partial u^\aG_I} \, .$$ In the finite order case they are vector fields along the map $\pi_{k,k-1}$ rather than on a single jet manifold. We also use the symbol $\partial_i$ to indicate $\partial/ \partial x^i$ as a vector field along the map $\pi_k$. On a nonholonomic jet manifold we need to distinguish between the two levels of jet coordinates, and we use juxtaposition, with a dot to indicate when a particular index is missing. So on $J^1\pi_1$ the coordinates are $(x^i, u^\aG_{\cdot\cdot}, u^\aG_{i \cdot}, u^\aG_{\cdot j}, u^\aG_{ij})$ and on $J^1\pi_{k-1}$ they are $(x^i, u^\aG_{I\cdot}, u^\aG_{Ij})$. Finally we note that $\pi_{1,0}: J^1\pi\to E$ is an affine bundle, modelled on the vector bundle $\pi^* T^* M \otimes V\pi$, so that the vertical bundle $V\pi_{1,0}$ is canonically isomorphic to $\pi_1^* T^* M \otimes \pi_{1,0}^* V\pi$; the inverse of this isomorphism may be regarded as a tensor field $$S = \partial_i \otimes \theta^\aG \otimes \frac{\partial}{\partial u^\aG_i} \, ,$$ a section of the bundle $\pi_1^* TM \otimes T^* J^1\pi\otimes TJ^1\pi$ over $J^1\pi$. We shall call this the *first order vertical tensor*. # Background {#Sback} Many of the results in the geometrical calculus of variations can be described in terms of *source forms* and *Lepage equivalents* (see [@KKS10] for a useful summary and historical references). A source form is a form $\varepsilon\in \Omega^{m+1}(J^l\pi)$ with the properties that it is horizontal over $E$, and maximally horizontal over $M$, so that in coordinates it appears as $\varepsilon= \varepsilon_\aG \theta^\aG \wedge \omega_0$. The zero set of a source form is a submanifold of $J^l\pi$ representing a family of partial differential equations; if $\lambda= L \, \omega_0 \in \Omega^m(J^k\pi)$ is a horizontal $m$-form, a Lagrangian, then it gives rise to a source form $\varepsilon_\lambda\in \Omega^{m+1}(J^{2k}\pi)$, the *Euler--Lagrange form* of $\lambda$, incorporating the Euler--Lagrange equations of the variational problem defined by $\lambda$: $$\varepsilon_\lambda= \sum_{\lvert I \rvert=0}^k (-1)^{\lvert I \rvert} d_I \biggl( \frac{\partial L}{\partial u^\aG_I} \biggr) \theta^\aG \wedge \omega_0 \, .$$ A Lepage form is a form $\vartheta\in \Omega^m(J^l\pi)$ with the property that $(d\vartheta)^{(1)}$, the $1$-contact component of its exterior derivative, is a source form. A Lepage equivalent of a Lagrangian $\lambda\in \Omega^m(J^k\pi)$ is a Lepage form $\vartheta_\lambda\in \Omega^m(J^l\pi)$ with $l \ge k$ such that the difference $\vartheta_\lambda- \pi_{l,k}^* \lambda$ is a contact form; the corrresponding source form $(d\vartheta_\lambda)^{(1)}$ is then just the Euler--Lagrange form $\varepsilon_\lambda$. Different Lepage equivalents of the same Lagrangian give the same Euler--Lagrange form. If $m = 1$ then each Lagrangian $\lambda$ gives rise to a unique globally-defined Lepage equivalent $\vartheta_\lambda\in \Omega^1(J^{2k-1}\pi)$, the Cartan form of the Lagrangian. However, complications arise when $m \ge 2$, and these concern both existence and uniqueness. Clearly if $\vartheta_\lambda$ is a Lepage equivalent of $\lambda$ then so is $\vartheta_\lambda+ d\psi + \omega$ where $\omega$ is at least $2$-contact, and in fact the converse is true: if $\vartheta_\lambda$, $\vartheta_\lambda^\prime$ are both Lepage equivalents of $\lambda$ then $\theta_\lambda^\prime - \theta_\lambda= d\psi + \omega$. As far as existence is concerned, if we initially consider forms which are at most $1$-contact then locally $$\vartheta_\lambda= L \, \omega_0 + \sum_{\lvert J \rvert=0}^{k-1} \sum_{\lvert K \rvert=0}^{k-\lvert J \rvert-1} \frac{(-1)^{\lvert J \rvert} (J+K+1_j)! \lvert J \rvert! \lvert K \rvert!}{(\lvert J \rvert+\lvert K \rvert+1)! J! K!} d_J \biggl( \frac{\partial L}{\partial u^\aG_{J+K+1_j}} \biggr) \theta^\aG_K \wedge \omega_j$$ is a Lepage equivalent, known as the *principal Lepage equivalent* of $\lambda$. When $k=1$ this is just the Poincaré--Cartan form of $\lambda$, $$\vartheta_\lambda= L \, \omega_0 + \frac{\partial L}{\partial u^\aG_j} \theta^\aG \wedge \omega_j$$ and is defined globally; it is the unique Lepage equivalent of $\lambda$ which is both at most $1$-contact and also horizontal over $E$. When $k=2$ we obtain $$\vartheta_\lambda= L \, \omega_0 + \biggl( \biggl( \frac{\partial L}{\partial u^\aG} - \frac{1}{\#(ij)} d_i \biggl( \frac{\partial L}{\partial u^\aG_{(ij)}} \biggr) \biggr) \theta^\aG + \frac{1}{\#(ij)} \frac{\partial L}{\partial u^\aG_{(ij)}} \theta^\aG_i \biggr) \wedge \omega_j$$ which again, perhaps surprisingly, is invariant under a fibred change of coordinates $\tilde{x}= \tilde{x}^j(x^i)$, $\tilde{u}^\beta= \tilde{u}^\beta(x^i, u^\aG)$ and is therefore also defined globally. For $k \ge 3$, however, there is no such Lepage equivalent invariant under coordinate changes [@HK83]; choices need to be made in order to obtain a globally defined form. Several authors (see, for instance, [@FF83; @GM83]) have used connections of various kinds for this purpose Another approach [@Sau87] has been to 'pretend' that the $k$-th order Lagrangian is really first order by using a tubular neighbourhood of $J^k\pi$ in $J^1\pi_{k-1}$ and 'spreading out' the Lagrangian using the neigbourhood's projection. By repeating this process, a global Lepage equivalent may be constructed. In fact only infinitesimal projections are needed, mapping $T_{J^k\pi}J^1\pi_{k-1}$ to $TJ^k\pi$, and we shall see in Section [7](#Sproj){reference-type="ref" reference="Sproj"} that a symmetric linear connection on $M$ determines a suitable family of projections. In the second order case, it may be seen that only the restriction of the projection to the semiholonomic manifold $\widehat{J}^2\pi$ is needed, explaining why symmetrization projection $\mathrm{p}_2$ may be used to give a global Lepage equivalent in this case. The Lepage equivalents described so far have all been at most $1$-contact. There have, however, been important examples of Lepage equivalents involving higher contact terms. One such, defined for a nonvanishing first order Lagrangian, is the *Carathéodory form* [@Car29] $$\vartheta_\lambda= \frac{1}{L^{m-1}} \bigwedge_{j=1}^m \biggl( L \, dx^j + \frac{\partial L}{\partial u^\aG_j} \theta^\aG \biggr) \, ;$$ this decomposable form is again defined globally and indeed is invariant, not just under a fibred change of coordinates, but under a general change $\tilde{x}= \tilde{x}^j(x^i, u^\aG)$, $\tilde{u}^\beta= \tilde{u}^\beta(x^i, u^\aG)$. A similar form for a nonvanishing second order Lagrangian, $$\vartheta_\lambda= \frac{1}{L^{m-1}} \bigwedge_{j=1}^m \biggl( L \, dx^j + \biggl( \frac{\partial L}{\partial u^\aG_j} - \frac{1}{\#(ij)} d_i \biggl( \frac{\partial L}{\partial u^\aG_{(ij)}} \biggr) \biggr) \theta^\aG + \frac{1}{\#(ij)} \frac{\partial L}{\partial u^\aG_{(ij)}} \theta^\aG_i \biggr)$$ was described in [@Olv93] (see also [@CS04]). The Lepage equivalent of particular interest in the present paper, again involving higher contact terms, is the *fundamental Lepage equivalent* of a first order Lagrangian [@Bet84; @Kru77] $$\vartheta_\lambda= \sum_{p=0}^{\min\{m,n\}} \frac{1}{(p!)^2} \, \frac{\partial^p L}{\partial u^{\aG_1}_{j_1} \cdots \partial u^{\aG_p}_{j_p}} \, \theta^{\aG_1} \wedge \cdots \wedge \theta^{\aG_p} \wedge \omega_{j_1 \cdots j_p} \, .$$ This satisfies the *closure property*, that $d\vartheta_\lambda= 0$ precisely when the Lagrangian is null: that is, when the Euler--Lagrange form $\varepsilon_\lambda$ is zero. (Of course any individual form $\vartheta_\lambda$ is either closed or not closed; the closure property applies to the procedure mapping $\lambda$ to $\vartheta_\lambda$.) Once again this form is defined globally, and indeed is invariant under a general change of coordinates $\tilde{x}= \tilde{x}^j(x^i, u^\aG)$, $\tilde{u}^\beta= \tilde{u}^\beta(x^i, u^\aG)$ [@CS05]. The content of the closure property lies in the requirement that $d\vartheta_\lambda= 0$ when $\lambda$ is null; for any Lepage equivalent $\vartheta_\lambda$ it is obvious that the converse holds, that $\lambda$ is null when $d\vartheta_\lambda= 0$. In the next Section we consider how the construction of the fundamental Lepage equivalent might be generalised for higher order Lagrangians. # The closure property {#Sclose} The question of whether it possible to find a procedure for constructing a Lepage equivalent which satisfies the closure property, although solved in 1977 for first order Lagrangians, has been an open problem for higher order Lagrangians (see [@Pal22; @Urb22] and the references therein). An original solution to this problem was given in [@Voi22], using the Vainberg--Tonti Lagrangian of a source form $\varepsilon$, the horizontal $m$-form $\lambda_\varepsilon$ obtained locally in coordinates from $\varepsilon= \varepsilon_\aG \theta^\aG \wedge \omega_0$ by the fibred homotopy operator $$\lambda_\varepsilon= u^\aG \int_0^1 \varepsilon_\aG(x^i, tu^\beta_I) dt \, .$$ Typically $\lambda_\varepsilon$ has the same order as $\varepsilon$. If in fact $\varepsilon= \varepsilon_\lambda$, so that the source form is the Euler--Lagrange form of a given Lagrangian $\lambda$, then the Vainberg--Tonti Lagrangian $\lambda_{\varepsilon_\lambda}$ and the pullback of $\lambda$ have the same Euler--Lagrange equations so that they differ by $h(d\aG)$ for some horizontal $(m-1)$-form $\aG$. Then, taking $\vartheta_{\lambda_{\varepsilon_\lambda}}$ to be the principal Lepage equivalent of the Vainberg--Tonti Lagrangian in the given coordinates and writing $\vartheta^\mathrm{F}= \vartheta_{\lambda_{\varepsilon_\lambda}}+ d\aG$ we find that, to within pullbacks, $$(d\vartheta^\mathrm{F})^{(1)} = d\vartheta_{\lambda_{\varepsilon_\lambda}}^{(1)} = \varepsilon_{\lambda_{\varepsilon_\lambda}} = \varepsilon_\lambda$$ so that $\vartheta^\mathrm{F}$ is a source form, and that $$h(\vartheta^\mathrm{F}) = h(\vartheta_{\lambda_{\varepsilon_\lambda}}) + h(d\aG) = \lambda_{\varepsilon_\lambda} + h(d\aG) = \lambda$$ so that $\vartheta^\mathrm{F}$ is a Lepage equivalent of $\lambda$, and finally that $$d\vartheta^\mathrm{F}= d\vartheta_{\lambda_{\varepsilon_\lambda}}$$ so that if $\lambda$ is a null Lagrangian then $\varepsilon_\lambda= 0$ and therefore $d\vartheta^\mathrm{F}= 0$. This procedure therefore satisfies the closure property. It is not, though, a generalisation of the fundamental Lepage equivalent for first order Lagrangians, because it is always at most $1$-contact, whereas the Fundamental Lepage equivalent is obtained from the Poincaré--Cartan form by adding higher contact terms. We shall, instead, define an alternative procedure which uses homotopy operators for the horizontal differential of the variational bicomplex to add the higher contact terms. Recall that this bicomplex is defined for forms of globally finite order on the infinite jet manifold $J^\infty \pi$, as shown in the diagram below (note that the squares with vertical arrows labelled $\pi_\infty^*$ commute, whereas those with vertical arrows labelled $d_\mathrm{v}$ anticommute.) The rows and columns are all locally exact, and indeed all the $d_\mathrm{h}$ rows apart from the first are globally exact [@And89; @Tak79; @Tuj82; @Tul80; @Vin78]. Any Lagrangian $\lambda\in \Omega^m(J^k\pi)$ will have a pullback $\pi_{\infty,k}^* \lambda\in \Omega^{0,m}$ on $J^\infty \pi$ which for simplicity we shall continue to denote by $\lambda$ without the pullback map. $$\begin{tikzcd} & & 0 \arrow{d} & 0 \arrow{d} & 0 \arrow{d} && 0 \arrow{d} \\ 0 \arrow{r} & \mathbb{R}\arrow{r} & \Omega^0(M) \arrow{r}{d} \arrow{d}{\pi_\infty^*} & \Omega^1(M) \arrow{r}{d} \arrow{d}{\pi_\infty^*} & \Omega^2(M) \arrow{d}{\pi_\infty^*} \arrow{r}{d} & \cdots \arrow{r}{d} & \Omega^m(M) \arrow{d}{\pi_\infty^*} \\ 0 \arrow{r} & \mathbb{R}\arrow{r} & \Omega^{0,0} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{0,1} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{0,2} \arrow{d}{d_\mathrm{v}} \arrow{r}{d_\mathrm{h}} & \cdots \arrow{r}{d_\mathrm{h}} & \Omega^{0,m} \arrow{d}{d_\mathrm{v}} \\ & 0 \arrow{r} & \Omega^{1,0} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{1,1} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{1,2} \arrow{d}{d_\mathrm{v}} \arrow{r}{d_\mathrm{h}} & \cdots \arrow{r}{d_\mathrm{h}} & \Omega^{1,m} \arrow{d}{d_\mathrm{v}} \\ & 0 \arrow{r} & \Omega^{2,0} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{2,1} \arrow{r}{d_\mathrm{h}} \arrow{d}{d_\mathrm{v}} & \Omega^{2,2} \arrow{d}{d_\mathrm{v}} \arrow{r}{d_\mathrm{h}} & \cdots \arrow{r}{d_\mathrm{h}} & \Omega^{2,m} \arrow{d}{d_\mathrm{v}} \\ & & \vdots & \vdots & \vdots && \vdots \end{tikzcd}$$ The variational bicomplex Let $\vartheta_\lambda$ denote the pullback to $J^\infty \pi$ of any local Lepage equivalent of $\lambda$ which is at most $1$-contact, so that $\vartheta_\lambda^{(1)} = \vartheta_\lambda- \lambda\in \Omega^{1,m-1}$, and let $P$ denote any local homotopy operator for the $d_\mathrm{h}$ rows (apart from the first) of the variational bicomplex. Define the *extension of $\vartheta_\lambda$ by $P$* to be the $m$-form defined locally by $$\begin{aligned} \vartheta^\mathrm{F}& = \vartheta_\lambda+ \sum_{p=1}^{m-1} (-Pd_\mathrm{v})^p \vartheta_\lambda^{(1)} \\ & = \lambda+ \vartheta_\lambda^{(1)} - (Pd_\mathrm{v}) \vartheta_\lambda^{(1)} + (Pd_\mathrm{v})^2 \vartheta_\lambda^{(1)} - \cdots + (-Pd_\mathrm{v})^{m-1} \vartheta_\lambda^{(1)} \\ & \in \Omega^{0,m} \oplus \Omega^{1,m-1} \oplus \Omega^{2,m-2} \oplus \Omega^{3,m-3} \oplus \cdots \oplus \Omega^{m,0} \, ,\end{aligned}$$ so that $\vartheta^\mathrm{F}$ is another Lepage equivalent of $\lambda$. We shall show that this method of constructing $\vartheta^\mathrm{F}$ satisfies the closure property, by diagram chasing. Suppose that $\lambda$ is a null Lagrangian, so that $$0 = \varepsilon_\lambda= (d\vartheta^\mathrm{F})^{(1)} = d_\mathrm{v}\vartheta_\lambda^{(0)} + d_\mathrm{h}\vartheta_\lambda^{(1)} = d_\mathrm{v}\lambda+ d_\mathrm{h}\vartheta_\lambda^{(1)} \, .$$ Then $$\begin{aligned} (d\vartheta^\mathrm{F})^{(2)} & = d_\mathrm{v}(\vartheta^{\mathrm{F}(1)}) + d_\mathrm{h}(\vartheta^{\mathrm{F}(2)}) \\ & = d_\mathrm{v}\vartheta_\lambda^{(1)} - d_\mathrm{h}P d_\mathrm{v}\vartheta_\lambda^{(1)} \\ & = P d_\mathrm{h}d_\mathrm{v}\vartheta_\lambda^{(1)} = - P d_\mathrm{v}d_\mathrm{h}\vartheta_\lambda^{(1)} = P d_\mathrm{v}d_\mathrm{v}\lambda= 0\end{aligned}$$ using the homotopy property $d_\mathrm{h}\circ P + P \circ d_\mathrm{h}= \mathop{\mathrm{id}}$, and in a similar way $$\begin{aligned} (d\vartheta^\mathrm{F})^{(p+1)} & = d_\mathrm{v}(\vartheta^{\mathrm{F}(p)}) + d_\mathrm{h}(\vartheta^{\mathrm{F}(p+1)}) \\ & = d_\mathrm{v}(\vartheta^{\mathrm{F}(p)}) - d_\mathrm{h}P d_\mathrm{v}(\vartheta^{\mathrm{F}(p)}) \\ & = P d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(p)}) = - P d_\mathrm{v}d_\mathrm{h}(\vartheta^{\mathrm{F}(p)}) = P d_\mathrm{v}d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) = 0\end{aligned}$$ for $2 \le p \le m-1$, where the penultimate equality arises recursively from $$d_\mathrm{h}(\vartheta^{\mathrm{F}(p)}) + d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) = (d\vartheta^{\mathrm{F}})^{(p)} = 0 \, .$$ Thus $d\vartheta^\mathrm{F}= (d\vartheta^\mathrm{F})^{(m+1)}$, and we may see that this maximal contact component also vanishes by traversing the diagram in the opposite direction. For $2 \le p \le m$ we have $$\begin{aligned} d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(p)}) & = - d_\mathrm{h}d_\mathrm{v}P d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) \\ & = d_\mathrm{v}d_\mathrm{h}P d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) \\ & = d_\mathrm{v}d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) - (d_\mathrm{v}P) d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) \\ & = - (d_\mathrm{v}P) d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(p-1)}) \, ;\end{aligned}$$ but $$d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(1)}) = d_\mathrm{h}d_\mathrm{v}\vartheta_\lambda^{(1)} = - d_\mathrm{v}d_\mathrm{h}\vartheta_\lambda^{(1)} = d_\mathrm{v}d_\mathrm{v}\lambda= 0$$ so that $d_\mathrm{h}d_\mathrm{v}(\vartheta^{\mathrm{F}(m)}) = 0$. As $d_\mathrm{h}: \Omega^{m+1,0} \to \Omega^{m+1,1}$ is injective by exactness, we see finally that $(d\vartheta^\mathrm{F})^{(m+1)} = d_\mathrm{v}(\vartheta^{\mathrm{F}(m)}) = 0$. We shall describe suitable local homotopy operators for $d_\mathrm{h}$, constructed using vertical endomorphisms, in the next Section; by using them we obtain the following result. **Theorem 1**. *Let $\lambda$ be the pullback to $J^\infty \pi$ of a Lagrangian of any order, and let $\vartheta_\lambda$ be the pullback to $J^\infty \pi$ of any local Lepage equivalent of $\lambda$ which is at most $1$-contact. A local homotopy operator $P$ then defines a local Lepage equivalent $\vartheta^\mathrm{F}$ which is an extension of $\vartheta_\lambda$ and which satisfies the closure property, that $d\vartheta^\mathrm{F}= 0$ precisely when $\lambda$ is null.* We can also consider a global version of this result, noting that the diagram chasing above would apply equally to global operators as it does to local ones. We have seen that additional structures, such as connections or nonholonomic projections, are needed to construct a global Lepage equivalent when the order of the Lagrangian is greater than two. A global homotopy operator for the horizontal differential on $J^\infty \pi$ has also been found [@And89 Theorem 5.56] and again this uses a symmetric linear connection on the base manifold $M$. **Theorem 2**. *Let $\lambda$ be the pullback to $J^\infty \pi$ of a Lagrangian of any order, and let $\vartheta_\lambda$ be the pullback to $J^\infty \pi$ of any global Lepage equivalent of $\lambda$ which is at most $1$-contact, constructed using additional data as appropriate. A global homotopy operator $P$, such as the one described in [@And89] using a symmetric linear connection, then defines a global Lepage equivalent $\vartheta^\mathrm{F}$ which is an extension of $\vartheta_\lambda$ and which satisfies the closure property, that $d\vartheta^\mathrm{F}= 0$ precisely when $\lambda$ is null.* We remark that in fact there is no requirement for the homotopy operators in each term to be the same, and we could generalise the formula to $$\vartheta^\mathrm{F}= \vartheta_\lambda+ \sum_{p=1}^{m-1} (-1)^p (P_{p+1}d_\mathrm{v}P_p d_\mathrm{v}\cdots P_2 d_\mathrm{v}) \vartheta_\lambda^{(1)}$$ where $P_p$ is a homotopy operator for the $p$-contact row of the variational bicomplex. # Vertical endomorphisms The most basic example of a 'vertical endomorphism' is the almost tangent structure on a tangent manifold $TM$. This is simply a tensorial expression of the isomorphism between a vector space and its tangent space at any point, applied to the tangent spaces to a manifold, and may be regarded as a $1$-form taking values in the sub-bundle of $TTM \to TM$ containing the vertical vectors. A similar object may be defined using a more complicated procedure on a higher order tangent manifold $T^k M$ [@CCS86], now giving a $1$-form taking its values in the sub-bundle of vertical vectors in $TT^k M \to T^k M$. Vertical endomorphisms $S^\eta$ on jet manifolds $J^k\pi$, where $\eta \in \Omega^1(M)$ is a closed $1$-form, were defined in [@Sau87]. The construction started with a point $j^k_p\phi\in J^k\pi$ and a tangent vector $\xi$ at $j^{k-1}_p\phi\in J^{k-1}\pi$ vertical over $M$. Any such vector may be represented by a $1$-parameter family of local sections $\phi_t$ where $\phi_0 = \phi$ and $\xi$ is the tangent vector at $t=0$ to the curve $t \mapsto j^{k-1}_p\phi_t$. Given a function $f$ on $M$ defined in a neighbourhood of $p$, the *vertical lift* of $\xi$ to $j^k_p\phi$ in the direction specified by $df$ then used the $1$-parameter family of local sections $\psi_t : q \mapsto \phi_{tf(q)}(q)$ to define a curve $j^k_p\psi_t$ in $J^k\pi$ and therefore a tangent vector at $j^k_p\phi$. The vertical endomorphism $S^\eta$ at any point $j^k_p\phi\in J^k\pi$ was then defined by starting with any tangent vector in $T_{j^k_p\phi}J^k\pi$, projecting it to $T_{j^{k-1}_p\phi}J^{k-1}\pi$, taking the vertical representative using the contact structure, and then applying the vertical lift (using any function $f$ satisfying $f(p)=0$ and $df = \eta$ in a neighbourhood of $p$) to give a new tangent vector in $T_{j^k_p\phi}J^k\pi$. It may be shown that this construction is well defined, and so independent of the choices of $\phi_t$ and $f$, and that it gives a tensor field $S^\eta \in \Omega^1(J^k\pi) \otimes \mathfrak{X}(J^k\pi)$ expressed in coordinates[^1] as $$S^\eta = \sum_{\lvert J \rvert+\lvert K \rvert \le k-1} \frac{(J+K+1_i)!}{J!\,K!\,(\lvert K \rvert+1)} \frac{\partial^{\lvert K \rvert} \eta_i}{\partial x^{K}} \, \theta^\aG_J \otimes \frac{\partial}{\partial u^\aG_{J+K+1_i}} \, .$$ It is evident from this formula that, when acting on forms, the operators $S^\eta$ on $J^k\pi$ and on $J^l\pi$ with $l>k$ are related by the pullback map $\pi_{l,k}^*$, so that we may define a similar operator acting on forms on $J^\infty \pi$ without ambiguity. Given local coordinates $(x^i)$ on $U \subset M$, we write $S^i$ rather than $S^{dx^i}$ for the operators on $U^\infty = \pi_\infty^{-1}(U)$. These local operators have the rather simpler coordinate description $$S^i = \sum_{\lvert I \rvert=0}^\infty \bigl( I(i)+1 \bigr) \theta^\aG_I \otimes \frac{\partial}{\partial u^\aG_{I+1_i}}$$ and may be used to construct local homotopy operators for the horizontal differential $d_\mathrm{h}$ on $J^\infty \pi$. One such homotopy operator, involving an ordering of the coordinates $x^i$, was given in [@Tul80]. Other homotopy operators, not using such an ordering, may be constructed from two different repeated actions of $S^i$ on forms: these are $$\tilde{S}^J = i_{S^{j_1} \circ S^{j_2} \circ \cdots \circ S^{j_r}} \, , \qquad \hat{S}^J = i_{S^{j_1}} \circ i_{S^{j_2}} \circ \cdots \circ i_{S^{j_r}}$$ where $\lvert J \rvert=r$ and $J = 1_{j_1} + 1_{j_2} + \cdots + 1_{j_r}$. Note that the first action is a derivation, whereas the second is not if $r>1$; the multi-index notation is justified because operators $S^i$ and $S^j$ commute. The following result was obtained in [@CS09 Theorem 1]. **Proposition 3**. *Define the differential operators $\tilde{P}, \hat{P}: \Omega^{p,q}(U^\infty) \to \Omega^{p,q-1}(U^\infty)$, with $p \ge1$ and $1 \le q \le m$, by $\tilde{P}(\omega) = i_{d/dx^i} \bigl( \tilde{P}^i(\omega) \bigr)$, $\hat{P}(\omega) = i_{d/dx^i} \bigl( \hat{P}^i(\omega) \bigr)$ where $$\begin{aligned} \tilde{P}^i(\omega) & = \sum_{I=0}^\infty \frac{(-1)^{\lvert I \rvert}(m-q)! \lvert I \rvert!}{p(m-q+\lvert I \rvert+1)! I!} d_I \tilde{S}^{I+1_i} \omega\, , \\ \hat{P}^i(\omega) & = \sum_{I=0}^\infty \frac{(-1)^{\lvert I \rvert}(m-q)! \lvert I \rvert!}{p^{\lvert I \rvert+1}(m-q+\lvert I \rvert+1)! I!} d_I \hat{S}^{I+1_i} \omega\, .\end{aligned}$$ Then both $\tilde{P}$ and $\hat{P}$ are homotopy operators for $d_\mathrm{h}$.* In general the operators $\tilde{P}$ and $\hat{P}$ are different, although they are equal when acting on forms projectable to $J^1\pi$, and also when acting on forms in $\Omega^{1,q}(U^\infty)$. In the latter case, writing the operator as $P$, [@CS09 Theorem 2] gives $$\omega- d_\mathrm{h}P \omega= \theta^\aG \wedge \sum_{\lvert I \rvert=0}^\infty (-1)^{\lvert I \rvert} d_I \bigl( i_{\partial/ \partial u^\aG_I} \omega\bigr)$$ for any $\omega\in \Omega^{1,m}$, so that $\omega- d_\mathrm{h}P \omega$ is a source form. If we write $\vartheta_\lambda= \lambda- P d_\mathrm{v}\lambda$ then $$(d\vartheta_\lambda)^{(1)} = d_\mathrm{v}(\vartheta_\lambda^{(0)}) + d_\mathrm{h}(\vartheta_\lambda^{(1)}) = (d_\mathrm{v}\lambda) - d_\mathrm{h}P (d_\mathrm{v}\lambda)$$ so that in particular $(d\vartheta_\lambda)^{(1)}$ is a source form. Thus $\vartheta_\lambda$ is a Lepage form, and it is clearly a local Lepage equivalent of $\lambda$. In coordinates with $\lambda= L \, \omega_0$ $$\vartheta_\lambda= L \, \omega_0 + \sum_{\lvert J \rvert, \lvert K \rvert = 0}^\infty \frac{(-1)^{\lvert J \rvert}(J+K+1_j)! \, \lvert J \rvert! \, \lvert K \rvert!}{(\lvert J \rvert+\lvert K \rvert+1)! \, J! \, K!} d_J \biggl( \frac{\partial L}{\partial u^\aG_{J+K+1_j}} \biggr) \theta_K \wedge \omega_j \, ,$$ so that it is the pullback to $J^\infty \pi$ of the principal Lepage equivalent of $\lambda$. We obtain the above formula from those for $S$ and $P$ by using the multi-index Leibniz' rule and the identity for weighted sums of binomial coefficients $$\sum_{0 \le K \le I} \frac{(-1)^{\lvert K \rvert}I!}{(\lvert K \rvert+p+1) K! (I-K)!} = \frac{p! \lvert I \rvert!}{(\lvert I \rvert+p+1)!}$$ obtained by first evalating the integral $\int_0^1 x^p (x-1)^r dx$ in two different ways, and then using the Vandermonde identity for the convolution of scalar binomial coefficients. We can also use the homotopy operators $P$ to give a simple proof of the result mentioned earlier, that the difference between two Lepage equivalents for the same Lagrangian is the sum of a closed form and a form which is at least $2$-contact. Let $\vartheta_\lambda$ and $\vartheta^\prime_\lambda$ be two Lepage equivalents for the Lagrangian $\lambda$, and put $\vartheta= \vartheta_\lambda- \vartheta^\prime_\lambda$, so that $\vartheta^{(0)} = 0$. As $\vartheta_\lambda$ and $\vartheta^\prime_{\lambda}$ give rise to the same Euler--Lagrange form, we see that $$d_\mathrm{h}(\vartheta^{(1)}) + d_\mathrm{v}(\vartheta^{(0)}) = (d\vartheta)^{(1)} = 0$$ so that $d_\mathrm{h}(\vartheta^{(1)}) = 0$ and therefore locally $\vartheta^{(1)} = d_\mathrm{h}P (\vartheta^{(1)})$. Then $$\vartheta^{(1)} = d P (\vartheta^{(1)}) - d_\mathrm{v}P (\vartheta^{(1)})$$ where $d_\mathrm{v}P (\vartheta^{(1)}) \in \Omega^{2,m-2}$, so that $$\begin{aligned} \vartheta& = d P (\vartheta^{(1)}) + \bigl( \vartheta^{(2)} - d_\mathrm{v}P (\vartheta^{(1)}) \bigr) + \cdots + \vartheta^{(m)} \\ & \in d\Omega^{1,m-2} \oplus \Omega^{2,m-2} \oplus \cdots \oplus \Omega^{m,0} \, .\end{aligned}$$ Finally in this Section we apply these operators to a first order Lagrangian $\lambda= L\, \omega_0$. As $d_\mathrm{v}\lambda= d\lambda$ is also first order, we see that $$\lambda- Pd_\mathrm{v}\lambda= L \, \omega_0 + S^i \biggl( \frac{\partial L}{\partial u^\aG} \theta^\aG + \frac{\partial L}{\partial u^\aG_j} \theta^\aG_j \biggr) \omega_i = L \, \omega_0 + \frac{\partial L}{\partial u^\aG_i} \theta^\aG \wedge \omega_i \, ,$$ the local expression of the Poincaré--Cartan form. This is also first order, and we then see that each successive operator $P^i$ is simply a multiple of $S^i$. We obtain $$(-Pd_\mathrm{v})^p \lambda= \frac{1}{(p!)^2} \frac{\partial^p L}{\partial u^{\aG_1}_{i_1} \cdots \partial u^{\aG_p}_{i_p}} \theta^{\aG_1} \wedge \cdots \wedge \theta^{\aG_p} \wedge \omega_{i_1 \cdots i_p} \, ,$$ showing that $\sum_{p=0}^m (-Pd_\mathrm{v})^p \lambda$ gives the local expression of the standard fundamental Lepage equivalent of a first order Lagrangian. # Connections and vertical tensors {#Sconn} We have seen that, for a first order Lagrangian, the fundamental Lepage equivalent may be constructed locally using the homotopy operators $S^i$, and also that it is a global object which may be constructed using the first order vertical tensor $S$. These are two different facets of the same construction and they arise because, in the first order case, the formulation of a vertical endomorphism $S^\eta$ does not in fact require the $1$-form $\eta$ to be closed. From the coordinate description $$S^\eta = \eta_i \theta^\aG \otimes \frac{\partial}{\partial u^\aG_i}$$ it is clear that at any point $j^1_p\phi\in J^1\pi$ the value of $S^\eta$ depends only on the cotangent vector $\eta|_p$ and not on the values of $\eta$ at any other points. Thus, given any cotangent vector $\eta|_p \in T^*_p M$, we may choose a closed $1$-form $\zeta$ in a neighbourhood of $p$ satisfying $\zeta|_p = \eta|_p$, for example the form given in coordinates centred on $p$ by $\zeta= d(\eta_i(p) x^i)$, and put $$S^\eta|_{j^1_p\phi} = S^\zeta|_{j^1_p\phi} \in (T^* J^1\pi\otimes TJ^1\pi)_{j^1_p\phi} \, .$$ Doing this at each point of $J^1\pi$ gives a well defined vertical endomorphism $S^\eta$ for an arbitrary $1$-form $\eta \in \Omega^1(M)$, and it is clear that the mapping $\eta \mapsto S^\eta$ is just that given by the vertical tensor $S$. The same approach will not work directly for higher order vertical endomorphisms. For example, the coordinate description of $S^\eta$ on $J^2\pi$ is $$S^\eta = \eta_i \theta^\aG \otimes \frac{\partial}{\partial u^\aG_i} + \frac{1}{\#(ij)} \frac{\partial\eta_i}{\partial x^j} \theta^\aG \otimes \frac{\partial}{\partial u^\aG_{(ij)}} + \frac{2}{\#(ij)}\eta_i \theta^\aG_j \otimes \frac{\partial}{\partial u^\aG_{(ij)}}$$ and at any point $j^2_p\phi\in J^2\pi$ the value of $S^\eta$ depends, not just on the cotangent vector $\eta_p$, but also on the derivative of the $1$-form $\eta$ at $p$. We can, however, deal with this problem by supposing that we are given a symmetric linear connection $\nabla$ on $M$; the infinitesimal parallel translation defined by $\nabla$ will then provide enough information to specify the derivative of $\eta$. The vertical endomorphism defined by the $1$-form $\eta$ (not necessarily closed) and the connection $\nabla$ will be given in coordinates as $$S^\eta_\nabla= \eta_i \theta^\aG \otimes \frac{\partial}{\partial u^\aG_i} + \frac{1}{\#(hj)} \eta_i \Gamma^i_{hj} \theta^\aG \otimes \frac{\partial}{\partial u^\aG_{(hj)}} + \frac{2}{\#(ij)} \eta_i \theta^\aG_j \otimes \frac{\partial}{\partial u^\aG_{(ij)}}$$ where $\Gamma^i_{hj}$ are the connection coefficients of $\nabla$, so that the mapping $\eta \to S^\eta_\nabla$ will define a second order vertical tensor $S_\nabla$, a section of the bundle $\pi_2^* TM \otimes T^* J^2\pi\otimes T J^2\pi$ over $J^2\pi$. Formally, as the $1$-form $\eta$ is a section of the cotangent bundle $\tau : T^* M \to M$, we regard the connection $\nabla$ as a linear Ehresmann connection $\Gamma: T^* M \to J^1\tau$, a section of the jet bundle $\tau_{1,0}: J^1\tau\to T^* M$, so that the connection coefficients $\Gamma^i_{hj}$ are just the jet coordinates of $\Gamma$. (Of course the connection $\nabla$ also defines a linear Ehresmann connection on the tangent bundle, but there the jet coordinates are $-\Gamma^i_{hj}$.) For each $j^2_p\phi\in J^2\pi$ we may choose a closed $1$-form $\zeta$ in a neighbourhood of $p$ satisfying $j^1_p\zeta= \Gamma(\eta|_p)$, for example the form given in coordinates centred on $p$ by $\zeta= d \bigl( \eta_i(p) x^i + \tfrac{1}{2} \eta_i(p) \Gamma^i_{hj}(p) x^h x^j \bigr)$, and put $$S_\nabla^\eta|_{j^2_p\phi} = S^\zeta|_{j^2_p\phi} \in (T^* J^2\pi\otimes TJ^2\pi)_{j^2_p\phi} \, .$$ Doing this at each point of $J^2\pi$ now gives a well defined vertical endomorphism $S_\nabla^\eta$ for an arbitrary $1$-form $\eta \in \Omega^1(M)$, and so we can construct a second order vertical tensor with coordinate expression $$\label{Snabla} S_\nabla= \partial_i \otimes \biggl( \theta^\aG \otimes \frac{\partial}{\partial u^\aG_i} + \frac{1}{\#(hj)} \Gamma^i_{hj} \theta^\aG \otimes \frac{\partial}{\partial u^\aG_{(hj)}} + \frac{2}{\#(ij)} \theta^\aG_j \otimes \frac{\partial}{\partial u^\aG_{(ij)}} \biggr) \, .$$ A similar procedure may be carried out for higher order vertical endomorphisms, but requires the use of semiholonomic jets to allow for symmetrization. For example, in the third order case we use the connection map $\Gamma: T^* M \to J^1\tau$, regarded as a bundle morphism $\tau \to \tau_1$ over the identity on $M$, and its prolongation $j^1\Gamma: J^1\tau\to J^1\tau_1$. The composition $j^1\Gamma\circ \Gamma$ then takes its values in the semiholonomic manifold $\widehat{J}^2\tau\subset J^1\tau_1$ [@Sau89 Section 5.3], so that if $\mathrm{p}_2 : \widehat{J}^2\tau\to J^2\tau$ is the symmetrization projection then we may use $$\Gamma_2 = \mathrm{p}_2 \circ j^1\Gamma\circ \Gamma: T^* M \to J^2\tau$$ as the map which allows us to specify the first and second derivatives at $p$ of the closed local $1$-form $\zeta$ by setting $j^2_p\zeta= \Gamma_2(\eta|_p)$. More generally, we construct the maps $\Gamma_l$ recursively. Suppose we have the map $\Gamma_{l-1} : T^* M \to J^{l-1}\tau$, and that it is a section of $\tau_{l-1,0}$ with the property that $j^1\Gamma_{l-1} \circ \Gamma$ takes its values in the semiholonomic manifold $\widehat{J}^l\tau\subset J^1\tau_{l-1}$, so that we may set $$\Gamma_l = \mathrm{p}_l \circ j^1\Gamma_{l-1} \circ \Gamma: T^* M \to J^l\tau\, .$$ We note first that $$\begin{aligned} \tau_{l,l-1}\circ \Gamma_l & = (\tau_{l-1})_{1,0}\circ i_{1,l-1} \circ \Gamma_l \\ & = (\tau_{l-1})_{1,0}\circ i_{1,l-1} \circ \mathrm{p}_l \circ j^1\Gamma_{l-1} \circ \Gamma\\ & = (\tau_{l-1})_{1,0}\circ j^1\Gamma_{l-1} \circ \Gamma\\ & = \Gamma_{l-1} \circ \tau_{1,0}\circ \Gamma\\ & = \Gamma_{l-1} \, ,\end{aligned}$$ so that $$\begin{aligned} \tau_{l,0}\circ \Gamma_l = \tau_{l-1,0}\circ \tau_{l,l-1}\circ \Gamma_l = \tau_{l-1,0}\circ \Gamma_{l-1} = \mathop{\mathrm{id}}_{T^* M}\end{aligned}$$ and therefore that $\Gamma_l$ is a section of $\tau_{l,0}: J^l\tau\to T^* M$. We must also check that $j^1\Gamma_l \circ \Gamma$ takes its values in the semiholonomic manifold $\widehat{J}^{l+1}\tau$, the submanifold of $J^1\tau_l$ given by equality of the two maps $j^1\tau_{l,l-1}$ and $i_{1,l-1} \circ (\tau_l)_{1,0}$ to $J^1\tau_{l-1}$ [@Sau89 Section 6.2]; but at any point $j^1_p\omega\in J^1\tau$ we know that $$\begin{aligned} \bigl( j^1\tau_{l,l-1}\circ j^1\Gamma_l \bigr) (j^1_p\omega) & = j^1(\tau_{l,l-1}\circ \Gamma_l) (j^1_p\omega) \\ & = j^1\Gamma_{l-1}(j^1_p\omega) \intertext{and} \bigl( i_{l-1,1} \circ (\tau_l)_{1,0}\circ j^1\Gamma_l \bigr) (j^1_p\omega) & = \bigl( i_{l-1,1} \circ (\tau_l)_{1,0}\bigr) \bigl( j^1_p(\Gamma_l \circ \omega) \bigr) \\ & = i_{l-1,1} \bigl( \Gamma_l (\omega(p)) \bigr) \\ & = j^1\Gamma_{l-1} \bigl( \Gamma(\omega(p)) \bigr) \, ,\end{aligned}$$ so that if $j^1_p\omega$ is in the image of $\Gamma$ then $j^1_p\omega= \Gamma(\omega(p))$ and $$\bigl( j^1\tau_{l,l-1}\circ j^1\Gamma_l \bigr) \bigl( \Gamma(\omega(p)) \bigr) = j^1\Gamma_{l-1} \bigl( \Gamma(\omega(p)) \bigr) = \bigl( i_{l-1,1} \circ (\tau_l)_{1,0}\circ j^1\Gamma_l \bigr) \bigl( \Gamma(\omega(p)) \bigr)$$ as required. We may therefore define $\Gamma_{l+1} = \mathrm{p}_{l+1} \circ \Gamma_l \circ \Gamma$ and continue the process. The recursion starts with $l=2$ and $\Gamma_1 = \Gamma: T^* M \to J^1\tau$, or even degenerately with $l=1$ and $\Gamma_0 = \mathop{\mathrm{id}}_{T^* M} : T^* M \to J^0\tau= T^* M$. To find a coordinate expression for these maps, let $(x^i, y_j)$ be the coordinates on $T^* M$, so that the jet coordinates on $J^1\tau$ are $y_{ij}$ and on $J^l\tau$ are $y_{iJ}$. As the connection is linear and symmetric, we see that $y_{ij} \circ \Gamma= y_h \Gamma^h_{ij}$ with $y_{ji} \circ \Gamma= y_{ij} \circ \Gamma$, and in general if $y_{iJ} \circ \Gamma_l = y_h \Gamma^h_{J+1_i}$ then $$y_{iJj} \circ j^1\Gamma_l = y_h \frac{\partial\Gamma^h_{J+1_i}}{\partial x^j} + y_{hj} \Gamma^h_{J+1_i}$$ so that $$y_{iJj} \circ j^1\Gamma_l \circ \Gamma= y_g \biggl( \frac{\partial\Gamma^g_{J+1_i}}{\partial x^j} + \Gamma^g_{hj} \Gamma^h_{J+1_i} \biggr) \, ;$$ the coordinates $y_{iJ+1_j} \circ \Gamma_{l+1}$ may then be obtained by symmetrization. In the degererate case, the coordinates of $\Gamma_0$ are of course $\Gamma^h_i = \delta^h_i$. We have, therefore, obtained the following result. **Theorem 4**. *Let $\pi : E \to M$ be a fibred manifold, and let $\nabla$ be a symmetric linear connection on $M$. On any jet manifold $J^k\pi$ there is a canonical vertical tensor $S_\nabla$ defined in the following way. If $\eta \in \Omega^1(M)$ and $j^k_p\phi\in J^k\pi$, let $\zeta$ be any local closed $1$-form defined in a neighbourhood of $p$ satisfying $j^{k-1}_p\zeta= \Gamma_{k-1}(\eta|_p)$ (for example, a $1$-form defined using a polynomial in coordinates $x^i$ centred on $p$) and put $S^\eta_\nabla|_{j^k_p\phi} = S^\zeta|_{j^k_p\phi}$. Then $S^\eta_\nabla|_{j^k_p\phi}$ is independent of the choice of $\zeta$. The resulting map $j^k_p\phi\mapsto S^\eta_\nabla|_{j^k_p\phi}$ is a vertical endomorphism depending at each point $j^k_p\phi$ only on the cotangent vector $\eta|_p$ and so defines a vertical tensor $\eta \mapsto S^\eta_\nabla$.* The coordinate expression of $S_\nabla$ is $$S_\nabla= \partial_h \otimes \sum_{\lvert J \rvert+\lvert K \rvert \le k-1} \frac{(J+K+1_i)!}{J!\,K!\,(\lvert K \rvert+1)} \Gamma^h_{K+1_i} \theta^\aG_J \otimes \frac{\partial}{\partial u^\aG_{J+K+1_i}} \, ,$$ and combining the sums over the index $i$ and the multi-index $K$ in the usual way then gives $$S_\nabla= \partial_h \otimes \sum_{\substack{\lvert J \rvert+\lvert K \rvert \le k\\ \lvert K \rvert>0}} \frac{(J+K)!}{J!\,K!} \Gamma^h_K \theta^\aG_J \otimes \frac{\partial}{\partial u^\aG_{J+K}} \, .$$ A similar formula, without an upper bound on the length of the multi-indices, may be used on $J^\infty \pi$ for the map $\pi_\infty^* T^* M \otimes T^* J^\infty \pi\to TJ^\infty \pi$. # Infinitesimal nonholonomic projections {#Sproj} As mentioned earlier, two possible approaches to defining global Lepage equivalents for higher order Lagrangians involve using either connections, or tubular neighbourhoods of holonomic jet manifolds inside nonholonomic jet manifolds. We have remarked that the latter approach really involves only the infinitesimal projection defined by the tubular neighbourhood at points of the holonomic submanifold, and we can now see that the existence of vertical tensors allows the two approaches to be related: a symmetric linear connection on the base manifold will define an infinitesimal nonholonomic projection $TJ^1\pi_{k-1}\to TJ^k\pi$ for $k \ge 2$. The simplest example is in the second order case, where $\mathrm{i}_{1,1}: J^2\pi\to J^1\pi_1$ is the canonical inclusion. We start with a point $j^2_p\phi\in J^2\pi$ and a tangent vector $\xi \in T_{j^2_p\phi} J^1\pi_1$ which is vertical over $J^1\pi$, so that $\xi \in V_{j^2_p\phi}(\pi_1)_{1,0}$. We then apply the isomorphism $$\mathsf{A}: V(\pi_1)_{1,0}\to (\pi_1)_1^* T^* M \otimes (\pi_1)_{1,0}^* V\pi_1$$ arising from the affine structure of $(\pi_1)_{1,0}: J^1\pi_1\to J^1\pi$ (restricted to points of $J^2\pi$) and follow this by $S_\nabla$, giving a map $$p_\nabla= S_\nabla\circ \mathsf{A}: V_{J^2\pi}(\pi_1)_{1,0} \to V\pi_{2,0}$$ so that $p_\nabla(\xi) \in V_{j^2_p\phi}\pi_{2,0}$. There are, of course, many possible extensions of $p_\nabla$ to a map $T_{J^1\pi}J^1\pi_1\to TJ^1\pi$; but there is precisely one such extension satisfying the requirement that $p_\nabla\circ T\mathrm{i}_{1,1}= \mathop{\mathrm{id}}_{TJ^2\pi}$. We may see this by looking at coordinate representations. At any point $\mathrm{i}_{1,1}(j^2_p\phi) \in J^1\pi_1$ $$\mathsf{A}\biggl( \frac{\partial}{\partial u^\aG_{\cdot j}} \biggr) = dx^j \otimes \frac{\partial}{\partial u^\aG} \, , \qquad \mathsf{A}\biggl( \frac{\partial}{\partial u^\aG_{ij}} \biggr) = dx^j \otimes \frac{\partial}{\partial u^\aG_i} \, ,$$ and composing with $S_\nabla$ as presented in formula [\[Snabla\]](#Snabla){reference-type="eqref" reference="Snabla"} in the previous Section gives $$p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{\cdot j}} \biggr) = \frac{\partial}{\partial u^\aG_j} + \frac{1}{\#(ik)} \Gamma^j_{ik} \frac{\partial}{\partial u^\aG_{(ik)}} \, , \qquad p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{ij}} \biggr) = \frac{1}{\#(ij)} \frac{\partial}{\partial u^\aG_{(ij)}} \, .$$ (Nominally the image of, say, $$\frac{\partial}{\partial u^\aG_{ij}}\biggr|_{\mathrm{i}_{1,1}(j^2_p\phi)} \mapsto dx^j|_p \otimes \frac{\partial}{\partial u^\aG_i}\biggr|_{j^1_p\phi} \in T^*_p M \otimes T_{j^1_p\phi}J^1\pi$$ is not directly in the domain of $S_\nabla$; but as $S_\nabla$ incorporates the projection $T\pi_{2,1}: TJ^2\pi\to TJ^1\pi$ we may represent that image by an element of $T^*_p M \otimes T_{j^2_p\phi}J^2\pi$ without ambiguity.) Noting now that $$\begin{aligned} T\mathrm{i}_{1,1}\biggl( \frac{\partial}{\partial x^i} \biggr) & = \frac{\partial}{\partial x^i} & T\mathrm{i}_{1,1}\biggl( \frac{\partial}{\partial u^\aG_i} \biggr) & = \frac{\partial}{\partial u^\aG_{i\cdot}} + \frac{\partial}{\partial u^\aG_{\cdot i}} \\ T\mathrm{i}_{1,1}\biggl( \frac{\partial}{\partial u^\aG} \biggr) & = \frac{\partial}{\partial u^\aG_{\cdot\cdot}} & T\mathrm{i}_{1,1}\biggl( \frac{\partial}{\partial u^\aG_{(ij)}} \biggr) & = \frac{\partial}{\partial u^\aG_{ij}} \\\end{aligned}$$ we see that necessarily we must have $$p_\nabla\biggl( \frac{\partial}{\partial x^i} \biggr) = \frac{\partial}{\partial x^i} \, , \qquad p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{\cdot\cdot}} \biggr) = \frac{\partial}{\partial u^\aG} \, , \qquad p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{i\cdot}} \biggr) = - \frac{1}{\#(jk)} \Gamma^i_{jk} \frac{\partial}{\partial u^\aG_{(jk)}} \, .$$ A direct calculation confirms that these coordinate formulæ for $p_\nabla$ are invariant under fibred coordinate transformations on $\pi$ and their prolongations to $J^1\pi_1$ and $J^2\pi$; we also see that the restriction of $p_\nabla$ to the semiholonomic submanifold $\widehat{J}^2\pi\subset J^1\pi_1$ is the symmetrization map $\mathrm{p}_2$, as we would expect. Note that $p_\nabla$ is a bundle morphism over $j^1\pi_{1,0}\to \pi_{2,1}$, not over $(\pi_1)_{1,0} \to \pi_{2,1}$. $$\begin{tikzcd} J^1\pi_1\arrow[swap]{d}{j^1\pi_{1,0}} \arrow[near start]{drr}{(\pi_1)_{1,0}} && J^2\pi\arrow{d}{\pi_{2,1}} & & TJ^1\pi_1\arrow{r}{p_\nabla} \arrow[swap]{d}{Tj^1\pi_{1,0}} & TJ^2\pi\arrow{d}{T\pi_{2,1}} \\ J^1\pi\arrow[leftrightarrow]{rr} && J^1\pi& & TJ^1\pi\arrow[leftrightarrow]{r} & TJ^1\pi \end{tikzcd}$$ We apply the same approach in the general case to obtain the infinitesimal projection $p_\nabla: TJ^1\pi_{k-1}\to TJ^k\pi$, with coordinate expressions $$\begin{aligned} p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{Ij}} \biggr) & = \sum_{\lvert K \rvert>0}^{k-\lvert I \rvert} \frac{(I+K)!}{I! \, K!} \Gamma^j_K \frac{\partial}{\partial u^\aG_{I+K}} \intertext{obtained by composing $\mathsf{A}$ with $S_\nabla$, with the necessary consequence that} p_\nabla\biggl( \frac{\partial}{\partial u^\aG_{J\cdot}} \biggr) & = (1 - \lvert J \rvert) \frac{\partial}{\partial u^\aG_J} - \sum_{I+1_j = J} \biggl( \sum_{\lvert K \rvert=2}^{k-\lvert I \rvert} \frac{(I+K)!}{I! \, K!} \Gamma^j_K \frac{\partial}{\partial u^\aG_{I+K}} \biggr) \\ p_\nabla\biggl( \frac{\partial}{\partial x^i} \biggr) & = \frac{\partial}{\partial x^i} \, .\end{aligned}$$ This gives us the following result. **Theorem 5**. *Let $\pi : E \to M$ be a fibred manifold, and let $\nabla$ be a symmetric linear connection on $M$. For each nonholonomic jet manifold $J^1\pi_{k-1}$ there is a unique infinitesimal projection $p_\nabla$ satisfying $p_\nabla\circ T\mathrm{i}_{1,k-1}= \mathop{\mathrm{id}}_{J^k\pi}$ and $p_\nabla|_{\widehat{J}^k\pi} = \mathrm{p}_k$, constructed by composing the isomorphism $$\mathsf{A}: V(\pi_{k-1})_{1,0}\to (\pi_{k-1})_1^* T^* M \otimes (\pi_{k-1})_{1,0}^* V\pi_{k-1}$$ arising from the affine structure of $(\pi_{k-1})_{1,0}: J^1\pi_{k-1}\to J^{k-1}\pi$ (restricted to points of $J^k\pi$) with the vertical tensor $S_\nabla$ on $J^k\pi$.* # Homotopy operators for the horizontal differential {#Shom} We have seen that homotopy operators for the horizontal differential play an important part in the construction of Lepage equivalents satisfying the closure condition (and, indeed, of at most $1$-contact Lepage equivalents in general), and that locally such homotopy operators can be constructed using vertical endomorphisms. We have also noted that global homotopy operators (depending on a choice of a symmetric linear connection) have been shown to exist, but the construction in [@And89] uses a quite different method, relating the horizontal differential on forms to an operator acting on evolutionary vector fields. It is therefore of some interest to see whether a global homotopy operator can be constructed directly for differental forms by using vertical differentials. I conjecture that this can be done, and offer a possible method of doing so. The proposed formula has been checked for small values of the parameters $p$, $q$ and $r$ (see the Appendix for an example calculation); although the general result might be amenable to a direct calculation, there may well be a more geometric method of approaching it. There are three ingredients in the proposed formula, which mimics the local formula described above. The vertical tensor $S_\nabla$, regarded as a map $\Omega^{p,q} \to \mathfrak{X}(M) \otimes \Omega^{p,q}$, has already been specified, and this can be iterated to give a map $S_\nabla^r : \Omega^{p,q} \to \odot^r \mathfrak{X}(M) \otimes \Omega^{p,q}$, where $\odot^r \mathfrak{X}(M)$ denotes the symmetric multivector fields on $M$. We shall also need a covariant version of the horizontal differential, which we shall denote $d_{\mathrm{h}\nabla}$; this will be a map $\odot^r \mathfrak{X}(M) \otimes \Omega^{p,q} \to \odot^r \mathfrak{X}(M) \otimes \Omega^{p,q+1}$, given on basis tensors by $$d_{\mathrm{h}\nabla}(X \otimes \omega) = \nabla X \wedge \omega+ X \otimes d_\mathrm{h}\omega$$ and extended by multilinerarity, symmetry and skewsymmetry. The final ingredient will be an operator $\mathsf{C}: \odot^r \mathfrak{X}(M) \otimes \Omega^{p,q} \to \odot^{r-1} \mathfrak{X}(M) \otimes \Omega^{p,q-1}$ contracting a vector component with a form component, again taking advantage of symmetry and skewsymmetry. The proposed homotopy operator is then $P_\nabla: \Omega^{p,q} \to \Omega^{p,q-1}$ where $$P_\nabla\omega= \sum_{r=0}^\infty \frac{(-1)^r (m-q)!}{p(m-q+r+1) r!} \bigl( \mathsf{C}\circ d_{\mathrm{h}\nabla}\bigr)^r \mathsf{C}\bigl( S_\nabla^{r+1} \omega\bigr) \, .$$ # Discussion One of the features of the approach taken in this paper is that it combines the use of finite and infinite jets. Variational problems are by theie nature of finite order, and the various differential forms involved in their analysis are normally defined on a finite order jet manifold. Indeed, as we have seen, the properties of Lepage equivalents of first order and second order Lagrangians are rather different from those of higher order Lagrangians. On the other hand, the variational bicomplex is best considered on the infinite jet manifold. In [@And89] a subcomplex called the Jacobian subcomplex which is projectable to a finite order jet manifold is shown after lengthy calculations to be locally exact; but no mention is made of a homotopy operator acting on forms which are not $d_\mathrm{h}$-closed. The operators $\hat{P}$ and $\tilde{P}$ described earlier, although acting on all the forms on each finite order jet manifold, generally increase their order. It seems to be the case that the complexity of ascertaining a bound on the order of the forms obscures the homotopy structure of the problem, and indeed the potential for a global solution. The alternative approach in [@Voi22], which involves a single homotopy operator for the variational derivative (and thus, essentially, for the vertical differential) avoids this problem, but then cannot reduce to the classical fundamental Lepage equivalent for first order Lagrangians; in addition, global versions are likely to be constrained by topological considerations. The investigations in the second half of the paper suggest that vertical endomorphisms, when glued together as a vertical tensor using a symmetric linear connection, could be a significant part of the geometry of the jet bundle structure on a fibred manifold. If the conjecture that they define a global homotopy operator for $d_\mathrm{h}$ is correct, then the simple formula $$\vartheta_{\lambda, \nabla} = \sum_{p=0}^m (-P_\nabla d_\mathrm{v})^p \lambda$$ will give a Lepage equivalent of the Lagrangian $\lambda$ satisfying the closure property without the need for a separate choice of $\vartheta_\lambda^{(1)}$ to start the recursion. There will, though, be the question of whether the truncated form $\lambda- (P_\nabla d_\mathrm{v}) \lambda$ is the same as the Poincaré--Cartan form constructed using the infinitesimal projections $p_\nabla$. A final observation is that we have not explicitly addressed the question of whether it is possible to find, for second order Lagrangians, a geometrical construction of a Lepage equivalent satisfying the closure condition independently of any connection, as can be done for the Poincaré--Cartan form and the Carathéodory form. I suspect that this will not be the case. # Acknowledgements {#acknowledgements .unnumbered} I should like to acknowledge correspondence with Nicoleta Voicu which encouraged me to return to this topic after a number of years. Some results from this paper were presented at a meeting in Torino in honour of Marco Ferraris in June 2023, and at the International Summer School on Global Analysis and Applications in Prešov in August 2023. # Appendix: An example calculation {#appendix-an-example-calculation .unnumbered} We consider the form $\omega= f^i_{\aG m} dx^m \otimes \theta^\aG_i$ where $p = q = 1$ and the form is projectable to $J^2\pi$, so that the formula is $$P_\nabla\omega= \sum_{r=0}^\infty \frac{(-1)^r (m-1)!}{(m+r) r!} \bigl( \mathsf{C}\circ d_{\mathrm{h}\nabla}\bigr)^r \mathsf{C}\bigl( S_\nabla^{r+1} \omega\bigr) \, .$$ We obtain $$d_\mathrm{h}\omega= (d_l f^i_{\aG m}) dx^l \wedge dx^m \wedge \theta^\aG_i + f^i_{\aG m} dx^l \wedge dx^m \wedge \theta^\aG_{(il)}$$ so that $$\begin{aligned} S_\nabla(d_\mathrm{h}\omega) & = (d_l f^i_{\aG m}) \partial_i \otimes dx^l \wedge dx^m \wedge \theta^\aG + f^i_{\aG m} \Gamma^k_{il} \partial_k \otimes dx^l \wedge dx^m \wedge \theta^\aG \\ & \qquad + f^h_{\aG m} \partial_k \otimes dx^k \wedge dx^m \wedge \theta^\aG_h + f^k_{\aG m} \partial_k \otimes dx^h \wedge dx^m \wedge \theta^\aG_h \, , \\[1ex] \mathsf{C}S_\nabla(d_\mathrm{h}\omega) & = (d_i f^i_{\aG j}) dx^j \wedge \theta^\aG - (d_j f^i_{\aG i}) dx^j \wedge \theta^\aG + f^i_{\aG j} \Gamma^k_{ik} dx^j \wedge \theta^\aG - f^i_{\aG k} \Gamma^k_{ij} dx^j \wedge \theta^\aG \\ & \qquad + m \, \omega- f^i_{\aG i} dx^j \wedge \theta^\aG_j \intertext{and} S_\nabla^2 (d_\mathrm{h}\omega) & = 2 \partial_i \otimes \partial_l \otimes \bigl( f^i_{\aG m} dx^l \wedge dx^m \wedge \theta^\aG \bigr) \, , \\[1ex] \mathsf{C}S_\nabla^2 (d_\mathrm{h}\omega) & = 2 m \partial_i \otimes \bigl( f^i_{\aG j} dx^j \wedge \theta^\aG \bigr) - 2 \partial_j \otimes \bigl( f^i_{\aG i} dx^j \wedge \theta^\aG \bigr) \, , \\[1ex] \tfrac{1}{2} d_{\mathrm{h}\nabla}\mathsf{C}S_\nabla^2 d_\mathrm{h}\omega & = m \Gamma^k_{ih} \partial_k \otimes dx^h \wedge \bigl( f^i_{\aG j} dx^j \wedge \theta^\aG \bigr) + m \partial_i \otimes \bigl( (d_k f^i_{\aG j}) dx^k \wedge dx^j \wedge \theta^\aG \bigr) \\ & \qquad + m \partial_i \otimes \bigl( f^i_{\aG j} dx^k \wedge dx^j \wedge \theta^\aG_k \bigr) - \Gamma^k_{jh} \partial_k \otimes dx^h \wedge \bigl( f^i_{\aG i} dx^j \wedge \theta^\aG \bigr) \\ & \qquad - \partial_j \otimes \bigl( (d_k f^i_{\aG i}) dx^k \wedge dx^j \wedge \theta^\aG \bigr) - \partial_j \otimes \bigl( f^i_{\aG i} dx^k \wedge dx^j \wedge \theta^\aG_k \bigr) \, , \\[1ex] \tfrac{1}{2} \mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_\nabla^2 (d_\mathrm{h}\omega) & = m \Gamma^k_{ik} f^i_{\aG j} dx^j \wedge \theta^\aG - m \Gamma^k_{ij} dx^j \wedge f^i_{\aG k} \theta^\aG \\ & \qquad + m (d_i f^i_{\aG j}) dx^j \wedge \theta^\aG + m \omega - (d_j f^i_{\aG i}) dx^j \wedge \theta^\aG - f^i_{\aG i} dx^j \wedge \theta^\aG_j \, .\end{aligned}$$ On the other hand, $$\begin{aligned} S_\nabla\omega& = \partial_i \otimes (f^i_{\aG m} dx^m \wedge \theta^\aG ) \, , \\[1ex] \mathsf{C}S_\nabla\omega& = f^i_{\aG i} \theta^\aG \, , \\[1ex] d_\mathrm{h}(\mathsf{C}S_\nabla\omega) & = (d_j f^i_{\aG i}) dx^j \wedge \theta^\aG + f^i_{\aG i} dx^j \wedge \theta^\aG_j\end{aligned}$$ so that $$\begin{aligned} \tfrac{1}{2} \mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_\nabla^2 (d_\mathrm{h}\omega) + d_\mathrm{h}(\mathsf{C}S_\nabla\omega) & = m \Gamma^k_{ik} f^i_{\aG j} dx^j \wedge \theta^\aG - m \Gamma^k_{ij} dx^j \wedge f^i_{\aG k} \theta^\aG \\ & \qquad + m (d_i f^i_{\aG j}) dx^j \wedge \theta^\aG + m \omega\, .\end{aligned}$$ But from $$\begin{aligned} \mathsf{C}S_\nabla(d_\mathrm{h}\omega) & = (d_i f^i_{\aG j}) dx^j \wedge \theta^\aG + f^i_{\aG j} \Gamma^k_{ik} dx^j \wedge \theta^\aG - f^i_{\aG k} \Gamma^k_{ij} dx^j \wedge \theta^\aG + m \, \omega- d_\mathrm{h}(\mathsf{C}S_\nabla\omega)\end{aligned}$$ we see that $$\tfrac{1}{2} \mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_\nabla^2 (d_\mathrm{h}\omega) = m \, \mathsf{C}S_\nabla d_\mathrm{h}\omega- m(m-1) \omega+ (m-1) d_\mathrm{h}\mathsf{C}S_\nabla\omega$$ so that $$\omega= \biggl( \frac{1}{m-1} \mathsf{C}S_\nabla- \frac{1}{2m(m-1)} \mathsf{C}d_{\mathrm{h}\nabla}\mathsf{C}S_\nabla^2 \biggr) d_\mathrm{h}\omega + d_\mathrm{h}\biggl( \frac{1}{m} \mathsf{C}S_n \omega\biggr) \, .$$ 99 I.M. Anderson: *The variational bicomplex* Technical Report, Utah State University 1989; available at <http://www.math.usu.edu/~fg_mp/> D. Betounes: Extension of the classical Cartan form *Phys.Rev.D* **29(4)** (1984) 599--606 D. Betounes: Global shift operators and the higher order calculus of variations *J.Geom.Phys.* **10(2)** (1993) 185--201 C. Carathéodory: Über die Variationsrechnung bei mehrfachen Integralen *Acta Szeged Sect.Sci.Math.* **4** (1929) 193--216 M. Crampin, W. Sarlet and F. Cantrijn: Higher-order differential equations and higher-order Lagrangian mechanics *Math.Proc.Camb.Phil.Soc.* **99** (1986) 565--587 M. Crampin and D.J. Saunders: The Hilbert--Carathéodory and Poincaré--Cartan forms for higher-order multiple-integral variational problems *Houston J.Math.* **30(3)** (2004) 657--689 M. Crampin and D.J. Saunders: On null Lagrangians *Diff.Geom.Appl.* **22** (2005) 131--146 M. Crampin and D.J. Saunders: Homotopy operators for the variational bicomplex, representations of the Euler--Lagrange complex, and the Helmholtz--Sonin conditions *Lobachvskii J.Math.* **30(2)** (2009) 107--123 M. Ferraris and M. Francaviglia: Global formalisms in higher order calculus of variations *In: Proc.Conf.Diff.Geom.Appl. 1983*, (J.E. Purkyně  Univ., Brno, 1984) 93--117 P.L. García and M. Muñoz: On the geometrical structure of higher order variational calculus *In: Modern developments in analytical mechanics Vol I*, (Acta Acad.Sci.Taurinensis, 1982) 128--147 M. Horák and I. Kolář: On the higher-order Poincaré--Cartan forms *Czech.Math.J.* **33(108)** (1983) 467--475 D. Krupka: A map associated to the Lepagian forms on the calculus of variations in fibred manifolds *Czech.Math.J.* **27(102)** (1977) 114--118 D. Krupka, O. Krupková and D.J. Saunders: The Cartan form and its generalizations in the calculus of variations *Int.J.Geom.Meth.Mod.Phys.* **7(4)** (2010) 631--654 P. Libermann: Introduction to the theory of semi-holonomic jets *Arch.Math.(Brno)* **33** (1997) 173--189 P.J. Olver: *Applications of Lie Groups to Differential Equations* (Springer, 1986) P.J. Olver: Equivalence and the Cartan form *Acta Appl.Math.* **31** (1993) 99--136 M. Palese, O. Rossi and F. Zanello: Geometric integration by parts and Lepage equivalents *Diff.Geom.Appl.* **81** (2022) 101866 D.J. Saunders: An alternative approach to the Cartan form in Lagrangian field theories *J.Phys.A.(Math.Gen.)* **20** (1987) 339--349 D.J. Saunders: *The Geometry of Jet Bundles* (Cambridge, 1989) F. Takens: A global version of the inverse problem of the calculus of variations *J.Diff.Geom.* **14** (1979) 543--562 T. Tsujishita: On variation bicomplexes associated to differential equations *Osaka J.Math.* **19** (1982) 311--363 W.M. Tulczyjew: The Euler--Lagrange resolution *In: Lecture Notes in Mathematicsd 836: Differential geometric methods in mathematical physics*, (Springer, 1980) 22--48 Z. Urban and J. Volná: The fundamental Lepage form in two independent variables: a generalization using order-reducibility *Mathematics* **10(8)** (2022) 1211 A.M. Vinogradov: A spectral sequence associated with a non--linear differential equation, and algebro--geometric foundations of Lagrangian field theory with constraints *Sov.Math.Dokl.* **19** (1978) 144--148 N. Voicu, S. Garoiu and B. Vasian: On the closure property of Lepage equivalents of Lagrangian *Diff.Geom.Appl.* **81** (2022) 101852 D.J. Saunders\ Lepage Research Institute\ Email: [david\@symplectic.email](david@symplectic.email) [^1]: The numerical coefficient given in [@Sau87 eqn 3.4] and repeated in [@Sau89] after definition 6.5.6 is incorrect as it does not take account of the use of an individual index $i$ in a multi-index formula.

