Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

TL;DR

This paper extends the understanding of elliptic operators in 1-sided chord-arc domains, showing Carleson measure estimates imply the elliptic measure is in A_infinity, with applications to perturbation and symmetry conditions.

Contribution

It generalizes previous results to non-symmetric operators, simplifies proofs, and establishes new equivalences between elliptic measure properties and domain geometry.

Findings

Carleson measure estimates imply elliptic measure in A_infinity.

Extension of perturbation results to non-symmetric coefficients.

Equivalence of A_infinity membership for operators and their transposes under Carleson conditions.

Abstract

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition then $\omega_L\in A_\infty$ if and only if $\omega_{L^\top}\in A_\infty$. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $L$ to the class $A_\infty$ yields that the domain is indeed a chord-arc domain.

Equations

$$\underset{\begin{array}{c}x\in \partial \Omega \\ 0<r<\infty \end{array}}{\sup }\frac{1}{r^n}\int {\int }_{B(x,r)\cap \Omega }|\nabla u(X){|}^2\delta (X)dX\le C\Vert u{\Vert }_{L^{\infty }(\Omega )}^2.$$

$$\varrho (A_1,A_0)(X):=\underset{Y\in B(X,\delta (X)/2)}{\sup }|A_1(Y)-A_0(Y)|,X\in \Omega ,$$

$$\underset{\begin{array}{c}x\in \partial \Omega \\ 0<r<\mathrm{diam}(\partial \Omega )\end{array}}{\sup }\frac{1}{\sigma (B(x,r)\cap \partial \Omega )}\int {\int }_{B(x,r)\cap \Omega }\frac{\varrho (A_1,A_0)(X{)}^2}{\delta (X)}dX<\infty .$$

$$\operatorname{div}_{C}(A-A^{\top})(X)=\bigg{(}\sum_{i=1}^{n+1}\partial_{i}(a_{i,j}-a_{j,i})(X)\bigg{)}_{1\leq j\leq n+1},\qquad X\in\Omega.$$

$$\sup_{\begin{subarray}{c}x\in\partial\Omega\\ 0<r<\operatorname{diam}(\partial\Omega)\end{subarray}}\frac{1}{\sigma(B(x,r)\cap\partial\Omega)}\int\mkern-13.5mu\int_{B(x,r)\cap\Omega}\big{|}\operatorname{div}_{C}(A-A^{\top})(X)\big{|}^{2}\delta(X)\,dX<\infty.$$

$$\sup_{\begin{subarray}{c}x\in\partial\Omega\\ 0<r<\operatorname{diam}(\partial\Omega)\end{subarray}}\frac{1}{\sigma(B(x,r)\cap\partial\Omega)}\int\mkern-13.5mu\int_{B(x,r)\cap\Omega}\big{|}\nabla A(X)\big{|}^{2}\delta(X)\,dX<\infty.$$

$$\sup_{\begin{subarray}{c}x\in\partial\Omega\\ 0<r<\operatorname{diam}(\partial\Omega)\end{subarray}}\frac{1}{\sigma(B(x,r)\cap\partial\Omega)}\int\mkern-13.5mu\int_{B(x,r)\cap\Omega}\big{|}\nabla A(X)\big{|}\,dX<\infty,$$

$$\omega_{L}\in A_{\infty}(\partial\Omega)\qquad\Longrightarrow\qquad\Omega\mbox{ is a $\mathrm{CAD}$ (cf. Definition \ref{def-1CAD})}.$$

$$C^{-1}{r}^n\le H^n(E\cap B(x,r))\le C{r}^n,0<r<\mathrm{diam}(E),x\in E.$$

$$\mathbb{D}_{k}:=\big{\{}Q_{j}^{k}\subset\partial\Omega:\>j\in\mathcal{J}_{k}\big{\}},$$

$$\mathbb{D}(E):=\bigcup _k{\mathbb{D}}_k,$$

$$\Delta (x_Q,2r_Q)\subset Q\subset \Delta (x_Q,C{r}_Q)$$

$$B_Q:=B(x_Q,r_Q),{\Delta }_Q:=\Delta (x_Q,r_Q),$$

$${\tilde{B}}_Q:=B(x_Q,C{r}_Q),{\tilde{\Delta }}_Q:=\Delta (x_Q,C{r}_Q),$$

$$\delta (X_Q)\approx \mathrm{dist}(X_Q,Q)\approx \mathrm{diam}(Q).$$

$$\mathbb{D}_{Q}:=\big{\{}Q^{\prime}\in\mathbb{D}(E):\,Q^{\prime}\subset Q\big{\}}.$$

$${\mathbb{D}}_{\mathcal{F}}:=\mathbb{D}(E)\setminus \bigcup _{Q_i\in \mathcal{F}}{\mathbb{D}}_{Q_i}.$$

$${\mathbb{D}}_{\mathcal{F},Q}:={\mathbb{D}}_Q\setminus \bigcup _{Q_i\in \mathcal{F}}{\mathbb{D}}_{Q_i}={\mathbb{D}}_{\mathcal{F}}\cap {\mathbb{D}}_Q.$$

$$4\mathrm{diam}(I)\le \mathrm{dist}(4I,\partial \Omega )\le \mathrm{dist}(I,\partial \Omega )\le 40\mathrm{diam}(I),\forall I\in \mathcal{W},$$

$$\mathrm{diam}(I_1)\approx \mathrm{diam}(I_2),\text{ whenever }{I}_1\text{ and }{I}_2\text{ touch}.$$

$$U_Q:=\bigcup _{I\in {\mathcal{W}}_Q^{\ast }}{I}^{\ast },$$

$$k(Q)-k^{\ast }\le k_I\le k(Q)+k^{\ast },\forall I\in {\mathcal{W}}_Q^{\ast },$$

$$X(I){\to }_{U_Q}{X}_Q,\forall I\in {\mathcal{W}}_Q^{\ast },$$

$$\mathrm{dist}(I,Q)\le K_0{2}^{-k(Q)},\forall I\in {\mathcal{W}}_Q^{\ast }.$$

$$T_{Q}:=\operatorname{int}\bigg{(}\bigcup_{Q^{\prime}\in\mathbb{D}_{Q}}U_{Q^{\prime}}\bigg{)}.$$

$$\Omega_{\mathcal{F},Q}=\operatorname{int}\bigg{(}\bigcup_{Q^{\prime}\in\mathbb{D}_{\mathcal{F},Q}}U_{Q^{\prime}}\bigg{)}=\operatorname{int}\bigg{(}\bigcup_{I\in\mathcal{W}_{\mathcal{F},Q}}I^{*}\bigg{)},$$

$$T_{Q}^{*}:=\operatorname{int}\bigg{(}\bigcup_{Q^{\prime}\in\mathbb{D}_{Q}}U_{Q^{\prime}}^{*}\bigg{)},\qquad\Omega^{*}_{\mathcal{F},Q}:=\operatorname{int}\bigg{(}\bigcup_{Q^{\prime}\in\mathbb{D}_{Q}}U_{Q^{\prime}}^{*}\bigg{)},\qquad U_{Q}^{*}:=\bigcup_{I\in\mathcal{W}_{Q}^{*}}I^{**}.$$

$${\kappa }_1{B}_Q\cap \Omega \subset T_Q\subset T_Q^{\ast }\subset T_Q^{\ast \ast }\subset \overline{T_Q^{\ast \ast }}\subset {\kappa }_0{B}_Q\cap \overline{\Omega }=:\frac{1}{2}{B}_Q^{\ast }\cap \overline{\Omega },$$

$${\Lambda }^{-1}|\xi {|}^2\le A(X)\xi \cdot \xi ,|A(X)\xi \cdot \zeta |\le \Lambda |\xi ||\zeta |,$$

$$\int {\int }_{\Omega }A(X)\nabla u(X)\cdot \nabla \phi (X)dX=0,\forall \phi \in C_c^{\infty }(\Omega ).$$

$$\frac{{\omega }_L^{X_{{\Delta }_0}}(F)}{{\omega }_L^{X_{{\Delta }_0}}(\Delta )}\le \alpha ⟹\frac{\sigma (F)}{\sigma (\Delta )}\le \beta .$$

Full text

The first author was partially supported by “la Caixa”-Severo Ochoa international PhD Programme. The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D” (SEV-2015-0554). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The second author was supported by NSF grant DMS-1664047. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics and by NSF grant DMS-1664867.

1. 1 Introduction and Main results

2. 2 Preliminaries

3. 2.1 Notation and conventions

4. 2.2 Some definitions

5. 2.3 Dyadic grids and sawtooths

6. 2.4 PDE estimates

7. 3 Proof of Theorem 1.1

8. 3.1 The Carleson measure condition implies $A_{\infty}$

9. 3.2 $A_{\infty}$ implies the Carleson measure condition

10. 4 Proof of Theorems 1.3 and 1.6

1. Introduction and Main results

F. and M. Riesz showed in [RR] that harmonic measure is absolutely continuous with respect to the surface measure for any simply connected domain in the complex plane whose boundary is rectifiable. Since then, one can find many references in the literature studying how the previous result, or its quantitative version obtained by Lavrentiev [Lav], can be extended to higher dimensions. In doing that, some kind of “strong” connectivity hypotheses is needed (as shown by the counter example in [BJ]). Dahlberg in [Dah] established that harmonic measure satisfies a quantitative version of absolute continuity with respect to the surface measure for every Lipschitz domain. That quantitative version says that harmonic measure is in the Muckenhoupt class of weights $A_{\infty}$, and more precisely it belongs to $RH_{2}$, the class of weights satisfying a reverse Hölder condition with exponent $2$.

Jerison and Kenig [JK] introduced a new class of domains called NTA (non-tangentially accessible). These domains satisfy interior and exterior Corkscrew conditions (these are quantitative versions of the fact that the domain and its exterior are open sets). They also satisfy an interior Harnack Chain condition (which is a quantitative version of the path-connectivity). In this class of domains they developed the boundary regularity theory for harmonic functions, they also established the properties of the harmonic measure, and the Green function. NTA domains whose boundary is Ahlfors regular are called of type chord-arc. In this class of domains which include Lipschitz domains David-Jerison [DJ] and independently Semmes [Sem] proved that the harmonic measure is an $A_{\infty}$ weight with respect to surface measure to the boundary. It belongs to some class $RH_{p}$ with $p>1$.

Recently a big effort has been made to understand in what domains and for what operators the elliptic measure is an $A_{\infty}$ weight with respect to surface measure to the boundary of the domain. One context where the theory has been satisfactorily developed is that of $1$-sided chord-arc domains. These are open sets $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, whose boundaries $\partial\Omega$ are $n$-dimensional Ahlfors regular (cf. Definition 2.3), and which satisfy interior (but not exterior) Corkscrew and Harnack Chain conditions see Definitions 2.1 and 2.2 below). In [HM3, HMUT] the authors show that in the setting of 1-sided chord-arc domains, harmonic measure is in $A_{\infty}(\partial\Omega)$ (cf. 2.13) if and only if $\partial\Omega$ is uniformly rectifiable (a quantitative version of rectifiability). It was shown later in [AHMNT] that under the same background hypothesis, if $\partial\Omega$ is uniformly rectifiable then $\Omega$ satisfies an exterior corkscrew condition and hence $\Omega$ is a chord-arc domain. All these together and, additionally, [AHMNT] in conjunction with [DJ] or [Sem], give a characterization of chord-arc domains, or a characterization of the uniform rectifiability of the boundary, in terms of the membership of harmonic measure to the class $A_{\infty}(\partial\Omega)$. For other elliptic operators $Lu=-\operatorname{div}(A\nabla u)$ with variable coefficients it was shown recently in [HMT2] that the same characterization holds provided $A$ is locally Lipschitz and has appropriately controlled oscillation near the boundary.

This paper is the second part of a series of two articles where we consider perturbation of real elliptic operators in the setting of 1-sided chord-arc domains. In the first paper of the series [CHM] we worked with symmetric operators and studied perturbations that preserve the $A_{\infty}(\partial\Omega)$ property extending the work of [FKP, MPT1, MPT2] (see also [HL], [HM2, HM1]) to the setting of 1-sided chord-arc domains. It was shown that if the disagreement between two elliptic symmetric matrices satisfies certain Carleson measure condition, then one of the associated elliptic measures is in $A_{\infty}(\partial\Omega)$ if and only if the other one is in $A_{\infty}(\partial\Omega)$. In other words, the property that the elliptic measure belongs to $A_{\infty}(\partial\Omega)$ is stable under Carleson measure type perturbations. That result was proved using the so-called extrapolation of Carleson measures, which originated in [LM] (see also [HL, AHLT, AHMTT]), in the form developed in [HM2, HM1] (see also [HM3]). The method is a bootstrapping argument, based on the Corona construction of Carleson [Car] and Carleson and Garnett [CG], that, roughly speaking, allows one to reduce matters to the case in which the perturbation is small in some sawtooth subdomains. Implicit in the proof of the perturbation result in [CHM] one can find the treatment of the case in which the perturbation is small, and this allowed the authors to obtain that for sufficiently small perturbations, not only the class $A_{\infty}$ is preserved but one can also keep the same exponent in the corresponding reverse Hölder class.

