contrast, in the case of Besov spaces, they are connected by the trace theorem of
Barton and Mayboroda [22, Theorem 3.9].

**Remark 7.4.** The boundary condition in $(R\mathbf{X})_A^\mathbf{P}$ is equivalent to the Dirichlet condition

$$ \lim_{t \to 0} u(t, \cdot) = g \in \mathbf{X}_{\perp}^{\mathbf{P}+1} $$

where $\nabla_{\parallel} g = f$, since $\nabla_{\parallel}$ is an isomorphism from $\mathbf{X}_{\perp}^{\mathbf{P}+1}$ onto $\mathbf{X}_{\parallel}^{\mathbf{P}}$. Therefore $(R\mathbf{X})_A^\mathbf{P}$ could be thought of as a Dirichlet problem $(D\mathbf{X})_A^{\mathbf{P}+1}$. The limit above is taken in the topology of $\mathbf{X}_{\perp}^{\mathbf{P}+1}$, which we defined as a subspace of $\mathcal{Z}'$, but which also embeds in the space of Schwartz distributions modulo constants when $\theta(\mathbf{p}) \in [-1, 0]$. When $\mathbf{p}$ is finite, we can be more precise. When $\theta(\mathbf{p}) = -1$, $\mathbf{X}_{\perp}^{\mathbf{P}+1}$ also embeds in the space of Schwartz distributions as the Lebesgue space $L^i(\mathbf{p})$. In this case, there is also convergence in $L^i(\mathbf{p})$ having fixed the constant (recall that $\nabla u \in X^\mathbf{P}$ so $u$ is determined up to a constant), see [16, Corollary 1.4]. The case $\theta(\mathbf{p}) = 0$ is also treated in [16, Corollary 1.2]. In the intermediate case, we can proceed similarly. First, Theorems 2.52 and 2.53 show that when $\mathbf{p}$ is finite these spaces can be realised as contained (not embedded) in $L_{\text{loc}}^1$. We can then impose the limit in $L_{\text{loc}}^1$, which fixes the floating constant in the determination of $u$ in terms of $g$. It can be shown that this limit holds a priori in $L_{\text{loc}}^1$. We omit further details.

We use the first representation theorem (Theorem 6.11) to characterise the well-posedness of $(R\mathbf{X})_A^\mathbf{P}$ and $(N\mathbf{X})_A^\mathbf{P}$ for $\mathbf{p}$ in the classification region $J(\mathbf{X}, DB)$ (see Definition 6.10). Let $N_\perp$ and $N_\parallel$ denote the projections from $\mathbf{X}_D^\mathbf{P}(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ onto $\mathbf{X}_\perp^\mathbf{P}$ and $\mathbf{X}_\parallel^\mathbf{P}$ respectively. For $\mathbf{p} \in J(\mathbf{X}, DB)$ we have defined $\mathbf{X}_{DB}^{\mathbf{P},+}(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ as a subspace of $\mathbf{X}_D^\mathbf{P}(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$, and through this identification we define

$$ N_{\mathbf{X},DB,\parallel}^{\mathbf{P}} : \mathbf{X}_{DB}^{\mathbf{P},+} \to \mathbf{X}_{\parallel}^{\mathbf{P}} \quad \text{and} \quad N_{\mathbf{X},DB,\perp}^{\mathbf{P}} : \mathbf{X}_{DB}^{\mathbf{P},+} \to \mathbf{X}_{\perp}^{\mathbf{P}} $$

as the restrictions of $N_\perp$ and $N_\parallel$ respectively.

**Theorem 7.5 (Characterisation of well-posedness).** Let $B = \hat{A}$, and suppose that $\mathbf{p} \in J(\mathbf{X}, DB)$. Then $(R\mathbf{X})_A^\mathbf{P}$ (resp. $(N\mathbf{X})_A^\mathbf{P}$) is well-posed if and only if $N_{\mathbf{X},DB,\parallel}^\mathbf{P}$ (resp. $N_{\mathbf{X},DB,\perp}^\mathbf{P}$) is an isomorphism.

**PROOF.** The results for $\theta(\mathbf{p}) = 0$ and $\theta(\mathbf{p}) = -1$ correspond to [16, Theorems 1.5 and 1.6], so we need only consider $\theta(\mathbf{p}) \in (-1, 0)$. Theorem 6.11 tells us that $\nabla_A u = C_{DB}^+ F_0$ for a unique $F_0 \in \mathbf{X}_{DB}^{\mathbf{P},+}$, with $F_0 = \lim_{t \to 0} \nabla_A u(t, \cdot)$ in $X^\mathbf{P}$ (here is where we use that $\mathbf{p} \in J(\mathbf{X}, DB)$). Since $\nabla_A = [\partial_{\nu_A}, \nabla_{\parallel}]$, the boundary conditions for $(R\mathbf{X})_A^\mathbf{P}$ and $(N\mathbf{X})_A^\mathbf{P}$ can be rewritten as

$$ N_{\parallel}(F_0) = f \in \mathbf{X}_{\parallel}^{\mathbf{P}} \quad \text{and} $$

$$ N_{\perp}(F_0) = f \in \mathbf{X}_{\perp}^{\mathbf{P}} $$

respectively. The result follows. $\square$

<figure><img src="image_41.png" /></figure>

Define the energy exponent $e = (2, -1/2)$. For all $A$, the Lax–Milgram theorem guarantees well-posedness of the problems $(R\mathbf{X})_A^e$ and $(N\mathbf{X})_A^e$ (see [13, Theorems 3.2 and 3.3]).

**Definition 7.6.** Suppose $\mathbf{p}$ and $\mathbf{q}$ are exponents with $\theta(\mathbf{p}), \theta(\mathbf{q}) \in [-1, 0]$. We say that the boundary value problems $(P\mathbf{X})_A^\mathbf{P}, (P\mathbf{X})_A^\mathbf{q}$ are mutually well-posed if they
are both well-posed, and if for all boundary data $f \in \mathbf{X}_{\bullet}^{\mathbf{P}} \cap \mathbf{X}_{\bullet}^{\mathbf{q}}$ (where $\bullet$ is either $\parallel$ or $\perp$ depending on the choice of boundary condition) the solutions to $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ with boundary data $f$ are equal. We say that $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ is compatibly well-posed if $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ (where $\mathbf{e}$ is the energy exponent defined above) are mutually well-posed.

If $B = \hat{A}$ and $\mathbf{p}, \mathbf{q} \in J(\mathbf{X}, \mathcal{D}\mathcal{B})$, then Theorem 7.5 says that $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ are mutually well-posed if and only if $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}}$ and $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{q}}$ are isomorphisms and $(N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}})^{-1} = (N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{q}})^{-1}$ on $\mathbf{X}_{\bullet}^{\mathbf{P}} \cap \mathbf{X}_{\bullet}^{\mathbf{q}}$. Evidently mutual well-posedness is transitive: if $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ are mutually well-posed, and if $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ are mutually well-posed, then $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ are mutually well-posed.

For finite exponents in the classification region we can interpolate mutual well-posedness; 'mutuality' is required in order to interpolate invertibility.

**Theorem 7.7 (Interpolation of mutual well-posedness).** Let $B = \hat{A}$, and suppose $\mathbf{p}, \mathbf{q} \in J(\mathbf{X}, \mathcal{D}\mathcal{B}) \cap \mathcal{E}_{\text{fin}}$. If $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ are mutually well-posed, then the problems $((P_{\mathbf{X}})_{A}^{[\mathbf{p},\mathbf{q}]_{\eta}})_{\eta \in [0,1]}$ are pairwise mutually well-posed. Furthermore, if $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$ and $\mathbf{X} = \mathbf{H}$, then the problems $((P_{\mathbf{B}})_{A}^{[\mathbf{p},\mathbf{q}]_{\eta}})_{\eta \in (0,1)}$, $(P_{\mathbf{H}})_{A}^{\mathbf{P}}$ and $(P_{\mathbf{H}})_{A}^{\mathbf{q}}$ are pairwise mutually well-posed.

**PROOF.** We use the interpolation result for smoothness spaces, Theorem 2.56. By the previous discussion, we have

$$(N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}})^{-1} = (N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{q}})^{-1}$$

on the intersection $\mathbf{X}_{\bullet}^{\mathbf{P}} \cap \mathbf{X}_{\bullet}^{\mathbf{q}}$. Since this intersection is dense in $\mathbf{X}_{\bullet}^{\mathbf{P}}$ and $\mathbf{X}_{\bullet}^{\mathbf{q}}$ (here is where we use finiteness of $\mathbf{p}$ and $\mathbf{q}$), we have a well-defined operator

$$\tilde{N}: \mathbf{X}_{\bullet}^{\mathbf{P}} + \mathbf{X}_{\bullet}^{\mathbf{q}} \to \mathbf{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{P},+} + \mathbf{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{q},+}$$

which restricts to $(N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}})^{-1}$ and $(N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{q}})^{-1}$ on $\mathbf{X}_{\bullet}^{\mathbf{P}}$ and $\mathbf{X}_{\bullet}^{\mathbf{q}}$ respectively. By complex interpolation, for each $\eta \in [0,1]$, $\tilde{N}$ restricts to a bounded operator $\tilde{N}_{\eta}: \mathbf{X}_{\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}} \to \mathbf{X}_{\mathcal{D}\mathcal{B}}^{[\mathbf{p},\mathbf{q}]_{\eta,+}}$. Since $\tilde{N}_{\eta}$ is equal to $(N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}})^{-1}$ on $\mathbf{X}_{\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}} \cap \mathbf{X}_{\bullet}^{\mathbf{P}}$, and since $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}}$ is equal to $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}}$ on $\mathbf{X}_{\mathcal{D}\mathcal{B}}^{[\mathbf{p},\mathbf{q}]_{\eta,+}} \cap \mathbf{X}_{\mathcal{D}\mathcal{B}}^{\mathbf{P}}$, we find that $\tilde{N}_{\eta}$ is the inverse of $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}}$. Therefore $N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}}$ is an isomorphism, and since $[\mathbf{p},\mathbf{q}]_{\eta}$ satisfies the assumptions of Theorem 7.5 (by Theorem 4.32), $(P_{\mathbf{X}})_{A}^{[\mathbf{p},\mathbf{q}]_{\eta}}$ is well-posed. Furthermore, since $\tilde{N}_{\eta} = (N_{\mathbf{X},\mathcal{D}\mathcal{B},\bullet}^{\mathbf{P}})^{-1}$ on $\mathbf{X}_{\bullet}^{[\mathbf{p},\mathbf{q}]_{\eta}} \cap \mathbf{X}_{\bullet}^{\mathbf{P}}$, the problems $(P_{\mathbf{X}})_{A}^{[\mathbf{p},\mathbf{q}]_{\eta}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ are mutually well-posed. Transitivity of mutual well-posedness completes the proof of the first statement. When $\mathbf{X} = \mathbf{H}$ and $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$, applying real interpolation with the same argument yields the second statement. $\square$