arxiv_math

--- abstract: | Consider the Schrödinger--Bopp--Podolsky system $$\tag{$\mathrm{SBP}_\epsilon$}\label{SBP_epsilon} \begin{cases} -\epsilon^2\Delta u+\parens{V+K\phi}u=u\abs{u}^{p-1};\\ \Delta^2\phi-\Delta\phi=4\pi K u^2 \end{cases} ~\text{in}~\mathbb{R}^3$$ for sufficiently small $\epsilon>0$, where $V,K\colon\mathbb{R}^3\to\coi{0,\infty}$; $p\in\ooi{1,5}$ are fixed and we want to solve for $u,\phi\colon\mathbb{R}^3\to\mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$. **Keywords.** Elliptic systems, Schrödinger--Bopp--Podolsky equations, standing wave solutions, semiclassical states, perturbation methods. **2010 Mathematics Subject Classification.** 35J48, 35B25, 35Q60. address: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo SP, Brazil author: - Gustavo de Paula Ramos bibliography: - bibliography.bib title: Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential --- # Introduction The behavior of a free quantum particle with mass $m$ and charge density $K$ subject to a potential $\tilde{V}$ and an electric potential $\phi$ may be modeled by $$\label{Equation:NSE} i\hbar\partial_t\psi = -\frac{\hbar^2}{2m}\Delta_x\psi+\parens{K\phi+\tilde{V}}\psi-\psi|\psi|^{p-1},$$ where $$\mathbb{R}^3\times\mathbb{R} \ni \parens{x,t} \mapsto \psi(x,t) \in \mathbb{C}$$ denotes the wave function of the particle and $\hbar\in\ooi{0,\infty}$ is a fixed constant. By restricting ourselves to search for *standing wave* solutions to [\[Equation:NSE\]](#Equation:NSE){reference-type="eqref" reference="Equation:NSE"} with a known energy $E\in\mathbb{R}$, i.e., wave functions of the form $\psi(x,t)=u(x)e^{iEt/\hbar}$ for an unknown $u\colon\mathbb{R}^3\to\mathbb{R}$, it becomes sufficient to solve $$-\epsilon^2 \Delta u + \parens{V + K \phi} u - u \abs{u}^{p - 1} = 0 ~\text{in}~ \mathbb{R}^3,$$ where $V:=\tilde{V}+E$ and $\epsilon := \parens{2m}^{-1/2} \hbar$. Let us suppose that $\epsilon$ may be arbitrarily small and that $\phi$ denotes the electric potential generated by the particle itself. In the trivial case $K\equiv 0$, this scenario is modeled by the *Nonlinear Schrödinger* equation $$\tag{$\mathrm{NLS}_\epsilon$} \label{NLS_epsilon} -\epsilon^2 \Delta u + V u = u \abs{u}^{p - 1} ~\text{in}~ \mathbb{R}^3.$$ If we consider Maxwell's theory of electromagnetism, then we obtain the *Schrödinger--Poisson--Slater* system $$\tag{$\mathrm{SPS}_\epsilon$}\label{SPS_epsilon} \begin{cases} -\epsilon^2\Delta u+\parens{V+K\phi}u=u|u|^{p-1};\\ -\Delta\phi=4\pi K u^2\\ \end{cases} ~\text{in}~\mathbb{R}^3.$$ Finally, we obtain the *Schrödinger--Bopp--Podolsky* system [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} if we consider the Bopp--Podolsky theory of electromagnetism instead (for more details, see [@dAveniaSiciliano2019 Section 2]). The study of the Schrödinger--Bopp--Podolsky system has gained traction in the mathematical literature only recently after the publication of [@dAveniaSiciliano2019], where d'Avenia and Siciliano introduced the problem; presented the appropriate functional framework; established the existence/non-existence of solutions to $$\begin{cases} -\Delta u+\omega u + q^2 \phi u = u\abs{u}^{p-2};\\ \Delta^2 \phi - a^2 \Delta \phi = 4 \pi u^2 \end{cases} ~\text{in}~\mathbb{R}^3$$ according to the values of the parameters $q, p$ for fixed $a, \omega \in \ooi{0, \infty}$ and proved that its solutions tend to solutions of the Schrödinger--Poisson--Slater system as $a \to 0^+$. In particular, there is a large and increasing number of studies about this system in the presence of a potential $V \colon \mathbb{R}^3 \to \coi{0, \infty}$. For instance, [@ChenTang2020; @LiPucciTang2020; @Zhang2022; @HuWuTang2023; @MascaroSiciliano2023; @FigueiredoSiciliano2023]. Among those, the ones which considered contexts closest to ours are [@MascaroSiciliano2023; @FigueiredoSiciliano2023], where the authors established multiplicity results for solutions in the *semiclassical limit*, i.e., when $\epsilon > 0$ is sufficiently small. In [@MascaroSiciliano2023], Mascaro and Siciliano use a technique due to Benci and Cerami to estimate the number of solutions in function of the Lusternik--Schnirelmann category of $$M:=\set{x \in \mathbb{R}^3 : V \parens{x}=\inf V>0}.$$ Among other results, Figueiredo and Siciliano employed the Lusternik--Schnirelmann and Morse theories in [@FigueiredoSiciliano2023] to establish multiplicity results of negative solutions. On the other hand, we know of few studies which considered the Schrödinger--Bopp--Podolsky system in the presence of a (possibly non-constant) charge density $K \colon \mathbb{R}^3 \to \coi{0, \infty}$, that is, [@TengYan2021; @JiaLiChen2022; @Peng2022]. To the best of our knowledge, [@Peng2022] is the only paper which simultaneously considered a potential $V$ and a charge density $K$ in the context of the Schrödinger--Bopp--Podolsky system. In this paper, Peng established the existence of ground-state solutions/infinite high energy solutions and studied the existence/multiplicity of sign-changing solutions by considering multiple techniques. We remark that, unlike the present text, Peng did not consider semiclassical states or used any perturbative techniques. In this context, the goal of this paper is to show that concentration results which Ianni and Vaira proved to hold for [\[SPS_epsilon\]](#SPS_epsilon){reference-type="eqref" reference="SPS_epsilon"} in [@IanniVaira2008] also hold for [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"}. We also estimate the number of solutions to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} analogously as in Ambrosetti, Malchiodi and Secchi's result for [\[NLS_epsilon\]](#NLS_epsilon){reference-type="eqref" reference="NLS_epsilon"} in [@AmbrosettiMalchiodiSecchi2001]. Let us state our hypotheses on $V, K \colon \mathbb{R}^3 \to \coi{0, \infty}$. We suppose throughout the text that $V$ satisfies 1. [\[V_1\]]{#V_1 label="V_1"} $V$ is of class $C^2$ and $\norm{V}_{C^2}<\infty$; 2. [\[V_2\]]{#V_2 label="V_2"} $\inf V>0$. Similarly, we always suppose that 1. [\[K_1\]]{#K_1 label="K_1"} $K$ is continuous and bounded. Before stating our results about concentrated solutions, we need to introduce a certain solution to $$\label{Equation:SimpleProblem} -\Delta u+u=u\abs{u}^{p-1}~\text{in}~\mathbb{R}^3.$$ It is well-known that there exists a unique positive spherically symmetric function that vanishes at infinity and solves [\[Equation:SimpleProblem\]](#Equation:SimpleProblem){reference-type="eqref" reference="Equation:SimpleProblem"} (see the theorem at [@Kwong1989 p. 23]), which we will always denote by $$U \colon \mathbb{R}^3 \to \ooi{0, \infty}.$$ Furthermore, it follows from [@GidasNiNirenberg1981 Proposition 4.1] that $$\label{Equation:ExponentialDecay} \text{there exists}~ r\in\ooi{0,\infty} ~\text{such that}~ U\parens{x}, \abs{\nabla U\parens{x}} \lesssim \frac{e^{-\abs{x}}}{\abs{x}} ~\text{whenever}~ \abs{x}>r,$$ and so $U\in H^1$. Let us state our first result, which concerns the existence of a family of solutions concentrated around non-degenerate critical points of $V$. **Theorem 1**. *If $x_0$ is a non-degenerate critical point of $V$, then there exist $\epsilon_0\in\ooi{0,1}$ and $\set{u_\epsilon}_{\epsilon\in\ooi{0,\epsilon_0}}\subset H^1$ such that given $\epsilon\in\ooi{0,\epsilon_0}$, $\parens{u_\epsilon,\varphi_\epsilon}$ is a weak solution to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} and $$\label{Equation:Concentration} \norm*{ u_\epsilon - U_{\lambda_{x_0}}\parens*{ \frac{1}{\epsilon}\parens{\cdot - x_0} } }_{H^1} \to 0 ~\text{as}~ \epsilon \to 0,$$ where $\lambda_{x_0} := V \parens{x_0}^{1/2}$; $U_{\lambda} := \lambda^{2 / \parens{p - 1}} U \parens{\lambda \cdot}$ and $\varphi_\epsilon:=\phi_{\epsilon, u_\epsilon^2}$ is defined in [\[Equation:phi\]](#Equation:phi){reference-type="eqref" reference="Equation:phi"}.* Due to [\[Equation:Concentration\]](#Equation:Concentration){reference-type="eqref" reference="Equation:Concentration"}, we say that the family of solutions provided by Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} is *concentrated* around $x_0$. We remark that if a family of solutions to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} is concentrated around $x_0 \in \mathbb{R}^3$, then $\nabla V \parens{x_0}=0$ (see Proposition [Proposition 21](#Proposition:Concentration){reference-type="ref" reference="Proposition:Concentration"}). Our next result establishes a sufficient condition for the existence of a family of solutions concentrated around a degenerate critical point of $V$. Technically, this is done by following Ianni and Vaira's approach of comparing how many derivatives of $V$ and $K$ vanish at the same point. Let us introduce further hypotheses on $V, K$ before proceeding to our next result. Consider the hypotheses 3. [\[V_3\]]{#V_3 label="V_3"} $x_0$ is a critical point of $V$ and either - - $n\geq 4$ is an even integer; - $V$ is of class $C^n$; - $\norm{V}_{C^n}<\infty$; - $\parens{ \partial^n_1 V \parens{x_0}, \partial^n_2 V \parens{x_0}, \partial^n_3 V \parens{x_0} } \neq 0$ but the mixed partial derivatives of $V$ with order $n$ vanish at $x_0$ and - $\mathrm{D}_{x_0}^l V=0$ for $l \in \set{1, \ldots, n - 1}$ or - $V$ is smooth; every derivative of $V$ is bounded and every derivative of $V$ vanishes at $x_0$, in which case we set $n=\infty$ and 3. [\[K_2\]]{#K_2 label="K_2"} either - - $m\geq 2$ is an even integer; - $K$ is of class $C^m$; - $\norm{K}_{C^m}<\infty$; - $\parens{ \partial^m_1 K \parens{x_0}, \partial^m_2 K \parens{x_0}, \partial^m_3 K \parens{x_0} } \neq 0$ but the mixed partial derivatives of $K$ with order $m$ vanish at $x_0$ and - $\mathrm{D}_{x_0}^l K = 0$ for $l \in \set{0, \ldots, m - 1}$ or - $K$ is smooth; every derivative of $K$ is bounded and every derivative of $K$ vanishes at $x_0$, in which case we set $m=\infty$. We can finally state our second result. **Theorem 2**. *Suppose that [\[V_3\]](#V_3){reference-type="ref" reference="V_3"}, [\[K_2\]](#K_2){reference-type="ref" reference="K_2"} hold and $\gamma:=\min\parens{n,2m+3}<\infty$. If $n>2m+3$, then suppose further that $\ker \mathrm{D}_{x_0} g = 0$, where $g \colon \mathbb{R}^3 \to \mathbb{R}^3$ is defined as $$g\parens{\xi} = \frac{1}{\parens{m!}^2} \sum_{ \substack{0 \leq \alpha, \beta \leq m; \\ 1 \leq i, j, l \leq 3} } \brackets*{ \partial_j^m K\parens{x_0} \partial_l^m K\parens{x_0} \binom{m}{\alpha} \binom{m}{\beta} \xi_j^{m-\alpha} \xi_l^{m-\beta} \tilde{C}_{\alpha,\beta,i,j,l} }e_i$$ and $$\tilde{C}_{\alpha,\beta,i,j,l} := \int\int x_j^\alpha y_l^\beta U_{\lambda_{x_0}} \parens{x}^2 U_{\lambda_{x_0}} \parens{y} \partial_i U_{\lambda_{x_0}} \parens{y} \mathrm{d}x\mathrm{d}y$$ for every $i, j, l \in \set{1, 2, 3}$; $\alpha, \beta \in \set{0, \ldots, m}$. In this situation, the conclusion of Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} holds once again.* Analogously as in [@IanniVaira2008 Remark 1.2], the condition $$\parens{ \partial^m_1 K \parens{x_0}, \partial^m_2 K \parens{x_0}, \partial^m_3 K \parens{x_0} } = \parens{1, \delta, \delta}$$ for $\delta \in \mathbb{R}$ with sufficiently small absolute value is sufficient to have $\ker \mathrm{D}_{x_0} g = 0$ in the context of Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"}. Let us recall two notions before stating our multiplicity result. The first is that of (non-degenerate) critical manifolds. **Definition 3**. Let $X$ be a Banach space and let $I\in C^2\parens{X}$. We say that $Y\subset X$ is a *critical manifold* of $I$ when $Y$ is a submanifold of $X$ and $\nabla I\parens{y}=0$ for every $y\in Y$. If $Y$ is a critical manifold of $I$, then it is said to be *non-degenerate* when given $y\in Y$, $\mathrm{T}_yY=\ker\mathrm{D}^2_yI$ and $$X \ni x \mapsto \mathrm{D}^2_y I \parens{x, \cdot} \in X'$$ is a Fredholm operator with index $0$. The second is the definition of the cup-length. **Definition 4**. Suppose that $M \subset \mathbb{R}^3$. If $\check{H}^* \parens{M} = 0$, then the *cup-length* of $M$ is defined as $\mathop{\mathrm{cupl}}\parens{M} = 1$, where $\check{H}^* \parens{M}$ denotes the Alexander--Spanier cohomology of $M$. Otherwise, we set $$\mathop{\mathrm{cupl}}\parens{M} = \sup \set{ k \in \mathbb{N}: \exists \alpha_1, \ldots, \alpha_k \in \check{H}^* \parens{M};~ \alpha_1 \cup \ldots \cup \alpha_k \neq 0 },$$ where $\cup$ denotes the cup product. In a similar spirit to [@MascaroSiciliano2023 Theorem 1.1], our last theorem estimates the multiplicity of concentrated solutions around points in a compact non-degenerate critical manifold of $V$ in function of its cup-length. **Theorem 5**. *If $M$ is a compact non-degenerate critical manifold of $V$ and $\epsilon\in\ooi{0,1}$ is sufficiently small, then [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} has at least $\mathop{\mathrm{cupl}}\parens{M} + 1$ weak solutions that concentrate around points of $M$.* Similarly as in [@AmbrosettiMalchiodi2006; @IanniVaira2008], our results are obtained through an application of the classical Lyapunov--Schmidt reduction technique (for a quick overview, see Section [2.2](#LyapunovSchmidt){reference-type="ref" reference="LyapunovSchmidt"}). To the best of our knowledge, this is the first time that this approach was considered to construct solutions to the Schrödinger--Bopp--Podolsky system. In order to avoid unnecessary repetition of non-crucial arguments, our reasoning along the text relies upon results in the aforementioned references. We remark that Theorem [Theorem 5](#Theorem:MultiplicitySolutions){reference-type="ref" reference="Theorem:MultiplicitySolutions"} is analogous to the proposed generalization of [@IanniVaira2008 Theorem 1.1] in the last paragraph of [@IanniVaira2008 Section 4] (see also [@AmbrosettiMalchiodi2006 Theorem 8.5]). Theorems [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"}, [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"} are respectively analogous to [@IanniVaira2008 Theorems 4.1, 4.2], whose proofs are sketched in [@IanniVaira2008 p. 593]. Let us introduce a notation before proceeding to more technical considerations. If $f \colon \mathbb{R}^3 \to \ooi{0,\infty}$ satisfies $0 < \inf f \leq \sup f < \infty$, then we define the Hilbert space $H^1_f$ as the completion of $C_c^\infty$ with respect to $$\angles{u \mid w}_{H^1_f} := \int\parens{\nabla u \cdot \nabla w + f u w}.$$ Note that it is easy to check that $\norm{\cdot}_{H^1_f}$ is equivalent to $\norm{\cdot}_{H^1}$, so we will canonically identify $H^1_f$ with $H^1$. We explain in Section [2.1](#VariationalFramework){reference-type="ref" reference="VariationalFramework"} that in order to obtain weak solutions to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"}, it suffices to search for critical points of the *energy functional* $E_\epsilon \colon H^1 \to \mathbb{R}$ given by $$\label{Equation:EnergyFunctional} E_\epsilon\parens{u} = \frac{1}{2} \norm{u}_{H^1_{V_\epsilon}}^2 + \frac{\epsilon^3}{4} \int \parens{ K_\epsilon \phi_{\epsilon, u^2} u^2 } - \frac{1}{p+1} \norm{u}_{L^{p+1}}^{p+1},$$ where $$\label{Equation:phi} \phi_{\epsilon, u w}:=\parens{K_\epsilon u w} \ast \kappa_\epsilon$$ for every $u, w \in H^1$; $\ast$ denotes the convolution; $\kappa\parens{x} := \abs{x}^{-1}\brackets{1-\exp\parens{-\abs{x}}}$ for every $x\in\mathbb{R}^3\setminus\set{0}$; $V_\epsilon := V \parens{\epsilon \cdot}$ and similarly for $K_\epsilon$, $\kappa_\epsilon$. Similarly, critical points of the functionals $I_\epsilon, J_\epsilon \in C^2\parens{H^1}$ given by $$I_\epsilon\parens{u} = \frac{1}{2} \norm{u}_{H^1_{V_\epsilon}}^2 - \frac{1}{p+1} \norm{u}_{L^{p+1}}^{p+1};$$ $$J_\epsilon\parens{u} = \frac{1}{2} \norm{u}_{H^1_{V_\epsilon}}^2 + \frac{\epsilon^2}{4} \int \cbrackets*{ K_\epsilon \brackets*{\parens{K_\epsilon u^2} \ast \abs{\cdot}^{-1}} u^2 } - \frac{1}{p+1} \norm{u}_{L^{p+1}}^{p+1}$$ are respectively associated to weak solutions to ([\[NLS_epsilon\]](#NLS_epsilon){reference-type="ref" reference="NLS_epsilon"}; [\[SPS_epsilon\]](#SPS_epsilon){reference-type="ref" reference="SPS_epsilon"}). On one hand, we can interpret both functionals $E_\epsilon$ and $J_\epsilon$ as perturbations of $I_\epsilon$. On the other hand, unlike $\abs{\cdot}^{-1}$, $\kappa$ is bounded and not $(-1)$-homogeneous, so $E_\epsilon$ is obtained from $I_\epsilon$ by summing a perturbation of order $\epsilon^3$ (see Lemma [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"}), while $J_\epsilon$ is obtained by summing a perturbation of order $\epsilon^2$. This discrepancy is the reason why the sufficient condition at Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"} differs from that in [@IanniVaira2008 Theorem 4.2]. Indeed, it follows from the proof of Lemma [Lemma 30](#Lemma:ExpansionDegenerate){reference-type="ref" reference="Lemma:ExpansionDegenerate"} that we have to compare $n$ with $2m+3$, so the case $n=2m+3$ is automatically ruled out. The analogous comparison in [@IanniVaira2008 Theorem 4.2] is done between $n$ and $2m+2$, so one also has to consider the case $n=2m+2$. Let us finish with a comment on the organization of the text. The preliminaries are done in Section [2](#Preliminaries){reference-type="ref" reference="Preliminaries"}: we explain our notion of weak solutions; we develop a variational characterization for them; we do a quick review of the Lyapunov--Schmidt reduction and we highlight a related problem. In Section [3](#MultiplicityResult){reference-type="ref" reference="MultiplicityResult"}, we develop the details of the Lyapunov--Schmidt reduction to prove Theorem [Theorem 5](#Theorem:MultiplicitySolutions){reference-type="ref" reference="Theorem:MultiplicitySolutions"}. Similarly, Section [4](#ConcentrationResults){reference-type="ref" reference="ConcentrationResults"} is concerned with Theorems [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} and [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"}. ## Notation {#notation .unnumbered} Given functions $f,g\colon X\to[0,\infty[$, we write $f\parens{x}\lesssim g\parens{x}$ for every $x\in X$ when there exists $C\in\ooi{0,\infty}$ such that $f\parens{x}\leq Cg\parens{x}$ for every $x\in X$. We denote the canonical basis of $\mathbb{R}^3$ by $e_1, e_2, e_3 \in \mathbb{R}^3$. Given $r\in\ooi{0,\infty}$, we define $B_r=\set{x\in\mathbb{R}^3: \abs{x}<r}$. We only consider functional spaces over $\mathbb{R}^3$, so the considered domain is omitted from the notation. Likewise, integrals are implied to be taken in $\mathbb{R}^3$ unless denoted otherwise. The functional spaces $H^1$ and $\mathcal{X}$ are obtained as the respective Hilbert space completions of $C_c^\infty$ with respect to the inner products $$\angles{u \mid v}_{H^1} := \int\parens{ \nabla u\cdot\nabla v + uv } \quad \text{and} \quad \angles{u \mid v}_{\mathcal{X}} := \int\parens{ \Delta u\Delta v + \nabla u\cdot\nabla v }.$$ ## Acknowledgements {#acknowledgements .unnumbered} This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. # Preliminaries {#Preliminaries} ## Variational framework {#VariationalFramework} Let us begin with some considerations about [\[Equation:phi\]](#Equation:phi){reference-type="eqref" reference="Equation:phi"}. **Remark 6**. Consider a fixed $\epsilon \in \ooi{0, 1}$. We have - $\phi_{\epsilon, u w} \in \mathcal{X}$ for every $u, w \in H^1$ and - the application $$\parens{u_1, u_2, u_3, u_4} \in \parens{H^1}^4 \mapsto \int \parens{ K_\epsilon \phi_{\epsilon, u_1 u_2} u_3 u_4 } = \int \parens{ K_\epsilon \phi_{\epsilon, u_3 u_4} u_1 u_2 }$$ is multilinear. We proceed to a discussion about our notion of weak solutions. Given $u \in H^1$, we define a *weak solution* to $$\label{Equation:Electrostatic} \Delta^2\phi-\Delta\phi=4\pi K u^2~\text{in}~\mathbb{R}^3$$ as a $\phi\in\mathcal{X}$ such that $$\int \parens{ \Delta \phi \Delta w + \nabla \phi \cdot \nabla w } = 4 \pi \int \parens{K u^2 w}$$ for every $w\in C_c^\infty$. Analogously, we say that $\parens{u,\phi}$ is a *weak solution* to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"} when $u\in H^1$, $\phi$ is a weak solution to [\[Equation:Electrostatic\]](#Equation:Electrostatic){reference-type="eqref" reference="Equation:Electrostatic"} and $$\int \brackets{ \epsilon^2 \nabla u \cdot \nabla w + \parens{V + K \phi} u w } = \int \parens{u \abs{u}^{p-1} w}$$ for every $w\in C_c^\infty$. It follows from the Riesz Theorem that $\phi_{1,u^2}$ is the unique weak solution to [\[Equation:Electrostatic\]](#Equation:Electrostatic){reference-type="eqref" reference="Equation:Electrostatic"}. Therefore, we naturally associate solutions to $$\begin{cases} \int \brackets{ \epsilon^2 \nabla u \cdot \nabla w + \parens{V + K \phi_{1,u^2}} u w } - \int \parens{u \abs{u}^{p-1} w} = 0 ~\text{for every}~ w\in C_c^\infty; \\ u\in H^1 \end{cases}$$ to weak solutions to [\[SBP_epsilon\]](#SBP_epsilon){reference-type="eqref" reference="SBP_epsilon"}. By considering the change of variable $x\mapsto \epsilon x$, we see that the previous problem is equivalent to $$\begin{gathered} \label{Equation:WeakSolution} \begin{cases} \angles{u \mid w}_{H^1_{V_\epsilon}} + \int \parens{ \epsilon^3 K_\epsilon \phi_{\epsilon,u^2} u w } - \int \parens{u \abs{u}^{p-1} w} = 0 ~\text{for every}~ w\in C_c^\infty; \\ u \in H^1. \end{cases}\end{gathered}$$ In view of this discussion, we will restrict ourselves to search for solutions to [\[Equation:WeakSolution\]](#Equation:WeakSolution){reference-type="eqref" reference="Equation:WeakSolution"} along the text. As $\kappa\leq\abs{\cdot}^{-1}$ in $\mathbb{R}^3\setminus\set{0}$, it follows from the Hardy--Littlewood--Sobolev Inequality (see [@LiebLoss2001 Section 4.3]) that $$\label{Equation:HLS} \abs*{ \int \phi_{1,u} f } \leq \abs*{ \int\int \frac{u\parens{y}^2f\parens{x}}{\abs{x-y}} \mathrm{d}y\mathrm{d}x } \lesssim \norm{u}_{L^{12/5}}^2 \norm{f}_{L^{6/5}} ~\text{for every}~ f\in L^{6/5}.$$ The following estimate follows from Hölder's Inequality and the Sobolev embeddings $\mathcal{X},H^1\hookrightarrow L^6$; $H^1\hookrightarrow L^{3/2}$. **Lemma 7**. *We have $$\int\abs{ K_\epsilon\phi_{\epsilon,u_1 u_2}u_3u_4 } \lesssim \norm{u_1}_{H^1} \norm{u_2}_{H^1} \norm{u_3}_{H^1} \norm{u_4}_{H^1}$$ for every $u_1, \ldots, u_4 \in H^1$ and $\epsilon \in \ooi{0,1}$.* The regularity of $E_\epsilon$ follows as a corollary. **Lemma 8**. *The functional $E_\epsilon$ is of class $C^2$. Moreover, $$\mathrm{D}_u E_\epsilon \parens{w_1} = \angles{u \mid w_1}_{H^1_{V_\epsilon}} + \epsilon^3 \int \parens{ K_\epsilon \phi_{\epsilon,u^2} u w } - \int\parens{u \abs{u}^{p-1} w_1}$$ and $$\begin{gathered} \mathrm{D}_u^2 E_\epsilon \parens{w_1,w_2} = \angles{w_1 \mid w_2}_{H^1_{V_\epsilon}} + \epsilon^3 \int \brackets*{ K_\epsilon \parens{ \phi_{\epsilon, u^2} w_1 w_2 + 2 \phi_{\epsilon, u w_1} u w_2 }} + \\ - p \int \parens*{\abs{u}^{p-1} w_1 w_2}\end{gathered}$$ for every $u, w_1, w_2 \in H^1$.* Due to the previous lemma, we can characterize solutions to [\[Equation:WeakSolution\]](#Equation:WeakSolution){reference-type="eqref" reference="Equation:WeakSolution"} as critical points of $E_\epsilon$. To finish, we list some elementary inequalities that will be often used throughout the text. **Lemma 9**. *We have $$\abs*{ \parens{a + b}^q - a^q - q a^{q-1} b } \lesssim \begin{cases} \abs{b}^q &\text{if}~q \leq 2; \\ \abs{b}^2+\abs{b}^q &\text{if}~q > 2; \end{cases}$$ $$\begin{gathered} \abs*{ \parens{a + b_1}^q - \parens{a + b_2}^q - q a^{q-1} \parens{b_1 - b_2} } \lesssim \\ \lesssim \begin{cases} \abs{b_1 - b_2} \parens{\abs{b_1}^{q-1} + \abs{b_2}^{q-1}} &\text{if}~q \leq 2; \\ \abs{b_1 - b_2} \parens{ \abs{b_1}^{q-1} + \abs{b_2}^{q-1} + \abs{b_1} + \abs{b_2} } &\text{if}~q > 2;\end{cases}\end{gathered}$$ and $$\abs*{ \parens{a + b}^{q-1} - a^{q-1} } \lesssim \begin{cases} \abs{b}^{q-1} &\text{if}~q \leq 2; \\ \abs{b} + \abs{b}^{q-1} &\text{if}~q > 2 \end{cases}$$ for every $a \in \cci{-1, 1}$ and $b, b_1, b_2 \in \mathbb{R}$.* ## Lyapunov--Schmidt reduction {#LyapunovSchmidt} Let us provide an overview of how the *Lyapunov--Schmidt procedure* lets us rewrite the problem $$\label{Equation:CriticalPoint} \nabla E_\epsilon\parens{u}=0; \quad u\in H^1$$ as a critical point equation for a functional on a finite-dimensional manifold. Suppose that $\mathcal{Z}_\epsilon$ is a *manifold of pseudo-critical points* of $E_\epsilon$, i.e., $\mathcal{Z}_\epsilon$ is a finite-dimensional submanifold of $H^1$ and we can conveniently bound $\norm{\nabla E_\epsilon \parens{z}}_{H^1}$ for $z\in\mathcal{Z}_\epsilon$. In particular, we can associate any $z\in\mathcal{Z}_\epsilon$ to an orthogonal decomposition of $H^1$, $$H^1 = \mathrm{T}_z \mathcal{Z}_\epsilon \oplus \parens{\mathrm{T}_z \mathcal{Z}_\epsilon}^\perp,$$ where $\mathrm{T}_z \mathcal{Z}_\epsilon$ denotes the tangent space to $\mathcal{Z}_\epsilon$ at $z$. We then rewrite the critical point equation [\[Equation:CriticalPoint\]](#Equation:CriticalPoint){reference-type="eqref" reference="Equation:CriticalPoint"} as the system of equations \ \_,z = 0; [\[Equation:Auxiliary\]]{#Equation:Auxiliary label="Equation:Auxiliary"}\ = 0; [\[Equation:Bifurcation\]]{#Equation:Bifurcation label="Equation:Bifurcation"}\ uH\^1, where $\Pi_{\epsilon,z} \colon H^1 \to \parens{\mathrm{T}_z \mathcal{Z}_\epsilon}^\perp$ is an orthogonal projection and we respectively name [\[Equation:Auxiliary\]](#Equation:Auxiliary){reference-type="eqref" reference="Equation:Auxiliary"}, [\[Equation:Bifurcation\]](#Equation:Bifurcation){reference-type="eqref" reference="Equation:Bifurcation"} the *auxiliary* and *bifurcation* equations. The next step consists in using the Implicit Function Theorem to prove that if $\epsilon$ is sufficiently small, then we can associate each $z\in\mathcal{Z}_\epsilon$ to a $w_{\epsilon, z} \in \parens{\mathrm{T}_z \mathcal{Z}_\epsilon}^\perp$ such that $\parens{z + w_{\epsilon, z}}$ solves the auxiliary equation. To conclude, it suffices to show that the newly obtained manifold, $$\tilde{\mathcal{Z}}_\epsilon := \set{z + w_{\epsilon, z}: z\in\mathcal{Z}_\epsilon} \subset H^1,$$ is a natural constraint of $E_\epsilon$, i.e., a critical point of $E_\epsilon|_{\tilde{\mathcal{Z}}_\epsilon}$ is a critical point of $E_\epsilon$. ## A related problem {#RelatedProblem} Note that Problem [\[Equation:SimpleProblem\]](#Equation:SimpleProblem){reference-type="eqref" reference="Equation:SimpleProblem"} is closely related to $$\label{P_lambda}\tag{$P_\lambda$} -\Delta u + \lambda^2 u = u\abs{u}^{p-1}~\text{in}~\mathbb{R}^3$$ for any $\lambda\in\ooi{0,\infty}$. Indeed, if $u\colon\mathbb{R}^3\to\mathbb{R}$ is a solution to [\[Equation:SimpleProblem\]](#Equation:SimpleProblem){reference-type="eqref" reference="Equation:SimpleProblem"}, then a straightforward computation shows that the function $x\mapsto\lambda^{2/\parens{p-1}}u\parens*{\lambda x}$ solves [\[P_lambda\]](#P_lambda){reference-type="eqref" reference="P_lambda"}. We finish by remarking that it suffices to search for critical points of the functional $\overline{I}_\lambda \in C^2\parens{H^1}$ given by $$\overline{I}_\lambda\parens{u} = \frac{1}{2} \norm{u}_{H^1_\lambda}^2 - \frac{1}{p+1} \norm{u}_{L^{p+1}}^{p+1}$$ to obtain weak solutions to [\[P_lambda\]](#P_lambda){reference-type="eqref" reference="P_lambda"}. # Multiplicity result {#MultiplicityResult} ## The manifold of pseudo-critical points {#PseudoMult} We begin with a Taylor expansion. **Remark 10**. In view of [\[V_1\]](#V_1){reference-type="ref" reference="V_1"}, $$\abs*{ V \parens{\epsilon x} - V \parens{\epsilon \xi} } \lesssim \epsilon \abs{x - \xi} \quad\text{and}\quad \abs*{ V \parens{\epsilon x} - V \parens{\epsilon \xi} - \epsilon \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} } \lesssim \epsilon^2 \abs{x - \xi}^2$$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$ and $x \in \mathbb{R}^3$. Considering the discussion in Section [2.3](#RelatedProblem){reference-type="ref" reference="RelatedProblem"}, we define $z_{\epsilon, \xi} \colon \mathbb{R}^3 \to \ooi{0, \infty}$ as $$z_{\epsilon, \xi}\parens{x} = U_{\lambda_{\epsilon \xi}}\parens{x - \xi},$$ so that $z_{\epsilon, \xi}$ solves $(P_{\lambda_{\epsilon \xi}})$, i.e., $$\label{Equation:PDE_z_epsilon,xi} -\Delta z_{\epsilon, \xi} + \lambda_{\epsilon \xi}^2 z_{\epsilon, \xi} = z_{\epsilon, \xi}^p ~\text{in}~\mathbb{R}^3,$$ where we recall that $\lambda_{\epsilon \xi} = V \parens{\epsilon \xi}^{1/2}$. Let us state some estimates involving the functions $z_{\epsilon, \xi}$. **Lemma 11**. *We have $$\label{Equation:z_epsilon,xi:1} 1 \lesssim \norm{z_{\epsilon, \xi}}_{H^1}, \norm{\partial_i z_{\epsilon, \xi}}_{H^1} \lesssim 1;$$ $$\label{Equation:z_epsilon,xi:2} \norm{\dot{z}_{\epsilon,\xi,i} + \partial_i z_{\epsilon,\xi}}_{H^1} \lesssim \epsilon$$ and $$\label{Equation:z_epsilon,xi:3} \abs{\angles{ \dot{z}_{\epsilon, \xi, i} \mid \dot{z}_{\epsilon, \xi, j} }_{H^1}} \lesssim \epsilon ~\text{if}~ i \neq j$$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$ and $i \in \set{1, 2, 3}$, where $\dot{z}_{\epsilon, \xi, i}$ denotes the $i$^th^ partial derivative of $\mathbb{R}^3 \ni \zeta \mapsto z_{\epsilon, \zeta} \in H^1$ evaluated at $\xi$.* *Proof.* *Proof of [\[Equation:z_epsilon,xi:1\]](#Equation:z_epsilon,xi:1){reference-type="eqref" reference="Equation:z_epsilon,xi:1"}.* Clearly, $$\parens{\inf V}^{1/\parens{p-1}} U \parens*{ \lambda_{\epsilon \xi} \parens{x-\xi} } \leq z_{\epsilon, \xi} \parens{x} \leq \parens{\sup V}^{1/\parens{p-1}} U \parens*{ \lambda_{\epsilon \xi} \parens{x-\xi} }.$$ Therefore, $$\parens{\inf V}^{1/\parens{p-1}} \leq \lambda_{\epsilon \xi} \frac{\norm{z_{\epsilon, \xi}}_{H^1}}{\norm{U}_{H^1}} \leq \parens{\sup V}^{1/\parens{p-1}},$$ so it follows from [\[V_1\]](#V_1){reference-type="ref" reference="V_1"} and [\[V_2\]](#V_2){reference-type="ref" reference="V_2"} that $1 \lesssim \norm{z_{\epsilon, \xi}} \lesssim 1$. It follows from the differentiation of [\[Equation:PDE_z\_epsilon,xi\]](#Equation:PDE_z_epsilon,xi){reference-type="eqref" reference="Equation:PDE_z_epsilon,xi"} that $$\norm{\partial_i z_{\epsilon, \xi}}_{H^1} = p \norm*{ z_{\epsilon, \xi}^{p - 1} \partial_i z_{\epsilon, \xi} }_{L^{p / \parens{p - 1}}} = p \lambda_{\epsilon \xi}^{p / \parens{p - 1}} \norm{U \partial_i U}_{L^{p / \parens{p - 1}}},$$ thus $$p\parens{\inf V}^{p/\parens{p-1}} \leq \frac{\norm{\partial_i z_{\epsilon, \xi}}_{H^1}}{ \norm{U \partial_i U}_{L^{p / \parens{p - 1}}} } \leq p\parens{\sup V}^{p/\parens{p-1}}.$$ In this situation, the other estimate also follows from [\[V_1\]](#V_1){reference-type="ref" reference="V_1"} and [\[V_2\]](#V_2){reference-type="ref" reference="V_2"}. *Proof of [\[Equation:z_epsilon,xi:2\]](#Equation:z_epsilon,xi:2){reference-type="eqref" reference="Equation:z_epsilon,xi:2"}.* Differentiating $\zeta \mapsto z_{\epsilon, \zeta}$, we obtain $$\dot{z}_{\epsilon, \xi, i} \parens{x} = \epsilon \partial_i V \parens{\epsilon \xi} \brackets*{ \frac{1}{p-1} V \parens{\epsilon \xi}^{-1} z_{\epsilon, \xi} \parens{x} + \frac{1}{2} V \parens{\epsilon \xi}^{-1/2} \partial_i z_{\epsilon, \xi} \parens{x} } - \partial_i z_{\epsilon, \xi} \parens{x}.$$ The conclusion follows from [\[V_1\]](#V_1){reference-type="ref" reference="V_1"}, [\[V_2\]](#V_2){reference-type="ref" reference="V_2"} and [\[Equation:z_epsilon,xi:1\]](#Equation:z_epsilon,xi:1){reference-type="eqref" reference="Equation:z_epsilon,xi:1"}. *Proof of [\[Equation:z_epsilon,xi:3\]](#Equation:z_epsilon,xi:3){reference-type="eqref" reference="Equation:z_epsilon,xi:3"}.* Due to the previous result, $$\abs*{ \angles{ \dot{z}_{\epsilon, \xi, i} \mid \dot{z}_{\epsilon, \xi, j} }_{H^1} - \angles{ \partial_i z_{\epsilon, \xi} \mid \partial_j z_{\epsilon, \xi} }_{H^1} } \lesssim \epsilon.$$ To conclude, it suffices to prove that $\angles{ \partial_i z_{\epsilon, \xi} \mid \partial_j z_{\epsilon, \xi} }_{H^1} = 0$. It is clear that $$\angles{ \partial_i z_{\epsilon, \xi} \mid \partial_j z_{\epsilon, \xi} }_{H^1} = \lambda_{\epsilon \xi}^{\parens{5 - p}/\parens{p - 1}} \angles{ \partial_i U \mid \partial_j U }_{H^1},$$ so it suffices to show that $\angles{ \partial_i U \mid \partial_j U }_{H^1} = 0$. As $$-\Delta \parens{\partial_i U} + \partial_i U = p U^{p-1} \partial_i U ~\text{in}~ \mathbb{R}^3,$$ we deduce that $\angles{ \partial_i U \mid \partial_j U }_{H^1} = p \int \brackets{ \parens{\partial_i U} \parens{\partial_j U} U^{p-1} }$. The function $U$ is spherically symmetric, while $x \mapsto \partial_i U \parens{x} \partial_j U \parens{x}$ is odd in the $i$^th^ and $j$^th^ variable, hence the result. ◻ We proceed to define $\mathcal{Z}_\epsilon = \set{z_{\epsilon, \xi}: \xi \in \mathbb{R}^3}$, which is easily seen to be a non-compact submanifold of $H^1$ and we remark that its tangent spaces are given by $$\mathrm{T}_{z_{\epsilon, \xi}} \mathcal{Z}_\epsilon = \mathrm{span} \set{ \dot{z}_{\epsilon, \xi, 1}, \dot{z}_{\epsilon, \xi, 2}, \dot{z}_{\epsilon, \xi, 3} } \subset H^1.$$ The manifold $\mathcal{Z}_\epsilon$ will act as our manifold of pseudo-critical points of $E_\epsilon$ due to the result that follows. **Lemma 12**. *We have $\norm{\nabla E_\epsilon \parens{z_{\epsilon, \xi}}}_{H^1} \lesssim \epsilon\abs{\nabla V\parens{\epsilon\xi}} + \epsilon^2$ for every $\parens{\epsilon,\xi}\in\ooi{0,1}\times\mathbb{R}^3$.* *Proof.* Clearly, $$\mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{u} = \mathrm{D}_{z_{\epsilon, \xi}} I_\epsilon \parens{u} + \epsilon^3 \int \parens*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi} u }.$$ Due to Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"} and [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}, $\epsilon^3 \int \abs{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi} u } \lesssim \epsilon^3 \norm{u}_{H^1}$. The result then follows from [@AmbrosettiMalchiodi2006 Lemma 8.8]. ◻ ## Solving the auxiliary equation {#AuxMult} Following the steps of the Lyapunov--Schmidt reduction, our next task consists in solving the auxiliary equation. More precisely, the goal of this section is to prove the following lemma. **Lemma 13**. *There exists $\epsilon_0 \in \ooi{0, 1}$ and an application of class $C^1$, $$\label{Equation:wMapMultiplicity} \ooi{0, \epsilon_0} \times \mathbb{R}^3 \ni \parens{\epsilon, \xi} \mapsto w_{\epsilon,\xi}\in H^1,$$ such that given $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$, $$\Pi_{\epsilon, \xi}\parens{ \nabla E_\epsilon \parens{z_{\epsilon, \xi} + w_{\epsilon, \xi}} } = 0 \quad\text{and}\quad w_{\epsilon, \xi} \in W_{\epsilon, \xi} := \parens{ \mathrm{T}_{z_{\epsilon,\xi}}\mathcal{Z}_\epsilon }^\perp.$$ Moreover, $$\norm{w_{\epsilon, \xi}}_{H^1} \lesssim \epsilon \abs{\nabla V \parens{\epsilon \xi}} + \epsilon^2; \quad \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} \lesssim \brackets*{ \epsilon \abs{\nabla V \parens{\epsilon\xi}} + \epsilon^2 }^\mu$$ for every $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$ and $i\in\set{1,2,3}$, where $\mu:=\min\parens{1,p-1}$.* We need several preliminary results to prove Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}. Let us begin by showing that the second derivative of $E_\epsilon$ is coercive on a certain subspace of $H^1$ if $\epsilon$ is sufficiently small. **Lemma 14**. *There exists $\epsilon_0 \in \ooi{0, 1}$ such that $\mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{u, u} \gtrsim \norm{u}_{H^1}^2$ for every $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$ and $u \in \parens{ \mathrm{span}\set{z_{\epsilon, \xi}} \oplus \mathrm{T}_{z_{\epsilon, \xi}} \mathcal{Z}_\epsilon }^\perp$.* *Proof.* Due to Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"}, [Lemma 8](#Lemma:EnergyClassC2){reference-type="ref" reference="Lemma:EnergyClassC2"} and [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}, $$\abs*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{u, u} - \mathrm{D}^2_{z_{\epsilon, \xi}} I_\epsilon \parens{u, u} } \leq \epsilon^3 \int \abs*{ K_\epsilon \parens{ \phi_{\epsilon, z_{\epsilon, \xi}^2} u^2 + 2 \phi_{\epsilon, z_{\epsilon, \xi} u} z_{\epsilon, \xi} u }} \lesssim \epsilon^3 \norm{u}_{H^1}^2.$$ In view of this estimate, the result follows from [@AmbrosettiMalchiodi2006 Lemma 8.9]. ◻ Let $R_\epsilon \colon \parens{H^1_{V_\epsilon}}'\to H^1_{V_\epsilon}$ denote the Riesz isomorphism and let $A_{\epsilon, \xi} \colon H^1 \to H^1$, $L_{\epsilon, \xi} \colon W_{\epsilon, \xi} \to W_{\epsilon, \xi}$ be given by $$A_{\epsilon, \xi} \parens{u} = R_\epsilon \parens*{ \mathrm{D}_{z_{\epsilon, \xi}}^2 E_\epsilon \parens{u, \cdot} } \quad\text{and}\quad L_{\epsilon, \xi} \parens{w} = \Pi_{\epsilon, \xi} \circ A_{\epsilon, \xi} \parens{w}.$$ We proceed to a sufficient condition for the invertibility of $L_{\epsilon, \xi}$ with arguments loosely based on the proof of [@AmbrosettiMalchiodi2006 Lemma 8.10] (see also [@IanniVaira2008 Lemma 3.2]). **Lemma 15**. *There exists $\epsilon_0 \in \ooi{0, 1}$ such that $L_{\epsilon, \xi}$ is invertible and $\norm{L_{\epsilon, \xi}^{-1}}_{\mathcal{L}\parens{W_{\epsilon, \xi}}} \lesssim 1$ for every $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$.* *Proof of Lemma [Lemma 15](#Lemma:BoundNormInverseL){reference-type="ref" reference="Lemma:BoundNormInverseL"}.* *First step.* Let us prove that $\norm{ z_{\epsilon, \xi} - \Pi_{\epsilon, \xi}\parens{z_{\epsilon, \xi}} }_{H^1} \lesssim \epsilon$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$. Due to Lemma [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}, $$\begin{gathered} \abs*{ \angles{ z_{\epsilon, \xi} \mid \dot{z}_{\epsilon, \xi, i} }_{H^1_{\lambda_{\epsilon \xi}}} } \leq \abs*{ \angles{ z_{\epsilon, \xi} \mid \dot{z}_{\epsilon, \xi, i} + \partial_i z_{\epsilon, \xi} }_{H^1_{\lambda_{\epsilon \xi}}} } + \underbrace{ \abs*{ \angles{ z_{\epsilon, \xi} \mid \partial_i z_{\epsilon, \xi} }_{H^1_{\lambda_{\epsilon \xi}}} } }_{=0} \lesssim \epsilon.\end{gathered}$$ *Second step.* We want to show that $\norm{ A_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} + \parens{p-1} z_{\epsilon, \xi} }_{H^1} \lesssim \epsilon$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$. Indeed, in light of Lemma [Lemma 8](#Lemma:EnergyClassC2){reference-type="ref" reference="Lemma:EnergyClassC2"} and [\[Equation:PDE_z\_epsilon,xi\]](#Equation:PDE_z_epsilon,xi){reference-type="eqref" reference="Equation:PDE_z_epsilon,xi"}, $$\begin{gathered} \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{z_{\epsilon, \xi}, u} + \parens{p - 1} \angles{ z_{\epsilon, \xi} \mid u }_{H^1_{\lambda_{\epsilon \xi}}} = \\ = \int \cbrackets{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} z_{\epsilon, \xi} u } + 3\epsilon^3 \int \parens*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi} u }.\end{gathered}$$ On one hand, it follows from [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"} and Remark [Remark 10](#Remark:TaylorMult){reference-type="ref" reference="Remark:TaylorMult"} that $$\int \abs{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} z_{\epsilon, \xi} u } \lesssim \epsilon \int \abs{x - \xi} z_{\epsilon, \xi} \parens{x} \abs{u \parens{x}} \mathrm{d}x \lesssim \epsilon \norm{u}_{H^1}.$$ On the other hand, it suffices to argue as in the proof of Lemma [Lemma 14](#Lemma:Coercive){reference-type="ref" reference="Lemma:Coercive"} to prove that $\int \abs{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi} u } \lesssim \epsilon^3 \norm{u}_{H^1}$. *Conclusion.* Clearly, $$\begin{gathered} L_{\epsilon, \xi} \circ \Pi_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} = -\parens{p-1} \Pi_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} + \Pi_{\epsilon, \xi} \parens*{ A_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} + \parens{p-1} z_{\epsilon, \xi} } + \\ + \Pi_{\epsilon, \xi} \circ A_{\epsilon, \xi} \parens*{ \Pi_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} - z_{\epsilon, \xi} }.\end{gathered}$$ In view of the first and second steps, $$\abs*{ L_{\epsilon, \xi} \circ \Pi_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} + \parens{p-1} \Pi_{\epsilon, \xi} \parens{z_{\epsilon, \xi}} } \lesssim \epsilon.$$ Therefore, $$\norm{L_{\epsilon, \xi} + \parens{p-1} \mathop{\mathrm{Id}}_{V_{\epsilon, \xi, 1}}}_{ \mathcal{L} \parens{V_{\epsilon, \xi, 1}, W_{\epsilon, \xi}} } \lesssim \epsilon \quad\text{and}\quad \norm{ L_{\epsilon, \xi} + \Pi_{\epsilon, \xi} \circ A_{\epsilon, \xi} }_{ \mathcal{L} \parens{V_{\epsilon, \xi, 2}, W_{\epsilon, \xi}} } \lesssim \epsilon,$$ where $$V_{\epsilon,\xi,1} := \mathrm{span} \set{\Pi_{\epsilon, \xi} \parens{z_{\epsilon,\xi}}} \quad\text{and}\quad V_{\epsilon,\xi,2} := \parens*{ \mathrm{span} \set{z_{\epsilon,\xi}} \oplus \mathrm{T}_{z_{\epsilon, \xi}} \mathcal{Z}_\epsilon }^\perp.$$ Finally, the result follows from Lemma [Lemma 14](#Lemma:Coercive){reference-type="ref" reference="Lemma:Coercive"}. ◻ We still need two lemmas before proceeding to the proof of Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}. **Lemma 16**. *We have $$\norm*{ \mathrm{D}_{\parens{z_{\epsilon, \xi} + w}} E_\epsilon - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon }_{H^{-1}} \lesssim \norm{w}_{H^1} + \norm{w}_{H^1}^p;$$ $$\norm*{ \mathrm{D}_{\parens{z_{\epsilon, \xi} + w}} E_\epsilon - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w, \cdot} }_{H^{-1}} \lesssim \norm{w}_{H^1}^2 + \norm{w}_{H^1}^p$$ and $$\norm*{ \mathrm{D}^2_{\parens{z_{\epsilon, \xi} + w}} E_\epsilon \parens{u, \cdot} - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{u, \cdot} }_{H^{-1}} \lesssim \parens*{ \norm{w}_{H^1} + \norm{w}_{H^1}^2 + \norm{w}^{p-1}_{H^1} } \norm{u}_{H^1}$$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$ and $u, w \in H^1$.* *Proof.* The estimates are obtained similarly, so we only prove that the first one holds. In view of Remark [Remark 6](#Remark:phi){reference-type="ref" reference="Remark:phi"} and Lemma [Lemma 8](#Lemma:EnergyClassC2){reference-type="ref" reference="Lemma:EnergyClassC2"}, $$\begin{gathered} \mathrm{D}_{\parens{z_{\epsilon, \xi} + w}} E_\epsilon \parens{u} - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{u} = \angles{w \mid u}_{H^1_{V_\epsilon}} + \epsilon^3 \int \parens{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} u w } + \\ + 2\epsilon^3 \int \brackets*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi} w} u \parens{z_{\epsilon, \xi} + w} } + \epsilon^3 \int \brackets*{ K_\epsilon \phi_{\epsilon, w^2} u \parens{z_{\epsilon, \xi} + w} } + \\ - \int \cbrackets*{ \brackets*{ \parens{z_{\epsilon, \xi}+w} \abs{z_{\epsilon, \xi}+w}^{p-1} - z_{\epsilon, \xi}^p } u }.\end{gathered}$$ On one hand, it follows from Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"} and [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"} that $$\begin{gathered} \left| \epsilon^3 \int \parens{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} u w } + 2\epsilon^3 \int \brackets*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi} w} u \parens{z_{\epsilon, \xi} + w} } + \right. \\ \left. + \epsilon^3 \int \brackets*{ K_\epsilon \phi_{\epsilon, w^2} u \parens{z_{\epsilon, \xi} + w} } \right| \lesssim \epsilon^3 \norm{w}_{H^1} \norm{u}_{H^1}.\end{gathered}$$ On the other hand, the Sobolev embeddings and Lemma [Lemma 9](#Lemma:Elementary){reference-type="ref" reference="Lemma:Elementary"} imply $$\abs*{\int \cbrackets*{ \brackets*{ \parens{z_{\epsilon, \xi}+w} \abs{z_{\epsilon, \xi}+w}^{p-1} - z_{\epsilon, \xi}^p } u }} \lesssim \parens*{ \norm{w}_{H^1} + \norm{w}_{H^1}^p } \norm{u}_{H^1},$$ hence the result. ◻ We proceed to the last preliminary result. **Lemma 17**. *It holds that $\norm{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } }_{H^{-1}} \lesssim \epsilon$ for every $\parens{\epsilon, \xi} \in \ooi{0, 1} \times \mathbb{R}^3$ and $i \in \set{1, 2, 3}$.* *Proof.* As $\nabla \overline{I}_{\lambda_{\epsilon \xi}} \parens{z_{\epsilon, \xi}} = 0$ for every $\xi \in \mathbb{R}^3$, we deduce that $\mathrm{D}^2_{z_{\epsilon, \xi}} \overline{I}_{\lambda_{\epsilon \xi}} \parens{\dot{z}_{\epsilon, \xi, i}, \cdot} = 0$. Therefore, it follows from Lemma [Lemma 8](#Lemma:EnergyClassC2){reference-type="ref" reference="Lemma:EnergyClassC2"} that $$\begin{gathered} \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, u } = \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, u } - \mathrm{D}^2_{z_{\epsilon, \xi}} \overline{I}_{\lambda_{\epsilon \xi}} \parens{ \dot{z}_{\epsilon, \xi, i}, u } = \\ = \int \cbrackets{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} \dot{z}_{\epsilon, \xi, i} u } + \epsilon^3 \int \brackets*{ K_\epsilon \parens{ \phi_{\epsilon, z_{\epsilon, \xi}^2} \dot{z}_{\epsilon, \xi, i} u + 2 \phi_{\epsilon, z_{\epsilon, \xi} \dot{z}_{\epsilon, \xi, i}} z_{\epsilon, \xi} u }}.\end{gathered}$$ On one hand, it follows from [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"} and Remark [Remark 10](#Remark:TaylorMult){reference-type="ref" reference="Remark:TaylorMult"} that $$\abs*{ \int \cbrackets{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} \dot{z}_{\epsilon, \xi, i} u } } \lesssim \epsilon \int \abs{x - \xi} \dot{z}_{\epsilon, \xi, i} \parens{x} u \parens{x} \mathrm{d}x \lesssim \epsilon \norm{u}_{H^1}.$$ On the other hand, Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"} and [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"} imply $$\epsilon^3 \abs*{ \int \brackets*{ K_\epsilon \parens{ \phi_{\epsilon, z_{\epsilon, \xi}^2} \dot{z}_{\epsilon, \xi, i} u + 2 \phi_{\epsilon, z_{\epsilon, \xi} \dot{z}_{\epsilon, \xi, i}} z_{\epsilon, \xi} u }}} \lesssim \epsilon^3 \norm{u}_{H^1}.$$ ◻ Let us finally prove Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} with arguments loosely based on those in the proof of [@AmbrosettiMalchiodi2006 Proposition 8.7]. *Proof of Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}.* *Setup.* Take $\epsilon_0 \in \ooi{0, 1}$ as in Lemma [Lemma 15](#Lemma:BoundNormInverseL){reference-type="ref" reference="Lemma:BoundNormInverseL"} and fix $\bar{C} \in \ooi{0, \infty}$. We define $\mathcal{H} \colon \mathcal{O} \to H^1 \times \mathbb{R}^3$ as the application of class $C^1$ given by $$\mathcal{H}\parens{\epsilon, \xi, w, \alpha} = \begin{pmatrix} \nabla E_\epsilon\parens{z_{\epsilon, \xi}+w} - \sum_{1\leq i\leq 3}\alpha_i\dot{z}_{\epsilon, \xi, i} \\ \sum_{1 \leq i \leq 3} \angles{w \mid \dot{z}_{\epsilon, \xi, i}}_{H^1} e_i \end{pmatrix},$$ where $$\mathcal{O} := \set*{ \parens{\epsilon, \xi, w, \alpha} \in \coi{0, \epsilon_0} \times \mathbb{R}^3 \times H^1 \times \mathbb{R}^3: \norm{w}_{H^1} \leq \bar{C} \brackets*{ \epsilon \abs{\nabla V \parens{\epsilon \xi}} + \epsilon^2 } }.\footnote{ The reasoning for such a choice of a domain is explained in \cite[Section 3.3.2]{IanniVaira2008} }$$ and $E_0 := \overline{I}_1$. Note that the auxiliary equation $$\Pi_{\epsilon, \xi} \parens{ \nabla E_\epsilon\parens{z_{\epsilon,\xi} + w} } = 0; \quad w \in W_{\epsilon, \xi}$$ is solved if, and only if, $\mathcal{H}\parens{\epsilon,\xi,w,\alpha}=0$ for a certain $\alpha\in\mathbb{R}^3$. *A preliminary result.* We claim that, up to shrinking $\epsilon_0$, $$\mathrm{D}_{\parens{w,\alpha}} \mathcal{H}_{\epsilon, \xi} \colon H^1 \times \mathbb{R}^3 \to H^1 \times \mathbb{R}^3$$ is invertible and $$\norm*{ \mathrm{D}_{\parens{w,\alpha}} \mathcal{H}_{\epsilon, \xi} \parens{v, \beta} }_{H^1 \times \mathbb{R}^3} \gtrsim \norm{v}_{H^1} + \abs{\beta}$$ for every $\parens{\epsilon, \xi, w, \alpha} \in \mathcal{O}$ and $\parens{v, \beta} \in H^1 \times \mathbb{R}^3$. Indeed, it follows from Lemma [Lemma 16](#Lemma:ApproxDer){reference-type="ref" reference="Lemma:ApproxDer"} that $$\begin{gathered} \norm*{ \mathrm{D}_{\parens{w,\alpha}} \mathcal{H}_{\epsilon, \xi} \parens{v, \beta} - \begin{pmatrix} A_{\epsilon, \xi} \parens{v} - \sum_{1\leq i\leq 3}\beta_i\dot{z}_{\epsilon, \xi, i} \\ \sum_{1 \leq i \leq 3} \angles{v \mid \dot{z}_{\epsilon, \xi, i}}_{H^1} e_i \end{pmatrix} }_{H^1 \times \mathbb{R}^3} \lesssim \\ \lesssim \parens*{ \norm{w}_{H^1} + \norm{w}_{H^1}^2 + \norm{w}^{p-1}_{H^1} } \norm{v}_{H^1} \lesssim \parens*{ \epsilon \abs{\nabla V \parens{\epsilon \xi}} + \epsilon^2 }^\mu \norm{v}_{H^1}.\end{gathered}$$ By considering the linear isomorphism $H^1 \times \mathbb{R}^3 \to W_{\epsilon, \xi} \times \mathrm{T}_{z_{\epsilon, \xi}}\mathcal{Z}_\epsilon \times \mathbb{R}^3$ and Lemma [Lemma 17](#Lemma:Estimate2ndDer){reference-type="ref" reference="Lemma:Estimate2ndDer"}, we obtain $$\norm*{ \mathrm{D}_{\parens{w,\alpha}} \mathcal{H}_{\epsilon, \xi} \parens{v, \beta} - \begin{pmatrix} L_{\epsilon, \xi}\parens{v^\perp} \\ -\sum_{1\leq i\leq 3}\beta_i\dot{z}_{\epsilon, \xi, i} \\ \sum_{1 \leq i \leq 3} \angles{v^\parallel \mid \dot{z}_{\epsilon, \xi, i}}_{H^1} e_i \end{pmatrix} }_{H^1 \times \mathbb{R}^3} \lesssim \epsilon^\mu \norm{v}_{H^1},$$ where $\parens{v^\perp, v^\parallel} \in W_{\epsilon, \xi} \times \mathrm{T}_{z_{\epsilon, \xi}}\mathcal{Z}_\epsilon$ and $v = v^\perp + v^\parallel$. The result then follows from Lemma [Lemma 15](#Lemma:BoundNormInverseL){reference-type="ref" reference="Lemma:BoundNormInverseL"}. *The mapping [\[Equation:wMapMultiplicity\]](#Equation:wMapMultiplicity){reference-type="eqref" reference="Equation:wMapMultiplicity"}, its regularity and estimation of $\norm{w_{\epsilon, \xi}}_{H^1}$.* In light of the preliminary result and the equality $\mathcal{H} \parens{0, \cdot, 0, 0} \equiv 0$, we can use the Implicit Function Theorem to fix an application of class $C^1$, $$\coi{0, \epsilon_0} \times \mathbb{R}^3 \ni \parens{\epsilon, \xi} \mapsto \parens{w_{\epsilon, \xi}, \alpha_{\epsilon, \xi}} \in H^1 \times \mathbb{R}^3,$$ such that $$\label{Equation:IFT} \mathcal{H} \parens{ \epsilon, \xi, w_{\epsilon, \xi}, \alpha_{\epsilon, \xi} } = 0$$ and $\alpha_{0, \xi} = w_{0, \xi} = 0$ for every $\parens{\epsilon, \xi} \in \coi{0, \epsilon_0} \times \mathbb{R}^3$. The estimate on $\norm{w_{\epsilon, \xi}}_{H^1}$ follows from the definition of $\mathcal{O}$. *Estimation of $\norm{\dot{w}_{\epsilon,\xi,i}}_{H^1}$.* By differentiating [\[Equation:IFT\]](#Equation:IFT){reference-type="eqref" reference="Equation:IFT"}, we deduce that $$0 = \partial_i\mathcal{H}_{ \epsilon, w_{\epsilon, \xi}, \alpha_{\epsilon, \xi} } \parens{\xi} + \mathrm{D}_{\parens{w_{\epsilon, \xi}, \alpha_{\epsilon, \xi}}} \mathcal{H}_{\epsilon, \xi} \parens{ \dot{w}_{\epsilon, \xi, i}, \dot{\alpha}_{\epsilon, \xi, i} }.$$ It follows from the preliminary result that $$\begin{gathered} \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} \leq \norm*{ \brackets*{ \mathrm{D}_{\parens{w_{\epsilon, \xi}, \alpha_{\epsilon, \xi}}} \mathcal{H}_{\epsilon, \xi} }^{-1} \parens*{ \partial_i\mathcal{H}_{ \epsilon, w_{\epsilon, \xi}, \alpha_{\epsilon, \xi} } \parens{\xi} } }_{H^1 \times \mathbb{R}^3} \lesssim \\ \lesssim \norm*{ \partial_i\mathcal{H}_{ \epsilon, w_{\epsilon, \xi}, \alpha_{\epsilon, \xi} } \parens{\xi} }_{H^1 \times \mathbb{R}^3}.\end{gathered}$$ Clearly, $$\begin{gathered} \norm*{ \partial_i\mathcal{H}_{ \epsilon, w_{\epsilon, \xi}, \alpha_{\epsilon, \xi} } \parens{\xi} }_{H^1 \times \mathbb{R}^3} = \\ = \norm*{ \begin{pmatrix} R_\epsilon\parens*{ \mathrm{D}^2_{\parens{ z_{\epsilon, \xi}+w_{\epsilon, \xi} }} E_\epsilon \parens{\dot{z}_{\epsilon, \xi, i}, \cdot} } - \sum_{1 \leq j \leq 3} \alpha_{\epsilon, \xi, j} \ddot{z}_{\epsilon, \xi, i, j} \\ \sum_{1 \leq j \leq 3} \angles{ w_{\epsilon, \xi} \mid \ddot{z}_{\epsilon, \xi, i, j} }_{H^1} e_j \end{pmatrix} }_{H^1 \times \mathbb{R}^3} \lesssim \\ \lesssim \norm*{ \mathrm{D}^2_{\parens{ z_{\epsilon, \xi}+w_{\epsilon, \xi} }} E_\epsilon \parens{\dot{z}_{\epsilon, \xi, i}, \cdot} }_{H^{-1}} + \abs{\alpha_{\epsilon, \xi}} + \norm{w_{\epsilon, \xi}}_{H^1}.\end{gathered}$$ Considering Lemmas [Lemma 16](#Lemma:ApproxDer){reference-type="ref" reference="Lemma:ApproxDer"} and [Lemma 17](#Lemma:Estimate2ndDer){reference-type="ref" reference="Lemma:Estimate2ndDer"}, we obtain $$\begin{gathered} \norm*{ \mathrm{D}^2_{\parens{z_{\epsilon, \xi}+w_{\epsilon, \xi}}} E_\epsilon \parens{\dot{z}_{\epsilon, \xi, i}, \cdot} }_{H^{-1}} \leq \\ \leq \norm*{ \mathrm{D}^2_{\parens{z_{\epsilon, \xi} + w_{\epsilon, \xi}}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } } + \norm*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } }_{H^{-1}} \lesssim \\ \lesssim \epsilon + \norm{w_{\epsilon, \xi}}_{H^1} + \norm{w_{\epsilon, \xi}}_{H^1}^2 + \norm{w_{\epsilon, \xi}}_{H^1}^{p-1} \lesssim \brackets*{ \epsilon \abs{\nabla V \parens{\epsilon \xi} + \epsilon^2} }^\mu.\end{gathered}$$ In view of [\[Equation:IFT\]](#Equation:IFT){reference-type="eqref" reference="Equation:IFT"}, Lemmas [Lemma 12](#Lemma:PseudoCriticalPoints){reference-type="ref" reference="Lemma:PseudoCriticalPoints"} and [Lemma 16](#Lemma:ApproxDer){reference-type="ref" reference="Lemma:ApproxDer"}, we conclude that $\abs{\alpha_{\epsilon, \xi}} \lesssim \epsilon \abs{\nabla V \parens{\epsilon \xi}} + \epsilon^2$. ◻ ## The reduced functional Consider the context of Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}. Given $\epsilon \in \ooi{0, \epsilon_0}$, we define the *reduced functional* $\Phi_\epsilon \colon \mathbb{R}^3 \to \mathbb{R}$ as $$\Phi_\epsilon \parens{\xi} = E_\epsilon \parens{z_{\epsilon, \xi} + w_{\epsilon,\xi}}.$$ In particular, $\Phi_\epsilon$ is of class $C^1$ as a composition of mappings of class $C^1$. Let us prove that, up to shrinking $\epsilon_0$, critical points of $\Phi_\epsilon$ yield critical points of $E_\epsilon$. **Lemma 18**. *Up to shrinking $\epsilon_0$, the following implication holds: if $\epsilon \in \ooi{0, \epsilon_0}$ and $\nabla \Phi_\epsilon \parens{\xi} = 0$, then $\nabla E_\epsilon \parens{z_{\epsilon, \xi} + w_{\epsilon, \xi}} = 0$.* *Proof.* By construction, $$\nabla E_\epsilon \parens{z_{\epsilon, \xi} + w_{\epsilon, \xi}} = \sum_{1 \leq i \leq 3} c_{\epsilon, \xi, i} \dot{z}_{\epsilon, \xi, i} \in \mathrm{T}_{z_{\epsilon, \xi} + w_{\epsilon, \xi}} \mathcal{Z}_\epsilon.$$ In particular, $$\partial_i \Phi_\epsilon \parens{\xi} = \angles{ \dot{z}_{\epsilon, \xi, i} + \dot{w}_{\epsilon, \xi, i} \mid \nabla E_\epsilon \parens{z_{\epsilon, \xi} + w_{\epsilon, \xi}} }_{H^1} = \sum_{1 \leq j \leq 3} c_{\epsilon, \xi, j} \angles{ \dot{z}_{\epsilon, \xi, i} + \dot{w}_{\epsilon, \xi, i} \mid \dot{z}_{\epsilon, \xi, j} }_{H^1}$$ for every $i \in \set{1, 2, 3}$. In view of the previous paragraph, $$M_{\epsilon, \xi} \begin{pmatrix} c_{\epsilon, \xi, 1} \\ c_{\epsilon, \xi, 2} \\ c_{\epsilon, \xi, 3} \end{pmatrix} = 0, ~\text{where}~ M_{\epsilon, \xi} := \parens*{\angles{ \dot{z}_{\epsilon, \xi, i} + \dot{w}_{\epsilon, \xi, i} \mid \dot{z}_{\epsilon, \xi, j} }_{H^1}}_{1 \leq i, j \leq 3}.$$ It follows from Lemmas [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"} and [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} that, up to shrinking $\epsilon_0$, $M_{\epsilon, \xi}$ is non-singular for every $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$. In this situation, the only solution for the previous linear system is $c_{\epsilon, \xi, 1} = c_{\epsilon, \xi, 2} = c_{\epsilon, \xi, 3} = 0$. ◻ Let us introduce a decomposition of $\Phi_\epsilon$ inspired by [@AmbrosettiMalchiodiSecchi2001 (32)]. **Lemma 19**. *We have $$\Phi_\epsilon \parens{\xi} = C_0 V \parens{\epsilon \xi}^\theta + \Lambda_\epsilon \parens{\xi} + \Omega_\epsilon \parens{\xi} + \Psi_\epsilon \parens{\xi}$$ for every $\xi\in\mathbb{R}^3$, where $$C_0:=\parens*{\frac{1}{2}-\frac{1}{p+1}}\norm{U}_{L^{p+1}}^{p+1}; \quad \theta:=\frac{p+1}{p-1}-\frac{3}{2};$$ $$\Lambda_\epsilon\parens{\xi} := \frac{1}{2} \int\cbrackets{ \brackets{ V_\epsilon-V_\epsilon\parens{\xi} } z_{\epsilon,\xi}^2 } + \int\cbrackets{ \brackets{ V_\epsilon-V_\epsilon\parens{\xi} } z_{\epsilon,\xi} w_{\epsilon,\xi} };$$ $$\Omega_\epsilon \parens{\xi} := \frac{\epsilon^3}{4} \int\brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_{\epsilon,\xi}+w_{\epsilon,\xi}}^2 } \parens{z_{\epsilon,\xi}+w_{\epsilon,\xi}}^2 }$$ and $$\Psi_\epsilon \parens{\xi} := \frac{1}{2} \norm{w_{\epsilon,\xi}}_{H^1_{V_\epsilon}}^2 - \frac{1}{p + 1} \int\brackets*{ \abs{z_{\epsilon,\xi}+w_{\epsilon,\xi}}^{p+1} - z_{\epsilon,\xi}^{p+1} - \parens{p+1} z_{\epsilon,\xi}^p w_{\epsilon,\xi} }.$$* *Proof.* Clearly, $$\begin{gathered} \Phi_\epsilon \parens{\xi} = \frac{1}{2} \norm{ z_{\epsilon,\xi} + w_{\epsilon,\xi} }_{H^1_{\lambda_{\epsilon \xi}}}^2 + \frac{1}{2} \int \cbrackets*{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} \parens{z_{\epsilon,\xi} + w_{\epsilon,\xi}}^2 } + \\ + \frac{\epsilon^3}{4} \int \brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_{\epsilon,\xi} + w_{\epsilon,\xi}}^2 } \parens{z_{\epsilon,\xi} + w_{\epsilon,\xi}}^2 } - \frac{1}{p+1} \norm{ z_{\epsilon,\xi} + w_{\epsilon,\xi} }^{p+1}_{L^{p+1}}.\end{gathered}$$ By developing the squares; summing and subtracting $\norm{z_{\epsilon, \xi}}_{L^{p+1}}^{p+1}/\parens{p+1}$, we obtain $$\begin{gathered} \Phi_\epsilon \parens{\xi} = \frac{1}{2} \norm{z_{\epsilon,\xi}}_{H^1_{\lambda_{\epsilon \xi}}}^2 - \frac{1}{p+1} \norm{z_{\epsilon, \xi}}_{L^{p+1}}^{p+1} + \\ + \underbrace{ \frac{1}{2} \int\cbrackets{ \brackets{V_\epsilon-V_\epsilon\parens{\xi}} z_{\epsilon,\xi}^2 } + \int\cbrackets{ \brackets{ V_\epsilon-V_\epsilon\parens{\xi} } z_{\epsilon,\xi} w_{\epsilon,\xi} } }_{= \Lambda_\epsilon \parens{\xi}} + \\ + \underbrace{ \frac{\epsilon^3}{4} \int \brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_{\epsilon,\xi} + w_{\epsilon,\xi}}^2 } \parens{z_{\epsilon,\xi} + w_{\epsilon,\xi}}^2 } }_{= \Omega_\epsilon \parens{\xi}} + \\ + \frac{1}{2} \int\cbrackets{ \brackets{ V_\epsilon-V_\epsilon\parens{\xi} } w_{\epsilon,\xi}^2 } + \frac{1}{2} \norm{w_{\epsilon,\xi}}_{H^1_{\lambda_{\epsilon \xi}}}^2 + \\ - \frac{1}{p+1} \norm{ z_{\epsilon,\xi} + w_{\epsilon,\xi} }^{p+1}_{L^{p+1}} + \frac{1}{p+1} \norm{z_{\epsilon, \xi}}_{L^{p+1}}^{p+1} + \angles{ z_{\epsilon,\xi} \mid w_{\epsilon,\xi} }_{H^1_{\lambda_{\epsilon \xi}}}.\end{gathered}$$ Due to [\[Equation:PDE_z\_epsilon,xi\]](#Equation:PDE_z_epsilon,xi){reference-type="eqref" reference="Equation:PDE_z_epsilon,xi"}, $$\norm{z_{\epsilon,\xi}}_{H^1_{\lambda_{\epsilon \xi}}}^2 = \norm{z_{\epsilon,\xi}}_{L^{p+1}}^{p+1} \quad \text{and} \quad \angles{ z_{\epsilon, \xi} \mid w_{\epsilon, \xi} }_{H^1_{\lambda_{\epsilon \xi}}} = \int\parens{ z_{\epsilon, \xi}^p w_{\epsilon, \xi} }.$$ The previous equalities imply $$\Phi_\epsilon \parens{\xi} = \parens*{\frac{1}{2}-\frac{1}{p+1}} \norm{z_{\epsilon, \xi}}_{L^{p+1}}^{p+1} + \Lambda_\epsilon \parens{\xi} + \Omega_\epsilon \parens{\xi} + \Psi_\epsilon \parens{\xi}.$$ In this situation, the result follows from the definition of $z_{\epsilon, \xi}$. ◻ To finish, we use the previous decomposition to expand $\Phi_\epsilon$ and $\nabla \Phi_\epsilon$ by arguing as in the proof of [@AmbrosettiMalchiodi2006 Lemma 8.11]. **Lemma 20**. *We have $$\abs*{\Phi_\epsilon \parens{\xi} - C_0 V \parens{\epsilon \xi}^\theta} \lesssim \epsilon \quad\text{and}\quad \abs*{ \nabla \Phi_\epsilon \parens{\xi} - \epsilon a \parens{\epsilon \xi} \nabla V \parens{\epsilon \xi} } \lesssim \epsilon^{1+\mu}$$ for every $\parens{\epsilon,\xi} \in \ooi{0, \epsilon_0} \times \mathbb{R}^3$, where $a \parens{\epsilon \xi} := \theta C_0 V \parens{\epsilon \xi}^{\theta - 1}$.* *Proof.* Consider the decomposition in Lemma [Lemma 19](#Lemma:DecompMult){reference-type="ref" reference="Lemma:DecompMult"}. *Expansion of $\Phi_\epsilon$.* *-Estimation of $\abs{\Lambda_\epsilon}$.* In light of Remark [Remark 10](#Remark:TaylorMult){reference-type="ref" reference="Remark:TaylorMult"} and the Cauchy--Schwarz Inequality, $$\abs{\Lambda_\epsilon \parens{\xi}} \lesssim \epsilon \abs{\nabla V \parens{\epsilon \xi}} \cbrackets*{ \int \parens*{\abs{x - \xi} z_{\epsilon, \xi}^2} + \norm{w_{\epsilon, \xi}}_{H^1}^2 \brackets*{ \int \parens*{\abs{x - \xi}^2z_{\epsilon,\xi}^2} }^{1/2} }.$$ It then follows from [\[V_1\]](#V_1){reference-type="ref" reference="V_1"}, [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"} and Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} that $\abs{\Lambda_\epsilon \parens{\xi}} \lesssim \epsilon$. *-Estimation of $\abs{\Omega_\epsilon}$.* Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"}, [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"} and [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} imply $\abs{\Omega_\epsilon \parens{\xi}} \lesssim \epsilon^3$. *-Estimation of $\abs{\Psi_\epsilon}$.* Due to Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}, $\norm{w_{\epsilon, \xi}}_{H^1_{V_\epsilon}}^2 \lesssim \epsilon^2$. In view of Lemma [Lemma 9](#Lemma:Elementary){reference-type="ref" reference="Lemma:Elementary"} and the Sobolev embeddings, $$\abs*{ \int\brackets*{ \abs{z_{\epsilon,\xi}+w_{\epsilon,\xi}}^{p+1} - z_{\epsilon,\xi}^{p+1} - \parens{p+1} z_{\epsilon,\xi}^p w_{\epsilon,\xi} } } \lesssim \norm{w_{\epsilon, \xi}}_{H^1}^2 + \norm{w_{\epsilon, \xi}}_{H^1}^{p + 1}.$$ By considering Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} once again, we obtain $$\abs*{ \int\brackets*{ \abs{z_{\epsilon,\xi}+w_{\epsilon,\xi}}^{p+1} - z_{\epsilon,\xi}^{p+1} - \parens{p+1} z_{\epsilon,\xi}^p w_{\epsilon,\xi} } } \lesssim \epsilon^2.$$ *Expansion of $\nabla \Phi_\epsilon$.* Clearly, $$\begin{gathered} \partial_i \Phi_\epsilon \parens{\xi} = \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i} } + \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{w}_{\epsilon, \xi, i} } + \\ + \brackets*{ \mathrm{D}_{z_{\epsilon, \xi} + w_{\epsilon, \xi}} E_\epsilon \parens{\dot{z}_{\epsilon, \xi, i}} - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{\dot{z}_{\epsilon, \xi, i}} - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{z}_{\epsilon, \xi, i}} } + \\ + \brackets*{ \mathrm{D}_{z_{\epsilon, \xi} + w_{\epsilon, \xi}} E_\epsilon \parens{\dot{w}_{\epsilon, \xi, i}} - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{\dot{w}_{\epsilon, \xi, i}} - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{w}_{\epsilon, \xi, i}} } + \\ + \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{z}_{\epsilon, \xi, i}} + \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{w}_{\epsilon, \xi, i}}.\end{gathered}$$ Therefore, $$\begin{gathered} \label{Equation:Aux} \abs*{ \partial_i \Phi_\epsilon \parens{\xi} - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i} } } \leq \norm*{\mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon}_{H^{-1}} \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} + \\ + \norm*{\mathcal{R}_{\epsilon, \xi}}_{H^{-1}} \parens*{ \norm{\dot{z}_{\epsilon, \xi, i}}_{H^1} + \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} } + \\ + \norm*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } }_{H^{-1}} \norm{w_{\epsilon,\xi}}_{H^1} + \abs*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{w}_{\epsilon, \xi, i}} },\end{gathered}$$ where $$\mathcal{R}_{\epsilon, \xi} := \mathrm{D}_{z_{\epsilon, \xi} + w_{\epsilon, \xi}} E_\epsilon - \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon - \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \cdot} \in H^{-1}.$$ Let us estimate the terms on the right-hand side of [\[Equation:Aux\]](#Equation:Aux){reference-type="eqref" reference="Equation:Aux"}. Due to Lemmas [Lemma 12](#Lemma:PseudoCriticalPoints){reference-type="ref" reference="Lemma:PseudoCriticalPoints"} and [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}, $$\norm*{\mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon}_{H^{-1}} \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} \lesssim \epsilon^{1 + \mu}.$$ In view of Lemmas [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}, [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} and [Lemma 16](#Lemma:ApproxDer){reference-type="ref" reference="Lemma:ApproxDer"}, $$\norm*{ \mathcal{R}_{\epsilon, \xi} }_{H^{-1}} \parens*{ \norm{\dot{z}_{\epsilon, \xi, i}}_{H^1} + \norm{\dot{w}_{\epsilon, \xi, i}}_{H^1} } \lesssim \epsilon^{1 + \mu}.$$ Lemmas [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"}, [Lemma 17](#Lemma:Estimate2ndDer){reference-type="ref" reference="Lemma:Estimate2ndDer"} imply $$\norm*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i}, \cdot } }_{H^{-1}} \norm{w_{\epsilon,\xi}}_{H^1} \lesssim \epsilon^2.$$ Similar arguments show that $$\abs*{ \mathrm{D}^2_{z_{\epsilon, \xi}} E_\epsilon \parens{w_{\epsilon,\xi}, \dot{w}_{\epsilon, \xi, i}} } \lesssim \epsilon^{1 + \mu}.$$ To conclude, it suffices to show that $$\abs*{ \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i} } - a \parens{\epsilon \xi} \partial_i V \parens{\epsilon \xi} } \lesssim \epsilon^2.$$ Indeed, $$E_\epsilon \parens{z_{\epsilon, \xi}} = C_0 V \parens{\epsilon \xi}^\theta + \frac{1}{2} \int \cbrackets*{ \brackets{V_\epsilon - V \parens{\epsilon \xi}} z_{\epsilon, \xi}^2 } + \frac{\epsilon^3}{4} \int \parens*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi}^2 }.$$ Differentiating, we obtain $$\begin{gathered} \label{Equation:Aux2} \mathrm{D}_{z_{\epsilon, \xi}} E_\epsilon \parens{ \dot{z}_{\epsilon, \xi, i} } - \epsilon a \parens{\epsilon \xi} \partial_i V \parens{\epsilon \xi} = \\ = \int \brackets{ V_\epsilon \parens{x} - V \parens{\epsilon \xi} - \epsilon \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} } z_{\epsilon, \xi} \parens{x} \dot{z}_{\epsilon, \xi, i} \parens{x} \mathrm{d}x + \\ + \epsilon \int \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} z_{\epsilon, \xi} \parens{x} \brackets*{ \dot{z}_{\epsilon, \xi, i} \parens{x} + \partial_i z_{\epsilon, \xi} \parens{x} } \mathrm{d}x + \\ - \frac{\epsilon}{2} \brackets*{ 2 \int \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} z_{\epsilon, \xi} \parens{x} \partial_i z_{\epsilon, \xi} \parens{x} \mathrm{d}x + \partial_i V \parens{\epsilon \xi} \int \parens{z_{\epsilon, \xi}^2} } + \\ + \epsilon^3 \int \parens*{ K_\epsilon \phi_{\epsilon, z_{\epsilon, \xi}^2} z_{\epsilon, \xi} \dot{z}_{\epsilon, \xi, i} }.\end{gathered}$$ Let us estimate the terms on the right-hand side of [\[Equation:Aux2\]](#Equation:Aux2){reference-type="eqref" reference="Equation:Aux2"}. It follows from Remark [Remark 10](#Remark:TaylorMult){reference-type="ref" reference="Remark:TaylorMult"} and Lemma [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"} that the first term may be estimated by $\epsilon^2$. Due to [\[V_1\]](#V_1){reference-type="ref" reference="V_1"} and Lemma [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}, the second term is estimated by $\epsilon^2$. Consider the third term. An integration by parts shows that $$\begin{gathered} 2 \int_0^\infty \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} z_{\epsilon, \xi} \parens{x} \partial_i z_{\epsilon, \xi} \parens{x} \mathrm{d}x_i + \partial_i V \parens{\epsilon \xi} \int_0^\infty z_{\epsilon, \xi} \parens{x}^2 \mathrm{d}x_i = \\ = \int_0^\infty \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi} z_{\epsilon, \xi} \parens{x}^2 \mathrm{d}x_i = 0\end{gathered}$$ because $x_i \mapsto z_{\epsilon, \xi} \parens{x}$ is even and $x_i \mapsto \nabla V \parens{\epsilon \xi} \cdot \parens{x - \xi}$ is odd. Finally, the last term is estimated by $\epsilon^3$ due to Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"} and [Lemma 11](#Lemma:z_eps,xi){reference-type="ref" reference="Lemma:z_eps,xi"}. ◻ ## Proof of Theorem [Theorem 5](#Theorem:MultiplicitySolutions){reference-type="ref" reference="Theorem:MultiplicitySolutions"} {#proof-of-theorem-theoremmultiplicitysolutions} We need two preliminary results for the proof. The first result is that concentration necessarily occurs around critical points of $V$. **Proposition 21**. *Suppose that $x_0 \in \mathbb{R}^3$, $\epsilon_0 \in \ooi{0, 1}$ and $$\set*{ u_\epsilon \in H^1: \nabla E_\epsilon \parens{u_\epsilon} = 0; \epsilon \in \ooi{0,\epsilon_0} }$$ are such that $$\label{Equation:ConcentrationOccurs} \norm*{ u_\epsilon - U_{\lambda_{x_0}} \parens{\cdot-x_0} }_{H^1}\to 0 ~\text{as}~ \epsilon \to 0^+.$$ We conclude that $\nabla V\parens{x_0}=0$.* In fact, the previous proposition is analogous to [@IanniVaira2008 Theorem 5.1] and may be proved accordingly. The second preliminary result is [@AmbrosettiMalchiodiSecchi2001 Theorem 1], which reads as follows. **Theorem 22**. *Let $f \in C^2 \parens{\mathbb{R}^3}$, $\mathcal{M}$ be a compact non-degenerate critical manifold of $f$, $\mathcal{N}$ be a neighborhood of $\mathcal{M}$ and $g \in C^1 \parens{\mathcal{N}}$. We conclude that if $\norm{f-g}_{C^1}$ is sufficiently small, then $g$ has at least $\mathop{\mathrm{cupl}}\parens{\mathcal{M}}+1$ critical points in $\mathcal{N}$.* We proceed to the proof, based on the arguments in [@AmbrosettiMalchiodi2006 Proof of Theorem 8.5]. *Proof of Theorem [Theorem 5](#Theorem:MultiplicitySolutions){reference-type="ref" reference="Theorem:MultiplicitySolutions"}.* *Multiplicity of solutions.* It suffices to apply Theorem [Theorem 22](#Theorem:Temp){reference-type="ref" reference="Theorem:Temp"}. Indeed, let $f = C_0 V_\epsilon^\theta$; $\mathcal{M} = M$ and $\mathcal{N}$ be a bounded neighborhood of $\mathcal{M}$ in $\mathbb{R}^3$. It follows from Lemma [Lemma 20](#Lemma:ExpansionMultiplicity){reference-type="ref" reference="Lemma:ExpansionMultiplicity"} that $\norm{f - \Phi_\epsilon}_{C^1} \to 0$ as $\epsilon \to 0$, hence the result. *Concentration around points of $M$.* Up to shrinking $\mathcal{N}$, we can suppose that if $x \in \overline{\mathcal{N}}$ and $\nabla V\parens{x}=0$, then $x \in M$. Due to the previous result, we can fix $\epsilon_0 \in \ooi{0, 1}$ such that given $\epsilon \in \ooi{0, \epsilon_0}$, $u_\epsilon:=z_{\epsilon, \xi_\epsilon}+w_{\epsilon_n, \xi_\epsilon}$ satisfies $\nabla E_{\epsilon} \parens{u_\epsilon} = 0$. Suppose that $x_0 \in \overline{\mathcal{N}}$ is an accumulation point of $\set{\xi_\epsilon}_{\epsilon \in \ooi{0, \epsilon_0}}$. We deduce from the estimate on Lemma [Lemma 13](#Lemma:AuxMult){reference-type="ref" reference="Lemma:AuxMult"} that [\[Equation:ConcentrationOccurs\]](#Equation:ConcentrationOccurs){reference-type="eqref" reference="Equation:ConcentrationOccurs"} holds up to subsequence. It follows from Proposition [Proposition 21](#Proposition:Concentration){reference-type="ref" reference="Proposition:Concentration"} that $x_0 \in M$, hence the result. ◻ # Concentration results {#ConcentrationResults} Our present goal is to prove Theorems [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} and [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"}, so let us suppose throughout this section that $x_0$ is a critical point of $V$. To simplify the notation, we suppose further that $V\parens{x_0}=1$ and $x_0=0$. ## Lyapunov--Schmidt reduction {#Section:LyapunovConcentration} As in Section [3.1](#PseudoMult){reference-type="ref" reference="PseudoMult"}, we begin with a Taylor expansion. **Remark 23**. Due to [\[V_1\]](#V_1){reference-type="ref" reference="V_1"}, $$\abs*{ V_\epsilon \parens{x} - 1 - \frac{\epsilon^2}{2} \mathrm{D}^2_0 V \parens{x, x} } \lesssim \epsilon^3 \abs{x}^3$$ if the hypotheses of Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} are satisfied; due to [\[V_3\]](#V_3){reference-type="ref" reference="V_3"}, $$\abs*{ V_\epsilon \parens{x} - 1 - \frac{\epsilon^n}{n!} \mathrm{D}^n_0 V \parens{x, \ldots, x} } \lesssim \epsilon^{n+1} \abs{x}^{n+1}$$ if the hypotheses of Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"} are satisfied and $n < \infty$, where the estimates hold for every $\parens{\epsilon, x} \in \ooi{0, 1} \times \mathbb{R}^3$. The following estimate is analogous to [@IanniVaira2008 Lemma 3.1] and may be proved accordingly. **Lemma 24**. *Suppose that $s$ is a positive integer and $\mathrm{D}^i_0K=0$ for every $i\in\set{0,\ldots,s-1}$. Given $\eta \in \ooi{0, \infty}$, we conclude that $$\int\abs{ K_\epsilon \phi_{\epsilon, u z_\xi} w z_\xi } \lesssim \epsilon^{2s}\norm{u}_{H^1}\norm{w}_{H^1}$$ for every $u,w\in H^1$ and $\xi\in B_\eta$.* As Problem [\[Equation:SimpleProblem\]](#Equation:SimpleProblem){reference-type="eqref" reference="Equation:SimpleProblem"} is invariant by translation, we deduce that $$\mathcal{Z} := \set*{ z_\xi:=U\parens{\cdot-\xi}: \xi\in\mathbb{R}^3 }$$ is a non-compact critical manifold of $\overline{I}_1$. Furthermore, [@AmbrosettiMalchiodi2006 Lemma 4.1] shows that $\mathcal{Z}$ is non-degenerate. The next result shows that we can bound the derivative of $E_\epsilon$ at points in $\mathcal{Z}$ similarly as in Lemma [Lemma 12](#Lemma:PseudoCriticalPoints){reference-type="ref" reference="Lemma:PseudoCriticalPoints"}. **Lemma 25**. *Given $\eta\in\ooi{0,\infty}$, there exists $\epsilon_\eta\in\ooi{0,1}$ such that $$\norm{\nabla E_\epsilon\parens{z_\xi}}_{H^1} \lesssim \begin{cases} \epsilon^2 &\text{if the hypotheses of Theorem \ref{Theorem:SolutionsNonDegenerate} are satisfied}; \\ \epsilon^\gamma &\text{if the hypotheses of Theorem \ref{Theorem:SolutionsDegenerate} are satisfied} \end{cases}$$ for every $\parens{\epsilon,\xi} \in \ooi{0,\epsilon_\eta} \times B_\eta$.* The previous lemma is proved very similarly as [@IanniVaira2008 Propositions 3.1, 3.2], so we omit its proof. In view of Lemma [Lemma 25](#Lemma:PseudoCriticalConcentration){reference-type="ref" reference="Lemma:PseudoCriticalConcentration"}, we can prove the result that follows by arguing as in Section [3.2](#AuxMult){reference-type="ref" reference="AuxMult"}. **Lemma 26**. *Given $\eta\in\ooi{0,\infty}$, there exist $\epsilon_\eta\in\ooi{0,1}$ and an application of class $C^1$, $$\ooi{0, \epsilon_\eta} \times B_\eta \ni \parens{\epsilon, \xi} \mapsto w_{\eta, \epsilon,\xi} \in H^1,$$ such that given $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_\eta} \times B_\eta$, $$\Pi_\xi \parens{ \nabla E_\epsilon \parens{z_\xi + w_{\eta, \epsilon, \xi}} } = 0 \quad\text{and}\quad w_{\eta, \epsilon, \xi} \in W_\xi := \parens{ \mathrm{T}_{z_\xi} \mathcal{Z} }^\perp.$$ Furthermore, $$\norm{w_{\eta,\epsilon,\xi}}_{H^1} \lesssim \epsilon^2; \quad \quad \quad \quad \norm{\dot{w}_{\eta,\epsilon,\xi,i}}_{H^1} \lesssim \epsilon^{2\mu}$$ if the hypotheses of Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} are satisfied and $$\norm{w_{\eta,\epsilon,\xi}}_{H^1} \lesssim \epsilon^\gamma; \quad \quad \quad \quad \norm{\dot{w}_{\eta,\epsilon,\xi,i}}_{H^1} \lesssim \epsilon^{\gamma \mu}$$ if the hypotheses of Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"} are satisfied, where the estimates hold for every $\parens{\epsilon,\xi}\in\ooi{0,\epsilon_\eta}\times B_\eta$ and $i\in\set{1,2,3}$.* Fix $\eta\in\ooi{0,\infty}$; let $\epsilon_\eta\in\ooi{0,1}$ be such that the previous lemmas hold and fix $\epsilon\in\ooi{0,\epsilon_\eta}$. In this situation, we define the *reduced functional* $\Phi_{\eta, \epsilon} \colon B_\eta \to \mathbb{R}$ as $$\Phi_{\eta, \epsilon} \parens{\xi} = E_\epsilon \parens{z_\xi + w_{\eta,\epsilon,\xi}}.$$ Once again, $\Phi_{\eta, \epsilon}$ is of class $C^1$. **Lemma 27**. *Up to shrinking $\epsilon_\eta$, the following implication holds: if $\epsilon \in \ooi{0, \epsilon_\eta}$ and $\nabla \Phi_{\eta, \epsilon} \parens{\xi} = 0$, then $\nabla E_\epsilon \parens{z_\xi + w_{\eta, \epsilon, \xi}}=0$.* *Proof.* On one hand, $$\partial_i \Phi_{\eta, \epsilon} \parens{\xi} = \mathrm{D}_{\parens{z_\xi + w_{\eta,\epsilon,\xi}}} E_\epsilon \parens{ \dot{z}_{\xi,i} + \dot{w}_{\eta,\epsilon,\xi,i} }$$ for every $i\in\set{1,2,3}$. On the other hand, [@AmbrosettiMalchiodi2006 Theorem 2.25] shows that $$\tilde{\mathcal{Z}}_\eta:=\set{ z_\xi+w_{\eta,\epsilon,\xi}: \xi \in B_\eta }$$ is a natural constraint of $E_\epsilon$, hence the result. Alternatively, one may argue as in the proof of Lemma [Lemma 18](#Lemma:NaturalConstraintMultiplicity){reference-type="ref" reference="Lemma:NaturalConstraintMultiplicity"}. ◻ To finish the section, we decompose $\Phi_{\eta,\epsilon}$ similarly as in Lemma [Lemma 19](#Lemma:DecompMult){reference-type="ref" reference="Lemma:DecompMult"}. **Lemma 28**. *We have $$\Phi_{\eta,\epsilon}\parens{\xi} = C_0 + \Lambda_{\eta, \epsilon} \parens{\xi} + \Omega_{\eta, \epsilon} \parens{\xi} + \Psi_{\eta, \epsilon} \parens{\xi}$$ for every $\xi\in B_\eta$, where $C_0$ was defined in Lemma [Lemma 19](#Lemma:DecompMult){reference-type="ref" reference="Lemma:DecompMult"}; $$\Lambda_{\eta, \epsilon}\parens{\xi} := \frac{1}{2} \int \brackets*{\parens{V_\epsilon-1} z_\xi^2} + \int \brackets*{\parens{V_\epsilon-1} z_\xi w_{\eta,\epsilon,\xi}};$$ $$\Omega_{\eta, \epsilon} \parens{\xi} := \frac{\epsilon^3}{4} \int\brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_\xi+w_{\eta,\epsilon,\xi}}^2 } \parens{z_\xi+w_{\eta,\epsilon,\xi}}^2 }$$ and $$\Psi_{\eta, \epsilon} \parens{\xi} := \frac{1}{2}\norm{w_{\eta,\epsilon,\xi}}_{H^1_{V_\epsilon}}^2 - \int\brackets*{ \abs{z_\xi+w_{\eta,\epsilon,\xi}}^{p+1} - z_\xi^{p+1} - \parens{p+1} z_\xi^p w_{\eta,\epsilon,\xi} }.$$* ## Proof of Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"} {#proof-of-theorem-theoremsolutionsnondegenerate} Fix $\eta \in \ooi{0, \infty}$ such that $0$ is the only critical point of $V$ in $\overline{B_\eta}$ and let $\epsilon_\eta \in \ooi{0, 1}$ be such that the lemmas in the previous section hold. Let us proceed to an expansion of $\nabla \Phi_{\eta, \epsilon}$. **Lemma 29**. *There exists $\delta\in\ooi{0,\infty}$ such that $$\abs{ \nabla \Phi_{\eta, \epsilon} \parens{\xi} - \epsilon^2 \Gamma_1 \parens{\xi} } \lesssim \epsilon^{2+\delta}$$ for every $\parens{\epsilon,\xi}\in\ooi{0,\epsilon_\eta}\times B_\eta$, where $$\Gamma_1 \parens{\xi} := \sum_{1 \leq i \leq 3} \brackets*{ \partial_i^2 V \parens{0} \xi_i \int x_i U \parens{x} \partial_i U \parens{x} \mathrm{d}x } e_i.$$* *Proof.* Consider the decomposition in Lemma [Lemma 28](#Lemma:DecompositionOfPhiConcentration){reference-type="ref" reference="Lemma:DecompositionOfPhiConcentration"}. *Expansion of $\partial_i \Lambda_{\eta, \epsilon}$.* It is clear that $$\partial_i\Lambda_{\eta, \epsilon}\parens{\xi} = \int\brackets{ \parens{V_\epsilon-1} z_\xi \dot{z}_{\xi,i} } + \int\brackets{ \parens{ V_\epsilon-1 } \dot{z}_{\xi,i} w_{\epsilon,\xi} } + \int\brackets{ \parens{ V_\epsilon-1 } z_{\xi} \dot{w}_{\epsilon,\xi,i} }.$$ *- Expansion of $\int \parens{V_\epsilon-1} z_\xi \dot{z}_{\xi,i}$.* Due to [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"} and Remark [Remark 23](#Remark:Taylor){reference-type="ref" reference="Remark:Taylor"}, $$\begin{gathered} \abs*{ \int\brackets{ \parens{V_\epsilon-1} z_\xi \dot{z}_{\xi,i} } - \frac{\epsilon^2}{2} \int \mathrm{D}^2_0V\parens{x,x} z_{\xi}\parens{x} \dot{z}_{\xi,i}\parens{x} \mathrm{d}x } \lesssim \\ \lesssim \epsilon^3 \int \abs{x}^3 z_\xi\parens{x} \abs{\dot{z}_{\xi,i}\parens{x}} \mathrm{d}x \lesssim \epsilon^3.\end{gathered}$$ It follows from a change of variable that $$\int \mathrm{D}^2_0 V\parens{x,x} z_\xi \parens{x} \dot{z}_{\xi,i} \parens{x} \mathrm{d}x = \int \mathrm{D}^2_0 V\parens{x + \xi,x + \xi} U \parens{x} \partial_i U \parens{x} \mathrm{d}x.$$ Considering that $U$ is an even function and $\partial_i U$ is an odd function, we arrive at $$\int \mathrm{D}^2_0 V\parens{x,x} z_\xi \parens{x} \dot{z}_{\xi,i} \parens{x} \mathrm{d}x = 2 \partial_i^2 V \parens{0} \xi_i \int x_i U \parens{x} \partial_i U \parens{x} \mathrm{d}x.$$ *- Estimation of $\int\brackets{ \parens{V_\epsilon-1} \dot{z}_{\xi,i} w_{\epsilon,\xi} }$, $\int\brackets{ \parens{V_\epsilon-1} z_{\xi} \dot{w}_{\epsilon,\xi,i} }$.* In view of [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"}, [\[V_1\]](#V_1){reference-type="ref" reference="V_1"} and Lemma [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"}, $$\int\abs{ \parens{V_\epsilon-1} \dot{z}_{\xi,i} w_{\epsilon,\xi} } \lesssim \epsilon^2 \norm{w_{\epsilon,\xi}}_{H^1} \brackets*{ \int \mathrm{D}^2_0V\parens{x,x}^2 \dot{z}_{\xi,i}\parens{x}^2 \mathrm{d}x }^{1/2} \lesssim \epsilon^4$$ and $\int\abs{ \parens{V_\epsilon-1} z_{\xi} \dot{w}_{\epsilon,\xi,i} } \lesssim \epsilon^{2 \parens{1+\mu}}.$ *Estimation of $\abs{\partial_i \Omega_{\eta, \epsilon}}$.* Due to Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"} and [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"}, we obtain $\abs{\partial_i \Omega_{\eta, \epsilon} \parens{\xi}} \lesssim \epsilon^3$. *Estimation of $\abs{\partial_i \Psi_{\eta, \epsilon}}$.* Clearly, $$\begin{gathered} \abs{\partial_i\Psi_{\eta, \epsilon}\parens{\xi}} \leq \norm{w_{\epsilon,\xi}}_{H^1} \norm{\dot{w}_{\epsilon,\xi,i}}_{H^1} + \int \abs*{ \parens{V_\epsilon-1} w_{\epsilon,\xi} \dot{w}_{\epsilon,\xi,i} } + \\ + \parens{p+1} \int\abs*{ \parens*{ \parens{z_\xi+w_{\epsilon,\xi}} \abs{z_\xi+w_{\epsilon,\xi}}^{p-1} - z_\xi^p - p z_\xi^{p-1} w_{\epsilon,\xi} } \dot{z}_{\xi,i} } + \\ + \parens{p+1} \int\abs*{ \brackets*{ \parens{z_\xi+w_{\epsilon,\xi}} \abs{z_\xi+w_{\epsilon,\xi}}^{p-1} - z_\xi^p } \dot{w}_{\epsilon,\xi,i} }.\end{gathered}$$ As $V$ is bounded, we can use Lemma [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"} to conclude that $$\norm{w_{\epsilon,\xi}}_{H^1} \norm{\dot{w}_{\epsilon,\xi,i}}_{H^1} + \int \abs*{ \parens{V_\epsilon-1} w_{\epsilon,\xi} \dot{w}_{\epsilon,\xi,i} } \lesssim \epsilon^{2\parens{1+\mu}}.$$ In view of Lemmas [Lemma 9](#Lemma:Elementary){reference-type="ref" reference="Lemma:Elementary"} and [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"}, $$\int\abs*{ \parens*{ \parens{z_\xi+w_{\epsilon,\xi}} \abs{z_\xi+w_{\epsilon,\xi}}^{p-1} - z_\xi^p - p z_\xi^{p-1} w_{\epsilon,\xi} } \dot{z}_{\xi,i} } \lesssim \epsilon^{\min\parens{4,2p}}$$ and $$\int\abs*{ \brackets*{ \parens{z_\xi+w_{\epsilon,\xi}} \abs{z_\xi+w_{\epsilon,\xi}}^{p-1} - z_\xi^p } \dot{w}_{\epsilon,\xi,i} } \lesssim \epsilon^{2 \parens{1+\mu}}.$$ ◻ Now, we argue similarly as in the proof of [@AmbrosettiMalchiodi2006 Theorem 2.17] to prove Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"}. *Proof of Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"}.* *Existence of critical point.* The only root of $\Gamma_1$ in $B_\eta$ is $0$, so $\deg \parens{\Gamma_1, B_\eta, 0}\neq 0$, where $\deg$ denotes the topological degree. In view of the continuity of $\deg \parens{\cdot, B_\eta, 0}$ and Lemma [Lemma 29](#Lemma:ExpansionNonDegenerate){reference-type="ref" reference="Lemma:ExpansionNonDegenerate"}, we deduce that, up to shrinking $\epsilon_\eta$, $\deg \parens{\nabla \Phi_{\eta, \epsilon}, B_\eta, 0} \neq 0$ for every $\epsilon \in \ooi{0, \epsilon_\eta}$. We conclude that given $\epsilon \in \ooi{0, \epsilon_\eta}$, $\Phi_{\eta,\epsilon}$ has a critical point $\xi_\epsilon \in B_\eta$ and $\nabla E_\epsilon \parens{u_\epsilon} = 0$, where $u_\epsilon := z_{\xi_\epsilon} + w_{\eta, \epsilon, \xi_\epsilon}$. *Concentration around $0$.* It suffices to argue as in the concentration part in the proof of Theorem [Theorem 5](#Theorem:MultiplicitySolutions){reference-type="ref" reference="Theorem:MultiplicitySolutions"} by taking $\mathcal{M}=\set{0}$; $\mathcal{N} = B_\eta$; $f = \epsilon^2 \Gamma_1$ and $g = \Phi_{\eta, \epsilon}$. ◻ ## Proof of Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"} {#proof-of-theorem-theoremsolutionsdegenerate} We begin with an expansion of $\nabla \Phi_{\eta, \epsilon}$. **Lemma 30**. *Fix $\eta \in \ooi{0, \infty}$ and let $\epsilon_\eta$ be furnished by Lemma [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"}. We conclude that there exists $\delta\in\ooi{0,\infty}$ such that $$\abs{ \nabla \Phi_{\eta, \epsilon} \parens{\xi} - \epsilon^\gamma \Gamma_2 \parens{\xi} } \lesssim \epsilon^{\gamma+\delta}$$ for every $\parens{\epsilon,\xi}\in\ooi{0,\epsilon_\eta}\times B_\eta$, where $$\Gamma_2\parens{\xi} := \begin{cases} f\parens{\xi} &\text{if}~\gamma=n<2m+3; \\ g\parens{\xi} &\text{if}~n>2m+3=\gamma; \end{cases}$$ and $$f \parens{\xi} := \frac{1}{n!} \sum_{\substack{0\leq\alpha\leq n \\ 1\leq i\leq 3}} \brackets*{ \partial_i^n V \parens{0} \binom{n}{\alpha} \xi_i^{n-\alpha} \int x_i^\alpha U \parens{x} \partial_i U \parens{x} \mathrm{d}x } e_i.$$* *Proof.* The difference with the proof of Lemma [Lemma 29](#Lemma:ExpansionNonDegenerate){reference-type="ref" reference="Lemma:ExpansionNonDegenerate"} is that the lowest order term of the expansion of $\partial_i\Phi_{\eta,\epsilon}$ will come from the expansion of $\partial_i \Lambda_{\eta, \epsilon}$ if $n<2m+3$ and from the expansion of $\partial_i \Omega_{\eta, \epsilon}$ if $n>2m+3$. *Expansion of $\partial_i \Lambda_{\eta, \epsilon}$.* Suppose that $n = \infty$. Due to [\[V_1\]](#V_1){reference-type="ref" reference="V_1"}, $$\abs{V_\epsilon \parens{x} - 1} = \epsilon^L \abs*{ \int_0^1 t^L \mathrm{D}_{t \epsilon x}^L V \parens{x, \ldots, x} \mathrm{d}t } \lesssim \epsilon^L.$$ for every $x \in \mathbb{R}^3$ and $L \in \mathbb{N}$. Therefore, Lemma [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"} implies $\abs{\partial_i \Lambda_{\eta, \epsilon} \parens{\xi}} \lesssim \epsilon^L$ for every $L \in \mathbb{N}$ and $\parens{\epsilon, \xi} \in \ooi{0, \epsilon_\eta} \times \mathbb{R}^3$. Suppose that $n < \infty$. Let us prove that $$\abs*{ \partial_i \Lambda_{\eta, \epsilon} \parens{\xi} - f \parens{\xi} \cdot e_i } \lesssim \epsilon^{n + \min\parens{1, \gamma \mu}}.$$ Due to [\[Equation:ExponentialDecay\]](#Equation:ExponentialDecay){reference-type="eqref" reference="Equation:ExponentialDecay"} and Remark [Remark 23](#Remark:Taylor){reference-type="ref" reference="Remark:Taylor"}, $$\begin{gathered} \abs*{ \int\brackets{ \parens{V_\epsilon-1} z_\xi \dot{z}_{\xi,i} } - \frac{\epsilon^n}{n!} \int \mathrm{D}^n_0V\parens{x, \ldots, x} z_{\xi}\parens{x} \dot{z}_{\xi,i}\parens{x} \mathrm{d}x } \lesssim \\ \lesssim \epsilon^{n+1} \int \abs{x}^{n+1} z_\xi\parens{x} \abs{\dot{z}_{\xi,i}\parens{x}} \mathrm{d}x \lesssim \epsilon^{n+1}.\end{gathered}$$ We can argue as in the analogous situation in the proof of Lemma [Lemma 29](#Lemma:ExpansionNonDegenerate){reference-type="ref" reference="Lemma:ExpansionNonDegenerate"} to deduce that $$\int \mathrm{D}^n_0V\parens{x, \ldots, x} z_{\xi}\parens{x} \dot{z}_{\xi,i}\parens{x} \mathrm{d}x = f \parens{\xi} \cdot e_i.$$ Let us estimate the remaining term. We have $$\int\abs{ \parens{V_\epsilon-1} \dot{z}_{\xi,i} w_{\epsilon,\xi} } \lesssim \epsilon^n \norm{w_{\epsilon,\xi}}_{H^1} \brackets*{ \int \mathrm{D}^n_0 V\parens{x, \ldots, x}^2 \dot{z}_{\xi,i}\parens{x}^2 \mathrm{d}x }^{1/2} \lesssim \epsilon^{n + \gamma}$$ and $\int\abs{ \parens{V_\epsilon-1} z_{\xi} \dot{w}_{\epsilon,\xi,i} } \lesssim \epsilon^{n + \gamma \mu}.$ *Expansion of $\partial_i \Omega_{\eta, \epsilon}$.* Suppose that $m = \infty$. Due to Lemma [Lemma 24](#Lemma:EstimateForK){reference-type="ref" reference="Lemma:EstimateForK"}, we can treat this case similarly as the case $n = \infty$ in the expansion of $\partial_i \Lambda_{\eta, \epsilon}$. Suppose that $m < \infty$. Let us prove that $$\abs*{ \partial_i \Omega_{\eta, \epsilon} \parens{\xi} - g \parens{\xi} \cdot e_i } \lesssim \epsilon^{2m+\min\parens{\gamma \mu, 1}}.$$ Due to Remark [Remark 6](#Remark:phi){reference-type="ref" reference="Remark:phi"}, $$\partial_i \Omega_{\eta, \epsilon} \parens{\xi} = \epsilon^3 \int\brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_\xi+w_{\eta,\epsilon,\xi}}^2 } \parens{z_\xi+w_{\eta,\epsilon,\xi}} \parens{\dot{z}_{\xi, i}+\dot{w}_{\eta, \epsilon, \xi, i}} }.$$ Clearly, $$\begin{gathered} \label{Equation:Temporary} \int\brackets*{ K_\epsilon \phi_{ \epsilon, \parens{z_\xi+w_{\eta,\epsilon,\xi}}^2 } \parens{z_\xi+w_{\eta,\epsilon,\xi}} \parens{\dot{z}_{\xi, i}+\dot{w}_{\eta, \epsilon, \xi, i}} } = \int\parens{ K_\epsilon \phi_{\epsilon, z_\xi^2} z_\xi \dot{z}_{\xi, i} } + \\ + \cbrackets*{ \int\brackets*{ K_\epsilon \phi_{\epsilon, \parens{z_\xi+w_{\eta,\epsilon,\xi}}^2} \parens{z_\xi+w_{\eta,\epsilon,\xi}} \parens{\dot{z}_{\xi, i}+\dot{w}_{\eta, \epsilon, \xi, i}} } - \int\parens{ K_\epsilon \phi_{\epsilon, z_\xi^2} z_\xi \dot{z}_{\xi, i} } }.\end{gathered}$$ Let us expand the first term on the right-hand side of [\[Equation:Temporary\]](#Equation:Temporary){reference-type="eqref" reference="Equation:Temporary"}. By definition, $$\int\parens{ K_\epsilon \phi_{\epsilon, z_\xi^2} z_\xi \dot{z}_{\xi, i} } = \int \int K_\epsilon \parens{x} K_\epsilon \parens{y} \kappa_\epsilon \parens{x - y} z_\xi \parens{y}^2 z_\xi \parens{x} \dot{z}_{\xi, i} \parens{x} \mathrm{d}x \mathrm{d}y.$$ Similarly as in Lemma [Remark 23](#Remark:Taylor){reference-type="ref" reference="Remark:Taylor"}, the Taylor expansion of $K, \kappa$ around $0$ shows that $$\begin{gathered} \left| \int\parens{ K_\epsilon \phi_{\epsilon, z_\xi^2} z_\xi \dot{z}_{\xi, i} } - \frac{\epsilon^{2m}}{\parens{m!}^2} \int \int \mathrm{D}^m_0 K \parens{x, \ldots, x} \mathrm{D}^m_0 K \parens{y, \ldots, y} \times \right. \\ \left. \times z_\xi \parens{y}^2 z_\xi \parens{x} \dot{z}_{\xi, i} \parens{x} \mathrm{d}x \mathrm{d}y \right| \lesssim \epsilon^{2m+1}.\end{gathered}$$ To finish, it suffices to do a change of variable. An application of Lemmas [Lemma 7](#Lemma:FourTerms){reference-type="ref" reference="Lemma:FourTerms"}, [Lemma 26](#Lemma:SolutionAuxiliaryConcentration){reference-type="ref" reference="Lemma:SolutionAuxiliaryConcentration"} shows that second term on the right-hand side of [\[Equation:Temporary\]](#Equation:Temporary){reference-type="eqref" reference="Equation:Temporary"} may be estimated up to multiplicative constant by $\epsilon^{2m+\gamma \mu}$. *Estimation of $\abs{\partial_i \Psi_{\eta, \epsilon}}$.* Done as in the proof of Lemma [Lemma 29](#Lemma:ExpansionNonDegenerate){reference-type="ref" reference="Lemma:ExpansionNonDegenerate"}. ◻ We finish by proving the theorem. *Proof of Theorem [Theorem 2](#Theorem:SolutionsDegenerate){reference-type="ref" reference="Theorem:SolutionsDegenerate"}.* Fix $\eta\in\ooi{0,\infty}$ such that $0$ is the only root of $\Gamma_2$ in $B_\eta$ and let $\epsilon_\eta \in \ooi{0,1}$ be such that the lemmas in Section [4.1](#Section:LyapunovConcentration){reference-type="ref" reference="Section:LyapunovConcentration"} hold. At this point, it suffices to follow the general argument used to prove Theorem [Theorem 1](#Theorem:SolutionsNonDegenerate){reference-type="ref" reference="Theorem:SolutionsNonDegenerate"}. ◻