In the present paper we work in the same setting of 1-sided chord-arc domains and consider real not necessarily symmetric elliptic operators. Our first goal is to establish that for any real elliptic operator non-necessarily symmetric $L$, the property that all bounded solutions of $L$ satisfy Carleson measure estimates yields $\omega_{L}\in A_{\infty}(\partial\Omega)$. This extends the work [KKPT] where they treated bounded Lipschitz domains and domains above the graph of a Lipschitz function. That the converse is true (hence both properties are equivalent) follows from [HMT1] where a more general estimate is obtained. Indeed, assuming that $\omega_{L}\in A_{\infty}(\partial\Omega)$ then it is shown that the conical square function is controlled by the non-tangential maximal function in every $L^{p}(\partial\Omega)$ for every $1<p<\infty$ where both are applied to solutions of $L$. Applying this estimate with $p=2$ to a bounded solution one obtains the desired Carleson. Here, nevertheless, we present a simpler and novel argument for the latter fact. The precise result is as follows:

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{n+1}$ be a 1-sided $\mathrm{CAD}$ and let $Lu=-\operatorname{div}(A\nabla u)$ be a real (not necessarily symmetric) elliptic operator (cf. Definition 2.12). The following statements are equivalent:

• $(a)$

Every bounded weak solution of $Lu=0$ satisfies a Carleson measure estimate, that is, there exists $C$ such that every $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ with $Lu=0$ in $\Omega$ in the weak sense, satisfies the Carleson measure condition

[TABLE]

• $(b)$

$\omega_{L}\in A_{\infty}(\partial\Omega)$* (cf. Definition 2.13).*

Our second goal is to use the previous characterization to extend the “large” constant perturbation result from [CHM] to the non-symmetric case:

Theorem 1.3**.**

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be a 1-sided $\mathrm{CAD}$ (cf. Definition 2.4). Let $L_{1}u=-\operatorname{div}(A_{1}\nabla u)$ and $L_{0}u=-\operatorname{div}(A_{0}\nabla u)$ be real (not necessarily symmetric) elliptic operators (cf. Definition 2.12). Define the disagreement between $A_{1}$ and $A_{0}$ in $\Omega$ by

[TABLE]

where $\delta(X):=\operatorname{dist}(X,\partial\Omega)$, and assume that it satisfies the Carleson measure condition

[TABLE]

Then, $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L_{1}}\in A_{\infty}(\partial\Omega)$ (cf. Definition 2.13).

To prove this result we use a novel approach which is interesting on its own right and is conceptually simpler. The bottom line is that assuming that $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$ and based on Theorem 1.1 we just need to establish that all bounded solutions for $L_{1}$ satisfy the aforementioned Carleson measure estimates, rather than trying to establish the “more delicate” condition $\omega_{L_{1}}\in A_{\infty}(\partial\Omega)$. In doing this we exploit the fact that $\omega_{L_{0}}\in A_{\infty}(\Omega)$ to find a sawtooth domain whose boundary has with ample contact with $\partial\Omega$, where the averages of $\omega_{L_{0}}$ are essentially constant. Hence in (1.2) one can replace $\delta$ by $G_{L_{0}}$ in a sawtooth with ample contact. This in turn allows us to perform some integrations by parts to conclude the desired estimate. We would like to emphasize that this approach cannot be used to get the “small” constant perturbation since that requires to directly show that the two elliptic measures are in the same reverse Hölder class without passing through the Carleson measure estimates.

Our last main result establishes a connection between the elliptic measures of an operator and its adjoint assuming that the derivative of the antisymmetric part of the matrix defining the operator satisfies some Carleson measure condition:

Theorem 1.6**.**

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be a 1-sided $\mathrm{CAD}$ (cf. Definition 2.4). Let $Lu=-\operatorname{div}(A\nabla u)$ be a real (not necessarily symmetric) elliptic operator (cf. Definition 2.12), let $L^{\top}$ denote the transpose of $L$ (i.e, $L^{\top}u=-\operatorname{div}(A^{\top}\nabla u)$ with $A^{\top}$ being the transpose matrix of $A$), and let $L^{\rm sym}=\frac{L+L^{\top}}{2}$ be the symmetric part of $L$. Assume that $(A-A^{\top})\in{\rm Lip}_{\rm loc}(\Omega)$ and let

[TABLE]

Assume that the following Carleson measure estimate holds

[TABLE]

Then $\omega_{L}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L^{\top}}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L^{\rm sym}}\in A_{\infty}(\partial\Omega)$ (cf. Definition 2.13).

As an immediate consequence of the previous result we obtain the following:

Corollary 1.9**.**

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be a 1-sided $\mathrm{CAD}$ (cf. Definition 2.4). Let $Lu=-\operatorname{div}(A\nabla u)$ be a real (not necessarily symmetric) elliptic operator (cf. Definition 2.12). Assume that $A\in{\rm Lip}_{\rm loc}(\Omega)$, $|\nabla A|\,\delta\in L^{\infty}(\Omega)$ and the following Carleson measure estimate

[TABLE]

Then $\omega_{L}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L^{\top}}\in A_{\infty}(\partial\Omega)$.

In particular, if one further assumes that

[TABLE]

then

[TABLE]

The first part of Corollary 1.9 follows from Theorem 1.6. For the second part, we notice that once $\omega_{L}\in A_{\infty}(\partial\Omega)$ implies, after using the first part, that $\omega_{L^{\top}}\in A_{\infty}(\partial\Omega)$. In turn, we can then invoke [HMT2, Theorem 1.5] to conclude that $\Omega$ is a CAD. Note that comparing this with [HMT2, Theorem 1.5] what we are proving is that with the given background hypotheses one just needs to assume $\omega_{L}\in A_{\infty}(\partial\Omega)$, and the assumption $\omega_{L^{\top}}\in A_{\infty}(\partial\Omega)$ is redundant.

The organization of the paper is as follows. In Section 2 we present some of the needed preliminaries, notations, definitions and some of the PDE estimates which will be needed throughout the paper. Section 3 contains the proof of Theorem 1.1. Theorems 1.3 and 1.6 are proved in Section 4, as a matter of facts both results are particular cases of the much more general Theorem 4.13.

2. Preliminaries

2.1. Notation and conventions

• $\bullet$

Our ambient space is $\mathbb{R}^{n+1}$, $n\geq 2$.

• $\bullet$

We use the letters $c$, $C$ to denote harmless positive constants, not necessarily the same at each occurrence, which depend only on dimension and the constants appearing in the hypotheses of the theorems (which we refer to as the “allowable parameters”). We shall also sometimes write $a\lesssim b$ and $a\approx b$ to mean, respectively, that $a\leq Cb$ and $0<c\leq a/b\leq C$, where the constants $c$ and $C$ are as above, unless explicitly noted to the contrary. Moreover, if $c$ and $C$ depend on some given parameter $\eta$, which is somehow relevant, we write $a\lesssim_{\eta}b$ and $a\approx_{\eta}b$. At times, we shall designate by $M$ a particular constant whose value will remain unchanged throughout the proof of a given lemma or proposition, but which may have a different value during the proof of a different lemma or proposition.

• $\bullet$

Given a domain (i.e., open and connected) $\Omega\subset\mathbb{R}^{n+1}$, we shall use lower case letters $x,y,z$, etc., to denote points on $\partial\Omega$, and capital letters $X,Y,Z$, etc., to denote generic points in $\mathbb{R}^{n+1}$ (especially those in $\Omega$).

• $\bullet$

The open $(n+1)$-dimensional Euclidean ball of radius $r$ will be denoted $B(x,r)$ when the center $x$ lies on $\partial\Omega$, or $B(X,r)$ when the center $X\in\mathbb{R}^{n+1}\setminus\partial\Omega$. A “surface ball” is denoted $\Delta(x,r):=B(x,r)\cap\partial\Omega$, and unless otherwise specified it is implicitly assumed that $x\in\partial\Omega$. Also if $\partial\Omega$ is bounded, we typically assume that $0<r\lesssim\operatorname{diam}(\partial\Omega)$, so that $\Delta=\partial\Omega$ if $\operatorname{diam}(\partial\Omega)<r\lesssim\operatorname{diam}(\partial\Omega)$.

• $\bullet$

Given a Euclidean ball $B$ or surface ball $\Delta$, its radius will be denoted $r(B)$ or $r(\Delta)$ respectively.

• $\bullet$

Given a Euclidean ball $B=B(X,r)$ or surface ball $\Delta=\Delta(x,r)$, its concentric dilate by a factor of $\kappa>0$ will be denoted by $\kappa B=B(X,\kappa r)$ or $\kappa\Delta=\Delta(x,\kappa r)$.

• $\bullet$

For $X\in\mathbb{R}^{n+1}$, we set $\delta_{\partial\Omega}(X):=\operatorname{dist}(X,\partial\Omega)$. Sometimes, when clear from the context we will omit the subscript $\partial\Omega$ and simply write $\delta(X)$.

• $\bullet$

We let $H^{n}$ denote the $n$-dimensional Hausdorff measure, and let $\sigma_{\partial\Omega}:=H^{n}\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}$ denote the “surface measure” on $\partial\Omega$. For a closed set $E\subset\mathbb{R}^{n+1}$ we will use the notation $\sigma_{E}:=H^{n}\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,E}$. When clear from the context we will also omit the subscript and simply write $\sigma$.

• $\bullet$

For a Borel set $A\subset\mathbb{R}^{n+1}$, we let $\mathbf{1}_{A}$ denote the usual indicator function of $A$, i.e., $\mathbf{1}_{A}(x)=1$ if $x\in A$, and $\mathbf{1}_{A}(x)=0$ if $x\notin A$.

• $\bullet$

For a Borel set $A\subset\mathbb{R}^{n+1}$, we let $\operatorname{int}(A)$ denote the interior of $A$, and $\overline{A}$ denote the closure of $A$. If $A\subset\partial\Omega$, $\operatorname{int}(A)$ will denote the relative interior, i.e., the largest relatively open set in $\partial\Omega$ contained in $A$. Thus, for $A\subset\partial\Omega$, the boundary is then well defined by $\partial A:=\overline{A}\setminus\operatorname{int}(A)$.

• $\bullet$

For a Borel set $A\subset\mathbb{R}^{n+1}$, we denote by $C(A)$ the space of continuous functions on $A$ and by $C_{c}(A)$ the subspace of $C(A)$ with compact support in $A$. Note that if $A$ is compact then $C(A)\equiv C_{c}(A)$.

• $\bullet$

For a Borel set $A\subset\partial\Omega$ with $0<\sigma(A)<\infty$, we write \mathchoice{{\vbox{\hbox{\textstyle- }}\kern-7.63756pt}}{{\vbox{\hbox{\scriptstyle- }}\kern-5.75256pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.63129pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.14378pt}}\!\int_{A}f\,d\sigma:=\sigma(A)^{-1}\int_{A}f\,d\sigma.

• $\bullet$

We shall use the letter $I$ (and sometimes $J$) to denote a closed $(n+1)$-dimensional Euclidean cube with sides parallel to the co-ordinate axes, and we let $\ell(I)$ denote the side length of $I$. We use $Q$ to denote a dyadic “cube” on $E\subset\mathbb{R}^{n+1}$. The latter exists, given that $E$ is $\mathrm{AR}$ (cf. [DS1], [Chr]), and enjoy certain properties which we enumerate in Lemma 2.5 below.

2.2. Some definitions

Definition 2.1** (Corkscrew condition).**

Following [JK], we say that an open set $\Omega\subset\mathbb{R}^{n+1}$ satisfies the “Corkscrew condition” if for some uniform constant $c\in(0,1)$ and for every surface ball $\Delta:=\Delta(x,r)=B(x,r)\cap\partial\Omega$ with $x\in\partial\Omega$ and $0<r<\operatorname{diam}(\partial\Omega)$, there is a ball $B(X_{\Delta},cr)\subset B(x,r)\cap\Omega$. The point $X_{\Delta}\in\Omega$ is called a “corkscrew point” relative to $\Delta$. Note that we may allow $r<C\operatorname{diam}(\partial\Omega)$ for any fixed $C$, simply by adjusting the constant $c$.

Definition 2.2** (Harnack Chain condition).**

Again following [JK], we say that $\Omega\subset\mathbb{R}^{n+1}$ satisfies the Harnack Chain condition if there is a uniform constant $C$ such that for every $\rho>0$, $\Theta\geq 1$, and every pair of points $X,X^{\prime}\in\Omega$ with $\delta(X),\delta(X^{\prime})\geq\rho$ and $|X-X^{\prime}|<\Theta\rho$, there is a chain of open balls $B_{1},\dots,B_{N}\subset\Omega$, $N\leq C(\Theta)$, with $X\in B_{1}$, $X^{\prime}\in B_{N}$, $B_{k}\cap B_{k+1}\neq\mbox{{\O}}$ and $C^{-1}\operatorname{diam}(B_{k})\leq\operatorname{dist}(B_{k},\partial\Omega)\leq C\operatorname{diam}(B_{k})$. The chain of balls is called a “Harnack Chain”.

Definition 2.3** (Ahlfors regular).**

We say that a closed set $E\subset\mathbb{R}^{n+1}$ is $n$-dimensional $\mathrm{AR}$ (or simply $\mathrm{AR}$), if there is some uniform constant $C=C_{\mathrm{AR}}$ such that

[TABLE]

Definition 2.4** (1-sided chord-arc domain and chord-arc domain).**

We say that $\Omega\subset\mathbb{R}^{n+1}$ is a “1-sided chord-arc domain” (1-sided CAD for short) if it satisfies the Corkscrew and Harnack Chain conditions and if $\partial\Omega$ is $\mathrm{AR}$. Analogously, we say that $\Omega\subset\mathbb{R}^{n+1}$ is a “chord-arc domain” (CAD for short) if it is a 1-sided CAD and additionally $\Omega_{\rm ext}=\mathbb{R}^{n+1}\setminus\overline{\Omega}$ also satisfies the Corkscrew condition.

2.3. Dyadic grids and sawtooths

We give a lemma concerning the existence of a “dyadic grid”:

Lemma 2.5** (“Dyadic grid”** [DS1, DS2], [Chr]).

Suppose that $E\subset\mathbb{R}^{n+1}$ is $n$-dimensional $\mathrm{AR}$. Then there exist constants $a_{0}>0$, $\eta>0$ and $C<\infty$ depending only on dimension and the $\mathrm{AR}$ constant, such that for each $k\in\mathbb{Z}$ there is a collection of Borel sets (“cubes”)

[TABLE]

where $\mathcal{J}_{k}$ denotes some (possibly finite) index set depending on $k$, satisfying:

• $(a)$

$E=\bigcup_{j}Q_{j}^{k}$* for each $k\in\mathbb{Z}$.*

• $(b)$

If $m\geq k$ then either $Q_{i}^{m}\subset Q_{j}^{k}$ or $Q_{i}^{m}\cap Q_{j}^{k}=\mbox{{\O}}$.

• $(c)$

For each $j,k\in\mathbb{Z}$ and each $m>k$, there is a unique $i\in\mathbb{Z}$ such that $Q_{j}^{k}\subset Q_{i}^{m}$.

• $(d)$

$\operatorname{diam}(Q_{j}^{k})\leq C\,2^{-k}$.

• $(e)$

Each $Q_{j}^{k}$ contains some “surface ball” $\Delta(x_{j}^{k},a_{0}2^{-k})=B(x_{j}^{k},a_{0}2^{-k})\cap E$.