Well-posedness can be extrapolated, preserving mutuality, within the classification region $J(\mathbf{X}, \mathcal{D}\mathcal{B})$, however we must restrict to exponents $\mathbf{p}$ with $\theta(\mathbf{p}) \in (-1,0)$. This is a consequence of Šneïberg's extrapolation theorem [70]; the proof uses the same argument as that of Theorem 5.18.

**Theorem 7.8 (Extrapolation of well-posedness).** Let $B = \hat{A}$, and suppose $\mathbf{p} \in J(\mathbf{X}, \mathcal{D}\mathcal{B})$ with $\theta(\mathbf{p}) \in (-1,0)$. If $\mathbf{X} = \mathbf{H}$ then further assume that $j(\mathbf{p}) \neq 0$, and if $\mathbf{X} = \mathbf{B}$ then assume $i(\mathbf{p}) > 1$. Suppose that $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ is well-posed. Then there exists a $(j, \theta)$-neighbourhood $O_{\mathbf{p}}$ of $\mathbf{p}$ such that for all $\mathbf{q} \in O_{\mathbf{p}}$ with $\theta(\mathbf{q}) \le 0$, $(P_{\mathbf{X}})_{A}^{\mathbf{q}}$ and $(P_{\mathbf{X}})_{A}^{\mathbf{P}}$ are mutually well-posed.
**Remark 7.9.** For $\mathbf{X} = \mathbf{H}$, the restriction $j(\mathbf{p}) \neq 0$ rules out BMO-Sobolev spaces, which are not in the interior of any of our complex interpolation scales. Also with $\mathbf{X} = \mathbf{H}$, a corresponding result is true for $\theta(\mathbf{p}) \in \{-1, 0\}$, but well-posedness is only obtained for $\mathbf{q}$ near $\mathbf{p}$ with $\theta(\mathbf{q}) = \theta(\mathbf{p})$. The proof uses the same argument. For $\mathbf{X} = \mathbf{B}$, the restriction $i(\mathbf{p}) > 1$ is due to the lack of a complex interpolation theorem for quasi-Banach $\mathcal{Z}$-spaces.

This leads to an interesting corollary: a topological characterisation of mutual well-posedness.

**Corollary 7.10.** Let $B = \hat{A}$, and let $\mathbf{p}, \mathbf{q} \in J(\mathbf{X}, DB) \cap \mathbf{E}_{\text{fin}}$ with $\theta(\mathbf{p}), \theta(\mathbf{q}) \in (-1, 0)$. Then $(P_{\mathbf{X}})^{\mathbf{p}}_{A}$ and $(P_{\mathbf{X}})^{\mathbf{q}}_{A}$ are mutually well-posed if and only if $\mathbf{p}$ and $\mathbf{q}$ are in the same connected component of $WP(P_{\mathbf{X}})_{A} \cap J(\mathbf{X}, DB)$ (the set of all exponents in the classification region for which $(P_{\mathbf{X}})_{A}$ is well-posed).

**PROOF.** If $(P_{\mathbf{X}})^{\mathbf{p}}_{A}$ and $(P_{\mathbf{X}})^{\mathbf{q}}_{A}$ are mutually well-posed, then Theorem 7.7 (interpolation of mutual well-posedness) implies that $\mathbf{p}$ and $\mathbf{q}$ lie in the same connected component of $WP(P_{\mathbf{X}})_{A} \cap J(\mathbf{X}, DB)$. On the other hand, Theorem 7.8 (extrapolation of well-posedness) shows that the sets

$$ \{\mathbf{r} \in J(\mathbf{H}, DB) : (P_{\mathbf{H}})^{\mathbf{r}}_{A} \text{ and } (P_{\mathbf{H}})^{\mathbf{p}}_{A} \text{ mutually well-posed, } j(\mathbf{r}) \neq 0, \theta(\mathbf{r}) \in (-1, 0)\}, $$
$$ \{\mathbf{r} \in J(\mathbf{B}, DB) : (P_{\mathbf{B}})^{\mathbf{r}}_{A} \text{ and } (P_{\mathbf{B}})^{\mathbf{p}}_{A} \text{ mutually well-posed, } \theta(\mathbf{r}) \in (-1, 0)\}, $$

and their complements in $WP(P_{\mathbf{X}})_{A} \cap J(\mathbf{X}, DB)$, are both open. Therefore if $(P_{\mathbf{X}})^{\mathbf{p}}_{A}$ and $(P_{\mathbf{X}})^{\mathbf{q}}_{A}$ are not mutually well-posed, then $\mathbf{p}$ and $\mathbf{q}$ lie in different connected components of $WP(P_{\mathbf{X}})_{A} \cap J(\mathbf{X}, DB)$. $\square$

Now we present a $\heartsuit$-duality principle for well-posedness.

**Theorem 7.11 ($\heartsuit$-duality of well-posedness).** Let $B = \hat{A}$, and suppose that $\mathbf{p}, \mathbf{q} \in J(\mathbf{X}, DB) \cap \mathbf{E}_{\text{fin}}$. If $(P_{\mathbf{X}})^{\mathbf{p}}_{A}$ and $(P_{\mathbf{X}})^{\mathbf{q}}_{A}$ are mutually well-posed, then $(P_{\mathbf{X}})^{\heartsuit}_{A*}$ and $(P_{\mathbf{X}})^{\mathbf{q}}_{A*}$ are mutually well-posed.

**Remark 7.12.** If $i(\mathbf{p}) \in (1, \infty)$, then this statement is an equivalence. Furthermore, if $(P_{\mathbf{X}})^{\mathbf{p}}_{A}$ is compatibly well-posed, then $(P_{\mathbf{X}})^{\heartsuit}_{A*}$ is also compatibly well-posed, since $e^{\heartsuit} = e$.

We point out the case where $\mathbf{p} = (1, s)$ with $s \in (-1, 0]$: in this case the result says that well-posedness of a problem with coefficients $A$ and boundary data in the Hardy–Sobolev space $\dot{H}_{s}^{1}$ (resp. the Besov space $\dot{B}_{s}^{1,1}$) implies well-posedness of the corresponding problem for $A^*$ with boundary data in the image of BMO–Sobolev space $BMO_{-s}$ (resp. the Hölder space $\hat{A}_{-s}$) under $D$. We elaborate on this in Remark 7.13 below.

**PROOF OF THEOREM 7.11.** We will be sketchy because all the important details of this argument have already been done by the second author, Mourgoglou, and Stahlhut (see [19, §12.2] and [16, §13]). Recall from Remark 5.21 that $\hat{A}^* = NB^*N =: \hat{B}$. When $\mathbf{p}$ is finite, the pairing

$$ \langle f, g \rangle_{X_{DB}^{\mathbf{p}}}^{N} := \langle f, Ng \rangle_{X_{DB}^{\mathbf{p}}} $$
is a duality pairing between $X_{DB}^P$ and $X_{\tilde{B}D}^{P'}$. By Proposition 5.6 we have that
$$ \|Dg\|_{X_{\tilde{B}D}^{\mathcal{P}'}} \simeq \|g\|_{X_{\tilde{B}D}^{\mathcal{P}'}} \text{ whenever } g \in \mathcal{D}(D) \cap X_{\tilde{B}D}^{\mathcal{P}'}, \text{ and so the pairing} $$

$$ (7.1) \qquad \langle f, g \rangle_{X_{DB}^{\mathcal{P}'}} := \langle f, N D^{-1} g \rangle_{X_{DB}^{\mathcal{P}'}} $$

is a duality pairing between $X_{DB}^P$ and $\mathcal{R}(D) \cap X_{\tilde{D}\tilde{B}}^{\mathcal{P}'}$. Since $\mathbf{p} \in J(\mathbf{X}, DB)$, we have identified $\mathbf{X}_D^P = \mathbf{X}_{DB}^P$ and $\mathbf{X}_D^{\mathcal{P}'} = \mathbf{X}_{\tilde{D}\tilde{B}}^{\mathcal{P}'}$ as completions of $X_{DB}^P$ and $X_{\tilde{D}\tilde{B}}^{\mathcal{P}'}$ respectively (using a simple modification of Proposition 5.13 to make the second identification), and by density the pairing (7.1) extends to a duality pairing between $\mathbf{X}_{DB}^P$ and $\mathbf{X}_{\tilde{D}\tilde{B}}^{\mathcal{P}'}$. As in the proof of [16, Lemma 13.3], this pairing realises $\mathbf{X}_{\tilde{D}\tilde{B}}^{\mathcal{P}'} \not\equiv$ as the dual of $\mathbf{X}_{\tilde{D}\tilde{B}}^{\mathcal{P}, \pm}$, $\mathbf{X}_{\perp}^{\mathcal{P}'}$ as the dual of $\mathbf{X}_{\parallel}^{\mathcal{P}}$, and $\mathbf{X}_{\parallel}^{\mathcal{P}'}$ as the dual of $\mathbf{X}_{\perp}^{\mathcal{P}}$. The remainder of the argument precisely follows the proof of [16, Theorem 1.6]. $\square$