arxiv_math

--- abstract: | Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp. 441--467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution. address: - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China - Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China author: - Wei Jiang - Chunmei Su - Ganghui Zhang title: A second-order in time, BGN-based parametric finite element method for geometric flows of curves --- Parametric finite element method, geometric flow, shape metrics, BGN scheme, high-order in time. # Introduction Geometric flows, which describe the evolution of curves or surfaces over time based on the principle that the shape changes according to its underlying geometric properties, such as the curvature, have been extensively studied in the fields of computational geometry and geometric analysis. In particular, second-order (e.g., mean curvature flow, which is also called as curve-shortening flow for curve evolution) and fourth-order (e.g., surface diffusion flow) geometric flows have attracted considerable interest due to their wide-ranging applications in materials science [@Zhao-Jiang-Wang-Bao; @Zhao-Jiang-Bao], image processing [@Aubert06], multiphase fluids [@Garcke23] and cell biology [@BGN08C]. For more in-depth information, readers can refer to the recent review articles [@BGN20; @DDE2005], and references provided therein. In this paper, we focus on three different types of geometric flows of curves: curve-shortening flow (CSF), area-preserving curve-shortening flow (AP-CSF) and surface diffusion flow (SDF). First, assume that $\Gamma(t)$ is a family of simple closed curves in the two-dimensional plane. We consider that the curve is governed by the three geometric flows, i.e., its velocity is respectively given by $$\label{Geometric equation} {\mathcal V}=\begin{cases} -\kappa\mathbf{n}, &\quad \text{CSF}, \\ (-\kappa+\left<\kappa \right>)\mathbf{n},&\quad \text{AP-CSF}, \\ (\partial_{ss}\kappa) \mathbf{n}, &\quad \text{SDF}, \end{cases}$$ where $\kappa$ is the curvature of the curve, $s$ is the arc-length, $\left<\kappa \right>:=\int_{\Gamma(t)}\kappa \mathrm{d}s/\int_{\Gamma(t)}1 \mathrm{d}s$ is the average curvature and $\mathbf{n}$ is the outward unit normal to $\Gamma$. Here, we use the sign convention that a unit circle has a positive constant curvature. By representing the curves $\Gamma(t)$ as a parametrization $\mathbf{X}(\cdot,t):\mathbb{I}\rightarrow {\mathbb R}^2$, where $\mathbb{I}:=\mathbb{R}/\mathbb{Z}$ is the "periodic\" interval $[0, 1]$, Barrett, Garcke and Nürnberg [@BGN07A; @BGN20] creatively reformulated the above equations [\[Geometric equation\]](#Geometric equation){reference-type="eqref" reference="Geometric equation"} into the following coupled forms: $$\label{Coupled equation} \begin{split} \partial_t\mathbf{X}\cdot \mathbf{n} &= \begin{cases} -\kappa, &\quad \text{CSF},\\ -\kappa+\left<\kappa \right>,&\quad \text{AP-CSF},\\ \partial_{ss}\kappa, &\quad \text{SDF}, \end{cases} \\ \kappa \mathbf{n}&= -\partial_{ss}\mathbf{X}. \end{split}$$ Based on the above equations and the corresponding weak formulations, a series of numerical schemes (the so-called BGN schemes) were proposed for solving different geometric flows, such as mean curvature flow and surface diffusion [@BGN07A; @BGN07B], Willmore flow [@BGN08C], anisotropic geometric flow [@Bao-Jiang-Li], solid-state dewetting [@Zhao-Jiang-Wang-Bao; @Zhao-Jiang-Bao] and geometric flow for surface evolution [@BGN08B]. Recently, based on the BGN formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}, structure-preserving schemes have been proposed for axisymmetric geometric equations [@Bao-Garcke-Nurnberg-Zhao] and surface diffusion [@Bao-Zhao; @Bao-Jiang-Li], respectively. In practical simulations, ample numerical results have demonstrated the high performance of the BGN scheme, due to inheriting the variational structure of the original problem and introducing an appropriate tangential velocity to help mesh points maintain a good distribution. However, for the original BGN scheme, because its formal truncation error is $\mathcal{O}(\tau)$, where $\tau$ is the time step size, the temporal convergence order of the scheme is limited to the first-order. This has been confirmed by extensive numerical experiments [@Bao-Zhao; @BGN07A; @BGN07B; @Zhao-Jiang-Wang-Bao]. Therefore, how to design a temporal high-order scheme which is based on the BGN formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"} is challenging and still open. It is also worth noting that rigorous numerical analysis for BGN schemes remains an open problem [@BGN20]. In this paper, based on the BGN formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}, we propose a novel temporal second-order parametric finite element method for solving geometric flows of curves, i.e., CSF, AP-CSF and SDF. Specifically, to discretize the same continuous-in-time semi-discrete formulation as the classical BGN scheme [@BGN07A], we begin by fixing the unit normal as that on the current curve $\Gamma^m$ and then discretize other terms using the Crank-Nicolson leap-frog scheme [@Hurl16]. The resulting scheme is a second-order semi-implicit scheme, which only requires solving a system of linear algebraic equations at each time step. Furthermore, the well-posedness and mild energy stability of the fully discrete scheme can be established under suitable assumption conditions. Numerical results have demonstrated that the proposed scheme achieves second-order accuracy in time, as measured by the shape metrics, outperforming the classical BGN scheme in terms of accuracy and efficiency. It is worth mentioning that there exist several temporal higher-order numerical schemes based on other formulations which have been proposed for simulating geometric flows. For the specific case of curve-shortening flow, a Crank-Nicolson-type scheme combined with tangential redistribution [@Balazovjech-Mikula] and an adaptive moving mesh method [@Mackenzie-Nolan-Rowlatt-Insall] have been developed. Both of the schemes are convergent quadratically in time and fully implicit, requiring to solve a system of nonlinear equations at each time step. Recently, an evolving surface finite element method together with linearly implicit backward difference formulae for time integration for simulating the mean curvature flow has been proposed in [@KLL2019; @KLL2020]. In comparison to these existing approaches, our newly proposed scheme is based on the BGN formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}, then it inherits the variational structure of the original geometric flows, and has very good property with respect to mesh distribution. The new scheme exhibits comparable computational cost to the classical BGN scheme while surpassing it in terms of accuracy. Furthermore, it can be extended easily to other geometric flows with applications to various fields. The main reason why we have successfully proposed a temporal high-order, BGN-based parametric finite element method for solving geometric flows lies in the following two key points: (1). we choose an appropriate metric (i.e., shape metrics) to measure numerical errors of the proposed schemes; (2). we use the classical first-order BGN scheme as "a good partner" of the proposed scheme to help mesh points maintain a good distribution without sacrificing the accuracy. How to measure the errors of numerical solutions for geometric flows is an important issue. A natural approach is to use classical Sobolev norms, such as $L^2$-norm, $L^\infty$-norm or $H^1$-norm, which are widely used in the numerical analysis for geometric flows [@Dziuk1994; @Dziuk1999; @KLL2019; @KLL2020]. However, when it comes to numerical schemes that involve in tangential movements, these function norms may not be suitable for quantifying the differences between two curves/surfaces. To address this issue, we consider an alternative approach using shape metrics, such as manifold distance (as used in [@Bao-Zhao; @Zhao-Jiang-Bao2021]) and Hausdorff distance [@Bai2011]. These metrics provide a measure of how similar or different two curves/surfaces are in terms of their shape characteristics. Extensive numerical experiments have been conducted, and the results demonstrate that our proposed scheme achieves second-order accuracy when measured using shape metrics. On the other hand, the quality of mesh distribution is always a major concern when simulating geometric flows using parametric finite element methods. It is important to note that the original flow [\[Geometric equation\]](#Geometric equation){reference-type="eqref" reference="Geometric equation"} requires the curve to evolve only in the normal direction, thus the numerical methods based on [\[Geometric equation\]](#Geometric equation){reference-type="eqref" reference="Geometric equation"} which prevent tangential movement of mesh points might lead to mesh distortion or clustering during the evolution. To address this issue, various approaches have been proposed in the literature to maintain good mesh quality, e.g., artificial mesh regularization method [@Bansch-Morin-Nochetto], reparametrization by introducing a tangential velocity [@Deckelnick-Dziuk; @Mikula-Sevcovic; @Elliott-Fritz; @Kimura; @Mikula-Sevcovic2004]. On the contrary, the BGN formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"} does not enforce any condition on the tangential velocity, which allows for an intrinsic tangential motion of mesh points, as demonstrated by the standard BGN scheme [@BGN07A; @BGN07B] constructed based on this formulation [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}. Though the semi-discrete scheme of [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}, where only spatial discretization is performed, results in precise equidistribution of mesh points, our proposed fully discrete second-order BGN-based scheme exhibits oscillations in terms of mesh ratio and other geometric quantities, which may lead to instability in certain situations. To overcome this problem, we employ the classical first-order BGN scheme as a mesh regularization procedure to improve mesh quality once poorly distributed polygonal approximations are observed. Extensive numerical experiments indicate that this mesh regularization remedy enhances the stability of the new scheme and improves mesh quality significantly. Fortunately, numerous numerical experiments have also demonstrated that this mesh regularization only occurs a few times during the evolution, thus not compromising the temporal second-order accuracy of the proposed scheme. The remaining of the paper is organized as follows. In Section 2, taking CSF as an example, we begin by recalling the standard BGN scheme, and then propose a second-order in time, BGN-based parametric finite element method for solving CSF. Section 3 is devoted to explaining the importance of using shape metrics, such as manifold distance and Hausdorff distance, to accurately measure the errors of two curves. We extend the proposed second-order scheme to other geometric flows such as AP-CSF and the fourth-order flow SDF in Section 4. Extensive numerical results are provided to demonstrate the accuracy and efficiency of the proposed schemes in Section 5. Finally, we draw some conclusions in Section 6. # For curve shortening flow (CSF) In this section, we propose a parametric finite element method with second-order temporal accuracy for numerically solving the CSF. The same idea can be easily extended to other geometric flows (cf. Section 4). To provide a comprehensive understanding, we first review the classical first-order BGN scheme proposed by Barrett, Garcke and Nürnberg [@BGN07A; @BGN07B; @BGN20]. ## Weak formulation and BGN scheme To begin with, we rewrite the CSF into the following formulation as presented in Eqs. [\[Coupled equation\]](#Coupled equation){reference-type="eqref" reference="Coupled equation"}: $$\label{CSF:Coupled equation} \begin{split} \partial_t \mathbf{X}\cdot \mathbf{n} &=-\kappa,\\ \kappa \mathbf{n}&=-\partial_{ss}\mathbf{X}. \end{split}$$ We introduce the following finite element approximation. Let $\mathbb{I}=[0,1]= \bigcup_{j=1}^N I_j$, $N\ge 3$, be a decomposition of $\mathbb{I}$ into intervals given by the nodes $q_j$, $I_j=[\rho_{j-1},\rho_j]$. Let $h=\max\limits_{1\le j\le N} |\rho_j-\rho_{j-1}|$ be the maximal length of a grid element. Define the linear finite element space as $$V^h:=\{u\in C(\mathbb{I}): u|_{I_j} \,\,\, \mathrm{is\,\,\,linear,\,\,\,} \forall j=1,2,\ldots,N;\quad u(\rho_0)=u(\rho_N) \}\subseteq H^1(\mathbb{I}).$$ The mass lumped inner product $(\cdot,\cdot)_{\Gamma^h}^h$ over the polygonal curve $\Gamma^h$, which is an approximation of $(\cdot,\cdot)_{\Gamma^h}$ by using the composite trapezoidal rule, is defined as $$(u,v)_{\Gamma^h}^h:=\frac{1}{2}\sum_{j=1}^N|\mathbf{X}^h(\rho_j,t)-\mathbf{X}^h(\rho_{j-1},t)|\left[(u\cdot v)(\rho_j^-)+(u\cdot v)(\rho_{j-1}^+) \right],$$ where $u, v$ are two scalar/vector piecewise continuous functions with possible jumps at the nodes $\{\rho_j\}_{j=1}^N$, and $u(\rho_j^{\pm})=\lim\limits_{\rho\rightarrow \rho_j^{\pm}}u(\rho)$. Subsequently, the semi-discrete scheme of the formulation [\[CSF:Coupled equation\]](#CSF:Coupled equation){reference-type="eqref" reference="CSF:Coupled equation"} is as follows: given initial polygon $\Gamma^h(0)$ with vertices lying on the initial curve $\Gamma(0)$ clockwise, parametrized by $\mathbf{X}^h(\cdot,0)\in [V^h]^2$, find $(\mathbf{X}^h(\cdot,t),\kappa^h(\cdot,t))\in [V^h]^2\times V^h$ such that $$\label{CSF:Semi-discrete} \begin{cases} \left(\partial_t\mathbf{X}^h\cdot \mathbf{n}^h,\varphi^h \right)_{\Gamma^h}^h+\left( \kappa^h,\varphi^h\right)^h_{\Gamma^h}=0,\quad \forall\ \varphi^h\in V^h,\\ \left(\kappa^h,\mathbf{n}^h\cdot \bm{\omega}^h\right)^h_{\Gamma^h}-\left(\partial_s \mathbf{X}^h,\partial_s\bm{\omega}^h \right)_{\Gamma^h}=0,\quad \forall\ \bm{\omega}^h\in [V^h]^2, \end{cases}$$ where we always integrate over the current curve $\Gamma^h$ described by $\mathbf{X}^h$, the outward unit normal $\mathbf{n}^h$ is a piecewise constant vector given by $$\mathbf{n}^h|_{I_j}=-\frac{\mathbf{h}_j^\perp}{|\mathbf{h}_j|}, \quad \mathbf{h}_j=\mathbf{X}^h(\rho_j,t)-\mathbf{X}^h(\rho_{j-1},t),\quad j=1,\ldots, N,$$ with $\cdot^\perp$ denoting clockwise rotation by $\frac{\pi}{2}$, and the partial derivative $\partial_s$ is defined piecewisely over each side of the polygon $\partial_s f|_{I_j}=\frac{\partial_\rho f}{|\partial_\rho \mathbf{X}^h|}|_{I_j}=\frac{(\rho_j-\rho_{j-1})\partial_\rho f|_{I_j}}{|\mathbf{h}_j|}$. It was shown that the scheme [\[CSF:Semi-discrete\]](#CSF:Semi-discrete){reference-type="eqref" reference="CSF:Semi-discrete"} will always equidistribute the vertices along $\Gamma^h$ for $t>0$ if they are not locally parallel (see Remark 2.4 in [@BGN07A]). For a full discretization, we fix $\tau>0$ as a uniform time step size for simplicity, and let $\mathbf{X}^m\in [V^h]^2$ and $\Gamma^m$ be the approximations of $\mathbf{X}(\cdot,t_m)$ and $\Gamma(t_m)$, respectively, for $m=0,1,2,\ldots$, where $t_m:=m\tau$. We define $\mathbf{h}_j^m:=\mathbf{X}^m(\rho_j)-\mathbf{X}^m(\rho_{j-1})$ and assume $|\mathbf{h}_j^m|>0$ for $j=1,\ldots,N$, $\forall\ m>0$. The discrete unit normal vector $\mathbf{n}^m$, the discrete inner product $(\cdot,\cdot)^h_{\Gamma^m}$ and the discrete operator $\partial_s$ are defined similarly as in the semi-discrete case. Barrett, Garcke and Nürnberg used a formal first-order approximation [@BGN07A; @BGN07B] to replace the velocity $\partial_t \mathbf{X}$, $\kappa$ and $\partial_s\mathbf{X}$ by $$\begin{split} \partial_t \mathbf{X}(\cdot, t_m)&= \frac{\mathbf{X}(\cdot,t_{m+1})-\mathbf{X}(\cdot, t_m)}{\tau}+\mathcal{O}(\tau), \\ \kappa(\cdot,t_m)&=\kappa(\cdot,t_{m+1})+\mathcal{O}(\tau), \\ \partial_s\mathbf{X}(\cdot,t_m)&= \partial_s \mathbf{X}(\cdot, t_{m+1})+\mathcal{O}(\tau), \end{split}$$ and the fully discrete semi-implicit BGN scheme (denoted as BGN1 scheme) reads as: (**BGN1, First-order in time BGN scheme for CSF**): For $m\ge 0$, find $\mathbf{X}^{m+1}\in [V^h]^2$ and $\kappa^{m+1}\in V^h$ such that $$\label{CSF:BGN1} \begin{cases} \left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^m}{\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \kappa^{m+1},\varphi^h \right)_{\Gamma^m}^h=0,\quad \forall\ \varphi^h\in V^h,\\ \left(\kappa^{m+1},\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\partial_s \mathbf{X}^{m+1},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0,\quad \forall\ \bm{\omega}^h\in [V^h]^2. \end{cases}$$ The well-posedness and energy stability were established under some mild conditions. In practice, numerous numerical results show that the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} converges quadratically in space [@BGN07B] and linearly in time (cf. Fig. [1](#Fig:CSF_EOC1){reference-type="ref" reference="Fig:CSF_EOC1"} in Section [5.1](#sec:order 2, illu){reference-type="ref" reference="sec:order 2, illu"}). ## A second-order in time, BGN-based scheme {#sec:order 2} Instead of using the first-order Euler method, we apply the Crank-Nicolson leap-frog time stepping discretization in [\[CSF:Semi-discrete\]](#CSF:Semi-discrete){reference-type="eqref" reference="CSF:Semi-discrete"} based on the following simple calculation $$\label{app} \begin{split} \partial_t\mathbf{X}(\cdot,t_m)&= \frac{\mathbf{X}(\cdot, t_{m+1})-\mathbf{X}(\cdot, t_{m-1})}{2\tau}+\mathcal{O}(\tau^2),\\ \kappa(\cdot,t_m)&= \frac{\kappa(\cdot, t_{m+1})+\kappa(\cdot, t_{m-1})}{2}+\mathcal{O}(\tau^2),\\ \partial_s\mathbf{X}(\cdot,t_m)&= \frac{\partial_s \mathbf{X}(\cdot, t_{m+1})+\partial_s \mathbf{X}(\cdot, t_{m-1})}{2}+\mathcal{O}(\tau^2), \end{split}$$ then the corresponding second-order scheme (denoted as BGN2 scheme) is as follows: (**BGN2, Second-order in time BGN-based scheme for CSF**):  For $\mathbf{X}^0 \in [V^h]^2$, $\kappa^0\in V^h$ and $(\mathbf{X}^1,\kappa^1)\in [V^h]^2\times V^h$ which are the appropriate approximations at the time levels $t_0=0$ and $t_1=\tau$, respectively, find $\mathbf{X}^{m+1}\in [V^h]^2$ and $\kappa^{m+1}\in V^h$ for $m\ge 1$ such that $$\label{CSF:BGN2} \begin{cases} \left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^{m-1}}{2\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \frac{\kappa^{m+1}+\kappa^{m-1}}{2},\varphi^h \right)_{\Gamma^m}^h=0,\\ \vspace{-3mm}\\ \left(\frac{\kappa^{m+1}+\kappa^{m-1}}{2},\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\frac{\partial_s \mathbf{X}^{m+1}+\partial_s \mathbf{X}^{m-1}}{2},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0, \end{cases}$$ for all $(\varphi^h, \bm{\omega}^h)\in V^h\times [V^h]^2$. The scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} is semi-implicit and the computational cost is comparable to that of the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"}. Moreover, as a temporal discretization of the semi-discrete version [\[CSF:Semi-discrete\]](#CSF:Semi-discrete){reference-type="eqref" reference="CSF:Semi-discrete"}, it can be easily derived from [\[app\]](#app){reference-type="eqref" reference="app"} that the truncation error is of order $\mathcal{O}(\tau^2)$. **Remark 1**. *To begin the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}, we need to first prepare the data $\kappa^0$ and $(\mathbf{X}^1,\kappa^1)$. In practical simulations, this can be easily achieved without sacrificing the accuracy of the scheme by utilizing the standard BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} to get $(\mathbf{X}^1,\kappa^1)$, and the following formula of discrete curvature was proposed in [@BGN07A Page 461] to prepare $\kappa^0$ (note the the sign convention of the curvature is opposite to [@BGN07A]) $$\label{kappa formula} \kappa^0=(N_0^\top N_0)^{-1}N_0^\top A_0 \textbf{X}^0,$$ where $N_0$ is a $2N\times N$ matrix, $\textbf{X}^0$ is a $2N\times 1$ vector and $A_0$ is a $2N\times 2N$ matrix given by $$\begin{aligned} &N_0 = \begin{pmatrix} (\varphi_i,(\mathbf{n}^0)^{[1]} \varphi_j)_{\Gamma^0}^h\\ (\varphi_i,(\mathbf{n}^0)^{[2]} \varphi_j)_{\Gamma^0}^h \end{pmatrix}, \quad \textbf{X}^0=\begin{pmatrix} \textbf{x}^0\\ \textbf{y}^0 \end{pmatrix},\\ &A_0=\begin{pmatrix} (\partial_s\varphi_i,\partial_s\varphi_j)_{\Gamma^0} & 0\\ 0 & (\partial_s\varphi_i,\partial_s\varphi_j)_{\Gamma^0} \end{pmatrix},\end{aligned}$$ where $\varphi_i, 1\le i\le N$ are the standard Lagrange basis over $\mathbb{I}$, and $\mathbf{a}^{[1]},\mathbf{a}^{[2]}$ are the first and second component of vector $\mathbf{a}\in {\mathbb R}^2$, and $\textbf{x}_j^0=(\mathbf{X}^0)^{[1]}(\rho_j)$, $\textbf{y}_j^0=(\mathbf{X}^0)^{[2]}(\rho_j)$ for $j=1,\ldots, N$. Note that this formula can be derived by solving the finite element approximation of the equation $\kappa \mathbf{n}=-\partial_{ss}\mathbf{X}$ and using the least square method. We can summarize the process as Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"}, which outlines the steps to prepare the required data $\kappa^0$ and $(\mathbf{X}^1,\kappa^1)$. Once we have obtained these data, we can directly apply the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} to calculate $(\mathbf{X}^m,\kappa^m)$, for $m\ge 2$.* **Algorithm 1**. ***(Preparation for the initial data of BGN2 for CSF)** [\[CSF:BGN initial data 1\]]{#CSF:BGN initial data 1 label="CSF:BGN initial data 1"}* *$\bm{Step~0.}$ Given the initial curve $\Gamma(0)$, the number of grid points $N$ and the time step size $\tau$. We choose the polygon $\Gamma^0$ with $N$ vertices lying on $\Gamma(0)$ such that $\Gamma^0$ is (almost) equidistributed, i.e., each side of the polygon is (nearly) equal in length. We parameterize $\Gamma^0$ with $\mathbf{X}^0\in [V^h]^2$ and the grid points $\rho_j$ can be determined correspondingly.* *$\bm{Step~1.}$ Using $\mathbf{X}^0$ as the input, we compute $\kappa^0$ using the discrete curvature formula [\[kappa formula\]](#kappa formula){reference-type="eqref" reference="kappa formula"}.* *$\bm{Step~2.}$ Using $\mathbf{X}^0$ as the input, we obtain $(\mathbf{X}^1,\kappa^1)$ by solving the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} for one time step.* **Remark 2**. *When dealing with an initial curve which is not regular, an alternative approach for initialization is to solve the BGN1 scheme twice and start the BGN2 scheme from $m=2$. Specifically, for given $\mathbf{X}^0$, we can compute $(\mathbf{X}^1, \kappa^1)$ and $(\mathbf{X}^2, \kappa^2)$, which are the appropriate approximations at time levels $t_1=\tau$ and $t_2=2\tau$, by solving the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} twice. These approximations can be used as initial values to implement the BGN2 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} for $m\ge 2$. For the superiority of this approach, see Fig. [7](#Fig:flower_geo_MRandnoMRb){reference-type="ref" reference="Fig:flower_geo_MRandnoMRb"} in Section 5.3.* Similar to the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"}, we can show the well-posedness of the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} under some mild conditions as follows. **Theorem 1** (Well-posedness). *For $m\ge 0$, we assume that the following two conditions are satisfied:* 1. *There exist at least two vectors in $\{\mathbf{h}_j^m\}_{j=1}^{N}$ which are not parallel, i.e., $$\mathrm{dim}\left( \mathrm{Span}\left\{\mathbf{h}_j^m \right\}_{j=1}^{N}\right)=2.$$* 2. *No degenerate vertices exist on $\Gamma^m$, i.e., $$\min_{1\le j\le N}| \mathbf{h}_j^m|>0.$$* *Then the full discretization [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} is well-posed, i.e., there exists a unique solution $(\mathbf{X}^{m+1},\kappa^{m+1})\in [V^h]^2\times V^h$ of [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}.* *Proof.* It suffices to prove the following algebraic system for $(\mathbf{X},\kappa)\in [V^h]^2\times V^h$ has only zero solution, $$\begin{cases} \left(\frac{\mathbf{X}}{\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \kappa ,\varphi^h \right)_{\Gamma^m}^h=0,\quad \forall\ \varphi^h\in V^h,\\ \left(\kappa,\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\partial_s \mathbf{X},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0,\quad \forall\ \bm{\omega}^h\in [V^h]^2. \end{cases}$$ Indeed, the stiffness matrix is exactly the same as the standard BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} and thus the same argument in [@BGN07B Theorem 2.9] yields the conclusion under the assumptions (1) and (2). ◻ Subsequently, we prove the following energy stability property, assuming a mild condition regarding the upper bound of the mesh ratio. Furthermore, the numerical results presented in Section [5](#sec:illust){reference-type="ref" reference="sec:illust"} demonstrate that the proposed BGN2 scheme remains stable even for very large time step $\tau$. **Theorem 2** (Mild energy stability). *Assume that the mesh ratio of $\mathbf{X}^{m}$ satisfies $$\label{Mesh assumption} \Psi^m:=\frac{\max_j|\mathbf{h}^m_j| }{\min_j|\mathbf{h}^m_j|}\le c,$$ where $c$ is a constant independent of $\tau$, $h$ and $m$. Then for any $\tau>0$ and $m\ge 1$, the energy stability holds in the following sense, $$E^{m+1}\le c E^{m-1},$$ where $E^{m+1}:=\sum\limits_{j=1}^N |\mathbf{h}^{m+1}_j|^2.$* *Proof.* Taking $\bm{\omega}^h=\frac{\mathbf{X}^{m+1}-\mathbf{X}^{m-1}}{2\tau}$ and $\varphi^h=\frac{\kappa^{m+1}+\kappa^{m-1}}{2}$ in [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}, we get $$\begin{aligned} &\left( \frac{\kappa^{m+1}+\kappa^{m-1}}{2},\frac{\kappa^{m+1}+\kappa^{m-1}}{2}\right)_{\Gamma^m}^h\\ &=-\left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^{m-1}}{2\tau},\left(\frac{\kappa^{m+1}+\kappa^{m-1}}{2}\right) \mathbf{n}^m \right)^h_{\Gamma^m}\\ &=-\left(\frac{\partial_s \mathbf{X}^{m+1}+\partial_s \mathbf{X}^{m-1}}{2},\frac{\partial_s \mathbf{X}^{m+1}-\partial_s \mathbf{X}^{m-1}}{2\tau}\right)_{\Gamma^m}\\ &=-\frac{1}{4\tau}\left(\left(\partial_s \mathbf{X}^{m+1},\partial_s \mathbf{X}^{m+1}\right)_{\Gamma^m}-\left(\partial_s \mathbf{X}^{m-1},\partial_s \mathbf{X}^{m-1}\right)_{\Gamma^m} \right). \end{aligned}$$ Noticing $$\begin{aligned} \left(\partial_s \mathbf{X}^{m+1},\partial_s \mathbf{X}^{m+1}\right)_{\Gamma^m} &=\sum_{j=1}^N\frac{|\mathbf{h}_j^{m+1}|}{|\mathbf{h}_j^{m}|}\frac{|\mathbf{h}_j^{m+1}|}{|\mathbf{h}_j^{m}|}|\mathbf{h}_j^{m}|=\sum_{j=1}^N\frac{|\mathbf{h}_j^{m+1}|^2}{|\mathbf{h}_j^{m}|}, \end{aligned}$$ we can estimate for any $\tau>0$, $$\begin{aligned} E^{m+1}-cE^{m-1} &=\sum_{j=1}^N |\mathbf{h}^{m+1}_j|^2-c\sum_{j=1}^N |\mathbf{h}^{m-1}_j|^2\\ &\le \left(\max_{j}|\mathbf{h}_j^{m}|\right)\sum_{j=1}^N\frac{|\mathbf{h}_j^{m+1}|^2}{|\mathbf{h}_j^{m}|}-\left(\min_{j}|\mathbf{h}_j^{m}|\right)c\sum_{j=1}^N\frac{|\mathbf{h}_j^{m-1}|^2}{|\mathbf{h}_j^{m}|}\\ &\le \left(\max_{j}|\mathbf{h}_j^{m}|\right)\left(\sum_{j=1}^N\frac{|\mathbf{h}_j^{m+1}|^2}{|\mathbf{h}_j^{m}|}-\sum_{j=1}^N\frac{|\mathbf{h}_j^{m-1}|^2}{|\mathbf{h}_j^{m}|} \right)\\ &= -4\tau \max_{j}|\mathbf{h}_j^{m}| \left( \frac{\kappa^{m+1}+\kappa^{m-1}}{2},\frac{\kappa^{m+1}+\kappa^{m-1}}{2}\right)_{\Gamma^m}^h\le 0, \end{aligned}$$ and the proof is completed. ◻ ## Mesh regularization As was mentioned earlier, the semi-discrete scheme [\[CSF:Semi-discrete\]](#CSF:Semi-discrete){reference-type="eqref" reference="CSF:Semi-discrete"} possesses the mesh equidistribution property [@BGN20 Theorem 79]. In practice, the fully-discrete BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} can maintain the asymptotic long-time mesh equidistribution property. However, the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} may have oscillating mesh ratio due to the structure of two-step method, which can potentially amplify the mesh ratio and cause mesh distortion or clustering during the evolution, especially for some initial curves which are not so regular, e.g., a 'flower' curve (see the second row of Fig. [8](#Fig:flower_evo_MRandnoMR){reference-type="ref" reference="Fig:flower_evo_MRandnoMR"}). Therefore, a mesh regularization procedure is necessary in real simulations to help the mesh maintain a good distribution property during the evolution, when the mesh ratio exceeds a given threshold value. Inspired by the good mesh distribution property of the BGN1 scheme, we utilize the BGN1 scheme as the mesh regularization technique. In the following, we denote $n_{\rm{MR}}$ as the threshold value chosen initially. If the mesh ratio $\Psi^{m}>n_{\rm{MR}}$, then we use the mesh regularization procedure to improve the mesh distribution. We present a summary of the complete algorithm of BGN2 scheme for solving the CSF in Algorithm [\[Full algorithm\]](#Full algorithm){reference-type="ref" reference="Full algorithm"}. **Algorithm 2**. ***(**BGN2 scheme for CSF**)** [\[Full algorithm\]]{#Full algorithm label="Full algorithm"}* *$\bm{Step~0.}$ Given the initial curve $\Gamma(0)$, and $N,T,n_{\rm{MR}}$, $\tau$, compute $\mathbf{X}^0$ as in *Step 0* in Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"}.* *$\bm{Step~1.}$ Using $\mathbf{X}^0$ as the input, we compute $\kappa^0$ using the discrete curvature formula [\[kappa formula\]](#kappa formula){reference-type="eqref" reference="kappa formula"} and solve $(\mathbf{X}^1, \kappa^1)$ via the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"}. Set $m=1$.* *$\bm{Step~2.}$ Calculate the mesh ratio $\Psi^m$ of $\mathbf{X}^m$, $m\ge 1$.* *$\bm{Step~3.}$ If the mesh ratio $\Psi^m>n_{\rm{MR}}$, then replace $(\mathbf{X}^m,\kappa^m)$ with the solution of the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} with $\mathbf{X}^{m-1}$ as the input by one run; otherwise, skip this step.* *$\bm{Step~4.}$ Use the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} to obtain $(\mathbf{X}^{m+1},\kappa^{m+1})$.* *$\bm{Step~5.}$ Update $m=m+1$. If $m< T/\tau$, then go back to $\textbf{Step~2}$; otherwise, stop the algorithm and output the data.* As shown in $\emph{\textbf{Step~3}}$ of Algorithm [\[Full algorithm\]](#Full algorithm){reference-type="ref" reference="Full algorithm"}, if the mesh ratio $\Psi^{m}>n_{\rm{MR}}$, we replace $(\mathbf{X}^m,\kappa^m)$ with the solution of the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} with $\mathbf{X}^{m-1}$ as the input by one run, to help us realize the mesh regularization. Extensive numerical experiments suggest that the mesh regularization procedure is very effective, and the mesh ratio decreases immediately to a small value after this procedure (cf. Fig. [5](#Fig:tube_evo_and_Geo){reference-type="ref" reference="Fig:tube_evo_and_Geo"}(d) in Section 5). The BGN2 scheme with the aid of the BGN1 scheme as the mesh regularization is very efficient and stable in practical simulations. The reason comes from that the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} can intrinsically lead to a good mesh distribution property, which was explained in [@BGN07A; @BGN20], but a more convincing explanation needs further rigorous numerical analysis for the scheme. One concern that may arise is whether the BGN2 scheme with necessary mesh regularization can still achieve second-order accuracy, considering that the BGN1 scheme is only first-order accurate. It is important to note that for certain smooth initial curves, such as elliptic curves, the mesh regularization procedure is never required during the evolution. In such cases, the numerical evolution remains remarkably stable and the mesh ratio remains bounded. While for certain special initial curves, like a 'flower' curve or a 'tube' curve, the mesh regularization procedure may be needed only a few times (cf. Section [5.3](#sec:long time, illu){reference-type="ref" reference="sec:long time, illu"}). Nevertheless, this does not compromise the temporal second-order accuracy of the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}. # Shape metric is a better choice {#sec:Different error} As we are aware, it is an interesting and thought-provoking problem to determine how to quantify the difference between two curves in 2D or two surfaces in 3D. Given two closed curves $\Gamma_1$ and $\Gamma_2$, we assume that the two curves are parametrized by $\mathbf{X}(\rho)$ and $\mathbf{Y}(\rho)$, respectively, over the same interval $\mathbb{I}$. Consequently, we can define the following four metrics for measurement: - (**$L^2$-error**)  The $L^2$-norm between the parameterized functions $\mathbf{X}(\rho)$ and $\mathbf{Y}(\rho)$ is defined in a classical way $$A(\mathbf{X},\mathbf{Y}) :=\|\mathbf{X}(\rho)-\mathbf{Y}(\rho)\|_{L^2(\mathbb{I})}.$$ - (**$L^\infty$-error**) The $L^\infty$-norm between the parameterized functions $\mathbf{X}(\rho)$ and $\mathbf{Y}(\rho)$ is defined as $$B(\mathbf{X},\mathbf{Y}) :=\|\mathbf{X}(\rho)-\mathbf{Y}(\rho)\|_{L^\infty(\mathbb{I})}.$$ - (**Manifold distance**)  The manifold distance between the curves $\Gamma_1$ and $\Gamma_2$ is defined as [@Zhao-Jiang-Bao2021] $$\begin{aligned} \mathrm{M}\left(\Gamma_1,\Gamma_2\right) &: = |(\Omega_1\setminus\Omega_2)\cup (\Omega_2\setminus\Omega_1) | =|\Omega_1 |+|\Omega_2 |-2 |\Omega_1\cup \Omega_2 |,\end{aligned}$$ where $\Omega_1$ and $\Omega_2$ represent the regions enclosed by $\Gamma_1$ and $\Gamma_2$, respectively, and $|\Omega|$ denotes the area of $\Omega$. - (**Hausdorff distance**)  The Hausdorff distance between the curves $\Gamma_1$ and $\Gamma_2$ is defined as [@Bai2011] $$H(\Gamma_1,\Gamma_2) = \max\{\widetilde{H}(\Gamma_1,\Gamma_2),\widetilde{H}(\Gamma_2,\Gamma_1)\},$$ where $\widetilde{H}(\Gamma_1,\Gamma_2) = \max\limits_{a\in \Gamma_1}\min\limits_{b\in \Gamma_2}d(a,b)$, and $d$ is the Euclidean distance. **Remark 3**. *The $L^2$-error and $L^\infty$-error fall within the domain of *function metrics*, which rely on the parametrization of curves. On the other hand, as demonstrated in [@Zhao-Jiang-Bao2021 Proposition 5.1] and [@Bai2011], it has been easily proven that both manifold distance and Hausdorff distance fulfill the properties of symmetry, positivity and the triangle inequality. Therefore, they belong to the category of *shape metrics* and not influenced by the specific parametrization.* **Remark 4**. *It should be noted that the aforementioned shape metrics can be easily calculated using simple algorithms. As the numerical solutions are represented as polygons, it is very easy to calculate the area of the symmetric difference region, i.e., the manifold distance, between two polygonal curves. Additionally, a polygon-based approach proposed in the literature [@Bai2011] can be employed to calculate the Hausdorff distance between planar curves.* In order to test the convergence rate of numerical schemes, for example, we consider the evolution of the CSF with an initial ellipse defined by $$\{(x,y)\in \mathbb{R}^2: \quad x^2+4y^2=4\}.$$ This initial ellipse is approximated using an equidistributed polygon $\mathbf{X}^0$ with $N$ vertices. Here, we simulate the CSF by using three different numerical schemes: Dziuk's scheme [@Dziuk1994 Section 6], BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} and BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}. Since the exact solution of the CSF for an elliptical curve is unknown, we first compute a reference solution $\mathbf{X}_{\mathrm{ref}}$ by Dziuk's scheme (to test the convergence of Dziuk's scheme) or the BGN2 scheme (to test the convergence of BGN-type schemes) with a fine mesh and a tiny time step size, e.g., $N=10000$ and $\tau=10^{-1}*2^{-11}$. To test the temporal error, we still take a large number of grid points, e.g., $N=10000$, such that the spatial error is ignorable. The numerical error and the corresponding convergence order are then determined as follows $$\label{eqn:errordef1} \mathcal E_{\mathcal{M}}:=\mathcal E_{\tau}(T)= \mathcal{M} (\mathbf{X}^k_{\tau}, \mathbf{X}_{\mathrm{ref}}), \quad \text{Order}=\log\Big(\frac{\mathcal E_{\tau}(T)}{\mathcal E_{\tau/2} (T)} \Big)\Big/ \log 2,$$ where $k=T/\tau$, and $\mathcal{M}$ represents any one of the four metrics defined above. ------------------------------------------------------------------------ $\tau=\tau_0$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ ------------------- --------------- ------------ -------------- -------------- -- $L^2$-norm 1.17E-2 6.31E-3 3.26E-3 1.62E-3 Order -- 0.89 0.95 1.01 $L^{\infty}$-norm 3.05E-2 1.63E-2 8.41E-3 4.19E-3 Order -- 0.90 0.96 1.00 6.89E-2 3.65E-2 1.86E-2 9.17E-3 Order -- 0.92 0.97 1.02 3.04E-2 1.62E-2 8.29E-3 4.09E-3 Order -- 0.91 0.97 1.02 : Numerical errors quantified by various metrics for Dziuk's scheme [@Dziuk1994 Section 6], with the parameters $N=10000, \tau_0=1/40$, and $T=0.25$. ------------------------------------------------------------------------ [\[Tab:Different_norm_ellipse_Dziuk\]]{#Tab:Different_norm_ellipse_Dziuk label="Tab:Different_norm_ellipse_Dziuk"} Tables [1](#Tab:Different_norm_ellipse_Dziuk){reference-type="ref" reference="Tab:Different_norm_ellipse_Dziuk"}-[3](#Tab:Different_norm_ellipse_BGN2){reference-type="ref" reference="Tab:Different_norm_ellipse_BGN2"} display the numerical errors at time $T=0.25$ measured by the four different metrics for Dziuk's scheme [@Dziuk1994], the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} and the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}, respectively. As anticipated, we easily observe linear convergence in time for Dziuk's scheme across all four different metrics. While linear and quadratic convergence for both shape metrics (i.e., the manifold distance and Hausdorff distance) are observed for the BGN1 scheme in Table [2](#Tab:Different_norm_ellipse_BGN1){reference-type="ref" reference="Tab:Different_norm_ellipse_BGN1"} and the BGN2 scheme in Table [3](#Tab:Different_norm_ellipse_BGN2){reference-type="ref" reference="Tab:Different_norm_ellipse_BGN2"}, respectively. ------------------------------------------------------------------------ $\tau=\tau_0$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ ------------------- --------------- ------------ -------------------------- -------------------------- -- $L^2$-norm 4.25E-3 3.98E-3 4.05E-3 4.15E-3 Order -- 0.10 $-$`<!-- -->`{=html}0.03 $-$`<!-- -->`{=html}0.03 $L^{\infty}$-norm 1.00E-2 9.17E-3 9.47E-3 9.79E-3 Order -- 0.12 $-$`<!-- -->`{=html}0.05 $-$`<!-- -->`{=html}0.05 3.11E-2 1.58E-2 7.96E-3 4.00E-3 Order -- 0.98 0.99 0.99 8.23E-3 4.18E-3 2.11E-3 1.06E-3 Order -- 0.98 0.99 0.99 : Numerical errors quantified by various metrics for the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"}, with the parameters $N=10000,\tau_0=1/40$, $T=0.25$. ------------------------------------------------------------------------ [\[Tab:Different_norm_ellipse_BGN1\]]{#Tab:Different_norm_ellipse_BGN1 label="Tab:Different_norm_ellipse_BGN1"} ------------------------------------------------------------------------ $\tau=\tau_0$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ ------------------- --------------- ------------ -------------- -------------- -- $L^2$-norm 1.49E-2 1.45E-2 1.45E-2 1.43E-2 Order -- 0.04 0.00 0.02 $L^{\infty}$-norm 3.32E-2 3.30E-2 3.29E-2 3.29E-2 Order -- 0.01 0.00 0.00 8.44E-4 2.11E-4 5.27E-5 1.32E-5 Order -- 2.00 2.00 1.99 2.00E-4 4.98E-5 1.26E-5 3.29E-6 Order -- 2.01 1.98 1.94 : Numerical errors quantified by various metrics for the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}, with the parameters $N=10000,\tau_0=1/40$, $T=0.25$. ------------------------------------------------------------------------ [\[Tab:Different_norm_ellipse_BGN2\]]{#Tab:Different_norm_ellipse_BGN2 label="Tab:Different_norm_ellipse_BGN2"} It is worth noting that unlike Dziuk's scheme, the convergence of the BGN1 scheme and BGN2 scheme under function metrics (the $L^2$-norm and $L^\infty$-norm) is not as satisfactory. This is not surprising since the error in classical Sobolev space depends on the specific parametrization of the curve. In contrast, the BGN formulation [\[CSF:Coupled equation\]](#CSF:Coupled equation){reference-type="eqref" reference="CSF:Coupled equation"} allows tangential motion to make the mesh points equidistribute, which indeed affects the parametrization while preserving the shape of the curve. Thus it is not appropriate to use the classical function metrics to quantify the errors of the BGN-type schemes which are based on the BGN formulation. Instead, as observed from Tables [2](#Tab:Different_norm_ellipse_BGN1){reference-type="ref" reference="Tab:Different_norm_ellipse_BGN1"} and [3](#Tab:Different_norm_ellipse_BGN2){reference-type="ref" reference="Tab:Different_norm_ellipse_BGN2"}, the shape metrics are much more suitable for quantifying the numerical errors of the schemes that allow intrinsic tangential velocity. In the remaining of the article, we will employ the manifold distance or the Hausdorff distance when measuring the difference between two curves. # Applications to other geometric flows In this section, we extend the above proposed BGN2 scheme to other geometric flows. ## For area-preserving curve-shortening flow (AP-CSF) As is known, the AP-CSF can be viewed as the $L^2$-gradient flow with respect to the length functional under the constraint of total area preservation [@BGN20; @Jiang23]. Similar to [\[CSF:Coupled equation\]](#CSF:Coupled equation){reference-type="eqref" reference="CSF:Coupled equation"}, we rewrite the AP-CSF as the following coupled equations $$\label{AP-CSF:Coupled equation} \begin{split} \partial_t \mathbf{X}\cdot \mathbf{n} &=-\kappa+\left<\kappa \right> ,\\ \kappa \mathbf{n}&=-\partial_{ss}\mathbf{X}, \end{split}$$ where the average of curvature is defined as $\left<\kappa\right>:=\int_{\Gamma(t)}\kappa \mathrm{d}s /\int_{\Gamma(t)}1 \mathrm{d}s$. The fully-discrete, first-order in time semi-implicit BGN scheme for AP-CSF reads as [@BGN20]: (**BGN1 scheme for AP-CSF**): For $m\ge 0$, find $\mathbf{X}^{m+1}\in [V^h]^2$ and $\kappa^{m+1}\in V^h$ such that $$\label{AP-CSF:BGN1} \begin{cases} \left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^m}{\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \kappa^{m+1}-\left< \kappa^{m+1}\right>_{\Gamma^m} ,\varphi^h \right)_{\Gamma^m}^h=0,\\ \left(\kappa^{m+1},\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\partial_s \mathbf{X}^{m+1},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0, \end{cases}$$ for all $(\varphi^h, \bm{\omega}^h)\in V^h\times [V^h]^2$, where $\left<\kappa^{m+1}\right>_{\Gamma^m}:=\left(\kappa^{m+1},1 \right)_{\Gamma^m}^h/\left(1,1 \right)_{\Gamma^m}^h$. Based on the same spirit, we can propose the following second-order BGN2 scheme. (**BGN2 scheme for AP-CSF**):  For $m\ge 1$, find $(\mathbf{X}^{m+1}, \kappa^{m+1})\in [V^h]^2\times V^h$ such that $$\label{AP-CSF:BGN2} \begin{cases} \big(\frac{\mathbf{X}^{m+1}-\mathbf{X}^{m-1}}{2\tau},\varphi^h \mathbf{n}^m \big)^h_{\Gamma^m}=-\big( \frac{\kappa^{m+1}+\kappa^{m-1}}{2}-\big<\frac{\kappa^{m+1}+\kappa^{m-1}}{2} \big>_{\Gamma^m},\varphi^h \big)_{\Gamma^m}^h,\\ \big(\frac{\kappa^{m+1}+\kappa^{m-1}}{2},\mathbf{n}^m\cdot \bm{\omega}^h\big)_{\Gamma^m}^h-\big(\frac{\partial_s \mathbf{X}^{m+1}+\partial_s \mathbf{X}^{m-1}}{2},\partial_s\bm{\omega}^h\big)_{\Gamma^m}=0, \end{cases}$$ for all $(\varphi^h, \bm{\omega}^h)\in V^h\times [V^h]^2$. Similarly, the stiffness matrix of the linear system to be solved in [\[AP-CSF:BGN2\]](#AP-CSF:BGN2){reference-type="eqref" reference="AP-CSF:BGN2"} is exactly the same as the BGN1 scheme [\[AP-CSF:BGN1\]](#AP-CSF:BGN1){reference-type="eqref" reference="AP-CSF:BGN1"}, whose well-posedness has been established in [@BGN20 Theorem 90]. Additionally, a mild energy stability can be easily obtained under the same conditions as stated in Theorem [Theorem 2](#CSF:Mild energy stability){reference-type="ref" reference="CSF:Mild energy stability"}. ## For surface diffusion flow (SDF) We consider the fourth-order flow---SDF, which can be viewed as the $H^{-1}$-gradient flow with respect to the length functional [@BGN20; @Bao-Zhao]. In a similar fashion, we rephrase the SDF as the subsequent system of equations $$\label{SD:Coupled equation} \begin{split} \partial_t \mathbf{X}\cdot \mathbf{n} &=\partial_{ss} \kappa ,\\ \kappa \mathbf{n}&=-\partial_{ss}\mathbf{X}. \end{split}$$ The fully discrete, first-order in time semi-implicit BGN scheme for SDF reads as [@BGN07A]: (**BGN1 scheme for SDF**): For $m\ge 0$, find $\mathbf{X}^{m+1}\in [V^h]^2$ and $\kappa^{m+1}\in V^h$ such that $$\label{SD:BGN1} \begin{cases} \left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^m}{\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \partial_s \kappa^{m+1}, \partial_s\varphi^h \right)_{\Gamma^m}=0,\quad \forall\ \varphi^h\in V^h,\\ \vspace{-3mm} \left(\kappa^{m+1},\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\partial_s \mathbf{X}^{m+1},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0,\quad \forall\ \bm{\omega}^h\in [V^h]^2. \end{cases}$$ In line with the same approach, we can put forward the subsequent second-order BGN2 scheme: (**BGN2 scheme for SDF**):  For $m\ge 1$, find $(\mathbf{X}^{m+1}, \kappa^{m+1})\in [V^h]^2\times V^h$ such that $$\label{SD:BGN2} \begin{cases} \left(\frac{\mathbf{X}^{m+1}-\mathbf{X}^{m-1}}{2\tau},\varphi^h \mathbf{n}^m \right)^h_{\Gamma^m}+\left( \frac{\partial_s\kappa^{m+1}+\partial_s\kappa^{m-1}}{2},\partial_s\varphi^h \right)_{\Gamma^m}=0,\\ \vspace{-5mm}\\ \left(\frac{\kappa^{m+1}+\kappa^{m-1}}{2},\mathbf{n}^m\cdot \bm{\omega}^h\right)_{\Gamma^m}^h-\left(\frac{\partial_s \mathbf{X}^{m+1}+\partial_s \mathbf{X}^{m-1}}{2},\partial_s\bm{\omega}^h\right)_{\Gamma^m}=0, \end{cases}$$ for all $(\varphi^h, \bm{\omega}^h)\in V^h\times [V^h]^2$. The well-posedness and energy stability of the above scheme can be shown similarly under certain mild conditions. For the schemes [\[AP-CSF:BGN2\]](#AP-CSF:BGN2){reference-type="eqref" reference="AP-CSF:BGN2"} and [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"}, we consistently set $\mathbf{X}^0\in [V^h]^2$ as specified in Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"}, that is, $\mathbf{X}^0$ is a parametrization of an (almost) equidistributed interpolation polygon with $N$ vertices for the initial curve $\Gamma(0)$. Similar as the case of CSF, to start the BGN2 schemes, we need to prepare the initial data $\kappa^0$ and $(\mathbf{X}^1,\kappa^1)$, which can be achieved by using the similar approach as Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"} by using the corresponding BGN1 scheme. A complete second-order scheme can be obtained as in Algorithm [\[Full algorithm\]](#Full algorithm){reference-type="ref" reference="Full algorithm"} with the corresponding BGN1 scheme as a mesh regularization when necessary. # Numerical results {#sec:illust} ## Convergence tests {#sec:order 2, illu} In this subsection, we test the temporal convergence of the second-order schemes [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"}, [\[AP-CSF:BGN2\]](#AP-CSF:BGN2){reference-type="eqref" reference="AP-CSF:BGN2"} and [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"} for solving the three geometric flows: CSF, AP-CSF and SDF, respectively. As previously discussed in Section [3](#sec:Different error){reference-type="ref" reference="sec:Different error"}, we quantify the numerical errors of the curves using the shape metrics, such as the manifold distance and Hausdorff distance. For the following simulations, we select four distinct types of initial shapes: - (**Shape 1**): a unit circle; - (**Shape 2**): an ellipse with semi-major axis $2$ and semi-minor axis $1$; - (**Shape 3**): a 'tube' shape, which is a curve comprising a $4 \times 1$ rectangle with two semicircles on its left and right sides; - (**Shape 4**): a 'flower' shape, which is parameterized by $$\mathbf{X}(\rho)=((2+\cos(12\pi\rho))\cos(2\pi\rho),(2+\cos(12\pi\rho))\sin(2\pi\rho)),\quad \rho\in \mathbb{I}=[0,1].$$ We note that for the CSF with Shape 1 as its initial shape has the following true solution, i.e., $$\mathbf{X}_{\mathrm{true}}(\rho,t)=\sqrt{1-2t}(\cos(2\pi\rho),\sin(2\pi\rho)),\quad \rho\in \mathbb{I},\quad t\in [0,0.5).$$ For this particular case, we compute the numerical error by comparing it with the true solution. However, for all other cases, we utilize the reference solutions which are obtained by the BGN2 scheme with large $N$ and a tiny time step size $\tau$. In addition, the mesh regularization threshold is consistently set to $n_{\text{MR}}=10$. ![Log-log plot of the numerical errors at time $T=0.25$ measured by the manifold distance for BGN1 [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} and BGN2 [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} schemes for solving the CSF with two different initial curves: (a) Shape 1 and (b) Shape 2, respectively, where the number of nodes is fixed as $N=10000$.](CSF_EOC1.eps){#Fig:CSF_EOC1 width="5.3in" height="2in"} We begin our test by calculating the convergence of the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} for the CSF with either Shape 1 or Shape 2 as initial data. Fig. [1](#Fig:CSF_EOC1){reference-type="ref" reference="Fig:CSF_EOC1"} presents a log-log plot of the numerical errors at time $T=0.25$, measured by the manifold distance. The errors for the Hausdorff distance, which are similar, are not included here for brevity. To ensure a fair comparison, we also include the numerical results of the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} under the same computational parameters, with a fixed number of grid points $N=10000$. As clearly shown in Fig. [1](#Fig:CSF_EOC1){reference-type="ref" reference="Fig:CSF_EOC1"}, the numerical error of the BGN2 scheme reduces very rapidly with second-order accuracy in time, while the BGN1 scheme only achieves first-order convergence. {#Fig:APCSFandSDF_EOC1 width="5.3in" height="2in"} {#Fig:APCSFandSDF_EOC2 width="5.3in" height="2in"} Fig. [2](#Fig:APCSFandSDF_EOC1){reference-type="ref" reference="Fig:APCSFandSDF_EOC1"} shows the temporal errors of the BGN2 scheme [\[AP-CSF:BGN2\]](#AP-CSF:BGN2){reference-type="eqref" reference="AP-CSF:BGN2"} for solving the AP-CSF and SDF with Shape 2 as initial data. It is clear that the numerical error of the BGN2 scheme converges quadratically, whereas the BGN1 scheme [\[AP-CSF:BGN1\]](#AP-CSF:BGN1){reference-type="eqref" reference="AP-CSF:BGN1"} converges only linearly. Moreover, since both the AP-CSF and SDF eventually evolve into a circle, we also investigate the convergence of the BGN2 scheme over long-time simulations. As illustrated in Fig. [3](#Fig:APCSFandSDF_EOC2){reference-type="ref" reference="Fig:APCSFandSDF_EOC2"}, the numerical errors at three different times $T=0.25,0.5,2$ of the BGN2 scheme all display quadratic convergence. ## Comparison of computational costs {#sec:cost} In order to show that the computational cost of the proposed BGN2 scheme is comparable to that of the BGN1 scheme, we present two examples about solving the CSF and SDF, respectively. The numerical codes were written by using MATLAB 2021b, and they were implemented in a MacBook Pro with 1.4GHz quad-core Intel Core i5 and 8GB RAM. BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} ------------------------------------------------------------------------------------ --------------------- --------------------- -------- ------------------------------------------------------------------------------------ --------------------- --------------------- -------- $N$ $\mathcal E_{M}(T)$ $\mathcal E_{H}(T)$ $N$ $\mathcal E_{M}(T)$ $\mathcal E_{H}(T)$ 320 5.61E-4 1.25E-4 0.350s 160 8.35E-4 2.02E-4 0.200s 640 3.34E-4 6.37E-5 1.70s 320 2.09E-4 5.04E-5 0.430s 1280 1.81E-4 3.22E-5 9.85s 640 5.20E-5 1.27E-5 2.30s 2560 9.38E-5 1.62E-5 110s 1280 1.29E-5 3.20E-6 12.9s 5120 4.78E-5 8.16E-5 1893s 2560 3.08E-6 8.16E-7 130s : Comparisons of the CPU times (seconds) and the numerical errors measured from the manifold distance $\mathcal E_{M}(T)$ and Hausdorff distance $\mathcal E_{H}(T)$ for the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} and the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} applied to CSF, where the initial shape is chosen as Shape 1, with $\tau=0.5/N$ and $T=0.05$. [\[tab:CPU time\]]{#tab:CPU time label="tab:CPU time"} Table [4](#tab:CPU time){reference-type="ref" reference="tab:CPU time"} displays a comparison of CPU times in seconds and numerical errors at time $T=0.05$, as measured by the manifold distance $\mathcal E_{M}(T)$ and Hausdorff distance $\mathcal E_{H}(T)$, using the BGN2 scheme [\[CSF:BGN2\]](#CSF:BGN2){reference-type="eqref" reference="CSF:BGN2"} and the BGN1 scheme [\[CSF:BGN1\]](#CSF:BGN1){reference-type="eqref" reference="CSF:BGN1"} for solving the CSF, where the initial shape is chosen as Shape 1. Table [5](#tab:CPU2){reference-type="ref" reference="tab:CPU2"} provides similar results for solving the SDF with Shape 3 as its initial shape. Based on the findings presented in Tables [4](#tab:CPU time){reference-type="ref" reference="tab:CPU time"} and [5](#tab:CPU2){reference-type="ref" reference="tab:CPU2"}, the following conclusions can be drawn: (i) On the same mesh, the computational cost of the BGN2 scheme is slightly higher compared to the BGN1 scheme, as it involves additional calculations for the initial values and the right-hand side of the linear system at each time level. However, the numerical solution obtained using the BGN2 scheme is significantly more accurate than the BGN1 scheme. (ii) Achieving the same level of accuracy requires a much higher computational cost for the BGN1 scheme. For instance, as demonstrated in Table [4](#tab:CPU time){reference-type="ref" reference="tab:CPU time"}, when comparing the results of the BGN1 scheme with $N=5120$ and the BGN2 scheme with $N=1280$, it is evident that the BGN2 scheme is not only more accurate but also more than $100$ times faster than the BGN1 scheme. Similar trends can be observed in Table [5](#tab:CPU2){reference-type="ref" reference="tab:CPU2"} for the SDF. BGN1 scheme [\[SD:BGN1\]](#SD:BGN1){reference-type="eqref" reference="SD:BGN1"} BGN2 scheme [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"} --------------------------------------------------------------------------------- --------------------- --------------------- -------- --------------------------------------------------------------------------------- --------------------- --------------------- -------- $N$ $\mathcal E_{M}(T)$ $\mathcal E_{H}(T)$ $N$ $\mathcal E_{M}(T)$ $\mathcal E_{H}(T)$ 320 4.73E-3 6.91E-4 0.470s 160 7.51E-3 2.62E-3 0.260s 640 2.24E-3 3.38E-4 2.03s 320 2.53E-3 1.14E-3 0.610s 1280 1.10E-3 1.67E-4 12.6s 640 8.28E-4 4.17E-4 2.270s 2560 5.53E-4 8.34E-5 132.6s 1280 2.30E-4 1.12E-4 15.1s 5120 2.80E-4 4.16E-5 2180s 2560 5.42E-5 2.82E-5 153s : Comparisons of the CPU times (seconds) and the numerical errors measured by the manifold distance $\mathcal E_{M}(T)$ and Hausdorff distance $\mathcal E_{H}(T)$ using the BGN2 scheme [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"} and the BGN1 scheme [\[SD:BGN1\]](#SD:BGN1){reference-type="eqref" reference="SD:BGN1"} for SDF, where the initial shape is chosen as Shape 3, with $\tau=0.5/N$, and $T=0.05$. [\[tab:CPU2\]]{#tab:CPU2 label="tab:CPU2"} ## Applications to the curve evolution {#sec:long time, illu} As is well-known, the AP-CSF and SDF possess some structure-preserving properties, such as the perimeter decreasing and area conserving properties [@JiangLi21; @Jiang23; @Bao-Zhao]. In this subsection, we investigate the structure-preserving properties of the proposed BGN2 schemes [\[AP-CSF:BGN2\]](#AP-CSF:BGN2){reference-type="eqref" reference="AP-CSF:BGN2"} and [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"} applied to AP-CSF and SDF, respectively. As an example, we mainly focus on the SDF here. Moreover, we will discuss the importance of the mesh regularization procedure. {#Fig:Ellipse_evo_and_Geo width="5in" height="3.5in"} Fig. [4](#Fig:Ellipse_evo_and_Geo){reference-type="ref" reference="Fig:Ellipse_evo_and_Geo"} (a) illustrates the evolution of an initially elliptic curve, referred to as Shape 2, driven by SDF towards its equilibrium state. Fig. [4](#Fig:Ellipse_evo_and_Geo){reference-type="ref" reference="Fig:Ellipse_evo_and_Geo"}(b)-(d) show the evolution of various geometric quantities during the process: the relative area loss $\Delta A(t)$, the normalized perimeter $L(t)/L(0)$, and the mesh distribution function $\Psi(t)$, which are defined respectively as $$\Delta A(t)|_{t=t_m}=\frac{A^m-A^0}{A^0},\quad \left.\frac{L(t)}{L(0)}\right|_{t=t_m}=\frac{L^m}{L^0},\quad \Psi(t)|_{t=t_m}=\Psi^m,\quad m\ge 0,$$ where $A^m$ is the area enclosed by the polygon determined by $\mathbf{X}^m$, $L^m$ represents the perimeter of the polygon, and the mesh ratio $\Psi^m$ is defined in [\[Mesh assumption\]](#Mesh assumption){reference-type="eqref" reference="Mesh assumption"}. As depicted in Fig. [4](#Fig:Ellipse_evo_and_Geo){reference-type="ref" reference="Fig:Ellipse_evo_and_Geo"}(b), the area loss exhibits a weakly oscillating behavior, which may result from the two-step structure of the BGN2 scheme. It is worth noting that despite the oscillations, the normalized area loss remains very low, consistently below $0.01\%$. By employing a smaller grid size, the area loss can be further reduced, and it is significantly lower than that of the BGN1 scheme under the same computational parameters. Furthermore, Fig. [4](#Fig:Ellipse_evo_and_Geo){reference-type="ref" reference="Fig:Ellipse_evo_and_Geo"}(c) shows the BGN2 scheme preserves the perimeter-decreasing property of the SDF numerically. Furthermore, in Fig. [4](#Fig:Ellipse_evo_and_Geo){reference-type="ref" reference="Fig:Ellipse_evo_and_Geo"}(d), it can be observed that the mesh distribution function $\Psi(t)$ remains lower than $1.2$ during the evolution. This indicates that the mesh distribution remains well-maintained and almost equidistributed during the process. Therefore, in this scenario, there is no need to perform the mesh regularization procedure because $\Psi(t)$ is always smaller than the chosen threshold $n_{\text{MR}}$ (here we choose it as $10$) in the simulations. {#Fig:tube_evo_and_Geo width="5in" height="3.5in"} To provide a more comprehensive comparison, we conduct simulations of evolution of Shape 3 curve driven by the SDF. Fig. [5](#Fig:tube_evo_and_Geo){reference-type="ref" reference="Fig:tube_evo_and_Geo"}(b)-(c) demonstrates that the BGN2 scheme effectively preserves two crucial geometric properties of the SDF: the conservation of area and the reduction of perimeter properties [@JiangLi21; @Bao-Zhao]. It should be noted that Fig. [5](#Fig:tube_evo_and_Geo){reference-type="ref" reference="Fig:tube_evo_and_Geo"}(d) reveals that without the implementation of mesh regularization, the mesh distribution function $\Psi(t)$ can become very large. Therefore, in our algorithm, when $\Psi(t)$ exceeds a threshold $n_{\text{MR}}$, we employ the BGN1 scheme [\[SD:BGN1\]](#SD:BGN1){reference-type="eqref" reference="SD:BGN1"} for a single run to perform mesh regularization, similar to $\text{Step~3}$ of Algorithm [\[Full algorithm\]](#Full algorithm){reference-type="ref" reference="Full algorithm"}. As clearly shown in Fig. [5](#Fig:tube_evo_and_Geo){reference-type="ref" reference="Fig:tube_evo_and_Geo"}(d), following this step, the mesh ratio rapidly decreases to a low value, which makes the method more stable. Importantly, this mesh regularization procedure is only required four times throughout the entire evolution, without sacrificing the accuracy of the BGN2 scheme (cf. Table [5](#tab:CPU2){reference-type="ref" reference="tab:CPU2"}). ![Evolution of the three geometrical quantities when the initial data is prepared as in Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"}: (a) the normalized area loss, (b) the normalied perimeter, (c) the mesh distribution function $\Psi(t)$, with mesh regularization procedure.](flower_geo_MR_initialkappa.eps){#Fig:flower_geo_MRandnoMR width="5.1in" height="1.6in"} {reference-type="ref" reference="iniitalp"}: (a) the normalized area loss, (b) the normalized perimeter, (c) the mesh distribution function $\Psi(t)$, with mesh regularization procedure (shown in the top row) and without mesh regularization procedure (shown in the bottom row).](flower_geo_MRandnoMRb.eps){#Fig:flower_geo_MRandnoMRb width="5.2in" height="3.2in"} Next, we proceed to simulate the evolution of a nonconvex curve, referred to as Shape 4. Fig. [6](#Fig:flower_geo_MRandnoMR){reference-type="ref" reference="Fig:flower_geo_MRandnoMR"} and Fig. [7](#Fig:flower_geo_MRandnoMRb){reference-type="ref" reference="Fig:flower_geo_MRandnoMRb"} (top row) show the evolution of the geometric quantities based on two different initial data preparations: Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"} and Remark [Remark 2](#iniitalp){reference-type="ref" reference="iniitalp"}, respectively. A comparison of the results reveals the superiority of the latter approach for several reasons: (i) the magnitude of area loss is significantly lower when using the approach in Remark [Remark 2](#iniitalp){reference-type="ref" reference="iniitalp"}; (ii) the perimeter-decreasing property is preserved while the perimeter oscillates at the beginning when using Algorithm [\[CSF:BGN initial data 1\]](#CSF:BGN initial data 1){reference-type="ref" reference="CSF:BGN initial data 1"}; (iii) the number of mesh regularization implementations is smaller with the approach in Remark [Remark 2](#iniitalp){reference-type="ref" reference="iniitalp"}. Thus we recommend preparing the data for a nonconvex initial curve following the approach outlined in Remark [Remark 2](#iniitalp){reference-type="ref" reference="iniitalp"}. Fig. [7](#Fig:flower_geo_MRandnoMRb){reference-type="ref" reference="Fig:flower_geo_MRandnoMRb"} (bottom row) illustrates the evolution of the same quantities without implementation of mesh regularization. In this case, all three quantities exhibit significant oscillations after a certain time period, and the area loss and mesh ratio of the polygon becomes excessively large, resulting in the breakdown of the BGN2 scheme. Notably, mesh clustering has happened at $t=1$ (see Fig. [8](#Fig:flower_evo_MRandnoMR){reference-type="ref" reference="Fig:flower_evo_MRandnoMR"} (c2)), eventually leading to mesh distortion at $t=2$ (see Fig. [8](#Fig:flower_evo_MRandnoMR){reference-type="ref" reference="Fig:flower_evo_MRandnoMR"} (d2)). These issues can be avoided by implementing mesh regularization (see [7](#Fig:flower_geo_MRandnoMRb){reference-type="ref" reference="Fig:flower_geo_MRandnoMRb"} (a1)-(c1) and Fig. [8](#Fig:flower_evo_MRandnoMR){reference-type="ref" reference="Fig:flower_evo_MRandnoMR"} (a1)-(d1)). This demonstrates the essential role of mesh regularization in the effectiveness of the BGN2 scheme and the BGN1 scheme can greatly improve the mesh distribution. ![Evolution of the curve driven by SDF starting with Shape 4 as initial data by using the BGN2 scheme [\[SD:BGN2\]](#SD:BGN2){reference-type="eqref" reference="SD:BGN2"} with mesh regularization procedure (shown in the top row), and without mesh regularization procedure (shown in the bottom row). The simulations are conducted with a grid number of $N=80$ and a time step size $\tau=1/160$.](flower_evo_MRandnoMRb.eps){#Fig:flower_evo_MRandnoMR width="5.2in" height="2.8in"} We close this section by simulating the evolution of a nonconvex initial curve [@Mikula-Sevcovic2004; @Balazovjech-Mikula; @Mackenzie-Nolan-Rowlatt-Insall] driven by CSF, AP-CSF and SDF using the BGN2 schemes. The initial curve can be parametrized as $$\mathbf{X}_0(\rho)=( \cos(2\pi \rho), \sin(\cos(2\pi\rho ))+\sin(2\pi \rho)(0.7+\sin(2\pi \rho)\sin^2(6\pi \rho))),$$ for $\rho\in \mathbb{I}=[0,1]$. The numerical results are depicted in Fig. [9](#Fig:Mikula_evo){reference-type="ref" reference="Fig:Mikula_evo"}. As shown in this figure, the CSF initially transforms the intricate curve into a circle before it disappear. Both the AP-CSF and SDF drive the curve to evolve into a perfect circle as its equilibrium shape. {#Fig:Mikula_evo width="5.3in" height="3in"} # Conclusions We proposed a novel temporal second-order, BGN-based parametric finite element method (BGN2 scheme) for solving geometric flows of curves such as CSF, AP-CSF and SDF. Based on the BGN formulation and the corresponding semi-discrete FEM approximation [@BGN07A; @BGN07B; @BGN20], our numerical method employs a Crank-Nicolson leap-frog method to discretize in time and the key innovation lies in choosing a discrete inner product over the curve $\Gamma^m$, such that the time level $t_m$ coincides with when all quantities have approximations with an error of $\mathcal{O}(\tau^2)$. We established the well-posedness and mild energy stability of the fully-discrete scheme, subject to suitable assumptions. We emphasized the use of shape metrics (manifold distance and Hausdorff distance) rather than function norms (e.g., $L^2$-norm, $L^{\infty}$-norm) to measure the numerical errors of the BGN-based schemes. In the case of certain initial curves, such as a 'flower' shape, we found that the BGN2 scheme, in conjunction with the BGN1 scheme for mesh regularization, exhibited remarkable efficiency and stability in practical simulations. Extensive numerical experiments demonstrated that the proposed BGN2 scheme achieves second-order accuracy in time, as measured by the shape metrics, outperforming the BGN1 scheme in terms of accuracy. Furthermore, it is worth mentioning that the approach we have presented for constructing a temporal high-order BGN-based scheme can be readily extended to address various other problems, such as anisotropic geometric flows [@Bao-Jiang-Li], Willmore flow [@BGN08C], two-phase flow [@Garcke23], solid-state dewetting [@Zhao-Jiang-Bao2021] and geometric flows in 3D [@Zhao-Jiang-Bao]. In our future research, we will further investigate the development of structure-preserving temporal high-order BGN-based schemes [@Bao-Zhao; @JiangLi21] and conduct the numerical analysis of the BGN-based schemes with respect to the shape metric. These investigations will contribute to enhancing the overall understanding and applicability of the BGN type scheme in different contexts. # CRediT authorship contribution statement {#credit-authorship-contribution-statement .unnumbered} **Wei Jiang**: Conceptualization, Methodology, Supervision, Writing. **Chunmei Su**: Conceptualization, Methodology, Supervision, Writing. **Ganghui Zhang**: Methodology, Numerical experiments, Visualization and Writing. # Declaration of competing interest {#declaration-of-competing-interest .unnumbered} The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. # Data availability {#data-availability .unnumbered} No data was used for the research described in the article. # Acknowledgement {#acknowledgement .unnumbered} This work was partially supported by the NSFC 12271414 and 11871384 (W. 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arxiv_math

--- abstract: | We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by Jothilingam in 1985 ([@Jothilingam]) which involves listing quadratic residues in some order. On the way, we also observe some more congruences for the Fermat quotient that generalize Eisenstein's classical congruence (Lemma 1). Using such polynomial identities, we obtain some sums involving harmonic numbers. We also prove formulae for binomial sums of harmonic numbers of higher order (Theorem 2).\ **Keywords:** Polynomial identities, Fermat quotients, Harmonic numbers author: - | Takao Komatsu\ Department of Mathematical Sciences, School of Science\ Zhejiang Sci-Tech University\ Hangzhou 310018 China\ `komatsu@zstu.edu.cn`\ \ B. Sury\ Stat-Math Unit\ Indian Statistical Institute\ 8th Mile Mysore Road\ Bangalore 560059 India\ `sury@isibang.ac.in` date: " MR Subject Classifications: Primary 11B65; Secondary 11A07, 05A10, 11B50, 11B73 " title: Polynomial identities and Fermat quotients --- Throughout, we use the standard notation $\frac{a}{b} \equiv \frac{c}{d}$ modulo $m$ for a positive integer $m$ relatively prime to $bd$ if, $\frac{a}{b}-\frac{c}{d}=\frac{mp}{q}$ with $\gcd(m,q)=1$. # Generalizing Eisenstein's congruence for Fermat quotients Eisenstein had proved for an odd prime that, modulo $p$, we have $$\frac{2^{p-1}-1}{p} \equiv \sum_{j:odd,~j<p-1} \frac{1}{j}.$$ It is possible to obtain this and some more congruences by exploiting an elementary polynomial identity as follows. \ For any integer $m \geq 1$, we have the evident polynomial identity $$\frac{x^m-x}{m} =\sum_{r=1}^{m-1} {m \choose r} \frac{(x+1)^r(-1)^{m-r}}{m} +\frac{(x+1)^m-x+(-1)^m}{m}. \label{eq:0}$$ As a consequence, we observe: **Lemma 1**. *For any odd prime $p$, we have the following congruence modulo $p$: $$\begin{aligned} \frac{2^{p-1}-1}{p} &\equiv \frac{1}{2} \sum_{j=0}^{p-2}\frac{1}{{p-2 \choose j}}. \label{eq:E}\end{aligned}$$ Further, for any integer $n \geq 2$ and any odd prime $p$, we have the following congruences modulo $p$: $$\frac{n^p-n}{p} \equiv - \sum_{r=1}^{p-1} \frac{1^r+2^r + \cdots +n^r}{r}. \label{eq:A}$$ $$\frac{n^p-n}{p} \equiv - \sum_{r=1}^{p-1} \frac{(-1)^r(1^r+2^r + \cdots+ (n-1)^r)}{r}. \label{eq:AA}$$ In particular, modulo $p$, we have: $$\begin{aligned} \frac{2^{p-1}-1}{p} &\equiv - \frac{1}{2} \sum_{j=1}^{p-1}\frac{2^j}{j}, \label{eq:B}\\ \frac{2^{p-1}-1}{p} &\equiv - \frac{1}{2} \sum_{j=1}^{p-1}\frac{(-1)^j}{j}. \label{eq:C}\\ \frac{2^{p-1}-1}{p} &\equiv \sum_{j{\rm :odd},~j<p-1}\frac{1}{j} \equiv - \frac{1}{2} \sum_{j=1}^{(p-1)/2} \frac{1}{j} \label{eq:D}\end{aligned}$$* It follows from ([\[eq:0\]](#eq:0){reference-type="ref" reference="eq:0"}) for $m=p$ that: $$\frac{a^p-a}{p} = \frac{(a+1-1)^p-a}{p} =\sum_{r=1}^{p-1} {p \choose r} \frac{(a+1)^r(-1)^{p-r}}{p} +\frac{(a+1)^p-(a+1)}{p}.$$ We claim that, modulo $p$, $$\frac{a^p-a}{p} \equiv \sum_{r=1}^{p-1}\frac{(a+1)^r}{r} + \frac{(a+1)^p-(a+1)}{p}. \label{spadesuit}$$ In the above, we used the observation that if $p$ is an odd prime and $0< r <p$, then the integer $\frac{1}{p} {p \choose r} \equiv \frac{(-1)^{r-1}}{r}\mod p$. This is so because $$\frac{1}{p} {p \choose r} = \frac{(p-1)(p-2) \cdots (p-r+1)}{r!} \equiv \frac{(-1)^{r-1}(r-1)!}{r!} = \frac{(-1)^{r-1}}{r}.$$ Putting $a=0$ gives the well-known congruence $\sum_{r=1}^{p-1} \frac{1}{r} \equiv 0\mod p.$\ Thus, congruence (6) is an equivalent version of Eisenstein's congruence.\ Putting $a=1$ gives congruence ([\[eq:B\]](#eq:B){reference-type="ref" reference="eq:B"}).\ Putting $a=-2$ gives the congruence ([\[eq:C\]](#eq:C){reference-type="ref" reference="eq:C"}).\ Inductively, from ([\[spadesuit\]](#spadesuit){reference-type="ref" reference="spadesuit"}), one gets then that $$\frac{n^p-n}{p} \equiv - \sum_{r=1}^{p-1} \frac{2^r + \cdots +n^r}{r}\mod p.$$ When $p$ is an odd prime, $\sum_{r=1}^{p-1} \frac{1}{r}\equiv 0\mod p$ (indeed, it is even zero modulo $p^2$ when $p>3$ by Wolstenholme's theorem). Thus, we have the more symmetric form asserted as ([\[eq:A\]](#eq:A){reference-type="ref" reference="eq:A"}). Finally, ([\[eq:AA\]](#eq:AA){reference-type="ref" reference="eq:AA"}) is gotten similarly to ([\[eq:A\]](#eq:A){reference-type="ref" reference="eq:A"}) inductively from ([\[spadesuit\]](#spadesuit){reference-type="ref" reference="spadesuit"}) by putting $a = -2, -3, -4$ etc.\ Clearly, using the fact that $\sum_{j \leq p-1} \frac{1}{j} \equiv 0\mod p$, ([\[eq:C\]](#eq:C){reference-type="ref" reference="eq:C"}) implies the two congruences in ([\[eq:D\]](#eq:D){reference-type="ref" reference="eq:D"}). To prove the congruence ([\[eq:E\]](#eq:E){reference-type="ref" reference="eq:E"}), let us use the following identities which were proved in [@Sury93]: $$\sum_{r=0}^n \frac{1}{{n \choose r}} = \frac{n+1}{2^n} \sum_{i=0}^n \frac{2^i}{i+1} = \frac{n+1}{2^n} \sum_{j{\rm :odd}} {n+1 \choose j} \frac{1}{j}.$$ If we put in $n=p-2$ for a prime $p \geq 3$, the second expression becomes $$\frac{p-1}{2^{p-2}} \sum_{i=0}^{p-2} \frac{2^i}{i+1} = \frac{p-1}{2^{p-1}} \sum_{j=1}^{p-1} \frac{2^j}{j} \equiv -\sum_{j=1}^{p-1} \frac{2^j}{j},$$ which gives the congruence ([\[eq:E\]](#eq:E){reference-type="ref" reference="eq:E"}) on using ([\[eq:B\]](#eq:B){reference-type="ref" reference="eq:B"}). This completes the proof of the lemma. # New congruences for Fermat quotient modulo $p^2$ We prove some polynomial identities which are then used to obtain two different congruences for Fermat quotients modulo $p^2$; these are simpler than the ones obtained in [@Jothilingam]. **Lemma 2**. *For any odd positive integer $n$, $$\sum_{r=1}^n \frac{(-1)^{r-1}x^r}{r} = \sum_{r=1}^n {n \choose r} \frac{(-1)^{r-1}(x+1)^r}{r} + \sum_{r=1}^n \frac{(-1)^r}{r} {n\choose r}.$$* Consider the evident identity $$x^n = \sum_{r=1}^n {n \choose r}(x+1)^r (-1)^{r-1}-1.$$ As $n$ is odd, we have $$1-x+x^2- \cdots + x^{n-1}= \frac{x^n+1}{x+1}= \sum_{r=1}^n {n\choose r} (-1)^{r-1} (x+1)^{r-1}.$$ Integration gives $$x - \frac{x^2}{2} + \cdots + \frac{x^n}{n} = \sum_{r=1}^{n} {n \choose r} (-1)^{r-1} \frac{(x+1)^r}{r} + c$$ for some constant $c$. Putting $x=0$, we obtain $c= \sum_{r=1}^n\frac{(-1)^r}{r} {n \choose r}$. Thus, we have the asserted polynomial identity: $$\sum_{r=1}^n \frac{(-1)^{r-1}x^r}{r} = \sum_{r=1}^n {n \choose r} \frac{(-1)^{r-1}(x+1)^r}{r} + \sum_{r=1}^n \frac{(-1)^r}{r} {n\choose r}.$$ Note in passing that by comparing coefficients of $x^2$ and $x^3$, we get $$\begin{aligned} n &= \sum_{r=1}^{n-1} (-1)^r {n \choose r} (r-1);\\ (n-1)(n-2) &= 2 + \sum_{r=1}^{n-1} (-1)^r {n \choose r}(r-1)(r-2).\end{aligned}$$ Now, we use the above polynomial identity to obtain two different congruences for the Fermat quotient modulo $p^2$ for an odd prime $p$. **Proposition 1**. *For an odd prime $p$, $$\frac{2^{p-1}-1}{p} \equiv \sum_{r=1}^{p-1} \frac{-2^{r-1}}{r} \mod {p^2}.$$* *Proof.* In the polynomial identity $$\sum_{r=1}^n \frac{(-1)^{r-1}x^r}{r} = \sum_{r=1}^n {n \choose r}\frac{(-1)^{r-1}(x+1)^r}{r} + \sum_{r=1}^n \frac{(-1)^r}{r} {n\choose r},$$ take $x=-2$. We obtain $$-\sum_{r=1}^p \frac{2^r}{r} = -\sum_{r=1}^{P-1} {p \choose r} \frac{1}{r} - \frac{2}{p} + \sum_{r=1}^{p-1} {p \choose r} \frac{(-1)^r}{r}.$$ Rewriting this, we have $$\frac{2^p-2}{p} + \sum_{r=1}^{p-1} \frac{2^r}{r} = 2 \sum_{r{\rm :odd},~r<p} {p \choose r} \frac{1}{r}.$$ Firstly, for $p=3$, the proposition follows by direct computation. Therefore, the proposition will follow if we show that for $p \geq 5$, $$\sum_{r{\rm :odd},~ r<p} {p \choose r} \frac{1}{pr} \equiv 0\mod p.$$ Now for $1 \leq r \leq p-2$ with $r$ odd, we have, modulo $p$, $$\frac{1}{pr} {p \choose r} = \frac{(p-1)(p-2) \cdots(p-r+1)}{r^2} \equiv \frac{1}{r^2}$$ since $r$ is odd. Therefore, $$\sum_{r{\rm :odd},~ r<p} {p \choose r} \frac{1}{pr} \equiv \sum_{r{\rm :odd},~r<p} \frac{1}{r^2} \equiv \frac{1}{2} \sum_{r=1}^{p-1} \frac{1}{r^2}$$ since $(p-r)^2 \equiv r^2\mod p$ and $p-r$ runs through the even integers $<p$ when $r$ runs through the odd integers $<p$. But, $$\sum_{r=1}^{p-1} \frac{1}{r^2} \equiv \sum_{d=1}^{p-1} d^2 = \frac{(p-1)p(2p-1)}{6} \equiv 0$$ if $p \geq 5$. Therefore, the proposition is proved. ◻ We have already proved a congruence for $\frac{2^p-2}{p}$ modulo $p^2$. In 1985, Jothilingam [@Jothilingam] had proved a congruence that involves an ordered choice of quadratic residues. Below, we prove a different, simpler congruence. **Lemma 3**. *$\sum_{r=0}^{n-1} \frac{(1-x)^r}{r+1} = \sum_{r=0}^{n-1} {n \choose r+1} \frac{(-1)^r}{r+1} \frac{x^{r+1}-1}{x-1}.$* Start with the elementary polynomial identity $$- \sum_{k=1}^n (1-x)^{k-1} = \sum_{k=1}^n {n \choose k} (-1)^k x^{k-1}.$$ Integrating this, we have $$\sum_{k=1}^n \frac{(1-x)^k}{k} = \sum_{k=1}^n {n \choose k}\frac{(-1)^k (x^k-1)}{k}.$$ The above identity has been written after finding the constant of integration by putting $x=1$. Rewriting the above identity by taking $k=r+1$, we have the asserted polynomial identity: $$\sum_{r=0}^{n-1} \frac{(1-x)^r}{r+1} = \sum_{r=0}^{n-1} {n \choose r+1} \frac{(-1)^r}{r+1} \frac{x^{r+1}-1}{x-1}. \label{heartsuit}$$ As an application, we get: **Theorem 1**. *For any odd prime $p$, we have $$\frac{2^{p-1}-1}{p} \equiv \sum_{r{\rm :odd},~r<p} \frac{1}{r} - p \sum_{r=1}^{p-1} \frac{2^{r-1}}{r^2} \mod{p^2}.$$* *Proof.* For any $n\geq 1$, integrating ([\[heartsuit\]](#heartsuit){reference-type="ref" reference="heartsuit"}) we have $$\sum_{r=0}^{n-1} \frac{(1-x)^{r+1}}{(r+1)^2} = \sum_{r=0}^{n-1} {n\choose r+1} \frac{(-1)^{r+1}}{r+1} \left(x+ \frac{x^2}{2}+ \cdots +\frac{x^{r+1}}{r+1}\right) + C,$$ where the constant $C$ is obtained by putting $x=1$. We obtain $C = \sum_{r=0}^{n-1} {n \choose r+1}\frac{(-1)^{r}}{r+1} (1+ \frac{1}{2}+ \cdots + \frac{1}{r+1})$.\ Let us consider $x=-1$ and $n=p$, for an odd prime $p$. Since ${p\choose r+1} \equiv 0\mod p$ for $r<p-1$, we have $$\sum_{r=0}^{p-1} \frac{2^{r+1}}{(r+1)^2} \equiv \frac{2}{p} \sum_{d{\rm :odd},~ d \leq p} \frac{1}{d}\mod p.$$ Clearly, this is the congruence $$\frac{2-2^p}{p^2} \equiv \sum_{r=1}^{p-1} \frac{2^r}{r^2} -\frac{2}{p} \sum_{r{\rm :odd},~ r<p} \frac{1}{r}\mod p,$$ which gives, on multiplying by $p$, the asserted congruence modulo $p^2$ in the proposition. ◻ **Corollary 1**. *$$\sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k+1} =\frac{n}{n+1}.$$ $$H_n:=\sum_{k=1}^n \frac{1}{k} = \sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k}.$$* The polynomial identity $$\sum_{k=1}^n \frac{(1-x)^k}{k} = \sum_{k=1}^n {n \choose k}\frac{(-1)^k (x^k-1)}{k}.$$ can be integrated to yield $$\sum_{k=1}^n \frac{(1-x)^{k+1}}{k(k+1)} = \sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k} \left(\frac{x^{k+1}}{k+1} - x\right) + C,$$ where we get $C = \sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k+1}$ by putting $x=0$. The value at $x=0$ implies the first identity: $$\sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k+1} =\frac{n}{n+1}.$$ Equating the coefficients of $x$ on both sides of the polynomial identity gives us the second identity: $$H_n:=\sum_{k=1}^n \frac{1}{k} = \sum_{k=1}^n {n \choose k} \frac{(-1)^{k+1}}{k}.$$ # Sums of higher harmonic numbers We obtained an identity for harmonic numbers $H_n$ above. In this section, we prove more general identities for the harmonic numbers of higher order. Let $$H_k^{(r)}:=\sum_{i=1}^k 1/i^r$$ be the $k$-th harmonic number of order $r$. In [@Mneimneh], for a positive integer $n$ and $0\le q\le 1$, it is shown that $$\sum_{k=0}^n H_k\binom{n}{k}(1-q)^k q^{n-k}=H_n-\sum_{j=1}^n\frac{q^j}{j}\,. \label{eq:mneimneh}$$ This relation is derived by the author from an interesting probabilistic analysis. In this section, we obtain formulae generalizing ([\[eq:mneimneh\]](#eq:mneimneh){reference-type="ref" reference="eq:mneimneh"}). **Theorem 2**. *$$\begin{aligned} &\sum_{k=0}^n H_k^{(r)}\binom{n}{k}(1-q)^k q^{n-k}\notag\\ &=H_n^{(r)}-\sum_{j=1}^n\left(\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^{r-1}}\right)\frac{q^j}{j}\,. \label{eq:101}\end{aligned}$$ [\[th:rrr\]]{#th:rrr label="th:rrr"}* In particular, when $r=1$, we find the following relation. Thus, the formula ([\[eq:mneimneh\]](#eq:mneimneh){reference-type="ref" reference="eq:mneimneh"}) is recovered. **Lemma 4**. *$$\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}=1\,. \label{eq:1r1}$$ [\[lem:r1\]]{#lem:r1 label="lem:r1"}* When $r=2$, we find the following relation. Here, $(n)_j=n(n-1)\cdots(n-j+1)$ ($j\ge 1$) is the falling factorial with $(n)_0=1$, and $\left[n\atop k\right]$ denotes the (unsigned) Stirling number of the first kind, arising from the relation $(x)_n=\sum_{k=0}^n(-1)^{n-k}\left[n\atop k\right]x^k$. **Lemma 5**. *$$\begin{gathered} \sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{n-l}=H_n-H_{n-j}\\ =\frac{1}{(n)_j}\sum_{\nu=0}^{j-1}(-1)^{j-\nu-1}(\nu+1)\left[j\atop\nu+1\right]n^\nu\,. \label{eq:1r2}\end{gathered}$$ [\[lem:r2\]]{#lem:r2 label="lem:r2"}* Note that $$\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{n-l}\ne \frac{l+1}{(n)_j}\left[j\atop l+1\right]n^l\,.$$ Hence, we have the following formula. **Corollary 2**. *$$\begin{gathered} \sum_{k=0}^n H_k^{(2)}\binom{n}{k}(1-q)^k q^{n-k}=H_n^{(2)}-\sum_{j=1}^n(H_n-H_{n-j})\frac{q^j}{j}\\ =H_n^{(2)}-\sum_{j=1}^n\left(\frac{1}{(n)_j}\sum_{\nu=0}^{j-1}(-1)^{j-\nu-1}(\nu+1)\left[j\atop\nu+1\right]n^\nu\right)\frac{q^j}{j}\,. \label{eq:104}\end{gathered}$$ [\[cor:r2\]]{#cor:r2 label="cor:r2"}* *Proof of Theorem [\[th:rrr\]](#th:rrr){reference-type="ref" reference="th:rrr"}..* We shall show $$\begin{aligned} &\sum_{k=0}^n H_k^{(r)}\binom{n}{k}(1-q)^k q^{n-k}\notag\\ &=H_n^{(r)}-\sum_{j=1}^n\binom{n}{j}\left(\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}\frac{1}{(n-l)^r}\right)q^j\,. \label{eq:100}\end{aligned}$$ We have $$\begin{aligned} &\sum_{k=0}^n H_k^{(r)}\binom{n}{k}(1-q)^k q^{n-k} =\sum_{k=0}^n H_k^{(r)}\binom{n}{k}\sum_{l=0}^k(-1)^{k-l}\binom{k}{l}q^{n-l}\\ &=\sum_{l=0}^n q^{n-l}\binom{n}{l}\sum_{k=l}^n(-1)^{k-l}\binom{n-l}{n-k}H_k^{(r)}\\ &=\sum_{j=0}^n\binom{n}{j}q^j\sum_{\nu=0}^j(-1)^{j-\nu}\binom{j}{\nu}H_{n-\nu}^{(r)}\\ &=H_n^{(r)}-\sum_{j=1}^n(-1)^{j-1}\binom{n}{j}q^j\sum_{\nu=0}^j(-1)^{\nu}\binom{j}{\nu}\sum_{l=0}^{n-1}\frac{1}{(n-l)^r}\,.\end{aligned}$$ Since $$\sum_{\nu=0}^l(-1)^\nu\binom{j}{\nu}=(-1)^l\binom{j-1}{l}\quad(\text{proved by induction on $l(\ge 0)$})$$ and $$\sum_{\nu=0}^j(-1)^\nu\binom{j}{\nu}=(1-1)^j=0\,,$$ we have $$\begin{aligned} &\sum_{\nu=0}^j(-1)^{\nu}\binom{j}{\nu}\sum_{l=0}^{n-1}\frac{1}{(n-l)^r}\\ &=\sum_{l=0}^{j-1}\left(\sum_{\nu=0}^l(-1)^\nu\binom{j}{\nu}\right)\frac{1}{(n-l)^r}+\sum_{l=j}^{n-1}\left(\sum_{\nu=0}^j(-1)^\nu\binom{j}{\nu}\right)\frac{1}{(n-l)^r}\\ &=\sum_{l=0}^{j-1}(-1)^l\binom{j-1}{l}\frac{1}{(n-l)^r}\,.\end{aligned}$$ By ([\[eq:100\]](#eq:100){reference-type="ref" reference="eq:100"}), it is straightforward to get ([\[eq:101\]](#eq:101){reference-type="ref" reference="eq:101"}). ◻ *Proof of Lemma [\[lem:r1\]](#lem:r1){reference-type="ref" reference="lem:r1"}..* Put $$\begin{aligned} A(n,j)&=\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\,,\\ B(n,j)&=\sum_{l=0}^{j}(-1)^{j-l}\binom{n-l}{n-j}\binom{n}{l}\,.\end{aligned}$$ Since, $$\begin{aligned} B(n+1,j)&=\frac{n+1}{n-j+1}B(n,j)=\frac{(n+1)n}{(n-j+1)(n-j)}B(n-1,j)\\ &=\cdots=\frac{(n+1)n\cdots(j+1)}{(n-j+1)!}B(j,j)\\ &=\frac{(n+1)!}{(n-j+1)!j!}\sum_{l=0}^j(-1)^{j-l}\binom{j}{l}\\ &=\binom{n+1}{j}(1-1)^j=0\,,\end{aligned}$$ we have $$A(n,j+1)-A(n,j)=B(n,j)=0\,.$$ Hence, we obtain $$A(n,j)=A(n,j-1)=\cdots=A(n,1)=(-1)^0\binom{n-1}{n-1}\binom{n}{0}=1\,.$$ ◻ *Proof of Lemma [\[lem:r2\]](#lem:r2){reference-type="ref" reference="lem:r2"}..* The formula ([\[eq:1r2\]](#eq:1r2){reference-type="ref" reference="eq:1r2"}) is yielded from the definition of the Stirling numbers of the first kind: $$\begin{aligned} (x)_j&=\sum_{k=0}^j(-1)^{j-k}\left[j\atop k\right]x^k\\ &=\sum_{\nu=0}^{j-1}(-1)^{j-\nu-1}\left[j\atop\nu+1\right]x^{\nu+1}\quad(\text{if}\, j\ge 1)\,.\end{aligned}$$ Differentiating both sides with respect to $x$ gives $$(x)_j\sum_{l=0}^{j-1}\frac{1}{x-l}=\sum_{\nu=0}^{j-1}(-1)^{j-\nu-1}(\nu+1)\left[j\atop\nu+1\right]x^\nu\,.$$ Thus, the right-hand side of ([\[eq:1r2\]](#eq:1r2){reference-type="ref" reference="eq:1r2"}) is equal to $$\sum_{l=0}^{j-1}\frac{1}{n-l}=H_n-H_{n-j}\,.$$ Put the left-hand side of ([\[eq:1r2\]](#eq:1r2){reference-type="ref" reference="eq:1r2"}) as $$C(n,j):=\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{n-l}\,.$$ Then $$\begin{aligned} &C(n,j)-C(n,j-1)\\ &=\sum_{l=0}^{j-2}(-1)^{j-l-1}\left(\binom{n-l-1}{n-j}\binom{n}{l}+\binom{n-l-1}{n-j+1}\binom{n}{l}\right)\frac{1}{n-l}\\ &\quad +\binom{n}{j-1}\frac{1}{n-j+1}\\ &=\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l}{n-j+1}\binom{n}{l}\frac{1}{n-l}\\ &=\binom{n}{j-1}\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}\frac{1}{n-l}\,.\end{aligned}$$ Now, $$\begin{aligned} &\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}\frac{1}{n-l}\\ &=\left.\int\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}x^{n-l-1}d x\right|_{x=1}\\ &=\left.\int x^{n-1}\left(1-\frac{1}{x}\right)^{j-1}d x\right|_{x=1}\\ &=\left.\left(1-\frac{1}{x}\right)^{j}\frac{{}_2 F_1(-j+1,n-j+1;n-j+2;x)}{(x-1)^j(n-j+1)}\right|_{x=1}\\ &=\frac{\Gamma(j)\Gamma(n-j+2)}{(n-j+1)\Gamma(n+1)}=\frac{(j-1)!(n-j)!}{n!}\,,\end{aligned}$$ where ${}_2 F_1(a,b;c;z)$ is the Gauss hypergeometric function. Hence, $$C(n,j)-C(n,j-1)=\frac{1}{n-j+1}\,.$$ Therefore, $$\begin{aligned} C(n,j)&=C(n,j-1)+\frac{1}{n-j+1}\notag\\ &=C(n,j-2)+\frac{1}{n-j+2}+\frac{1}{n-j+1}\notag\\ &=\cdots\notag\\ &=C(n,1)+\frac{1}{n-1}+\cdots+\frac{1}{n-j+2}+\frac{1}{n-j+1}\notag\\ &=\sum_{l=0}^{j-1}\frac{1}{n-l}\,. \label{eq:r333}\end{aligned}$$ ◻ ## The case $r=3$ When $r=3$, we have the following. **Proposition 2**. *$$\begin{aligned} &\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^2}\\ &=\frac{(H_n-H_{n-j})^2}{2}+\frac{H_n^{(2)}-H_{n-j}^{(2)}}{2}\,. %&=H_n^{(2)}-H_{n-j}^{(2)}-\sum_{m=0}^{j-1}\frac{H_{j-m-1}-H_m}{n-m}\\ %&=H_n^{(2)}-H_{n-j}^{(2)}-\sum_{m=0}^{\fl{j/2}-1}\frac{h_{j-2 m-1}^{(m+1)}}{\binom{j-m-1}{m}}\left(\frac{1}{n-m}-\frac{1}{n-j+m+1}\right)\,,\end{aligned}$$ [\[prp:r3\]]{#prp:r3 label="prp:r3"}* Therefore, we have the following formula. **Corollary 3**. *$$\begin{aligned} &\sum_{k=0}^n H_k^{(3)}\binom{n}{k}(1-q)^k q^{n-k}\\ &=H_n^{(3)}-\sum_{j=1}^n\left(\frac{(H_n-H_{n-j})^2}{2}+\frac{H_n^{(2)}-H_{n-j}^{(2)}}{2}\right)\frac{q^j}{j}\,. %&=H_n^{(3)}-\sum_{j=1}^n\left(H_n^{(2)}-H_{n-j}^{(2)}-\sum_{m=0}^{j-1}\frac{H_{j-m-1}-H_m}{n-m}\right)\frac{q^j}{j}\\ %&=H_n^{(3)}\\ %&\quad -\sum_{j=1}^n\left(H_n^{(2)}-H_{n-j}^{(2)}-\sum_{m=0}^{\fl{j/2}-1}\frac{h_{j-2 m-1}^{(m+1)}}{\binom{j-m-1}{m}}\left(\frac{1}{n-m}-\frac{1}{n-j+m+1}\right)\right)\frac{q^j}{j}\,.\end{aligned}$$ [\[cor:r2\]]{#cor:r2 label="cor:r2"}* *Proof of Proposition [\[prp:r3\]](#prp:r3){reference-type="ref" reference="prp:r3"}..* Put $$D(n,j):=\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^2}\,.$$ Then $$D(n,j)-D(n,j-1)=\binom{n}{j-1}\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}\frac{1}{(n-l)^2}\,. \label{eq:rr44}$$ We shall prove that $$D(n,j)-D(n,j-1) =\frac{1}{n-j+1}(H_n-H_{n-j})\,. %=\frac{1}{(n-j+1)^2}-\sum_{\ell=0}^{j-2}\frac{1}{(j-\ell-1)(n-\ell)}+\frac{H_{j-1}}{n-j+1}\,. \label{eq:rr4}$$ By ([\[eq:rr4\]](#eq:rr4){reference-type="ref" reference="eq:rr4"}), we get $$\begin{aligned} D(n,j)&=\left(\frac{H_n}{n-j+1}-\frac{H_{n-j}}{n-j+1}\right)+\left(\frac{H_n}{n-j+2}-\frac{H_{n-j+1}}{n-j+2}\right)\\ &\quad +\cdots+\left(\frac{H_n}{n-1}-\frac{H_{n-2}}{n-1}\right)+\left(\frac{H_n}{n}-\frac{H_{n-1}}{n}\right)\\ &=H_n\left(\frac{1}{n-j+1}+\frac{1}{n-j+2}+\cdots+\frac{1}{n-1}+\frac{1}{n}\right)\\ &\quad-\left(\frac{H_{n-j}}{n-j+1}+\frac{H_{n-j+1}}{n-j+2}+\cdots+\frac{H_{n-2}}{n-1}+\frac{H_{n-1}}{n}\right)\\ &=H_n(H_n-H_{n-j})-\left(\frac{H_n^2-H_n^{(2)}}{2}-\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2}\right)\\ &=\frac{(H_n-H_{n-j})^2}{2}+\frac{H_n^{(2)}-H_{n-j}^{(2)}}{2}\,. %=\frac{1}{(n-j+1)^2}+\frac{1}{(n-j+2)^2}+\cdots++\frac{1}{(n-1)^2}+D(n,1)\\ %&\quad -\left(\frac{1}{j-1}+\frac{1}{j-2}+\cdots+1\right)\frac{1}{n}\\ %&\quad -\left(\frac{1}{j-2}+\frac{1}{j-3}+\cdots+1-H_1\right)\frac{1}{n-1}\\ %&\quad -\left(\frac{1}{j-3}+\frac{1}{j-4}+\cdots+1-H_2\right)\frac{1}{n-2}\\ %&\quad -\cdots\\ %&\quad -\left(\frac{1}{2}+1-H_{j-3}\right)\frac{1}{n-j+3}\\ %&\quad -\left(1-H_{j-2}\right)\frac{1}{n-j+2}\\ %&\quad +\frac{H_{j-1}}{n-j+1}\\ %&=\sum_{\ell=0}^{j-1}\frac{1}{(n-\ell)^2}-\sum_{m=0}^{j-1}\frac{H_{j-m-1}-H_m}{n-m}\\ %&=H_n^{(2)}-H_{n-j}^{(2)}-\sum_{m=0}^{j-1}\frac{H_{j-m-1}-H_m}{n-m}\,.\end{aligned}$$ In order to prove ([\[eq:rr4\]](#eq:rr4){reference-type="ref" reference="eq:rr4"}), we put $$E(n,j)=(n-j+1)\bigl(D(n,j)-D(n,j-1)\bigr)\,.$$ Then by ([\[eq:rr44\]](#eq:rr44){reference-type="ref" reference="eq:rr44"}) and Lemma [\[lem:r1\]](#lem:r1){reference-type="ref" reference="lem:r1"} ([\[eq:1r1\]](#eq:1r1){reference-type="ref" reference="eq:1r1"}), we have $$\begin{aligned} &E(n,j)-E(n,j-1)\\ &=\frac{1}{n-j+1}\sum_{l=0}^{j-2}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}+\frac{1}{n-j+1}\binom{n}{j-1}\\ &=\frac{1}{n-j+1}\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}=\frac{1}{n-j+1}\,.\end{aligned}$$ Hence, by $D(n,1)=1/n^2$, we get $$\begin{aligned} &D(n,j)-D(n,j-1)= \frac{E(n,j)}{n-j+1}\\ &=\frac{1}{n-j+1}\left(\frac{1}{n-j+1}+\frac{1}{n-j+2}+\cdots+\frac{1}{n-1}+E(n,1)\right)\\ &=\frac{1}{n-j+1}(H_n-H_{n-j})\,. %\\ %&=\frac{1}{n-j+1}\left(\frac{1}{n-j+1}-\sum_{\ell=0}^{j-2}\left(\frac{1}{j-\ell-1}-\frac{1}{n-\ell}\right)+H_{j-1}\right)\\ %&=\frac{1}{(n-j+1)^2}-\sum_{\ell=0}^{j-2}\frac{1}{(j-\ell-1)(n-\ell)}+\frac{H_{j-1}}{n-j+1}\,,\end{aligned}$$ which is the right-hand side of ([\[eq:rr4\]](#eq:rr4){reference-type="ref" reference="eq:rr4"}). ◻ ## The case $r=4$ When $r=4$, we have the following. **Proposition 3**. *$$\begin{aligned} &\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^3}\\ &=\frac{(H_n-H_{n-j})^3}{6}+\frac{(H_n-H_{n-j})(H_n^{(2)}-H_{n-j}^{(2)})}{2}+\frac{H_n^{(3)}-H_{n-j}^{(3)}}{3}\,.\end{aligned}$$ [\[prp:r4\]]{#prp:r4 label="prp:r4"}* Similarly to the case $r=3$, put $$\mathfrak D(n,j):=\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^3}\,.$$ Then $$\mathfrak D(n,j)-\mathfrak D(n,j-1)=\binom{n}{j-1}\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{j-1}{l}\frac{1}{(n-l)^3}\,.$$ Put $$\mathfrak E(n,j)=(n-j+1)\bigl(\mathfrak D(n,j)-\mathfrak D(n,j-1)\bigr)\,.$$ Then by ([\[eq:r333\]](#eq:r333){reference-type="ref" reference="eq:r333"}), we have $$\begin{aligned} &\mathfrak E(n,j)-\mathfrak E(n,j-1)\\ &=\frac{1}{n-j+1}\sum_{l=0}^{j-1}(-1)^{j-l-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{n-l}\\ &=\frac{C(n,j)}{n-j+1}=\frac{1}{n-j+1}\sum_{\ell=0}^{j-1}\frac{1}{n-\ell}=\frac{H_n-H_{n-j}}{n-j+1}\,.\end{aligned}$$ Thus, by $\mathfrak E(n,1)=1/n^2$, we get $$\begin{aligned} &\mathfrak E(n,j)\\ &=\frac{H_n-H_{n-j}}{n-j+1}+\frac{H_n-H_{n-j+1}}{n-j+2}+\cdots+\frac{H_n-H_{n-2}}{n-1}+E(n,1)\\ &=H_n(H_n-H_{n-j})-\left(\frac{H_n^2-H_n^{(2)}}{2}-\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2}\right)\,.\end{aligned}$$ Hence, by $\mathfrak D(n,1)=1/n^3$, we have $$\begin{aligned} &\mathfrak D(n,j)-\mathfrak D(n,j-1)= \frac{\mathfrak E(n,j)}{n-j+1}\\ &=\frac{H_n(H_n-H_{n-j})}{n-j+1}-\frac{1}{n-j+1}\left(\frac{H_n^2-H_n^{(2)}}{2}-\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2}\right)\,.\end{aligned}$$ Therefore, $$\begin{aligned} &\mathfrak D(n,j)\\ &=\frac{H_n^2+H_n^{(2)}}{2(n-j+1)}-\frac{H_n H_{n-j}}{n-j+1}+\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2(n-j+1)}\\ &\quad +\frac{H_n^2+H_n^{(2)}}{2(n-j+2)}-\frac{H_n H_{n-j+1}}{n-j+2}+\frac{H_{n-j+1}^2-H_{n-j+1}^{(2)}}{2(n-j+2)}\\ &\quad +\cdots\\ &\quad +\frac{H_n^2+H_n^{(2)}}{2(n-1)}-\frac{H_n H_{n-2}}{n-1}+\frac{H_{n-2}^2-H_{n-2}^{(2)}}{2(n-1)}\\ &\quad +\mathfrak D(n,1)\\ &=\frac{(H_n^2+H_n^{(2)})(H_n-H_{n-j})}{2}-H_n\left(\frac{H_n^2-H_n^{(2)}}{2}-\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2}\right)\\ &\quad +\sum_{1\le i_1<i_2<i_3\le n}\frac{1}{i_1 i_2 i_3}-\sum_{1\le i_1<i_2<i_3\le n-j}\frac{1}{i_1 i_2 i_3}\,.\end{aligned}$$ Since for $n\ge 3$ $$\begin{aligned} H_n^3&=H_n^{(3)}+6\sum_{1\le i_1<i_2<i_3\le n}\frac{1}{i_1 i_2 i_3}\\ &\quad +3\left(\frac{1}{1^2}\left(H_n-\frac{1}{1}\right)+\frac{1}{2^2}\left(H_n-\frac{1}{2}\right)+\cdots+\frac{1}{n^2}\left(H_n-\frac{1}{n}\right)\right)\\ &=H_n^{(3)}+6\sum_{1\le i_1<i_2<i_3\le n}\frac{1}{i_1 i_2 i_3}+3(H_n^{(2)}H_n-H_n^{(3)})\,.\end{aligned}$$ we have $$\sum_{1\le i_1<i_2<i_3\le n}\frac{1}{i_1 i_2 i_3}=\frac{1}{6}(H_n^3+2 H_n^{(3)}-3 H_n^{(2)}H_n)\,.$$ Therefore, $$\begin{aligned} &\mathfrak D(n,j)\\ &=\frac{(H_n^2+H_n^{(2)})(H_n-H_{n-j})}{2}-H_n\left(\frac{H_n^2-H_n^{(2)}}{2}-\frac{H_{n-j}^2-H_{n-j}^{(2)}}{2}\right)\\ &\quad +\frac{1}{6}(H_n^3+2 H_n^{(3)}-3 H_n^{(2)}H_n-H_{n-j}^3-2 H_{n-j}^{(3)}+3 H_{n-j}^{(2)}H_{n-j})\\ &=\frac{(H_n-H_{n-j})^3}{6}+\frac{(H_n-H_{n-j})(H_n^{(2)}-H_{n-j}^{(2)})}{2}+\frac{H_n^{(3)}-H_{n-j}^{(3)}}{3}\,.\end{aligned}$$ ## Speculations on harmonic numbers and partitions In general, let us put $$\mathcal D_r(n,j):=\sum_{l=0}^{j-1}\binom{n-l-1}{n-j}\binom{n}{l}\frac{1}{(n-l)^{r-1}}.$$ Then, arguing similarly to the above cases where $1\le r\le 4$, we have $$\begin{aligned} &\mathcal D_5(n,j)=\frac{(H_n-H_{n-j})^4}{4!}+\frac{(H_n-H_{n-j})^2(H_n^{(2)}-H_{n-j}^{(2)})}{4}\\ &\quad +\frac{(H_n-H_{n-j})(H_n^{(3)}-H_{n-j}^{(3)})}{3}+\frac{(H_n^{(2)}-H_{n-j}^{(2)})^2}{4}+\frac{H_n^{(4)}-H_{n-j}^{(4)}}{8}\,,\\ &\mathcal D_6(n,j)=\frac{(H_n-H_{n-j})^5}{5!}+\frac{(H_n-H_{n-j})^3(H_n^{(2)}-H_{n-j}^{(2)})}{12}\\ &\quad +\frac{(H_n-H_{n-j})^2(H_n^{(3)}-H_{n-j}^{(3)})}{6}+\frac{(H_n-H_{n-j})(H_n^{(2)}-H_{n-j}^{(2)})^2}{8}\\ &\quad +\frac{(H_n-H_{n-j})(H_n^{(4)}-H_{n-j}^{(4)}}{4}+\frac{(H_n^{(2)}-H_{n-j}^{(2)})(H_n^{(3)}-H_{n-j}^{(3)})}{6}\\ &\quad +\frac{H_n^{(5)}-H_{n-j}^{(5)}}{5}\,,\\ &\mathcal D_7(n,j)=\frac{(H_n-H_{n-j})^6}{6!}+\frac{(H_n-H_{n-j})^4(H_n^{(2)}-H_{n-j}^{(2)})}{48}\\ &\quad +\frac{(H_n-H_{n-j})^3(H_n^{(3)}-H_{n-j}^{(3)})}{18}+\frac{(H_n-H_{n-j})^2(H_n^{(2)}-H_{n-j}^{(2)})^2}{16}\\ &\quad +\frac{(H_n-H_{n-j})^2(H_n^{(4)}-H_{n-j}^{(4)})}{8}+\frac{(H_n-H_{n-j})(H_n^{(2)}-H_{n-j}^{(2)})(H_n^{(3)}-H_{n-j}^{(3)})}{6}\\ &\quad +\frac{(H_n-H_{n-j})(H_n^{(5)}-H_{n-j}^{(5)})}{5}+\frac{(H_n^{(2)}-H_{n-j}^{(2)})^3}{48}\\ &\quad +\frac{(H_n^{(2)}-H_{n-j}^{(2)})(H_n^{(4)}-H_{n-j}^{(4)})}{8}+\frac{(H_n^{(3)}-H_{n-j}^{(3)})^2}{18}+\frac{H_n^{(6)}-H_{n-j}^{(6)}}{6}\,.\end{aligned}$$ It is interesting to observe that the number of terms of each of the right-hand sides of $\mathcal D_r(n,j)$ is equal to the number of partitions of $r$ ($r=1,2,3,4,5,6,7$), respectively. In addition, the same terms of generalized harmonic numbers appear in [@Choi; @Hoffman]: $$\begin{aligned} &\sum_{n=1}^\infty\frac{H_n}{(n+1)(n+2)}=1\,,\\ &\sum_{n=1}^\infty\frac{(H_n)^2-H_n^{(2)}}{2(n+1)(n+2)}=1\,,\\ &\sum_{n=1}^\infty\frac{(H_n)^3-3 H_n H_n^{(2)}+2 H_n^{(3)}}{3!(n+1)(n+2)}=1\,,\\ &\sum_{n=1}^\infty\frac{(H_n)^4-6(H_n)^2 H_n^{(2)}+8 H_n H_n^{(3)}+3(H_n^{(2)})^2-6 H_n^{(4)}}{4!(n+1)(n+2)}\,,\\ &\sum_{n=1}^\infty\frac{1}{5!(n+1)(n+2)}\biggl((H_n)^5-10(H_n)^3 H_n^{(2)}+20(H_n)^2 H_n^{(3)})^2\\ &\quad +15 H_n(H_n^{(2)}-30 H_n H_n^{(4)}-20 H_n^{(2)}H_n^{(3)}+24 H_n^{(5)}\biggr)=1\,,\\ &\sum_{n=1}^\infty\frac{1}{6!(n+1)(n+2)}\biggl((H_n)^6-15(H_n)^4 H_n^{(2)}+40(H_n)^3 H_n^{(3)}\\ &\quad +45(H_n)^2(H_n^{(2)})^2-90(H_n)^2 H_n^{(4)}-120 H_n H_n^{(2)}H_n^{(3)}+144 H_n H_n^{(5)}\\ &\quad -15(H_n^{(2)})^3+90 H_n^{(2)}H_n^{(4)}+40(H_n^{(3)})^2-120 H_n^{(5)} \biggr)=1\,.\end{aligned}$$ No simple closed form has been found, but $\mathcal D_r(n,j)$ ($r\ge 2$) can be expressed by a combinatorial sum ([@Hoffman Proposition 1 (17)]): $$\begin{gathered} \mathcal D_{r+1}(n,j) =\sum_{i_1+2 i_2+3 i_3+\cdots=r}\frac{1}{i_1!i_2!i_3!\cdots}\\ \times\left(\frac{H_n-H_{n-j}}{1}\right)^{i_1}\left(\frac{H_n^{(2)}-H_{n-j}^{(2)}}{2}\right)^{i_2}\left(\frac{H_n^{(3)}-H_{n-j}^{(3)}}{3}\right)^{i_3}\cdots\end{gathered}$$ and in terms of the determinant ([@MacDonald Ch. I §2]): $$\begin{gathered} \mathcal D_{r+1}(n,j)=\frac{1}{r!}\\ \times\left| \begin{array}{ccccc} H_n-H_{n-j}&-1&0&\cdots&0\\ H_n^{(2)}-H_{n-j}^{(2)}&H_n-H_{n-j}&-2&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ H_n^{(r-1)}-H_{n-j}^{(r-1)}&H_n^{(r-2)}-H_{n-j}^{(r-2)}&H_n^{(r-3)}-H_{n-j}^{(r-3)}&\cdots&-r+1\\ H_n^{(r)}-H_{n-j}^{(r)}&H_n^{(r-1)}-H_{n-j}^{(r-1)}&H_n^{(r-2)}-H_{n-j}^{(r-2)}&\cdots&H_n-H_{n-j} \end{array} \right|\,.\end{gathered}$$ See also [@ChCh; @CoCa]. # Acknowledgments {#acknowledgments .unnumbered} This work was done during the first author's visit to the Indian Statistical Institute Bangalore, India in July-August 2023. He is grateful for his second author's hospitality. 99 X. Chen and W. Chu, *The Gauss ${}_2 F_1(1)$-summation theorem and harmonic number identities*, Integral Transforms Spec. Funct. **20** (2009), 925--935. J. Choi, *Summation formulas involving binomial coefficients, harmonic numbers, and generalized harmonic numbers*, Abst. Appl. Anal. **2014** (2014), art. 501906 (10 pp). M. Coppo and B. Candelpergher, *The Arakawa-Kaneko zeta function*, Ramanujan J. **22** (2010), 153--162. M. E. Hoffman, *Harmonic-number summation identities, symmetric functions, and multiple zeta values*, Ramanujan J. **42** (2017), No.2, 501--526. P. Jothilingam, *Generalization of Eisenstein's congruence*, Acta Math. Hung. **46** (1985), 265--267. I. G. MacDonald, *Symmetric Functions and Hall Polynomials*, 2nd ed., Clarendon Press, Oxford, 1995. S. Mneimneh, *A binomial sum of harmonic numbers*, Discrete Math. **346** (2023), Article No.113075, 11 pp. B. Sury, *Sum of the reciprocals of the binomial coefficients*, Eur. J. Comb. **14** (1993), No.4, 351--353.