• $(f)$

H^{n}\big{(}\big{\{}x\in Q_{j}^{k}:\,\operatorname{dist}(x,E\setminus Q_{j}^{k})\leq\tau 2^{-k}\big{\}}\big{)}\leq C\tau^{\eta}H^{n}(Q_{j}^{k}), for all $j,k\in\mathbb{Z}$ and for all $\tau\in(0,a_{0})$.

A few remarks are in order concerning this lemma.

• $\bullet$

In the setting of a general space of homogeneous type, this lemma has been proved by Christ [Chr], with the dyadic parameter $1/2$ replaced by some constant $\delta\in(0,1)$. In fact, one may always take $\delta=1/2$ (cf. [HMMM, Proof of Proposition 2.12]). In the presence of the Ahlfors regularity property, the result already appears in [DS1, DS2].

• $\bullet$

We shall denote by $\mathbb{D}(E)$ the collection of all relevant $Q_{j}^{k}$, i.e.,

[TABLE]

where, if $\operatorname{diam}(E)$ is finite, the union runs over those $k\in\mathbb{Z}$ such that $2^{-k}\lesssim\operatorname{diam}(E)$.

• $\bullet$

For a dyadic cube $Q\in\mathbb{D}_{k}$, we shall set $\ell(Q)=2^{-k}$, and we shall refer to this quantity as the “length” of $Q$. It is clear that $\ell(Q)\approx\operatorname{diam}(Q)$. Also, for $Q\in\mathbb{D}(E)$ we will set $k(Q)=k$ if $Q\in\mathbb{D}_{k}$.

• $\bullet$

Properties $(d)$ and $(e)$ imply that for each cube $Q\in\mathbb{D}(E)$, there is a point $x_{Q}\in E$, a Euclidean ball $B(x_{Q},r_{Q})$ and a surface ball $\Delta(x_{Q},r_{Q}):=B(x_{Q},r_{Q})\cap E$ such that $c\ell(Q)\leq r_{Q}\leq\ell(Q)$, for some uniform constant $c>0$, and

[TABLE]

for some uniform constant $C>1$. We shall denote these balls and surface balls by

[TABLE]

[TABLE]

and we shall refer to the point $x_{Q}$ as the “center” of $Q$.

• $\bullet$

Let $\Omega\subset\mathbb{R}^{n+1}$ be an open set satisfying the Corkscrew condition and such that $\partial\Omega$ is $\mathrm{AR}$. Given $Q\in\mathbb{D}(\partial\Omega)$ we define the “corkscrew point relative to $Q$” as $X_{Q}:=X_{\Delta_{Q}}$. We note that

[TABLE]

Following [HM3, Section 3] we next introduce the notion of “Carleson region” and “discretized sawtooth”. Given a cube $Q\in\mathbb{D}(E)$, the “discretized Carleson region” $\mathbb{D}_{Q}$ relative to $Q$ is defined by

[TABLE]

Let $\mathcal{F}=\{Q_{i}\}\subset\mathbb{D}(E)$ be a family of disjoint cubes. The “global discretized sawtooth” relative to $\mathcal{F}$ is the collection of cubes $Q\in\mathbb{D}(E)$ that are not contained in any $Q_{i}\in\mathcal{F}$, that is,

[TABLE]

For a given $Q\in\mathbb{D}(E)$, the “local discretized sawtooth” relative to $\mathcal{F}$ is the collection of cubes in $\mathbb{D}_{Q}$ that are not contained in any $Q_{i}\in\mathcal{F}$ or, equivalently,

[TABLE]

We also introduce the “geometric” Carleson regions and sawtooths. In the sequel, $\Omega\subset\mathbb{R}^{n+1}$ ($n\geq 2$) will be a 1-sided $\mathrm{CAD}$. Given $Q\in\mathbb{D}(\partial\Omega)$ we want to define some associated regions which inherit the good properties of $\Omega$. Let $\mathcal{W}=\mathcal{W}(\Omega)$ denote a collection of (closed) dyadic Whitney cubes of $\Omega\subset\mathbb{R}^{n+1}$, so that the cubes in $\mathcal{W}$ form a pairwise non-overlapping covering of $\Omega$, which satisfy

[TABLE]

and

[TABLE]

Let $X(I)$ denote the center of $I$, let $\ell(I)$ denote the sidelength of $I$, and write $k=k_{I}$ if $\ell(I)=2^{-k}$.

Given $0<\lambda<1$ and $I\in\mathcal{W}$ we write $I^{*}=(1+\lambda)I$ for the “fattening” of $I$. By taking $\lambda$ small enough, we can arrange matters, so that, first, $\operatorname{dist}(I^{*},J^{*})\approx\operatorname{dist}(I,J)$ for every $I,J\in\mathcal{W}$, and secondly, $I^{*}$ meets $J^{*}$ if and only if $\partial I$ meets $\partial J$ (the fattening thus ensures overlap of $I^{*}$ and $J^{*}$ for any pair $I,J\in\mathcal{W}$ whose boundaries touch, so that the Harnack Chain property then holds locally in $I^{*}\cup J^{*}$, with constants depending upon $\lambda$). By picking $\lambda$ sufficiently small, say $0<\lambda<\lambda_{0}$, we may also suppose that there is $\tau\in(1/2,1)$ such that for distinct $I,J\in\mathcal{W}$, we have that $\tau J\cap I^{*}=\mbox{{\O}}$. In what follows we will need to work with dilations $I^{**}=(1+2\lambda)I$ or $I^{***}=(1+4\lambda)I$, and in order to ensure that the same properties hold we further assume that $0<\lambda<\lambda_{0}/4$.

For every $Q\in\mathbb{D}(\partial\Omega)$ we can construct a family $\mathcal{W}_{Q}^{*}\subset\mathcal{W}$, and define

[TABLE]

satisfying the following properties: $X_{Q}\in U_{Q}$ (actually, $X_{Q}$ can be taken to be the center of some Whitney cube $I\in\mathcal{W}_{Q}^{*}$), and there are uniform constants $k^{*}$ and $K_{0}$ such that

[TABLE]

Here, $X(I)\rightarrow_{U_{Q}}X_{Q}$ means that the interior of $U_{Q}$ contains all balls in a Harnack Chain (in $\Omega$) connecting $X(I)$ to $X_{Q}$, and moreover, for any point $Z$ contained in any ball in the Harnack Chain, we have $\operatorname{dist}(Z,\partial\Omega)\approx\operatorname{dist}(Z,\Omega\setminus U_{Q})$ with uniform control of the implicit constants. The constants $k^{*},K_{0}$ and the implicit constants in the condition $X(I)\rightarrow_{U_{Q}}X_{Q}$, depend on at most allowable parameters and on $\lambda$. Moreover, given $I\in\mathcal{W}$ we have that $I\in\mathcal{W}_{Q_{I}}^{*}$, where $Q_{I}\in\mathbb{D}(\partial\Omega)$ satisfies $\ell(Q_{I})=\ell(I)$, and contains any fixed $\widehat{y}\in\partial\Omega$ such that $\operatorname{dist}(I,\partial\Omega)=\operatorname{dist}(I,\widehat{y})$. The reader is referred to [HM3] for full details.

For a given $Q\in\mathbb{D}(\partial\Omega)$, the “Carleson box” relative to $Q$ is defined by

[TABLE]

For a given family $\mathcal{F}=\{Q_{i}\}$ of pairwise disjoint cubes and a given $Q\in\mathbb{D}(\partial\Omega)$, we define the “local sawtooth region” relative to $\mathcal{F}$ by

[TABLE]

where $\mathcal{W}_{\mathcal{F},Q}:=\bigcup_{Q^{\prime}\in\mathbb{D}_{\mathcal{F},Q}}\mathcal{W}_{Q^{\prime}}^{*}$. Analogously, we can slightly fatten the Whitney boxes and use $I^{**}$ to define new fattened Whitney regions and sawtooth domains. More precisely, for every $Q\in\mathbb{D}(\partial\Omega)$,

[TABLE]

Similarly, we can define $T_{Q}^{**}$, $\Omega^{**}_{\mathcal{F},Q}$ and $U^{**}_{Q}$ by using $I^{***}$ in place of $I^{**}$.

Given a pairwise disjoint family $\mathcal{F}\subset\mathbb{D}$ (we also allow $\mathcal{F}$ to be the null set) and a constant $\rho>0$, we derive another family $\mathcal{F}(\rho)\subset\mathbb{D}$ from $\mathcal{F}$ as follows. Augment $\mathcal{F}$ by adding cubes $Q\in\mathbb{D}$ whose sidelength $\ell(Q)\leq\rho$ and let $\mathcal{F}(\rho)$ denote the corresponding collection of maximal cubes. Note that the corresponding discrete sawtooth region $\mathbb{D}_{\mathcal{F}(\rho)}$ is the union of all cubes $Q\in\mathbb{D}_{\mathcal{F}}$ such that $\ell(Q)>\rho$. For a given constant $\rho$ and a cube $Q\in\mathbb{D}$, let $\mathbb{D}_{\mathcal{F}(\rho),Q}$ denote the local discrete sawtooth region and let $\Omega_{\mathcal{F}(\rho),Q}$ denote the geometric sawtooth region relative to it.

Given $Q\in\mathbb{D}(\partial\Omega)$ and $0<\varepsilon<1$, if we take $\mathcal{F}_{0}=\mbox{{\O}}$, one has that $\mathcal{F}_{0}(\varepsilon\ell(Q))$ is the collection of $Q^{\prime}\in\mathbb{D}(\partial\Omega)$ such that $\varepsilon\ell(Q)/2<\ell(Q^{\prime})\leq\varepsilon\ell(Q)$, hence $\mathbb{D}_{\mathcal{F}_{0}(\varepsilon\ell(Q)),Q}=\{Q^{\prime}\in\mathbb{D}_{Q}:\,\ell(Q^{\prime})>\varepsilon\ell(Q)\}$. We then introduce $U_{Q,\varepsilon}=\Omega_{\mathcal{F}_{0}(\varepsilon\ell(Q)),Q}$ , which is a Whitney region relative to $Q$ whose distance to $\partial\Omega$ is of the order of $\varepsilon\ell(Q)$. For later use, we observe that given $Q_{0}\in\mathbb{D}(\partial\Omega)$, the sets $\{U_{Q,\varepsilon}\}_{Q\in\mathbb{D}_{Q_{0}}}$ have bounded overlap with constant that may depend on $\varepsilon$. Indeed, suppose that there is $X\in U_{Q,\varepsilon}\cap U_{Q^{\prime},\varepsilon}$ with $Q,Q^{\prime}\in\mathbb{D}_{Q_{0}}$. By construction $\ell(Q)\approx_{\varepsilon}\delta(X)\approx_{\varepsilon}\ell(Q^{\prime})$ and $\operatorname{dist}(Q,Q^{\prime})\leq\operatorname{dist}(X,Q)+\operatorname{dist}(X,Q^{\prime})\lesssim_{\varepsilon}\ell(Q)+\ell(Q^{\prime})\approx_{\varepsilon}\ell(Q)$. The bounded overlap property, with constants depending on $\varepsilon$, follows then at once.

Following [HM3], one can easily see that there exist constants $0<\kappa_{1}<1$ and $\kappa_{0}\geq\max\{2C,4/c\}$ (with $C$ the constant in (2.8), and $c$ such that $c\ell(Q)\leq r_{Q}$), depending only on the allowable parameters, so that

[TABLE]

where $B_{Q}$ is defined as in (2.7).

2.4. PDE estimates

Next, we recall several facts concerning the elliptic measures and the Green functions. For our first results we will only assume that $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, is an open set, not necessarily connected, with $\partial\Omega$ satisfying the $\mathrm{AR}$ property. Later we will focus on the case where $\Omega$ is a 1-sided $\mathrm{CAD}$.

Definition 2.12**.**

Let $Lu=-\operatorname{div}(A\nabla u)$ be a variable coefficient second order divergence form operator with $A(X)=(a_{i,j}(X))_{i,j=1}^{n+1}$ being a real (not necessarily symmetric) matrix with $a_{i,j}\in L^{\infty}(\Omega)$ for $1\leq i,j\leq n+1$, and $A$ uniformly elliptic, that is, there exists $\Lambda\geq 1$ such that

[TABLE]

for all $\xi,\zeta\in\mathbb{R}^{n+1}$ and almost every $X\in\Omega$.

In what follows we will only be working with this kind of operators, we will refer to them as “elliptic operators” for the sake of simplicity. We write $L^{\top}$ to denote the transpose of $L$, or, in other words, $L^{\top}u=-\operatorname{div}(A^{\top}\nabla u)$ with $A^{\top}$ being the transpose matrix of $A$.

We say that a function $u\in W^{1,2}_{\text{loc}}(\Omega)$ is a weak solution of $Lu=0$ in $\Omega$, or that $Lu=0$ in the weak sense, if

[TABLE]

Associated with $L$ and $L^{\top}$ one can respectively construct the elliptic measures $\{\omega_{L}^{X}\}_{X\in\Omega}$ and $\{\omega_{L^{\top}}^{X}\}_{X\in\Omega}$, and the Green functions $G_{L}$ and $G_{L^{\top}}$ (see [HMT1] for full details). We next present some definitions and properties that will be used throughout this paper.

Definition 2.13**.**

Let $\Omega\subset\mathbb{R}^{n+1}$ be a 1-sided $\mathrm{CAD}$ and let $L$ be a real (non-necessarily symmetric) elliptic operator. We say that the elliptic measure $\omega_{L}\in A_{\infty}(\partial\Omega)$ if there exist constants $0<\alpha,\beta<1$ such that given an arbitrary surface ball $\Delta_{0}=B_{0}\cap\partial\Omega$, with $B_{0}=B(x_{0},r_{0})$, $x_{0}\in\partial\Omega$, $0<r_{0}<\operatorname{diam}(\partial\Omega)$, and for every surface ball $\Delta=B\cap\partial\Omega$ centered at $\partial\Omega$ with $B\subset B_{0}$, and for every Borel set $F\subset\Delta$, we have that

[TABLE]

It is well known (see [GR], [CF]) that since $\sigma$ is a doubling measure (recall that $\partial\Omega$ satisfies the $\mathrm{AR}$ condition), $\omega_{L}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L}\ll\sigma$ in $\partial\Omega$ and there exists $1<q<\infty$ such that for every $\Delta_{0}$ and $\Delta$ as above

[TABLE]

where $k_{L}^{X_{\Delta_{0}}}=d\omega_{L}^{X_{\Delta_{0}}}/d\sigma$ is the Radon-Nikodym derivative. Moreover since $\Omega$ is a 1-sided $\mathrm{CAD}$ the latter is equivalent to the scale invariant estimate (see [HMT1])

[TABLE]

for every surface ball $\Delta_{0}$.