**Remark 7.13.** The restriction to finite exponents in Theorem 7.11 appears because the duality $\mathbf{X}^{P'} \simeq (\mathbf{X}^P)'$ only holds when $\mathbf{p}$ is finite. When $\mathbf{p} = (\infty, s; 0)$, for example, we do not have $(\mathbf{B}\dot{M}\mathbf{O}_s)' \simeq \dot{H}_{-s}^1$. However, we do have

$$ \mathrm{VMO}' \simeq \dot{H}^1, $$

where VMO, the space of functions of vanishing mean oscillation, is the norm closure of $C_c^\infty$ in BMO.<fnref n="1" /> In this long remark, we will sketch how ‘fractional VMO’ spaces could be included in our results; for lack of concrete applications, we will keep this at the level of a sketch. Given our omission of proofs, the reader should consider this as an informed conjecture.

For $s \in \mathbb{R}$, define $\mathbf{VMO}_s$ to be the norm closure of $C_c^\infty$ in $\mathbf{B}\dot{M}\mathbf{O}_s$. Then we have the duality

$$ \mathbf{VMO}'_s \simeq \dot{H}_{-s}^1. $$

For an operator $A$ satisfying the Standard Assumptions, we may define $A$-adapted fractional pre-VMO spaces $\mathbf{VMO}_{A,s}$ analogously to the spaces $\mathbb{H}_A^{(\infty, s; 0)}$ by using vanishing tent spaces $vT_{s;0}^\infty$. These may be defined as the norm closure of $T^2 \cap T_{s;0}^\infty$ in $T_{s;0}^\infty$ (the weak-star closure returns $T_{s;0}^\infty$), and their properties have been established for example by Jiang and Yang [53] when $s = 0$. In particular the duality

$$ (vT_{s;0}^\infty)' \simeq T_{-s;0}^1 $$

holds for all $s \in \mathbb{R}$.

For auxiliary functions $\psi \in \Psi_\infty^\infty$, define canonical completions $\psi\mathbf{VMO}_{DB,s}$ as the closures of $\mathcal{Q}_{\psi, DB}\mathbf{VMO}_{DB,s}$ in $vT_{s;0}^\infty$ (or equivalently, as the norm closures of $\mathcal{Q}_{\psi, DB}\mathbf{VMO}_{DB,s}$ in $T_{s;0}^\infty$, or even as the norm closures of $\mathcal{Q}_{\psi, DB}\mathbb{H}_{DB}^{(\infty, s; 0)}$ in $T_{s;0}^\infty$). Then the duality

$$ (\psi\mathbf{VMO}_{DB,s})' \simeq \tilde{\psi}\mathbf{H}_{B^*D}^{(1,-s;0)} $$

follows from the arguments of Section 4.3.

One can follow the arguments of Theorem 6.9 and characterise solutions to $(\mathrm{CR})_{DB}$ in $vT_{s;0}^\infty$ (with decay at infinity) as generalised Cauchy extensions of functions in $\mathbf{VMO}_{s, DB}^+$ when $(\infty, s; 0)^{\mathcal{P}} = (1, -s; 0) \in I(\mathbf{H}, DB^*)$. As in this section, one can then characterise well-posedness of Regularity and Neumann problems with

<footnote n="1">¹This is the VMO space of Coifman and Weiss [28, §4] rather than that first introduced by Sarason [69].</footnote>
$\dot{\text{VMO}}_s$ boundary data in terms of a corresponding map $N_{\text{VMO},DB,\bullet}$ being an isomorphism.

The $\heartsuit$-duality argument of Theorem 7.11 then says that for coefficients $A$, when $(1, -s; 0) \in I(\mathbf{H}, D\hat{A}^*)$, well-posedness of a Regularity or Neumann problem for $L_A$ with data in $\dot{\text{VMO}}_s$ implies well-posedness of the adjoint problem (i.e. that for $L_{A^*}$) with data in $\dot{H}_{-s}^1$. Furthermore, Theorem 7.11 applied to $L_{A^*}$ for the exponent $(1, -s; 0)$ then implies that the original problem is, in fact, well-posed with data in $\dot{\text{BMO}}_s$. This is a priori stronger, so we deduce that well-posedness (of a problem) with data in $\dot{\text{BMO}}_s$ is equivalent to well-posedness with data in $\dot{\text{VMO}}_s$. Thus we can generalise observations made by the second author and Mourgoglou [15] (where $s = 0$), which followed the observation of Dindos, Kenig, and Pipher [31] that BMO solvability for real coefficient equations is equivalent to VMO solvability.

## 7.2. Regions of well-posedness for certain classes of coefficients

The results above show that from well-posedness of a boundary value problem for an exponent $\mathbf{p} \in J(\mathbf{X}, DB)$ with $B = \hat{A}$, we may deduce well-posedness for a larger range of exponents by $\heartsuit$-duality, interpolation, and extrapolation. As an application of this principle we consider various classes of coefficients for which we already have some information on well-posedness.

**7.2.1. Real coefficients, $m = 1$.** First we consider the regularity problems $(R_X)_A^p$ in the real scalar case: this is a good way of exhibiting the main features of our theory, and for the case of Hardy–Sobolev boundary data these results seem to be new.

Suppose that $m = 1$ (so that $L_A u = 0$ is a single equation rather than a system) and that the entries of $A$ are real. In this setting, there exists a number $\alpha \in (0, 1]$ such that for every Euclidean ball $B = B(X_0, 2r)$ in $\mathbb{R}_+^{1+n}$ and every solution $u$ to $L_A u = 0$ in $B$, we have

$$ (7.2) \qquad |u(X) - u(X')| \lesssim \left( \frac{|X - X'|}{r} \right)^{\alpha} \left( \iint_B |u|^2 \right)^{1/2} $$

for all $X, X'$ in the smaller ball $B(X_0, r)$. In this case say that the coefficients $A$ satisfy the De Giorgi–Nash–Moser condition of exponent $\alpha$. The adjoint matrix $A^*$ will also satisfy a De Giorgi–Nash–Moser condition of (possibly different) exponent $\alpha^*$.

The second author and Stahlhut [19, Corollary 13.3] show that in this case<fnref n="2" /> we have

$$ \left( \frac{n}{n + \alpha}, p_+(DB) \right) \subset I_0(\mathbf{H}, DB), \quad \left( \frac{n}{n + \alpha^*}, p_+(D\tilde{B}) \right) \subset I_0(\mathbf{H}, D\tilde{B}), $$

where $\tilde{B} = \hat{A}^*$ (note that $\tilde{B} \neq B^*$) and where $p_+(DB), p_+(D\tilde{B}) > 2$. Therefore by $\heartsuit$-duality (see Proposition 5.13 and Remark 5.21), we have

$$ (p^+(D\tilde{B})', \infty) \subset I_{-1}(\mathbf{H}, DB), \quad (p^+(DB)', \infty) \subset I_{-1}(\mathbf{H}, D\tilde{B}). $$

By interpolation (Theorem 4.32) we then have that $I(\mathbf{H}, DB)$ contains the region pictured in Figure 8, and $I(\mathbf{B}, DB)$ contains the interior of that region. The point

<footnote n="2">²In fact, a weaker assumption is needed there, and the parameters $\alpha, \alpha^*$ are not necessarily those from (7.2). Still, such parameters exist.</footnote>
<figure><img src="image_42.png" /><figcaption>FIGURE 8. Exponents $\mathbf{p} \in I(H, DB)$, when $m = 1$ and $A$ is real, with $B = \hat{A}$.</figcaption></figure>

$\mathbf{x}_A$ here is defined as the pictured intersection, which is a function of $n, \alpha$, and $p^+(D\tilde{B})$ that we need not compute explicitly.

There is also a corresponding diagram for $\tilde{B}$ that we have not pictured, including a corresponding exponent $\mathbf{x}_{A^*}$. By applying $\heartsuit$-duality to the exponents $\mathbf{p} \in I(H, D\tilde{B})$ with $i(\mathbf{p}) \in (1, 2)$, and another application of interpolation, we can increase these ranges to that pictured in Figure 9.