arxiv_math

--- abstract: | We study the asymptotic behavior and the scattering from infinity problem for the massive Maxwell-Klein-Gordon system in the Lorenz gauge, which were previously only studied for the massless system. For a general class of initial data, in particular of nonzero charge, we derive the precise asymptotic behaviors of the solution, where we get a logarithmic phase correction for the complex Klein-Gordon field, a combination of interior homogeneous function, radiation fields to null infinity and an exterior charge part for the gauge potentials. Moreover, we also derive a formula for charge at infinite time, which shows that the charge is concentrated at timelike infinity, a phenomenon drastically different from the massless case. After deriving the notion of the asymptotic profile, we prove the scattering from infinity by constructing backward solutions given the scattering data. We show that one can determine the correct charge contribution using the information at timelike infinity, which is important for us to obtain solutions not only for the reduced equations in the Lorenz gauge but also for the original physical system. address: Johns Hopkins University, Department of Mathematics, 3400 N.Charles St., Baltimore, MD 21218, USA author: - Xuantao Chen bibliography: - reference.bib title: Asymptotics and Scattering for massive Maxwell-Klein-Gordon equations --- # Introduction In this paper, we study both the forward (asymptotic behavior) and backward (scattering from infinity) problem of the massive Maxwell-Klein-Gordon (mMKG) equations on $\mathbb{R}^{3+1}$:[^1] $$\label{mMKG} \begin{split} &\partial^\nu F_{\mu\nu}=J[\phi]_\mu=\Im(\phi \cdot\overline{D_\mu\phi}),\\ &D^\mu D_\mu\phi-m^2\phi=0. \end{split}$$ This is a coupled system of an electromagnetic field and a complex scalar field. We study the case $m\neq 0$, and for simplicity we can normalize the mass so that $m^2=1$. The covariant derivative is defined as $$D_\mu =\partial_\mu+iA_\mu,$$ where $A_\mu$ is the connection $1$-form, and the curvature $2$-form $F_{\mu\nu}$ is given by $$F_{\mu\nu}=(dA)_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$ The commutator of the covariant derivative on the complex scalar field is $$(D_\mu D_\nu -D_\nu D_\mu)\phi=i F_{\mu\nu}\phi.$$ The current $1$-form $J[\phi]_\mu=\Im(\phi\, \overline{D_\mu\phi})$, where $\Im$ denotes the imaginary part of complex numbers. One can easily derive the conservation law from the equations: $$\partial^\mu J_\mu=0.$$ The system is gauge invariant, in the sense that $(A-d\psi, e^{i\psi}\phi)$ solves the same equation [\[mMKG\]](#mMKG){reference-type="eqref" reference="mMKG"} for any potential function $\psi$. Define $\lambda=\partial^\mu A_\mu$. Then the system can be rewritten as $$\begin{split} &-\Box A_\mu=\Im(\phi \cdot\overline{D_\mu\phi})-\partial_\mu \lambda,\\ &-\Box\phi+\phi=i\lambda\phi+2iA^\mu \partial_\mu\phi-A^\mu A_\mu\phi, \end{split}$$ where $\Box=\partial^\mu \partial_\mu=-\partial_t^2+\triangle_x$ is the D'Alembertian operator. The Lorenz gauge condition says that $\lambda=\partial^\mu A_\mu=0$. Under this gauge condition, the system becomes the reduced mMKG system: $$\label{reducedMKG} \begin{split} &-\Box A_\mu=\Im(\phi \cdot\overline{D_\mu\phi}),\\ &-\Box\phi+\phi=2iA^\mu \partial_\mu\phi-A^\mu A_\mu\phi. \end{split}$$ One can verify that for $(A_\mu,\phi)$ satisfying [\[reducedMKG\]](#reducedMKG){reference-type="eqref" reference="reducedMKG"}, the quantity $\lambda$ satisfies its own equation $$\label{eqoflambda} \Box \lambda=|\phi|^2\lambda.$$ Therefore, if the Lorenz gauge condition holds at an initial time slice, it holds everywhere in the spacetime. The global wellposedness of the mMKG equations was first established by the pioneering works [@eardley1982global1; @eardley1982global2] of Eardly-Moncrief. Later Klainerman-Machedon [@KlainermanMachedon] used bilinear estimates to generalize the result to data only with bounded energy in the massless case ($m=0$). Regarding long-time behaviors, Lindblad-Sterbenz [@LindbladSterbenz] proved the global existence of massless MKG equations with global decay estimates, for small initial data with nonzero charge, which is previously outlined by Shu [@shu1991asymptotic] (see also a simplified proof by Bieri-Miao-Shahshahani [@bieri2014asymptotic]). The massless system was also studied for large initial data [@yang2016decay; @yang2018global; @YangYu]. For the massive case which is what we discuss in this work, the global existence with decay estimates is first studied by Psarelli [@psarelli1999asymptotic] with restriction on compactly supported initial data. The restriction is then removed by Klainerman-Wang-Yang [@KlainermanWangYang], in which they constructed global solutions and derived decay estimates of the mMKG system in the exterior of the light cone. Then the result without assuming compactly supported data is established in Fang-Wang-Yang [@FangWangYang]. In this paper, we study the precise asymptotic behavior of solutions of the mMKG system. We are interested in two types of problems, which are closely related: 1. *Asymptotics* (forward problem): We want a precise description of the asymptotic behaviors of the solution. We show that given the initial data, one can use a set of functions to describe the asymptotic profile of the solution at (causal) infinity; we may call this set of functions the "scattering data\"; 2. *Scattering from infinity* (backward problem): Now given the scattering data at infinity (satisfying some compatibility conditions), we study if there always exists a global solution of the mMKG system such that its behavior at infinity is exactly the one given by this scattering data. In [@CKL], Candy-Kauffman-Lindblad studied the forward problem for the massless MKG system in the Lorenz gauge. Later He [@LiliHe21] established a refinement of the result, and proved a scattering from infinity result for the same system. The method of constructing backward solutions originates in the work of Lindblad-Schlue [@LindbladSchlue1] on the scattering from infinity for wave equations modeling the Einstein equation in wave coordinates. Recently the result in [@LiliHe21] was extended to large scattering data in [@DaiMeiWeiYang] in a gauge invariant setting. The massive Maxwell-Klein-Gordon system, on the other hand, presents significantly different behavior compared with the massless system. Due to the presence of nonzero mass, the complex scalar field presents the behavior of the Klein-Gordon fields rather than massless waves. Therefore one needs to deal with the coupling of the electromagnetic field, which behaves similarly to wave fields, and the complex scalar field which satisfies a Klein-Gordon equation. This coupling of fields with different asymptotics results in that many techniques used in the massless case break down. Unlike the wave equation, the Klein-Gordon equation does not have scale invariance, which is one of the important ingredients of the vector-field method introduced by Klainerman [@Klainerman1985vectorfield]. Instead, Klainerman [@Klainerman1985KG] found a variation of the vector-field method, which only requires the use of Lorenz boost vector fields. This is called the hyperboloidal foliation method, which is later used by LeFloch-Ma [@LeFLochMabook] and Wang [@WangEKG] in the study of the Einstein-Klein-Gordon equations. While the hyperboloidal foliation is by its nature only applicable to compactly supported data, one can combine it with estimates in the exterior as in the work [@KlainermanWangYang] we mentioned above (see also [@LeflochMafull] for the Einstein-Klein-Gordon system). Nevertheless, the hyperboloidal version of the vector-field method itself only provides energy bound and decay estimates, hence does not directly give a precise expansion of the solution, which is the problem we are interested in. In [@CL], we considered this problem for the wave-Klein-Gordon model introduced by LeFloch-Ma [@LeFlochMamodel] and Wang [@Wangmodel], and derived both the asymptotics and scattering from infinity results. In this paper, we show that we can also give concrete answers to the forward and backward problems to the massive Maxwell-Klein-Gordon equations, in particular with nonzero charges. ## Main results We consider the initial value problem, which can be formulated in the Lorenz gauge by the set $(a_i,\dot a_i,\phi_0,\dot\phi_0)$. Using the equation and the gauge condition, one can then determine other components, hence the value of $A_\mu$, $\partial_t A_\mu$, $\phi$, $\partial_t\phi$ at the initial slice. In this work, we consider the initial time slice to be $\{t=t_0=2\}$. The charge is then given by $$\mathbf{q}_0=\frac{1}{4\pi}\int_{t=t_0} \Im(\phi\, \overline{D_0\phi})\, dx.$$ We let the initial data satisfy similar bounds as in [@KlainermanWangYang; @FangWangYang] (where the assumptions are written in a gauge invariant way, see [\[ExteriorCondition\]](#ExteriorCondition){reference-type="eqref" reference="ExteriorCondition"}). To state the results, we define the hyperboloidal coordinates $\tau=\sqrt{t^2-r^2}$, where $r=|x|$, and $y=x/t$. We will use $y$ as the variable on the hyperboloids $\{t^2-r^2=\tau^2\}$. We also define $q=r-t$, and the angular variable $\omega\in\mathbb{S}^2$. The non-increasing smooth cutoff function $\chi(s)$ is defined so that $\chi(s)=1$ when $s\leq 1/2$ and $\chi(s)=0$ when $s\geq 3/4$. **Theorem 1** (Asymptotics). *Consider the initial value problem of [\[mMKG\]](#mMKG){reference-type="eqref" reference="mMKG"} with suitable initial data (less than $\varepsilon$ in some weighted norms), which admits global solutions by the existence result [@KlainermanWangYang; @FangWangYang]. Then, there exist functions $a_+(y)$, $a_-(y)$, $U_\mu(y)$, defined for $|y|\leq 1$, and radiation fields $F^{I}_\mu(q,\omega)$, $F^R_\mu(q,\omega)$, so that in the Lorenz gauge, the solution $(A_\mu,\phi)$ satisfies $$\begin{split} A_\mu&=\frac{U_\mu(y)}\tau+\chi(\frac{\langle q\rangle}r)\frac{F^R_\mu(q,\omega)}r+A_\mu^{Linear}+O(\varepsilon\langle t+r\rangle^{-2+\delta}),\quad t-r\geq 4,\\ A_\mu&=\frac{F^{I}_\mu(q,\omega)}{r}+\mathbf{q}_0 r^{-1} \delta_{0\mu}+A_\mu^{Linear}+O(\varepsilon\langle t+r\rangle^{-2+\delta}),\quad t-r\leq 4,\\ \phi&=\tau^{-\frac 32}(e^{i\tau+iU(y)\ln\tau+ih}a_+(y)+e^{-i\tau-iU(y)\ln\tau-ih}a_-(y))+O(\varepsilon\langle t+r\rangle^{-\frac 52+\delta}), \end{split}$$ where $U_\mu(y)$ can be determined by $a_\pm(y)$, $U(y)=U^\tau(y):=-\frac{x_\mu}\tau U_\mu(y)$, $h$ is a uniformly bounded function, and $A^{Linear}_\mu$ is the solution of a linear wave equation (with the given initial data written in the Lorenz gauge). We have the properties[^2] $$|(F_\mu,F_\mu^R)(q,\omega)|\lesssim \varepsilon\langle q\rangle^{-1+\delta}, \quad |a_\pm(y)|\lesssim \varepsilon (1-|y|^2)^{\frac 54-\delta}, \quad (1-|y|^2)^{-\frac 12}U_\mu(y)\rightarrow \mathcal U_\mu(\omega)\text{ as $|y|\rightarrow 1$},$$ where $F_\mu(\omega)$ are functions of $\omega$. Moreover, we have the following relation on the radiation fields $$rA_L\rightarrow \mathbf{q}_0\quad \text{as $r\rightarrow\infty$ with $q=r-t$ fixed,}$$ which implies $\mathcal U_L(\omega)=\mathbf{q}_0$ as a corollary. Lastly, we can express the charge using only the information at timelike infinity: $$\label{q0intro} \mathbf{q}_0=\frac{1}{4\pi}\int_{|y|\leq 1} |a_-(y)|^2-|a_+(y)|^2 \, dH_1$$ where $dH_1$ is the standard induced measure on the unit hyperboloid.* **Remark 1**. *We extract the part $A^{Linear}_\mu$ just to improve the decay of the remainder. We can also include them in the remainder by replacing the power $-2+\delta$ by $-1-\frac{\gamma_0'-1}2$ where $\gamma_0'>1$ comes from the initial data.* **Remark 2**. *The result says that $A_\mu\sim U_\mu(y)/\tau$ towards timelike infinity, and $\phi$ behaves like a linear expansion with a logarithmic phase correction. Towards null infinity, all terms (apart from the remainder) in the expansion contribute to radiation fields, so neither $F^R_\mu$ nor $F^I_\mu$ is the total radiation field of $A_\mu$. In fact, $F_\mu^R$ is the radiation field of lower order terms in the interior, and $F^I_\mu$ is $F_\mu^R$ combined with the radiation fields that originate from the interior leading behavior $U_\mu(y)/\tau$.* We now state our next result, in which we study the backward problem. We define the scattering data by the set $(a_+(y),a_-(y),F_\mu(q,\omega))$. We impose the decay conditions $$|(1-|y|^2)^{|I|}\nabla_y^I a_\pm(y)|\lesssim \varepsilon(1-|y|^2)^\alpha,\quad |(\langle q\rangle \partial_q)^k \partial_\omega^\beta F_\mu(q,\omega)|\lesssim \varepsilon\langle q\rangle^{-1+\gamma},\quad |I|\leq N,\, k+|\beta|\leq N$$ for some $N\geq 5$, $\alpha\geq \frac 74$, $\gamma<1/2$. Motivated by [\[q0intro\]](#q0intro){reference-type="eqref" reference="q0intro"}, we define the charge at infinity $\mathbf{q}_\infty$ by $$\label{defqinftyintro} \mathbf{q}_\infty :=\frac{1}{4\pi}\int_{|y|\leq 1} |a_-(y)|^2-|a_+(y)|^2 \, dH_1$$ In the forward problem, we have that towards null infinity $rA_L\rightarrow \mathbf{q}_0$. This is in fact from the Lorenz gauge condition, so here we need to have a similar property without assuming the condition. We will prove the following important observation: **Proposition 1**. *Given $a_\pm(y)$, we determine functions $U_\mu(y)$ by [^3] $$-\Box(\frac{U_\mu(y)}\tau)=\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2),$$ which is equivalent to an elliptic equation on the unit hyperboloid. Then we have $$r\frac{U_\mu(y)}\tau \rightarrow \mathbf{q}_\infty,\quad\text{as $r\rightarrow\infty$, with $q=r-t<0$ fixed,}$$ where $\mathbf{q}_\infty$ is defined by $a_\pm(y)$ by [\[defqinftyintro\]](#defqinftyintro){reference-type="eqref" reference="defqinftyintro"}.* We note that while in the forward problem, the analogous relation is just a corollary of the Lorenz gauge condition, in the backward problem one needs to prove it for arbitrary $a_\pm(y)$ without assuming the gauge condition. This is a crucial step in our proof. We now define the approximate solutions: $$\begin{split} A^{(0)}_\mu&=A^M_\mu+\chi(\frac{\langle q\rangle}r) \left(\frac{F_\mu(q,\omega)}r+\frac{F^{(1)}_\mu(q,\omega)}{r^2}\right)+\mathbf{q}_\infty \chi_{ex}(q) \delta_{0\mu} r^{-1},\\ {\phi^{(0)}}&=\tau^{-\frac 32}\left(e^{i\tau+i\int (A^{(0)})^\tau d\tau} a_+(y)+e^{-i\tau-i\int (A^{(0)})^\tau d\tau} a_-(y)\right), \end{split}$$ where $A^M_\mu$ is exactly $U_\mu(y)/\tau$ when $t-r\geq 4$ and suitably modified near the light cone, and $F_\mu^{(1)}$ is a second-order approximation introduced for technical reasons. We then impose the condition on $F_\mu$, which can be viewed as the Lorenz gauge condition at null infinity: $$\label{admissiblecondintro} F_L+F^M_L+\mathbf{q}_\infty \chi_{ex}(q)=\mathbf{q}_\infty$$ where $F^M_\mu$ is the radiation field of $A^M_\mu$. Note that it would not be possible to impose this without Proposition [Proposition 1](#propintro){reference-type="ref" reference="propintro"}, because $F_\mu$ decay to zero as $q\rightarrow -\infty$. In fact, by Proposition [Proposition 1](#propintro){reference-type="ref" reference="propintro"} we know that $F^M_L=\mathbf{q}_\infty$ when $q=r-t\leq -4$. Therefore $F_L$ is actually compactly supported. We are now ready to state our next result. **Theorem 2** (Scattering from infinity). *Consider a set of scattering data $(a_+(y),a_-(y),F_\mu(q,\omega))$ which satisfies the condition [\[admissiblecondintro\]](#admissiblecondintro){reference-type="eqref" reference="admissiblecondintro"}. Then, there exists a global solution to the reduced mMKG system with the exact same asymptotic behavior described by the scattering data, which satisfies the Lorenz gauge condition, hence also a solution to the original mMKG system. More precisely, for the approximate solution $(A^{(0)}_\mu,{\phi^{(0)}})$ defined above, there exists $(v_\mu,w)$ such that $(A^{(0)}_\mu+v_\mu,{\phi^{(0)}}+w)$ solves the mMKG system, with $v_\mu$ and $w$ being lower order terms.* We note that while the forward result gives the asymptotic behavior of the solution using $a_\pm(y)$ and radiation fields, it does not answer the question of whether arbitrarily given scattering data admits a global solution, and it is the backward result that gives the affirmative answer. ## Related works We also mention other related results on equations apart from the massive or massless Maxwell-Klein-Gordon equations. In the forward direction, the radiation field of a wave equation is first studied by Friedlander [@F62]. The asymptotics of nonlinear wave equations were studied in [@BB15; @BVW15], and the Einstein equation in higher dimensions in [@W14; @W13]. In [@L17], Lindblad studied the asymptotics of the $1+3$ dimensional Einstein equation in wave coordinates with a logarithmically modified behavior at null infinity and also an interior asymptotics. In recent years, the asymptotics of the quasilinear wave equation were also studied in [@DengPusateri; @Yuforward]. Polyhomogeneous expansions were derived by Hintz and Vasy [@HVMin], for the Einstein equation in generalized wave coordinates. For the linear Klein-Gordon equation, the asymptotics can be found in [@H97]. For nonlinear Klein-Gordon equations in one spatial dimension, the asymptotic behavior is derived by Delort [@D1; @D2] (see also [@LindbladSoffer05forward]). With variable coefficients on nonlinearities, the problem was studied by Lindblad-Soffer [@LS15] and Sterbenz [@sterbenz2016dispersive], with further results in [@LLS20; @LLS21; @LLSS22; @LuhrmannSchlag21]. For the backward problem, i.e. scattering from infinity, Flato-Simon-Taflin [@flato1987global] proved the existence of the modified wave operators for the Maxwell-Dirac equations. Lindblad-Soffer [@LindbladSoffer05backward] studied the scattering from infinity for 1D nonlinear Klein-Gordon equations. The scattering problem for blackhole spacetimes was studied in [@Dafermos2013scattering; @Dafermos2018scattering; @angelopoulos2020non]. Yu [@Yubackward] established the existence of modified wave operators for a scalar field equation satisfying the weak null condition. Apart from [@LindbladSchlue1], Lindblad-Schlue recently studied scattering from infinity for wave equations with sources modeling slow decay along the light cone [@LindbladSchlue2]. We also note that one can study similar problems on the Fourier side. For the related wave-Klein-Gordon models, Ionescu-Pausader [@IPmodel] showed the global existence using the spacetime resonance method and also derived the modified scattering (in the forward direction; then they also did it for the Einstein-Klein-Gordon equations [@IPEKG]). Later Ouyang [@Ouyang] studied the corresponding backward problem. ## Idea of proof ### Global existence We note two important results on the global existence and decay estimates of solutions of the mMKG equations [@KlainermanWangYang; @FangWangYang]. The work [@KlainermanWangYang] constructs solutions in the exterior of a light cone. This resolves the problem that previously, the hyperboloidal foliation method can only deal with compactly supported initial data. With the existence and estimates of the exterior solution, one gets control of the energy on the light cone which makes it possible to apply the hyperboloidal foliation in the interior. The interior part is done in [@FangWangYang] (see also an earlier work on the interior assuming compact initial data [@psarelli1999asymptotic]). In this paper, we shall make use of the results to get preliminary decay estimates which will be used when deriving asymptotic behaviors. Since the Klein-Gordon field concentrates more in the interior, the corresponding work [@FangWangYang] is more relevant to our work. (Nevertheless, we note that the result in [@KlainermanWangYang] also plays a fundamental role in making it possible for us to study this much more general, non-compactly supported initial data.) We will give a brief review of the estimates in the interior. The only adaptation we make here is that we do the calculation of higher order commutators and higher order estimates of the flux on the null cone so that we can apply them to our case, where we need more than two orders of vector fields. ### Asymptotics In the Lorenz gauge $\partial^\mu A_\mu=0$, the mMKG system becomes $$\begin{split} -\Box A_\mu&=\Im(\phi \cdot\overline{D_\mu\phi}),\\ -\Box\phi+\phi&=2iA^\mu \partial_\mu\phi-A^\mu A_\mu\phi. \end{split}$$ This is a wave-Klein-Gordon type coupled system. With the assumption of compactly supported initial data, we studied a similar model in [@CL]. Here due to the effect of charge, generally one cannot expect compactly supported initial data. However, we will see that one still has a relatively good decay estimate of the complex scalar field $\phi$ near the light cone, which serves, together with the initial data in the interior, as an "initial condition\" for us to derive the asymptotics of $\phi$ in the interior. To derive the asymptotics, we first decompose the Klein-Gordon equation. The main term on the right hand side is $\sim 2i A^\tau \partial_\tau\phi$, where $A^\tau=-\frac{x_\mu}\tau A_\mu$ and $\partial_\tau=\frac{x^\mu}\tau \partial_\mu$, since derivatives tangent to the hyperboloids decay better, and cubic terms are ignorable. So approximately, using $-\Box=\partial_\tau^2+3\tau^{-1}\partial_\tau-\tau^{-2}\triangle_y$, we have for $\Phi=\tau^\frac 32 \phi$ that $$\partial_\tau^2 \Phi+\Phi=2i A^\tau \partial_\tau \Phi+\cdots$$ where the remainder decays better in $\tau$. Now define $\Phi_\pm=e^{\mp i\tau} (\partial_\tau\Phi\pm i\Phi)$, so $\partial_\tau \Phi=\frac 12(e^{i\tau}\Phi_++e^{-i\tau}\Phi_-)$. Therefore $$\partial_\tau \Phi_\pm=e^{\mp i\tau} i A^\tau (e^{i\tau}\Phi_++e^{-i\tau}\Phi_-),$$ so for example for $\Phi_+$ we get $\partial_\tau \Phi_+=iA^\tau \Phi_+$ plus oscillating part. The oscillating part has extra cancellation after integration, so it also affects little. Therefore we expect $\Phi_+$ to be $e^{i\int A^\tau d\tau}$ multiplied by a function of $y$ (and similarly for $\Phi_-$), and hence obtain $$\begin{split} \phi&\sim \tau^{-\frac 32}(e^{i\tau+i\int A^\tau d\tau} a_+(y)+e^{-i\tau-i\int A^\tau d\tau} a_-(y)),\\ \partial_\tau \phi &\sim i\tau^{-\frac 32}(e^{i\tau+i\int A^\tau d\tau} a_+(y)-e^{-i\tau-i\int A^\tau d\tau} a_-(y)). \end{split}$$ One can plug this expansion in the right hand side of the wave equations. In [@CL] we studied nonlinearities like $\phi^2$ and $(\partial_t\phi)^2$, and the expansion gives $$2\tau^{-3} a_+(y)a_+(y)+\tau^{-3}e^{2i\tau+2i\int A^\tau d\tau}(a_+(y))^2+\tau^{-3}e^{-2i\tau-2i\int A^\tau d\tau}(a_-(y))^2,$$ where we see that the last two terms are oscillating. Then we performed an integration by part argument using the representation formula, which requires a tedious calculation. Remarkably, the special structure of the nonlinearity $\Im(\phi\, \overline{D_\mu\phi})$ here ensures that there will be no such oscillating terms: Modulo lower order terms, the nonlinearity $\sim -\frac{x_\mu}\tau\Im(\phi\, \overline{\partial_\tau\phi})$, and plugging in the expansions we get $$\Im(\phi\, \overline{\partial_\tau\phi})\sim \tau^{-3}(|a_-(y)|^2-|a_+(y)|^2).$$ Therefore, the main terms on the right hand side of the wave equations of $A_\mu$ are $-\frac{x_\mu}\tau \tau^{-3} (|a_-(y)|^2-|a_+(y)|^2)$.[^4] These are still of the form $-\Box \psi=\tau^{-3} Q(y)$, which is studied in [@CL]: One can find a function $\Psi(y)$ which satisfies the equation $$-\Box(\frac{\Psi(y)}\tau)=\tau^{-3} Q(y),$$ or equivalently, an elliptic equation on the unit hyperboloid $$\triangle_y\Psi+\Psi=Q(y).$$ In practice, to avoid the jump along the light cone, we proceed in this way: we start solving the wave equation at a positive time, say $t=2$, with the source $\tau^{-3}Q(y)$. One can then show that when $t-r\geq 4$, the solution exactly equals $\Psi(y)/\tau$. This in fact comes from the strong Huygens' principle. Applying this to the equations of $A_\mu$, we see that $$A_\mu\sim U_\mu(y)/\tau$$ in the interior. This also gives the main part of the phase correction on the expansion of the scalar field: $\int A^\tau d\tau\sim U(y)\ln\tau$, where $U(y)=U^\tau(y)$. For the wave fields, we still need to describe the behaviors at null infinity. It is not hard to show the existence of radiation fields, defined as the limit of $rA_\mu$ as $r\rightarrow\infty$ with $q=r-t$ and the angular variable $\omega$ fixed. However, there are additional structures. If we decompose the Lorenz gauge condition $\partial^\mu A_\mu=0$ in the null frame, we see that $\partial_q A_L$ behaves like derivatives tangent to the outgoing light cone, which are expected to decay better towards null infinity. This means that near null infinity we have $$\partial_q A_L\sim 0.$$ Note that due to the charge effect, there is a $\mathbf{q}_0 r^{-1}$ tail of $A_\mu$ (we can make it only on $A_0$) at spacelike infinity. The above relation implies that, this effect on the component $A_L$ will be preserved along the null infinity, all the way to the timelike infinity. One can show that $$rA_L\rightarrow \mathbf{q}_0\quad \text{as $r\rightarrow\infty$, for every $q=r-t$ and $\omega$ fixed.}$$ We see that the $L$ component of the radiation fields must be $\mathbf{q}_0$, which is the charge of the system given by the initial data. This structure comes from the gauge condition itself, and was first observed in [@CKL] for the massless MKG system. However, the charge itself behaves rather differently for the massive MKG system. In the massless system, the scalar field satisfies a wave equation, and then mainly propagates along the null directions. Using the conservation law $\partial^\mu J_\mu$, one can express the charge using the asymptotics of the scalar field at null infinity. For the massive field we study here, it decays much more towards null infinity, and one can verify that the flux is zero at null infinity when we integrate the divergence identity. This is not a contradiction to the conservation law, but the reflection of the fact that the charge is concentrated at timelike infinity. We can show that $$\mathbf{q}_0=\int_{|y|\leq 1} |a_-(y)|^2-|a_+(y)|^2\, dH_1.$$ This will also be extremely important for the backward problem. ### Scattering from infinity We now discuss the backward problem. We shall study the problem for the reduced mMKG system first, and then show that the constraint we impose on the scattering data ensures that the solution we construct also solves the original mMKG system. First, for scattering data, we should give the pair $a_\pm(y)$, which determines the size of the Klein-Gordon (scalar) field. It should also give the leading behavior of the nonlinearities that appeared in the wave equation of $A_\mu$. In fact, as in the forward problem, we can consider a part $A_\mu^M$ satisfying the equation $$-\Box A_\mu^M=\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)$$ with vanishing data at $\{t=2\}$. Then we have $A^M_\mu=U_\mu(y)/\tau$ for some $U_\mu(y)$ which should be the leading behavior of $A_\mu$ towards timelike infinity. The other piece of scattering data is the radiation field $F_\mu(q,\omega)$. We know that $A^M_\mu$ contributes to the radiation field, so what we impose should be from the remaining part, and we expect (and in fact, also need) them to decay in $\langle q\rangle$. Then we might be tempted to define the approximate solution as $$A^{(0)}_\mu=A^M_\mu+\chi(\frac {\langle q\rangle}r)\frac{F_\mu(q,\omega)}r,\quad \phi=\tau^{-\frac 32}(e^{i\tau+i\int (A^{(0)})^\tau d\tau} a_+(y)+e^{-i\tau-i\int (A^{(0)})^\tau d\tau} a_-(y)),$$ which is similar to [@CL], so one can apply the method used in that work to get a global solution that scatters to the asymptotic behavior given by $(a_+(y),a_-(y),F_\mu(q,\omega))$. However, the solution we get in this way will not have the $r^{-1}$ tail at spacelike infinity, hence not exactly what we want. Now since $\chi_{ex}(q)r^{-1}$ is an exact solution of the linear wave equation, we can then consider adding a part $C_{ex}\chi_{ex}(q)r^{-1}$, where $C_{ex}$ is a constant, $\chi_{ex}(q)$ a cutoff equal to $1$ when $q\geq -1/2$ and $0$ when $q\leq -1$, to the approximate solution, say the $A^{(0)}_0$ component, so that there is a part that represents the effect of charge. We see that one can find backward solutions of the reduced mMKG system for any $C_{ex}$. However, we still need the Lorenz gauge condition to hold, which requires the approximate solutions to satisfy this condition approximately. Recall that the Lorenz gauge condition written in the null frame reads $\partial_q A_L\sim 0$, so similar to the forward problem, we expect the $L$ component of the (total) radiation fields to be conserved along $\partial_q$ at null infinity, this time from the timelike infinity to the spacelike infinity. Therefore, $C_{ex}$ should be determined from the information at timelike infinity. This raises one of the key questions in this work: What is the $L$ component of the radiation fields of $A^M_\mu$, or equivalently $U_\mu(y)/\tau$, as $q\rightarrow -\infty$? Recall that $U_\mu(y)$ is determined by $a_\pm(y)$ by a wave equation or equivalently an elliptic equation on the unit hyperboloid. It is not hard to see that the radiation field of $U_\mu(y)/\tau$ is independent of $q$, and equals $$\lim_{|y|\rightarrow 1} (1-|y|^2)^{-\frac 12} U_\mu(y),$$ so what we are asking is $$\lim_{|y|\rightarrow 1} (1-|y|^2)^{-\frac 12}U_L(y)=\, ?$$ Up to now, it is not clear at all whether this limit as $q=r-t\rightarrow -\infty$ will be independent of the angular variable $\omega$, but we need it to be because otherwise with the exterior part being $C_{ex}\chi_{ex}(q)r^{-1}$, there is no hope that the Lorenz gauge condition will be true. We however show in this paper, through a careful calculation of the limiting measure, that the limit, is indeed a constant independent of $\omega$, and is nothing but the integral $$\frac{1}{4\pi}\int_{|y|\leq 1} |a_-(y)|^2-|a_+(y)|^2 \, dH_1$$ which is the same expression as in the relation derived in the forward problem (of course, the $a_\pm(y)$ there is determined from initial data, and here $a_\pm(y)$ is where we start). This is the result of Proposition 1. We denote the value of this integral as $\mathbf{q}_\infty$ to represent the charge computed at infinity. Then we are ready to write down the corrected expression of $A^{(0)}$: $$A^{(0)}_\mu=A^M_\mu+\chi(\frac{\langle q\rangle}r) \left(\frac{F_\mu(q,\omega)}r+\frac{F^{(1)}_\mu(q,\omega)}{r^2}\right)+\mathbf{q}_\infty \chi_{ex}(q) \delta_{0\mu} r^{-1},$$ where the $F^{(1)}$ is a second order approximation for technical reasons. Now, to make the approximate Lorenz gauge condition at null infinity hold, it suffices to impose the constraint that the $L$ component of the radiation fields of $A^{(0)}_\mu$ is always equal to $\mathbf{q}_\infty$. Since when $q\leq -4$ and $q\geq 2$ it is already $\mathbf{q}_\infty$ without $F_L$, this constraints implies that $F_L$ is compactly supported in $-4\leq q\leq 2$. While we have discussed the Lorenz gauge condition towards null infinity in detail, and argued that $A^{(0)}_\mu$ along with ${\phi^{(0)}}$ are now a good approximate solution for both the reduced and original system, we still have to ensure that the condition is also approximately satisfied in the interior. Since the $A^M_\mu$ (or $U_\tau(y)/\tau$) are the leading terms in the interior, we should look at the behavior $\partial^\mu A^M_\mu$ towards timelike infinity. To do this we can commute the wave equations of $A^M_\mu$ with $\partial^\mu$ (i.e. $-\partial_t$, $\partial_i$). Since $\partial_\mu x^\mu=4$, $x_\mu \partial^\mu=\tau \partial_\tau$, we have $$\begin{split} \Box \partial^\mu A^M_\mu&=(\partial^\mu x_\mu)\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2)+x_\mu \partial^\mu (\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2))\\ &=4\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2)-\tau\cdot 4\tau^{-5} (|a_-(y)|^2-|a_+(y)|^2)=0, \end{split}$$ so $\partial^\mu A^M_\mu$ satisfies a linear homogeneous wave equation. Now since at $t=2$ we have $\partial^\mu A^M_\mu$ compactly supported in $\{|x|\leq 2\}$, by strong Huygens' principle we see that $\partial^\mu A^M_\mu$ is supported in $\{0\leq t-r\leq 4\}$, a region that should be categorized as near light cone rather than interior. Therefore the Lorenz gauge condition is also reasonably satisfied in the interior. Then we can use similar backward energy estimates introduced in [@CL] to obtain a backward solution, which can be shown to also satisfy the Lorenz gauge condition. ## Acknowledgements {#acknowledgements .unnumbered} The author would like to thank his advisor, Hans Lindblad, for many helpful discussions and his encouragement on studying this problem. # Notations, Preliminaries ## Hyperboloidal coordinates We introduce the hyperboloidal coordinates $\tau=\sqrt{t^2-r^2}$ where $r=|x|$, and $y=x/t$, so $|y|=|x|/t$. We also define the angular variables $\omega\in\mathbb{S}^2$ and $\omega_i=x_i/|x|$. The $\tau$-hyperboloid is defined by $H_\tau=\{(t,x)\colon t^2-|x|^2=\tau^2\}$. In $(\tau,y_i)$ coordinates, we have[^5] $$\partial_\tau=\frac{t}{\tau}\partial_t+\frac{x^i}\tau \partial_i,\quad \partial_{y_i}=\frac{\tau}{(1-|y|^2)^{\frac 32}}\left((y_i y_j+\delta_{ij})\partial_j+y_i \partial_t\right).$$ We also have the dual frame $$d\tau=\frac t\tau dt-\frac{x^i}\tau dx^i,\quad dy^i=-\frac{x_i}{t^2} dt+\frac 1t dx^i.$$ Then, for a vector field $V^\mu$, we can define the quantities $V^\tau$ and $V^{y_i}$. For example, in this paper we frequently use $A^\tau:=-\frac{x_\mu}\tau A_\mu$. We define the hyperboloidal orthonormal frame $\{\partial_\tau,\tau^{-1}\Omega_{0r},e_1,e_2\}$ where $\Omega_{0r}=\omega^i \Omega_{0i}$, and $\{e_1,e_2\}$ is a set of orthonormal frames on the $2$-sphere. We will also use truncated versions of hyperboloids in this paper: $\hat H_\tau=H_\tau\cap \{t-r\geq 1\}$, and $\widetilde H_\tau=H_\tau\cap \{t-r\geq r^\frac 12\}$. ## Vector fields We use $Z$ to denote the vector fields in the set $$\mathcal{Z}=\{\partial_\mu,\Omega_{0i}=t\partial_i+x_i\partial_t,\Omega_{ij}=x_i\partial_j-x_j\partial_i\}.$$ It is straightforward to verify the schematic commutation relation $[Z,Z]=\pm Z$. We also use $\Omega$ to denote boost and rotation vector fields. Sometimes we also use the scaling vector fields $S=t\partial_t+x^i\partial_i$. We use the notation $(Z,S)$ to denote the set above added with $S$. We use the multi-index notation: for $I=(\beta_1,\beta_2,\cdots,\beta_n)$, we define $$Z^I=Z_1^{\beta_1} Z_2^{\beta_2} \cdots Z_n^{\beta_n}$$ where $Z_1,\cdots Z_n \in \mathcal{Z}$, The size of a multi-index $I=(\beta_1,\beta_2,\cdots,\beta_n)$ is defined as $|I|:=\sum_{i=1}^n \beta_i$. Note that the rotation and boost vector fields are tangent to the hyperboloids $H_\tau$, and we have the following relation between $\partial_{y_i}$ and vector fields: $$\label{dy} \partial_{y_i}=\frac 1{1-|y|^2} \left(\Omega_{0i}+\frac{x_j}{t} \Omega_{ij}\right).$$ Moreover, the Laplacian-Beltrami operator on the unit hyperboloid $\triangle_y$ can be expressed by vector fields: $$\label{laplaciany} \triangle_y=\sum_{i=1}^3 \Omega_{0i}^2+\sum_{j=1}^3 \sum_{i=1}^j \Omega_{ij}^2.$$ We also use $\triangle_x$ and $\triangle_\omega$ to denote the Laplacian on $\mathbb{R}^3$ and $\mathbb{S}^2$ respectively. ## Integration Apart from the standard integration with respect to induced volume elements, it is convenient to consider the integral of a function $u(t,x)$ on hyperboloids with respect to the measure $dx$. $$\int_{H_\tau} u \, dx:=\int_{\mathbb{R}^3}u(\sqrt{\tau^2+|x|^2},x)\, dx,$$ In this paper, we define the $L^2$ norm on the hyperboloid using the measure $dx$ instead: $$||u||_{L^2(H_\tau)}=\left(\int_{H_\tau} |u|^2\, dx\right)^\frac 12.$$ We also use similar definitions for the $L^2$ norm on truncated hyperboloids $\hat H_\tau$, $\widetilde H_\tau$. ## Null frame Besides the hyperboloidal frame, we also need the null frame, which is fundamental in the analysis of wave equations. It is defined by the set $\{L,{\underline{L}},e_1,e_2\}$, where $L=\partial_t+\partial_r$, ${\underline{L}}=\partial_t-\partial_r$, with $\partial_r=\omega^i\partial_i$, and $\{e_1,e_2\}$ is an orthonormal frame on the sphere. We also extensively use the function $q=r-t$, and the derivative $\partial_q=-\frac 12(\partial_r-\partial_t)$. We also use the tangential derivatives ${\slashed{\partial}}_i:=\partial_i-\omega_i\partial_r$. ## Lie derivatives We define the Lie derivative of tensor fields with respect to a vector field by $$\begin{gathered} {\mathcal L}_Z K^{\alpha_1\alpha_2\cdots \alpha_s}_{\beta_1\beta_2\cdots \beta_r}=Z\, K^{\alpha_1\alpha_2\cdots \alpha_s}_{\beta_1\beta_2\cdots \beta_r}-(\partial_\nu Z^{\alpha_1})\, K^{\nu\alpha_2\cdots \alpha_s}_{\beta_1\beta_2\cdots \beta_r}-\cdots-(\partial_{\nu} Z^{\alpha_s})\, K^{\alpha_1\alpha_2\cdots \nu}_{\beta_1\beta_2\cdots \beta_r}+(\partial_{\beta_1}Z^\nu)\, K^{\alpha_1\alpha_2\cdots \alpha_s}_{\nu\beta_2\cdots \beta_r}\\ +\cdots+(\partial_{\beta_r} Z^\nu)\, K^{\alpha_1\alpha_2\cdots \alpha_s}_{\beta_1\beta_2\cdots \nu}.\end{gathered}$$ In particular, if $X$, $Y$ are vector fields, we have ${\mathcal L}_X Y=[X,Y]$. In this paper, since the background metric is Minkowski, and the vector fields we use will be Killing, the Lie derivatives commute with contractions (we will see later that they also commute with partial derivatives). The factors of the form $\partial Z$ in the definition above will also be constants. ## Curvature components We can write components of the curvature $2$-form $G$ in the null frame, denoted as $$\alpha_B[G]:=G_{Le_B},\quad \underline{\alpha}_B[G]:=G_{{\underline{L}}e_B},\quad \rho[G]:=\frac 12 G_{L{\underline{L}}},\quad \sigma[G]:=G_{e_1 e_2},\quad B=1,2.$$ We can also consider its components in the hyperboloidal frame. We have $$G(\partial_\tau, \tau^{-1}\Omega_{0r})=\rho[G],\quad G(\partial_\tau,e_B)=\frac{t+r}\tau \alpha[G]+\frac{t-r}\tau \underline{\alpha}[G],\quad G(\tau^{-1}\Omega_{0r},e_B)=\frac{t+r}\tau \alpha[G]-\frac{t-r}\tau \underline{\alpha}[G].$$ Later, as in [@FangWangYang], we will see that the components in the hyperboloidal frame have similar bounds, so we denote all components in this frame by $G^H$. # The existence results In this section, we brief review the results in [@KlainermanWangYang; @FangWangYang] with slight adaptations to our case. This part is done in a gauge independent way. ## Higher order commutators Here we view $D_X\phi=X^\mu D_\mu\phi$ as a scalar field.[^6] Define the trilinear form $$Q(F,\phi,Z):=2i Z^\nu F_{\mu\nu} D^\mu \phi+i\partial^\mu(Z^\nu F_{\mu\nu}) \phi.$$ We have $$\phi=Q(F,\phi,Z),$$ $$\phi=Q(F,D_X \phi,Y)+Q(F,D_Y \phi,X)+Q({\mathcal L}_X F,\phi,Y)+Q(F,\phi,[X,Y])-2F_{X\mu} F^{\mu}_{\, \, Y}\phi.$$ The key of the proof is the following relation $$\begin{split} D_X Q(F,\phi,Y)&= D_X (2i Y^\nu F_{\mu\nu} D^\mu \phi+i\partial^\mu(Y^\nu F_{\mu\nu}) \phi)\\ &=i X^\lambda (\partial_\lambda+iA_\lambda)(2Y^\nu F_{\mu\nu} D^\mu\phi+\partial^\mu(Y^\nu F_{\mu\nu})\phi)\\ &=2i({\mathcal L}_X Y)^\nu F_{\mu\nu}D^\mu\phi+2i Y^\nu {\mathcal L}_X F_{\mu\nu} D^\mu\phi+2i Y^\nu F_{\mu\nu} ({\mathcal L}_X+iA_X)D^\mu\phi\\ &\quad +i{\mathcal L}_X (\partial^\mu (Y^\nu F_{\mu\nu}))\phi+i\partial^\mu(Y^\nu F_{\mu\nu})D_X \phi\\ &=2i({\mathcal L}_X Y)^\nu F_{\mu\nu}D^\mu\phi+2i Y^\nu {\mathcal L}_X F_{\mu\nu} D^\mu\phi+2iY^\nu F_{\mu\nu} D^\mu D_X\phi-2F_{X\mu}F^{\mu}_{\,\, Y}\phi\\ &\quad +i\partial^\mu (({\mathcal L}_X Y^\nu) F_{\mu\nu})\phi+i\partial^\mu(Y^\nu{\mathcal L}_X F_{\mu\nu})\phi+i\partial^\mu(Y^\nu F_{\mu\nu})D_X\phi\\ &=Q(F,\phi,[X,Y])+Q({\mathcal L}_X F,\phi,Y)+Q(F,D_X\phi,Y)-2F_{X\mu}F^{\mu}_{\,\, Y}\phi \end{split}$$ where we used $$\label{lieDcommutator} \begin{split} ({\mathcal L}_X+iA_X) D^\mu\phi&=D_X D^\mu\phi-D^\lambda\phi \partial_\lambda X^\mu=D^\mu (D_X\phi)+iF_{X}^{\, \, \mu}\phi-(\partial^\mu X^\lambda)D_\lambda \phi-D^\lambda\phi \partial_\lambda X^\mu\\ &=D^\mu D_X\phi+iF_{X}^{\, \, \mu}\phi \end{split}$$ since $X$ is Killing. We can further calculate the third-order commutator $$\begin{split} [\Box_A, D_X D_Y D_Z]\phi&=\Box_A D_X D_Y D_Z \phi-D_X \Box_A D_Y D_Z \phi+D_X \Box_A D_Y D_Z\phi -D_X D_Y D_Z \Box\phi\\ &=[\Box_A, D_X]D_Y D_Z\phi+D_X [\Box_A,D_Y D_Z]\phi\\ &=Q(F,D_Y D_Z\phi,X)+D_X ([\Box_A,D_Y D_Z]\phi) \end{split}$$ Using the formula for the second-order commutator and the relation above, we see that the third-order commutator is of a similar form. Inductively, using ${\mathcal L}_Z Z=\pm Z$, one can show the schematic relation (where we omit all constant factors) $$\phi= \sum_{|I_1|+|I_2|\leq |I|} Q({\mathcal L}_Z^{I_1} F,D_Z^{I_2}\phi,Z) +\sum_{|I_3|+|I_4|+|I_5|<|I|} Z^\lambda {\mathcal L}_Z^{I_3} F_{\lambda\mu} {\mathcal L}_Z^{I_4} F^{\mu}_{\, \, \nu} Z^\nu D_Z^{I_5}\phi.$$ Note that since $Z$ are Killing, ${\mathcal L}_Z m=0$ and ${\mathcal L}_Z$ commutes with raising and lowering indices. We also need to deal with the Lie derivatives of the current $1$-form. Using [\[lieDcommutator\]](#lieDcommutator){reference-type="eqref" reference="lieDcommutator"} we have $${\mathcal L}_Z J_\mu[\phi]=\Im (D_Z\phi\cdot\overline {D_\mu\phi})+\Im(\phi\cdot\overline{D_\mu D_Z\phi+i F_{Z\mu}\phi}).$$ We can use this inductively and show that schematically, $$\begin{split} {\mathcal L}_Z^I J_{\mu}[\phi]&=\sum_{|I_1|+|I_2|\leq |I|}\Im(D_Z^{I_1} \phi\cdot\overline{D_\mu D_Z^{I_2} \phi})+\sum_{|J|<|I|}{\mathcal L}_Z^{J} \Im(\phi\cdot\overline{iF_{Z\mu}\phi})\\ &=\sum_{|I_1|+|I_2|\leq |I|}\Im(D_Z^{I_1} \phi\cdot\overline{D_\mu D_Z^{I_2} \phi})+\sum_{|I_3|+|I_4|+|I_5|<|I|}\Im(D_Z^{I_3}\phi\cdot\overline{i{\mathcal L}_Z^{I_4} F_{Z\mu} D_Z^{I_5}\phi}) \end{split}$$ using $[Z,Z]=\pm Z$. ## Null cone flux We expect a part of $F$ governed by the charge. The chargeless part of $F$ is defined by $$\tilde F=F-\mathbf{q}_0 r^{-2} \chi_{\{t-r>2\}} dt\wedge dr.$$ This decomposition is crucial in the exterior, but less important in the interior, where one may just do the estimate for the original Maxwell field $F$. However, the energy bounds from the exterior are all on the chargeless part $\tilde F$. Therefore, to do the energy estimates in the interior, we still need to establish the bound of the energy flux of $F$ on the null cone. First we note that $|\rho[F]-\rho[\tilde F]|\lesssim |\mathbf{q}_0| r^{-2}$, and other components are the same. When we use the multiplier $S$, the energy flux of $F$ on the cone $C=\{t-r=1\}\cap\{t\geq 2\}$ is like $$\int_{C} \rho[F]^2+\sigma[F]^2+r|\alpha[F]|^2 dvol,$$ so the difference, i.e. the integral with $F$ replaced by $F-\tilde F$, is integrable. Now we need to estimate the difference with vector fields, and we will show that the difference is also integrable for any number of vector fields. Denote $\tau_+=t+r$, $\tau_-=t-r$. We now compute various commutators of vector fields. First we have $$=\frac tr{\slashed{\partial}}_i-\partial_i-\omega_i\partial_t=-\omega_i L+\frac{\tau_-}{r}{\slashed{\partial}}_i,\quad [\Omega_{ij},L]=0.$$ In order to apply more Lie derivatives on $L$ and ${\underline{L}}$, we need the identities $$=-r^{-1}(\delta_{ij}-\omega_i\omega_j)\omega^k \Omega_{0k}, \quad [\Omega_{ij},{\slashed{\partial}}_k]=-(\delta_{ik}-\omega_i\omega_k){\slashed{\partial}}_j+(\delta_{jk}-\omega_j \omega_k){\slashed{\partial}}_i.$$ We also have $\Omega^I(\omega_i)=O(1)$, $\Omega^I (r)=O(r)$ when $t\approx r$ for any muli-index $I$. Therefore $${\mathcal L}_\Omega^I {\slashed{\partial}}_i=O(r^{-1})\Omega+O(1)\slashed\partial+O(r^{-1})\partial=O(1)L+O(r^{-1}\langle\tau_-\rangle){\underline{L}}+O(1)\slashed\partial.$$ We also have $\Omega^I (\tau_-)=O(\tau_-)$. Therefore using the $[\Omega,L]$ formulas we have $${\mathcal L}_\Omega^I L=O(1) L+O(r^{-1}\tau_-)\slashed\partial+O(r^{-2}\tau_-^2){\underline{L}}.$$ For partial derivatives, we have $[\partial_t,L]=0$ and $[\partial_i,L]=r^{-1}{\slashed{\partial}}_i$. Therefore we have $${\mathcal L}_Z^I L=O(1) L+O(r^{-1}\tau_-)\slashed\partial+O(r^{-2}\tau_-^2){\underline{L}}.$$ Similarly we have $$=-\omega_i{\underline{L}}+\frac{\tau_+}r {\slashed{\partial}}_i,\quad [\Omega_{ij},{\underline{L}}]=0$$ which yields $${\mathcal L}_Z^I {\underline{L}}=O(1){\underline{L}}+O(1)(\slashed\partial,L).$$ We shall make the following induction assumptions $$|(\alpha,\sigma)[{\mathcal L}_Z^{I}(F-\tilde F)]|\lesssim |\mathbf{q}_0|\langle\tau_-\rangle r^{-3}, \quad |\rho[{\mathcal L}_Z^{I}(F-\tilde F)]|\lesssim |\mathbf{q}_0| r^{-2},\quad |\underline\alpha[{\mathcal L}_Z^{I}(F-\tilde F)]|\lesssim |\mathbf{q}_0|r^{-2}$$ for $|I|\leq k-1$. Denote $G=F-\tilde F$. For $i=1,2,3$, we have $$\alpha[{\mathcal L}_Z^I G]({\slashed{\partial}}_i)={\mathcal L}_Z^I G_{L{\slashed{\partial}}_i}=Z^I (G_{L{\slashed{\partial}}_i})-\sum_{\substack{|I_1|+|I_2|+|I_3|=|I|\\|I_1|<|I|}} ({\mathcal L}_Z^{I_1} G)({\mathcal L}_Z^{I_2} L,{\mathcal L}_Z^{I_3} {\slashed{\partial}}_i).$$ Using the estimates above and the induction assumptions, we have for $|I|\leq k-1$ ($\slashed\partial={\slashed{\partial}}_i$, $i=1,2,3$) $$\begin{split} |\alpha[{\mathcal L}_Z^I G]|&\lesssim \sum_{|J|\leq k-1}\tau_- r^{-1}|{\mathcal L}_Z^J G(L,{\underline{L}})|+|{\mathcal L}_Z^J G(L,\slashed\partial)|+r^{-2}\tau_-^2 |{\mathcal L}_Z^J G({\underline{L}},\slashed\partial)|+r^{-1}\tau_- |{\mathcal L}_Z^J G(\slashed\partial,\slashed\partial)|\\ &\lesssim |\mathbf{q}_0| \tau_- r^{-3}. \end{split}$$ Similarly, $$\begin{split} |\rho[{\mathcal L}_Z^I G]|&\lesssim |Z^I (G_{L{\underline{L}}})|+\sum_{|J|\leq k-1} |{\mathcal L}_Z^J G\left(O(1)(L,\slashed\partial)+O(r^{-1}\tau_-){\underline{L}},O(1)({\underline{L}},L,\slashed\partial)\right)|\\ &\lesssim\sum_{|J|\leq k-1}|{\mathcal L}_Z^J G(L,{\underline{L}})|+|{\mathcal L}_Z^J G(L,\slashed\partial)|+|{\mathcal L}_Z^J G({\underline{L}},\slashed\partial)|+|{\mathcal L}_Z^J G(\slashed\partial,\slashed\partial)|\\ &\lesssim |\mathbf{q}_0| r^{-2}, \end{split}$$ $$\begin{split} |\sigma[{\mathcal L}_Z^I G]|&\lesssim \sum_{|J|\leq k-1} |{\mathcal L}_Z^J G\left(O(1)L+O(r^{-1}\langle\tau_-\rangle){\underline{L}}+O(1)\slashed\partial,O(1)L+O(r^{-1}\langle\tau_-\rangle){\underline{L}}+O(1)\slashed\partial\right)|\\ &\lesssim\sum_{|J|\leq k-1}r^{-1}\langle\tau_-\rangle|{\mathcal L}_Z^J G(L,{\underline{L}})|+|{\mathcal L}_Z^J G(L,\slashed\partial)|+r^{-1}\langle\tau_-\rangle|{\mathcal L}_Z^J G({\underline{L}},\slashed\partial)|+|{\mathcal L}_Z^J G(\slashed\partial,\slashed\partial)|\\ &\lesssim |\mathbf{q}_0| \langle\tau_-\rangle r^{-3}, \end{split}$$ $$\begin{split} |\underline\alpha[{\mathcal L}_Z^I G]|&\lesssim \sum_{|J|\leq k-1} |{\mathcal L}_Z^J G\left(O(1)({\underline{L}},L,\slashed\partial),O(1)L+O(r^{-1}\langle\tau_-\rangle){\underline{L}}+O(1)\slashed\partial\right)|\\ &\lesssim\sum_{|J|\leq k-1}|{\mathcal L}_Z^J G(L,{\underline{L}})|+|{\mathcal L}_Z^J G(L,\slashed\partial)|+|{\mathcal L}_Z^J G({\underline{L}},\slashed\partial)|+|{\mathcal L}_Z^J G(\slashed\partial,\slashed\partial)|\\ &\lesssim |\mathbf{q}_0| r^{-2}. \end{split}$$ Therefore we prove the estimates for $k$, hence the estimates for arbitrary number of vector fields. As a result, we obtain the same bounds of the null cone flux for $F$. ## Bootstrap arguments In this subsection, we briefly review the bootstrap argument which establishes the existence in the interior. As is mentioned above, with the slight generalization to higher orders in previous section, the argument is now essentially the same (in fact a bit easier) as in [@FangWangYang]. We make no claim of originality on this part. First we recall the exterior result [@KlainermanWangYang]. While the original result is stated with no greater than two order vector fields, it was pointed out, and can be also seen from above, that one can easily generalize it to higher orders. The initial value problem in gauge invariant form is formulated as follows: We consider the electric part $E_i=F_{0i}$ and magnetic part $H_i=\prescript{\star}{}{F_{0i}}$, where $\star$ is the Hodge dual operator. The initial data is the set $(E,H,\phi,\dot\phi_0)$ which prescribes the value of $E$, $H$, $\phi$ and $D_0\phi$, but the constraint equations need to be satisfied: $$\text{div}E=\Im(\phi_0\, \overline{\dot\phi_0}),\quad \text{div}H=0.$$ We define the chargeless part of $E$ by $\tilde E_i=E_i-\mathbf{q}_0\chi_{ex}(r)r^{-1}\omega_i$. We now state the result, mainly the part that we need: **Theorem 3** (Exterior solution, [@KlainermanWangYang]). *Consider the Cauchy problem for ([\[mMKG\]](#mMKG){reference-type="ref" reference="mMKG"}) with the admissible initial data set $(E,H,\phi_0,\dot\phi_0)$ (satisfying constraint equations) at $\{t=2\}$. Then if for small $\varepsilon>0$ and some $1<\gamma_0<2$, $$\label{ExteriorCondition} \sum\limits_{|I| \leq N}\int_{r\geq 1}(1+r)^{\gamma_0+2|I|}(|\bar D\bar D^I\phi_0|^2+|\bar D^I\dot\phi_0|^2+|\bar D^I\phi_0|^2+|\bar\partial^I \tilde{E}|^2+|\bar\partial^I H|^2)dx\leq \varepsilon,$$ where $\bar D$ and $\bar \partial$ are projections of $D$ and $\partial$ to $\mathbb{R}^3$ respectively, then the unique local solution $(F, \phi)$ of the mMKG system can be extended to the whole exterior region $\{t-r\leq 1\}$ with the energy bounds $$q_+^{\gamma_0-1}\int_{C_q}r|D_L D_Z^I\phi|^2+|\slashed DD_Z^I\phi|^2+|D_Z^I\phi|^2+ \int_{C_q} r^{\gamma_0}|\alpha[{\mathcal L}_Z^I \tilde{F}]|^2+q_+^{\gamma_0} \int_{C_q} |\rho[{\mathcal L}_Z^I \tilde F|^2+|\sigma[{\mathcal L}_Z^I F]|^2\leq C\varepsilon$$ for $k\leq N$, where $C_q=\{r-t=q\}$ are outgoing null hypersurfaces. We also have the decay estimates of the scalar field $|D_Z^I\phi| \leq C \varepsilon r^{-\frac 32}q_+^{-\frac{\gamma_0}2}$ for $k\leq N-2$.* Here we let $N\geq 5$. This in particular implies the bound of the energy flux on the cone $\{t-r=1\}$, thanks to the estimates in last subsection: $$C_k[F,\phi]=\int_{t-r=1} |(\rho,\sigma)[{\mathcal L}_Z^I F]|^2+r|\alpha[{\mathcal L}_Z^I F]|^2+|D_L D_Z^I\phi|^2+|\slashed D D_Z^I\phi|^2+|D_Z^I\phi|^2 \leq C\varepsilon.$$ Now we use $E^H_{S}(F,\tau)$ and $E^H_{KG}(\phi,\tau)$ to denote the energy of the Maxwell field and the scalar field on truncated hyperboloids $\hat H_\tau$, induced by the multiplier $S$ and $\partial_t$ respectively. We also define $E^H_{S,k}=\sum_{|I|\leq k}E^H_S({\mathcal L}_Z^I F,\tau)$, $E^H_{KG,k}=\sum_{|I|\leq k}E^H_{KG}(D_Z^I\phi,\tau)$. We have the comparability $$E^H_{S}(\tau)\approx \sum_{|I|\leq k}\int_{\hat H_\tau} \frac{\tau^2}t|{\mathcal L}_Z^I F^H|^2 dx,\quad E_{KG}^H(\tau)\approx \sum_{|I|\leq k}\int_{\hat H_\tau} (\frac \tau t)^2 |D_{\partial_t}\phi|^2+(\frac \tau t)^2|D_R\phi|^2+|\slashed D\phi|^2+|\phi|^2 dx,$$ where $R=\tau^{-1}\Omega_{0r}$. We set up the following bootstrap assumptions: $$E_{S,k}^H(\tau)^\frac 12\leq C_b \varepsilon,\quad E_{KG,k}^H(\tau)^\frac 12\leq C_b \varepsilon\, \tau^\delta,\quad k\leq N.$$ We do not include $C_b$ in the implicit constants from "$\lesssim$\". Using the Klainerman-Sobolev type estimate on hyperboloids, we immediately get the decay estimate of the scalar field: $$\frac{\tau}t|DD_Z^I\phi|+|D_Z^I\phi|\lesssim C_b\varepsilon t^{-\frac 32}\tau^\delta,\quad |I|\leq N-2.$$ To obtain the decay of components of the Maxwell field one can also decompose the tensor to electric and magnetic parts under the hyperboloidal frame as in [@FangWangYang] and then apply the Sobolev estimate to get $$|({\mathcal L}_Z^I F)^H|\lesssim C_b\varepsilon t^{-1}\tau^{-1}, \quad |I|\leq N-2.$$ ### The Maxwell field For the Maxwell field, we have no commutator terms, so we have $$\begin{split} E^H_{S,k}(\tau)\lesssim E^H_{S,k}(\tau_0)+C_k(\tau)+\sum_{|I|\leq k}\left|\iint_{D_{\tau_0}^\tau} S^\mu {\mathcal L}_Z^I F_{\mu\nu} {\mathcal L}_Z^I J^\nu[\phi]\right| \end{split}$$ where $D_{\tau_0}^\tau$ is the region enclosed by $\hat H_{\tau_0}$, $\hat H_\tau$, and part of the cone $\{t-r=1\}$ between the two hyperboloids, denoted by $C_{\tau_0}^\tau$. The last integral is bounded by the sum of terms like ($|I|=k$) $$\sum_{|I_1|+|I_2|\leq |I|}\left|\iint_{D_{\tau_0}^\tau} S^\mu {\mathcal L}_Z^{I} F_{\mu\nu} \Im (D_Z^{I_1}\phi\cdot\overline{D^\nu D_Z^{I_2}\phi})\right|+\sum_{|I_3|+|I_4|+|I_5|<|I|} \left|\iint_{D_{\tau_0}^\tau} S^\mu {\mathcal L}_Z^{I}\Im(D_Z^{I_3}\phi\cdot\overline{i{\mathcal L}_Z^{I_4} F_{Z\mu} D_Z^{I_5}\phi})\right|$$ The second here contains an additional decay factor so we omit it. For the first term, when $|I_2|<|I|$, we can use the following estimate $$\left|\iint_{D_{\tau_0}^\tau} S^\mu {\mathcal L}_Z^{I} F_{\mu\nu} \Im (D_Z^{I_1}\phi\cdot\overline{D^\nu D_Z^{I_2}\phi})\right|\leq \iint_{D_{\tau_0}^\tau} \tau |{\mathcal L}_Z^I F^H| |D_Z^{I_1}\phi|\cdot\tau^{-1}|D_\Omega D_Z^{I_2}\phi|$$ since one component of ${\mathcal L}_Z^I F$ is $S$. Here we used the relation $|\slashed D\phi|+|D_R\phi|\lesssim \tau^{-1}|D_{\Omega_{0i}}\phi|$, which can be seen from the relation $t\slashed\partial_i=\Omega_{0i}-\omega_i\Omega_{0r}$. Note that here the volume element is $dvol=\frac{\tau}t dx d\tau$. The bootstrap assumption says $$\int_{\hat H_\tau} \frac{\tau^2}t|({\mathcal L}_Z^I F)^H|^2 dx\lesssim (C_b\varepsilon)^2,\quad |I|\leq k.$$ Therefore the integral can be controlled by $$\sum_{|I_1|+|I_2|\leq |I|} \int_{\tau_0}^\tau \left(\int_{\hat H_\tau} \frac{\tau^2}t |{\mathcal L}_Z^I F^H|^2 dx\right)^\frac 12 \left(\int_{\hat H_\tau} \tau^{-1} |D_Z^{I_1}\phi|^2 |D_\Omega D_Z^{I_2}\phi|^2 dx\right)^\frac 12 d\tau$$ One of the scalar field factor can be estimated by the decay, so e.g. we get $$\int_{\hat H_\tau} C_b\varepsilon t^{-\frac 32}\tau^{-\frac 12+\delta} \left(\int_{\hat H_\tau} \frac{\tau^2}t |{\mathcal L}_Z^I F^H|^2 dx\right)^\frac 12 \left(\int_{\hat H_\tau} |D_\Omega D_Z^{I_2}\phi|^2 dx\right)^\frac 12\leq C_b^3 \varepsilon^3.$$ When $|I_2|=|I|$, the integration by part argument in [@FangWangYang] plays a fundamental role. Since $\partial^\mu S^\nu=m^{\mu\nu}$, we have $$\begin{split} S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im &(\phi\cdot\overline{D^\mu D_Z^I\phi})\\ &=\partial^\mu (S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(\phi\cdot\overline{D_Z^I \phi}))-S^\nu (\partial^\mu {\mathcal L}_Z^I F_{\mu\nu}) \Im(\phi\cdot\overline{D_Z^I\phi})-S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(D^\mu\phi\cdot\overline{D_Z^I \phi}). \end{split}$$ Notice that $\partial^\mu{\mathcal L}_Z^I F_{\mu\nu}=-{\mathcal L}_Z^I J[\phi]_\nu$. This is quadratic, so we obtain an extra decay power. The last term can be treated similarly as above. Therefore it suffices to control the boundary flux $$\left|\int_{\hat H_\tau} \tau^{-1} S^\mu S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(\phi\cdot\overline{D_Z^I\phi})\right|+\left|\int_{C_{\tau_0}^\tau} L^\mu S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(\phi\cdot\overline{D_Z^I\phi})\right|+\left|\int_{\hat H_{\tau_0}} \tau_0^{-1} S^\mu S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(\phi\cdot\overline{D_Z^I\phi})\right|.$$ Using the anti-symmetry of ${\mathcal L}_Z^I F$, we see that the first and last term are zero. Also, using $S=\frac 12(\tau_+L+\tau_-{\underline{L}})$, the decay (boundedness) of $\phi$, and the bounds on the cone flux we get $$|\int_{C_{\tau_0}^\tau} L^\mu S^\nu {\mathcal L}_Z^I F_{\mu\nu} \Im(\phi\cdot\overline{D_Z^I\phi})|\lesssim |\int_{C_{\tau_0}^\tau} \rho[{\mathcal L}_Z^I F] \Im(\phi\cdot\overline{D_Z^I\phi})|\lesssim C_b^3\varepsilon^3,$$ Therefore we get $E_{S,k}^H(\tau)\lesssim \varepsilon^2+C_b^3 \varepsilon^3$, so making $C_b$ relatively big (compared with the implicit constants) and then $\varepsilon$ small we improve the boostrap assumption on $E_{S,k}^H(\tau)$. ### The scalar field For the scalar field, the right hand sides are commutators, which leads to the integral in the estimate of $E_{KG,k}^H(\tau)$: $$\sum_{|I_1|+|I_2|<|I|}\iint_{D_{\tau_0}^\tau} |D_0 D_Z^I\phi||Q({\mathcal L}_Z^{I_1}F,D_Z^{I_2}\phi,Z)|+\sum_{|I_3|+|I_4|+|I_5|<|I|}\iint_{D_{\tau_0}^\tau} |D_0 D_Z^I\phi||{\mathcal L}_Z^{I_3} F_{Z\mu} {\mathcal L}_Z^{I_4} F^\mu_{\, \, Z} D_Z^{I_5}\phi|.$$ The second term behaves better so we omit it. Since $\Omega_{0i}=\omega_i\Omega_{0r}+t{\slashed{\partial}}_i$, the main contribution of the trilinear form $Q$ can be estimated as $$|Z^\nu {\mathcal L}_Z^{I_1} F_{\mu\nu} D^\mu D_Z^{I_2}\phi|\lesssim t|{\mathcal L}_Z^{I_1} F^H|(|\slashed DD_Z^{I_2}\phi|+|D_{R}D_Z^{I_2}\phi|+|D_{\partial_\rho}D_Z^{I_2}\phi|)$$ We have $$|\slashed D\phi|+|D_R\phi|\lesssim \tau^{-1}|D_{\Omega_{0i}}\phi|,\quad |D_{\partial_\tau}\phi|\lesssim \frac \tau t |D_{\partial_t}\phi|+\tau^{-1}|D_{\Omega_{0i}}\phi|,$$ so the main term becomes $\tau |{\mathcal L}_Z^{I_1} F^H| |D_{\partial_t}\phi|$. We can then estimate $$\begin{split} \iint_{D_{\tau_0}^\tau} |D_0 D_Z^I\phi|&|Q({\mathcal L}_Z^{I_1}F,D_Z^{I_2}\phi,Z)|\frac\tau t dx d\tau \leq \iint_{D_{\tau_0}^\tau} \frac \tau t \cdot\tau|D_0 D_Z^I \phi||{\mathcal L}_Z^{I_1} F^H||D D_Z^{I_2}\phi| dx d\tau\\ &\leq \int_{\tau_0}^\tau s\left(\int_{\hat H_s} (\frac s t |D_0 D_Z^I \phi|)^2 dx\right)^\frac 12 \left(\int_{\hat H_s} |{\mathcal L}_Z^{I_1} F^H|^2 |D D_Z^{I_2}\phi|^2 dx\right)^\frac 12 ds\\ &\leq \int_{\tau_0}^\tau s^{1+\delta}\cdot C_b\varepsilon \left(\int_{\hat H_s} C_b^2 \varepsilon^2 t^{-2} \tau^{-2} |D D_Z^{I_2}\phi|^2+(C_b\varepsilon t^{-\frac 12}\tau^{-1+\delta})^2 |{\mathcal L}_Z^{I_1} F^H|^2 dx\right)^\frac 12 ds\\ &\leq \int_{\tau_0}^\tau C_b^2\varepsilon^2 s^{-1+\delta}\left(\int_{\hat H_s} t^{-2} s^{2} |D D_Z^{I_2}\phi|^2 dx+\int_{\hat H_s} s^{2\delta} \frac{s^2}{t}|{\mathcal L}_Z^{I_1} F^H|^2 dx \right)^\frac 12 ds\\ &\leq \int_{\tau_0}^\tau C_b^3 \varepsilon^3 s^{-1+2\delta}\, ds \lesssim C_b^3 \varepsilon^3 \tau^{2\delta}. \end{split}$$ Again, letting $C_b$ be relatively big and then $\varepsilon$ be small we can improve the bootstrap assumption on $E_{k,KG}^H(\tau)$. Therefore we have improved both assumptions and hence obtain the global existence, with global decay estimates. # Asymptotics ## Initial data Under the Lorenz gauge, the initial data needs to satisfy the gauge condition as well as the constraint equation, which comes from the Maxwell equation: $$\partial^\mu A_\mu=0,\quad -\triangle A_0=J_0-\partial^i(\partial_t A_i),$$ where $J_0=\Im(\phi\, \overline{D_0\phi})=\Im(\phi\, \overline{\partial_t\phi})-|\phi|^2 A_0$. Therefore the initial data set can be set to be $(A_i,\partial_t A_i,\phi,\partial_t\phi)|_{t=2}=(a_i,\dot{a}_i,\phi_0,\dot\phi_0)$ with $i=1,2,3$. The remaining components $(a_0,\dot a_0):=(A_0,\partial_t A_0)|_{t=2}$ can then be determined by the conditions above. Define the norm $$||\psi||^2_{H^{k,\gamma_0}}:=\sum_{|I|\leq k} \int_{\mathbb{R}^3} (1+r)^{\gamma_0+2|I|} |\partial_x^I \psi|^2 dx.$$ We impose the initial data set $(a_i,\dot a_i,\phi_0,\dot\phi_0)$ satisfying the bound with $\gamma_0>1$: $$||a_i||_{H^{k+1,\gamma_0-2}}+||\dot a_i||_{H^{k,\gamma_0}}+||(\phi_0,\partial_x \phi_0,\phi_1)||_{H^{k,\gamma_0}}\leq \varepsilon.$$ We want to verify the initial energy condition [\[ExteriorCondition\]](#ExteriorCondition){reference-type="eqref" reference="ExteriorCondition"} to obtain the existence. This requires also the information of $A_0$ initially. Since $\mathbf{q}_0=\frac 1{4\pi}\int_{\mathbb{R}^3} J_0$ is generally nonzero, we expect a $r^{-1}$ tail of $A_0$ since it satisfies an elliptic equation. Define $$\tilde a_0=a_0-\mathbf{q}_0 \chi_{ex}(r) r^{-1},$$ where $\chi_{ex}$ is a non-decreasing smooth cutoff function with $\chi_{ex}(s)=0$ when $s\leq -1$, and $\chi_{ex}(s)=1$ when $s\geq -1/2$. Then at the initial slice $E_i=F_{0i}=\partial_t a_i-\partial_i a_0=\partial_t a_i-\partial_i \tilde a_0+\mathbf{q}_0 \chi_{ex}(r) r^{-2}\omega_i-\mathbf{q}_0 \chi_{ex}'(r)\omega_i r^{-1}$. The last term vanishes when $r\geq 1$. Therefore $\tilde E_i=E_i-\mathbf{q}_0 \chi_{ex}(r) r^{-2} \omega_i$ coincides with the previous definition when $r\geq 1$. (To construct the exterior solution, we only need to care about the part where $r\geq 1$.) One can establish the estimate of $\tilde a_0$ using elliptic estimates, see details in [@CKL]: **Lemma 1**. *If $||a_i||_{H^{k+1,\gamma_0-2}}+||\dot a_i||_{H^{k,\gamma_0}}+||(\phi_0,\partial_x \phi_0,\dot\phi_0)||_{H^{k,\gamma_0}}\leq \varepsilon$ for some $\gamma_0\in (1,3)$, then for any $\gamma_0'<\gamma_0$ we have $$||a_0-\mathbf{q}_0 \chi_{ex}(r) r^{-1}||_{H^{k+1,\gamma_0'-2}}+||\dot a_0||_{H^{k,\gamma_0'}}\lesssim \varepsilon.$$* To verify the condition of the exterior theorem (with $\gamma_0$ replaced by $\gamma_0'$), we need to check the norm of $\tilde E$, $H$, $\phi_0$ and $\dot\phi_0$ on $\{t=2,r\geq 1\}$. When $r\geq 1$ we have $\tilde E_i=\dot a_i-\partial_i \tilde a_0$, so we can obtain the bound of $\tilde E_i$ using what we have for $\tilde a_0$. For the scalar field it suffices to replace partial derivatives by the covariant derivatives, which is because of the bounds of $A_\mu$ and Sobolev estimates (this is fairly easy and we leave it to the next subsection). Therefore the data satisfies the initial condition and we have a global solution with estimates in the previous section. ## Decay estimates Recall that under the Lorenz gauge we have $$-\Box A_\mu=J[\phi]_\mu=\Im (\phi\overline{D_\mu\phi}).$$ Similar to above we consider the part with the charge $\mathbf{q}_0 \chi_{ex}(r-t) r^{-1}\delta_{0\mu}$. Notice that this is an exact solution to the homogeneous wave equation. Therefore we have for $\tilde A_\mu=A_\mu-\mathbf{q}_0 \chi_{ex}(r-t) r^{-1}\delta_{0\mu}$ that $$-\Box \tilde A_\mu=\Im(\phi\overline{D_\mu \phi})$$ with initial data $(a_0,\dot a_0)$. From the existence part we have $|D_Z^I\phi|\lesssim \varepsilon(1+t+r)^{-\frac 32}\langle \tau\rangle^\delta q_+^{-\frac{\gamma_0'}2}$ (set $\tau=1$ in the exterior), so in particular we have $$|\phi|+|D\phi|\lesssim \varepsilon t^{-\frac 32}\tau^{2\delta} q_+^{-\frac{\gamma_0'}2}\lesssim \varepsilon t^{-3+2\delta}\mathbf{1}_{t\geq r-1}+\varepsilon r^{-\frac 32} q_+^{-{\gamma_0'}}\mathbf{1}_{t<r-1}.$$ Here $\mathbf 1$ is the indicator function. This gives an estimate of the right hand side of the wave equations. By linearity, we can analyze the effect of the inhomogeneous term and the initial data separately. For the inhomogeneous terms above, if we consider a wave equations with them on the right hand side with vanishing initial data, then the solution (denoted by $\tilde A_\mu^{source}$) has the decay $$|\tilde A_\mu^{source}|\lesssim \varepsilon^2 (1+t+r)^{-1+\delta} (1+q_+)^{-{\gamma_0'}}.$$ The estimate is standard, and here one can treat the interior and exterior sources separately. See [@CKL Appendix] for the radial estimate method and [@LeFlochMamodel] for alternative proof using the representation formula. For the initial data, from the initial bound of $A_\mu$ and the following Sobolev estimate we can derive the initial decay. **Lemma 2**. *Let $\psi$ be a function on $\mathbb{R}^3$. We have $$\sup_x\, \langle r\rangle^{\frac{\gamma_0'+1}2}|\psi|\leq C\sum_{|\beta|+k\leq 3} \left(\int_{\mathbb{R}^3} \langle r\rangle^{\gamma_0'-2} |(\langle r\rangle \partial_r)^k \partial_\omega^\beta\psi|^2 dx\right)^\frac 12.$$* Then applying this to the zeroth and first order derivative of $\tilde A_\mu$ at the initial slice, we get $|\tilde A_\mu||_{t=2}\lesssim \varepsilon (1+r)^{-\frac{\gamma_0'+1}2}$, and $|\partial\tilde A_\mu||_{t=2}\lesssim \varepsilon (1+r)^{-\frac{\gamma_0'+3}2}$. Now we need the following estimate, which is also standard: **Lemma 3**. *If $w$ is the solution of the homogeneous wave equation $-\Box w=0$ with $(w,\partial_t w)|_{t=0}=(w_0,w_1)$, then for any $\alpha\in (0,1)$ we have $$(1+t+r)(1+|t-r|)^\alpha|w(t,x)|\leq C\sup_x \left((1+|x|)^{2+\alpha} (|w_1(x)|+|\partial w_0(x)|)+(1+|x|)^{1+\alpha} |w_0(x)|\right).$$* Therefore, the contribution from the initial data is of the size $\varepsilon (1+t+r)^{-1} (1+|t-r|)^{-\frac{\gamma_0'-1}2}$. So by linearity we get the estimate $$|\tilde A_\mu|\lesssim \varepsilon t^{-1+2\delta}\mathbf{1}_{t\geq r}+\varepsilon(1+t+r)^{-1}(1+q_+)^{-\frac{\gamma_0'-1}2}\mathbf{1}_{t<r}.$$ **Remark 3**. *In the statement of Theorem [Theorem 1](#mainthm1){reference-type="ref" reference="mainthm1"}, we extract the linear part $A^{Linear}_\mu$, which is exactly the initial data part we discussed here. Due to the slow decay of the initial data, this term decays less than other lower order terms in the interior. Nevertheless, it does not affect the main behavior in the interior either.* It is also straightforward to show this with vector fields applied to $\tilde A_\mu$, i.e. $Z^I \tilde A_\mu$, by commuting the wave equation of $\tilde A_\mu$ with $Z^I$. This serves as a preliminary bound of $\tilde A_\mu$. Later we will also use improved bounds of the derivative of $\tilde A_\mu$, which is expected to be better than the decay of themselves. We already have better behaviors for derivatives of the initial data part (because in this case we can easily commute the linear wave equation with $(Z,S)$). For the source part, we decompose the source $$\Im(\phi \overline{D_\mu\phi})=\chi_{ex}(q)\Im(\phi \overline{D_\mu\phi})+(1-\chi_{ex}(q))\Im(\phi \overline{D_\mu\phi}).$$ We first note that the size is well-preserved when $Z$ or $S$ fall on the cutoff because $|(Z,S)q|\lesssim 1$. For the exterior part, we can commute the wave equation with $S$: Using $[\Box,S]=2\Box$, we have $|\Box \tilde A_\mu^{source}|\lesssim |S(\chi_{ex}(q)\Im(\phi \overline{D_\mu\phi}))|+|\chi_{ex}(q)\Im(\phi D_\mu\phi)|$. Then since $S=\Omega_{0r}+(t-r){\underline{L}}$, we have $|S(\chi_{ex}(q)\Im(\phi\overline{D_\mu\phi}))|\lesssim \varepsilon^2 t^{-3+2\delta}(1+|t-r|)(1+q_+)^{-\gamma_0'}$. We see that the exterior source is still decaying enough, so this part gives a solution $S\tilde A_\mu^{source,ex}$ satisfying $\tau|\partial_\tau A_\mu^{source,ex}|=|SA_\mu^{source,ex}|\lesssim \varepsilon^2\langle t+r\rangle^{-1+2\delta}$. For the interior source $(1-\chi_{ex}(q))\Im(\phi \overline{D_\mu\phi})$, this is now a problem supported in $\{t-r\geq 1\}$. We can simply use the conformal energy estimate on the hyperboloids as in [@CL] which deals with compactly supported data: $$E_{con}(Z^I \tilde A_\mu^{source,in},\tau)[H_\tau]^\frac 12\lesssim E_{con}(Z^I \tilde A_\mu^{source,in},\tau_0)[H_{\tau_0}]^\frac 12+\int_{\tau_0}^\tau \varepsilon s^{-\frac 12+\delta}\sum_{|J|\leq |I|+2}||D_Z^J \phi||_{L^2(H_s)}\lesssim \varepsilon \tau^{\frac 12+2\delta},$$ The norm $||\tau^2 t^{-1} \partial_\tau Z^I\tilde A_\mu^{source,in}||_{L^2(H_\tau)}$ is bounded by the conformal energy. Then by the Klainerman-Sobolev estimate on hyperboloids, this implies $|\tilde A_\mu^{source,in}|\lesssim \varepsilon^2 t^{-\frac 12}\tau^{-\frac 32+2\delta}$. Therefore we have $|\partial_\tau\tilde A_\mu^{source}|\lesssim \varepsilon t^{-\frac 32}\tau^{-\frac 12+2\delta}$. Combining these two source parts with the linear part we can get $|\partial_\tau \tilde A_\mu|\lesssim \varepsilon t^{-\frac 12}\tau^{-\frac 32+2\delta}$. We also need a weak decay of derivatives of the scalar field without the connection. For example, we have $|Z_1 Z_2 \phi|\leq |D_{Z_1} D_{Z_2} \phi|+|Z_1^\mu A_\mu Z_2\phi|+|Z_1(Z_2^\nu A_\nu)\phi|+|(Z_1^\mu A_\mu)(Z_2^\nu A_\nu)\phi|\lesssim \varepsilon\langle t+r\rangle^{-\frac 32+3\delta}$. Generally we have $|Z^I \phi|\lesssim \varepsilon \langle t+r\rangle^{-\frac 32+(2|I|+1)\delta}$. Note that $\delta$ is small, and we are not doing boostrap arguments in this section, so we will not be very careful on the coefficients in front of $\delta$. To derive the asymptotic behaviors of the Klein-Gordon field, we will integrate along the hyperboloidal rays ($r/t$ being constants less than $1$). This requires a "initial condition\" provided by the initial data and the decay of the field near the light cone. The decay of the scalar field $\phi$ on the cone, from the exterior result, are $$|D_Z^I\phi|\lesssim \varepsilon r^{-\frac 32}.