Lemma 2.16**.**

Suppose that $\Omega\subset\mathbb{R}^{n+1}$ is an open set such that $\partial\Omega$ satisfies the $\mathrm{AR}$ property. Let $L$ be an elliptic operator, there exist constants $c<1$ and $C>1$ (depending only on the $\mathrm{AR}$ constant and on the ellipticity of $L$) such that for every $x\in\partial\Omega$ and every $0<r<\operatorname{diam}(\partial\Omega)$, we have

[TABLE]

We refer the reader to [Bou, Lemma 1] for the proof in the harmonic case and to [HMT1] for general elliptic operators. See also [HKM, Theorem 6.18] and [Zha, Section 3].

The proofs of the following lemmas may be found in [HMT1]. We note that, in particular, the $\mathrm{AR}$ hypothesis implies that $\partial\Omega$ satisfies the Capacity Density Condition, hence $\partial\Omega$ is Wiener regular at every point (see [HLMN, Lemma 3.27]).

Lemma 2.17**.**

Suppose that $\Omega\subset\mathbb{R}^{n+1}$ is an open set such that $\partial\Omega$ satisfies the $\mathrm{AR}$ property. Given an elliptic operator $L$, there exist $C>1$ (depending only on dimension and on the ellipticity of $L$) and $c_{\theta}>0$ (depending on the above parameters and on $\theta\in(0,1)$) such that $G_{L}$, the Green function associated with $L$, satisfies

[TABLE]

Moreover, $G_{L}(\cdot,Y)\in W^{1,2}_{\rm loc}(\Omega\setminus\{Y\})$ for every $Y\in\Omega$, and satisfies $LG_{L}(\cdot,Y)=\delta_{Y}$ in the weak sense in $\Omega$, that is,

[TABLE]

Lemma 2.24**.**

Suppose that $\Omega\subset\mathbb{R}^{n+1}$ is a 1-sided $\mathrm{CAD}$. Let $L$ be an elliptic operator, there exist $C$, $0<\gamma\leq 1$ (depending only on dimension, the $1$-sided $\mathrm{CAD}$ constants and the ellipticity of $L$), such that for every $B_{0}=B(x_{0},r_{0})$ with $x_{0}\in\partial\Omega$, $0<r_{0}<\operatorname{diam}(\partial\Omega)$, and $\Delta_{0}=B_{0}\cap\partial\Omega$ we have the following properties:

• $(a)$

If $0\leq u\in W^{1,2}_{\rm loc}(B_{0}\cap\Omega)\cap C(\overline{B_{0}\cap\Omega})$ is a weak solution of $Lu=0$ in $B_{0}\cap\Omega$ such that $u\equiv 0$ in $\Delta_{0}$, then

[TABLE]

• $(b)$

If $B=B(x,r)$ with $x\in\partial\Omega$ and $\Delta=B\cap\partial\Omega$ is such that $2B\subset B_{0}$, then for all $X\in\Omega\setminus B_{0}$ we have that

[TABLE]

• $(c)$

If $X\in\Omega\setminus 4B_{0}$ then

[TABLE]

• $(d)$

If $B=B(x,r)$ with $x\in\partial\Omega$ and $\Delta:=B\cap\partial\Omega$ is such that $B\subset B_{0}$, then for every $X\in\Omega\setminus 2\kappa_{0}B_{0}$ with $\kappa_{0}$ as in (2.11), we have that

[TABLE]

Moreover, if $F\subset\Delta_{0}$ is a Borel set then

[TABLE]

3. Proof of Theorem 1.1

3.1. The Carleson measure condition implies $A_{\infty}$

To prove that : $(a)\Longrightarrow(b)$ we first introduce some notation.

Definition 3.1**.**

Let $E\subset\mathbb{R}^{n+1}$ be an $n$-dimensional $\mathrm{AR}$ set. Fix $Q_{0}\in\mathbb{D}(E)$ and let $\mu$ be a regular Borel measure on $Q_{0}$. Given $\varepsilon_{0}\in(0,1)$ and a Borel set $F\subset Q_{0}$, a good $\varepsilon_{0}$-cover of $F$ with respect to $\mu$, of length $k\in\mathbb{N}$, is a collection $\{\mathcal{O}_{\ell}\}_{\ell=1}^{k}$ of Borel subsets of $Q_{0}$, together with pairwise disjoint families $\mathcal{F}_{\ell}=\{Q_{i}^{\ell}\}\subset\mathbb{D}_{Q_{0}}$, such that

• $(a)$

$F\subset\mathcal{O}_{k}\subset\mathcal{O}_{k-1}\subset\cdots\subset\mathcal{O}_{2}\subset\mathcal{O}_{1}\subset Q_{0}$,

• $(b)$

$\mathcal{O}_{\ell}=\bigcup_{Q_{i}^{\ell}\in\mathcal{F}_{\ell}}Q_{i}^{\ell},\qquad 1\leq\ell\leq k$,

• $(c)$

$\mu(\mathcal{O}_{\ell}\cap Q_{i}^{\ell-1})\leq\varepsilon_{0}\,\mu(Q_{i}^{\ell-1}),\qquad\forall\,Q_{i}^{\ell-1}\in\mathcal{F}_{\ell-1},\quad 2\leq\ell\leq k$.

Lemma 3.2**.**

If $\{\mathcal{O}_{\ell}\}_{\ell=1}^{k}$ is a good $\varepsilon_{0}$-cover of $F$ with respect to $\mu$ of length $k\in\mathbb{N}$ then

[TABLE]

Proof.

Fix $1\leq\ell\leq k$ and we proceed by induction in $m$. If $m=\ell$ the estimate is trivial since $\mu(\mathcal{O}_{\ell}\cap Q_{i}^{\ell})=\mu(Q_{i}^{\ell})$. If $m=\ell-1$ (in which case necessarily $\ell\geq 2$) then (3.3) follows directly from $(c)$ in Definition 3.1. Assume next that (3.3) holds for some fixed $2\leq m\leq\ell$ and we prove it for $m-1$ in place of $m$. We first claim that for every $Q_{i}^{m-1}\in\mathcal{F}_{m-1}$ there holds

[TABLE]

To see this, take $x\in\mathcal{O}_{\ell}\cap Q_{i}^{m-1}\subset\mathcal{O}_{m}$. Hence, there exists a unique $Q_{j}^{m}\in\mathcal{F}_{m}$ such that $x\in Q_{j}^{m}$ and consequently either $Q_{i}^{m-1}\subset Q_{j}^{m}$ or $Q_{j}^{m}\subsetneq Q_{i}^{m-1}$. If $Q_{i}^{m-1}\subset Q_{j}^{m}$ then $\mu(Q_{i}^{m-1})=\mu(\mathcal{O}_{m}\cap Q_{i}^{m-1})\leq\varepsilon_{0}\mu(Q_{i}^{m-1})$, by $(c)$ in Definition 3.1, and this is a contradiction since $0<\varepsilon_{0}<1$. Thus, $Q_{j}^{m}\subsetneq Q_{i}^{m-1}$ and (3.4) holds. Therefore

[TABLE]

where we have applied the induction hypothesis to the $Q_{j}^{m}$’s and the properties of the good $\varepsilon_{0}$-cover. ∎

Lemma 3.5**.**

Let $E\subset\mathbb{R}^{n+1}$ be an $n$-dimensional $\mathrm{AR}$ set and fix $Q_{0}\in\mathbb{D}(E)$. Let $\mu$ be a regular Borel measure on $Q_{0}$ and assume that it is dyadically doubling on $Q_{0}$, that is, there exists $C_{\mu}\geq 1$ such that $\mu(Q^{*})\leq C_{\mu}\mu(Q)$ for every $Q\in\mathbb{D}_{Q_{0}}\setminus\{Q_{0}\}$, with $Q^{*}\supset Q$ and $\ell(Q^{*})=2\ell(Q)$ (i.e., $Q^{*}$ is the “dyadic parent” of $Q$). For every $0<\varepsilon_{0}\leq e^{-1}$, if $F\subset Q_{0}$ with $\mu(F)\leq\alpha\mu(Q_{0})$ and $0<\alpha\leq\varepsilon_{0}^{2}/(2C_{\mu}^{2})$ then $F$ has a good $\varepsilon_{0}$-cover with respect to $\mu$ of length $k_{0}=k_{0}(\alpha,\varepsilon_{0})\in\mathbb{N}$, $k_{0}\geq 2$, which satisfies $k_{0}\,\approx\,\tfrac{\log{\alpha^{-1}}}{\log{\varepsilon_{0}^{-1}}}.$ In particular, if $\mu(F)=0$, then $F$ has a good $\varepsilon_{0}$-cover of arbitrary length.

Proof.

Fix $\varepsilon_{0}$, $F$ and $\alpha$ as in the statement and write $a:=C_{\mu}/\varepsilon_{0}>1$. Note that since $0<\alpha<\varepsilon_{0}^{2}/(2C_{\mu}^{2})=a^{-2}/2$ there is a unique $k_{0}=k_{0}(\alpha,\varepsilon_{0})\in\mathbb{N}$, $k_{0}\geq 2$, such that

[TABLE]

and our choice of $\varepsilon_{0}$ gives that

[TABLE]

Since $\mu(F)\leq\alpha\mu(Q_{0})$, by outer regularity there exists a relatively open set $U\subset E$ such that $F\subset U$ and $\mu(U\setminus F)<\alpha\mu(Q_{0})$. Set $\widetilde{F}:=U\cap Q_{0}\subset Q_{0}$ and define the level sets

[TABLE]

where $M_{\mu,Q_{0}}^{d}$ is the local dyadic maximal operator with respect to $\mu$ given by

[TABLE]

Clearly, $\Omega_{1}\subset\Omega_{2}\subset\dots\subset\Omega_{k_{0}}\subset Q_{0}$. Moreover, $\widetilde{F}\subset\Omega_{1}$. To see this fix $x\in\widetilde{F}$ and use that $U$ is relatively open to find $B_{x}=B(x,r_{x})$ with $r_{x}>0$ so that $B_{x}\cap E\subset U$. Take next $Q_{x}\in\mathbb{D}$ with $Q_{x}\ni x$ so that $\ell(Q_{x})<\ell(Q_{0})$ and $\operatorname{diam}(Q_{x})<r_{x}$. Since $x\in\widetilde{F}\cap Q_{x}\subset Q_{x}\cap Q_{0}$ and $\ell(Q_{x})<\ell(Q_{0})$ it follows that $Q_{x}\in\mathbb{D}_{Q_{0}}$. Also since $\operatorname{diam}(Q_{x})<r_{x}$ we easily see that $Q_{x}\subset B_{x}\cap E\subset U$ and eventually we have obtained that $Q_{x}\subset\widetilde{F}$ which in turn gives

[TABLE]

Hence, $x\in\Omega_{1}$ as desired.

All the previous observations show that $F\subset\widetilde{F}\subset\Omega_{1}\subset\Omega_{2}\subset\dots\subset\Omega_{k_{0}}\subset Q_{0}$ and in particular $\Omega_{k}\neq\mbox{{\O}}$ for every $k\geq 1$. Moreover, by our choice of $k_{0}$, we have that for every $1\leq k\leq k_{0}$

[TABLE]

Subdividing $Q_{0}$ dyadically we can then select a pairwise disjoint collection of cubes $\mathcal{F}_{k}=\{Q_{i}^{k}\}\subset\mathbb{D}_{Q_{0}}\setminus\{Q_{0}\}$ which is maximal with respect to the property that

[TABLE]

and also $\Omega_{k}=\bigcup_{Q_{i}^{k}\in\mathcal{F}_{k}}Q_{i}^{k}$ (note that $\mathcal{F}_{k}\neq\mbox{{\O}}$ since $\Omega_{k}\neq\mbox{{\O}}$). By the maximality of $\mathcal{F}_{k}$ as well as the dyadic doubling property of $\mu$ we obtain that

[TABLE]

where $(Q_{i}^{k})^{*}$ is the dyadic parent of $Q_{i}^{k}$.

Next we claim that for each $Q_{j}^{k+1}\in\mathcal{F}_{k+1}$ we have that $\mu(\Omega_{k}\cap Q_{j}^{k+1})\leq\varepsilon_{0}\mu(Q_{j}^{k+1})$. To see this we first observe that if $Q_{i}^{k}\cap Q_{j}^{k+1}\neq\mbox{{\O}}$, then necessarily $Q_{i}^{k}\subset Q_{j}^{k+1}$, for otherwise $Q_{j}^{k+1}\subsetneq Q_{i}^{k}$ and by the maximality of $\mathcal{F}_{k+1}$ using (3.7) we would have that $a^{-k}\mu(Q_{i}^{k})<\mu(\widetilde{F}\cap Q_{i}^{k})\leq a^{-k-1}\mu(Q_{i}^{k})$, which leads to a contradiction since $a>1$. Hence, $Q_{i}^{k}\subset Q_{j}^{k+1}$ whenever $Q_{i}^{k}\cap Q_{j}^{k+1}\neq\mbox{{\O}}$. Using this, (3.7), and (3.8) (for $Q_{j}^{k+1}$ and $k+1$ replacing $Q_{i}^{k}$ and $k$ respectively), we have that

[TABLE]

and this proves the claim.

To complete the proof of the lemma we define $\mathcal{O}_{k}:=\Omega_{k_{0}-k+1}$ and note that the sets $\{\mathcal{O}_{k}\}_{k=1}^{k_{0}}$ form a good $\varepsilon_{0}$-cover of $F$, with respect to $\mu$, of length $k_{0}$ which satisfies (3.6). Finally we observe that if $\mu(F)=0$, then $\alpha$ can be taken arbitrarily small, hence $k_{0}$, the length of the good $\varepsilon_{0}$-cover of $F$, can be taken as large as desired by (3.6). ∎

Given $Q_{0}\in\mathbb{D}(\partial\Omega)$ and for every $\eta\in(0,1)$ we define the modified non-tangential cone

[TABLE]

As already noted in Section 2, the sets $\{U_{Q,\eta^{3}}\}_{Q\in\mathbb{D}_{Q_{0}}}$ have bounded overlap with constant depending on $\eta$.