It has been shown that there exist $p_R(A) > 1$ (possibly small) and $0 < \alpha^\sharp \le \min(\alpha, \alpha^*)$ such that the Regularity problem $(R_H)_{A}^{(p,0)}$ is compatibly well-posed for all $p \in (n/(n + \alpha^\sharp), p_R(A)]$, and likewise for $A^*$ (we can choose $\alpha^\sharp \le \min(\alpha, \alpha^*)$ which works for both $A$ and $A^*$).$^3$ By the results of the previous paragraph, we have $(p, 0) \in I_0(H, DB) \cap I_0(H, D\tilde{B})$ for all such $p$,<fnref n="4" /> and so we may apply $\heartsuit$-duality and interpolation as in the previous argument to deduce compatible well-posedness of $(R_H)_A^\mathbf{p}$ for $\mathbf{p}$ in the region pictured in Figure 10, and of $(R_B)_A^\mathbf{p}$ in the interior of this region.<fnref n="5" />

We can expand this region slightly for Besov spaces: applying $\heartsuit$-duality to compatible well-posedness of $(R_B)_A^{\mathbf{p}*}$ for $\mathbf{p}$ in the open triangle with vertices $\mathbf{y}_{A^*}$ (defined pictorially in Figure 10), $(n+\alpha^\sharp/n, 0)$, and $(1, 0)$, we find that $(R_B)_{A}^{(\infty, \alpha; 0)}$ is compatibly well-posed for all $\alpha \in (-1, -1-\alpha^\sharp)$. Therefore (after another iteration

<footnote n="3">³The $p_R(A)$ endpoint of this result is due to Kenig and Rule in dimension $n+1=2$ [61, Theorem 1.4] and Hofmann, Kenig, Mayboroda, and Pipher in dimension $n+1 \ge 3$ [41, Corollary 1.2]. The other endpoint is an extrapolation result of the second author and Mourgoglou [15, §10.1].</footnote>

<footnote n="4">⁴It is possible that $p > p_+(DB)$ or $p > p_+(D\tilde{B})$, in which case we have to restrict to small $p$.</footnote>

<footnote n="5">⁵We can also deduce results for BMO-Sobolev spaces, which correspond to the unpictured $j(\mathbf{p}) = 0$ range.</footnote>
FIGURE 9. More exponents $\mathbf{p} \in I(H, DB)$, when $m = 1$ and $A$ is real, with $B = \hat{A}$. The dark shaded region corresponds to Figure 8.

<figure><img src="image_43.png" /></figure>

FIGURE 10. Exponents $\mathbf{p}$ for which $(R_H)^{1/n}$ is compatibly well-posed (the dark shaded region). The light shaded region is from Figure 9.

<figure><img src="image_44.png" /></figure>
<figure><img src="image_45.png" /><figcaption>FIGURE 11. Exponents $\mathbf{p}$ for which $(R_B)_A^\mathbf{p}$ is compatibly well-posed (the dark shaded region); this includes no exponents with $\theta(\mathbf{p}) = 0$ or $\theta(\mathbf{p}) = -1$. The light shaded region is from Figure 9.</figcaption></figure>

of interpolation) we have well-posedness of $(R_B)_A^\mathbf{p}$ for all $\mathbf{p}$ in the shaded region of Figure 11. This is the same region obtained by Barton and Mayboroda for compatible well-posedness of $(R_B)_A^\mathbf{p}$ in this setting [22, Figure 3.5].<fnref n="6" /> To recover the result of [22, Corollary 3.24], one need only apply Lemma 6.4 (which is valid for the region of $\mathbf{p}$ pictured in Figure 11, see Figure 5) to remove the decay assumption at infinity from $(R_B)_A^\mathbf{p}$, and the trace theorem [22, Theorem 6.3] to replace our boundary condition with a trace condition.

In the case of dimension $n + 1 = 2$, Neumann problems can be transposed to Regularity problems using the conjugate equation. This idea goes back to Morrey [67]. Thus the same results hold for Neumann problems in this case.

In the case that $A$ is symmetric and $n \ge 1$, in addition to the above assumptions, results of Kenig and Pipher [60] imply that we have the additional information $p_R(A) > 2$, and regions of compatible well-posedness of $(R_X)_A^\mathbf{p}$ can be expanded accordingly. Note that the results there are proven in the context of the unit ball instead of the upper-half space, but similar arguments can be used. An argument showing well-posedness at $p = 2$ in our context has been given by the second author, Axelsson and McIntosh [8, Theorem 2.2]. From this, Šneïberg's extrapolation gives $p_R(A) > 2$. The corresponding Neumann problems $(N_X)_A^\mathbf{p}$ are well-posed for an analogous range of $\mathbf{p}$ by the same arguments.

<footnote n="6">⁶The only difference is in the light shaded ‘region of applicability’: ours depends on exponent $p^+(DB)$ from [19], while that in Barton–Mayboroda is in terms of the exponent appearing in Meyers’ theorem [66, Theorem 2] (see also [22, Lemma 2.12]). It is not clear whether there is any relationship between these exponents.</footnote>
**7.2.2. Constant coefficients.** Suppose that A is constant. By results of the second author and Stahlhut [19, §3.4 and Theorem 5.1], we have $I_0(H, DB) = (n/(n+1), \infty)$, and therefore by $\heartsuit$-duality and interpolation we have

$$J(H, DB) = I_{\max},$$

$$J(B, DB) = I_{\max} \cap \{\mathbf{p} \in E : \theta(\mathbf{p}) \in (-1, 0)\}.$$

Furthermore, it was observed by the second author and Mourgoglou [16, §14.1] that the Regularity and Neumann problems $(P_H)^{\mathbf{p}}_A$ are compatibly well-posed for all $\mathbf{p} \in I_0(H, DB)$. By $\heartsuit$-duality and interpolation (arguing as we did for real symmetric scalar equations), we thus have compatible well-posedness of $(P_H)^{\mathbf{p}}_A$ for all $\mathbf{p} \in I_{\max}$, and of $(P_B)^{\mathbf{p}}_A$ for all $\mathbf{p} \in I_{\max}$ with $\theta(\mathbf{p}) \in (-1, 0)$. We note that these results may be more simply obtained by Fourier multiplier theory; our framework is overkill for constant coefficients.

**7.2.3. Self-adjoint coefficients.** Suppose that A is self-adjoint, $A = A^*$, but not necessarily real and with $m = 1$. Then [8, Theorem 2.2] shows that both Regularity and Neumann problems $(P_H)^{(2,0)}_A$ are compatibly well-posed, and by $\heartsuit$-duality $(P_H)^{(2,-1)}_A$ is also compatibly well-posed. Interpolation then gives compatible well-posedness of $(P_H)^{(2,s)}_A$ and $(P_B)^{(2,s)}_A$ for all $s \in (-1, 0)$, and extrapolation (via Theorem 7.8) gives compatible well-posedness for exponents $\mathbf{p}$ sufficiently close to this line.

**7.2.4. Diagonal block coefficients.** Suppose that A has the form

$$A = \begin{bmatrix} A_{\perp\perp} & 0 \\ 0 & A_{\parallel\parallel} \end{bmatrix}.$$

By results of the second author and Mourgoglou [16, §14.2], the problems $(R_H)^{\mathbf{p}}_A$ and $(N_H)^{\mathbf{p}}_A$ are compatibly well-posed for all $\mathbf{p} \in I_0(H, DB)$. By $\heartsuit$-duality, we also have compatible well-posedness of these problems for all $\mathbf{p} \in I_0(H, DB^*)^\heartsuit$. Interpolating between the intervals $I_0(H, DB)$ and $I_0(H, DB^*)^\heartsuit \cap E_{\text{fin}}$ then yields a set of exponents $\mathbf{p}$ such that the problems $(P_H)^{\mathbf{p}}_A$ and $(P_B)^{\mathbf{p}}_A$ are all compatibly well-posed.

**7.2.5. Block triangular coefficients.** Suppose that A has the lower triangular block form

$$A = \begin{bmatrix} A_{\perp\perp} & 0 \\ A_{\parallel\perp} & A_{\parallel\parallel} \end{bmatrix}.$$

Then results of the second author with McIntosh and Mourgoglou [13, Theorem 6.2] imply that the Neumann problem $(N_H)^{(2,0)}_A$ is compatibly$^7$ well-posed. On the other hand, if A has the upper triangular form

$$A = \begin{bmatrix} A_{\perp\perp} & A_{\perp\parallel} \\ 0 & A_{\parallel\parallel} \end{bmatrix}$$

then by [13, Theorem 6.4] the Regularity problem $(R_H)^{(2,0)}_A$ is compatibly well-posed. Since taking adjoints interchanges lower triangular and upper triangular block form matrices, we find by $\heartsuit$-duality that $(R_H)^{(2,-1)}_A$ is compatibly well-posed when A is lower triangular, and $(N_H)^{(2,-1)}_A$ is compatibly well-posed when A is

<footnote>$^7$The compatibility is not explicitly detailed there, but the argument is not too difficult. See [9, Theorem 2.32]. Likewise for the Regularity problem below.</footnote>
upper triangular. As in the previous examples we can use interpolation and extrapolation to deduce compatible well-posedness of these problems for larger open regions of exponents.

**7.2.6. An example with incompatible well-posedness.** Let $m = n = 1$. Since we are working in the plane, results of the second author and Stahlhut [19, Proposition 3.11 and Theorem 5.1], along with the usual $\heartsuit$-duality and interpolation arguments, yield $J(\mathbf{H}, DB) = I_{\max}$ and $J(\mathbf{B}, DB) = I_{\max} \cap \{\theta(\mathbf{p}) \in (-1, 0)\}$. For all $k \in \mathbb{R}$ define the coefficient matrix

$$A_k(x) := \begin{bmatrix} 1 & k \operatorname{sgn}(x) \\ -k \operatorname{sgn}(x) & 1 \end{bmatrix}.$$

The following well-posedness result was proven by Axelsson [20, Theorem 1.2].

**Theorem 7.14 (Axelsson).** Suppose $p \in (1/2, \infty)$. Then

* the Regularity problem $(R_H)_{A_k}^{(p,0)}$ is well-posed if $k \neq -\tan(\pi/2p)$, and

* the Neumann problem $(N_H)_{A_k}^{(p,0)}$ is well-posed if $k \neq \tan(\pi/2p)$.