$$ We remark here this decay on the cone can be improved. In [@LeFLochMa2022model], LeFloch-Ma observed the following identity: $$-\partial_t^2\phi+\triangle_x\phi=\frac{r^2-t^2}{t^2}\partial_t^2\phi-\frac{2x^i}{t^2}\partial_t \Omega_{0i}\phi+t^{-2}\Omega_{0i}\Omega_{0i}\phi+\frac{x^i}{t^3}\Omega_{0i}\phi-t^{-1}(3+\frac{r^2}{t^2})\partial_t\phi$$ where we take the sum on repeated spatial indices. This yields the estimate $$|\Box\phi|\lesssim \varepsilon t^{-1}(1+|t-r|)\sum_{|I|\leq 2} |Z^I\phi|.$$ Therefore, using the bounds we get and the equation $$\phi=\Box\phi+2iA^\mu \partial_\mu\phi-A^\mu A_\mu\phi,$$ we obtain $$|\phi|\lesssim \varepsilon t^{-\frac 52+\delta}(1+(t-r))\lesssim \varepsilon\tau^{-\frac 32+\delta} (1-|y|^2)^{\frac 74-\frac\delta 2}+\varepsilon\tau^{-\frac 52+\delta} (1-|y|^2)^{\frac 54-\frac \delta 2} , \quad\text{when }t>r.$$ So in particular $\phi$ decays at the rate $r^{-\frac 52+\delta}$ along the light cone. Similar bounds hold with some vector fields. ## Asymptotics ### The scalar field Recall that the scalar field equation can be written under the Lorenz gauge as $$-\Box \phi+\phi=2iA^\mu \partial_\mu \phi-A^\mu A_\mu \phi.$$ The charge part of $A_\mu$, i.e. $\mathbf{q}_0 \chi_{ex}(r-t) r^{-1}\delta_{0\mu}$, is supported around the light cone and contribute little to our analysis below. Therefore we focus on the chargeless part and with a slight abuse of notation denote $\tilde A_\mu$ by $A_\mu$ in this subsection. We decompose $A^\mu \partial_\mu\phi$ in the hyperboloidal frame: $$A^\mu \partial_\mu\phi=(A^\tau \partial_\tau \phi+R_{tan,1})\chi_{LC}(q)+(1-\chi_{LC}(q))A^\mu \partial_\mu\phi.$$ The cutoff function $\chi_{LC}(q)$ is $1$ when $q\leq -1$ and $0$ when $q\geq -\frac 12$. We introduce this cutoff because the decomposition in the hyperboloidal frame becomes irregular close to the the light cone. The term $R_{tan,1}$ involves derivative tangential to hyperboloids, and is of the size $O(t^{-1}|A||\nabla_y \phi|)$. Then, in the support of $\chi_{LC}(q)$, we have $$\begin{split} |R_{tan,1}|&\lesssim \varepsilon t^{-1} t^{-1+\delta} (1-|y|^2)^{-1} |\Omega\phi|\lesssim \varepsilon^2 t^{-2+\delta} (1-|y|^2)^{-1} t^{-\frac 52+\delta}(t-r)\\ &\lesssim \varepsilon^2\tau^{-\frac 72+2\delta} (1-|y|^2)^{\frac 74-\delta} \end{split}$$ which is good. We also denote $R_{LC,1}=(1-\chi_{LC}(q))A^\mu \partial_\mu \phi$, and $|R_{LC,1}|\lesssim \varepsilon t^{-\frac 72+2\delta}$. The decay of $A^\mu A_\mu\phi$ is even better and we omit it. Using $-\Box=\partial_\tau^2+3\tau^{-1}\partial_\tau-\tau^{-2}\triangle_y$, one can write the Klein-Gordon equation in the hyperboloidal coordinates $$\partial_\tau^2 (\tau^\frac 32\phi)+\tau^\frac 32\phi=\tau^\frac 32(2i A^\mu \partial_\mu\phi-A^\mu A_\mu\phi)+\tau^{-\frac 12}(\triangle_y \phi+\textstyle\frac 34\phi)$$ Now let $\Phi=\tau^\frac 32\phi$, and $\Phi_\pm=e^{\mp i\tau}(\partial_\tau\Phi\pm i\Phi)$. Using the decay above we have $|\Phi_\pm|\lesssim \varepsilon\tau^{-1+\delta}(1-|y|^2)^{\frac 54-\delta}+\varepsilon\tau^\delta (1-|y|^2)^{\frac 94-\delta}$. We have $$\label{dtauPhipm} \partial_\tau \Phi_\pm=e^{\mp i\tau}\left(\tau^\frac 32 (2iA^\mu\partial_\mu\phi-A^\mu A_\mu \phi)+\tau^{-\frac 12}(\triangle_y \phi+\textstyle\frac 34\phi)\right).$$ Since $A^\mu \partial_\mu \phi=(A^\tau \partial_\tau \phi+R_{tan,1})\chi_{LC}(q)+R_{LC,1}=(\tau^{-\frac 32} A^\tau \partial_\tau \Phi-\frac 32 \tau^{-1} A^\tau \phi+R_{tan,1})\chi_{LC}(q)+R_{LC,1}$, and $\partial_\tau\Phi=\frac 12(e^{i\tau}\Phi_+ +e^{-i\tau}\Phi_-)$, we get $$\partial_\tau\Phi_\pm =i\chi_{LC}(q)A^\tau \Phi_\pm+i e^{\mp 2i\tau}\chi_{LC}(q)A^\tau \Phi_\mp+e^{\mp i\tau}\tau^\frac 32 R,$$ where $R=2i\chi_{LC}(q)R_{tan,1}+\chi_{LC}(q)R_{tan,2}+2iR_{LC,1}+R_{LC,2}+R_{good}$, with $R_{tan,2}:=\tau^{-2}(\triangle_y \phi+\frac 34\phi)$, $R_{LC,2}=(1-\chi_{LC}(q))R_{tan,2}$, and $R_{good}=-A^\mu A_\mu\phi-3i\tau^{-1} A^\tau\phi$. We have the estimate $|\chi_{LC}(q)R_{tan,2}|\lesssim \varepsilon t^{-\frac 52+\delta}(t-r)\tau^{-2}\lesssim \varepsilon\tau^{-\frac 72+\delta} (1-|y|^2)^{\frac 94-\frac\delta 2}$, and $|R_{LC,2}|\lesssim \varepsilon\tau^{-2} t^{-\frac 52+\delta}\lesssim \varepsilon\tau^{-\frac 72+\delta} (1-|y|^2)^{\frac 54-\frac \delta 2}$. This gives, e.g., for $\Phi_+$ that (for simplicity denote $\chi=\chi_{LC}(q)$) $$\partial_\tau \left(e^{-i\int \chi A^\tau d\tau}(\Phi_+ +\frac 12 \chi A^\tau e^{-2i\tau}\Phi_-)\right)=\frac 12 e^{-2i\tau} \partial_\tau (e^{-i\int \chi A^\tau d\tau} \chi A^\tau \Phi_-)+\tau^\frac 32 e^{-i\tau-i\int \chi A^\tau d\tau} R.$$ We have $$\begin{split} |\partial_\tau(\chi A^\tau)\Phi_-|&=|(\partial_\tau \chi_{LC}(q))A^\tau \Phi_-|+|\chi_{LC}(q)\frac{x_\mu}\tau \partial_\tau A_\mu \Phi_-|\lesssim \varepsilon^2 t\tau^{-1}\cdot t^{-\frac 12} \tau^{-\frac 32+2\delta}\cdot\tau^\frac 32 t^{-\frac 52+\delta}(t-r)\\ &\lesssim \varepsilon^2\tau^{-2+3\delta} (1-|y|^2)^{\frac 32-\frac\delta 2} \end{split}$$ using the bounds we established for $\partial_\tau A_\mu$. When $\partial_\tau$ falls on $\Phi_-$, we can use [\[dtauPhipm\]](#dtauPhipm){reference-type="eqref" reference="dtauPhipm"} to see it behaves well. Overall, from all the estimates, we see that the right hand side of the equation can be bounded by $\varepsilon^2\tau^{-\frac 32+2\delta}(1-|y|^2)^{\frac 32-\delta}+\varepsilon\tau^{-2+2\delta}(1-|y|^2)^{\frac 54-\delta}$. Then, integrating this equation, we get $\Phi_+=e^{i\int \chi A^\tau d\tau} b_+(y)+O(\varepsilon\tau^{-1+3\delta}(1-|y|^2)^{\frac 54-\delta})$, where $$\begin{gathered} b_+(y)=\int_{\tau_0}^\infty \frac 12 e^{-2i\tau} \partial_\tau (e^{-i\int \chi A^\tau d\tau} \chi A^\tau \Phi_-)+\tau^\frac 32 e^{-i\tau-i\int \chi A^\tau d\tau} R d\tau\\ +\left(e^{-i\int \chi A^\tau d\tau}(\Phi_+ +\frac 12 \chi A^\tau e^{-2i\tau}\Phi_-)\right)|_{\tau=\tau_0}.\end{gathered}$$ Note that along the hyperboloid $\{t^2-r^2=\tau_0^2\}$ we have $t^{-2}\sim (1-|y|^2)$. Then since we have $|Z^I\phi|\lesssim \varepsilon t^{-\frac 52+\delta}$ when $|t-r|$ is bounded, we have that along the initial hyperboloid $|\Phi_\pm|\lesssim \varepsilon(1-|y|^2)^{\frac 54-\delta}$. Therefore we can derive $|b_+(y)|\lesssim \varepsilon (1-|y|^2)^{\frac 54-\delta}$. **Remark 4**. *In view of the expression of $b_+(y)$ and the fact that $\Omega_{ij}$ and $\Omega_{0i}$ are tangent to the hyperboloids (and behaves well when falling on the cutoff $\chi_{LC}(q)$), we can derive similar bounds of derivatives of $b_+(y)$: $$(1-|y|^2)^{|I|} \nabla_y^I\, b_+(y)\lesssim \varepsilon (1-|y|^2)^{\frac 54-\delta}.$$* Similarly we have $\Phi_- =e^{-i\int A^\tau d\tau}b_-(y)+O(\tau^{-1+3\delta} (1-|y|^2)^{\frac 54-\frac \delta 2})$ with $b_-(y)$ satisfying the same bounds. Then using $\Phi=-\frac i2(e^{i\tau}\Phi_+-e^{-i\tau}\Phi_-)$ and $\partial_\tau\Phi=\frac 12(e^{i\tau}\Phi_+ +e^{-i\tau}\Phi_-)$, we have $$\begin{split} &\phi\sim -\frac i2\tau^{-\frac 32}(e^{i\tau+i\int \chi A^\tau d\tau} b_+(y)-e^{-i\tau-i\int A^\tau d\tau} b_-(y)),\\ &\partial_\tau \phi\sim \frac 12\tau^{-\frac 32}(e^{i\tau+i\int \chi A^\tau d\tau} b_+(y)+e^{-i\tau-i\int A^\tau d\tau} b_-(y)). \end{split}$$ In order to make the expansion of $\phi$ look simpler, we define $a_\pm(y):=\mp\frac i2 b_\pm(y)$. Then $$\begin{split} &\phi\sim \tau^{-\frac 32}(e^{i\tau+i\int \chi A^\tau d\tau} a_+(y)+e^{-i\tau-i\int \chi A^\tau d\tau} a_-(y)),\\ &\partial_\tau \phi\sim i\tau^{-\frac 32}(e^{i\tau+i\int \chi A^\tau d\tau} a_+(y)-e^{-i\tau-i\int \chi A^\tau d\tau} a_-(y)). \end{split}$$ ### The gauge potential We now turn to the wave equations $-\Box \tilde A_\mu=\Im(\phi\cdot\overline{D_\mu\phi})$ and derive the asymptotics of $A_\mu$. We have $$\Im (\phi\cdot\overline{D_\mu\phi})=-\frac{x_\mu}\tau \Im (\phi\cdot\overline{\partial_\tau \phi})+R_{cub}+R_{tan,w},$$ where $R_{cub}=O(|A|\cdot |\phi|^2)$, $R_{tan,w}=O(|\phi|\cdot t^{-1}|\nabla_y \phi|)$. Plugging the asymptotics of the scalar field above, we have $$\begin{split} \Im&(\phi\cdot\overline{\partial_\tau\phi}) \\ &\approx \Im\left(-i\tau^{-3}(e^{i\tau+i\int A^\tau d\tau} a_+(y)+e^{-i\tau-i\int A^\tau d\tau} a_-(y))(e^{-i\tau-i\int A^\tau d\tau} \overline{a_+(y)}-e^{i\tau+i\int A^\tau d\tau} \overline{a_-(y)})\right)\\ &=\Im\left(-i\tau^{-3} \left(|a_+(y)|^2-|a_-(y)|^2-e^{2i(\tau+\int A^\tau d\tau)}a_+(y)\overline{a_-(y)}+e^{-2i(\tau+\int A^\tau d\tau)}a_-(y)\overline{a_+(y)}\right)\right)\\ &=-\tau^{-3}(|a_+(y)|^2-|a_-(y)|^2), \end{split}$$ where the remainder (from the "$\approx$\") $R_{KG}=O(\varepsilon^2\tau^{-3} (1-|y|^2)^{\frac 54-\delta}\tau^{-1+3\delta} (1-|y|^2)^{\frac 54-\frac \delta 2})=O(\varepsilon^2 t^{-4+3\delta})$. Also, while the expansion is only applicable when $\tau\geq \tau_0$, we use this expression everywhere when $t\geq r$, as one can see that the error $R_{source}$ is supported in a region near the light cone $\{t\geq r,\, t-r\leq \tau_0 (t+r)^{-1}\}$, and hence decaying at the rate better than $t^{-\frac 72}$ using the boundedness of $t-r$. Therefore the equation becomes $$-\Box \tilde A_\mu=\frac{x_\mu}\tau \tau^{-3} (|a_+(y)|^2-|a_-(y)|^2)+R_{cub}+R_{tan,w}+R_{KG}+R_{source}.$$ Again, by linearity, we consider vanishing initial data first. Notice that $\frac{x_\mu}\tau$ is only dependent on $y$ variables. Therefore the analysis of the main terms is in fact on the equation $$-\Box A^M_\mu=t^{-3} F_\mu(y)$$ with vanishing data (at $t=2$), where $F_\mu(y)=(\frac t\tau)^3 \frac{x_\mu}\tau (|a_+(y)|^2-|a_-(y)|^2)$. These functions in $y$ are defined for $|y|\leq 1$, and we understand them as zero when $|y|>1$. We have studied this equation in detail in [@CL]: Using the representation formula, the solution can be written as $$\label{lambdaformula} \begin{split} A^M_\mu(t,x)&=\frac 1{4\pi}\int_2^t \frac{1}{t-s} \int_{|\bar{z}|=t-s} s^{-3} F_\mu\left (\frac{x-\bar z}{s}\right ) \, d\sigma(\bar z) ds\\ &=\frac 1{4\pi} \int_{\frac 2t}^1 \frac{1}{1-\lambda} \int_{|\bar z|=(1-\lambda)t} (\lambda t)^{-3} F_\mu\left (\frac{x-\bar z}{\lambda t}\right )\, d\sigma(\bar z) d\lambda\\ &=\frac 1{4\pi}\int_{\frac 2t}^1 \int_{\mathbb{S}^2} (1-\lambda)t^{-1}\lambda^{-3} F_\mu\left (\frac{x/t-(1-\lambda)\eta}{\lambda}\right )\, d\sigma(\eta) d\lambda \end{split}$$ Notice that if $\lambda$ near $0$ satisfies $1-\lambda-r/t>\lambda$, i.e., $\lambda<\frac 12(1-\frac rt)$, then the integrand is zero because of the support of $F_\mu$. Therefore, if $2/t<\frac 12(1-r/t)$, i.e., $t-r>4$, we can replace the lower bound of the integral $2/t$ by $\frac 12(1-r/t)$. As a result, in the region $\{t-r>4\}$ we have $A^M_\mu=t^{-1} \widetilde U_\mu(x/t)$, where $$\widetilde U_\mu(y):=\frac 1{4\pi}\int_{\frac 12(1-|y|)}^1 \int_{\eta\in \mathbb{S}^2} (1-\lambda) \lambda^{-3} F_\mu\left (\frac{y-(1-\lambda)\eta}{\lambda}\right )\, d\sigma(\eta) d\lambda.$$ Therefore, for $U_\mu(y)=\frac \tau t \widetilde U_\mu(y)$, we have $$A^M_\mu=\frac{U_\mu(y)}\tau, \quad t-r\geq 4.$$ For the remaining inhomogeneous terms, we consider $$-\Box A^R_\mu=R_{cub}+R_{tan,w}+R_{good}+R_{LC}$$ with vanishing data. Using the decay estimates we can bound all of them by $\varepsilon^2(1+t+r)^{-4+3\delta}$. This implies $$|A^R_\mu|\lesssim \varepsilon^2 (1+t+r)^{-1}(1+|t-r|)^{-1+3\delta}, \text{ so } |A^R_\mu|\lesssim \varepsilon^2\tau^{-2+3\delta}(1-|y|^2)^{\frac {3}2 \delta} \text{ when $t-r\geq 1$.}$$ Now for the initial data part, we have already derived the estimate of its contribution $\varepsilon(1+t+r)^{-1}(1+|t-r|)^{-\frac{\gamma_0'-1}2}\lesssim \varepsilon\tau^{-\frac{\gamma_0'+1}2}(1-|y|^2)^{\frac{3-\gamma_0'}4}$. So combining these together we have (note that $\gamma_0'>1$) $$A_\mu=\frac{U_\mu(y)}\tau+O(\varepsilon\tau^{-\frac{\gamma_0'+1}2}(1-|y|^2)^{\frac{3-\gamma_0'}4}), \quad t-r\geq 4.$$ Therefore, we also have $$A^\tau=\frac{U(y)}\tau+O(\varepsilon\tau^{-\frac{\gamma_0'+1}2}(1-|y|^2)^{-\frac{\gamma_0'-1}2}),\quad t-r\geq 4,$$ so for the phase in the expansion of the complex scalar field, we have $\int \chi_{LC}(q) A^\tau d\tau= U(y)\ln\tau+h$, where $h=O(\tau^{-1}(1+(t-r))^{-\frac{\gamma_0'}2})$ is uniformly bounded in the region $\{t-r\geq 4\}$. Note that if we denote the initial data part (linear part) by $A_\mu^{Linear}$ and write them as an independent part in the expansion, then the remainder has better estimates as in Theorem [Theorem 1](#mainthm1){reference-type="ref" reference="mainthm1"}. ### Radiation fields Now we discuss the behavior of $A_\mu$ towards null infinity. While the lower order terms above decay well towards timelike infinity, one cannot expect the decay to be better than $r^{-1}$ along the light cone. In fact, since each of them satisfies a wave equation, the radiation fields all exist. For example we have **Proposition 2**. *The limit $\lim_{r\rightarrow \infty} rA^M_\mu(r,q,\omega)$ exists for each $q(=r-t)$ and $\omega$, which we denote by $F^M_\mu(q,\omega)$. We also have the bound $$|rA^M_\mu-F^M_\mu|\lesssim \varepsilon r^{-1} q_-$$ where $q_-=\max\{-q,0\}$.* *Proof.* Recall the equation $-\Box A^M_\mu=\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)$. Since the bound of $a_+(y)$ is similar with vector fields, we can commute the equation with $\triangle_\omega=\sum\Omega_{ij}^2$ to derive the decay of $\triangle_\omega A^M_\mu$. Then one can use the decomposition $-\Box A^M_\mu=r^{-1}(\partial_t-\partial_r)(\partial_t+\partial_r)(r A^M_\mu)+r^{-2}\triangle_\omega A^M_\mu$. Integrating in $-(\partial_t-\partial_r)$ direction to the initial slice gives (when $t<r$ the source is zero) $$|(\partial_t+\partial_r)(rA^M_\mu)|\lesssim \varepsilon r^{-1} (1+t+r)^{-1} q_-.$$ Now integrating in $\partial_t+\partial_r$ direction for $r\in [r_1,r_2]$, we get $$|r_1 A^M_\mu(r_1,q,\omega)-r_2 A^M_\mu(r_2,q,\omega)|\lesssim \varepsilon r_1^{-1} q_-,$$ which shows the existence of the limit. Letting $r_2\rightarrow\infty$ also gives $$|r A^M_\mu-F^M_\mu(q,\omega)|\lesssim \varepsilon r^{-1} q_-$$ as required. ◻ Since $A^M_\mu=U_\mu(y)/\tau$ when $t-r\geq 4$, the proposition above also shows the existence of the limit $\lim_{|y|\rightarrow 1^-}(1-|y|^2)^{-\frac 12} U_\mu(y)$. For each $\mu=0,1,2,3$, the limit is a function of $\omega$ and we denote it as $\mathcal U_\mu(\omega)$. Then we have $F^M_\mu(q,\omega)=\mathcal U_\mu(\omega)$ when $q<-4$. One can similarly prove the existence of the radiation field for other parts ($A^R_\mu$ and the initial data part). However, due to additional decay in $\langle t-r\rangle$, these radiation fields is decaying in $\langle q \rangle$ as $q\rightarrow \pm\infty$. Then, also taking into account of the charge part $\mathbf{q}_0 \chi_{ex}(r-t) r^{-1}\delta_{0\mu}$, we have the radiation field of $A_\mu$, denoted by $F^{total}_\mu(q,\omega)$, satisfies $$F^{total}_\mu(q,\omega)=\mathcal U_\mu(\omega)\mathbf 1_{q<0}+\mathbf{q}_0 \delta_{0\mu}\mathbf 1_{q>0}+O(\varepsilon\langle q\rangle^{-\frac{\gamma_0'-1}2}).$$ **Remark 5**. *We can also compute the convergence rate to the radiation fields for other parts. For example for the linear (initial data) part, we can proceed similarly as above to get $$|(\partial_t-\partial_r)(\partial_t+\partial_r)(A^{Linear}_\mu)|\lesssim \varepsilon r^{-1}(1+t+r)^{-1}(1+|q|)^{-\frac{\gamma_0'-1}2}$$ Integrating this to the initial slice means integrating $q$ between $r-t$ and $r+t$, which gives $$|(\partial_t+\partial_r)(rA^{Linear}_\mu)|\lesssim \varepsilon r^{-1}(1+t+r)^{-1}((1+|q|)^{1-\frac{\gamma_0'-1}2}+(1+t+r)^{1-\frac{\gamma_0'-1}2})$$ which gives $$|A^{Linear}_\mu-\chi(\frac{\langle q\rangle}r)F^{Linear}_\mu(q,\omega)|\lesssim \varepsilon (1+t+r)^{-1-\frac{\gamma_0'-1}2}.$$* Nevertheless, this is not the whole story. The gauge condition itself implies additional structure: if we decompose the Lorenz gauge condition $\partial^\mu A_\mu$ in the null frame $\{L,{\underline{L}},e_1,e_2\}$, we see that ${\underline{L}}A_L\sim -L A_{{\underline{L}}}+2(e_B A_\mu)(e_B)^\mu$. Therefore, the ${\underline{L}}$ derivative of some components of $A$, generally expected to be the "bad\" derivative, behaves like tangential derivatives (to the outgoing light cone). Therefore, the $A_L$ component may decay at a rate better than $r^{-1}$. This is observed in [@CKL] for the massless MKG system in the Lorenz gauge. We now prove that this also holds for the massive system. **Proposition 3**. *The radiation fields $F_\mu(q,\omega)$ satisfies $F_L(q,\omega)=\mathbf{q}_0$ for all $q$ and $\omega$.* *Proof.* We have $$\begin{split} L(r{\underline{L}}(rA_L)+r\tilde A_{\underline{L}})&=rL{\underline{L}}(rA_L)+{\underline{L}}(r A_L)+L(rA_{{\underline{L}}}-\mathbf{q}_0 \chi_{ex}(r-t))\\ &=-r^2 (\Box A_\mu)L^\mu+(\triangle_\omega A_\mu)L^\mu+r{\underline{L}}A_L+rLA_{{\underline{L}}}-A_\mu (\partial_r)^\mu\\ &=r^2 J_L+2r(e_B A_\mu)(e_B)^\mu+(\triangle_\omega A_\mu)L^\mu-A_\mu(\partial_r)^\mu. \end{split}$$ Notice that $A_\mu (\partial_r)^\mu=\tilde A_\mu (\partial_r)^\mu$, $e_B A_\mu=e_B\tilde A_\mu$, $\triangle_\omega A_\mu=\triangle_\omega\tilde A_\mu$, and $|re_B \tilde A_\mu|\lesssim \sum_{i<j}|\Omega_{ij} A_\mu|$. Then using the decay estimates we have $$|L(r{\underline{L}}(rA_L)+r\tilde A_{\underline{L}})|\lesssim \varepsilon(1+t+r)^{-1+\delta}(1+q_+)^{-\frac{\gamma_0'-1}2}.$$ Integrating in $\partial_t+\partial_r$ direction backward to $r=0$ or $t=0$ gives $$|r{\underline{L}}(r A_L)+r\tilde A_{\underline{L}}|\lesssim \varepsilon(1+t+r)^{\delta} (1+q_+)^{-\frac{\gamma_0'-1}2}.$$ Since $|\tilde A_L|\lesssim \varepsilon(1+t+r)^{-1+\delta}(1+q_+)^{-\frac{\gamma_0'-1}2}$, we get $$|{\underline{L}}(rA_L)|\lesssim \varepsilon r^{-1}(1+t+r)^{\delta}(1+q_+)^{-\frac{\gamma_0'-1}2}.$$ Now integrate in $\partial_r-\partial_t$ direction to the initial slice. Recall that at the initial slice $$|rA_L-\mathbf{q}_0|\lesssim \varepsilon (1+r)^{-\frac{\gamma_0'-1}2},$$ so we have $$\begin{split} |rA_L-\mathbf{q}_0|&\leq |rA_L(t,r)-rA_L(0,t+r)|+|rA_L(0,t+r)-\mathbf{q}_0|\\ &\lesssim \int_{r-t}^{r+t} \varepsilon r^{-1}(1+t+r)^{\delta}(1+q_+)^{-\frac{\gamma_0'-1}2} dq+\varepsilon (1+t+r)^{-\frac{\gamma_0'-1}2}\\ &\leq \varepsilon r^{-1}(1+t+r)^\delta \int_{r-t}^{r+t} (1+q_+)^{-\frac{\gamma_0'-1}2} dq+\varepsilon (1+t+r)^{-\frac{\gamma_0'-1}2}\\ &\lesssim \varepsilon r^{-1}(1+t+r)^{1+\delta-\frac{\gamma_0'-1}2}+\varepsilon (1+t+r)^{-\frac{\gamma_0'-1}2}. \end{split}$$ Note that $\delta$ is a small number, so we have $\delta<\frac{\gamma_0'-1}2$. Therefore, we have $rA_L\rightarrow \mathbf{q}_0$ as $r\rightarrow\infty$ for every $q=r-t$ fixed. This shows that the radiation fields $F^{total}_\mu$ satisfies $F^{total}_L=\mathbf{q}_0$. ◻ ## Charge at infinity Recall the charge conservation identity $\partial^\mu J_\mu=0$. We can use this to derive an expression of charge using the information at infinity. Consider the region enclosed by the initial slice, a part of an incoming null cone $\underline C_{\tau^2}=\{t+r=\tau^2,\, t-r\leq 1\}$, and the hyperboloid $\hat H_\tau$ (truncated at $t-r=1$) to get $$\int_{\hat H_\tau} \Im (\phi\cdot\overline{D_\tau \phi})\, dH_\tau+\int_{\underline C_{\tau^2}} \Im(\phi\cdot\overline{D_{{\underline{L}}} \phi})\, dvol=4\pi \mathbf{q}_0.$$ Using the decay estimates, it is easy to see that the second term on the left is decaying to zero as $\tau \rightarrow\infty$. On the other hand, we have shown above that in the interior $$\Im(\phi\cdot\overline{D_\tau\phi})=-\tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)+O(t^{-\frac 72+\delta}),$$ so using $dH_\tau=\tau^3 dH_1$, we have for the first integral $$\begin{split} \int_{\hat H_\tau} \Im(\phi\cdot\overline{D_\tau\phi})\, dH_\tau&=\int_{\hat H_\tau} (\tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)+O(t^{-\frac 72+\delta})) \tau^3 dH_1\\ &\rightarrow\int_{H_1} |a_-(y)|^2-|a_+(y)|^2 \, dH_1 \quad \text{ as }\tau\rightarrow\infty. \end{split}$$ Therefore $$\label{q0atinfinity} \mathbf{q}_0=\frac{1}{4\pi}\int_{H_1} |a_-(y)|^2-|a_+(y)|^2 \, dH_1.$$ # Scattering from infinity The goal of this section is to study the scattering from infinity problem of the mMKG system. We will first consider the system in the Lorenz gauge: $$-\Box A_\mu=\Im(\phi\cdot\overline{D_\mu\phi}),\quad -\Box\phi+\phi=2iA^\mu\partial_\mu\phi-A^\mu A_\mu\phi,$$ and then show that for admissible scattering data, the solution we get in fact also solves the original mMKG system. In the forward problem, we showed that $$A_\mu \sim U_\mu(y)/\tau, \quad \phi\sim \tau^{-\frac 32}(e^{i\tau+iU(y)\ln \tau+ih}a_+(y)+e^{-i\tau-iU(y)\ln\tau-ih}a_-(y))$$ in the interior, where $U(y):=U^\tau (y)=-\frac{x_\mu}\tau U_\mu(y)$. The functions $U_\mu(y)$ are in fact determined by $a_\pm(y)$, and $R$ is uniformly bounded in $\{t-r\geq 1\}$. We also have the radiation fields $F_\mu(q,\omega)=\lim_{r\rightarrow\infty} rA_\mu(r,q,\omega)$. In particular in the forward problem we have $rA_L\rightarrow \mathbf{q}_0$ where $\mathbf{q}_0$ is the charge defined by the initial data. Now for the backward problem, we consider similar type of asymptotic behaviors to be given. Since we want the Lorenz gauge condition to hold, we expect a similar relation on the radiation fields of the solution we construct. In view of the charge relation [\[q0atinfinity\]](#q0atinfinity){reference-type="eqref" reference="q0atinfinity"}, we define the charge at infinity using the scattering data $$\mathbf{q}_\infty :=\frac{1}{4\pi}\int_{H_1} |a_-(y)|^2-|a_+(y)|^2 dH_1.$$ As in the forward problem, $a_+(y)$ and $a_-(y)$ determine the main part of the wave components in the interior. Hence we define $A^M_\mu$ to solve $$-\Box A^M_\mu=\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)$$ with vanishing data at $\{t=2\}$. Then we know that when $t-r>4$ we have $A^M_\mu=U_\mu(y)/\tau$, so $U_\mu(y)$ are determined by $a_+(y)$ and $a_-(y)$. **Remark 6**. *One may also add a $\chi_{LC}(q)$ cutoff factor on the right hand side of $-\Box A^M_\mu$ (we used $\chi_{LC}(q)$ in the forward problem on the phase; it is $1$ when $q\leq -1$ and $0$ when $q\geq -1/2$). It is not hard to see that this differs from the one without cutoff by a part supported near the light cone, which decays very well and affects little in the analysis. Also, we can see that when we commute the equation with vector fields, terms with vector fields falling on $\chi_{LC}(q)$ gives similarly good terms supported in $-1\leq q\leq 0$ as $|Zq|\leq |t-r|\leq 1$. For similar reasons, one can also add this cutoff factor on the phase correction in front of $(A^{(0)})^\tau$ below. The advantage of doing this is that the approximate solution behaves more regularly near the light cone. However, to make things look simpler we here proceed without the cutoff, but these modifications are straightforward.* We can now define the approximate solutions $$A^{(0)}_\mu=A^M_\mu+\chi(\frac{\langle q\rangle}r) \left(\frac{F_\mu(q,\omega)}r+\frac{F^{(1)}_\mu(q,\omega)}{r^2}\right)+\mathbf{q}_\infty \chi_{ex}(q) \delta_{0\mu} r^{-1},$$ $${\phi^{(0)}}=\tau^{-\frac 32}(e^{i\tau+i\int (A^{(0)})^\tau d\tau} a_+(y)+e^{-i\tau-i\int (A^{(0)})^\tau d\tau} a_-(y)).$$ Here $\chi_{ex}(q)$ supported in $\{q\geq 1\}$ is $1$ when $q\geq 2$. The fields $F_\mu(q,\omega)$ should be understand as the total radiation fields subtracted by the contribution of $A^M_\mu$ and $\chi_{ex}(q)\, \mathbf{q}_\infty \delta_{0\mu}r^{-1}$, and we require them to decay in $\langle q\rangle$: $$\sum_{k+|\beta|\leq N}|(\langle q\rangle \partial_q)^k \partial_\omega^\beta F_\mu(q,\omega)|\lesssim \varepsilon\langle q\rangle^{-1+\gamma},\quad \gamma<1/2,\, N\geq 6.$$ Later we will see that $F_\mu(q,\omega)$ needs to satisfy a compatibility condition [\[admissiblecondition\]](#admissiblecondition){reference-type="eqref" reference="admissiblecondition"} which essentially comes from the Lorenz gauge condition. The second order approximation $F^{(1)}(q,\omega)$ is defined similarly as in [@LindbladSchlue1] and [@CL]: $$2\partial_q F^{(1)}_\mu(q,\omega)=\triangle_\omega F_\mu(q,\omega).$$ We also impose the decay assumptions on $a_\pm(y)$: $$\sum_{|I|\leq N}|(1-|y|^2)^{|I|}\nabla_y^I a_\pm(y)|\lesssim \varepsilon(1-|y|^2)^\alpha, \quad \alpha\geq 7/4,$$ which implies that $a_\pm(y)$ decay at the same rate when applied with rotation and boost vector fields. **Remark 7**. *When we apply partial derivatives to $a_\pm(y)$, using this condition, we get an additional $\tau^{-1}(1-|y|^2)^{-\frac 12}$. This quantity is bounded when $t-r\geq 1$, and we see that one advantage of the cutoff in the previous remark is that we can allow more partial derivatives in the estimate.* In the forward problem, we see that the Lorenz gauge condition is closely related to the fact that $L$ component of the radiation fields to be constant (the charge). From the expression of $A^{(0)}$ we see that the value is $\mathbf{q}_\infty$ when $q\rightarrow +\infty$. Therefore, to make the approximate solutions a good candidate not only for the reduced system but also for the original system, a crucial step is to show that when we consider the radiation fields of $A^M_\mu$, denoted by $F^M_\mu(q,\omega)$, we have $F^M_L(q,\omega)\rightarrow \mathbf{q}_\infty$ as $q\rightarrow -\infty$. We shall address this question in the following subsection, and show that $rA^M_L\rightarrow \mathbf{q}_\infty$ as $r\rightarrow \infty$ with $q=r-t<-4$ fixed. In later subsections, We shall first construct solutions to the reduced system, and then show that the solutions also solve the original mMKG system. The latter step relies heavily on the decay property of $\partial^\mu A^{(0)}_\mu$, which requires, again, the $L$ component of the radiation field is always $\mathbf{q}_\infty$. Therefore, the coefficient of the exterior part must be $\mathbf{q}_\infty$, because otherwise one would not be able to get a solution to the original system. ## Computing the radiation field of the interior solution {#sectionradiationfieldcharge} Recall that $$-\Box A^M_\mu= \frac{x_\mu}\tau \tau^{-3} (|a_+(y)|^2-|a_-(y)|^2)=:x_\mu t^{-4} P(y)$$ and the data is vanishing at $t=2$.[^7] We also denote $F(y):=|a_+(y)|^2-|a_-(y)|^2=\tau^4 t^{-4} P(y)=(1-|y|^2)^2 P(y)$. When $q<-4$, we have $A^M_\mu=U_\mu(y)/\tau$. The form implies that the radiation fields $F^M_\mu(q,\omega)$ will in fact be independent of $q$ when $q<-4$. Therefore, we only need to fix one $q<-4$ in the computation. When $t-r>4$, the representation formula gives $$A^M_\mu=\frac{1}{4\pi}\int_2^t \frac 1{t-s}\int_{\mathbb{S}^2} \left(s,x-(t-s)\eta\right)_\mu\, s^{-4} P\left(\frac{x-(t-s)\eta}{s}\right)(t-s)^2 d\sigma(\eta) ds$$ Here $(s,x-(t-s)\eta)_0=-s$, and $(s,x-(t-s)\eta)_i=(x-(t-s)\eta)_i$. Since $x=r\omega$, contracting with $L^\mu$ we get $$A^M_L=\frac{1}{4\pi}\int_2^t \int_{\mathbb{S}^2} (-s+r-(t-s)\langle\eta,\omega\rangle)\, s^{-4} P\left(\frac{x-(t-s)\eta}{s}\right)(t-s) d\sigma(\eta) ds.$$ Recall that this is an integral on the part of a backward light cone with the tip at $(t,x)$ restricted in the part $t\geq 2$. In the case when $t-|x|>4$, for all angles, the backward cone already hits the support boundary $\{t=|x|\}$ at some time greater than $2$. We can compute the value for each $\eta$: Consider the distance $R=R(\eta)$ so that $(t,r,0,0)-(R,R\eta_1,R\eta_2,R\eta_3)$ is on the cone $\{t=r\}$. Then $$(t-R)^2=|x-R\eta|^2\quad \implies \quad R(\eta)=\frac{t^2-r^2}{2(t-r\langle\eta,\omega\rangle)}.$$ We now consider the change of variable $R=t-s$. This gives $$A^M_L=\frac{1}{4\pi}\int_{\mathbb{S}^2} \int_0^{R(\eta)} (R-t+r-R\langle \eta,\omega\rangle)\, (t-R)^{-4} P\left(\frac{x-R\eta}{R}\right) R\, dR d\sigma(\eta).$$ We want to study the limit of $tA_L^M$ as $t\rightarrow\infty$ with $q=r-t$ fixed. When $q<-4$, we have $$tA^M_L=\frac{1}{4\pi}\int_{\mathbb{S}^2} \int_0^{R(\eta)} tR(R+q-R\langle\eta,\omega\rangle)\, (t-R)^{-4} P\left(\frac{x-R\eta}{R}\right) dR d\sigma(\eta).$$ **Lemma 4**. *Given $t$ and $x$ with $|x|<t$. Then for a function defined on the unit hyperboloid (equipped with the induced metric from Minkowski) $P(y)$ and $F(y)=(1-|y|^2)^2 P(y)$, we have the following relation $$\int_{H_1} F(y) \, dH_1=\int_{\mathbb{S}^2} \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \frac{R^2(t-r\langle\eta,\omega\rangle)}{(R-t)^4} dR d\sigma(\eta)$$ where $R(\eta)=(t^2-r^2)/(2(t-r\langle\eta,\omega\rangle))$ is defined as above.* *Proof.* Without loss of generality, we can assume $\omega=(1,0,0)$ in proving this lemma, so $\langle \eta,\omega\rangle=\eta_1$. We first write the integral on the unit hyperboloid $H_1$. In the coordinate $(|y|,\theta,\varphi)$, the integral can be written as $$\int_{\mathbb{S}^2} \int_0^1 F(y) \frac 1{(1-|y|^2)^{2}} |y|^2 d|y| d\mathbb{S}^2=\int_{\mathbb{S}^2} \int_0^1 P(y) |y|^2 d|y| d\mathbb{S}^2.$$ Now if we denote the standard induced Euclidean metric on unit ball (including interior) by $dy$, then the integral equals $$\int_{|y|\leq 1} P(y)\, dy.$$ We wish to switch to a coordinate system more adapted to the integral we need to deal with. For every given $x=(r,0,0)$ and $t$, all points in the unit ball can be uniquely expressed in $R$ and $\eta$ as $$y=\frac{x-R\eta}{t-R}$$ where $0\leq R \leq R(\eta)$. Now we want to write the integral in $R$ and $\eta$. We use the coordinate $(R,\eta_2,\eta_3)$, so we need to split the case $\eta_1\geq 0$ and $\eta_1<0$. When $\eta_1\geq 0$, we have $\eta_1=\sqrt{1-\eta_2^2-\eta_3^2}$. Then the coordinate change reads $$y_1=\frac{r-R\sqrt{1-\eta_2^2-\eta_3^2}}{t-R},\quad y_2=-\frac{R\eta_2}{t-R},\quad y_3=-\frac{R\eta_3}{t-R}.$$ One can then compute the Jacobian $$\det\frac{\partial(y_1,y_2,y_3)}{\partial(R,\eta_2,\eta_3)}=\frac{R^2(r\sqrt{1-\eta_2^2-\eta_3^2}-t)}{(R-t)^4 \sqrt{1-\eta_2^2-\eta_3^2}}.$$ So the part $\eta_1\geq 0$ can be written as $$\int_0^1 \int_0^1 \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \left|\frac{R^2(r\sqrt{1-\eta_2^2-\eta_3^2}-t)}{(R-t)^4 \sqrt{1-\eta_2^2-\eta_3^2}}\right|\, dR d\eta_2 d\eta_3$$ The case when $\eta_1<0$ in fact gives the same expression. Also $\frac 1{\sqrt{1-\eta_2^2-\eta_3^2}}d\eta_2 d\eta_3=d\mathbb{S}^2=d\sigma(\eta)$, so the integral becomes $$\int_{\mathbb{S}^2} \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \frac{R^2(t-r\eta_1)}{(R-t)^4} dR d\sigma(\eta)$$ which concludes the proof. ◻ Recall $$\mathbf{q}_\infty=\frac{1}{4\pi}\int \left(|a_-(y)|^2-|a_+(y)|^2\right)\, dH_1=-\frac{1}{4\pi}\int_{H_1} F(y)\, dH_1.$$ Then using the lemma, we have $$tA^M_L-\mathbf{q}_\infty=\frac{1}{4\pi} \int_{\mathbb{S}^2} \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \frac{1}{(R-t)^4}(tR(R+q-R\langle\eta,\omega\rangle)+R^2(t-r\langle\eta,\omega\rangle)) dR d\sigma(\eta)$$ $$=\frac{1}{4\pi}\int_{\mathbb{S}^2} \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \frac{1}{(R-t)^4}\left(2tR^2(1-\langle\eta,\omega\rangle)+qR(t-R\langle\eta,\omega\rangle)\right)\, dR d\sigma(\eta).$$ Now we need to consider the integral $$\int_{\mathbb{S}^2} \int_0^{R(\eta)} P\left(\frac{x-R\eta}{t-R}\right) \frac{1}{(R-t)^4}\left(2tR^2(1-\langle\eta,\omega\rangle)+qR(t-R\langle\eta,\omega\rangle)\right)\, dR d\sigma(\eta).$$ First we point out that the part where $R\leq 1$ will have a lot of decay in $t$ due to the $(R-t)^{-4}$ factor, hence converging to zero as $t\rightarrow\infty$. Therefore, in the analysis below we only consider the part $R\geq 1$. Using $|P(y)|\leq \varepsilon$, we seek to estimate the absolute integral $$\mathcal I=\int_{\mathbb{S}^2} \int_0^{R(\eta)} \frac{1}{(R-t)^4}\left(2tR^2(1-\langle\eta,\omega\rangle)+|q|R(t-R\langle\eta,\omega\rangle)\right)\, dR d\sigma(\eta).$$ Since the integrand is now nonnegative, we can change the order of integration. Moreover, the integral is now independent of $\omega$, so we may just take $\omega=(1,0,0)$. For fixed $R$, which varies from $0$ to $(t+r)/2$ (note from above, however, we only care about values greater than $1$), in view of the expression of $R(\eta)$, the range of $\langle \eta,\omega\rangle=\eta_1$ becomes $$\frac{2tR-(t^2-r^2)}{2rR}\leq \eta_1\leq 1.$$ Also, for the angular variable, the integrand now is only depedent on $\eta_1$, so we can rewrite it as an integral in $\eta_1$. In this case we have $d\sigma(\eta)=2\pi |d\eta_1|$. So the integral (the $R\geq 1$ part) becomes $$2\pi\int_1^{\frac{t+r}2} \int_{\frac{2tR-(t^2-r^2)}{2rR}}^1 \frac{1}{(R-t)^4}\left(2tR^2(1-\eta_1)+|q|R(t-R\eta_1)\right)\, d\eta_1 dR.$$ We have $$1-\eta_1\leq \frac{(2R-(t+r))q}{2rR},\quad t-R\eta_1\leq t(1-\eta_1)+(t-R)\eta_1,$$ so can proceed the estimate as follows $$|\mathcal I|\lesssim \int_1^{\frac{t+r}2} \int_{\frac{2tR-(t^2-r^2)}{2rR}}^1 \frac{1}{(R-t)^4}\left(2tR^2 (1-\eta_1)+|q|R\left(t(1-\eta_1)+(t-R)\eta_1\right)\right)\, d\eta_1 dR$$ $$\lesssim \int_1^{\frac{t+r}2} \frac{1}{(R-t)^4} \left((2tR^2+|q|Rt) \frac{(2R-(t+r))q}{2rR}+|q|R(t-R)\right) \frac{(2R-(t+r))q}{2rR}\, dR$$ $$\lesssim \int_1^{\frac{t+r}2} (\frac{2t|q|^2}{r^2}+\frac{t|q|^3}{r^2 R}) {(R-t)^{-4}}(R-\frac{t+r}2)^2\, dR+\int_1^{\frac{t+r}2} \frac{|q|^2}{r}{(t-R)^{-3}}(\frac{t+r}2-R) dR$$ ($R=\frac{t+r}2-s$; $R\geq 1$) $$\lesssim \int_0^{\frac{t+r}2-1} \frac{t(2|q|^2+|q|^3)}{(t+q)^2}(s+\frac{|q|}2)^{-4}s^2\, ds+\int_0^{\frac{t+r}2-1} \frac{|q|^2}{r}{(s+\frac{|q|}2)^{-3}}s\, ds$$ $$\lesssim \frac{t(2|q|^2+|q|^3)}{(t+q)^2} \int_0^\infty {(s+\frac{|q|}2)^{-4}}s^2\, ds+\frac{|q|^2}{(t+q)}\int_0^\infty {(s+\frac{|q|}2)^{-3}}s\, ds.$$ Note that $q$ is a fixed constant here. Now since both integrals are convergent, $|\mathcal I|$ goes to zero as $t\rightarrow \infty$. Therefore we have proved that $$tA^M_L \rightarrow \mathbf{q}_\infty,\quad t\rightarrow\infty \quad \text{with $q=r-t<-4$ fixed,}$$ which, equivalently, means that $$(1-|y|^2)^\frac 12 U_L(y)\rightarrow \mathbf{q}_\infty.$$ This is actually the conclusion of Proposition [Proposition 1](#propintro){reference-type="ref" reference="propintro"}. Later we will need an estimate on how fast $tA^M_\mu$ (or $rA^M_\mu$) converges to the radiation field $F^M_\mu$. We have discussed the same problem in the forward part, and one can show that $$\label{convergencerate} |rA^M_\mu-F^M_\mu|\leq C_q \varepsilon \, r^{-1}.$$ Here $C_q$ is a constant that may grow in $\langle q\rangle$, but we will only use this estimate when $q$ is in a compact interval. Moreover, using the argument in [@H97 Section 6.2], we have that the radiation field of $Z^I A^M_\mu$ equals $rZ^I (r^{-1} F^M_\mu(q,\omega))$. Then by commuting the equation of $A^M_\mu$ with vector fields, one can also prove the convergence rate with vector fields: $$\label{convergencerateZ} |rZ^I A^M_\mu-rZ^I(r^{-1} F_\mu(q,\omega))|\leq C_q \varepsilon\, r^{-1}.$$ ## Constructing the approximate solutions ### Condition on the scattering data Given scattering data $(a_+(y),a_-(y),F_\mu(q,\omega))$, we can define the approximate solution $$A^{(0)}_\mu=A^M_\mu+\chi(\langle q\rangle/r) (F_\mu(q,\omega) r^{-1}+F^{(1)}_\mu(q,\omega)r^{-2})+\chi_{ex}(q)\mathbf{q}_\infty r^{-1}\delta_{0\mu}.$$ where $F_\mu(q,\omega)$ satisfies $$\sum_{k+|\beta|\leq N} |(\langle q\rangle \partial_q)^k \partial_\omega^\beta F_\mu(q,\omega)|\lesssim \varepsilon \langle q\rangle^{-1+\gamma}.$$ Similar to [@LiliHe21], we need the scattering data to satisfy an *asymptotic Lorenz gauge condition*, which says that $\partial^\mu A^{(0)}_\mu$ decays well, e.g., at the rate of $O(\langle t+r\rangle^{-2})$. Now we compute this quantity. Recall the wave equations of $A^M_\mu$. Commuting them with $\partial^\mu$, we get $$\Box \partial^\mu A^M_\mu=(\partial^\mu x_\mu)\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2)+x_\mu \partial^\mu (\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2))$$ Notice that $x^\mu \partial_\mu=S=\tau\partial_\tau$. Then the right hand side becomes $$4\tau^{-4} (|a_-(y)|^2-|a_+(y)|^2)-\tau\cdot 4\tau^{-5} (|a_-(y)|^2-|a_+(y)|^2)=0.$$ We know that $A^M_\mu$ and the time derivatives are zero at $\{t=2\}$. From the equation $$\Box A^M_0=-\frac t\tau \tau^{-3} (|a_-(y)|^2-|a_+(y)|^2)$$ we can also determine $$\partial_t^2 A^M_0|_{t=2}=\frac t\tau \tau^{-3} (|a_-(y)|^2-|a_+(y)|^2)|_{t=2}=2(4-|x|^2)^{-2}(|a_-(x/2)|^2-|a_+(x/2)|^2).$$ Then we have $(\partial^\mu A^M_\mu)|_{t=2}=0$, $\partial_t(\partial^\mu A^M_\mu)|_{t=2}=-\partial_t^2 A^M_0|_{t=2}$ which is compactly supported in $|x|\leq 2$. However, by strong Huygen's principle, the effect of this initial data part will only be present when $-4\leq q\leq 0$. Therefore, we have $\partial^\mu A^M_\mu=0$ when $q<-4$. Then using $\partial_\mu q=L_\mu$ and [\[convergencerateZ\]](#convergencerateZ){reference-type="eqref" reference="convergencerateZ"}, we have $$\begin{split} \partial^\mu A^{(0)}_\mu&=\partial^\mu A^M_\mu \mathbf{1}_{|q|\leq 4}+\partial^\mu \left(\chi(\frac {\langle q\rangle}r)F_\mu(q,\omega) r^{-1}\right)+\partial^\mu (\chi_{ex}(q)\, \mathbf{q}_\infty r^{-1} \delta_{0\mu})\\ &=\left(\partial^\mu (A^M_\mu-r^{-1} F^M_\mu)+\partial^\mu (r^{-1}F^M_\mu)\right)\mathbf{1}_{|q|\leq 4}+\partial^\mu \left(\chi(\frac{\langle q\rangle}r) F_\mu(q,\omega)\right) r^{-1}+\chi_{ex}'(q)\mathbf{q}_\infty r^{-1}\\ &\quad +O(\varepsilon\langle t+r\rangle^{-2})\\ &=\left(\partial_q F^M_L\right) r^{-1}+\partial_q \left(\chi(\frac{\langle q\rangle}r) F_L\right) r^{-1}+\chi_{ex}'(q) \mathbf{q}_\infty r^{-1}+O(\varepsilon\langle t+r\rangle)^{-2}). \end{split}$$ Therefore we derive the exact condition we need to impose on the scattering data: $$\partial_q \left(F^M_L+\chi(\frac{{\langle q\rangle}}{r})F_L+\chi_{ex}(q) \mathbf{q}_\infty\right)=0.$$ Note that $F_\mu$ decays to zero as ${\langle q\rangle}\rightarrow \infty$, and $F^M_L(q,\omega)=\mathbf{q}_\infty$ whenever $q\leq -4$. Therefore, the quantity in the parenthesis should always be $\mathbf{q}_\infty$. In view of the support of $F^M_L-\mathbf{q}_\infty$ and $\chi_{ex}(q)$, the term $\chi(\frac{\langle q\rangle}r)F_L$ must vanish at points where $\chi(\frac{\langle q\rangle}r)$ is not $1$. So one can drop the cutoff function factor $\chi(\frac{\langle q\rangle}r)$ above, and it must hold that $$\label{admissiblecondition} F_L=\mathbf{q}_\infty-\mathbf{q}_\infty \chi_{ex}(q)-F^M_L.$$ The condition ensures that $\partial^\mu A^{(0)}_\mu$ decays like $(t+r)^{-2}$. Now we consider the case with vector fields. Towards null infinity we have $$\begin{split} rZ(A^M_L)&=rA^M_\mu (ZL^\mu)+r(ZA^M_\mu)L^\mu\\ &\rightarrow F^M_\mu(q,\omega)(ZL^\mu)+r(Z(r^{-1}F^M_\mu(q,\omega)))L^\mu=rZ(r^{-1}F_\mu(q,\omega)L^\mu)\\ &=rZ(r^{-1}\mathbf{q}_\infty) \end{split}$$ and from the previous subsection, the convergence rate is $O_q(\varepsilon r^{-1})$. Using $\partial^\mu A^M_\mu=0$ when $t-r>4$ again, we have $$\begin{split} Z \partial^\mu A^{(0)}_\mu&=Z \partial^\mu A^M_\mu \mathbf{1}_{|q|\leq 4}+Z\partial^\mu \left(\chi(\frac{\langle q\rangle}r)F_\mu(q,\omega) r^{-1}\right)+Z\partial^\mu (\chi_{ex}(q)\, \mathbf{q}_\infty r^{-1} \delta_{0\mu})\\ &=\left(Z\partial^\mu (A^M_\mu-F^M_\mu r^{-1})+Z\partial^\mu (F^M_\mu r^{-1})\right)\mathbf{1}_{|q|\leq 4}+\left(Z\partial^\mu (F_\mu \chi(\frac {\langle q\rangle}r))\right) r^{-1}+\partial^\mu\left(\chi(\frac{\langle q\rangle}r) F_\mu\right) Z(r^{-1})\\ &\quad +Z(\chi_{ex}'(q)\mathbf{q}_\infty r^{-1})+O(\varepsilon\langle t+r\rangle^{-2})\\ &=\partial_q F^M_L Z(r^{-1})+Z(\partial_q F^M_L)r^{-1}+\left(Z \partial_q (F_L \chi(\frac{\langle q\rangle}r))\right) r^{-1}+\partial_q(\chi(\frac{\langle q\rangle}r) F_L) Z(r^{-1})\\ &\quad +Z(\chi_{ex}'(q)) \mathbf{q}_\infty r^{-1}+\chi_{ex}'(q)\mathbf{q}_\infty Z(r^{-1})+O(\varepsilon\langle t+r\rangle^{-2})\\ &=Z(r^{-1})\left(\partial_q F^M_L+\partial_q(\chi(\frac {\langle q\rangle}r) F_L)+\mathbf{q}_\infty \chi_{ex}'(q)\right)\\ &\quad +Z\left(\partial_q F^M_L+\partial_q(\chi(\frac {\langle q\rangle}r) F_L)+\mathbf{q}_\infty \chi_{ex}'(q)\right)r^{-1}+O(\langle t+r\rangle^{-2})=O(\varepsilon\langle t+r\rangle^{-2}). \end{split}$$ One can apply more vector fields and derive similar estimates. In conclusion, once the scattering data satisfies the admissble condition [\[admissiblecondition\]](#admissiblecondition){reference-type="eqref" reference="admissiblecondition"}, we have $$Z^I \partial^\mu A^{(0)}_\mu=O(\varepsilon\langle t+r\rangle^{-2}).$$ ## Estimates of the approximate solutions In this subsection we derive estimates of the approximate solutions. First we have (For $A^M_\mu$, we can commute its wave equation with vector fields to derive the estimate) $$\begin{split} |(Z,S)^I A^{(0)}_\mu|&\lesssim |Z^I A^M_\mu|+|Z^I(\chi({\langle q\rangle}/r) (F(q,\omega) r^{-1}+F^{(1)}(q,\omega) r^{-2}))|+|Z^I(\chi_{ex}(q) \mathbf{q}_\infty r^{-1} \delta_{0\mu})|\\ &\lesssim \varepsilon\langle t+r\rangle^{-1}. \end{split}$$ Using the definition of ${\phi^{(0)}}$, we also have ($\Omega$ being rotation or boosts) $$|\partial^I \Omega^J {\phi^{(0)}}|\lesssim \tau^{-\frac 32-|I|} (\ln\tau)^{|J|} (1-|y|^2)^{-\frac{|I|}2}.$$ In view of the definition of the second order approximate radiation field $F^{(1)}(q,\omega)$, as in [@LindbladSchlue1; @CL], we have the following estimate $$|\Box\Big(\chi(\frac{\langle q\rangle}r) \big(\frac{F_\mu(q,\omega)}{r}+\frac{F_\mu^{(1)}(q,\omega)}{r^2}\big)\Big)|\lesssim \varepsilon\langle t+r\rangle^{-4}\langle q\rangle^{\gamma}.$$ Then since $\chi_{ex}(q)\mathbf{q}_\infty r^{-1}\delta_{0\mu}$ are exact solutions to the linear wave equation, we obtain the estimate $$|\Box Z^I(A^{(0)}_\mu)-Z^I(\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)|\lesssim \varepsilon\langle t+r \rangle^{-4}\langle q\rangle^\gamma.$$ Now we turn to Klein-Gordon field. We have $$\begin{split} -\Box{\phi^{(0)}}+{\phi^{(0)}}&-2i(A^{(0)})^\mu\partial_\mu{\phi^{(0)}}=\tau^{-\frac 32}(\partial_\tau^2+1)(\tau^\frac 32 {\phi^{(0)}})-2i(A^{(0)})^\tau\partial_\tau{\phi^{(0)}}+R_{tan}\\ &=\tau^{-\frac 32}(\partial^2_\tau+1)(e^{i\tau+i\int (A^{(0)})^\tau d\tau}a_+(y)+e^{-i\tau-i\int(A^{(0)})^\tau d\tau} a_-(y))\\ &\quad -2i(A^{(0)})^\tau \tau^{-\frac 32}(ie^{i\tau+i\int (A^{(0)})^\tau d\tau}a_+(y)-ie^{-i\tau-i\int(A^{(0)})^\tau d\tau} a_-(y))+R_{good,1}+R_{tan}\\ &=\tau^{-\frac 32} (-2(A^{(0)})^\tau e^{i\tau+i\int (A^{(0)})^\tau d\tau} a_+(y)-2(A^{(0)})^\tau e^{-i\tau-i\int (A^{(0)})^\tau d\tau}a_-(y))\\ &\quad -2i(A^{(0)})^\tau \tau^{-\frac 32}(ie^{i\tau+i\int (A^{(0)})^\tau d\tau}a_+(y)-ie^{-i\tau-i\int(A^{(0)})^\tau d\tau} a_-(y))\\ &\quad +R_{\tan}+R_{good,1}+R_{good,2}+R_{good,3}\\ &=R_{\tan}+R_{good,1}+R_{good,2}+R_{good,3}. \end{split}$$ where $$\begin{split} R_{tan}&=-\tau^{-2}(\triangle_y {\phi^{(0)}}+\frac 34{\phi^{(0)}})-2i(A^{(0)})^{y_i}\partial_{y_i}{\phi^{(0)}},\\ R_{good,1}&=2((A^{(0)})^\tau)^2 \tau^{-\frac 32} (e^{i\tau+i\int (A^{(0)})^\tau d\tau}a_+(y))-2((A^{(0)})^\tau)^2 \tau^{-\frac 32} (e^{-i\tau-i\int (A^{(0)})^\tau d\tau}a_-(y)),\\ R_{good,2}&=-((A^{(0)})^\tau)^2{\phi^{(0)}},\\ R_{good,3}&=i\tau^{-\frac 32}\partial_\tau(A^{(0)})^\tau e^{i\tau+i\int (A^{(0)})^\tau d\tau}a_+(y)-i\tau^{-\frac 32}\partial_\tau(A^{(0)})^\tau e^{-i\tau-i\int (A^{(0)})^\tau d\tau}a_-(y). \end{split}$$ These remainder terms enjoy good decay properties. We have $$|Z^I (R_{tan}+R_{good,1}+R_{good,2}+R_{good,3})|\lesssim \varepsilon\tau^{-\frac 72}(1-|y|^2)^{\alpha}.$$ ## Backward energy estimate We are now ready to derive the backward energy estimate which constructs the solution. This part is very similar to [@CL]. We consider the following system $$\Box (A^{(0)}_{\mu}+v_{\mu})=\Im\left(({\phi^{(0)}}+w)\overline{(\partial_\mu({\phi^{(0)}}+w)+i(A^{(0)}_\mu+v_{\mu})({\phi^{(0)}}+w))}\right),$$ $$-\Box({\phi^{(0)}}+w)+({\phi^{(0)}}+w)=2im^{\mu\nu}(A^{(0)}_\mu+v_{\mu})\partial_\nu ({\phi^{(0)}}+w)-m^{\mu\nu} (A^{(0)}_\mu+v_{\mu}) (A^{(0)}_\nu+v_{\nu}) ({\phi^{(0)}}+w),$$ i.e., let $(A^{(0)}_\mu+v_\mu,{\phi^{(0)}}+w)$ solve the reduced system. In view of the properties of the approximate solution, this gives $$\label{eqofvT} \begin{split} \Box v_{\mu}&=\Box(\chi (F_\mu r^{-1}+F_\mu^{(1)} r^{-2}))+R_\mu\\&+\Im(w\overline{\partial_\mu w}+w\overline{w\cdot iv_{\mu}}) +\Im({\phi^{(0)}}\overline{\partial_\mu w}+w\overline{\partial_\mu{\phi^{(0)}}})-|{\phi^{(0)}}|^2 v_\mu+\Im({\phi^{(0)}}\overline{iA^{(0)}_\mu w})\\ &+\Im(w\overline{iA^{(0)}{\phi^{(0)}}}) +\Im({\phi^{(0)}}\overline{iv_\mu w})-|w|^2A^{(0)}_\mu+\Im(w\overline{iv_\mu {\phi^{(0)}}}). \end{split}$$ $$\label{eqofwT} \begin{split} -\Box w+w&=%2i((\az)^\mu-(A^M)^\mu)\pa_\mu\phiz+ 2im^{\mu\nu} v_{\mu}\partial_\nu w-m^{\mu\nu}v_{\mu} v_{\nu}w+R\\ &+2i ((A^{(0)})^\mu \partial_\mu w)+2i v^\mu \partial_\mu{\phi^{(0)}}-2(A^{(0)})^\mu v_\mu ({\phi^{(0)}}+w)-v^\mu v_\mu {\phi^{(0)}}. \end{split}$$ where $R_\mu=\Im({\phi^{(0)}}\overline{iA^{(0)}_\mu{\phi^{(0)}}})+\Im({\phi^{(0)}}\overline{\partial_\mu{\phi^{(0)}}})-\frac{x_\mu}\tau \tau^{-3}(|a_+(y)|^2-|a_-(y)|^2)$, $R=\Box{\phi^{(0)}}-{\phi^{(0)}}+2i (A^{(0)})^\mu \partial_\mu{\phi^{(0)}}$. These remainders are independent of $v_\mu$ and $w$, and we have good estimates of them. We have already obtained $$|Z^I R|\lesssim \varepsilon\tau^{-\frac 72}(1-|y|^2)^{\alpha}.$$ We also have $$R_\mu=-|{\phi^{(0)}}|^2 A^{(0)}_\mu+\Im\left({\phi^{(0)}}\overline{\partial_\mu\left(\tau^{-\frac 32} e^{i\int (A^{(0)})^\tau d\tau}a_+(y)\right)e^{i\tau}+\partial_\mu\left(\tau^{-\frac 32} e^{-i\int (A^{(0)})^\tau d\tau}a_-(y)\right)e^{-i\tau}}\right).$$ Recall that $\partial_t=\frac t\tau\partial_\tau-t^{-1}y\cdot\nabla_y$, $\partial_i=-\frac{x_i}\tau\partial_\tau+t^{-1}\partial_{y_i}$. One can then express $y$-derivatives using vector fields as [\[laplaciany\]](#laplaciany){reference-type="eqref" reference="laplaciany"} to get $$|Z^I R_\mu|\lesssim %\langle t+r\rangle^{-4} \mathbf{1}_{t\geq r} \varepsilon\tau^{-4} (1-|y|^2)^{2\alpha-\frac 12}.$$ We consider $(v_{\mu,T},w_T)$ solving the equations above, with the whole right hand side multiplied by $\tilde\chi(t/T)$. The cutoff $\tilde\chi(s)$ is non-increasing with $\tilde\chi(s)=1$ when $s\leq 1/4$, and $\tilde\chi(t/T)=0$ when $s\geq 1/2$. We solve the equations backward, and let the data at $t=T$ vanish. We aim to show the limits of $v_{\mu,T}$ and $w_T$ exist as $T\rightarrow \infty$. First we note that the vector field applied to the cutoff essentially gives the same thing. **Lemma 5**. *$|Z^I(\tilde\chi(t/T))|\lesssim \tilde\chi(t/2T)$ in the relevant region. By relevant region we mean the place where the right hand side of [\[eqofvT\]](#eqofvT){reference-type="eqref" reference="eqofvT"} and [\[eqofwT\]](#eqofwT){reference-type="eqref" reference="eqofwT"} are nonzero.* To see this, notice that all terms in the equations which are independent of $v_{\mu,T}$ and $w_T$, are supported in $\{t\geq r/5\}$. Then since $v_{\mu,T}$ and $w_T$ are zero near $t=T$, by finite speed of propagation, we see that $v_{\mu,T}$ and $w_T$ will be supported in $\{t+r\leq 6T\}$. Therefore the relevant region is contained in $\{t+r\leq 6T\}$, and the lemma follows by the expression of vector fields. ### Bootstrap assumptions, $L^2$ estimate of approximate solutions As in [@CL], we work on the foliation with the part when $t-r\geq r^\frac 12$ is a truncated hyperboloid $\widetilde H_\tau$, and is extended to the exterior by a constant time slice denoted by $\Sigma_\tau^e$. One can compute that on $\Sigma_\tau^e$, $t\approx \tau^\frac 43$. One also have $dt\approx \tau^\frac 13 d\tau$ in the exterior. Then we have the energy estimate $$\begin{split} E_{w}(Z^I v_{\mu,T},\tau_1)^\frac 12& \lesssim E_{w}(Z^I v_{\mu,T},\tau_2)^\frac 12+\int_{\tau_1}^{\tau_2} ||Z^I(\Box v_{\mu,T})||_{L^2(\widetilde H_\tau)}+\tau^\frac 13 ||Z^I(\Box v_{\mu,T})||_{L^2(\Sigma_\tau^e)} d\tau,\\ E_{KG}(Z^I w_T,\tau_1)^\frac 12\lesssim &\, E_{KG}(Z^I w_T,\tau_2)^\frac 12+\int_{\tau_1}^{\tau_2} ||Z^I(-\Box+1) w_T||_{L^2(\widetilde H_\tau)}+\tau^\frac 13 ||Z^I(-\Box+1)w_T||_{L^2(\Sigma_\tau^e)} d\tau, \end{split}$$ where the energy reads $$\begin{split} E_w(\psi,\tau)&=\int_{\widetilde H_\rho} |(\tau/t) \partial_t \psi|^2+|t^{-1}\Omega_{0i} \psi|^2 dx+\int_{\Sigma_\rho^e} |\partial\psi|^2 dx,\\ E_{KG}(\psi,\tau)=\int_{\widetilde H_\rho} &|(\tau/t) \partial_t \psi|^2+|t^{-1}\Omega_{0i} \psi|^2+|\psi|^2 dx+\int_{\Sigma_\rho^e} |\partial\psi|^2+|\psi|^2 dx. \end{split}$$ We also define the higher order version $E_{w,k}(\psi,\tau)$ by $E_{KG,k}(\psi,\tau)$ by applying at most $k$ vector fields in $\mathcal Z$ to $\psi$. We have derived estimates of the approximate solutions above. Now we need estimate them at $L^2$ level. Denote $R'_\mu=R_\mu+\Box(\chi({\langle q\rangle}/r)(F_\mu r^{-1}+F_\mu^{(1)} r^{-2}))$. Then $$\begin{split} ||Z^I R'_\mu||_{L^2(\widetilde H_\tau)}&\lesssim ||\varepsilon\langle t+r\rangle^{-4}\langle q\rangle^\gamma||_{L^2(\widetilde H_\tau)}\lesssim \varepsilon\tau^{2\gamma}\left(\int_{\tau/\sqrt 3}^\infty \langle t+r\rangle^{-8-2\gamma} r^2 dr\right)^\frac 12\lesssim \varepsilon\tau^{-\frac 52+\gamma},\\ %这个决定了整个的size ||Z^I R'_\mu||_{L^2(\Sigma^e_\tau)}&\lesssim ||\varepsilon\langle t+r\rangle^{-4}\langle q\rangle^\gamma||_{L^2(\Sigma^e_\tau)}\lesssim \varepsilon\left(\int_{-t\leq t-r\leq 2t^\frac 12} \frac{r^2}{\langle t+r\rangle^8} \langle q\rangle^{2\gamma} d\omega dr\right)^\frac 12\\ &\lesssim \varepsilon\left(\int_{-2t^\frac 12}^t t^{-6} \langle q\rangle^{2\gamma} dq\right)^\frac 12\lesssim \varepsilon t^{-\frac 52+\gamma}\lesssim \varepsilon\tau^{-\frac{10}3+\gamma},\\ %这个如果不加二阶逼近将会是出问题的项，加了之后变成不主要的了 ||Z^I R||_{L^2(\widetilde H_\tau)}&\lesssim ||\varepsilon\langle t+r\rangle^{-\frac 72}||_{L^2(\widetilde H_\tau)}\lesssim \varepsilon\left(\int_0^{2\tau^\frac 43} \langle t+r\rangle^{-7} r^2 dr\right)^\frac 12\lesssim \varepsilon\tau^{-4}\left(\int_0^{2\tau^\frac 43} \langle t+r\rangle^{-3} r^2 dr\right)^\frac 12\\ &\lesssim \varepsilon\tau^{-2}(\ln\tau)^\frac 12,\\ ||Z^I R||_{L^2(\Sigma^e_\tau)}&\lesssim ||\varepsilon\langle t+r\rangle^{-\frac 72}||_{L^2(\Sigma^e_\tau)}\lesssim \varepsilon(t^{-7}\cdot t^\frac 12)^\frac 12\lesssim \varepsilon t^{-\frac {13}4}\lesssim \varepsilon\tau^{-\frac{13}3}. \end{split}$$ We now make the bootstrap assumption that for all $\tau\geq T^*$, $$E_{w,k}(v_T,\tau)^\frac 12 \leq C_b\varepsilon\tau^{-\frac 32+\gamma+k\delta},\quad E_{KG,k}(w_T,\tau)^\frac 12\leq C_b \varepsilon \tau^{-1+\gamma+k\delta}.$$ This clearly holds for all $\tau\geq T$. We will improve the bounds and hence show that one can take $T^*=1$. ### Decay estimates With bootstrap assumptions we can obtain decay estimates of $v_{\mu,T}$ and $w_T$. Using Sobolev embeddings, we have $$|Z^I v_{\mu,T}|\lesssim C_b \varepsilon t^{-\frac 12+(|I|+2)\delta}\tau^{-\frac 32+\gamma+(|I|+2)\delta},\quad |Z^I w_T|\lesssim C_b \varepsilon t^{-\frac 32} \tau^{-1+\gamma+(|I|+2)\delta},\quad \text{when }t-r\leq r^\frac 12,$$ $$|Z^I v_{\mu,T}|\lesssim C_b\varepsilon r^{-\frac 12}\tau^{-\frac 32+\gamma+(|I|+2)\delta},\quad |Z^I w_T|\lesssim C_b \varepsilon r^{-1} \tau^{-1+\gamma+(|I|+2)\delta}, \quad\text{when }t-r\leq r^\frac 12.$$ (See proof of the embeddings in [@CL Lemma 6.6, 6.8].) To derive the $L^2$ bound of $v_{\mu,T}$ themselves, we can use the Hardy-type estimate ([@CL Lemma 6.9]) to get $$||r^{-1}Z^I v_{\mu,T}||_{L^2(\widetilde H_\tau)}+||r^{-1} Z^I v_{\mu,T}||_{L^2(\Sigma^e_s)}\leq 8E_w(v_{\mu,T},\tau)^\frac 12\lesssim C_b \varepsilon \tau^{-\frac 32+\gamma+|I|\delta}.$$ ### Energy estimates We start with the energy estimate of the wave (gauge potential) components. For shorthand notations, we write $v_\mu=v_{\mu,T}$, $w=w_T$. Recall the equation of $v_{\mu,T}$, i.e., [\[eqofvT\]](#eqofvT){reference-type="eqref" reference="eqofvT"} with the right hand side multiplied by $\tilde\chi(t/T)$. There are many terms on the right hand side, but starting from $-|{\phi^{(0)}}|^2 v_\mu$, all terms are lower order terms and can be omitted. Also, using the decay of $a_\pm(y)$, the terms with ${\phi^{(0)}}$ (including $R_\mu$) contribute little in the exterior, and thus can also be omitted. For $|I|\leq k$, the estimate reads $$\begin{gathered} E_{w}(Z^I v_\mu,\tau_1)^\frac 12\lesssim E_{w}(Z^I v_\mu,\tau_2)^\frac 12+\int_{\tau_1}^{\tau_2} (||Z^I R'_\mu||_{L^2(\widetilde H_\tau)} +||Z^I(\Im(w\overline{\partial_\mu w}))||_{L^2(\widetilde H_\tau)}\\ +||Z^I(\Im({\phi^{(0)}}\partial_\mu w+w\partial_\mu{\phi^{(0)}}))||_{L^2(\widetilde H_\tau)} +\int_{\tau_1}^{\tau_2} \tau^{\frac 13} (||Z^I R'_\mu||_{L^2(\Sigma^e_\tau)}+||Z^I(\Im(w\overline{\partial_\mu w}))||_{L^2(\Sigma^e_\tau)} d\tau\\ \lesssim \varepsilon\tau_1^{-\frac 32+\gamma+k\delta}+\int_{\tau_1}^{\tau_2} \varepsilon \tau^{-\frac 32} E_{KG,k}(w,\tau)^\frac 12+\varepsilon \tau^{-\frac 32}\ln\tau\, E_{KG,k-1}(w,\tau)^{\frac 12}+C_b^2 \varepsilon^2 \tau_1^{-2+\frac 34\gamma+(k+2)\delta} d\tau\\ +\int_{\tau_1}^{\tau_2} C_b \varepsilon\tau^{-2+\gamma}E_{KG,k}(w,\tau)^\frac 12 d\tau \lesssim (\varepsilon+C_b^2\varepsilon^2)\tau_1^{-\frac 32+\gamma+k\delta}.\end{gathered}$$ Now we turn to the Klein-Gordon equation. Again, one can ignore terms involving ${\phi^{(0)}}$ (including $R$) in the exterior. We have $$\begin{gathered} E_{KG}(\tau_1,Z^I w)^\frac 12\lesssim E_{KG}(\tau_2,Z^I w)^\frac 12+\int_{\tau_1}^{\tau_2} ||Z^I R||_{L^2(\widetilde H_\tau)}+||Z^I((A^{(0)})^\mu \partial_\mu w)||_{L^2(\widetilde H_\tau)}\\ +||Z^I(v\cdot \partial w)||_{L^2(\widetilde H_\tau)}+||Z^I(v^\mu \partial_\mu{\phi^{(0)}})||_{L^2(\widetilde H_\tau)} +\tau^\frac 13\left(||Z^I((A^{(0)})^\mu \partial_\mu w)||_{L^2(\Sigma^e_\tau)}+||Z^I(v\cdot \partial w)||_{L^2(\Sigma^e_\tau)}\right) d\tau\\ \lesssim \int_{\tau_1}^{\tau_2} \varepsilon\tau^{-2}\ln\tau+\varepsilon\tau^{-1}E_{KG,k}(w,\tau)^\frac 12+\varepsilon\tau^{-\frac 32}E_{w,k}(v_\mu,\tau)^\frac 12\\ +\varepsilon\tau^{-\frac 32}\ln\tau E_{w,k-1}(v_\mu,\tau)^\frac 12+C_b^2\varepsilon^2\tau^\frac 13 \tau^{-\frac 32+\gamma+k\delta} \tau^{-1+\gamma+2\delta}d\tau \\ \lesssim \varepsilon\tau_1^{-1}\ln\tau_1+C_b\varepsilon^2 \tau_1^{-1+\gamma+k\delta}+C_b^2\varepsilon^2\tau_1^{-\frac 76+2\gamma+(k+2)\delta},\end{gathered}$$ where we used $|Z^I((A^{(0)})^\mu \partial_\mu w)|\lesssim \tau^{-1} \frac \tau t |\partial Z^J w|$ on $\widetilde H_\tau$, and $\tau^\frac 13|Z^I((A^{(0)})^\mu \partial_\mu)|\lesssim \tau^\frac 13 t^{-1}|\partial Z^J w|\lesssim \tau^{-1}|\partial Z^J w|$ on $\Sigma^e_\tau$. When more vector fields fall on $v$, we have used the bounds from Hardy estimate, for example $$\tau^\frac 13 ||Z^I v\cdot\partial w||_{L^2(\Sigma^e_\tau)}\lesssim \tau^\frac 13 ||r^{-1} Z^{I_1}v_\mu||_{L^2(\Sigma^e_\tau)} ||rw||_{L^\infty(\Sigma^e_\tau)}\lesssim C_b^2\varepsilon^2\tau^{\frac 13} \tau^{-\frac 32+\gamma+k\delta} \tau^{-1+\gamma+2\delta}.$$ Therefore for given $\gamma<1/6$, we improve the boostrap bounds, so the bootstrap bounds on $v_{\mu,T}$ and $w_T$ hold for all $\tau\geq T^*$. ## Taking the limit We want to show that the limit as $T\rightarrow \infty$ exists. Let $T_2>T_1$. We denote $v_{\mu,i}:=v_{\mu,T_i}$, $w_i:=w_{T_i}$. Consider the difference $\hat v_\mu:=v_{\mu,2}-v_{\mu,1}$ and $\hat w:=w_2-w_1$ ($i=1,2$). We have $$-\Box \hat v_\mu=\tilde\chi(t/T_2)N_{\mu}(v_{\mu,2},w_2) -\tilde\chi(t/T_1)N_\mu(v_{\mu,1},w_1),$$ $$-\Box \hat w+\hat w=\tilde\chi(t/T_2)N(v_{\mu,2},w_2) -\tilde\chi(t/T_1)N(v_{\mu,1},w_1).$$ Here $N_\mu$ and $N$ are the terms on the right hand side of [\[eqofvT\]](#eqofvT){reference-type="eqref" reference="eqofvT"} and [\[eqofwT\]](#eqofwT){reference-type="eqref" reference="eqofwT"} respectively. Then we get $$-\Box {\hat v}_\mu=(\tilde\chi(t/T_2)-\tilde\chi(t/T_1))N_\mu(v_{\mu,1},w_1)\\ +\tilde\chi(t/T_2)(N_\mu(v_{\mu,2},w_2)-N_\mu(v_{\mu,1},w_1)),$$ $$-\Box {\hat w}+\hat w=(\tilde\chi(t/T_2)-\tilde\chi(t/T_1))N(v_{\mu,1},w_1)\\ +\tilde\chi(t/T_2)(N(v_{\mu,2},w_2)-N(v_{\mu,1},w_1)).$$ We then consider the energy estimate between $\Sigma_\tau$ and $\Sigma_{T_1}$ with $\tau<T_1$. Note that $v_{\mu,1}$ and $w_1$ vanish near $\Sigma_{T_1}$. We also have established the bounds $$E_{KG,k}(w_1,\tau)^\frac 12+E_{KG,k}(w_2,\tau)^\frac 12\lesssim \varepsilon \tau^{-1+\gamma+k\delta},\quad E_{w,k}(v_{\mu,1},\tau)^\frac 12+E_{w,k}(v_{\mu,2},\tau)^\frac 12\lesssim \varepsilon\tau^{-\frac 32+\gamma+k\delta}%还有decay$$ for all $\tau\leq T_2$, as well as the corresponding decay estimates. Since $\tilde\chi(t/T_2)-\tilde\chi(t/T_1)$ is nonzero only when $\tau\geq\frac{1}{16}T_1^\frac 34$, the integration in $\tau$ in the energy estimate starts at this value. Then the integral of the first term in the energy estimate for both equations decays is $T_1$ since the integrands are essentially the same as the estimates above. For the second term, because of the difference form, all terms can be written as a factor times $\hat v_\mu$ or $\hat w$. Using the bounds we established above, we have the following estimates, by noticing that the structures are the same as the estimate above ($|I|\leq N-2$): $$\begin{gathered} \int_\tau^{T_1} ||Z^I(N_\mu(v_{\mu,2},w_2)-N_\mu(v_{\mu,1},w_1))||_{L^2(\widetilde H_s)}+s^\frac 13 ||Z^I(N_\mu(v_{\mu,2},w_2)-N_\mu(v_{\mu,1},w_1))||_{L^2(\Sigma_s^e)} ds\\ \lesssim\int_\tau^{T_1} ||Z^I(\partial_\mu{\phi^{(0)}},{\phi^{(0)}})(\hat w,\partial\hat w)||_{L^2(\widetilde H_s)}+s^\frac 13||Z^I(\partial_\mu{\phi^{(0)}},{\phi^{(0)}})(\hat w,\partial\hat w)||_{L^2(\Sigma_s^e)}\\ +||Z^I((\partial_\mu w_i) (\hat w,\partial_\mu \hat w))||_{L^2(\widetilde H_s)} +s^\frac 13 ||Z^I((\partial_\mu w_i) (\hat w,\partial_\mu \hat w))||_{L^2(\Sigma_s^e)}+(\varepsilon s^{-1} E_{w,k}(s,\hat v_\mu)^\frac 12) ds\\ \lesssim \int_\tau^{T_1} \varepsilon (s^{-\frac 32}+s^{-2+\alpha+\delta}) E_{KG,k}(s,\hat w)^\frac 12+\varepsilon s^{-\frac 32}(\ln s)\, E_{KG,k-1}(s,\hat\omega)^\frac 12+(\varepsilon s^{-1} E_{w,k}(s,\hat v_\mu)^\frac 12)\, ds,\end{gathered}$$ and $$\begin{gathered} \int_\tau^{T_1} ||Z^I(N(v_{\mu,2},w_2)-N(v_{\mu,1},w_1))||_{L^2(\widetilde H_s)}+s^\frac 13 ||Z^I(N(v_{\mu,2},w_2)-N(v_{\mu,1},w_1))||_{L^2(\Sigma_s^e)}\\ \lesssim\int_\tau^{T_1} ||Z^I(v_{\mu,i}(\hat w,\partial_\mu \hat w))||_{L^2(\widetilde H_s)}+||Z^I((\partial_\mu w_i,w_i)\hat v_{\mu}||_{L^2(\widetilde H_s)}\\ +s^\frac 13(||Z^I(v_{\mu,i}(\hat w,\partial_\mu \hat w))||_{L^2(\Sigma_\tau^e)}+||Z^I((\partial_\mu w_i,w_i)\hat v_{\mu}||_{L^2(\Sigma_\tau^e)})\\ +||Z^I(A^{(0)}\cdot \partial\hat w)||_{L^2(\widetilde H_s)}+s^\frac 13 ||Z^I(A^{(0)}\cdot \partial\hat w)||_{L^2(\Sigma^e_s)} +||Z^I(\partial{\phi^{(0)}}\cdot\hat v_\mu)||_{L^2(\widetilde H_s)}+s^\frac 13 ||Z^I(\partial{\phi^{(0)}}\cdot\hat v_\mu)||_{L^2(\Sigma^e_s)} ds\\ \lesssim \int_\tau^{T_1} \varepsilon s^{-1}E_{KG,k}(s,\hat w)^\frac 12+\varepsilon (s^{-\frac 12}+s^{-\frac 23+\gamma+2\delta}) E_{w,k}(s,\hat v_\mu)^\frac 12+\varepsilon s^{-\frac 12}(\ln s) E_{w,k-1}(s,\hat v_\mu)^\frac 12\, ds.\end{gathered}$$ These estimates are similar to the energy estimate we did above. Now define $E_k(\tau)^\frac 12=E_{w,k}(\hat v,\tau)^\frac 12+\tau^{\frac 12} E_{KG;k}(\hat v,\tau)^\frac 12$. We have for $\tau\leq T_1$ that $$E_k(\tau)^\frac 12 \lesssim \varepsilon (T_1)^{-\frac 12+\gamma+k\delta}+\varepsilon((T_1)^\frac 34)^{-\frac 12+\gamma+k\delta} +\int_\tau^{T_1} \varepsilon s^{-1} E_k(s)^\frac 12+\varepsilon s^{-1}(\ln s)^k E_{k-1}(s)^\frac 12 ds.$$ When $k=0$, the $E_{k-1}$ term does not appear, so using Grönwall's inequality we have $$E_0(\tau)^\frac 12 \lesssim \varepsilon (T_1/\tau)^{C\varepsilon} (T_1)^{-\frac 38+\frac 34\gamma}\rightarrow 0\quad\text{as }T_1\rightarrow \infty.$$ Then it is straightforward to show by induction that $$E_k(\tau)^\frac 12\lesssim \varepsilon (T_1/\tau)^{C\varepsilon} (T_1)^{-\frac 38+\frac 34\gamma+\frac 34 k\delta},\quad k\leq N-2.$$ Then by Sobolev embeddings, we have for $|I|\leq N-4$ that $$\sup_{\tau(t,x)\leq T_1}|Z^I \hat w|+|Z^I \hat v|\lesssim \varepsilon (T_1)^{-\frac 38+\frac 34\gamma+\frac 34(k+2)\delta+C\varepsilon}$$ which converges uniformly to zero as $T_2>T_1\rightarrow \infty$. This shows the existence of the limit $w=\lim_{T\rightarrow\infty} w_T$ and $v_\mu=\lim_{T\rightarrow\infty} v_{\mu,T}$, and that $(A^{(0)}_\mu+v_{\mu},{\phi^{(0)}}+w)$ gives a solution of the reduced mMKG system, with the asymptotic behavior being exactly the one given by the scattering data. ## The original system {#sectionoriginalsystem} In this subsection, we show that the solution we obtained above also satisfies the Lorenz gauge condition, hence solves the original mMKG system. Since $A^{(0)}_\mu+v_{\mu,T}$ and ${\phi^{(0)}}+w_T$ solve the reduced system for each $T$, using [\[eqoflambda\]](#eqoflambda){reference-type="eqref" reference="eqoflambda"}, we have that $\lambda_T=\partial^\mu(A^{(0)}_\mu+v_{\mu,T})$ satisfies the equation $$\Box\lambda_T=|{\phi^{(0)}}+w_T|^2\lambda_T.$$ Recall we have $Z^I\lambda_T=Z^I\partial^\mu (A^{(0)}_\mu+v_{\mu,T})=O(\varepsilon\langle t+r\rangle^{-2})+Z^I\partial^\mu v_{\mu,T}$. One can then do the energy estimate $$E_{w}(Z^I\lambda_T,\tau)^\frac 12\lesssim E_w(Z^I\lambda_T,T)^\frac 12+\int_\tau^T ||Z^I(|{\phi^{(0)}}+w_T|^2\lambda_T)||_{L^2(\widetilde H_\tau)} +s^\frac 13||Z^I(|{\phi^{(0)}}+w_T|^2\lambda_T)||_{L^2(\Sigma^e_s)}ds.$$ We have shown that $$|\partial Z^I \partial^\mu A_\mu^{(0)}|\lesssim \varepsilon\langle t+r\rangle^{-2},$$ and we know that $v_{\mu,T}$ vanish near $\tau=T$. Therefore we have $$\begin{split} E_{w} (Z^I\lambda_T,T)^\frac 12 &\lesssim (\int_{T\leq t,\, 0\leq r\leq 2T^\frac 43} \varepsilon^2(t+r)^{-4} dx)^\frac 12+(\int_{t=T^\frac 43,r\geq t/2} \varepsilon^2(t+r)^{-4} dx)^\frac 12+E_w(Z^I\partial v_{\mu,T},T)^\frac 12\\ &\lesssim \varepsilon T^{-\frac 12}, \end{split}$$ so for all $|I|\leq N-2$, the integral can be estimated as $$\begin{split} \int_\tau^T ||\, Z^I&(|{\phi^{(0)}}+w_T|^2\lambda_T)||_{L^2(\widetilde H_s)}+s^\frac 13||\, Z^I(|{\phi^{(0)}}+w_T|^2\lambda_T)||_{L^2(\Sigma^e_s)}d s\\ &\lesssim \sum_{|I_1|\leq |I|}\int_\tau^T ||\varepsilon t^{-3}Z^{I_1}\lambda_T||_{L^2(\widetilde H_s)}+\varepsilon\tau^\frac 13 \tau^{-2+2\gamma}r^{-1}||r^{-1} Z^{I_1}\lambda_T||_{L^2(\Sigma^e_s)} ds\\ &\lesssim \int_\tau^T \varepsilon t^{-2}||r^{-1}Z^{I_1}\lambda_T||_{L^2(\widetilde H_s)}+\varepsilon \tau^{-2}||r^{-1} Z^{I_1} \lambda_T||_{L^2(\Sigma^e_s)} ds\\ &\lesssim \sum_{|I_1|\leq |I|}\int_\tau^T \varepsilon\tau^{-2} E_w(Z^{I_1} \lambda_T,\tau)^\frac 12 ds. \end{split}$$ Then using Grönwall's inequality for the estimate, we get $$E_w(Z^I\lambda_T,\tau)^\frac 12\lesssim \varepsilon T^{-\frac 12}.$$ Therefore, $E_w(Z^I\lambda_T,\tau)^\frac 12$ is bounded by $T^{-1}$ for all $\tau\leq T$. We also have that $\lambda_T$ vanishes as $r\rightarrow 0$ at each time slice. Then, letting $T\rightarrow \infty$, we see that the Lorenz gauge condition is satisfied everywhere for the solution we get. ## The case when the radiation fields decay slower We have made the assumption that $F_\mu(q,\omega)$ is decaying at the rate of $\langle q\rangle^{-1+\gamma}$ where $\gamma<1/6$. In this section we discuss the case when they decay even less. They still have to, of course, satisfy the condition [\[admissiblecondition\]](#admissiblecondition){reference-type="eqref" reference="admissiblecondition"}. One can see from above that, in some sense, this part of radiation fields, compared with the parts governed by the Klein-Gordon source and the charge, corresponds to the part of the wave components satisfying the linear wave equation. In fact, we can first find linear solutions $A_\mu^{Linear}$ that scatter to the radiation fields $F_\mu(q,\omega)$. This is a linear scattering problem, and [@LindbladSchlue1] it was shown that for all $\gamma<1/2$ one can find linear solutions $A_\mu^{Linear}$ satisfying $$|(Z,S)^I \left(A_\mu^{Linear}-\chi({\langle q\rangle}/r)(F_\mu(q,\omega)r^{-1}+F^{(1)}_\mu(q,\omega))\right)|\lesssim \varepsilon\langle t+r\rangle^{-1} t^{-\gamma}.$$ Then we can modify our definition of the approximate solution $A^{(0)}$: $$A^{(0)}_\mu=A^M_\mu+A^{Linear}_\mu+\mathbf{q}_\infty\chi_{ex}'(q)r^{-1}\delta_{0\mu}.$$ Since $-\Box A_\mu^{Linear}=0$, the estimate of perturbations are in fact easier. However, we need to be more careful about the Lorenz gauge condition. The above condition implies $$|Z^I\partial^\mu \left(A_\mu^{Linear}-\chi({\langle q\rangle}/r)(F_\mu(q,\omega)r^{-1}+F^{(1)}_\mu(q,\omega))\right)|\lesssim \varepsilon \langle t+r\rangle^{-1}t^{-\gamma}(1+|q|)^{-1}\lesssim \varepsilon\langle t+r\rangle^{-1-\gamma},$$ which is worse than $O(\varepsilon\langle t+r\rangle^{-2})$, so the new $\partial^\mu A^{(0)}_\mu$ is only a $O(\varepsilon\langle t+r\rangle^{-1-\gamma})$ term. But if we review the proof in Section [5.6](#sectionoriginalsystem){reference-type="ref" reference="sectionoriginalsystem"} above, we see that this decay is still enough for $\gamma<1/2$, and the bound of $E_w(Z^I\lambda_T,T)^\frac 12$ there becomes $\varepsilon T^{-\gamma+\frac 12}$, still allowing us to prove the result. [^1]: Throughout this paper, we raise and lower indices using the standard Minkowski metric $m_{\mu\nu}=\mathrm{diag}\{-1,1,1,1\}$, and we use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean. [^2]: *We use the notation $a\lesssim b$ to denote that there exists a constant $C>0$ such that $a\leq Cb$.* [^3]: *Of course, there is a smoothness problem for the solution when $t=r$, but it does not matter in view of the strong Huygens' principle.* [^4]: We remark that for the wave-Klein-Gordon model, the pair $a_\pm(y)$ are in fact conjugate to each other. Here this is nonzero because $\phi$ is complex. [^5]: We use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean. [^6]: In some literatures, the notation $D_X D_Y \phi$ could mean the $X,Y$-component of the tensor $D_\mu D_\nu\phi$, but here it represents $D_X(Y^\mu D_\mu \phi)$. [^7]: One can pick any positive number $\varepsilon>0$ instead of $2$ here, and then the part where $q<-\varepsilon$ will be exactly $U_\mu(y)/\tau$. This is simply a consequence of strong Huygens' principle.