Lemma 3.10**.**

There exist $0<\eta\ll 1$, depending only on dimension, the $1$-sided $\mathrm{CAD}$ constants and the ellipticity of $L$, and $\alpha_{0}\in(0,1)$, $C_{\eta}\geq 1$ both depending on the same parameters and additionally on $\eta$, such that for every $Q_{0}\in\mathbb{D}$, for every $0<\alpha<\alpha_{0}$, and for every Borel set $F\subset Q_{0}$ satisfying $\omega_{L}^{X_{Q_{0}}}(F)\leq\alpha\omega_{L}^{X_{Q_{0}}}(Q_{0})$, there exists a Borel set $S\subset Q_{0}$ such that the bounded weak solution $u(X)=\omega^{X}_{L}(S)$ satisfies

[TABLE]

Assuming this result momentarily, we can now prove Theorem 1.1.

Proof of Proof of Theorem 1.1: $(a)\Longrightarrow(b)$.

Our first goal is to see that given $\beta\in(0,1)$ there exists $\alpha\in(0,1)$ so that for every $Q_{0}\in\mathbb{D}$ and every Borel set $F\subset Q_{0}$, we have that

[TABLE]

Fix $\beta\in(0,1)$ and $Q_{0}\in\mathbb{D}$, and take a Borel set $F\subset Q_{0}$ so that ${\omega_{L}^{X_{Q_{0}}}(F)}\leq\alpha\omega_{L}^{X_{Q_{0}}}(Q_{0})$ where $\alpha\in(0,1)$ is to be chosen. Applying Lemma 3.10, if we assume that $0<\alpha<\alpha_{0}$, then $u(X)=\omega^{X}_{L}(S)$ satisfies (3.11) and therefore

[TABLE]

where we have used that $\Gamma_{Q_{0}}^{\eta}(x)\subset T_{Q_{0}}\subset B_{Q_{0}}^{*}\cap\Omega$ (see (2.11)), and Fubini’s theorem. To estimate the inner integral we fix $Y\in B_{Q_{0}}^{*}\cap\Omega$ and $\widehat{y}\in\mathbb{D}(\partial\Omega)$ such that $|Y-\widehat{y}|=\delta(Y)$. We claim that

[TABLE]

To show this let $x\in Q_{0}$ be such that $Y\in\Gamma_{Q_{0}}^{\eta}(x)$. Then there exists $Q\in\mathbb{D}_{Q_{0}}$ such that $x\in Q$ and $Y\in U_{Q,\eta^{3}}$. Hence, there is $Q^{\prime}\in\mathbb{D}_{Q}$ with $\ell(Q^{\prime})>\eta^{3}\ell(Q)$ such that $Y\in U_{Q^{\prime}}$ and consequently $\delta(Y)\approx\operatorname{dist}(Y,Q^{\prime})\approx\ell(Q^{\prime})$. Then,

[TABLE]

thus $x\in\Delta(\widehat{y},C\eta^{-3}\delta(Y))$ as desired. If we now use (3.14) and the $\mathrm{AR}$ property we conclude that for every $Y\in B_{Q_{0}}^{*}\cap\Omega$

[TABLE]

Plugging this into (3.13) and using (1.2), since $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ with $Lu=0$ in the weak sense in $\Omega$, we obtain

[TABLE]

where we have used that $\Delta_{Q_{0}}^{*}=B_{Q_{0}}^{*}\cap\partial\Omega$, that $0\leq u(X)\leq\omega^{X}(\partial\Omega)\leq 1$ and that $\partial\Omega$ is AR. Rearranging the terms we see that $\sigma(F)/\sigma(Q_{0})\leq\beta$ provided $0<\alpha<\min\{\alpha_{0},e^{-CC_{\eta}^{2}\eta^{-3n}\beta^{-1}}\}$ and (3.12) follows.

Next we see that (3.12) implies that $\omega_{L}\in A_{\infty}(\partial\Omega)$. To see this we first obtain a dyadic-$A_{\infty}$ condition. Fix $Q^{0},Q_{0}\in\mathbb{D}$ with $Q_{0}\subset Q^{0}$. Lemma 2.24 parts $(c)$ and $(d)$, Harnack’s inequality and Lemma 2.16 gives for every $F\subset Q_{0}$

[TABLE]

With all these in hand we fix $\beta\in(0,1)$ and take the corresponding $\alpha\in(0,1)$ so that (3.12) holds. We are going to see that

[TABLE]

Assuming that the first estimate holds we see that (3.15) yields $\frac{\omega_{L}^{X_{Q_{0}}}(F)}{\omega_{L}^{X_{Q_{0}}}(Q_{0})}\leq\alpha$. Thus we can apply (3.12) to obtain that $\frac{\sigma(F)}{\sigma(Q_{0})}\leq\beta$ as desired.

To complete the proof we need to see that (3.16) gives (2.14). We show its contrapositive. Fix $\beta\in(0,1)$ and a surface ball $\Delta_{0}=B_{0}\cap\partial\Omega$, with $B_{0}=B(x_{0},r_{0})$, $x_{0}\in\partial\Omega$, and $0<r_{0}<\operatorname{diam}(\partial\Omega)$. Take an arbitrary surface ball $\Delta=B\cap\partial\Omega$ centered at $\partial\Omega$ with $B=B(x,r)\subset B_{0}$, and let $F\subset\Delta$ be a Borel set such that ${\sigma(F)}>\beta{\sigma(\Delta)}$. Consider the pairwise disjoint family $\mathcal{F}=\{Q\in\mathbb{D}:Q\cap\Delta\neq\mbox{{\O}},\frac{r}{4\,C}<\ell(Q)\leq\frac{r}{2\,C}\}$ where $C$ is the constant in (2.6). In particular, $\Delta\subset\cup_{\mathcal{F}}Q\subset 2\Delta$. The pigeon-hole principle yields that there is a constant $C^{\prime}>1$ depending just on the Ahlfors regularity constant of $\sigma$ so that $\frac{\sigma(F\cap Q_{0})}{\sigma(Q_{0})}>\frac{\beta}{C^{\prime}}$ for some $Q_{0}\in\mathcal{F}$. Let $Q^{0}\in\mathbb{D}$ be the unique dyadic cube such that $Q_{0}\subset Q^{0}$ and $\frac{r_{0}}{2}<\ell(Q^{0})\leq r_{0}$. We can then invoke (3.16) with $\frac{\beta}{C^{\prime}}$ to find $\alpha\in(0,1)$ such that by Lemma 2.24, and Harnack’s inequality

[TABLE]

In short, we have obtained that for every $\beta\in(0,1)$ there exists $\widetilde{\alpha}\in(0,1)$ such that

[TABLE]

which is the contrapositive of (2.14). This completes the proof of Theorem 1.1 modulo the proof of Lemma 3.10. ∎

Before proving Lemma 3.10 we need some notation and some estimates. Let $\eta=2^{-k_{*}}<1$.

[TABLE]

Using this notation we have the following estimates which will be used later:

[TABLE]

where $C$ depends on dimension, the $1$-sided $\mathrm{CAD}$ constants and the ellipticity of $L$ and $\gamma$ is the parameter in Lemma 2.24. To see this, keeping in mind the notation introduced in (2.6), let $\varphi(X)=\varphi_{0}((X-x_{Q})/r_{Q})$ where $\varphi_{0}\in C_{c}(\mathbb{R}^{n+1})$ with $\mathbf{1}_{B(0,1)}\leq\varphi_{0}\leq\mathbf{1}_{B(0,2)}$. Note that $\varphi\in C_{c}(\mathbb{R}^{n+1})$ with $0\leq\varphi\leq 1$, $\operatorname{supp}(\varphi)\subset 2B_{Q}$, and $\varphi\equiv 1$ in $B_{Q}$. In particular, $\varphi\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}\leq\mathbf{1}_{2\Delta_{Q}}\leq\mathbf{1}_{Q}$ and hence

[TABLE]

Note that $v\in W^{1,2}_{\rm loc}(\Omega)\cap C(\overline{\Omega})$ is a weak solution with $0\leq v\leq 1$ and $v\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}=\varphi\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}\equiv 1$ in $B_{Q}$. Thus, $\widetilde{v}=1-v\in W^{1,2}_{\rm loc}(\Omega)\cap C(\overline{\Omega})$ is a weak solution with $0\leq\widetilde{v}\leq 1$ and $\widetilde{v}\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}=1-\varphi\,\rule[-6.0pt]{0.38pt}{12.0pt}_{\,\partial\Omega}\equiv 0$ in $B_{Q}$. Thus we can use (3.19) and part $(a)$ in Lemma 2.24 to see that

[TABLE]

where the last estimate follows from

[TABLE]

since $x_{Q}\in\widetilde{Q}$ and $X_{\widetilde{Q}}$ is a corkscrew point relative to $\widetilde{Q}$.

We also claim that there exists $c_{0}\in(0,1)$ depending only on the $\mathrm{AR}$ constant and on the ellipticity of $L$ so that if $\eta$ is small enough (depending only on $n$ and the AR constant) then

[TABLE]

The first inequality follows at once from Lemma 2.16 and Harnack’s inequality. For the second one we claim that if $\eta$ is small enough we can find $\widetilde{Q}^{\prime}\in\mathbb{D}$ with $\ell(\widetilde{Q}^{\prime})=\ell(\widetilde{Q})$, $\widetilde{Q}^{\prime}\cap\widetilde{Q}=\mbox{{\O}}$ and $\operatorname{dist}(\widetilde{Q},\widetilde{Q}^{\prime})\lesssim\ell(\widetilde{Q})$. Indeed, if we write $\widetilde{Q}^{j}$ for the $j$-th ancestor of $\widetilde{Q}$ (that is, the unique cube satisfying $\ell(\widetilde{Q}^{j})=2^{j}\ell(\widetilde{Q})$ and $\widetilde{Q}\subset\widetilde{Q}^{j}$) then $\sigma(\widetilde{Q}^{j})\gtrsim\ell(\widetilde{Q}^{j})^{n}=2^{jn}\ell(\widetilde{Q})^{n}>\sigma(\widetilde{Q})$ for $j$ large enough depending on the $\mathrm{AR}$ constant. Note that in the previous estimates we are implicitly using that $\ell(\widetilde{Q})\lesssim\operatorname{diam}(\partial\Omega)$, fact that follows by choosing $\eta$ small enough depending on the $\mathrm{AR}$ constant. Once $j$ has been chosen we must have $\widetilde{Q}\subsetneq\widetilde{Q}^{j}$, and we can easily pick $\widetilde{Q}^{\prime}\in\mathbb{D}_{\widetilde{Q}^{j}}$ with all the desired properties. In turn by Harnack’s inequality and Lemma 2.16 one can see that $\omega^{X_{\widetilde{Q}}}(\widetilde{Q}^{\prime})\gtrsim\omega^{X_{\widetilde{Q}^{\prime}}}(\widetilde{Q}^{\prime})\geq C^{-1}$ with $C>1$ and consequently

[TABLE]

which is the desired estimate.

Proof of Lemma 3.10.

Let $\eta=2^{-k_{*}}<1$ be a small dyadic number to be chosen and such that (3.18) and (3.21) hold. Fix $Q_{0}\in\mathbb{D}$ and note that $\omega:=\omega_{L}^{X_{Q_{0}}}$ is a regular Borel measure on $\partial\Omega$ which is dyadically doubling with constants $C_{0}$ (depending only on dimension, the $1$-sided $\mathrm{CAD}$ constants and the ellipticity of $L$) by part $(c)$ of Lemma 2.24 and Harnack’s inequality. Let $0<\varepsilon_{0}<e^{-1}$ and $0<\alpha<\varepsilon_{0}^{2}/(2C_{0}^{2})$, sufficiently small to be chosen later, and let $F\subset Q_{0}$ be a Borel set such that $\omega(F)\leq\alpha\omega(Q_{0})$. By Lemma 3.5 applied to $\mu=\omega$, it follows that $F$ has a good $\varepsilon_{0}$-cover of length $k\approx\tfrac{\log{\alpha^{-1}}}{\log{\varepsilon_{0}^{-1}}}$, with $k\geq 2$. Let $\{\mathcal{O}_{\ell}\}_{\ell=1}^{k}$ be the corresponding collection of Borel sets so that $F\subset\mathcal{O}_{k}\subset\cdots\subset\mathcal{O}_{1}\subset Q_{0}$ and $\mathcal{O}_{\ell}=\bigcup_{Q_{i}^{\ell}\in\mathcal{F}_{\ell}}Q_{i}^{\ell}$, with disjoint families $\mathcal{F}_{\ell}=\{Q_{i}^{\ell}\}\subset\mathbb{D}_{Q_{0}}\setminus\{Q_{0}\}$. Now, using the notation above (see (3.1)) we define $\widetilde{\mathcal{O}}_{\ell}:=\bigcup_{Q_{i}^{\ell}\in\mathcal{F}_{\ell}}\widetilde{Q}_{i}^{\ell}$ and consider the Borel set S:=\bigcup_{j=2}^{k}\big{(}\widetilde{\mathcal{O}}_{j-1}\setminus\mathcal{O}_{j}\big{)}. Note that the union of sets comprising $S$ is disjoint, hence

[TABLE]

Now we introduce some notation. For each $y\in F$ and $1\leq\ell\leq k$, there exists a unique $Q_{i}^{\ell}(y)\in\mathcal{F}_{\ell}$ such that $y\in Q_{i}^{\ell}(y)$. Let $P_{i}^{\ell}(y)\in\mathbb{D}_{Q_{i}^{\ell}(y)}$ be the unique cube verifying $y\in P_{i}^{\ell}(y)$ and $\ell(P_{i}^{\ell}(y))=\eta\ell(Q_{i}^{\ell}(y))$. Associated with $P_{i}^{\ell}(y)$ we can construct $\widetilde{P}_{i}^{\ell}(y)$ as above (see (3.1)), that is, $\widetilde{P}_{i}^{\ell}(y)\in\mathbb{D}_{P_{i}^{\ell}(y)}$ satisfies $\ell(\widetilde{P}_{i}^{\ell}(y))=\eta\ell(P_{i}^{\ell}(y))$ and $x_{P_{i}^{\ell}(y)}\in\widetilde{P}_{i}^{\ell}(y)$, where $x_{P_{i}^{\ell}(y)}$ is the center of $P_{i}^{\ell}(y)$. As usual we write $X_{\widetilde{Q}_{i}^{\ell}(y)}$ and $X_{\widetilde{P}_{i}^{\ell}(y)}$ to denote, respectively, the corkscrew points associated to $\widetilde{Q}_{i}^{\ell}(y)$ and $\widetilde{P}_{i}^{\ell}(y)$.