However, results of Kenig, Koch, Pipher, and Toro [59] and of Kenig and Rule [61] (see [20, Theorem 1.1]) show that $(R_H)_{A_k}^{(p,0)}$ (resp. $(N_H)_{A_k}^{(p,0)}$) is not *completely* well-posed when $k < -\tan(\pi/2p)$ (resp. $k > \tan(\pi/2p)$). We will use our theory to deduce some results for more general exponents. Let us consider only Regularity problems, as Neumann problems can be handled by the same arguments.

Rewriting the conditions above in terms of $p$, we find that $(R_H)_{A_k}^{(p,0)}$ is compatibly well-posed for all $p \in (1/2, \infty)$ when $k > 0$, and for all $p \in (1/2, \frac{\pi}{2\arctan(-k)})$ when $k < 0$. Furthermore, when $k < 0$, $(R_H)_{A_k}^{(p,0)}$ is (incompatibly) well-posed for all $p > \frac{\pi}{2\arctan(-k)}$. Note that $\frac{\pi}{2\arctan(-k)} > 1$.

Let us fix $k < 0$ for convenience. Since $A_k^* = A_{-k}$, $\heartsuit$-duality tells us that $(R_H)_{A_k}^{(p,-1)}$ is compatibly well-posed for all $p \in (1, \infty)$. Therefore, by two iterations of $\heartsuit$-duality and interpolation, as we carried out in Subsection 7.2.1, we deduce that $(R_H)_{A_k}^{\mathbf{p}}$ and $(R_H)_{A_{-k}}^{\mathbf{p}}$ are compatibly well-posed for the exponents $\mathbf{p}$ pictured in Figures 12 and 13. Real interpolation then yields compatible well-posedness for the problems $(R_B)_{A_k}^{\mathbf{p}}$ for $\mathbf{p}$ in the interior of this region. One can argue similarly to find regions of well-posedness for Neumann problems.

**Remark 7.15.** We would like to extrapolate and find more exponents for which $(R_H)_{A_k}^{\mathbf{p}}$ is well-posed (but not compatibly well-posed). However, our extrapolation theorem (Theorem 7.8) only holds for exponents $\mathbf{p}$ with $\theta(\mathbf{p}) \in (-1, 0)$, and we would need to extrapolate from exponents with $\theta(\mathbf{p}) = 0$. We conjecture that there is a region of exponents $\mathbf{p}$, relatively open with respect to $I_{\max}$, such that $(R_H)_{A_k}^{\mathbf{p}}$ is incompatibly well-posed. Such a region is pictured in Figure 14.

### 7.3. Stability of well-posedness under perturbation of coefficients

In this section, we turn to studying stability in the coefficients of well-posedness, that is, how well-posedness results are affected by perturbing the coefficients of the system $L_A u = 0$ in $L^\infty$.

**Theorem 7.16 (Stability of well-posedness in coefficients).** Suppose that $A_0$ and $A$ are coefficients, let $B_0 = \tilde{A}_0$ and $B = \hat{A}$, and suppose $\mathbf{p} \in J(\mathbf{X}, DB)$ for
FIGURE 12. Exponents $\mathbf{p}$ for which $(R_H)_{A_k}^{\mathbf{p}}$ is compatibly well-posed (the dark shaded region), with $k < 0$. The thick dashed line consists of exponents $\mathbf{p}$ such that $(R_H)_{A_k}^{\mathbf{p}}$ is well-posed, but not compatibly well-posed. The exponent $\mathbf{x}_{-k}$ is defined geometrically in Figure 13. The region $j(\mathbf{p}) > 1$ is not to scale.

<figure><img src="image_46.png" /></figure>

all $B$ with $\|B - B_0\|_{\infty}$ sufficiently small, with uniform implicit constants in $B$. If $\mathbf{p} \in \text{WP}(P_{\mathbf{X}})_{A_0}$, and if $\|A_0 - A\|_{\infty}$ is sufficiently small, then $\mathbf{p} \in \text{WP}(P_{\mathbf{X}})_A$.

**Remark 7.17.** Note that in this theorem we make a rather strong a priori assumption that $\mathbf{p}$ is in the classification region $J(\mathbf{X}, DB)$ for all $B$ sufficiently close to $B_0$ (which includes $B = B_0$ in particular), and we also assume that the implicit constants in the norm equivalence $X_{DB}^{\mathbf{p}} = X_D^{\mathbf{p}}$ are uniform in $B$. We know that this assumption holds for certain $\mathbf{p}$ in certain cases; we return to this point in Remark 7.19.

**PROOF OF THEOREM 7.16.** By Theorem 7.5 it suffices to show that

$$N_{\mathbf{X},DB,\bullet}^{\mathbf{p}} : \mathbf{X}_{DB}^{\mathbf{p},+} \to \mathbf{X}_{\bullet}^{\mathbf{p}}$$

is an isomorphism, where $\bullet$ denotes either $\perp$ or $\|\|$ depending on the boundary value problem in question. Recall that $N_{\mathbf{X},DB,\bullet}^{\mathbf{p}}$ is the restriction of $N_{\bullet}: \mathbf{X}_D^{\mathbf{p}} \to \mathbf{X}_{\bullet}^{\mathbf{p}}$ to $\mathbf{X}_{DB}^{\mathbf{p},+} \subset \mathbf{X}_D^{\mathbf{p}}$, which is defined provided that $B$ is sufficiently close to $B_0$ thanks to our assumptions. Furthermore, since $DB$ has bounded $H^{\infty}$ functional calculus on $\mathbf{X}_D^{\mathbf{p}}$ for all $B$ sufficiently close to $B_0$, uniformly in $B$ (using $\mathbf{p} \in I(\mathbf{X}, DB)$ and Proposition 4.24, along with the assumptions), we get

$$(7.3) \qquad \|\chi^+(DB) - \chi^+(DB_0)\|_{L(\mathbf{X}_D^{\mathbf{p}})} \lesssim \|B - B_0\|_{\infty}$$
<figure><img src="image_47.png" /><figcaption>FIGURE 13. Exponents p for which $(R_H)_{A-k}^p$ is compatibly well-posed (the dark shaded region), with $k < 0$. The thick dashed line consists of exponents p such that $(R_H)_{A-k}^p$ is well-posed, but not compatibly well-posed. The exponent $x_{-k}$ is defined as the pictured intersection. The region $j(p) > 1$ is not to scale.</figcaption></figure>

by the results of Axelsson, Keith, and McIntosh [21, §6].<fnref n="8" />

Write

$$N_{\mathbf{X},DB,\bullet}^{\mathbf{P}} = N_{\bullet}\chi^{+}(DB_{0})\chi^{+}(DB) + N_{\bullet}(\chi^{+}(DB) - \chi^{+}(DB_{0})).$$

Note that

$$\begin{aligned} \chi^{+}(DB_{0})\chi^{+}(DB) &= (\chi^{+}(DB_{0}) - \chi^{+}(DB) + \chi^{+}(DB))\chi^{+}(DB) \\ &= I - (\chi^{+}(DB) - \chi^{+}(DB_{0})) \end{aligned}$$

on $\mathbf{X}_{DB}^{\mathbf{P},+}$, so by the estimate (7.3), $\chi^{+}(DB_{0})\chi^{+}(DB): \mathbf{X}_{DB}^{\mathbf{P},+} \to \mathbf{X}_{DB_{0}}^{\mathbf{P},+}$ is invertible by a Neumann series if $\|B - B_{0}\|_{\infty}$ is sufficiently small. Since $N_{\mathbf{X},DB_{0},\bullet}^{\mathbf{P}}: \mathbf{X}_{DB_{0}}^{\mathbf{P},+} \to \mathbf{X}_{\bullet}^{\mathbf{P}}$ is an isomorphism (by Theorem 7.5 and the assumptions), the composition

$$M := N_{\bullet}\chi^{+}(DB_{0})\chi^{+}(DB) = N_{\mathbf{X},DB_{0},\bullet}^{\mathbf{P}}\chi^{+}(DB_{0})\chi^{+}(DB): \mathbf{X}_{DB_{0}}^{\mathbf{P},+} \to \mathbf{X}_{\bullet}^{\mathbf{P}}$$

is also an isomorphism. One can then write

$$N_{\mathbf{X},DB,\bullet}^{\mathbf{P}} = M(I + M^{-1}N_{\bullet}(\chi^{+}(DB) - \chi^{+}(DB_{0})))$$

and note that the second factor is also invertible by a Neumann series if $\|B - B_{0}\|_{\infty}$ is sufficiently small. Hence $N_{\mathbf{X},DB,\bullet}^{\mathbf{P}}$ is an isomorphism, and we are done. $\square$

<footnote n="8">⁸More precisely, this gives the corresponding estimate on $\overline{\mathbb{R}}(D)$. Proposition 4.24 then extends the estimate to $\mathbf{X}_{D}^{\mathbf{P}}$.</footnote>
<figure><img src="image_48.png" /><figcaption>FIGURE 14. A *conjectural* extrapolation of Figure 12. Two disjoint mutual well-posedness classes for $(R_H)^p_{A_k}$, with $k < 0$. Again, the region $j(\mathbf{p}) > 1$ is not to scale.</figcaption></figure>

**Remark 7.18.** The proof above shows that the inverse of $N_{X, DB, \bullet}^p$ is given by an expression coming from multiple Neumann series involving the operators $(N_{X,DB_0,\bullet}^p)^{-1}$ and $\chi^+(DB) - \chi^+(DB_0)$. Therefore, if the assumptions of Theorem 7.16 are satisfied for two exponents $\mathbf{p}, \mathbf{q}$, and if $(P_X)_{A_0}^p$ and $(P_X)_{A_0}^q$ are mutually well-posed, it follows that $(P_X)_A^p$ and $(P_X)_A^q$ remain mutually well-posed for $\|A - A_0\|_\infty$ sufficiently small.