arxiv_math

--- abstract: | Given a configuration of pebbles on the vertices of a graph $G$, a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph $G$ is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of $G$ through some sequence of pebbling moves. Through constructing a non-tree weight function for $Q_4$, we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the $n$-dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions. author: - "Marshall Yang Carl Yerger[^1] Runtian Zhou [^2]" title: Lollipop and Cubic Weight Functions for Graph Pebbling --- # Introduction Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. The game is composed of a sequence of pebbling moves, where each pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The graph pebbling model was first introduced by Chung $\cite{Chung}$. The graph pebbling problem originated as a proof technique to prove a zero-sum theorem, introduced by Erdős et al. $\cite{ZivTheoremIT}$. Proof details can be found in $\cite{Chung}$. Pebbling has been extensively studied in the last $30$ years and there have been many interesting results $\cite{page}$. Calculating pebbling numbers is difficult: Clark and Milans $\cite{milans_complexity_2006}$ proved that determining whether the pebbling number is at most $k$ is $\Pi_{2}^{p}$-complete. Hurlbert $\cite{hurlbert2011linear}$ introduced a linear programming technique based on weight functions on trees, in the hope of more efficiently computing bounds on pebbling numbers. Weight functions on trees might not always be able to give tight bounds for the pebbling numbers of particular graphs. For instance, Chung proved in $\cite{Chung}$ that $\pi(Q_3)=8$, but Hurlbert $\cite{hurlbert2011linear}$ claimed that the best upper bound of $\pi(Q_3)$ that could be obtained using only weight functions on trees is $9$. Later, Cranston et al. $\cite{cranston2015modified}$ proved $\pi(Q_3)\leq 8$ but had to employ non-tree weight functions. In this paper, we construct a set of new non-tree weight functions and use them to prove $\pi(Q_4) = 16$. Recently, Flocco et al. $\cite{flocco}$ used mixed-integer linear programming (MILP) to offer certificates of pebbling bounds based on weight functions of an ideal set of subtrees in a graph. Flocco's work employs computational techniques to obtain weight functions using a new open-source computational toolkit.[^3] The authors of $\cite{hurlbert2011linear}$ and $\cite{flocco}$ only worked on weight functions of trees. With additional programming, we anticipate that our non-tree weight functions will strengthen computational results that only employ weight functions of trees. Specifically, each non-tree weight function acts as an additional set of stronger constraints that can now be applied to any graph that has this structure as a subgraph when computing bounds for its pebbling number. Therefore, using our non-tree weight functions, the techniques such as MILP may be used to provide certificates of better upper bounds for pebbling numbers. As seen in $\cite{cranston2015modified}$, incorporating more involved weight functions can significantly improve known pebbling bounds, and we see the potential for these improvements as a major consequence of this work. ## Notation and Terminology We denote the vertex and the edge sets of a graph $G$ by $V(G)$ and $E(G)$ respectively. The **distance** between two distinct vertices $u,v\in V(G)$, denoted by $d(u,v)$, is the length of the shortest path between $u$ and $v$. The **diameter** of $G$ is defined as $\max_{u,v\in V(G)}d(u,v)$. A (pebble) **configuration** $p$ on $G$ is a function $p: V(G) \rightarrow \mathbb N \cup \{0\}$, where $p(v)$ is the number of pebbles on $v$ for each $v\in V(G)$. A **weight function** on graph $G$ is a map $w: V(G)\rightarrow \mathbb{R}^+\cup\{0\}$. A **pebbling move** from vertex $u$ to an adjacent vertex $v$ consumes $2$ pebbles on $u$ and places $1$ pebble on $v$. Let $1_G$ be the configuration on $G$ where each vertex has exactly $1$ pebble. In all figures of this paper that describe weight functions, each vertex is presented by a small circle, the weight of this vertex is presented within the circle, and the label of the vertex is presented next to the circle. Let $G'$ be a subgraph of $G$ and $p$ be a configuration on $G$. Let $w$ be a weight function on $G$. Then, the size of $p$, denoted by $|p|$, the number of pebbles on $G'$, denoted by $p(G')$, the weight of $p$, denoted by $w(p)$, and the restriction of $w$ on $G'$, denoted by $w_{G'}$, are defined as $$\begin{aligned} |p|&=\sum_{v\in V(G)}p(v)\\ p(G')&=\sum_{v\in V(G')}p(v)\\ w(p)&=\sum_{v\in V(G)}p(v)\cdot w(v)\\ w_{G'}&\equiv w|_G.\end{aligned}$$ If $G$ is "rooted" at some vertex $r\in V(G)$ and there exists a sequence of pebbling moves started from $p$ that could put $1$ pebble on $r$, we say $p$ is **$r$-solvable**. If no such sequence exists, we say $p$ is $r$-unsolvable. A weight function $w$ for a graph $G$ rooted at $r$ is **valid** if and only if $r$ is the only vertex with weight $0$ and every $r$-unsolvable configuration $p$ satisfies $w(p)\leq w(1_G)$. The pebbling number of a graph $G$ rooted at $r$, denoted by $\pi(G,r)$, is the smallest integer such that any configuration with at least $\pi(G,r)$ pebbles could reach $r$ through some sequence of pebbling moves. The pebbling number of $G$, denoted by $\pi(G)$, is defined as $$\pi(G)=\max_{r\in V(G)}\pi(G,r).$$\ ## Background In this subsection we introduce some helpful lemmas that will be used in later proofs. **Lemma 1**. *Let $w$ be a valid weight function on a graph $G$ rooted at $r$. Let $m=\min_{v\in V(G)-\{r\}}w(v)$. Then, $$\pi(G,r)\leq \left \lfloor \frac{w(1_G)}{m} \right \rfloor +1.$$* *Proof.* Given any configuration $p$ with $|p|= \left \lfloor \frac{w(1_G)}{m} \right \rfloor +1$ and $p(r)=0$, we have $$w(p)\geq m\cdot \left (\left \lfloor \frac{w(1_G)}{m} \right \rfloor +1 \right )>w(1_G).$$ Thus by validity of $w$, configuration $p$ is $r$-solvable. Because $p$ is arbitrary, $\pi(G,r)\leq |p|=\left \lfloor \frac{w(1_G)}{m} \right \rfloor +1$. ◻ **Lemma 2**. *For any $k\geq 1$, let $P_{k+1}$ be the path $(v_0,v_1,v_2,\ldots, v_{k-1},r)$ rooted at $r$. Let $w$ be a weight function on $P_{k+1}$ such that $w(r)=0$ and $w(v_i)=2^i$ for each $i$ from $0$ to $k-1$. Let $p$ be an arbitrary configuration on $P_{k+1}$. Then, $p$ is $r$-solvable if and only if $w(p)\geq 2^k$.* *Proof.* For the first direction, suppose $S$ is a sequence of pebbling moves that solves $p$. Thus the last pebbling move on $S$ removes $2$ pebbles on $v_{k-1}$ to place $1$ pebble on $r$. Let $p'$ be the configuration before this pebbling move is performed. Observe that, under $w$, any pebbling move on $P_{k+1}$ either does not change the weight of a configuration or decreases it. Hence $$w(p)\geq w(p')\geq w(v_{k-1})\cdot p'(v_{k-1})\geq 2^k.$$ For the other direction, suppose $p$ is $r$-unsolvable. For $i$ from $0$ to $k-2$, perform the following operation in sequence: move pebbles from $v_{i}$ to $v_{i+1}$ until at most $1$ pebble is left at $v_{i}$. Finally we have a new configuration, denoted by $p'$. Clearly $w(p)=w(p')$. Notice that $p$ being $r$-unsolvable implies $p'(v_{k-1})\leq 1$ and $p'(r)=0$. Since every vertex in $p'$ contains at most $1$ pebble, we conclude that $$w(p)=w(p')\leq \sum_{i=0}^{k-1}2^i=w(1_{P_{k+1}}).$$ ◻ Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"} gives an idea on how weight function techniques can be used to calculate pebbling numbers. The next lemma is a generalized form of Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"}. **Lemma 3**. *Let $w$ be a weight function on a tree $T$ rooted at $r$. For each $v\in V(T)-\{r\}$, let $v^+$ be the parent of $v$, the neighbor of $v$ that is closer to $r$. If $w(r)=0$ and $w(v^+)\geq 2w(v)$ for every $v$ not adjacent to $r$, then $w$ is valid.* Lemma [Lemma 3](#l3){reference-type="ref" reference="l3"} presents the first set of weight functions described in the pebbling literature. Hurlbert $\cite{hurlbert2011linear}$ devised these weight functions and Cranston et al. $\cite{cranston2015modified}$ made improvements upon them. **Lemma 4** (Covering Lemma). *For a graph $G$ rooted at $r$, let weight function $w$ on $G$ be a conic combination of valid weight functions for connected subgraphs of $G$ rooted at $r$. Then $w$ is valid.* *Proof.* Let $w_1,w_2,\ldots,w_m$ be valid weight functions on $G_1,G_2,\ldots,G_m$ respectively, where $G_1,G_2,\ldots,G_m$ are all connected subgraphs of $G$ rooted at $r$. For each $i$ from $1$ to $m$ and for each $v\in V(G)-V(G_i)$, define $w_i(v)=0$. Let $a_1,a_2,\ldots,a_m$ be non-negative real numbers such that for each $v\in V(G)$, $w(v)=\sum_{i=1}^{m}a_iw_i(v)$. Let $p$ be an arbitrary $r$-unsolvable configuration. Then, the validity of $w_1,w_2,\ldots,w_m$ implies that: $$\begin{aligned} w(p)=\sum_{i=1}^{m}a_iw_i(p)\leq \sum_{i=1}^{m}a_iw_i(1_G)=w(1_G).\end{aligned}$$ Since $p$ is arbitrary, $w$ is a valid weight function on $G$. ◻ When studying large graphs, it is not easy to construct valid weight functions directly. A common method is studying subgraphs of the original graph, constructing valid weight functions on those smaller subgraphs, and finally applying Lemma [Lemma 4](#l4){reference-type="ref" reference="l4"} to get a valid weight function on the original graph.\ Here is an example of applications of Lemma [Lemma 1](#l1){reference-type="ref" reference="l1"},[Lemma 2](#l2){reference-type="ref" reference="l2"},[Lemma 3](#l3){reference-type="ref" reference="l3"},[Lemma 4](#l4){reference-type="ref" reference="l4"}. **Theorem 5**. *For any $k\in \mathbb N$, $\pi(C_{2k+1})=2\left \lfloor \frac{2^{k+1}}{3}\right \rfloor +1$.* This theorem is proved in $\cite{opg}$. Here we show a shorter proof using weight functions. *Proof.* Notice that the weight function in Figure [\[f3\]](#f3){reference-type="ref" reference="f3"} is the addition of $3$ copies of the weight function in Figure [\[f2\]](#f2){reference-type="ref" reference="f2"}. In particular, the $3$ induced subgraphs of all vertices with $1$ on the first, second, or third index are the graphs that these $3$ copies of the weight function in Figure [\[f2\]](#f2){reference-type="ref" reference="f2"} lie on. Thus by Lemma [Lemma 4](#l4){reference-type="ref" reference="l4"}, the weight function in Figure [\[f3\]](#f3){reference-type="ref" reference="f3"} is valid. ◻ **Lemma 6**. *The weight function in Figure $4$ is valid.* *Proof.* We begin with terminology helpful for our proof. Let the graph in Figure [\[f4\]](#f4){reference-type="ref" reference="f4"} be $G$. Let the subgraph induced by $\{v_1,v_2,v_3,m_1,m_2,m_3,k\}$ be $G'$. Let the weight functions in Figure [\[f4\]](#f4){reference-type="ref" reference="f4"} and Figure [\[f3\]](#f3){reference-type="ref" reference="f3"} be $w$ and $w'$, respectively. Let $M=\{m_1,m_2,m_3\}$ and $V=\{v_1,v_2,v_3\}$. Given any configuration $p$, since $G$ is symmetric, we can assume $p(m_1)\geq p(m_2)\geq p(m_3)$. Let $p$ be an arbitrary configuration such that $w(p)>w(1_G)=15$. By definition of validity in Section 1.1, it suffices to prove that $p$ is $r$-solvable. For purpose of contradiction, assume $p$ is $r$-unsolvable. Clearly $p(u)\leq 1$.\ **Case $1$**: When $p(u)=1$. If $p(u)=1$, then $w(p)-w(u)>15-4=11$. By Proposition [\[p1\]](#p1){reference-type="ref" reference="p1"} we can use pebbles on $\{v_1,v_2,v_3,m_1,m_2,m_3,k\}$ to move another pebble to $u$, then from $u$ a pebble can be moved to $r$. This shows $p$ is $r$-solvable. In rest cases, suppose $p(u)=0$.\ **Case $2$**: When $p(v_i)\geq 2$ for some $v_i\in V$. Move $1$ pebble from $v_i$ to $u$ to get a new configuration $p'$. Clearly $w(p')=w(p)>15$. By Case $1$, $p'$ is $r$-solvable. Thus $p$ is also $r$-solvable. In rest cases, suppose $p(u)=0$ and $p(v_i)\leq 1$ for each $v_i\in V$.\ **Case $3$**: When $p(v_i)=1$ for each $v_i\in V$. Observe that if we can move $2$ pebbles from $M\cup \{k\}$ to $V$, then $p$ is immediately $r$-solvable. Suppose we can move at most $1$ pebble from $M$ to $V$. Then, $p(M)\leq 1\cdot2+3\cdot 1=5$. Thus $$p(k)\geq {w(p)-5\cdot \frac{4}{3}-6\over 1}>2$$ so we can move $1$ pebble from $k$ to any vertex of $M$. But $p(k)>2$ implies $p(M)\leq 3$, otherwise we could move $2$ pebbles from $M\cup\{k\}$ to $V$. Following similar logic, $p(M)\leq 3$ implies $p(k)\geq 6$, $p(k)\geq 6$ implies $p(M)= 0$, $p(M)= 0$ implies $p(k)\geq 8$. Finally, $p(k)\geq 8$ implies we can move two pebbles from $k$ to $V$, which shows $p$ is $r$-solvable. In rest cases, assume $p(v_i)=0$ for some $v_i\in V$.\ **Case $4$**: When $w(p)\geq 16$. **Subcase $4.1$**: When $p(v_1)=p(v_2)=1$ and $p(v_3)=0$. By assumption, $p(m_i)\leq 1$ for each $m_i\in M$. Thus $p(k)\geq 16-4-3\cdot \frac{4}{3}=8$. From $k$ we can move $1$ pebble to each of $v_1$ and $v_2$ and clearly $p$ is $r$-solvable.\ **Subcase $4.2$**: When $p(v_i)=1$ and one child of $v_i$ has at least $2$ pebbles for some $v_i\in V$.  Clearly, we can remove these $3$ pebbles to place a pebble on $u$ to form a new configuration $p'$. Notice that $w_{G'}(p')=w(p)-2-2\cdot\frac{4}{3}>11$. By Proposition [\[p1\]](#p1){reference-type="ref" reference="p1"}, we can move another pebble from $G'$ to $u$, so $p$ is $r$-solvable. In the rest $2$ subcases, assume that for each $v_i\in V$, if $p(v_i)=1$, then any of its children has at most $1$ pebble.\ **Subcase $4.3$**: When $p(v_i)=0$ for each $v_i\in V$. Because $p$ is $r$-unsolvable, the following two implications are clear: $$\begin{aligned} p(m_1)\geq 6&\Rightarrow p(m_2)\leq 1\textnormal{ and } p(m_3)\leq 1\\ p(m_1)\geq 4&\Rightarrow p(m_2)+p(m_3)\leq 4.\end{aligned}$$ This two implications further imply that $M$ has at most $9$ pebbles. Since $w(k)=1$, we know $p(k)\geq \left \lceil 16-9\cdot\frac{4}{3}\right \rceil =4$ and we can move $2$ pebbles from $k$ to any vertices of $M$. If $p(m_1)+p(m_2)=6$, moving $2$ pebbles from $k$ to $M$ can make $p(m_2)$ and $p(m_1)$ both even and they can move $\frac{6+2}{2}=4$ pebbles to their common parent, and $p$ is clearly $r$-solvable. Hence $p(m_1)+p(m_2)<6$. Since $p(m_1)\geq p(m_2)\geq (m_3)$, we also know $p(m_3)\leq 2$. Notice that the only possiblity for $p(m_1)+p(m_2)=5$ and $p(m_3)=2$ is $p(m_1)=3,p(m_2)=p(m_3)=2$. But since $p(k)=4$, $p$ is $r$-solvable. Hence $p(M)\leq 6$ and $p(k)\geq \left \lceil 16-6\cdot \frac{4}{3}\right \rceil=8$. If $p(M)\geq 4$, moving $4$ pebbles from $k$ to $M$ could make each $p(m_i)$ even and $p(M)=8$ forces $p$ to be $r$-solvable. Thus $p(M)\leq 3$. Following similar logic, $p(M)\leq 3$ implies $p(k)\geq 12$, $p(k)\geq 12$ implies $p(M)\leq 1$, $p(M)\leq 1$ implies $p(k)\geq 14$, $p(k)\geq 14$ implies $p(M)=0$. Finally, $p(M)=0$ implies $p(k)\geq 16$, forcing $p$ to be $r$-solvable.\ **Subcase $4.4$**: When vertex $v_i\in V$ has $1$ pebble and the two other vertices of $V$ have $0$ pebbles. Let $m_i,m_j,m_t\in M$ such that $m_i,m_j$ are both adjacent to $v_i$ and $m_t$ not adjacent to $v_i$. We know by assumption that $p(m_i)\leq 1,p(m_j)\leq 1$. Thus $w(m_t)+w(k)\geq 16-2-2\cdot\frac{4}{3}=\frac{34}{3}$. If $p(m_i)=p(m_j)=0$, then $w(m_t)+w(k)\geq 14$. Observe that $p(v_i)=1$ implies $p(k)\leq 11$, so $p(m_t)\geq \left \lceil \frac{14-11}{\frac{4}{3}}\right \rceil=3$. This implies $w(m_t)+w(k)\geq 14-3\cdot\frac{4}{3}=10$. Move pebbles from $k$ to $m_t$ until at most $1$ pebble is left at $k$. Since $2>\frac{4}{3}$, $m_t$ now has at least $3+\left \lceil \frac{10-1}{2}\right \rceil=8$ pebbles and $p$ is $r$-solvable. If $p(m_i)=1$ and $p(m_j)=0$, then $p(k)\leq 9$, so $w(m_t)\geq 16-2-9-\frac{4}{3}=\frac{11}{3}$ and $p(m_t)\geq 3$. Following similar logic, $p(m_t)\geq 3$ implies $p(k)\leq 3$. Finally, $p(k)\leq 3$ implies $p(m_t)\geq 8$, which forces $p$ to be $r$-solvable. Following similar logic, $p(m_i)=p(m_j)=1$ implies $p(k)\leq 7$, $p(k)\leq 7$ implies $p(m_t)\geq 4$, $p(m_t)\geq 4$ implies $p(k)\leq 1$. Finally, $p(k)\leq 1$ implies $p(m_t)\geq 8$, forcing $p$ to be $r$-solvable.\ Notice that $w(p)$ is a multiple of $\frac{1}{3}$ and we are left with only two possibilities.\ **Case $5$**: When $w(p)=15\frac{1}{3}$ or $15\frac{2}{3}$. **Subcase $5.1$**: When $p(v_i)=p(v_j)=1$ for $v_i,v_j\in V$ and the other vertex of $V$ has $0$ pebbles. First, suppose $w(p)=15\frac{1}{3}$. It follows that $w(M)\cup\{k\})=11\frac{1}{3}$. If $p(m_i)=1$ and $p(k)=10$ for some $m_i\in M$, we can move $5$ pebbles from $k$ to $m_i$, then move $3$ to any of its parent with $1$ pebble and it follows that $p$ is $r$-solvable. If $p(M)=4$ and $p(k)=6$, clearly $p(m_1)= 2$. If $p(m_2)=p(m_1)=2$, move $1$ pebble from $\{m_2,m_1\}$ to each of $\{v_i,v_j\}$ and it follows that $p$ is $r$-solvable. If $p(m_2)<2$, then $p(m_2)=p(m_3)=1$. We can move $1$ pebble from $k$ to each of $\{m_2,m_3\}$ and then $1$ pebble from $\{m_2,m_3\}$ to each of $\{v_i,v_j\}$, which forces $p$ to be $r$-solvable. If $p(M)=7,p(k)=2$, clearly $p(m_1)\leq 5$, so $p(m_2)\geq 1$. We can move $1$ pebble from $k$ to $m_2$, then move $1$ pebble from $\{m_2,m_1\}$ to each of $\{v_i,v_j\}$, which implies $p$ is $r$-solvable. Second, suppose $w(p)=15\frac{2}{3}$. If $p(M)=2$ and $p(k)=9$, remove $4$ pebbles from $k$ to add one on $v_i$ and remove another $4$ pebbles from $k$ to add $1$ pebble on $v_j$. It follows that $p$ is $r$-solvable. If $p(M)=5,p(k)=5$, by the pigeonhole principle $p(m_1)\geq 2$. Remove $2$ pebbles from $m_1$ and add $1$ pebble to one of $m_1$'s parent from $\{v_i,v_j\}$ and remove $4$ pebbles from $k$ to add $1$ pebble to the other vertex of $\{v_i,v_j\}$. This shows $p$ is $r$-solvable. If $p(M)=8,p(k)=1$, observe that when $p(M)\geq 6$ and all $p(m_i)$ is even, $p$ is solvable. When $p(M)=8$, since at most $2$ vertices of $M$ has an odd number of pebbles, ignoring $1$ pebble from each vertex of the form $m_i$ with odd pebbles on it implies that $p$ is $r$-solvable.\ **Subcase $5.2$**: When $p(v_i)=1$ for $v_i\in V$ and two other vertices of $V$ have $0$ pebbles. Let $k'=\left \lfloor \frac{p(k)}{2}\right \rfloor$, $m=p(M)$. Let $v_i$ be adjacent to $m_i,m_j\in M$ and not adjacent to $m_t\in M$. Notice that we can remove $2k'$ pebbles from vertex $k$ and distribute $k'$ pebbles $M$. First, suppose $k'\geq 2$, it follows that $$m_i+m_j\leq 5-k',m_i+m_t\leq 7-k',\textnormal{ and }m_j+m_t\leq 7-k'.$$ Then $2m\leq 19-3k'$. If $w(p)=15\frac{1}{3}$, then $k'=\frac{p(k)}{2},2m=20-3k'$. It follows that $20\leq 19$, forming a contradiction. If $w(p)=15\frac{2}{3}$, then $p(k)= 2k'+1$ and $2m=\frac{41-3p(k)}{2}\geq 19-3k'$. Thus $2m=19-3k'$, which implies that every inequality here takes equality. Thus $p(m_i)=p(M)-(p(m_j)+p(m_t))=\frac{19-3k'}{2}-7+k'=\frac{5-k'}{2}$. Similarly $p(m_j)=\frac{5-k'}{2}$ and $p(m_t)=\frac{9-k'}{2}$. But $p(k)$ can only be $7$ or $11$. Readers can easily verify that $p(m_i)=p(m_j)=1,p(m_t)=3,p(k)=7$ and $p(m_i)=p(m_j)=0,p(m_t)=2,p(k)=11$ are two $r$-solvable configurations. Second, suppose $k'<2$. If $w(p)=15\frac{1}{3}$, then $p(M)=10$ and $p(k)=0$. By subcase $4.3$, $p(M)$ is at most $9$, forming a contradiction. If $w(p)=15\frac{2}{3}$, then $p(M)=8$ and $p(k)=3$. Thus from $k$ we can move a pebble to $M$. Observe that when $p(M)=8$ and $p(m_a)$ is even for each $m_a\in M$, the configuration is solvable. When $p(M)=8$, either each $p(m_a)$ is even or exactly $2$ of them are odd. In the latter case, we can move one pebble from $k$ to a vertex $m_a\in M$ where $p(m_a)$ is odd and ignore a pebble on another vertex $m_b\in M$ where $p(m_b)$ is odd. This shows $p$ is $r$-solvable.\ All possible configurations $p$ are covered and we conclude that $w$ is valid. ◻ **Corollary 7**. *Let $w$ be a weight function on $Q_4$ where $w(v)=1$ for each $v\neq r$ and $w(r)=0$. Then $w$ is valid, which implies $\pi(Q_4)=16$.* *Proof.* Let $w$ be the weight function in Figure [\[f6\]](#f6){reference-type="ref" reference="f6"} and $U_n$ be the graph in Figure [\[f6\]](#f6){reference-type="ref" reference="f6"}. Let $P$ be the path $(v,v_2,\ldots,v_n,r)$ and $U$ be vertex set $\{u_1,u_2,\ldots,u_{2^{n+1}},k\}$. Let $p$ be an arbitrary $r$-unsolvable configuration on $U_n$. Note that for each $v_i\in P$ with $i\neq n$, we have $w(v_{i+1})=2w(v_i)$. Let $p$ be an arbitrary configuration on $U_n$ with $$w(p)\geq w(1_{U_n})+1=2^{n+2}.$$ To prove $w$ is valid, it suffices to show $p$ is $r$-solvable. For purpose of contradiction suppose $p$ is not $r$-solvable. By definition of section $1.1$, we have $$\begin{aligned} w(p)&=\sum_{v\in V(U_n)}p(v)\cdot w(v)\\ w_P(p)&=\sum_{v\in V(P)}p(v)\cdot w(v)\\ p(U)&=\sum_{v\in U}p(v).\end{aligned}$$ Since $w(v_1)=2$, applying Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"} on $P$ shows that if $w_P(p)\geq 2^{n+1}$ then $p$ will be immediately $r$-solvable. Suppose $w_P(p)<2^{n+1}$. For simplicity of notation, let $w_P(p)=x$ and $p(k)=2b+y$ where $b\geq 0$ and $y$ equals $0$ or $1$. Let the number of vertices of the form $u_i$ with an odd number of pebbles be $a$. Clearly $x$ is even and $a\leq 2^{n+1}$. Since $w(v)=1$ for each $v\in U$, we know $$p(U)={w(p)-w_P(p)\over 1}\geq 2^{n+2}-x.$$ Lemma [Lemma 2](#l2){reference-type="ref" reference="l2"} shows that it suffices to prove we can move at least $\frac{2^{n+1}-x}{2}$ pebbles from $U$ to $v_1$.\ **Case 1:** When $a\leq b$. We will do the following pebbling moves in sequence: - Move $1$ pebble from $k$ to each vertex of the form $u_i$ that has an odd number of pebbles. Now each vertex of the form $u_i$ has an even number of pebbles. - For each vertex of the form $u_i$, move pebbles from it to $v_1$ until there is no pebbles on $u_i$. - If $k$ still has at least $4$ pebbles, move two pebbles from $k$ to $u_1$ and then move $1$ pebble from $u_1$ to $v_1$. Repeat this step until $k$ has less than $4$ pebbles. In the pebbles moves described above, there are three ways for us to add a pebble onto $v_1$: use $2$ pebbles from $k$ and $1$ pebble from some vertex of the form $u_i$, use $2$ pebbles from some vertex of the form $u_i$, or use $4$ pebbles from $k$. Hence we conclude that adding $1$ pebble onto $v_1$ uses at most $4$ pebbles. Observe that after these pebbling moves, each vertex of form $u_i$ has no pebbles and $k$ has at most $3$ pebbles. Thus at least $p(U)-3$ pebbles are consumed in order to add pebbles onto $v_1$. Hence, the number of pebbles we added to $v_1$ through these pebbling moves is at least$$\left \lceil {p(U)-3\over 4}\right \rceil\geq \left\lceil {2^{n+2}-x-3\over 4}\right\rceil\geq {2^{n+1}-x\over 2}$$where the last inequality holds because $x$ is even.\ **Case 2:** When $a>b$. We will do the following pebbling moves in sequence: - Pick $b$ vertices of the form $u_i$ that has an odd number of pebbles. Move $1$ pebble from $k$ to each of them. - For each vertex of the form $u_i$, move pebbles from it to $v_1$ until $u_i$ has at most $1$ pebble. Similar to Case 1, the number of pebbles we added to $v_1$ through these pebbling moves is exactly $${p(U)-a-y\over 2}\geq \left\lceil {2^{n+2}-a-x-y\over 2}\right\rceil={2^{n+1}-x\over 2}+\left\lceil{(2^{n+1}-a-y)\over 2}\right\rceil\geq {2^{n+1}-x\over 2}$$where the last inequality holds because $a+y\leq 2^{n+1}+1$.\ Hence in all cases we can move at least ${2^{n+1}-x\over 2}$ pebbles from $U$ to $v_1$ and we conclude that $w$ is valid. ◻ In fact, this weight function can be further generalized to contain $m$ vertices of the form $u_i$, where $m\geq 2^{n+1}$. Here is the brief idea for proving its validity. Denote this graph as $G$ and let $p$ be an arbitrary configuration on $G$ such that $w(p)>w(1_G)$. Then, we first move pebbles from each vertex of the form $u_i$ to $v_1$ until at most $1$ pebble is left on $u_i$. This gives us a new configuration $p'$. Clearly $w(p)=w(p')$. Then, ignore $m-2^{n+1}$ arbitrary vertices of the form $u_i$ and let the subgraph induced by the rest vertices be denoted by $G'$. Notice that $w_{G'}(p')\geq w(p)-1\cdot (m-2^{n+1})>w(1_{G'})$. By Theorem [\[t3\]](#t3){reference-type="ref" reference="t3"}, configuration $p'$ on $G'$ is $r$-solvable, which implies configuration $p$ on $G$ is also $r$-solvable. # Open questions 1. Can more efficient proof techniques be developed to prove the validity of weight functions? 2. Is it possible to modify weight functions to characterize other variations of pebbling problems? For example, how could techniques like weight functions be used to characterize configurations that could place at least $2$ pebbles on the root node $r$? The authors believe that such modifications could help prove Conjecture [\[c1\]](#c1){reference-type="ref" reference="c1"}. 3. In this paper we constructed weight functions for odd cycles. Can we find weight functions for (generalized) lollipop graphs with odd cycles analogous to what was presented in $\cite{cranston2015modified}$? 4. Can we use these new weight functions to improve the bound of the pebbling number of $L\square L$ (the Cartesian product of two Lemke Graph $\cite{cranston2015modified}$)? Note that our results have more implications than simply constructing other valid weight functions or computing the pebbling numbers of a few graphs. We believe that they can be coupled with computational tools (extending work of Flocco et al. $\cite{flocco}$) to improve computational techniques for computing pebbling numbers of any graph that has hypercubes or lollipops as induced subgraphs. 100 F. Chung, Pebbling in Hypercubes, *SIAM Journal on Discrete Mathematics* **2** (1989), 467--472. D. Cranston, L. Postle, C. Xue, C. Yerger, Modified linear programming and Class 0 bounds for graph pebbling, *Journal of Combinatorial Optimization* **34** (2017), 114--132. P. Erdos, A. Ginzburg, A. Ziv, Theorem in additive number theory, *Israel Research and Development Nat. Council Bull.* **10F** (1961), 41--43. D. Flocco, J. Pulaj, C. Yerger, Automating weight function generation in graph pebbling, (2022), preprint, <https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4293654>. G. Hurlbert, Hurlbert's pebbling page, [ http://www.people.vcu.edu/\~ghurlbert/pebbling/pebb.html ]( http://www.people.vcu.edu/~ghurlbert/pebbling/pebb.html ). G. Hurlbert, The weight function lemma for graph pebbling, *Journal of Combinatorial Optimization* **34** (2017), 343--361. G. Hurlbert, Two pebbling theorems, *Congressus Numerantium* **135** (2000), 3--6. K. Milans, B. Clark, The Complexity of Graph Pebbling, *SIAM Journal on Discrete Mathematics* **20** (2006), 769--798. L. Patcher, H. Snevily, B. Voxman, On pebbling graphs, *Congressus Numerantium* **107** (1995), 65--80. N. Watson, The complexity of pebbling and cover pebbling, <https://arxiv.org/abs/math/0503511> (2005). [^1]: Davidson College, Davidson, North Carolina 28035 E-mail: `cayerger@davidson.edu` [^2]: Davidson College, Davidson, North Carolina 28035 Email: `dazhou@davidson.edu` [^3]: https://github.com/dominicflocco/Graph$\underline{~}$Pebbling

arxiv_math

--- abstract: | Anisotropic diffusion processes with a diffusion tensor are important in image analysis, physics, and engineering. However, their numerical approximation has a strong impact on dissipative artefacts and deviations from rotation invariance. In this work, we study a large family of finite difference discretisations on a $3\times3$ stencil. We derive it by splitting 2-D anisotropic diffusion into four 1-D diffusions. The resulting stencil class involves one free parameter and covers a wide range of existing discretisations. It comprises the full stencil family of Weickert et al. (2013) and shows that their two parameters contain redundancy. Furthermore, we establish a bound on the spectral norm of the matrix corresponding to the stencil. This gives time step size limits that guarantee stability of an explicit scheme in the Euclidean norm. Our directional splitting also allows a very natural translation of the explicit scheme into ResNet blocks. Employing neural network libraries enables simple and highly efficient parallel implementations on GPUs. author: - Karl Schrader - Joachim Weickert - Michael Krause bibliography: - myrefs.bib title: | Anisotropic Diffusion Stencils:\ From Simple Derivations over Stability Estimates to ResNet Implementations [^1] --- # Introduction Anisotropic diffusion models with a diffusion tensor have numerous applications in physics and engineering. Moreover, they also play a fundamental role in image analysis [@We97], where they are used for denoising, enhancement, scale-space analysis, and various interpolation tasks such as inpainting and superresolution. Sophisticated nonlinear models with appropriate directional behaviour can close interrupted structures and maintain or create sharp edges. However, to achieve results with only few dissipative artefacts and good rotation invariance, appropriate numerical approximations are needed. They should also come with provable stability guarantees and lead in a natural way to efficient implementations. Ideally they should also exploit the impressive parallelisation potential of modern GPUs. The goal of our contribution is to address these numerical issues. **Our Contributions.** Motivated by image analysis applications, where one has a regular pixel grid and aims at simple numerical algorithms, we consider finite difference approximations on a $3 \times 3$ stencil. However, our results are also useful for anisotropic diffusion problems in other areas. Our contributions are threefold: First, we study space discretisations of a general anisotropic diffusion operator on a $3 \times 3$ stencil. They split the 2-D anisotropic process into four 1-D diffusions. This class has one free parameter that can be used for quality optimisation. It covers the two-parameter stencil family of Weickert et al. [@WWW13], while removing its parameter redundancy and offering a simpler derivation. Moreover, it subsumes many previous discretisations with second-order consistency. Our second contribution consists of a detailed stability analysis, where we establish fairly tight bounds on the spectral norm of the matrix associated with the stencil family. It allows to derive time step size restrictions for the corresponding explicit scheme (and accelerations that rely on it). Last but not least, our stencil derivation based on a directional splitting enables the translation of the explicit anisotropic diffusion scheme into a ResNet block [@HZRS16], which is a highly popular component of neural networks. This showcases that ideas are often shared between numerical schemes and neural architectures. More importantly, it allows simple and fast parallel implementations of anisotropic diffusion on GPUs using neural network libraries such as PyTorch. **Related Work.** Many finite difference discretisations for anisotropic diffusion processes exist in the literature. Often they use spatial discretisations on a $3 \times 3$ stencil with consistency order two. The stencil class of Weickert et al. [@WWW13] comprises seven of them. Our findings offer a simpler derivation and representation of this family. Moreover, we extend the results from [@WWW13] by establishing concrete time step size limits for explicit schemes, connecting these algorithms to neural networks, and exploring simple and efficient parallelisations. Our stencil family originates from a splitting 2-D anisotropic diffusion into four 1-D diffusions along fixed directions. Earlier splittings of this type intended to derive discretisations that are stable in the maximum norm [@We97; @MN01a]. In general this is only possible for fairly mild anisotropies [@We97]. We consider stencils that offer stability in the Euclidean norm for all anisotropies. Recent works [@ASAPW21; @ASWPA22; @RDF20; @RH20] connect explicit schemes for partial differential equations (PDEs) to the ResNet [@HZRS16] architecture. For example, Alt et al. [@ASAPW21] show that evolutions of discretised 1-D diffusion models with a scalar-valued diffusivity can be represented as ResNet blocks. In [@ASWPA22], they also explore the 2-D anisotropic case. However, their methodology is limited to evolution equations that arise as gradient descent of an energy functional. This excludes popular methods like edge-enhancing diffusion [@We97], for which Welk [@We21] has shown that no energy functional exists. We can translate these methods as well. **Organisation of the Paper.** In Section 2, we derive a class of finite difference discretisations on a $3\times3$ stencil. We establish stability results for the corresponding explicit scheme in Section 3. The fourth section shows how our splitting into 1-D diffusions leads to a translation of this scheme to a ResNet architecture, and it analyses its performance on a GPU. We conclude our paper in Section 5. # Discretising Anisotropic Diffusion with the $\bm \delta$-Stencil In this section, we study a simple and fairly general approach for a space discretisation of anisotropic diffusion on a $3 \times 3$ stencil. It is based on a directional splitting into four 1-D diffusion processes, which we discuss first. **1-D Diffusion.** To denoise a 1-D signal $f(x): [a, b] \to \mathbb{R}$, one can create simplified versions $\{u(x, t)\,|\, t \ge 0\}$ of it with the nonlinear diffusion process [@PM90] $$\begin{aligned} \partial_t u &\;=\; \partial_x\Big(g\big((\partial_x u)^2\big)\, \partial_x u\Big) & (t> 0), \label{eq:diffusion_1d_cont}\\ u(x,0) &\;=\; f(x)\;.\end{aligned}$$ Larger diffusion times $t$ correspond to more pronounced simplifications. The diffusivity $g: \mathbb{R}\to (0,1]$ is a function that decreases in its argument $(\partial_x u)^2$ in order to preserve discontinuities. At the domain boundaries $a$ and $b$, we impose reflecting boundary conditions. To prepare for the later translation to a neural architecture, we introduce the flux function $\,\Phi(\partial_x u) = g((\partial_x u)^2)\,\partial_x u$. It leads to the evolution equation $\,\partial_t u = \partial_x(\Phi(\partial_x u))$. A finite difference discretisation of this 1-D process serves as building block for discretising anisotropic diffusion. To obtain a discrete signal $\bm u = (u_i) \in \mathbb{R}^N$, we sample $u$ with grid size $h$. We discretise the derivatives with a forward difference in time and for the inner spatial derivative, and a backward difference for the outer one. This leads to the explicit scheme $$\frac{u^{k+1}_i - u^k_i}{\tau} \;=\; \frac{1}{h}\left( \Phi\left(\frac{u^k_{i+1} - u^k_{i}}{h}\right) - \Phi\left(\frac{u^k_{i} - u^k_{i-1}}{h}\right) \right), \label{eq:iso_explicit}$$ where $\tau>0$ is the time step size, $i$ denotes the location, and $k$ the time level. **Anisotropic Diffusion.** In image analysis, anisotropic diffusion with a diffusion tensor [@We97] creates filtered versions $u(\bm x, t)$ of a scalar-valued (i.e. greyscale) image $f(\bm x)$ by evolving it with the PDE $$\begin{aligned} \partial_t u &\;=\; \mbox{\rm{\bf{div\,}}}(\bm D \, \bm\nabla u), \qquad \bm D \;=\; \begin{pmatrix}a \,&\, b \\ b \,&\, c \end{pmatrix} \label{eq:aniso_continuous}\end{aligned}$$ where we initialise $u(\bm x, 0)$ with $f(\bm x)$ and use reflecting boundary conditions. The diffusion tensor $\bm D \in \mathbb{R}^{2 \times 2}$ is symmetric, positive semidefinite with at least one positive eigenvalue. $\bm D$ may depend on Gaussian-smoothed first order derivatives of the evolving image $u$. This allows to enhance edges and coherent flow-like structures by smoothing along them, but not perpendicular to them [@We97]. **The $\bm\delta$-Stencil.** The discrete setting considers images $\bm f, \bm{u}^k \in \mathbb{R}^N$ obtained by sampling $f$ and $u(.,k\tau)$ with a grid size of $h$ and arranging the pixel values into column vectors. The key idea of our discretisation is the decomposition of an anisotropic 2-D diffusion process into a sum of four nonlinear 1-D diffusions along the axial and diagonal directions $$\bm e_0 = \begin{pmatrix}1\\0\end{pmatrix}, \quad \bm e_1 = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}, \quad \bm e_2 = \begin{pmatrix}0\\1\end{pmatrix}, \quad \bm e_3 = \frac{1}{\sqrt{2}}\begin{pmatrix}-1\\1\end{pmatrix}.$$ We determine directional diffusivities $w_0, \dots, w_3$ for the corresponding directions by solving the system of three equations with four unknowns arising from $$\mbox{\rm{\bf{div\,}}}\left(\begin{pmatrix}a\,&\,b\\b\,&\,c\end{pmatrix}\bm\nabla u\right) \stackrel{!}{\;=\;} \sum_{i=0}^{3} \partial_{\bm e_i}\left(w_i\, \partial_{\bm e_i} u\right).$$ Its solution has one free parameter which we call $\delta$: $$w_0 = a-\delta, \qquad w_1 = \delta+b, \qquad w_2 = c-\delta, \qquad w_3 = \delta-b \,.$$ All four 1-D diffusion processes can be discretised as before in [\[eq:iso_explicit\]](#eq:iso_explicit){reference-type="eqref" reference="eq:iso_explicit"}. Each direction uses three pixels of its $3\times 3$ neighbourhood. Discretising e.g. $\partial_{\bm e_1}(w_1\,\partial_{\bm e_1} u)$ in the pixels $(i\!-\!1, j\!-\!1)$, $(i,j)$, and $(i\!+\!1, j\!+\!1)$ at distance $h\sqrt{2}$ gives $$\frac{1}{h\sqrt{2}}\left((\delta+b)_{i+\frac{1}{2}, j+\frac{1}{2}} \frac{u_{i+1, j+1}-u_{i,j}}{h\sqrt{2}} - (\delta+b)_{i-\frac{1}{2}, j-\frac{1}{2}} \frac{u_{i, j}-u_{i-1,j-1}}{h\sqrt{2}}\right).$$ Incorporating all four directions yields the following **${\bm \delta}$-stencil** for $\mbox{\rm{\bf{div\,}}}(\bm D \, \bm \nabla u)$: $$\scalebox{.88}{\mbox{\ensuremath{\displaystyle \kern-.6em \frac{1}{h^2} \cdot \, \begin{array}{|c|c|c|} \hline \mbox{\rule[-4ex]{0pt}{8ex}} \begin{array}{l} \tfrac{1}{2}\,(\delta-b)_{i-\frac{1}{2},j+\frac{1}{2}} \end{array} & \begin{array}{l} (c-\delta)_{i,j+\frac{1}{2}} \end{array} & \begin{array}{l} \tfrac{1}{2}\,(\delta+b)_{i+\frac{1}{2},j+\frac{1}{2}} \end{array} \\ \hline \mbox{\rule[-10ex]{0pt}{20ex}} \begin{array}{l} (a-\delta)_{i-\frac{1}{2},j} \end{array} ~ & \begin{array}{c} {}-(a-\delta)_{i+\frac{1}{2},j}\,-\,(a-\delta)_{i-\frac{1}{2},j}\\ {}-\tfrac{1}{2}\,(\delta+b)_{i+\frac{1}{2},j+\frac{1}{2}} \,-\,\tfrac{1}{2}\,(\delta+b)_{i-\frac{1}{2},j-\frac{1}{2}}\\ {}-(c-\delta)_{i,j+\frac{1}{2}}\,-\,(c-\delta)_{i,j-\frac{1}{2}}\\ {}-\tfrac{1}{2}\,(\delta-b)_{i-\frac{1}{2},j+\frac{1}{2}} \,-\,\tfrac{1}{2}\,(\delta-b)_{i+\frac{1}{2},j-\frac{1}{2}} \end{array} ~ & \begin{array}{l} (a-\delta)_{i+\frac{1}{2},j} \end{array} ~ \\ \hline \mbox{\rule[-4ex]{0pt}{8ex}} \begin{array}{l} \tfrac{1}{2}\,(\delta+b)_{i-\frac{1}{2},j-\frac{1}{2}} \end{array} & \begin{array}{c} (c-\delta)_{i,j-\frac{1}{2}} \end{array} & \begin{array}{l} \tfrac{1}{2}\,(\delta-b)_{i+\frac{1}{2},j-\frac{1}{2}} \end{array} \\ \hline \end{array}}}}\label{eq:delta_stencil}$$ where the $x$-axis points to the right, and the $y$-axis to the top. We assume that the diffusion tensor $\bm D$ is available in the staggered grid locations $\left(i \pm \frac{1}{2}, j \pm \frac{1}{2}\right)$. This is fairly natural if it relies on first-order derivatives, which can be computed with central differences in a $2 \times 2$ neighbourhood [@WWW13]. We obtain values in $\left(i \pm \frac{1}{2}, j\right)$ and $\left(i, j \pm \frac{1}{2}\right)$ by averaging: $$(a-\delta)_{i \pm \frac{1}{2}, j} \;=\; \frac{1}{2}\left((a-\delta)_{i \pm \frac{1}{2}, j + \frac{1}{2}} \;+\; (a-\delta)_{i \pm \frac{1}{2}, j - \frac{1}{2}}\right) \label{eq:ave}$$ and similar for $(c-\delta)_{i, j \pm \frac{1}{2}}$. Then the $\delta$-stencil family has consistency order two. **Incorporation of the Stencil Family of Weickert et al. [@WWW13].** With [\[eq:ave\]](#eq:ave){reference-type="eqref" reference="eq:ave"} and $\delta = \alpha a + \beta b + \alpha c$, the $\delta$-stencil family comprises that of Weickert et al. [@WWW13] that uses two parameters $\alpha$ and $\beta$. This shows that the parameters of the latter contain redundancy which we remove with the $\delta$-stencil. Moreover, our stencil derivation is simpler than the one in [@WWW13] that has been obtained by discrete energy minimisation. In [@WWW13] it is shown that these stencils comprise seven discretisations from the literature. Since we are not aware of any second-order accurate discretisations on a $3 \times 3$ stencil that is not covered by this class, the $\delta$-stencil family may even be more general. # Stability Theory for the $\bm\delta$-Stencil Let the matrix $\bm A = \bm A(\bm u^k)$ act on an image $\bm u^k$ locally by applying the space-variant $\delta$-stencil. Weickert et al. [@WWW13] have already established that $\bm A$ is negative semidefinite for $\alpha \le \frac{1}{2}$ and $|\beta| \leq 1-2\alpha$. They have also replaced $\beta$ by a parameter $\gamma$ such that $\beta = \gamma (1\!-\!2\alpha)\,\text{sgn}(b)$ and $|\gamma|\leq 1$. Choosing $\alpha$ close to $\frac{1}{2}$ and $\gamma$ close to $1$ improves rotation invariance and reduces dissipativity in experiments [@WWW13]. In practice, parameters $\alpha < 0$ are irrelevant and make a stability analysis more complicated. Thus, we exclude them from now on. Consider an explicit anisotropic diffusion scheme $\,\bm u^{k+1} = (\bm I +\tau \bm A(\bm u^k))\,\bm u^k\,$ with unit matrix $\bm I$, time step size $\tau>0$, and a negative semidefinite matrix $\bm A$ with spectral norm $\rho(\bm A)>0$. Then stability in the Euclidean norm in terms of $\,\|\bm u^{k+1}\|_2 \leq \|\bm u^k\|_2\,$ holds if $$\tau \;\leq\; \frac{2}{\rho(\bm A)} \,. \label{eq:tau_limit}$$ We can bound $\rho(\bm A)$ as follows: [\[th:a_norm\]]{#th:a_norm label="th:a_norm"} Let the eigenvalues of $\bm D$ be given by $\lambda_1 \ge \lambda_2 \ge 0$. Assume that $\,\delta = \alpha(a\!+\!c) + \beta b\,$ where $\,\beta=\gamma(1\!-\!2\alpha)\,\textup{sgn}(b)\,$ for $\alpha \in [0, \frac{1}{2}]$ and $|\gamma|\leq 1$. Then the spectral norm of the matrix $\bm A$ satisfies $$\rho(\bm A) \;\leq\; \frac{4\,(1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) \,+\, 2\,(1-\gamma\,(1\!-\!2\alpha))\,(\lambda_1\!-\!\lambda_2)}{h^2}\;.$$ *Proof.* For the considered choice of $\alpha$ and $\beta$, we know from [@WWW13] that the symmetric matrix $\bm A$ is negative semidefinite. Thus, its spectral norm is determined by its smallest eigenvalue $\lambda_{\min}$ as $\,\rho(\bm A) = -\lambda_{\min}(\bm A)\,$. Let us now bound $\lambda_{\min}(\bm A)$ with Gershgorin's circle theorem [@We18]. As $\bm A$ applies the $\delta$-stencil, this theorem states that the smallest eigenvalue of $\bm A$ is bounded from below by the central stencil entry, minus the sum of absolute values of all other entries. Using [\[eq:ave\]](#eq:ave){reference-type="eqref" reference="eq:ave"} and grouping all terms by the four diffusion tensor locations $(i \pm \frac{1}{2}, j \pm \frac{1}{2})$ gives $$\scalebox{0.86}{\mbox{\ensuremath{\displaystyle \begin{matrix*}[l] \rho(\bm A) \;\leq\; \frac{1}{2h^2}\max\limits_{\smash{a,b,c}} &\big( &((a-\delta) + |a-\delta| + (\delta+b) + |\delta+b| + (c-\delta) + |c-\delta|)_{i-\frac{1}{2},j-\frac{1}{2}} \vspace{-.5mm} \\ &+&((a-\delta) + |a-\delta| + (\delta-b) + |\delta-b| + (c-\delta) + |c-\delta|)_{i-\frac{1}{2},j+\frac{1}{2}} \\ &+&((a-\delta) + |a-\delta| + (\delta-b) + |\delta-b| + (c-\delta) + |c-\delta|)_{i+\frac{1}{2},j-\frac{1}{2}} \\ &+&((a-\delta) + |a-\delta| + (\delta+b) + |\delta+b| + (c-\delta) + |c-\delta|)_{i+\frac{1}{2},j+\frac{1}{2}} \, \big). \end{matrix*}\label{eq:rho_limit_1}}}}$$ Next, we bound the right hand side from above by assuming that the diffusion tensors at the different locations are independent. The notation $$M_\pm \;\coloneqq\; (a-\delta) + |a-\delta| + (\delta\pm b) + |\delta\pm b| + (c-\delta) + |c-\delta|$$ allows us to rewrite the bound as $$\rho(\bm A) \;\leq\; \frac{1}{h^2}\big(\max\limits_{a,b,c}(M_+) + \max\limits_{a,b,c} (M_-)\big)\,. \label{eq:rho_limit_2}$$ In the following, we determine the maximum of $M_+$. Calculations for $M_-$ are analogous. Notice that in $M_+$, the three terms $\,a-\delta$, $\,\delta + b$, and $\,c - \delta$ appear pairwise with their absolute values. This will simplify the calculation of the maximum. Consider the sum of the three terms: $$m_+ \;\coloneqq\; (a-\delta) \,+\, (\delta+b) \,+\, (c-\delta) \;=\; a+b+c-\delta \,. \label{eq:mplus}$$ If $m_+$ has a maximum in which $a - \delta$, $\delta + b$, and $c - \delta$ are all nonnegative, then $\max(M_+) = 2\,\max(m_+)$, since $\,x+|x|=2x$ for $x \ge 0$. We now proceed to show that such a maximum of $m_+$ exists. To this end, we rewrite the entries $a$, $b$, and $c$ of the positive semidefinite diffusion tensor $\bm D$ in terms of its normalised eigenvectors $(u, v)^\top$, $(v, -u)^\top$ and their eigenvalues $\lambda_1 \geq \lambda_2 \geq 0$: $$\begin{aligned} a = \lambda_1 u^2 + \lambda_2 v^2\,, \qquad b = (\lambda_1\!-\!\lambda_2)\, uv\,, \qquad c = \lambda_2 u^2 + \lambda_1 v^2\,. \label{eq:abc_decomp}\end{aligned}$$ The possible ranges for eigenvalues may differ between diffusion models. Therefore, we determine the maximum of $m_+$, and by extension our limit on $\rho(\bm A)$, as a function of $\lambda_1$ and $\lambda_2$. This leaves the entries $u, v$ of the eigenvectors as the only variables to maximise $m_+$ over. Using [\[eq:abc_decomp\]](#eq:abc_decomp){reference-type="eqref" reference="eq:abc_decomp"} in [\[eq:mplus\]](#eq:mplus){reference-type="eqref" reference="eq:mplus"} gives $$\begin{aligned} \max_{u,v}\, (m_+) &\;=\; \max_{u,v}\big((1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) \,+\, (1\!-\!\beta)(\lambda_1\!-\!\lambda_2)\,uv\big) \nonumber\\ &\hspace{-14mm} \;=\; \max_{u,v}\big((1\!-\!\alpha)\, (\lambda_1 \!+\! \lambda_2) \,+\, (1\!-\!\gamma(1\!-\!2\alpha)\,\text{sgn}(uv))\, (\lambda_1\!-\!\lambda_2)\,uv \big) \nonumber\\ &\hspace{-14mm} \;=\; \max_{u,v}\begin{cases} (1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) + \underbrace{(1-\gamma(1\!-\!2\alpha))}_{\geq 0} \underbrace{(\lambda_1\!-\!\lambda_2)}_{\geq 0} \underbrace{uv}_{> 0} & \mbox{for }\,uv > 0,\\ (1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) + \underbrace{(1+\gamma(1\!-\!2\alpha))}_{\geq 0} \underbrace{(\lambda_1\!-\!\lambda_2)}_{\geq 0} \underbrace{uv}_{\leq 0} & \mbox{for }\,uv\leq 0.\\ \end{cases}\end{aligned}$$ The case where $uv> 0$ gives always larger results than the second one. Thus, $$\max_{u,v}\, (m_+) \;=\; \max_{u,v} \big((1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) \,+\, (1-\gamma(1\!-\!2\alpha))\,(\lambda_1\!-\!\lambda_2)\,uv\big)\,.$$ We maximise the second term by maximising $uv$. For our normalised eigenvectors, $u^2+v^2=1$ holds. Hence, $\max(uv)=\frac{1}{2}$ for $u=v=\pm \frac{1}{\sqrt{2}}$. Since we only need the maximal function value, we can consider only $u=v=\frac{1}{\sqrt{2}}$. This gives $$\max_{u,v}\, (m_+) \;=\; (1\!-\!\alpha)\,(\lambda_1 \!+\! \lambda_2) \,+\, \frac{1}{2}\,(1-\gamma(1\!-\!2\alpha))\,(\lambda_1\!-\!\lambda_2)\,. \label{eq:mplusMax}$$ We are not able to draw conclusions about the maximum of $M_+$ from the maximum of $m_+$ yet. It remains to show that $a - \delta$, $\delta + b$, and $c - \delta$ are all nonnegative in our maximum with $u=v=\frac{1}{\sqrt{2}}$. We start with $a-\delta$ and use [\[eq:abc_decomp\]](#eq:abc_decomp){reference-type="eqref" reference="eq:abc_decomp"}: $$\begin{aligned} (a-\delta)|_{u=v=\frac{1}{\sqrt{2}}} &\;=\; \big(a - \beta b - \alpha(a+c)\big)|_{u=v=\frac{1}{\sqrt{2}}} \nonumber\\ &\;=\; \frac{1}{2} \, \big((\lambda_1 + \lambda_2) -\gamma\underbrace{(1-2\alpha)}_{\geq0}\underbrace{(\lambda_1-\lambda_2)}_{\geq 0} \,-\, 2\alpha(\lambda_1 + \lambda_2) \big)\nonumber\\[-0.5mm] &\;\geq\; \frac{1}{2} \, \big((\lambda_1 + \lambda_2) -(1-2\alpha)(\lambda_1-\lambda_2) -2\alpha(\lambda_1 + \lambda_2) \big) \nonumber\\[1mm] &\;=\; (1-2\alpha)\, \lambda_2 \;\geq\; 0\,.\end{aligned}$$ In a similar way, one shows $(c-\delta)|_{u=v=\frac{1}{\sqrt{2}}} \geq 0$. For $b + \delta$ we verify $$\begin{aligned} (b+\delta)|_{u=v=\frac{1}{\sqrt{2}}} &\;=\; \big((1+\beta)\,b + \alpha(a+c)\big)|_{u=v=\frac{1}{\sqrt{2}}} \nonumber\\ &\;=\; \frac{1}{2}\,\underbrace{(1+\gamma(1-2\alpha))}_{\geq 0}\underbrace{(\lambda_1-\lambda_2)}_{\geq 0} \,+\; \alpha \, (\lambda_1+\lambda_2) \;\geq\; 0\,.\end{aligned}$$ As all three terms are nonnegative in the maximum, we can conclude that $$\max_{u,v}(M_+) \,=\, 2\max_{u,v}(m_+) \,=\, 2\,(1\!-\!\alpha)\,(\lambda_1\!+\!\lambda_2) + (1-\gamma(1\!-\!2\alpha))\,(\lambda_1\!-\!\lambda_2)\,.$$ Analogous computations lead to the same maximum for $M_-$. Inserting both into [\[eq:rho_limit_2\]](#eq:rho_limit_2){reference-type="eqref" reference="eq:rho_limit_2"} produces the claimed bound on the spectral norm. 0◻ ◻ Using Theorem [\[th:a_norm\]](#th:a_norm){reference-type="ref" reference="th:a_norm"} within [\[eq:tau_limit\]](#eq:tau_limit){reference-type="eqref" reference="eq:tau_limit"} directly gives the following time step size limit: [\[th:ex\]]{#th:ex label="th:ex"} An explicit anisotropic diffusion scheme $\,\bm u^{k+1} = \left(\bm I +\tau \bm A(\bm u^k)\right)\bm u^k$, where $\bm A$ satisfies the assumptions of Theorem [\[th:a_norm\]](#th:a_norm){reference-type="ref" reference="th:a_norm"} with $\lambda_1>0$, is stable in the Euclidean norm for $$\tau \;\le\; \frac{h^2}{2\,(1\!-\!\alpha)\,(\lambda_1\!+\!\lambda_2) \,+\, (1-\gamma\,(1\!-\!2\alpha))\,(\lambda_1\!-\!\lambda_2)}\,. \label{eq:bound}$$ While our proof does not guarantee that this bound is strict, our practical experience suggests that it is. Corollary [\[th:ex\]](#th:ex){reference-type="ref" reference="th:ex"} covers two important special cases: 1. In the **homogeneous diffusion case** with $\lambda_1=\lambda_2=1$, this time step size limit simplifies to $\,\tau \le \frac{h^2}{4(1-\alpha)}$. Moreover, setting $\alpha \coloneqq 0$ turns the $\delta$-stencil into the standard five point approximation of the Laplacian, which leads to the well-known 2-D time step size limit $\,\tau \le \frac{h^2}{4}$; see e.g. [@MM05]. 2. In the **maximally anisotropic case** with $\lambda_1=1$, $\lambda_2=0$, and $\gamma=1$, one performs 1-D diffusion along one eigendirection of $\bm D$. Then [\[eq:bound\]](#eq:bound){reference-type="eqref" reference="eq:bound"} becomes $\,\tau \le \frac{h^2}{2}$. In spite of being in a 2-D setting, this coincides with the typical 1-D time step size limit [@MM05], which is less restrictive. This shows that our scheme takes full advantage of the anisotropy. In image analysis, this result is relevant for coherence-enhancing nonlinear diffusion filters [@We97]. Similar findings have also been made with a recent numerical scheme for a maximally anisotropic backward parabolic PDE [@SW22]. # Translating Anisotropic Diffusion into ResNets Let us now interpret our explicit scheme in the context of neural networks. This extends the result of Alt et al. [@ASAPW21] from the 1-D setting to the 2-D anisotropic case. We start by presenting their translation of 1-D diffusion into a ResNet block, and then build the anisotropic ResNet block from there. **ResNets.** Residual networks (ResNets) [@HZRS16] are very popular neural network architectures. They use ResNet blocks that compute an output $\bm u^{k+1}$ from an input $\bm u^k$ by $$\bm u^{k+1} \;=\; \sigma_2\left(\bm u^k + \bm K_2\, \sigma_1 \left(\bm K_1\bm u^k + \bm b_1\right)+\bm b_2\right) \label{eq:resnet_block}$$ for discrete convolution kernels $\bm K_1, \bm K_2$, bias vectors $\bm b_1, \bm b_2$, and nonlinear activation functions $\sigma_1, \sigma_2$ such as the ReLU function $\sigma(x)=\max\{x,0\}$. Adding the input $\bm u^k$ before applying the second activation $\sigma_2$ helps to avoid vanishing gradients and to improve stability. This allows to train very deep networks. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![**(a) Left**: Translation of 1-D nonlinear diffusion into a ResNet block. Adapted from [@ASAPW21]. **(b) Right**: Anisotropic diffusion as a ResNet block with a sum of four 1-D divergence blocks. The blocks $\bm{B}_0,...,\bm{B}_3$ correspond to the directions $\bm{e}_0,...,\bm{e}_3$.](tikz_figures/isotropic_block/isotropic_block.pdf){#fig:isotropicblock height="60mm"} ![**(a) Left**: Translation of 1-D nonlinear diffusion into a ResNet block. Adapted from [@ASAPW21]. **(b) Right**: Anisotropic diffusion as a ResNet block with a sum of four 1-D divergence blocks. The blocks $\bm{B}_0,...,\bm{B}_3$ correspond to the directions $\bm{e}_0,...,\bm{e}_3$.](tikz_figures/aggregate_block/aggregate_block.pdf){#fig:isotropicblock height="60mm"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- **Translating 1-D Diffusion into ResNets.** The basic translation of 1-D diffusion into ResNets is surprisingly simple [@ASAPW21; @RH20]: In vector notation, the explicit scheme [\[eq:iso_explicit\]](#eq:iso_explicit){reference-type="eqref" reference="eq:iso_explicit"} becomes $$\bm u^{k+1} \;=\; \bm u^k + \tau \bm D_h^-\left(\Phi\left(\bm D_h^+ \bm u^k\right)\right). \label{eq:exflux}$$ Here, $\bm D_h^+$ and $\bm D_h^-$ represent matrices computing forward and backward first order derivative approximations with grid size $h$. Comparing [\[eq:exflux\]](#eq:exflux){reference-type="eqref" reference="eq:exflux"} to the ResNet block [\[eq:resnet_block\]](#eq:resnet_block){reference-type="eqref" reference="eq:resnet_block"} reveals that both perform the same computations when identifying $$\bm K_1 = \bm D_h^+, \qquad \sigma_1 = \tau\Phi, \qquad \bm K_2 = \bm D_h^-, \qquad \bm b_1 = \bm b_2 = \bm 0, \qquad \sigma_2 = \text{Id}\,.$$ The computational graph for this is shown in Figure [2](#fig:isotropicblock){reference-type="ref" reference="fig:isotropicblock"}(a). Alt et al. [@ASAPW21] use this connection to advocate ResNet architectures with mirrored kernels $\bm K_1$ and $\bm K_2$ to guarantee stability in the Euclidean norm. Moreover, their experiments show advantages of nonmonotone activation functions. **Translating 2-D Anisotropic Diffusion into ResNets.** Our directional splitting allows also a natural translation of anisotropic diffusion into ResNets. We split the divergence term of 2-D anisotropic diffusion into a sum of four divergence terms of 1-D diffusion processes, and use the previous translation for each. This is illustrated in Figure [2](#fig:isotropicblock){reference-type="ref" reference="fig:isotropicblock"}(b). By appropriately concatenating the 2-D convolution kernels into 4-D tensors, we match the ResNet structure precisely. **Experiments.** Implementing numerical schemes for GPUs using CUDA can be labour-intensive and requires expertise. However, deep learning frameworks are capable of fully automatic and efficient parallelisation of user code. As we were able to decompose our discretisation into neural network primitives, we can use these frameworks to obtain an efficient implementation with little effort. We consider 10 iterations of an explicit scheme for edge-enhancing image diffusion (EED) [@We97] and compare three implementations. The first uses C and runs on the CPU. It computes entries of the $\delta$-stencil before applying it to the image. The second is an implementation in the PyTorch framework which follows the same strategy. As this style is uncommon in most neural networks, the implementation is fairly involved. The third also uses PyTorch, but follows our ResNet translation. It only requires two convolutions, one activation function, and a summation. This leads to a concise and simple implementation. For an image with $2048 \times 2048$ pixels, our C code takes $1.6\,s$ on an AMD 5800X CPU. Both PyTorch implementations perform one order of magnitude faster at $0.16\,s$ and $0.15\,s$ respectively on an Nvidia 3090 GPU. This demonstrates that our ResNet translation is able to significantly accelerate EED with a straightforward parallel implementation. It is even as fast as the much more involved stencil-based GPU implementation. This behaviour is consistent across image and batch sizes, provided that the total pixel count is sufficiently large. # Conclusions We have explored three aspects of anisotropic diffusion stencils. The first was an intuitive derivation of a large second-order stencil family based on directional splitting. While it covers the full stencil class of Weickert et al. [@WWW13], its derivation is simpler, and it requires only one free parameter ($\delta$) instead of two. Therefore, we call it the $\delta$-stencil family. Secondly, we have established a rigorous spectral norm estimate of the matrix associated to this stencil family. It allows to derive fairly tight time step size limits of explicit schemes to guarantee stability in the Euclidean norm. We have restricted ourselves to explicit schemes, since they are structurally similar to feedforward neural networks. Moreover, they form the backbone of acceleration methods based on super time stepping [@WGSB16] and extrapolation concepts [@HOWR16]. Further multigrid-like acceleration options may arise from multiscale network features such as pooling operations and U-net structures [@ASAPW21]. Thirdly, the directional splitting from our derivation has been instrumental in linking anisotropic diffusion to ResNets. It paves the road to an effortless and efficient parallelisation with libraries such as PyTorch. This also illustrates the usefulness of neural networks outside their original field of machine learning. In our ongoing work, we also address the reverse direction: We investigate how neural architectures can benefit from the integration of anisotropic diffusion. **Acknowledgements.** We thank Kristina Schaefer for careful proofreading. [^1]: This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 741215, ERC Advanced Grant INCOVID).