Let $u(X):=\omega^{X}_{L}(S)$ then

[TABLE]

The following lemma contains a lower bound for the oscillation of $u$. Here $\eta$ is as in (3.1) and $F$ which was used to construct $S$ (as above) has a good $\varepsilon_{0}$-cover.

Lemma 3.24**.**

If $\eta$ and $\varepsilon_{0}$ are taken sufficiently small (depending only on $n$, the $1$-sided $\mathrm{CAD}$ constants and the ellipticity of $L$), then for each $y\in F$, and each $1\leq\ell\leq k-1$, we have that

[TABLE]

where $c_{0}$ is the constant in (3.21)

Assume this result for now and continue the proof of Lemma 3.10. Fix $\eta$ and $\varepsilon_{0}$ as in Lemma 3.24. Fix also $y\in F$, $1\leq\ell\leq k-1$, and write $Q_{i}^{\ell}:=Q_{i}^{\ell}(y)\in\mathbb{D}_{Q_{0}}$, and $P_{i}^{\ell}:=P_{i}^{\ell}(y)\in\mathbb{D}_{Q_{i}^{\ell}}$ using the notation above. By construction $X_{\widetilde{Q}_{i}^{\ell}}\in U_{\widetilde{Q}_{i}^{\ell}}$ and $X_{\widetilde{P}_{i}^{\ell}}\in U_{\widetilde{P}_{i}^{\ell}}$, hence we can find Whitney cubes $I_{\widetilde{Q}_{i}^{\ell}}\in\mathcal{W}_{\widetilde{Q}_{i}^{\ell}}^{*}$ and $I_{\widetilde{P}_{i}^{\ell}}\in\mathcal{W}_{\widetilde{P}_{i}^{\ell}}^{*}$ so that $X_{\widetilde{Q}_{i}^{\ell}}\in I_{\widetilde{Q}_{i}^{\ell}}$ and $X_{\widetilde{P}_{i}^{\ell}}\in I_{\widetilde{P}_{i}^{\ell}}$.

Also, note that $\ell(\widetilde{Q}_{i}^{\ell})=\eta\ell(Q_{i}^{\ell})$ and $\ell(\widetilde{P}_{i}^{\ell})=\eta^{2}\ell(Q_{i}^{\ell})$ which imply $\ell(\widetilde{Q}_{i}^{\ell})>\ell(\widetilde{P}_{i}^{\ell})>\eta^{3}\ell(Q_{i}^{\ell})$ since $\eta<1$. On the other hand, $\widetilde{Q}_{i}^{\ell}\subset Q_{i}^{\ell}$ and $\widetilde{P}_{i}^{\ell}\subset P_{i}^{\ell}\subset Q_{i}^{\ell}$, which in turn yield that $I_{\widetilde{Q}_{i}^{\ell}}^{*}$ and $I_{\widetilde{P}_{i}^{\ell}}^{*}$ are both contained in $U_{Q_{i}^{\ell},\eta^{3}}$. Using (3.25), the notation [u]_{U_{Q_{i}^{\ell},\eta^{3}}}:=\mathchoice{{\vbox{\hbox{\textstyle- }}\kern-7.63756pt}}{{\vbox{\hbox{\scriptstyle- }}\kern-5.75256pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.63129pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.14378pt}}\!\int\mkern-11.5mu\mathchoice{{\vbox{\hbox{\textstyle- }}\kern-7.63756pt}}{{\vbox{\hbox{\scriptstyle- }}\kern-5.75256pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.63129pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.14378pt}}\!\int_{U_{Q_{i}^{\ell},\eta^{3}}}udX, Moser’s “local boundedness” estimates and the previous observations we can obtain

[TABLE]

where the last estimate follows from the Poincaré’s inequality in [HMT2, Lemma 3.1], and the fact that $\delta(Y)\approx_{\eta}\ell(Q_{i}^{\ell})$ for every $Y\in U_{Q_{i}^{\ell},\eta^{3}}$. Summing up the above estimate, taking into account that the sets $\{U_{Q,\eta^{3}}\}_{Q\in\mathbb{D}_{Q_{0}}}$ have bounded overlap with constant depending on $\eta$, and using Lemma 3.5, we obtain if $\alpha$ is small enough

[TABLE]

This completes the proof of Lemma 3.10. ∎

Proof of Lemma 3.24.

Fix $y\in F$ and write $Q_{i}^{\ell}:=Q_{i}^{\ell}(y)$, $P_{i}^{\ell}:=P_{i}^{\ell}(y)$. Our first goal is to estimate $u(X_{\widetilde{Q}_{i}^{\ell}})$. By (3.18) and using (3.23) we have

[TABLE]

For $1\leq\ell\leq k-1$ we have that $Q_{i}^{\ell}\subset\mathcal{O}_{\ell}\subset\mathcal{O}_{j}$ for each $2\leq j\leq\ell$ and hence by (3.22) we have

[TABLE]

with the understanding that if $\ell=k-1$ then $\mathrm{I}_{1}=0$.

Next, we claim that $\mathrm{I}_{1}\leq C_{\eta}\varepsilon_{0}$. This is clear if $\ell=k-1$. For $1\leq\ell\leq k-2$, using Harnack’s inequality to move from $X_{\widetilde{Q}_{i}^{\ell}}$ to $X_{Q_{i}^{\ell}}$ (with constants depending on $\eta$), Lemma 2.24 parts $(c)$ and $(d)$ (recall that $\omega=\omega_{L}^{X_{Q_{0}}}$), we have that

[TABLE]

where the next-to-last estimate follows from Lemma 3.2 with $\mu=\omega$, and the last one uses that $\varepsilon_{0}<e^{-1}$. Let us now focus on $\mathrm{I}_{2}$. Note that $Q_{i}^{\ell}\cap\widetilde{\mathcal{O}}_{\ell}=\widetilde{Q}_{i}^{\ell}$, hence (3.21) yields

[TABLE]

Collecting this with (3.26), (3.27), (3.28), we conclude that

[TABLE]

by choosing first $\eta$ small enough so that $C\eta^{\gamma}<c_{0}/8$ and then $\varepsilon_{0}$ small enough so that $C_{\eta}\varepsilon_{0}<c_{0}/8$.

To get a lower bound for $u(X_{\widetilde{Q}_{i}^{\ell}})$ we use that $Q_{i}^{\ell}\cap\widetilde{\mathcal{O}}_{\ell}=\widetilde{Q}_{i}^{\ell}$ and (3.21):

[TABLE]

Using Harnack’s inequality to move from $X_{\widetilde{Q}_{i}^{\ell}}$ to $X_{Q_{i}^{\ell}}$ (with constants depending on $\eta$), Lemma 2.24 parts $(c)$ and $(d)$ (recall that $\omega=\omega_{L}^{X_{Q_{0}}}$), we have that

[TABLE]

where the last estimate follows from Lemma 3.2 with $\mu=\omega$ and since $1\leq\ell\leq k-1$. Assuming further that $C_{\eta}\varepsilon_{0}<c_{0}/4$ we arrive at

[TABLE]

Let us now focus on estimating $u(X_{\widetilde{P}_{i}^{\ell}})$ and we consider two cases:

Case 1: $P_{i}^{\ell}\cap\widetilde{Q}_{i}^{\ell}=\mbox{{\O}}$. Much as before by (3.18)

[TABLE]

For $1\leq\ell\leq k-1$ we have that $P_{i}^{\ell}\subset Q_{i}^{\ell}\subset\mathcal{O}_{\ell}\subset\mathcal{O}_{j}$ for each $2\leq j\leq\ell$ and hence

[TABLE]

with the understanding that if $\ell=k-1$ then $\widehat{\mathrm{I}}_{1}=0$. The estimate for $\widehat{\mathrm{I}}_{1}$ (when $\ell\leq k-2$) follows from that of $\mathrm{I}_{1}$ since using Harnack’s inequality to move from $X_{\widetilde{P}_{i}^{\ell}}$ to $X_{\widetilde{Q}_{i}^{\ell}}$ and the fact that $P_{i}^{\ell}\subset Q_{i}^{\ell}$ we easily obtain from (3.28)

[TABLE]

On the other hand, note that $P_{i}^{\ell}\cap(\widetilde{\mathcal{O}}_{\ell}\setminus\mathcal{O}_{\ell+1})=(P_{i}^{\ell}\cap\widetilde{Q}_{i}^{\ell})\setminus\mathcal{O}_{\ell+1}\subset P_{i}^{\ell}\cap\widetilde{Q}_{i}^{\ell}=\mbox{{\O}}$ and hence $\widehat{\mathrm{I}}_{2}=0$. Thus (3.32), (3.33), and (3.34) yield

[TABLE]

by choosing first $\eta$ small enough so that $C\eta^{\gamma}<c_{0}/8$ and then $\varepsilon_{0}$ small enough so that $C_{\eta}\varepsilon_{0}<c_{0}/8$. This estimate along with (3.31) give at once

[TABLE]

which is the desired estimate.

Case 2: $P_{i}^{\ell}\cap\widetilde{Q}_{i}^{\ell}\neq\mbox{{\O}}$. Notice that since both cubes have the same sidelength it follows that $P_{i}^{\ell}=\widetilde{Q}_{i}^{\ell}$. Our goal is to get a lower bound for $u(X_{\widetilde{P}_{i}^{\ell}})$. We use that $P_{i}^{\ell}\cap\widetilde{\mathcal{O}}_{\ell}=\widetilde{Q}_{i}^{\ell}\cap\widetilde{\mathcal{O}}_{\ell}=\widetilde{Q}_{i}^{\ell}=P_{i}^{\ell}$ and (3.18):

[TABLE]

Moreover, using Harnack’s inequality to move from $X_{\widetilde{P}_{i}^{\ell}}$ to $X_{\widetilde{Q}_{i}^{\ell}}$ (with constants depending on $\eta$) and (3.30) we observe that

[TABLE]

Collecting the obtained estimates we conclude that

[TABLE]

if we choose first $\eta$ small enough so that $C\eta^{\gamma}<c_{0}/8$ and then $\varepsilon_{0}$ small enough so that $C_{\eta}\varepsilon_{0}<c_{0}/8$. If we now gather (3.29) and (3.36) we eventually obtain the desired estimate

[TABLE]

This completes the proof. ∎

3.2. $A_{\infty}$ implies the Carleson measure condition

The proof of Theorem 1.1: $(b)\Longrightarrow(a)$ requires some additional notation and several auxiliary results.

Let $Q_{0}\in\mathbb{D}$ and $\alpha=\{\alpha_{Q}\}_{Q\in\mathbb{D}_{Q_{0}}}$ be a sequence of non-negative numbers indexed by the dyadic cubes in $\mathbb{D}_{Q_{0}}$. For any collection $\mathbb{D}^{\prime}\subset\mathbb{D}_{Q_{0}}$, we define the associated discrete “measure”

[TABLE]

We say that $\mathfrak{m}_{\alpha}$ is a discrete “Carleson measure” (with respect to $\sigma$) in $Q_{0}$, if

[TABLE]

The following result reduces the desired Carleson measure estimate to a discrete one:

Lemma 3.39**.**

Let $\Omega\subset\mathbb{R}^{n+1}$ be a 1-sided $\mathrm{CAD}$ and let $Lu=-\operatorname{div}(A\nabla u)$ be a real (not necessarily symmetric) elliptic operator. Let $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ satisfy $Lu=0$ in the weak sense in $\Omega$ and define

[TABLE]

Suppose that there exist $C_{0},M_{0}\geq 1$ such that $\|\mathfrak{m}_{\alpha}\|_{\mathcal{C}(Q)}\leq C_{0}\|u\|_{L^{\infty}(\Omega)}^{2}$ for every $Q\in\mathbb{D}(\partial\Omega)$ verifying $\ell(Q)<\operatorname{diam}(\partial\Omega)/M_{0}$. Then,

[TABLE]

where $C$ depends only on dimension, the 1-sided $\mathrm{CAD}$ constants, and the ellipticity of $L$.

Proof.

By homogeneity we may assume that $\|u\|_{L^{\infty}(\Omega)}=1$. First, we claim that

[TABLE]

Given $Q_{0}\in\mathbb{D}(\partial\Omega)$ such that $\ell(Q_{0})<\operatorname{diam}(\partial\Omega)/M_{0}$, we have that

[TABLE]

Otherwise, if $\ell(Q_{0})\geq\operatorname{diam}(\partial\Omega)/M_{0}$ (this happens only if $\operatorname{diam}(\partial\Omega)<\infty)$, there exists a unique $k_{0}\geq 1$ so that

[TABLE]

As observed before if $\operatorname{diam}(\partial\Omega)<\infty$ then $\ell(Q_{0})\lesssim\operatorname{diam}(\partial\Omega)$ hence $2^{k_{0}}\lesssim M_{0}$. Define the disjoint collection \mathcal{D}_{0}:=\big{\{}Q^{\prime}\in\mathbb{D}_{Q_{0}}:\>\ell(Q^{\prime})=2^{-k_{0}}\ell(Q_{0})\big{\}} and let

[TABLE]

Note that

[TABLE]

Note that if $Q\in\mathbb{D}_{Q_{0}}^{\rm small}$, there exists a unique $Q^{\prime}\in\mathcal{D}_{0}$ such that $Q\in\mathbb{D}_{Q^{\prime}}$, hence

[TABLE]

where we have used our hypothesis since $\ell(Q^{\prime})=2^{-k_{0}}\ell(Q_{0})<\operatorname{diam}(\partial\Omega)/M_{0}$. For the second term, since $\delta(X)\approx\ell(Q)$ for $X\in U_{Q}$, we write

[TABLE]

where we have used Caccioppoli’s inequality, the fact that the family $\{U_{Q}^{*}\}_{Q\in\mathbb{D}}$ has bounded overlap, the normalization $\|u\|_{L^{\infty}(\Omega)}=1$, (2.11), the $\mathrm{AR}$ property, and that $2^{k_{0}}\lesssim M_{0}$. Combining (3.43), (3.44), and (3.45) we have that (3.42) holds.