**Remark 7.19.** There are three situations in which we can guarantee that the hypotheses of Theorem 7.16 are satisfied.

First, for any coefficients $A$, the hypotheses are satisfied provided that $\mathbf{p} \in I_{\min}$ (see Theorem 5.17; in the Besov space case we also require $\theta(\mathbf{p}) \in (-1, 0)$). The only thing which is not obvious here is uniformity of implicit constants: when $\theta(\mathbf{p}) = 0$ this is contained in the proof of [19, Proposition 7.1], and the general case follows by $\heartsuit$-duality and interpolation (as in the derivation of the region $I_{\min}$).

Second: whenever $n=1$ the hypotheses are satisfied for all coefficients and for all $\mathbf{p} \in I_{\max}$ (with $\theta(\mathbf{p}) \in (-1, 0)$ in the Besov space case): this follows from [19, Proposition 3.11 and Theorem 5.1], $\heartsuit$-duality, and interpolation.

Finally, write $A$ in the transversal/tangential decomposition as in (1.7), and suppose that $A_{\|\|}^*$ satisfies the De Giorgi–Nash–Moser condition of exponent $\alpha$ in dimension $n$ (see (7.2)). It is shown in [19, §13] that the hypotheses of Theorem 7.16 are satisfied for the exponent $(p, 0)$ whenever $p \in (n/(n+\alpha), p_+(DB))$, where $p_+(DB) > 2$. If in addition $A_{\|\|}$ satisfies the De Giorgi–Nash–Moser condition of exponent $\alpha$, then it follows by $\heartsuit$-duality and interpolation that the hypotheses of
Theorem 7.16 hold for all $\mathbf{p}$ in the region pictured in Figure 9 (in the Besov space case, we again require $\theta(\mathbf{p}) \in (-1, 0)$).

## 7.4. The method of layer potentials

We conclude the monograph by explaining how the first-order approach relates to the method of layer potentials. In solving boundary value problems for $L_A$, it has been a standing question whether solutions constructed by different methods agree, and in particular whether solutions are always given by layer potentials when their defining integrals exist. A consequence of Rosén's identification of the layer potentials in terms of Cauchy operators for $DB$ having assumed the De Giorgi-Nash-Moser condition on $A$ and $A^*$, is that solutions constructed via Cauchy operators coincide with solutions constructed by the method of layer potentials, and satisfy a layer potential representation

$$ (7.4) \qquad u(t,x) = S_t(\partial_{\nu_A} u|_{t=0})(x) - D_t(u|_{t=0})(x) $$

in some appropriate sense (modulo constants) [68]. At the time, Rosén's identification applied when $X^P = L_\theta^2$ for $-1 \le \theta \le 0$. As a consequence of our results, we obtain a third representation theorem for exponents in the classification region. We do not yet know what happens outside the classification region.

**Theorem 7.20 (Layer potential representation).** For $\mathbf{p} \in J(X, DB)$, any solution $u$ of $L_A u = 0$ with the interior control $\nabla u \in \tilde{X}^P$ is given by the layer potential representation (7.4) modulo constants.

We shall prove this theorem (and more) later in the section. In our general setting the operators in (7.4) are generalised layer potential operators which we define below. When $A$ and $A^*$ satisfy the De Giorgi-Nash-Moser condition, these are the more familiar integral operators that we review next. We will also explain what we mean by 'the method of layer potentials'. This method has a long history in the case of the Laplace equation, or of equations with smooth coefficients. In the case of elliptic equations $L_A u = 0$ with non-smooth coefficients, solvability by means of layer potentials was first successfully developed in [2]. See the references therein for historical background.

Suppose, for the moment, that $A$ and $A^*$ both satisfy the De Giorgi-Nash-Moser condition (7.2) of some exponent. Then for all $(t, x) \in \mathbb{R}^{1+n}$ there exists a fundamental solution $\Gamma_{(t,x)}$ for $L_{A^*}$ in $\mathbb{R}^{1+n}$ with pole at $(t, x)$.<fnref n="9" /> The fundamental solution $\tilde{\Gamma}_{(t,x)} = (\tilde{\Gamma}_{(t,x)}^{\beta, \alpha})$ is a $\mathbb{C}^{m \times m}$-valued locally integrable function on $\mathbb{R}^{1+n}$ satisfying

$$ \text{div } A^* \nabla \tilde{\Gamma}_{(t,x)} = \delta_{(t,x)} \mathbf{1} \quad \text{in } \mathbb{R}^{1+n} $$

in the usual weak sense, where $\delta_{(t,x)}$ is the Dirac mass at $(t, x)$ and $\mathbf{1}$ is the identity matrix. The assumption on $A^*$ guarantees existence and uniqueness of fundamental solutions, with various decay estimates. A formal application of Green's formula tells us that

$$ u^{\alpha}(t, x) = \sum_{\beta=1}^{m} \int_{\mathbb{R}^n} \tilde{\Gamma}_{(t,x)}^{\beta, \alpha}(0, y) \partial_{\nu_A} u^{\beta}(0, y) dy - \sum_{\beta=1}^{m} \int_{\mathbb{R}^n} \overline{\partial_{\nu_{A^*}} \tilde{\Gamma}_{(t,x)}^{\beta, \alpha}(0, y)} u^{\beta}(0, y) dy $$

<footnote n="9">⁹Fundamental solutions were constructed in dimension $n+1 \ge 3$ by Hofmann and Kim [43] at least for systems having pointwise accretivity bounds, and in dimension $n+1 \ge 2$ and accretivity in our sense by Rosén [68].</footnote>
when $\alpha = 1, \dots, m$, with our convention of the normal vector pointing in the $t$-direction (that is, inward for the upper half-space).

For a (reasonable) function $f: \mathbb{R}^n \to \mathbb{C}^m$ and for $(t, x) \in \mathbb{R}^{1+n}$, with $t \neq 0$, one is led to define the double layer potential $\mathcal{D}_t f$ of $f$ by

$$ \mathcal{D}_t f(x)^\alpha := \int_{\mathbb{R}^n} \left( \partial_{\nu_{A^*}} \tilde{\Gamma}_{(t,x)}^{\alpha}(0, y), f(y) \right) dy \quad (\alpha = 1, \dots, m), $$

and the single layer potential $S_t f$ of $f$ by

$$ S_t f(x)^\alpha := \int_{\mathbb{R}^n} (\tilde{\Gamma}_{(t,x)}^{\alpha}(0, y), f(y)) dy \quad (\alpha = 1, \dots, m). $$

Here, $(z, \zeta) = \bar{z} \cdot \zeta$ for $z, \zeta \in \mathbb{C}^m$. The first question is whether these operators extend boundedly to spaces of boundary data: on $L^2(\mathbb{R}^n : \mathbb{C}^m)$ for $\mathcal{D}_t$, and from $L^2(\mathbb{R}^n : \mathbb{C}^m)$ into $L_1^2(\mathbb{R}^n : \mathbb{C}^m)$ for $S_t$ (uniformly in $t > 0$). This was asked by Hofmann in [40]. Another formal observation is that $(\tilde{\Gamma}_{(t,x)}(s, y))^* = \Gamma_{(s,y)}(t, x)$, where $\Gamma_{(s,y)}$ is the fundamental solution for $L_A$ with pole at $(s, y)$. Note that $\Gamma_{(0,y)}$ is a solution to the equation $L_A u = 0$ away from the pole $(0, y)$, so that the integrals 'defining' $\mathcal{D}_t$ and $S_t$ may be used to construct solutions. Proving the invertibility of such operators is one way to solve boundary value problems (there are other ways; see below). More precisely, one can solve Dirichlet problems for $L_A$ in $\mathbb{R}_+^{1+n}$ with boundary data $\varphi$ by solving the double layer equation

$$ \lim_{t \searrow 0} \mathcal{D}_t f = \varphi, $$

and likewise one can solve Neumann problems for $L_A$ in $\mathbb{R}_+^{1+n}$ with boundary data $\varphi$ by solving the single layer equation

$$ \lim_{t \searrow 0} \partial_{\nu_A} S_t f = \varphi. $$

The corresponding solutions $u$ are then given by $u(t, x) = \mathcal{D}_t f(x)$ and $u(t, x) = S_t f(x)$ respectively.

It was shown by Rosén [68] that these layer potential operators fall within the scope of the first-order framework, hence answering Hofmann’s question on layer potential boundedness in the positive. Keeping the De Giorgi–Nash–Moser assumption on $A$ and $A^*$, and writing $B = \hat{A}$ as usual, for all $f$ in a dense class in $L^2(\mathbb{R}^n : \mathbb{C}^m)$ and $t \in \mathbb{R} \setminus \{0\}$, Rosén shows that

$$ (7.5) \qquad \mathcal{D}_t f = - \operatorname{sgn}(t) \left( \mathbb{P}_D e^{-tBD} \chi^{\operatorname{sgn}(t)}(BD) \mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_\perp $$

and

$$ (7.6) \qquad \nabla_A S_t f = \operatorname{sgn}(t) e^{-tDB} \chi^{\operatorname{sgn}(t)}(DB) \begin{bmatrix} f \\ 0 \end{bmatrix}, $$

where the vectors $\begin{bmatrix} f \\ 0 \end{bmatrix}$ are written with respect to the transversal/tangential splitting. In terms of Cauchy operators, on $\mathbb{R}_\pm^{1+n}$ this amounts to

$$ (7.7) \qquad \mathcal{D}f = \mp \left( \mathbb{P}_D C_{BD}^{\pm} \mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_\perp, \qquad \nabla_A Sf = \pm C_{DB}^{\pm} \begin{bmatrix} f \\ 0 \end{bmatrix}. $$
Remark that vectors in the form $[f]$ belong to $\overline{\mathcal{R}(D)}$ but not to $\overline{\mathcal{R}(BD)}$, hence the presence of the projection $\mathbb{P}_{BD}$.<fnref n="10" />

Now, we reverse the point of view. The right hand sides of the expressions in (7.7) are defined for all coefficients $A$, whether or not the De Giorgi–Nash–Moser assumptions are satisfied. Thus we may define abstract double and single layer potentials.