arxiv_math

--- abstract: | We present the development and analysis of a reinforcement learning (RL) algorithm designed to solve continuous-space mean field game (MFG) and mean field control (MFC) problems in a unified manner. The proposed approach pairs the actor-critic (AC) paradigm with a representation of the mean field distribution via a parameterized score function, which can be efficiently updated in an online fashion, and uses Langevin dynamics to obtain samples from the resulting distribution. The AC agent and the score function are updated iteratively to converge, either to the MFG equilibrium or the MFC optimum for a given mean field problem, depending on the choice of learning rates. A straightforward modification of the algorithm allows us to solve mixed mean field control games (MFCGs). The performance of our algorithm is evaluated using linear-quadratic benchmarks in the asymptotic infinite horizon framework. author: - "Andrea Angiuli[^1]" - "Jean-Pierre Fouque[^2]" - "Ruimeng Hu[^3]" - "Alan Raydan[^4]" bibliography: - reference.bib title: Deep Reinforcement Learning for Infinite Horizon Mean Field Problems in Continuous Spaces --- **Keywords:** Actor-critic, Linear-quadratic control, Mean field game, Mean field control, Mixed mean field control game, Score matching, Reinforcement learning, Timescales. # Introduction *Mean field games* (MFG) and *mean field control* (MFC)---collectively dubbed mean field problems---are mathematical frameworks used to model and analyze the behavior and optimization of large-scale, interacting agents in settings with varying degrees of cooperation. Since the early 2000s, with the seminal works [@lions2007; @malhame2006], MFGs have been used to study the equilibrium strategies of competitive agents in a large population, accounting for the aggregate behavior of the other agents. Alternately, MFC, which is equivalent to optimal control of McKean-Vlasov SDEs [@mckean1966; @mckean1967], focuses on optimizing the behavior of a central decision-maker controlling the population in a cooperative fashion. Cast in the language of stochastic optimal control, both frameworks center on finding an optimal control $\alpha_t$ which minimizes a cost functional objective $J(\alpha)$ subject to given state dynamics in the form of a stochastic differential equation. What distinguishes mean field problems from classical optimal control is the presence of the mean field distribution $\mu_t$, which may influence both the cost functional and the state dynamics. The mean field is characterized by a flow of probability measures that emulates the effect of a large number of participants whose individual states are negligible but whose influence appears in the aggregate. In this setting, the state process $X_t$ models a representative player from the crowd in the sense that the mean field should ultimately be the law of the state process: $\mu_t = \mathcal{L}(X_t)$. The distinction between MFG and MFC, a competitive game versus a cooperative governance, is made rigorous by precisely how we enforce the relationship between $\mu_t$ and $X_t$. We will address the details of the MFG/MFC dichotomy in greater depth in . MFG and MFC theories have been instrumental in understanding and solving problems in a wide range of disciplines, such as economics, social sciences, biology, and engineering. In finance, mean field problems have been applied to model and analyze the behavior of investors and markets. For instance, MFG can be used to model the trading strategies of individual investors in a financial market, taking into account the impact of the overall market dynamics. Similarly, MFC can help optimize the management of large portfolios, where the central decision-maker seeks to maximize returns while considering the average behavior of other investors. For in-depth examples of mean field problems in finance, we refer the reader to [@carmona2015; @carmona-mfg; @carmona-lecture]. Although traditional numerical methods for solving MFG and MFC problems have proceeded along two avenues, solving a pair of coupled partial differential equations (PDE) [@carmona2021b] or a forward-backward system of stochastic differential equations (FBSDE) [@angiuli2019], there has been growing interest in solving mean field problems in a model-free way [@unified_q_learning; @capponi_lehalle_2023; @guo2019; @perrin2021; @lauriere2022; @carmona2021]. With this in mind, we turn to *reinforcement learning* (RL), an area of machine learning that trains an agent to make optimal decisions through interactions with a "black box" environment. RL can be employed to solve complex problems, such as those found in finance, traffic control, and energy management, in a model-free manner. A key feature of RL is its ability to learn from trial-and-error experiences, refining decision-making policies to maximize cumulative rewards. *Temporal difference* (TD) methods [@sutton1988] are a class of RL algorithms that are particularly well-suited for this purpose. They estimate value functions by updating estimates based on differences between successive time steps, combining the benefits of both dynamic programming and Monte Carlo approaches for efficient learning without requiring a complete model of the environment. For a comprehensive overview of the foundations and numerous families of RL strategies, consult [@Sutton1998]. *Actor-critic* (AC) algorithms---the modern incarnations of which were introduced in [@degris2012]---are a popular subclass of TD methods where separate components, the actor and the critic, are used to update estimates of both a policy and a value function. The actor is responsible for selecting actions based on the current policy, while the critic evaluates the chosen actions and provides feedback to update the policy. By combining the strengths of both policy- and value-based approaches, AC algorithms achieve more stable and efficient learning. The mean field term itself poses an interesting problem regarding how to numerically store and update a probability measure on a continuous space in an efficient manner. Some authors have chosen to discretize the continuous space, which leads to a vectorized representation as in [@unified_q_learning; @capponi_lehalle_2023], while others have looked towards deep learning and deep generative models [@perrin2021]. We extend the latter avenue by considering a method of distributional learning known as *score-matching* [@hyvarinen2005] in which a probability distribution is represented by the gradient of its log density, i.e., its score function. A parametric representation of the score function is updated using samples from the underlying distribution and allows us to compute new samples from the distribution using a discrete version of Langevin dynamics. We explain how to modify the score-matching procedure for our online regime in . Building off of the work of [@unified_q_learning; @capponi_lehalle_2023], in which the authors adapt tabular Q-learning [@watkins1989] to solve discrete-space MFG and MFC problems, this paper introduces an AC algorithm in the style of advantage actor-critic [@pmlr-v48-mniha16] for solving continuous-space mean field problems in a unified manner. That is to say, for a given mean field problem, we use the *same* algorithm to solve for both the MFG and MFC solutions simply by adjusting the relative learning rates of the parametric representations of the actor, critic, and mean field distribution. Our method combines the mean field with the actor-critic paradigm by concurrently learning the score function of the mean field distribution along with the optimal control, which we derive from the policy learned by the actor. The rest of the paper is organized as follows. In , we review the infinite horizon formulation for asymptotic mean field problems and recall the relevant background from RL, respectively. In we modify the Markov decision process setting of RL to apply to mean field problems and present our central algorithm. Numerical results and comparisons with benchmark solutions are presented in . As a concluding application, we alter the algorithm in to apply to *mean field control games* (MFCG), an extension of mean field problems combining both MFG and MFC to model multiple large homogeneous populations where interactions occur not only within each group, but also between groups. # Infinite Horizon Mean Field Problems {#sec: mf problems} In this section, we introduce the framework of mean field games and mean field control in the continuous-time infinite horizon setting. We further emphasize the mathematical distinction between the two classes of mean field problems, highlighting that they yield distinct solutions despite the apparent similarities in their formulation. In both cases, the mathematical setting is a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}= (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$ satisfying the usual conditions which supports an $m$-dimensional Brownian motion $(W_t)_{t \geq 0}$. The measurable function $f: \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^{d}) \times \mathbb{R}^k \to \mathbb{R}$ is known as the running cost, and $\beta > 0$ is a discount factor. For the state dynamics we have drift $b: \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^{d}) \times \mathbb{R}^k \to \mathbb{R}^d$ and volatility $\sigma: \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^{d}) \times \mathbb{R}^k \to \mathbb{R}^{d \times m}$. We will focus on the asymptotic formulation of the infinite horizon mean field problem. In this formulation, we seek a control in the feedback form that depends solely on the state process of the representative player $X: [0, \infty) \times \Omega \to \mathbb{R}^d$, with no explicit time dependency. In other words, the control function is of the form $\alpha: \mathbb{R}^d \to \mathbb{R}^k$, and the trajectory of the control will be given by $\alpha_t = \alpha(X_t)$. This choice allows us to frame the problem more naturally in terms of the Markov decision process setting of reinforcement learning (see ), which is naturally formulated with time-independent policies. ## Mean Field Games {#subsec: mfg} The solution of a mean field game, known as a mean field game equilibrium, is a control-mean field pair $$(\hat{\alpha}, \hat{\mu}) \in \mathbb{A} \times \mathcal{P}(\mathbb{R}^{d}),$$ where $\mathbb{A}$ is the set of measurable functions $\alpha: \mathbb{R}^d \to \mathbb{R}^k$, satisfying the following conditions: 1. $\hat{\alpha}$ solves the stochastic optimal control problem $$\label{eq: mfg cost} \inf _{\alpha \in \mathbb{A}} J_{\hat{\mu}}(\alpha)=\inf _{\alpha \in \mathbb{A}} \mathbb{E}\left[ \int_0^\infty e^{-\beta t} f\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu}, \alpha(X_t^{\alpha, \hat{\mu}})\right)\, \ensuremath{\mathrm{d}}t \right],\quad \beta>0,$$ subject to $$\label{eq: mfg dynamics} \ensuremath{\mathrm{d}}X_t^{\alpha, \hat{\mu}}=b\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu}, \alpha(X_t^{\alpha, \hat{\mu}})\right) \, \ensuremath{\mathrm{d}}t+\sigma\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu}, \alpha(X_t^{\alpha, \hat{\mu}})\right)\, \ensuremath{\mathrm{d}}W_t, \quad X_0^{\alpha, \hat{\mu}}=\xi;$$ 2. $\hat{\mu} = \lim_{t \to \infty} \mathcal{L}(X_t^{\hat{\alpha}, \hat{\mu}})$, where $\mathcal{L}(X^{\hat{\alpha}, \hat{\mu}}_t)$ refers to the law of $X^{\hat{\alpha}, \hat{\mu}}_t$. This problem models a scenario in which an infinitesimal player seeks to integrate into a crowd of players already in the asymptotic regime as time tends toward infinity. The resulting stationary distribution represents a Nash equilibrium under the premise that any new player entering the crowd sees no benefit in diverging from this established asymptotic behavior. ## Mean Field Control {#subsec: mfc} A mean field control problem solution is a control $\alpha^* \in \mathbb{A}$ which satisfies an optimal control problem with McKean-Vlasov dynamics: $$\label{eq: mfc cost} \inf _{\alpha \in \mathbb{A}} J(\alpha)=\inf _{\alpha \in \mathbb{A}} \mathbb{E}\left[ \int_0^\infty e^{-\beta t} f\left(X_t^{\alpha}, \mu^\alpha, \alpha(X_t^{\alpha})\right)\, \ensuremath{\mathrm{d}}t \right],$$ subject to $$\label{eq: mfc dynamics} \ensuremath{\mathrm{d}}X_t^{\alpha}=b\left(X_t^{\alpha}, \mu^\alpha, \alpha(X_t^{\alpha})\right) \, \ensuremath{\mathrm{d}}t+\sigma\left(X_t^{\alpha}, \mu^\alpha, \alpha(X_t^{\alpha})\right)\, \ensuremath{\mathrm{d}}W_t, \quad X_0^{\alpha}=\xi,$$ using the notation $\mu^\alpha = \lim_{t \to \infty} \mathcal{L}(X_t^{\alpha})$. We will also adopt the notation $\mu^*$ to refer to $\mu^{\alpha^*}$---the limiting distribution for the mean field distribution under the optimal control. In this alternate scenario, we are considering the perspective of a central organizer. Their objective is to identify the control which yields the best possible stationary distribution, ensuring that the societal costs incurred are the lowest possible when a new individual integrates into the group. Although the initial distribution $\xi$ is specified in both cases, under suitable ergodicity assumptions, the optimal controls $\hat{\alpha}$ and $\alpha^*$ are independent of this initial distribution. For an in-depth treatment of infinite horizon mean field problems, with explicit solutions for the case of linear dynamics and quadratic cost, refer to [@malhame2020]. ## Mean Field Game/Control Distinction We summarize the crucial mathematical distinction between MFG and MFC. In the former, one must solve an optimal control problem depending on an arbitrary distribution $\mu$ and then recover the mean field $\hat{\mu}$, which yields the law of the optimal limiting state trajectory. If we consider the map $$\Phi(\mu) = \lim_{t \to \infty} \mathcal{L}(X^{\tilde{\alpha}, \mu}_t),$$ where $\tilde{\alpha} = \arg \min J_{\mu}(\alpha)$, then the MFG equilibrium arises as a fixed point of $\Phi$ in the sense that $$\hat{\mu} = \Phi(\hat{\mu}).$$ In the latter case, the mean field is explicitly the law of the state process throughout the optimization and should be thought of as a pure control problem in which the law of the state process influences the state dynamics. Note that in the MFC case, the distribution $\mu^\alpha$ "moves" with the choice of control $\alpha$, while in the MFG case, it is "frozen" during the optimization step and then a fixed point problem is solved. These interpretations play a key role in guiding the development of this paper's central algorithm, detailed in . Crucially, we conclude this section by noting that, in general, $$(\hat{\alpha}, \hat{\mu}) \neq (\alpha^*, \mu^*),$$ for the same choice of running cost, discount factor, and state dynamics. Indeed, we will encounter examples of mean field problems with differing solutions when we test our algorithm against benchmark problems in . # Reinforcement Learning and Actor-Critic Algorithms {#sec: rl_ac} Reinforcement learning is a family of machine learning strategies aimed at choosing the sequence of actions which maximizes the long-term aggregate reward from an environment in a model-free way, i.e., assuming no explicit knowledge of the state dynamics or the reward function. Intuitively, one should imagine a black box environment in which an autonomous agent makes decisions in discrete time and receives immediate feedback in the form of a scalar reward signal. At stage $n$, the agent is in a state $X_{t_n}$ from a given set of states $\mathcal{X}$ and selects an action $A_{t_n}$ from a set of actions $\mathcal{A}$. The environment responds by placing the agent in a new state $X_{t_{n+1}}$ and bestowing it with an immediate reward $r_{t_{n+1}} \in \mathbb{R}$. The agent continues choosing actions, encountering new states, and obtaining rewards in an attempt to maximize the total expected discounted return $$\label{eq: cummul reward} \mathbb{E}\left[ \sum_{n=0}^\infty \gamma^n r_{t_{n+1}} \right],$$ where $\gamma \in (0,1)$ is a discount factor specifying the degree to which the agent prioritizes immediate reward over long-term returns. The case in which we seek to minimize cost instead of maximize reward as in most financial applications, can be recast in the above setting by taking $r_{t_{n+1}} = -c_{t_{n+1}}$ where $c_{t_{n+1}}$ is the immediate cost incurred at the $n^{th}$ time-step. The expectation in [\[eq: cummul reward\]](#eq: cummul reward){reference-type="ref" reference="eq: cummul reward"} refers to the stochastic transition from $X_{t_n}$ to $X_{t_{n+1}}$ and, eventually, to the randomness in the choice of $A_{t_n}$. When the new state $X_{t_{n+1}}$ and immediate reward $r_{t_{n+1}}$ only depend on the preceding state $X_{t_n}$ and action $A_{t_n}$, the above formulation is known as a Markov decision process (MDP). The agent chooses its actions according to a policy $\pi: \mathcal{X} \to \mathcal{P}(\mathcal{A})$, which defines the probability that a certain action should be taken in a given state. The goal of the agent is then to find an optimal policy $\pi^*$ satisfying $$\pi^* \in \mathop{\mathrm{arg\,max}}_{\pi} \mathbb{E}_{\pi} \left[ \sum_{n=0}^\infty \gamma^n r_{t_{n+1}} \right].$$ As the reward $r_{t_{n+1}} = r(X_{t_n}, A_{t_n})$ is a function of the current state and current action, the value to be maximized does indeed depend on the policy $\pi$. For a given policy $\pi$, two quantities of interest in RL are the so-called *state-value function* $v_\pi:\mathcal{X} \to \mathbb{R}$ and the *action-value function* $q_\pi: \mathcal{X} \times \mathcal{A} \to \mathbb{R}$ given by $$\begin{aligned} v_\pi(x) &= \mathbb{E}_{\pi} \left[ \sum_{n=0}^\infty \gamma^n r(X_{t_n}, A_{t_n}) \mid X_{t_0} = x \right],\\ q_\pi(x, a) &= \mathbb{E}_{\pi} \left[ \sum_{n=0}^\infty \gamma^n r(X_{t_n}, A_{t_n}) \mid X_{t_0} = x, A_{t_0} = a \right].\end{aligned}$$ The state-value function defines the expected return obtained from beginning in an initial state $x$ and following the policy $\pi$ from the get-go, whereas the action-value function defines the expected return starting from $x$, taking an initial action $a$, and then proceeding according to $\pi$ after the first step. Moreover, $v_\pi$ and $q_\pi$ are related to each other via the following: $$v_{\pi}(x) = \sum_{a \in \mathcal{A}} \pi(a \mid x) q_\pi(x, a).$$ The action-value function is integral to many RL algorithms since, assuming that the action-value function $q_*$ corresponding to an optimal policy is known, one can derive an optimal policy by taking the uniform distribution over the actions that maximize $q_*$: $$\pi^*(\cdot \mid x) = \mathop{\mathrm{unif}}\left(\mathop{\mathrm{arg\,max}}_{a\in \mathcal{A}} q_*(x, a)\right).$$ However, since this paper makes far more use of $v_\pi$ than $q_\pi$, we will henceforth refer to the former simply as the "value function\" and the latter as the "Q-function\" as is common in the literature. When referring to both $v_\pi$ and $q_\pi$, we may refer to them jointly as the value functions associated with $\pi$. ## Temporal Difference Methods {#subsec: td} In the search for an optimal policy, one often begins with an arbitrary policy, which is improved as the RL agent gains experience in the environment. A key factor in improving a policy is an accurate estimate of the associated value function since this allows us to quantify precisely how much better one policy is over another. The value function satisfies the celebrated *Bellman equation*, which relates the value of the current state to that of the successor state: $$\label{eq: bellman} v_\pi(x) = \mathbb{E}_\pi [r_{t_{n+1}} + \gamma v_\pi(X_{t_{n+1}}) \mid X_{t_n} = x].$$ Since the transition from $X_{t_n}$ to $X_{t_{n+1}}$ is Markovian, [\[eq: bellman\]](#eq: bellman){reference-type="ref" reference="eq: bellman"} holds for all $n \geq 0$, not just the initial state. Importantly, the Bellman equation uniquely defines the value function for a given $\pi$, a fact which underlies all algorithms under the umbrella of *dynamic programming*. Solving the Bellman equation for $v_\pi$ is impossible without knowing the reward function and state transition dynamics, so an alternative strategy is needed for our model-free scenario. Temporal difference methods center around iteratively updating an approximation $V$ to $v_{\pi}$ in order to sufficiently minimizes the TD error $\delta_n$, $$\label{eq: td error} \delta_n \coloneqq r_{t_{n+1}} + \gamma V(X_{t_{n+1}}) - V(X_{t_n})$$ at each timestep. TD methods use estimates of the value at future times to update the value at the current time, a strategy known as "bootstrapping". More importantly, the TD methods we reference here require only the immediate transition sequence $\{X_{t_n}, r_{t_{n+1}}, X_{t_{n+1}}\}$ and no information regarding the MDP model. ## Actor-Critic Algorithms {#subsec: ac} Actor-critic algorithms form a subset of TD methods in which explicit representations of the policy (the actor) and the value function (the critic) are stored. Often, the representation of the policy is a parametric family of density functions in which the parameters are the outputs of another parametric family of functions, e.g., linear functions, polynomials, and neural networks. In the implementation discussed in , our actor is represented by a feedforward neural network which outputs the mean and standard deviation of a normal distribution. The action $A$ is then sampled according to this density. The benefit of a stochastic policy such as this is that it allows for more exploration of the environment so that the agent does not myopically converge to a suboptimal policy. Since the value function simply outputs a scalar, it may be represented by any sufficiently rich family of real-valued functions. Let $$\Pi_{\psi} \approx \pi \qquad \text{and} \qquad V_{\theta} \approx v$$ be the parametric approximations of $\pi$ and $v$, both differentiable in their respective parameters $\psi$ and $\theta$. The goal for AC algorithms is to converge to an optimal policy by iteratively updating the actor to maximize the value function and updating the critic to satisfy the Bellman equation. For the critic, this suggests minimizing the following loss function at the $n^{th}$ step: $$L_V(\theta) \coloneqq \left( r_{n+1} + \gamma V_\theta(X_{t_{n+1}}) - V_\theta(X_{t_n}) \right)^2 \eqqcolon \delta_n^2.$$ Note that the terms inside the square are precisely the TD error $\delta_n$ from [\[eq: td error\]](#eq: td error){reference-type="ref" reference="eq: td error"}. The traditional gradient TD update treats the term $y_{t_{n+1}} = r_{n+1} + \gamma V_{\theta}(X_{t_{n+1}})$---known as the *TD target*---as a constant and only considers the term $-V_{\theta}(X_{t_n})$ as a function of $\theta$. This yields faster convergence from gradient descent as opposed to treating both terms as variables in $\theta$ [@vanHasselt2012]. With this in mind, the gradient of $L_V$ is then $$\nabla_{\theta} L_V(\theta) = -2 \delta_n \nabla_{\theta} V_{\theta}(X_{t_n}),$$ meaning that, with some learning rate $\rho_V > 0$, we can update the parameters of the critic iteratively using gradient descent: $$\theta' = \theta + 2\rho_V \delta_n \nabla_{\theta} V_{\theta}(X_{t_n}).$$ Updating the actor, on the other hand, is not so obvious since updating $\psi$ to maximize $V_\theta$ requires somehow computing $\nabla_\psi V_\theta$. Since the connection between $\Pi_\psi$ and $V_\theta$ is not explicit, it is not clear how to compute this gradient a priori. Thankfully, the desired relation comes in the form of the *policy gradient theorem* [@sutton1999], which relates a parameterized policy and its value function via the following: $$\label{eq: pgt} \nabla_\psi v_{\Pi_\psi}(x) \propto \mathbb{E}_{\Pi_\psi}\left[q_{\Pi_\psi}(X_{t_n}, A_{t_n}) \nabla_\psi \log \Pi_\psi(A_{t_n} \mid X_{t_n})\right]$$ for any initial state $x \in \mathcal{X}$, where $v_{\Pi_{\psi}}$ and $q_{\Pi_{\psi}}$ are the true value functions associated with the parameterized policy $\Pi_{\psi}$. As a result of the Bellman equation, we have the identity $q_{\pi}(x, a) = \mathbb{E}_{\pi} [ r_{t_{n+1}} + \gamma v_\pi(X_{t_{n+1}})]$ given that $r_{t_{n+1}}$ was the reward obtained by taking the action $a$ in the state $x$. Moreover, adding an arbitrary \"baseline\" value $\lambda$ to $q_{\Pi_\psi}(X_{t_n}, A_{t_n})$ does not alter the gradient in [\[eq: pgt\]](#eq: pgt){reference-type="ref" reference="eq: pgt"} as long as $\lambda$ does not depend on the action $A_{t_n}$. A common baseline value demonstrated to reduce variance and speed up convergence is $\lambda = -v_{\Pi_\psi}(X_{t_n})$ [@vanHasselt2012]. With this in mind, we can replace $q_{\Pi_{\psi}}(X_{t_n}, A_{t_n})$ in [\[eq: pgt\]](#eq: pgt){reference-type="ref" reference="eq: pgt"} with the TD error $\delta_n = r_{n+1} + \gamma v_{\Pi_\psi}(X_{t_{n+1}}) - v_{\Pi_\psi}(X_{t_n})$ which allows us to reuse $\delta_n$ from its role in updating the critic. As a whole, this suggests the following loss function for the actor: $$L_{\Pi}(\psi) \coloneqq -\delta_n \log \Pi_\psi(A_{t_n} \mid X_{t_n})$$ For a learning rate $\rho_\Pi > 0$, the gradient descent step would then be $$\psi' = \psi + \rho_\Pi \delta_n \nabla_{\psi} \log \Pi_\psi(A_{t_n} \mid X_{t_n}).$$ In practical applications, updating the actor and critic in the above fashion at each step generally yields convergence to an optimal policy and value function, respectively. While convergence has been proven in the case of linearly parameterized actor and critic [@konda2003], convergence in the general case is still an open problem. ### Relative Learning Rates for Actor and Critic Since the gradient descent learning rates play a crucial role in the development of our fundamental algorithm presented in , we briefly comment on the choice of learning rates in AC algorithms. The AC framework alternates between two key steps: refining the critic to accurately approximate the value function associated with the actor's policy---known as *policy evaluation*---and updating the actor to maximize the value returned by the critic---known as *policy improvement*. As the policy improvement step relies on the policy gradient theorem ([\[eq: pgt\]](#eq: pgt){reference-type="ref" reference="eq: pgt"}), a sufficiently precise critic is required for its success. Hence, the learning rates for the actor and critic are traditionally chosen such that $$\rho_{\Pi} < \rho_V.$$ This constraint prompts the critic to learn at a quicker pace compared to the actor, thereby ensuring that the value function from the policy evaluation phase closely aligns with the policy's true value function. # Unified Mean Field Actor-Critic Algorithm for Infinite Horizon {#sec: mf_rl} In this section, we introduce a novel *infinite horizon mean field actor-critic* (IH-MF-AC) algorithm for solving both MFG and MFC problems in continuous time and continuous space. Although there have been significant strides in recasting the MDP framework for continuous-time using the Hamiltonian of the associated continuous-time control problem as an analog of the Q-function [@JMLR:v23:21-0947; @JMLR:v23:21-1387; @jia2023qlearning; @JMLR:v21:19-144], we instead take the classical approach of first discretizing the continuous-time problem and then applying the MDP strategies discussed in . As our focus is aimed at identifying the stationary solution of the infinite horizon mean field problems, discretizing time does not meaningfully depart from the original continuous-time problem presented in ; While in our ongoing work [@angiuli2024] where we tackle the finite horizon regime, the time-discretization must be treated with more care since the mean field becomes a flow of probability distributions parameterized by time, and the optimal control also becomes time-dependent in this context. In the sequel, we will first recast the mean field setting from as a discrete MDP parameterized by the distribution $\mu$ and then lay out the general procedure of the algorithm before addressing the continuous-space representation of $\mu$ via score functions in . addresses the justification for alternating between the MFG and MFC solutions using the actor, critic, and mean field learning rates. To begin, we fix a small step size $\Delta t > 0$ and consider the resulting time discretization $(t_0, t_1, t_2, \dots)$ where $t_{n} = n\Delta t$. We then rewrite the cost objectives in [\[eq: mfg cost,eq: mfc cost\]](#eq: mfg cost,eq: mfc cost){reference-type="ref" reference="eq: mfg cost,eq: mfc cost"} as the Riemann sum $$\mathbb{E}\left[ \sum_{n=0}^\infty e^{-\beta t_n} f(X_{t_n}, \mu, A_{t_n}) \Delta t \right]$$ and the state dynamics in [\[eq: mfg dynamics,eq: mfc dynamics\]](#eq: mfg dynamics,eq: mfc dynamics){reference-type="ref" reference="eq: mfg dynamics,eq: mfc dynamics"} as $$\label{eq: discrete dynamics} X_{t_{n+1}} = X_{t_n} + b(X_{t_n}, \mu, A_{t_n}) \Delta t + \sigma(X_{t_n}, \mu, A_{t_n}) \Delta W_n, \qquad \Delta W_n \sim \mathcal{N}(0,\Delta t).$$ This reformulation is directly in correspondence with the MDP setting presented in ---albeit, parameterized by $\mu$. Observe that $r_{t_{n+1}} = -f(X_{t_n}, \mu, A_{t_n}) \Delta t$, $\gamma = e^{-\beta \Delta t}$, and the state transition dynamics are given by [\[eq: discrete dynamics\]](#eq: discrete dynamics){reference-type="ref" reference="eq: discrete dynamics"}. In the style of the AC method described in , our algorithm maintains and updates a policy $\Pi_\psi$ and a value function $V_{\theta}$ which are meant as stand-ins for the control $\hat{\alpha}$ (resp. $\alpha^*$) of the MFG (resp. MFC) and the cost functional $J$, respectively. Both are taken to be feedforward neural networks. The third component is the mean field distribution $\mu$, which is updated simultaneously with the actor and critic at each timestep to approximate the law of $X_t$. The procedure at the $n^{th}$ step is as follows: the agent is in the state $X_{t_n}$ as a result of the dynamics in [\[eq: discrete dynamics\]](#eq: discrete dynamics){reference-type="ref" reference="eq: discrete dynamics"} where $\mu$ is replaced with the current estimate of the mean field $\mu_{n-1}$. The value of $X_{t_n}$ is then used to update the mean field, yielding a new estimate $\mu_n$ (see section for details). Using the actor's policy, the agent samples an action $A_{t_n} \sim \Pi_{\psi_n}(\cdot \mid X_{t_n})$ and executes it in the environment. It receives a reward which, unbeknownst to the agent, is given by $r_{t_{n+1}} = -f(X_{t_n}, \mu_n, A_{t_n})\Delta t$. The environment places the agent in a new state $X_{t_{n+1}}$ according to [\[eq: discrete dynamics\]](#eq: discrete dynamics){reference-type="ref" reference="eq: discrete dynamics"} using the distribution $\mu_n$ and the action $A_{t_n}$ while $\Pi_{\psi_n}$ and $V_{\theta_n}$ are updated according to the update rules from . To mimic the infinite horizon regime, we iterate this procedure for a large number of steps until we achieve convergence to the limiting distribution $\hat{\mu}$ (resp. $\mu^*$) and the equilibrium (resp. optimal) control $\hat{\alpha}$ (resp. $\alpha^*$). The complete pseudocode is presented in . Initial distribution $\xi$; number of time steps $N \gg 0$; discrete time step size $\Delta t$; neural network learning rates for actor $\rho_\Pi$, critic $\rho_V$, and score $\rho_\Sigma$; Langevin dynamics step size $\epsilon$. Initialize neural networks:\ **Actor** $\Pi_{\psi_0}: \mathbb{R}^d \to \mathcal{P}(\mathbb{R}^k)$\ **Critic** $V_{\theta_0}: \mathbb{R}^d \to \mathbb{R}$\ **Score** $\Sigma_{\varphi_0}: \mathbb{R}^d \to \mathbb{R}^d$ Sample $X_{t_0} \sim \xi$ Compute score loss: $\quad L_\Sigma (\varphi_n) = \mathop{\mathrm{tr}}\left( \nabla_x \Sigma_{\varphi_n}(X_{t_n}) \right) + \frac{1}{2}\left\lVert\Sigma_{\varphi_n}(X_{t_n})\right\rVert_2^2$ Update score with SGD: $\quad \varphi_{n+1} = \varphi_n -\rho_\Sigma \nabla_{\varphi} L_\Sigma (\varphi_n)$ Generate mean field samples $S_{t_n} = \left(S_{t_n}^{(1)}, S_{t_n}^{(2)}, \dots, S_{t_n}^{(k)}\right)$ from $\Sigma_{\varphi_{n+1}}$ using Langevin dynamics ([\[eq: langevin\]](#eq: langevin){reference-type="ref" reference="eq: langevin"}) with step size $\epsilon$ and compute $\overline{\mu}_{S_{t_n}} \coloneqq \frac{1}{k} \sum_{i=1}^k \delta_{S_{t_n}^{(i)}}$. Sample action: $\quad A_{t_n} \sim \Pi_{\psi_n}(\cdot \mid X_{t_n})$ Observe reward from the environment: $\quad r_{n+1} = -f(X_{t_n}, \overline{\mu}_{S_{t_n}}, A_{t_n}) \Delta t$ Observe the next state from the environment:\ $\quad X_{t_{n+1}} = b(X_{t_n}, \overline{\mu}_{S_{t_n}}, A_{t_n})\Delta t + \sigma(X_{t_n}, \overline{\mu}_{S_{t_n}}, A_{t_n}) \sqrt{\Delta t} \,z_n, \qquad z_n \sim \mathcal{N}(0,1)$ Compute TD target: $\quad y_{n+1} = r_{n+1} + e^{-\beta \Delta t} V_{\theta_n}(X_{t_{n+1}})$ Compute TD error: $\quad \delta_{\theta_n} =y_{n+1} - V_{\theta_n}(X_{t_n})$ Compute critic loss: $\quad L_V(\theta_n) = \delta_{\theta_n}^2$ Update critic with SGD: $\quad \theta_{n+1} = \theta_n - \rho_V \nabla_{\theta} L_V(\theta_n)$ Compute actor loss: $\quad L_{\Pi}(\psi_n) = -\delta_{\theta_n} \log \Pi_{\psi_n}(A_{t_n} \mid X_{t_n})$ Update actor with SGD: $\quad \psi_{n+1} = \psi_n - \rho_{\Pi} \nabla_{\psi} L_\Pi(\psi_n)$ $(\Pi_{\psi_N}, \Sigma_{\varphi_N})$ ## Representation of $\mu$ via Score-matching {#subsec: score-matching} The question of how to represent and update $\mu$ in the continuous-space setting deserves special consideration in this work. In [@capponi_lehalle_2023; @unified_q_learning], the authors deal with the discrete-space mean field distribution in a natural way, using a normalized vector containing the probabilities of each state. Each individual state is modeled as a one-hot vector (a Dirac delta measure), and the approximation $\mu_n$ is updated at each step using an exponentially weighted update of the form $\mu_{n+1} = \mu_{n} + \rho_\mu (\delta_{X_{t_{n}}} - \mu_{n})$ with the mean field learning rate $\rho_\mu > 0$. [@frikha2023] uses a similar update in the context of an AC algorithm for solving only MFC problems, while focusing on a more in-depth treatment of the continuous-time aspect. The authors in [@perrin2021] tackle continuous state spaces for the MFG problem using the method of normalizing flows, which pushes forward a fixed latent distribution, such as a Gaussian, using a series of parameterized invertible maps [@rezende2015]. There is reason to believe that other deep generative models, such as generative adversarial networks (GANs) or variational auto-encoders (VAEs), may yield successful representations of the population distribution with their own drawbacks and advantages. In our case, partly due to its simplicity of implementation, we opt for the method known as *score-matching* [@hyvarinen2005], which has been successfully applied to generative modeling [@song2019]. If $\mu$ has a density function $p_\mu: \mathbb{R}^d \to \mathbb{R}$, then its score function is defined as $$s_\mu(x) = \nabla \log p_\mu(x).$$ The score function is a useful proxy for $\mu$ in the sense that we can use $s_\mu$ to generate samples from $\mu$ using a Langevin Monte Carlo approach. Given an initial sample $x_0$ from an arbitrary distribution and a small step size $\epsilon > 0$, the sequence defined by $$\label{eq: langevin} x_{m+1} = x_m + \frac{\epsilon}{2} s_\mu(x_m) + \sqrt{\epsilon}\,z_m, \qquad z_m \sim \mathcal{N}(0,1)$$ converges to a sample from $\mu$ as $m \to \infty$. From the standpoint of parametric approximation, if $(\Sigma_\varphi)_{\varphi \in \Phi}$ is a sufficiently rich family of functions from $\mathbb{R}^d \to \mathbb{R}^d$, the natural goal is to find the parameters $\varphi$ which minimize the residual $\mathbb{E}_{x \sim \mu} [ \left\lVert\Sigma_{\varphi}(x) - s_\mu(x)\right\rVert_2^2 ]$. Although we do not know the true score function, a suitable application of integration by parts yields an expression that is proportional to the previous residual but independent of $s_\mu$: $$\label{eq: expected score loss} \mathbb{E}_{x \sim \mu} \left[ \mathop{\mathrm{tr}}(\nabla_x \Sigma_{\varphi}(x)) + \frac{1}{2} \left\lVert\Sigma_\varphi(x)\right\rVert^2_2 \right].$$ We adapt the above expression for our online setting in the following way. At the $n^{th}$ step, we have a sample $X_{t_n}$ of the state process and a score representation $\Sigma_{\varphi_n}$. We take the loss function for $\Sigma$ to be $$L_\Sigma (\varphi_n) \coloneqq \mathop{\mathrm{tr}}\left( \nabla_x \Sigma_{\varphi_n}(X_{t_n}) \right) + \frac{1}{2}\left\lVert\Sigma_{\varphi_n}(X_{t_n})\right\rVert_2^2.$$ Assuming $\Sigma$ is differentiable with respect to $\varphi$, we then update the parameters using the gradient descent step $$\label{eq: score update} \varphi_{n+1} = \varphi_n - \rho_\Sigma \nabla_{\varphi} L_\Sigma (\varphi_n)$$ where $\rho_\Sigma > 0$ is the mean field learning rate. Now we can generate samples from $\Sigma_{\varphi_{n+1}}$ and take $\mu_n$ to be the empirical distribution of these samples. More concretely, let $S_{t_n} = \left(S_{t_n}^{(1)}, S_{t_n}^{(2)}, \dots, S_{t_n}^{(k)}\right)$ be the $k$ samples generated from $\Sigma_{\varphi_{n+1}}$ using the Langevin Monte Carlo algorithm in [\[eq: langevin\]](#eq: langevin){reference-type="ref" reference="eq: langevin"}, and let $$\mu_n = \overline{\mu}_{S_{t_n}},$$ where the notation $\overline{\mu}_S \coloneqq \frac{1}{k} \sum_{i=1}^k \delta_{S^{(i)}}$ denotes the empirical distribution of the points $S = (S^{(1)}, S^{(2)}, \dots, S^{(k)})$. By the law of large numbers, $\overline{\mu}_{S_{t_n}}$ converges to the true distribution corresponding to $\Sigma_{\varphi_{n+1}}$ as $k \to \infty$. In the context of generative modeling, the gradient descent update in [\[eq: score update\]](#eq: score update){reference-type="ref" reference="eq: score update"} is usually evaluated with several mini-batches of independent samples all from a single distribution. This contrasts with our online approach in which each update is done with the current state $X_{t_n}$, which is generated from a different distribution than the previous state. We justify this as a form of bootstrapping in which we attempt to learn a target distribution that is continuously moving, but ultimately converging to the limiting distribution of the MFG or MFC. Since our updates depend on individual samples, we expect the loss $L_\Sigma$ to be a noisy estimate of the expectation in [\[eq: expected score loss\]](#eq: expected score loss){reference-type="ref" reference="eq: expected score loss"}, which may slow down convergence. Rather than updating at every timestep, another option would be to perform a batch update after every $m > 1$ timesteps using all samples $(X_{t_n}, X_{t_{n+1}}, \dots, X_{t_{n+(m-1)}})$ generated along the state trajectory, which may accelerate convergence by reducing variance. It is important to acknowledge that the $m$ samples will come from different distributions, so the batch update will also introduce bias into the gradient estimate. This may be mitigated by instead running multiple trajectories in parallel and updating the score function at each step using the samples $(X^{(1)}_{t_n}, X^{(2)}_{t_n}, \dots, X^{(m)}_{t_n})$ from the same timestep. ## Unifying Mean Field Game and Mean Field Control Problems {#subsec: unifying} Having laid out the general algorithm, we now address the issue of unifying the MFG and MFC formulations in the style of [@unified_q_learning; @capponi_lehalle_2023]. The intuitions presented in regarding the difference between MFG and MFC suggest that the interplay between the learning rates $\rho_{\Pi}$, $\rho_{V}$, and $\rho_{\Sigma}$ may be used to differentiate between the two solutions of the mean field problem. Taking $\rho_\Sigma < \min\{\rho_\Pi, \rho_V\}$ emulates the notion of solving the classical control problem corresponding to a fixed (frozen) $\mu$---or, in this case, a slowly moving $\mu$---and then updating the distribution to match the law of the sate process in an iterative manner. This matches the strategy discussed in for finding an MFG equilibrium. Conversely, taking $\rho_\Sigma > \max\{\rho_\Pi, \rho_V\}$ is more in keeping with simultaneous optimization of the mean field and the policy, which should yield the MFC solution as discussed in . For a more rigorous justification of the correspondence between the learning rates and mean field problem solutions in the vein of Borkar's two timescale approach [@borkar1997; @borkar2008], consult [@unified_q_learning; @capponi_lehalle_2023; @angiuli2023b]. # Numerical Results {#sec: numerical} ## A Linear-Quadratic Benchmark We test our algorithm on a 1-dimensional linear-quadratic (LQ) mean field problem where we wish to optimize $$\label{eq: lq cost} \mathbb{E}\left[ \int_0^\infty e^{-\beta t} \left(\frac{1}{2} \alpha_t^2+c_1\left(X_t-c_2 m\right)^2+c_3\left(X_t-c_4\right)^2+c_5 m^2\right) \, \ensuremath{\mathrm{d}}t \right]$$ with state dynamics $$\label{eq: lq dynamics} \ensuremath{\mathrm{d}}X_t = \alpha_t\, \ensuremath{\mathrm{d}}t + \sigma \, \ensuremath{\mathrm{d}}W_t, \qquad t \in [0, \infty)$$ where $m = \int x \,\mu(\ensuremath{\mathrm{d}}x)$ so that the mean field dependence is only through the first moment of the asymptotic distribution $\mu$. Note that the state dynamics depend only linearly on the control $\alpha$, and the running cost function depends on $\alpha$, $X$, and $m$ quadratically, hence the name linear-quadratic. The various terms in [\[eq: lq cost\]](#eq: lq cost){reference-type="ref" reference="eq: lq cost"} have the following interpretations: the first and last terms penalize $\alpha$ and $m$ from being too large, the second term addresses the relationship between the state process and the mean field distribution, which penalizes $X$ from deviating too far from $c_2m$, and the third term penalizes $X$ for being far from $c_4$. The coefficients $c_1$, $c_3$, and $c_5$ determine the relative influence of each term on the total cost. Both the MFG and MFC problems corresponding to [\[eq: lq cost,eq: lq dynamics\]](#eq: lq cost,eq: lq dynamics){reference-type="ref" reference="eq: lq cost,eq: lq dynamics"} have explicit analytic solutions, which we state now using the notation consistent with the full derivations in [@unified_q_learning]. ## Solution for Asymptotic Mean Field Game Define the constants $$\hat{\Gamma}_2 =\frac{-\beta+\sqrt{\beta^2+8\left(c_1+c_3\right)}}{4} \qquad \text{and} \qquad \hat{\Gamma}_1 = - \frac{2 \hat{\Gamma}_2 c_3 c_4}{\hat{\Gamma}_2 ( \beta + 2 \hat{\Gamma}_2 ) - c_1 c_2}.$$ Then the optimal control for the MFG is $$\label{eq: mfg control} \hat{\alpha}(x) = -\left(2 \hat{\Gamma}_2 x + \hat{\Gamma}_1\right).$$ Substituting [\[eq: mfg control\]](#eq: mfg control){reference-type="ref" reference="eq: mfg control"} into [\[eq: lq dynamics\]](#eq: lq dynamics){reference-type="ref" reference="eq: lq dynamics"} yields the Ornstein-Uhlenbeck process $$\ensuremath{\mathrm{d}}\hat{X}_t = -\left( 2\hat{\Gamma}_2 \hat{X}_t + \hat{\Gamma}_1 \right)\, \ensuremath{\mathrm{d}}t + \sigma \, \ensuremath{\mathrm{d}}W_t,$$ whose limiting distribution $\hat{\mu} = \lim_{t \to \infty} \mathcal{L}(\hat{X}_t)$ is $$\hat{\mu} = \mathcal{N}\left( -\frac{\hat{\Gamma}_1}{2 \hat{\Gamma}_2}, \frac{\sigma^2}{4 \hat{\Gamma}_2} \right).$$ Since the mean field interaction for the LQ problem is only through the mean $\hat{m} = \int x \,\hat{\mu}(\ensuremath{\mathrm{d}}x)$, we note that a simplified form of $\hat{m}$ is $$\label{eq: mfg mean} \hat{m} = -\frac{\hat{\Gamma}_1}{2 \hat{\Gamma}_2} = \frac{c_3 c_4}{c_1 + c_3 - c_1 c_2}.$$ ## Solution for Asymptotic Mean Field Control Proceeding as above, we define the constants $$\Gamma^*_2 =\frac{-\beta+\sqrt{\beta^2+8\left(c_1+c_3\right)}}{4} \qquad \text{and} \qquad \Gamma^*_1 = - \frac{2 \Gamma_2^* c_3 c_4}{\Gamma^*_2(\beta + 2 \Gamma^*_2) + c_5 - c_1 c_2(2 - c_2)}.$$ Then the optimal control for the MFC is $$\label{eq: mfc control} \alpha^*(x) = -\left(2 \Gamma_2^* x + \Gamma^*_1\right).$$ Substituting [\[eq: mfc control\]](#eq: mfc control){reference-type="ref" reference="eq: mfc control"} into [\[eq: lq dynamics\]](#eq: lq dynamics){reference-type="ref" reference="eq: lq dynamics"} yields the Ornstein-Uhlenbeck process $$\ensuremath{\mathrm{d}}X^*_t = -\left(2 \Gamma_2^* X^*_t + \Gamma^*_1\right)\, \ensuremath{\mathrm{d}}t + \sigma \, \ensuremath{\mathrm{d}}W_t,$$ whose limiting distribution $\mu^* = \lim_{t \to \infty} \mathcal{L}(X^*_t)$ is $$\mu^* = \mathcal{N}\left( -\frac{\Gamma^*_1}{2 \Gamma^*_2}, \frac{\sigma^2}{4 \Gamma^*_2} \right).$$ Since the mean field interaction is only through the mean $m^* = \int x \, \mu^*(\ensuremath{\mathrm{d}}x)$, we note that an equation for $m^*$ which only depends explicitly on the running cost coefficients is $$\label{eq: mfc mean} m^* = -\frac{\Gamma^*_1}{2 \Gamma^*_2} = \frac{c_3 c_4}{c_1 +c_3 + c_5 - c_1 c_2 (2- c_2)}.$$ ## Hyperparameters and Numerical Specifics {#subsec: mf hyperperams} For our numerical experiment, we test our algorithm on two different sets of values for the running cost coefficients $c_1$ to $c_5$ and volatility $\sigma$ as listed in . The discount factor is fixed in both cases to $\beta = 1$, and the continuous time is discretized using step size $\Delta t = 0.01$. The critic and score functions are both feedforward neural networks with one hidden layer of 128 neurons and a `tanh` activation function. The actor is also a feedforward neural network that outputs the mean and standard deviation of a normal distribution from which an action is sampled. Its architecture consists of a shared hidden layer of size 64 neurons and a `tanh` activation followed by two separate layers of size 64 neurons for the mean and standard deviation. The standard deviation layer is bookended by a `softmax` activation function to ensure its output is positive. The actor is meant to converge to a deterministic policy---also known as a pure control---over time, so in order to ensure a minimal level of exploration, we add a baseline value of $10^{-5}$ to the output layer. This straightforwardly mimics the notion of entropy regularization detailed in [@JMLR:v21:19-144]. Refer to for the learning rates used by the actor, critic, and score networks. summarizes the total parameter count for each neural network. For the Langevin Monte Carlo iterations, we pick a step size $\epsilon = 5 \times 10^{-2}$ as shown in . Rather than beginning the iterations at the $n^{th}$ step with samples $x_0 = (x_0^{(1)}, x_0^{(2)}, \dots, x_0^{(k)})$ from an arbitrary distribution, we take $x_0 = S_{t_{n-1}}$, the samples generated from the Langevin dynamics in the previous step, to accelerate convergence. We run 200 iterations at each step using $k=1000$ samples. The results of the algorithm applied to the LQ benchmark problem after $N=10^6$ iterations are displayed in [\[fig: results 1,fig: results 2\]](#fig: results 1,fig: results 2){reference-type="ref" reference="fig: results 1,fig: results 2"} with different sets of parameters along with the corresponding analytic solutions. We observe many of the same insights alluded to by [@capponi_lehalle_2023; @unified_q_learning] regarding the differences in recovering the MFG versus the MFC solution. Specifically, convergence to the MFG solution is more stable and faster than convergence to the MFC solution, as evidenced by the convergence plots in [\[fig: means 1,fig: means 2\]](#fig: means 1,fig: means 2){reference-type="ref" reference="fig: means 1,fig: means 2"}. Further, in both cases, there were certain runs in which instability was amplified by the AC algorithm, in which case we saw the weights of the neural networks diverge to numerical overflow. In order to combat this, we imposed a bound on the state space during the first 200,000 iterations, truncating all states to the interval $[-5, 5]$. We removed the artificial truncation following the initial iterations and were able to mitigate the instability issues leading to overflow. Observe that the optimal control is particularly well-learned within the support of the learned distribution. We postulate that a more intricate exploration scheme, perhaps along the lines of entropy regularization [@JMLR:v21:19-144], may aid in learning the control in a larger domain. We conclude by noting that for all the numerical results in this paper, the gradient descent updates of (steps 5, 13, and 15) were computed using the Adam optimization update [@kingma2015] rather than the stochastic gradient descent update suggested in the pseudocode. MFG MFC ----------------- -------------------- ----------------------------- $\rho_\Pi$ $5 \times 10^{-6}$ $5 \times 10^{-6}$ $\rho_V$ $10^{-5}$ $10^{-5}$ $\rho_{\Sigma}$ $\mathbf{10^{-6}}$ $\mathbf{5 \times 10^{-4}}$ $\epsilon$ $5\times 10^{-2}$ $5\times 10^{-2}$ : Choice of learning rates used to obtain results in all figures seen in this work---$\rho_\Pi$ for the actor, $\rho_V$ for the critic, $\rho_\Sigma$ for the score function, and $\epsilon$ for the Langevin dynamics. Boldface values indicate a difference in the learning rate between the MFG and MFC regimes. [\[tab: learning rates\]]{#tab: learning rates label="tab: learning rates"} Actor Critic Score --------------- -------- -------- -------- \# parameters 258 385 385 activation `tanh` `ELU` `tanh` : Parameter counts and activation functions for the actor $\Pi_{\psi}$, critic $V_{\theta}$, and score $\Sigma_{\varphi}$ neural networks used to obtain results in all of the figures seen in this work. [\[tab: architectures\]]{#tab: architectures label="tab: architectures"} $c_1$ $c_2$ $c_3$ $c_4$ $c_5$ $\sigma$ ------- ------- ------- ------- ------- ---------- 0.25 1.5 0.5 0.6 1.0 0.3 : Running cost coefficients and volatility for [\[eq: lq cost,eq: lq dynamics\]](#eq: lq cost,eq: lq dynamics){reference-type="ref" reference="eq: lq cost,eq: lq dynamics"}. The results for this parameter set are displayed in [\[fig: results 1,fig: means 1\]](#fig: results 1,fig: means 1){reference-type="ref" reference="fig: results 1,fig: means 1"}. [\[tab: coeffs 1\]]{#tab: coeffs 1 label="tab: coeffs 1"} {#fig: mfg results1 width=".99\\linewidth"} {#fig: mfc results1 width=".99\\linewidth"} {#fig: mfg mean 1 width=".98\\linewidth"} {#fig: mfc mean 1 width=".98\\linewidth"} $c_1$ $c_2$ $c_3$ $c_4$ $c_5$ $\sigma$ ------- ------- ------- ------- ------- ---------- 0.15 1.0 0.25 1.0 2.0 0.5 : Running cost coefficients and volatility for [\[eq: lq cost,eq: lq dynamics\]](#eq: lq cost,eq: lq dynamics){reference-type="ref" reference="eq: lq cost,eq: lq dynamics"}. The results for this parameter set are displayed in [\[fig: results 2,fig: means 2\]](#fig: results 2,fig: means 2){reference-type="ref" reference="fig: results 2,fig: means 2"}. [\[tab: coeffs 2\]]{#tab: coeffs 2 label="tab: coeffs 2"} {#fig: mfg results width=".99\\linewidth"} {#fig: mfc results width=".99\\linewidth"} {#fig: mfg mean 2 width=".98\\linewidth"} {#fig: mfc mean 2 width=".98\\linewidth"} # Actor-Critic Algorithm for Mean Field Control Games (MFCG) {#sec: mfcg} As observed in [@angiuli2023] in the case of tabular Q-learning, our IH-MF-AC algorithm () can easily be extended to the case of mixed mean field control game problems that involve two population distributions, a local one and a global one. This type of game corresponds to competitive games between a large number of large collaborative groups of agents. The local distribution is the "representative\" agent's group distribution, while the global distribution is the distribution of the entire population. We refer to [@angiuli2023; @angiuli2022c] for further details on MFCG, including the limit from finite player games to infinite player games. Note that the solution gives an approximation of the Nash equilibrium between the competitive groups. The solution of an infinite horizon mean field control game is a control-mean field pair $(\hat{\alpha}, \hat{\mu}) \in \mathbb{A} \times \mathcal{P}(\mathbb{R}^{d})$ satisfying the following: 1. $\hat{\alpha}$ solves the McKean-Vlasov stochastic optimal control problem $$\label{eq: mfcg cost} \inf _{\alpha \in \mathbb{A}} J_{\hat{\mu}}(\alpha)=\inf _{\alpha \in \mathbb{A}} \mathbb{E}\left[ \int_0^\infty e^{-\beta t} f\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu},\mu^{\alpha, \hat{\mu}}, \alpha(X_t^{\alpha, \hat{\mu}})\right)\, \ensuremath{\mathrm{d}}t \right],\quad \beta>0,$$ subject to $$\label{eq: mfcg dynamics} \ensuremath{\mathrm{d}}X_t^{\alpha, \hat{\mu}}=b\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu}, \mu^{\alpha, \hat{\mu}}, \alpha(X_t^{\alpha, \hat{\mu}})\right) \, \ensuremath{\mathrm{d}}t+\sigma\left(X_t^{\alpha, \hat{\mu}}, \hat{\mu}, \mu^{\alpha, \hat{\mu}}, \alpha(X_t^{\alpha, \hat{\mu}})\right)\, \ensuremath{\mathrm{d}}W_t, \quad X_0^{\alpha, \hat{\mu}}=\xi,$$ where $\mu^{\alpha, \hat{\mu}}=\lim _{t \rightarrow \infty} \mathcal{L}(\mathrm{X}_t^{\alpha, \hat{\mu}})$; 2. fixed point condition: $\hat{\mu} = \lim_{t \to \infty} \mathcal{L}(X_t^{\hat{\alpha}, \hat{\mu}})$. Note that conditions 1 and 2 above imply that $\hat{\mu} = \mu^{\hat{\alpha}, \hat{\mu}}$. We modify into our *infinite horizon mean field control game actor-critic* (IH-MFCG-AC) algorithm such that the global score function $\Sigma_\varphi$ represents the global distribution $\hat{\mu}$ and the local score function $\widetilde{\Sigma}_\xi$ represents the local distribution $\mu^{\alpha, \hat{\mu}}$. This is meant to mimic the parallel between the mean field game solution with the global distribution, and the mean field control solution with the local distribution. Following our intuition from , our choice of the now four learning rates will be chosen according to $$\label{eq: mfcg lr relations} \rho_{\Sigma} < \min\{\rho_\Pi, \rho_V\} < \max\{\rho_\Pi, \rho_V\} < \rho_{\widetilde{\Sigma}}.$$ Refer to for the complete pseudocode. Initial distribution $\xi$; number of time steps $N \gg 0$; discrete time step size $\Delta t$; neural network learning rates for actor $\rho_\Pi$, critic $\rho_V$, global score $\rho_\Sigma$, and local score $\rho_{\widetilde{\Sigma}}$; Langevin dynamics step size $\epsilon$. Initialize neural networks:\ **Actor** $\Pi_{\psi_0}: \mathbb{R}^d \to \mathcal{P}(\mathbb{R}^k)$\ **Critic** $V_{\theta_0}: \mathbb{R}^d \to \mathbb{R}$\ **Global Score** $\Sigma_{\varphi_0}: \mathbb{R}^d \to \mathbb{R}^d$\ **Local Score** $\widetilde{\Sigma}_{\xi_0}: \mathbb{R}^d \to \mathbb{R}^d$ Sample $X_{t_0} \sim \xi$ Compute score loss for $\Sigma$: $\quad L_\Sigma (\varphi_n) = \mathop{\mathrm{tr}}\left( \nabla_x \Sigma_{\varphi_n}(X_{t_n}) \right) + \frac{1}{2}\left\lVert\Sigma_{\varphi_n}(X_{t_n})\right\rVert_2^2$ Update $\Sigma$ with SGD: $\quad \varphi_{n+1} = \varphi_n -\rho_\Sigma \nabla_{\varphi} L_\Sigma (\varphi_n)$ Compute score loss for $\widetilde{\Sigma}$: $\quad L_{\widetilde{\Sigma}} (\xi_n) = \mathop{\mathrm{tr}}\left( \nabla_x \widetilde{\Sigma}_{\xi_n}(X_{t_n}) \right) + \frac{1}{2}\left\lVert\widetilde{\Sigma}_{\xi_n}(X_{t_n})\right\rVert_2^2$ Update $\widetilde{\Sigma}$ with SGD: $\quad \xi_{n+1} = \xi_n -\rho_{\widetilde{\Sigma}} \nabla_{\xi} L_{\widetilde{\Sigma}} (\xi_n)$ Generate mean field samples $S_{t_n} = \left(S_{t_n}^{(1)}, S_{t_n}^{(2)}, \dots, S_{t_n}^{(k)}\right)$ from $\Sigma_{\varphi_{n+1}}$ and $\widetilde{S}_{t_n} = \left(\widetilde{S}_{t_n}^{(1)}, \widetilde{S}_{t_n}^{(2)}, \dots, \widetilde{S}_{t_n}^{(k)}\right)$ from $\widetilde{\Sigma}_{\xi_{n+1}}$ using Langevin dynamics ([\[eq: langevin\]](#eq: langevin){reference-type="ref" reference="eq: langevin"}) with step size $\epsilon$ and compute $\overline{\mu}_{S_{t_n}} \coloneqq \frac{1}{k} \sum_{i=1}^k \delta_{S_{t_n}^{(i)}}$ and $\overline{\mu}_{\widetilde{S}_{t_n}} \coloneqq \frac{1}{k} \sum_{i=1}^k \delta_{\widetilde{S}_{t_n}^{(i)}}$. Sample action: $\quad A_{t_n} \sim \Pi_{\psi_n}(\cdot \mid X_{t_n})$ Observe reward from the environment: $\quad r_{n+1} = -f(X_{t_n}, \overline{\mu}_{S_{t_n}}, \overline{\mu}_{\widetilde{S}_{t_n}} A_{t_n}) \Delta t$ Observe next state from environment:\ $\quad X_{t_{n+1}} = b(X_{t_n}, \overline{\mu}_{S_{t_n}}, A_{t_n})\Delta t + \sigma(X_{t_n}, \overline{\mu}_{S_{t_n}}, A_{t_n}) \sqrt{\Delta t} \,z_n, \qquad z_n \sim \mathcal{N}(0,1)$ Compute TD target: $\quad y_{n+1} = r_{n+1} + e^{-\beta \Delta t} V_{\theta_n}(X_{t_{n+1}})$ Compute TD error: $\quad \delta_{\theta_n} =y_{n+1} - V_{\theta_n}(X_{t_n})$ Compute critic loss: $\quad L_V(\theta_n) = \delta_{\theta_n}^2$ Update critic with SGD: $\quad \theta_{n+1} = \theta_n - \rho_V \nabla_{\theta} L_V(\theta_n)$ Compute actor loss: $\quad L_{\Pi}(\psi_n) = -\delta_{\theta_n} \log \Pi_{\psi_n}(A_{t_n} \mid X_{t_n})$ Update actor with SGD: $\quad \psi_{n+1} = \psi_n - \rho_{\Pi} \nabla_{\psi} L_\Pi(\psi_n)$ $(\Pi_{\psi_N}, \Sigma_{\varphi_N}, \widetilde{\Sigma}_{\xi_N})$ ## A Linear-Quadratic Benchmark We test on the following linear-quadratic MFCG. We wish to minimize $$\label{eq: mfcg lq cost} \begin{split} \mathbb{E}\Biggl[\int_0^{\infty} e^{-\beta t}\biggl(\frac{1}{2} \alpha_t^2+c_1\left(\mathrm{X}_t^{\alpha, \mu}-c_2 m\right)^2+c_3\left(\mathrm{X}_t^{\alpha, \mu}-c_4\right)^2\\ \quad {}+\tilde{c}_1\left(\mathrm{X}_t^{\alpha, \mu}-\tilde{c}_2 m^{\alpha, \mu}\right)^2+\tilde{c}_5\left(m^{\alpha, \mu}\right)^2\biggr) \mathrm{d} t\Biggr] \end{split}$$ subject to the dynamics $$\label{eq: mfcg lq dynamics} \mathrm{d}X_t^{\alpha, \mu}=\alpha_t \,\mathrm{d} t+\sigma \, \mathrm{d}W_t, \qquad t \in [0, \infty)$$ where $m = \int x \, \ensuremath{\mathrm{d}}\mu(x)$ and $m^{\alpha,\mu} = \int x \, \ensuremath{\mathrm{d}}\mu^{\alpha, \mu}(x)$ and the fixed point condition $m=\lim _{t \rightarrow \infty} \mathbb{E}(X_t^{\hat{\alpha}, \mu})=m^{\hat{\alpha}, \mu}$ where $\hat{\alpha}$ is the optimal action. We present the analytic solution to the MFCG problem using notation consistent with the derivation in [@angiuli2023]. Define $$\Gamma_2=\frac{-\beta+\sqrt{\beta^2+8\left(c_1+c_3+\tilde{c}_1\right)}}{4} \qquad \text{and} \qquad \Gamma_1 = -\frac{2 \Gamma_2 c_3 c_4}{c_1\left(1-c_2\right)+\tilde{c}_1\left(1-\tilde{c}_2\right)^2+c_3+\tilde{c}_5}.$$ Then the optimal control for the MFCG is $$\label{eq: mfcg optimal control} \hat{\alpha}(x) = -(2 \Gamma_2 x + \Gamma_1).$$ Substituting [\[eq: mfcg optimal control\]](#eq: mfcg optimal control){reference-type="ref" reference="eq: mfcg optimal control"} into [\[eq: mfcg lq dynamics\]](#eq: mfcg lq dynamics){reference-type="ref" reference="eq: mfcg lq dynamics"} yields the Ornstein-Uhlenbeck process $$\ensuremath{\mathrm{d}}X_t = -\left(2 \Gamma_2 X_t + \Gamma_1\right)\, \ensuremath{\mathrm{d}}t + \sigma \, \ensuremath{\mathrm{d}}W_t$$ whose limiting distribution is $$\hat{\mu} = \mu^{\hat{\alpha}, \hat{\mu}} = \mathcal{N}\left( -\frac{\Gamma_1}{2\Gamma_2}, \frac{\sigma^2}{4 \Gamma_2} \right).$$ We note that an equation for $\hat{m}$ and $m^{\hat{\alpha}, \hat{\mu}}$ that only depends on the running cost coefficients is $$\label{eq: mfcg mean} m \coloneqq \hat{m} = m^{\hat{\alpha}, \hat{\mu}} = \frac{c_3 c_4}{c_1\left(1-c_2\right)+\tilde{c}_1\left(1-\tilde{c}_2\right)^2+c_3+\tilde{c}_5}.$$ ## Hyperparameters and Numerical Specifics {#hyperparameters-and-numerical-specifics} For the LQ benchmark problem, we consider the following choice of parameters: $c_1 = 0.5$, $c_2 = 1.5$, $c_3 = 0.5$, $c_4 = 0.25$, $\tilde{c}_1 = 0.3$, $\tilde{c}_2 = 1.25$, $\tilde{c}_5 = 0.25$, discount factor $\beta = 1$, and volatility $\sigma = 0.5$. The time discretization is again $\Delta t = 0.01$. Our intention was to modify as few of the numerical hyperparameters from as possible, including the neural network architectures for the actor and critic. The global and local score networks both inherit the architecture from the score network described in and . The learning rates for the networks are taken directly from with the global and score network learning rates assuming the values used to obtain the MFG and MFC results, respectively, from . This is to say, $(\rho_\Pi, \rho_V, \rho_\Sigma, \rho_{\widetilde{\Sigma}}) = (5\times 10^{-6}, 10^{-5}, 10^{-6}, 5 \times 10^{-4}$), which satisfy $\rho_\Sigma < \rho_\Pi < \rho_V < \rho_{\widetilde{\Sigma}}$, the learning rate inequality proposed in [\[eq: mfcg lr relations\]](#eq: mfcg lr relations){reference-type="ref" reference="eq: mfcg lr relations"}. The global and local distribution samples are computed at each time step using Langevin dynamics with $\epsilon = 5 \times 10^{-2}$ for 200 iterations using $k=1000$ samples. The results of the IH-MFCG-AC algorithm () are presented in [\[fig: mfcg results,fig: mfcg means\]](#fig: mfcg results,fig: mfcg means){reference-type="ref" reference="fig: mfcg results,fig: mfcg means"}. As expected, the learning of the global and local distributions reflects that of the optimal MFG distribution and the optimal MFC distribution, respectively. We observe that the global score is learned faster and with more accuracy than the local score, which is prone to outliers and instability. The optimal control is learned well within the support of the optimal distribution, but could possibly be expanded with a more advanced exploration strategy. ![The blue curve is a rolling average of the absolute error of the mean of samples produced from the global score function $\Sigma_{\varphi_n}$ (left)---denoted $\hat{m}_{t_n}$---and the local score function $\widetilde{\Sigma}_{\xi_n}$---denoted $m^{\alpha, \hat{\mu}}_{t_n}$---compared to the optimal mean $m$ from [\[eq: mfcg mean\]](#eq: mfcg mean){reference-type="ref" reference="eq: mfcg mean"}. Large jumps are due to random outliers which result from the stochasticity of our algorithm.](plots/mfcg_results.png){#fig: mfcg means width="0.98\\linewidth"} ![The blue curve is a rolling average of the absolute error of the mean of samples produced from the global score function $\Sigma_{\varphi_n}$ (left)---denoted $\hat{m}_{t_n}$---and the local score function $\widetilde{\Sigma}_{\xi_n}$---denoted $m^{\alpha, \hat{\mu}}_{t_n}$---compared to the optimal mean $m$ from [\[eq: mfcg mean\]](#eq: mfcg mean){reference-type="ref" reference="eq: mfcg mean"}. Large jumps are due to random outliers which result from the stochasticity of our algorithm.](plots/mfcg_mean.png){#fig: mfcg means width="0.98\\linewidth"} # Conclusion We have introduced a novel AC algorithm for solving infinite horizon mean field games and mean field control problems in continuous spaces. This algorithm, called IH-MF-AC, uses neural networks to parameterize a policy and value function, from which an optimal control is derived, as well as a score function, which represents the optimal mean field distribution on a continuous space. The MFG or MFC solution is arrived at depending on the choice of learning rates for the actor, critic, and score networks. We test our algorithm against a linear-quadratic benchmark problem and are able to recover the analytic solutions with a high degree of accuracy. Finally, we propose and test a modification of the algorithm, called IH-MFCG-AC, to solve the recently developed mixed mean field control game problems. # Acknowledgment {#acknowledgment .unnumbered} J.F. was supported by NSF grant DMS-1953035. R.H. was partially supported by the NSF grant DMS-1953035, the Regents' Junior Faculty Fellowship at UCSB, and a grant from the Simons Foundation (MP-TSM-00002783). Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 1720256) at UC Santa Barbara. R.H. is grateful to Jingwei Hu for the useful discussions. [^1]: Prime Machine Learning Team, Amazon. 320 West lake Ave N, SEA83, Seattle, WA, 98109, *aangiuli\@amazon.com*. The work presented here does not relate to this author's position at Amazon. [^2]: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, *fouque\@pstat.ucsb.edu*. [^3]: Department of Mathematics, and Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3080, *rhu\@ucsb.edu*. [^4]: Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, *alanraydan\@ucsb.edu*.