Our next goal is to see that (3.42) yields (3.41). For $x\in\partial\Omega$ and $0<r<\infty$. Set

[TABLE]

Given $I\in\mathcal{I}$, let $Z_{I}\in I\cap B(x,r)$ and note that by (2.9)

[TABLE]

Set

[TABLE]

with the understanding that $\mathcal{I}^{\rm big}=\mbox{{\O}}$ if $\operatorname{diam}(\partial\Omega)=\infty$. Then,

[TABLE]

here we understand that $\mathrm{II}=0$ if $\mathcal{I}^{\rm big}=\mbox{{\O}}$.

To estimate $\mathrm{I}$ we set $r_{0}=\min\{r,\operatorname{diam}(\partial\Omega)/4\}$ and pick $k_{2}\in\mathbb{Z}$ so that $2^{k_{2}-1}\leq r_{0}<2^{k_{2}}$. Set

[TABLE]

Given $I\in\mathcal{I}^{\rm small}$ we pick $y\in\partial\Omega$ so that $\operatorname{dist}(I,\partial\Omega)=\operatorname{dist}(I,y)$. Hence there exists a unique $Q_{I}\in\mathbb{D}$ so that $y\in Q_{I}$ and $\ell(Q_{I})=\ell(I)<r_{0}\leq\operatorname{diam}(\partial\Omega)/4$ by the definition of $\mathcal{I}^{\rm small}$ and our choice of $r_{0}$. This as mentioned above implies that $I\in\mathcal{W}_{Q_{I}}^{*}$. On the other hand by (3.46)

[TABLE]

hence there exists a unique $Q\in\mathcal{D}_{1}$ so that $y\in Q$. Since $\ell(Q_{I})<r_{0}<2^{k_{2}}=\ell(Q)$ we conclude that $Q_{I}\subset Q$ and consequently $I\subset\operatorname{int}(U_{Q_{I}})\subset T_{Q}$. In short we have shown that if $I\in\mathcal{I}^{\rm small}$ then there exists $Q\in\mathcal{D}_{1}$ so that $I\subset T_{Q}$. Thus,

[TABLE]

where we have used that the Whitney boxes have non-overlapping interiors, (3.42), the fact that $\mathcal{D}_{1}$ is a pairwise disjoint family, that $\bigcup_{Q\in\mathcal{D}_{1}}Q\subset\Delta(x,Cr)$ ($C$ depends on $n$ and the AR constant), and that $\partial\Omega$ is Ahlfors regular.

We now estimate $\mathrm{II}$ using (2.9), Caccioppoli’s inequality and our assumption $\|u\|_{L^{\infty}(\Omega)}=1$:

[TABLE]

To estimate the last term we observe that if $Y\in I\in\mathcal{I}^{\rm big}$ we have by (2.9)

[TABLE]

This and the fact that Whitney boxes have non-overlapping interiors imply

[TABLE]

Therefore,

[TABLE]

Collecting the estimates for $\mathrm{I}$ (3.48) and $\mathrm{II}$ (3.49) we obtain (3.41). ∎

Proof of Theorem 1.1: $(b)\Longrightarrow(a)$.

Let $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ be so that $Lu=0$ in the weak sense in $\Omega$. Our goal is to prove that (1.2) holds. By homogeneity we may assume, without loss of generality, that $\|u\|_{L^{\infty}(\Omega)}=1$. On the other hand, by Lemma 3.39 we can reduce matters to establish that $\|\mathfrak{m}_{\alpha}\|_{\mathcal{C}(Q)}\leq C_{0}$, for every $Q\in\mathbb{D}(\partial\Omega)$ such that $\ell(Q)<\operatorname{diam}(\partial\Omega)/M_{0}$ and where $\alpha$ is given in (3.40). To show this we fix $M_{0}>2\kappa_{0}/c$, where $c$ is the corkscrew constant and $\kappa_{0}$ as in (2.11). We also fix a cube $Q^{0}\in\mathbb{D}(\partial\Omega)$ with $\ell(Q^{0})<\operatorname{diam}(\partial\Omega)/M_{0}$. Applying [HMT2, Lemma 3.12] it suffices to show that for every $Q_{0}\in\mathbb{D}_{Q^{0}}$ we can find some pairwise disjoint family $\mathcal{F}_{Q_{0}}\subset\mathbb{D}_{Q_{0}}\setminus\{Q_{0}\}$ satisfying

[TABLE]

and prove that

[TABLE]

With all the previous reductions our main goal is to find $\mathcal{F}_{Q_{0}}$ so that (3.50) holds and establish (3.51). Having these in mind we let $B_{Q_{0}}:=B(x_{Q_{0}},r_{Q_{0}})$ with $r_{Q_{0}}\approx\ell(Q_{0})$ as in (2.6). Let $X_{0}:=X_{M_{0}\Delta_{Q_{0}}}$ be the corkscrew point relative to $M_{0}\Delta_{Q_{0}}$ (note that $M_{0}r_{Q_{0}}\leq M_{0}\ell(Q_{0})<\operatorname{diam}(\partial\Omega)$). By our choice of $M_{0}$, it is clear that $Q_{0}\subset M_{0}\Delta_{Q_{0}}$ and also that $\delta(X_{0})\geq cM_{0}r_{Q_{0}}>2\kappa_{0}r_{Q_{0}}$. Hence, by (2.11),

[TABLE]

On the other hand, $\delta(X_{Q_{0}})\approx\ell(Q_{0})$, $\delta(X_{0})\approx M_{0}\ell(Q_{0})\geq\ell(Q_{0})$, and $|X_{0}-X_{Q_{0}}|\lesssim M_{0}\ell(Q_{0})$. Using Lemma 2.16 and Harnack’s inequality, there exists $C_{0}\geq 1$ depending on the 1-sided $\mathrm{CAD}$ constants, the ellipticity of $L$, and on $M_{0}$ (which is already fixed), such that $\omega_{L}^{X_{0}}(Q_{0})\geq C_{0}^{-1}$.

Next, we define the normalized elliptic measure and Green function as

[TABLE]

Note the fact that $\omega_{L}^{X_{0}}(\partial\Omega)\leq 1$ implies

[TABLE]

Recall that we have assumed that $\omega_{L}\in A_{\infty}(\partial\Omega)$ and, as observed above, this means after passing to the previous renormalization that $\omega_{0}\ll\sigma$ and we write $k_{0}=d\omega_{0}/d\sigma$ for the Radon-Nikodym derivative. Using (2.15) we have that there exists $q>1$ such that since $Q_{0}\subset M_{0}\Delta_{Q_{0}}$, we have

[TABLE]

In particular, for any Borel set $F\subset Q_{0}$, using Hölder’s inequality we obtain

[TABLE]

Hence we can apply [HMT2, Lemma 3.5] to $\mu=\omega_{0}$, and extract a pairwise disjoint family $\mathcal{F}_{Q_{0}}=\{Q_{j}\}\subset\mathbb{D}_{Q_{0}}\setminus\{Q_{0}\}$ verifying (3.50), as well as

[TABLE]

with $K_{1}=(4K_{0})^{1/\theta}$, $K_{0}=\max\{C_{0},C_{2}\}$, and $\theta=1/q^{\prime}$.

We next observe that if $I\in\mathcal{W}_{Q}^{*}$ with $Q\in\mathbb{D}_{\mathcal{F}_{Q_{0}},Q_{0}}$ then $2B_{Q}\subset B_{Q_{0}}^{*}$ (see (2.11)). Hence, using Harnack’s inequality, parts $(b)$ and $(c)$ of Lemma 2.24, (3.54) and the $\mathrm{AR}$ property we have

[TABLE]

where $X_{I}$ is the center of $I$.

At this point, we are looking for $M_{1}$ independent of $Q_{0}$ and $Q^{0}$ such that (3.51) holds. Recalling (3.40) we note that

[TABLE]

where we have used Harnack’s inequality, (3.55), and the bounded overlap of the family $\{U_{Q}\}_{Q\in\mathbb{D}}$.

As in Section 2.3 for every $N\geq 1$ we can consider the pairwise disjoint collection \mathcal{F}_{N}:=\mathcal{F}_{Q_{0}}\big{(}2^{-N}\ell(Q_{0})\big{)} which is the family of maximal cubes of the collection $\mathcal{F}_{Q_{0}}$ augmented by adding all of the cubes $Q\in\mathbb{D}_{Q_{0}}$ such that $\ell(Q)\leq 2^{-N}\ell(Q_{0})$. In particular, $Q\in\mathbb{D}_{\mathcal{F}_{N},Q_{0}}$ if and only if $Q\in\mathbb{D}_{\mathcal{F}_{Q_{0}},Q_{0}}$ and $\ell(Q)>2^{-N}\ell(Q_{0})$. Clearly, $\mathbb{D}_{\mathcal{F}_{N},Q_{0}}\subset\mathbb{D}_{\mathcal{F}_{N^{\prime}},Q_{0}}$ if $N\leq N^{\prime}$, and therefore $\Omega_{\mathcal{F}_{N},Q_{0}}\subset\Omega_{\mathcal{F}_{N^{\prime}},Q_{0}}\subset\Omega_{\mathcal{F}_{Q_{0}},Q_{0}}$. This and the monotone convergence theorem give that

[TABLE]

We now formulate an auxiliary result that will lead us to the desired estimate, namely (3.51).

Proposition 3.58**.**

Given $C_{1}\geq 1$, one can find $C$ such that if $\mathcal{F}_{N}\subset\mathbb{D}_{Q_{0}}$, $N\in\mathbb{N}$, is a family of pairwise disjoint dyadic cubes satisfying

[TABLE]

then

[TABLE]

Here, $C$ depends only on dimension, the 1-sided $\mathrm{CAD}$ constants, and the ellipticity of $L$.

Assuming this result momentarily, (3.54) and the construction of $\mathcal{F}_{N}$ give (3.59). Next, we combine (3.56), (3.57) and (3.60) to conclude (3.51). This completes the proof of $(b)\Longrightarrow(a)$ Theorem 1.1, modulo obtaining the just stated proposition. ∎

Proof of Proposition 3.58.

We introduce an adapted cut-off function which can be obtained from a straightforward modification of [HMT2, Lemma 4.44] by simply replacing $\lambda$ by $2\lambda$ (recall that $\lambda$ appearing in Section 2.3 can be chosen arbitrarily small).

Lemma 3.61**.**

There exists $\Psi_{N}\in C_{c}^{\infty}(\mathbb{R}^{n+1})$ such that

• $(a)$

$\mathbf{1}_{\Omega_{\mathcal{F}_{N},Q_{0}}}\lesssim\Psi_{N}\leq\mathbf{1}_{\Omega^{*}_{\mathcal{F}_{N},Q_{0}}}$.

• $(b)$

$\sup_{X\in\Omega}|\nabla\Psi_{N}(X)|\delta(X)\lesssim 1$.

• $(c)$

Set

[TABLE]

Then

[TABLE]

with implicit constants depending only on the allowable parameters but uniform in $N$.

Taking then $\Psi_{N}$ as above, Leibniz’s rule leads us to

[TABLE]

Note that $u\,\mathcal{G}_{0}\,\Psi_{N}^{2}\in W^{1,2}_{0}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$ since $\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}}$ is a compact subset of $\Omega$ (indeed by construction $\operatorname{dist}(\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}},\partial\Omega)\gtrsim 2^{-N}\ell(Q_{0})$), $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$, $\mathcal{G}_{0}\in W^{1,2}_{\rm loc}(\Omega\setminus\{X_{0}\})$, $\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}}\subset\overline{T_{Q_{0}}^{**}}\subset\frac{1}{2}B_{Q_{0}}^{*}$ (cf. (2.11)), and (3.52). Moreover, since $u\in W^{1,2}_{\rm loc}(\Omega)$ it follows that $u\in W^{1,2}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$. All these plus the fact that $Lu=0$ in the weak sense in $\Omega$ easily give

[TABLE]

Moreover as above $u^{2}\,\Psi_{N}^{2}\in W^{1,2}_{0}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$. Also, Lemma 2.17 (see in particular (2.23)) gives at once that $\mathcal{G}_{0}\in W^{1,2}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$ and $L^{\top}\mathcal{G}_{0}=0$ in the weak sense in $\Omega\setminus\{X_{0}\}$. Thus, we easily obtain

[TABLE]

Using ellipticity, (3.63), (3.64), (3.65), the fact that $\|u\|_{L^{\infty}(\Omega)}=1$, and Lemma 3.61, we have

[TABLE]

To estimate $\mathrm{I}$ we use Lemma 3.61, Caccioppoli’s and Harnack’s inequalities, and the fact that $\|u\|_{L^{\infty}(\Omega)}=1$:

[TABLE]

where $X_{I}$ is the center of $I$. Note that for every $I\in\mathcal{W}_{N}^{\Sigma}$ there is $Q\in\mathbb{D}_{\mathcal{F}_{N},Q_{0}}$ such that $I\in\mathcal{W}_{Q}^{*}$. Hence we can use (3.55) and (3.62) to obtain

[TABLE]

Plugging (3.68) into (3.66) we get (3.60) and complete the proof of Lemma 3.58. ∎

4. Proof of Theorems 1.3 and 1.6

We will prove Theorems 1.3 and 1.6 by showing that all bounded weak solutions satisfy the Carleson measure estimate (1.2), in which case Theorem 1.1 will give the $A_{\infty}$ properties. First we prove an integration by parts identity.

Lemma 4.1**.**

Let D=(d_{i,j}\big{)}_{i,j=1}^{n+1}\in L^{\infty}(\Omega)\cap{\rm Lip}_{\rm loc}(\Omega) be an antisymmetric real matrix and set for $X\in\Omega$

[TABLE]

which is the vector formed by taking the divergence operator acting on the columns of $D$. Then,

[TABLE]

for every $u\in W^{1,2}_{\rm loc}(\Omega)$ and every $v\in W^{1,2}(\Omega)$ such that $K=\operatorname{supp}(v)\subset\Omega$ is compact.

Proof.