We first define the abstract double layer potential. We need only consider $f$ in a dense class. For $f \in L^2(\mathbb{R}^n : \mathbb{C}^m)$, (7.5) is well-defined by the functional calculus of the bisectorial operator $BD$. When $f$ belongs to the classical Sobolev space $W_1^2(\mathbb{R}^n : \mathbb{C}^m)$, $[f] \in \mathbb{X}_D^{\mathbf{P}+1}$ with $\mathbf{p} = (2, 0)$ (Hardy-Sobolev and Besov spaces are the same in this case). It follows from Proposition 5.23 and Corollary 5.25 that (7.5) is well-defined as an element of $(\mathbb{X}_{BD}^{\mathbf{P}+1} \cap \mathcal{D}(D))_{\perp} = (\mathbb{X}_D^{\mathbf{P}+1} \cap \mathcal{D}(D))_{\perp}$. Furthermore, by the calculations in the proof of Theorem 6.13, we have that $\mathcal{D}f$ is a weak solution of the equation $L_A u = 0$ on each half-space.

Next, we turn to the abstract single layer potential. Assume $f \in L^2(\mathbb{R}^n : \mathbb{C}^m)$. Then $h = [f] \in \overline{\mathcal{R}(D)}$, and we can define

$$ (7.8) \qquad Sf = \mp \left( D^{-1} C_{DB}^{\pm} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp} $$

as an element of $L^\infty(\mathbb{R}^\pm : \mathbf{X}_\perp^{(2,1)})$, where $D^{-1}$ is the inverse of the isomorphism $D : \mathbf{X}_D^{(2,1)} \to \overline{\mathcal{R}(D)}$ (a nice Fourier multiplier). By Corollary 5.24, we then have

$$ C_{DB}h = DC_{BD}\mathbb{P}_{BD}D^{-1}h = D\mathbb{P}_{D}C_{BD}\mathbb{P}_{BD}D^{-1}h. $$

Applying $D^{-1}$ and mapping into $\mathbf{X}_D^{(2,1)}$ yields

$$ D^{-1}C_{DB}h = \mathbb{P}_{D}C_{BD}\mathbb{P}_{BD}D^{-1}h, $$

hence

$$ (D^{-1}C_{DB}h)_{\perp} = (\mathbb{P}_{D}C_{BD}\mathbb{P}_{BD}D^{-1}h)_{\perp} = (C_{BD}\mathbb{P}_{BD}D^{-1}h)_{\perp}, $$

where the last equality follows from Corollary 5.25. Taking the sign function into account, this shows that

$$ Sf = \mp(C_{BD}^{\pm}\mathbb{P}_{BD}D^{-1}h)_{\perp} $$

on $\mathbb{R}^\pm$. It follows by the calculation of Theorem 6.13 that $Sf$ is also a solution to $L_A u = 0$ in each half-space and that $\nabla_A Sf$ satisfies the equality in (7.7).

Having defined the layer potentials abstractly on dense subspaces, the results of Section 5.2 yield bounded extensions of $S$ and $\mathcal{D}$ on the classical smoothness spaces $\mathbf{X}^\mathbf{P}$ when $\mathbf{X}_{BD}^\mathbf{P} = \mathbf{X}_D^\mathbf{P}$. More precisely, we can compute them as

$$ Sf = \mp \left( D^{-1} C_{DB}^{\pm} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp} \quad (f \in \mathbf{X}^{\mathbf{P}}) $$

<footnote n="10">¹⁰This projection, and $\mathbb{P}_D$, were systematically forgotten in [19] by the abuse of notation explained in section 5.6, as this does not affect the definition in $L^2$. The stated results for the extensions there were correct. There is also a typo there: the four formulae (81-85) are written with $e^{+t\cdots}$ instead of $e^{-t\cdots}$ when $t < 0$.</footnote>
and

$$ (7.9) \qquad \mathcal{D}f = \pm \left( \mathbb{P}_D C_{BD}^+ \mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp} \quad (f \in \mathbf{X}^{\mathbf{P}+1}) $$

using the appropriate extensions from Section 5.2 and 5.4. Note that it is necessary to use $\mathbb{P}_D$ in (7.9) for arbitrary $f \in \mathbf{X}^{\mathbf{P}+1}$, although this does not matter for $f \in L^2$. Thus for $\mathbf{p} \in J(\mathbf{X}, \mathcal{D}\mathcal{B})$, we get the uniform bounds

$$ (7.10) \qquad \sup_{t \neq 0} \|\nabla_A S_t f\|_{\mathbf{X}^p} + \|S_t f\|_{\mathbf{X}^{p+1}} \lesssim \|f\|_{\mathbf{X}^p} $$

and

$$ (7.11) \qquad \sup_{t \neq 0} \|\nabla_A \mathcal{D}_t f\|_{\mathbf{X}^p} + \|\mathcal{D}_t f\|_{\mathbf{X}^{p+1}} \lesssim \|f\|_{\mathbf{X}^{p+1}}. $$

Compare these bounds with those of Barton and Mayboroda [22, Chapter 3], in particular (3.11-16). The inequalities (7.10) and (7.11) encapsulate a number of concrete inequalities; see the table in Section 2.7.

We also obtain limits for these operators as $t \to 0^+$ (in $\mathbf{X}^p$ or $\mathbf{X}^{p+1}$ accordingly, and in the strong or the weak-star topology depending on whether $\mathbf{p}$ is finite). In particular we can also recover the jump relations with this formalism. The equation

$$ (7.12) \qquad \nabla_A S_{0+} f - \nabla_A S_{0-} f = (\chi^+(DB) + \chi^-(DB)) \begin{bmatrix} f \\ 0 \end{bmatrix} = \begin{bmatrix} f \\ 0 \end{bmatrix} $$

encodes both the jump relation of the conormal derivative of $S_t$ and the continuity of $S_t$ across the boundary. Here we used that $\chi^+(DB) + \chi^-(DB) = I$ on $\mathbf{X}_D^p$, and that $\begin{bmatrix} f \\ 0 \end{bmatrix} \in \mathbf{X}_D^p$ when $f \in \mathbf{X}^p$. For the double layer operator, we have

$$ (7.13) \qquad \begin{aligned} \mathcal{D}_{0+} f - \mathcal{D}_{0-} f &= - \left( \mathbb{P}_D (\chi^+(BD) + \chi^-(BD)) \mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp} \\ &= - \left( \mathbb{P}_D \mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp} = -f. \end{aligned} $$

This time we used that $\chi^+(BD) + \chi^-(BD) = \mathbb{P}_{BD}$ on $\mathbf{X}_{BD}^{p+1}$ and that (5.10) applies to $\begin{bmatrix} f \\ 0 \end{bmatrix} \in \mathbf{X}_D^{p+1}$ when $f \in \mathbf{X}^{p+1}$.

For all exponents $\mathbf{p} \in J(\mathbf{X}, \mathcal{D}\mathcal{B})$, we have proven the tent/Z-space estimates

$$ \|C_{DB}^+ h\|_{X^p} \lesssim \|h\|_{X_D^p} $$

(see Theorems 4.34 and 5.26 and those in [19, §12.3].) Applied to $h = \begin{bmatrix} f \\ 0 \end{bmatrix}$ and taking extensions, these immediately yield

$$ (7.14) \qquad \|\nabla_S f\|_{X^p} \lesssim \|f\|_{X^p} $$

and

$$ (7.15) \qquad \|\nabla \mathcal{D} f\|_{X^p} \lesssim \|f\|_{X^{p+1}}. $$

Similar bounds for layer potentials on the lower half-space corresponding to (7.14) and (7.15) can also be derived. Compare these results with those of Barton and Mayboroda [22, Theorem 3.1] concerning $S$ and $\mathcal{D}$: again, we recover their Besov
space results (without the De Giorgi–Nash–Moser assumption that they use) and we obtain new Hardy–Sobolev space results.

With these results at hand, the proof of Theorem 7.20 is easy. We mostly reproduce calculations of [15, Lemma 8.1]. By Theorem 6.11 and the assumption on $\mathbf{p}$, we have the semigroup representation for the conormal gradient $\nabla_A u(t, x) = e^{-tDB}\chi^+(DB)h(x)$, with $h \in \mathbf{X}_{DB}^{\mathbf{p}_+}$ being the trace of $\nabla_A u$ at the boundary. Thus $h_{\perp} = \partial_{\nu_A} u|_{t=0}$ and $h_{\parallel} = \nabla_{\parallel} u|_{t=0}$. Write

$$e^{-tDB}\chi^+(DB)h = e^{-tDB}\chi^+(DB)\begin{bmatrix} h_{\perp} \\ 0 \end{bmatrix} + e^{-tDB}\chi^+(DB)\begin{bmatrix} 0 \\ h_{\parallel} \end{bmatrix}$$

and, writing $u|_{t=0} = f$, we know that

$$e^{-tDB}\chi^+(DB)\begin{bmatrix} 0 \\ h_{\parallel} \end{bmatrix} = \nabla_A \left( \mathbb{P}_D e^{-tBD}\chi^+(BD)\mathbb{P}_{BD} \begin{bmatrix} f \\ 0 \end{bmatrix} \right)_{\perp}$$

as in the proof of Theorem 6.13. The second term is precisely $-\nabla_A D_t f$.