arxiv_math

--- abstract: | We consider non-local energy forms of fractional Laplace type on quasicircles and prove that they can be approximated by similar energy forms on polygonal curves. The approximation is in terms of generalized Mosco convergence along a sequence of varying Hilbert spaces. The domains of the energy forms are the natural trace spaces, and we focus on the case of quasicircles of Hausdorff dimension greater than one. The jump in Hausdorff dimension results in a mismatch of fractional orders, which we compensate by a suitable choice of kernels. We provide approximations of quasidiscs by polygonal $(\varepsilon,\infty)$-domains with common parameter $\varepsilon>0$ and show convergence results for superpositions of Dirichlet integrals and non-local boundary energy forms. address: - $^1$Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy - $^2$Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany - $^3$Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy author: - Simone Creo$^1$ - Michael Hinz$^2$ - Maria Rosaria Lancia$^3$ title: "Non-local boundary energy forms for quasidiscs: Codimension gap and approximation" --- # Introduction We investigate planar domains with fractal boundaries and non-local boundary energy forms of fractional Laplace type, explicitly defined as double integrals with hypersingular kernels. We prove that such non-local energy forms on a fractal boundary can be approximated by similar non-local energy forms on approximating polygonal curves. Since polygonal curves are one-dimensional, but the fractal boundary typically has Hausdorff dimension greater than one, the approximating energy forms have to be rescaled and modified appropriately to achieve convergence. Our aim is to provide quantitative information on how such rescalings and modifications can be chosen; this is motivated by recent studies of boundary value problems in [@CefaloCreoLanciaVernole19; @CreoLanciaNazarov20; @CreoLanciaNazarovVernole19]. A simple but non-trivial effect, which we call a "codimension gap", obstructs a naive solution. We overcome this effect by an averaging technique and the choice of suitable kernels for the approximating non-local forms; the method might be seen as a non-local variant of singular fractal homogenization, [@MoscoVivaldi07]. In addition, we consider superpositions of energy forms which are the sum of the Dirichlet integral on the domain plus the non-local boundary energy form. We show that such forms are limits of similar energy forms on approximating polygonal domains. As applications we obtain convergence results for solutions to associated elliptic and parabolic problems. Non-local energy forms on boundaries appear naturally in connection with trace spaces. Recall that if $B=B(0,1)\subset \mathbb{R}^2$ is the unit disk, then the unique weak solution $u\in H^1(B)$ to the Dirichlet problem for the Laplacian on $B$ with boundary values $\varphi\in H^{1/2}(\partial B)$ on the unit circle $\partial B$ is the Poisson integral $u=\mathbb{H}\varphi$ of $\varphi$. The boundary energy form $\mathcal{Q}^1_{\partial B}(\varphi):=\int_B |\nabla \mathbb{H}\varphi|^2dx$, $\varphi\in H^{1/2}(\partial B)$, on $L^2(\partial B)$ can explicitly be written as Douglas' integral, [@Douglas31; @Rado30], $$\mathcal{Q}^1_{\partial B}(\varphi)=\frac{1}{8\pi}\int_0^{2\pi}\int_0^{2\pi}\frac{(\varphi(\theta)-\varphi(\theta'))^2}{\sin^2((\theta-\theta')/2)}d\theta' d\theta \asymp \int_{\partial B}\int_{\partial B}\frac{(\varphi(x)-\varphi(y))^2}{|x-y|^2}dxdy,\quad \varphi\in H^{1/2}(\partial B),$$ see [@FOT94 Section 1.2]. Here $\asymp$ means comparability of quadratic forms. The infinitesimal generator of the boundary energy form $\mathcal{Q}^1_{\partial B}$ is the Dirichlet-to-Neumann operator, [@ArendtMazzeo07; @ArendtterElst11; @CaffarelliSilvestre07], which in the case of more general domains is a particularly interesting non-local operator of fractional Laplace type related to inverse problems and shape optimization. In the context of dynamical boundary conditions non-local boundary energy forms describe a jump-type transport along the boundary. The most classical dynamical boundary conditions are local Wentzell conditions involving a diffusion term; they were introduced by Feller, [@Feller52; @Feller54], partially motivated by questions in genetics, [@Feller51]. Wentzell, [@Ventsell59], then studied all possible boundary conditions that could occur for the generator of a diffusion process in a multidimensional domain. Many authors subsequently investigated dynamical boundary conditions from analytical and probabilistic point of view, see for instance [@ArendtKunkelKunze16; @BonyCourregePriouret68; @Galakhov01; @LanciaVernole14; @NicaiseLiMazzucato17; @Skubachevskii97; @Taira14; @VogtVoigt03] respectively [@GrothausVosshall17; @Ikeda61; @IkedaWatanabe89; @Ishikawa89; @SatoUeno65]. In applications Wentzell boundary conditions appear in connection with water waves, [@Korman83], and hydraulic fracturing, [@CannonMeyer71]; they are also closely related to effective boundary conditions in ferromagnetics, [@HaddarJoly01]. Boundary conditions of Robin-Wentzell type, [@Engel03; @Favini02; @MugnoloRomanelli06; @Nittka11; @Warma13], are known to be relevant in acoustics, [@GalRuizGoldsteinGoldstein03; @Mugnolo06]. Certain non-local boundary conditions have applications in plasma physics, [@Bitsadze69; @Gurevich12]. Non-local operators of integro-differential type appeared already in early references as summands or special cases, [@Ikeda61; @Ishikawa89; @SatoUeno65], and received increasing attention in more recent papers [@CefaloCreoLanciaVernole19; @CreoLanciaNazarovVernole19; @GalWarma16a; @GalWarma16b; @Gurevich12; @LanciaVelezVernole16], where non-local Robin and Wentzell boundary conditions have been considered. Here we consider purely non-local boundary conditions of, roughly speaking, fractional Laplace type. For simplicity we concentrate on a clarification of approximation results in this situation and do not add any diffusion term; mixed dynamics will be considered in follow-up studies. We are specifically interested in fractal boundaries. On the one hand most of the existing literature on dynamical boundary conditions requires at least Lipschitz regularity, so it seems desirable to provide generalizations to non-Lipschitz domains. On the other hand generalized Wentzell conditions appear in transmission problems, [@FilocheSapoval00; @Sapoval94], at highly conductive layers, [@Lancia02; @Mosco94; @PhamHuy74], and numerical experiments support the conjecture that enhanced heat draining may be achieved if the layer is fractal, [@CefaloCreoLanciaRodriguez23; @LanciaCefaloDellAcqua12]. Problems with local Wentzell conditions on Koch snowflake boundaries were studied in [@HLVT18; @LanciaVernole14], similar problems involving non-local boundary terms in [@LanciaVelezVernole16]. From a practical perspective, fractal boundaries are idealized limit objects, and only suitable non-fractal shapes approximating them will be computationally tractable. This makes approximations highly relevant; for equations on fractal state spaces the same idea had been emphasized in [@HinzMeinert20; @HinzMeinert22]. In [@LanciaVernole14] the convergence of problems with local Wentzell boundary conditions on polygonal pre-fractal domains to limit Wentzell problems on Koch snowflake domains was proved. The papers [@CefaloCreoLanciaVernole19; @CreoLanciaNazarov20; @CreoLanciaNazarovVernole19] contain regularity results and numerical approximations for problems with Wentzell boundary conditions having an additional non-local term on polygonal boundaries. This motivates the question how to provide pre-fractal-to-fractal approximations similar to [@LanciaVernole14] for such boundary conditions; there specifically adapted function spaces were used, [@LanciaVernole14 formulas (3.6) and (3.14)]. Here we take a different route and consider non-local boundary energy forms of fractional Laplace type whose domains are standard trace spaces. Then the intended approximations run into an obstacle: Suppose that $\Gamma\subset \mathbb{R}^2$ is the Koch snowflake curve and $\mu$ is the natural arc-wise self-similar Borel probability measure on $\Gamma$. The quadratic forms $$\mathcal{Q}^1(\varphi)=\int_\Gamma\int_\Gamma\frac{(\varphi(x)-\varphi(y))^2}{|x-y|^d\mu(B(x,|x-y|))}\mu(dx)\mu(dy) \asymp \int_\Gamma\int_\Gamma\frac{(\varphi(x)-\varphi(y))^2}{|x-y|^{2d}}\mu(dx)\mu(dy)$$ are comparable to the analog of Douglas' integral for the Koch snowflake domain $\Omega$ enclosed by $\Gamma$. Here $d>1$ is the Hausdorff dimension of $\Gamma$ and the functions $\varphi$ are recruited from the trace space $B^{2,2}_1(\Gamma)$ of $H^1(\mathbb{R}^2)$ on $\Gamma$, [@Jonsson94]. If $\Gamma_n$ are the natural pre-fractal polygonal boundaries enclosing the domains $\Omega_n$ and approximating $\Gamma$ as in [@LanciaVernole14], then they all have finite and nonzero one-dimensional Hausdorff measure $\mathcal{H}^1$ and the rescaled measures $\mu_n:=\frac{3^n}{4^n}\mathcal{H}^1(\cdot\cap \Gamma_n)$ converge weakly to $\mu$. For smooth functions $u$ on $\mathbb{R}^2$ we can observe that $$\int_{\Gamma_n}\int_{\Gamma_n}\frac{(u(\xi)-u(\eta))^2}{|\xi-\eta|^{d'}\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta)\to \int_\Gamma\int_\Gamma\frac{(u(x)-u(y))^2}{|x-y|^{d'}\mu(B(x,|x-y|))}\mu(dx)\mu(dy)$$ as $n\to \infty$. If the approximating boundary energy forms are supposed to have the natural trace spaces $H^{1/2}(\Gamma_n)$ of $H^1(\Omega_n)$ as domains, then $d'=1$ must be chosen and the desirable $d'=d>1$ is out of bounds. The parameter $d'$ should be viewed as a smoothness parameter; this can be seen from well-known results for non-local Dirichlet forms on metric measure spaces, [@ChenKumagai08]. The smaller codimension $2-d$ of the fractal curve $\Gamma$ in comparison to $2-1$ in the case of the polygons $\Gamma_n$ results in a higher smoothness parameter for the trace spaces, [@Jonsson94; @JonssonWallin84], and this is what we mean by "codimension gap". Here our workaround is to approximate $\mathcal{Q}^1$ by boundary energy forms $$\begin{gathered} \mathcal{Q}^1_n(\varphi)=\frac{3^n}{4^n}\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,C\: 3^{-n})^c}\frac{(\varphi(\xi)-\varphi(\eta))^2}{|\xi-\eta|^{1+d}}\mathcal{H}^1(d\xi)\mathcal{H}^1(d\eta)\notag\\ +\frac{3^n}{4^n}\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,C\: 3^{-n})}\frac{(\varphi(\xi)-\varphi(\eta))^2}{|\xi-\eta|^{2}}\mathcal{H}^1(d\xi)\mathcal{H}^1(d\eta);\end{gathered}$$ where $C>0$ is a fixed constant. For large interaction ranges the desired exponent $d'=d$ appears in the kernel, for short range interactions we keep $d'=1$. In the limit for $n\to \infty$ the first summand takes over, the second vanishes, and we recover $\mathcal{Q}^1$. Instead of $\mathcal{Q}^1$ and $\mathcal{Q}^1_n$, defined in terms of double integrals with explicit kernels, one could of course study the exact analogs of Douglas' integral, abstractly defined using Dirichlet form theory, [@BH91; @FOT94]. Then the convergence can actually be proved much easier. Such an abstract point of view is very suitable for compactness and stability results, [@HR-PT21; @HR-PT23], but in general it cannot be expected to give much explicit information. Numerical investigations, on the other hand, require a high level of detail with truly explicit formulas, as for instance achieved in [@Achdou06; @LanciaCefaloDellAcqua12; @NicaiseLiMazzucato17] for other mathematical questions. Our results here are a middle ground, maybe somewhat similar to [@Mosco13]: We study boundary energy forms that are only comparable to the exact analogs of Douglas' integral, but explicitly defined as double integrals. As illustrated, this reveals appropriate rescalings and the asymptotics of constants, which to understand was our aim. In the special case of the Koch snowflake our results are explicit. However, for general quasidiscs they are only "semi-explicit" in the sense that the polygonal curves do not necessarily follow any hierarchy and the rescaled measures $\mu_n$ might not admit a very simple expression, see Sections [3](#S:Polygons){reference-type="ref" reference="S:Polygons"} and [6](#S:Disks){reference-type="ref" reference="S:Disks"}. In Theorem [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"} we prove a result that contains the Mosco convergence, [@Mosco94], of the forms $\mathcal{Q}^1_n$ to $\mathcal{Q}^1$ as a very special case. Since the boundaries and measures give rise to varying Hilbert spaces, we formulate it in the framework of [@KuwaeShioya03], see [@KuwaeShioya03 Definition 2.11] or Appendix [10](#S:Notions){reference-type="ref" reference="S:Notions"}. The proof of Theorem [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"} is based on a variant of an averaging method originally used in [@Hinz09] for $d$-sets, related results can be found in [@HinzTeplyaev15; @MoscoVivaldi07; @PostSimmer21]. Here we extend the method from [@Hinz09] beyond $d$-sets. In particular, $\Gamma$ does not have to be a Koch snowflake curve: In [@Rohde01 Theorem 1.2] it was shown how to construct measures with refined doubling conditions for any quasicircle, see [@Wu98] for related arguments. Because these refined doubling conditions are just what a suitable trace theorem requires, [@Jonsson94 Theorem 1], we can formulate Theorem [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"} for non-local energy forms on more general quasicircles. Its proof does not make a direct use of quasiconformal parametrizations, but of metric consequences, [@Ahlfors63; @Lehto87; @MartioSarvas79; @Vaisala88]. The approximation of energy forms on closed quasidiscs, [@Gehring82], by similar energy forms on closed polygonal domains is verified in Theorem [Theorem 4](#T:MoscoWentzell){reference-type="ref" reference="T:MoscoWentzell"}. Intermediate results are Theorem [Theorem 2](#T:adhoc){reference-type="ref" reference="T:adhoc"}, where we provide a possible construction of approximating polygonal $(\varepsilon,\infty)$-uniform domains with common parameter $\varepsilon>0$, and Theorem [Theorem 3](#T:Mosco){reference-type="ref" reference="T:Mosco"}, where we show the Mosco convergence of Dirichlet integrals on these domains. The convergence of solutions to elliptic and parabolic problems with non-local boundary conditions on approximating polygonal domains to corresponding solutions on limit quasidiscs is observed in Theorems [Theorem 5](#T:ellipticstability){reference-type="ref" reference="T:ellipticstability"} and [Theorem 6](#T:parabolicstability){reference-type="ref" reference="T:parabolicstability"} respectively. We proceed as follows: In Section [2](#S:Quasi){reference-type="ref" reference="S:Quasi"} we state and discuss our standing assumptions. In Section [3](#S:Polygons){reference-type="ref" reference="S:Polygons"} we investigate sequences of approximating polygonal curves and measures defined by averaging. We introduce non-local energy forms on quasicircles and polygonal curves in Section [4](#S:Non-local){reference-type="ref" reference="S:Non-local"} and prove their Mosco-convergence in Section [5](#S:Moscoboundary){reference-type="ref" reference="S:Moscoboundary"}. Section [6](#S:Disks){reference-type="ref" reference="S:Disks"} contains an approximation scheme for quasidiscs in terms of polygonal $(\varepsilon,\infty)$-uniform domains. In Sections [7](#S:Mosco){reference-type="ref" reference="S:Mosco"}, [8](#S:MoscoWentzell){reference-type="ref" reference="S:MoscoWentzell"} and [9](#S:Apps){reference-type="ref" reference="S:Apps"} we provide results on the Mosco-convergence of Dirichlet integrals and superpositions and show brief applications to elliptic and parabolic equations. By $\mathcal{L}^2$ and $\mathcal{H}^s$ we denote the $2$-dimensional Lebesgue measure and the $s$-dimensional Hausdorff measure on $\mathbb{R}^2$, respectively. Given a Borel measure $\mu$ on $\mathbb{R}^2$, a Borel set $E\subset \mathbb{R}^2$ with $\mu(E)>0$ and a Borel function $f$ that is $\mu$-integrable over $E$, we use the notation $\fint_E f\:d\mu:=\frac{1}{\mu(E)}\int_E f\:d\mu$. # Quasicircles and measures {#S:Quasi} Recall that a *quasicircle* $\Gamma\subset \mathbb{R}^2$ is the image of a circle under a quasiconformal map of the plane $\mathbb{R}^2$ onto itself, [@Ahlfors63; @Gehring82; @Jones81; @Lehto87]. Our standing assumptions in this article are that $\Gamma\subset \mathbb{R}^2$ is a quasicircle, that $$\label{E:sd} 1\leq d\leq s<2$$ and $c_{\mu}>1$ are constants and that $\mu$ is a Borel probability measure on $\mathbb{R}^2$ with $\mathop{\mathrm{supp}}\mu=\Gamma$ satisfying $$\label{E:refineddoubling} c_{\mu}^{-1}k^d\mu(B(x,r))\leq \mu(B(x,kr)\leq c_{\mu} k^s\mu(B(x,r)),\quad x\in \Gamma,\ r>0,\ k>1,\ kr\leq \mathop{\mathrm{diam}}\Gamma,$$ and $$\label{E:lowerboundmu} \mu(B(x,1))\geq c_{\mu}^{-1},\quad x\in \Gamma.$$ *Examples 1*. [\[Ex:measure\]]{#Ex:measure label="Ex:measure"} 1. If $\Gamma$ is rectifiable, then ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) and ([\[E:lowerboundmu\]](#E:lowerboundmu){reference-type="ref" reference="E:lowerboundmu"}) hold for $\mu=\mathcal{H}^1(\Gamma\cap \cdot)/\mathcal{H}^1(\Gamma)$ with $s=d=1$ in ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}). 2. If $\Gamma$ is the classical arc-wise self-similar Koch snowflake curve with contraction ratio $1/3$, [@Falconer90; @Mattila95], then the arc-wise self-similar probability measure $\mu$ satisfies ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) with $d=s=\log 4/\log 3$. For variants with contraction ratio $1/4\leq p< 1/2$ the same is true with $d=s=\log 4/(-\log p)$. If $\Gamma$ is a homogeneous scale irregular snowflake-like curve [@Capitanelli10; @Mosco02; @Rohde01] and a law of large numbers holds for the sequence of scaling factors, then there is a Borel probability measure $\mu$ satisfying ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) suitable $1\leq d=s< 2$, [@Capitanelli10 p. 1224]. Also ([\[E:lowerboundmu\]](#E:lowerboundmu){reference-type="ref" reference="E:lowerboundmu"}) holds in all these cases. 3. In [@Rohde01 Theorem 1.2 and its proof] it was shown that for any quasicircle $\Gamma$ one can construct a Borel probability measure $\mu$ on $\mathbb{R}^2$ with $\mathop{\mathrm{supp}}\mu=\Gamma$ such that ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) and ([\[E:lowerboundmu\]](#E:lowerboundmu){reference-type="ref" reference="E:lowerboundmu"}) hold with $d=1$ and some $1\leq s<2$ in ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}). As pointed out, [@Rohde01 p. 645], the measures $\mu$ constructed there are generally not canonical in any way. A particularly interesting class of quasicircles are the general snowflake-like curves in [@Rohde01]; any quasicircle is the image of such a curve under a bi-Lipschitz map of the plane onto itself, [@Rohde01 Theorem 1.1]. *Remark 1*. 1. Property ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) implies that $\mu$ is volume doubling in the usual sense, more precisely, $$\label{E:mudoubling} \mu(B(x,2r))\leq c_{\mu,D}\mu(B(x,r)),\quad x\in \Gamma,\quad 0<r\leq (\mathop{\mathrm{diam}}\Gamma)/2,$$ with a constant $c_{\mu,D}>1$. 2. Property ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) also implies that $$\label{E:uplowreg} c^{-1}r^s\leq \mu(B(x,r))\leq c\:r^d,\quad x\in \Gamma,\quad 0<r\leq \mathop{\mathrm{diam}}\Gamma,$$ with a suitably readjusted constant $c>1$. In particular, the Hausdorff dimension of $\Gamma$ is bounded by $s$ and bounded below by $d$. 3. If $\mu$ satisfies ([\[E:uplowreg\]](#E:uplowreg){reference-type="ref" reference="E:uplowreg"}) with $d=s$, then it is said to be *$d$-regular*; in this case it also satisfies ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) with $d=s$. The measures in Examples [\[Ex:measure\]](#Ex:measure){reference-type="ref" reference="Ex:measure"} (i) and (ii) are $d$-regular. 4. The scaling properties in ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) allow the local dimension of $\Gamma$ to fluctuate continuously. They have been investigated by various authors, see for instance [@Assouad80; @BylundGudayol00; @Dynkin84; @LukkainenSaksman98; @VolbergKonyagin87]. In [@Jonsson94] they have been used as geometric hypotheses for trace and extension results for Besov spaces. 5. Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} ensures that arcs in one quasi partition have about the same diameter. This metric uniformity makes our approximation method in Section [5](#S:Moscoboundary){reference-type="ref" reference="S:Moscoboundary"} tractable, and it seems more important for this method than a uniform local dimension. # Polygonal approximations and measures {#S:Polygons} We agree to take all subarcs $a$ of $\Gamma$ to be closed in the relative topology of $\Gamma$. We call two subarcs of $\Gamma$ *disjoint* if their interiors are disjoint. To a collection of disjoint subarcs $a_n$ of $\Gamma$ that cover $\Gamma$ we refer as a *partition* of $\Gamma$. *Assumption 1*. [\[A:basicass\]]{#A:basicass label="A:basicass"} We assume that $0<p,q<1$, $M>1$, $\omega=(\omega_1,\omega_2,...)\in \{p,q\}^\mathbb{N}$ and $(\mathcal{I}_n)_{n\geq 0}$ is a sequence of finite partitions $\mathcal{I}_n=\{a_{n,j}\}_j$ of $\Gamma$ such that for all $n$ and $j$ we have $$\label{E:basicass} \omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma\leq \mathop{\mathrm{diam}}a_{n,j}<M\:\omega_1\cdots \omega_n \mathop{\mathrm{diam}}\Gamma.$$ *Examples 2*. 1. For homogeneous snowflake-like curves, [@Rohde01], one can choose natural partitions. Given $1/4\leq p< 1/2$, consider the polygonal curve in $\mathbb{R}^2$ having four edges of length $p$ and vertices $(0,0)$, $(p,0)$, $(1/2,h_p)$, $(1-p,0)$ and $(1,0)$, where $h_p^2=p^2-(1/2-p)^2$. To any scaled copy of this polygonal curve we refer as a *Koch type segment with parameter $p$*. Suppose that $\Gamma$ is the classical arc-wise self-similar Koch snowflake, [@Falconer90; @LanciaVernole14], and that it is constructed starting from an equilateral triangle with edge length one by successive and simultaneous replacement of edges by 'outward pointing' Koch type segments with parameter $3^{-n}$. At stage $n$, let $V_n$ be the vertex set of the resulting polygon with edges of length $3^{-n}$ and let $\mathcal{I}_n$ be the natural partition of $\Gamma$ into the arcs connecting two neighboring points from $V_n$. Then Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} holds with $p=q=1/3$ and these very $\mathcal{I}_n$. Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} is also satisfied for the obvious modification using a more general parameter $p$ as indicated. If $1/4\leq q< p<1/2$ and at each stage $n$ all edges are replaced by a Koch type segment with the same parameter $\omega_n\in \{p,q\}$, then we recover the homogeneous scale irregular snowflake-like curves in [@Capitanelli10; @Mosco02] and [@Rohde01], and with an analogous natural partition Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} is satisfied. 2. Recall that a *Jordan curve* $\Gamma\subset \mathbb{R}^2$ is the image of a circle under an injective continuous map into $\mathbb{R}^2$. For any Jordan curve $\Gamma\subset \mathbb{R}^2$ and sufficiently small $0<q<1$ one can find a sequence of finite partitions $(\mathcal{I}_n)_{n\geq 0}$ of $\Gamma$ into arcs having the same diameter, [@AltBeer35; @Schoenberg40], see [@HerronMeyer12 Section 2.5] for detailed comments. This suffices to satisfy Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} with $p=q$ and $M=2$. Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be satisfied and let the $\mathcal{I}_n=\{a_{n,j}\}_j$ be as there. For fixed $n$ and $j$, let $e_{n,j}\subset \mathbb{R}^2$ denote the line segment connecting the two endpoints of the arc $a_{n,j}$ and let $\Gamma_n:=\bigcup_j e_{n,j}$ denote the closed polygonal curve obtained as the union of these line segments. Since each edge $e_{n,j}$ of $\Gamma_n$ is contained in the convex hull of $a_{n,j}$, we have $$\label{E:convex} \max_{x\in a_{n,j}, w\in e_{n,j}}|x-w|\leq \mathop{\mathrm{diam}}a_{n,j}.$$ Recall that since $\Gamma$ is a quasicircle, it satisfies the *bounded turning condition* (also called *arc condition*) $$\label{E:3point} S:=\sup_{x,y\in \Gamma,\ x\neq y}\frac{\mathop{\mathrm{diam}}a(x,y)}{|x-y|}<+\infty,$$ where $a(x,y)$ denotes a subarc of $\Gamma$ of minimal diameter connecting $x$ and $y$. It is well known that this condition characterizes quasicircles within the class of planar Jordan curves, [@Ahlfors63], see for instance [@Lehto87 Chapter I, Section 6.5] for details. Given two closed subsets $F_1$ and $F_2$ of $\mathbb{R}^2$, their Hausdorff distance is defined as $$d_H(F_1,F_2):=\inf\left\lbrace \varepsilon>0: F_1\subset (F_2)_\varepsilon\ \text{and}\ F_2\subset (F_1)_\varepsilon\right\rbrace,$$ where for each closed $F\subset \mathbb{R}^2$ and each $\varepsilon>0$ we write $(F)_\varepsilon:=\{x\in\mathbb{R}^2: \mathop{\mathrm{dist}}(x,F)\leq \varepsilon\}$ for the closed $\varepsilon$-parallel set of $F$. **Lemma 1**. *[\[L:Hausdorffconvbd\]]{#L:Hausdorffconvbd label="L:Hausdorffconvbd"} Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be in force.* 1. *The polygonal curves $\Gamma_n$ converge to $\Gamma$ in the Hausdorff sense, more precisely $$\lim_{n\to\infty} d_H(\Gamma_n,\Gamma)<M\:\omega_1\cdots \omega_n \mathop{\mathrm{diam}}\Gamma.$$* 2. *For each $n$ and each edge $e_{n,j}$ of $\Gamma_n$ we have $$\label{E:edgelength} S^{-1}\mathop{\mathrm{diam}}a_{n,j}\leq \mathcal{H}^1(e_{n,j})\leq \mathop{\mathrm{diam}}a_{n,j}$$* *Proof.* Item (i) follows from ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and ([\[E:convex\]](#E:convex){reference-type="ref" reference="E:convex"}), item (ii) from ([\[E:convex\]](#E:convex){reference-type="ref" reference="E:convex"}) and ([\[E:3point\]](#E:3point){reference-type="ref" reference="E:3point"}). ◻ *Examples 3*. If $\Gamma$ is the classical Koch snowflake curve, endowed with the natural partitions $\mathcal{I}_n$ as mentioned in Example [Examples 2](#Ex:basicass){reference-type="ref" reference="Ex:basicass"} (i), then the polygonal curve $\Gamma_n$ is the familiar pre-fractal at stage $n$. Since $\Gamma$ is a quasicircle, there is some $0<\theta\leq 1$ such that for all $\xi\in \Gamma$ and $r>0$ $$\label{E:theta} \text{only the connected component of $\Gamma\cap \overline{B(\xi,r)}$ that contains $\xi$ intersects $B(\xi,\theta r)$.}$$ This is another of the many conditions that characterize quasicircles within the class of planar Jordan curves, see for instance [@MartioSarvas79 2.28. Remark]. We call two subarcs $a$ and $a'$ in a partition $\mathcal{I}_n$ *adjacent* if they share an endpoint. By ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}), ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) two adjacent subarcs have comparable measure. **Lemma 2**. *Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be in force. Then for any $n\geq 1$ and any adjacent $a,a'\in \mathcal{I}_n$ we have $$\label{E:unicompare} \mu(a')\leq c_\mu\Big(\frac{8MS}{\theta}\Big)^s\:\mu(a).$$* *Proof.* Suppose that $a\in \mathcal{I}_n$ and $x,y\in V_n$ are such that $a=a(x,y)$. Let $z\in a$ be such that $2r:=|z-x|=|z-y|$. Then the connected component of $\Gamma\cap \overline{B(z,r)}$ containing $z$ is $a\cap \overline{B(z,r)}$, and by ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) it follows that $\Gamma\cap B(z,\theta r)=a\cap B(z,\theta r)$. Consequently $$\mu(B(z,\theta r))=\mu(a\cap B(z,\theta r))\leq \mu(a).$$ Now suppose that $a'\in\mathcal{I}_n$ is adjacent to $a$. Then $$a\cup a'\subset B(z,2M\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma)\quad \text{and}\quad \frac{1}{4S} \omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma\leq r$$ by ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}), and by ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) therefore $$\mu(a')\leq \mu(a\cup a')\leq \mu(B(z,8MSr))\leq c_\mu\Big(\frac{8MS}{\theta}\Big)^s\:\mu(B(z,\theta r))\leq c_\mu\Big(\frac{8MS}{\theta}\Big)^s\:\mu(a).$$ ◻ Generalizing the notation used in Examples [Examples 2](#Ex:basicass){reference-type="ref" reference="Ex:basicass"} (i), let $V_n\subset \Gamma$ denote the finite set of all endpoints of arcs $a_{n,j}$ in $\mathcal{I}_n$; clearly $V_n\subset \Gamma_n$. We write $V_\ast:=\bigcup_{n\geq 0} V_n$. *Remark 2*. In the case of the homogeneous snowflake-like curves with natural partitions as in Examples [Examples 3](#Ex:Kochagain){reference-type="ref" reference="Ex:Kochagain"} (i) the sets of endpoints increase monotonically, $V_n\subset V_{n+1}$, $n\geq 0$. Also the partitions used in [@Rohde01 Theorem 1.2] to construct measures $\mu$ satisfying ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) have this property. Here we do not insist on it. Due to ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}) and ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) the measure $\mu$ is atom free and in particular, $\mu(V_\ast)=0$. We introduce Borel probability measures $\mu_n$ on the approximating polygonal curves $\Gamma_n$ by $$\label{E:disint} \mu_n(B):=\sum_{j}\frac{\mu(a_{n,j})}{\mathcal{H}^1(e_{n,j})}\mathcal{H}^1(B\cap e_{n,j}),\quad \text{$B\subset \Gamma_n$ Borel}.$$ *Remark 3*. The measures $\mu_n$ are comparable to $\mathcal{H}^1(\cdot \cap\Gamma_n)$: Since $0<\min_j\mu(a_{n,j})\leq \mu(a_{n,j})\leq 1$ for all $j$, ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and ([\[E:edgelength\]](#E:edgelength){reference-type="ref" reference="E:edgelength"}) imply that $$\label{E:absolute} \frac{\min_j \mu(a_{n,j})}{M\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma}\mathcal{H}^1(\cdot \cap \Gamma_n)\leq \mu_n\leq \frac{\#(\mathcal{I}_n) S}{\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma}\mathcal{H}^1(\cdot \cap \Gamma_n),$$ where $\#(\mathcal{I}_n)$ denotes the cardinality of $\mathcal{I}_n=\{a_{n,j}\}_j$. In particular, the density $d\mu_n/d\mathcal{H}^1(\cdot \cap\Gamma_n)$ is bounded and bounded away from zero. The comparison in ([\[E:absolute\]](#E:absolute){reference-type="ref" reference="E:absolute"}) implies that for any $0\leq p\leq +\infty$ we have $L^p(\Gamma_n,\mu_n)=L^p(\Gamma_n,\mathcal{H}^1(\cdot\cap\Gamma_n))$. *Examples 4*. If $\Gamma$ is the classical Koch snowflake curve, endowed with the natural partitions $\mathcal{I}_n$ and $\mu$ is the natural arc-wise self-similar probability measure, then $\mu(a_{n,j})=4^{-n}$, $\mathcal{H}^1(e_{n,j})=3^{-n}$, and consequently $$\label{E:munKochcase} \mu_n=\frac{3^n}{4^n}\:\mathcal{H}^1(\cdot \cap \Gamma_n).$$ The edge-wise averaging inherent to the measures $\mu_n$ can be shifted to functions. For each point $x\in \Gamma\setminus V_\ast$ and each $n$ let $a_n(x)$ be the unique arc in $\mathcal{I}_n$ such that $x\in a_n(x)\setminus V_n$ and $e_n(x)$ the edge of $\Gamma_n$ having the same endpoints as $a_n(x)$. We can rewrite ([\[E:disint\]](#E:disint){reference-type="ref" reference="E:disint"}) as $$\label{E:defmun} \mu_n(B)=\int_\Gamma \frac{1}{\mathcal{H}^1(e_n(x))}\:\mathcal{H}^1(B\cap e_n(x))\:\mu(dx),\quad \text{$B\subset \Gamma_n$ Borel}.$$ Given $\varphi\in L^1(\Gamma_n,\mu_n)$, we define a function $[\varphi]_n$ on $\Gamma\setminus V_\ast$ by edge-wise averaging on $\Gamma_n$, $$\label{E:deffn} [\varphi]_n(x):=\fint_{e_n(x)}\varphi(w)\:\mathcal{H}^1(dw),\quad x\in \Gamma\setminus V_\ast.$$ Identity ([\[E:defmun\]](#E:defmun){reference-type="ref" reference="E:defmun"}) and standard approximation then imply that $$\label{E:trading} \int_{\Gamma_n} \varphi(w)\:\mu_n(dw)=\int_\Gamma[\varphi]_n(x) \mu(dx).$$ We have $\lim_{n\to \infty}\mu_n= \mu$ in the weak sense: If $u\in C_b(\mathbb{R}^2)$, then $u(x)=\lim_{n\to \infty} [u]_n(x)$ at any $x\in \Gamma\setminus V_\ast$, and ([\[E:trading\]](#E:trading){reference-type="ref" reference="E:trading"}) and bounded convergence give $$\label{E:weakconveval} \lim_{n\to \infty} \int_{\Gamma_n} u\:d\mu_n=\int_\Gamma u\:d\mu.$$ Let $E_\Gamma:\mathop{\mathrm{Lip}}(\Gamma)\to \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$ denote the Whitney extension operator acting on the space $\mathop{\mathrm{Lip}}(\Gamma)$ of Lipschitz functions on $\Gamma$ and taking values in the space $\mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$ of bounded Lipschitz functions on $\mathbb{R}^2$, [@Stein70 Chapter VI, Theorem 3]. Using ([\[E:weakconveval\]](#E:weakconveval){reference-type="ref" reference="E:weakconveval"}) we find the following result on the convergence of Hilbert spaces in the sense of [@KuwaeShioya03 Section 2.2], see Appendix [10](#S:Notions){reference-type="ref" reference="S:Notions"}. **Lemma 3**. *Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be satisfied. Then for any $\varphi\in \mathop{\mathrm{Lip}}(\Gamma)$ we have $$\lim_{n\to \infty} \left\| E_\Gamma \varphi\right\|_{L^2(\Gamma_n,\mu_n)}=\left\|\varphi\right\|_{L^2(\Gamma,\mu)}.$$ With identification maps $\varphi\mapsto (E_\Gamma \varphi)|_{\Gamma_n}$, $\varphi\in \mathop{\mathrm{Lip}}(\Gamma)$, the sequence of Hilbert spaces $(L^2(\Gamma_n,\mu_n))_n$ converges to $L^2(\Gamma,\mu)$.* For later use we prove a lemma that refines ([\[E:absolute\]](#E:absolute){reference-type="ref" reference="E:absolute"}) in the sense that although the original measure $\mu$ may have local dimensional fluctuations, the measures $\mu_n$ are locally $1$-regular and volume doubling in a *uniformly scale dependent sense*. To formulate it, recall from [@Rohde01 Lemma 4.1] that there are constants $c>0$ and $\gamma>0$ such that for any $R>0$ any arc $a\subset \Gamma$ contains at most a number $$\label{E:NR} N(R)\leq c\:R^{\gamma}$$ of disjoint subarcs $a_1',...,a_{N(R)}'$ of diameter at least $(\mathop{\mathrm{diam}}a)/R$. Given $n\geq 1$ and $\xi\in \Gamma_n\setminus V_n$, let $e_n(\xi)$ be the unique edge of $\Gamma_n$ containing $\xi$ and let $a_n(\xi)$ be the arc in $\mathcal{I}_n$ having the same endpoints as $e_n(\xi)$. We use the shortcut notation $$\label{E:rn} r_n:=\frac{1}{2MS}\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma,\quad n\geq 1.$$ Note that $\lim_{n\to \infty} r_n=0$ by Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}. **Lemma 4**. *Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be satisfied. Given $C>1$, let $N=N(C+2MS)$ be as in ([\[E:NR\]](#E:NR){reference-type="ref" reference="E:NR"}). There is some $n_0\geq 1$ such that $$\label{E:mundreg} \frac{\mu(a_n(\xi))}{2CM^2S r_n}\:r\leq \mu_n(B(\xi,r))\leq \frac{N c_\mu^N \Big(\frac{8MS}{\theta}\Big)^{Ns} \mu(a_n(\xi))}{Mr_n}\:r$$ for all $n\geq n_0$, $\xi\in \Gamma_n\setminus V_n$ and $0<r<CM\:r_n$. In particular, $$\label{E:mundoubling} \mu_n(B(\xi,2r))\leq c_{D}\mu_n(B(\xi,r))$$ for all $n\geq n_0$, $\xi\in \Gamma_n\setminus V_n$ and $0<r<CM\:r_n/2$ with a constant $c_{D}>1$ that does not depend on $n$, $\xi$ or $r$.* *Examples 5*. If $\Gamma$ is the classical Koch snowflake, endowed with the natural partitions $\mathcal{I}_n$ the natural arc-wise self-similar probability measure $\mu$, then $\omega_1\cdots \omega_n=4^{-n}$ and $\mu(a_n(\xi))=3^{-n}$ for all $\xi\in \Gamma_n\setminus V_n$, and ([\[E:mundreg\]](#E:mundreg){reference-type="ref" reference="E:mundreg"}) reduces to $$c^{-1}\frac{4^n}{3^n} r\leq \mu_n(B(\xi,r))\leq c\frac {4^n}{3^n} r,\quad 0<r<3^{-n},$$ as expected from ([\[E:munKochcase\]](#E:munKochcase){reference-type="ref" reference="E:munKochcase"}). *Proof of Lemma [Lemma 4](#L:mundoubling){reference-type="ref" reference="L:mundoubling"}..* It suffices to verify the two inequalities in ([\[E:mundreg\]](#E:mundreg){reference-type="ref" reference="E:mundreg"}), condition ([\[E:mundoubling\]](#E:mundoubling){reference-type="ref" reference="E:mundoubling"}) then follows. If $0<r<\frac{1}{2S}\omega_1\cdots\omega_n\mathop{\mathrm{diam}}\Gamma$, then $r<\mathcal{H}^1(e_n(\xi))/2$ by ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and ([\[E:edgelength\]](#E:edgelength){reference-type="ref" reference="E:edgelength"}). This gives $\mathcal{H}^1(B(\xi,r)\cap e_n(\xi))\geq r$. If now $0< r<CMr_n$, then we can use ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}), ([\[E:edgelength\]](#E:edgelength){reference-type="ref" reference="E:edgelength"}) and ([\[E:disint\]](#E:disint){reference-type="ref" reference="E:disint"}) to see that $$\mu_n(B(\xi,r))\geq \mu_n(B(\xi,r/C))\geq \frac{\mu(a_n(\xi))}{M\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma}\mathcal{H}^1(B(\xi,r/C)\cap e_n(\xi))\geq \frac{\mu(a_n(\xi)) r}{CM\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma},$$ which is the lower bound in ([\[E:mundreg\]](#E:mundreg){reference-type="ref" reference="E:mundreg"}). To verify also the upper bound in ([\[E:mundreg\]](#E:mundreg){reference-type="ref" reference="E:mundreg"}), let $n_0$ be large enough to have $\max\{Cr_n,d_H(\Gamma,\Gamma_n)\}\leq \mathop{\mathrm{diam}}\Gamma/100$ for all $n\geq n_0$. Let such $n$ be fixed and $0< r<CMr_n$. Then there must be some $z\in \Gamma_n\setminus B(\xi,r)^c$. Starting at $z$, follow $\Gamma_n$ clockwise. Let $x$ be the last point of $V_n$ visited by $\Gamma_n$ after starting at $z$ and before entering $B(\xi,r)$ for the first time. Let $y$ be the first point of $V_n$ visited by $\Gamma_n$ after leaving $B(\xi,r)$ for the last time before returning to $z$. By ([\[E:3point\]](#E:3point){reference-type="ref" reference="E:3point"}) the arc $a(x,y)\subset\Gamma$ from $x$ to $y$ has diameter $$\mathop{\mathrm{diam}}a(x,y)\leq S|x-y|\leq S(2r+2\max_j\mathop{\mathrm{diam}}e_{n,j})\leq (C+2MS)\omega_1\cdots\omega_n\mathop{\mathrm{diam}}\Gamma.$$ By ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and the remarks preceding the lemma, the arc $a(x,y)$ can contain at most $N=N(C+2MS)$ subarcs $a'_1,...,a'_{N}$ from $\mathcal{I}_n$. Since $x,y\in V_n$, the corresponding line segments $e_1',...,e_N'\subset \Gamma_n$ are the only segments of $\Gamma_n$ that intersect $B(\xi,r)$. From ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}), ([\[E:edgelength\]](#E:edgelength){reference-type="ref" reference="E:edgelength"}) and ([\[E:unicompare\]](#E:unicompare){reference-type="ref" reference="E:unicompare"}) it now follows that $$\mu_n(B(\xi,r))\leq \sum_{i=1}^N\frac{\mu(a_i')}{\mathcal{H}^1(e_i')}\mathcal{H}^1(e_i'\cap B(\xi,r))\leq \frac{N c_\mu^N \Big(\frac{8MS}{\theta}\Big)^{Ns} \mu(a_n(\xi))\:2r}{S^{-1}\omega_1\cdots\omega_n\mathop{\mathrm{diam}}\Gamma}.$$ ◻ # Non-local energy forms on curves {#S:Non-local} Given $$\label{E:constellation} \frac{2-d}{2}<\alpha<1+\frac{2-s}{2}$$ and a Borel function $\sigma_\alpha:\mathbb{R}^2\times (0,+\infty)\to (0,+\infty)$ such that $$\label{E:symbol} c^{-1}r^{2\alpha-2}\mu(B(x,r))\leq \sigma_\alpha(x,r)\leq c\:r^{2\alpha-2}\mu(B(x,r)),\quad x\in \Gamma,\quad 0<r\leq \mathop{\mathrm{diam}}\Gamma$$ with $c>1$ independent of $x$ and $r$, we can define a quadratic form $\mathcal{Q}^\alpha$ on $L^2(\Gamma,\mu)$ by $$\label{E:boundaryenergy} \mathcal{Q}^\alpha(\varphi):=\int_\Gamma\int_\Gamma \frac{(\varphi(x)-\varphi(y))^2}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\mu(dx)\mu(dy),\quad\varphi\in L^2(\Gamma,\mu).$$ We follow [@Jonsson94] and write $B_\alpha^{2,2}(\Gamma)$ for the Hilbert space $(B_\alpha^{2,2}(\Gamma),\left\|\cdot\right\|_{B_\alpha^{2,2}(\Gamma)})$ of all $\varphi\in L^2(\Gamma,\mu)$ such that $$\label{E:Besovnorm} \left\|\varphi\right\|_{B_\alpha^{2,2}(\Gamma)}:=\left(\left\|\varphi\right\|_{L^2(\Gamma,\mu)}^2+ \mathcal{Q}^\alpha(\varphi)\right)^{1/2}$$ is finite. Since $\mu$ satisfies conditions ([\[E:refineddoubling\]](#E:refineddoubling){reference-type="ref" reference="E:refineddoubling"}) and ([\[E:lowerboundmu\]](#E:lowerboundmu){reference-type="ref" reference="E:lowerboundmu"}), [@Jonsson94 Theorem 1 and Proposition 2] show that $B_\alpha^{2,2}(\Gamma)$ is the trace space of $H^\alpha(\mathbb{R}^2)$ on $\Gamma$: Given $f\in H^\alpha(\mathbb{R}^2)$, the limit $$\widetilde{f}(x):=\lim_{r\to 0}\fint_{B(x,r)} f(y)dy$$ of $f$ exists at $H^\alpha(\mathbb{R}^2)$-quasi every $x\in \mathbb{R}^2$. Setting $$\label{E:traceasop} \mathrm{Tr}_\Gamma f:=\widetilde{f}$$ gives a bounded linear operator $\mathrm{Tr}_\Gamma: H^\alpha(\mathbb{R}^2)\to B_\alpha^{2,2}(\Gamma)$, and there is a bounded linear extension operator $\mathrm{E}_\Gamma:B_\alpha^{2,2}(\Gamma)\to H^\alpha(\mathbb{R}^2)$ of Whitney type such that $\mathrm{Tr}_\Gamma\circ \mathrm{E}_\Gamma$ is the identity. Since the space $\mathop{\mathrm{Lip}}_c(\mathbb{R}^2)$ of compactly supported Lipschitz functions on $\mathbb{R}^2$ is dense in $H^\alpha(\mathbb{R}^2)$, the space $\mathop{\mathrm{Lip}}(\Gamma)$ of Lipschitz functions on $\Gamma$ is dense in $B_{\alpha}^{2,2}(\Gamma)$. From $\mathcal{Q}^\alpha$ as defined in ([\[E:boundaryenergy\]](#E:boundaryenergy){reference-type="ref" reference="E:boundaryenergy"}), we obtain a symmetric bilinear (energy) form $(\mathcal{Q}^\alpha, B_{\alpha}^{2,2}(\Gamma))$ by polarization; it is a regular Dirichlet form on $L^2(\Gamma,\mu)$ in the sense of [@FOT94]. *Remark 4*. If $\mu$ is $d$-regular, then [@Jonsson94 Theorem 1] recovers a special case of [@JonssonWallin84 Chapter VI, Theorem 2], where $B_\alpha^{2,2}(\Gamma)$ (up to norm equivalence) is denoted by $B_{\alpha'}^{2,2}(\Gamma)$ with $$\label{E:alphaprime} \alpha'=\alpha-\frac{(2-d)}{2}.$$ See also [@Jonsson94 Example 1]. In this case $0<\alpha'<1$ and $c^{-1}r^{2\alpha'}\leq \sigma_\alpha(x,r)\leq c\:r^{2\alpha'}$ for all $x\in \Gamma$ and $0<r\leq \mathop{\mathrm{diam}}\Gamma$. *Examples 6*. Suppose that $\Gamma$ is the classical Koch snowflake curve, $\mu$ is the natural arc-wise self-similar probability measure and $\sigma_\alpha(x,r)\equiv r^{2\alpha'}$ with $\alpha'$ as in ([\[E:alphaprime\]](#E:alphaprime){reference-type="ref" reference="E:alphaprime"}). Then $$\mathcal{Q}^\alpha(\varphi)=\int_\Gamma\int_\Gamma \frac{(\varphi(x)-\varphi(y))^2}{|x-y|^{2\alpha'}\mu(B(x,|x-y|))}\mu(dx)\mu(dy),\quad \varphi\in B_\alpha^{2,2}(\Gamma).$$ Now let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be in force and let $$\label{E:constellation2} \frac12<\alpha<1+\frac{2-s}{2};$$ this implies ([\[E:constellation\]](#E:constellation){reference-type="ref" reference="E:constellation"}). Let $\sigma_\alpha$ be as in ([\[E:constellation\]](#E:constellation){reference-type="ref" reference="E:constellation"}) and ([\[E:symbol\]](#E:symbol){reference-type="ref" reference="E:symbol"}), and let $$A\geq \max\{128\:S,M\}.$$ For each $n$ and $\xi\in \Gamma_n$ set $$\varrho_n(\xi,r):=\begin{cases} \sigma_\alpha(\xi,r) & r> Ar_n,\\ r^{2\alpha-1} & r\leq Ar_n,\end{cases}$$ and consider the quadratic form $\mathcal{Q}^\alpha_n$ on $L^2(\Gamma_n,\mu_n)$ defined by $$\label{E:boundaryenergyn} \mathcal{Q}^\alpha_n(\varphi):=\int_{\Gamma_n}\int_{\Gamma_n} \frac{(\varphi(\xi)-\varphi(\eta))^2}{\varrho_n(\xi,|\xi-\eta|)\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta),\quad\varphi\in L^2(\Gamma_n,\mu_n).$$ Since $1/2<\alpha<3/2$ by ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}), the maximal domain $\{\varphi\in L^2(\Gamma_n,\mu_n): \mathcal{Q}^\alpha_n(\varphi)<+\infty\}$ of $\mathcal{Q}^\alpha_n$ agrees with the trace space $B_{\alpha}^{2,2}(\Gamma_n)$ of $H^{\alpha}(\mathbb{R}^2)$ on $\Gamma_n$ as discussed in the last section, and $$\varphi\mapsto \left(\left\|\varphi\right\|_{L^2(\Gamma_n,\mu_n)}^2+\mathcal{Q}^\alpha_n(\varphi)\right)^{1/2}$$ is a Hilbert space norm on $B_{\alpha}^{2,2}(\Gamma_n)$ equivalent to $\left\|\cdot\right\|_{B_{\alpha}^{2,2}(\Gamma_n)}$. As before, polarization gives a symmetric bilinear (energy) form $(\mathcal{Q}^\alpha_n, B_{\alpha}^{2,2}(\Gamma_n))$; it is a regular Dirichlet form on $L^2(\Gamma_n,\mu_n)$ in the sense of [@FOT94]. *Examples 7*. Suppose that $\Gamma$ is the classical Koch snowflake curve, the $\mathcal{I}_n$ are the natural partitions, $\mu$ is the natural arc-wise self-similar probability measure and $\sigma_\alpha(x,r)\equiv r^{2\alpha'}$ with $\alpha'$ as in ([\[E:alphaprime\]](#E:alphaprime){reference-type="ref" reference="E:alphaprime"}). Then there is some $c>0$ such that $r_n=c\:3^{-n}$, and we find that $$\begin{gathered} \mathcal{Q}^\alpha_n(\varphi)=\frac{3^n}{4^n}\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,Ac\: 3^{-n})^c}\frac{(\varphi(\xi)-\varphi(\eta))^2}{|\xi-\eta|^{2\alpha+d-1}}\mathcal{H}^1(d\xi)\mathcal{H}^1(d\eta)\notag\\ +\frac{3^n}{4^n}\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,Ac\: 3^{-n})}\frac{(\varphi(\xi)-\varphi(\eta))^2}{|\xi-\eta|^{2\alpha}}\mathcal{H}^1(d\xi)\mathcal{H}^1(d\eta).\end{gathered}$$ # Mosco convergence of boundary energy forms {#S:Moscoboundary} We make the following additional assumption. *Assumption 2*. For all $x\in \Gamma$ and all $0<r<\mathop{\mathrm{diam}}\Gamma$ we have $\mu(\partial B(x,r))=0$. *Remark 5*. If $d>1$, then by upper $d$-regularity and density comparison, [@Mattila95 Theorem 6.9], Assumption [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} is always satisfied. In this section we prove the following convergence in the KS-generalized Mosco sense, see Appendix [10](#S:Notions){reference-type="ref" reference="S:Notions"} for the notion. **Theorem 1**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} be satisfied, let $\alpha$ be as in ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}) and let $\mathcal{Q}^\alpha$ and $\mathcal{Q}^\alpha_n$ be as in ([\[E:boundaryenergy\]](#E:boundaryenergy){reference-type="ref" reference="E:boundaryenergy"}) and ([\[E:boundaryenergyn\]](#E:boundaryenergyn){reference-type="ref" reference="E:boundaryenergyn"}) respectively. Then $$\lim_{n\to\infty} \mathcal{Q}^\alpha_n=\mathcal{Q}^\alpha$$ in the KS-generalized Mosco sense with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}.* One ingredient for the proof of Theorem [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"} is the following result on the pointwise convergence of energy forms on Lipschitz functions. Given a function $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$, we use the abbreviation $\mathcal{Q}^\alpha(u):=\mathcal{Q}^\alpha(u|_\Gamma)$, and similarly for the forms $\mathcal{Q}^\alpha_n$. **Proposition 1**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} be satisfied, let $\alpha$ be as in ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}) and let $\mathcal{Q}^\alpha$ and $\mathcal{Q}^\alpha_n$ be as in ([\[E:boundaryenergy\]](#E:boundaryenergy){reference-type="ref" reference="E:boundaryenergy"}) and ([\[E:boundaryenergyn\]](#E:boundaryenergyn){reference-type="ref" reference="E:boundaryenergyn"}) respectively. Then $\lim_{n\to \infty} \mathcal{Q}^\alpha_n(u)=\mathcal{Q}^\alpha(u)$ for all $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$.* To prove Proposition [Proposition 1](#P:convLip){reference-type="ref" reference="P:convLip"}, note that $\mathcal{Q}_n^\alpha(u)=\mathcal{Q}_n^{\alpha,s}(u)+\mathcal{Q}^{\alpha,\ell}(u)$, $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$, where $$\mathcal{Q}^{\alpha,s}_n(u):=\int_{\Gamma_n}\int_{\Gamma_n}\mathbf{1}_{\{|\xi-\eta|\leq Ar_n\}}\frac{(u(\xi)-u(\eta))^2}{|\xi-\eta|^{2\alpha-1}\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta).$$ and $$\mathcal{Q}_n^{\alpha,\ell}(u):=\int_{\Gamma_n}\int_{\Gamma_n}\mathbf{1}_{\{|\xi-\eta|> Ar_n\}}\frac{(u(\xi)-u(\eta))^2}{\sigma_\alpha(\xi,|\xi-\eta|))\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta).$$ We claim that under the hypotheses of Proposition [Proposition 1](#P:convLip){reference-type="ref" reference="P:convLip"}, $$\label{E:claimcores} \lim_{n\to \infty}\mathcal{Q}_n^{\alpha,s}(u)=0$$ and $$\label{E:claimcorel} \lim_{n\to \infty}\mathcal{Q}_n^{\alpha,\ell}(u)=\mathcal{Q}^\alpha(u)$$ for any $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$. If so, then Proposition [Proposition 1](#P:convLip){reference-type="ref" reference="P:convLip"} follows. We first prove claim ([\[E:claimcores\]](#E:claimcores){reference-type="ref" reference="E:claimcores"}). *Proof of ([\[E:claimcores\]](#E:claimcores){reference-type="ref" reference="E:claimcores"})..* Using the Lipschitz property of $u$ and Lemma [Lemma 4](#L:mundoubling){reference-type="ref" reference="L:mundoubling"} with $C=2A/M$, $$\begin{aligned} \mathcal{Q}^{\alpha,s}_n(u)&\leq \mathop{\mathrm{Lip}}(u)^2\int_{\Gamma_n}\int_{\overline{B(\eta,Ar_n)}}\frac{|\xi-\eta|^{3-2\alpha}}{\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta)\notag\\ &\leq c_{D}\:\mathop{\mathrm{Lip}}(u)^2\int_{\Gamma_n}\int_{\overline{B(\eta,Ar_n)}}\frac{|\xi-\eta|^{3-2\alpha}}{\mu_n(B(\eta,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta).\notag\end{aligned}$$ For fixed $\eta\in \Gamma_n$ we rewrite the inner integral as a Stieltjes integral with respect to $r\mapsto \mu_n(B(\eta,r))$; and since $2\alpha<3$ by ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}) and ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}), this gives $$\begin{aligned} \int_0^{Ar_n}\frac{r^{3-2\alpha}}{\mu_n(B(\eta,r))}d\mu_n(B(\eta,r))&=\sum_{j=0}^\infty \int_{2^{-j-1}Ar_n}^{2^{-j}Ar_n}\frac{r^{3-2\alpha}}{\mu_n(B(\eta,r))}d\mu_n(B(\eta,r))\notag\\ &\leq (A r_n)^{3-2\alpha} \sum_{j=0}^\infty 2^{-j(3-2\alpha)}\frac{\mu(B(\eta,2^{-j}Ar_n))}{\mu(B(\eta,2^{-j-1}Ar_n))}\notag\\ &\leq \frac{c_{D}(A r_n)^{3-2\alpha}}{1-2^{3-2\alpha}}.\notag\end{aligned}$$ Since this goes to zero uniformly in $\eta$ and $\mu_n$ is a probability measure, ([\[E:claimcores\]](#E:claimcores){reference-type="ref" reference="E:claimcores"}) follows. ◻ The proof of ([\[E:claimcorel\]](#E:claimcorel){reference-type="ref" reference="E:claimcorel"}) uses two observations. The first is the following comparison lemma. **Lemma 5**. *Let Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} be satisfied. Suppose that $n\geq 1$, $x,y\in \Gamma$ with $|x-y|>Ar_n/2$ and $w\in e_n(x)$, $z\in e_n(y)$.* 1. *We have $$\label{E:difference} ||x-y|-|w-z||\leq 16\:S\: r_n.$$* 2. *For any $\alpha$ as in ([\[E:constellation\]](#E:constellation){reference-type="ref" reference="E:constellation"}) we have $$\label{E:sigmacompare} c^{-1}\sigma_\alpha(B(x,|x-y|))\leq \sigma_\alpha(B(w,|w-z|)\leq c\:\sigma_\alpha(B(x,|x-y|))$$ with $c:=c_{\mu,D}^2\max(2^{2\alpha-2},2^{2-2\alpha})$.* 3. *We have $$\label{E:mumuncompare} c_{\mu,D}^{-1}\mu(B(x,|x-y|))\leq \mu_n(B(w,|w-z|)\leq c_{\mu,D}\mu(B(x,|x-y|)).$$* *Proof.* By ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}) and ([\[E:convex\]](#E:convex){reference-type="ref" reference="E:convex"}) we have $$\label{E:convex2} |x-w|\leq 8\:S\:r_n\quad \text{and}\quad |y-z|\leq 8\:S\: r_n$$ whenever $w\in e_n(x)$ and $z\in e_n(y)$, so that the triangle inequality gives ([\[E:difference\]](#E:difference){reference-type="ref" reference="E:difference"}). Since $x$ and $y$ are far apart, it follows that $$\label{E:easy} |x-y|/2\leq |w-z|\leq 2|x-y|$$ and that $B(x,|x-y|/4)\subset B(w,|w-z|)$ and $B(w,|w-z|)\subset B(x,4|x-y|)$. Together with ([\[E:mudoubling\]](#E:mudoubling){reference-type="ref" reference="E:mudoubling"}) this gives ([\[E:sigmacompare\]](#E:sigmacompare){reference-type="ref" reference="E:sigmacompare"}). Using ([\[E:defmun\]](#E:defmun){reference-type="ref" reference="E:defmun"}) and ([\[E:convex2\]](#E:convex2){reference-type="ref" reference="E:convex2"}) we obtain $$\begin{gathered} \mu_n(B(w,|w-z|))=\int_\Gamma\fint_{e_n(x')}\mathbf{1}_{B(w',|w-z|)}(w)\mathcal{H}^1(dw')\mu(dx')\geq \int_\Gamma \mathbf{1}_{B(x',|w-z|-8S r_n)}(w)\mu(dx')\notag\\ =\mu(B(w,|w-z|-8S r_n))\geq \mu(B(x,|w-z|-16Sr_n))\notag\\ \geq \mu(B(x,|x-y|-32S r_n))\geq \mu(B(x,|x-y|/2)),\end{gathered}$$ and by ([\[E:mudoubling\]](#E:mudoubling){reference-type="ref" reference="E:mudoubling"}) the lower bound in ([\[E:mumuncompare\]](#E:mumuncompare){reference-type="ref" reference="E:mumuncompare"}) follows. The upper bound is seen similarly. ◻ The second observation is a continuity lemma. Under Assumption [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} we have $$\label{E:openorcloset} \mu(B(x,r))=\mu(\overline{B(x,r)}),\quad x\in \Gamma,\quad 0<r\leq \mathop{\mathrm{diam}}\Gamma.$$ It is convenient to use the notation $\mu(B(x,0)):=0$, $x\in \Gamma$; then $(x,r)\mapsto \mu(B(x,r))$ may be seen as a function on $\Gamma\times [0,\mathop{\mathrm{diam}}\:\Gamma]$. **Lemma 6**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} be satisfied.* 1. *The function $(x,r)\mapsto \mu(B(x,r))$ is uniformly continuous on $\Gamma\times [0,\mathop{\mathrm{diam}}\:\Gamma]$.* 2. *If $\alpha$ is as in ([\[E:constellation\]](#E:constellation){reference-type="ref" reference="E:constellation"}), then we have $$\label{E:sigmacont} \lim_{n\to \infty}\sup_{w\in e_n(x), z\in e_n(y)}|\sigma_\alpha(w,|w-z|)-\sigma_\alpha(x,|x-y|)|=0,\quad (x,y)\in \Gamma\times \Gamma\setminus \mathop{\mathrm{diag}},$$ and for any $\delta>0$ this convergence is uniform on $\{(x,y)\in \Gamma\times\Gamma:\ |x-y|\geq\delta\}$.* 3. *We have $$\label{E:mumuncont} \lim_{n\to \infty}\sup_{w\in e_n(x), z\in e_n(y)}|\mu_n(B(w,|w-z|))-\mu(B(x,|x-y|))|=0,\quad (x,y)\in \Gamma\times \Gamma\setminus \mathop{\mathrm{diag}},$$ and for any $\delta>0$ this convergence is uniform on $\{(x,y)\in \Gamma\times\Gamma:\ |x-y|\geq\delta\}$.* *Proof.* Given $x\in \Gamma$ and $0<r\leq \mathop{\mathrm{diam}}\Gamma$ the continuity of $\mu$ from below gives $$\mu(B(x,r))=\mu\left(\bigcup_{n=1}^\infty B(x,r_n)\right)=\lim_{n\to \infty} \mu(B(x,r_n))$$ for any sequence $(r_n)_n$ with $r_n\uparrow r$. Given $x\in \Gamma$ and $0\leq r\leq \mathop{\mathrm{diam}}\Gamma$ the continuity of $\mu$ from above and ([\[E:openorcloset\]](#E:openorcloset){reference-type="ref" reference="E:openorcloset"}) give $$\mu(B(x,r))=\mu\left(\bigcap_{n=1}^\infty \overline{B(x,r_n)}\right)=\lim_{n\to \infty} \mu(\overline{B(x,r_n)})=\lim_{n\to \infty} \mu(B(x,r_n))$$ for any sequence $(r_n)_n\subset [0,\mathop{\mathrm{diam}}\Gamma]$ with $r_n\downarrow r$. Given $(x,r),(x',r')\in \Gamma\times [0,\mathop{\mathrm{diam}}\Gamma]$ with $|(x,r)-(x',r')|<\varepsilon$, we have $B(x',r')\subset B(x,r+2\varepsilon)$ and therefore $$\mu(B(x',r'))-\mu(B(x,r))\leq \mu(B(x,r+2\varepsilon))-\mu(B(x,r)).$$ For $r=0$ the left-hand side is nonnegative. For $r>0$ we may choose $\varepsilon$ small enough to have $B(x,r-2\varepsilon)\subset B(x',r')$ and therefore $$\mu(B(x,r))-\mu(B(x',r'))\leq \mu(B(x,r))-\mu(B(x,r-2\varepsilon)).$$ Together with the preceding, this implies that $(x,r)\mapsto \mu(B(x,r))$ is continuous; the compactness of $\Gamma\times [0,\mathop{\mathrm{diam}}\:\Gamma]$ implies the uniform continuity as claimed in (i). If $n$ is large enough, then for any $w\in e_n(x)$ and $z\in e_n(y)$ we have $$\begin{gathered} |\sigma_\alpha(w,|w-z|)-\sigma_\alpha(x,|x-y|)|\notag\\ \leq \sup_{w\in e_n(x), z\in e_n(y)}||w-z|^{2\alpha-2}-|x-y|^{2\alpha-2}|\mu(B(w,|w-z|))\notag\\ +|x-y|^{2\alpha-2}\sup_{w\in e_n(x), z\in e_n(y)}|\mu(B(w,|w-z|))-\mu(B(x,|x-y|))|.\notag\end{gathered}$$ The first summand goes to zero by ([\[E:mumuncompare\]](#E:mumuncompare){reference-type="ref" reference="E:mumuncompare"}) and the uniform continuity of $(w,z)\mapsto |w-z|^{2\alpha-2}$ outside a neighborhood of the diagonal. Since for large enough $n$ we have $$B(x,|x-y|-24S r_n)\subset B(w,|w-z|)\subset B(x,|x-y|+24S r_n)$$ for all $w\in e_n(x)$ and $z\in e_n(y)$, the second summand goes to zero by (i). This gives the first claim in (ii). The second claim follows from the uniform continuity in (i), note that for large $n$ we have $|x-y|\geq \delta>24S r_n$. To see (iii), suppose that $(x,y)\in \Gamma\times\Gamma\setminus \mathop{\mathrm{diag}}$, $w\in e_n(x)$ and $z\in e_n(y)$. If $x'\in \Gamma\setminus V_\ast$ and $w'\in e_n(x')$, then $$\begin{aligned} \big|\mathbf{1}_{B(w,\left|w-z\right|)}(w')&-\mathbf{1}_{B(x,\left|x-y\right|)}(x')\big|\notag\\ &=\left|\mathbf{1}_{B(w',\left|w-z\right|)}(w)-\mathbf{1}_{B(x,\left|x-y\right|)}(x')\right|\notag\\ &=\mathbf{1}_{B(w',\left|w-z\right|)}(w)\mathbf{1}_{B(x,\left|x-y\right|)^c}(x')+\mathbf{1}_{B(w',\left|w-z\right|)^c}(w)\mathbf{1}_{B(x,\left|x-y\right|)}(x').\notag\end{aligned}$$ Since $$B(w',|w-z|)\subset B(w',|x-y|+16S r_n)\subset B(x',|x-y|+24S r_n)$$ by ([\[E:difference\]](#E:difference){reference-type="ref" reference="E:difference"}) and ([\[E:convex2\]](#E:convex2){reference-type="ref" reference="E:convex2"}) and similarly $$B(x',|x-y|- 24S r_n)\subset B(w',|x-y|-16S r_n) \subset B(w',|w-z|),$$ the preceding is bounded by $$\begin{aligned} &\mathbf{1}_{B(x',\left|x-y\right|+ 24S r_n)}(w)\mathbf{1}_{B(x,\left|x-y\right|)^c}(x')+\mathbf{1}_{B(x',\left|x-y\right|-24S r_n))^c}(w)\mathbf{1}_{B(x,\left|x-y\right|))}(x')\notag\\ &=\mathbf{1}_{B(w,\left|x-y\right|+24S r_n)}(x')\mathbf{1}_{B(x,\left|x-y\right|)^c}(x')+\mathbf{1}_{B(w,\left|x-y\right|-24S r_n)^c}(x')\mathbf{1}_{B(x,\left|x-y\right|)}(x')\notag\\ &\leq \mathbf{1}_{B(x,\left|x-y\right|+32S r_n)}(x')\mathbf{1}_{B(x,\left|x-y\right|)^c}(x')+\mathbf{1}_{B(x,\left|x-y\right|-32S r_n)^c}(x')\mathbf{1}_{B(x,\left|x-y\right|)}(x').\notag\end{aligned}$$ Therefore $$\begin{aligned} \big|\mu_n(B(w,\left|w-z\right|))&-\mu(B(x,\left|x-y\right|))\big|\notag\\ &=\left|\int_{\Gamma_n}\mathbf{1}_{B(w,\left|w-z\right|)}(w')\mu_n(dw')-\int_\Gamma\mathbf{1}_{B(x,\left|x-y\right|)}(x')\mu(dx')\right|\notag\\ &\leq \int_\Gamma \fint_{e_n(x')}\left|\mathbf{1}_{B(w,\left|w-z\right|)}(w')-\mathbf{1}_{B(x,\left|x-y\right|)}(x')\right|\:\mathcal{H}^1(dw')\mu(dx')\notag\\ &\leq \int_K \mathbf{1}_{B(x,\left|x-y\right|+32S r_n)}(x')\mathbf{1}_{B(x,\left|x-y\right|)^c}(x')\mu(dx')\notag\\ &\hspace{80pt}+\int_K \mathbf{1}_{B(x,\left|x-y\right|-32S r_n)^c}(x')\mathbf{1}_{B(x,\left|x-y\right|)}(x')\mu(dx')\notag\\ &=\mu(B(x,\left|x-y\right|+32S r_n))-\mu(B(x,\left|x-y\right|))\notag\\ &\hspace{80pt}+\mu(B(x,\left|x-y\right|))-\mu(B(x,\left|x-y\right|-32S r_n))\notag\\ &=\mu(B(x,\left|x-y\right|+32S r_n))-\mu(B(x,\left|x-y\right|- 32S r_n)),\notag\end{aligned}$$ and (iii) follows using (i). ◻ We can now prove claim ([\[E:claimcorel\]](#E:claimcorel){reference-type="ref" reference="E:claimcorel"}). *Proof of ([\[E:claimcorel\]](#E:claimcorel){reference-type="ref" reference="E:claimcorel"})..* Our first claim is that, given $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$ and $(x,y)\in \Gamma\times\Gamma\setminus \mathop{\mathrm{diag}}$, we have $$\begin{gathered} \label{E:firstclaim} \lim_{n\to \infty}\mathbf{1}_{\{|x-y|>(A\pm 16S)r_n\}}\fint_{e_n(y)}\fint_{e_n(x)}\frac{(u(w)-u(z))^2}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\mathcal{H}^1(dw)\mathcal{H}^1(dz)\\ =\frac{(u(x)-u(y))^2}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}.\end{gathered}$$ Clearly the indicator becomes one for large enough $n$, and $$\lim_{n\to \infty}\fint_{e_n(y)}\fint_{e_n(x)}(u(w)-u(z))^2\mathcal{H}^1(dw)\mathcal{H}^1(dz)=(u(x)-u(y))^2,$$ because $(w,z)\mapsto (u(w)-u(z))^2$ is uniformly continuous on $\mathbb{R}^2$. To verify ([\[E:firstclaim\]](#E:firstclaim){reference-type="ref" reference="E:firstclaim"}), it therefore suffices to show that $$\begin{gathered} \label{E:suffclaim} \lim_{n\to \infty} \fint_{e_n(y)}\fint_{e_n(x)}(u(w)-u(z))^2 \Big(\frac{1}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\\ -\frac{1}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\Big)\mathcal{H}^1(dw)\mathcal{H}^1(dz)=0\end{gathered}$$ for any fixed $(x,y)\in \Gamma\times\Gamma\setminus \mathop{\mathrm{diag}}$. This is a consequence of ([\[E:sigmacont\]](#E:sigmacont){reference-type="ref" reference="E:sigmacont"}) and ([\[E:mumuncont\]](#E:mumuncont){reference-type="ref" reference="E:mumuncont"}): We have $$\begin{gathered} \label{E:diffofreciprocals} \Big|\frac{1}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}-\frac{1}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\Big|\\ \leq \Big|\frac{1}{\sigma_\alpha(w,|w-z|)}-\frac{1}{\sigma_\alpha(x,|x-y|)}\Big|\frac{1}{\mu_n(B(w,|w-z|))}\\ +\frac{1}{\sigma_\alpha(x,|x-y|)}\Big|\frac{1}{\mu_n(B(w,|w-z|))}-\frac{1}{\mu(B(x,|x-y|))}\Big|.\end{gathered}$$ By ([\[E:sigmacompare\]](#E:sigmacompare){reference-type="ref" reference="E:sigmacompare"}) and ([\[E:mumuncompare\]](#E:mumuncompare){reference-type="ref" reference="E:mumuncompare"}) the first summand on the right-hand side is bounded by $$\label{E:sigmabound} c\:\frac{|\sigma_\alpha(w,|w-z|)-\sigma_\alpha(x,|x-y|)|}{\sigma_\alpha(x,|x-y|)^2\mu(B((x,|x-y|))}$$ and the second by $$\label{E:mumunbound} c\:\frac{|\mu(B(x,|x-y|))-\mu_n(B(w,|w-z|))|}{\sigma_\alpha(x,|x-y|)\mu(B((x,|x-y|))^2},$$ here $c>1$ is a constant independent of $w$, $x$, $y$ and $z$. Since $x$ and $y$ are fixed, ([\[E:suffclaim\]](#E:suffclaim){reference-type="ref" reference="E:suffclaim"}) now follows using Lemma [Lemma 6](#L:cont){reference-type="ref" reference="L:cont"} (ii) and (iii). By Lemma [Lemma 5](#L:compare){reference-type="ref" reference="L:compare"} (ii) and (iii) and ([\[E:easy\]](#E:easy){reference-type="ref" reference="E:easy"}) we can find a constant $c'>0$ such that for any $x,y\in \Gamma\times\Gamma\setminus \mathop{\mathrm{diag}}$, any $u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$ and any $n$ we have $$\begin{gathered} \label{E:majorant} \mathbf{1}_{\{|x-y|>(A\pm 16S)r_n\}}\fint_{e_n(y)}\fint_{e_n(x)}\frac{(u(w)-u(z))^2}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\mathcal{H}^1(dw)\mathcal{H}^1(dz)\\ \leq \frac{c'\mathop{\mathrm{Lip}}(u)^2|x-y|^2}{\sigma_\alpha(x,|x-y|)\mu(B((x,|x-y|))}.\end{gathered}$$ By ([\[E:symbol\]](#E:symbol){reference-type="ref" reference="E:symbol"}), ([\[E:uplowreg\]](#E:uplowreg){reference-type="ref" reference="E:uplowreg"}) and ([\[E:mudoubling\]](#E:mudoubling){reference-type="ref" reference="E:mudoubling"}), we have $$\begin{gathered} \int_\Gamma \frac{|x-y|^2}{\sigma_\alpha(x,|x-y|)\mu(B((x,|x-y|))}\mu(dx)\leq c\: \int_\Gamma \frac{|x-y|^{4-2\alpha-s}}{\mu(B((y,|x-y|))}\mu(dx)\notag\\ =c\:\int_0^{\mathop{\mathrm{diam}}\Gamma}\frac{r^{4-2\alpha-s}}{\mu(B(y,r))}d\mu(B(y,r))\leq c\:\sum_{j=0}^\infty 2^{-j(4-2\alpha-s)}\frac{\mu(B(y,2^{-j}\mathop{\mathrm{diam}}\Gamma))}{\mu(B(y,2^{-j-1}\mathop{\mathrm{diam}}\Gamma))}\leq \frac{c\:c_{\mu,D}}{1-2^{4-2\alpha-s}}\notag\end{gathered}$$ for any $y\in \Gamma$ and with constants $c>0$ independent of $y$. Since $\mu$ is a probability measure, this shows that the right-hand side of ([\[E:majorant\]](#E:majorant){reference-type="ref" reference="E:majorant"}) is in $L^1(\Gamma\times\Gamma,\mu\otimes\mu)$, so that by ([\[E:firstclaim\]](#E:firstclaim){reference-type="ref" reference="E:firstclaim"}) and dominated convergence, $\mathcal{Q}^{\alpha}(u)$ is seen to equal $$\lim_{n\to \infty} \int\int_{\{|x-y|>(A\pm 16S)r_n\}}\fint_{e_n(y)}\fint_{e_n(x)}\frac{(u(w)-u(z))^2}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\mathcal{H}^1(dw)\mathcal{H}^1(dz)\mu(dx)\mu(dy).$$ Since by ([\[E:convex2\]](#E:convex2){reference-type="ref" reference="E:convex2"}) we have $$\begin{gathered} \{(x,y,w,z)\in \Gamma^2\times \Gamma_n^2: |x-y|>(A+16S)r_n, w\in e_n(x), z\in e_n(y)\}\notag\\ \subset \{(x,y,w,z)\in \Gamma^2\times \Gamma_n^2: w\in e_n(x), z\in e_n(y), |w-z|>Ar_n\}\notag\\ \subset \{(x,y,w,z)\in \Gamma^2\times \Gamma_n^2: |x-y|>(A-16S)r_n, w\in e_n(x), z\in e_n(y)\},\end{gathered}$$ monotonicity implies that $$\begin{aligned} \mathcal{Q}^{\alpha}(u)&=\lim_{n\to \infty} \int_\Gamma\int_\Gamma\fint_{e_n(y)}\fint_{e_n(x)}\frac{\mathbf{1}_{\{|w-z|>Ar_n\}}(u(w)-u(z))^2}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\mathcal{H}^1(dw)\mathcal{H}^1(dz)\mu(dx)\mu(dy)\notag\\ &=\lim_{n\to \infty}\int_{\Gamma_n}\int_{\Gamma_n}\mathbf{1}_{\{|\xi-\eta|>Ar_n\}}\frac{(u(\xi)-u(\eta))^2}{\sigma_\alpha(\xi, |\xi-\eta|))\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta)\notag\\ &=\lim_{n\to \infty}\mathcal{Q}^{\alpha,\ell}_n(u).\notag\end{aligned}$$ ◻ **Proposition 2**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} be satisfied, let $\alpha$ be as in ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}) and let $\mathcal{Q}^\alpha$ and $\mathcal{Q}^\alpha_n$ be as in ([\[E:boundaryenergy\]](#E:boundaryenergy){reference-type="ref" reference="E:boundaryenergy"}) and ([\[E:boundaryenergyn\]](#E:boundaryenergyn){reference-type="ref" reference="E:boundaryenergyn"}) respectively. For any $\varphi\in L^2(\Gamma,\mu)$ there is a sequence $(\varphi_n)_n$ of elements $\varphi_n\in L^2(\Gamma_n,\mu_n)$ that converges KS-strongly to $\varphi\in L^2(\Gamma,\mu)$ and satisfies $$\limsup_{n\to\infty} \mathcal{Q}^\alpha_n(\varphi_n) \leq \mathcal{Q}^\alpha(\varphi).$$* *Proof.* Let $\varphi\in L^2(\Gamma,\mu)$. We may assume that $\varphi\in B_\alpha^{2,2}(\Gamma)$, because otherwise $\mathcal{Q}^\alpha(\varphi)=+\infty$ and there is nothing to prove. By density we can find a sequence $(\psi_n)_n\subset \mathop{\mathrm{Lip}}(\Gamma)$ such that $$\label{E:L2conv} \lim_{n\to\infty}\left\|\psi_n-\varphi\right\|_{L^2(\Gamma,\mu)}=0$$ and $$\label{E:energiesconv} \lim_{n\to\infty}\mathcal{Q}^\alpha(\psi_n)=\mathcal{Q}^\alpha(\varphi).$$ For each $n\geq 1$ the Whitney extension $E_\Gamma\psi_n$ is an element of $\mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$; we denote it again by $\psi_n$. Given $j\geq 1$ we can find some $n_j\geq 1$ such that $$\Big|\left\|\psi_j-\psi_i\right\|_{L^2(\Gamma_n,\mu_n)}-\left\|\psi_j-\psi_i\right\|_{L^2(\Gamma,\mu)}\Big|<2^{-j},\ i=1,2,...,j,$$ and $$\label{E:energiesclose} \Big|\mathcal{Q}_n^\alpha(\psi_j)-\mathcal{Q}^\alpha(\psi_j)\Big|<2^{-j};$$ this follows from ([\[E:weakconveval\]](#E:weakconveval){reference-type="ref" reference="E:weakconveval"}) and Proposition [Proposition 1](#P:convLip){reference-type="ref" reference="P:convLip"}. From ([\[E:L2conv\]](#E:L2conv){reference-type="ref" reference="E:L2conv"}) it follows that for all $k\geq 1$ we can find some $i_k$ such that $\|\psi_i-\psi_j\|_{L^2(\Gamma,\mu)}<2^{-k}$ if $i,j\geq i_k$. Combining, it follows that $$\left\|\psi_j-\psi_{i_k}\right\|_{L^2(\Gamma_n,\mu_n)}\leq \left\|\psi_j-\psi_{i_k}\right\|_{L^2(\Gamma,\mu)}+2^{-j}\leq 2^{-k}+2^{-j}$$ for all $k$, all $j>i_k$ and all $n\geq n_j$. Now set $\varphi_n:=\psi_j$ for all $n_j< n\leq n_{j+1}$ and $\widetilde{\chi}_k:=\psi_{i_k}$. Then $$\limsup_{n\to\infty}\left\|\varphi_n-\widetilde{\chi}_k\right\|_{L^2(\Gamma_n,\mu_n)}\leq 2^{-k}$$ for all $k\geq 1$. Since on the other hand we have $\lim_{k\to\infty}\left\|\varphi-\widetilde{\chi}_k\right\|_{L^2(\Gamma,\mu)}=0$ by ([\[E:L2conv\]](#E:L2conv){reference-type="ref" reference="E:L2conv"}), it follows that $\lim_{n\to\infty}\varphi_n=\varphi$ KS-strongly. Combining ([\[E:energiesconv\]](#E:energiesconv){reference-type="ref" reference="E:energiesconv"}) and ([\[E:energiesclose\]](#E:energiesclose){reference-type="ref" reference="E:energiesclose"}) gives $$\lim_{n\to \infty} \mathcal{Q}^\alpha_n(\varphi_n)=\mathcal{Q}^\alpha(\varphi).$$ ◻ **Proposition 3**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} be satisfied, let $\alpha$ be as in ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}) and let $\mathcal{Q}^\alpha$ and $\mathcal{Q}^\alpha_n$ be as in ([\[E:boundaryenergy\]](#E:boundaryenergy){reference-type="ref" reference="E:boundaryenergy"}) and ([\[E:boundaryenergyn\]](#E:boundaryenergyn){reference-type="ref" reference="E:boundaryenergyn"}) respectively. If $(\varphi_n)_n$ is a sequence of elements $\varphi_n\in L^2(\Gamma_n,\mu_n)$ that converges KS-weakly to $\varphi\in L^2(\Gamma,\mu)$, then we have $$\mathcal{Q}^\alpha(\varphi)\leq \liminf_{n\to\infty} \mathcal{Q}^\alpha_n(\varphi_n).$$* *Proof.* Let $\delta>0$. We will show that $$\mathcal{Q}^{\alpha,\delta}(\psi):=\int_\Gamma\int_\Gamma\mathbf{1}_{\{|x-y|\geq \delta\}}\frac{(\psi(x)-\psi(y))^2}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\mu(dx)\mu(dy),\quad \psi\in L^2(\Gamma,\mu),$$ satisfies $$\label{E:Qdeltaclaim} \mathcal{Q}^{\alpha,\delta}(\varphi)\leq \liminf_{n\to\infty} \mathcal{Q}^\alpha_n(\varphi_n).$$ Since $\delta>0$ was arbitrary, this proves the lemma. To prepare the proof of ([\[E:Qdeltaclaim\]](#E:Qdeltaclaim){reference-type="ref" reference="E:Qdeltaclaim"}), let $$\begin{gathered} \mathcal{Q}^{\alpha,\delta}_n(\psi):=\int_{\Gamma}\int_{\Gamma}\mathbf{1}_{\{|x-y|\geq \delta\}}\fint_{e_n(y)}\fint_{e_n(x)}\mathbf{1}_{\{|w-z|>Ar_n\}}\frac{(\psi(w)-\psi(z))^2}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}\times\notag\\ \times\mathcal{H}^1(dw)\mathcal{H}^1(dz)\mu(dx)\mu(dy),\quad \psi\in L^2(\Gamma_n,\mu_n);\end{gathered}$$ note that for large enough $n$ we have $\mathbf{1}_{\{|w-z|>Ar_n\}}\equiv 1$ in the inner integral. Since $\delta>0$, Lemma [Lemma 6](#L:cont){reference-type="ref" reference="L:cont"} (ii) and (iii) and the estimate in ([\[E:diffofreciprocals\]](#E:diffofreciprocals){reference-type="ref" reference="E:diffofreciprocals"}), ([\[E:sigmabound\]](#E:sigmabound){reference-type="ref" reference="E:sigmabound"}) and ([\[E:mumunbound\]](#E:mumunbound){reference-type="ref" reference="E:mumunbound"}) imply that given $\varepsilon>0$, $$\sup_{|x-y|\geq \delta}\ \sup_{w\in e_n(x), z\in e_n(y)}\Big|\frac{1}{\sigma_\alpha(w,|w-z|)\mu_n(B(w,|w-z|))}-\frac{1}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\Big|<\varepsilon$$ whenever $n$ is large enough. Accordingly, we find that $$\begin{aligned} &\mathcal{Q}^{\alpha,\delta}_n(\varphi_n) \notag\\ &\geq \int_\Gamma\int_\Gamma\mathbf{1}_{\{|x-y|\geq \delta\}}\left(\frac{1}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}-\varepsilon\right)\times\notag\\ &\hspace{150pt} \times\fint_{e_n(y)}\fint_{e_n(x)}(\varphi_n(w)-\varphi_n(z))^2\mathcal{H}^1(dw)\mathcal{H}^1(dz)\mu(dx)\mu(dy)\notag\\ &\geq \int_\Gamma\int_\Gamma \mathbf{1}_{\{|x-y|\geq \delta\}}\left(\frac{1}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}-\varepsilon\right)([\varphi_n]_n(x)-[\varphi_n]_n(y))^2\mu(dx)\mu(dy)\notag\\ &\geq \int_\Gamma\int_\Gamma\mathbf{1}_{\{|x-y|\geq \delta\}}\frac{([\varphi_n]_n(x)-[\varphi_n]_n(y))^2}{\sigma_\alpha(x,|x-y|)\mu(B(x,|x-y|))}\mu(dx)\mu(dy)\notag\\ & \hspace{200pt} -\varepsilon\int_\Gamma\int_\Gamma ([\varphi_n]_n(x)-[\varphi_n]_n(y))^2\mu(dx)\mu(dy)\notag\\ &\geq \mathcal{Q}^{\alpha,\delta}([\varphi_n]_n)-4\varepsilon\left\|[\varphi_n]_n\right\|_{L^2(\Gamma,\mu)}^2\notag\end{aligned}$$ for any such $n$, note that Jensen's inequality and ([\[E:deffn\]](#E:deffn){reference-type="ref" reference="E:deffn"}) give $$\begin{aligned} \fint_{e_n(y)}\fint_{e_n(x)}(\varphi_n(w)-\varphi_n(z))^2\mathcal{H}^1(dw)\mathcal{H}^1(dz)&\geq \left(\fint_{e_n(y)}\fint_{e_n(x)}(\varphi_n(w)-\varphi_n(z))\mathcal{H}^1(dw)\mathcal{H}^1(dz)\right)^2\notag\\ &=([\varphi_n]_n(x)-[\varphi_n]_n(y))^2.\notag\end{aligned}$$ In a similar manner we can see that $$\label{E:supfinite} \sup_n \left\|[\varphi_n]_n\right\|_{L^2(\Gamma,\mu)}\leq \sup_n\left\|\varphi_n\right\|_{L^2(\Gamma_n,\mu_n)}<+\infty,$$ the finiteness is due to the KS-weak convergence. Now let $(\varphi_{n_k})_k\subset (\varphi_n)_n$ be such that $$\lim_{k\to\infty}\mathcal{Q}^\alpha_{n_k}(\varphi_{n_k})=\liminf_{n\to\infty}\mathcal{Q}^\alpha_n(\varphi_n);$$ we may assume that this quantity is finite. By ([\[E:supfinite\]](#E:supfinite){reference-type="ref" reference="E:supfinite"}) there is a subsequence $(n_{k_j})_j$ of $(n_k)_k$ such that the sequence of averages $([\varphi_{n_{k_j}}]_{n_{k_j}})_j$ converges to some $\psi$ weakly in $L^2(\Gamma,\mu)$, and by the Banach-Saks theorem, [@RieszNagy56 Section 38], we may assume that its arithmetic means converge to $\psi$ strongly in $L^2(\Gamma,\mu)$. Now the KS-weak convergence implies that $\psi=\varphi$, because $$\begin{aligned} \left\langle\psi,\chi\right\rangle_{L^2(\Gamma,\mu)}&=\lim_{j\to\infty}\big\langle [\varphi_{n_{k_j}}]_{n_{k_j}},\chi\big\rangle_{L^2(\Gamma,\mu)}\notag\\ &=\lim_{j\to\infty}\int_\Gamma\fint_{e_n(x)}\varphi_{n_{k_j}}(w)\mathcal{H}^1(dw)E_\Gamma\chi(x)\mu(dx)\notag\\ &=\lim_{j\to\infty}\int_\Gamma\fint_{e_n(x)}\varphi_{n_{k_j}}(w)E_\Gamma\chi(w)\mathcal{H}^1(dw)\mu(dx)\notag\\ &=\lim_{j\to\infty}\big\langle \varphi_{n_{k_j}}, E_\Gamma\chi\big\rangle_{L^2(\Gamma_{n_{k_j}},\mu_{n_{k_j}})}\notag\\ &=\left\langle \varphi,\chi\right\rangle_{L^2(\Gamma,\mu)}\notag\end{aligned}$$ for any $\chi\in \mathop{\mathrm{Lip}}(\Gamma)$. Since the quadratic form $\mathcal{Q}^{\alpha,\delta}$ is continuous on $L^2(\Gamma,\mu)$, it follows that $$\label{E:Cesaroconv} \lim_{N\to\infty}\mathcal{Q}^{\alpha,\delta}\Big(\frac{1}{N}\sum_{j=1}^N [\varphi_{n_{k_j}}]_{n_{k_j}}-\varphi\Big)=0.$$ Now $$\liminf_{n\to\infty}\mathcal{Q}^\alpha_n(\varphi_n)=\lim_{j\to\infty} \mathcal{Q}_{n_{k_j}}^\alpha(\varphi_{n_{k_j}})\geq \limsup_{j\to\infty}\mathcal{Q}^{\alpha,\delta}([\varphi_{n_{k_j}}]_{n_{k_j}})-4\varepsilon\sup_j\big\|[\varphi_{n_{k_j}}]_{n_{k_j}}\big\|_{L^2(\Gamma,\mu)}^2.$$ Since the left-hand side does not depend on $\varepsilon$, we arrive at $$\begin{aligned} \liminf_{n\to\infty}\mathcal{Q}^\alpha_n(\varphi_n)^{1/2}&\geq \limsup_{j\to\infty}\mathcal{Q}^{\alpha,\delta}\big([\varphi_{n_{k_j}}]_{n_{k_j}}\big)^{1/2}\notag\\ &\geq \limsup_{j\to\infty}\frac{1}{N}\sum_{j=1}^N\mathcal{Q}^{\alpha,\delta}\big([\varphi_{n_{k_j}}]_{n_{k_j}}\big)^{1/2}\notag\\ &\geq \lim_{j\to\infty}\mathcal{Q}^{\alpha,\delta}\Big(\frac{1}{N}\sum_{j=1}^N [\varphi_{n_{k_j}}]_{n_{k_j}}\Big)^{1/2}\notag\\ &=\mathcal{Q}^{\alpha,\delta}(\varphi)^{1/2};\notag\end{aligned}$$ here we have used the domination of the limsup over the Cesàro limsup, the triangle inequality for the seminorm $(\mathcal{Q}^{\alpha,\delta})^{1/2}$ and ([\[E:Cesaroconv\]](#E:Cesaroconv){reference-type="ref" reference="E:Cesaroconv"}). This is ([\[E:Qdeltaclaim\]](#E:Qdeltaclaim){reference-type="ref" reference="E:Qdeltaclaim"}). ◻ Propositions [Proposition 2](#P:limsupboundary){reference-type="ref" reference="P:limsupboundary"} and [Proposition 3](#P:liminfboundary){reference-type="ref" reference="P:liminfboundary"} now give Theorem [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"}. # Approximation of quasidiscs {#S:Disks} For homogeneous snowflake-like curves $\Gamma\subset \mathbb{R}^2$ the natural partitions $\mathcal{I}_n$ lead to polygonal curves $\Gamma_n$ that are Jordan curves. For general quasicircles $\Gamma$ and partitions we make this an assumption on the partitions $\mathcal{I}_n$. *Assumption 3*. For any large enough $n$ the polygonal curve $\Gamma_n$ is a Jordan curve. Given an arbitrary quasicircle $\Gamma\subset \mathbb{R}^2$, we can use ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) to actually construct finite partitions $\mathcal{I}_n$ satisfying Assumption [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} by an ad hoc procedure. We do not claim any kind of optimality. **Lemma 7**. *Let $0<p,q<1$ and $\omega=(\omega_1,\omega_2,...)\in \{p,q\}^\mathbb{N}$. Then, for any large enough $n$ there is a finite partition $\mathcal{I}_n$ of $\Gamma$ satisfying Assumption [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} with $M=5$ and Assumption [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"}. Moreover, for any large enough $n$ and any $\xi\in \Gamma_n$ and $r<\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma$ the set $\Gamma_n\cap \overline{B(\xi,r)}$ is connected.* *Proof.* We consider $\Gamma$ oriented clockwise. We use $\theta$ as in ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) and write $$\label{E:rnprime} r_n':=4\omega_1\cdots \omega_n\mathop{\mathrm{diam}}\Gamma.$$ Choose some $x_0\in \Gamma$ and let $n$ be large enough to have $\Gamma\setminus B(x_0,r_n'/\theta)\neq \emptyset$. Let $x_1\in \partial B(x_0,r_n')$ be the point where $\Gamma$ exits $B(x_0,r_n')$ for the last time before exiting $\overline{B(x_0,r_n'/\theta)}$. Let $e_1$ be the line segment from $x_0$ to $x_1$. Now suppose that $x_j\in \Gamma$ has been determined. If $x_j$ is visited by $\Gamma$ before its re-entry into $B(x_0,r'_n)$, let $x_{j+1}$ be the point on $\partial B(x_j,r_n')$ where $\Gamma$ exits $B(x_j,r_n')$ the last time before leaving $\overline{B(x_j,r_n'/\theta)}$. If $x_{j+1}$ is not in $B(x_0,r'_n)$, then let $e_{j+1}$ be the line segment from $x_j$ to $x_{j+1}$ and repeat this step with $x_{j+1}$ in place of $x_j$. After a finite number of steps we obtain a point $x_k\in \Gamma\cap B(x_0,r'_n)$. By ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) we have $x_{j+1}\notin \bigcup_{i=0}^j B(x_i,r_n')$ for all $j=0,...,k-2$, and we still have $x_k\notin \bigcup_{i=1}^{k-1} B(x_i,r_n')$. We have $|x_{j+1}-x_j|=r_n'$, $j=0,...,k-1$; therefore the corresponding arcs $a(x_j,a_{j+1})$ all satisfy $$r_n'\leq \mathop{\mathrm{diam}}a(x_j,x_{j+1})\leq S\:r_n'$$ and the resulting line segments $e_{j+1}$, $j=0,...,k-1$, have length $r_n'$. Two neighboring line segments share an endpoint, but no other point. Line segments that do not share an endpoint cannot intersect: To intersect a line segment $e_{i+1}$ with $i<j-1$, the segment $e_{j+1}$ would have to cross the union $B(x_i,r_n')\cup B(x_{i+1},r_n')$, but this would require $e_{j+1}$ to have length at least $\sqrt{3}r_n'>r_n'$. It remains to discuss the situation of $x_k\in \Gamma\cap B(x_0,r'_n)$. Although some line segment $e_{i+1}$ with $i\leq k-2$ may intersect $B(x_0,r'_n)$, the point $x_k$ could never lie in the closure of the resulting disk segment cut off by $e_{i+1}$: Otherwise the maximal possible distance between $x_k$ and $e_{i+1}$ would be $r_n'-\frac12\sqrt{4r_n'^2-r_n'^2}=(1-\sqrt{3}/2)r_n'$, and since this is less than $(\sqrt{3}/2)r_n'$, the point $x_k$ would have to be an element of $B(x_i,r_n')\cup B(x_{i+1},r_n')$, which we know is impossible. Now recall that $x_k\notin B(x_1,r_n')$. In the case that $x_k\in B(x_0,r'_n)\setminus B(x_0,\frac14 r_n')$ we define $e_{k+1}$ to be the line segment from $x_k$ to $x_0$; it has length $\frac14 r_n'<\mathcal{H}^1(e_{k+1})< r_n'$, and by the preceding it cannot cross any other segment. This gives the Jordan curve $\Gamma_n:=e_1\cup e_2\cup ...\cup e_{k+1}$. In the case that $x_k\in B(x_0,\frac14 r'_n)$ we discard $x_0$ and redefine $e_1$ to be the line segment from $x_k$ to $x_1$; its length is $r_n'\leq \mathcal{H}^1(e_1)<\frac54 r_n'$, and again it cannot cross any other segment. This gives the Jordan curve $\Gamma_n:=e_1\cup e_2\cup ...\cup e_{k}$. The last statement is immediate from the construction. ◻ Recall that the interior (bounded) planar domain enclosed by a Jordan curve is called a *Jordan domain* and a domain bounded by a quasicircle is called a *quasidisc*, [@Gehring82; @Lehto87]. We write $\Omega$ to denote the interior (bounded) quasidisc bounded by $\Gamma$ and, with Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} in force, $\Omega_n$ to denote the Jordan domains bounded by the Jordan curves $\Gamma_n$, respectively. The following approximation result is easily seen. **Lemma 8**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} be satisfied. Then we have $\lim_{n\to\infty} \mathbf{1}_{\Omega_n}=\mathbf{1}_\Omega$ in $L^p(\mathbb{R}^2)$, $p\in [1,+\infty)$.* *Proof.* Given $\varepsilon>0$ we have $\Gamma\subset (\Gamma_n)_\varepsilon$ and $\Gamma_n\subset (\Gamma)_\varepsilon$ for all sufficiently large $n$ by Lemma [\[L:Hausdorffconvbd\]](#L:Hausdorffconvbd){reference-type="ref" reference="L:Hausdorffconvbd"}. For such $n$ the set $\overline{\Omega}_n=\Omega_n\cup \Gamma_n$ is contained in $(\overline{\Omega})_\varepsilon$, therefore $\Omega_n\setminus \Omega\subset (\Gamma)_\varepsilon$ and $\mathcal{L}^2(\Omega_n\setminus \Omega)\leq\mathcal{L}^2((\Gamma)_\varepsilon)$. The bounded domain $(\Omega^c \cup (\Gamma)_\varepsilon)^c$ enclosed by the "inner" boundary $\partial(\Gamma)_\varepsilon\cap \Omega$ of $(\Gamma)_\varepsilon$ is contained in $\Omega_n$, consequently $\Omega\setminus \Omega_n\subset (\Gamma)_\varepsilon$, and similarly as before we obtain $\mathcal{L}^2(\Omega\setminus \Omega_n)\leq \mathcal{L}^2((\Gamma)_\varepsilon)$. Since $\mathcal{L}^2(\Gamma)=0$ by ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}), the value $\mathcal{L}^2((\Gamma)_\varepsilon)$ can be made arbitrarily small. This means that $$\lim_{n\to\infty} \mathcal{L}^2((\Omega_n\setminus \Omega)\cup (\Omega\setminus \Omega_n))=0.$$ ◻ For later purposes it is desirable to have some uniform control on the geometry of the approximating Jordan curves $\Gamma_n$. At larger scales their behaviour with respect to condition ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) is uniformly controlled by that of $\Gamma$. **Lemma 9**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} be in force. Let $M$ be as in ([\[E:basicass\]](#E:basicass){reference-type="ref" reference="E:basicass"}), $\theta$ as in ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}) and $r_n'$ as in ([\[E:rnprime\]](#E:rnprime){reference-type="ref" reference="E:rnprime"}). For any $n$, any $x\in \Gamma_n$ and any $r\geq \frac{M}{\theta}r_n'$ only the connected component of $\Gamma_n\cap \overline{B(x,\frac32r)}$ that contains $x$ intersects $B(x,\frac{\theta}{4}r)$.* *Proof.* Assume that there is a connected component of $\Gamma_n\cap \overline{B(x,\frac32 r)}$ that does not contain $x$ but intersects $B(x,\frac{\theta}{4} r)$. Then there are $y\in \Gamma_n\setminus \overline{B(x,\frac32 r)}$ and $z\in \Gamma_n\cap B(x,\frac{\theta}{4} r)$ such that, following $\Gamma_n$ in a fixed orientation, $y$ is visited between visiting $x$ and $z$. By Lemma [\[L:Hausdorffconvbd\]](#L:Hausdorffconvbd){reference-type="ref" reference="L:Hausdorffconvbd"} (i) we have $d_H(\Gamma_n,\Gamma)<\frac{M}{4}\:r_n'$; consequently there are $\xi,\eta,\zeta\in \Gamma$ such that $\max(|x-\xi|,|y-\eta|,|z-\zeta|)<\frac{M}{4} r_n'\leq \frac{\theta}{4} r$ and $\eta$ is visited by $\Gamma$ between visiting $\xi$ and $\zeta$. But then $|\xi-\zeta|<\theta r$ and since $\overline{B(\xi,r)}\subset B(x,(\frac32-\frac{\theta}{4})r)$ also $\eta\in \Gamma\setminus \overline{B(\xi,r)}$. This contradicts ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}). ◻ For the polygonal Jordan curves $\Gamma_n$ constructed in Lemma [Lemma 7](#L:adhoc){reference-type="ref" reference="L:adhoc"} a uniform control at all scales is quickly seen. **Corollary 1**. *Let $\theta$ be as in ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}), $0<p,q<1$, $\omega=(\omega_1,\omega_2,...)\in \{p,q\}^\mathbb{N}$ and let $\mathcal{I}_n$ be the finite partitions of $\Gamma$ constructed in Lemma [Lemma 7](#L:adhoc){reference-type="ref" reference="L:adhoc"}.* 1. *For any large enough $n$, any $x\in \Gamma_n$ and any $r>0$ only the connected component of $\Gamma_n\cap \overline{B(x,r)}$ that contains $x$ intersects $B(x,\frac{\theta}{6}r)$.* 2. *There is a constant $S'=S'(\theta)\geq 1$, depending only on $\theta$, such that for any sufficiently large $n$ and any distinct $x,y\in \Gamma_n$ we have $\mathop{\mathrm{diam}}a_n(x,y)\leq S'|x-y|$, where $a_n(x,y)$ denotes a subarc of $\Gamma_n$ of minimal diameter connecting $x$ and $y$. In particular, each $\Gamma_n$ is a quasicircle and each $\Omega_n$ a quasidisc.* *Proof.* Let $r_n'$ be as in ([\[E:rnprime\]](#E:rnprime){reference-type="ref" reference="E:rnprime"}). For large enough $n$ and any $r<\frac14 r_n'$ the set $\Gamma_n\cap \overline{B(x,r)}$ is connected by Lemma [Lemma 7](#L:adhoc){reference-type="ref" reference="L:adhoc"}. The same is true for $\frac14 r_n'\leq r<\frac{15}{2\theta} r_n'$: If there would be a connected component $C_n$ of $\Gamma_n\cap \overline{B(x,r)}$ other than that through $x$, the set $C_n\cap \overline{B(x,\frac{\theta}{20}r)}$ would be a connected component of $\Gamma_n \cap \overline{B(x,\frac{\theta}{30}r)}$ not containing $x$, but since $\frac{\theta}{30} r<\frac14 r_n'$, Lemma [Lemma 7](#L:adhoc){reference-type="ref" reference="L:adhoc"} prevents this from happening. For $r\geq \frac{15}{2\theta} r_n'$ Lemma [Lemma 9](#L:controlatlargscales){reference-type="ref" reference="L:controlatlargscales"} states that only the $x$-component of $\Gamma_n\cap \overline{B(x,r)}$ can hit $B(x,\frac{\theta}{6}r)$. This shows (i). Statement (ii) now follows from [@MartioSarvas79 2.25. Lemma] and its proof. ◻ Given $\varepsilon>0$, a domain $\Omega\subset \mathbb{R}^2$ is said to be an *$(\varepsilon,\infty)$-uniform domain* if for any $x,y\in \Omega$ there is a rectifiable arc $\gamma\subset \Omega$ of length $\ell(\gamma)$ connecting $x$ and $y$ such that $\ell(\gamma)\leq\frac{|x-y|}{\varepsilon}$ and $\mathop{\mathrm{dist}}(z,\partial\Omega)\geq \varepsilon\:\frac{|x-z||y-z|}{|x-y|}$ for all $z\in \gamma$, [@Jones81]. A number of different but equivalent definitions can be found in [@Vaisala88]. It is well-known that a Jordan domain is a quasidisc if and only if it is an $(\varepsilon,\infty)$-domain for some $\varepsilon>0$, see [@Jones81 Theorem C] or [@MartioSarvas79 2.33. Corollary]. Actually, both implications in this equivalence are quantitative, and we will use one of them in a quantitative fashion: Any bounded quasidisc $\Omega$ with boundary $\Gamma=\partial\Omega$ and $S$ as in ([\[E:3point\]](#E:3point){reference-type="ref" reference="E:3point"}) is an $(\varepsilon,\infty)$-domain with parameter $\varepsilon=\varepsilon(S)>0$ depending only on $S$. This can for instance be concluded by combining [@Lehto87 Theorem 6.6 and its proof] with [@Lehto87 Theorems 6.2 and 6.3] or [@MartioSarvas79 2.15. Theorem]. *Remark 6*. To see that any bounded $(\varepsilon,\infty)$-domain in $\mathbb{R}^2$ is a quasidisc with $S=S(\varepsilon)\geq 1$ in ([\[E:3point\]](#E:3point){reference-type="ref" reference="E:3point"}) depending only on $\varepsilon$, one can combine [@Lehto87 Theorems 6.4 and 6.5]. The preceding discussion gives the following result on approximating polygonal domains. **Theorem 2**. *Let $\theta$ be as in ([\[E:theta\]](#E:theta){reference-type="ref" reference="E:theta"}), $0<p,q<1$, $\omega=(\omega_1,\omega_2,...)\in \{p,q\}^\mathbb{N}$. Let $\mathcal{I}_n$ be the finite partitions of $\Gamma$ constructed in Lemma [Lemma 7](#L:adhoc){reference-type="ref" reference="L:adhoc"} and for any large enough $n$, let $\Omega_n$ be the quasidisc enclosed by the polygonal Jordan curve $\Gamma_n$. There is some $\varepsilon=\varepsilon(\theta)>0$, depending only on $\theta$, such that for any large enough $n$ the quasidisc $\Omega_n$ is an $(\varepsilon,\infty)$-uniform domain.* # Mosco convergence of Dirichlet energy forms {#S:Mosco} Suppose that Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} are satisfied. On the domains $\Omega$ and $\Omega_n$ we consider the classical Dirichlet (energy) forms $$\mathcal{D}(u)=\int_\Omega |\nabla u|^2\:dx,\quad u\in H^1(\Omega),$$ and $$\mathcal{D}_n(u)=\int_{\Omega_n} |\nabla u|^2\:dx,\quad u\in H^1(\Omega_n).$$ A priori they are densely defined quadratic forms on $L^2(\Omega)$ and $L^2(\Omega_n)$, respectively. However, it is convenient to view $H^1(\Omega)$ and $H^1(\Omega_n)$ as dense subspaces of $L^2(\mathbb{R}^2)$ by saying that $u\in L^2(\mathbb{R}^2)$ is an element of $H^1 (\Omega)$ if $u|_\Omega$ is, and similarly for $\Omega_n$. With this agreement in mind we extend the definitions of $\mathcal{D}$ and $\mathcal{D}_n$ to all of $u\in L^2(\mathbb{R}^2)$ by setting $\mathcal{D}(u):=+\infty$ for $u\in L^2(\mathbb{R}^2)$ such that $u|_\Omega\notin H^1(\Omega)$ and $\mathcal{D}_n(u):=+\infty$ for $u\in L^2(\mathbb{R}^2)$ such that $u|_{\Omega_n}\notin H^1(\Omega_n)$. We now assume the following. *Assumption 4*. There is some $\varepsilon>0$ such that for large any enough $n$, the domain $\Omega_n$ is an $(\varepsilon,\infty)$-uniform domain. Note that Assumption [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} is satisfied for the approximating quasidiscs $\Omega_n$ in Theorem [Theorem 2](#T:adhoc){reference-type="ref" reference="T:adhoc"}. Under Assumption [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} the convergence of the Dirichlet forms can be seen using standard arguments; the result and its proof are similar to earlier results in [@CapitanelliVivaldi2011; @HR-PT21; @HR-PT23; @LanciaVernole14]. **Theorem 3**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}, [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} and [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} be satisfied. Then we have $$\lim_{n\to\infty} \mathcal{D}_n=\mathcal{D}$$ in the Mosco sense on $L^2(\mathbb{R}^2)$.* We give a proof for the convenience of the reader. Since $\Omega$ is an $(\varepsilon,\infty)$-domain, there is a bounded linear extension operator $E_\Omega:H^1(\Omega)\to H^1(\mathbb{R}^2)$ by [@Jones81 Theorem 1]. The same theorem together with Assumption [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} ensures that for all large enough $n$ there are linear extension operators $E_{\Omega_n}:H^1(\Omega_n)\to H^1(\mathbb{R}^2)$ whose operator norms are uniformly bounded with respect to $n$. *Proof.* To prove the $\liminf$-condition, suppose that $(u_n)_n\subset L^2(\mathbb{R}^2)$ and $u\in L^2(\mathbb{R}^2)$ are such that $\lim_{n\to\infty} u_n=u$ weakly in $L^2(\mathbb{R}^2)$. We may assume that $L:=\liminf_{n\to\infty} \mathcal{D}_n(u_n)<+\infty$ and can find a sequence $(n_k)_k$ with $n_k\uparrow +\infty$ such that $\lim_{k\to\infty} \mathcal{D}_{n_k}(u_{n_k})=L$. Since $\sup_n\|u_n\|_{L^2(\mathbb{R}^2)}<+\infty$ by weak convergence, we find that $\sup_k\|u_{n_k}\|_{H^1(\Omega_{n_k})}<+\infty$, so that $\sup_k\|E_{\Omega_{n_k}}u_{n_k}\|_{H^1(\mathbb{R}^2)}<+\infty$ by [@Jones81 Theorem 1]. Passing to subsequences if necessary, we may assume that $(E_{\Omega_{n_k}}u_{n_k})_k$ converges to some $u^\ast$ weakly in $H^1(\mathbb{R}^2)$ with convex combinations converging strongly to $u^\ast$ in $H^1(\mathbb{R}^2)$, [@RieszNagy56 Section 38]. This implies that $u^\ast=u$. We may also assume that $(\nabla E_{\Omega_{n_k}}u_{n_k})_k$ converges weakly in $L^2(\mathbb{R}^2,\mathbb{R}^2)$ with convex combinations converging strongly; by the preceding its limit must be $\nabla u$. Using Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"} we find that $$\lim_{k\to\infty} \mathbf{1}_{\Omega_{n_k}}\nabla u_{n_k}=\lim_{k\to\infty}\mathbf{1}_{\Omega_{n_k}}\nabla E_{\Omega_{n_k}}u_{n_k}=\mathbf{1}_\Omega \nabla u$$ weakly in $L^2(\mathbb{R}^2,\mathbb{R}^2)$ and accordingly, $$\mathcal{D}(u)=\int_\Omega |\nabla u|^2\:dx\leq \liminf_{k\to\infty}\int_{\Omega_{n_k}}|\nabla u_{n_k}|^2\:dx=\lim_{k\to\infty} \mathcal{D}_{n_k}(u_{n_k})=L.$$ To prove the $\limsup$-condition, suppose that $u\in L^2(\mathbb{R}^2)$. We may assume that $\mathcal{D}(u)<+\infty$. It follows that $u\in H^1(\Omega)$. Consequently $E_\Omega u$ is an element of $H^1(\mathbb{R}^2)$, and setting $u_n:=(E_\Omega u)|_{\Omega_n}$, we can use Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"} and bounded convergence to see that $$\lim_{n\to\infty} u_n=u\quad \text{in $L^2(\mathbb{R}^2)$}$$ and $$\limsup_{n\to\infty}\mathcal{D}_n(u_n)=\lim_{n\to\infty}\int_D\mathbf{1}_{\Omega_n}|\nabla E_\Omega u|^2dx=\int_D\mathbf{1}_\Omega |\nabla E_\Omega u|^2dx=\mathcal{D}(u).$$ ◻ # Mosco convergence of superpositions {#S:MoscoWentzell} Consider the Borel measure $$m:=\mathcal{L}^2|_\Omega+\mu.$$ The space $L^2(\overline{\Omega},m)$ is isometrically isomorphic to the orthogonal direct sum $L^2(\Omega)\oplus L^2(\Gamma,\mu)$ under the linear map $u\mapsto (u|_\Omega,u|_\Gamma)$, and we identify these spaces. In particular, $$\label{E:sps} \left\langle u,v\right\rangle_{L^2(\overline{\Omega},m)}=\left\langle u|_\Omega,v|_\Omega\right\rangle_{L^2(\Omega)}+\left\langle u|_\Gamma,v|_\Gamma\right\rangle_{L^2(\Gamma,\mu)},\quad u,v\in L^2(\overline{\Omega},m).$$ Now suppose that Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} are satisfied. Let $\Gamma_n$, $\mu_n$ and $\Omega_n$ be as before and set $$m_n:=\mathcal{L}^2|_{\Omega_n}+\mu_n.$$ Then analogous statements as above are true for each $L^2(\overline{\Omega}_n,m_n)$. Since $H^1(\Omega)$ is dense in $L^2(\Omega)$ and $\mathop{\mathrm{Lip}}(\Gamma)$ is dense in $L^2(\Gamma,\mu)$, the direct sum $H^1(\Omega)\oplus \mathop{\mathrm{Lip}}(\Gamma)$ is dense in $L^2(\overline{\Omega},m)$. Using Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}, Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"}, bounded convergence and ([\[E:sps\]](#E:sps){reference-type="ref" reference="E:sps"}), we can conclude that the Hilbert spaces converge in the sense of [@KuwaeShioya03 Section 2.2], see Appendix [10](#S:Notions){reference-type="ref" reference="S:Notions"}. **Lemma 10**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} be satisfied. For any $u\in H^1(\Omega)\oplus \mathop{\mathrm{Lip}}(\Gamma)$ we have $$\lim_{n\to \infty} \left\| (E_\Omega(u|_\Omega))|_{\Omega_n}+(E_\Gamma(u|_\Gamma))|_{\Gamma_n}\right\|_{L^2(\overline{\Omega}_n,m_n)}=\left\|u\right\|_{L^2(\overline{\Omega},m)}.$$ The sequence of Hilbert spaces $L^2(\overline{\Omega}_n,m_n)$ converges to $L^2(\overline{\Omega},m)$ with identification maps $u\mapsto (E_\Omega(u|_\Omega))|_{\Omega_n}+(E_\Gamma(u|_\Gamma))|_{\Gamma_n}$, $u\in H^1(\Omega)\oplus \mathop{\mathrm{Lip}}(\Gamma)$.* Given an element $u$ of $L^2(\Omega)$ or $L^2(\Omega_n)$, we write $u^\circ$ for its continuation by zero to all of $\mathbb{R}^2$. The following will be used later on. **Lemma 11**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} be satisfied.* 1. *Let $v\in L^2(\mathbb{R}^2)$ and let $\varphi_n\in L^2(\Gamma_n,\mu_n)$ and $\varphi\in L^2(\Gamma,\mu)$ be such that $\lim_{n\to\infty} \varphi_n=\varphi$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}. Then $\lim_{n\to\infty} (v|_{\Omega_n}+\varphi_n)=v|_\Omega+\varphi$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}.* 2. *Let $u_n\in L^2(\overline{\Omega}_n,m_n)$ and $u\in L^2(\overline{\Omega},m)$ be such that $\lim_{n\to\infty} u_n=u$ KS-weakly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}. Then $\lim_{n\to\infty} (u_n|_{\Omega_n})^\circ=(u|_\Omega)^\circ$ weakly in $L^2(\mathbb{R}^2)$ and $\lim_{n\to\infty} u|_{\Gamma_n}=u|_\Gamma$ KS-weakly with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}.* *Proof.* Let $(\widetilde{\varphi}_m)_m\subset \mathop{\mathrm{Lip}}(\Gamma)$ be such that $$\label{E:bdapp} \lim_{m\to\infty}\limsup_{n\to\infty}\|(E_\Gamma\widetilde{\varphi}_m)|_{\Gamma_n}-\varphi_n\|_{L^2(\Gamma_n,\mu_n)}=0\quad\text{and}\quad \lim_{m\to\infty} \|\widetilde{\varphi}_m-\varphi\|_{L^2(\Gamma,\mu)}=0$$ and $(\widetilde{v}_m)_m\subset H^1(\Omega)$ such that $$\label{E:bulkapp} \lim_{m\to\infty}\|\widetilde{v}_m-v|_\Omega\|_{L^2(\Omega)}=0.$$ Then for each $m$ we have $\widetilde{v}_m+\widetilde{\varphi}_m\in H^1(\Omega)\oplus \mathop{\mathrm{Lip}}(\Gamma)$ and $$\lim_{m\to\infty} \|\widetilde{v}_m+\widetilde{\varphi}_m-(v|_\Omega+\varphi)\|_{L^2(\overline{\Omega},m)}=0$$ by ([\[E:sps\]](#E:sps){reference-type="ref" reference="E:sps"}). Since $$\|\mathbf{1}_{\Omega_n}(E_\Omega\widetilde{v}_m)-\mathbf{1}_{\Omega_n}v\|_{L^2(\mathbb{R}^2)}\leq \|(\mathbf{1}_{\Omega_n}-\mathbf{1}_\Omega)E_\Omega\widetilde{v}_m\|_{L^2(\mathbb{R}^2)}+\|(\mathbf{1}_{\Omega_n}-\mathbf{1}_{\Omega})v\|_{L^2(\mathbb{R}^2)}+\|\widetilde{v}_m-v|_\Omega\|_{L^2(\Omega)},$$ we can use ([\[E:sps\]](#E:sps){reference-type="ref" reference="E:sps"}), the subadditivity of $\limsup$, Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"} and bounded convergence to see that $$\begin{aligned} \limsup_{n\to\infty}\|(E_\Omega\widetilde{v}_m)|_{\Omega_n} &+(E_\Gamma\widetilde{\varphi}_m)|_{\Gamma_n}-(v|_{\Omega_n}+\varphi_n)\|_{L^2(\overline{\Omega}_n,m_n)}\notag\\ &\leq \limsup_{n\to\infty}\|(E_\Omega\widetilde{v}_m)|_{\Omega_n}-v|_{\Omega_n}\|_{L^2(\Omega_n)}+\limsup_{n\to\infty}\|(E_\Gamma\widetilde{\varphi}_m)|_{\Gamma_n}-\varphi_n\|_{L^2(\Gamma_n,\mu_n)}\notag\\ &\leq \|\widetilde{v}_m-v|_{\Omega}\|_{L^2(\Omega)}+\limsup_{n\to\infty}\|(E_\Gamma\widetilde{\varphi}_m)|_{\Gamma_n}-\varphi_n\|_{L^2(\Gamma_n,\mu_n)}.\notag\end{aligned}$$ Using ([\[E:bdapp\]](#E:bdapp){reference-type="ref" reference="E:bdapp"}) and ([\[E:bulkapp\]](#E:bulkapp){reference-type="ref" reference="E:bulkapp"}) we arrive at $$\lim_{m\to\infty}\limsup_{n\to\infty}\|(E_\Omega\widetilde{v}_m)|_{\Omega_n}+(E_\Gamma\widetilde{\varphi}_m)|_{\Gamma_n}-(v|_{\Omega_n}+\varphi_n)\|_{L^2(\overline{\Omega}_n,m_n)}=0.$$ This shows (i). To see (ii), let $w\in L^2(\mathbb{R}^2)$. Then $\lim_{n\to\infty}\mathbf{1}_{\Omega_n}w=\mathbf{1}_\Omega w$ in $L^2(\mathbb{R}^2)$ by Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"} and bounded convergence. By (i), applied with $\varphi_n= 0$ and $\varphi=0$, it follows that $\lim_{n\to\infty}w|_{\Omega_n}=w|_{\Omega}$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}. Consequently $$\lim_{n\to\infty}\big\langle (u_n|_{\Omega_n})^\circ,w\big\rangle_{L^2(\mathbb{R}^2)}=\lim_{n\to\infty}\big\langle u_n,w|_{\Omega_n} \big\rangle_{L^2(\Omega_n)}= \big\langle u,w|_{\Omega} \big\rangle_{L^2(\Omega)}= \big\langle (u|_{\Omega})^\circ,w\big\rangle_{L^2(\mathbb{R}^2)}.$$ This give the first claim in (ii). For the second, let $\varphi_n\in L^2(\Gamma_n,\mu_n)$ and $\varphi\in L^2(\Gamma,\mu)$ be such that $\lim_{n\to\infty} \varphi_n=\varphi$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}. Item (i), applied with $v=0$, gives the same convergence KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"} and therefore $$\lim_{n\to\infty} \big\langle u_n|_{\Gamma_n},\varphi_n\big\rangle_{L^2(\Gamma_n,\mu_n)}=\big\langle u|_\Gamma,\varphi\big\rangle_{L^2(\Gamma,\mu)},$$ which shows the second claim. ◻ We choose $\alpha=1$ in ([\[E:constellation\]](#E:constellation){reference-type="ref" reference="E:constellation"}) and ([\[E:constellation2\]](#E:constellation2){reference-type="ref" reference="E:constellation2"}), note that by ([\[E:sd\]](#E:sd){reference-type="ref" reference="E:sd"}) this choice is admissible. For simplicity, we take $\sigma_1(x,r):=\mu(B(x,r))$ in ([\[E:symbol\]](#E:symbol){reference-type="ref" reference="E:symbol"}). Then $$\label{E:Q1} \mathcal{Q}^1(\varphi)=\int_\Gamma\int_\Gamma\frac{(\varphi(x)-\varphi(y))^2}{\mu(B(x,|x-y|))^2}\mu(dx)\mu(dy),\quad \varphi\in B_1^{2,2}(\Gamma),$$ and $$\begin{gathered} \mathcal{Q}^1_n(\varphi)=\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,A\:r_n)^c}\frac{(\varphi(\xi)-\varphi(\eta))^2}{\mu(B(\xi,|\xi-\eta|))\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta)\notag\\ +\int_{\Gamma_n}\int_{\Gamma_n\cap B(\xi,A\:r_n)}\frac{(\varphi(\xi)-\varphi(\eta))^2}{|\xi-\eta|\mu_n(B(\xi,|\xi-\eta|))}\mu_n(d\xi)\mu_n(d\eta),\quad \varphi\in B_1^{2,2}(\Gamma_n).\end{gathered}$$ Composing $E_\Omega$ with ([\[E:traceasop\]](#E:traceasop){reference-type="ref" reference="E:traceasop"}) we obtain a bounded linear trace operator $$\mathrm{Tr}_{\Omega,\Gamma}:=E_\Omega\circ \mathrm{Tr}_{\Gamma}:H^1(\Omega)\to B_1^{2,2}(\Gamma).$$ The space $$V(\overline{\Omega}):=\{v+\mathrm{Tr}_{\Omega,\Gamma}v:\ v\in H^1(\Omega)\}$$ is a subspace of the vector space $H^1(\Omega)\oplus B_1^{2,2}(\Gamma)$; note that for an element $u=v+\mathrm{Tr}_{\Omega,\Gamma}v$ of $V(\overline{\Omega})$ we have $u|_\Omega=v$ and $u|_\Gamma=\mathrm{Tr}_{\Omega,\Gamma}v$. Since the space $V(\overline{\Omega})$ contains all restrictions of $\mathop{\mathrm{Lip}}_c(\mathbb{R}^2)$-functions to $\overline{\Omega}$, it is dense in $L^2(\overline{\Omega},m)$ by Stone-Weierstrass. The space $\mathop{\mathrm{Lip}}_c(\mathbb{R}^2)|_{\overline{\Omega}}$ of such restrictions to $\overline{\Omega}$ is a dense subspace of $V(\overline{\Omega})$; this is straightforward from the density of $\mathop{\mathrm{Lip}}_c(\mathbb{R}^2)$ in $H^1(\mathbb{R}^2)$ and the continuity of the trace operator ([\[E:traceasop\]](#E:traceasop){reference-type="ref" reference="E:traceasop"}). Now let Assumption [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} be in force. Composition of the extension operators $E_{\Omega_n}$ with ([\[E:traceasop\]](#E:traceasop){reference-type="ref" reference="E:traceasop"}) gives linear trace operators $$\mathrm{Tr}_{\Omega_n,\Gamma_n}:=E_{\Omega_n}\circ \mathrm{Tr}_{\Gamma_n}:H^1(\Omega_n)\to B_1^{2,2}(\Gamma_n).$$ Similarly as before, the space $V(\overline{\Omega}_n):=\{(u,\mathrm{Tr}_{\Omega_n,\Gamma_n}u):\ u\in H^1(\Omega_n)\}$ is dense in $L^2(\overline{\Omega}_n,m_n)$. We consider the quadratic forms $$\mathcal{E}(u):=\mathcal{D}(u|_\Omega)+\mathcal{Q}^1(u|_\Gamma),\quad u\in V(\overline{\Omega}),$$ and $$\mathcal{E}_n(u):=\mathcal{D}_n(u|_{\Omega_n})+\mathcal{Q}^1_n(u|_{\Gamma_n}),\quad u\in V(\overline{\Omega}_n);$$ they define Dirichlet forms $(\mathcal{E},V(\overline{\Omega}))$ and $(\mathcal{E}_n,V(\overline{\Omega}_n))$ on $L^2(\overline{\Omega},m)$ and $L^2(\overline{\Omega}_n,m_n)$, respectively. The Hilbert space norm $\|\cdot\|_{V(\overline{\Omega})}$ in $V(\overline{\Omega})$ is determined by $$\|u\|_{V(\overline{\Omega})}^2= \mathcal{E}(u)+ \|u|_\Omega\|_{L^2(\Omega)}^2 +\|u|_{\Gamma}\|^{2}_{L^2(\Gamma,\mu)},$$ similarly for $V(\overline{\Omega}_n)$. We set $$\mathcal{E}(u):=+\infty\quad \text{for $u\in L^2(\overline{\Omega},m)\setminus V(\overline{\Omega})$}\quad \text{and }\quad \mathcal{E}_n(u):=+\infty\ \text{for $u\in L^2(\overline{\Omega}_n,m_n)\setminus V(\overline{\Omega}_n)$.}$$ We conclude the convergence of these Dirichlet forms in the KS-generalized Mosco sense. **Theorem 4**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}, [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"}, [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} and [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} be satisfied. Then we have $$\lim_{n\to\infty} \mathcal{E}_n=\mathcal{E}$$ in the KS-generalized Mosco sense with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}.* To verify Theorem [Theorem 4](#T:MoscoWentzell){reference-type="ref" reference="T:MoscoWentzell"} we use the following result, which is immediate from Proposition [Proposition 1](#P:convLip){reference-type="ref" reference="P:convLip"}, Lemma [Lemma 8](#L:convcharfcts){reference-type="ref" reference="L:convcharfcts"} and bounded convergence. **Proposition 4**. *Let Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}, [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} be in force. Then we have $$\lim_{n\to \infty} \mathcal{E}_n(u)=\mathcal{E}(u),\quad u\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2).$$* We prove Theorem [Theorem 4](#T:MoscoWentzell){reference-type="ref" reference="T:MoscoWentzell"}. *Proof.* To see the $\liminf$-condition, suppose that $u_n\in L^2(\overline{\Omega}_n,m_n)$ and $u\in L^2(\overline{\Omega},m)$ be such that $\lim_{n\to\infty} u_n=u$ KS-weakly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}. By Lemma [Lemma 11](#L:inherit){reference-type="ref" reference="L:inherit"} (ii) we have $\lim_{n\to\infty} (u_n|_{\Omega_n})^\circ=(u|_\Omega)^\circ$ weakly in $L^2(\mathbb{R}^2)$ and $\lim_{n\to\infty} u_n|_{\Gamma_n}=u|_\Gamma$ KS-weakly with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}. By Theorems [Theorem 3](#T:Mosco){reference-type="ref" reference="T:Mosco"} and [Theorem 1](#T:Moscoboundary){reference-type="ref" reference="T:Moscoboundary"} therefore $\mathcal{D}(u|_\Omega)\leq \liminf_{n\to\infty} \mathcal{D}_n(u_n|_{\Omega_n})$ and $\mathcal{Q}^1(u|_\Gamma)\leq \liminf_{n\to\infty} \mathcal{Q}^1_n(u_n|_{\Gamma_n})$, and the superadditivity of $\liminf$ gives $$\mathcal{E}(u)\leq \liminf_{n\to\infty} \mathcal{E}_n(u_n).$$ To see the $\limsup$-condition, suppose that $u\in L^2(\overline{\Omega},m)$. We may assume $\mathcal{E}(u)<+\infty$, so that $u\in V(\overline{\Omega})$. By density we can find a sequence $(v_m)_m\subset \mathop{\mathrm{Lip}}_c(\mathbb{R}^2)|_{\overline{\Omega}}$ such that $$\lim_{m\to\infty} \mathcal{E}(v_m)=\mathcal{E}(u)\quad \text{and}\quad \lim_{m\to\infty}\|v_m-u\|_{L^2(\overline{\Omega},m)}=0.$$ One can now use Proposition [Proposition 4](#P:convLipWentzell){reference-type="ref" reference="P:convLipWentzell"} and analogous arguments as in the proof of Proposition [Proposition 2](#P:limsupboundary){reference-type="ref" reference="P:limsupboundary"} to construct a sequence $(u_m)_m$ converging to $u$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"} and satisfying $$\limsup_{n\to\infty} \mathcal{E}_n(u_n)= \mathcal{E}(u).$$ ◻ # Application to elliptic and parabolic problems {#S:Apps} To fix notation we write $(\mathcal{L}^1_\Gamma,\mathcal{D}(\mathcal{L}^1_\Gamma))$ for the infinitesimal generator of $(\mathcal{Q}^1,B^{2,2}_1(\Gamma,\mu))$ as in ([\[E:Q1\]](#E:Q1){reference-type="ref" reference="E:Q1"}), that is, the unique non-positive self-adjoint operator on $L^2(\Gamma,\mu)$ such that $$\mathcal{Q}^1(\varphi,\psi)=-\left\langle \mathcal{L}^1_\Gamma \varphi,\psi\right\rangle_{L^2(\Gamma,\mu)},\quad \varphi\in \mathcal{D}(\mathcal{L}^1_\Gamma),\ \psi\in B^{2,2}_1(\Gamma,\mu).$$ It has the representation $$\mathcal{L}_\Gamma^1\varphi(x)=\int_\Gamma(\varphi(y)-\varphi(x))\left\lbrace \mu(B(x,|x-y|))^{-2}+\mu(B(y,|x-y|))^{-2}\right\rbrace\mu(dy),\quad \varphi\in \mathcal{D}(\mathcal{L}^1_\Gamma).$$ The infinitesimal generator of $(\mathcal{D},H^1(\Omega))$ is is the Neumann Laplacian $(\mathcal{L}_\Omega,\mathcal{D}(\mathcal{L}_\Omega))$ for $\Omega$, it is the unique non-positive self-adjoint operator on $L^2(\Omega)$ satisfying $$\mathcal{D}(u,v)=-\left\langle \mathcal{L}_\Omega u,v\right\rangle_{L^2(\Omega)},\quad u\in \mathcal{D}(\mathcal{L}_\Omega),\quad v\in H^1(\Omega).$$ Given $\lambda>0$, $f\in L^2(\Omega)$ and $\varphi\in L^2(\Gamma,\mu)$, we call $u\in V(\overline{\Omega})$ a *weak solution* to the elliptic boundary value problem $$\label{E:elliptic} \begin{cases} (\lambda-\mathcal{L}_\Omega)(u|_\Omega) =f & \text{on $\Omega$},\\ (\lambda-\mathcal{L}_\Gamma^1)(u|_\Gamma) =\varphi & \text{on $\Gamma$} \end{cases}$$ if $$\mathcal{E}(u,v)+\lambda\int_{\overline{\Omega}} uv\:dm=\int_\Omega fv\:dx+\int_\Gamma \varphi v\:d\mu,\quad v\in V(\overline{\Omega}).$$ Under Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} we can also consider ([\[E:elliptic\]](#E:elliptic){reference-type="ref" reference="E:elliptic"}) with $\Omega_n$, $\Gamma_n$ and $\mu_n$ in place of $\Omega$, $\Gamma$ and $\mu$. The infinitesimal generator $(\mathcal{L}^1_{\Gamma_n},\mathcal{D}(\mathcal{L}^1_{\Gamma_n}))$ of $(\mathcal{Q}^1_n,B^{2,2}_1(\Gamma_n,\mu_n))$ is of the form $$\mathcal{L}^1_{\Gamma_n}\varphi(\xi)=\int_{\Gamma_n}(\varphi(\eta)-\varphi(\xi))j_n(\xi,\eta)\mu_n(d\eta),\quad \varphi\in \mathcal{D}(\mathcal{L}^1_{\Gamma_n}),$$ where $$\begin{gathered} j_n(\xi,\eta)=\mathbf{1}_{B(\xi,Ar_n)^c}(\eta)\left\lbrace \mu(B(\xi,|\xi-\eta|))^{-1}\mu_n(B(\xi,|\xi-\eta|))^{-1}\right.\notag\\ \left.+\mu(B(\eta,|\xi-\eta|))^{-1}\mu_n(B(\eta,|\xi-\eta|))^{-1} \right\rbrace\notag\\ +\mathbf{1}_{B(\xi,Ar_n)}(\eta)|\xi-\eta|^{-1}\left\lbrace \mu_n(B(\xi,|\xi-\eta|))^{-1}+\mu_n(B(\eta,|\xi-\eta|))^{-1}\right\rbrace.\end{gathered}$$ The Riesz representation theorem gives the following. **Lemma 12**. *Let $\lambda>0$. Then for any $f\in L^2(\Omega)$ and $\varphi\in L^2(\Gamma,\mu)$ there is a unique weak solution $u$ to ([\[E:elliptic\]](#E:elliptic){reference-type="ref" reference="E:elliptic"}). If Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} are satisfied, then for any large enough $n$ and any $f\in L^2(\Omega_n)$ and $\varphi\in L^2(\Gamma_n,\mu_n)$ there is a unique weak solution $u_n$ to the corresponding problem on $\Omega_n$.* *Remark 7*. The requirement to have $n$ large enough stems from Assumption [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"}, which was formulated that way and makes sure the $\Gamma_n$ are boundaries of domains $\Omega_n$. Theorem [Theorem 4](#T:MoscoWentzell){reference-type="ref" reference="T:MoscoWentzell"}, [@KuwaeShioya03 Theorem 2.4], Lemma [Lemma 11](#L:inherit){reference-type="ref" reference="L:inherit"} (i) and Lemma [Lemma 13](#L:traceconv){reference-type="ref" reference="L:traceconv"} below give the next stability result. **Theorem 5**. *Suppose that Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}, [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"}, [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} and [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} are in force. Let $\lambda>0$, $f\in L^2(\mathbb{R}^2)$ and $g\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$, and let $u$ and $u_n$ denote the unique weak solutions to ([\[E:elliptic\]](#E:elliptic){reference-type="ref" reference="E:elliptic"}) on $\Omega$ and $\Omega_n$ with $\mathrm{Tr}_\Gamma g$ and $\mathrm{Tr}_{\Gamma_n} g$ in place of $\varphi$, respectively. Then we have $$\lim_{n\to\infty} u_n=u$$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}.* To see Theorem [Theorem 5](#T:ellipticstability){reference-type="ref" reference="T:ellipticstability"} we can use the following consequence of Lemma [\[L:Hausdorffconvbd\]](#L:Hausdorffconvbd){reference-type="ref" reference="L:Hausdorffconvbd"} (i) and the fact that the Whitney extension operator $E_\Gamma$ preserves the Lipschitz constant, [@Stein70 Chapter VI, Theorem 3]. **Lemma 13**. *Suppose that Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"} are in force and let $g\in \mathop{\mathrm{Lip}}_b(\mathbb{R}^2)$. Then $\lim_{n\to\infty} \mathrm{Tr}_{\Gamma_n} g=\mathrm{Tr}_\Gamma g$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 3](#L:KSconvbd){reference-type="ref" reference="L:KSconvbd"}.* In a similar manner related Cauchy problems can be studied. Let $(\mathcal{L},\mathcal{D}(\mathcal{L}))$ be the infinitesimal generator of $(\mathcal{E},V(\overline{\Omega}))$; it is the superposition of $\mathcal{L}_\Omega$ and $\mathcal{L}_\Gamma^1$. Given $\lambda>0$ and $u_0\in L^2(\overline{\Omega},m)$, we call a function $u:[0,+\infty)\to L^2(\overline{\Omega},m)$ a *solution* to the Cauchy problem $$\label{E:parabolic} \begin{cases} \frac{du}{dt}(t)&=(\mathcal{L}-\lambda)u(t), \quad t>0,\\ u(0)&=u_0\end{cases}$$ if $u$ is an element of $C^1((0,+\infty),L^2(\overline{\Omega},m))\cap C([0,+\infty),L^2(\overline{\Omega},m))$, $u(t)\in \mathcal{D}(\mathcal{L})$ for all $t>0$ and ([\[E:parabolic\]](#E:parabolic){reference-type="ref" reference="E:parabolic"}) holds. Under Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} we can again consider the analogous problem on $\Omega_n$, we use similar notation. Existence and uniqueness of solutions are clear by semigroup theory. **Lemma 14**. *Let $\lambda>0$. Then for any $u_0\in L^2(\overline{\Omega},m)$ there is a unique solution $u$ to ([\[E:parabolic\]](#E:parabolic){reference-type="ref" reference="E:parabolic"}). If Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"} and [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} are satisfied, then for any large enough $n$ and any $u_{0,n}\in L^2(\overline{\Omega}_n,m_n)$ there is a unique solution $u_n$ to the corresponding problem on $\Omega_n$.* Stability now follows from Theorem [Theorem 4](#T:MoscoWentzell){reference-type="ref" reference="T:MoscoWentzell"} and [@KuwaeShioya03 Theorem 2.4]. **Theorem 6**. *Suppose that Assumptions [\[A:basicass\]](#A:basicass){reference-type="ref" reference="A:basicass"}, [Assumption 2](#A:boundaryless){reference-type="ref" reference="A:boundaryless"}, [Assumption 3](#A:Jordan){reference-type="ref" reference="A:Jordan"} and [Assumption 4](#A:uniformeps){reference-type="ref" reference="A:uniformeps"} are in force. Let $\lambda>0$ and let $u_0\in L^2(\overline{\Omega},m)$ and $u_{0,n}\in L^2(\overline{\Omega}_n,m_n)$ be such that $\lim_{n\to\infty} u_{0,n}=u_0$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}. Let $u$ and $u_n$ denote the unique solutions to ([\[E:elliptic\]](#E:elliptic){reference-type="ref" reference="E:elliptic"}) on $\Omega$ and $\Omega_n$ with initial conditions $u_0$ and $u_{0,n}$, respectively. Then for any $t>0$ we have $$\lim_{n\to\infty} u_n(t)=u(t)$$ KS-strongly with respect to the convergence of Hilbert spaces in Lemma [Lemma 10](#L:KSconv){reference-type="ref" reference="L:KSconv"}.* # Notions of convergence {#S:Notions} For the convenience of the reader we briefly recall some basic notions from [@KuwaeShioya03] in a form adapted to our text. Let $H$ and $H_1, H_2, ...$ be separable Hilbert spaces, $\mathcal{C}$ a dense subspace of $H$ and $\Phi_n:\mathcal{C}\to H_n$ linear maps. If $$\label{E:KSdef} \lim_{n\to\infty} \|\Phi_n u\|_{H_n}=\|u\|_H,\quad u\in \mathcal{C},$$ then we say that the sequence *$(H_n)_n$ converges to $H$ with identification maps $\Phi_n$, $n\geq 1$*. Suppose now that this is the case. A sequence $(u_n)_n$ with $u_n\in H_n$ is said to *converge KS-strongly to $u\in H$* with respect to ([\[E:KSdef\]](#E:KSdef){reference-type="ref" reference="E:KSdef"}) if there is a sequence $(\widetilde{u}_m)_m\subset \mathcal{C}$ such that $\lim_{m\to\infty} \widetilde{u}_m=u$ in $H$ and $$\lim_{m\to \infty}\limsup_{n\to\infty} \|\Phi_n \widetilde{u}_m-u_n\|_{H_n}=0,$$ see [@KuwaeShioya03 Definition 2.4]. A sequence $(u_n)_n$ with $u_n\in H_n$ is said to *converge KS-weakly to $u\in H$* with respect to ([\[E:KSdef\]](#E:KSdef){reference-type="ref" reference="E:KSdef"}) if $$\lim_{n\to\infty}\left\langle u_n,v_n\right\rangle_{H_n}=\left\langle u,v\right\rangle_H$$ for any sequence $(v_n)_n$ with $v_n\in H_n$ and any $v\in H$ such $\lim_{n\to\infty} v_n=v$ KS-strongly with respect to ([\[E:KSdef\]](#E:KSdef){reference-type="ref" reference="E:KSdef"}), see [@KuwaeShioya03 Definition 2.4]. Suppose that $Q:H\to [0,+\infty]$ and $Q_n:H_n\to [0,+\infty]$ are quadratic forms. We say that the sequence $(Q_n)_n$ *converges to $Q$ in the KS-generalized Mosco sense* with respect to ([\[E:KSdef\]](#E:KSdef){reference-type="ref" reference="E:KSdef"}) if 1. for any sequence $(u_n)_n$ with $u_n\in H_n$ and any $u\in H$ such that $\lim_{n\to\infty} u_n=u$ KS-weakly with respect to ([\[E:KSdef\]](#E:KSdef){reference-type="ref" reference="E:KSdef"}), we have $$Q(u)\leq \liminf_{n\to\infty} Q_n(u_n),$$ 2. for any $u\in H$ there is a sequence $(u_n)_n$ with $u_n\in H_n$ such that $\lim_{n\to\infty} u_n=u$ KS-strongly and $$Q(u)\geq \limsup_{n\to\infty} Q_n(u_n).$$ See [@KuwaeShioya03 Definition 2.11]. As in the case of Mosco convergence on a single Hilbert space, [@Mosco94 Definition 2.1.1, Theorem 2.4.1 and Corollary 2.6.1], the KS-generalized Mosco convergence is equivalent to the strong convergence of resolvent and semigroup operators, [@KuwaeShioya03 Theorem 2.4.1]. **Acknowledgements.** S. C. and M. R. 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