We first consider the case $u,v\in C_{c}^{\infty}(\Omega)$. Using Leibniz’s rule and the fact that $D$ is antisymmetric we have that

[TABLE]

Using this we integrate by parts to obtain

[TABLE]

To obtain the general case let $u\in W^{1,2}_{\rm loc}(\Omega)$ and $v\in W^{1,2}(\Omega)$ such that $K=\operatorname{supp}(v)\subset\Omega$ is compact. It is standard to see, using for instance the Whitney covering, that we can find $\Phi_{K}\in C_{c}^{\infty}(\Omega)$ so that $\Phi_{K}\equiv 1$ in $K$. Write $K^{*}=\operatorname{supp}(\Phi_{K})$ which is a compact subset of $\Omega$ and define

[TABLE]

which satisfies $\operatorname{dist}(\overline{U},\partial\Omega)\geq\operatorname{dist}(K^{*},\partial\Omega)/2>0$, hence $\overline{U}$ it is also a compact subset of $\Omega$. Since $u\in W^{1,2}_{\rm loc}(\Omega)$ we clearly have that $u\Phi_{K}\in W^{1,2}_{0}(U)$ and hence we can find $\{u_{j}\}_{j}\subset C^{\infty}_{c}(U)$ so that $u_{j}\to u\Phi_{K}$ in $W^{1,2}(U)$. Also, since $v\in W^{1,2}(\Omega)$ verifies $K=\operatorname{supp}(v)\subset\Omega$ it is also easy to see that $v\in W^{1,2}_{0}(U)$ and hence we can find $\{v_{j}\}_{j}\subset C^{\infty}_{c}(U)$ so that $v_{j}\to v$ in $W^{1,2}(U)$. Notice that extending the $u_{j}$’s and $v_{j}$’s as [math] outside of $U$ one sees that $\{u_{j}\}_{j},\{v_{j}\}_{j}\subset C^{\infty}_{c}(\Omega)$. Thus, we can use (4.3) and for every $j$

[TABLE]

Note that using that $\operatorname{supp}(v_{j}),\operatorname{supp}(v)=K\subset U$ and that $\Phi_{K}\equiv 1$ in $K\subset U$ we have

[TABLE]

and the last term converges to [math] as $j\to\infty$ since $D\in L^{\infty}(\Omega)$ and the $v_{j}$’s are uniformly bounded in $W^{1,2}(U)$. Analogously,

[TABLE]

which also converges to 0 as $j\to\infty$ since $D\in{\rm Lip}_{\rm loc}(\Omega)$ and the $v_{j}$’s are uniformly bounded in $W^{1,2}(U)$. Combining (4.5), (4.9) and (4.4) yields (4.3). ∎

We show that Theorems 1.3 and 1.6 follow from the following more general result which is interesting on its own right:

Theorem 4.13**.**

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be a 1-sided $\mathrm{CAD}$ (cf. Definition 2.4). Let $L_{1}u=-\operatorname{div}(A_{1}\nabla u)$ and $L_{0}u=-\operatorname{div}(A_{0}\nabla u)$ be real (not necessarily symmetric) elliptic operators (cf. Definition 2.12). Suppose that $A_{0}-A_{1}=A+D$ where $A,D\in L^{\infty}(\Omega)$ are real matrices satisfying the following conditions:

• $(i)$

Define for $X\in\Omega$

[TABLE]

where $\delta(X):=\operatorname{dist}(X,\partial\Omega)$, and assume that it satisfies the Carleson measure condition

[TABLE]

• $(ii)$

$D\in{\rm Lip}_{\rm loc}(\Omega)$* is antisymmetric and suppose that $\operatorname{div}_{C}D$ defined in (4.2) satisfies the Carleson measure condition*

[TABLE]

Then, $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L_{1}}\in A_{\infty}(\partial\Omega)$ (cf. Definition 2.13).

Assuming this result we can easily prove Theorems 1.3 and 1.6:

Proof of Theorem 1.3.

For $L_{0}$ and $L_{1}$ as in the statement of Theorem 1.3 we set $A=A_{0}-A_{1}$ and $D=0$. Thus, it suffices to check that $A$ and $D$ satisfy the required conditions in Theorem 4.13. For $(i)$ notice that $a=\varrho(A_{1},A_{0})$ (cf. (4.14) and (1.4)), hence (1.5) gives immediately (4.15). On the other hand since $D=0$ we clearly have all the conditions in $(ii)$. With all these in hand, Theorem 4.13 gives at once the desired conclusion. ∎

Proof of Theorem 1.6.

Set $A_{0}=A$, $A_{1}=A^{\top}$, $\widetilde{A}=0$ and $D=A-A^{\top}$ so that $A_{0}-A_{1}=\widetilde{A}+D$. As before we can easily see that $\widetilde{A}$ and $D$ satisfy the required conditions in Theorem 4.13. This time $(i)$ is trivial. For $(ii)$ notice that by assumption $D=A-A^{\top}\in{\rm Lip}_{\rm loc}(\Omega)$ and also that (1.8) yields (4.16) since (1.7) agrees with (4.2). As a result, we can invoke Theorem 4.13 obtaining that $\omega_{L}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L^{\top}}\in A_{\infty}(\partial\Omega)$.

On the other hand, if we let $A_{0}=A$, $A_{1}=A^{\rm sym}=\frac{A+A^{\top}}{2}$, $\widetilde{A}=0$ and $D=\frac{A-A^{\top}}{2}$ so that $A_{0}-A_{1}=\widetilde{A}+D$, the same argument yields that $\omega_{L}\in A_{\infty}(\partial\Omega)$ if and only if $\omega_{L^{\rm sym}}\in A_{\infty}(\partial\Omega)$.

∎

Besides the previous results one can easily get other interesting perturbation results from Theorem 4.13. For instance suppose that $L_{0}u=-\operatorname{div}(A_{0}\nabla u)$ has an associated elliptic measure satisfying $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$. Let $D$ be a real antisymmetric matrix with locally Lipschitz coefficients and assume that $\|D\|_{L^{\infty}(\Omega)}<\lambda_{0}$ where $\lambda_{0}>0$ is so that $A(X)\xi\cdot\xi\geq\lambda_{0}\,|\xi|^{2}$ for all $\xi\in\mathbb{R}^{n+1}$ and a.e. $X\in\Omega$. The latter ensures that $A_{1}=A_{0}+D$ is uniformly elliptic and hence if we assume that $\operatorname{div}_{C}D$ satisfies (4.16) then Theorem 4.13 gives immediately that $\omega_{L_{1}}\in A_{\infty}(\partial\Omega)$ where $L_{1}u=-\operatorname{div}(A_{1}\nabla u)$. In particular, the $A_{\infty}$ property is preserved under perturbations by antisymmetric “sufficiently small” matrices $D$ with locally Lipschitz coefficients so that $|\nabla D|^{2}\delta$ satisfies a Carleson measure condition.

Proof of Theorem 4.13.

By symmetry it suffices to assume that $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$ and prove that $\omega_{L_{1}}\in A_{\infty}(\partial\Omega)$. By Theorem 1.1 it suffices to show that given $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ with $L_{1}u=0$ in the weak sense in $\Omega$ then (1.2) holds. As before, by homogeneity we may assume without loss of generality that $\|u\|_{L^{\infty}(\Omega)}=1$. We can now follow closely the proof of $(b)\Longrightarrow(a)$ in Theorem 1.1 with the following changes. Here we are assuming that $\omega_{L_{0}}\in A_{\infty}(\partial\Omega)$ and hence (3.53) needs to be replaced by

[TABLE]

where $X_{0}:=X_{M_{0}\Delta_{Q_{0}}}$ is chosen as before so that (3.52) holds.

Notice that in the present situation $u$ satisfies $L_{1}u=0$ (as opposed to what happened above where both $u$ and $\mathcal{G}_{0}$ where associated with the same operator). Other than that, and keeping in mind (4.17), all estimates (3.54)–(3.57) hold. Thus it is straightforward to see that everything reduces to obtain the following analog of Proposition 3.58:

Proposition 4.18**.**

Given $C_{1}\geq 1$, one can find $C$ such that if $\mathcal{F}_{N}\subset\mathbb{D}_{Q_{0}}$, $N\in\mathbb{N}$, is a family of pairwise disjoint dyadic cubes satisfying

[TABLE]

then

[TABLE]

Here, $C$ depends only on dimension, the 1-sided $\mathrm{CAD}$ constants, the ellipticity of $L_{0}$ and $L_{1}$, and on $C_{A}$ and $C_{D}$.

The proof of Theorem 4.13 follows from Proposition 4.18 as the proof in section 3.2 follows from Proposition 3.58. ∎

Proof of Proposition 4.18.

Take $\Psi_{N}$ from Lemma 3.61 and write $\mathcal{E}(X):=A_{1}(X)-A_{0}(X)$. Then Leibniz’s rule leads us to

[TABLE]

Note that $u\,\mathcal{G}_{0}\,\Psi_{N}^{2}\in W^{1,2}_{0}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$ since $\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}}$ is a compact subset of $\Omega$ (indeed by construction $\operatorname{dist}(\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}},\partial\Omega)\gtrsim 2^{-N}\ell(Q_{0})$), $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$, $\mathcal{G}_{0}\in W^{1,2}_{\rm loc}(\Omega\setminus\{X_{0}\})$, $\overline{\Omega^{**}_{\mathcal{F}_{N},Q_{0}}}\subset\overline{T_{Q_{0}}^{**}}\subset\frac{1}{2}B_{Q_{0}}^{*}$ (cf. (2.11)), and (3.52). Moreover, since $u\in W^{1,2}_{\rm loc}(\Omega)$ it follows that $u\in W^{1,2}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$. Thus since $L_{1}u=0$ in the weak sense in $\Omega$ we have

[TABLE]

On the other hand, much as before $u^{2}\,\Psi_{N}^{2}\in W^{1,2}_{0}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$. Also, Lemma 2.17 (see in particular (2.23)) gives at once that $\mathcal{G}_{0}\in W^{1,2}(\Omega^{**}_{\mathcal{F}_{N},Q_{0}})$ and $L_{0}^{\top}\mathcal{G}_{0}=0$ in the weak sense in $\Omega\setminus\{X_{0}\}$. Thus, we easily obtain

[TABLE]

Using ellipticity, (4.21), (4.22), (4.23), the fact that $\|u\|_{L^{\infty}(\Omega)}=1$, and Lemma 3.61, we have

[TABLE]

Much as in (3.67) and (3.68) we can show that $\mathrm{I}\lesssim\sigma(Q_{0})$. To estimate $\mathrm{II}$ note that since $\mathcal{E}=A_{1}-A_{0}=-(A+D)$ it follows that

[TABLE]

For the term $\mathrm{II}_{1}$ we use that $A\in L^{\infty}(\Omega)$ and the fact that $\|u\|_{L^{\infty}(\Omega)}=1$ to obtain

[TABLE]

For $\mathrm{III}_{1}$ we note that $\sup_{I^{**}}|A|\leq\inf_{I^{*}}a$ for every $I\in\mathcal{W}$, since $I^{**}\subset\{Y\in\Omega:\,|Y-X|<\delta(X)/2\}$ for every $X\in I^{*}$ (see (2.9)). Hence, Lemma 3.61, Caccioppoli’s and Harnack’s inequalities, (3.55), the fact that the family $\{I^{**}\}_{I\in\mathcal{W}}$ has bounded overlap, and (2.11) yield

[TABLE]

where in the last estimate we have used (4.15) and AR along with the fact that $r(B_{Q_{0}}^{*})=2\kappa_{0}r_{Q_{0}}\leq 2\kappa_{0}\ell(Q_{0})\leq 2\kappa_{0}\operatorname{diam}(\partial\Omega)/M_{0}<\operatorname{diam}(\partial\Omega)$ by our choice of $M_{0}$. On the other hand, we observe that

[TABLE]

where we have used Lemma 3.61, Harnack’s inequality, the normalization $\|u\|_{L^{\infty}(\Omega)}=1$ and the last estimate follows as in (3.68).

Let us now turn our attention to estimating $\mathrm{II}_{2}$. Note that $u^{2}\in W^{1,2}_{\rm loc}(\Omega)$ since $u\in W^{1,2}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$; $\operatorname{supp}(\mathcal{G}_{0}\,\Psi_{N}^{2})\subset\overline{\Omega^{*}_{\mathcal{F}_{N},Q_{0}}}$ which is a compact subset of $\Omega$ since by construction $\operatorname{dist}(\overline{\Omega^{*}_{\mathcal{F}_{N},Q_{0}}},\partial\Omega)\gtrsim 2^{-N}\ell(Q_{0})$; and finally $\mathcal{G}_{0}\,\Psi_{N}^{2}\in W^{1,2}(\Omega)$ since $\mathcal{G}_{0}\in W^{1,2}_{\rm loc}(\Omega\setminus\{X_{0}\})$, $\overline{\Omega^{*}_{\mathcal{F}_{N},Q_{0}}}\subset\overline{T_{Q_{0}}^{*}}\subset\frac{1}{2}B_{Q_{0}}^{*}$ (cf. (2.11)), and (3.52). Thus we can invoke Lemma 4.1 to see that

[TABLE]

where we have used $\|u\|_{L^{\infty}(\Omega)}=1$ and the last estimate is obtained as follows:

[TABLE]

where we have used Harnack’s inequality, (3.55), the fact that the family $\{I^{**}\}_{I\in\mathcal{W}}$ has bounded overlap, (2.11), and the last estimate follows from (4.16), the fact that $r(B_{Q_{0}}^{*})=2\kappa_{0}r_{Q_{0}}\leq 2\kappa_{0}\ell(Q_{0})\leq 2\kappa_{0}\operatorname{diam}(\partial\Omega)/M_{0}<\operatorname{diam}(\partial\Omega)$ by our choice of $M_{0}$, and the Ahlfors regularity of $\partial\Omega$.

At this point we can collect (4.24)–(4.29) and use Young’s inequality to conclude that

[TABLE]

The last term is finite since $\operatorname{supp}(\Psi_{N})\subset\overline{\Omega^{*}_{\mathcal{F}_{N},Q_{0}}}$ which is a compact subset of $\Omega$, $u\in W^{1,2}_{\rm loc}(\Omega)$, $\mathcal{G}_{0}\in L^{\infty}_{\rm loc}(\Omega\setminus\{X_{0}\})$, (3.52), and (2.11). Hence we can hide it and use Lemma 3.61 to conclude as desired that

[TABLE]

This completes the proof, see (4.20). ∎