Putting everything together, this gives us (7.4).

Let us make some comments on the use of invertibility of layer potentials to solve boundary value problems. As mentioned, one can use invertibility of the operator $D_{0+}$ for Dirichlet problems and that of the operator $\partial_{\nu_A} S_{0+}$ for Neumann problems. This is obtained (for example) for self-adjoint real equations ($m = 1$) for $\mathbf{p} = (2, 0)$, i.e. spaces of $L^2$ Dirichlet/Neumann data, in [2, Theorem 1.13], and for the lower half-space as well (looking at the $0^-$ limits). Prior to this was the work of Verchota for Laplace's equation on Lipschitz domains [79].

Such invertibility is not necessary for well-posedness as characterised in Theorem 7.5. But this is not the only way to prove well-posedness via layer potentials. For regularity problems, one can use invertibility of the layer potential $S_0$, as is done in [2, Theorem 1.13] for real symmetric equations with $L^2$ regularity data, in [41] for real equations with $L^p$ regularity data (for some $p > 1$), and in [15] for $H^p$ regularity data for some $p < 1$. For Neumann problems, one can use invertibility of $\partial_{\nu_A} D_0$. We give a quick sketch and refer to the proof of [15, Theorem 1.11] for details. The operator $\operatorname{sgn}(DB)$, written in matrix form according to the transversal/tangential decomposition, is

$$\operatorname{sgn}(DB) = \begin{bmatrix} \cdot & -2T \\ 2\nabla_x S_0 & \cdot \end{bmatrix}.$$

The operator $T$ will be discussed below. The operator $S_0$ is the single layer potential at $t = 0$, given by

$$S_0 f := \lim_{t \to 0} S_t f,$$

recalling that (7.12) shows that the limits $t \to 0^+$ are the same for this operator. For $\mathbf{p} \in J(\mathbf{X}, DB)$, well-posedness of $(R_{\mathbf{X}})_{\mathbf{A}}^{\mathbf{p}}$ on both half-spaces is equivalent to invertibility of $\nabla_x S_0: \mathbf{X}_{\perp}^{\mathbf{p}} \to \mathbf{X}_{\parallel}^{\mathbf{p}}$ (a multiple of the bottom left component of $\operatorname{sgn}(DB)$), or equivalently, invertibility of the single layer potential $S_0$ from $\mathbf{X}^{\mathbf{p}}$ (which is the same here as $\mathbf{X}_{\perp}^{\mathbf{p}}$) onto $\mathbf{X}^{\mathbf{p}+1}$ as in [15, Theorem 1.11]. Then, for $f \in \mathbf{X}^{\mathbf{p}+1}$, the solutions of the regularity problems with data $\nabla_{\parallel} f$ (that is, $f$ is the Dirichlet data) on the upper and lower half-spaces are respectively given by

$$u^{\pm} = S_t S_0^{-1} f \quad (t \in \mathbb{R}^{\pm}).$$

There is also a notion of mutual well-posedness on both half-spaces, given by the conjunction of mutual well-posedness on each half-space separately. As we did
before, this is akin to demanding the agreement of inverses of the single layer potential for different $\mathbf{p}$. We leave this to the reader.

There are corresponding statements for Neumann problems, using the invertibility of the top right component $-2T$ of $\operatorname{sgn}(DB)$. A calculation shows that $T(\nabla f) = \partial_{\nu_A} D_0 f$ (there is no jump here) and so invertibility of $T$ corresponds to invertibility of the conormal derivative of the double layer potential $\partial_{\nu_A} D_0: \mathbf{X}^{p+1} \to \mathbf{X}^p$ (again the latter space should be thought of as the transversal component). The solutions to the Neumann problems in the upper and lower half-spaces with Neumann data $g$ are given respectively by

$$u^{\pm} = D_t(\partial_{\nu_A} D_0)^{-1}g \quad (t \in \mathbb{R}^{\pm}).$$

We remark that for the block diagonal case (see Section 7.2.4) both $S_0$ and $\partial_{\nu_A} D_0$ are invertible in the range where well-posedness was established: this is because $\operatorname{sgn}(DB)$ is an involution with zero diagonal entries (and this is because $A$ is diagonal if and only if $\hat{A}$ is diagonal, and then in this case $\operatorname{sgn}(DB) = DB[DB]^{-1}$ has zero diagonal entries).

For the triangular block form, it is either the invertibility of $\partial_{\nu_A} D_0$ or that of $S_0$ that holds, depending on the boundary value problem and on the value of $\mathbf{p}$ in the range of well-posedness. The results of [13] show this explicitly for the Regularity and Neumann problems with $L^2$ data.

We mention without proof that our results on stability in the coefficients (see Section 7.3) allow one to perturb invertibility of boundary layer operators. This was already observed in [2] in the following form: invertibility of a collection of boundary layer operators is stable under perturbation. Here, we observe that invertibility is preserved for each individual boundary layer operator, which is a little stronger.

In conclusion, we see that there are many different ways to obtain well-posedness (compatible or not) for boundary value problems associated with $L_A u = 0$ under our current assumptions on $A$. The first order method essentially covers all results in the literature except for that in [42], concerning a Dirichlet problem with data in $L^p$ for large $p$, which uses harmonic measure methods for real equations. We still lack the understanding to establish well-posedness for more general coefficients, with any currently available approach.
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Index

atom

Tent space, 22

Z-space, 27

atomic decomposition
of tent spaces, 22
of Z-spaces, 27

bisectorial operator, 45

boundary value problem, 2, 125
Dirichlet, 2, 126
Neumann, 4, 125
Regularity, 3, 125

Calderón reproducing formula, 58

Calderón siblings, 58

Carleson operator, 21, 23

Cauchy operator
on DB-adapted BHS spaces, 83
on adapted BHS spaces, 69
on $\overline{R}(A)$, 48

Cauchy-Riemann system, 8
anti-, 93
classification of solutions, 11, 96
equivalence with elliptic equation, 8
completion
canonical, 64
of D-adapted spaces, 75
of a quasinormed space, 64
of adapted BHS spaces, 64
weak-star, 64

conormal derivative, 4

conormal gradient, 9

contraction operator, 55

De Giorgi-Nash-Moser condition, 95, 130, 138, 139

Dirac operator, 8, 73
perturbed, 8, 73
satisfies Standard Assumptions, 51

downward shift, 32

duality

♥-, 20

of adapted BHS spaces, 62, 66

of canonical completions, 66

of exponents, 19

of identification regions, 78

of inclusion regions, 78

of slice spaces, 35

of tent spaces, 23

of well-posedness, 128

of Z-spaces, 26, 29

duality pairing

$L^2$, 23

dyadic characterisation
of slice space, 36
of Z-space, 25

elliptic equation, 1
basic properties of solutions, 93
boundary behaviour of solutions, 118
classification of solutions, 97, 98
decay of solutions, 94
equivalence with Cauchy-Riemann
system, 8
representation of solutions
by layer potentials, 139
solution space, 117

embedding
of adapted BHS spaces, 62
of exponents, 20
of slice spaces, 35
of tent spaces, 23
of tent- and Z-spaces, 32

exponent, 19
energy, 126
finite, 19
infinite, 19

extension operator, 55

extrapolation
of identification region, 80
of well-posedness, 127

functional calculus
bounded $H^\infty$, 47
holomorphic, 45
on adapted BHS spaces, 63, 67
on canonical completions, 67
similarity, 75
$I_{\max}$, 78

$I_{\min}$, 79

interpolation

of adapted BHS spaces, 68

of canonical completions, 68

of exponents, 20

of inclusions of adapted BHS spaces, 69

of mutual well-posedness, 127

of smoothness spaces, 39

of solution spaces, 116

of tent spaces, complex, 23

of tent spaces, real, 25, 30

of Z-spaces, complex, 31

of Z-spaces, real, 31

layer potential, 139

double, 140

jump relations, 142

representation by Cauchy operator, 140

single, 140

Lusin operator, 21

nondegenerate

holomorphic function, 46

off-diagonal estimates, 49

region

classification, 97

identification, 77

openness, 80

inclusion, 68

well-posedness, 125

for specific coefficients, 130

Riesz potential, 37

space

adapted Besov-Hardy-Sobolev, 60

characterisation by Cauchy operator,

71, 83

characterisation by contraction maps,

61

D-adapted, 73

positive and negative, 60, 63

Besov, 36

BMO-Sobolev, 38

Hölder-Lipschitz, 38

Hardy-Sobolev, 36

slice, 34

smoothness, 39

solution, 117

tent, 21

vanishing, 129

VMO, 129

Wiener amalgam, 31, 36

X-, 31

Z-, 24

spectral projection, 48

spectral subspaces, 48

Standard Assumptions, 51

tempered distributions modulo

polynomials, 36

well-posedness, 125

characterisation, 126

compatible, 127

incompatible, 135

mutual, 126

topological characterisation, 128

stability in coefficients, 135

Whitney average, 24

Whitney cube, 25

Whitney grid, 25

Whitney parameter, 24

Whitney region, 24